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1909.00401v1.Switchable_Josephson_current_in_junctions_with_spin_orbit_coupling.pdf

Switchable Josephson current in junctions with spin-orbit coupling B. Bujnowski,1,R. Biele,2, 3and F. S. Bergeret1, 4,y 1Donostia International Physics Center (DIPC) - Manuel de Lardizabal 5, E-20018 San Sebasti an, Spain 2Max Bergmann Center of Biomaterials, TU Dresden, 01062 Dresden, Germany 3Institute for Materials Science, TU Dresden, 01062 Dresden, Germany 4Centro de F sica de Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU - 20018 Donostia-San Sebastian, Basque Country, Spain (Dated: September 4, 2019) We study the Josephson current in two types of lateral junctions with spin-orbit coupling and an exchange eld. The rst system (type 1 junction) consists of superconductors with heavy metal interlayers linked by a ferromagnetic bridge, such that the spin-orbit coupling is nite only at the superconductor/heavy metal interface. In the second type (type 2) of system we assume that the spin orbit coupling is nite in the bridge region. The length of both junctions is larger than the magnetic decay length such that the Josephson current is carried uniquely by the long-range triplet component of the condensate. The latter is generated by the spin-orbit coupling via two mechanisms, spin precession and inhomogeneous spin-relaxation. We show that the current can be controlled by rotating the magnetization of the bridge or by tuning the strength of the spin-orbit coupling in type 2 junctions., and also discuss how the ground-state of the junction can be tuned from a 0 to aphase dierence between the superconducting electrodes. In leading order in the spin-orbit coupling, the spin precession dominates the behavior of the triplet component and both junctions behave similarly. However, when spin relaxation eects are included junction of type 2 oers a wider parameter range in which 0- transitions take place. I. INTRODUCTION The interplay between superconductivity and ferromagnetism leads to triplet superconducting correlations1{4. The simplest setup for the generation of a triplet component is a superconductor (S)-ferromagnet (F) heterostructure with a homogeneous exchange eld. The superconducting singlet Copper pairs can penetrate the ferromagnet, and due to the local exchange eld, are partially converted into triplet pairs with the total spin projection zero with respect to the local exchange eld. Oscillations of the triplet correlations in the F region lead to the well understood eect of the sign reversal of the critical current, the so-called 0  transition5{9. In a diusive monodomain F, both singlet and triplet correlations decay on the magnetic length scaleh=p D=h, wherehis the magnitude of the exchange eld and Dis the diusion constant. For conventional Ss and typical exchange eld strengths, h is much shorter than the thermal length scale of decay !p D=T in a non-magnetic system. On the other hand, triplet components with non-zero spin projection, are not aected by its pair breaking eect and would decay over a length scale comparable to !. Such long-range triplet components (LRTC) can be generated due to inhomogeneities of the exchange eld1,2,4or due to the presence of spin-orbit coupling (SOC) and a homogeneous exchange eld10,11. The prediction of LRTC in S/F hybrid structures has stimulated multiple experimental works12{22. More re- cently, transverse vertical heterostructures with in-plane magnetic elds and SOC materials have been experimen- tally explored but the long-range correlations due to SOC have not been observed23{25. In accordance with previ-ous theoretical works10,11in vertical multilayered SFS junctions the condition for the generation of a LRTC is quite restrictive. More suitable for the observation of LRTC induced by the SOC are lateral structures where currents have also a component owing in the direction parallel to the hybrid interface26{28. In this work we present a study of the Josephson cur- rent in lateral geometries with SOC of Rashba and Dres- selhaus type and how to control it via external elds in the diusive regime. We focus on two types of junc- tions: One consists of two superconducting electrodes on top of a ferromagnetic lm, see Fig. 1a). Between the two materials we assume there is an interlayer with a - nite SOC. Hereafter we refer to this junction as type 1 junction. The junction of type 2, Fig. 1b), consists of a similar lateral geometry, but the SOC is nite in the bridge region. Whereas type 1 junctions may correspond to junctions with a heavy metal interlayer, type 2 junc- tions describe, for example, a lateral Josephson junction made of a 2D electron gas in the presence of a Zeeman eld. We assume that in both junctions the distance be- tween the superconductor electrodes is larger than the magnetic decay length, such that the Josephson current is only carried by LRTC. The latter is generated by the SOC via two mechanisms: spin precession and inhomo- geneous spin-relaxation and the current strongly depends on the direction of the exchange or Zeeman elds. In ad- dition, in type 2 junctions the Josephson current can also be tuned by a voltage gate that controls the strength of the Rashba SOC. We focus on the control of possible 0- transitions. With the help of an analytical solution for type 1 junction in the case of small SOC, we rst show that in leading order the LRTC is generated only by the spin-precessionarXiv:1909.00401v1 [cond-mat.supr-con] 1 Sep 20192 term and the junction remains in the 0-state indepen- dently of the direction of the exchange eld. The next leading order contribution to the current is due to the in- homogeneous spin relaxation with a negative sign, such that for certain directions of the exchange eld the junc- tion can switch to the -state. In junctions of type 1 this only occurs if both, the Rashba and Dresselhaus SOC are nite. In a second part we present numeric calculations of the current for arbitrary SOC strength that conrm these ndings. In addition these calculations reveal that type 2 junctions allow for 0 transitions in a wider range of SOC parameters. Specically the transition can be induced by a pure Rashba or Dresselhaus SOC by changing their strengths. This is a new possibility to in- duce 0-transition by tuning the Rashba SOC strength, which is experimentally achievable by gating the SOC active material. Besides the interesting applications of such lateral junctions as 0- switchers, they can also be used to detect the LRTC by measuring the changes of the Josephson current as a function of the direction of the applied eld or magnetization in a single junction. The work is organized as follows: In the next sec- tion II we present the basic equations describing diu- sive Josephson junctions and we adapt these equations to the lateral junctions type 1 and 2. In section III we derive the analytical expression for the Josephson current in junction type 1 perturbatively, up to second order in the SOC parameter for semi-innite leads. In section IV we present numerical results for the Josephson current for both types of junctions and compare them to the an- alytical results. Conclusions are given in section V. II. BASIC EQUATIONS FOR DIFFUSIVE JOSEPHSON JUNCTIONS WITH SOC We consider two spatially separated superconducting electrodes on top of a non-superconducting material with either an intrinsic exchange eld, as in a ferromagnet, or a Zeeman eld induced by an external magnetic eld. We distinguish two dierent types of junctions: one with SOC active layers just below the superconductors, Fig.1a), that we refer to as junction type 1. The other junction with SOC in the bridge region is referred to as junction type 2, and is shown in Fig.1b). We assume that the proximity eect, i.e.the induced superconducting correlations in the bridge, is weak and that the system is in the diusive regime. In this case spectral and transport properties of the junction can be accurately described by the linearized Usadel equation29 generalized to linear in momentum SOC. This equation provides the spatial dependence of the induced supercon- ducting correlations in the non-superconducting region which is described in terms of the anomalous Green's function ^f10,1130: D~r2 k^f+ 2j!nj^fisign(!n)n ^h;^fo = 0: (1) x<latexit sha1_base64="Xb2gHMoGwRhrW/OgQvHRwb1cQmM=">AAACIXicbVDLSsRAEJz4XONbj14GF8GDLIkIehS9eNwFdxU2QSaTjg7OTMJMR11CvsCrfoFf4028iT/j7LqCr4KGoqqb7q6kkMJiELx5E5NT0zOzjTl/fmFxaXllda1n89Jw6PJc5uY8YRak0NBFgRLOCwNMJRLOkuvjoX92A8aKXJ/ioIBYsUstMsEZOqlzd7HSDFrBCPQvCcekScZoX6x6fpTmvFSgkUtmbT8MCowrZlBwCbUflRYKxq/ZJfQd1UyBjavRpTXdckpKs9y40khH6veJiilrBypxnYrhlf3tDcWdRP1n90vMDuJK6KJE0PxzV1ZKijkdfk5TYYCjHDjCuBHuXMqvmGEcXT5+pOGW50oxnVYRQF1FCHdYQV3/tIT4skRdu/TC31n9Jb3dVhi0ws5e8/BonGODbJBNsk1Csk8OyQlpky7hBMg9eSCP3pP37L14r5+tE954Zp38gPf+ARNRpJY=</latexit><latexit 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sha1_base64="lscsJX+797AmIDsRbdxbCT3JFfk=">AAAB6nicbVA9SwNBEJ3zM8avqKXNYhCswl0aLYNp7IzEfEByhL3NXLJkb+/Y3RPCkZ9gY6GIrb/Izn/jJrlCEx8MPN6bYWZekAiujet+OxubW9s7u4W94v7B4dFx6eS0reNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoJJfe53nlBpHstHM03Qj+hI8pAzaqzUbN7XB6WyW3EXIOvEy0kZcjQGpa/+MGZphNIwQbXueW5i/Iwqw5nAWbGfakwom9AR9iyVNELtZ4tTZ+TSKkMSxsqWNGSh/p7IaKT1NApsZ0TNWK96c/E/r5ea8MbPuExSg5ItF4WpICYm87/JkCtkRkwtoUxxeythY6ooMzadog3BW315nbSrFc+teA/Vcu02j6MA53ABV+DBNdTgDhrQAgYjeIZXeHOE8+K8Ox/L1g0nnzmDP3A+fwDVm417</latexit><latexit sha1_base64="lscsJX+797AmIDsRbdxbCT3JFfk=">AAAB6nicbVA9SwNBEJ3zM8avqKXNYhCswl0aLYNp7IzEfEByhL3NXLJkb+/Y3RPCkZ9gY6GIrb/Izn/jJrlCEx8MPN6bYWZekAiujet+OxubW9s7u4W94v7B4dFx6eS0reNUMWyxWMSqG1CNgktsGW4EdhOFNAoEdoJJfe53nlBpHstHM03Qj+hI8pAzaqzUbN7XB6WyW3EXIOvEy0kZcjQGpa/+MGZphNIwQbXueW5i/Iwqw5nAWbGfakwom9AR9iyVNELtZ4tTZ+TSKkMSxsqWNGSh/p7IaKT1NApsZ0TNWK96c/E/r5ea8MbPuExSg5ItF4WpICYm87/JkCtkRkwtoUxxeythY6ooMzadog3BW315nbSrFc+teA/Vcu02j6MA53ABV+DBNdTgDhrQAgYjeIZXeHOE8+K8Ox/L1g0nnzmDP3A+fwDVm417</latexit>FIG. 1. Schematic view of the two junction types considered in the text. a) The junction of type 1 consists of two super- conductors contacted to a ferromagnet (F) via a material with strong SOC. The magnetization of F, and hence the exchange eldh, lies in the xyplane. b) For the junction of type 2 the bridge region connecting the two superconductors has a sizable SOC. . HereDis the diusion constant, !nis the Matsubara fre- quency, ~rk=@ki[^Ak;:::] is the covariant derivative with the SU(2) vector potential, ^Ak=^a 2Aa k, describing the SOC and ^h= ^ahais the exchange eld. Symbols with a ^ stand for operators in spin space and ^ aare the Pauli matrices. We use the Einstein summation conven- tion and sum over repeated indices. The general form of the condensate function in spin space is ^f=fs^1 +fa t^a(2) wherefsis the singlet component and fa tare the triplet components. In our representation the short (long)-range triplet component corresponds to the component parallel (orthogonal) to the exchange or Zeeman eld. In order to describe hybrid interfaces between the su- perconductor and a substrate one needs boundary con- ditions for the Green's functions. We use here the Kupriyanov-Lukichev ones31generalized for materials with SOC. In its linearized form at an S/X interface they read10,11: Nih ~ri^fi S=X= fBCS^1; (3) whereXdenotes any non-superconductor material. Here Niis thei-th component of the interface normal, is the interface transparency, fBCS =
ei'ip
2!2nis the anoma- lous Green's function in the bulk S region with the am- plitude of the superconducting order parameter
and3 its phase'iin thei-th electrode. At the interface with vacuum (V) no current ows and the boundary condition reads: Nih ~ri^fi X=V= 0: (4) Below we determine the Josephson current density in the bridge region for the two setups depicted in Fig.1. It can be expressed as32: j=ieN 0DTX !nTrn ^f~r^f^f~r^fo (5) whereN0is the density of states and^f= ^y^f^y. We now re-write Eqs.(1) and (3) for the specic case of junction of type 1 and 2. Both junctions are assumed to be translational invariant in the y-direction. The order parameter is a step-like-function along the x-direction, with amplitude
at the S electrodes and zero at the bridge. We denote with 'the phase dierence between the superconductors, such that
(x;z) = (z(W+d))(jxjL=2)
ei' 2sign(x):(6) The SOC elds are nite only in the SOC layers thus for the junction type 1 (Fig.1a) Aa k(x;z) = (W+dz)(zW)(jxjL=2)Aa k(7) and for the junction type 2 with the SOC in the bridge region (Fig.1b) Aa k(x;z) = (Wz)(zW)(L=2jxj)Aa k(8) with constantAa k. We restrict ourselves to SOC of the Rashba and Dresselhaus type dened by the fol- lowing vector potential: ^Ax==2^x=2^yand ^Ay==2^x=2^y. Rashba SOC corresponds to terms proportional to , while Dresselhaus SOC corresponds to terms proportional to . For both junctions the ex- change eld has only nite components in the xyplane and is present in the region F, ^h(x;z) =h(cos#^x+ sin#^y)(z)(Wz);(9) whereh=p haha. To distinguish components that are parallel and per- pendicular to the exchange eld, i.e.short - and long- range components, it is convenient to rotate Eqs.(1) and (3) by the unitary transformation U=ei^x# 2. After the rotation the exchange eld is xed along the x-axis, U^hUy=h^x: (10) Thus in our notation the long-range triplet components are those polarized in yandzdirection. Assuming for simplicity, that the thickness dof the SOC interlayers (if present) and the bridge Wis small against the typical length on which ^fchanges, we can integrate the Usadelequation along the z-direction11. This reduces the initial two dimensional problem to an eective one dimensional one. Here we illustrate how the z-integration is carried out. Besides the rst term in Eq.(1) , all other terms do not contain a spatial derivative in the z-direction and there- fore the integration results simply in the averaged value off. Integration of the rst term of Eq.1 leads to ZW+d 0~r2 k^fdz=ZW+d 0dz @2 x+@2 z
^f 2ih ^Ax;@x^fi h ^Ak;h ^Ak;^fii  fBCS + (W+d)@2 x^f2idh ^Ax;@x^fi dh ^Ak;h ^Ak;^fii :(11) In the rst step we have use the translational invariance iny-direction and our choice of the SOC, which is step- wise constant. In the second step we use the continuity of^fin thez-direction and the boundary condition Eq.(3) at the interface at z=W+das well as the boundary condition with the vacuum, Eq.(4). The z-integration causes an averaging of the couplings that diers for the two junction types. We therefore present the nal equa- tions separately. A. Usadel equations for type 1 lateral junction After performing the z-integration, and the rotation of Eq.(1), the resulting system of equations for the rotated anomalous Green's function^~f=U^fUyis Dh @2 x~fsi 2j!nj~fs2isign(!n)h~fx t= D fBCSeisign(x)' 2; (12) Dh @2 x~fa t+ 2Cab x @x~fb ti 2j!nj~fa tDab~fb t= x;a2isign(!n)h~fs; (13) where we introduced the Kronecker-Delta i;j, the com- ponents of the averaged spin precession tensor Cab k="acbAc kd=(W+d) (14) and averaged Dyakonov-Perell (DP) spin relaxation ten- sor ab= Ac kAc ka;bAa kAb k
d=(W+d): (15) The averaged coupling constants are dened as ha= haW=(W+d), =d=(W+d),=d=(W+d) and  = =(W+d). The spatial dependence of the SOC elds, exchange eld and order parameter in x-direction is not explicitly written, and is dened in Eqs.(6), (7) and (9). In the rotated system the non vanishing spin precession tensor elements are Cxz x=Czx x=cos(#)sin(#); (16) Cyz x=Czy x= sin(#)cos(#): (17)4 0.00.51.01.5jc/|jc,0|×103 a)¯β= 0 ¯αξ0=0.01 ¯αξ0=0.02 ¯αξ0=0.03 ¯αξ0=0.04 ¯αξ0=0.05 0.00.51.01.5×103 b)¯α= 0 ¯βξ0=0.01 ¯βξ0=0.02 ¯βξ0=0.03 ¯βξ0=0.04 ¯βξ0=0.05 0123×103 c)¯α=¯β ¯αξ0=0.01 ¯αξ0=0.02 ¯αξ0=0.03 ¯αξ0=0.04 ¯αξ0=0.05 0.00 0.25 0.50 0.75 1.00 ϑ/π0123jc/|jc,0|×104 d)¯αξ0=0.1 ¯αξ0=0.2 ¯αξ0=0.3 ¯αξ0=0.4 ¯αξ0=0.5 0.00 0.25 0.50 0.75 1.00 ϑ/π0123×104 e)¯βξ0=0.1 ¯βξ0=0.2 ¯βξ0=0.3 ¯βξ0=0.4 ¯βξ0=0.5 0.00 0.25 0.50 0.75 1.00 ϑ/π−4−20×105 f)¯αξ0=0.1 ¯αξ0=0.2 ¯αξ0=0.3 ¯αξ0=0.4 ¯αξ0=0.5 FIG. 2. Numerical results for the critical current as function of the orientation of an in-plane exchange eld for junction of type 1. Dierent curves correspond to dierent values of the SOC parameters. In all plots we set h= 10
,L= 50,T= 0:01
and the thickness of the SOC and F layer are chosen such that d=W = 1: The non zero elements of the DP spin relaxation tensor are xx(#) =yy(#) = 2+2+ sin(2#)
W+d d (18) zz(#) =xx(#) +yy(#); (19) xy(#) =yx(#) = 2cos (2#)W+d d(20) The solution of Eqs. (12)-(13) and its covariant deriva- tive are continuous at the boundaries x=L=2 between the dierent regions thus: @x~fs x=L 2+0=@x~fs x=L 2+0+; (21) @x~fa t x=L 2+0=h @x~fa t+Cab x~fb ti x=L 2+0: (22) The equations (12) and (13) together with the bound- ary conditions Eq.(21) and Eq.(22) fully determine the condensate within the limits of the mentioned approxi- mations. Finally the current in the bridge region is given by j= 4eN 0DTX !nImh ~f s@x~fs(~fi t)(@x~fi t)i :(23) B. Usadel equations for type 2 lateral junction For the junction type 2 the SOC coupling and the exchange eld are nite over the whole bridge. Conse-quently ha=h, =,=and  = =W . Thus the z-integrated Usadel equation is like in Eqs.(12) and (13) where now Cab k=Cab k="acbAc k (24) and the DP spin relaxation tensor ab= ab=Ac kAc ka;bAa kAb k: (25) The spatial dependence of the SOC elds is now given by Eq.(8). The solution of this system of equations is continuous and fullls @x~fs x=L 2+0=@x~fs x=L 2+0+; (26) h @x~fa t+Cab x~fb ti x=L 2+0=@x~fa t x=L 2+0: (27) Thus for type 2 junctions, the condensate function is de- termined from Eqs.(12)-(13) and Eqs.(26)-(27). Finally the current through the junction is given by j= 4eN 0DTX !nImh ~f s@x~fs(~fi t)(@x~fi t) +(~fz t) ~fx t+~fy ti : (28) III. THE JOSEPHSON CURRENT IN TYPE 1 JUNCTIONS: ANALYTICAL SOLUTION Her we focus on type 1 junctions in the case when the exchange interaction is the dominant energy scale,5 D2;D2;D;Th. The junction is larger than the magnetic length, h, and hence the current is solely de- termined by the LRTC, ~fy tand ~fz t. The other two com- ponents decay over hin the F region. We solve Eqs.(12-13) perturbatively up to second order in the SOC elds, Aa k. In zeroth order only the singlet and triplet component parallel to the eld, ~fx t;0, are nite. Their explicitly form is given in the appendix, Eq.(A13). In rst order in the SOC the component ~fz tappears as a consequence of the precession term. Specically it is determined by @2 x~fz t;12j!nj D~fz t;1=2Czx x @x~fx t;0 : (29) The component ~fy tappears in second order of the SOC and satises: @2 x~fy t;22j!nj D~fy t;2=2Cyz x @x~fz t;1 +yx(#)~fx t;0:(30) The explicit expressions for these components are given in the appendix, Eqs.(A20), (A24). From these solutions we obtain the current density in the F region. The cur- rent density Eq.(5) is only due to the contribution of the long-range components ~fz tand ~fy tin Eq. (28). The max- imum value of the Josephson current, i.e. the critical currentjc, is obtained at '==2: jc=j '= 2 =eN 0DTX !njfb xj2e!L cos#+sin#
2 2!822cos22# 3!! (31) where we dene !=p 2j!nj=Dand fb xi sign(!n)2 h 2fBCS (32) is the value of ~fx tfor zero SOC, in the F region below the superconducting electrodes far from the bridge region. The rst term in the second line of Eq.(31) is the lowest correction in the SOC which stems from the precession term in Eq. (29) which generates the LRTC ~fz tfrom ro- tation of the short-range ~fx t;0. It is a positive contribution (0-junction) and as expected depends on the direction of the eld. For #= 0, it is only nite if the Rashba SOC is nonzero ( cf.with the numerical results shown in Fig.2). In the next order of the SOC the contribution to the current is negative, second term in the second line of Eq. (31), and it is due to the spin relaxation term yx in Eq. (30), that leads to a nite ~fy tcomponent. This contribution is only nite if both Rashba and Dresselhaus type of SOC are present. This explains why in the case of a pure Rashba or Dresselhaus SOC the current does not change sign as a function of #(see numerical results shown in Fig.2a-b,d-e). Thus, the sign and magnitude of the critical current is determined by two competing contributions, namelyspin precession and anisotropic spin relaxation11, which in turn depend strongly on the direction of the ap- plied Zeeman eld. For example the contribution due to spin precession is zero whenever the SU(2) electric eld strength in transport direction Fx;0(#) =ih ^Ax;^h(#)i vanishes. This is in accordance with previous theoreti- cal investigations that identied Fk;0as the generator of the LRTC10,11. According to Eq. (31), Fx;0(#) = 0, for #0= arctan   +n. For this value of #the second negative term dominates provided that cos 2 #06= 0, and leads to a change of sign of the critical current, a 0-  transition. For either pure Rashba or pure Dresselhaus SOC the dependence of the critical current on #is simply shifted by=2 for the same magnitude of the SOC parameter. This can be already inferred from Eq. (1), which is sym- metric when interchanging $andx$yfor the coordinate labels in spin space. Eq. (31) is valid for a symmetric junction, i.e. a junc- tion of type 1 with the same SOC at both electrodes. In the case that the left (L) and right (R) electrodes have dierent values for the Rashba and Dresselhaus SOCs L=RandL=Rit is possible to obtain a 0- transition solely due to spin precession eects. Namely, the critical current up to rst order in the SOC elds reads jc=eN 0DTX !njfb xj2 !e!L (Lcos#+Lsin#)(Rcos#+Rsin#): (33) By inspecting Eq.(33) we see that the current reversal appears every-time the SU(2) electric eld strength dis- appears in the left or right lead FL=R x;0= 0, as long as LR6=RL. When all couplings are non vanishing this takes place at the angles #L=R 0= arctan L=R L=R + n. The interval, where the current is reversed with re- spect to the symmetric case, is maximized when there is only Rashba SOC in one lead and only Dresselhaus SOC in the other as then jc/sin(2#). To summarize this section, for low SOC strength, the long-range supercurrent is mainly determined by the spin precession. If the S-electrodes are symmetric and only one type of SOC is active, the current can be switched on and o by rotating the exchange eld in the xy plane, but no 0- transition takes place. A reversal of the current only appears if both SOC types are nite and originates in a competition of the spin precession- and the spin relaxation eects. A current reversal due to spin precession eects can only be achieved by choosing leads with dierent SOC parameters. IV. NUMERICAL RESULTS In this section we compute numerically the Joseph- son current for both types of junctions with nite S-6 012jc/|jc,0|×103 a)¯β= 0 ¯αξ0=0.01 ¯αξ0=0.02 ¯αξ0=0.03 ¯αξ0=0.04 ¯αξ0=0.05 012×103 b)¯α= 0 ¯βξ0=0.01 ¯βξ0=0.02 ¯βξ0=0.03 ¯βξ0=0.04 ¯βξ0=0.05 024×103 c)¯α=¯β ¯αξ0=0.01 ¯αξ0=0.02 ¯αξ0=0.03 ¯αξ0=0.04 ¯αξ0=0.05 0.00 0.25 0.50 0.75 1.00 ϑ/π−2−1012jc/|jc,0|×104 d)¯αξ0=0.1 ¯αξ0=0.2 ¯αξ0=0.3 ¯αξ0=0.4 ¯αξ0=0.5 0.00 0.25 0.50 0.75 1.00 ϑ/π−2−1012×104 e)¯βξ0=0.1 ¯βξ0=0.2 ¯βξ0=0.3 ¯βξ0=0.4 ¯βξ0=0.5 0.00 0.25 0.50 0.75 1.00 ϑ/π−4−202×104 f)¯αξ0=0.1 ¯αξ0=0.2 ¯αξ0=0.3 ¯αξ0=0.4 ¯αξ0=0.5 FIG. 3. Numerical results for the critical current as function of the orientation of an in-plane exchange eld for in junction of type 2. Dierent curves correspond to dierent values of the SOC parameters. In all plots we set h= 10
,L= 50,T= 0:01
. 0.0 0.2 0.4 0.6 0.8 1.0 ϑ/π−0.250.000.250.500.751.001.251.50jc/|jc,0|×103L=5ξ0,T=0.01∆,¯h=10.0∆,¯αL= ¯αR=0.04ξ0 ¯βξ0=0.0 ¯βξ0=0.01 ¯βξ0=0.02 ¯βξ0=0.03 ¯βξ0=0.04 FIG. 4. Numerical results for the critical current as function of the orientation of an in-plane exchange eld for an asym- metric junction of type 1. electrodes. The total length of the system is Ltot= 2LS+L, whereLSis the length of the S-electrode, and is set toLtot= 10L. The systems of equations (12), (13) are complemented by the boundary condition Eq.(4) at the outer interfaces: ~rx^fjx=Ltot=2= 0: (34) The resulting critical current density for the junction type 1 is shown in Fig.2 a)-f) and for junction type 2 inFig.3 a)-f). For low SOC strengths and any of the studied SOC types and junction types, the current vanishes when the SU(2) electric eld strength vanishes in accordance with previous theories10. Indeed, the critical current for both setups and small SOC show qualitatively identical behavior (Fig.2 a)-c), 3 a)-c)), in very good agreement with the analytical result of Eq.(31). This implies that at the level of spin-precession eects both junctions be- have similarly. As expected the critical current curves for the case of pure Rashba or Dresselhaus SOC are shifted by=2 when comparing curves of corresponding SOC strengths. When increasing the SOC strength for j type 1 junc- tion we observe the competition between the two LRTC generating mechanisms. Comparing the upper and lower panels of Fig.2 we see that the current changes sign at suciently large SOC strengths, only when both Rashba and Dresselhaus SOC are nite, as expected. For the special case when =and exchange eld orienta- tion#==4 there is no 0transition possible as the spin relaxation contribution to the current vanishes. At#= 3=4spin precession and spin relaxation contri- butions vanish simultaneously, as can be seen in Fig.2f). By further increase of the SOC the numerical results shown in Fig.(2) f) dier qualitatively from the analytic ones: there is a strong increase of the critical current in two negative dips around #==4. The two negative dips move closer to #==4 by increasing the SOC strength. Also there is a attening of the curve at #= 3=4. The 0transition due to spin precession eect in7 a) 0.0 0.1 0.2 0.3 0.4 0.5 αξ0−2−1012jc/|jc,0|×104L=5ξ0,T=0.01∆,¯h=10.0∆,¯βξ0=0.0 ϑ/π=0.0 ϑ/π=0.125 ϑ/π=0.25 ϑ/π=0.375 ϑ/π=0.5 b) 0.0 0.1 0.2 0.3 0.4 0.5 αξ0−2−1012jc/|jc,0|×104L=5ξ0,T=0.01∆,¯h=10.0∆,¯βξ0=0.1 ϑ/π=0.0 ϑ/π=0.125 ϑ/π=0.25 ϑ/π=0.375 ϑ/π=0.5 c) 0.0 0.1 0.2 0.3 0.4 0.5 αξ0−5−4−3−2−1012jc/|jc,0|×104L=5ξ0,T=0.01∆,¯h=10.0∆,¯βξ0=0.5 ϑ/π=0.0 ϑ/π=0.125 ϑ/π=0.25 ϑ/π=0.375 ϑ/π=0.5 FIG. 5. Critical current for a junction of type 2 as function of the Rashba SOC strength , for dierent orientations of the exchange eld. The Dresselhaus SOC parameter is in- creased from panel a) to panel c). junction type 1 with asymmetric SO interaction obtained analytically in the previous section is conrmed by the numerics as shown in Fig.4. In particular, the points of current reversal as function of #are in agreement with the analytical result, Eq.(33). The case of large SOC in type 2 junctions are shown in Fig.3 d)-f). We clearly see that 0 transitions are possible for any choice of SOC. The case when ;6= 0 is qualitatively similar to junction type 1. In contrast, forjunction 2, 0transitions are possible for pure Rashba or Dresselhaus SOC when increasing the SOC strength, as shown in Fig.3 c),d) and Fig.5. Our results, regard- ing the current sign reversal, are similar to the results of Ref.27, where a one dimensional junction with a pure Rashba has been studied. Similarly to the one dimen- sional case, our results for two dimensional SOC, show that the direction of the current can be inverted by tun- ing the strength of the Rashba SOC, which can be done by a voltage gate if the bridge region is a semiconduc- tor. Such a gate has also been suggested in Ref.26for creation of a long ranged spin-triplet helix in a ballistic ferromagnetic Josephson junction. V. CONCLUSION We present a study of the eects of Rashba and Dres- selhaus SO interaction in two types of diusive lateral Josephson junctions. In the rst type the bridge link- ing the superconducting electrodes is a ferromagnet and the SOC elds originated from heavy metal interlayers placed between the S leads and the F bridge. In the second geometry the exchange elds and SOC elds are nite over the whole bridge. In a realistic setup this can be realized by a a 2D semiconducting bridge in an ex- ternal magnetic eld. In both cases we determine the long-range triplet Josephson current. We show how the magnitude and sign of the supercurrent can be controlled by varying the direction of the exchange eld as well as tuning the strengths of the SOC. Besides their relevance for application as supercurrent valves such lateral junc- tions can be used as a unequivocal way of detecting the long-range triplet component of the condensate in lateral setups. Note added: During the preparation of the manuscript we became aware of the very recent work Ref.28that studies junction type 1 in great detail. Our work conrms and extends the analytical and numerical results therein as the authors mainly focus on pure Rashba SOC. ACKNOWLEDGEMENTS BB and FSB acknowledge funding by the Spanish Min- isterio de Ciencia, Innovacin y Universidades (MICINN) under the project FIS2017-82804-P and by the Transna- tional Common Laboratory QuantumChemPhys . RB ac- knowledges funding from the European Unions Horizon 2020 research and innovation program under the Marie Skodowska-Curie grant agreement No. 793318.8 Appendix A: Basic equations After performing the z-integration, the resulting sys- tem of dierential equations for the transformed anoma- lous Green's function^~f=U^fUyforjxj>L= 2 is: Dh @2 x~fsi 2j!nj~fs2isign(!n)h~fx t= D fBCSeisign(x)' 2 (A1) Dh @2 x~fx t+ 2Cxb x @x~fb ti 2j!nj~fx tDxb~fb t= 2isign(!n)~fsh (A2) Dh @2 x~fy t+ 2Cyb x @x~fb ti 2j!nj~fy tDyb~fb t= 0 (A3) Dh @2 x~fz t+ 2Czb x @x~fb ti 2j!nj~fz tDzb~fb t= 0: (A4) In the barrier region jxj<L= 2 we get D@2 x~fs2j!nj~fs2isign(!n)h~fx t= 0 (A5) D@2 x~fx t2j!nj~fx t2isign(!n)h~fs= 0 (A6) D@2 x~fy t2j!nj~fy t= 0 (A7) D@2 x~fz t2j!nj~fz t= 0: (A8) Thez-integration causes a averaging of the couplings as described in the main text. The solution of this system of equations are continuous and fulll @x~fs x=L 2+0=@x~fs x=L 2+0+ (A9) @x~fa t x=L 2+0=h @x~fa t+Cab x~fb ti x=L 2+0;(A10) at the boundaries between the dierent regions. The spin precession tensor components Cab kand DP tensor compo- nents abin the rotated system are determined from the transformed elds ^~Ax=^x 2(#)(#)^y 2; (A11) ^~Ay=^x 2(#)(#)^y 2; (A12) with(#) = cos(#) +sin(#) and(#) =sin(#) + cos(#). The equations (A1)-(A10) fully determine the junction system within the limits of the approximations mentioned in the main text. 1. Zeroth order correction As in the main text we consider the junction type 1 assuming semi-innite leads. Solving the above system of equations for vanishing SOC gives the following zerothorder solution for the function ~fx t, ~fx t;0= 8 >< >:AL 1 +e+x+AL 2 ex+fb xei' 2;x<L 2 B1 +e+xB2 +e+x+B3 exB4 ex AR 1 +e+xAR 2 ex+fb xei' 2;x>L 2:(A13) where=q 2j!nj Di2sgn(!n)h D, AL 1=2=fb sfb x 2sinhLi' 2 (A14) B1=2=+ 4 fb s+fb x
expL+i' 2 (A15) B3=4= 4 fb sfb x
expLi' 2 (A16) AR 1=2=fb sfb x 2sinhL+i' 2 (A17) and the bulk solutions for the singlet and triplet xcom- ponent fb s=D fBCS 2j!nj j!nj2+h2 j!nj2 ~h 2hfBCS (A18) fb x=iD fBCS 2sign(!n)h j!nj2+h2i sign(!n)2 h 2fBCS: (A19) withh=p D=h. 2. First order correction The Ansatz for the solution of Eq.(29) reads ~fz t;1(x) =8 >< >:K1e!x+ZL 1e+x+ZL 2ex; x<L 2 K2e!x+K3e!x;jxj<L 2 K4e!x+ZR 1e+x+ZR 2ex; x>L 2(A20) where ZL 1=(#)AL 1 (+22!); ZL 2=(#)AL 2 (22!);(A21) ZR 1=(#)AR 1 (+22!); ZR 2=(#)AR 2 (22!);(A22) Keeping only leading order terms when h T;maxab and assuming Lhwe nd 0 B@K1 K2 K3 K41 CAfb x !(#) 20 BB@sinh(L!i' 2) 1 2eL! 2ei' 2 1 2eL! 2ei' 2 sinh(L!+i' 2)1 CCA(A23)9 3. Second order correction The ansatz for the solution of Eq.(30) reads ~fy t;2(x) = 8 >< >:(L1+xYL 1)e!x+YL 2e+x+YL 3ex+YL 4; x<L 2 L2e!x+L3e!x;jxj<L 2 (L4+xYR 1)e!x+YR 2e+x+YR 3ex+YR 4; x>L 2 (A24) with YL=R 1 =(#)K1=4; (A25) YL=R 2 =2+2(#)ZL 1+yxAL=R 1 +(+22!); (A26) YL=R 3 =22(#)ZL 2+yxAL=R 2 (22!); (A27) YL=R 4 =yxfb xei' 2 2!: (A28)Considering only leading order terms when h T;maxab consistent with the rst order correction and assuming that Lhgives for the relevant coe- cients inside the bride  L2 L3 =1 2eL 2 YR 4 YL 4 : (A29) bogusz.bujnowski@gmail.com yfs.bergeret@csic.es 1F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Phys. Rev. Lett. 86, 4096 (2001). 2F. S. Bergeret, A. F. Volkov, and K. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005), 0506047. 3M. Eschrig, Phys. Today 64, 43 (2011), 1509.02242. 4J. Linder and J. W. Robinson, Nat. Phys. 11, 307 (2015). 5L. Bulaevskii, V. Kuzii, and A. Sobyanin, JETP Lett. 25, 290 (1977). 6S. V. Panyukov, L. N. Bulaevskil, and A. I. Buzdin, Sov. Phys. JETP 35, 178 (1982). 7V. V. Ryazanov et al. , Phys. Rev. Lett. 86, 2427 (2001), 0008364. 8A. F. Volkov, K. B. Efetov, and F. S. Bergeret, Phys. Rev. B64, 1 (2001). 9T. Kontos et al. , Physical Review Letters 89, 1 (2002). 10F. S. Bergeret and I. V. Tokatly, Phys. Rev. Lett. 110, 1 (2013), 1211.3084. 11F. S. Bergeret and I. V. Tokatly, Phys. Rev. B 89, 134517 (2014), 1402.1025. 12J. Robinson, J. Witt, and M. Blamire, Science 329, 59 (2010). 13M. S. Anwar, F. Czeschka, M. Hesselberth, M. Porcu, and J. Aarts, Phys. Rev. B 82, 2 (2010). 14X. L. Wang et al. , Phys. Rev. B 89, 3 (2014). 15E. C. Gingrich et al. , Phys. Rev. B 86, 1 (2012). 16J. W. A. Robinson, F. Chiodi, M. Egilmez, G. B. Hal asz, and M. G. Blamire, Sci. Rep. 2, 699 (2012).17F. Chiodi et al. , EPL 101, 37002 (2013). 18A. Pal, Z. Barber, J. Robinson, and M. Blamire, Nat. Commun. 5, 3340 (2014). 19J. W. A. Robinson, N. Banerjee, and M. G. Blamire, Phys. Rev. B 89, 104505 (2014). 20Y. Kalcheim, O. Millo, M. Egilmez, J. W. A. Robinson, and M. G. Blamire, Phys. Rev. B 85, 1 (2012). 21N. Banerjee et al. , Nat. Commun. 5, 3048 (2014). 22T. S. Khaire, M. A. Khasawneh, W. P. Pratt, and N. O. Birge, Phys. Rev. Lett. 104, 137002 (2010). 23N. Satchell and N. O. Birge, Phys. Rev. B 97, 1 (2018). 24N. Banerjee et al. , Phys. Rev. B 97, 1 (2018). 25N. Satchell, R. Loloee, and N. O. Birge, Phys. Rev. B 99, 1 (2019), arXiv:1904.08798v2. 26X. Liu, J. K. Jain, and C. X. Liu, Phys. Rev. Lett. 113, 1 (2014). 27J. Arjoranta and T. T. Heikkil a, Phys. Rev. B 93, 1 (2016). 28J. R. Eskilt, M. Amundsen, N. Banerjee, and J. Linder, arxiv (2019), 1906.07725. 29K. Usadel, Phys. Rev. Lett. 25, 507 (1970). 30Eq. (1) is written in the strict diusive limit and do not take into account charge-spin conversion terms which are higher order in the momentum relaxation rate32,33. 31M. Y. Kupriyanov and V. F. Lukichev, Sov. Phys. JETP 67, 1163 (1988). 32F. Konschelle, I. V. Tokatly, and F. S. Bergeret, Phys. Rev. B 92(2015), 1506.02977. 33F. S. Bergeret and I. V. Tokatly, Epl 110(2015).

1306.4904v1.Induced_spin_filtering_in_electron_transmission_through_chiral_molecular_layers_adsorbed_on_metals_with_strong_spin_orbit_coupling.pdf

1 Induced spin filtering in electr on transmission through chiral molecular layers ad sorbed on metals with st rong spin-orb it coupling Joel Gersten(1), Kristen Kaasbjerg(2) and Abraham Nitzan(2) (1) Department of Physics, City College of th e City University of New York, New York, NY 10031, U.S.A., jgersten@ccny.cuny.edu (2) School of Chemistry, the Sackler Faculty of Science, Tel Aviv University, Tel Aviv, 69978, Israel, nitzan@post.tau.ac.il Abstract Recent observations of considerable spin polarization in photoemission from metal surfaces through monolayers of chiral molecules were followed by several efforts to rationalize the results as the effect of spin-orbit interactio n that accompanies electronic motion on helical, or more generally strongly curved, potential surfaces. In this paper we (a) argue, using simple models, that motion in curved forc e-fields with the typical energi es used and the characteristic geometry of DNA cannot account for such observations; (b) introduce the concept of induced spin filtering, whereupon selectivity in the transmission of the electron orbital angular momentum can induce spin selectivity in the transmission process provided there is strong spin- orbit coupling in the substrate; and (c) show that the spin polarizability in the tunneling current as well as the photoemission current from gold covered by helical adsorbates can be of the observed order of magnitude. Our results can account for most of the published observations that involved gold and silver substrates, however rece nt results obtained with an aluminum substrate can be rationalized within the present model only if strong spin-orbit coup ling is caused by the built-in electric field at the molecule-metal interface. 2 1. Introduction Recent observations[1-3] [4] of spin-selec tive electron transmission through double- strand DNA monolayers adsorbed on gold substr ates have attracted considerable interest stemming from the surprising appearance of an apparently large spin-orbit coupling effect in an environment where such large coupling has not been previously observed. Indeed, the following observations need to be rationalized: (a)[1] High longitudinal (normal to the surface) spin polarization, up to ~6 0 % A (the “-“ sign indicates that the majority spins are antiparallel to the ejecting electron velocity, that is, pointing towards the surface) is observed in photoelectrons ejected from gold covered by self-assembled monolayer of dsDNA at room temperature, largely independent of the polarization of the incident light. The light ( 5.84 eV with pulse duration ~200 ps) was incident normal to the sample. (b)[1] Using four different lengths of the dsDNA (26, 40, 50 and 78 base pairs) the spin polarization observed on polycrysta lline gold surface appears to in crease linearly with molecular length. The ~ -60% polarization was obtained with the 78 base-pair monolayer. (c)[1] ssDNA monolayers show essentially no sp in filtering effect (or rather, a small positive polarization which is barely detectab le above the experimental noise). (d)[1] The observed spin polarization is independ ent of the final kinetic energy of the ejected electron in the range 0…1.2 eV provided by the ejecting light. (Note that the kinetic energy of the emitted electrons reflects the shift in the metal work function caused by the adsorbed monolayer). (e)[2] Spin selectivity appears to play a role al so in the current voltage response of junctions comprising one or a small number of dsDNA oligomers bridging between a nickel substrate and a gold nanoparticle, probed by a conducting AFM (Platinum coated tip) at room temperature. The voltage threshold for conduction and the conduction itself are sensitive to the direction of magnetization induced in the Ni substrate by an underlying magnet, indicating that transmission through the chiral monolayer is sensitive to th e spin of the transmitte d electron. The effect disappears when a non-chiral layer is used or when the Ni substrate is replaced by gold. (f)[2] While the I/V behavior appears in these experiments to depend on the length (number of base pairs) of the DNA bridge, the small number of samples used (26, 40 and 50 base pairs) and the statistical noise that characterize single molecule junctions make it impossible to reach a firm 3 conclusion about length dependence of the asymmetry under reversal of magnetization in the Ni substrate. (g)[4] Spin selectivity appears also in junction s involving dsDNA adsorbed on silver. Here an open circuit configuration was used and the spin-selective electron transfer across the DNA monolayer was inferred from the magnetic field dependence of the voltage induced between the silver substrate and an underlying magnetized nickel, following electron transfer across the DNA from the silver to an optically excited dye attached on the other side of the DNA molecule. The asymmetry is strongly temperature dependent (inc reasing at lower temperature) and at room temperature appears to be substantially smaller than that the effects described above on gold, although direct comparison cannot be made because of the different experimental configurations used. As in [1], a linear dependence on the DNA chain length is found. The account of these observations should be supplemented by the well-known fact that photoemission from surface states of solids such as gold characterized by strong spin-orbit coupling using circularly polarized light shows a marked spin polarization that depends on the light polarization.[5] Photoemission from Au(111) with light incident normal to the surface shows[1] electron spin polarization of 22% A whose sign reverses with the orientation of the circularly polarized light. No such asymmetry is observed with linearly polarized light. Earlier observations of the overall (not spin resolved) photoemission[6] or photo-induced transmission[7] through chiral molecules induced by circularly polarized light show yields that depend on the combination of molecular helicity and light polarization. This can be rationalized[3] as consistent with the later observations desc ribed above: the different spin components transmit through the chiral molecular layer with different efficiencies and, consequently, the overall transmission of the asymmetric spin distribution photoemitted from the gold under circularly polarized illumination depends on the matching between the molecular helicity and the light polarization. These experimental observations were followed by several attempts to provide theoretical rationalization of these data. It is natural to suspect the implication of spin orbit coupling in the helical molecule as the source for spin selectivit y. Indeed, while the atomic spin orbit coupling in carbon is rather small, as evidenced by the very small effect measured in electronic collisions with chiral molecules in the gas phase[8-10] there are experimental and theoretical indications[11-15] that the curvature and torsion imposed on the electron path in helical 4 structures such as carbon nanotubes leads to larg er spin-orbit coupling than in their linear or planar counterparts[16] due to the overlap of neighboring carbon p orbitals of different symmetries. It is also interesting to note that me asured spin-orbit coupling in carbon nanotubes is found[14, 15] to be considerably larger (~3 meV) than is indicated by tight-binding based theoretical calculations that take curvature into account. It is therefore tempting to associate the observations of Refs. [1-4] with spin filtering ideas such as proposed in Refs. [17] and [18]. Recent theoretical efforts[19-25] have pursued this path in somewhat different ways. The authors of Refs. [19] and [20] have considered spin dependent electron scattering by the helical potential in analogy to earlier gas-phase scattering calculations[8-10], while those of Refs. [21, 23, 24] have focused on band motion in a tight-binding helical chain. In both approaches, rather strong, and in our opinion questionable, assumptions are needed to account for the magnitude of the observed effects: Medina et al[20] invoke the density of scattering centers as a source of magnification, but do it by imposing unphysica l normalization on the electron wavefunctions while still considering only a few (usually two) scattering events. Gutierrez and coworkers [21] suggest that the origin of the strong observed effe ct is a strong internal electric field experienced by the electron moving along the helix axis, but do not support this assumption by actual calculations. Furthermore, Guo and coworkers[23] have argued that the model used in Ref. [21] (electron transmission through a single simple helix) should not yield any spin dependent transmission. Instead Guo and coworkers invoke a more complex model, a double helix with interchain interactions in the pr esence of dephasing, to get asy mmetric spin transmission, still without accounting for the magnitude of the effect. Note that the required dephasing appears to stand in contrast to the observation[4] that the magnitude of spin polarization increases at low temperature. Also, unlike the experimental observation, the spin polarization obtained from these calculations appears to be quite sensitive to th e electron energy. Note that these tight-binding calculations do not readily account for the observed chain-length dependence (the calculation in Ref. [23] does come close for particular choices of injection energy and dephasing rate), while in the scattering calculation of Ref. [20] this obse rvation is attributed to incoherent additive accumulation of the scattering probability. Finall y, Rai and Galperin have shown that pure spin current can be obtained in such tight-binding models from the combined effects of external AC electromagnetic field and DC magnetic field.[25] 5 In the present paper we examine the possible contribution of induced spin filtering to the transmitted spin polarization observed in Refs. [1 -4]. As explained below, such contribution to spin filtering by the helical molecular layer re flects the combined effect of orbital angular momentum filtering that characterizes electron transmission through helical molecules and strong spin-orbit coupling at th e metal surface. The latter may reflect the intrinsic spin-orbit coupling property of the substrate, and/or Rashba-t ype coupling associated with built-in electric field at the molecule-metal interface. For example, in the isolated gold atom the energies of the lowest-lying excited states of electronic configurations (5d)9(6s)2 2D3/2 and (5d)9(6s)2 2D5/2, with energies of 1.136 eV and 2.658 eV above the gr ound state, respectively, reflect a spin-orbit splitting of 1.522 eV that results from the intens e electric field in the inner core of the atom which is in turn caused by the large atomic number and the short-range screening of the electric field by the core electrons.[26] In gold metal, band-structure calculations of the partial density of states for the d electrons[27-29] show that the spin-orbit splitting in gold and silver are 2.65 eV and 0.79 eV, respectively. These calculations are in agreement with high-resolution x-ray photoemission measurements.[30] The mechanism considered is similar in spirit to the mechanism for spin polarized photoemission by circularly polarized light[31]. It is simplest to make the point for photoionization of single atoms. Circularly polarized light couples specific eigenstates of the electronic orbital angular momentum, denoted ,llm for a given quantization axis. In the presence of spin-orbit coupling, the atomic angular momentum eigenstates ,jjm correspond to the total angular momentum and its azimuthal proj ection. Still, the information encoded in the selection rules for coupling between the ,llm states affects the transitions between ,jjm states (through the corresponding mixing or Clebsc h-Gordan coefficients) so as to affect the spin distribution of the ejected or transmitted electrons. The same argument holds for photoemission, in particular when the electrons originate from relatively narrow bands that maintain to some extent the local atomic symmetry. The orbital angular momentum “filtering” in photoemission by circularly polarized light is thus translat ed at the metal surface to spin filtering. The proposed mechanism also have some conceptual similarity to a recent suggestion by Vager and Vager,[32] who argue that curvature induced spin orbit coupling leads to correlation between the spin and orbital currents that results in transmitted spin selectivity in any curved 6 path irrespective of the curvature. In our opinion, this correlation implies that the transmission of up-spin with momentum k and that of down-spin with momentum kk are equally probable, with 0 k when the curvature vanishes, so it can account for spin selectivity only by fine- tuning a narrow emitted energy window (see also [33]). The essence of our proposal is that helical molecules can act similarly to circularly polarized light in affecting angular momentum filtering. This is based on the observation that under suitable conditions electron transfer can have the characteristic of current transfer, [34-36] that is, the transferred electron can carry information about its linear and/or angular momentum. Such a picture was used previously[34] to interpret the observations,[6, 7] already mentioned above, that when electron transfer or transmission induced by circularly polarized light take place through chiral molecules, their efficiencies are larger when the light polarization matches the molecular helicity than when it does not. Similarly, in the present case, opposite angular momentum lm states couple differently to the molecular helix and, provided the substrate surface is characterized by strong SO coupling, this orbital angular momentum filtering translates into spin filtering during the injectio n process. This picture implies that the spin filtering observed in References [1-3] [4] may reflect the spin-orbit coupling at the metal- molecule interface in addition to any spin filtering in the molecular layer itself. An immediate consequence of this model is the prediction that the effect will be smaller for interfaces with weaker spin-o rbit coupling, which seems to be consistent with the weaker effect found on silver,[4] but not with recent results obtained on Aluminum.[37] It should be kept in mind, however, that Rashba spin-orbit co upling can result from strong interfacial fields at metal-molecule interfaces that in turn depend on the electronic chemical potential difference between the metal and the adsorbate layer and are made stronger because of the short electron screening length in the metal. In this paper we explore other implications of this picture, using several different models for the electron propag ation through the molecular environment. We start in Section 2 by considering the effect of SO coupling in the helical molecular structure. We analyze two models for electron transport through a helical structure where the SO coupling is derived from the helical potential, and show that such models cannot account for the observed spin polarization. In Section 3 we introduce and discuss the concept of induced filtering. Sections 4 and 5 consider angular momentum selectivity an d the consequent spin filtering for different transmission models: One (Section 4) considers electron transmission through a helical tight-7 binding chain and the other (Section 5) describes on electron scattering by the molecular helical potential. The first model seems to represent the situation incurred for electron tunneling transmission with energy well below the vacuum leve l, while the other is more suitable for the description of photoemission, where the electron energy is larger than the vacuum level. We calculate the spin filtering associated with each of these models and compare its properties as compared with the experimental observations. Section 5 concludes. 2. Spin orbit coupling induced by motion through the helix In this Section we analyze the implication of electron motion through the helical structure on its spin evolution caused by the ensuing spin orbit coupling. We find that the predicted effect is small. While the actual motion of the electron should be obtained by solving the Schrödinger equation under the effect of the electron-molecule coupling, we expect that a reasonable order- of-magnitude estimate can be obtained by considering two limiting cases. In one, the electron is assumed to travel in a 1-dimensional path along the helix. In the other the unperturbed electron is assumed to be a plane wave travelling in the z (axial) direction and to be scattered by the helical potential. 2a. Spin rotation duri ng helical motion . Consider an electron moving along a 1-dimensional helical path embedded in 3- dimensional space. The spin degrees of freedom will be treated quantum mechanically and the translational motion will be treated cla ssically. Denote the helix radius by a, the pitch by p and the speed along the axis of the helix by v (see Fig. 1). For a right-handed helix the location of the electron as a function of time is 2 2zxa c o sp zya s i np zt v ( 1 ) Fig. 1. radius a move w The vel o and By Ne w where m is simp l spin or b A generic h a, pitch p an with constan t ocity and a c 2 2caxp ayp z v v v 2 2 0xap yap z v wton's seco n Fm r m is the ele c lified in th a bit interacti o 4SOH m helical stru c nd length L t speed v al cceleration c 2sin 2cosz p z p v 2 22cos 2sinp p v v nd law, the f V ctron's mas s at the prese n on (includin 2r mc cture refer r (the latter n long the axi component s z p z p force respo n s and V is th nce of disc r g the Tho m 24V mc8 red to in th e not indicat e al direction s are, respe c nsible for th e he potential rete molec u mas 1/2 fact o rF e present s e ed). The tra n . ctively, e accelerati o confining t h ular groups or[38]) is gi vection. It i s nsmitting e l on is he electron . along the h ven by s characteri z lectron is a s (2) (3) (4) . Note that t helix is ign o (5) zed by its ssumed to this model ored. The 9 Here c should be taken to be the speed of light in the adsorbate medium. Thus, 2 2 222 2 2 4SO z x yttHa a s i n c o spp p p c vv v vv (6) These equations are the same in form as those used to describe electron spin resonance or nuclear spin resonance (see e.g. Ref. [39]). The Heisenberg equations of motion for the dynamical Pauli matrices are (),( )SOdt iH tdt ( 7 ) (Other terms in the Hamiltonian commute with ߪԦ). Introducing the vector 32 2 2 22 222 2 2 2 44ˆ ˆˆ 4aa t a tAt k sin i cos jpp p p p cc c vv v v v v v (8) the equations of motion become the familiar Bloch equations for the precession of spin around a time-dependent vector () 2() ()dtAttdt ( 9 ) In terms of the scaled time 2t pv ( 1 0 ) and the dimensionless parameters 2 2ab pcv ; agcpv ( 1 1 ) the equations of motion become 2 22 2 ()x zy y zx z yxdbc o s gd dbs i n gd dbs i n c o sd ( 1 2 ) In particular, the timescale for changing z is seen to be 221232 scpct av ( 1 3 ) 10 The corresponding length scale is ݖ௦ൌݐݒ௦ or 2 2scz pc pav ( 1 4 ) Using the parameters c = 3x108 m/s, a = 10 nm, p = 3.4 nm, and v= 5.9x105 m/s (1eV) one gets 4~4 1 0sczp . Thus, only little rotation of the spin ca n be expected when an electron traverses a helix consisting of several turns. This consideration is also supported by a simple perturbation calculation (see Appendix A). The same magnitude of the effect is expected for a cross-coupled double helix structure. 2b. Electron scattering by a helical potential . In an alternative picture, consider an electron moving in the outwards z direction while interacting with the helix potential, and the spin polarization that results from the Rashba interaction. Again, the velocity of the electron in the z direction is taken to be constant, so ztv . The motion in the x- and y- directions will be treated quantum mechanically. The unperturbed Hamiltonian is taken to be 2 0 ,, ; ˆ 2ˆx ypH Vx yt p p i p jm (15) where the time dependent helical potential experienced by the electron in its rest frame is modeled as 0 0,,V x y t V xcos t ysin t a xsin t ycos t V xC yS a xS yC (16) where a is the radius of the coil, V0 is the strength of the interaction and 2, cos , ttC t S s i n tp v (17) It is convenient to use a rotating coordinates frame by defining ' 'xxcos t ysin t xC yS y xsin t ycos t xS yC , (18) in terms of which the model potential is given by 0 '',' ' , Vxyt V x a y . ( 1 9 ) The magnetic field is given by 11 221 EB V ce c vv ( 2 0 ) where c is the speed of light in the medium . The spin-orbit interaction Hamiltonian is soH B ( 2 1 ) where ߤԦ is the magnetic moment associated with the spin 22 2eegsgmm ( 2 2 ) Eqs. (20)-(22) lead to 24sogH V mc v ( 2 3 ) Let ݒԦൌ݇ݒ (where ݇ is a unit vector in the direction of the z axis) and take g = 2. Introducing also the Thomas factor (1/2), the spin-orbit interaction becomes 2ˆ 4v soH kV mc ( 2 4 ) In Appendix B we use time-dependent perturbation theory (first order) to calculate the amplitude for making a transition from an initial state ሺ݇ሬԦୄ,ݏሻ to a final state ሺ݇ሬԦ′ୄ,′ݏሻ, where ݇ሬԦୄ corresponds to motion in the xy-plane and the spatial parts of the initial and final wavefunctions are ߰ൌ݁ሬԦ∙ோሬԦ/√ܣ and ߰ൌ݁ሬԦᇲ∙ோሬԦ/√ܣ ,A being the normalization surface in the xy plane (the resulting probability is multiplied below by the number of adsorbed helical molecules in the normalization area, so that the transition cross-section will be proportional to the density h of such molecules). The result for the transition amplitude is 2 † 0 ' ,' 2 ' 00 1 0 2 4N iqacos s s ksq iV pce dq A mc v v (25) where s and 's are the initial and final spin vectors, ' qk k and x y qq i q . In the case that N is an integer one obtains † 0 0 ' ',' 20 1 0 4s s ksq iNpVcJ q aq A mc (26) 12 Spin flip transiti ons occur when ( s,s') = (1,-1) or (-1,1). The transiti on probability in either case is obtained from the square of the amplitude calculated from (26) multiplied by the density of final states, 2/2A 22 00 00 22() () 1() 16 16hNV pqJ qa NV pqJ qaPqA mc mc (27) The expression on the right is obtained by fu rther multiplying by the number of helical molecules, hhNA , adsorbed in the normalization area, where h is the surface density of such molecules. The transition probability (27) depends on the s quare of the molecular length. More importantly, there is no difference between positive and negative helicity states and hence no spin selectivity occurs for a given q. The total transition probability is obtained by integrating over all wave-vector transfers 2 2 0 23 2 0 22 0 22222 2 2 00 1 111 2(4 ) 422 2 2 2 2 1 23k totalVN pP d qP q dqq J qaA mc akka J ka kaJ ka J ka k a J ka (28) where the parameter χ is 2 0 22 22(4 )hVN p mc a ( 2 9 ) For V0 = 10 eV nm2, Np = 50 nm, mc2 = 511 keV (for this estimate we use the speed of light in vacuum), a = 1 nm, and 20.3 nmh this yields 910 . For an electron energy of 1 eV the value of k is 6x109 so ka ൎ 6 and the order of magnitude of the result is not changed by much. Furthermore, the transition probability is symmetric for positive and negative helicities and equivalently, as noted above, for and spin transitions. It therefore does not lead to a net spin polarization. In Appendix B we furthe r show that a model with two helical molecules yields essentially similar results. In conclusion, the axial motion of the electron through the helix is not a good model for explaining the spin polar ization of the electrons that pass through the DNA molecule. It should be noted that because motion in the z direction has be taken classical, the calculation that lead to Eq. (28) does not take into account possible constructive interference in the diffraction of the electronic wavefunction from the periodic helix structure (see Section 5, 13 Eq. (88) and the discussion following it for a tr eatment that takes this constructive interference into account). Removing this shortcoming still leads to a negligibly small contribution.[40] 3. Induced filtering We define induced filtering (or induced selectivity) as a process where geometry or symmetry-imposed selec tivity in one variable A causes selectivity in another variable B that is coupled to it. Specifically, let ˆˆˆH AB where ˆˆ,0AB. Then eigenstates of the Hamiltonian can be written as products ,ab ab of eigenstates of the ˆA and ˆB operators. When in such a state, the probability to observe the system in eigenstate state b’ of ˆB, is obviously ,'bb . Consider the transformation ,' ,ab a b induced by some external or internal perturbation represented by a hermitian operator ˆV that couples only states in the a sub-space, so that ',ˆab ab V (e.g., in optical transitions ˆV is often the dipole operator) with ',|| || 1ab. As indicated, the state in the B subspace is not affected by this coupling, so the probability of observing a particular value of the B variable is the same before and after the transition. Indeed, ', , , ' , ' '' T r ' || ' ' ||b A ab ab bb bb aPb V V b a V a a V a (30) In the presence of coupling between ˆA and ˆB the eigenstates of the system Hamiltonian are no longer simple products of a and b, but can be expanded in the form ab abca b ( 3 1 ) For a system in this state the reduced density matrix in the B subspace is * ' ,'TrB A ab ab bb acc ( 3 2 ) and in particular, the probability to observe the system in state b is 2 ba b aPc ( 3 3 ) 14 The transformed state is now ˆˆab abVc b V a and the reduced density matrix in the B subspace is * '' ,' ", ' * '' ,'ˆˆ Tr " ' " 'B Aa b a b bb aa a ab a b aaVV c c a V a a V a cc a V V a (34) The probability to observe b is then * ' ,''ba b a b aaPc c a V V a ( 3 5 ) Comparing to (33) we see that the final distribu tion in the B-subspace is now affected by the transition - any selectivity expressed by the '| |aV Va matrix elements is partly imparted into this distribution. As a variation of this theme we note that instead of TrA in (34) we often need to sum only over states on an energy shell, so that the quantity of interest is * '' " ,' ", ' * '' ,' ,'"' " 1 2B ab a b a bb aa a ab a b a a aaEc c a V a a V a E E cc E (36) where ,' " "2' | | " "aa a aE aVa aV a E E (37) Eqs. (35) and (36) are manifestations of induced selectivity. In what follow we consider two concrete examples. Induced selectivity in transmission . Consider a junction in which a molecular bridge M connects two free electron reservoirs, L (left) and R (right), as seen in Fig. 2. The transmission function E is given by the Landauer formula †TrLRE EG E EGE (38) where G is the molecular retarded Green function and K , , KLR is twice the imaginary part of the self-energy of the bridge associated with its coupling to the reservoir K and the trace is over the bridge subspace. In the basis of eigenstates of the bridge Hamiltonian 15 ,','2K nk k n knnkEV V E , (39) where the sum is over the free electron states of energies kin the reservoir K. The subscript k enumerates the states in the reservoirs, and is usually associated with eigenstates of operators that appear in the Hamiltonian and commute with it. Consider now the situation where the single electron states in R are characterized, in addition to their energy, by quantum numbers l, s associated with independent operators ˆL and ˆS that commute with the Hamiltonian, while in L these operators are coupled by some internal single electron force field, so that only some combined operator ˆJ commutes with the Hamiltonian. For example, if the left and right electronic reservoirs are metals with single electron states described by Bloch wavefunctions i nn eukr kkrr , the quantum numbers n (or a set of such numbers) characterizing different bands will have atomic character if the bands are narrow relative to the spacing between the parent atomic levels. In such a case, n can stand for the quantum numbers ( l,s) of the orbital and spin angular momenta in a metal w ith no spin-orbit coupling, while in the presence of spin orbit coupling only the state j of the total angular momentum is meaningful. (Note that in reality we should also consider the projections of these vector operators on some axis, and the corresponding quantum numbers ,lsmm and jm. This is done in the application discussed in Section 4). Equation (39) can then be recast in more detailed forms ,, ',' ,';2LL j L j nj k j kn j knn nnjkEV V E (40a) ,, ',' ,',;2RR l s R l s n lsk lsk n lsknn nnls kEV V E (40b) 16 Fig. 2. A schematic view of a helical molecular bridge connecting two metal leads, left and right, characterized by the electronic electrochemical potentials, ,LR, respectively. Eqs. (40) are identical to what will be obtained in a multi-terminal junction, where each of the Lj and Rls groups of states represent different terminals. The transmission fluxes between two such terminals take the forms[41] †TrLj Rls Lj Rls E EG E EGE (41) Whether such fluxes are measurable or not depend on the energetic details of the system. For example, if the bands j are energetically distinct, it is possible in principle, by tuning the voltage bias window, to focus on the flux associated with a particular “ j terminal”. If the electrons emerging on the right can be also analyzed and their quantum state ( l,s) can be determined, we are in a position to determine the flux associated with the transmission function of Eq. (41). In the application considered in Section 4 we are interested in the transmission into a particular eigenstate s of the operator ˆS (that is, in an experiment where ˆS is monitored in the outgoing electronic flux in R). This corresponds to the transmission function †=TrLj Rs Lj Rs Lj Rls l Rs Rls lE EE G E E G E (42) Suppose now that the bridge Hamiltonian, as well as the Hamiltonian of the R reservoir and the couplings between the bridge and the reservoirs, do not depend on the operator ˆS. In this case (cf. Eq. (40)) ,, ','2Rls Rl nl k l kn l knnkVV E as well as Rs defined in Eq. (42) do not depend on s. In particular, Rs will be denoted R below. On the left, writing (as in (31)), ,jl s lsj cl s ( 4 3 ) we find * ,, ',' , ',''2Lj Ljs jl s jl s nl k lkn j knnkll s scc VV E (44a) 17 * ,, ',' , ',''2Ljs jls j l s n lk l k n jknnkllcc VV E (44b) If in addition we disregard in the Green functions in Eq. (42) terms that make them non spin- diagonal, then the separability of Eq. (44) into its s components make it possible to write the transmission function for the flux from the L terminal into a particular state s in the outgoing flux on the right in the form †TrLjs R Lj Rs E EG E EGE (45) Two comments regarding this result are in order: First, although we have made the assumption the bridge Hamiltonian does not depend on S, the Green functions GE are in general non spin-diagonal because of the self-energy terms associated with the coupling between the bridge and the left reservoir in which strong S-L coupling exists. Such terms can couple different s states of the bridge through their interaction with the same j-state on the left reservoir. Such couplings have been disregarded in obtaining Eq. (45) - a reasonable approximation when the molecule-lead coupling is not too strong. Second, the appearance of the subscript j reflects our assumption that transmission out of the L reservoir is dominated by a particular band whose atomic origin is indicated by the quantum number j. Equation (45) constitutes our final result for th is case. To see its significance consider, for example, the transmission from L to R as would be realized if the bias is such that (a) a particular band j in L is the source, and (b) the L states are occupied while their R counterparts are vacant. The probability to measure a value s for the observable ˆS in the source terminal is, from Eq. (43) 2 ,Lj s js l lPc ( 4 6 ) while the probability for this measuremen t in the exit terminal is given by Lj Rs R s Lj Rs sEPE ( 4 7 ) Obviously, R Lj ssPP , implying that the bridge acts as a ˆS filter although its transmission properties do not depend on ˆS. As already noted, in Section 4 we will replace j, l, and s by 18 ,jjm, ,llm and ,ssm - the quantum numbers that characterize the total, orbital and spin atomic angular momenta and their projections, respectively. Finally, we note two simplified special cases. First, in the case of a single state bridge, or when the coupling between the bridge and the left terminal is channeled through a single state of the bridge (denoted by 1), Eqs. (47) and (45) lead to Ljs R sLjs sP ; * ,, ' 1 ,' , 1 '2Ljs jls j l s lk l k jk kllcc V V E (48) Second, as will be seen below, sometimes the sum over l,l’ is dominated by the diagonal 'll contributions, in which case 2 2 ,1 , 2Ljs jls lk jk klcV E (49) Induced selectivity in photoemission . We consider photoemission fr om a simple atomic lattice model, where the electronic bands are narrow relative to the energy separation between the electronic levels of the constituent atoms. Photoemission then reflects the symmetry property of ionization from a single atom with one differenc e - the existence of the solid-vacuum interface. Accordingly, we consider a one-electron atom located at the origin and positioned at a distance a to the left of this interface, represented by a planar surface, za. To the right of this surface is vacuum. The interface is simply treated as a step potential given by 0 0 Vi fzaVzifza ( 5 0 ) where 00 V. Consider next the atomic state in the absence of the interfacial wall. It is taken to be an eigenstate of total angular momentum operator ˆj and its azimuthal projection ˆzj, with quantum numbers j an d mj, respectively. In terms of eigenstate s of the orbital angular momentum and spin operators, the corresponding wavefunction takes the form ,, , ,|( ) ( , )j ls lss jml s j j l l m m mmu r lm sm jm r Y v (51) 19 Here ,(, )llmY are eigenfunctions of the angular momentum operator, ,()jlr v are their radial counterparts and ߯ೞ௦ are spin wavefunctions - two-component spinors. The symbols |ls jlm sm jm are Clebsch-Gordan coefficients. We assume that this wavefunction was obtained by absorbing a photon, so its energy E is positive and the state is 21j degenerate.[42] This atomic wavefunction is embedded in a co ntinuum of states associated with the semi- infinite spaces to the right and left of the wall at z = a . In the vacuum, za, the Schrödinger equation is 22,, 0R kR z ; 222 km E (52) where m is the (effective) electron mass. The relevant solutions may be expr essed in the form of a sum of Bessel transforms , ,0,,l ls l s lsiqz im s Rm m m m mmRz A Q JQ R e d Q (53) where ݍൌඥ݇ଶെܳଶ for ܳ൏݇ and ݍൌ݅ඥܳଶെ݇ଶ for ܳ݇ . In the former case these are outgoing waves into vacuum whereas the latter case describes evanescent waves in the vacuum side of the interface. In the solid, za, the free Schrodinger equation is 22,, 0LRz ( 5 4 ) where 22 0 2 km V . The outgoing solution, i.e., a le ft-travelling wave, may also be expressed as a sum of Bessel transforms ' , ,0,,l ls l s lsiq z im s Lm m m m mmRz B Q JQ R e d Q (55) where ݍᇱൌඥߢଶെܳଶ. We will approximate the total wave function in the solid with the atom as a linear combination of an atomic wave function and the free wave function. By using the free wave function rather than the more general solution of the solid-plus-atom potential we are neglecting final-state interactions.[43] In this a pproximation, the total wave function for za is , ,, ,,j Lj m R zR z u r ( 5 6 ) 20 whereas in vacuum, za, the total wave function is simply ,, ,,R Rz Rz ( 5 7 ) This wavefunction represents a scattering ‘in-state’. A similar construction may be used to obtain the scattering ‘out-state’. The expansion coefficients A and B in (53) and (55) can be found by matching the wave functions at the surface za, using the continuity of and its normal derivative at this surface. This is done in Appendix C, leading to ,', ' 1 ' ' ls lsiqa iqamm iq a iq ammAQ qe i e qq BQ qe ie (58) where 22 ,, 0| Θ,0ll ls j m j l l m Ql m s mj m J Q R R a Y R d R v (59) and 22,, 0|( , 0 )ll ls j m j l l m rR asinQl m s mj m J Q R c o s rY R d Rrr v (60) The coefficients ܣ,ೞሺܳሻ and ܤ,ೞሺܳሻ are seen to be simply proportional to the Clebsch- Gordan coefficients. Eqs. (53), (57), and (58)-(60) give an explicit expression for the outgoing solution outside the solid. The emitted electron flux in th e direction normal to the surface is ** 2zR R R RJmi z z ( 6 1 ) The total current is obtained by integrating the curr ent density over an area in vacuum parallel to the surface 21 2222 , , 00 0 2 222 2 ,, 2 , 00 2 ,, 02 2|' Θ,0 ' (, 0 )ls ls ll ls llk zz m m mm k ls j m j l l m mm mj l l m rR aqId R R d J d Q A QmQ qQlm sm jmd Q qJ Q R R a Y R d Rm qq sinJQ Rc o s r Y R d Rrr v v (62) To get (62) we have used the identities ' ,' 02ll llim m mm ed ( 6 3 ) and (orthogonality for spinors) † ', 's ss ss mms mm ( 6 4 ) Also, the upper limit of the Q-integration has been changed from ∞ to k since for Q > k the variable q is imaginary and there is no contribution to the current. As before (Eqs. (45) and (44b)), the appearance here of the Clebsch-Gordan coefficients in the emitted current implies that if an lm-filter was in effect, induced filtering of sm could result. In particular, lm selectivity can be imposed by circularly polarized light. Indeed it should be noted that our treatment is an analogue of the Fano theory of spin-polarized photoemission from atoms characterized by strong spin-orbit coupling,[31] generalized to the presence of the solid-vacuum interface. 4. Induced spin filtering in tunneling through a molecular helix Here we implement the results from Section 3, Eqs. (44), (45) and (47), to calculate the induced spin selectivity in a model that incorporates a metal substrate and an adsorbed helical molecule. While we keep the calculation at a gene ric level, we use the band structure of gold and the structure of the DNA helix to choose specific parameters when needed. It should be emphasized that the actual behavior of electron tunneling between substrate and adsorbate depends on details of the electronic structure as manifested mainly in the alignment between 22 their levels and in their electronic coupling. As we are not using such data but instead make assumptions and take shortcuts in order to simp lify the calculation,[44] the results obtained below should be regarded as suggestive of the order of magnitude of the spin polarization effect, rather than conclusive. In order to get such estimate, the following assumptions are made (a) The tunneling electrons originate primarily from the relatively narrow d-band of gold. More specifically, this band split into a higher energy 2 52D and a lower energy 2 32D band which are somewhat overlapping,[27] and we assume that the tunneling current is dominated by the 2 52D sub-band. This spectroscopic term reflects the atomic parenthood of these states, of orbital angular momentum 2l and total angular momentum 52j . (b) The DNA molecule is represented by a tight binding helical chain with nearest neighbor intersite coupling V and axis normal to the gold surface, taken below as the z direction.[45] (c) The DNA-substrate coupling is dominated by the substrate atom at position Ar nearest to the DNA (see Fig. 3). We disregard crystal-field distortion of the atomic wavefunctions, so the relevant coupling results from the overlap between the 2l wavefunctions of this atom and the DNA site orbitals. In the calculation below we assume that this coupling, between the atomic wavefunction 2,l lm A rr centered at Ar and a DNA site wavefunction centered at nr is proportional to 2,l lmn A rr. Otherwise, the substrate density of states in the energy range relevant for the tunneling process is assumed constant. Atomic wavefunctions used are hydrogenic wavefunctions for the 5n (outer gold) shell, calculated with effective atomic number Z=2 to account for screening by inner shell electrons. 23 z A12NN-1 p a Fig. 3. The DNA-substrate model, with the D NA described as a tight-binding chain while the DNA-substrate coupling is assumed to be domi nated by a substrate atom A nearest to the molecule. To evaluate the transmission probability we need to specify the substrate density of states , the position 1 Arr of the helix site 1 nearest to the surface atom (Fig. 3), the self-energy of the helix associated with its coupling to the reservoirs near positions 1 and N, and the geometrical and electronic structures of the helix as expressed by the relative position of the helix sites and the intersite coupling. Only the last two intrinsic helix properties affect the resulting spin polarization of th e electronic wave injected into the helix, however the overall spin polarization at a detector placed outside the far end of the helix, as expressed by the analog of Eq. (47), also depends on the transmission prope rties at the two interfaces (see below). In cylindrical coordinates the positio n of a helix site is written ,,za, where a is the helix radius. The surface atom is placed at 0, , 0AA Azr so that Ar measures its distance from the symmetric position on the axis, the first helix site is placed at 11,, 0za and subsequent sites are positioned so that two nearest neighbors are positioned at ,,za and ,, 2 /p p zp Na N so that the nearest neighbor distance is 24 2221 c o s 2nn p pda N p N . In our calculation we use typical DNA values for the helix: radius, 1nma , pitch, 3.4nmp and number of sites per pitch, 11pN. The intersite coupling is set to 1V and is used in what follows as our unit of energy. Because of the Kramers degeneracy, substrate states that belong to a given j band must appear as degenerate pairs jm . From jlsmm m , it follows that for each substrate state with a given sm parent there is a substrate state of the same energy with the opposite, sm, parent. Therefore, if the transmission process is by itself spin independent, without S-L coupling both spin orientations will be expressed in the transmitted flux in equal amounts. As seen in Section 3, spin selectivity can be affected in the transmission process by the combined effect of (a) dependence of the transmission on the orbita l motion and (b) the spin-orbit coupling in the substrate or at the substrate surface. The calculation proceeds by rewriting Eqs. (44b) and (45) so as to take into account the actual selection rules. For our problem the ope rators of interest are the orbital angular momentum ˆL, the spin ˆS and the total angular momentum ˆJ as well as their projections ˆˆ,zzLS and ˆzJ. The corresponding quantum numbers are 1/2s and 5/2j and 2l as determined by our assumption concerning the incoming electrons. Also, the spin projection, ms, is determined by the final measurement that checks whether sm is +1/2 or -1/2. The expressions for the transmission function equivale nt to Eqs. (45) and (44b) are 2 52† '''s sm N Dm nnE nE n n G E N E N G E n (65) ,' , ',* ''15 152 , ,, , 2 , ' ,, ,22 222s ll jllm nm m nnnlsj l sj mm mmm m m V V mm (66) where ,, , | ,ls jlm sm jm are Clebsch-Gordan coefficients. Since these coefficients vanish unless jlsmm m , Eq. (66) can be simplified. We get ,, ','2152, , , ,222s ll lm nm m nnnls l s mVV mmm m (67) 25 In obtaining this result we have assumed that all states of the 2d5/2 sub-band of gold contribute equally to the transmission. Other models could be considered. For example, a more careful study of the density of states of the 2d5/2 sub-band for gold shows peaks in the density of states that arise from Stark splitting of the different jm atomic states by the crystal electric field.[27] The jm states with the highest energy fill the energy interval within a depth of 2.8 eV below the Fermi level. If we assume that only this group of jm states contributes to the photoemission signal, the calculation described above will be modified. For example, if these states correspond to 5 2jm , that is, only these values of mj contribute to Eq. (67), this equation becomes ,2 2 , ',2 '12 152, 2, , 1 2 ,5 2222llnm m nnnVV (68) The model should be supplemented by the self-energies that account for the coupling of the helix to its environment. For the self-energy at the far end (site N) of the molecular helix we consider two models. In one, we take a complete ly transparent boundary, in effect assuming that the helix extends to infinite length, by associating with end site the exact self-energy of a tight binding lattice, 2 2 00 4 22NN NV i (69) where ε0 and V are the site energy and nearest neighbor coupling of the molecular tight-binding model. In the other model, we as sume that the space outside site N is characterized by a wide- band spectrum, and associate with this site a constant damping rate ΓN (i.e. /2 )NN i . On the surface side, one contribution to the se lf-energy comes from the coupling to the surface atom that dominates the electron injection. For a given 1/2sm state this is ,' ,'/2s sm m Ann nni ( 7 0 ) In addition, we assign a self-energy to site 1 of the helix that accounts for electron flux losses all other available states of the substrate. For this self-energy, 1 we take again one of the two 26 models used for N, that is, either the tight binding expression (69) or a constant 1 /2i . Finally, the Green functions that appear in Eq. (65) are obtained by inverting the Hamiltonian matrix of the helix, including the relevant self-energies 1 1 () = () ()r NA GI H (71) where H is the nearest-neighbor tight-binding Hamiltonian of the helix. Results of these calculations are shown in Figures 4a-c, which show the asymmetry factor 22 52 52 22 52 521/2 1/2 1/2 1/2ss ssDm Dm Dm DmE E EE E (72) as a function of the transmission energy. Here 12sm corresponds to spin projection pointing towards the positive z direction, that is, away from the surface. Figure 4a shows the asymmetry factor in a model where 1 and N are both given by Eq. (69), while Fig. 4b show similar results for the model with / 2 with 2 for 1,jj j ij N . The full (blue) line is the result for a calculation based on Eq . (67), that is, assuming that all jm states of the 5/2j band contribute equally, while the dashed (green) line corresponds to Eq. (68) that singles out the contribution of the 5/2jm states. In these calculations the substrate atom A is placed on the helix axis, in cartesian position ,, 0 . , 0 . , 0 . 1AA Axyz nm, while the position of the nearest helix site is 11 , 1,0 . , 1 . , 0 .xy z nm. Fig. 4c shows the effect of breaking this axial symmetry, taking , , 0.,0.5, 0.1AAAxyz nm. The following observations should be pointed out: Fig. 4a. intersit e The sel f a calcu l dashed ( not dep e The asym m e coupling o f-energies a t lation that t (green) lin e end on the h metry facto r on the heli x t sites 1 an d takes all co n e correspon d helix length r, Eq. (72), x. Note that d N are take ntributions ds to the ca s . 27 plotted aga the transm i n from Eq. associated w se where o n inst the tra n ission vani s (69). The f u with the j nly 5jm nsmission e n shes at the b ull (blue) li n 5/2 subs /2 contrib unergy (in u n band edges , ne shows t h strate band, utes. These nits of the , 2 E ). he result of while the results do Fig. 4b. with pitches ) configu r Fig. 4 c ,A A xy (a) Co n substra t the sub polariz a (b) The electro n bounda r observe approp r which t h Same as 4 a 2. The h e ) considere d rations. c. Same a s ,0 . , 0Az nsiderable s te through a strate toge t ation can be magnitude n spin is p r ry conditio n d in the riate for tu n he sign of t h a, except th a elix-length d d. The ins e s Figs 4a, b 0.5, 0.1 nm spin polari z a helical mo ther with o r substantial of the eff e referably p o ns and the i photemissi o nneling tra n he spin pol a at the self-e n dependence et shows t h b , exept m. zations can lecule by a rbital angu l and remai n ect as well olarized in interfacial g on experi m nsmission,[ 2 arization ca n 28 nergies at s i of these re s he transmi s that the i n be obtaine mechanis m lar momen t ns so even w as its sign ( the direct i geometry. W ments,[1] h 2] (see Sec t nnot be det e ites 1 and N sults is neg l ssion func t njecting at d for elect r m that relie s tum selecti v when the ax i (positive a s ion out of We recall t h owever th e tion 5 for t r ermined. N are taken t o ligible in t h tion for th e om is po s ron transm i s on strong s vity impos e ial symmetr y symmetry f a the surfac e hat negativ e e present reatment o f o be 1,N he length ra n e two out g sitioned of f ission out o spin-orbit c o ed by the h y is broken actor impli e e) is sensit i e asymmetr y calculation f photoemi s(/ 2 )i , nge (a few going spin f axis, at of a metal oupling in helix. This (Fig. 4c). es that the ive to the y factor is is more ssion), for 29 (c) We have found (not shown) that when 1 is set to zero, the asymmetry factor, Eq. (72), becomes practically zero. It should be pointed out that the effect of reflection is expected to be less pronounced in pulse experiments if the signal is over before appreciable reflection sets in, see, e.g. Ref. [34]. (d) In the reflectionless case, the length of the helix does not a ffect the resulting spin polarization. In the presence of reflection (Fig. 4b) the length dependence is still very small for lengths in the range of a few helix pitches. We no te that the effect of molecular length observed in the tunneling transmission experiment [2] is not very pronounced above the experimental noise. We conclude that this simple model of tunneling transmission can account for the observed spin polarization for tunneling out of gold. The computed polarization is positive and essentially independent of molecular length. It is however sensitive to reflections, and it should be kept in mind that reflections by structural irregularities, which are disregarded here, can translate into length dependence. We defer such considerations to future work. 5. Induced spin filtering in a scattering mo del for photoemission through a monolayer of helical molecules In this section we examine a different mechanism for induced spin filtering by the molecular helix, perhaps better suited to account for over-barrier transmission such as takes place in photoemission. The electron is assumed to have been excited by the light to a free particle state moving in the z (outward, normal to the surface) direction with enough energy to exit. We further assume that elastic collisions with the molecular adsorbate are the primary source for filtering electrons away from the outgoing flux. The calculation is simplified by an additional, rather strong, assumption, that a single collision with a molecular helix makes this electron lost to the detector (the actual process may involve c onsecutive collisions). Our goal is to determine the cross section for such collision and its dependence on the azimuthal quantum number ml. To this end we start with the Schrodinger equation 22 22mkr V r r ( 7 3 ) 30 where 222 km E. The relevant solution to Eq. (73) is expressed as a sum of an incident plane wave and a scattered wave ikz s re r ( 7 4 ) The incident wave, a solution of the homogeneous Helmholtz equation, represents the electron emitted by a photoexcited atom. Using a plane wave moving in the z direction (normal to the surface) is a choice based on our expectation that such waves are most likely to emerge through the adsorbed molecular layer both because they travel parallel to the molecular chains and because they carry the highest available energy in the exit direction. In terms of the Green function that satisfies the inhomogeneous Helmholtz equation with a point source, 22,' ' kG r r rr ( 7 5 ) the scattered wavefunction satisfies 22,' ' ' 'smrG r r V r r d r (76) and in the first Born approximation 1 ' 22,' ' 'B ikz smrG r r V r e d r (77) We are interested in the asymptotic form, r of this function. To this end we use the asymptotic form of the Green function in cylindrical coordinates ,,Rz (see Appendix D) ' 01,' c o s [ ] ' ) '(4ikrn r ikz cos nn neGrr e n i J k Rs i nr (78) where n is given by Eq. (148). Using this in Eq. (77) yields 1(,)ikrB serfr ( 7 9 ) where the scattering amplitude is given by 2 '(1 ) 2 0 00,' ' ' ' c o s ' ' ' 2n ikz cos nn nmf id z d R R dn e J k R s i n V r ( 8 0 ) 31 As a model for the DNA molecule we represent the scattering potential 'Vr as a helical delta-function potential 0 '2' 2''c o s' s i n Θ'Θ 'zzVr V x a y a z L zpp (81) where a is the radius of the helix, L is its length, p is the pitch and 0V is a constant of dimension energy x length2. In cylindrical coordinates this translates into '' ' 0 2'Θ'Θ ''V zVr R a z L zRp (82) which, when used with Eq. (80), leads to '(1 ) 0 2 0 02',' 2L n ikz cos nn nmV zf id z c o s n e J k a s i np (83) Evaluating the integral and using the identities n nii and n nnJx J x yields ,in n nf fe ( 8 4 ) where 21 0 21 2 21nik c o s Lpn nnmV efi J k a s i n nik c o sp (85) The differential scattering cross section is ' ( '2 *) ' ,Ωin n nn nndff f ed (86) and the total scattering cross-section is 2 002in (s, dd f . Using 2 ' ,'02in n nn de , the total cross-section is obtained in the form n n ( 8 7 ) where 32 2 2 2 0 2 02122( )21nnnLsin k cosp mVds i nJ k a s i nnkc o sp (88) For the set of (positive) n values that satisfy 01 n kp ( 8 9 ) the denominator in Eq. (88) can vanish and the partial cross-sections given by Eq. (88) are particularly large. They can be evaluated for large L by using 22ws i n w L d w L to make the approximation 2 2()sin wLL w w ( 9 0 ) so that, for n in this range 2 22 2 0 4211nnmV L nJk akp k (91) The resonant condition, vanishing of the denominat or in Eq. (88) may be given a simple physical interpretation. Consider an electron that is incident along the helix and follows two paths, labeled 1 and 2 in Fig. 5. Path 1 is longer than path 2 by an amount l p pcos . The condition for constructive interference is 2 ln n k . So the resonance condition becomes 12 0kc o s n p , which is precisely the form of the denominator. For those angles which satisfy this condition constructive interference results in strong scattering. 33 Fig. 5. Diffractive scattering from an helix: In terpretation of Eq. (88). For n outside the range in Eq. (89), including all 0n, the cross section for scattering remains small, and becomes independent of L for large L. To see this we note that the rapidly oscillating sin2 function in Eq. (88) can in this case be approximated by it average ½, so Eq. (88) becomes, for n outside the range of Eq. (89) 2 2 2 0 2 01()21nnmVd sin J kasinnkc o sp (92) These observations are confirmed by numerical evaluation of the full expression (88). As an example, the reduced partial cross-section, 12 0 22nnmV , is shown as a function of L in Fig. 6, using typical DNA parameters: a = 1.0 nm, p = 3.4 nm and energy 2220 . 5 e Ve Em k (me = electron mass). The different modes of L dependence in the 1 n cases are clearly shown. Fig. 6. T eV T are effe c linearly The am the cor r place. The reduce d Thus, in thi ctively scat t with L. Ind 0 n nL d ount of filt e relation bet w d cross-sec t s model on l tered and t h deed 21 1nkJ ka Jdsin ering is see n ween orbita l tion n plo ly those wa v herefore filt e 21 2nn kp Jkasin coskp n to grow l i l and spin a 34 otted agains t ves with az i ered out of t 2 2 n kp ; 0 inearly wit h angular mo m t the molec u imuthal qu a the transmi t 1 n kp h the molec u menta impli e ular length antum num b tted beams, ular length es that spin L for ener g bers n that s a and this ef f (93) L. Finally, selectivity gy E = 0.5 atisfy (89) fect grows as before, also takes 35 In what follows we make some drastic simplifications in order to estimate the resulting spin filtering effect. First, noting that the azimuthal quantum number n that can contribute to the incoming wave considered above in a given energy region corresponds to the values of the quantum numbers lm that can be obtained by the photoexcitation of the substrate metal, we assume at the outset that the magnitudes lm of these values fall in the range (89). We further assume that an electron that is scattered by the DNA is lost to the detector. Denote by N the number of DNA molecules absorbed per unit area. The probability that an electron with azimuthal quantum number ݉ will pass through molecular layer without scattering is ; ;1 111 0ll ll lmm mm mNN TNN (94) In this model it may be possible for the DNA monolayer to be opaque to one value of ݉ and yet allow electrons with other values of ݉ through. Consider now the expected degree of spin polarization of electrons photoemitted from gold covered by a monolayer of DNA molecules. For energies close to the photoemission threshold the electrons originate primarily from the relatively narrow d-band of gold and are promoted to the broad p-band conduction band. Those electrons with energies above the vacuum level can pass into the vacuum. Noting again that the d-bands in gold are split into a higher energy 2 52D and a lower energy 2 32D band (although there is some overlap between the bands),[27] it will be assumed that the energy of the incident light is sufficiently low that only the 2 52D band contributes to the photoemitted flux. Since these d-bands are narrow, we will treat them as being atomic-like. Thus, in this si mplified picture, photoemission originates from essentially atomic states characterized by total angular momentum quantum number j = 5/2 with azimuthal projection quantum numbers jm associated with the parent 2l orbital angular momentum state. (The quantization axis is taken to be normal to the metal surface). The set of jm states that contribute to the observed photoemission may be further restricted by energy considerations brought about, for example, by crystal fields in the solid. We examine photoemission by unpolarized incident light and consider separately the contribution from its right- and left-hande d components whose handedness is denoted 1. 36 The selection rules for optically allowed dipole transitions involve the orbital angular momentum ݈ and its azimuthal projection ݉. They are 1, , 0ls lms m ( 9 5 ) where s and sm are the quantum numbers for the spin and its azimuthal projection. To implement these selection rules for an initial 2, 5 / 2,j l j m state we expand it in terms of the eigenstates ,, ,lslm sm that characterize an atom without spin-orbit coupling, keeping 2l. Since for the optical d-band p-band transition 1l , it follows that 11l lm l . For ߤൌ 1 we therefore have െ2 ݉ 0, while for ߤൌെ 1 we have 0݉ 2. The conditional probability to observe a final spin projection sm for a given μ is denoted s Pm . Since the probability of having either component is 12 P , the overall probability to observe a final spin projection using unpolarized light is 11 1 1 1 1 (1 / 2)s ss s s P m Pm P P m P Pm P m (96) We first assume (other assumptions wi ll be considered later) that all the ݉ states associated with 2, 5 / 2lj are degenerate with each other, so their relative contribution to the photoemission process is not restricted by their energy. The probability to observe a final spin projection for 1 is then 2 0 11 2152, , , ,22l ljsm l s j mmPm T m m m (97) where ,, , | ,ls jlm sm jm are Clebsch-Gordan (CG) coefficients. The sum over jm can be dropped because j ls mm m has to be satisfied (that is, CG=0 unless this is so). Eq. (97) becomes 20 11 2 22 2 10 1152, , , ,22 15 15 152, 2, , , 2 2, 1, , , 1 2, 0, , , 022 22 22l lsm l s l ms ss ss s sP m T mmm Tm m T m m mm Tm (98) Similarly for 1 37 22 11 0 22 2 10 1152, , , ,22 15 15 152, 0, , , 0 2,1, , ,1 2, 2, , , 222 22 22l lsm l s l m ss s s sss Pm T m m m Tm m T m m T mm m (99) The needed Clebsch-Gordan coefficients are listed in Appendix E. Using these, Eqs. (98) and (99) lead to 11 0 111 2 3 25 5 5PT T T (100) 11 0 114 3 25 5PT T T (101) 11 0 113 4 25 5PT T T (102) 11 0 113 2 1 25 5 5PT T T (103) Together the contributions from the two polarization states gives, for spin up 11 10112 12 23 41225 5 5PPPT T T (104) Similarly, for spin down 11 10112 12 43 21225 5 5PPPT T T (105) Finally, the spin polarization asymmetry ratio is 11 10112 12 12 12 3PP TT PP T T T (106) From Figure 5 we see that for large L, ߪ≫ߪଵ≫ߪିଵ. If, for the sake of quick estimate, we invoke Eq. (94) to assume that 11 T while 01 0 TT, we get the polarization ratio 13 , to be compared with the observed polarization ~0 . 6 . This rough estimate should be regarded more as an example of what can be estimated from such arguments rather than a theoretical prediction. Other quick estimates may be attempted. For example, if we assume as in Section 4 that only ݉ states with the highest energy 38 contribute to the photoemission signal, and if these states correspond to 5 2jm , Eqs. (98) and (99) become ,1 22 11152, 2, , , 222s ss s m Pm T m m (107) 122 1, 1152, 2, , , 222s ss s m Pm T m m (108) and Eqs. (100)-(105) are replaced by 11 111;022PT P (109) 11 1110;22PP T (110) 11 112 12 11222PPPT (111) 11 112 12 11222PPPT (112) The spin polarization for this case is 11 1112 12 12 12PP TT P PT T (113) and for 110; 1TT we get 1 , that is full polarization towards to surface. Obviously, the observed result, 2/3 , can be obtained in intermediate situations. 6. Summary and conclusions Three issues were discussed in this paper: We have first argued that spin orbit coupling induced by electron motion through a helical structure cannot, by itself, account for recent observations of large spin selectivity in photoemission through such structures. Second, and most important, we have introduced the concept of induced selectivity or induced filtering - selectivity in the dynamical evolution of one observable can induce selectivity in another observable that is coupled to it. We have demonstrated such induced filtering in transmission between two 39 reservoirs: one in which two such dynamical variables are coupled and another reservoir where they are not, through a bridge whose transmission properties depend only on the state of one of these variables. Another example is electron photoemission from surfaces characterized by strong spin-orbit coupling using circularly polarized light. Third, we have applied this theoretical framework to the interpretation of recent experimental observations of large spin selectivity in electron photoemission and tunneling process through DNA and other chiral molecules, where at least one of the metals involved is gold or si lver - metals characterized by strong spin-orbit coupling. We have studied two models: in one, appropriate for tunneling situations, we have estimated the spin polarizability in the transmission current calculated from the Landauer formula. This model predicts positive spin polar izability in the transmitted current and does not show molecular length dependence of the effect in the absence of dephasing processes. In another, more suitable for over-barrier transmis sion as in photoemission, we have studied the spin selectivity induced by the orbital angular momentum dependence of electron scattering by helical structures. This model yields negative spin polarization that increases linearly with the helix length in the range studied. In either case we considered only elastic process. It will be of interest to consider the possible consequences of energy losses in futu re studies, but at first glance it seems that such effects are small in the energy range relevant to current experimental results, that is below the electronic excitation spectrum of DNA. Another consideration for future study is the possibility that the adsorbed helical layer affects the nature of the incident light, perhaps inducing some circular polarization character that is expressed in the photoexcitation process. Both models considered yield spin polarization of the observed order of magnitude using reasonable parameters for the system geometry and its electronic structure. These results should be regarded as estimates only and should be repeated with more detailed structural data for the specific systems used in the experiments. In particular, we have used the bulk electronic structure of gold as a guide for our arguments, while, obviously, the surface electronic structure should also be considered in rigorous calculations. While our results seem to be in accord with published experimental results on gold and silver, recent observation of considerable spin polarization in the photoemission from bacteriorhodopsin covered aluminum bring up new questions. By itself, aluminum is a low spin-orbit coupling mate rial, so the mechanism discussed in this paper can be relevant only provided such coupling is caused at the molecule-metal 40 interface by the interfacial built-in potential. Obviously this may also be an indication that another mechanism, yet unknown, is at play. These issues will be subjects of future studies. 41 Appendix A. Perturbative es timate of spin rotation. The initial conditions for the dynamical Pauli matrices are the usual Pauli spin matrices ˆ ˆˆ01 0 1 0010 0 0 1ii j ki (114) Assuming that the speed v varies in the range 105-107 m/s , the parameters g and b of Eq. (11) assume values in the ranges ~g 3 x10-4 - 3 x10-2 and b ~ 1. x10-7 - 1.x10-3. Note that b is roughly the same size as g2. Thus both parameters are small in magnitude and this suggests that a perturbation solution of the equations of motion would suffice. To lowest order in both g and b the solutions of Eqs. (12) are 2 2 2 212 12 (1 ) 2 2( 1 ) 1( 1 ) (1 ) 1x y i zibsin ig ig bsin bc o s i g ig bc o s ib e ib e (115) If the initial state is one of positive spin projection, that is 1 0 , the expectation values of the Pauli spin matrix components are 0 0, 0 0, 0 1xy z . After traversing some length of the helix the expectation values become , ( 1), 1xy z bsin b cos (116) If the helix consists of N turns (where N need not be an integer) then ߠൌ2 ܰߨ .Since b is a small number the spin projection does not change much from its starting value. In this approximation, when N is an integer the expectation values return to their original values. Appendix B. Spin flip by spin-orbit scattering off a helical potential Here we start from Eqs. (17)-(24) and derive Eq. (25). First note that 42 ˆˆyx xy kV k V V V (117) Thus, Eq. (24) becomes 20 0 4yx so yxVi V HVi V mc v (118) The off-diagonal matrix elements connect spin-up states, ቀ1 0ቁ, and spin-down states, ቀ0 1ቁ, and causes spin flipping. Carrying out the derivatives gives 0 0' 'xV V cos xcos ysin a xsin ycos V sin xcos ysin a xsin ycos (119) and 0 0' 'yV V sin xcos ysin a xsin ycos V cos xcos ysin a xsin ycos (120) As noted in the main text, th e effect of the helix is on the xy-motion of the electron. We next use time-dependent perturbation theory (first order) to calculate the amplitude for making a transition from an initial state ሺ݇ሬԦୄ,ݏሻ to a final state ሺ݇ሬԦ′ୄ,′ݏሻ, i.e., /ik R is eA and ' ' /ik R fs eA , where ݇ሬԦୄ corresponds to motion in the xy-plane, A is the normalization area and s is the spin vector. Later it will be assumed that the scattering is elastic, i.e., it changes only the direction of ݇ሬԦୄ but not its magnitude. (Affecting the spin is also not an energetic issue in the absence of a magnetic field). In what follows we disregard scattering by V(x,y,t) and only take into account the magnetic coupling, that is , consider scattering by Hso only.[46] The transition amplitude is ' ', ' 0', ' | | , i TEEt so ksice k s H k s d t (121) where T is the transit time. Thus, introducing the wave-vector transfer ݍԦൌ݇ሬԦୄെ′݇ሬሬሬԦୄ, we get 2 ' 201', ' | | , 4 0xyiq r sos s xyii V ks H k s d r eA mc ii V v (122) Note that 43 0 '' ' ' ''i xyiV V e x a y ixa y (123) and 0 '' ' ' ''i xyi VV e xa y ixa y (124) It is convenient to introduce rota ted wave-vector transfer components 'xx yqq c o s q s i n (125) and 'yx yqq s i n q c o s (126) and to recall that 'xxcos ysin (127) and 'yx s i n y c o s (128) Note that this is a time-depende nt transformation. The variables x and y are defined in a fixed coordinate system, while x’ and y’ are defined according to a coordinate system that rotates in time. Then ''qr qr (129) and 22'dr dr (130) Integration by parts yields ' '' 2'' ' 'xiq a iq r x dr e x a y i q e (131) and ' ' 2 '' '' 'xia iq r yqdr e x a y i q e (132) So ' † 0 ' 2 † 0 ' 20 1', ' | | , 4 '0 0 1 0 4' x xyi iq a s s si iqc o s qs i n a s sqe Vks H k s eA mc qe q Veq A mc v v (133) where ݍേൌݍ௫േݍ݅௬. The transition amplitude becomes 44 ' ΩΩ † 0 ' ,' 2 0'0 1 0 4vxyi TEEt iqc o s t qs i n ta s s ksq iVce e d tq A mc (134) and Ωൌଶగ௩ . It makes sense to take E’ = E since we are interested mainly in processes that affect spin, not orbital motion. This assumes that changing the spin did not affect the energy (i.e., recall that k’ = k, whereas there is direction change ݍԦൌ݇ሬԦୄെ݇ሬԦୄᇱ). Let the length of the helix be L = Np, where N is the length in units of the pitch (which need not be an integer). The relevant transit time is /v TL , so /Ω 'Ω † 0 ,' 2 ' 00 1 0 4vv xyNpiqc o s t qs i n ta s s ksq iVce d tq A mc (135) Introduce polar coordinates ሺߠ,ݍሻ in place of the Cartesian coordinates ൫ݍ௫,ݍ௬൯ so 2 † 0 ' ',' 2 00 1 0 2 4v vN iqacos s s ksq iV pce dq A mc (136) We can, without loss of generality, take ߠൌ0 . This simply means that we define the origin of the cylindrical angle ߮ by the direction of ݍԦ . This leads to Eq. (25). Finally we note that the introduction of a s econd helix does not change the result by very much. The potential may be written as 0 0, '' ' 'V x y V xcos ysin a xsin ycos V xcos ysin a xsin ycos (137) Where ߮ᇱൌ߮െߜ and ߜ is an offset angle distinguishing the second helix from the first. The amplitudes turn out to be just twice what they were before for a single helix. Appendix C. Derivation of Eqs. (58)-(60) The expansion coefficients A and B in (53) and (55) can be found by matching the wave functions at the surface za. The continuity of the total wave function at this surface can be expressed in terms of of the radial distance from the z-azis, R as 45 22 ,, , ' , ,0 , ,0|( ) ( Θ,0)l ls ls l ls l s ls l ls l s lsim s ls jj l l m m mm iq a im s mm m m mm iqa im s mm m m mmlm sm jm R a Y e B QJ Q Re d Q A QJ Q Re d Q v (138) where 1 22Θacos Ra . Similarly, the component of the gradient of the wave function in the direction normal to the surface must be continuous. Using sincoszr r (139) leads to ,, , ' , ,0 , ,0|( ) ( θ,0) 'l ls ls l ls l s ls l ls l s lsim s ls j j l l m m mm iq a im s mm m m mm iqa im s mm m m mmsinlm sm jm cos r Y err BQ J Q R i q e d Q AQ J Q R i q e d Q v (140) where 22rR a and ߠൌΘ . It follows that , 0 22 ' ,, , 0| Θ,0ls l ll s liqa mm m iq a ls jj l l m m m mAQ J Q R e d Q lm sm jmR a Y B Q J Q R e d Q v (141) and , 0 ,, ' , 0|( ) ( , 0 ) (' )ls l l ls liqa mm m ls j j l l m iq a mm mA Q J QR iqe dQ sinlm sm jm cos r Yrr BQ J Q R i q e d Q v (142) 46 Multiplying Eq. (141) through by the Bessel function 'lmRJ Q R , integrating over R using the relation 01''llmmJQ R JQ R R d R Q QQ (143) leads to , 22 ' ,, , 0| Θ,0ls ll l siqa mm iq a ls j m j l l m m mAQ e Ql m s mj m J Q R R a Y R d R B Qe v(144) Similarly from (142), ', ,, 0 ,|( , 0 ) 'ls ll lsiqa mm ls j m j l l m iq a mmiqA Q e sinQl m s mj m J Q R c o s rY R d Rrr iq B Q e v (145) Solving Eqs. (144) and (145) for the coefficients ܣ,ೞሺܳሻ and ܤ,ೞሺܳሻ yields the results (58)-(60) for the A and B coefficients. Appendix D. The asymptotic Green function, Eq. (78) In what follows we will use the following expr ession for the Green function in cylindrical coordinates ,,Rz [47] 01,' ,, ' , ' c o s [ ' ]4n nH nGrr G kR R z z n (146) where 2 22'' 2 ' c o s ( ) '' 2 2201,, , c o s '' 2 ' c o s ( )ik R R z z RR n HeGk R R z z nd RR z z R R (147) and 1 0 2 0nif n if n (148) 47 For the scattering process we are interested in the asymptotic form of the Green function for large R and |ݖ|. If ' Vr is localized in space, the values of R’ and |′ݖ| remain bounded and we may expand 2 22 2' ''' 2 ' c o s ( ) 1 ''zz RR cosR Rz z R R r r z c o s R s i n c o s r (149) where 22rR z , zcosr and Rsinr. Note that ሺ߶,ߠ,ݎሻ are the spherical coordinates of the point ,,Rz expressed in cylindrical coordinates. From Eqs. (147) and (149) we get '' 0 '1,, , ' c o s '( ) (' )ikr n ikz cos ikR sin cos H ikrn ikz cos neGk R R z z n e dr eie J k R s i nr (150) where we have used the integral representation of the Bessel function 0cosn izcos niJz e nd (151) Using this in (146), the asymptotic Green function is obtained in the form (78). Appendix E. Relevant Cleb sch-Gordan coefficients The following table summarizes the Clebsch-Gordan coefficients needed for the evaluation of the scattering cross-sections, Eqs. (98) and (99) l ml s ms j mj ,, , |lslm sm j m 2 -2 1/2 1/2 5/2 -3/2 1/5 2 -1 1/2 1/2 5/2 -1/2 2/5 2 0 1/2 1/2 5/2 1/2 3/5 2 1 1/2 1/2 5/2 3/2 4/5 2 2 1/2 1/2 5/2 5/2 1 2 -2 1/2 -1/2 5/2 -5/2 1 2 -1 1/2 -1/2 5/2 -3/2 4/5 2 0 1/2 -1/2 5/2 -1/2 3/5 48 2 1 1/2 -1/2 5/2 1/2 2/5 2 2 1/2 -1/2 5/2 3/2 1/5 Acknowledgements . We thank Ron Naaman, Rafael Gutierrez and Guido Burkard for helpful discussions, and R. Naaman for disclosing experimental data prior to publication. The research of A. N. is supported by the Israel Scienc e Foundation, the Israel-US Binational Science Foundation and the European Research Council under the European Union's Seventh Framework Program (FP7/2007-2013; ERC grant agreement no. 226628). K. K. acknowledges support from the Villum Kann Rasmussen Foundation. 49 References [1] B. Gohler et al. , Science 331 (2011). [2] Z. Xie et al. , Nano Letters 11 (2011). [3] R. Naaman, and D. H. Waldeck, J. Phys. Chem. Lett. 3 (2012). [4] K. S. Kumar et al. , To be published (2013). [5] See, e.g., F. Meier, and D. Pescia, Physical Review Letters 47 (1981). [6] K. Ray et al. , Science 283 (1999). [7] J. J. Wei et al. , J. Phys. Chem. B 110 (2006). [8] D. M. Campbell, and P. S. Farago, Journal of Physics B: Atomic and Molecular Physics 20 (1987). [9] S. Mayer, and J. Kessler, Physical Review Letters 74 (1995). [10] C. Nolting, S. Mayer, and J. Kessler, Journal of Physics B: Atomic, Molecular and Optical Physics 30 (1997). [11] T. Ando, J. Phys. Soc. 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2211.05978v2.Orbital_Origin_of_Intrinsic_Planar_Hall_Effect.pdf

Orbital Origin of Intrinsic Planar Hall Effect Hui Wang,1, 2,∗Yue-Xin Huang,1, 3, 4, ∗Huiying Liu,5, 1Xiaolong Feng,1, 6 Jiaojiao Zhu,1Weikang Wu,7, 1Cong Xiao,8, 9, 10, †and Shengyuan A. Yang8 1Research Laboratory for Quantum Materials, Singapore University of Technology and Design, Singapore 487372, Singapore 2Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore 3School of Sciences, Great Bay University, Dongguan 523000, China 4Great Bay Institute for Advanced Study, Dongguan 523000, China 5School of Physics, Beihang University, Beijing 100191, China 6Max Planck Institute for Chemical Physics of Solids, N¨ othnitzer Strasse 40, D-01187 Dresden, Germany 7Key Laboratory for Liquid-Solid Structural Evolution and Processing of Materials, Ministry of Education, Shandong University, Jinan 250061, China 8Institute of Applied Physics and Materials Engineering, University of Macau, Macau, China 9Department of Physics, The University of Hong Kong, Hong Kong, China 10HKU-UCAS Joint Institute of Theoretical and Computational Physics at Hong Kong, Hong Kong, China Recent experiments reported an antisymmetric planar Hall effect, where the Hall current is odd in the in-plane magnetic field and scales linearly with both electric and magnetic fields applied. Existing theories rely exclusively on a spin origin, which requires spin-orbit coupling to take effect. Here, we develop a general theory for the intrinsic planar Hall effect (IPHE), highlighting a previ- ously unknown orbital mechanism and connecting it to a band geometric quantity — the anomalous orbital polarizability (AOP). Importantly, the orbital mechanism does not request spin-orbit cou- pling, so sizable IPHE can occur and is dominated by orbital contribution in systems with weak spin-orbit coupling. Combined with first-principles calculations, we demonstrate our theory with quantitative evaluation for bulk materials TaSb 2, NbAs 2, and SrAs 3. We further show that AOP and its associated orbital IPHE can be greatly enhanced at topological band crossings, offering a new way to probe topological materials. The Hall effects are of fundamental importance in con- densed matter physics [1–3]. In nonmagnetic materials, Hall effect appears under an applied magnetic field. For Bfield out of the transport plane, i.e., the plane formed by the driving Efield and the measured Hall current jH, this is the ordinary Hall effect due to Lorentz force [4]. Recently, a new type of Hall effect was found in exper- iments on certain bulk nonmagnetic materials, where a Hall response was induced by an in-plane Bfield and scales as jH∼EB[5, 6]. Note that this effect is distinct from many previously reported planar Hall effects [7–16], where the Hall current is an even function in Band hence is not a genuine Hall response but rather represents an off-diagonal anisotropic magnetoresistance [5]. There have been theoretical studies on this effect [17– 19]. However, the current understanding is far from com- plete for the following reasons. First, existing theories are exclusively based on spin-orbit coupling (SOC) and Zee- man coupling between spin and in-plane Bfield. The pre- dicted response vanishes when SOC is neglected. Such a treatment misses the orbital degree of freedom of Bloch electrons, which also couples to Bfield [20] and can re- sult in Hall transport regardless of SOC. Second, previ- ous works are mostly focused on specific models, such as Rashba model [17], modified Luttinger model [19], and honeycomb lattice model [18]. It is urgent to develop a general theory that can be implemented in first-principles calculations for real materials. Third, from the scaling re-lation and symmetry constraint, one can easily see that the configuration allows an important intrinsic contribu- tion, i.e., an intrinsic planar Hall effect (IPHE), which is independent of scattering and manifests the inherent property of a material. To understand IPHE and make it a useful tool, one must clarify what intrinsic band ge- ometric properties are underlying the effect. In this work, we address the above challenges by for- mulating a general theory of IPHE. We show that be- sides Berry curvature and spin/orbital magnetic moment, there are two new band geometric quantities coming into play — the anomalous spin polarizability (ASP) [21] and the anomalous orbital polarizability (AOP). Apart from spin contribution, we reveal a previously unknown or- bital contribution (connected to AOP) to IPHE , which can generate a large response and dominate the effect in systems with weak SOC. We express the response tensor in terms of the intrinsic band structure of a material. We clarify the symmetry character of the effect and obtain the form of response tensor for each of the 32 crystal classes. Combining our theory with first-principles cal- culations, we perform quantitative evaluation of IPHE in three concrete materials TaSb 2, NbAs 2, and SrAs 3. We demonstrate that while spin and orbital contributions are comparable in TaSb 2(with relatively large SOC), the orbital mechanism completely dominates in NbAs 2and SrAs 3(with weaker SOC). In addition, for SrAs 3, the large response can be further attributed to the enhancedarXiv:2211.05978v2 [cond-mat.mes-hall] 11 Mar 20242 AOP surrounding the gapped topological nodal loop at Fermi level, suggesting IPHE as a new way to probe topo- logical band structures. Band origin of IPHE. We consider a general three- dimensional (3D) system. The IPHE response can be derived within the extended semiclassical theory [22–25], which incorporates field corrections to the band quan- tities. The resulting intrinsic Hall current is found to bejint H=−R f0(˜ε)(EטΩ) (details in the Supplemental Material [26]), where we take e=ℏ= 1, f0is the Fermi distribution function, and the tilde in band energy ˜ εand Berry curvature ˜Ωindicates that these quantities include corrections by external fields [22]. To compute the IPHE scaling as ∼EB, we just need to retain corrections to ˜ ε and˜Ωwhich are linear in B. For Berry curvature, we have ˜Ω=Ω+ΩB, where Ωis the (unperturbed) Berry curvature in the absence of external fields, and ΩB∼Bis the B-field-induced correction [26]. One may write ΩB=∇k×ABin terms of the B-field-induced Berry connection [22, 25]: AB b(k) =Ba[FS ab(k) +FO ab(k)], (1) where the subscripts aandbdenote the Cartesian com- ponents, Einstein summation convention is adopted, and the coefficients FS abandFO abare the ASP and AOP, re- spectively. For a particular band with index n, they are expressed as FS ab(k) =−2 ReX m̸=nMS,nm aAmn b εn−εm, (2) and FO ab(k) =−2 ReX m̸=nMO,nm aAmn b εn−εm−1 2ϵacd∂cGn db.(3) Here, Anm b=⟨un|i∂b|um⟩is the unperturbed interband Berry connection, ∂b≡∂kb,|un⟩is the unperturbed cell- periodic Bloch state with energy εn;MS,mn=−gµBsmn andMO,mn=P ℓ̸=n(vmℓ+δℓmvn)×Aℓn/2 are the in- terband spin and orbital magnetic moments, respectively, withsmn(vmℓ) being the matrix elements of spin (veloc- ity) operator, µBis the Bohr magneton, and gis the g- factor for spin; Gn db= ReP m̸=nAnm dAmn bis known as the quantum metric tensor [39], and ϵabcis the Levi-Civita symbol. Before proceeding, we have several comments on ASP and AOP. First, they are gauge-invariant quantities, as can be directly checked from (2) and (3). It follows that ABis also gauge invariant. Physically, it represents a positional shift of a Bloch wave packet induced by a B field [22]. Since AB bEbthen corresponds to an energy change of the wave packet, one can see that FS abEband FO abEbactually give the anomalous spin and orbital mag- netic moments induced by an Efield [21, 25, 40]. This is why FS(FO) is termed as ASP (AOP). Second, thetwo quantities have distinct dependence on SOC. ASP is allowed only when SOC is nonzero, which can be readily understood from its physical meaning discussed above. In contrast, AOP is not subjected to this constraint. This distinction leads to important consequences in the resulting IPHE, as will be discussed below. Third, one notices that the expression for AOP resembles ASP ex- cept for the second term in (3) with the quantum metric tensor. Intuitively, the quantum metric tensor measures the distance between states at different wave vectors in Hilbert space [23, 39]. Hence, its appearance in AOP can be understood, because unlike spin operator, the orbital moment operator is nonlocal in k-space. As for the field-corrected band energy, we have (band index nsuppressed here) ˜ ε=ε−B·(MS+MO), where MS(MO) is the intraband spin (orbital) magnetic mo- ment [20]. Substituting ˜Ωand ˜εinto the expression for jint Hand collecting terms of order O(EB), we obtain the IPHE current jint a=χint abcEbBc, (4) with the response tensor χint abc=Z [dk]f′ 0h ΘS abc(k) + ΘO abc(k)i , (5) where [ dk]≡P ndk/(2π)3, and in the integrand of (5) we have explicitly separated spin and orbital contribu- tions, with Θi abc(k) =vaFi cb−vbFi ca+ϵabdΩdMi c (6) andi= S,O. Equations (4-6) give the general formula for the IPHE tensor. We have the following observations. First, due to the f′ 0factor in (5), IPHE is a Fermi surface effect, as it should be. Second, as an intrinsic effect, χint abcis deter- mined solely by the intrinsic band structure of a ma- terial. Particularly, our formula reveals its connection to the band geometric quantities ASP and AOP. The first two terms in (6) may be called the ASP (AOP) dipole, similar to the definition of Berry curvature dipole [41]. Third, our theory reveals the orbital contribution to IPHE, which was not known before. As discussed, ASP and AOP have distinct dependence on SOC. As a result, sizable IPHE can still appear and is dominated by orbital contribution in systems with weak SOC. Fur- thermore, from Eqs. (2) and (3), we see that like Berry curvatures, ASP and AOP are enhanced around small- gap regions in a band structure. Roughly speaking, ASP scales as 1 /(∆ε)2and AOP scales as 1 /(∆ε)3, where ∆ ε is the local gap. Therefore, AOP can be more enhanced than ASP with a decreased gap; and one can expect pro- nounced IPHE in topological semimetal states. Symmetry property. From Eqs. (4-6), one sees that χint abcis clearly antisymmetric with respect to its first two3 TABLE I. Constraints on the tensor elements of Xfrom point group symmetries. As Xis time-reversal ( T) even, symmetry operations OandOTimpose the same constraints. Cz n, Sz 4,6, σzCx n, Sx 4,6, σxCy n, Sy 4,6, σyP Xzx × × ✓ ✓ Xzy × ✓ × ✓ indices. Therefore, we have jint aEa= 0, which indeed de- scribes a dissipationless Hall current. We may define a corresponding IPHE charge conductivity σint ab≡χint abcBc, which satisfies the requirement σab(B) =−σab(−B) for a genuine Hall conductivity. In comparison, the previ- ously studied planar Hall effect satisfies σab=σbaand σab(B) =σab(−B), so it represents a kind of anisotropic magnetoresistance. By virtue of the antisymmetric character, χint abccan be reduced to a time-reversal ( T) even rank-two tensor Xdc=ϵabdχint abc/2. (7) The IPHE current can be expressed as jint=E×σH with a T-odd Hall pseudovector σH d=XdcBc. In a typ- ical experimental setup for IPHE, the Bfield is applied within the transport plane, taken as x-yplane. Then, the effect is specified by only two tensor elements, Xzxand Xzy. Generally, it is convenient to choose a coordinate system that fits the crystal structure, which simplifies the form of Xtensor. In Table I, we list the constraints of common point-group operations on the two relevant tensor elements. One finds that IPHE is forbidden by an out-of-plane rotation axis and by the horizontal mir- ror, but is allowed by in-plane rotation axes and vertical mirrors. The most general matrix forms of Xin 32 crys- tallographic point groups are presented in Supplemental Material [26]. With the coordinate axes and Xfixed, assume the in- plane E(B) field makes an angle ϕ(φ) from the xaxis, i.e.,E=E(cosϕ,sinϕ,0) and B=B(cosφ,sinφ,0). Then, the IPHE current will flow in the direction of (−sinϕ,cosϕ,0), with a magnitude given by jint=XHEB, (8) where XH=Xzxcosφ+Xzysinφ. (9) One observes that as a linear-in- Eintrinsic Hall response, the magnitude jintdoes not depend on the E-field direc- tion. Meanwhile, it does depend on the B-field direction and generally exhibits a 2 πperiodicity. Application to TaSb 2and NbAs 2.To better under- stand features of IPHE, especially the relative impor- tance of spin and orbital contributions, we first apply our theory to two real materials: TaSb 2and NbAs 2, belonging to the family of transition metal dipnictides. ;' ň :'- 9* * B C D 5B
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F7 UPUBM TQJO PSCJU G H [YZ [YZ <><>FIG. 1. (a,b) The common crystal structure of TaSb 2and NbAs 2. The cleavage plane ( ¯201) is taken as the x-yplane. (c) shows the Brillouin zone. (d,e) Calculated band structures (SOC included) for (d) TaSb 2and (e) NbAs 2. (f,g) Calculated IPHE coefficient Xzxfor (f) TaSb 2and (g) NbAs 2, as a func- tion of Fermi energy µ. Here, besides the total result, we also separately plot the spin and orbital contributions. The two materials are isostructural, but they differ in the strength of SOC: with heavier elements, TaSb 2has a larger SOC strength than NbAs 2. Their common lat- tice structure is shown in Fig. 1(a) and 1(b), which has space group C2/m(No. 12) and point group C2h, with the twofold rotation axis along the ydirection here. The two materials have been synthesized by chemical vapor transport method [42, 43], and their transport and opti- cal properties have been studied in several recent exper- iments [42, 44–46]. We perform first-principles calculations on these ma- terials (details are given in Supplemental Material [26]). The obtained band structures are shown in Fig. 1(d) and 1(e), which agree with previous studies [47]. One can see that both materials are metallic, and they have a band near-degeneracy region around Fermi level along the L-I path. The smallest local vertical gap is around 61.2 (15.6) meV for TaSb 2(NbAs 2). In fact, without SOC, the local gap would close and the two bands would cross [47]. The smaller local gap in NbAs 2reflects its weaker SOC than TaSb 2. Recent experiments show the ( ¯201) plane as the cleav- age surface [48, 49], so we take it as the transport plane, which corresponds to the x-yplane in Fig. 1(a). Accord-4 TABLE II. Calculated IPHE coefficient Xzx(in unit of Ω−1m−1T−1) for three concrete materials. For TaSb 2and NbAs 2, the transport plane is set as ( ¯201). For SrAs 3, the transport plane is set as (001). The spin and orbital contri- butions are also listed separately. Xzx TaSb 2 NbAs 2 SrAs 3 spin 5.32 3.00 4.56 orbital 6.13 206.92 -143.45 total 11.45 209.92 -138.89 ing to Table I, this setup allows only a single Xzxelement, and the response can be captured by XH=Xzxcosφ. (10) This angular dependence can be directly verified in ex- periment. Since our theory is formulated in terms of intrinsic band quantities, it can be easily implemented in first- principles calculations to evaluate the Xtensor. Fig- ures 1(f) and 1(g) show the calculated Xzxfor TaSb 2 and NbAs 2, as a function of Fermi energy µ. One can see that the effect is pronounced when µapproaches the aforementioned small-gap regions, where the band geo- metric quantities are enhanced. At the intrinsic Fermi level (i.e., µ= 0), we obtain Xzx= 11.45 Ω−1m−1T−1 for TaSb 2andXzx= 209 .92 Ω−1m−1T−1for NbAs 2. These values are quite large and are definitely detectable in experiment [6]. To assess the relative importance of spin and orbital contributions to IPHE, in Figs. 1(f) and 1(g) we also separately plot the two contributions. One finds that for TaSb 2with a stronger SOC, the two contributions are comparable near the intrinsic Fermi level. In contrast, for NbAs 2with a weaker SOC, the orbital contribution is overwhelmingly dominating over the spin part. The specific values at µ= 0 are listed in Table II. These results demonstrate that (1) orbital contribution to IPHE is significant; (2) beyond previous theories, large IPHE can occur in materials with weak SOC and is dominated by the orbital mechanism. Application to SrAs 3.The other example we wish to discuss is the topological semimetal SrAs 3. It also has space group C2/m, and its crystal structure is shown in Fig. 2(a) and 2(b). Previous works showed that in the absence of SOC, SrAs 3possesses a nodal loop across the Fermi level in the Γ- Y-Splane [Fig. 2(c)] [50–52], pro- tected by the mirror symmetry. Inclusion of SOC will gap out the nodal loop, but since the SOC strength in SrAs 3is weak, the opened gap is small. From our calcu- lated band structure in Fig. 2(d), we find the gap values being 32.0 meV and 5.9 meV on S-YandY-Γ paths, respectively, which agree with previous calculations [52]. In addition, by scanning around the original nodal loop, we find the smallest gap opened is ∼2.2 meV. Such small : 5 :4 ň4S"T B C D 


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F7 UPUBM TQJO PSCJU F [YZ YZ [ G LYL[<> <> FIG. 2. (a,b) Crystal structure of SrAs 3. Here, the (001) plane is taken as the x-yplane. (c) shows its Brillouin zone. The red line illustrates the nodal loop in the absence of SOC. (d) Calculated band structure of SrAs 3(SOC included). (e) Calculated Xzxas a function of µ. (f)k-resolved contribution toXzx(the integrand of Eq. (5)) on the intrinsic Fermi surface in the Γ- Y-S(ky= 0) plane. The black lines show the Fermi surface. The green dotted line indicates the original nodal loop in the absence of SOC. The unit of colormap is µB˚A2/eV. gaps near the Fermi level are expected to generate a large orbital contribution to IPHE. We take the (001) plane ( x-yplane in Fig. 2(a) and 2(b)), which is a cleavage plane of SrAs 3[52], to be trans- port plane. Similar to TaSb 2and NbAs 2, for IPHE, there is only one nonzero element Xzx, and the angular depen- dence follows Eq. (10). Our calculation result for Xzxis plotted in Fig. 2(e). Again, one observes that in such a material with weak SOC, the effect is dominated by the orbital contribution. Moreover, we find that the orbital contribution is mostly from the AOP dipole [26]. At the intrinsic Fermi level, we get Xzx=−138.89 Ω−1m−1T−1, which is comparable to the value in NbAs 2. To correlate this large IPHE with the band topology, in Fig. 2(f), we plot the integrand of Eq. (5) on the Fermi surface in the ky= 0 plane, i.e., the plane that contains the original nodal loop. Here, the Fermi surface is marked by the black lines, forming two figure-eight parts, and the green dotted loop indicates the original nodal loop in this plane. One can see that the large contribution is indeed concentrated at regions where the Fermi surface touches the nodal loop. Discussion. We have developed a general theory for the IPHE and, importantly, discovered the orbitally in-5 duced contribution to the effect. The finding greatly ex- tends the scope of IPHE, which was previously limited only to spin-orbit-coupled systems. From our calculation results, orbital IPHE can be comparable to spin contri- bution in systems with strong SOC, and it dominates in systems with weak SOC. Our theory clarifies the band origin of IPHE, especially highlights the role of AOP and ASP, making the effect a new intrinsic property to char- acterize materials and to probe band geometric quanti- ties. Orbital IPHE can be directly probed in materials with weak SOC, such as NbAs 2and SrAs 3discussed here. In fact, our analysis [26] shows that orbital IPHE is very likely also dominating the signals reported for ZrTe 5[5] and VS 2-VS [6], although there are still some uncertain- ties about these two materials that need to be clarified in experiment, as discussed in Supplemental Material [26]. In this work, we focus on the intrinsic effect, which can be quantitatively evaluated for each material and serves as benchmark for experiment. There should also exist extrinsic contributions arising from disorder scattering. By adopting certain disorder models, they may be evalu- ated by approaches similar to the anomalous Hall effect [2, 53]. Experimentally, extrinsic effects can be separated by their different scaling with respect to system parame- ters, such as temperature and disorder strength [54–58]. Recently, there were also studies on nonlinear planar Hall effect, which scales as ∼E2B[59–61]. Its symme- try property is different from IPHE here. In practice, the responses with different Edependence can be readily dis- tinguished by applying a low-frequency modulation and using the lock-in technique [58, 62, 63]. Finally, although our theory is developed for nonmag- netic materials, it also applies to magnetic systems. 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2403.12701v1.Unraveling_the_dynamics_of_magnetization_in_topological_insulator_ferromagnet_heterostructures_via_spin_orbit_torque.pdf

Unraveling the dynamics of magnetization in topological insulator-ferromagnet heterostructures via spin-orbit torque Taekoo Oh∗and Naoto Nagaosa∗ RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan E-mail: taekoo.oh@riken.jp; nagaosa@riken.jp Abstract Spin-orbit coupling stands as a pivotal determinant in the realm of condensed mat- ter physics. In recent, its profound influence on spin dynamics opens up a captivating arena with promising applications. Notably, the topological insulator-ferromagnet het- erostructure has been recognized for inducing spin dynamics through applied current, driven by spin-orbit torque. Building upon recent observations revealing spin flip sig- nals within this heterostructure, our study elucidates the conditions governing spin flips by studying the magnetization dynamics. We establish that the interplay between spin-anisotropy and spin-orbit torque plays a crucial role in shaping the physics of magnetization dynamics within the heterostructure. Furthermore, we categorize var- ious modes of magnetization dynamics, constructing a comprehensive phase diagram across distinct energy scales, damping constants, and applied frequencies. This re- search not only offers insights into controlling spin direction but also charts a new pathway to the practical application of spin-orbit coupled systems. 1arXiv:2403.12701v1 [cond-mat.mes-hall] 19 Mar 2024Keywords Spintronics, Magnetization dynamics, Topological insulator, Spin-orbit torque. Introduction Spin-orbit coupling (SOC), acknowledged as one of the fundamental interactions in ma- terials,1reveals a fertile landscape within condensed matter physics. A standout illustration of its impact is evident in the burgeoning field of spintronics, where SOC-driven spin dynam- ics unveils a promising horizon for practical applications. Accordingly, recent attention has been drawn to topological insulator-ferromagnet (TI-FM) heterostructures, as evinced by notable reports.2–7The surge in interest stems from their remarkable efficiency in catalyzing spin dynamics, primarily attributed to the spin-orbit torque (SOT) induced by an applied current.8,9 These heterostructures exhibit intriguing phenomena that closely follow its intricate dy- namics of spins. For instance, an emergent inductance from the motive force in spiral mag- nets10–12was extended to the TI-FM heterostructures.13–15Nonreciprocal transport phenom- ena associated with the SOC was explored.16,17The significant increase of Curie temperature was reported as well.18–21Notably, it has been acknowledged that the SOT at the TI-FM in- terface is robust enough to flip magnetization and induce the sign change of Hall Effect.22–26 Such properties hold promise for applications in spintronics devices, particularly in utilizing topological SOTs for magnetic memories.27–32 Motivated by such intriguing phenomena due to the SOC and SOT, this study explores magnetization dynamics in TI-FM heterostructures with an applied current, aiming to un- veil the conditions for magnetization flip. Considering both spin-anisotropy and SOT, we establish the equilibrium state of magnetization and develop a model describing its dynam- ics under direct current (DC) or alternating current (AC). In the absence of damping, we identify oscillating, faltering, and flipping modes for DC, with the latest inducing magne- 2a 𝑚⃗𝐼⃗𝜎TIFM̂𝑧'𝑥'𝑦 𝑚'𝑦̂𝑧'𝑥𝜙𝜃b 𝑥=𝜎/𝐾01𝜃=arcsin𝑥Figure 1: Magnetization dynamics of TI-FM heterostructure. (a) The schematics of the TI-FM heterostructure. (b) The equilibrium state of magnetization in the different relative strengths of SOT x=σ/K. tization flip. The choice between modes is determined by the relative strengths of SOT to spin-anisotropy. With the introduction of damping, the occurrence of spin flip hinges on the duration elapsed until the crossover from flipping to oscillating modes takes place. For AC, on the other hand, we discern three final states—adiabatic, resonating, and chaotic. We classify five distinct modes in the adiabatic state at low frequencies, which is derived from DC modes. By providing the phase diagrams of adiabatic modes, we illus- trate that the modes are determined by SOT, spin-anisotropy, driving frequency, and initial driving phase. Lastly, we explore their transition to resonating and chaotic states at higher frequencies in the viewpoint of the Fourier transform, revealing that the periodic array of peaks gives rise to the chaotic state. By overhauling the dynamics, we provide insights into the complex dynamics of magnetization in TI-FM heterostructures. 3Results Model and its equilibrium The physical configuration is illustrated in Fig. 1(a), where a current flows along the x-axis, the itinerant electron spin aligns with the y-axis, and the ferromagnet’s magnetization lie along the + z-axis at t= 0. The polar and azimuthal angles with respect to the y-axis are denoted as θandϕ. We represent the magnetization as ⃗ m=m(sinθsinϕ,cosθ,sinθcosϕ) and the itinerant electron spin as ⃗ σ=σ(t)(0,1,0). For simplicity, we exclusively focus on the dynamics of the single ⃗ m, and ignore that of ⃗ σ. This is justified by the stronger spin- momentum locking of the TI surface state compared to anisotropy.3,15,33–36For instance, Ni has the easy-axis anisotropy energy about 2 .7µeV/atom in the bulk37and 4 .0 meV /atom in the monolayer,38while the spin-orbit coupling of Bi 2Se3is estimated as ∼1.2 eV/atom. Accordingly, we set our unit of energy to be 1 .0µeV∼2.4 GHz and the unit of time to be (2 .4 GHz)−1≈0.42 ns. The spin flip is then denoted as the switching of spin direction between + zand−zhalf space. Two crucial potential energies emerge: VA=−K 2m2 z(K > 0), representing magnetic anisotropy in the ferromagnet, and VS=−γ ⃗ m·⃗ σsignifying the Rashba-field-like39,40(or domain-damping-like)41SOT. The equilibrium state of magnetization is obtained, showcased in Fig. 1(b) for fixed ⃗ σ. Normalizing m=γ= 1, key energy scales become the anisotropy energy Kand SOT σ. The dimensionless parameter x=σ Kis defined. As xincreases from 0, the equilibrium states initially align along ±z-axis and gradually rotate towards the y-axis. In terms of angles, sin θ=x, ϕ=±π/2. Before reaching the y-axis, the equilibrium states are twofold degenerate. For x≥1, the degenerate equilibrium states converge at the y-axis. To describe magnetization dynamics out of equilibrium, the Lagrangian density is con- sidered: L=LB−VA−VS,LB=m(1−cosθ)˙ϕ. (1) 4Note that LBis the Berry phase term, making polar and azimuthal angles conjugate.42,43 Thus, as magnetization rotates from the z-axis to its equilibrium state, deviations from theyz-plane induce complex motion. Introducing Gilbert damping through the Rayleigh dissipation function,44R=η 2∂ ⃗ m ∂t·∂ ⃗ m ∂t, with the dimensionless damping constant η, yields the equations of motion: (1 +η2)˙θ=−K 2sinθsin 2ϕ+ηK 2sin 2θcos2ϕ−ησ(t) sinθ,and (2) (1 +η2)˙ϕ=−Kcosθcos2ϕ+σ(t)−ηK 2sin 2ϕ. (3) The initial conditions are set at θ=π/2 and ϕ= 0. This system of equations governs physics for both DC and AC scenarios. The numerical computation is performed by Mathematica. It should be noted that the field-like SOT σ(t) precesses the magnetization around y- axis while the damping-like SOT ησ(t) cants the magnetization toward y-axis. This is consistent with the Landau-Lifshitz-Gilbert equation. It is noteworthy that the relation between the damping constant ηand the Gilbert damping constant αisα=ηγ, where γis the gyromagnetic ratio.44We already set γ= 1, resulting in α=η. As a consequence, the field-like SOT is responsible for the spin flip, whereas the damping-like SOT deters it. Magnetization dynamics under DC In the exploration of magnetization dynamics under DC, we initiate with a straightforward scenario where α= 0 and σ(t) = 0 for t <0 and σ(t) =σfort≥0. Lacking damping, we anticipate permanent magnetization precession, described by the simplified equations: ˙θ=−Ksinθsinϕcosϕ,˙ϕ=−Kcosθcos2ϕ+σ. (4) These equations are intricate to solve analytically, but insight can be gained by examining two limiting cases, x≪1 and x≫1 with α= 0. In the first limit, we find oscillatory 5behavior: θ=π 2+x(cos(Kt)−1), ϕ=xsin(Kt). (5) Both θandϕoscillate with amplitude xand frequency K, termed the oscillating mode . In the second limit: θ=π 2+1 4x(cos(2 σt)−1), ϕ=σt. (6) Here, θoscillates with amplitude 1 /(4x) and frequency 2 σ, while ϕmonotonically increases. This implies the spin precession about the y-axis and a spin flip occurring at period of π/σ, termed the flipping mode . Comparing the frequency of θdynamics in each mode, we speculate that the transition from the oscillating to the flipping modes might occur at x= 1/2 where K= 2σ. The guess can be verified by numerical computations of frequency in units of Kfor σ(t) =σfort≥0 and α= 0 in Fig. 2(a). The frequency variation exhibits ωDC≈K= 1 forx≪1 and ωDC≈2σforx≫1. The singularity at x= 1/2 indicates the expected transition. Only at this transition point, the spins are faltering between zaxis and xy-plane in period, so we call this faltering mode. For DC, we focus on the oscillating and flipping modes, as the faltering mode requires a fine tuning. When damping αis introduced under DC, the magnetization eventually attains an equi- librium state. For x≥1, the equilbrium state aligns with y-axis, so the spin flip is absent. However, since two equilibrium states exists for x <1, the selection between them occurs, which can flip the spin. The selection is determined by both xandα. Figure 2(b) displays two examples of dynamics leading to an even-flipped ( x= 0.6, α= 0.05) and odd-flipped (x= 0.6, α= 0.1) final state, respectively. The magnetization in the even-flipped state stays in +zhalf space, while that in the odd-flipped state stays in −zhalf space. In both exam- ples, owing to the damping effect, the crossover from the flipping to the oscillating mode 6a b𝑥=0.60𝜂=0.05𝑥=0.60𝜂=0.10𝑡 (0.42 ns)050𝜔!"𝑥0.51.01.52.01234c 0.00.51.00.010.05𝛼0.020.030.04𝑥No / Even FlipsOdd FlipsOscillatingFlipping 25 Faltering FlippingOscillating FlippingOscillatingFigure 2: The magnetization dynamics under DC. (a) Computed frequencies of os- cillating ( x < 0.5), faltering ( x= 0.5), and flipping modes ( x > 0.5) under DC without damping in varying x=σ/K. (b) Dynamics of magnetization at x= 0.6, α= 0.05 (lower) andx= 0.6, α= 0.1 (upper) in time. Chromatically, the red indicates + z, the green indi- cates in-plane, and the blue indicates −zdirections of magnetization. The direction inward the paper is the ydirection. (c) A 2D phase diagram of the final states in varying xandα. The dark blue indicates the even-flipped final state, and the yellow indicates the odd-flipped final state. occurs after a specific time τc. In Fig. 2(b), for the former, after τc≈10.8 ns, while for the latter, τc≈4.2 ns. Empirically, we find that τc∝e4.53xα−1for small α. [See Supporting Information (SI).] These show that the variation in τccaused by the interplay of xandα serves as a determining factor for the final state. We further delve into the relation of the final state with xandαby a 2D phase diagram in Fig. 2(c). When x <1/2, exclusively the even-flipped final state manifests, whereas for x≥1/2, both odd-flipped and even-flipped states emerge, forming a fan-like configuration. It should be noted that the fan-like configuration is not clear in α <0.01 due to the resolution. The exclusive appearance of even-flipped state for x <1/2 is due to the absence of spin flip 7in the oscillating mode. However, in the case of x≥1/2, the spin flip can occur multiple times before the crossover from the flipping to the oscillating modes takes place. If the crossover time exceeds half of the flipping mode period, the spin flips to −zhalf space. With a longer crossover time surpassing a full flipping mode period, the spin flips twice, returning to +zhalf space. By extending the crossover time further, the spin flips repeatedly. Even (odd) numbers of spin flips lead to an even-flipped (odd-flipped) final state, giving rise to the distinctive fan-like feature in x≥1/2. This pattern persists until x= 1, where the equilibrium state converges to y-axis. Magnetization dynamics under AC Based on above results, we here explore the magnetization dynamics under AC, which is given byσ(t) =σsin(ωACt+δ) for t≥0 and σ(t) = 0 for t <0.ωACis the driving frequency, and δis the initial phase of AC. The initial state of the magnetization is again aligned with + z- axis. Owing to the damping α, the relative strength of SOT x, and driving frequency ωAC, the initial state overcomes the irregular dynamics and transits to the final states after some time. We identify three distinct final states in Fig. 3(a): adiabatic, resonating, and chaotic states. Both adiabatic and resonating states are the steady states coming after the decay of initial state. At each time t,σ(t) determines an equilbrium state as shown in Fig. 1(b). The adiabatic state denotes that the magnetization mostly adheres to the equilibrium state at each time. The resonating state denotes that the magnetization dynamics is periodic but detach from the equilibrium state at each time. On the other hand, the chaotic state is evolved from the initial state, retaining its irregularity, in which the magnetization never has a periodic motion. We primarily observe the adiabatic states for low-frequency or strong damping regime. We observe five distinct modes in the adiabatic states: I) an even-flipped oscillating mode, II) an odd-flipped oscillating mode, III) an even-flipped faltering mode, IV) an odd-flipped faltering mode, and V) a periodically flipping mode. These modes are depicted in Fig. 3(b). 8The even-flipped (odd-flipped) oscillating mode or Mode I (II) is the oscillation of magne- tization within + z(−z) half space. The even-flipped (odd-flipped) faltering mode or Mode III (IV) is the repeated faltering of magnetization between xy-plane and + z(−z) half space. The periodically flipping mode or Mode V is the repeated magnetization flip. Notably, Modes I and II (or III and IV) are almost identical, but differ in the number of spin flips before decaying to the adiabatic state. Only Mode V shows the continuous spin flip in time in its adiabatic state. As one can expect from their names, the adiabatic modes originate from DC modes. Modes I and II come from oscillating mode, Modes III and IV come from the faltering mode, and Mode V comes from the flipping mode. The modes are chosen by the interplay of x,ωAC,δ, and α. Both xandωACunderscore their importance after decaying to adiabatic states while δandαplay a pivotal role during the decay process. Primary investigation is performed by phase diagrams for the modes in xandωACatα= 0.03 with different δpresented in Figs. 3(c,d). For x <1, Modes I and II appear, while for x≥1, Modes III, IV, and V manifest. This transition occurs around x∼1 since x= 1 is where two equilibrium states meet at the y-axis. Specifically, for x <1, as the equilibrium position does not reach y-axis at any time, the magnetization oscillates within either + zor−zhalf space. For x≥1, however, as the equilibrium reaches y-axis, the magnetization either falters on the xy-plane or flips its position between + zand−zhalf spaces periodically. ωACalso plays a role in determining the modes. Primarily, ωACchooses the faltering and flipping modes when x≥1, since it determines the time duration τythat the equilibrium state at each time stays at the y-axis. One can obtain the duration by finding the maximum τysatisfying σ(t)≥1 int∈[t0, t0+τy]. In the case of sinusoidal σ(t), this can be expressed asτy=1 ωAC(π−2ζ), where ζ= arcsin(1 /x)∈(0, π/2]. For τy, as the equilibrium state is at they-axis, the magnetization modulates near the y-axis. After τy, the equilibrium position is divided again and deviates away from the y-axis. Then, depending on its modulated position, the magnetization chooses one of the equilibrium state, which leads to either faltering or 9Mode IMode IIMode IIIMode IVMode Vbcd 𝛿𝛿ef𝜔!" 𝜔!"𝜋/200.020.040.060.080.10 0.020.040.060.080.10 𝜋/3𝜋/6𝜋/20𝜋/3𝜋/6 No FlipFlip onceNo FlipFlip twiceFlip once𝑥𝑥𝜔!"𝜔!"0120120.010.050.030.020.040.010.050.030.020.04 aFinal StateDriving FrequencyPeriodicityAdhere to equilibrium at tAdiabaticSmallOOResonatingIntermediateOXChaoticLargeXX Figure 3: The modes in the adiabatic state under AC. (a) The classification and comparison of final states. The final states transit with the driving frequency of AC, from adiabatic to resonating, and to chaotic state in sequence. While adiabatic and resonating states have periodicity, chaotic state does not. While adiabatic state adheres to the equi- librium state at each t, others do not. (b) Schematics of distinct AC adiabatic modes after decay. (c-d) 2D phase diagrams of adiabatic modes in xandωACwith α= 0.03, at (c) δ= 0 and (d) δ=π/2. (e-f) 2D phase diagrams of adiabatic modes in δandωACat (e) x= 0.6 and (f) x= 0.8. The violet denotes Mode I, the red denotes Mode II, the blue denotes Mode III, the yellow denotes Mode IV, and the green denotes Mode V. 10flipping modes. Additionally, the window of xfor every mode at higher ωACis opened up wider than that at lower ωAC, showing the fan-like feature in Figs. 3(c,d). Unlike the faltering mode under DC, the window of Modes III and IV, originating from the faltering mode, expands to the finite range of x. Lastly, it is noteworthy that the phase of ωAC= 0 in Figure 3(d) is the same as the phase in α= 0.03 line of Fig. 2(c). This substantiates that adiabatic modes can be derived from DC modes. On the other hand, δacts as a switch to turn on the spin flip during the decay process to adiabatic states. Specifically, comparing Figs. 3(b) to (c), Modes II and IV barely appear when δ= 0, which means that the spin flip is turned off near δ= 0. This happens because the average of σ(t) during the decay time is small, so the spin cannot flip before decaying the adiabatic state. This can be supported by Figs. 3(e-f). In Fig. 3(e), we present phase diagrams for the modes in δandωACatx= 0.6. The transition from Modes I to II occurs once during the increase of δ. This happens because the spin flip is turned on by finite δ. Increasing x, the spin flips more times just as in DC case, so the repeated transition between Modes I and II can also be observed in Fig. 3(f). We should note that the increase ofαreduces the decay time. Above arguments holds also for the triangular wave instead of sinusoidal wave. [See SI.] We move on to high- ωACand low- αregime to investigate ferromagnetic magnetoresonance (FMR). One could expect that the resonance frequency is closely related to the frequencies ωDCin Fig. 2(a). In fact, when ωACapproaches ωDC, the oscillating amplitude becomes larger, and eventually the adiabatic state changes to the resonating state and to the chaotic state sequentially. By checking the final state after t∼1 ms, we present 2D phase diagrams inωACandαatx= 0.3 and 0 .7 in Fig. 4(a). Both diagrams show similar behavior. It begins with the adiabatic state at low frequencies, confronts the transition to the resonating state, and reaches the chaotic state at a certain range of ωAC. For x= 0.7, the window of chaotic states is opened up widely as SOT becomes large. We further illuminate the transition of states by Fourier transform. The transition is 11c 𝜔log!"𝜔log!"𝜙𝜔𝜔iiiiiiiviviviv𝜔!"𝜔#2𝜔#−𝜔!"2𝜔!"2𝜔!"2𝜔!"2𝜔!"𝜔iv𝜔 𝑥=0.7 𝜔!" 10𝛼×104𝛼×104𝑥=0.3ba iiiiiiiiiiiiiii𝜔#𝜔!"𝜔#−𝜔!"0.00.51.01.52.010-610-510-40.0010.0100.100iiiiviviviviiiiv 0.00.51.01.52.010-610-510-40.0010.0100.100iviiiiiiiiiiiiiii ~1/𝜔𝜔!"=0.60𝜔!"=0.53𝜔!"=0.10iviv2𝜔!"2𝜔!" log!"𝜔0.01.02.00.01.02.00.01.0-1-6-2-3-4-5-1-6-2-3-4-5-3-7-4-5-6Adiabatic ChaoticResonating 2.0log!"𝜙𝜔log!"𝜙𝜔 log!"𝜙𝜔log!"𝜙𝜔log!"𝜙𝜔Figure 4: The transition from adiabatic to resonating and chaotic states under AC. (a) 2D phase diagrams of the final states in ωACandα. The adiabatic state is in dark blue, the resonating state is in green, and the chaotic state is in dark red. The top panel is at x= 0.3 and the bottom panel is at x= 0.7. (b) The schematics of change in Fourier transform in for each state. (c) The corresponding examples to each state of (b) at x= 0.3 andα= 0. illustrated in Fig. 4(b), where the change in |ϕ(ω)|byωACis schematically presented. Here, ϕ(ω) corresponds to the Fourier transform of azimuthal angle ϕ(t) from t= 0 to t= 0.21 ms (5×105unit times). Figure 4(c) showcases representative examples of Fourier transforms at x= 0.3 and α= 0. For adiabatic and resonating states, four distinct peaks are observed in the top panels: i) a main peak at ω1derived from ωDC, ii) another main peak driven by AC atω2=ωAC, iii) induced peaks from i) and ii) at ω3=ω1+n(ω1−ωAC), and iv) subpeaks at ω4=|ω1,2,3±2nωAC|(n∈N). Notably, as the time interval is elongated from 0.21 ms, Peaks i and iii decay while Peaks ii and iv gain intensity. [See SI.] This means that Peaks i and iii are related to the decay process while Peaks ii and iv are related to the stable state after decaying. Increasing ωACto the higher frequency, the change only occurs to the distance between peaks, as the frequency of Peak i decreases and that of Peak ii increases. Thus, adiabatic and resonating states are indistinguishable solely by Fourier transform. This is substantiated by the phase diagrams in Fig. 4(a), where the transition from adiabatic and resonating states is not distinctly delineated. This behavior is consistent for all modes in 12Fig. 3(b). Near the chaotic state, the peaks form array in a period of ωAC−ω1as shown in the middle panels. As ωACincreases more, chaos sets in, destroying all peaks and being ϕ(ω)∝1/ωas shown in the bottom panels. This behavior does not change although the range of time is elongated. Discussion We address here about the typical values of parameters in the realistic systems. The typical value of ferromagnetic anisotropy is 1 −10µeV/atom,26that of the current density in the experiments is ∼107A·cm−2, and that of the itinerant spin polarization by Rashba-Edelstein Effect is estimated about 10−4ℏper unit cell.45The typical value of Gilbert damping constant is∼10−3−10−2,46and that of the unit of AC frequency is estimated about 1 −10 GHz. So far, we discuss the magnetization dynamics at the interface of TI-FM heterostructure. Under DC, we observe the oscillating and flipping modes without damping, which is deter- mined by the relative strength of anisotropy and SOT. With damping, the number of spin flips is determined by the duration of the crossover from flipping to the oscillating mode. Under AC, we observe five distinct modes in the low frequency regime and their evolution to the resonating and chaotic states in the high frequency regime. Although we mainly discuss the TI-FM heterostructure due to its high efficiency, our work can be applied to the general systems with strong Rashba spin-orbit coupling. This is because the assumption un- derlying in our work is only that the current carries finite spin due to the Rashba-Edelstein Effect. Our work offers insights into the spin control by spin-orbit coupling, underscoring the practical aspects of the world of spintronics. Acknowledgement We thank Wataru Koshibae for the fruitful discussions. This work was supported by JST, CREST Grant Number JPMJCR1874, Japan. 13Supporting Information Available This material is available free of charge via the Internet at https://pubs.acs.org/ •The dependence of crossover time from flipping to oscillating mode on xandα, the dependence of decay time to adiabatic modes on α, the comparison between triangular and sinusoidal waves, and the time evolution of peaks in Fourier transform. 14Supporting information for “Unraveling the dynamics of magnetization in topological insulator-ferromagnet heterostructures via spin-orbit torque” Taekoo Oh∗and Naoto Nagaosa∗ RIKEN Center for Emergent Matter Science (CEMS), Wako, Saitama 351-0198, Japan E-mail: taekoo.oh@riken.jp; nagaosa@riken.jp Table of contents •The dependence of crossover time of DC in xandα. •The dependence of decay time of AC in α. •The comparison of triangular and sinusoidal waves. •The time evolution of peaks in Fourier transform. 1The dependence of crossover time of DC in xand α ab 𝛼𝛼𝛼 Figure 1: τcinxand α.(a)τcinxwith various α. (b) τcinαwith various x. Under DC, the system shows flipping, faltering, and oscillating modes without damping. When damping is introduced, the crossover from flipping or faltering mode to oscillating mode occurs. We acquire the crossover time τcempirically. Figure 1(a) shows the relation ofxandτc, while Figure 1(b) exhibits the relation of αandτc. For small α, one could note that the empirical relation of τctoxandαisτc∝e4.53xα−1. The dependence of decay time of AC in α Under AC and damping, the system decays into an resonating or an adiabatic state or evolves into a chaotic state. The decay time is defined as the duration of time reaching to resonating or adiabatic states from the initial state. We empirically observe the decay time at different x,ωACandαin Fig. 2 One could note that the decay time is proportional to α−1. 2Figure 2: The decay time in α.The decay time in αwith different xandωAC. The comparison of triangular and sinusoidal waves We mainly discuss the magnetization dynamics under sinusoidal waves of AC. One could replace the sinusoidal waves with triangular waves. The triangular wave is given by σ(t) =|mod[2ωACσ π(t−t0),4σ]−2σ| −σ. (1) The amplitude of this function is σ, and the frequency of this function is ωAC. At t0= π/(2ωAC), the function becomes sine-like, while at t0= 0, the function becomes cosine-like, as shown in Fig. 3(a). We set ωAC= 2π/240≈0.026,α= 0.03, and compare the adiabatic modes for sinusoidal and triangular waves in Fig. 3(b). In the manuscript, we describe that Modes II and IV does not appear under sine waves. The argument is also consistent with triangular waves, since the phase diagram under sine-like triangular waves does not show Modes II and IV as well. The difference between sinusoidal and triangular waves is that the window for each mode widens up for triangular waves. 3a𝜎(𝑡) 𝜎(𝑡)𝑡/𝑇𝑡/𝑇b𝑥0.51.01.52.0𝑥0.51.01.52.0SineSine-like triangular 𝑥0.51.01.52.0𝑥0.51.01.52.0CosineCosine-like triangularSine-like triangular Cosine-like triangularIIIVIIIIVFigure 3: The magnetization dynamics under triangular waves. (a) (upper panel) a sine-like triangular wave, (lower panel) a cosine-like triangular wave. (b) The adiabatic mode diagrams at ωAC= 2π/240 and α= 0.03 for sine (first), sine-like triangular (second), cosine (third), and cosine-like triangular (fourth) waves. The violet denotes Mode I, the red denotes Mode II, the blue denotes Mode III, the yellow denotes Mode IV, and the green denotes Mode V. The time evolution of peaks in Fourier transform In our manuscript, we describe four types of peaks observed in the Fourier transform of ϕ(t). These include: i) the main peak originating from the DC modes denoted as ω1, ii) a peak corresponding to the AC driven frequency ω2=ωAC, iii) induced peaks resulting from the frequency difference between i) and ii), given by ω3=ω1+n(ω1−ω2), and iv) subpeaks represented by ω4=|ω1,2,3±2nωAC|. Peaks i) and iii) are associated with the decaying process, whereas peaks ii) and iv) represent the stable state after decay. Figure 4 illustrates this phenomenon. In Fig. 4(a), the Fourier transform is performed from t= 0 to 1 .26µs, while in Fig. 4(b), it is performed from t= 0.12474 to 0 .126 ms. As time progresses, Peaks i and iii diminish due to damping, while Peaks ii and iv remain robust owing to the driven frequency. 40.00.51.01.52.010-510-40.0010.010 0.00.51.01.52.010-510-40.0010.010iiviviviviviiiiviviiiiiviviviab𝜙𝜔𝜙𝜔𝜔𝜔Figure 4: The time evolution of peaks in Fourier transform of ϕ(t) (a) The Fourier transform of ϕ(t) int= 0 to 1 .26µs. (b) The Fourier transform of ϕ(t) near t= 0.12474 to 0.126 ms. 5References (1) Witczak-Krempa, W.; Chen, G.; Kim, Y. B.; Balents, L. Correlated quantum phe- nomena in the strong spin-orbit regime. Annu. Rev. Condens. Matter Phys. 2014 ,5, 57–82. (2) Hasan, M. Z.; Kane, C. L. Colloquium: topological insulators. Reviews of modern physics 2010 ,82, 3045. (3) Qi, X.-L.; Zhang, S.-C. Topological insulators and superconductors. Reviews of Modern Physics 2011 ,83, 1057. (4) Pi, U. H.; Won Kim, K.; Bae, J. Y.; Lee, S. C.; Cho, Y. J.; Kim, K. S.; Seo, S. Tilting of the spin orientation induced by Rashba effect in ferromagnetic metal layer. Applied Physics Letters 2010 ,97. (5) Liu, L.; Moriyama, T.; Ralph, D.; Buhrman, R. Spin-torque ferromagnetic resonance induced by the spin Hall effect. Physical review letters 2011 ,106, 036601. (6) Kim, J.; Sinha, J.; Hayashi, M.; Yamanouchi, M.; Fukami, S.; Suzuki, T.; Mitani, S.; Ohno, H. 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EPJ Web of Conferences. 2013; p 18003. 25TOC Graphic Topological InsulatorFerromagnet⃗𝐼⃗𝜎𝑀 26

1607.01692v2.Orbital_mapping_of_energy_bands_and_the_truncated_spin_polarization_in_three_dimensional_Rashba_semiconductors.pdf

1 Orbital mapping of energy bands and the truncated spin polarization in three-dimensional Rashba semiconductors Qihang Liu1,*, Xiuwen Zhang1, J. A. Waugh2, D. S. Dessau1,2 and Alex Zunger1,* 1University of Colorado, Boulder, Renewable and Sustainable Energy Institute Colorado 80309, USA 2Department of Physics, University of Colorado, Boulder, Colorado 80309, USA *Email: qihang.liu85@gmail.com, alex.zunger@colorado.edu. PACS: 71.70.Ej, 75.70.Tj, 75.10.Dg Abstract Associated with spin-orbit coupling (SOC) and inversion symmetry breaking, Rashba spin polarization opens a new avenue for spintronic applications that was previously limited to ordinary magnets. However, spin polarization effects in actual Rashba systems are far more complicated than what conventional single-orbital models would suggest. By studying via first-principles DFT and a multi-orbital k • p model a 3D bulk Rashba system (free of complications by surface effects) we find that the physical origin of the leading spin polarization effects is SOC-induced hybridization between spin and multiple orbitals, especially those with nonzero orbital angular momenta. In this framework we establish a general understanding of the orbital mapping, common to the surface of topological insulators and Rashba system. Consequently, the intrinsic mechanism of various spin polarization effects, which pertain to all Rashba systems even those with global inversion symmetry, is understood as a manifestation of the orbital textures. This finding suggests a route for designing high spin-polarization materials by considering the atomic-orbital content. 2 The coupling between the motion of electrons and spins leading to spin polarization without external magnetic field is the focus of the emerging field of spin-orbitronics [1, 2], a branch of spintronics [3] that encompasses many interesting areas such as the Dresselhaus [4] and the Rashba [5] effects, spin-orbital toque [6, 7], topological insulation [8], and Majorana fermions [9]. The idea of control of spin degree of freedom even without external magnetic field is based on the fact that in a non- centrosymmetric system, spin-orbit coupling (SOC) sets up an effective internal magnetic field that creates spin splitting 𝐸"𝒌,↑−𝐸'(𝒌,↓) between spin-up and spin-down components in bands 1 and 2 away from the time-reversal invariant wavevector K*. Specifically, Rashba spin splitting provides a classical scenario of spin topology encoded already in a simplified single-orbital (e.g., one s band) Hamiltonian 𝐻=ℏ𝟐𝒌𝟐/2𝑚∗+𝜆∇𝑉 ×𝒌∙𝝈, where fully spin-polarized bands form two oppositely rotating spin loops at Fermi surface. However, in real materials spin can couple to multiple orbitals that make up the band eigenstates. The leading SOC spin effects that deviate from the classical single-orbital picture include: (i) the spin polarization Sn(k) of each spin-split band (n, k) appears to be truncated below its maximum value 100%; (ii) Each branch of the pair of spin-split bands (that are degenerate without SOC) experiences different degrees of spin truncation away from the time-reversal invariant wavevector K*, resulting in a net spin polarization for the band pair (iii) The Rashba bands with two loops of energy contours can have identical helicities of spin texture. These effects were studied primarily in two-dimensional (2D) metallic surfaces — the classic Rashba systems [10-16]. Effect (i) and (ii) were theoretically discussed for freestanding Au(111) films [10], and in BiAg2 metallic surface alloys [11, 16], while effect (iii) was predicted in the unoccupied bands of Bi/Cu(111) [12] and Bi/Ag(111) [14] surface alloys. The physics behind these intriguing spin effects was briefly touched upon in the context of 2D Rashba metallic films as being a consequence of the coupling between spin and different in-plane orbitals [12, 16]. However, in all 2D Rashba platform noted above there is a need for a free surface to observe the effects, so ordinary surface effects (such as broken bonds and surface band bending [17]) can cloud the intrinsic mechanism of these spin effects to be well established [13]. By extending the spin effects (i)-(iii) from the originally studied 2D metallic Rashba systems to 3D surface-free bulk Rashba semiconductors, we provide two pertinent 3 generalizations: (a) We use the construct of orbital texture [a k-space map 𝐼<=,>(𝑘@,𝑘A) of the content of orbital ml in band n, see Supplementary Eq. (S1)] familiar from topological insulators [18-22]. We point out that the effect of switch in orbital texture between two bands is common to topological insulators and to 3D Rashba semiconductors. Specifically, in the topologically-trivial bulk Rashba semiconductor the orbital textures of different Rashba bands switches from “radial” to “tangential” (with respect to the energy contour) character at the band crossing wavevector K*, in full analogy with the phenomena previously observed [19, 22, 23] and calculated [21, 22] at the surfaces of 3D topological insulator (TI) Bi2Se3. Thus, the switch of orbital texture and indeed effects (i)-(iii) are not specific to topological or Rashba effects, but originate fundamentally from the fact that energy bands in complex solids invariably show a mixture of different azimuthal total orbital angular momentum (OAM) mj, and that SOC can induce hybridization specifically between spin and multiple orbitals especially those with nonzero ml, respectively. (b) We show that the spin polarization truncated by multiple orbital hybridization can be generalized even to systems with global inversion symmetry, manifesting the “hidden spin polarization effect” [24]. The understanding of effects (i)-(iii) and their reflection in the switch in orbital texture could provide better design guidelines for material selection and for spin manipulation in actual material application, e.g., electron confinement induced by spin-flip backscattering [16] and spin-galvanic effect [25, 26]. We reached these conclusions by applying density functional theory (DFT) and a multi-band k • p model to a 3D bulk Rashba compound BiTeI [27]. We find that within the six energy bands near the Fermi level (EF) there are (i) large spin truncation per band at the band crossing wavevector K* with the residual spin polarization ranging from 0% to 85% far greater than the ~5 % seen in Au [111] surface; (ii) a net spin polarization of band pairs up to 50% for the top two valence bands, and (iii) identical spin-rotating loops at the occupied bands that can be examined by future angle-resolved photon emission spectroscopy (ARPES) measurements. Truncated spin polarization, net spin polarization and spin texture in BiTeI: This compound is a 3D bulk semiconductor [space group P3m1, see Fig. 1(a)] that manifest 4 strong SOC, the ensuing orbital hybridization, and a polar field due to the positively charged Bi-Te layer that connects to the negatively charged I layer. The consequent Rashba spin-split bands [27, 28] from DFT calculations are shown in Fig. 1(b) (see Supplementary Materials for the DFT methods). We focus on the top four hole bands VB1-VB4 (going down from EF) and bottom two electron bands CB1-CB2 (going up from EF) around the K* = A(0,0,0.5) wavevector. The spin polarization Sn(k) (n = VB1-VB4, CB1-CB2) along ky direction is shown in Fig. 1(c). In what follows we discuss spin effects (i)-(iii) in bulk BiTeI: (i) Truncation of single band spin polarization: The band-by-band spin polarization is calculated as the expectation value of the spin operator in each of the six spin-split bands at the wavevector K*. We find that the magnitude of spin polarization is below the maximal magnitude of 100%. For VB1, VB2 and CB1, CB2 the spin polarization is ±85% and ±51%, respectively, while for VB3, VB4 the spin polarization is 0, i.e., a complete quenching of spin. Away from the band crossing wavevector K* the spin polarization of VB1 and VB2 evolves quite differently with ky. VB1 is highly spin polarized in the considered momentum range up to SVB1 = -96%. while VB2 loses its spin polarization rapidly with increasing ky down to SVB2 = 46%. (ii) Net spin polarization of pairs of spin-split bands: If we sum SVB1(k) and SVB2(k) (which would add up to zero in the absence of SOC), we find a net spin up of -50% at ky = 0.06 Å-1. On the other hand, the sum of SVB3(k) and SVB4(k) reaches 48% at k = 0.06 Å-1, while SCB1(k) and SCB2(k) change slightly with k. We note that the position of the net polarization peak does not overlap with the energy peak in the band structure which locates at k = 0.09 Å-1 in ky direction, suggesting that the spin polarization is not a reflection of the eigenvalue dispersion. (iii) Identical directions of spin-rotation in the helical spin texture bands: The band pair VB1+VB2 and the pair CB1+CB2 show the classical Rashba-type spin texture, i.e., opposite helicities of spin loops. However, as shown in Fig. 1(c) before SVB4 falls below 0, SVB3 and SVB4 have the same sign, implying two spin loops with the same helicity in the area of |k//| < 0.10 Å-1(see Supplementary Fig. S1-S3 the spin textures of all the six bands). 5 All three spin effects discussed above are absent in the conventional single-orbital model and thus reflect a manner of the complex interplay between spin and various OAM under the regime of SOC. To get a full picture it is useful to consider the orbital texture, i.e., the k-space map 𝐼<=,>(𝑘@,𝑘A) of the content of orbital ml in band n. Orbital texture and its behavior for different bands: In real solids the orbital content generally varies with the wavevector and band index, reflecting the changing symmetry. Our base DFT calculation in this paper involves a vibrational calculation including all occupied states below EF in the solid via DFT total energy minimization, assuring physically realistic electronic structure. The orbital intensity is obtained by projecting the SOC-relevant band eigenstate (n, k) onto local orbitals on atomic sites as shown in Supplementary Eq. (S1). This approach is different from the usual model Hamiltonian approaches that assume at the outset which orbitals will participate in given energy bands in a given crystal structure, notwithstanding the question if such orbitals would result in a total energy that supports the stability of the said crystal structure. Figures 2(a)-2(d) show the DFT calculated orbital texture given by py orbital intensity at different energy contours relative to K*, for VB1 and VB2. We find that for VB1 the calculated py orbital texture component is maximal along the ky direction and minimal along kx (where the px orbital dominates the in-plane states). On the other hand, for VB2 the py orbital texture component is minimal along ky and maximal along kx. Thus, the orbital texture of VB1 and VB2 are different from each other and dominated, by radial and tangential in-plane orbital patterns, respectively. This difference leads to a radial-tangential orbital texture switch. To trace the switch between these two bands we follow Ref. [22] to define the in-plane orbital polarization λ as a function of momentum k and band index n as 𝜆𝑛,𝒌=CDEFC(DG)CDEHC(DG), where 𝐼𝑝@,A denotes the calculated orbital intensity of px,y. Figure 2e shows λ as a function of the in-plane azimuth angle θ (defined in Fig. 2b), confirming the switch of the intensity distribution in going from VB1 to VB2. Moreover, the intensity variation fits very well to a sin2θ or cos2θ distribution, with a period of π. As shown in Fig. 2f, λ changes the sign as the momentum ky, passing through K*, indicating that the radial-tangential switch happens exactly at the band crossing wavevector. For VB3 and VB4 the orbital textures also have a switch between tangential 6 and radial characters at K* (see Supplementary Fig. S4). On the other hand, in CB1 and CB2 both Bi and Te atoms have considerable px,y components, but with different orbital textures. For px,y orbitals of Bi atom there is a radial-tangential switch from CB1 to CB2, while for px,y orbitals of Te atom the orbital switch has an opposite trend, i.e., tangential-radial (see Supplementary Fig. S5). This observation agrees closly with the recent ARPES measurement by King et al. on the conduction surface state of BiTeI [29], and further confirms that such intriguing behavior comes from the intrinsic bulk state rather than any surface effects. Universality of orbital texture in bulk Rashba and surface of TI revealed by k • p modeling of DFT: The orbital texture switch between two bands at K* in the topologically-trivial semiconductor bares an interesting analogy to the recently observed angle-resolved photon emission spectroscopy (ARPES) measurements [19, 22] and DFT calculations [21] at the surfaces of TI Bi2Se3. Here we use a multi-orbital k • p model to illustrate how mixing orbitals of different ml and mj couple with spin and lead to the orbital texture switch and the spin polarization effects (i)-(iii). The crucial basis set represented in terms of mj is obtained in DFT; we now explicitly isolate it from all other DFT bands in k • p model below. Taking VB1 and VB2 as an example, we consider the SOC Hamiltonian as a perturbative form 𝐻J=𝛼(𝜎A𝑘@−𝜎@𝑘A) that is valid for both Rashba bulk and TI surface [21], and thus write the wavefunctions in the vicinity of K* as: |𝑉𝐵1,𝑘="'(1−𝜔QR'+𝜇QR𝑘)|𝑝T⨂|𝐿𝐻+[X'(−1−𝜔QR'+𝜐QR𝑘)|𝑝Z+(𝜔QR−𝜉QR𝑘)|𝑍]⨂|𝑅𝐻 (1) |𝑉𝐵2,𝑘="'(−1−𝜔QR'+𝜇QR𝑘)|𝑝T⨂|𝑅𝐻+[X'(1−𝜔QR'+𝜐QR𝑘)|𝑝Z+(𝜔QR+𝜉QR𝑘)|𝑍]⨂|𝐿𝐻 (2) Where the in-plane p orbital basis are tangential 𝑝T=−sin𝜃𝑝@+cos𝜃|𝑝A and radial 𝑝Z=cos𝜃𝑝@+sin𝜃|𝑝A; |𝑍=𝜔e|𝑠+𝜔g|𝑝g with 𝜔e'+𝜔g'=1; the spin basis are eigenstates of 𝐻J, i.e., LH and RH helical spin states |𝐿𝐻="'𝑖𝑒FXj1 and |𝑅𝐻="'−𝑖𝑒FXj1; and 𝜔QR, 𝜐QR, 𝜉QR are the wavefunction coefficients that are 7 band-dependent. More details on these wavefunctions can be found in Supplementary Materials. By calculating the difference of pt and pr intensity and omitting the higher-order k term we find that [𝐼𝑛𝑡𝑝T− 𝐼𝑛𝑡𝑝Z]|QR"(QR')=±𝑘[1−𝜔𝑉𝐵'𝜇𝑉𝐵∗−𝜐𝑉𝐵∗+𝑐.𝑐.] (3) From the modeling wavefunctions Eq. (1) and (2) and the difference of pt and pr intensity shown in Eq. (3), clear evidence is provided that the dominant in-plane orbital is different for the two spin-split valence bands, in other words the radial-tangential orbital texture switches at k = 0, i.e., the band crossing point K*. The model also reveals that the symmetry lowering away from K* permits the mixing of new mj = ±3/2 (|+⨂|↑ and |−⨂|↓, or “heavy hole” like) components into the K* wavefunctions (mj = ±1/2 for the VB1 and VB2 of BiTeI), leading to the orbital texture switch. Benefiting from the truncated basis set represented in terms of mj, as distilled from the all-orbital DFT representation, we conclude that the underlying physical origin of the common behavior in both TI and non-TI materials is the SOC symmetry-enforced hybridization of different azimuthal total OAM mj components into band eigenstates.. This hybridization was absent at the high symmetry K* point. Thus, this effect is not limited to topological insulators but is far more general and applies also to Rashba compounds that are topological-trivial bulk semiconductors such as BiTeI [29] (see Supplementary Materials for more details on the comparison between Rashba bulk and TI surface). Understanding the spin polarization effects: Eq. (1) and (2) show that each orbital component couples with a certain spin state, forming orbital-dependent spin textures [see Supplementary Eq. (S2)]. Maximal spin magnitude arises when every orbital-dependent spin texture co-aligns, i.e., has the same helicity. This requires that the band eigenstate be composed exclusively of orbitals with the same azimuthal quantum number ml. In real materials where SOC mixes orbitals with different ml in one eigenstate (n, k), the corresponding spin polarization is truncated relative to its maximal value. Specifically, the tangential in-plane orbital (pt) always couples opposite spin texture to that of radial in-plane orbital (pr), s and pz orbital. At the wavevector A with k → 0, pt and pr 8 components have the same intensity but opposite spin pattern and thus cancel each other, making the spin polarization S(k→0) = 𝜔QR' all contributed by s and pz orbital. This scenario gives the truncated spin polarization at K* for all the bands shown in Fig. 1f. Especially, VB3 and VB4 are mj = ±3/2 states with the corresponding wavefunctions at K* containing only in-plane p components:|𝑝H⨂|↑ and |𝑝F⨂|↓, leading to the complete quenching of spin. The total spin polarization summing over all bands is equivalent to the value obtained from the contributions of ml = 0 states, e.g., s, pz, and dz2, etc. This statement is valid also in the traditional 2D Rashba systems like Au(111) surface [10], in which the surface Rashba bands are nearly exclusively composed by the s and pz states, and thus have nearly 100% spin polarization. Due to the orbital texture switch, we find from Eq. (1) and (2) that the dominat in-plane p orbitals of a pair of Rashba bands couple to spin textures with the same helicity. This fact is confirmed by DFT calculation showing that the dominating radial orbital for VB1 and tangential orbital for VB2 both have RH spin texture (see Fig. 3). Therefore, by summing the in-plane orbital contribution and the s+pz orbital contribution to the spin textures, the total spin polarization shows different degree of truncation for VB1 and VB2, as shown in Fig. 1f (see Supplementary Materials for more details on orbital-dependent spin textures for different bands and the effects of SOC strength). For VB3 and VB4, pt and pr dominate the whole state around K*, respectively. Consequently, the total spin textures form two LH helical spin loops (see Fig. S3 and S4). Unlike the case in Bi/Cu(111) [12] and Bi/Ag(111) [14] surface alloys, these identical spin-rotating loops occur at the occupied bands that is detectable by ARPES measurements. Generalizing to systems with inversion symmetry: The truncated spin polarization and the net polarization effect are illustrated above in a Rashba semiconductor BiTeI. However, we are not using any special feature of this orbitally-hybridized compound other than SOC-induced (Rashba) spin splitting, and thus expect our finding to pertain to a very broad range of such compounds, whether inversion symmetry is present or not. In centrosymmetric crystals the spin bands are degenerate in E vs k momentum space, but this does not mean that the spins are mixed in position space. Correspondingly, the centrosymmetric systems with lower-symmetry sectors could manifest a “hidden” form 9 of spin polarization named R-2 effect [24, 30]. Like two oppositely stacked Rashba layers, in R-2 system there is still finite spin polarization localized on each atomic site i that feels inversion-asymmetric environment, and such polarization is compensated in k space by another atom forming inversion partner of site i. Therefore, if we consider the local spin polarization in real space, i.e., localized on one inversion-asymmetric atoms or sectors, the spin truncation effects also happen when spin is coupled by multi-orbitals with different ml. We choose a centrosymmetric R-2 material LaOBiS2 (using the reported space group P4/nmm) to illustrate the truncation effects and find that the hole spin has the spin polarization ~90%, while the electron spin is only ~30% polarized (see Supplementary Materials for more details). It is noticeable that the orbital-dependent spin texture is robust again small perturbation that breaks the global inversion symmetry such as electric fields, which is highly feasible for the detection on experiments. Discussion and design implications: In the past few years wide areas of physics and material science that related to SOC build up the new field spin-orbitronics. By generalizing the previously observed orbital texture switch in TI surface to a bulk Rashba semiconductor, we unveil the deeper mechanism of various spin polarization effects that is unexpected in the simple Rashba model, and provide a clear picture of the delicate interplay between spin and multi-orbitals. Our work is also expected to open a route for designing by the atomic-orbital feature high spin-polarization materials that are of vital importance for nonmagnetic spintronic applications [3]. For example, in Rashba splitting the in-plane spin drives a current perpendicular to the direction of spin polarization induced by the asymmetric Elliot-Yafet spin-relaxation, named spin-galvanic effect [25, 26]. Since the current is proportional to the average magnitude of the spin polarization, the effects of spin truncation and that different branches experience different degrees of spin polarization could have a more complex impact on the conversion process between spin and current in a Rashba system, which calls for further investigation. Acknowledgement We are grateful for helpful discussions with Dr. Yue Cao. This work was supported by NSF Grant No. DMREF-13-34170. This work used the Extreme Science and Engineering 10 Discovery Environment (XSEDE), which is supported by National Science Foundation grant number ACI-1053575. References [1] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, Nat Mater 14, 871 (2015). [2] C. Ciccarelli et al., Nat Nano 10, 50 (2015). [3] I. Žutić, J. Fabian, and S. Das Sarma, Reviews of Modern Physics 76, 323 (2004). [4] G. Dresselhaus, Phy. Rev. 100, 580 (1955). [5] Y. A. Bychkov and E. I. Rashba, Soviet Journal of Experimental and Theoretical Physics Letters 39, 78 (1984). [6] A. Manchon, Nat Phys 10, 340 (2014). [7] KurebayashiH et al., Nat Nano 9, 211 (2014). [8] M. Z. Hasan and C. L. Kane, Reviews of Modern Physics 82, 3045 (2010). [9] V. Mourik et al., Science 336, 1003 (2012). [10] J. Henk, A. Ernst, and P. Bruno, Physical Review B 68, 165416 (2003). [11] G. Bihlmayer, S. Blügel, and E. V. Chulkov, Physical Review B 75, 195414 (2007). [12] H. Mirhosseini et al., Physical Review B 79, 245428 (2009). [13] H. Hirayama, Y. Aoki, and C. Kato, Physical Review Letters 107, 027204 (2011). [14] S. N. P. Wissing et al., Physical Review Letters 113, 116402 (2014). [15] E. E. Krasovskii, Journal of Physics: Condensed Matter 27, 493001 (2015). [16] S. Schirone et al., Physical Review Letters 114, 166801 (2015). [17] M. S. Bahramy et al., Nat. Commun. 3, 1159 (2012). [18] I. Zeljkovic et al., Nat Phys 10, 572 (2014). [19] Z. Xie et al., Nat Commun 5, 3382 (2014). [20] Z. Zhiyong, C. Yingchun, and S. Udo, New Journal of Physics 15, 023010 (2013). [21] H. Zhang, C.-X. Liu, and S.-C. Zhang, Physical Review Letters 111, 066801 (2013). [22] Y. Cao et al., Nat Phys 9, 499 (2013). [23] Z. H. Zhu et al., Phys. Rev. Lett. 110, 216401 (2013). [24] X. Zhang, Q. Liu, J.-W. Luo, A. J. Freeman, and A. Zunger, Nat. Phys. 10, 387 (2014). [25] S. D. Ganichev et al., Physical Review B 68, 081302 (2003). [26] S. D. Ganichev et al., Nature 417, 153 (2002). [27] K. Ishizaka et al., Nat. Mater. 10, 521 (2011). [28] M. S. Bahramy, R. Arita, and N. Nagaosa, Physical Review B 84, 041202 (2011). [29] L. Bawden et al., Science Advances 1, e1500495 (2015). [30] Q. Liu et al., Physical Review B 91, 235204 (2015). 11 Fig. 1: (a) Crystal structure and (b) DFT band structure of BiTeI along the high-symmetric line L(0,0.5,0.5) – A(0,0,0.5) – H(1/3,1/3,0.5). (c) Spin polarization Sn(k) of the six bands VB1-4 and CB1,2 along ky direction at the kx = 0 cut. Note that Spin polarization at ky < 0 fulfills S1(-k) = -S2(k) due to Kramer’s degeneracy. 12 Fig. 2: (a-d) Orbital texture indicated by py intensity at different energy contours relative to the band crossing point, (a,b) for VB1 and (c,d) for VB2. (e,f) In-plane orbital polarization λ for (a) different energy contours as a function of the azimuth angle θ defined in panel b, and for (f) different spin-splitting bands as a function of the momentum ky at the kx = 0 cut. Note that the orbital polarization switch signs exactly at the band crossing point ky = 0. 13 Fig. 3: (a,b) The in-plane p-dependent spin textures (white arrows) for VB1 (a) and VB2 (b). The background color indicates the py intensity. (c,d) Schematic plots for the radial orbital texture and the corresponding orbital-dependent spin texture of VB1 (c) and the tangential orbital texture and the corresponding orbital-dependent spin texture of VB2 (d).

1401.4554v1.Position_and_Spin_Control_by_Dynamical_Ultrastrong_Spin_Orbit_Coupling.pdf

arXiv:1401.4554v1 [cond-mat.mes-hall] 18 Jan 2014Position and Spin Control by Dynamical Ultrastrong Spin-Or bit Coupling C. Echeverr´ ıa-Arrondo1and E. Ya. Sherman1,2 1Department of Physical Chemistry, Universidad del Pa´ ıs Va sco UPV/EHU, 48080 Bilbao, Spain 2IKERBASQUE, Basque Foundation for Science, Bilbao, Spain (Dated: May 24, 2021) Focusing on the efficient probe and manipulation of single-pa rticle spin states, we investigate the coupled spin and orbital dynamics of a spin 1/2 particle i n a harmonic potential subject to ultrastrong spin-orbit interaction and external magnetic field. The advantage of these systems is the clear visualization of the strong spin-orbit coupling i n the orbital dynamics. We also investigate the effect of a time-dependent coupling: Its nonadiabatic ch ange causes an interesting interplay of spin and orbital motion which is related to the direction and magnitude of the applied magnetic field. This result suggests that orbital state manipulation can be realized through ultrastrong spin-orbit interactions, becoming a useful tool for handli ng entangled spin and orbital degrees of freedom to produce, for example, spin desirable polarizati ons in time interesting for spintronics implementations. PACS numbers: 72.25.Rb,73.63.Kv,71.70.Ej Spin-orbitinteractionshavebeenprovenveryusefulfor realization of spintronics [1] with electrons in nanosys- tems. On one hand, it has been demonstrated in the- ory and experiment that spins can be tuned by various electric means.[2–7] On the other hand, since spin-orbit coupling entangles spin and orbital motion, spin read- out is reachable by electric means.[8] Such combination points to the possibility of probing and manipulating spins hosted by semiconductor quantum dots [9, 10] us- ing only electric fields.[5] The spin-orbit control of qubits is a promising tool that suggests to investigate the ul- trastrong spin-orbit coupling regime to see all features of this technique. Extreme spin-orbit interactions can be achieved at the surfaces of semiconductors coated with heavy metals (see, e.g. [11, 12]), which allow for spin ma- nipulation by electric fields.[13, 14] Rashba performed a detailed analysis of two-dimensional quantum dots with the ultrastrong spin-orbit coupling [15]. Very recently, it was recognized that fully controllable strong interac- tions, greatly beyond the range reachable in semicon- ductors, can be produced in ultracold atomic Bose and Fermi gases by optical means.[16, 17] Similar to electrons in quantum dots, these systems are located in harmonic traps and can demonstrate changeable in time spin-orbit coupling, opening new venues for studies of related dy- namics. The spin dynamics in these systems can be studied theoreticallybyanalyzingtheinteractionsbetweenahar- monic oscillatorand a two-levelspin, makingit similar to the Jaynes-Cummings model in quantum optics, as sug- gested by Debald and Emary.[18] This is a wide-purpose model (see e.g. [19, 21, 22]) applied in different fields of condensed matter and quantum optics such as cavity quantum electrodynamics, trapped ions, and supercon- ducting qubits (see [23, 24] for recent results). In ad- dition, carbon nanotubes holding electron spins deeply coupled to the vibrational modes [25] can be describedFIG. 1: (Color online) (a) Setup scheme for external gen- eration of spin-orbit coupling through a bias voltage. Bold solid curves indicate the confining parabolic potential. (b ) A non-adiabatic time-dependent field can cause electron dis - placement with spin rotation. with the Jaynes-Cummings model. The systems with spin-orbit couplings have several advantages not applicable elsewhere. We mention just two. First, the coordinate dependence of the spin density makes it possible to visualize the effects of strong cou- pling in terms of particle position and measurable spin densities. Second, spin-orbit interaction and the Zeeman field can be made time-dependent [26–28] makingthe rel- evant dynamicsboth in spin and coordinatespacesacces- sible. Theseeffects providestronglynontrivialextensions of the conventional Jaynes-Cummings model. In a quantum dot, a spin-orbit strength α=ξeEz, whereξis the material- and structure-dependent con- stant, can be induced by applying an external electric fieldEz, [29]; aschemefor the experimentalsetup is given in Fig. 1. In cold atomic gases this time-dependent mod-2 ification can be reached by changing amplitudes and ge- ometries of the corresponding laser fields. In this Letter, we present a description of the spin dy- namics of an electron in a semiconductor quantum dot or confined coldatoms, both systemssubject to strongspin- obit interactions. We focus on a one-dimensional har- monic oscillator with spin degree of freedom, capturing main physics of the systems of interest. This oscillator is underanappliedmagneticfieldwhichonecanrotatewith respect to the coordinate axes. We consider two types of spin-orbit couplings, constant and time-dependent; the results given below thereby acquiring wider application. The eigenstates can be obtained from the following Hamiltonian: ˆH(t) =/planckover2pi12k2/2m+mω2x2/2+α(t)σxk+∆ 2σ∆,(1) wheremis the particle effective mass, ∆ and θare the Zeeman splitting and tilt angle of the magnetic field as applied inxzplane, respectively, σ∆=σxsinθ+σzcosθ is the corresponding spin projection, and σx,σzare the Pauli matrices. In the second quantization this Hamilto- nian reads ˆH=/planckover2pi1ω(ˆa†ˆa+1/2)+i/planckover2pi1ωg(t)(ˆa†−ˆa)σx+∆ 2σ∆,(2) where ˆa†and ˆaare the creation and annihilation opera- tors, respectively, and g=α/radicalbig m/2/planckover2pi13ωis a dimensionless coupling constant, which can be understood as the ratio of the characteristic anomalous spin-dependent velocity α//planckover2pi1to the characteristic quantum velocity spread in the ground state of the harmonic oscillator/radicalbig /planckover2pi1ω/m, or as the ratio of the quantum oscillator length l0=/radicalbig /planckover2pi1/mω to the spin precession length /planckover2pi12/mα.We use the basis of spin orbitals |n/angbracketright|σ/angbracketright, composed of the eigenstates of ˆ a†ˆa, |n/angbracketright, and those of σz,|σ/angbracketright ≡ |↑/angbracketrightzand|↓/angbracketrightzwith respect to thez-axis. Numerical values of gstrongly vary from system to system. For InSb-based quantum dots, where αcan reach 10−5meVcm,mis of the order of 0.02 of the free electron mass, and ω∼1012s−1, one can expect g≈1. For cold fermions such as40K, whereα//planckover2pi1can be of the order of 10 cm/s and ω∼103s−1,gcan be of the order of 10. To make connection to previous works on the ultra- strong regime (see, e.g. [19]), first we investigate the effect of a constant coupling. The eigenstates of the full Hamiltonian |φi/angbracketrighthave the form/summationtext n|n/angbracketright(cu n|↑/angbracketrightz+cd n|↓/angbracketrightz) with the expansion coefficients cu nandcd n; the normalized orbitals are expressed in the x-representation as /angbracketleftx|n/angbracketright=4/radicalBigg 1 πl2 022n(n!)2exp/bracketleftbigg −x2 2l2 0/bracketrightbigg Hn/bracketleftbiggx l0/bracketrightbigg ,(3) whereHnis then-th order Hermite polynomial. In the limit of a very weak coupling, g≪1, in an arbitrar- ily directed magnetic field, |φi/angbracketrightcontains five main con- tributions. The main one is the direct product of |n/angbracketrightFIG. 2: (Color online) Nonzerospin densities ρx,zfor different gvalues when the electron is at the ground state. Note that ρxandρzmerge for smaller gcoupling. Here and below we use a truncated Hilbert space of 64 states, sufficient to study ultrastrong spin-orbit coupling. The Zeeman splitting in a ll calculations is taken as ∆ = 0 .5/planckover2pi1ω. and eigenstate of σ∆. The other four are perturbative terms of direct products of |n±1/angbracketrightand eigenstates of σ∆. Atθ= 0, corresponding to the conventional Jaynes- Cummings model, spin selection rules exclude two of these four states. Moreover, at θ= 0, in the eigenstates the contributions with different spatial parities have op- posite spins. The expectation value of the velocity in the eigenstates is zero, /angbracketleftv/angbracketright=d/angbracketleftx/angbracketright/dt=0; however, the mean momentum is finite: /angbracketleftv/angbracketright=i /planckover2pi1/angbracketleft[H,x]/angbracketright=/planckover2pi1 m/angbracketleftk/angbracketright+α /planckover2pi1/angbracketleftσx/angbracketright= 0, /angbracketleftk/angbracketright=−√ 2g l0/angbracketleftσx/angbracketright. (4) Correspondingly, for the coordinate /angbracketleftφi|x|φi/angbracketright= 0. Equa- tion (4) corresponds to zero mechanical momentum /planckover2pi1k− Afor the gauge [20] A=−mασx//planckover2pi1. The total spectrum results from the magnetic-field mixing of two parabolic branches, that for /angbracketleftk/angbracketright=−√ 2g/l0when the spin state is |↑/angbracketrightx, and that for /angbracketleftk/angbracketright=√ 2g/l0when it is |↓/angbracketrightx. To show the advantages of systems with spin-orbit coupling, we calculate the spatially resolved spin den- sitiesρj(x),j=x,y,z, providing valuable info about the system.[33] For an arbitrary state |ψ/angbracketright, presented in the form/summationtext n|n/angbracketright(au n|↑/angbracketrightz+ad n|↓/angbracketrightz), these functions are defined as ρj(x) =/summationdisplay n,m/angbracketleftn|x/angbracketright/angbracketleftx|m/angbracketright(au∗ n,ad∗ n)σj/parenleftbiggau m ad m/parenrightbigg .(5) We focus on a particle in the ground state and take θ=π/4 as an example. The densities ρj(x) are presented in Fig. 2. The integrals of ρj(x) over thex-coordinate are the spin expectation values.3 FIG. 3: (Color online) Spin densities (a) ρxand (b) ρyas a function oftime for an electron spin antiparallel totheapp lied magnetic field with θ=π/4; here we take the marginal case g=1. Next, weanalyzethe coupleddynamicsofasystemput away from an eigenstate. For this purpose, we choose as a typical example an eigenstate of σ∆antiparallel to the magnetic field: |ψ(0)/angbracketright=|0/angbracketright(−sin(θ/2)|↑/angbracketrightz+cos(θ/2)|↓/angbracketrightz),(6) The time dependence is obtained from |ψ(t)/angbracketright=/summationtext iζi|φi/angbracketrighte−iEit//planckover2pi1,whereζiare the corresponding expan- sion coefficients. We study the dynamics of spin densi- ties forθ=π/4 andg= 1. The particle oscillations are shown in Fig. 3(a). The Gaussian-like shape of ρxis ro- bustagainsttime; however,thoseof ρyandρz(notshown in the Figure) are fully changed. As a consequence, we see a strong correlation between spin state and position of the particle even in the dynamical regime. One of the main advantages of quantum dots and cold atoms is the ability to manipulate the strength of spin- orbit coupling and, thus, to cause dynamics in the or- bital and spin channels. The time-dependent spin-orbit coupling can be used for generation of spin currents in two-dimensional electron gas [30] and spin separation in two-electron quantum dots similar to that predicted in Ref.[31]. Here we take a single-period perturbationg(t) =g0sin(Ωt) at frequency Ω, fast enough to yield an appreciable nonadiabatic behavior, but being sufficiently slow to allow using the available electrical and optical means to generate the spin-orbit interaction. To clearly see the effects of strong spin-orbit coupling, we compare belowthe obtained numericallyexactresult with the per- turbation theory. To use the perturbation theory we take the basis of first four eigenstates ψ1=|0/angbracketright| ↓/angbracketrightz, ψ2=|0/angbracketright| ↑/angbracketrightz, ψ3=|1/angbracketright| ↓/angbracketrightz, ψ4=|1/angbracketright| ↑/angbracketrightz (7) where time-dependent wavefunction becomes ψ(t) =a1(t)ψ1+a3(t)ψ3e−iωt+a4(t)ψ4e−i(ω+∆)t.(8) The assumed time-dependence yields the expansion co- efficients a3(t) =−g0sinθ/integraldisplayt 0sin(Ωτeiωτdτ (9) a4(t) =g0cosθ/integraldisplayt 0sin(Ωτ)ei(ω+∆)τdτ.(10) The expectation values of coordinate and spin projec- tion onto the magnetic are expressed as: /angbracketleftx(t)/angbracketright ≡ /angbracketleftψ(t)|/hatwidex|ψ(t)/angbracketright=1√ 2/parenleftbig a3(t)e−iωt+c.c./parenrightbig (11) /angbracketleftσ∆(t)/angbracketright ≡ /angbracketleftψ(t)|σ∆|ψ(t)/angbracketright=−1+2/vextendsingle/vextendsinglea2 4(t)/vextendsingle/vextendsingle.(12) These perturbation theory formulas show the role of the directionofmagneticfieldonthespinandspatialdynam- ics, not present in the conventional Jaynes-Cummings model and allowing to extend the abilities for coordinate and spin manipulation. We treat the problem numerically for a single period T= 2π/Ω [32], beginning with θ= 0, where Zeeman and spin-orbit fields are orthogonal, similar to the Jaynes- Cummings model. Figure 4 demonstrates time depen- denceof/angbracketleftσ∆(t)/angbracketrightfordifferentcouplings g0andcomparison with the perturbation result for g0= 0.5 in the inset. As it can be seen in Fig. 4, spin projection at the magnetic field changes strongly with time, corresponding to the spin rotation due to the spin-orbit coupling, and remains constant, as expected, after the change stops. The value of this projection after the end of the perturbation cor- responds to the degree of nonadiabaticity of this process. It is interesting to mention the appearance of plateaus at strong spin-orbit coupling regime. These plateaus show that even a low-frequency dynamics is strongly nonadi- abatic. The reason is the following. In the presence of spin-orbit coupling and magnetic field with θ= 0, the splitting of the ground-state doublet, ∆exp( −2g2)≪∆, is small due to weakly overlapping eigenstates in the mo- mentum space, as discussed after Eq.(4). As a result, even very slow changes in the system parameters cannot4 FIG. 4: (Color online) Dynamics of spins under a time- dependent g(t)=g0sin(Ωt). Lines are marked by correspond- ingg0values. Inset shows comparison of the exact (dashed line) and perturbation theory (solid line) results for g0= 0.5. be treated adiabatically. Here spatial motion does not occur (/angbracketleftx(t)/angbracketright= 0), in agreement with Eq.(11). Next, we take the state of Eq.(6), with θ=π/4 as the initial oneto demonstratethe qualitativeroleofthe mag- netic field direction. First qualitativeeffect is nonzeroos- cillations of the coordinate, nonmonochromatic at t<T, asdepicted in Fig.5. The oscillationsin /angbracketleftx(t)/angbracketrightarecaused mainly by transitions between ground and first excited eigenstates of the model Hamiltonian; these states get mixed by the strong coupling. The amplitude of oscilla- tionsincreaseswith g0, asexpected, but alsowith Ω. The behavior strongly depends on the frequency and ampli- tude of spin-orbit coupling, as can be seen by comparison ofadiabaticandnonadiabaticplots. After the endofper- turbation, the coordinate oscillates at the frequency ω. Next qualitative difference is in the behavior of /angbracketleftσ∆(t)/angbracketright. Here, in the case of a strong static spin-orbit coupling, the splitting of the lowest doublet is ∆cos θ. We see the effects of nonadiabatic perturbation and strong coupling in Fig. 6. Dependent on g0and Ω, different regimes in the dynamics of /angbracketleftσ∆(t)/angbracketrightare possible: the system either returns to the initial state after the end of perturbation or switches to other states. The switching effect depends on the frequency of the field. At a small frequency, the dynamics is more adiabatic and perturbative, while for the larger frequency the switching can occur. The in- set shows that the perturbation theory works at short time, while at longer times more states participate, and the results become different. From this plot we can for- mulate the condition of strong spin-orbit coupling for a given system as a transition from peak-like (“weak” or “moderate” coupling) to step-like (“strong”) evolution of /angbracketleftσ∆(t)/angbracketright. It is interesting to mention that the transition and, therefore, the definition of the coupling strength inFIG. 5: (Color online) Dynamics of particle mean coordi- nate under time-dependent strength g(t)=g0sin(Ωt). Lines are marked by corresponding g0values. Inset shows compari- son of the exact (dashed line) and perturbation theory (soli d line) results for g0= 0.5. a dynamical system depends on the frequency of the ap- plied external field. In summary, we have investigated how strong dynam- ical spin-orbit couplings can be applied to probe and manipulate spins of electrons in semiconductor quantum dotsandcoldatomsinparabolicconfinementthroughthe correlated spin and orbital motion. We reveal the impor- tanceofthetiltangleoftheappliedmagneticfield, theef- fect strongly beyond the conventional Jaynes-Cummings model. The obtained dynamics shows that, under a strong constant coupling, a particle oscillates in correla- tion with its spin orientation. The motion of the particle can be influenced by time-dependent coupling with the result strongly dependent on all parameters. We observe a transition from periodic to step-like behavior of spin component parallel to the magnetic field with increas- ing the coupling strength. This fact clarifies the way to define qualitative effect if the ultrastrong spin-orbit coupling. The present work widens the applicability of the spin-orbit control, as it covers different strengths for the induced interaction and tilt angles for the applied magnetic field. It also emphasizes the usability of elec- tric and optical fields for spin probe and manipulation, which are crucial for spintronics. These results may also be of interest for quantum optics and quantum informa- tion realizations. We gratefully acknowledge fruitful discussions with E. Il’ichev, G. Romero, E. Solano, and, especially, with J. C. Retamal. We acknowledge support of the MINECO of Spain (grant FIS 2009-12773-C02-01), the Govern- ment of the Basque Country (grant ”Grupos Consolida- dos UPV/EHU del Gobierno Vasco” IT-472-10), and the UPV/EHU (program UFI 11/55).5 FIG. 6: (Color online) Dynamics of /angbracketleftσ∆/angbracketrightunder time- dependent strength g(t)=g0sin(Ωt). 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Zwierlein, Phys. Rev. Lett. 109, 095302 (2012). [18] S. Debald and C. Emary, Phys. Rev. Lett. 94, 226803 (2005). [19] J. Casanova, G. Romero, I. Lizuain, J. J. Garc´ ıa-Ripol l, and E. Solano, Phys. Rev. Lett. 105, 263603 (2010) and references therein. [20] I. L. Aleiner and V. I. Fal’ko, Phys. Rev. Lett. 87, 256801 (2001). [21] J. Clarke and F.K. Wilhelm, Nature 453, 1031 (2008). [22] D. Braak, Phys. Rev. Lett. 107, 100401 (2011). [23] S. Schmidt and G. Blatter, Phys. Rev. Lett. 104216402 (2010) and references therein. [24] Y. Hu and L. Tian, Phys. Rev. Lett. 106, 257002 (2011) and references therein. [25] A. P´ alyi, P. R. Struck, M. Rudner, K. Flensberg, and G. Burkard, Phys. Rev. Lett. 108, 206811 (2012). [26] O.Z. Karimov, G.H. John, R.T. Harley, W.H. Lau, M.E. Flatt´ e, M. Henini, and R. Airey, Phys. Rev. Lett. 91, 246601 (2003). [27] P.S. Eldridge, W.J.H. Leyland, P.G. Lagoudakis, R.T. Harley, R.T. Phillips, R. Winkler, M. Henini, and D. Taylor, Phys. Rev. B 82, 045317 (2010). [28] A. Balocchi, Q. H. Duong, P. Renucci, B. L. Liu, C. Fontaine, T. Amand, D. Lagarde, and X. Marie, Phys. Rev. Lett. 107, 136604 (2011). [29] R. Winkler, Spin-orbit coupling effects in two- dimensional electron and hole systems , Springer Tracts in Modern Physics (Springer, Berlin, 2003). [30] A. G. Mal′shukov, C. S. Tang, C. S. Chu, and K. A. Chao, Phys. Rev. B 68233307 (2003). [31] M. P. Nowak and B. Szafran, arXiv:1303.0211. [32] The dynamics are computed using a 4th-order Runge- Kuttamethodtill t= 2π/Ω, andusingthedecomposition |ψ(t)/angbracketright=/summationtext iζi|φi/angbracketrighte−iEit//planckover2pi1beyond this time. [33] D. V. Khomitsky and E. Ya. Sherman, EPL 9027010 (2010).

1101.5301v1.Magnetic_excitations_in_one_dimensional_spin_orbital_models.pdf

arXiv:1101.5301v1 [cond-mat.str-el] 27 Jan 2011Magnetic excitations in one-dimensional spin-orbital mod els Alexander Herzog and Peter Horsch Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany Andrzej M. Ole´ s Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany and Marian Smoluchowski Institute of Physics, Jagellonian Uni versity, Reymonta 4, PL-30059 Krak´ ow, Poland Jesko Sirker∗ Department of Physics and Research Center OPTIMAS, University of Kaiserslautern, D-67663 Kaiserslautern, Ge rmany and Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany (Dated: November 6, 2018) We study the dynamics and thermodynamics of one-dimensiona l spin-orbital models relevant for transition metal oxides. We show that collective spin, orbi tal, and combined spin-orbital excitations with infinite lifetime can exist, if the ground state of both s ectors is ferromagnetic. Our main focus is the case of effectively ferromagnetic (antiferromagneti c) exchange for the spin (orbital) sector, respectively, and we investigate the renormalization of sp in excitations via spin-orbital fluctuations using a boson-fermion representation. We contrast a mean-fi eld decoupling approach with results obtained by treating the spin-orbital coupling perturbati vely. Within the latter self-consistent ap- proach we find a significant increase of the linewidth and addi tional structures in the dynamical spin structure factor as well as Kohn anomalies in the spin-w ave dispersion caused by the scattering of spin excitations from orbital fluctuations. Finally, we a nalyze the specific heat c(T) by compar- ing a numerical solution of the model obtained by the density -matrix renormalization group with perturbative results. At low temperatures Twe find numerically c(T)∼Tpointing to a low-energy effective theory with dynamical critical exponent z= 1. PACS numbers: 75.10.Pq, 75.30.Et, 05.10.Cc, 05.70.Fh I. INTRODUCTION In condensed matter systems the coupling between different degrees of freedom often plays an important role. The electron-phonon coupling, for example, can lead to the formation of renormalized quasiparticles, so- called polarons,1,2as well as to phase transitions like the Peierls instability.3In recent years, the coupling between fermionic and bosonic degrees of freedom has also been intensely studied in Bose-Fermi (BF) mixtures of ultra- cold quantum gases.4–8 Coupled degrees of freedom also seem to be important in certain transition metal oxides where the low-lying electronic states (termed “orbitals”) are not completely quenchedsothattemperatureordopingcanleadto asig- nificant redistribution of the valence electron density. In insulating materials with partly filled degenerate orbitals the superexchange between the magnetic degrees of free- dom then becomes a function of the orbital occupation. This leads to models of coupled spin and orbital degrees of freedom as, for example, the Kugel-Khomskii model where the orbitals are represented by a pseudospin.9 LaMnO 3,10–13LaTiO 3,14–17and LaVO 3or YVO 3,18–25 are well-known examples for compounds believed to be described by effective spin-orbital models. They exhibit a wide range of fascinating effects ranging from colossal magnetoresistance12to temperature-induced magnetiza- tion reversals.18,19Common to all these transistion metal oxides is a lift- ing of the fivefold degeneracy of the dorbitals into two eg orbitals ( x2−y2and 3z2−r2) and three t2gorbitals ( xy, yz, andxz). This splitting is due to the perovskite struc- ture where oxygen ions, O2−, form octahedra around the transition metal ions which are therefore exposed to an approximately cubic crystal field. As a consequence, the orbitals pointing towards the oxygen ions are energeti- cally unfavorable. In YVO 3thet2gorbitals are occupied by two electrons forming an effective spin S= 1 due to large Hund’s rule coupling. The material is an insulator with an interest- ing phase diagram.18–20,22At temperatures below 77 K the system is in a G-type antiferromagnetic (AF) phase, i.e., AF in all three directions. In a range of higher tem- peratures, 77 K < T <116 K, the magnetic structure is C-type with spins ordering antiferromagnetically in the (a,b) plane and ferromagnetically along the caxis. The surprising fact that the ferromagnetic (FM) exchange in- tegral in this phase is much larger than the AF exchange interactions in the ( a,b) plane22was explained by strong orbital fluctuations along the c-axis chains that trigger ferromagnetism.21In theC-type phase a neutron scat- tering study revealed that the magnon dispersion along the FMc-axis chainsconsists of twobranches. This split- ting has been interpreted as due to a periodic modula- tion of the FM exchange along these chains caused by an entropy gain of fluctuating orbital occupations.22,23 Support for an orbital Peierls effect in this material was2 . . .j−1jj+ 1. . .|FS,Fτ/angbracketright=(a) . . .j−1jj+ 1. . .S− j|FS,Fτ/angbracketright=(b) . . .j−1jj+ 1. . .S− jτ− j|FS,Fτ/angbracketright=(c) FIG. 1: (a) Ground state of the FM spin-orbital model, Eq. (1.1), (b) a spin excitation, and (c) a coupled spin-orbi tal excitation. The two orbitals per site are assumed to be de- generate (the splitting is only for clarity of presentation ). given by numerical investigations23,24and a mean-field (MF) decoupling approach.26 However, the dynamics in such systems cannot easily be studied numerically and a MF decoupling is unable to explain important features of coupled spin-orbital de- grees of freedom27as can be seen in the following ex- ample. Consider the one-dimensional (1D) spin-orbital Hamiltonian28–30 H=J/summationdisplay j(Sj·Sj+1+x)(τj·τj+1+y),(1.1) withferromagnetic superexchange interaction J <0, whereSjandτjare spin Sand pseudospin τoperators at sitej, respectively, and xandyare constants. For generalx,ythe model has an SU(2) ⊗SU(2) symmetry and exhibits an additional Z2symmetry, interchanging spin and orbital sectors, if x=y. ForS=τ= 1/2 and x=y= 1/4 the symmetry is enlarged to SU(4).31 In the following we discuss the case S=τ= 1/2 and we choose xandysuch that the ground state/vextendsingle/vextendsingleFS,Fτ/angbracketrightbig is given by fully polarized spin and orbital sec- tors as illustrated in Fig. 1(a). Using the equation of motion method we find that the state S− j/vextendsingle/vextendsingleFs,Fτ/angbracketrightbig shown in Fig.1(b) is always an elementary excitation with dis- persionωS(q) =|J|(1 + 4y)(1−cosq)/4. Analogously, the orbital flip is also an elementary excitation with ωτ(q) =|J|(1+4x)(1−cosq)/4. However,thesecollective excitations are not the only undamped elementary exci- tations of the Hamiltonian, Eq. ( 1.1). In addition, a cou- pled spin-orbital excitation S− jτ− j, as shown in Fig. 1(c) may exist. To investigate this issue we again apply theequation of motion method leading to /braceleftbig/bracketleftbig H,S− jτ− j/bracketrightbig −|J|[Cj(x,y)+Dj(x,y)]/bracerightbig |FS,Fτ/an}bracketri}ht= 0, (1.2) where Cj(x,y) = (x+y)S− jτ− j−1 4/parenleftbig S− j−1τ− j−1+S− j+1τ− j+1/parenrightbig is the coherent part, and Dj(x,y) =−1 2/bracketleftbigg/parenleftbigg x−1 4/parenrightbigg/parenleftbig S− jτ− j−1+S− jτ− j+1/parenrightbig +/parenleftbigg y−1 4/parenrightbigg/parenleftbig S− j−1τ− j+S− j+1τ− j/parenrightbig/bracketrightbigg (1.3) contains terms which lead to a spatial decoherence of the excitation. Hence in order to have a fully confined spin-orbital excitation, Dj(x,y) has to vanish, which ob- viously is the case if x=y= 1/4. From this we draw the conclusion that confined spin-orbital excitations are rather the exception than the rule, relying on the par- ticular value of the constants. If we introduce the Bloch states ΨSτ(q) =1 N/summationdisplay jeijqS− jτ− j|FS,Fτ/an}bracketri}ht,(1.4) the dispersion of the coupled spin-orbital excitation for x=y= 1/4 is given by ωSτ(q) =|J|(1−cosq)/2. (1.5) Thus for x=y= 1/4 we have ωS(q) =ωτ(q) =ωSτ(q), i.e., the dispersions of all three elementary excitations are degenerate.32Interestingly, they all lie withinthe continuum of spin-orbital excitations given by γ(q,p) = ωS(q/2+p) +ωτ(q/2−p). The Hamiltonian, however, does not allow for a decay of these three elementary ex- citations in the ferromagnetic case. For the case of the coupled spin-orbitalexcitation we see from Eq. ( 1.3) that such a decay becomes possible once we move away from the special point x=y= 1/4. Our conclusions partly differ from the ones presented in Ref. 30, where the cou- pledspin-orbitalexcitationisconsideredasaboundstate below the spin-orbital continuum.33 There are several ways to generalize the S=τ= 1/2 casetoarbitraryspin-andpseudospinquantumnumbers. Ifwe startagainfrom fully polarizedspin and orbitalsec- tors and only demand that S− jτ− j/vextendsingle/vextendsingleFS,Fτ/angbracketrightbig stays confined, we find the condition x=S(1−S) andy=τ(1−τ). Another way of generalizing the S=τ= 1/2 case to ar- bitrarySandτrelies on the fact that the Hamiltonian, Eq. (1.1), withx=y= 1/4 is equivalent to H=J 4/summationdisplay jDS=1 2 j,j+1Dτ=1 2 j,j+1. (1.6) whereDσ=1 2 j,lwithσ∈ {S,τ}is Dirac’sexchangeoperator forσ= 1/2.34A generalization of this exchange operator3 to arbitrary spin has been discussed by Schr¨ odinger.35 For instance for σ= 1 the spin (pseudospin) exchange operator is given by Dσ=1 j,l= (σj·σl)2+σj·σl−1.36 The Hamiltonian in Eq. ( 1.6) with arbitrary spin (pseu- dospin) quantum number does not only keep the sin- gle spin-orbital flip confined as in the generalization dis- cussed above but rather all spin-orbital excitations of the type (S− j)mS(τ− j)mτ/vextendsingle/vextendsingleFS,Fτ/angbracketrightbig wheremSandmτare the multiplicities forspin and pseudospin, respectively. Since a MF decoupling solution treats the spin-orbital chain as two separate chains with effective exchange parame- ters determined self-consistently, the physics of coupled spin-orbital excitations cannot be captured within this approach. The purpose of this paper is to study the importance ofcoupled spin-orbitalexcitations in aspin-orbitalmodel withantiferromagnetic superexchange and anisotropic orbital exchange. This case is intriguing as spin and or- bital degrees of freedom may be expected to be strongly entangled.27In fact, it has been shown that compos- ite spin-orbital excitations have to be analyzed together with spin waves in systems with active egorbitals, such as for instance KCuF 3.37,38This follows from the non- conservation of the orbital flavor in hopping processes which implies that spin excitations are not independent and may occur in general together with an orbital flip. Here we will consider an anisotropic generalization of the spin-orbital model ( 1.1) with parameters x,ysuch that thespinsstillorderferromagneticallyinthegroundstate. The orbital sector, however, will no longer be in a fully polarized state due to the AF superexchange which fa- vors orbital alternation. Independent spin, orbital, and coupled spin-orbital excitations of collective type, as dis- cussed above, therefore can no longer exist. We will fo- cus, in particular, on the question how spin excitations are modified by the presence of orbitals in this case. The paper is organizedas follows: In Sec. IIwe present a generalization of the spin-orbital model, Eq. ( 1.1), to a modelwithanisotropicorbitalexchange. Fortheextreme quantumlimit oftheorbitalsectorinteractingviaanXY- typecouplingwethen deriveaneffective BFmodelwhich resembles models considered in the context of ultracold BF gases. By using a density matrix renormalization groupalgorithmapplied totransfermatrices(TMRG) we exemplarily investigate numerically the crossover from AF to FM correlations. In Sec. IIIwe discuss the MF decoupling approach. We allow for a dimerization in both sectorsanddiscusstheobtainedMF phasediagram. In Sec.IVwe summarize the results for the dynamical spin structure factor S(q,ω) obtained within the modi- fied spin-wave theory (MSWT) for the uniform FM spin chain.39,40In Sec.Vwe formulate an approach where the coupling between spins and orbitals is treated per- turbatively. The approach is based on representing the spins by bosons using the MSWT41,42and the orbitals by Jordan-Wigner fermions. Finally, in Sec. VI, we consider the effects of coupled spin-orbital degrees of freedom on the thermodynamics of the system. We focus, in partic-ular, on the specific heat as a function of temperature and compare perturbative results with numerical data obtained by TMRG. In Sec. VIIwe summarize and dis- cuss our results. The Appendix provides details of the perturbative approach. II. SPIN-ORBITAL MODEL AND MAPPING ONTO A BOSON-FERMION MODEL A. One-dimensional spin-orbital model We will focus here on the physical situation realized in the vanadium perovskites, such as YVO 3, where the superexchange interactions are antiferromagnetic. In YVO3the twodelectrons occupy the lower lying t2gor- bitals while the egorbitals are empty. From electronic structure calculations43–45it is concluded that the t2g orbitals are split into a lower-lying xyorbital level and a higher-lying doublet of xzandyzorbitals. Therefore thexyorbital will always be occupied by one electron, controlling the AF correlations in the ( a,b) planes. The large Hund’s coupling JH(normalized to the interatomic Coloumb interaction U) present in YVO 3will support parallel alignment of electronic spins at V3+ions ind2 configurations,46leading to S= 1 spins. Therefore, the remaining electron will be placed in one of the two other orbitals{xz,yz}which constitutes the τ= 1/2 orbital degree of freedom. On this basis, an effective spin-orbital superexchange model for YVO 3with spins S= 1 has been derived.21Here we shall study the 1D spin-orbital model extracted from it for the c-axis.26 The simplest Hamiltonian for the c-axis FM chains in YVO3takingJHinto account is given by Eq. ( 1.1) with J >0,x= 1, and y= 1/4−γH. HereγHis proportional to Hund’s coupling JH, supporting FM correlations in the spin sector. For S= 1 and realistic values of Hund’s coupling for vanadates, γH∼0.1, numerical investiga- tions ofthis model showedstrongbut short-rangeddimer correlations in a certain finite temperature range caused by the related entropy gain although the ground state is uniformly FM.24The same Hamiltonian was also studied using a MF decoupling scheme.26Within this approach a finite temperature phasewith dimer orderin both sectors wasfound. However,asdiscussedintheintroduction, the MF decoupling approach has severe limitations as it does not take the coupled spin-orbital dynamics into account. We start from a generalizationof Eq. ( 1.1) which reads HSτ(Γ) =J/summationdisplay j(Sj·Sj+1+x)/parenleftbig [τj·τj+1]Γ+y/parenrightbig ,(2.1) with [τj·τj+1]Γ≡τj·τj+1−Γτz jτz j+1.(2.2) The calculations presented here are valid for general S andx, but we will, unless stated otherwise, only address4 -1.5-1-0.500.51<SiSi+1> 0 1 T/J-0.4-0.200.20.4<τx iτx i+1+τy iτy i+1>y=0.4y=-0.3 y=0.4 y=-0.3 FIG. 2: (Color online) Nearest-neighbor spin and orbital co r- relation functions for the spin-orbital model (2.1) with Γ = 1 as a function of temperature Tin units of J(we setkB= 1). In both panels y= 0.4, 0.3, 0.2, 0.15, 0.1, 0.05, 0.0, −0.1, −0.2,−0.3 in arrow direction. The spin correlations switch from AF to FM at y= 0.1 (dashed lines). The dotted lines in the upper (lower) panel correspond to the limiting values 1 and−1.4015 (−1/πand 1/π), respectively. the caseS= 1,x= 1 relevant for YVO 3in the following. For Γ = 1 the pseudospin sectorreduces to an XY model. In Fig.2the nearest-neigbor spin and orbital corre- lation functions for Γ = 1 as a function of temperature for various parameters yobtained by TMRG are shown. This method allows us to obtain thermodynamic quan- tities for 1D quantum systems directly in the thermody- namic limit.47–49Fory/lessorsimilar0.1 the ground state has ferro- magneticallyalignedspins. Thephasetransitionbetween thefullypolarizedFMstateandastatewithAFspincor- relations at y≈0.1 is first order. In the limit y≫1 the value/an}bracketle{tSjSj+1/an}bracketri}ht ≃ −1.4015 for a Haldane S= 1 Heisen- berg chain is reached,50while the orbital correlations ap- proach/an}bracketle{tτx jτx j+1+τy jτy j+1/an}bracketri}ht →1/π. We note that the FM ground state is lost at y≃0.1 both in the model with Γ = 1 investigated here as well as in the model with an isotropic pseudospin sector (Γ = 0).24,51For the model with Γ = 1, however, we have a direct phase transition from the FM to the Haldane phase while for the isotropic model an orbital valence bond phase is intervening be- tween these two phases.51 The isotropic model ( 2.1), Γ = 0, has also been in- tensely studied for S=τ= 1/2. Here the phase diagram is more complex than in the S= 1,τ= 1/2 case.52,53Forx= 1 the FM spin state is again found to be stable fory/lessorsimilar0.1. However, now the transition at y∼0.1 is to a gapless “renormalized SU(4)” phase followed by a fur- ther phase transition at larger yinto a dimer phase. The phase with ferromagnetically polarized orbitals is absent because/an}bracketle{tSj·Sj+1+x/an}bracketri}ht>0 forx= 1 but is again present for large yifx/lessorsimilarln2−1/4. B. Boson-fermion model For the 1D spin-orbital model, Eq. ( 2.1), at the point Γ = 1, we will now derive an effective BF model which will be used as a starting point for the perturbative approach. First, applying the Jordan-Wigner transfor- mation the orbital part is mapped onto a free fermion model (for Γ /ne}ationslash= 1 the pseudospins map onto interacting fermions). The spin part of the spin-orbital Hamiltonian (2.1) will be represented by bosons. Concentrating on the case where the spin part is ferromagnetically polar- ized in the ground state, we can treat the spin sector by the MSWT.41,42To this end, we introduce bosonic op- erators by a Dyson-Maleev transformation. If we retain bosonic operators only up to quadratic order we end up with H≡ HSτ(1)−JN(S2+x)y≃H0+H1.(2.3) HereH0is already diagonal H0=/summationdisplay kωB(k)b† kbk+/summationdisplay qωF(q)f† qfq,(2.4) withf† qandfq(b† kandbk) being the fermionic (bosonic) creation and annihilation operators, respectively. The magnon dispersion is given by ωB(k) = 2JS|y|(1−cosk), (2.5) and the fermion dispersion reads ωF(q) =J(S2+x)cosq. (2.6) The spinless fermions fill up the Fermi sea between the Fermi points at kF=±π/2. For FM spin chains usual spin-wave theory has to be modified by a Lagrange multiplier µacting as a chemical potential which enforces the Mermin-Wagner theorem of vanishing magnetization at finite temperature41 S=1 N/summationdisplay k/angbracketleftBig b† kbk/angbracketrightBig . (2.7) Thermodynamic quantities calculated with this method are in excellent agreement with the exact Bethe ansatz solution for the uniform chain as well as with numerical TMRGdataforthedimerizedFMchainfortemperatures up toT∼ |Jeff|S2, withJeffbeing the effective exchange constant of the model under consideration.26,41,42,545 The interacting part couples bosons and fermions and reads H1=1 N/summationdisplay k1,k2,qωBF(k1,k2,q)b† k1bk2f† qfk1−k2+q,(2.8) with the vertex ωBF(k1,k2,q)≡JS[cos(k2−q)+cos(k1+q) −cos(k1−k2+q)−cosq].(2.9) The Hamiltonian H=H0+H1withH0andH1given by Eqs. ( 2.4) and (2.8), supplemented by the constraint (2.7), is an effective BF representation valid at low tem- peratures. We will investigate this model in Sec. Vtreat- ing the BF coupling perturbatively. III. MEAN-FIELD DECOUPLING The spin-orbital model ( 2.1) contains rich and inter- esting physics. A first attempt to understand the prop- erties of the model is to apply a MF decoupling which neglects the coupled spin-orbital degrees of freedom and treats the spin-orbital chain as two separate chains with effective coupling constants which have to be determined self-consistently. Note, however,that this treatment does not involve site variables as in the classical Weiss-MF theory but takes the correlations on a bond as relevant variables. Interestingly, these expectation values never vanish, which makes them useful particularly in cases without long-range order. A. Decoupling into spin and orbital chain Applying a MF decoupling and allowing for a dimer- ization in both sectors26,27we obtain from Eq. ( 2.1) HSτ(Γ)≃HMF S+HMF τ(Γ), (3.1) with the spin and orbital Hamiltonians HMF S=JSN/summationdisplay j=1{1+(−1)jδS}Sj·Sj+1, HMF τ(Γ) =JτN/summationdisplay j=1{1+(−1)jδτ}[τj·τj+1]Γ.(3.2) Within this approximation the effective superexchange constants and dimerization parameters are given by Jτ=J∆+ SS+2x 2, δ τ=∆− SS ∆+ SS+2x, JS=J∆+ ττ+2y 2, δ S=∆− ττ ∆+ττ+2y,(3.3)where we have defined ∆± SS=/an}bracketle{tS2j·S2j+1/an}bracketri}ht±/an}bracketle{tS2j·S2j−1/an}bracketri}ht, ∆± ττ=/angbracketleftbig [τ2j·τ2j+1]Γ/angbracketrightbig ±/angbracketleftbig [τ2j·τ2j−1]Γ/angbracketrightbig .(3.4) Here ∆− σσwithσ=S(σ=τ) is an order parameter for the spin (orbital) dimerization, respectively. Thus, the exchangeconstantsand dimerization parametersfor each sectoraredeterminedbythe nearest-neighborcorrelation functions in the other sector, making a self-consistent calculation necessary. In the following we want to solve Eqs. (3.2)-(3.4) for the spin exchange being effectively FM, i.e. JS<0. B. Dimerized orbital correlations Numerical investigations of the model with isotropic orbital exchange, HSτ(0), have shown orbital-singlet for- mation in the ground state51fory/greaterorsimilar0.1. Moreover, although the ground state consists of a fully spin polar- ized FM state for y/lessorsimilar0.1 with AF orbital correlations, it has been shown that a tendency towards orbital sin- glet formation is still present but has to be activated by thermal fluctuations.24In Ref. 26 the model ( 3.1) was studied in the FM regime with x= 1,y=1 4−γH, Γ = 0 and γH= 0.1 in order to address the ques- tion whether this orbital-Peierls effect can be captured within a MF decoupling approach. A dimerized phase for 0.10/lessorsimilarT/J/lessorsimilar0.49 (we set kB=/planckover2pi1= 1) was found with the dimerization amplitude in the spin sector being much larger than in the orbital sector. We now want to compare this result with the case where we set Γ = 1 in Eq. ( 3.1) so that the self- consistent Eqs. ( 3.3) can be solved analytically by ap- plying a Jordan-Wigner transformation and MSWT. In- troducing fermionic operators f(†) j,eif the index jis even andf(†) j,oifjis odd for the pseudospins, we rewrite HMF τ≡HMF τ(Γ = 1) in Fourier representation. Finally introducing new fermionic operators φ(†) qandϕ(†) qwhich diagonalize the Hamiltonian HMF τ, we find HMF τ=/summationdisplay qωMF F(q,δτ)(φ† qφq+ϕ† qϕq),(3.5) with the fermionic dispersion55 ωMF F(q,δτ)≡ Jτ/radicalBig cos2q+δ2τsin2q.(3.6) We can now calculate ∆± ττ, as given in Eq. ( 3.4) straightforwardly and obtain ∆− ττ=2δτ N/summationdisplay q/braceleftbig 2nF[ωMF F(q,δτ)]−1/bracerightbig sin2q/radicalBig cos2q+δ2τsin2q, ∆+ ττ=2 N/summationdisplay q/braceleftbig 2nF[ωMF F(q,δτ)]−1/bracerightbig cos2q/radicalBig cos2q+δ2τsin2q,(3.7)6 wherenF(x) ={exp(βx)+1}−1is the Fermi function andβ= 1/T. C. Dimerized spin correlations Next we turn to the spin part of Eq. ( 3.1) to which we apply the MSWT.41,42We introduce two bosonic op- eratorsb(†) j,e[b(†) j,o] forjeven [odd] by means of a Dyson- Maleev transformation. Retaining only terms bilinear in the bosonic operators we can diagonalize the resulting Hamiltonian by a Bogoliubov transformation leading to HMF S=/summationdisplay k/braceleftBig ωMF B,−(k,δS)α† kαk+ωMF B,+(k,δS)β† kβk/bracerightBig +JSNS2, (3.8) with the two magnon branches ωMF B,±(k,δS) = 2|JS|S/parenleftbigg 1±/radicalBig cos2k+δ2 Ssin2k/parenrightbigg .(3.9) The constraint of vanishing magnetization at finite tem- perature ( 2.7) now reads S=1 N/summationdisplay k/braceleftbig nB[ζ−(k,δS)]+nB[ζ+(k,δS)]/bracerightbig ,(3.10) wherenB(x) ={exp{βx}−1}−1is the Bose function and ζ±(k,δS) =ωMF B,±(k,δS)−µ(δS). To calculate the nearest-neighborcorrelationfunctions B±≡/angbracketleftbig Sj·Sj±1/angbracketrightbig it is necessaryto go beyond linear spin- wave theory. Taking terms of quartic order into account and using Eq. ( 3.10) we obtain26 B±= 1 N/summationdisplay kf±(k,δS)/summationdisplay σ∈{±}σnB[ζσ B(k,δS)] 2 . (3.11) Here we have defined f±(k,δS)≡cos2k±δSsin2k/radicalBig cos2k+δ2 Ssin2k.(3.12) From these expressions we can obtain ∆± SSwhich, com- bined with Eq. ( 3.7), allows us to solve Eqs. ( 3.1)-(3.4) self-consistently. D. Mean-field phase diagram We first discuss the ground state phase diagram of the Hamiltonian ( 3.1) for Γ = 1. Depending on the sign of the effective coupling constant JSwe find/an}bracketle{tSjSj+1/an}bracketri}ht= 1,−1.4015 with the latter value being the approximate result for the S= 1 AF Haldane chain. In the following, we restrict our discussion to −1< x <1.4015 so that0.1 0.15 0.2y00.10.20.30.4 T/Jdimerizeduniform uniform(a) ytpT2T1 tricritical point 0.2 0.3 0.4 T/J00.51Dimerization parameters(b)δS δτ T2/J T1/Jdimerized FIG. 3: (a) Phase diagram of the Hamiltonian (3.1) with Γ = 1 and x= 1 in mean-field decoupling. The shaded area represents the dimerized phase. The phase transition at T2 is first order whereas the transition at T1is of second order. The two transition lines merge at the tricritical point ytp. (b) Dimerization parameters δSandδτforx= 1 and y= 0.14. The lines are guides to the eye. The shaded area marks the temperature range where the dimerization is nonzero. /an}bracketle{tSjSj+1/an}bracketri}htandJτ=J(/an}bracketle{tSjSj+1/an}bracketri}ht+x) always have the same sign. For the orbital sector we obtain, on the other hand,/an}bracketle{tτx jτx j+1+τy jτy j+1/an}bracketri}ht=±1/π.y >1/πimpliesJS>0 and the ground state is therefore certainly AF (Haldane phase) whereas JS<0 fory <−1/πleading to a FM state. In the regime −1/π < y < 1/πthe self-consistent equationshave twosolutions with energies EAF 0≈(1/π+ y)(−1.4015+x) andEFM 0= (−1/π+y)(1+x) and a first order phase transition between the FM and AF states occurs where the energies cross. For the case x= 1 we are focussing on here, this happens at yc≈0.212 and the FM stateis stablefor y < yc. Comparedto the numerical solution where yc≈0.1 (see Fig. 2) the range of stability of the FM state is therefore increased in the MF solution. Next, we investigate the possibility of a finite tem- perature dimerization for x= 1 in that part of the phase diagram where the ground state is FM. As shown7 in Fig.3(a) we find that a dimerized phase at finite temperatures does indeed exist in MF decoupling for ytp≈0.128/lessorsimilary/lessorsimilaryc≈0.212 where ytpdenotes the tricrictal point. As in the model with an isotropic pseu- dospin sector,26the temperature range where the dimer- ized phase is stable depends on y. At the onset temper- atureT1the phase transition is of second order whereas at the reentrance temperature T2it is of first order, see Fig.3(b). As in the case Γ = 0, the dimerization in the spin sector is always much larger than in the orbital sector. As pointed out before, the MF decoupling suffers from severe limitations and it is expected to be an even worse approximation in the extreme quantum case Γ = 1 than in the case Γ = 0 studied previously.26In particular the coupling between spin and orbital degrees of freedom is completely lost within this approach. In the following sections we will therefore develop an alternative pertur- bative treatment of the spin-orbital coupling. IV. DYNAMICAL SPIN STRUCTURE FACTOR FOR THE UNIFORM FERROMAGNETIC CHAIN In order to investigate coupled spin-orbital degrees of freedom and, in particular, their implications on the spin dynamics of the spin-orbital chain, a detailed under- standing of the spin dynamics of a FM chain is useful. We shall avoid the complications of the dimerized chain and focus our study on the uniform 1D ferromagnet.56 In doing so we neglect the coupling between spin and pseudospin operators for a moment and consider HS=JS/summationdisplay jSj·Sj+1, (4.1) withJS<0. It is well-known that MSWT does not respect the SU(2) symmetry of the FM Heisenberg chain Eq. (4.1). We therefore directly calculate the full spin correlation function39,40 G(r,τ)≡ −/an}bracketle{tT[Sj(0)·Sj+r(τ)]/an}bracketri}ht.(4.2) In Fourier space we obtain G(q,ων,B)=1 N/summationdisplay k(1+nB[ζ(k)])nB[ζ(q−k)]1−e−βǫq(k) iων,B−ǫq(k), (4.3) where we have used the bosonic Matsubara frequencies ων,Bandǫq(k)≡ζ(k)−ζ(q−k) withk∈[−π,π]. The reduced magnon dispersion reads ζ(k) = 2JSS(1−cosk)−µ. (4.4) In Fig.4(a) the dynamical spin structure factor, S(q,ω) = 2nB(−ω)ImGret(q,ω),(4.5)0 1 2 3 4 ω/|JS|050100 |JS|S(q,ω) 3 3.5 ω/|JS|02040|JS|S(q,ω) 3 3.5 3.8025 4 4048 |JS|ρ(ω)q=0q=π/5q=2π/5q=3π/5q=4π/5q=π(a) (b) ω4π/5max FIG. 4: (Color online) (a) Dynamical spin structure factor S(q,ω) as obtained for T/|JS|= 0.1 and 0 ≤q≤π. The dashed line indicates the upper boundary of the two magnon continuum. The dots are projections of the peak positions onto the ( q,ω) plane. They are connected by the dotted line which is a guide to the eye. (b) Dynamical spin structure factorS(q,ω) for the same parameters at q= 4π/5 (solid line) and the corresponding density of states (dashed line) . is shown for the uniform FM chain at ω >0, where ImGret(q,ω) =π N/summationdisplay k(1+nB[ζ(k)])nB[ζ(q−k)] ×/parenleftBig e−βǫq(k)−1/parenrightBig δ(ω−ǫq(k))(4.6) is the imaginary part of the retarded Green’s function obtained from Eq. ( 4.3) by analytical continuation. Up to a factor of 2 π, as a matter of definition, we obtain the result previouslygivenbyTakahashi.39The structure factor fulfills detailed balance, S(q,ω) = eβωS(q,−ω).8 The symbols in Fig. 4show the peak positions projected onto the ( q,ω) plane. They follow the reduced disper- sion Eq. ( 4.4). Also shown in Fig. 4(a) as a dashed curve is ωmax q= 4|JS|Ssinq 2corresponding to the up- per boundary of the two magnon continuum ǫq(k) above whichS(q,ω) is zero in this approximation. At the edge of the two magnon continuum S(q,ω) has a singularity. In Fig. 4(b) the dynamical spin structure factor for the same parameters as used in Fig. 4(a) is shown at q= 4π/5 together with the density of states which is given by ρq(ω) = 1//radicalBig (ωmaxq)2−ω2. Right be- low the singularityat ωmax qthe density of states to lowest order reads ρq(ωmax q−δω)∼1/√ δω, i.e.,S(q,ω) shows a square root divergence at the upper threshold. If the edge singularity and the central peak are well separated then the spectral weight of the edge singularity is much smaller than the spectral weight of the central peak. If, on the other hand, the edge singularity is close to the central peak then the shape of the latter is strongly af- fected by the occurence of the edge singularity. In this case the edge singularity gives a significant contribution. It is instructive to analyze S(q,ω) in the limit of small q. If the edge singularity and the peak of the struc- ture factor are well separated, the lineshape of the peak can be obtained approximately. To this end, for small qbut|JS|S2q/T≫1 we only retain the leading terms of Eq. (4.3). Performing a saddle point approximation to lowest order we find S(q,ω)∼nB(−ω)(a(q,ω)− a(q,−ω)), with a(q,ω)≈2S|JS|Sq ξ (ω−JSSq2)2+/parenleftBig JSSq ξ/parenrightBig2.(4.7) This Lorentzian lineshape is only valid for low temper- atures. Here ξ≈ |JS|S2/Tis the correlation length in the low-temperaturelimit.39–42Finally, we want to stress that forT→0 the peaks will reduce to δ-functions, i.e., only thermal broadening is included in this approxima- tion. V. PERTURBATION THEORY In this section we intent to go beyond the MF decou- pling approach treating the influence of the BF inter- action, Eq ( 2.8), on the spin-wave dispersion perturba- tively. Naively one would expect that the magnon should be able to couple to the fermionic degrees of freedom if it lies inside the fermionic two-particle continuum. The upper and lower boundary of the latter are given by ǫmax F(q) = 2J(S2+x)sin(q/2), ǫmin F(q) =J(S2+x)sinq, (5.1) respectively. The continuum and the magnon dispersion ωB(q) fory=−1 are shown in Fig. 5. One would there- fore expect that in this case ωB(q) is unaffected by the0 0.5 1 q/π01234 ω/JωB(q) FIG. 5: (Color online) Magnon dispersion ωB(q) (dashed line) fory= 1 andfermionic two-particle continuum(shaded area). presenceofthe fermionsfor q < π/2, sinceit can not cou- ple to these degrees of freedom. However for higher mo- menta the spin wave may couple to the fermionic degrees offreedomandthus abroadeningof S(q,ω)shouldoccur. Moreover by choosing different values for ythe point at which the magnon enters the fermionic two-particle con- tinuum is changed. Thus the momentum at which the spin wave is affected by the coupling to the fermionic de- grees of freedom depends directly on the parameter y. These argumentsgive the qualitatively correct picture, i.e., we find indeed that the coupling of the magnon to the fermionic degrees of freedom has strong effects on the dynamical spin structure factor at intermediate and high momenta and that the onset of these effects can be well estimated by our simple argument. However, there arealsocertain aspects which can not be captured within this picture. Forinstance, for 2 S|y|>(S2+x) it suggests that the spin wave may leave the fermionic two-particle continuum at a certain momentum qland thus should be unaffected by the BF coupling for q > ql. The detailed calculation, however, revealsthat this is not true because the spin wave decays into a fermionic particle-hole anda remainingspin waveaswill becomeclearinthe following. A. General formulation Here we want to study the Hamiltonian Hgiven by Eq. (2.3), with its noninteracting part H0and interact- ing part H1defined by Eqs. ( 2.4) and (2.8), by treat- ing the BF interaction perturbatively, i.e., in the limit |x|,|y| ≫1. As explained in the appendix, we start by performing a MF decoupling for the interaction H1. Cor- rections to this solution are then taken into account per- turbatively. Here we adress the bosonic Green’s function at zero temperature GB(q,t) =−i/angbracketleftbig Tt/bracketleftbig bq(t)b† q(0)/bracketrightbig/angbracketrightbig . (5.2)9 q qk1 k1k2 q qq−k1+k2 k2k1q qq−k1+k2 k2 k1(c) (a) (b) FIG. 6: Diagrams which contribute to a renormalization of th e magnon in a perturbation theory: (a,b) Diagrams with momen - tum exchange between the magnon and the fermions, and (c) dia gram without momentum exchange. All diagrams are second order. Fermionic propagators are shown by solid lines, wher eas bosonic propagators are shown as dashed lines. q qq−k1+k2 k2k1 FIG. 7: Second order diagram for a system of interacting fermions with momentum exchange. In Fig.6all distinct, connected diagrams beyond the MF decoupling up to second order are shown. We calculate the Green’s function from the Dyson equation GB(q,ω) =1 /braceleftBig G(0) B(q,ω)/bracerightBig−1 −Σ(q,ω),(5.3) with /braceleftBig G(0) B(q,ω)/bracerightBig−1 =ω−ζ(q), (5.4) whereζ(q) is the reduced magnon dispersion defined in Eq. ( 4.4) withJS=J(y−1/π), and the self- energy Σ( q,ω) is approximated by the proper self-energy Σ2(q,ω) obtained by summing up the diagrams which can be composed of the diagrams shown in Fig. 6. The diagrams shown in Figs. 6(a) and6(b) are of par- ticular interest because they are the lowest order dia- grams where bosons and fermions exchange momentum. They describe the part of spin-orbital dynamics which cannot be captured within the MF decoupling approach discussed in section III. The diagram shown in Fig. 6(b) has to be thermally activated, i.e., it does not give any contribution at zero temperature. The same is true for the diagram shown in Fig. 6(c). Thus at T= 0 the only secondorderdiagramwhichcontributestotheselfenergyis the one shown in Fig. 6(a) leading to Σ+ BF(q,ω) =−1 N2/summationdisplay k1,k2ω2 BF(q,k1,k2) ×Θ[ωF(q−k1+k2)]Θ[−ωF(k2)] ω−Ω+q(k1,k2)+i0+,(5.5) where we have abbreviated57 Ω± q(k1,k2)≡ ±ζ(k1)+ωF(q−k1+k2)−ωF(k2),(5.6) withk1,k2∈[−π,π]. For systems of interacting fermions we know that per- turbation theory in one dimension often leads to infrared divergencies.58,59Such divergencies occur, for example, for the fermionic analogon of the diagram with momen- tum exchange, see Fig. 7. These problems can be over- come by the Dzyaloshinski-Larkin solution or bosoniza- tion techniques. For the model considered here, however, we find no divergencies within the considered diagrams. One reason for this behavior is a lack of nesting. While for a fermionic interaction as shown in Fig. 7all the dispersions in the denominator of Eq. ( 5.5) are approxi- mately linear at low energies here one of the dispersions is approximately quadratic so that nesting only occurs for singular points. As a further check, we have evalu- ated the integrals in Eq. ( 5.5) for a constant vertex at smallqandωand did not find any infrared divergencies. For finite temperatures the Matsubara formalism can be applied straightforwardly. The self-energy Eq. ( 5.5) now reads Σ+ BF(q,ων,B) =−1 N2/summationdisplay k1,k2ω2 BF(q,k1,k2) iων,B−Ω+q(k1,k2) ×N+ F,B(k1,k2,ων,B,T)NF,F(q,k1,k2,ων,B,T), (5.7) where we have abbreviated N± F,B(k1,k2,ων,B,T)≡nB[ζ(k1)]±nF[ωF(k2)], NF,F(q,k1,k2,ων,B,T)≡nF[ωF(q−k1+k2)] −nF[ωF(k2)−ζ(k1)].(5.8)10 0102030JS(q,ω) 0 2 4 68 ω/J10-810-4100-ImΣ(q,ω)q=3π/10 q=5π/10 q=7π/10 q=9π/10-1-0.500.51 q/π051015ω/J(a) (b)0204060JS(q,ω) 0 5 10 15 ω/J10-810-4100-ImΣ(q,ω)q=3π/10 q=5π/10 q=7π/10 q=9π/10-1-0.500.51 q/π051015ω/J(c) (d) FIG. 8: (Color online) Perturbative results for the BF model at zero temperature with x= 1. In the left (right) panel y=−1 (y=−2), respectively. (a),(c) S(q,ω) with the inset showing the region for which ImΣ+ BF(q,ω) is nonzero as a shaded area and the renormalized spin-wave dispersion as a dotted line. WhileS(q,ω) is sharply peaked at low momenta, a significant broadening occurs at higher q. Moreover we find additional structures which, as explained in the text, are due to coupled spin-orbital excitations. (b),(d) −ImΣ+ BF(q,ω) as given in Eq. (5.5) for the corresponding values of qshown in (a) and (c), respectively. The coupled spin-orbital excitations show u p as peaks and edges in ImΣ+ BF(q,ω) (notice the logarithmic scale). Note that at finite temperatures both, the reduced spin- wave dispersion ζ(q) as well as the fermionic dispersion is renormalized due to the MF decoupling applied to Eqs. (2.8). The respective expressions are given in the appendix, see Eqs. ( A.1) and (A.2). At finite temperatures also the diagram shown in Fig.6(b) contributes and is given by Σ− BF(q,ων,B) =−1 N2/summationdisplay k1,k2ω2 BF(q,k1,k2) iων,B−Ω−q(k1,k2) ×(1+N− F,B(k1,k2,ων,B,T))NF,F(q,k1,k2,ων,B,T). (5.9)B. Dynamical spin structure factor Below we present the results obtained by summing up the diagrams shown in Fig. 6in a Dyson series, but re- placingtheexternallegsbytheSU(2)symmetricfunction given in Eq. ( 4.3). While the perturbative results can, strictly speaking, only be valid for |x|,|y| ≫1 we extend the results here to more physical values |x|,|y| ∼ O(1) where we still expect perturbation theory to give at least a qualitatively correct picture. Numerical results ob- tained for the dynamical spin structure factor within this perturbative approach are shown in Fig. 8forT= 0 and y=−1 andy=−2. In both cases S(q,ω) is sharply peaked at small momenta whereas a significant broad- ening occurs at higher momenta. Note, that within the11 0 0.2 0.4 0.60.8 q/π0246810 ω/J 0.5 0.52 0.54 q/π22.22.42.6ω/J y=-0.75y=-1y=-2 FIG. 9: (Color online) Renormalized magnon dispersions ωq in the FM chain for x= 1 and selected values of y. Inset: The most pronounced Kohn anomalies occur at qnearπ/2. MSWTS(q,ω) is always a δ-function for the pure spin model at T= 0, i.e., the broadening here is solely due to the coupling to orbital excitations. By extracting the central peaks of the dynamical spin structurefactoratvariousmomenta, weobtaintherenor- malized spin-wave dispersion ωqwithin the perturbative approach. The result of this is shown in the insets of Fig.8(a,c) and in more detail in Fig. 9. The magnon dis- persion is renormalized and small kinks are visible close toq=π/2, which may be interpreted as Kohn anoma- lies (see below). The inset of Fig. 9shows the Kohn anomalies with a higher resolution. For itinerant ferro- magnets Kohn anomalies are well-known. Here the inter- action between the spins of localized ions is mediated by an exchange with the conduction electrons.60–66These Kohn anomalies can thus be used to gain information about the Fermi surface of the conduction electrons.60–62 However to the best of our knowledge Kohn anomalies in the spin-wavedispersionforinsulatingmaterialshavenot been adressed so far. As we will show below, the Kohn anomalies in our case are caused by coupled spin-orbital degrees of freedom. Apart from extracting the effective spin-wave disper- sion from S(q,ω), we also want to discuss the magnon bandwidth (full width at half maximum (FWHM)) Γ q of the central peaks. A broadening of the zero temper- ature peaks occurs whenever the imaginary part of the self-energy, Im Σ+ BF(q,ω) =−π N2/summationdisplay k1,k2ω2 BF(q,k1,k2)Θ[−ωF(k2)] ×Θ[ωF(q−k1+k2)]δ(ω−Ω+ q(k1,k2)), (5.10) is non-zero at ω= Ω+ q(k1,k2). The contributions within the sums are now determined by the argument of the δ- function as well as by the constraints given by the Heav- iside functions. This procedure, for a given set of param-0 0.2 0.4 0.60.8 1 q/π00.10.20.30.40.5Γq/Jy=-0.75 y=-1 y=-2 FIG. 10: (Color online) Magnon linewidth Γ q(FWHM of S(q,ω)) at zero temperature (data points), as obtained for x= 1 and the representative values of yindicated in the plot. The lines are guides to the eye. etersx,y,andS, effectively yields a region within which the spin wave may scatter on fermion pairs. Henceforth wecallthisregiontheBFcontinuum. TheBFcontinuum is shown in the insets of Figs. 8(a) and8(c) as shaded ar- eas. The upper boundary of the BF continuum (solid lines), which is periodic with a period of 2 π, is given by −4JS(y−1/π) + 2J(S2+x) forq= 0, and decreases monotonously from this value with increasing |q|. The lower boundary (dashed lines) is periodic with a period ofπ. To obtain the FWHM of the structure factor, the mag- nitude of the contributions to the sums given in Eq. (5.10) are essential. Here not only the ( k1,k2)-region which contributes to the summation but also the magni- tude of the vertex ωBFis of importance. We observe that the vertex is small at small momenta but increases at in- termediate and high momenta. This leads to a strong increase of the magnitude of the imaginary part of the self-energyas shownin panels (b) and (d) ofFig. 8. From the insets of Fig. 8it becomes clear that the spin-wave dispersion enters the BF continuum depending on y. For higher values of |y|the spin wave enters at lower mo- menta. However, since the vertex gives smaller contribu- tions at smaller momenta, the broadening of the central peaks of the dynamical spin structure factor turns out to be smaller the smaller the momenta are at which the spin wave enters the BF continuum. This can be seen in Fig.10where the magnon linewidth Γ qis shown. The onset of a finite Γ qsignals the entrance of the spin-wave dispersion into the BF continuum and depending on the momentum at which the entrance occurs the increase of Γqis either smooth (entranceat lowmomentum) orsteep (entrance at high momentum). In addition, we observe that Γ qhas a maximum at the boundary of the Brillouin zone for stronger interactions ( y=−0.75 andy=−1 in Fig.10respectively), whereas for smaller interactions we observe the maximum at smaller momenta followed12 0 0.2 0.4 0.60.8 1 q/π0246810 ω/Jωq FIG. 11: (Color online) Effective dispersion of the spin waveωqfory=−1 (red solid line) together with coupled spin-orbital excitations deduced from −ImΣ2(q,ω), shown by (green) dots within the Bose-Fermi continuum (shadded area). by a decrease of the FWHM towards the zone boundary (y=−2 in Fig. 10). Interestingly, the coupling to the orbital degrees of freedom does not only give rise to a featureless broaden- ing ofS(q,ω) but produces additional structures. These additional structures aremost obviousin Fig. 8(a). From Fig.8(b) it becomes clear that these structures are dom- inated by local extrema as well as edges in the imagi- nary part of the self-energy. Eq. ( 5.10) shows that such extrema can occur if Ω+ q(k1,k2), Eq. (5.6), becomes sta- tionaryas a function ofthe momenta k1,k2as long as the Heaviside functions in Eq. ( 5.10) for these momenta are non-zero. The position of the local maxima in the imag- inary part of the self-energy is therefore approximately given by the values Ω+ q(k1,k2) at these stationary points. These values correspond to the energy of spin-orbital ex- citations into which the initial spin wave can decay, see Fig.6(a) and which are stable against small redistribu- tions of momenta. We conclude that while we do not have completely sharp spin-orbital excitations any more as in the Hamil- tonian (1.6) with FM exchange considered in the intro- duction, there are still characteristic spin-orbital excita- tions of finite width within the spin-orbital continuum. AsshowninFig. 11wecanextractthe dispersionofthese characteristic excitations and find that the coupled spin- orbital excitations are gapless for the parameters con- sidered here. However, as can be clearly seen in Figs. 8(b) and8(d), the weights of the low-energy excitations are orders of magnitude smaller than the excitations lo- cated at higher energies. Hence the excitations at high energies give the most dominant contribution to the dy- namical spin structure factor. We therefore expect that these excitations will generate additional entropy in the corresponding temperature range which should show up,0 2 4 6 ω/J0102030JS(q,ω)q=3π/10 q=5π/10 q=7π/10 q=9π/10 FIG. 12: (Color online) Dynamical spin structure factor S(q,ω) calculated perturbatively for the spin-orbital chain at temperature T/J= 0.1 withy=−1. for example, in the specific heat which will be studied in the next section. Moreover, we find that these coupled spin-orbtial exci- tationsareresponsiblefortheKohnanomaliesmentioned above. We observe that the Kohn anomalies at interme- diate momenta occur when the energy of the spin wave coincides with that of a characteristic spin-orbital exci- tation. This is different from the Kohn anomaly in the spin-wave dispersion of itinerant ferromagnets. In this case the interaction between the localized spins given by the lattice ions is induced by scattering with conduction electrons and hence the Kohn anomaly is determined by the shape of the Fermi surface.60–66The Kohn anomaly we find within the present context is also due to interac- tioneffects, wherethenatureoftheinteractions-coupled spin-orbital degrees of freedom - is distinct from the ones of the itinerant ferromagnets. For the case x=−y= 1 the spin-wave dispersion has a discontinuity of the order ∆ω≃0.01Jat the point q≃0.509π(see inset of Fig. 9). However, for the crossing points of the magnon and the coupled spin-orbital excitation located at q≃0.37πand q≃0.76π(see Fig. 11) no Kohn-anomaly could be re- solved. We believe that this is a consequence of the weight of the coupled spin-orbital excitations: Whereas atq≃0.509πthe imaginary part of the self-energy dis- plays a steep increase of several magnitudes, at the other crossing points the slope towards the local maxima is far more moderate. At finite temperatures two effects contribute to the broadening of the central peaks of the dynamical spin structure factor. First, there is a broadening due to ther- mally excited magnons which is already present in the 1D Heisenberg chain discussed in Sec. IV. This is com- bined with the broadening due to the interaction with the orbital degrees of freedom. Here the BF continuum is smeared out by thermal fluctuations compared to the zero temperature case. Results for the structure factor13 0 0.2 0.4 0.60.8 1 q/π00.10.20.30.40.5Γq/JT/J=0.2 T/J=0.1 T/J=0 FIG. 13: (Color online) Magnon linewidth Γ q(FWHM of S(q,ω)) as a function of q, as obtained at y=−1 and tem- peratures T/J= 0, 0.1 and 0.2. The lines are guides to the eye. at finite temperatures are shown in Fig. 12. The broadening at small momenta is dominated by thermal fluctuations and the lineshape is very similar to that of the pure spin model discussed in Sec. IV. A further strong broadening in going from q= 0.3πto q= 0.5πsignals the relevance of coupled spin-orbital degrees of freedom on the spin dynamics at intermedi- ate and high momenta. Again, additional structures in S(q,ω) are visible related to the spin-orbital excitations discussed above. Finally, we analyze the variation of the FWHM with increasing temperature for a representative value of y= −1, see Fig. 13. AtT >0 the thermal broadening at small momenta is clearly visible. As in the zero tempera- ture case another strong increase of Γ qbetween q= 2π/5 andq=π/2 is observed due to coupled spin-orbital de- grees of freedom. For T/J= 0.1 andT/J= 0.2 we observe that Γ qhas a temperature dependent maximum from where Γ qdecreases towards the boundary of the Brillouin zone. This is due to the fact that the ther- mal broadening of the central peaks of the dynamic spin structure factor decreases from intermediate to high mo- menta (see Fig. 4). Actually without coupling to any or- bital degrees of freedom we expect Γ qto be very small at the boundary of the Brillouin zone. Thus a large band- width at q=πmakes the spin-orbital model distinct from a pure 1D Heisenberg ferromagnet. VI. THERMODYNAMICS Coupled spin-orbital degrees of freedom will not only influence the spin dynamics but also the thermodynam- ics of the system. We expect, in particular, that the spin-orbital excitations which were shown to affect the/Bullet /Bullet /Bullet /Bullet FIG.14: Diagramatic representations ofthesecond orderco n- tributions to the free energy as given by Eq. (6.6). dynamical spin structure factor in the previous section will also become observable in thermodynamic quanti- ties when comparing the MF decoupling and the per- turbative solution. In order to investigate this issue we rewrite Eq. ( 2.1) as HSτ(1) =HMF+δH, (6.1) withδH=HSτ(1)−HMF. The MF part reads HMF=N/summationdisplay j=1/braceleftbig Jτ[τj·τj+1]1+JSSj·Sj+1 −/an}bracketle{tSj·Sj+1/an}bracketri}htMF/angbracketleftbig [τj·τj+1]1/angbracketrightbig MF/bracerightbig .(6.2) The exchange constants JS,τare defined in Eqs. ( 3.3).67 We use the Hamiltonian ( 6.1) to determine the free en- ergy per site perturbatively, following the expansion, f=fS MF+fτ MF+1 N/an}bracketle{tδH/an}bracketri}htc MF−1 2NT/angbracketleftbig δH2/angbracketrightbigc MF+..., (6.3) wherefS MF(fτ MF) is the expression for the free energy per site stemming from the spin (pseudospin) sector within the MFdecouplingsolution. The subscriptindicatesthat the respective correlation functions are calculated with HMF. Moreover the superscript cmeans that the above expansion of the free energy is restricted to connected diagrams. We note that Eq. ( 6.3) is a high temperature expansion valid if |x|T/J≫1 and|y|T/J≫1. A straightforward calculation shows that the first or- der contribution only shifts the free energy, Eq. ( 6.3), and will not show up in thermodynamic observables ob- tained by taking derivatives of the free energy. For the second order contribution we have to evaluate two- and four-point correlationfunctions both for the spin and the pseudospin part. We use the abbreviations /an}bracketle{t(Sj·Sj+1)(Sl·Sl+1)/an}bracketri}ht=a+b(j,l) (6.4) for the spins and /angbracketleftbig [τj·τj+1]1[τl·τl+1]1/angbracketrightbig =c+d(j,l) (6.5) for the pseudospins. Here the site-independent quanti- tiesaandcstand for the disconnected parts of the four- point correlation functions whereas b(j,l) andd(j,l) fol- low from the connected ones. One finds after a straight- forward calculation that only the product of the con- nected parts contributes to the second order correction,14 leading to /angbracketleftbig δH2/angbracketrightbig MF=J2N/summationdisplay j,l=1b(j,l)d(j,l).(6.6) To proceed further, we again apply the MSWT to the spin and a Jordan-Wigner transformation to the pseu- dospin part. The evaluation of d(j,l) is again straight- forward and yields d(j,l) =1 2N2/summationdisplay k1,k2nF[ωMF F(k1,0)]{1−nF[ωMF F(k2,0)]} ×ei(k1−k2)(j−l){1+cos(k1+k2)} (6.7) withωMF F(q,δ) as given in Eq. ( 3.6). The evaluation of Eq. ( 6.4) is more involved. We first apply a Dyson-Maleev transformation and treat the ob- tained expressionsusing Wick’s theorem. In addition, we also have to account for the constraint of nonzero mag- netization at T >0 imposed by the MSWT. The cor- responding diagrams are shown in Fig. 14. Within this approximation the specific heat per site reads c=cMF+c2+... , (6.8) with c2=J2T∂2 ∂T2N/summationtext j,l=1b(j,l)d(j,l) 2NT.(6.9) We calculate the first term in Eq. ( 6.8) within the MF decoupling, cMF=cS MF+cτ MF, from the internal energy which is determined by the respective nearest-neighbor correlation functions allowing us to keep terms up to quartic order in the bosonic operators.54This strategy makes it possible to obtain reliable results for cS MFup to T/(|JS|S2)≤1. The second order correction c2given in Eq. (6.9) is obtained using the Dyson-Maleev transfor- mation so that quartic terms are also included and the order of approximation is the same. Since we are using a high temperature expansion, Eq. ( 6.3), in combination with the MSWT to evaluate the diagrams, our results are only valid in an intermediate temperature regime. If we restrict ourselves to parameters x=−y >0 then this temperature range is given by 1 /x≪T/J≪xS2. In the following we therefore only consider the case x≫1 and compare the results from perturbation theory with numerical data obtained by TMRG. As shown in Fig. 15, the specific heat c/(Jx)2exhibits a broad maximum which corresponds to the character- istic energies of spin and fermionic particle-hole excita- tions. In the temperature range where the perturbative approach is valid we find excellent agreement with the numerical solution. In particular, the perturbative cor- rectionc2(see Fig. 16) correctlycapturesthe weight shift from low to intermediate temperatures visible when com- paring the numerical and the MF decoupling solution. In0 0.2 0.4 0.60.8 1 T/(Jx)00.20.40.6c/(4J2)00.20.40.6c/(100J2) 0 0.2 0.4 0.60.8 1 T/(10J)00.20.40.6c/(100J2) 0 0.2 0.4 0.60.8 1 T/(2J)00.20.40.6c/(4J2)(a) (b)cMFScMFτcMF cMFScMFτcMF FIG. 15: Specific heat per site c/(Jx)2as a function of T/Jx for (a)x= 10 and (b) x= 2. The circles denote the nu- merical data from TMRG with the solid line obtained by a low-temperature fit of the TMRG data for the inner en- ergy. The dashed lines correspond to MF decoupling while the dashed-dotted lines are the results obtained by perturb a- tion theory. The perturbative results are expected to be val id for 1/x2≪T/(Jx)≪1. Insets: Specific heat cMFwithin the MF solution (dashed-dotted line) with the contributions fr om the spin ( cS MFsolid line) and the orbital ( cτ MFdashed line) sector shown separately. spiteofthisweightshift, theMFdecouplingyieldsoverall a very reasonabledescription of the specific heat for both cases shown in Fig. 15. Forx= 10, (Fig. 15(a)) the spe- cific heat chas a broad maximum at T/(Jx)≃0.4. This maximum results from a distinct maximum in the orbital contribution cτ MF, see insets in Fig. 15. In contrast the spin contribution cS MFincreases steadily with increasing temperature, in agreement with the higher energy scale for spin excitations. As a result, the total specific heat has only a weaker and broader maximum than suggested by the orbital part. The MF decoupling does seem to fail, however, at very low temperatures. Here the MF solution predicts that spin excitations give the dominant contribution leading15 0 0.2 0.4 0.60.8 1 T/(Jx)-0.0300.03c2/(Jx)2 FIG. 16: (Color online) Perturbative contribution to the sp e- cific heat per site c2/(Jx)2, Eq. (6.9), as a function of T/Jx forx= 10 (solid line) and x= 2 (dashed line). to ac(T)∼√ Tbehavior. This is in contrast to an ex- trapolation of the numerical data shown as dot-dashed lines in Fig. 15which suggests an approximately linear dependenceontemperature. Thisdiscrepancycomesasa surprise because our perturbative calculations of the dy- namical spin-structure factor in Sec. VBlead us to the conclusion that the magnons survive as sharp quasipar- ticles at low energies. More generally, one might argue that the ground state does not show any entanglement between the two sectors because of the classical nature of the FM state thus allowing for spin-wave excitations at low energies. From the point of view of a low-energy effective field theory, however, the situation is much less clear. While a FM chain is described by a low-energy effective theory with dynamical critical exponent z= 2, the fermionic orbital chain has z= 1. A coupling of spatial spin deviations to time-dependent orbital fluc- tuations then seems to require that the low-energy ef- fective theory for the coupled system has z= 1. As a consequence the temperature dependence of c(T) would indeed be linear. However, such an approach leaves open the role and treatment of the Berry phase terms. Within the perturbative approach the open question is whether or not higher order contributions to the self- energy might induce a significant broadening of the dy- namical spin-structure factor also at low energies. In this regard we note that the vertex responsible for the broad- ening of S(q,ω) studied in Sec. VBdoes not play any rolefor the thermodynamics ofthe system. Here the con- straint of vanishing magnetization means that such dia- grams do not contribute to staticcorrelation functions so that the lowestordercorrectionsarecaused by the vertex shown in Fig. 14.VII. CONCLUSIONS In summary, we have investigated coupled spin-orbital degrees of freedom in a one-dimensional model. For fer- romagnetic exchange we have shown that the considered modelatspecialpointsinparameterspacecanbewritten in terms ofDiracexchangeoperatorsfor spin Sand pseu- dospinτ. As a consequence, three collective excitations of spin, orbital and coupled spin-orbital type do exist. In particular, we discussed the case of Dirac exchange oper- ators for S=τ= 1/2 where the dispersions of all three elementary excitations are degenerate and lie within the spin-orbital continuum. While the spin-orbital excita- tions stay confined in this case, a decay becomes possible once we move away from this special point. For antiferromagnetic exchange the one-dimensional spin-orbital model captures fundamental aspects of physics relevant for transition metal oxides and, as we have shown, sharp excitations do not exist. To address the question how the spin dynamics is influenced by fluc- tuatingorbitalsweconsideredtheextremequantumlimit of orbitals interacting via an XY-type coupling. This al- lowed us to map the orbital sector onto free fermions using the Jordan-Wigner transformation. In spin-wave theory the spin sector is described by bosons so that our model corresponds to an effective boson-fermion model which applies for low temperatures. An analytic calcu- lation of the properties within a mean-field decoupling approachis then straightforward. Compared to a numer- ical phase diagram based on the density-matrix renor- malization group we find that the regime with ferromag- netically polarized spins is stabilized by the decoupling procedure. Furthermore, the mean-field decoupling gives rise to a finite temperature dimerized phase for certain parameterswhen starting from the ferromagnetic ground state. Whileaphasewithlong-rangedimerorderatfinite temperatures is not possible in a purely one-dimensional model, the mean-field approach also completely ignores any kind of coupled spin-orbital excitations. Thus we developed a self-consistent perturbative scheme to explore the role played by spin-orbital cou- pling. In perturbation theory the boson-fermion inter- action does not produce any infrared divergencies in the one-dimensional model due to the lack of nesting. This makes a perturbative calculation of the spin structure factorS(q,ω) possible. At large momenta q, we find that S(q,ω) shows a significant broadening due to scattering of magnons by orbital excitations. For small momenta, on the other hand, no broadening in this lowest order perturbative approach is observed because the magnon cannot scatter on these excitations. The onset of the broadening occurs at momenta where the magnon enters the boson-fermion spectrum. This point, as well as de- tails of the full width at half maximum is determined by the strength of interaction. Most interestingly, S(q,ω) does show additional peaks and shoulders corresponding tocharacteristicspin-orbitalexcitations. Atpointswhere the renormalized spin-wave dispersion and the dispersion16 of these excitations cross, Kohn anomalies do occur. Furthermore, we compared numerical data for the spe- cific heat of the spin-orbital model with the mean-field decoupling solution and an approach where we also took the second order correction to the mean-field result into account. Overall, we found that the mean-field decou- pling does describe the specific heat reasonably well. A redistribution of entropic weight from low to intermedi- ate temperatures observed when comparing the numeri- cal data and the mean-field solution is very well captured by the second order perturbative correction. An interest- ing open point is the behavior of the specific heat c(T) at low temperatures. While the mean-field solution predicts c(T)∼√ Tdue to spin-wave excitations, the numerical data suggest instead that c(T)∼T. We have speculated that a coupling of the two sectors might indeed lead to a low-energy effective theory with dynamical critical expo- nentz= 1 but details of such a theory need to be worked out in the future. In conclusion, we have shown that while collective spin-orbital excitations with infinite lifetime do not ex- ist forantiferromagneticsuperexchangethe coupled spin- orbital degrees of freedom have a strong influence on the spin excitation spectrum as well as on the thermody- namic properties of the system. However, the treatment of spin-orbital systems beyond the range of validity of mean-field decoupling and perturbative schemes remains an open problem in theory. Acknowledgments The authors thank O. P. Sushkov for valuable dis- cussions. A.M.O. acknowledges support by the Foun- dation for Polish Science (FNP) and by the Polish Min- istry of Science and Higher Education under Project No. N202069639. J.S. acknowledgessupport by the graduate school of excellence MAINZ (MATCOR). Appendix: Details of the perturbative approach for the boson-fermion model We start bya MF decoupling, rewritingthe interaction as H1=H1,MF+(H1−H1,MF)/bracehtipupleft/bracehtipdownright/bracehtipdownleft/bracehtipupright δH, with H1,MF=1 N2/summationdisplay kωBF(k,k,q)2/bracketleftBig ˜nb,kf† qfq+ ˜nf,qb† kbk/bracketrightBig . We treat δHas perturbation. The averages ˜ na,p=/angbracketleftbig a† pap/angbracketrightbig are determined self-consistently within this MF scheme. This leads to a renormalization of the magnon and fermion dispersions. We find ζ(q) = 2JS/parenleftBigg |y|−1 N/summationdisplay kcosk˜nf,k/parenrightBigg (1−cosq)−µ, (A.1)and ωF(q) =/parenleftBigg S2+x+2 N/summationdisplay k(1−cosk)˜nb,k/parenrightBigg cosq.(A.2) ForT= 0 only the magnon dispersion is renormalized to ζ(q) = 2JS(|y|+1/π)(1−cosq). By virtue of Dyson’s equation we may calculate the bosonic Green’s function at T= 0.68Since the MF de- coupling already takes the first order contributions into account, the lowest order diagrams we obtain are of sec- ond order. The self energy is thus approximated by the proper self-energy obtained by summing up those diagrams which may be composed by the second order diagrams. From this we have Σ( q,ω)≈Σ2(q,ω)≡ Σ+ BF(q,ω)+Σ− BF(q,ω)+Σ2,1(q) where the diagrams are givenby Σ+ BF(q,ω) (Fig.6(a)), Σ− BF(q,ω) (Fig.6(b)), and Σ2,1(q) (Fig.6(c)). A straigthforward calculation reveals that at zero temperature Σ 2,1(q) and Σ− BF(q,ω) vanish. For Σ+ BF(q,ω) we find the expression given in Eq. ( 5.5). For finite temperatures we calculate the imaginary time Green’s function. 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B 67, 012407 (2003). 67Performing the MF decoupling with the Hamiltonian ( 2.1)also takes the quartic orders of the Dyson-Maleev transfor- mation into account. These higher orders are neglected in the MF decoupling performed in Sec. V. For low temper- atures, i.e. the temperature region we have used in Sec. V the contributions from these higher order terms are small. 68A. L. Fetter and J. D. Walecka, Quantum Theory of Many- Particle Sytems (McGraw-Hill, Inc., New York, 1971).

0902.3853v1.Polaron_formation_in_the_presence_of_Rashba_spin_orbit_coupling__implications_for_spintronics.pdf

arXiv:0902.3853v1 [cond-mat.str-el] 23 Feb 2009Polaron formation in the presence of Rashba spin-orbit coup ling: implications for spintronics Lucian Covaci and Mona Berciu Department of Physics and Astronomy, University of British Columbia, Vancouver, BC, Canada, V6T 1Z1 (Dated: October 31, 2018) We study the effects of the Rashba spin-orbit coupling on the p olaron formation, using a suitable generalization of the Momentum Average approximation. Con trary to previous investigations of this problem, we find that the spin-orbit interaction decreases t he effective electron-phonon coupling. It is thus possible to lower the effective mass of the polaron by i ncreasing the spin-orbit coupling. We also show that when the spin-orbit coupling is large as compa red to the phonon energy, the polaron retains only one of the spin polarized bands in its coherent s pectrum. This has major implications for the propagation of spin-polarized currents in such mate rials, and thus for spintronic applications. PACS numbers: 71.38.-k, 71.70.Ej In the field of spintronics there is a continued effort to develop means of efficiently manipulating the spin of electrons [1]. One widely studied approach is to use spin- orbit coupling (SO) to lift the spin degeneracy, for exam- ple the Rashba SO coupling [2] that arises in a confined systemifthequantumwelllacksinversionsymmetry. Ex- perimentally, the Rashba effect has been seen in many systems, e.g.semiconductor heterostructures like GaAs and InAs, surface states of metals like Au(111)[3] and surface alloys like Bi/Ag [4] or Pb/Ag [5]. Confined two-dimensional (2D) systems may also cou- ple strongly to optical phonons of the substrate. Tuning of these electron-phonon (el-ph) interactions was shown to be experimentallyviable in organicsingle-crystaltran- sistors [6]. Strong el-ph coupling is interesting because it leads to polaron creation, whereas the coherent quasipar- ticle is an electron surrounded by a phonon cloud. The polaron dispersion and effective mass can be significantly renormalized from those of the bare band electron [7]. An interesting question is whether the el-ph and the SOcouplingcanbeusedinconjuctiontotailordifferently the properties of the two bands with different spin. The interplayofthe RashbaSO and the el-ph interactionshas been considered previously for continuous systems with parabolic bands and weak el-ph coupling, using the self- consistent Born approximation [8]. The main conclusion was that SO enhances the effective el-ph coupling due to a topological modification of the Fermi surface, namely aneffectivereduceddimensionalityofthelowenergyelec- tronicdensity ofstates. Inthis Letter, we investigatethis problemusingasuitablegeneralizationoftheMomentum Average (MA) approximation [9], which allows us to in- vestigate non-parabolic bands (tight-binding models) for any el-ph coupling strength. This is because this method is accurate for any coupling strength, being exact in the asymptotic limits of both very weak and very strong cou- pling. We demonstrate that the conclusions of Refs. 8 do not apply in the low density limit for a tight-binding dis- persion. In fact, the exact opposite holds, namely the effective el-ph coupling is suppressed by SO coupling.We also calculate spin-dependent spectral functions and show that in a certain regime, the coherent polaron band is dominated by contributions from only one SO elec- tronic band ( the “-” band). This has major implications for spin dependent transport through such materials, al- lowingforthe possibilitytomanipulatethe effectivemass and the spin polarization of quasiparticles by tuning the el-ph and SO interactions. We consider a single electron on a two dimensional squarelatticewith SOcoupling, whichalsointeractswith optical phonons of energy Ω ( /planckover2pi1= 1) through a local Holstein-type interaction [10]. The Hamiltonian can be written in terms of k-space spinors as: H=/summationdisplay k/parenleftBig c† k↑c† k↓/parenrightBig/parenleftbigg ǫkφk φ∗ kǫk/parenrightbigg/parenleftbigg ck↑ ck↓/parenrightbigg +Ω/summationdisplay qb† qbq +g√ N/summationdisplay k,q/parenleftBig c† k−q↑c† k−q↓/parenrightBig/parenleftbiggb† q+b−q0 0b† q+b−q/parenrightbigg/parenleftbiggck↑ ck↓/parenrightbigg whereǫk=−2t[cos(kxa) + cos(kya)] is the 2D free- electron dispersion for nearest-neighbor hopping and φk= 2Vs[isin(kxa)+sin(kya)] describes the Rashba SO coupling. Different dispersions and/or SO couplings can be studied similarly. As usual, c† k,σis the creation oper- ator for an electron with momentum kand spin σ, while b† qcreates a phonon of momentum q. Both electron spin channels interact with the phonons through the local lat- ticedisplacement, withanel-phcouplingconstant g. The lattice has a total of Nsites and periodic boundary con- ditions, and in the end we let N→ ∞. Momentum sums are over the Brillouin zone (BZ). The 2×2 Green’s functions for the non-interacting system (no el-ph coupling) are defined as: ¯G0(k,ω) =/angbracketleft0|/parenleftbigg ck↑ ck↓/parenrightbigg ˆG0(ω)/parenleftBig c† k↑c† k↓/parenrightBig |0/angbracketright =/parenleftbiggG0+(k,ω)eiξkG0−(k,ω) e−iξkG0−(k,ω)G0+(k,ω)/parenrightbigg (1) whereˆG0(ω) = [ω− H0+iη]−1is the resolvent cor- responding to H0=H|g=0. The symmetric and anti-2 symmetric Green’s functions are: G0±(k,ω)=1 2/parenleftbigg1 ω+iη−ǫk+|φk|±1 ω+iη−ǫk−|φk|/parenrightbigg and the phase factor is ξk=φk/|φk|. To avoid confusion with scalars, all 2 ×2 matrices will be identified by a bar, hence the ¯G0(k,ω) notation. The full Green’s function, defined as usually: ¯G(k,ω) =/angbracketleft0|/parenleftbiggck↑ ck↓/parenrightbigg ˆG(ω)/parenleftBig c† k↑c† k↓/parenrightBig |0/angbracketright,(2) can in principle be calculated exactly by applying the equation of motion method. The full Green’s function can be related to the non-interacting one by using the Dyson equation, ˆG(ω) =ˆG0(ω) +ˆG(ω)VˆG0(ω) - here V=H−H0isthe el-ph interaction. As shownpreviously for the Holstein Hamiltonian [9, 11, 12], the repeated use of the Dyson identity generates a system of equations in- volving generalized Green’s functions with various num- bers of phonons. In the presence of SO interactions, the equations of motion are similar, except they are in terms of 2×2 Green’s functions. As a result, we can implement the MA approximations in the same way, and all conclu- sions regarding accuracy for all el-ph coupling strengths, sum rules obeyed exactly by the spectral weight, etc. re- main true. A somewhat analogous procedure was used to compute the 2 ×2 Green’s function describing rippled graphene [13], however there the matrices are related to different sublattices, not to different spin projections. For all levels of the MA approximation, the Green’s function can be written in the standard form: ¯G(k,ω) = [¯G0(k,ω)−1−¯Σ(k,ω)]−1, (3) where the self-energy ¯Σ(k,ω) has different expressions depending on the level of MA approximation used. For the simplest, least accurate MA(0)level [9, 11], the self- energy has no kdependence and is given by an infinite continued fraction ¯ΣMA(0)(ω) =g2¯A1(ω), defined by ¯An(ω) =n¯g0(ω−nΩ)[1−g2¯g0(ω−nΩ)¯An+1(ω)]−1,(4) where ¯g0(ω) =1 N/summationdisplay k¯G0(k,ω) =/parenleftbigg g0+(ω) 0 0g0+(ω)/parenrightbigg (5) is the momentum average of the noninteracting Green’s function over the BZ. Note that because the off-diagonal partisanti-symmetric, itsaverageovertheBrillouinzone vanishes, thus ¯ g0(ω) and all ¯An(ω) matrices are diagonal. As discussed extensively in Refs. 11 and 12, MA(0)is accurate for ground state properties but it fails to prop- erly predict the polaron+one phonon continuum. As a result, it overestimatesthe polaronbandwith. This prob- lem is fixed at the MA(1)level, where a phonon is allowedto appear away from the polaron cloud. For the Holstein model (and by extension, in the presence of SO coupling) both these approximations predict k-independent self- energies. Here, they are diagonal as well, i.e.phonon emission/absorption is not allowed to scatter the elec- tron between the two spin-polarized bands. These, of course, are approximations, although it is worth point- ingthat all non-crosseddiagramsgive k-independent and diagonal contributions to the self-energy. This is why the self-consistent Born approximation prediction for the self-energy is also k-independent and diagonal [8]. In MA, however, the effect of non-crossed diagrams is included for MA(2)and higher levels (in variational terms, these allow two or more phonons to appear away from the main polaron cloud and the order of their cre- ation/annihilation now becomes relevant). We therefore report MA(2)results here. Following a similar procedure to that described in detail in Ref. [12], the MA(2)self- energy is found as: ¯ΣMA(2)(k,ω) = ¯x(0), (6) given by the solution of the system of coupled equations for the unknown 2 ×2 matrices ¯ x(i): /summationdisplay j¯Mi,j(k,ω)¯x(j) =eikRig2¯G0(−i,˜˜ω).(7) The sum is over lattice sites i= (ix,iy) located at Ri= ixaˆx+iyaˆy. The 2×2 matrices ¯Mi,j(k,ω) are: ¯M00=1−g2¯g0(˜ω)¯g0(˜˜ω)/parenleftbig 2¯a−1 31−¯a−1 21/parenrightbig ,(8) ¯Mi0=−g2¯g0(˜ω)eik·Ri¯G0(−i,˜˜ω)/parenleftbig 2¯a−1 31−¯a−1 21/parenrightbig (9) fori/negationslash= 0, and for both i,j/negationslash= 0: ¯Mij= ¯a21δi,j1−g2eik·Ri¯G0(j,˜ω) ×/bracketleftbig (¯A2−¯A1)δi,−j+¯G0(−i−j,˜˜ω)¯a−1 21/bracketrightbig .(10) Here we defined ¯ aij=1−g2¯g0(˜ω)(¯Ai−¯Aj) where ¯A1≡ ¯A1(ω−2Ω),¯A2≡¯A2(ω−Ω),¯A3≡¯A3(ω) are continuos fractions defined by Eq. (4), and ˜ ω=ω−2Ω−g2¯A1|(1,1), ˜˜ω=ω−g2¯g0(˜ω)|(1,1)(¯a−1 21)|(1,1)(since the ¯A,¯a,¯g0matri- ces are proportional to 1, the (2,2) diagonal matrix ele- ment can be used just as well in the definitions of ˜ ω,˜˜ω). Finally, the real space Green’s functions appearing in the inhomogeneous terms are given, as usual, by: ¯G0(i,ω) =1 N/summationdisplay keikRi¯G0(k,ω). (11) It is important to note that for i/negationslash= 0,¯G0(i,ω) acquires off-diagonal components, which lead to the off-diagonal contributions in ¯ΣMA(2)(k,ω). Because below the free- electron continuum the ¯G0(i,ω) decrease exponentially as|Ri|increases, the system in Eq. (7) can be truncated at a small |i|. We truncate at |Ri| ≃10a, such that the relative error of the spectral function is less that 10−3.3 -5-4-3-2-1 0 0 1 2 3 4 5 6(EGS-E0)/t Vs/tλ=0.75 λ=1.25 λ=2 0 2 4 6 0 2 4 6ln(m*/m) Vs/t FIG. 1: (color online) The ground state energy as a function of Rashba SO coupling for three values of the el-ph coupling. The inset shows the effective mass on a logarithmic scale as a function of Rashba SO coupling. Once the self-energy is known, the full Green’s func- tion can be calculated and in turn provide accurate esti- mates for spectral weights, the ground state energy, the effective mass etc. In the non-interacting case ( g= 0) the ground state consists of four degenerate points in k- space,(±kmin,±kmin), wherekmina= arctan[ Vs/(√ 2t)], with the ground-state energy E0=−4tcos(kmina)− Vs√ 8|sin(kmina)|. On the other hand, in the absence of SO coupling ( Vs= 0), as the effective el-ph coupling λ=g2/(4tΩ) is turned on, there is a crossover from a light, large-polaron to a very heavy, small-polaron at λ∼1. In Fig. 1 we show the ground state energy mea- sured from E0for weak, medium and strong effective el- ph couplings, as a function of the Rashba SO coupling. For large SO coupling, the renormalization of the en- ergy and effective mass (shown in the inset) is strongly suppressed, indicating large-polaron behavior even when λ= 2. This contradicts reported results for a parabolic band dispersion [8], which are based on the fact that the density of states at the band edge has a square root singularity, because in a continuum model the locus of momenta defining the ground state is a circle of radius kmin. This is not generically true for a tight binding dis- persion, where the Van Hove singularity is shifted from the 4-point degenerate ground state to higher energies. As expected, though, the results for a parabolic disper- sion can be recovered by our tight-binding model in the regime where both λandVs/tare very small. Indeed, forλ= 0.75, the effective mass increases slightly with Vs at small Vs, before decreasing at larger Vsvalues. Such behavior is more apparent as λ→0 [14]. Our results can be understood by noting that the free- electron bandwidth increases with increasing Vs. This results in an effective el-ph coupling, which compares the polaron binding energy to this renormalized bandwidth,that effectively decreases. As a result, away from the limit where both λandVs/tare very small, an increase in the SO coupling leads to a drop in the effective mass, making it possible to tune the mass of the polaron be- tween heavy and light. Light polarons are thus found for either small λirrespective of Vs/t, or at large λand large enoughVs/t. As we show next, however, their nature and spin-character may be very different. The MA(2)approximation is quantitatively accurate not only for the ground state, but also for high energy states, as it satisfies exactly the first 8 spectral weight sum rules and with good accuracy all other ones. It is thuspossibletohaveanaccuratedepictionofthehighen- ergy states by calculating the spectral function. Because we expect to have spin polarized bands it is necessary to calculate a spin dependent spectral function: /vectorA(k,ω) =−1 πIm(Tr[/vector σ¯G(k,ω)]), (12) where/vector σare the Pauli matrices. The direction of /vectorA(k,ω) gives the direction of the expectation value of the spin, while its magnitude gives the density of states with mo- mentum k. We know that in the non-interacting case as we go around the Γ-point, eigenstates have spin perpen- dicular to their momentum direction and rotating clock- wise for one band and anti-clockwise for the other. For the coupled system we observe similar spin eigenstates and thus choose to plot the following quantity: ˜A(k,ω) = [/vector uk×/vectorA(k,ω)]·/vector uz, (13) where/vector ukand/vector uzare unit vectors parallel to k, respec- tivelyz-axis. The two spin polarized bands will now cor- respond to opposite signs of ˜A(k,ω). Since the polaron bandwidth cannot exceed Ω, we ex- pect the character of the polaron band to depend on the relationbetween Ω and the energy difference between the spin-split electron bands. In order to exemplify this we plot˜A(k,ω) in Fig. 2 for λ= 1, Ω = 0 .8tandVs= 0.4t. The spectral function is shown along the (0 ,0)-(π,π) line in order to intersect the ground state at ( kmin,kmin). In thiscaseΩ /(E0−4t)>1andweseetwocoherentpolaron bands corresponding to the two spin polarizations, with similar quasiparticle weights. We conclude that when Ω is larger than the SO splitting, the polaronic quasipar- ticles are rather similar to the non-interacting electrons, except for the renormalized mass and supressed quasi- particle weight. Of course, the spectrum above EGS+Ω becomes incoherent due to el-ph scaterring. In Fig. 3 we plot ˜A(k,ω) for Ω = 0 .2t,λ= 1.0 and Vs= 0.8t. Now Ω/(E0−4t)<1 and because the polaron bandwidth cannot exceed Ω, it is dominated by the “-” band. There is a large difference between the quasiparti- cle weights of the two coherent polaron bands (note the different positive and negative scales for the the contour plot), and the effective mass of the dominant “-” band4 FIG. 2: (color online) Spin dependent spectral function ˜A(k,ω), forλ= 1, Ω = 0 .8tandVs= 0.4t. The right panel shows only the ”+” band for clarity. is much smaller than that of the low-weight “+” band. Higher energy states have small weights and are highly incoherent, i.e.short lived. Consider now injection of a current in such a system. WhereasinaregimelikethatdepictedinFig.2weexpect the spin to precess between the two coherent polaronic bands, like it does for non-interacting electrons [15, 16], in aregimelikein Fig.3onlythe spin-componentparallel to the “-” band can be efficiently transmitted through the system, which therefore acts as an intrinsic “spin- polarizer”. This becomes more and more apparent as one moves further into the asymptotic limit where both the SO and the el-ph couplings are large compared to Ω, i.e. FIG. 3: (color online) Spin dependent spectral function ˜A(k,ω), forλ= 1, Ω = 0 .2tandVs= 0.8t. The right-hand side panel shows only the ”+” band for clarity.(E0−4t)≫Ω andλ≫1. In this limit the “-” band in the coherent polaron spectrum becomes lighter and has a higher quasiparticle weight, whereas the “+” band es- sentially vanishes from the coherent spectrum (its quasi- particle weight is extremely low and its effective mass is extremely large). The resulting light polaron is thus very differentfromtheonewefindinthesmall λregime,which has both bands in the coherent spectrum with roughly equal quasiparticle weights and effective mass. This demonstrates that an interplay between SO and el-ph couplings allows indeed for different tailoring of the properties of the two spin-polarized bands, whereas one is well described by a long-lived, light quasiparticle while the other becomes highly incoherent. This will natu- rally lead to very different conductivities for the two spin polarizations, making such a material ideal as a source and/ordetector of spin-polarizedcurrents – these areim- portant components needed for many spintronics appli- cations. These conclusions are based on the use of the MA approximation which is known to be highly accurate for all el-ph couplings. Moreover, at the MA(2)level we use here, it results in a non-diagonal, k-dependent self- energy, therefore our results properly include phonon- mediated scattering between the two electronic bands. Acknowledgments: This work was supported by CI- FAR Nanoelectronics and NSERC. Discussions with Frank Marsiglio are gratefully acknowledged. [1] I.ˇZuti´ c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [2] E. I. Rashba, Sov. Phys. Solid State 2, 1224 (1960). [3] S. LaShell, B. A. McDougall, and E. Jensen, Phys. Rev. Lett.77, 3419 (1996). [4] C. R. Ast, et al., Phys. Rev. Lett. 98, 186807 (2007). [5] D. Pacil´ e, et al., Phys. Rev. B 73, 245429 (2006). [6] I. H. Hulea et al., Nat. Mater. 5, 982 (2006). [7] for a review see H. Fehske and S. A. Trugman, in Po- larons inAdvancedMaterials, editedbyA.S.Alexandrov (Canopus, Bath/Springer-Verlag, Bath, 2007). [8] E. Cappelluti, C. Grimaldi and F. Marsiglio, Phys. Rev. B76, 085334 (2007); ibid, Phys. Rev. Lett. 98, 167002 (2007); F. Marsiglio (private communications). [9] M. Berciu, Phys. Rev. Lett, 97, 036402 (2006). [10] T. Holstein, Ann. Phys. (N.Y.) 8, 325 (1959); ibid8, 343 (1959). [11] G. L. Goodvin, M. Berciu and G. A. Sawatzky, Phys. Rev. B74, 245104 (2006). [12] M. Berciu and G. L. Goodvin, Phys. Rev. B 76, 165109 (2007); M. Berciu, Phys. Rev. Lett. 98, 209702 (2007). [13] L. Covaci and M. Berciu, Phys. Rev. Lett. 100, 256405 (2008). [14] L. Covaci and M. Berciu, unpublished. [15] B. Srisongmuang et al., Phys. Rev. B 78, 155317 (2008). [16] C.A. Perroni et al., J. Phys.: Cond. Matt., 19, 186227 (2007).

1108.2260v1.Erratum__Engineering_a_p_ip_Superconductor__Comparison_of_Topological_Insulator_and_Rashba_Spin_Orbit_Coupled_Materials__Phys__Rev__B_83__184520__2011__.pdf

arXiv:1108.2260v1 [cond-mat.mes-hall] 10 Aug 2011Erratum: Engineering a p+ip Superconductor: Comparison of Topological Insulator and Rashba Spin-Orbit Coupled Materials [Phys. Rev. B 83, 18 4520 (2011)] Andrew C. Potter and Patrick A. Lee Department of Physics, Massachusetts Institute of Technol ogy, Cambridge, Massachusetts 02139 In Ref. 1 (henceforth referred to as I), we analyzed the effects of disorder on proposals to create an effective p+ipsuperconductorfromamagnetizedtwo-dimensional electron gas with Rashba spin–orbit coupling (SOC) by placing it on the surface of an ordinary bulk supercon- ductor. This problem was previously treated numerically in Refs. 2 and 3. In I, we pointed out that, since time- reversalsymmetry is broken in the surface layer, disorder is generically pair-breaking and tends to suppress the in- duced superconductivity (SC). For this system there are three distinct types of disorder: 1) impurities which re- side in the SOC surface state, 2) interface disorder and 3) impurities in the bulk superconductor. Recently the problem of bulk-impurities was addressed again in Ref. 4. They agree with our general finding that impurity types 1) and 2) are pair breaking but conclude that type 3) is not. This led us to re–examine our analysis, and we now conclude that our finding concerning type 3) im- purities in Iis incorrect. As will be explained below, the pair-breaking rate due to bulk impurities is actually negligible. This removes one barrier to the realization of proximity induced SC in semiconducting nanowires, but our previous conclusion that disorder in the wire and in the interface is detrimental remains the same. The pair-breakingeffects of bulk-impurities come from processes like that in Fig. 1. In this process, an electron tunnels from the surface into the bulk, where it is scat- tered by a bulk-impurity before returning to the surface. In our analysis in Appendix A of I, we assumed the same Fermi-surface geometry for the surface layer and bulk su- perconductor. Morerealistically, the surface-layeris two- dimensional whereas the bulk superconductor is three- dimensional. Here we show this mixed dimensionality strongly constrains the available phase-space for these scattering processes, and that the pair-breaking effects of bulk-impurities is negligible. While we only consider the case of a 2D electron gas with SOC, the following argument can also be applied to a 1D (or quasi-1D) wire in contact with a 3D bulk superconductor. For a clean interface, the components of momentum parallel to the surface–bulk interface ( xandycompo- nents) are conserved whereas the perpendicular ( z) com- ponent is not. An electron initially in the surface-layer with momentum k/bardblcan tunnel into any bulk states with momentum k= (k/bardbl,kz), but pays a largeenergycost un- lesskzis within ∼γ/vFof the bulk Ferm-surface (FS). HerevFis the bulk Fermi-velocity and γ=πNB|Γ|2 whereNBis the bulk tunneling density of states and Γ is the surface–bulk tunneling amplitude. Once in the bulk the electron can scatter to any momentum k+qwithin ∼1/τvFof the bulk FS, where τ−1is the bulk disorderΓΓB S B S SB B k+qk|| (k+q)||k'k||ΓΓ k k'+q FIG. 1. Diagrammatic depiction of the pair-breaking pro- cess due to bulk impurities. The geometrical constraints on scattering due to the different dimensionality of the surfac e and bulk (see Fig. 2) suppress these processes by a factor of γ/εF≪1. Circles with Γ show surface–bulk tunneling (S and B label surface or bulk Green’s functions), bulk impurities are denoted by ×, and the dashed line indicates that both ×refer to the same impurity. FIG. 2. Momentum space geometry for surface–bulk tun- neling. The 2D surface Fermi-surface (FS) is extended into a cylinder since tunneling does not conserve the momentum perpendicular to the interface (in the z-direction). Surface– bulktunnelingeventsinvolveonlystates neartheintersec tion, I, of the surface and bulk FS’s. scatteringrate. However, in orderto subsequently return to the surface-layer, the in-plane component of k+q must again be within γ/vFof the surface FS. There- fore, the available phase-space for such scattering is ≈ (2πkF)(1 τvF)(γ vF). In contrast, the phase space available for arbitrary bulk impurity scattering is ≈4πk2 F(1 τvF). The pair-breaking scattering rate τ−1 pbis smaller than the bulk impurity scattering rate τ−1by the ratio of these two phase-space volumes: τ−1 pb/τ−1≈γ 2vFkF∼γ εF≪1 whereεFis the bulk-Fermi energy. In a typicalsupercon- ducting metal, εFwill greatly exceed γ, hence the pair breaking due to bulk disorder can be safely neglected. Before concluding, we would like to emphasize the dis- tinction between the scattering rate, τ−1 sb, for surface-2 ΓS SB k+q k|| k||Γk k FIG. 3. Diagrammatic depiction of a non-pair breaking scat- tering process for surface-electrons due to bulk impuritie s. Unlike the pair-breaking process shown in Fig. 1, this pro- cess has an unconstrained phase space. electronsfrom bulk-impurities and the pair-breakingrate τ−1 pb. The scattering rate, τ−1 sbincludes all possible bulk- disorder processes, and is dominated by processes likethe one shown in Fig. 3, where an electron tunnels from surface to bulk, scatters from a bulk impurity and then continues to propagate in the bulk. This type of scat- tering is not pair breaking since, after scattering, the electron propagates only in the bulk where time-reversal symmetry is intact and pairing is not disrupted. Such processes do not suffer the same phase-space restrictions described above, and consequently τ−1 sbcanbe quite large even though the pair-breaking rate τ−1 pbis small. There- fore, it is not that the surface-electrons are largely unaf- fected by bulk impurities, but ratherthat scatteringfrom these impurities is predominantly non-pair breaking. Acknowledgements – We thank Roman Lutchyn and Jason Alicea for helpful discussions. 1A.C. Potter and P.A. Lee, Phys. Rev. B 83, 184520 (2011) 2A.C. Potter and P.A. Lee, Phys. Rev. Lett. 105, 227003 (2010); 3R.M. Lutchyn, T.D. Stanescu, S. Das Sarma, Phys. Rev.Lett. 106, 127001 (2011); 4T. Stanescu, R. M. Lutchyn, S. Das Sarma arXiv:1106.3078 (2011)

2201.10845v2.Magnetoconductance_Anisotropies_and_Aharonov_Casher_Phases.pdf

Magnetoconductance Anisotropies and Aharonov-Casher Phases R. I. Shekhter,1O. Entin-Wohlman,2,M. Jonson,1and A. Aharony2 1Department of Physics, University of Gothenburg, SE-412 96 G oteborg, Sweden 2School of Physics and Astronomy, Tel Aviv University, Tel Aviv 69978, Israel (Dated: July 19, 2022) The spin-orbit interaction (SOI) is a key tool for manipulating and functionalizing spin-dependent electron transport. The desired function often depends on the SOI-generated phase that is accumu- lated by the wave function of an electron as it passes through the device. This phase, known as the Aharonov-Casher phase, therefore depends on both the device geometry and the SOI strength. Here we propose a method for directly measuring the Aharonov-Casher phase generated in an SOI-active weak link, based on the Aharonov-Casher-phase dependent anisotropy of its magnetoconductance. Specically we consider weak links in which the Rashba interaction is caused by an external electric eld, but our method is expected to apply also for other forms of the spin-orbit coupling. Mea- suring this magnetoconductance anisotropy thus allows calibrating Rashba spintronic devices by an external electric eld that tunes the spin-orbit interaction and hence the Aharonov-Casher phase. Introduction. The spin-orbit interaction (SOI), which allows for an interplay between charge and spin currents in all-electrical devices [1, 2], substantially aects the transport properties of two- and three-dimensional con- ductors. The spin-Hall eect [3], the electrical control of the magnetization in magnetic heterostructures by inter- facial spin-orbit toques [4], and the spin relaxation yield- ing quantum anti weak-localization in impure conductors [5, 6] are examples of such an in uence. The SOI is due to the interaction between the magnetic moment of a particle moving with velocity vthrough a static electric eld Eand the magnetic eld Bso= (Ev)=c2seen in the rest frame of the particle. This relativistic eect adds a term Hso(k) =
Bso=
(Ev)=c2: (1) to the Hamiltonian, which can be removed by a gauge transformation, !exp(i^AC) , that adds a phase fac- tor to the wave function. Here ^AC=1 ~Zt 
Bsodt0=1 ~c2Zr (E)
dr0;(2) where the integration is over the path of the particle, is known as the Aharonov-Casher phase [7, 8]. We use the notation ^ACfor the phase operator and ACfor its eigenvalue, to be discussed below. One notes from Eqs. (1) and (2) that the strength of the SOI, which can be readily measured [9, 10], is a local property while, importantly, the Aharonov-Casher phase depends in addition on the geometry of the device, usually not precisely known, and is therefore a global property of the device. The Aharonov-Casher phase gives rise to quite a num- ber of new spintronic functionalities in devices formed by weak links bridging bulk conductors in congurations where the spin-orbit interaction is active solely in the weak link [11]. Such devices are capable of spin-selective electric transport [12{14]. They allow for spin accumula- xyzkEBsoBqLRFIG. 1. (color online) Sketch of the system considered. A spin-orbit active one-dimensional weak link of length dis at- tached to two leads, LandR. An external electric eld ap- plied in the negative y-direction interacts with an electron moving along the x-direction with momentum kand gener- ates a pseudo-magnetic eld Bsoin thez-direction. An exter- nal magnetic eld Bis applied in the yz-plane, forming an anglewith thez-direction. tion and generation of spin currents [15{17] and for elec- tromotive forces [18] and spin-polarization of supercon- ducting Cooper pairs [19]. These eects are more striking in quantum networks built of spin-orbit active weak links [20, 21] where the Aharonov-Casher phase dominates the interference of the electronic spinors [22] and even yield spin ltering [23]. Because of its importance it would be useful to be able to measure and tune the Aharonov-Casher phase without separate knowledge of the SOI strength and device geom- etry. The purpose of this Letter is to propose a method for achieving just that. Our proposal is based on mag- netoconductance measurements and does not require ob- serving interference patterns in multiply connected sys- tems. Originally, the Aharonov-Casher phase [7] had been predicted for a neutral particle possessing an electric mo- ment, which moves in a magnetic eld, and thereforearXiv:2201.10845v2 [cond-mat.mes-hall] 18 Jul 20222 the earlier demonstrations were accomplished on neutron and atomic interferometers (see e.g., Ref. 24). More re- cently this phase has been shown to induce changes in the Aharonov-Bohm oscillations observed in the magne- toconductance data taken on ring structures fabricated from HgTe/HgCdTe quantum wells, where continuously adjustable spin-orbit coupling is available [20]. However, the interpretation of such experiments relies on model calculations, and the deduction of that phase is rather indirect [25]. In this Letter we show that the magnetoconductance anisotropy (with respect to the direction of the magnetic eld) of a single weak link allows for a detailed calibra- tion of the Aharonov-Casher phase itself. By a weak link we here mean a pseudo-one-dimensional conductor that connects two bulk conductors and dominates the elec- trical resistance of the system. The system we have in mind is sketched in Fig. 1. Electrons can propagate ei- ther ballistically or by tunneling through the weak link. We mainly consider the ballistic case, which is more use- ful in the present context, and defer a detailed discussion of the tunneling case to the supplemental material [26]. We focus specically on weak links where the Rashba interaction [27] is controlled by an external electric eld. In the absence of a magnetic eld this interaction, which preserves time-reversal symmetry, does not aect the conductance [13]. This is because the two spin projec- tions on the direction of the pseudo magnetic eld, gen- erated by the spin-orbit interaction, are good quantum numbers and hence constants of motion of the traversing electrons. As a result, the electron transmission ampli- tude is diagonal in spin space and therefore the conduc- tance does not depend on the Aharonov-Casher phase. However, by switching on an external magnetic eld, which aects the dynamics of the electrons through the Zeeman interaction, time-reversal symmetry is broken and the above spin projections are no longer good quan- tum numbers. This makes the transmission amplitude non-diagonal and mixes the two spin projections, which correspond to dierent Aharonov-Casher phase factors. The ensuing interference re ects the relative orientation of the applied magnetic eld Band the pseudo eld Bsogenerated by the SOI, giving rise to an anisotropic magnetoconductance, which depends on the product of the strength of the spin-orbit interaction and the length of the weak link, forming the Aharonov-Casher phase. Hence, measurements of the anisotropy of the magneto- conductance singles out the Aharonov-Casher phase. Two-terminal quantum conductance of a weak link. The conductance of a conductor coupled to two terminals is given by the Landauer-B uttiker expression, G=g0Trfttyg; (3) g0=e2=hbeing the quantum unit of conductance [28]. In general the dimension of the matrix tis twice the number of channels in the link (including the two spinprojections). For a one-dimensional weak link the trans- mission amplitude tis a (22) matrix in spin space t= 1 2 LGi(d)1 2 R; (4) where Gi(d) is the Green's function (aka propagator) of the link, of length d, and L(R) is the coupling to the left (right) terminal, in energy units [29] (these are scalars when the terminals are free electron gases). Magnetoconductance for ballistic transport. The explicit expression for the propagator describing ballistic transport through the weak link reads [30] Gi(d) =d 2Z dkeikdh EF+i0+H(k)i1 ;(5) whereH(k) is the Hamiltonian of an electron moving along the weak link (assumed to lie on the ^xaxis) with momentum k=k^x(using ~= 1 units) H(k) =k2 2mkkso mzB
: (6) Here we have used that for electrons =B, where Bis the Bohr magneton and is the vector of the Pauli matrices. The Zeeman eld B(in energy units since we letB= 1) lies in the yzplane, i.e., B=B^nand ^n= sin()^y+ cos()^z, and the pseudo magnetic eld induced by the spin-orbit interaction, of strength ksoin momentum units, is along ^z. Comparing Eqs. (1) and (6) we note that ksois proportional to the total electric eld felt by the electron. It is common to extract its value from experiments, see for instance Ref. [9]. The energy of the propagating electron is EF=k2 F=(2m),mbeing the eective mass of the electron. An important point of our discussion concerns the di- rection of the external magnetic eld, B. The Hamil- tonian (6) shows that the eect of the spin-orbit cou- pling can be viewed as that of a pseudo Zeeman eld which depends on the momentum and does not break time-reversal symmetry. We nd below that when the external Zeeman eld is parallel to this pseudo eld (i.e., when= 0) the magnetoconductance of the weak link is totally devoid of the Aharonov-Casher phase. As op- posed, when the Zeeman eld deviates away from the ^z direction, the magnetoconductance becomes anisotropic, with the anisotropy mainly determined by the Aharonov- Casher phase. The integral in Eq. (5) is carried out using the Cauchy theorem [26, 30]. It is illuminating to examine the result- ing propagator for zero Zeeman eld, Gi(d;B= 0) =imdp k2 F+k2soe[idp k2 F+k2so+iksodz]:(7) As seen, there are two ngerprints of the spin-orbit cou- pling. One is of rather minor importance, the wave vector of the propagating electron is modied, kF)p k2 F+k2so.3 The second is the accumulation of the Aharonov-Casher phase factor [7], exp( i^AC), where ^AC=ksodzis an operator in spin space. As explained in physical terms above, and as is obvious from Eqs. (3), (4), and (7), the phase disappears from the conductance in the absence of an external Zeeman eld and one nds that G(B= 0)=(g0LR) = 2(md)2=(k2 F+k2 so):(8) A Zeeman eld with a component perpendicular to the pseudo eld induced by the spin-orbit coupling is needed for this coupling to have any eect on the conductance. For a magnetic eld along a general direction in the yzplane, the Cauchy integration in Eq. (5) requires some care and is carried out in detail in Ref. [26]. Up to second order in By[31] the propagator is given by two poles at k=kso(1Ri=2) +eq; (9) where eq2 =kF+k2 so2mBz+Ri(k2 somBz); (10) andRi= (mBy)2=[(mBz)2(kFkso)2]. For zero spin- orbit coupling the propagator, and consequently the con- ductance, depends solely on B= [B2 y+B2 z]1=2, and is isotropic with respect to the direction of the magnetic eld. Forkso6= 0, it becomes anisotropic. Below, the conductance is calculated up to second order in the two magnetic eld components, in particular assuming that BykFkso=m=vFkso=
so=2. The energy split caused by the spin-orbit interaction,
so, is45 meV [9, 32]. As the magnetic energy (with appropriate con- stants reinstated) is gBByg0:06(By=T) meV, this inequality might be satised for reasonable values of the g-factor. (Note that the magnetic-eld strength is an externally-controlled parameter.) Following Eqs. (3) and (4), the conductance is ob- tained by tracing over the matrix product Gi(d)Gy i(d). To order (Bz;y=EF)2one nds that TrfGi(d)Gy i(d)g=2(md)2 k2 F+k2so 1 +hBz EFi2k4 F (k2 F+k2so)2 +hBy EFi24k2 F+ 3k2 so 4(k2 F+k2so)1 2hBy EFi2 sin2(AC) ;(11) where the Aharonov-Casher phase, equal to AC=ksod, is the eigenvalue of ^AC. The normalized magnetocon- ductance is (assuming ksokF) GG(B= 0) G(B= 0)h 12k2 so k2 FihB EFi2 +7k2 so 4k2 FhBy EFi2 1 2hBy EFi2 sin2(AC): (12) Once the Rashba interaction is switched-on, as can be done by applying gate voltages [9] or external electric FIG. 2. (color online) A contour plot of the anisotropy of the magnetoconductance for the ballistic case, as dened by Eq. (13), plotted as a function of the Aharonov-Casher phase AC=ksod(horizontal axis) and the angle between an ex- ternal magnetic eld and the pseudo-magnetic eld generated by the spin-orbit interaction (vertical axis). The anisotropy has a maximum in the center of the contour plot, where AC===2, and vanishes on the borders, where AC oris either 0 or . By measuring the anisotropy at dierent anglesone may determine AC. elds [10], the magnetoconductance becomes anisotropic , as demonstrated by the last two terms in Eq. (12). These arise from dierent sources. The rst one is determined by the strength of the spin-orbit coupling in the particu- lar material forming the weak link. It is governed by the ratio (kso=kF)2, which is typically small: For the values ofkF=p2nsreported in Refs. [9, 32] ( nsis the elec- tron density) kF(36)108m1whileksois about two orders of magnitude smaller. Thus this material- dependent anisotropy is minute and will be neglected in what follows. In contrast, the last term on the right hand-side of Eq. (12) provides a neat possibility to measure the Aharonov-Casher phase ACdirectly from the anisotropy of the magnetoconductance. To do so we normalize the measured magnetoconductance for an arbitrary an- glebetween the external magnetic eld [ By=Bsin(), Bz=Bcos()] and the pseudo eld induced by the spin- orbit interaction to the measured magnetoconductance for= 0. One nds that G(B;)G(0;0) G(B;0)G(0;0)= 11 2sin2() sin2(AC);(13) which is a result that only depends on the Aharonov- Casher phase ACand the angle . This function is shown in Fig. 2. Magnetoconductance for tunneling electrons . The discussion above pertains to weak links through which electrons propagate ballistically. One may wonder what4 form the magnetoconductance takes for a weak link through which electron transport is by tunneling. In- terestingly, the end result for the magnetoconductance anisotropy in the two cases is formally not much dier- ent. The conductances, however, are quite disparate, im- plying detrimental experimental consequences. Here we oer a heuristic comparison and explanations of the two scenarios. Viewing a tunnel junction as a potential barrier whose height exceeds the energy of the impinging electrons by E0= 1=(2ma2 0), wherea0can be thought of as a tunnel- ing length, the expression for the propagator is modied. We refer to Ref. 26 for a detailed derivation of the prop- agator Ge(d) and for the form of Tr fGe(d)Gy e(d)gfor this case. To order ( By;z=E0)2and forBy=E0ksoa01 one nds TrfGeGy eg= 2(ma0d)2e2d=a0 n 1 +(Bd)2 2(E0a0)2h 1 +5 2a0 d+ 2a2 0 d2i +(Byd)2 2(E0a0)2ha0 dsin(AC) AC2 +sin2(AC) (AC)21io : (14) The magnetoconductance is again anisotropic due to the term on the second line (which as expected vanishes ifBy= 0). The magnetoconductance, normalized as in Eq. (13), becomes, G(B;)G(0;0) G(B;0)G(0;0)= 1sin2() 1 + (5=2)(a0=d) + 2(a0=d)2 nh 1sin(AC) AC2i + 2a0 dh 1sin(2AC) 2ACio :(15) In order for the anisotropy to be determined by the single parameterAC=ksod, the Aharonov-Casher phase, as in the ballistic case, we need a0=d1, in which case the magnetoconductance, normalized as in Eq. (13), is G(B;)G(0;0) G(B;0)G(0;0)= 1sin2()h 1sin2(AC) (AC)2i :(16) However, the assumption that a0=d1 is unrealis- tic since that would make the conductance, which Eq. (14) shows is proportional to exp( 2d=a0), too small to be measured. A realistic smallest value of exp( 2d=a0) would be about 104, corresponding to a0=d0:2, which is a number that does not justify neglecting terms pro- portional to a0=dand (a0=d)2in Eq. (15). Comparing the expressions for the magnetoconduc- tance for ballistic- and tunneling transport through a weak link, Eqs. (11) and (14), one observes that the renormalization of the electronic propagator by an exter- nal magnetic eld is qualitatively dierent for the two transport regimes. In the ballistic transport regime the magnetic eld modies the Fermi momentum and conse- quently the orbital phase factors, turning exp[ ikFd] intoexp[ikd] [see Eqs. (7), (9), and (10)]. These orbital phase factors have no eect on the conductance. In a tunneling device, on the other hand, the magnetic eld normalizes the tunneling length, introducing additional exponential dependence on the width dof the tunnel junction. Such a modication aects the modulus of the propagator, and reduces further the tunneling conduc- tance. Discussion. We have presented an analysis of the mag- netoconductance of one-dimensional weak links through which electrons propagate either ballistically (ballistic transport regime) or by tunneling (tunneling transport regime). Whether the eigenstates of the electrons in the weak link are plane waves or evanescent tunneling modes, they acquire a phase factor due to the Aharonov-Casher eect [7]. The propagator matrix, written in terms of these eigenstates, is diagonal as long as the external mag- netic eld is parallel to the pseudo magnetic eld induced by the spin-orbit coupling. In this case the conductance, given by Eqs. (3) and (4), is determined by the sum of the squared moduli of the diagonal elements of the propaga- tor and therefore becomes independent of the Aharonov- Casher phase. An external magnetic eld that has a component per- pendicular to the pseudo eld, on the other hand, has the eect that spin is no longer a good quantum num- ber, the propagator matrix becomes non-diagonal, and interference between its non-diagonal elements leads to a magnetoconductance that depends on the Aharonov- Casher phase. One can think of this as a manifestation of the Aharonov-Casher phase in the magnetoconductance due to an internal interference between spin-up and spin- down states. Our main result is Eq. (13), showing that in the ballistic transport regime this Aharonov-Casher phase can be deduced by comparing measurements of the magnetoconductance of the weak link in an external magnetic eld oriented in dierent directions. We acknowledge the hospitality of the PCS at IBS, Daejeon, Korea, where part of this work was done with support from IBS Funding No. IBS-R024-D1. orawohlman@gmail.com [1] S. Sahoo, T. Kontos, J. Furer, C. Homana, M. Gr aber, A. Cottet, and C. Sch onenberger, Electric eld control of spin transport, Nature Phys. 1, 99 (2005). [2] A. Hirohata, K. Yamada, Y. Nakatani, I.-L. Prejbeanu, B. Di eny, P. Pirro, and B. Hillebrands, Review on spin- tronics: Principles and device applications, J. Magn. Magn. Mater. 509, 166711 (2020). [3] J. Wunderlich, B. Kaestner, J. Sinova, and T. 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2009.06956v1.Spin_orbital_polarization_of_Majorana_edge_states_in_oxides_nanowires.pdf

Spin-orbital polarization of Majorana edge states in oxides nanowires J. Settino,1F. Forte,1, 2C. A. Perroni,3, 4V. Cataudella,3, 4M. Cuoco,1, 2and R. Citro1, 2 1CNR-SPIN c/o Universit a degli Studi di Salerno, I-84084 Fisciano (Sa), Italy 2Dipartimento di Fisica \E. R. Caianiello", Universit a degli Studi di Salerno, I-84084 Fisciano (Sa), Italy 3CNR-SPIN c/o Universit a degli Studi di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy 4Physics Department "Ettore Pancini", Universit a degli Studi di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, I-80126 Napoli, Italy We investigate a paradigmatic case of topological superconductivity in a one-dimensional nanowire with dorbitals and a strong interplay of spin-orbital degrees of freedom due to the competition of orbital Rashba interaction, atomic spin-orbit coupling, and structural distortions. We demonstrate that the resulting electronic structure exhibits an orbital dependent magnetic anisotropy which aects the topological phase diagram and the character of the Majorana bound states (MBSs). The inspection of the electronic component of the MBSs reveals that the spin-orbital polarization generally occurs along the direction of the applied Zeeeman magnetic eld, and transverse to the magnetic and orbital Rashba elds. The competition of symmetric and antisymmetric spin-orbit coupling remarkably leads to a misalignment of the spin and orbital moments transverse to the orbital Rashba elds, whose manifestation is essentially orbital dependent. The behavior of the spin-orbital polarization along the applied Zeeman eld re ects the presence of multiple Fermi points with inequivalent orbital character in the normal state. Additionally, the response to variation of the electronic parameters related with the degree of spin-orbital entanglement leads to distinctive evolution of the spin-orbital polarization of the MBSs. These ndings unveil novel paths to single-out hallmarks relevant for the experimental detection of MBSs. I. INTRODUCTION One-dimensional topological superconductivity of intrinsic1{7or articial8{17p-wave superconductors can harbor the so-called Majorana bound states (MBSs) which are pinned to zero-energy and are charge neu- tral. Recent experimental observations18{25have brought evidences for the presence of MBSs in articial topo- logical superconductors, which are based on nano- sized chains with magnetic atoms deposited on top of a superconducting substrate. In these experiments, while the MBSs occur in an eective spinless regime, their observed ngerprints arise from the nontrivial spin structure of the corresponding MBS wavefunction. There, the spin polarization of the MBS conguration can be accessed by means of scanning tunneling mi- croscopy (STM) through a measurement of spin-selective conductance26{28. This physical scenario applies also to semiconducting nanowires proximity-coupled with an s-wave spin-singlet superconductor29{33and networks34 where the electronic spin orientation and spatially re- solved texture of the MBS can exhibit ngerprints that depend on the relative strength of Rashba and Dressel- haus interactions35. On a general ground, due to sym- metry constraints a Kramers pair of MBSs is marked by an Ising spin, i.e., the spin density is nonvanishing only along a specic direction36. On the contrary, the spin density for the case of MBSs protected by a sublat- tice (chiral) symmetry is identically zero35,37. However, the electron projection of the spin-density is generally nonvanishing even for eective spinless topological super- conductors, where the spin polarization is locked along a given orientation and this can be probed by STM orcharge transport measurements. Along this line, a radi- cally dierent situation can occur in intrinsic quasi one- dimensional topological superconductors, where the elec- tron spin is an active degree of freedom in setting out the topological behavior38{40, and chiral protected mul- tiple MBSs at the edges can manifest both an Ising type behavior and a spin texture with characteristic spatial patterns and orientations37. Moving to a broader physical scenario, one can ask whether for superconducting materials having an elec- tronic structure with nontrivial spin-orbital entangle- ment, the electron spin and orbital moment are active degrees of freedom in the MBSs and can leave a unique imprint on spin-resolved and, potentially, on orbitally- resolved local spectroscopic probes. Such appealing per- spective can naturally occur when considering supercon- ducting materials with atomic multi-orbital degrees of freedom. These microscopic elements are commonly en- countered in oxides materials, where d-bands can lead to fascinating spin-orbital correlated phenomena which, apart from fundamental physical challenges, can also lead to tantalizing solutions for emergent technologies41. In this framework, a prototype superconductivity with electronic components having intrinsically coupled spin- orbital degrees of freedom is provided by the two dimen- sional electron gas (2DEG) forming at the interface of oxide band insulators. Oxide 2DEGs, indeed, are char- acterized by the simultaneous presence of strong spin- orbit coupling42and superconductivity43, both widely tunable by electric eld eect42,44, while 2D magnetism, coexisting with superconductivity,45,46can be induced by opportune atomic engineering of the heterostructures47. Hence, the combination of magnetism, superconductivityarXiv:2009.06956v1 [cond-mat.supr-con] 15 Sep 20202 A-block B-block C-block FIG. 1. Schematic view nearby the point in the Brillouin zone of the bands arising from the considered atomic ( dxy; dxz; dyz) congurations. The tetragonal crystal eld potential splits the dxywith respect to ( dxz; dyz) lowering the energy of the dxy state (a). Then, the spin-orbit coupling leads to a conguration with nontrivial combination of spin ( s) and orbital ( l) angular momentum as schematically illustrated for the point Kramers states in (a),(b) and (c). We notice that the states in (b) have only z-component of l, while the congurations in (a) and (c) have dominant lxandlzcomponents, respectively. There, the degree of mixing can be qualitatively extracted from the inspection of the angular distribution. The application of a magnetic eld splits the Kramers states at the point but due to the spin-orbit coupling the splitting amplitude depends on the orbital character. For a magnetic eld applied along the nanowire direction ( x) the lowest energy band has a larger splitting compared with the other bands due to the dominant spin-orbital polarization along x. Once the topological state is achieved for a given electron lling (dotted line indicates the chemical potential) the Majorana bound states (MBSs) occur at the edges of the nanowire with a characteristic spin-orbital content as sketched in (g), (h), and (i). We notice that the spin and orbital polarizations of the MBS lie in the xzplane coplanar to the direction of the applied magnetic eld and perpendicular to the orbital Rashba eld direction (i.e. y). The behavior of the components are orbital dependent. and inversion asymmetry provides a quite unique plat- form for the realization of multi-orbital topological su- perconducting phases. Recently, quasi-2D electron gases formed at the interface between LaAlO 3and SrTiO 3 (LAO/STO)48have been theoretically proposed as pos- sible candidates for the realization of topological super- conducting phases in two-dimensional49{52and various topological scenarios have been explored in eective quasi one-dimensional models53{59. In this work we aim to assess the spin-orbital charac- ter of MBSs occurring in a topological superconducting phase that is induced by an applied Zeeman magnetic eld for a one-dimensional nanowire with dorbitals (dxy;dxz;dyz) and strong interplay of spin-orbital degrees of freedom. As the d-orbitals have a nontrivial angu- lar momentum and an anisotropic spatial distribution,the nature of the electronic states is signicantly sen- sitive to spin-orbit coupling and system's dimensional- ity. Here, we focus on a typical electronic situation in low-dimensional materials where the interplay of spin- orbit coupling and tetragonal distortions breaks the spin and orbital rotational invariance resulting into a charac- teristic atomic spin-orbital distribution (Figs. 1(a)-(c)). Additionally, inversion asymmetry yields orbital Rashba- type interaction that together with the spin-orbit cou- pling sets out a non-trivial momentum dependent spin- orbital splitting. In this framework, apart from the non- standard spin-orbital texture naturally occurring at the Fermi level60, the response to a Zeeman magnetic eld is highly anisotropic and orbital dependent (Figs. 1(d)- (f)). Thus, once a topological superconducting phase is obtained, the emergent MBSs may exhibit unique nger-3 prints of the underlying spin-orbital electronic substrate from which they arise. This is indeed the key outcome of the paper and we nd distinct characteristics of the spin-orbital content of the MBSs that we summarize in Figs. 1(g)-(i). The inspection of the electronc component of the MBSs reveals that the spin-orbital polarization has always a planar orientation. Moreover, the compo- nents along the direction of the applied Zeeeman mag- netic eld and orthogonal to the magnetic-and-orbital Rashba elds are strongly sensitive to a variation of the spin-orbit strength. The emerging trend is to have a tunable orientation which is dependent on the orbital character of the bands where the topological pairing sets in. We also nd that the competition of symmetric and antisymmetric spin-orbit coupling remarkably leads to a misalignment of the spin and orbital moment orientations for the MBSs whose manifestation is inequivalent for the dxycompared to the dxz;dyzbased bands. Addition- ally, even in a regime where the spin-momentum locking substantially deviates from that due to the spin Rashba coupling, we nd that the spin-orbital polarization has a planar orientation. We also investigate the behavior of the electron spin-orbital polarization along the applied Zeeman eld across the topological phase transition and show that it re ects the presence of multiple Fermi points with inequivalent orbital character in the normal state. These ndings unveil a rich scenario concerning the spin- orbital content of the MBSs and nonstandard paths to single-out hallmarks which may be relevant for the ex- perimental detection of MBSs. The paper is organized as follows. In Section II, the model hamiltonian for oxides nanowires is presented. In Section III, we introduce the orbital dependent Majo- rana fermion polarization and present the topological phase diagram resulting from the application of a Zee- man magnetic eld. In Section IV we provide the key ngerprints of the orientation and spatial prole of the electron spin and orbital polarization of the MBSs. In particular, we focus on the behavior nearby the topologi- cal phase transition for the various bands and discuss dif- ferences with respect to the canonical spin-Rashba model employed to study topological phase transitions in semi- conducting nanowires. Finally conclusions are given in Section V. The Appendix A is devoted to the derivation of the Majorana polarization for the case of multi-orbital topological superconductors, while in Appendix B we re- port the characterization of the spin-orbital polarization of the states at the Fermi level in the normal phase. II. MODEL AND METHODOLOGY In transition metal oxides with perovskite structure the transition metal (TM) elements are surrounded by oxygen (O) in an octahedral environment. Owing to the crystal eld potential generated by the oxygen around the TM, the vefold orbital degeneracy is removed and d orbitals split into the t2gsector, i.e., yz,zx, andxy, andtheegsector, i.e., x2y2and 3z2r2. For a tetragonal symmetry the low-energy electronic structure can be de- scribed by a model having only the t2g-orbitals contribut- ing to the Fermi level. Additionally, for weak octahedral distortions, the TM-O bond angle is almost ideal and thus the three t2g-bands are mainly directional and ba- sically decoupled, e.g., an electron in the dxy-orbital can predominantly hop along the yorxdirection through the intermediate px- orpy-orbitals. Similarly, the dyz- and dzx-bands are quasi one-dimensional when considering a square geometry for the 2D TM-O bonding network. Furthermore, the atomic spin-orbit interaction (SO) mixes thet2gorbitals thus competing with the quench- ing of the orbital angular momentum resulting from the crystal eld potential. Out-of-plane buckling modes of the TM-O bond are very important in 2DEGs forming at the interface of insulating oxide materials, as they cause an orbital mixing, which is odd in space, of dxyanddyz ordzx-orbitals along the yorxdirections, respectively. Indeed, the inversion symmetry breaking is primarily af- fecting the orbital degrees of freedom and leads to an orbital momemtum locking through the so-called orbital Rashba interaction while the spin-momentum coupling derives from the atomic spin-orbit. The atomic spin-orbit interaction (SO) is then a crucial term to be included into the electronic description both for the setting out of the spin-orbital texture in the reciprocal space60, and for the natural mixing of the spin-orbital degrees of the t2g-states in competition with the quenching of the orbital angular momentum due to the crystal potential. Since we are interested in the analysis of topological phase that it established as a consequence of time re- versal symmetry breaking due to an external magnetic eld, the model Hamiltonian we are going to consider includes both a coupling of electron spin and orbital mo- ments to the magnetic eld and a superconducting pair- ing term. The conditions to achieve a topological non- trivial superconducting phase for a quasi one-dimensional nanowire were already discussed in Ref.56. In partic- ular for the chosen geometry of the nanowire oriented along thex-axis, the optimal magnetic eld direction for achieving a topological phase is to be perpendicular to they-direction, that is the orientation of the orbital Rashba eld. The model Hamiltonian, including the t2ghopping terms, the atomic spin-orbit coupling, the inversion symmetry breaking term, and the external magnetic eld can be generally expressed as52,60{63 H=X k^D(k)yH(k)^D(k); (1) H(k) =H0+HSO+HZ+HM; (2) where ^Dy(k) =h cy yz"k;cy zx"k;cy xy"k;cy yz#k;cy zx#k;cy xy#ki is a vector whose components are associated with the elec- tron creation operators for a given spin (= [";#]), orbital(= [xy;yz;zx ]), and momentum kin the Brillouin zone. Then H0;HSO;HZandHMrepresent4 the kinetic energy, the spin-orbit, the inversion symmetry breaking and the Zeeman interaction term, respectively. In the spin-orbital basis, H0(k) is given by H0= ^"k ^0; (3) ^"k=0 @"yz0 0 0"zx0 0 0"xy1 A; "yz= 2t2(1coskx); "zx= 2t1(1coskx); "xy= 2t1coskx+
t; where ^0is the unit matrix in spin space and t1andt2 are the orbital dependent hopping amplitudes.
tde- notes the crystal eld potential as due to the symmetry lowering from cubic to tetragonal also related to inequiv- alent in-plane and out-of-plane transition metal-oxygen bond lengths. The symmetry reduction yields a level splitting between dxy-orbital and dyz=dzx-orbitals.HSO denotes the atomic l
sspin-orbit coupling, HSO=
SOh ^lx ^x+^ly ^y+^lz ^zi ;(4) with ^i(i=x;y;z ) being the Pauli matrix in spin space and^l(=x;yz) are the projections of the l= 2 angular momentum operator onto the t2gsubspace, i.e., ^lx=0 @0 0 0 0 0i 0i01 A; (5) ^ly=0 @0 0i 0 0 0 i0 01 A; (6) ^lz=0 @0i0 i0 0 0 0 01 A; (7) assumingfdyz;dzx;dxygas orbital basis. As mentioned above, the breaking of the mirror plane, due to the out-of-plane oset of the positions of the TM and O atoms, results into an inversion asymmetric orbital Rashba coupling that is described by the term HZ(k): HZ=  ^ly ^0sinkx : (8) This contribution gives an inter-orbital process, due to the broken inversion symmetry, that mixes dxyanddyz ordzx. Finally, we consider the eects of a magnetic eld lying into the plane of the 2D electron gas. The resulting Zeeman-type interaction is described by the Hamiltonian HM, which characterizes the coupling of the electron spin and orbital moments to the magnetic eld64: HM=Mxh ^lx ^0+^l0 ^xi +Myh ^ly ^0+^l0 ^yi + (9) Mzh ^lz ^0+^l0 ^zi ; (10)with ^l0being the unit matrix in the orbital space. We no- tice that the inclusion of the orbital coupling to the eld can be neglected because the atomic spin-orbit coupling, once the spin symmetry is broken by the Zeeman eld, also acts to orbital polarize the electronic conguration along the same direction56. Concerning the superconducting pairing, we assume that the interaction is local, with spin-singlet symmetry and active only for electrons sharing the same orbital symmetry56,65{68. Hence, the superconducting term HP can be expressed as HP=gX i;ni;"ni;#; (11) wheregis the pairing interaction, ni;=cy i;;ci;is the local spin-density operator for the polarization ,and theorbital, at a given position iin the square lattice. We then employ the usual decoupling scheme for the pair- ing term using a mean-eld approach for the spatial and orbital degrees of freedom: HP=X i;
i;h cy i;;"cy i;;#+ci;;#ci;;"i +gX i;D2 i;; with the pairing amplitude Di;=hci;;#ci;;"iand the order parameter
i;=gD i;are taken in a gauge such as to have a real amplitude. For the determination of the topological phase diagram and the spin-orbital character of the Majorana bound states we compute the spectrum and the corresponding eigenvectors of the Bogoliubov- De Gennes Hamiltonian both in the momentum and in real space by exploring dierent electronic regimes concerning the orbital lling and the spin-orbit strength. The numerical tight bind- ing hamiltonian is implemented by using KWANT69and solved with the help of NumPy routines70. In the follow- ing sections we set the parameters of the Hamiltonian in units of the main hopping term t1, specically as: the weaker hopping amplitude t2= 0:1, the orbital Rashba interaction = 0:2, the tetragonal crystal eld potential
t=0:5, the superconducting pairing
i;= 0:003, in- dependent of iand, and the atomic spin-orbit coupling
SOvarying from 0 :01 to 0:1. This set of parameters is representative of a physical regime with an electronic hierarchy of the energy scales such that
t> >
SO. III. TOPOLOGICAL PHASE DIAGRAM In this section we present the topological phase dia- gram as a function of the applied Zeeman magnetic eld and the strength of the spin-orbit coupling for three rep- resentative electron llings corresponding to the chemical potential crossing the bands nearby the point. For clar- ity and convenience we indicate as A, B and C each sector of two bands, associated with the -point Kramers dou- blet at zero eld, which occur when moving from lower to higher energies in the spectrum (Fig. 1(a),(b),(c)).5 A-block B-block C-block C-blocka) b) c) d) FIG. 2. Topological phase diagram evaluated by means of the Majorana polarization P(!= 0) (see main text for the denition) with a value of about 1 or 0 to signal the onset of a topological or trivial superconducting conguration, respectively. For convenience we indicate as A, B and C the physical cases for a given electron lling that refer to the two bands sector, associated with the -point Kramers doublet at zero eld. The sectors A, B and C occur when moving from lower to higher energies in the spectrum as depicted in Figs. 1(a),(b),(c). (a)-(c) topological phase diagram in the spin-orbit coupling/magnetic eld plane related to the bands for the A, B, C sectors assuming a magnetic eld Mxoriented along the nanowire direction. (d) topological phase diagrama for refers the bands belonging to the block C for an out-of-plane magnetic eld Mz. The chemical potential has been selected to be pinned at the energy lying in the middle of the split Kramers doublet for each block to distinctively follow the topological behavior of the corresponding orbital sectors. We vary the amplitude of the spin-orbit coupling
SOand the applied magnetic eld Mto search for the boundary separating the topological and trivial superconducting phase. The black dashed line schematically indicates the transition from a topological to trival superconducting phase as obtained by looking at the gap closing in the momentum space. The gap amplitude for the various orbitals is
= 0:003 in unit of t1. All the energies and electronic parameters are in units of t1. In particular, according to the selected range of param- eters for the spin-orbit coupling and crystal eld poten- tial, the block A refers to the bands with a dominant dxycharacter and sub-dominant ( dxz;dyz) contributions (Fig. 1(a)). The block B at intermediate energies cor- responds to bands arising from Kramers doublets with concomitant highest values of the spin and orbital com- ponents (Fig. 1(b)). Finally, the high energy bands set out the block C composed of states with dominant (dxz;dyz) character and a sub-dominant dxycontribution (Fig. 1(c)). The main purpose is to compare the topo- logical phase diagram for the various bands, with the aim to assess the role of the spin-orbital anisotropy and of the degree of spin-orbital entanglement. There are various approaches to identify a topological phase transition where Majorana bound states then occur at the edges of the quantum chain.7,12,71As discussed in Ref. [35], the Majorana polarization is one of them being a suitable indicator for detecting the topological phases and it can be considered as a sort of order parameter. In analogy with the case of a single band electronic model, one can dene the Majorana polarization for a multiband system as follows PL(R)(!) =X nPL(R) n(!)[(!en) +(!+en)]: (12) with PL(R) n(!) = 2N=2(N)X j=1(N=2+1)X ;un;j;;vn;j;;:(13) HereNis the size of the chain, enis a given eigen-energy of the BdG Hamiltonian and un;j;; andvn;j;; are A-blocka) b) c) d)FIG. 3. x- and zelectron components of the spin s((a) and (c)) and orbital l((b) and (d)) angular momentum eval- uated nearby the topological phase transition point ( MT x) on the lowest energy excited state in the trivial superconducting phase ( Mx< MT x) and for the MBS in the topological cong- uration ( Mx> MT x). The behavior refers to the topological phase diagram for the electronic states belonging to the low- est energy sector (block A) in the presence of a magnetic eld oriented along the nanowire. Solid and dashed lines refers to the MBS spin-orbital component at the two edges of the nanowire. The component collinear to the magnetic eld owes the same sign and amplitude at the two edges of the nanowire. The transverse spin and orbital components with respect to the applied eld have opposite sign at the two edges but equal amplitude. respectively the electronic and hole components of the eigenfunctions of the hamiltonian (more details of the derivation are reported in the Appendix A). The Majorana polarization has been then used to sin-6 gle out the topological superconducting phase in response to a change of the spin-orbit coupling, by considering dierent representative electron llings and an applied Zeeman eld along the directions perpendicular to the orbital Rashba eld. These eld orientations are those more favorable to yield a topological nontrivial state. Hence, in Fig. 2 we show the topological phase diagram for electron densities corresponding to a chemical poten- tial that uniquely crosses the energy bands in the blocks A, B and C (Fig. 1). A common aspect for the phase diagrams linked to the blocks A and C is that the trivial- topological boundary is weakly dependent on the ampli- tude of the spin-orbit coupling if the eld is applied along the easy magnetic axis ( xandz, respectively), with the critical threshold for the applied magnetic eld of the order of the superconducting gap. On the other hand, for the block B and C one needs to apply a magnetic eld which is signicantly larger than the superconduct- ing gap to induce a topological phase if the magnetic eld is applied along the hard magnetic axis. Additionally, the boundary line for the hard magnetic direction (i.e. xfor the blocks B and C), as expected, is more sensitive to the strength of the spin-orbit coupling. Although the phase diagram of the A and C sectors for a eld applied along the easy magnetic axis is substantially unchanged by a variation of the spin-orbit coupling, the character of the MBS in the topological phase is strongly dependent on the strength of spin-orbital interaction as we will discuss below in the Sect. IV. We notice that the bands A, as pointed out in Ref.56, show a spin Rashba-like transition with the in-plane mag- netic eld along the direction of the nanowire ( Mx) and a transition point approximately determined by MT xp
2+2 0, where0is the energy dierence between the chemical potential and the bottom of the band. Surpris- ingly, a similar behaviour with an out-of-plane magnetic eld is also observed for the band C although the inver- sion symmetry splitting deviates from the canonical spin Rashba prole. Also in this case we nd that the transi- tion point is essentially determined by MT zp
2+2 0. The validity of using the Majorana polarization to signal the trivial-to-topological phase transition is con- rmed by the correspondence of the critical values with those obtained by evaluating the position of the gap clos- ing in the reciprocal space (black dashed lines in Fig. 2). IV. SPIN-ORBITAL POLARIZATION OF MAJORANA BOUND STATES In this section we consider the behavior of the elec- tron spin-orbital polarization of the MBSs by focusing on the orientation, the spatial prole and the changeover across the phase transition going from in-gap fermionic states to Majorana edge modes. Following the schematic structure of the energy bands, we determine the spin- orbital polarization of the MBSs arising from each band by varying the strength of the spin-orbit coupling. There B-blockFIG. 4. x- and zelectron components of the spin s((a) and (c)) and orbital l((b) and (d)) angular momentum eval- uated across the topological phase transition point ( MT x) for the lowest energy excited state in the trivial superconducting phase (i.e. for Mx< MT x) and for the MBS in the topological side (i.e. for Mx> MT x). The behavior refers to the topo- logical phase diagram due to a magnetic eld oriented along the nanowire and considering the electronic states of the in- termediate energy sector for the normal state spectrum (i.e. bands of the B-block). Solid and dashed lines refers to the MBS spin-orbital component at the two edges of the nanowire. The amplitude is the same for the two MBS localized at the edges while concerning the relative orientation, the compo- nent collinear to the magnetic eld are parallel while those transverse are anti-aligned. are various questions we aim to address. The rst issue to account for is about the dependence of the spin-orbital polarization of MBSs on the strength of the spin-orbital coupling, the orientation of the magnetic eld, the spin- orbital anisotropy and on the character of the orbitals which are involved in the pairing at the Fermi level. Ad- ditionally, we aim to provide distinctive ngerprints re- garding the spin-orbital orientation and the spatial tex- ture of the MBSs at the edge of the superconductor. An- other relevant aspect in the problem upon examination is to assess whether the spin-orbital polarization of the electronic states at the Fermi level in the normal state is imprinting the behavior of the spin-orbital content of the MBS. Due to the intricate spin-orbital character of the electronic states, we expect that the components of the spin-orbital polarization of the MBS are strongly sus- ceptible to the variation of the spin-orbit coupling or the crystal eld amplitude in a way that can be markedly orbital dependent. Along this line, the outcome of the analysis unveils striking behaviors of the components of the spin-polarization that are collinear or transverse to the applied magnetic eld. Let us start by considering the topological phase due to an applied magnetic eld along the x-direction for an electron pairing in the band belonging to the block A with dominant xyorbital character (Fig. 3). We observe that the spin-orbital polarization is nonvanishing only for thexandzspatially averaged components. However, the response to a variation of the spin-orbit coupling reveals7 C-block FIG. 5. x- and zelectron components of the spin s((a) and (c)) and orbital l((b) and (d)) angular momentum evaluated across the topological phase transition point ( MT x) for the lowest energy excited state in the trivial region ( Mx< MT x) and for the MBS in the topological side ( Mx> MT x). The behavior refers to the topological phase diagram with an ap- plied magnetic eld along the nanowire and for the electronic states belonging to the intermediate energy sector (i.e. block C). Solid and dashed lines refers to the MBS spin-orbital com- ponent at the two edges of the nanowire. C-block FIG. 6. x- and zelectron components of the spin s((a) and (c)) and orbital l((b) and (d)) angular momentum evaluated nearby the topological phase transition point ( MT z) for the lowest energy excited state in the trivial region ( Mz< MT z) and for the MBS in the topological conguration ( Mz> MT z). The behavior refers to the topological phase diagram with an out-of-plane eld oriented along z-direction and for the electronic states belonging to the intermediate energy sector (i.e. block C). Solid and dashed lines refers to the MBS spin- orbital component at the two edges of the nanowire. a remarkable behavior when considering the collinear and transverse to the magnetic eld components for the MBS. Indeed, while the sxspin density gets reduced in am- plitude when the spin-orbit coupling increases, the sz value is essentially unchanged for any spin-orbit value. Furthermore, the szspin density has opposite sign com- ponents on the two sides of the nanowire while the sx component has the same sign and value. The fact thattheszcomponent has a constant amplitude means that any small variation in the spin-orbit coupling leads to a change in the orientation of the electron spin moment of the MBS at each edge of the nanowire. When considering the electron orbital component of the MBS, the resulting outcome is completely uncorre- lated to that of the spin density. The lxcomponent turns out to be less variable with respect to a change in the spin-orbit coupling unveiling a subtle non monotonous behavior. On the contrary, the lzprojection of the or- bital angular moment grows in amplitude with the in- crease of the spin-orbit strength. As for the spin com- ponent, the orbital part has an opposite sign at the two edges for the z-orientation while it is collinear for the x-direction along the applied magnetic eld. It is worth pointing out that, although the spin and orbital polar- izations are substantially locked at the Fermi level in the normal state conguration through the combination of the orbital Rashba coupling and the spin-orbit interac- tion (see Appendix B for details), the character of the MBSs unveils a completely opposite behavior. The spin and orbital orientations are essentially misaligned and the misalignment is nontrivially tuned by a change in the strength of the spin-orbital coupling. Another rel- evant observation of our analysis is that, although the electronic states for the sector A and the resulting topo- logical behavior might be well described by an eective single band spin-Rashba model, the spin-orbital content of the MBS unveils an intricate interplay of the quantum spin and orbital constituents. Moving to the topological phase for the states in the B sector, we start by observing that the pairing involves more than one Fermi point and that, due to the mag- netic anisotropy, the amplitude of the applied eld along thex-axis to reach the topological transition has to be larger than that employed for the congurations in the A sector. These elements completely alter the behav- ior of thesxandlxcomponents resulting into a smooth changeover across the topological transition (Fig. 4). For the transverse projection ( z) one has an opposite behav- ior if compared to the MBSs arising from the bands in the sector A. Indeed, the lzcomponent is essentially un- aected by the spin-orbit coupling while the szdensity exhibits an increase in the amplitude. This is a remark- able reconstruction of the MBS spin content with an un- expected enhancement of the spin density for a stronger value of the spin-orbit coupling. A similar behavior is also obtained when considering the MBSs arising from the topological conguration in the high-energy C block. Again, the orbital component transverse to the applied eld is independent of the spin-orbit coupling, while the spin density has a monotonous prole. We point out that, for the bands of the sector C hav- ing an easy orbital axis along the transverse z-direction, the spin and orbital contents of the MBSs in Fig. 6 are substantially analogous to the ones of the A sector, with the transverse spin component being slightly dependent on the spin-orbit coupling. We attribute this behavior8 A-block A-blockB-block B-block C-blockC-blocka)b) c) d) e) f) FIG. 7. Spatial prole of the x- and z- component of the spin density for the MBSs arising from the pairing of electronic states belonging to the sector A ((a) and (d)), B ((b) and (e)), C ((c) and (f)) at a given amplitude of the Zeeman magnetic eld along the x-direction corresponding to the topological side of the phase diagram. The computation has been performed for a quantum chain with 4000 sites. to the dierent character of the quantum congurations involved in the block C as compared to the A-sector. In- deed, in the block C the two quasi-degenerate orbitals (xz,yz) with larger amplitude in the electronic congu- ration are spin-orbit active for the lzangular momentum component. On the other hand, for the block A the main component of the electronic state is associated only to thexyorbital. Another remark is that for the bands of the sector B and C having an easy orbital axis along the transversez-direction, the electron orbital content of the MBSs is substantially constant and robust to changes of the spin-orbital coupling strength. Finally, we discuss the spatial pattern of the electron spin density of the MBS at the edges of the nanowire (Fig. 7). As suggested by the amplitude of the Majo- rana polarization, the MBSs wavefunctions are localized at the two edges of the nanowire with a characteristic decaying length of the order of few hundred inter-atomic distances. As expected, the spin and orbital polariza- tions of the MBSs are non-vanishing only close to the edge where the MBS has a maximal amplitude. We nd that the strength of the spin component is comparable for all the band sectors but it can exhibit a sign change as a function of the position thus leading to a non-trivial evolution of the spin orientation in the xzplane. For in- stance, for the A-block (Fig. 7(a),(d)) the sxcomponent changes sign moving away from the edge while the sz component is always positive. Hence, although the spin orientation is pinned in the xzplane, the spatial pattern exhibits a texture with a signicant rotation of the spin polarization. This behavior is observed only for the MBS in the A-sector while for the B and C-type MBSs we have a less variable modication of the spin-orientation as a function of the position in the nanowire. It is also worth noticing that some oscillations appear in the real space prole of nonzero components of sandl. These oscilla-tions re ect the characteristic length scales of the Fermi wave-vectors of the normal state congurations. Indeed, the MBSs arising from the A sector show only one har- monic, while for B and C block the MBSs display multiple components for the amplitude modulation in the spatial prole ofsx;zandlx;z, that can be directly associated to the multiple Fermi points occurring at the corresponding electron lling. This is also conrmed by the analysis of the spectrum in the normal state as discussed in the Appendix B. V. DISCUSSION AND CONCLUSIONS We have unveiled the spin-orbital character of MBSs occurring in magnetic-eld driven topological supercon- ductors where the electron spin and angular momen- tum are strongly entangled due to the interplay of atomic spin-orbit interaction and inversion asymmetric couplings (e.g. spin and orbital Rashba). Taking the paradigmatic example of an electronic system in one- dimension with anisotropic d-orbitals (i.e. dxy,dzx,dyz), which is of direct impact for multi-orbitals superconduc- tivity at oxides interface or surface, we nd that the resulting MBSs display a rich variety of striking spin- orbital hallmarks. A general nding refers to the orien- tation of the MBS spin-orbital polarization. It is planar and lies in the plane set by the direction of the applied magnetic eld and the direction that is transverse to the magnetic and orbital-Rashba elds. While the orienta- tion's plane is common for the spin and angular momen- tum of the MBS, the spin and orbital polarizations are typically misaligned with an angle that is sensitive to a variation of the electronic interactions. This behavior is in stark contrast with that of the spin-orbital congu- rations in the normal state, for instance at a given mo-9 mentum in the Brillouin zone, where the spin and orbital components are essentially collinear. Such observation can be potentially relevant for distinguishing the occur- rence of MBSs from conventional in-gap bound states at the edge of the superconductor as induced by inhomo- geneities or extrinsic eects. The presented analysis allowed us to understand the fundamental interrelation among the spin-orbital polar- ization of the MBS, the strength of the coupling between the spin and orbital degrees of freedom and the mag- netic and orbital anisotropy of the electronic states that contribute to the formation of the topological pairing. For this aim, we have essentially employed the atomic spin-orbit interaction as a knob to modify the degree of the spin-orbital entanglement in order to assess the con- sequences on the MBS. In this respect, the analysis pro- vides direct access to the spin-orbital susceptibility of the MBS with respect to a variation of the electronic param- eters. We nd that when the magnetic eld is applied along an easy axis for the corresponding electronic states at the Fermi level, the transverse to the magnetic eld MBS spin-polarization is more resilient to variation of the spin-orbit coupling. We qualitatively attribute this behavior to the fact that the induced transverse compo- nent is uniquely tied to the structure of the MBS, since there are no corresponding contributions in the normal state, and that being a hard axis it is weakly activated by a change in the spin-orbit coupling. A completely dif- ferent behavior is achieved when the topological phase is obtained by applying a magnetic eld along a hard axis direction for the paired electrons. In that case, we nd that the transverse orbital component to the mag- netic eld gets substantially unaected by a modication of the spin-orbit coupling. We argue that this strongly asymmetric behavior can be attributed to the inequiva- lent orbital susceptibility of the electronic states at the Fermi level. Indeed, the states with dominant xyorbital character the easy axis is in plane, while for those with mixedxz,yzorbital congurations the easy axis is along the out-of-plane direction. However, the energy sepa- ration of the orbitals due to the crystal eld potential makes the out-of-plane direction easier to activate than the in-plane one. Hence, there is a clear separation in the behavior of the spin and orbital degrees of freedom of the MBS and this outcome is expected to occur for spin- orbital correlated superconductors in a regime where the spin-orbit interaction competes with the structural cou- plings. Although the analysis focused on the role of the spin- orbit interaction we argue that similar response can also occur when other electronic parameters are varied es- pecially if considering the crystal eld potential associ- ated to the structural distortions. Indeed, the tetragonal splitting of the d-orbitals typically tends to quench the orbital angular momentum and thus competes with the spin-orbit coupling by indirectly reducing its eective- ness. In this context, we point out that a modication of the structural conguration through local strains or byapplying electric eld would manifest into striking eects in terms of reorientation of the spin-orbital polarization of the MBS. Remarkably, the rearrangement of the spin- orbital MBS is dierent for the spin and orbital compo- nents and it also manifests with a complete restructuring of the spatially resolved textures. From an experimental point of view, we argue that the identied signatures of the MBSs can be accessed in spin-selective transport probes and would manifest in a strong anisotropic response of the conductance. Since the averaged spin-polarization of the MBSs can be sen- sitive to structural changes, we also expect a signicant strain driven magnetic anisotropy to occur in the zero bias tunneling conductance. A similar response would be detected in spin resolved STM experiments where the atomic prole of the MBS spin polarization can be di- rectly accessed. Our prediction of spatial dependent ori- entation of the spin-polarization of the MBS with a gradi- ent that is sensitive to the orbital character of the paired electrons can be employed to assess the nature of the topological phases in multi-orbital superconductors. Fi- nally, concerning the orbital polarization of the MBS, we point out that it is much challenging to design an experi- ment to directly access the orbital angular momentum of the MBS. While the anisotropy of the d-orbitals would naturally manifest into a characteristic angular depen- dence of the tunneling or STM conductance, in order to pin point the orbital polarization one would device a spe- cic lter of orbital-selective angular momentum. One possible way out is to design a tunnel barrier with tun- able inversion asymmetric interactions (e.g. Rashba and Dresselhaus) that, due to the orbital momentum locking, can allow the injected control of electrons with selected orbital polarization along a given direction. This setup denitely requires a high degree of control of interface and materials engineering. Advancements along this di- rection might open the path to a fully spin-orbital spec- troscopic probe of in-gap modes of topological supercon- ductors and contribute to single-out distinctive signature for the experimental detection of MBS with strong inter- play of spin-orbital degrees of freedom. ACKNOWLEDGMENTS We acknowledge discussions with Anton Akhmerov, Marco Salluzzo, Francesco Romeo and Alfonso Maiel- laro. This work was supported by the project Quan- tox of QuantERA-NET Cofund in Quantum Technolo- gies, implemented within the EU-H2020 Programme and the project \Two-dimensional Oxides Platform for SPIN- orbitronics nanotechnology (TOPSPIN)" funded by the MIUR-PRIN Bando 2017 - grant 20177SL7HC.10 Appendix A: Majorana polarization The real space creation and annihilation operators can be expressed in the basis of single particle eigenfunctions of the BdG Hamiltonian in the following form: ^ j;;(t) =X n un;j;; ^cn;;(t) +vn;j;; ^cy n;;(t)
(A1) ^ y j;;(t) =X n u n;j;;^cyn;;(t) +v n;j;; ^cn;;(t) ; (A2) whereun;j;; andvn;j;; are, respectively, the electronic and hole component of the n-th eigenfunction in the or- bitaland spin, calculated at the position j. The most generic Majorana operators can be written as ^ a j;;(t)=e{'^ j;;(t) +e{'^ y j;;(t) (A3) ^ b j;;(t)={(e{'^ j;;(t)e{'^ y j;;(t)):(A4) Majorana polarization has been introduced in72,73in order to detect the topological phases35,74{77. It can be interpreted as the dierence of the probabilities of having a Majorana modes ^ aand^ b, at position j, in the orbital, with spin and energy !. In the language of Green's functions, it is related to the two local spectral functions: Pj;;(!) =Aa j;;(!)Ab j;;(!) (A5) with Aa;b j;;(!)= 1 Im {Z1 1e{!t(t)Dn ^ a;b j;;(t);^ a;b j;;(0)oE : We can write the anti-commutator of Eq. A6 using Eq.s A3 and A4, obtaining:  ^ a j;;(t);^ a j;;(0) =n ^ j;;(t);^ y j;;(0)o + n ^ y j;;(t);^ j;;(0)o +e2{'n ^ j;;(t);^ j;;(0)o + e2{'n ^ y j;;(t);^ y j;;(0)o and  ^ b j;;(t);^ b j;;(0) =n ^ j;;(t);^ y j;;(0)o + n ^ y j;;(t);^ j;;(0)o + e2{'n ^ j;;(t);^ j;;(0)o + e2{'n ^ y j;;(t);^ y j;;(0)o : The two non-anomalous terms are wiped out by the dierence in Eq. A5, which results in Pj;;(!) =1 Im {Z1 12e{!t(t) e2{'Dn ^ j;;(t);^ j;;(0)oE +e2{'Dn ^ y j;;(t);^ y j;;(0)oE : (A6) Both anti-commutators can be rewritten in terms of single particle eigenfunction of the hamiltonian: Dn ^ j;;(t);^ j;;(0)oE =X nun;j;;vn;j;; e{ent+e{ent
(A7)and Dn ^ y j;;(t);^ y j;;(0)oE =X nu n;j;;v n;j;; e{ent+e{ent
: (A8) By performing the integration, we can write: Pj;;(!) =2 Im(X ne2{'un;j;;vn;j;; P1 !en +P1 !+en {((!en) +(!+en)) + e2{'u n;j;;v n;j;; P1 !en +P1 !+en {((!en) +(!+en)) (A9) and therefore Pj;;(!) =2 Im[X n;;Re e2{'un;j;;vn;j;; P1 !en +P1 !+en {((!en) +(!+en)) ]:11 Then, by taking the imaginary part, one obtains Pj;;(!) = 2X nRe e2{'un;j;;vn;j;; [(!en) +(!+en)]: (A10) The integral of MP in the whole Hilbert space of a closed system is zero77, but if two Majorana states are spatially separated, the integral in each separated region is equalto 1. For this reason, in the main text we have calculated the integral of MP in the left and right half of the wire, by summing on spin and orbital degrees of freedom PL(R)(!) =X n2 Re2 4N=2(N)X j=1(N=2+1)X ;e2{'un;j;;vn;j;;3 5[(!en) +(!+en)] =X nPL(R) n(!)[(!en) +(!+en)]: (A11) The quantity Pn(!) is the real part of a complex num- ber whose phase depends on the particular choice of the global wavefunction phase factor. Therefore, the only physically relevant quantity, for each eigenstate labelled byn, is the modulus of Pn(!), resulting in the nal def- inition: PL(R)(!) =X nPL(R) n(!)[(!en) +(!+en)]: (A12) with PL(R) n(!) = 2N=2(N)X j=1(N=2+1)X ;un;j;;vn;j;;:(A13) Appendix B: Spin-orbital polarization at the Fermi level in the normal phase In this Appendix we investigate the spin-orbital char- acter of the electronic states at the Fermi level in the normal phase of the model described by the Hamilto- nian in Eq. 1. We solve such a model by imposing peri- odic boundary conditions along the wire direction x. The emerging electronic band structure is made up by three blocks with inequivalent orbital character, A, B and C, each forming a Kramers doublet at the point, as shown in Fig. 1(a),(b),(c). Depending on the choice of the electron lling, one or several bands cut the Fermi level, thus leading to the presence of multiple Fermi points KF. Here we will focus on the representative case correspond- ing to the chemical potential crossing the bands of the B block nearby the point. In such a case, the Fermi points occur at characteristic vectors dened as Ks, KFb1andKFb2, which arise from the the lowest of the eld split bands of the B sector, and from the highest and lowest bands of of the A sector, respectively. Thiscircumstance is graphically depicted in Fig.8 i). Figure 8 shows a comprehensive overview of the evolu- tion of the average spin polarization in the ( x;z) plane at the dierent KFvectors, upon variation of the spin-orbit coupling
SOand of the Zeeman eld Mx. We recall that the intricate spin-orbital entanglement of the elec- tronic states yields a strong anisotropy for the magnetic response of the bands under consideration. In particular, in the adopted regime of parameters for the spin-orbit coupling and crystal eld potential, the Kramers doublet of the A block is marked by a nonvanishing average spin density both along xandzdirections, while the B states are characterized by the highest values of the spin com- ponents along z. This implies the existence of hard/easy spin directions, specically xis easy from the bands of block A while it is hard for the bands of block B. Thesxpolarization, i.e. collinear to the applied eld is shown in panels a), b) and c) in Fig.8. From the inspec- tion of the gure, we observe that the spin component along the direction of the eld has always a monotonous evolution with
SOandMx. It is evident that for all the states at each KF, thesxcomponent grows in ab- solute value with the Zeeman eld and is strongly sensi- tive to the variation of the spin-orbit strength, reducing in amplitude with increasing
SO. Beyond such simi- lar monotonous behavior, we point out some important dierences which markedly depend on the specic spin- orbital sector. We notice that the spin polarization is more susceptible to the variation of the spin-orbit cou- pling for the B state at KFs. Moreover, we observe that the states of the A block are characterized by an opposite sign of the spin polarization, being parallel and antipar- allel to the applied eld. Theszcomponent, which is orthogonal to the Zeeman eld, is zero by symmetry. In our calculations, we con- sider a small symmetry breaking eld along this direction and observe that the value of szis essentially independent on the spin-orbit coupling, as shown in Fig.8. g) and h).12 FIG. 8. Density plots representing the dependence upon the spin-orbit coupling
SOand the Zeeman eld Mxof the average spin polarization at the Fermi level in the normal state, for a choice of the chemical potential which is depicted in panel i). In panels d), e) and f) the sign of the product of the components sxlxis reported for the three distinct Fermi points. The spin density arising from the bands of the A block is always vanishing. On the other hand, in the B block we distinguish two regimes: the polarization is maximum in a small window below a threshold of the magnetic eld which is almost independent on
SO, while its is vanish- ing above this window. Finally, in panels d), e) and f) we display the sign of the product ( sxlx), which is representative of the rela- tive orientation of the components of the spin and or- bital angular momenta collinear to the magnetic eld. It is evident that the sign is uniform in the parameterspace, but shows unalike behavior for the distinct states at the Fermi level. At KFsthe spin and angular momen- tum turn out to be parallel, while they have an opposite sign in the case of the states belonging to the A block. Such behavior re ects the dierent orbital susceptibiliy of the states belonging to the A and B sectors; lxis un- quenched and substantial, also being antiparallel to sxat the point, within the Kramers doublet of the A block. For the states of the B block, lxis identically zero at the point, and the eect of the eld is to induce a small nonvanishing component, which is parallel to the spin polarization. 1M. Sigrist and K. Ueda, Rev. Mod. 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1906.08566v1.Effects_of_spin_orbit_coupling_on_the_neutron_spin_resonance_in_iron_based_superconductors.pdf

Effects of spin-orbit coupling on the neutron spin resonance in iron-based superconductors Daniel D. Scherer1and Brian M. Andersen1 1Niels Bohr Institute, University of Copenhagen, Lyngbyvej 2, DK-2100 Copenhagen, Denmark The so-called neutron spin resonance consists of a prominent enhancement of the magnetic response at a particular energy and momentum transfer upon entering the superconducting state of unconventional super- conductors. In the case of iron-based superconductors, the neutron resonance has been extensively studied experimentally, and a peculiar spin-space anisotropy has been identified by polarized inelastic neutron scatter- ing experiments. Here we perform a theoretical study of the energy- and spin-resolved magnetic susceptibility in the superconducting state with s+-wave order parameter, relevant to iron-pnictide and iron-chalcogenide superconductors. Our model is based on a realistic bandstructure including spin-orbit coupling with electronic Hubbard-Hund interactions included at the RPA level. Spin-orbit coupling is taken into account both in the generation of spin-fluctuation mediated pairing, as well as the numerical computation of the spin susceptibility in the superconducting state. We find that spin-orbit coupling and superconductivity in conjunction can repro- duce the salient experimentally observed features of the magnetic anisotropy of the neutron resonance. This includes the possibility of a double resonance, the tendency for a c-axis polarized resonance, and the existence of enhanced magnetic anisotropy upon entering the superconducting phase. I. INTRODUCTION For unconventional superconductors, i.e., superconductors supported by non-phonon mediated Cooper pairing, the role of magnetic fluctuations has been extensively discussed in the literature1,2. From an empirical perspective this is mo- tivated by the fact that the superconducting phase most of- ten exists in close proximity to a magnetically ordered state, and significant magnetic fluctuations remain in, and poten- tially even generete, superconductivity. From a theoretical perspective, spin-fluctuation mediated pairing has been thor- oughly studied and applied to various candidate systems in- cluding cuprates, iron-based superconductors, Sr 2RuO 4, or- ganic Bechgaard salts, and heavy fermion materials1,2. This line of research goes back to the seminal theoretical work by Berk and Schrieffer in 19663, and its extensions found in Refs. 4–6. An important experimental fingerprint of the coupling be- tween magnetic excitations and superconductivity is given by the so-called neutron spin resonance7. Originally discovered in the hole-doped cuprate material YBa 2Cu3O6+x8, the neu- tron resonance manifests itself as an enhanced scattering cross section in the superconducting phase at a material-specific momentum and at an energy transfer of the order of the su- perconducting gap scale. While the origin of the neutron resonance has been intensely discussed,7its most natural ex- planation is given in terms of an interaction-driven collective spin-1 excitation allowed by the gapped particle-hole excita- tions of the superconductor. Importantly, within this scenario, a sign-changing order parameter is necessary to expose the resonance. Therefore, the existence of a neutron resonance is often taken as evidence for unconventional pairing. We stress that within this picture, the neutron resonance is not evidence for pairing caused by magnetic fluctuations per se , but rather a feed-back effect of superconductivity on the magnetic exci- tations. For iron-based superconductors (FeSCs), the neutron reso- nance was detected early on in K-doped BaFe 2As29. Subse- quently the resonance and its associated spin-gap were alsofound in the superconducting state of other FeSCs, and es- tablished to reside at the antiferromagnetic wavevector Q= (;0). For a detailed overview of the neutron resonance stud- ies of FeSCs we refer to the review articles in Refs. 10–12. As one of the hallmarks of sign-changing gap functions, the existence of the neutron resonance was considered strong evi- dence fors+-wave pairing with sign-reversed superconduct- ing gaps on Fermi pockets connected by Q= (;0)13–17. This interpretation, however, was challenged, and other scenarios for the emergence of the neutron resonance feature were pro- posed18,19. This motivated many additional studies into the properties of the neutron spin resonance including the detailed spin anisotropy of the neutron scattering resonance11,12. Even in a non-magnetic phase significant spin anisotropy of the magnetic fluctuations exists, and can be as- cribed to sizable spin-orbit coupling (SOC) in the iron- based superconducting materials20,21. In the supercon- ducting state, the low-energy magnetic excitations, in- cluding the neutron resonance, tend to be c-axis polar- ized as determined by spin-flip neutron scattering measure- ments11. This is reported for a large series of compounds including LiFeAs, Ba 1xKxFe2As2, BaFe 2(As1xPx)2, BaFe 2xCoxAs2, BaFe 2xNixAs2, Fe(Se,Te), FeSe, and Sr1xNaxFe2As2.22–35The general property of leading c-axis polarized low-energy susceptibility is opposite to e.g. un- doped BaFe 2As2in the normal phase where the a-axis polar- ization dominates, in agreement with magnetic moments be- ing aligned along the aaxis in the spin-density wave (SDW) phase at low temperatures. The doping-induced crossover to a dominant c-axis oriented susceptibility is also in agree- ment with out-of-plane oriented moments observed in the C4- symmetric double-Q phase of Na-doped BaFe 2As236,37. Thus, broadly summarized, doping appears to catalyze a transition from in-plane to out-of-plane dominated low-energy magnetic fluctuations in the normal state, which are further enhanced in the superconducting state. The spin anisotropy tends to vanish in the overdoped regime.22,38 Before proceeding we point out two additional aspects of the neutron resonance of FeSCs: 1) The possibility of a double resonance, and 2) the enhancement of spin anisotropy uponarXiv:1906.08566v1 [cond-mat.supr-con] 20 Jun 20192 entering the superconducting state. The double resonance is seen, for example, in Co-doped BaFe 2As2and in Co-doped NaFeAs, as two separate neutron resonance peaks at low en- ergies26,39. The magnetic anisotropy of the double resonance was found to be much more pronounced for the lowest en- ergy peak, which was strongly c-axis dominated, as com- pared to the higher energy peak, which was nearly isotropic in spin space26,29. Regarding the second point above, it was found, for example, in optimally doped BaFe 2(As1xPx)2 that the normal state appears isotropic in spin space whereas upon entering the superconducting state significant spin-space anisotropy was induced33. This is similar to BaFe 2xNixAs2 where the spin anisotropy was also observed to become fur- ther enhanced upon entering the superconducting state.27. The preferred fluctuation direction of the low-energy spin excitations is of great interest because of the potential impor- tance of these fluctuations in determining the superconducting state. For an explicit theoretical demonstration where SOC and the associated spin anisotropy is strong enough to push the leading superconducting instability from a spin-singlet state into a spin-triplet phase, we refer to a recent theoretical study relevant to Sr 2RuO 440. For the case of FeSCs, a recent ex- perimental study of Sr 1xNaxFe2As2proposed that the pre- ferred out-of-plane c-axis polarized low-energy fluctuations may also be important for the competition between supercon- ductivity and ordered in-plane magnetic SDW phases35. Thus, these works highlight the importance of understanding the po- larization of the magnetic fluctuations and their role in deter- mining the properties of the superconducting state. We note that understanding the effects of SOC in FeSCs has been also recently emphasized in terms of possible topological phases existing in these materials41,42. From a theoretical perspective any detailed understanding of the behavior of the magnetic anisotropy of the neutron res- onance is lacking at present. The rather complex behavior of the observed anisotropy has been argued to be evidence for the importance of orbital ordering tendencies of FeSCs27,43. In addition, since the ordered magnetic state exhibits c-axis dominated low-energy spin fluctuations, it was been suggested that the role of antiferromagnetic SDW order may be impor- tant44. The presence of SDW order could also explain the ex- istence of a double resonance44. The existence of the double resonance and the salient features of the magnetic anisotropy of the resonance do not, however, seem to be tied to the ex- istence of static magnetic order. Finally we note a theoretical study of the magnetic neutron resonance comparing transverse and longitudinal fluctuations between s++ands+supercon- ductivity, within a simplified three-band model including only part of the SOC in the bandstructure45. Here, we perform a theoretical study of the magnetic exci- tations in the superconducting phase from an itinerant weak- coupling RPA perspective, i.e., we apply a realistic ten-band model including SOC relevant to iron-based materials, and electron interactions incorporated via the multi-orbital RPA framework. In our previous paper20, we focussed on the para- magnetic phase and found overall agreement between the the- oretical results and the experimental data in terms of material- variability, doping-, temperature-, and energy-dependence ofthe polarization of the low-energy magnetic fluctuations. This remarkable variability of the spin anisotropy, despite a con- stant atomic SOC, is a natural consequence of itinerant sys- tems close to nesting conditions as explained in detail in Ref. 20. Here, extending this procedure to the supercon- ducting state, we find that the main experimental findings of the magnetic anisotropy summarized above are naturally ex- plained within our theoretical framework. In particular, the emergence of a double resonance, the tendency for a c-axis dominated neutron resonance, and the possibility of enhanced spin anisotropy in the superconducting phase all follow from properly including SOC in the bandstructure and the super- conducting pairing. We stress that SOC is included both in the bandstructure and in the pairing kernel generated from spin- fluctuation mediated superconductivity. The paper is structured as follows. In Sec. II, we intro- duce the normal-state Hamiltonian for FeSCs and discuss de- tails pertaining to the inclusion of SOC. While Sec. III only briefly describes the approximations we employed in solving the quantum many-body problem for spin-fluctuation induced pairing in the presence of SOC and determination of the neu- tron scattering amplitude of a spin-orbit coupled supercon- ductor, elaborations on these topics can be found in Ref. 46 and the supplementary material, Sec. S1- S4, respectively. We then move to the presentation of our numerical results in Sec. IV and conclude with a discussion and possible future directions in Sec. V. II. MODEL In the following, we briefly define the Hamiltonian de- scribing the normal metallic state of the spin-orbit coupled electronic system. For details regarding the Bogoliubov-de- Gennes (BdG) Hamiltonian describing the superconducting system, we refer the reader to Sec. S1. We model the electronic degrees of freedom of the 3dshell of iron relevant for the low-energy properties of the FeSC ma- terials by a multiorbital Hubbard Hamiltonian H=H0 0N+HSOC+Hint. Here,H0is the hopping Hamiltonian encoding both the electronic bandstructure in the absence of SOC and the orbital character of single-particle states and the electronic filling is fixed by the chemical potential 0, with Ndenoting the total particle number operator. Defining the fermionic operators cy li,clito create and destroy, respec- tively, an electron on sublattice lat siteiin orbitalwith spin polarization ,H0can be written as H0=X X l;l0;i;jX ;cy lit li;l0jcl0j; (1) where hopping matrix elements t li;l0jare material specific. The indices l;l02 fA;Bgdenote the 2-Fe sublattices, corresponding to the two inequivalent Fe-sites in the 2-Fe unit cell due to the pnictogen(Pn)/chalcogen(Ch) staggering about the FePn/FeCh plane, and the indices i;jrun over the unit cells of the square lattice. On sublattice l=A, the indices ; specifying the 3d-Fe orbitals run over the3 setfdxz;dyz;dx2y2;dxy;d3z2r2g, while on sublattice l= B, we pick the gauge fdxz;dyz;dx2y2;dxy;d3z2r2g. We note, that while without SOC, a 1-Fe description (with one iron site and correspondingly 5-orbitals per unit cell) is possible, through the introduction of SOC the 2-Fe and 1-Fe descriptions are no longer unitarily equivalent. In the above-defined phase-staggered basis, the site-local SOC- Hamiltonian becomes HSOC= 2X l;iX ;X ;0cy lil]
0cli0;(2) withthe vector of Pauli matrices acting in spin space. The components of the angular momentum operator [Ll]sat- isfy[Lx;y A]=[Lx;y B],[Lz A]= [Lz B]. The reason for the breakdown of the unitary equivalence of 1-Fe and 2- Fe descriptions can be found in the properties of the angu- lar momentum operator. The unitary transformation, which in the absence of SOC block-diagonalizes Eq. (1) (with two blocks, where each gives rise to a 5-orbital Hamiltonian in distinct regions of momentum space), does not produce a block-diagonal angular momentum operator, i.e., Eq. (2) is not block-diagonalized by the same transformation. Thus, the proper inclusion of SOC necessitates either the use of a 2- sublattice representation (2-Fe description) or the introduction of a (momentum-space) non-local angular momentum opera- tor (1-Fe description). In this work, we will use hopping pa- rameterst li;l0jas specified in Ref. 47. The effect of SOC on the Fermi surface and the orbital composition of Fermi sur- face states for two different chemical potentials is shown in Fig. 1, where we model doping by a rigid band shift. Without SOC, the Fermi surfaces feature three hole pockets around the point and two electron pockets around the Mpoint. The inner and outer hole pockets are mostly composed of dxzand dyzorbitals, while the middle hole pocket is dominated by thedxyorbital. The approximate nesting of hole and elec- tron pockets with nesting vectors Q1andQ2leads to strong spin fluctuations at these wavevectors, which in an itinerant weak coupling picture eventually gives rise to an SDW in- stability and the condensation of SDW order. In terms of spin-fluctuation mediated superconductivity, these spin fluctu- ations will naturally give rise to a so-called s+pairing state, where, due to the spin-mediated, repulsive interpocket inter- action, the gap features a -phase between hole and electron pockets. Typically, the superconducting instability emerges upon either hole or electron doping, when the SDW order- ing tendencies are sufficiently suppressed. For strongly doped systems, the Fermi surface topology will eventually change due to the vanishing of electron or hole pockets15. In these more extreme cases, different pairing states than the s+can be realized due to the concomitant changes in the momentum structure of the spin-fluctuation mediated interaction. Return- ing to the influence of SOC on the electronic states, it is obvi- ous from Figs. 1(b),(c) and 1(e),(f) that it leads to a splitting of states at the 2-Fe BZ boundary. Correspondingly, it tends to mix and equalize the orbital character of the electron pockets. The same effect seems to occur for the hole pockets. We note here that the superconducting gap scale and the SOC-strength are treated as free parameters in our model, FIG. 1. Normal state Fermi surfaces in the 1-Fe BZ ( (Kx;Ky)de- notes momenta in the 1-Fe BZ coordinate system) extracted from the orbitally resolved contributions to the electronic spectral func- tion with (a)-(c) 0= 0eV and (d)-(e) =45meV for increasing . The 2-Fe BZ is indicated by the dashed square. The colors refer todxz(red),dyz(green) anddxy(blue) orbital contributions. SOC leads to a splitting of the states at the 2-Fe BZ boundary. and have been varied in a quite generous parameter interval in order to obtain a complete picture of the possible SOC- induced spectral features of the neutron resonance mode of the superconducting system as emerging from the sign-changing s+state. While the use of as large as 100meV might strictly speaking not be realistic, it is worth emphasizing that the bandstructure entering our calculations has a bandwidth of about 5eV . While for 1111 FeSCs the bandwidth renormal- ization of DFT-LDA bands due to correlation effects is rather weak, these renormalizations can reduce the bandwidth by a factor 2-3 for 122 FeSCs. As it presently seems unclear, how the effective SOC energy scale is affected by inclusion of cor- relations on top of DFT bandstructures, we deem it a sensible strategy to explore a wide window of parameter values in or- der to clearly expose the role of the SOC on the neutron spin resonance. Completing the discussion of the model Hamiltonian, we finally turn to the interactions of the 3dstates, which are mod-4 eled by a local Hubbard-Hund interaction term Hint=UX l;i;nli"nli#+ U0J 2X l;i;<;;0nlinli0(3) 2JX l;i;<Sli
Sli+J0 2X l;i;6=; cy licy liclicli+ h:c: : The Hamiltonian Eq. (3) is parametrized by an intraorbital HubbardU, an interorbital coupling U0, Hund’s coupling J and pair hopping J0, satisfyingU0=U2J,J=J0due to orbital rotational invariance of the Coulomb matrix ele- ments with respect to the Wannier basis functions. The op- erators for local charge and spin are nli=nli"+nli#with nli =cy licli andSli= 1=2P 0cy li0cli0, respectively. In the following, we will further constrain the value of the Hund’s coupling to J=U=4in order to re- duce the number of parameters. Below, we will treat interac- tion effects at the level of the RPA. The bare interaction ver- tex defined by the interaction Hamiltonian above will provide the corresponding approximation to the 2-particle irreducible (2PI) vertex in the particle-hole channel. III. NEUTRON SCATTERING AMPLITUDE To model the superconducting properties of the system, we proceed as follows. First, in order to take into account SOC already at the level of the spin-fluctuation mediated pairing mechanism, we construct the 2PI vertex in the particle-particle channel by performing the RPA resummation of particle-hole diagrams, employing the bare 2PI particle-hole vertex and normal-state Greens functions including the effect of SOC. We then determine the leading solution of the correspond- ing Fermi-surface projected linearized gap equation (LGE) (Bethe-Salpeter equation in the particle-particle channel) in the presence of SOC (for details, see Ref. 46) for given and interaction parameters U,J. We note that we solve the LGE in the static approximation, i.e., we take the 2PI particle-particle vertex at vanishing energy arguments. Due to the locking of spin and orbital degrees of freedom by virtue of the SOC Hamiltonian, the pairing problem for Fermi surface states is most naturally formulated for Cooper pairs composed of the states in a Kramer’s doublet, i.e., single- particle states at momenta kandkwhich are related by time-reversal. In fact, due to time-reversal and inversion sym- metry in the normal state, each band is still doubly degener- ate. Due to breaking of continuous spin-rotational symmetry by SOC, spin no longer represents a good quantum number to label the states in this degenerate subspace. It is possible, however, to define a pseudo-spin degree of freedom, which coincides with physical spin as !0, and which indeed has the transformation properties of a spin-1/2 degree of freedom, even for finite SOC. The solutions of the Fermi surface pro- jected LGE can therefore be classified as even-parity pseudo- spin singlet and odd-parity pseudo-spin triplet solutions. As mentioned in Sec. II, the strong spin fluctuations at the nest- ing vectors tend to drive a Cooper instability to a supercon-ducting state with even-parity s+gap structure upon dop- ing. In this work, we will therefore exclusively be concerned with the neutron-scattering signatures of s+gap solutions. By restricting the doping range such that the topology of the Fermi surface does not drastically change, the spin fluctua- tions at wavevectors Q1andQ2are indeed dominant and en- tail a leading Cooper instability of even-parity s+-type. The LGE solutions obtained for fixed and interaction pa- rameterUthen serve as the starting point for defining a BdG Hamiltonian (see Sec. S1) and the corresponding Nambu- Gorkov Greens function (see. Sec. S2) for the superconduct- ing state. Here, we make use of the procedure described in Ref. 48 in order to obtain gap solutions throughout the entire Brillouin zone from the Fermi-surface projected LGE. We de- note the corresponding pseudo-spin singlet gap function by ^
b 0(k) =
0gb(k); (4) with
0a parameter fixing the gap amplitude, and gb(k)di- mensionless functions (one for each band b) defined on the 2-Fe BZ, describing the gap structure obtained from the LGE. The functions gb(k)are normalized, such that
0is the max- imum value ofj^
b 0(k)j(where maximization is carried out overkandb). We note that we do not model a temperature- dependent
0, but instead use
0as a free parameter. Again approximating the 2PI particle-hole vertex by the bare vertex defined by Eq. (3), we finally determine the RPA spin susceptibility of the superconducting state, see Sec. S4 for details. In the following, we will refer to this approxi- mation as BCS+RPA49,50. To this end, we compute the con- nected, imaginary-time spin-spin correlation function in the superconducting state (here i;jrefer to the spatial directions x;y;z ) ij(i!n;q)=g2Z 0dei!nhTSi q()Sj q(0)ic;BCS+RPA;(5) withg= 2and the Fourier-transformed electron spin operator (in the imaginary-time Heisenberg picture) for the 2-Fe unit cell given as Si q() =1p NX k;l;;;0cy kql()i 0 2ckl0():(6) The symbolTin Eq. (5) denotes the time-ordering opera- tor for the imaginary time variable andcy kl,ckldenote the Fourier-transformed fermionic creation and annihilation operators, respectively, and i!ndenotes a bosonic Matsubara frequency. The momentum kis an element of the 2-Fe BZ, andNcounts the number of 2-Fe unit cells. Performing an- alytic continuation i!n!!+ iofij(i!n;q)in Eq. (5) with>0, we gain access to the momentum- and frequency- resolved spectral density of magnetic excitations with differ- ent spin-space polarization as probed by polarized neutron scattering. Due to the proportionality of the experimentally accessible neutron scattering amplitude and the spectral den- sity of magnetic excitations, the theoretical determination of the magnetic susceptibility in the superconducting state is a valuable tool in unveiling the underlying electronic structure.5 (a)= 0 meV (b)= 25 meV (c)= 100 meV (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) FIG. 2. (a)-(c) (Pseudo-)Spin singlet s+gap function ^
b 0(k)in the 2-Fe BZ as obtained from the linearized gap equation with a 2PI pairing vertex in RPA approximation for various values of spin-orbit coupling strength at chemical potential 0= 0 meV . Red and blue color correspond to positive and negative gap values, respectively. (d)-(o) Imaginary part of the susceptibility at the nesting vector Q1as a function of energy for varying and gap amplitude
0. The SOC-strength is fixed in each column and corresponds to the values in (a)-(c). The gap amplitude
0varies as (d)-(f)
0= 0meV , (g)-(i)
0= 25 meV , (j)-(l)
0= 50 meV and (m)-(o)
0= 75 meV . Blue color corresponds to polarization i=x, red toi=yand green to i=z. The black dashed curves in (d), (g), (j), (m) show the non-interacting susceptibilities (scaled up for visibility by a factor of 75). IV . RESULTS In the following, we will present the results for the neutron scattering amplitude as extracted from the imaginary part of the spin susceptibility determined in BCS+RPA. As we are interested in the spectral splitting and magnetic anisotropy of the neutron resonance mode of the spin fluctuation induced s+state in the presence of SOC, we focus on the frequency dependence of the neutron scattering amplitude at wavevec-torQ1= (;0). Here we choose a coordinate system with x=a,y=bandz=c, where the cartesian axes are aligned with the orthorhombic crystal axes. We further note that in tetragonal states, which we consider here, the magnetic re- sponse at the wavevector Q2= (0;)is related to the re- sponse at Q1by aC4rotation in the abplane. The cross-terms withi6=jin Eq. 5 vanish for tetragonal systems. We first discuss the results obtained for a chemical poten- tial0= 0eV , for which the system has an electronic filling6 ofn= 6 (withnthe number of filled states per unit cell). We obtained the LGE solutions for a range of Uandval- ues. The leading instability in the Cooper channel was, as expected from the momentum structure of the susceptibility, always found to be of even parity s+type. The resulting gap structures gb(k)forU= 0:80eV and various values for are shown in Fig. 1(a)-(c). The value of the interaction was chosen to bring the system close to the SDW instability. Eval- uating the susceptibility for the BCS Hamiltonian with the corresponding (pseudo-)spin singlet, intraband order param- eter^
b 0(k), we clearly observe the formation of neutron reso- nance features in the spectral density of magnetic excitations upon going from the normal to the superconducting state, see Fig. 2. We note that we took the liberty to tune the interac- tion parameter in the superconducting state to a slightly larger value ofU= 0:88eV in order to enhance the resonance sig- natures. Qualitatively, however, the results remain unchanged compared to using the interaction value used in the construc- tion of the pairing vertex entering the LGE. In Fig. 2(d)-(o) we vary
0from 0(d)-(f) over 25(g)-(i) and 50(j)-(l) to 75meV (m)-(o). The low-energy neutron scattering amplitude in the superconducting state is decreased compared to the normal state. Due to the sign-changing nature of the s+state, how- ever, the emergence of a bound state in the form of the neutron resonance mode is possible for energies !<2
0. Momentarily focussing on the case without SOC ( = 0), the neutron scattering amplitude is isotropic, i.e., as expected it exhibits no difference between the different x,yandzpolar- ization channels, see Fig. 2(d),(g),(j),(m). We further observe an apparent splitting of the resonance mode upon increasing
0, while the positions of the resonance peaks simultaneously shift to higher energies. Increasing SOC in the normal state, the polarization re- solved scattering amplitudes eventually split, Fig. 2(d)-(f). The splitting increases with SOC and Fig. 2(f) clearly demon- strates that the paramagnetic spin excitations with xpolar- ization have become almost gapless, while paramagnon ex- citations with yandzpolarization reside at larger energies with fluctuations polarized along yexhibiting the highest en- ergy paramagnon branch. Moving on to the superconducting state we observe qualitatively different behavior of the neutron scattering amplitude with energy in the small and large SOC regimes. With
0= 25 meV , only a small polarization de- pendent differentiation is visible for = 25 meV , much like in the paramagnetic state (Fig. 2(h)). The large SOC case, however, features a two-peak structure in the z-polarization channel, with a third peak in the x-polarization channel situ- ated between the two z-polarization peaks on the energy axis (Fig. 2(i)). This feature grows more pronounced as the su- perconducting gap amplitude is increased, see Fig. 2(l),(o). On the small SOC side, increasing the gap amplitude beyond 25meV eventually leads to several split peaks as well. Here, thex- andy-polarized excitation modes occur with almost equal spectral weight in a double-peak structure, while the z- polarized mode also features a double peak structure, which is squeezed between the peaks of the other two polarization channels, see Fig. 2(k),(n). It seems rather noteworthy that the magnetic anisotropy of the neutron resonance mode(s) isnot simply inherited from the paramagnetic state, but instead seems to emerge from the interplay of the s+state and SOC. Before we move to the results obtained for negative chemi- cal potential, we emphasize that the case of = 0eV is rele- vant for both parent and doped FeSC materials in terms of the magnetic anisotropy of the normal state. Previsouly, we have identified that dominating xpolarization of the normal state is a robust feature in a certain window around = 0eV20. Moving to the case of chemical potential 0=0:45meV , resembling a hole-doped system with dominating zpolar- ization in the normal state, we obtained LGE solutions for an interaction parameter U= 0:70eV , which are shown in Fig. 3(a)-(c). In contrast to the = 0eV system, hole doping leads to a stronger gap-anisotropy, in that the gap on the inner hole pocket now has a markedly larger maximal amplitude than the middle and outer hole pockets. The gap on the inner hole pocket also turns out larger than on the electron pockets. The gap on the outer electron pocket features a much larger gap variation than the inner electron pocket as the Fermi mo- mentum paces out the pocket shape. Increasing SOC seems to have two effects: While the interpocket anisotropy eventually decreases for the hole pockets, the gap amplitude on the outer electron pocket develops a more pronounced maximum (with the gap on the inner electron pocket being basically unaffected by SOC). Performing the same analysis as before (where we tuned the interaction to U= 0:80eV for the same purpose as above), we observe the emergence of a resonance feature upon in- creasing the gap amplitude, see Fig. 3(d),(g),(j),(m). In con- trast to the = 0 eV system, however, in the absence of SOC we observe only a single peak in the magnetic excita- tion spectrum. Switching on SOC, the normal state features dominant and almost gapless z-polarized paramagnon excita- tions. The splitting in paramagnon gaps is, however, rather small. Consequently, the energy-dependent spectral weight of the three polarization channels is almost the same, see Fig. 3(e),(f). We note that we have chosen the chemical poten- tial to provide the largest possible splitting between x- andz- polarized paramagnons. Increasing the gap amplitude leads to a stronger splitting of the spectral weight contributions, even for fixed SOC. In the case of a small SOC energy scale, the x- andy-polarization channels behave almost identically, cf. Fig. 3(h),(k),(n), while larger SOC leads to a more pronounced differentiation of these two channels, cf. Fig. 3(f),(i),(l),(o). We note further, that in the case of large SOC, apparently no resonance forms in the y-polarization channel. It is only the z- andx-polarized spectral weight curves that feature more or less well formed peaks, with the peak corresponding to z- polarized excitations systematically occuring at lower ener- gies. V . DISCUSSION AND CONCLUSIONS Having established, that a weak coupling BCS+RPA cal- culation can produce a variety of realizations of neutron- resonance features, all depending on the detailed values of chemical potential, SOC strength and gap amplitude, we at-7 (a)= 0 meV (b)= 25 meV (c)= 100 meV (d) (e) (f) (g) (h) (i) (j) (k) (l) (m) (n) (o) FIG. 3. (a)-(c) (Pseudo-)Spin singlet s+gap function ^
b 0(k)in the 2-Fe BZ as obtained from the linearized gap equation with a 2PI pairing vertex in RPA approximation for various values of spin-orbit coupling strength at chemical potential 0=45meV , corresponding to a hole- doped system. Red and blue color correspond to positive and negative gap values, respectively. (d)-(o) Imaginary part of the susceptibility at the nesting vector Q1as a function of energy for varying and gap amplitude
0. The SOC-strength is fixed in each column and corresponds to the values in (a)-(c). The gap amplitude
0varies as (d)-(f)
0= 0 meV , (g)-(i)
0= 25 meV , (j)-(l)
0= 50 meV and (m)-(o)
0= 75 meV . Blue color corresponds to polarization i=x, red toi=yand green to i=z. The black dashed curves in (d), (g), (j), (m) show the non-interacting susceptibilities (scaled up for visibility by a factor of 75). tempt to uncover hints pointing to the mechanism behind the generation of the observed anisotropy. Here it is worth emphasizing that the interaction Hamil- tonianHintcannot generate anisotropy on its own, as it re- spects rotational symmetry in both orbital and spin space. This leaves as the only two factors influencing the anisotropy in the neutron scattering amplitude i) the interplay of SOC with the bandstructure in the superconducting system and ii) the spin structure of the superconducting solutions. As a de-tailed understanding of i) turned out to be a rather involved problem even in the normal state20, we do not attempt to pro- vide a full answer here. Rather, we try to assess the explana- tory possibilities of ii), essentially by testing for the effect of the spin-triplet component of the superconducting order pa- rameter in orbital spin space on the magnetic anisotropy. To this end, we performed additional numerical calculations, where we either projected out the spin-triplet component in8 the^
b 0(k)^
b0 0(kq)prefactor of the FF contribution to the irreducible bubble in the particle-hole channel but left the electronic structure of the superconductor unaltered, or where we removed the spin-triplet component entirely, thereby also altering the gap structure. In both cases, it turns out that the removal of the spin-triplet component does not lead to quali- tatively different results than those summarized in Fig. 2 and Fig. 3. This observation can partly be explained by noting that the spin-triplet component of the superconducting order parameter is basically driven by SOC. Since SOC (compared to the electronic bandwidth) is a rather small energy scale for the systems under consideration, the largest gap amplitudes of the spin-triplet component turn out to be suppressed by at least one order of magnitude with respect to the spin-singlet component. As the spin-triplet component and its associated d-vector do apparently not play a decisive role in determining the mag- netic anisotropy of the superconducting state at the nesting vectors Q1andQ2, we conclude that it must be the sub- tle interplay of SOC and the bandstructure of the s+state that gives rise to an inherent tendency of the s+state to fa- vorz-axis polarized excitations at low energies. Indeed, as a comparison of the results for = 0meV and=45meV systems shows, a large SOC is required to render the low- energy magnetic excitations to be zpolarized, if the normal state paramagnons at low energies are preferably xpolarized. For the case of dominantly zpolarized paramagnons (with x, ypolarized excitations having larger, yet similarly sized exci- tation gaps) the s+order parameter and increased SOC have the tendency to enhance the normal state magnetic anisotropy. We emphasize once more, that, as the anisotropy is not gen- erated by the interaction, the results we found are – at least at a qualitative level – robust with respect to variations of the interaction strength. We do, however, find pronounced changes in the relative spectral weight distribution of the dif- ferent polarization channels, when the superconducting sys- tem approaches an SDW instability by tuning the interaction appropriately. Typically, the lowest-energy peak grows con- siderably in size, while simultaneously moving to lower en- ergies. The remaining peaks only show a moderate increase in spectral weight and are more or less locked in place on the energy axis. In the introduction we emphasized a number of experimen-tal observations that broadly summarized the main findings of the spin anisotropy of the neutron resonance. These included the possibility of a double resonance, the tendency for a c-axis dominated resonance, the enhancement of spin anisotropy be- lowTc, and the diminishing anisotropy for large enough dop- ing levels. While the latter point has not been investigated nu- merically in this work, we do know from our earlier study that there is a clear tendency of the normal state spin anisotropy to vanish for sufficiently large doping levels20. We expect the same to hold true for the anisotropy of the neutron resonance. Regarding the former three points, however, they seem to be reasonably well captured by the BCS+RPA approach, includ- ing SOC, as presented in this work. It is simply an inherent property of the bandstructure of the FeSCs and the form of the SOC in these materials, that the low-energy resonance favors c-axis polarization. Further, it is a property of a large enough superconducting gap to secure the possible existence of a dou- ble resonance. Finally, the interplay of both SOC and super- conductivity cause the enlarged spin anisotropy upon entering the superconducting phase. From this qualitative agreement between the model and the experimental situation we con- clude that the methodology presented in this work is sufficient to explain the properties of the low-energy spin anisotropy of FeSCs. An interesting future direction includes more material- specific modelling, and to address the challenge of quantita- tive agreement between measurements and theory. This re- quires not only the necessity to go beyond BCS+RPA in terms of electronic interaction effects, but also ab initio develop- ments allowing for quantitatively matching bandstructures as a stating point. Another interesting study would be to investi- gate superconductors with larger SOC than in FeSCs. 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B 79, 134520 (2009). 49N. Bulut and D. J. Scalapino Phys. Rev. Lett. 67, 2898 (1991). 50N. Bulut and D. J. Scalapino, Phys. Rev. Lett. 68, 706 (1992). 51D. D. Scherer, I. Eremin, and B. M. Andersen, Phys. Rev. B 94, 180405(R) (2016).1 Supplementary Material: “Effects of spin-orbit coupling on the neutron spin resonance in iron-based superconductors” S1. BDG HAMILTONIAN In the following, we will provide details on the Bogoliubov-de-Gennes (BdG) Hamiltonian for a spin-orbit coupled supercon- ductor and the corresponding Nambu-Gorkov Greens function entering the computation of maginary-time correlation functions in the superconducting state. We start out with transforming the normal-state Hamiltonian to momentum space, where we define the Fourier-transformed fermionic operators ckl=1p NX ieik
(ri+l)cli; cy kl=1p NX ieik
(ri+l)cy li; (S1) where the sum runs over lattice sites, ridenotes the lattice vector corresponding to the ith unit cell and ldenotes an intra unit cell vector referencing the relative positions of the two Fe atoms. The non-interacting Hamiltonian H0+HSOC, see Eq. (1) and Eq. (2), can then be written as H0+HSOC=X k;l;l0;;0cy kl(k)]l;l000ckl000; (S2) withh(k)denoting a 2020k-dependent Bloch-matrix acting in the single-particle Hilbert space of sublattice orbital spin. The Bloch-matrix h(k)is diagonalized by a unitary transformation ckl=X b;[U(k)]l;b kb; cy kl=X b;[U(k)] l;b y kb(S3) with the unitary matrix U(k)comprising the eigenvectors of h(k), with corresponding eigenvalues b(k). Here, the index b labels electronic bands, while 2f+;glabels a pair of degenerate states, that is guaranteed to exist at a given kfor a time- reversal-invariant and inversion-symmetric system. In the solution of the pairing problem with spin-orbit coupling, the -degree of freedom plays the role of a pseudospin. With a judicious choice of the eigenbasis in the degenerate subspaces, the pseudospin indeed has the transformation properties of a spin-1/2 degree of freedomS1. Moving to the superconducting state by adding a Hamiltonian describing the coupling of electrons to the pairing field, H
, the full BCS Hamiltonian containing the superconducting order parameter can be written as HBCS=H00N+HSOC+H
=1 2X k y kl[hBdG(k)]l;l00 kl00; (S4) where we introduced the Nambu-Gorkov spinors kl, y kl. In terms of the original electron operators in the orbital Wannier basis, these are defined as kl= ckl";ckl#;cy kl";cy kl#T ; y kl= cy kl";cy kl#;ckl";ckl# ; (S5) where the transposition turns the row- into a columnvector, but acts trivially on the Fock-space operators. For each pair of indices l;andl0;0,[hBdG(k)]l;l00is a44matrix acting in particle-hole spin-space. We can compactly write hBdG(k) =P+ (h(k)0 1) +P hT(k) +0 1
++
(k) + 
(k); (S6) whereP=1 2( 1z),=1 2(xiy)withi,i=x;y;z Pauli matrices acting in particle-hole space. We keep a general orbital and spin structure for the pairing fields
(k),
(k). The spin structure can be decomposed into spin-singlet ( &= 0) and spin-triplet ( &=x;y;z ) as [
(k)]l;l000=p 2X &s&[
&(k)]l;l00[&];0;[
(k)]l;l000=p 2X &[
&(k)]l;l00[&];0; (S7) where we introduced the Balian-Werthamer (BH) spin matrices 0=1p 21iy;x=1p 2xiy;y=1p 2yiy;z=1p 2ziy; (S8)2 characterizing the spin structure of singlet and triplet Cooper pairs, and s&= +1 for&2fx;zgands&=1for&2f0;yg. By applying the transformation Eq. (S3) to Eq. (S6), the BdG Hamiltonian can be brought into the band-space representation. Here, we simply note the following transformation rules that connect the orbital- and band-space representations of the pairing fields: [^
(k)]b;b00=X l;l0X ;0X ;0[U(k)] l;b [U(k)] l000;b00[
(k)]l;l000; (S9) [^
(k)]b;b00=X l;l0X ;0X ;0[U(k)]l;b [U(k)]l000;b00[
(k)]l;l000: (S10) The band-space pairing fields in turn can be decomposed into pseudospin-singlet and pseudospin-triplet components: [^
(k)]b;b00=p 2X &s&[^
&(k)]b;b0[^&];0;[^
(k)]b;b00=p 2X &[^
&(k)]b;b0[^&];0; (S11) where we notationally distinguish BH matrices in pseudospin space by a hat. The pseudospin basis is constructed such that for vanishing SOC it coincides with the physical spin. Correspondingly, in this limit the unitary transformation diagonalizing Eq. (S2) factorizes as [U(k)]l;b = [U(k)]l;b0;. For finite SOC, the discussion of inter- and intraband pairing is thus necessarily tied to the pseudospin degree of freedom. S2. NAMBU-GORKOV GREENS FUNCTION Having defined the BdG Hamiltonian in Eq. (S6), the corresponding imaginary-time Nambu-Gorkov Green function can be obtained as G(i!n;k) =Z 0dei!nhT k() y k(0)iBCS; (S12) where the expectation value is evalueated with respect to a thermal Gibbs state of inverse temperature = 1=kBTof the BCS Hamiltonian Eq. (S4) and !n=2 (n+ 1=2),n2 Zdenotes a fermionic Matsubara frequency. We then decompose the Nambu-Gorkov Green function in the same way as the BdG Hamiltonian to obtain G(i!n;k) =P+ G+(i!n;k)P G(i!n;k) ++ F(i!n;k) + F(i!n;k): (S13) In the following, we will specialize to purely intraband superconductivity, i.e., we will assume [^
(k)]b;b00= 0 forb6=b0. In order to arrive at the expressions for the particle-hole components of the Nambu-Gorkov Greens function in the orbital representation, we first define a series of auxiliary quantities. We define Eb;(k) =b;(k)0,^
b &(k) = [ ^
&(k)]b;b, 0(k) =P &^
b &(k)^
b &(k), &(k) = (1) &(k) + (2) &(k)(for&6= 0),~ &(k) = (1) &(k) (2) &(k), where (1) &(k) = ^
b &(k)^
b 0(k) +^
b 0(k)^
b &(k)and (2) &(k) = iP i;j&ij^
b i(k)^
b j(k), as well asEb;(k) =q E2 b;(k) + 0(k). Equipped with these definitions, we obtain the following general expressions valid for intraband superconductors [G+(i!n;k)]l;l000=X b;;0[U(k)]l;b [U(k)] l000;b0(i!n+Eb;(k))h !2 n+E2 b;(k) ^1+P0 &s&~ &(k)^&i ;0  !2n+E2 b;(k)2 P0 &~ 2&(k); (S14) [F(i!n;k)]l;l000=X b;;0[U(k)]l;b [U(k)]l000;b0h !2 n+E2 b;(k) ^1+P0 &s&~ &(k)^&ihP &0p 2s&0^
b &0(k)^&0i ;0  !2n+E2 b;(k)2 P0 &~ 2&(k); (S15) [F(i!n;k)]l;l000=X b;;0[U(k)] l;b [U(k)] l000;b0h !2 n+E2 b;(k) ^1P0 & &(k)^&ihP &0p 2^
b &0(k)^&0i ;0  !2n+E2 b;(k)2 P0 &~ 2&(k); (S16)3 whereP &(:::)runs over&= 0;x;y;z andP0 &(:::)is restricted to the pseudospin triplet components x;y;z . We also have G(i!n;k) = [G+(i!n;k)]T. Specializing further to a pseudospin singlet superconductor, i.e., ^
b 0(k)6= 0, while ^
b x(k) =^
b y(k) =^
b z(k) = 0 throughout the Brillouin zone, the expressions Eqs. (S14)-(S16) simplify to [G+(i!n;k)]l;l000=X b;;0[U(k)]l;b [U(k)] l000;b0i!n+Eb; !2n+E2 b;(k)[^1];0; (S17) [F(i!n;k)]l;l000=p 2X b;;0[U(k)]l;b [U(k)]l000;b0^
b 0(k) !2n+E2 b;(k)[^0];0; (S18) [F(i!n;k)]l;l000= +p 2X b;;0[U(k)] l;b [U(k)] l000;b0^
b 0(k) !2n+E2 b;(k)[^0];0: (S19) S3. BARE PARTICLE-HOLE CORRELATION FUNCTION OF A SPIN-ORBIT COUPLED SUPERCONDUCTOR Here we collect results on the generalized, connected correlation function (also referred to as irreducible bubble) evaluated for a BCS Hamiltonian with SOC for an intraband, pseudospin singlet superconductor. We define [0(i!n;q)]l111;l222 l333;l444=1 NZ 0dei!nX k;k0hTcy kql111()ckl222()cy k0+q0l333(0)ck0l444(0)ic;BCS;(S20) with!nnow denoting a bosonic Matsubara frequency2 n,n2 Z. Applying Wick’s theorem, we arrive at the following representation of the irreducible bubble in terms of the components of the Nambu-Gorkov Greens function: [0(i!n;q)]l111;l222 l333;l444=1 NX k2 Z;k[G+(ik;k)]l222;l333[G+(iki!n;kq)]l444;l111 1 NX k2 Z;k[F(ik;k)]l111;l333[F(iki!n;kq)]l444;l222: (S21) Using Eqs. (S17)-(S19) and performing Matsubara summations, we eventually arrive at [0(i!n;q)]l111;l222 l333;l444=1 NX kX b;b0[MGG b;b0(k;q)]l111;l222 l333;l444LGG(i!n;Eb(k);Eb0(kq);Eb(k);Eb0(kq)) 1 NX kX b;b0[MFF b;b0(k;q)]l111;l222 l333;l444LFF(i!n;Eb(k);Eb0(kq);^
b 0(k);^
b0 0(kq));(S22) where we dropped the pseudospin index on energy arguments, as we assume time-reversal-invariance and inversion-symmetry, implyingEb;+(k) =Eb;(k)andEb;+(k) =Eb;(k). The coefficients and Lindhard factors are defined as [MGG b;b0(k;q)]l111;l222 l333;l444=X 1;2;3;4[U(kq)] l111;b01[U(k)]l222;b2[U(k)] l333;b3[U(kq)]l444;b04[^1]1;4[^1]2;3; [MFF b;b0(k;q)]l111;l222 l333;l444=X 1;2;3;4[U(k)] l111;b1[U(k+q)]l222;b02[U(k)] l333;b3[U(kq)]l444;b04s0[^0]1;3[^0]4;2 and LGG( ;E1;E2;E1;E2) =1 4(E1+E1) (E2+E2) E1E2nF(E2)nF(E1) +E2E1+1 4(E1E1) (E2+E2) E1E21nF(E2)nF(E1) +E2+E1+ 1 4(E1+E1) (E2E2) E1E2nF(E2)nF(E1)1 E2E1+1 4(E1E1) (E2E2) E1E2nF(E2) +nF(E1) E2+E1; LFF( ;E1;E2;
1;
2) =1 4
1
2 E1E2nF(E2)nF(E1) +E2E1+1nF(E2)nF(E1) +E2+E1+nF(E2)nF(E1)1 E2E1+nF(E2) +nF(E1) E2+E1 :4 Above, we also introduced the Fermi-Dirac distribution function nF(E) = 1=(exp(E) + 1) . In view of the SU(2) ‘gauge freedom’ related to the degenarcy of eigenstates, which is adressed in detail in Ref. S1, it is important to note, that for a fixed Wannier gauge, the irreducible bubble can indeed be shown to be gauge invariant, as long as each quadruplet of states at kandkis constructed from a single state through time-reversal, inversion, and the combined application of time-reversal and inversion. S4. RPA CORRELATION FUNCTION OF A SPIN-ORBIT COUPLED SUPERCONDUCTOR Following Refs. S2 and S3, where we developed the RPA in the absence of spin SU(2) symmetry (either due to non-collinear magnetic order or SOC), we briefly describe the RPA formalism required to analyze the magnetic anisotropy of the neutron resonance in the presence of SOC. The goal is to compute the connected imaginary-time spin-spin correlation function (where i;jrefer to the spatial directions x;y;z ) ij(i!n;q) =g2Z 0dei!nhTSi q()Sj q(0)ic;BCS+RPA; (S23) with the Fourier transformed electron spin operator for the 2-Fe unit cell given as Si q() =1p NX k;l;;;0cy kql()i 0 2ckl0(): (S24) We note that we typically specify the transfer momentum qwith respect to the coordinate system of the 1-Fe Brillouin zone. It is then understood that ‘ kq’ refers to subtraction of the two vectors in a common coordinate system. Here Tdenotes the time-ordering operator with respect to the imaginary-time variable 2[0;), withthe inverse temperature and i 0thei-th Pauli matrix. From the imaginary part of ij(!;q)(after having analytically continued from imaginary to real frequencies), we can extract the spectrum of spin excitations that are probed by neutron scattering. To ease notation we introduce a combined index X(l;; )by collecting sublattice, orbital and spin indices. In the absence of interactions, the correlation function [(i!n;q)]X1;X2 X3;X4[(i!n;q)]l111;l222 l333;l444reduces to [0(i!n;q)]X1;X2 X3;X4=1 NZ 0dei!nX k;k0hTcy kqX1()ckX2()cy k0+q0X3(0)ck0X4(0)ic;BCS; (S25) which was discussed extensively in Sec. S3. Ignoring anomalous vertices, the RPA equation for the generalized particle-hole correlation function reads as (see Refs. S2 and S3) [(i!n;q)]X1;X2 X3;X4= [0(i!n;q)]X1;X2 X3;X4+ [0(i!n;q)]X1;X2 Y1;Y2[U]Y1;Y2 Y3;Y4[(i!n;q)]Y3;Y4 X3;X4: (S26) Repeated indices are summed over in Eq. (S26). The bare fluctuation vertex [U]X1;X2 X3;X4[U]l111;l222 l333;l444originates from the Hubbard-Hund interaction and describes how electrons scatter off a collective excitation in the particle-hole channel. The electronic Hubbard-Hund interaction Hamiltonian (with normal ordering implied) can be compactly written as Hint=1 2X iX flng;fjg;fkg[U]l111;l222 l333;l444cy l1i11cl2i22cy l3i33cl4i44: (S27) The bare interaction vertex [U]11;2233;44defined above can be decomposed into charge and spin vertices in the particle-hole channel as [U]l111;l222 l333;l444=1 2l1;l2l2;l3l3;l4 [Uc]12 341234[Us]12 3412
34
; (S28) which in turn are defined by [Us] =U; [Us] =U0;[Us] =J;[Us] =J0;with6=; (S29) and [Uc] =U; [Uc] = 2JU0;[Uc] = 2U0J;[Uc] =J0;with6=; (S30) and zero otherwise. The onsite interaction is parametrized by an intraorbital Hubbard- U, an interorbital coupling U0, Hund’s couplingJand pair hopping J0.5 Solving Eq. (S26), we obtain the RPA approximation for the particle-hole correlation function. The RPA approximation for the spin susceptibility tensor ij(!;q)can be obtained by forming the appropriate linear combinations of components of the particle-hole correlation function and performing analytic continuation i!n!!+ i,>0: ij(!;q) =g2X l;l0X ;X 1;:::; 4i 12 2j 34 2[(i!n!!+ i;q)]l1;l2 l03;l04: (S31) [S1] D. D. Scherer and B. M. Andersen, unpublished. [S2] D. D. Scherer, I. Eremin, and B. M. Andersen Phys. Rev. B 94, 180405(R) (2016). [S3] D. D. Scherer and B. M. Andersen, Phys. Rev. Lett. 121, 037205 (2018).

1701.03673v1.Bose_Einstein_condensates_in_the_presence_of_Weyl_spin_orbit_coupling.pdf

arXiv:1701.03673v1 [cond-mat.quant-gas] 13 Jan 2017Bose-Einstein condensates in the presence of Weyl spin-orb it coupling Ting Wu and Renyuan Liao Fujian Provincial Key Laboratory for Quantum Manipulation and New Energy Materials, College of Physics and Energy, Fujian Normal University, Fu zhou 350108, China and Fujian Provincial Collaborative Innovation Center for Opt oelectronic Semiconductors and Efficient Devices, Xiamen, 361005, China (Dated: August 10, 2018) We consider two-component Bose-Einstein condensates subj ect to Weyl spin-orbit coupling. We obtain mean-field ground state phase diagram by variational method. In the regime where inter- species coupling is larger than intraspecies coupling, the system is found to be fully polarized and condensed at a finite momentum lying along the quantization a xis. We characterize this phase by studying the excitation spectrum, the sound velocity, the q uantum depletion of condensates, the shift of ground state energy, and the static structure facto r. We find that spin-orbit coupling and interspecies coupling generally leads to competing effects . I. INTRODUCTION The creation of synthetic gauge fields in ultracold atomic gases provides fascinating opportunities for ex- ploring quantum many-body physics [1]. Of particular interest is the realization of non-Abelian spin-orbit cou- pling (SOC) [2–4]. Spin-orbit coupling is crucial for re- alizing intriguing phenomena such as the quantum spin Hall effect [5], new materials classes such as topologi- cal insulators and superconductors [6–8]. In bosonic sys- tems, the presence of SOC may lead to novel ground states that have no known analogs in conventional solid- state materials [9–11]. In cold atomic gases, spin-orbit coupling can be implemented by Raman dressing of atomic hyperfine states [12, 13]. The tunability of the Raman coupling parameters promises a highly flexible experimental platform to explore interesting physics re- sulting from spin-orbit coupling [14]. Recently, two- dimensional SOC has been experimentally realized in cold atomic gases [15, 16]. In anticipation of immediate experimental relevance, intense theoretical attention has been paid to the physics of ultracold atomic gases in the presence of SOC [3, 4]. In the absence of interparticle interactions, the low- lying density of states is two-dimensional for Rashba- type SOC [9]. In particular, the single-particle energy minimum featured a Rashba-ring, which has important consequences on the ground state and finite-temperature propertiesofSOC Bose gases[17–26], asthe roleofquan- tum fluctuations gets enhanced due to huge degenera- cies at the lowest-lying states. The three-dimensional analog of Rashba-type SOC is interesting because it is expected to stabilize a long-sought skyrmion mode in the ground state of trapped Bose-Einstein condensates (BECS) [19, 27, 28]. This Weyl-type SOC can be imple- mented following the proposals [29–31] by using power- ful quantum technology. Although there is currently no evidence for Weyl fermions to exist as fundamental par- ticles in our universe, Weyl-like quasiparticles have been detected recently in condensed-matter systems [32, 33]. In light of these discoveries, the study of Weyl SOC in ul- tracoldatomsystemsbecomesparticularlyrelevant,sincethe ability of manipulate the Weyl-SOC strength creates interestingopportunitiesforthe explorationofeffects not predicted in the realm of particle physics. In addition, the study of the effects of SOC may reveal some inter- esting physics unexplored in conventional binary Bose condensates [34, 35]. In this work, we shall examine the physics of two-component Bose gases subject to Weyl- typeSOC. Firstly, wewill introducethe modeland deter- mine the mean-field ground state by variation approach. Secondly, we will set out to study a particular realization of ground state where quantum fluctuation plays an es- sential role. Specially, we will investigate the interplay of spin-orbit coupling and interspecies interaction upon the ground state properties of the system. Finally, we will come to a summary. II. MODEL AND FORMALISM We consider a 3D homogeneous interacting two- component Bose gas subject to Weyl-type spin-orbit cou- pling, described by the Hamiltonian H=H0+HI, with H0=/integraldisplay d3rΨ†(r)/bracketleftbigg −/planckover2pi12∇2 2m+λ/vector σ·ˆp/bracketrightbigg Ψ(r),(1a) HI=/integraldisplay d3r/bracketleftigg g/summationdisplay σnσ(r)2+2g↑↓n↑n↓/bracketrightigg .(1b) Here Ψ(r) = (ψ↑,ψ↓)Tis a two-component spinor field, /vector σ= ˆxσx+ˆyσy+ˆzσz,ˆpis the momentum operator, nσ= ψ† σψσis the density for component σ∈ {↑,↓},λis the strength of the spin-orbit coupling, and the strength for the intraspecies interaction and interspecies interaction isgandg↑↓, respectively. For brevity, we set /planckover2pi1= 2m= 1 from now on. Diagonalization of H0yields the two-branch single- particle energy spectrum E±(p) =p2±λp, and the cor- responding eigenfunctions are given by Φ±(p) =/parenleftbigg sin[(π−2θp±π)/4]e−iϕp cos[(π−2θp±π)/4]/parenrightbiggeip·r √ V,(2)2 whereVis the volume of the system. The lowest-energy state for a given propagating direction parameterized by θpandϕpis from the “-” branch and occurs at momen- tump=λ 2(sinθpcosϕp,sinθpsinϕp,cosθp). To determine the ground state of an interacting sys- tem, as routinely done in the literature [10, 23, 36–38], we assume that the system has condensed into a coher- ent superposition of two plane-wave states with opposite momenta with magnitude p=λ/2. Thus the conden- sate wave function adopts the form Φ 0=C+Φ−(p) + C−Φ−(−p), whereC+andC−are two complex num- bers to be determined and subject to normalization con- dition|C+|2+|C−|2=n0. Without loss of generality, the normalizationconditionsuggeststhe parametrization |C+|2=n0cos2(α/2) and|C−|2=n0sin2(α/2), with α∈[0,π]. Upon substitution into EG=<Φ0|H|Φ0>, the variational ground state energy per particle is evalu- ated as EG n0V=−λ2 4+gn0+(g↑↓−g)n0 2f(θp,α),(3) wheref(θp,α) = sin2θp+sin2α−3sin2θpsin2α/2. Min- imization of the ground state energy with respect to θp andα, one obtains the ground state phase diagram, sum- marized in Fig.1. When g↑↓−g>0, the system is found to be in the phase of PW-Polar, which is a fully po- larized phase with condensation momentum lying along the quantization axis. When g↑↓−g <0, at mean-field level, there aretwodegeneratephases: one is unpolarized PW-Axial phase, which is condensed at one plane-wave with momentum lying in the x-yplane; the other one is the SP-Polar phase, which is striped phase mixing of two opposite momentum along the z-axis. There exists a critical point when g↑↓−g= 0. In this case the system enjoysa SU(2) pseudo-spin rotationsymmetry. To deter- mine which phase the system prefers requires calculation going beyond mean field, and in principle it is believed to lead to a unique ground state via the mechanism of “order from disorder” [18, 39]. Within the imaginary-time field integral, the par- tition function of the system may be cast as [40] Z=/integraltext D[ψ∗ σψσ]e−S[ψ∗ σ,ψσ], with the action S=/integraltextβ 0dτ[dr/summationtext σψ∗ σ(∂τ−µ)ψσ+H(ψ∗ σ,ψσ)], whereβ= 1/T is the inverse temperature and µis the chemical poten- tial introduced to fix the total particle number. Here, for simplicity, we restrict ourself to studying the PW-Polar phase. Without lossofgenerality, wefurther assumethat the condensation occurs at momentum /vector κ= (0,0,−λ/2), then the ground state wave function is determined as Φ0=√n0(1,0)Te−iλz/2. It is a fully polarized phase with condensation momentum aligning antiparallel withthe quantization axis. We split the Bose field into the mean-field part φ0σand the fluctuating part φqσ asψqσ=φ0σδq/vector κ+φqσ. After substitution, the ac- tion can be formally written as S=S0+Sf, where S0=βV/bracketleftig (−λ2 4−µ)n0+gn2 0/bracketrightig is the mean-field con- tribution and Sfdenotes a contribution from the fluc- 0PW-Axial/SP-PolarPW-Polar (a) (b)z y x2λ g g ↑↓ − FIG. 1. (color online) Mean-field ground state phase diagram . Panel (a): for g↑↓−g >0, the ground state is in PW-Polar phase, where it is a plane wave with the condensation mo- mentum being parallel to the z-axis. For g↑↓−g <0, the system may be in the phase of either PW-Axial or SP-Polar. Here PW-Axial phase stands for one plane wave with the con- densation momentum lying in the x-yplane, and SP-Polar phase stands for the condensation at two opposite momenta along the z-axis. Panel (b): schematic representation of the PW-Polar and PW-Axial phases. For one plane-wave con- densation at either the north pole or the south pole is called the PW-Polar phase, it is a fully polarized phase as only one component is allowed. For one plane-wave condensation in thex-yplane is called PW-Axial, it is an unpolarized phase. tuating fields. The chemical potential may be deter- mined via saddle point condition ∂S0/∂n0= 0, yield- ingµ=−λ2 4+2gn0. At this point, the action is exact. However, it contains terms of cubic and quartic orders in fluctuating fields. To proceed, we resort to the cele- brated Bogoliubov approximation, where only terms of zeroth and quadratic orders in the fluctuating fields are retained. By defining a four-dimensional column vec- tor Φq= (φ/vector κ+q↑,φ/vector κ+q↓,φ∗ /vector κ−q↑,φ∗ /vector κ−q↓), we can bring the fluctuating part of the action into the compact form Sf≈/summationtext q,iwn1 2Φ† qG−1(q,iwn)Φq−β 2/summationtext q,σǫqσ,where wn= 2πn/βis the bosonic Matsubar frequencies, and the inverse Green’s function G−1(q,iwn) is defined as G−1= −iwn+ǫq↑Rq 2gn0 0 R∗ q−iwn+ǫq↓0 0 2gn0 0iwn+ǫ−q↑R∗ −q 0 0 R−qiwn+ǫ−q↓ , (4)3 −2 0 204812 qz/√gn0ω±/gn0 ω+ ω− 0 1 2379 q⊥/√gn0ω±/gn0 ω+ ω−(a) (b) FIG. 2. (color online) Two branches of excitation spectrum ω±in the momentum space: (a) along the z-axis, and (b) in thex-yplane. Here we set interspecies coupling η= 2.0 and spin-orbit coupling λ=√gn0. whereǫq↑=q2+2gn0,ǫq↓=q2−2λqz+λ2+2(g↑↓−g)n0, andRq=λ(qx−iqy). Throughout our calculation, we will choose gn0as a basic energy scale and√gn0as the corresponding momentum scale. To characterize the strength of interspecies coupling, we define a dimension- less parameter η=g↑↓/g. III. CALCULATION AND RESULTS The excitation spectrum ofthe system can be found by examiningthepolesoftheGreen’sfunction G(q,iwn). To achievethis , one proceedsby evaluating the determinant ofG−1(q,iwn), det[G−1] = (iw2 n−ω2 10)/bracketleftbig (iwn+2λqz)2−ω2 20/bracketrightbig −2λ2q2 ⊥F, F=iwn(iwn+2λqz)+(q2+2gn0)ω20−λ2q2 ⊥ 2, (5) whereω10=q/radicalbig q2+4gn0andω20=λ2+q2+2(g↑↓− g)n0, andq⊥=q2 x+q2 y. By solving the secular equation DetG−1(q,ωqs) = 0, one finds two branches of excita- tion spectrum ωq±. As seen from Eq. (5), the excita- tion spectrum enjoys the azimuthal symmetry. There- fore we only plot the spectrum along two typical direc- tions in Fig. 2. Along the z-axis, the lower branch show the features of roton-maxon structure, indication of the tendency toward crystallization [36]. Such roton-maxon spectrum has been detected in recent Braggspectroscopy experiments [41–43], and the spectrum is asymmetrical with respect to reversing the direction. In the x-yplane, the two branches are well separated as the upper branch is gapped while the lower branch becomes gapless as it approaches the origin q= (0,0,0). Aside from the roton mode discussed above, the lower branch of the excitation spectrum also contains impor- tant information about the photon mode. Along zdi- rection where q⊥= 0, it is straight forward to ana-0 0.5 11.31.82 θq/πvs/√gn0 0 0.5 1012 θq/πvs/√gn0 λ= 0.5 λ= 1.0 λ= 1.5η= 1.0 η= 1.2 η= 1.4 (a) (b) FIG. 3. (color online) Polar angle θqdependence of the sound velocity vs: (a) for different spin-orbit coupling strength λat η= 2.0; (b) for different interspecies coupling ηatλ=√gn0. lytically derive two branches of solutions from Eq. (5): ω−=ω10andω+=−2λqz+ω20. The sound velocity alongthis direction is vs z= 2√gn0. In thex-yplane, low- energy expansion around the gapless point (0 ,0,0) yields ω−≈vs ⊥q⊥+O(q2 ⊥) with in-plane isotropic sound veloc- ity given by vs ⊥=√2gn0/radicalbig 2(η−1)/[2(η−1)+λ2/gn0]. Numerically we compute the sound velocity via vs(q) = limq→0ω−(q)/q. We find that the sound velocity varies with the polar angle θq, as shown in Fig. 3. The sound velocity enjoys a symmetry of vs(θq) =vs(π−θq), with the maximum sound velocity achieved along z-axis and the minimum one in the x-yplane. Away from the criti- cal point where η= 1, the spin-orbit coupling suppresses the sound velocity along any polar direction except for θq= 0 andπ, as indicated in Fig. 3( a). Interestingly, as seen in Fig. 3( b), suppression of sound velocity due to spin-orbit coupling could be mitigated by increasing the interspecies coupling, an indication of competing effects of spin-orbit coupling and interspecies coupling. Being an intrinsic property of a BEC, the quantum depletion of the condensates provides vital information concerning the robustness of the superfluid state. The number density of exited particles can be evaluated by employing the quasi-particle’s Green’s function nex=/summationdisplay q,iwn[G11(q,iwn)+G22(q,iwn)].(6) We show the density of the excited particles out of the condensates due to quantum fluctuation in Fig. 4. At a fixed interspecies coupling η, the quantum depletion is monotonically enhanced by spin-orbit coupling, and it reduces to the case of spinless Bose gases with nex= (gn0)3/2/(3π2) in the absence of spin-orbit coupling [44], as seen in Fig. 4( a). At a fixed spin-orbit coupling4 0 2 450.030.070.110.13 λ/√gn0nex η= 1.0 η= 1.5 η= 2.0 1 3 50.030.040.05 ηnex λ= 0.5 λ= 1.0 λ= 1.5(a)(b) FIG. 4. (color online) Density of the excited particles due to quantum fluctuation nex[in units of ( gn0)3/2]: (a) as a function of spin-orbit coupling strength λfor three typical in- terspecies coupling strength η= 1.0,η= 1.5 andη= 2.0; (b) as a function of interspecies coupling strength ηfor three typ- ical spin-orbit coupling strength λ= 0.5√gn0,λ= 1.0√gn0 andλ= 1.5√gn0. strength, the interspecies coupling actually suppresses quantum depletion, signifying the competing effects of spin-orbit coupling and interspecies coupling upon quan- tum depletion. When the spin-orbit coupling is small, the effect of interspecies coupling decreases as well, as indicated in Fig. 4( b). This is quite remarkable, because there is only one species of condensation. In the absence of spin-orbit coupling, we do not expect that the the in- terspecies coupling plays any role in quantum depletion. We attribute this behavior to stemming from quantum fluctuation enhanced by spin-orbit coupling. The thermodynamic potential of this system is given by Ω = −lnZ/β= Ω0+ Ωf, where the mean-field part is Ω 0=−Vgn2 0and the fluctuating part is Ω f= 1 2βTrlnG−1−1 2/summationtext qσǫqσ. The thermodynamic potential Ω possesses an ultraviolet divergence, an artifact of zero range interaction, which can be removed either by re- placing the bare interaction g with a Tmatrix [45] or by subtracting counter-terms [46]. At zero temperature, the ground-state energy becomes EG= Ω +µN, renormal- ized as EG=EMF+/summationdisplay qs=±/bracketleftbiggωqs−(ǫq↑+ǫq↓)/2 2+g2n2 0 2q2/bracketrightbigg .(7) HereEMF=V(gn2 0−λ2 4) is the mean-field energy. We show the shift of ground state energy due to quantum fluctuation ∆ EG=EG−EMFin Fig. 5. As seen in panel (a), at a fixed interspecies coupling η, the shift of the ground state energy ∆ EGdecreases monotonically with the strengthofspin-orbitcoupling λ. In the absence ofthe spin-orbitcouplingand interspeciesinteraction, we have checked that the ground state energy EGrecovers the well-known Lee-Huang-Yang result [47] for spinless and weakly-interacting Bose gases with EG/V=µn 2(1+ 128 15√π)√ na3, whereais the scattering length. While, for0 2 45−0.08−0.020.040.10.12 λ/√gn0∆EG 1 3 50.080.090.10.110.11 η∆EG η= 1.0 η= 1.5 η= 2.0λ= 0.5 λ= 1.0 λ= 1.5(a) (b) FIG. 5. (color online) The fluctuation shift of ground state energy ∆ EG=EG−EMF[measured in units of V(gn0)5/2]: (a) as a function of spin-orbit coupling strength λfor three typical interspecies coupling strength η= 1.0,η= 1.5 and η= 2.0; (b) as a function of interspecies coupling strength η for three typical spin-orbit coupling strength λ= 0.5√gn0, λ= 1.0√gn0andλ= 1.5√gn0. 0 2 400.51 q⊥/√gn0S(q) 0 2 400.51 q⊥/√gn0S(q) 0 0.5 10.250.350.45 θq/πS(q) 0 0.5 10.30.40.5 θq/πS(q) λ= 0.5 λ= 1.0 λ= 1.5η= 1 η= 2 η= 3 λ= 0.5 λ= 1.0 λ= 1.5η= 1 η= 2 η= 3λ= 2.0 q= 1.0λ= 2.0q= 1.0η= 1.5 η= 1.5(b) (c)(d)(a) FIG. 6. (color online) Distribution of the static structure fac- torS(q) in the momentum space. Upper panel: as a function of in-plane momentum q⊥for (a) different spin-orbit coupling strength λand (b) different interspecies strength η. Lower panel: as a function of polar angle θqfor (c) different spin- orbit coupling strength λand (d) different interspecies cou- pling strength η. a finite spin-orbit coupling, the shift of the ground state energy increases with interspecies coupling η, evidently shown in panel (b). The static structure factor S(q) probes density fluctu- ations of a system. It provides information on both the spectrum of collective excitations, which could be inves- tigated at low momentum transfer, and the momentum distribution, which characterizes the behavior of the sys- tem at high momentum transfer, where the response is dominated by single-particle effects. At the Bogoliubov5 level, it can be evaluated as NS(q) =<δρ† qδρq> =N0/summationdisplay iwn(G11+G33+G13+G31) =N0/summationdisplay iwn−2q2A(q,iwn) det[G−1(q,iwn)], (8) whereA(q,iwn) = (iwn+ 2λqz)2−ω2 20+λ2ω20sin2θq. It is quite clear that the static structure factor possesses the cylindrical symmetry S(q) =S(q,θq). Atq⊥= 0, the static structure factor adopts a close form as follows S(q⊥= 0,qz) =N0 Nq2 ω10cothβω10(q) 2.(9) In this case, it recovers the Feynman relation [48, 49], which connects the static structure factor to the exci- tations spectrum of a Bose system with time-reversal symmetry. We show the behavior of the static struc- ture in Fig. 6. In the upper panel, we show the in-plane static structure factor S(q,θq=π/2) in terms of in-plane momentum q⊥. It decreases as the spin-orbit coupling strength is increased, but increases as the interspecies coupling is increased. Such reversing trend signifies that spin-orbit coupling and interspecies coupling act with re- versal role in the density response of the system. In the lower panel, we show angular dependence of the static structure factor at q=√gn0. It is interesting to notice thatS(q) is also symmetrical with reflection about the x-yplane, namely S(q,θq) =S(q,π−θq). The staticstructure factor develops its minimum along θq=π/2. The spin-orbit coupling suppresses the density response greatly in the x-yplane, as seen in panel (c). In turn, the interspecies coupling enhances the density response greatly along the direction of θq=π/2. IV. SUMMARY AND CONCLUSIONS To sum up, wehavestudied two-componentBosegases in the presence of Weyl-type SOC. We obtain the phase diagram via a variational approach. We find competing effects between spin-orbit coupling and interspecies cou- pling strength upon various properties of the PW-Polar phase. There is one crucial difference between them: spin-orbit coupling allows the process of pseudospin flip- ping process, while interspecies interaction does not per- mit that. This has far-reaching consequence in the quan- tum depletion of the condensates. 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0709.3521v2.Spin_orbit_mediated_anisotropic_spin_interaction_in_interacting_electron_systems.pdf

arXiv:0709.3521v2 [cond-mat.mes-hall] 23 Apr 2008Spin-orbit mediated anisotropic spin interaction in inter acting electron systems Suhas Gangadharaiah, Jianmin Sun and Oleg A. Starykh Department of Physics, University of Utah, Salt Lake City, U T 84112 (Dated: August 21, 2021) We investigate interactions between spins of strongly corr elated electrons subject to the spin-orbit interaction. Our main finding is that of a novel, spin-orbit m ediated anisotropic spin-spin coupling of the van der Waals type. Unlike the standard exchange, this interaction does not require the wave functions to overlap. We argue thatthis ferromagnetic inte raction is important in theWigner crystal state where the exchange processes are severely suppressed . We also comment on the anisotropy of the exchange between spins mediated by the spin-orbital cou pling. PACS numbers: 71.70.Ej, 73.21.La, 71.70.Gm Introduction. Studies of exchange interaction between localized electrons constitutes one of the oldest topics in quantum mechanics. Strong current interest in the pos- sibility to control and manipulate spin states of quantum dots has placed this topic in the center of spintronics and quantum computation research. As is known from the papers of Dzyaloshinskii [1] and Moriya [2], in the pres- ence of the spin-orbital interaction (SOI) the exchange is anisotropic in spin space. Being a manifestation of quantum tunneling, the ex- change is exponentially sensitive to the distance between electrons [3]. This smallness of the exchange leads to a large spin entropy of the Wigner crystal state, as compared to the Fermi liquid state, of diluted two- dimensional electron gas in semiconductor field-effect transistors [4]. The consequence of this, known as the Pomeranchukeffect, isspectacular: Wignercrystalphase is stabilized by a finite temperature. In this work we show that when subjected to the spin-orbit interaction, as appropriate for the structure- asymmetric heterostructures and surfaces [5], interacting electrons acquire a novel non-exchange coupling between the spins. The mechanism of this coupling is very similar to that of the well-known van der Waals (vdW) interac- tion between neutral atoms. This anisotropic interaction is of the ferromagnetic Ising type. It lifts extensive spin degeneracy of the Wigner crystal and leads to the long- range ferromagnetic order. We also re-visit and clarify the role of spin-orbit interaction in lowering the sym- metry of the exchange coupling between spins. Particu- larly, we point out that the exchange Hamiltonian, de- spite its anisotropic appearance, retains spin-rotational invariance to the second order in the spin-orbital cou- pling. We argue that spin-rotational symmetry is broken only in the forthorder in SOI coupling. Calculation of the vdW coupling. To illuminate the origin of the vdW coupling, we consider the toy prob- lem of two single-electron quantum dots described by the double well potential [6, 7], see Figure 1, /tildewideV(xj,yj) =mω2 x 2a2(x2 j−a2 4)2+mω2 y 2y2 j,(1)(+a 2,0,0)x(−a 2,0,0)z yσ1σ2 FIG. 1: (color online) Two-dot potential (1). Blue (darkgre y) arrows indicate electron’s spins. whereωx/yare confinement frequencies along x/ydirec- tions. The electrons, indexed by j= 1,2, are subject to SOI of the Rashba type[5] with coupling αR HSO=/summationdisplay j=1,2αR/vector pj×/vector σj·ˆz, (2) where/vector σiare the Pauli matrices and ˆ zis normal to the plane of motion. Finally, electrons experience mutual Coulomb repulsion so that the total Hamiltonian reads H=/summationdisplay j=1,2[/vector p2 j 2m+/tildewideV(xj,yj)]+e2 |/vector r1−/vector r2|+HSO.(3) At large separation between the two dots the exchange is exponentially suppressed and the electrons can be treated as distinguishable particles. One then expects that Coulomb-induced correlations in the orbital motion of the electrons in two dots translate, via the spin-orbit interaction, into correlation between their spins. Con- sider the distance between the dots, a, much greater than the typical spread of the electron wave functions, 1/√mωx. In this limit the electrons are centered about different wells, and the potential can be approximated as V(/vector r1,/vector r2)≈1 2mω2 x((x1−a/2)2+(x2+a/2)2) +1 2mω2 y(y2 1+y2 2). (4)2 At this stage it is crucial to perform a unitary transfor- mation [8, 9] which removes the linear spin-orbit term from (3) U= exp[imαRˆz·(/vector r1×/vector σ1+/vector r2×/vector σ2)].(5) Owing to the non-commutativity of Pauli spin matrices, SOI can not be eliminated completely, resulting in higher order in the Rashba coupling αRcontributions as given by/tildewideH=UHU†below /tildewideHSO=/summationdisplay j=1,2[−mα2 R˜Lz j˜σz j+4 3m2α3 R(yj˜σy j+xj˜σx j)˜Lz j +2 3im2α3 R(yj˜σx j−xj˜σy j)]+O(α4 R). (6) Here˜Lz jis the angular momentum of the jthelectron, ˜Lz=x˜py−y˜px, andtildedenotesunitarilyrotatedopera- tors. The calculationis easiest when the confining energy is much greater than both the Coulomb energy e2/aand the spin-orbit energy scale√mωαR. In terms of the new (primed) coordinates /vectorr′1=/vector r1−/vector a/2 and/vectorr′2=/vector r2+/vector a/2 centered about ( a/2,0) and (−a/2,0), respectively, the interaction potential e2/|/vectorr′1−/vectorr′2+/vector a|is expanded in powers of 1 /akeeping terms up to second order in the dimensionless relative distance ( /vectorr′1−/vectorr′2)/a. The linear term,e2(x′ 1−x′ 2)/a2, slightly renormalizes the equilib- rium distance between the electrons and can be dropped from further considerations. In terms of symmetric (S) and anti-symmetric (A) coordinates: xS/A=x′ 1±x′ 2√ 2;yS/A=y′ 1±y′ 2√ 2,(7) the quadratic term e2(2(x′ 1−x′ 2)2−(y′ 1−y′ 2)2)/2a3 renormalizes the anti-symmetric frequency ωAx2→ω2 x+ 4e2/(ma3) andω2 Ay=ω2 y−2e2/(ma3) while leaving the symmetric ones unmodified, ωSx2=ω2 xandω2 Sy=ω2 y. Quite similarly to the textbook calculation of the vdW force[10], the resultingHamiltonian /tildewideH=/tildewideHS+/tildewideHA+/tildewideHSO becomes that of harmonic oscillators /tildewideHS/A=/vector˜p2 S/A 2m+m 2(ω2 xS/Ax2 S/A+ω2 yS/Ay2 S/A) (8) perturbed by /tildewideHSO=/tildewideH(2) SO+δ/tildewideH(2) SO+O(α3 R), where /tildewideH(2) SO=−mα2 R 2[(xS˜pyS−yS˜pxS)+S↔A](˜σz 1+ ˜σz 2) −mα2 R 2[(xS˜pyA−yA˜pxS)+S↔A](˜σz 1−˜σz 2),(9) δ/tildewideH(2) SO=−mα2 Ra 2√ 2[˜pyS(˜σz 1−˜σz 2)+ ˜pyA(˜σz 1+ ˜σz 2)].(10) It is evident from Eqn. (9,10) that the leading correc- tions to the ground state energy is obtained either by the excitation of a single y-oscillator (through (10)) and bythe simultaneous excitation of oscillators in both the x andydirections (through (9)), ∆E=−/summationdisplay i,j=S,A|/an}bracketle{t0|δ/tildewideH(2) SO|1yi/an}bracketri}ht|2 ωiy+|/an}bracketle{t0|/tildewideH(2) SO|1xi1yj/an}bracketri}ht|2 ωix+ωjy. It is easy to see that the spin-dependent contributions fromδ/tildewideH(2) SOcancel exactly while those originating from /tildewideH(2) SOdo not, resulting in the novel spin interaction HvdW=1 8m2α4 R˜σz 1˜σz 2/parenleftBig φ(ωSy,ωSx)+φ(ωAy,ωAx) −φ(ωAy,ωSx)−φ(ωSy,ωAx)/parenrightBig , (11) where the function φis given by a simple expression φ(x,y) =(x−y)2 xy(x+y). (12) In case of cylindrically symmetric dots, ωx=ωy, HvdW=−α4 Re4 4a6ω5x˜σz 1˜σz 2. (13) Thephysicsofthisnovelinteractionisstraightforward: it comes from the interaction-induced correlation of the or- bital motion of the two particles, which, in turn, induces correlations between their spins via the spin-orbit cou- pling. The net Ising interaction would have been zero if not for the shift in frequency of the anti-symmetric mode due to the Coulomb interaction. Note that the coupling strengthexhibitsthesamepower-lawdecaywithdistance as the standard van der Waals interaction [10]. From (11), it follows that in the extreme anisotropic limit ofωy→ ∞, or equivalently, the one-dimensional (1D)limit, thereisnocouplingbetweenspins. Thisresult is understood by noting that 1D version of SOI, given by αR/summationtext jσy jpx j, can be gauged away to all orders in αRby a unitary transformation U1D= exp[imαR(x1σy 1+x2σy 2)]. Hence the absence of the spin-spin coupling in this limit. However, either by including magnetic field (Zeeman in- teraction, see below) in a direction different from σy, or by increasing the dimensionality of the dots by reducing the anisotropy of the confining potential, the spin-orbital Hamiltonian acquiresadditionalnon-commutingspin op- erators. The presence of the mutually non-commuting spin operators (for example, σxandσyin (2)) makes it impossible to gaugethe SOI completely, opening the pos- sibility of fluctuation-generated coupling between distant spins, as in equation (13). Effect of the magnetic field. For simplicity, we neglect orbital effects and concentrate on the Zeeman coupling, HZ=−∆z/summationtext jσz j/2,where∆ z=gµB. Unitarytransfor- mation(5)changesitto HZ−∆zmαRa(σx 1−σx 2)/2+δ/tildewideHZ. Here δ/tildewideHZ=−/summationdisplay j=1,2mαR∆z(x′ j˜σx j+y′ j˜σy j) (14)3 describes the coupling between the Zeeman and Rashba terms. In the basis (7) it reduces to δ/tildewideHZ=−m∆zαRyS(σy 1+σy 2)+xS(σx 1+σx 2)√ 2, −m∆zαRyA(σy 1−σy 2)+xA(σx 1−σx 2)√ 2.(15) For sufficiently strong magnetic field, ∆ z≫√mωαR, /tildewideHSOcan be neglected in comparison with δ/tildewideHZ. Calcu- lating second order correction to the ground state energy of the two dots, represented as before by /tildewideHS+/tildewideHA, and extracting the spin-dependent contribution, we obtain ∆EZ=−∆2 zα2 Re2 a3(2σx 1σx 2 ω4x−σy 1σy 2 ω4y).(16) Intheextremeanisotropiclimit ωy→ ∞thedotsbecome 1D and we recoverthe result ofRef. 11. For the isotropic limitωx=ωy, the coupling of spins acquires a magnetic dipolar structure identical to that found in Ref. 12. Anisotropy of the exchange. Next, we allow for the electron tunneling between the dots. The spin dynamics of the electrons is now described by the sum of exchange and the van der Waals interactions, H=HEx+HvdW. Here the exchangecoupling, HEx, containsboth isotropic andpossibleanisotropicinteractions,while HvdWisgiven by (11) and (13). In the absence of spin-orbit interac- tion, the total spin is conserved and the Hamiltonian is SU(2) invariant. As such, the only spin interaction al- lowed has the well known isotropic form HEx∼/vector σ1·/vector σ2. The anisotropy of the exchange is mediated by the spin- rotational symmetry breaking SOI (2). When the tun- neling is no longer spin-conserving, electron spins precess while exchanging their respective positions, giving rise to the anisotropic terms. As a result [13, 14, 15] HEx=J 4/parenleftBig b/vector σ1·/vector σ2+Dˆd·/vector σ1×/vector σ2+Γ(ˆd·/vector σ1)(ˆd·/vector σ2)/parenrightBig ,(17) whereˆdistheunit Dzyloshinskii-Moriyavector, ofampli- tudeD, with odd dependence on the spin-orbit coupling αR. Coefficients band Γ have even dependence on the spin-orbit coupling [2, 16], while the exchange integral J, independent of αRin this representation, sets the over- all energy scale. The direction of the DM vector can be understood as follows. As the D-term must be even un- der exchange operation P: 1↔2, its amplitude must be odd with respect to inter-spin distance /vector a=/vector r1−/vector r2=aˆx, henceˆd∼ˆa= ˆx. In addition, as ˆ z→ −ˆztransformation in (2) changes sign of αR, it must be that ˆd∼ˆzas well. Thus, it must be that ˆd= ˆz׈a= ˆy. In the simplest approximation one neglects the “rem- nants” of SOI (6) altogether and writes the only possible exchange coupling /tildewideH(0) Ex=J 4/vector˜σ1·/vector˜σ2in terms of unitarily transformed spin operators /vector˜σj. The meaning of this in- teraction is understood in the original basis by undoingthe unitary transformation, H(0) Ex=U†/tildewideH(0) ExU. Using (5) and replacing /vector r1, /vector r2by their respective average values, a/2ˆxand−a/2ˆx, one observes that spin 1 (2) is rotated about ˆyaxis by the angle θ=mαRain clockwise (coun- terclockwise) direction. As a result, one immediately ob- tains Eq.(17) with parameters b0= cos2θ,D0= sin2θ,Γ0= 1−cos2θ,ˆd= ˆy.(18) As it originated from the SU(2)-invariant scalar product /vector˜σ1·/vector˜σ2, the Hamiltonian (17) with parameters (18) does notbreak spin-rotational SU(2) symmetry, despite its asymmetric appearance. Because of its “non-diagonal” nature, the D-term affects the eigenvalues only in D2∼ θ2order, and must always be considered together with the Γ-term. In the current situation (18), the two contri- butions compensate each other exactly. This important observation, made in Ref.16 (see also [17]), was over- looked in several recent calculations of the DM term [13, 18, 19]. It is thus clear that the symmetry-breaking DM term must originate from so far omitted /tildewideHSO(6). To cap- ture it, we set up the exchange problem calculation along the lines of the standard Heitler-London (HL) approach. Despite its well-known shortcomings [20, 21, 22], this approach offers conceptually simple way to estimate ex- change splitting [7] and the structure of anisotropic spin coupling. Ourbasis set is formed bythe antisymmetrized two-particle wave function |/tildewideψ/an}bracketri}ht=|ψ/an}bracketri}ht−P|ψ/an}bracketri}ht, |ψ/an}bracketri}ht=ϕ(1,2){c1| ↑↑/an}bracketri}ht+c2| ↑↓/an}bracketri}ht+c3| ↓↑/an}bracketri}ht+c4| ↓↓/an}bracketri}ht}(19) is written in terms of unknown coefficients c1−4. Here ϕ(1,2) =f(x1−a/2)f(y1)f(x2+a/2)f(y2) describes spatial wave function of distinguishable particles lo- calized near ( a/2,0) and ( −a/2,0), respectively, and f(x−x0) denotes the ground state wave function of one- dimensional harmonic oscillator centered around x=x0. As constructed, ϕ(1,2) is the lowest energy eigenstate of two particles moving in the potential profile (4). The rest of the confining potential, Eq.(1), together withthe SOI (6), forms the perturbation Vpert(1,2) =/summationdisplay j=1,2/tildewideV(xj,yj)−V(/vector r1,/vector r2)+/tildewideHSO,(20) which is responsible for removing spin degeneracy of states contributing to (19). The eigenvalue problem (H0+Vpert)|/tildewideψ/an}bracketri}ht=E|/tildewideψ/an}bracketri}ht, (21) whereH0is the sum of kinetic energy and confinement potential (4), is formulated as a 4 ×4 matrix problem by multiplying (21) by the bra /an}bracketle{ts1s2|ϕ(1,2) from the left (heresj=1,2=↑or↓) and integrating the result over the whole space. The obtained exchange Hamiltonian for the rotatedspins/vector˜σis of the form (17) with J=3 2mω2 xa2e−mωxa2/2,D=32mα3 R 9ωxωya,(22)4 whileb= 1,Γ = 0to this order. The calculationsketched is valid in the large separation limit, a≫1/√mωx, and its mostimportantfeatureisthe scaling D∼α3 Rbetween the DM coupling and the spin-orbital one. This result is due to the fact that O(α2 R) term in (6) excites both x andyoscillators. Since the wave function (19) contains only the ground states of the oscillators, the O(α2 R) term drops out and the first asymmetric correction originates inO(α3 R) terms of (6). We checked that this crucial fea- tureisnotanartifactoftheHLapproximationandisalso obtained from a more reliable “median-plane” approach [19, 21, 22, 23], which we initiated. Noting that the DM term Dˆy·/vector˜σ1×/vector˜σ2affects the eigen- value of the two-spin problem only in D2order, we con- clude that exchange asymmetry due to the spin-orbit in- teraction may appear only in α4 Ror higher order. This is because the effect of Γ-term in (17) on the eigenval- ues is of first order in Γ, and our calculation shows that Γ∼O(α4 R). Being proportional to J, see (22), this con- tribution is also exponentially small. We then conclude that the leading source of spin anisotropy is provided by the vdW contribution (11) and (13), which does not con- tain an exponential smallness of the exchange. Estimate of the vdW coupling. We now turn our at- tention to physical manifestations of the vdW spin cou- pling in the Wigner crystal. Neglecting the exchange interaction for the moment, we consider a two-electron problem within the frozen lattice approximationin which all other electrons are assumed fixed in their equilibrium lattice positions. The potential energy then is just that of four harmonic oscillators [24] with frequencies ωξ,η=/radicalbig (γ∓2)/(m2aBa3) andωu,v=/radicalbig (γ∓1)/(m2aBa3), in notations of Ref.24. Here γ≈5.52,[24]aB=κ/(me2) is the Bohr radius, κis the dielectric constant and a is the lattice constant of the electron crystal, inversely proportional to the electron density n:a= (2/√ 3n)1/2. Repeating the steps that led to (11) we obtain for the Wigner crystal problem Hwigner vdW=m2α4 RB˜σz 1˜σz 2=gvdW˜σz 1˜σz 2(23) whereB= [φ(ωξ,ωv) +φ(ωη,ωu)−φ(ωξ,ωu)− φ(ωη,ωv)]/8 =−3.75·10−3√ m2aBa3. The spin-orbit mediated ferromagnetic coupling removes extensive spin degeneracy of the crystal, suppressing the Pomeranchuk effect physics [4]. Being of non-frustrated nature, it es- tablisheslong-rangemagneticorderofIsingtypewiththe ordering temperature of the order of the vdW constant gvdW(23). It should be compared with the much stud- ied Heisenberg exchange Jwc=c(rs)exp[−1.612√rs], expressed in Rydbergs R= 1/(2ma2 B). Herers= 1//radicalbig πa2 Bnis the dimensionless measure of the interac- tion strength, and the pre-factor c(rs) is a smooth func- tion of it [3]. We find that gvdWdominates the exchangeforrs> r∗ s≈20 in InAs, which has αR≈1.6·104m/s [25]. For GaAs, with αR≈300m/s [26], more diluted situation is required, r∗ s≈90. Given that multi-particle ring-exchangeprocesseson the triangularlattice strongly frustrateany orderingtendencies due to the exchange[3], it appears that our estimate is just a lower bound on the critical density below which spin-orbit-inducedferromag- netic state should be expected. We would like to thank L. Balents, L. Glazman, D. Maslov, E. Mishchenko, M. Raikh, O. Tchernyshyov, and, especially, K. Matveev, for productive discussions. O.A.S. is supported by ACS PRF 43219-AC10. [1] I.E. Dzyaloshinskii, J. Phys. Chem. Solids 4, 241 (1958). [2] T. Moriya, Phys. Rev. Lett. 4, 228 (1960); Phys. Rev. 120, 91 (1960). [3] B. Bernu, L. Candido, and D.M. Ceperley, Phys. Rev. Lett.86, 870 (2001). [4] B. Spivak and S.A. Kivelson, Phys. Rev. B 70, 155114 (2004). [5] Yu. A. Bychkov and E.I. Rashba, J. Phys. C 17, 6039 (1984). [6] G. Burkard, D. Loss, and D.P. DiVincenzo, Phys. Rev. B59, 2070 (1999). [7] M.J. Calderon, B. Koiller, and S. Das Sarma, Phys. Rev. B74, 045310 (2006). [8] T.V. Shahbazyan and M.E. Raikh, Phys. Rev. Lett. 73, 1408 (1994). [9] I.L. Aleiner and V.I. Fal’ko, Phys. Rev. Lett. 87, 256801 (2001). [10] D.J. Griffiths, Introduction to Quantum Mechanics , (Pearson, Upper Saddle River, 2005), 2nd ed., p. 286. [11] C. Flindt, A.S. Sorensen, and K. Flensberg, Phys. Rev. Lett.97, 240501 (2006). [12] M.Trif, V.N. Golovach, and D. Loss, Phys. Rev. B 75, 085307 (2007). [13] K.V. Kavokin, Phys. Rev. B 69, 075302 (2004). [14] D. Stepanenko et al., Phys. Rev. B 68, 115306 (2003). [15] H. Imamura, P. Bruno, and Y. Utsumi, Phys. Rev. B 69, 121303 (2004). [16] L. Shekhtman, O. Entin-Wohlman, and A. Aharony, Phys. Rev. Lett. 69, 836 (1992). [17] A. Zheludev et al., Phys. Rev. B 59, 11432 (1999). [18] K.V. Kavokin, Phys. Rev. B 64, 075305 (2001). [19] L.P. Gor’kov and P.L. Krotkov, Phys. Rev. B 67, 033203 (2003); Phys. Rev. B 68, 155206 (2003). [20] C. Herring, Rev. Mod. Phys. 34, 631 (1962) [21] C. Herring and M. Flicker, Phys. Rev. 134, A362 (1964) [22] L.P. Gor’kov and L.P. Pitaevskii, Sov. Phys. Dokl. 8, 788 (1964). [23] I.V. Ponomarev, V.V. Flambaum, and A.L. Efros, Phys. Rev. B60, 15848 (1999). [24] V.V. Flambaum, I.V. Ponomarev, and O.P. Sushkov, Phys. Rev. B 59, 4163 (1999). [25] D. Grundler, Phys. Rev. Lett. 84, 6074 (2000). [26] C.P. Weber et al., Phys. Rev. Lett. 98, 076604 (2007).

0801.0924v1.Spin_orbit_and_tensor_mean_field_effects_on_spin_orbit_splitting_including_self_consistent_core_polarizations.pdf

arXiv:0801.0924v1 [nucl-th] 7 Jan 2008Spin-orbit and tensor mean-field effects on spin-orbit split ting including self-consistent core polarizations M. Zalewski,1J. Dobaczewski,1,2W. Satu/suppress la,1and T.R. Werner1 1Institute of Theoretical Physics, University of Warsaw, ul . Ho˙ za 69, 00-681 Warsaw, Poland. 2Department of Physics, P.O. Box 35 (YFL), FI-40014 Universi ty of Jyv¨ askyl¨ a, Finland (Dated: October 25, 2018) A new strategy of fitting the coupling constants of the nuclea r energy density functional is pro- posed, which shifts attention from ground-state bulk to sin gle-particle properties. The latter are analyzed in terms of the bare single-particle energies and m ass, shape, and spin core-polarization effects. Fit of the isoscalar spin-orbit and both isoscalar a nd isovector tensor coupling constants directly to the f5/2−f7/2spin-orbit splittings in40Ca,56Ni, and48Ca is proposed as a practical realization of this new programme. It is shown that this fit re quires drastic changes in the isoscalar spin-orbit strength and the tensor coupling constants as co mpared to the commonly accepted values but it considerably and systematically improves basic sing le-particle properties including spin-orbit splittings and magic-gap energies. Impact of these changes on nuclear binding energies is also discussed. PACS numbers: 21.10.Hw, 21.10.Pc, 21.60.Cs, 21.60.Jz I. INTRODUCTION In this work we propose and explore two new ideas per- taining to the energy density functional (EDF) methods. First, we suggest a necessity of shifting attention and fo- cus of these methods from ground-state bulk properties (e.g. total nuclear masses) to single-particle (s.p.) prop- erties, and to look for a spectroscopic-quality EDFs that would correctly describe nuclear shell structure. Proper positions of s.p. levels are instrumental for good descrip- tion of deformation, pairing, particle-core coupling, and rotational effects, and many other phenomena. On the one hand, careful adjustment of these posi- tions were at the heart of tremendous success of phe- nomenological mean-field (MF) models, like those of Nils- son, Woods-Saxon, or folded Yukawa [1]. On the other hand, similarly successful phenomenological description of nuclear masses, within the so-called microscopic- macroscopic method [2], relies on the liquid-drop mass surface, which is entirely decoupled from the s.p. struc- ture. Up two now, methods based on using EDFs, in any of its variants like local Skyrme, non-local Gogny, or relativistic-mean-field (RMF) [3] approach, were mostly using adjustments to bulk nuclear properties. As a result, shell properties were described poorly. After so many years of investigations, a further increase in precision and predictability of all methods based on the EDFs may re- quire extensions beyond forms currently in use. Before this can be fully achieved, we propose to first take care of the s.p. properties, and come back to precise adjustment of bulk properties once these extensions are implemented. Second, we propose to look at the s.p. properties of nuclei through the magnifying glass of odd-even mass differences. This idea has already been put forward in a seminal paper by Rutz et al. [4], where calculations performed within the RMF approach were presented. In this paper we perform analogous analysis for the EDFsbased on the Skyrme interactions. On the one hand, it was recognized long time ago, cf., e.g., Refs. [5, 6], that the theoretical s.p. energies, defined as eigenvalues of the MF Hamiltonian, cannot be directly compared to experiment, because they are strongly renormalized by the particle-core coupling. On the other hand, procedures used to deduce the s.p. ener- gies from experiment [7, 8, 9] require various theoretical assumptions, by which these quantities cease to result from direct experimental observation. In the past, these theoretical and experimental caveats hampered the use of s.p. energies for proper adjustments of EDFs. However, the odd-even mass differences carry very similar physi- cal information to that given by s.p. energies, and have advantage of being clearly defined, both experimentally and theoretically. Indeed, experimental difference in mass between an odd nucleus and its lighter even-even neighbor, i.e., the particle separation energy, is an easily available and un- ambiguous piece of data, which reflects the physical role of the s.p. energy, with all polarizations and couplings taken into account. Similarly, differences of masses be- tween the low-lying excited states in an odd system and lighter even-even neighbor may illustrate effective posi- tions of higher s.p. states. Physical connections between these mass differences and s.p. energies are closest in semi-magic nuclei, which will be studied in the present paper. In theory, the primary goal of the EDF methods is to describe ground-state energies of fermion systems, i.e., in nuclear-physics applications – masses of nuclei. For odd systems, the EDF methods should give masses of ground states and of several low-lying excited states; the lat- ter being obtained by blocking specific s.p. orbitals. We stress here that in an odd system, separate self-consistent calculations have to be performed for each of the blocked states, so as to allow the system for exploring all possible polarizations exerted by the odd particle on the even-even2 core. Again, this procedure is clearly defined and entirely within the scope and spirit of the EDF method, which is supposed to provide exact energies of correlated states. Note, that although one here calculates total masses of odd and even systems, the comparison with experiment involves the differences of masses , for which many effects cancel out. Therefore, one can confidently attribute cal- culated differences of masses to properties of effective s.p. energies, with all polarization effects taken into account, like in the experiment. Again, in semi-magic nuclei such a connection is most reliable. Following the above line of reasoning, in the present paper we study experimental and theoretical aspects of the spin-orbit (SO) splitting between the s.p. states. In particular, we analyze the role of the SO and tensor MFs in providing the correct values of the SO split- tings across the nuclear chart. The paper is organized as follows: In Sect. II we briefly recall basic theoretical building blocks related to the SO and tensor terms of the EDFs and interactions. In Sect. III we discuss in details three major sources of the core polarization: the mass, shape, and spin polarizabilities. In Sect. IV we present a novel method that allows for a firm adjustment of the SO and tensor coupling constants arguing that currently used functionals require major revisions con- cerning strengths of both these terms. The analysis is based on the f7/2−f5/2SO splittings in three key nuclei, including spin-saturated isoscalar40Ca, spin-unsaturated isoscalar56Ni, and spin-unsaturated isovector48Ca sys- tems, and is subsequently verified by systematic calcula- tions of the SO splittings in magic nuclei. In Sect. V we discuss an impact of these changes on binding energies in magic nuclei. Conclusions of the paper are presented in Sect. VI. II. SPIN-ORBIT AND TENSOR ENERGY DENSITIES, MEAN FIELDS, AND INTERACTIONS We begin by recalling the form of the EDF, which will be used in the present study. In the notation defined in Ref. [10] (see Ref. [11] for more details and extensions), the EDF reads E=/integraldisplay d3rH(r), (1) where the local energy density H(r) is a sum of the kinetic energy, and of the potential-energy isoscalar ( t= 0) and isovector ( t= 1) terms, H(r) =/planckover2pi12 2mτ0+H0(r) +H1(r), (2) with Ht(r) =Heven t(r) +Hodd t(r), (3)and Heven t =Cρ tρ2 t+C∆ρ tρt∆ρt+ (4) Cτ tρtτt+CJ tJ2 t+C∇J tρt∇·Jt, Hodd t=Cs ts2 t+C∆s tst·∆st+ (5) CT tst·Tt+Cj tj2 t+C∇J tst·(∇×jt). For the time-even, ρt,τt, andJt, and time-odd, st,Tt, andjt, local densities we follow the convention introduced in Ref. [12], see also Refs. [3, 11] and references cited therein. In particular, the SO density Jis the vector part of the spin-current tensor density J, i.e., Jµν=1 3J(0)δµν+1 2εµνηJη+J(2) µν, (6) with J2≡/summationdisplay µνJ2 µν=1 3(J(0))2+1 2J2+/summationdisplay µν(J(2) µν)2.(7) In the context of the present study, the time-even ten- sor and SO parts of the EDF, HT=CJ 0J2 0+CJ 1J2 1, (8) HSO=C∇J 0ρ0∇·J0+C∇J 1ρ1∇·J1, (9) are of particular interest. In the spherical-symmetry limit, the scalar J(0)and symmetric-tensor J(2) µνparts of the spin-current tensor vanish, and thus HT=1 2CJ 0J2 0(r) +1 2CJ 1J2 1(r), (10) HSO=−C∇J 0J0(r)dρ0 dr−C∇J 1J1(r)dρ1 dr,(11) where the SO density has only the radial component, Jt=r rJt(r). Variation of the tensor and SO parts of the EDF over the radial SO densities J(r) gives the spherical isoscalar ( t= 0) and isovector ( t= 1) SO MFs, WSO t=1 2r/parenleftbigg CJ tJt(r)−C∇J tdρt dr/parenrightbigg L·S,(12) which can be easily translated into the neutron ( q=n) and proton ( q=p) SO MFs, WSO q=1 2r/braceleftbigg (CJ 0−CJ 1)J0(r) + 2CJ 1Jq(r) (13) −(C∇J 0−C∇J 1)dρ0 dr−2C∇J 1dρq dr/bracerightbigg L·S. Although below we perform calculations without assum- ing spherical and time-reversal symmetries, here we do not repeat general expressions for the SO mean-fields, which can be found in Refs. [11, 12]. We also note that, in principle, in this general case, one could use different coupling constant multiplying each of the three terms in3 Eq. (7). In the present exploratory study, we do not implement this possible extension of the EDF, and we use unique tensor coupling constants CJ t, as defined in Eq. (4). Identical potential-energy terms of the EDF, Eqs. (4) and (5), are obtained by averaging the Skyrme effective interaction within the Skyrme-Hartree-Fock (SHF) ap- proximation [3]. By this procedure, the EDF coupling constants Ctcan be expressed through the Skyrme-force parameters, and one can use parameterizations existing in the literature. It is clear that one can study tensor and SO effects entirely within the EDF formalism, i.e., by considering the corresponding tensor and SO parts of the EDF, Eqs. (8) and (9), and coupling constants CJ t andC∇J t, respectively. However, in order to link this approach to those based on the Skyrme interactions, we recall here expressions based on averaging the zero-range tensor and SO forces [13, 14], see also Refs. [11, 15, 16] for recent analyses. Namely, in the spherical-symmetry limit, one has HT=5 8/bracketleftbig teJn(r)Jp(r) +to(J2 0(r)−Jn(r)Jp(r))/bracketrightbig ,(14) HSO=1 4/bracketleftbigg 3W0J0(r)dρ0 dr+W1J1(r)dρ1 dr/bracketrightbigg , (15) where in Eq. (15) two different coupling constants, W0 andW1, were introduced following Ref. [17]. The corresponding SO MFs read WSO q=1 2r/braceleftbigg 5 8((te+to)J0(r)−(te−to)Jq(r)) (16) +1 4/parenleftbigg (3W0−W1)dρ0 dr−2W1dρq dr/parenrightbigg/bracerightbigg L·S, [note that in Ref. [15], the factor of1 2was missing at the W0term of Eq. (4)]. By comparing Eqs. (13) and (16), one obtains the following relations between the coupling constants: CJ 0=5 16(3to+te), (17) CJ 1=5 16(to−te), (18) C∇J 0=−3 4W0, (19) C∇J 1=−1 4W1. (20) For further discussion of the Skyrme forces and their rela- tion to tensor components we refer the reader to an exten- sive and complete recent discussion presented in Ref. [16]. In this exploratory work, we base our considerations on the EDF method and deliberately break the connec- tion between the functional (4), and the Skyrme central, tensor, and SO forces. Nevertheless, in the time-even sector, our starting point is the conventional Skyrme- force-inspired functional with coupling constants fixed at the values characteristic for either SkP [18], SLy4 [19], or SkO [20] Skyrme parameterizations. However, poorly known coupling constants in the time-odd sector (thosewhich are not related to the time-even ones through the local-gauge invariance [10]) are fixed independently of their Skyrme-force values. For this purpose, the spin cou- pling constants Cs tare readjusted to reproduce empirical values of the Landau parameters, according to the pre- scription given in Refs. [21, 22], and C∆s tare set equal to zero. These variants of the standard functionals are below denoted by SkP L, SLy4 L, and SkO L. Strictly pragmatic reasons, like technical complexity and lack of firm experimental benchmarks, made the majority of older Skyrme parameterizations simply dis- regard the tensor terms, by setting CJ t≡0. Recent experimental discoveries of new magic shell-openings in neutron-reach light nuclei, e.g., around N= 32 [23, 24], and their subsequent interpretation in terms of tensor in- teraction within the shell-model [25, 26, 27], caused a re- vival of interest in tensor terms within the MF approach [15, 16, 28, 29, 30, 31], which is naturally tailored to study s.p. levels. Indeed, as shown in Ref. [15], the ten- sor terms mark clear and unique fingerprints in isotonic and isotopic evolution of s.p. levels and, in particular, in the SO splittings. In this paper, we perform systematic study of the SO splittings. The goal is to resolve contributions to the SO MF (12) due to the tensor and SO parts of the EDF, and to readjust the corresponding coupling constants CJ t andC∇J t. It is shown that this goal can be essentially achieved by studying the f7/2−f5/2SO splittings in three key nuclei:40Ca,48Ca, and56Ni. Before we present in Sec. IV details of the fitting procedure and values of the obtained coupling constants, first we discuss effects of the core polarization and its influence on the calculated and SO splittings. III. CORE-POLARIZATION EFFECTS In spite of the fact that the s.p. levels belong to basic building blocks of the MF methods, there is still a vivid debate concerning their physical reality. The question whether they constitute only a set of auxiliary quanti- ties, or represent real physical entities that can be in- ferred from experimental data, was never of any con- cern for methods based on phenomenological one-body potentials. Indeed, these potentials were bluntly fitted directly to reproduce the s.p. levels deduced, in one way or another, from empirical information around doubly- magic nuclei, see, e.g., Refs. [9, 32]. In turn, these potentials also appear to properly (satisfactorily) repro- duce the one-quasiparticle band-heads in open-shell nu- clei, see, e.g., Ref. [33]. This success seems to legitimate the physical reliability of the theoretical s.p. levels within the microscopic-macroscopic approaches. The debate concerns mostly the self-consistent MF ap- proaches based on the EDF methods or two-body ef- fective interactions. The arguments typically brought forward in this context underline the fact that the self- consistent MFs are most often tailored to reproduce bulk4 nuclear properties like masses, densities, radii, and cer- tain properties of nuclear matter. Consequently, the un- derlying interactions are non-local with effective masses m⋆/m∼0.8 [34], what in turn artificially lowers the den- sity of s.p. levels around the Fermi energy. It was pointed out by several authors [5, 6, 35] that restitution of phys- ical density of the s.p. levels around the Fermi energy can be achieved only after inclusion of particle-vibration coupling, i.e., by going beyond MF. Within the effective theories, however, these arguments do not seem to be fully convincing. First of all, an ef- fective theory with properly chosen data set used to fit its parameters should automatically select physical value of the effective mass and readjust other parameters (cou- pling constants) to this value. Examples of such implicit scaling are well known for the SHF theory including: ( i) explicit effective mass scaling of the coupling constants CsandCTthrough the fit to the spectroscopic Landau parameters [21]; ( ii) direct dependence of the isovector coupling constants Cτ 1andCρ 1on the isoscalar effective mass through the fit to the (observable) symmetry en- ergy strength [36]; ( iii) numerical indications for the m⋆ scaling of the SO interaction inferred from the f7/2-d3/2 splittings in A∼44 nuclei [22]. The differences between various parameterizations of the Skyrme-force (or func- tional) parameters rather clearly suggest that such an implicitm⋆dependence of functional coupling constants is a fact, which is, however, neither well recognized nor understood so far. Secondly, the use of effective interaction with parame- ters fitted at the MF level is not well justified in beyond- MF approximation. Rather unavoidable double count- ing results in such a case, and quantitative estimates of level shifts resulting from such calculations need not be very reliable. It is, therefore, quite difficult to accept the viewpoint that unsatisfactory spectroscopic proper- ties of, in particular, modern Skyrme forces can be cured solely by going beyond MF. On the contrary, the magni- tude of discrepancies between the SHF and experimental s.p. levels (see below) rather clearly suggest that: ( i) the data sets used to fit the force (or functional) parameters are incomplete and ( ii) the interaction/functional should be extended. This is exactly the task undertaken in the present exploratory work. We extend the conventional EDFs based on Skyrme interactions by including tensor terms, and fit the corresponding coupling constants to the SO splittings rather than to masses. The prelimi- nary goal is to improve spectroscope properties of func- tionals, even at the expense of the quality in reproducing the binding energies. Empirical energy of a given s.p. neutron orbital can be deduced from the difference between the ground-state en- ergy of the doubly-magic core with Nneutrons, E0(N), and energies, Ep(N+ 1) orEh(N−1), of its odd neigh- boring isotopes having a single-particle (p) or single-hole(h) occupying that orbital, i.e., ǫp(N) =Ep(N+ 1)−E0(N), (21) ǫh(N) =E0(N)−Eh(N−1), (22) see, for example, Refs. [7, 8, 9] and referenced cited therein. Note that the total energies above are nega- tive numbers and decrease with increasing numbers of particles, Eh(N−1)> E0(N)> Ep(N+ 1). Similarly, we can define a measure of the neutron shell gap as the difference between the lowest single-particle and highest single-hole energy, ∆ǫgap(N) =ǫp(N)−ǫh(N) (23) =Ep(N+ 1) +Eh(N−1)−2E0(N). Single-particle energies of proton orbitals and proton shell gaps are defined in an analogous way. Consistently with the empirical definitions, in the present paper, the same procedure is used on the the- oretical level [4]. It means that we determine the total energies of doubly-magic cores and their odd neighbors by using the EDF method, and then we calculate the s.p. energies as the corresponding differences (21) or (22). In this way, we avoid all the ambiguities related to questions of what the s.p. energies really are and how they can be extracted from data. Our methodology simply amounts to a specific way of comparing measured and calculated masses of nuclei. In odd nuclei, one has to ensure that particular s.p. orbitals are occupied by odd particles, but still the calculated total energies correspond to masses of their ground and low-lying excited states. Since here we consider only doubly-magic nuclei and their odd neigh- bors, the pairing correlations are neglected. In our calculations, both time-even and time-odd po- larizations exerted by an odd particle/hole on the doubly- magic core are evaluated self-consistently. In order to discuss these polarizations, let us recall that the extreme s.p. model, in which ground-state energies are sums of fixed s.p. energies of occupied orbitals, is the model with no polarization of any kind included. In this model, the differences of ground-state energies, Eqs. (21) and (22), are, of course, exactly equal to the model s.p. energies. Having this background model in mind, we can distin- guish three kinds of polarization effects: A. Mass polarization effect (time-even) This effect can be understood as the self-consistent rearrangement of all nucleons, which is induced by an added or subtracted odd particle, while the shape is con- strained to the spherical point and the time-odd mean fields are neglected. One has to keep in mind, that none of the s.p. orbitals of a spherical multiplet is spher- ically symmetric, and therefore, when a particle in such a state is added to a closed core, the resulting state of an odd nucleus cannot be spherically symmetric, and thus cannot be self-consistent. A self-consistent solution can5 only be realized within the so-called filling approxima- tion, whereupon one assumes the occupation probabili- ties of all orbitals belonging to the spherical multiplet of angular momentum jand degeneracy 2 j+ 1, to be equal tov2 j= 1/(2j+ 1). The mass polarization effect is exactly zero when eval- uated within the first-order approximation. Indeed, as expressed by the Koopmans theorem [37], the linear term in variation of the total energy with respect to adding or subtracting a particle is exactly equal to the bare single- particle energy of the core. However, the obtained results do not agree with Koopmans theorem. This is illustrated in Fig. 1, where we compare bare and polarized s.p. ener- gies of the νf7/2andνd3/2orbitals in40Ca. One can see that for both orbitals there is a quite large and positive mass polarization effect. The reason for this disagreement lies in the fact that, because of the center-of-mass correction, the standard EDF calculations are not really variational with respect to adding or subtracting a particle. Indeed, in these cal- culations, the total energy of an A-particle system is cor- rected by the center-of-mass correction [38, 39], Ec.m.≃Edir c.m.=−1 AEkin, (24) whereEkinis the average kinetic energy of the system. The factor of1 Ais not varied when constructing the stan- dard mean-field Hamiltonian, and therefore, the Koop- mans theorem is violated. Had this variation been in- cluded, it would have shifted the mean-field potential and thus all bare s.p. energies in given nucleus by a constant, ǫ′ i=ǫi+1 A2Ekin. (25) In40Ca, this shift equals to 0.40 MeV and almost ex- actly corresponds to the entire mass polarization effect shown in Fig. 1. The remaining mass polarization effect can be attributed to higher-order polarization effects (be- yond linear) and to the non-selfconsistency of the filling approximation. A few remarks about the shift in Eq. (25) are here in order. First, the effect is independent on whether the two-body or one-body center-of-mass correction [39] is used, and on whether this is done before or after vari- ation. Only the detailed value of the shift may depend on a particular implementation of the center-of-mass cor- rection. Second, the shift induces an awkward result of the mean-field potential going asymptotically to a posi- tive constant and not to zero. Although this may seem to be a trivial artifact, which does not influence the s.p. wave functions and observables, it shows that the stan- dard center-of-mass correction should be regarded as an ill-defined theoretical construct. This fact shows up as an acute problem in fission calculations [40]. Third, when- ever the bare s.p. energies are compared to empirical data, this shift must by taken into account. Alternatively, as advocated in the present study, one should compare directly the calculated and measured mass differences.-9.5 -9.4 -9.3 -9.2 -9.1 particle state -15.3 -15.2 -15.1 -15.0 -14.9 hole state 1f7/2 1d3/2 -1Energy [MeV] bare bare mass mass mass & shape mass, shape & spin mass, shape & spin mass & shape FIG. 1: Bare, mass polarized, mass and shape polarized (time-even) and mass, shape and spin polarized s.p. ener- gies of the νf7/2particle and the νd−1 3/2hole in40Ca. The polarized s.p. levels were deduced from binding energies in one particle/hole nuclei according to the formulas (21) and (22). Note that attractive (repulsive) polarization corre ctions to the binding energies shift the particle state down (up) an d the hole state up (down), respectively. Finally, one should note that the shift is irrelevant when differences of the s.p. energies are considered, such as the SO splittings discussed in Sec. IV. B. Shape polarization effect (time-even) This effect is well known both in the MF and particle- vibration-coupling approaches. Within a deformed MF theory (with the time-odd mean fields neglected), it cor- responds to a simple fact that the s.p. energies (eigen- energies in a deformed potential) depend on the defor- mation in specific way, which is visualized by the stan- dard Nilsson diagram [1]. Indeed, in an axially deformed potential, a spherical multiplet of angular momentum jsplits into j+ 1/2 orbitals according to moduli of the angular-momentum projections K=|mj|. Unless j= 1/2, for prolate and oblate deformations, orbitals withK= 1/2 decrease and increase in energy, respec- tively, while those with maximum K=jbehave in an opposite way. Therefore, both for prolate and oblate de- formations, and for j >1/2, the lowest orbitals have the energies that are lower than those at the spherical point. Hence, a j >1/2 particle added to a doubly-magic core always polarizes the core in such a way that the total energy decreases. On the other hand, the energy of a j= 1/2,K= 1/2 orbital does not depend on deforma- tion (in the first order), and thus such an orbital does not exert any shape polarization (in this order). Exactly the same result is obtained in a particle- vibration-coupling model, in which a j >1/2 particle can be coupled with either 0+or 2+state of the core, [ j⊗0+]j or [j⊗2+]j, and the repulsion of these two configurations decreases the energy of the ground state with respect to the unperturbed spherical configuration [ j⊗0+]j. As before, for j= 1/2, the configuration [ j⊗2+]jdoes not6 exist, and the ground state is not lowered. The above reasoning can be repeated for hole states, with the result that the j > 1/2 holes added to the doubly-magic core always polarize the core in such a way that the total energy also decreases. As a consequence, the shape polarization effect decreases the s.p. energies of particle states (21) and increases those of hole states (22), and thus decreases the shell gap (24). This effect is clearly illustrated in Fig. 1. C. Spin polarization effect (time-odd) When an odd particle or hole is added to the core, and the time-odd fields are taken into account, it exerts po- larization effects both in the time-even (mass and shape) and time-odd (spin) channels. It means that a non-zero average spin value of the odd particle induces a time-odd component of the mean field, which influences average spin values of all particles, leading to a self-consistent amplification of the spin polarization. It should be noted at this point that the spin polar- ization effect dramatically depends on the assumed sym- metries and choices made for the occupied orbital, see discussion in the Appendix of Ref. [41]. Indeed, without the time-odd fields, in order to occupy the odd parti- cle or hole, one can use any linear combination of states forming the Kramers-degenerate pair. The total energy is independent of this choice, because the time-even den- sity matrix does not depend on it. This allows for mak- ing specific additional assumptions about the conserved symmetries, e.g., in the standard case, one assumes that the odd state is an eigenstate of the signature (or sim- plex) symmetry with respect to the axis perpendicular to the symmetry axis. However, in order to fully allow for the spin polarization effects through time-odd fields, one has to release all such restrictive symmetries and al- low for alignment of the spin of the particle along the symmetry axis. This requires calculations with broken signature symmetry and only the parity symmetry be- ing conserved. Therefore, calculations of this kind are more difficult than those performed within the standard cranking model. D. Total polarization effect In Tables I and II we list the neutron s.p. energies calculated in six doubly-magic nuclei for the SLy4 Lin- teraction. The bare s.p. energies (a) are compared to those calculated from total energies, Eqs. (21) and (22), with the mass and shape (b) or mass, shape, and spin (c) polarizations included. In order to remove ambigu- ities associated with occupancy of the valence particle (hole), binding energies of odd- Anuclei were calculated by blocking the lowest (highest) K=jorbitals at oblate (prolate) shape for particle (hole) orbitals. The blocked orbitals were selected by performing cranking calculationwith angular-frequency vector parallel to the symmetry axis. Such a cranking does not affect total energy or wave function, but splits spherical multiplets into orbitals hav- ing good projections of the angular momentum on the symmetry axis. Calculations were performed by using the code HFODD (v2.30a) [42, 43, 44, 45] for the spher- ical basis of Nsh= 14 harmonic-oscillator shells. As seen by comparing columns (b) and (a) of Tables I and II, the energy shifts caused by the time-even polar- ization effects with respect to bare s.p. spectra are almost always positive, both for particle and hole states. A few exceptions occur only for large- junfavored ( j=ℓ−1/2) SO partners in heavy nuclei. These shifts clearly de- crease in magnitude with increasing mass, from about 1 MeV in16O to below 0.25 MeV in208Pb. As dis- cussed in Sec. III A, they are mainly caused by the mass- polarization effects related to the center-of-mass correc- tion. Indeed, shifts of s.p. energies (25), calculated for the six doubly-magic nuclei of Tables I and II, are 0.87, 0.40, 0.36, 0.20, 0.14, and 0.09 MeV, respectively. It is also clearly visible that shifts of particle states are systematically smaller than those of hole states, i.e., the time-even polarizations tend to slightly decrease shell gaps. The time-odd polarization effects systematically shift the hole states down and particle states up in energy, i.e., they result in an increase of shell gaps, cf. also Fig. 1. This result is at variance with that obtained within the RMF approach [4], where the time-odd fields corre- sponded to magnetic properties driven by the Lorentz in- variance, while here they are determined by experimental values of the Landau parameters [21, 22]. We note here that in recent derivations of the time-odd coupling con- stants within the relativistic point-coupling model [46], one obtains values of the Landau parameters compat- ible with experimental values. Shifts of s.p. energies due to the time-odd polarization effects also decrease with mass, from about −0.7(+0.5) MeV in16O to below −0.1(0.15)MeV in208Pb for hole (particle) states. The total effect of combined time-even and time-odd polarizations results in adding up the shifts for particle states and subtracting those for hole states. In this way, the total shifts of particle and hole states become mostly positive and (apart from light nuclei) comparable in mag- nitude, with quite small net effects on shell gaps. They also decrease with increasing mass, from up to 1.5 MeV in 16O to below 0.25 MeV in208Pb. Altogether, polarization effects turn out to be significantly smaller than those ob- tained in previous estimates. Although in quantitative analysis they cannot at all be neglected, discrepancies with experimental data (last columns in Tables I and II) are still markedly larger in magnitude. Therefore, bare s.p. energies can be safely used, at least in all studies that do not achieve any better overall agreement with data.7 TABLE I: Neutron s.p. energies in16O,40Ca, and48Ca (in MeV). The columns show: (a) bare unpolarized s.p. ener gies in doubly-magic cores, (b) self-consistent s.p. energies obt ained from binding energies in one-particle/hole nuclei, E qs. (21) and (22), with time-even mass and shape polarizations included , (c) as in (b), but with time-odd spin polarizations include d in addition, and (d) experimental data taken from Ref. [8]. Tim e-even, (b) −(a), time-odd, (c) −(b), and total, (c) −(a) polarizations are also shown, along with the differences between the self-c onsistent and experimental spectra, (c) −(d). Positive s.p. energies are shown only to indicate that particular orbitals are unbo und in calculations; their values are only very approximate ly related to positions of resonances. All results have been calculate d using the Sly4 Lfunctional. bare T-even T-even T-even T-odd total exp. theory pol. & T-odd pol. pol. [8] −exp (a) (b) (b) −(a) (c) (c) −(b) (c)−(a) (d) (c) −(d) 16O 1νp3/2−20.57−19.61 0.96 −20.29−0.68 0.28 −21.84 1.55 1νp1/2−14.54−13.55 0.99 −13.86−0.31 0.68 −15.66 1.80 1νd5/2−6.75−5.83 0.92 −5.43 0.40 1.32 −4.22−1.21 2νs1/2−3.78−2.79 0.99 −2.30 0.49 1.48 −3.35 1.05 1νd3/2 0.39 1.19 0.80 1.37 0.18 0.98 1.50 −0.13 40Ca 1νd5/2−22.01−21.55 0.46 −21.87−0.32 0.14 −22.39 0.52 2νs1/2−17.25−16.93 0.32 −17.51−0.58−0.26−18.19 0.68 1νd3/2−15.31−14.84 0.47 −14.98−0.14 0.33 −15.64 0.66 1νf7/2−9.58−9.22 0.36 −9.00 0.22 0.58 −8.62−0.38 2νp3/2−5.24−4.85 0.39 −4.67 0.18 0.57 −6.76 2.09 2νp1/2−3.06−2.66 0.40 −2.55 0.11 0.51 −4.76 2.21 1νf5/2−1.38−1.10 0.28 −0.99 0.11 0.39 −3.38 2.39 48Ca 1νd5/2−22.60−22.02 0.58 −22.21−0.19 0.39 −17.31 −4.90 2νs1/2−17.60−17.26 0.34 −17.78−0.52−0.18−13.16 −4.62 1νd3/2−16.55−15.97 0.58 −16.02−0.05 0.53 −12.01 −4.01 1νf7/2−9.79−9.23 0.56 −9.34−0.11 0.45 −9.68 0.34 2νp3/2−5.54−5.25 0.29 −5.12 0.13 0.42 −5.25 0.13 2νp1/2−3.54−3.21 0.33 −3.11 0.10 0.43 −3.58 0.47 1νf5/2−1.33−1.25 0.08 −1.21 0.04 0.12 −1.67 0.46 IV. SPIN-ORBIT SPLITTINGS Before proceeding to readjustments of coupling con- stants so as to improve the agreement of the SO splittings with data, we analyze the influence of time-even (mass and shape) and time-odd (spin) polarization effects on the neutron SO splittings. Based on results presented in the preceding Section, we calculate the SO splittings as ∆ǫnℓ SO=ǫnℓj<−ǫnℓj>. (26) Figure 2 shows the SO splittings calculated using SLy4L— the functional based on the original SLy4 [19] functional with spin fields readjusted to reproduce em- pirical Landau parameters according to the prescription given in Refs. [21, 22]. Plotted values correspond to re- sults presented in Tables I and II. The results are la- beled according to the following convention: open sym- bols mark results computed directly from the s.p. spectra in doubly-magic nuclei (bare s.p. energies). These bare values contain no polarization effect. Gray symbols label the SO splittings involving polarization due to the time- even mass- and shape-driving effects, i.e., those obtained with all time-odd components in the functional set equalto zero. Black symbols illustrate fully self-consistent re- sults obtained for the complete SLy4 Lfunctional. Gray and black symbols are shifted slightly to the left-hand (right-hand) side with respect to the doubly-magic core in order to indicate the hole (particle) character of the SO partners. Mixed cases involving the particle-hole SO partners are also shifted to the right. The impact of polarization effects on the SO splittings is indeed very small, particularly for the cases where both SO partners are of particle or hole type. Indeed, for these cases, the effect only exceptionally exceeds 200 keV, re- flecting a cancellation of polarization effects exerted on thej=ℓ±1/2 partners. The smallness of polariza- tion effects hardly allows for any systematic trends to be pinned down. Nevertheless, the self-consistent results show a weak but relatively clear tendency to slightly en- large or diminish the splitting for hole or particle states, respectively. The situation is clearer when the SO partners are of mixed particle ( j=ℓ−1/2) and hole ( j=ℓ+ 1/2) character. In these cases, the shape polarization tends to diminish the splitting quite systematically by about 400–500keV. This behavior follows from naive deformed8 TABLE II: Same as in Table I but for90Zr,132Sn, and208Pb. bare T-even T-even T-even T-odd total exp. theory pol. & T-odd pol. pol. [8] −exp (a) (b) (b) −(a) (c) (c) −(b) (c)−(a) (d) (c) −(d) 90Zr 1νf7/2−23.16−22.83 0.33 −22.94−0.11 0.22 −14.76 −8.18 1νf5/2−17.07−16.72 0.35 −16.74−0.02 0.33 −13.05 −3.69 2νp3/2−17.52−17.30 0.22 −17.44−0.14 0.08 −12.74 −4.70 2νp1/2−15.46−15.26 0.20 −15.35−0.09 0.11 −12.37 −2.98 1νg9/2−12.08−11.75 0.33 −11.81−0.06 0.27 −11.69 −0.12 2νd5/2−6.73−6.59 0.14 −6.52 0.07 0.21 −7.20 0.68 3νs1/2−4.93−4.70 0.23 −4.44 0.26 0.49 −5.78 1.34 2νd3/2−3.99−3.62 0.37 −3.59 0.03 0.40 −4.77 1.18 1νg7/2−3.75−3.75 0.00 −3.74 0.01 0.01 −4.62 0.88 132Sn 2νd5/2−11.72−11.48 0.24 −11.56−0.08 0.16 −9.10−2.46 3νs1/2−9.46−9.28 0.18 −9.59−0.31−0.13−7.55−2.04 1νh11/2−7.66−7.30 0.36 −7.33−0.03 0.33 −7.42 0.09 2νd3/2−9.11−8.91 0.20 −8.95−0.04 0.16 −7.17−1.78 2νf7/2−2.01−2.00 0.01 −1.95 0.05 0.06 −2.29 0.34 3νp3/2 0.17 0.26 0.09 0.31 0.05 0.14 −1.31 1.62 1νh9/2 0.95 0.79 −0.16 0.77 −0.02−0.18−0.91 1.68 3νp1/2 0.97 1.08 0.11 1.12 0.04 0.15 −0.72 1.84 2νf5/2 0.82 0.84 0.02 0.88 0.04 0.06 −0.35 1.23 208Pb 2νf7/2−12.02−11.85 0.17 −11.90−0.05 0.12 −9.96−1.94 1νi13/2−9.52−9.29 0.23 −9.30−0.01 0.22 −8.92−0.38 3νp3/2−9.23−9.09 0.14 −9.17−0.08 0.06 −8.12−1.05 2νf5/2−9.03−8.89 0.14 −8.91−0.02 0.12 −7.78−1.13 3νp1/2−8.11−8.01 0.10 −8.06−0.05 0.05 −7.72−0.34 2νg9/2−3.19−3.19 0.00 −3.16 0.03 0.03 −3.73 0.57 1νi11/2−1.53−1.65−0.12−1.67−0.02−0.14−3.11 1.44 3νd5/2−0.50−0.46 0.04 −0.43 0.03 0.07 −2.22 1.79 4νs1/2 0.56 0.65 0.09 0.80 0.15 0.24 −1.81 2.61 2νg7/2 0.08 0.10 0.02 0.11 0.01 0.03 −1.35 1.46 3νd3/2 0.69 0.76 0.07 0.78 0.02 0.09 −1.33 2.11 Nilsson model picture where the highest- Kmembers of thej=ℓ−1/2 (j=ℓ+1/2) multiplet slopes down (up) as a function of the oblate (prolate) deformation parameter. As discussed in the previous Section, the time-odd fields act in the opposite way, tending to slightly enlarge the gap. The net polarization effect does not seem to exceed about 300 keV. In these cases, however, we deal with large ℓorbitals having also quite large SO splittings of the order of ∼8 MeV. Hence, the relative corrections due to polarization effects do not exceed about 4%, i.e., they are relatively small – much smaller than the effects of tensor terms discussed below and the discrepancy with data, which in Fig. 2 is indicated for the neutron 1f SO splitting in40Ca. These results legitimate the direct use of bare s.p. spectra in magic cores for further studies of the SO splittings, which considerably facilitates the calculations.As already mentioned, empirical s.p. energies are es- sentially deduced from differences between binding en- ergies of doubly-magic core and their odd- Aneighbors. Different authors, however, use also one-particle transfer data, apply phenomenological particle-vibration correc- tions and/or treat slightly differently fragmented levels. Hence, published compilations of the s.p. energies, and in turn the SO splittings, differ slightly from one another depending on the assumed strategy. The typical uncer- tainties in the empirical SO splittings can be inferred from Table III, which summarizes the available data on the SO splittings based on three recent s.p. level compi- lations published in Refs. [7, 8, 9]. Instead of large-scale fit to the data (see, e.g., Refs. [16, 31]), we propose a simple three-step method to adjust three coupling constants C∇J 0,CJ 0, andCJ 1. The entire idea of this procedure is based on the observation that9 Nucleus orbitals Ref. [7] Ref. [8] Ref. [9] 16Oν1p−1 3/2−ν1p−1 1/26.17 6.18 −− ν1d5/2−ν1d3/25.08 5.72 5.08 π1p−1 3/2−π1p−1 1/26.32 6.32 −− π1d5/2−π1d3/25.00 4.97 5.00 40Caν2p3/2−ν2p1/22.00 2.00 1.54 ν1f7/2−ν1f5/24.88 5.24 5.64 ν1d−1 5/2−ν1d−1 3/26.00 6.75 6.75 π2p3/2−π2p1/22.01 1.72 1.69 π1f7/2−π1f5/24.95 5.41 6.05 π1d−1 5/2−π1d−1 3/26.00 5.94 6.74 48Caν2p3/2−ν2p1/2−− 1.67 1.77 ν1f−1 7/2−ν1f5/2−− 8.01 8.80 ν1d−1 5/2−ν1d−1 3/2−− 5.30 3.08 π2p3/2−π2p1/2−− 2.14 1.77 π1f7/2−π1f5/2−− 4.92 −− π1d−1 5/2−π1d−1 3/2−− 5.01 5.29 56Niν2p3/2−ν2p1/2−− 1.88 1.12 ν1f−1 7/2−ν1f5/2−− 6.82 7.16 π2p3/2−π2p1/2−− 1.83 1.11 π1f−1 7/2−π1f5/2−− 7.01 7.50 90Zrν2d5/2−ν2d3/2−− 2.43 −− ν1g−1 9/2−ν1g7/2−− 7.07 −− ν2p−1 3/2−ν2p−1 1/2−− 0.37 −− ν1f−1 7/2−ν1f−1 5/2−− 1.71 −− π2d5/2−π2d3/2−− 2.03 −− π1g9/2−π1g7/2−− 5.56 −− π2p−1 3/2−π2p−1 1/2−− 1.50 −− π1f−1 7/2−π1f−1 5/2−− 4.56 −− 100Snν2d5/2−ν2d3/21.93 −− 1.93 ν1g−1 9/2−ν1g7/27.00 −− 7.00 π1g−1 9/2−π1g7/26.82 −− 6.82 π2p−1 3/2−π2p−1 3/22.85 −− 2.85 132Snν2f7/2−ν2f5/22.00 1.94 −− ν3p3/2−ν3p1/20.81 0.59 1.15 ν1h−1 11/2−ν1h9/26.53 6.51 6.68 ν2d−1 5/2−ν2d−1 3/21.65 1.93 1.66 π2d5/2−π2d3/21.48 1.83 1.75 π1g−1 9/2−π1g7/26.08 5.33 6.08 208Pbν3d5/2−ν3d3/20.97 0.89 0.97 ν2g9/2−ν2g7/22.50 2.38 2.50 ν1i−1 13/2−ν1111/25.84 5.81 6.08 ν3p−1 3/2−ν3p−1 1/20.90 0.90 0.89 ν2f−1 7/2−ν2f−1 5/22.13 2.18 1.87 π2f7/2−π2f5/21.93 2.02 1.93 π3p3/2−π3p1/20.85 0.45 0.52 π1h−1 11/2−π1h9/25.56 5.03 5.56 π2d−1 5/2−π2d−1 3/21.68 1.62 1.46 TABLE III: Empirical SO splittings. Compilation is based on the empirical s.p. levels taken from Refs. [7, 8, 9].2468 exp 1357Δε SO [MeV] 16 O48 Ca 40 Ca 90 Zr 132 Sn 208 Pb 1f 1p -11d 1d -1 2p 2p 1f ph 1d -1 2p -12d 1f -11g ph 2d -1 3p 2f 1h ph 1i ph 3d 2f -12g SLy4 L FIG. 2: Neutron SO splittings (26) calculated using the SLy4Lfunctional. White, gray, and black symbols mark bare, mass and shape polarized (time-even), and mass, shape, and spin polarized (time-even and time-odd) results, respecti vely. Results for hole (particle and particle-hole) orbitals are shifted to the left (right) with respect to the core (open symbols). A typical discrepancy with experiment is shown by the arrow in 40Ca. the empirical 1 f7/2−1f5/2SO splittings in40Ca,56Ni, and48Ca form very distinct pattern, which cannot be reproduced by using solely the conventional SO interac- tion. The readjustment is done in the following way. First, experimental data in the spin-saturated (SS) nucleus 40Ca are used in order to fit the isoscalar SO coupling constant C∇J 0. One should note that in this nucleus, the SO splitting depends only onC∇J 0, and not on CJ 0(be- cause of the spin saturation), nor on C∇J 1(because of the isospin invariance at N=Z), nor on CJ 1(because of both reasons above). Therefore, here one experimental number determines one particular coupling constants. Second, once C∇J 0is fixed, the spin-unsaturated (SUS) N=Znucleus56Ni is used to establish the isoscalar tensor coupling constant CJ 0. Again here, because of the isospin invariance, the SO splitting is independent of either of the two isovector coupling constants, C∇J 1 orCJ 1. Finally, in the third step,48Ca is used to adjust the isovector tensor coupling constant CJ 1. Such a pro- cedure exemplifies the focus of fit on the s.p. properties, as discussed in the Introduction. It turns out that current experimental data, and in particular lack of information in48Ni, do not allow for adjusting the fourth coupling constant, C∇J 1. For this10 4567 456740 Ca 0.7 0.8 0.9 1f7/2 -f5/2 f7/2 -d3/2 56 Ni f7/2 -f5/2 f7/2 -d3/2 -40 -30 -20 -10 0 C0J 2468 -80 -60 -40 -20 0f7/2 -f5/2 f7/2 -d3/2 from binding energies 48 Ca C1Jfrom binding energies Single-particle levels splittings [MeV]fJΔ FIG. 3: Figure illustrates the three-step procedure used to fit the isoscalar SO coupling constant C∇J 0in40Ca (up- per panel), the isoscalar tensor strength CJ 0in56Ni (middle panel), and the isovector tensor strength CJ 1in48Ca (lowest panel). These particular calculations have been done for th e SkP functional, but the pattern is common for all the an- alyzed parameterizations including SLy4 and SkO. See text for further details. reason, in the present study we fix it by keeping the ratio ofC∇J 0/C∇J 1equal to that of the given standard Skyrme force. In the process of fitting, all the remaining time- even coupling constants Ctare kept unchanged. Variants of the standard functionals obtained in this way are be- low denoted by SkP T, SLy4 T, and SkO T. When the time-odd channels, modified so as to reproduce the Lan- dau parameters, are active, we also use notation SkP LT, SLy4LT, and SkO LT. For the SkP functional, the procedure is illustrated in Fig. 3. We start with the isoscalar N=Znucleus40Ca. The evolution of the SO splittings in function of f∇J, which is the factor scaling the original SkP coupling con- stantC∇J 0, is shown in the upper panel of Fig. 3. As it is clearly seen from the Figure, fair agreement with data requires about 20% reduction in the conventional SO in- teraction strength C∇J 0. It should be noted also that the reduction in the SO interaction considerably improves the 1f7/2−1d3/2and 1f7/2−2p3/2splittings but slightly spoils the 2 p3/2−2p1/2SO splitting. Qualitatively, sim- ilar results were obtained for the SLy4 and SkO interac- tions. Reasonable agreement to the data requires ∼20% reduction of the original C∇J 0in case of the SkO inter- action and quite drastic ∼35% reduction of the original C∇J 0in case of the SLy4 force. Having fixed C∇J 0in40Ca we move to the isoscalar nu-Skyrme C∇J 0C∇J 0/C∇J 1CJ 0 CJ 1 force [MeV fm5] [MeV fm5] [MeV fm5] SkPT−60.0 3 −38.6 −61.7 SLy4T−60.0 3 −45.0 −60.0 SkOT−61.8 −0.78 −33.1 −91.6 TABLE IV: Spin-orbit C∇Jand tensor isoscalar CJ 0and isovector CJ 1functional coupling constants adopted in this work and subsequently used in Figs. 4, 5, and 6, where global calculations of the SO splittings are presented. cleus56Ni. This nucleus is spin-unsaturated and there- fore is very sensitive to the isoscalar CJ 0tensor coupling constant. The evolution of theoretical s.p. levels versus CJ 0is illustrated in the middle panel of Fig. 3. As shown in the Figure, reasonable agreement between the empiri- cal and theoretical 1 f7/2−1f5/2SO splitting is achieved forCJ 0∼ −40 MeV fm5, which by a factor of about five exceeds the original SkP value for this coupling constant. It is striking that a similar value of CJ 0is obtained in the analogical analysis performed for the SLy4 interaction. Finally, the isovector tensor coupling constant C∇J 1is established in N/negationslash=Znucleus48Ca. The evolution of the- oretical neutron s.p. levels versus CJ 1is illustrated in the lowest panel of Fig. 3. As shown in the Figure, the value ofC∇J 1∼ −70 MeV fm5is needed to reach reasonable agreement for the 1 νf7/2−1νf5/2SO splitting in this case. For this value of the CJ 1strength one obtains also good agreement for the proton 1 πf7/2−1πf5/2SO split- ting (see Figs. 4 and 5 below), without any further read- justment of the C∇J 1strength. Again, very similar value for theCJ 1strength is deduced for the SLy4 force. Note also the improvement in the 1 f7/2−1d3/2splitting caused by the isovector tensor interaction. Dotted lines show re- sults obtained from the mass differences, i.e., with all the polarization effects included. During the fitting procedure all the remaining func- tional coupling constants were kept fixed at their Skyrme values. The ratio of the isoscalar to the isovector cou- pling constant in the SO interaction channel was locked to its standard Skyrme value of C∇J 0/C∇J 1= 3. Since no clear indication for relaxing this condition is seen (see also Figs. 4 and 5 below), we have decided to investi- gate the isovector degree of freedom in the SO interac- tion (see [17]) by performing our three-step fitting process also for the generalized Skyrme interaction SkO [20], for whichC∇J 0/C∇J 1∼ −0.78. All the adopted functional coupling constants result- ing from our calculations are collected in Table IV. Note, that the procedure leads to essentially identical SO inter- action strengths C∇J 0for all three forces irrespective of their intrinsic differences, for example in effective masses. The tensor coupling constants in both the SkP and the SLy4 functionals are also very similar. In the SkO case, one observes rather clear enhancement in the isovector11 tensor coupling constant which becomes more negative to, most likely, counterbalance the non-standard positive strength in the isovector SO channel. The functionals were modified using only three specific pieces of data on the neutron 1 f7/2−1f5/2SO splittings. In order to verify the reliability of the modifications, we have performed systematic calculations of the experimen- tally accessible SO splittings. The results are depicted in Figs. 4, 5, and 6 for the SkP, SLy4, and SkO function- als, respectively. Additionally, Fig. 7 shows neutron and proton magic gaps (24) calculated using the SkP func- tional. In all these Figures, estimates taken from Ref. [8] are used as reference empirical data. This global set of the results can be summarized as follows: •Theℓ= 1,p3/2−p1/2, SO splittings are slightly better reproduced with original rather than modi- fied functionals. •The 1dSO splittings (16O,40,48Ca) are rather poorly reproduced by both the original and modi- fied functionals. These splitting are also subject to relatively big empirical uncertainties as shown in Table III and, therefore, need not be very conclu- sive. In particular, the 1 dSO splittings in N=Z 16O and40Ca nuclei deduced from Ref. [8] and de- picted in the figures show surprisingly large isospin dependence. •The 2dand 3dsplittings are quite well reproduced by both the original and modified functionals with slight preference for the modified functional, in par- ticular for the SLy4 interaction. •All theℓ≥3 SO splittings are reproduced consid- erably better by the modified functionals. •Magic gaps are also better (although not fully sat- isfactorily) reproduced by the modified functionals. Without any doubt the SO splittings are better de- scribed by the modified functionals. It should be stressed that the improvements were reached using only three ad- ditional data points without any further optimization. The tensor coupling constants deduced in this work and collected in table IV should be therefore considered as reference values. Indeed, direct calculations show that variations in CJ twithin±10% affect the calculated SO splittings only very weakly. The price paid for the im- provements concerns mostly the binding energies, which for the nuclei56Ni,132Sn, and208Pb become worse as compared to the original values. This issue is addressed in the next section. V. BINDING ENERGIES Parameters of the Skyrme functional have been fitted to reproduce several physical quantities, with emphasisNucleus SLy4 SLy4 TSLy4Tmin 40Ca−2.197−1.830 −5.775 48Ca−1.912 5.039 −0.279 56Ni 0.625 15.138 9.127 90Zr−1.845 7.492 −3.032 132Sn−0.660 19.898 2.222 208Pb 0.822 24.910 −3.048 TABLE V: Differences between calculated and experimental ground-state energies, Ecalc−Eexp, (in MeV) for a set of spherical nuclei. Column denoted as SLy4 Tminshows results obtained after (local) minimization with respect to parame - tersti,xi, i= 0,1,2,3. See text for details. on the masses of magic nuclei. Therefore it is not sur- prising that dramatic modifications of the SO and tensor terms of the functional, described in Sec. III, while im- proving the agreement between the calculated and mea- sured single-particle properties, can destroy the quality of the mass fit. Hence, it is interesting to know whether this disagreement is significant and whether it can be healed by refitting the remaining parameters of the functional. Table V shows differences between calculated and ex- perimental (Ref. [47]) ground-state energies, Ecalc−Eexp, (in MeV) for a set of spherical nuclei. Negative val- ues mean that nuclei are overbound. Results given in the second column, denoted as SLy4, correspond to the standard SLy4 [19] parametrization. The third column, denoted as SLy4 T, illustrates significant deterioration of the quality of fit when parameters C∇J 0,CJ 0andCJ 1are modified (see Sec. III). Values presented in the last col- umn, SLy4 Tmin, were obtained by minimizing the rmsof relative discrepancies between the calculated and mea- sured masses Values of C∇J 0,CJ 0andCJ 1were kept fixed at their SLy4 Tvalues, while minimization was performed by varying the remaining parameters of the functional ti andxi. Note that for the standard SLy4 functional, the tensor coupling constants are set equal to zero indepen- dently of the values of tiandxi. For the minimization, we have used the same methodology, namely, the influ- ence of parameters tiandxion tensor coupling constants was disregarded. It should be emphasized that no attempt has been made to find the global minimum — the minimization was purely local, in the vicinity of the standard SLy4 values of the parameters tiandxi. One can see that even this very limited procedure can lead to significant reduction of discrepancies, down to quite reasonable val- ues (with an exception of56Ni nucleus). It is worth noting that the resulting modifications of thetiandxiparameters turned out to be very small. Table VI shows the values of the tiandxiparameters in the standard SLy4 parametrization (second column) and those obtained as the result of the minimization pro- cedure (third column). The last column shows relative12Spin-orbit splittings [MeV] 1357 16 O40 Ca 48 Ca 56 Ni 90 Zr 132 Sn 208 Pb 16 O40 Ca 48 Ca 56 Ni 90 Zr 132 Sn 208 Pb p pn n 1h1h1i d5/2 -d3/2 p3/2 -p1/2 d5/2 -d3/2 p3/2 -p1/2 f7/2 -f5/2 g9/2 -g7/2 f7/2 -f5/2 g9/2 -g7/2 SkP LT 1357 FIG. 4: Experimental [8] (black symbols) and theoretical SO splittings calculated using the original SkP functional (g ray symbols) and our modified SkP LTfunctional (white symbols) with the SO and tensor coupling c onstants given in Table IV. Upper left and right panels show neutron SO splittings for lo w-ℓ= 1,2 (pandd) and high- ℓ≥3 orbitals, respectively. Analogical information but for proton SO splittings is depicted in the l ower panels. changes of parameters (in percent). As one can see, they are at most of the order of one percent. Nevertheless, even such small changes were sufficient to improve sig- nificantly the agreement between calculated and experi- mental masses. We stress again that the refitting procedure is used here only for illustration purposes and the global fit to masses must probably include extended functionals and improved methodology. For example, the Wigner energy correction [48] was not included in the fit, as it was nei- ther included in the fit of the SLy4 parametrization. This correction alone may change the balance of discrepancies obtained for the N=ZandN/negationslash=Znuclei, and strongly impact the results. Systematic studies of these effects will be performed in the near future. VI. CONCLUSIONS Applicability of functional-based self-consistent mean- field or energy-density-functional methods to nuclear structure is hampered by their unsatisfactory s.p. proper- ties. This fact seems to be a mere consequence of strate- gies used to select datasets that were applied in the pro- cess of adjusting free parameters of these effective theo- ries. In spite of the fact that the s.p. energies are at the heart of these methods, the datasets are heavily oriented towards reproducing bulk nuclear properties in large- N limit, with only a marginal influence of the s.p. levels orparam. SLy4 SLy4 Tminchange (%) t0−2488.913 −2490.00300 0.04 t1 486.818 486.78460 −0.01 t2−546.395 −545.35849 −0.19 t3 13777.000 13767.77776 −0.07 x0 0.834 0.83257 −0.17 x1−0.344 −0.34227 −0.50 x2−1.000 −0.99798 −0.20 x3 1.354 1.36128 0.54 TABLE VI: Skyrme force parameters ti,xi, i= 0,1,2,3 of the standard SLy4 parametrization (second column) com- pared with those obtained from the minimization procedure described in the text (third column). The last column shows relative change of parameters (in percent). level splittings in finite nuclei. In this work we suggest a necessity of shifting attention from bulk to s.p. properties and to look for spectroscopic- quality EDF, even at the expense of deteriorating its quality in reproducing binding energies. Such a strategy requires well-defined empirical input related to the s.p. energies, to be used directly in the fitting process. We ar- gue that odd-even mass differences around magic nuclei not only provide unambiguous direct information about nuclear s.p. energies but are also well anchored within13Spin-orbit splittings [MeV] 16 O40 Ca 48 Ca 56 Ni 90 Zr 132 Sn 208 Pb 16 O40 Ca 48 Ca 56 Ni 90 Zr 132 Sn 208 Pb p pn n 1h1h1i d5/2 -d3/2 p3/2 -p1/2 d5/2 -d3/2 p3/2 -p1/2 f7/2 -f5/2 g9/2 -g7/2 f7/2 -f5/2 g9/2 -g7/2 SLy4 LT 1357 1357 FIG. 5: Same as in Fig. 4 but for the SLy4 functional.Spin-orbit splittings [MeV] 1357 16 O40 Ca 48 Ca 56 Ni 90 Zr 132 Sn 208 Pb 16 O40 Ca 48 Ca 56 Ni 90 Zr 132 Sn 208 Pb p pn n 1h1h 1i d5/2 -d3/2 p3/2 -p1/2 d5/2 -d3/2 p3/2 -p1/2 f7/2 -f5/2 g9/2 -g7/2 f7/2 -f5/2 g9/2 -g7/2 SkO LT 1357 FIG. 6: Same as in Fig. 4 but for the SkO functional. the spirit of the EDF formalism. Indeed, the theorems due to Hochenberg and Kohn [49] and Levy [50], see also Refs. [51], imply existence of universal EDF capable, at least in principle, treating ground-states energies of nu- clei exactly. One can, therefore, argue that this implies essentially exact treatment of at least the lowest s.p. lev- els forming ground states in one-particle (one-hole) odd-Anuclei with respect to even-even cores or, alternatively, almost exact description of core-polarization phenomena caused by odd single-particle (single-hole). An attempt to refit the EDF is preceded by a system- atic analysis of the s.p. energies and self-consistent core- polarization effects within the state-of-the-art Skyrme- force-inspired EDF. Three major sources of core-14 246810 p nSkP SkP T 820 28 50 82 126 56 Ni p1/2 d5/2 d3/2 f7/2 f7/2 p3/2 g9/2 d5/2 f7/2 d3/2 g9/2 p1/2 820 28 40 50 82 48 Ca p1/2 d5/2 d3/2 f7/2 f7/2 p3/2 g9/2 g7/2 g9/2 p1/2 h9/2 s1/2 Magicgaps[MeV]exp FIG. 7: Experimental [8] (dots) and theoretical values of ma gic gaps (24), calculated using the original SkP functional (open triangles) and the SkP Tfunctional (full triangles) with the SO and the tensor coupl ing constants from Table. IV. The gaps were computed using the bare unpolarized s.p. spectra. polarization, including mass, shape and spin (time-odd) effects, are identified and discussed. The analysis is per- formed for even-even doubly-magic cores and the lowest s.p. states in odd- Aone-particle(hole) nuclei. The discus- sion is supplemented by analysis of the s.p. SO splittings. New strategy in fitting the EDF is applied to the SO and tensor parts of the nuclear EDF. Instead of large- scale fit to binding energies we propose simple and in- tuitive three-step procedure that can be used to fit the isoscalar strength of the SO interaction as well as the isoscalar and isovector strengths of the tensor interac- tion. The entire idea is based on the observation that the f7/2−f5/2SO splittings in spin-saturated isoscalar40Ca, spin-unsaturated isoscalar56Ni, and spin-unsaturated isovector48Ca form distinct pattern that can neither beunderstood nor reproduced based solely on the conven- tional SO interaction. 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1309.3852v1.Spin_Hot_Spots_in_Single_Electron_GaAs_based_Quantum_Dots.pdf

arXiv:1309.3852v1 [cond-mat.mes-hall] 16 Sep 2013Spin Hot Spots in Single-Electron GaAs-based Quantum Dots Martin Raith1, Thomas Pangerl1, Peter Stano2,3, and Jaroslav Fabian1 1Institute for Theoretical Physics, University of Regensbu rg, D-93040 Regensburg, Germany 2RIKEN Center for Emergent Matter Science, 2-1 Hirosawa, Wak o, Saitama, 351-0198 Japan 3Institute of Physics, Slovak Academy of Sciences, 845 11 Bra tislava, Slovakia Spin relaxation of a single electron in a weakly coupled doub le quantum dot is calculated numer- ically. The phonon assisted spin flip is allowed by the presen ce of the linear and cubic spin-orbit couplings and nuclear spins. The rate is calculated as a func tion of the interdot coupling, the magnetic field strength and orientation, and the dot bias. In an in-plane magnetic field, the rate is strongly anisotropic with respect to the magnetic field or ientation, due to the anisotropy of the spin-orbit interactions. The nuclear spin influence is negl igible. In an out-of-plane field, the nuclear spins play a more important role due selection rules imposed on the spin-orbit couplings. Our theory shows a very good agreement with data measured in [Srinivasa , et al., PRL 110, 196803 (2013)], allowing us to extract information on the linear spin-orbit interactions strengths in that experiment. We estimate that they correspond to spin-orbit lengths of ab out 5-15 µm. PACS numbers: 72.25.Rb, 03.67.Lx, 71.70.Ej, 73.21.La I. INTRODUCTION Semiconductor heterostructure based quantum dots with confined electronic spins are amongthe most promi- nent platforms of spitronics1,2and quantum informa- tion related technology.3–6The lifetime of information stored in a quantum dot spin qubit is limited by the spin relaxation7,8and decoherence.9,10Whereas the lat- ter, mostly due to nuclear spins,11can be suppressed by spin echo protocols,12the former is fundamentally lim- ited by the relaxation through phonons.13–16 Phonons do not couple to the electron spin di- rectly. The spin relaxation is enabled by the spin-orbit interactions13,17–21ornuclearspins.11,22Sincethese spin- dependent interactions are weak, compared to the con- finement energy, the spin relaxation is very slow (may reach even seconds), which was one of the original moti- vations to consider spin qubits. The exception happens at points in the parameter space where levels (anti)cross. Here the spin relaxation rate is strongly enhanced, by or- ders of magnitude. Such points are called spin hot-spots. The important influence that the spin hot-spots might imply on the spin relaxation was recognized in bulk metals23and in quantum dots.24,25In the latter this in- fluence is predicted to result in a very strong anisotropy in the spin lifetimes and the exchange interaction, which should be present generally, for variousdot materials and chargeoccupations.26–29However, it is only recently that spin hot spots were experimentally established in gated Si and GaAs quantum dots.30,31 Motivated by these recent experiments, here we in- vestigate the spin relaxation in a single electron biased weakly coupled double dot in GaAs.32–38This comple- ments our studies of single electron unbiased double dots15,25,26and two electron biased double dots.28We investigate the relaxation rates anisotropy with respect to the in-plane magnetic field orientation, and compare the spin-orbit and nuclear fields effectiveness to induce the electron spin relaxation in in-plane and out-of-planeE || dγy xδB d FIG. 1. The orientation of the potential dot minima (denoted as the two circles) with respect to the crystallographic axe s (x= [100] and y= [010]) is defined by the angle δ. The magnetic field orientation is given by the angle γ. The electric fieldEis parallel to d. magnetic fields. We explain the observed relaxation rate intricate behavior by examining different channels that contribute to the total rate. Finally, we extract typical spin-orbit lengths in a GaAs quantum dot by fitting data from a recent experiment.31 We organize the paper as follows. The model of the double dot, material parameters, and the numerical technique used for computation are outlined in Sec. II. Sec. III contains the numerical results for the relaxation rate for an in-plane magnetic field, and perpendicular field, comparison of the spin-orbit and nuclear effective- ness, and the fit of the experimental data from Ref. 31.2 II. MODEL We consider a GaAs/AlGaAs heterojunction with growth direction ˆ z= [001]. The electrons at the inter- face are further confined by the electrostatic field of top gates. Using the envelope function approximation, the two-dimensional Hamiltonian of a single electron in a bi- ased double dot reads as H=T+V+HZ+Hso+Hnuc. (1) HereT=P2/2mis the kinetic energy with the elec- tron effective mass m, and the kinematic momentum P=−i/planckover2pi1∇+eA, whereeis the proton charge. The two dimensional vector potential reads A=−(yBz/2)ˆx+ (xBz/2)ˆy, where ˆx= [100] and ˆy= [010]. The mag- netic field is B=/parenleftbig B/bardblcosγ,B/bardblsinγ,Bz/parenrightbig , whereγis the angle between the in-plane component of the magnetic field and the [100]-direction. The orbital effects of the in-plane magnetic field are neglected.15The in-plane po- sition vector is r= (x,y). The double dot is defined by the bi-quadratic confinement potential,39–41 V=/planckover2pi12 2ml4 0min/braceleftBig (r−d)2,(r+d)2/bracerightBig +eE·r.(2) For zero electric field Ethe potential minima are located at±d, and we call 2 d/l0the (dimensionless) interdot dis- tance. The angle between dand [100] is denoted as δ. The potential strength is characterized by the confine- ment energy E0=/planckover2pi12/ml2 0, with the confinement length l0. The electric field Eapplied along dleads to an energy offset between the potential minima, ǫ= 2eEd, which we call bias in further. The geometry is summarized in Fig. 1. The Zeeman term reads HZ=g 2µBB·σ, (3) wheregis the effective conduction band gfactor,µB is the Bohr magneton, and σis the vector of the Pauli matrices. The spin-orbit coupling, Hso=Hbr+Hd+Hd3, consists of three terms, the Bychkov-Rashba, the lin- ear, and the cubic Dresselhaus spin-orbit coupling.1,2 The Bychkov-Rashba Hamiltonian, arising from the het- erostructure asymmetry, reads as42 Hbr=/planckover2pi1 2mlbr(σxPy−σyPx), (4) where the strength is parameterized by the spin-orbit lengthlbr. The bulk inversion asymmetry of the zinc- blende structure enables the Dresselhaus interaction.43 It consists of two terms: linear, and cubic (referring to the power of the momentum operator), Hd=/planckover2pi1 2mld(−σxPx+σyPy), (5) Hd3=γc 2/planckover2pi13/parenleftbig σxPxP2 y−σyPyP2 x/parenrightbig +H.c.,(6)respectively. The linear term is parameterized by the spin-orbit length ld, andγcis a material parameter. ThelastterminEq.(1)describesthehyperfineinterac- tion of the confined electron with the lattice’s nuclei,44,45 Hnuc=β/summationdisplay nIn·σδ(R−Rn), (7) whereβis a constant, and InandRnare the spin and the position of the n-th nucleus. Here the vectors of position are three-dimensional, R= (r,z). The electron wavefunction along the growth direction, Ψ( z), defines an effective width hz=/parenleftbig/integraltext dz|Ψ(z)|4/parenrightbig−1.46We assume Ψ(z) to be the ground state of a hard-wall confinement of width w, and get hz= 2w/3. The relaxation is enabled by acoustic phonons. The electron-phonon interaction Hamiltonian reads as Hph=i/summationdisplay Q,λ/radicalBigg /planckover2pi1Q 2ρVcλVQ,λ/parenleftBig b† Q,λeiQ·R−bQ,λe−iQ·R/parenrightBig , (8) withλ=l,t1,t2 denoting the polarization of the phonons (one longitudinal and two transverse). The three-dimensional phonon wave vector is Q. The phonon creation and annihilation operator is given by band b†, respectively. The mass density of the crystal is ρ, its volume is V, and the sound velocities are cλ. The deformation potential is VQ,λ=σeδλ,land the piezo- electric potential is VQ,λ=−ieh14Nλ/Q3withNλ= 2/parenleftbig qxqyˆeλ z+qzqxˆeλ y+qyqzˆeλ x/parenrightbig . The unit polarization vec- tor isˆ eλ. The relaxation rate for the transition from state |i/angbracketrightto |f/angbracketrightis calculated using the Fermi’s Golden Rule in the zero temperature limit, Γif=π ρV/summationdisplay Q,λQ cλ|VQ,λ|2|Mif|2δ(Eif−EQ),(9) whereMif=/angbracketleftbig i/vextendsingle/vextendsingleeiQ·R/vextendsingle/vextendsinglef/angbracketrightbig is the transition matrix ele- ment, and Eifis the energy difference between |i/angbracketrightand |f/angbracketright. To incorporatenuclei, we averagethe relaxationrate in Eq. (9) over several (typically 50) random configura- tions of an unpolarized nuclear bath—see Ref. 28 for de- tails. In numerics we use the material parameters of bulk GaAs:m= 0.067me, wheremeis the free electron mass, g=−0.44,ρ= 5300 kg/m3,cl= 5290 m/s, ct= 2480 m/s,γc= 27.5 eV˚A3,σe= 7 eV, eh14= 1.4×109 eV/m,β= 1µeVnm3, andI= 3/2. The quantum dot parameters are l0= 34nm ( E0= 1meV), lbr= 2.42 µm andld= 0.63µm.47We use the coupling strength of 2d/l0= 4.35, corresponding to a tunneling energy of t= 0.01meV. The orientation of the dots is along [110], i.e.δ= 45◦, unless stated otherwise. Since the energy spectrum of the Hamiltonian in Eq. (1) cannot be solved for analytically, we treat it nu- merically using the finite differences method with Dirich- let boundary conditions48including the magnetic field3 050100150200250 ε [µeV]0.811.2E [meV]' ' FIG. 2. (Color online) Calculated energy spectrum of a GaAs double dot as a function of the bias ǫin an in-plane magnetic fieldBof 7 T with δ=γ= 45◦andt= 0.01 meV. The states are labeled according to their spin orientation and parity a t ǫ= 0 (unprimed for even, primed for odd). The inset magni- fies the anticrossing. The thin arrows denote transitions th at contribute to the measured relaxation rate. via the Peierl’s phase.49The resulting eigenvalue prob- lem is then solved using the Lanczos algorithm.50In the numerics we use grid dimensions of typically around 200×200 grid points. The relative error is below 10−5. III. RESULTS A. In-plane magnetic field anisotropy Let us first look at the dot energy spectrum. Figure 2 shows the lowest four levels as a function of the bias for a weakly coupled double dot. States are denoted according to their spatial inversion parity at zero bias: states with a prime have odd, and states without a prime have even parity. AtadetuningenergyofabouttheZeemanenergy, ǫ= 0.178meV, the states |↑/angbracketright′and|↓/angbracketrightform an anisotropic anticrossing due to spin-orbit coupling. The anticrossing energy is maximal for an orientation of γ= 45◦, and absent if γ= 135◦. This special point in the spectrum is the spin hot spot.23,25Herethe spin orientationsmoothly changes from an up to a down state and vice versa. We define the ”spin relaxation rate” Γ according to what is measured in corresponding experiments.31The initial state for the transition is the lowest spin down state,|↓/angbracketright, while the transition is considered as completed if the lowest state, |↑/angbracketright, is detected. Since the transitions between spin alike states are much faster than a dura- tion of the measurement cycle, they can be considered instantaneous and we have Γ≈Γ|↓/angbracketright→|↑/angbracketright+Γ|↓/angbracketright→|↑′/angbracketright. (10) The individual transition rates for a weakly coupled dou- ble dot are plotted as a function of the bias in Fig. 3.0 0.1 0.2 0.3 Detuning ε [meV]105106107108Relaxation rate [1/s] Γ|↓〉→|↑〉 Γ|↓〉→|↑〉’Γtotal FIG. 3. Calculated spin relaxation rate, resolved into chan - nels, of a double dot as a function of detuning for δ=γ= 45◦ withB= 7 T,t= 0.01 meV and T= 0 K. The hyperfine cou- pling is neglected. The dotted, solid, and dashed line gives Γ|↓/angbracketright→|↑/angbracketright , Γ|↓/angbracketright→|↑′/angbracketright, and Γ, respectively. The relaxation rate between the two lowest Zeeman split statesΓ |↓/angbracketright→|↑/angbracketrightis,apartfromtheanticrossing,notvarying much, due to the energy difference being constant. On the other hand, the transition into the first excited state |↑/angbracketright′is highly non-monotonic. Initially it grows, since, as the two states become closer in energy, it is easier to ad- mix the spin-opposite component into the states. If the energy difference becomes too small, the rate drops, as now the diminishing density of states of phonons takes over the trend. For detunings beyond the anticrossing, the second term of the right hand side of Eq. (10) will be suppressedat low temperatures, which contributes to the strong asymmetry of the relaxation rates as a function of the bias with respect to the position of the anticrossing. The spin-orbit enabled relaxation rate as a function of detuning and orientation of an in-plane magnetic field is plotted in Fig. 4. It shows the anisotropic relaxation landscape and the existence of two principal axes for the in-plane magnetic field orientation: parallel ( γ= 45◦) andperpendicular( γ= 135◦) tothedot mainaxis d. For smalldetunings, andinthevicinityofthespinhotspotat ǫ= 0.178meV, the relaxation rate is strongly suppressed ifγ= 135◦. On the other hand, the relaxation rate for large detunings is minimal if γ= 45◦, as here the sys- tem has single dot character. This directional switch of the axis of minimal relaxation has previously been found in two-electron double dots28,51and can be understood from the effective, spin-orbit induced, magnetic field.26 It is only for γ= 135◦that changing between unbiased and highly biased configurations can be achieved with- out passing through a regime of strongly enhanced spin relaxation. This feature is known as an easy passage.264 0 30 60 90 120 150 180 γ [deg] 0 0.1 0.2 0.3 0.4 0.5Detuning ε [meV] 104105106107108 6⋅1055⋅105 FIG. 4. (Color online) Calculated spin relaxation of a doubl e dot as a function of detuning and orientation of the in-plane magnetic field B= 7 T, with δ= 45◦,t= 0.01 meV and T= 0 K. The hyperfine coupling is neglected. The corresponding energy spectrum (at γ= 45◦) is shown in Fig. 2. The rate is plotted according to the color scale on the right in invers e seconds. The labeled contours represent equirelaxation li nes. B. Spin-orbit vs nuclear fields Let us now comment on the role of nuclei and how the presented results are altered in the presence of hyperfine- induced spin relaxation. For the double quantum dot considered above, we find that the relaxation rates due to the nuclei are typically 2 orders of magnitude smaller than the rates given in Fig. 4. The exception occurs at the spectral anticrossing ( ǫ≈0.178 meV) because the states |↓/angbracketrightand|↑/angbracketright′are always coupled by the nuclear spins irrespective of the orientation of the in-plane mag- netic field. Thus, the hyperfine-induced spin relaxation becomesdominant at the anticrossingalongthe easypas- sage, where the spin-orbit contribution to the relaxation is of the order of 105s−1. However, the impact here is rather small, as we show now. We present the spin relaxation rates enabled by either spin-orbit coupling or hyperfine coupling for the double dot with parameters given above in Fig. 5 and Fig. 6, respectively. The in-plane magnetic field orientation for this comparison is chosen to be γ= 45◦, i.e. away from the easy passage. We see that the spin-orbit contribution is dominant over the whole parameter range. The spike inthe relaxationratemapofFig.6becomesrelevantonly in the easy passage configuration (or for magnetic fields below 2T). However,we find that the impact onthe total relaxation rate is rather weak because of the small width of the spike. Particularly, the spike is hardly visible for magnetic fields of 6T or more. 0 2 4 6 8 10 Magnetic field B|| [T] 0 0.1 0.2 0.3Detuning ε [meV] 100102104106108 106 FIG. 5. (Color online) Calculated spin relaxation of a doubl e dot as a function of detuning and magnitude of the in-plane magnetic field with t= 0.01 meV, B= 7 T,δ=γ= 45◦, andT= 0 K. The hyperfine coupling is neglected. The rate is plotted according to the color scale on the right in invers e seconds. The labeled contours represent equirelaxation li nes. 0 2 4 6 8 10 Magnetic field B|| [T] 0 0.1 0.2 0.3Detuning ε [meV] 100102104106108 5⋅105 FIG. 6. (Color online) Same as Fig. 5, but now only hyperfine coupling is considered (no spin-orbit coupling). The cellu lar structure of the plot around the anticrossing is a finite reso - lution artifact, not a physical effect. C. Perpendicular magnetic field Forcompleteness, we now considerasymmetric double quantum dot in an external magnetic field perpendicular to the dot plane. We focus on the dependence of the spin relaxationon the magnetic field magnitudeand the inter- dot distance, and compare the impact of spin-orbit and hyperfine coupling on the relaxation rates. For more on biased dots in perpendicular magnetic fields, see Ref. 52. Perpendicular magnetic field has also orbital effects,5 0 2 4 6 8 10 Magnetic field B⊥ [T] 0 1 2 3 4 5Interdot distance [2d /l0] 1001021041061081010Tunneling energy T [meV]0.480.350.210.090.023E-3 107 FIG. 7. (Color online) Spin relaxation of a double dot as a function of perpendicular magnetic field and interdot cou- pling. The rate is given in s−1. The solid lines represent equirelaxation lines. The relaxation rate was calculated c on- sidering only the spin-orbit coupling. resulting in an effective confinement length25lB=/parenleftbig l−4 0+B2e2/4/planckover2pi12/parenrightbig−1/4. This effectively changes the in- terdot coupling, and the tunneling energydecreases. The positions of the level crossings in the energy spectrum are therefore strongly dependent on both the interdot distance and the magnetic field strength. We plot the spin relaxation rates of a double dot in a perpendicular magnetic field for the cases of either spin- orbit coupling or hyperfine coupling in Figs. 7 and 8, respectively. Without the nuclear field (Fig. 7), the first hot spots of the single dot ( d= 0) are at B≈4.5T, B≈7.9T, etc. We find that the spikes in the relaxation rate map in Fig. 7 generally become less pronounced for stronger magnetic fields. Except for the first anticross- ingatB≈4.5T, the levelcrossingsforinterdotdistances 2d/l0/greaterorsimilar2 (T/lessorsimilar0.2meV) are found at a constant mag- netic field. Switching off the spin-orbit coupling and considering only the coupling to the nuclei (Fig. 8), we find more spikes in the relaxation rate map as compared to Fig. 7. The difference between the influence of nuclei and spin- orbit coupling is due to the chosen confinement profile. Namely, in a parabolic well, the linear-in- pspin-orbit in- teractions couple only Fock-Darwin states with orbital momentum differing by 1.53This leads to a very strong suppression of the width of higher anticrossings induced by the spin-orbit interaction. On the other hand, there is no such selection rule for unpolarized nuclei and in this case the widths of consecutive anticrossings decay much slower. Since the parabolic confinement is believed to be a good description of the low lying part of the spectrum, we conclude that the hyperfine-induced spin relaxation plays a more important role if the external magnetic field is perpendicular. 0 2 4 6 8 10 Magnetic field B⊥ [T] 0 1 2 3 4 5Interdot distance [2d /l0] 1001021041061081010Tunneling energy T [meV]0.480.350.210.090.023E-3 106 FIG. 8. (Color online) Spin relaxation of a double dot as a function of perpendicular magnetic field and interdot cou- pling. The rate is given in s−1. The solid lines represent equirelaxation lines. The relaxation rate was calculated c on- sidering only the hyperfine coupling. D. Extracting spin-orbit lengths from Ref. 31 Having available a quantitatively faithful theory, we fit the data measured in Ref. 31, an experiment on the spin relaxation in a single electron weakly coupled GaAs dou- ble dot. We aim at extraction of the spin-orbit lengths. Despite their crucial importance for spintronics applica- tions and theory, their values are not reliably established insmall(occupiedbyfew electrons)quantumdots, where the strong confinement may renormalize the values ex- trapolated from measurements in quantum wells or bulk. Ref. 31 gives the following experimentally accessible parameters: the confinement energy of 1 meV, the tun- neling energy of 8 µeV, the magnetic field of 6.5 T, ap- plied alongthe dot main axis, γ=δ, and the anticrossing occurring at the detuning of 0.136 meV. They translate into the following parameters of our model: l0= 34 nm, g=−0.364, and d= 76.5 nm. For m,γcand phonon characteristics we use the bulk values, as given below Eq.(9). Finally, we use a finite temperature of 0.25 K, and neglect nuclear spins. We keep the spin-orbit lengths lbr,ld, and the dot ori- entation δ(the angle between the dot main axis and the crystallographic axis [100]) as fitting parameters. We adopt a standard procedure54and fit by minimizing the χ2measure χ2=/summationdisplay ǫi(log[Γ⋆(ǫi)]−log[Γ(ǫi,lbr,ld,γ)])2.(11) Hereǫilabels different measurements, and Γ⋆are mea- sured values (100 data points). Since the rates vary over orders of magnitude, we use a logarithmic scale. Because of a highly non-linear shape of the relaxation rate curves, the figure of merit of the fit, the function χ2,6 -300 -200 -100 0 100 200 300 ε [µeV]102103104105106107Γ [1/s] FIG. 9. Comparison of calculated (solid line; parameters fr om the first line of Tab. I) spin relaxation rates to the ones mea- sured in Ref. 31 (symbols). The dashed line is the result of the minimization with a fixed dot orientation, δ= 45◦=γ, which gave α= 0.84 meV ˚A, andβ= 0.47 meV ˚A. setδ α [meVA] β[meV ˚A]lbr[µm]ld[µm]χ2 min 1 307◦-1.34 1.51 -4.2 3.8 9.66 2 60◦0.65 0.89 8.8 6.4 9.69 3 62◦0.55 0.82 10.3 7.0 9.71 4 203◦0.34 0.67 16.7 8.4 9.76 5 294◦0.48 -0.45 11.8 -12.6 9.87 TABLE I. Fitted spin-orbit lengths lbrandldand the dot orientation δ(angle between the main dot axis and [100]). Each set corresponds to a local minimum of χ2, Eq. (11), in the parameter space. The spin-orbit lengths are also given in alternative units through α=/planckover2pi12/2mlbrandβ=/planckover2pi12/2mld. Our definition of the spin-orbit lengths lsoin Eqs. (4) and (5) is such that in a one dimensional model these Hamiltonians induce a rotation of the spin by an angle 2 πr/lsoupon a spatial displacement of the electron by a distance r. has many local minima in the fitting parameters space.We give several examples in Tab. I. The rather small differences in values of χ2in these minima mean that all these parameter sets fit the data almost equally well. This also gives a very crude estimate on the reliability of the extracted values of the spin-orbit strengths—their relative sign remains unknown and their magnitudes can not be established better than within a factor of 3. The fit corresponding to the parameters in the first line of the table is plotted together with the measured data in Fig. 9. We also plot a result of minimization with a fixed dot orientation, δ= 45◦=γ, which might have been the case in the experiment.55This would also make the values in lines 2 and 3 of the Tab. I more probable than others. IV. SUMMARY We have calculated phonon induced spin relaxation rates enabled via spin-orbit coupling and hyperfine cou- pling of single electron states in biased double quantum dots. We find strong anisotropies in the relaxation rate, due to anisotropy of the underlying spin-orbit interac- tions, and the relatedspin hots and easypassages,known fromworksonunbiaseddots. Forthespin-orbitstrengths of the order of 1 µm, chosen by fitting data measured in Ref. 7, we find that the contribution of nuclear spins is negligible. Fitting data from a different experiment of Ref. 31, we extract the spin-orbit lengths of the order of 10µm, for which nuclear spin contribution is roughly comparable to that of the spin-orbit interactions. To nail down the spin-orbit interactions strengths with bet- ter confidence, the measurement of the rate as a function ofthe magneticfield orientationis calledfor. In addition, such a measurement is ideal for separate identification of the two linear spin-orbit strengths. ACKNOWLEDGMENTS This work was supported by DFG under grant SPP 1285 and SFB 689. 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1502.05400v1.Quenching_of_dynamic_nuclear_polarization_by_spin_orbit_coupling_in_GaAs_quantum_dots.pdf

arXiv:1502.05400v1 [cond-mat.mes-hall] 18 Feb 2015Quenching of dynamic nuclear polarization by spin-orbit co upling in GaAs quantum dots John M. Nichol,1Shannon P. Harvey,1Michael D. Shulman,1Arijeet Pal,1Vladimir Umansky,2Emmanuel I. Rashba,1Bertrand I. Halperin,1and Amir Yacoby1 1Department of Physics, Harvard University, Cambridge, MA, 02138, USA 2Braun Center for Submicron Research, Department of Condens ed Matter Physics, Weizmann Institute of Science, Rehovot 76100 Israel The central-spin problem, in which an electron spin interac ts with a nuclear spin bath, is a widely studied model of quantum decoherence [1]. Dynamic nuclear p olarization (DNP) occurs in central spin systems when electronic angular momentum is transferr ed to nuclear spins [2] and is exploited in spin-based quantum information processing for coherent electron and nuclear spin control [3]. However, the mechanisms limiting DNP remain only partially understood [4]. Here, we show that spin-orbit coupling quenches DNP in a GaAs double quantum do t [5], even though spin-orbit cou- pling in GaAs is weak. Using Landau-Zener sweeps, we measure the dependence of the electron spin-flip probability on the strength and direction of in-pl ane magnetic field, allowing us to distin- guish effects of the spin-orbit and hyperfine interactions. T o confirm our interpretation, we measure high-bandwidth correlations in the electron spin-flip prob ability and attain results consistent with a significant spin-orbit contribution. We observe that DNP i s quenched when the spin-orbit compo- nent exceeds the hyperfine, in agreement with a theoretical m odel. Our results shed new light on the surprising competition between the spin-orbit and hyperfin e interactions in central-spin systems. Dynamic nuclear polarization occurs in many con- densed matter systems, and is used for sensitivity en- hancement in nuclear magnetic resonance [6] and for de- tecting and initializing solid-state nuclear spin qubits [7]. DNP also occurs in two-dimensional electron systems [8] via the contact hyperfine interaction. In both self- assembled [9–13] and gate-defined quantum dots [3, 14– 16], for example, DNP is exploited to create stabilized nuclearconfigurationsforimprovedquantuminformation processing. Closed-loop feedback [15] based on DNP, in particular, is a key-component in one- and two-qubit op- erations in singlet-triplet qubits [3, 17, 18]. Despite the importance of DNP, it remains unclear what factors limit DNP efficiency in semiconductor spin qubits [4]. In particular, the relationship between the spin-orbit and hyperfine interactions [19–21] has been overlooked in previous experimental studies of DNP in quantum dots. In this work we show that spin-orbit coupling competes with the hyperfine interaction and ultimately quenches DNP in a GaAs double quantum dot [5, 17], even though the spin orbit length is much larger than the interdot spacing. We use Landau-Zener (LZ) sweeps to characterizethe static and dynamic prop- erties of ∆ ST(t), the splitting between the singlet Sand ms= 1 triplet T+, and the observed suppression of DNP agrees quantitatively with a new theoretical model. Figure 1(a) shows the double quantum dot used in this work [5, 17]. The detuning, ǫ, between the dots determines the ground-state charge configuration, which is either (1,1) [one electron in each dot], or (0,2) [both electrons in the right dot] as shown in Fig. 1(b). To mea- sure ∆ ST(t), the electrons are initialized in |(0,2)S∝angbracketright,ǫis swept through the S−T+avoided crossing at ǫ=ǫST, andtheresultingspinstateismeasured[Fig.2(a)]. Intheabsenceof noise, slowsweeps causetransitions with near- unity probability. For large magnetic fields, however, we find maximum transition probabilities of approximately 0.5. This reduction is a result of rapid fluctuations in the sweep rate arising from charge noise (see Supple- mentary Information). Even in the presence of noise, however, the average LZ probability ∝angbracketleftPLZ(t)∝angbracketrightcan be ap- proximated for fast sweeps as2π/angbracketleft|∆ST(t)|2/angbracketright /planckover2pi1β(see Supple- mentary Information). Here ∝angbracketleft···∝angbracketrightindicates an average over the hyperfine distribution and charge fluctuations, andβ=d(ES−ET+)/dtis the sweep rate, with ES andET+the energies of the SandT+levels. To ac- curately measure σST≡/radicalbig ∝angbracketleft|∆ST(t)|2∝angbracketright, we therefore fit ∝angbracketleftPLZ∝angbracketrightvsβ−1to a straight line for values of βsuch that 0<∝angbracketleftPLZ∝angbracketright<0.1. [Fig. 2 (a)]. We first measure σSTvsφatB= 0.5 T [Fig. 2(b)], whereφis the angle between the magnetic field Band the z axis [Fig. 1(a)]. σSToscillates between its extreme values at 0◦and 90◦with a periodicity of 180◦. Fixing φ= 0◦and varying B, we find that σSTdecreasesweakly with with B, but when φ= 90◦,σSTincreases steeply withB, reaching values greater than 10 times that for φ= 0◦, as shown in Fig. 2(c). We interpret these results by assuming that both the hyperfine and spin-orbit interactions contribute to ∆ST(t) and by considering the charge configuration of the singlet state at ǫST[Figs. 1(b) and (c)]. The matrix element between SandT+can be written as ∆ ST(t) = ∆HF(t)+∆SO. ∆HF(t) =g∗µBδB⊥(t) is the hyperfine contribution, which arises from the difference in perpen- dicular (relative to B) hyperfine field, δB⊥(t), between the two dots [22]. (In the following, we set g∗µB= 1.) ∆HF(t), which is a complex number, couples |(1,1)S∝angbracketright to|(1,1)T+∝angbracketrightwhen the two dots are symmetric. ∆ SO2 ε250 nm energy (0,2) S (0,2) S (1,1) T - (1,1) S (1,1) T 0 (1,1) T + S T+ (1,1) S ∆ST (t) |g*| µBB detuning (ε)(a) (c)(b) φ B xz (1,1) T +(1,1) S (0,2) S Hyperfine Spin orbit0εST B ΩSO B ΩSO φ=0° φ=90° FIG. 1. Experimental setup. (a) Scanning electron micro- graph of the double quantum dot. A voltage difference be- tween the gates adjusts the detuning ǫbetween the potential wells, and a nearby quantum dot on the left senses the charge state of the double dot. The gate on the right couples the double dot to an adjacent double dot, which is unused in this work. The angle between Band the zaxis isφ. (b) En- ergy level diagram showing the two-electron spin states and zoom-in of the S−T+avoided crossing. (c) The hyperfine interaction couples |(1,1)S/angbracketrightand|(1,1)T+/angbracketrightwhen the two dots are symmetric, regardless of the orientation of B, and the spin-orbit interaction couples |(0,2)S/angbracketrightand|(1,1)T+/angbracketrightwhenB has a component perpendicular to ΩSO= ΩSOˆz, the effective spin-orbit field experienced by the electrons during tunnel ing. is the spin-orbit contribution, which arises from an ef- fective magnetic field ΩSO= ΩSOˆzexperienced by the electron during tunneling [19]. Only the component of ΩSO⊥Bcauses an electron spin flip. ∆ SOtherefore couples|(0,2)S∝angbracketrightto|(1,1)T+∝angbracketrightwhenφ∝negationslash= 0◦, and Ω SO is proportional to the double-dot tunnel coupling [19], which is 23.1 µeV here. At ǫST, the singlet state |S∝angbracketrightis a hybridized mixture: |S∝angbracketright= cosθ|(1,1)S∝angbracketright+ sinθ|(0,2)S∝angbracketright, where the singlet mixing angle θapproaches π/2 asB increases (see Supplementary Information). Taking both θandφinto account, we write [19] ∆ST(t) = ∆HF(t)+∆SO =δB⊥(t)cosθ+ΩSOsinφsinθ.(1) The data in Fig. 2(b) therefore reflect the dependence of ∆ST(t) onφin equation (1). The data in Fig. 2(c) re- flect the dependence of ∆ ST(t) onθ. AsBincreases, θalso increases, and |S∝angbracketrightbecomes more |(0,2)S∝angbracketright-like, causing ∆ HF(t) to decrease. When φ= 0◦, ∆SO= 0 for allB, but when φ= 90◦, ∆SO= ΩSOsinθ, andφ=0° φ=90°(a) (c) (b)time load sweep meas. (0,2) (1,1) εPLZ φ (degrees) σST /h (MHz) σST /h (MHz) (h/β) (x10-3 µs/GHz) (h β)-1 (x10 -4 µs/GHz) εST B=0.5 TB=1 T φ=90° B (T) 0 0.5 1020 40 60 80 100 -180 -90 090 180010 20 30 40 50 60 70 0 2 4 6 800.20.40.60.81 0 1 2 300.050.1PLZ FIG. 2. Measurements of σST. (a) Data for a series of LZ sweeps with varying rates, showing reduction in maximum probability due to charge noise. The horizontal axis is pro- portional to the sweep time. Upper inset: Data and linear fit for fast sweeps such that 0 </angbracketleftPLZ/angbracketright<0.1. Lower Inset: In a LZ sweep, a |(0,2)S/angbracketrightstate is prepared, and ǫis swept through ǫST(dashed line) with varyingrates. Here h= 2π/planckover2pi1is Planck’s constant. (b) σSTvsφ(dots) and simulation (solid line). (c) σSTvsBforφ= 0◦andφ= 90◦(dots) and fits to equation (1) (solid lines). Error bars are fit errors. σSTincreases with B. Fitting the data in Fig. 2(c) al- lows a direct measurement of the spin-orbit and hyper- fine couplings (see Supplementary Information). We find/radicalbig ∝angbracketleft|δB2 ⊥(t)|∝angbracketright= 34±1 neV and Ω SO= 461±10 neV, cor- responding to a spin-orbit length λSO≈13µm [19], in goodagreementwith previousestimates in GaAs[23–25]. We further verify that ∆ ST(t) contains a signifi- cant spin-orbit contribution by measuring the dynam- ical properties of PLZ(t). A key difference between the spin-orbit and hyperfine components is that ∆ SO is static, while ∆ HF(t) varies in time because it arises from the transverse Overhauser field, which can be con- sidered a precessing nuclear polarization in the semi- classical limit [22]. To distinguish the components of ∆ST(t) through their time-dependence, we develop a high-bandwidth technique to measure the power spec- trum ofPLZ(t). Instead of measuring the two-electron spin state after a single sweep, ǫis swept twice through ǫSTwith a pause of length τbetween sweeps [Fig. 3(a)] (See Supplemen- tary Information). Assuming that St¨ uckelberg oscilla- tions rapidly dephase during τ[18, 23], and after sub- tracting a background and neglecting electron spin re-3 (a) (b) (c) (d) time (0,2) load τmeas. (1,1) εRPP (τ) f71 Ga -f 69 Ga f69 Ga -f 75 As f71 Ga -f 75 As f75 As f69 Ga f71 Ga sweep sweep SP(ω) φ (degrees) φ=0°φ=25°φ=80°εST 0 10 20 30 40 50 60 70 80 -0.1-0.0500.050.1 τ ( µs) 0510 15 020 40 60 80 ω/2 π (MHz) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 B=0.1T , φ=0° FIG. 3. Correlations and power spectrum of PLZ(t). (a) Pulse sequence to measure RPP(τ) using two LZ sweeps. (b) RPP(τ)forφ= 0◦andB= 0.1T.Thedataextendto τ= 200 µs, but for clarity are only shown to 75 µs here. (c) SP(ω) vsφobtained by Fourier-transforming RPP(τ). Atφ= 0◦, the differences between the nuclear Larmor frequencies are evident, but for |φ|>0◦, the absolute Larmor frequencies ap- pear, consistent with a spin-orbit contribution to σST. The reduction in frequency with φis likely due to the placement of the device slightly off-center in our magnet (see Supplemen- tary Information). (d) Line cuts of SP(ω) atφ= 0◦, 25◦, and 80◦. laxation, the time-averaged triplet return probability is proportional to RPP(τ)≡ ∝angbracketleftPLZ(t)PLZ(t+τ)∝angbracketright, the au- tocorrelation of the LZ probability [Fig. 3(b)]. Taking a Fourier-transform therefore gives SP(ω), the power spec- trum of PLZ(t) [Figs. 3(c) and 3(d)]. For PLZ(t)≪1, PLZ(t)∝ |∆ST(t)|2, soSP(ω)∝S|∆ST|2(ω), the power spectrum of |∆ST(t)|2. This two-sweep technique allows us to measure the high-frequency components of SP(ω), because the maximum bandwdith is not limited by the quantum dot readout time. Becauseitarisesfromtheprecessingtransversenuclear polarization, ∆ HF(t) contains Fourier components at the Larmor frequencies of the69Ga,71Ga, and75As nuclei in the heterostructure, i.e., ∆ HF(t) =/summationtext3 α=1∆αe2πifαt+θα, whereα= 1,2,or3indexesthethreenuclearspecies, and theθαare the phases of the nuclear fields. Without spin-orbit interaction, |∆ST(t)|2=|/summationtext3 α=1∆αe2πifαt+θα|2 containsonlyFouriercomponentsatthedifferencesofthe nuclear Larmor frequencies. With a spin-orbit contribu- tion, however, |∆ST(t)|2=|∆SO+ ∆HF(t)|2contains cross-terms like ∆ SO∆αe2πifαt+θαthat give |∆ST(t)|2 Fourier components at the absolute Larmor frequencies. A signature of the spin-orbit interaction would therefore be the presence of the absolute Larmor frequencies in SP(ω) forφ∝negationslash= 0◦[26]. Figure 3(b) shows RPP(τ) measured with B= 0.1 T andφ= 0◦. Figure 3(c) shows SP(ω) for 0◦≤φ≤90◦. Atφ= 0◦, only the differences between the Larmor fre- quencies are evident, but as φincreases, the absolute nu- clear Larmor frequencies appear, as expected for a static spin-orbit contribution to ∆ ST(t). These results, includ- ing the peak heights, which reflect isotopic abundances and relative hyperfine couplings, agree well with simula- tions (see Supplementary Information). Having established the importance of spin-orbit cou- pling at the S−T+crossing, we next investigate how the spin-orbit interaction affects DNP. Previous research has shown that repeated LZ sweeps through ǫSTincrease both the averageand differential nuclear longitudinal po- larization in double quantum dots [3]. However, the rea- sons for left/right symmetry breaking, which is needed for differential DNP (dDNP), and the factors limiting DNP efficiency in general are only partially understood. Here, we measure dDNP precisely by measuring δBz, the differentialOverhauserfield, usingrapidHamiltonian learning strategies [27] before and after 100 LZ sweeps to pump the nuclei with rates chosen such that ∝angbracketleftPLZ∝angbracketright= 0.4 (see Supplementary Information) [Fig. 4(a)]. Figure 4(b) plots the change in δBzper electron spin flip forB= 0.2 T and B= 0.8 T for varying φ. In each case, the dDNP decreases with |φ|. Because the spin-orbit interaction allows electron spin flips without corresponding nuclear spin flops, dDNP is suppressed as |∆SO|=|ΩSOsinφsinθ|increases with |φ|. The reduc- tion in dDNP occurs more rapidly at 0.8 T because ∆ SO is larger at 0.8 T than at 0.2 T. We gain further insight into this behavior by plotting the data against σHF/σST, whereσHF≡/radicalbig ∝angbracketleft|∆HF(t)|2∝angbracketright[Fig 4(c)]. Plotted in this way, the two data sets show nearly identical behavior, suggesting that the size of the hyperfine interaction rela- tive to the total splitting primarily determines the DNP efficiency. Based on theoretical results and experimental data, to be presented elsewhere, we expect that the dDNP should be proportional to the total DNP, with a constant of proportionality that depends on B, but not βorφ. We thereforeexplainourmeasurementsofdDNPusingathe- oretcal model in which we have computed the average angular momentum ∝angbracketleftδm∝angbracketrighttransfered to the ensemble of4 (b) (a) (c)-60 -50 -40 -30 -20 -10 0-500 0500 1000 1500 2000 2500 φ (degrees) σHF /σST dDNP (Hz per flip) 0 0.2 0.4 0.6 0.8 100.5 1Normalized dDNP time (0,2) δBz x 120 Probe Pump Probe load load meas. meas. δBz x 120 (1,1) εload meas. sweep x 100 εST B=0.8T B=0.2T B=0.8T B=0.2T FIG. 4. DNP quenching by spin-orbit coupling. (a) Protocol to measure DNP. δBzis measured before and after 100 LZ sweeps by evolving the electrons around δBz. (b) dDNP vs φat fixed /angbracketleftPLZ/angbracketright= 0.4 forB= 0.8 T and B= 0.2 T and theoretical curves (solid lines). dDNP is suppressed for |φ|> 0 because of spin-orbit coupling. (c) Data and theoretical curves for fixed /angbracketleftPLZ/angbracketrightcollapse when normalized and plotted vsσHF/σST. Vertical error bars are statistical uncertainties, and horizontal error bars are fit errors. nuclear spins following a LZ sweep as: ∝angbracketleftδm∝angbracketright ∝σ2 HF/angbracketleftbiggP′ LZ(∆ST) |∆ST|/angbracketrightbigg , (2) whereP′ LZ(∆ST) is the derivative of the LZ probabil- ity with respect to the magnitude of the splitting. (See Supplementary Information for more details.) Neglect- ing charge noise, we have the usual Landau-Zener for- mula [23] PLZ(∆ST) = 1−exp/parenleftbigg −2π|∆ST|2 /planckover2pi1β/parenrightbigg ,(3) and equation (2) reduces to ∝angbracketleftδm∝angbracketright ∝σ2 HF2π /planckover2pi1β∝angbracketleft1−PLZ∝angbracketright. (4)The data in Figs. 4(b) and (c) can therefore be under- stood in light of equation(4) because asthe splitting σST increases with |φ|, the sweep rate βwas also increased to maintain a constant ∝angbracketleftPLZ∝angbracketright. Because the hyperfine con- tribution σHFis independent of φ,∝angbracketleftδm∝angbracketrighttherefore de- creases. The data collapse in Fig. 4(c) can also be under- stood from equation (4), assuming a constant splitting and fixed probability. In this case, β∝ |∆ST|2, as fol- lows from equation (3), and hence ∝angbracketleftδm∝angbracketright ∝σ2 HF/|∆ST|2. Measurements with fixed rate βalso exhibit a similar suppression of dDNP (see Supplementary Information). In this case ∝angbracketleftPLZ∝angbracketrightincreases with |φ|, because of the in- creasing spin-orbit contribution to σST, and according to equation (4), ∝angbracketleftδm∝angbracketrighttherefore decreases. Thetwotheoreticalcurvesin Figs.4(b) and(c) arecal- culated using equation (4) multiplied by fitting constants C, which are different for the two fields, and agree well with the data. As discussed in the Supplementary In- formation, we do not expect charge noise to modify the agreement between theory and data in Figs. 4(b) and (c) beyond the experimental accuracy. Interestingly, the peak dDNP is less at B= 0.2 T than at B= 0.8 T, perhaps because the electron-nuclear coupling becomes increasingly asymmetric with respect to the center of the quantum dots at higher fields [28]. Finally, the peak dDNP value also approximately agrees with a simple cal- culation (see Supplementary Information) based on mea- sured properties of the double dot. In summary, we have used LZ sweeps to measure the S−T+splitting in a GaAs double quantum dot. We find that the spin-orbit coupling dominates the hyperfine in- teractionandquenchesDNPforawiderangeofmagnetic field strengths. A misalignment of BtoΩSOby only 5◦ atB= 1 T can reduce the DNP rate by a factor of two, and DNP is completely suppressed for a misalignment of 15◦. The techniques developed here are directly ap- plicable to other quantum systems such as InAs or InSb nanowires and SiGe quantum wells, where the spin-orbit and hyperfine interactions compete. On a fundamental level, our findings suggest avenues of exploration for im- provedS−T+qubit operation [23] and underscore the importance of the spin-orbit interaction in the study of nuclear dark states [29, 30] and other mechanisms that limit DNP efficiency in central-spin systems. This researchwasfunded by the United States Depart- ment of Defense, the Office of the Director of National Intelligence,IntelligenceAdvancedResearchProjectsAc- tivity, and the Army Research Office grant W911NF-11- 1-0068. S.P.H was supported by the Department of De- fense through the National Defense Science Engineering Graduate Fellowship Program. 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Rashba, Physical Review Letters 109, 236803 (2012).arXiv:1502.05400v1 [cond-mat.mes-hall] 18 Feb 2015Supplementary Information for Quenching of dynamic nuclear polarization by spin-orbit co upling in GaAs quantum dots John M. Nichol,1Shannon P. Harvey,1Michael D. Shulman,1Arijeet Pal,1Vladimir Umansky,2Emmanuel I. Rashba,1Bertrand I. Halperin,1and Amir Yacoby1 1Department of Physics, Harvard University, Cambridge, MA, 0 2138, USA 2Braun Center for Submicron Research, Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 76100 Israel 1. MEASURING σST Here we describe the fitting procedure to extract σST. The experimentally measured quantity is the average triplet occupation probability ∝an}bracketle{tPT∝an}bracketri}ht, which we interpret as the aver- age Landau-Zener (LZ) probability ∝an}bracketle{tPLZ∝an}bracketri}ht, at the end of a sweep. Here ∝an}bracketle{t···∝an}bracketri}htindicates an average over the hyperfine distribution and charge fluctuations f or the same nominal sweep parameters. We calibrate the rate β=d(ES−ET+)/dtusing the spin-funnel technique [1] and assume a linear change in the S−T+splitting near the avoided crossing. ∆HF(t) varies in time because of the nuclear Larmor precession and statis tical fluctua- tions in the magnitude of the nuclear polarizations. We argue that bo th types of hyperfine fluctuations occur on time scales much longer than LZ transitions an d can be treated as quasi-static. In typical experiments, the S−T+splitting is swept through approximately 5 GHz in less than 1 µs. For splittings of order 10 MHz, the total time spent near the avoided crossing is less than 10 ns, which is much faster than the nuc lear Larmor period at 1 T, roughly 100 ns. Furthermore, during 1 µs, the nuclear polarization diffuses by approx- imately 7 kHz [2], which is 3 orders of magnitude smaller than σHF. We therefore assume that the splitting is constant during a single sweep. Numerical simulat ions discussed below also support the hypothesis that nuclear Larmor precession does not significantly affect ∝an}bracketle{tPT∝an}bracketri}ht for the sweep rates used here [Fig. S1]. In the absence of hyperfine or charge fluctuations, the probabilit y for a transition is given2 by the LZ formula: PLZ(t) = 1−exp(−2π|∆ST(t)|2/(/planckover2pi1β)) [3]. Neglecting high-frequency charge noise, the exact form of the LZ probability averaged over t he hyperfine distribution can be computed. Let the total splitting be ∆ ST= ∆HF+ ∆SO. We take ∆ SOto be the constant, real spin-orbit part and ∆ HFthe complex hyperfine contribution. Assuming that the real andimaginary parts of ∆ HF(uandv, respectively) areGaussian-distributed around zero such that the root-mean-square hyperfine splitting is σHF, the probability distribution for the splitting to have magnitude ∆ = |∆ST|is p(∆) =1 πσ2 HF/integraldisplay∞ −∞du/integraldisplay∞ −∞dv e−u2+v2 σ2 HFδ/parenleftBig ∆−/radicalbig (∆SO+u)2+v2/parenrightBig (S1) =2∆ σ2 HFe−∆2+∆2 SO σ2 HFI0(2∆∆SO/σ2 HF), (S2) whereI0isthezeroth-ordermodifiedBessel functionofthefirstkind. Not ethatwhen∆ SO= 0, equation (S2) reduces to the familiar distribution p(∆) =2∆ σ2 HFe−∆2/σ2 HF[4]. Integrating the LZ probability over this distribution yields the average LZ probab ility∝an}bracketle{tPLZ∝an}bracketri}ht: ∝an}bracketle{tPLZ∝an}bracketri}ht=/integraldisplay∞ 0d∆/parenleftbigg 1−exp/parenleftbigg −2π∆2 /planckover2pi1β/parenrightbigg/parenrightbigg p(∆) (S3) = 1−Qexp/parenleftbigg −2π∆2 SO /planckover2pi1βQ/parenrightbigg , (S4) with Q=1 1+2πσ2 HF /planckover2pi1β. (S5) Note that this result agrees with another derivation [5]. Note also th at to leading order in β−1,∝an}bracketle{tPLZ∝an}bracketri}ht ≈2π(∆2 SO+σ2 HF)//planckover2pi1β. The average triplet return probability ∝an}bracketle{tPT∝an}bracketri}htmay be modified due to effects of charge noise on the defining gates or in the two-dimensional electron gas its elf. High-frequency charge noise in double quantum dots has recently been identified as a major source of decoherence [6]. In the current setting, corrections to ∝an}bracketle{tPT∝an}bracketri}htshould occur, because charge fluctuations lead to time-dependent variations in S−T+detuning ES−ET+, on top of the linear time-dependence due to the prescribed sweep rate β. Additionally, charge fluctuations can add noise to the off-diagonal coupling ∆ ST(t) =δB⊥(t)cosθ+ΩSOsinφsinθ, because the singlet mixing angle θ= tan−1/parenleftBig ǫ+√ ǫ2+4t2 2t/parenrightBig depends on ǫ. (Heret= 23.1µeV is the3 0 0.5 100.20.40.60.81B=0.1 T, φ=90° B=0.3 T, φ=90° B=0.5 T, φ=90° B=0.7 T, φ=90° B=0.9 T, φ=90° B=0.1 T, φ=0° B=0.3 T, φ=0° B=0.5 T, φ=0° B=0.7 T, φ=0° B=0.9 T, φ=0°0 0.05 0.100.20.40.60.81 0 0.02 0.0400.20.40.60.81 0 0.01 0.0200.20.40.60.81 0 0.005 0.0100.20.40.60.81 0 0.5 1 1.500.20.40.60.81 0 0.5 1 1.500.20.40.60.81 0 0.5 1 1.500.20.40.60.81 0 1 200.20.40.60.81 0 1 2 300.20.40.60.81(h/β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) (h/β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) (h/ β) (µs/GHz) P LZ P LZ P LZ Data Analytic Simulation with charge noise, hyperfine averaging, Larmor precessionSimulation with hyperfine averaging, Larmor precession FIG. S1. Comparison of LZ data and simulations. Each panel sh ows data and simulations for a different magnetic field strength and orientation. Red curves are experimental data for a series of LZ sweeps with varying rates. Blue curves are simulated data including charge noise, hyperfine averaging, and nuclear Larmor precession for the calculate d value of the splitting corresponding to the red curves. Green curves are simulated data with hyper fine averaging and Larmor pre- cession for the same value of the splitting as the blue curves . Black curves are calculated via equation (S4) using the same value of the splitting. In all pa nels, the y axis is ∝an}bracketle{tPLZ∝an}bracketri}ht, and the x axis ish/β(µs/GHz). Here h= 2π/planckover2pi1is Planck’s constant. double-dot tunnel coupling.) As discussed below, however, the nois e in ∆ STshould have much less effect than the detuning noise for the magnetic fields stud ied here. We observe that for high magnetic fields and slow sweeps, the maximu m LZ probability falls to 0.5 as shown in Fig. S1. It was previously noted that strong de tuning noise can have such an effect [7]. To confirm that charge noise causes the pro bability reduction, we have performed Monte Carlo simulations of the Schr¨ odinger equat ion for symmetric double dots undergoing LZ sweeps, including the effects of wide-band char ge noise, nuclear Larmor4 precession, and averaging over the hyperfine distribution. The re sults of the simulations and experimental data are shown in Fig. S1. We generate random ch arge noise with power spectrum 14 ×10−14V2 Hz/parenleftBig 1Hz f/parenrightBig0.7 forf <1 GHz, and 0 otherwise. We generate the Fourier transform of the charge noise time record by picking the amplitude c orresponding to the chosen power spectrum and a random phase for each frequency fin the desired range. We then perform an inverse Fourier transform to obtain the charge n oise time record. The spectrum we chose corresponds to a noise amplitude of 3 nV/√ Hz atf= 1 MHz, which is approximately the measured level of charge noise in the double dot u sed here. Note that we have extrapolated the f−0.7frequency dependence that was previously measured to f= 1 MHz in ref. [6] up to f= 1 GHz in these simulations. However, one expects the results to be most sensitive to noise in the range of 10-100 MHz, correspon ding to the size of the splitting. The ǫ-dependent Hamiltonian used in these simulations was H(ǫ) = ǫ 2−B δB ⊥(t)cosθ+ΩSOsinφsinθ δB∗ ⊥(t)cosθ+ΩSOsinφsinθ −1 2√ ǫ2+4t2 (S6) in the{|T+∝an}bracketri}ht,|S∝an}bracketri}ht}basis. Linear ǫsweeps through the S−T+crossingǫST=B2−t2 Bwere used in the simulation to replicate the actual experiments. For each stre ngth and orientation of themagneticfield, θwascalculated at ǫSTusing themeasuredtunnel coupling, andthefitted values of the spin-orbit and hyperfine couplings from the main text w ere used to compute the splitting. We assumed a lever arm of 10 to convert the voltage no ise on the quantum dot gates to ǫnoise. The simulated LZ curves with charge noise agree well with the data as shown in Fig. S1. Thesamesimulationsincludingaveragingoverthehyperfinedistribut ionandnuclearLarmor precession, but without chargenoise, show very littlereduction inp robabilitycompared with the analytic result, equation (S4), supporting the hypothesis tha t charge noise is responsible for most of the observed probability reduction. A key feature in th ese experiments is the decreasing maximum probability with increasing magnetic field. We can u nderstand that this trend occurs because the effect of charge noise on the Landa u Zener probability is controlled by the fluctuation in the energy splitting δE(ǫ) produced by a given fluctuation in the detuning ǫ, which is proportional todE(ǫ) dǫ|ǫ=ǫST. SinceE(ǫ) =ǫ 2−B+1 2√ ǫ2+4t2, the magnitude ofdE(ǫ) dǫ|ǫ=ǫSTincreases sharply with increasing magnetic field.5 φ=0° φ=90° 0.2 0.4 0.6 0.8 1-25-20-15-10-5 05 Field (T) Systematic Error (%) FIG.S2. Fittingerror. Wecomputethefittingerrorbysimula ting∝an}bracketle{tPLZ∝an}bracketri}htforthecalculated splitting at each of the magnetic field configurations in the presence of charge noise. The simulated ∝an}bracketle{tPLZ∝an}bracketri}htvs β−1is fitted to a straight line for 0 <∝an}bracketle{tPLZ∝an}bracketri}ht<0.1, and the fitted value of the splitting is subtracted from the value chosen for the simulation. The difference is the n divided by the simulated value of the splitting. Error bars are fit errors. Even in the presence of noise, however, the average LZ probability in the limit of fast sweeps is still 2 π|∆ST(t)|2//planckover2pi1β, which is identical to the leading order behavior of the usual LZ formula, as shown in section 3.1 of ref. [7]. Replacing the LZ formula in equation (S4) by its leading order behavior, and performing the integration over the quasi-static distribution gives∝an}bracketle{tPLZ∝an}bracketri}ht ≈2π(∆2 SO+σ2 HF)//planckover2pi1β. Such a result can be understood because the effect of detuning noise is reduced on short time scales. Figure S1 demonstra tes this idea because the analytic curves deviate significantly from the data for ∝an}bracketle{tPLZ∝an}bracketri}ht/greaterorsimilar0.2, but for 0 <∝an}bracketle{tPLZ∝an}bracketri}ht<0.1, the analytic results agrees well with the data. Based on additional s imulations, we estimate the systematic error in the deduced value of σSTas obtained by fitting measured values of ∝an}bracketle{tPLZ∝an}bracketri}htto a straight line for values of βsuch that 0 <∝an}bracketle{tPLZ∝an}bracketri}ht<0.1 to be small for most of the experimental conditions as shown in Fig. S2. Wenotethatthecoupling∆ ST(t) =δB⊥(t)cosθ+ΩSOsinφsinθdependson ǫthroughthe singlet mixing angle θ. This dependence means that during a LZ sweep, the coupling ∆ ST(t) variesbothdueto thelinear ǫsweep aswell aschargenoise. Weestimate thatdE(ǫ) dǫ≥40dσST dǫ for the fields studied here. We therefore expect detuning fluctua tions to be the dominant noise source. Furthermore, when |E(ǫ)|< σST,σSTchanges by only a few percent during the sweep and is likely not a significant source of error in the measure ment of ∆ ST(t). Additionally, we note that the simulations in Fig. S1, which include ǫ-depending coupling,6 demonstrate that the fitting procedure described above allows an accurate measurement ofσST. Finally, we have also performed additional simulations, taking into ac count the measured values of E(ǫ), which deviate slightly from the values predicted by assuming a constant tunnel coupling, and we observe no significant change in o ur results. 2. DIRECTION OF Ω SO The double quantum dot axis is aligned within ≈5◦of either the [ ¯110] or [110] axes of the crystal, but we do not know which. In the later case, both th e Rashba and Dres- selhaus spin-orbit fields are aligned with the z axis, and their magnitud es add [8]. In the former case, the Rashba and Dresselhaus contributions are also a ligned with the z axis, but their magnitudes subtract. The techniques used here could be emp loyed to distinguish the Rashba and Dresselhaus spin-orbit contributions by measuring σSTwith double quantum dots fabricated on different directions with respect to the crysta l axes. 3. FITTING σSTVS B AND φ We fit the data in Fig. 2(c) in the main text to a function of the form σST= /radicalbig ∆2 SOsin2θsin2φ+σ2 HFcos2θ, with ∆ SOandσHFas fit parameters. The singlet mixing angleθis computed by assuming that the (1 ,1) and (0 ,2) singlet branches are a two-level system with constant tunnel coupling, as discussed above. ∆SOis held at 0 when fitting data for φ= 0◦to determine the hyperfine coupling. We also exclude data points for B <0.2 T in the fit, as the hyperfine contribution appears to decrease at very low fields. We determine the spin orbit length using e quation (28) of Ref. [8], where the spin-orbit field is computed as Ω SO=4t 3λDQD λSO, whereλDQD≈200 nm is the interdot spacing, and λSOis the spin-orbit length. The simulation in Fig. 2(b) in the main text is generated using the same equation with the fitted values of t he ∆SOandσHF. 4. MEASURING RPP(τ) Here we derive the triplet return probability after two consecutive LZ sweeps with a pause of length τin between. In experiments, both sweeps were in the same direction , and7 ǫwas held in the (0 ,2) region between sweeps, as shown in Fig. 3(a) in the main text. Suppose the first LZ sweep takes place at time twith probability PLZ(t). The probability for the two electrons to be in the T+state isPLZ(t), while the probability to be in the Sstate is 1 −PLZ(t). Then, the detuning is quickly swept into the (0 ,2) region. Here, electron spin dephasing occurs rapidly, and there is very little T+occupation in thermal equilibrium because the SandT+states are widely separated in energy. Thus, after a wait of length τ, but before the second sweep, the triplet population is PLZ(t)e−τ/T1, and the singlet population is 1 −PLZ(t)e−τ/T1, whereT1is the electron relaxation time. After the second sweep, the triplet occupation probability is PT(t+τ) =/parenleftbig 1−PLZ(t)e−τ/T1/parenrightbig PLZ(t+τ)+PLZ(t)e−τ/T1(1−PLZ(t+τ)) (S7) =−2PLZ(t)PLZ(t+τ)e−τ/T1+PLZ(t+τ)+PLZ(t)e−τ/T1. (S8) The second and third terms in equation (S8) vary slowly with τ. These terms are found by fitting the measured triplet probability to an exponential with an offs et and are subtracted. WhenT1≫τ, relaxationcanbeneglected, andthepredicted time-averaged sig nal is∝an}bracketle{tPT(t+ τ)∝an}bracketri}ht ∝RPP(τ), whereRPP(τ)≡ ∝an}bracketle{tPLZ(t)PLZ(t+τ)∝an}bracketri}ht, theautocorrelationoftheLZprobability. Whenφ= 0◦,T1≫τmax= 200µs, where τmaxis the largest value of τmeasured. The shortest relaxation time T1≈100µs in these experiments time occurs when φ= 90◦, which is consistent with spin-orbit-induced relaxation [9]. The effect of T1relaxation is to multiply the measured correlation by an exponentially- decaying window, which reduces the spectral resolution of the Fou rier transform, but does not shift the frequency of the observed peaks. We expect statis tical fluctuations in the amplitude of the hyperfine field to affect the spectrum in a similar way, although we expect this effect to be less than that of electron relaxation. The raw data , [Fig. 3(b) in the main text] consisting of 667 points (each a result of two sweeps with a 40 % chance of a LZ transition) spaced by 300 ns, were zero-padded to a size of 169 1 points to smooth the spectrum, and a Gaussian window with time constant 150 µs was applied to reduce the effects of noise and ringing from zero-padding before Fourier tran sforming. The magnetic resonance frequencies in Fig. 3(c) appear to decrea se withφ. The inhomo- geneity of the x-coil in our vector magnet is 1.6 % at 0.6 cm offset from the center. Thus, the field could easily be reduced by more than 3 % for a misplacement of the sample by 1 cm from the magnet center. We have simulated the data in Fig. 3(c) in the main text based8 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6020 40 60 80 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6020 40 60 80 φ (degrees) φ (degrees) ω/2 π (MHz) Experiment Simulation ω/2 π (MHz) (a) (b) FIG. S3. Simulations of SP(ω). (a) Experimental data. (b) Theoretical simulation takin g into account known sweep rates, nuclear magnetic resonance freq uencies, hyperfine couplings, and a 4.4% reduction in field in the x direction. The expected frequ encies at B= 0.1 T are f69Ga= 1.0248 MHz, f71Ga= 1.302 MHz, and f75As= 0.7315 MHz. on the measured hyperfine and spin-orbit couplings and the known s weep rates. Assuming a 4.4 % reduction in the field from the x-coil, we obtain good agreement b etween theory and experiment [Figs. S3(a) and (b)]. Wearguedinthe maintext that onlythe difference frequencies shou ld appear inthe spec- trumSP(ω)withoutspin-orbitcoupling byconsidering thetime-dependence of |∆ST(t)|2and becauseSP(ω)∝S|∆ST|2(ω) whenPLZ(t)≪1. Since PLZ(t) contains only even powers of |∆ST(t)|,SP(ω) can generally be expressed in terms of differences of the resonan ce frequen- cies, but will not contain the absolute frequencies in the absence of spin-orbit coupling, regardless of the value of PLZ(t).9 5. DERIVATION OF NUCLEAR POLARIZATION CHANGE ∝an}bracketle{tδm∝an}bracketri}ht Here we derive equations 2 and 4 in the main text. Let ∆ ST= ∆SO+∆HFwhere ∆ SO is real and ∆HF=/summationdisplay jλjI+ j, (S9) whereI+ jis the raising operator for the jthnuclear spin, and the λjare individual coupling constants. We assume that there are many nuclear spins, so that each coupling constant is small. Also, σ2 HF≡ ∝an}bracketle{t|∆2 HF|∝an}bracketri}ht=2 3I(I+1)/summationdisplay jλ2 j=5 2/summationdisplay jλ2 j, (S10) whereI=3 2is the spin of the nuclei, and the angular brackets refer to an avera ge over the distribution of nuclear spins. We pick one of the nuclear spins, j, and we wish to compute ∝an}bracketle{tδmj∝an}bracketri}ht, the mean value of the change in Iz jafter one sweep. Let PLZ(∆ST) be the probability of an S−T+transition for a fixed value of ∆ HF. Clearly, PLZdepends on |∆ST|. We calculate δmjas follows. Write ∆ST=a+beiθj, (S11) whereaincludes the contributions of spin orbit and of all nuclei other than t he nucleus j, and the second term represents the contribution (of order λj) from nucleus j. According to equation (31) of Ref. [4], the value of δmjfor this configuration should be given by δmj=1 2π/contintegraldisplay dθjPLZ(∆ST)dϕ dθj, (S12) whereϕ= arctan(Im(∆ ST)/Re(∆ST)) specifies the orientation of ∆ STin the complex plane. Without loss of generality, we may suppose that ais real. Then we have, ignoring terms that are higher order in b/a, dϕ dθj=b acosθj (S13) PLZ(∆ST) =PLZ(a)+bP′ LZ(a)cosθj (S14) δmj=b2 2aP′ LZ(a), (S15) whereP′ LZ(a) is the derivative of PLZ(a) with respect to a. Averaging over nuclear configu- rations, we obtain ∝an}bracketle{tδmj∝an}bracketri}ht=∝an}bracketle{tb2∝an}bracketri}ht/angbracketleftbiggP′ LZ(a) 2a/angbracketrightbigg , (S16)10 with∝an}bracketle{tb2∝an}bracketri}ht= (5/2)λ2 j. In the case of no charge noise, we have PLZ(∆ST) = 1−exp/parenleftbigg −2π|∆ST|2 /planckover2pi1β/parenrightbigg , (S17) so P′ LZ(a) 2a=2π /planckover2pi1β(1−PLZ(a)) (S18) and ∝an}bracketle{tδmj∝an}bracketri}ht=2π /planckover2pi1β∝an}bracketle{tb2∝an}bracketri}ht∝an}bracketle{t1−PLZ(a)∝an}bracketri}ht. (S19) Finally, we sum over all nuclear spins and make the replacement a≈ |∆ST|, obtaining ∝an}bracketle{tδm∝an}bracketri}ht=2π /planckover2pi1βσ2 HF∝an}bracketle{t1−PLZ(∆ST)∝an}bracketri}ht. (S20) The collapse demonstrated in Fig. 4(c) in the main text can be unders tood from equa- tion (S20), assuming constant ∆ STand fixed probability. In this case, β∝ |∆ST|2from equation (S17), and hence ∝an}bracketle{tδm∝an}bracketri}ht ∝σ2 HF/|∆ST|2. In the case of a fixed splitting, equation (S20) reduces to ∝an}bracketle{tδm∝an}bracketri}ht=2π /planckover2pi1βσ2 HFexp/parenleftbigg −2π|∆ST|2 /planckover2pi1β/parenrightbigg . (S21) In equation (S21), ∝an}bracketle{tδm∝an}bracketri}ht →0 for both β→0 andβ→ ∞. In practice however, experiments necessarily average over the hyperfine distribution. Thus, using e quation (S4) with ∆ SO= 0 to compute ∝an}bracketle{t1−PLZ(∆ST)∝an}bracketri}ht, we have ∝an}bracketle{tδm∝an}bracketri}ht=2π /planckover2pi1βσ2 HFQ (S22) =2πσ2 HF /planckover2pi1β 1+2πσ2 HF /planckover2pi1β. (S23) According to equation (S23), in the limit of slow sleeps, where β→0,∝an}bracketle{tδm∝an}bracketri}ht →1, and in the limit of fast sweeps, where β→ ∞,∝an}bracketle{tδm∝an}bracketri}ht →0, as expected. The theory curves in Figs. 4(c) and (d) in the main text were genera ted by computing equation (S20). For each field angle φ, the parameters θ, ∆SO, andσHFwere calculated using the fitted values of the spin-orbit and hyperfine couplings as w ell as the measured tunnel coupling. Equation (S4) was then solved using the calculated parameters to find the rate βsuch that ∝an}bracketle{tPLZ∝an}bracketri}ht= 0.4. In order to compare with data on the dDNP rate, the11 φ (degrees) φ (degrees) dDNP rate (Hz per flip) B=0.8T B=0.2T B=0.8T B=0.2T (a) (b) -60 -50 -40 -30 -20 -10 0-50005001000150020002500 -60 -50 -40 -30 -20 -10 00.30.40.50.60.70.80.9PLZ FIG. S4. DNP quenching with fixed sweep rate. (a) dDNP vs φatB= 0.2 T and B= 0.8 T. For each field, the sweep rate βwas chosen to give ∝an}bracketle{tPLZ∝an}bracketri}ht= 0.4 atφ= 0◦and then was held constant forφ∝ne}ationslash= 0◦. (b) As |φ|increases, σSTincreases. As a result, ∝an}bracketle{tPLZ∝an}bracketri}htalso increases and DNP is suppressed, according to equation (S20). Error bars are sta tistical uncertainties. Lines between points serve as a guide to the eye. theoretical curves for ∝an}bracketle{tδm∝an}bracketri}htwere multiplied by fitting constants C, which are different for the two curves. As explained in the main text, and further discussed be low, we expect the ratio between the dDNP rate and ∝an}bracketle{tδm∝an}bracketri}htto depend on the magnetic field but to be independent of the sweep rate. Data taken at fixed sweep rate βalso show a suppression of DNP, as shown in Fig. S4(a). In this case, ∝an}bracketle{tPLZ∝an}bracketri}htincreases with |φ|because of spin-orbit coupling [Fig. S4(b)], and ∝an}bracketle{tPLZ∝an}bracketri}ht therefore increases, causing ∝an}bracketle{tδm∝an}bracketri}htto decrease, according to equation (S20). ToaddresstheeffectofchargenoiseondDNP,werecomputeequa tion(S20)inthelimitof strongnoiseusingtheresultsofRef.[7],makingthereplacement P(a) =1 2/parenleftBig 1−exp/parenleftBig −4πa2 /planckover2pi1β/parenrightBig/parenrightBig forPLZ(a) both in the derivation leading to equation (S20) and in equation (S4) for the computation of β. The expected dDNP in the presence of strong noise is shown in Fig. S 5, and it does not significantly deviate from the case without noise, at le ast at the level of the12 B=0.8T B=0.2T No charge noise Charge noise -60 -50 -40 -30 -20 -10 0-50005001000150020002500 φ (degrees) dDNP rate (Hz per flip) FIG. S5. The effect of charge noise on dDNP. The data and solid li nes are the same as in Fig. 4 in the main text, and the dashed lines are the theoretical est imates for dDNP in the presence of charge noise. The dashed and solid lines are normalized to th e same values at φ= 0◦. Error bars are statistical uncertainties. experimental accuracy. 6. MEASURING δBz Wemeasure δBzby first initializing the double dot inthe |(0,2)S∝an}bracketri}htstateand thenseparat- ing the electrons by rapidly changing ǫto a large negative value [10]. When the electrons are separated, the exchange energy isnegligible, andthe magneticfield gradient δBzdrives oscil- lations between |S∝an}bracketri}htand|T0∝an}bracketri}ht. In our experiments, we measure the two-electron spin state for 120 linearly increasing values of the separation time. The resulting sin gle-shot measurement record is thresholded, zero padded, and Fourier transformed. T he frequency corresponding to the peak in the resulting Fourier transform is chosen as the value ofδBz. This technique is related to a previously described rapid Hamiltonian estimation techn ique [2]. 7. EXPECTED DNP RATE In this section we give a simple calculation to explain the value of the pea k (φ= 0◦) dDNP rate, as shown in Fig. 4 of the main text. Additional measureme nts were carried out to measure the pumping rate of the sum hyperfine field, ( Br+Bl)/2, where BrandBl denote the longitudinal hyperfine fields in the right and left dots. Th is rate was determined by measuring the location of ǫSTbefore and after a series of LZ sweeps to polarize the nuclei13 atB= 0.2 T. We observe that the sum field is pumped roughly twice as efficiently as the difference field, δBz=Br−Bl. Setting ( ˙Br+˙Bl)/2 = 2(˙Br−˙Bl), where ˙Bl(r)indicates the pumping rate of the left(right) dot, we have ˙Bl= (3/5)˙Br, meaning that the left dot is pumped 3 /5 as often as the right dot. Under these conditions, the average g radient builds up at a rate (per electron spin flip) of ( ˙Br−˙Bl)/(˙Br+˙Bl) that is only 1/4 the rate that would occur if nuclear spin flips occurred in only one dot. To determine the expected change in δBz, we require the approximate number of spins overlapped by the electronic wave function in the double dot. We hav e measured the inho- mogeneous dephasing time of electronic oscillations around δBzand find T∗ 2= 18 ns [10]. This dephasing time corresponds to a rms value of the gradient σδBz≡/radicalbig ∝an}bracketle{t|δBz|2∝an}bracketri}ht= h//parenleftbig |g∗|µB√ 2πT∗ 2/parenrightbig = 2 mT, where his Planck’s constant. The total number of spins over- lapped by thewavefunction is N= (h1/σδBz)2≈3×106, whereh1= 4.0 T[11]. If all nuclear spins were fullypolarized, then thedots would experience a hyperfin e fieldof h0= 5.3T [11], and if the nuclear spins in the two dots were fully polarized in opposite d irections, the gra- dient would be 2 h0. Therefore, the expected change in the gradient per electron sp in flip, corresponding to a change in nuclear angular momentum of /planckover2pi1, is2π /planckover2pi1×2|g∗|µBh0 2I(N/2)= 12 kHz, whereI= 3/2 is the nuclear spin. The average dDNP under actual conditions is 1/ 4 of this value, or 3 kHz, in reasonable agreement with our observations. In addition, we note the reasonable agreement between the measured valueof σδBz= 2 mT andtheroot-mean-square hyperfine gap/radicalbig ∝an}bracketle{t|δB⊥(t)|2∝an}bracketri}ht ≈34 neV/ ( |g∗|µB) = 1.5 mT. [1] J. R. Petta, H. Lu, and A. C. Gossard, Science 327, 669 (2010). [2] M. D. Shulman, S. P. Harvey, J. M. Nichol, S. D. Bartlett, A . C. Doherty, V. Umansky, and A. Yacoby, Nature Communications 5, 5156 (2014). [3] S. Shevchenko, S. Ashhab, and F. Nori, Physics Reports 492, 1 (2010). [4] I. Neder, M. S. Rudner, and B. I. Halperin, Physical Revie w B89, 085403 (2014). [5] C. Dickel, S. Foletti, V. Umansky, and H. Bluhm, (2014), a rXiv:1412.4551. [6] O. E. Dial, M. D. Shulman, S. P. Harvey, H. Bluhm, V. Umansk y, and A. Yacoby, Physical Review Letters 110, 146804 (2013). [7] Y. Kayanuma, Journal of the Physical Society of Japan 53, 108 (1984).14 [8] D. Stepanenko, M. S. Rudner, B. I. Halperin, and D. Loss, Physical Review B 85, 075416 (2012). [9] P. Scarlino, E. Kawakami, P. Stano, M. Shafiei, C. Reichl, W. Wegscheider, and L. M. K. Vandersypen, (2014), arXiv:1409.1016. [10] J. R. Petta, A. C. Johnson, J. M. Taylor, E. Laird, A. Yaco by, M. D. Lukin, C. M. Marcus, M. P. Hanson, and A. C. Gossard, Science 309, 2180 (2005). [11] J. M. Taylor, J. R. Petta, A. C. Johnson, A. Yacoby, C. M. M arcus, and M. D. Lukin, Physical Review B 76, 035315 (2007).

1704.00532v1.Inverse_spin_galvanic_effect_in_the_presence_of_impurity_spin_orbit_scattering__a_diagrammatic_approach.pdf

Article Inverse spin galvanic effect in the presence of impurity spin-orbit scattering: a diagrammatic approach Amin Maleki and Roberto Raimondi? Dipartimento di Matematica e Fisica, Università Roma Tre, Via della Vasca Navale 84, 00146 Roma, Italy. *Correspondence: email: roberto.raimondi@uniroma3.it; Tel. +39 06 5733 7032 Version March 4, 2022 submitted to Condens. Matter Abstract: Spin-charge interconversion is currently the focus of intensive experimental and theoretical research both for its intrinsic interest and for its potential exploitation in the realization of new spintronic functionalities. Spin-orbit coupling is one of the key microscopic mechanisms to couple charge currents and spin polarizations. The Rashba spin-orbit coupling in a two-dimensional electron gas has been shown to give rise to the inverse spin galvanic effect, i.e. the generation of a non-equilibrium spin polarization by a charge current. Whereas the Rashba model may be applied to the interpretation of experimental results in many cases, in general in a given real physical system spin-orbit coupling also occurs due other mechanisms such as Dresselhaus bulk inversion asymmetry and scattering from impurities. In this work we consider the inverse spin galvanic effect in the presence of Rashba, Dresselhaus and impurity spin-orbit scattering. We find that the size and form of the inverse spin galvanic effect is greatly modified by the presence of the various sources of spin-orbit coupling. Indeed, spin-orbit coupling affects the spin relaxation time by adding the Elliott-Yafet mechanism to the Dyakonov-Perel and, furthermore, it changes the non-equilibrium value of the current-induced spin polarization by introducing a new spin generation torque. We use a diagrammatic Kubo formula approach to evaluate the spin polarization-charge current response function. We finally comment about the relevance of our results for the interpretation of experimental results. Keywords: Spin-orbit coupling; Spin transport; 2DEG 1. Introduction The spin galvanic effect and its inverse manifestation have been intensively investigated over the past decade both for their intrinsic fundamental interest [1] and for their application potential in future generation electronic and spintronics technology [2,3]. The non-equilibrium generation of a spin polarization perpendicular to an externally applied electric field is referred to as the inverse spin galvanic effect (ISGE), whereas the spin galvanic effect (SGE) is its Onsager reciprocal, whereby a spin polarization injected through a nonmagnetic material creates a charge current in the direction perpendicular to the spin polarization. As an all-electrical method of generating and detecting spin polarization in nonmagnetic materials, both these effects may be used for applications such as spin-based field effect transistors [4–7] and magnetic random access memory (MRAM) [8,9]. The ISGE, also known as Edelstein effect or current-induced spin polarization, was originally proposed by Ivchenko and Pikus [10], and observed by Vorob’ev et al. in tellurium [11]. Later the ISGE was theoretically analyzed by Edelstein in a two-dimensional electron gas (2DEG) with Rashba spin-orbit coupling (SOC) [12] and also by Lyanda-Geller and Aronov[13]. Notice that the SGE in the spin-charge conversion is sometimes referred to as the inverse Rashba-Edelstein effect. The SGE Submitted to Condens. Matter , pages 1 – 14 www.mdpi.com/journal/condensedmatterarXiv:1704.00532v1 [cond-mat.mes-hall] 3 Apr 2017Version March 4, 2022 submitted to Condens. Matter 2 of 14 has been observed experimentally in GaAs QWs by Ganichev et al. [14], where the spin polarization was detected by measuring the current produced by circularly polarized light. In semiconducting structures the ISGE can be measured by optical methods such as Faraday rotation, linear-circular dichroism in transmission of terahertz radiation and time resolved Kerr rotation [1,15–17]. Very recently, a new way of converting spin to charge current has been experimentally developed by Rojas-Sánchez et al., where, by the spin-pumping technique, the non-equilibrium spin polarization injected from a ferromagnet into a silver (Ag)/Bismuth (Bi) interface yields an electrical current [18]. Successively, the SGE has also been observed in many interfaces with strong spin-orbit splitting, including metals with semiconductor giant SOC or insulator such as Fe/GaAs [19], Cu/Bi2O3[20]. Generally speaking, the SGE can be understood phenomenologically by symmetry arguments. Electrical currents and spin polarizations are polar and axial vectors, respectively. In centro-symmetric systems, polar and axial vectors transform differently and no SGE effect is expected. In restricted symmetry conditions, however, polar and axial vectors components may transform similarly. Consider, for instance, the case of electrons confined in the xy plane with the mirror reflection through the yz plane. Under such a symmetry operation, the electrical currents along the the x and y directions transform as Jx! Jxand Jy!Jy. The spin polarizations transform as the components of angular momentum, and we have Sy! Syand Sx!Sx. Hence, one expects a coupling between JxandSyor between JyandSx. Such a coupling is the SGE. At microscopic level the strength of the coupling is due to the SOC. Usually the SOC is classified as extrinsic and intrinsic, depending on the origin of the electrical potential. The intrinsic SOC arises due to the crystalline potential of the host material or due the confinement potential associated with the device structure. On the other hand, the extrinsic SOC is due to the atomic potential of random impurities, which determine the transport properties of a given material. The majority of the studies on SGE/ISGE has focused on the Rashba SOC (RSOC) for electrons moving in the xy plane, which was originally introduced by Rashba [21] to study the properties of the energy spectrum of non-centrosymmetric crystals of the CdS type and later successfully applied to the interpretation the two-fold spin splitting of electrons and holes in asymmetric semiconducting heterostructures [22]. RSOC is classified as due to structure inversion asymmetry (SIA), which is responsible for the confinement of electrons in the xy plane. In addition one may also consider the SOC arising from the bulk inversion asymmetry (BIA), usually referred to as Dresselhaus SOC (DSOC) [23]. Both RSOC and DSOC modify the energy spectrum by introducing a momentum-dependent spin splitting. This also can be understood quite generally on the basis of symmetry considerations. In a solid spin degneracy for a couple of states with opposite spin and with cristalline wave vector kis the result of both time reversal invariance and parity (space inversion invariance). By breaking the parity, as for instance, in a confined two-dimensional electron gas, the spin degeneracy is lifted and the Hamiltonian acquires an effective momentum-dependent magnetic field, which is the SOC. As a result electron states can be classified with their chirality in the sense that their spin state depends on their wave vector. In a such a situation, scalar disorder, although not directly acting on the spin state, influences the spin dynamics by affecting the wave vector of the electrons and holes. Spin relaxation arising in this context is usually referred to as the Dyakonov-Perel (DP) mechanism. Extrinsic SOC originates from the potential which is responsible for the scattering from an impurity. In this case, before and after the scattering event, there is no direct connection between the wave vector and the spin of the electron. The scattering amplitude can be divided in spin-independent and spin-dependent contributions Sp,p0=A+ˆpˆp0
sB, (1) where ˆpand ˆp0are the unit vector along the direction of the momentum before and after the scattering and sis the vector of the Pauli matrices. As explained in Ref.[24], different combinations of the amplitudes Aand Bcorrespond to specific physical processes. The jAj2+jBj2describes the total scattering rate, whereas jBj2is associated to the Elliott-Yafet (EY) spin relaxation rate. Interference terms between the two amplitudes yield coupling among the currents. More in detail, theVersion March 4, 2022 submitted to Condens. Matter 3 of 14 combination AB+ABdescribes the skew scattering, which is responsible for the coupling between the charge and spin currents, whereas ABABgives rise to the swapping of spin currents. As noted in Ref.[1], when both intrinsic and extrinsic SOC is present, the non-equilibrium spin polarization of the ISGE depends on the ratio of the DP and EY spin relaxation rates. This was analyzed in Ref.[25] by means of the Keldysh non-equilibrium Green function within a SU(2) gauge theory-description of the SOC. Successively, a parallel analysis by standard Feynman diagrams for the Kubo formula was carried out in Ref.[26]. These theoretical studies indeed confirmed that the ratio of DP to EY spin relaxation is able to tune the value of the ISGE. Such tuning is also affected by the value of the spin Hall angle due to the fact that spin polarization and spin current are coupled in the presence of intrinsic RSOC. Recently, it has been shown theoretically [27] that the interplay of intrinsic and extrinsic SOC gives rise to an additional spin torque in the Bloch equations for the spin dynamics and affects the value of the ISGE. This additional spin torque, which is proportional to both the EY spin relaxation rate and to the coupling constant of RSOC, in Ref.[27] has been derived in the context of the SU(2) gauge theory formulation mentioned above. Although the SU(2) gauge theory is a very powerful approach, in order to emphasize the physical origin of this new torque it is very useful to show also how the same result can be obtained independently by using the diagrammatic approach of the Kubo linear response theory. This is the aim of the present paper. In this paper we obtain an analytical formula of the ISGE in the presence of the Rashba, Dresselhaus and impurity SOC. In a 2DEG we will show that the intrinsic and extrinsic SOC act in parallel as far as relaxation to the equilibrium state is concerned. The model Hamiltonian for a 2DEG in the presence of SOC reads H=p2 2m+a(pysxpxsy) +b(pxsxpysy) +V(r)l2 0 4rV(r)p
s, (2) where p= (px,py)is the vector of the components of the momentum operator, s= (sx,sy,sz) andrare the Pauli matrices and the coordinate operators. mis the effective mass, aand bare the Rashba and Dresselhaus SOC constants. V(r)represents a short-range impurity potential and finally l0is the effective Compton wave length describing the strength of the extrinsic SOC. We assume the standard model of white-noise disorder potential with hV(r)i=0 and Gaussian distribution given byhV(r)V(r0)i=niv2 0d(rr0) = ( ¯h/(2pN0t0))d(rr0).N0=m/2¯h2p,niand v0are the single-particle density of states per spin in the absence of SOC, the impurity concentration and the scattering amplitude, respectively. t0is the elastic scattering time at the level of the Fermi Golden Rule. From now on we work with units such that ¯ h=1. The layout of the paper is as follows. In the next Section we formulate the ISGE (the SGE can be obtained similarly by using the Onsager relations) in terms of the Kubo linear response theory. In Section 3 we derive an expression for the ISGE in the presence of the RSOC and extrinsic SOC. This case with no DSOC, whereas it is important by itself, allows to understand the origin of the additional spin torque in a situation which technically simpler to treat with respect to the general case when both RSOC and DSOC are different from zero. In Section 4, we expand our result to the specific case when the both RSOC and DSOC, as well as SOC from impurities, are present. We show how our result can be seen as the stationary solution of the Bloch equations for the spin dynamics. We comment briefly on the relevance of our result for the interpretation of the experiments. Finally, we state our conclusions in Section 5. 2. Linear response theory In this Section we use the standard Kubo formula of linear response theory to derive the ISGE in the presence of extrinsic and intrinsic SOC. The in-plane spin polarization to linear order in the electric fields is given by Si=sij ECEj,i,j=x,y, (3)Version March 4, 2022 submitted to Condens. Matter 4 of 14 where Eiis the external electric fields with frequency wandsij ECis the frequency-dependent "Edelstein conductivity"[28] given by the Kubo formula [29] sij EC(w) =(e) 2på pTr[GA(e+w)Ui(e,w))GR(e)Jj], (4) where the trace symbol includes the summation over spin indices. We keep the frequency dependence ofsij EC(w)in order to obtain the Bloch equations for the spin dynamics. In Eq.(4), Ui(e,w)is the renormalized spin vertex relative to a polarization along the iaxis, required by the standard series of ladder diagrams of the impurity technique [30,31]. Jjare the bare number current vertices. In the plane-wave basis their matrix element from state p0to state pread Jx=dp,p0px masy+bsx +djx,pp0, (5) Jy=dp,p0py m+asxbsy +djy,pp0. (6) The latter term dJj,pp0in Eqs.(5-6), which depends explicitly on disorder, is of order l2 0and originates from the last term in the Hamiltonian of Eq.(2). Such a term gives rise to the side-jump contribution to the spin Hall effect [32,33] due to the extrinsic SOC. The side-jump and skew-scattering contributions to the spin Hall effect in the presence of RSOC have been considered in Ref.[25,34,35]. A similar analysis of the side-jump and skew-scattering contributions to the ISGE has been carried out within the SU(2) gauge theory formualtion in Ref.[25] and, more recently, in Ref.[26] by standard Kubo formula diagrammatic methods. For this reason we will not repeat such an analysis here, where instead we concentrate on the contributions generated by the first term on the right hand side of Eq.(5-6). Within the self-consistent Born approximation, the last two terms of the Hamiltonian (2) yield an effective the self-energy when averaging over disorder. The self-energy is diagonal in momentum space and has two contributions due to the spin independent and spin dependent scattering [28,36] SR tot(p)SR 0(p) +SR EY(p) =niv2 0å p0GR p0+niv2 0l4 0 16å p0szGR p0sz(pp0)2 z. (7) Whereas the imaginary part of the first term gives rise to the standard elastic scattering time ImSR 0(p) =i2pN0niv2 0=i 2t0, (8) The second one is responsible for the EY spin relaxation. From the point of view of the scattering matrix introduced in the previous Section (cf. Eq.(1)), the two self-energies contributions correspond to the Born approximation for the jAj2andjBj2, respectively. Given the self-energy (7), the retarded Green function is also diagonal in momentum space and can be expanded in the Pauli matrix basis in the form GR p=GR 0s0+GR xsx+GR ysy, (9)Version March 4, 2022 submitted to Condens. Matter 5 of 14 where GR 0=GR++GR 2 GR x= ( aˆpy+bˆpx)GR+GR 2g GR y=(aˆpx+bˆpy)GR+GR 2g. (10) In the above GR(e) = ( ep2 2mgp+i 2t)1is the Green function corresponding to the two branches in which the energy spectrum splits due to the SOC. The factor g2=a2+b2+2absin(2f)with ˆpx=cos(f)and ˆpy=sin(f)describes the dependence in momentum space of the SOC, when both RSOC and DSOC are present. Notice that inversion in the two-dimensional momentum space ((px,py)!(px,py)) leaves the factor ginvariant, since it corresponds to f!f+p. As a consequence, Gx,y! Gx,y, whereas G0is invariant. This observation will turn out to be useful later when evaluating the renormalization of the spin vertices. The advanced Green function is easily obtained via the relation GA= (GR). In the expression for GR,1 2tis a band-dependent time relaxation and plays an important role in our analysis. In order to obtain this term we note that, after momentum integration over p0in Eq.(7), the imaginary part of the retarded self-energy reads SR =i1 2t0i l2 0 4!21 4t0p2 Fp2 i 2t(11) Above, we indicate with pFthe Fermi momentum without RSOC and DSOC and with pthe g-dependent momenta of the two spin-orbit split Fermi surfaces. To lowest order in the spin-orbit splitting we have p=pF(1g vF), (12) where vF=pF/m. The momentum factors originate from the square of the vector product in the second term of Eq.(7). The factor p2 Fis due to the inner p0momentum, which upon integration is eventually fixed at the Fermi surface in the absence of RSOC and DSOC. More precisely, when evaluating the momentum integral, one ends up by summing the contributions of the two spin-orbit split bands in such a way that the a- and b-dependent shift of the two Fermi surfaces cancels in the sum. However, the outer pmomentum remains unfixed. Its value will be fixed by the poles of the Green function in a successive integration over the momentum. Then, the g-dependent relaxation times of the two Fermi surfaces read 1 t=1 t(1t tEYg vF), (13) where 1 t=1 t0+1 2tEY, (14) with the standard expression for the EY spin relaxation rates 1 tEY=1 t0l0pF 24 . (15) In order to evaluate Eq.(4), we need the renormalized spin vertex Uiwhich has an expansion in Pauli matrices Ui=år=0,1,2,3Ur isr, with the bare spin vertices U(0) i=si. We have dropped the explicit dependence Ui(e,w)for simplicity’s sake. For vanishing RSOC or DSOC, symmetry tells that the renormalized spin vertices share the same matrix structure of the bare ones Uisi. However,Version March 4, 2022 submitted to Condens. Matter 6 of 14 when both RSOC and DSOC are present, symmetry arguments again indicate that UxandUyare not simply proportional to sxandsy, but acquire both sxandsycomponents. By following the standard procedure [36], after projecting over the Pauli matrix components, the vertex equation reads Ur i=dri+1 2å mulImuTr[srsmslsu]Ul i+1 2å mulJmuTr[srszsmslsusz]Ul i, (16) where Imu=1 2pN0t0å p0GA m(e+w)GR u(e),Jmu=t0 2tEYImu. (17) Once the spin vertices are known, the "Edelstein conductivities" from Eq.(4) can be put in the form sij EC=Ur iPrj (18) with the bare "Edelstein conductivities" given by Prj=(e) 2på pTr[GA(e+w)sr 2GR(e)Jj]. (19) The bare "Edelstein conductivities" are those one would obtain by neglecting the vertex corrections due to the ladder diagrams. It is useful to point that one could have adopted the alternative route to renormalize the number current vertices and use the bare spin vertices. Indeed, this was the route followed originally by Edelstein [28]. Since, the renormalized number current vertices, in the DC zero-frequency limit, vanish [31], the evaluation of the Edelstein conductivity reduces to a bubble with bare spin vertices and the current vertices in absence of RSOC and DSOC. 3. Inverse spin-galvanic effect in the Rashba model To keep the discussion as simple as possible, in this Section we confine first to the case when only RSOC is present. We will derive the spin polarization, Sy, when an external electric field is applied along the xdirection. Then in the next Section we will evaluate the Bloch equation in the more general case when both RSOC and DSOC are present. In the case b=0, the renormalized spin vertex Uyis simply proportional to sy, which means Uy=Uy ysy. Upon the integration over momentum in Eq.(16), only I00is non-zero and other eight possibilities of (m,n)inIm,nare zero. The cases (0,x/y),(x/y, 0), (x,y)and(y,x)vanish because of angle integration, whereas the two other cases (x,x)and(y,y) cancel each other after taking the trace in Eq.(16). As a result we finally obtain (in the diffusive approximation wt1) Uy=Uy ysy=1 1I00+J00sy=14iwt t tsiwtsy(20) where the integral I00has been evaluated in the appendix A I00= 13iwtt ta 14iwt!t t0 (21) with the total spin relaxation rate being1 ts=1 tEY+1 ta. Here 1/ ta= (2ma)2Ddefines the DP spin relaxation rate due to the RSOC. Notice that, in the absence of SOC the vertex becomes singular by sending to zero the frequency, signaling the spin conservation in that limit. One sees that the EY and DP relaxation rates simply add up. This gives then syx=Uy yPyx. Physically, in the zero-frequency limit, the factor Uy y=ts/tcounts how many impurity scattering events are necessary to relax the spin. In the diffusive regime tst, i.e. many impurity scattering events are necessary to erase the memory of the initial spin direction.Version March 4, 2022 submitted to Condens. Matter 7 of 14 By neglecting the contribution from the extrinsic SOC in the expression (5) for the current vertex, the bare conductivity Pyxnaturally separates in two terms P(A) yxandP(B) yxdue to the components px/m andasyof the number current vertex. The expression for P(A) yxreads P(A) yx = (e)1 2på pTr GA(e+w)sy 2GR(e)px m =e 4pmå pp 2h GA +(e+w)GR +(e)GA (e+w)GR (e)i =e 4m p+N+ iw+1 t+pN iw+1 t! . (22) In the above p,Nandtrefer to the Fermi momentum, density of states and quasiparticle time in the-band. To order a/vF, one has p=pF(1a/vF),N=N0(1a/vF). (23) By including the contribution of the quasiparticle time in the -band from Eq.(13), one gets P(A) yx=S0 1t 2tEYiwt 12iwt! , (24) where S0=eN0at. The evaluation of P(B) yxis more direct. It gives P(B) yx=ea 2på pTr GA(e+w)sy 2GR(e)sy =ea 2på p GA 0(e+w)GR 0(e) =S0 1t ta3iwt 14iwt! (25) Combining both contributions with accuracy up to order wtgives Pyx=P(A) yx+P(B) yx=S0 t tat 2tEY 16iwt! (26) By combining the vertex correction Eq.(20) and the bare conductivity Pyxin Eq.(18), we get following contribution to the frequency-dependent spin polarization (Sy)(1)=1 (t tsiwt)14iwt 16iwt Sx at tat 2tEY . (27) with Sx a=eN0atEx. This is not the full story yet as we are to explain. What we have learned up to now is that the momentum dependence of the EY self-energy on the two spin-split Fermi surfaces yields an extra term to the Edelstein polarization. Such a momentum dependence can also modify the vertex corrections (the integrals Jmuin Eq.(17)), which lead to the renormalized spin vertex. To appreciate this aspect we notice that in evaluating such integrals in the absence of the RSOC, the moduli of pandp0are taken at the unsplit Fermi surface. We emphasize that, instead, taking into account the momentum dependence on the Rashba-split Fermi surfaces one gets an extraVersion March 4, 2022 submitted to Condens. Matter 8 of 14 Figure 1. The diagram needed to evaluate the extra vertex correction to the ISGE due to extrinsic SOC. The left and right vertices denote the spin vertex Syand the component (px/m)of the number current vertex Jx, whereas, the crosses on the top and bottom Green functions line stand for i(l2 0/4)p0p andi(l2 0/4)pp0, respectively. contribution. Consider the diagram of Fig. 1. After integration over p0, the left side part of the diagram gives (l2 0/4)2p2 Fp2 2t0t=t 2tEYp2 p2 F. If we set p=pF, we would recover the standard diagrammatic calculation in the absence of intrinsic RSOC. By combining the above left side with the rest of the diagram, one gets an additional contribution to the bare conductivity (dP) =t 2tEY e 2på pp2 p2 FTr GA(e+w)sy 2GR(e)px m! =t 2tEY(e 4mp2 F) p3+N+ iw+1 t+p3N iw+1 t! . (28) To this expression we must subtract the one obtained by replacing p=pF, which is already accounted for in the ladder summation. Hence the extra vertex part ( dP) modifies the spin polarization to give the second contribution (Sy)(2)=1 t tsiwt14iwt 16iwt Sx a t 2tEY . (29) Hence, by summing the above result to Eq.(27), the total spin polarization reads Sy=1 1 tsiw 1+2iwt 16iwt Sx a1 ta1 tEY 1 1 tsiwSx a1 ta1 tEY . (30) In the diffusive regime, terms in wtin the second round brackets on the right hand side of Eq.(30) which are responsible for higher-order frequency dependence, can be neglected. In the zero-frequency limit, the Eq.(30) has two main contributions described by the two terms in the last round brackets. The first term is responsible for the Edelstein result [28] due to the intrinsic SOC, whereas the second one, which arises to order l4 0, is an additional contribution to the spin polarization due to the extrinsic SOC. In the Rashba model without extrinsic SOC, only the first term is present and, indeed, Eq.(30) reduces to it when l0=w=0. After Fourier transforming, the above equation can be written in the form of the Bloch equation ¶tSy=1 ta+1 tEY Sy+1 ta1 tEY Sx a. (31)Version March 4, 2022 submitted to Condens. Matter 9 of 14 The terms on the right hand side describe the various torques controlling the spin dynamics. The first term, which includes DP and EY contributions, is the spin relaxation torques, wheres the second term represent the spin generation torques. The above result coincides with that obtained in Ref.[27] by the SU(2) gauge theory formulation. We have then succeeded in showing by diagrammatic methods the origin of the EY-induced spin torque discussed by Ref.[27]. In the next Section we will generalize this result to the case when both RSOC and DSOC are present. 4. Inverse spin-galvanic effect in the Rashba-Dresslhaus model As we have seen in the previous Section, the size and form of the ISGE is greatly modified by the presence of the EY spin relaxation due to the extrinsic SOC. To analyze this fact more generally we focus here on the model with RSOC and DSOC as well as SOC from impurities. In order to evaluate Eq.(4) for the Edelstein conductivity, we need the renormalized spin vertex Ui. For vanishing RSOC or DSOC, the renormalized spin vertices share the same matrix structure of the bare ones Uisi. However, when both RSOC and DSOC are explicitly taken into account, UxandUyare not only simply proportional to sxandsy, but also acquire components on both sxandsy. By following the procedure shown in Eq.(16) and upon integration over momentum, the vertex equation for Uyreduces to 1I00+J002(IyxJyx) 2(IxyJxy)1I00+J00! Uy y Ux y! = 1 0! (32) while that for Uxis 1I00+J002(IxyJxy) 2(IyxJyx)1I00+J00! Uy x Ux x! = 0 1! , (33) where 1I00+J00' iw+h1 tgi+1 tEY 14iwt! t (34) 2(IxyJxy)'1iwt 14iwt 1t tEY2t tab, whereh. . .iindicated the average over the momentum directions. The technical points of the calculation in Eq.(34) are given in appendix A at the end of the paper. In the diffusive regime, 1 tg= (2mg)2Dand1 tab= (2m)2abDare the Dyakonov-Perel (DP) relaxation rates due to the total intrinsic spin-orbit strength and the interplay of RSOC/DSOC, respectively. For vanishing DSOC, the Eq.(34) reduce to the same expression in Eq.(20) as expected in the Rashba model. However, with both RSOC and DSOC, spin relaxation is anisotropic and one needs to diagonalize the matrix in the left hand side of Eqs.(32-33). Such a matrix then identifies the spin eigenmodes. Having in mind to derive the Bloch equations governing to spin dynamics, we rewrite Eq.(3) by using Eq.(18) Sx Sy! = Ux xUy x Ux yUy y! å j Pxj Pyj! Ej (35) where, by virtue of Eqs.(32-33) Ux xUx y Uy xUy y!1 =t 14iwt iw+h1 tgi+1 tEY2 tab(1iwt) 2 tab(1iwt)iw+h1 tgi+1 tEY! . (36) In the diffusive regime we can safely neglect the factor wtwith respect to unity in the denominator in front of the matrix and in the off diagonal elements of the matrix. The quantities Prjappearing in the right hand side of Eq.(35) can be evaluated by standard techniques. However, some care is requiredVersion March 4, 2022 submitted to Condens. Matter 10 of 14 when evaluating the momenta due to the extrinsic SOC at the spin-split Fermi surfaces, as we did in Eq.(28). The final result for the bare conductivities reads Pxx=tSx b 16iwth1 tg1 tEY2 tga2 g2i, (37) Pxy=tSy a 16iwth1 tg1 tEY2 tgb2 g2i, (38) Pyx=tSx a 16iwth1 tg1 tEY2 tgb2 g2i, (39) Pyy=tSy b 16iwth1 tg1 tEY2 tga2 g2i, (40) with Sx b=eN0tbEx (41) Sy a=eN0taEy (42) Sx a=eN0taEx (43) Sy b=eN0tbEy.. (44) We take the angular average over the DP relaxation rates in Eqs.(36-40) 2pZ 0df 2p1 tg=1 ta+1 tb(45) (2)(a2orb2)2pZ 0df 2p1 tg1 g2=2 taor2 tb. (46) where1 ta= (2ma)2D,1 tb= (2mb)2Dare the DP relaxation rates due to RSOC and DSOC in the diffusive approximation. By inserting the above expression into Eqs.(37-40) and vertex correction in Eq.(36) and using Eq.(35), we may write the expression of the ISGE components in a form reminiscent of the Bloch equations iw+1 ta+1 tb+1 tEY2 tab 2 tabiw+1 ta+1 tb+1 tEY! Sx Sy! = Sy a(1 ta1 tb1 tEY)Sx b(1 ta+1 tb1 tEY) Sx a(1 ta1 tb1 tEY) +Sy b(1 ta+1 tb1 tEY)! , (47) Indeed, by performing the anti-Fourier transform with respect to the frequency w, Eq.(47) can be written as ¶tS=(ˆGDP+ˆGEY)S+ (ˆGDPˆGEY)N0 2B, (48) where Brepresents the internal SOC field induced by the electric current. The ˆGDPand ˆGEYare the DP and EY relaxation matrix B=2et bEx+aEy (aEx+bEy)! ,ˆGDP= 1 ta+1 tb2 tab 2 tab1 ta+1 tb! ,ˆGEY= 1 tEY0 01 tEY! . (49) Eq.(48) is the main result of our paper. It shows that the intrinsic and extrinsic SOC act in parallel as far as relaxation to the equilibrium state is concerned, i.e. the DP and EY spin relaxation matrices add up. However, as far as the spin generation torques are concerned, DP and EY processes haveVersion March 4, 2022 submitted to Condens. Matter 11 of 14 opposite sign. This is in full agreement with the result of Ref.[27] once we take into account also the spin generation torque due to side-jump and skew-scattering processes discussed diagramatically in Ref.[26]. This is simply obtained by multiplying the DP relaxation matrix ˆGDPin the second term in the right hand side of Eq.(48) by the factor 1 +qsH ext/qsH int, where qsH extandqsH intare the spin Hall angles for extrinsic and intrinsic SOC. To develop some quick intuition, one may notice that again for b=l0=0 and Ey=w=0, Eq.(47) reproduces the Edelstein result for the Rashba model [12]. Furthermore, when also w6=0 it reproduces the frequency-dependent spin polarization for the Rashba model as shown in the previous Section. When l06=0 and b=0, we see that the ISGE, due to the interplay of the extrinsic and intrinsic SOC, gets an additional spin torque, suggesting that the EY spin-relaxation is detrimental to the Edelstein effect. The diagrammatic analysis reported here provides the following interpretation. The EY spin relaxation depends on the Fermi momentum. When there are two Fermi surfaces with different Fermi momenta, the one with the smaller momentum undergoes less spin relaxation of the EY type than the one with larger momentum. On the other hand, the ISGE arises precisely because there is an unbalance among the two Fermi surfaces with respect to spin polarization. For a given momentum direction, the larger Fermi surface contributes more to the Edelstein polarization than the smaller Fermi surface. Hence, the combination of these two facts suggests a negative effect from the interplay of Edelstein effect and EY spin relaxation. By neglecting the EY relaxation, one sees that the DP terms can cancel each other if the RSOC and DSOC strengths are equal. This cancellation or anisotropy of the spin accumulation could be used to determine the absolute values of the RSOC and DSOC strengths under spatial combination of spin dependent relaxation. Finally, we comment on the relevance of our theory with respect to existing experiments Ref.[37]. The latter show that the current-induced spin polarization does not align along the internal magnetic field Bdue to the SOC. According to our Eq.(48) this may occur due to the presence of the extrinsic SOC both in the spin relaxation torque and in the spin generation torque. Indeed when the extrinsic SOC is absent, the spin polarization must necessarily align along the Bfield. Hence, our theory could, in principle, provide a method to measure the relative strength of intrinsic and extrinsic SOC. 5. Conclusions In this present work, we showed how the interplay of intrinsic and extrinsic spin-orbit coupling modifies the current-induced spin polarization in a 2DEG. This phenomenon, known as the inverse spin galvanic effect, is the consequence of the coupling between spin polarization and electric current, due to restricted symmetry conditions. We derived the frequency-dependent spin polarization response, which allowed us to obtain the Bloch equations governing the spin dynamics of carriers. We identified the various sources of spin relaxation. In fact, the precise relation between the non-equilibrium spin polarization and spin-orbit coupling depends on ratio of the DP and EY spin relaxation rates. More precisely, the spin-orbit coupling affects the spin relaxation time by adding the EY mechanism to the DP and, furthermore, it changes the non-equilibrium value of the current-induced spin polarization by introducing an additional spin torque. Our treatment, which is valid at the level of Born approximation and was obtained by diagrammatic technique agrees with the analysis of Ref.[27], derived via the quasiclassical Keldysh Green function technique. Finally, to make comparison between theory and experiments, we found that the spin polarization and internal magnetic field will not be aligned if the EY is strong enough.Version March 4, 2022 submitted to Condens. Matter 12 of 14 Appendix Integrals of products involving pairs of retarded and advanced Green functions To perform the calculations of the renormalized spin vertex in Eq.(34) and also in all the analysis, we encounter the following kind of integrals, which are evaluated to first order ing vFandwt å ppnGR (e+w)GA (e)2pNpn 1 iw+1 t(50) å ppnGR (e+w)GA (e)2pN0pn 1 iw2igpF+1 t(51) where n=0, 1. We can then evaluate the I00integral as I00=1 2pN0t0å p0GA 0(e+w)GR 0(e) =1 2pN0t0å p01 4 GA +(e+w)GR +(e) +GA +(e+w)GR (e) +GA (e+w)GR +(e) +GA (e+w)GR (e) =1 4N0t0hN+ iw+1 t++N iw+1 t+N0 iw+2ipFg+1 t+N0 iw2ipFg+1 ti (t t0) 13iwtht tgi 14iwt! (52) and the same calculations for 2 Ixy=2Iyxyields 2Ixy=2 2pN0t0å p0GA x(e+w)GR y(e) =2 2pN0t0ab 4g2 å p0 GA +(e+w)GR +(e)GA +(e+w)GR (e)GA (e+w)GR +(e) +GA (e+w)GR (e) (4t t0)(2t tg)ab 4g2 (1iwt 14iwt) =2t tab(1iwt 14iwt). (53) Acknowledgments: We thank Cosimo Gorini, Ilya Tokatly, Ka Shen and Giovanni Vignale for discussions. A.M. thanks Juan Borge for help received during the initial stages of this work. Bibliography 1. Ganichev, S.D.; Trushin, M.; Schliemann, J. Spin polarisation by current. ArXiv e-prints 2016 , [arXiv:cond-mat.mes-hall/1606.02043]. 2. Ando, Y.; Shiraishi, M. Spin to Charge Interconversion Phenomena in the Interface and Surface States. Journal of the Physical Society of Japan 2017 ,86, 011001, [http://dx.doi.org/10.7566/JPSJ.86.011001]. 3. Soumyanarayanan, A.; Reyren, N.; Fert, A.; Panagopoulos, C. Emergent phenomena induced by spin–orbit coupling at surfaces and interfaces. Nature 2016 ,539, 509. 4. Gardelis, S.; Smith, C.G.; Barnes, C.H.W.; Linfield, E.H.; Ritchie, D.A. Spin-valve effects in a semiconductor field-effect transistor: A spintronic device. Phys. Rev. B 1999 ,60, 7764–7767. 5. Sarma, S.D.; Fabian, J.; Hu, X.; ˘Zuti´ c, I. Spin electronics and spin computation. Solid State Communications 2001 ,119, 207–215. 6. Sugahara, S.; Tanaka, M. 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Ganichev, S.; Danilov, S.; Schneider, P .; Bel’kov, V .; Golub, L.; Wegscheider, W.; Weiss, D.; Prettl, W. Electric current-induced spin orientation in quantum well structures. Journal of Magnetism and Magnetic Materials 2006 ,300, 127–131. The third Moscow International Symposium on Magnetism 2005The third Moscow International Symposium on Magnetism 2005. 17. Yang, C.L.; He, H.T.; Ding, L.; Cui, L.J.; Zeng, Y.P .; Wang, J.N.; Ge, W.K. Spectral Dependence of Spin Photocurrent and Current-Induced Spin Polarization in an InGaAs/InAlAs Two-Dimensional Electron Gas. Phys. Rev. Lett. 2006 ,96, 186605. 18. Sánchez, J.C.R.; Vila, L.; Desfonds, G.; Gambarelli, S.; Attané, J.P .; Teresa, J.M.D.; Magén, C.; Fert, A. Spin-to-charge conversion using Rashba coupling at the interface between non-magnetic materials. Nature Commun. 2013 ,4, 2944. 19. Chen, L.; Decker, M.; Kronseder, M.; Islinger, R.; Gmitra, M.; Schuh, D.; Bougeard, D.; Fabian, J.; Weiss, D.; Back, C.H. 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The Edelstein effect in the presence of impurity spin-orbit scattering. arXiv preprint arXiv:1610.08258 2016 . 27. Gorini, C.; Maleki, A.; Shen, K.; Tokatly, I.V .; Vignale, G.; Raimondi, R. Theory of current-induced spin polarizations in an electron gas. arXiv preprint arXiv:1702.04887 2017 . 28. Edelstein, V . Spin polarization of conduction electrons induced by electric current in two-dimensional asymmetric electron systems. Solid State Communications 1990 ,73, 233–235. 29. Shen, K.; Vignale, G.; Raimondi, R. Microscopic theory of the inverse Edelstein effect. Physical Review Letters 2014 ,112, 096601. 30. Schwab, P .; Raimondi, R. Magnetoconductance of a two-dimensional metal in the presence of spin-orbit coupling. The European Physical Journal B - Condensed Matter and Complex Systems 2002 ,25, 483–495.Version March 4, 2022 submitted to Condens. Matter 14 of 14 31. Raimondi, R.; Schwab, P . Spin-Hall effect in a disordered two-dimensional electron system. Phys. Rev. B 2005 ,71, 033311. 32. Engel, H.A.; Halperin, B.I.; Rashba, E.I. Theory of Spin Hall Conductivity in n-Doped GaAs. Phys. Rev. Lett. 2005 ,95, 166605. 33. Tse, W.K.; Das Sarma, S. Spin Hall Effect in Doped Semiconductor Structures. Phys. Rev. Lett. 2006 , 96, 056601. 34. Raimondi, R.; Schwab, P . Tuning the spin Hall effect in a two-dimensional electron gas. EPL (Europhysics Letters) 2009 ,87, 37008. 35. Raimondi, R.; Schwab, P . Interplay of intrinsic and extrinsic mechanisms to the spin Hall effect in a two-dimensional electron gas. Physica E: Low-dimensional Systems and Nanostructures 2010 ,42, 952–955. 18th International Conference on Electron Properties of Two-Dimensional Systems. 36. Shen, K.; Raimondi, R.; Vignale, G. Theory of coupled spin-charge transport due to spin-orbit interaction in inhomogeneous two-dimensional electron liquids. Phys. Rev. B 2014 ,90, 245302. 37. Norman, B.; Trowbridge, C.; Awschalom, D.; Sih, V . Current-induced spin polarization in anisotropic spin-orbit fields. Physical Review Letters 2014 ,112. c 2022 by the authors. Submitted to Condens. Matter for possible open access publication under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

2202.13896v2.Observation_of_long_range_orbital_transport_and_giant_orbital_torque.pdf

Observation of long-range orbital transport and giant orbital torque Hiroki Hayashi,1Daegeun Jo,2Dongwook Go,3, 4Tenghua Gao,1, 5Satoshi Haku,1Yuriy Mokrousov,3, 4Hyun-Woo Lee,2and Kazuya Andoa1, 5, 6 1Department of Applied Physics and Physico-Informatics, Keio University, Yokohama 223-8522, Japan 2Department of Physics, Pohang University of Science and Technology, Pohang 37673,Korea 3Peter Gr unberg Institut and Institute for Advanced Simulation, Forschungszentrum J ulich and JARA, 52425 J ulich, Germany 4Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany 5Keio Institute of Pure and Applied Sciences, Keio University, Yokohama 223-8522, Japan 6Center for Spintronics Research Network, Keio University, Yokohama 223-8522, Japan aCorrespondence and requests for materials should be addressed to ando@appi.keio.ac.jp 1arXiv:2202.13896v2 [cond-mat.mes-hall] 6 Feb 2023Abstract Modern spintronics relies on the generation of spin currents through spin-orbit coupling. The spin-current generation has been believed to be triggered by current-induced orbital dynamics, which governs the angular momentum trans- fer from the lattice to the electrons in solids. The fundamental role of the orbital response in the angular momentum dynamics suggests the importance of the or- bital counterpart of spin currents: orbital currents. However, evidence for its existence has been elusive. Here, we demonstrate the generation of giant orbital currents and uncover fundamental features of the orbital response. We exper- imentally and theoretically show that orbital currents propagate over longer distances than spin currents by more than an order of magnitude in a ferro- magnet and nonmagnets. Furthermore, we nd that the orbital current enables electric manipulation of magnetization with eciencies signicantly higher than the spin counterpart. These ndings open the door to orbitronics that exploits orbital transport and spin-orbital coupled dynamics in solid-state devices. Introduction Since the discovery of the giant magnetoresistance, the concept of spin currents has played a key role in the development of condensed matter physics and spintronics applications1{3. Of particular recent interest is the spin Hall eect (SHE), which generates spin currents from charge currents through spin-orbit coupling (see Fig. 1a)4,5. The spin current can interact with local spins, triggering magnetic dynamics in magnetic heterostructures6. The current- induced magnetic dynamics lies at the foundation of a variety of spintronics phenomena, providing a way to realize a plethora of spin-based devices, such as nonvolatile memories, nanoscale microwave sources, and neuromorphic computing devices7. Although spin transport has been central to spintronics, both spin and orbital angular momentum can be carried by electrons in solids8{11. An important theoretical prediction is that the SHE is a secondary eect arising from the orbital Hall eect (OHE) in combination with the spin-orbit coupling10. The OHE is a phenomenon that generates an orbital current owing perpendicular to an applied electric eld (see Fig. 1b)8{20, which stems from nonequi- librium interband superpositions of Bloch states with dierent orbital characters induced by the electric eld11. This process triggers the transfer of the angular momentum from the 2lattice to the orbital part of the electron system, and the orbital angular momentum can be further transferred to the spin part of the electrons by the spin-orbit coupling21. This mechanism illustrates the primary role of the orbital response in the angular momentum dynamics in solids, suggesting that the orbital transport is more fundamental than the spin transport. Despite the signicance of the orbital response, however, experimental detection of orbital currents remains a major challenge. The behavior of orbital currents is predicted to be fundamentally dierent from that of spin currents in ferromagnets (FMs), providing a way to probe the orbital transport22,23. When a spin current is injected into a FM, its transverse component to the magnetization precesses rapidly in the real space because of the spin splitting, which induces the dierence in the wavevectors of the majority and minority spins at the Fermi surface (see Fig. 1a)24. The precession wavelength is dierent depending on the incident angles of the electrons, resulting in the short spin decay length, less than 1 nm. In contrast, a recent theory predicts that an orbital current does not precess rapidly and decays over much longer distances than a spin current in FMs does (see Fig. 1b, the physical picture is explained in Supplementary Note 1)23. This dierence has been attributed to the unique feature of the orbital current that its constituent orbital states in FMs can remain nearly degenerate in limited regions of the momentum space, which form hot-spots for the orbital response23. In the FMs, the angular momentum of the injected spin and orbital currents is transferred to the local spins, giving rise to torques on the magnetization: spin and orbital torques22. Although recent experimental studies have suggested the presence of the orbital torque25{34, the fundamental properties of the orbital torque and orbital transport are still elusive. In fact, in the previous works, the observed torque eciency is much lower than the spin counterpart despite the fundamental role of the orbital response in the angular momentum transport. Furthermore, experimental evidence for the long-range orbital transport in FMs is lacking. In this work, we report the observation of the long-distance orbital transport and giant orbital torques, revealing the fundamental properties of orbital currents. We show that sizable current-induced torques are generated in Ni/Ti bilayers despite the weak spin-orbit coupling of Ti. We nd that the torque eciency increases with increasing the Ni-layer thickness and disappears by replacing the Ni layer with Ni 81Fe19. The unconventional torque, which cannot be attributed to the SHE, is also observed in Ni/W and Ni 81Fe19/W bilayers. We show that these observations are consistent with semirealistic tight-binding calculations, 3demonstrating that the observed torques originate from the OHE. The experimental and theoretical results evidence that orbital currents propagate over longer distances than spin currents by more than an order of magnitude both in the FM and nonmagnets (NMs). Furthermore, we demonstrate that the orbital torque eciency exceeds the spin torque eciency of exotic materials, such as topological insulators, as well as Pt, which exhibits the strongest SHE among single element crystals, by an order of magnitude. We also nd that the power consumption of the orbital devices can be lower than that of the representative spin-orbitronic devices. These ndings provide unprecedented opportunities for advancing the understanding of the angular momentum dynamics in solids. Results Evidence and characteristics of orbital transport First, we provide evidence for the existence of orbital currents and orbital torques originat- ing from the OHE in FM/NM structures. To capture the orbital transport, we chose a light metal, Ti, as a source of orbital currents. In Ti, the spin transport plays a minor role be- cause of the weak spin-orbit coupling. In fact, the spin Hall conductivity in Ti is vanishingly small35,Ti SH=1:2 (~=e) 1cm1, which is more than three orders of magnitude smaller than the prediction of the intrinsic orbital Hall conductivity Ti OH:jTi OH=Ti SHj1. Another important feature is that the sign of the spin Hall conductivity is opposite to that of the or- bital Hall conductivity32:Ti OH>0 andTi SH<0. These distinct dierences between the SHE and OHE make it possible to distinguish between the spin and orbital Hall currents from the magnitude and sign of the current-induced torques. We also note that a recent study reports the detection of current-induced orbital accumulation in Ti by an optical technique32. These facts make this light metal a promising platform for revealing the fundamental properties of the orbital transport and orbital torques. To investigate the orbital transport, we measure current-induced torques for Ni/Ti and Ni 81Fe19/Ti lms using spin-torque ferromagnetic resonance (ST-FMR). In these het- erostructures, as shown in Fig. 1b, the OHE in the Ti layer generates an orbital Hall current, kzLy, carrying the ycomponent of the orbital angular momentum, Ly, by an electric eld applied along the xdirection, where kzis thezcomponent of the wavevector. When the orbital current is injected into the FM layer, the orbital angular momentum interacts with the local spins along the xdirection by a combined action of the spin-orbit coupling and 4spin exchange coupling between the conduction-electron spins and local spins, inducing the zcomponent of the orbital angular momentum, Lz. The induced Lzpropagates in the FM layer without oscillation through the degenerate orbital hot spots in the momentum space. The propagating Lzinteracts with the local spins at each site by a combined action of the spin-orbit coupling and spin exchange coupling, exerting a damping-like (DL) orbital torque on the magnetization in the FM layer (for details, see Supplementary Note 1)23. This process indicates that the generation of the orbital torque relies on the spin-orbit coupling in the FM layer, as well as the OHE in the NM layer. Thus, in the presence of both SHE and OHE, the DL-torque eciency per unit electric eld can be expressed as E DL=Tint SHNM SH+FMTint OHNM OH, whereNM SH(OH) is the spin(orbital) Hall conductivity in the NM layer and Tint SH(OH) is the spin(orbital) transparency at the FM/NM interface. Here, FM represents the eective coupling between the orbital angular momentum and magnetization originating from the spin-orbit coupling and spin exchange coupling in the FM layer. SinceFMoriginates from the orbital-to-spin conversion due to the spin-orbit correlation near the Fermi energy in the FM layer22, the orbital torque is sensitive to the electronic structure of the FM layer21. Among the conventional 3 dFMs, Ni is predicted to show the strongest orbital-to-spin conversion31. In the following, we assume that FMin Ni 81Fe19is much weaker than that in Ni: jNi=Ni81Fe19j 1. This assumption is supported by the fact that the physical origin of the strong orbital-to-spin conversion in Ni is in the optimal electronic occupation of dorbital shells such that the Fermi energy is located in the energy gap induced by the spin-orbit coupling21. This is manifested by the strong SHE in Ni, which results from the orbital-to-spin conversion as a result of the combined eect of the OHE and the spin-orbit coupling in the same material12. The previous work has shown that the spin Hall conductivity in Ni exhibits a sharp spike at the Fermi energy, and the value drops signicantly even if the Fermi energy is slightly varied12. This implies that in a situation like Ni 81Fe19, the eciency of the orbital-to-spin conversion can be strongly aected by the change of the electronic occupation by Fe doping. Under the assumption of jNi=Ni81Fe19j1 with the theoretical prediction of jTi OH=Ti SHj1, the orbital transport and orbital torque are expected to be pronounced in the Ni/Ti bilayer. Figure 2a shows ST-FMR spectra for the Ni 81Fe19/Ti and Ni/Ti bilayers, measured by applying a radio-frequency (RF) current with a frequency of fand an external magnetic eldH(see Methods). The measured spectra are consistent with the prediction of the 5direct-current (DC) voltage due to the ST-FMR:36,37 VDC=VsymW2 (0H0Hres)2+W2+VantisymW(0H0Hres) (0H0Hres)2+W2; (1) whereWis the linewidth and Hresis the FMR eld. Here, the symmetric component Vsym is proportional to the DL eective eld HDL, while the antisymmetric component Vantisym is proportional to the sum of the Oersted eld HOeand eld-like eective eld HFL. We conrmed that magnetic eld angle dependence of VsymandVantisym is consistent with the prediction of the ST-FMR model (see Supplementary Note 2). Notable is that the ST- FMR spectral shape is clearly dierent between the Ni 81Fe19/Ti and Ni/Ti bilayers. In the ST-FMR spectra for the Ni 81Fe19/Ti bilayer, the symmetric component Vsymis vanishingly small, consistent with previous reports that demonstrate negligible DL torque in Ti-based structures38,39. In contrast, Vsymis clearly observed for the Ni/Ti bilayer, demonstrating the generation of a sizable DL torque in this system. Here, the observed Vsymsignals cannot be attributed to spin-pumping and thermoelectric signals (see Supplementary Note 2). In Fig. 2b, we show Ti-layer thicknesses tTidependence of the DL-torque eciency per unit electric eld, E DL=(2e=~)0MstFMHDL=E, determined from the ST-FMR for the Ni81Fe19/Ti and Ni/Ti bilayers, where Msis the saturation magnetization, tFMis the thick- ness of the FM layer, and Eis the applied electric eld (for details, see Supplementary Note 3). Here, =1 for FM/NM/substrate structures and = 1 for NM/FM/substrate structures. Figure 2b shows that E DLof the Ni/Ti bilayer increases with increasing tTi. We conrmed that the variation in E DLwithtTiis not induced by a possible change of the magnetic property of the Ni layer; the eective demagnetization eld Meis independent of tTi, as shown in Fig. 2c. Thus, the clear increase in E DLwithtTiindicates that the observed DL-torque originates from the bulk eects, the SHE or OHE, in the Ti layer; self-induced torques in the Ni layer and interfacial spin-orbit coupling eects are not the source of the observed DL torque (see also Supplementary Notes 3 and 4). The negligible contribution from the interfacial spin-orbit coupling is consistent with a recent study39. We also note that the sizable DL torque in the Ni/Ti bilayer is supported by second-harmonic Hall resistance measurements (see Supplementary Note 5). The unconventional torque in the Ni/Ti bilayer is consistent with the orbital torque originating from the OHE in the Ti layer. We note that the sign of the DL-torque in the Ni/Ti layer is opposite to the prediction of the SHE but is consistent with that of the OHE in the Ti 6layer. Furthermore, the observed DL-torque eciency is more than two orders of magnitude higher than the spin Hall conductivity of Ti. These results provide clear evidence that the SHE in the Ti layer is not responsible for the observed DL torque. Figure 2b also shows that the DL-torque eciency E DLof the Ni/Ti bilayer is more than an order of magnitude larger than that of the Ni 81Fe19/Ti bilayer, demonstrating that the electronic structure of the FM layer plays a crucial role in generating the observed torque. The distinct dierence in the DL- torque eciency between the Ni/Ti and Ni 81Fe19/Ti devices is consistent with the scenario of the orbital torque with the assumption of jNi=Ni81Fe19j 1. We also note that the orbital transparency Tint OHcan also be dierent between the Ni/Ti and Ni 81Fe19/Ti devices. Since the spin Hall conductivity in Ti is vanishingly small35,Ti SH=1:2 (~=e) 1cm1, the DL torque due to the SHE is negligible regardless of tTi, resulting in the OHE-dominated torque with E DL>0 over the thickness range investigated in the Ni/Ti bilayer. We also demonstrate magnetization switching by the orbital torque (see Supplementary Note 6). We nd that the DL-torque eciency in the Ni/Ti bilayer is further enhanced by increas- ing the Ni-layer thickness tNi, demonstrating the long-range orbital transport in the Ni layer. Figure 2d shows that E DLincreases with increasing tNiup totNi= 20 nm despite the fact that the magnetic property is unchanged as shown in Fig. 2e. In the scenario of spin torques, since spin currents decay within 1 nm due to the spin dephasing, the DL-torque eciency is independent of the FM-layer thickness tFMwhentFM>1 nm. In fact, we conrmed that E DLis independent of tNiin Ni/Pt bilayers, where the DL-torque is dominated by the SHE in the Pt layer, as shown in Fig. 2f (see also Supplementary Note 7). In contrast, since orbital currents can propagate over much longer distances than the spin dephasing length in FMs, the DL-torque eciency increases with tFM23. The observed tNidependence of E DL demonstrates that the orbital currents responsible for the DL torque propagate over longer distances than the spin dephasing length by an order of magnitude in the Ni layer. Here, the suppression of E DLin the Ni/Ti device when tNi>20 nm can be attributed to a self-induced torque in the Ni layer. As shown in Fig. 2d, the DL-torque, whose sign is opposite to that of the Ni/Ti bilayer, is non-negligible in a Ni single-layer lm when tNi>10 nm. This result is consistent with the scenario of the self-induced torque, which is non-negligible only when the magnetic layer is thicker than the exchange length (8.4 nm for Ni)40. In contrast to the self-induced torque, which increases with tNiand becomes sizable especially at tNi>20 nm, the orbital torque tends to saturate with increasing tNi. The dierent tNidependences result 7in the suppression of E DLin the Ni/Ti bilayer at tNi>20 nm. Crossover between spin and orbital torques Next, we investigate the competition between spin and orbital torques by replacing the light metal Ti with a heavy metal W. The two metals are dierent in terms of the strength of the spin-orbit coupling. In the Ti-based device, because of the weak spin-orbit coupling, the spin transport and spin torques are negligible. In contrast, in the W-based system, the SHE contributes to the DL torque due to the strong spin-orbit coupling. Although the spin and orbital torques coexist in the W-based system, it is still possible to clarify the dominant mechanism of the angular momentum transport, spin or orbital channels, because the sign of the orbital Hall conductivity in W10,W OH>0, is opposite to that of the spin Hall conductivity41,W SH =785 ( ~=e) 1cm1in-W andW SH =1255 ( ~=e) 1cm1in -W. In Figs. 3a and 3b, we show W-thickness tWdependence of the DL-torque eciency E DL, determined by the ST-FMR, for Ni 81Fe19/W and Ni/W lms (see the black circles and also Supplementary Notes 2 and 3). We rst focus on E DLaroundtW= 20 nm, where E DL>0 in both Ni 81Fe19/W and Ni/W lms. Figures 3a and 3b show that E DLof the Ni/W bilayer is more than an order of magnitude higher than that of the Ni 81Fe19/W bilayer, and E DL increases with tWin both Ni 81Fe19/W and Ni/W bilayers. These unconventional features are consistent with the scenario of the orbital torque, observed for the FM/Ti devices (see also Supplementary Note 8). In fact, the sign of the DL torque, E DL>0, is consistent with the OHE and opposite to the SHE in W, indicating that the DL torque is dominated by the orbital transport around tW= 20 nm. Here, the validity of the determined DL-torque eciency is supported by second-harmonic Hall resistance measurements (see Supplementary Note 5). The observed tWdependence of E DLfor the FM/W bilayers demonstrates the crossover from the SHE-dominant regime to the OHE-dominant regime, illustrating dierent length scales of the spin and orbital transport. In the Ni 81Fe19/W bilayer, the sign of E DLchanges from negative to positive by increasing the W thickness tW(see the black circles in Fig. 3a). The sign of E DLin the Ni/W bilayer, where the OHE contribution is more pronounced due to the larger FM, also changes from negative to positive around tW= 8 nm. In the W-based devices, the spin Hall contribution to the DL torque with E DL;SH=Tint SHW SH<0 8should be saturated when tWis larger than the spin diusion length 1:5 nm in W42. In contrast, the orbital Hall contribution with E DL;OH=FMTint OHW OH>0 increases as tW increases and becomes comparable to the long orbital decay length, as evidenced by the clear increase in E DLwithtWof around 20 nm in the FM/W bilayers. Because of the dierent tWdependence, the ratio between the orbital and spin torque eciencies, jE DL;OH=E DL;SHj= jFMTint OHW OH=Tint SHW SHj, increases with tW. The observed variation in E DLwithtWin the FM/W bilayers is consistent with the competition between the spin Hall and orbital Hall contributions. In fact, Fig. 3a shows that E DLis almost saturated around tW= 2 nm, which is comparable to the spin diusion length in W, and remains so until the orbital Hall contribution shows up at large tW. We also note that the sign and magnitude of the saturation value of E DLis consistent with the spin Hall conductivity of W, supporting the dominant role of the SHE in generating the DL torque in the Ni 81Fe19/W device with tW5 nm. Figure 3b demonstrates that the orbital torque eciency in the Ni/W bilayer is compa- rable or larger than the DL-torque eciency originating from the SHE in the Ni/Pt bilayer (see Fig. 2f). The ecient torque generation is a unique feature of -W. Here, the resistivity Wof the W layer, shown in Fig. 3c, indicates that the W layer is low-resistivity -W when tW>10 nm, while the W layer is a mixture of low-resistivity -phase and high-resistivity -phase when tW<10 nm (for details, see Methods). To test the role of the structural phase in generating the DL torque, we also fabricated Ni/ -W and Ni 81Fe19/-W lms with tW= 5 and 25 nm by changing the sputtering condition (see the blue diamonds in Figs. 3a-c and Methods). Figure 3b shows that the high DL-torque eciency of the Ni/ -W lm at tW= 25 nm is signicantly suppressed by replacing the -W layer with the -W layer, suggesting that that jTint OHW OH=Tint SHW SHjof the-W device is much lower than that of the -W device. The result for the -W devices shows E DL>0 only in the Ni/ -W device with tW= 25 nm, which highlights the larger FMin Ni, as well as the dierent tWdependence of E DL;SHandE DL;OH. In the Ni 81Fe19/-W devices, the SHE provides the dominant contri- bution toE DLbecause of the following two reasons. First, the ratio jTint OHW OH=Tint SHW SHjof the-W devices is much lower than that of the -W devices. Second, FMof Ni 81Fe19 is smaller than that of Ni. The combination of these two features results in negligible jE DL;OH=E DL;SHj=jFMTint OHW OH=Tint SHW SHjin the Ni 81Fe19/-W devices; the SHE provides the 9dominant contribution to E DLeven attW= 25 nm. Here, the eective spin Hall angles obtained from E DLfor the Ni 81Fe19/(+)-W (black circles) and Ni 81Fe19/-W (blue dia- monds) devices with tW= 5 nm are (+)W SH;e =E DL(+)W=0:02 for (+)-W and W SH;e=E DLW=0:65 for-W. These values are consistent with literature6,41, support- ing the dominant role of the SHE in the Ni 81Fe19/W devices. In contrast, the OHE contri- bution is non-negligible in the Ni/ -W device due to the larger FMof Ni; although the SHE provides the dominant contribution to E DLattW= 5 nm, the contribution from the OHE exceeds that from the SHE in the Ni/ -W device at tW= 25 nm because jE DL;OH=E DL;SHj increases with tWdue to the long orbital decay length, resulting in the sign reversal of E DL. Semirealistic calculation We perform the numerical calculation of the DL torque. The rst-principles calculation is limited to very thin systems with the thickness of a few nm. To examine experimental situations with much thicker systems, we combine the rst-principles calculation scheme43 with the tight-binding scheme. The resulting semirealistic calculation scheme23(for details, see Supplementary Notes 9 and 10) allows the torque calculation up to 10 nm thick systems for Ni(12)/Ti( N) and Ni(12)/W( N) bilayers (the integers in the parentheses denote the number of atomic layers). For various thicknesses of the nonmagnet, N, we obtain the electronic structures and calculate the DL torque under an external electric eld Ex^xby evaluating the Kubo formula21, TDL=e~Ex NkX kX m6=n(fnkfmk)Imhunkj(TXC)yjumkihumkjvx(k)junki (EnkEmk+i)2 ; (2) wheree > 0 is the elementary charge, Nkis the number of kpoints used to sample the Brillouin zone,junkiis the periodic part of the Bloch state with energy eigenvalue Enk,fnk is the Fermi-Dirac distribution function, vx(k) is thex-component of the velocity operator, is the energy broadening, and TXC=2B ~S XCis the exchange torque operator with the Bohr magneton B, spin operator S, and exchange eld operator XC. Figure 4a shows the torque eciency cal DL= (e=~)TDL=(AcellEx) (Acellis the unit cell area) as a function of Nfor Ni(12)/Ti(N). The torque eciency increases as Ti layer becomes thicker and it saturates at the thickness of approximately 10 nm, where cal DL>400 1cm1is of the same order of magnitude as our experimental values for Ni/Ti (Fig. 2b). We note that this cal DLis dominated 10by the orbital torque while the SHE gives an insignicant contribution (Supplementary Note 9). Figure 4b shows cal DLas a function of Nfor Ni(12)/W( N) and it exhibits a signicant thickness dependence with a characteristic length &10 nm which is an order of magnitude longer than the spin diusion length 1:5 nm in W42. We also nd that this long-range behavior stems from the orbital torque which has a positive sign, whereas the spin Hall contribution has a negative sign with a much shorter length scale (Supplementary Note 9). As a result of the competition between OHE and SHE, the sign of the torque changes from negative to positive as the W layer becomes thicker (Fig. 4b), which is also observed in our experiment (Fig. 3). Hence, our theoretical calculations provide further evidence of the positive long-range orbital torque and support our experimental observations. Orbital torque in clean limit Finally, we demonstrate exceptionally high orbital torque eciencies beyond the prediction of the intrinsic OHE. The evidence for this is obtained by further increasing the NM-layer thicknesstNM, a situation that is dicult to capture by semirealistic tight-binding calcula- tions. Figures 5a and 5b show ST-FMR spectra for Ti(60 nm)/Ni and -W(70 nm)/Ni bi- layers, where the stacking order is changed from FM/NM/SiO 2-substrate to NM/FM/SiO 2- substrate to minimize the interfacial roughness in the thicker devices. Figures 5a and 5b show that the sign of the ST-FMR voltage for the NM/Ni devices is opposite to that for the Ni/NM devices (see Fig. 2a), as expected for the reversed stacking order. From the ST-FMR result, we obtain tTiandtWdependence of E DL, as shown in Figs. 5c and 5d. The results fortNM<30 nm are qualitatively consistent with E DLof the Ni/NM bilayers, shown in Figs. 2b and 3b, supporting that the observed torques are dominated by the OHE. Here, the thickness of the Ni layer, 8 nm, in the NM/Ni devices is thin enough to neglect the contribution from the self-induced torque (see also Fig. 2d). Figures 5c and 5d show that E DLincreases with tNMeven in the very thick devices ( tNM> 30 nm), revealing the exceptionally high orbital torque eciencies. When we assume that the orbital Hall conductivity is independent of tNM, thetNMdependence of the orbital torque eciencyE DL=FMTint OHNM OHcan be expressed as E DL(tNM) =E DL;0[1sech (tNM=NM)], whereNMis the orbital decay length. By tting the results in Figs. 5c and 5d, we obtain Ti= 4711 nm andE DL;0= 2:4103 1cm1for the Ti/Ni bilayer and W= 6816 nm andE DL;0= 34103 1cm1for the-W/Ni bilayer. The corresponding eective orbital 11Hall angles in the bulk limit, NM OH;e=E DL;0bulk NM, areTi OH;e= 0:13 andW OH;e= 0:45, wherebulk NMis the bulk-limit resistivity of the NM layer: bulk Ti= 54:07 cm andbulk W= 13:30 cm (see Supplementary Note 11). These values are signicantly larger than the spin-diusion length sd NM, the spin Hall conductivity NM SH, and the spin Hall angle NM SHof Ti and-W:sd Ti13 nm,Ti SH=1:2 (~=e) 1cm1, andTi SH=3:6104in Ti35; sd W1:5 nm,W SH =785 ( ~=e) 1cm1, andjW SHj<0:07 in-W41,42,44. Furthermore, the orbital torque eciency of -W,E DL104 1cm1, is an order of magnitude larger than that of the Ni/Pt bilayer (see Fig. 2f and Supplementary Note 12), demonstrating the giant orbital torque. The observation of the high orbital torque eciency suggests the existence of a mechanism that generates orbital torques beyond the intrinsic OHE. One possible mechanism is an extrinsic OHE originating from the skew scattering, theory of which has not been developed to date. The skew scattering, which relies on disorder scattering, has been shown to lead to exceptionally high anomalous Hall conductivities beyond the intrinsic mechanism45{47. This mechanism becomes dominant in the clean limit because the skew-induced Hall conductivity is proportional to the electric conductivity, while the intrinsic Hall conductivity is insensitive to the electric conductivity45. The high conductivity of the -W layers,105 1cm1, shows that the -W layers are possibly in this clean regime. To examine the possibility of the extrinsic mechanism, in Fig. 5e, we plot E DLfor all the devices with dierent tNMinvestigated in this study against the electric conductivity NM of the NM layer in each device. Figure 5e shows that the high orbital torque eciency, /Tint OHNM OH, of the W devices increases with NM, consistent with the extrinsic mechanism. However, the scaling relation between E DLandNMdeviates from the linear scaling predicted from the conventional skew scattering theory. This suggests that to understand the result based on the skew scattering, it is necessary to further take into account other possibilities, such as the increase in Tint OHwithtNMdue to the long orbital decay length, suggested by the present experimental results and numerical calculations, and abnormal scaling behavior of the skew scaling mechanism of the OHE. The skew scattering scenario is one of the possibilities to explain the experimental observation, and clarifying the exact mechanism remains to be a challenge for future study. Discussion 12Over the past three decades, extensive eorts have been directed towards discovering and understanding phenomena arising from spin currents and spin torques, leading to the rapid and exciting development of spintronics. In contrast, the exploration of the physics of orbital transport has only just begun, and the fundamental properties of orbital currents and orbital torques have been elusive. Thus, in the present work, we focus on revealing the fundamental properties of orbital currents rather than demonstrating and optimizing the ability of orbital currents for specic applications, such as magnetization switching devices. This is the reason why we focus on the characterization of the torque eciency in the devices based on Ni, which is predicted to show the strongest orbital response among the conventional 3 dFMs. The present work provides evidence for the orbital response by the systematic measure- ments of the current-induced torque for the dierent combinations and thicknesses of the NM and FM layers with the theoretical calculations. In particular, the present work pro- vides unambiguous demonstration of the long-range transport of angular momentum in Ni, which is a unique feature of the orbital transport. In metallic FMs, the spin transport length is limited to be less than 1 nm by spin dephasing. The present work demonstrates a means of long-range angular momentum transport in a metallic FM, as well as in NMs, originating in the electronic structure of a material. This result suggests the possibility to realize spintronic functionalities beyond magnetization switching, such as transmitting the signals between dierent components in the array of diverse elements, which may be termed orbitronic interconnect . In a spin-based metallic interconnect, only NMs with weak spin-orbit coupling can be used. In contrast, in the orbital interconnect, even metallic FMs and NMs with strong spin-orbit coupling can be used, providing a new degree of freedom in device design. Further experiments, such as nonlocal transport measurements48, are neces- sary for accurate determination of the orbital decay length and direct demonstration of the long-range transmission of orbital angular momentum. The gigantic torque eciency in the W/Ni devices far exceeds the eciency of spin- orbitronic devices based not only on Pt but also on exotic materials, such as topological insulators, by an order of magnitude, and the power consumption of the orbital devices can be lower than that of the representative spin-orbitronic devices (see Supplementary Note 12). The present work also demonstrates the ecient torque generation by using Ti, which is light, environment-friendly, abundant on earth, and cheap. We also note that the ability of orbital currents to increase the torque eciency by tuning the FM-layer thickness and 13materials provides more room for optimizing device parameters, such as the resistance and power consumption. In contrast, spin-based devices do not have this degree of freedom, and the eciency is xed by the choice of the spin-Hall layer. This suggests that the tunability of the orbital response oers a unique advantage in device applications. These results imply that our work has an impact not only on the research in academia but also on the development of devices by industries. To realize orbital-based devices, it is important to explore FMs that exhibit strong spin-orbit correlation and are compatible with the current spintronics technology. At this stage, orbital torques are not optimized for magnetization switching devices, as thick NM and FM layers are necessary to maximize the orbital torque eciency. Our results show that the switching power consumption can be further reduced by reducing the orbital transport length. This point has not been recognized previously because the orbital transport has been believed to be short-ranged due to orbital quenching. Although the relaxation mechanism of orbital currents is not clear at this stage, it is possible that nearly degenerate states are responsible for the long-range orbital transport23. This suggests that the orbital decay length in FMs and NMs can be controlled by engineering the band structures, such as by alloying. The FM-layer thickness can also be minimized by using a mechanism of the orbital-torque generation that is dierent from the mechanism observed in the present study. In this study, the process of the orbital-torque generation is of the second order in the spin-orbit coupling of the FM layer (see Supplementary Note 1). This process is associated with the long-range orbital transport, which is responsible for the torque generation, in the FM layer. We note that orbital torques can also be generated by the injection of orbital currents through a process that is of the rst order in the spin-orbit coupling of the FM layer22. In this process, the injected orbital current is converted into a spin current by the spin-orbit coupling in the FM layer. Since the angular momentum responsible for the torque generation is carried by the spin current, the torque eciency is saturated in the scale of spin dephasing length in this scenario. These two mechanisms are predicted to be sensitive to the band structure of the FM layer, implying that the optimum FM layer thickness can be controlled by material design. We therefore believe that our discovery of the long-range orbital transport and gigantic orbital torque eciencies provides important information for the material design of orbital-based devices, which will stimulate further experimental and theoretical studies and lead to the fundamental understanding of 14the physics of orbital currents for practical applications. 15Methods Devices. The ferromagnet (FM) and nonmagnet (NM) layers in the FM( tFM)/NM(tNM) and NM(tNM)/FM(tFM) structures (FM = Ni or Ni 81Fe19, NM = Ti or W) were fabricated on SiO 2substrates by radio frequency (RF) magnetron sputtering under 6N-purity-Ar at- mosphere. Here, tFMandtNMrepresent the thickness of the FM and NM layers, respectively. The surface of the lms was covered by 4-nm-thick SiO 2. Prior to the lm deposition, Ti was sputtered in the chamber (at least 5 min, 0.4 Pa, 120 W) to reduce residual hydrogen and oxygen contents. The resulting base pressure in the chamber was better than 5 :0107Pa. We used a linear shutter, moving at a constant speed ( <0:05 mm/s), during the sputtering to vary the thickness of the lm in each substrate. All sputtering process was performed at room temperature. The materials characterization is described in Supplementary Notes 13-16. The resistivity of the FM layer, measured using the four-probe method, is 17.6  cm for Ni and 41.4  cm for Ni 81Fe19. Since the self-induced torque is non-negligible only when the FM layer is much thicker than the exchange length40, we have chosen the thicknesses of the Ni and Ni 81Fe19layers so that we can neglect the self-induced torque; the thicknesses of the Ni and Ni 81Fe19layers are thinner than the exchange length in all the devices used in this study, except for the experiment in Fig. 2d. For the spin-torque ferromagnetic resonance (ST-FMR) measurement, the lms were pat- terned into rectangular strips with a width of 10 m and a length of 150 m by conventional photolithography followed by Ar milling. On the edges of the strip, Au(200 nm)/Ti(2 nm) electrodes were deposited by the sputtering and patterned by the photolithography and lift- o technique to form a ground-signal-ground (GSG) contact that guides an RF current into the device. Spin-torque ferromagnetic resonance. For the ST-FMR measurement, an RF current with a frequency of fand a power of Pwas applied along the longitudinal direction of the device. An in-plane magnetic eld Hwas applied with an angle of from the longitudinal direction of the device. The RF current excites FMR through the current-induced damping- like (DL) and eld-like (FL) torques, as well as an Oersted eld. Under the FMR, the magnetization precession changes the resistance of the device at the frequency of fdue to the anisotropic magnetoresistance (AMR), generating a direct current (DC) voltage through the mixing of the RF current and oscillating resistance36,37. We measured magnetic eld 16Hdependence of the DC voltage VDCusing a bias tee at room temperature. Here, we determined the RF current owing in the devices by measuring the resistance change due to Joule heating induced by application of DC and RF currents (see Supplementary Note 3). The determined values of the RF current have been used to obtain the applied electric eld. Electric resistivity and crystal structure of W. To characterize the crystal structure of the W layer, we plot thicknesses tWdependence of the electric resistivity Wfor W lms, as shown in Fig. 3c. For tW>10 nm, the tWdependence Wis consistent with W(tW) =at1 W+bulk W, whereat1 Wrepresents the resistivity due to the surface scattering and bulk Wis the resistivity in the bulk limit. The extracted bulk resistivity bulk W= 9:96 cm indicates that the W layer is low-resistivity -W in this thickness range. This result is supported by X-ray diraction measurements (see Supplementary Note 14). By decreasing the W thickness, the measured resistivity Wdeviates from the tting, which indicates that the W layer with tW<10 nm is a mixture of low-resistivity -phase and high-resistivity -phase49. 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Giant enhancement of the intrinsic spin Hall conductivity in -tungsten via substitutional doping. Phys. Rev. B 96, 241105 (2017). 42Wang, T.-C., Chen, T.-Y., Wu, C.-T., Yen, H.-W. & Pai, C.-F. Comparative study on spin- orbit torque eciencies from W/ferromagnetic and W/ferrimagnetic heterostructures. Phys. Rev. Materials 2, 014403 (2018). 43http://www.flapw.de . 44Pai, C.-F. et al. Spin transfer torque devices utilizing the giant spin Hall eect of tungsten. Appl. Phys. Lett. 101, 122404 (2012). 45Nagaosa, N., Sinova, J., Onoda, S., MacDonald, A. H. & Ong, N. P. Anomalous Hall eect. Rev. Mod. Phys. 82, 1539{1592 (2010). 2046Yang, S.-Y. et al. Giant, unconventional anomalous Hall eect in the metallic frustrated magnet candidate, KV 3Sb5.Sci. Adv. 6, eabb6003 (2020). 47Fujishiro, Y. et al. Giant anomalous Hall eect from spin-chirality scattering in a chiral magnet. Nat. Commun. 12, 317 (2021). 48Gorbachev, R. V. et al. Detecting topological currents in graphene superlattices. Science 346, 448{451 (2014). 49Lee, J.-S., Cho, J. & You, C.-Y. Growth and characterization of and-phase tungsten lms on various substrates. J. Vac. Sci. Technol. 34, 021502 (2016). 50Li, W. et al. Ni layer thickness dependence of the interface structures for Ti/Ni/Ti trilayer studied by X-ray standing waves. ACS Appl. Mater. Interfaces. 5, 404{409 (2013). 21Correspondence and requests for materials should be addressed to K.A. (ando@appi.keio.ac.jp) Acknowledgements This work was supported by JSPS KAKENHI (Grant Number 22H04964, 19H00864), JST FOREST Program (Grant Number JPMJFR2032), Canon Foundation, and Spintronics Re- search Network of Japan (Spin-RNJ). H.H. is supported by JSPS Grant-in-Aid for Research Fellowship for Young Scientists (DC1) (Grant Number 20J20663). D.J. and H.-W.L. ac- knowledge the nancial support by the SSTF (Grant No. BA-1501-51). D.G. and Y.M. ac- knowledge Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - TRR 173/2 - 268565370 - Spin+X (Project A11), and TRR 288 - 422213477 (Project B06), for funding. We also gratefully acknowledge the J ulich Supercomputing Centre and RWTH Aachen University for providing computational resources under projects ji40 and jara0062. Competing interest The authors declare no competing interests. Author contributions H.H., T.G., and S.H. fabricated the devices, performed the experiments, and analyzed the data. D.J. performed the numerical calculations with the help from H.W.L., D.G., and Y.M. K.A. wrote the manuscript with the help from H.H., D.J., H.W.L, D.G, Y.M, T.G., and S.H. All authors discussed results and reviewed the manuscript. K.A. supervised the study. 22a b zxy Figure 1.Spin and orbital transport in ferromagnets. a Schematic illustration of the spin Hall eect and spin transport in a ferromagnetic/nonmagnetic bilayer. The black arrow represents the local spins in the ferromagnetic layer. The blue and red arrows denote the up and down spins, respectively. The injected spin, the blue arrow, precesses around the local spin due to the spin exchange coupling in the ferromagnetic layer. bSchematic illustration of the orbital Hall eect and orbital transport in a ferromagnetic/nonmagnetic bilayer. The blue and red arrows, associated with the motion of the electrons, denote the orbital angular momentum. The orbital Hall eect generates the orbital current carrying the ycomponent of the orbital angular momentum Ly. The orbital angular momentum, the blue arrow, injected into the ferromagnetic layer induces zcomponent of the orbital angular momentum, Lz, through a combined action of the spin-orbit coupling and spin exchange coupling. The induced Lz ows without oscillation through the hot-spots in the momentum space in the ferromagnetic layer. 23-50050 -300 -150 0 150 300-50050VDC (μV) VDC (μV) 0H (mT)μ0H (mT)μNi81Fe19/Ti(5.4 nm) Ni/Ti(5.4 nm)f = 4 GHz 6 GHz12 GHz8 GHz 10 GHz14 GHz f = 4 GHz6 GHz 12 GHz8 GHz10 GHzexp. fitting exp. fittingNi81Fe19/Ti(tTi)Ni/Ti(tTi)TiFM 10 5 06 3 0EξDL (102 Ω-1cm-1) tTi (nm) f10 5 0 -5 30 20 10 01.0 0.5 30 20 10 0EξDL (102 Ω-1cm-1) ξ tNi (nm)Ni(tNi)/Ti Ni(tNi)tNi (nm) tNi (nm)EξDL (103 Ω-1cm-1) ξa 10 5 0tTi (nm)1.0 0.50Meff (T) Ni(tNi)/TiNi81Fe19/Ti(tTi) Ni/Ti(tTi)b c de 4 2 10 5 0Ni(tNi)/Ptμ0Meff (T) μFigure 2.Current-induced torque generated by Ti. a Magnetic eld Hdependence of the DC voltage VDCfor the Ni 81Fe19(5 nm)/Ti(5.4 nm) (upper) and Ni(8 nm)/Ti(5.4 nm) (lower) lms with an applied RF power of 100 mW at dierent frequencies f. The solid circles are the experimental data and the solid curves are the tting result. bTi-layer-thickness tTidepen- dence of the DL-torque eciency per unit electric eld E DLfor the Ni(8 nm)/Ti( tTi) (red) and Ni81Fe19(5 nm)/Ti(tTi) (black) bilayers. Error bars, one-standard-deviation uncertainties from the tting, are smaller than the symbols. The negligibly small but nonzero positive torque eciency attTi= 0 nm is consistent with the result for the Ni single-layer lm within the experimental uncertainty due to random error. cTi-layer-thickness tTidependence of the eective demagne- tization eld Mefor the Ni(8 nm)/Ti( tTi) (red) and Ni 81Fe19(5 nm)/Ti( tTi) (black) bilayers. Mewas determined from the fdependence of the resonance eld Hresusing the Kittel formula: (2f= ) =p 0Hres(0Hres+0Me), where is the gyromagnetic ratio. dNi-layer-thickness tNi dependence of E DLfor the Ni(tNi)/Ti(8 nm) bilayer (red) and Ni( tNi) (black) single-layer lms. We have conrmed that the thickness of a magnetic dead layer, tdead, is less than half a nanometer by measuring tNidependence of the magnetic moment per unit area for the Ni( tNi)/Ti lms. This result shows that tdeadis more than an order of magnitude smaller than tNi. Thus, the eective thickness of the Ni layer is tNitdead'tNi; the magnetic dead layer can be neglected in the analysis of the DL-torque eciencies. Here, the range of tNiis above the critical thickness, 1.9 nm, for the amorphous-to-crystalline transition of Ni on Ti50.etNidependence of Mefor the Ni(tNi)/Ti(8 nm) bilayer. fNi-layer-thickness tNidependence of E DLfor the Ni( tNi)/Pt(8 nm) bilayer. 2420 10 0tW (nm)6 0EξDL (103 Ω-1cm-1) ξ35 -5EξDL (102 Ω-1cm-1) ξ0 101102103 ρ W (μΩcm)ca bNi81Fe19/ -W(tW)βNi81Fe19/W(tW) 20101 0tW (nm) Ni/ -W(tW)βNi/W(tW) βα-W ( + )-Wαβ-W fitting20101 0tW (nm)0Meff (T) μ0Meff (T) μFigure 3.Current-induced torque generated by W. a W-layer-thickness tWdependence of E DLfor the Ni 81Fe19(5 nm)/W( tW) (black circles) and Ni 81Fe19(5 nm)/-W(tW) (blue diamonds) bilayers. The inset shows tWdependence of Me. Error bars, one-standard-deviation uncertainties from the tting, are smaller than the symbols. Here, E DL(blue diamonds) of the Ni 81Fe19/-W with tW= 5 nm is smaller than that of the Ni 81Fe19/-W withtW= 25 nm, even though both torques are dominated by the spin Hall eect in the -W layer. This dierence can be attributed to the suppression of the intrinsic spin Hall conductivity in the dirty-metal regime due to the shortening of the carrier lifetime induced by decreasing tW. This interpretation is supported by the fact that the resistivity of the -W lm increases with decreasing the thickness from tW= 25 nm to tW= 5 nm, as shown in Fig. 3c. btWdependence of E DLfor the Ni(8 nm)/W( tW) (black circles) and Ni(8 nm)/ -W(tW) (blue diamonds) bilayers. ctWdependence of the resistivity Wfor the W (black circles) and -W (blue diamonds) lms. The resistivity shows that the W lm (black circles) is-W whentW>10 nm and the mixture of - and-phase W when tW<10 nm. The red curve is the tting result using W(tW) =at1 W+bulk W, whereat1 Wrepresents the resistivity due to the surface scattering and bulk Wis the resistivity in the bulk limit. 25a b N (atomic layers)10 20 30 40 50 0 N (atomic layers)10 20 30 40 50 0Thickness (nm) 246810 0 3.03.54.0Thickness (nm) 246810 0 -3.003.0 -6.0calξDL (102 Ω-1cm-1) ξ calξDL (102 Ω-1cm-1) ξFigure 4.Theoretical calculations of the current-induced torque. Torque eciency from the theoretical calculations cal DLfor (a) Ni(12)/Ti( N) and ( b) Ni(12)/W( N) bilayer structures. N is the number of atomic layers of the nonmagnet and the corresponding thickness is indicated on the top axis. 26c d0 150 300-50050 0 150 300050 0H (mT)μ0H (mT)μ0H (mT)μ0H (mT)μ VDC (μV)VDC (μV)Ti(60 nm)/Ni 4 GHz6 GHz 8 GHz10 GHz 12 GHzexp.fittinga b -W(70 nm)/Niα e Ti/Ni/sub. -W/Ni/sub. Ni/Cr/sub.Ni/Ta/sub.α Ni/ -W/sub. ( tW > 10 nm) α Ni/Ti/sub. θeff = 0.01θeff = 0.1 θeff = 0.001θeff = 1 100101102103104105 102103104105EξDL (Ω-1cm-1) ξ σNM (Ω-1cm-1)3 2 140 20 50 0 100 50 0 tW (nm) tTi (nm) EξDL (103 Ω-1cm-1) ξEξDL (103 Ω-1cm-1) ξexp.fittingexp.fittingTi(tTi)/Ni -W(tW)/Niα Ti NiW NiFigure 5.Relation between orbital torque eciency and electric conductivity. Mag- netic eldHdependence of the DC voltage VDCfor the ( a) Ti(60 nm)/Ni(8 nm) and ( b)- W(70 nm)/Ni(8 nm) lms with an applied RF power of 100 mW at dierent frequencies f.c Ti-layer-thickness tTidependence of E DLfor the Ti(tTi)/Ni(8 nm) bilayer. dW-layer-thickness tW dependence of E DLfor the-W(tW)/Ni(8 nm) bilayer. The solid circles are the experimental data and the solid curves are the tting result assuming that the orbital Hall conductivity is independent oftNM.eThe DL-torque eciency E DLas a function of the longitudinal electric conductivity NM for all the devices investigated in this study. The data for the Ni/Ta and Ni/Cr structures are taken from published papers30,31. The dotted lines are the eective Hall angles e=E DL=NM. Error bars, one-standard-deviation uncertainties from the tting, are smaller than the symbols. 27

1106.5046v1.Spin_orbit_coupling_induced_Mott_transition_in_Ca___2_x__Sr___x__RuO___4____0_x_0_2_.pdf

arXiv:1106.5046v1 [cond-mat.str-el] 24 Jun 2011Spin-orbit coupling induced Mott transition in Ca 2−xSrxRuO4(0≤x≤0.2) Guo-Qiang Liu Max-Planck-Institut f¨ ur Festk¨ orperforschung, D-70569 Stuttgart, Germany (Dated: October 5, 2018) We propose a new mechanism for the paramagnetic metal-insul ator transition in the layered perovskite Ca 2−xSrxRuO4(0≤x≤0.2). The LDA+ Uapproach including spin-orbit coupling is used to calculate the electronic structures. In Ca 2RuO4, we show that the spin-orbit effect is strongly enhanced by the Coulomb repulsion, which leads to an insulat ing phase. When Ca is substituted by Sr, the effective spin-orbit splitting is reduced due to th e increasing bandwidth of the degenerate dxzanddyzorbitals. For x= 0.2, the compound is found to be metallic. We show that these res ults are in good agreement with the experimental phase diagram. PACS numbers: 71.30.+h, 71.15.Mb, 71.27.+a, 71.20.-b The layered perovskite Ca 2−xSrxRuO4(CSRO) has been intensely studied during recent years since this se- ries of compounds exhibits a variety of interesting physi- cal properties as a function of the Sr concentration x.1–8 Sr2RuO4is a p-wave superconductor1,9with a K 2NiF4- type structure. The substitution of Ca for Sr causes the RuO6octahedra to rotate, and start to tilt at x= 0.5.5 Following with the structure distortion, CSRO under- goes a series of phase transition from a paramagnetic metal (0.5 <x<2) to a magnetic metal (0.2 <x<0.5), and finally to a Mott insulator (0 <x<0.2).4It is unusual that in the Mott insulating regime the metal-insulator tran- sition temperature ( TMI) is higher than the N´ eel tem- perature ( TN) of the antiferromagnetic (AFM) phase, which shows that a paramagnetic (PM) insulating phase exists between these transition temperatures.5,10,11For pure Ca 2RuO4, the PM insulating regime extends from TN= 110 K to TMI= 357 K.2,3,10This property makes Ca2−xSrxRuO4(0<x<0.2) different from other AFM Mott insulators. Recently, Qi et al.12found that the substitution of the lighter Cr for the heavier Ru strongly depresses TMIin Ca2Ru1−yCryO4(0<y<0.13), which implies a possible influence of the relativistic spin-orbit (SO) coupling on the Mott transition as pointed out by the authors. It is well known that SO coupling plays an important role in 5dtransition-metal oxides. For example, Kim et al.13 found that Sr 2IrO4is aJeff= 1/2 Mott insulator, and they showed that the unusual insulating state can be ex- plained by the combined effect of the SO coupling and Coulomb interaction. In the 4 doxides, the importance of SO coupling is under debate. Mizokawa et al.,14ob- served strong SO coupling in Ca 2RuO4from their pho- toemission experiment. Based this finding, they argued that the strong SO coupling in Ca 2RuO4would cause a complex electronic configuration. Theoretical studies re- vealed strong SO effects in Sr 2RuO4and Sr 2RhO4,15,16 which seemingly support the photoemission experiment. However, Fang et al.17,18reported an LDA+ Ustudy of Ca2RuO4. They found the AFM state has a rather sim- ple configuration xy↑↓xz↑yz↑without much influence of the SO coupling. These seemingly inconsistent view- points raise a question: what role does the SO couplingplay in CSRO? In this paper we present electronic structure calcula- tions for Ca 2−xSrxRuO4using the LDA+ Umethod in- cludingtheSOcoupling. Weshowthecombinationofthe SO coupling and Coulomb repulsion opens a band gap in PM Ca 2RuO4. The appearance of the Mott insulating phase is strongly dependent on the tilting of the RuO 6 octahedra, which naturally explains the PM Mott tran- sition in the experimental phase diagram. On the other hand, we find SO has much less influence on the AFM order. We show that these phenomena can be explained by a simple formalism. All the calculations in this work were performed with thefull-potentiallinearaugmentedplanewave(FLAPW) within the local-density approximation (LDA), as im- plemented in package WIEN2K.19Two experimental structure5were considered in this work. For Ca 2RuO4, we used the structure at 180 K, with the space group Pbca, lattice constant a=5.394, b=5.600, and c=11.765 ˚A.5For Ca 1.8Sr0.2RuO4, we used the experimental struc- ture at 10 K, but the substitution of Sr for Ca is only taken into account via the structural changes. Ca1.8Sr0.2RuO4also has the space group Pbca but with lattice constant a=5.330,b=5.319, and c=12.409 ˚A.5For the AFM state, we considered the ’A-centered’ mode.3 The LDA+ Ucalculations were performed with U= 3.0 eV, which is similar to the value used by Fang et al.17,18 We will show that this U value can reproduce the mea- sured band gap in Ca 2RuO4. In Fig. 1, we present our theoretical band structures for paramagnetic Ca 2RuO4using different approxima- tions. The LDA band structure is well known20,21: the bands crossing the Fermi level are from Ru t2gorbitals, containing four delectrons. Our LDA band structure shown in Fig. 1a is consistent with the previous study.20 The inclusion of the SO coupling (Fig. 1b) only shows some slight changes on the band structure. This is not surprising since the SO coupling constant ζin Ca2RuO4 is presumably similar to the one in Sr 2RuO4, where it is only about 93 meV.15The inclusion of Coulomb inter- action (Fig. 1c) also shows little influence on the band structure since U does not break the orbital symmetry in theparamagneticstate. Surprisingly,thecombinedinter-2 -2-1 0 1Energy (eV)LDA (a) LDA+SO (b) -2-1 0 1 Γ X M ΓEnergy (eV)LDA+U (c) Γ X M ΓLDA+U+SO (d) FIG. 1: Theoretical band structures for paramagnetic Ca2RuO4using different approximations, (a) LDA, (b) LDA+SO, (c) LDA+ U, and (d) LDA+ U+SO. The LDA+ U and LDA+ U+SO band structures are calculated with U= 3.0 eV. action of the SO coupling and Coulomb repulsion gives a very different band structure compared to the LDA, LDA+SO or LDA+ Uresults. The LDA+ U+SO band structure shows an insulating phase with a gap about 0.2 eV wide. The band gap obtained from the chosen U is in good agreement with the experimental data.2,22 Similar combined effect of the SO coupling and U has been found in Sr 2RhO4,16where it wastermed Coulomb - enhanced spin -orbit splitting . Sr2RhO4has a simi- lar crystal structure to Ca 2−xSrxRuO4, and it can be regarded as a two-band ( xzandyz) system since the xyband is below the Fermi level due to the RhO 6 rotation.23,24The simpler problem of Sr 2RhO4can help us to understand the LDA+ U+SO band structure of Ca2RuO4. In Sr 2RhO4, the SO coupling splits the de- generate xzandyzbands to the higher χ±3/2bands, and lowerχ±1/2bands, where χ3/2= (xz+iyz)↑, χ−3/2= (xz−iyz)↓ χ1/2= (xz+iyz)↓, χ−1/2= (xz−iyz)↑. This splitting happens aroundthe Fermi level, and there- fore the occupancies of the χ±3/2andχ±1/2states are changed: ( n1/2+n−1/2)−(n3/2+n−3/2) =p >0, where n1/2=n−1/2andn3/2=n−3/2. When the Coulomb interaction is taken into account, the SO splitting is en- hanced due to the different occupancies of the χ±3/2 andχ±1/2states. The interplay the SO coupling and Coulomb interaction can be represented by an effective SO constant16 ζeff=ζ+1 2(U−J)p, (1) whereJis the Hund’s coupling and pis determined self- consistently. 0 0.2 0.4 0.6 0.8 1 1.2 1.4 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5PDOS (states/eV) Energy (eV)χ+−1/2 χ+−3/2 xy x2-y2 3z2-r2 FIG. 2: Ru- dPDOS for paramagnetic Ca 2RuO4calculated by LDA+ U+SO. The problem of Ca 2RuO4is more complicated than Sr2RhO4since the xyorbital is also involved. Fig. 2 presents the partial density of states (PDOS) for the Ru- dorbitals calculated by LDA+ U+SO. Here we present the PDOS for the χ±3/2andχ±1/2orbitals instead of xzandyz. The PDOS shows that the unoccupied t2g bands (0.2-0.9 eV) are dominated by the χ±3/2states while the χ±1/2states are nearly fully occupied. The well separated χ±3/2andχ±1/2states indicate a large effective spin-orbit splitting in Ca 2RuO4. Therefore, A simple explanation for the PM Mott transition is that the Coulomb-enhanced spin-orbit splitting opens a gap between the χ±3/2andχ±1/2bands, leading to an insu- latingphasewithtwoholesresidingonthe χ±3/2orbitals. Inthisexplanation,the xystateisassumedtobefullyoc- cupied. However, in the experimental structure, the xy, xzandyzorbitals hybridize with each other due to the structural distortion. As may be seen, the weight of the xystateintheunoccupied t2gbandsisnotsmallasshown in Fig. 2. The relative hole population shown in Fig. 2 isxy:χ±1 2:χ±3 2=19:13:68, while this ratio is 21:39.5:39.5 within LDA approximation. This shows that the inclu- sion of SO and U hardly changes the occupancy of the xy orbital. We may conclude that the band gap is mainly due to the splitting of χ±3/2andχ±1/2orbitals although thexyorbital is also involved in the Mott transition. Experimental research has found that the Mott tran- sition in CSRO is accompanied by an structural phase transition from the high temperature L-Pbcaphase to the low temperature S-Pbcaphase,5,11where L (S) in- dicates a long (short) c-axis. The phase transition tem- perature TSis a function of Sr concentration x, which decreases from 357 K at x= 0 to 0 K at x∼0.25,11. Forx≥0.2, CSRO is metallic and only has the L-Pbca phase. AsindicatedbyFriedt etal.,5, thestructuraltran- sition from the L-PbcatoS-Pbcaphase is characterized by an increase in the tilting angle of RuO 6octahedra. We will show that the tilting angle of RuO 6plays an im- portant role in the Mott transition. To illuminate the3 0 0.5 1PDOS (states/eV) (a) Ca1.8Sr0.2RuO4 χ+−1/2χ+−3/2 xy 0 0.5 1PDOS (states/eV) (c) 0 0.5 1 1.5 -1 -0.5 0 0.5PDOS (states/eV) Energy (eV)(e) LDA(b) Ca2RuO4LDA+SO(d) -1 -0.5 0 0.5 1LDA+U+SO Energy (eV)(f) FIG. 3: Ru- t2gPDOS for paramagnetic state using different approximations. The left panels are for Ca 1.8Sr0.2RuO4, and right panels for Ca 2RuO4. relation between the Mott transition and the structural phase transition, we apply the LDA+ U+SO calculation to Ca1.8Sr0.2RuO4. Fig. 3 presents our calculated PDOS for x= 0.2 (L- Pbca)andx= 0(S-Pbca). Fig. 3aand3bshowtheLDA PDOS for x= 0.2 andx= 0. The xz/yzbandwidth is about 1.6 eV in Ca 1.8Sr0.2RuO4, and it is reduced to 1.1 eV in Ca 2RuO4. The narrower xz/yzband in Ca 2RuO4 isduetoitslargertilting angle. Thetiltingangleisabout 12◦in Ca2RuO4, and it is 6◦in Ca1.8Sr0.2RuO4.5The tilting of the in-plane Ru-O can significantly reduces the interaction between Ru- dxz/yzand O-pz. Consequently, thexz/yzbandwidth decreases from x= 0.2 tox= 0, while the xybandwidth is less influenced. With the nar- rowerxz/yzband, Ca 2RuO4shows much higher xz/yz PDOS around the Fermi level than Ca 1.8Sr0.2RuO4. Fig. 3c and 3d show the LDA+SO PDOS for x= 0.2 and x= 0. As may be seen, the occupancy difference be- tween the χ±3/2andχ±1/2states is larger in Ca 2RuO4 than in Ca 1.8Sr0.2RuO4. This is understandable if we consider the higher PDOS in Ca 2RuO4. Eq. (1) shows theeffectiveSOsplittingisproportionaltotheoccupancy difference p. Then the lager occupancy difference pin Ca2RuO4will cause larger SO splitting when Coulomb interaction is taken into account. This is confirmed by the LDA+ U+SO PDOS shown in Fig. 3e and 3f. Using Eq. (1), we get ζeff=0.9 eV for Ca 2RuO4, and 0.6 eV for Ca 1.8Sr0.2RuO4. The larger SO splitting in Ca 2RuO4-1.5-1-0.5 0 0.5 1 1.5 -1.5 -1-0.5 0 0.5 1PDOS (states/eV) Energy (eV)(a) χ+−1/2 χ+−3/2 xy -1.5 -1-0.5 0 0.5 1 Energy (eV)(b) FIG. 4: Ru- t2gPDOS for AFM Ca 2RuO4using different ap- proximations, (a) LDA+ U, and (b) LDA+ U+SO. leads an insulating phase, while Ca 1.8Sr0.2RuO4remains metallic. Therefore, we have shown that the PM Mott transition in CSRO can be explained by the interplay of SO coupling, Coulomb interaction and structural distor- tion. CSRO is a single-layer system, where the t2gbands are split into the singly degenerate xyband and doubly degenerate xz/yzbands. The bare SO coupling mainly influences the degenerate bands since Ru has a moderate SO constant. Therefore, the SO induced Mott transition is strongly orbital dependent. Fig. 3 indicates that the Mott transition in PM CSRO is driven by the narrowing of thexzandyzbands, while the xyorbital plays a lessimportantrole. Thisstrongorbital-dependencecould also be seen from Eq. (1). The bare SO parameter ζ is constant for each orbital, but the Coulomb enhanced SO parameter ζeffis a function of orbital occupancies. This suggests that the strong orbital-dependence is an inherent feature of the SO induced Mott transition. Our calculation has shown that the PM insulating phaseof CSROoriginatesin the strongeffective SO split- ting. This picture supports the photoemission measure- ment by Mizokawa et al.14The suppression of TMIin Ca2Ru1−yCryO412can also be understood within this picture. Cr has a much smaller atomic SO constant than Ru due to its smaller mass. And therefore the substitu- tion of Cr for Ru will reduce the SO splitting, leading to the observed decrease of TMI. As mentioned above, Fang et al.17,18found that SO coupling has no much influence on the electronic con- figuration. They however pointed out that the pho- toemission measurement was done above the N´ eel tem- perature, while they applied the LDA+ Umethod to the low temperature AFM state. To clarify if the SO coupling is less important in AFM state, we ap- ply the LDA+ Uand LDA+ U+SO calculation to AFM Ca2RuO4. Our LDA+ Ucalculation gives a magnetic moment of mRu=1.25µB, which is consistent with Fang et al.’s calculation17; while SO reduces the moment to4 1.21µB, showing a weak SO effect. The AFM PDOS for Ca 2RuO4are presented in Fig. 4. In contrast to the PM state, Fig. 4 shows that there is no Coulomb- enhanced SO splitting in the AFM state. The relative weak SO splitting in the AFM state can be explained by Eq. (1). The LDA+ Ucalculation produces an insulat- ing phase for AFM Ca 2RuO4as shown in Fig. 4a. Since there is no density of states around the Fermi level, SO coupling can not change the orbital occupancies, which givesp= 0. Then we get ζeff=ζ, showing no enhanced SO splitting. It is noticeable that our calculations give an AFM ground state for Ca 2RuO4, which is consistent with the experimental phase diagram.5 Comparing the PM and AFM insulating phases in CSRO, we may find that the two kinds of Mott transi- tion are similar. They both have an interaction to break the orbital symmetry. The interaction is SO coupling in the PM state and spin polarization in the AFM state. The breaking of orbital symmetry lifts the degenerate bands and changes the orbital occupancies. When the Coulomb interaction is taken into account, the orbital splitting, which is SO splitting in the PM state or ex- change splitting in the AFM state, is enhanced. If the enhanced splitting is large enough, it will lead to an in-sulating phase. The PM-AFM transition at TNcan be regarded as the competition of the Coulomb-enhanced SO splitting and the Coulomb-enhanced exchange split- ting. In the AFM state, the SOenhancement isquenched bythelargeexchangesplitting, whichcausesthe verydif- ferent electronic configuration from the PM state. In summary, we have applied LDA+ U+SO calcula- tions to CSRO. We find the Coulomb enhanced SO splitting produces an insulating phase in PM Ca 2RuO4. This finding is consistent with the photoemission ex- periment, and also explains the recent experiment on Ca2Ru1−yCryO4. We show that the SO induced Mott transition in CSRO is driven by the change of the xz/yz bandwidth. For x= 0.2, the compound is found to be metallic. On the other hand, we find that SO cou- pling has much less influence on the AFM state, which is in agreement with the previous LDA+ Ustudy. The above picture shows that SO coupling plays a very sub- tle role in the correlated systems. The interplay of SO coupling, electron correlation and crystal structure dis- tortion would cause very rich physical phenomena. The author gratefully acknowledges Ove Jepsen for helpful discussions and useful comments. 1Y. Maeno, H. Hashimoto, K. Yoshida, S. Nishizaki, T. Fu- jita, J. G. Bednorz, and F. Lichtenberg, Nature (London) 372, 532 (1994). 2G. Cao, S. McCall, M. Shepard, J. E. Crow, and R. P. Guertin, Phys. Rev. B 56, R2916 (1997). 3M. Braden, G. Andre, S. Nakatsuji, and Y. Maeno, Phys. Rev. B58, 847 (1998). 4S. Nakatsuji and Y. Maeno, Phys. Rev. Lett. 84, 2666 (2000). 5O. Friedt, M. Braden, G. Andre, P. Adelmann, S. Nakat- suji, and Y. Maeno, Phys. Rev. B 63, 174432 (2001). 6O. Friedt, P. Steffens, M. Braden, Y. Sidis, S. Nakatsuji, and Y. Maeno, Phys. Rev. Lett. 93, 147404 (2004). 7M. Kriener, P. Steffens, J. Baier, O. Schumann, T. Zabel, T. Lorenz, O. Friedt, R. Muller, A. Gukasov, P. G. Radaelli, P. Reutler, A. Revcolevschi, S. Nakatsuji, Y. Maeno, and M. Braden, Phys. Rev. Lett. 95, 267403 (2005). 8E. Gorelov, M. Karolak, T. O. Wehling, F. Lechermann, A. I. Lichtenstein, and E. Pavarini, Phys. 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2308.12335v1.Spin_pumping_from_a_ferromagnetic_insulator_into_an_altermagnet.pdf

Spin pumping from a ferromagnetic insulator into an altermagnet Chi Sun1and Jacob Linder1 1Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway A class of antiferromagnets with spin-polarized electron bands, yet zero net magnetization, called altermagnets is attracting increasing attention due to their potential use in spintronics. Here, we study spin injection into an altermagnet via spin pumping from a ferromagnetic insulator. We find that the spin pumping behaves qualitatively different depending on how the altermagnet is crystallographically oriented relative the interface to the ferromagnetic insulator. The altermagnetic state can enhance or suppress spin pumping, which we explain in terms of spin-split altermagnetic band structure and the spin-flip probability for the incident modes. Including the effect of interfacial Rashba spin-orbit coupling, we find that the spin-pumping effect is in general magnified, but that it can display a non-monotonic behavior as a function of the spin-orbit coupling strength. We show that there exists an optimal value of the spin-orbit coupling strength which causes an order of magnitude increase in the pumped spin current, even for the crystallographic orientation of the altermagnet which suppresses the spin pumping. Introduction. – Spin pumping is a mechanism for generating spin currents in which the precessing magnetization in a mag- netic material transfers angular momentum into its adjacent nonmagnetic layers [ 1–5]. Compared with metals, magnetic insulators can function as efficient spin-current sources with low dissipation and reduced energy loss [ 4], in which the ferro- magnetic insulator (FI) YIG demonstrates the lowest known spin dissipation with an exceptionally low Gilbert damping [6,7]. In conventional FI/normal metal (NM) heterostructures, the injected spin current affects the magnetization dynamics in the FI and creates a spin accumulation in the NM, resulting in a measurable damping increase in the linewidth of a ferro- magnetic resonance (FMR) signal, which has been extensively investigated [ 3,8–10]. When the NM is replaced by another material such as a superconductor, the spin pumping effect is considerably modulated by various superconducting gap properties and interfacial effects [11–18]. Recently, a new magnetic phase dubbed altermagnetism [ 19– 22] has attracted increasing attention. Such materials exhibit a large momentum-dependent spin-splitting and vanishing net macroscopic magnetization at the same time, thus combining features from conventional ferromagnets and antiferromagnets [23–26]. The spin splitting in the altermagnet (AM), which is of a strong non relativistic origin, is protected by the broken sym- metries of the spin arrangements on the crystal, distinct from ferromagnetic and relativistically spin-orbit coupled (SOC) systems [ 23,24,27]. It is predicted that AM can span a large range of materials, from insulators like FeF 2and MnF 2, semi- conductors like MnTe, metals like RuO 2, to superconductors like La 2CuO 4[23,28–30]. These novel properties make AM a fascinating material platform to investigate superconducting [25, 31–35] and spintronics phenomena [36–40]. In this work, we theoretically determine spin pumping from a FI into a metallic AM in a FI/AM bilayer (see Fig. 1). To cover different crystallographic orientations of the interface relative to the spin-polarized lobes of the altermagnetic Fermi surface, two representative metallic AMs, as shown in Figs. 1(a) and 1(b), are studied in detail. In addition to the non relativistic interfacial effect induced by the AM, a relativistic Rashba SOC is included at the FI/AM interface in our model. We find that the spin pumping current can be enhanced or suppressed by FIG. 1. (Color online) Spin pumping is considered in a bilayer consisting of a ferromagnetic insulator (FI) and an altermagnet (AM). The magnetization 𝑴(𝑡)in the FI is precessing around the 𝑧axis at the FMR frenquency Ω. Different interface orientations are also considered, effectively rotating the spin-resolved Fermi surface in the AM for𝑒↑(red ellipse) and 𝑒↓(blue ellipse) spin carriers. For notation simplicity, the two AM orientations are referred as AM1 and AM2, respectively. altermagnetism, depending on the interface orientation, thus offering versatility. This is explained in terms of the spin-split altermagnetic band structure and the spin-flip probability for the incident modes toward the interface. In addition, the spin pumping current shows a non-monotonic behavior as a function of the interfacial SOC strength. We show that the interfacial SOC can, in a certain range, increase the spin pumping current in a FI/AM bilayer by more than an order of magnitude. Theory. – The effective low-energy Hamiltonian for the AM shown in Fig. 1(a), using an electron field operator basis 𝜓=[𝜓↑,𝜓↓]𝑇, is given by 𝐻AM=−ℏ2▽2 2𝑚𝑒−𝜇+𝛼𝜎𝑧𝑘𝑥𝑘𝑦, (1) in which𝛼is the parameter characterizes the altermag- netism strength, 𝜎𝑧denotes the Pauli matrix, 𝑚𝑒is the electron mass and 𝜇is the chemical potential. By solving the stationary Schr ¨odinger equation as an eigen- value problem (see SM for details), the 𝑥-components of the wave vectors in the AM with energy 𝐸are givenarXiv:2308.12335v1 [cond-mat.mes-hall] 23 Aug 20232 by𝑘𝑒↑(↓),±=±ℏ−1√︃ 2𝑚𝑒(𝜇+𝐸)−ℏ2𝑘2𝑦+𝛼2𝑚2𝑒𝑘2𝑦/ℏ2∓′ 𝛼𝑚𝑒𝑘𝑦/ℏ2, in which the±sign denotes the propagation di- rection along the±𝑥,𝑒↑(↓) describes electron with spin up (down), and∓′=−(+) for↑(↓) . Here we assume translational invariance in the 𝑦-direction with belonging momentum 𝑘𝑦of the incident particle. On the other hand, the Hamiltonian for the FI has the form 𝐻FI=−ℏ2▽2 2𝑚𝑒+𝑈+𝐽ˆ𝝈·𝑴(𝑡), (2) in which ˆ𝝈denotes the Pauli matrix vector and 𝐽is the ex- change interaction. Here the potential 𝑈is larger than 𝜇 in the nearby AM to ensure the ferromagnet to be insulat- ing. The normalized magnetization is defined as 𝑴(𝑡)= (𝑚cosΩ𝑡,𝑚sinΩ𝑡,√ 1−𝑚2), where𝑚∈[0,1]is the mag- netization oscillation amplitude and Ωdenotes the FMR fre- quency for spin pumping. By employing a wavefunction with the structure(𝑒−𝑖Ω𝑡 2,𝑒𝑖Ω𝑡 2)𝑇for its additional time-dependence, the non-stationary Schr ¨odinger equation can be solved as an eigenvalue problem (see SM for details). The two eigen- pairs are obtained as: 𝐸1=𝐸+with(𝑎+,𝑏+)𝑇and𝐸2=𝐸− with(𝑎−,𝑏−)𝑇, in which𝐸±=𝑈+ℏ2(𝑘2 𝑥+𝑘2 𝑦) 2𝑚𝑒±𝐽𝑅with 𝑅=(1−2𝛽√ 1−𝑚2+𝛽2)1/2and𝛽=ℏΩ/2𝐽. To study the spin pumping effect, we first consider an 𝑒↑ incident electron with excitation energy 𝐸from the AM side based on the FI/AM bilayer. The wavefunctions are given by ΨAM,𝑒↑= 1 0 𝑒𝑖𝑘𝑒↑,−𝑥+𝑟 1 0 𝑒𝑖𝑘𝑒↑,+𝑥 𝑒−𝑖𝐸𝑡 ℏ +𝑟′ 0 1 𝑒𝑖𝑘′ 𝑒↓,+𝑥𝑒−𝑖𝐸′𝑡 ℏ, (3) ΨFI,𝑒↑=𝑡 𝑎+𝑒−𝑖Ω𝑡 2 𝑏+𝑒𝑖Ω𝑡 2! 𝑒−𝑖𝑘F1,𝑒↑𝑥𝑒−𝑖𝐸1𝑡 ℏ +𝑝 𝑎−𝑒−𝑖Ω𝑡 2 𝑏−𝑒𝑖Ω𝑡 2! 𝑒−𝑖𝑘F2,𝑒↑𝑥𝑒−𝑖𝐸2𝑡 ℏ, (4) in which𝑟and𝑟′are coefficients describing reflection with- out and with spin-flip in the AM, respectively, and 𝑡and 𝑝are transmission coefficients in the FI. To differentiate it from the incident energy 𝐸, the energy after the spin- flip in the AM due to spin pumping is denoted as 𝐸′. By matching the time-dependence of the wavefunction compo- nents on the AM and FI sides, we obtain 𝐸′=𝐸−ℏΩ and𝐸1=𝐸2=𝐸−ℏΩ 2. In terms of 𝐸, the correspond- ing𝑥-component of the two wave vectors in the FI are ex- pressed as𝑘F1,𝑒↑=ℏ−1√︃ 2𝑚𝑒[𝐸−𝑈−𝐽(𝑅+𝛽)]−ℏ2𝑘2𝑦and 𝑘F2,𝑒↑=ℏ−1√︃ 2𝑚𝑒[𝐸−𝑈+𝐽(𝑅−𝛽)]−ℏ2𝑘2𝑦. Note that the wave numbers in the FI possess imaginary values due to a large potential 𝑈, ensuring evanescent electron states in the FI. Details of the wave functions induced by an 𝑒↓incident partice with excitation energy 𝐸from the AM can be found in the SM, in which we have 𝐸′=𝐸+ℏΩ.Appropriate boundary conditions are required to solve the reflection and transmissions coefficients in the wavefunctions. Here we consider a Rashba spin-orbit coupled interface with the Hamiltonian 𝐻𝐼=[𝑈0+𝑈SO 𝑘𝐹ˆ𝒙·(ˆ𝝈×𝒌)]𝛿(𝑥)=[𝑈0−𝑈SO 𝑘𝐹𝑘𝑦𝜎𝑧]𝛿(𝑥),(5) in which𝑈0is the interfacial energy barrier, 𝑈SOdescribes the Rashba SOC, 𝑘𝐹=√︁ 2𝑚𝑒𝜇/ℏis the Fermi wave vector and ˆ𝒙 denotes the interface normal. On the other hand, to derive the boundary condition, antisymmetrization of the altermagnetic term𝛼𝑘𝑥𝑘𝑦𝜎𝑧→𝛼𝑘𝑦 2{𝑘𝑥,Θ(𝑥)}𝜎𝑧is necessary to ensure hermiticity of the Hamilton-operator, where Θ(𝑥)is the step function and 𝑘𝑥=−i𝜕𝑥. Combing all related Hamiltonian contributions in the FI/AM system, we obtain ΨAM,𝑒↑ 𝑥=0= ΨFI,𝑒↑ 𝑥=0=(𝑓,𝑔)𝑇and 𝜕𝑥ΨAM,𝑒↑ 𝑥=0−𝜕𝑥ΨFI,𝑒↑ 𝑥=0= 𝑘𝛼,+1𝑓 𝑘𝛼,−1𝑔 , (6) where𝑘𝛼,𝜎=2𝑚𝑒 ℏ2[𝑈0−(i𝛼 2+𝑈SO 𝑘𝐹)𝑘𝑦𝜎]with𝜎=+1(−1). Here the imaginary number iappears in𝑘𝛼,𝜎since we consider 𝑘𝑦invariance (unlike 𝑘𝑥=−i𝜕𝑥). Note that the boundary conditions for 𝑒↓incident from the AM side have the same forms as𝑒↑with different explicit expressions of 𝑓and𝑔in the wave functions. The longitudinal quantum mechanical spin current polarized along the𝑧axis in the AM is given by 𝑗𝑠𝑧,𝑒↑(↓)=ℏ2 2𝑚𝑒(ℑm{𝑓∗∇𝑓}−ℑm{𝑔∗∇𝑔})+𝛼𝑘𝑦 2(|𝑓|2+|𝑔|2). (7) Integrating over all energies and all possible transverse modes via∫ 𝑑𝑘𝑥=∫ 𝑑𝐸(𝑑𝑘𝑥/𝑑𝐸)and∫ 𝑑𝑘𝑦, the spin pumping current is calculated as 𝐼𝑠,𝑒↑(↓)=∫ 𝑑𝑘𝑦∫ 𝑑𝐸𝑑𝑘𝑥 𝑑𝐸𝑗𝑠𝑧,𝑒↑(↓)𝑓0(𝐸), (8) in which𝑓0(𝐸)denotes the Fermi-Dirac distribution. Note that𝑑𝑘𝑥/𝑑𝐸plays the role of 1D DOS in the AM instead of 2D DOS since here∫ 𝑑𝑘𝑦is included separately. Including contributions from both 𝑒↑and𝑒↓incidents, the total spin pumping current is 𝐼𝑠=𝐼𝑠,𝑒↑+𝐼𝑠,𝑒↓. In general, a backflow spin current exists due to a spin accumulation that is built up in the material connected to the precessessing FI [ 1], which diminishes the magnitude of the total spin current flowing across the interface. The backflow spin current can safely be neglected in the present case of a ballistic large AM reservoir. To show how the crystallographic orientation of the interface between the materials affects the spin pumping, the AM corresponding to a45degree rotation of the interface, as shown in Fig. 1(b), is modeled by replacing 𝛼𝑘𝑥𝑘𝑦→𝛼(𝑘2 𝑥−𝑘2 𝑦)/2in𝐻AM. This leads to different expressions for the wavevectors, boundary conditions and quantum mechanical spin pumping current (see SM for details). Our model can also be expanded to a AM with arbitrary rotation by combination of the established 0 and 45 degree cases, i.e., using 𝛼1𝑘𝑥𝑘𝑦𝜎𝑧+𝛼2(𝑘2 𝑥−𝑘2 𝑦)𝜎𝑧/2in𝐻AM with the arbitrary angle determined by 𝜃𝛼=1 2arctan(𝛼1/𝛼2).3 Results: Altermagnetism dependence. – For notation sim- plicity, we refer the altermagnetic Fermi surface structures shown in Figs. 1(a) and 1(b) as AM1 and AM2, respectively, corresponding to different interface orientations by effectively rotating 45 degree of the spin-resolved Fermi-surfaces. To ensure each spin-polarized lobe of the altermagnetic Fermi surface described by 𝐻AMdefines a closed integral path or ellipse rather than a hyperbola, 𝛼 <ℏ2/𝑚𝑒≡𝛼𝑐should be satisfied (see SM for details). The semi-major (minor) axis 𝑎 (𝑏) of the ellipse can be obtained as 𝑎=√︄ 2𝑚𝑒(𝜇+𝐸) ℏ2−𝑚𝑒𝛼, 𝑏=√︄ 2𝑚𝑒(𝜇+𝐸) ℏ2+𝑚𝑒𝛼, (9) based on which 𝑎(𝑏) increases (decreases) with 𝛼. In the absence of Rashba SOC, the dimensionless param- eter𝑍=𝑚𝑒𝑈0 ℏ2𝑘𝐹characterizes the quality of electric contact between the FI and AM. To model high-transparent to tun- neling interfaces, we investigate the spin pumping current 𝐼𝑠 with𝑍=0,1,3in Fig. 2. As is reasonable, 𝐼𝑠decreases as 𝑍 increases. More importantly, we find that 𝐼𝑠increases with 𝛼in FI/AM1 [Fig. 2(a)] while it decreases with 𝛼in FI/AM2 [Fig. 2(b)], indicating the crucial role of the interface orientation in FI/AM for spin pumping. To understand the altermagnetism dependence behavior, it is instructional to consider the altermagnetic Fermi sur- faces and energy bands. For simplicity, let us focus on par- ticles close to normal incidence, 𝑘𝑦→0, which contribute the most to the transport across the junction. In AM1, the wavevectors of the 𝑒↑and𝑒↓incident particles are the same, i.e.,𝑘𝑒↑(↓),±=±ℏ−1√︁ 2𝑚𝑒(𝜇+𝐸), just like the NM case. This analogy also applies when integrating over all possible𝑘𝑦values, i.e., the total spin polarization of the in- cident particles cancels since spin- ↓is the majority carrier for𝑘𝑦>0and spin-↑is the majority carrier for 𝑘𝑦<0and the two spin bands contribute equally. On the other hand, in AM2, the wavevectors can be strongly mismatched even for𝑘𝑦→0, i.e.,𝑘𝑒↑,±=±ℏ−1√︁ 2𝑚𝑒(𝜇+𝐸)/(ℏ2+𝑚𝑒𝛼)and 𝑘𝑒↓,±=±ℏ−1√︁ 2𝑚𝑒(𝜇+𝐸)/(ℏ2−𝑚𝑒𝛼). This is similar to the ferromagnetic metal (FM) case, in which a large mis- match between these wavevectors is induced by a (momentum- independent) spin-splitting or exchange energy 𝐽exby consider- ing the Hamiltonian 𝐻FM=−ℏ2▽2 2𝑚𝑒−𝜇+𝐽ex𝜎𝑧. Therefore, it is useful to compare the spin pumping current based on FI/NM and FI/FM, as shown in Figs. 2(c) and 2(d), respectively. The total spin current is determined by the spin-flip proba- bility between 𝑒↑and𝑒↓states induced by spin pumping, and also the number of available 𝑘𝑦modes for spin-flip. Let us first consider the altermagnetism dependence of the number of 𝑘𝑦 modes. As discussed before, 𝑎(𝑏) increases (decreases) with 𝛼. In AM1, the allowed number of 𝑘𝑦mode or|𝑘𝑦|maximum for both𝑒↑and𝑒↓bands increases with 𝛼as the semi-major axis 𝑎increases, giving rise to more available transverse 𝑘𝑦modes in which the the spin-flip between 𝑒↑and𝑒↓can be realized. Note that the asymmetry between incident spin 𝑒↑and𝑒↓ is broken by the spin pumping FMR frequecy Ω. Therefore, the total spin current 𝐼𝑠, which includes contributions from 00.20.40.60.81 /c10-210-1100IS/IS0FI/AM1 (a) Z=0 1 3 00.20.40.60.81 /c10-410-310-210-1100IS/IS0FI/AM2 (b) 00.20.40.60.81 /010-310-210-1100IS/IS0FI/NM (c) 00.20.40.60.81 Jex/010-510-410-310-210-1100IS/IS0FI/FM (d)FIG. 2. (Color online) Normalized spin pumping current 𝐼𝑠/𝐼𝑠0as a function of altermganetism for FI/AM1 and FI/AM2 in (a) and (b), respectively. (c) 𝐼𝑠/𝐼𝑠0as a function of chemical potential 𝜇for FI/NM. (d)𝐼𝑠/𝐼𝑠0as a function of exchange energy 𝐽exfor FI/FM. In the absence of Rashba SOC, different interfacial barriers 𝑍=0,1,3 are considered. Here 𝑚=0.2andℏΩ= 0.5meV are utilized. 𝐼𝑠0 corresponds to the spin pumping current for FI/NM with 𝜇/𝜇0=1. both𝑒↑and𝑒↓incidents, is enhanced when integrating over 𝑘𝑦. This is consistent with the trends shown in Fig. 2(a). Similarly, the allowed 𝑘𝑦range for spin-flip can be increased by increasing 𝜇in the NM, giving rise to an enhanced 𝐼𝑠with a high-transparent 𝑍=0interface [see blue curve in Fig. 2(c)]. However, it can be seen that the trends change for large 𝑍, indicating a difference between increasing 𝛼and𝜇, although in both cases the number of 𝑘𝑦states that carry spin current increases. This can be explained by considering the spin-flip probability for each 𝑘𝑦mode, which we will get back to. On the other hand, in AM2, the allowed 𝑘𝑦modes increase with increasing 𝛼and semi-major axis 𝑎for the𝑒↑band while they decrease with increasing 𝛼and decreasing semi-minor axis𝑏for the𝑒↓band. This results in an enhanced mismatch between the spin-bands at a given value of 𝑘𝑦, and therefore less transverse modes available to realize spin-flip between the two bands. This corresponds to the trend that 𝐼𝑠is suppressed with𝛼, as shown in Fig. 2(b). The same mechanism applies for FM in Fig. 2(d), in which the mismatch between available 𝑘𝑦modes for𝑒↑and𝑒↓bands is enhanced with increasing 𝐽ex, confirming the similarity between AM2 and FM. Next, we turn to the spin-flip probability at a fixed 𝑘𝑦, in particular small|𝑘𝑦|close to normal incidence which contribute the most. As calculated in detail in the SM (see Fig. 4), it is found that the spin-flip probability increases (decreases) with altermagnetism for FI/AM1(AM2) , which corresponds to the trends shown in Fig. 2. The spin-flip probability behavior can be understood by considering the magnitude of momentum transfer (along 𝑥), e.g., when a (spin-flip) reflection requires a large momentum transfer, its probability is diminished [ 41,42]. In AM1 (AM2), the magnitude of the momentum transfer [e.g., between𝑘𝑒↑,−and𝑘′ 𝑒↓,+in Eq. (3)] at fixed 𝑘𝑦decreases (increases) with altermagnetism. Similarly, in FI/NM, the4 0 1 2 3 ZSOC10-210-1100101IS/IS0FI/NM (c)0 1 2 3 ZSOC10-210-1100101IS/IS0FI/AM1 (a) 0 1 2 3 ZSOC10-210-1100101102IS/IS0FI/AM2 (b) 0 1 2 3 ZSOC10-210-1100101IS/IS0FI/FM (d) Z=0 1 3 FIG. 3. (Color online) Normalized spin pumping current 𝐼𝑠/𝐼𝑠0 as a function of Rashba 𝑍SOC for FI/AM1 and FI/AM2 in (a) and (b), respectively, in which 𝛼/𝛼𝑐=0.6. (c)𝐼𝑠/𝐼𝑠0as a function of 𝑍SOC for FI/NM. (d) 𝐼𝑠/𝐼𝑠0as a function of 𝑍SOC for FI/FM with 𝐽ex/𝜇0=0.6. Different interfacial barriers 𝑍=0,1,3are considered. Here𝑚=0.2andℏΩ= 0.5meV are utilized. 𝐼𝑠0corresponds to the spin pumping current for FI/NM with 𝜇/𝜇0=1in the absence of Rashba SOC, the same as 𝐼𝑠0used in Fig. 2. magnitude of momentum transfer for spin-flip increases as 𝜇, which suppresses the spin-flip probability. This compensates the fact that more 𝑘𝑦modes are available when 𝜇increases, as discussed before, giving a total suppression of spin current for large𝑍in Fig. 2(c). Results: Spin-orbit dependence. – Similar to the barrier 𝑍=𝑚𝑒𝑈0 ℏ2𝑘𝐹, the interfacial Rashba SOC can be characterized by introducing the dimensionless parameter 𝑍SOC=𝑚𝑒𝑈SO ℏ2𝑘𝐹, based on which 𝑘𝛼,𝜎in Eq. (6) can be written as 𝑘𝛼,𝜎= 2𝑍𝑘𝐹−2𝑍SOC𝑘𝑦𝜎−i𝛼𝑚𝑒𝑘𝑦 ℏ2𝜎with𝜎=+1(−1). In Fig. 3, the spin pumping current is plotted as a function of 𝑍SOCfor different bilayers with gradually increasing interface barrier 𝑍=0,1,3. A non-monotonic behavior with a maximum whose position can be shifted with 𝑍is achieved in all setups. This is related to the effective spin-dependent barrier induced bySOC in the form of 𝑘𝑦𝜎in𝑘𝛼,𝜎. When𝑍SOCis present and 𝑍is fixed, there exists an optimal value of 𝑍SOCwhere the barrier is strongly reduced for many angles of incidence (i.e., 𝑘𝑦modes) of a given spin type due to the 𝑘𝑦𝜎dependence in the boundary condition, resulting in enhanced spin-flip and spin current. When 𝑍SOCcontinues to increase, the total barrier then increases again which causes less spin-flip and reduces the spin current. Note that the Fermi-level mismatch between the two layers also results in normal reflection and acts as an effective barrier even when 𝑍=0[43], which can thus be compensated by 𝑍SOCto achieve the optimal spin current via the argument above. In the absence of 𝑍SOC, it is shown in Fig. 2 that FI/AM1 produces a larger spin pumping current compared with FI/AM2, indicating that AM1 is the spin pumping-enhanced-orientation. However, this changes when 𝑍SOCis present. FI/AM2 with the spin pumping-suppressed-orientation can in that case generate a much larger spin current compared with FI/AM1 when 𝑍SOC is tuned to its optimal value, as shown in Fig. 3(b). Similar behavior can be observed in FI/FM [Fig. 3(d)] but with a smaller spin pumping current maximum compared with FI/AM2. The suppression of spin current due to interfacial Rashba interaction via spin memory loss and spin current absorption has been studied previously [27] within a perturbative framework. Concluding remarks. – We investigate spin pumping from a FI to an AM by considering two representative AMs with 0 and 45-degree rotation relative to the interface. We find the spin pumping current can be both enhanced and suppressed by altermagnetism depending on the interface orientation. 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(A2) Applying𝐸1=𝐸2=𝐸, the𝑥-components of the wave vectors in the AM are given by 𝑘𝑒↑,±=±1 ℏ√︄ 2𝑚𝑒(𝜇+𝐸)−ℏ2𝑘2𝑦+𝛼2𝑚2𝑒𝑘2𝑦 ℏ2−𝛼𝑚𝑒𝑘𝑦 ℏ2, (A3) 𝑘𝑒↓,±=±1 ℏ√︄ 2𝑚𝑒(𝜇+𝐸)−ℏ2𝑘2𝑦+𝛼2𝑚2𝑒𝑘2𝑦 ℏ2+𝛼𝑚𝑒𝑘𝑦 ℏ2, (A4)6 in which the±sign in the subscript denotes the propagation direction along the ±𝑥. Here we assume translational invariance in the 𝑦-direction with belonging conserved momentum 𝑘𝑦. The momentum 𝑘𝑦of the incident particle appearing in Eqs. (A3,A4) is determined by the Fermi surface of the incident particle, which is described as follows. Consider an 𝑒↑particle in the AM. We then have 𝐸=𝐸+=ℏ2(𝑘2 𝑥+𝑘2 𝑦) 2𝑚𝑒−𝜇+𝛼𝑘𝑥𝑘𝑦in Eq. (A2), which defines an elliptical Fermi surface in the 𝒌-space when 𝛼<ℏ2/𝑚𝑒≡𝛼𝑐. On the other hand, Eq. (A2) corresponds to a hyperbola when 𝛼>𝛼𝑐, which can not define a closed integral path. Therefore, we confine 𝛼<𝛼𝑐in this work. The general equation of the ellipse is given by ℏ2𝑘2 𝑥 2𝑚𝑒+𝛼𝑘𝑥𝑘𝑦+ℏ2𝑘2 𝑦 2𝑚𝑒−(𝜇+𝐸)=0, (A5) from which the semi-major (minor) axis can be obtained as 𝑎1=√︄ 2𝑚𝑒(𝜇+𝐸) ℏ2−𝑚𝑒𝛼, 𝑏 1=√︄ 2𝑚𝑒(𝜇+𝐸) ℏ2+𝑚𝑒𝛼. (A6) Consequently, the wave vectors on the Fermi surface of 𝑒↑in the AM are described by 𝑘𝑦,𝑒↑=𝑟1sin𝜃, 𝑘𝑥,𝑒↑=𝑟1cos𝜃, 𝑟 1=𝑎1𝑏1√︃ 𝑏2 1cos2(𝜃+𝜋/4)+𝑎2 1sin2(𝜃+𝜋/4), (A7) in which𝜃is the incident angle in the AM with respect to the 𝑥-axis. Similarly, we can obtain the wave vectors on the Fermi surface of 𝑒↓particle in the AM, i.e., 𝑘𝑦,𝑒↓=𝑟2sin𝜃, 𝑘𝑥,𝑒↓=𝑟2cos𝜃, 𝑟2=𝑎2𝑏2√︃ 𝑏2 2cos2(𝜃−𝜋/4)+𝑎2 2sin2(𝜃−𝜋/4), 𝑎 2=√︄ 2𝑚𝑒(𝜇+𝐸) ℏ2−𝑚𝑒𝛼, 𝑏 2=√︄ 2𝑚𝑒(𝜇+𝐸) ℏ2+𝑚𝑒𝛼. (A8) Consider the 𝑒↑incident from the AM side based on the FI/AM bilayer, we have ΨAM,𝑒↑= 1 0 𝑒𝑖𝑘𝑒↑,−𝑥+𝑟 1 0 𝑒𝑖𝑘𝑒↑,+𝑥 𝑒−𝑖𝐸𝑡 ℏ+𝑟′ 0 1 𝑒𝑖𝑘′ 𝑒↓,+𝑥𝑒−𝑖𝐸′𝑡 ℏ, (A9) in which we use 𝑘𝑦=𝑘𝑦,𝑒↑given in Eq. (A7). 𝑟and𝑟′are coefficients describing reflection without and with spin-flip, respectively. These coefficients can be determined by applying appropriate boundary conditions, which we will get back to. To differentiate it from the incident energy 𝐸, the energy after the spin-flip is denoted as 𝐸′. Similarly,𝑘′ 𝑒↑,±and𝑘′ 𝑒↓,±have the same forms as shown in Eqs. (A3,A4) with respect to 𝐸′. Consider the 𝑒↓incident from the AM side based on the FI/AM bilayer, we have ΨAM,𝑒↓= 0 1 𝑒𝑖𝑘𝑒↓,−𝑥+𝑟 0 1 𝑒𝑖𝑘𝑒↓,+𝑥 𝑒−𝑖𝐸𝑡 ℏ+𝑟′ 1 0 𝑒𝑖𝑘′ 𝑒↑,+𝑥𝑒−𝑖𝐸′𝑡 ℏ, (A10) in which we use 𝑘𝑦=𝑘𝑦,𝑒↓given in Eq. (A8). Appendix B: Expressions in the FI In the FI, the Hamiltonian for electron-like quasiparticles has the form 𝐻FI=−ℏ2▽2 2𝑚𝑒+𝑈+𝐽ˆ𝝈·𝑴(𝑡), (B1) in which ˆ𝝈denotes the Pauli matrix vector. The potential 𝑈is larger than the chemical potential 𝜇in the nearby AM. 𝐽decribes the exchange interaction in the ferromagnet between the localized spin magnetization and the itinerant electrons. The normalized magnetization is defined as 𝑴(𝑡)=(𝑚cosΩ𝑡,𝑚sinΩ𝑡,√︁ 1−𝑚2), (B2)7 where𝑚∈[0,1]is the magnetization oscillation amplitude and Ωdenotes the FMR frequency for spin pumping. By employing a wavefunction with the structure 𝑒−𝑖𝐸𝑡 ℏ(𝑒−𝑖Ω𝑡 2,𝑒𝑖Ω𝑡 2)𝑇for its time-dependence, the non-stationary Schr ¨odinger equation can be solved as an eigenvalue problem. The two eigenpairs are obtained as: 𝐸1=𝐸+with(𝑎+,𝑏+)𝑇and𝐸2=𝐸−with(𝑎−,𝑏−)𝑇. In terms of the adiabaticity parameter 𝛽=ℏΩ 2𝐽, the eigenenergies are given by 𝐸±=𝑈+ℏ2(𝑘2 𝑥+𝑘2 𝑦) 2𝑚𝑒±𝐽√︃ 1−2𝛽√︁ 1−𝑚2+𝛽2. (B3) The corresponding eigenstates are described by the coefficients 𝑎±=𝜂±√︁ 𝜂2 ±+1, 𝑏±=1√︁ 𝜂2 ±+1, (B4) 𝜂±=√ 1−𝑚2−𝛽±√︃ 1−2𝛽√ 1−𝑚2+𝛽2 𝑚, (B5) which satisfy 𝑎−=−𝑏+, 𝑏−=𝑎+. (B6) Note when𝑚=0, we have𝑎+=1,𝑏+=0,𝑎−=0 and𝑏−=1. Based on the above, the total wavefunction in the FI is constructed as ΨFI=𝑡 𝑎+𝑒−𝑖Ω𝑡 2 𝑏+𝑒𝑖Ω𝑡 2! 𝑒−𝑖𝑘F1𝑥𝑒−𝑖𝐸1𝑡 ℏ+𝑝 𝑎−𝑒−𝑖Ω𝑡 2 𝑏−𝑒𝑖Ω𝑡 2! 𝑒−𝑖𝑘F2𝑥𝑒−𝑖𝐸2𝑡 ℏ, (B7) where𝑡and𝑝are transmission coefficients to be determined by applying appropriate boundary conditions. Consider the 𝑒↑incident from the AM side based on the FI/AM bilayer, in order to match the time-dependence of the wavefunction components on the FI and AM sides, we can obtain 𝐸′=𝐸−ℏΩand𝐸1=𝐸2=𝐸−ℏΩ 2. In terms of 𝐸, the corresponding 𝑥component of the two wave vectors in the FI are expressed as 𝑘F1,𝑒↑=√︂ 2𝑚𝑒[𝐸−𝑈−𝐽(√︃ 1−2𝛽√ 1−𝑚2+𝛽2+𝛽)]−ℏ2𝑘2𝑦 ℏ, (B8) 𝑘F2,𝑒↑=√︂ 2𝑚𝑒[𝐸−𝑈+𝐽(√︃ 1−2𝛽√ 1−𝑚2+𝛽2−𝛽)]−ℏ2𝑘2 𝑦 ℏ. (B9) Based on the above, we write down ΨFI,𝑒↑=𝑡 𝑎+𝑒−𝑖Ω𝑡 2 𝑏+𝑒𝑖Ω𝑡 2! 𝑒−𝑖𝑘F1,𝑒↑𝑥𝑒−𝑖𝐸1𝑡 ℏ+𝑝 𝑎−𝑒−𝑖Ω𝑡 2 𝑏−𝑒𝑖Ω𝑡 2! 𝑒−𝑖𝑘F2,𝑒↑𝑥𝑒−𝑖𝐸2𝑡 ℏ. (B10) Consider the 𝑒↓incident from the AM side based on the FI/AM bilayer, in order to match the time-dependence of the wavefunction components on the FI and AM sides, we can obtain 𝐸′=𝐸+ℏΩand𝐸1=𝐸2=𝐸+ℏΩ 2. In terms of 𝐸, the corresponding 𝑥component of the two wave vectors in the FI are expressed as 𝑘FI1,𝑒↓=√︂ 2𝑚𝑒[𝐸−𝑈−𝐽(√︃ 1−2𝛽√ 1−𝑚2+𝛽2−𝛽)]−ℏ2𝑘2𝑦 ℏ, (B11) 𝑘F2,𝑒↓=√︂ 2𝑚𝑒[𝐸−𝑈+𝐽(√︃ 1−2𝛽√ 1−𝑚2+𝛽2+𝛽)]−ℏ2𝑘2𝑦 ℏ. (B12) Based on the above, we write down ΨFI,𝑒↓=𝑡 𝑎+𝑒−𝑖Ω𝑡 2 𝑏+𝑒𝑖Ω𝑡 2! 𝑒−𝑖𝑘F1,𝑒↓𝑥𝑒−𝑖𝐸1𝑡 ℏ+𝑝 𝑎−𝑒−𝑖Ω𝑡 2 𝑏−𝑒𝑖Ω𝑡 2! 𝑒−𝑖𝑘F2,𝑒↓𝑥𝑒−𝑖𝐸2𝑡 ℏ. (B13) Note that all wave numbers in the FI possess imaginary values since a large potential 𝑈is required to ensure the ferromagnet to be insulating. To ensure this, we use 𝑈=2𝜇throughout this work.8 Appendix C: Wavefunctions in the AM and FI Here we summarize the wavefunctions in the AM and FI, in which the time-dependence is omitted since we have applied equal time-dependence on both sides. Consider the 𝑒↑incident from the AM side based on the FI/AM bilayer, we have ΨAM,𝑒↑= 1 0 𝑒𝑖𝑘𝑒↑,−𝑥+𝑟 1 0 𝑒𝑖𝑘𝑒↑,+𝑥+𝑟′ 0 1 𝑒𝑖𝑘′ 𝑒↓,+𝑥, (C1) ΨFI,𝑒↑=𝑡 𝑎+ 𝑏+ 𝑒−𝑖𝑘F1,𝑒↑𝑥+𝑝 𝑎− 𝑏− 𝑒−𝑖𝑘F2,𝑒↑𝑥, (C2) in which𝐸′=𝐸−ℏΩ. Consider the 𝑒↓incident from the AM side based on the FI/AM bilayer, we have ΨAM,𝑒↓= 0 1 𝑒𝑖𝑘𝑒↓,−𝑥+𝑟 0 1 𝑒𝑖𝑘𝑒↓,+𝑥+𝑟′ 1 0 𝑒𝑖𝑘′ 𝑒↑,+𝑥, (C3) ΨFI,𝑒↓=𝑡 𝑎+ 𝑏+ 𝑒−𝑖𝑘F1,𝑒↓𝑥+𝑝 𝑎− 𝑏− 𝑒−𝑖𝑘F2,𝑒↓𝑥, (C4) in which𝐸′=𝐸+ℏΩ. Appendix D: Boundary conditions We consider a planar FI in contact with AM1 through a Rashba spin-orbit coupled interface. This interfacial contribution to the Hamiltonian takes the form 𝐻𝐼=[𝑈0+𝑈SO 𝑘𝐹ˆ𝒏·(ˆ𝝈×𝒌)]𝛿(𝑥) =[𝑈0−𝑈SO 𝑘𝐹𝑘𝑦𝜎𝑧]𝛿(𝑥), (D1) in which we take 𝒏=𝒙and𝑘𝐹=√︁ 2𝑚𝑒𝜇/ℏis the Fermi wave vector. This 𝛿-function will influence the boundary conditions that the scattering wavefunctions have to satisfy. Consequently, the Hamiltonian of the bilayer system becomes 𝐻=−ℏ2∇2 2𝑚𝑒+𝐻𝐼+𝛼𝑘𝑦 2{𝑘𝑥,Θ(𝑥)}𝜎𝑧, (D2) in which only the terms affecting the boundary conditions are included. Note here antisymmetrization of the altermagnetic term 𝛼𝑘𝑥𝑘𝑦𝜎𝑧→𝛼𝑘𝑦 2{𝑘𝑥,Θ(𝑥)}𝜎𝑧is necessary to ensure hermiticity of the Hamilton-operator, where Θ(𝑥)is the step function. Above,𝑘𝑥=−i𝜕𝑥. Eq. (D2) can be rewritten as 𝐻=−ℏ2∇2 2𝑚𝑒+(𝑈0−𝑈SO 𝑘𝐹𝑘𝑦𝜎)𝛿(𝑥)+𝛼𝑘𝑦𝜎 2{𝑘𝑥,Θ(𝑥)}, (D3) where𝜎=+1(−1)for𝑒↑(↓) . In Eq. (D3), we have {𝑘𝑥,Θ(𝑥)}Ψ=𝑘𝑥[Θ(𝑥)Ψ]+Θ(𝑥)(𝑘𝑥Ψ) =−i[Ψ𝜕𝑥Θ(𝑥)+Θ(𝑥)𝜕𝑥Ψ]−iΘ(𝑥)𝜕𝑥Ψ =−i𝛿(𝑥)Ψ−2iΘ(𝑥)𝜕𝑥Ψ.(D4) Apply𝐻Ψ=𝐸Ψand integrate over[−𝜖,𝜖]with𝜖→0, we have ∫+𝜖 −𝜖𝜕2 𝑥Ψ𝑑𝑥=2𝑚𝑒 ℏ2∫+𝜖 −𝜖[𝑈0−(i𝛼 2+𝑈SO 𝑘𝐹)𝑘𝑦𝜎]𝛿(𝑥)Ψ𝑑𝑥 −2𝑚𝑒 ℏ2∫+𝜖 −𝜖i𝛼𝑘𝑦𝜎Θ(𝑥)𝜕𝑥Ψ𝑑𝑥−2𝑚𝑒 ℏ2∫+𝜖 −𝜖𝐸Ψ𝑑𝑥. (D5)9 Consequently, the remaining nonzero terms are 𝜕𝑥Ψ +𝜖−𝜕𝑥Ψ −𝜖=2𝑚𝑒 ℏ2[𝑈0−(i𝛼 2+𝑈SO 𝑘𝐹)𝑘𝑦𝜎]Ψ +𝜖(D6) withΨ +𝜖=Ψ −𝜖and𝜎=+1(−1)for𝑒↑(↓) . For notation convenience, we rewrite the boundary conditions for 𝑒↑incident from the AM side based on the FI/AM bilayer as ΨAM 𝑥=0=Ψ FI 𝑥=0= 𝑓 𝑔 , (D7) 𝜕𝑥ΨAM 𝑥=0−𝜕𝑥ΨFI 𝑥=0= 𝑘𝛼,+1𝑓 𝑘𝛼,−1𝑔 , (D8) where𝑘𝛼,𝜎=2𝑚𝑒 ℏ2[𝑈0−(i𝛼 2+𝑈SO 𝑘𝐹)𝑘𝑦𝜎]with𝜎=+1(−1). The boundary conditions for 𝑒↓incident from the AM side have the same forms as Eqs. (D7,D8) with different explicit expressions for 𝑓and𝑔. Appendix E: 1D DOS and 2D DOS in the AM For𝑒↑incident from the AM1 side based on the FI/AM1 bilayer, we have 𝐸=𝐸+=ℏ2(𝑘2 𝑥+𝑘2 𝑦) 2𝑚𝑒−𝜇+𝛼𝑘𝑥𝑘𝑦, (E1) based on which the 1D density of states (DOS) can be calculated as 𝑑𝑘𝑥/𝑑𝐸=(ℏ2𝑘𝑥 𝑚𝑒+𝛼𝑘𝑦)−1. (E2) On the other hand, the general expression for 2D DOS is given by 𝑁(𝐸)=1 4𝜋2∫𝑑𝑙 |∇𝒌𝐸(𝒌)|, (E3) which can be used for anisotropic DOS. In Eq. (E3), we can use 𝑑𝑙=√︂ (𝑑𝑘𝑥 𝑑𝜃)2+(𝑑𝑘𝑦 𝑑𝜃)2𝑑𝜃, (E4) |∇𝒌𝐸(𝒌)|=√︄ (𝜕𝐸 𝜕𝑘𝑥)2+(𝜕𝐸 𝜕𝑘𝑦)2 =√︄ (ℏ2𝑘𝑥 𝑚𝑒+𝛼𝑘𝑦)2+(ℏ2𝑘𝑦 𝑚𝑒+𝛼𝑘𝑥)2. (E5) Insert𝑘𝑥=𝑘𝑥,𝑒↑and𝑘𝑦=𝑘𝑦,𝑒↑in Eq. (A7) into Eqs. (E4) and (E5), |∇𝒌𝐸(𝒌)|is expressed in terms of 𝐸and𝜃, i.e., |∇𝒌𝐸(𝒌)|=𝐾(𝐸,𝜃). Consequently, Eq. (E3) can be rewritten as 𝑁(𝐸)=∫2𝜋 0𝑁(𝐸,𝜃)𝑑𝜃, (E6) 𝑁(𝐸,𝜃)=1 4𝜋2√︃ (𝑑𝑘𝑥,𝑒↑/𝑑𝜃)2+(𝑑𝑘𝑦,𝑒↑/𝑑𝜃)2 𝐾(𝐸,𝜃)(E7) in which𝑁(𝐸,𝜃)corresponds to the DOS at a given incident angle 𝜃. Following the same procedure as described above, the DOS in the AM for 𝑒↓incident with 𝐸=𝐸−can be calculated. Note that 1D DOS instead of 2D DOS is utilized in the main text since∫ 𝑘𝑦is included separately.10 Appendix F: Backflow spin-current In general, a backflow spin current exists due to a spin accumulation that is built up in the material connected to the precessing FI [1]. This backflow current diminishes the magnitude of the total spin current flowing across the interface. Assuming that the material which the spin current is pumped into act as a highly conductive reservoir which drains the spin current, the backflow spin current may be neglected. For a ferromagnet/normal metal bilayer with a Rashba spin-orbit coupled interface, as in the present system, Ref. [ 27] derived a backflow factor 𝜉∝(𝜆sd/𝑙mfp)coth(𝑑𝑁/𝜆sd)where𝜆sdis the spin diffusion length, 𝑙mfpis the electronic mean free path, and 𝑑𝑁is the thickness of the normal layer. For ballistic, large reservoirs, 𝜉→0. Appendix G: 45-degree rotated AM2 Here we summarize the useful equations for the rotated Hamiltonian of AM2 shown in Fig. 1(b) in the main text, i.e., 𝐻AM=−ℏ2▽2 2𝑚𝑒−𝜇+𝛼 2(𝑘2 𝑥−𝑘2 𝑦)𝜎𝑧, (G1) which corresponds to a 45 degree rotation of the FI/AM1 interface. •1: eigenpairs: The two eigenpairs are obtained as: 𝐸1=𝐸+with(1,0)𝑇for𝑒↑and𝐸2=𝐸−with(0,1)𝑇for𝑒↓. The eigen-energies are described by 𝐸±=ℏ2(𝑘2 𝑥+𝑘2 𝑦) 2𝑚𝑒−𝜇±𝛼 2(𝑘2 𝑥−𝑘2 𝑦). (G2) •2: wave vectors in the AM to construct the wave functions: 𝑘𝑒↑,±=±√︄ 2𝑚𝑒(𝜇+𝐸+𝛼𝑘2𝑦/2)−ℏ2𝑘2𝑦 ℏ2+𝑚𝑒𝛼, (G3) 𝑘𝑒↓,±=±√︄ 2𝑚𝑒(𝜇+𝐸−𝛼𝑘2𝑦/2)−ℏ2𝑘2𝑦 ℏ2−𝑚𝑒𝛼. (G4) •3: wave vectors on the AM Fermi surface: 𝑘𝑦,𝑒↑=𝑟1sin𝜃, 𝑘𝑥,𝑒↑=𝑟1cos𝜃, 𝑟1=𝑎1𝑏1√︃ 𝑏2 1cos2(𝜃+𝜋/2)+𝑎2 1sin2(𝜃+𝜋/2), 𝑎 1=√︄ 2𝑚𝑒(𝜇+𝐸) ℏ2−𝑚𝑒𝛼, 𝑏 1=√︄ 2𝑚𝑒(𝜇+𝐸) ℏ2+𝑚𝑒𝛼(G5) 𝑘𝑦,𝑒↓=𝑟2sin𝜃, 𝑘𝑥,𝑒↓=𝑟2cos𝜃, 𝑟2=𝑎2𝑏2√︃ 𝑏2 2cos2𝜃+𝑎2 2sin2𝜃, 𝑎 2=√︄ 2𝑚𝑒(𝜇+𝐸) ℏ2−𝑚𝑒𝛼, 𝑏 2=√︄ 2𝑚𝑒(𝜇+𝐸) ℏ2+𝑚𝑒𝛼. (G6) •4: boundary conditions: ΨAM 𝑥=0=Ψ FI 𝑥=0= 𝑓 𝑔 , (G7)  (1+𝑚𝑒𝛼/ℏ2)𝜕𝑥𝑓AM (1−𝑚𝑒𝛼/ℏ2)𝜕𝑥𝑔AM 𝑥=0−𝜕𝑥ΨFI 𝑥=0= 𝑘𝛼,+1𝑓 𝑘𝛼,−1𝑔 (G8) for𝑒↑and𝑒↓incidents, in which 𝑘𝛼,𝜎=2𝑚𝑒 ℏ2[𝑈0−𝑈SO 𝑘𝐹𝑘𝑦𝜎]with𝜎=+1(−1). To get the above boundary conditions, we follow the similar procedure as described in Sec. D by considering the Hermitian Hamiltonian of the bilayer system as 𝐻=−ℏ2∇2 2𝑚𝑒+𝐻𝐼+𝛼 2[𝑘𝑥Θ(𝑥)𝑘𝑥−𝑘𝑦Θ(𝑥)𝑘𝑦]𝜎𝑧, (G9) in which𝑘𝑥=−i𝜕𝑥.11 •5: The longitudinal quantum mechanical spin current polarized along the 𝑧axis in the AM: 𝑗𝑠𝑧,𝑒↑(↓)=ℏ2 2𝑚𝑒(ℑm{𝑓∗∇𝑓}−ℑ m{𝑔∗∇𝑔})+𝛼 2(ℑm{𝑓∗∇𝑓}+ℑ m{𝑔∗∇𝑔}). (G10) Appendix H: Spin-flip probability Here we consider FI/AM1 as an example. If we write the wave function in the form of Ψ=(𝑓,𝑔)𝑇, the probability current in the AM is given by 𝑗AM 𝑃=ℏ 𝑚𝑒[ℑm{𝑓∗∇𝑓}+ℑ m{𝑔∗∇𝑔}]+𝛼𝑘𝑦 ℏ(|𝑓|2−|𝑔|2). (H1) Consider the 𝑒↑incident from the AM side based on the FI/AM bilayer, we have ΨAM,𝑒↑with𝑓=𝑒𝑖𝑘𝑒↑,−𝑥+𝑟𝑒𝑖𝑘𝑒↑,+𝑥and 𝑔=𝑟′𝑒𝑖𝑘′ 𝑒↓,+𝑥, we have ℑm{𝑓∗∇𝑓}=𝑘𝑒↑,−𝑒−2ℑm[𝑘𝑒↑,−]𝑥+𝑘𝑒↑,+|𝑟|2𝑒−2ℑm[𝑘𝑒↑,+]𝑥+Re[(𝑘𝑒↑,++𝑘∗ 𝑒↑,−)𝑟𝑒𝑖(𝑘𝑒↑,+−𝑘∗ 𝑒↑,−)𝑥], ℑm{𝑔∗∇𝑔}=𝑘′ 𝑒↓,+|𝑟′|2𝑒−2ℑm[𝑘′ 𝑒↓,+]𝑥, |𝑓|2=𝑒−2ℑm[𝑘𝑒↑,−]𝑥+|𝑟|2𝑒−2ℑm[𝑘𝑒↑,+]𝑥+2Re[𝑟𝑒𝑖(𝑘𝑒↑,+−𝑘∗ 𝑒↑,−)𝑥], |𝑔|2=|𝑟′|2𝑒−2ℑm[𝑘′ 𝑒↓,+]𝑥.(H2) Therefore, the probability current is given by 𝑗AM,𝑒↑ 𝑃=(ℏ𝑘𝑒↑,− 𝑚𝑒+𝛼𝑘𝑦 ℏ)𝑒−2ℑm[𝑘𝑒↑,−]𝑥 +(ℏ𝑘𝑒↑,+ 𝑚𝑒+𝛼𝑘𝑦 ℏ)|𝑟|2𝑒−2ℑm[𝑘𝑒↑,+]𝑥+ℏ 𝑚𝑒Re[(𝑘𝑒↑,++𝑘∗ 𝑒↑,−)𝑟𝑒𝑖(𝑘𝑒↑,+−𝑘∗ 𝑒↑,−)𝑥]+2𝛼𝑘𝑦 ℏRe[𝑟𝑒𝑖(𝑘𝑒↑,+−𝑘∗ 𝑒↑,−)𝑥] +(ℏ𝑘′ 𝑒↓,+ 𝑚𝑒−𝛼𝑘𝑦 ℏ)|𝑟′|2𝑒−2ℑm[𝑘′ 𝑒↓,+]𝑥.(H3) On the other hand, the probability current in the FI is given by 𝑗FI 𝑃=ℏ 𝑚𝑒[ℑm{𝑓∗∇𝑓}+ℑ m{𝑔∗∇𝑔}]. (H4) ForΨFI,𝑒↑, we have𝑓=𝑡𝑎+𝑒−𝑖𝑘F1,𝑒↑𝑥+𝑝𝑎−𝑒−𝑖𝑘F2,𝑒↑𝑥and𝑔=𝑡𝑏+𝑒−𝑖𝑘F1,𝑒↑𝑥+𝑝𝑏−𝑒−𝑖𝑘F2,𝑒↑𝑥. Since the wavevectors in the FI are imaginary, we apply 𝑘F1,𝑒↑=𝑖𝜅1and𝑘F2,𝑒↑=𝑖𝜅2where𝜅1and𝜅2are real. Consequently, we have 𝑓=𝑡𝑎+𝑒𝜅1𝑥+𝑝𝑎−𝑒𝜅2𝑥and 𝑔=𝑡𝑏+𝑒𝜅1𝑥+𝑝𝑏−𝑒𝜅2𝑥. ℑm{𝑓∗∇𝑓}=ℑm{𝜅1|𝑡|2|𝑎+|2𝑒2𝜅1𝑥+𝜅2|𝑝|2|𝑎−|2𝑒2𝜅2𝑥+(𝜅1𝑎+𝑎∗ −𝑡𝑝∗+𝜅2𝑎∗ +𝑎−𝑡∗𝑝)𝑒(𝜅1+𝜅2)𝑥}. (H5) It is obvious that the first two terms in Eq. (H5) are zero since 𝜅1and𝜅2are real. If𝜅1=𝜅2, we can haveℑm{𝑓∗∇𝑓}=0. However, we should have 𝜅1≠𝜅2according to the exchange 𝐽in Eqs. (B8,B9) of FI. Therefore, we have ℑm{𝑓∗∇𝑓}=ℑm{(𝜅1𝑎+𝑎∗ −𝑡𝑝∗+𝜅2𝑎∗ +𝑎−𝑡∗𝑝)𝑒(𝜅1+𝜅2)𝑥}. (H6) Similarly, we have ℑm{𝑔∗∇𝑔}=ℑm{(𝜅1𝑏+𝑏∗ −𝑡𝑝∗+𝜅2𝑏∗ +𝑏−𝑡∗𝑝)𝑒(𝜅1+𝜅2)𝑥}. (H7) Consequently, the probability current is given by 𝑗FI,𝑒↑ 𝑃=ℏ 𝑚𝑒ℑm{[𝜅1(𝑎+𝑎∗ −+𝑏+𝑏∗ −)𝑡𝑝∗+𝜅2(𝑎∗ +𝑎−+𝑏∗ +𝑏−)𝑡∗𝑝]𝑒(𝜅1+𝜅2)𝑥}. (H8) Note here there are no separate terms regarding the transmission coefficients 𝑡and𝑝but the mixing terms between them.12 -5000 05000012310-4FI/NM Z=0 0.3*0 0.9*0 -5000 05000 E (meV)00.20.40.60.8110-6FI/NM Z=3-5000 0500001234Spin-flip prob10-4FI/AM1 Z=0 0.3*c 0.9*c 129513001.4251.4310-4 -5000 05000 E (meV)051015Spin-flip prob10-7FI/AM1 Z=3 129513004.34.3510-7-5000 050000123410-4FI/AM2 Z=0 0.3*c 0.9*c -5000 05000 E (meV)00.20.40.60.8110-6FI/AM2 Z=3 FIG. 4. (Color online) Spin-flip probability for different spin pumping bilayers for 𝑍=0and𝑍=3at a fixed𝑘𝑦mode. Here a small 𝑘𝑦=0.1is utilized. Apply𝑗FI,𝑒↑ 𝑃=𝑗AM,𝑒↑ 𝑃and insert Eqs. (H3,H8), 1=𝐴(𝐸)+𝐵(𝐸)+𝐶(𝐸), 𝐴(𝐸)=−(ℏ𝑘𝑒↑,+ 𝑚𝑒+𝛼𝑘𝑦 ℏ)|𝑟|2𝑒−2ℑm[𝑘𝑒↑,+]𝑥+ℏ 𝑚𝑒Re[(𝑘𝑒↑,++𝑘∗ 𝑒↑,−)𝑟𝑒𝑖(𝑘𝑒↑,+−𝑘∗ 𝑒↑,−)𝑥]+2𝛼𝑘𝑦 ℏRe[𝑟𝑒𝑖(𝑘𝑒↑,+−𝑘∗ 𝑒↑,−)𝑥] (ℏ𝑘𝑒↑,− 𝑚𝑒+𝛼𝑘𝑦 ℏ)𝑒−2ℑm[𝑘𝑒↑,−]𝑥, 𝐵(𝐸)=−(ℏ𝑘′ 𝑒↓,+ 𝑚𝑒−𝛼𝑘𝑦 ℏ)|𝑟′|2𝑒−2ℑm[𝑘′ 𝑒↓,+]𝑥 (ℏ𝑘𝑒↑,− 𝑚𝑒+𝛼𝑘𝑦 ℏ)𝑒−2ℑm[𝑘𝑒↑,−]𝑥, 𝐶(𝐸)=ℏ 𝑚𝑒ℑm{[𝜅1(𝑎+𝑎∗ −+𝑏+𝑏∗ −)𝑡𝑝∗+𝜅2(𝑎∗ +𝑎−+𝑏∗ +𝑏−)𝑡∗𝑝]𝑒(𝜅1+𝜅2)𝑥} (ℏ𝑘𝑒↑,− 𝑚𝑒+𝛼𝑘𝑦 ℏ)𝑒−2ℑm[𝑘𝑒↑,−]𝑥=0,(H9) in which𝐴(𝐸),𝐵(𝐸)and𝐶(𝐸)are the probability coefficients regarding reflection without spin-flip ( 𝑟), reflection with spin-flip (𝑟′) and transmission ( 𝑡and𝑝), respectively. Note that 𝐶(𝐸)becomes zero when Eq. (B6) is employed. Next, we will focus on the spin-flip probability regarding 𝐵(𝐸), which plays an important role in spin pumping. In Fig. 4, the spin-flip probability is plotted for different spin pumping bilayers including FI/AM1(AM2) for 𝑍=0and𝑍=3, in which it is found that the spin-flip probability increases (decreases) with altermagnetism in FI/AM1(AM2). On the other hand, it is shown that the spin-flip probability decreases with 𝜇in FI/NM for large 𝑍. These observations are consistent with the spin pumping current behavior shown in Fig. 2 in the main text.13 Appendix I: Arbitrary-angle rotated AM The arbitrary-angle rotated AM can be modeled based on the combination of our established AM1 and AM2 cases, i.e., a more general Hamiltonian is 𝐻AM=−ℏ2▽2 2𝑚𝑒−𝜇+𝛼1𝑘𝑥𝑘𝑦𝜎𝑧+𝛼2(𝑘2 𝑥−𝑘2 𝑦)𝜎𝑧/2, (I1) in which two different altermagnetism strength parameters 𝛼1and𝛼2are introduced and the arbitrary angle is determined by 𝜃𝛼=1 2arctan(𝛼1/𝛼2). Following the same procedure as introduced before, the eigenvalues and wave vectors can be solved from the Hamiltonian, e.g., 𝐸±=ℏ2(𝑘2 𝑥+𝑘2 𝑦) 2𝑚𝑒−𝜇±𝛼1𝑘𝑥𝑘𝑦±𝛼2 2(𝑘2 𝑥−𝑘2 𝑦), (I2) 𝑘𝑒↑,±=±1 ℏ+𝛼2𝑚𝑒/ℏ√︄ 2𝑚𝑒(𝜇+𝐸)(1+𝛼2𝑚𝑒 ℏ2)−ℏ2𝑘2𝑦+(𝛼2 1+𝛼2 2)𝑚2𝑒𝑘2𝑦 ℏ2 −𝛼1𝑚𝑒𝑘𝑦 ℏ2+𝑚𝑒𝛼2, (I3) which reveal features of both the AM1 and AM2 cases. To ensure that the energy dispersion corresponds to an elliptical energy surface rather than a hyperbola, the altermagnetism parameters should satisfy ¯𝛼≡√︃ 𝛼2 1+𝛼2 2<𝛼𝑐≡ℏ2/𝑚𝑒. The corresponding semi-major and semi-minor axes are 𝑎=√︃ 2𝑚𝑒(𝜇+𝐸) ℏ2−𝑚𝑒¯𝛼and𝑏=√︃ 2𝑚𝑒(𝜇+𝐸) ℏ2+𝑚𝑒¯𝛼for electron incidents, based on which the DOS can be calculated. Similarly, the boundary conditions and spin current expressions can be derived from the Hamiltonian with all necessary details included in our previous explanation for the AM1 and AM2 cases.

0904.1999v1.Semiclassical_spin_transport_in_spin_orbit_coupled_systems.pdf

arXiv:0904.1999v1 [cond-mat.mes-hall] 13 Apr 2009 1Semiclassical spin transport in spin-orbit-coupled systems May 26, 2018 Dimitrie Culcer1 Advanced Photon Source, Argonne National Laboratory, Argon ne, IL 60439 Northern Illinois University, De Kalb, IL 60115 Abstract This article discusses spin transport in systems with spin- orbit inter- actions and how it can be understood in a semiclassical pictu re. I will first present a semiclassical wave-packet description of spin tr ansport, which explains howthemicroscopic motion ofcarriers gives rise t oaspincurrent. Due to spin non-conservation the definition of the spin curre nt has some arbitrariness. In the second part I will briefly review the ph ysics from a density matrix point of view, which makes clear the relation ship between spin transport and spin precession and the important role of scattering. Contents 1 Definition of the subject and its importance 3 2 Introduction 3 3 Spin currents in electric fields 5 3.1 Wave-packet picture of spin transport . . . . . . . . . . . . . . . 5 3.2 Density matrix picture of spin transport . . . . . . . . . . . . . . 11 4 Future Directions 14 Glossary •Extrinsic effect an effect which has an explicit dependence on the form or strength of the disorder potential. 1Current affiliation: Condensed Matter Theory Center, Depart ment of Physics, University of Maryland, College Park MD 20742. 2•Intrinsic effect an effect which does not depend explicitly on the form and strength of the disorder potential. •Semiclassical theory atheoryinwhichaparticle’spositionandmomentum are considered simultaneously. •Spin-orbit interaction a relativistic interaction between the spin of a par- ticle and its momentum (which is associated with its orbital motion.) •Steady-state spin current a flow of spins induced by an electric field. •Steady-state spin density a net spin density induced by an electric field. 1 Definition of the subject and its importance Spin transport refers to the physical movement of spins across a sample and, if spin were a conserved quantity, one could make a straightforwar d distinction between spin-up and spin-down charge currents. The recent ups urge of inter- est in spin transport is, however, motivated by systems in which spin is not conserved due to the presence of spin-orbit interactions, which g ive rise to spin precession. Here, due to non-conservation of spin the spin curre nt is not well defined [1, 2, 3]. Spin transport in these cases usually does not involv e charge transport as the charge currents in the direction of spin flow canc el out. Finally, in certain materials, spin currents are accompanied by steady-sta te spin densi- ties. The appearance of a spin density is not a transport phenomen on, but it is a steady-state process and is intimately connected to spin transp ort. The word semiclassical as used in this work refers to theories which consider the position and momentum of a particle simultaneously. Semiclassical pictures are intuitive and useful in descriptions of transport, particularly in inhomoge- neous systems and in spatially dependent fields, which typically vary o n length scales much larger than atomic size. In recent years, steady progress has been made towards realiza tion of conve- nient semiconducting ferromagnets and spin injection into semicond uctors from ferromagnetic metals [4, 5, 6, 7] yet spin injection from a ferromag netic metal into a semiconductor is hampered by the resistivity mismatch betwee n the two [8]. This is one factor, in addition to basic science, motivating the sear ch for an understanding of the way spins are manipulated electrically. The last few years haveseen many experimental advances in spin transport, and spin currentshave been measured directly [9, 10] and indirectly [11, 12, 13, 14, 15]. 2 Introduction Novel physical phenomena that may lead to improved memory device s and ad- vances in quantum information processing are closely related to spin -orbit in- teractions. [16] Spin-orbit interactions are present in the band st ructure and in potentials due to impurity distributions. Spin-orbit coupling is in princ iple 3always present in impurity potentials and gives rise to skew scatterin g. Band structure spin-orbit coupling may arise from the inversion asymmet ry of the underlying crystal lattice [17] (bulk inversion asymmetry), from th e inversion asymmetry of the confining potential in two dimensions [18] (struc ture inversion asymmetry), and may be present also in inversion symmetric system s. [19] Although many observations in this entry are general, the discussio n will fo- cusonnon-interactingspin-1/2electronsystems, whicharepeda gogicallyeasier. The Hamiltonian of these systems typically contains a kinetic energy t erm and a spin-orbit coupling term, Hk=/planckover2pi12k2 2m∗+Hso k, where m∗is the electron ef- fective mass. In spin-1/2 electron systems, band structure spin -orbit coupling can always be represented as a Zeeman-like interaction of the spin w ith a wave vector-dependenteffective magnetic field Ωk, thusHso k= (1/2)σ·Ωk. Common examples of effective fields are the Rashba spin-orbit interaction, [ 18] which is often dominant in quantum wells with inversion asymmetry, and the Dr essel- haus spin-orbit interaction, [17] which is due to bulk inversion asymm etry. The spin operator is given by sσ= (/planckover2pi1/2)σσ, whereσσis a Pauli spin matrix. The spin current operator in these systems will be taken to be ˆJσ i= (1/2){sσ,vi}, where the velocity operator is vi= (1//planckover2pi1)∂Hk/∂ki. An electron spin at wave vector kprecesses about the effective field Ωkwith frequency Ω k//planckover2pi1≡ |Ωk|//planckover2pi1and is scattered to a different wave vector within a characteristic momentum scattering time τp. I will assume in this work that εFτp//planckover2pi1≫1, where εFistheFermienergy,whichisequivalenttothe assumption that the carrier mean free path is much larger that the de Broglie wa velength. Within this range, the relative magnitude of the spin precession freq uency Ω k and inverse scattering time 1 /τpdefine three qualitatively different regimes. In the ballistic (clean) regime no scattering occurs and the temperatu re tends to absolute zero, so that εFτp→ ∞and Ωkτp//planckover2pi1→ ∞. The weakscattering regime is characterized by fast spin precession and little momentum scatte ring due to, e.g., a slight increase in temperature, yielding εFτp//planckover2pi1≫Ωkτp//planckover2pi1≫1. In the strong momentum scattering regime εFτp//planckover2pi1≫1≫Ωkτp//planckover2pi1. I will concentrate on effects originating in the band structure, the observation of wh ich requires the assumption that the materials under study are in the weak mome ntum scattering regime. Electric fields will be assumed uniform. The first part of this article will present a semiclassical theory of sp in trans- port, identifying the terms responsible for spin currents in the micr oscopic dy- namics of carriers. Spin non-conservation as a result of spin prece ssion leads to several possible definitions of the spin current, which emerge ou t of the spin equation of continuity. The second part presents a different point of view, which explains aspects not easily captured in the semiclassical approach. The steady- state density matrix is shown to contain a contribution due to prece ssing spins and one due to conserved spins. Steady state corrections ∝τpare associated with the absenceof spin precession and give rise to spin densities in external fields. [20, 21, 22, 23, 24, 26, 25, 27, 28] Steady state correctio ns independent of τpare associated with spin precession and give rise to spin currents in e xternal fields. [1, 2, 3, 9, 10, 11, 12, 13, 14, 15, 29, 30, 31, 32, 33, 34, 35 , 36, 37, 38, 439, 40, 41, 42, 43, 44, 45, 46] Scattering between these two dist ributions induces significant corrections to steady-state spin currents. 3 Spin currents in electric fields 3.1 Wave-packet picture of spin transport This section presents a semiclassical theory of spin transport valid for a general spin-orbit system. The semiclassical method is a suitable approach t o the study of transport, because, typically, in the relevant systems the ext ernal fields vary smoothly on atomic length scales. All information about the system is taken to be contained in the band structure, thus allowing a description of sp in transport which does not make reference to the detailed form of the spin-orb it interaction. The system under study is regarded as as a collection of carriers, w hose semiclassical dynamics in a non-degenerate band iare described by a wave packet [47], with its charge centroid having coordinates ( rc,kc) |wi∝an}b∇acket∇i}ht=/integraldisplay d3k a(k,t)eik·ˆr|ui(rc,k,t)∝an}b∇acket∇i}ht. (1) In the above, the function a(k,t) is a narrow distribution sharply peaked at kc, the phase of which specifies the center of charge position rc, while|ui(rc,k,t)∝an}b∇acket∇i}ht are lattice-periodic Bloch wave functions. The size of the wave pack et in mo- mentum space must be considerably smaller than that of the Brillouin z one. In real space, this implies that the wave packet must stretch over ma ny unit cells. The external electric field drives the center of the wave packet in k-space according to the semiclassical equations of motion /planckover2pi1˙rc=∂εi ∂kc−qE×Ωi (2) /planckover2pi1˙kc=qE, withqthe charge of the carriers, εithe band energy, and the Berry curvature Ωi=i∝an}b∇acketle{t∂ui ∂k|×|∂ui ∂k∝an}b∇acket∇i}ht. (3) The electric field also gives rise to an adiabatic correction to the wave functions, which mixes the states making up the wave packet. The wave functio ns|ui∝an}b∇acket∇i}ht therefore have the following form: |ui∝an}b∇acket∇i}ht=|φi∝an}b∇acket∇i}ht−/summationdisplay j∝an}b∇acketle{tφj|i/planckover2pi1d dt|φi∝an}b∇acket∇i}ht εi−εj|φj∝an}b∇acket∇i}ht, (4) where the φiare the unperturbed Bloch eigenstates. The |ui∝an}b∇acket∇i}htform a complete set and retain the Bloch periodicity. 5r rCharge Spin c s Figure 1: For a particle of finite extent the charge and spin distribut ions in real space are in general do not coincide. The same is true of the charge and spin distributions in reciprocal space. The distribution of carriers is described by a function f. When scattering is present, the distribution function satisfies the following equation: ∂f ∂t+˙rc·∂f ∂rc+˙kc·∂f ∂kc=/parenleftbiggdf dt/parenrightbigg coll, (5) where (df dt)collis the usual collision term. In independent bands, in the relax- ation time approximation, the collision term takes the formf0−f τp, withf0the equilibrium distribution and τpthe momentum relaxation time. In the Boltz- mann theory, the change in the distribution function with time arises through the drift terms, which are determined from the semiclassical equat ions of mo- tion, as well as through scattering with other carriers, with localize d impurities or with phonons. For transport in a non-degenerate band, it is con sistent to ignore interband scattering effects in the weak scattering limit. In t his case the relaxation time is a scalar quantity. The effects of interband cohere nce due to scattering will be explored in the next section. In order to obtain expressions for macroscopic quantities of inter est, such as densities and currents, one needs to carry out a coarse graining b y averaging over microscopic fluctuations. In classical dynamics this coarse gr aining is per- formed by means of a sampling function, which is smooth and has a sign ificant magnitude only in a finite range [48]. This range is large compared to ato mic dimensions, butsmallcomparedtothescaleofvariationofthe distr ibutionfunc- tion. Moreover, it has a rapidly converging Taylor expansion over dis tances of atomic dimensions, and its form does not need to be specified. This me thod has a close analog in wavepacket dynamics, where the sampling function is replaced by aδ-function. It is crucial to recognize that, in general, the center of spin and th e center of charge are distinct, since the wave packet samples a range of wave vectors and the spin is usually a function of k. Following the line of thought outlined above, 6 t > 0 t = 0 Figure 2: In the presence of spin-orbit interactions the spin distrib ution of a particle changes in time. The horizontal axis may represent position or wave vector. the spin density is defined to be (henceforth kcwill be abbreviated to k) Sσ(R,t) =/integraldisplay /integraldisplay d3kd3rcf(rc,k,t)∝an}b∇acketle{tδ(R−ˆ r)ˆsσ∝an}b∇acket∇i}ht, (6) wherethe bracketindicates quantum mechanicalaveragingovert he wavepacket with charge centroid ( rc,k). As the δ-function has operator arguments, it will be regarded as a sampling operator , whose expectation value yields a spatial average, evaluated at position r. To account for the fact that spin is not con- served, a new quantity is introduced, which will be referred to as th e torque density, defined by Tσ(R,t) =/integraldisplay /integraldisplay d3kd3rcf(rc,k,t)∝an}b∇acketle{tδ(R−ˆ r)ˆτσ∝an}b∇acket∇i}ht. (7) ˆτσin the above stands for the rate of change of the spin operator, g iven by i /planckover2pi1[ˆH,ˆsσ], and symmetrization of products of non-commuting operators ha s been assumed. Finally, the microscopic spin current density is defined as: Jσ(R,t) =/integraldisplay /integraldisplay d3kd3rcf(rc,k,t)∝an}b∇acketle{tδ(R−ˆr)ˆsσˆv∝an}b∇acket∇i}ht. (8) We obtain the following continuity equation for the spin density and cu rrent: ∂Sσ ∂t+∇·Jσ=Tσ+Fσ. (9) The equation of continuity contains a bulk source term, which coincid es with the torque density and acts as a mechanism for spin generation. Sim ilar source terms are associated with nonconserved quantities, for example, in quantum electrodynamics and in Maxwell’s equations. The last term in (9) repre sents the scattering contribution, which will be discussed further below. Let us discuss the terms in the equation of continuity, beginning with the spin density. The argument of the sampling operator can be expres sed as [r− rc−(ˆr−rc)], and, as the second term is of atomic dimensions, the sampling 7operator can be written as a Taylor expansion about ( ˆr−rc). The density can therefore be re-expressed, in terms of macroscopic quantities, as Sσ(R,t) =ρsσ(R,t)−∇R·Psσ(R,t), (10) where summation over repeated indices has been assumed. In the a bove, the monopole density is given by ρsσ(R,t) =/integraldisplay /integraldisplay d3kd3rcf(rc,k,t)∝an}b∇acketle{tˆsσ∝an}b∇acket∇i}htδ(R−rc) =/integraldisplay d3kf∝an}b∇acketle{tˆsσ∝an}b∇acket∇i}ht|rc=R,(11) wherefin the second line, and henceforth, is to be understood as f(R,k,t), and the dipole density is Psσ(R,t) =/integraldisplay /integraldisplay d3kd3rcf∝an}b∇acketle{t(ˆr−rc)ˆsσ∝an}b∇acket∇i}htδ(R−rc) =/integraldisplay d3kfpsσ|rc=R.(12) The average spin of the wave packet has been denoted by ∝an}b∇acketle{tˆsσ∝an}b∇acket∇i}ht, and the spin- dipole is defined to be psσ=∝an}b∇acketle{t(ˆr−rc)ˆsσ∝an}b∇acket∇i}ht|rc=R. It will be seen that the first term in the density is the average of a monopole density located at rc, while the dipole term is the average of a point dipole density located at rc, and similarly for higher orders. The dipole must be understood as the ave rage of the quantum mechanical dipole operator, as an exact analogy with t he electric dipole of classical electrodynamics cannot be made. The density can thus be viewed as a collection of point multipoles, located at the centroid of ea ch wave packet. The microscopic distribution of spin is important at the molec ular level, but at the macroscopic level the effect of this molecular distribution is replaced by a sum of multipoles. Since the center of spin is different from the ce nter of charge, in principle all multipoles are present. Following a similar manipulation and using the Boltzmann equation, the torque density is re-expressed as: Tσ(R,t) =ρτσ(R,t)−∇·Pτσ(R,t) (13) with the torque monopole density ρτσ(R,t) =/integraldisplay /integraldisplay d3kd3rcf(rc,k,t)∝an}b∇acketle{tˆτσ∝an}b∇acket∇i}htδ(R−rc) =/integraldisplay d3kf∝an}b∇acketle{tˆτσ∝an}b∇acket∇i}ht|rc=R,(14) and the torque dipole density Pτσ(R,t) =/integraldisplay /integraldisplay d3kd3rcf(rc,k,t)∝an}b∇acketle{t(ˆR−rc)ˆτσ∝an}b∇acket∇i}ht|rc=R=/integraldisplay d3kfpτσ|rc=R.(15) In analogy with the spin dipole, the torque dipole has been defined as pτσ= ∝an}b∇acketle{t(ˆr−rc)ˆτσ∝an}b∇acket∇i}ht|rc=R. The torque density is therefore also a sum of multipole moments, that is, the moments of a point spin source located at rc. Even in the casewhen the center of ∝an}b∇acketle{tˆsσ∝an}b∇acket∇i}htcoincides with the center ofcharge, ∝an}b∇acketle{tˆτσ∝an}b∇acket∇i}htmaynot be centered at rc, with the result that the higher order terms in the torque density 8are in general present. The second and higher terms of Tσcancel exactly the analogous terms in the continuity equation which come from the curr ent. Since only the gradient of the spin current appears in the equation o f conti- nuity, in the expansion of the sampling operator we keep the leading t erm Js(R,t) =/integraldisplay d3kf∝an}b∇acketle{tˆvˆsσ∝an}b∇acket∇i}ht|rc=R. (16) Keeping terms to first order in ( ˆ r−rc), the current can be decomposed into the following: Jsσ=Csσ+Dsσ−Pτσ(17) The convective term Csσrepresents the spin being transported along with the wave packet Csσ(R,t) =/integraldisplay /integraldisplay d3kd3rcf(rc,k,t)˙ rc∝an}b∇acketle{tˆsσ∝an}b∇acket∇i}htδ(R−rc) =/integraldisplay d3kfcsσ|rc=R,(18) whileDsσcomes from the rate of change of the spin dipole, which has already been introduced. It has the form: Dsσ(R,t) =/integraldisplay d3kfdpsσ dt|rc=R (19) Pτσis the torque dipole introduced above. The corresponding monopole term appears in the source term of the continuity equation, as will emerg e below. The presence of the torque dipole here is to be contrasted with the absence of an analogous term in classical electrodynamics. There, an electric d ipole arises from the placement of two charges a small distance from each othe r, but the charge itself is conserved. Finally, we come to the source term in (9). The first part, composed of the torque density, has already been discussed. The second term, de noted by Fσ, becomes, in the relaxation time approximation Fσ=Sσ 0−Sσ τp=1 τp/integraldisplay d3k(f0−f)∝an}b∇acketle{tˆsσ∝an}b∇acket∇i}ht, (20) whereτpis the momentum relaxation time and f0the equilibrium distribution, which is usually the Fermi-Dirac distribution function. Based on the continuity equation alone, there is some flexibility in defin ing the current and the source. In systems in which spin is conserved, the torque density becomes, to first order in ( ˆ r−rc), a pure divergence, which can be incorporated into a redefinition of the spin current. This current, henceforth referred to as the spin transport current, is only due to the conv ective and spin dipole contributions: Jtσ(R,t) =Csσ(R,t)+Dsσ(R,t) (21) 9With respect to this spin transport current, the continuity equat ion takes the following form: ∂Sσ ∂t+∇·Jtσ=d dt/integraldisplay d3kf∝an}b∇acketle{tˆsσ∝an}b∇acket∇i}ht (22) In the steady stateunder a constantelectric field, the distributio n function is composedofanequilibriumpart,independent ofthefield, andanon- equilibrium part, which is first order in the field. Henceforth, terms in the spin c urrent and source which depend on the equilibrium distribution function will be ref erred to as intrinsic, whereas the terms depending on the nonequilibrium sh ift in the distribution will be referred to as extrinsic. For example, the integr and in Eq. (16) can be decomposed into a zero order spin-velocity, v∝an}b∇acketle{tˆsσ∝an}b∇acket∇i}ht, wherevis the usual group velocity of the band, and a first order correction. Th erefore, there will be a contribution to the current from the non-equilibrium part of the dis- tribution and the zero order spin-velocity, which has been discusse d extensively in previous work [29, 30, 31, 32, 33]. There will also be a contribution from the equilibrium distribution and the first order correction to the spin -velocity, which is referred to as the intrinsic contribution. In the wave packe t formalism presented here this effect arises from the change in wave function s induced by the electric field, rather than from the change in distribution funct ions that is responsible for most conventional transport effects. The intrins ic spin current is calculated from (16) using the equilibrium distribution and the expec tation values of the spin and spin dipole operators in a Bloch state perturbe d to first order in E. In its turn, the source in (9) can be decomposed into intrinsic and ex trinsic contributions. The present entryconsidershomogeneoussyste ms, sothat all the gradient terms vanish, and the torque density is simply f∝an}b∇acketle{tˆτσ∝an}b∇acket∇i}ht. The zeroth order contribution to this term is null, as the Bloch wave functions are eigen states of the Hamiltonian. Thus, to first order in the electric field, we find th at∝an}b∇acketle{tˆτσ∝an}b∇acket∇i}ht is simply given by ( eE//planckover2pi1)·(∂∝an}b∇acketle{tˆsσ∝an}b∇acket∇i}ht/∂k). One is thus justified in replacing f by its equilibrium value f0, in which case this term is purely intrinsic. The second term in the source, Fσ, which depends on the nonequilibrium shift in the distribution function, is entirely extrinsic. The extrinsic source term takes into account the effect of scatte ring, and is a term which usually appears in the equation of continuity. The intrins ic source accountsforthe effect ofall spin-nonconservingterms, and mus t be present even in a clean system, if the Hamiltonian contains spin-dependent contrib utions. In general, in addition to the rateofchangeof spin arisingfrom the spin -dependent terms in the Hamiltonian, scattering processes may alter the orient ation of the spin, with the result that any one spin component is not conserved, and the orientation of spins is randomized over a longer time period. For a unif orm steady-state system, the current is constant and the intrinsic s ource term must vanish. However, near the boundary of the system, or at an inter face with a different semiconductor with (for example) weaker spin-orbit inter actions, the spin current driven by an electric field will vary spatially and Tmust reach a non-zero value. 10Let us take a closer look at the spin dipole and torque dipole, which are seen to be the main mechanisms responsible for generating the spin c urrent. Because of its narrow distribution in k, the mean spin of the wave packet is ∝an}b∇acketle{twi|ˆsσ|wi∝an}b∇acket∇i}ht=∝an}b∇acketle{tui|ˆsσ|ui∝an}b∇acket∇i}ht, where it is understood that the wavevectorofthe Bloch function is set at kcand ˆsσisan arbitraryprojectionofthe spin vectoroperator. The spin dipole of the wave packet, defined relative to the charge ce nter of the wave packet is given, in terms of Bloch functions, by the expression : psσ i=i 2[∝an}b∇acketle{tui|ˆsσ|∂ui ∂k∝an}b∇acket∇i}ht−∝an}b∇acketle{t∂ui ∂k|ˆsσ|ui∝an}b∇acket∇i}ht]−∝an}b∇acketle{tui|i∂ui ∂k∝an}b∇acket∇i}ht∝an}b∇acketle{tui|ˆsσ|ui∝an}b∇acket∇i}ht(23) Interestingly, the spin dipole is independent of the wave packet widt h. The expression is also invariant under a local gauge transformation, in t he sense that if|ui∝an}b∇acket∇i}htis modified by a phase factor eiα(k)the spin dipole is unchanged. The torque dipole term has a special interpretation in the case of sp in trans- port. The rate of change of spin is equivalent to a torque, and the t orque dipole represents the moment exerted by this torque about the center of the wave packet. The semiclassical expression for the torque moment is pτσ i=i 2[∝an}b∇acketle{tui|˙ˆsσ|∂ui ∂k∝an}b∇acket∇i}ht−∝an}b∇acketle{t∂ui ∂k|˙ˆsσ|ui∝an}b∇acket∇i}ht]−∝an}b∇acketle{tui|i∂ui ∂k∝an}b∇acket∇i}ht∝an}b∇acketle{tui|˙ˆsσ|ui∝an}b∇acket∇i}ht.(24) The torque moment has the same gauge invarianceproperties as th e spin dipole, and like the spin dipole it also does not depend on the wave packet width . It is important to note that the spin current, Jσ, can be simplified to: Jσ(R,t) =/integraldisplay d3kftr∝an}b∇acketle{tui|ˆsσˆv|ui∝an}b∇acket∇i}ht, (25) which is the semiclassical equivalent of the Kubo formula for spin curr ents. 3.2 Density matrix picture of spin transport Semiclassical theory provides a straightforward, intuitive picture of the way spin currents arise in the course of carrier dynamics in an electric fie ld. The theory was developed for independent bands. It turns out that in terband co- herence arising from scattering is crucial in spin transport, and is d ifficult to treat semiclassically. Although the semiclassical theory can be gene ralized to multiple bands [49, 50], it is more instructive to examine spin transport from a different point of view that is closer in outlook to the philosophy under lying the Kubo formula (with which the semiclassical theory agrees.) This will sh ed some light on additional issues, such as the relationships between spin cur rents and spin precession, between spin currents and spin densities, the com plex effect of disorder and the vanishing of spin current in certain systems. A large, uniform system of non-interacting spin-1/2 electrons is re presented by a one-particle density operator ˆ ρ. The expectation value of an observable represented by a Hermitian operator ˆOis given by tr(ˆ ρˆO) and ˆρsatisfies the quantum Liouville equation dˆρ dt+i /planckover2pi1[ˆH+ˆU,ˆρ] = 0. (26) 11The Liouville equation is projected onto a set of time-independent st ates of definite wave vector {|ks∝an}b∇acket∇i}ht}, which are not assumed to be eigenstates of the Hamiltonian ˆH. The matrix elements of ˆ ρin this basis will be written as ρkk′≡ ρss′ kk′=∝an}b∇acketle{tks|ˆρ|k′s′∝an}b∇acket∇i}ht. Spin indices are not shown explicitly, and ρkk′is a matrix in spin space, referred to as the density matrix. In this work we req uire the expectation values of operators which are diagonal in wave vector , and will thus requirethepartofthedensitymatrixdiagonalinwavevector, ρkk≡fk=nk11+ Sk. In the presence ofa constant uniform electric field E,fk=f0k+fEk, where the equilibrium density matrix f0kis given by the Fermi-Dirac distribution, and the correction fEkis due to the E. We subdivide f0k=n0k11 +S0kand fEk=nEk11+SEk. The spin-dependent part of the nonequilibrium correction to the density matrix SEkis interpreted as the spin density induced by E. The equations governing the time evolution of nEkandSEkis [51] ∂nEk ∂t+ˆJ0(nEk) =eE /planckover2pi1·∂n0k ∂k ∂SEk ∂t+i /planckover2pi1[Hk,SEk]+ˆJ0(SEk) =eE /planckover2pi1·∂S0k ∂k−ˆJs(nEk)≡ΣEk,(27) where the scalar part of the scattering operator ˆJ0and its spin-dependent part ˆJshave been defined in Ref. [51]. The equation for nEkhas the well-known solution nEk= (eEτp//planckover2pi1)·(∂n0k/∂k), in other words, nEkdescribes the shift of the Fermi sphere in the presence of the electric field E, with the momentum relaxation time τp. It is seen from Eq. (27) that spin-dependent scattering gives rise to a renormalization of the driving term in the equation for SEk. This renormalization has no analog in charge transport. Weneedtofindtheexpectationvalueofthespincurrentoperator ˆJσ idefined in the introduction. In the systems understudy the spin currento peratorcan be written as ˆJσ i=/planckover2pi1kisσ/m∗+(1/4/planckover2pi1)∂Ωσ/∂ki11. We need to determine SEk. To this end we remember that an electron spin at wave vector kprecesses about an effective magnetic field Ωk. The spin can be resolved into components parallel and perpendicular to Ωk. In the course of spin precession the component of the spin parallel to Ωkis conserved, while the perpendicular component is continually changing. Corresponding to this decomposition of the sp in is an analogous decomposition of the spin distribution SEkinto a part representing conserved spin and a part representing precessing spin, denoted bySEk/bardbland SEk⊥respectively. There is an analogous decomposition of the source on the RHS of Eq. (27) into Σ Ek/bardbland Σ Ek⊥. This decomposition is carried out by introducing projection operators P/bardblandP⊥as described in Ref. [51], giving for SEk/bardblandSEk⊥in the weak momentum scattering limit ∂SEk/bardbl ∂t+P/bardblˆJ0(SEk) = ΣEk/bardbl, (28a) ∂SEk⊥ ∂t+i /planckover2pi1[Hk,SEk⊥] = ΣEk⊥−P⊥ˆJ0(SEk). (28b) Equation (28b) shows that scattering mixes the distributions of co nserved and 12precessing spins. This is so because when one spin at wave vector kand pre- cessing about Ωkis scattered to wave vector k′and precesses about Ωk′, its conserved component changes, a process which alters the distrib utions of con- served and precessing spin. Equations (28) can be solved straight forwardly if one assumes the impurity potential to be short-ranged, obtaining [51] SEk/bardbl= ΣEk/bardblτp+P/bardbl(1−¯P/bardbl)−1¯ΣEk/bardblτp, (29a) SEk⊥=Ωk×(ΣEk⊥τp+P⊥¯SEk/bardbl)·στp 2/planckover2pi1(1+Ω2 kτ2p//planckover2pi12)−(ΣEk⊥τp+P⊥¯SEk/bardbl) 1+Ω2 kτ2p//planckover2pi12.(29b) The correction SEk/bardbldoes not give rise to a spin current. Inspection of Eq. (29a) shows that integrals of the form/integraltext dθˆJσ iSEk/bardblcontain an odd number of pow- ers ofkand are therefore zero. It can, however, give rise to a nonequilibr ium spin density, since integrals of the form/integraltext dθˆsσSEk/bardblcontain an even number of powers of kand may be nonzero. Similarly SEk⊥does not lead to a nonequi- librium spin density. The expectation value of the spin operator yields integrals of the form/integraltextdθˆsσS(0) Ek⊥, which involve odd numbers of powers of kand are therefore zero. This term does, however, give rise to nonzero sp in currents, since integrals if the form/integraltext dθˆJσ iSEk⊥contain an even numbers of powers ofkand may be nonzero. Therefore, in the absence of spin-orbit coup ling in the scattering potential, nonequilibrium spin currents arise from sp in precession (as outlined by Sinova et al.[35]), and nonequilibrium spin densities from the absence of spin precession. The dominant contribution to the none quilibrium spin density in an electric field exists because in the course of spin pre cession a component of each individual spin is preserved. For an electron wit h wave vectork, this spin component is parallel to Ωk. In equilibrium the average of these conserved components is zero. When an electric field is app lied the Fermi surface is shifted and the average of the conserved spin co mponents may be nonzero, as illustrated in Fig. 1. This argument explains why the no nequi- librium spin density ∝τ−1 pandrequires scattering to balance the drift of the Fermi surface. Although spin densities in electric fields require band structure spin-orbit interactions and therefore spin precession, the domina nt contribution arises as a result of the absence of spin precession. Systems in which Ωkis linear in kare special, in that the spin current as defined in this section vanishes [39, 40, 41, 42, 43, 44, 45, 46]. This is because of the renormalization of the driving term on the RHS of Eq. (28b) fo rSEk⊥, in other words because of scattering from the conserved spin dist ribution to the precessing spin distribution. In Eq. (29b) it is also clear that if Σ Ek⊥τp+ P⊥¯SEk/bardblvanishes, then all the contributions to SEk⊥also vanish. Since ¯SEk/bardbl effectively represents a steady-state spin density, we see that t he presence of this spin density tends to diminish the spin current. In systems with e nergy dispersion linear in kit cancels the spin current completely. 13kxky kxky 0 00 0E=0 E>0 (b) (a) Figure 3: Effective field Ωkat the Fermi energy in the Rashba model [18] (a) without ( E= 0) and (b) with an external electric field ( E >0). 4 Future Directions Whereasthe community appearsto be in agreementthat spin curre ntsexist and aremeasurable,manyquestionsremainunanswered. Theoretically ,intrinsicand extrinsic effects (such as due to skew scattering and side jump) ha ve not been studied on the same footing for an arbitrary form of band structu re spin-orbit interactions. The relative magnitude of intrinsic and extrinsic spin cu rrents in such a general system remains to be determined. Also, different de finitions of the spin current give results that often differ by a sign [1, 2]. The rela tionship between spin current and spin accumulation at the boundary is not c lear, again thanks to the non-conservation of spin. It appears that what ha ppens at the boundary is sensitive to the type of boundary conditions assumed. Thus so far as quantitative interpretation of experimental data is concer ned, theory has some way to go. Despite tremendous progress, experiment is still searching for a r eliable way tomeasure, as opposed to detect, spin currents directly. Practically, the ques- tion of what to do with spin once it has been transported/generate d remains. 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2207.10360v1.Spin_orbit_transitions_in_the_N_______3P__J_A______H__2____rightarrow__NH______X_2Π_____4Σ______H___2S___reaction__using_adiabatic_and_mixed_quantum_adiabatic_statistical_approaches.pdf

Submitted to Journal of Chemical Physics (2022), in press. Spin-orbit transitions in the N+(3PJA) + H 2!NH+(X2P,4S)+ H(2S) reaction, using adiabatic and mixed quantum-adiabatic statistical approaches Susana Gómez-Carrasco Facultad de Farmacia, Universidad de Salamanca, Campus Miguel de Unamuno, C. Lic. Méndez Nieto, s/n, 37007-Salamanca, Spain Daniel Félix-González and Alfredo Aguado Unidad Asociada UAM-CSIC, Departamento de Química Física Aplicada, Facultad de Ciencias M-14, Universidad Autónoma de Madrid, 28049, Madrid, Spain Octavio Roncero† Instituto de Física Fundamental (IFF-CSIC), C.S.I.C., Serrano 123, 28006 Madrid, Spain The cross section and rate constants for the title reaction are calculated for all the spin-orbit states of N+(3PJA) using two statistical approaches, one purely adiabatic and the other one mixing quantum capture for the entrance channel and adiabatic treatment for the products channel. This is made by using a symmetry adapted basis set combining electronic (spin and orbital) and nuclear angular momenta in the reactants channel. To this aim, accurate ab initio calculations are performed separately for reactants and products. In the reactants channel, the three lowest electronic states (without spin-orbit couplings) have been diabatized, and the spin-orbit cou- plings have been introduced through a model localizing the spin-orbit interactions in the N+atom, which yields accurate results as compared to ab initio calculations including spin-orbit couplings. For the products, eleven purely adiabatic spin-orbit states have been determined with ab initio calculations. The reactive rate constants thus obtained are in very good agreement with the available experimental data for several ortho-H 2fractions, assuming a thermal initial distribution of spin-orbit states. The rate constants for selected spin-orbit JAstates are obtained, to provide a proper validation of the spin-orbit effects to obtain the experimental rate constants. I. INTRODUCTION The formation of hydrides can be considered as the first step of chemistry in space and determines the abundances of more complex molecules arising in chemical networks from them. The study of the evolution of abundances of molec- ular species allows the probe of physical conditions along the stellar evolution, from the parent molecular cloud to the star system, passing through the intermediate stages such as cold and hot cores, protoplanetary disk, etc. Among the most abundant elements, nitrogen plays a singular role, because its more abundant forms are thought to be N 2and atomic nitro- gen, which are difficult to be observed because they have no permanent dipole moment, specially in cold cores. The abun- dance of nitrogen is then established by other molecules, such as its hydrides NH n, CN, HCN/HNC, N 2H+, etc, requiring the construction of increasingly more accurate chemical networks [1, 2]. Nitrogen hydrides are particularly interesting and ammonia is among the first polyatomic molecules detected in the inter- stelar medium (ISM) [3]. The ortho/para ratios observed for NH 2and NH 3[4] and their deuteration enrichment [5] serve as sensitive probes to check gas-phase chemistry models [2]. In this regard, hydrides present a comparably small number of reactions in the chemical networks. In photodissociation regions (PDR), hydrides are normally formed from the atoms Electronic address: susana.gomez@usal.es †Electronic address: octavio.roncero@csic.es(neutral and/or cations, depending on their ionization poten- tial, as compared to atomic hydrogen) by successive addition of hydrogen atoms, followed by dissociative recombination with electrons in the case of cations. The ionization step in nitrogen in PDR is improbable difficult because its ioniza- tion energy is larger than that of hydrogen, unlike most of other metal atoms, and the density of N+is therefore smaller. Therefore other neutral reactions of N atoms with OH and CH are alternative routes to form nitrogen hydrides [2]. The rate constants involved in the first steps of the chemi- cal networks have an enormous influence in the relative abun- dances, ortho/para ratios and deuteration fractions of many of the nitrogen-bearing molecules. For these reasons many ex- periments have been performed to study the following reac- tion [6–9]: N+(3PJA)+H2(X1S+ g)!NH+(2P;4S)+H(2S) (1) These experiments are performed in different conditions, which raises questions about the reactivity associated to each fine structure state of N+(3PJA), since the exact thermalization conditions are not known. Theoretical dynamical calculations have been performed on the ground adiabatic electronic state potential [10–13], both classical [13, 14] and quantum [15, 16] ones, without tak- ing into account the fine structure of nitrogen. These studies demonstrate that the reaction dynamics in the ground adia- batic state is mediated by many long lived resonances due to the deep insertion well of the potential energy surface (PES). These calculations suggest that the reaction proceeds statis- tically, but none of them describe any electronic transition among spin-orbit states.arXiv:2207.10360v1 [physics.chem-ph] 21 Jul 20222 Several statistical simulations have been recently per- formed including the fine structure [17, 18]. However, in these statistical simulations only long range interactions are included, within the assumption that only the first 3 adiabatic fine-structure states can react. However, the inclusion of tran- sitions among the different spin-obit states in the entrance channel may include important variations of the experimen- tally determined rate constants, specially at low temperature, as it has been discussed by Zymak et al. [8] and Fanghanel [9]. The main goal of this work is simulating the transitions between the fine structure N+(3PJA) states, determining the cross sections and rates for each of them individually. Since the problem involves 9 spin-orbit states, some of them show- ing deep insertion wells, complete quantum calculations are not feasible. For this reason, in this work a detailed poten- tial model is developed separately for reactants and products, all based on accurate ab initio calculations. In the N+(3PJA) + H 2reactants channel, a diabatic model is developed allow- ing to include the couplings among the spin-orbit states. In the products channel, pure adiabatic spin-orbit potentials are calculated. These diabatic states are used to build total elec- tronic and angular basis sets allowing the study of the corre- lation of angular momenta, electronic and nuclear, to prop- erly describe spin-orbit transitions[19]. These basis set func- tions are then used within an adiabatic statistical (AS) approx- imation [20] and mixed description of the AS and a quantum statistical (QS) [21–24] (denoted by the acronym QAS). The AS approximation has been recently applied to the study of many reactions and inelastic processes for many systems and is widely used[25]. A precedent of mixing quantum capture in the entrance channel and statistical approaches to describe the reaction probability has been proposed previously for four atom complex-forming reactions[26]. In this work, the calcu- lation of quantum capture probabilities is done with a time- independent method based on a renormalized Numerov prop- agation scheme developed to this aim and presented in the Appendix A. This work is organized as follows. A detailed ab initio study of the system will be described in section II, treat- ing separately reactants, N+(3PJA) + H 2(X1S+ g), and prod- ucts, H(2S)+ NH+(2P;4S) channels, including spin-orbit couplings. In the case of the reactants, a diabatic model is built for the different N+(3PJA) states, which is necessary to include the transitions among them. The PESs will be used to calculate the capture probabilities needed in the quantum and adiabatic statistical methods, and described in section III, paying special attention to the transitions among different fine structure states. Also cross sections and rate coefficients for each individual state will be presented in section IV. Finally, in section V, some conclusion will be extracted. II. POTENTIAL ENERGY SURFACES An overall picture of the electronic states of reactants and products of this system is displayed in Fig. 1. In the reactant region, the N(4S) + H+ 2(X2S+ g) channel is located about 1 eV 0 1 2 3 4 5Energy / eV N(4S) + H2+ (2Σg+) N+(3P) + H2 (X1Σg+)N(4S) + H++ H (2S)N+(3P) + H (2S) + H (2S)REACTANTS PRODUCTS 0 1 2 3 4 5 Energy / eV NH(3Σ-) + H+ NH+(4Σ-) + H(2S) NH+(X2Π) + H(2S)FIG. 1: Asymptotic electronic states for reactants and products above the N+(3PJA) + H 2(X1S+ g) one so, the former channel will not be populated at the energy range used in this work. Regarding the product region, the lowest channels are NH+(X 2P, a4S) + H(2S) and NH(3S) + H+ones. A. Ab initio calculations for the reactant channel The N+(3Pg) +H2(X1S+ g)reactant channel has been cal- culated using a state-average complete active space self- consistent field/multireference configuration interaction (SA- CASSCF/MRCI) method with a VTZ-F12 explicitly cor- related atomic basis set as implemented in the MOLPRO program[27]. Without taking spin-orbit coupling into account, three adiabatic electronic states,3Pand3S, correlate with the reactants in the C ¥vgroup of symmetry. The calculations have been done in the Cspoint group of symmetry so that the state average multiconfigurational wave function has included two3A00and one3A0states, with the molecule lying on the y-z plane. Subsequent MRCI energies have been obtained at the geometries described in the Supplementary Information (SI). The ab initio points have been interpolated using a 3D cubic spline method. Finally, the long-range terms, charge-induced dipole and quadrupole [28–30], have been included for R> 15 a0, using the following switching function of R centered at 20 a0: f(R) =1tanh(0:5[R20:0]) 2(2) The long-range terms included are described in detail in the Supplementary Information (SI), together with some figures describing the main features of the PESs. Here we shall use a non-relativistic atomic basis set (here- after called diabatic basis set) jLLSSi, where LandSare the modula of the electronic orbital and spin angular momenta of N+, andLandStheir projections, respectively, on the Jacobi body-fixed z-axis. In this basis, the non-relativistic electronic matrix takes the form [31] H=0 @E1V0 V E 0V 0V E 11 A with E1=E1 (3)3 60 120 Θ (degrees)135R (Å) −1.5−0.5 0.5 0.5 1 1.5 r (Å) −1.5−0.5 0.5 135 −1.5−0.5 0.5 −1.5−0.5 0.5 135 −1.5−0.5 0.5 −1.5−0.5 0.5 3Π 60 120 Θ (degrees)1353Π r (Å) 3Σ 1353Σ VΣΠ 135 VΣΠ FIG. 2: Contour plots of the PESs for the diabatic3S,3Pelec- tronic states, and the SPcoupling, obtained at the H 2equilibrium distance, r= 0.7 Å as a function of the Jacobi distance, R, and the angle g(left panels) and at g= 90oas a function of R and r Jacobi distances. Energies are in eV , and the contour lines are at 0, 0.5 and 1 eV . whose eigenvalues correspond to the 13A00, 13A0and 23A00adi- abatic electronic energies. The three unknown E1;E0andVin Eq. (3) can then be expressed in terms of the ab initio energies as [31] E1=E13A0 E0=E13A00+E23A00E13A0 (4) V=r (E13A00E23A00)2(E1E0)2 8: These diabatic energies are represented in Fig. 2, and the cou- pling Vin top panels reveal that the coupling between the S andPstates become larger in the repulsive parts or the adia- batic PESs, where the 13A0and 23A00differ the most. The spin-orbit basis set, jJAWAi, is expressed in terms of the diabatic representation jLLSSidefined above as jJAWAi=å L;S(1)LS+WAp 2JA+1) L S J A L SWA jLLSSi; (5) where





 are 3-j symbols. Since H 2is closed-shell, the -0.15-0.1-0.050 4.0 6.0 8.010.0 12.0 14.0N+ (3P0) + H2N+ (3P2) + H2 RN+ -----H2 / a. u.SO1 SO4 SO7-0.15-0.1-0.050 N+ (3P1) + H2N+ (3P2) + H2Energy / eV. SO2 SO5 SO8-0.15-0.1-0.050 N+ (3P1) + H2N+ (3P2) + H2 SO3 SO6 SO9FIG. 3: Energy profiles for the 9 spin-orbit electronic states correlat- ing with N+(3PJA=2;1;0)+H 2as a function of the R Jacobi coordinate for r= 1.4 a.u. and g=90 degrees. The energy curves are distributed in three panels (1, 4 and 7 in the bottom panel, 2, 5 and 8 in the middle panel and 3 ,6 and 9 in the top panel) to show more clearly the differences between ab initio (points) and the diabatic+atomic spin-orbit model (lines), using the ab initio spin-orbit splittings. total orbital ( L) and spin ( S) electronic angular momenta cor- respond to atom N+(3PJA), with JA=L+S, so that we shall consider that H SOdoes not depend on the distance R, and has eigenvectorsjJAWAi, whose EJAeigenvalues are (2JA+1) degenerate. Following the treatment of Jouvet and Beswick [19], summarized in the Supplementary Information for com- pleteness, the electronic Hamiltonian is expressed as H= HSO+Hel, with Hel=Ho el+H1 elwith lim R!¥H1 el=0; (6) i:e:,H1 eldescribes the non-relativistic interaction between H 2 and N+, while H0 eldescribes the two fragment at infinity. The4 matrix elements of H1 elare defined as (see SI and Ref [19]) JAWAH1 elJ0 AW0 A =å L;L0å Så k(1)WAW0 A (7) VK LL0(r;R) YKL0L(g;0)q (2JA+1)(2J0 A+1)  L S J A L SWA L S J0 A L0SW0 A ; In thejJAWAibasis set, the atomic spin-orbit Hamiltonian, HSO, is diagonal. The experimental atomic spin-orbit split- tings are 48.7 and 130.8 cm1from NIST[32]. The 9 spin-orbit electronic states correlating with N+(3PJA=0;1;2)+H 2have been calculated at MRCI level using the Breit-Pauli operator. At very long distances between N+ and H 2, the ab initio calculations yield 40.2 cm1and 120.6 cm1for the energy of the N+(3P1)and N+(3P2)spin-orbit levels, respectively, respect to the energy of the ground spin- orbit state N+(3P0). These results are in good agreement with the experimental values. In Fig. 3, the adiabatic spin-orbit energies obtained in the ab initio calculations are compared to those obtained diagonalizing the Hel+HSOHamiltonian (see SI for more information), in which the spin-orbit term is considered to only affect N+(3PJA)subsystem, using the ab initio spin-orbit splittings. The agreement is fairly good specially at long distances, and only some discrepancies are found in the region of the bottom of the well. This validates the approximation of considering the spin-orbit term only for the N+atom in the entrance channel. Within this approximation, electronic transitions between all the spin-orbit states in the entrance channel will be considered in the statistical calculations presented below, using the experimental splittings. B. Ab initio calculations for the product channels Looking at the products side in Fig. 1, the three lowest channels that could be energetically accessible at the collision energies used in this work correlate with NH+(X2P)+H(2S), NH+(a4S) +H(2S)andNH(3S) +H+asymptotes. The NH+(A2S) +H(2S)andNH+(b4P) +H(2S)channels are too high in energy. As done for reactants, a SA-CASSCF/MRCI method has been used to calculate the product channels. The electronic states correlating with the three lowest channels without tak- ing into account spin-orbit coupling (in C¥vpoint group of symmetry) are shown in Table I. Since the ab initio calculations are done in the C ssymmetry, the state average CASSCF wavefunction has included one3A0, three3A00, one1A0, one1A00and one5A00states. In the products channel, we shall use the adiabatic spin- orbit ab initio states, without considering the couplings among them, in contrast with the treatment described above for the reactants channel. We have focused on the states correlat- ing with the two lowest channels, i.e., NH+(X2P) +H(2S) andNH+(4S)+H(2S)(see Table I). That involves a total ofProduct asymptote C ¥v NH+(X2P)+ H(2S)3P;1P NH+(a4S)+ H(2S)5S;3S NH ( X3S) + H+ 3S TABLE I: Electronic states correlating with the three lowest product channels. 16 SO states. However, since we need to know the symme- try of the spin-orbit states under the reflection respect to the molecular plane, A0orA00, the quintuplet electronic states have not been included because they are repulsive and the sym- metry treatment is not yet implemented in the Molpro 2015 program. In any case, we have checked that their omission does not affect much the accuracy of the calculations. So, finally, 11 adiabatic spin-orbit energies have been obtained, which are shown in Figs. 4 (see also SI). Among those 11 states, 8 of them correlate with the lowest product channel, NH+(X2P3=2;1=2) +H(2S), and the other 3 connect with the NH+(a4S)+H(2S)asymptote. Fig 4 shows the energy pro- files of the 11 SO-states as a function of the R product Jacobi coordinate, for even and odd symmetries with respect to re- flection through the plane of the molecule. These curves show several crossings among the spin-orbit states, which do not oc- cur for all the angles. The anisotropy of the potential depend a lot on the existence or not of such crossings. Thus, the lowest spin orbit states on each symmetry are clearly connected to the deep insertion well for g>60o. However, those intermediates presenting a crossing about g=90o, present a narrower well only in the 60 <g<120o, and this will reduce the capture probabilities, as discussed below. Finally, the higher states do not present wells and they will be neglected in the statistical calculations presented in this work. Another issue which is not yet clear for this system, it is the ergicity of this reaction. Experimentally this reaction has been found to be endoergic by 18 2 meV [6]. Gerlich[33] compared the measured temperature dependencies on the rate constants with a statistical theory for n-H 2and proposed an endoergicity of 17 meV . Our calculations yield an endoergic- ity of 80 meV , including zero point energies of reactants and products. Below, we shall use the value of 17 meV . III. QUANTUM STATISTICAL CALCULATIONS The thermal reaction rate constant is defined as K(T) =å q;b=1wq;b=1(T)å q0b0Kbq;b0q0(T) (8) with wq;b=(2Ibc+1)(2jbc+1)(2JA+1)eEbq=kBT åq00b00(2Ibc+1)(2jbc+1)(2JA+1)eEb00q00=kBT where the sum is over all vibrational, rotational and electronic states of the reactants, H 2(X1S+, v j) + N+(3PJA), of energy Ebq. In these expressions, b;q;mare collective quantum num- bers specifying the particular state of reactants and products.5 -7-6-5-4-3-2-1 0 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Energy/eVγ=180o SO1+ SO2+SO3+ SO4+SO5+ SO6+-0.05 0 0.05 4 5 6NH+ (2Π1/2) + HNH+ (2Π3/2) + HNH+ (4Σ-) + H -7-6-5-4-3-2-1 0 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5γ=180o SO7- SO8-SO9- SO10-SO11--0.05 0 0.05 4 5 6NH+ (2Π1/2) + HNH+ (2Π3/2) + HNH+ (4Σ-) + H -7-6-5-4-3-2-1 0 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5Energy/eV RNH+---H/a.u.γ=90o SO1+ SO2+SO3+ SO4+SO5+ SO6+-0.1-0.05 0 0.05 34567NH+ (2Π1/2) + HNH+ (2Π3/2) + HNH+ (4Σ-) + H -7-6-5-4-3-2-1 0 1 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 RNH+---H/a.u.γ=90o SO7- SO8-SO9- SO10-SO11--0.1-0.05 0 0.05 34567NH+ (2Π1/2) + HNH+ (2Π3/2) + HNH+ (4Σ-) + H FIG. 4: Energy profiles for the 11 spin-orbit electronic states (6 even, left panels, and 5 odd, right panels, with respect to the reflection through the plane of the molecules) correlating with NH+(X2P;a4S)+H(2S) products as a function of the R Jacobi coordinate for r= 2 a.u. and g=90 (bottom panels) and 180 (top panels) degrees. Energies are in eV . b=1;2;3 denote the arrangement channel, H 2+N+,and the two equivalent H + HN+and NH++H channels of products, respectively. q=ebc;vbc;jbc;Ibc;JAare the electronic, vibra- tional, rotational and nuclear spin quantum numbers of the BC fragment ( Ibc=0 and 1 for para/ortho H 2), while JAdenotes the electronic angular momentum of the atomic fragment. Finally, m=Wbc;WAare the projections of the angular momentum of the diatomic and atomic fragment in the body-fixed z-axis, respectively, in each rearrangement channel. Kb=1q;b0q0(T) are the state-to-state rate constants, which correspond to the Boltzmann average over the translation energy, E, of the reac- tion state-to-state cross section Kbq;b0q0(T) =s 8 pm(kBT)3Z dE E sbq;b0q0(E)eE=kBT:(9) The cross section is obtained under the partial wave sum- mation over the total angular momentum, J, and parity under inversion of spatial coordinates, p, as sbq;b0q0(E) =p (2jbc+1)(2JA+1)k2 bq(E) å Jpå mm0(2J+1)PJp bqm;b0q0m0(E);(10)where kbq=q 2m(EEbq)=¯h(with mbeing the H 2+ N+ reduced mass), and Ebeing the total energy. PJp bqm;b0q0m0(E) =jSJp bqm;b0q0m0j2(E)are the state-to-state re- action probability from a particular initial state (b;q;m)of the reactants to a final state of products ( b0q0;m0). This quan- tity can be calculated with different methods, exact and ap- proximate, quantum and classical. In the statistical approach [34, 35] the state-to-state reaction probability is calculated as PJp bqm;b0q0m0(E) =CJp bqm(E)BJp b0q0m0(E) (11) with the branching ratio matrix, BJp b0q0m0, being defined as BJp b0q0m0(E) =CJp b0q0m0(E) åb00q00m00CJp b00q00m00(E)(12) where the sum in the denominator runs over all the accessi- ble states of reactants and products. For J=j=0 there are many forbidden channels. This factorization, allows to define6 a capture cross section as sC q;b=1=pk2 v jJA(E) (2j+1)(2JA+1)å Jpå m(2J+1)CJp bqm(E): (13) In the case of very exothermic reactions, the capture cross sec- tion coincides with the reactive cross section. In this factor- ization, we could define approximately the cross section as sbq;b0q0(E)sC bq(E)Bb0q0(E) (14) with Bb0q0(E) =å m0å JpBJp b0q0m0(E); (15) which would only be accurate when the individual BJp b0q0m0(E) do not strongly depend on Jandp. Otherwise it can only be taken as an approximation for complex forming reactions. The different statistical approaches depend on the proce- dure followed to calculate the CJp bqmcapture probabilities. In the present work we use the quantum statistical [21–24], and the adiabatic statistical [20, 36, 37] approaches. In the quantum statistical approximation, a set of inelas- tic close-coupled equations is solved for each rearrangement channel independently imposing complex boundary condi- tions at short distances as described in the Appendix A. For doing so, we have developed here a new program based in the Renormalized Numerov method (called aZticc), as described in the Appendix. The original coupled nuclear-electronic di- abatic basis set used for N+(3PJA)+ H 2reactants is that of Ref.[19],jJMWjJAWApi, which are linear combinations of functions jJMWjJAWAi=r 2J+1 8p2DJ MW(f;q;c)YjWWA(g;0)jJAWAi; (16) with parity p=1 with respect to inversion of spatial coordi- nates. The treatment is described in the SI for completeness, where the matrix elements of the different terms of the Hamil- tonian are also shown. For products, described in an adiabatic spin-orbit approximation, no correlation among electronic and nuclear angular momenta is considered, and we treat them as a particular case with L=S=0. The adiabatic statistical approach [20, 36, 37] uses a classi- cal approach for the capture probability, i:e: CJp bqm=( 1 when E>Eb 0 when E<Eb; (17) where Ebis the energy at the top of the barrier associated to the corresponding adiabatic eigenvalue of the matrix V(R)ap- pearing in the close-coupling equations, in Eq. (A2). 0 0.2 0.4 0.6 0.8 1 0.25 0.3 0.35j=0, JA=1, ΩA=0J=0,p=+1Capture Probability E (eV) 0 0.2 0.4 0.6 0.8 1 j=0, JA=0, ΩA=0 j=0, JA=2, ΩA=0J=0,p=-1 FIG. 5: Quantum capture probabilities obtained for the N+(3PJA;JA=0;1;2) +H 2(v=0,j=0) for J= 0, and p = +1, bottom panel, and p =1, top panel. Blue and red probabilities correspond to R c= 3 bohr, while green and orange to R c= 4 bohr IV . RESULTS AND DISCUSSIONS A. Quantum versus classical capture probabilities We start by showing the quantum capture probabilities cal- culated with the aZticc program, described in the appendix. The details of the numerical calculations are described in the Supplementary Information. In Fig. 5 the capture probabil- ities obtained for j=0,J=0 and p=1 are shown, where the full spin-orbit fine structure of N+(3PJA) is considered with the symmetry restrictions introduced by the treatment of Jouvet and Beswick[19] (in the SI). The diabatic channel JA=0;WA=0 and JA=2;WA=0 (appearing for p=1) are directly connected to the insertion well and they show a larger capture probability. The diabatic channel JA=1;WA=0 presents a barrier, but it presents a non-negligible capture probability, and this is only possible to non-adiabatic transi- tion. It is important to note that the quantum capture probabilities7 are rather different from the classical ones, which are 1 above the barrier. These results are obtained with a capture radius of Rc= 3 and 4 bohr, as indicated in the caption of Fig. 5. The captures probabilities depend on the Rc[24, 38]. To consider other Rc, we can not do it by simply setting Rmin=Rc, because the repulsive electronic states are still open for some ener- gies. Instead, we set the adiabatic-to-adiabatic transfer matrix Si j=1 below Rc, in Eq. (A16). By setting Rc=4 bohr, the quantum capture probabilities increase a lot, becoming very close to 1, i:e:very similar to the classical capture probabil- ities using in the adiabatic statistical approach. This demon- strate that capture probabilities decreases because of the tran- sition among different channels, which reflect back part of the incoming flux. The quantum capture converges rapidly, and forRc= 3.5 bohr the results are nearly indistinguishable to those shown for Rc= 3 bohr in Fig. 5. It is important to note here that, depending on the parity, p, and total angular momentum, J, not all the spin orbit-states of the atom, JA;WA, exist due to symmetry restrictions. This is particularly important for j=0, for which only WWA= 0 exists. For J=0;p= +1,j=0, in the top panel of Fig. 5, only the functions with JA=0;WA=0 and JA=2;WA=0 exist, while for J=0;p=1,j=0, only JA=1;WA=0 appears. As Jand jincreases, and thefore W, more JA;WAstates partici- pate. This makes appear contributions from the three values of the JA=0, 1, and 2 to the reactive cross section. This oc- curs in the quantum as well as in the pure adiabatic statistical approaches, as a consequence of using a coupled basis set for electronic and nuclear angular momenta. This is not the case of previous treatments [17, 18], where it is assumed that only the three lower adiabatic spin-orbit states of N+(3PJA), cor- relating to JA=0 and 1 react, while the six higher adiabatic spin-orbit states do not react. In the products channel describing the NH+(2S+1LW) + H collision, independent adiabatic spin-orbit states are consid- ered in this work. The capture probabilities calculated with the quantum and adiabatic (or classical) approaches are pre- sented in Fig. 6. In general the capture probabilities for a sin- gle adiabatic state are larger and with less structure. Narrow resonances are in general absent. For states 3 and 8, the quan- tum capture is nearly 1 and constant, as in the adiabatic case. This is an indication that the PES anisotropy and anharmonic- ity do not change from NH+products along the channel up to capture. For states 1 and 2, the quantum probability oscillates slightly around 0.95, i.e. is rather constant, and the error of the adiabatic capture is of the order of 5%. The most extreme cases are states 4 and 9, for which the quantum capture prob- ability is in the interval 0.7-0.75, so that we consider that in these cases the adiabatic capture produces a relatively large error, of30%, but nearly constant with energy. This trends persist for higher J, and one possible approximation could be to multiply the adiabatic capture probability by a correction factor, depending on the electronic state and energy indepen- dent, and this is done below for the mixed quantum-adiabatic statistical approach. All these results demonstrate that quantum capture proba- bilities are in general lower than the classical ones, which take a value of 1 for all the adiabatic electronic states. This reduc- 0.70.80.91 0.3 0.4SO1+: NH+(2Π1/2)Capture probability Total Energy (eV)Quantum Adiabatic 0.3 0.4SO2+: NH+(2Π1/2) Quantum Adiabatic0.70.80.91 SO3+: NH+(2Π3/2) Quantum Adiabatic SO4+: NH+(2Π3/2) Quantum Adiabatic0.70.80.91 SO8-: NH+(2Π1/2) Quantum Adiabatic SO9-: NH+(2Π3/2) Quantum AdiabaticFIG. 6: Capture probabilities obtained for the NH+(e;v=0;j=0) +H for J= 0, and p = +1, with R c= 3 bohr. Those corresponding to SO7- is equal to SO2+, SO5+ is SO4+, and SO6+, SO10- and SO11- are repulsive with no capture. tion is particularly important when several electronic states are considered, for which electronic transitions occur specially at the crossings. When only one electronic state is considered, as it is the case for product arrangement, the curves associated to different channels are nearly parallel, what reduces consid- erably the transitions among them before being captured. In these cases, the quantum capture probabilities are much closer to one, in general, close to the adiabatic statistical approxi- mation. It should be noted, that the anisotropy of the single adiabatic potential (see SO4+ and SO9- in Fig. 6) introduces crossings among rotational channels that can also reduce the capture. B. Total reactive cross section The reaction cross section for this reaction was measured by Sunderlin and Armentrout [7] in a rather broad collision8 10 100 0.01 0.1Reactive cross section (Å2) Collision Energy (eV)AS 305 K QAS3 305 K QAS4 305 K Exp. 305 K 10 100 0.01 0.1 AS 105 K QAS3 105 K QAS4 105 K Exp. 105 K FIG. 7: Thermal Reactive cross section for H 2+N+(3PJA), includ- ing the thermalized at 305 K (bottom panel) and 105 K (top panel), obtained with the Adiabatic Statistical (AS) and mixed- Quan- tum/adiabatic statistical (QAS) methods. The experimental values are from Ref. [7]. The results obtained with the two approaches are also convoluted with the Doppler broadening, according to Chantry[39] shown with dotted lines. energy interval. In these experiments, the H 2reactants are considered at two temperatures 105 and 305 K, and the results are broadened by the ion energy spread and Doppler broaden- ing [7]. In Fig. 7, the experimental results at 305 K and 105 K are compared with those obtained in this work with the AS and the QAS methods. The theoretical results convoluted with a gaussian accounting for the Doppler broadening according with the method of Chantry[39] are also shown in the figure, showing a slight increase of the cross section. However, this increase is not enough to match the experimental results. The QAS results, with Rc=3 and 4 bohr (QAS 3and QAS 4, respectively), are always below the AS results, because the quantum capture probabilities are lower than one, as describedabove. At collision energies below 0.03 eV , the AS results at 305K match very well with the experimental results[7]. This is not the case for 105K. Above 0.03 eV , however, the AS and QAS 4results are above the experimental results, while the QAS 3are below. In fact, AS/QAS 3cross section difference increases with energy, because quantum captures continue de- creasing, while adiabatic captures are always one above the barrier. Above 0.03 eV (for both temperatures), the experi- mental results are in between the AS and QAS 3results, being the QAS 4probably the best matching the experimental results. At 0.2 eV and below, the main contributions arise from SO1+, SO2+ and SO7-, while the other contributions are minor. The contribution of the more excited states is relatively small at these energies, and even if only the SO1+, SO2+ and SO7- are included, the cross section at 0.2 eV obtained with the AS and QAS 4methods are always slightly larger than the experi- mental measurements. However, the QAS 3is below in all the energy interval considered here. The AS treatment considers that all the flux overpassing the effective barrier is trapped, and therefore is treated sta- tistically. However, when considering a quantum capture ap- proach, we have demonstrated that it strongly depends on the capture radius [24, 38]. The problem is therefore to determine the trapping region, without introducing artificial bias among different channels. In fact, considering too short capture ra- dius includes inelastic transitions in the so-called trapping re- gion, but only within the same rearrangement channel, while in the pure statistical spirit it should be considered among all rearragement channels. To avoid this bias, here we used the AS results as a benchmark to determine the best capture ra- dious, without including any unbalance among the different rearrangement channels, what leads to the optimal value of Rc=4 bohr in this case, close to the average possition of the effective barrier used in the AS method. It is worth mentioning, that AS and QAS 4results above 0.2 eV also overestimate the reaction cross section. The reason for this is attributed to the large mass mismatch between N+and H2subsystems, which reduces the energy transfer probability. Statistical asumption, however, implies that energy is com- pletely redistributted among all degrees of freedom, yield- ing to an overstimation of the reaction cross section. This is demonstrated in the SI, where statistical results are compared with complete quantum calculations performed with the wave packet code MADWA VE3[40, 41] using the single adiabatic potential energy surface, PES IV of Ref. [14]. The simulated cross sections change a lot varying the tem- perature from 105 to 305 K. The temperature mainly affects the rotational distribution of H 2in the cell. The cross sec- tions for the individual initial states of the reactants show that H2(j=0) is closed for JA=0 and 1 below 0.01 eV , while it is open for all J Aand for H 2(j=1) at all collision energies. This clearly explains why theoretical thermal cross section varies so much from 105 to 305 K. These changes, however, are not so important in the experimental results, which show a good agreement at 305 K with the AS and QAS 4results, while the agreement is much worse at 105 K. In order to improve the experimental/theoretical agreement, different exothermicities have been considered. This was also9 done by Grozdanov and McCarrol[17], who increased the en- dothermicity from 18.45 meV to 23.45 meV to reduce their cross section, which was slightly overestimated in their ap- proach as compared to the experimental thermal cross sec- tion. However, the variation of the endoergicity, in all cases considered in this work, yield rate constants in considerably worse agreement with the available experimental measure- ments, performed in several studies with different techniques. We therefore conclude that the cross sections measured by Sunderlin and Armentrout [7] at 105 K are also affected by the ion energy spread, as discussed by these authors, which is not accounted for in this work because the exact conditions of those experiments are not known. We also conclude that the endothermicity of 17 meV is the best choice, as shown below. C. Rate constants The thermal rate constants for ortho-H 2fraction f=0.005 and 0.75, of H 2are shown in Fig. 8 and compared with the available experimental data, for the AS (bottom panel), QAS 4 (middle panel) and QAS 3(top panel) methods. There is a rather good qualitative agreement between the two simulated rate constants (AS and QAS 3and QAS 4) and the experimen- tal results. The QAS 3results for f=0.005 agree very well with the experimental measurements of Zymak and et al.[8], and for f=0.75 lies in between the three sets of experimental re- sults for temperatures below 50 K. However, for T50 K and f=0.75, the QAS 3results are considerably lower than any set of experimental results. The QAS 4and AS results are in be- tween all the sets of experimental data in the whole tempera- ture interval considered here, being in general closer to those of Fanghanel[9]. The difference between experimental results allows to establish a certain error, probably due to the exact ortho-H 2fraction f. The variation of the rate constants for more values of f are shown in Fig. 9, for the two best theoretical results, QAS 4 (top panel) and AS(bottom panel). For f=0 (para-H 2), the experimental results of Zymak et al[8] (which were extrapo- lated) are in better agreement with the QAS results than with the pure AS. However, for f=1 (ortho-H 2) the agreement at higher temperatures is better for the AS results than for the QAS. This is probably because the AS results are larger at 0.01 eV than the QAS, and in better agreement with the ex- perimental cross sections, in Fig. 7. The overall agreement of the two simulations, AS and QAS, is in general excellent for temperatures between 20 and 100 K, and the increase of the error for T <20 K could be attributted to small contam- ination of ortho/para ratios, as well as to inaccuracies of the simulations. The agreement between the two sets of experimental rate constants also show some discrepancies. These discrepancies are similar in magnitude to that between simulations and ex- periments. It is important to note the large variation of the rate constant as a function of the ortho-H 2fraction, f, due to the fact that the reaction is exothermic for ortho-H 2(j=1), while it is closed for para-H 2(j=0), whose ratios may change slightly. Moreover, a similar situation holds for the spin-orbit states 1×10−12 1×10−11 1×10−10 1×10−9 0 50 100 150 200ASf=0.75 f=0.005Rate constant (cm3/s) T (K) 1×10−12 1×10−11 1×10−10 1×10−9 0 50 100 150 200QAS4f=0.75 f=0.005 1×10−12 1×10−11 1×10−10 1×10−9 0 50 100 150 200QAS3f=0.75 f=0.005 FIG. 8: Thermal Reactive rate constants for H 2+N+(3PJA), obtained for the two limiting experimental o-H 2fractions, f=0.005 and f=0.75 (n-H 2), as a funtion of temperature. Symbols are the experimental results: open circles by Zymak and et al.[8], open square are taken from Table II of Marquette et al.[6] (the value list at 27 K for n-H 2 has been corrected to read 2.7 1012) and open triangles are the results of Fanghanel[9]. Lines are the simulated rate constants with QAS 3(top panel), QAS 3(middle panel) and the AS (bottom panel) described in this work. of N+(3PJA): for H 2(j=1) all JA=0 and 1 states are open, while for H 2(j=0) only JA=2 is open. The individual rate constants for each JAspin-orbit state and different ortho-H 2fractions are shown in Fig. 10. In the two formalisms, AS and QAS, the rate constants for the 3 spin-orbit states are non-zero. Such situation may introduce changes in the experimental determi- nations of the state specific rate constants, as discussed by Zymak et al. [8] and Fanghanel[9]. AS and QAS methods yields to rather different rate con- stants for each individual JAspin-orbit state. The AS method tends to produce a progression JA= 0, 1 and 2, with the rate10 1×10−13 1×10−12 1×10−11 1×10−10 1×10−9 0 2 4 6 8 1010050 20 10 AS f=1 (j=1) f=0.75f=0.38 f=0.22 f=0.13 f=0.033 f=0.005f=0 (j=0)Rate constant (cm3/s) 100/T (K−1) 1×10−13 1×10−12 1×10−11 1×10−10 1×10−9 0 2 4 6 8 1010050 20 10 QAS4 f=1 (j=1) f=0.75f=0.38 f=0.22 f=0.13 f=0.033 f=0.005f=0 (j=0) T (K) FIG. 9: Thermal Reactive rate constants for H 2+N+(3PJA), obtained for different o-H 2fractions, as described by Zymak and et al.[8]. Open circles are the experimental results extracted from Fig. 4by Zy- mak and et al.[8] (note that for f=0 and 1, their values are extrap- olated). Full circles are taken from Table II of Marquette et al.[6]. Triangles are the experimental results of Fanghanel[9]. Top panel show the mixed Quantum and Adiabatic statistical results for R c= 4 bohr (QAS 4), bottom panel show the pure Adiabatic Statistical re- sults (AS). In all cases, the population of the J Aspin-orbit states cor- respond to a Boltzmann distribution. forJA=2 being the larger, simply because it correspond to the most endothermic case. The situation varies a lot for the QAS results, for which the rates for all JAare closer and their relative importance varies with temperature. This result is a consequence of the explicit treatment of transitions among spin-orbit states, using correlated electronic-nuclear diabatic 04 10−108 10−10 0 100 200p−H2 AS Temperature (K)JA=0 JA=1 JA=204 10−108 10−10 n−H2 ASRate constant (cm3/s) JA=0 JA=1 JA=204 10−108 10−10 o−H2 AS JA=0 JA=1 JA=2 0 100 200p−H2 QAS4 JA=0 JA=1 JA=2 n−H2 QAS4 JA=0 JA=1 JA=2 o−H2 QAS4 JA=0 JA=1 JA=2FIG. 10: Thermal Reactive rate constants for H 2+N+(3PJA), for or- tho, natural and para H 2for each N+(3PJA)spin-orbit state, obtained with the AS (left panels) and QAS 4(right panels) methods. basis set . Since this is accounted for more exactly in the QAS method, in contrast to the AS one, we conclude that the QAS JA-dependent rate constants are more accurate. The nu- merical values of the JA-dependent rate constants are given in the SI. Our results are in general in better agreemnet with the experimental results of Fanghanel[9], where the reactivity of N+(3PJA=2)is considered to be non-zero, as it is demonstrated in this work. The accurate determination of the reaction rate constants is important to improve the accuracy of astrophysical mod- els. The rate constant available in the Kida Data base for this reaction at low temperatures corresponds to the value re- ported by Marquette et al.[6] for the n-H 2(corresponding to an ortho-fraction of f=0.75). These experimental values are compared with the present results in Figs. 8 and 9. This reac- tion, however, strongly depends on the initial rotational state of H 2and also on the spin-orbit state of N+(3PJA), as shown in this work. In detailed astronomical models, it is impor- tant to incorporate the specific rate, at least for ortho and para hydrogen. For this reason we provide in the Supplementary11 Information the parameters obtained in a fit of the numerical rate constants obtained in this work, and shown in Fig. 10, for each ortho-fraction of H 2and each electronic JAvalue for N+, listed in a Table. V . CONCLUSIONS In this work we have studied the spin-orbit depen- dence of the rate constants for the N+(3PJA)+ H 2!H + NH+(2P1=2;3=2). The potential energy surfaces on re- actants and products channels have been calculated sepa- rately, using accurate ab initio methods. In the reactants N+(3PJA)+ H 2channel, the couplings among the spin-orbit states have been calculated, using a diabatization method to- gether with a model based on atomic spin-orbit localized in the N+cation. This method has been compared with accu- rateab initio calculations showing excellent agreement. The NH+(2P1=2;3=2;4S)+H products potential energy surfaces have been calculated in the adiabatic spin-orbit approxima- tion. To account explicitly for the spin-orbit couplings, the treat- ment of Jouvet and Beswick[19] have been implemented within two statistical models: an adiabatic statistical (AS) model and a mixed quantum and adiabatic statistical (QAS) method. A variation of the renormalized Numerov method has been developed to treat open-quantum boundary condi- tions, needed to calculate quantum capture probabilities, used in the mixed quantum-adiabatic statistical method. It is worth noting, that the AS model provide quite accurate rates for all spin-orbit states of N+(3PJA),JA= 0, 1 and 2, when the basis is formed by proper symmetry functions combining electronic (spin and orbital) and nuclear angular momenta. On the contrary, when the adiabatic approximation is done at the spin-orbit electronic states alone first, only the first 3 spin orbit states (correlating to JA= 0 and 1) can contribute to the reactive cross section and rate constants. Thermal cross section and rate constants have been calcu- lated and compared with the available experimental measure- ments. The calculated thermal rate constants for different or- tho fractions of H 2show reasonable good agreement with the experimental measurements of Marquette et al. [6], Zymak et al.[8] and Fanghanel[9], confirming an endothermicity of 17 meV . We find that the three JAspin-orbit states have an appre- ciable contributions rate constants for all the o-H 2fractions, f, measured. In particular, the possible effect of JA=2 in the determination of the rate constants for f= 0 and 1 (not directly measured) was not taken into account by Zymak et al. [8] and it was included and discussed by Fanghanel[9]. We demon- strate here, that it is important to be included for this system, since there are many different energy thresholds, for reactants (JAand jvalues) and products, and are of particular interest for astro physics models of cold molecular clouds.VI. SUPPLEMENTARY MATERIAL See supplementary material for detailed description of the ab initio calculations for the reactants and product channels, for the computational details of the dynamical calculations, the treatment used to treat the collisions of open shell atoms with closed shell diatomic molecules, and the state-specific rate constants for the different spin-orbit states of N+(3PJA) and ortho and para H 2are described and provided in separate files. VII. ACKNOWLEDGEMENTS We want to thank Prof. P. Armentrout for providing us the experimental values of the cross section measurements. The research leading to these results has received funding from MICIYU under grant No. PID2021-122549NB-C2. The calculations have been performed in Trueno-CSIC and CCC- UAM. VIII. DATA A VAILABILITY The data that support the findings of this study are available from the authors upon reasonable request. Appendix A: Quantum capture method 1. Diabatic representation The method used here to evaluate the quantum capture probabilities is very similar to that previously described by Rackham et al. [21]. Expanding the total wave function in a diabatic basis set as Yb(R;x) =å nFb a(R)ja(x) (A1) The close-coupling equations can be written as ¶2Fb a(R) ¶R2=2m ¯h2å a0fVaa0(R)Edaa0gFb a0(R)(A2) F00(R) =2m ¯h2 V(R)1E F(R) where a;bdenotes the collections of quantum numbers needed to specify the channels, and Fis a vector and Vis a matrix. Eq.(A2) are solved here using a Numerov-Fox- Goodwin or renormalized Numerov method [42, 43], in which each of the quantities is discretized in a radial grid of N equidistant Ripoints, with D=Ri+1Ri. Denoting (R= Ri) =iandV(R=Ri) =Vi, and doing a Taylor expansion of the coefficients and their second derivatives, a three points Numerov relationship is found ai1Fi1+biFi+gi+1Fi+1= 0 (A3)12 where ai1=1D2 122m ¯h2(Vi1E1¯) bi=2110D2 122m ¯h2(ViE1¯) (A4) gi+1=1D2 122m ¯h2(Vi+1E1) ; with an error proportional to D6. The Fox-Goodwin algorithm consists in defining Fi=RiFi+1 (A5) so that imposing the boundary condition at i=1, theRiis prop- agated according Ri= ai1Ri1+bi1 gi+1 until i=N, where the second boundary conditions of incoming plus outgoing waves are imposed. Usually, a real Fiis propagated to simplify the calculation because the potential is also real and a regular solution with Fi=0=0 is imposed because V1>Efor all the channels in- volved. On the contrary, in the case of capture in a well, it is as- sumed that the V1<Efor some of the channels. In order to impose the boundary condition, a transformation to a new adi- abatic basis is first done by diagonalizing the potential matrix ati=1 as ViiT=DiiT; (A6) where Dis a diagonal matrix with the eigenvalues and Tare the transformation matrix. In this adiabatic representation the coefficients are denoted Yto be distinguished from those of the original “diabatic” basis, and the boundary outgoing con- ditions are applied as 1Fa(R<R1) =8 >>< >>:0 if Da>E iq 2m p¯h2kaST a(E)eikaRifDa<E(A7) where it is being assumed that Vi=V1fori<1,i:e:that the potential is constant at Rdistances shorter than R1, the cap- ture distance. In this expression ka=p 2m(Eda)=¯handST a(E)2is the capture probability, since it correspond to the flux going to R<R1. Under this assumptions in the adiabatic representations we have A= [1F1]1 0F0=8 >< >:ejkajDifDa>E eikaDifDa<E(A8) Transforming back to the diabatic representation in which the integration is performed, we get (i=1) Ri=iT1AiT: (A9)After defining the propagation matrix in the first point of the grid, Riis iteratively propagated from i=2 to N, where the usual incoming/outgoing boundary conditions are imposed as Fa(RN) =8 >>>>>< >>>>>:ejkajRifVa;a(RN)>E iq 2m p¯h2kah ei(kaR`p=2)db;a SR ab(E)ei(kaR`p=2)i ifVa;a(RN)<E(A10) In the usual procedure, Riis real and real boundary condi- tions are imposed to calculate the symmetric reaction matrix, K, and from it the S-matrix which is unitary. In the present case,R1is complex, all this procedure is done in the complex plane, and assuming that the integration is done until suffi- ciently long distance, Eq.(A10) is also fulfilled at RN1, and the S-matrix is directly obtained from the propagation matrix as SR= RN1PNPN11  MN1RN1MN (A11) whereMiandPiare diagonal matrices with elements defined as Maa(Ri) =8 >>< >>:ejkajRifVa;a(Ri)>E q 2m p¯h2kaei(kaR`p=2)ifVa;a(RN)<E (A12) Paa(Ri) =8 >>< >>:0 if Va;a(Ri)>E q 2m p¯h2kaei(kaR`p=2)ifVa;a(RN)<E The resulting SR(E), in Eq.(A10) is not, in general, unitary. This is evident by inspection of Eq.(A7), since for those chan- nels with Da<Ethere is a flux that is trapped at distances R<R1. If all Da>Ethe normal situation is got, and the SR- matrix becomes unitary. The capture probability for a given initial channel is then obtained as Cb(E) =jST bbj2=1å ajSR baj2: (A13) 2. Adiabatic-by-sectors representation The number of channels increases very rapidly with total angular momentum, specially with many electronic states, as considered here. In order to reduce the number of channels we have implemented a variant of the adiabatic-by-sectors method [44–47]. In brief, this method consists in diagonal- izing the Vimatrix in the close-coupling equations, Eq. (A2) as in Eq. (A6). The new adiabatic functions,iAp, depend on the collision coordinate R i, are expressed in the original dia-13 batic basis set as jiApi=å ajjaiiTapiA=jiT jjai=å pjiApiiTapij=AiT1: The total wave function, Eq. A1, in the new basis set takes the form Yb(R;x) =å pjiApiiFb p!iF=iTF (A14) . The Numerov auxiliary matrices in Eq. (A3) can be re- expressed in the adiabatic representation as i1ai1=i1T1ai1i1T=1D2 122m ¯h2(Di1E1¯) (A15) iai1=iT1ai1iT=Sii1i1ai1S1 ii1; and similarly for biandgi+1matrices, where the transfer ma- trix has being defined as Si j=iT1jT: (A16) Doing some algebra, the recurrence equation of the propa-gation matrix, Eq. (A6) becomes i+1Ri= i+1ai1i+1Ri1+i+1bi1 i+1gi+1 (A17) wherei+1Riis the propagation matrix conecting the function iandi+1, represented in the adiabatic basis at i+1, and i+1Ri1=Si+1iiRi1S1 i+1i (A18) withiRi1being obtained in the previous iteration. Eq. (A17) can be iteratively solved analogously to procedure in the dia- batic representation with the extra-effort of transforming the matrices from one point to the following one. In this adia- batic representation, the first value is that1R0=A, defined in Eq. (A8). Also, the diabatic and adiabatic representation coin- cide for i=N, so thatNRN1RN1and the outer boundary conditions are imposed as in Eq. (A10). 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2308.02911v1.Optical_vortex_harmonic_generation_facilitated_by_photonic_spin_orbit_entanglement.pdf

1 Optical v ortex harmonic generation facilitated by photonic spin- orbit entanglement Chang Kyun Ha, Eun Mi Kim , Kyoung Jun Moon , and Myeong Soo Kang* Department of Physics, Korea Advanced Institute of Science and Technology (KAIST) 291 Daehak -ro, Yuseong- gu, Daejeon 34141, Republic of Korea *mskang@kaist.ac.kr Photon s can undergo spin-orbit coupling , by which the polarization (spin) and spatial profile (orbit) of the electromagnetic field interact and mix. Strong photonic spin-orbit coupling may reportedly arise from light propagation confine d in a small cross -section , where the optical mode s feature spin-orbit entangle ment . However, w hile photonic Hamiltonians generally exhibit nonlinear ity, the role and implication of spin-orbit entanglement in nonlinear optics have received little attention and are still elusive. Here, we report the first experimental demonstration of nonlinear optical frequency conversion , where spin-orbit entangle ment facilitates spin-to-orbit transfer among different optical frequencies . By pumping a multimode optical nanofiber with a spin-polarized Gaussian pump beam, we produce an optical vortex at the third harmonic, which has long been regarded as a forbidden process in isotropic media . Our finding s offer a unique and powerful means for efficient optical vortex generation that only incorporates a single Gaussian pump beam , in sharp contrast to any other approaches employ ing structured pump field s or sophisticatedly designed medi a. Our work opens up new possibilities of spin-orbit -coupling subwavelength waveguides , inspiring fundamental studies of nonlinear optics involving various types of structured light , as well as paving the way for the realization of hybrid quantum systems comprised of telecom photonic network s and long- lived quantum memor ies. 2 Introduction Spin- orbit coupling (SOC)—the interaction and mixing between the spin and orbital degrees of freedom (DOFs) of a particle—arises in diverse disciplines. In individual atom s and molecules , SOC occurs via the orbital motion of spin- 1/2 electron s in the nucleus -generating elect rostatic potential, contributing to the formation of fine structures [ 1]. An analogous energy level splitting takes place inside an atomic nucl eus through the cooperation of the orbital motion and strong nuclear force for protons and neutrons [2]. In condensed matter s, SOC plays a crucial role in the emergence of various fundamental phenomena i nvolving electronic spins, such as t he spin Hall effect and anomalous Hall effect [3], Rashba effect [4], antisymmetric exchange [5,6], and spin-momentum locking [7]. These phenomena have been investigated for the generat ion and control of spin currents [3], spin -orbit torque s [8], and magnetic skyrmions [9], as well as the implementat ion of functional spintronic devices for faster and more efficient information storage and processing [10]. Photons also possess spin and orbital DOFs , which are manifested in the polarization and spatial distribution , respectively, of the electromagnetic field, and can accordingly carry spin angular momentum (SAM) and orbital angular momentum (OAM) [1 1] like other particles . In particular, the optical bea m possessing a nonzero OAM exhibits a helically twisted wavefront and a donut -shaped intensity profile with an intensity null at the beam center, so it is referred to as an optical vortex [ 12]. Moreover, t he two DOFs of light can interact and mix in some circumstances . Such photonic SOC has been exploited primarily to generate and manipulate optical vortices through spin- to-orbit transfer , being at the heart of the remarkable advances in optical communications [ 13], optical tr apping [14], and optical imaging [ 15,16] , as well as the recent advent of novel technologies, such as nonclassical light generation [ 17,18 ], complex lattice pattern generation [ 19], spiral phase contrast imaging with invisible illumination [ 20], and high- dimensional photonic quantum information processing [ 21,22]. More recently, it has been pointed out that p hotonic SOC can arise from simple light confinement in a small cross -section al area. When light is strongly focused by a high- numerical -aperture lens [23,24] or tightly confined within a subwavelength waveguide [25– 27], the longitudinal field component become s significant, so the light field loses the transversality. Then , the propagation eigenmode s of photons cannot be represented as a simple product of spin and orbital states (i.e., SAM/OAM eigenmode ) any longer but appear as spin- orbit -entangled states [25,26] , akin to the total angular momentum (TAM) eigen states in the 3 spin- orbit -coupled atomic fine structure [1]. Since such spin- orbit entanglement ( SOE) is a purely geometrical effect, it can occur in a broad range of micro/nano- photonic s ystems made of any material s, including even isotopic ones (e.g., fused silica glass) . While the photonic Hamiltonian given in terms of the electric permittivity of a medium can involve nonlinear ity, which is a sharply distinctive feature to the intrinsically linear counterparts of other quantum particles , the role and implication of the confinement -induced SOC and the resulting SOE in nonlinear optics have been rarely studied and thus still elusive. In this paper, we experimentally reveal that nonlinear optical frequency conversion can accompany spin -to-orbit transfer among different optical frequencies when the nonlinear photonic Hamiltonian is created by spin -orbit -entangled light. By pumping a multimode optical nanofiber [ 28] that tightly confines optical fields with a circularly polarized Gaussian beam (possessing a spin ±1 but no OAM) , we generat e an optical vortex having a nonzero OAM at the third harmonic optical frequency . This observation indicates that spin-to-orbit conversion occurs from the pump to the third harmonic in the third -harmonic generation (THG) process, even though the nanofiber is axially symmetric and made entirely of isotropic material (fused silica glass) . This unprecedented process , which we term third -harmonic optical vortex generation (TH -OVG), cannot be observed in isotropic bulk media in the paraxial regime (i.e., without SOC ) and weakly guiding conventional optical fibers . In reality , THG has long been perceived to be forbidden in such media when pumped with a circularly polarized beam [29– 31]. Our full-vectorial nonlinear coupled- mode theory convinces us that the multimode nanofiber exhibit ing confinement -induced photonic SOC supports spin- orbit -entangled guided eigenmodes , which plays an essential role in facilitating the TH-OVG process. Our result also offer s a new mechanism of all-optical vortex generation, which operates with a single Gaussian pump beam only and relies exclusively on the confinement -induced SOC, not requir ing any structured light , specially designed medium , or particular class of material . Our scheme is thus simple and cost-effective while equipped with ultrafast all-optical switching at the same time and should be distinguished from any other approaches that employ sophisticatedly structured components (e.g., spiral phase plates [ 32], Q plates [ 33], computer -generated holograms [ 34], spatial light modulators [ 35], microcavities [3 6], on- chip gratings [3 7], and metasurfaces [38,39]) or nonlinear optic al phenomena in structured media [40,41] or with complex pumping configurations using multiple /structured laser beams [42–46] . We focus on harmonic generation here, but our concept can be applied to any nonlinear optical wave mixing processes. 4 Principle and theoretical analysis An o ptical waveguide with a circular cross -section supports a family of optical vor tices as the guided eigenmodes . Each optical vortex mode ( OVM), which we denote here by ,OVlmσ, can be formed by linear superposition of the even and odd hybrid modes with the phase difference of 90º as below [27,47]. even odd , 1, 1, even odd , 1, 1,OV HE HE , OV EH EH ,l m lm lm l m lm lmi i± ±++ ±−−= ± = ± (1) where l and σ represent the canonical OAM and SAM numbers , respectively , in the paraxial regime , and m is the radial mode number. For instance, the circularly polarized Gaussian -like fundamental HE 11 mode is designated as the 0,1OV± mode, which carries an SAM of 1=±σ but no OAM ( 0l=). (See Supplementary Note 1 for more detail) In the paraxial regime, each OVM is a simultaneous OAM and SAM eigenmode , both the OAM and SAM eigenvalues being integer s [25,26] . In this case, the OAM and SAM should be individually conserved in any nonlinear optical processes , so their mutual exchange (e.g., spin- to-orbit conversion and vice versa) is prohibited [43]. The OVMs in weakly guiding conventional optical fibers also have t hese propert ies approximately. In contrast, when the OVM is confined within a sufficiently small cross -sectional area, it loses the field transversality and exhibits significant SOC associated with the longitudinal field component [25–27] , which can be expressed by the normalized field distribution of the OVM as follows. , ,, , ,,ˆ ˆ ˆ () () () , ˆ ˆ ˆ () () () ,ij i i jm jm jm ij i i jm jm jme iA r e iB r e C r e iB r e iA r e C rφφ φ φφ φ+− + + +− −− + − +−= ++ = ++ez ez (2) where jlσ= + is the TAM number of the OVM, and +e and −e correspond to the even+i odd OVM of left -handedness and the even–i odd OVM of right -handedness, respectively. ˆˆ ˆ ( )2i±= ±xy are the transverse unit vectors along the left -handed ( ˆ+) and right -handed (ˆ−) circular polarizations, while ˆz is the axial unit vector, and (, , )rzφ are the cylindrical coordinates . (See Supplementary N ote 3 for more detail) Equation ( 2) indicates that the OVM is no longer the OAM or SAM eigenmode but become s a spin- orbit -entangled TAM eigenmode . 5 In this scenario , neither the OAM nor SAM is necessarily pre served in a nonlinear optical process, while the TAM must still be conserved [27]. This feature of spin- orbit -entangled OVMs may allow optical vortex generation through spin- to-orbit -converting nonlinear wave mixing of OVMs . In this work, we exploit the characteristics of the strongly spin- orbit -entangl ed OVMs tightly confined in a multimode optical nanofiber to reveal the optical -vortex -creating unconventional nonlinear frequency conversion process . Let us focus on THG pumped by a circularly polarized Gaussian beam having no OAM ( lp = 0) but an SAM of σp = ±1, as illustrated in Fig. 1(a). Since a single third -harmonic photon is created at the expense of the annihilation of three pump photons in the THG process , the TAM of the third -harmonic photon should be js = ±3 because that of each pump photon is j p = ±1 (Fig. 1(b)). One might think that this THG process can occur naturally and lead to optical vortex creation via spin-to-orbit conversion, as the SAM of the third- harmonic photon can be σs = ±1 only . However, the THG process is forbidden in isotropic bulk media in the paraxial regime and weakly -guiding conventional optical fibers [29–31 ], which is related to the fact mentioned earlier that the OAM and SAM should be preserved separately in the nonlinear frequency conversion. In fact, it has long been perceived that THG cannot occur when pumped by a circularly polarized beam, and thus most previous THG experiments employed a linearly polarized pump beam exclusively [29–31] . In sharp contrast, when the circularly polarized pump beam is launched into the Gaussian -like 0,1OV± mode of a multimode nanofiber , a third harmonic can be generated efficiently if the phase- matching condition is satisfied. The phase -match ing nanofiber diameter [28,48] is so small (717 nm in our case in Fig. 1( c)) that the guided OVMs can exhibit strong SOC , and either the OAM or the SAM does not have to be conserved in the THG process . We will show that, a s a consequence of the SOC -induced SOE , third -harmonic photons in an other OVM with the TAM of j s = ±3 can be generated efficiently , which can eventually evolve in to a circularly polarized optical vortex with an integer OAM ( ls = ±2) and SAM ( σs = ±1) in the weakly guiding or paraxial regime . We perform numerical modeling of the OVMs to verify the experimental feasibility of the TH-OVG. The 1550 -nm pump in the fundamental 0,1OV± mode is phase -matche d with the third harmonic in the higher -order OVMs at specific diameters of silica glass optical nanofiber (Fig. 1(c)). Among various third -harmonic OVM s, we concentrate on the 2,1OV± ± mode ( HE 31 hybrid 6 mode ) because it has a TAM of ±3 and thus may be generated through TH -OVG . The phase - match ing nanofiber diameter is calculated to be 717 nm. Figure 1(d) displays the field profiles of the OVMs and the constituent hybrid modes in the 717- nm-thick nanofiber, where SOC appears as the nonuniformity in the polarization and phase patterns on the beam cross -sections. (See Supplementary Figure S1 , which also displays the OVMs of right -handedness ) We note that the degeneracy between the 2,1OV± ± and the 2,1OV± mode (EH 11 hybrid mode) is broken significantly , as seen in Fig. 1(c). These two modes are the spin- orbit aligned and anti -aligned OVMs, respectively, of the same OAM number of ±2, and the evident non- degeneracy between them verifies the action of strong SOC. To reveal and elucidate the role of the SOC -induced SOE in the TH -OVG process , we derive the full -vectorial nonlinear coupled- mode equatio ns that describe the TH -OVG in the OVM basis as below . s, (3) 3 0 T,OVM p,9,16iz xxxxdai kZ J a edzβχ± ∆ ± = (3) ()()()() ()** T,OVM p, p, p, s, p, p, p, s, sp2 3 cos 3 ,J dA d jjφφ++ ++ −− −−= ⋅ ⋅+⋅ ⋅ ∝−∫ ∫ee ee ee ee (4) Here, T,OVMJ is the effective overlap integral that represent s the spatial overlap between the pump OVM and the third -harmonic OVM for TH-OVG. a is the slowly varying field amplitude of each OVM , and the subscript s p and s stand for the pump and the third harmonic , respectively. (See Supplementary Note s 2 and 3 for more detail ) Equation (4) indicates that T,OVMJ is nonzero in the presence of SOC only if the TAM can be conserved, i.e., sp3jj= , whereas the requirement for the simultaneous conservation of both OAM and SAM is released. Consequently , the THG in the 2,1OV± ± mode with s3 j=± is permitted with a circularly polarized pump beam in the fundamental 0,1OV± mode having p1 j=± . In contrast, T,OVMJ becomes identically zero for the 2,1OV± and the 0,2OV± modes with s1 j=± , so THG in those OVMs is forbidden. (See Supplementary Table 1 for t he calculated values of overlap integrals for three third -harmonic OVMs , the 2,1OV± ± (HE 31), the 2,1OV± (EH 11), and the 0,2OV± (HE 12) modes ) On the other hand, without SOC , the spin and orbital DOFs of an OVM are separated 7 from each other , as only either ,()jmAr or ,()jmBr is nonzero in Eq. ( 2). T,OVMJ then takes the form of ()sp T,OVM ,3 s p cos 3 0, J d llσσδφ φ ∝ −=∫ (5) where δ is the Kronecker delta , which is also valid approximately i n the weakly guiding regime with negligible SOC. (See Supplementary Note 3 for more detail ) Equation (5) implies that T,OVMJ is always zero in the absence of SOC , as the SAM cannot be preserve d (sp3σσ≠ ), which corroborates that the SOC -induced SOE is crucial in facilitating the TH-OVG p rocess . The SOC -induced SOE is a confinement effect that generally strengthens as light is trapped in a smaller cross -section. To figure out the influence of the effect on TH -OVG more quantitatively, we calculate T,OVMJ for our experimental scenario where a 17-μm-thick cladding- etched silica optical fiber is tapered, as shown in Fig. 2(a) . T,OVMJ increases monotonically as the cladding diameter falls. Such an increase of T,OVMJ can be contributed by two factors. One is the reduction of the effective mode areas [49] of the pump and third harmonic. This contribution also commonly affects other nonlinear optical effects, such as the self-phase modulation ( SPM ) of the pump and the pump- driven cross -phase modulation ( XPM ) of the third harmonic , monotonically increas ing the respective overlap integrals P,OVMJ and X,OVMJ . (See Supplementary Note 3 for the definitions of P,OVMJ and X,OVMJ ) However, the enhancement of T,OVMJ by a factor of ~6,000 during the entire course of tapering is highly substantial , compared to only ~100 or smaller for those of P,OVMJ and X,OVMJ and even that of T1J for THG in the HE 12 mode [ 28,48 ]. (See Supplementary Note 2 for the definition of T1J) The dramatic enhancement of T,OVMJ arises dominantly from another contribution of the SOC -induced SOE, whereas the increases in other J’s are governed predominantly by the reduction in the effective mode area. We also calculate the canonical OAMs and SAMs of the pump and third- harmonic OVMs according to the cladding diameter . As the OVMs exhibit SOC and SOC -induced SOE more strongly , they become more deviated from the OAM/SAM eigenmodes for the paraxial regime, and their canonical OAM s and SAM s differ more from the nominal integer values [ 25,26] . Accordingly, w e may employ the resulting change of the canonical OAM/SAM of the OVM as a quantitative measure of the SOC strength or the degree 8 of SOE. As the fiber gets tapered down, the SOC -induced shift in the canonical OAM/SAM value increases monotonically, as shown in Fig. 2(b) , which indicates the reinforcement of SOC. The strengthen ed SOC -induced SOE boosts up T,OVMJ cooperatively with the decrease in the effective mode area. We note that even in the presence of SOC each OVM is kept as a TAM eigenmode with a fixed integer TAM eigenvalue regardless of the cladding diameter variation . (See Supplementary Figure S4) Experiment al results Observation and characterization of TH-OVG For experimental observation of TH-OVG , we first fabricate silica multimode adiabatic submicron tapers ( MASTs ) from conventional telecom single -mode fiber (SMF) employing the recently developed two -step process [ 28]. We deeply wet -etch the SMF cladding below 20 μm and then taper the etched SMF down to submicron diameter using the conventional flame - brushing and pulling technique [ 50]. The MAST waist acts as a multimode nanofiber guiding spin- orbit -entangled OVMs, and w e adjust the MAST waist diameter close to ~717 nm to achieve the phase matching for intermodal THG in the 2,1OV± ± (HE 31) mode (Fig. 1( c)). The waist has a 5 mm length and is connected to untapered fiber sections via a pair of ~30 -mm- long exponential transition s. During the whole tapering process, we check that the 532- nm LP 21 mode ( a combination of the HE 31 and the EH 11 mode) transmits adiabatically through the fiber taper. (See Ref. [ 28] for the detailed description of MAST fabricati on and its working principle ) The final transmission s of the 532-nm LP21 mode and the 1550- nm LP01 mode are measured to be 58% and 94% , respectively. We package the MAST in an acrylic box to prevent its contamination for subsequent long -term experiments. In our experiments, w e use an all- fiber master oscillator power amplifier (MOPA) system as a pump source for THG (Fig. 3 (a)). It incorporates a homemade wavelength -tunable narrow - linewidth passively mode -locked fiber laser that emits a 110 ps pulse train at a 2.1 MHz repetition rate over the wavelength tuning range of 1535–1563 nm [51]. We control the polarization state of the MOPA output with a fiber polarization controller and launch the pump beam into the fabricated MAST. We observe that the spatial profile of the third -harmonic output signal is generally in a mixture of multiple higher -order modes such as the LP02, the LP21, and a donut mode , depend ing on the polarization state of the pump beam . By adjust ing 9 the pump polarization state , we can make the third -harmonic output signal closely resembl e the purely donut -shaped mode. Such the donut mode can be created with the highest purity at two mutually orthogonal pump polarization states . We measure the output power of the donut - shaped third-harmonic signal over the entire wavelength tuning range of the MOPA system (1535–1563 nm) at a fixed average input pump power of 0.25 W . The THG in the donut -shaped mode is most efficien t at a specific pump wavelength (1543 nm for Fig. 3(b)) , which is close to the design wavelength of 1550 nm , while other pump wavelengths also generate smaller amounts of donut -shaped third -harmonic signal (Fig. 3(c)) mainly due to the waist diameter nonuniformity [28]. We also measure the third -harmonic output powers over a range of pump powers, while fixing the pump wavelength a t 1543 nm. The third -harmonic powers are almost identical for the two mutually orthogonal pump polarization state s that yield the purest third -harmonic donut mode , exhibiting a theor etically predicted cubic dependence on the pump power , as shown in Fig. 3(d). The maximum THG conversion efficiencies are measured as 1.54×10-6 and 1.58×10- 6 for the two pump polarizations at 0.37 W pump power. We measure the topological charge of the third -harmonic signal based on the astigmatic field transform through the beam focusing at a cylindrical lens (focal length : 50 mm) [ 52]. For one pump polarization state (denoted by ‘A’), the transformed third -harmonic field displays a counter -clockwise slanted array of three lobes, as shown in Fig. 3(e), verifying together with the donut -shaped intensity profile that the generated third -harmonic field is an optical vortex with a topological charge of ls = +2. On the contrary, for another pump polarization state (denoted by ‘B’), an array of three lobes appears again but slant s clockwise, which indicates an optical vortex with a topological charge of ls = -2. The se optical vortex structure s are maintained over a broad range of pump power s, though their purit ies become slightly degraded as the pump power increases beyond ~0.37 W, mainly due to the unwanted THG in the HE 12 mode associated with the nonlinear pump polarization change [ 28,53] . The measured third -harmonic spectra in Figs. 3(f) and 3(g) show that the two optical vortices with the respective topological charges of +2 and - 2 have identical wavelengths (514.4 nm), which almost equal one -third of the pump wavelength (1543 nm in Fig. 3(h)). As the pump power rises beyond ~0.37 W, a pair of sidebands appear in the spect ra of both the pump and third harmonic. The sideband locations are the same for the third harmonic and the pump, distant from the main spectral peaks by ~2 THz. In addition, both the pump in the fundamental 10 0,1OV± (HE 11) mode and the third harmonic in the 2,1OV± ± (HE 31) mode are in the normal dispersion regime in the MAST waist. Therefore, the creation of the spectral sidebands can be attributed to XPM -induced modulation instability [ 28,54,55] . In principle , the third -harmonic OVM s of left -handed (2,1OV+ +) and right -handed (2,1OV− −) quasi -circular polarizations are generated from the pump beam of left -handed (0,1OV+) and right -handed (0,1OV−) quasi -circular polarizations, respectively , in the MAST waist . In practice, however, the polarization s vary in the lead fiber sections to the MAST output and input port s due to the residual birefringence. Moreover, the residual stress and/or elliptical deformation can induce significant coupling of the HE 31 mode to the nearly degenerate EH 11 mode in the weakly guiding lead fiber [ 56]. These two effects hinder the precise experimental identification of the polarization states of the pump and third harmonic in the MAST waist. Nevertheless, we experimentally examine their polarization profiles at the output port using wave plates and a linear polarizer in addition to the existing setup in Fig. 3(a) . We confirm that the polarization states of the two output third -harmonic optical vorti ces are always almost orthogonal to each other, and so are the corresponding pump polarization states simultaneously , although they generally appear elliptical. In addition, the polarization state of the output third harmonic is almost uniform over the entire beam cross -section, which supports that the third harmonic is close to an OVM made up of a balanced (50:50) superposition of the even and odd hybrid modes as in Eq. (1), rather than a v ector beam with a nonuniform polarization distribution [57,58] consisting of a significantly unbalanced combination of the two hybrid modes . Finally, the topological charge of the third -harmonic output is measured as purely either +2 or - 2 regardless of the elliptical polarization , which supports the existence of the mode mixing between the nearly degenerate HE 31 and EH 11 modes , whereas the circular birefringence is negligible . It is because such mode mixing always appears as a change of polarization (SAM) only, while preserv ing the OAM unless there is circular dichroism [56]. One might think of an alternative type of birefringence- induced mode coupling that occurs between the even and odd modes. However, such even -odd mode mixing gives rise to the simultan eous sign reversal of the OAM and SAM according to Eq. (1) and consequently yields a mixture of two third - harmonic OAMs of +2 and -2 at the output , which is contradictory to our experimental observation. 11 Effect of m odal birefringence in MAST on TH -OVG When the MAST cross -section is perfectly circular , the pump 0,1OV+ and 0,1OV− modes that are left-handed and right -handed quasi -circularly polarized yield TH-OVG in the 2,1OV+ + and 2,1OV− − mode , respectively, in a symmetric fashion while completely forbidding THG in the 0,2OV± (HE 12) mode . However, we observe in some cases that the third harmonic in the HE 12 mode is not entir ely suppressed through the pump polarization control , which degrades the vortex purity and the symmetry of TH-OVG with respect to the handedness of pump polarization . We attribute the non- ideal TH-OVG performance to the non-circular MAST cross -section , which arise s primarily in the deep wet -etching step of the MAST fabrication process . (See Supplementary Note 4 and Supplementary Figure 5 for more detail) The overall TH-OVG process can be viewed as a linear superposition of four THG processes involving different combinations of even/odd hybrid modes of the pump and third harmonic (2 for pump and 2 for third harmonic), as illustrated in Fig. 4( a). (See Suppl ementary Note 4 for more detail .) The TH-OVG then work s properly when the even and odd hybrid modes are degenerate for both the pump and third harmonic. However , the deviation of the MAST cross -section from the circular one breaks the modal degeneracy , which hinders the perfect operation of TH -OVG. We model the non- circular MAST cross -section as an elliptically deformed one, as illustrated in Fig. 4( b), and investigate the influence of the deformation- induced birefringence on TH -OVG based on the full-vectorial finite element analysis of the spatial modes in the MAST . First, w e calculate the beat length between the even and odd hybrid modes for the pump and the third harmonic according to the eccentricity of the MAST cross -section . As the eccentricity rises, the degeneracy becomes broken more strongly , which appears as the decrease in the beat length , as shown in Figs. 4(c) and 4( d). In particular , the degeneracy is broken more s eriously for the pump HE 11 mode and the third -harmonic HE 12 mode because their transverse electric field s are almost parallel to the major or minor axis of the cross -section [56]. The beat length s can be even shorter than the MAST waist length (5 mm in our experiments) at relatively small eccentricit ies below 0.1. Then, the relative phase between the even and odd modes varies as the pump and third- harmonic fields propagate along the MAST, which can deteriorate the performance and controllability of TH -OVG. For instance, the THG in the HE 12 mode cannot be fully suppressed as the pump polarization state varies during propagation. Furthermore, w hile the four constituent THG processes in Fig. 4(a) are phase 12 matched simultaneously for a perfectly circular MAST with degenerate even and odd hybrid modes, their phase -matching conditi ons become deviated from each other as the eccentricity rises (Figs. 4(e) and 4( f)). The phase -match ing nanofiber diameter s for the four processes of THG in the HE 12 mode can differ by even tens of nanometers. S uch a large deviation of phase match ing conditions also hinders the suppress ion of the THG in the HE 12 mode. In contrast, the third -harmonic HE 31 mode ha s a nonuniform polarization profile , and consequently , the degree of lifting the degeneracy is less significant, resulting in relatively longer beat lengths. Hence, the symmetry in the TH -OVG with respect to the pump polarization is more robust against the elliptical deformation compared to the vortex purity with the suppression of the THG in the HE 12 mode. Discussion We have r evealed that SOC -induced photonic SOE in a multimode optical nanofiber facilitates the efficient nonlinear frequency conversion from the spin- polarized Gaussian -like pump beam to the harmonic optical vortex with a nonzero OAM . This remarkable observation contrast s sharply with a myriad of previous experiment s employing conventional isotropic media , where TH-OVG is forbidden. Furthermore, our full -vectorial nonlinear coupled- mode theory unveils the crucial role of SOE in creating spin- to-orbit -converting nonlinear polarization, which offers a unique and powerful means of all -optical generation of an optical vortex without the requirement of any structured pump bea ms, sophisticatedl y designed media or particular types of materials . We emphasize that our MAST is a unique and ideal platform for spin- orbit - interacting nonlinear optics, as providing adiabatic guidance of multiple spatial modes , including spin- orbit -entangled OVMs, together with the ultra high nonlinearity and broad dispersion controllability owing to the tight light confinement. All these features are essential to achieve phase- matched efficient nonlinear frequency conversion among different OVMs . The new research paradigm of spin- orbit -interacting nonlinear wave mixing of spin -orbit - entangled light is not limited to TH -OVG . It can also be extended to other intermodal nonlinear frequency conversion processes, such as different kinds of harmonic generation [ 59–61] , FWM [62], and stimulated Raman scattering [63], as well as a combination of two or more nonlinear optical processes , which could generate optical vortices of topological charges other than ±2 or in the superposition/entanglement of different topological charges. We also anticipate that a coherent acoustic vortex might be generated in an optical nanofiber via stimulated Brillouin 13 scattering by pumping with a spin- orbit -entangled OVM [27]. Conversely , exciting and controlling torsional acoustic vibration in an optical nanofiber [6 4,65] could implement nonlinear switch ing of chirality and optical vortices . Furthermore , recently realized exotic kinds of structured light, such as polarization vortices [ 57,58] , polarization Möbius strips [ 66], optical vortex knots [ 67], non-integer vortices [ 68], optical vortices with multiple phase singularities [69], and optical wheels having t ransverse OAMs [70], might also be all -optical ly created and manipulated via spin- orbit -coupling nonlinear wave mixing . Although residual THG in the unwanted HE 12 mode currently degrades the vortex purity, we believe that such m ixing of undesired modes might be resolved by employing specially designed optical waveguides w ith proper modal dispersions, such as vortex fibers [64], microstructured fibers [71], and integrated on- chip waveguides [72] . In addition, t he TH-OVG conversion efficiency achieved so far is on the order of ~10-6, limited primarily by short effective interaction length. The effective indices of tightly confined modes are sensitive to the waveguide geometry , the nanofiber diameter variation as small as only a few n anometers disturbing the phase match ing significantly [28,73] . The nanofibers made out of other glasses with larger optical nonlinearit ies (e.g., chalcogenide glass [74]) or the existing silica MAST coated with such materials [75] m ight enhance the conversion efficiency. Finally, it is worth while to mention some potential key application s of our work . As nonlinear optical frequency conversion processes usually produce strong correlation among multiple photonic degrees of freedom (e.g., frequency, propagation direction, position/timing, as well as SAM/OAM) of different optical frequencies owing to the requirement of conservation of physical quantities (i.e., total energy and wavevector, as well as TAM). Hence, our scheme could immediately apply to multi- dimensional photonic quantum information processing. In addition, the guided light along the MAST waist can efficiently interact with other photonic systems in the vicinity via the evanescent fields. The visible optical vortices generated via TH -OVG in the MAST waist, together with the quasi -circularly polarized pump beam at the telecom wavelengths , provide powerful tools for sophisticated manipulation of atoms/molecules or nano particles trapped near the MAST waist via their strong evanescent fields [76,77] . Moreover , the harmonic optical vortex can be coupled to various nanophotonic systems [78–81] for efficient photonic information processing. In recent noticeable experiment s [82], for instance, the guided light is couple d to long-lived quantum memories (e.g., nitrogen -vacancy centers) imbedded in optical nanofibers for storage and retrieval of the 14 information encoded in the polarization state of telecom photons via THG. Our work might be combined with conventional telecom optical networks [83] to implement multi- dimensional hybrid quantum networks , where the OAMs of visible photons are connected to the SAMs of telecom photons and the quantum memor ies. Acknowledgements This work was supported by the National Research Foundation of Korea (NRF) grants funded by the Korea government ( NRF -2020K1A3A1A19088178, NRF -2022R1H1A2092792) and the KAIST C2 Project. Competing interests The authors declare no competing interests. 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The input pump beam has no orbital angular momentum (OAM , lp = 0) but only a spin of σp = ±1, so its total angular momentum (TAM) is jp = ±1. The output third harmonic has an OAM of ls = ±2 and a spin of ±1 with the same handedness, so its TAM is js = ±3. This TH-OVG process is facilitated by the SOE manifested in the pump and the third harmonic, which arises from the confinement-induced spin-orbit coupling in the multimode nanofiber. In sharp contrast, the TH -OVG is forbidden in isotropic bulk media, as THG cannot occur with a CP pump beam. (b) Energy level diagram for TH- 24 OVG displaying the simultaneous conservation of energy and TAM in the process . (c) Calculated effective indices of some optical vortex modes (OVMs) in silica -glass multimode nanofibers as a function of the nanofiber diameter . The red curve corresponds to the pump beam in the fundamental OVM (even odd 0,1 11 11 OV HE HE i±= ± ) at 1550 nm wavelength, whereas the others are the third harmonic in the higher -order OVMs. (d) Pump and third -harmonic OVMs that are phase -matched for the TH-OVG in the 2,1OV± ± mode in a 717-nm-thick nanofiber indicated by the magenta dashed circle in (c). The profiles of the field amplitude and phase of each OVM are shown, together with the field profiles of the constituent even and odd hybrid modes —the pump HE 11 mode s and the third -harmonic HE 31 mode s. The white arrows and black dashed circles represent the polarization profile and the nanofiber cross -section, respectively. The phase pattern in the rightmost column displays the phase of the horizontal (x) field component relative to the longitudinal (z) one . See Supplementary Note 1 for the case of right -handed CP OVMs. 25 FIG. 2. T heoretical analysis of spin -orbit coupling (SOC) and third- harmonic optical vortex generation (TH -OVG) in optical microfibers and nanofibers. (a) Calculated overlap integrals for third -order nonlinear optical processes involving optical vortex modes (OVMs) according to the cladding diameter in the course of tapering of an etched silica optical fiber. The turquoise solid curve (T,OVMJ ) corresponds to TH -OVG in the 2,1OV± ± OVM from the 1550 nm pump in the 0,1OV± OVM . The red curve (P,OVMJ ) corresponds to the self -phase modulation of the pump OVM, and the blue one (X,OVMJ ) to the cross -phase modulation of the third -harmonic OVM by the pump OVM. The three overlap integrals are defined in Eqs. (S28) and (S29) in Supplementary Note 3. On the other hand, the turquoise dashed curve (T1,HE12J defined in Eq. (S14) in Supplementary Note 2) represents the third -harmonic generation (THG) in the HE 12 hybrid mode from the pump in the HE 11 hybrid mode of the same even/odd parity [28]. Note that T,OVM 0 J= for the THG in the even odd 0,2 12 12 OV HE HE i±= ± mode, so it does not appear in this plot of the current logarithmic vertical scale. (b) Calculated SOC -induced change of the canonical orbital angular momenta (OAMs) of the 0,1OV± mode at the 1550 nm pump wavelength and the 2,1OV± ± mode at the third harmonic, according to the cladding diameter. The SOC -induced changes of the canonical spin angular momenta (SAMs) occur with the opposite signs, so the total angular momenta are preserved regardless of the cladding diameter variation. See Supplementary Note 3 and Supplementary Figure 4 for more detail. In both ( a) 26 and ( b), the initial core [cladding] diameter and numerical aper ture of the etched fiber prior to tapering are set as the experimental values of 8.7 [17] μ m and 0.13, respectively, and t he grey vertical dashed lines indicate the phase -matching nanofiber diameter of 717 nm for the TH - OVG process. 27 FIG. 3. Experimental observation of third -harmonic optical vortex generation (TH- OVG) . (a) Schematic diagram of the experimental setup. The output from a homemade wavelength -tunable mode- locked fiber laser [51] is amplified by an erbium -doped- fiber optical amplifier. This pump beam is launched into a multimode adiabatic submicron taper (MAST) after its polarization is adjusted with a fiber polarization controller (FPC) to generate a third - harmonic optical vortex. AL, aspheric lens; SPF , short- pass filter ; CL, cylin drical lens; CMOS, complementary metal -oxide- semiconductor. ( b) Measured third -harmonic output powers at a fixed average pump power of 0.25 mW over the pump wavelength tuning range of 1535–1563 28 nm. The polarization state at each pump wavelength is adjusted to minimize the TH G in the HE 12 mode. The inset shows the scanning electron micrograph of the MAST waist , where the white scale bar corresponds to 1 µm. (c) Far -field profiles of the third -harmonic output observed without the CL at different pump wavelengths (white number s in nanometers ). (d) Measured third -harmonic output powers over a range of average pump power s at a fixed pump wavelength of 1543 nm yielding the maximum TH -OVG conversion efficiency , as seen in ( b). The blue and purple correspond to the two mutually orthogonal pump polarization states that yield optimum TH-OVG, which we denote by pump polarization A and B. The turquoise line is the cubic fit to the measurements . (e) Far-filed profiles of the third -harmonic output observed with the pump polarization state s A and B without and with the CL. The white number s indicate the average pump power s in watts. The far -field profiles obtained with the CL reveal the topological charges l s = +2 (upper) and - 2 (lower) of the third- harmonic output signals. ( f,g) Optical spectra of the third -harmonic output signals with ls = +2 (f) and - 2 (g) obtained at pump polarization state s A and B , respectively . (h) Optical spectra of the output pump beam . In (f– h), the color of each curve corresponds to the average pump power displayed in (h), 0.17 (grey), 0.28 ( dark yellow ), 0.34 ( dark magenta ), and 0.37 W (dark cyan). 29 FIG. 4. Effect of the deformation- induced nanofiber birefringence on the phase matc hing for third -harmonic optical vortex generation . (a) Energy level diagrams representing four third -harmonic generation (THG) processes that involve different combinations of the pump and third harmonic hybrid modes, where JTi’s (i = 1 to 4) are the corresponding overlap integrals. (See Supplementary Note 2 for their definitions) The red and turquoise stand for the pump and third harmonic, and the solid (‘e’) and dashed (‘o’) arrows represent the even and odd hybrid modes. ( b) Model of the cross -section of birefringent nanofiber with elliptical deformation. The wet-etched pure- silica cladding is elliptical, whereas the germanium-doped- silica core is perfectly circular. a and b are the major and minor axis , respectively , of the elliptical cladding. ( c,d) Calculated beat length s between the even and odd hybrid modes for the pump HE 11 mode (red curves) and the third harmonic HE 31 (c) and HE 12 (d) modes (turquoise curves) over a range of cladding eccentricity defined as ( )21ba− . The arithmetic mean of a and b, d0 = (a+b)/2, is fixed as the phase- matching diameters of 717 and 762 nm for THG in the HE 31 and the HE 12 mode, respectively, in the perfectly circular nanofibers. ( e,f) Calculated phase- match ing d0 for THG in the HE 31 (e) and the HE 12 mode ( f) with four different 30 combinations of even/odd hybrid modes defined in (a) over a range of cladding eccentricity. In (c–f), the pump wavelength is fixed at 1550 nm, and the diameter of the germanium -doped- silica core is kept smaller than d0 by a factor of 8.7/17, the same as the experimental value. 31 Supplementary Information Supplementary Note 1: Optical vortex modes (OVMs) in optical nanofiber An o ptical waveguide with a circular cross -section supports OVMs as the guided eigenmodes . Each OVM designated by ,OVlm can be formed by linear superposition of the even and odd hybrid modes with the phase difference of 90º as belo w [1,2]. even odd , 1, 1, even odd , 1, 1,OV HE HE , OV EH EH ,l m lm lm l m lm lmi i (S1) where l and represent the canonical orbital angular momentum (OAM) and spin angular momentum (SAM) numbers , respectively , under the paraxial approximation, and m is the radial mode number. Figure S1 displays the electric field profiles of four OVMs, the even odd 0,1 1,1 1,1OV HE HE i modes at 1550 nm wavelength and the even odd 2,1 3,1 3,1OV HE HE i modes at the third harmonic, in a silica glass optical nanofiber of 717 nm diameter, where the OVMs are phase matched for third -harmonic optical vortex generation. The ,OVlm modes composed of the HE hybrid modes have the OAM and SAM in the same handedness, so they are referred to as the spin -orbit aligned modes in some literature [3]. On the contrary, the ,OVlm modes consisting of the EH hybrid modes are the spin -orbit anti -aligned modes, in the sense that the OAM and SAM are in the opposite handedness. 32 FIG. S1. Electric field profiles of four optical vortex modes (OVMs) in an optical nanofiber. The even odd 0,1 1,1 1,1OV HE HE i modes at 1550 nm wavelength and the even odd 2,1 3 ,1 3,1OV HE HE i modes at the third harmonic in a silica glass optical nanofiber with the diameter of 2 d = 717 nm are displayed. The dark grey dashed circles correspond to the air - silica interface. The white arrows indicate the polarization distributions and are superimposed upon color maps of normalized field amplitudes. The color maps without any white arr ows show the profiles of the phase of the x -field component relative to the z -field component on the +x -axis. 33 Supplementary Note 2: Derivation of the full-vectorial nonlinear coupled -mode equatio ns describing the intermodally phase -matched third -harmonic generation (THG) We derive the full-vectorial nonlinear coupled- mode equations that describ e the experimentally observed intermodally phase -matched THG process . Here, w e treat the pump field (pE and pH ) and the third -harmonic field (sE and sH ) propagating along the z -axis as linear combinations of the corresponding even and odd hybrid modes as below. p,e p,o p,e p,o s,e s,o1/2 p 0 p,e p,e p,o p,o 1/2 p 0 p,e p,e p,o p,o 1/2 s 0 s,e s,e s,o s,o(; ) (, ) () (, ) () , (; ) (, ) () (, ) () , (; 3 ) (, ) () (, ) ()iz iz iz iz iz izZ xya ze xya ze Z xya ze xya ze Z xya ze xya ze Er e e Hr h h Er e e s,e s,o 1/2 s 0 s,e s,e s,o s,o, (; 3 ) (, ) () (, ) () ,iz izZ xya ze xya ze Hr h h (S2) 3 ps 3 ps1( , ) ( ; ) ( ;3 ) c.c. ,2 1( , ) ( ; ) ( ;3 ) c.c. ,2it i t it i tte e te e Er E r E r Hr H r H r (S3) where (,)tEr and (,)tHr are the total electric and magnetic fields, is the optical angular frequency of the pump field, and 001( ) Zc is the impedance of free space, where 0 and c are the electric permittivity and speed of light, respectively, in a vacuum. (,)qxye and (,)qxyh are the normalized electric and magnetic field distribution s of the mode q, and ()qaz and q are the slowly varying complex field amplitude and propagation constant of the mode . The subscripts p and s represent the pump and third -harmonic field s, and e and o stand for the even and odd modes. By substituting Eqs. (S2) and (S3) into Maxwell ’s equation s and applying the reciprocity theorem for electromagnetic fields [4], we obtain 34 p,e p,o s,e s,op,e 1/2 * (3) 0 p,e p p,o 1/2 * (3) 0 p,o p s,e 1/2 * (3) 0 s,e s s,o 1/2 * (3) 0 s,o s() (; ) ,4 () (; ) ,4 () 3( ;3 ) ,4 () 3( ;3 ) ,4iz iz iz izda z iZ e dAdz da z iZ e dAdz da z iZ e dAdz da z iZ e dAdz e Pr e Pr e Pr e Pr (S4) (3) (3) (3) 3 NL p s1( , ) ( ; ) ( ;3 ) c.c. ,2it i tte e Pr Pr Pr (S5) where we neglect the dispersion terms as we are interested in the (quasi -)continuous -wave regime. The surface integrals in Eq. (S4) and all the following equations are performed within the glass part of the waveguide cross -section. (3) NL(,)t Pr is the thi rd-order nonlinear polarization , which brings about various types of third -order nonlinear optical processes involving the four hybrid modes (2 for the pump and 2 for the third harmonic) . We consider self-phase modulation (SPM), cross -phase modulation (XPM), a nd four -wave mixing (FWM), as well as THG and degenerate third- order parametric down- conversion (TPDC). We obtain (3) p,sP for THG and degenerate TPDC, which take the form of (pump) (third harmonic)(3) (3) * * p,TPDC 0 s p p (3) * * * * 0 s pp p ps (3) (3) s , T H G 0 ppp (3) 0 p pp3(; ) ( ; 3 , , ) () () ()4 12 , 4 1(; 3 ) ( 3 ; , , ) () () ()4 1. 4xxxx xxxx P r E rE rE r EEE E EE P r Er Er Er E EE (S6) (3) p,sP for SPM is written as [5] 35 (pump) (third harm(3) (3) * p,SPM 0 p p p (3) * * 0 p pp p pp (3) (3) * s , S P M 0 sss (3) * * 0 ss s ss s3(; ) ( ; , , ) () () ()4 12 , 4 3(; 3 ) ( 3 ; 3 , 3 , 3 ) () () ()412 , 4xxxx xxxx P r Er Er Er E EE E EE P r E rE rE r EEE EE E onic) (S7) whereas for XPM combined with FWM [5], (pump)(3) (3) * p,XPM 0 s s p (3) * * * 0 s sp p ss p ss (3) (3) * s , X P M 0 pps (3) * * * 0 p ps s pp s pp3(; ) ( ; 3 , 3 , ) () () ()2 1, 2 3(; 3 ) ( 3 ; , , 3 ) () () ()2 1 2xxxx xxxx P r E rE rE r EEE E EE E EE P r E rE rE r E EE EEE EEE (third harmonic). (S8) Here, (3) is the third -order nonlinear susceptibility tensor. In the derivation of Eqs. (S6) – (S8), we utilize the fact that for waveguides made of isotropic m aterials such as fused silica glass, the elements of (3) that originates from non -resonant electronic response have the following properties [5] . (3) (3) (3) (3) (3) (3) (3) (3) ( ,,, ,,) , 3,ijkl xxyy ij kl xyxy ik jl xyyx il jk xxyy xyxy xyyx xxxxijkl xyz (S9) where is the Kronecker delta. Furthermore, in the absence of birefringence, we can set p,e p,o p and s,e s,o s . By inserting Eqs. (S6)–(S8) into Eq. (S4) , we obtain the full-vectorial nonlinear coupled- mode equations for the slowly varying complex field amplitudes as follows . p,e (3) * *2 * * * * *2 0 T1 p,e s,e T2 p,e p,o s,o T3 p,o s,e 222* P1 p,e p,e P2 p,o p,e P3 p,o p,e 22** X1 s,e p,e X2 s,o p,e F1 s,e s,o p,o F2 s,e s,o p,o3216 2 2iz xxxxda i k Z Jaa Jaaa Jaa edz J a a J a a Jaa J a a J a a Jaaa Jaaa , (S10) 36 p,o (3) * *2 * * * * *2 0 T4 p,o s,o T3 p,e p,o s,e T2 p,e s,o 22*2* P4 p,o p,o P2 p,e p,o P3 p,e p,o 22* * ** X3 s,e p,o X4 s,o p,o F1 s,e s,o p,e F2 s,e s,o p,e3216 2 2iz xxxxda i k Z Jaa Jaaa Jaa edz J a a J a a Jaa J a a J a a Jaaa Jaaa , (S11) s,e (3) 3 2 0 T1 p,e T3 p,e p,o 222* S1 s,e s,e S2 s,o s,e S3 s,o s,e 22* ** X1 p,e s,e X3 p,o s,e F1 p,e p,o s,o F2 p,e p,o s,o91 16 3 2 2iz xxxxda i kZ J a J a a edz J a a J a a Jaa J a a J a a Jaaa Jaaa , (S12) s,o (3) 3 2 0 T4 p,o T2 p,e p,o 22*2* S4 s,o s,o S2 s,e s,o S3 s,e s,o 22** * X2 p,e s,o X4 p,o s,o F1 p,e p,o s,e F2 p,e p,o s,e91 16 3 2 2iz xxxxda i kZ J a J a a edz J a a J a a Jaa J a a J a a Jaaa Jaaa , (S13) where kc is the pump wavenumber in a vacuum, and ps3 is the wavenumber mismatch for THG and degenerate TPDC. J’s in Eqs. (S10) –(S13) are the overlap integrals representing the strengths of nonlinear optical processes involving the four interacting modes. TiJ are the overlap integrals for THG and degenerate TPDC : * T1 p,e p,e p,e s,e ** T2 p,e p,o p,e s,o p,e p,e p,o s,o ** T3 p,e p,o p,o s,e p,o p,o p,e s,e * T4 p,o p,o p,o s,o, 12,3 12,3 .J dA J dA J dA J dA ee ee ee ee ee ee ee ee ee ee ee ee (S14) Here, T1J [T4J] are for THG/TPDC driven by the even [odd] pump mode only, while T2J and T3J are for THG/TPDC excited by the pump field in a mixture of the even and odd modes. PiJ and SiJ are for SPM o f the pump and the third harmonic, respectively : 37 42 P1 p,e p,e p,e 2 2 22* P2 p,e p,o p,e p,o p,e p,o 2* ** P3 p,e p,o p,e p,e p,o p,o 42 P4 p,o p,o p,o12,3 1,3 12,3 12.3J dA J dA J dA J dA e ee ee ee e e ee ee ee e ee (S15) 42 S1 s,e s,e s,e 2 2 22* S2 s,e s,o s,e s,o s,e s,o 2* ** S3 s,e s,o s,e s,e s,o s,o 42 S4 s,o s,o s,o12,3 1,3 12,3 12.3J dA J dA J dA J dA e ee ee ee e e ee ee ee e ee (S16) Here, P1J [S1J] and P4J [S4J] correspond to the SPM of the pump [third- harmonic] field in the even and the odd mode, respectively. On the other hand, P2J [S2J] and P3J [S3J] can be viewed as intermodal XPM and intermodal conversion, respectively, between the even and odd modes of the pump [third harmonic]. XiJ and FiJ are for XPM and FWM, respectively, between the pump and the third harmonic : 2 2 22* X1 p,e s,e p,e s,e p,e s,e 2 2 22* X2 p,e s,o p,e s,o p,e s,o 2 2 22* X3 p,o s,e p,o s,e p,o s,e 2 2 22* X4 p,o s,o p,o s,o p,o s,o1,3 1,3 1,3 1 3J dA J dA J dA J ee ee e e ee ee e e ee ee e e ee ee e e .dA (S17) * * * * ** F1 p,e p,o s,e s,o p,e s,o p,o s,e p,e s,e p,o s,o ** * * ** F2 p,e p,o s,e s,o p,e s,o p,o s,e p,e s,e p,o s,o1, 3 1. 3J dA J dA ee ee eeee eeee ee ee eeee eeee(S18) 38 In Eq. (S17), X1J [X4J] is for XPM between the even [odd] pump mode and the even [odd] third -harmonic mode. On the other hand, X2J [X3J] is for XPM between the even [odd] pump mode and the odd [even] third -harmonic mode. In Eq. (S18), F1J and F2J are for FWM among four modes (2 for the pump and 2 for the third harmonic). F1J causes intermodal conversion in the same direction (either e ven to odd or odd to even) f or the pump and the third harmonic, whereas F2J in the opposite direction. To appreciate the physical meanings of all the overlap integrals graphically, we illustrate the energy -level diagram for the third -order nonlinear optical process corresponding to each overlap integral in Fig. S2. We also numerically calculate the overlap integrals for three different third -harmonic hybrid modes , the HE 12, the EH 11, and the HE 31 modes, in silica glass optical nanofibers with respective phase- matching diameters, as summarized in Supplementary Table 1. For optical waveguides of perfectly circular cross -sections that preserve the degeneracy of the even and odd hybrid modes, the overlap integrals for SPM have the following symmetr ies. P1 P4 S1 S4, .JJ JJ (S19) For XPM/FWM and THG/TPDC, the symmetry properties of the overlap integrals depend on the azimuthal number of the third -harmonic hybrid mode. For third- harmonic modes having the azimuthal number 1 (e. g., the HE 12 and the EH 11 mode s), T1 T2 T3 T4 X1 X4 X2 X333 , , ,J J JJ JJ JJ (S20) whereas for third -harmonic mode s with the azimuthal number 3 (e.g., the HE 31 mode), T1 T2 T3 T4 X1 X2 X3 X4 F1 F2, , .JJ J J JJJJ JJ (S21) We note that the symmetry relations among TiJ’s in Eqs. ( S20) and ( S21) are closely related to the total angular momentum (TAM) conservation in the THG process, which we will discuss in Supplementary Note 3. 39 F IG. S2. Energy level diagrams for third -order nonlinear optical processes involving hybrid modes in an optical nanofiber. The red and turquoise represent the pump and third- harmonic photons, and the solid and dashed arrows correspond to the even (‘e ’) and odd (‘o’) hybrid modes. Each overlap integral J is defined as Eqs. (S14)–(S18). The diagrams for the self-phase modulation of the third harmonic (SiJ) have the same configurations as those of the pump (PiJ) ( to1 4i ). The diagrams for J can be obtained by inverting or 180º-rotating those for J. Note that JJ because J is real -valued. 40 Supplementary Table 1. Calculated overlap integrals for third -order nonlinear optical processes in silica glass optical nanofibers driven by the pump beam in the HE 11 mode at 1550 nm wavelength. J T,OVM is defined as Eq. (S28) in Supplementary Note 3. Third - harmonic modes Hybrid mode HE 12 EH 11 HE 31 Optical vortex mode 0,2OV± 2,1OV± 2,1OV± ± Total angular momentum ±1 ±1 ±3 Phase -matching nanofiber diameter (nm) 762 624 717 Overlap integrals (μm-2) THG and degenerate TPDC JT1 0.3703 0.0043 0.0767 JT2 0.1234 0.0014 0.0767 JT3 0.1234 0.0014 -0.0767 JT4 0.3703 0.0043 -0.0767 JT,OVM 0 0 0.1534 SPM of the pump JP1 0.9642 0.3332 0.7516 JP2 0.3635 0.1243 0.2830 JP3 0.2372 0.0845 0.1855 JP4 0.9642 0.3332 0.7516 SPM of the third harmonic JS1 3.8831 4.3102 4.0835 JS2 1.5385 1.5774 1.7437 JS3 0.8062 1.1553 0.5961 JS4 3.8831 4.3102 4.0835 XPM JX1 1.4354 0.7598 1.0060 JX2 0.6440 0.6528 1.0060 JX3 0.6440 0.6528 1.0060 JX4 1.4354 0.7598 1.0060 FWM JF1 0.3499 0.0020 -0.1584 JF2 0.4415 0.1050 0.1584 41 Supplementary Note 3: Role of the spin -orbit coupling (SOC) and the resulting spin- orbit entanglement (SOE) in third -harmonic optical vortex generation (TH -OVG) In the previous analysis in Supplementary Note 2, the electric fields are expressed in the basis of the even and odd hybrid modes. Here, we rewrite the nonlinear coupled- mode equations in Eqs. (S10) –(S13) and the overlap integrals J’s in Eqs. (S14) –( S18) in the OVM basis to elucidate the role of SOC and SOE in TH -OVG. The normalized field distribution of each OVM is related to those of the constituent even and odd hybrid modes as below. p, p,e p,o s, s,e s,o1, 2 1. 2i i e ee e ee (S22) Considering this relationship, we express the slowly varying complex field amplitude of each OVM as follows. p, p,e p,o s, s,e s,o1, 2 1. 2a a ia a a ia (S23) Eqs. (S10) and (S11) can then be combined into 2 P1 P2 P3 P4 p, p, 2 P1 P3 P4 p, p, p, (3) 2 0 P1 P2 P3 P4 p, p, 222 P1 P4 p, p, p, p, p, p,42 22 31,42 16 4 2xxxxJ J J Ja a J J Ja a da i kZJ J J J aa dz J Jaa aa a a (S24) whereas Eqs. (S12) and (S13) into 42 3 T1 T2 T3 T4 p, 2 T1 T2 T3 T4 p, p, 2 T1 T2 T3 T4 p, p, 3 T1 T2 T3 T4 p, 22 s, X1 X2 X3 X4 p, p, (3) 01333 1333 91 16 4 2iz xxxxJ J J Ja J J J J aa eJ J J J aa J J J Ja da J J J Ja a i kZdz s, X1 X2 X3 X4 p, p, p, p, s, X1 X2 X3 X4 p, p, p, p, s, 22 X1 X2 X3 X4 p, p, s, 22 F1 F2 p, p, s, F1 F2 p, p, p, p, s,2 2a J J J J aa aa a J J J J aa aa a JJ J Ja aa J Ja a a J J aa aa a , (S25) where we use the undepleted -pump approximation [5] and assume spaa so as to take into account only THG, SPM of the pump, and XPM of the third harmonic combined with FWM driven by the pump. In Eqs. (S24) and (S25), several terms vanish due to the symmetry relations among the overlap integrals in Eqs. (S19) –(S21). In particular, since P1 P40 JJ according to Eq. (S19), the last term in Eq. (S24) is zero, so SPM -induced conversion between the two pump OVMs of mutually opposite spins is inhibited. Moreover, in Eq. (S25), the last THG term and the last XPM term also vanish because T1 T2 T3 T433 0 J J JJ and X1 X2 X3 X40 JJJJ owing to the relations in Eqs. (S20) and (S21), which indicates that the pump in one OVM cannot create the third harmonic in another of the opposite spin. Equation (S25) suggests that a pure third- harmonic OVM can be generated by the pump in a single OVM, i.e., when either p,a or p,a is zero. In this scenario, Eqs. (S24) and (S25) are simplified as 2p, (3) 0 P,OVM p, p,3,16xxxxda i kZ J a adz (S26) 2s, (3) 3 0 T,OVM p, X,OVM p, s,912, 16 3iz xxxxda i kZ J a e J a a dz (S27) 43 T,OVM T1 T2 T3 T4 ** p, p, p, s, p, p, p, s,1334 2.J J J JJ dA ee ee ee ee (S28) P,OVM P1 P2 P3 P4 X,OVM X1 X2 X3 X4 F1 F2142 ,4 12.4J J J JJ J JJJ J JJ (S29) For the 0,2OV and the 2,1OV third -harmonic modes, T,OVMJ vanishes because of the symmetry relations of TiJ’s in Eq. (S20), so the THG in those OVMs by the pump in the 0,1OV mode is forbidden. In strong contrast, T,OVMJ is nonzero for the 2,1OV third - harmonic mode according to Eq. (S21), so the TH -OVG is permitted. These results are also anticipated from the TAM conservation. We emphasize that the nonzero T,OVMJ is facilitated by SOC that releases the requirement of the simultaneous conservation of the OAM and SAM. The normalized field distributio n of the OVM, ,OVlm, in Eq. (S22) may be expressed in an alternative form of [4,6] ,L ,R ,z(, ) (, ) (, ),ur u r ur ez (S30) ( 1) , ,L ( 1) ,R , ,z ,() () , ()ij jm ij jm ij jmiA r e u u iB r e u C re (S31) ( 1) , ,L ( 1) ,R , ,z ,() () , ()ij jm ij jm ij jmiB r e u u iA r e u C re (S32) where ( )2 i xy are the transverse unit vectors along the left -handed ( ) and right -handed ( ) circular polarizations, whereas z is the axial unit vector, and (, , )rz are the cylindrical coordinates . l and 1 jl are the canonical OAM and TAM numbers of the modes , respectively, in the paraxial regime, where 1 is the canonical SAM number. We note that j is the same as the azimuthal mode number of the hybrid mode that composes 44 the OVM. In the presence of SOC, all the three real -valued functions, ,()jmAr , ,()jmBr , and ,()jmCr , are non- negligible, as can be seen in Fig. S3. Then, the OVM in Eq. (S30) becomes a spin- orbit entangled state, its orbital and spin degrees of freedom being coupled to each other. The polarization gets deviated from perfectly circular (Fig. S1(d)), and the ca nonical OAM and SAM are not integers [7], i.e., the OVM is no longer an eigenmode of either the OAM or SAM, while it is still a TAM eigenmode with an integer TAM value, as shown in Fig. S4. In this case, by inserting Eqs. (S30) –(S32) into Eq. (S28), we find that the azimuthal integral for T,OVMJ takes the form of T,OVM s pcos 3 , J d jj (S33) where pj and sj are the TAM numbers of the pump and the third- harmonic OVM, respectively. This result indicates that T,OVMJ is nonzero, and thus TH -OVG can take place, only if the TAM can be conserved, i.e., sp30 jj j . Otherwise, TH -OVG is forbidden. On the other hand, the simulta neous conservation of the OAM and SAM is no longer required. When the pump is in the fundamental 0,1OV mode (p1 j ), T,OVM0 J for THG in the 0,2OV and the 2,1OV mode with s1 j and 0j , whereas T,OVMJ is nonzero for THG in the 2,1OV mode having s3 j and 0j . These results were also obtained from the previous analysis of T,OVMJ using the symmetry relations in Eqs. (S20) and (S21), as shown in Supplementary Table 1. In strong contrast, light propagating in the paraxial or weakly guiding regime exhibits radically different behaviors of TH -OVG. In general, the wave equation for the transverse ( e) and longitudinal (ze) electric field components of a guided mode in an optical waveguide can be derived from Maxwell ’s equations [4]: 2 22 2 2 0 z( , ) (ln ) ,ˆk n xy ni e eez (S34) where 0k and (,)nxy are the wavevector in a vacuum and the refractive index profile of the waveguide, respectively, and is the propagation constant of the mode. is the 45 transverse gradient. For weakly guiding optical waveguides with the tiny refractive index difference between the core and cladding, the right -hand side of Eq. (S34) can be neglected: 2 22 2 z( , ) 0, 0.kn xy e e (S35) In this case, the electric field distribution of an OVM in Eqs. (S30) –(S32) can be simplified, as can be noticed from Fig. S3. For the spin- orbit aligned OVMs composed of the HE hybrid modes, ,()jmBr and ,()jmCr become negligible, so ,ˆ ()il jmiA r e e . On the other hand, for the spin- orbit anti -aligned OVMs consisting of the EH hybrid modes, ,()jmAr and ,()jmCr are negligible, so ,ˆ ()il jmiB r e e [4,6]. Consequently, the SOC disappears, and the OVM becomes a product state rather than a spin- orbit entangled state, i.e., its orbital and spin degrees of freedom are completely separated from each other. The polarization is perfectly circular, and the canonical OAM and SAM have integer values, i.e., the OVM becomes a simultaneous eigenmode of the OAM and SAM, as well as the TAM. The azimuthal integral for T,OVMJ in Eq. (S28) can then be expressed as sp** T,OVM p, p, p, s, p, p, p, s, ,3 s p2 cos 3 0.J dA d ll ee ee ee ee (S36) Here, is the Kronecker delta. Eq. (S36) implies that T,OVMJ is nonzero only when both the OAM and SAM are conserved individually during the THG process, which is, however, impossible because the SAM conservati on cannot be achieved with p,s1 (sp3 ). Therefore, TH -OVG is forbidden without SOC, e.g., in the paraxial regime and under the weakly guiding approximation. 46 FIG. S3. Comparison of the electric field components of optical vortex modes (OVMs) between a weakly guiding fiber (WGF) and an optical nanofiber (ONF). We consider four OVMs in Fig. S1, the 0,1OV mode at 1550 nm wavelength and the 2,1OV mode at the third harmonic. The WGF is a standard step -index fiber having a core diameter of 8.7 µm and a numerical aperture of 0.13. The ONF has a diameter of 717 nm . LE and RE are the left - handed and right -handed circularly polarized field components, whereas zE is the longitudinal one. The grey dashed circles indicate the core- cladding boundary for the WGF ( 2d = 8.7 µm) and the air -silica interface for the ONF ( 2d = 717 nm ). The lowermost row for each OVM stands for the phase profile of each field component relative to the phase of zE on the +x-axis, which is identical for the WGF and ONF. 47 FIG. S4. Calculated canonical angular momenta of optical vortex modes (OVMs) according to the cladding diameter in the course of tapering of an etched silica fiber. ( a, b) Orbital angular momentum (OAM, cyan), spin angular momentum (SAM, magenta), and total angular momentum (TAM, dark yellow) of the 0,1OV+ mode at the 1550 nm pump wavelength (a) and the 2,1OV+ + mode at the third harmonic ( b), which interact through third - harmonic optical vortex generation (TH -OVG) . As the cladding diameter reduce s to the strongly guiding regime, the canonical OAM and SAM deviate from the integer values for the paraxial regime, while the TAM is conserved as an integer regardless of the cladding diameter variation . For the 0,1OV− and the 2,1OV− − modes, all the OAM, SAM, and TAM have the opposite signs and the same magnitudes as those of the 0,1OV+ and the 2,1OV+ + modes. T he initial core [cladding] diameter and numerical aperture of the cladding -etched fiber are set as the experimental values of 8.7 [17] μ m and 0.13, respectively, and t he grey vertical dashed lines indicate the phase -matching nanofiber diameter of 717 nm for the TH- OVG process. 48 Supplementary Note 4: Effect of birefringence on third -harmonic optical vortex generation (TH -OVG) When the cross -section of an optical nanofiber deviates from the perfectly circular one, the degeneracy between the hybrid even and odd modes can be broken. Then, TH -OVG can b e viewed as a coherent combination of four distinct THG processes of different wavenumber mismatches. In this case, the THG part in the differential equation in Eq. (S27) is modified as follows: 123 4s, (3) 3 0 T1 T2 T3 T4 p,9 11 1,16 4 3 3iz iz iz iz xxxxda i kZ J e J e J e J e adz (S37) where i (i = 1 to 4) is the wavenumber mismatch for each THG process defined as 1 p,e s,e 2 p,e p,o s,o 3 p,e p,o s,e 4 p,o s,o3, 2, 2, 3. (S38) In our case, such an imperfection of the nanofiber cross -section can arise during the deep wet - etching step of the nanofiber fabrication process. We observe the cross -section of deep cladding- wet-etched fiber with a scanning electron microscope (SEM ) while one end is polished with focused ion beam (FIB) milling. Since the sample orientation with respect to the electron detector in the FIB -SEM system is difficult to identify precisely, it is almost impossible to directly determine the fiber cladding geometry. Instead, assuming that the germanium -doped fiber core has a perfectly circular cross -section, we compare the eccentricity between the cladding and the core that appear in the scanning electron micrograph to determine the cladding eccentricity (0.05 in the case of Supplementary Fig. S5). Here, the deformation is approximated to be elliptical, and its eccentricity is defined as 2 1ba , where a and b are the major and minor axes, respectively, of the elliptical cross -section. The phase -matching conditions for TH-OVG depend on the deformation of the nanofiber cross -section, as the four constituent THG processes are generally phase matched at different nanofiber diameters according to Eq. (S38) . For efficient TH -OVG with high suppression of THG in the unwanted HE 12 mode, the pump polarization should be maintained circular along 49 the nanofiber, which is, however, hard to achieve in the presence of the deformation -induced nanofiber birefringence. Then, THG in the HE 12 mode cannot be fully suppressed with input pump polarization control. Furthermore, the degeneracy between the third- harmonic HE 31 even and odd modes is also broken by the nanofiber deformation, which alters the relative phase between the two modes during their propagation along the nanofiber. THG in the HE 12 mode is then hard to suppress even at the input pump polarization that yields optimum TH-OVG at the output, which can deteriorate the controllability of the third- harmonic topological charge with input pump polarization adjustment. While it is practically difficult to characterize the deformation of nanofiber cross -section, we observe incomplete suppression of THG in the HE 12 mode in some cases [8], which indicates elliptical deformation of nanofiber cross -section. The experimental results in Fig. S6 show a typical example of non- ideal TH -OVG with limited topological charge c ontrollabilit y due to the deformation of the nanofiber cross -section, although other key features of TH -OVG are reproduced. The most efficient TH -OVG is observed around a certain pump wavelength (1558 nm in the case of Fig. S6(a)), and a donut - shaped third- harmonic fie ld pattern can be observed with the THG in the HE 12 mode significantly suppressed ( Fig. S6(b)). The cubic pump power dependence of the third- harmonic output power is also obtained, as shown in Fig. S6(c), where a maximum TH -OVG conversion efficiency is measured to be ~10-6 at 0.38 W pump power. However, only a single topological charge of either +2 or - 2 (+2 in the case of Fig. S6(d)) is revealed via pump polarization control, whereas another with the opposite sign of topological charge is hardly excited wi th sufficiently suppressing the THG in the undesired HE 12 mode. 50 F IG. S5. Observation of the cross -section of a cladding wet -etched fiber using a focused ion beam (FIB) -scanning electron microscope (SEM) system. One end of the cladding - etched fiber is polished by FIB milling, and then the end facet is in- situ observed with the SEM. The SEM image on the right displays the end facet, where the slightly darker ellipse is the core - cladding boundary. Note that the cross- section appears generally elliptical on the SEM image because it is detected at an oblique angle q from the fiber axis in the FIB -SEM system. The white scale bar in the SEM image corresponds to 5 µm. 51 FIG. S6. Non -ideal third -harmonic optical vortex generation (TH -OVG) with limited topological charge controllability. ( a) Measured TH -OVG conversion efficiencies over a pump wavelength range of 1535–1563 nm, while the average pump power is fixed at 0.25 W. The pump polarization state is adjusted to minimize the third harmonic in the undesired HE 12 mode. ( b) Far -field profiles of the third -harmonic output signals recorded at different pump wavelengths (white numbers in nanometers). ( c) Measured third -harmonic (TH) output power s over a range of pump powers at a fixed pump wavelength of around 1558 nm. The turquoise line is a cubic fit to the measurement (blue solid s quares). ( d) Far -field profiles of the third - harmonic output signals without (upper row) and with (lower row) the a stigmatic beam transforms via focusing at a cylindrical lens (CL). The numbers in white are the average pump powers in Watt. 52 Supplementary References 1. P. Z. Dashti, F. Alhassen, and H. P. Lee, “Observation of orbital angular momentum transfer between acoustic and optical vortices in optical fiber, ” Phys. Rev. Lett. 96, 043604 (2006). 2. D. S. Han and M. S. Kang, “ Reconfigurable generation of optical vortice s based on forward stimulated intermodal Brillouin scattering in subwavelength -hole photonic waveguides, ” Photon. Res. 7, 754–761 (2019). 3. P. Gregg, P. Kristensen, A. Rubano, S. Golowich, L. Marrucci, and S. Ramachandran, “Enhanced spin orbit interaction of light in highly confining optical fibers for mode division multiplexing,” Nat. Commun. 10, 4707 (2019). 4. A. W. Snyder and J. D. Love, Opti cal Waveguide Theory , Chapman and Hall (1983). 5. R. W. Boyd, Nonlinear Optics , 4th Edition , Academic Press ( 2020) . 6. K. Okamoto, Fundamentals of Optical Waveguides , Academic Press (2005). 7. M. F. Picardi, K. Y . Bliokh, F. J. Rodr íguez -Fortuño, F. Alpeggiani, and F. Nori, “ Angular momenta, helicity, and other properties of dielectric -fiber and metallic -wire modes, ” Optica 5, 1016–1026 (2018). 8. C. K. Ha, K. H. Nam, and M. S. Kang, “ Efficient harmonic generation in an adiabatic multimode submicron tapered optical fiber,” Commun. Phys. 4, 173 (2021).

1608.06445v2.The_effect_of_spin_orbit_coupling_on_the_effective_spin_correlation_in_YbMgGaO4.pdf

The eect of spin-orbit coupling on the eective-spin correlation in YbMgGaO 4 Yao-Dong Li1, Yao Shen1, Yuesheng Li2;3, Jun Zhao1;4, and Gang Chen1;4 1State Key Laboratory of Surface Physics, Center for Field Theory and Particle Physics, Department of Physics, Fudan University, Shanghai 200433, People's Republic of China 2Department of Physics, Renmin University of China, Beijing 100872, People's Republic of China 3Experimental Physics VI, Center for Electronic Correlations and Magnetism, University of Augsburg, 86159 Augsburg, Germany and 4Collaborative Innovation Center of Advanced Microstructures, Nanjing, 210093, People's Republic of China (Dated: October 12, 2018) Motivated by the recent experiments on the triangular lattice spin liquid YbMgGaO 4, we explore the eect of spin-orbit coupling on the eective-spin correlation of the Yb local moments. We point out the anisotropic interaction between the eective-spins on the nearest neighbor bonds is sucient to reproduce the spin-wave dispersion of the fully polarized state in the presence of strong magnetic eld normal to the triangular plane. We further evaluate the eective-spin correlation within the mean-eld spherical approximation. We explicitly demonstrate that, the nearest-neighbor anisotropic eective-spin interaction, originating from the strong spin-orbit coupling, enhances the eective-spin correlation at the M points in the Brillouin zone. We identify these results as the strong evidence for the anisotropic interaction and the strong spin-orbit coupling in YbMgGaO 4. I. INTRODUCTION The rare earth triangular lattice antiferromagnet YbMgGaO 4was recently proposed to be a candidate for quantum spin liquid (QSL)1{5. In YbMgGaO 4, the Yb3+ ions form a perfect two-dimensional triangular lattice. For the Yb3+ions, the strong spin-orbit coupling (SOC) entangles the orbital angular momentum, L(L= 3), with the total spin, s(s= 1=2), leading to a total moment, J(J= 7=2)1,2. Like the case in the spin ice material Yb2Ti2O76, the crystal electric eld in YbMgGaO 4fur- ther splits the eight-fold degeneracy of the Yb3+total moment into four Kramers' doublets. The ground state Kramers' doublet is separated from the excited doublets by a crystal eld energy gap. At the temperature that is much lower than the crystal eld gap, the magnetic prop- erties of YbMgGaO 4are fully described by the ground state Kramers' doublets2. The ground state Kramers' doublet is modeled by an eective-spin-1/2 local mo- ment S. Therefore, YbMgGaO 4is regarded as a QSL with eective-spin-1/2 local moments on a triangular lat- tice1{4. The existing experiments on YbMgGaO 4have involved thermodynamic, neutron scattering, and SR measure- ments1,3{5. The system was found to remain disor- dered down to 0.05K in the recent SR measurement5. The thermodynamic measurement nds a constant mag- netic susceptibility in the zero temperature limit. In the low temperature regime, the heat capacity1,4behaves as CvconstantT0:7. The inelastic neutron scattering measurements from two research groups have found the presence of broad magnetic excitation continuum3,4. In particular, the inelastic neutron scattering results from Yao Shen et al clearly indicate the upper excitation edge and the dispersive continuum of magnetic excitations3. Both neutron scattering results found a weak spectral peak at the M points in the Brillouin zone3,4. Based on the existing experiments, we have proposed that the zYb3+O2- Mg2+/Ga3+(a) a1a2 a3 xy z(b)FIG. 1. (Color online.) (a) The crystal structure of YbMgGaO 4. Mg and Ga ions form the non-magnetic layer. (b) The Yb triangular layer. spinon Fermi surface U(1) QSL gives a reasonable de- scription of the experimental results3. Previously, two organic triangular antiferromagnets, -(ET) 2Cu2(CN) 3and EtMe 3Sb[Pd(dmit) 2]2, were pro- posed to be QSLs7{10. These two materials are in the weak Mott regime, where the charge uctuation is strong. It was then suggested that the four-spin ring exchange interaction due to the strong charge uctuation may destabilize the magnetic order and favor a QSL ground state11,12. Unlike the organic counterparts, YbMgGaO 4 is in the strong Mott regime1,2. The 4felectrons of thearXiv:1608.06445v2 [cond-mat.str-el] 11 Sep 20162 Yb3+ion is very localized spatially. As a result, the phys- ical mechanism for the QSL ground state in this new material is deemed to be quite dierent. The new in- gredient of the new material is the strong SOC and the spin-orbit entangled nature of the Yb3+local moment. It was pointed out that the spin-orbit entanglement leads to highly anisotropic interactions between the Yb local mo- ments2,13{15. The anisotropic eective-spin interaction is shown to enhance the quantum uctuation and suppress the magnetic order in a large parameter regime where the QSL may be located2. On the fundamental side, it was recently argued that, as long as the time reversal symmetry is preserved, the ground state of a spin-orbit- coupled Mott insulator with odd number of electrons per cell must be exotic16. This theoretical argument implies that the spin-orbit-coupled Mott insulator can in princi- ple be candidates for spin liquids. YbMgGaO 4falls into this class and is actually the rst such material. More recently, Ref. 4 introduced the XXZ exchange interactions on both nearest-neighbor and next-nearest- neighbor sites to account for the spin-wave dispersion in the strong magnetic eld and the weak peak at the M points in the eective-spin correlations. The authors further suggested the further neighbor competing ex- change interactions as the possible mechanism for the QSL in YbMgGaO 4. In this paper, however, we fo- cus on the anisotropic eective-spin interactions on the nearest-neighbor sites. After carefully justifying the un- derlying microscopics that supports the nearest-neighbor anisotropic model, we demonstrate that the nearest- neighbor model is sucient to reproduce the spin-wave dispersion of the polarized state in the strong magnetic eld. With the nearest-neighbor anisotropic model, we further show that the eective-spin correlation also de- velops a peak at the M points. Therefore, we think the nearest-neighbor anisotropic model captures the essential physics for YbMgGaO 4. The remaining part of the paper is outlined as fol- lows. In Sec. II, we describe some of the details about the microscopics of the interactions between the Yb lo- cal moments. In Sec. III, we compare the spin-wave dispersion of the nearest-neighbor anisotropic interac- tions in a strong eld with the existing experimental data. In Sec. IV, we evaluate the eective-spin correla- tion from the eective-spin models with and without the anisotropic interaction. Finally in Sec. V, we conclude with a discussion. II. THE ANISOTROPIC INTERACTION FOR THE EFFECTIVE SPINS Compared to the organic spin liquid candidates7{10, YbMgGaO 4is in the strong Mott regime, and the charge uctuation is rather weak. Therefore, the four- spin ring exchange, that is a higher order perturba- tive process than the nearest-neighbor pairwise interac- tion, is strongly suppressed. In the previous work1,2,we have proposed that the generic pairwise eective- spin interaction for the nearest-neighbor Yb moments in YbMgGaO 4, H=X hrr0iJzzSz rSz r0+J(S+ rS r0+S rS+ r0) +J( rr0S+ rS+ r0+  rr0S rS r0) iJz 2 (  rr0S+ r rr0S r)Sz r0 +Sz r(  rr0S+ r0 rr0S r0) ; (1) whereS r=Sx riSy r, and rr0= r0r= 1;ei2=3;ei2=3 are the phase factors for the bond rr0along the a1,a2, a3directions (see Fig. 1). The JandJzterms of Eq. (1) are anisotropic interactions arising naturally from the strong SOC. Due to the SOC, the eective-spins in- herit the symmetry operation of the space group, hence there are bond-dependent JandJzinteractions. Our generic model in Eq. (1) contains the contribu- tion from all microscopic processes that include the di- rect 4f-electron exchange, the indirect exchange through the intermediate oxygen ions, and the dipole-dipole in- teraction. The further neighbor interaction is neglected in our generic model. Like the ring exchange, the fur- ther neighbor superexchange usually involves higher or- der perturbative processes via multiple steps of electron tunnelings than the nearest-neighbor interactions. Even though the further neighbor superexchange interaction can be mediated by the direct electron hoppings between these sites, the contribution should be very small due to the very localized nature of the 4 felectron wavefunc- tion. The remaining contribution is the further neighbor dipole-dipole interaction. For the next-nearest neighbors, the dipole-dipole interaction is estimated to be 0.01- 0.02K and is thus one or two orders of magnitude smaller than the nearest-neighbor interactions. Therefore, we can safely neglect the further neighbor interactions and only keep the nearest neighbor ones in Eq. (1). The large chemical dierence prohibits the Ga or Mg contamination in the Yb layers. The Yb layers are kept clean, and there is little disorder in the exchange in- teraction. Although there exists Ga/Mg mixing in the nonmagnetic layers, the exchange path that they involve would be Yb-O-Ga-O-Yb or Yb-O-Mg-O-Yb (see Fig. 1). This exchange path is a higher order perturbative pro- cess than the Yb-O-Yb one and thus can be neglected. We do not expect the Ga/Mg mixing in the nonmagnetic layers to cause much exchange disorder within the Yb layers. The Ga/Mg disorder in YbMgGaO 4is dierent from the Cu/Zn disorder in herbertsmithite17{20. In the latter case, the Cu disorder carries magnetic moment and directly couples to the spin in the Cu layers. The XXZ limit of our generic model has already been studied in some of the early works21,22. It was shown that the magnetic ordered ground state was obtained for all parameter regions in the XXZ limit. To obtain a disor- dered ground state for the generic model, it is necessary to have the JandJzinteractions. In Ref. 2, we have3 a Γ1 K M Γ2M YM X00.511.522.5E(meV ) 318 518 622 805 832 980 1004 108602.8655.738.59511.4614.325318 518 622 805 832 980 1004 1086 02.8655.738.59511.4614.325b Γ1 K M Γ2M YM X00.511.522.5E/(6.1 Jzz) Γ1Γ2 KMK/prime Y X FIG. 2. (Color online.) (a) The experimental spin-wave dispersion in the presence of external eld along z-direction with eld strength 7.8T at 0.06K (adapted from Ref. 4). According to Ref. 4, the white circles indicate the location of the max- imum intensity. The error bar, however, was not indicated in the plot. The red lines show a t to the spin-wave dispersion relation that is obtained after including both nearest-neighbor and next-nearest-neighbor XXZ exchange interactions4. (b) The theoretical spin-wave dispersion according to the nearest-neighbor anisotropic exchange model Eq. (1), where we set J=Jzz= 0:66;J=Jzz= 0:34, andh=J zz= 10:5. The analytical expression of the dispersion is given in Eq. (2). The inset of (b) is the Brillouin zone. shown that the 120-degree magnetic order in the XXZ limit is actually destabilized by the enhanced quantum uctuation when the anisotropic JandJzinterac- tions are introduced. III. SPIN-WAVE DISPERSION IN THE STRONG MAGNETIC FIELD The nearest-neighbor interaction between the Yb local moments are of the order of several Kelvins1; as a result, a moderate magnetic eld in the lab is sucient for po- larizing the local moment2. Under the linear spin-wave approximation, the spin-wave dispersion in the presence of the strong external magnetic eld is given as2 !z(k) =n gzBBz3Jzz+ 2J3X i=1cos(k
ai)2 4J2 cos(k
a1) +ei2 3cos(k
a2) +ei2 3cos(k
a3)2o1=2 ; (2) wheregzandBzare Land e factor and magnetic eld alongz-direction, respectively. Notice that the dispersionin Eq. (2) is independent of Jz; this is an artifact of the linear spin-wave approximation. In the recent experiment in Ref. 4, a magnetic eld of 7.8T normal to the Yb plane at 0.06K, a gapped magnon band structure is observed. In Fig. 2, we compare our theoretical result with a tentative choice of exchange couplings in Eq. (2) with the experimental results from Ref. 4. Since the error bar is not known from Ref. 4, judg- ing from the extension of the bright region in Fig. 2a, we would think that the agreement between the theoretical result and the experimental result is reasonable. Here, we have to mention that the dispersion that is plotted in Fig. 2 is not quite sensitive to the choice of J. There- fore, we expect it is better to combine the spin-wave dis- persion for several eld orientations and to extract the exchange couplings more accurately. For an arbitrary external eld in the xzplane, the Hamiltonian is given by Hxz=HX rB gxBxSx r+gzBzSz r : (3) Sincegx6=gz, the uniform magnetization, mhSri, is generally not parallel to the external magnetic eld. ForBxBsinandBzBcos, the magnetization is4 given by m=m(^xsin0+ ^zcos0); (4) where tan0= (gx=gz) tan. At a suciently large mag- netic eld, all the moments are polarized along the direc- tion dened by 0. In the linear spin-wave theory for this polarized state, we choose the magnetization to be the quantization axis for the Holstein-Primako transforma-tion, Sr
m jmj1 2ay rar; (5) Sr
^y1 2(ar+ay r); (6) Sr
(m jmj^y)1 2i(aray r); (7) whereay r(ar) is the creation (annihilation) operator for the Holstein-Primako boson. In the linear spin-wave approximation, we plug the Holstein-Primako transfor- mation intoHxzand keep the quadratic part of the the Holstein-Primako bosons. The spin-wave dispersion is obtained by solving the linear spin-wave Hamiltonian and is given by !xz(k)=nh gxBBxsin0+gzBBzcos06Jsin203Jzzcos20+ cos( k
a1)J 2(3 + cos 20) Jsin20+Jzz 2sin20 + cos( k
a2)J 2(3 + cos 20) +J 2sin20+Jzz 2sin20 +p 3 4Jzsin 20 + cos( k
a3)J 2(3 + cos 20) +J 2sin20+Jzz 2sin20p 3 4Jzsin 20i2 cos(k
a1) Jsin20J(1 + cos20)iJzsin0Jzz 2sin20 +cos( k
a2) Jsin20+J 4(3 + cos 204ip 3 cos0)Jzz 2sin20+Jz 4(2isin0p 3 sin 20) +cos( k
a3) Jsin20+J 4(3 + cos 20+ 4ip 3 cos0)Jzz 2sin20+Jz 4(2isin0+p 3 sin 20)2o1=2 :(8) Likewise, for the eld within the xyplane, the Hamiltonian is given by Hxy=HX rB gxBxSx r+gyBySy r : (9) Now because of the three-fold on-site symmetry, gx=gy. The magnetization is parallel to the external magnetic eld. ForBxBcosandByBsin, the magnetization is m=m(^xcos+ ^ysin), and the corresponding spin-wave dispersion in the strong eld limit is given by !xy(k)=nh gxBBxcos+gyBBysin6J+ cos( k
a1) J+Jzz 2Jcos 2
+cos( k
a2) J+Jzz 2+Jcos(2 3) + cos( k
a3) J+Jzz 2+Jcos(2+ 3)i2 cos(k
a1) JJzz 2Jcos 2+iJzcos + cos( k
a2) JJzz 2+Jcos(2+ 3) iJzcos( 3) + cos( k
a3) JJzz 2+Jcos(2 3)iJzcos(+ 3)2o1=2 : (10) IV. EFFECTIVE-SPIN CORRELATION In both Ref. 3 and Ref. 4, a weak spectral peak at the M points is found in the inelastic neutron scattering data. This result indicates that the interaction between the Yb local moments enhances the correlation of the eective- spins at the M points. Actually in Ref. 2, we have alreadyshown that, the anisotropic JandJzinteractions, if they are signicant, would favor a stripe magnetic order with an ordering wavevector at the M points23. This theoretical result immediately means that the anisotropic JandJzinteractions would enhance the eective- spin correlation at the M points. In the following, we demonstrate explicitly that the generic model in Eq. (1)5 a h0 1 -1J±=0.66 J±±=0 Jz±=0 kBT=0.25 b h0 1 -1012 −1 −2 h+2kJ±=0.66 J±±=0.34 Jz±=0 kBT=0.25 c h0 1 -1J±=0.66 J±±=0 Jz±=0.6 kBT=0.25 Intensity (arb.unit) 0 1 d h0 1 -1012 −1 −2 h+2kJ±=0.66 J±±=0.34 Jz±=0.6 kBT=0.25 FIG. 3. (Color online.) Contour plot of eective-spin cor- relation hS+ kS kiin the momentum space. The correla- tion function is computed from the nearest-neighbor model in Eq. (1), with parameters in units of Jzzindicated. Without the anisotropic exchanges, the spectral weight peaks around K. The anisotropic JandJzinteractions can switch the peak to M. with the anisotropic nearest-neighbor interactions does enhance the eective-spin correlation at the M points. We start from the mean-eld partition function of the system, Z=Z D[Sr]Y r(S2 rS2)eH =Z D[Sr]D[r]eH+P rr[S2 rS2] Z D[Sr]D[r]eSeff[; r]; (11) whereHis given in Eq. (1), Seis the eective action that describes the eective-spin interaction, and ris the lo- cal Lagrange multiplier that imposes the local constraint withjSrj2=S2. Although this mean-eld approximation does not gives the quantum ground state, it does provide a qualitative understanding about the relationship be- tween the eective-spin correlation and the microscopic spin interactions. To evaluate the eective-spin correlation, we here adopt a spherical approximation24by replacing the lo- cal constraint with a global one such thatP rjSrj2= NsiteS2, whereNsiteis the total number of lattice sites. This approximation is equivalent to choosing a uniform Lagrange multiplier with r. It has been shown thatthe spin correlations determined from classical Monte Carlo simulation are described quantitatively within this scheme24. In the momentum space, we dene S r1pNsiteX k2BZS keik
r; (12) and the eective action is given by Se[;] =X k2BZ J(k) +
() S kS k Nsite
()S2; (13) where we have placed 
() in a saddle point ap- proximation, ;=x;y;z , andJ(k) is a 33 exchange matrix that is obtained from Fourier transforming the exchange couplings. Note the XXZ part of the spin in- teractions only appears in the diagonal part of J(k) while the anisotropic JandJzinteractions are also present in the o-diagonal part. Hence the eective-spin correlation is given as hS kS ki=1  J(k) +
()1331 ; (14) where 133is a 33 identity matrix. The saddle point equation is obtained by integrating out the eective-spins in the partition function and is given by X k2BZX 1 [J(k) +
()133]1 =NsiteS2; (15) from which, we determine
( ) and the eective-spin correlation in Eq. (14). The results of the eective-spin correlations are pre- sented in Fig. 3. In the absense of the JandJz interactions, the correlation function is peaked at the K points. This result is understood since the XXZ model favors the 120-degree state would simply enhance the eective-spin correlation at the K points that correspond to the ordering wavevectors of the 120-degree state. Af- ter we include the JandJzinteractions, the peak of the correlation function is switched to the M points (see Fig. 3d). This suggests that it is sucient to have the JandJzinteractions in the nearest-neighbor model to account for the peak at the M points in the neutron scattering results. V. DISCUSSION Instead of invoking further neighbor interaction in Ref. 4, we have focused on the anisotropic spin inter- action on the nearest-neighbor bonds to account for the spin-wave dispersion of the polarized state in the strong magnetic eld and the eective-spin correlation in YbMgGaO 4. The bond-dependent interaction is a6 natural and primary consequence of the strong SOC in the system. As for the importance of further neigh- bor interaction, it might be possible that some physical mechanism, that we are not aware of, may suppress the anisotropic spin interactions between the nearest neigh- bors but enhance the further neighbor spin interactions. We have recently proposed that the spinon Fermi sur- face U(1) QSL provides a consistent explanation for the experimental results in YbMgGaO 4. We pointed out that the particle-hole excitation of a simple non-interacting spinon Fermi sea already gives both the broad continuum and the upper excitation edge in the inelastic neutron scattering spectrum. In the future work, we will varia- tionally optimize the energy against the trial ground state wavefunction that is constructed from a more generic spinon Fermi surface mean-eld state. It will be ideal to directly compute the correlation function of the localmoments with respect to the variational ground state. To summarize, we have provided a strong evidence of the anisotropic spin interaction and strong SOC in YbMgGaO 4. In particular, the nearest-neighbor spin model, as used throughout the paper, proves to be the appropriate description of the system. Acknowledgements. |G.C. sincerely acknowledges Dr. Martin Mourigal for discussion and appologizes for not informing him properly about adapting Fig. 2a. We thank Dr. Zhong Wang at IAS of Tsinghua University and Dr. Nanlin Wang at ICQM of Peking University for the hospitality during our visit in August 2016. This work is supported by the Start-up Funds of Fudan Uni- versity (Shanghai, People's Republic of China) and the Thousand-Youth-Talent Program (G.C.) of People's Re- public of China. gangchen.physics@gmail.com, gchen physics@fudan.edu.cn 1Yuesheng Li, Gang Chen, Wei Tong, Li Pi, Juanjuan Liu, Zhaorong Yang, Xiaoqun Wang, and Qingming Zhang, \Rare-Earth Triangular Lattice Spin Liquid: A Single- Crystal Study of YbMgGaO4," Phys. Rev. Lett. 115, 167203 (2015). 2Yao-Dong Li, Xiaoqun Wang, and Gang Chen, \Anisotropic spin model of strong spin-orbit-coupled trian- gular antiferromagnets," Phys. Rev. B 94, 035107 (2016). 3Yao Shen, Yao-Dong Li, Hongliang Wo, Yuesheng Li, Shoudong Shen, Bingying Pan, Qisi Wang, H. C. Walker, P. Steens, M Boehm, Yiqing Hao, D. L. Quintero-Castro, L. W. 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2401.08373v1.Active_Control_of_Ballistic_Orbital_Transport.pdf

1 Active Control of Ballistic Orbital Transport Sobhan Subhra Mishra , James Lourembam, Dennis Jing Xiong Lin , Ranjan Singh* S. Mishra , Prof. R. Singh Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, 21 Nanyang Link, Singapore 637371, Singapore E-mail: ranjans@ntu.edu.sg S. Mishra , Prof. R. Singh Center for Disruptive Photonic Technologies, The Photonics Institute, Nanyang Technological University, Singapore 639798, Singapore E-mail: ranjans@ntu.edu.sg Dr. J. Lourembam , D.J.X. Lin Institute of Materials Research and Engineering A*STAR (Agency for Science, Technology and Research) , 2 Fusionopolis Way, Innovis, #08-03, Singapore 138364, Singapore 2 Abstract Orbital current, defined as the orbital character of Bloch states in solids, can ballistically travel with larger coherence length through a broader range of materials than its spin counterpart, facilitating a robust, higher density and energy efficient in formation transmission. Hence, active control of orbital transport plays a pivotal role in propelling the progress of the evolving field of quantum information technology. Unlike spin angular momentum, orbital angular momentum (OAM), couples to phonon angu lar momentum (PAM) efficiently via orbital - crystal momentum (L -k) coupling, giving us the opportunity to control orbital transport through crystal field potential mediated angular momentum transfer. Here, leveraging the orbital dependant efficient L -k coupling, we have experimentally demonstrated the active control of orbital current velocity using THz emission spectroscopy. Our findings include the identification of a critical energy density required to overcome collisions in orbital transport, enabling a swifter flow of orbital current. The capability to actively control the ballistic orbital transport lays the groundwork for the development of ultrafast devices capable of efficiently transm itting information over extended distance. 3 Introduction The interconversion of spin current and charge current has been extensively explored in the field of spintronic s research1,2. Notable instances of this conversion encompass phenomena such as spin Hall2 and inverse spin Hall effects3,4 observed in heavy metals, spin momentum locking in topological insulators5,6, and the Rashba -Edelstein effect along with its inverse counterpart in two -dimensional electron gases7–9. In recent times, these effects have ga rnered substantial traction as a potent method for harnessing the spin angular momentum carried by electrons10,11, allowing for generat ion of ultrafast charge current , thereby enabling the emission of broadband THz pulses12,13. Recent research has brought to light the significance of orbital angular momentum of electrons , giving rise to the emerging field of orbitronics14,15. It can possess a substantially greater value compared to its spin counterpart16. Additionally, the conversion between orbital and charge current do not require a heavy metal17,18, thereby expanding the range to include light metals available for utilization. One of the limitations of these systems is the absence of direct source of orbital current ; however, this limitation can be eliminated by various orbital pumping techniques through magnetization and lattice dynamics19. One of the most used methods of generating orbital current is by using a high Spin Orbit coupling (SoC) ferromagnet li ke Ni or Co to facilitate the interconversion of spin current and orbital current within the ferromagnet making them indirect sources of orbital current14–16. The converted orbital current transports to the adjacent nonmagnetic metal layer where it converts to an accelerated charge current generating THz waves as shown Fig 1 (a)20,21. Unlike spin transport, which is super diffusive in nature22, orbital transport is ballistic and hence can propagate over long distance reaching up to 80 nm21. Recent study has proposed that the orbital current velocity can be measured up to 0.14 nm/fs in Ni/W heterostructures21. 4 Here , we pr opose an active method to control the orbital current velocity through the applied optical fluence in a Ni/Pt heterostructure. Our findings demonstrate that upon ultrafast photoexcitation, the Ni/Pt heterostructure emits THz radiation primarily due to ballistic transport of orbital current within the Pt layer, which is subsequently convert ed to charge current . Harnessing electron -phonon coupling which is dependent on orbital angular momentum but independent of spin , we can actively control the orbital transport through laser fluence. Our findings show tunable orbital current velocity from 0.14 nm/fs to 0.18 nm/fs exhibiting a direct control of velocity of orbital transport. We also determine the critical energy densit y necessary to overcome collisions, thereby enabling swifter movement of orbital current within the metal layer . Results and Discussion While spin and orbital angular momentum have similar symmetry properties , they exhibit distinct behaviors in ultrafast timescales21,23,24. Photoexcitation creates spin accumulation μS25,26 which is proportional to the difference in instantaneous magnetization and equilibrium magnetization resulting in release of spin current jS at a rate proportional to μS. In a similar way photoexcitation can also induce orbital accumulation μLresulting in orbital angular momentum transport proportional to μL. As the S - type and L - type ultrafast magnetization dynamics is similar for ferromagnets (FM) like Ni27,28, μS and μL will also be proportional to each other resulting in linear relation between jS and jL. However, despite identical driving dynamics, spin and orbital current evolution can vary with thickness and time in the nonmagnetic metal (NM) layer as the transport of both the currents can be different at the interfaces and in the bulk of NM . Additionally, due to applied electric field perturbation because of laser fluence, the angular momentum exchange between the lattice and orbital wave function can also control the transport of orbital angular momentum .19. 5 As depicted in Figure 1(a), when the FM/NM heterostructure is subjected to ultrafast photoexcitation, the initially generated ultrafast spin current partially transforms into an ultrafast orbital current due to LS correlation in ferromagnet layer resulting in the injection of both spin (𝑗𝑆) and orbital current (𝑗𝐿) into the nonmagnetic metal layer. Through the processes of LCC and SCC, a ccelerated charge current is induced, serving as the source of THz radiation. The resultant THz electric field is directly proportional to the sheet charge current (𝐼𝐶(𝑡)), as described by the following equation12,13,29. 𝐸(𝑡)∝𝐼𝐶(𝑡)∝∫ 𝑑𝑧[𝜃𝐿𝐶(𝑧)𝑗𝐿(𝑧,𝑡)+𝜃𝑆𝐶(𝑧)𝑗𝑆(𝑧,𝑡)]𝑑𝑁𝑀 −𝑑𝐹𝑀 (1) Here 𝑑𝐹𝑀 and 𝑑𝑁𝑀 denotes the thickness of the ferromagnet and nonmagnetic metal layer respectively and 𝜃𝐿𝐶 and 𝜃𝑆𝐶 denote the orbital Hall and spin Hall angle which govern the efficiency of orbital to charge (LCC) and spin to charge conversion (SCC) . The equation does not include the minor process of THz emission like ultrafast demagnetization and anomalous hall effect due to the ferromagnet as those can be separated out from the photocurrent mechanisms experimentally30. Relative sign of spin hall and orbital hall angle depends on the LS correlation 〈L.S〉 as shown in Fig 1(b) and 1(c). If L.S <0, the spin and orbital transport will have opposite sign whereas if L.S>0, spin and orbital transport will have same sign31. To facilitate a direct comparison between orbital transport and spin transport, we study two different types of FM/NM heterostructure s. In the first case, Ni was chosen as the FM layer to illustrate orbital transport as Ni has higher efficiency of generating orbital current from spin current because of its higher L.S correlation near fermi level32. In the second case, to demonstrate spin transport , NiFe is chosen as FM due to its higher spin current generation efficiency12. In both the cases, NM chosen was Pt due to its high Orbital Hall Conductivity and Spin Hall Conductivity33. An in-6 plane constant magnetic field of 128 mT was applied to keep the system at saturated magnetization state. Figure 2(a) displays the emitted THz radiation from a Ni(3 nm)/Pt(x nm) heterostructure, when photoexcited by a constant fluence of 1270 J/cm2, with variable Pt thickness (x = 3 , 6, 9, 18 ). It is observ ed that as the Pt thickness increases , the emitted THz pulse encounters a delayed arrival, whereas in case of NiFe (3 nm)/ Pt (x nm) with x = 1, 2, 4, 6, 8 , there is no delay in THz pulse as illustrated in Figure 2(b). This comparison serves to establish the prevalence of different photocurrent mechanism of THz emission from Ni/Pt compared to NiFe/Pt. In case of NiFe /Pt where spin transport dominates the THz emission, the increase in Pt thickness does not delay the arrival of THz due to lower relaxation length of spin current . A very small delay of around 2 fs can be induced (See supplementary for calculations) due to the THz refractive index of platinum. Therefore, the delay in Ni/Pt could be attributed to a long-distance transport of orbital angular momentum. The shift in the peak THz pulse is graphically represented in Figure 2(c), demonstrating the linear correlation between Pt thickness and the delay obtained in THz peak which validates the ballistic nature of orbital transport and provid es us with an orbital current velocity of approximately 0. 18 nm/fs , as opposed to the zero delay in THz peak value when increasing the thickness of Pt in the case of NiFe/Pt, where spin transport predominates . Furthermore, due to its larger relaxation length, orbital current will experience more significant angular dispersion at higher thicknesses, resulting in the widening of the THz pulse depicted in Figure 2(d) . The pulse width measured from peak -to-peak time increases at larger thickness of Pt in case of Ni/Pt . However, in case of NiFe/Pt where spin transport dominates , due to the very short relaxation time , the spin current does not travel enough distance to have large angular dispersion resulting in constant pulse width with increase in thickness of Pt. 7 The orbital angular momentum of the nonlocalized electrons interacts with lattice through the crystal field potential with following continuity equation34,35 𝜕 𝜕𝑡〈𝐿〉= 〈𝐹𝐿〉+1 𝑖ħ〈[𝐿,𝑉𝐶𝐹]〉+〈𝑆×𝐿〉 (2) Here 𝐹𝐿 is the orbital flux term, 〈[𝐿,𝑉𝐶𝐹]〉 describes the transfer of angular momentum between orbital and crystal , VCF is crystal field potential and 𝑆×𝐿 describes the mutual transfer of spin and orbital angular momentum within a single electron. The electric field of the applied fluence interacts with the orbital, causing a perturbation in the orbital wave function which gives rise to a non -equilibrium orbital wave function. The electric field perturbed non - equilibrium orbital wave function extracts angular momentum from the lattice, thereby increasing the velocity of orbital transport. Figure 3 offers a comprehensive illustration of the active control of orbital transport within the Pt lay er. The thickness of the ferromagnetic material Ni remains constant at 3 nm throughout the experiment. In Figures 3(a -d), the emitted THz pulse is depicted for Ni(3 nm) and Pt(x nm), where x takes values of 3, 6, 9, and 18, respectively, at different fluence levels. As the fluence increases, a noticeable shift in the emitted THz pulse is observed. Initially, the pulse shifts towards the right, indicating a delayed arrival of the THz puls e. However, beyond a certain fluence threshold referred to as the critical fluence , there is an increase in orbital current velocity . Consequently, beyond this critical fluence , there is left shift in the arrival time of the THz pulse. Similar behaviour was also observed in other orbital transport -based emission systems such as Ni/Ru (See supplementary section). Figure 3(e) demonstrates that critical fluence is contingent on the thickness of the nonmagnetic Pt layer. As the thickness is increased, the carriers require more energy to overcome collisions to move ballistically over a larger distance , resulting in a higher critical fluence. Similar behaviour was also observed when the Peak -to- peak time was recorded as shown in Fig 3(f). 8 Figure 4(a) elucidates the schematic representation of the orbital transport mechanism both prior to and post reaching critical fluence. Non localized electrons, bearing information about orbital angular momentum, traverse through orbital hopping between n uclei within a solid. An increase in fluence increases the number of charge carriers thereby enhancing the number of collisions that impede the transport process. Nevertheless, beyond the critical fluence, nonlocalized electrons perturbed by laser fluence absorb additional angular momentum from the lattice according to equation 219,34,35, as illustrated in Figure 4(b). As we apply laser fluence, the magnetic moment of localized electrons couples with the nonlocalized electron through exchange interaction10, thus creating a spin current. Due to high spin orbit correlation of Ni near Fermi level, there is angular momentum transfer between spin and orbital which can be explained by th e cross product of S and L (〈𝑆×𝐿〉) as shown in equation 2 . Additionally due to the electric fi eld of the applied laser fluence, a perturbation in the orbital wave function is induced, thus creating a non -equilibrium state. Consequently, the non -equilibrium orbital wave function takes angular momentum from the lattice through the crystal field potential VCF explained by 1 iħ〈[𝐿,𝑉𝐶𝐹]〉 in the equation 2 , enabling the orbital current to surpass collisions, facilitating a more rapid transport. Evidently, the critical fluence is linearly corelated with the thickness of the heavy metal layer. Figure 4( c) illustrates that as the thickness increases, the critical fluence also increases proportionally. The slope of th e linear relation can be called as critical energy density denoted by 𝜀𝐶 and quantified as 343 J/cm3 in case of Ni/Pt heterostructure representing the energy required per unit volume to overcome the collision and facilitate a complete ballistic orbital transport in Pt . For different orbital converter s, 𝜀𝐶 can be different depending on the intrinsic properties of the materials, and further investigation on this is required. Finally, t he orbital current velocit ies at different fluences were extracted by recording the delay in the THz peak with respect to the Ni (3 nm)/ Pt (3 nm) heterostructure. Given the ballistic nature of orbital 9 transport, the slope of the linear relationship between delay and the increase in heavy metal thickness was utilized to calculate the velocity . As shown in Fig 4 (d), the orbital current velocity was extracted at 191 J/cm2 and found out to be 0.14 nm/fs. As we increase the fluence, the slope started increasing and at 382 J/cm2 the orbital current velocity was 0.16 nm/fs and at 1270 J/cm2, the velocity was found to be 0.18 nm/fs. The linear relation also proves the ballistic nature of the orbital transport in Pt layer over a larger distance than spin transport. In summary, leveraging THz emission spectroscopy, we demonstrate THz emission from optically excited Ni/Pt heterostructure predominantly from long range ballistic orbital transport. Furthermore, the orbital transport can be controlled through the electric field of the applied fluence. Absorption of higher energy initially leads to more charge carrier formation enhancing the collision thus delaying the transport. However, after a critical fluence, carriers overcome the collision and the orbital transport ha ppen at a swifter pace due to absorption of phonon angular momentum . Exploiting this phenomenon, we have also highlighted the active enhancement of experimentally calculated orbital current velocity from 0.14 nm/fs to 0.18 nm/fs through increase in fluence enabling longer and faster orbital transport . Our findings establish an approach to control the long -distance ballistic L transport, thus creating new opportunities to design future ultrafast devices with orbitronics materials . Additionally, because of the orbital dependent efficient p honon orbital coupling , it is also po ssible to have lattice assisted orbital pumping called Orbital Angular Position (OAP)19,36. Thus, integrating twistronics and orbitronics towards THz emission, we envision an Orbitronic Terahertz Emitter (OTE) without the application of magnetic field . 10 4. Methods Sample preparation : The FM/ NM films were deposited on 1 mm - Quartz substrates by d .c. magnetron sputtering at room temperature, using a Chiron ultrahigh vacuum system with a vacuum base pressure of 1× 10−8torr. THz emission experiment : The THz radiation emitted is captured using a 1 mm thick ZnTe crystal oriented along the <110> axis, known for its nonlinear properties. The femtosecond laser pulse, which illuminates the orbitronics heterostructure, has a wavelength of 800 nm, correspondi ng to a laser energy of 1.55 electron volts (eV). It has a pulse width of 35 femtoseconds (fs) and operates at a repetition rate of 1 kHz. A beam splitter is employed to divide it into two parts. The higher intensity portion is dir ected towards photoexcitation of the emitter, while the lower intensity portion serves as a probe for detection. Precise time matching is ensured by a mechanical delay stage. The details about the THz emission spectro scopy set up can be found in supplementary section. As the emitted THz pulse , collected by parabolic mirrors, focuses on the ZnTe <110> detector, it induces birefringence in the crystal. Simultaneously, the time-matched probe beam traverses through the crystal and encounters a change in its polarization, directly proportional to the birefringence . For the electro -optic detection37, a quarter -wave plate and a Wollaston prism are employed to distinguish the s and p polarization of the laser. The rotational changes in the probe laser are subsequently identified using a balanced photodiode that measure s the intensity difference between the s and p polarized light. The electrical signal thus detected undergoes initial pre -amplification before being input into the lock -in amplifier to enhance the signal -to-noise ratio. The resultant signal from the lock -in amplifier is utilized to generate the THz electric field through a homemade LabVIEW code. 11 Data availability The data supporting this study’s findings are available from the corresponding author upon reasonable request. Supplementary information is linked to the online version of the pape r. Acknowledgments The authors thank T homas Tan and Baolong Zhang for valuable discussions and suggestions. R.S. and S.M. would like to acknowledge the Ministry of Education (MoE) , Singapore, for the support through MOE -T2EP50121 -0009 . J.L. and D.J.X.L would like to acknowledge funding support from SpOT -LITE programme (Agency for Science, Technology and Research, A*STAR Grant No. A18A6b0057) through RIE2020 funds from Singapore . Author Contributions SM, J.L and R.S. conceived the project. S.M and R. S designed the experiments. S.M. performed all the THz measurements and experimental analysis. D.J.X.L and J.L. fabricated the orbitronic emitter. All the authors analysed and discussed the results. S.M. and R.S. wrote the manuscript with inputs from J.L. R.S. lead the overall project. Author Information Reprints and permissions information is available online. The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to Prof. Ranjan Singh ( ranjans@ntu.edu.sg ). 12 Figure 1: Pumping and Detection of Terahertz Orbital (L) and Spin(S) currents (a) Upon ultrafast photoexcitation of the FM, spin currents are generated and partially convert ed to orbital currents due to L.S correlation, injecting both spin current (jS) having shorter relaxation length and orbital current (jL) having longer relaxation length into metal layer. Orbital -to- charge conversion ( LCC ) and spin-to-charge conversion ( SCC ) processes generate a n accelerated charge current (jC) to emit a THz radiation . (b) L.S correlation dependance of direction of orbital current and spin current propagation in Orbital Hall effect (c) Sign of spin orbit coupling and the orbital and spin hall conductivity for Ni, NiFe, Pt and W. 13 Figure 2: Ballistic Orbital Transport in Ni/Pt heterostructure (a) Terahertz signal emitted from Ni(3 nm) /Pt (x nm) heterostructure with variable Pt thickness x = 3, 6 , 9 and 18; Inset shows the zoomed in peak of the THz pulse indicating a right shift as we increase the Pt thickness (b) Terahertz signal emitted from NiFe (3 nm) /Pt (x nm) heterostructure with variable Pt thickness with x = 1, 2, 4, 6 and 8 indicating no right shift in the peak. (c) Delay in the emitted THz pulse with increase in Pt thickness demonst rating the ballistic orbital transport in Ni/Pt heterostructure in contrast to spin transport in NiFe/Pt heterostructure (d) Increase in peak -to-peak width indicating widening of the emitted THz pulse with increase in Pt thickness in Ni/Pt heterostructure (Orbital Transport) providing evidence of the increase in the angular dispersion proving the longer relaxation length of orbital current in contrast to spin transport in NiFe/Pt heterostructure 14 Figure 3: Active Control of Ballistic Orbital Transport . THz emission from Ni (3nm)/Pt ( x nm) at different fluence when (a) x = 3 nm, (b) x = 6 nm, (c) x = 9 nm, (d) x = 18 nm, A clear shift in THz peak was observed as we change the fluence Extracted (e) T ime delay for different fluence is shown for Ni (3 nm)/ Pt (x nm) with x = 3 , 6, 9, 18 . Initially the shift is towards right 15 indicating the decrease in orbital velocity and after a critical fluence, delay starts decreasing with increase in fluence showing swifter orbital transport; Similar shift is not seen in spin transport as shown in NiFe (3 nm)/ Pt (3 nm). (f) Peak -to-peak time difference of THz pulse, for different fluence is shown for Ni (3 nm)/ Pt (x nm) with x = 3 , 6, 9 and 18 . Similar behavior as (e) can be seen 16 Figure 4: Active Control of Orbital Current Velocity . (a) Schematic of transport of orbital angular momentum, before critical fluence collision between charge carriers make the transport slower, after critical fluence, swifter transport of orbital angular momentum takes place (b) Mechanism of increase in orb ital current velocity due to angular momentum transfer between lattice and electrons . c) Extracted critical fluence with variation in thickness. The slope of the fitted straight line indicates the “critical energy density” ( d) Extracted time delay with thickness at different fluences. The slope of the linear fitted line can be termed as orbital current velocity which could be tuned from 0.14 nm/fs to 0.18 nm/fs as the fluence was increased from 191 J/cm2 from 1270 J/cm2 17 References 1. Wolf, S. A. et al. Spintronics: A Spin -Based Electronics Vision for the Future. Science 294, 1488 –1495 (2001). 2. Hirsch, J. E. Spin Hall effect. Phys. Rev. Lett. 83, 1834 –1837 (1999). 3. Miao, B. F., Huang, S. Y., Qu, D. & Chien, C. L. Inverse Spin Hall Effect in a Ferromagnetic Metal. Phys. Rev. Lett. 111, 066602 (2013). 4. Saitoh, E., Ueda, M., Miyajima, H. & Tatara, G. Conversion of spin current into charge current at room temperature: Inverse spin -Hall effect. Appl. Phys. Lett. 88, 182509 (2006). 5. Luo, S., He, L. & Li, M. Spin -momentum locked interaction between guided photons and surface electrons in topological insulators. Nat. 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1211.0771v2.Bright_solitons_in_spin_orbit_coupled_Bose_Einstein_condensates.pdf

arXiv:1211.0771v2 [cond-mat.quant-gas] 24 Jan 2013Bright solitons in spin-orbit-coupled Bose-Einstein cond ensates Yong Xu(徐勇),1,2Yongping Zhang,3and Biao Wu(吴飙)2,∗ 1Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China 2International Center for Quantum Materials, Peking Univer sity, Beijing 100871, China 3The University of Queensland, School of Mathematics and Phy sics, St. Lucia, Queensland 4072, Australia (Dated: September 5, 2018) We study bright solitons in a Bose-Einstein condensate with a spin-orbit coupling that has been realized experimentally. Both stationary bright solitons and moving bright solitons are found. The stationary bright solitons are the ground states and posses s well-defined spin-parity, a symmetry involving both spatial and spin degrees of freedom; these so litons are real valued but not positive definite, and the number of their nodes depends on the strengt h of spin-orbit coupling. For the moving bright solitons, their shapes are found to change wit h velocity due to the lack of Galilean invariance in the system. PACS numbers: 03.75.Lm, 03.75.Kk, 03.75.Mn, 71.70.Ej INTRODUCTION Solitons are one of the most interesting topics in non- linearsystems. Themost fascinatingand well-knownfea- ture of this localized wave packet is that it can propagate without changing its shape as a result of the balance between nonlinearity and dispersion [1]. The achieve- ment of Bose-Einstein condensation in a dilute atomic gas has offered a clean and parameter-controllable plat- form to study the properties of solitons [2]. In a Bose- Einstein condensate (BEC), the nonlinearity originates from the atomic interactions and is manifested by the nonlinear term in the Gross-Pitaevskii equation (GPE), which is the mean-field description of BEC [3]. With at- tractive and repulsive interatomic interactions, the GPE can have bright and dark solitons solutions, respectively. Such dark and bright solitons in BECs have been studied extensively both theoretically [4–10] and experimentally [11–18]. The developments with two-component BECs have further enriched the investigation of solitons in matter waves. The two-component BECs not only introduce more tunable parameters, for example, the interaction between the two species, but also bring in novel nonlin- ear structures which have no counterparts in the scalar BEC, such as dark-bright solitons (one component is a dark soliton while the other is bright) [19–22], dark-dark solitons [23], bright-bright solitons [24–26], and domain walls [27–29]. Recently, in a landmark experiment, the Spielman group at NIST have engineered a synthetic spin-orbit coupling (SOC) for a BEC [30]. In the experiment, two Raman laser beams are used to couple a two-component BEC. The momentum transfer between lasers and atoms leads to synthetic spin-orbit coupling [31–40]. This kind of spin-orbit coupling has subsequently been realized for neutral atoms in other laboratories[41–44]. These exper- imental breakthroughs [30, 44, 45] have stimulated ex- tensive theoretical investigation of the properties of spin-orbit-coupled BECs [46–67], which includes some early studies on solitons. For example, bright-soliton solutions were found analytically for spin-orbit-coupled BECs by neglecting the kinetic energy [68]. Dark solitons for such a system were studied in a one-dimensional ring [69]. In thisworkweconductasystematicstudyofbrightsolitons for a BEC with attractive interactions and the experi- mentally realized SOC [30, 42–44]. By solving the GPE both analyticallyandnumerically, wefind that thesesoli- tonspossessanumberofnovelpropertiesduetotheSOC. In particular, we find that the stationary bright soli- tons that are the ground state of the system have nodes in their wave function. For a conventional BEC without SOC, its ground state must be nodeless thanks to the “no-node” theorem for the ground state of a bosonic sys- tem [70]. Furthermore, these solitons are found to have well-defined spin-parity, a symmetry that involves both spatial and spin degrees of freedom, and can exist in sys- tems with SOC. We have also found solutions for moving bright soli- tons. They have the very interesting feature that their shapes change dramatically with increasing velocity. For a conventional BEC, the shape of a soliton does not changewith velocitydue to the Galileaninvarianceofthe system. In other nonlinear systems, such as Korteweg-de Vries KdV systems, the shape of a soliton changes only in height and width with velocity [71]. In stark contrast, bright solitons in a BEC with SOC can change shape dramatically from nodeless to having many nodes with varying velocity. The new feature arises because of the lack of Galilean invariance due to SOC [62]. It is worth- while to note that a similar model was proposed a long time ago in the context of nonlinear birefringent fibers [72]. This shows that our results will find applications in nonlinear optics.2 MODEL EQUATION A BEC with the experimentally realized SOC is de- scribed by the following GPE i¯h∂Ψ ∂t=/bracketleftBig1 2m(px+¯hκσy)2+¯h∆σz−gΨ†·Ψ/bracketrightBig Ψ,(1) where the spinor wavefunction Ψ = (Ψ 1,Ψ2)T, and Ψ†·Ψ =|Ψ1|2+|Ψ2|2with Ψ 1for up-spin and Ψ 2 for down-spin. The nonlinear coefficient −g <0 is for attractive interatomic interactions, and we have taken g11=g22=g12for simplicity. The SOC is realized ex- perimentally by two counter-propagating Raman lasers that couple two hyperfine ground states Ψ 1and Ψ 2. The strength of SOC κdepends on the relative incident an- gle of the Raman beams and can be changed [65]. The Rabi frequency ∆ can be tuned easily by modifying the intensity of the Raman beams. σare Pauli matrices. A bias homogeneous magnetic field is applied along the y direction. We consider the case that the radial trapping frequency is large and, therefore, the system is effectively one dimensional [14, 15]. For numerical simulation, we rewrite Eq. (1) in a di- mensionless form by scaling energy with ¯ h∆ and length with/radicalbig ¯h/m∆. The dimensionless GPE is i∂tΦ =/bracketleftbig −1 2∂2 x+iασy∂x+σz−γΦ†·Φ/bracketrightbig Φ.(2) The dimensionless parameters α=−κ/radicalbig ¯h∆/mand γ=Ng/radicalbig m/(¯h∆)/¯hwithNbeing the total number of atoms. The dimensionless wavefunctions Φ satisfy/integraltext dx(|Φ1|2+|Φ2|2) = 1. The SOC term iασy∂xin Eq. (2) indicates that spin σyonly couples the momentum in thexdirection. The energy functional of our system is E=/integraldisplay dx/bracketleftBig1 2|∂xΦ1|2+1 2|∂xΦ2|2+|Φ1|2−|Φ2|2 +αΦ∗ 1∂xΦ2−αΦ∗ 2∂xΦ1 (3) −γ 2(|Φ1|4+|Φ2|4+2|Φ1|2|Φ2|2)/bracketrightBig . STATIONARY BRIGHT SOLITONS We focus on the simplest stationary bright solitons, which are the ground states of the system. To find these solitons, we solve Eq. (2) by using an imaginary time- evolution method. Two typical bright solitons are shown in Fig. 1. One interesting feature is immediately noticed. There are “nodes” in these ground-state bright solitons. This is very different from the conventional BEC, where there are no nodes in this kind of ground state soliton as demanded by the no-node theorem for the ground state of a boson system. Our results confirm that this no-node theorem does not hold for systems with SOC [70].−0.400.40.8Φ2 −0.1500.15Φ1(b) (a) −8 08−0.400.40.8 xΦ2 −8 08−0.400.4 xΦ1(c) (d) FIG. 1: Stationary bright solitons at γ= 1.0. The solid lines are numerical results and the circles are from the variation al method. In (a) and (b), α= 1.0. In (c) and (d), α= 2.0. There can exist a unique symmetry for systems with SOC, spin-parity, which involves both spatial and spin degrees of freedom. The operator for spin-parity is de- fined as P=Pσz, (4) wherePis the parity operator. It is easy to verify that our system is invariant under the action of spin-parity P. By direct observation, one can see that the bright solitons shown in Fig. 1 satisfy P/parenleftbigg Φ1(x) Φ2(x)/parenrightbigg =−/parenleftbigg Φ1(x) Φ2(x)/parenrightbigg , (5) as the up component Φ 1has odd parity while the other component Φ 2is even. Therefore, these bright solitons have spin-parity −1. In fact, all the ground state bright solitons that we have found have spin-parity −1. That the eigenvalue of Pfor these solitons is −1 and not 1 can be understood in the following manner. When the strength of SOC αdecreases to zero, the up component Ψ1shrinks to zero and only the down component sur- vives. Since the system becomes a conventional BEC without SOC, the no-node theorem demands that the surviving down component has even symmetry. As the SOC is turned up continuously and slowly, the symme- try of the second component should remain and the spin parity has to be −1. For a more detailed analysis of these bright solitons, we attempt to find an analytical approximation for the wavefunctions using the variational method. Motivated by the features of the stationary bright solitons shown in3 0.5 1 1.5 20.511.522π/J γ=1.0 γ=1.5 γ=2.0 (a) 0.5 11.5 20.511.52 S2π/J α=2.0 α=1.5 α=1.0 (b) FIG. 2: The relation between the number of soliton nodes 2π/Jand the soliton width S. (a) Circles, squares, and stars are forγ= 1.0,1.5,2.0, respectively. Here αincreases from 0.5 to2.0 (the arrow direction). (b) Circles, squares, andstars are forα= 1.0,1.5,2.0. Hereγincreases from 1 .0 to 4.0 (the arrow direction). The open circle and square correspond to the bright solitons in Figs. 1(a), and 1(b) and Figs. 1(c), an d 1(d), respectively. Fig. 1, we propose the following trial wave functions for these solitons: Φ =/parenleftbiggAsin(2πx/J) Bcos(2πx/J)/parenrightbigg sech(x/S). (6) The parameters A,B,J, andSare determined by min- imizing the energy functional in Eq. (3) with the con- straining normalization. The results of the trial wave functions arecompared with the numerical results in Fig. 1, where it can be seen that they are in excellent agree- ment. It is clear from Eq. (6) that the parameter 2 π/Jcan be regarded roughly as the number of nodes in the brightsolitons, while Sis for the overall width of the soliton. Both of them depend on the SOC strength αand the interaction strength γ. In Fig. 2 we have plotted the relation between 2 π/JandS, demonstrating how the number of nodes is related to the soliton width for dif- ferent values of αandγ. As shown in the Fig. 2, for solitons with the same number of nodes, they are wider for smaller interaction strength γ[Fig. 2(a)]; the solitons with the same width have more nodes for larger SOC strength α[Fig. 2(b)]. 420 012−1−0.8−0.6−0.4<σz> γα FIG. 3: The spin polarization /angbracketleftσz/angbracketrightas a function of αandγ. It has been reported that there exists a quantum phase transition for the ground states in the spin-orbit-coupled system with repulsive interaction [65]. It is interesting to check whether such a phase transition exists for the case of attractive interaction. For this purpose, we have computed the spin polarization /angbracketleftσz/angbracketright=/integraltextdx(Φ2 1−Φ2 2) for these bright solitons and the results are plotted in Fig. 3. We see that for a given γ, it changes smoothly with the SOC strength α. For the cases of repulsive interac- tion and no interaction, the spin polarization is found to change sharply with α, indicating a quantum phase tran- sition [65]. The smooth behavior of Fig. 3 suggests there is no quantum phase transition. MOVING BRIGHT SOLITONS After the study of stationary bright solitons, we turn our attention to moving bright solitons. For a conven- tional BEC without SOC, it is straightforward to find a moving bright soliton from a stationary soliton: if the wave function Φ sdescribes a stationary soliton, then exp(ivx)Φs(x−vt) is the wave function, up to a trivial phase, for a soliton moving at speed v. This is due to the invariance of the system under Galilean transformations. However, Galilean invariance is violated for a spin- orbit-coupled BEC [62]. To see this explicitly, we assume4 moving solitons having the following form: ΦM(x,t) = Φv(x−vt,t)exp(ivx−i1 2v2t),(7) whereΦ visalocalizedfunction. Substitution ofΦ M(x,t) into Eq. (2) yields i∂tΦv=/bracketleftbig −1 2∂2 x+ασy(i∂x−v)+σz−γΦ† v·Φv/bracketrightbig Φv.(8) Compared to Eq. (2), this dynamical equation has an additionalterm αvσy, indicatingtheviolationofGalilean invariance. This violation means that it is no longer a trivial task to find a moving bright soliton for a BEC with SOC. 0.20.40.60.8|Φ2|2 0.010.020.03|Φ1|2 α=1.0 v=0.1(a1) (a2) 0.10.20.30.4|Φ2|2 0.020.040.060.08|Φ1|2 α=1.0 v=1.0(b1) (b2) 0.10.20.30.4|Φ2|2 0.050.10.150.2|Φ1|2α=2.0 v=0.001(c1) (c2) −8 080.050.10.150.2 x|Φ2|2 −8 080.050.1 x|Φ1|2α=2.0 v=0.01(d1) (d2) FIG. 4: Moving bright soliton profiles Φ v(x) from the numeri- cal calculation (solid line) andthevariational method(ci rcles) with Eq. (9). γ= 1.0. In (a) and (b), α= 1.0. In (c) and (d)α= 2.0. To find moving bright solitons, we numerically solve Eq. (8) usingtheimaginarytime-evolutionmethod. Two−2 02 x−2 02 xt=0(a) |Φ1|2(b) |Φ2|2 v=0.01v=0.28 v=0t=1900 FIG. 5: Dynamical evolution of a bright soliton under the influence of a small linear potential V(x) =ηx.α= 1.7, γ= 1.0,η= 0.001. For better comparison of shapes, the centers of the soliton densities have all been shifted to the zero point. The velocity of the soliton is labeled at the righ t of (b). typical moving bright solitons are shown in Fig. 4, where we see clearly that the shapes of moving bright solitons change with their velocities. As seen in Figs. 4(a) and 4(b), when the velocity vis changed from 0.1 to 1, the density of the up component changes from having two peaks to having only one. At a larger SOC strength, such asα= 2 in Figs. 4(c) and 4(d), a small change in velocityleads to adramatic changein the solitonprofiles. Similar to the stationary soliton, these moving bright solitons can also be found with the variational method by minimizing the energy functional with the following trial wave function: Φv(x) =/parenleftbiggA/bracketleftbig sin2πx J+ρ1icos2πx J/bracketrightbig B/bracketleftbig cos2πx J+ρ2isin2πx J/bracketrightbig/parenrightbigg sechx S,(9) with two new parameters ρ1andρ2. Whenρ1=ρ2= 0, we recover the stationary soliton in Eq. (6). The solu- tions obtained with the variational method are plotted in Fig. 4 and they agree well with the numerical results. It is clear from the trial wave function that the moving bright soliton has no well-defined spin-parity P. These moving bright solitons are adiabatically linked to the stationary bright solitons. To see this, we slowly accelerate the stationary bright soliton by adding a small linear potential in Eq. (2) integrating with a stationary bright soliton as the initial condition. The dynamical evolution of this soliton is shown in Fig. 5, where we see clearly how a stationary soliton is developed into a moving soliton with its shape changing constantly. Note that the centers of solitons in Fig. 5 have all been shifted to the zero point for better comparison between shapes. We note that there are other solitons, which also change their shapes with velocity, for example, dark soli- tonsin aBECandbrightsolitonsintheKdVsystem[71]. This change is also caused by the lack of the Galilean5 invariance in the system. For the dark soliton, the con- stant background provides a preferred reference frame and breaks the Galilean invariance. In the KdV system, the violation is caused by the non-quadratic linear dis- persion. However, in these systems, the change in shape with velocity is not as dramatic: there are only changes in the height and width of the solitons. With a spin- orbit-coupled BEC, the number of peaks in the solitons can change with a slight change in velocity. CONCLUSION We have systematically studied both stationary and moving bright solitons in a spin-orbit-coupled BEC. These bright solitons have features not present without spin-orbit coupling, for example, the existence of nodes and spin-parity in the stationary bright solitons, and the change in shape with velocity in the moving bright soli- tons. Although there are multiple peaks in the soliton profiles, the bright solitons that we have found are sin- gle solitons. It would be very interesting to seek out multiple-solitonssolutionsforthisspin-orbit-coupledsys- tem. These bright solitons should be able to be observed in experiment. One can apply the same Raman laser setup to generate the synthetic spin-orbit coupling for an optical dipole-trapped7Li condensate where the in- teratomic interaction is attractive by nature. 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1710.03228v2.Tunable_superconducting_critical_temperature_in_ballistic_hybrid_structures_with_strong_spin_orbit_coupling.pdf

Tunable superconducting critical temperature in ballistic hybrid structures with strong spin-orbit coupling Haakon T. Simensen and Jacob Linder Center for Quantum Spintronics, Department of Physics, Norwegian University of Science and Technology, NO-7491 Trondheim, Norway (Dated: February 28, 2018) We present a theoretical description and numerical simulations of the superconducting transition in hybrid structures including strong spin-orbit interactions. The spin-orbit coupling is taken to be of Rashba type for concreteness, and we allow for an arbitrary magnitude of the spin-orbit strength as well as an arbitrary thickness of the spin-orbit coupled layer. This allows us to make contact with the experimentally relevant case of enhanced interfacial spin-orbit coupling via atomically thin heavy metal layers. We consider both interfacial spin-orbit coupling induced by inversion asymmetry in an S /F-junction, as well as in-plane spin-orbit coupling in the ferromagnetic region of an S /F/S- and an S /F-structure. Both the pair amplitudes, local density of states and critical temperature show dependency on the Rashba strength and, importantly, the orientation of the exchange field. In general, spin-orbit coupling increases the critical temperature of a proximity system where a magnetic field is present, and enhances the superconducting gap in the density of states. We perform a theoretical derivation which explains these results by the appearance of long-ranged singlet correlations. Our results suggest that Tcin ballistic spin-orbit coupled superconducting structures may be tuned by using only a single ferromagnetic layer. I. INTRODUCTION In recent years, di erent ways to exert spin-control over the superconducting state and properties has garnered increasing interest [ 1,2]. This includes phenomena such as spin-polarized supercurrents [ 3,4] where, traditionally, magnetic inhomo- geneities have played a key role in this endeavour as they pro- vide a source to spin-polarized Cooper pairs [ 5–7]. However, more recently the focus has shifted to exploiting spin-orbit interactions as a way to achieve a spin-dependent coupling to the superconducting state. E ects such as magnetoanisotropic supercurrents [ 8–10], anisotropic and paramagnetic Meissner eects [ 11], thermospin e ects [ 12,13], and spin-galvanic couplings [ 14,15] have very recently been investigated in this context. We note in particular that a recent experiment [16] reported a spin-valve e ect on the superconducting tran- sition temperature Tcin a layered Nb /Pt/Co/Pt structure. This contrasts previous superconducting spin-valve measurements where two ferromagnets were used [ 17–19] instead of a single magnetic layer. The spin-valve e ect is made possible due to the thin Pt layers which provide Rashba spin-orbit interactions due to interfacial inversion symmetry breaking. Motivated by this experiment and the interesting physics arising in spin-orbit coupled hybrid structures including super- conducting elements, we here present a study of the critical temperature, local density of states, and the induced pairing correlations in such systems. We use a fully quantum mechani- cal treatment and solve the BdG-equations in the ballistic limit. With a non-zero exchange field in the ferromagnetic region, both the pair amplitude, local density of states and critical temperature show dependency on the strength and, importantly, the orientation of the exchange field. In general, spin-orbit coupling increases the critical temperature of the system, and strengthens the superconducting gap in the density of states. We also present results for the same observable quantities for in-plane spin-orbit coupling in the ferromagnetic region of an S/F/S- and an S /F-structure. The results are similar to in- terfacial spin-orbit coupling, although the e ect is in generalstronger. Additionally, this type of spin-orbit coupling gives rise to a stronger anisotropy in the dependence on the exchange field direction. Our results demonstrate how Tcmay be con- trolled with a single ferromagnetic layer in ballistic spin-orbit coupled superconducting hybrids. II. THEORY AND METHODS A. Spin-orbit coupling A commonly used model for the spin-orbit Hamiltonian in systems where structural inversion asymmetry is broken, for instance by interfaces, is the Rashba Hamiltonian given by [20, 21] HSO=R(ˆnˆ)
k; (1) whereRis the Rashba parameter, ˆnis the unit vector point- ing in the direction of the broken inversion symmetry, and ˆis the vector of Pauli matrices. We will refer to ˜hSO R(ˆnk)as the SOC-induced field. To ensure that we use a Hermitian Hamiltonian, we symmetrize it by letting R(x)kx!1 2fR(x);kxg, and the Hamiltonian thus becomes HSO=1 2fR;kg
(ˆnˆ);wheref:::gis an anticommutator. This procedure is necessary in hybrid structures, as considered in this paper, where SOC exists only in certain layers. In order to test the physical validity of this Hamiltonian in an actual system, one could for instance do spin-resolved ARPES mea- surements to test how the crystal momentum of the electrons correlate with their spin orientation. B. Psuedospin and Cooper pairs Let us briefly explain how Cooper pairs are formed in sys- tems with both SOC and a magnetic field present. For simplic- ity, we start by defining a Hamiltonian in which the magneticarXiv:1710.03228v2 [cond-mat.supr-con] 27 Feb 20182 field is perpendicular to the SOC-induced fields. Let the mag- netic field be h=h0ˆz, and restrict the SOC-induced field to be parallel to the x-axis and proportional to ky. This type of SOC may be physically realized by for instance interfacial SOC between two regions in a two-dimensional system spanning theyz-plane. By writing out the Pauli matrices explicitly, the Hamiltonian follows as H= h0Rˆky Rˆkyh0! : (2) As the Hamiltonian includes SOC, spin is no longer conserved, and we refer to pseudospin, 0, as the new well-defined quan- tum number. It can be shown that the singlet s-wave Cooper pair projects onto the new eigenstate basis as ky;"Eky;#E ky;#Eky;"E =cos(SO)"ky;"0Eky;#0E ky;#0Eky;"0E# sin(SO)"ky;"0Eky;"0E +ky;#0Eky;#0E#:(3) where cos(SO)=h0 ˜h,sin(SO)=Rky ˜h, and ˜h=q h2 0+2 Rjkyj2. In the presence of only a magnetic field, the singlet state does not transform at all since we in this limit have =0and SO=0. With SOC present however, it is evident that the singlet state projected onto the eigenbasis results in both a pseudospin-singlet and a pseudospin-triplet component. Since Cooper pairs are comprised of electrons of approx- imately equal energy, Eq. (3)must be modified in order to reflect the real pairings. We have to pair electrons of di erent momenta, such that the momentum shift cancels the energy dierence caused by the SOC and the magnetic field. As pseudospin by definition defines the two possible eigenstates of the Hamiltonian in Eq. (2)for a given momentum ky, it is apparent that electrons with equal pseudospin and jkyjare found at the same energy level, while the ones with opposite pseudospin and equal jkyjare found at di erent energy levels. We thus treat pseudospin just as we treat spin with only mag- netic fields present. That is, we define a shifted momentum k y=ky+(
k), where theapplies for pseudospin up /down. (
k)is defined such that the di erent single-particle pseu- dospin states involved in the two-particle states have equal energy. By using the notationk y;0E =ky;0E ei(
k)x, we can express the s-wave singlet Cooper pair wave function as ?(x)cos(SO)(k+ y;"0Ek y;#0E ei[(
k)+(
k)]x k y;#0Ek+ y;"0E ei[(
k)+(
k)]x) sin(SO)(k+ y;"0Ek+ y;"0E +k y;#0Ek y;#0E); (4)where we have neglected the change in SOdue to momen- tum shift, and where the ?indicates that the magnetic field and SOC-induced fields are orthogonal to each other. One may observe that the pseudospin-singlet component of the sin- glet state gains a phase shift, whereas the pseudospin-triplet component does not. We choose to adapt the terminology which is frequently used on spin-triplet pairs, and name the pseudospin-triplet pair which is not subject to a pair-breaking phase a long-ranged pair. From this analysis, it is evident that a fraction of the s-wave Cooper singlet pair can adapt a long-ranged behaviour in a system featuring both Rashba SOC and a magnetic exchange field. It therefore follows that the singlet pair in such a system is partly long-ranged and partly short-ranged. In contrast, in the absence of SOC, the s-wave singlet pair would have adapted only short-ranged behaviour. If we define the system such that the magnetic field and SOC-induced fields are parallel, spin would still be a good (con- served) quantum number. As a consequence, no pseudospin- triplet component of the singlet state will appear, and the sin- glet state remains purely short-ranged. If we had made the anal- ysis completely general, that is include all intermediate angles, the algebra would have become messy. However, the analysis would have revealed that the transition between orthogonal and parallel setup happens smoothly and gradually, and the parallel and orthogonal setup therefore represents two extrema. Another consequence of an arbitrary magnetization alignment is that both the singlet states and all the triplet states would have been projected onto the eigenbasis as linear combinations of both the pseudospin-singlet and all the pseudospin-triplets. A consequence of this is that mixing between the Cooper pair spin-states may occur. C. Solving the BdG equations The Hamiltonian for a system which includes a magnetic exchange field h, and Rashba SOC reads [20, 22, 23] H=X Z drˆ y(r;)h Heh(r)
ˆ+iR(r)(ˆˆn)
riˆ (r;) +Z dr
(r)ˆ (r;#)ˆ (r;")+
(r)ˆ y(r;")ˆ y(r;#) :; (5) where ˆ yandˆ are electronic creation and annihilation opera- tors respectively, and He=~2k2 2m+V(r), that is the combined kinetic energy and non-magnetic potential energy. We have assumed that the SOC-term of the Hamiltonian is Hermitian. If this is not the case, the symmetrization procedure presented in the last section must be applied. We will now specialize this Hamiltonian to a two-dimensional system spanning the xy-plane. We follow closely the technical procedure presented in many papers by K. Halterman and various coauthors, for instance in Ref. [ 24]. The system is translationally invariant in they-direction, and of length din the x-direction. We first perform a Bogoliubov transformation of the elec- tronic operators,3 ˆ (x;")=X n un;"(x) nv n;"(x) y n ; ˆ (x;#)=X n un;#(x) n+v n;#(x) y n ;(6) where un;"(x) and vn;"(x) are quasielectron and quasihole wave functions, respectively. We have assigned a label nto each state, denoting they are energy eigenstates. This operator trans- formation should by definition transform the Hamiltonian into the form H=P n;En; y n; n;. These quasiparticle ampli- tudes are found by solving the Bogoliubov-de Gennes (BdG) equations. It is however convenient to first expand these am- plitudes as Fourier series. With n[un;";un;#;vn;";vn;#], the Fourier expansion follows as n(x)=r 2 d1X q=1ˆ nqsin kqx ; (7)where kq=q=d, and where the components of ˆ nq= [ˆu" nq;ˆu# nq;ˆv" nq;ˆv# nq]Tare the Fourier components of the expan- sion. By expanding the wave functions in sine-functions, the boundary conditions are automatically satisfied. It is moreover useful to define new spin-orbit operators, hi SO, ˜hi SOun;hi SOun;; ˜hi SOvn;hi SOvn;:(8) where the particle /hole-dependence is isolated in the sign con- vention. With these definitions, the BdG equations in Fourier space are given by 0BBBBBBBBBBBBB@ˆHeˆhzˆhz SOˆhx+iˆhyˆhx SO+iˆhy SO0 ˆ
ˆhxiˆhyˆhx SOiˆhy SOˆHe+ˆhz+ˆhz SOˆ
0 0 ˆ
(ˆH eˆhz+ˆhz SO)ˆhxiˆhy+ˆhx SO+iˆhy SOˆ
0ˆhx+iˆhy+ˆhx SOiˆhy SO(ˆH e+ˆhzˆhz SO)1CCCCCCCCCCCCCA0BBBBBBBBBBBBBB@ˆu" n ˆu# n ˆv" n ˆv# n1CCCCCCCCCCCCCCA=En0BBBBBBBBBBBBBB@ˆu" n ˆu# n ˆv" n ˆv# n1CCCCCCCCCCCCCCA; (9) where we have defined ˆu n=[ˆu n1;ˆu n2;ˆu n3;:::] and ˆv n= [ˆv n1;ˆv n2;ˆv n3;:::]. The BdG equations determine the quasipar- ticle amplitudes, as well as the energy spectrum. The matrix elements appearing above are defined as ˆHe(q;q0)=2 dZd 0dxsin kq0x"~2 2mq d2 +V(x)+E?EF# sin kqx ; ˆ
(q;q0)=2 dZd 0dxsin kq0x
(x) sin kqx ; ˆhi(q;q0)=2 dZd 0dxsin kq0x hi(x) sin kqx ; ˆhi SO(q;q0)=2 dZd 0dxsin kq0x hi SO(x) sin kqx ; where i2fx;y;zgin the two last definitions. One of the main goals of solving the BdG equations is finding the superconducting energy gap,
. It is defined as
(r)=VSC(x)Dˆ (x;")ˆ (x;#)E ; (10)where VSCis a coupling strength between electrons inside the energy interval [EF~!D;EF+~!D]. By insertion of the Bo- goliubov transformation in Eq. (6), and VSC(x)=(x)=D2(EF), where the weak-coupling constant is finite inside supercon- ductors and zero elsewhere, while D2(EF)=m ~2is the energy- independent density of states per area in two dimensions, we obtain
(x)=(x)EF 4kFP0 n[un;"(x)v n;#(x)+un;"(x)v n;#(x)] tanh (En=2kBT); (11) The sum over nis a sum over all energy eigenstates, which formally is a sum over all eigenstates of Eq. (9)for every possible value of E?. The primed summation indicates that this is a constrained sum over energy levels within the energy interval where s-wave singlet Cooper pairing occurs. This opens up for using a self-consistent approach. We start out byguessing an initial
. The closer the initial guess is to the actual
, the fewer iterations through the BdG equations are necessary. Before starting this procedure, we make
a dimensionless quantity by letting
(x)=
0!
(x), where
0 is the bulk value of the superconducting energy gap within a clean superconductor.
(x) should therefore presumably be constrained toj
(x)j1. For most situations, using a zeroth order approach by guessing
= 1 inside superconducting regions, and
=0 elsewhere, is a su ciently accurate starting point. Solve the BdG equations in (9)for this
, and obtain a set of eigenvectors n. Use this set of eigenvectors to define a new
using Eq. (11), and repeat this procedure until
4 converges towards the true superconducting gap. In this paper, we stopped the procedure when
at no point had a relative change of more than 103between two consecutive iterations. D. Pair amplitudes The singlet energy gap captures the singlet correlation within the superconducting regions of a system. Outside of these re- gions, the amplitude is by definition identically zero due to (x) being zero. To provide information on the proximity e ect, that is how far into non-superconducting regions superconducting order penetrates, we define the s-wave singlet pair amplitude f0(x)=
(x) (x): (12) This amplitude is chosen to be normalized to jf0j1. We furthermore define the s-wave triplet amplitudes, that is the odd-frequency triplets, as [25] f1(x;)=1 2Dˆ (x;;")ˆ (x;0;#)+ˆ (x;;#)ˆ (x;0;")E ;(13) f2(x;)=1 2Dˆ (x;;")ˆ (x;0;")ˆ (x;;#)ˆ (x;0;#)E ;(14) f3(x;)=1 2Dˆ (x;;")ˆ (x;0;")+ˆ (x;;#)ˆ (x;0;#)E ;(15) (16) whereis the relative time coordinate. We name the triplets captured by the f1-amplitude ( sz=0)-pairs, reflecting that these have zero spin projection along the z-axis, thus being z-eigenstates. We name the Cooper pairs responsible for the f2- and f3-amplitudes ( sz=1)-triplets. We underline that these are not z-eigenstates, and hence have no well defined szas they are linear combinations of two-particle states with sz=1. By insertion of the Bogoliubov transformations in Eq. (6), and utilizing the identities which were used in the derivation of Eq. (11), we obtain f1(x;)=1 2X nh un;"(x)v n;#(x)un;#(x)v n;"(x)i n();(17) f2(x;)=1 2X nh un;"(x)v n;"(x)+un;#(x)v n;#(x)i n();(18) f3(x;)=1 2X nh un;"(x)v n;"(x)un;#(x)v n;#(x)i n();(19) wheren(t)=sinEn ~ icosEn ~ tanhEn 2kBT . In this paper, the triplet pair amplitudes have been normalized with the same prefactor as f0. Additionally, we only plot the real part of the pair amplitudes. E. LDOS The local density of states (LDOS), N(E;x), provides infor- mation on the distribution of states as a function of energy andposition. Its interpretation is that N(E;x)dEequals the num- ber of quantum states within the infinitesimal energy interval [E;E+dE] at position x. It can be expressed as [23] N(E;x)=X nX n jun(x)j2(EEn)+jvn(x)j2(E+En)o ; (20) where the-function is the Dirac delta function. As all the energy levels are discretized, N(E;x) will be a discrete dis- tribution function. To smoothen out the density of states, we perform a convolution with a Gaussian of width 0:02
0. In this paper, the LDOS is normalized to be 1 in the normal metal limit, that is several times
0away from EF. F. Critical temperature For the calculation of the critical temperature, we follow closely the procedure of Ref. [ 26], where Tcis found by treating
as a small first-order perturbation. We can then solve the BdG equations once to zeroth order, that is with
=0, and use perturbation theory to define a finite
from the eigenvectors. This first-order
will be T-dependent, and we findTcby identifying the point where
=0 is the only possible solution. The complete derivation of this perturbative approach is performed in Appendix A. Such a procedure assumes that the superconducting transition is not a first order one, since in that case
cannot be made arbitrarily close to zero. The result is a matrix eigenvalue problem,
(1) l=X kJlk(T)
(1) k; (21) where the matrix elements Jlkare defined by the formula Jlk(T)=2EF kFd3X nX mkX p;qKpql ( v(0)y mqJ2u(0) npP i;ju(0)y niJ2v(0) m jKi jk Ep nEhmtanh Ep n 2kBT! +v(0)y nqJ2u(0) mpP i;ju(0)y miJ2v(0) n jKi jk EhnEp mtanh Eh n 2kBT!) ;(22) where vnj=[v" n j;v# n j]Tand un j=[u" n j;u# n j]Tare vec- tors of quasiholes and quasielectrons, (0)-superscipt de- notes zeroth order, J2is the (22)exchange matrix (de- fined in Appendix A), and Eh nand Ep nare the zeroth or- der energy spectra of quasiholes and quasielectrons, re- spectively. To simplify notation, we have introduced Ki jk=Rd 0dx(xx0) sin(kix)sin kjx sin(kkx). The sums over i,j,pandqgo over the Fourier wave numbers. The constrained sum over ngoes over the kinetic energy contribu- tions from all directions. The sum over mkgoes over all kinetic energy contributions from the x-direction, with m?=n?im- plied.5 Eq.(21) is a matrix eigenvalue equation. It has one obvious solution, the trivial solution, that is
(x)=0. This solution is of no particular interest, since it implies that superconductivity is absent. If we assume
(x),0 however, the equation has a solution if and only if the matrix J(T) has an eigenvalue which is 1. Since superconductivity is sensitive to temperature, one should therefore expect only the trivial solution to remain if T>Tc, where Tcis the critical temperature where supercon- ductivity breaks down. This involves that all the eigenvalues of J(T) falls below 1. The critical temperature is therefore found by identifying at which temperature the largest eigenvalue of J(T) drops below 1. III. RESULTS AND DISCUSSION SOC lifts spin-rotational symmetry, and thus the simulta- neous presence of SOC and a magnetic field should reveal spin-anisotropic behaviour of superconductivity. This is what motivates us to explore SOC in F /S-structures. We will look at two types of spin-orbit coupling. First, we will explore the highly localized interfacial SOC, of which results are given in Sec. III A. We will thereafter look at in-plane SOC inside F-regions, of which results are given in Secs. III B and III C. We use ~!D=EF=0:04 for all calculations, as well as T=0 in all calculations for the pair amplitudes and LDOS. The exchange field strength is chosen up to h0=EF=0:3, most realistically realized by placing transition metal ferromagnets such as Fe, Ni, or Co in the ferromagnetic region. The Rashba strength is varied between up to RkF=EF=0:5. Large SOC eects are probably easiest to realize at the surface of heavy metals such as Au or Pt, as it has been reported that these structures give a Rashba e ect two orders of magnitude larger than in semiconducting 2DEGs [ 21]. As we in this paper would like to explore the general e ects of combining exchange fields and the Rashba e ect, we vary Rover a relatively wide range. The dimensions used in this paper coincide with the routinely achieved experimental dimensions in heterostructures. In all systems, we assume that the Fermi level EFis equal and constant in all regions. Furthermore, we assume that the eective masses, work functions and densities are equal in all regions, in addition to there being no scattering potential in the interface between the regions. This is clearly a crude simplifi- cation, and in order to provide a more a realistic description of specific materials one should rather use parameters obtained from experiments. In this paper, we have chosen not to include these parameters, as doing so would result in more undeter- mined parameters that would complicate the analysis. The main purpose of this paper is to show the qualitative eect of combining exchange fields and SOC, and we therefore choose the simplest approximation, namely that all of these param- eters are constant throughout the system. Despite the crude approximation, similar routines (excluding SOC) have previ- ously provided results which coincide well with experimental results [ 27]. For future work, it is straightforward to include Fermi level mismatch and interface scattering potentials in the numerical method, and thereby obtain results which are closer to specific materials.A. Interfacial SOC in an F /S-structure We start by looking at interfacial SOC in an F /S-stucture. The system is two-dimensional, of length d=1:230in the x-direction, and is translationally invariant in the y-direction. The length of the F-region has been set to 0 :20, and the super- conductor’s length has been set to 0;where0is the coherence length of the superconductor. In between these regions, there is a SOC-layer of width 0 :030. The system is illustrated in Fig. 1. The SOC-potential has been Gaussian distributed in- side this region, that is R(x)N(x;SO;SO), where the expectation value of the distribution, SO, is in the middle of the SOC-region. The variance of the distribution is 2 SO, and 4SO=0:030covers most of the distribution. We use a Rashba coupling strength of RkF=EF=0:5. In the F-region, we define a magnetic field h=h0sin(h)ˆx+cos(h)ˆz
, in which we set h0=EF=0:3. z x −yθhh ddS 4σSOdFˆ n FIG. 1. An illustration of the F /S-structure with SOC in the junction. The system considered is in reality not of restricted length along the y- axis, but is of infinite extent in this direction. Moreover, the structure is of zero height, that is of no extent in the z-direction. 1. Pair amplitudes The s-wave singlet amplitude is plotted for five di erent magnetization angles, h, in Fig. 2. The upper plot shows the results forh=0, and the magnetization angle is increased by=8 for every plot downwards. It can be observed that the singlet correlation, and thus the superconducting pair potential, grows by increasing the magnetization angle. At h=0, its maximum before the oscillations at the boundary is approx- imately 0:1. Growing steadily by increasing magnetization angle, this maximum doubles as happroaches =2. Hence, it seems as though a magnetization perpendicularly aligned to the SOC-induced fields results in best conditions for supercon- ductivity to exist. If the magnetic field and SOC-induced fields are aligned in thez-direction, szis still a conserved quantum number. For the s-wave singlet Cooper pairs, the SOC then mimics a potential6 0 0.25 0.5 0.75 x/ 9000.20.4 3h = 0 0 0.25 0.5 0.75 x/ 9000.20.4 3h = :/8 0 0.25 0.5 0.75 x/ 9000.20.4 3h = :/4 0 0.25 0.5 0.75 x/ 9000.20.4 3h = 3 :/8 -0.23 0 0.25 0.5 0.75 1 x/ 9000.20.4 3h = :/2f0(x) FIG. 2. The singlet pair amplitude plotted for five di erent magne- tization angles, h, for an F /S-structure with SOC in a thin layer at the interface. The SOC-layer is Gaussian distributed within the blue dotted lines (which cover a width of 4 SO). barrier, causing the F- and S-regions to be partly decoupled. In this case, we would therefore in general expect SOC to pro- tect the superconducting state to some extent by damping the proximity e ect. If the magnetic field and SOC-induced fields are not aligned however, we cannot precisely make qualitative predictions by evaluating sz-states. We therefore turn to the pseudospin eigenstates, derived in Sec. II B. The main result of this section was that if the SOC-induced field is perpen- dicular to a magnetic field, a component of the singlet state becomes long-ranged. That is, if we project the singlet state onto the eigenbasis, it will in general be a linear combination of a pseudospin-singlet and a ( s0=1)-pseudospin-triplet, the latter of which do not gain a relative phase throughout the sys- tem due to having zero CoM. As a consequence of this e ect, the leakage of singlets is reduced, allowing for a larger singlet amplitude to sustain. This e ect of SOC is h-dependent, and will therefore increase as hincreases. The results obtained by numerical calculations seem to support this analysis. Another prediction from Sec. II B was that mixing between the triplet pairs should occur at intermediate angles, due to they using a common set of pseudospin channels through the system. This is verified by Fig. 3, where all triplet amplitudes are plotted for five di erent relative times =!Dt. In the absence of SOC, the f3-amplitude would not have appeared by rotating the magnetic field in the xz-plane. With SOC however, it clearly appears, and this must therefore be due to Cooper pair spin-mixing caused by SOC.s At h=0,szis a conserved quantum number, and no ( sz=1)-amplitudes may be produced. For increasing magnetization angles, the spin-mixing e ect seem to grow. At h==2 however, the 00.250.50.75 -1.5-0.75 0 0.75 1.5 3h = 0 00.250.50.75 -1.5-0.75 0 0.75 1.5 3h = 0 00.250.50.75 -7.5-3.75 0 3.75 7.5 3h = 0 00.250.50.75 -1.5-0.75 0 0.75 1.5 3h = :/8 00.250.50.75 -1.5-0.75 0 0.75 1.5 3h = :/8 00.250.50.75 -7.5-3.75 0 3.75 7.5 3h = :/8 00.250.50.75 -1.5-0.75 0 0.75 1.5 3h = :/4 00.250.50.75 -1.5-0.75 0 0.75 1.5 3h = :/4 00.250.50.75 -7.5-3.75 0 3.75 7.5 3h = :/4 00.250.50.75 -1.5-0.75 0 0.75 1.5 3h = 3 :/8 00.250.50.75 -1.5-0.75 0 0.75 1.5 3h = 3 :/8 00.250.50.75 -7.5-3.75 0 3.75 7.5 3h = 3 :/8 -0.2300.250.50.751 x/ 90 -1.5-0.75 0 0.75 1.5 3h = :/2 -0.2300.250.50.751 x/ 90 -1.5-0.75 0 0.75 1.5 3h = :/2 -0.2300.250.50.751 x/ 90 -7.5-3.75 0 3.75 7.5 3h = :/2 = = 0 = = 3 = = 6 = = 9 = = 12f1(x) f2(x) f3(x) ( # 10-2) ( # 10-2) ( # 10-4)FIG. 3. The triplet amplitudes for five di erent magnetization angles, h, for the F /S-structure with a SOC-layer in the junction. Each plot contains triplet correlations for five di erent relative times, . shift of spin basis does not cause spin mixing, as predicted by the discussion in Sec. II B. 2. LDOS As a consequence of the analysis so far, we expect the band gap to be more developed for higher magnetization angles, h. This is due to the creation of long-ranged singlets, which should imply fewer triplet states relative to singlet states, thus reducing the number of states within the band gap. When h=0, this e ect does not occur, and the plots should be qualitatively rather equal to a clean F /S-junction. For h= =2, the e ect should be at its maximum, creating the most prominent band gap. The LDOS at four di erent positions are plotted in Fig. 4, both inside the F- and S-region. The plots show very clearly that the superconducting gap becomes much more prominent for higher magnetization an- gles. Forh=0, one can in fact almost not spot any gap at all. As we rotate hfurther towards =2, this gap grows, and it is almost a complete gap for h==2. This applies to all positions in the system, both inside the F-region and inside the S-region. As the energy gap grows with h, this indicates that the fraction of singlet states grows, and that superconductivity is thus being strengthened. This is just in accordance with the analytical derivation in Sec. II B, where the existence of long-ranged singlets was predicted.7 N(E)x/ 90 = -0.115 x/ 90 = 0.115 x/ 90 = 0.515 x/ 90 = 0.915-1 -0.5 0 0.5 100.51 -1 -0.5 0 0.5 100.511.52 -1 -0.5 0 0.5 1 E/ "000.511.5 -1 -0.5 0 0.5 1 E/ "000.511.53 h = 0 3 h = :/8 3 h = :/4 3 h = 3 :/8 3 h = :/2 FIG. 4. The LDOS for the F /S-structure with SOC in the interface plotted at four di erent positions, as indicated above each plot. At each position, the LDOS is plotted for five di erent magnetization angles,h. The results are obtained with N?=2000. 3. Critical temperature We have so far seen that the closer hcomes to=2, the stronger is the enhancing e ect on superconductivity. In order to reveal the exact angular dependence, we have plotted the critical temperature with respect to the magnetization angle in Fig. 5. The analysis is done for three di erent Rashba coupling strengths. The magnetic field is as before, h0=EF=0:3. Firstly, 0 0.2 0.4 0.6 0.8 1 3h/ :00.050.10.150.2Tc/Tc0,RkF/EF = 0.5 ,RkF/EF = 0.3 ,RkF/EF = 0.1 FIG. 5. The critical temperature of an F /S-structure with SOC in the interface plotted for three di erent Rashba parameters, as function of the magnetization angle, h. these results confirm that the Hamiltonian is invariant under the transformation h!h, as the plot is symmetric about =2. Furthermore, the results clearly indicate that the closer the magnetization angle is to =2, the more robust is the super- conducting state. This is an interesting result, as we are able to control the critical temperature by adjusting a macroscopic pa- rameter. Although not directly comparable, these results show similar behaviour as obtained by a quasiclassical approach in the di usive limit in Ref. [ 28,29]. In these works, it was shown that for equal weights of Rashba and Dresselhaus SOC, rotating the magnetic field over an interval of =2 causes thecritical temperature to go from minimum to maximum. This is just what we found for the F /S-structure with interfacial SOC studied here, with a fully quantum mechanical approach. In summary, the F /S-structure with interfacial SOC shows interesting properties. Firstly, it allows for controlling the critical temperature by adjusting macroscopic factors such as the magnetic field. Secondly, it may be used to control triplet amplitudes. Such a structure therefore serves as a promising alternative to magnetic multilayers for the purpose of achieving superconducting spin-valve e ects. B. In-plane SOC in an S /F/S-structure Our observations from the previous section, and the predic- tions made in Sec. II B, motivates us to look at the case of in-plane SOC. That is, the combined presence of SOC and mag- netic fields gives rise to long-ranged singlet pairs. The analysis so far predicts that the closer these interactions are in space, the more prominent the e ects will be. In-plane SOC inside a ferromagnet maximizes the spatial copresence of the spin-orbit interaction and the exchange field’s e ect on the electrons, and we thus expect a larger relative amount of long-ranged singlet pairs as compared to with interfacial SOC. We start out by looking at a S /F/S-structure, with in-plane SOC in the F-region. One way to realize such a setup, a so-called Rashba ferromagnet, is to use a thin film of a strong transition metal ferromagnet, e.g.Fe, Co or Ni. The Rashba e ect could be fur- ther enhanced by adding a thin layer ( 1 nm) of a heavy metal. This setup mimics a situation where SOC and ferromagnetic order coexist, albeit not entirely homogeneously. As SOC in general is expected to protect a fraction of the sin- glets, we need not define a very long system for superconductiv- ity to sustain. Therefore, the full system length is only defined to be 1:10. The S-regions are of length dS1=0=dS2=0=0:5, which leaves the F-region with SOC of length dF=0=0:1 be- tween the S-regions. The system is illustrated in Fig. 6. There is no applied phase-di erence between the superconductors. As opposed to in the case of interfacial SOC, where a spin- rotational symmetry remained in the system, the spin-rotational symmetry is completely broken by the combined e ect of the magnetic field and SOC. We have to keep the magnetic field completely general, and write it as h=h0 cos()sin()ˆx+sin()sin()ˆz+cos()ˆy ;(23) whereis the azimuthal angle and is the polar angle of slightly modified spherical coordinates, that is with yandz having changed roles. Moreover, the SOC Hamiltonian be- comes HSO=Rh kxzkzxi +Rz 2ih (xxL)(xxR)i ;(24) where xLandxRare the x-coordinates of the left and right boundaries of the SOC-region respectively, and where the posi- tion dependence of the Rashba parameter has been suppressed8 y x zθh ddF dS2 dS1φˆ n FIG. 6. An illustration of the S /F/S-structure with in-plane SOC in the F-region. The system considered is in reality not of restricted length along the z-axis, but is of infinite extent in this direction. Moreover, the structure is of zero height, that is of no extent in the y-direction. in the notation. Interestingly, we see that the requirement of rendering the Hamiltonian Hermitian leads to an e ective bar- rier term at the interfaces which looks like a spin-dependent scattering potential with an imaginary amplitude. This term may seem like an unwanted term. It introduces complex num- bers on the diagonal of HSO, which in general could cause complex eigenvalues, resulting in complex, unphysical ener- gies. However, the actual matrix elements entering the diagonal of the BdG equations in Eq. (9) remain purely real even in the presence of the additional -function term, as can be verified by direct insertion. For this analysis, we set magnetic field strength to h0=EF=0:1, and the Rashba coupling strength is set toRkF=EF=0:4. 1. Pair amplitudes The singlet amplitudes for magnetization along x-,y- and z-axis are plotted in Fig. 7. The qualitative behaviour of the singlet amplitudes in these magnetization setups are all approx- imately the same. There is however a significant quantitative dierence between the di erent setups. The superconducting state seem to prefer the y-alignment of the magnetic field, and is most suppressed by an x-aligned field. Hence, as with inter- facial SOC, in-plane SOC introduces a prominent dependence upon the direction of the magnetic field. If SOC was switched o, the singlet amplitudes would drop to zero no matter the magnetization direction, implying that SOC once again shows an enhancing e ect on superconductivity. The triplet amplitudes for the same magnetization setups are plotted in Fig. 8. Note that the axes are scaled di erently, and the graphical amplitudes are thus not directly compara- ble between the di erent plots. If SOC was switched o , the rotation of the magnetic field would only cause the triplet am- plitudes to rotate between each other. An x-aligned, y-aligned andz-aligned magnetization would have given a non-zero f2- 0.25 0.50.6 0.85 x/ 900.20.40.60.81h k ^ x 0.25 0.50.6 0.85 x/ 900.20.40.60.81h k ^ y 0 0.25 0.50.6 0.85 1.1 x/ 9000.20.40.60.81h k ^ zf0(x)FIG. 7. The singlet amplitude plotted for the S /F/S-structure with in-plane SOC in the F-region, for magnetization along the x-,y- and z-axis. The dotted blue lines indicate the junctions between the F- and S-regions. The Rashba parameter has been set to RkF=EF=0:4. amplitude, f3-amplitude and f1-amplitude, respectively. For each magnetization configuration, all other than the mentioned triplet amplitude would have been identically zero. With SOC switched on however, all triplet amplitudes appear for the x- andy-aligned fields, while only the f1-amplitude remains non- zero for the z-aligned field. There are two reasons to the appearance of other triplet am- plitudes. Firstly, as explained in Sec. II B, SOC induces mag- netic impurities in the junction between the S- and F-regions. These cause an inhomogeneous magnetization configuration when the magnetic field points along the x- ory-axis, which alone would result in two triplet amplitudes to appear. Sec- ondly, SOC introduces spin-mixing when the magnetic field is not either orthogonal or parallel to the SOC-induced field. These e ects combined generally cause all triplet amplitudes to be present, except for hkˆz, where no spin mixing occurs and thus only one non-zero triplet amplitude appears. 2. LDOS As magnetization in either the x-,y- orz-directions clearly give di erent pair amplitudes, it makes an interesting analysis to take a closer look at the configuration of states around the Fermi energy for each case. The LDOS at four di erent posi- tions have therefore been plotted in Fig. 9. The upper two plots show the density of states at two di erent positions inside the left S-region, while the two lower plots do the same for inside the F-region. Inside the S-region, there is a fully developed9 0.250.550.85 -1-0.5 0 0.5 1h k ^ x 0.250.550.85 x/ 90-2-1 0 1 2h k ^ x 0.250.550.85 x/ 90-2-1 0 1 2h k ^ x 0.250.550.85 -1-0.5 0 0.5 1h k ^ y 0.250.550.85-4-2 0 2 4h k ^ y 0.250.550.85-4-2 0 2 4h k ^ y 00.250.550.851.1 x/ 90 02.5 57.5 h k ^ z 00.250.550.851.1 x/ 90-4-2 0 2 4h k ^ z 00.250.550.851.1 x/ 90-4-2 0 2 4h k ^ z = = 0 = = 3 = = 6 = = 9 = = 12f1(x) f2(x) f3(x) ( # 10-2) ( # 10-2) ( # 10-2) FIG. 8. The triplet amplitudes for the S /F/S-structure with in-plane SOC in the F-region for magnetization along the x-,y- and z-axis. The results are plotted for five di erent relative times , as indicated by the legend. The black dotted lines indicate the junctions between the di erent regions. Note that the axes are scaled di erently, and the graphical amplitudes are thus not directly comparable. -1 -0.5 0 0.5 1 E/ "000.250.50.751h k ^ x h k ^ y h k ^ zN(E)x/ 90 = 0.2 x/ 90 = 0.4 x/ 90 = 0.525 x/ 90 = 0.55-1 -0.5 0 0.5 10123 -1 -0.5 0 0.5 10123 -1 -0.5 0 0.5 1 E/ "000.250.50.751 FIG. 9. The LDOS at four di erent positions inside the S /F/S-structure with in-plane SOC in the F-region. The positions are indicated in the plots, and the colour coding is indicated by the legend. The results are obtained with N?=2000. energy gap for magnetization in the y- and z-directions, with hkˆygiving the largest gap. The gap is much less developed for the x-aligned magnetic field. Inside the F-region, the am- plitudes are being suppressed for all system setups, with an average of about 0 :25 outside the band gap region. This is both an e ect of the magnetic field, which suppresses certain spin-configurations, as well as due to SOC suppressing states dependent upon both their momentum and spin.The band gap is fully developed at both positions inside the F-region for both the y- and z-aligned fields. Once again, the gap is widest for the y-aligned field. These results are consistent to the results obtained for the singlet amplitudes. In general, the band gap seems to be wider and more prominent forhkˆy, and weakens for hkˆzand hkˆx, in that order. 3. Critical temperature The analysis so far has been performed with a constant Rashba parameter. We follow up this analysis by investigating how varying the Rashba parameter a ects the physics of the system. This analysis is once again performed for the magnetic field pointing in both the x-,y- and z-directions. The results are plotted in Fig. 10. 0 0.1 0.2 0.3 0.4 0.5 RkF/EF00.10.20.30.40.50.60.70.8Tc/Tc0 FIG. 10. The critical temperature plotted with respect to the Rashba parameter for the S /F/S-structure with in-plane SOC in the F-region. The magnetic field strength is set to h0=EF=0:1. Each line represents a magnetic field orientation along an axis, orthogonal to the others, as indicated by the legend. The critical temperature increases with increasing Rfor all three magnetic field configurations. The degree to which the temperature rises di er between all configurations, as we should expect after the analysis so far. The critical tempera- ture is generally higher for a magnetic field pointing in the y-direction. For magnetic field configurations in the xz-plane, that is in the plane which is spanned by the physical system, the critical temperature is di erent for small Rashba parameters. The critical temperature is in general higher when the magnetic field is pointing in the z-direction, but this di erence vanishes almost entirely as RkFapproaches 0 :5EF. These results are consistent with what observed for the singlet amplitude and for the LDOS. However, this analysis also brings some new and interesting observations, which can help us understand the physics better. We start by looking into the easily visible di erence be- tween magnetization along the y-direction and in the xz-plane. It is obvious from Fig. 10 that superconductivity is most re- sistant to thermal e ects with a magnetic field along the y- axis. This result is rather straightforward to understand. The superconductivity-enhancing e ect caused by interfacial SOC10 was observed to be most prominent at h==2, as is also predicted by theory. For the S /F/S-system considered here, the SOC-induced fields are always parallel to the xz-plane, perpen- dicular to the y-axis. Thus, for magnetization in the y-direction, the requirement for maximal e ect of SOC is always satisfied, which explains the higher Tc. In a pure F /F/S-structure, a perpendicular relative orientation of neighbouring magnetic field regions causes a lower critical temperature than a parallel alignment [ 30]. This is due to the long-range triplet production, which e ectively causes another channel of triplet leakage to occur. We can use this result to explain the-dependence of Tc, that is the di erence between thex- and z-directions. When ==2, all magnetic fields in the system are either parallel or antiparallel. Hence, in this con- figuration, only the short-range triplet channel is open. When is decreased however, the production of long-range triplet pairs is increased. This e ect reaches its maximum at =0. Another channel of leakage is thus opened by the magnetic field configuration when 0  < = 2, which implies lower critical temperature for an x-aligned magnetic field than for a z-aligned field. For an increasing Rashba parameter however, the amount of short-ranged triplets are reduced, which further implies less leakage into other triplet channels. Thus for in- creasingR,Tcshould be less sensitive to changes in . As is evident from the results, full -invariance seems to occur at RkF=EF0:5. In order to make the analysis complete, and in order to reveal the exact angular dependence, the critical temperature as func- tion of the magnetization angles ( ;) is plotted in Fig. 11. The plot contains three graphs, each of which corresponds to rota- tion in either the xy-,yz- orzx-plane. The Rashba parameter has been set to RkF=EF=0:4. The results are consistent with 0 0.1 0.2 0.3 0.4 0.5 /0.30.350.40.450.50.550.60.650.7Tc/Tc0 FIG. 11. The critical temperature in the S /F/S-structure with in-plance SOC plotted with respect to di erent magnetization angles, with RkF=EF=0:4.andhave been rotated between 0 and =2 in thexy-plane (blue), zy-plane (red) and zx-plane (green). The angle represents either or, and is specified by the legend for each individual line. the analysis made in the discussion of magnetization in the x-, y- orz-direction. We also observe that the graphs are all strictly increasing or decreasing, and contain thus no local minima or maxima. The transition between the di erent extrema, namelymagnetization along the coordinate axes, happens smoothly. There are no intermediate angles at which e ects other than those discussed up until now occur. Fig. 11 shows that the largest change in Tcby rotating the magnetic field happens for rotation in the xy-plane. The di er- ence between =0 and==2 is almost 0 :4T0 c. This structure thus has a great potential in controlling Tcby adjusting both the SOC-strength and the magnetization angles. It also serves a candidate for controlling the triplet production, as magnetiza- tion along the x-,y- and z-axis all give di erent properties for the triplet amplitudes. Additionally, these e ects are obtainable for a structure of only 1 :10, which is generally shorter than required for clean ferromagnet-superconductor-structures. C. In-plane SOC in an S /F-structure We will now look at in-plane SOC in an S /F-structure. The structure will have equal dimensions as the previous S /F/S- structure, only with the F-region to the far right side of the system. That is, the S-region is of length dS=0, while the F-region is of length dF=0:10. We still use RkF=EF=0:4 andh0=EF=0:1. The system is illustrated in Fig. 12. The F + S O C Sh ˆ n φy zxθ dS dF d FIG. 12. An illustration of the S /F-structure with in-plane SOC in the F-region. The system considered is in reality not of restricted length along the z-axis, but is of infinite extent in this direction. Moreover, the structure is of zero height, that is of no extent in the y-direction. qualitative di erence between this structure and the S /F/S- structure is that this is a bilayer structure rather than a trilayer, and that the SOC-region now forms a boundary region. The results are very similar to the S /F/S-structure, and we will therefore not give a complete treatment of this structure. We will restrict the analysis to include the critical temperature plots analogous to the ones given for the S /F/S-structure, and the discussion will mainly focus on the di erences.11 1. Critical temperature The critical temperature with respect to the Rashba coupling strength,R, is plotted in Fig. 13. There are several qualitative similarities to the corresponding plot for the S /F/S-structure, given in Fig. 10. Firstly, we observe that the x-aligned mag- netic field has a clearly visible suppressed critical temperature compared to with the magnetic field pointing in either the y- orz-directions. With an increasing Rashba parameter, we also observe that the critical temperature is strictly increasing for all of the three magnetization configurations. The most interesting 0 0.1 0.2 0.3 0.4 0.5 RkF/EF00.10.20.30.40.50.60.7Tc/Tc0 FIG. 13. The critical temperature plotted with respect to the Rashba parameter for the S /F-structure with in-plane SOC in the F-region. The magnetic field strength is set to h0=EF=0:1. Each line represents a magnetic field orientation along an axis, orthogonal to the others, as indicated by the legend. observations might however be the di erences. We see that for all configurations, the critical temperature is non-zero at a lower Rashba parameter than in the S /F/S-structure. This may be explained from the fact that the superconductor length here is twice the size of the lengths of the two S-regions in the S/F/S-structure. With no SOC, few singlet pairs may tunnel through the F-region, and the two regions are in that sense more or less decoupled. In the S /F-structure in comparison, the doubled length of the S-region makes superconductivity arise at an earlier stage. As we increase the Rashba coupling strength however, the critical temperature of the S /F/S-system eventually passes that of the S /F-system. At this point, more pairs may pass through the F-region in the S /F/S-structure, and we might say that the F-region gradually adapts normal metal properties. While in the S /F-structure, the F-region is at the boundary, and the singlet pairs cannot simply tunnel through into a new S-region, but rather has to be reflected and pass through the F-region once more before re-entering the S- region. This results in more triplet conversion, which explains why the critical temperature of the S /F/S-region eventually becomes larger than that of the S /F-region. In Fig. 14, the critical temperature is plotted with respect to the magnetization angles ( ;) in the xy-xz- and yz-plane. This should be compared to the results for the S /F/S-structure plotted in Fig. 11. We observe that for the rotations in the xz- and yz-plane, the local extrema of Tcseem generally not 0 0.1 0.2 0.3 0.4 0.5 /0.10.20.30.40.50.60.7Tc/Tc0FIG. 14. The critical temperature in the S /F-structure with in-plance SOC plotted with respect to di erent magnetization angles, with RkF=EF=0:4.andhave been rotated between 0 and =2 in thexy-plane (blue), zy-plane (red) and zx-plane (green). The angle represents either or, and is specified by the legend for each individual line. to be where the magnetization is along one of the coordinate axes, but rather shifted somewhat from this point. This is interestingly just what we observe when rotating the relative magnetization direction in a clean F /F/S-structure. In the S /F/S- structure with SOC, there was no such e ect. This is likely due to the fact this e ect occurs on both sides of the F-region in the S/F/S-structure, e ectively adding together to produce results where the extrema are found at andequal to 0 and =2. The S /F-structure with in-plane SOC may therefore, at least to some extent, serve the same purpose as an F /F/S-structure. Put in other words, the F-region with in-plane SOC may to some extent serve as a substitute for an F /F-region. IV . CONCLUSION The e ects of strong Rashba spin-orbit coupling in ferromagnet-superconductors-structures have been analyzed with both an analytical and a numerical approach. We first did a theoretical analysis in which the s-wave spin-singlet state was projected onto the pseudospin eigenbasis in a system where a spin-orbit field and an exchange field coexist throughout the entire non-superconducting material. The analysis showed that the spin-singlet state is projected onto the eigenbasis as a linear combination of a of a short-ranged pseudospin-singlet state and a long-ranged pseudospin-triplet state, depending upon both the relative orientation and the strengths of the exchange and spin-orbit fields. The spin-singlet therefore gains a long-ranged component, which can traverse through the system with a slow decay. The theoretical analysis predicts that the copresence of spin-orbit coupling and magnetic fields can raise the critical temperature compared to if spin-orbit coupling is absent. The numerical calculations support the predictions of the theoretical analysis. We explored interfacial spin-orbit cou- pling between a ferromagnet and a superconductor, as well as in-plane spin-orbit coupling in the ferromagnetic region of12 an S/F/S- and an S /F-structure. Both the pair amplitudes, lo- cal density of states and the critical temperature showed to be strongly dependent upon the direction of the exchange field. By rotating the relative orientation between the spin-orbit coupling and exchange fields, or by adjusting either the magnetic field or Rashba coupling, one may therefore control the superconduct- ing properties of the system, making these structures possible candidates for use in cryogenic spintronics components. ACKNOWLEDGMENTS M. Amundsen, N. Banerjee, S. Jacobsen, J. A. Ouassou, and V . Risingård are thanked for useful discussions. We thank in particular K. Halterman for useful correspondence. J.L. acknowledges funding via the Outstanding Academic Fellows program at NTNU, the NV-Faculty, and the Research Council of Norway Grant numbers 216700 and 240806. This work was partially supported by the Research Council of Norway through the funding of the Center of Excellence ”QuSpin” project no. 262633. Appendix A: Finding Tcwith perturbation theory We start by defining particle- /hole-amplitude vectors un(x)= un;"(x) un;#(x)! ; vn(x)= vn;"(x) vn;#(x)! ; (A1) and the matrices ¯He= He0 0He! ; ¯
=J2
; (A2) where J2is the (22) exchange matrix, sometimes also referred to as the backward identity matrix, J2= 0 1 1 0! : (A3) This matrix will also be of use later to express cross-coupling terms like un;vn;. Furthermore define as the vector of Pauli matrices, and a slightly altered vector of Pauli matrices ˜=[x;y;z]. By using this notation, the BdG equations take the form ¯Heh
hSO
 ¯
¯
h¯Heh
˜+hSO
˜i! un vn! =En un vn! ; (A4) where we remind ourselves that hSOis a momentum-dependent operator, while and ˜act in spin space. We now do a perturbation expansionun=u(0) n+u(1) n+O(2); (A5) vn=v(0) n+v(1) n+O(2); (A6) En=E(0) n+E(1) n+O(2); (A7) ¯
=0+¯
(1)+O(2); (A8) whereis an arbitrary perturbation parameter, which eventu- ally will be set to 1. u(1) nis conventionally assumed to be an orthogonal function to u(0) n, that isRd 0dxu(1)y n(x)u(0) n(x)=0, and v(1) nis likewise assumed to be or- thogonal to v(0) n. We have defined the superconducting band gap such that it first enters the equations at order O(). To zeroth order, Eq. (A4) is diagonal, meaning unandvnare com- pletely decoupled. This implies that u(0) nandv(0) nhave separate energy spectra, Ep nandEh nrespectively, where pandhdenote particle and hole, and are found by solving the zeroth order BdG equations: ¯Heh
hSO
 u(0) n(x)=Ep nu(0) n(x); (A9) ¯Heh
˜+hSO
˜ v(0) n(x)=Eh nv(0) n(x): (A10) To order first order, O(), the BdG equations read ¯Heh
hSO
 u(1) n+¯
(1)v(0) n=E(1) nu(0) n+E(0) nu(1) n; (A11) ¯Heh
˜+hSO
˜ v(1) n+¯
(1)u(0) n=E(1) nv(0) n+E(0) nv(1) n: (A12) Now operate on Eq. (A11) withP mk,nkRd 0dx0u(0) m(x)u(0)y m(x0), and on Eq. (A12) withP mk,nkRd 0dx0v(0) m(x)v(0)y m(x0). Use the orthogonality and completeness relations of unand vn, and the following formulas for the first order corrections are then obtained: u(1) n(x)=X mk,nkRd 0dx0u(0)y m(x0)¯
(1)(x0)v(0) n(x0) E(0) nEp mu(0) m(x);(A13) v(1) n(x)=X mk,nkRd 0dx0v(0)y m(x0)¯
(1)(x0)u(0) n(x0) E(0) nEhmv(0) m(x);(A14) where it is implied in the notation that the perpendicular energy quantum number is equal for all involved wave functions, that ism?=n?. The sum over mk,nkis a sum over a complete set of one-dimensional eigenfunctions, with the exception of mk=nk, which is not included due to the assumption that the first order corrections are orthogonal to the zeroth order functions. Keep in mind that for the perturbation expansion to be valid, the fractions in Eqs. (A13) and(A14) have to be 1.13 We now want to derive an expression for the first order correction to
(x), that is
(1)(x), by using the first order re- sults for the wave functions. First expand
(x) in its Fourier components,
(x)=X q
qsin kqx ; (A15) where, as previously, kq=q=d. Equivalently, we may write
l=2 dZd 0dx
(x) sin(klx): (A16)
(x) is non-zero only inside intrinsic superconductors. Thus for a system in the x-direction with a non-superconducting region on the interval [0 ;x0), and a superconducting material on the interval [ x0;d], we may also write
(x)= (xx0)
(x)= (xx0)X q
qsin kqx ;(A17) where (xx0) is the unit step function. It seems as though introducing this step function is unnecessary, but it will come of use quite soon. We now insert the definition of
(x), given in Eq. (11), into Eq. (A16), and obtain
l=EF 2kFdX nZd 0dxvy n(x)J2un(x) sin(klx)tanh(En=2kBT); (A18) where J2is the exchange matrix defined in Eq. (A3). We now do a perturbation expansion of the Fourier coe cients of
. Since we are working to first order, we want to find
(1) l, and
(0) l=0 by assumption. By inserting the perturbation expansion of
lto first order on the left hand side, and the perturbation expansions of unandvnin Eqs. (A5) and(A6) on the right hand side of Eq. (A18), we obtain 
(1) l=EF 2kFdX nZd 0dx v(0)y n(x)+v(1)y n(x) J2  u(0) n(x)+u(1) n(x) sin(klx)tanh(En=2kBT):(A19) We observe that there is no term of order O(0) on either side of the equation, which is consistent. Now insert the first order results from Eqs. (A13) and(A14) into(A19) , expand unand vnas in Eq. (7), and expand
(1)(x) as in Eq. (A17) . We neglect the terms of order O(2) which appear on the right hand side of the equation, and we get the following matrix equation
(1) l=X kJlk(T)
(1) k; (A20) where the matrix elements Jlkare defined by the formula Jlk(T)=2EF kFd3X nX mkX p;qKpql ( v(0)y mqJ2u(0) npP i;ju(0)y niJ2v(0) m jKi jk Ep nEhmtanh Ep n 2kBT! +v(0)y nqJ2u(0) mpP i;ju(0)y miJ2v(0) n jKi jk EhnEp mtanh Eh n 2kBT!) :(A21) [1] J. Linder and J. W. A. Robinson, Nat. Phys. 11, 307 (2015). [2]D. Beckmann, Journal of Physics: Condensed Matter 28, 163001 (2016). [3]J. W. A. Robinson, J. D. S. Witt, and M. G. Blamire, Science 329, 59 (2010). [4]T. S. Khaire, M. A. Khasawneh, W. P. Pratt, and N. O. Birge, Phys. Rev. Lett. 104, 137002 (2010). [5]F. S. Bergeret, A. F. V olkov, and K. B. Efetov, Phys. Rev. Lett. 86, 4096 (2001). [6]F. S. Bergeret, A. F. V olkov, and K. B. Efetov, Rev. Mod. Phys. 77, 1321 (2005). [7] M. Eschrig, Physics Today 64, 43 (2011). [8]S. H. Jacobsen, I. Kulagina, and J. Linder, Sci. Rep. 6, 23926 (2016). [9]A. Costa, P. H ¨ogl, and J. Fabian, Phys. Rev. B 95, 024514 (2017). [10] P. H¨ogl, A. Matos-Abiague, I. Zutic, and J. Fabian, Phys. Rev. Lett. 115, 116601 (2015). [11] C. Espedal, T. Yokoyama, and J. Linder, Phys. Rev. Lett. 116, 127002 (2016).[12] I. V . Bobkova and A. M. Bobkov, Phys. Rev. B 96, 104515 (2017). [13] I. V . Bobkova and A. M. Bobkov, Phys. Rev. B 95, 184518 (2017). [14] F. Konschelle, I. V . Tokatly, and F. S. Bergeret, Phys. Rev. B 92, 125443 (2015). [15] M. Amundsen and J. Linder, Phys. Rev. B 96, 064508 (2017). [16] N. Banerjee, J. Ouassou, Y . Zhu, N. Stelmashenko, J. Linder, and M. Blamire, (2017), arXiv:1709.03504. [17] J. Y . Gu, C.-Y . You, J. S. Jiang, J. Pearson, Y . B. Bazaliy, and S. D. Bader, Phys. Rev. Lett. 89, 267001 (2002). [18] I. C. Moraru, W. P. Pratt, and N. O. Birge, Phys. Rev. Lett. 96, 037004 (2006). [19] P. Leksin, N. Garif’yanov, I. Garifullin, Y . V . Fominov, J. Schu- mann, Y . Krupskaya, V . Kataev, O. G. Schmidt, and B. B ¨uchner, Phys. Rev. Lett. 109, 057005 (2012). [20] Y . A. Bychkov and E. I. Rashba, JETP Letters 39, 78 (1984). [21] A. Manchon, H. C. Koo, J. Nitta, S. M. Frolov, and R. A. Duine, Nature Materials 14, 871 (2015). [22] K. Fossheim and A. Sudbø, Superconductivity Physics and Ap-14 plications (John Wiley & Sons Ltd., 2004) p. 424. [23] K. Halterman and O. T. Valls, Phys. Rev. B 65, 014509 (2001). [24] K. Halterman, O. T. Valls, and C.-T. Wu, Phys. Rev. B 92, 174516 (2015). [25] K. Halterman, O. T. Valls, and P. H. Barsic, Phys. Rev. B 77, 174511 (2008). [26] P. H. Barsic, O. T. Valls, and K. Halterman, Phys. Rev. B 75, 104502 (2007).[27] K. Halterman and O. T. Valls, Phys. Rev. B 66, 224516 (2002). [28] S. H. Jacobsen, J. A. Ouassou, and J. Linder, Phys. Rev. B 92, 024510 (2015). [29] J. A. Ouassou, A. Di Bernardo, J. W. A. Robinson, and J. Linder, Sci. Rep. 6, 29312 (2016). [30] C.-T. Wu, O. T. Valls, and K. Halterman, Phys. Rev. B 86, 014523 (2012).

1406.4816v1.Engineering_spin_orbit_coupling_for_photons_and_polaritons_in_microstructures.pdf

1 Engineering spin-orbit coupling for photons and pol aritons in microstructures V. G. Sala 1,2 , D. D. Solnyshkov 3, I. Carusotto 4, T. Jacqmin 1, A. Lemaître 1, H. Terças 3, A. Nalitov 3, M. Abbarchi 1,5 , E. Galopin 1, I. Sagnes 1, J. Bloch 1, G. Malpuech 3, A. Amo 1 1Laboratoire de Photonique et Nanostructures, LPN/CNRS, Route de Nozay, 91460 Marcoussis, France 2Laboratoire Kastler Brossel, Université Pierre et Mari e Curie, École Normale Supérieure et CNRS, UPMC Case 74, 4 place Jussieu, 75252 Paris Cedex 05, France 3Institut Pascal, PHOTON-N2, Clermont Université, Uni versity Blaise Pascal, CNRS, 24 avenue des Landais, 63177 Aubière Cedex, France 4INO-CNR BEC Center and Dipartimento di Fisica, Univer sità di Trento, I-38123, Povo, Italy 5Laboratoire Pierre Aigrain, École Normale Supérieure, CNRS (UMR 8551), Université Pierre et Marie Curie, Université D. Diderot, 75231 Paris Cedex 05, France One of the most fundamental properties of electroma gnetism and special relativity is the coupling between the spin of an electron and it s orbital motion 1. This is at the origin of the fine structure in atoms, the spin Hal l effect in semiconductors 2, and underlies many intriguing properties of topological insulators, in particular their chiral edge states 3,4 . Configurations where neutral particles experience an effective spin- orbit coupling have been recently proposed and real ized using ultracold atoms 5,6 and photons 7-10 . Here we use coupled micropillars etched out of a semiconductor microcavity to engineer a spin-orbit Hamiltonian fo r photons and polaritons in a microstructure. The coupling between the spin and o rbital momentum arises from the polarisation dependent confinement and tunnelling o f photons between micropillars arranged in the form of a hexagonal photonic molecu le. Dramatic consequences of the spin-orbit coupling are experimentally observed in these structures in the wavefunction of polariton condensates, whose helica l shape is directly visible in the spatially resolved polarisation patterns of the emi tted light. The strong optical nonlinearity of polariton systems suggests exciting perspectives for using quantum fluids of polaritons 11 for quantum simulation of the interplay between in teractions and spin-orbit coupling. Spin-orbit (SO) coupling is the coupling between th e momentum and spin of a particle. It gives rise to the fine structure in atomic spectra and it is present in some bulk materials. A prominent example is semiconductors without inversi on symmetry, in which static electric fields present in the crystal Lorentz transform to a magnetic field in the reference frame of a moving electron, which then couples to the electron spin. The resulting SO coupling leads to a number of exciting phenomena like the spin Hall e ffect 2,12 , the persistent spin helix 13 , or topological insulation in the absence of any extern al magnetic field 3. In semiconductors the SO coupling is determined by the crystalline structure being, therefore, hard to manipulate and often difficult t o separate from other effects. Novel 2 systems, like ultracold atomic gases under suitably designed optical and/or magnetic field configurations 6,14 , and photons in properly designed structures 15 , allow for a great flexibility and control of the system Hamiltonian. Even though particles without magnetic moment, such as photons, cannot experience the usual SO cou pling, the engineering of an effective Hamiltonian acting on photons in structured media h as led, for instance to the observation of the photonic analogue of the spin Hall effect in pl anar structures 16-18 , and unidirectional photon transport in lattices with topological prote ction from disorder scattering 7,19-21 . Effective SO couplings have been induced in arrays of photoni c ring resonators using the spin-like degree of freedom associated to the rotation direct ion of photons in the resonator 8,21 . A promising perspective is to use the intrinsic photo n spin: the polarisation degree of freedom 10 . In combination with the strong spin-dependent int eractions naturally present in microcavity-polariton devices and the possibility o f scaling up to lattices of arbitrary geometry 22-24 , the realization of such a coupling will open the way to the simulation of many- body effects in a new quantum optical context11 . Some envisioned examples would be the controlled nucleation of fractional topological exc itations 25,26 , the self-organisation of polarisation patterns 27 , the simulation of spin models using photons 28 , or the generation of fractional quantum Hall states for light 9. In this Letter we engineer the coupling between the polarisation (spin) and the momentum (orbital) degrees of freedom for photons and polari tons using a photonic microstructure with a ring-like shape. The structure is a hexagonal cha in of overlapping micropillars as shown in Figure 2a. Each individual micropillar shows discre te confined photonic modes, the lowest one with cylindrical s symmetry. Thanks to the spatial overlap of adjacen t micropillars the confined photons can tunnel between neighbouring si tes 29,30 . As it was shown in Ref. 31, the tunnelling amplitude through a given link is differ ent for the polarisation states parallel and orthogonal to the link direction (Fig. 1a,b). We sh ow here that when extended to the hexagonal structure, this polarisation dependent tu nnelling together with on-site energy splittings, results in an effective SO coupling for photons. In our system, photons are strongly coupled to quantum-well excitons giving rise to pol ariton states, which hold the same spin/polarisation properties as the confined photon s. We show that the engineered SO coupling drives the Bose-Einstein condensation of p olaritons into states with complex spin textures. Within a tight-binding formalism, the system can be modelled along the lines sketched in Fig. 1. For each link connecting the pillars
↔
+1 , we can define a pair of (real) unit vectors  and   , respectively longitudinal and transverse to the l ink direction (see Fig. 1c). In the absence of particle-particle inter actions, the tight-binding Hamiltonian describing the six coupled pillars reads:  = −ℏ ⋅  ⋅+ℏ ⋅    ⋅+ℎ..+ ! + ∆$ %&⋅ +' (& +' ⋅(−&⋅  + ' (&  + ' ⋅(), (1) 3 Fig. 1. Spin-orbit split states . a Scheme of two coupled pillars showing different tu nnelling probabilities for photons polarised longitudinal (l ight blue) and transverse (pink) to the link. b This results in polarisation splitting of the bonding an d anti-bonding states. c Scheme of the unit vectors perpendicular and transverse to each link used to c alculate the SO-coupled eigenstates of Hamiltonian (1). d Polarisation dependent classification of the energ y states in the absence of SO coupling ( = and ∆* = 0 ) as a function of their total momentum , =-+. . States located on each horizontal line have a well-defined orbital momentu m - and form multiplets. e Same as d for ≳ . The SO coupling is evidenced by the anticrossing of the bands, at , = 0 and , =3 . f, g, Fine structure and polarisation pattern of the molecular eigenstat es corresponding to the multiplets - =1 , f, and - = 2, g, split by the SO coupling for ∆ = 30 µeV, ∆* = 0 . Note that 2,2′≪1 . The salmon coloured molecules show the states in which condensation tak es place in Figs. 3 and 4. The insets in f,g show the relative energy of the top and bottom states of each multiplet when ∆* 60 . where =78 9,7+7'8 9,7' is the vector field operator for polaritons. For e ach site
on the ring, the 8 9,77' operators destroy a polariton in the .+.− circular polarisation basis.  and  are the tunnelling amplitudes for photons with lin ear polarisation oriented along and transverse to the link direction, respect ively (see Fig. 1b). Finally, the ∆* terms provide an on-site splitting between linearly polar ised states oriented azimuthally and radially to the hexagon. These terms account for the wavegui de-like geometry of the molecule 32 : if instead of a hexagonal chain we would have a unifor m ring guide, these terms would be the dominant contribution to the SO coupling effects. N ote that in Hamiltonian (1) the rigid rotation of the unit vectors  ,   while going around the hexagon will be crucial to describe the SO coupling. In the case ∆ ≡− =0 and ∆* =0 , the spin and the momentum are decoupled and Hamiltonian (1) reduces to  =−∑ℏ< =>⋅+⋅? ! . This is analogous to the Hamiltonian that describes the tunnelling between e lectronic @ states of the C 6H6 benzene molecule. The eigenfunctions are delocalised over t he six micropillars and can be classified in terms of the orbital angular momentum - =0, A1, A2, 3 , which determines the relative phase of the lobe of each micropillar (Fig. 2c): 8 9B,7=∑6' =⁄EFG %⁄8 9,7 ! . The wavefunction of the - =0 state presents a constant phase over all the pilla rs, while - =A1,A2 states contain phase vortices of topological charge - (the phase changes by 2@- when going around the molecule). Finally, the - =3 state presents a phase change of @ from pillar to pillar. The eigenergies depend on - as follows: *-,. =−2ℏ HI 2@- 6⁄ . 4 In the general case, when − ≠0 and/or ∆* ≠0 , spin and orbital degrees of freedom are coupled. As it happens to electrons in atoms, in th e presence of a finite SO coupling neither the spin nor the orbital angular momentum are good quantum numbers: Hamiltonian (1) is no longer symmetric under separate orbital or spin rot ations. Nevertheless, combined spin and orbital rotations remain symmetry elements, and the corresponding conserved quantity is the total angular momentum , =-+. . The eigenstates are then better labelled in terms of ,, which remains a good quantum number. Figure 1d,e sh ows the dispersion of the eigenstates as a function of , for negligible (Fig. 1d) and significant values of − 60 (Fig. 1d) as predicted by Hamiltonian (1), which can be rewritte n in ,-space as:  ≅−∑L2ℏ HI 2@M,−.N6⁄ 8 9B,78 9B,7O B,7 −∑ℏ ∆< =HIFB %+∆$ =L8 9B,7'8 9B,7+ℎ..O B . (2) As compared to the --dependent dispersion, the ,-dependent dispersions for the two .A spin states are shifted by A1 units of angular momentum. The crossing that is vi sible in Fig. 1d at , =0 and , =3 for − =0 and ∆* =0 , is lifted by the mixing of the two spin components by the SO coupling when − ≠0 and/or ∆* ≠0 (Fig. 1e), giving rise to a fine structure in the energy spectrum. At , =0 (respectively , =3 ), the magnitude of the splitting between the two states is ∆*PQ =|+ℏ∆+∆* | (respectively ∆*PQ =|−ℏ∆+∆* |). With respect to the orbital form, the two , =0 eigenstates are symmetric/antisymmetric combinations of states with opposite circular polar isation and with opposite orbital angular momentum - =A1 : S, =0,A =2' =⁄&E'TUV W|, =0 - =−1 ,.+ XAETUV W|, =0 - = 1 ,.−X(. In real space, the lower (upper) state correspond s to a polarisation texture oriented in the azimuthal (radial) directions, respectively, as represented in Fig. 1f. Note that the two remaining states with |-|=1 (|- =+1,.+ X, and |- =−1,.− X) are far away in energy from their “partner” of same k and opposite spin. They a re thus practically unaffected by the SO coupling, and they remain located in between the sp lit apart states. An analogue situation takes place at , =3 : the resulting eigenstates are symmetric and antisymmetric combinations of orbital states with o pposite circular polarisation and opposite angular momentum momentum - =A2 : S, =3,A =2' =⁄&E'TUV W|, =3 - =2 ,.+ X∓ ETUV W|, =3 - =−2 ,.− X(. For small ∆* (<ℏ ∆ ), the ordering in energy is exchanged with respect to the , =0 doublet, giving an azimuthally (respectively, radi ally) polarised lower (respectively upper) state. As sketched in the inse t of Fig. 1g, for increasing positive ∆*, the SO-splitting can be cancelled ( ∆* =ℏ ∆ ) and its sign reversed ( ∆* 6ℏ ∆ ). Note that an alternative description of the SO coupling in terms of an effective magnetic field acting on the pseudo-spin of the photon is given in the Methods a nd Supplementary Material together with a detailed solution of Hamiltonian (1). 5 In order to experimentally evidence this engineered SO coupling we use a polaritonic structure fabricated from an AlGaAs [ 2⁄ microcavity with 12 GaAs quantum wells, a Rabi splitting of 15 meV, and a polariton lifetime of 50 ps (see Methods). We engineer hexagonal molecules made out of six overlapping round micropi llars (Fig. 2a). Each micropillar provides a quasi-cylindrical 0D confinement and the overlap results in a first neighbour tunnel coupling 29 ≳ =0.3 meV. According to the measurements reported in Ref . 31 for two coupled micropillars, the polarisation-dependent tu nnelling term dominates over the on-site splitting (∆~30 ]eV6 |∆*|), which we will neglect in the following. The samp le is kept at 10 K and excited out of resonance with a Ti:Sapph c w laser, focused in a 12 µm diameter spot, entirely covering a single molecule. The ener gy resolved photoluminescence is recorded by a CCD camera and its polarisation is an alysed in the six Stokes polarisation components: linear vertical (V), horizontal (H), di agonal (D) and antidiagonal (A), and circular .+, and .− (see Methods). Fig. 2. The photonic molecular states . a Scanning electron microscope image of the polarito nic molecule. b Emission spectrum in momentum space at ,`=0.2 µm-1 and low excitation power (1.5 mW), showing the four molecular states ( *0,…,*3 ) arising from the coupling of the lowest energy mode of each single micropillar. In this low excita tion power regime, the spin-orbit coupling is small as compared to the emission linewidth and cannot be re solved. c Scheme of the phase winding of the eigenfunctions of the *- states without accounting for the spin. d, e Measured real space, d, and momentum space, e, emission. f Simulation of the momentum space emission obtained by the Fourier transform of the tight binding model eigenfunctions with a Gaussian distribution over each lobe. Solid lines in d show the contours of the structure. 6 At low excitation density, the incoherent relaxatio n of carriers in the structure populates all the molecular levels (Fig. 2b). As the natural pola riton linewidth of 100 µeV is much larger than the polarisation dependent tunnelling ∆, SO coupling effects are not revealed and the spontaneous emission is effectively unpolarised. Th e eigenstates can be described as the spinless eigenstates defined above when = and ∆* =0 , with four energy levels (Fig. 2b) and the orbital structure depicted in Fig . 2c. Each molecular eigenmode has six lobes centred on each micropillar, as evidenced in the energy resolved real-space images in Fig. 2d. The orbital phase structure gives rise to distinct patterns in momentum space as shown in Fig. 2e,f and can be directly imaged by pe rforming interferometric measurements 33 (see Supplementary Information). Fig. 3. Condensation in b=c state . a, b Real space emission in the .+ and .− circular polarisations for a polariton condensate in the sta te 2' =⁄&E'TUV W|0,.+X−ETUV W|0,.−X(, observed at a pumping intensity of 84 mW. Each polarisation compo nent contains a vortical current in the clockwise (.+) and counterclockwise ( .−) direction, evidenced in the fork-like dislocation s apparent in the interferometric measurements shown in c and d, respectively, and in the extracted phase gradient s in e and f. g The green traces show the plane of linear polarisa tion of the emission measured locally, superimposed to the total emitted intensity of the molecule. The condensate shows azimuthal linear polarisation. Solid lines depict the contour of the structure. h Power dependence of the emitted intensity at the energy of the state described in a-g, and at the energy of the state described in Fig. 4 (grey points). 7 At high excitation density, polariton condensation takes place and the population ends up concentrating in a single quantum state with a redu ced emission linewidth 34 . As usual in non- equilibrium systems, condensation does not necessar ily occur in the ground state and the steady state is determined by a complex interplay b etween pumping, relaxation and decay (see Supplementary Information). In our structure, in particular, we observe two non-linear thresholds in the emission intensity as a function of pumping intensity, which correspond to condensation in two different states (Fig. 3h). The existence of two thresholds arises from the non-linear power dependence of the polariton relaxa tion channels 35 . The combination of a reduced linewidth and condensation in excited state s grants access to the fine structure caused by SO coupling in the polaritonic benzene mo lecule. In the case shown in Fig. 3 (excitation intensity of 84 mW), polaritons condens e in the lowest state arising from the |-|=1 quadruplet, that is, into the lowest state at , =0 (Fig. 1f). This fact is evidenced when mapping the linear polarisation of the emission (Fi g.3g): the polarization is directed in the azimuthal direction around the molecule as predicte d in Fig. 1f. The polarisation-selective interferometric analysis of the emission in the cir cular basis shown in Fig. 3c-f (see Methods) reveals the underlying helical orbital structure of the state, consisting of the linear superposition of two states of opposite orbital vor ticity - =A1 and opposite spins. For each spin state .∓, the phase winds by A2@ while looping around the molecule. Fig. 4. Condensation in b=d state . a and b Real space emission in the .+ and .− circular polarisations for a polariton condensate in the sta te 2' =⁄&E'TUV W|3,.+X+ETUV W|3,.−X( observed at 25 mW. Each polarisation component contains a vorti cal current of double phase charge in the counterclockwise ( .+) and clockwise ( .−) directions, evidenced in the double fork-like dis locations apparent in the interferometric measurements shown in c and d, respectively, and in the extracted phase gradients in e and f. g The green traces show the plane of linear polarisa tion of the emission measured locally, superimposed to the total emitted intensity of the molecule. The condensate shows a linear polarisation pattern pointing radially in the direction of the micropillars. Solid lines depi ct the contour of the structure. 8 Condensation in other SO coupled states can be obse rved by varying the excitation conditions (Fig. 3h). For a 25 mW excitation densit y, polariton condensation takes place in the lowest energy state of the |-|=2 quadruplet. Within the pillars, the emission is li nearly polarised along the radial direction (Fig.4g) as pr edicted in the lowest state of Fig.1g. On the other hand, the polarisation-selective interferomet ric images of Fig.4c-f evidence the underlying orbital structure of the state, consisti ng in the linear superposition of two states of opposite spin and opposite orbital angular momentum . In contrast to the case of Fig. 3, the extracted phase (Fig. 4e,f) changes from zero to A4@ while looping around the molecule. In this experiment, polariton condensation occurs s electively into two particular states with the polarisation structures shown in Figs. 3 and 4, evidencing the SO coupling we have engineered for polaritons. The polarisation pattern of the other states should also be accessible by resorting to a resonant excitation sc heme as opposed to the non-resonant one used here. The SO coupling reported here for polari tons originates in the polarisation dependence of the photonic confinement and of the p hoton tunnelling amplitude, which can both be engineered with a suitable design of the st ructure. For instance, by asymmetrising the micropillars we can enhance ∆* such that the SO coupling is cancelled in the mult iplet |-|=2, or its sign reversed, as sketched in the inset of Fig. 2g. Notice that the same SO coupling engineering could be implemented for pure photons, either choosing a larger exciton-photon detuning or processing an empty cavi ty. Further promising developments are expected to occu r when the SO coupling is scaled up to larger systems such as two-dimensional lattices, wh ere photonic quantum spin Hall states and spin topological insulators 3 can be realized. Exciting new features are anticip ated in systems with a high degree of phase frustration, li ke the optically accessible flat bands recently reported in a honeycomb lattice of micropi llars 24 . At strong light intensities, our system appears to be an excellent platform to study the effect of SO coupling on non-linear topological excitations like vortex solitons 36 and non-linear ring states 37 . When the strong polariton nonlinearities are brought towards the si ngle-particle level, new quantum features are expected to originate while polaritons enter a strongly correlated state 11 . Methods Sample and experimental set-up . The microcavity sample is grown by Molecular Beam Epitaxy and it is made out of Bragg reflectors with 29 and 40 pairs of alternating Al 0.95 Ga 0.01 As/Al 0.20 Ga 0.80 As λ/4 layers that define a λ/2 cavity ( λ=787nm). Three sets of four GaAs quantum wells of 70 Å in width are distributed at the three central maxima of the confined electromagnetic field. The coupling betwee n the quantum well 1s excitons and the fundamental longitudinal cavity mode results in a R abi splitting of 15 meV at 10 K. The planar structure is dry etched into a hexagonal photonic molecule made out of six coupled micropillars. The diameter of each micropil lar is 3 µm, and the centre-to-centre separation is 2.4 µm, resulting in a photonic nearest neighbours coupl ing 29 of 0.3 meV. The homogeneity of the emitted intensity over all the m icropillars seen in Fig. 2d demonstrates 9 negligible defects in the structure. We work at zer o photon-exciton detuning, where we measure a polariton lifetime of about 50 ps, corres ponding to a photon quality factor of 66000. Excitation is performed with a cw monomode laser at a wavelength of 735 nm, above the first reflectivity minimum of the stop band defined by th e Bragg mirrors. The laser is focused in a spot of 12 µm in diameter pumping the whole molecule. The photo luminescence is collected with a microscope objective and it is energy resolv ed in a 0.5 m spectrometer attached to a CCD camera. We use a long-wavelength pass filter to remove the stray light from the laser. A set of λ/4 and λ/2 waveplates and linear polarisers allows us selec ting the different polarisation components of the emission. With this information we can reconstruct the direction of the linear polarisation of the emissio n at each point of the molecule (see Supplementary Information). The relative phase of the emission coming from diff erent micropillars is measured using a modified Michelson interferometer. The real space e mission of the molecule interferes in the CCD camera at zero time delay with a magnified imag e coming from only one of the micropillars, providing a homogeneous phase referen ce. The two images reach the CCD at different angles, resulting in the interference pat terns shown in Fig. 3c,d and Fig. 4 c,d. We extract the spatial phase distribution (shown in Fi g. 3e,f and Fig.4e,f) from a Fourier transform analysis 33 . Spin-Orbit Hamiltonian in operator and effective fi eld forms Hamiltonians (1) and (2) can be expressed in the fo rm of an operator acting on a spinor Sfg
=MS
,S'
N , with
=1,…,6 the micropillar index and +,- the two elements of the circular polarisation basis. For this purpose we in troduce the diagonal part of the spinor Hamiltonian hi=h∆ =0, ∆* =0 . Eigenstates of hi are described by the quantum number -: *-,. =−2ℏ HI 2@- 6⁄ We can introduce an operator jh=kT$ kGT=HI 2@- 6⁄ allowing us to write the complete Hamiltonian: h=hi−∆* 2&0 E'=lm E=lm0(−ℏ∆ 2njh&0 E'=lm E=lm0(+&0 E'=lm E=lm0(jho, where p=
@ 3⁄. In the context of SO coupling in semiconductors, it is meaningful to express the Hamiltonian in terms of an effective magnetic field acting on t he spin of the particles. In the case of Hamiltonian (1)-(2), this can be done in cylindrica l coordinates the following way: qSfg=−ℏ= 2rs=t= tp=Sfg−Ωffg∙Sfg where R is the mean radius of the molecule and Ωffg is the effective field: 10 ℏwffg ==ℏ ∆< =HIFB %+∆$ =&0 E'=lm E=lm0(. The polarisation patterns in the eigenmodes can the n be understood as those arising from the alignment and antialignement of the photon pseu dospin with respect to the effective field Ωffg. Acknowledgements We thank J. W. Fleischer, G. Molina-Terriza, A. Pod dubny, P. Voisin and M. Wouters for fruitful discussions. 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Hafezi, M., Mittal, S., Fan, J., Migdall, A. & Tay lor, J. M. Imaging topological edge states in silic on photonics. Nat. Photon. 7, 1001-1005 (2013). 22. Kim, N. Y. et al. Dynamical d-wave condensation of exciton-polaritons in a two-dimensional square-l attice potential. Nature Phys. 7, 681-686 (2011). 23. Cerda-Méndez, E. A. et al. Dynamic exciton-polari ton macroscopic coherent phases in a tunable dot la ttice. Phys. Rev. B 86 , 100301 (2012). 24. Jacqmin, T. et al. Direct Observation of Dirac Cones and a Flatband in a Honeycomb Lattice for Pola ritons. Phys. Rev. Lett. 112 , 116402 (2014). 25. Hivet, R. et al. Half-solitons in a polariton q uantum fluid behave like magnetic monopoles. Nature Phys. 8, 724-728 (2012). 26. Manni, F. et al. Dissociation dynamics of singl y charged vortices into half-quantum vortex pairs. Nature Comms. 3, 1309 (2012). 27. Sinha, S., Nath, R. & Santos, L. Trapped Two-Dimen sional Condensates with Synthetic Spin-Orbit Couplin g. Phys. Rev. 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Amo 1 1Laboratoire de Photonique et Nanostructures, LPN/CNRS, Route de Nozay, 91460 Marcoussis, France 2Laboratoire Kastler Brossel, Université Pierre et Mar ie Curie, École Normale Supérieure et CNRS, UPMC Case 74, 4 place Jussieu, 75252 Paris Cedex 05, France 3Institut Pascal, PHOTON-N2, Clermont Université, Uni versity Blaise Pascal, CNRS, 24 avenue des Landais, 63177 Aubière Cedex, France 4INO-CNR BEC Center and Dipartimento di Fisica, Univer sità di Trento, I-38123, Povo, Italy 5Laboratoire Pierre Aigrain, École Normale Supérieure, CNRS (UMR 8551), Université Pierre et Marie Curie, Université D. Diderot, 75231 Paris Cedex 05, France Table of contents A. Phase structure of the molecular eigenstates dis regarding the spin degree of freedom B. Polarisation structure of the
=  and
=  condensed states C. Condensation kinetics D. Model A Phase structure of the molecular eigenstates disr egarding the spin degree of freedom The polariton benzene molecule employed in our stud ies is made out of six overlapping cylindrical micropillars. In a single micropillar t he three-dimensional confinement of polaritons results in discrete energy levels whose spacing is determined by the diameter of the molecule and the exciton-photon detuning. In our st ructure, with micropillars of 3 µm and zero photon-exciton detuning, the splitting 1 between the lowest and first excited state is on the order of 2 meV. The spatial overlapping of the micropillars creates a coupling between the ground states of adjacent pillars of 0.3 meV, r esulting in the molecular energy levels ,  = 0,1,2,3 , accessible in energy-resolved photoluminescence a t low power as shown in Fig. 1b. The value of 0.3 meV was measured in two c oupled micropillars with the same diameter and overlap as the micropillars in the hex agonal molecule 2. In the actual hexagonal molecule, next-nearest neighbour tunnelling also pl ays a role in the energy difference between the  energy levels, resulting in a splitting smaller th an 0.3 meV between  and . This smaller splitting is well reproduced by the 2D Schrödinger equation simulations shown in Supplementary Figure 6. 2 Spin orbit coupling effects result in fine structur e splittings that are smaller than the emission linewidth observed in the spontaneous emission regi me. These effects can only be evidenced in the condensation regime under high exc itation density. In this section of the Supplementary Information we concentrate in the low power emission and we disregard the spin. The form of the molecular wavefunctions can be calc ulated with a tight binding approach in which the system is described by six sites with nea rest-neighbour tunnelling. In this simple approximation, the stationary wavefunction of the  level takes the form:  =  6 ⁄  ,|!" with #,= $%! 3 ⁄, where |!" is a wavefunction spatially localised in the ! = 0,… ,5 micropillar. All the eigenfunctions present equal probability amplitudes in the six sites, resulting in equal emission intensities at the centre of each micropil lar, as observed in Fig. 1d. The main difference is in the pillar to pillar relative phas e, which can be parameterised by the angular momentum quantum number $ = ± . The ground state presents a constant phase over a ll the pillars ( # ,= 0), the levels  = 1 and  = 2 are both doubly degenerate with eigenvalues $ = +1,−1 and $ = +2,−2 , respectively. In these eigenstates the phase of t he wavefunction changes from 0 to ±2%$ when circumventing the molecule, defining a phase vortex of charge ±$. The  = 3 state is non-degenerate (the periodicity of the ef fective potential creates a combination of the states $ = ±3 , with a phase that changes by % from pillar to pillar). The phase structure we just described can be access ed experimentally by an interferometric experiment at low excitation density (conditions of Fig. 1). We use the technique explained in the Methods: the emission from the ensemble of the molecule interferes with a magnified image of the emission from one of the micropillars, which provides a flat phase reference. Both images reach the entrance slit of an imaging s pectrometer at an angle. In the spectrometer, each emission line is resolved in ene rgy reaching a different position in the CCD camera. Thus, in the CCD, we observe the interf erence between the reference phase and the emission from the whole molecule, for each energy level. By performing an energy- resolved, two-dimensional, real-space tomography we can reconstruct the real-space interferometric images corresponding to the emissio n from each of the  , … ,  , energy levels. These images are shown in Supplementary Fig ure 1a-d in the conditions of Fig. 1. We then perform a Fourier Transform and filter out the off diagonal elements of the real and imaginary part. We Fourier transform back the filte red images and we obtain both the visibility of the fringes and the spatial changes o f the phase, shown in Supplementary Figure 1e-l. More details of this technique can be found in Ref. 3. The ground state,  , non-degenerate, shows a homogeneous relative phas e at the centre of the pillars, as can be seen by following the dotted line in Supplementary Figure 1l (the radial variation of the phase is an artefact arising from the geometry of the interferometric experiment). 3 Supplementary Figure 1 . Interference (a-d), visibility of fringes (e-h) a nd extracted phase (j-l) of the  = 0,1,2,3 levels measured in the spontaneous emission regime at low excitation density (conditions of Fig. 1 in the main text). The  energy level is double degenerate, with the emissi on being a superposition of states with $ = +1,−1 : |"=∑12  ⁄ ./ ,⁄|!"+ 12  ⁄ / ,⁄.0 |!" , where 1 is a relative phase whose value, in the spontaneous emission regi me, changes randomly in time. The interference pattern shown in Supplementary Figure 1c arises from the superposition between the emission from the whole molecule and a reference corresponding to an enlarged image of the micropillar located at the ar row ( ! = 0 ). In our time integrated images, interference fringes are only visible at the locati on of the reference pillar ( ! = 0 , self- 4 interference) and the one located in front ( ! = 3 ), which has a fixed relative phase difference % for both $ = +1 , −1 substates. This phase difference is well observed in Supplementary Figure 1k. At intermediate positions, the phase dif ference respect to the emission from the ! = 0 micropillar is scrambled when integrating the imag es due to the random values of 1, resulting in a low visibility (Supplementary Figure 1g). A similar situation is observed for the degenerate states $ = +2,−2 (): |"=∑12  ⁄ ./ , ⁄|!"+ 12  ⁄ / , ⁄.0 |!" . As the phase now increases with the angle twice as fast, there are four positions in th e molecule in which the phase difference with respect to the reference ! = 0 micropillar is the same for both substates (an int eger multiple of %), thus resulting in a high fringe visibility (Supp lementary Figure 1b,f). For those four positions, a phase jump of % is well observed (Supplementary Figure 1j). The non-degenerate  = 3 state shows a high fringe visibility for all the m icropillars (Supplementary Figure 1a,e), with a phase jump of % from pillar to pillar (follow dotted lines in Supplementary Figure 1i). B Measurement of the polarisation structure of the
=  and
=  condensed states The polarisation of light emitted at a particular p oint of the sample can be described in the paraxial approximation as in Born and Wolf 4. We can define an arbitrary polarisation state in the following way: 23= 456789: ;<= + > 567? 2@= 456789: ;<= + > 567? where 4,567 describes the amplitude of the electric field alon g the A and B directions in the plane of the molecule at a given point 6, < is the frequency of light and >,567 are fixed phases for each component. The polarisation state i s fully determined by the ratio 45674567 ⁄ and by the phase difference >567− >567. The condition for linear polarisation is > = > 567− >567= C% , with C = 0,±1, ±2,… , with the ratio 45674567 ⁄ defining the direction of polarisation. Circular p olarisation is obtained under the condition 4567= 4567, > = > 567− >567= C% 2 ⁄, with C = ±1, ±3, ±5,… Other values of 45674567 ⁄ and > result in elliptical polarisations. A basis of interest is that of circular polarisatio n: DE.= 4E.56789: ;<= + > E.567?A − F 4 E.56789: ;<= + > E.567?B DE= 4E56789: ;<= + > E567?A + F 4 E56789: ;<= + > E567?B 5 In this basis, a linear polarised state corresponds to 4E.567= 4E567; the phase difference >E567= >E.567− >E567 sets the direction of the linear polarisation: 1 = − 5>E.567− >E567 72⁄, with 1 the clockwise angle from the A direction. Experimentally, the polarisation state of the emiss ion can be fully described by the Stokes coefficients: G = HI G=H3− H@ G G=H.JK − HJK G G,=HE.− HE G where H3,@,.JK,JK,E.,E is the emitted intensities when detecting the line arly polarised emission along the x, y axis, the +45º, -45º direct ions with respect to the x axis, and the L+,L − circularly polarisation, respectively. G, G, G,, are defined such that they correspond to the degree of polarisation along the same axes. The relation to the 4E./4E ratio and 1 is: 4E. 4E=G,+ 1 G,− 1 1 =1 2arctan SG GT Graphically, the polarisation state corresponds to a point in the Poincaré sphere determined by the G, G, G, axis: Supplementary Figure 2 . Poincaré sphere representing the polarisation state of the emitted light. Note that # = arccosG ,. 6 In order to reconstruct the linear polarisation pat terns shown in Figs. 3g and 4g, we measure the Stokes coefficients G and G by using a linear polariser in combination with a half- waveplate. Due to constraints in the experimental s et-up, in order to reconstruct the linear polarisation map shown in Figs. 3g and 4g we measur e G and G along the polarisation axes oriented 22.5º/112.5º and 67.5º/157.5º with respect to the x direction (horizontal) in Figs. 3 and 4. We can redefine the coefficients G and G along those axes: GW=H.K− H.K G GW=HYZ.K− HKZ.K G G, is measured by using a quarter-waveplate and a lin ear polariser. Supplementary Figure 3 and 4 show the polarisation emission filtered in the different projections needed to reconstruct GW, GW, and G, corresponding to the situations described in Figs. 3 and 4 of the main text, respectively. The a ngle [ shows the orientation of the polarizers used to analyse the emission with respec t to the horizontal axis of the molecule, as defined in the inset. From images (a) through (d ) in Supplementary Figures 3 and 4 we can extract the direction of the linear polarisatio n of the emission plotted in Figs. 3g and 4g. The angle 1 setting the direction of linear polarisation at ea ch point in the plane of the figures is calculated from the following expression: tan \2 ∙ 51 + 22.5∘7_= GWGW⁄ where 1 increases counterclockwise and 0 corresponds to the horizonta l positive direction. Note that the shallow diagonal traces observed in Supplementary Figures 3 and 4 arise from an artefact due to the presence of a neutral density filter in the detecti on path. Supplementary Figure 3. Linear polarisation tomography of the  = 1 condensed state shown in Fig. 3 of the main text. From the degree of linear polarisation reported in (g)-(h), we extract the ov erall linear polarisation direction shown in Fig. 3g. 7 Supplementary Figure 4. Linear polarisation tomography of the  = 1 condensed state shown in Fig. 4 of the main text. From the degree of linear polarisation reported in (g)-(h), we extract the ov erall linear polarisation direction shown in Fig. 4. C Condensation kinetics In our structure, we observe two condensation thres holds at two different excitation powers as shown in Fig. 3h. This can be more clearly obser ved when looking at the angle resolved spectra at different powers, shown in Supplementary Figure 5. Between 7 and 17mW, below the first threshold, we o bserve a blueshift related to the continuous increase of reservoir excitons in the sy stem. At 25 mW, condensation takes place in the E2 level. At higer power, 57mW, we observe s imultaneous condensation in the E1 and E2 levels. Finally, above 84mW only the E1 level re mains highly occupied. Note that the redshift observed for the E1 level between 57 and 1 10mW arises from the heating of the sample due to the large absorbed optical density. 8 Supplementary Figure 5 . Experimental spectra at different excitation dens ities. In an open dissipative system with pumping and life time the Bose Einstein condensation does not necessarily occur in the ground state sinc e relaxation kinetics has to be taken into account 5. Within the simplest approximation, the equation d escribing the occupation of a confined state reads: given by ( ) ( ) 1 1/ in out dN W N N W dt τ = + − + , (1) where τ is the state lifetime. ,in out W are the scattering rates toward and outward the st ate. They verify: / / b b E k T E k T in out cR W W e WY e − − = = , where cR Y is the overlap integral between the state density and the exciton reservoir distribution. W depends on the system parameters, but since the ma in relaxation mechanism is based on exciton-exciton interaction, W is a growing function of the carrier density and of the pumping power. Equation 1 can be recast as ( ) ( ) 1in out dN W N N W dt = + − ɶ where '/ / b b E k T E k T in out cR W W e WY e − − = = ɶ which yields 9 1' ln 1 b cR E E k T WY τ = + + (2) W, Y and τ are in general functions of E. The meaning of the previous development is that the state occupation in our pump-dissipative s ystem can be expressed as thermal-like distribution function if one defines a new energy s cale that includes the effect of particle life time. This is what is expressed in Eq. 2, where we can see that the state with the lower effective energy 'E is not necessarily the original ground state of th e system. For instance, an excited state with a long lifetime τ can become the state with lowest effective 'E, and thus, the most favoured for condensation to take pl ace. The exact value of 'E depends on the relaxation efficiency, pumping power, overlap i ntegral between the reservoir and the state, lifetime. In the rest of the text we are goi ng to analyse which is the most favoured state in the molecule depending on the excitation conditi ons. The increase of W with pumping reduces the impact of the kinetics on the determina tion of the ground state. Lifetime As a first approximation, the polariton lifetime in this system depends on the absolute value of the angular momentum. Indeed, the decay of the particles is given by the extension of the wave function out of the structure . This extension is maximal for bound states 0l=, it decreases for higher values of l and is minimal for the anti-bound states 6 3l= with a variation which we estimated numerically of the order of 20 %. As a result, condensation in states with 0l= is not favoured by kinetics because of their short lifetime. There are two additional mechanisms that result in the increase of the polariton lifetime with increasing energy. First of all the exciton content increases with energy, resulting in an enhanced lifetime. Second, the wavefunction of high er energy modes presents zeroes at the constrictions between the pillars, where the defect density is higher. Thus, higher energy modes are more protected against the non-radiative losses associated to defects. The combination of these effects results in condensatio n in the E3 energy level with the lowest threshold. Overlap with the reservoir, spin-anisotropic intera ction . As said above, the degenerate state can be represented either, as spatially homog eneous circularly polarized states or as linearly polarized spatially inhomogeneous states. The spin-anisotropic interaction of polaritons makes that it is favorable energetically for a condensate to be linearly polarized 7. On the other hand, linear combination of states res ulting in amplitude inhomogeneities show a reduced overlap with the excitonic reservoir, whi ch is homogeneous all over the molecule in our excitation conditions. For these reasons, th e states which favour condensation are the spatially homogeneous states resulting from the cou pling between 1 l+= − and 1l−= and 2l+=, 2 l−= − . 10 Summary. At low pumping, condensation is expected to occur o n the more kinetically favoured states, namely the states resulting from t he coupling 2l+=, 2 l−= − . It is also reasonable to expect that the system will choose th e lowest of these two states. Going to higher power, the effective energies of the state e volve and the effective ground state should become the lowest of the two states resulting from the coupling between 1 l+= − , 1l−=, namely, the state with an azimuthal polarisation. Numerical simulations. In order to confirm this finding, we use the model based on self- consistent coupled semi-classical Boltzmann and non linear Schrödinger equations 2, and find a good agreement with the experimental measurements . To be able to carry out a direct comparison with th e experiment, we calculate the emission from the quantized states of our benzene molecule i n the reciprocal space, taking into account the occupation numbers found from the simul ations described above. The results are shown in Supplementary Figure 6. Below threshol d (left panel), all states are approximately equally populated. Main emission come s from the ground state which can be identified by a maximum of emission at k=0. At high er pumping (right panel), condensation occurs at the most favoured state which is the stat e with 1l= and azimuthal polarization. The emission from this state dominates the spectrum . Supplementary Figure 6 . Simulated emission pattern in the reciprocal spac e below (left panel) and above threshold (right panel). Condensation occurs on the state with angular momentum 1 and azimuthal polarization. 0 0 1 1 -1 -1 k□(µm□)-1k□(µm□)-1E□(meV) 01211 References 1. Bajoni, D. et al. Polariton Laser Using Single M icropillar GaAs-GaAlAs Semiconductor Cavities. Phys. Rev. Lett. 100 , 047401 (2008). 2. Galbiati, M. et al. Polariton Condensation in Ph otonic Molecules. Phys. Rev. Lett. 108 , 126403 (2012). 3. Manni, F., Lagoudakis, K. G., Liew, T. C. H., An dré, R. & Deveaud-Plédran, B. Spontaneous Pattern Formation in a Polariton Condensate. Phys. Rev. Lett. 107 , 106401 (2011). 4. Born, M. & Wolf, E. Principles of optics electromagnetic theory of prop agation, interference and diffraction of light (Pergamon, Oxford, 1980). 5. Sanvitto, D. et al. Exciton-polariton condensati on in a natural two-dimensional trap. Phys. Rev. B 80 , 045301 (2009). 6. Aleiner, I. L., Altshuler, B. L. & Rubo, Y. G. R adiative coupling and weak lasing of exciton- polariton condensates. Phys. Rev. B 85 , 121301 (2012). 7. Shelykh, I. A., Rubo, Y. G., Malpuech, G., Solny shkov, D. D. & Kavokin, A. Polarization and Propagation of Polariton Condensates. Phys. Rev. Lett. 97 , 066402 (2006). Supplementary Information for "Engineering spin-orbit coupling for photons and polaritons in microstructures" D.- Model V.G. Sala,1, 2D. D. Solnyshkov,3I. Carusotto,4T. Jacqmin,1A. Lema^ tre,1H. Ter cas,3 A. Nalitov,3M. Abbarchi,1, 5E. Galopin,1I. Sagnes,1J. Bloch,1G. Malpuech,3and A. Amo1 1Laboratoire de Photonique et Nanostructures, CNRS/LPN, Route de Nozay, 91460 Marcoussis, France 2Laboratoire Kastler Brossel, Universit e Pierre et Marie Curie, Ecole Normale Sup erieure et CNRS, UPMC Case 74, 4 place Jussieu, 75252 Paris Cedex 05, France 3Institut Pascal, PHOTON-N2, Clermont Universit, Universit Blaise Pascal, CNRS, 24 avenue des Landais, 63177 Aubire Cedex, France 4INO-CNR BEC Center and Dipartimento di Fisica, Universit a di Trento, I-38123, Povo, Italy 5Laboratoire Pierre Aigrain, Ecole Normale Sup erieure, CNRS (UMR 8551), Universit e Pierre et Marie Curie, Universit e D. Diderot, 75231 Paris Cedex 05, France (Dated: April 29, 2014) I. TIGHT-BINDING DERIVATION OF THE SPIN-ORBIT HAMILTONIAN OF PHOTONIC BENZENE In this section we present the tight-binding derivation of the spin-orbit Hamiltonians (1) and (2) of the main text. We consider a chain of cylindrical micropillar cav- ities arranged at the vertices of a regular polygon of M sides, as sketched in Fig. 1 for the M= 6 case of the benzene experiments. The polygon is assumed to sit on thexyplane. FIG. 1. Sketch of the system under consideration. On each site, a single orbital mode is available, with an approximately cylindrically symmetric wavefunction. Each orbital mode has a twofold spin degeneracy corre- sponding to the two polarization directions on the xand ydirections. On the jth site centered at position Rj, the vector electric (or polaritonic) eld operator has the form: ^~E(r) =(rRj)[ex^aj;x+ey^aj;y] = =(rRj)[e+^aj;++e^aj;y];(1)where the ^ax, ^ayand ^a+, ^aare two among the possible basis on which the vector electric eld can be expanded. ex;yis a cartesian basis, while e= (exiey)=p 2 is the circular basis. In the following, it will be convenient to use the com- pact and basis-independent vector notation ^ai=ex^aj;x+ey^aj;y=e+^aj;++e^aj;y:(2) For both the cartesian and the circular basis, the com- mutators satisfy bosonic commutation rules. For each link connecting the j!j+ 1 sites, we can dene the real unit vectors e(j) Lande(j) Trespectively par- allel and orthogonal to the link direction Ri+1Ri. The main assumption of our model is that tunneling along the j!j+1 link occurs with dierent amplitudes tLandtT for photons polarized along the e(i) Lande(i) Tdirections, respectively. According to S. Michaelis de Vasconcellos et al., Appl. Phys. Lett. 99, 101103 (2011),jtLj&jtTj. In second quantization terms, the many-body Hamil- tonian then reads: H=MX j=1f[~tL(^ay j+1
e(j) L)(e(j)y L
^aj)+ +~tT(^ay j+1
e(j) T)(e(j)y T
^aj)] + h.c.g:(3) Physically, the e(j)y L;T
^ajexpression selects the component of the vector ^ajeld on the jth side along the e(j) L;Tunit vector. Given the periodic boundary conditions of the system, in the sum the ( M+1)th site has to be identied with the 1st and the 0th with the Mth. As the many-body Hamiltonian (3) does not involve any interaction terms, it is straightforward to derive a Schr odinger equation for the single-particle C-number vector wavefunction jthat denes states on the one- body subspace: j 1i=X j[j
^ay j]jvaci (4)2 As usual, the Schr odinger equation can be obtained from the Heisenberg equation for the eld operators ^aj i~d dt^aj= [^aj;H] (5) by replacing the eld operators ^ajwith the one-body wavefunction j. In this way, one obtains idj dt=tLe(j1) L(e(j1)y L
j1)+tTe(j1) T(e(j1)y T
i1)+ +tLe(j) L(e(j)y L
j+1) +tTe(j) T(e(j)y T
j+1):(6) IfMis even, the symmetry operation Sdened as: Sy^ajS= (1)j^aj (7) anticommutes with the Hamiltonian SyHS=H: (8) As a result, ifj iis an eigenstate of the Hamiltonian Hof energyE, thenSj iis an eigenstate of opposite energyE. At the single-particle level, the action of the operatorSis to invert the sign of the eld on the odd sites (S)j= (1)jj; (9) so that the tunneling energy of each link changes sign. The standard translation operator Tis dened as Ty^ajT=^aj+1: (10) physically, the photon eld on the translated state Tj i at sitejis equal to the original eld on the site j+ 1. In the general case tL6=tT, the operator Tdoes not commute with the Hamiltonian: translation changes the relative orientation of the photon eld ^ajand the link frame dened by the pair e(j) L;T. One has to dene a new translation operator ~Tas fol- lows Ty^ajT=R2=M^aj+1: (11) translation by one site has to be combined with a rota- tion by an angle 2=M so to compensate the dierent orientation of the link. This new operator ~Tcommutes with the Hamiltonian H. It is immediate to check that after a full round trip around the chain, it gives back the identity ~TM=1. In physical terms, the operator ~Tde- scribes discrete rotations by 2 =M around the chain, and therefore corresponds to the total angular momentum of the state. Let's now restrict to the single-particle space. As ~T commutes with H, eigenstates can be found with well dened quantum number kas compared to generalized translations, that is, such that ~Tj i=e2ik=Mj i (12)withk= 1;:::;M . It is important to note that the wavevector kdoes not straightforwardly correspond to the orbital part of the angular momentum. This is easy to see oncircularly polarized states such that j=j 1 i : (13) In this case, if the wavefunction has orbital angular mo- mentum is l
=3 (againl= 1;:::;M ),j=oeijl= 3, the total angular momentum of the state j=oeijl= 3 1 i : (14) contains a spin contribution, k=l1. On the other hand, an experiment where light is selected, is sensi- tive to the orbital wavefunction only. FIG. 2. Sketch of the eigenstates of a M= 6 photonic ben- zene system. Red (blue) points indicate the +() polarized states. States are labeled in terms of the total angular mo- mentumk. Let's begin to diagonalize Hin thetL=tT=tcase. In this case, on each site one has e(j) Le(j)y L+e(j) Te(j)y T=1: (15) The energy of the state is therefore given by the orbital energy only, equal to El=2~tcos(2l=M ): (16) Even though the spin decouples from the orbital motion, it is instructive to see the behavior of the generalized translation operators ~T. The circularly polarized states are eigenstates of the rotation operator R=ei: (17) As a result, the energy of =polarized states of total angular momentum kis E(k;) =2~tcos[2(k)=M] (18) and its dispersion is sketched in Fig. 2. The presence of a spin-orbit coupling term is apparent in (18), where the3 dispersion of the is laterally shifted by
k=1. In Hamiltonian terms, we can write H=X k;E(k;)^by k;^bk;; (19) where ^bk;=1p MX je2i(k)j=M^aj; (20) is the destruction operator for a photon of total angular momentum kand spin In the general case tL6=tT, the orbital angular mo- mentumlis no longer a good quantum number, but k remains so: the tLtTcoupling does not break rota- tional invariance. To qualitatively understand the degen- eracies of the eigenstates in the experimentally relevant casejtLtTjtL;Tit is useful to draw the dispersion of thestates as a function of kand include the eect oftLtTonly at the crossing points of the bands. As one can see in Fig. 2, for even values of Msuch crossings only occur at k= 0;M=2. For all other values, the bands are non-degenerate. As a result, the ground state is two-fold degener- ate and is spanned by the eigenstates j+;k= 1iand j;k=1i(as usual, because of the discrete rotational symmetry the quantum number kis only dened modulo M, sok=1 is equivalent to k=M1). Its energy is'2t. Correspondingly, thanks to the Ssymmetry mentioned above (the Soperation sends a state with orbital momentum linto a state of orbital momentum l+M=2), the highest energy state is two-fold degenerate at energy'2tand is spanned by j+;k=M=2 + 1iand j;k=M=21i. The rst excited manifold at E'2~tcos(2=M ) = ~twas four-fold degenerate for tL=tT. The two ex- ternal states ( +;k= 2) and (;k=2) have no other available state at the same k, so remain degenerate at E=~t. On the otehr hand, the two other states j+;k= 0iandj;k= 0ican be mixed by the angular momentum conserving tLtTterms. To understand the form of the new eigenstates, it is useful to give an explicit expression for the
t=tLtT terms in the basis; from now on we will assume for clarity
t>0. Rewriting the Hamiltonian H=MX j=1f~t 2[^ay j+1^aj+^ay j^aj+1]+ ~
t 2[(^ay j+1
e(j) L)(e(j)y L
^aj)+ (^ay j+1
e(j) T)(e(j)y T
^aj)] + h.c.g:(21) and expressing it in terms of the k-space operators, onegets to H=MX j=1f~t 2[^ay j+1^aj+^ay j^aj+1]+ ~
t 2X kcos(k 3) [^ay k;^ak;+e4i=M+ h.c.]:(22) From this expression it is immediate to see that the
tterm still conserves angular momentum and that the j+;k= 0iandj;k= 0igive rise to new eigenstates at energies E'2~tcos(2=M )~
t. Given the phase factor in the last term of (28), the lowest eigenstate at energy ~(2tcos(2=M )
t) is of the form (A)=1p 2[e2i=Mj0;+i+e2i=Mj0;i]; (23) while the highest at energy ~(2tcos(2=M ) +
t) has the form (B)=1p 2[e2i=Mj0;+ie2i=Mj0;i]: (24) These are the upper and lower states sketched in Fig. 1f of the main text. It is interesting to get insight on the spa- tial polarization structure of the A;Bstates. Replacing the explicit expression of the j0;istates, one obtains for the lower Astate (A) j=1 2[e2i(j1)=M 1 i +e2i(j1)=M 1 i ] = = cos(2(j1)=M) sin(2(j1)=M) (25) which is an azymuthal polarization: For instance, for j= 1 the polarization is horizontal. Analogously for the higherBstate, (B) j=1 2[e2i(j1)=M 1 i e2i(j1)=M 1 i ] = =i sin(2(j1)=M) cos(2(j1)=M) (26) which is a radial polarization. These features are visible in the polarization patterns of the dierent eigenstates that are plotted in Fig. 3. In particular, the Astate is the one at E=1:002 (we have used ~t= 1 and ~
t= 0:002) on the third column (from left) of the top row. TheBstate is the one at E=0:998 on the second column of the second row. The structure of the second highest energy manifold atE=~t(corresponding to Fig. 1g of the main text) is directly obtained via the Ssymmetry that sends the total angular momentum k!k+=M while keeping the polarization pattern intact. As a result, the highest state A0of this manifold will have a azimuthal polarization, while the lowest one B0will have a radial polarization.4 FIG. 3. Calculated polarization pattern for the dierent eigenstates of a photonic benzene molecule. Eigenstates are sorted for growing eigenenergy, indicated in the center of each panel (bottom-up on each column). The calculation is performed by diagonalizing the Hamiltonian (3) within the one-particle subspace. Parameters: tL= 1:001,tT= 0:999,
E= 0. This is again visible in Fig. 3 by comparing the state at E= 1:002 on the second column of the lowest row and the state at E= 0:998 on the third column of the second row. In addition to the polarisation depenendent tunnelling we have considered up to now ( tLtT) our photonic structures may show an onsite splitting
Ebetween modes linearly polarised azimuthal and radially with re- spect to the ring geometry. This onsite splitting accounts for the waveguide shape of the structure. Formally, this splitting can be introduced in the tight-binding model a Hamiltonian term of the form: HLT=
EMX j=1h (^ay j
e(j) L)(e(j)y L
^aj)(^ay j
e(j) T)(e(j)y T
^aj)i (27) with
E=~!R~!A. Expressing it in terms of the k-space operators, one gets a term of the form HAR=
E 2X k[^ay k;^ak;+e4i=M+ h.c.] (28)with exactly the same form as the
tcorrection in (21). Equation (28) thus takes the form: H=MX j=1f~t 2[^ay j+1^aj+^ay j^aj+1]+ X k~
t 2cos(k 3) +
E 2 [^ay k;^ak;+e4i=M+h.c.]: (29) As the result, the eigenstates maintain the same form with the replacement
tcos(k 3)!
tcos(k 3)
AR=2: (30) The lowerAeigenstate of the k= 0 manifold remains azimuthal polarized as long as ~
t>
AR=2. The situ- ation is slightly dierent for the k= 3 manifold, where the radially polarized state keeps a lower energy as long5 as
t >
AR=2. as sketched in the insets of Fig. 1f-g of the main text. The relative magnitude of the two ef- fects has to be determined case by case on each specic structure. For instance, if instead of a hexagonal chain we would have a uniform ring guide,
ARwould be the dominant contribution to the SO coupling. II. SPIN-ORBIT HAMILTONIAN IN MATRIX AND OPERATOR FORM We can gain insights on the emergence of the spin-orbit coupling from the polarisation dependent tunneling and onsite splittings by doing a matricial treatment of the problem. We again consider the basis of single pillar states with polarisations oriented longitudinal ( eL) and transverse ( eT) to the link between jandj+1:jj;L=Ti. The polarisation dependent tunneling is described by sin- gle polariton Hamiltonian matrix elements: tL=D j;L^Hj+ 1;LE tT=D j;T^Hj+ 1;TE (31)while D j;L^Hj+ 1;TE =D j;L^Hj+ 1;LE = 0:(32) In order to include the onsite splitting between modes polarised in the direction radial and azimuthal to the ring geometry of the molecule, it is convenient to change to the polarisation basis n,depicted in Fig. 1: jj;ni=p 3 2jj;Ti1 2jj;Li jj;ti=1 2jj;Tip 3 2jj;Li jj+ 1;ni=p 3 2jj+ 1;Ti1 2jj+ 1;Li jj+ 1;ti=1 2jj+ 1;Ti+p 3 2jj+ 1;Li:(33) In this basis, the tight binding Hamiltonian reads: ^H=0 BBBBBBBBBBBBBBBBB@1;n1;2;n2;3;n3;4;n4;5;n5;6;n6; 1;n En0tnntn 0 0 0 0 0 0 tnntn 1; 0Etnt0 0 0 0 0 0 tnt 2;ntnntnEn0tnntn 0 0 0 0 0 0 2; tnt0Etnt0 0 0 0 0 0 3;n 0 0tnntnEn0tnntn0 0 0 0 3; 0 0tnt0Etnt0 0 0 0 4;n 0 0 0 0 tnntnEn0tnntn0 0 4; 0 0 0 0 tnt0Etnt0 0 5;n 0 0 0 0 0 0 tnntnEn0tnntn 5; 0 0 0 0 0 0 tnt0Etnt 6;ntnntn 0 0 0 0 0 0 tnntnEn0 6;tnt0 0 0 0 0 0 tnt0E1 CCCCCCCCCCCCCCCCCA; (34) where the onsite energy splitting
E=EnE, and tnn=1 4(3tTtL) t=1 4(3tLtT) tn=p 3 4(tT+tL):(35) In order to nd the explicit spin-orbit coupling terms of Hamiltoinan matrice2, it is convenient to change to the circular polarization basis (( j+i;ji)) instead of radial-azimuthal (( jni;ji)) via the transformation: jj;ni=1p 2 exp(ij1 3)jj;+i+exp(ij1 3)jj;i jj;ti=1p 2 exp(i(j1 3+1 2))jj;+i+exp(i(j1 3+1 2))jj;i :(36) Next, for the spatial component of the wavefunction, we change to the basis of orbital angular momentum jli (l= 0;1;2;3) via the transformation6 jl;i=6X j=1exp(il(j1) 3)jj;i: (37) In the orbital-circular polarisation basis, the Hamiltonian takes the form: 0 BBBBBBBBBBBBBBB@E2J 0 0 0 0 0 0
E 2~
t 20 0 0 0 0 E2J 0 0 0 0 0 0
E 2~
t 20 0 0 0 0 EJ 0 0 0 0 0 0 0 0
E 2+~
t 2 0 0 0 EJ
E 2~
t 0 0 0 0 0 0 0 0 0 0
E 2~
t E J 0 0 0 0 0 0 0 0 0 0 0 0 EJ 0 0 0 0
E 2+~
t 20 0 0 0 0 0 0 E+J 0 0
E 2+~
t 0 0
E 2~
t 20 0 0 0 0 0 E+J 0 0 0 0 0
E 2~
t 20 0 0 0 0 0 E+J 0 0 0 0 0 0 0 0 0
E 2+~
t 20 0 E+J 0 0 0 0 0 0 0
E 2+~
t 20 0 0 0 E+2J 0 0 0
E 2+~
t 20 0 0 0 0 0 0 0 E+2J1 CCCCCCCCCCCCCCCA;0+ 0 +1+ +1 1+ 1 +2+ +2 2+ 2 3+ 3 (38) where E=1 2(E+En) t=1 2(tL+tT)(39) This matricial form of the Hamiltonian is equivalent to Hamiltonian (29), showing the coupling between states of opposite orbital momentum and spin. This Hamiltonian can also be expressed in operator form acting on a spinor [ +(j) (j)]T, wherejplays a role of generalized integer coordinate, j= 1;:::;6. For this we introduce the diagonal part of the Hamiltonian ^H0=^H(
AR=~t= 0). The eigenstates of ^H0can be classied in terms of the orbital angular momentum land produce the basis of Hamiltonian (38). Its eigenvalues Elare El=E2Jcosl 3 (40) We can introduce an operator ^M=@2El @l2=cosl 3
, which allows us to rewrite the Hamiltonian in the operator form: ^H=^H0
E 2 0e2i'j e2i'j 0 +~
t 2(^M 0e2i'j e2i'j 0 + 0e2i'j e2i'j 0 ^M); (41) where'j=j=3. It is also possible to represent the same Hamiltonian in a more compact form using an operator ^Kwhich returns the cosine of the sum of orbital momentum and spin: D l^KlE =cos(l+): (42) Then, Hamiltonian (38) can be expressed ^H=^H0
E 2 0e2i'j e2i'j 0 +~
t 2^K 0e2i'j e2i'j 0 : (43)

1707.04518v1.Insights_into_the_orbital_magnetism_of_noncollinear_magnetic_systems.pdf

Insights into the orbital magnetism of noncollinear magnetic systems Manuel dos Santos Dias1,and Samir Lounis1,y 1Peter Gr unberg Institut and Institute for Advanced Simulation, Forschungszentrum J ulich & JARA, D-52425 J ulich, Germany (Dated: July 17, 2017) Abstract The orbital magnetic moment is usually associated with the relativistic spin-orbit interaction, but recently it has been shown that noncollinear magnetic structures can also be its driving force. This is important not only for magnetic skyrmions, but also for other noncollinear structures, either bulk-like or at the nanoscale, with consequences regarding their experimental detection. In this work we present a minimal model that contains the eects of both the relativistic spin-orbit interaction and of magnetic noncollinearity on the orbital magnetism. A hierarchy of models is discussed in a step-by-step fashion, highlighting the role of time-reversal symmetry breaking for translational and spin and orbital angular motions. Couplings of spin-orbit and orbit-orbit type are identied as arising from the magnetic noncollinearity. We recover the atomic contribution to the orbital magnetic moment, and a nonlocal one due to the presence of circulating bound currents, exploring dierent balances between the kinetic energy, the spin exchange interaction, and the relativistic spin-orbit interaction. The connection to the scalar spin chirality is examined. The orbital magnetism driven by magnetic noncollinearity is mostly unexplored, and the presented model contributes to laying its groundwork. 1arXiv:1707.04518v1 [cond-mat.mes-hall] 14 Jul 2017I. INTRODUCTION Magnetic skyrmions1are a kind of topological twist in a ferromagnetic structure, with unusual properties.2They have been found in bulk samples and in thin lms, also at room temperature3{9. When electrons travel throught the noncollinear magnetic structure of the skyrmion, they experience emergent electromagnetic elds.10{12This strong coupling between the electronic and magnetic degrees of freedom leads to very ecient motion of skyrmions with electric currents.13It also generates a topological contribution to the Hall eect, a transport signature of a skyrmion-hosting sample,14,15and was shown to enable the electrical detection of an isolated skyrmion.16,17The link between the magnetic structure and orbital electronic properties was explored for other kinds of magnetic systems before,18{23with renewed interest since the experimental discovery of skyrmions.24{30 Inspired by the investigations of nanosized skyrmions in the PdFe/Ir(111) system,31,32 we uncovered another manifestation of their topological nature: a new kind of orbital mag- netism. In Ref. 27, magnetic trimers and skyrmion lattices were compared, and the orbital magnetic moment was shown to have two contributions: a spin-orbit-driven one and a scalar- chirality-driven one. The minimum number of magnetic atoms needed for a non-vanishing scalar chirality is three, n1
(n2n3)6= 0, with nithe orientation of their respective spin magnetic moments. This means that the magnetic structure is noncoplanar, a requirement for the appearance of this new kind of orbital magnetism. Magnetic trimers were analyzed in detail within density functional theory (DFT), before considering a skyrmion lattice meant to mimic PdFe/Ir(111). Although some calculations were feasible with DFT, to address larger skyrmion sizes a minimal tight-binding model was constructed from the DFT data. This model reproduced both contributions to the local orbital moment, and showed that the sum of all scalar-chirality-driven contributions leads to a topological orbital magnetic moment for a skyrmion lattice. The goal of the present paper is to get more insight into the physical mechanisms driving the orbital magnetism of systems in which both the relativistic spin-orbit interaction (RSOI) and a noncollinear magnetic structure coexist. To this end, we reprise our tight-binding model of Ref. 27 but now applied to magnetic trimers, and present a walkthrough of the dierent sources of orbital magnetism in this model. In classical physics, the orbital magnetic moment arises from the presence of bound currents in a given material. This is a consequence of the continuity equation for the electronic charge density in equilibrium: 0 =@ @t=r
j=)j(r) =rm(r): (1) From the quantum-mechanical point of view, if the ground state supports such a nite bound current, time-reversal symmetry must be broken. Magnetic materials naturally break time-reversal symmetry, due to the existence of ordered spin magnetic moments stabilized by exchange interactions. As long as correlation eects are only moderately important, the orbital magnetic moment is usually assumed to be due to the RSOI, which can be introduced either in atomic or in Rashba-like form,33 HSOI/L
or (Ep)
: (2) Here L=rpis the atomic orbital angular momentum operator, the vector of Pauli matrices, Ethe electric eld and p=i~rthe linear momentum operator. For an atom both forms are equivalent. It is the coupling between the nite spin moment in a magnetic 2material and the orbital degrees of freedom depending on porLthat leads to the nite orbital moment.34 Recently another way of coupling spin and orbital degrees of freedom has been identied and explored.10,12Consider the non-interacting electron hamiltonian consisting of a kinetic term and a spin exchange coupling to an underlying magnetic structure: H=p2 2m0+Jm(r)
: (3) Here0is the unit spin matrix and Jis the strength of the exchange coupling. The magne- tization eld is m(r) =m(r)n(r), with its spatially varying magnitude m(r) and direction n(r). For collinear magnetic systems, such as ferromagnets or simple antiferromagnets, m(r) =m(r)nz, where nzis the direction of the ferromagnetic or staggered magnetization, respectively (accordingly, we allow m(r) to be negative). Then the eigenstates of the system can be labelled with the usual `up' and `down' eigenspinors of z, and we get two decoupled hamiltonians, one for spin-up and another for spin-down. In this way the magnetic order decouples from the orbital degrees of freedom, if the RSOI is not considered. If a common spin quantization axis cannot be chosen, i.e. the system has a noncollinear magnetic structure, such a coupling is indeed present. To see how this arises, consider the unitary transformation that at every point in space diagonalizes the exchange term, Uy(r)m(r)
U(r) =m(r)z=)U(r) =eiw(r)
: (4) The vector w(r) describes the spin rotation in the axis-angle representation, but its explicit form is not required for the present argument, only the fact that it must have a spatial dependence for a noncollinear magnetic structure. If this unitary transformation is applied to the whole hamiltonian, it eects a SU(2) gauge transformation35{37with the result H0=UyHU=2 2m+Jm(r)z; =p0+X (~@w)=p0+X A: (5) The kinetic momentum  consists of the canonical momentum pand a vector potential A that couples to the spin, with ;=x;y;z . The kinetic energy now has four contributions: X 2 =X p2 0+ 2X A(r)pi~X (@A(r))+X  A(r)
20:(6) The rst term is the usual spin-independent contribution, the second term is a spin-orbit interaction (coupling spin to linear momentum), the third is a Zeeman-like contribution, and the fourth is a spin-independent potential-like contribution. We thus see that noncollinear magnetic structures lead to emergent elds that couple the spin and orbital degrees of freedom. This line of reasoning has been very successful in explaining the emergent electro- dynamics of slowly-varying magnetic textures.2,12 When both kinds of spin-orbit interaction are at play, the one of relativistic origin and the one arising from a noncollinear magnetic structure, they compete with each other, and a unied picture can only be given for special limiting cases. First-principles electronic structure calculations provide both qualitative and quantitative insights. Here we adopt the tight-binding model of Ref. 27 to analyze the smallest system for which both contributions to the orbital moment are present: a magnetic trimer. 3This paper is organized as follows. Section II introduces the model and its ingredients, and then a step-by-step construction of the eigenstates with broken time-reversal symmetry is provided. First, in Sec. III we explore how a magnetic eld breaks the translational symmetry of the trimer. Then we show in Sec. IV how a noncollinear magnetic structure produces orbital eects analogous to those of an external magnetic eld. Finally, we combine noncollinear magnetism and the atomic spin-orbit interaction in Sec. V, drawing parallels between the relativistic and the noncollinear sources of orbital magnetism. We discuss our results and present our conclusions in Sec. VI. II. TIGHT-BINDING MODEL FOR NONCOLLINEAR MAGNETIC STRUC- TURES In Ref. 27 the following minimal tight-binding model was introducted, to describe two magneticd-bands experiencing the eects of the relativistic spin-orbit interaction and of a noncollinear magnetic structure: H=Hkin+Hmag+Hsoi: (7) The kinetic energy is given by Hkin=X i;j6=iX mm0scy imstim;jm0cjm0s: (8) Herecy imscreates an electron on atomic site iand on the d-orbital labelled m, with spin projections. The detailed form of the hopping matrices tim;jm0is presented later. The coupling to a background magnetic structure is described by Hmag=JX iX mss0cy imsni
ss0cims0; (9) withJthe strength of the coupling, and nithe unit vector describing the direction of the background magnetic structure on every atomic site. We see that Hkin+Hmagis the tight-binding equivalent of the model of Eq. 3 that was discussed in the introduction. The RSOI is considered in atomic form, Hsoi=X iX mss0cy imsLmm0
ss0cims0; (10) withits coupling strength and Lmm0the matrix elements of the atomic orbital angular momentum operator for the two d-orbitals in the model. Time reversal symmetry is usually described by the antiunitary operator T= iyK.38 HereKtakes the complex conjugate of the spinor wavefunction it is applied to, while i yen- sures that the spin is also reversed. The hamiltonian is time-reversal invariant if it commutes with it, [T;H] = 0. The action of Tis illustrated in the following example: k"(r) =eik
r 1 0 =) T k"(r) =eik
r 0 1 1 0 1 0 =eik
r 0 1 = k#(r): (11) 4It is more helpful to think of Tas reversing the state of motion. We could then write this operator asT=TPTS, whereTPreverses the orbital part of the motion ( k)k in the example) and TSreverses the spin angular momentum ( ")# in the example).39 Hamiltonians describing magnetic systems are typically not time-reversal invariant, either due to the presence of external magnetic elds or to the exchange interactions that stabilize the magnetic ground state. Whether they might still be invariant under reversal of the orbital motion,TP, is only clear in the momentum representation, as it amounts to H(p;) = H(p;). An alternative is to choose basis functions that are not invariant under either TP orTS, as shown by the combination of a plane-wave with a spinor in the example.39This is the strategy that will be employed in the following. III. THE TRIMER WITH SIMPLE HOPPING Consider three identical atomic sites forming an equilateral triangle with sides taken as the unit of length, a= 1, as shown in Fig. 1(a). To uncover the role of the electronic motion around the trimer, we rst consider a simplied model with one orbital per site and no spin dependence. Let jiibe a basis state for one electron being on atom i. In this basis, the hamiltonian is given by Hkinj1i j2i j3i h1j0t t h2jt0t h3jt t 0=) H kin=tR0+tRy 0: (12) t=jtjeiis the hopping amplitude for counterclockwise hops around the triangle, and its complex conjugate tis the one for clockwise hops. The complex hopping breaks the symmetry of translational motion, and could be due to the presence of a magnetic ux threading the triangle. The operatorR0generates the counterclockwise hops, R0=0 @0 0 1 1 0 0 0 1 01 A;R0jii=ji+1i;Ry 0=R1 0;Ry 0jii=ji1i;(13) cos!= 1/2N (cos!= 0)cos!= –1/2F (cos!= 1)"F (cos!= –1)#(a)(b) 123 xy FIG. 1. The magnetic trimer. (a) Atomic structure and choice of coordinate axes, with golden spheres representing the atomic sites. (b) Some magnetic structures described by Eq. (23), with the choice of angles discussed in the text. The red arrows the local orientation of the magnetic structure. The structures with cos =1 are ferromagnetic, cos = 0 is the antiferromagnetic N eel structure, and the others are noncollinear structures. 5and its spectral representation is R0=X kei2k 3jkihkj;jki=1p 3 j1i+ei2k 3j2i+ei2k 3j3i ; k2f0;1g: (14) Note that these basis states are not invariant under reversal of the translational motion, Tjki=TPjki=jki, as the clockwise motion is the time-reversed form of the counter- clockwise motion. The state k= 0 corresponds to no overall translational motion, so it equals itself under TP. The hamiltonian commutes with R0, so fromHkinjki=Ekjkiwe nd the eigenenergies Ek= 2jtjcos2k 3 : (15) As this model is equivalent to a linear chain of three atoms with periodic boundary condi- tions, we shall call kthe ring momentum, which characterizes the translational motion of each eigenstate. We can also dene momentum raising and lowering operators, which will be very useful in the following sections: R=0 @1 0 0 0ei2 30 0 0ei2 31 A;Rjki=jk1i; (R)3jki=jki: (16) The last equality is due to the periodicity of the phase, k3 =k. A uniform magnetic eld perpendicular to the plane of the trimer is a simple example of broken time-reversal symmetry. In the symmetric gauge, with the origin at the center of the triangle, the vector potential is given by A(r) =B 2rnz: (17) The Peierls substitution40{42provides the phase acquired by an electron hopping from site jto sitei: ij=2 0Zri rjdr
A(r) =ij2 0S 3B=ij2 3B 0=ij=)tij=jtjeiij;(18) with the path integral evaluated on the straight line connecting the sites. S=p 3a2=4 0:1 nm2is the area of the triangle, for typical bond lengths. The sign of the result is ij= 1 ifiis a neighbor of jin the counterclockwise sense, and ij=1 for the clockwise sense. 0=h=e4103T nm2is the magnetic ux quantum, and  Bthe actual magnetic ux threading the triangle. Due to the magnetic eld, hopping in a clockwise sense is no longer equivalent to hopping in a counterclockwise sense, and this leads to Ek6=Ek, as already derived above. Asis proportional to the magnetic eld B, we dene the orbital magnetic moment operator as M=@H @=ijtj eiR0eiRy 0 : (19) 6(a) (b) -101ΦB / Φ0-202EkE+E0E− -101ΦB / Φ0-202MkM+M0M− FIG. 2. Properties of the trimer model with one orbital per site and no spin dependence, Eq. (12), as a function of the relative magnetic ux. Here jtj= 1. (a) Eigenenergies, Eq. (15). (b) Orbital magnetic moment of each eigenstate, Eq. (20). The dotted lines indicate the values p 3, the orbital moment for k=1 whenB= 0. It represents the net current owing around the triangle, and can be evaluated for each eigenstate using the eigenfunctions given in Eq. (14), Mk=2jtjsin2k 3 =@Ek @: (20) The last equality is a nice illustration of the Hellmann-Feynman theorem.43,44The following properties will be useful to simplify certain matrix elements appearing in the next sections: Ek1+Ek=Ek+1; Mk1+Mk=Mk+1; (21a) Ek1Ek=p 3Mk+1; Mk1Mk=p 3Ek+1: (21b) The eigenvalues and the corresponding orbital moments are plotted in Fig. 2, as a function of the relative magnetic ux. Both quantities are simple periodic functions of the magnetic ux. Whenever two eigenstates are degenerate in energy, see Fig. 2(a), their orbital moments are equal in magnitude and opposite in sign, cancelling each other, while the third eigenstate (which is non-degenerate) has zero orbital moment, as seen in Fig. 2(b). Thus, for an arbitrary electron lling in thermal equilibrium (0 < N e<3), a nite net orbital moment requires lifting of the energy degeneracy. For nanosized trimers and for realistic laboratory magnetic elds,  B=01, so the previous discussion might seem fanciful. However, once spin exchange to a noncollinear magnetic structure is considered, such seemingly unrealistic eective magnetic elds do emerge. We analyze this case in the next section. IV. NONCOLLINEAR MAGNETISM We now extend the model from the previous section by including the spin exchange coupling, H=Hkin+Hmag=X i;j6=iX scy istijcjs+JX iX ss0cy isni
ss0cis0; (22) 7withs=1 the spin projection on an arbitrary quantization axis. We refer to spin-up and spin-down by the associations "= +1 and#=1. In tandem with the orbital impact of the magnetic eld, accounted for by the complex hopping parameters tij, the standard spin Zeeman coupling will also be considered. The atomic structure is invariant under 2 =3 rotations in real space around the central axis of symmetry of the triangle. We similarly require the atomic plus magnetic structure to be invariant under the combination of a spatial rotation and a spin rotation, both by an angle of 2 =3, around the respective rotation axes. The local direction of the magnetic structure is thus chosen to be ni= sin(cos'inx+ sin'iny) + cosnz; (23) in spherical coordinates with respect to the spin quantization axis nz, and the azimuthal angles are'1= 0,'2= 2=3, and'3=2=3, see Fig. 1(b). We single out the following magnetic structures: ferromagnetic pointing along + z(F"); planar triangular N eel structure (N); and ferromagnetic pointing along z(F#). For these magnetic structures the scalar spin chirality takes the form n1
(n2n3) =3p 3 2sin2cos=3p 3 2C(): (24) This quantity is expected to play an important role as a driver of orbital magnetism.18{23,25,27 We take as basis states the tensor product of the ring states from Eq. (14) with the spin-up and spin-down eigenstates of z: jksi=1p 3 j1si+ei2k 3j2si+ei2k 3j3si : (25) The basis states are not invariant under either reversal of translational motion, TPjksi= jksi, or of spin angular motion, TSjksi=sjksi. Dening J?=JsinandJz= Jcos, the spin exchange coupling is then expressed in this basis as Hmag=JX ini
=J? R++R+
+Jzz; (26) with the spin raising and lowering operators = (xiy)=2, and the ring momentum raising and lowering operators dened in Eq. (16). Eq. (26) shows that the spin- ip part of the magnetic coupling exchanges spin angular momentum with ring momentum. When the spin on every site is decreased ( ), the ring momentum kincreases by one unit ( R+), and vice-versa. This is the spin-orbit interaction driven by the noncollinear structure, in the present model. The basis states couple pairwise, forming the hamiltonian blocks: Hapairsj1"iwith j0#i;Hbpairsj0"iwithj+1#i; andHcpairsj+1"ipairs withj1#i. Their matrix elements are Hjk1"i j k#i hk1"jEk1+Jz+B J? hk#jJ?EkJzB; 2fa;b;cg; (27) whereEkare the eigenenergies dened in Eq. (15), and the spin Zeeman coupling to the external magnetic eld was included, Bz. Each block has then the eigenenergies (see 8Appendix A and Eq. (21)) E=Ek+1 2r 3 4 Mk+1
2+p 3 Jz+B
Mk+1+ Jz+B
2+J2 ?: (28) For=B= 0 and introducing x=3jtj 2J, this yields Ea=jtj 2Jp 12xcos+x2; E b=jtj 2Jp 1 + 2xcos+x2; E c=jtjJ : (29) The competition between kinetic and magnetic energies is encoded in the parameter x. The magnetic moment associated with each of the eigenstates can be calculated directly from the eigenenergies. It has two contributions (treating Bandas independent): M=@E @B@E @=MS+MP: (30) MSis the spin magnetic moment, arising from the Zeeman interaction, and signals the broken symmetry under reversal of spin angular momentum ( TS).MParises from the broken symmetry under reversal of the translational motion ( TP), due to the currents owing around the trimer. This is the orbital magnetic moment already encountered in the previous section. For=B= 0 we nd the spin moments: MSa=cosxp 12xcos+x28 < : cosxsin23x2 2C()
; x1  1 +1 2x2sin2+1 x3C()
; x1;(31a) MSb=cos+xp 1 + 2xcos+x28 < : cos+xsin23x2 2C()
; x1  +11 2x2sin2+1 x3C()
; x1;(31b) MSc=cos : (31c) MStells us about the spin character of an eigenstate. A positive sign indicates ", a neg- ative one#, and if it vanishes it has an equal amount of each character. The adiabatic approximation, valid for Jjtj, makes the electron spin collinear with the direction of the magnetic structure. This would lead to a cos dependence, which is the rst term in the x1 expansion. The expansions were carried out up to the rst term with the angular dependence of the scalar chirality, Eq. (24). The orbital moments can then be shown to be simply related to the spin moments: MPa=MmaxMSa+ 1 2; M Pb=MmaxMSb1 2; M Pc=MmaxMSc;(32) withMmax=p 3jtjthe maximum value of the orbital magnetic moment for this model. First let us consider the case of the magnetic exchange dominating the kinetic energy, i.e.Jjtj, allowing a comparison with the adiabatic approximation. Fig. 3 displays the results fort= 1 andJ= 3 (x= 1=2). The eigenenergies form two groups, separated by the exchange splitting 2 J, as seen in Fig. 3(a). The magnetic noncollinearity eectively reduces the kinetic energy, evidenced by the shrinking `bandwidth' of each group when going from 9(a) (b) (c) F↑NF ↑cosθ-404EξEa+Eb+Ec+Ea−Eb−Ec−E+E− F↑NF ↑cosθ-101MSξMa+Mb+Mc+Ma−Mb−Mc−M+M− F↑NF ↑cosθ-101MPξ / MmaxMa+Mb+Mc+Ma−Mb−Mc−M+M− FIG. 3. Trimer with one orbital per site and a noncollinear magnetic structure, Eq. (22), in the strong exchange coupling regime, t= 1 andJ= 3 (x= 1=2). (a) Eigenenergies, Eq. (29). (b) Spin magnetic moment of each eigenstate, Eq. (31). (c) Orbital magnetic moment of each eigenstate, Eq. (32). The magnetic structures are dened by Eq. (23) and illustrated in Fig. 1(b). The curves are labelled with the states for F "taken as reference: the color labels the value of MP; solid lines and dashed lines indicate the sign of MS. The following combinations of eigenstates are also plotted: ( ) = (a) + (b) + (c), with=. the F"structure to the N structure. Fig. 3(b) shows the spin magnetic moments. The spin moments for the eigenstates labelled (c) follow perfectly the adiabatic approximation, seen as the linear behavior, while those for the eigenstates labelled (a) and (b) show deviations from the linear behavior. Fig. 3(c) shows the orbital magnetic moments. For the F "structure, the eigenstates for each spin projection are decoupled and are just the ring states previously discussed, with the same orbital moments. The variation of the orbital moments with the magnetic structure reveals the presence of the emergent magnetic eld that it generates. Going from F"to N, we arrive at a new energy degeneracy. Comparison of the evolution of the curves with those in Fig. 2 lets us assign  B=0=1=2 to the eigenstates evolving from spin-up, and  B=0= +1=2 for those evolving from spin-down. The net orbital moment is zero for the ferromagnetic structures, but not for the noncollinear ones. To illustrate this, we sum all the contributions corresponding to the + and bands, which corresponds to placing three electrons in the three upper or lower eigenstates, see Fig. 3(a). The average spin for these combinations follows the magnetic structure almost linearly, see Fig. 3(b), the behavior expected in the adiabatic limit. From Fig. 3(c) we observe that the average orbital moments are indeed zero for the F endpoints and for the N structure, but are nite for the noncollinear structures. A comparison with the x1 expansion in Eq. 31 shows thatMP/C(), the scalar spin chirality, to leading order. Next consider the case of the magnetic exchange being comparable to the kinetic energy, i.e.Jjtj. Fig. 4 displays the results for t= 1 andJ= 1 (x= 3=2). As 2J < 3t, the eigenergies for spin-up and spin-down overlap, see Fig. 4(a). Comparing with the previous case, we see that the ordering of the states for F "has changed, as indicated by the sequence of colors in the gure. This has a dramatic impact on the behavior of the magnetic moments, Fig. 4(b,c). On the one hand, the eigenstates labelled (c) still follow the linear behavior. On the other hand, both the spin and the orbital moments for the eigenstates labelled (a) and (b) are only weakly modied by the magnetic structure. This has a simple explanation: the ring states coupled in Hcare degenerate in energy for J= 0, and so the coupling to the magnetic structure is always non-perturbative, while HaandHbeach couple states split by 3 t 10(a) (b) (c) F↑NF ↑cosθ-202EξEa+Eb+Ec+Ea−Eb−Ec−E+E− F↑NF ↑cosθ-101MSξMa+Mb+Mc+Ma−Mb−Mc−M+M− F↑NF ↑cosθ-101MPξ / MmaxMa+Mb+Mc+Ma−Mb−Mc−M+M− FIG. 4. Trimer with one orbital per site and a noncollinear magnetic structure, Eq. (22), in the weak exchange coupling regime, t= 1 andJ= 1 (x= 3=2). (a) Eigenenergies, Eq. (29). (b) Spin magnetic moment of each eigenstate, Eq. (31). (c) Orbital magnetic moment of each eigenstate, Eq. (32). The magnetic structures are dened by Eq. (23) and illustrated in Fig. 1(b). The curves are labelled with the states for F "taken as reference: the color labels the value of MP; solid lines and dashed lines indicate the sign of MS. The following combinations of eigenstates are also plotted: (+) = (c+) + (c ) and () = (a+) + (a) + (b+) + (b). forJ= 0, and so the exchange coupling has only a perturbative eect. To visualize whether there is a net orbital moment also in this case, we sum all the contributions corresponding either to the four lower or to the two higher eigenstates, see Fig. 4(a). Although the net spin moment is zero for the F and N structures, surprisingly it acquires a nite value for the noncollinear structures, see Fig. 4(b). Fig. 4(c) shows that the net orbital moment has a similar behavior. A comparison with the x1 expansion in Eq. 31 shows that MP/MS/C(), the scalar spin chirality, to leading order. We have thus seen how a noncollinear magnetic structure can lead to orbital magnetic eects in the absence of the RSOI. The impact on the electronic structure depends crucially on whether the states which become coupled by the magnetic exchange are initially degen- erate in energy or not. For the former the adiabatic approximation is always valid, while for the latter the exchange coupling must overcome the dierence in kinetic energy between the states for it to have a strong in uence. The picture is also very dierence if each eigenstate is considered by itself or if a group of eigenstates is considered together. In the next section the RSOI is introduced in the model, and its consequences analyzed. V. INTERPLAY BETWEEN NONCOLLINEAR MAGNETISM AND THE REL- ATIVISTIC SPIN-ORBIT INTERACTION We extend our model one nal time, by taking the orbital character of the electrons on every site into account. Following Ref. 27, we consider two d-orbitals to be present on every site, namelyjxyiandjx2y2i, assumed to be initially degenerate in energy. We shall work with their complex counterparts, which are eigenstates of Lz: ji=1p 2 jx2y2iijxyi
; Lzj2i=2j2i: (33) The RSOI in atomic form reduces to L
=Lzz, as the other angular momentum operators vanish when restriced to these two orbitals. To label the states, we make the identications 11+2 = and2 =. However, the kinetic hamiltonian now has to describe the directionality of the orbitals. For zero magnetic eld, the hopping matrix in either the real or complex basis is given by tijjx2y2i jxyi hx2y2jt(1 + cos ij)tsin ij hxyjtsin ijt(1cos ij)ortijj i ji h jt tei ij hjtei ijt; (34) where ij=4 is the angle between the bond and the x-axis. We have ij= ji(mod 2), and 12= 0, 23= 2=3 and 31=2=3. This encompasses the fourfold symmetry of the orbitals, and their directionality. For example, if two orbitals are along the x-axis, hopping can only occur if they are both of jx2y2itype. We can encode the action of the hopping matrix on the orbitals using a new set of Pauli matrices (to be distinguished from the ones used for spin), tij=jtjeiij 0+ei ij++ei ij
=ty ji; (35) where the magnetic eld was restored via the Peierls phase, see Eq. (18). Our basis states are the tensor product of the ring states k= 0;1, of the two orbitals m=2, and of the spinors s=1: jkmsi=1p 3 j1msi+ei2k 3j2msi+ei2k 3j3msi : (36) These basis functions are ideal to describe time-reversal symmetry breaking: the time- reversed counterpart of each basis state is Tjkmsi=sjkmsi, which corresponds to reversing translational, orbital and spin motions, i.e., reversing each of the variables describing the state of motion. The action of the hamiltonian can then be separated into a diagonal part (that leaves the basis state unchanged), and dierent kinds of o-diagonal terms, according to what change they eect on the basis state. The diagonal part of the hamiltonian is H0=tR0+tRy 0+Jzz+zz+Bz: (37) The new terms are the RSOI ( term), and the orbital Zeeman coupling ( Bterm). This part of the hamiltonian acts on a basis state as, recalling Eq. (15), H0jkmsi= Ek+sJz+ sgn(m) (s+B)
jkmsi: (38) We have already seen from the previous section that there are two terms that exchange spin angular momentum and ring momentum, HS=J?R;HSjkmsi=J?jk1ms1i: (39) These terms led to the spin-orbit interaction generated by the noncollinear magnetic struc- ture. The remaining piece of the kinetic term generates two terms that exchange orbital angular momentum and ring momentum, HL= tR0R+tRRy 0
; (40) 12recall Eq. (16), with the result HLjkmsi= tei2(k1) 3+tei2k 3 jk1m2si=Ek1ei2 3jk1m2si; (41) according to the denition in Eq. (15). This might be called an orbit-orbit interaction, as the translational motion and the local orbital motion are coupled. Our hamiltonian can now be written as initially presented in Eq. (7), H=Hkin+Hmag+Hsoi=H0+HS++HS+HL++HL; (42) and represents a 12 12 matrix, composed of three 4 4 blocks. The basis states that can be coupled by the hamiltonian are limited by ( HS)2jkmsi= 0 and (HL)2jkmsi= 0, as the spin and the atomic angular momentum cannot be raised or lowered more than once. We then have the following chain of coupled states: jk1 "i ! HLjk"i ! HSjk+1#i ! HL+jk #i ! HS+jk1 "i: (43) The three blocks are generated by the three possible starting values of k, and can be orga- nized as follows:Hacouplesj0 "i,j+1"i,j+1 #i, andj1#i;Hbcouplesj+1 "i, j1"i,j1 #i, andj0#i; andHccouplesj1 "i,j0"i,j0 #i, andj+1#i. The matrix elements for these blocks have the form Hjk1 "i j k"ijk #i j k+1#i hk1 "jEk1+Jz++B Ek+1ei2 3 J? 0 hk"jEk+1ei2 3Ek+JzB 0 J? hk #jJ? 0EkJz+B E k1ei2 3 hk+1#j 0 J? Ek1ei2 3Ek+1Jz+B: (44) The case of a noncollinear magnetic structure is analytically cumbersome, as the hamilto- nian blocks are 44 matrices. If the magnetic exchange is much stronger than all the other terms, we can adopt the frequently used adiabatic approximation.12The spin projectors that diagonalize the magnetic exchange interaction are (see Appendix A) P=1 2 0(cosz+ sinx)
: (45) As the results for s=1 can be obtained from those for s= +1 by the replacements J!Jand cos! cos, we sets= +1 and drop the spin label in the following. Tracing over the spin components, we dene an eective hamiltonian by eH=h+jHj+i= TrP+H=H;""+H;## 2+ cosH;""H;## 2+ sinJ? 0;(46) which can be written using the orbital Pauli matrices as eH=J0+1 4 Ekp 3Mkcos
0p 3Mk+ 3Ekcos4 (cos+B)
z 1 2 Ekp 3Mkcos
ei2 3++ei2 3
: (47) 13The only role played by Jis to dene the energy zero, so we will also set J= 0 from now on. The eigenergies for the eective hamiltonian blocks eHare E=1 4 Ekp 3Mkcosjtjp D
; (48) with the discriminants jtj2D=p 3Mk+ 3Ekcos4 (cos+B)
2+ 4 Ekp 3Mkcos
2: (49) The orbital moments can be decomposed into two contributions (taking Bandto be independent), M=@E @B@E @=ML+MP: (50) MLis the atomic orbital moment, stemming from the orbital Zeeman interaction, and signals the broken symmetry under reversal of the local orbital motion ( TL). In the previous section we already encountered MP, the contribution to the orbital motion from the currents circulating around the trimer. In the adiabatic approximation MS= cosby construction, so it does not merit further consideration. For=B= 0 and setting y= 4=jtj, we obtain the eigenergies Ea=jtj 4 1 + 3 cosq 136 1y
cos+ 45 + 6y+y2
cos2 ; (51a) Eb=jtj 4 13 cosq 13 + 6 1y
cos+ 45 + 6y+y2
cos2 ; (51b) Ec=jtj 4 2q 16 + 6y
2cos2 : (51c) The atomic orbital moments are MLa=(y+ 3) cos+ 3pDa; M Lb=(y+ 3) cos3pDb; M Lc=(y6) cospDc: (52) MLtells us about the atomic orbital character of an eigenstate. A positive sign indicates , a negative one , and if it vanishes it has an equal amount of each character. The orbital moments arising from the circulating currents are (recall Mmax=p 3jtj) MPa=Mmax 4 cos+ 11 + (2 +y) cos+ 3 (1y) cos2pDa ; (53a) MPb=Mmax 4 cos1+1 + (2 +y) cos3 (1y) cos2pDb ; (53b) MPc=Mmax 2 12 +ypDc cos : (53c) They can be used to characterize the translational motion, as in Sec. III. For a ferromagnetic structure,MP=Mmax1 can be associated with k=1, andMP=Mmax0 withk= 0. 14(a) (b) (c) F↑NF ↑cosθ-4-202EξEa+Eb+Ec+Ea−Eb−Ec−E+E− F↑NF ↑cosθ-101MLξMa+Mb+Mc+Ma−Mb−Mc−M+M− F↑NF ↑cosθ-0.600.6MPξ / MmaxMa+Mb+Mc+Ma−Mb−Mc−M+M− FIG. 5. Trimer with two orbitals per site and a noncollinear magnetic structure in the adiabatic approximation, Eq. (47), and with no relativistic spin-orbit interaction ( t= 1 and= 0). (a) Eigenenergies, Eq. (51). (b) Atomic orbital magnetic moment, Eq. (52). (c) Orbital magnetic moment arising from the circulating currents, Eq. (53). The magnetic structures are dened by Eq. (23) and illustrated in Fig. 1(b). The curves are labelled with the states for F "taken as reference: the color labels MP, similarly to Mkin Fig. 2(b); solid lines and dashed lines indicate the sign ofML. The following combinations of eigenstates are also plotted: ( ) = (a)+(b)+(c), with=. Note that there is no simple relation between MLandMP, in contrast to the results of the previous section. Now that the analytical expressions have been derived, let us explore the physics. We begin by examining what happens when the RSOI is turned o ( = 0), with the results gathered in Fig. 5. Consider rst the ferromagnetic structures. There are two pairs of degenerate eigenenergies, and another is non-degenerate, see Fig. 5(a). They can be charac- terized by their atomic orbital moments, Fig. 5(b), and by their translational motion MP, Fig. 5(c). The degenerate eigenstates have strong ororbital character ( ML1 or1, respectively), and k=1 character ( MP=Mmax0:5). The non-degenerate eigenstates are orbitally mixed ( ML= 0), withk= 0 character ( MP= 0). There is overall no net orbital magnetic moment, as non-degenerate eigenstates have zero orbital moment, and degenerate ones have orbital moments with opposite values, thus cancelling out. As already seen in the simpler model of Sec. IV, the noncollinear magnetic structures lift the energy degeneracies and modify the orbital moments of each eigenstate, enabling a net orbital moment without the RSOI being present. To illustrate this, we sum all the contributions corresponding to the + andbands, which corresponds to placing three electrons in the three upper or lower eigenstates. There is a net atomic orbital moment, see Fig. 5(b), with a C()-like angular dependence (Eq. (24)), but no net current, see Fig. 5(c). We nally bring the RSOI into play. If it is weak when comparing to the kinetic hopping, jtj, the picture is qualitatively similar to the previous one. One major dierence is that it lifts the energy degeneracies in the ferromagnetic structures, thus allowing net orbital moments. This is the well-known role of the RSOI in ferromagnetic systems. We focus on the opposite limit, jtj, to see how it counteracts the kinetic term. The results for = 5 andt= 1 are shown in Fig. 6. In the adiabatic approximation, the RSOI is projected onto the local magnetization direction. Combined with our choice of orbitals, this results in a simple cosdependence, as seen in Eq. (47). All the results show the same behavior, except for a small window around the N eel magnetic structure, where cos = 0, and the kinetic 15(a) (b) (c) F↑NF ↑cosθ-606EξEa+Eb+Ec+Ea−Eb−Ec−E+E− F↑NF ↑cosθ-101MLξMa+Mb+Mc+Ma−Mb−Mc−M+M− F↑NF ↑cosθ-101MPξ / MmaxMa+Mb+Mc+Ma−Mb−Mc−M+M− FIG. 6. Trimer with two orbitals per site and a noncollinear magnetic structure in the adiabatic approximation, Eq. (47), and with strong relativistic spin-orbit interaction ( t= 1 and= 5). (a) Eigenenergies, Eq. (51). (b) Atomic orbital magnetic moment, Eq. (52). (c) Orbital magnetic moment arising from the circulating currents, Eq. (53). The magnetic structures are dened by Eq. (23) and illustrated in Fig. 1(b). The curves are labelled with the states for F "taken as reference: the color labels MP, similarly to Mkin Fig. 2(b); solid lines and dashed lines indicate the sign ofML. The following combinations of eigenstates are also plotted: ( ) = (a)+(b)+(c), with=. term becomes important. The eigenenergies are thus linear in cos , Fig. 6(a), and the atomic orbital moments are almost saturated to the atomic limit, see Fig. 6(b). The tlimit also modies how the electrons move around the trimer, revealed in the behavior of MP, Fig. 6(b). For the ferromagnetic structures we nd values close to those of the model without orbital dependence, MP=Mmax0;1, compare with Fig. 3(b), and a linear departure from those values when the magnetic structure departs from the ferromagnetic ones. In this limit the trimer approximately decouples into two separate orbital channels, each behaving as described in Sec. IV. When the magnetic structure is close to the N eel structure, there is some subtle behavior. To illustrate this, we sum all the contributions corresponding to the + andbands, which corresponds to placing three electrons in the three upper or lower eigenstates. The average atomic orbital moment is featureless, see Fig. 6(b), but the net current changes sign before vanishing at the N structure, see Fig. 6(c). VI. DISCUSSION AND CONCLUSIONS In this paper we discussed a sequence of related models for a trimer, to ascertain how magnetic noncollinearity leads to orbital magnetism, even in the absence of the usual RSOI. The simplest model was introduced in Sec. III, and an external magnetic eld was used to dene the orbital magnetic moment arising from currents circulating around the trimer. It was augmented with the spin degrees of freedom in Sec. IV, and a family of noncollinear magnetic structures was found to lead to the same kind of orbital moment, even without an external magnetic eld. The model was nally endowed with orbital degrees of freedom in Sec. IV, enabling the appearance of the RSOI. The adiabatic approximation was adopted, and the competition between the bond-forming tendencies of the orbital-dependent hopping, and the favoring of current-carrying states by the magnetic noncollinearity and the RSOI was analyzed. Trimer-like structures have been considered in the seminal work of Ref. 18 ( Jjtj) 16and of Refs. 19{21 ( J jtj), where the appropriate limits of our model are indicated. Those works established the scalar spin chirality C(), see Eq. 24, as the smoking gun of the non-RSOI-driven orbital eects. It vanishes for ferromagnetic structures and for the triangular antiferromagnetic N eel structure. Our results show that the orbital magnetism of an individual eigenstate is not proportional to C() (for instance, some have a pure cos  dependence), but that considering a full `shell' or `band' does yield this angular dependence, both in the Jjtjand in the Jjtjlimits. We thus expect partial electron llings to lead to non- C() angular behavior, as we already found in DFT calculations for magnetic trimers.27 We also analyzed separately the behavior of the two contributions to the orbital magnetic moment, the atomic one and the one due to circulating (bound) currents. The former is derived from the atomic orbital Zeeman interaction, while the latter follows from the Peierls phase acquired by the hopping amplitudes. In general such a separation is also possible, as established by the modern theory of orbital magnetization.45,46They give access to two aspects of the persistent (bound) current owing around the trimer: whether it swirls locally around each atomic site (the local orbital moment), and whether there is a net current circulating around the trimer (the nonlocal orbital moment). Our previous work in Ref. 27 focused on the atomic orbital moment in trimers but also in a skyrmion lattice, where a topological contribution was identied, and found to be separable from the RSOI-driven one. As this arose from the magnetic noncollinearity being of a special type for a skyrmion, as encoded in its topological charge,2we expect that also the nonlocal orbital moment of skyrmions should also contain such a topological contribution.30The orbital magnetic moment can be measured independently of the spin magnetic moment with x-ray magnetic circular dichroism,47{49and there is a theoretical proposal for how to separate the local and nonlocal contributions to the orbital moment.45 Recent advances in atomic scale manipulation with the tip of a scanning tunneling micro- scope have enabled  a la carte assembly of magnetic nanostructures, including trimers.50{52 The several physical regimes explored in our model can be realized experimentally: the in- terplay between Jandjtjcan be tuned by changing the separating between the magnetic atoms, or by assembling them on metallic or (semi-)insulating surfaces, while the strength of the RSOI, , can be manipulated by choosing a surface with strong RSOI, or by work- ing with heavy magnetic atoms.53Detection of the orbital magnetism at the atomic scale remains challenging, but recent progress in very sensitive magnetometers utilizing nitrogen vacancies in nanodiamonds might open a way forward.54 For a long time the experimental and theoretical study of the orbital magnetic moment has been neglected in favor of its spin counterpart. This is natural, as the spin moment in most cases determines most of the total magnetic moment in a solid, and the magnetic structures and dynamics are governed by the interatomic spin exchange interactions. Orbital interactions are well-known to be important for transport measurements, as can be seen from the large family of Hall eects. The recent focus on the coupling between the itinerant electrons and the spin moments, described by emergent electromagnetic elds, is part of that.2,10We hope that our work helps bringing the humble orbital magnetic moment back to the limelight. 17Appendix A: Eigenvectors for the two-dimensional problem We wish to diagonalize the following matrix with real parameters w,x,yandz, A= a+bzbxiby bx+ ibyabz =a0+b
: (A1) The eigenvalues and the associated eigenspace projectors are then =ap b
b; P=1 2 0b
p b
b : (A2) The corresponding eigenvectors can be parametrized as j+i= c ei's ;ji= ei's c ; (A3) with c=r 1 + cos 2; s =r 1cos 2; (A4) and the angles cos=bzp b
b; 2[0;]; tan'=by bx2[0;2]: (A5) ACKNOWLEDGMENTS We thank Juba Bouaziz, Phivos Mavropoulos, Yuriy Mokrousov and Stefan Bl ugel for insightful discussions, and Julen Iba~ nez-Azpiroz and Sascha Brinker for a critical reading of the manuscript. We would also like to acknowledge the software package Mathematica55 for its assistance with ensuring the correctness of the sometimes cumbersome analytical expressions. This work is supported by the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (ERC-consolidator grant 681405 { DYNASORE). m.dos.santos.dias@fz-juelich.de ys.lounis@fz-juelich.de 1AN Bogdanov and DA Yablonskii, \Thermodynamically stable vortices in magnetically ordered crystals. the mixed state of magnets," Sov. Phys. JETP 68, 101{103 (1989). 2Naoto Nagaosa and Yoshinori Tokura, \Topological properties and dynamics of magnetic skyrmions," Nat. Nanotechnol. 8, 899{911 (2013). 3S. M uhlbauer, B. Binz, F. Jonietz, C. 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1707.09847v1.Spin_orbit_torques_from_interfacial_spin_orbit_coupling_for_various_interfaces.pdf

arXiv:1707.09847v1 [cond-mat.mes-hall] 31 Jul 2017Spin-orbit torques from interfacial spin-orbit coupling f or various interfaces Kyoung-Whan Kim,1, 2, 3Kyung-Jin Lee,4, 5Jairo Sinova,1, 6Hyun-Woo Lee,7,∗and M. D. Stiles2,† 1Institut f¨ ur Physik, Johannes Gutenberg Universit¨ at Mai nz, Mainz 55128, Germany 2Center for Nanoscale Science and Technology, National Inst itute of Standards and Technology, Gaithersburg, Maryland 20899, USA 3Maryland NanoCenter, University of Maryland, College Park , Maryland 20742, USA 4Department of Materials Science and Engineering, Korea Uni versity, Seoul 02841, Korea 5KU-KIST Graduate School of Converging Science and Technolo gy, Korea University, Seoul 02841, Korea 6Institute of Physics, Academy of Sciences of the Czech Repub lic, Cukrovarnick´ a 10, 162 53 Praha 6 Czech Republic 7PCTP and Department of Physics, Pohang University of Scienc e and Technology, Pohang 37673, Korea (Dated: August 1, 2017) We use a perturbative approach to study the e ffects of interfacial spin-orbit coupling in magnetic multil ayers by treating the two-dimensional Rashba model in a fully thre e-dimensional description of electron transport near an interface. This formalism provides a compact analyt ic expression for current-induced spin-orbit torques in terms of unperturbed scattering coe fficients, allowing computation of spin-orbit torques for var ious contexts, by simply substituting scattering coe fficients into the formulas. It applies to calculations of spin -orbit torques for magnetic bilayers with bulk magnetism, those with inter face magnetism, a normal metal /ferromagnetic insulator junction, and a topological insulator /ferromagnet junction. It predicts a dampinglike component of spin-orbit torque that is distinct from any intrinsic contr ibution or those that arise from particular spin relaxation mechanisms. We discuss the e ffects of proximity-induced magnetism and insertion of an add itional layer and provide formulas for in-plane current, which is induced by a perpendicular bias, anisotropic magnetoresistance, and spin memory loss in the same formalism. I. INTRODUCTION Broken inversion symmetry in magnetic multilayers allows for physics that is forbidden in symmetric systems, enrich- ing the range of their physical properties and their relevan ce to spintronic device applications. Spin-orbit coupling co m- bined with inversion symmetry breaking is a core ingredient of emergent phenomena, such as the intrinsic spin Hall e ffect [1– 6], spin-orbit torques [7–17], Dzyaloshinskii-Moriya int erac- tions [18–24], chiral spin motive forces [25, 26], perpendi cu- lar magnetic anisotropy [27–29], and anisotropic magnetor e- sistance [16, 30–34]. The contributions from an interface a re frequently modeled by a two-dimensional Rashba model [35] while those from bulk are modeled by incorporating the spin Hall effect [6] into a drift-di ffusion formalism [36] in three dimensions. Both interface and bulk contributions have the same symmetry since they originate from equivalent symme- try breaking making it di fficult to distinguish mechanisms, particularly when di fferent mechanisms are treated by di ffer- ent models. Recent theoretical studies generalize the two-dimensiona l Rashba model in order to take into account three-dimensiona l transport of electrons near an interface. The two-dimensio nal Rashba model assumes electrons near the interface behave like a two-dimensional electron gas, thus allows only for in - plane electronic transport. Haney et al. [37] generalize this model to three-dimensions by including a delta-function-l ike Rashba interaction at the interface and compute interfacia l contributions to in-plane -current-driven spin-orbit torques. They show that some results are qualitatively di fferent from those of the two-dimensional model. Chen and Zhang [38] ∗hwl@postech.ac.kr †mark.stiles@nist.govtreat spin pumping with this model using a Green function approach. Studies of spin-orbit torques [39] and anisotrop ic magnetoresistance [40] in magnetic tunnel junctions under perpendicular bias give contributions that are at least second order in the spin-orbit coupling strength due to in-plane sy m- metry. Refs. [28] and [33] calculate respectively magnetic anisotropy and anisotropic magnetoresistance from the thr ee- dimensional Rashba model in particular contexts. Refs. [41 ] and [42] incorporate interfacial spin-orbit coupling e ffects into the drift-diffusion formalism by modifying the boundary con- ditions. Doing so treats both interfacial and bulk spin-orb it coupling in a unified picture. Ref. [43] reports the solution of the drift-diffusion equation in the non-magnetic layer to capture the spin Hall e ffect coupled to a quantum mechani- cal solution in the ferromagnetic layer to capture the e ffects of interfacial spin-orbit coupling. The results of each of these theories are model specific. Studying physical consequences for a variety of systems re- quires recomputing them for each system, such as metallic ferromagnets in contact with heavy metals, those with insul at- ing ferromagnets, and topological insulators in contact wi th a magnetic layer. Even for a single system, a work function dif - ference between the two layers forming the interface and pos - sible existence of proximity-induced magnetism [44] at the interface may complicate the analysis. General analytic ex - pressions that are applicable for a variety of interfaces wo uld make it easier to understand trends within systems and di ffer- ences between systems. In this paper, we develop analytic expressions for interfac e contributions to current-induced spin-orbit torques by tr eating the interfacial spin-orbit coupling as a perturbation. The form of the analytic expression is independent of the details of t he interface [Eq. (16)], written in terms of the scattering amp li- tudes of the interface. All details unrelated to spin-orbit cou- pling are captured by those scattering amplitudes, similar ly to2 magnetoelectric circuit theory [45, 46]. This approach all ows for the computation of spin-orbit torques either by computi ng scattering amplitudes for a given interface or using the sca t- tering amplitudes as fitting parameters. It is possible to co m- pute the scattering amplitudes through first-principles ca lcu- lations or by solving the Schr¨ odinger equation for toy mode ls. Adopting the latter approach allows us to compute spin-orbi t torques for various types of interfaces. We use the same for- malism to find expressions for in-plane current, which is in- duced by perpendicular bias (like inverse spin Hall e ffect [47– 49]), anisotropic magnetoresistance (like spin Hall magne - toresistance [50–55]), and spin memory loss [56] in terms of scattering amplitudes. These are presented in Appendix A. The three-dimensional model for interfacial spin-orbit co u- pling reveals effects which are absent in the two-dimensional electron gas model. In the two-dimensional model, Rashba spin-orbit coupling generates mostly fieldlike component o f spin-orbit torque [11, 12], while the dampinglike componen t becomes noticeable only when one considers an extremely re- sistive [13–16] system or a non-quadratic dispersion [17]. In contrast, the three-dimensional model of interfacial spin -orbit coupling reveals that in metallic magnetic bilayers a curre nt flowing in the normal metal generates fieldlike and damp- inglike components of spin-orbit torque of the same order of magnitude, and in some parameter regimes the dampinglike component can even be larger than the fieldlike component. The dampinglike contribution that we obtain here is distinc t from those due to previously suggested mechanisms. For in- stance, an intrinsic mechanism is independent of the scatte ring time and vanishes for a quadratic dispersion [4, 5], while ou r result is proportional to the scattering time (thus the cond uc- tivity) and survives even for a quadratic dispersion. A deta iled discussion of the distinctions is presented in Sec. VI A. Another result is a generalization of previous approaches to systems with different-Fermi-surfaces, for example, a finite exchange interaction. Previous theories [33, 37, 41, 42], a s- sume that all band structures are the same so that the wave vectors in the normal metal and the ferromagnet are identi- cal, significantly simplifying the computation. However, e ven in the simplest model of bulk ferromagnetism, the exchange splitting Jσ·mintroduces three di fferent wave vectors; one defined in the normal metal and one each for the majority and minority bands in the ferromagnet. In this work, we carefull y take into account the di fferent Fermi surfaces and all the re- sulting evanescent modes, and demonstrate that proper trea t- ment of the evanescent modes is crucial for accurate calcula - tion of spin-orbit torques. Indeed, they can yield significa nt contributions since the amplitudes of reflected states are l arge at the interface and their velocities are slow where the ener gy is close to the barrier. This makes the interaction time for t hese electrons quite long so they can be more strongly a ffected by interfacial fields. We indeed demonstrate a significant cont ri- bution to spin-orbit torque from the evanescent modes with a toy model [See Fig. 4, Eq. (21), and related discussions]. In the previous theories, some of e ffects like the anisotropic mag- netoresistance [33] originate from a di fference between the relaxation times in majority and minority electrons in the f er- romagnet, but we show that existence of the bulk magnetismby itself can also cause such e ffects. We compute the current-induced contribution to the spin- orbit torque, in distinction to the electric-field-induced con- tribution. The former is proportional to the scattering tim e, thus is extrinsic. A recent paper [17] reports the existence of intrinsic spin-orbit torque from the Berry phase, which is p er- pendicular to the extrinsic component. This contribution c an be an explanation for dampinglike spin-orbit torque for jun c- tions with a ferromagnetic insulator or topological insula tor (See Sec. V D). We leave the calculation of intrinsic spin-or bit torque (induced by the Berry phase) in the same formalism for future work. This paper is organized as follows. In Sec. II, we summa- rize the core results. In Sec. III, we develop a general per- turbation theory of scattering matrices. First we define sca t- tering matrices (Sec. III A) and calculate them (Sec. III B). Then we derive expressions for modified scattering matrices due to interfacial spin-orbit coupling in a perturbative re gime (Sec. III C). The resulting scattering matrices allow us to w rite down electronic eigenstates. In Sec. IV, we derive an expres - sion for spin-orbit torque from these eigenstates, by calcu lat- ing the angular momentum transfer (spin current) to the fer- romagnet. We assume that an in-plane electrical current is applied along the xdirection. Since the expression is writ- ten in terms of unperturbed scattering matrices, it allows u s to compute spin-orbit torque by calculating unperturbed sc at- tering matrices for a given interface. Therefore, in Sec. V, we apply our theory to various types of interfaces and var- ious situations, such as magnetic bilayers with bulk mag- netism, those with interface magnetism, a normal metal in contact with a ferromagnetic insulator, and topological in - sulator in contact with a metallic ferromagnet. Calculatin g unperturbed scattering matrices is straightforward by sol ving the one-dimensional Schr¨ odinger equation. We plot fieldli ke and dampinglike components of spin-orbit torque with vary- ing parameters and discuss the results in each subsection. I n Sec. VI, we make some general remarks on our theory. We compare our theory with the two-dimensional Rashba model. We also discuss how our result can be generalized when mul- tilayer structures are considered. We discuss how proximit y- induced magnetization can be considered in our theory. In Sec. VII, we summarize our results. Appendices include sup- plementary calculations that are not necessary for the main results. II. SUMMARY OF THE RESULTS The purpose of this section is to summarize the behavior of the spin-orbit torques presented in Sec. V A to Sec. V D, be- fore showing the general perturbation theory. Here we focus on the existence and relative magnitudes of spin-orbit torq ues generated by interfacial spin-orbit coupling saving detai led discussions for later sections. The systems under consider - ation are normal metal /ferromagnetic metal junctions, normal metal/ferromagnetic insulator junctions, and topological insu- lator/ferromagnet junctions. Throughout this paper, we refer to these as magnetic bilayers, ferromagnetic insulators, a nd3 topological insulators in short, respectively. We describ e the results below and summarize them in Table I. When an in-plane current is applied to a bilayer junction, two components of spin-orbit torque can act on the magne- tization m. Both are perpendicular to magnetization. When a torque is odd (even) in m, it is called fieldlike (damping- like) [57]. For instance, for a constant vector y,m×yis field- like and m×(y×m) is dampinglike.1The names can be under- stood by their behaviors under time reversal: Fieldlike con tri- butions are conservative while dampinglike contributions are dissipative. In fact, a dampinglike spin torque can act as an anti-dampinglike source, thus the terminology ‘dampingli ke’ does not mean an energy loss but originates from irreversibi l- ity. In contrast to the two-dimensional Rashba model, we show that magnetic bilayers show both fieldlike and dampinglike components even without the Berry phase [17] contribution and a spin relaxation mechanism [13–16]. The relative mag- nitude depends on the details of the system. We first consider a magnetic bilayer where the magnetism is dominated by an exchange splitting in the ferromagnetic bulk (not at the int er- face). In experiments, people usually apply a current in the normal metal side. We show that a current flowing in the nor- mal metal ( jN) generates dampinglike and fieldlike spin-orbit torques that are of the same order of magnitude. The current also flows in the ferromagnet ( jF), generating a large field- like spin-orbit torque that can be the dominant contributio n. Therefore, if jN≫jF, the dampinglike and fieldlike compo- nents are on the same order of magnitude. But if jN≈jF, the fieldlike component tends to dominate. If magnetism at the interface plays a more important role than the bulk magnetism considered above, both components have similar orders of magnitude. As for the bulk magnetism case, there are two sources of spin-orbit torque, jNand jF. We demonstrate in Sec. V B that the dampinglike contribu- tions are mostly subtractive and the fieldlike contribution s are mostly additive. Therefore, the current in the ferromagnet tends not to change the total fieldlike spin-orbit torque, bu t tends to reduce the dampinglike spin-orbit torque. For systems with a ferromagnetic insulator or a topological insulator, the dampinglike component is found to be absent. III. PERTURBATION OF SCATTERING AMPLITUDES A. Definition of the scattering matrices We consider a nonmagnet ( z<0)/ferromagnet ( z>0) in- terface at z=0 where zis the interface normal direction. Ei- ther materials could be insulating. We define scattering am- plitudes by Fig. 1. The scattering of electronic states inci dent from the normal metal define reflection and transmission am- plitudes randt. Those incident from the ferromagnet define 1These are indeed the directions of spin-orbit torque induce d by an applied current along x. [See Eq. (16)]System Source FLT DLT Magnitude 2D Rashba model jF/check× Magnetic bilayer jN/check/check FLT/lessorsimilarDLT (bulk magnetism) jF/check/checkFLT≫DLT Magnetic bilayer jN/check/check FLT/greaterorsimilarDLT (interface magnetism) jF/check/check FLT/greaterorsimilarDLT Ferromagnetic insulator jN/check× Topological insulator jF/check× TABLE I. Behaviors of in-plane-current-induced spin-orbi t torques for various systems. FLT and DLT refer to fieldlike torque and damp- inglike torque respectively, and /check(×) refers to their existence (ab- sence). Our analytic calculation allows for the separation of con- tributions from a pure charge current ( jN) and a pure spin current (jF) separately. A magnetic bilayer is a normal metal /ferromagnetic metal junction. Bulk magnetism originates from an exchange split- ting in the ferromagnetic bulk and interface magnetism orig inates from a spin-dependent scattering at the interface. A ferrom agnetic (topological) insulator is assumed to be attached to a norma l (fer- romagnetic) metal where the applied current flows. For all ca ses, interfacial spin-orbit coupling is present right at the int erface. We present the behavior of the two-dimensional (2D) Rashba mod el as a reference. Since the intrinsic contribution to spin-orbit torque is not taken into account in our theory, the dampinglike component in 2D Rashba model is not considered here. r′andt′. In the ferromagnet, there is an exchange splitting energy J>0. At the interface, we assume an interface po- tential HI=(/planckover2pi12/2me)ˆκδ(z), where ˆκis a 2×2 matrix in spin space. The delta-function-like potential describes physi cs on a length scale shorter than the mean free path. The e ffects of lat- tice mismatch, interface magnetism [44], and interfacial s pin- orbit coupling are examples. In our perturbative approach, we first ignore any interfacial spin-orbit coupling. After s olv- ing a boundary matching problem of the Schr¨ odinger equa- tion at z=0, we obtain the scattering matrices in terms of the interface potential ˆ κ[Eq. (6)]. We then add a Rashba-type interfacial spin-orbit coupling potential ∆ˆκ=hRˆσ·(k×z) to obtain a perturbative expansion of the scattering matric es, such as ˆ rk=ˆr0 k+∆ˆrk, where ˆ rkis the 2×2 reflection ma- trix in spin space2for momentum k, ˆr0 kis the unperturbed re- flection matrix (without interfacial spin-orbit coupling) , and ∆ˆrkis its correction due to interfacial spin-orbit coupling [Eq. (10)]. Then, the electronic eigenstates are computabl e analytically and play a crucial role in computing spin-orbi t torque in Sec. IV. We now mathematically write down electronic states as def- initions of scattering amplitudes illustrated in Fig. 1. Th e wave incident from the nonmagnet having the momentum k=(kx,ky,kz) and spinσis (1/√ V)eik·rξσwhere Vis the vol- ume of the system, r=(x,y,z) is the position vector, and ξσis the spinor for the spin σstate. The spin quantization axis we use here is the direction of magnetization in the ferromagne t 2Throughout this paper, we denote any matrix in spin space by a symbol with a hat, ˆ·.4 F NISOC (z=0) ISOC (z=0)incident wavereflected wave rtransmitted wave t z x F Nincident wave reflected wave r' transmitted wave t'z x(a) (b) FIG. 1. Scattering matrices at the interface of a nonmagnet( z<0, denoted by N)/ferromagnet ( z>0, denoted by F) structure. The interface is at z=0.randtrefer to reflection and transmission matrices when the electronic state is incident from the norm al metal layer. r′andt′are those when the electronic state is incident from the ferromagnetic layer. When the transverse mode is conserved , these are 2×2 matrices in spin space. In our model, the interfacial spin- orbit coupling (ISOC) is present only at z=0 being delta-function- like, although the thickness of the region drawn with finite t hickness for illustration. andσ=±1 corresponds to the minority /majority bands (with the higher/lower exchange energy). When the plane wave hits the interface at z=0, it scatters out. For simplicity, we as- sume translational symmetry over the xyplane. Then, the transverse momentum is conserved, thus the scattering matr i- ces are diagonal in transverse modes. The reflected wave has the momentum ¯k=(kx,ky,−kz). Here we denote the reflec- tion amplitude rσ′σ kby the amplitude of the scattering process (k,σ)N→(¯k,σ′)N, where the Roman subscript /superscript N refers to the nonmagnet. The scattering state in the nonmagn et is ψN kσ(z<0)=1√ Veik·rξσ+1√ Vei¯k·r/summationdisplay σ′rσ′σ kξσ′. (1a) The transmission matrix is defined in a similar way. Given the energy of the electronic state, the momentum kzin the fer- romagnet is different from that in the nonmagnet due to the exchange splitting J. We denote kσ zfor the corresponding mo- mentum for spin σband. For instance, if the kinetic energy is given by/planckover2pi12|k|2/2meand the exchange interaction is given byJˆσ·mwhere mis the unit vector along the magnetiza- tion,/planckover2pi12k2 z/2me=/planckover2pi12(k+ z)2/2me+J=/planckover2pi12(k− z)2/2me−Jdefines the relation between kzandkσ z. Then we denote the transmis- sion amplitude tσ′σ kby the amplitude of the scattering process (k,σ)N→(kσ′,σ′)F, where the roman subscript /superscript F refers to the ferromagnet. Therefore, the scattering stat e inthe ferromagnet is ψN kσ(z>0)=1√ V/summationdisplay σ′/radicaligg |kz| |kσ′ z|eikσ′·rtσ′σ kξσ′. (1b) Here the prefactor/radicalbig |kz|/|kσ′ z|is introduced, in order to make the conservation of electrical charge equivalent to the uni tar- ity of the scattering amplitudes [58]. The absolute value is introduced for cases where kσ′ zis imaginary so that the trans- mitted wave function is evanescent. Since evanescent waves do not contribute to unitarity, this convention is arbitrar y, but the choice should not a ffect the final expressions for physical quantities. Now we introduce a compact matrix notation. Since the scattering amplitudes have two indices ( σ′,σ), they are 2×2 matrices in spin space. We define the matrix ˆ rk=/summationtext σ′σξσ′rσ′σ kξ† σ, and similarly ˆtkwith tσ′σ k. The wave functions in this notation are ψN kσ(z<0)=eikxx+ikyy √ V(eikzzˆ1k+e−ikzzˆrk)ξσ, (2a) ψN kσ(z>0)=eikxx+ikyy √ V/radicalig |kz||ˆKz|−1eiˆKzzˆtkξσ, (2b) where ˆKz=k+ zu++k− zu−is a diagonal 2×2 matrix consist- ing of momenta in the ferromagnet for each spin band and uσ=ξσξ† σis the projection matrix to the spin σband. Equa- tion (2) defines the electronic state depicted in Fig. 1(a). ˆ1k is essentially the identity matrix, but slightly di fferent, as we explain in the next paragraph. In a similar way, the scatter- ing states derived from waves incident from the ferromagnet define ˆ r′ kandˆt′ kmatrices in Fig. 1(b). ψF kσ(z<0)=eikxx+ikyy √ V/radicalig |kz|−1e−ikzzˆt′ k/radicalig |ˆKz|ξσ, (3a) ψF kσ(z>0)=eikxx+ikyy √ V/radicalig |ˆKz|−1(e−iˆKzzˆ1′ k+eiˆKzzˆr′ k)/radicalig |ˆKz|ξσ. (3b) The matrices ˆ1kand ˆ1′ lare the projection matrices to the Hilbert space . These matrices are introduced to prevent un- physical states (not in the Hilbert space) from contributin g to any physical quantities that we compute. ˆ1kand ˆ1′ kare the identity matrices (the scalar 1) in the Hilbert space, but ar e zero, out of the Hilbert space. An electronic state is out of the Hilbert space when the incident wave is evanescent. For instance, an electronic state written as Eq. (2a) with an ima g- inary kzis not in the Hilbert space in the scattering theory, so it should not contribute to any physical quantity. Thus, we define ˆ1kby the following projection operator: ˆ1k=1 if kzis real, 0 if kzis imaginary .(4a) In this paper, we define ˆ rk=ˆtk=0 if the electronic state is out of the Hilbert space. Then, one can see that ˆ rk=ˆrkˆ1kand5 ˆtk=ˆtkˆ1k. In a similar way, ˆ1′ kis defined by, ˆ1′ k=1 if both k± zare real, u−if only k− zis real, 0 if both k± zare imaginary .(4b) ˆ1′ k=1 if both majority and minority bands are propagating, ˆ1′ k=u−(projection to the majority band) if only majority band is propagating, and ˆ1′ k=0 if both bands are evanescent. Similarly, ˆ r′ k=ˆr′ kˆ1′ kandˆt′ k=ˆt′ kˆ1′ k. Defining the projection matrices is crucial when we con- sider the continuity of the wave functions at z=0. If the Hilbert space is not properly considered, matching Eqs. (2a ) and (2b) at z=0 gives 1+ˆrk=/radicalig |kz||ˆKz|−1ˆtk. However, it does not hold when kzis imaginary so ˆ rk=ˆtk=0. When we project this equation to the Hilbert space by multiplying ˆ1k, ˆ1k+ˆrk=/radicalig |kz||ˆKz|−1ˆtkis the correct continuity condition. In a similar way, the continuity at z=0 of Eqs. (3a) and (3b) is given by ˆ1′ k+ˆr′ k=/radicalig |ˆKz||kz|−1ˆt′ k. Therefore, with the pro- jection matrices, we can write down a single equation which holds regardless of the reality of the perpendicular moment a. Another place where the projection matrices are crucial is the unitarity relation of the scattering amplitudes. It hol ds only for physical states in the Hilbert space. Therefore, th e unitarity relation in our notation can be subtle. We derive t he unitarity relation for the scattering amplitudes in Append ix B. B. Relation to the interface potential and introduction of extended scattering matrices In this section, we derive explicit expressions for scatter ing matrices for a given interface potential. This allows defin- ingextended scattering matrices (ˆ rk,ex,ˆtk,ex,ˆr′ k,ex,ˆt′ k,ex) which even satisfy the continuity relation without projection. T he ex- tended scattering matrices remove the singularity of the sc at- tering matrices,3which is a main obstacle of our perturbation theory. The explicit expressions for the scattering matrices are given by the interface potential. We start from the followin g interface potential at z=0 HI=/planckover2pi12 2meˆκδ(z), (5) where ˆκis a 2×2 matrix in spin space and has the dimension of the inverse of length. Solving the Schr¨ odinger equation gi ves the scattering matrices in terms of ˆ κ. The boundary condition for the delta-function-like potential is given by the deriv ative mismatching condition, ˆ κψz=0=∂zψz=+0−∂zψz=−0. After some 3Note that, for some momenta, the scattering matrices are zer o or propor- tional to u−, thus are not invertible matrices.algebra, we obtain the scattering matrices as ˆtk,ex=2ikz/radicalig |ˆKz||kz|−1(iˆKz+ikz−ˆκ)−1, (6a) ˆt′ k,ex=(iˆKz+ikz−ˆκ)−12iˆKz/radicalig |kz||ˆKz|−1, (6b) ˆrk,ex=(iˆKz+ikz−ˆκ)−1(ikz−iˆKz+ˆκ), (6c) ˆr′ k,ex=/radicalig |ˆKz|(iˆKz+ikz−ˆκ)−1(iˆKz−ikz+ˆκ)/radicalig |ˆKz|−1,(6d) where we call the matrices with the subscript ‘ex’ the extend ed matrices, discuss their meaning below. The expressions in Eq. (6) are nonzero even when the incident wave is evanes- cent. (For instance, ˆ rk|kz=iqz/nequal0.) In our convention, the scat- tering matrices are zero if the electronic state is evanesce nt because the scattering matrices capture the asymptotic beh av- ior of the scattering process. Therefore, the scattering ma tri- ces are obtained from the extended matrices by projecting th e latter to the Hilbert space. ˆrk=ˆrk,exˆ1k,ˆtk=ˆtk,exˆ1k, ˆr′ k=ˆr′ k,exˆ1′ k,ˆt′ k=ˆt′ k,exˆ1′ k. (7) Now the expressions satisfy ˆ rk=ˆrkˆ1kand similar relations for the others. The introduction of the extended matrices is purely mathe- matical. As far as physical quantities are concerned, the pa rts of the extended matrices out of the Hilbert space are com- pletely arbitrary and cannot a ffect any physical quantity. In this paper, there are three reasons why we choose the conven- tion in Eq. (6). First, it provides a natural way to write down analytic expressions valid for any momenta (even imaginary ). Equation (6) is the result from boundary matching at z=0 of the Schr¨ odinger equation whether or not all wave vectors ar e real. Second, it satisfies the generalized continuity relat ions 1+ˆrk,ex=/radicalig |kz||ˆKz|−1ˆtk,exand 1+ˆr′ k,ex=/radicalig |ˆKz||kz|−1ˆt′ k,ex even out of the Hilbert space. This supports the idea that Eq. (6) is the most natural way to define the extended matri- ces. Third and most importantly, the extended matrices have well-defined inverses. The singularity of ˆ1kandˆ1′ kfor some momenta complicates the development of a perturbation the- ory, and the extended matrices give one way to resolve this difficulty. Three remarks are in order. First, although we claim that Eq. (6) is the most natural form to extend out of the Hilbert space, this form depends on the normalization convention in Eqs. (2) and (3) for evanescent states. But we again em- phasize that the mathematical convention cannot a ffect cal- culation of physical quantities. Second, the four matrices in Eq. (6) are not independent since they are defined by a single matrix ˆκ. There are three relationships between the matrices. Two of them are the generalized continuity relations presen ted above. Another relationship that is derived from Eq. (6) is (1+ˆrk,ex)k−1 z=/radicalig |ˆKz|−1(1+ˆr′ k,ex)/radicalig |ˆKz|ˆK−1 z. Third, if ˆ κis a spin-conserving Hamiltonian, ˆ κand ˆKzcommute with each other. For instance,/radicalig |ˆK|zand/radicalig |ˆK|−1zin ˆr′ k,excancel so that the expression becomes simpler. The last constraint become s6 simpler (1+ˆrk,ex)k−1 z=(1+ˆr′ k,ex)ˆK−1 z. These features are useful for simplifying unperturbed contributions, which we consi der spin conserving in the next section. C. Perturbation of scattering matrices To focus on the effects of interfacial spin-orbit coupling, we use a perturbative approach. Let the interface potential be ˆκ=ˆκ0+hRˆσ·(k×z), (8) where the first term is the unperturbed interface potential and the second term is the interface Rashba interaction only present at z=0. Here ˆσis the vector of the spin Pauli ma- trices, zis the unit vector along the interface normal direction z, and hRis the dimensionless Rashba parameter. We treat hR perturbatively. Ref. [37] shows that the numerically compu ted spin-orbit torques are mostly linear in hR, supporting this per- turbative approach. We also assume that ˆ κ0is spin conserv- ing in the sense that it is diagonal in spin space. Therefore, [ˆκ0,uσ]=[ˆκ0,ˆKz]=0. Examples of spin-conserving poten- tials are spin-independent barriers and interface exchang e po- tentials in the form of umˆσ·m. One interpretation of interface magnetism ( um) is proximity-induced magnetism [44], which is discussed in Sec. VI C in more detail. The success of the conventional magnetoelectric circuit theory [45, 46] impl ies that assuming a spin conserving interface potential is reas on- able. To develop a perturbation theory, we denote unperturbed scattering matrices by a superscript 0. For instance from Eq. (6), ˆt0 k,ex=2ikz/radicalig |ˆKz||kz|−1(iˆKz+ikz−ˆκ0)−1. It is straight- forward after some algebra to show that the exact scattering matrix in the presence of hRis related to the unperturbed scat- tering matrix as follows; ( ˆt0 k,ex)−1ˆtk,ex=1+(hR/2ikz) ˆσ·(k× z)(1+ˆrk,ex).4By multiplying ˆt0 k,exon both sides, ˆtk,ex=ˆt0 k,ex+hR 2ikzˆt0 k,exˆσ·(k×z)/radicalig |kz||ˆKz|−1ˆtk,ex,(9) which allows a perturbative expansion with respect to hRin an iterative way. For instance, replacing ˆtk,exin the right-hand side by ˆt0 k,exgives the first order perturbation result for ˆtk,ex. From ˆtk,ex, the three constraints mentioned in the previous sec- tion give the rest of the extended matrices immediately. The n, projecting to the Hilbert space by multiplying by ˆ1kand ˆ1′ k 4Note that the invertibility of extended matrices is crucial for deducing this.gives our central result for the scattering matrices. ˆtk=ˆt0 k+hR 2ikzˆt0 k,exˆσ·(k×z)(ˆ1k+ˆr0 k), (10a) ˆt′ k=ˆt′0 k+hR 2ikz(1+ˆr0 k,ex) ˆσ·(k×z)ˆt′0 k, (10b) ˆrk=ˆr0 k+hR 2ikz(1+ˆr0 k,ex) ˆσ·(k×z)(ˆ1k+ˆr0 k), (10c) ˆr′ k=ˆr′0 k+hR 2ikzˆt0 k,exˆσ·(k×z)ˆt′0 k. (10d) With Eq. (10) in combination with Eqs. (2) and (3), one can write down the electronic wave functions for nonzero hR. Then, physical quantities can be written in terms of unper- turbed scattering matrices, as we present in the next sectio n and Appendix A. These expressions in terms of reflection and transmission coefficients can be used for general interfaces with spin-nonconserving Hamiltonians of the Rashba type.5 By computing the unperturbed scattering matrices with first - principles calculations or toy models, our theory enables c om- puting interfacial spin-orbit coupling contributions for various types of interfaces. This approach is similar to the way that one computes the spin mixing conductance [45, 46] in mag- netoelectric circuit theory. Three remarks are in order. First, one may notice that Eq. (10) includes 1 /2ikzfactors only, but there is no 1 /2iˆKz factor in ˆ r′ kandˆt′ k. The absence of 1 /2iˆKzseems asymmet- ric since we consider all the waves incident from the normal metal and the ferromagnet. This is simply because we used the constraint (1+ˆrk,ex)0k−1 z=(1+ˆr′ k,ex)0ˆK−1 zto convert all 1/2iˆKzto 1/2ikzfor simplicity. Therefore, our result does not break the symmetry in the expressions. Second, in the pres- ence of interfacial spin-orbit coupling, a bound state that does not correspond to any unperturbed state could arise. In Ap- pendix C, we demonstrate that a bound state is not present in the perturbative regime that we consider here. Third, the presence of the extended matrices in Eq. (10) is purely math- ematical. This is similar to the Born approximation in scatt er- ing theory. In the Born approximation, the mathematical ex- pression of scattering states contains virtual transition s which are not allowed due to the conservation of energy. However, such a treatment allows us to calculate the scattering state s in a perturbative regime. Similarly, the presence of the ex- tended matrices does not mean a physical transition but a mathematical artifact of the perturbative approach. Physi cal quantities do not depends on the extended space. The relatio n (1+ˆr0 k,ex)k−1 z=(1+ˆr′0 k,ex)ˆK−1 zis helpful for this purpose. For instance, when we need to project ˆ rk,exbyˆ1′ kto compute a physical quantity, the relation allows expressing ˆ rk,exin terms of ˆr′ k,ex, so the projection by ˆ1′ kis given by the natural relation ˆr′ k,exˆ1′ k=ˆr′ k. 5Even if the perturbing Hamiltonian is not in the Rashba type, our approach is still valid when one replaces hRσ·(k×z) by the perturbing Hamiltonian.7 IV . EXPRESSION OF SPIN-ORBIT TORQUE We consider a situation that an external current is applied. In the absence of spin-orbit coupling, angular momentum con - servation suggests that the total angular momentum injecte d into the ferromagnet is equal to the spin current at the inter - face. However, in the presence of interfacial spin-orbit co u- pling, the spin current at z=+0 is not equal to the spin-orbit torque, because some of the angular momentum is transferred to the lattice. Thus, it requires a careful separation of the an- gular momentum flow [41, 42] (See Fig. 2). To develop an expression for the torque, we first ignore magnetism at the interface and restore it later. Then, the to tal spin-orbit torque is computed by the spin current right at th e interface in the ferromagnet ,z=+0. For illustration, we first compute the spin current at z=−0 and how much angular mo- mentum changes at the interface due to interfacial spin-orb it coupling. We compute the charge and spin current density at z=−0 by ˆjz(r)=−eTrkRe[ρδ(rop−r)vz], (11) where Re[ A]=(A+A†)/2 refers to the real part of the given matrix A, Tr kis the partial trace over konly (the re- sult is 2×2 matrix in spin space), ˆjz=(je z+ˆσ·js z)/2,js z is the spin current flowing along zwith the direction of the vector denoting the direction of spin, je zis the charge cur- rent along z,vz=(/planckover2pi1/mei)∂zis the velocity operator, ρ=/summationtext kσ′σ,a=N/Ffa k,σ′,σ|kσ′;a∝an}b∇acket∇i}ht∝an}b∇acketle{tkσ;a|is the density matrix, fN/F k,σ′,σ is the reduced density matrix for a given k,ropis the po- sition operator, and ris a c-number indicating the position at which the current density is evaluated. Here |kσ; N/F∝an}b∇acket∇i}ht refers to a scattering state incident from the nonmagnet (fe r- romagnet) that has momentum kin the nonmagnet and spin σ, that is,ψa kσ(r)=∝an}b∇acketle{tr|kσ;a∝an}b∇acket∇i}ht. The current is written by jz(r)=(−e/2)/summationtext kσ′σ,a=N/Ffa kσ′σ∝an}b∇acketle{tkσ;a|{vz,δ(rop−r)}ˆσ|kσ′;a∝an}b∇acket∇i}ht, and similarly for the charge current. Since we know the elec- tronic wave functions Eqs. (2a) and (3a), we can calculate th is analytically. The delta function enables computing the ma- trix element without performing any integration. After som e algebra, ˆjz|z=−0=−eL hV/summationdisplay k⊥/integraldisplay dE(ˆfN kˆ1k−ˆrkˆfN kˆr† k−ˆt′ kˆfF kˆt′† kˆ1k).(12) Here, ˆfN/F k=/summationtext σ′σξσ′fN/F k,σ′,σξ† σis the matrix representation of the reduced density matrix, Lis the thickness of the system along zdirection, h=2π/planckover2pi1is the Planck constant, the sum- mation over k⊥refers to the summation over all transverse momenta, and Eis the energy of the electron. In order to compute the contribution from ˆfF k, we assume that ˆfF kis diag- onal inσ, so that the electrons in the ferromagnet has no spin component perpendicular to the magnetization, as assumed i n the magnetoelectric circuit theory. In order to convert/summationtext kto/summationtext k⊥/integraltext dE, we use/summationtext k=/summationtext k⊥/summationtext kzand/summationtext kz=(L/2π)/integraltext dkz= (meL/2π/planckover2pi12)/integraltext dE/kz. We use ˆ1kˆfN kˆ1=ˆfN kandˆ1′ kˆfF kˆ1′=ˆfF kby their definition.6These relations play a role in projecting the extended matrices in Eq. (10) when computing physical quantities. Equation (12) has the same form as the core result of the conventional magnetoelectric circuit theory [45, 46]. An evanescent contribution from a wave incident from the ferro - magnet with σ=−1 cannot contribute to ˆjz|z=−0(see addi- tional ˆ1kfactor in the last term). But in Appendix A, we show that an evanescent contribution plays an important role in a n in-plane current flow in the presence of interfacial spin-orbit coupling. Applying an external field shifts the distribution function . In linear response regime, we approximate the Fermi sur- face contribution by defining chemical potentials ∆ˆfN/F k= e∆ˆµN/Fδ(E−EF) where EFis the Fermi level. The delta func- tion allows us to perform integration over Ein Eq. (12) by taking E=EF. In the presence of an electrical (charge) cur- rent along xdirection, it shifts the electron distribution func- tion with a finite momentum relaxation time τNin the non- magnet,τ↑/↓in the ferromagnet. Here ↑and↓refer to the majority (σ=−1) and minority ( σ=1) bands. That is, ∆ˆµN=(Ex/me)/planckover2pi1kxτNˆ1kand∆ˆµF=(Ex/me)/planckover2pi1kxˆτFˆ1′ kwhere Exis the applied electric field, and ˆ τF=τ↓u++τ↑u−is a 2×2 matrix of the relaxation times in the ferromagnet. Then the nonequilibrium current is ˆjz|z=−0=−e2L hV/summationdisplay k⊥/bracketleftig ∆ˆµNˆ1k−ˆrk∆ˆµNˆr† k−ˆt′ k∆µFˆt′† kˆ1k/bracketrightig E=EF =e2LEx 2πmeV/summationdisplay k⊥/bracketleftig kxˆrkτNˆr† k+kxˆt′ kˆτFˆt′† kˆ1k/bracketrightig E=EF.(13) Since the expression is given by quantities at the Fermi leve l, we from now on omit [ ···]|E=EFand implicitly assume that the nonequilibrium current is evaluated at the Fermi level. The simple formula Eq. (13) gives the nonequilibrum spin and charge currents right at the interface in the nonmagnet. Wit h- out spin-orbit coupling, the system has the rotational symm e- try around z, so ˆrkis an even function of kxandky. Therefore, Eq. (13) vanishes identically after summing up over all tran s- verse modes. Thus, we reproduce the well-known result that there is no conventional spin-transfer torque induced by an in-plane charge current. However, the existence of interfacial spin-orbit coupling changes the situation drastically. Since Eq. (10) includes a term which is odd in k, Eq. (13) gives rise to a finite contribu- tion. Putting Eq. (10) into Eq. (13), we obtain ˆjz|z=−0=−hRe2LEx 4πmeVIm/summationdisplay k⊥k2 ⊥ kz(ˆ1k+ˆr0 k) ˆσyˆt′0 k(ˆτF−τN)ˆt′0† k, (14) where Im[ A]=(A−A†)/2irefers to the imaginary part of the given matrix Aandk⊥=/radicalig k2x+k2y. Here we use the unitarity 6There is no incident electron out of the Hilbert space.8 relation ˆ r0 kˆr0† k+ˆt′0 kˆt′0† kˆ1k=ˆ1kwhich is derived in Appendix B. In addition, we perform an angular average of the summand: For any unit vector u,/summationtext k⊥(k·u) ˆσ·(k×z)=/summationtext k⊥(k2 ⊥/2) ˆσ·(u× z) after taking average of contributions from all directions of k⊥=kxx+kyywith the same magnitude. Taking u=xyields ˆσyin Eq. (14). Now we compute the discontinuity at the interface. The derivative mismatch condition hRˆσ·(k×z)ψz=0=∂zψz=+0− ∂zψz=−0, allows us to compute ∆ˆjz≡ˆjz|z=+0−ˆjz|z=−0in terms of the wave function at z=0. From Eq. (11), jz|z=+0−jz|z=−0= −hR(e/planckover2pi1/me) Tr kIm[ρδ(rop−r) ˆσˆσ·(k×z)], and similarly for the charge current. Here ris the position at the interface, so it does not have a z-component. Since the expression is already proportional to hR, we can replace ˆ rkin the wave function by ˆr0 k. After some algebra, ∆ˆjz=hRe2LEx 4πmeVIm/summationdisplay k⊥k2 ⊥ |kz|ˆσy[(ˆ1k+ˆr0 k)τN(ˆ1k+ˆr0† k)+ˆτ′0 kˆτFˆτ′0† k]. (15) The physical meaning of Eq. (15) is the angular momentum absorption or emission at the interface due to spin-orbit co u- pling. Therefore, Eq. (15) amounts to how much angular mo- mentum is transferred from the lattice at the interface. The expression for the spin-orbit torque is given by the Pauli components of ˆjz|z=+0perpendicular to the magnetiza- tion m, and ˆjz|z=+0is given by the sum of Eqs. (14) and (15). Explicitly, TR=−(/planckover2pi1V/2eL) Trσ[ ˆσ⊥ˆjz|z=+0], where ˆσ⊥=ˆσ−m( ˆσ·m) is the transverse part of the Pauli ma- trix vector to mand Trσis the trace over the 2 ×2 spin space. After some algebra, TR=Im[TR]m×(y×m)+Re[TR]m×y, (16a) TR=TN R+TF R, (16b) TN R=hR/planckover2pi1eExτ 8πme/summationdisplay k⊥k2 ⊥ kz(1−r↑∗ kr↓ k)(r↑ k−r↓∗ k) (16c) TF R=−hR/planckover2pi1eEx 8πme/summationdisplay k⊥k2 ⊥ kz(r↓ k|t′↑ k|2τ↑−r↑∗ k|t′↓ k|2τ↓) +hR/planckover2pi1eEx 8πme/summationdisplay k⊥,k2z<0k2 ⊥ |kz|(|t′↑ k|2τ↑−|t′↓ k|2τ↓), (16d) where we expanded ˆ r0 k=r↓ ku++r↑ ku−andˆt′0 k=t′↓ ku++t′↑ ku−as done in magnetoelectric circuit theory. Here ↑is assigned for σ=−1 sinceσ=−1 is majority in our model. Equation (16) is the central result of this paper. The terms in Eq. (16) are t he dampinglike spin-orbit torque and fieldlike spin-orbit tor que respectively. TN Ris the contribution from a current flowing in the non- magnet. By the Drude model, the applied current is written asnNe2Exτ/mewhere nNis the electron density in the non- magnet. Therefore, τmultiplied by Exis proportional to the applied current. Similarly, TF Ris the contribution from the current flowing in the ferromagnet. Especially, the second term inTF Ris an evanescent contribution in the nonmagnet (seek2 z<0). Although the wave function in the nonmagnet is evanescent, incident waves from the ferromagnet can be prop - agating, giving rise to a finite amount of spin-orbit torque.F Nincoming angular momentum (1)to lattice (2)to interface magnetization (4)to bulk magnetization (3) z x Angular momentum conservation (boundary condition) (1)-(2) = (3)+(4) FIG. 2. (color online) Illustration of conservation of angu lar momen- tum at the interface. The incoming angular momentum (1) spli ts into three drains; to the lattice by interfacial spin-orbit coup ling (2), to the bulk magnetism (3), and to the interface magnetism (4). The c onser- vation of angular momentum implies that (1) =(2)+(3)+(4), which is captured by the boundary condition of the Schr¨ odinger eq uation. Equation (16) is computed by (1) −(2), which is, by the conservation of angular momentum, (3) +(4), the total spin-orbit torque to magne- tization. Such a contribution is crucial for topological insulators w here the nonmagnet is insulating. We also demonstrate in Sec. V A that this contribution can also be the dominant contributio n in magnetic bilayers. In the case of torques due to the spin Hall e ffect in the interior of the layer, the spin Hall current proportional to θSHExcreates a spin current into the ferromagnet, where θSH is the spin Hall angle. For this mechanism, the real part of the spin mixing conductance G↑↓contributes to the damping- like torque and the imaginary part contributes to the fieldli ke torque [45, 46]. Comparing this result to Eq. (16) suggests thatθSHExG↑↓in the spin Hall effect contribution corresponds toiTRin the interfacial spin-orbit coupling contribution (up to a constant factor). Now we restore the possibility of interface magnetism at the interface and argue that Eq. (16) is unchanged. When we add interfacial magnetism umˆσ·m, the boundary condition changes to umˆσ·mψz=0+hRˆσ·(k×z)ψz=0=∂zψz=+0−∂zψz=−0. This is nothing but the angular momentum conservation re- lation at the interface. The terms in the left-hand side cor- respond to the angular momenta transferred from the inter- face magnetization and the lattice. The terms in the right- hand side correspond to the incoming and outgoing angu- lar momenta. The first term in the left-hand side and the first term in the right-hand side correspond to the (negative of) spin-transfer torque to the interfacial magnetism and t he spin-transfer torque to the bulk. Therefore, the total spin - transfer torque is computed by the sum of the interfacial spi n- transfer torque and the bulk spin-transfer torque, which co r- responds to ∂zψz=+0−umˆσ·mψz=0. This is the same as ∂zψz=−0+hRˆσ·(k×z)ψz=0, which is exactly what we express in Eq. (16). Conservation of angular momentum is summarized in Fig. 2.9 V . SPIN-ORBIT TORQUE FOR V ARIOUS TYPES OF INTERFACES In this section, we use Eq. (16) to compute spin-orbit torques for various types of interfaces. Examples includes magnetic bilayers, ferromagnetic insulators, and topolog ical insulators as presented in Table I. A. Magnetic bilayers - Bulk magnetism We start from the following unperturbed Hamiltonian. H=−/planckover2pi12 2me∇2+Jˆσ·mΘ(z), (17) whereΘ(z) is the Heaviside step function representing that the bulk ferromagnetism Jis present only in z>0. The potential energy profile is presented in Fig. 3. Here we assume EF>0, otherwise the normal metal is insulating. Since we have no interface potential other than spin-orbit coupling, we use Eq. (6) by putting ˆ κ=0. r↑/↓ k=kz−k∓ z kz+k∓zifkzis real, 0 if kzis imaginary ,(18a) t′↑/↓ k=2i/radicalbigk∓z|kz| ikz+ik∓zifk∓ zis real, 0 if k∓ zis imaginary ,(18b) When we define k2 F=2meEF//planckover2pi12and∆2=2meJ//planckover2pi12, each momentum has the following relations: k2 ⊥+k2 z=k2 Fandk2 ⊥+ (k± z)2±∆2=k2 F. There always are evanescent contributions from any scattered wave regardless of EFandJ, since k⊥can be arbitrarily close to kF. The total spin-orbit torque is given by the sum of Eq. (16c) and Eq. (16d). Putting Eq. (18a) into Eq. (16c) gives spin-or bit F NPotential profile z z=02JMinority band Majority bandEF 0 FIG. 3. (color online) The potential profile (blue lines) for the model Eq. (17). The energy profile is spin-independent for z<0 while it has a 2Jgap between the majority and minority bands for z>0. Here the red line denotes the Fermi level. The figure shows a typical si tuation where EF>Jso that the spin polarization at the ferromagnet is incomplete, but the theory covers the whole range of positiv eEF.torque generated by a current flowing in the normal metal. Re[TN R]=−hR/planckover2pi1eExτ 2πme/summationdisplay k2 ⊥<k2 Fk2 ⊥kz(k− z)2−|k+ z|2 (kz+k−z)2|kz+k+z|2,(19a) Im[TN R]=−hR/planckover2pi1eExτ 2πme/summationdisplay k2 ⊥<k2 Fk2 ⊥kz2k− zIm[k+ z] (kz+k−z)2|kz+k+z|2,(19b) where/summationtext k2 ⊥<k2 Frefers to the summation over all transverse mode satisfying k2 ⊥<k2 Fthus making kzreal. Here, kzand k− zare real and positive. The evanescent contribution Im[ k+ z] is crucial for the dampinglike component Im[ TN R]. We re- mark that the real and imaginary parts of Eq. (19) have the same sign. This implies that the dampinglike and the fieldlik e component of Eq. (16c) in this model have the same sign. Putting Eq. (18b) into Eq. (16d) gives spin-orbit torque gen - erated by a current flowing in the ferromagnet. Although the situation is slightly more complicated than above, the expl icit expressions are similar: Re[TF R]=−hR/planckover2pi1eExτ↑ 2πme/summationdisplay k2 ⊥<k2 Fk2 ⊥kzk2 z−|k+ z|2 (kz+k−z)2|kz+k+z|2, −hR/planckover2pi1eExτ↓ 2πme/summationdisplay k2 ⊥+∆2<k2 Fk2 ⊥k+ zk− z−kz (kz+k+z)2(kz+k−z) +hR/planckover2pi1eExτ↑ 2πme/summationdisplay k2 F<k2 ⊥<k2 F+∆2k2 ⊥k− z ∆2, (20a) Im[TF R]=hR/planckover2pi1eExτ↑ 2πme/summationdisplay k2 ⊥<k2 Fk2 ⊥k− z2kzIm[k+ z] (kz+k−z)2|kz+k+z|2.(20b) Here, terms proportional to τ↑andτ↓are contributions from majority and minority electron flows respectively. We remar k that evanescent modes are crucial for the existence of damp- inglike components. The first two terms of the real part are majority and minority counterparts of Re[ TN R]. The imagi- nary part has also the same form as Im[ TN R], but only majority electrons contribute because minority electrons do not mak e any transition to an evanescent state in this model. The last term in the real part has no counterpart in Eq. (19) since it originates from the imbalance between majority and minorit y states due to a nonzero J. This term originates from transition of majority electrons in the ferromagnet to evanescent stat es in the normal metal. We show below that this contribution is very large and can dominate the other contributions mak- ing the consideration of a finite Jand the resulting evanescent modes very important. Converting the summations in Eqs. (19) and (20) to integra- tions allows us to compute the spin-orbit torque as a functio n ofEF/J. To do this, we convert/summationtext k2 ⊥<k2 Fto (A/4π)/integraltextk2 F 0d(k⊥)2, where A=V/Lis the are of the interface, and similarly for the other summations. To express all momenta in terms of k⊥, we usek2 z=k2 F−k2 ⊥and ( k± z)2=k2 F∓∆2−k2 ⊥. There are two regimes. For EF≤J,k+ zis imaginary for the whole interval of the integration 0 <k2 ⊥<k2 F. On the other hand, for EF>J, it is necessary to consider the intervals 0 <k2 ⊥<k2 F−∆2and10 1 2 3 4 5 6 EF/J20 15 10 5 0 -10 -15SOT (normalized)From FM Majority From NMFLT - conventional MinorityDLTFLT - majority to evanescent FIG. 4. (color online) Spin-orbit torque (SOT) to the bulk ma gnetism in magnetic bilayer described by Eq. (17). The red lines are c ontri- butions from a current flowing in the normal metal, and the oth er lines are those from a current flowing in the ferromagnet. Amo ng these, the blue lines are contributions from majority elect rons flow and the cyan line represents contributions from minority el ectron flow. The dashed and dot-dashed lines are the real part, thus r epre- senting fieldlike torque (FLT), while the solid lines are the imaginary part, thus representing dampinglike torque (DLT). The dot- dashed line represents the third term in Re[ TF R] in Eq. (20), which originates from a nonzero value of Jand transitions to resulting the evanescent states. The spin-orbit torque from electrons in normal meta l, majority electrons, and minority electrons are divided by 2 hR/planckover2pi1eExτAk3 F/πme, 2hR/planckover2pi1eExτ↑Ak3 F/πme, and 2 hR/planckover2pi1eExτ↓Ak3 F/πme, respectively, and the results are dimensionless. Thus, the total spin-orbit torq ue is given by the weighted sum of all the contributions with the weighti ng fac- tors (τ,τ↑,τ↓). k2 F−∆2<k2 ⊥<k2 Fseparately, since k+ zis imaginary for the for- mer and is real for the latter. Thus, it has di fferent properties when taking the absolute value. In both cases, k− zis always real, and kzis imaginary only when k2 F<k2 ⊥<k2 F+∆2. The integration can be performed fully analytically, however, we present only numerical results due to complexity of the ex- pressions. Figure 4 presents (normalized) contributions of spin-orbi t torques as a function of EF/J. The values are divided by fac- tors proportional to Exτ,Exτ↑, and Exτ↓for electrons in nor- mal metal, majority electrons in the ferromagnet, and minor ity electrons in the ferromagnet, respectively. In most experi men- tal situations, people apply an electrical current mainly i n the normal metal. Thus we discuss the spin-orbit torque origina t- ing from a current in the normal metal first and consider the effects of a current leaking to the ferromagnet. Red lines in Fig. 4 represent fieldlike (dashed line) and dampinglike (solid line) components of spin-orbit torque i n- duced by a current flowing in the normal metal. Unlike the two-dimensional Rashba model [11, 12], the damping- like component has the same order of magnitude as the field- like component and even larger for wide range of EF. As we remark above, each component has the same sign. If the current leaking to the ferromagnet is su fficiently small or the ferromagnet is more resistive than the normal metal, we can consider the current to flow mainly in the normal metal. In this case, the dampinglike torque is comparable or even larger than fieldlike torque implying that experimental res ults for the dampinglike spin-orbit torque due to the spin Hall ef - fect [8–10, 59–61] should be carefully analyzed due to thepossibility of the contributions from interfacial spin-or bit cou- pling [62, 63]. Now we consider the contributions from the current flow- ing in the ferromagnet. Neglecting the unconventional term from a finite J(dot-dashed line in Fig. 4), the dampinglike component (solid blue line) has the opposite sign of the nor- mal metal contribution because the dampinglike component is generated by angular momentum carried by a spin current, which has opposite directions for these two contributions. On the other hand, since the fieldlike component originates fro m the current-induced spin-orbit field, the contributions fr om currents in both sides can act additively. Thus, the dashed blue line and the cyan line have the same sign as the dashed red line in wide range of EF/J.7 One remarkable result of this calculation is that the third term in Re[TF R] in Eq. (20) is larger than the other contribu- tions. It is around five times larger than the dampinglike com - ponents (solid lines) at EF≈J(not shown). The origin of this term is the finite magnitude of Jand the resulting evanescent states. Ifτ↑has the same order of magnitude as τ, this term can be the dominant contribution, illustrating the importa nce of accounting for the di fferent Fermi surface. If the current flowing in the ferromagnet is at least comparable to that in the normal metal, the total spin torque is approximated by th e third term in Re[TF R]. The summation is performed in a simple analytic form: TR≈hR/planckover2pi1eExτ↑ 2πme/summationdisplay k2 F<k2 ⊥<k2 F+∆2k2 ⊥k− z ∆2=hReExτ↑A∆ 15πh(5EF+2J). (21) Since the number of electrons in the majority band remains finite when EF→0, the contribution does not vanish in this limit, unlike the other contributions in Fig. 4.8 B. Magnetic bilayers - Interface magnetism We start from the following unperturbed Hamiltonian. H=−/planckover2pi12 2me∇2+/planckover2pi12 2me(u0+umˆσ·m)δ(z), (22) where u0and umare parameters for spin-independent and spin-dependent interface potentials. The former refers to an interface barrier and the latter refers to an interface mag- netism. The interface magnetism is a possible simple model for the proximity-induced magnetism at the interface [44]. Since all the Fermi wave vectors are the same, there are no evanescent waves. The potential energy profile is presented 7This argument is only valid when the propagating contributi ons are domi- nant. If EFis close to J, this argument is not guaranteed as shown in Fig. 4 (blue and cyan dashed lines). We consider a model not contain ing any evanescent mode in Sec. V B, giving a clearer example of this c onclusion. 8In Fig. 4, the dot-dashed line seems divergent when EF→0. However, this is a result of the normalizing factor ∼k3 F. Figure 4 does not present the dependence on EF, but EF/J.11 F NPotential profile z z=0u0EF 0um um FIG. 5. (color online) The potential profile (blue lines) for the model Eq. (22). Here the spin-independent potential u0and spin-dependent potential umare present at z=0. The red line denotes the Fermi level. In this figure, a delta functions is represented as a square fu nction with a finite height and a finite width for illustration. in Fig. 5. Each side around z=0 is symmetric, so there is no explicit difference between the normal metal and the fer- romagnet. Here we model the ferromagnet by assigning dif- ferentτ↑andτ↓values and assuming that angular momentum right at z=+0 are absorbed into the ferromagnet and make a contribution to spin-orbit torque. In other words, we impli c- itly assume a vanishingly small magnitude of magnetism in the ferromagnetic bulk. We use Eq. (6) by putting ˆKz=kzand ˆκ=u0+umˆσ·m, to obtain r↑↓ k=u0∓um 2ikz−(u0∓um),t′↑↓ k=1+r↑↓ k. (23) Putting this into Eq. (16), Re[TR]=hR/planckover2pi1eEx πme/summationdisplay k2 ⊥<k2 F2u0um(τ+τe F)+(u2 0+u2 m)τs F Dk(u0,um),(24a) Im[TR]=−hR/planckover2pi1eEx πme/summationdisplay k2 ⊥<k2 Fkz2u0(τ−τe F)−2umτs F Dk(u0,um), (24b) where Dk(u0,um)=[4k2 z+(u0−um)2][4k2 z+(u0+um)2]/k2 ⊥kz. τe/s F=(τ↑±τ↓)/2 amounts to charge /spin current flowing in the ferromagnet. We observe the following features. First, considering only charge current contributions, the fieldlike component is ad di- tive [τ+τe Fin Eq. (24a)] and the dampinglike component is subtractive [ τ−τe Fin Eq. (24b)]. This observation is consis- tent with the discussion in Sec. V A. We can observe it here more clearly since there are no evanescent contributions in this model. Second, the charge current contributions are all zer o when u0=0. This means that, if an applied current is mostly flowing in the normal metal, interfacial spin-orbit torque i n- duced by the current is proportional to the spin-independent barrier at the interface. Third, the spin-orbit torque does not vanish even when there is no magnetism; um=0. The spin- orbit torque contribution without magnetism is attributed to our assumption that there is a vanishingly small magnetism FLT Charge current Spin currentDLT 0.0 1.0 u0/kFum/kF 0.00.0 2.02.0 -2.0 FIG. 6. Magnitude of spin-orbit torque in the presence of int erface magnetism. We plot spin-orbit torque as a function of u0/kFand um/kF. To compare magnitude clearly, we plot absolute values, dis - carding the signs. The upper two panels represent fieldlike c om- ponents [Eq. (24a)] and the lower two panels represent dampi ng- like components [Eq. (24b)]. The left two panels represent c harge- current-induced contributions (proportional to τ±τe F). The right two panels represent spin-current-induced contributions (pr oportional to τs F). The values are divided by hR/planckover2pi1eExA/24πmekFfor all panels, and additionally divided by τ±τe Ffor the charge-current-induced con- tributions (+for FLT and−for DLT) and τs Ffor the spin-current- induced contributions. The resulting values are dimension less. The black lines near u0=0 orum=0 are regions where spin-orbit torque is not computed due to numerical instability. in the ferromagnetic bulk. When angular momentum is trans- ferred from the lattice through interfacial spin-orbit cou pling, a finite amount of spin current at z=+0 is generated. In our approach, we assume, even when exchange in the bulk is not explicitly included, that dephasing transfers the spin ang ular momentum from the spin current to the bulk magnetism, giv- ing rise to a torque. The size of the spin-orbit torque to the bulk is then determined by the conservation of angular mo- mentum, so that the total spin torque absorbed into the fer- romagnet is determined by the spin current at z= +0, no matter how small the bulk exchange coupling strength is in the model. In conclusion, the contributions proportional t oum are spin-orbit torque to the interface magnetism um, while the other contributions are spin-orbit torque to the ferromagn etic bulk. To compare the relative magnitude of the fieldlike and12 F NPotential profile z z=02J UMinority band Majority bandEF 0 FIG. 7. (color online) The potential profile (blue lines) for the model Eq. (25). In the ferromagnet, there is a spin-independent ba rrier U that makes the ferromagnet insulating. The Fermi level (red line) is less than U−J, thus both majority and minority bands are evanescent. dampinglike components, we convert the summations in Eq. (24) to integrations as in Sec. V A. We plot the absolute values of each contribution in Fig. 6. Normalization factor s, ∝Ex(τ±τe F) or∝Exτs F, are introduced. The total spin-orbit torque is given by the weighted sum of each panel. We ob- serve that the fieldlike and dampinglike components are on the similar order of magnitude, but the fieldlike component i s in general larger than dampinglike component in wide range of parameters. The spin-current-induced fieldlike contrib ution is the largest. The same model has been studied by Haney et al. [37] without a perturbative approach. They show that the fieldlike component is in general larger than the dampinglik e component, which is consistent with our approach. C. Ferromagnetic insulators We start from the following unperturbed Hamiltonian. H=−/planckover2pi12 2me∇2+(U+Jˆσ·m)Θ(z), (25) where Uis a spin-independent potential that makes the fer- romagnet ( z>0) insulating. Thus, the Fermi level should be below U−J. Without loss of generality, we can assume U>J, otherwise there are no occupied electronic states. The potential energy profile is presented in Fig. 7. Since the ferromagnet is insulating, k± zare all imaginary. We define q± z=−ik± z, which is real and positive. The reflec- tion amplitudes are given by the same formula as in Sec. V A. r↑↓ k=ikz+q∓ z ikz−q∓z. (26) Since all the other momenta are imaginary, there is no contri - bution fromTF R. Thus, we only need to compute Eq. (16c): TR=hR/planckover2pi13eExτ 2πm2eJ U2−J2/summationdisplay k2 ⊥<k2 Fk2 ⊥kz. (27) SinceTRis real, only fieldlike spin-orbit torques can survive. SOT (normalized) 0.2 0
   0.8 1.0 J/U0.080.100.120.14 0.06 0.04 0.02 FIG. 8. Fieldlike spin-orbit torque in ferromagnetic insul ators di- vided by 8 hR√2meeExτA/15π2/planckover2pi12×[EF/(U−J)]5/2/U3/2. The result is dimensionless. We now perform the summation in the same way described in Sec. V A. After some algebra, TR=hR8√2meeExτA 15π2/planckover2pi12JE5/2 F U2−J2. (28) The spin-orbit torque vanishes at EF=0 since there are no occupied states. It increases as EFincreases. Equation (28) has a singularity at J=U, but it does not diverges at J= Ubecause as Japproaches U,EFapproaches to zero since EF<U−J. The maximum of Eq. (28) occurs at EF=U−J; TR∝J(U−J)5/2/(U2−J2), which has a maximum at J=U/3. Therefore, TR≤hR8√meeExτA 45√ 3π2/planckover2pi12U3/2, (29) which is finite. To see the numerical behavior of TR, we parameterize EF, U, and Jwith two parameters. Since U>Jand 0<EF≤ U−J, we put J=αUandEF=β(U−J) with dimensionless parameters αandβsatisfying 0≤α,β≤1. Then, TR=hR8√2meeExτA 15π2/planckover2pi12×β5/2 U3/2×α(1−α)3/2 1+α. (30) The first factor is an overall factor proportional to the appl ied current. The second factor shows a simple dependence of spin torque as a function of βandU. We plot the last factor in Fig. 8. As we discuss above the maximum occurs at J/U= 1/3. This plot confirms again that the spin-orbit torque is finit e even when Japproaches U. We add the interface potential ( u0+umˆσ·m)δ(z) atz=0 and briefly see how results change. The reflection amplitudes change to r↑↓ k=ikz+(q∓ z+u0∓um) ikz−(q∓z+u0∓um). (31) The spin-orbit torque is then TR=hR/planckover2pi1eExτ 2πme/summationdisplay k⊥k2 ⊥kz(q− z+q+ z+2u0)(q− z−q+ z−2um) Dk(u0,um), (32)13 F NPotential profile z z=0EF 0 2JMinority band Majority bandU FIG. 9. (color online) The potential profile (blue lines) for the model Eq. (33). Here the spin-independent potential u0and spin-dependent potential umare present at z=0. The Fermi level (red line) is below the barrier U. where Dk(u0,um)=[k2 z+(q− z+u0−um)2][k2 z+(q+ z+u0+um)2]. TRis still real. It is consistent with the observation in Sec. V A that evanescent waves in the nonmagnet are crucial to get a dampinglike component. In this model, since the ferromagne t is insulating, there are no evanescent waves in the nonmagne t, thus only the fieldlike component can survive, regardless of an additional interface potential. D. Topological insulators in contact with a ferromagnet For the case of topological insulators in contact with a fer- romagnet, the nonmagnet is insulating. Thus a current flows along the ferromagnet only and the Rashba-type interaction at the interface z=0 gives rise to spin-orbit torque. Thus, we start from the following unperturbed Hamiltonian [64, 65].9 H=−/planckover2pi12 2me∇2+UΘ(−z)+Jˆσ·mΘ(z). (33) Here the barrier U(>EF) makes the nonmagnetic layer insu- lating. Without loss of generality, we can assume that U>J andEF>−J, otherwise there are no occupied states. The potential profile of Eq. (33) is presented in Fig. 9. In the non - magnet, the wave vector is imaginary, so we define qz=−ikz. Then, q2 z+(k± z)2=2me(U∓J)//planckover2pi12. From Eq. (6), the trans- mission amplitude is given by t′↑↓ k=2i/radicalbigRe[k∓z]qz ik∓z−qz. (34) Since the normal metal is insulating, Eq. (16d) gives spin- orbit torque. Since there is no incident wave in normal metal , r↑↓ k=0, so the second term in Eq. (16d) contributes only. After some algebra, TR=hR/planckover2pi13eEx 4πm2e/summationdisplay k⊥k2 ⊥/parenleftiggRe[k− z] U+Jτ↑−Re[k+ z] U−Jτ↓/parenrightigg . (35) 9The model can be an oversimplication of topological surface states, but we present this model for pedagogical reasons.Here k− zis real since majority waves should be propagating. Butk+ zcan be imaginary depending on k⊥. To perform the summation, we convert it to an integration as in Sec. V A. After some algebra, TR=hR√2meeExA 15π2/planckover2pi12Re/bracketleftigg(EF+J)5/2τ↑ U+J−(EF−J)5/2τ↓ U−J/bracketrightigg . (36) Here taking the real part (Re) eliminates contribution from states out of the Hilbert space. Explicitly, Re[( EF−J)5/2]= (EF−J)5/2Θ(EF−J), thus the second term does not contribute when EF<J. Equation (36) shows that spin-orbit torques exist even when τ↑=τ↓. A similar observation is made in Appendix A that an anisotropic magnetoresistance can arise even without di ffer- ence between τ↑andτ↓, unlike Ref. [33]. This is because, in our theory, k+ F/nequalk− F. Since we break a symmetry, we obtain a torque originating from the asymmetry. As a passing remark, similarly to Sec. V C, U−Jin the denominator in the second term in Eq. (36) does not yield any singularity at J=U. This is because EF≤U. When Japproaches to U, there must be a point where Jbecomes equal or larger than EFat which the second term does not contribute. Since Eq. (36) is real, only a fieldlike component can sur- vive. This is not an artifact of the particular Hamiltonian t hat we choose [Eq. (33)]. We remark that the second term in Eq. (16d) is always real, regardless of any detail of a model. On the other hand, recent experiments [66–68] report siz- able dampinglike spin-orbit torques in topological insula tors in contact with a ferromagnetic layer, contrary to our resul ts. One possible cause of the dampinglike torque is that in real materials, the location of topological surface states is sh ifted on a nanometer scale when attached to a ferromagnet [69– 71]. This displacement can be a cause of a finite dampinglike torque. Another possible cause is intrinsic spin-orbit tor que. In our theory, we only consider extrinsic contributions tha t are proportional to scattering times. However, in the two- dimensional Rashba model, the intrinsic spin-orbit torque is perpendicular to extrinsic spin-orbit torque [17]. If simi lar contributions exist in our three-dimensional model, they c ould cause a dampinglike component. VI. DISCUSSION A. Comparison to the two-dimensional Rashba model Magnetic bilayers with bulk magnetism can behave quite differently from two-dimensional Rashba models. The two- dimensional Rashba model shows only fieldlike compo- nents [11, 12] unless one takes into account intrinsic spin- orbit torque from the Berry phase [17] or a spin relaxation mecha- nism [13–16]. However, in our approach, a dampinglike spin- orbit torque with a similar order of magnitude arises if the c ur- rent mostly flows in the normal metal layer. If a current flow- ing in the ferromagnet has a similar order of magnitude to tha t in the normal metal, the result shows mostly fieldlike contri - butions, dominated by the dot-dashed line in Fig. 4. However ,14 the behavior is still very di fferent from the two-dimensional model. The contribution only comes from the majority elec- trons in the ferromagnet and thus is proportional to τ↑only. On the other hand, the fieldlike spin-orbit torque derived by the two-dimensional Rashba model is proportional to the spi n polarization∝(τ↑−τ↓)/(τ↑+τ↓). Therefore, the dominant contribution is not the counterpart of the fieldlike spin-or bit torque from the two-dimensional Rashba model. Magnetic bilayers with interface magnetism behave simi- larly to the two-dimensional Rashba model. As in Fig. 6, the fieldlike spin-orbit torque from spin current is the largest . In the two-dimensional Rashba model, the imbalance between the numbers of electrons in majority and minority is the pri- mary source of spin-orbit field and the resulting fieldlike sp in- orbit torque. In our model, it is modeled by τ↑/nequalτ↓since we do not have a finite exchange splitting explicitly. Therefor e, the upper right panel in Fig. 6 corresponds to the traditiona l contribution from the two-dimensional Rashba model. How- ever, the spin-orbit torque contributions driven by pure ch arge currentsτe F(left panels in Fig. 6) is a unique feature of the three-dimensional model. Systems with a ferromagnetic insulator or a topological in- sulator show only fieldlike spin-orbit torques. This is simi - lar to the two-dimensional Rashba model. We observe that a current flowing at z=0 (where the Rashba interaction ex- ists) results in fieldlike spin-orbit torque while a current across z=0 results in dampinglike spin-orbit torque. If one of the layers is insulating, there is no propagating wave from one to another. The situation is the same as the two-dimensional Rashba model. In the two-dimensional model, the Rashba in- teraction is present over the whole sample, and only in-plan e electron transport is allowed. Hence, there is no possibili ty for electrons to cross a Rashba region, eliminating the possibi lity of a dampinglike contribution. We do not consider intrinsic contributions from the Berry phase [17]. In general, spin torques have two di fferent con- tributions; extrinsic and intrinsic. The former is proport ional to scattering times, while the latter is independent of scat ter- ing times. In this sense, the latter is an electric-field-ind uced spin torque, not a current-induced one. Ref. [72] highlight s the subtle difference between them. The origin of an extrin- sic spin torque is the change of distribution functions in th e presence of an applied electric field. On the other hand, the origin of an intrinsic spin torque is the change of electroni c wave functions due to an applied electric field. In the two- dimensional Rashba model, intrinsic spin torque was found t o be larger than the extrinsic one in some contexts. But, it was also shown that intrinsic contributions are completely can - celed out by vertex corrections [4, 5] in metallic systems wi th an ideal quadratic dispersion. Therefore, the relative mag - nitude of the extrinsic and intrinsic spin torques depends o n the situation. Similar studies would be possible in the thre e- dimensional model, but is beyond the scope of this paper. We defer this question for future work.B. Multilayer generalization The starting point of our approach is a normal metal( z< 0)/ferromagnet( z<0) bilayer. In metallic systems consist- ing of layers of thicknesses larger than the mean free path, one can describe each interface separately and solve the bul k property by the spin drift-di ffusion equation. Therefore, a bi- layer model is sufficient to describe a multilayer system. How- ever, if any of the layers has a thickness not much greater tha n the mean free path, or the system includes an insulating in- sertion layer at which the spin drift-di ffusion equation cannot be written down, one needs to consider a multilayer situatio n quantum mechanically. The results of our theory will change depending on the situation. However, we here show that when we consider a normal metal ( z<0)/any underlying structure (0<z<L) with a Rashba interaction at z=0, the reflection matrix ˆ rkis independent of the details of the structure under- neath. We start from the interface Hamiltonian Eq. (5) with ˆ κ= ˆκ0+hRˆσ·(k׈z). Since we do not know any details for z>0, we use the transfer matrix formalism to focus on the interfac e atz=0 only. We write the wave function near z=0 by ψ0(z<0)=eikx+ikyy √ V/summationdisplay σ(eikzzaR σξσ+e−ikzzaL σξσ), (37a) ψ0(z>0)=eikx+ikyy √ V/summationdisplay σ(eikσ zzbR σξσ+e−ikσ zzbL σξσ),(37b) where RandLrefer to right-going and left-going states re- spectively. Here and from now on we neglect subscripts kσ indicating electronic states for simplicity. We define colu mn vectors ˆ aR/L=(aR/L +,aR/L −)TandˆbR/L=(bR/L +,bR/L −)T. Ap- plying the boundary conditions at z=0 given by ψ0(+0)= ψ0(−0) andψ0′(+0)−ψ0′(−0)=(2me/planckover2pi12)HIψ0(0), we obtain the following linear relation between ˆbR/Land ˆaR/L: ˆbR ˆbL=M0ˆaR ˆaL, (38a) M−1 0=1 2ikzikz+iˆKz−ˆκikz−iˆKz−ˆκ ikz−iˆKz+ˆκikz+iˆKz+ˆκ. (38b) Here the physical meaning of M0is the transfer matrix at z=0 (from z=−0 toz=+0). We now express the wave function at z=L+0 by the matrices ˆ cR/Lwith a suitable basis determined by the structure in z<L. By solving the Schr¨ odinger equa- tion, we can also write down the following transfer matrix. ˆcR ˆcL=M0→LˆbR ˆbL. (39) Here M0→Lis the transfer matrix from z=0 toz=L+0 of which the detailed form is unnecessary here. We consider a situation that a wave ˆψiis incident from z<0 and it splits up to reflected ( ˆψr) and transmitted ( ˆψt) parts. In the language of the transfer matrices, ˆψt 0=M0→LM0ˆψi ˆψr. (40)15 Inverting this, ˆψi ˆψr=1 2ikzikz+iˆKz−ˆκikz−iˆKz−ˆκ ikz−iˆKz+ˆκikz+iˆKz+ˆκM−1 0→Lˆψt 0 ≡1 2ikzˆmi −ˆmrˆψt, (41) where ˆ mi/rare 2×2 matrices. Now the reflection matrix is given by ˆ rex=−ˆmrˆm−1 i. To expand ˆ mr/i=ˆm(0) r/i+ˆm(1) r/i, we split ˆκ=ˆκ0+ˆκR. Here ˆκRis essentially the Rashba Hamiltonian, but written in the ˆ ·space where the magnetization direction is along z. Explicitly, ˆκR=hRξ† +ˆσ·(k×z)ξ+ξ† +ˆσ·(k×z)ξ− ξ† −ˆσ·(k×z)ξ+ξ† −ˆσ·(k×z)ξ−. (42) Then, ˆm(0) i ˆm(0) r=ikz+iˆKz−ˆκ0ikz−iˆKz−ˆκ0 −ikz+iˆKz−ˆκ0−ikz−iˆKz−ˆκ0M−1 0→L1 0, (43a) ˆm(1) i ˆm(1) r=−ˆκR1 1 1 1M−1 0→L1 0. (43b) Equation (43) allows us to compute ˆ m(1) r/iin terms of un- perturbed quantities. A typical way would be expressing M−1 0→L(1,0)Tin terms of ˆ m(0) i/rby inverting the matrix in front of it. Putting this into Eq. (43b) will give ˆ m(1) i/rin terms of ˆ m(0) i/r. However, the matrix inversion is very complicated. Instead , it is useful to observe from Eq. (43a) that 1−1 1−1ˆm(0) i ˆm(0) r=2ikz1 1 1 1M−1 0→d1 0. (44) Comparing with Eq. (43b), we obtain ˆm(1) i=ˆm(1) r=−1 2ikzˆκR( ˆm(0) i−ˆm(0) r). (45) Note that this expression is not perturbative since we have n ot assumed a small hRup to this point. From Eq. (45), we calculate ˆ rex=−ˆmrˆm−1 i. First, we per- turbatively expand ˆ rexby ˆrex=ˆr(0) ex+ˆr(1) ex+ˆr(2) ex···, where ˆ r(n) ex is the n-th order Rashba contribution. After some algebra, we obtain ˆr(n) ex=2ikzˆG(ˆκRˆG)n, (46a) ˆG=1 2ikz(1+ˆr(0) ex). (46b) Here we used ˆ m−1 i=( ˜m(0) i)−1−( ˜m(0) i)−1˜m(1) i( ˜m(0) i)−1+ ( ˜m(0) i)−1˜m(1) i( ˜m(0) i)−1˜m(1) i( ˜m(0) i)−1+···and ˆκRˆG=−ˆm(1) i( ˆm(0) i)−1 actively. Taking n=1 gives the same result as Eq. (10c). Since we do not assume anything about the underlying struc- ture in z>0, our result on the reflection matrix holds for arbitrary underlying structures.Three remarks are in order. First, although the same ex- pression holds only for the reflection matrix, it is very usef ul for some situations. If one looks into a response of the nor- mal metal induced by a current flow in the normal metal, the expression only includes ˆ rk. The anisotropic magnetoresis- tance calculated in Appendix A is an example. Second, this derivation is a mathematical result, so the results holds in the extended space. Projecting Eq. (46a) by ˆ1 does indeed give Eq. (10c). Third, the derivation by the transfer matrix is so me- what more abstract than the scattering formalism in Sec. III , but it allows easily generalizing our result up to any higher order contributions from hR. The second order term is used in Appendix A. C. Effects of proximity-induced magnetism We model proximity-induced magnetism as magnetism right at the interface in Sec. V B and V C. In this section, we present how one can treat e ffects of interface magnetism more generally. We first consider a situation without interface magnetism, and then treat umseparately. Let ˆ rum=0 ex be the reflection ma- trix in the absence of interface magnetism. Then, it would be valuable to see how the scattering coe fficients change in the presence of um. The transfer matrix formalism in Sec. VI B allows calculating the contributions from umperturbatively. When we replace ˆ κRin Eq. (46a) by umˆσz, we obtain ˆ rex= ˆrum=0 ex+ˆr(1) ex+ˆr(2) ex+···where ˆ r(n) ex=−2ikzˆG(umˆσzˆG)nand ˆG=(−1/2ikz)(1+ˆrum=0 ex). Here we use the fact that any two di- agonal matrices commute with each other. The result is given by the sum of a geometric series. After some algebra, ˆrex=ˆrum=0 ex−1+ˆrum=0 ex 2ikzu−1mˆσz(1+ˆrum=0 ex)−1+1. (47) This expression is of course consistent with Eqs. (23) and (31). The other scattering matrices are given by the con- straints 1+ˆrex=/radicalig |kz||ˆKz|−1ˆtex, 1+ˆr′ ex=/radicalig |ˆKz||kz|−1ˆt′ ex, and (1+ˆrex)k−1 z=(1+ˆr′ ex)ˆK−1 z. Equation (47) allows for the exploration of interface mag- netism effects up to any higher order in umor 1/um. By focus- ing on consequences of second term in Eq. (47), one can look into the effects of proximity-induced magnetism on a given expression. VII. CONCLUSION In summary, we develop a perturbation theory for scatter- ing matrices to compute interfacial spin-orbit coupling e ffects in magnetic bilayers. We extend the two-dimensional Rashba model by embedding it in three-dimensional transport of ele c- trons. We explicitly show that spin or charge current can be generated perpendicularly to an applied bias. Using this fa ct, we calculate current-induced (extrinsic) spin-orbit torq ue in terms of scattering amplitudes. For a given spin-orbit cou- pling Hamiltonian (like the Rashba form in our study), the16 resulting expressions from our theory are independent of de - tails of the interface, so they are easily applicable for wid e range of contexts. As demonstrations, we apply our formulas to various types of interfaces such as magnetic bilayers wit h bulk magnetism, those with interface magnetism, ferromag- netic insulators in contact with a nonmagnet, and topologic al insulators in contact with a ferromagnet. For magnetic bilayers, we show that a dampinglike com- ponent can be on the same order of or larger than a fieldlike component, even without taking into account the Berry phase contribution and spin relaxation mechanisms. For the syste ms with insulating layers, we found that only a fieldlike com- ponent can arise, since a dampinglike component originates from a current across the interface. We also demonstrate tha t for finite bulk exchange coupling, the evanescent states tha t become important for the mismatched Fermi surfaces can give rise to the dominant contribution to spin-orbit torque. Although we express the systems by analytic toy models, combining with first-principles calculations would enrich the implications of our theory significantly. We provide some re - marks on possible generalization of our theory and future di - rections. Furthermore, we present other spin-orbit coupli ng phenomena, such as an in-plane current generation by a per- pendicular bias (similar to the inverse spin Hall e ffect), a spin memory loss at the interface, and an anisotropic magnetore- sistance (similar to the spin Hall magnetoresistance) in th e appendices below. Our theory helps to characterize feature s of spin-orbit coupling phenomena for a given interface and further it provides insight on separating the roles of multi ple sources of spin-orbit coupling e ffects such as spin Hall e ffect, interfacial spin-orbit coupling, and the magnetic proximi ty ef- fect. Note During preparation of the manuscript, we found a re- cent report [73] which uses a similar scattering formalism t o our theory and describes several interface spin-orbit coup ling phenomena, but focuses on a particular context, metallic bi - layers without interface magnetism. ACKNOWLEDGMENTS The authors acknowledge J. McClelland, P. Haney, and O. Gomonay, for critical reading of the manuscript. K.W.K ac- knowledges V . Amin, and D.-S. Han for fruitful discussion. K.W.K. was supported by the Cooperative Research Agree- ment between the University of Maryland and the National Institute of Standards and Technology, Center for Nanoscal e Science and Technology (70NANB10H193), through the Uni- versity of Maryland. K.W.K also acknowledges support by Basic Science Research Program through the National Re- search Foundation of Korea (NRF) funded by the Ministry of Education (2016R1A6A3A03008831). K.W.K and J.S. are supported by Alexander von Humboldt Foundation, the ERC Synergy Grant SC2 (No. 610115), and the Transregional Col- laborative Research Center (SFB /TRR) 173 SPIN+X. K.J.L was supported by the National Research Foundation of Korea (2015M3D1A1070465, 2017R1A2B2006119). H.W.L. was supported by the SBS Foundation.Appendix A: Other physical consequences of interfacial spin-orbit coupling 1. In-plane current induced by a perpendicular bias The spin-orbit torque derived in the main text is essentiall y perpendicular spin current generation by in-plane charge c ur- rent flow. Here we derive its Onsager counterpart. When a perpendicular bias (chemical potential di fference) is applied, an in-plane current can be generated. Suppose first that there is no spin-orbit coupling and note that the current operator is proportional to k. If the system has rotational symmetry around xyplane, all the scattering matri- ces must satisfy ˆ rk=ˆr−kfor any in-plane kvector and similar relations for ˆ r′,ˆt, and ˆt′. Thus, even if there is a perpendicular bias, any contribution from kto an in-plane current is canceled out by the opposite state −k. Therefore, there is no in-plane current generation by a perpendicular bias. However, the situation drastically changes when interfaci al spin-orbit coupling is introduced. Here we present the pert ur- bation result in the main text again. ˆtk=ˆt0 k+hR 2ikzˆt0 k,exˆσ·(k×z)(ˆ1k+ˆr0 k), (A1a) ˆt′ k=ˆt′0 k+hR 2ikz(1+ˆr0 k,ex) ˆσ·(k×z)ˆt′0 k, (A1b) ˆrk=ˆr0 k+hR 2ikz(1+ˆr0 k,ex) ˆσ·(k×z)(ˆ1k+ˆr0 k), (A1c) ˆr′ k=ˆr′0 k+hR 2ikzˆt0 k,exˆσ·(k×z)ˆt′0 k. (A1d) The Rashba contributions are odd in k. When they are multi- plied by the current operator, the contributions from kand−k are no longer canceled out, thus an in-plane current can aris e. Since the Rashba contribution is also proportional to the Pa uli matrix vector, a charge bias will generate an in-plane spin c ur- rent and a spin bias will generate an in-plane charge current . The latter has the same symmetry as the inverse spin Hall ef- fect, implying that one needs to be careful when analyzing experiments [47, 49, 56, 74–76] using the inverse spin Hall effect as highlighted in Ref. [63]. Here we derive explicit expressions of the current density at the interface. We do not assume kto be in-plane, thus our result will also recover the results of the magnetoelectric cir- cuit theory. The current density at the normal metal along a unit vector uis calculated by ˜ju(z<0)=−e 2Trσ[ρ{vu,δ(rop−r)}ˆ˜σ], (A2) where vu=(/planckover2pi1/mei)∂uandρis the density matrix, ropis the position operator, and ris the position c-number. ˜σ=(1,σ) is the four-dimensional Pauli matrix vector. ˜juis a four- dimensional vector whose zeroth component is the charge current along uand the other three components are the spin current along uwith spin x,y,zdirections. Here and from now on, we denote any four-dimensional vector by the ˜ ·no- tation. As we develop in the main text, each of the eigen- states is written by a wave incident from the normal metal17 or a wave incident from the ferromagnet. Thus, they al- low writing down the density matrix by a block-diagonal form. Using the notation of direct summation, ρ=ρN+ρF, whereρN/Fare the density matrices block consisting of elec- trons incident from the normal metal /ferromegnet side. Thus we split the current into two terms: ˜ju=˜jN u+˜jF u, where ˜jN/F u=−e 2Trσ[ρN/F{vu,δ(rop−r)}ˆ˜σ]. LetρN=/summationtext kσ′σfN k,σ′σ|kσ′∝an}b∇acket∇i}ht∝an}b∇acketle{tkσ|where fN k,σ′σis the 2×2 re- duced density matrix. In a matrix form ˆfN k, each component is given by fN k,σ′σ=ξ† σ′ˆfN kξσ. Since we consider a noncollinear spin injection from the normal metal, we allow for ˆfN khaving an off-diagonal component. By its definition, ˆ1kˆfN kˆ1k=ˆfN k, since there is no incident electrons out of the Hilbert space . The current at the normal metal from ρNis then calculated by the wave function Eq. (2a). After some algebra, ˆjN u(z<0)=−e/planckover2pi1 meV/summationdisplay k(k·uˆfN k+k·¯uˆrkˆfN kˆr† k), (A3) where ¯u=(ux,uy,−uz).ˆjN/F uis a 2×2 matrix whose Pauli components are (1 /2)˜jN/F u, that is, ˜jN/F u=Trσ[ˆ˜σˆjN/F u]. When a perpendicular bias is applied, the distribution func - tion shifts. In the linear response regime, the distributio n shift occurs only near the Fermi surface. To focus on the nonequilibrium current, we replace ˆfN k=e∆ˆµNˆ1kδ(E−EF) where∆ˆµNis the shift of the chemical potential of the normal metal due to the bias. To deal with the delta function eas- ily, we convert the summation in Eq. (A3) to an integration:/summationtext k→(L/2π)/summationtext k⊥/integraltext dkz. By using dE=(/planckover2pi12/me)kzdkz, we convert the summation to an integration over energy. Due tothe delta function, the energy integration is nothing but th e integrand evaluated at the Fermi level. As a result, we obtai n ˆjN u(z<0)=−e2L hV/summationdisplay k⊥1 kz[k·u∆ˆµNˆ1k+k·¯uˆrk∆ˆµNˆr† k]E=EF, (A4) where Lis the length along zdirection. We use a similar method to obtain ˆjF u. There are three differences. First, we assume that there are no o ff- diagonal elements in ˆfF kdue to strong dephasing. Second, when we convert the summation by an integration,/summationtext k→ (L/2π)/summationtext k⊥/integraltext dkσ z, instead of dkz, because the wave func- tion is normalized by the incident wave. And then, we use (/planckover2pi12/me)kσ zdkσ z=dE. Third, the intervals of the integrations are different. The integral interval for ˆjN uis 0<E<EF. However, in this case, the integral interval is σJ<E<EF. However, since we focus on the Fermi surface contributions only, the lower bound of the energy does not matter. Omitting the algebra, we obtain ˆjF u(z<0)=−e2L hVRe/summationdisplay k⊥/bracketleftiggk·¯u |kz|e2 Im[ kz]zˆt′ k∆ˆµFˆt′† k/bracketrightigg E=EF.(A5) Now, the current right in the normal metal near the interface is given by the Pauli components of ˆju(z<0)=ˆjN u(z<0)+ˆjF u(z<0). (A6) In a similar way, we obtain the expression of the current in the ferromagnet near the interface. ˆju(z>0)=ˆjN u(z>0)+ˆjF u(z>0), (A7a) where ˆjN u(z>0)=−e2L hVRe/summationdisplay k⊥ˆK·ueiˆKzz /radicalig |ˆKz|ˆtk∆ˆµNt† ke−iˆK∗ zz /radicalig |ˆKz| E=EF, (A7b) ˆjF u(z>0)=−e2L hVRe/summationdisplay k⊥ˆK·eiˆKzz /radicalig |ˆKz|/bracketleftig ¯u∆ˆµFˆ1k′+uˆr′ k∆ˆµFˆr′† ke−2 Im[ ˆKz]z+(¯ue−2iˆKzz∆ˆµFˆr′† k+uˆr′ k∆ˆµFe2iˆKzz)/bracketrightige−iˆKzz /radicalig ˆKz E=EF,(A7c) where ˆK=(kx,ky,ˆKz) is a vector consisting of 2 ×2 matri- ces. From Eqs. (A4)–(A7), one can compute the current near the interface for given (spin /charge) chemical potential excita- tion. As in the main text, we from now on omit the [ ···]E=EF and implicitly assume that the expressions are evaluated at the Fermi level. We now simplify the expressions more. In Eq. (A5), the Im[kz] contribution originates from transmitted evanescent waves incident from the ferromagnet. For perpendicular tra ns- port, since k·¯u/|kz|is imaginary, there is no contribution from evanescent modes to a perpendicular current, consis-tently with the conservation of charge current. However, fo r in-plane transport, such a contribution can be nonzero. Not e that the evanescent contribution dies after 1 /kzlength scale. Since 1/kFis shorter than the mean free path scale, the cur- rent is almost unmeasurable in experimental resolution. Th us, we neglect decaying contributions in Eqs. (A5) and (A7).10,11 We also neglect highly oscillating terms in Eq. (A7). This is 10The approximation is exact for perpendicular transport. 11Ifk⊥is sufficiently close to kF, this does not hold. Therefore, this approx-18 a common approximation to take into account dephasing of a transverse component to min the ferromagnet. Then, we obtain ˆju(z<0)=−e2L hV/summationdisplay k⊥ˆ1k kz[k·¯u(ˆrk∆ˆµNˆr† k+ˆt′ k∆ˆµFˆt′† k) +k·u∆ˆµN], (A8a) ˆju(z>0)=−e2L hV/summationdisplay k⊥ˆ1k′ ˆKz[ˆK·uDiag[ˆ r′ k∆ˆµFˆr′† k+ˆtk∆ˆµNˆt† k]] +ˆK·¯u∆ˆµF, (A8b) where Diag[···]=/summationtext sus[···]usis the spin-diagonal part of a matrix. Physical meaning of this operation is the dephasing of a transverse component of spin in the ferromagnet. We first take u=zto see that Eq. (A8) is consistent with the conventional magnetoelectric circuit theory. It is eas y to show that ˆjz(z<0)=−e2L hV/summationdisplay k2 ⊥<k2 F[∆ˆµN−(ˆrk∆ˆµNˆr† k+ˆt′ k∆ˆµFˆt′† k)],(A9) which is exactly the result of the magnetoelectric circuit t he- ory. One can also show with Eq. (A8) and the unitarity con- straint developed in Sec. B that ˆjz(z>0)=Diag[ ˆjz(z<0)]. This implies the continuity of electrical current across th e in- terface, thus we do not need to keep ( z>0) or ( z<0). Following the procedure of the magnetoelectric circuit the - ory [45, 46], we take ∆ˆµN=∆µN 0−ˆσ·s∆µN swhere sis the direction of spin magnetic moment in the normal metal. scan be deviated from mwhen one considers noncollinear spin in- jection. We also take ∆ˆµF=∆µF 0−ˆσ·m∆µF s.12The scattering matrices are taken by Eq. (A1). However, all the first order Rashba contributions are canceled out after summing over al l transverse modes. They are odd in in-plane momentum kxor ky, thus are canceled by an opposite contribution from −kx or−ky. Therefore, we can discard the Rashba contributions and take only unperturbed scattering matrices, which are ex - panded by ˆ r0 k=r↓ ku++r↑ ku−and so on. Then, spin and chargecurrents are given by V LTrσ[ˆjz]=(G↑↑+G↓↓)(∆µF 0−∆µN 0) +(G↑↑−G↓↓)(∆µF s−m·s∆µN s),(A10a) −V LTrσ[ ˆσˆjz]=(G↑↑−G↓↓)(∆µF 0−∆µN 0)m +(G↑↑+G↓↓)(∆µF s−m·s∆µN s)m −2 Re/bracketleftig G↑↓[m×(s×m)+is×m]/bracketrightig ∆µN s, (A10b) where Gss′=(e2/h)/summationtext k2 ⊥<k2 F(1−rs krs′∗ k).G↑↑/↓↓is the interface conductance for spin majority /minority electrons and G↑↓is the spin mixing conductance. Eq. (A10) recovers all the re- sults in the traditional theory. We now take an in-plane u. As we discuss above, non- Rashba contributions cannot generate an in-plane current. Us- ing Eq. (A1) and collecting the first order contributions to hR, ˆju(z<0)=−hRe2L hVIm/summationdisplay k2 ⊥<k2 Fk·u k2z(1+ˆr0 k) ˆσ·(k×z) ×[(1+ˆr0 k)∆ˆµNˆr0† k+ˆt0 k∆ˆµFˆt′0† k]. (A11) When summing up over all transverse modes, one should con- sider all possible angles of k⊥for a given magnitude. If the system has rotational symmetry, we can take an angle average of (k·u)[ ˆσ·(k×z)] by integrating over the in-plane angle from−πtoπand dividing the result by 2 π. Then, we obtain (k2 ⊥/2) ˆσ·(u×z). After the angle average, ˆju(z<0)=−hRe2L 2hVIm/summationdisplay k2 ⊥<k2 FE⊥ EF−E⊥(1+ˆr0 k) ˆσ·(u×z) ×[(1+ˆr0 k)∆ˆµNˆr0† k+ˆt0 k∆ˆµFˆt′0† k]. (A12) Here E⊥=/planckover2pi12k2 ⊥/2meand we used k2 ⊥/k2 z=E⊥/(EF−E⊥). Then, the in-plane spin and charge currents are given by its Pauli components. V LTrσ[ˆju(z<0)]=−(u×z) 2i·/bracketleftig (G↑↑↑ Rt−G↓↓↓ Rt)(∆µF 0−∆µN 0)m+(G↑↑↑ Rt+G↓↓↓ Rt)(∆µF s−m·s∆µN s)m +(G↑↓↓ Rr−G↓↑↑∗ Rr)∆µN s[m×(s×m)+is×m]/bracketrightig +c.c., (A13a) imation is a crude approximation even for a low experimental resolution. However, we present this here since the approximation simpl ifies the ex- pressions a lot. 12In our model, the spin magnetic moment is antiparallel to the electron spindirection. Thus, we take ∆µN/F sto have a negative sign to make a consistent notation with the previous theories.19 −V LTrσ[ ˆσˆju(z<0)]=−m(u×z) 2i·/bracketleftig (G↑↑↑ Rt+G↓↓↓ Rt)(∆µF 0−∆µN 0)m+(G↑↑↑ Rt−G↓↓↓ Rt)(∆µF s−m·s∆µN s)m −(G↑↓↓ Rr+G↓↑↑∗ Rr)∆µN s[m×(s×m)+is×m]/bracketrightig −1 2i/bracketleftig (G↑↓↓ Rt−G↓↑↑∗ Rt)∆µF 0−(G↑↓↓ Rt+G↓↑↑∗ Rt)∆µF s+(G↑↓↓ Rr−G↓↑↑∗ Rr)∆µN 0−m·s(G↑↓↓ Rr+G↓↑↑∗ Rr)∆µN s/bracketrightig ×{m×[(u×z)×m]−im×(u×z)}−(u×z)·m 2i(G↑↑↓ Rr+G↓↓↑∗ Rr)∆µN s[m×(s×m)+is×m]+c.c. (A13b) where c.c. refers to complex conjugate of all terms in front of it. In the conductances Gss′s′′ RrandGss′s′′ Rt, the subscript R refers to Rashba contributions, and randtrefer to contribu- tions from reflection and transmission. The explicit expres - sions in terms of unperturbed scattering matrices are Gss′s′′ Rr=−hRe2 2h/summationdisplay k⊥E⊥ EF−E⊥(1+rs k)(1+rs′ k)rs′′∗ k,(A14a) Gss′s′′ Rt=−hRe2 2h/summationdisplay k⊥E⊥ EF−E⊥(1+rs k)t′s′ kt′s′′∗ k. (A14b) Equation (A13a) clearly demonstrates a charge current is generated by a spin chemical potential bias. For the case of collinear transport ( s=m), the charge current along u=xis proportional to my. Since the inverse spin Hall e ffect is also dependent on my, it requires a careful analysis. Simultaneous description of the inverse spin Hall e ffect and interfacial spin-orbit coupling would be a future challenge. As passing remarks, we present some properties of Gss′s′′ Rr/t. First, there are three indices. This is because there is inte rfa- cial spin-orbit coupling as an additional spin scattering s ource. A complete description requires three indices; first one for the incident spin, second one for the scattering due to interfac ial spin-orbit coupling, and third one for the transmitted spin . Second, Gss′s′′ Rr/tis symmetric under exchange between sand s′. Third, the unitarity constraint in Sec. B implies Im[ Gsss Rr+ Im[Gsss Rt]=0. This is shown by using |rs k|2+|t′s k|2=1 to derive Gsss Rr+Gsss Rt=−(hRe2/2h)/summationtext k2 ⊥<k2 F[(E⊥)/(EF−E⊥)]|1+rs k|2. This constraint guarantees the absence of a charge current gener a- tion at equilibrium. We also remark that in-plane current can be discontinu- ous at the interface in the presence of interfacial spin-orb it coupling. To see this, we define a discontinuity matrix by ∆ˆju=ˆju(z>0)−Diag[ ˆju(z<0)]. This is zero for u=z, not for an in-plane u. After some algebra, we obtain V LTrσ[∆ˆju]=(u×z) 2i·/bracketleftig (∆G↑↑ h−G↓↓ R∆)(∆µF 0−∆µN 0)m+(G↑↑ R∆+G↓↓ R∆)(∆µF s−m·s∆µN s)m−2G↑↓ R∆m×(s×m)∆µN s/bracketrightig +c.c., (A15a) −V LTrσ[ ˆσ∆ˆju]=(u×z) 2i·/bracketleftig (G↑↑ R∆+G↓↓ R∆)(∆µF 0−∆µN 0)m+(G↑↑ R∆−G↓↓ R∆)(∆µF s−m·s∆µN s)m+2iG↑↓ R∆s×m∆µN s/bracketrightig +c.c.,(A15b) where Gss′ R∆=−hRe2 2h/summationdisplay k2 ⊥<k2 FE⊥ EF−E⊥(1+rs k)(1+rs′ k). (A16) Before closing the section, we mention that the currents are proportional to L, the size of the system. In reality, the contri- butions will relax on the length scale of the mean free path λ. Thus, if L≫λ,Lin the above expressions should be replaced byλwhen one takes into account the bulk scattering. In the main text and this section, we have demonstrated that a bias can generate a current perpendicular to the applied bi as direction, not affecting its longitudinal transport. These re- sults are first order perturbation theory. In the following t wo sections, we calculate second order e ffects in hRto examine the effects of interfacial spin-orbit coupling on longitudinal transport . Each section deals with a perpendicular bias to theinterface and an in-plane bias respectively. 2. Spin memory loss and spin torque from collinear spin injection Equation (A10) describes the generation of a perpendicular current to the interface in the presence of a perpendicular b ias. Spin flip at the interface due to interfacial spin-orbit coup ling vanishes due to rotation symmetry around the xyplane. How- ever, if we consider second order contributions, the result can change. Below we demonstrate, a spin-up (down) current can be generated by a spin-down (up) bias. This means that in- terfacial spin-orbit coupling flips the spin at the interfac e even when we consider a collinear transport. We interpret this as spin memory loss at the interface. We start from Eq. (A9). The second order expansion is20 given by Eq. (46a). For simplicity, we consider only colline ar transport with perpendicular magnetization ( s=m=z). For s/nequalm, the coefficient of spin torque in Eq. (A10) will also change. For m/nequalz, the result will depend on the direction of the magnetization, leading to a current-perpendicular mag ne- toresistance. Below we make more remarks on this case. Writing down the expression explicitly, one realizes that the Rashba contributions arises in the form of ˆκRˆAˆκR. For instance, there is a contribution proportional to 4k2 zˆG[ˆκRˆG∆ˆµNˆG†ˆκR]ˆG†. The angle average over all transverse modes simplifies such expressions a lot. From Eq. (42) and some algebra, angle average of ˆ κRˆAˆκRfor an arbitrary diago- nal matrix ˆAis also diagonal: ˆκRa10 0a2ˆκR→h2 Rk2 ⊥a20 0a1. (A17) We emphasize that the diagonal components are exchanged. As we show below, this exchange results in mixing of spin-up and spin-down components. As a result, a spin-up (down) bias can generate spin-down (up) current. To see spin flip at the interface clearly, we use up /down (↑/↓) notations rather than the spin /charge (0/s) notations used in Sec. A 1. We expand ∆ˆµN=∆µN ↓u++∆µN ↑u−and similarly for∆ˆµF.13We also define I↑↓=(V/L) Trσ[u∓ˆjz], which are spin-up/down currents. We simply denote ∆µ↑/↓,∆µF ↑↓−∆µN ↑↓, the spin chemical potential di fference across the interface. Af- ter some algebra, I↑ I↓=G↑↑−2hRReG↓↑↑ RrGflip Gflip G↓↓−2hRReG↑↓↓ Rr∆µ↑ ∆µ↓, (A18a) Gflip=h2 Re2 2h/summationdisplay k2 ⊥<k2 FE⊥ EF−E⊥|1+r↑ k|2|1+r↓ k|2. (A18b) Here Gssare the unperturbed interface conductances derived from the magnetoelectric circuit theory. The corrections t o the diagonal terms are the second order corrections14from inter- facial spin-orbit coupling. These diagonal terms simply de - scribes a spin-up/down current generation by a spin-up /down bias. On the other hand, Gflipdescribes spin-flipping contri- butions: A spin-up (down) bias generates spin down (up) cur- rent. Ifm/nequalz, Eq. (A17) does not hold. Even the result is not diagonal.15The existence of an o ff-diagonal element implies that one cannot use the two-current model and there arises a spin current whose spin direction is transverse to the mag- netization. This means that a spin torque (depending on mz) 13The relation between ∆µN ↑↓and∆µN 0/sis∆µN ↑↓=∆µN 0±∆µN sand a similar relation holds for∆ˆµFandI↑↓. 14Since Gss′s′′ Rris already in first order in hR, the corrections are in second order. 15In Sec. A 3, we demonstrate how an angle average can have o ff-diagonal components for a general m.can be generated even for a collinear spin injection. Due to complexity of the expressions, we do now show it here, but the expression is given by exactly the same procedure [excep t Eq. (A17)]. Although this is a second order e ffect, it would be valuable if this spin torque is experimentally realized. The spin memory loss and spin torque we illustrate above do not violate the conservation of angular momentum. The source of the angular momentum is nothing but the lattice at the interface. Spin-orbit coupling at the interface pump s orbital angular momentum from (to) the lattice to (from) the spin-magnetization system. 3. Anisotropic magnetoresistance In the previous section, we examine second order e ffects for perpendicular transport to the interface. In this secti on, we consider in-plane transport. We below show that the elec- trical resistance depends on the direction of magnetizatio n. When a charge current is flowing along x, the resistance in- cludes terms proportional to m2 xandm2 y. The former has the same symmetry with the conventional anisotropic magnetore - sistance from ferromagnetic bulk, but can have the opposite sign. The latter has the same symmetry as the spin Hall mag- netoresistance [50–55]. For simplicity we consider a current mainly flowing in the normal metal. When a charge current is flowing in the nor- mal metal, the distribution function is shifted: fN k=f0,N k+ (eE/planckover2pi1/me)kxδ(E−EF) where xis the direction of the current flow. Or equivalently, ∆ˆµN=(E/planckover2pi1τ/me)kx. Putting this and u=xinto Eq. (A4) and taking trace over the spin space gives the charge current due to the distribution shift. Trσ[ˆjN x]=−e2EτL 2πmeVTrσ/summationdisplay k2 ⊥<k2 Fk2 x kz(1+ˆrkˆr† k). (A19) The first term in the parenthesis (1) is the conventional elec tri- cal current and the second term (ˆ rkˆr† k) is the Rashba contribu- tion. The second order expansion is given by Eq. (46a). First order Rashba contributions are odd in k⊥so are all canceled out after summing up over all transverse modes. To compute second order contributions, by the same reason in Sec. A 2, we take angle average of the expression ˆ κRˆAˆκRfor a diagonal ˆA. k2 xˆκRa10 0a2ˆκR→h2 Rk4 ⊥ 2a20 0a1 +(a2−a1)h2 Rk4 ⊥mz 8m∝ba∇dbl(m2 x+3m2 y) ˆσx +(a2−a1)h2 Rk4 ⊥ 4m∝ba∇dblmxmyˆσy +(a2−a1)h2 Rk4 ⊥ 8(m2 x+3m2 y) ˆσz,(A20) after angle average. Here m∝ba∇dbl=/radicalig m2x+m2y.21 In charge transport, the result is given by taking the trace. Thus the diagonal terms (the first and last terms) only matter . Due to the existence of ( m2 x+3m2 y) contribution in front of ˆ σz, a magnetoresistance proportional to ( m2 x+3m2 y) can arise. If the reflection matrix is spin independent, the magnetoresistan ce is zero since Tr[ σz]=0. However, if the reflection matrix is spin dependent, the contribution can survive. In a previo us work [33], a magnetoresistance proportional to ( m2 x+3m2 y) is reported, but is proportional to τ↑−τ↓, thus a current should flow in the ferromagnet. This is because their model assumes an equal-Fermi-surface model. On the other hand, we take into account more generalized situation and demonstrate th at a magnetoresistance ∝(m2 x+3m2 y) can arise if r↑ k/nequalr↓ k. Another remark is that the term 3 m2 yhas the same symmetry as the spin Hall magnetoresistance. Therefore, the spin Hal l magnetoresistance should also be carefully analyzed since there is an interface contribution. There are several theor eti- cal and experimental reports [16, 32–34] that interface Ras hba effect can give rise to a magnetoresistance having the same symmetry as the spin Hall magnetoresistance. Another dif- ference is that the Rashba contribution has a m2 xcontribution as well. The contribution has the same symmetry as the con- ventional anisotropic magnetoresistance, but below we sho w that it can have the opposite sign (we call it a negative magne- toresistance). Therefore, observing the spin-Hall-like m agne- toresistance and a negative anisotropic magnetoresistanc e of a similar orders of magnitude will shed light on separating th e bulk effects and the interfacial e ffects. To complete our analysis, we explicitly calculate the curre nt [Eq. (A19)] with Eqs. (46a) and (A20). After some algebra, Trσ[ˆjN x]=jconv+jnonMR+jMR(m2 x+3m2 y), (A21a) jconv=−e2EτL 4πmeV/summationdisplay k2 ⊥<k2 F,s=↑,↓k2 ⊥ kz(1+|rs k|2), (A21b) jnonMR=−h2 Re2EτL 8πmeV ×Re/summationdisplay k2 ⊥<k2 Fk4 ⊥ k3zρ↑ kρ↓ k(ρ↑∗ kρ↓∗ k+ρ↓ kr↓∗ k+ρ↑ kr↑∗ k), (A21c) jMR=−h2 Re2EτL 32πmeVRe/summationdisplay k2 ⊥<k2 F[(|ρ↑ k|2−|ρ↓ k|2)2 −2(ρ↑ k−ρ↓ k)(ρ↑2 kr↑∗ k−ρ↓2 kr↓∗ k)], (A21d) whereρ↑↓ k=1+r↑↓ k. Here jconvis the non-Rashba contribution, jnonMR is the second order Rashba correction that is indepen- dent of m, and jMRis the magnetoresistance. If the summand injMRis positive, the magnitude of the current increases when m2 xincreases, which is the negative magnetoresistance. Below we discuss when a negative magnetoresistance arises. For magnetic bilayers, if bulk magnetism is dominant, the reflection matrix is real [See Eq. (18a)]. The reality of r↑↓ kallows simplifying the summand in jMRas (|ρ↑ k|2−|ρ↓ k|2)2−2 Re[(ρ↑ k−ρ↓ k)(ρ↑2 kr↑∗ k−ρ↓2 kr↓∗ k)]) =(r↑ k−r↓ k)2(2−r↑2 k−r↓2 k)≥0. (A22) For ferromagnetic insulators, one can deduce jMR=0. This is consistent with the Landauer-B¨ uttiker formalism. Sinc e the ferromagnet is insulating, the conductance (thus resistan ce) is determined by the number of transverse modes in the normal metal, and is independent of the magnetization direction. T his is a general result which holds unless the translational sym me- try along the current flowing direction is broken or there is a spin-dependent scattering source (for example, the spin Ha ll effect generating spin Hall magnetoresistance). The case for the topological insulators cannot be described by Eq. (A21) since we assume that a current is flowing in the normal metal. But we mention that it was studied in a previous report [33]. For complex r↑↓ kin metallic magnetic bilayers, we found no a priori argument which guarantees the sign of jMR. Appendix B: Unitarity constraint of the scattering amplitu des Provided that all waves are propagating, charge conserva- tion implies the following unitarity constraint: For the fo llow- ing scattering matrix Sk=ˆrkˆt′ k ˆtkˆr′ k, (B1) S† kSk=1 holds. Up to this point, the expressions are conven- tional. However, this works only if all waves are propagatin g, thus it must be generalized to the extended space. In this sec - tion, we omit the subscript ‘ex’ for simplicity. Let theσ= +1 band be evanescent in the ferromagnet. As far as unitarity is concerned, some of matrices should be projected into σ=−1. Note that Skconnects incoming waves to outgoing waves. ψN,out k ψF,out k=SkψN,in k ψF,in k (B2) The transmitted outgoing waves for σ= +1 electrons are dropped. Thus ˆtk→u−ˆtkand ˆr′ k→u−ˆr′ k. In addition, there is no contribution incident from FM for σ=+1 electrons. Thus ˆt′ k→ˆt′ ku−andu−ˆr′ k→u−ˆr′ ku−. Thus the projected scattering matrix is Sproj k=ˆrk ˆt′ ku− u−ˆtku−ˆr′ ku−. (B3) Lastly, the unitarity constraint is given by the conservati on of electrical charge. Neglecting σ=+1 in the ferromagnet, the unitarity constraint is given by (Sproj k)†Sproj k=1 0 0u−. (B4)22 Let onlyσ=−1 band be propagating. There is no contri- bution from ˆ rk,ˆtk, and ˆt′ kfor the unitarity constraint. Sproj k=0 0 0u−ˆr′ ku−, (B5) and the unitarity constraint is given by (Sproj k)†Sproj k=0 0 0u−. (B6) The above relations also hold when the order of ( Sproj k)†and Sproj kis reversed. The three expressions can be combined by means of the projection matrices. Sproj k=ˆ1kˆrkˆ1kˆt′ k ˆ1′ kˆtkˆ1′ kˆr′ k, (B7) (Sproj k)†Sproj k=Sproj k(Sproj k)†=ˆ1k0 0ˆ1′ k. (B8) Explicitly calculating each component gives the unitarity con- straint of scattering matrices in our theory. ˆr† kˆ1kˆrk+ˆt† kˆ1′ kˆtk=ˆ1k, (B9a) ˆr′† kˆ1′ kˆr′ k+ˆt′† kˆ1kˆt′ k=ˆ1′ k, (B9b) ˆr† kˆ1kˆt′ k+ˆt† kˆ1′ kˆr′ k=0, (B9c) and ˆ1k(ˆrkˆr† k+ˆt′ kˆt′† k)ˆ1k=ˆ1k, (B10a) ˆ1′ k(ˆr′ kˆr′† k+ˆtkˆt† k)ˆ1′ k=ˆ1′ k, (B10b) ˆ1k(ˆrkˆt† k+ˆt′ kˆr′† k)ˆ1′ k=0. (B10c) Appendix C: Absence of a bound state We show that the delta-function spin-orbit coupling poten- tial at z=0 does not create a bound state unless its magnitudeis beyond a perturbative regime. For mathematical simplici ty, we take a simpler model in which electrons are subject to the largest magnitude of the delta function. A bound state is mos t likely to exist in this situation. The maximum magnitude of the delta function is ( /planckover2pi12/2me)hRkmax ⊥where kmax ⊥is the maxi- mum value of/radicalig k2x+k2y. Then, the Hamiltonian becomes H=−/planckover2pi12∇2 2me±JΘ(z)−/planckover2pi12 2mehRkmax ⊥δ(z), (C1) where±refers to each spin band. Let the bound state wave function be eq1zforz<0 and e−q2zforz>0. Then both q1 andq2should be positive. The relation between q1andq2is −/planckover2pi12q2 1/2me=±J−/planckover2pi12q2 2/2me, or equivalently (q2−q1)(q2+q1)=±2meJ /planckover2pi12. (C2) Now, the derivative mismatching condition from the delta- function potential is /planckover2pi12 2me(q2+q1)=/planckover2pi12 2mehRkmax ⊥. (C3) Combining the two conditions, 2q1/2=hRkmax ⊥±Jme /planckover2pi12kmax ⊥hR, (C4) where the choice of the sign ±that corresponds to 1 or 2 is ambiguous but it is not necessary to be determined. A necessary condition for q1andq2being positive is that q1q2is positive. 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1312.3292v1.Spin_orbit_coupling_in_quantum_gases.pdf

Spin-orbit coupling in quantum gases Victor Galitski1,2 and Ian B. Spielman1 1Joint Quantum Institute, National Institute of Standards and Technology, and University of Maryland, Gaithersburg, Maryland, 20899, USA. 2Condensed Matter Theory Center, University of Maryland, College Park, Maryland, 20742, USA Spin-orbit coupling links a particle's velocity to its quantum mechanical spin, and is essential in numerous condensed matter phenomena, including topological insulators and Majorana fermions. In solid-state materials, spin-orbit coupling originates from the movement of electrons in a crystal's intrinsic electric field, which is uniquely prescribed. In contrast, for ultracold atomic systems, the engineered “material parameters” are tuneable: a variety of synthetic spin-orbit couplings can be engineered on demand using laser fields. Here we outline the current experimental and theoretical status of spin-orbit coupling in ultracold atomic systems, discussing unique features that enable physics impossible in any other known setting. A particle’s spin is quantized. In contrast to the angular momentum of an ordinary, i.e. classical, spinning top which can take on any value, measurements of an electron’s spin angular momentum (or just “spin”) along some direction can result in only two discrete values: ±ħ/2, commonly referred to as spin-up or spin-down. This internal degree of freedom has no classical counterpart; in contrast, a quantum particle’s velocity is directly analogous to a classical particle’s velocity. It is therefore no surprise that spin is a cornerstone to a variety of deeply quantum materials like quantum magnets1 and topological insulators2. Spin-orbit coupling (SOC) intimately unites a particle’s spin with its momentum, bringing quantum mechanics to the forefront; in materials, this often increases the energy scale at which quantum effects are paramount. The practical utility of any material is determined, not only by its intrinsic functional behavior, but also by the energy or temperature scale at which that behavior is present. For example, the quantum Hall effects – rare examples of truly quantum physics where the spin is largely irrelevant – are relegated to highly specialized laboratories because these phenomena manifest themselves only under extreme conditions – at liquid-helium temperatures and high magnetic fields3,4. The integer quantum Hall effect (IQHE) was the first observed topological insulator (TI), but it has a broken time-reversal symmetry. This is in contrast with a new class of topological insulators (see Box 1), which rely on SOC instead of magnetic fields for their quantum properties, and are expected to retain their quantumness up to room temperature2. As fascinating and unusual as the already existing topological world of spin-orbit-coupled systems is, all this physics is largely based on a non-interacting picture of independent electrons filling up a prescribed topological landscape. But there is clearly physics beyond this, as suggested by the fractional quantum Hall effect (FQHE) materials, where interactions between electrons yield phenomena qualitatively different from those encountered in IQHE. In FQHE systems, the charged excitations are essentially just fraction of an electron – with fractional charge – a new type of emergent excitation with no analogue elsewhere in physics. Furthermore, even non-Abelian excitations are possible: a system can be in one of many states of equal energy in which “non-Abelions,” are at the same location, and differ only by the sequence of events that created them. At zero magnetic field, strong interactions and strong spin-orbit coupling can also give rise to fractionalization in TIs: the emergence of excitations that are fundamentally different from the constituent particles. We currently know little about these fractional topological insulators, but we do know they should exist and also expect them to be stable at a much larger range of parameters and experimental temperatures than the FQHE: perhaps even up to room temperature in solids. It is ironic then, that we focus on the most fundamental behavior of spin-orbit coupled systems using ultracold atoms at nano-Kelvin temperatures. These nominally low temperatures are often a deception, as what matters is not an absolute temperature scale, but rather the temperature relative to other energy scales in the system (e.g., the Fermi energy), and from this perspective, ultracold atom systems are often not that cold5. However, ultracold atomic systems are among the simplest and most controllable of quantum many-body systems. While only one type of SOC has been experimentally realized to date, realistic theoretical proposals to create a range of SOCs abound, many of which have no counterpart in material systems6,7,8,9,10. The laser-coupling technique first experimentally implemented by our team11,12 – now implemented in laboratories around the world – is well suited to realize topological states with one-dimensional atomic systems13. In sharp contrast to solid-state systems, in which we do not control or even know with certainty all details of the complicated material structure, ultracold atoms are remarkable in that most aspects of their environment can be engineered in the laboratory. Also, their tunable interactions and their single-particle potentials are both well characterized: the full atomic Hamiltonian is indeed known. This provides a level of control unprecedented in condensed matter and allows one to address basic physics questions at the intersection of material science and many-body theory. To study material systems, theorists create “spherical-cow” models of real materials, while in cold atom physics experimentalists actually make spherical cows. Interactions – even the simple, contact interactions present in cold atom systems – enrich the physics of quantum systems by engendering new phases and phenomena. For example, when combined with SOC, the celebrated superfluid-Mott insulator transition14,15, gives rise to numerous magnetic phenomena both in the insulating and superfluid phases16,17. Such interacting systems are often impossible to treat exactly with current theoretical techniques, but cold-atom experiments can directly realize these systems and shed light on the complicated and often exotic physics mediated by the strong interactions. Likewise, by asking basic questions such as how do strong interactions destroy TIs - or create them - we can understand the mechanisms underlying fractional TIs. These exotic quantum states have not yet been observed, but are present in realistic theoretical descriptions of ultracold atoms with SOC18,19. Ultracold atoms with synthetic spin-orbit couplings8 can not only shed light on the outstanding problems of condensed matter physics, but also yield completely new phenomena with no analogue elsewhere in physics. A notable example of such a unique system is spin-orbit coupled bosons with just two spin states: a synthetic spin-1/2 system. The existence of such particles with real spin-1/2 is prohibited in fundamental physics due to Pauli’s spin-statistics theorem, but synthetic symmetries – imposed by restricting the states available to the atoms – relax these constraints allowing bosons with pseudospin-1/2 to exist20. Spin-orbit coupling also results in a wide array of new many-body quantum states including a zoo of exotic quantum spin states in spin-orbit-coupled Mott insulators21,16,17, unusual spin-orbit-coupled Bose-Einstein condensates, with a symmetry protected degenerate ground state22, and perhaps even strongly correlated composite fermion phases analogous to the FQHE states in electron systems18. These are just a few examples of phenomena from a veritable treasure trove of exciting physics that is just waiting to be uncovered in this emerging and fast-developing field. Basics of spin-orbit coupling In any context, SOC requires symmetry breaking since the coupling strength is related to velocity as measured in a preferred reference frame (such as an electron’s velocity with respect to its host crystalline lattice, or an atom’s velocity with respect to a reference frame defined by its illuminating laser beams). Conventional SOC thus results from relativistic quantum mechanics, where the spin is a fundamental and inseparable component of electrons as described by the Dirac equation. In the non-relativistic limit, the Dirac equation reduces to the familiar Schrödinger equation, with relativistic corrections including an important term coupling the electron’s spin to its momentum and to gradients of external potentials. This is the fundamental origin of SOC, which underlies both the L∙S coupling familiar in atomic and molecular systems and all spin-orbit (SO) phenomena in solids. SOC can most simply be understood in terms of the familiar –µ∙B Zeeman interaction between a particle’s magnetic moment µ parallel to the spin, and a magnetic field B present in the frame moving with the particle. SOC is most familiar in traditional atomic physics where it gives rise to atomic fine-structure splitting, and it is from this context that it acquires its name: a coupling between an electrons spin and its orbital angular momentum about the nucleus. The electric field produced by the charged nucleus gives rise to a magnetic field in the reference frame moving with an orbiting electron (along with an anomalous factor of two resulting from the electron’s non-inertial trajectory encircling the atomic center of mass), leading to a momentum-dependent effective Zeeman energy. In materials, the connection to a momentum-dependent Zeeman energy is particularly clear. For example, the Lorentz-invariant Maxwell’s equations dictate that a static electric field E = E0ez in the lab frame (at rest) gives a magnetic field BSO = (E0 ħ/mc2) (kxey – kyex) in the frame of an object moving with momentum k, where c is the speed of light in vacuum and m is the particle’s mass. The resulting momentum-dependent Zeeman interaction –µ∙B ~ σxky – σykx is known as Rashba SOC23. This often arises from the built-in electric field in two-dimensional semiconductor heterostructures resulting from asymmetries of the confining potential24, and is depicted in Fig. 1. Figure 1d plots a typical spin-orbit dispersion relation, where the minima for each spin state (red or blue) is displaced from zero; in the case of Rashba SOC, this dispersion is axially symmetric, meaning that this double-well structure is replicated for motion in any direction in the ex-ey plane. Because of the momentum-dependent Zeeman interaction, the equilibrium alignment of a particle's magnetic moment depends on its velocity. Quantum mechanically, this implies that the quantum mechanical eigenstates are generally momentum-dependent superpositions of the initial and spin states. In most condensed matter systems, electrons move in a crystal potential and when there is a potential gradient on the average, effective SO interactions arise. These usually originates from: a lack of mirror symmetry in two-dimensional systems leading to the Rashba SOC described above23; or a lack of inversion symmetry in bulk crystals leading to other forms of spin-orbit coupling such as the linear Dresselhaus SOC25, described by a Zeeman interaction –µ∙B ~ σxkx – σyky reminiscent to that of Rashba SOC. SOC phenomena are ubiquitous in solids and have been known to exist since the early days of quantum mechanics and band theory. However, these phenomena and the field of spintronics26 have moved to the forefront of condensed matter research only recently. This renewal of interest was stimulated by a number of exciting proposals for spintronic devices, whose functionality hinges on an electric field dependent coupling between the electron spin and its momentum. Apart from these potential useful applications, SO coupled systems turned out to display an amazing variety of fundamentally new and fascinating phenomena: spin-Hall effects27,28, topological insulators2, Majorana13 and Weyl fermions29, exotic spin textures in disordered systems30, to name just a few. The problem of synthesizing Majorana fermions stands out as perhaps the most active and exciting area of research combining both profound fundamental physics and a potential for applications. Indeed Kitaev noticed that a Majorana fermion, being a linear combination of a particle and a hole, should not couple much to external sources of noise and as such should be protected from its debilitating effects and decoherence31. Furthermore, when many such Majorana entities are put together, they form a non-Abelian network that is capable of encoding and processing topological quantum information that may be ideal for quantum computing applications32. Spin-orbit-coupled superconductors in a magnetic field can host Majorana fermions33, and creating such topological fermionic superfluids in spin-orbit-coupled quantum gases appears to be within experimental reach, and perhaps cold atoms may become the first experimental platform to create and manipulate non-Abelian quantum matter. Synthetic SOC in cold atomic gases As we have seen, SOC links a particle’s spin to its momentum and in conventional systems, it is a relativistic effect originating from electrons moving through a material’s intrinsic electric field. This physical mechanism for creating SOC –requiring electric fields at the trillions of V/m level for significant SOC – is wildly inaccessible in the laboratory. Such fields exist inside atoms and materials, but not in laboratories. Instead, we engineer SOC in systems of ultracold atoms, using two photon Raman transitions – each driven by a pair of laser beams with wavelength λ – that change the internal atomic “spin.” Physically, this Raman process corresponds to the absorption of a single photon from one laser beam and its stimulated re-emission into the second. Each of these photons carries a tiny momentum with magnitude pR=h/λ called the photon recoil momentum (h is Planck’s constant). Conservation of momentum implies that the atom must acquire the difference of these two momenta (equal to 2pR for counter-propagating laser beams). In most materials, the photon recoil is negligibly small; indeed, in conventional condensed matter systems, the “optical transitions” are described as having no momentum change. Ultracold atoms, however, are at such low temperatures that the momentum of even a single optical photon is quite large. Thus as first put forward by Higbie and Stamper-Kurn34, Raman transitions can provide the required velocity dependent link between the spin and momentum: because the Raman lasers resonantly couple the spin states together, when an atom is moving, its Doppler shift effectively tunes the lasers away from resonance, altering the coupling in a velocity-dependent way. Remarkably, nearly all SOC phenomena present in solids can potentially be engineered with cold atoms (and some already have), but in contrast to solids where SOC is an intrinsic material property, synthetic SOC in cold atoms can be controlled at will. Furthermore, unlike the common electron, laser-dressed atoms with their pseudo-spins are not constrained by fundamental symmetries; this leads to a remarkably broad array of “synthetically-engineered” physical phenomena not encountered anywhere else in physics. Figure 2 depicts the currently implemented technique for creating SOC in ultracold atoms12,35,36,37,38. The first step, shown in Fig. 2a, is to select from the many available internal atomic states a pair of states which we will associate with pseudo-spins states and that together comprise the atomic “spin.” Two counterpropagating laser beams, which here define the x axis, couple this selected pair of atomic states to the atoms’ motion along ex. Reminiscent of the case for Rashba SOC shown in Fig. 1d, this coupling alters the atom’s energy-momentum dispersion, although here only motion along the x direction is affected (Fig. 2c). In the standard language, both Rashba and Dresselhaus SOC are present, and have equal magnitude, giving the effective Zeeman shift –µ∙B ~ -σykx. In solids, this symmetric combination of the Rashba and Dresselhaus coupling goes by the name of “persistent spin-helix symmetry point,” where it on one hand allows spin control via SOC, but on the other minimizes the undesirable effect of spin memory loss30. Since, SOCs effect on a single particle is equivalent to that of a momentum-dependent Zeeman magnetic field, the particle's dispersion relation (e.g., the familiar kinetic energy mv2/2 = p2/2m for a free particle) is split into two sub-bands corresponding to two spin-split components, now behaving differently (measured in Fig. 2c). For the linear SOC on which we focus, the band splitting simply shifts the minimum of the dispersion relation by an amount depending on the particle’s internal state and the laser coupling strength. This effect, depicted in Fig. 2b, was first measured indirectly in Ref. 12 where a BEC was prepared in a mixture of and in each of the two minima of the dispersion, and the momentum of the two spins was measured as a function of laser intensity. More recently, the full dispersion curve was measured spectroscopically38, clearly revealing the spin-orbit coupled structure as a function of momentum (Fig. 2c). A panoply of SOCs can be created with additional lasers linking together additional internal states. Figure 3 shows a realistic example where three internal atomic states can be coupled, producing a tunable combination of Rashba and Dresselhaus SOC39. In these cases, one of the three initial atomic states is shifted by a large energy, leaving behind two pseudo-spins comprising a two-level system6. A further extension can generate an exotic 3D analog to the Rashba SOC, which we call Weyl SOC, that cannot exist in materials10 or to types of SOC with more than the usual two spin states9. Many-body physics An example of a unique quantum phenomenon made possible in ultracold atomic systems is spin-orbit coupled Bose-Einstein condensates. The main ingredient of these exotic many-body states are laser-dressed bosons with states and that create a synthetic spin-1/2 system. Because the Pauli spin-statistics theorem prohibits the existence of bosons with real spin-1/2, this is a weird and interesting entity by itself, but when many such entities are brought together in a SO coupled system, the degree of weirdness further increases. As the temperature is lowered, the bosons tend to condense, but in contrast to the conventional BEC, where the zero-momentum state is the unique state with lowest energy (the ground state is non-degenerate), SO bosons can have energy-momentum dispersion with several lowest energy states (the ground state is degenerate). For example, for Rashba=Dresselhaus SOC (Fig. 2c) there are two such minima; for pure Rashba SOC there is a continuous ring of minima (Fig. 1d); for the Weyl-type spin-orbit coupling there is a sphere of minima10. This is in contrast with the more conventional case of spinor BECs that include two or more spin states, but do not alter the energy-momentum dispersion relation. The bosons’ indecisiveness about what state to condense into is partially resolved by their interactions, which limits which states have lowest energy. But unless the interactions break a “synthetic time-reversal” (Kramers) symmetry, some degeneracy must remain, leading to the possibility of exotic states. For example, repulsive bosons with a non-equal combination of Rashba and Dresselhaus SOC are predicted to condense into a strongly entangled many-body “cat” state, where the whole condensate simultaneously is in a superposition of states with equal and opposite momentum. Such many-body cat states have long been sought in various experiments, but have never been convincingly observed. The SO BECs, existing in a double-well “potential” in momentum space (e.g., Fig. 1d) are promising in this regard because robust arguments support the existence of many-body cats22: (i) the symmetry protection of the exact spin degeneracy from splitting and (ii) an argument based on the Heisenberg uncertainty relation, which suggests that in order for the repulsive bosons to stay as far as possible from each other in real space, they should be as close as possible in dual momentum space. An experimental realization of such a many-body cat state would be a major scientific development. On the experimental front, there are already exciting developments, which include the first realization of an Abelian spin-orbit coupling (corresponding to the persistent spin helix symmetry point, where Rashba and Dresselhaus SOCs are identical, see supporting text box for a discussion of the connection to Abelian and non-Abelian gauge fields) and observation of a spin-orbit-coupled Bose-Einstein condensate with rubidium atoms12,35,36. Exactly as expected, the time-of-flight images of cold SO coupled bosons feature two peaks that correspond to left- and right-moving condensates flying apart in the opposite directions. They however do not represent a cat state (where all the atoms are either in the left-moving or all in the right-moving condensate), but rather are either in a “striped” state (where all of the atoms are in the same state, which involves both positive and negative momenta), or in a phase separated state of the right- and left-moving condensates in the Abelian SO system12,40,41,42 see Fig. 2b. Spin-orbit-coupled ultracold fermions are intriguing8: even the behavior of two interacting fermions is fundamentally altered with the addition of SOC. Without SOC and in one spatial dimension, any attraction between two fermions, no matter how weak, always gives rise to the formation of a molecule. In two dimensions, the resulting molecular pairing is suppressed but not absent, with an exponentially small binding energy; and in three dimensions there is a threshold below-which there is no molecular state. However, in systems with many fermions, many-body effects guarantee the formation of Cooper pairs in any dimension, as long as attraction is present. The crossover between a BEC of molecules to a Bardeen-Cooper-Schrieffer (BCS) condensate of pairs is a smooth transition between physics described in terms of simple “native” molecules to the truly many-body physics of Cooper pairs43,44. SOC provides a completely different avenue for enhancing the pairing between two fermions. The ground-state of the Rashba SOC Hamiltonian consists of a one-dimensional ring in momentum and that of Weyl SOC is a two-dimensional sphere. This reduces the effective dimensionality and thereby strongly enhances molecular pairing. This ensures that there is no threshold for molecular formation in such SO systems and that the BEC-BCS crossover is strongly modified10,45,46,47. The many body physics of the BCS side is greatly affected as well. The main difficulty in realizing topological fermionic superfluids is creating the unconventional pairing mechanism between the atoms48,49. Such topological pairing has been proven difficult to achieve using p-wave Feshbach resonances due to debilitating effects of three-body losses50. SOC also can create effective interactions: for example, in analogy to the d-wave interactions recently demonstrated between colliding BECs51, stable p-wave interactions generated by synthetic spin-orbit are expected and pave the way to atomic topological superfluids52,53,54. Experimentally, SOC in atomic Fermi systems has been realized in two labs37,38, where the basic physical phenomena at the single particle level were confirmed. Outlook Spin-orbit coupled cold atoms represent a fascinating and fast-developing area of research significantly overlapping with traditional condensed matter physics, but importantly containing completely new phenomena not realizable anywhere else in nature. The potential for new experimental and theoretical understanding abounds. Spin-orbit-coupled Bose-Einstein condensates and degenerate Fermi gases have now been realized in a handful of laboratories: the experimental study of these systems is just beginning. The immediate outlook centers on implementing the full range of SOCs that currently exist only in theoretical proposals: to date just one form of SOC has been engineered in the lab. To realize the true promise of these systems, a central experimental task is to engineer SOCs that link spin to momentum in two and three dimensions (non-abelian, and without analog in material systems). An unfortunate reality of light induced gauge fields, as currently envisioned, is the presence off-resonant light scattering – spontaneous emission – that leads to atom-loss, heating, or both. In the alkali atoms, this heating cannot be fully mitigated by selecting different laser parameters (like wavelength), as a result, an important direction of future research is finding schemes, or selecting different atomic species, where this problem is mitigated or absent. Another thrust of research with synthetic SO-coupled fermions is to realize topological insulating states in optical lattices. A recent breakthrough in condensed matter physics is the understanding that the quantum Hall states represent just a tip of the iceberg of a zoo of topological states. A complete classification of those has by now been achieved for fermion systems in thermodynamic limit55,56. This leads to fundamentally different classes of Hamiltonians. For example, no non-trivial insulators exist in three dimensions if time-reversal invariance is allowed to be broken, but the by-now famous Z2 classification exists otherwise57. There are nine symmetry classes in each spatial dimension, however not all of them have been realized in solids. In materials, the symmetries are usually “non-negotiable” while in “synthetic” SO systems, the symmetries and lack thereof can be controlled at will, opening the possibility to create and control topological states, including topological phases that are not realizable in solids. The ability to tune synthetic couplings suggests that a larger class of non-Abelian gauge structures is within the immediate experimental reach. These structures do not have analogues, or even names in solid state physics, but are most appropriately characterized as SU(3)-spin-orbit couplings. These can be created by focusing on a three-level manifold of dressed states, as opposed to two-level manifold corresponding to spin-up and down states for the usual SOC. The general coupling of the three internal dressed degrees of freedom to particle motion can not be spanned by three spin matrices, but requires 3x3 Gell-Mann matrices58, which form generators of the SU(3) group well studied in the context of elementary particle physics. The algebraic structure, geometry, and topology of this complicated group are much different from the familiar spin case, and these differences will have profound observable manifestations. A completely different way to create such topological matter is related to the non-equilibrium physics of spin-orbit-coupled systems. It is easy to experimentally engineer dynamic synthetic SOC and gauge fields with a prescribed time-dependence, which gives the opportunity to realize interesting dynamic structures, such as Floquet topological insulators59 and Floquet Majorana fermions60. We expect that the most exciting physics in atomic SOC systems will rely on interactions, and lie at the intersection of experiment and theory. What is the physics of spin-orbit coupled Mott insulators and the corresponding superfluid-to-insulator phase transition? What is the ground state of the Rashba bosons, which were recently argued to undergo a statistical transmutation into fermions. How is the BEC-BCS transition altered by SOC? Each of these questions can only be answered in a partnership between experiment and theory: the underlying physics is so intricate that the correct answer is difficult to anticipate without direct measurement, and the meaning of these measurements can be inexplicable without theoretical guidance. d Dispersion of resulting Rashba SOCc Model system in electron’s frameb Model system in lab framea Structural inversion-symmetry breaking in a materialdonor layere-e-Electric fieldv = v0exE=E0ez B(v)eyexeyez e-v = 0E=E0ez exeyezEnergyMomentum Figure 1 | Physical origin of SOC in conventional systems. a In materials, SOC requires a broken spatial symmetry. For example, the growth profile of two-dimensional GaAs electron (or hole) systems can create an intrinsic electric field breaking inversion symmetry. b The effective model system consists of an electron confined in the ex-ey plane (in this example moving along ex) in the presence of an electric field along ez. c In the rest frame of the electron, the Lorentz-transformed electric field generates a magnetic field along ey (generating a Zeeman shift ) which depends linearly on the electron’s velocity. d For such systems the spin orbit coupling is linear, and the usual free particle mv2/2 = p2/2m dispersion relation is altered in a spin dependent way. In this case, pure Rashba SOC shifts the free-particle dispersion relations for each spin state away from zero (red and blue curves). The crossing point of these curves can be split by an applied magnetic field (grey curve). a Typical level diagram c Dispersion measured in 6LiExperimental Rashba = Dresselhaus configuration b Minima location43210Raman Coupling -1.00.01.0Quasimomentum q/kL /EL = 0 ER = 1.72 ER Zeeman shiftbetweenground statesExcited state Figure 2 | Laser coupling schemes. a. In current experiments, a pair of lasers – often counter propagating – couple together a selected pair of atomic states labeled by and that together comprise the atomic “spin.” These lasers are arranged in a two-photon Raman configuration that uses an off-resonant intermediate state (grey). These lasers link atomic motion along the x direction to the atom’s spin creating a characteristic spin-orbit coupled energy-momentum dispersion relation. b. Measured location of energy minimum or minima, where as a function of laser intensity the characteristic double minima of SOC dispersion move together and finally merge12. c. Complete dispersion before and after laser coupling measured in a 6Li Fermi gas (Data reproduced with permission of M. Zwierlein, from Ref. 38), compared with the predicted dispersion (white dashed curves), showing typical spin-orbit dispersion relations depicted in Fig. 1d. a Coupling scheme c Coupled dispersion b Laser geometry -2002-20Energy in units of EL481216 Momentum qy/kLMomentumqx/kLCold atoms Figure 3 | Generalized SOC. a-c. Going beyond current experiments, more complicated forms of SOC may be created. These require both more laser beams and more internal states. a Each state is coupled by a two photon Raman transition, each produced by a pair of the beams shown in b. The depicted configuration could realize a tunable combination of Rashba and Dresselhaus SOC in the alkali atoms39; the outcome is equivalent to that of the well-known tripod configuration6 with detuning, but practical in the alkali atoms. BOX 1 Topological matter Topological insulators2 are strongly spin-orbit-coupled materials that have seemingly mutually exclusive properties: they are both insulating and metallic at the same time. In their interior (bulk), electrons cannot propagate, while their surfaces are highly conducting. To get an insight into the complicated theory of these exotic materials, let us recall that electrons in an insulator fully occupy a certain number of allowed energies (bands) in such a way that the highest occupied state is separated from the lowest empty one by a gap of forbidden energies. Hence, a non-zero energy is required to excite an electron across the gap (i.e., to make it move) and small perturbations have almost no effect on the insulator. From this perspective, it is as good as vacuum: nothing moves inside. It may seem that any two such insulators (“vacua”) should be indistinguishable, but it is not so! If we ask whether one insulator can be smoothly deformed into another without breaking certain symmetries or turning it into a metal along the way, we find that it is not always possible. Insulators are divided into qualitatively different categories, including trivial insulators (which are much like vacuum) and topological insulators, characterized by a non-zero integer topological index, related to the momentum-dependent spin in spin-orbit coupled materials such as occurs in HgTe/CdTe quantum wells2 (preserving time-reversal symmetry) or from the magnetic field in quantum Hall systems (breaking time-reversal symmetry). Integers can not change smoothly one into another, but whenever we have a surface of a topological insulator, i.e., a boundary with a true vacuum, we effectively do enforce a transition between the media characterized by different integers, say 1 and 0, and the only way to cross between them is to either break symmetries or close the gap abruptly, that is to create a boundary metal. This is why the topological boundary states are so robust: they are squeezed in between the two vacua (the usual vacuum and the twisted one – the topological insulator) and have nowhere to go. As different as superconductors are from insulators in their electromagnetic properties, the characterizations of their excitation spectra are closely related. A superconductor is a condensate of electron pairs (Cooper pairs) behaving like a superfluid. Since it is energetically favorable to form electron pairs in a superconductor, it takes energy to break a pair to create single electrons, just like it takes energy to move an electron across an energy gap in an insulator. So, a superconductor is an insulator for its fermionic excitations and as such can be characterized by topological integers with similar consequences, including boundary states. But the latter are unusual at the edges of a topological superconductor, which get filled by weird chargeless and spinless entities - linear combinations of an electron and a hole (an absentee electron). Under certain circumstances these can also become Majorana fermions – zero energy particles that are their own antiparticle – which were predicted in spin-orbit-coupled systems and might have been observed there33. BOX 2 Connection to gauge fields The forms of SOC discussed in this review are all examples of static gauge fields, which can be mathematically included in the atomic Hamiltonian as . The most elementary example of a gauge field is the electromagnetic vector potential defined by , where q is the electric charge of the particle. This vector potential defines magnetic and electric fields though its spatial and temporal properties; a uniform time-independent vector potential is of no physical consequence. A gauge field is non-Abelian when the components of the vector are non-commuting operators, for example . Such non-Abelian gauge potentials are generic in problems ranging from nuclear magnetic resonance to molecular collisions61. Using techniques related to those discussed here62, it is possible to engineer artificial magnetic11 and electric fields in ultracold atoms. 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1106.3665v1.Orbital_order_and_magnetism_of_FeNCN.pdf

arXiv:1106.3665v1 [cond-mat.str-el] 18 Jun 2011Orbital order and magnetism of FeNCN Alexander A. Tsirlin,1,∗Klaus Koepernik,2and Helge Rosner1,† 1Max Planck Institute for Chemical Physics of Solids, N¨ othn itzer Str. 40, 01187 Dresden, Germany 2IFW Dresden e.V., P.O. Box 270116, D-01171 Dresden, Germany Based on density functional calculations, we report on the o rbital order and microscopic magnetic model of FeNCN, a prototype compound for orbital-only model s. Despite having a similar energy scale, the spin and orbital degrees of freedom in FeNCN are on ly weakly coupled. The ground-state configuration features the doubly occupied d3z2−r2(a1g) orbital and four singly-occupied dorbitals resulting in the spin S= 2 on the Fe+2atoms, whereas alternative ( E′ g) configurations are about 75 meV/f.u. higher in energy. Calculated exchange coupling s and band gap are in good agreement with the available experimental data. Experimental effects arising from possible orbital excitations are discussed. PACS numbers: 71.20.Ps, 75.30.Et, 71.70.Ch The relationship between spin and orbital degrees of freedom in transition-metal compounds is well estab- lished on both phenomenological and microscopic levels. Phenomenologically, the orbital states are described by pseudospin operators and can be treated using diverse techniques developed for spin Hamiltonians [1]. Micro- scopically, the orbital pattern determines the superex- change couplings that are the main driving force of mag- netism in insulators. The opposite effect, the influence of magnetism on the orbital state, is less universal [2] than initially expected [1]. Particularly, recent computational studies of model orbitally-ordered materials [3] indicated the important roleoflattice distortionsin stabilizing spe- cific orbital states. Although spin and orbital degrees of freedom are inherently tangled, there has been consid- erable theoretical effort in exploring orbital-only models with no spin variables involved [4–6]. In this paper, we will present a compound that features intrinsically weak coupling between spins and orbitals, and may be a feasi- ble experimental probe for such orbital-only models. Asamodelcompound, weconsidertherecentlydiscov- ered iron carbodiimide FeNCN containing Fe+2cations that form layers of close-packed FeN 6octahedra in the abplane [7]. The layers are connected via linear NCN units (Fig. 1). Experimental information on FeNCN is ratherscarce. The compound isa colored(dark-red[7] or brown[8])antiferromagneticinsulatorwith theN´ eeltem- perature of 345 K [7]. The electronic structure of FeNCN was studied by Xiang et al.[8] who arrived at a puzzling conclusiononthedramaticfailureofconventionaldensity functionaltheory(DFT)+ Umethodsthatwereunableto reproduce the experimentally observed insulating ground state. We will show that this failure is caused by a sub- tle effect of competing orbital states. Such orbital states are readily elucidated in a careful DFT-based study, and reveal an unusually weak coupling to the magnetism. Our DFT calculations are performed in a full-potential code with a local-orbital basis set (FPLO) [9]. We used the experimental crystal structure, the local density ap- proximation (LDA) with the exchange-correlation po-a1gt2gaa bb ccC NN /c97 /c98egJabJcJc’ eg’ FIG.1. (Color online)Leftpanel: crystalstructureofFeNC N. Right panel: close-packed layer of FeN 6octahedra (top), the single octahedronsqueezedalongthethree-fold caxis(bottom left), and a sketch of the ground-state d6electronic configu- ration of Fe+2(bottom right). tential by Perdew and Wang [10], and well-converged k meshes of 1500 −2000points in the symmetry-irreducible parts of the first Brillouin zone. The application of a generalized-gradient-approximation (GGA) exchange- correlation potential led to quantitatively similar results. TheLDAenergyspectrumforFeNCN(Fig.2) strongly resemblesthat of ironoxides. Nitrogen2 pstates form va- lence bands below −2 eV, whereas Fe 3 dstates are found in the vicinity of the Fermi level. The energy spectrum is metallic due to the severe underestimation of electronic correlations in LDA. Thelocalpicture ofthe electronicstructurestems from the crystal-field levels of Fe+2with the electronic config- urationd6. The octahedral local environment induces the conventional splitting of five dstates into t2gand eglevels. This primary effect is accompanied by a weak trigonal distortion that further splits the t2gstates into thea1gsinglet and e′ gdoublet (Fig. 1, bottom). The2 5 /c4511 0 M K A L H A /c71 /c71Energy□(eV)/c458 /c454 4 0Total CFeLDA N Energy□(eV)01015 DOS□(eV )/c451 FIG. 2. (Color online) Top: LDAdensityofstates for FeNCN. Bottom: Band structure (thin light lines) and the fit with the tight-binding model (thick dark lines). The Fermi level is a t zero energy. balance between the a1gande′ gstates is determined by fine features of the local environment. According to sim- ple electrostatic arguments, the squeezing of the octahe- dronalongthethree-foldaxisshouldslightlyfavorthe a1g state (note the N–Fe–N angles α= 95.9◦andβ= 84.1◦ in Fig. 1), which is represented by d3z2−r2orbital in the conventional coordinate system ( zalong the caxis). To quantify the orbital energies, we fit the Fe 3 d bandswithatight-bindingmodelbasedonWannierfunc- tions adapted to specific orbital symmetries (Fig. 3). The fit yields εa1g=−0.29 eV,εe′g=−0.30 eV, and εeg= 0.58 eV. The t2g−egsplitting of about 0.9 eV is typical for 3 dsystems, whereas the energy separation of 10 meV between the a1gande′ gstates is very small and opposite to the naive crystal-field picture. The difference may arise from covalency effects and/or long-range in- teractions inherent to solids (note a similar example in Ref. [11]). Similar to iron oxides, FeNCN is expected to feature the high-spin state of Fe+2(the non-magnetic low-spin state apparently contradicts the experimental magnetic response reported in [7]). Therefore, in the local pic- ture five out of six delectrons occupy each of the dor- bitals, whereas the sixth electron takes any of the a1gor e′ gorbitals, thereby creating orbital degrees of freedom. Since the Mott-insulating state implies integerorbitaloc- cupations, the ground state of FeNCN should feature one doubly-occupied and four singly-occupied orbitals. The nature of the doubly occupied orbital is determined by the energy difference between a1gande′ gand, more im- portantly, by correlation effects. a1gc ba eg' FIG. 3. (Color online) Wannier functions based on the a1g ande′ gorbitals. To account for correlation effects in FeNCN, we use the DFT+ Umethod that treats strong electronic corre- lationsin amean-fieldapproximationvalidforinsulators. IncontrasttoRef.[8]reportingthehalf-metallicsolution, wereadilyobtainedinsulatingsolutionsbyexplicitlycon- sidering orbitally-ordered configurations. The failure of the previous computational work is likely related to spu- rious solutions arising from random starting configura- tions. Such solutions are not converged in charge due to the charge shuffling effect in a (half)-metal. To over- come this problem, we first performed calculations with a fixed occupation matrix (i.e., fixed the orbital configu- ration) and later released this matrix to allow for a fully self-consistent procedure. A similar approach has been used in previous computational studies [11, 12], and was shown to be vital for the proper treatment of systems with orbital degrees of freedom. The input parameters of the DFT+ Umethod, the on- site Coulomb repulsion ( Ud) and exchange ( Jd), are eval- uated in a constrained LDA procedure [13] implemented in the TB-LMTO-ASA code [14]. We find ULMTO d= 6.9 eV and Jd= 0.9 eV, whereas a comparative calcula- tion for FeO yields similar values of ULMTO d= 7.1 eVand Jd= 0.9 eV. The magnitude of electronic correlations in FeNCN is, therefore, the same as in Fe+2oxides. By contrast, a somewhat reduced Udparameter was found in CuNCN (6.6 eV vs. 9 −10 eV in Cu+2oxides) and as- cribed to sizable hybridization between the Cu 3 dand N 2pstates [15]. In FeNCN, such a hybridization is rather weak (Fig. 2). Since the two Fe sites in the unit cell of FeNCN are crystalographically equivalent, we restrict ourselves to ferro-type orbital configurations featuring the same dou- bly occupied orbital on both Fe sites. Starting from dif- ferent occupation matrices, we were able to stabilize sev- eral solutions [16]. The ground-state configuration fea- tures the doubly occupied a1gorbital and is further re- ferred as A1g. TheE′ gconfigurations with two electrons on either of the e′ gorbitals lie higher in energy for about 75 meV/f.u. This energydifference is nearlyindependent ofthe specific Udvalue. It is alsopossible toput twoelec- trons on one of the egorbitals, but such configurations are highly unfavorable (0.81 eV/f.u. above the ground3 TABLE I. Exchange couplings (in K) calculated for the ground-state ( A1g) and one of the higher-lying ( E′ g) configu- rations. Interatomic distances are given in ˚A (see also Fig. 1). The on-site Coulomb repulsion parameter is Ud= 7 eV. Distance A1g E′ g Jab 3.27 −11 −6 Jc 4.70 51 58 J′ c 5.73 2 0 state) in agreement with the large t2g−egsplitting of 0.9 eV in LDA. The artificial low-spin configuration with all six electrons on the t2gorbitals has an even higher energy of 3.6 eV/f.u. above the A1gground state. The lowest-energy spin configurations are antiferro- magnetic, irrespectiveofthe A1gorE′ gorbital state. The energy spectra are quite similar [17], although the band gap (Egap) for the A1gorbital configuration is systemat- ically higher than for any of the E′ gconfigurations: for example, at Ud= 7 eVEgap= 2.95eV and 2 .30−2.50eV forA1gandE′ g, respectively. The change in Udcauses a systematic shift of the band gaps [18]. While the lack of the experimental optical data prevents us from tuning theUdparameter against the experimental band gap, we note that the calculated Egapvalues agree well with the dark-red color of FeNCN. We now investigate the interplay between spin and or- bital degrees of freedom in FeNCN. To evaluate magnetic couplings, we doubled the unit cell in the abplane, and calculated total energies for several spin configurations. These total energies were further mapped onto the clas- sical Heisenberg model yielding individual exchange in- tegralsJi. We evaluated the nearest-neighbor coupling Jabin theabplane as well as the nearest-neighbor and next-nearest-neighbor interplane couplings JcandJ′ c, re- spectively (Fig. 1). Further couplings are expected to be weak due to negligible long-range hoppings in our tight- binding model. Surprisingly, the calculated exchange couplings listed in Table I depend only weakly on the orbital order. BothA1gandE′ gorbital configurations induce lead- ing antiferromagnetic (AFM) exchange Jcvia the NCN groups. The intraplane coupling Jabis ferromagnetic (FM), whereas J′ cis AFM. The resulting spin lattice is non-frustrated, and features AFM long-range order with parallel spins in the abplane and antiparallel spins in the neighboring planes. This prediction awaits its verifi- cation by a neutron scattering experiment. To test our microscopic magnetic model against the available exper- imental data, we simulated the magnetic susceptibility using the quantum Monte-Carlo loopalgorithm [19] im- plemented in the ALPS package [20]. The susceptibility wascalculatedforathree-dimensional L×L×Lfinite lat- tice with periodic boundary conditions and L= 12 [21].6experiment simulation 0 100 200 300TN=□345□K 400 Temperature□(K)57 /c99(10 m /mol)/c458 3 FIG. 4. (Color online) Experimental magnetic susceptibili ty of FeNCN [7] and the simulated curve for Jc= 46.5 K,Jab= −16.3 K, and J′ c= 2.3 K. The deviations at low temperatures are due to an impurity contribution and/or anisotropy effect s. The simulated magnetic susceptibility was compared to the experimental data from Ref. [7] (Fig. 4). The data above 100 K and, particularly, the transition anomaly at TN= 345 K are perfectly reproduced with Jc= 46.5 K, Jab=−16.3 K, and J′ c= 2.3 K in remarkable agreement with the calculated exchange couplings for the A1gor- bital configuration (Table I) [22]. The low-temperature upturn of the experimental susceptibility violates the be- havior expected for a Heisenberg antiferromagnet, and signifies an impurity contribution or effects of exchange anisotropy that are not considered in our minimum mi- croscopic model. The three-dimensional magnetism of FeNCN is some- what unexpected considering the seemingly layered na- ture of the crystal structure (Fig. 1). The leading ex- change is antiferromagnetic and runs between the lay- ers, although the nearest-neighbor interlayer distance of 4.70˚A is much longer than the intralayer distance of 3.27˚A. The unusually strong interlayer exchange orig- inates from the peculiar nature of the NCN units that feature a strong π-bonding and mediate hoppings be- tween the neighboring layers. This effect has been illus- trated by sizable contributions of both nearest-neighbor and second-neighbor nitrogen atoms to Wannier func- tions in CuNCN [15]. Similar contributions are found for theegorbitals in FeNCN. The intralayer interaction is a conventional combina- tion of the direct exchange and Fe–N–Fe superexchange that result in a weakly ferromagnetic coupling (the Fe– N–Fe angles are 95 .9◦, i.e., close to 90◦). The spin lattice of FeNCN reminds of another transition-metal carbodi- imide, CuNCN, where the long-range antiferromagnetic superexchange mediated by the NCN groups was also re- ported [15]. The difference between the two compounds is the strong Jahn-Teller distortion in CuNCN that splits the close-packedlayersoftransition-metaloctahedrainto structural chains running along a, with the ferromag-4 netic coupling resembling Jabin FeNCN. By contrast, FeNCN is not subjected to a Jahn-Teller distortion, and theground-stateorbitalconfigurationislargelystabilized by electronic correlations. Indeed, there is no clear pref- erence for a certain orbital state on the LDA level. According to our results, the energy scales for the spin and orbital degrees of freedom in FeNCN are compara- ble, about 52 meV/f.u. and 75 meV/f.u., respectively. However, magnetic couplings weakly depend on the or- bital configuration keeping spins and orbitals nearly de- coupled. This effect is explained by different dstates responsible for the orbital and magnetic effects. The or- bitaldegreesoffreedomareoperativeinthe t2gsubspace, whereas magnetic couplings are largely determined by theegorbitals featuring stronger intersite hoppings. The decoupling of spin and orbital variables along with the low energy scale of the competing orbital states suggest that the orbital-only physics can be probed in FeNCN. The characteristic energy scale of 75 meV/f.u. corre- sponds to temperatures around 850 K, and implies that theE′ gorbital states may emerge at elevated tempera- tures. Although FeNCN, alike all transition-metal car- bodiimides, is thermodynamically unstable [23], it can be maintained up to at least 680 K, which is the prepa- ration temperature reported in Ref. [7]. Other options of activatingthe E′ gorbitalstatesofFeNCNaretheapplica- tion ofpressureandlaserirradiation. The latterhasbeen successfully used for melting the orbital order in several prototype orbital systems [24] and could be applied to FeNCN as well. If the switching of the orbital state is possible, the anticipated effect is a sizable reduction in the bandgap(foratleast0.4eV),while thestructurewill probably adjust to the new orbital state. However, it is more likely that severalcompeting E′ gstates will form an orbital liquid, thereby maintaining the high symmetry of the crystal structure. To probe such effects, further ex- perimental work on FeNCN is highly desirable. We also mention an isostructural compound CoNCN [25], where orbital degrees of freedom arising from Co+2(d7) are ex- pected. In summary, we have shown that FeNCN presents an unusual example of weakly coupled spin and orbital de- grees of freedom acting on a similar energy scale. The ground-state orbital configuration features two electrons on thea1gorbital, in agreement with the simple electro- static arguments, but in contrast to the LDA-based ex- pectations. The calculated properties, such as the band gap, exchange couplings, and N´ eel temperature, are in very good agreement with the experiment. We have also remedied the failure of the recent computational work [8] and confirmed the remarkable performance of DFT+ U techniques applied to Mott insulators with orbital de- grees of freedom. We are grateful to Deepa Kasinathan and Oleg Jan- son forfruitful discussions. We alsoacknowledgeRichard Dronskowskiand Andrey Tchougr´ eefffor drawingour at-tention to FeNCN. A.T. was funded by Alexander von Humboldt Foundation. ∗altsirlin@gmail.com †Helge.Rosner@cpfs.mpg.de [1] K. I. Kugel and D. I. Khomskii, Sov. Phys. JETP 37, 725 (1973); Sov. Phys. Usp. 25, 231 (1982). [2] M. V. Mostovoy and D. I. Khomskii, Phys. Rev. Lett. 92, 167201 (2004). [3] E. Pavarini, E. Koch, and A. I. Lichtenstein, Phys. Rev. Lett.101, 266405 (2008); E. Pavarini and E. Koch, 104, 086402 (2010). [4] J. van den Brink, New J. Phys. 6, 201 (2004). [5] A. van Rynbach, S. Todo, and S. Trebst, Phys. Rev. Lett.105, 146402 (2010). [6] S. Wenzel and A. M. L¨ auchli, Phys. Rev. Lett. 106, 197201 (2011). [7] X. Liu, L. Stork, M. Speldrich, H. Lueken, and R. Dron- skowski, Chem. Europ. J. 15, 1558 (2009). [8] H. Xiang, R. Dronskowski, B. Eck, and A. L. Tchougr´ eeff, J. Phys. Chem. A 114, 12345 (2010). [9] K. Koepernik and H. Eschrig, Phys. Rev. B 59, 1743 (1999). [10] J. P. Perdew and Y. Wang, Phys. Rev. B 45, 13244 (1992). [11] D.Kasinathan, K.Koepernik, andH.Rosner,Phys.Rev. Lett.100, 237202 (2008). [12] D. Kasinathan, K. Koepernik, and W. E. Pickett, New J. Phys. 9, 235 (2007); D. Kasinathan, K. Koepernik, U. Nitzsche, and H. Rosner, Phys. Rev. Lett. 99, 247210 (2007); O.Janson, J. Richter, P. Sindzingre, andH.Ros- ner, Phys. Rev. B 82, 104434 (2010). [13] O. Gunnarsson, O. K. Andersen, O. Jepsen, and J. Za- anen, Phys. Rev. B 39, 1708 (1989). [14] O. K. Andersen, Z. Pawlowska, and O. Jepsen, Phys. Rev. B34, 5253 (1986). [15] A. A. Tsirlin and H. Rosner, Phys. Rev. B 81, 024424 (2010). [16] We use the ground-state antiferromagnetic spin configu - ration with parallel spins in the abplane and antiparallel spins in the neighboring layers. [17] See Supplementary information for representative ene rgy spectra calculated with DFT+ U. [18] Compare to Egap= 2.45 eVand Egap= 1.55−1.85 eVfor thea1gande′ gconfigurations, respectively, at Ud= 5 eV. [19] S. Todo andK. Kato, Phys. Rev.Lett. 87, 047203 (2001). [20] A. F. Albuquerque et al., J. Magn. Magn. Mater. 310, 1187 (2007). [21]L= 12 is sufficient to eliminate finite-size effects for the magnetic susceptibility within the temperature range un- der consideration. [22] The fitted g-value is g= 1.98. [23] M. Launay and R. Dronskowski, Z. Naturforsch. B 60, 437 (2005). [24] S. Tomimoto, S. Miyasaka, T. Ogasawara, H. Okamoto, and Y. Tokura, Phys. Rev. B 68, 035106 (2003); D. A. Mazurenko, A. A. Nugroho, T. T. M. Palstra, and P. H. M. van Loosdrecht, Phys. Rev. Lett. 101, 245702 (2008). [25] M. Krott et al., Inorg. Chem. 46, 2204 (2007).5 Supplementary information for “Orbital order and magnetism of FeNCN” Alexander A. Tsirlin, Klaus Koepernik, and Helge Rosner 224Total Total FeA1gorbital□configuration Eg'orbital□configuration Fe N N C C466 1.5 1.0 /c451.5 /c451.0 /c453.0 /c452.00.0 0.0 0 0 Energy□(eV) Energy□(eV)4a1g a1g 4 /c454 /c454 /c458 /c4580 08 DOS□(eV )/c451 DOS□(eV )/c451 eg' eg' eg eg FIG. 5.Atomic- and orbital-resolved density of states for the A1g(left panel) and one of the E′ g(right panel) orbital configurations of FeNCN. The Fermi level is at zero energy. The on -site Coulomb repulsion parameter is Ud= 7 eV.

2401.12320v1.Finite_momentum_and_field_induced_pairings_in_orbital_singlet_spin_triplet_superconductors.pdf

Finite-momentum and field-induced pairings in orbital-singlet spin-triplet superconductors Jonathan Clepkens1and Hae-Young Kee1, 2,∗ 1Department of Physics and Center for Quantum Materials, University of Toronto, 60 St. George St., Toronto, Ontario, M5S 1A7, Canada 2Canadian Institute for Advanced Research, Toronto, Ontario, M5G 1Z8, Canada (Dated: January 24, 2024) Finite-momentum pairing in a Pauli-limited spin-singlet superconductor arises from the pair- breaking effects of an external Zeeman field, a mechanism which is not applicable in odd-parity spin-triplet superconductors. However, in multiorbital systems, the relevant bands originating from different orbitals are usually separated in momentum space, implying that orbital-singlet pairing is a natural candidate for a finite-momentum pairing state. We show that finite-momentum pairing arises in even-parity orbital-singlet spin-triplet superconductors via the combination of orbitally- nontrivial kinetic terms and Hund’s coupling. The finite-momentum pairing is then suppressed with an increasing spin-orbit coupling, stabilizing a uniform pseudospin-singlet pairing. We also examine the effects of the magnetic field and find field-induced superconductivity at large fields. We apply these findings to the multiorbital superconductor with spin-orbit coupling, Sr 2RuO 4and show that a finite-momentum pseudospin-singlet state appears between the uniform pairing and normal states. Future directions of inquiry relating to our findings are also discussed. I. INTRODUCTION Superconductivity is conventionally understood as the condensation of electron pairs near the Fermi energy with momentum kand−k, i.e., a zero center-of-mass momen- tum, referred to as uniform pairing. A term in the Hamil- tonian that breaks the degeneracy of the paired electrons generally suppresses the pairing. For instance, in the case of a spin-singlet superconductor (SC), the Zeeman field acts as a pair breaker and destroys the superconductiv- ity. However, before the normal state is reached, a finite center-of-mass momentum, q, (finite-q) pairing known as the Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) state [1, 2] is found between the uniform pairing and normal states as the Zeeman field increases. Unlike the spin-singlet pairing, a finite-q pairing in a spin triplet may require a different mechanism, because the Zeeman field acting on the triplet pairs will suppress only the component of thed-vector parallel to the field. If the d-vector is not pinned along a certain direction, it can rotate under the field and thus an FFLO state is not expected. A natural question is what the potential mechanisms for finite-q pairing in spin-triplet SCs are. Finite-q pair- ing that occurs without the explicit breaking of time- reversal symmetry is often referred to as a pair-density wave (PDW) [3], and has been conjectured to be present in a wide variety of correlated materials of interest [4–10]. The putative spin-triplet SC, UTe 2, is one of the materi- als which has shown evidence for PDW order [7, 8]. The Fe-based SCs and Sr 2RuO 4have displayed signatures of finite-q pairing [9–14] and an interorbital spin-triplet or- der parameter has been argued as a plausible option for both of them [15–25]. While the possibility of finite-q ∗hy.kee@utoronto.caspin-triplet pairing remains a relevant topic for these cor- related SCs, various theoretical models have focused on spin-singlet finite-q pairings [26–39], and finite-q pairing involving spin-triplet SCs remains relatively less explored Here, we explore a mechanism for generating finite-q pairing in multiorbital systems. In these systems, vari- ous types of superconducting order parameters exist. No- tably, the even-parity orbital-singlet spin-triplet (OSST) pairing, which implies an antisymmetric wavefunction under the exchange of two orbitals, has been proposed for multiorbital SCs such as Sr 2RuO 4and Fe-based SCs [15– 25]. As orbital degeneracy is essential for uniform orbital- singlet pairing, we anticipate that terms that break the orbital degeneracy, serving as pair-breakers, would hin- der the uniform pairing and finite-q pairing may emerge before reaching the normal state. To illustrate this concept, we first examine a generic two-orbital model featuring Hund’s coupling, resulting in an attractive interaction in the OSST channel within mean-field (MF) theory. In this model, the orbital- dependent potential, known as orbital polarization, along with the interorbital hopping termed orbital hybridiza- tion, gives rise to finite-q pairing. We also study the effects of spin-orbit coupling (SOC) and a Zeeman field, and find the appearance of a Fulde-Ferrell (FF) state for small fields and large SOC. For large fields, we find a field-induced finite-q and uniform pairing phase, which survives for a small range of SOC values. We then inves- tigate the application to a SC involving three t2gorbitals, Sr2RuO 4. Both the SOC and Hund’s interaction have been recognized as playing an important role in under- standing the superconductivity in Sr 2RuO 4[40–46]. The OSST finite-q pairing with a wavevector along the x-axis emerges between the uniform pairing and normal states with the introduction of the in-plane magnetic field. The paper is organized as follows. Since the finite-q pairing requires a perturbation that hinders the uniformarXiv:2401.12320v1 [cond-mat.supr-con] 22 Jan 20242 pairing, we discuss possible pair breakers in the next sec- tion. A generic microscopic model for two orbitals, in- cluding the kinetic terms and SOC, along with the MF pairing terms is introduced in Sec. III. The phase dia- gram for OSST pairing as a function of both the orbital polarization and SOC is obtained. We also obtain the phase diagram for uniform and finite-q phases as a func- tion of magnetic field and SOC in Sec. IV. In Sec. V, we use a three-orbital model to discuss the potential applica- tions of our findings to Sr 2RuO 4, and we discuss further directions of inquiry in the last section. II. PAIR BREAKERS FOR UNIFORM OSST PAIRING In this section, we provide an intuitive discussion on generating finite-q OSST pairing before proceeding to the MF calculations. Considering that the Zeeman field acts as a pair breaker for the spin-singlet by polarizing the spins, any term disrupting the degeneracy of orbitals serves as a pair-breaker and may induce finite-q pairing before reaching the normal state. To formulate such effects, let us consider a two-orbital OSST pairing and pair-breaking term (denoted by hP) in the following Hamiltonian Hmodel : Hmodel =c† k,α[iη2]α,βc† −k,β+c† k,α[hP·η]α,βck,β,(1) where c† k,α(β)creates an electron with momentum kand orbital or spin α(β), and ηis a vector of Pauli matri- ces in orbital or spin space. The first term represents the superconducting pairing. If α, βare spin-indices, the pairing is an orbital-triplet spin-singlet (OTSS), and any direction of the magnetic field, hP=−h, suppresses the pairing by breaking the spin degeneracy. If α, βare orbital indices, the pairing is an OSST. Within a MF theory, the Hund’s coupling can generate an attractive interaction for the OSST pairing [15–17, 20, 47, 48]. The second term represented by hPbreaks the orbital de- generacy. The orbital field in the z-direction, hz P, is the SC gap Pair-breaker OTSS: ∆ 0τ0iσ2 Zeeman field: - τ0(h·σ) OSST: iτ2dab·σiσ2 Orbital polarization: hz Pτ3σ0 Orbital hybridization: hx Pτ1σ0 TABLE I. Terms in the normal state Hamiltonian, hij(k)τiσj, with τi(σj) the Pauli matrices in the orbital(spin) space, which have a pair-breaking effect on a SC gap matrix, ∆( k) = ∆ij(k)τiσjiσ2. The second row is for pair-breakers of OTSS pairing and the third and fourth row are pair-breakers for OSST pairing.orbital polarization corresponding to an orbital depen- dent potential. Another example that generates the or- bital polarization is the nematic order, i.e., a sponta- neous rotational symmetry breaking in the density of the two orbitals, such as ⟨nyz−nxz⟩in (dyz, dxz) orbitals on a square lattice. Alternatively, hz Pcan have a k- dependence due to an anisotropy in the dispersions of the orbitals. The x-component of hPacts similarly, and corresponds to an orbital hybridization via interorbital hopping. If α, βrepresent two layers of a layered system, then hx Pcorresponds to the inter-layer coupling, and the same considerations hold for the odd inter-layer pairing. The OSST pairing along with the respective terms in the Hamiltonian that act as pair-breakers are summa- rized in Table I. For comparison, the OTSS is also listed. Note that the Zeeman field does not have a pair-breaking effect on OSST pairing in the absence of a pinning of the d-vector (for example, via SOC), as discussed previously. III. FINITE-Q OSST PAIRING IN ZERO FIELD To elucidate the mechanism for generating finite-q pairing from the uniform OSST pairing, we will consider the case of two orbitals related by C4, such as ( dyz, dxz) or (px, py). We introduce the basis, ψ† k= (ca† k↑, ca† k↓, cb† k↑, cb† k↓), which consists of creation operators for an electron in one of the two orbitals a, bwith spin σ=↑,↓. We use the Pauli matrices, τiandσi(i= 1, ...3), associated with the orbital and spin degrees of freedom respectively, and useτ0/σ0to denote the identity matrices in orbital/spin space. The normal-state Hamiltonian is then given by h0(k) =ξ+ k 2τ0σ0+ξ− k 2τ3σ0+tkτ1σ0, (2) corresponding to a minimal model for two orbitals on a square lattice respecting time-reversal and inversion symmetries. Taking the symmetry of the orbitals to be (dyz, dxz)-like, the orbital dispersions are ξa/b k= −2t1cosky/x−2t2coskx/y−µwhere t1andt2are the nearest neighbor (n.n.) hoppings along the y/x- and x/y- axis for dyz/dxzorbitals. We have defined ξ± k=ξa k±ξb k, and thus ξ− kplays the role of the k-dependent orbital polarization, similar to hz Plisted in Table I. The or- bital hybridization originating from interorbital hopping istk=−4tabsinkxsinky. Using the Nambu spinor, Ψ† k= (ψ† k, ψT −k+q), the MF Hamiltonian including the SC order parameter matrix denoted by ∆( k) is given by, HMF= Ψ† k h(k) ∆( k) ∆†(k)−h∗(−k+q) Ψk, (3) with h(k) =h0(k). Note that we assume only a single center-of-mass momentum wave vector, q, for simplicity.3 FIG. 1. Phase diagram for OSST pairing as a function of the orbital polarization, δt, within the two-orbital model de- scribed in the main text. Here, the orbital polarization arising from the ( dyz, dxz)-like electronic dispersions generates a k- dependent splitting which gives rise to a transition between the uniform and finite-q OSST pairing. The first transition occurs at ( δt)1≈0.016 and the transition to the normal state at (δt)2≈0.046. The representative Fermi surface (FS) in each phase is also shown. The OSST pairing is represented by ∆( k) =iτ2dab(q)· σiσ2, where dab(q) =−V 8NX kσσ′[iσ2σ]σσ′⟨ca −k+qσcb kσ′−cb −k+qσca kσ′⟩. (4) Here an attractive interaction Vcan be obtained from the Hund’s coupling within the MF theory [15–17, 20, 47, 48], as mentioned previously. We show how the finite-q pairing state arises by numer- ically solving for the three components of the spin-triplet order parameter given in Eq. 4, at zero temperature, us- ing the eigenvectors and eigenvalues of Eq. 3. The energy units are defined by 2 t1= 1 and we set µ=−0.4 and tab= 0.015. While a non-zero value for tabis important to split apart the remaining orbital degeneracy in the re- gions of k-space where ξ− kvanishes, the precise value is not important for our results. Here, we investigate the effect of a k-dependent splitting of the bands via the hop- ping anisotropy, quantified by δt≡t1−t2, i.e., we tune the strength of ξ− k. Similar results can be obtained by adding a rigid shift of the orbitals instead, as discussed previously. The ordering wave vector is fixed to be of the form, q=qˆx, as an examination of alternative wave vectors such as q=q(ˆx+ ˆy) did not yield a MF solution with a lower free energy. The interaction is fixed at V= 0.8, and for each value of δt, the self-consistent solution and free energy are obtained as a function of qto find the ground state solution. The resulting phase diagram as a function ofδtis shown in Fig. 1, with ( δt)1≈0.016 and ( δt)2≈ 0.046. For δt < (δt)1, the uniform, i.e. q= 0, OSST pairing is favoured due to the significant regions of orbital degeneracy in k-space. This uniform state gives way to the finite-q pairing, q̸= 0, for ( δt)1< δt < (δt)2, before the normal state is reached for δt > (δt)2. The three components of the d-vector given by Eq. 4 are non-zero and degenerate in both phases. The order parameters as FIG. 2. Phase diagram for OSST pairing in the two-orbital model as a function of both the orbital polarization, δt, and SOC strength, λ. The three components of the d-vector are degenerate for both q= 0 and q̸= 0 phases for λ= 0. For finite SOC, the uniform pairing (grey) corresponds to dab(q) = (0 ,0, dz ab(0)) and is characterized by the dominant intraband pseudospin-singlet pairing component. The smaller finite-q pairing region (green) corresponds to (0 ,0, dz ab(q)), while the larger finite-q pairing region (red) corresponds to (dx ab(q), dy ab(q),0). Both states have a large interband pair- ing component at the Fermi energy. The (0 ,0, dz ab(q)) state is differentiated from the ( dx ab(q), dy ab(q),0) state by the mix- ture of singlet and triplet pairing in the band basis due to the SOC. The blue arrow represents the value of δtused for the phase diagram shown in Fig. 3. Further details are discussed in the Appendix, including the order parameters for two cuts through the parameter space. a function of δtare shown in the Appendix (see Fig. A1). Effect of SOC The uniform OSST pairing is favoured when the or- bital degeneracy in k-space is large, and gives way to the finite-q pairing as the degeneracy is reduced. How- ever, the introduction of SOC allows for an intraband pseudospin-singlet component of pairing which can sta- bilize the uniform pairing even for large orbital splitting [17, 20, 49]. In light of this, one expects that the SOC and orbital splitting have opposite effects on the finite-q pair- ing. To demonstrate this, we now include the effects of SOC by modifying h(k) in Eq. 3 to, h(k) =h0(k)−λτ2σ3. The SOC takes the form corresponding to the zcompo- nent of the atomic SOC, λLzSz. We tune the strength of the SOC, λ, in addition to the orbital polarization, δt, with the resulting phase di- agram shown in Fig. 2. The y-axis corresponds to the phase diagram shown in Fig. 1, while the pairing regions for non-zero SOC are now split into those where either thedz abor{dx ab, dy ab}order parameters in the orbital ba- sis are non-zero. The uniform pairing state at non-zero4 SOC corresponds to only the z-component of the d-vector being non-zero due to the σ3dependence of the SOC, which generates the intraband pairing component. This is labelled ‘ q= 0 pseudospin-singlet’ in Fig. 2 since, for large orbital polarization, it is the intraband pseudospin- singlet component which is not only dominant, but also supports the uniform pairing over the finite-q pairing. The finite-q pairing appears in a dome-like shape due to competing effects of δtand SOC. There are two separate finite-q pairing regions, with thed-vector rotating from the out-of-plane, ( z), to in- plane, ( x/y), direction when the orbital polarization is large enough. This can be explained by the dx ab/dy aborder parameters projecting mostly to interband triplet pair- ing, while the dz aborder parameter contributes to both in- traband singlet and interband singlet and triplet pairing. Indeed, the (0 ,0, dz ab(q)) state occurs near the boundary of the uniform (0 ,0, dz ab(0)) state dominated by intra- band pairing. As shown in Fig. B1 of the Appendix, the (0,0, dz ab(q)) phase gives rise to both interband gaps at the Fermi energy, and intraband gaps close by, in contrast to the purely interband gaps in the ( dx ab(q), dy ab(q),0) phase. Additional details, including the MF order pa- rameters for a representative cut through the parameter space, are given in the Appendix. The finite-q OSST pairing will therefore be favoured when the SOC is either zero, or intermediate in strength, depending on the value of the orbital polarization. As the SOC increases, the OSST pairing is no longer well- defined as the orbitals and spins are strongly coupled, and the pairing is described by an (intraband) pseudospin singlet, even though the microscopic interaction leading to the attractive interaction is defined on the original orbital and spin basis, such as the Hund’s coupling. This type of spin-triplet pairing was referred to as a shadowed triplet [25, 50]. IV. MAGNETIC FIELD INDUCED PHASES Since the SOC leads to a mixture of singlet and triplet pairing, the question of how these pairing states respond to the magnetic field naturally arises. To investigate this, we now include the effects of an in-plane Zeeman field by modifying h(k) in Eq. 3 to, h(k) = h0(k)−λτ2σ3− hxτ0σ1, and treat hxas a tuning parameter. The in- plane direction for the field is chosen, as we focus on the Zeeman coupling and neglect the coupling of the orbital angular momentum to the field. Starting from zero SOC, and fields that are small rela- tive to the band splitting, the field merely suppresses the x-component of the triplet order parameter. When the field is of the order of the band splitting, a re-entrant or field-induced pairing state with only the xcomponent of thed-vector may appear. Therefore, we consider a value of the orbital polarization such that there is no supercon- ductivity, corresponding to δt= 0.05, as shown by the blue arrow in Fig. 2, and study the effects of the field and FIG. 3. Phase diagram as a function of magnetic field, hx, and SOC, λ. For λ≳0.022, the uniform state (grey) appears, and is destabilized by the field in favour of a finite-q state with a qvector connecting the spin-split bands (green). The order parameters as a function of hxatλ= 0.05 are shown in the right inset. There is also an orders of magnitude smaller dy abcomponent with a relative phase ofπ 2induced by the in-plane field which is omitted for clarity. At large fields, there is a field-induced uniform pairing (blue) due to the field compensating for the orbital splitting and bringing oppositely spin-split bands together. The uniform pairing is surrounded by a finite- qpairing (pink), mirroring the behaviour at low- field. The order parameters are shown for λ= 0.005 on the left, and the FS for ( λ, hx) = (0 ,0.067)) is shown in Fig. C1, along with the interband pairing amplitude. SOC together. The resulting phase diagram is shown in Fig. 3. The magnetic field is increased from zero to a value on the same order as the band splitting on the y-axis, and the SOC is increased from zero along the x-axis. As shown in Fig. 3, there are four distinct pairing states. For val- ues of the SOC larger than λ≳0.022, a uniform pair- ing state with order parameter, (0 ,0, dz ab(0)), arises due to the SOC generating an intraband pseudospin-singlet pairing component. This pairing state behaves similarly to a true spin-singlet and is therefore suppressed by the magnetic field, giving way to a finite-q pairing state with a small wave vector, q=qˆx, matching the field split- ting. This phase resembles a field-driven FF phase, but originates from OSST pairing. It is distinct from the zero-field PDW phase, which corresponds to interband pairing between two Kramer’s degenerate bands, rather than between two spin-split bands. The order parameters are plotted as a function of the magnetic field in the inset on the right for λ= 0.05. A jump corresponding to the transition from q= 0 to q̸= 0 can be seen. In addition to the primary out-of-plane spin-triplet order parameter, an orders of magnitude smaller dy abcomponent with a relativeπ 2phase is induced by the in-plane field, which we omit for clarity. When the magnetic field approaches the band splitting5 caused by the orbital polarization and hybridization, the two bands which are oppositely spin-polarized approach each other in kspace. This is shown by the represen- tative FS for ( λ, hx) = (0 ,0.067) in the Appendix (see Fig. C1). When the SOC is small, the field reduces the band splitting enough to drive a transition to the uniform pairing with order parameter, ( dx ab(0),0,0), i.e., pairing between up- and down-spin along the xdirection. As the SOC is increased, due to the non-commuting nature of the Zeeman field proportional to σ1and the SOC propor- tional to σ3, the bands do not approach each other suf- ficiently close enough for the uniform pairing to emerge. Instead, a window of finite-q pairing with order parame- ter, (dx ab(q),0,0), is obtained, where qis given by the mis- match between the two oppositely spin-polarized bands. This phase is labelled by PDW′in Fig. 3 since it occurs in the presence of time-reversal symmetry breaking but is distinct from an FF phase. As the SOC is increased further, this pairing phase is also suppressed. The or- der parameters as a function of the field are shown in the inset on the left in Fig. 3 for λ= 0.005. The order parameters correspond to a cut through the region with the uniform pairing surrounded by two finite-q pairing states. A similar field-induced uniform interband pairing has been found recently in Refs. [32, 51], starting from purely spin-singlet pairing. In Ref. [32], a single-band system was considered, with the field-induced phase requiring the presence of altermagnetism. Ref. [51] considered an effective two-band conventional spin-singlet SC with both intra and interband pairing. Here, we have shown that OSST pairing exhibits a similar uniform field-induced phase, and that such a phase can survive inclusion of the SOC up to a critical value which will depend on the exact microscopic model. However, in addition to the uniform field-induced phase, we also find a finite-q pairing state which is induced by a large magnetic field. The tran- sitions between the finite-q and uniform phases at large field mirror the FF-like pairing phase transition for larger values of SOC. For the former, the pairing wave vec- tor connects oppositely spin-polarized bands that orig- inate from two distinct zero-field Kramer’s degenerate bands. Concretely, the pairing in the band basis, de- fined by fermionic operators, f† i,k,s, in bands i= 1,2 with pseudospin s= +/−, corresponds to interband pairing, ⟨f† 1,k,−f† 2,−k+q,+⟩ ̸= 0. With negligible SOC, the spins +/−are defined by the direction of the field. In the FF- like pairing state, the dominant pairing component in the band basis for small fields corresponds to intraband pair- ing, i.e., ⟨f† i,k,−f† i,−k+q,+⟩ ̸= 0, which is possible due to the SOC. V. FINITE-Q PAIRING IN Sr 2RuO 4 To show the applicability of finite-q OSST pairing be- yond a two-orbital model, we consider three t2gorbitalson a square lattice, along with sizeable SOC and Hund’s coupling. Depending on the tight-binding parameters used, such a model can describe Sr 2RuO 4, and other ma- terials such as the iron-pnictide SCs [52]. Here, we con- sider the application to Sr 2RuO 4and show that OSST pairing can give rise to a finite-q pairing state. We consider an effective interaction Hamiltonian con- taining an attractive channel for OSST pairing origi- nating from the on-site Hubbard-Kanamori interaction terms [15–17, 20, 47, 48], Heff=−2NVX {a̸=b}ˆd† a/b(q)·ˆda/b(q), (5) where Nis the number of sites and {a̸=b}represents a sum over the unique pairs of orbital indices. The nine order parameters, da/b, are related to those in Eq. 4 by da/b=−2 Vdab. The interaction strength, V=JH−U′, is given in terms of the renormalized low-energy interorbital Hubbard ( U′) and Hund’s interaction ( JH) parameters [17, 20]. As before, we consider only a single wave vector, q, in the MF theory to simplify the calculation, as the qualitative physics is not expected to change from this choice. We use the normal state Hamiltonian described in Ref. [22], and described in the Appendix, which includes a sizeable value of SOC, λ= 0.085, appropriate for Sr2RuO 4[43]. We express energies in units of 2 t3= 1, where t3is the nearest-neighbour hopping of the dxyor- bital. An external in-plane Zeeman field is included, HZeeman =−ghxP iSx i, with g= 2 assumed for sim- plicity. Fixing the interaction strength at V= 0.6, we self-consistently solve for the order parameters, da/b(q), at zero temperature as a function of q. As before, wave vectors of the form, q=qˆx, are found to be more stable thanq=q(ˆx+ ˆy). The resulting order parameters, along with the asso- ciated wave vectors, are shown in Fig. 4. Starting from zero, the in-plane field, hx, is increased along the x-axis. At zero field, the atomic SOC, L·S, favours a uniform pairing state with the three primary non-zero order pa- rameters, ( dx xz/xy(0), dy xy/yz(0), dz yz/xz(0)). The in-plane magnetic field also induces the dz xy/yzanddy yz/xzorder parameters to become finite with a relative phase ofπ 2, however they are orders of magnitudes smaller. The in- traband pseudospin-singlet component of the pairing is apparent from the suppression of all three components of the d-vector as the field is increased from zero. At hx≈0.003, the pair-breaking effect of the field is large enough such that a finite-q pairing develops, and there is a transition to an FF phase. At this point, the xcompo- nent of the d-vector is noticeably suppressed compared to the ycomponent, as expected for the field along the xdirection for a triplet order parameter. In the band basis where orbtials and spins are mixed, the intraband pseudospin-singlet is dominant, as discussed above. The wave vector increases with the field until the normal state is reached at hx≈0.006.6 FIG. 4. MF order parameters, da/b, and ordering wave vec- tor,q, for the three-orbital model [22] as a function of in- plane magnetic field, hx. Of the nine order parameters, only the three non-zero ones are shown for clarity. The in-plane magnetic field along the x-direction also induces the dz xy/yz anddy yz/xzorder parameters to become non-zero with a rela- tive phase ofπ 2, however they are orders of magnitude smaller and thus omitted. The results shown here with three orbitals are similar to the low-field pairing phases shown in Fig. 3 for the sim- pler two-orbital model. The presence of finite-q supercon- ductivity in the case of moderately large SOC is therefore robust to additional complexity introduced when describ- ing a real multiorbital material such as Sr 2RuO 4. As shown by Fig. 3, the high-field phases are expected to be relevant for a material with sizable band degeneracy and small SOC. The application of these parts of the pa- rameter space, as well as those found in Fig. 1 for other multiorbital SCs such as the Fe-based SCs are left for future study. VI. SUMMARY AND DISCUSSION In summary, we have shown how finite-q pairing, re- ferred to as PDW or FF phases depending on the origin of the finite center-of-mass momentum, can be generated from uniform OSST pairing. With zero external field, the PDW state arises due to the pair-breaking effects of the orbital polarization and/or hybridization on the uniform state. Using a simple model of two orbitals on a square lattice, we demonstrate the appearance of the PDW upon increasing the orbital polarization. When the SOC is in- troduced, the PDW region splits into two separate phases with different d-vector directions due to the competition between purely interband pairing characterized by the d- vector in the x-y plane, versus a combination of intraband and interband pairing characterized by the d-vector along the z-axis. An external in-plane magnetic field inducesthe uniform OSST pairing at high fields for small SOC, which becomes a finite-q pairing for larger SOC. Upon increasing SOC, the uniform pseudospin-singlet pairing is found at low fields, which becomes the FF phase by increasing the field. We have applied this picture to the three-orbital SC, Sr 2RuO 4, which has exhibited signs of an FFLO phase in recent nuclear magnetic resonance ex- periments [14]. Within a realistic three-orbital model for the t2gorbitals and SOC, the uniform OSST pairing evolves into an FF phase with an in-plane Zeeman field. This phase is characterized by three distinct spin-triplet components with the associated d-vectors. Due to the strong SOC, the dominant pairing can be described as an intraband pseudospin-singlet. A few aspects concerning the limitations of the cur- rent study and topics for future investigation warrant some discussion. For a two-orbital model, we have used the component of the atomic SOC, LzSz, relevant for the dxzanddyzorbitals. In this case, the phase diagram de- pends on the direction of the field. We have presented the phase diagram for the field along the x(or equivalently y)-direction. When the field aligns with the z-direction, the uniform field-induced phase may extend to larger val- ues of SOC. In this scenario, a transition between multi- ple uniform and finite-q phases separated by a large field interval may be possible. However, if this corresponds to the out-of-plane direction, the orbital limiting of the pairing will have to be considered. We note that in the spin-triplet SC, UTe 2, there is a complicated phase dia- gram as a function of magnetic field, which exhibits low- and high-field pairing phases [53], in addition to recent evidence for finite-q superconductivity [7, 8]. The ap- plication of OSST pairing to this problem is left as an interesting future direction. Here we limit the analysis to the finite-q pairing with a single wave vector. Further study considering both qand −qcomponents simultaneously with an OSST order pa- rameter is desirable to compare to the nuclear magnetic resonance Knight shift measurement in the FFLO state. An examination of the differences between the finite-q OSST pairing and one arising from a more conventional spin-singlet order parameter, as well as the effect of dif- ferent pairing symmetries are also interesting future di- rections. We leave a more detailed examination of these specific questions for understanding Sr 2RuO 4for future study. An additional candidate for finite-q pairing originating from an OSST order parameter is the family of Fe-based SCs. The OSST pairing state has been proposed previ- ously to explain the superconductivity of the iron pnic- tide SCs [16, 20, 23], due to the strong Hund’s coupling and significant SOC [54]. Indeed, a PDW state based on OSST pairing was also proposed using a two-band model for LaFeAsO 1−xFx[55]. However, the SOC and associated intraband pairing was neglected in Ref. [55]. We have shown that the finite-q PDW phase within our model can survive the inclusion of SOC, and leads to two different PDW states. Furthermore, we have found the7 presence of finite-q and uniform field-induced phases for a range of SOC values up to 3% of the largest n.n. hop- ping parameter. The SOC strength for the Fe-based SCs may be of an appropriate size for these phases to become relevant. However, whether an OSST PDW phase ap- pears in the iron-pnictides taking into account the SOC and competition with other order parameters such as a spin-density wave remains as an interesting future study. ACKNOWLEDGMENTS This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) Dis- covery Grant No. 2022-04601, the Center for Quantum Materials at the University of Toronto, the Canadian In- stitute for Advanced Research, and the Canada Research Chairs Program. Computations were performed on the Niagara supercomputer at the SciNet HPC Consortium. SciNet is funded by the Canada Foundation for Innova- tion under the auspices of Compute Canada, the Gov- ernment of Ontario; Ontario Research Fund - Research Excellence, and the University of Toronto. APPENDIX A: MF ORDER PARAMETERS The MF order parameters corresponding to the three components of Eq. 4 are shown for two different cuts through the phase diagram of Fig. 2 in Fig. A1. The top left (right) panels of Fig. A1 correspond to fixed values ofλ= 0(0 .005) respectively, and zero magnetic field, hx= 0. A representative FS for λ= 0 and a value of δt= 0.035, where the q̸= 0 state is stabilized, is shown as an inset in the left panel. The ordering wave vector is of the form, q=qˆx, connecting the two bands. Since the SOC is zero, the OSST pairing corresponds to purely interband spin-triplet pairing in the band basis for q= 0. The right panel shows the two separate q̸= 0 phases with different d-vector directions that appear in Fig. 2 for small SOC. FIG. A1. MF order parameters corresponding to cuts through Fig. 2 from the main text.Note that the difference between the in-plane ( dx/y ab) and out-of-plane ( dz ab) order parameters is that, for q= 0, the in-plane one projects to purely interband spin-triplet pairing, while the out-of-plane one projects to both inter- band spin-triplet, interband spin-singlet, and intraband spin-singlet. When q̸= 0, both states may also acquire an additional small intraband pairing component due to contributions from kinetic terms which are odd under k→ −k+q. For small values of orbital polarization, δt, the out-of-plane triplet order parameter allows for an intraband component of pairing to exist near the Fermi energy as SOC is increased (see the quasiparticle dis- persion below in Fig. B1). However once the splitting between bands is large enough, δt≈0.025, the in-plane pairing state becomes more favourable due to a larger interband pairing component. APPENDIX B: QUASIPARTICLE DISPERSION The quasiparticle dispersion obtained by diagonalizing Eq. 3 of the main text for the MF solution with λ= 0.005 andδt= 0.02 (left) as well as δt= 0.025 (right) are shown in Fig. B1. The former corresponds to the state with out-of-plane d-vector, while the latter corresponds to the in-plane state. The top row of Fig. B1 shows the dispersion plotted over the direction in k-space parallel to the wave vector, q=qˆx, i.e., the X′−Γ−X direction as depicted in the inset FS. The finite qgives rise to an overlap between a particle f1-band and the corresponding shifted hole f2-band at the Fermi energy for both positive and negative k. This opens up a gap originating from the interband pairing. However, due to the single- q, an unpaired set of bands arise for both positive and negative k, leading to the depaired regions [1]. The intraband crossings are shifted away from the Fermi energy, and are gapped (gapless) for δt= 0.02(0.025) as shown by the left(right) panels in the top row of Fig. B1. This is due to the intraband singlet component of pairing occurring only for the out-of-plane d-vector pairing state. The bottom row of Fig. B1 shows the direction per- pendicular to q, i.e., the Γ −Y direction. Only the pos- itivekdirection is shown for clarity since the disper- sion is symmetric. Since this direction is perpendicular toq, the two distinct bands near the Y point are not connected to each other after being shifted, but rather mostly overlap with themselves. Therefore, the interband gaps occur away from the Fermi energy, similar to the q= 0 case. The intraband crossings are shifted slightly away from the Fermi energy and are gapped (gapless) for δt= 0.02(0.025).8 FIG. B1. Quasiparticle dispersion for the MF solutions corresponding to λ= 0.005 and δt= 0.02(0.025) in the left(right) panels. The top row shows the X′−Γ−X direction, which is parallel to the pairing wave vector, q=qˆx. The bottom row shows the Γ −Y direction, which is perpendicular to q, as shown by the inset FS in the top left panel. Only the positive k direction is shown for the y-direction since the dispersion is symmetric. APPENDIX C: FIELD-INDUCED PAIRING The field-induced pairing region shown in Fig 3 of the main text arises due to the oppositely spin-polarized bands approaching each other in k-space as the magnetic field is increased. For this to occur, the pairing in the orbital basis must project to interband pairing. For zero SOC, the uniform OSST order parameters projects to purely interband pairing, making the field-induced phase robust. To show this, the interband pairing amplitude, ⟨f2,k,+f1,−k,−⟩is shown in Fig. C1 for a representative field value for zero SOC in the uniform field-induced region, ( λ, δt, h x) = (0 ,0.05,0.067), with order parameter, ( dx ab(0),0,0). The underlying normal-state FS is shown by the grey dashed line. The pairing is maximal on the two overlapping oppositely spin-polarized bands, f2,k,+, f1,k,−, and reaches a maximal value of 0.5. This can be compared to the zero-temperature BCS value for the pairing amplitude evaluated on the FS, ⟨ck,↑c−k,↓⟩=∆k 2√ ξ2 k+|∆k|2FS− − →1 2∆k |∆k|. FIG. C1. Interband pairing amplitude for the field-induced pairing state with order parameter, ( dx ab(0),0,0), plotted over the normal-state FS shown by the grey dashed lines, for ( λ, δt, h x) = (0 ,0.05,0.067).9 t1 t2 t3 t4 t5 λ µ 1 µ2 0.45 0.05 0.5 0.2 0.025 0.085 0.531 0.631 TABLE D1. Tight-binding parameters for the three-orbital model from Ref. [22] used in Sec. V of the main text. All parameters are in units of 2 t3= 1. APPENDIX D: THREE-ORBITAL TIGHT-BINDING MODEL The normal state Hamiltonian for the three-orbital model in Sec. V of the main text is, H0=X k,σ,mξm kcm† kσcm kσ+X k,σtkcyz† kσcxz kσ+ H.c. + iλX k,l,m,nϵlmncl† kσcm kσ′σn σσ′, (D1) including the orbital dispersions, hybridization between ( dyz, dxz) orbitals, and atomic SOC respectively. 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2204.11664v3.Harmonic_generation_in_bent_graphene_with_artificially_enhanced_spin_orbit_coupling.pdf

Harmonic generation in bent graphene with articially-enhanced spin-orbit coupling Arttu Nieminen1and Marco Ornigotti1 1Tampere University, Photonics Laboratory, Physics Unit, FI-33720 Tampere, Finland We theoretically investigate the nonlinear response of bent graphene, in the presence of articially- enhanced spin-orbit coupling, which can occur either via adatom deposition, or by placing the sheet of bent graphene in contact with a spin-orbit active substrate. We discuss the interplay between the spin-orbit coupling and the articial magnetic eld generated by the bending, for both the cases of Rashba and intrinsic spin-orbit coupling. For the latter, we introduce a spin-eld interaction Hamiltonian addressing directly the electron spin as a degree of freedom. Our ndings reveal that in this case, by controlling the amount of spin-orbit coupling, it is possible to signicantly tune the spectrum of the nonlinear signal, achieving, in principle, ecient conversion of light from THz to UV region. I. INTRODUCTION Since its discovery in 2004 [1], graphene has attracted a lot of interest in the scientic community, mainly due to its exquisite, and unexpected, electronic, mechanical, and thermal properties [2, 3], but also fascinating opti- cal properties, such as universal absorption [4], ultrafast broadband response [5], and large nonlinear optical re- sponses [6, 7], to name a few. Most of these properties derive directly from the presence, in the band structure of graphene, of Dirac cones, i.e., points in k-space, where valence and conduction band touch, thus giving rise to a gapless linear dispersion [8]. Monolayers of graphene also admit spin-orbit coupling (SOC), in the form of both in- trinsic (
I) and Rashba (
R) coupling, the former orig- inating as a true SOC due to the relativistic nature of electrons in graphene, while latter occurring only in the presence of an external electric eld [8]. For the partic- ular case of
I>
R=2, Kane and Mele discovered, for a nite monolayer of graphene, that the gap opened by SOC at the Dirac points sustains topologically protected edge states near its boundary [9], where spin-dependent impurity backscattering is strongly suppressed, resulting in the so-called quantum spin Hall (QSH) edge states. The discovery of Kane and Mele, moreover, was so in u- ential, that it gave birth to the exciting eld of topolog- ical insulators [10], which, soon after its discovery in the context of condensed matter physics, started contaminat- ing other elds of physics, giving rise to new ideas, such as topological photonics [11, 12], topological mechanics [13], and topological atomic physics [14]. The QSH eect, however, is quite hard to be observed experimentally in pristine graphene, since the value of the intrinsic SOC is too small, to allow its experimen- tal verication [15{17]. The QSH state, however, has been experimentally observed in other systems, such as HgTe quantum wells [18], InAs/GaSb quantum wells [19], and WTe 2[20], to name a few. To overcome the prob- lem of small SOC in graphene, several dierent strategies have been proposed, ranging from increasing the Rashba marco.ornigotti@tuni.coupling by depositing graphene on Ni surfaces [21, 22], to a signicant increase of the intrinsic SOC by adatom deposition of dierent compounds [23], such as Indium, Thallium, or Bi 2Te3nanoparticles, the latter represent- ing the rst experimental evidence of the occurrence of QSH eect in graphene with articially enhanced SOC [24]. Contextually, several works investigated the eect of Rashba [25{27] and intrinsic SOC on the electronic struc- ture of graphene, in the presence of magnetic elds [28]. The latter, in particular, have attracted considerable at- tention in the last years, because they can be realised by applying strain or bending to single and multilayered 2D materials [29, 30], resulting in high articial magnetic elds, which, contrary to the real ones, cannot break time-reversal symmetry. A comprehensive review on the topic can be found in Ref. [31]. Interestingly, though, only few works investigated the eects of (pseudo)magnetic elds in the nonlinear op- tical response of graphene, and they have been mainly focused on estimating how the third-order nonlinear sus- ceptibility of graphene depends on the applied magnetic eld, in the limit of strong magnetic eld [32]. In a re- cent work, moreover, the role of a constant, out-of-plane magnetic eld in the nonlinear response of graphene has been thoroughly investigated, revealing the possiblity to use the magnetic eld strength to control the frequency conversion, up to the visible range [33]. None of them, though, investigated the role SOC might have in shap- ing the nonlinear optical response of graphene, and, more generally, 2D materials. In this work, we investigate the eect of both intrin- sic and Rashba SOC on the nonlinear signal generated by an ultrashort electromagnetic pulse impinging upon a ake of bent graphene. To do so, we extend the formal- ism recently developed by one of the authors in Ref. 33 to explicitly account for the presence of a nonzero SOC coupling, and accomodate explicitly spin dynamics in the model. For the case of intrinsic SOC, in particular, we exploit the fact that the spin degeneracy of the electronic bands is lifted and we introduce a spin-eld interaction Hamiltonian and compare its action with the standard sub-lattice (i.e., minimal coupling) interaction Hamilto-arXiv:2204.11664v3 [cond-mat.mes-hall] 9 Sep 20222 nian. Our results show, that by addressing directly the spin degree of freedom of Dirac electrons in graphene it is possible, by controlling the level of intrinsic SOC, to signicantly broaden the spectrum of the nonlinear sig- nal, allowing ecient conversion of light from THz to the UV region. This work is organised as follows: in Sect. II we present the basic model used in this work, namely the Hamilto- nian for a 2D material (graphene, in this specic case) in the presence of SOC. Section III is then dedicated to the case of bent graphene, and to deriving its eigen- states and eigenvalues, for both the cases of intrinsic and Rashba SOC. In Section IV, then, we brie y discuss how to introduce the spin-eld interaction in the graphene Hamiltonian, and discuss explicitly the cases of linear and circular polarisation. Section V is then dedicated to the discussion of the nonlinear signal in the presence of SOC. Finally, conclusions are drawn in Sec. VI. II. GRAPHENE HAMILTONIAN WITH SOC Electron dynamics in graphene are typically de- scribed using a 4-component spinor, ( r;t) = (A K;B K;A K0;B K0)T, wherefA;Bgrefer to the sub- lattice site (associated to the so-called pseudospin degree of freedom), referring to the two carbon atoms per unit cell, and the indices fK;K0gindicate the two nonequiv- alent valleys in k-space [8]. If SOC is present, the spin degeneracy of the two Dirac bands in each valley is lifted, leading to a spin-resolved 4 band system in each valley (see Fig. 1). In this case, then, electron dynamics are completely described by means of a 8-component spinor (r;t)("(r;t);#(r;t))T, wheref";#gis the spin index. To write the Hamiltonian for graphene in the presence of SOC, notice that ( r;t) depends on three in- dependent degrees of freedom, namely pseudospin (sub- lattice), spin, and valley, each spanning one of three dif- ferent two-dimensional subspaces, i.e., ( r;t)2 H  Hvalley HAB Hspin, where each individual subspace His spanned by its own set of Pauli matrices  x;y;z, and dimfHg = 8. Since SOC does not mix the valley degree of freedom, we can factor out the valley degree of freedom and reorganise the elements of ( r;t) by introducing the spin-resolved valley spinor = ( ";A; ";B; #;A; #;B)T, with=f1;1gfK;K0gbeing the valley index, so that ( r;t) = (K;K0)T. By doing so, we can then write the single-valley Hamiltonian in the presence of SOC as ^H=^H 0+^H I+^H R; (1) where ^H 0=vf(pxIs x+pyIs y); (2) is the free Hamiltonian, Isis the identity matrix in spin subspace,x;yare the Pauli matrices spanning the two- D 62&JDSEFIG. 1. Band structure of graphene in the vicinity of one Dirac valley (for example, K), for both the case of no SOC (a), where the usual gapless, linear dispersion relation appears, and in the presence of SOC (b). Depending on the kind of SOC considered, a gap can be opened at the Dirac point (intrinsic SOC), and the spin-degeneracy of both valence and conduction bands can be lifted (Rashba SOC). dimensional sub-lattice space HAB, ^H I=
Isz z; (3) (withsx;ybeing the Pauli matrices spanning the two- dimensional spin space Hspin) describes intrinsic SOC, which accounts for the case where the electron spin is oriented perpendicular to the graphene plane [8], and ^H R=
R(sy xsx y); (4) is the Rashba Hamiltonian, describing the case in which the electron spin is oriented in the plane of graphene, and it is responsible for the spin-momentum locking [8]. In graphene, the magnitude of both SOC terms is gen- erally small, with the Rashba term being
R'10eV per V/nm [34], when an electric eld is applied, and the intrinsic SOC term
I'24eV [35]. These ef- fects, however, can be articially enhanced by suitable adatom deposition, such as Indium and Thallium [23], or by putting the graphene sheet in contact with tellurites- based nanoparticles [24]. This results in an increase of the SOC of graphene of many orders of magnitude. It is worth noticing, that although in the remaining of this manuscript we will refer only to articially enriched graphene, the model presented in this work can be eas- ily adapted to any 2D material in the presence of SOC, such as, for example, transition metal dichalcogenides (TMDs). III. BENT GRAPHENE IN THE PRESENCE OF SOC The discussion above is valid for an unstrained single layer of graphene. When strain, or bending, is taken into account, an articial gauge eld (AGF) emerges, whose explicit expression depends on the nature of the strain3 (a) L W (b) <latexit sha1_base64="0fXTsyU99XsyV5GfDDvNE2buPA8=">AAACB3icbVDLSsNAFJ3UV62vqEtBBovgQkpSiroRSt24rGAf0IQymU7aoZNJmJkINWTnxl9x40IRt/6CO//GSRtBWw8MnDnnXu69x4sYlcqyvozC0vLK6lpxvbSxubW9Y+7utWUYC0xaOGSh6HpIEkY5aSmqGOlGgqDAY6Tjja8yv3NHhKQhv1WTiLgBGnLqU4yUlvrmoRMgNfL8pJFeNpzTn58zQiq5T9O+WbYq1hRwkdg5KYMczb756QxCHAeEK8yQlD3bipSbIKEoZiQtObEkEcJjNCQ9TTkKiHST6R0pPNbKAPqh0I8rOFV/dyQokHISeLoy21POe5n4n9eLlX/hJpRHsSIczwb5MYMqhFkocEAFwYpNNEFYUL0rxCMkEFY6upIOwZ4/eZG0qxX7rFK7qZXr1TyOIjgAR+AE2OAc1ME1aIIWwOABPIEX8Go8Gs/Gm/E+Ky0Yec8++APj4xt/dpmw</latexit>B=Bˆz <latexit sha1_base64="beMwGDWsaodeuvKWTwaa/Ro2Ojk=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHYJUY8kXjxCIo8ENmR2aGBkdnYzM6shG77AiweN8eonefNvHGAPClbSSaWqO91dQSy4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS0eJYthkkYhUJ6AaBZfYNNwI7MQKaRgIbAeT27nffkSleSTvzTRGP6QjyYecUWOlxlO/WHLL7gJknXgZKUGGer/41RtELAlRGiao1l3PjY2fUmU4Ezgr9BKNMWUTOsKupZKGqP10ceiMXFhlQIaRsiUNWai/J1Iaaj0NA9sZUjPWq95c/M/rJmZ446dcxolByZaLhokgJiLzr8mAK2RGTC2hTHF7K2FjqigzNpuCDcFbfXmdtCpl76pcbVRLtUoWRx7O4BwuwYNrqMEd1KEJDBCe4RXenAfnxXl3PpatOSebOYU/cD5/AOLTjPQ=</latexit>w <latexit sha1_base64="E/FyVENwGYI/8ue/djPIcsqt61U=">AAAB63icbVBNS8NAEJ3Ur1q/qh69LBbBU0lKUY8FLx4r2FZoQ9lsJ+3S3STsboQS+he8eFDEq3/Im//GTZuDtj4YeLw3w8y8IBFcG9f9dkobm1vbO+Xdyt7+weFR9fikq+NUMeywWMTqMaAaBY+wY7gR+JgopDIQ2Aumt7nfe0KleRw9mFmCvqTjiIecUZNLAxRiWK25dXcBsk68gtSgQHtY/RqMYpZKjAwTVOu+5ybGz6gynAmcVwapxoSyKR1j39KIStR+trh1Ti6sMiJhrGxFhizU3xMZlVrPZGA7JTUTverl4n9ePzXhjZ/xKEkNRmy5KEwFMTHJHycjrpAZMbOEMsXtrYRNqKLM2HgqNgRv9eV10m3Uvat6875ZazWKOMpwBudwCR5cQwvuoA0dYDCBZ3iFN0c6L86787FsLTnFzCn8gfP5AwuEjjQ=</latexit>` <latexit sha1_base64="jFNh2GS2fgIx94gYk5idwj9W+o8=">AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHYJUY8kXjyCkUcCGzI79MLI7OxmZtaEEL7AiweN8eonefNvHGAPClbSSaWqO91dQSK4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS8epYthksYhVJ6AaBZfYNNwI7CQKaRQIbAfj27nffkKleSwfzCRBP6JDyUPOqLFS475fLLlldwGyTryMlCBDvV/86g1ilkYoDRNU667nJsafUmU4Ezgr9FKNCWVjOsSupZJGqP3p4tAZubDKgISxsiUNWai/J6Y00noSBbYzomakV725+J/XTU1440+5TFKDki0XhakgJibzr8mAK2RGTCyhTHF7K2EjqigzNpuCDcFbfXmdtCpl76pcbVRLtUoWRx7O4BwuwYNrqMEd1KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AKq/jM8=</latexit>R<latexit sha1_base64="hQ37zUqG6tV8KvoinkttyAAQzw0=">AAAB6XicbVDLTgJBEOzFF+IL9ehlIjF6IruEqEcSLx6RyCMBQmaHWZgwO7uZ6TUhG/7AiweN8eofefNvHGAPClbSSaWqO91dfiyFQdf9dnIbm1vbO/ndwt7+weFR8fikZaJEM95kkYx0x6eGS6F4EwVK3ok1p6Eveduf3M399hPXRkTqEacx74d0pEQgGEUrNRqXg2LJLbsLkHXiZaQEGeqD4ldvGLEk5AqZpMZ0PTfGfko1Cib5rNBLDI8pm9AR71qqaMhNP11cOiMXVhmSINK2FJKF+nsipaEx09C3nSHFsVn15uJ/XjfB4LafChUnyBVbLgoSSTAi87fJUGjOUE4toUwLeythY6opQxtOwYbgrb68TlqVsnddrj5US7VKFkcezuAcrsCDG6jBPdShCQwCeIZXeHMmzovz7nwsW3NONnMKf+B8/gALII0A</latexit>R0 FIG. 2. (a) Pictorial representation of a rectangular ake of gr- pahene, deformed into an arc. The radii of the upper, and lower edges are, respectively, RandR0. (b) Flattened equivalent geome- try of the bent graphene ake in panel (a). The curvature induced by the bending is replaced with an articial gauge eld As, which gives rise to a uniform pseudomagnetic eld B=B^zparallel to thez-direction. The width wand length `of the attened ake can be calculated from the bent structure, and might be extended to innity for making calculations easier, as this operation does not change the essential role the pseudomagnetic eld has in light- matter interaction. or bending applied to the monolayer [36]. In the absence of out-of-plane modulations, such induced AGF can be generally written as A(s)=C(uxxuyy)^x2Cuxy^y; (5) whereCis a suitable constant, and u=@u@u is the strain tensor [37]. Assuming a strain prole as the one described in Ref. 33, and pictorially represented in Fig. 2, we obtain an AGF A(s)=Byx, correspond- ing to a uniform magnetic eld oriented perpendicular to the graphene plane, i.e., B=B^z. The interaction of electrons in graphene with the AGF dened above can be introduced through minimal coupling, namely by re- placing the kinetic momentum pof the electron, with the canonical momentum, i.e., p7!p+eA(s), in Eq. (2). This is equivalent to introducing a magnetic interaction Hamiltonian ^H B=evFA(s) xIs x=vFeByIs x;(6) into Eq. (2). Since we are also considering the electron spin as a degree of freedom, simply adding the interaction term above to Eq. (2) is not enough, as we have to add an extra term to account for interaction of the electron spin with the synthetic magnetic eld, i.e., a Zeeman term of the following form [38]: ^HZ=
Zsz IAB; (7) whereIABis the identity matrix in sub-lattice space, and
Z=gsBB=2 (withgsbeing the gyromagnetic factor, andBthe Bohr magneton). The total, single-valley Hamiltonian for SOC in the presence of an AGF is then given by ^H tot=^H+^H B+ ^H Z, and we can use it to the Dirac equation for electrons in the presence of SOC and under the action of an AGF as follows i~@ @tj (r;t)i=X =1^H totj (r;t)i: (8)To solve the above equation, we write j (r;t)ias as a lin- ear combination of the instantaneous eigenstates of ^H tot, with time dependent expansion coecients [39]. To do so, despite ^H totadmits closed-form eigenstates, in the form of parabolic cylinder functions for the general case of both instrinsic and Rashba SOC present in the system [28], in this work we discuss the two cases separately. This will provide a much easier framework, and will al- low us to gain better insight on the role each of these two mechanisms has in the nonlinear optical response of graphene. A. Instantaneous eigenstates of ^H totfor intrinsic SOC only To start with, we neglect Rasbha coupling, and set
R= 0. As intrinsic SOC cannot lift the spin degener- acy of the bands near the Dirac point, but only introduces a nonzero gap proportional to
I, the two spin states fj"i;j#ig can be treated independently, and the spin index=1f";#gonly enters as a parameter (and not as a quantum number) in the expression of the eigen- states and eigenvalues of ^H tot. Thanks to this, we can then write the single valley-single spin Hamiltonian as ^H =vf(pxx+pyyeByx) +(
Iz+
Z); (9) and then ^H tot=^H " ^H #, and thereforej (r;t)i= j  "(r;t)i j  #(r;t)i. Written in the form above, it is easy to recognise Eq. (9) as a gapped Landau Hamiltonian, whose eigenvalues and eigenstates can be written in terms of harmonic os- cillator eigenstates in the ydirection as [40] j  n;(y;t;px)i= N n;ei px ~xE n; ~t! j n;()i;(10) with j; n;()i=^O() n;n1() n() ; (11) where ^O() =Ifor= 1 (K valley), and ^O() =zx for=1 (K' valley). Here, n() are normalised one- dimensional harmonic oscillator eigenstates [41] (with 1() = 0),= (y+ 2`2 Bpx)=`B, (with`B=p ~=eB being the magnetic length), N n;is a normalisation con- stant,  n;=E n;E 0 ~!cpn; (12) andE n;=~!cp n+ (
I=~!c)2+
zare the eigen- values of Eq. (9) (with !c=vFp 2=`Bbeing the cy- clotron frequency). n2N0indicates the Landau lev- els in the conduction ( = 1) and valence ( =1)4 FIG. 3. First few Landau energy levels in the vicinity of the Dirac point, in the presence of intrinsic SOC. As it can be seen, the main eect of the intrinsic SOC is to open a gap between valence and conduction bands (lifting the band degeneracy at the Dirac point). The lifting of the spin degeneracy of each level is due to the Zeeman coupling. The dashed lines, depicting the band structure of unperturbed, pristine graphene, have been added to help the visualisation of the band structure. band, respectively. Notice, moreover, that since the to- tal Hamiltonian factors in spin space, there cannot be a common n= 0 Landau level, as in the case of no in- trinsic SOC [33], but rather two separate levels, with spin-dependent eigenvalues E0;=(
I
Z), and spin-polarized eigenstates j 0;i=1 21=4 0 0() ; (13) The corresponding band structure in the vicinity of the Dirac point for the rst few Landau levels is shown schematically in Fig. 3. B. Instantaneous eigenstates of ^H totfor Rashba SOC only We now turn our attention to the case, where only Rashba SOC is present, i.e., we set
I= 0 =
Zin the total Hamiltonian. Since the Rashba coupling term
R introduces a coupling between the spin states, we cannot factor ^H totin block diagonal form anymore. We then need to deal with the full four-dimensional Hamiltonian ^H tot=^H+^H R. Its eigenstates and eigenvalues, how- ever, can still be computed analytically and they can still be expressed in terms of harmonic oscillator states [28] as follows j ; n;(y;t;px)i= N n;ei px ~xE n; ~t j n;()i;(14) FIG. 4. Landau energy levels in the vicinity of the Dirac point, for the case of nonzero Rashba coupling, and for n=f0;1g.Contrary to the case of intrinsic SOC, in this case we observe the presence of a doubly degenerate zero state.The dashed lines, depicting the band structure of unperturbed, pristine graphene, have been added to help the visualisation of the band structure. where now n2Z, and j n;()i=0 BBBBBB@in
Ran1n1() i
RE n;ann() E n;2 n ann()  (E n;)2n E n; an+1n+1()1 CCCCCCA;(15) wherean=p n! , andE n;are the eigenvalues of ^H tot, whose explicit expression is in general a complicated function of
Rand~!c(see Ref. 28 for details). For the particular case "=
R=~!c1, i.e., small Rashba coupling, however, we can nd the following approximate analytical expression for the eigenvalues E n;+=~!c 1 +"2 2p n+ 1 +O "4
; (16a) E n;=~!c 1"2 2pn+O "4
: (16b) Notice that in this limit ^H totadmits two zero energy states, one corresponding to n= 0 (j; 0;i), and the other one to n=1 (j; 1;+i). Notice, moreover, that for
R= 0, i.e.,"= 0 in Eq. (16), the condition E m1;+=E m;holds, which implies a degeneracy of Landau levels. This degeneracy is then lifted for
R>0, and in this case (and this case only) it makes sense to label the eigenstates with the index =, which, however, is not to be associated with a genuine spin index, since for the case of Rashba coupling, the bands are still degenerate in spin. The band structure in the vicinity of the Dirac point for the case of nonzero (but small) Rashba coupling, in- cluding the two zero energy states, and the states corre- sponding to n=f0;1gin both valence and conduction band, is reported in Fig. 4.5 051015202500.20.40.60.81 0 1 2 3 400.10.20.30.4(a) (b) FIG. 5. (a) Spectrum of emitted radiation by articially enhanced bent graphene, for the case of impinging x-polarisation, and (b) zoom on the low-frequency part of the spectrum, i.e., 0 !4!1. The black, dashed line indicates the position of the fundamental frequency (i.e.,!=!L=!1). The solid, blue line corresponds to the case of no SOC, and serves as a reference for the discussion. The red and green dashed lines represent, respectively, small, and high intrinsic SOC achievable with usual enhancement processes, while the pink, dashed line represents the same situation, but for the Rashba coupling. For these gures, the following parameters have been used: = 50 fs, !L= 78 THz, B= 2 T, and EL= 107V/m. IV. INTERACTION WITH THE ELECTROMAGNETIC FIELD We now consider the interaction of a ake of bent graphene in the presence of SOC, with an external elec- tromagnetic eld, and we then calculate its nonlinear response. To account for such an interaction, we em- ploy minimal coupling, and simply make the electro- magnetic vector potential A(t) appear via the mini- mal substitution p7! p+eA(s)+eA(t)
, so that both the eects of an actual (electromagnetic pulse) and ar- ticial (bending) gauge eld are described within the same formalism. Throughout this whole section, we as- sume the electromagnetic vector potential to be writ- ten as A(t) =A(t)ei!Lt^f+ c.c., where ^fis a suitable polarisation vector, !Lis the pulse carrier frequency, and the pulse shape is assumed Gaussian, i.e., A(t) = ELexp (tt0)2=2
, withELbeing the pulse ampli- tude, andits duration. We moreover assume, for sim- plicity, that the electromagnetic eld impinges normally on the bent graphene sheet. This assumption is justied by the fact, thatour model Hamiltonian for graphene au- tomatically takes into account of a re-centering of k-space around the position of the Dirac points. Accounting for all the aforementioned assumption, in this section we will therefore consider the following form of single valley Dirac equation i~@ @tj (x;y;t )i=h ^H tot+^Hint(t)i j (x;y;t )i;(17) where the interaction term ^Hint(t) will assume dierent explicit forms, depending on the kind of interaction we are considering, as it is discussed below.In general, however, the interaction term ^Hint(t) can be gauged away by means of the phase transformation j (x;y;t )i=Z dpxeipx ~xei^G(t)j (y;t;px)i;(18) where ^G(t) =~1Rt 0ds^Hint(s). Following the reasoning from Ref. [33], we can calculate the instantaneous eigen- states of the equation above, which we dene as j; n;(t)i, for which we have j; n;(t)i=j; n;(t)ifor the case of intrinsic SOC, and j; n;(t)i=j; n;(t)ifor the case of Rashba coupling. To solve Eq. (17), we then employ the Ansatz j (y;t;px)i=ei^G(t)X n;;c; n;(t)j; n;(y;t;px)i;(19) which amounts to considering the general electron dy- namics for SOC bent graphene interacting with an ex- ternal eld as a weighted superposition of instantaneous Landau eigenstates. Substituting this Ansatz into Eq. (17) leads to solving a dierential equation system for the expansion coecients c n;(t): _c;0 m;0(t) =iX nX ;h;0 m;0j^Hint(t)j; n;ic; n;(t);(20) where;=f1;1g. Notice, that the states j; n;iare orthogonal with respect to the index n, and the number of eigenstates involved in the sum above is regulated, essentially, by the initial conditions. We then consider two dierent regimes of light-matter interaction: rst, we consider the usual sub-lattice coupling and assume that6 051015202500.20.40.60.81 0 1 2 3 400.10.20.30.40.50.6(a) (b) FIG. 6. (a) Spectrum of emitted radiation by articially enhanced bent graphene, for the case of impinging x-polarisation, and (b) zoom on the low-frequency part of the spectrum, i.e., 0 !4!1. The black, dashed line indicates the position of the fundamental frequency (i.e.,!=!L=!1). The solid, blue line corresponds to the case of no SOC, and serves as a reference for the discussion. The dashed, red line corresponds to
I= 15 meV, and depicts the same situation of the green, dashed line of Fig. 5. The dot-dashed green, and magenta lines correspond, respectively, to
I= 30 meV and
I= 50 meV. For these gures, the following parameters have been used: = 50 fs, !L= 78 THz, B= 2 T, and EL= 107V/m. the impinging electromagnetic eld is linearly polarised along thex-direction (namely, the one indicated by `in Fig. 2), i.e., we choose ^f=^x. The minimal coupling Hamiltonian in this case reads ^Hint(t) =evFIs Ax(t)x: (21) Secondly, we introduce a spin-eld interaction term, which couples the photon and electron spin directly, thus granting us access to spin dynamics. To do so, we con- sider the case of a circularly polarised pulse, for which we choose ^f=^h, where ^h= (^x+i^y)=p 2 is the helicity basis [42], and =1 is the photon spin an- gular momentum (SAM), corresponding to left-handed (= +1) or right-handed ( =1) circular polarisation. In this case, since the impinging pulse is carrying SAM, we can introduce an extra interaction Hamiltonian that describes the SAM-electron spin interaction, rather than the usual sub-lattice interaction described by Eq. (21) of the form ^HSF int(t) =evFA(t)s IAB; (22) where the superscript SFstands for spin-eld , to em- phasise the nature of the interaction, and distinguish the above interaction Hamiltonian from Eq. (21). Notice, that ^HSF int(t) makes only sense as interaction Hamiltonian when the electron spin can be addressed as an independent degree of freedom, namely only in the intrinsic SOC case, with nonzero Zeeman eect. In all other cases, where the spin degeneracy is not lifted, the interaction Hamiltonian cannot be written in this form, as one couldn't dene Pauli matrices for the spin states, since spin is not an available degree of freedom.A. Dirac Current and Nonlinear Signal The nonlinear response of graphene can be estimated by rst calculating the electric current generated by the interaction of the electromagnetic eld with the graphene layer, namely J (t) =Z dxdyh j^Jj i; (23) where ^Jis a suitable current operator, whose deni- tion depends on the kind of interaction described above, namely ^J=Is for the interaction described by Eq. (21), and ^J=s IABfor the spin-eld interaction described by Eq. (22). From the Dirac current, we can then calculate the spectrum of the emitted radiation, the so-called nonlinear signal, as a function of frequency, as I(!)/j!~J(!)j2; (24) where ~J(!) is the Fourier transform of the Dirac current J(t). V. NONLINEAR SIGNAL We now have all the tools needed to investigate the nonlinear response of bent graphene, in the presence of articially enhanced SOC, considering the eects of both the sub-lattice and spin-eld interactions. We as- sume a pseudomagnetic eld of magnitude B= 2 T, and an impinging electromagnetic pulse with amplitude EL= 107V/m, and duration of = 50 fs, with a carrier frequency of !L= 78 THz, fully resonant with7 0 5 10 1500.20.40.60.81 FIG. 7. Spectrum of emitted radiation in the case of left-handed circularly polarised impinging pulse, for the case of low intrinsic SOC values. The black, dashed line indicates the position of the fundamental frequency (i.e., !=!L=!1). For these gures, the following parameters have been used: = 50 fs,!L= 160 THz, B= 2 T, and EL= 107V/m. the transition from between the lowest- and the zeroth state in the case of no SOC, and nearly resonant be- tween the lowest and second-lowest states in the case when SOC is present. As initial conditions, we as- sume, for both the cases of intrinsic and Rashba SOC, cT(0) = (1 0 0 1 0 0)T=p 2. For the case of intrinsic SOC, this initial condition corresponds to assuming equal pop- ulation of the lowest spin-up and spin-down states, i.e., cT(0) = cV 1;"(0)c0;"(0)cC 1;"(0)cV 1;#(0)c0;#(0);cC 1;#(0)T . For the case of Rashba coupling on the other hand, since spin is not a viable quantum number anymore, the initial condition above just reduces to assuming equal popula- tion in the + and states of valence band, since cT(0) = cV 1;(0)c0;(0)cC 1;(0)cV 0;+(0)c1;+(0)cC 0;+(0)
T. A. Linear Polarisation We start by rst discussing the case of sub-lattice cou- pling with an x-polarised impinging eld. To this aim, we mainly consider the eect of intrinsic (
I) SOC, as, as we will show, the in uence of Rashba (
R) SOC is not so signicant. To corroborate this statement, we have rst performed simulations using typical values of both intrin- sic and Rashba SOC, for the case of articially-enhanced graphene, and chose
I= 5 meV, and
R= 15 meV. The result of these simulations is reported in Fig. 5. As it can be seen from panel (a), the presence of SOC of any kind does not drastically change the nonlinear response of bent graphene, but rather introduces small changes, mainly in the spectral region around the fundamental frequency, and in the high-harmonics region. Let us rst discuss the impact of the Rashba coupling(pink line in Fig. 5). We see from Fig. 5(b), that the main spectral region where Rashba SOC aects the non- linear signal is the low-frequency region, around the fun- damental frequency !L=!1). In this region, in fact, the nonlinear spectrum near the fundamental is slightly deformed. Overall, however, Fig. 5 clearly shows how even a big chosen value of
R= 15 does not introduce signicant changes in the nonlinear spectrum. This ob- servation then allows us to conclude, that Rashba SOC does not really contribute signicantly to the nonlinear signal, and cannot therefore be used as an active control parameter to shape and engineer the nonlinear response of bent graphene. On the other hand, we clearly see from Fig. 5 (b), that the intrinsic SOC has a much higher impact on the non- linear response of bent graphene. In fact, although from panel (a) we can see that a signicant change in magni- tude of the SOC from
I= 5 meV (dashed, red line) to
I= 15 meV (dashed, green line) does not have great impact on the high frequency side of the spectrum, where it only introduces a slight redistribution of energy [see, for example, the
I-dependent peak modulation around the 18th harmonic in panel (a)], it has quite a signicant impact on the low frequency part of the nonlinear signal. A careful investigation of panel (b), in fact, reveals, that although the usual harmonic-oscillator-like structure of low harmonics [33] is preserved also in the case of SOC, higher values of
Itend to redistribute energy around the 2nd harmonic in a more ecient way than lower (or absent) values of SOC. For
I= 15 meV, in particular, we see the emergence of a train of almost equally spaced non-integer harmonics, covering the range [ !L;4!L]. This observation sparks the interesting question, whether this trend can be pushed forward, and intrin- sic SOC can be used to eectively control the shape of the nonlinear response, at least in some frequency region. If this would be the case, articially enhancing graphene could be a viable choice to tune its optical properties, even in \real time". To this aim, in Fig. 6 we compare the nonlinear signal produced by articially enhanced graphene with progres- sively increasing values of intrinsic SOC, i.e.,
I= 15 meV (dashed, red line in Fig. 6),
I= 30 meV (green, dot-dashed line in Fig. 6), and
I= 50 meV (magenta, dot-dashed line in Fig. 6). As it can be seen, increasing the intrinsic SOC has dierent eects in dierent parts of the nonlinear spectrum. For high harmonics ( !15!1), increasing the amount of SOC in the system has the direct result of progressively broadening the harmonic peaks, until, for
I= 50 meV, we obtain a single, broad peak, with higher intensity than the initial harmonics (blue line in Fig. 6). For the range 5 !115!1, on the other hand, no signicant changes seem to appear, as the basic structure of the nonlinear spectrum remains essen- tially the same, up to a small redistribution of energy be- tween the harmonics. The situation for frequencies, i.e., !5!1, however, is quite dierent, as can be seen from Fig. 6 (b). While for small values of
Ithe characteristic8 equally-spaced spectrum is still visible, although slightly red-shifted, for high values of
Ithe situation changes drastically, as the harmonic-oscillator-like peaks, typical of the nonlinear response in this region [33], disappears, and a single, intense peak appears at approximately the non-integer frequency != 5!=2. Moreover, comparing the blue curve (no SOC) with all the others reveal how the presence of SOC makes the peak at the fundamental frequency!=!1disappear, meaning that SOC encour- ages a total redistribution of energy from the pump pulse to the dierent harmonics created in the material. This is an indication on how SOC can be used to tune the nonlinear response of articially enhanced graphene, to generate devices for ecient conversion of light to spe- cic frequencies, that do not need being integer multiples of some fundamental frequency. To conclude this subsection, we would like to point out, that although this situation is highly unlikely to be observed in graphene, where even with articial enrich- ment the SOC values achievable still remain in the meV regime, our results could be useful to describe the eect of SOC on the nonlinear signal of 2D materials in gen- eral, whose electronic properties can be described by a graphene-like Hamiltonian. TMDs, in particular, are a good example of that, since SOC is already quite big in such materials, and it could be enhanced even further with techniques similar to those utilised for graphene. B. Circular Polarisation We now turn our attention to the spin-eld interac- tion Hamiltonian and investigate what kind of nonlinear signal such an interaction reveals. Here, as we did in the previous subsection, we only concentrate on intrinsic SOC, and discuss the cases for both small and large val- ues of
I, whose results are depicted in Figs. 7 and 8, respectively. In both gures, the laser parameters are the same that those used for the case of linear polarisation, except the carrier frequency, which in Fig. 8 changes with increasing SOC, to match the dierent values of SOC used for the simulations, so that the impinging electro- magnetic pulse will always be resonant with an actual transition between a level in the valence band, and one in the conduction band, and no detuning will be present. For small values of
I, as it can be seen from Fig. 7, the situation is similar to the case of linear poarisation, and no appreciable changes can be observed. However, for values of
I15 meV (see the green line in Fig. 7), we start observing a signicant blue shift of the nonlinearresponse. Notice, moreover, that the blue shift is not con- stant over the whole spectrum, but it is chirped in such a way, that lower frequencies experience a smaller shift, than the higher ones. This blue shift can be interpreted, with the help of Fig. 3, as the progressive increase of the gap between the (still equally spaced) Landau levels in valence and conduction bands, which become bigger as
Iincreases. If we increase
Ieven further, as shown in Fig. 8, we witness a progressive broadening of the harmonic spec- trum and the possibility, at very high SOC values [see Fig. 8 (c)], of eciently generating high harmonics. In particular, in the case shown in Fig. 8 (c), we can excite the 28th harmonic of the fundamental frequency !L= 217 THz, corresponding to a wavelength of about 28= 310 nm, well within the UV region of the electro- magnetic spectrum. VI. CONCLUSIONS In this work, we theoretically investigated the eect of spin-orbit coupling (SOC) on the nonlinear response of a sheet of graphene under the action of an articial magnetic eld, for both the cases of Rashba and intrinsic SOC. The latter, in particular, allows for direct spin-eld coupling, via the Hamiltonian in Eq. (22), which shows interesting features. Our results show, that although for realistic values of both intrinsic and Rashba SOC in graphene no signicant changes are induced in the nonlinear signal by SOC, con- trolling the amount of intrinsic SOC in 2D materials in the presence of an articial magnetic eld can result in a broadening of the spectrum of harmonics, and can even allow ecient conversion of light from the THz to the UV regions of the electromagnetic spectrum. This might lead to novel ways of generating spatially-varying fre- quency generation devices based on 2D materials, which could be achieved, for example, by inhomogeneously de- positing SOC-active compounds over the surface of the 2D material, thus de-facto creating a gradient of intrinsic SOC through the material surface. ACKNOWLEDGEMENTS The authors acknowledge the nancial support of the Academy of Finland Flagship Programme (PREIN - de- cisions 320165). [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov. Electric eld eect in atomically thin carbon lms. Science , 306(5696):666{669, 2004.[2] A.C. Ferrari et al. Science and technology roadmap for graphene, related two-dimensional crystals, and hybrid systems. Nanoscale , 7:4598{4810, 2015. [3] K. S. Novoselov, V. I. Fal'ko, L. Colombo, P. R. Gellert, M. G. Schwab, and K. Kim. A roadmap for graphene.9 051015202530 051015202530 0 5 10 15 2000.20.40.60.81 (a) (b) (c) Δ = 30 meVI ω= 180 THz 1Δ = 40 meVI ω= 198 THz 1Δ = 50 meVI ω= 218 THz 1 FIG. 8. Spectrum of emitted radiation in the case of left-handed circularly polarised impinging pulse, for the case of high SOC values. The laser frequency !Lhas been adjusted to the various levels of intrinsic SOC, to ensure that the impinging pulse has no detuning. Notice, how the spectrum broadens with increasing values of
I, reaching, in panel (c), almost the 30th harmonic. The black, dashed line indicates the position of the fundamental frequency (i.e., !=!L=!1). For these gures, the following parameters have been used: = 50 fs,B= 2 T, and EL= 107V/m. Nature , 490:192, 2012. [4] R. R. Nair, P. Blake, A. N. Grigorenko, K. S. Novoselov, T. J. Booth, T. Stauber, N. M. R. Peres, and A. K. Geim. Fine structure constant denes visual transparency of graphene. Science , 320(5881):1308{1308, 2008. [5] Z. Sun, T. Hasan, F. Torrisi, D. Popa, G. Privitera, F. Wang, F. Bonaccorso, D. M. Basko, and A. C. Fer- rari. Graphene mode-locked ultrafast laser. ACS Nano , 4:803, 2010. [6] Z. Sun, A. Martinez, and F. Wang. Optical modulators with 2d layered materials. Nature Photon. , 10:227, 2016. [7] X. Liu, Q. Guo, and J. Qiu. Emerging low-dimensional materials for nonlinear optics and ultrafast photonics. Adv. mater. , 29:1606128, 2017. [8] Mikhail I. Katsnelson. Graphene: Carbon in Two Di- mensions . 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1112.1644v1.Singlet_triplet_splitting_in_double_quantum_dots_due_to_spin_orbit_and_hyperfine_interactions.pdf

arXiv:1112.1644v1 [cond-mat.mes-hall] 7 Dec 2011Singlet-triplet splitting in double quantum dots due to spi n orbit and hyperfine interactions Dimitrije Stepanenko,1Mark Rudner,2Bertrand I. Halperin,2and Daniel Loss1 1Department of Physics, University of Basel, Klingelbergst rasse 82, CH-4056 Basel, Switzerland 2Department of Physics, Harvard University, 17 Oxford St., 5 Cambridge, MA 02138, USA (Dated: December 8, 2011) We analyze the low-energy spectrum of a two-electron double quantum dot under a potential bias in the presence of an external magnetic field. We focus on the r egime of spin blockade, taking into account the spin orbit interaction and hyperfine coupling of electron and nuclear spins. Starting from a model for two interacting electrons in a double dot, we derive a perturbative, effective two- level Hamiltonian in the vicinity of an avoided crossing bet ween singlet and triplet levels, which are coupled by the spin-orbit and hyperfine interactions. We evaluate the level splitting at the anticrossing, and show that it depends on a variety of parame ters including the spin orbit coupling strength, the orientation of the external magnetic field rel ative to an internal spin-orbit axis, the potentialdetuningofthedots, andthedifferencebetweenhy perfinefieldsinthetwodots. Weprovide a formula for the splitting in terms of the spin orbit length, the hyperfine fields in the two dots, and the double dot parameters such as tunnel coupling and Cou lomb energy. This formula should prove useful for extracting spin orbit parameters from tran sport or charge sensing experiments in such systems. We identify a parameter regime where the spin o rbit and hyperfine terms can become of comparable strength, and discuss how this regime might be reached. PACS numbers: 73.23.-b, 73.63.Kv, 73.21.La, 75.76.+j, 85. 35.Be I. INTRODUCTION Electron spins in gated quantum dots have been ex- tensively studied for their possible use in quantum in- formation processing [1–3]. In this context the main in- terest lies in the study of coherent quantum evolution of electron spins in a network of coupled quantum dots in the presence of external magnetic fields. A double quantum dot (DQD) populated by two electrons is the smallest such network in which all of the steps necessary for quantum computation can be demonstrated. In ad- dition, a DQD can host encoded two-spin qubits which require less resources for control than the single-electron spins in quantum dots. In DQDs, the spins can be ma- nipulated exclusively by electric fields in the presence of a constant magnetic field, taking advantage of the spin orbit and/or nuclear hyperfine interactions [2, 4–13]. High precision requirements for the control of spin qubits have prompted the detailed study of the inter- actions of electron spins in quantum dots. The DQDs give experimental access to the coherent spin dynamics. Studies of transport through DQDs in the spin block- ade regime [14] have been particularly useful for probing the electron and nuclear spin dynamics. In this regime, charge transfer between the two dots of a DQD can take place only when the electrons form a singlet state with total spin zero. This allows weak spin non-conserving interactions to be studied via charge sensing [15] or by chargetransport measurements [14, 16], even in the pres- ence of much stronger spin-conserving interactions. The most important spin non-conserving interactions are the spin orbit interaction and the hyperfine interaction be- tween the electron spins and a collection of nuclear spins inside the DQD [17].In this work, we investigate the hyperfine and spin- orbit mediated coupling between electronic singlet and triplet spin states of a DQD in the spin-blockade regime. We show that the spin-orbit and hyperfine contributions to this splitting can be tuned by a number of parameters. We derive an explicit formula that gives this splitting as a function of a homogeneous external magnetic field and the detuning between the ground-state energies of the dots in a DQD. These parameters can be varied in an ex- periment. In addition, the splitting depends on the spin- orbitcouplinginteractionandtheinhomogeneousnuclear Overhauser field, as well as on the dot parameters such as the hopping amplitude between the dots in a DQD, Coulomb repulsion between the electrons, and the direct exchange interaction. Further, we describe how the de- pendence of the singlet-triplet splitting on these param- eters might be used to extract the intrinsic strengths of the spin orbit and hyperfine couplings from charge sens- ing measurements in which the DQD is swept through a singlet-triplet level crossing in the presence of spin-orbit interaction and a fluctuating nuclear field. Recently, it was predicted that the angular momentum transferredbetween electron and nuclearspins in both dc transport [18] and Landau-Zener-type gate sweep exper- iments [19, 20] can show extreme sensitivity to the ratio of spin-orbit and hyperfine couplings. Our result gives an explicit dependence of this ratio on the detuning and externalmagnetic field, thus showing howall regimes can potentially be reached. The paper is organized as follows. In Sec. II we intro- duce a model of a DQD, and describe its energy levels as a function of detuning. In Sec. III we find the matrix ele- mentsofthespinorbitinteractionin thespaceofrelevant low-energy states. In Sec. IV we study the orbital and2 spin structureofthe singletand tripletstateswhich nom- inally intersect for particular combinations of the DQD potentialdetuningandexternalmagneticfield. InSec.V, wedefineaneffectiveHamiltonianwhichdescribestheac- tionofthespin-orbitandhyperfinecouplingsinthecorre- sponding two-level subspace. Then, in Sec. VI we study the dependence of the resulting singlet-triplet splitting on external parameters and show how this dependence can be used to extract the spin orbit interaction strength and the size of Overhauser field fluctuations from charge sensing measurements. In Sec. VII we discuss how the DQD can be tuned between the regimes of spin-orbit- dominated splitting and the hyperfine-dominated split- ting. Finally, we summarize our results in Sec. VIII. II. MODEL HAMILTONIAN FOR DOUBLE QUANTUM DOTS In a DQD, electrons are confined near the minima of a double-well potential VDQD, created by electrical gating of a two-dimensional electron gas (2DEG) in, e.g. GaAs, see Fig. 1. For the case of a deep potential, we treat the two local minima of the double-well as isolated harmonic wells with ground state wave functions ϕ1,2. In order to define an orthonormal basis of single particle states for building-up the two-electron states of the DQD, we form the Wannier orbitals Φ Land ΦR, centered in the left and right dots, respectively [21]: ΦL,R=1/radicalbig 1−2sg+g2(ϕ1,2−gϕ2,1),(1) wheres=∝an}bracketle{tϕ1|ϕ2∝an}bracketri}ht= exp[−(a/aB)2] is the overlap of the harmonic oscillator ground state wave functions of the two wells,aB=/radicalbig /planckover2pi1/mω0is the Bohr radius of a single quantum dot, /planckover2pi1ω0is the single-particle level spacing, and 2a=lis the interdot distance. The mixing factor of the Wannier states is g= (1−√ 1−s2)/s. The two electrons in the DQD are coupled by the Coulomb interaction, C=1 4πκe2 |r1−r2|, (2) wherer1(r2) is the position of electron 1 (2) and κis the dielectric constant of the host material. In this work we consider the regime where the single-particle level spacing is the largest energy scale, in particular /planckover2pi1ω0≫ e2/(4πκa). In this case, and assuming that the hyperfine and spin-orbit interactions are also weak, single-particle orbital excitations can be neglected. Therefore, the rel- evant part of the two-electron Hilbert space is approx- imately spanned by Slater determinants involving the Wannier orbitals Φ L,R. Including spin, and using second quantization notation where c† L(c† R) creates an electron in the Wannier state Φ L(ΦR), we define the two-electronbasis states: |(2,0)S∝an}bracketri}ht=c† L↑c† L↓|0∝an}bracketri}ht, (3) |(0,2)S∝an}bracketri}ht=c† R↑c† R↓|0∝an}bracketri}ht, (4) |(1,1)S∝an}bracketri}ht=1√ 2/parenleftig c† L↑c† R↓−c† L↓c† R↑/parenrightig |0∝an}bracketri}ht,(5) |T+∝an}bracketri}ht=c† L↑c† R↑|0∝an}bracketri}ht, (6) |T0∝an}bracketri}ht=1√ 2/parenleftig c† L↑c† R↓+c† L↓c† R↑/parenrightig |0∝an}bracketri}ht,(7) |T−∝an}bracketri}ht=c† L↓c† R↓|0∝an}bracketri}ht. (8) The orbital parts of the basis states with single occu- pancy in each well, i.e. the spin-singlet |(1,1)S∝an}bracketri}htand the spin-triplet |T0,±∝an}bracketri}ht, are given by Ψs ±(r1,r2) =1√ 2[ΦL(r1)ΦR(r2)±ΦR(r1)ΦL(r2)],(9) while the orbital parts of the two states |(0,2)S∝an}bracketri}htand |(2,0)S∝an}bracketri}htwith double occupation of the right and left wells, respectively, are given by Ψd L,R(r1,r2) = ΦL,R(r1)ΦL,R(r2).(10) The orbital functions Ψs +and Ψd L,Rare symmetric under exchange of particles, and therefore must be associated with the antisymmetric singlet spin wave function (total spinS= 0), while Ψs −is antisymmetric under exchange, and is associated with the symmetric triplet spin wave function (total spin S= 1). The electrostatic gates that create the potential VDQD can, in addition, tune the energies of the electrons in the potential minima by creatingan additional bias potential Vbias. We model this bias as a simple detuning ε, which gives an energy difference for an electron occupying the left or the right dot, ε=∝an}bracketle{tΦL|eVbias|ΦL∝an}bracketri}ht−∝an}bracketle{tΦR|eVbias|ΦR∝an}bracketri}ht.(11) In the symmetric case, ε= 0, the voltages on the elec- trostatic gates are set so that, in the absence of electron- electron interactions, an electron would have the same energy in either well. The Coulomb repulsion, Eq.(2), penalizes the states |(0,2)S∝an}bracketri}htand|(2,0)S∝an}bracketri}htwith double occupation of either well by an amount U, given by U=∝an}bracketle{tΨd L,R|C|Ψd L,R∝an}bracketri}ht. (12) Therefore, for a symmetric potential, ε≈0, the lowest energystates oftwoelectronswill be primarilycomprised of singly occupied orbitals. When the detuning is large enough to overcome the on-site electron-electron repul- sion in one well, |ε|>U, the doubly occupied state with both electrons on the dot with lower potential becomes the ground state. Varying the gate voltages to increase the detuning from large and negative to large and pos- itive values then tunes the occupation numbers of the two dots in the ground state of the DQD through the3 sequence (2 ,0)→(1,1)→(0,2). Since the states with the charge configurations (2 ,0) and (0,2) are singlets, while those with the (1 ,1) charge configuration can be either singlet or triplet, the measurement of charge as a function of detuning can reveal the spin states. The strong spin-independent interactions of the elec- trons with the confinement potential, VDQD, Coulomb repulsionC, as well as the kinetic energy are, at low en- ergies and in the limit of tight confinement, described by the matrix elements between Slater-determinant-type states in which the two electrons are loaded into some combination of the the Wannier orbitals (see Eq.(12) and Ref. ([21])): t=∝an}bracketle{tΦL,R|h0 1,2|ΦR,L∝an}bracketri}ht−1√ 2∝an}bracketle{tΨs +|C|Ψd L,R∝an}bracketri}ht,(13) V±=∝an}bracketle{tΨs ±|C|Ψs ±∝an}bracketri}ht, (14) X=∝an}bracketle{tΨd L,R|C|Ψd R,L∝an}bracketri}ht. (15) Here, we have used h0 1,2=Hosc+VDQD(r)−Vh(r∓aeξ) to label the part of Hamiltonian that includes the ki- netic energy Tand the harmonic part of the potential VDQDnear the dot centers. The dots are displaced by ±aalong the axis with the unit vector eξ, see Fig. 1. Thus,h0 1,2−Hoscdescribes the tunneling due to the mis- match between the true double-dot potential VDQDand the potential of the harmonic wells [21]. The matrix el- ementXdescribes coordinated hopping of two electrons from one quantum dot to the other, tis the renormalized single electron hopping amplitude between the two dots, which includes contributions of both the single-particle tunneling amplitude and the Coulomb interaction, and V+(V−) is the Coulomb energy in the singlet (triplet) state with one electron in each well. The confinement potential VDQD, the Coulomb inter- actionC, and the detuning εprovide the largest en- ergy scales in a DQD. These terms add up to the spin- independent Hamiltonian H0=T+VDQD+C+Vbias, which within the space of the six lowest energy dot or- bitals, using the basis defined in Eq. (3) - Eq. (8), is represented by H0=/parenleftbigg HSS0 0HTT,0/parenrightbigg , (16) where the singlet Hamiltonian in the basis |(0,2)S∝an}bracketri}ht, |(2,0)S∝an}bracketri}ht,|(1,1)S∝an}bracketri}htis HSS= U−ε X −√ 2t X U+ε−√ 2t −√ 2t−√ 2t V+ ,(17) and the triplet Hamiltonian is diagonal, HTT,0=V−. In addition to the terms described above, there are three sources of spin-dependent interactions: Zeeman couplingtoanexternalmagneticfield, hyperfinecoupling between electron spins and nuclear spins in a quantum dot, and the spin orbit interaction. For now we neglecta)SLΩSRb εξxyz ϕ ε = 0VhVh εVhVhb) FIG. 1: a) double quantum dot model and the coordinate system. SLandSRdenote the spin-1 /2 of the electron in the right and in the left quantum dot, respectively. The dots lie in theξz-plane and are tunnel-coupled along the ξ-direction (perpendicular to the z-axis). They can be detuned by the externally applied voltage ε/e. The spin orbit field Ωpoints along thez-axis, defining the first quantization axis for the triplet states |Tz,±∝angbracketrightand|Tz0∝angbracketright(see main text). The effective magnetic sum-field ¯bdefines the second quantization axis for the triplet states |T±∝angbracketrightand|T0∝angbracketright. We choose the mutually orthogonal axes x,y,zso that¯blies in thexz-plane. b) effect of detuning on the quantum dot levels. At zero detuning ε= 0, an electron has the same energy on the left and right quantumdot. For nonzero detuning, the energy of an electron on the left dot is εhigher than the energy of an electron on the right dot. the spin orbit interaction, and analyze it in detail in the next section. The direct coupling of the electron spins to a uniform external magnetic field Bis described by the Zeeman term HZ=−gµeB·(SL+SR), (18) wheregis the electron g−factor and µeis the electron magnetic moment. In addition, the Fermi contact hyper- fine interaction between electron and nuclear spins reads Hnuc=/summationdisplay ihi·Si, (19) wherehi,i=L,R, is the Overhauser field of the quan-4 tum doti, given by [22] hi=/summationdisplay jAj|Ψi(Rj)|2Ij. (20) HereAjisthe hyperfinecouplingconstantforthe nuclear species at site j, with typical size of the order of 90 µeV for GaAs [23], Ψ iis the electron orbital envelope wave function in the right ( i=R) and left (i=L) dot,Rjis the position of the jth nucleus in the quantum dot, and Ijis the corresponding nuclear spin. Because the Zeeman and hyperfine interactions, Eqs. (18) and (19), have similar forms, we combine them into a single effective field that acts on the electron spin in each dot: Hnuc+HZ=−bL·SL−bR·SR.(21) The effective fields bL,R=gµBBL,R−hL,Rinclude the contributions of the external and Overhauser fields, with all coupling constants absorbed in the field definitions. Theenergylevelsarisingfromthespin-conservingHamil- tonian, Eq. (16), along with coupling to a uniform effec- tive field, Eq. (21) with bL=bR, are shown in the left panel of Fig. 2. Below we will be interested in transitions that change the total spin of the pair of electrons in the DQD. To facilitate the discussion, we separate the total field into a sum-field ¯b= (bL+bR)/2 and a difference field δb= (bL−bR)/2: Hnuc+HZ=−¯b·(SL+SR)−δb·(SL−SR).(22) The symmetric component ¯bconserves the magnitude of the total spin,/bracketleftig ¯b,(SL+SR)2/bracketrightig = 0, while the anti- symmetric component δbdoes not. We include the spin- conservingfield ¯bintotheunperturbedHamiltonian, and define HTT=HTT,0−¯b·(SL+SR). (23) Below we will investigate the role of the Overhauser fields in causing spin transitions near a singlet-triplet level crossing in a two-electron DQD. While the exter- nal magnetic field Bis a classical variable, the Over- hauser fields hL,Rare, in principle, quantum operators that involvea largenumber ofnuclear spins, see Eq. (20). Hyperfine-induced electron spin transitions may be ac- companied by nuclear spin flips, and the dynamical, quantum nature of the Overhauser field may be impor- tant. However, due to the large number of nuclear spins, the time scale for the Overhauser fields hL,Rto change appreciably can be much longer than the time spent near the avoided crossing, where spin transitions are possible. Thus we will treat the fields ¯bandδbas quasi-static classical variables, including a discussion of the averag- ing which occurs due to nuclear Larmor precession and statistical (thermal) fluctuations.-1 0 1 2 ε / U-2-101234energy / U S- - T+ anticrossing(1, 1)S - (0, 2)S anticrossing(0, 2)S (2, 0)S T- T0 T+(1, 1)Sb ε*HST = 0 HST = 0 FIG. 2: Energy levels of the double quantum dot system ob- tained from exact numerical diagonalization of Hgiven in Eq. (30) and plotted as a function of the detuning εmea- sured in units of the Coulomb on-site repulsion energy U. Of particular interest here are the crossings and anti-crossi ngs of singlet and triplet states due to spin orbit and hyper- fine interactions. For HST= 0 (see Eq. (42)), i.e. van- ishing singlet-triplet mixing (left-hand side of plot), th e pa- rameter values chosen are ( U,t,p,X,V −,V+,¯b,δby,δb·e,δb· e′,ϕ) = (1,0.1,0,0,0.05,0.04,0.3,0,0,0,0). In this case, the singlets|(1,1)S∝angbracketrightand|(0,2)S∝angbracketrightanti-cross (left oval) and a fi- nite gap opens, whereas the singlets and triplets only cross (no gap). For HST∝negationslash= 0 additional gaps open (right-hand side of plot), in particular at the lower singlet-triplet an ti- crossing around ε=ε∗(right oval) with an energy split- ting ∆ STthat depends on magnetic field, detuning, spin orbit and hyperfine interactions (see main text and figures below). The singlet S−is a superposition of |(1,1)S∝angbracketrightand |(0,2)S∝angbracketright(see Eq. (37)). The parameter values chosen for the right plot are ( U,t,p,X,V −,V+,¯b,δby,δb·e,δb·e′,ϕ) = (1,0.1,0.01,0,0.05,0.04,0.3,0.02,0.02,0.01,π/2). III. SPIN ORBIT INTERACTION In addition to the external and hyperfine fields, elec- tronspins in aDQD arealsoinfluenced by orbitalmotion due to the spin-orbit interaction. Here we describe how the bulk spin-orbitcoupling ofthe 2DEGis manifested in the confined DQD system. In GaAs quantum wells, the spin-orbit interaction is caused by the inversion asymme- try of the interface that forms the quantum well [24–26] and the inversion asymmetry of host material [27]. With the 2DEG being much thinner than the lateral quantum dot dimensions, both spin orbit interactions are linear in the in-plane momenta of the confined electrons, and together are given by HSO=α(px′σy′−py′σx′)+β(−px′σx′+py′σy′),(24) where the Rashba and Dresselhaus spin orbit interaction constantsαandβ, respectively, depend on the thickness and shape of the confinement in the growthdirection and5 onthematerialpropertiesofthe heterostructureinwhich the 2DEG is fabricated. This form of spin-orbit coupling appears in a quantum well fabricated in the (001) plane of GaAs crystal, and the x′- andy′-axes point along the crystallographic directions [100] and [010], respectively. Within the space of low-energy single-electron orbitals in the DQD, the action of the spin orbit interaction can be expressed in terms of a spin-orbit field Ω, HSO=i 2Ω·/summationdisplay α,β=↑↓/parenleftig c† LασαβcRβ−h.c./parenrightig ,(25) where the field iΩ=∝an}bracketle{tΦL|pξ|ΦR∝an}bracketri}htaΩ (26) depends on the orientation of the dots with respect to crystallographic axes through the vector aΩ[28]. For a 2DEG in the (001) plane, aΩis given by aΩ= (β−α)cosθe[¯110]+(β+α)sinθe[110],(27) where the angle between the eξdirection and the [110] crystallographicaxisisdenotedby θ. Thematrixelement ofpξ, the momentum component along the ξ-direction that connects the two dots, see Fig. 1, is taken between the corresponding Wannier orbitals, and it depends on the envelope wave function and the double dot bind- ing potential. The spin orbit field Ωaccounts for the spin rotation when the electron hops between the dots. Therefore, the spin orbit interaction enables transitions between triplet stateswith singleoccupationofeachwell, to the singlet states with double occupation of either the left or the right well. The matrix element in Eq. (26) can be calculated ex- plicitly in a model potential [21, 28], giving Ω=4t 3l ΛSOaΩ |aΩ|, (28) wherelis the interdot distance. The numerical prefactor is model-dependent, but the dependence on other param- eters is generic. The hopping amplitude t, and the inter- dot separation ldepend on the geometry of the double dot system, whereas the spin-orbit length Λ SOis deter- mined by material properties (Rashba and Dresselhaus spin orbit strength) and by the orientation of the DQD with respect to the crystallographic axes. In particular, if the 2DEG lies in the (001) plane, it is given by 1 ΛSO=/radicaligg/parenleftbiggcosθ λ−/parenrightbigg2 +/parenleftbiggsinθ λ+/parenrightbigg2 , (29)whereλ±=/planckover2pi1/[m∗(β±α)] [29], with m∗being the effec- tive band mass of the electron. In the special case β= 0, θ= 0, this definition reduces to the Rashba spin orbit length Λ SO|β=θ=0=λSO=/planckover2pi1/m∗α. One of the main goals in the following is to derive the dependence of the energy splitting at the anticrossingbe- tween the lowest-energy electron spin triplet and singlet states (see Fig. 2, S−−T+anticrossing) in terms of this spin-orbitlength Λ SO. Detailed understandingofthis de- pendence may then be used to extract the value of Λ SO, e.g. from measurements of the singlet-triplet transition probability in gate-sweep experiments. Within the model used above for explicit calculation, the components of Ωare real, even in the presence of magnetic fields. This is due to the high symme- try of the ground state orbitals of the quantum dot in the model, and remains true even after the replacement pξ→pξ−(e/c)Aξin the spin-orbit Hamiltonian HSO, Eq. (24). However, this fact is not essential for the physics described below. IV. SINGLET-TRIPLET TRANSITIONS AND THE CHOICE OF SPIN QUANTIZATION AXIS Transitions between singlet and triplet states can be mediated by the spin orbit interaction or by an inhomo- geneouseffective magneticfield (externalplus hyperfine), δb. In [19] it was shown that the transfer of angular mo- mentum between electrons and the nuclei strongly de- pends on the relative size and phase of the electron spin flipmatrixelementsinducedbyspinorbitinteractionand by the difference of the Overhauserfields in the two dots. Using our model of a detuned DQD, we will study these matrix elements in the following in detail and in partic- ular focus on the singlet-triplet level splitting, see Fig. 2, right panel. The homogeneous field ¯bacts only within the spin triplet subspace, while the inhomogeneous field δbmixes singlet S= 0 and triplet S= 1 states. Representing the total Hamiltonian in the basis {(|(0,2)S∝an}bracketri}ht,|(2,0)S∝an}bracketri}ht,|(1,1)S∝an}bracketri}ht,|Tz+∝an}bracketri}ht,|Tz0∝an}bracketri}ht,|Tz−∝an}bracketri}ht)}, where thez-axis is taken along Ω, see Fig. 1, we find6 H= U−ε X −√ 2t 0 −i√ 2Ω 0 X U+ε−√ 2t 0 −i√ 2Ω 0 −√ 2t−√ 2t V + −√ 2(δbx−iδby) 2δbz√ 2(δbx+iδby) 0 0 −√ 2(δbx+iδby)V−+2¯bz¯bx√ 2 0 i√ 2Ωi√ 2Ω 2 δbz¯bx√ 2V−¯bx√ 2 0 0√ 2(δbx−iδby) 0 ¯bx√ 2V−−2¯bz . (30) The Hamiltonian H, Eq. (30), is the starting point for all of our further calculations. Our results will show the dependence of the singlet-triplet splitting on the param- eters that enter H. This Hamiltonian describes a dou- ble quantum dot with single orbital per dot, i.e. in the Hund-Mulliken approximation, and it is valid as long as the dot quantization energy is the largest energy scale in the problem. It can be applied to double quantum dots of various kinds, for example the gated lateral or vertical dots in III-V semiconductor materials, quantum dots in nanowires, or self assembled quantum dots. We illustrate the spectrum of Hin Fig. 2, for a set of parameters that emphasizes anticrossings of the levels. The spectrum is obtained by exact diagonalization of H, and it is given as a function of detuning ǫ. For other types of quantum dots, the parameter values would change, but the overall structure of the spectrum remains the same. V. EFFECTIVE HAMILTONIAN NEAR THE SINGLET-TRIPLET ANTICROSSING In the limit of large detuning, |ε| ≫U,V+,V−,|¯b|, the ground state is a spin singlet with both electrons in ei- ther the left or right dot, depending on whether ε>0 or ε<0. In the region of weak detuning, the ground state has one electron in each of the dots. For a sufficiently strong sum-field, |¯b|>U−V+, the singlet ground state exhibits an avoided level crossing with the lowest energy triplet state, i.e. the S= 1 state oriented along ¯b, see Fig. 1, at a detuning where the potential energy gained by the singlet’s double occupancy of the lower well com- pensates the Zeeman energy gained by the spin-polarized triplet. Here, the residual splitting is determined by the spin non-conserving interactions. The behavior near this anticrossinghas been the focus of many recent studies on the interaction of electron spins with the nuclei [11, 30– 33]. The role of spin orbit interaction has received less attention than that of the nuclei, and will be analyzed in the following sections. The orbital structure ofthe levelsnear the anticrossing is determined by the spin-independent interactions and by the direction and amplitude of the sum-field ¯b. The singlet subspace acted on by the Hamiltonian in Eq. (17) includes a state with single occupation of the two dots, and two states which feature double occupation of either the left or the right dot. Generically, the state that takespart in the anticrossing includes amplitudes of all three singlet states. However, because U≫J, whereJ≈ 4t2/(U−V+)∼0.01−0.1meVisthesplittingbetweenthe lowest-energy triplet and the lowest-energy singlet state, andU∼1meV, admixture of at least one of the singlets will always be suppressed at the anticrossing by a large energy denominator (note that t≈0.01−0.1meV≪ U−V+). Let us now construct the effective Hamiltonian which acts in the two-level subspace spanned by the levels near the anticrossing. First, the spin-conserving part of the full (6×6) Hamiltonian reads Hsc=/parenleftbigg HSS0 0HTT/parenrightbigg , (31) where the block HSSacts in the singlet subspace, HTT acts in the triplet subspace, and the off-diagonal blocks vanish due to spin conservation. Explicitly, the block HSSis given byEq. (17) in the ba- sisEq.(3)-(8). Thetripletblock HTTisgiveninEq.(23). SinceU≫t≫Xin a typical quantum dot, the state at the anticrossing can at most include significant con- tributions from two out of three basis singlets. We will consider the anticrossing at positive values of the detun- ingε(the anticrossing at negative voltage is analogous) so that the |(2,0)S∝an}bracketri}htstate with the energy U+εis far detuned from the other two singlets. The remaining two singlets can be close in energy. In order to include the possibility of near degeneracy, we will introduce a mix- ing angleψthat parametrizes the hybridization of the |(0,2)S∝an}bracketri}htand|(1,1)S∝an}bracketri}htstates. In the restricted subspace of these hybridized states, the singlet Hamiltonian reads: Hr=U−ε+V+ 2+τ·n/radicalbigg (U−ε−V+)2 4+2t2,(32) whereτ= (τx,τy,τz) is the vector of Pauli matrices. The pseudospin τdescribes the components of the an- ticrossing singlet state, |τz= 1∝an}bracketri}ht=|(1,1)S∝an}bracketri}ht,|τz= −1∝an}bracketri}ht=|(0,2)S∝an}bracketri}ht, within the approximation that we ne- glect the remaining |(2,0)S∝an}bracketri}htcomponent (this is valid when|t/U| ≪1). In this case, n=ezcos2ψ+exsin2ψ is a unit vector parametrized by the mixing angle ψthat describes the relative size of mixing and splitting of τz eigenstates. Notethatthe y-componentof nvanishesdue to the choiceof phases in the quantum dot ground states, which guaranteesthat the spin-independent hopping ma- trix element tis purely real. We remarkagain that in our7 DQD set-up the hopping matrix element tstaysreal even in the presence of magnetic fields, [21]. The mixing angle of the doubly and singly occupied states at the S= 0 an- ticrossing, |(0,2)S∝an}bracketri}htand|(1,1)S∝an}bracketri}ht, respectively, is defined by cos2ψ=U−V+−ε/radicalig (U−V+−ε)2+8t2,(33) sin2ψ=2√ 2t/radicalig (U−V+−ε)2+8t2.(34)InthebasisofeigenstatesofEq.(32)thesingletHamil- tonianHSSis given by HSS= U+ε X cosψ−√ 2tsinψ−Xsinψ−√ 2tcosψ Xcosψ−√ 2tsinψ E S+ 0 −Xsinψ−√ 2tcosψ 0 ES− , (35) where ES±=U−ε+V+ 2±/radicalbig (U−ε−V+)2/4+2t2(36) are the eigenvalues of HSSin the spin-conserving sector. The basis vectors used here are the far-detuned singlet |(2,0)S∝an}bracketri}ht(Eq. (3)) and the singlets |S±∝an}bracketri}htdefined by |S+∝an}bracketri}ht= sinψ|(1,1)S∝an}bracketri}ht−cosψ|(0,2)S∝an}bracketri}ht,(37) |S−∝an}bracketri}ht= cosψ|(1,1)S∝an}bracketri}ht+sinψ|(0,2)S∝an}bracketri}ht.(38) In the limit U≫t,|S±∝an}bracketri}htbecome eigenstates of HSSwith energiesES±given in Eq. (36). While the mixing of orbital states belonging to sin- glets does not affect the triplet Hamiltonian HTT, it will change the form of coupling between the singlet and triplet states near the anticrossing. In the following, we first diagonalize the triplet sector in order to find the ex- plicit form of the triplet state at the anticrossing, and then find the effective Hamiltonian of the singlet-triplet coupling. The triplets are Zeeman split by the sum-field ¯b. We have chosen the z-axis of spin quantization so that the spin-orbit interaction couples |(0,2)S∝an}bracketri}htand|(2,0)S∝an}bracketri}htto the|S= 1,Sz= 1∝an}bracketri}htstate. We will now diagonalize the triplet part of the spin-conserving Hamiltonian, given by HTT=V−+2¯b −cosϕ1√ 2sinϕ0 1√ 2sinϕ01√ 2sinϕ 01√ 2sinϕcosϕ ,(39) where we have used ¯b=|¯b|, cosϕ=¯bz/¯b, and sinϕ= ¯bx/¯b(see Fig. 1). The unitary transformation UtHTTU† t that diagonalizes HTTis Ut= cos2ϕ 2−1√ 2sinϕsin2ϕ 2 −1√ 2sinϕ−cosϕ1√ 2sinϕ sin2ϕ 21√ 2sinϕcos2ϕ 2 .(40)Wedenotethebasisstatesby |T+∝an}bracketri}ht,|T0∝an}bracketri}ht, and|T−∝an}bracketri}ht, where, now, the quantization axis is given by the sum-field ¯b. With the diagonalization of the triplet block of the spin-conserving Hamiltonian, and the preceding ap- proximate diagonalization of the spin-conserving singlet Hamiltonian, Eq. (35), we are able to describe the spin- conserving interaction near the anticrossing in a conve- nient form. We will use |S−∝an}bracketri}htand|T+∝an}bracketri}htas basis vectors, and study the effective Hamiltonian in the vicinity of the anticrossing that can emerge from spin non-conserving interactions. The full Hamiltonian is H=Hsc+HSO−δb·(SL−SR),(41) and reads in block form H=/parenleftbigg HSSHST HTSHTT/parenrightbigg . (42) The diagonal blocks HSSandHTTgive the spin- conserving part, denoted by Hsc, while the off-diagonal blocksHSTandHTS=H† STinduce singlet-triplet transi- tions. In the basis ( |(2,0)S∝an}bracketri}ht,|S+∝an}bracketri}ht,|S−∝an}bracketri}ht,|T+∝an}bracketri}ht,|T0∝an}bracketri}ht,|T−∝an}bracketri}ht), the diagonal blocks take a simple form. The singlet block isHSSgiven in Eq. (35). The triplet block HTTis diag- onal and reads HTT= diag/parenleftbig V−−|¯b|,V−,V−+|¯b|/parenrightbig ,(43) where we used that |T+∝an}bracketri}htis the lowest-energy triplet. The effective Hamiltonian near the anticrossing is de- termined by the spin-conserving terms t,U,X,ε, ¯b, and by the spin non-conserving terms Ω, arising from spin- orbit coupling, and δb, the effective difference field. All these interactions can be treated perturbatively in dots with weak overlap of the orbitals. The off-diagonalterms are denoted by HST=HSO ST+Hδb ST, andHTS=H† ST, where8 HSO ST=iΩ −sinϕ−√ 2cosϕ sinϕ cosψsinϕ√ 2cosψcosϕ−cosψsinϕ −sinψsinϕ−√ 2sinψcosϕsinψsinϕ , (44) Hδb ST= 0 0 0√ 2(iδby+δb·e′)sinψ2(δb·e)sinψ√ 2(iδby−δb·e′)sinψ√ 2(iδby+δb·e′)cosψ2(δb·e)cosψ√ 2(iδby−δb·e′)cosψ , (45) where the unit vector e=exsinϕ+ezcosϕ (46) points in the direction of the homogeneous field ¯b, and the vector e′=−excosϕ+ezsinϕ (47) lies in thexz-plane, which contains Ωand¯b, and points in the direction normal to ¯b. In the vicinity of the anticrossing, the DQD behaves as an effective two-level system, with the dynamics de- scribed by an effective Hamiltonian denoted by Hcr. Up to first order in spin non-conserving interactions and, af- ter neglecting the high-energy state |(2,0)S∝an}bracketri}ht, we find H(1) cr=/parenleftbigg ES−H34 H43ET+/parenrightbigg , (48) whereH34=∝an}bracketle{tS−|H|T+∝an}bracketri}htis the matrix element of H, Eq. (42), between the anticrossing states, and ET+= V−−¯b. VI. SINGLET-TRIPLET SPLITTING AT THE ANTICROSSING The singlet-triplet splitting ∆ STat theS−−T+anti- crossing, see Fig. 2, can be accessed in the spin blockade regime by transport measurements or by charge sensing. This splitting gives valuable information about the prop- erties of the quantum dots and the nuclear polarization. We shall derive now explicit expressions for ∆ STin terms of the experimentally relevant quantities ε,¯b,δb, andΩ. We proceed by perturbation expansion in HST,t/U, and X/U. First, we focus on the first-ordercontributions and afterward address the higher order corrections which be- comerelevantaroundthepointswheretheleadingcontri- butions can be tuned to zero by the control parameters. AstheDQDdetuning εisvariedwithotherparameters held fixed, there is a special value ε∗where the energy ES−of the lowest singlet, Eq. (36), becomes equal to the energyET+=V−−¯bof the triplet |T+∝an}bracketri}ht, Eq. (43). The detuning at which this crossing occurs is controlled by the amplitude ¯bof the sum-field, as well as the tunnelcouplingt. For the unperturbed case, described by H0, ε∗is the solution of the equation ES−(ε∗) =V−−¯b. (49) When the spin non-conservinginteractionsaretakeninto account, the crossing of singlet ( S= 0) and triplet ( S= 1) states is avoided due to state mixing (hybridization). Up to first order in HST,t/U, andX/U, the splitting follows from Eq. (48) and reads ∆ST(ε,¯b,ϕ) = 2/radicalig |H34|2+(ES−−ET+)2.(50) For a fixed value of ¯b, the splitting attains its minimum value ∆∗ STatε∗, ∆∗ ST≡∆ST(ε∗(¯b),¯b,ϕ) = min ε∆ST(ε,¯b,ϕ),(51) whereε∗implicitly depends on ¯b, as well as other DQD parameters. From Eqs. (48), (44), and (45), the splitting is equal to 2 |H34|, ∆∗ ST= 2/vextendsingle/vextendsingle/vextendsingle−iΩsinϕsinψ+√ 2(δb·e′+iδby)cosψ/vextendsingle/vextendsingle/vextendsingle. (52) Notethat∆∗ STcontainscontributionsfromboththespin- orbit coupling and the difference field δb. The relative importance of each of the two terms depends on the de- tuningε∗through the mixing angle ψ, as well as on the geometry through the angle ϕbetween the effective field ¯band the spin-orbit field Ω. When varying the detuning εfrom large to small values, ψdecreases from ψ≈π/2 at strong detuning, ε≫U−V+, toψ≈0 at|ε| ≪t. For a mixing angle ψ≈π/2, the contribution to ∆∗ STcoming from the spin orbit interaction dominates the one from the difference-field, and vice versa for ψ≈0. Reaching the ψ≈0 regime requires weak magnetic fields, i.e. ¯b≪t, and in this casethe energysplittings be- tween the triplet states are not large enough to make use of the simple model for a two-level anticrossing. On the other hand, reaching the regime with ψ≈π/2 requires that the detuning ε∗at which |S−∝an}bracketri}htand|T+∝an}bracketri}htanticross is far away from ε12, the detuning at the anticrossing of |(1,1)S∝an}bracketri}htand|(0,2)S∝an}bracketri}htsinglets, cf. Fig. 2. The width of the|(1,1)S∝an}bracketri}ht−|(0,2)S∝an}bracketri}htanticrossing is of the order t, so the requirement is |ε∗−ε12| ≫t. Therefore, the Zee- man energy of |T+∝an}bracketri}htmust be larger than t, sogµBB≫t,9 which gives B≫0.2T for typical values t∼10µeV and |g|= 0.4. These considerations show that, at least in principle, the relativestrengths ofthe spin-orbit and hyperfine con- tributions to the singlet-triplet coupling can be tuned through a wide range of values using a combination of gate voltages and magnetic field strength and direction. What situation do we expect for typical GaAs dots? Us- ing a value of 3-5 mT for the random hyperfine field (see e.g. Ref. [2] and references therein), and an electron g- factor|g|= 0.4, we estimate |δb| ≈70−120 neV. For the spin-orbit coupling Ω, see Eq.(28), using t= 10µeV, an interdot separation l= 50 nm, and a spin-orbit length ΛSOin the range 6-30 µm (see e.g. Refs.[10, 34]), we find |Ω| ≈20−110 neV. Parameters may vary from device to device, but it appears that the spin-orbit and hyper- fine couplings are generally of similar orders of magni- tude, with the spin-orbit coupling typically a few times weaker. Thus adjustments of the matrix elements over a reasonable range of ψmay be sufficient to explore both the hyperfine and spin-orbit dominated regimes. Similar analysis can be performed for devices in other materi- als, such as InAs or InSb nanowires, where the natural balance between hyperfine and spin-orbit couplings may shift. A. Singlet-triplet splitting ∆∗ STforδb= 0 Let us first consider the special case of vanishing dif- ference field, δb= 0, and finite uniform field, ¯b∝ne}ationslash= 0. In this case, the splitting depends not only on Ωbut also on the detuning at the minimal splitting, ε∗, which itself is implicitly determined by the sum-field amplitude ¯b. The geometry of the system enters through the angle ϕbe- tween¯band the spin-orbit field Ω. In Fig. 3 we show a plot of the level splitting ∆ ST(ε∗,¯b,ϕ), given in Eq. (52) as a function of its variables and for δb= 0. At any fixed angle ϕ∝ne}ationslash= 0, ∆∗ STshows a dependence on the detuning ε∗at the anticrossing due to the mixing of |(0,2)S∝an}bracketri}ht, which is coupled to |T+∝an}bracketri}htvia the spin-orbit in- teraction, and |(1,1)S∝an}bracketri}ht, whichdoesnotcoupleto |T+∝an}bracketri}htvia spin-orbit coupling in the first order, see Fig.4. At large values of detuning ε∗≫U−V+, the splitting reaches a saturation value 2Ωsin ϕ. For typical GaAs quantum dots, reaching this regime requires strong magnetic fields ofB≫t. At lower values of the detuning ε < U−V+, the mixing of the singlet |(1,1)S∝an}bracketri}htbecomes significant, and the spin-orbit coupling value Ω cannot be read off directly from the splitting. The maximal splitting ∆∗ STcaused by spin-orbit inter- action is ∆∗ ST= 2Ω|sinϕ|, forψ=π/2. From Eq. (28) and Eq. (29), we find that Ω is set by the material prop- erties (Dresselhaus ( β) and Rashba ( α) spin orbit inter- actions) and the geometry of the dots. Assuming that the magnetic field is strong enough to separate the |T0∝an}bracketri}ht and|T−∝an}bracketri}htstatesfromthe anticrossing, U≫¯b≫t≫ |δb|,0.00.10.20.30.40.50.00.20.40.60.81.0 Ε/Star/Slash1U/CurlyPhi/Slash1/LParen12Π/RParen1 00.05ST*∆/U 0.01 0 FIG. 3: Singlet-triplet level splitting ∆∗ ST= ∆ST(ε∗,¯b,ϕ), Eq. (52), for δb= 0 (no nuclear field), as a function of the detuningε∗at which the crossing occurs (cf. Fig. 2) and of the angleϕbetween the spin-orbit field Ωand the magnetic sum-field ¯b(cf. Fig. 1). Parameters used for this plot are U= 1,t= 0.01,V+= 0.75,δb= 0. the maximal splitting is ( |sinψ|= 1) ∆∗ ST=4t 3l ΛSO|sinϕ|,(δb= 0),(53) wherelis the interdot distance. The numeric factor (of order unity) is non-universal and depends on the spe- cific dot geometry. Formula (53) is one of the main results of this paper. It provides a simple but useful relation between quantities that can be determined ex- perimentally, such as ∆∗ ST,t,l, andϕ, and a quantity of interest - the spin orbit length Λ SO. This relation could allow the strength of spin-orbit coupling to be measured experimentally[17, 33], though the geometry and detuning-dependence must be carefully taken into account in order to obtain an accurate estimate. Let us remark here briefly on the special case of zero detuning, i.e. ε= 0, and weak magnetic fields. In this case, the splitting is not described by our calculations which require sufficiently large separation of the triplets in energy. However, in a slightly different system – a single quantum dot containing two electrons – singlet- triplet coupling which is forbidden by time-reversal sym- metry can be generated by applying a magnetic field, ∆∗ ST≈(aB/λSO)EZ[29]. This case cannot be recovered from our DQD model with one orbital per site. Indeed, here we have seen that in weak fields, ¯b≪t, the cou- pling of the two states with single occupation in each well|(1,1)S∝an}bracketri}htand|T+∝an}bracketri}htdue to the spin-orbit interaction involves doubly occupied states which are higher in en- ergy due to the on-site repulsion. On the other hand, for a pair of electrons in a single quantum dot, the on-site10 0.0 0.1 0.2 0.3 0.4 0.5 ε / U246810103∆ST* / Uϕ = π / 4 ϕ = 3π / 8 ϕ = 5π / 8 FIG. 4: Singlet-triplet level splitting ∆∗ ST= ∆ST(ε∗,¯b,ϕ), Eq. (52), for δb= 0 (no nuclear polarization), as a function of the detuning ε∗at which the crossing occurs (cf. Fig. 2) and of the angle ϕbetween the spin-orbit field Ωand the magnetic sum-field ¯b(cf. Fig. 1). At small detuning ε∗<t, the splitting becomes rather small, while it saturates at la rge detuningε∗> U. The saturation value is ∝ |sinϕ|, shown forϕ=π/4 (full line), ϕ= 3π/8 (dashed line) and ϕ= 5π/8 (dashed-dottedline). Parameters used for this plot are U= 1, t= 0.01,V+= 0.75,δb= 0. repulsion is approximately the same for both states, sin- glet and triplet. We note that for DQDs in weak fields, the Zeeman energy EZ, occurring in the splitting for a single dot [29], gets replaced by the exchange energy Jif the mixing of triplets due to δb∝ne}ationslash= 0 is neglected. B. Singlet-triplet splitting ∆∗ STforδb∝negationslash= 0 In addition to spin-orbit coupling of the anticrossing triplet to |(0,2)S∝an}bracketri}ht, the anticrossing triplet is coupled to the singlet |(1,1)S∝an}bracketri}htthrough the difference-field δb. The previous considerations show that the contributions from thedifference-field δbtothesplittingcannotbeneglected for anglesϕ∼0,π, or for field strengths where ε∗< U−V+, which is often be the case. Therefore, we now discuss the splitting in the presence of both, the spin- orbit field Ωandδb. The splitting ∆∗ STas a function of detuningε∗and direction of ¯bis shown in Fig. 5. When both sourcesof splitting are present, generically, the gap ∆∗ STremains open. The spin-orbit contribu- tion to Eq. (52) is always purely imaginary, while the δb-contribution has both a real part, coming from the component lying in the xz-plane, and an imaginary part, coming from the perpendicular component δby. For any fixedvalueofthe spin-orbitcouplingstrength, closingthe gap would require fine tuning of δb, both in amplitude and direction. As a function ofthe direction of ¯bthe con- tributions to ∆∗ STcompete, and, in addition, the relative size of the competing terms will change as a function0.00.10.20.30.40.50.00.20.40.60.81.0 Ε/Star/Slash1U/CurlyPhi/Slash12Π 00.16ST*∆/U 00.015 FIG. 5: The same plot of the singlet-triplet splitting as in Fig. 3 except for finite nuclear polarization chosen to beδb= (−0.0006,0.0008,0.0012). In the strong detuning regime,ε∗> U−V+on the right-hand side of the plot, the splitting is determined by spin orbit interaction, vSO> vHF and resembles the same area in Fig. 3. In the weak detuning regime,ε∗< U−V+on the left-hand side of the plot, the hyperfine interaction increases the splitting. The regime w ith similar strengths of the interactions, vSO∼vHFcan be iden- tified in the region ε∗≈U−V+= 0.25U. Forε∗on the left- hand side of the vertical dashed line, and ϕ≈0,π, the split- ting is dominated by hyperfine interaction vHF> vSO. Near the dotted line, and for the angles ϕ≈π/2,3π/2 the sizes of spin orbit and hyperfine interactions are similar vSO≈vHF. ofε∗. Indeed, the spin-orbit term is strongest at large detuningε∗≫U−V+, while the difference-field term becomes significant at small detuning ε∗∼t. This com- petition affects the form of the ϕ-dependent splitting, see Fig.6. In the limit |Ωsinψ| ≪ |δbcosψ|, the splitting ∆∗ ST is caused mostly by the inhomogeneous field. In this case, the splitting is proportional to the size of compo- nentδb⊥=δb−e(δb·e) ofδbwhich is normal to the homogeneous field ¯b=|¯b|e. With the leading spin-orbit coupling correction, the splitting is [see Eq.(52)] ∆∗ ST= 2√ 2|δb⊥cosψ|−2Ωδbysinψcosψ |δb⊥cosψ|.(54) C. Measuring the singlet-triplet coupling The singlet-triplet coupling ∆∗ STis manifested experi- mentally, for example, in the spin flip probability when the system is taken through the level crossing during a time-dependent gate sweep [35]. In such experiments, the system is initialized to its ground state at large ǫ, the (0,2) singlet. When ǫis then ramped to take the sys-11 0 0.2 0.4 0.6 0.8 1 ϕ / 2π024681012103∆ST* / Uε* = 0.1 U ε* = 0.2 U ε* = 0.4 U FIG. 6: First order singlet-triplet splitting ∆∗ ST= ∆ST(ε∗,¯b,ϕ), Eq. (52), for finite nuclear polarization δb∝negationslash= 0, plotted as function of the angle ϕ(cf. Fig. 1), for various detuningsε∗(cf. Fig. 2). The parameters used for the plot areU= 1,t= 0.01,V+= 0.75,V−= 0.74, Ω = 0.005, andδb= (−0.0006,0.0008,0.0012). The curves correspond toε∗= 0.1 (full line), ε∗= 0.2 (dashed), ε∗= 0.4 (dashed- dotted). In the strong detuning regime (dashed-dotted line ) the angular dependence reflects the |sinϕ|dependence of the spin-orbit term (see Eq. (53). The regime of weaker detuning (full and dashed line) shows the hyperfine effects. tem through the singlet-triplet crossing, the two-electron spin state may change, with a probability determined by a combination of the coupling ∆∗ STand the sweep rate. Thefinalspinstatecanthenbereadoutbyquicklyramp- ing back to large ǫ, where the singlet and triplet states have discernibly different charge distributions, which can be detected by a nearby charge sensor. Even with single-shot spin detection [11], determin- ing the spin flip probability requires building up statis- tics over many experimental runs. Within each run, the parameters in Eq. (52) may be considered fixed. How- ever, the hyperfine field components are in general a pri- oriunknown: under typical experimental conditions, the temperature is high compared with all intrinsic energy scales within the nuclear spin system, and the equilib- rium state is nearly completely random. Depending on the measurement timescale, the nuclear fields on subse- quent experimental runs may either remain constant or may change. While the correlation time for the longi- tudinal component of the nuclear field (parallel to the external field) may be quite long, the transverse compo- nents change on the timescale of nuclear Larmor preces- sion, which formoderatefieldsofafew hundredmillitesla can reach the sub-microsecond timescale. The coherence time associated with this precession may reach several hundred microseconds to one millisecond. LetP(∆∗ ST)betheprobabilitythatthesystemmakesa transition to the triplet state in a single sweep, when the value of ∆∗ STis specified. In an experiment where mea- surementsofthesingletandtriplet fractionsareaveraged0 0.2 0.4 0.6 0.8 1 ϕ / 2π051015202530< ∆ST*2 > / Ω2ψ = π / 8 ψ = π / 4 ψ = 3 π / 8σ = 2 Ω σ = Ω σ = 0.5Ω σ = 0.1Ω FIG. 7: Square of the splitting ∝angbracketleft∆∗2 ST∝angbracketright, Eq. (52), averaged over Gaussian fluctuations of δbwith zero mean and standard deviationσ. The plots show the dependence of ∝angbracketleft∆∗2 ST∝angbracketrighton the angleϕfor various mixing angles ψ. We have assumed isotropic Gaussian fluctuations with a standard deviation σ. Plots are for the values σ/Ω = 0.1,0.5,1,2 and illustrate the effects of various strengths of fluctuations. The curves are found by numerical averaging over the fields δb. over a time long compared to all nuclear spin relaxation times, one obtains an averaged probability ∝an}bracketle{tP(∆∗ ST)∝an}bracketri}ht, where∝an}bracketle{tA∝an}bracketri}htdenotes the mean value of quantity A, av- eraging over a Gaussian distribution of δb, while other parameters such as B,t,¯b,φand the sweep rate are held fixed. If the measurements are averaged over a shorter period, which is long compared to the time for phase re- laxation of the nuclear spins, but short compared to the longitudinal relaxation times, then the Gaussian average should be taken only over the transverse components of δb, while the component parallelto the applied magnetic field is held fixed. When the sweep rate through the S-T transition is rapid, the probability P(∆∗ ST) should be proportional to (∆∗ ST)2, so an average value of P(∆∗ ST) will measure the mean value of (∆∗ ST)2. For lower values of the sweep rate,Pwill have corrections due to (∆∗ ST)4, etc.. There- fore, measurements of the averaged value ∝an}bracketle{tP(∆∗ ST)∝an}bracketri}htfor a wide range of sweep rates should, in principle, yield av- erage values of all powers of (∆∗ ST)2, and thus allow one to deduce the probability distribution for (∆∗ ST)2. Here we concentrate on the mean value of (∆∗ ST)2, and dis- cuss predictions for this mean value as a function of the parameters B,φ,andt. We illustrate the dependence of ∝an}bracketle{t(∆∗ ST)2∝an}bracketri}hton the angle ϕand mixing ψin Fig. 7 and Fig. 8. The dependence of the splitting on the angle ϕ, inherent in the non-averaged splitting, see Eq. (52), remains visible when the splitting is dominated by spin-orbit interaction. As expected, the dependence ofthe splitting on the angle ϕ, Fig. 7, is most visiblein the caseofweakfluctuationsof δb, i.e. forweak hyperfine coupling. In addition, the angular dependence12 0 0.1 0.2 0.3 0.4 0.5 ψ / π0102030< ∆ST*2 > / Ω2ϕ = π / 8 ϕ = π / 4 ϕ = 3 π / 8σ = 2Ω σ = Ω σ = 0.5Ωσ = 0.1Ω FIG. 8: Average square of the splitting ∝angbracketleft∆∗2 ST∝angbracketright, in the fluctu- ating nuclear field δb. The parameters are chosen as in Fig. 7, and we illustrate the dependence on the mixing angle ψ. is more pronounced for larger mixing angles, since the spin-orbit induced splitting depends on Ωsin ψ, Fig. 8. The fluctuating difference field, besides changing the average∝an}bracketle{t(∆∗ ST)2∝an}bracketri}ht, introduces noise in the splitting. We find that the standard deviation σ∆of the splitting, in the limit of weak fluctuations |Ωsinϕsinψ| ≫σis σ∆,G= 2σ|cosψ|, (55) so that it also shows dependence on the mixing angle ψ. Fluctuations in δbsmear the splitting at the anticross- ing ∆∗ ST. The average value and noise in the splitting can be used to measure the strengths of spin-orbit cou- pling Ω and the hyperfine field δb. The relative size of fluctuations in ∆∗ ST, as a function of ϕ, has minima at ϕ≈π/2,3π/2. D. Higher order corrections So far, our analysis of the |S−∝an}bracketri}ht-|T+∝an}bracketri}htanticrossing was based on the assumption that the largest contribution tothe splitting ∆∗ ST, Eq. (52), results from the direct cou- pling of the two states via HSO STandHδb ST. However, if the detuningε∗is not large enough to make the influence of the levelsthat areenergeticallyfurther awayfromthe an- ticrossing completely negligible, higher order terms that describe virtual transitions to such higher levels and thus involvemorethanonetransitionbetweenthesingletsand the triplets become important. To study this regime, we derive an effective Hamil- tonian in the vicinity of the anticrossing by a second order Schrieffer-Wolff (SW) transformation[36, 37]. We divide the Hilbert space of the DQD into a relevant part which includes the anticrossing states, and an aux- iliary part which contains the remaining 4 states. The time-independent perturbation series is then performed in powers of the spin-non-conserving interactions. The spin-conserving Hamiltonian Hsc, Eq. (31), is taken as the unperturbed part, while H−Hscis the perturbation. Inreorderingthebasis,wechoosethecrossingstates |S−∝an}bracketri}ht and|T+∝an}bracketri}htofHscto be the first two basis states. Then, the Hamiltonian has a block-diagonal form denoted by H=/parenleftbiggA C C†B/parenrightbigg , (56) whereAis a 2×2 matrix that describes the an- ticrossing states, Bis a 4×4 matrix of the states with energies far from the anticrossing and the 2 × 4 matrixCrepresents the coupling between of the subspaces controlled by AandB. In the basis (|S−∝an}bracketri}ht,|T+∝an}bracketri}ht,|T0∝an}bracketri}ht,|T−∝an}bracketri}ht,|(2,0)S∝an}bracketri}ht,|S+∝an}bracketri}ht), with the two anti- crossing level at the positions 1 and 2, we can read off the blocks from Eq. (56): A=/parenleftbigg ES− −iΩssinψ+√ 2δ+cosψ iΩssinψ+√ 2δ−cosψ E T+/parenrightbigg , (57) B= V− 0 i√ 2Ωc −i√ 2Ωccosψ+2δsinψ 0 V−+¯b −iΩsiΩscosψ−√ 2δ+sinψ −i√ 2Ωc iΩs U+ε −√ 2tsinψ−Xcosψ i√ 2Ωccosψ+2δsinψ−iΩscosψ−√ 2δ−sinψ−√ 2tsinψ−Xcosψ E S+ ,(58) C=/parenleftbigg −i√ 2Ωcsinψ+2δcosψ iΩssinψ−√ 2δ−cosψ−√ 2tcosψ+Xsinψ 0 0 0 iΩs −iΩscosψ+√ 2δ−sinψ/parenrightbigg ,(59)13 where we have used the abbreviations δ=δb·e,δ±= δb·e′±iδby, Ωs= Ωsinϕ, Ωc= Ωcosϕ, and the unit vectorse′andeare defined in Eq. (47) and Eq. (46), respectively. As a result of the SW transformation on Eq. (56), the off-diagonalblock Ciseliminatedup tosecondorderin C and the transformed block ASW− − →A+δAbecomes then the Hamiltonian of an effective two-level system. There-fore, the first-order Hamiltonian H(1) crfrom Eq. (48) be- comesmodifiedbysecondorderterms, HcrSW− − →Hcr+δA, whereδAis the second order correction to A. The diag- onal matrix elements δA11andδA22describe the renor- malization of the energy levels, and their effect is to shift the detuning ε∗at which the anticrossing occurs. The explicit expressions of these corrections are δA11=1 ES−−V−/bracketleftig 2(Ωcsinψ)2+4δ2cos2ψ/bracketrightig +1 ES−−U−ε/parenleftig√ 2tcosψ−Xsinψ/parenrightig2 + +1 ES−−V−−¯b/bracketleftbigg 2(δb·e′)2cos2ψ+/parenleftig√ 2δbycosψ+Ωssinψ/parenrightig2/bracketrightbigg ,(60) δA22=1 ET+−U−εΩ2 s+1 ET+−ES+/bracketleftbigg/parenleftig Ωscosψ+√ 2δbysinψ/parenrightig2 +2(δb·e′)2sin2ψ/bracketrightbigg , (61) δA12=−i 2Ωs/parenleftig√ 2tcosψ+Xsinψ/parenrightig/parenleftbigg1 ES−−U−ε+1 ET+−U−ε/parenrightbigg . (62) In experiments that probe the electron spin dynam- ics, the most important terms are the off-diagonal ones, δA12=δA∗ 21. They lead to a modification of the first order singlet-triplet splitting Eq. (51), i.e. ∆∗ STSW− − →2|H34+δA12|. Thus, up to second order, the splitting at the anticrossing becomes ∆∗ ST= 2/vextendsingle/vextendsingle/vextendsingle/vextendsingle−iΩsinϕ/bracketleftbigg sinψ−/parenleftbiggt√ 2cosψ−X 2sinψ/parenrightbigg/parenleftbigg1 ES−−U−ε∗+1 ET+−U−ε∗/parenrightbigg/bracketrightbigg +√ 2(δb·e′+iδby)cosψ/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (63) We seenowthat the newcorrectiontermsin ∆∗ STbecome significant for weak detuning, when ψ < π/2, because the spin-independent tunneling contribution, which is ∝cosψ, can alter the first-order result, which is ∝sinψ. We compare the splitting in the second order, Eq. (63), with the first order splitting and the result of exact nu- merical diagonalization of H, Eq. (30), in Fig. 9. The higher order corrections are small, but they do become significant for the magnetic field normal to the spin or- bit parameter, ϕ=π/2,3π/2, due to stronger effective strength Ωsin ϕof spin-orbit coupling. For other consid- ered values of detuning, ε∗= 0.1Uandε∗= 0.2U, the change of splitting is smaller than for the ε∗= 0.4Ucase. The limitψ=π/2 requires strong magnetic fields, B∼1T for a typical GaAs DQD. It is reasonable to as- sume that the experiments can be performed both in this limit and away from it, so that the dependence of ∆∗ ST onψcan be probed. In materials with larger g-factors such as InAs, InSb, SiGe, the limit is reached at lower fields. In addition, we have obtained similar results for a model DQD with t= 0.1U, Ω = 0.1t,|δb| ≈0.5Ω, thatdescribesa smallerDQD with morepronouncedhopping. VII. RATIO OF SPIN-ORBIT AND HYPERFINE TERMS As pointed out before, the spin non-conserving Hamil- tonian in the vicinity of the anticrossing can be used to get experimental access to the spin orbit interaction and nuclear polarization in the difference-field δb. Being a functionofcontrollableparameters ǫand¯b, thisHamilto- nian can be altered by applying voltages to electrodes in the vicinity of the quantum dots, adjusting the strength of an external magnetic field, or changing the direction of the field. The effect of the competition between spin orbit and hyperfine induced spin flips on the efficiency of angular momentum transfer between electron and nuclear spins was recently studied theoretically, both in the context of dc transport experiments [18] and in the context of14 0 0.2 0.4 0.6 0.8 1 ϕ / 2 π0246103∆ST* / Ufirst order second order numeric ε = 0.4 U FIG. 9: Comparison of the splitting ∆∗ STobtained from the perturbation in first order (full line), Eq. (52), and in the second order (dashed line), Eq. (63) to the exact numerical result (dashed-dotted line), obtained by the direct numeri cal diagonalization of H, Eq. (30). The plots show the splitting ∆∗ STas a function of the angle ϕfor the anticrossing at the detuningǫ∗= 0.4U. The parameters used in this plot are U= 1,t= 0.02,V+= 0.75,V−= 0.74, Ω = 0.005, and δb= (−0.0006,0.0008,0.0012). gate-sweep experiments [19, 20]. References [18] and [19] revealed striking sensitivities of the polarization transfer efficiency on the ratio of spin orbit and hyperfine cou- pling strengths. In those works, the coupling strengths were treated as phenomenological parameters. Here we provide explicit expressions for them, and discuss how they can be tuned. Following the notation of Ref. [19] we write vϑ=vSO+eiϑvHF, (64) wherevSOandvHFstand for the transitions caused by spin orbit and hyperfine interactions, respectively. Com- paring with Eq. (48), we can identify vϑwithH34. Then, adjusting the overall phase to make the spin-orbit part vSOreal, we identify in lowest order vSO=|Ωsinϕsinψ| (65) vHF=|cosψ|/radicalig (δb·e′)2+δb2y, (66) ϑ= arctan/bracketleftbigg−δb·e′ δby/bracketrightbigg . (67) The explicit expression for ϑshows that the phase of the matrix element can be adjusted not only by changing the direction of δb, but also by rotating the external magnetic field, which changes e′, and also controls the effective spin-orbit coupling strength. Our results show that, in principle, it is possible to switch between the two regimes of hyperfine-dominated and spin-orbit-dominated behavior, by changing the ex- ternal magnetic field strength and direction. In the strongfieldregime,withsum-field ¯bbeinglarge, ε∗isalsolarge, and thus the spin-orbit terms become dominant ( ψ approaches π/2). This behavior is illustrated in Fig. 5. Note that the large values of ε∗require strong ¯bfields. On the other hand, switching to the regime vHF>vSOis always possible by rotating the direction of the magnetic field so that it coincides with ±Ω/|Ω|givingϕ≈0. In this case, the term vSOis negligible and vHFdominates the splitting. Higher-order corrections to the effective Hamiltonian at the anticrossing point do not alter this basic picture of the splitting, but they do change the val- ues of the parameters ψandϕat which the switching occurs. The switching between the regimes dominated either by spin orbit or by hyperfine interactions can potentially be achieved as follows. For the regime vSO> vHF, the sum-field ¯bshould point along the spin orbit field Ω, see Eqs. (26) and (27), in order to maximize |sinϕ|. Also, the applied field should be as strong as possible, in order to maximize the amplitude of the singlet |(0,2)S∝an}bracketri}ht(con- tributing to the anticrossing singlet |S−∝an}bracketri}ht, see Eq. (37)). On the other hand, the opposite regime, vHF>vSO, can be reached by orienting ¯balongΩ, and thus reaching sinϕ= 0. If sin ϕ= 0 cannot be achieved, vSOcan be reduced by decreasing ¯band thereby increasing the am- plitude of the singlet state |(1,1)S∝an}bracketri}htin the|S−∝an}bracketri}ht-singlet at the anticrossing. VIII. CONCLUSIONS We havederivedaneffective twolevel Hamiltonian Hcr for a detuned two-electron double quantum dot in an external magnetic field. Our effective Hamiltonian de- scribes the dynamics of the electron spins for the values of detuning ε≈ε∗close to the anticrossing of the lowest energyS= 0and the lowestenergy S= 1 state. We have shown how Hcrcan be used in the interpretation of ex- periments that probe electron spin interactions by charge sensing and transport in the Coulomb blockade regime. The dependence of Hcron the detuning and magnetic fields can also be used to switch the spin dynamics in a double quantum dot between the spin-orbit dominated regime, and the hyperfine-dominated regime. The spin dynamics at the anticrossing is governed by the spin-orbit and nuclear hyperfine interactions. In a double quantum dot, these two interactions act differ- ently on the orbital electronic states. On one hand, the spin-orbit interaction causes hopping of an electron be- tween the quantum dots accompanied by a spin rotation, thus changing the occupation of the quantum dots. On the other hand, the nuclear hyperfine interaction acts as an inhomogeneous magnetic field, and causes spin rota- tions that are local to the dots, leaving the charge state unchanged. Due to this distinction, the detuning εcon- trols the relative strength of the two interactions in Hcr, in addition to the ratio |Ω|/|δb|, or|Ω|/σ. In the limit of detuning much stronger than the on-site repulsion of the dots,ε≫U,Hcrdescribes mostly the spin-orbit in-15 teraction, with negligible hyperfine effects. In the case of weaker detuning, the effective hyperfine interactions can be of the size comparable to the effective spin orbit interactions. In addition, we find that the orientations of both the double quantum dot and the external magnetic field, de- scribed in the Hcrby the spin-orbit field Ωand the sum field¯b, influence the effective spin orbit interaction. In particular, by having ¯bpointing along Ω, we can sup- press the spin orbit effects completely (in leading order). The splitting of the anticrossing states is accessible to experiments. It can be calculated from Hcr, and we find the dependence of this splitting on detuning and the strength and direction of the sum field, ∆ ST(ε,¯b,ϕ). Of particular interest is the splitting of levels at the anti- crossing. We calculate this quantity, ∆∗ ST(ε∗,ϕ), as a function of the detuning at the anticrossing point, ε∗, and the orientation of the sum field, given by the angle ϕ. Both the spin orbit interaction strength and the in- homogeneity in the hyperfine coupling can be deduced by measuring the splitting and using our formulas for ∆∗ ST(ε∗,ϕ).The relative strength of the spin orbit and hyper- fine terms in Hcrhas a profound effect on the coupled dynamics of electron and nuclear spins. The value of the average angular momentum transfer to nuclear spins as an electron tunnels through a spin-blockaded DQD changes sharply as the interaction goes from the spin- orbit-dominatedtothe hyperfine-dominatedregime. The spin orbit interaction is dominant in the limit of strong detuningε∗≫U−V+. The regime dominated by nu- clear hyperfine interaction is reached when the detuning is weakerε∗/lessorsimilartand the orientation of the sum field is along Ω. 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1809.07582v1.Multiple__Q__magnetic_orders_in_Rashba_Dresselhaus_metals.pdf

arXiv:1809.07582v1 [cond-mat.str-el] 20 Sep 2018Multiple- Qmagnetic orders in Rashba-Dresselhaus metals Ken N. Okada, Yasuyuki Kato, and Yukitoshi Motome Department of Applied Physics, University of Tokyo, Tokyo 1 13-8656, Japan We study magnetic textures realized in noncentrosymmetric Kondo lattice models, in which local- ized magnetic moments weakly interact with itinerant elect rons subject to Rashba and Dresselhaus spin-orbit couplings. By virtue of state-of-the-art numer ical simulations as well as variational cal- culations, we uncover versatile multiple- Qorderings under zero magnetic field, which are found to originate in the instabilities of the Fermi surface whose sp in degeneracy is lifted by the spin-orbit couplings. In the case with equally-strong Rashba and Dress elhaus spin-orbit couplings, which is known to realize a persistent spin helix in semiconductor qu antum wells, we discover a sextuple- Q magnetic ordering with a checkerboard-like spatial patter n of the spin scalar chirality. In the pres- ence of either Rashba or Dresselhaus spin-orbit coupling, w e find out another multiple- Qordering, which is distinct from Skyrmion crystals discussed under th e same symmetry. Our results indi- cate that the cooperation of the spin-charge and spin-orbit couplings brings about richer magnetic textures than those studied within effective spin models. Th e situations would be experimentally realized, e.g., in noncentrosymmetric heavy-fermion comp ounds and heterostructures of spin-orbit coupled metals and magnetic insulators. I. INTRODUCTION Over a decade noncoplanar spin configurations in met- als have been gathering growing interest as a source of topological transport phenomena. In general noncopla- narityoflocalizedspinsischaracterizedbythespinscalar chirality, defined as Sj·Sk×Slfor three spins spanned by sitesj,k, andl. In the spin-charge coupled systems the spin scalar chirality is imprinted on conduction elec- trons as a fictitious magnetic field, coined as an emergent magnetic field, through the so-called Berry curvature in real space [1, 2]. The emergent magnetic field gives rise to a peculiar Hall effect named the topological Hall ef- fect, distinguished from the conventional anomalous Hall effect in the presence of a ferromagnetic order. The topo- logical Hall effect has been first observed in a pyrochlore magnet [3] and recently in metallic compounds hosting magnetic Skyrmion crystals (SkXs) [4, 5]. Noncoplanarspinconfigurationsareoftendescribed by superpositions of spin helices running in different direc- tions. These are called multiple- Qmagnetic orderings. One of the latest examples is a magnetic SkX mentioned above, which can be described as a double- or triple- Q ordering [2, 6–8]. In real space such a SkX forms a two- dimensional periodic array of spin-swirling nanometric objects, called Skyrmions. Each Skyrmion is character- ized by a topological invariant defined by the integra- tion of the spin scalar chirality, which guarantees the topological stability. To date, SkXs have been experi- mentally identified in various noncentrosymmetric mag- nets, includingchiralmetalssuchas B20-typealloys MX (M=Mn, Fe, Co; X=Si, Ge)[6,7]and β-Mn-typeCo-Zn- Mn alloys [9] as well as heterostructures like a monolayer of Fe on Ir substrates [8]. Notably SkXs not only bring about peculiar transport of conduction electrons such as the topological Hall effect [4, 5], but also show their own intriguing dynamics driven by an electric current flow, resulting in current-induced motion with a remarkably- lowthreshold[10]andtheSkyrmionHalleffect[11]. SuchhighmobilityofSkyrmionswouldbepotentiallyharnessd to future memory devices. There are several known mechanisms for the forma- tion of multiple- Qorderings. For SkXs observed in noncentrosymmetric 3 d-electron systems listed above, the Dzyaloshinskii-Moriya(DM) interaction described as D·Sj×Skplays a crucial role, which originates in the spin-orbit coupling (SOC) under broken spatial inversion symmetry. Indeed, magnetic-field–temperature phase di- agrams in those compounds including the SkX phase can be qualitatively explained by using localized spin mod- els with ferromagnetic and DM interactions between the neighboring spins [7, 12]. On the other hand, recent theoretical studies havepro- posedadistinct mechanismforthe formationofmultiple- Qorderings in centrosymmetric itinerant magnets [13– 16]. They revealed that in centrosymmetric Kondo lat- tice models, in which conduction electrons are coupled to localized spins, multiple- Qorderings could be driven by the Fermi surface instability, irrespective of lattice types and electron fillings [13–16]. Perturbation analy- ses up to fourth order with respect to the spin–charge coupling strength [13–15] as well as unbiased numerical simulations [15, 16] showed that when partial nesting oc- curs on the Fermi surface at multiple wave vectors, in other words, when portions of the Fermi surface are con- nected to each other, multiple- Qorders are ubiquitously favored rather than single- Qorderings in the weak cou- pling regime. We note that recently a SkX has been discovered in a centrosymmetric f-electron compound Gd2PdSi3, whose origin might be closely related to this mechanism [17]. Consideringthe abovearguments, a question naturally arises; can multiple- Qorderings also show up, or if so what kind, when both the SOC and the Fermi surface instability cooperate under broken inversion symmetry? Thus it is an intriguing task to uncover magnetic order- ings in noncentrosymmetric Kondo lattice models with conduction electrons subject to the SOC. Recently, this2 issue was addressed in the case of Rashba SOC by one of the authors and his coworker, by deriving an effective spinmodelbythesecond-orderperturbationwithrespect to the spin-charge coupling [18]. Nonetheless, it would be important to solve the original noncentrosymmetric Kondo lattice model beyond the second-order perturba- tion, when considering the fact that in the Kondo lattice model without the SOC higher-order contributions may stabilizedistinct magnetic texturesfromthosein the per- turbative regime [13–15]. Moreover, it would be interest- ing to study the effects of other types of SOC, e.g., the Dresselhaus SOC, in the Kondo lattice model. In this work we study the noncentrosymmetric Kondo lattice model, while fully incorporating the effect of con- duction electrons subject to the SOC, by virtue of a recently-developed efficient numerical simulation tech- nique [15, 16, 19, 20]. We introduce the SOCs of Rashba and Dresselhaus types, whose coupling constants are de- noted as αandβ, respectively. The Rashba SOC stems from breaking of the mirror symmetry, e.g., at the in- terface of a heterostructure, while the Dresselhaus SOC from breaking of the space inversion symmetry in a bulk crystal structure, e.g., in the zincblende structure. Specifically we focus on three cases on a square lat- tice: (i) the case with both the Rashba and Dresselhaus SOCs with equal strength ( α=β∝ne}ationslash= 0), (ii) the case with only the Rashba SOC ( α∝ne}ationslash= 0,β= 0), and (iii) the case with only the Dresselhaus SOC ( α= 0,β∝ne}ationslash= 0). The case (i) was discussed to stabilize a peculiar spin texture called the persistent spin helix [21]. The situation was realized on heterostructures of zincblende-type semicon- ductor GaAs [22], and a long-living transient spin he- lix was observed by spin injection, e.g., through optical means[22,23], whichmayfindapplicationstospintronics and quantum information. In our work we treat a local- ized spin system coupled with conduction electrons char- acterized with α=β, and find out sextuple- Qorderings reflecting the peculiar spin-split Fermi surface. This sit- uation might be potentially applied to the semiconductor quantum wells doped with magnetic impurities, though in our model the magnetic moments are positioned at every site. The case (ii) belongs to C4vpoint group sym- metry, which would be a more general and common sys- tem with broken mirror symmetry at heterointerfaces. In the case (ii) we discover multiple- Qorderings distinct from those discussed in localized spin systems under the same symmetry. Meanwhile, the case (iii) belongs to D2d point group symmetry, which is also widely encountered, not only in nonmagnetic materials like zincblende- and chalcopyrite-type semiconductors [24] but also in itiner- ant magnets such as a family of Heusler compounds [25]. In the case (iii) we also find multiple- Qorderings, which are related with those in the case (ii) by a simple global rotation. The remaining of the paper is organized as follows. In Sec. II we introduce the Kondo lattice model with the Rashba and Dresselhaus SOCs and derive an effective spin interactions given by the bare magnetic suscepti-bility. We also discuss a unique spin-dependent gauge transformation applicable to the α=βcase as well as the exchangebetween αandβ. In Sec. III we explain the details of the numerical simulation and variational calcu- lations. The results are described in Sec. IV. We devote Secs. IVA-IVC to the aforementioned three cases (i)- (iii) with different types of SOCs, respectively. In these sections we discuss the magnetic orderings obtained by the simulation, comparing them with the bare magnetic susceptibility and the results of the variational calcula- tions. Finally, in Sec. V, we summarize our results. II. MODEL A. Hamiltonian In this paper we study a Kondo lattice model on a square lattice with Rashba and Dresselhaus SOCs. The Hamiltonian is given by H=−/summationdisplay jj′stjj′c† jscj′s+/summationdisplay jj′ss′igjj′·c† jsσss′cj′s′ −J/summationdisplay jss′Sj·c† jsσss′cjs′.(1) Herecjs(c† js) is the electron annihilation (creation) op- erator at site jwith spin s(=↑,↓), andSj=t(Sx j,Sy j,Sz j) describes a localized spin at site j, which is treated as a classical spin with the normalized length |Sj|= 1 for simplicity. σis a vector of Pauli matrices, defined as σ=t(σx,σy,σz).tjj′represents the hopping amplitude of electrons form site j′to sitej(tjj′=tj′j), andJ the spin-charge–coupling strength. The SOCs are imple- mented in the second term, in which gjj′reads gjj′= −αjj′ey jj′+βjj′ex jj′ αjj′ex jj′−βjj′ey jj′ 0 . (2) Hereαjj′andβjj′denote the strength of Rashba and Dresselhaus SOCs, respectively, which work on an elec- tron hopping between the sites jandj′(αjj′=αj′j andβjj′=βj′j).ejj′= (ex jj′,ey jj′) is a normalized displacement vector from jtoj′, represented as ejj′= (rj′−rj)/|rj′−rj|with lattice position vectors rjand rj′; we denote rj= (nj,mj), where njandmjare in- tegers with the unit lattice constant. In the following calculations, we consider the electron hopping processes betweenthe nearest-neighbor(NN) sites andbetween the third-nearest-neighbor (TNN) sites. We denote the hop- ping amplitudes tjj′between the NN and TNN sites as t andt3, respectively. Likewise, we represent the Rashba (Dresselhaus) SOC αjj′(βjj′) between the NN and TNN sites asα(β) andα3(β3), respectively. In the following we taketas energy unit ( t= 1).3 In the momentum-space representation the Hamilto- nian in Eq. (1) is described as H=/summationdisplay kss′c† ksH0 ss′(k)cks′−J/summationdisplay kqss′Sq·c† ksσss′ck+qs′.(3) Herecksis defined by the Fourier transform of cjsas cks≡1√ N/summationtext je−ik·rjcjs, whereN=L2is the number of sites (L: the linear dimension of the system). Sqis the Fourier transform of Sjdefined as Sq=1 N/summationdisplay jeiq·rjSj, (4) in which the spin normalization ( |Sj|= 1) leads to the sum constraint of/summationtext q/summationtext ρ|Sρ q|2= 1 (ρ=x,y,z).H0(k) is a 2×2 matrix defined as H0(k) =ǫ0 kI+dk·σ, (5) in which Iis the identity matrix, and ǫ0 kanddkaregiven by ǫ0 k=−2t(coskx+cosky)−2t3(cos2kx+cos2ky) (6) and dk= 2 αsinky−βsinkx+α3sin2ky−β3sin2kx −αsinkx+βsinky−α3sin2kx+β3sin2ky 0 . (7) B. Generalized RKKY interaction To get insight into the magnetic instability by the spin-charge coupling in the weak Jregime, it is useful to derive an effective spin Hamiltonian by the second- orderperturbationanalysisontheHamiltonianinEq.(3) with respect to J[18]. This gives a generalization of the Ruderman-Kittel-Kasuya-Yosida (RKKY) inter- actions [26–28]. The effective Hamiltonian reads Heff=−J2N/summationdisplay q/summationdisplay ρρ′Sρ qχρρ′ q(Sρ′ q)∗,(8) in which the bare magneticsusceptibility χρρ′ qis obtained as χρρ′ q=T N/summationdisplay ωn/summationdisplay ktr[G0(k,iωn)σρG0(k+q,iωn)σρ′], (9) by using the noninteracting 2 ×2 Green function G0(k,iωn) = 1/(iωn−H0(k)+µ);ωnrepresentsthe Mat- subara frequency and µis the chemical potential. More explicitly, Eq. (9) is written down as χρρ′ q=−1 N/summationdisplay k/summationdisplay ττ′∝an}bracketle{tkτ|σρ|k+qτ′∝an}bracketri}ht∝an}bracketle{tk+qτ′|σρ′|kτ∝an}bracketri}ht ×f(ǫkτ)−f(ǫk+qτ′) ǫkτ−ǫk+qτ′. (10)Hereǫkτand|kτ∝an}bracketri}htare the eigenvalue and eigenstate of H0(k) with the band index τ.f(ǫ) is the Fermi distri- bution function expressed as f(ǫ) = 1/(1+e(ǫ−µ)/kBT), wherekBis the Boltzmann constant and Tis the tem- perature. In the presence of SOC, in general, the bare magnetic susceptibility χρρ′ qin Eq. (10) has nonzero off-diagonal components. To examine the dominant magnetic insta- bility, therefore, it is useful to diagonalize the effective spin Hamiltonian in Eq. (8) in the form: Heff=−J2N/summationdisplay q/summationdisplay ξλξ q|S′ξ q|2. (11) Here we define the eigenvaluesand eigenvectorsof χρρ′ qin Eq.(10)as λξ qanduξ q(ξ= 1−3),respectively,formulated as χquξ q=λξ quξ q. (12) Note that we sort the eigenvalues as λ1 q≧λ2 q≧λ3 q.S′ q is a transformed spin Fourier component, given by Sq= U∗ qS′ qwithUq= [u1 q,u2 q,u3 q]. Note the sum constraint also holds for S′ qas/summationtext q/summationtext ξ|S′ξ q|2= 1. The diagonalized form of the effective spin Hamilto- nian in Eq. (11) gives us important information on mag- netic instability. Suppose the largest eigenvalue λ1 qtakes the maxima at a set of wave vectors {Qν}. Then, under the sum constraint of/summationtext q/summationtext ξ|S′ξ q|2= 1, we find that the largestenergygainofthe RKKYHamiltonianin Eq. (11) is earned for multiple- or single- Qmagnetic orderings characterizedwith the wavevectors {Qν}with the corre- sponding spin Fourier components of SQν∝(u1 Qν)∗(see alsoSec. IIIA). Therefore, analyzingthe qprofileof λ1 qis important to figure out the inherent magnetic instability in the weak Jregime. Meanwhile, it should be noted that the generalized RKKY interactions leave degeneracy; the single- and multiple- Qorderings specified by {Qν}and the corre- sponding modes SQν∝(u1 Qν)∗are energetically degen- erate. Higher-order contributions play a crucial role in selectingoutthe lowest-energymagneticstate, asdemon- strated in the absence of SOC [13–15]. This motivates us to study the originalmodel in Eq. (1) or(3) by numerical simulation that treats the spin-charge coupling and the SOC on an equal footing. C. Spin-dependent gauge transformation for α=β In the case of α=βwith only the NN terms ( t3= α3=β3= 0), the Fermi surfaces have peculiar prop- erties [21]. The Fermi surfaces, which have spin degen- eracy in the absence of SOC, are unidirectionally split along the [ ¯110] direction by the SOC, and moreover, all the states in each Fermi surface have the same spin polarization parallel or antiparallel to the [110] direc- tion [for example, see Fig. 1(c)]. The shift vector con- necting the spin-split Fermi surfaces, Qs, is given by4 Qs= 2tan−1(√ 2α)(−1,1). Importantly, this peculiar nature of the Fermi surfaces indicates that the SOC with α=βcan be effectively taken away through a certain spin-dependentgaugetransformation,whichaddsorsub- tracts half of the shift vector, Qs/2, to or from the elec- tron momenta, depending on the spin directions [21]. For the annihilation operators, the gauge transformation can be formulated as ˜cj=/parenleftbigg eiQs·rj/20 0e−iQs·rj/2/parenrightbigg V0cj,(13) wherecj=t(cj↑,cj↓) and V0=1√ 2/parenleftbigg eiπ 41 eiπ 4−1/parenrightbigg . (14) This transformation adds spin-dependent gauges with the quantization axis to [110]. Then, by using the newly- defined annihilation and creation operators, ˜cjand˜c† j, the original Hamiltonian in Eq. (1) for α=βwith only the NN terms ( t= 1) is written into the form with effectively-vanishing SOC: H=−/radicalbig 1+2α2/summationdisplay jj′s˜c† js˜cj′s−J/summationdisplay jss′˜Sj·˜c† jsσss′˜cjs′.(15) Here the new spin frame ˜Sjis defined through the rota- tion by the amount of Qs·rjalong the [110] direction on the original spin frame as ˜Sj= cosQs·rjsinQs·rj0 −sinQs·rjcosQs·rj0 0 0 1 0 0 1 1√ 2−1√ 20 1√ 21√ 20 Sj. (16) Theseanalysesimplythatthemagneticorderingsfor α= β∝ne}ationslash= 0 are related to those without SOC through the site- dependent rotation in Eq. (16). We use this property in the discussion in Sec. IVA. D. Exchange between αandβ We also remark that the exchange between αandβ leads to a simple uniform rotation of magnetic orderings. Byapplyinga πrotationalongthe[110]axisinspinspace for conduction electrons as given by cj= exp/parenleftbigg −iπ 2σx+σy√ 2/parenrightbigg cj, (17) and likewise to the spin frame for the localized spins as Sj= 0 1 0 1 0 0 0 0−1 Sj, (18)the original Hamiltonian in Eq. (1) is transformed to the one with exchanged αjj′andβjj′: H=−/summationdisplay jj′stjj′c† jscj′s +/summationdisplay jj′ss′igjj′(αjj′↔βjj′)·c† jsσss′cj′s′ −J/summationdisplay jss′Sj·c† jsσss′cjs′.(19) This indicates that the magnetic orderings for the Rashba-only case in Sec. IVB also applies to the Dresselhaus-only case in Sec. IVC through the global rotations in Eqs. (17) and (18). We utilize this nature in Sec. IVC. III. METHOD A. KPM-LD To reveal the ground-state magnetic orderings for the Kondo lattice model with Rashba and Dresselhaus SOCs we employ a state-of-the-art large-scale numeri- cal simulation combining the kernel polynomial method (KPM) [29] with Langevin dynamics (LD) [19]. This recently-developed method, called KPM-LD, costs only O(N) (N: number of lattice sites), allowing us to run the simulation for the system sizes of up to ∼104sites. Herewe employthe modified versionofthe KPM-LD[20] making use of a probing method [30] and the stochastic Landau-Lifshitz-Gilbert equation in the LD. We perform the KPM-LD at zero temperature on the square lattice of N= 962. In the KPM, we expand the density of states in a series of Chebyschev polynomials up to the 2000th order, in which 144 random vectors are chosen by a probing technique [30] for calculation of the Chebyschev moments. In Sec. IVA the KPM-LD is initiated from a random spin configuration, aiming at an unbiased search for the ground state. On the other hand, in Sec. IVB, we start the KPM-LD from some given ansatzes for the configu- ration of localized spins, because for α∝ne}ationslash= 0 and β= 0 we found that random configurations often fail to con- verge to a homogeneous state and end up with a mixing of different ordering domains. This can be attributed to keen energy competitions of multiple magnetic orders originating in a considerable number of sharp peaks in λ1 q[see Fig. 5(f)]. Consequently, in Sec. IVB we use the KPM-LD as an “ansatz optimizer” rather than an unbi- ased simulation. Below we describe how we prepare the initial ansatzes used in Sec. IVB. The ansatzes we employ are single- Q helical states that maximize the energy gain of the gen- eralized RKKY Hamiltonian in Eq. (11) and multiple- Q superpositions of them. As discussed later in Sec. IVB, forα∝ne}ationslash= 0 and β= 0,λ1 qtakes the largest value at four5 wave vectors denoted as Qa ν(ν= 1−4) among all the characteristic wave vectors [see Fig. 5(d)]. {Qa ν}are re- lated with each other by C4andσvsymmetry opera- tions. Moreover, the C4vsymmetry dictates that the corresponding eigenvectors u1 Qa νare simply described as u1 Qa 1=t(ux,uy,iuz), (20a) u1 Qa 2=t(uy,ux,iuz), (20b) u1 Qa 3=t(−uy,ux,iuz), (20c) u1 Qa 4=t(−ux,uy,iuz), (20d) whereux,uy, anduzare real numbers. As mentioned in Sec. IIB, multiple- Qorderings maximizing the RKKY energy gain in Eq. (11) under the sum constraint of/summationtext q/summationtext ρ|Sρ q|2= 1 are characterized with the spin Fourier components of SQa ν∝(u1 Qaν)∗. As a result, in this Rashba-only case the multiple- Qstates are given by su- perpositions of symmetry-related helices with the spin rotation plane perpendicular to the xy-plane, which are given by Sj=ˆN4/summationdisplay ν=1Aν u1x QaνcosQa ν·rj u1y QaνcosQa ν·rj −u1z Qa νsinQa ν·rj .(21) Here the sum constraint/summationtext νA2 ν= 4 holds for Aνand ˆNrepresents the normalization factor so that |Sj|= 1. We note that without the normalization factor ˆNall the ansatzes described by Eq. (21) gain the same amount of the RKKY energy in Eq. (11). Among those multiple- Q orderings we take double- or single- Qorderings for the initial ansatzes in the KPM-LD for simplicity. For the double-Qorderings, we set Aν=√ 2 forν= 1 and 2, or ν= 1 and 3. Likewise, for the single- Qordering, we set Aν= 2 for one νand otherwise Aν= 0. In Sec. IVA we employ the periodic boundary condi- tion as in the previous works [15, 16], while in Sec. IVB we use the open boundary condition. This is because in the latter case it turns out that the KPM-LD yields in- commensurate magnetic orderings with a large magnetic unit cell [see Fig. 6(c)], which would be attributed to the wave vectors with the largest λ1 q,{Qa ν}, deviating from commensurate wave vectors [see Fig. 5(d)]. In order to exclude the boundary effects we extract the square with 642sites in the middle of the whole system with 962sites for analyzing the spin textures. For the spin textures obtained in the KPM-LD we cal- culate|Sq|[see Eq. (4)], which is proportional to the squarerootofthe spin structurefactor. We alsocompute the spin scalar chirality κpfor each square plaquette pas κp=1 4(Sj·Sk×Sl+Sk·Sl×Sm+Sl·Sm×Sj+Sm·Sj×Sk), (22) where the sites j,k,l, andmcorrespond to the bottom- left, bottom-right, top-right and top-left vertices of the square plaquette p, respectively. In the same manner as |Sq|, we define |κq|=|1 N/summationtext peiq·rpκp|.B. Variational calculation In Secs. IVA and IVB we also perform variational cal- culations. For given spin configurations we calculate the total energy by using the exact diagonalization of the one-body Hamiltonian and compare the values to deter- mine the ground state. The calculations are done for the system sizes of N= 962and 4802. IV. RESULTS A. Case with α=β
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kxky
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kxky(b) (a) (c) -
Q1 Q1 Q2 Q6QsQ3 Q4 Q5Q2 Q1 Q2
FIG. 1. Fermi surfaces and bare magnetic susceptibilities forµ∼ −1.4 (near quarter filling) (a,b) without the SOC (α=β= 0) and (c,d) with the equally-strong Rashba and Dresselhaus SOCs ( α=β= 0.2). (a,c) and (b,d) show the Fermi surfaces and the largest eigenvalues of the bare mag- netic susceptibility λ1 q[see Eq. (12)], respectively. Note that all the TNN terms are set to zero ( t3=α3=β3= 0). First, we discuss the Fermi surface instabilities for the case with equally-strong Rashba and Dresselhaus SOCs, namely, α=β, along with the case without the SOCs. In this section we consider only the NN terms in the Hamiltonian in Eq. (1) and neglect the TNN terms (t3=α3=β3= 0). Herewesetthechemicalpotentialas µ∼ −1.4, correspondingto nearquarterfilling ( n∼0.5). Figures 1(a) and 1(b) display the Fermi surface and the largest eigenvalue of the bare magnetic susceptibility in the absence ofSOC. The Fermi surface is partiallynested by the commensurate wave vectors Q1= (π/2,π) and Q2= (π,π/2) at this filling, as illustrated in Fig. 1(a). Reflecting the partial nesting, the susceptibility takes the largest value at two inequivalent positions on the edge6 (a) (b) -

0 0 -

(c) (d) -

0 0 -

qxqy qxqy qxqy qxqyqxqyqxqy (e) (f) (g) (h) -

0 0 -

(i) (j) (k) (l) -

0 0 -

-

0 0 -

-

0 0 -

FIG. 2. The results of the KPM-LD simulations for α=β= 0 and α=β= 0.2. (a-d) correspond to α=β= 0. (a) and (b) represent a real-space spin texture and the norm of its Fo urier transform, |Sq|, of 2Q-uudd for J= 0.1, respectively, while (c) and (d) correspond to those of 1 Q-uudd for J= 0.4. (e-l) correspond to α=β= 0.2. (e) and (f) are a spin texture and |Sq|of 6QforJ= 0.1, while (g) and (h) display those of 3 QforJ= 0.4. (i) shows the real-space pattern of the spin scalar chirality for 6 Qcorresponding to (e), with the absolute value of its Fourier transform, |κq|, represented in (j). (k) and (l) are those for 3 Q. The real-space textures of spin and scalar chirality in (a) , (c), (e), (g), (i), and (k) are shown for a part of the whole system with N= 962system for clarity. In (a), (c), (e), and (g), the arrows deno te the directions of the localized spins in thexyplane and the color represents the zcomponent. of the Brillouin Zone (BZ), Q1andQ2, as shown in Fig. 1(b). Whenαandβare introduced with equal strength, the spin-degenerate Fermi surfaces are split along the [ ¯110] direction, eachofwhichhasthe uniformspin polarization parallel or antiparallel to the [110] direction. Figure 1(c) shows the spin-split Fermi surfaces for α=β= 0.2. The partial nesting of the shifted Fermi surfaces yield additionalmaximain λ1 qatfourwavevectors, Q3=Q1+ Qs,Q4=Q1−Qs,Q5=Q2+Qs, andQ6=Q2−Qs, whereQsis the shift vector of the Fermi surfaces shown in Fig. 1(c). As a result, the bare magnetic susceptibility shows the largest value at totally six independent wave vectors as shown in Fig. 1(d). With theFermisurfaceinstabilitiesatthesewavenum- bers in mind we discuss the spin textures obtained by the KPM-LD simulations. We begin with the resultsforJ= 0.1. In the absence of SOC we obtain the noncollinear but coplanar double- Qordering [Figs. 2(a) and 2(b)], as reported in the previous work [31]. The two wave vectors characterizing the magnetic texture are identified as Q1andQ2, which coincide with those in Figs. 1(a) and 1(b). Since the spin components are mod- ulated in the up-up-down-down manner, hereafter we re- fer to this double- Qorder as 2 Q-uudd [31]. On the other hand, when the Rashba and Dressel- haus SOCs are introduced with the equal strength of α=β= 0.2, we find a complex noncoplanar spin tex- ture characterized with six wave vectors, as shown in Figs. 2(e) and 2(f). These wave vectors coincide with the ones giving the largest value in λ1 q,{Qν}(ν= 1−6) in Fig. 1(d). It is noteworthy that to the best of our knowledge there is no other theoretical or experimen- tal report showing stabilization of any magnetic ordering7 with more than three wave vectors in two-dimensional systems. Remarkably, we find that this sextuple- Qor- dering (6 Q) exhibits a checkerboard-like pattern of the spin scalar chirality [Fig. 2(i)], characterized with multi- ple wave vectors specified by ( π/2,0), (0,π/2), (π,π/2), and (π/2,π) [Fig. 2(j)]. While increasing JtoJ= 0.2 and 0.3, we find that the sameorderingpatternsareobtainedintheKPM-LD:2 Q- uudd without SOC and 6 Qwithα=β= 0.2. The result indicates that the Fermi surface instabilities govern the magnetic textures in the weak coupling regime. ForJ= 0.4, however, we find that the spin texture changes into a less complex one. Without the SOC ap- pears a simple single- Qstate composed of Q1(orQ2, de- pending on the initial configuration), which is a collinear up-up-down-down ordering [Figs. 2(c) and 2(d)]. In the same way as 2 Q-uudd, we denote this single- Qorder as 1Q-uudd [31]. With α=β= 0.2 we obtain the triple- Qordering characterized by the three ordering vectors Q1,Q2, andQ3(orQ4,Q5, andQ6) [Figs. 2(g) and 2(h)]. The 3 Qstate also shows the density wave of the spin scalar chirality as shown in Fig. 2(k), although it is only characterized by a single wave vector as shown in Fig. 2(l) in contrast to four in Fig. 2(j). As we mentioned in Sec. IIC, the spin-dependent gauge transformation guarantees the exact mapping of the model for α=β∝ne}ationslash= 0 to the SOC-free one in Eq. (15). Hence, the magnetic orderings stabilized for α=β∝ne}ationslash= 0 are related with those for α=β= 0 through the transformation in Eq. (16). Indeed, we have confirmed that 6Qand 3Quncovered in the KPM-LD simulations are obtained by applying the site-dependent rotation in Eq. (16) to 2 Q-uudd and 1 Q-uudd, respectively, after certain global rotations. E 2Q -
 E1Q-uudd
=  = 0 FIG. 3. Energy difference between 2 Q-uudd and 1 Q-uudd atµ∼ −1.4 in the absence of SOC, estimated by variational calculations for N=L2= 962and 4802. We also verified the results of the KPM-LD by varia- tional calculations. Figure 3 shows the energy difference between 2 Q-uudd and 1 Q-uudd obtained in the absence of SOC ( α=β= 0). Here we take the variational statesas Sj= cos(Q1·rj−π 4) cos(Q2·rj−π 4) 0 , (23) for 2Q-uudd, and Sj= √ 2cos(Q1·rj−π 4) 0 0 , (24) for 1Q-uudd [31]. Note that we do not need the normal- ization factor for the spin lengths as the wave numbers are commensurate. As shown in Fig. 3, 2 Q-uudd is more stable compared to 1 Q-uudd for J < J0 c∼0.33 and vice versa for J > J0 c. The variational result looks consistent with the KPM-LD results. By using the spin-dependent gauge transformation, we can derive the critical value of Jfor nonzero α=βasJc=J0 c√ 1+2α2. Q   6Q
) FIG. 4. Ground-state phase diagram for the Kondo lat- tice model with equally-large Rashba and Dresselhaus SOCs (α=β) forµ∼ −1.4, determined by the KPM-LD and variational calculations. The circles and triangles repre sent the parameters for which the KPM-LD simulations are per- formed. In the absence of SOC ( α=β= 0) 2Q-uudd is favored for J < J0 c∼0.33, while 1 Q-uudd is stabilized for J > J0 c(see Fig. 3). For finite SOCs ( α=β/negationslash= 0) 6Qap- pears on the red-shaded region, while 3 Qshows up on the blue-shaded region. The dashed line is the phase boundary given by Jc=J0 c√ 1+2α2. Combining the results by the KPM-LDand variational calculations, we summarize the J-αphase diagram for equally-large αandβin Fig.4. Thered-andblue-shaded regions correspond to 6 Qand 3Q, respectively, and the dashed line shows the phase boundary Jcdetermined by the variational calculations. The phase diagram in Fig. 4 indicates that the exotic sextuple- Qorderings are stabi- lized in a wide parameter range of αandJin the present spin-charge and spin-orbit coupled system.8 FIG. 5. Fermi surfaces and bare magnetic susceptibilities f or µ= 0.98 (a,b) without the SOCs ( α=β= 0) and (c-f) with the Rashba SOC ( α= 0.2,β= 0). Note that the TNN terms are introduced with t3=−0.5 andα3=−0.5α. (a,c,e) and (b,d,f) show the Fermi surfaces and the largest eigenvalues of the bare magnetic susceptibility, λ1 q[see Eq. (12)], respec- tively. (f) is the magnified view of (d). In (a) and (b) the arrows indicate the wave vectors that give the largest mag- netic susceptibility in the absence of SOC. In (c) and (d) the black arrows denote the wave vectors that give the largest λ1 q in the presence of the Rashba SOC. In (d) the white arrows correspond to the ordering vectors of 1 Q′, which is found for J= 0.2−0.4 in the KPM-LD simulations (see Fig. 8). In (e) and (f) the other characteristic wave vectors, which give th e comparably large λ1 q, are shown. B. Case with α/negationslash= 0andβ= 0 Next, wediscussthemagneticorderingsinthepresence of only Rashba SOC ( α∝ne}ationslash= 0 and β= 0). First of all, we showthe Fermisurfaceinstabilities. In this sectionwein- troduce the TNN terms with t3=−0.5 andα3=−0.5α, and set the chemical potential as µ= 0.98, following the previous study on the SOC-free case [15]. Figures 5(a) and 5(b) show the Fermi surface and the bare magnetic susceptibility in the absence of SOC. The Fermi surfaces show rather strong partial nesting with Q1= (π/3,π/3) andQ2= (−π/3,π/3) as shown in Fig. 5(a), which leadsto the distinct peaks in the susceptibility at the same wave vectors as shown in Fig. 5(b). When the Rashba SOC is introduced, the spin degen- eracy of the Fermi surface is lifted and accordingly the peaks in the susceptibility are split in a complicated way. Figures5(c)-5(f)displaythespin-splitFermisurfacesand the largest eigenvalues of the bare magnetic susceptibil- ityλ1 qforα= 0.2 andβ= 0. Due to the spin-splitting of the Fermi surface, the two peaks in the SOC-free suscep- tibility at Q1andQ2split into totally fourteen distinct peaks with almost equal amplitudes [Figs. 5(d) and 5(f)]. As shown in Fig. 5(f) we denote these wavevectorsas Qη ν withν= 1−4 forη=a,b,candν= 1,2 forη=d. Note that a set of wave vectors indexed with a superscript η, {Qη ν}, are related with each other by C4andσvsym- metry operations, yielding the exactly identical value of λ1 q. As displayed in Figs. 5(c) and 5(e) we can assign {Qa ν}(/braceleftbig Qb ν/bracerightbig ) to the wave vectors connecting two por- tions within the outer (inner) Fermi surfaces, while {Qc ν} (/braceleftbig Qd ν/bracerightbig ) to the wave vectors connecting from one in the inner (outer) Fermi surface to the other in the outer (in- ner). After closely comparing the competing heights of those peaks for large system sizes, we find out that λ1 qat {Qa ν}are slightly larger than the others [Figs. 5(c) and 5(d)]. In the following we discuss the results of the KPM-LD simulations. Figures 6 show the simulation results for J= 0.1. As already reported in details in the previous study [15], without the SOC a noncoplanar double- Qor- dering appears, characterizedwith Q1andQ2[Figs. 6(a) and 6(b)]. The double- Qordering, named 2 Q-vortex, shows a stripe of the spin scalar chirality [Figs. 6(e) and 6(f)] [15]. With the introduction of the Rashba SOC with α= 0.2, wefind amorecomplexmultiple- Qstate, asshownin Figs. 6(c) and 6(d). As stated in Sec. IIIA, for α= 0.2 we performed the KPM-LD by adopting several differ- ent spin ansatzes as the initial spin configurations, which are double- or single- Qorderings constructed from {Qa ν} [see Eq. (21)]. In Figs. 6(c) and 6(d) we only show the re- sults obtained from one of the initial ansatzes. We stress that for the other ansatzes we have also confirmed simi- lar multiple- Qorderingswith the same energy within the resolution of the KPM-LD, which are characterized with the same set of wave vectors as in Fig. 6(d), although the weight distributions among them vary to some extent. As seen in Fig. 6(d), the multiple- Qordering appears to be dominantly formed by {Qa ν}as well as other closely- located wave vectors such as Qc 3andQc 4[see Fig. 5(f)]. We find that the multiple- Qorder exhibits a modu- lated stripe of the spin scalar chirality whose net com- ponent vanishes, as shown in Figs. 6(g) and 6(h). The spatial pattern of the spin scalar chirality makes the multiple- Qorder distinct from SkXs, which, in general, show a nonzero net value of the scalar chirality. The discovery of such a complex multiple- Qordering is re- markable as compared with localized or continuum spin models with only NN interactions under the same sym-9 (b) (d) -

0 0 -

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0 0 -

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0 0 -

qy qx qy qxqy qxqy qx (e) (f) (g) (h) -

0 0 -

(  (c) FIG. 6. The results of the KPM-LD simulations for J= 0.1: (a,b,e,f) without the SOC ( α=β= 0) and (c,d,g,h) with the Rashba SOC ( α= 0.2 andβ= 0). (a) and (b) represent a typical spin pattern and the norm of its Fourier transform, |Sq|, of the 2Q-vortex state, while (c) and (d) correspond to those of the mu ltiple-Qstate. (e) and (f) are the real-space pattern of the spin scalar chirality and the absolute value of its Fourier t ransform, |κq|, for the 2 Q-vortex state in (a). (g) and (h) display those of the multiple- Qstate in (c). The real-space textures of spin and scalar chir ality in (a), (c), (e), and (g) are shown for a part of the whole system with N= 962system for clarity. In (a) and (c), the arrows denote the dire ctions of the localized spins in the xyplane, and the color represents the zcomponent. metry, in which simpler multiple- Qorderings like SkX are normally found [32, 33]. Although the effective spin model describing the RKKY interaction also predicts the stabilization of multiple- Qorderings [18], our study on the original Kondo lattice model indicates potential fo- mation of further complex multiple- Qorderings charac- terized with more than two wave vectors, which would be attributed to the full integration of the conduction electrons to the simulations.  (b)
-
0
-
0 qxqy FIG. 7. (a) Spin texture and (b) |Sq|of the coplanar 2 Q-flux order, constructed from Eq. (26). The dashed square in (a) denotes the manetic unit cell. Then we discuss the evolution of the magnetic order- ings while increasing J. The KPM-LD simulations reveal0.2 0.1 FIG. 8. Ground-state phase diagram for the Kondo lattice model with the Rashba SOC ( α/negationslash= 0 and β= 0), determined by the KPM-LD simulations. that without the SOC the noncoplanar 2 Q-vortex state is favored for J= 0.1 and 0.2, while it is replaced by a coplanar double- Qordering for J= 0.3 and 0.4, as represented in Fig. 7. We refer to the latter ordering as 2Q-flux. For α= 0.2 the complex multiple- Qorders are stabilized for J= 0.1 and 0.15, whereas a single- Q ordering is favored for J= 0.2−0.4. We note that the ordering vector of the single- Qordering found for large J, named 1 Q′, is not any of {Qη ν}but another relatively large wave vector denoted as the white arrows in Fig.10 5(d), around which the susceptibility takes a broad peak with a sizable height. We summarize the results in the J-αphase diagram in Fig. 8. (b) E - E1Q0.1 0.2 0      J  -1 -2 -3        ! " # $% & ' )* +J2Q-Q1Q3 2Q-Q1Q2 1Q'aa aaE - E1Q 2Q-vortex 2Q-flux-0.08 -0.16(a) (10-4) (10-4) FIG. 9. Jdependence of the energies for several ansatzes, estimated by variational calculations with (a) α=β= 0 and (b) α= 0.2 andβ= 0. (a) represents the energies for 2Q-vortex and 2 Q-flux, measured from that of the 1 Qhelical ordering. The inset of (a) is the same plot in the small J region. (b) shows the energies for two ansatzes for 2 Qstates and 1Q’, measured from that of the 1 Qhelical ordering. The calculations are done for N= 4802. Complementary to the KPM-LD we perform varia- tional calculations. In the absence of the Rashba SOC we compare the energiesof 2 Q-vortexand 2 Q-flux in Fig. 9(a). 2Q-vortex is described as [15] Sj= /radicalig 1−b2sin2(Q2·rj)cos(Q1·rj)/radicalig 1−b2sin2(Q2·rj)sin(Q1·rj) bsin(Q2·rj) ,(25) while 2Q-flux is found to be represented as Sj=ˆN cos(Q1·rj) cos(Q2·rj) 0 . (26) In Fig. 9(a) we set the variational parameter b, which describesthenoncoplanarity,at b= 0.6forthe2 Q-vortex ansatz in Eq. (25). Note that in Fig. 9(a) we subtractthe energy of the single- Qhelical ordering corresponding to theb= 0 case in Eq. (25). Figure 9(a) shows that forJ/lessorsimilar0.04, 2Q-vortex has lower energy than 2 Q-flux and vice versa for J/greaterorsimilar0.04. We also see that the helical ordering is unfavored in the whole range of Jstudied here. Thus, the variational calculations verify the trend in the KPM-LD that 2 Q-vortex transitions to 2 Q-flux while increasing J. The critical value of Jis considerably different between the two calculations, which might be attributed to the energy resolution of the KPM-LD or the incompleteness of the variational ansatzes. Figure 9(b) shows the energy comparison among sev- eral ansatzes for α= 0.2 andβ= 0. Since the multiple- Qstates discovered in the KPM-LD, e.g., Figs. 6(c) and 6(d), are too complicated to deduce the corresponding ansatzes, we simply employ the double- Qorderings that are used for the initial spin configurations in the KPM- LD [see Eq. (21)], which maximize the energy gain of the generalized RKKY Hamiltonian in Eq. (11) without the normalization factor. In Fig. 9(b) we denote the double- Qorderings formed by Qa ν1andQa ν2as 2Q-Qa ν1Qa ν2. Here the energies are measured from that of the single- Qor- dering formed by Qa ν, named 1 Q. Although the ansatzes for the multiple- Qstates are approximate ones, it turns out that they qualitatively reproduce the Jdependence obtained by the KPM-LD shown in Fig. 8: the double- Q orderings are favored up to J∼0.15 and replaced by 1 Q’ forJ/greaterorsimilar0.15. C. Case with α= 0andβ/negationslash= 0 Finally we discuss the case with the Dresselhaus SOC only (α= 0 and β∝ne}ationslash= 0). As stated in Sec. IID magnetic ordersfor the Dresselhaus-onlycase are identical to what are obtained by a πrotation of those for the Rashba-only case along the [110] axis [see Eq. (18)]. Hence the phase diagram for the Rashba-only case presented in Fig. 8 is common to the Dresselhaus-only case, with the simple global rotation applied to the magnetic orders. (a) , . / FIG. 10. (a) Multiple- Qorder with J= 0.1 forα= 0 and β= 0.2. This is produced by applying the πrotation along the [110] axis to the spin texture for α= 0.2 andβ= 0 shown in Fig. 6(c). (b) Spin scalar chirality of the multiple- Qorder in (a), which is identical to the Rashba-only case shown in Fig. 6(g).11 Figure 10(a) shows the multiple- Qorder in the Dresselhaus-only case with J= 0.1, which is obtained by applying the πrotation to the one for the Rashba-only case in Fig. 6(c). The uniform rotation leads to the same spatial pattern of the spin scalar chirality as the Rashba- only case, as shown in Fig. 10(b). We also remark on the distinction of the multiple- Qordering here from those expected in localized spin models describing only NN in- teractions with the same D2dsymemtry; in the latter case shows up a periodic array of antiskyrmions, which arecharacterizedwiththeoppositesignofthetopological invariant to conventional Skyrmions [25]. V. CONCLUSIONS To summarize, we have studied magnetic orderings generated by itinerant electrons subject to the Rashba (α) andDresselhaus( β) SOCsbymeansofthelarge-scale numerical simulations as well as the variational calcula- tions based on the perturbation analyses. We discovered the complex multiple- Qorderings under zero magnetic field, depending on the nature of the spin-split Fermi surfaces induced by the SOCs. For the equal strength of both SOCs ( α=β∝ne}ationslash= 0) the exotic spin texture is un- veiled in a broad range of J, characterized with as many as six wave vectors. Notably this sextuple- Qorderingshows a checkerboard-like pattern of the spin scalar chi- rality. In the case that only Rashba or Dresselhaus SOC exists (αorβ= 0) we found another type of complex multiple- Qstates, which aredistinct from those expected in localized spin systems under the same symmetry. Our findings suggest that the combination of the spin-charge and spin-orbit couplings under broken spatial inversion symmetry gives rise to richer multiple- Qmagnetic or- dersthanthe competition betweenthe ferromagneticand DM interactions in localized spin systems. Our theory would be potentially applicable to noncentrosymmetric f-electron compounds as well as heterostuctures of spin- orbit coupled metals and magnetic materials. ACKNOWLEDGMENTS We thank K. Barros and R. Ozawa for providing us with the code for the KPM-LD simulations. We are also grateful to R. Ozawa and S. Iino for fruitful discussions. K.N.O. acknowledges Y. Tserkovnyak for his incisive comments. The KPM-LD simulations were carried out at the Supercomputer Center, Institute for Solid State Physics, University of Tokyo. K.N.O. is supported by the Japan Society for the Promotion of Science through a research fellowship for young scientists. [1] K. Ohgushi, S. Murakami, and N. Nagaosa, Phys. Rev. B62, R6065 (2000). 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1805.00001v1.Spin_1_Bosons_in_the_Presence_of_Spin_orbit_Coupling.pdf

Spin-1 Bosons in the Presence of Spin-orbit Coupling Lin Xin1 1School of Physics, Georgia Institute of Technology, Atlanta, GA 30332, U.S.A (Dated: May 2, 2018) In this paper, I'm going to talk about the theoretical and experimental progress in studying spin- orbit coupled spin-1 bosons. Realization of spin-orbit coupled quantum gases opens a new avenue in cold atom physics. In particular, the interplay between spin-orbit coupling and inter-atomic interaction leads to many intriguing phenomena. Moreover, the non-zero momentum of ground states can be controlled by external elds, which allows for good quantum control. PACS numbers: 34.50.Cx 67.85.Hj 75.25.Dk Spin-orbit (SO) coupling is named because it's the in- teraction between a quantum particle's spin and its an- gular momentum. It comes from the conversion between electric eld to magnetic eld when changing into the moving frame of particles. It is important because when dealing with neutral atoms, the interaction between spin and magnetic eld dominates. A key point of this paper is to emphasize that a nearly isotropic SO coupling will dramatically enhance the eects of inter-particle interac- tions. There are two types of interaction that have been stud- ied extensively in condensed matter - Rashba ( xky ykx) [1] and Dresselhaus ( xky+ykx) [2] SO cou- pling.These two coupling have explained lots of phenom- ena for bulk materials. The Rashba SO coupling is a momentum-dependent splitting of spin bands in two- dimensional condensed matter systems (heterostructures and surface states) similar to the splitting of particles and anti-particles in the Dirac Hamiltonian. It can be derived in the framework of the k
pperturbation theory or from the point of view of a tight binding approxima- tion. The Dresselhaus SO coupling can be derived in a similar way. The most general expression of Rashba SO coupling is presented below: HSO= 2(~ ~k)
~ v: (1) Herevis the unit vector perpendicular to the surface, kis the momentum of particle and is the vector of Pauli matrices. There is a naive derivation of Rashba Hamiltonian for particles moving in static electric elds. For example, E=E0^z. When we use the frame moving with atoms, wherev=~ m(kx;ky;kz). By applying the Maxwell func- tion in a moving frame. ~E0=~E+~ v~B (2) ~B0=~B~ v~E c2(3)In the new frame, we have ~E0=E0^z (4) ~B0=E0 mc2(ky;kx;0): (5) Thus, the Hamiltonian has a term between the spin and interaction with the magnetic eld caused by mo- mentum. HSO=~ 
~BSO(k) =gsBS ~
E0 mc2(ky;kx;0) (6) =gsBE0 ~mc2(xkyykx) (7) wheregsis Landeg factor and Bis Bohr magneton. As a result the total Hamiltonian for a single boson only under the presence of electric eld is : H=~2 2m(~k2+ 2(~ ~k)
^z): (8) Here all the constants are combined in . In general situation with arbitrary ~Epoint in ^ndirection and with atomic units ~!1;me!1 H=1 2m(~k2+ 2(~ ~k)
^n): (9) The electric eld given by a nucleus in the rest frame of the electron is: E=jE rjr: (10) Additionally, since jEj=@V @r B=1 mec21 rrp@V @r=1 mec21 r@V @rL: (11) As a result, H=
B (12) =1 mec21 r@V @rL
gsBS ~=gsB mec2r~@V @rL
S: (13)arXiv:1805.00001v1 [cond-mat.quant-gas] 28 Apr 20182 FIG. 1: (color online) Pseudo-spin rotation rotates the real spin by 90oanticlockwise. The Larmor interaction energy is in the form of L
S coupling, that's the reason why it's called spin-orbital coupling. In the second quantized picture, the Hamiltonian is H=Z d2r y1 2m(~k2+ 2(~ ~k)
~ n) (14) In many other literature [3] people use pseudo-spin ro- tation0 x=yand0 y=xto simplify problem. This rotation will change the cross product between andk into a dot product, as shown in Fig. 1. Here, I follow this conversion. The Hamiltonian will be H=Z d2r y1 2m(~k2+ 2(~ 
~k)) (15) In the rst part of this paper, what I will focus on is the ground state for a two dimensional (2D) Bose-Einstein Condensates (BEC) and study dierent phases of them. Numerical Method In order to solve the ground state for general Hamil- tonian, using a numerical method is a very popular way to attack problem, given the advance of computer per- formance nowadays. The method used here is called the Imaginary Time Evolution method [4]. The physics be- hind that is straightforward. Decomposing the initial wave function into the eigenbasis, where 0is the ground state with lowest energy and nis thenthexcited state, I have j >=c0j0>+c1j1>+c2j2>+


(16) The time evolution of the wave function is j (t)>=c0eiE0 ~tj0>+c1eiE1 ~tj1>+


(17) FIG. 2: (color online) The energy dispersion for helicity  when a)>0 b)<0 If I change the time tinto imaginary time it, the time evolution will become j (t)>=c0eE0 ~tj0>+c1eE1 ~tj1>+


(18) The ground state will decay slowest since it has the lowest energy. As a result, after every discrete imaginary time step, if we renormalize j (t)>, it will have a larger and larger portion of j0>. When the energy coverages to a constant, we can consider j (t)>asj0>. In this Hamiltonian, I noticed that there are dier- ential operators that come from kinetic energy term K. In order to solve that, the fast Fourier transformation method instead of the time consuming nite dierence method is applied. When j (x)>is transformed into Fourier spacej (k)>,eiK ~tcan be multiplied directly since it's in the momentum space. After that I use the reverse transform to get eiK ~tj (k)>back to real space and multiply it with eiV ~t, which is the spatial depen- dent potential. The algorithm for computing time evolution is j (x;t+
t)>=eiV ~
tIFFT (eiK ~
tFFT (j (x;t)>)) (19) Here
tis the discrete time step between two iterations. Spin-1/2 BEC without Interaction Hamiltonian of spin-1/2 BEC without interaction is identical to single particle Hamiltonian. Without external potential H0=1 2m(~k2+ 2~ 
~k) (20) Spin is no longer a good quantum number here. In- stead, helicity is a good quantum number here. Helicity \" means that the spin direction is either parallel or anti-parallel to the momentum. For these two helicity3 branches, their dispersion relation are given by Ek=1 2m(jkj22jkj) (21) wherejkj=q k2x+k2y. For the case  > 0, the \" helicity branch has lower energy. The single particle minimum is locating atjkj=. This minimum has degenerate ground states with dierent azimuthal angles. Without SO coupling, the Hamiltonian has a unique ground state at k= 0. The weak interaction won't play a signicant role here because the ground state is unique. However, in the pres- ence of SO coupling the weak interaction plays a large part, since the ground states has a strong degeneracy. Assuming the atoms are in a harmonic potential V= 1 2m!2r2, since you need to trap the atom to observe it and one of the most common way is to use the dipole trap from a strong laser in experiment. In order to solve the Schrodinger equation with the Hamiltonian (15) for a many-body system, the mean-eld theory is used here, which replaces the eld operator with its expectation valuem=<^ m>. The energy of the system is in the form E=Z d2r1 2m(X mz=";# mz(k2+ ~!2r2)mz +2X mz;m0z=";#mz(xkx+ykx)m0z)(22) Here ~!=m!. Since what's important is the value of ~ ! and, I neglect the general factor1 2mhereafter. For the spin-1/2 case, the Pauli matrices look like this: x=0 1 1 0 ;y=0i i0 ;z=1 0 01 :(23) Writing the energy in explicit form of spin components, gives: E=Z d2rfX mz=";# mz(k2+ ~!2r2)mz +2f "(kxiky)#+ #(kx+iky)"g:(24) To solve it numerically, the main challenge comes from the two spin components (  m0zAmz;m0 z6=mz) be cou- pled together. ( Ais any operator). I can separated the coupled term Hsofrom non coupled term H0 j (t)>=eiH
tj > =eiHso
t(eiH0
tI)j >(25) whereeiHso
t=ei2(xkx+yky)
t. FIG. 3: (color online) The ground state density of both the spin up a) and down b) without scattering interaction, which has a strong degeneracy for dierent direction of k. TheeiHsothas an analytic formula that is solvable, because the exponential of a Pauli vector looks like: eia(^n
~ )=Icosa+i(^n
~ ) sina; (26) ei2(xkx+yky)
t=Icosa+i(^n
~ ) sina: (27) Here,a =2
tq k2x+k2yand ^n = (kx=q k2x+k2y;ky=q k2x+k2y) Spin-1/2 BEC with Interaction The interaction Hamiltonian term in a 2D spin-1/2 BEC is in the form of Hint=Z d2r y(c0 2n2+c2 2S2 z) (28) nis the operator of number of particles and Szis the op- erator of total spin z-component. This formula is derived from the contact interaction and the s-wave scattering. Now the time-dependent equation looks like i@ @t= (H0+HSO+Hint) (29) which is called Gross-Pitaevskii equation [5]. E=Z d2rfX mz=";# mz(r2+ ~!2r2)mz +2f "(kxiky)#+ #(kx+iky)"g +c0 2(j"j2+j#j2)2+c2 2(j"j2j#j2)2(30) Here below, all the numerical results are solved in atomic units (a.u.), which is ~= 1;me= 1. The matrix form of the interaction Hamiltonian is:4 Hint=  "  #c0+c2 2j"j2+c0c2 2j#j20 0c0+c2 2j#j2+c0c2 2j"j2" #: (31) eiHint
t= ei(c0+c2 2j"j2+c0c2 2j#j2)
t0 0 ei(c0+c2 2j#j2+c0c2 2j"j2)
t! (32) To simplify the expression, I use =c2=c0as the ratio between the dierent scattering lengths instead of using c0,c2as separate coecients. Combining these terms, I formulate the detailed equa- tion that will be used to calculate the ground state. (t+
t) =ei(H0+HSO+Hint)
t (t) =eiHSO
tei(H0+Hint)
t (t) = (Icosa+i(^n
~ ) sina)ei(H0+Hint)
t (t)(33) With: eiH0
t= ei(k2+~!2r2)
t0 0ei(k2+~!2r2)
t! (34) Icosa+i(^n
~ ) sina= cosa i (nxiny) sina i(nx+iny) sina cosa :(35) As a result, we can also solve the equation to get the ground state for the BEC with interaction. When >0, the ground state is called a plane wave (PW) phase. This is because the phase of condensate wave function looks like a plane wave in Fig. 5. When <0, the ground state is called standing wave (SW) phase. It's pretty straight- forward to understand it because the density looks pretty much like a standing wave (Fig. 6) [6]. Although in numerical simulation, the harmonic trap is included to avoid an artifact from a sharp boundary and also simulate the practical situation of a cold atom experiment, the results can be understood without ex- ternal potential. With SO coupling, Ek=jkj2=2jkj from the single particle Hamiltonian discussion above. Here the \" denotes dierent helicity (spin orthogonal to wave factor). When jj>0, the single particle ground state is in the negative helicity branch, with jkj=by minimizing the energy. The wave function is k=1p 2eikr1 ei'k (36) FIG. 4: (color online) The ground state density of both the spin"a) and#b) with scattering interaction c0= 5; = 1, which is the PW phase. From the gures we can see the vortices work as the domain wall. FIG. 5: (color online) The phase of both the spin "a) and #b) ground state with scattering interaction c0= 5; = 1. Since the phase increase from (dark regime) to (bright regime) periodically, we call it "plane wave phase" (PW) where'k=arg(kx+iky) +which is anti-parallel to momentum, and 'k='k+. Consider the wave function as a composition of two opposite wave vector states as '=1p 2eikr1 ei'k +2p 2eikr1 ei'k : (37) Using this wave function to minimize the interaction energy. When > 0, we get1= 1;2= 0 or1= 0;2= 1. '=1p 2eikr1 ei'k or1p 2eikr1 ei'k (38) The wave function is a single plane wave, corresponding to the PW phase. This state breaks the time-reversal,5 FIG. 6: (color online) The ground state density of both the spin"a) and#b) with scattering interaction c0= 5; =1, which is the SW phase. From the gures we can see both components oscillate periodically in the SW regime. FIG. 7: (color online) The phase of both the spin "a) and# b) ground state with scattering interaction c0= 5; =1. the rotational symmetry and the U(1) symmetry of the super uid phase. When <0, we get1=2= 1=p 2. '=1 2eikr1 ei'k +1 2eikr1 ei'k (39) Thus "/cos(kr);#/isin(kr) (40) Which corresponds to the SW phase (also named as a \stripe super uid"). This state breaks rotational sym- metry while keeping the U(1) symmetry of the super- uid phase, re ection symmetry and translation symme- try along stripe direction. In general, we should substitute the single kstate with a superposition of all wave vector states with dierent azimuthal angles: Z d'kkp 2eikr1 ei'k (41) If we minimize the energy with respect to k, we can nd the most favorable solution is alway single k or for a pair offk;kg. The reason of the symmetry breaking is because the interaction term has preference in specic spin vectors. This shows how a nearly can isotropic SO coupling enhance the eects of inter-particle interactions. FIG. 8: (color online) Numerical results for the spin-1 cases. a1-a3 are density of -1, 0, 1 components in the SW phase with = 0:2. b1-b3 are phase of -1, 0, 1 components in the PW phase with =0:2. [6] Spin-1 BEC without Interaction We can also derive the Hamiltonian for the spin-1 case. For this case, the Pauli matrices look like this: x=1p 20 @0 1 0 1 0 1 0 1 01 A;y=ip 20 @01 0 1 01 0 1 01 A; z=0 @1 0 0 0 0 0 0 011 A(42) The energy now changes into the form of E=Z d2rfX mz=1;0;1 mz(~2 2mr2+1 2m!2r2)mz +=p 2f 1(ky+ikx)0+ 0(kyikx)1 + 0(ky+ikx)1+ 1(ky+ikx)0g (43) E=Z d2rfX mz=1;0;1 mz(~2 2mr2+1 2m!2r2)mz +=p 2f 1(i@y@x)0+ 0(i@y+@x)1 + 0(i@y@x)1+ 1(i@y+@x)0g (44) This is also solvable. This case is a little bit compli- cated to program, so here I just present the results in Fig. 8 from [6].6 FIG. 9: (color online) a) A schematic of NIST experiment, in which two counter propagating Raman beams are applied. b) A schematic of how F= 1 levels are coupled by Raman beams. [3] RELEVANT EXPERIMENT Considering an experiment with a87Rb Bose-Einstein condensate (BEC), where a pair of Raman lasers cre- ate a momentum-sensitive coupling between two internal atomic states [7]. This SO coupling is equivalent to that of an electronic system with equal distribution of Rashba and Dresselhaus couplings, and with a uniform magnetic eld ofBin theyzplane. The derivation of the Hamil- tonian is shown below. The Raman resonance is a phenomenon which re- quires two-photon processes. As Fig. 9 a) shows, I have two counter-propogating beams shine on BEC. The blue beam isp =2ei(!1t+k1x)while the red beam isp =2ei(!2tk2x). The frequency dierence !1!2=!. Here!is the energy gap between two spin components. For this reason, the coupling term becomes: H12=p =2ei(!1t+k1x)p =2ei(!2tk2x) = =2ei(!t+(k1+k2)x)(45) At the situation where !k1;k2andk1k2=k0, the coupling term becomes H12= ei2k0x. The matrix form in the bases of j1;1>;j1;0> H= k2 x 2m+h 2 2ei2k0x 2ei2k0xk2 x 2mh 2! (46) By applying a unitary transformation with U=eik0x0 0eik0x (47) One reaches an eective Hamiltonian that describes FIG. 10: (color online) The dispersion relation for the Hamil- tonian in the experiment with = 0 and = 0 : 0 :1 : 1. SO coupling HSO=UHSOUy = (kx+k0)2 2m+h 2 2 2(kxk0)2 2mh 2! =1 2m(kx+k0z)2+ 2x+h 2z(48) Using the pseudo-spin rotation 0 x=z;0 z=x, and combining all constants in a single coecient, we have H=1 2m(k2+ z+x+kxx) (49) In this Hamiltonian, we get the last term kxxequiva- lent to an equal distribution of Rashba and Dresselhaus couplings. Thus, this Raman laser coupling system is equivalent to a 2D SO coupling: H=~2~k2 2m(~B+~BSO(k))
~ : (50) Since in realty, there is no spin 1 =2 bosons, we use F= 1 ground electronic manifold and label them with pseudo- spin-up and pseudo-spin-down: j">=jF= 1;mF= 0> andj#>=jF= 1;mF=1>. The initial BEC state is an equal population of j">andj#>. Let's do the same analysis, as the theory I mentioned previously. Without the interaction, we can get the dis- persion relation E=1 2m(k2 xp 2+2k2x+ 2kx+2) (51) where the eigenstates are u= 1 +kx 1 ;d= +1 +kx 1 (52)7 FIG. 11: (color online) The ground state density when is large, so there is only one minima and it has an equal distribution inj">x;j#>x. FIG. 12: (color online) The ground state momentum density when is large, since there is only one minima at kx= 0. It's easy to see the momentum density is centered at kx= 0. Here1=p 2+2k2x+ 2kx+2. When= 0 and I increase , the dispersion relation changes from two minima at kx6= 0 to a single minimum kx= 0 as shown in Fig. 10. Using the same numerical method as mentioned before, the ground state can be calculated, here the j">x;j#>x are spin up and down eigenstates in the xdirection. It's convenient to use this basis because of the two xterms in Hamiltonian. When = 0 and I change >0, the dispersion relation no longer has two global minima instead one of them changes into two local minima as shown in Fig. 13. Thus, the ground state will prefer the j#>xcomponents and change the average kx>0. If is not very strong, there will still be some fraction in j"x;kx<0>. This result is shown in Fig. 15 and Fig. 16. Similar things will happen when = 0 and I change >0. The ground state will prefer the j">xcomponents and change the average kx<0. If is not very strong, there will still be some fraction in j#x;kx>0>. This result is shown in Fig. 17 and Fig. 18. From the simulation results, the inter-particle inter- action doesn't play an important role here to determine the ground state phase. This is because the SO coupling here is not nearly isotropic. The kxxterm doesn't have a strong degeneracy here, thus there is no amplifying of the scattering term in the Hamiltonian. By changing dierent and , I can control the disper- sion minimum. I can realize the results of the simulations by varying the Raman laser coupling and z-direction FIG. 13: (color online) The dispersion relation for the Hamil- tonian in the experiment with small positive , which tilt the ground state has two local minimum j#x;kx>0>;j"x;kx< 0>. FIG. 14: (color online) The dispersion relation for the Hamil- tonian in the experiment with small negative , which tilt the ground state has two local minimum j#x;kx>0>;j"x;kx< 0>. magnetic eld adiabatically in experiment. To detect whether the state is mixed or at dierent phase, the time of ight method is applied. Through letting the con- densates evolve freely after some time, ground states has kx6= 0 at the double minimum phase will split. The total phase diagram in Fig. 19 is calculated in [7]. To sum up, I rst introduced the denition of spin- orbital coupling in this paper. Then, I discussed the SW and PW phases in a 2D Rashba SO coupled spin-1/2 BEC by controlling the sign of the scattering term [6] in the rst part. The ground state in the PW phase will show the domain wall formed by vortices. This reveals the fact that nearly isotropic SO coupling can amplify the inter-particle interaction. The next topic I discussed is the realization in an experiment by using the Raman coupling lasers in a87Rbspin-1 BEC[7]. By taking ad- vantage of this technique, I can study the eect of an equal contribution of Rashba and Dresselhaus coupling. By controlling the Raman coupling strength and mag-8 FIG. 15: (color online) The ground state density for small positive. Large fraction of ground state is in j#>xas shown in the Fig. 13. FIG. 16: (color online) The ground state momentum density whenis positive, since there are two local minimum at kx6= 0. Comparing to Fig. 15, the pairs j#x;kx>0>;j"x;kx< 0>are indicated clearly. netic eld, the ground state can be tuned adiabatically from a single spin component to double components, and from nonzero momentum to zero momentum. The basic method to do the analysis and numerical simulation is also included also to help understand the results. FIG. 17: (color online) The ground state density for small negative. Large fraction of ground state is in j ">xas shown in the Fig. 14. FIG. 18: (color online) The ground state momentum density whenis negative, since there are two local minimum at kx6= 0. Comparing to Fig. 17, the pairs j#x;kx>0>;j"x ;kx<0>are indicated clearly. FIG. 19: (color online) The ground state phase diagram cal- culated in [7].9 Electronic address: lxin9@gatech.edu [1] Y. A. Bychkov and E. I. Rashba, Journal of Physics C: Solid State Physics 17, 6039 (1984). [2] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [3] H. ZHAI, International Journal of Modern Physics B 26, 1230001 (2012).[4] H. Tal-Ezer and R. Koslo, The Journal of chemical physics 81, 3967 (1984). [5] E. P. Gross, Il Nuovo Cimento (1955-1965) 20, 454 (1961), ISSN 1827-6121. [6] C. Wang, C. Gao, C.-M. Jian, and H. Zhai, Phys. Rev. Lett. 105, 160403 (2010). [7] Y. J. Lin, K. Jim enez-Garc a, and I. B. Spielman, Nature 471, 83 (2011), ISSN 0028-0836.

1206.1060v2.Spin_Orbit_Coupling_in_LaAlO__3__SrTiO__3__interfaces__Magnetism_and_Orbital_Ordering.pdf

arXiv:1206.1060v2 [cond-mat.str-el] 15 Feb 2013Spin-Orbit Coupling in LaAlO 3/SrTiO 3interfaces: Magnetism and Orbital Ordering Mark H. Fischer1, Srinivas Raghu2, Eun-Ah Kim1 1Department of Physics, Cornell University, Ithaca, New York 148 53, USA 2Department of Physics, Stanford University, Stanford, Californ ia 94305, USA E-mail:mark.fischer@cornell.edu Abstract. Rashba spin-orbit coupling together with electron correlations in th e metallic interface between SrTiO 3and LaAlO 3can lead to an unusual combination of magnetic and orbital ordering. We consider such phenomena in th e context of the recent observation of anisotropic magnetism. Firstly, we show tha t Rashba spin-orbit coupling can account for the observed magnetic anisotropy, assu ming a correlation driven (Stoner type) instability toward ferromagnetism. Secondly , we investigate nematicity in the form of an orbital imbalance between d xz/ dyzorbitals. We find an enhanced susceptibility toward nematicity due to the van Hove sin gularity in the low-electron-density regime. In addition, the coupling between in-p lane magnetisation anisotropy and nematic order provides an effective symmetry brea king field in the magneticphase. Weestimatethiscouplingtobesubstantialinthe low -electron-density regime. The resulting orbital ordering can affect magneto transpo rt. PACS numbers: 75.70.Cn, 75.30.Gw, 75.25.DkSpin-Orbit Coupling in LaAlO 3/SrTiO 3 2 1. Introduction Rashbaspin-orbit-coupling(SOC)effectshavebeenmostlystudied inweaklyinteracting systems such as semiconductor heterostructures designed for spintronics applications[1], and occur in two-dimensional systems without mirror symmetry[2]. However, the effects of Rashba SOC in a two-dimensional system with strongly interacting electrons, found for example in interfaces, are emerging as a new frontier. There is t hus a pursuit for new emergent phases of matter in this regime, both theoretically[3] and experimentally[4], with the electron gas at the interface between the two non-magne tic insulators LaAlO 3 and SrTiO 3(LAO/STO) the widest-studied such example. The observed ferro magnetic instability atthis interface[5]could well beof Stoner type, which sug gests that electronic correlations may be enhanced due to low dimensionality and poor scre ening at low densities. Hence, combined with its demonstrated tunability[6, 7], t he LAO/STO interface is an ideal testbed for physics of Rashba SOC in correlate d electron systems. Recent observation of magnetic anisotropy may signal further ric hness in the phase diagram of the interface. Specifically, Bert et al.[8] and Li et al.[9] observed strong in-plane preference for magnetisation [see figure 1 (a)]. Bert et al.attributed this observed anisotropy to the shape anisotropy of the interface: e nergetic bias towards a certain magnetisation direction e.g., along the longest axis of an ellipso id, driven by an anisotropic demagnetisation field. However, for ultra thin films cons isting of only a few atomic layers this effect is subdominant next to microscopic effects[ 10, 11]. If the shape anisotropy is negligible in the interfaces, the dominant s ource of magnetic anisotropy would be spin-orbit coupling. However, the typ ical alignment of the spin with the largest angular momentum in itinerant d-electron systems[12] would predict an out-of-plane magnetisation upon occupation of d xzand dyzorbitals, in disagreement with experiments[13, 9, 8]. In this work, we show tha t Rashba spin- orbit coupling leads to the unusual circumstance of an anisotropic (spin) susceptibility . Assuming Stoner ferromagnetism close to a van Hove singularity nea r band edges due to SOC, we argue that this in turn leads to a magnetisation anisotrop y. Wefurtherinvestigatethepossibilityofnematicorderintheformof orbitalordering between the d xzand dyzorbitals, since the large density of states near the band edge also leads to an enhanced tendency towards such ordering. This or der can couple to an in-plane magnetisation anisotropy and we show here that a consequ ence of the Rashba SOC is a strong such coupling in the low-density regime near the band e dge. This implies that orbital ordering accompanies the magnetic phase and lea ds to an additional magnetisation anisotropy within the plane. 2. Anisotropic Susceptibility We use a three-band model for the Ti t 2gorbitals in the xyplane to describe the electronic structure of the interface[14]. For now, we ignore the a tomic spin-orbit coupling in order to gain more analytic insight. In the presence of an e xternal magneticSpin-Orbit Coupling in LaAlO 3/SrTiO 3 3 field/vectorH, the Hamiltonian reads H=H0+HR soc−µB/vectorH·/vectorS. (1) Here,H0is the hopping Hamiltonian H0=/summationdisplay l,k,sξ(0) lkc† lksclks (2) with bare dispersions ξ(0) lk=k2 x/2ml x+k2 y/2ml y−µlandc† lkscreates an electron in band l= (1,2,3)≡(dxz, dyz, dxy) with momentum k, and spin s. We use mass parameters from reference [15]: the light masses m=m1 x=m2 y=m3 x=m3 y= 0.7meand the heavy massesM=m1 y=m2 x= 15mewithmethe electron mass. The chemical potentials are related by µ=µ1=µ2=µ3+ ∆ and we use in the following ∆ = 50meV. In the Zeeman term [the last term in equation (1)], /vectorS=/summationtext l,k,s,s′c† lks/vector σss′clks′is the total spin with /vector σbeing the Pauli matrices. Finally, HR socis the Rashba SOC at the interface due to the absence of the in-plane mirror symmetry, which can phen omenologically be introduced as a relativistic effect due to an electric field /vectorEinzdirection: The spin of an electron moving with velocity /vector vcouples to an effective magnetic field ( /vectorE/c×/vector v). Since the velocity vl=∂ξ(0) lk/∂kfor an electron in band lisk-dependent, the Rashba SOC is HR soc=α/summationdisplay l,k,s,s′/vector glk·(c† lks/vector σss′clks′), (3) where/vector glk= (vl,y,−vl,x,0) and the overall scale α≈10−11eVm[16]. The Rashba termHR socchanges the zero-field bandstructure by splitting the spin degene racy of the individual bands, as shown in Fig. 1(b). Notice how different bands ha ve different band- edge configurations: a ring of lowest energy momenta for the two- dimensional (2D ) dxy band and a saddle point for the two quasi-one-dimensional (quasi-1 D ) bands stemming from the dxzanddyzorbitals. This difference is due to the isotropic (anisotropic) momentum dependence of the Rashba coupling /vector glkfor the 2D (quasi-1D ) bands and leads to different types of divergences in the density of states ρ(ε) at the respective band edges: A 1 /√εdivergence at the bottom of the dxyband and a logarithmic divergence nearthebottomofthe dxz,dyzbands. Intheregimeoflowelectrondensities, thissystem can thus have instabilities to broken-symmetry phases even for we ak interactions. Now we turn to the impact of HR socon the in-field ( /vectorH∝negationslash= 0) bandstructure, which leads to one of our main results: the anisotropy in the bare uniform s usceptibility near band edges. In order to calculate the bare susceptibility, we fi rst diagonalize the Hamiltonian (1) for each momentum kto obtain the in-field spectrum ξkν(/vectorH), where ν= 1,...,6. Then, the (diagonal) susceptibility is given through χi=∂2ω ∂H2 i/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vectorH=0, with ω= Ω/Nthe grand potential per lattice site, χi=1 N/summationdisplay ν,k/braceleftBig 1 4Tcosh[ξkν(/vectorH)/(2T)]2/bracketleftBig∂ξkν(/vectorH) ∂Hi/bracketrightBig2−nF[ξkν(/vectorH)]/bracketleftBig∂2ξkν(/vectorH) (∂Hi)2/bracketrightBig/bracerightBig |/vectorH=0,(4) whereTis the temperature and nF(ξ) is the Fermi distribution function. In the absence of the Rashba term, the Hamiltonian (1) is diagonal for the spin-qua ntisation directionSpin-Orbit Coupling in LaAlO 3/SrTiO 3 4 Figure 1. (a) The distribution of the dipole-moment direction in terms of the angle from the zaxis as observed in reference [8] (reproduced with permission). (b ) Dispersion ǫ(k) and density of states ρ(ε) of the three-band model equation (1) along kx. The two quasi-1D bands are shifted by ∆ = 50meV compared to the 2 D band. The inset shows a spin-texture along Rashba-spilt parabolic bands. parallel to /vectorHandξkν(/vectorH) is linear in /vectorH. Hence, only the first term in equation (4) contributes to the susceptibility: the usual Pauli susceptibility. Sin ce the Rashba term HR socdoes not commute with the Zeeman term, ξkν(/vectorH) is not linear in /vectorHonceHR socis present and the second term of equation (4), the so-called van Vle ck susceptibility, becomes non-zero. The direction-dependent balance between th e two contributions determines a possible anisotropy χz∝negationslash=χx. As the Hamiltonian in equation (1) is block diagonal in the orbital basis, the total bare susceptibility is the sum of contributions χi(l) from each orbital l. We start with the contribution from the quasi-1D orbitals. The two dxzbands have dispersions: ξxz k±(/vectorH) =k2 x 2m+k2 y 2M−µ±|(α/vector gxz,k−µB/vectorH)|. (5) They contribute to the total susceptibility through χP i(xz) =µ2 B/summationdisplay k,±(ˆgi xz,k)2 1 4Tcosh[ξxz k±(0)/(2T)]2(6) and χvV i(xz) =µ2 B/summationdisplay k[1−(ˆgi xz,k)2]nF[ξxz k−(0)]−nF[ξxz k+(0)] |/vector gxz,k|(7) with ˆgxz,kthe unit vector along /vector gxz,k. ForT→0, we substitute ˜ky= (M/m)kyand change the sums in equations (6) and (7) into (cylindrical) integrals. Forµ >0, we obtain χP i(xz) =µ2 BM 2π2/integraldisplay dφ(˜gi xz,φ)2 cos2φ+M msin2φ(8) and χvV i(xz) =µ2 BM 2π2/integraldisplay dφ1−(˜gi xz,φ)2 cos2φ+M msin2φ. (9)Spin-Orbit Coupling in LaAlO 3/SrTiO 3 5 Figure 2. In-plane (solid line) and out-of-plane (dashed line) total spin susce ptibility for the three-band model (a) without and (b) with atomic SOC ( αat= 5meV) with the gray area denoting the region with anisotropic susceptibility. Th e scale in these plots is given by χxy 0=µ2 Bm/πandχall 0=χxy 0+2µ2 B√ Mm/π(right vertical axes). In (a), the contributions of the two quasi-1D bands for χxare also shown separately. The arrows in the insets denote the preferred magnetisation direction , where in (b) also the anisotropy outside the ’Rashba’ regime due to atomic SOC is shown. with ˜gi xz,φ= (sinφ,−cosφ,0) andφis the angle relative to the crystalline x-axis. Clearly, the total d xzcontribution to the susceptibility χi(xz) =µ2 B√ mM/πis independent of the field direction ior the Rashba SOC strength α. On the other hand, near the band edge of the one-dimensional ban ds, i.e., −α2/2m < µ < 0, the susceptibilities in equations (6) and (7) yield χP i(xz) =µ2 BM 2π2/integraldisplay dφ(˜gi xz,φ)2 cos2φ+M msin2φ[1+2µm(cos2φ+M msin2φ)/α2]−1/2(10) and χvV i(xz) =µ2 BM 2π2/integraldisplay dφ1−(˜gi xz,φ)2 cos2φ+M msin2φ[1+2µm(cos2φ+M msin2φ)/α2]1/2.(11) The total contribution to the susceptibility is thus anisotropic at th e band edge. For the total bare susceptibility, we also need to consider the 2D or bital d xy contribution. We can read off χi(xy) from the quasi-1D contribution χi(xz) discussed above by setting M=mand shifting the band edge by ∆. Therefore, χi(xy) =χxy 0is isotropic for µ >−∆. When contributions from all the orbitals are combined, the totalSpin-Orbit Coupling in LaAlO 3/SrTiO 3 6 -4 -2 0 2 4¸(a.u.) ¹[meV] ® at=0meV ® at=5meV -6 Figure 3. Coupling constant λ∝χxxηfor coupling η=n1−n2toM2 x−M2 yas a function of chemical potential for different atomic SOC parameter s. Note that λfor atomic spin-orbit coupling only would in this plot not significantly deviate from the zero line (thin dotted line). susceptibility shows two regions in the chemical potential where χx=χy> χz: near the band edge of the 2D bands and that of the quasi-1D bands. Fig. 2(a ) shows the total susceptibility for the three bands near the quasi-1D band edge as a function of chemical potential, where we shaded the anisotropic region. Fig. 2(b) summarises two ways atomic SOC impacts the phase diagram : (i) it widens the region of anisotropic susceptibility (grey region), (ii) it ca uses anisotropy in the spin direction by aligning the spin with the largest angular momentu m when the susceptibility is isotropic (white region). The first effect results fro m adding the atomic SOC, HSOC=iαat 2/summationdisplay lmnǫlmn/summationdisplay k,s,s′c† lkscmks′σn ss′ (12) to the Hamiltonian (1) andevaluating the susceptibility (4) numerically . For the second effect, the atomic SOC is treated as a perturbation in the magnetic p hase[12]. This again predicts in-plane magnetisation below the grey region where on ly the 2D band is occupied, but a switch to out-of-plane magnetisation above this re gion. In order to compare our phase diagram with experiments, we need t o translate the chemical potential to a gate voltage. While such a translation is n on-trivial, Hall measurements under gate-voltage sweeps can offer hints as t o where the as-grown samples lie. Joshua et al.[13] showed that the interface acquires heavy carriers when under a gate voltage, which indicates that the as-grown samples ar e near the quasi-1D bandedge. ThoughthegreyregionisnarrowinFig.2(b), thedensit ychangesbyafactor of≈2 in this range. This pushes the density upper bound for in-plane mag netisation substantially, consistent with experiments[17].Spin-Orbit Coupling in LaAlO 3/SrTiO 3 7 3. Orbital Order and Magnetisation In the regime of low electron densities, the van Hove singularity can p romote broken symmetry phases in the presence of suitable interactions. While we f ocused so far on a magnetic instability, which could for instance be driven by repulsive in tra-orbital interactions, we now turn to orbital-ordering possibilities. Even tho ugh the three-fold degeneracy of the t2gorbitals is already broken due to the interface symmetry, the dxz anddyzorbitalsremaindegenerate. Aspontaneous orbitalsymmetry bre aking described by a non-vanishing order parameter η≡n1−n2, withn1/2the occupation of the dxz/dyz orbital, can be driven by inter-orbital repulsive interactions[18]. We first analyse the tendency toward such an instability by studying the (bare) nematic susceptibility. For this purpose, we introduce the field Hηconjugate to η, which enters the Hamiltonian (1) throughµ1/2=µ±Hη, i.e. it acts through an opposite shift of the chemical potential for the two orbitals. The nematic susceptibility then yields χη=∂2ω ∂H2η/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle Hη=0=∂2ω ∂µ2 1+∂2ω ∂µ2 2(13) and is given by the contribution of the dxzanddyzorbitals to the density of states. Hence, the van Hove singularity near the band edge of the quasi-1D bands[see Fig.1(b)] allows for a nematic order for sufficiently strong inter-orbital inter action[18]. Next, we consider the coexistence of nematic and magnetic order. While a system withC4symmetry does not allow for coupling of ηand|/vectorM|directly, there is an allowed tri-linear coupling between ηand an in-plane anisotropy M2 x−M2 y, as both acquire a factor of −1 under C 4rotation. Specifically, a coupling of the form λ(M2 x−M2 y)ηenters the free energy. Given the Hamiltonian (1), we can explicitly calculate the coupling constant λ. It is given by the generalised susceptibility λ∝χηxx=/parenleftBig∂3ω ∂Hη∂H2 x/parenrightBig |/vectorH|=0, (14) which measures the change in the spin susceptibility χxupon shifting the chemical potential of the two quasi-1D bands against each other. Figure 3 s hows the result with and without atomic SOC in the presence of Rashba SOC. (We find t he coupling to be negligible without Rashba SOC.) The coupling becomes substantia l in the above- identified density range with the in-plane preference in the bare spin susceptibility. This is due to the inequivalence between the two quasi-1D band contribut ions toχxshown as dotted and dash/dotted lines in Fig. 2(a), more specifically the differ ence in the slope of χx(xz) andχx(yz). The sign of λdetermines the relative sign between ηand (M2 x−M2 y) and whether the majority orbitals will be along or perpendicular to th e magnetisation axis. Notice how our result shown in Fig. 3 predicts a change of the sig n ofλover the chemical potential range exhibiting the magnetisation anisotropy. Giventheobserved magnetism, wefinallyinvestigatetheeffectsofc ouplingbetween magnetic order and nematic fluctuation associated with nearby nem atic phase. In particular, we may ask on the one hand how the proximity to a nematic instabilitySpin-Orbit Coupling in LaAlO 3/SrTiO 3 8 influences the in-plane magnetisation, and on the other hand, how a magnetisation effects the orbital ordering. Assuming an XY ferromagnet in the ab sence of a coupling to nematic order, the Landau free energy becomes f(Mx,My,η) =f0+a(T) 2|/vectorM|2+b 4|/vectorM|4+λ(M2 x−M2 y)η+aη 2η2+···,(15) whereMx(My) is the magnetisation along the crystalline x(y) axis. Since we assume no independent instability towards an orbital-ordered nematic, aη>0‡. Integrating out ηby minimising the free energy with η=−λ aη(M2 x−M2 y), (16) the free energy for the magnetisation becomes f(Mx,My) =f0+a(T) 2|/vectorM|2+b 4|/vectorM|4−λ2 aη(M2 x−M2 y)2. (17) For finite λ, the coupling to orbital order therefore locks the magnetisation a long one of the crystal axes, leading to an additional in-plane anisotropy. For the orbital ordering, the magnetisation anisotropy acts as a d riving field. For aηvery small, i.e., the system close to an instability, we should include addit ional terms to the free energy for η, f(η;/vectorM) =aη 2η2+bη 4η4+cη 6η6+λ(M2 x−M2 y)η+···. (18) Forbη<0, the system undergoes either a metanematic crossover, or a fir st-order transition at a critical magnetisation. This is in analogy to metamagne tic transitions observed in systems close to ferromagnetism[19] and could here ind irectly be driven by an applied magnetic field. 4. Concluding Remarks We have shown that the combination of Rashba SOC and atomic SOC lea ds to an electron-density dependent magnetisation anisotropy in LAO/STO interfaces. While experiments sofarappeartolieinthein-plane-magnetisationregion , wepredict aswitch to out-of-plane magnetisation at sufficiently high gate voltages. We have identified a regime near the band edge with anisotropic susceptibility as a non-tr ivial effect of the Rashba SOC in a low carrier density system. The high density of state s near the band edge in principle also allows for a spontaneous orbital ordering a nd we predict in this regime an enhanced coupling between the magnetisation directio n and this kind of nematic order. This coupling locks the in-plane magnetisation directio n to be along one of the crystal axes and promotes Ising nematicity. Next, we comment on the issue of heterogeneity detected in refer ence [8]. The observed heterogeneity is likely driven by both extrinsic and intr insic effects. ‡Even if strong correlation drives orbital order there will be no qualit ative change in the effect of the coupling λin driving the in-plane magnetisation anisotropy.Spin-Orbit Coupling in LaAlO 3/SrTiO 3 9 For instance, a recent study showed that strong Rashba SOC can promote phase separation[20]. The proposed magnetisation-nematicity coupling ha s important consequences in both extrinsic and intrinsic fronts. On the one han d, oxygen vacancies and other spatial inhomogeneities act as a random field for the Ising nematic and in turn cause a distribution of moment directions. On the other hand, the reduction of the magnetic order parameter symmetry due to the coupling cha nges the type of magnetic textures and their energetics. Furthermore, we expec t the coupling to cause non-trivial in-plane anisotropy in the magneto transport §. Moreover, the sign change in the coupling λcould be observed through the rotation in the dominant direction up on gate voltage sweep. Finally, we note that the range in the density wit h magnetisation- nematicity near the band edge of the quasi-one-dimensional bands has also been shown to exhibit critical scaling in recent Hall measurement[13]. Acknowledgments We are grateful to J. Bert, H. Hwang, B. Kalisky, and K. Moler for u seful discussions. MHF and E-AK acknowledge support from NSF Grant DMR-0955822 a nd from NSF Grant DMR-1120296 to the Cornell Center for Materials Research . SR acknowledges support fromtheLDRDprogramatSLAC andtheAlfred. P. SloanRe search fellowship. References [1] IgorˇZuti´ c, JaroslavFabian, and S. Das Sarma. Spintronics: Fundame ntals and applications. Rev. Mod. Phys. , 76:323–410, Apr 2004. [2] R Winkler. Spin-orbit coupling effects in two-dimensional electron an d hole systems , volume 191. Springer-Verlag Berlin, 2003. [3] Erez Berg, Mark S. Rudner, and Steven A. Kivelson. Electronic liq uid crystalline phases in a spin-orbit coupled two-dimensional electron gas. Phys. Rev. B , 85:035116, Jan 2012. [4] H. Y. Hwang, Y. Iwasa, M. Kawasaki, B. Keimer, N. Nagaosa, and Y. Tokura. Emergent phenomena at oxide interfaces. Nat Mater , 11(2):103–113, 02 2012. [5] A. Brinkman, M. Huijben, M. van Zalk, J. Huijben, U. Zeitler, J. C. Maan, W. G. van der Wiel, G. Rijnders, D. H. A. Blank, and H. Hilgenkamp. Magnetic effects at t he interface between non-magnetic oxides. Nat Mater , 6(7):493–496, 07 2007. [6] A. D. Caviglia, S. Gariglio, N. Reyren, D. Jaccard, T. Schneider, M . Gabay, S. Thiel, G. Hammerl, J. Mannhart, and J. M. Triscone. Electric field control of the LaAlO 3/SrTiO 3interface ground state.Nature, 456(7222):624–627, 12 2008. [7] N. Reyren, S. Thiel, A. D. Caviglia, L. Fitting Kourkoutis, G. Hamme rl, C. Richter, C. W. Schneider, T. Kopp, A.-S. Retschi, D. Jaccard, M. Gabay, D. A. Mu ller, J.-M. Triscone, and J. Mannhart. Superconducting interfaces between insulating oxid es.Science, 317(5842):1196– 1199, 2007. [8] JulieA. Bert, BeenaKalisky, ChristopherBell, Minu Kim, YasuyukiH ikita, HaroldY. Hwang, and Kathryn A. Moler. Direct imaging of the coexistence of ferromagne tism and superconductivity at the LaAlO 3/SrTiO 3interface. Nat Phys , 7(10):767–771, 10 2011. [9] Lu Li, C. Richter, J. Mannhart, and R. C. Ashoori. Coexistence o f magnetic order and two- dimensional superconductivity at LaAlO 3/SrTiO 3interfaces. Nat Phys , 7(10):762–766,10 2011. §Fischer et al., in preparationSpin-Orbit Coupling in LaAlO 3/SrTiO 3 10 [10] J A C Bland and BretislavHeinrich. Ultrathin magnetic structure I . Springer Berlin / Heidelberg, 2005. [11] H. J. G. Draaisma and W. J. M. de Jonge. Surface and volume anis otropy from dipole-dipole interactionsin ultrathin ferromagneticfilms. Journal of Applied Physics , 64(7):3610–3613,1988. [12] Gerrit van der Laan. Microscopic origin of magnetocrystalline an isotropy in transition metal thin films.Journal of Physics: Condensed Matter , 10(14):3239, 1998. [13] A.Joshua, S.Pecker,J.Ruhman, E.Altman, andS. Ilani. AUniv ersalCriticalDensity Underlying the Physics of Electrons at the LaAlO 3/SrTiO 3Interface. Nature Commun. , 3:1129, 2012 [14] Zoran S. Popovi´ c, Sashi Satpathy, and Richard M. Martin. O rigin of the two-dimensional electron gas carrier density at the LaAlO 3on SrTiO 3interface. Phys. Rev. Lett. , 101(25):256801, Dec 2008. [15] A. F. Santander-Syro, O. Copie, T. Kondo, F. Fortuna, S. Pa ilhes, R. Weht, X. G. Qiu, F. Bertran, A. Nicolaou, A. Taleb-Ibrahimi, P. Le Fevre, G. Herranz, M. Bibes, N . Reyren, Y. Apertet, P. Lecoeur, A. Barthelemy, and M. J. Rozenberg. Two-dimensiona l electron gas with universal subbands at the surface of SrTiO 3.Nature, 469(7329):189–193, 01 2011. [16] A. D. Caviglia, M. Gabay, S. Gariglio, N. Reyren, C. Cancellieri, and J.-M. Triscone. Tunable Rashba spin-orbit interaction at oxide interfaces. Phys. Rev. Lett. , 104:126803, Mar 2010. [17] Beena Kalisky, Julie A. Bert, Brannon B. Klopfer, Christopher B ell, Hiroki K. Sato, Masayuki Hosoda, Yasuyuki Hikita, Harold Y. Hwang, and Kathryn A. Moler. C ritical thickness for ferromagnetism in LaAlO 3/SrTiO 3heterostructures. Nat Commun , 3:922, 06 2012. [18] S. Raghu, A. Paramekanti, E A. Kim, R. A. Borzi, S. A. Grigera, A . P. Mackenzie, and S. A. Kivelson. Microscopic theory of the nematic phase in Sr 3Ru2O7.Phys. Rev. B , 79(21):214402, 2009. [19] R Z Levitin and A S Markosyan. Itinerant metamagnetism. Soviet Physics Uspekhi , 31(8):730, 1988. [20] S. Caprara, F. Peronaci, and M. Grilli. Intrinsic instability of elect ronic interfaces with strong Rashba coupling. Phys. Rev. Lett. , 109:196401, Nov 2012.

1803.07091v3.Spin_Dependent_Conductance_in_a_Junction_with_Dresselhaus_Spin_Orbit_Coupling.pdf

arXiv:1803.07091v3 [cond-mat.mes-hall] 1 Apr 2018Spin-Dependent Conductance in a Junction with Dresselhaus Spin-Orbit Coupling Daisuke Oshima,1Katsuhisa Taguchi,2and Yukio Tanaka1 1Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan 2Yukawa Institute for Theoretical Physics, Kyoto Universit y, Kyoto, 606-8502, Japan We studied spin-dependent conductance in a normal metal (NM )/NM junction with Dresselhaus spin-orbit coupling (DSOC) and magnetization along the out -of-plane direction. As a reference, we also studied the spin-dependent conductance in such a junct ion with Rashba spin-orbit coupling (RSOC). Using a standard scattering method, we calculated t he gate-voltage dependence of the spin-dependent conductances in DSOC and RSOC. In addition, we calculated the gate-voltage de- pendence of the conductances in a ferromagnetic metal (FM)/ NM junction with spin-orbit coupling and magnetization, which we call ferromagnetic spin-orbit metal (FSOM). From these results, we discuss the relation between these conductance in the prese nce of DSOC and that in the presence of RSOC. We found that conductance in DSOC is the same as that i n RSOC for the NM/FSOM junction. In addition, we found that in the FM/FSOM junction , the conductance in DSOC is the same as that in RSOC. I. INTRODUCTION Spin-dependent transport is a key issue in the context of spintronics. Recently, various spin-dependent trans- ports in the presence of spin-orbit couplings (SOCs) have been discussed. Among the spin-dependent transports due to the SOCs, Rashba and Dresselhaus spin-orbit coupling (RSOC and DSOC, respectively) have been in- triguingly studied1–20. RSOC and DSOC are caused by the structural and bulk inversion symmetry breaking, respectively21–25. In the context ofchargetransportin adiffusive regime, the transport as well as electromagnetic effects (e.g., Edelsteineffect)inthe presenceofRSOChavebeenstud- ied so far26and it depends on the spin textures: The spin texture at the Fermi surface in the momentum space, as shown in Fig. 1, is shifted by the applied electric field; this shift is proportionalto the electric field and the transport relaxation time of the diffusive regime. Hence, the spin polarization is generated by the applied electric fieldanditspolarizationisperpendicular(parallel)tothe applied electric field in the presence of RSOC (DSOC). FIG. 1. (Color online) Illustration of the spin texture (red arrows) of the (a) Rashba and (b) linear Dresselhaus-type spin-orbit coupling. In the context of a ballistic regime, spin-dependent tunneling conductance has been studied in two- dimensional systems1–3,5,7,8,10–18,20,27–34. For exam- ple, the spin-dependent conductance (e.g., magnetoresis- tance) depends on the RSOC spin texture, which is dis-cussedinthesurfacestateofathree-dimensionaltopolog- ical insulator with magnetism. However, the charge con- ductivity is not largely influenced by the spin textures, for example, in a normal metal (NM)/NM junction with RSOC and magnetization20. In this system, although such an NM with RSOC and magnetism has three types of spin textures depending on the Fermi level, the charge conductance could be independent of the spin texture of the RSOC. It is a natural to ask whether the charge conductance is independent of the structure of the spin texture and howthe spin-dependentconductancedepends onthespin textureintheballisticregimeinthepresenceofDSOC.In this work, in order to address this question, we studied the relation between the conductance and spin texture in an NM/NM junction with DSOC and magnetism. In this paper, for simplicity, we call an NM with SOC and magnetismaferromagneticspin-orbitmetal(FSOM).We calculated the charge and spin-dependent conductance whenthespintexturesoftheFSOMdepend ontheFermi level, which is tuned by an applied gate voltage in the NM/FSOM junction, as shown in Fig. 2. As a result, we show the relation between these conductances in the FIG. 2. Illustration of a two-dimensional NM (FM)/FSOM junction, where the FSOM has RSOC or DSOC and magneti- zation. The magnetization is along the out-of-plane direct ion. The Fermi level as well as the spin texture of the FSOM is changed by an applied gate voltage Vg. A bias voltage ( V) is applied on the NM (FM).2 DSOC and those in the RSOC. Furthermore, the spin- dependent conductance is discussed in a ferromagnetic metal (FM)/FSOM junction. We found that the charge conductance in the DSOC is the same as that in the RSOC for the NM/FSOM junction. The spin-dependent conductance in the DSOC is different from that in the RSOC, but there are some relations between them. Theorganizationofthispaperisasfollows. InSect. II, we describe the model and our obtained results. In Sect. III, we discussed the results through the transformation. In Sect. IV, we summarize the obtained results. II. MODEL AND RESULTS A. Model We first consider a two-dimensional NM/FSOM junc- tion as illustrated in Fig. 2, where the RSOC or DSOC of the FSOM is induced by the structural or bulk in- version symmetry breaking. A bias voltage V(to drive charge along the xdirection) and a gate voltage Vg(to change the Fermi level in the FSOM35) are applied on the NM and FSOM, respectively. The tunneling barrier at the interface is assumed to be a delta function13,15,32 for conservation of the ycomponent of the momentum. We also assume the case where the width of the junction along the ydirection is sufficiently large. The model Hamiltonian of the NM/FSOM junction can be described as3,15,20,21,36 H=HLθ(−x)+Uδ(x)+HRθ(x), (1) HL=/planckover2pi12k2 2mL, HR=/planckover2pi12k2 2mR+HSOC−MRσz+eVg, where HL(HR) is the Hamiltonian in the NM (FSOM). Here,θandδare the Heaviside step function and delta function, respectively. The first term in Eq. (1) denotes the kinetic energy, k2=k2 x+k2 y, and mL(mR)is the effec- tive mass of the electron in the NM (FSOM). The second term in HRdenotes the SOCs23,25: HSOC=/braceleftBigg α(kxσy−kyσx) ( RSOC), β(kxσx−kyσy)+o(k3) (DSOC),(2) whereσ=(σx,σy,σz)are Pauli matrices in the spin space.αandβare a coupling constant of the RSOC and DSOC, respectively. Here, we consider the linear DSOC, neglecting a term proportional to k3. The third term in HR,MRσz, denotes the exchange coupling of the magnetization, which is along the out-of-plane direction. eVg(≥0)is a potential caused by the gate voltage. The third term in Eq. (1), Uδ(x), indicates the tunneling barrier, where Uis the strength of the tunneling barrier.B. Transmission probability in an NM/FSOM junction Using a standard method, we calculated the angle- resolved transmission probability Tγ,s A=R,D(φ), whereφis an injection angle from the NM into the FSOM; it is de- fined as an angle between the wavevectorof the injection wave and the x-axis. Here, the superscript s(=↑,↓) de- notes the up- and down-spin along the spin polarization γ(=x,y,z)in the NM. The subscript A(=R,D)indicates the SOC (RSOC or DSOC). The detail of the derivation ofTγ,s Ais shown in the Appendix A. The transmission probability Tγ,s=↑,↓ Ais given by Tγ,s=↑,↓ A=1 2π/uni222B.dspπ 2 −π 2dφkFcosφ·Tγ,s A(φ),(3) where kF=√2mEF//planckover2pi1is the Fermi momentum in the NM.Tγ,s AandTγ,↑ A+Tγ,↓ Aare proportional to the spin- dependent conductance and the charge conductance, re- spectively (see Appendix A). It is found that at α=β, the energy dispersion of the FSOM of the RSOC is the same as the energy dispersion of the DSOC, but the spin texture of the RSOC is different from that of the DSOC (see Fig. 1). In order to show the relation between the conductances (i.e. transmission probabilities) and these spin textures, we simply set α=β >0,MR≥0, and mL=mR≡m. Fig. 3 shows the Vgdependence of Tγ,s A=R,DforMR=0 with several spin polarizations of the NM γ=x,y,z. It is found that at γ=x, the Vgdependence of Tx,s Ris dis- tinct at the kink eVg/EF=1; for eVg/EF≥1, the inner and outer spin textures are pointing along the counter- clockwise direction in the momentum space, whose spin texture is caused by the RSOC and the two spin-split bands; for eVg/EF≤1, the inner and outer spin tex- tures are pointing oppositely. The Tx,s Ris independent of s(=↑,↓)of the FSOM. On the other hand, the Vgdepen- dence of Tx,s Ddepends on s: theTx,↑ Dclearly has the kink ateVg/EF=1, but the Tx,↓ Ddoes not. Figure 3(b) indi- cates the Vgdependence of the Ty,s A=R,Dforγ=y. Then, it is noted that the Ty,s Rdepends on s, butTy,s Dis indepen- dent of s. Forγ=z, bothTz,s RandTz,s Dare equal and independent of s. Moreover, it is found that for all γ, the total transition probability Tγ,↑ R+Tγ,↓ Rin the RSOC case, which is proportional to the charge conductance, is equal to that in the DSOC. Then, it is independent of γ. Fromthesenumericalresults,itisfoundthatthetrans- mission probability for any γsatisfies the following rela- tion: Tγ,↑ R+Tγ,↓ R=Tγ,↑ D+Tγ,↓ D. (4) Furthermore, we note that the followingrelation between3 FIG. 3. (Color online) Vgdependence of the transmission probability of the RSOC Tγ,s R(dashed lines) and that of the DSOCTγ,s D(bold lines) in the NM/FSOM junction with sev- eral spin polarizations of the NM: (a) γ=x, (b)γ=y, and (c)γ=zatMR/EF=0. Red and blue lines correspond to the up- and down-spin polarized electron injection case fro m the NM into the FSOM, respectively. Black lines show the total transmission probability. Here, we set Eα/EF=0.55, UkF/EF=1.0, andα=β, where Eα=mα2/(2/planckover2pi12)is the Rashba energy13,32,36–41. Tγ,s RandTγ,s Dis satisfied for any Vg: Tx,↑(↓) D=Ty,↑(↓) R, Tx,↑ R=Tx,↓ R=Ty,↑ D=Ty,↓ D, Tz,↑ R=Tz,↓ R=Tz,↑ D=Tz,↓ D.(5) Thus, there are some relations of the transmission prob- abilities without the magnetization case ( MR=0). Next, we consider Tγ,s Ain the presence of MR/nequal0. Figure 4 shows the Vgdependence of Tγ,s Afor nonzero MR. In particular, under the RSOC with nonzero MR, there are two spin-split bands and three remarkable elec- tric states, the so-called normal Rashba metal (NRM), anomalous Rashba metal (ARM)15, and Rashba ring metal (RRM)20. In the presence of DSOC with nonzero MR, we can find three similar states. As a result, the Vgdependence of the probability Tγ,s Ahas two kinks at eVg/EF=0.5 and 1.5, whose kinks indicate the boundary between the states. For γ=y, the probability Tγ,s Rfor MR/nequal0 depends on sas well as that for MR=0. The magnitude of the probability for up-spin Ty,↑ Ris smaller than that for the down-spin Ty,↓ Rfor any Vg. In other words, under nonzero MR, the difference between the probabilities Tγ,↑ R− Tγ,↓ Rtakes a negative value. On the other hand, for γ=x,z, the probability Tγ,s RforMR/nequal0 depends on sand complicatedly depends on Vg, unlike FIG. 4. (Color online) Vgdependence of Tγ,s A(=R,D)in the NM/FSOM junction for MR/EF=0.5. (a)γ=x, (b)γ=y, and (c)γ=zatEα/EF=0.55,UkF/EF=1.0, andα=β. that for MR=0. Inγ=x, the difference between the probabilities takes a positive value in the NRM and a negative value in the ARM and RRM. In γ=z, the dif- ference takes a positive value in the NRM and ARM but it becomes a negative value in the RRM. Note that the Vgdependence of the sign of the difference depends on the parameters, and it does not completely correspond to the three states. We also numerically calculated Tγ,s DinMR/nequal0, as shown in Fig. 4. As a result, we found the following relation between Tγ,s DandTγ,s Rfor any Vg: Tx,s D=Ty,s R,Tx,↑(↓) R=Ty,↓(↑) D,Tz,s R=Tz,s D.(6) Furthermore, the total transition probability is indepen- dent ofγ: Eq. (4) is satisfied even for nonzero MR. C. Angle-resolved transmission probability In order to show the SOC dependence of the probabil- ity in more detail, we numerically calculated the angle- resolved transmission probability Tγ,s A(=R,D)(φ)defined in Eq. (3). Figure 5 shows the φdependence of Tγ,s R(φ)for MR/EF=0 and eVg/EF=0, i.e., in the NRM. The angle- resolved probability Tγ,s R(φ)complicatedly depends on φ. However, Tγ,s R(φ)implies a symmetric to φand s: Tx,↑ R(φ)=Tx,↓ R(−φ), Ty,s R(φ)=Ty,s R(−φ), Tz,↑ R(φ)=Tz,↓ R(−φ).(7) Thus, these relations are categorized by γ=x,y,z. We also found the relation between Tγ,s R(φ)and Tγ,s D(φ). The4 FIG. 5. (Color online) φdependence of Tγ,s A(=R,D)in the NM/FSOM junction for MR/EF=0: (a)γ=x, (b)γ=y, and (c)γ=zatEα/EF=0.55,UkF/EF=1.0,eVg/EF=0, andα=β. relations are given by Tx(y),s R(φ)=Ty(x),s D(φ),Tz,↑(↓) R(φ)=Tz,↓(↑) D(φ).(8) Furthermore, it is found that in the NRM, the totalprob- ability for each angle is given by Tγ,↑ R(φ)+Tγ,↓ R(φ)=Tγ,↑ D(φ)+Tγ,↓ D(φ).(9) We also confirmed these relations even in the RRM. The angle-resolved probability under nonzero MRis shown in Fig. 6. The φdependence of the angle-resolved transmission probability is highly complicated. However, we found that only in γ=y, the probability Ty,s R(φ)is symmetric for φ→ −φ: Ty,s R(φ)=Ty,s R(−φ), (10) The probability of the RSOC and that of the DSOC un- derMR/nequal0 satisfies the following relations: Tx,s D(φ)=Ty,s R(φ), Ty,↓(↑) D(φ)=Tx,↑(↓) R(−φ), Tz,s D(φ)=Tz,s R(−φ).(11) Thus, for nonzero MR,Tx,s R(φ)is equal to Ty,s D(φ)byφ→ −φand s=[↑ (↓)] → [↓ (↑)] , and Tz,s R(φ)is equal to Tz,s D(φ)by onlyφ→ −φ. It is noted that these relations are also numerically confirmed even in the case of ARM and RRM. FIG. 6. (Color online) φdependence of Tγ,s A(=R,D)in the NM/FSOM junction for MR/EF=0.5: (a)γ=x, (b)γ=y, and (c)γ=z. Here, we set Eα/EF=0.55,UkF/EF=1.0, eVg/EF=0, andα=β. D. Transmission probability in an FM/FSOM junction In the previous subsection IIB and IIC, we show the relation between Tγ,s RandTγ,s Din the NM/FSOM junc- tion. For MR/nequal0, the FSOM of the NM/FSOM junction breaks a time-reversal symmetry. Next, we consider an FM/FSOM junction without time-reversal symmetry in the whole of the junction. We studied the probability in the junction with the lower symmetry. The model of the FM/FSOM junction can be described as H=HLθ(−x)+Uδ(x)+HRθ(x), (12) HL=/planckover2pi12k2 2mL−ML·σ, HR=/planckover2pi12k2 2mR+HSOC−MRσz+eVg, whereMLdenotes the magnetization of the FM and ML/MLcorresponds to γas defined in the previous sec- tion. Then, the transition probability Tγ,s Ais given by Tγ,s A=1 2π/uni222B.dspπ 2 −π 2dφks Fcosφ·Tγ,s A(φ),(13) where ks=↑(↓) F=/radicalbig 2m[EF+(−)ML]//planckover2pi1is the momentum of up- (down-) spin electron in the FM at the Fermi level. Herein, we simply set mL=mR≡m. Figure 7 shows the Vgdependence of Tγ,s AinML/nequal0 and MR=0. The Vgdependence of Tγ,s Ras well as Tγ,s D is similar to that in Fig. 3. However, in contrast to Fig. 3,Tγ,s RandTγ,s Dalso depends on sin allγ. Furthermore,5 FIG. 7. (Color online) Vgdependence of Tγ,s A(=R,D)in the FM/FSOM junction for MR/EF=0: (a)ML/EF=(0.5,0,0), (b)ML/EF=(0,0.5,0), and (c) ML/EF=(0,0,0.5)corre- spond toγ=x,γ=y, andγ=z, respectively. Here, we set Eα/EF=0.55,UkF/EF=1.0, andα=βwith kF=√2mEF//planckover2pi1. forγ=z, the relation Tz,s R=Tz,s D, which is satisfied for ML=0, remains even in ML/nequal0. From these numerical results, the summary of the relation among Tγ,s R(Tγ,s D), s, andγfor any Vgis given by Tz,↑ R=Tz,↓ R,Tz,↑ D=Tz,↓ D. (14) In the FM/FSOM junction with MR=0, we find Tx,s D=Ty,s R,Tz,s D=Tz,s R. (15) The total probability has a relation only for γ=z: Tz,↑ R+Tz,↓ R=Tz,↑ D+Tz,↓ D. (16) We also numerically calculated the probability for ML/nequal0 and MR/nequal0, as shown in Fig. 8 (cf. Fig. 4). As a result, we found that the transmission probability has a unique Vgdependence: In γ=x, the Vgdependence ofTγ,s Ahas a kink at eVg/EF=0.5 and 1.5, whose prop- erties are similar to those in Fig. 4. However, the Vg dependence of the probability around the RRM in Fig. 8 monotonically decrease with increasing eVg/EF, unlike that in Fig. 4. In particular, in the RSOC, the kink be- tween NRM and ARM is clear but that between ARM and RRM disappears near eVg/EF=1.5. Forγ=y, the Vgdependence of Tγ,s Ais similar to that in Fig. 4. For γ=z, we found that Tγ,s RandTγ,s Dare equal. Then, Eq. (16) is satisfied even for nonzero MLand MRonly when γ=z. However, in the FM/FSOM junction with MR/nequal0, we cannot find the relation between Tγ,s RandTγ,s D, unlike in the case of an FM/FSOM junction with MR=0. FIG. 8. (Color online) Vgdependence of Tγ,s A(=R,D)in the FM/FSOM junction for MR/EF=0.5: (a) ML/EF= (0.5,0,0), (b)ML/EF=(0,0.5,0), and (c) ML/EF=(0,0,0.5). Here, we set Eα/EF=0.55,UkF/EF=1.0, andα=β. In Figs. 7 and 8, we can find that the sign of the differ- enceTx(y),↑ A−Tx(y),↓ Ais oppositeto that in the NM/FSOM junction (see Figs. 3 and 4). We can numerically confirm that this is caused by astrong ML>0. For small ML, the sign of the difference is the same as that in Figs. 3 and 4. For sufficiently large ML>0, the sign switches oppo- sitely. However, the Vgdependence of each Tγ,s Ain the FM/FSOM junction is similar to that in the NM/FSOM junction almost independently of the value of ML. Thus, we numerically indicated that the charge con- ductivity in the DSOC is the same as that in the RSOC intheNM/FSOMjunctionandFM/FSOMjunctionwith γ=z. Besides, at α=β, the energy dispersion of the DSOC is also the same as that of the RSOC, but the spin texture of the DOSC is different from that of the RSOC. These results implies that the charge conductiv- ity of the FSOM junctions are independent of the spin texture. This is main message in this paper, and it will be also discussed by using the transformation in the next section. III. DISCUSSION From the numerical calculations in the previous sec- tion, we will discuss the charge and spin-dependent con- ductance. In the previous sections, in order to clarify whether the charge and spin conductance depends on the spin texture of the SOCs, we calculated the transmission probabilities at α=β, since the energy dispersion of the FSOM is independent of the SOC type but the spin tex- ture depends on the type, RSOC or DSOC (see Fig. 1),6 atα=β. We found that the charge conductance in the RSOC is equal to that in the DSOC [see Eq. (4)], and the charge conductance is independent of the spin tex- tures as well as the magnetization MRin the NM/FSOM junction. On the other hand, the spin-dependent con- ductance Tγ,s Adepends on the spin textures as well as MR; however there are some relations between Tγ,s Rand Tγ,s D. The relation between Tγ,s RandTγ,s Dcan be understood by the following argument of the transformation. At α=β, the spin texture of the RSOC is changed by that of the DSOC via the spin transformation, σx→σy, σy→σx, andσz→ −σz8,24, which can be described as R=i/√ 2(σx+σy). This spin transformation can corre- spond to the relations between the spin-dependent con- ductance. In MR=0 in the NM/FSOM junction, Eq. (8) implied Ty,s R→Tx,s DandTz,↑(↓) R→Tz,↓(↑) Dasthe spintrans- formation Rviaσx→σy,σy→σx, andσz→ −σz. In MR/nequal0, TheTγ,s RandTγ,s Dhave the relation as Eq. (11), which indicates the transformation φ→ −φ,σx→ −σy, σy→σx, andσz→σz. The transformation can be described by Rand P.Pis the spin and momentum transformation as ky→ − ky,σy→ −σy, andσz→ −σz. Note that DSOC is even for this transformation P. Ap- plying Pafter R, in MR/nequal0 case, the junction in the RSOC corresponds to that in the DSOC with φ→ −φ, σx→ −σy,σy→σx, andσz→σz; it was shown in Eq. (11). We show that the conductance is invariant under the transformation Rand P(see Appendix B). In the FM/FSOM junction with MR=0, the junc- tion in the RSOC for γ=xandyare corresponded to that in the DSOC for γ=yand xrespectively by the transformation R, as shown in Tx,s D=Ty,s Rin Eq. (15). However, for γ=xory, since the FM/FSOM junction in the RSOC does not equal that in the DSOC for same γ, the charge conductance depended on SOC type and γ. Forγ=z, the junction in the RSOC corresponds to that in the DSOC for γ=zby the transformations Rand P regardless of MR, asTz,s D=Tz,s Rin Eq. (15). Then, the charge conductance is independent of the type of SOC only forγ=zin the FM/FSOM junction. Thus, we gain intuitively understand why the charge conductance is independent of the type of the RSOC and DSOC (or the spin texture of the SOCs) by using spin rotation symmetry. These results could imply that the charge conductance is independent of the spin texture rather than depends on the magnitude of the SOC ( α andβ). IV. CONCLUSION We have theoretically studied spin-dependent trans- port in NM/FSOM and FM/FSOM junctions, where the FSOM was applied by an electrical gate tuning the Fermi level, and the FSOM had also RSOC or DSOC with out- of-plane magnetization. We have shown the gate voltagedependence of the transmission probability in the RSOC and DSOC under the time-reversal symmetry or time- reversal symmetry breaking system. As a result, in the NM/FSOM junction, regardless of the value of MR, we have found the relations between the transmission prob- abilities in the RSOC and that in the DSOC, and the charge conductance is independent of the SOC type. In the FM/FSOM junction, the gate voltage dependence of the probabilities is similar to that in the NM/FSOM junction, as is the relation between the probabilities in the RSOC and that in the DSOC only when MR=0 or the magnetization in the FM is along the out-of-plane direction. However, the charge conductance depends on the SOCs and the direction of the magnetization in the FM, unlike that in the NM/FSOM junction. In this paper, we compared two systems with the same band structures and different spin textures; i.e., the sys- tem in the RSOC and that in the DSOC. Then, several relations between the transmission probabilities in the RSOC and the DSOC were derived. In particular, it was found that the charge conductance is independent of the type of SOC in some cases. This fact shows that in some cases, we can understand the character of the conduc- tance without the SOC details, such as the spin texture. We expect that this observation will be useful in future studies of spintronics and those involving properties like conductance as explored in our study. ACKNOWLEDGMENT This work was supported by a Grant-in-Aid for Sci- entific Research on Innovative Areas, Topological Mate- rial Science (Grants No. JP15H05851, No. JP15H05853 No. JP15K21717), a Grant-in-Aid for Challenging Ex- ploratory Research (Grant No. JP15K13498) from the Ministry of Education, Culture, Sports, Science, and Technology, Japan (MEXT), the Core Research for Evo- lutional Science and Technology (CREST) of the Japan Science and Technology Corporation (JST) (Grant No. JPMJCR14F1). Appendix A: Derivation of the Transmission Probability In order to obtain the transmission probability of the NM/FSOMjunctionunderthelow-temperaturelimit, we consider the wave function at the Fermi level. The wave function in NM, ψz,↑(↓)(x<0,y), is decomposed into the injected wave function ψz,↑(↓) inand the reflected one ψz,↑(↓) refas ψz,↑(↓)(x<0,y)=ψz,↑(↓) in+ψz,↑(↓) ref, (A1) ψz,↑(↓) in=χ↑(↓)ei(kxx+kyy), (A2) ψz,↑(↓) ref=/bracketleftBig rz,↑(↓) ↑χ↑+rz,↑(↓) ↓χ↓/bracketrightBig ei(−kxx+kyy),(A3)7 withχ↑=/parenleftbigg 1 0/parenrightbigg ,χ↓=/parenleftbigg 0 1/parenrightbigg ,kx=kFcosφand ky=kFsinφ. χ↑(↓),rz,↑(↓) ↑[rz,↑(↓) ↓] is the eigenfunction in NM, the re- flection coefficient of up [down] spin electrons with up (down) spin injection. Here, we assume that injected electrons are polarized along the zdirection. The trans- mitted wave function ψz,↑(↓)(x>0,y) ≡ψz,↑(↓) trais shown as20 ψz,↑(↓) tra=tz,↑(↓) 1χ1(k1)eik1·r+tz,↑(↓) 2χ2(k2)eik2·r,(A4) with χ1(k)=θ(∆)χ+(k)+θ(−∆)χ−(k),∆≡E−eVg+Ec, (A5) χ2(k)=χ−(k), (A6) χ+(k)=/parenleftbigg g+(k) 1/parenrightbigg , χ−(k)=/parenleftbigg 1 g−(k)/parenrightbigg . (A7) Here, tz,↑(↓) 1[tz,↑(↓) 2]denotes the transmission coefficient with up (down) spin injection. χ±is the eigenfunction for the eigenvalue in FSOM. k1=(k1,x,ky)andk2= (k2,x,ky)are the momentum in FSOM, which are defined fork2 1≤k2 2with k2 1(2)=k2 1(2),x+k2 y. We set g±as follows: g±(k)= −αi/parenleftbigkx∓iky/parenrightbig MR+/radicalBig α2k2+M2 R(RSOC), β/parenleftbig±kx+iky/parenrightbig MR+/radicalBig β2k2+M2 R(DSOC).(A8) From the energy dispersion of FSOM, k2 1(2)is given by k2 1(2)=2m /planckover2pi12/bracketleftbig (EF−eVg)+2ESOC −(+)/radicalBig 4ESOC(EF−eVg)+4E2 SOC+M2 R/bracketrightbigg . (A9) We define ESOC=mα2/(2/planckover2pi12)ormβ2/(2/planckover2pi12)corresponding to the SOCs. The signs of k1,xand k2,xare determined so that the velocity vxtakes a positive value, because the electron is injected along the xdirection. The velocity operator vx=∂H/(/planckover2pi1∂kx)is given by Eq. (1) as3,24,32 vx= /planckover2pi1kx m+α /planckover2pi1θ(x)σy(RSOC), /planckover2pi1kx m+β /planckover2pi1θ(x)σx(DSOC).(A10) When k1,x(k2,x) becomes an imaginary number; its sign is determined so that χ±→0 in the limit of x→ ∞. The boundary condition at the interface13,15,27,32,42,43 is given as follows: ψz,↑(↓)(+0,y)−ψz,↑(↓)(−0,y)=0, vx[ψz,↑(↓)(+0,y)−ψz,↑(↓)(−0,y)]=2U i/planckover2pi1ψz,↑(↓)(0,y).(A11)From this condition, we have a equation about the coef- ficients as follow: /parenleftbiggχ1 χ2−χ↑−χ↓ /parenleftbigvx−2U i/planckover2pi1/parenrightbigχ1/parenleftbigvx−2U i/planckover2pi1/parenrightbigχ2−vxχ↑−vxχ↓/parenrightbigg/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtAtz,↑(↓) 1 tz,↑(↓) 2 rz,↑(↓) ↑ rz,↑(↓) ↓/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA =/parenleftbigg χ↑(↓) vxχ↑(↓)/parenrightbigg . (A12) Solving the coefficient from this equation, we obtain the transmission probability at each angle Tz,↑(↓)(φ) as13,20,32,42 Tz,↑(↓)(φ)=Re/barex/barex/barex/barex/barexψz,↑(↓)† travxψz,↑(↓) tra ψz,↑(↓)† invxψz,↑(↓) in/barex/barex/barex/barex/barex=1−Re/barex/barex/barex/barex/barexψz,↑(↓)† refvxψz,↑(↓) ref ψz,↑(↓)† invxψz,↑(↓) in/barex/barex/barex/barex/barex =1−/parenleftBig |rz,↑(↓) ↑|2+|rz,↑(↓) ↓|2/parenrightBig . (A13) Using the Landauer formula, we describe an electric cur- rent between two leads, I, as follows13,20,32,38: I=e2V L 4π2/planckover2pi1/uni222B.dspπ/2 −π/2dφcosφ·kF[Tz,↑(φ)+Tz,↓(φ)].(A14) The first term in Eq. (A14) is proportional to Tz,↑, and the second term is Tz,↓. Thus,Tγ,↑+Tγ,↓is proportional to the conductance, because the conductance is propor- tional to the current at the low-temperature limit. Appendix B: Spin-dependent Conductance under the Unitary Transformation In this appendix, we will show that the conductance are invariant under an unitary transformation U, which is assumed kxU→kxand∂U/∂kx=0. Applying U, the Hamiltonian of the junction [in Eqs. (1) and (12)] and the eigenfunctions in the each side are changed as H→ ´H=UHU†andχ↑(↓,1,2)→´χ↑(↓,1,2)=Uχ↑(↓,1,2). Then, the velocity operator in ´Hbecomes ´vx=UvxU†. (B1) From the boundary condition in Eq. (A11) under U, the transmission and reflection coefficients, ´tz,↑(↓) 1[2]and´rz,↑(↓) ↑[↓] are given by: /parenleftbigg U0 0U/parenrightbigg /parenleftbiggχ1 χ2−χ↑−χ↓ /parenleftbigvx−2U i/planckover2pi1/parenrightbigχ1/parenleftbigvx−2U i/planckover2pi1/parenrightbigχ2−vxχ↑−vxχ↓/parenrightbigg ·/parenlefttpA/parenleftexA/parenleftexA/parenleftexA/parenleftexA/parenleftexA /parenleftbtA´tz,↑(↓) 1 ´tz,↑(↓) 2 ´rz,↑(↓) ↑ ´rz,↑(↓) ↓/parenrighttpA/parenrightexA/parenrightexA/parenrightexA/parenrightexA/parenrightexA /parenrightbtA=/parenleftbigg U0 0U/parenrightbigg /parenleftbigg χ↑(↓) vxχ↑(↓)/parenrightbigg . (B2) Note that Uis a 2×2 matrix and χ↑(↓,1,2)are two com- ponent vectors. Here, we notice that Eq. (B2) is the8 same as Eq. (A12), i.e., ´tz,↑(↓) 1[2]=tz,↑(↓) 1[2]and´rz,↑(↓) ↑[↓]=rz,↑(↓) ↑[↓]. Then, we obtain ´ψz,↑(↓) in[ref,tra]=Uψz,↑(↓) in[ref,tra], where´ψz,↑(↓) in[ref,tra] isthewavefunctionafterthetransformation. Asaresult, we find that the transmission probabilities are invariantabout the transformation U: Re/barex/barex/barex/barex/barexψz,↑(↓)† travxψz,↑(↓) tra ψz,↑(↓)† invxψz,↑(↓) in/barex/barex/barex/barex/barex=Re/barex/barex/barex/barex/barex´ψz,↑(↓)† traUvxU†´ψz,↑(↓) tra ´ψz,↑(↓)† inUvxU†´ψz,↑(↓) in/barex/barex/barex/barex/barex =Re/barex/barex/barex/barex/barex´ψz,↑(↓)† tra´vx´ψz,↑(↓) tra ´ψz,↑(↓)† in´vx´ψz,↑(↓) in/barex/barex/barex/barex/barex.(B3) Note that transformations Rand Pare satisfied the as- sumption of U. 1D. Grundler: Phys. Rev. B 63(2001) 161307. 2M. H. Larsen, A. M. Lunde, and K. Flensberg: Phys. Rev. B66(2002) 033304. 3P. Stˇ reda and P. ˇSeba: Phys. Rev. Lett. 90(2003) 256601. 4V. I. Perel’, S. A. Tarasenko, I. N. Yassievich, S. D. Ganichev, V. V. Bel’kov, and W. Prettl: Phys. Rev. B 67(2003) 201304. 5O. Krupin, G. Bihlmayer, K. Starke, S. Gorovikov, J. Pri- eto, K. D¨ obrich, S. Bl¨ ugel, and G. Kaindl: Phys. Rev. B 71(2005) 201403. 6L. G. Wang, W. Yang, K. 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1511.08868v1.Spin_orbit_torque_engineering_via_oxygen_manipulation.pdf

1 Spin-orbit torque engineering via oxygen manipulation Xuepeng Qiu1, Kulothungasagaran Narayanapillai1, Yang Wu1, Praveen Deorani1, Dong-Hyuk Yang2, Woo-Suk Noh2, Jae-Hoon Park2,3, Kyung-Jin Lee4,5, Hyun-Woo Lee6, and Hyunsoo Yang1,* 1Department of Electrical and Computer E ngineering, and NUSNNI-Nanocore, National University of Singapore, 117576, Singapore 2c_CCMR and Department of Physics, Pohang Uni versity of Science and Technology, Pohang 790-784, Korea 3Division of Advanced Materials Science, P ohang University of Science and Technology, Pohang 790-784, Korea 4Department of Materials Science and Engin eering, Korea University, Seoul 136-701, Korea 5KU-KIST Graduate School of Converging Scie nce and Technology, Korea University, Seoul 136-713, Korea 6PCTP and Department of Physics, Pohang Uni versity of Science and Technology, Pohang 790- 784, Korea *e-mail: eleyang@nus.edu.sg Spin transfer torques allow the electrical ma nipulation of the magnetization at room temperature, which is desirable in spintronic de vices such as spin transfer torque memories. 2 When combined with spin-orbit coupling, they give rise to spin-orbit torques which are a more powerful tool for magnetization control and can enrich device functionalities. The engineering of spin-orbit torques, based mostly on the spin Hall effect, is being intensely pursued. Here we report that the oxidation of spin-orbit torque devices triggers a new mechanism of spin-orbit torque, which is about two times stronger than that based on the spin Hall effect. We thus introduce a way to engineer spin-orbit torques via oxygen manipulation. Combined with electrical gatin g of the oxygen level, our findings may also pave the way towards reconfigurable logic devices. Controlling the magnetization direction via interaction between spins and charges is crucial for spintronic memory and logic devices 1-4. Magnetization switching using conventional current-induced spin transfer torque (STT) requir es a spin polarizer in a spin valve structure5,6. Recently, the combination of STT with spin-orb it coupling has led to a new type of torque, namely spin orbit torque (SOT). In magnetic bilayers, where an ultr athin ferromagnetic (FM) layer is in contact with a heavy metal (HM) la yer, SOT arises from an in-plane current and enables efficient manipulation of the magnetization7-14, in particular, low power magnetization switching7,8, fast domain wall motion9,15,16, and tunable nano-oscillators11,12.The high stability, simplicity, and scalability of SO T make it attractive for next-g eneration spintronic devices8,17. To fully realize the envisioned merits, enhanced control and design of SOT are desired. So far, most experimental studies8,16,18-20 have attributed SOT to the bulk spin Hall effect in the HM. According to this interpretation, the sign and magnitude of SOT are determined essentially by the spin Hall angle in the HM, while the characteristics of FM18,21,22 may only slightly affect the magnitude of SOT. Here we report experimental results whose interpretation requires additional elements beyond the bulk spin Hall effect. We examine the oxidation effect 3 on SOT in Pt/CoFeB/MgO/SiO 2. The Pt layer, which is the source of the bulk spin Hall effect, is hardly affected by oxygen in our experiment, and the oxygen effect is limite d to the CoFeB layer. We find that as the oxygen leve l in the CoFeB layer goes above a threshold level, SOT suddenly reverses its direction while its magnitude re mains roughly invariant. The bulk spin Hall interpretation of SOT may explain the magnitude change but is unable to explain the sign reversal. This result t hus implies that a new SOT mechanism is introduced by the oxidation. We estimate that the new mechanism in our sample can be two times stronger than the bulk spin Hall mechanism, resulting in greater tunability of SOT in addition to the bulk spin Hall effect. Capping layer thickness dependence of SOT The oxygen level can be controlled by the thickness of the SiO 2 capping layer in our layer structure. The sputter-deposited film structure of Pt/CoFeB/MgO/SiO 2 is shown in Fig. 1a, in which the thickness t of the SiO 2 layer is varied from 0 to 4 nm. For small t, oxygen can easily diffuse through both the SiO 2 and MgO layers, and reach the CoFeB layer. A scanning electron microscope (SEM) image of the patterned Ha ll bar is shown in Fi g. 1b. We find that t variation alters the SOT considerably. Figure 1c shows the anomalous Hall resistance ( RH) of the device as a function of the in-plane current ( I) applied to the device. In a ddition to the current, a small constant magnetic field of 40 mT is applied along the positive current direction to break the symmetry7,8 of the device and allow fo r selective magnetization switching by the in-plane current. Since RH probes the average z-component of the CoFeB magnetiz ation, the hystere tic switching of RH confirms that the current-induced SOT i ndeed switches the magnetization. The arrows represent the current sweep direction. Intere stingly, the resulting switching sequence is clockwise for t > 1.5 nm, but counterclockwise for t 1.5 nm. Only the sw itching sequence for 4 large t (i.e., low oxygen level) is cons istent with the pr eviously reported switching sequence for Pt/Co/AlOx7,23 and Pt/CoFe/MgO16. Thus, the switching sequence for small t is abnormal. The current-induced Oersted field cannot e xplain the sequence reversal. The switching sequence reversal is not due to the sign reversal of the relation between RH and the z-component of the magnetization either, since the purely ma gnetic-field driven magnetization switching (see Supplementary Figs. 2 and 6) does not exhi bit the switching seque nce reversal with t. The inset of Fig. 1e shows IS versus t, in which IS is defined as the current at which RH changes from a positive to a negative value (note the sign change of IS around t = 1.5 nm). Interestingly, this threshold thickness of 1.5 nm matc hes the native oxide thickness of Si for passivation. The main panel in Fig. 1e shows the ra tio between the anisotropy field Han and IS as a function of t, which provides a rough magnitude estimation of SOT17,23. Note the abrupt sign reversal of the ratio while its magnitude remains roughly the same befo re and after the sign reversal. This implies that a new mechanism of SOT is suddenly introdu ced by the oxidation, generating SOT that is two times stronger and of opposite sign. The abrupt and full sign reversal differs qualitatively from continuous and marginal sign reversal in a previous study21. For independent confirmation of the SOT sign reversal, we perform lock-in measurements of SOT13,21,24-26. We apply a small amplitude sinusoidal ac current with a frequency of 13.7 Hz to exert periodic SOT on the magnetization, so that the induced magnetiza tion oscillation around the equilibrium direction gene rates the second harmonic signal V 2ω. Depending on the measurement geometry, V2ω measures21,25,26 the damping-like or fiel d-like component of SOT, which are two mutually orthogonal vector components of SOT. As current induced magnetization switching is driven mainly by the damping-like SOT7,8,23, we present here the results for the damping-like SOT only, which can be probed by applying an external dc magnetic 5 field H along the current direction (tilted 4 degree away from the film plane) to tilt the equilibrium magnetization direction accordingly. Th e results for the field-like SOT are presented in Supplementary Fig. 4. The ma gnetization switching characterist ics have been measured from the first harmonic signals, and asymmetric 2V loops have been observed as shown in Fig. 1d. For t = 0 and 1.2 nm, there is a dip in 2V at a positive field and a peak at a negative field, while the opposite behaviour is observed for t = 1.8 and 3 nm. Opposite polarities in 2V prove that the damping-like SOT is pointing in opposite directions for small and large t, confirming the conclusion drawn from the switching sequence in Fig. 1c. For t = 1.5 nm, the 2Vsignal for positive field contains both a peak and a dip, which may indicate the coexistence of regions with opposite damping-like SOT directi ons. The extracted strengths of the damping-like effective field ( HL) are summarized in Fig. 1f for various t, indicating a sudden sign reversal of the damping-like SOT in agreement with Fig. 1e. Characterisation of oxidation In order to verify the oxidation for small t, we have carried out various measurements. Figure 2a shows the oxygen depth profiles obtai ned by secondary ion mass spectroscopy (SIMS) for devices with t = 0 and 2 nm. The depth profile for t = 0 nm shows a si gnificantly enhanced oxygen level in the CoFeB laye r compared to that of t = 2 nm, confirming the oxidation for small t. On the other hand, the two depth profiles are almo st identical in the Pt layer, indicating no oxidation of the Pt layer even for small t. This is natural since Pt has excellent resistance to oxidation, which is supported also by th e essentially indistinguishable Pt 4 f x-ray photoelectron spectroscopy (XPS) spectra in Fig. 2b for the t = 0 and 2 nm samples. We also use the x-ray absorption spectroscopy (XAS) to probe the electron ic structures of Fe and Co. The Fe and Co L2,3edge (2 p 3d) XAS spectra27 in Fig. 2c and in Suppleme ntary Fig. 10a exhibit spectral 6 features28 quite similar to those of -Fe 2O3 and CoO,29,30 indicating (Fe,Co) oxide formation with Fe3+ and Co2+, respectively. The XAS data show that the fraction of the oxidized atoms increases as the thickness t decreases (Figs. 2f and Supplementary Fig. 11). Since Fe3+ and Co2+ ions do not have net magnetic moments, the satu ration magnetization is expected to decrease with decreasing t, which is indeed the case as conf irmed (Fig. 2e) by vibrating sample magnetometry (VSM). The magnetic properties of Fe and Co can also be probed by the x-ray magnetic circular dichroism (X MCD). Both the Fe (Fig. 2d) and Co (Supplementary Fig. 10b) L2,3edge XMCD signals become weaker with decreasing t, which is consistent with the VSM measurements, since the XMCD signals arise from the ferromagnetic atoms that remain un- ionized. The orbital magnetic moments of the fe rromagnetic atoms can be evaluated from the XMCD sum rule.27 For the ferromagnetic Fe atoms, the ratio between the orbital to the spin magnetic moments is about 0.06 (Fig. 2f), which is about 40% larger than the bulk value 0.043 and comparable to the value for epitaxial Fe film at the two-dimensional percolation threshold27,31. Interestingly, as t decreases, the ratio increases ev en further to 0.065, indicating an enhancement of the orbital moment of th e FM at the interface with the oxide32. Considering that orbital moment enhancement typically occurs wh en ferromagnetic atoms are in an environment with broken symmetry33, this suggests that the ferromagnetic atoms are subject to more strongly broken symmetry as t decreases. Sign reversal of SOT by in-situ oxidation All these measurements support the oxi dation of the CoFeB layer for small t. However, we cannot yet rule out the possibil ity that the oxidation is merely correlated with instead of the cause of the SOT sign reversal. In order to ve rify that the oxidation of the FM is the key parameter for the sign reversal, we examine a sa mple with an oxidized CoFeB layer but with 7 large t (3 nm). To prepare such a sample, we intentionally oxidize the CoFeB layer with O 2 gas during its deposition before depositing the capping layers including 3 nm of SiO 2 as shown in Fig. 3a. In this case, even for t = 3 nm, we observe an abnormal anticlockwise switching loop as shown in Fig. 3b. Hence what is important for th e SOT sign reversal is the FM layer oxidation itself rather than the value of t. Small t is merely a method to induc e the oxidation. This result rules out other t-dependent changes such as strain from being key parameters of the SOT sign reversal. SOT beyond the spin Hall effect We now discuss the connection between the FM layer oxidation and the SOT sign reversal. The bulk spin Hall effect8 in the HM layer is an important mechanism of SOT. According to the spin Hall interpretation of SOT16,23, the sign of the damping-like SOT is determined by the bulk spin Hall angle of the HM layer. The FM affects the damping-like SOT through the real part of the spin mixing conductance34, which is always positive and cannot change its sign since its being negativ e implies a negative charge conductance.35 The spin Hall interpretation is thus inadequa te to explain the oxygen-induced sign reversal of the damping-like SOT, since the HM layer is not affected by oxidation (Fig. 2a and b). Furthermore, we have verified that the SOT change caused by the oxid ation is essentially independent of the Pt thickness (Supplementary Fig. 5). Hence our data necessitate a new source of SOT, other than the bulk spin Hall effect of the HM layer. One ca n think of two possibilities; one is the oxidized FM layer itself being a SOT source, and the othe r is the top or bottom in terfaces of the oxidized FM layer being a SOT source. To examine the first possibili ty, we change the thickness d CFB of the CoFeB layer for fixed t = 0 nm. Up to dCFB = 2 nm, the perpendicular magnetic anisotropy is well maintained, and 8 the saturation magnetization ( MS) values (Fig. 4a) are almost c onstant and do not increase with dCFB, implying an almost dCFB-independent oxidation level. The change of dCFB has a negligible effect on the current density, since the resistivity of CoFeB is much greater than that of Pt22. While the abnormal anti-clockwise switching seque nce is maintained (Fig. 4b) with changing dCFB, Han/IS and HL change almost linearly with 1/ dCFB (Fig. 4c). That is, HL dCFB, which is proportional to the total torque acting on the CoFeB layer, does not increase with dCFB. This implies that the bulk part of th e oxidized FM layer is not an SO T source. Hence, one can exclude the first possibility. To examine the second possibility, we elim inate the MgO layer from the device stack structure. The switching sequence reversal from the normal clockwise to abnormal anti- clockwise direction is still observed, when t changes from 4 to 1.2 nm as shown in Fig. 5a. This shows that the interface between the oxidized FM layer and the MgO layer is not the new SOT source. Next we eliminate the Pt layer instead. The current-induced switching itself (Fig. 5b) is not observed nor is the 2nd harmonic signal (Supplementary Fig. 13). This leads us to conclude that the interface between the oxidized CoFeB laye r and the Pt layer is the new SOT source. We further extend our experiments to devices in whic h the FM material CoFeB is replaced with Co. As shown in Fig. 5c, the curre nt-induced switching sequence s hows normal clockwise behaviour for t = 3 nm, but reverses to abnormal anti-clockwise behaviour for t = 0 nm, which is similar to the results from devices with CoFeB as the FM layer. Hence the observed sign reversal phenomenon is not restricted to a specifi c FM material, but can be universal. The most plausible mechanism consistent w ith our experimental data is then the interfacial spin-orbit coupling34,36-40, some signatures of which have been reported in earlier experiments by varying the degr ee of oxidation or changing the thickness of the Ta underlayer7,21. 9 If its contribution to the dampi ng-like SOT is of opposite sign to the spin Hall contribution and becomes larger with oxidation than the spin Ha ll contribution, the competition between these two contributions can explain the sign reversal of the damping-like SOT upo n oxidation. For this, the oxidation should enhance either (i ) the interfacial sp in-orbit coupling st rength or (ii) the efficiency to generate the damping-like torque fo r a given interfacial spin-orbit coupling strength. There is a well-known example of (i). The in terfacial spin-orbit coup ling at the magnetic Gd(0001) surface41 becomes three times stronger upon oxida tion, and interestingly reverses its sign. The enhanced strength is attributed to the en hanced internal electric field at the surface. An additional mechanism of (i) may arise from the at omic orbital degree of freedom. When atomic orbitals with angular momentum L are linearly superposed to ma ke a Bloch state with crystal momentum K , the quantum interference between orbita ls of neighbouring atoms generates an electric dipole moment42 towards the direction LK , which couples with the internal electric field E at the surface to generate a Coulomb energy proportional to ()ELK . When E is sufficiently strong and the orbital quenchi ng is weak, this energy tends to align43 L along the direction KE . Such L couples with spin S through the atomic spin-orbit coupling SOLS at the surface, where the coupling constant SO is large due to the hybridization between Pt 5 d orbitals and ferromagnetic 3 d orbitals44,45. Subsequently, the strong atomic spin-orbit coupling SOLS is converted to the strong in terfacial spin-o rbit coupling SOKES . It has been suggested42,44 that this orbital-based mechanism may enhance the Rashba-type interfacial spin- orbit coupling significantly. A recent first principles calculation45 for a Pt/Co bilayer confirms that a strong interfacial spin-orbit coupling can indeed arise at the HM/FM interface near the Fermi energy. 10 Regarding (ii), we are not awar e of any theoretical mechanism that predicts efficiency enhancement by oxidation. We remark, however , that the damping-like SOT caused by interfacial spin-orbit coupling has been significantly unde restimated in earlier theories34,36,37,39. It was later pointed out38,46 that due to the Berry phase effect, th e efficiency of the interfacial spin- orbit coupling mechanism is actually much higher and comparable to that of the spin Hall mechanism. The Berry phase effect has been confirmed46 for a bulk inversion symmetry broken material (Ga,Mn)As, but not yet for structural inversion symmetry broken interfaces. Previous observations11,15,16,18-20,23 of the damping-like SOT were attributed to the bulk spin Hall mechanism. Next we discuss the abruptness of the SOT sign reversal (Figs. 1e & 1f) despite the rather gradual changes of oxidation level (Figs. 2f & Supplementary Fig. 11). Concomitant with the sudden SOT sign reversal, the coercivity (Supplem entary Fig. 2) and the temperature dependence of RH (Supplementary Fig. 14) also change suddenl y. We suspect that such sudden changes may be manifestations of SOT instability. A possible origin of the SOT instability is the competition between the orbital ordering LKE and the orbital quenching common in transition metals. Although its origin is still unclear, the abrupt SO T reversal is of considerable value for device applications. When the oxidation level is near the threshold value, a tiny change of the oxidation level by electric gating47,48 can induce a large change of SOT. This takes the SOT engineering to a whole new level and may even pave the way towards reconfigurable logic devices. Our results may also be relevant to recent SOT experiments using Ta8,16,21,49 which is more susceptible to oxidation compared to Pt . Our results indicate that even for the exact same layer structure, very different SOTs can be obt ained depending on the detailed device preparation procedure, which may affect the oxygen content in the sample. Furthermore, we hope our work 11 initiates efforts to bridge the gap between metal spintronics and oxide electronics to combine the merits of the both fields50. Methods The stacked films were deposited on thermally oxidized Si substrates by magnetron sputtering with a base pressure < 2 × 10-9 Torr at room temperature. The structure of the t series films is substrate/MgO (2)/Pt (2)/Co 60Fe20B20 (0.8)/MgO (2)/SiO 2 (t) with t = 0 ~ 4, and that of the dCFB series films is substrate/MgO (2)/Pt (2)/Co 60Fe20B20 (dCFB)/MgO (2)/SiO 2 (t = 0 and 3) with dCFB = 0.8 ~ 2 (numbers are nominal thicknesses in nanometers). The bottom MgO layer is used to promote perpendicular anisotropy. The other film structures are schematically shown in Figs. 3 and 5. After deposition, except for the oxygen doped Co FeB sample in Fig. 3, all the other films were post-annealed at 300 C for 1 hour in a vacuum to obta in perpendicular anisotropy. The multilayers were coated with a ma-N 2401 negativ e e-beam resist and patterned into 600 nm width Hall bars by electron b eam lithography and Ar ion milling as shown in Fig. 1b. PG remover and acetone were used to lift-off the e- beam resist. Contact pads were defined by photolithography followed by the deposition of Ta (5 nm)/Cu (150 nm)/Ru (5 nm) which are connected to the Hall bars. Before the deposition of the contact pads, Ar ion milling was used to remove the SiO 2 and part of the MgO layer, in order to make low-resistance electrical contacts. All the devices for each batch were processed at the same time to ensure the same fabrication conditions, which was important fo r this study. Devices were wire -bonded to the sample holder and installed in a physical prope rty measurement system (Quantum Design) for the transport studies. We performed the measurements of cu rrent induced switching and the ac harmonic anomalous Hall voltage loops for the t and d CFB series devices at 200 K for the data set in Figs. 1 12 and 4, at which temperature all the devices retain desirable perpendicular anisotropy (Supplementary Figs. 1-3). The current induced sw itching was measured using a combination of Keithley 6221 and 2182A. A pulsed dc current of a duration of 50 μs was applied to the nanowires and the Hall voltage was measured si multaneously. An interval of 0.1 s was used for the pulsed dc current to eliminate the accumulated Joule heating effect. The x-ray absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD) measurements were performed at the 2A beamlin e in the Pohang Light S ource. All the spectra were measured at 200 K in the total electron yiel d mode, and the energy resolution was set to be ~ 300 meV. 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Nat.Nanotechnol. 8,411‐416(2013). 49 Yu,G.etal.Switching ofperpendicular magnetization byspin‐orbittorquesintheabsenceof externalmagnetic fields.Nat.Nanotechnol. 9,548‐554(2014). 50 Narayanapillai, K.etal.Current‐drivenspinorbitfieldinLaAlO3/SrTiO3 heterostructures. Appl. Phys.Lett.105,162405(2014). Acknowledgments This research is supported by the National Research Foundation, Prime Minister’s Office, Singapore under its Competitive Research Pr ogramme (CRP Award No. NRF-CRP12-2013-01 and NRF-CRP4-2008-06). K.L. acknowledges financial support by the NRF (NRF- 2013R1A2A2A01013188) and the MEST Pioneer Research Center Program (2011-0027905). H.W.L acknowledges financial suppor t by the NRF (NRF-2011-0030046 and NRF- 2013R1A2A2A05006237 ) and MOTIE (10044723). D.Y, W.N, and J.P acknowledge financial support by NCRI Program (2009-0081576) an d MPK Program (2011-0031558) through NRF funded by the MSIP. PAL is supported by POSTECH and MSIP. Author contributions X.Q. and H.Y. planned the study. X.Q. and K.N. fabricated devices. X.Q. measured transport properties. Y.W. helped characterization. D.Y, W. N, and J.P carried out x-ray measurements. All authors discussed the results. X.Q ., K.L., H.L., and H.Y. wrote the manuscript. H.Y. supervised the project. 16 Additional information Supplementary information accompanies this paper at www.nature.com/naturenanotechnology . Reprints and permission inform ation is available online at http://npg.nature.com/reprintsandpermissions/. Correspondence and requests for materials should be addressed to H. Y. Competing financial interests The authors declare no compe ting financial interests. 17 Fig. 1. Device structure and the effect of the SiO 2 capping layer thickness. a , Film structure of the multilayers. b, Scanning electron micrograph of the device and electr ical measurement schematic. Two electrodes for the current injecti on are labelled with I+ and I-. The other two electrodes for the Hall voltage measurements are labelled with V+ and V-. c, RH as a function of pulsed currents for different t at 200 K. t is indicated in each graph. A fixed 40 mT magnetic field is applied along the positive current direction. The width of pulsed currents is 50 μs, and RH is measured after a 16 μs delay. d, The first ( Vω) and second ( V2ω) harmonic anomalous Hall voltages as a function of magnetic field ( H). H is applied along the current direction with a tilt of 4° away from the film plane and Iac = 141.4 μA. e, Anisotropy field Han divided by switching current IS (defined as the switching current from a high to low RH) as a function of t. The inset shows IS. f, The extracted damping-like effective field ( HL) from the 2nd harmonic data at positive magnetic fields with different t. The planar Hall effect was taken into consideration for the extraction of HL using the method in ref. 26. The solid lines in e and f are the B-spline interpolations to the experimental data. Fig. 2. SIMS, XPS, XAS, VSM, and XMCD characterisation. a , SIMS depth profiles of oxygen for the films without ( t = 0 nm, solid line) and with ( t = 2 nm, dashed line) the SiO 2 capping layer. Expected layers with the etchi ng time are denoted at the bottom of the graph. b, Pt 4f XPS spectra for the films without ( t = 0 nm, solid line) and with ( t = 2 nm, dashed line) the SiO 2 capping. c, Fe L2,3edge XAS with various t at 200 K. Peaks of Fe and Fe 2O3 are indicated by arrows. d, Fe L2,3edge XMCD with various t at 200 K. e, The saturation magnetization data by VSM at 200 K. The solid line is a B-splin e interpolation to th e experimental data. f, The 18 orbital-to-spin moment ratio and the estimated Fe3+oxide site ratio as a function of t. The error bars correspond to uncertainties in the estimation of the ratios from the experimental data. Fig. 3. The CoFeB oxidation effect. a , Device structure where the CoFeB layer is intentionally oxidized with reactive sputtering (20 sccm Ar + 10 sccm O 2) for the first 14 seconds, then deposited using only Ar for the rest of the Co FeB layer. The whole 1.6 nm CoFeB layer requires 2 minutes for deposition. b, Current-induced switching measured by RH as a function of applied current at 300 K. A magnetic field of H = 4 mT is applied along th e positive current direction. Fig. 4. The CoFeB thickness effect. a , The saturation magnetization data at 200 K for various thicknesses d CFB of CoFeB for t = 0 nm (diamond symbols). Data for t = 3 nm (square symbols) are presented as a reference. The solid lines are the B-spline interpolations to the experimental data. b, Current-induced switching for the various dCFB measured by RH as a function of applied current at 200 K. A magnetic field of H = 40 mT is applied along th e positive current direction. c, Han/IS and HL versus 1/ dCFB. The solid lines show the linear fitting for the data. Fig. 5. Material and structural de pendence of the sign reversal. a , The MgO layer between the CoFeB and SiO 2 layers is eliminated from the stack. b, The Pt layer is eliminated. c, Co is utilized in the multilayer stack. The experimental setup is the same as that of Fig. 1c. The measurements were performed at 200 K for a and c, and at 300 K for b. 19 Fig. 1 -202 f ed cb t = 0 nma -202 t = 1.2 -101t = 1.5 RH () -404 t = 1.8 -2 -1 1 2-404 t = 3 I (mA)-202 t = 0 nm -303 t = 1.2 -202t = 1.5 V (102V) -404 t = 1.8 -2 0 2-303 t = 3 H (T)-202 -202 -101 V2 (V) -0.40.00.4 -0.50.00.5 01234-101 Han/IS (T/mA) t (nm)01234-2-1012 HL (mT) t (nm)024-1012 IS (mA)20 Fig. 2 02 0 0 4 0 0110100 700 710 720 730 740Si/SiO2 sub. MgOCoFeBPt MgO Intensity (counts) Etching time (s) 72 76 800.00.51.0 Binding Energy (eV) Intensity (a. u) 700 710 720 730 XAS Intensity (a. u) Photon Energy (eV)-Fe2O3t = 0.8 nmt = 1.5 nmt = 2.1 nmt = 3.0 nm FeFe2O3t = 3.0 nm t = 2.1 nm t = 1.5 nm t = 0.8 nm t = 0 nmt = 3.0 nm t = 2.1 nm t = 1.5 nm t = 0.8 nm t = 0 nmXMCD Signal (a. u) Photon Energy (eV) 01230.0600.0620.0640.066 ed c b t (nm)morb/mspina 012340.81.01.21.4 t (nm)f MS (106 A/m) 0.30.40.50.60.7 Fe3+ ratio21 Fig. 3 -2 -1 1 2-202 I (mA)b RH () a22 Fig. 4 -2 -1 1 2 RH ()dCFB = 0.8 nm ca dCFB = 2dCFB = 1.6 I (mA)dCFB = 1.2 10b 0.5 1.0 1.5 2.00.91.21.51.8 MS (106 A/m) dCFB (nm) 0.6 0.9 1.2024 Han/IS (T/mA) 1/dCFB (nm-1)-2-10 HL (mT)23 Fig. 5 -2 -1 1 2 t = 3 t = 0 nm 4 -2 2RH ()c a t = 4t = 1.2 nm I (mA)4 b -2 2 t = 3t = 0 nm 4

1411.6895v1.Angular_spin_orbit_coupling_in_cold_atoms.pdf

Angular spin-orbit coupling in cold atoms Michael DeMarco, and Han Pu Department of Physics and Astronomy, and Rice Quantum Institute, Rice University, Houston, TX 77251, USA (Dated: October 15, 2018) WeproposecouplingtwointernalatomicstatesusingapairofRamanbeamsoperatedinLaguerre- Gaussian laser modes with unequal phase windings. This generates a coupling between the atom’s pseudo-spin and its orbital angular momentum. We analyze the single-particle properties of the system using realistic parameters and provide detailed studies of the spin texture of the ground state. Finally, we consider a weakly interacting atomic condensate subject to this angular spin-orbit coupling and show how the inter-atomic interactions modifies the single-particle physics. PACS numbers: 03.75.Mn, 37.10.Vz, 67.85.Bc I. INTRODUCTION Spin-orbit (SO) coupling has traced a circuitous path through physics. Best known for the ~L
~Scoupling (here ~Land~Srepresent the orbital and spin angular momen- tum of the electron, respectively) that contributes to the atomic fine structure [1, 2], the term was soon applied to the Rashba [3] and Dresselhaus [4] coupling between electron spin ~Sand its linear momentum ~kpresent in cer- tainsolid statematerials. Inone ofthemanyrecent leaps in ultra-cold physics, Bose-Einstein Condensates (BECs) [5] and Fermi gases [6, 7] have been created with equal parts Rashba and Dresselhaus coupling, and proposals for more varied couplings abound [8–10]. In this paper, we bring SO coupling full circle by introducing a scheme to engineer a coupling between the atomic pseudo-spin and the orbital angular momentum of a cold atom. The key to creating this angular SO coupling is exchanging the two counter-propagating Gaussian Ra- man beams used in conventional SO coupling for two co-propagating Raman beams operated in Laguerre- Gaussian (LG) modes. LG beam modes carry orbital angular momentum along the direction of beam prop- agation [11]. By choosing beams with unequal phase windings, an orbital angular momentum change may be imparted to atoms transitioning between internal states while the linear momentum change used for conventional SO coupling is annulled by the use of co-propagating beams. Most excitingly, these systems are within experimen- tal reach. Through the use of holographic techniques or spiral wave plates [12–14], far-field LG beams can now be created with relative ease. Manipulating cold atoms with LG beams has been studied both experimentally and theoretically, in the context of quantum information storage [15], slow light propagation [16], synthetic gauge fields [17], etc. The experiments that directly motivated our investigations [18, 19] used LG beams to diabatically Current address: Department of Applied Mathematics and Theo- retical Physics, Wilberforce Road, University of Cambridge, Cam- bridge, CB3 0WA, UK χ1↑χ2↑χ1↓χ2↓ |↑/angbracketright |↓/angbracketrightI1I2 I1I2 (a) (b)FIG. 1: (Color online) (a) Schematic representation of the theoretical system. The atomic cloud interacts with two LG beams copropagating in the ^zdirection with phase windings n1andn2, intensities I1andI2, respectively. (b) Two atomic hyperfine ground states, labelled as j"iandj#i, are coupled by the pair of Raman beams. The beams also induce light shifts with strengths parameterized by the j. write phase windings and spin textures into a BEC, pro- ducing coreless vortices and skyrmions in the process. Our predictions are related to these results, though we focus on the adiabatic regime and consider the ground states of these systems. Here, we introduce a general formalism for analyzing angular SO coupled cold atoms and present results with low-order LG beams that showcase the unique and inter- esting properties of these systems. We derive the Hamil- tonian and discuss its symmetry properties in Section II, followed by a discussion of the single-particle spec- trum and the properties of the ground state in Section III. These single-particle studies form the basis for more challenging investigations about the many-body physics. As an example, we present our studies of a weakly- interacting BEC in Section IV. An outlook and conclud- ing remarks are presented in Section V. II. MODEL HAMILTONIAN Our theoretical system is schematically shown in Fig. 1. We consider atoms confined in a two-dimensional (2D) harmonic trap of frequency !extending in the xy- plane. Two LG beams copropagate in the ^zdirection, coaxial with the center of the trap. LG beam modes [11] are labelled by two indices n;m, and have complexarXiv:1411.6895v1 [cond-mat.quant-gas] 25 Nov 20142 electric field amplitudes at z= 0given by: E(~ r) =p 2I0einr wjnj Ljnj m2r2 w2 er2=w2; whereLjnj mis the associated Laguerre polynomial, wis the width of the beam, I0describes the intensity of the beam, and we have adopted the cylindrical coordinates ~ r= (r;z; ). The form forE(~ r)introduces a non-trivial intensity profile, I(r) =I0r w2jnj Ljnj m2r2 w2 er2=w22 ; while the phase winding einreflects the orbital angular momentum `z=n~carried by the beam. Viaatwo-photonRamanprocess[20], thelaserscouple two hyperfine states of the atom that we label as ";#. Under the rotating wave approximation, the following Hamiltonian can be derived [18, 20, 21]: ^H =p2 2m+L(r) +1 2!2r2+~ (r) ;(1) where L= L"0 0L# = 1"I1+2"I2+=2 0 0 1#I1+2#I2=2 ; withI1(r)andI2(r)being the intensity profiles of the two beams, encodes the light shifts characterized by the coefficients j(j= 1, 2 and=",#) and also includes the two photon Raman detuning , ~ = (r) 0ei(n1n2) ei(n1n2)0 ; represents the Raman coupling whose strength is charac- terized by the parameter 0with (r) = 0pI1I2, and finally = ( "; #)Tis the spinor wave function of the atom. By measuring mass in units of m, energy in units of~!,andlengthinunitsoftheoscillatorlengthp ~=m!, the Hamiltonian takes on the dimensionless form: ^H = 1 2r2+L(r) +1 2r2+~ (r) ;(2) and we have listed typical values of various parameters in Table I. Analogous to the procedure used in conventional SO coupling [5], we introduce new basis states ~ "=ein1 " and ~ #=ein2 . Rewriting the Schödinger equation for these states, and with the help of Pauli matrices iwhich act on the atomic pseudo-spin, we have: ~H~ = 1 2r2 r2~Lz+2 2r2+L+ x+1 2r2 ~ ; (3)TABLE I: Taking != 21kHz, the mass of87Rb M= 1:4431025kg, and ~as the unit measures of fre- quency, mass, and angular momentum, respectively, fixes the unit measures of length ( aosc=p ~=(m!)) and energy ( ~!). These are presented below in standard units along with typi- cal values of Iand 0Iin the units used in our calculation. These values are obtained for a typical laser intensity I= 1 mW/cm2and a single photon detuning of 1 GHz. aosc ~!jIj j 0Ij 0:34m 4:11012eV 1:8 3 :6 TABLE II: In order to reduce the dimensionality of the pa- rameter space to a more manageable size, we fix the values of the following parameters while allowing I10=I20=I0to vary. In the results presented, j, 0andIhave been rede- fined so as to be on the order of 1while leaving Iand 0I with values on the order of those presented in Table I. Couplings Beam Width Raman Detuning LG indices j 0w  n 1=n2mj 11 5aosc 0 1 0 where ~ = ( ~ ";~ #)T,~Lz=i@may be regarded as the quasi-angular momentum (QAM) operator, and = n10 0n2! =n1+n2 2+n1n2 2z; is a scaling matrix. Note that the equation is now rota- tionally invariant, i.e., the QAM operator ~Lzcommutes with the Hamiltonian ~H. Hence each eigenstate of the system possess definite values of QAM characterized by the corresponding quantum number ~`z, which is related to the the angular momentum of each spin component in the lab frame by `(";#) z =~`zn1;2. The coupling between atomic pseudo-spin and its orbital angular mo- mentum becomes explicit for n16=n2as the second term in the square bracket of Eq. (3) contains a term /(n1n2)z~Lz. The large number of parameters renders a full explo- ration of the parameter space beyond reach. To focus on the key features of the system, we may constrain a few parameters by consulting current experimental inter- est (these are tabulated in Table II for reference). Most experiments using LG beams involve only n=m= 0 (gaussian) andjnj= 1,m= 0beams, so we focus on the case ofn1= 1,n2=1, andm1=m2= 0with red single-photon detuning such that j= 0=1, and a beam width wider than the oscillator length w= 5. Lastly, we take the beams to have equal intensity coeffi- cientsI10=I20=I0and take the two-photon detuning = 0. As we shall see, the properties of the system are gov- erned by the interplay of the light shifts, the harmonic trap, and the Raman coupling. The light shifts and trap3 are static potentials (see Figure 2), but it is the Raman coupling that enforces the SO coupling. At low inten- sities, the SO coupling acts as a perturbation on the I0= 0simple harmonic oscillator (SHO) ground state, transferring small amounts of population within a given QAMcomponent. Forhighintensities,theLGlightshifts dominate and the condensate forms in a ring centered at r= 0, which in turn allows for the formation of clouds with higher order phase windings. A. Symmetries of the Hamiltonian If we consider the two spin components to experience the same light shifts (i.e., L"=L#), then we should expect the system to be invariant under the exchange of the spin labels. This action sends n!nas well as inverting the spin space, and is equivalent to reflecting the entire system across the xy-plane. For the spin- 1=2 Hilbert space, this can be represented by the action of x; for the spatial wave function we look for an operator that sends `z!`zwhile leaving ~ runaffected, that is, a time-reversing (antiunitary) operator. More precisely, we note that the time-reversal operator ^T=x^K; where ^Kdenotes complex conjugation, commutes with the Hamiltonian ^Hin Eqs. (1) and (2). This symmetry may be translated into the QAM frame by transforming ^Tunder the unitary matrix U= diag(ein1;ein2)to give an operator ~T=U^TU1=ei(n1+n2)x^K; thatcommuteswiththeHamiltonian ~HinEq.(3). These symmetries will play an important role in understanding the properties of the ground states in the following sec- tions. III. SINGLE-PARTICLE PHYSICS By fixing the QAM quantum number ~`z, we can math- ematically reduce the problem to the radial dimension only. For effective numerical simulation, the divergent terms can be removed by making the ansätz ~ =ei~`z rs"u"(r) rs#u#(r)! ; wheres"=j~`zn1jands#=j~`zn2j. The single- particle eigenstates may then be determined by applying the finite difference approximation to the equations for u";#and directly diagonalizing the resulting matrix. We now abandon our general analysis and focus on the specific case of n1= 1,n2=1,m1=m2= 0, and all other parameters as in Table II. The upper panel of Fig- ure2showsthedispersionrelations E(~`z)forthissystem,and the corresponding ground state wave function is dis- played in the lower panel. For the I0= 0cases, the band structure present is just that of the spinor simple har- monic oscillator (SHO) viewed in the QAM frame. The ground states found are the expected gaussian wavepack- ets populating the component with the lab-frame angular momentum `(";#) z=~`zn1;2= 0, or QAM ~`z=1. At low light intensity, the twin ground states are per- turbed versions of the original ~`z=1SHO ground states, with small amounts of population transferred into the previously vacant component. However, at I01:65, the system transitions to having a single ground state with ~`z= 0andj "j=j #j. We will see that this gives rise to a quantum phase transition in the many-particle BEC case. The symmetry of the bands about (n1+n2)=2is guar- anteed by the ~Tsymmetry discussed above. In particu- lar, the commutation relation [~T;~H] = 0; together with the anticommutation relation: f~Lz;~Tg= (n1+n2)~T; determines the effect of ~Ton the energy eigenstates: If ~ is an eigenstate of the system with QAM quantum number ~`z, then ~T~ is also an eigenstate with QAM quantum number n1+n2~`z. Put simply, ~Treflects the spectrum of the system about ~`z= (n1+n2)=2. Impor- tantly, this implies that any non-degenerate state must have ~`z= (n1+n2)=2, which is only possible if (n1+n2) is even. Hence if we choose, for example, n1= 1and n2= 0, then all eigenstates (including the ground state) will remain degenerate. A. Band Flattening From Fig. 2, one can see that as I0increases, in addi- tion to the ground state changing from two-fold degen- erate to non-degenerate, the low-lying dispersion bands becomes flattened. This phenomenon can be attributed to the fact that, for large I0, the atoms are confined to a ring-shaped region as a result of the light shifts induced by the red-tuned LG beams. Quantitatively, for a given QAM quantum number ~`z, each spin component is ex- posed to an effective static potential V";#(r) =~`2 z 2r2+z~`z r2+1 2r2+L";#+1 2r2;(4) examples of which are illustrated in the lower panel of Fig. 2. However, the spatially varying Raman coupling induces further energetic variation across the width of atomic cloud. For ~`z= 0and symmetric light shifts with L"=L#=L, the system may be decoupled by defining the symmetric and anti-symmetric superpositions of the4 0 5 10r0 5 10r−5 0 5−6−4−2024 ω+≈2.3312 ˜z 0 5 10r0 5 10r−5 0 513579 ˜z−5 0 502468 ˜z−5 0 5−60−55−50−45−40 ω+≈4.8781 ˜z |Ψ↑(r)||Ψ↓(r)| |Ψ↑(r)| |Ψ↓(r)| ˜z=1 ˜z=1 ˜z=1 ˜z=1I0=0 I0=25 I0=5 I0=1.25E 0 2 4 6 8 10r|Ψ↑(r)|=|Ψ↓(r)| 0 2 4 6 8 10r˜z=0 ˜z=0|Ψ↑(r)|=|Ψ↓(r)| FIG. 2: (Color online) Upper panel: Single-particle dispersion relations for I0= 0;1:25;5;25. For the case with I0= 0, the dispersion relation is simply that of the 2D Spinor SHO but viewed through the QAM formalism. The doubly degenerate states are shown with black squares. Other states are non-degenerate. The ring-trapped energy predictions of Eq. (5) are given by red dotted lines for I0= 5and 25. The spectrum is symmetric about (n1+n2)=2 = 0. At smallI0, the ground state is doubly degenerate with ~`z=1; at largeI0, the ground state is non-degenerate with ~`z= 0. The transition occurs at I01:65. Lower panel: Corresponding effective potentials V";#(r)(black dotted lines) and ground state wave functions (blue solid lines). two spin states: =1p 2(~ "~ #), which are governed by their respective Hamiltonian ~H= 1 2r2+1 2r2+L(r) (r) +1 2r2 ; with the corresponding effective potentials V(r) =1 2r2+L(r) (r) +1 2r2: For our choice of the parameters, (r)<0, hence we will now neglect the mode as it has higher energy than the +mode. In the limit of high intensity, the effective po- tentialV+(r)has a deep minimum at radius rmin6= 0, and the atomic density is concentrated in a thin annulus of radiusrmin. If we consider the ring trap to be infinitely thin, then the eigenenergies will be given by ~`2 z=(2r2 min). A recently work has explored this limit [22]. To account for the finite width of the ring, we may expand V+(r) aroundrminto second order such that V+(r)can be ap- proximated as a harmonic potential with freqeuncy !+. Therefore we conjecture that the low-lying eigenenergies can be represented as (apart from a constant shift) E(~`z;n+) =n+!++~`2 z 2r2 min; (5) wheren+= 1;2;3;:::representstheradialquantumnum- ber. The red dotted lines for I0= 5and 25 in the uppper panel of Fig. 2 represent the low-lying energy dispersion curves obtained using the above formula, and one can see that they fit with the numerical results extremely well. 234 rmin 0 5 10 15 20 2524 I0ω+(a) (b)FIG. 3: (Color online) Dependence of rmin(a) and!+(b) on the laser intensity I0. The dashed vertical lines indicate the critical laser intensity at which the ground state changes from two-fold degenerate to non-degenerate. Note that we obtained !+for states with ~`z= 0. For finite ~`z, the effective potential should contain extra con- tributions from the centrifugal term ~`2 z=(2r2). However, as one can see from Fig. 2, this extra term has negligible effect on the value of !+. From Eq. (5), we observe that for a given n+, the curvature of the dispersion is determined by the value ofrmin, and the spacing between adjacent bands (with
n+= 1) is given by !+. We plot in Fig. 3 how these two quantities vary as laser intensity I0changes. These5 results show that as I0increases,rmineventually satu- rates, whereas !+continues to increase. The infinitely thin ring limit is reached when !+is much larger than all other energy scales of the system. B. Spin Textures The SO coupling gives rise to intriguing spin textures. To characterize the ground state spin texture, we first define a normalized spin vector: ~ s= y~  2j j2: Previous studies of 2D Rashba SO coupled [23] BECs and BECs exposed to LG beams [19] have found that the spin texture contains a topological knot known as a 2D skyrmion. Obtained from their 3D siblings by stereo- graphicprojection,2DSkyrmionsareasubjectofinterest in BEC studies for the protection that arises from their topological nontriviality. In a skyrmion spin texture, the azimuthal and polar angles of the local spin may be writ- ten as (r)and(), respectively, which gives rise to the azimuthal n= (2)1()j2 =0and radial wind- ingsnr= cos (r)j1 r=0. The skyrmion number [24] is the topological invariant that distinguishes a skyrmion texture from that of the vacuum; in 2D it is given by: nskyrm =1 4 ~ s
(@x~ s@y~ s)d~ r; (6) or, in terms of the radial and azimuthal windings [25]: nskyrm =nrn: In Fig. 4, we present the ground state spin texture at four different values of I0. In our system, for the two-fold degenerate ground states with ~`z=1at small laser intensity ( I0>1:65), we have n= 2and nr=1=2, which corresponds to a half skyrmion. As I0increases from zero, population is transferred into the previously unoccupied component, and the radial wind- ing that forms the skyrmion texture approaches from r=1, asshowninFig.4. Thismannerofformationalso bypasses the topological protection usually enjoyed by skyrmions. At large r, the atomic density becomes very smallandhence, frombothanumericalandexperimental perspective, ~ sis ill-defined. For I0>1:65, the skyrmion spin texture persists in the ~`z=1components, but the ~`z= 0ground state cannot have a skyrmion spin tex- ture, as the ~Tsymmetry implies that j "(r)j=j #(r)j and hence the spin becomes planar and lies in the xy- plane. Correspondingly, the radial winding nrand the skyrmion number nskyrmall vanish. IV. WEAKLY INTERACTING BEC Our discussion so far has focused on the single-particle physics, which forms the foundation for further explo- FIG. 4: (Color online) Spin structure for the ground state. The arrows point in the direction of the local spin ~ s, and the color represents the spin along the z-axis. ForI0= 0, 0.5 and 1.65, the ground state is degenerate and we pick the one with ~`z= 1. ForI0= 2:5, the ground state is non-degenerate with ~`z= 0. The spin structure for I0= 0:5and 1.65 correspond to a half skyrmion with n= 2andnr= 1=2. ration of the many-body physics. Here, as a first at- tempt along this line, we consider a weakly interaction BEC in the mean-field regime. An interacting BEC of atomsexposedtothesameset-upisdescribedbyaGross- Pitaevksii Equation (GPE) that includes the single- particle Hamiltonian as well as an interaction term:  = ^H+G ; where G= gj "j2+g"#j #j20 0gj #j2+g"#j "j2! ; g"#characterizes the inter-species interaction strength, and we have taken the intra-species interaction strength g""=g##=gfor simplicity. To ensure the stability of the condensate, we consider the situation where all interaction strengths are positive. When we include interactions, the nonlinearity may spontaneously break the rotational symmetry. Hence when solving the GPE, we no longer assume that the system is rotationally symmetric and do the calculation in the 2Dxy-plane. We determine the ground state by applyingasplit-stepimaginarytimeevolution[26], treat- ing the kinetic energy, Raman coupling, and the remain- ing portions of the GPE in the momentum basis (~ p), QAM basis ~ (~ r), and the usual position basis (~ r), respectively. Figure 5 shows the ground state phase diagram in the parameter space spanned by the laser beam intensity I0 and inter-species interaction strength g"#for this system atg= 1, with insets depicting the representative den- sity profiles of the spin-up and spin-down components in6 III III /c108z~=1/c177/c108z~=0 /c108z~ indefiniteI02.5 2.01.51.00.5 0.5 0.75 1 1.25 1.5 gg /c173/c175/ FIG. 5: (Color online) Phase diagram for the weakly interact- ing BEC with g= 1. Insets depict the representative density profiles (j "j2in the left panel and j #j2in the right panel) of each phase. different phases. The three phases found are character- ized by their QAM. Phases II and III are the many-body analogs of the single-particle ground states at small and largeI0, respectively, and the weak interaction consid- ered here does not change the properties of these states in a qualitative way. Both these phases obey rotational symmetry with definite QAM. The ground state in Phase III is non-degenerate with ~`z= 0, while that in Phase II is two-fold degenerate with ~`z=1. By contrast, as is obvious from the density profiles, Phase I spontaneously breaks the rotational symmetry. Further analysis shows that the Phase I ground state can be regarded as an equal-weight superposition of the two single-particle ground states at ~`z=1, with an arbi- trary relative phase (it can be readily proved analytically that the energy of this equal-weight superposition state is independent of the relative phase). For each realiza- tion, this relative phase will be fixed through the mech- anism of spontaneous symmetry breaking. In this phase, each spin state can be regarded as a coherent superposi- tion of quantized vortices with different winding numbers [27]. Specifically, "in the lab frame is a superpositionof states with `z= 0and`z=2, while #is asuperpo- sition of states with `z= 0and`z= 2. However, unlike in previous proposals where such a superposition state is created dynamically [27], here the vortex superposition state represents the ground state of the system. Further- more, for a Phase I state, the density profiles of the two spin components completely overlap with each other, i.e., j "j2=j #j2. ThereforePhaseIoccurswhen g"#issmall. Increasing the laser intensity creates a more ring-shaped potential which tends to restore the rotational symme- try. This explains the reduced area of Phase I at higher intensities. V. OUTLOOK AND CONCLUSION In this work, we considered a situation where two hy- perfine ground states of an atom are Raman coupled by LG laser beams with different phase windings. This cre- ates a coupling between the atom’s pseudo-spin and its orbital angular momentum. Such a situation has already been realized in several experiments, although previous investigations have all focused on the dynamics, instead of the ground state properties that we explored in this work. Wehaveprovidedadetailedstudyofthesingle-particle physics using realistic parameters. Such studies will form the foundation for the exploration of many-body prop- erties involving a quantum gas. We have performed an investigation of a weakly interacting atomic BEC subject to this angular SO coupling under the mean-field frame- work. Already in this simple setting, the inter-atomic in- teractions lead to nontrivial effects. For example, under proper conditions, the interactions spontaneously break the rotational symmetry of the system. Future studies will be extended to stronger interactions which can in- duce more complicated spin textures [23, 28] and even lead to strongly-correlated beyond mean-field states [29], and also to systems of Fermi gases. Acknowledgment —This work is supported by the NSF, and the Welch Foundation (Grant No. C-1669). HP acknowledges useful discussions with Nick Bigelow. Note added — When writing the manuscript, we no- ticedapreprintbyHu et al.[30]thatconsideredasimilar system as ours. In places where we overlap, our results agree with each other. [1] C. Foot, Atomic Physics (Oxford Univ. Press, 2005). [2] Claude Cohen-Tannoudji, Advances in Atomic Physics: An Overview (World Scientific Publishing, 2011) [3] Y. A. Bychkov, and E. I. Rashba,J. of Phys. C: Solid State Physics 17,6039 ( 1984). [4] G. Dresselhaus, Phys. Rev. 100, 580 (1955). [5] Y. J. Lin, K. Jimenez-Garcia, and I. B. Spielman, Nature (London) 471, 83 (2011). [6] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huan, S. Chai, H. Zhai, and J. Zhang, Phys. Rev. Lett. 109, 095301 (2012).[7] L. W. Cheuk, A. T. Sommer, Z. Hadzibabic, T. Yefsah, W. S. Bakr, and M. W. Zwierlein, Phys. Rev. Lett. 109, 109, 095302 (2012). [8] J. Dalibard, F. Gerbier, G. Juzeli¯ unas, and P. Öhberg, Rev. Mod. Phys 83, 1523 (2011). [9] H. Zhai, Int. J. Mod. Phys. B 26, 1230001 (2012). [10] V. Galitski, and I. B. Spielman, Nature (London) 494, 49 (2013). [11] L.Allen, S.M.Barnett, andM.J.Padgett, Optical Angu- lar Momentum (Institute of Physics Publishing, Bristol,7 2003). [12] L. Marrucci, E. Karimi, S. Slussarenko, B. Piccirillo, E. Santamato, E. Nagali, F. Sciarrino, J. Opt. 13, 064001 (2011). [13] A. L. Gaunt, and Z. Hadzibabic, Sci. Rep. 2, 721 (2012). [14] M. Pasienski, and B. DeMarco, Opt. Express 16, 2176 (2008). [15] R. Pugatch, M. Shuker, O. Firstenberg, A. Ron, and N. Davidson, Phys. Rev. Lett. 98, 203601 (2007). [16] J. Ruseckas, G. Juzeli¯ unas, P. Öhberg, and S. M. Bar- nett, Phys. Rev A 76, 053822 (2007); J. Ruseckas, V. Kudriašov, I. A. Yu, and G. Juzeli¯ unas, Phys. Rev. A 72, 053632 (2005). [17] P. Öhberg, G. Juzeli¯ unas, J. Ruseckas, and M. Fleis- chhauer, Phys. Rev. A 72, 053632. [18] K. C. Wright, L. S. Leslie, A. Hansen, and N. P. Bigelow, Phys. Rev. Lett. 102, 030405 (2009). [19] L. S. Leslie, A. Hansen, K. C. Wright, B. M. Deutsch, and N. P. Bigelow Phys. Rev. Lett. 103, 250401 (2009). [20] B. W. Shore, The Theory of Coherent Atomic Excitation (Wiley-VCH, 1990).[21] K. C. Wright, L. S. Leslie, and N. P. Bigelow, Phys. Rev. A78, 053412 (2008). [22] K. Sun, C. Qu, and C. Zhang, arXiv:1411.1737. [23] H. Hu, B. Ramachandhran, H. Pu and X.-J. Liu, Phys. Rev. Lett. 108, 010402 (2012); B. Ramachandhran, B. Opanchuk, X.-J. Liu, H. Pu, P. D. Drummond, and H. Hu, Phys. Rev. A 85, 023606 (2012). [24] N. Manton, and P. Sutcliffe, Topological Soltions (Cam- bridge University Press 2004). [25] N. Nagaosa, and Y. Tokura, Nat Nano 8, 899 (2013). [26] W. Bao, and Q. Du, SIAM J. on Sci. Comp. 25, 1674 (2004). [27] K. T. Kapale and J. P. Dowling, Phys. Rev. Lett. 95, 173601 (2005); S. Thanvanthri, K. T. Kapale, and J. P. Dowling, Phys. Rev. A 77, 053825 (2008). [28] S. Sinha, R. Nath, and L. Santos, Phys. Rev. Lett. 107, 270401 (2011). [29] B. Ramachandhran, H. Hu, and H. Pu, Phys. Rev. A 87, 033627 (2013). [30] Y.-X. Hu, C. Miniatura, and B. Grémaud, arXiv:1410.8634.

2112.04765v1.Impact_of_Kondo_correlations_and_spin_orbit_coupling_on_spin_polarized_transport_in_carbon_nanotube_quantum_dot.pdf

Impact of Kondo correlations and spin-orbit coupling on spin-polarized transport in carbon nanotube quantum dot D. Krychowski, S. Lipi nski Institute of Molecular Physics, Polish Academy of Sciences M. Smoluchowskiego 17, 60-179 Pozna n, Poland (Dated: December 10, 2021) Spin polarized transport through a quantum dot coupled to ferromagnetic electrodes with non- collinear magnetizations is discussed in terms of nonequilibrium Green functions formalism in the nite-U slave boson mean eld approximation. The dierence of orientations of the magnetizations of electrodes opens o-diagonal spin-orbital transmission and apart from spin currents of longitu- dinal polarization also spin- ip currents appear. We also study equilibrium pure spin current at zero bias and discuss its dependence on magnetization orientation, spin{orbit coupling strength and gate voltage. Impact of these factors on tunneling magnetoresistance (TMR) is also undertaken. In general spin-orbit coupling weakens TMR, but it can change its sign. PACS numbers: 72.10.Fk, 72.25.-b, 73.21.La, 73.63.Fg I. INTRODUCTION Spin-dependent coherent electronic transport attracts great interest due to its potential applications in repro- grammable logic devices and quantum computers [1{3]. Carbon nanotubes (CNT) are very promising materials for spintronic applications due to their long spin life- times and because they can be contacted with ferromag- netic materials [4]. Additional to spin, orbital degrees of freedom corresponding to clockwise and counterclock- wise symmetry of wrapping modes in CNTs [5] open a new path for quantum manipulation. Due to the intrin- sic spin-orbit interaction (SO), enhanced by curvature, spin and orbital degrees of freedom are not independent [6]. Our main focus in this article is study of transport in strongly correlated regime. The energy of SO coupling in CNT quantum dot (CNTQD) is comparable to Kondo energy scale and therefore taking this perturbation into account is important when analyzing many-body eects. For spintronic applications of fundamental importance is tunnel magneto-resistance (TMR), the relative resistance change with the change of the orientation of polarized electrodes. Recently, there has been an increasing inter- est of generation of pure spin current (SC) without an accompanying charge current. The attractive attribute of spin current is that it is associated with a ow of an- gular momentum, which is a vector quantity. This fea- ture increases information capacity transferred through the system. In contrast to charge ow, spin transport is almost dissipativeless. The discussion of the dependence of TMR, spin currents and spin accumulation in CN- TQD on the strength of magnetic polarization, spin-orbit amplitude, gate voltage and the angle between magnetic moments of the electrodes is the subject of the present publication. We discuss transport through carbon nanotube quan- tum dot coupled to the noncollinear spin polarized elec- trodes (Fig. 1a). CNTQD exhibits fourfold shell struc- ture in the low energy spectrum. For short nanotubes with well separated energy levels it is enough to restrictat low temperatures to the single shell. The system is modelled by two-orbital Anderson model with equal in- traorbital and interorbital interaction parameters U and the strength of spin-orbit coupling  so: H=X lsElsnls+X klsEksnkls+X kls(tcy kLlsdls+ tss(')cy kRlsdls+tss(')cy kRlsdls+h:c:) + (1) UX lnl"nl#+UX ss0n1sn1s0; wherel=1,s=1 are the orbital and spin indexes. It is assumed that magnetization of the left electrode is oriented along the nanotube axis, whereas magnetiza- tion of the right electrode is tilted by angle '(Fig.1a). cy klsis the creation operator in the left or right elec- trode=L(R),tis hopping integral, nls=dy lsdls (nkls=cy klsckls) is the occupation number operator in CNTQD (lead), Ekls(Els=Ed(Vg) +lsso) denote energies of electrons in the lead (dot). For CNT with the wide bandgap, total spin-orbit contribution to energy (Zeeman-like and orbital-like) can be parametrized by a single eective SO parameter ( so) [6]. The tunneling elemets of the right electrode are given by tss=tcos(' 2), t11=tsin(' 2) andt11=tsin(' 2). To analyze strongly correlation eects in the system we use nite Uslave boson mean eld approach (SBMFA) of Kotliar and Ruckenstein [7]. Slave bosons e,pls,d20(02) ,dss0,tls andfare introduced to describe empty, single , doubly, triple and fully-four occupied states respectively. Six d bosons project onto states with double occupation of a given orbitalj2;0i,j0;2ior onto the states with single oc- cupation on each orbital js;s0i. The eective slave bosonarXiv:2112.04765v1 [cond-mat.mes-hall] 9 Dec 20212 FIG. 1: (Color online) (a) { Schematic view of CNTQD with non-colinear spin polarized electrodes, (b) TMR as a func- tion of gate voltage (c) Evolution of TMR with the increase of SO coupling (SOC). (d,e) - Gate-voltage dependencies of spin-orbital resolved conductances in parallel and antiparallel conguration of magnetization in the electrodes without and with SOC in CNTQD p= 1=2, (d)  so= 0, (e)  so= 0:05. Inset of Fig. 1e presents conductances for 1 ";1#(solid blue line) and 1#;1"(dashed blue line) in AP coniguration. Hamiltonian (2) reads: eH=X lsElsNls+X klsEksnkls+X kls(tcy kLlszlsfls+ tss(')cy kRlszlsfls+tss(')cy kRlszlsfls+h:c:) + UX lss0(dy ldl+dy ss0dss0) + 3UX lsty lstls+ 6Ufyf+ (2) X lsls(NlsQls) +(I1); whereNls=fy lsflsis the pseudofermion occupation operator (dy lszlsfy ls[7]).Qls=py lspls+dy ldl+ dy ssdss+dy ssdss+ty lstls+ty lstls+ty lstls+fyfandI= eye+P lspy lspls+P lss0(dy ldl+dy ss0dss0)+P lsty lstls+fyf are the conservation and completness realtions. zls= (eypls+py lsdl+py ls(l;1dss+l;1dss) +py lsdss+dy ltls+ dy sstls+(l;1dy ss+l;1dy ss)tls+ty lsf)=p Qls(1Qls) is the renormalization parameter of the coupling strength with left and right electrode. Couplings to electrodes are de- FIG. 2: (Color online) Transmissions of CNTQD: (a,c) for p= 0:5 and (b,d) for p= 0:95 (red, blue black, gray lines cor- respond to 1", 1#,1"and1#spin-orbital contributions). Upper and bottom gures are the transmissions for  so= 0 and  so= 0:05 respectively. Insets shows transmissions for antiparallel conguration. Bottom inset on Fig.2c presents the spin-orbital transmissions for P conguration close to the Fermi level (red dashed lines denote the value of the spin- orbital splitting). scribed by matrices L= [L l;s;s] and R= [R l;s;s0], where L l;s;s= (els=2)(1p), R l;s;s= (els=2)(1pcos(')) for s=1 respectively, where els= z2 lsand =t2=2D. The o-diagonal elements are R l;s;s= (elss=2)psin(') withelss= zlszls. 1=2Dis the rectangular density of states in the lead ( jEj<D andpdenotes polarization of electrodes). Throughout this paper we use relative en- ergy units chosen D=50 as the unit and we set jej=~= 1. The spin and charge currents are given by ( IX=Re[I+], IY=Im[I+],IZ,IC): I+=ih[eH;X licy kLl"ckLl#icy kRl"ckRl#]i(3) IC(Z)=I"I#=ih[eH;X lnLl"nRl" nLl#nRl#]i: The currents are expressed by electrode-lead correlation functions, which can be found using the nonequilibrium Green functions. Finally they are given by integrals of the diagonal and o-diagonal elements of the transmis- sion matrixT(E) = LeGR(E)ReGA(E), where eGR(A) are the retarded and advaced renornalized Green func- tion matrix with diagonal EElslsiP  l;s;s and o-diagonal elements iR l;s;s. Integrating the lesser3 Green functions, we can calculate the average spin com- ponentsSX;(Y;Z)=RP leG< ls;ls 0(E)dE 2i(eG<=eGRe<eGA, wheree< ls;ls0= 2iP  l;s;s0f(E),f(E) is the Fermi distribution function)[8]. The dierential conductance, tunnel magneto-resistance (TMR) and polarization of conductance are given by formulas: TMR (') = [G(0) G(')]=G(') andPC= [G"G#]=G(whereG=P sGs=P sdIs=dV = (e2=h)R+1 1(fLfR)P lsTls;ls LR(E)dE. All the calculations were performed for = 0 :05 andU= 6. II. RESULTS AND DISCUSSION The major issue of spintronics is a control of trans- port by a change of relative orientations of magnetic mo- ments of external leads. Fig. 1b shows the gate volt- age dependence of tunnel magnetoresistance of CNT-QD and Figures 1d,e present the corresponding conductances for P and AP congurations. In collinear conguration of magnetizations of the leads the decisive role in lin- ear TMR plays a competition of the central transmission peak (peak located closest to the Fermi level) of the di- agonal transmission for parallel (P) conguration (Fig. 2a,b) and central peak for antiparallel (AP) orientation (insets of Fig. 2a,b). The majority peak is wider and located closer to the Fermi level. In AP conguration ('=) only a single peak is visible with equal contribu- tions of both spin orientations. As it is seen from com- parison of Figs. 2a,b the disproportion between P and AP transmissions increases with increasing electrode po- larization P, which means an increase in TMR. For p= 1 AP transmission at the Fermi level disappears and trans- port is blocked (perfect spin valve eect). Without SO coupling ( so= 0) TMR reaches value only weakly de- pendent on gate voltage and determined by polarization of electrodesp2 1p2[9] and it increases up to Julli ere limit 2p2 1p2when approaches fully occupied or empty regions of the dot. For clarication of gate voltage dependencies of TMR we present on Figs.1d, ecorresponding conduc- tances for P and AP congurations. For  so= 0 AP conductance curve resembles dependence for p= 0, but with reduced plateau values for all occupations. This is a consequence of breaking of the left-right symmetry in individual spin channels. For P orientation dierences between spin-resolved conductances result from violation of spin symmetry. For n= 2 exchange splitting
is small and vanishes at e-h symmetry point (inset of Fig. 3b). Transmissions for both spins locate close to EFand both spin resolved conductances reach almost unitary limits in the centre of n= 2 valley. Outside this point they be- have dierently due to the dierent widths of transmis- sions. Forn= 1;3 transmission lines do not locate at EF and therefore the points of vanishing
do not re ect in equality of conductances. For  so6= 0 in P conguration conductances dier for dierent spin and orbital chan- nels. Breaking of SU(4) symmetry re ects most stronglyby suppression of conductances for n= 2, because in this region SU(4) Kondo temperature ( p= 0, so= 0) is low- est. In AP conguration there are only two dierent or- bital resolved conductances, which are mirror re ections with respect to the e-h symmetry line. Independent on the sign of spin splitting
= El++l+Ell conductance for parallel orientation of the magnetic mo- ments of electrodes dominates over conductance for an- tiparallel orientation and TMR remains positive in the whole range of gate potential. This changes for non-zero SO coupling. In transmission for P conguration in this FIG. 3: (Color online) (a,b) TMR and PC as a function of angle'presented for dierent SOC strengths. Inset of Fig.b shows gate-dependent exchange eld splitting (
=TSU(4) K ) and PC forp= 0:5;0:8 (solid and dashed, magenta and blue lines).(c) - gate dependencies of average spin components for  so= 0 (dotted lines) and for  so= 0:05 (solid lines) ('==2). d)hSX;(Y;Z)ias a function of 'for  so= 0 (solid lines) and for  so= 0:3 (dotted lines). (e) - An- gle dependencies of transverse equilibrium spin currents for so= 0;0:01;0:05;0:3 (red, blue, black and gray lines). In- set illustratesIXfor  so= 1. Dashed lines mark '==2 (p= 0:5). case (Fig. 2c,d), additional satellite peak appears located approximately around E=
soinN= 1eregion or E=
soforN= 3eandE=soforN= 2e. In the upper inset of Fig. 2d presenting transmission4 for AP orientation only satellite corresponding to the SO splitting is visible. The reconstruction of transmissions in P and AP congurations causes a suppression of TMR and for  so6= 0 even the sign of magnetoresistance may change in some cases (inverse TMR). Fig. 1c illustrates what a strong eect on TMR has weakening of the cou- pling of the leads to the dot. As it is seen with the decrease of inverse TMR( ') appears at lower value of SOC. This is a consequence of narrowing of perturbed Kondo peaks and in consequence the spin-orbit induced splitting of transmission peaks occurs for smaller values of so. Fig. 3a presents the angle dependence of TMR. When the directions of electrode magnetizations deviate from each other, TMR increases and reaches maximal value for'=. SO coupling decreases the amplitude of TMR oscillation and, as we already mentioned earlier, allows for its negative values. Inverse TMR is observed already for very small values of SOC ( so= 0:002) and maximal value reaches for  so= 0:2 for noncollinear con- guration ( '==2 and'= 3=2). Fig. 3b shows the oscillating, decreasing with the increase of  so, angle de- pendence of polarization of current with maximum for P conguration, what is a direct consequence of highest transmission for spin up channel for this conguration (Fig.2). Despite the fact that exchange eld changes sign when moving form N= 1eregion into N= 3e PC remains positive for all gate voltages (inset of Fig. 3b). Kondo temperature TSU(4) K changes with gate volt- age and it ranges from 105to 103. Coupling of CN- TQD to polarized electrodes induces spin accumulation. The average values of spin components plotted as a func- tions of gate voltage for  so= 0 and for nonvanishing SO coupling are shown on Fig. 3c. Due to exchange eld in- duced by polarized electrodes the dot magnetic moment is suppressed, but not quenched and local extrema occur at the boundaries of Coulomb blockades. The change of the sign of transverse spin components when cross- ing electron-hole symmetry point is the consequence of reversal of the eective exchange eld. The angle depen- dencies of average spin components are shown on Fig. 3d.hSZivanishes for antiparallel conguration ( '=), whereas transverse components hSX;Yivanish for'= 0 and'=. SO coupling diminishes the oscillation am- plitude. Recently there has been an increasing interest in generation of pure spin currents without an accompa- nying charge current. Fig. 3e presents the angle depen- dencies of equilibrium (zero bias) transverse component of spin currentIX(ESC). When magnetic moments of electrodes are deviating from the parallel orientation the transverse components of the exchange eld appear. For the geometry considered it is Ycomponent of the ef- fective eld ( HY) and spin torque HYspointing in Xdirection exerts on electrons owing through the dot (s="(#) in the global Z axis). Torque is oriented in the opposite directions for right and left moving electrons. In consequence, the charge ow in opposite directions is as- sociated with opposite x components of the spin. Fig. 4 concerns AP conguration. Fig. 4a presents dependen- FIG. 4: (Color online) (a) Equlibrium transverse spin currentsIXat the borders of Coulomb blockades ( Ed= U;2U;3U- red, black, blue curves). (b) ESC in Kondo range forN= 1;2;3e(red, black, blue lines). Insets present orbital spin- ip contributions to equilibrium spin currents (I1";1#,I1#;1",I1";1#andI1#;1"- red, blue, black, gray curves). c) The o-diagonal spin- ip orbital transmissions forN= 1e. Red dashed line marks  so= 0:05. d-f)IX as a function of bias voltage for N= 1e;2e;3eshown for so= 0;0:05;0:1;0:2 (red, blue, black and gray lines). In- sets present orbital spin- ip contributions to spin currents for  = 0:1 ('==2,p= 1=2). cies of ESC on SO coupling strength at the borders of Coulomb blockades and Fig. 4b the same dependencies for the regions of broken Kondo states for occupations N= 1;2;3. In the former case absolute values of IX decrease with the increase of SO coupling, whereas for the latter nonmonotonic dependencies are observed with predominantly increasing tendencies of IXfor small val- ues of so. Although the behavior of ESC is determined by a subtle interplay of integrated to the Fermi level o- diagonal in spin space transmissions, the increasing ten- dency in Kondo regime re ects the general property of increase of currents due to the increase of electrode-dot coupling with SO induced weakening of Kondo correla- tions. Fig. 4c shows examples of zero-bias partial o- diagonal transmissions for N= 1.T1";1# LR dominates atEFand so does the corresponding integral up to EF5 determining spin- ip I1";1#contribution toIX. Its largest and negative contribution is shown in the left up- per inset of Fig. 4b. Similar partial, orbital resolved spin- ip equilibrium spin currents for selected gate volt- ages are presented on the rest of insets of Figs. 4a,b. Figs. 4 d,e,f present bias voltage dependencies of IXspin current for dierent occupation regimes. Independent on the value of  so, the spin current remains negative for N= 2 andN= 3, whereas for N= 1 it changes sign at small voltages. For the detailed explanation of this observation the picture of the evolution of spin- ip trans- missions with voltage should be recalled, but even from the insight into the zero-bias transmissions (Fig. 4 and its inset) it is seen that by increasing the transport win- dow the positive contribution I1#;1"starts to dominate inIXover the negative. Summarizing, we have investigated the eect of spin- orbit coupling and Kondo correlations on spin currents and tunnel magnetoresistance in carbon nanotube quan- tum dot comparing also some calculations with the re-sults for the boundaries between the Coulomb valleys. Discussion of SO eects and noncollinearity of polariza- tions is important because the energy of SO coupling is comparable with the energy scale of Kondo eect and deviations of polarizations introduce real spin- ip pro- cesses. In consequence the many body resonances are formed due to the interplay between Kondo-like eective spin-orbital ips and real spin-orbit transitions. Apart from the current with spin component parallel to the global quantization axis, noncollinearity of polarizations induces also transverse components (spin- ip currents), which do not vanish even in equilibrium. ESC comes from the exchange coupling between the magnetic mo- ments of ferromagnetic electrodes and its direction is de- termined by the vector product of these magnetizations. The spin currents scale with coupling strengths to the leads and as such are much weaker in the Kondo range than in Coulomb charge uctuation regions. Spin-orbit coupling weakens TMR, but due to this interaction the inverse TMR eect may occur. [1] I. Zuti c, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76, 323-410 (2004). [2] D. Loss, P. DiVincenzo, Phys. Rev. A 57, 120 (1998). [3] D. D. Awschalom, D. Loss, N. Samarth, Semiconductor Spintronics and Quantum Computation (Springer-Verlag Berlin Heidelberg, 2002). [4] A.Cottet, T.Kontos, S. Sahoo, H.T. Man, M.-S. Choi, W. Belzig, C. Bruder, A. F. Morpugo, C. Schonenberger, Semicond. Sci. Technol. 21, 78 (2006). [5] W. Liang, M. Bockrath, H. Park, Semicond. Phys. Rev.Lett.88, 126801 (2002). [6] F. Kuemmeth, S. Ilani, D. C. Ralph, P.L. McEuen, Nature (London) 453, 448 (2008). [7] G. Kotliar, A.E. Ruckenstein, Phys. Rev. Lett. 57, 1362 (1986). [8] W. Rudzi nski, R. Swirkowicz, J. Barna s, M. Wilczy nski, J. Magn. Magn. Mat. 294, 1-9 (2007). [9] I. Weymann, R. Chirla, P. Trocha, C. P. Moca, Phys. Rev. B87, 085404 (2018).

2404.06097v1.Hydrostatic_pressure_control_of_the_spin_orbit_proximity_effect_and_spin_relaxation_in_a_phosphorene_WSe__2__heterostructure.pdf

Hydrostatic pressure control of the spin-orbit proximity effect and spin relaxation in a phosphorene-WSe 2heterostructure Marko Milivojevi´ c,1, 2,∗Marcin Kurpas,3Maedeh Rassekh,4Dominik Legut,5, 6and Martin Gmitra4, 7,† 1Institute of Informatics, Slovak Academy of Sciences, 84507 Bratislava, Slovakia 2Faculty of Physics, University of Belgrade, 11001 Belgrade, Serbia 3Institute of Physics, University of Silesia in Katowice, 41-500 Chorz´ ow, Poland 4Institute of Physics, Pavol Jozef ˇSaf´ arik University in Koˇ sice, 04001 Koˇ sice, Slovakia 5IT4Innovations, VSB-Technical University of Ostrava, 17. listopadu 2172/15, 708 00 Ostrava, Czech Republic 6Department of Condensed Matter Physics, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 3, 121 16 Prague 2, Czech Republic 7Institute of Experimental Physics, Slovak Academy of Sciences, 04001 Koˇ sice, Slovakia Effective control of interlayer interactions is a key element in modifying the properties of van der Waals heterostructures and the next step toward their practical applications. Focusing on the phosphorene-WSe 2heterostructure, we demonstrate, using first-principles calculations, how the spin-orbit coupling can be transferred from WSe 2, a strong spin-orbit coupling material, to phospho- rene and further amplified by applying vertical pressure. We simulate external pressure by changing the interlayer distance between bilayer constituents and show that it is possible to tune the spin- orbit field of phosphorene holes in a controllable way. By fitting effective electronic states of the proposed Hamiltonian to the first principles data, we reveal that the spin-orbit coupling in phos- phorene hole bands is enhanced more than two times for experimentally accessible pressures up to 17 kbar. Finally, we find that the pressure-enhanced spin-orbit coupling boosts the Dyakonov-Perel spin relaxation mechanism, reducing the spin lifetime of phosphorene holes by factor 4. I. INTRODUCTION Van der Waals (vdW) heterostructures [1–5] represent a unique class of materials and devices created by stack- ing together individual layers of two-dimensional mate- rials held together by weak van der Waals forces. Using different combinations of materials, each with distinct properties, it is possible for one material to inherit the properties of the other via the proximity effect [6, 7]. In the field of spintronics [8–12], focusing on the ma- nipulation of an electron’s spin, it has been shown that vdW heterostructures can be used to modify sizably the spin-orbit coupling strength [13–17] and induce mag- netism [18–22] into the material of interest using the proximity effect. Phosphorene [23–28, 52] is a semicon- ducting material with a sizable gap [30–33] and the po- tential to overtake graphene [12] in future spintronics de- vices, provided that a sizable spin-orbit coupling and/or magnetism can be induced in a system. Quite recently we have shown that combining mate- rials with different crystal structures provides an excel- lent platform for the engineering of exotic spin texture in a low symmetry [34, 35] or symmetry-free [36] en- vironment. Specifically, in a mixed-lattice heterostruc- ture made of phosphorene and WSe 2, a material with gi- ant spin-orbit coupling [38, 39], we have shown that the sizable spin-orbit coupling can be induced in phospho- rene holes, with the strengths exceeding the values when ∗marko.milivojevic@savba.sk; milivojevic@rcub.bg.ac.rs †martin.gmitra@upjs.skphosphorene is subjected to the perpendicular electrical field [37]. Furthermore, we have shown that by symmet- rical and asymmetrical encapsulation of phosphorene [35] by WSe 2monolayers, one can tune both the strength and the type of spin texture induced in phosphorene holes. In this paper, we investigate another knob for the fine- tuning of the spin-orbit proximity effect, namely, hydro- static pressure. It has been demonstrated to be a promis- ing approach for the modification of interactions between the vdW heterostructure constituents, resulting in a se- ries of novel phenomena predicted theoretically [40–42] and experimentally [43–50]. It has been shown that the spin-orbit proximity effects depend on the interlayer dis- tance between the heterostructure constituents [51], and being tunable with hydrostatic pressure. Here we study electronic properties of the zero twist angle phosphorene- WSe 2heterostructure showing that the spin-orbit cou- pling increases more than two times when assuming pres- sures up to 17 kbar. We note that such pressures are nowadays achievable in experimental setups [50]. We also show that the tunability of the spin-orbit field by the pressure can be traced to the decreased spin relax- ation times for the phosphorene holes. This represents an experimentally accessible way to indirectly trace the spin-orbit strength engineered by pressure [52, 53]. This paper is organized as follows. In Sec. II we present the geometry of the phosphorene-WSe 2heterostructure, introduce the notion of the vertical pressure applied to the heterostructure, and provide all the relevant numeri- cal details for the performed first-principles calculations. In Sec. III the effect of applied pressure on electronic bands of the phosphorene-WSe 2heterostructure is dis- cussed with a particular emphasis on the spin-orbit cou-arXiv:2404.06097v1 [cond-mat.mes-hall] 9 Apr 20242 P (a) (b)ГX YSkx ky ГKM x ky (c) (d)P WSe ykx FIG. 1. Side (a) and top (b) view of the phosphorene-WSe 2 heterostructure subjected to the vertical pressure. The het- erostructure is constructed in such a way that its x/ydirection corresponds to the zigzag/armchair edge of the phosphorene ML. In (c) and (d), the Brillouin zones of phosphorene and WSe 2MLs, respectively, are shown, with marked high sym- metry points. pling and spin relaxation of phosphorene holes close to the Γ point. Finally, Sec. IV summarizes the most im- portant findings. II. ATOMIC STRUCTURE AND FIRST PRINCIPLES CALCULATION DETAILS In Fig. 1 (a)-(b), we present top and side views of the atomic structure models of the phosphorene-WSe 2het- erostructure subjected to hydrostatic pressure P. The supercell of the considered heterostructure contains 20 phosphorene (5 unit cells), 8 tungsten, and 16 sele- nium atoms and was constructed using the CellMatch code [54]. Since we focus on the properties of phospho- rene, we keep its lattice vectors unstrained. A minor strain of 0.51% was applied to WSe 2to satisfy the com- mensurability condition required by the periodic simula- tion cell. Lattice vectors of phosphorene monolayer (ML) are equal to a=aex,b=bey, where a= 3.2986 ˚A andb= 4.6201 ˚A [55], while the lattice vectors of WSe 2 ML correspond to a1=aWex,a2=aW(−ex+√ 3ey)/2 (aW= 3.286˚A [56]). The corresponding Brillouin zones (BZ) of the heterostructure constituents are depicted in Fig. 1 (c) and (d). First-principles calculations of the phosphorene- WSe 2heterostructure were performed using the plane wave Q uantum ESPRESSO (QE) package [57, 58]. The Perdew–Burke–Ernzerhof exchange-correlation functional was employed [59]. Atomic relaxation was performed using the quasi-Newton scheme and scalar- relativistic SG15 optimized norm-conserving Vanderbilt (ONCV) pseudopotentials [60–63]. For ionic minimiza-tion, the force and energy convergence thresholds were set to 1 ×10−4Ry/bohr and 10−7Ry, respectively. Additionally, the Monkhorst-Pack scheme with 56 ×8 k-points mesh was used, small Methfessel-Paxton energy level smearing of 1 mRy [64], and kinetic energy cut-offs for the wave function and charge density 80 Ry and 320 Ry, respectively. We performed our calculations us- ing three different van der Waals corrections: Grimme’s DFT-D2 (D2) [65, 66], non-local rvv10 [67, 68], and Tkachenko-Scheffler (TS) [69]. In all cases, a vacuum of 20˚A in the z-direction was employed. For the relaxed structures, the average distance between the bottom phosphorene and the closest selenium plane in the z-direction is equal to 3 .31˚A, 3 .41˚A, and 3 .66˚A for the vdW corrections D2, rvv10, and TS, respectively. Application of the hydrostatic pressure affects mostly the interlayer distance between the phosphorene and WSe 2. The calculation procedure we adopted is as follows: we started from the equilibrium structure and reduced the vertical distance between phosphorene and WSe 2monolayer by dp. Next, we relaxed both the lattice parameters and atomic positions while fixing thez-components of the outermost phosphorene and selenium atoms. In this way, the relaxation of lateral lattice parameters lowers the in-plane stress in the heterostructure. Constraining the thickness by fixing the outermost atoms, a pressure with a dominant vertical component arises. In the case of noncolinear density functional theory (DFT) calculations with spin-orbit coupling, the k-points mesh and kinetic energy cutoffs for the wave function and charge density were the same as in the relaxation calcula- tions. We used fully relativistic SG15 ONCV pseudopo- tentials and increased the energy convergence threshold to 10−8Ry. Finally, the dipole correction [70] was taken into account to properly determine the energy offset due to dipole electric field effects. III. ANALYSIS OF THE PRESSURE MODULATED BAND STRUCTURE In Table I we present the calculated pressure values for dpranging from 0.1 ˚A to 0.5 ˚A. Since lateral lattice pa- rameters are free to change, we also present the relative changes δxandδyof the in-plane dimensions of the het- erostructure cell in the xandydirections. Note that for the minimal unit of phosphorene, the changes in the x andydirection correspond to δxandδy/5, respectively. The pressure increases roughly linearly when reducing the effective thickness of the heterostructure, with values that are in the range of the experimental setups (18 kbar in the graphene-WSe 2heterostructure [50]). The discrepancies in the calculated pressure using dif- ferent vdW corrections come solely from the fact that the initial interlayer distances of the relaxed heterostruc- ture using D2, rvv10, and TS vdW corrections and equal to 3.31 ˚A, 3.41 ˚A, and 3.66 ˚A, respectively. Thus, al-3 FIG. 2. Comparison of the hydrostatic pressure influence on the band structure in the phosphorene-WSe 2heterostructure with the fully relaxed one (a), unfolded to the XΓY path of the phosphorene Brillouin zone. In panels (b)-(d), the phosphorene-WSe 2 heterostructure was exposed under the hydrostatic pressure equal to 2.59 kbar, 9.02 kbar, and 16.50 kbar, respectively. In all cases, the heterostructures were relaxed using the QE code and the Grimme-D2 vdW correction. TABLE I. The modifications of the vertical pressure and the percentage changes of the in-plane dimensions of the het- erostructure cell in the xandydirection, δxandδy, upon reducing the vertical distance dp. We performed relaxation calculations using three different vdW corrections: Grimme- D2, non-local rvv10, and Tkachenko-Scheffler. vdW dp[˚A] P[kbar] δx[%] δy[%] D20.1 2.59 0.71 -0.42 0.2 5.57 0.77 -0.31 0.3 9.02 0.86 -0.17 0.4 12.59 0.95 -0.02 0.5 16.50 1.04 0.18 rvv100.1 2.04 0.96 -0.19 0.2 4.69 1.02 -0.10 0.3 7.94 1.10 -0.02 0.4 11.55 1.18 0.17 0.5 15.68 1.28 0.36 TS0.1 1.28 0.67 -0.72 0.2 2.56 0.70 -0.69 0.3 4.07 0.73 -0.65 0.4 4.95 0.74 -0.66 0.5 7.99 0.84 -0.52 though the absolute change of the heterostructure thick- ness is the same for all three vdW corrections, the relative change is not the same. That is why the calculated pres- sure in the heterostructure relaxed using the TS vdW correction, having the sizable larger interlayer distance when compared to the D2 and rvv10 cases, is much lower than in the other two cases. To analyze the changes induced by pressure on the elec- tronic properties of the phosphorene-WSe 2heterostruc- ture, in Fig. 2, we present the band structure unfolded to the XGY path of the Brillouin zone of phosphorene in four distinct cases (we assume D2 vdW correction in all 4 cases): (a) heterostructure without pressure, (b)-(d)heterostructure subjected to the pressure of 2.59, 9.02, and 16.50 kbar, respectively. The most prominent effect of the applied pressure is the increase of the direct band gap at the Γ point, which suggests a weakening of the interaction between the valence and conductance bands. The top valence phosphorene band dispersion around the Γ point remains almost unchanged. The K valley band of WSe 2maps along the ΓX direction, and the Γ valley band forms the lower-lying valence band. Under pres- sure, the lower-lying valence band moves closer to the top valence phosphorene band. Therefore, one can ex- pect an increased spin-orbit coupling proximity effect in phosphorene holes. Hybridization of the K valley WSe 2 band and phosphorene hole band at about −0.4 eV is re- sponsible for transferring the z-component of spin to the phosphorene. To get a quantitative insight into the changes in the spin-orbit proximity effect due to the applied pressure, in what follows we will analyze the spin-orbit field of phosphorene holes around the Γ point and compare it to the values obtained for the fully relaxed heterostructure configuration [34]. A. Effective parameters The spin physics of phosphorene holes around the Γ point can be well described in terms of the effective spin-orbit coupling Hamiltonian compatible with the C1v symmetry of the heterostructure [34, 35] Heff=λ1kxσy+λ2kyσx+λ3kxσz. (1) The Hamiltonian (1) includes terms linear in momenta, as well as the Pauli spin operators σi,i=x, y, z , which are connected to the spin operators Sithrough the rela- tionSi=ℏ/2σi. Whereas the λ1kxσy+λ2kyσxpart of4 0 2 4 6 8 10 12 14 16 18 0 0.1 0.2 0.3 0.4 0.5P (kbar) dP (Å)D2 RVV TS(a) 4 6 8 10 12 14 16 18 20 0 0.1 0.2 0.3 0.4 0.5λ1 (meVÅ) dP (Å)D2 RVV TS(b) 8 10 12 14 16 18 20 22 0 0.1 0.2 0.3 0.4 0.5λ2 (meVÅ) dP (Å)D2 RVV TS(c) −35−30−25−20−15−10−5 0 0.1 0.2 0.3 0.4 0.5λ3 (meVÅ) dP (Å)D2 RVV TS(d) FIG. 3. Dependence of the pressure (a) and spin-orbit cou- pling parameters of phosphorene holes (b-d) around the Γ point on the interlayer distance change dpfor three consid- ered vdW corrections. the total spin-orbit field corresponds to the in-plane spin components and can be created simply by applying the perpendicular electric field, the out-of-plane component λ3kxσzis a signature of the mixed-lattice heterostructure that triggers the effective in-plane electric field into the phosphorene monolayer. Although the effective form of the spin-orbit coupling Hamiltonian can be deduced from symmetry, the spin- orbit coupling parameters λ1,λ2, and λ3need to be de- termined by fitting the model (1) to the DFT data. This was done in the following way: for each studied configu- ration, the DFT data about the spin splitting energy and spin expectation values of the top valence band around the Γ point were fitted to the spin-orbit coupling Hamil- tonian model. For the fitting, we considered the XΓ and ΓY paths, up to the distance 0 .009˚A−1away from the Γ point. The obtained spin-orbit coupling parameters as a func- tion of the dpfor phosphorene holes are given Fig. 3(b-d). The pressure dependence on dp, given in Table I, is also plotted in Fig. 3(a). Comparing the spin-orbit coupling parameters values to the zero pressure case ( dp= 0) [34] one sees that the pressure enhances the spin-orbit cou- pling λparameters by a factor of three. This is expected since the acquired spin-orbit coupling of phosphorene holes comes due to the proximity effect, which depends 4 6 8 10 12λ1 (meVÅ)D2 RVV TS (a) 6 8 10 12 14λ2 (meVÅ)(b) −18−16−14−12−10−8−6 −1 −0.5 0 0.5 1λ3 (meVÅ) E (V/nm)(c)FIG. 4. Dependence of spin-orbit parameters of the model Hamiltonian (1) on the transverse electric field amplitudes for three considered vdW correction types. on the interlayer distance [34]. Thus, the presence of vertical pressure decreases the distance between the con- stituent MLs (strengthens the interlayer hopping) and at the same time increases the proximity-induced spin-orbit coupling. The controllability of the distance between the MLs through the vertical pressure suggests that pressure can be used as an experimentally accessible fine-tuning knob for the control of the spin physics in proximitized phosphorene. Although demonstrated in the example of phosphorene-WSe 2heterostructure with zero twist an- gle, the obtained results can be easily extended to the 60-degree twist angle case. As shown in [34], the domi- nant difference in the spin texture of phosphorene holes around the Γ point in the case of zero and 60-degree twist angle lies in the sign change of the λ3parameter only due to the discovered [34] valley-Zeeman nature of the spin-orbit field kxσz. Our results also suggest that the vertical pressure does not lead to a qualitative change in the type of spin texture induced due to the proxim- ity effect. Thus, the tunable increase of the spin-orbit proximity effect can be expected in phosphorene-based trilayer heterostructures, in which phosphorene is sym- metrically or asymmetrically encapsulated by two WSe 2 monolayers [35]. Finally, we discuss and compare the fine-tuning of the spin-orbit proximity effect using hydrostatic pressure with the electric field tuning of the spin-orbit coupling. As known from Ref. [37], the Rashba effect in phospho- rene leads to the appearance of the in-plane spin tex- ture only, triggered by the broken inversion symmetry. Besides that, in the phosphorene-WSe 2heterostructure,5 there is an additional possibility to tune the out-of-plane spin-orbit field with the electric field. By performing a se- ries of calculations for the relaxed heterostructure using the D2 vdW correction and the electric field strengths from -1 V/nm to 1 V/nm in the steps of 0.4 V/nm, we have determined the λparameters in all the studied cases, obtained after fitting the spin-orbit Hamiltonian model (1) to the DFT data, see Fig. 4. The calculations show that the negative electric field increases all the com- ponents of the spin-orbit field, although not as efficiently as when the vertical pressure is applied. Thus, pressure represents a promising alternative to the traditional ap- proaches of modulating the spin-orbit coupling strength in proximitized materials, such as the electric field tuning utilizing the Rashba effect. B. Pressure-modulated spin relaxation time of phosphorene holes Spin relaxation times represent an experimentally mea- surable quantity that is dependent on the spin-orbit cou- pling strength. While in pristine phosphorene the up- per limit of spin relaxation times [52, 53] comes from the Elliot-Yaffet mechanism [71], the proximity-induced spin-orbit coupling in phosphorene on a WSe 2monolayer triggers the Dyakonov-Perel (DP) spin relaxation mech- anism [72]. Within the DP regime, the spin relaxation time τican be calculated using the relation τ−1 i= Ω2 ⊥,iτp, (2) where Ω2 ⊥,i=⟨Ω2⟩ − ⟨ Ω2 i⟩corresponds to the Fermi surface average of the squared spin-orbit field compo- nent Ω2 k⊥,ithat is perpendicular to the spin orientation i={x, y, z}, while τpis the momentum relaxation time. From the first-principle calculations, one can directly ex- tract the spin-orbit field Ω k,ifor the spin-split top valence band of phosphorene at the given kpoint of the BZ using the relation [73] Ωk,i=∆so ℏsi |s|, (3) in which the ∆ socorresponds to the spin splitting value, while siis equal to the expectation value of the spin one- half operator at k. Using the Fermi contour averaging formula ⟨Ω2 i⟩=1 ρ(EF)SBZZ FCΩ2 k,i ℏ|vF(k)|, (4) in which SBZrepresent the Fermi surface area, ρ(EF) corresponds to the density of states per spin at the Fermi level, vF(k) is the Fermi velocity, while the integration goes over an isoenergy contour, one can calculate Ω2 ⊥,i for different hydrostatic pressure values, and correspond- ingly, analyze the influence of the hydrostatic pressure on DP spin relaxation. Results of such calculations for thetop valence band and three spin components are shown in Fig. 5. First, we notice that the spin relaxation times in all three directions are on the same time scale, which can be rationalized by the same energy scale of the λ parameters given in Fig. 3. For weak hole doping, when crystal momenta are relatively close to the Γ point, the ratio Ω2 ⊥,i(dp= 0.5˚A)/Ω2 ⊥,i(dP= 0˚A) is roughly 4 for all spin directions. It can be connected to factor 2 increase of λparameters with pressure, since Ω2 k,x/y/z∼s2 x/y/z∼(λ2/1/3)2. In- creasing the doping to EF≈30 meV, the ratio Ω2 ⊥,i(dp= 0.5˚A)/Ω2 ⊥,i(dp= 0˚A) reduces to approximately 3. One could explain this modification with the increased influ- ence of the higher in korder corrections to the spin-orbit field when moving from the Γ point, which is not included in (1). Thus, we have established a clear correspondence between the increase in the spin-orbit coupling strength and the decrease of τi. We believe that this correspon- dence can be exploited as a simple tool for the detection of the hydrostatic pressure-induced effects on spin-orbit coupling. IV. CONCLUSIONS We studied the influence of the vertical pressure in a mixed-lattice heterostructure made of phosphorene and a WSe 2monolayer, which is a promising platform for the transfer of spin-orbit coupling from WSe 2, a strong spin-orbit coupling material, to phosphorene, via the proximity effect. We focused on the spin physics of phosphorene holes around the Γ point, for which we have extracted the parameters of the model spin-orbit Hamiltonian, compatible with the C1vsymmetry of the phosphorene-WSe 2heterostructure, for several values of the hydrostatic pressure. We showed that for experi- mentally accessible pressure values, the increase in the spin-orbit coupling strength reaches a factor of two and is much more efficient than the prevalent electric-field tuning of the spin-orbit interaction. Finally, we have cal- culated the spin relaxation times of phosphorene holes related to the DP relaxation mechanism and showed that the gradual tuning of spin-orbit coupling strength by the hydrostatic pressure transfers to the tunable decrease of spin relaxation times, offering a simple route for indi- rectly determining the hydrostatic-pressure influence on the spin-orbit field in phosphorene using the spin relax- ation times, an experimentally available quantity in het- erostructures based on phosphorene. ACKNOWLEDGMENTS M.M. acknowledges the financial support provided by the Ministry of Education, Science, and Tech- nological Development of the Republic of Serbia. This project has received funding from the Euro-6 101102 -30 -20 -10 0τx-1/τp (ps-2) EF (meV)0.0 2.59 5.57 9.02 12.59 16.50(a) 100101102 -30 -20 -10 0τy-1/τp (ps-2) EF (meV)0.0 2.59 5.57 9.02 12.59 16.50(b) 100101 -30 -20 -10 0τz-1/τp (ps-2) EF (meV)0.0 2.59 5.57 9.02 12.59 16.50(c) FIG. 5. Calculated average spin relaxation rate τ−1 iof phosphorene hole states for spin components i={x, y, z }, in units of momentum relaxation time τp(ps) versus the position of the Fermi level EF, measured from the top of the valence band, and for different values of hydrostatic pressure, given in kbar units. In all studied cases, the phosphorene-WSe 2heterostructure was relaxed using the Grimme-D2 vdW correction. pean Union’s Horizon 2020 Research and Innovation Programme under the Programme SASPRO 2 CO- FUND Marie Sklodowska-Curie grant agreement No. 945478. M.K. acknowledges financial support provided by the National Center for Research and Development (NCBR) under the V4-Japan project BGapEng V4- JAPAN/2/46/BGapEng/2022. M.R. and M.G. acknowl- edge financial support provided by the Slovak Research and Development Agency under Contract No. APVV- SK-CZ-RD-21-0114. 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2004.05945v2.Theory_of_Current_Induced_Angular_Momentum_Transfer_Dynamics_in_Spin_Orbit_Coupled_Systems.pdf

Theory of Current-Induced Angular Momentum Transfer Dynamics in Spin-Orbit Coupled Systems Dongwook Go,1, 2, 3, 4,Frank Freimuth,1Jan-Philipp Hanke,1Fei Xue,5, 6Olena Gomonay,2 Kyung-Jin Lee,7, 8Stefan Bl ¨ugel,1Paul M. Haney,5,yHyun-Woo Lee,3and Yuriy Mokrousov1, 2,z 1Peter Gr ¨unberg Institut and Institute for Advanced Simulation, Forschungszentrum J ¨ulich and JARA, 52425 J ¨ulich, Germany 2Institute of Physics, Johannes Gutenberg University Mainz, 55099 Mainz, Germany 3Department of Physics, Pohang University of Science and Technology, Pohang 37673, Korea 4Basic Science Research Institute, Pohang University of Science and Technology, Pohang 37673, Korea 5Physical Measurement Laboratory, National Institute of Standards and Technology, Gaithersburg, MD 20899, USA 6Institute for Research in Electronics and Applied Physics & Maryland Nanocenter, University of Maryland, College Park, MD 20742 7Department of Materials Science and Engineering, Korea University, Seoul 02841, Korea 8KU-KIST Graduate School of Converging Science and Technology, Korea University, Seoul 02841, Korea (Dated: May 15, 2020) Motivated by the importance of understanding various competing mechanisms to the current-induced spin- orbit torque on magnetization in complex magnets, we develop a unified theory of current-induced spin-orbital coupled dynamics in magnetic heterostructures. The theory describes angular momentum transfer between dif- ferent degrees of freedom in solids, e.g., the electron orbital and spin, the crystal lattice, and the magnetic order parameter. Based on the continuity equations for the spin and orbital angular momenta, we derive equations of motion that relate spin and orbital current fluxes and torques describing the transfer of angular momentum between different degrees of freedom, achieved in a steady state under an applied external electric field. We then propose a classification scheme for the mechanisms of the current-induced torque in magnetic bilayers. Based on our first-principles implementation within the density functional theory, we apply our formalism to two different magnetic bilayers, Fe/W(110) and Ni/W(110), which are chosen such that the orbital and spin Hall effects in W have opposite sign and the resulting spin- and orbital-mediated torques can compete with each other. We find that while the spin torque arising from the spin Hall effect of W is the dominant mechanism of the current-induced torque in Fe/W(110), the dominant mechanism in Ni/W(110) is the orbital torque originating in the orbital Hall effect of the non-magnetic substrate. It leads to negative and positive effective spin Hall angles, respectively, which can be directly identified in experiments. This clearly demonstrates that our formalism is ideal for studying the angular momentum transfer dynamics in spin-orbit coupled systems as it goes beyond the “spin current picture” by naturally incorporating the spin and orbital degrees of freedom on an equal footing. Our calculations reveal that, in addition to the spin and orbital torque, other contributions such as the interfacial torque and self-induced anomalous torque within the ferromagnet are not negligible in both material systems. I. INTRODUCTION Spin-orbit coupling plays a central role in a plethora of phenomena occurring in magnetic multilayers [1]. Current- induced spin-orbit torque is one of the most important ex- amples, and is a workhorse in the field of spintronics [2, 3]. In contrast to spin-transfer torque in spin valve structures, a device utilizing spin-orbit torque does not require an ex- tra ferromagnetic layer to create spin polarized current. In- stead, nonequilibrium spin currents and spin densities are gen- erated in nonmagnetic materials due to spin-orbit coupling. The magnitude of spin-obit torque can be sufficient to in- duce magnetic switching, as demonstrated in magnetic bilay- ers consisting of a nonmagnet and a ferromagnet [4–8]. Spin- orbit torque also enables fast current-induced magnetic do- main wall motion [9–12]. Several microscopic mechanisms of current-induced spin-orbit torque have been proposed. How- ever, quantification of the individual contributions is challeng- ing both theoretically and experimentally. Moreover, our un- derstanding of the phenomenon based on the properties of the electronic structure is rather unsatisfactory yet. d.go@fz-juelich.de ypaul.haney@nist.gov zy.mokrousov@fz-juelich.deIn this work, we examine the fundamental physical na- ture of spin-orbit torque in the view of angular momentum exchange between different degrees of freedom in solids. The possible channels for angular momentum transfer among these degrees of freedom are schematically shown in Fig. 1. It is conceptually important to separate (i) angular momen- tum carried by a conduction electron angular momentum en- coded in its orbital and spin parts of the wave function, (ii) mechanical angular momentum of the lattice, and (iii) spin angular momentum encoded into the local magnetic moment emerging as a result of magnetic ordering. These degrees of freedom interact with each other and exchange angular mo- mentum. For example, spin-orbit coupling mediates an an- gular momentum transfer between spin and orbital degrees of the electron, crystal field potential leads to an orbital angu- lar momentum transfer between the electron and the lattice, and exchange interaction enables spin transfer between the conduction electron’s spin and local magnetic moment. In its most profound definition, the spin-orbit torque is understood as an angular momentum flow from the surrounding lattice harvested by the local magnetic moment a process which is mediated by spin-orbit entangled electrons. Here, taking this fundamental viewpoint as the foundation, we provide a unified and complete picture of possible scenarios on current-induced torque on the magnetization. Depending on the specifics of a particular angular momen-arXiv:2004.05945v2 [cond-mat.mes-hall] 13 May 20202 FIG. 1. Interactions between angular-momentum-carrying degrees of freedom in solids: spin and orbital of the electron, the crystal lattice, and the local magnetic moment. Orange arrows indicate microscopic interactions by which angular momentum is exchanged: the spin-orbit coupling for interaction between the spin and orbital momenta of an electron, crystal field potential for the interaction between the lattice and the orbital angular momentum of the electron, and exchange interaction for the interaction between the local magnetic moment and the spin of the electron. The orbital and lattice in the left column can have both electric and magnetic excitations, while the spin and local magnetic moment in the right column have only magnetic excitations. This property marks the orbital degree of freedom as an essential element in describing magnetoelectric responses, such as the current-induced torque. tum exchange transfer channel, which takes place in different parts of the solid e.g. in the bulk or at the interface, we can un- derstand various competing mechanisms in non-uniform mag- netic heterostructures in an unified manner. Here, we choose to consider a bilayer geometry comprising a nonmagnet ad- jecent to a ferromagnet which is most widely studied in experiments. Within our viewpoint, we classify the mecha- nisms of the current-induced torque into four different scenar- ios, which are schematically illustrated in Fig. 2. The clas- sification is based on two independent criteria: (1) the spatial origin of the spin-orbit interaction, and (2) the spatial origin of the current responsible for the angular momentum generation, which is absorbed by the magnetization. This classification is discussed in detail below, and demonstration of its relevance and completeness is the main goal of this work. In magnetic bilayers consisting of a nonmagnet and a fer- romagnet, the spin Hall effect arising from the nonmagnet is considered to be one of the main mechanisms for generating a torque on the magnetization of the ferromagnet [5, 6]. That is, an electrical current in the nonmagnet induces a transverse spin current, which is injected into the ferromagnet and re- sults in a torque (upper left panel in Fig. 2). In this picture, the spin Hall conductivity of the nonmagnet is assumed to be a bulk property, and the spin injection and resulting torque generation on the local magnetic moment is explained by thetheory of the spin-transfer torque [13, 14]. We denote such contribution due to spin injection from the nonmagnet as a spin torque. This analysis considers the spin-orbit coupling only in the nonmagnet and neglects the spin-orbit coupling at the nonmagnet/ferromagnet interface and in the ferromag- net. Moreover, current-induced effects from the ferromag- net are neglected. The spin-orbit coupling effect at the non- magnet/ferromagnet interface has been considered to be an- other dominant mechanism and intensively investigated [15– 23]. Since the Rashba-type interfacial states are formed at the nonmagnet/ferromagnet interface due to the broken inversion symmetry [24–26], scattering of electrons from the interface leads to finite spin density and current [22, 23], which inter- acts with and exerts a torque on the local magnetic moments of the ferromagnet (upper right panel in Fig. 2). We denote this contribution as interfacial torque. While the role of spin-orbit coupling in the ferromagnet has been considered to be negligible as compared to that of the spin-orbit coupling in the nonmagnet, which usually com- prises heavy atomic species, it has been found that spin-orbit coupling in the ferromagnet can induce a sizable amount of self-induced torque by the generation of the intrinsic spin cur- rent, e.g., via the spin Hall effect [27–29]. The corresponding torque contribution is called the anomalous torque in analogy to the anomalous Hall effect in the ferromagnet [28]. When3 FIG. 2. Classification of the mechanisms of the current-induced torque. The row represents the origin of spin-orbit coupling in either the nonmagnet or in the ferromagnet. The column represents the locality of the torque: i.e., whether the torque acting on the ferromagnet originates from the electrical current flowing in the nonmagnet (nonlocal) or in the ferromagnet itself (local). The red arrows represent the spin, and the blue arrows represent the orbital angular momentum. The local magnetic moment is represented with a big yellow arrow. inversion symmetry is present in a stand-alone ferromagnet, the net anomalous torque amounts to zero. However, in the nonmagnet/ferromagnet bilayer, where the inversion symme- try is broken at the interface, the anomalous torque may exert a finite torque (lower right panel in Fig. 2), comparable to the spin torque and interfacial torque. The above mechanisms (spin torque, interfacial torque, and anomalous torque) arise from spin-dependent scattering in the bulk or at the interface, and rely on the concept of spin current or spin density. Recently, a mechanism of the torque generation based on the orbital angular momentum injection has been pro- posed [30]. This mechanism is fundamentally different from the other mechanisms in that it requires the consideration of the orbital part of the electron’s angular momentum, rather than its spin. Called the orbital torque, it relies on two pro- cesses as described in lower left panel of Fig. 2. First, the orbital angular momentum or its current is generated, which can be achieved for instance by the orbital Hall effect [31– 34]. Second, the orbital angular momentum is injected into the ferromagnet and transfers its angular momentum to the local magnetic moment. In this process, the injected orbital angular momentum should couple to the spin of the conduction elec- tron, which interacts with the local magnetic moment via the exchange interaction. Thus, it requires the spin-orbit coupling within the ferromagnet. Since the orbital Hall conductivity can be truly gigantic, exceeding that of the spin Hall conduc-tivity of heavy elements [31, 32] by an order of magnitude, the orbital torque contribution to the current-induced torque can be substantial (note that in the remainder of the paper, we use the terms current-induced spin-orbit torque and current- induced torque interchangeably). Moreover, since the orbital Hall effect does not require the spin-orbit coupling, which is in contrast to the spin Hall effect, the orbital Hall conductivity is gigantic even in light elements [33, 34]. In nonmagnet/ferromagnet bilayers, the orbital Hall effect and spin Hall effect coexist in the nonmagnet, especially when the nonmagnet consists of heavy elements. Thus, depend- ing on the material combinations, the orbital torque and spin torque may add up or cancel each other [30]. To enhance the torque efficiency of the device, it is favorable to have the same sign of the orbital torque and spin torque. On the other hand, the case when the sign of the orbital torque and spin torque are opposite is of interest as well, because when the magnitude of the orbital torque is larger than that of the spin torque, the sign of the measured effective spin Hall angle in the nonmag- net/ferromagnet bilayer will be opposite to the sign promoted by the spin Hall conductivity of the nonmagnet. Considering that the sign is a more robust quantity than the magnitude in torque measurements, such a sign change can serve as the first hint of an active orbital torque mechanism. It turns out that all of the above mentioned mechanisms (spin torque, interfacial torque, anomalous torque, and orbital4 torque) contribute to both fieldlike torque and and damping- like torque, often with comparable magnitudes. The former gives rise to a precessional motion of the magnetization with respect to the spin accumulation direction, and the latter leads the magnetization to point away/toward an effective field di- rection. This complicates the analysis of the experiments. Since previous theoretical models have been developed as- suming a restricted setup and evaluated only specific contri- butions [22, 27], e.g., when the spin-orbit coupling exists only at the interface, it is hard to compare magnitudes of different contributions directly. On the other hand, first-principles ap- proaches often evaluate the total torque from linear response theory [35–40], which makes it difficult to assess contribu- tions by different mechanisms quantitatively. Thus, it is necessary to develop a unified theory within which different mechanisms of the current-induced torque are classified and can be separately evaluated for a given system. This would bridge the gap between the theoretical pictures set up by models and first-principles calculations of real materi- als. The main difficulty here lies in the nonlocality of magne- toelectric coupling [41, 42] and different sources of the spin- orbit coupling. The orbital torque mechanism [30] is highly nonlocal in nature, with the orbital current converted into the spin current in the ferromagnet. In view of the existing anal- ysis based on the spin current, the orbital torque mechanism appears abnormal as the spin current seems to emerge out of nowhere, while in fact it originates in the orbital current. This implies that tracing only the spin current inevitably fails to describe the orbital torque. In general, the spin is not con- served in the presence of spin-orbit coupling, and the spin current does not directly correspond to the spin accumulation or torque on the local magnetic moment [43]. However, it is important to realize that the angular momentum of the spin is not simply lost. Instead, it is transferred to other degrees of freedom. Therefore, in our theory, we track not only the flow of spin but also the flow of orbital angular momentum, as well as their interactions with other degrees of freedom in solids, such as crystal lattice and local magnetic moment. Detailed analysis of the transfer of angular momentum between these channels provides a long-sought insight into the microscopic nature of different competing mechanisms of current-induced torque. Recent theories imply that the current-induced dynamics and transport of the spin in the presence of spin-orbit cou- pling originate in the orbital degrees of freedom [32, 33]. For example, while the orbital Hall effect occurs regardless of the spin-orbit coupling, the spin Hall effect is a consequence of the orbital Hall effect by virtue of the spin-orbit coupling [33]. Depending on the correlation (or relative orientation) between the spin and orbital angular momentum, the relative sign of the orbital Hall effect and spin Hall effect may be the same or opposite, following Hund’s rule behavior [32, 33]. In this sense, the orbital Hall effect can be considered as a precur- sor to the spin Hall effect. Another example is a Rashba-type state, which is responsible for the interfacial torque genera- tion. It is well known that the Rashba state originates in a chiral orbital angular momentum texture [44–46]. Such an orbital Rashba effect persists even in the absence of spin-orbit coupling, which induces current-induced orbital dynam- ics and transport [47, 48]. Through spin-orbit coupling, the orbital Rashba state couples to the spin and the spin texture emerges, thus leading to spin dynamics. In general, such a hierarchy is expected to be a rather universal feature. The reason is the following: in the microscopic Hamiltonian of the electrons in solids, the spin cannot interact with an exter- nal electric field unless the spin-orbit coupling is present. On the other hand, the orbital degree of freedom, originated in the real-space behavior of the wave functions and distribution of charge, directly couples to an external electric field (see Fig. 1). Hence, under the perturbation by an external elec- tric field, the orbital dynamics is expected to occur prior to the spin dynamics and regardless of the spin-orbit coupling, and the spin dynamics becomes correlated with the orbital dy- namics due to the spin-orbit coupling. Therefore, the orbital degree of freedom should be explicitly incorporated into a the- oretical formulation to properly describe the current-induced torque, or magnetoelectric coupling phenomena in general. This will help to achieve clarity in resolving various contri- butions to the current-induced torques. In this paper, we develop a theoretical formalism that can track the flow and transfer of the angular momentum between spin and orbital degrees of freedom of electrons, the crys- tal lattice, and the local magnetic moment in the presence of an external electric field. Following the continuity equations for the spin and orbital angular momentum of the electron, which was outlined in Ref. [49], we clarify every channel for the angular momentum transfer: between spin-orbital, orbital- lattice, and spin-local magnetic moment. Then we derive equations of motion which hold in the steady state in the pres- ence of an external electric field. For the angular momentum transfer between electron’s spin and local magnetic moment, which is directly related to the current-induced torque, we propose criteria for classifying different microscopic mech- anisms based on physical properties: whether the magneto- electric coupling is of local or nonlocal nature and whether it originates in the atomic spin-orbit coupling of the nonmagnet or the ferromagnet. In this way, we classify the mechanism of the current-induced torque as spin torque, orbital torque, inter- facial torque, and anomalous torque, and separately evaluate them for a given system. As a proof of principle, we implement our formalism in the density functional theory framework, and perform first-principles calculations for two real material systems: Fe/W(110) and Ni/W(110), which are carefully chosen with the expectation that the spin torque and orbital torque have an opposite sign in these bilayers. We show that the current- induced torque in Fe/W(110) is dominated by the spin torque contribution, that is, the spin current flux in Fe equals the torque acting on the local magnetic moment. As a result, the effective spin Hall angle is negative, as it is well known for W. On the other hand, we find that the orbital torque is dom- inant over the spin torque in Ni/W(110). As a result, it leads to a positive sign of the effective spin Hall angle, which is opposite to the sign of the spin Hall conductivity in W. This peculiar result is due to a positive sign of the orbital Hall con- ductivity in W. In Ni/W(110) it is found that angular momen-5 tum transfer from the orbital to the spin channel is pronounced in the ferromagnet, which is a crucial requirement for the or- bital torque mechanism. We attribute the different behavior of Fe/W(110) and Ni/W(110) to the difference in the electronic structure, where the correlation between the spin and orbital angular momenta in the ferromagnet is more pronounced in Ni/W(110) than Fe/W(110) near the Fermi energy. In addi- tion, we find that the interfacial torque and anomalous torque are not negligible in both Fe/W(110) and Ni/W(110). These results clearly demonstrate the advantages of our theoretical formalism tracking the flow and transfer of the angular mo- mentum through various degrees of freedom. Moreover, a different sign of the effective spin Hall angle in two different systems can be readily measured in experiment. The paper is organized as follows. In Sec. II, we develop a theoretical formalism that describes angular momentum trans- fer between the spin and the orbital angular momentum of the electron, lattice, and local magnetic moment in the steady state under an external electric field. We propose a classifica- tion scheme for the mechanisms of current-induced torque and provide definitions of spin torque, orbital torque, interfacial torque, and anomalous torque. In Sec. III, we apply this for- malism to perform a first principles study of current induced torques in Fe/W(110) and Ni/W(110) bilayers. In Sec. IV, we further discuss the disentangling of the various mechanisms of current-induced torque and comment on several issues of orbital transport and dynamics. This includes similarity and difference between the orbital current and spin current, and implications on experiments. Finally, Sec. V summarizes and concludes the paper. II. THEORETICAL FORMALISM A. Overview In this section, we develop a theoretical formalism that de- scribes angular momentum transfer between different degrees of freedom to identify competing mechanisms of the current- induced torque separately. Before presenting detailed equa- tions, we provide a motivation and an overview of the for- malism that we aim to derive. Figure 1 shows interactions between spin and orbital momenta of the electron, lattice, and local magnetic moment, each of which carry angular momen- tum in solids. Considering microscopic interactions, the elec- tron’s spin interacts with the local magnetic moment via the exchange interaction, the electron’s orbital moment interacts with the lattice via the crystal field potential, and the electron’s spin and orbital momenta are coupled by the spin-orbit cou- pling. It is important to note that the local magnetic moment and the electron’s spin on the right column of Fig. 1 are re- lated to magnetic excitations, i.e., in the absence of spin-orbit coupling they do not respond to an electric field. On the other hand, the electron’s orbital and the crystal lattice, in the left column of Fig. 1, react to an application of an external elec- tric field, and their orbital dynamics couples to a magnetic field. Therefore, the electronic orbital degree of freedom is a core element in describing magnetoelectric coupling, e.g.,the current-induced torque. Note that a charge excitation of the ions in the lattice is efficiently screened by the electrons in metals, which are our main interest in this paper. More- over, we assume that the lattice degrees of freedom are frozen (absence of a phonon excitation) and we neglect a coupling between the ions and an external electric field. Therefore, according to this physical picture, the current-induced torque arises as follows: An external electric field excites the orbital dynamics, with which the spin dynamics is entangled by the spin-orbit coupling. The resulting spin dynamics alters the local magnetic moment by the exchange interaction. An exception to this picture is a noncollinear magnet, where the orbital angular momentum is associated with the scalar spin chirality [50, 51] or Skyrmion charge [52, 53]. Here, spin and orbital momenta may interact even without relativis- tic spin-orbit coupling [54]. Although such topological orbital angular momentum exhibits exotic dynamic phenomena asso- ciated with complex spin structures [55], we leave this case to future work. In the rest of this section, we first define different mecha- nisms of the current-induced torque in Sec. II B, which we aim to disentangle for a given magnetic bilayer system. To achieve this, we start from the effective single-particle Hamil- tonian to separately define the spin-orbit coupling, the crystal field potential, and the exchange interaction, which is adapted for the density functional theory framework (Sec. II C). Then we derive the continuity equations for the spin and orbital an- gular momentum in Sec. II D. In the continuity equations, rates for the changes of the spin and orbital angular mo- mentum are captured by the influxes of the spin and orbital angular momentum as well as torques describing the angu- lar momentum transfer between different degrees of freedom. To evaluate individual contributions appearing in the conti- nuity equations under an external electric field, we consider interband and intraband contributions within the Kubo for- mula (Sec. II E). However, we point out that the interband contribution does not satisfy the stationary condition in the steady state (Sec. II F). To resolve this problem, we propose a balance-type equation that describe a relation between the in- terband and intraband contributions in the steady state, which we call the interband-intraband correspondence. The appli- cation of the interband-intraband correspondence to the con- tinuity equations of the spin and orbital angular momentum leads to the equations of motion (Sec. II G), which is the main result of this section. Meanwhile, the intraband contribution satisfies the stationary condition by itself, for which we derive the equations of motion as well. B. Classifying Mechanisms of the Current-Induced Torque We aim to identify and disentangle various competing mechanisms for the current-induced torque with our formal- ism. Before presenting the detailed formalism, we define vari- ous mechanisms of the current-induced torque more precisely. We consider two independent criteria: (1) whether it is an ef- fect due to spin-orbit coupling in the nonmagnet or the ferro- magnet, and (2) whether it is due to electrical current flowing6 in the nonmagnet or in the ferromagnet. Figure 2 presents a table of the mechanisms of the current-induced torque, where the row classifies whether the spin-orbit coupling originates in the nonmagnet or in the ferromagnet, and the column clas- sifies whether the nature of the torque response is nonlocal or local. We define the nonlocal and local nature of the torque as the response arising in the ferromagnet from the electri- cal current flowing in the nonmagnet and the torque arising in the ferromagnet from the electrical current flowing in the non- magnet, respectively. Thus, we classify microscopic mecha- nisms of the current-induced torque as follows: Spin torque (nonlocal, spin-orbit coupling from non- magnet): Electric current flowing in the nonmagnet generates a transverse spin current via the spin Hall ef- fect. The spin current is injected to the ferromagnet and transferred to the local magnetic moment. Orbital torque (nonlocal, spin-orbit coupling from fer- romagnet): Electric current flowing in the nonmagnet generates a transverse orbital current via the orbital Hall effect. The orbital current is injected into the fer- romagnet and couples the spin in the ferromagnet via spin-orbit coupling. The converted spin or spin current generate a torque on the local magnetic moment. Interfacial torque (local, spin-orbit coupling from non- magnet): Electric current flowing in the ferromagnet scatters from the nonmagnet/ferromagnet interface. By the spin-orbit coupling of the nonmagnet, the interfa- cial scattering may alter the direction of the spin, i.e., by spin-orbit filtering or spin-orbit precession [23]. The reflected spin exerts torque on the local magnetic mo- ment. Anomalous torque (local, spin-orbit coupling from fer- romagnet): Electric current flowing in the ferromagnet induces transverse spin current via the spin Hall effect. As the inversion symmetry is broken by the nonmag- net/ferromagnet interface, spin accumulation at the top and at the bottom of the ferromagnet become asymmet- ric, leading to a finite torque on the local magnetic mo- ment. We remark that the our definition of the interfacial torque is restricted rather than general. For example, our definition ne- glect an effect of the current flowing in the nonmagnet in the proximity of the interface. The spin Hall or orbital Hall cur- rent in the nonmagnet may be enhanced near the interface, but we include this effect into the definition of the spin torque or the orbital torque, respectively. Thus, the definition of the in- terfacial torque agrees with the picture that spin-orbit effects in the ferromagnet originate in the proximity-induced spin- orbit coupling from the nonmagnet. Meanwhile, we empha- size that not only the orbital torque but also all the other mech- anisms involve an excitation of the orbital angular momentum or its current, because electric response of the spin follows the orbital response via spin-orbit coupling.C. Effective Single-Particle Hamiltonian Within the effective single-particle description, such as the Kohn-Sham treatment within the density functional theory, the general electronic Hamiltonian in a solid is formally writ- ten as H=Z d3r y(r)p2 2m+Ve(r) (r); (1) where (r)and y(r)are electron annihilation and creation field operators in the second quantization representation, re- spectively. Here, p=ihrris the momentum operator, his the reduced Plank constant, and mis the electron mass. The effective single-particle potential Ve(r)can be divided into the spin-orbit coupling VSO(r), the exchange interaction VXC(r), and the crystal field potential VCF(r): Ve(r) =VSO(r) +VXC(r) +VCF(r): (2) We defineVCF(r)such that it is independent of the spin. The spin-orbit coupling and exchange interaction are explic- itly written as VSO(r) =
rrVCF(r)p; (3) VXC(r) =B XC(r)
; (4) respectively. Here, is the vector of the Pauli matrices repre- senting the spin, = h=4m2c2with the speed of light c,Bis the Bohr magneton, and XC(r)is an effective magnetic field caused by the exchange interaction. We construct VSO(r)by neglectingVXC(r)as an approximation. Note that the degrees of freedom of the lattice and the local magnetic moment are implicitly included in this description, entering as coordinates in the respective potentials VXC(r)andVCF(r). In the eval- uation of operators we use symmetrized representations such that the hermiticity is kept in the numerical implementation. However, we present non-symmetrized forms throughout the paper for notational brevity. D. Continuity Equations for Spin and Orbital Angular Momenta The continuity equations for spin and orbital angular mo- mentum have been introduced by Haney and Stiles in Ref. [49]. Here, we derive the expression adapted for the first- principles calculation based on the density functional theory, starting from the general single particle Hamiltonian [Eqs. (1) and (2)]. In the Heisenberg picture (indicated by the hat sym- bol below), we define the orbital angular momentum and spin density operators as ^l(r;t) =^ y(r;t)L^ (r;t); (5a) ^s(r;t) =^ y(r;t)S^ (r;t): (5b) While the spin Sis represented by the vector of the Pauli ma- trices S= (h=2), evaluation of the orbital angular momen- tum is nontrivial in periodic solids because the position ris ill-defined under periodic boundary conditions. Nonetheless,7 we can calculate the orbital angular momentum with respect to the atomic spheres called muffin tins centered at the posi- tions of the atoms: L=X L; (6a) L= (Rr) (rp): (6b) Here, (x)is the Heaviside step function, is the index of an atom in the unit cell whose center is located at , r=ris the displacement from the atom center, and Ris the radius of the muffin tin. This method is called atom-centered approximation, and it gives a reliable result when orbital currents are associated with partially occupied dorfshells, which are localized around atomic centers. Thus, the usage of the atom-centered approximation is jus- tified in magnetic bilayers consisting of transition metal ele- ments, Fe/W(110) and Ni/W(110), which are in the focus of our study. Under the atom-centered approximation, the size of the region in real space which gives rise to the orbital angular moment is smaller than that of a wave packet, thus the orbital can be treated as an internal degree of freedom, similar to the spin (see Sec. IV B for the discussion). However, the atom- centered approximation neglects contributions from nonlocal currents, e.g., in Chern insulators and noncollinear magnets [56], and ultimately one should resort to the modern theory of orbital magnetization [57–59]. For the orbital angular momentum and spin densities de- fined in Eq. (5), we can derive continuity equations from the Heisenberg equations of motion. These are formally written as @^l(r;t) @t=1 ihh ^l(r;t);^H(t)i =rr
^QL(r;t) +^TL(r;t); (7a) @^s(r;t) @t=1 ihh ^s(r;t);^H(t)i =rr
^QS(r;t) +^TS(r;t); (7b) where=x;y;z . Here, ^QL(r;t) =1 2^ y(r;t)fL;vg^ (r;t); (8a) ^QS(r;t) =1 2^ y(r;t)fS;vg^ (r;t); (8b) are orbital and spin current operators, respectively, where v=ih 2m rL rrR r +rrVCF(r) (9) is the velocity operator ( rL randrR ract on the left and on the right, respectively), and ^TL(r;t) =1 ih^ y(r;t)[L;Ve(r)]^ (r;t); (10a) ^TS(r;t) =1 ih^ y(r;t)[S;Ve(r)]^ (r;t) (10b) are torque operators for the orbital angular momentum and spin, respectively.The appearance of the torques in Eq. (7) signals the fact that the orbital angular momentum and spin are not conserved. This implies that the angular momentum is transferred from the electron to other degrees of freedom as described in Fig. 1. The electrons exchange orbital angular momentum with the lattice and with the electron’s spin via the crystal field po- tentialVCF(r)and spin-orbit potential VSO(r), respectively. Thus, the torque acting on the orbital angular momentum of the electron is decomposed as ^TL(r;t) =^TL CF(r;t) +^TL SO(r;t); (11) where ^TL CF(r;t) =1 ih^ y(r;t)[L;VCF(r) +VXC(r)]^ (r;t);(12) ^TL SO(r;t) =1 ih^ y(r;t)[L;VSO(r)]^ (r;t): (13) We denote ^TL CF(r;t)as the crystal field torque and^TL SO(r;t) as the spin-orbital torque . Note that we included the effect ofVXC(r)in the definition of the crystal field torque, as it contains non-spherical component in general. On the other hand, the electron exchanges the spin angular momentum with the local magnetic moment and the electron’s orbital angular momentum via VXC(r)andVSO(r), respectively. Thus, the torque acting on the electron’s spin can be decomposed as ^TS(r;t) =^TS XC(r;t) +^TS SO(r;t); (14) where ^TS XC(r;t) =1 ih^ y(r;t)[S;VXC(r)]^ (r;t); (15) ^TS SO(r;t) =1 ih^ y(r;t)[S;VSO(r)]^ (r;t): (16) We denote ^TS XC(r;t)as the exchange torque and^TS SO(r;t)as thespin-orbital torque . Note that ^TL SO(r;t)and^TS SO(r;t)dif- fer, and we specify them as the spin-orbital torques acting on the orbital and spin, respectively. We have a few remarks on the different torques and their definitions. In the absence of the spin-orbit coupling, the spin-orbital torques vanish. Thus in a steady state, where h@^s(r;t)=@ti= 0, Eq. (7b) becomes hTS XC(r)i=rr
hQS(r)i. Here,h


i represents expectation value in the steady state. This implies that the spin current divergence is absorbed by the local magnetic moment. Thus, this corre- sponds to the spin-transfer torque in the absence of the spin- orbit coupling. If we consider the opposite situation where the spin current flux is absent, occurring e.g. in atomically thin magnetic films, where the spin current effect can be ne- glected along the perpendicular direction to the film plane, Eq. (7b) becomeshTS XC(r)i=hTS SO(r)i. Thus, the exchange torque amounts to the spin-orbital torque. This is related to the widely used terminology, spin-orbit torque [15]. How- ever, in our terminology, the net torque acting on the local magnetic moment is the exchange torque, which may differ from the spin-orbital torque due to the presence of the spin current flux. In general, both the spin current flux and spin- orbital torque contribute to the exchange torque.8 We obtain additional insight from explicitly evaluating the torques in a simplified situation. Let us first consider the ex- change torque. By using Eqs. (4) and (15), the exchange torque can be written as ^TS XC(r;t) =B^ y(r;t) [ XC(r)]^ (r;t)(17) in general. Thus, it describes a precession of the spin with respect to the direction of the exchange field. On the other hand, by using Eqs. (3) and (16), the spin-orbital torque acting on the spin is formally written as ^TS SO(r;t) =^ y(r;t) [frrVCF(r)pg]^ (r;t): (18) Since it depends on the spatial gradient of VCF(r), the dominant contribution to it is concentrated near the atom centers, where VCF(r)is almost spherical. Thus, within the muffin tins, we can approximately write rrVCF(r)P  (Rr) [@VCF(r)=@r]. Within this approxima- tion VSO(r)X ^ y(r;t) [(r)L
]^ (r;t):(19) Thus, the spin-orbital torque becomes ^TS SO(r;t)X (r) (L); (20) where (r) = rdVCF(r) dr(21) is the strength of the spin-orbit coupling for the -th atom. Therefore, Eq. (20) indicates that the spin-orbital torque de- scribes a mutual precession between the orbital angular mo- mentum and the spin. That is, ^TS SO(r;t)^TL SO(r;t): (22) While it is approximately true in most systems, we keep super- scripts SandLseparately, because ^TS SO(r;t)and^TL SO(r;t) differ in general due to nonspherical contributions to the VSO(r)although it is small. Meanwhile, the crystal field torque cannot be expressed in simple terms. In general, it describes an angular momentum transfer between the lattice and the electronic orbital angular momentum. It originates due to the breaking of the continuous rotation symmetry by the crystal field, which differentiates specific directions depending on the structure of the crystal, and leads to various anisotropic effects. E. Kubo Formula: Interband and Intraband Responses The current-induced torque corresponds to the response of the exchange torque to an electric field, [Eqs. (15) and (17)]. One of the most widely used approaches for its calculation is the linear response theory, where often interband and in- traband contributions are evaluated separately. The interbandcontribution originates in the change of a given state by a co- herent superposition of the eigenstates for a given k: in re- sponse to an external electric field E=Ex^xthe periodic part of the Bloch statejunkichanges as junki!junki+junki; (23) where junki=iheExX m6=njumkihumkjvx(k)junki (EnkEmk+i)2:(24) Here,e > 0is the absolute value of the charge of the elec- tron,kis the crystal momentum, Enkis the energy eigen- value for the periodic part of the n-th Bloch statejunki. The infinitesimally small number  > 0arises from the causal- ity relation. That is, in describing time-evolution of the state, the electric field is adiabatically turned on from t=1 to t= 0 by the vector potential A(t) =tet=hEx^xsuch that E=@A(t)=@t. As a result, the interband response of an observableOis given by hOiinter= 2X nkfnkRe [hunkjO(k)junki];(25) wherefnkis the Fermi-Dirac distribution function for the statejunki. By combining Eqs. (24) and (25) and manipu- lating the dummy indices nandm, we arrive at hOiinter=ehExX n6=mX k(fnkfmk) (26) ImhunkjO(k)jumkihumkjvx(k)junki (EnkEmk+i)2 : Here, we defineO(k) =eik
rOeik
rink-space. The inter- band contribution in Eq. (26) is also known as the intrinsic contribution since it depends only on the electronic structure, the eigenstates and their energy eigenvalues in the ground state. On the other hand, the intraband response arises due to a shift of the Fermi surface by disorder scattering. The leading contribution arises from the change of the occupation func- tion: hOiintra=X nk(fnk+
kfnk)hunkjO(k)junki; (27) which is also referred to as Boltzmann-like contribution . Here,
kx=eEx=his the shift of the Fermi surface caused by the electric field E=Ex^x, andis the momentum relaxation time. Up to linear order in
k, fnk+
kfnkh
kf0 nkhunkjvx(k)junki;(28) wheref0 nk=@fnk=@Enk. Thus, the intraband contribution is written as hOiintra=eExX nkf0 nkhunkjO(k)junki hunkjvx(k)junki:(29)9 Note that it is described by a single phenomenological param- eter, which is assumed to be state-independent. As in- creases, i.e., as the resistivity decreases, the intraband contri- bution linearly increases. In general, the momentum relax- ation time depends on the particular state in the electronic structure. In ferromagnets, for example, it is known that the momentum relaxation times of the majority and minority electrons are different, which plays an important role in un- derstanding various magnetotransport effects [60]. However, within the approach that we pursue here, as given by Eq. (29), we do not consider these effects. F. Stationary Condition in the Steady State A serious problem of the linear response described by Eqs. (26) and (29) is that the stationary condition is not satisfied. That is, dO dtintra +dO dtinter 6= 0; (30) wheredO=dt = [O;H]=ih. Thus, the continuity equa- tions (7) are not satisfied if one naively evaluates the sum of the interband and intraband contributions. This discrepancy is due to the inconsistent treatment of disorder scattering, which is only taken into account by the Fermi surface shift within the relaxation time approximation. In general, the effect of disorder scattering enters the equation via the self-energy cor- rection and vertex correction. It is known that a consistent treatment of the self-energy and vertex corrections up to the same order as the perturbation (which is a disorder potential in this case) makes the continuity equation satisfied. This is known as the Ward identity [61]. However, such treatment is computationally demanding, and it requires us to assume a specific model of the disorder potential. Instead, we propose a remedy by finding a nontrivial rela- tion between the interband and intraband contributions. This allows us to evaluate the response functions given by Eqs. (26) and (29) and retain the stationary condition. We find that the following relation holds: 1 hOiintra=dO dtinter (31) as long as the operator O(k)does not have k-dependence.The proof is presented in Appendix A. A physical interpre- tation of Eq. (31) is the following. The right hand side of the equation describes intrinsic pumping ofO, which de- pends only on the electronic structure. The left hand side of the equation is related to a relaxation process, which tend to suppress deviations from the equilibrium value of O. In the steady state, the intrinsic pumping and the relaxation rates are equal, thushOiintrais determined by the relaxation rate . Therefore, Eq. (31) describes a balance between a tendency to increaseOby the intrinsic process and a relaxation rate by the extrinsic process. For the spin operator, Eq. (31) holds precisely since it does not have k-dependence. On the other hand, the orbital angular momentum operator [Eq. (6)] de- pends on ksince it contains momentum operator p, which turns intoeik
rpeik
r=p+ hkink-space representation. However, the k-dependence of the local orbital momentum is usually very small within the atom-centered approximation as it is usually dominated by a k-independent contribution, i.e., L(k)L(0). In Secs. III D and III E, we verify that Eq. (31) is satisfied for the orbital angular momentum with high pre- cision, which implies that k= 0 contribution in L(k)dom- inates and determines overall behavior of the orbital angular momentum operator within the atom-centered approximation. Meanwhile, the intraband contribution alone satisfies the steady state condition: dO dtintra = 0: (32) A proof of the stationary condition for the intraband contribu- tion is given in Appendix B. Note that for the intraband contri- bution, the stationary condition does not rely on k-dependence ofO(k), which is in contrast to the interband-intraband cor- respondence [Eq. (31)]. Equations (31) and (32) are used to derive the equations of motion below. G. Steady State Equations of Motion for Spin and Orbital Angular Momenta By applying the interband-intraband correspondence [Eq. (31)] to the continuity equations [Eq. (7)], we arrive at the following equations: 1 hl(r)iintra=rr
QL(r)inter+D TL CF(r)Einter +D TL SO(r)Einter ; (33a) 1 hs(r)iintra=rr
QS(r)inter+D TS XC(r)Einter +D TS SO(r)Einter : (33b) Note that that the time dependence no longer appears since the equations describe the steady state. Also, the hat symbol for the Heisenberg picture is removed. Equation (33) relates thecurrent fluxes and torques of the intrinsic origin to the intra- band accumulation of the orbital angular momentum and spin. Application of Eq. (32) leads to constraints between intraband10 contributions for the current fluxes and torques of the orbital angular momentum and the spin: rr
QL(r)intra+D TL CF(r)Eintra +D TL SO(r)Eintra = 0; (34a) rr
QS(r)intra+D TS XC(r)Eintra +D TS SO(r)Eintra = 0: (34b) The above equations constitute equations of motion for the spin and orbital angular momenta, which are coupled by the spin-orbit coupling, in the steady state reached after an exter- nal electric field has been applied. This is one of the main results of our work. Previous theories on the current-induced torque have focused on evaluating linear response of the ex- change torque [Eq. (15)] [35–39, 62, 63]. In contrast, Eqs. (33) and (34) enable one to identify individual microscopic mechanisms responsible for current-induced torque, as we il- lustrate next. III. FIRST-PRINCIPLES CALCULATIONS In this section we apply the formalism presented in the pre- vious section to two specific systems: W/Fe and W/Ni bilay- ers. Before presenting an in-depth analysis of these systems based on the formalism presented in the previous section, it is useful to begin with an overview of the systems’ behav- ior. The angular momentum flows that we calculate for the two systems are illustrated schematically in Fig. 3. For the W/Fe system, the flux of orbital angular momentum into the ferromagnetic layer is mostly transferred to a torque on the lattice, while the flux of spin angular momentum is mostly transferred to a torque on the magnetization. This behavior is emblematic of the conventional spin Hall effect combined with spin transfer picture of spin-orbit torque in bilayer sys- tems. The W/Ni system exhibits qualitatively different behav- ior: the orbital angular momentum flux entering the ferromag- netic layer contributes substantially to the torque on the mag- netization, indeed a magnitude which exceeds the contribution from the spin current flux. In this case, the more prominent spin-orbit coupling in Ni enables a flow of angular momen- tum from orbital to spin degrees of freedom. The distinction between W/Fe and W/Ni is evident by a different sign of the current-induced torque on the magnetization in the two sys- tems (equivalently, a different sign of the effective spin Hall effect). In the following sections we begin with a description of the key differences in the electronic structure of the two sys- tems which underlie the difference in their magnetic response. We then briefly discuss the symmetry constraints on the sys- tem, and finally present an in-depth analysis of the terms en- tering the conservation of angular momentum in Eq. (33). FIG. 3. Schematics of the angular momentum flow in (a) W/Fe and (b) W/Ni. We note that (a) in W/Fe a torque on the magnetization is mostly coming from the spin current influx. (b) On the other hand, in W/Ni, there is a significant contribution of the spin-orbital torque to the magnetization torque. A. Motivation for Choice of Material Systems One of the main motivations in choosing a material sys- tem is to find a system with dominant orbital torque behav- ior, which has been elusive since the first theoretical predic- tion [30], and compare with a conventional system where the spin torque is dominant. To do this, consider a case in which the signs of the orbital torque and spin torque are opposite. The sign of the net torque acting on the local magnetic mo- ment will vary depending on whether the orbital torque is larger than the spin torque, or vice versa. This implies that when the orbital torque is dominant over the spin torque, the sign of the torque acting on the local moment can be opposite to that expected from the spin torque mechanism only. This situation can be realized either (1) when the spin Hall effect and orbital Hall effect in the nonmagnet have opposite signs and the spin-orbit correlation in the ferromagnet is positive11 FIG. 4. Competition between the orbital torque and the spin torque when the directions of the orbital Hall effect and spin Hall effect are opposite in the nonmagnet (NM). In the ferromagnet (FM), rotations of the angular momentum represent angular momentum transfer to the local magnetic moment by dephasing, whose directions are op- posite for the spin injection and orbital injection. or (2) when the spin Hall effect and orbital Hall effect in the nonmagnet have same sign and the spin-orbit correlation in the ferromagnet is negative. The spin-orbit correlation in the ferromagnet is important in the orbital torque mechanism be- cause the injected orbital angular momentum in the ferromag- net first couples to the spin and then exerts a torque on the local magnetic moment. For typical 3dferromagnets, such as Fe, Co, and Ni, the spin-orbit correlation is expected to be positive asdshells are more than half-filled, which tends to align the orbital and spin angular momenta along the same direction. Thus, we aim to achieve the case (1), which is schematically illustrated in Fig. 4. As the directions orbital Hall effect and spin Hall effect are opposite, the angular mo- mentum transfers by dephasing, which are represented as the rotation of the arrows in the ferromagnet in Fig. 4, are also opposite. One of the key features of the orbital torque mechanism is that it relies on the spin-orbit coupling of the ferromagnet, thus the orbital torque depends on the choice of the ferromag- net. Although the spin-orbit coupling strength is similar for typical 3dferromagnets such as Fe, Co, and Ni, the resulting effect of spin-orbit coupling depends on details of the elec- tronic structure, such as the band structure, band filling, mag- nitude of the exchange splitting, etc. This explains a notice- able difference of the spin Hall conductivities of Fe and Ni: Fe SH= 519 (h=e)( cm)1andNi SH= 1688 (h=e)( cm)1 [27]. Thus, even among 3dferromagnets the effective spin- orbit coupling strength which incorporates not only the spin-orbit coupling itself but also electronic structure effects can vary significantly. We expect that the effective spin- orbit coupling strength is much stronger in Ni than in Fe, and we show this by explicit calculations below. Therefore, we consider nonmagnet/ferromagnet bilayers where the nonmagnet exhibits an opposite sign of the orbital Hall effect and spin Hall effect, while the ferromagnet is var- ied such that the strength of effective spin-orbit coupling iscontrolled. This leads us to the choice of Fe/W and Ni/W bi- layerstwo prototypical systems that satisfy these criteria. For W, the orbital Hall conductivity is by an order of magni- tude larger than the spin Hall conductivity, with opposite sign [31]. A reason for choosing Fe and Ni as ferromagnets comes from the expectation that the orbital-to-spin conversion effi- ciency of the orbital torque mechanism is much larger in Ni than it is in Fe. Moreover, both materials can be grown epi- taxially along the [110] direction of the body-centered cubic (bcc) structure. We denote these systems as Fe/W(110) and Ni/W(110), respectively. Meanwhile, Fe/W(110) has been previously studied for the anisotropic Dzyaloshinskii-Moriya interactions for stabilizing the anti-Skyrmion [64]. Figures 5(a) and 5(b) respectively display side and top views of the ferromagnet/W(110) structure, where ferromag- net = Fe or Ni. We consider 8 layers of W and 2 layers of the ferromagnet. We denote the magnetic atom closest to the interface as Fe1 and Ni1, while the magnetic atom at the sur- face of the slab is marked as Fe2 and Ni2. For the bcc(110) stack of the W layers, we assume that the film follows the bulk lattice parameters of the bcc W, whose lattice constant isa= 6:028a0in the cubic unit cell convention, where a0is the Bohr radius. As a result, the distance between the neigh- boring layers of W is dWW=a=p 2 = 4:263a0. The in- plane unit cell is of a rectangular shape, whose length along the[001] and[110] directions are a= 6:028a0andb=p 2a= 8:525a0, respectively. The layer distances between W-ferromagnet and ferromagnet-ferromagnet were optimized in order to minimize the total energy: dWFe= 3:825a0and dFeFe= 3:296a0for Fe/W(110), and dWNi= 3:607a0 anddNiNi= 3:301a0for Ni/W(110). We assume that the local magnetic moment is oriented along the direction of ^z, where ^zis defined as the direction of [110] . The details of first-principles calculation are given in Appendix C. B. Spin-Orbit Correlation and Orbital Quenching The calculated electronic band structures of Fe/W(110) and Ni/W(110) are shown in Figs. 5(c) and 5(d), respectively. On top of each energy band Enk, the spin-orbit correlation in the ferromagnethL
SiFM nkis shown in color, which is defined as hL
SiFM nk=X z2FMh nkjPz(L
S)Pzj nki:(35) Here,j nkiis the Bloch state of band natk-point k, andPz is the projection operator onto a layer whose index is z. It can be seen that near the Fermi energy EF, the spin-orbit correla- tion is negligible in Fe/W(110). The hotspot of this quantity is located about 1.0 eV below the Fermi energy, whose ef- fect is negligible in the steady state transport. On the other hand, in Ni/W(110) the spin-orbit correlation is much more pronounced for states near the Fermi energy. The positive sign of this correlation tends to align the orbital angular momentum and the spin in the same direction. The difference in the spin-orbit correlation directly affects the orbital moment of the ferromagnet in equilibrium. In Figs. 5(e) and 5(f), spin and orbital magnetic moments are plot-12 FIG. 5. (a) Crystal structure of ferromagnet (FM)/W(110), where FM = Fe or Ni. Side and top views are displayed on the left and right, respectively. (b) First Brillouin zone and high symmetry points of bcc(110) film. Electronic energy dispersion Enkand the spin-orbit correlation in the ferromagnet hL
SiFM nkfor (c) Fe/W(110) and (d) Ni/W(110), which are represented by the line and color map, respectively. Note thathL
SiFM nkis much more pronounced in Ni compared to Fe near the Fermi energy EF. Layer-resolved plots of the spin (blue squares) and orbital (red stars) moments for (e) Fe/W(110) and (f) Ni/W(110). Comparing Fe/W(110) and Ni/W(110), the spin moment in Fe is much larger than that in Ni, but the relative ratio of the orbital moment over the spin moment is much larger in Ni. This implies that the orbital degree of freedom is not frozen in Ni/W(110), while it is quenched in Fe/W(110). ted in each layer for Fe/W(110) and Ni/W(110), respectively. Blue square symbols and red star symbols respectively indi- cate the spin and orbital moments. For Fe/W(110) [Fig. 5(e)], the magnitude of the spin moment is large: +2:259Band +2:856Bfor Fe1 and Fe2, respectively. On the other hand, the orbital moments of Fe1 and Fe2 are small: +0:069Band +0:079B, respectively. The ratio of the orbital moment over the spin moment is 3.06 % and 2.76 % for Fe1 and Fe2, re- spectively, which is fairly small. Thus, the orbital magnetism is strongly quenched in Fe. This implies that even though the orbital angular momentum may be injected into Fe, i.e., by the orbital Hall effect of W, it is likely that most of the orbital angular momentum is relaxed to the lattice through the crys- tal field torque [Eq. (12)] instead of being transferred to the angular momentum of the spin through the spin-orbital torque [Eq. (13)]. Therefore, in Fe/W(110), it is expected that the or- bital torque mechanism is not significant and the spin torque mechanism will be dominant, in accordance with common ex- pectation. Meanwhile, we find proximity magnetism in W8 by the hybridization with Fe, where the spin and orbital moments are0:114Band0:009B, respectively. In contrast to Fe/W(110), Ni atoms in Ni/W(110) exhibit much smaller spin moment but relatively large orbital mo-ment. The spin moments are +0:146B,+0:510Band the orbital moments are +0:023B,+0:070Bfor Ni1 and Ni2, respectively. Remarkably, the ratio of the orbital moment over the spin moment is 15.64 % and 13.80 % for Ni1 and Ni2, respectively. Thus, the orbital moment is far from be- ing quenched in Ni. Such electronic structure, which is prone to the formation of the orbital angular momentum, promotes the mechanism where an orbital Hall effect-induced orbital angular momentum can efficiently couple to the spin, result- ing in the torque on the local magnetic moment. Therefore, at this point we expect that the orbital torque can be signif- icantly larger than the spin torque in Ni/W(110), leading to the opposite effective spin Hall angle when compared to the Fe/W(110) bilayer. C. Symmetry Constraints Before presenting the results of first-principles calculations, we consider symmetry constraints on the electric response for quantities taking part in the equations of motion. We define ^xk[001] ,^yk[110], and ^zk[110] , and apply an external electric field along the ^xdirection. We consider a situation13 (a) (b)(c) (d) FIG. 6. Electric response (per unit cell) of LyandSycurrent influxes[QLy z]and[QSy z]and various torques TLy SO,TLy CF,TSy SO, and TSy XCarising from the interband processes and accumulation, and arising from the intraband processes (divided by ) in Fe/W(110). Spatial profiles for (a) orbital and (b) spin quantities at the true Fermi energy EF=Etrue F. Fermi energy dependence for (c) orbital and (d) spin quantities, summed over the ferromagnet layers (Fe1 and Fe2). Note that the sum of the interband responses of the orbital/spin current influx and the total torque ( TLy=TLy SO+TLy CFandTSy=TSy SO+TSy XCfor orbital and spin, respectively) matches with the intraband response of the orbital/spin accumulation divided by . when ^m=^z, for which the symmetry analysis reveals that only theycomponent is nonzero in Eq. (33). On the other hand, for the equations of motion of the intraband contribution [Eq. (34)], the xcomponent is the only non-zero component. Thus, we present the result for =yand=xin Eqs. (33) and (34), respectively. Details of the symmetry analysis are explained in Appendix D. The current-induced torque on the local magnetic moment is given by Tm= TS XCinter TS XCintra(36a) =^yD TSy XCEinter ^xD TSx XCEintra : (36b) We further decompose Tminto dampinglike ( TDL) and field- like (TFL) components: Tm=TDL^m(^m^y) +TFL^m^y =TDL^y+TFL^x; (37) By comparing Eqs. (36b) and (37), we have TDL=D TSy XCEinter ; (38a) TFL=D TSx XCEintra : (38b) Below, we present the analysis for LyandSycomponents of quantities from Eqs. (33a) and (33b), respectively, which is closely related to that of the dampinglike torque. The anal- ysis forLxandSxfrom Eqs. (34a) and (34b) is presented in the Appendix E. In order to perform the decomposition of the computed quantities into contributions coming from each atomic layer, we adopt the tight-binding representation of the equations of motion, as explained in detail in Appendix F. In the tight-binding representation, we denote orbital and spincurrent influxes, which correspond to the first terms in the right hand side of Eqs. (7a) and (7b), as [QLz]and[QSz]. D. Fe/W(110) In Fig. 6(a), spatial profiles of individual terms appearing in Eq. (33a) are shown for Ly. Note that the current influx and torque have the same dimension, thus we omit the labels for the current influx in the y-axes. We find that h[QLyz]iinter (blue squares) is negative near W1 and positive at W8, which corresponds to a positive sign of the orbital Hall conductiv- ity. In concurrence with h[QLyz]iinter,hTLy CFiinter(purple di- amonds) appears in the opposite sign. However, hTLy SOiinter (red stars) is much smaller than h[QLyz]iinterandhTLy CFiinter. This means that most of the the orbital current influx is ab- sorbed by the lattice. Meanwhile, the sum of h[QLyz]iinter and the total torque hTLyiinter=hTLy SOiinter+hTLy CFiinter (cyan crosses), which corresponds to the right hand side of Eq. (33a), matches hLyiintra=(black dashed line), which corresponds to the left hand side of Eq. (33a). This confirms the validity of the equation of motion Eq. (33a). Slight de- viations are due to a finite parameter assumed in the calcu- lation of the interband responses by Eq. (26) (Appendix C) andk-dependence of the orbital angular momentum operator (Appendix A). Analogously, spatial profiles of the individual terms appear- ing in Eq. (33b), related to the spin degree of freedom, are displayed in Fig. 6(b). We remark that the responses related to spin are an order of magnitude smaller than those related to the orbital channel in Fig. 6(a). This is natural since the spin dynamics is caused by the orbital dynamics that occurs first.14 From the sign ofh[QSyz]iinter(light blue squares), which is positive near W1 and negative near W8, we conclude that the sign of the spin Hall conductivity is negative. Only in Fe layers,hTSy XCiinter(orange circles) is sizable, where the ex- change interaction is dominant. The overall positive sign of hTSy XCiinterin Fe layers corresponds to a negative sign of the effective spin Hall angle. We observe a strong correlation be- tweenh[QSyz]iinterandhTSy XCiinter. This implies that the spin current influx is mostly transferred to the local magnetic mo- ment, which agrees with the spin torque mechanism. Mean- while,hTSy SOiinter(dark red stars) is much smaller, but not neg- ligible. The sum of h[QSyz]iinterand the total torque on the spinhTSyiinter=hTSy SOiinter+hTSy XCiinter(green crosses), the right hand side of Eq. (33b), corresponds to hSyiintra= on the left hand side (black dashed line). A pronounced value of h[QSyz]iinternear the Fe layers, compared to its value at W1, may seem anomalous [Fig. 6(b)]. However, it can be understood by looking at hSyiintra, which exhibits a much more pronounced magnitude in W1 and W2, as compared to its value in Fe1 and Fe2. That is, in Fe1 and Fe2, the spin current is efficiently absorbed by the ferromag- net instead of inducing the spin accumulation. The situation is opposite in W1 and W2, where such spin current absorp- tion is not possible, and the spin current simply results in spin accumulation. A similar behavior, where the spin current is strongly enhanced near the ferromagnet interface, has been also predicted in Co/Pt [36] and Py/Pt [65]. To understand the predicted behavior in terms of the elec- tronic structure, we present the Fermi energy dependence of the computed quantities in Figs. 6(c) and 6(d) for spin and orbital channels, respectively, where a superscript FM means that it is summed over Fe1 and Fe2 layers. To arrive at these plots, we intentionally varied the Fermi energy EF from2 eV to+2 eV with respect to the true Fermi energy Etrue F, assuming that the potential [Eq. (2)] remains invari- ant whenEFchanges. For the orbital channel [Eq. (33a) and Fig. 6(c)], we observe that h[QLyz]iinter(blue solid line) andhTLy CFiinter(purple solid line) tend to cancel each other. Meanwhile,hTLy SOiinter(red solid line) is smaller than the rest of the contributions. Thus, most of the orbital angular mo- mentum is transferred to the lattice instead of the spin. We find that the equation of motion [Eq. (33a)] is valid over the whole range of EF, where the sum of h[QLyz]iinterand hTLyiinter(cyan solid line) corresponds to hLyiintra=(black dashed line). The Fermi energy properties for the spin channel [Eq. (33b)] are shown in Fig. 6(d). Here, a strong correlation betweenh[QSyz]iinter(light blue solid line) and hTSy XCiinter (orange solid line) can be observed. We thus conclude that the spin torque mechanism is dominant over the whole range ofEF. At the same time, hTSy SOiinter(dark red solid line) is suppressed, which implies that the contribution to the current- induced torque caused by the spin-orbit coupling in the ferro- magnet, i.e., the orbital torque and anomalous torque mecha- nisms, is negligible. In order to clarify the microscopic mechanism of the current-induced torque better, we intentionally switch on and (a) (b)W-SOC on, Fe-SOC off W-SOC off, Fe-SOC on FIG. 7. Fermi energy dependence of interband responses (per unit cell) of the spin current influx [QSy z](light blue solid line), spin-orbital torque TSy SO(dark red solid line), and exchange torque TSy XC(orange solid line), which are summed over the Fe layers in Fe/W(110). (a) The result when spin-orbit coupling is on in W and off in Fe, and (b) the result when spin-orbit coupling is off in W and on in Fe. off the spin-orbit coupling in Fe or W atoms. When spin-orbit coupling is on in W and off in Fe [Fig. 7(a)], the Fermi en- ergy dependence of h[QSyz]iinter(light blue solid line) per- fectly matches that of hTSy XCiinterwith reversed sign (orange solid line), which supports the spin torque mechanism. On the other hand,hTSy SOiinter(dark red solid line) is essentially zero due to the absence of spin-orbit coupling in Fe. Mean- while, when spin-orbit coupling is off in W and on in Fe [Fig. 7(b)], all the responses become very small. Thus, any contri- bution arising from the spin-orbit coupling of the ferromagnet (orbital torque or anomalous torque) is negligible. E. Ni/W(110) In Figs. 8(a) and 8(b) we show the plots of layer-resolved individual terms appearing in the equation of motion [Eq. (33)] for the ycomponent of the orbital and spin parts, respec- tively, in Ni/W(110). In Fig. 8(a), we find that the orbital Hall conductivity is positive in sign according to h[QLyz]iinter (blue squares). As in the case of Fe/W(110), h[QLyz]iinter andhTLy CFiinter(purple diamonds) are only different in sign, implying that the orbital angular momentum is transferred to the lattice. Thus, hTLy SOiinter(red stars) is much smaller. These features are similar to those we found in Fe/W(110). The interband-intraband correspondence between hLyiintra= (black dashed line) and the sum of h[QLyz]iinterand total torquehTLyiinter(cyan crosses) is also preserved. On the other hand, as shown in Fig. 8(b), spatial pro-15 (a) (b)(c) (d) FIG. 8. Electric response (per unit cell) of LyandSycurrent influxes[QLy z]and[QSy z]and various torques TLy SO,TLy CF,TSy SO, and TSy XCarising from the interband processes and accumulation, and arising from the intraband processes (divided by ) in Ni/W(110). Spatial profiles for (a) orbital and (b) spin quantities at the true Fermi energy EF=Etrue F. Fermi energy dependence for (c) orbital and (d) spin quantities, summed over the ferromagnet layers (Ni1 and Ni2). Note that the sum of the interband responses of the orbital/spin current influx and the total torque ( TLy=TLy SO+TLy CFandTSy=TSy SO+TSy XCfor orbital and spin, respectively) matches with the intraband response of the orbital/spin accumulation divided by . files of spin quantitites are significantly different from those of Fe/W(110). First, we notice that h[QSyz]iinter(light blue squares) does not exhibit a close correlation with hTSy XCiinter (orange circles). Moreover, the sign of hTSy XCiinteris negative. This means positive effective spin Hall angle in Ni/W(110), which is opposite to the negative sign of the spin Hall con- ductivity in W. This is in contrast to the common interpre- tation that the spin Hall angle is a property of the nonmag- net, regardless of the ferromagnet. Second, hTSy SOiinter(dark red stars) is comparable to the rest of the contributions, in- dicating the importance of spin-orbit coupling in Ni. Mean- while, the interband-intraband correspondence stands with high precision (green crosses for the sum of h[QSyz]iinterand hTSyiinter, and a black dashed line for hSyiintra=). The Fermi energy dependence of the computed quantities, shown in Figs. 8(c) and 8(d) for orbital and spin channels respectively, provides a detailed information on the overall trend. Althoughh[QLyz]iinterandhTLy CFiinterhave opposite sign, their magnitudes differ we find that hTLy SOiinteris very pronounced near the Fermi energy, with corresponding peak indicated with a black arrow [Fig. 8(c)]. Since the response of the spin quantities is an order of magnitude smaller than that for the orbital channel, the pronounced spin-orbital torque, which is still much smaller than h[QLyz]iinterandhTLy CFiinter, can have a significant effect on the dynamics of spin. In con- currence with the increase of hTLy SOiinter,hTLy CFiinteris signif- icantly decreased near the Fermi energy. This implies that a channel for the orbital angular momentum transfer to the lat- tice is suppressed. As a result, the response of spin in Ni/W(110) exhibits a much more rich and complicated behavior when compared toFe/W(110) [Fig. 8(d)]. We first notice that the correlation betweenh[QSyz]iinter(light blue solid line) and hTSy XCiinter (orange yellow solid line) is no longer present. Moreover, with the negative drop of hTSy XCiinter, corresponding to the positive sign of the effective spin Hall angle, there is an as- sociated positive peak from hTSy SOiinter(dark red solid line), which is indicated with a black arrow. This indicates that the spin is transferred from the orbital rather than spin current in- flux. Therefore, the orbital angular momentum is responsible for the current-induced torque in Ni/W(110). Meanwhile, the interband-intraband correspondence (green solid line for the sum ofh[QSyz]iinterandhTSyiinterand black dashed line forhSyiintra=) is satisfied. As we have done for Fe/W(110), we switch on and off the spin-orbit coupling separately for W and Ni atoms in Ni/W(110) as well, showing the results in Fig. 9. In Fig. 9(a), the Fermi energy dependence of h[QSyz]iinter,hTSy SOiinter, andhTSy XCiinteris shown when the spin-orbit coupling of W is on and the spin-orbit coupling of Ni is off. First of all, we find thathTSy XCiinteris positive at the Fermi energy, which is opposite to the full-spin-orbit coupling case [Fig. 8(d)]. In this case, we find a strong correlation between h[QSyz]iinterand hTSy XCiinter. Thus, the negative sign of the effective spin Hall angle is caused by the spin injection from the spin Hall effect of W. However, such correlation is not as perfect as in the case of Fe/W(110) [Fig. 7(a)]. We attribute such difference to an interfacial mechanism, where the torque is generated regard- less of the spin current. Meanwhile, hTSy SOiinteris negligible since the spin-orbit coupling of Ni is off. When the spin-orbit coupling is off in W and on in Ni, nontrivial features show up in h[QSyz]iinter,hTSy SOiinter, and16 (a) (b)W-SOC on, Ni-SOC off W-SOC off, Ni-SOC on (c) W-SOC off, Ni-SOC on (E-field in W) W-SOC off, Ni-SOC on (E-field in Ni)(d) FIG. 9. Fermi energy dependence of interband responses (per unit cell) of the spin current influx [QSy z](light blue solid line), spin-orbital torque TSy SO(dark red solid line), and exchange torque TSy XC(orange solid line), which are summed over the Ni layers in Ni/W(110), for the case when (a) the spin-orbit coupling is on in W and off in Ni, and (b) the spin-orbit coupling is off in W and on in Ni. Both (c) and (d) show the results when the spin-orbit coupling is off in W and on in Ni, and the external electric field is applied only in (c) W and (d) Ni layers. hTSy XCiinter, which is in contrast to Fe/W(110) [Fig. 7(b)]. This is due to nontrivial spin-orbit correlation of Ni shown in Fig. 5(d). Moreover, hTSy XCiinteris negative at the Fermi energy. We find that nontrivial peak features [black arrows in Fig. 8(d)] are reproduced in this calculation. Thus, we confirm that the latter peaks originate in the spin-orbit cou- pling of Ni. To further clarify the microscopic mechanisms, we apply the external electric field in W only [Fig. 9(c)] or Ni only [Fig. 9(d)] when the spin-orbit coupling of the W is off and the spin-orbit coupling of Ni is on, which corre- spond to the orbital torque and the anomalous torque contri- butions, respectively (more details can be found in the Ap-pendix C). In both cases, hTSy XCiinterexhibits a negative drop nearEFEtrue F0:15 eV , which is correlated with a pos- itive peak ofhTSy SOiinter. This implies that for both cases the angular momentum transfer from the orbital channel to the spin channel is crucial. The difference is that for the orbital torque mechanism, Fig. 9(c), h[QSyz]iinterexhibits a positive peak at the Fermi energy (marked with a black arrow), which comes from the conversion of the orbital current into the spin current by the spin-orbit coupling of Ni. We find that it is cor- related with a shoulder feature of hTSy XCiinterat the Fermi en- ergy (marked with a black arrow). Such peak of h[QSyz]iinter implies that in the orbital torque mechanism, there are two different microscopic channels for the orbital-to-spin conver- sion: one for the spin converted from the orbital angular mo- mentum viahTSy SOiinter, and the other for the conversion of the orbital current into the spin current followed by the spin- transfer torque. Meanwhile, in Fig. 9(d), which corresponds to the anomalous torque mechanism, h[QSyz]iinteris not very pronounced, and only the peak of hTSy SOiinteris observed (in- dicated with a black arrow). The negative sign of hTSy XCiinter (positive sign of the effective spin Hall angle) is due to a pos- itive sign of the spin Hall conductivity in Ni. We note that, as expected, for the anomalous torque mechanism, the orbital- to-spin conversion via hTSy SOiinteris crucial since it originates in the spin-orbit coupling of the ferromagnet. Therefore, we conclude that in Ni/W(110) the orbital torque and anomalous torque are the first and the second dominant mechanisms for the torque generation on the local magnetic moment. IV . DISCUSSION A. Disentangling Different Microscopic Mechanisms In Sec. III, we found that the spin torque provides the dom- inant contribution to the current-induced torque in Fe/W(110) according to the correlation between the exchange torque and the spin current influx from W, which is reflected in the neg- ative effective spin Hall angle [Fig. 7(a)]. In Ni/W(110), on the other hand, the orbital torque is found to be the most dom- inant contribution. The evidence for the orbital torque is pro- vided by pronounced peaks in the spin-orbital torque and the spin current influx that suggests a positive effective spin Hall angle, associated with the exchange torque [Fig. 9(c)]. How- ever, we also observed that the anomalous torque can be asso- ciated with the spin-orbital torque [Fig. 9(d)] because the self- induced spin accumulation in the ferromagnet results from the current-induced orbital angular momentum. A crucial differ- ence between the orbital torque and anomalous torque is that while the orbital torque is due to an electrical current flowing in the nonmagnet, the anomalous torque is due to an electrical current passing through the ferromagnet. In this respect, only the orbital torque is important for memory applications where the ferromagnetic layer must be patterned to form a physically separate memory cell, whereas both orbital torque and anoma- lous torque are important for applications based on magnetic17 (a) (b) Fe/W(110) Ni/W(110) FIG. 10. Disentanglement of the dampinglike torque into the spin torque, orbital torque, interfacial torque, and anomalous torque in (a) Fe/W(11) and (b) Ni/W(110). Note that the spin torque and or- bital torque are the most dominant mechanisms in Fe/W(110) and Ni/W(110), respectively. We note that the interfacial torque and anomalous torque are not negligible neither in Fe/W(110) nor in Ni/W(110). textures (i.e., domain walls and Skyrmions) for which such patterning is not necessary. We can disentangle each of the contributions in the current- induced torque of Fe/W(110) and Ni/W(110), according to the classification scheme outlined in Sec. II B. The different con- tributions to the current-induced torque can be disentangled by modifying the system parameters “by hand” in the calcula- tion. To distinguish between local and nonlocal contributions to the torque, the electric field is selectively applied to only the ferromagnetic or nonmagnetic layer, respectively. We note, however, that this is an approximate measure since an electric current may flow in the ferromagnet(nonmagnet) although an electric field is applied only to the nonmagnet (ferromagnet) layer, as the electronic wave functions are delocalized across the film. For determining the spin-orbit coupling origin (non- magnet versus ferromagnet), we do not simply turn on and off the spin-orbit coupling because it causes significant change of the band structure. Instead, we change the sign of the the spin- orbit coupling in the relevant layer, which changes the sign of its contribution. For example, we rely on the property that the sign of the orbital torque and anomalous torque should be- come opposite after flipping the sign of spin-orbit coupling in the ferromagnet, while the spin torque and interfacial torque remain invariant. By computing the torque under different system configurations, the four contributions to the current- induced torque can be determined, as illustrated in Sec. II B and described in detail in Appendix G. By this way, the sum of spin torque, orbital torque, interfacial torque, and anomalous torque equals the net torque when the electric field applied to the whole layers with actual spin-orbit coupling strength of each atom. Although this classification scheme relies on com- putational handles with no experimental counterpart, it pro- vides a systematic basis for physically interpreting the results of calculations, which in turn enables the development of in- tuition about materials and system designs. In Figs. 10(a) and 10(b) we show the decomposition of the total dampinglike torque in Fe/W(110) and Ni/W(110), respectively, into separate contributions. In Fe/W(110), the spin torque is the most dominant contribution. However, our analysis reveals that the interfacial torque is not negligible, ac-counting for about 35 % of the spin torque. Overall, the spin torque and interfacial torque are larger than the orbital torque and anomalous torque, implying that the spin-orbit coupling in W is more important than that in Fe. In Ni/W(110), on the other hand, the orbital torque is the most dominant contribu- tion. The second largest contribution is the anomalous torque, which is comparable to a half of the orbital torque. The mag- nitude of the interfacial torque is not much smaller, reaching as much as 37 % of the magnitude of the orbital torque. Over- all, the orbital torque and anomalous torque are dominant over the spin torque and interfacial torque in Ni/W(110). This sug- gests that the spin-orbit coupling in Ni is more important than the spin-orbit coupling in W in this system, in contrast to an intuitive expectation that spin-orbit coupling in 3 dferromag- nets plays a minor role as compared to the spin-orbit coupling of the heavy element. These results are consistent with our analysis of the results presented in Figs. 7 and 9. B. Orbital Current versus Spin Current Although the orbital current [Eq. (8a)] and the spin cur- rent [Eq. (8b)] are defined in a similar way, there are con- ceptual differences. While the spin and its current can be locally defined everywhere in space, the orbital angular mo- mentum is nonzero only inside the muffin-tin within the atom- centered approximation. Thus, the atom-centered approxima- tion does not properly describe the interstitial region between muffin-tins, where the orbital information is supposed to be transported. Nonetheless, orbital current influx to a muffin-tin can be defined even within the atom-centered approximation, which is the reason why we evaluate the influx instead of the current itself throughout the manuscript. Heuristically, the or- bital angular momentum is encoded in a vorticity of the phase of a wave function, which is exists not only in the muffin-tin but also in the interstitial region. It is the vorticity of the wave function that is transported through the interstitial region. The wave function is properly described in our calculation, so that we can reliably compute the flux of vorticity into the muffin- tin. As the atom-centered approximation neglects the contribu- tion from interstitial region, the crystal field torque in our cal- culation [Eq. (12)] only describes angular momentum transfer from the orbital to the lattice within the muffin-tin, which is mostly concentrated near the surfaces and the interface [Figs. 6(a) and 8(a)]. In general, we expect that nonspherical compo- nent of the potential is more pronounced in the interstitial re- gion, which provides another channel for angular momentum transfer from the electronic orbital to the lattice. However, as thedcharacter electronic wave function of a transition metal is localized inside the muffin-tin, we expect that additional con- tribution to the crystal field torque from the interstitial region is small.18 C. Experiments and Materials Although the effective spin Hall angle measured in experi- ments is the sum of all contributions to the torque on the local magnetic moment, it has been assumed that it is a property of the nonmagnet in nonmagnet/ferromagnet bilayers, which can be incorrect. For example, we have shown that the current- induced torque depends on the choice of the ferromagnet in ferromagnet/W(110), where ferromagnet is Fe or Ni. In this case, it is due to an opposite sign of the orbital Hall effect and spin Hall effect in W, and the resulting orbital-to-spin con- version efficiencies are different for Fe and Ni. As a result, even the sign of the effective spin Hall angle changes: from negative for Fe/W(110) to positive for Ni/W(110). We believe that such change-of-sign behavior can be directly measured in experiment. More concretely, we suggest performing a spin- orbit torque experiment on an FeNi alloy in order to observe change of the effective spin Hall angle as the alloying ratio varies, with the effective spin Hall angle turning to zero at a certain critical concentration. We speculate that this behavior would be observed in other systems where the orbital Hall effect competes with the spin Hall effect. For example, among 5delements, Hf, Ta, and Re exhibit gigantic orbital Hall conductivity, whose sign is oppo- site to that of the spin Hall conductivity [32]. Such behavior holds in general for groups 4-7 among transition metals. For 3delements, such as Ti, V , Cr, and Mn, the spin Hall con- ductivity is much smaller than that of 5delements, while the orbital Hall conductivity is almost as large as in 5delements [34]. Thus, the orbital torque contribution is expected to be more pronounced than the spin torque contribution when the nominally nonmagnetic substrate is made of 3delements, as compared to the systems where the nonmagnet is made of 5d elements. Therefore, alloying not only the ferromagnet but also the nonmagnet provides a useful knob for observing com- peting mechanisms of the current-induced torque. Layer thickness dependence of the spin-orbit torque has been measured in Ta/CoFeB/MgO [66] and Hf/CoFeB/MgO [67], where the sign of the current-induced torque was found to change when the thickness of Ta or Hf was as small as 1 nm to2 nm . The origin of the sign change has been attributed to the competition between the bulk and interfa- cial mechanisms, which correspond to the spin torque and interfacial torque mechanisms in our terminology. Recently, such behavior has also been observed in a similar system Zr/CoFeB/MgO [68], where a 4delement Zr was used instead of a5delement. Due to a negligible spin Hall conductivity of Zr as compared to the orbital Hall conductivity, it has been proposed that the sign change occurs due to a competition be- tween the spin torque and orbital torque [68], instead of the competition between the spin torque and interfacial torque. Detailed investigation of these systems by our method may reveal the origin of the sign change. Another widely-studied system in spintronics is a Pt-based magnetic heterostructure. Due to a large spin Hall conduc- tivity of Pt [69], the spin torque is assumed to be the most dominant mechanism of the torque in Pt-based systems [5]. In Co/Pt, however, theoretical analysis revealed that the in-terfacial spin-orbit coupling contributes significantly to the fieldlike torque [19, 70]. On the other hand, the damping- like torque is attributed to the spin torque mechanism [35, 70], which is also supported by experiments [71]. Hiroki et al. compared Ni/Pt and Fe/Pt bilayers, finding that the current- induced torque strongly depends on the choice of the ferro- magnet [72]. According to their interpretation, while the bulk effect is dominant in Ni/Pt, a pronounced interface effect in Fe/Pt not only leads to fieldlike torque but also suppresses the spin current injection from Pt, which leads to a distinct ferro- magnet dependence of the torque [72]. A similar conclusion has also been drawn in an experiment by Zhu et al. , where the interfacial spin-orbit coupling has been varied by choos- ing different samples and annealing conditions [73]. Further investigation of the exact mechanism in these systems by the- ory is required. For the study of the interplay between the spin and orbital degrees of freedom transition metal oxides may present a very fruitful playground. In transition metal oxides, a strong enta- glement of the spin, orbital, and charge degrees of freedom has been intensively studied in the past [74–76]. For exam- ple, magnetic properties of transition metal oxides are heavily affected by the orbital physics not only via the effect of spin- orbit coupling but also because of the anisotropic exchange interactions caused by the shape of participating orbitals [74]. However, most studies on the transition metal oxides have fo- cused on their ground state properties, such as various com- peting magnetic phases. We expect that the investigation of the spin-orbital entangled dynamics would provide crucial in- sights into understanding the complex physics of transition metal oxides. V . CONCLUSION Motivated by various proposed mechanisms of the current- induced torques, which are challenging to disentangle both theoretically and experimentally, we developed a theory of current-induced spin-orbital coupled dynamics in magnetic heterostructures, which tracks the transfer of the angular mo- mentum between different degrees of freedom in solids: spin and orbital of the electron, lattice, and local magnetic moment. By adopting the continuity equations for the orbital and spin angular momentum [Eq. (7)], we derived equations for the an- gular momentum dynamics in the steady state reached when an external electric field is applied, which provide relations between interband and intraband contributions to the current influx, torques, and accumulation of the spin and orbital an- gular momentum [Eqs. (33) and (34)]. This formalism is particularly useful for the detailed study of the microscopic mechanisms of the current-induced torque and we used its first principles implementation to investigate the spin-orbit torque origins in Fe/W(110) and Ni/W(110) bi- layers. In Fe/W(110), we observe a strong correlation be- tween the spin current influx and the exchange torque, which is a key characteristic of the spin torque mechanism. On the other hand, such correlation is not observed in Ni/W(110). Instead, we observe a pronounced correlation between the ex-19 change torque and the spin-orbital torque, indicating the trans- fer of angular momentum from the orbital to the spin channel. Moreover, the spin current influx exhibits a sign opposite to that of the spin Hall effect in W. This leads us to a conclusion that the orbital torque is dominant in Ni/W(110). We further proposed a classification scheme of the different mechanisms of current-induced torque based on the criteria of whether the scattering source is in the nonmagnet-spin-orbit coupling or the ferromagnet-spin-orbit coupling, and whether the torque response is of local or nonlocal nature (Fig. 2). This analysis also confirms that the spin torque and orbital torque are the most dominant mechanisms in Fe/W(110) and Ni/W(110), respectively. However, we also find that the other contributions, interfacial torque and anomalous torque, are not negligible as well. Our formalism enables an analysis of the angular momentum transport and transfer dynamics in detail, which clearly goes beyond the “spin current picture”. Since it treats the spin and orbital degrees of freedom on an equal footing, it is ideal for systematically studying the spin-orbital coupled dynamics in complex magnetic heterostructures. ACKNOWLEDGMENTS D.G. thanks insightful comments from discussions with Gustav Bihlmayer, Filipe Souza Mendes Guimar ˜aes, Math- ias Kl ¨aui, OukJae Lee, Kyung-Whan Kim, and Daegeun Jo. K.-J. L., J.-P. H., and Y . M. acknowledge discussions with Mark D. Stiles. J.-P. H., and Y . M. additionally acknowledge discussions with Jairo Sinova. We thank Mark D. Stiles and Matthew Pufall for carefully reading the manuscript and pro- viding insightful comments. We gratefully acknowledge the J¨ulich Supercomputing Centre for providing computational resources under project jiff40. D.G. and H.-W. Lee were supported by SSTF (Grant No.BA-1501-07). F.X. acknowl- edges support under the Cooperative Research Agreement be- tween the University of Maryland and the National Institute of Standards and Technology Physical Measurement Laboratory, Award 70NANB14H209, through the University of Maryland. We also acknowledge funding under SPP 2137 “Skyrmionics”(project MO 1731/7-1) and TRR 173 268565370 (project A11) of the Deutsche Forschungsgemeinschaft (DFG, Ger- man Research Foundation). Appendix A: Interband-Intraband Correspondence Here we provide a proof of Eq. (31). We assume that the operatorOdoes not have position dependence, which leads to O(k) =eik
rOeik
r=O. From Eqs. (29), the left hand side of Eq. (31) is written as 1 hOiintra=eEx hX nk@kxfnkhunkjOjunki;(A1) where we used @fnk @kx=@fnk @Enk@Enk @kx=f0 nkhhunkjvx(k)junki:(A2) Application of integration by parts to the first term in Eq. (A1) leads to 1 hOiintra=eEx hX nkfnk[h@kxunkjOjunki +hunkjOj@kxunki]:(A3) It can be rewritten as 1 hOiintra=eEx hX n6=mX k(fnkfmk) Re [h@kxunkjumkihumkjOjunki]: (A4) By using identities h@kxunkjumki=hhunkjvx(k)jumki EnkEmk(A5) and humkjOjunki=ihhumkj(1=ih)[O;H(k)]junki EnkEmk; (A6) forn6=m, we have 1 hOiintra=ehExX n6=mX k(fnkfmk) Imhunkjvx(k)jumkihumkj(1=ih) [O;H(k)]junki (EnkEmk)2 (A7a) =ehExX n6=mX k(fnkfmk) Imhunkj(1=ih) [O;H(k)]jumkihumkjvx(k)junki (EnkEmk+i)2 (A7b) =dO dtinter : (A7c) This proves Eq. (31). In case when O(k)isk-dependent, the deviation is given by 1 hOideviation=eEx hfnhunkj@kxO(k)junki;(A8)such that 1 hOiintra+1 hOideviation=dO dtinter (A9)20 holds even whenO(k)isk-dependent. Appendix B: Stationary Condition of the Intraband Contribution For a proof of Eq. (32), we apply Eq. (29) to dO=dt: dO dtintra = (B1) eEx ih2X nk[@kxfnkhunkj[O(k);H(k)]junki]: Because hunkj[O(k);H(k)]junki= 0 (B2) for any Hermitian operator O, we have dO dtintra = 0: (B3) Appendix C: Computational Method First-principles calculation consists of three steps. The first step is calculation of the electronic structure from the density functional theory. In this step, we obtain Bloch states and their energy eigenvalues. The second step is to obtain maximally- localized Wannier functions (MLWFs) starting from the Bloch states obtained in the first step. Once the MLWFs are found, matrix elements of all relevant operators (Hamiltonian, po- sition, spin, and orbital) are expressed within the basis set of the MLWFs. Thus, a tight-binding model is obtained. The last step is evaluation of the interband and intraband responses of the individual terms in the equations of motion [Eqs. (33) and (34)] by solving the tight-binding model obtained from the second step. The electronic structure of ferromagnet/W(110) (ferromag- net=Fe or Ni), whose lattice structure is shown in Fig. 5, is calculated self-consistently in the film mode of the full- potential linearized augmented plane wave method [77] from the code FLEUR [78]. We use Perdew-Burke-Ernzerhof exchange-correlation functional within the generalized gra- dient approximation [79]. Muffin-tin radii of the ferromag- net and W atoms are set to 2:1a0and2:5a0, respectively, wherea0is the Bohr radius. The plane wave cutoff is set to3:8a1 0. The Monkhorst-Pack k-mesh of 2424are sam- pled from the first Brillouin zone. The spin-orbit coupling is treated self-consistently within the second variation scheme. The layer distances dFMFManddWFMare optimized such that the total energy is minimized. The optimized values for Fe/W(110) are dWFe= 3:825a0anddFeFe= 3:296a0, and those for Ni/W(110) are dWNi= 3:607a0anddNiNi= 3:301a0. In order to obtain MLWFs, we initially project the Bloch states ontodxy,dyz,dzx, andsp3d2trial orbitals for each atom, and minimize their spreads using the code WANNIER90 [80]. We obtain in total 180 MLWFs out of 360 Bloch states,that is, 18 MLWFs for each atom. For the disentanglement of the inner and outer spaces, we set the frozen window as 2 eV above the Fermi energy. The Hamiltonian, position, spin, and orbital operators, which are evaluated beforehand within the Bloch basis, are then transformed to the basis of MLWFs, and the tight-binding model is obtained. Individual terms appearing in the equations of motion [Eqs. (33) and (34)] are evaluated using Eqs. (26) and (29) for in- terband and intraband contributions, respectively. The inte- gration is performed over interpolated k-mesh of 240240. For the interband contributions, we set = 25 meV for con- vergence, which describes broadening of the spectral weight by disorders. In the intraband contribution, we set the momen- tum relaxation time as = h=2with = 25 meV , which corresponds to = 1:261014s. We set the temperature in the Fermi-Dirac distribution function as room temperature T= 300 K . For the application of an external electric field specifically onto ferromagnet or W layers, we replaced vxin Eq. (26) by vFM x=X z2FMPzvx+vxPz; (C1a) vW x=X z2WPzvx+vxPz; (C1b) wherePzis the projection onto the MLWFs located in a layer whose index is z. We confirm that the 18 MLWFs are well localized in each layer. Note that Eq. (C1) is defined such that vx=vFM x+vW x: (C2) Appendix D: Symmetry Analysis In Sec. III C, we state that only yandxcomponents are nonzero in Eqs. (33) and (34), respectively. Here, we prove this by symmetry argument. Two important symme- tries present in ferromagnet/W(110), where the magnetization is pointing the zdirection, areTMxandTMysymmetries. Here,Tis the time-reversal operator and Mx(y)is the mirror reflection operator along the direction of x(y). Since all the terms appearing in the same equation should transform in the same way, we consider only the response of a torque operator TJ=dJ dt(D1) for a general angular momentum operator J, which can be ei- ther orbital and spin origin. To find symmetry constraints on the interband [Eq. (26)] and intraband [Eq. (29)] responses, we first investigate how matrix elements of vxandTJtrans- form. We define UTandUMx(y)as Hilbert space representa- tions ofTandMx(y), respectively. Note that Ttransforms vxandTJas U1 TvxUT=vx; (D2) and U1 TTJUT= +TJ; (D3)21 respectively. On the other hand, MxandMysymmetries transformvxandTJas U1 MxvxUMx=vx; (D4a) U1 MyvxUMy= +vx; (D4b) and U1 MxTJxUMx= +TJx; (D5a) U1 MxTJyUMx=TJy; (D5b) U1 MxTJzUMx=TJz; (D5c) U1 MyTJxUMy=TJx; (D5d) U1 MyTJyUMy= +TJy; (D5e) U1 MyTJzUMy=TJz: (D5f) As a result,TMxandTMysymmetries transform vxand TJas U1 TMxvxUTMx= +vx; (D6a) U1 TMyvxUTMy=vx; (D6b) and U1 TMxTJxUTMx= +TJx; (D7a) U1 TMxTJyUTMx=TJy; (D7b) U1 TMxTJzUTMx=TJz; (D7c) U1 TMyTJxUTMy=TJx; (D7d) U1 TMyTJyUTMy= +TJy; (D7e) U1 TMyTJzUTMy=TJz; (D7f)whereUTMx(y)=UTUMx(y). Note thatTandMx(y)com- mute each other. We remark that UTandUMx(y)are anti-unitary and unitary operators, respectively. Thus, UTMx(y)is anti-unitary. For an arbitrary anti-unitary operator , a matrix element of an operatorOsatisfies hjOj i=hj 1O
j i: (D8) Thus, combining this result with Eqs. (D6) and (D7) pro- vides constraints on the interband [Eq. (26)] and intraband [Eq. (29)] contributions. As an illustration, let us demonstrate that both interband and intraband contributions vanishes for TJz. We consider TMxsymmetry at first. By this, matrix elements of vxand TJztransform as hUTMx mkjvxjUTMx nki= +h nk0jvxj mk0i; (D9) and hUTMx nkjTJzjUTMx mki=h mk0jTJzj nk0i; (D10) where k0= (+kx;ky;kz). On the other hand, TMysym- metry gives hUTMy mkjvxjUTMy nki=h nk00jvxj mk00i; (D11) and hUTMy nkjTJzjUTMy mki=h mk00jTJzj nk00i; (D12) where k00= (kx;+ky;kz). A constraint for the interband contribution for TJz[Eq. (26)] is given byTMysymmetry: TJzinter=ehExX n6=mX k(fnk00fmk00)ImhUTMy nkjTJzjUTMy mkihUTMy mkjvxjUTMy nki (Enk00Emk00+i)2 (D13a) =ehExX n6=mX k(fnk00fmk00)Imh mk00jTJzj nk00ih nk00jvxj mk00i (Enk00Emk00+i)2 (D13b) =ehExX n6=mX k(fmkfnk)Imh nkjTJzj mkih mkjvxj nki (EmkEnk+i)2 (D13c) = TJxinter(D13d) in the limit!0+. Thus,hTJziinteris forbidden byTMy symmetry. In Eq. (D13a), we used the fact that the linear response can also be written in terms of the transformed states.Note that we use the Bloch state representation instead of their periodic parts. For the intraband contribution, we have the following constraint by TMxsymmetry:22 (a) (b)(c) (d) FIG. 11. Electric response of current influxes [QLy z]and[QSy z]and various torques TLy SO,TLy CF,TSy SO, andTSy XCarising from the intraband process in Fe/W(110). Spatial profiles for (a) the orbital and (b) the spin at true Fermi energy EF=Etrue F. Fermi energy dependences for (c) the orbital and (d) the spin, which are summed over the ferromagnet layers (Fe1 and Fe2). (a) (b)(c) (d) FIG. 12. Electric response of current influxes [QLy z]and[QSy z]and various torques TLy SO,TLy CF,TSy SO, andTSy XCarising from the intraband process in Ni/W(110). Spatial profiles for (a) the orbital and (b) the spin at true Fermi energy EF=Etrue F. Fermi energy dependences for (c) the orbital and (d) the spin, which are summed over the ferromagnet layers (Ni1 and Ni2). TJzintra=eEx hX nk@k0x fnk0hUTMx nkjTJzjUTMx nki (D14a) = +eEx hX nk@k0x fnk0h nk0jTJzj nk0i (D14b) = TJzintra: (D14c) Therefore, both interband and intraband responses for TJz vanishes by the symmetries. By the procedure for different components of the torque, we arrive at the conclusion that the presence ofTMxandTMysymmetries allows only hTJyiinterandhTJxiintrato be nonzero.Appendix E: Intraband Response In Fig. 11, intraband contributions appearing in Eq. (34) are plotted for each layer of Fe/W(110). We confirm that the sum of the current influx and torques vanishes for the intra- band contributions, respectively for the orbital and spin, which confirms Eq. (34). For the orbital [Fig. 11(a)], we find that23 (a) (b) Fe/W(110) Ni/W(110) FIG. 13. Disentanglement of the fieldlike torque into the spin torque, orbital torque, interfacial torque, and anomalous torque in (a) Fe/W(11) and (b) Ni/W(110). In both systems, the spin torque is most dominant mechanism. We note that the anomalous torque is not negligible in Ni/W(110). h[QLxz]iintratends to cancel with hTLx CFiintraandhTLx SOiintra is small. Meanwhile, for the spin, not only h[QSxz]iintra andhTSx XCiintrabut alsohTSx SOiintraare of comparable mag- nitudes, which is distinct from the interband response [Fig. 6(b)]. However, near the Fe layers, hTSx SOiintrais small, and hTSx XCiintratends to cancel with h[QSxz]iintra. We attribute this behavior to small spin-orbit correlation in Fe [Fig. 5(c)], and quenching of the orbital moment. Fermi energy depen- dence plots in Fig. 11 also show the cancellation behaviors between the the orbital current influx and crystal field torque, and between the spin current influx and and the exchange torque. Although the spin-orbital torque is not particularly small in general, only near the true Fermi energy it is sup- pressed. Therefore, the fieldlike torque originates in the spin current injection (spin torque mechanism). In Ni/W(110), for the orbital, h[QLxz]iintraandhTLx CFiintra cancel each other, with small magnitude of hTLx SOiintra[Fig. 12(a)]. For the spin, on the other hand, as well as h[QSxz]iintra,hTSx SOiintracontributes tohTSx XCiintra, in com- parable magnitudes [Fig. 12(b)]. This is due to pronounced spin-orbit correlation of Ni at the Fermi energy [Fig. 5(d)]. The Fermi energy dependence plots in Figs. 12(c) and 12(d) also show that the spin-orbital torque is nonnegligible at the Fermi energy. Therefore, in Ni/W(110), the fieldlike torque is a combined effect of the spin injection and the spin-orbit cou- pling. Such behavior has also been observed in Pt/Co [70]. To clarify microscopic mechanisms of different origins, we disentangle the fieldlike torque into the spin torque, orbital torque, interfacial torque, and interfacial torque, analogously to Fig. 10. For Fe/W(110) [Fig. 13(a)], we find that the spin torque is the most dominant contribution, as expected. On the other hand, for Ni/W(110) [Fig. 13(b)], not only the spin torque but also the anomalous torque significantly contributes. This is due to pronounced spin-orbit correlation in Ni. Mean- while, we also find that the interfacial torque is not negligible. Appendix F: Tight-binding Representation of the Continuity Equation Here, we derive a tight-binding representation of the current influx and torque appearing in the continuity equation [Eq.(7)]. To do this, we first define Pzas a projection operator onto a set of MLWFs located near a layer whose index is z. Then, for the spin operator S, we define S(z) =1 2[SPz+PzS] (F1) as the spin operator at z, such that S=X zS(z): (F2) The Heisenberg equation of motion for S(z)is written as dS(z) dt=1 ih[S(z);H] (F3a) =1 2ih[SPz;PzS;H] (F3b) =1 2ihf[S;H]Pz+S[Pz;H] +[Pz;H]S+Pz[S;H]g (F3c) =TS(z) + [jS](z): (F3d) We define local torque operator at zby TS(z) =1 2ihfPz[S;H] + [S;H]Pzg (F4a) =1 2 TSPz+PzTS ; (F4b) where TS=1 ih[S;H] (F5) is the total torque operator, and we define [jS](z) =1 2ihn [Pz;H]S+S[Pz;H]o (F6) the spin current influx at z. Although [jS](z)may not seem intuitive, it corresponds to an usual definition of the spin current influx. To demon- strate this point, we consider the case where P=jrihrjand H=h2r2 r=2m, wherejriis an eigenket for the position operator r. Then [jS]becomes [jS] =1 2ihn jrihrjHSHjrihrjS +SjrihrjH SHjrihrjo : (F7) Thus, a matrix element between states and is written as hj[jS]j i=ih 2mn (r)S r2 r (r)  r2 r(r) S (r)o (F8a) =rr
hjjSj i; (F8b) where hjjSj i=ih 2mn (r)S[rr (r)] + [rr(r)]S (r)o : (F9) From Eq. (F9), we find that this is consistent with usual defini- tion of the spin current jS=S (p=m). Therefore, Eq. (F6) can be understood as an operator of the spin current influx to the subspace defined by the projection Pz.24 Appendix G: Disentangling Different Contributions of the Current-Induced Torque To disentangle different contributions of the torque (Figs. 10 and 13), we utilize a property that upon changing the sign of the spin-orbit coupling constant the orbital torque and anomalous torque flip their signs while the signs of the spin torque and interfacial torque remains invariant. That is, the total exchange torque is decomposed as the sum of the con- tribution driven by the spin-orbit coupling in the nonmagnet and the contribution driven by the spin-orbit coupling in the ferromagnet: TS XCtot= TS XCNMSOC+ TS XCFMSOC:(G1) In an auxiliary system where the sign of the spin-orbit cou- pling is flipped in the ferromagnet atoms, the exchange torque becomes TS XCaux= TS XCNMSOC TS XCFMSOC:(G2)Thus, the nonmagnet-spin-orbit coupling contribution is writ- ten as TS XCNMSOC=1 2h TS XCtot+ TS XCauxi ;(G3) and the ferromagnet-spin-orbit coupling contribution is writ- ten as TS XCFMSOC=1 2h TS XCtot TS XCauxi :(G4) Then, by applying the electric field only in the nonmagnet or ferromagnet layers by Eq. (C1), we can separately evaluate the spin torque, orbital torque, anomalous torque, and interfa- cial torque. [1] Frances Hellman, Axel Hoffmann, Yaroslav Tserkovnyak, Ge- offrey S. D. Beach, Eric E. Fullerton, Chris Leighton, Allan H. MacDonald, Daniel C. Ralph, Dario A. 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1308.1198v2.Chirality_from_interfacial_spin_orbit_coupling_effects_in_magnetic_bilayers.pdf

arXiv:1308.1198v2 [cond-mat.mes-hall] 12 Nov 2013Chirality from Interfacial Spin-Orbit Coupling Effects in M agnetic Bilayers Kyoung-Whan Kim1,2, Hyun-Woo Lee2,∗Kyung-Jin Lee3,4, and M. D. Stiles5 1Basic Science Research Institute, Pohang University of Sci ence and Technology, Pohang, 790-784, Korea 2Department of Physics, Pohang University of Science and Tec hnology, Pohang, 790-784, Korea 3Department of Materials Science and Engineering, Korea Uni versity, Seoul, 136-701, Korea 4KU-KIST Graduate School of Converging Science and Technolo gy, Korea University, Seoul 136-713, Korea 5Center for Nanoscale Science and Technology, National Inst itute of Standards and Technology, Gaithersburg, Maryland 20899-6 202, USA (Dated: March 2, 2022) As nanomagnetic devices scale to smaller sizes, spin-orbit coupling due to the broken structural inversion symmetry at interfaces becomes increasingly imp ortant. Here we study interfacial spin- orbit coupling effects in magnetic bilayers using a simple Ra shba model. The spin-orbit coupling introduces chirality into the behavior of the electrons and through them into the energetics of the magnetization. In the derived form of the magnetization dyn amics, all of the contributions that are linear in the spin-orbit coupling follow from this chira lity, considerably simplifying the analysis. For these systems, an important consequence is a correlatio n between the Dzyaloshinskii-Moriya interaction and the spin-orbit torque. We use this correlat ion to analyze recent experiments. Magnetic bilayers that consist of an atomically thin ferromagnetic layer (such as Co) in contact with a non- magnetic layer (such as Pt) with strong spin-orbit cou- pling have emerged as prototypical systems that exhibit very strong spin-orbit coupling effects. Strong spin-orbit coupling can enhance the efficiency of the electrical con- trol of magnetization. A series of recent experiments [1– 4] on magnetic bilayers report dramatic effects such as anomalously fast current-driven magnetic domain wall motion [2] and reversible switching of single ferromag- netic layers by in-plane currents [3, 4]. Strong spin-orbit couplingcanintroducechiralityintothemagneticground state [5, 6]. This chirality is predicted [7] to boost the electricalcontrolofmagneticdegreesoffreedomevenfur- ther as has been confirmed in two experiments [8, 9]. Interfaces lack structural inversion symmetry, allowing interfacial spin-orbit coupling to play an expanded role. In magnetic bilayers, it generates various effects includ- ing the Dzyaloshinskii-Moriya (DM) interaction [10–12] and the spin-orbit torque [13–18]. Here, we examine a simple Rashba model of the interface region. We com- pute the equation of motion for a magnetization texture ˆm(r) by integrating out the electron degrees of freedom. We report two main findings. The first is the correlation between the DM interaction and the spin-orbit torques. Spin-orbit torques arise from interfacial spin-orbit cou- pling but also from the bulk spin Hall effect, and the importance of each contribution is hotly debated [3, 19– 22]. The correlation we find opens a way to quantify the contribution from interfacial spin-orbit coupling by mea- suring the DM interaction, allowing one to disentangle the two contributions. The second finding is that all linear effects of the in- terfacial spin-orbit coupling, including the DM interac- tion and the spin-orbit torque, can be captured through a simple mathematical construct, which we call a chiral derivative. The chiral derivative also shows in the equa-tion of motion how each contribution that is linear in the spin-orbit coupling corresponds to a contribution that is present even in the absence of spin-orbit coupling. This correspondence provides a simple way to quantitatively predict and understand a wide variety of interfacial spin- orbit coupling effects allowed by symmetry [18]. In the last part of the Letter, we discuss briefly the extension to realistic situations, which go beyond the simple Rashba model. Our analysis begins with the two-dimensional (2D) Rashba Hamiltonian H=Hkin+HR+Hexc+Himp =p2 2me+αR /planckover2pi1σ·(p׈z)+Jσ·ˆm+Himp,(1) wherepis the 2D electron momentum in the xyplane, the vector σof the Pauli matrices represents the electron spin, and |ˆm(r)|= 1.His a minimal model [13–18] for electronic properties of the interface region between the ferromagnetic and nonmagnetic layers in magnetic bilay- ers, and captures the broken symmetries; Hexcbreaks the time-reversal symmetry, and HRbreaks the struc- tural inversion symmetry. The last term Himpdescribes the scattering by both spin-independent and quenched spin-dependent impurities. The latter part of Himpcon- tributes to the Gilbert damping and the nonadiabatic spin torque [23, 24]. Here, we focus on effects of HRon the equation of motion for the magnetization up to order αR. These effects include the DM interaction and the spin-orbit torque. We neglect effects of order α2 Rsuch as interface- induced magnetic anisotropy, contributions to Gilbert damping [25, 26], and to the nonadiabaticity parame- ter[27]. Weintroducethe unitarytransformation[28,29] U= exp[−ikRσ·(r׈z)/2], (2)2 where kR=2αRme /planckover2pi12(3) andr= (x,y).Urotates the electron spin around the ˆr׈zdirection by the angle kRr, where r=|r|. We also introduce the r-dependent 3 ×3 matrix R, which achieves the same rotation of a classical vector such as ˆm. Upon the unitary transformation, one finds (Supple- mentary Material [30]) U†HU=Hkin+Jσ·ˆm′+H′ imp+O(α2 R),(4) where ˆm′=R−1ˆm (5) andH′ imp=U†HimpU. We ignorethe lasttermin Eq.(4) as higher order. H′ impis not identical to Himpbut they share the same impurity expectation values up to O(αR), which implies that HRhas no effect to linear order on the Gilbert damping coefficient or the nonadiabaticity coefficient [23, 24]. Thus up to O(αR),H′ impmay be identified with Himp. Then the unitary transformation fromHtoU†HUhas eliminated HRat the expense of replacing ˆmbyˆm′. With this replacement, we compute the energy of the filled Fermi sea as a function of ˆm. Without HR, the energy can depend on ˆmonly through spatial deriva- tives∂uˆm(u=x,y) since the energy cannot depend on the direction of ˆmwhenˆmis homogeneous. For ˆmsmoothly varying over length scales longer than the Fermi wavelength, the energy density εmay be expressed as the micromagnetic exchange interaction density ε= A(∂xˆm·∂xˆm+∂yˆm·∂yˆm), where Ais the interfacial exchange stiffness coefficient. Equation (4) implies that inthepresenceof HR,εcanbeobtainedsimplybyreplac- ing∂uˆmwith∂uˆm′;ε=A(∂xˆm′·∂xˆm′+∂yˆm′·∂yˆm′). One then uses the relation (Supplementary Information) ∂uˆm′=∂u(R−1ˆm) =R−1˜∂uˆm, (6) where the chiral derivative ˜∂uis defined by ˜∂uˆm=∂uˆm+kR(ˆz׈u)׈m. (7) Hereˆuis the unit vector along the direction u. The secondterm in Eq. (7) arisesfrom the derivativeoperator acting on the r-dependent R−1.εin the presence of the interfacial spin-orbit coupling then becomes ε=A(∂xˆm·∂xˆm+∂yˆm·∂yˆm) (8) +D[ˆy·(ˆm×∂xˆm)−ˆx·(ˆm×∂yˆm)]+O(α2 R), with D= 2kRA. (9)Note that the second term in Eq. (8) is nothing but the interfacialDM interactionresponsibleforchiralmagnetic order addressed recently [7–9]. A few remarks are in order. First, this derivation shows that the DM inter- action is intimately related to the usual micromagnetic exchange interaction that exists even in the absence of interfacial spin-orbit coupling. This is the first example of the one-to-one correspondence and illustrates how the interfacial spin-orbit coupling generates a term in linear order from each term present in the absence of the spin- orbitcoupling. Second, this mechanismforthe DM inter- actionin anitinerantferromagnetis similartothat ofthe Ruderman-Kittel-Kasuya-Yosida interaction in nonmag- netic systems acquiring the DM-like character [31, 32] when conduction electrons are subject to interfacial spin- orbit coupling. Next, we demonstrate the correlation between the DM interaction and the spin-orbit torque. Although the spin- orbit torque has already been derived from Eq. (1) in previous studies [13–18], we present below a derivation of the spin-orbit torque that shows the relationship be- tween it and the DM interaction. Without HR, it is well known [33] that the total spin torque Tstinduced by an in-plane current density jconsists of the following two components, Tst=vs(ˆj·∇)ˆm−βvsˆm×(ˆj·∇)ˆm,(10) where the first and the second components are the adi- abatic [34] and nonadiabatic [35, 36] spin toques, re- spectively. Here ˆj=j/j,j=|j|,βis the nonadia- baticity parameter [35, 36], and the spin velocity vs= PjgµB/(2eMs), where Pis the polarization of the cur- rent,gis the Land´ e gfactor,µBis the Bohr magneton, Msis the saturation magnetization, and −e(<0) is the electron charge. In the presence of HR, Eqs. (4) and (6) imply that Tstchanges to Tst=vs(ˆj·˜∇)ˆm−βvsˆm×(ˆj·˜∇)ˆm,(11) where˜∇= (˜∂x,˜∂y). One then obtains from Eq. (7) Tst=vs(ˆj·∇)ˆm−βvsˆm×(ˆj·∇)ˆm (12) +τfvsˆm×(ˆj׈z)−τdvsˆm×[ˆm×(ˆj׈z)]. The two terms in the second line are the two components of the spin-orbit torque. The first (second) component in the second line is called the fieldlike (dampinglike) spin- orbit torque and arises from the adiabatic (nonadiabatic) torque in the first line. This is the second example of the one-to-one correspondence. The chiral derivative fixes the coefficients of the two spin-orbit torque components to τf=kR, τd=βkR. (13) When combined with Eq. (9), one finds τf=D/2A, τd=βD/2A. (14)3 This correlation between the DM coefficient Dand the spin-orbit torque coefficients τfandτdis a key result of this work. A recent experiment [8] examined current-driven do- main wall motion in the systems Pt/CoFe/MgO and Ta/CoFe/MgO and concluded that domain wall motion against (along) the electron flow in the former (latter) system is due to the product DτdPbeing positive (nega- tive). According to Eqs. (9) and (13), DτdP= 2βPAk2 R should be of the same sign as βPregardless of kRsince Ais positive by definition. Thus explaining the exper- imental results for Ta/CoFe/MgO within the interfacial spin-orbit coupling theory requires βPto be negative. Whereas βPcan be negative, in most models and pa- rameter ranges it is positive. We tentatively conclude thatτdin Ta/CoFe/MgO [8] has a different origin, the spin Hall effect being a plausible mechanism as argued in Ref. [8]. For Pt/CoFe/MgO, on the other hand, the reported sign is consistent with the sign determined from Eqs. (9) and (13) if βP >0. The Pt-based structure in Ref. [9] also gave the same sign as Ref. [8]. To investigate the origin of the spin-orbit torque in Pt/CoFe/MgO, we attempt a semiquantitative analysis. For the suggested valuesD= 0.5 mJ/m2,A= 10−11J/m in Ref. [8], Eq. (9) predicts kR= 2.5×108m−1. ForP= 0.5, β= 0.4,Ms= 3×105Am−1, which are again from Ref. [8], Eq. (13) predicts the effective transverse field −(τfvs/γ)ˆj׈zof the fieldlike spin-orbit torque and the effective longitudinal field ( τdvs/γ)(ˆm×(ˆm׈z)) of the dampinglike spin-orbit torque to have the magnitudes 1.3 mT and 0.52 mT, respectively, for j= 1011A/m2. Hereγis the gyromagnetic ratio. The former value is in reasonable agreement with the measured value 2 mT considering uncertainty in the parameter values quoted above, whereas the latter value is about an order of mag- nitude smaller than the measured value 5 mT in Ref. [8]. We thus conclude that the fieldlike spin-orbit torque of Pt/CoFe/MgO in Ref. [8] is probably due to the inter- facial spin-orbit coupling whereas the dampinglike spin- orbit torque is probably due to a different mechanism such as the bulk spin Hall effect. For the fieldlike spin- orbittorqueofPt/CoFe/MgO,the relativesignof τfwith respect to Dis also consistent with the prediction of the interfacial spin-orbit coupling if Pis positive. These two examples illustrate the idea that all linear effects of the interfacial spin-orbit coupling can be cap- tured through the chiral derivative ˜∂uˆm. To gain insight into its physical meaning, it is illustrative to take u=x and examine the solution of ˜∂xˆm= 0, which forms a left- handed (for kR>0) cycloidal spiral (Fig. 1), where ˆm precessesaroundthe −(ˆz׈x) axis [−(ˆz׈y) axis ifu=y] asxincreases with the precession rate dθ/dx=kR. This chiral precession gives the name, chiral derivative. Note that this precession is identical to the conduction elec- tron spin precession caused by HRin nonmagnetic sys- tems [37]. Moreover when ˜∂xˆm= 0,Hexcalso causesxz y θ FIG. 1: (color online) Chiral precession of magnetization ˆm. Chiral precession profile of ˆmwith˜∂xˆm= 0 forms a left- handed (for kR>0) cycloidal spiral. This profile is identical to the spin precession profile of conduction electrons mov- ing in the + xor−xdirection in nonmagnetic systems with HR[37]. the same conduction electron spin precessionas HRdoes. Thus effects of HRandHexcbecome harmonious and the one-dimensional “half” p2 x/2me−(αR//planckover2pi1)σypx+Jσ·ˆmof the 2D Hamiltonian (1) gets minimized when ˜∂xˆm= 0. Interestingly,thesumoftheexchangeenergyandtheDM interaction, namely A∂xˆm·∂xˆm+D(ˆz׈x)·(ˆm×∂xˆm), also gets minimized when ˜∂xˆm= 0. This is not a coin- cidence as this sum by definition should agree with the energy landscape of the Hamiltonian, which forces the valueDin Eq. (9). One consequence of deriving the spin-orbit torque us- ing the chiral derivative is that such a derivation shows that the spin-orbit torque is chiral when combined with theconventionalspintorquejust astheDM interactionis chiral when combined with the micromagnetic exchange interaction. For example, when jis along the xdirection, the total torque Tstin Eq. (11) vanishes even for finite j if˜∂xˆm= 0. As a side remark, the first and second terms in Eq. (11) are nothing but current-dependent correc- tions to the torques due to the total equilibrium energy density in Eq. (8) and the Gilbert damping, respectively. This identification is a straightforward generalization of a previously reported counterpart; when HRis absent, the adiabatic and nonadiabatic spin torques in Eq. (10) are the current-dependent corrections to the torques due to the micromagnetic exchange interaction [38] and the Gilbert damping [25]. The anomalously fast current-driven domain wall mo- tion demonstrated in Ref. [9] raises the possibility that chirally ordered magnetic structures [5, 6] such as topo- logical Skyrmion lattices may be very efficiently con- trolled electrically. Such motion would be similar to the highlyefficientelectricallydrivendynamicsofaSkyrmion lattice in a system with bulk spin-orbit coupling such as the B20 structure [39]. Flexible deformation of the Skyrmion lattice is proposed [40] as an important contri- bution to the high efficiency of current-driven dynamics in B20 structures. We expect Skyrmion lattices in mag- netic bilayers to behave similarly because both systems are similarly frustrated. The chiral derivative is noncom- mutative, ˜∂x˜∂yˆm/negationslash=˜∂y˜∂xˆm, so the energy landscape of the lattice structure is necessarily frustrated leading to4 the existence of many metastable structures with low ex- citation energies. In a Skyrmion lattice, another linear effect of the in- terfacial spin-orbit coupling becomes important. Con- sider a Skyrmion lattice without interfacial spin-orbit coupling. The spatial variation of ˆmintroduces a real space Berry phase [41], which can affect the electron transport through a Skyrmion lattice. It produces a fictitious magnetic field [42] B±=∓(h/e)ˆzb, where b= (∂xˆm×∂yˆm)·ˆm/4πis nothing but the Skyrmion number density [41]. Here the upper and lower signs apply to majority (spin antiparallel to ˆm) and minority (spin parallel to ˆm) electrons, and thus this field is spin dependent. An experiment [6] on Fe/Ir bilayer reported the Skyrmion spacing of 1 nm. For a Skyrmion density of (1 nm)−2,B±becomes of the order of 104T, which can significantly affect electron transport. In the presence of interfacial spin-orbit coupling, the Berry-phase-derived field becomes chiral. Following the same procedure as above, one finds that B±is now given by∓(h/e)ˆz˜b, where˜b= (˜∂xˆmט∂yˆm)·ˆm/4π=b+bR+ O(α2 R), where bR=kR∇·ˆm/4π. (15) We estimate the magnitude of bRfor the Mn/W bi- layer [5], for which left-handed cycloidal spiral with pe- riod 12 nm is reported. From the estimated value D= 23.8/(2π) nm meV per Mn atom and A= 94.2/(2π)2 nm2meV per Mn atom, we find kR= 0.794 nm−1from Eq. (9), and ( h/e)bRbecomes about 140 T. Thus for the left-handed cycloidal spiral, for which the Skyrmion den- sityb= 0, the effective magnetic field is governed by this interfacial spin-orbit coupling contribution. For completeness, we also discuss briefly the interfacial spin-orbit coupling contribution to the fictitious electric fieldE±, which is spin dependent and arises when ˆm varies in time. Without HR, it is known that E±= ±(h/4πe)(eadia+enon), wheretheso-calledadiabaticcon- tribution [42–44] is given by ( eadia)u= (∂tˆm×∂uˆm)·ˆm and the nonadiabatic contribution [45, 46] is given by (enon)u=β(∂uˆm·∂tˆm). In the presence of HR, correc- tionsarise. Recently someofus[26] reportedacorrection termeadia R,andRef.[47]reportedanothercorrectionterm enon R, which are given by (eadia R)u=−kR(ˆz׈u)·∂tˆm, (16) (enon R)u=βkR(ˆz׈u)·(ˆm×∂tˆm).(17) Here we point out that the previously reported correc- tions can be derived almost trivially using the chiral derivative since ( eadia+eadia R)u= (∂tˆmט∂uˆm)·ˆmand (enon+enon R)u=β(˜∂uˆm·∂tˆm). This derivation also re- veals the chiral nature of eadia Randenon R. For the drift motion of chiral magnetic structures at 100 m/s, the parameter values of the Mn/W bilayer [5] lead to the estimation that both ( h/4πe)(eadia) and (h/4πe)(eadia R)are of the order of 104V/m, which should be easily de- tectable. So far we focused on magnetic bilayers. But these re- sults should also be relevant for the high-mobility 2D electron gas formed at the interface between two dif- ferent insulating oxide materials. One example is the LaAlO 3/SrTiO 3interface [48], which has broken struc- tural inversion symmetry [49] and becomes magnetic [50] under proper conditions. Last, we briefly discuss how features of real systems might affect our conclusions. Two differences in realis- tic band structures, are that the energy-momentum dis- persion is not parabolic and that there are multiple en- ergy bands [51]. Another difference is that magnetic bi- layers are not strictly 2D systems, unlike systems such as LaAlO 3/SrTiO 3. To test the effects of more realis- tic band structures, in the Supplementary Material [30], we examine a tight-binding versionof H, which generates nonparabolicenergy bands, and find that the relation (9) remains valid despite the nonparabolic dispersion. The two dimensionality is tested in a recent publication by some of us [52]. There, we perform a three-dimensional Boltzmann calculation to address the interfacial spin- orbit coupling effect on the spin-orbit torque and obtain results, which are in qualitative agreement with those of the 2D Rashba model. On the basis of these observa- tions, we expect that predictions of the simple Rashba model will survive at least qualitatively even in realistic situations and thus can serve as a good reference point for more quantitative future analysis. To conclude, we examined effects of interfacial spin- orbit coupling using the Rashba model. We found that all linear effects of the interfacial spin-orbit coupling can be derived by replacing spatial derivatives with chiral derivatives. This allows these effects to be understood in terms of chiral generalizations of effects in the absence of spin-orbit coupling. 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2103.06033v1.Magnetic_phases_for_two__t__2g___holes_with_spin_orbit_coupling_and_crystal_field.pdf

Magnetic phases for two t2gholes with spin-orbit coupling and crystal eld Pascal Strobel,1Friedemann Aust,1and Maria Daghofer1, 2 1Institut f ur Funktionelle Materie und Quantentechnologien, Universit at Stuttgart, 70550 Stuttgart, Germany 2Center for Integrated Quantum Science and Technology, University of Stuttgart, Pfaenwaldring 57, 70550 Stuttgart, Germany (Dated: March 11, 2021) We investigate two holes in the the t2glevels of a square-lattice Mott insulator with strong spin- orbit coupling. Exact diagonalization of a spin-orbital model valid at strong onsite interactions, but arbitrary spin-orbit coupling and crystal eld is complemented by an eective triplon model (valid for strong spin-orbit coupling) and by a semiclassical variant of the model. We provide the magnetic phase diagram depending on crystal eld and spin-orbit coupling, which largely agrees for the semiclassical and quantum models, as well as excitation spectra characterizing the various phases. I. INTRODUCTION The interplay between spin-orbit coupling (SOC) and correlated electrons as a driving force of physical prop- erties in transition metal compounds has gathered sig- nicant interest in the last decade [1, 2]. The manifold of competing interactions in these materials has led to a plethora of interesting properties like topological Mott insulators, superconductivity and spin liquids [3]. Focus was rst on materials with one hole in the t2g manifold and strong SOC in addition to sizable correla- tions, as realized in 4 dand 5dstates. SOC couples spin S=1/slash.left2 and orbital L=1 degrees of freedom to a total angular moment J=1/slash.left2, so that the model in the end can be described by an eective half-lled model. In addition to similarities to high- TCcuprates and the potential re- alization [4] of the exactly solvable Kitaev model [5] in a honeycomb lattice, which have stimulated extensive re- search on these compounds [6, 7], potential applications in spintronics have been proposed more recently [8]. Interest was then extended to other llings [9, 10], and we will here focus on the Mott-insulating state for two holes. For dominant SOC (as possibly in Ir), the system is in thej-jlimit and the groundstate is thus likely a non- magnetic ground state [11, 12] given by two holes lling thej=1/slash.left2 states. For weaker SOC, e.g. in ruthenates, L-Scoupling is more appropriate, where SOC couples L=1 and S=1 toJ=0, again leading to a nonmagnetic ground state for a single ion [13]. However, energy scales are here rather dierent with a much smaller splitting between the singlet and triplet states. When going from an isolated ion to a compound with a lattice, compet- ing processes can overcome the splitting. Superexchange mixes in states from the J=1 level, which can lead to a magnetic ground state. This phenomenon is also known as excitonic or Van- Vleck magnetism [14], and has for instance been pro- posed to provide a route to a bosonic Kitaev-Heisenberg model [15, 16] and to explain magnetic excitations of Ca2RuO 4[17]. In one dimension, density-matrix renor- malization group has been applied to a spin-orbit coupled and correlated t2gmodel with two holes, and antiferro-magnetic (AFM) order has been found [18, 19] both for intermediate correlations (of a more 'standard' excitonic type with intersite pairs) and for strong correlations (of the 'onsite' type discussed in Ref. [14]). Similarly, dy- namical mean-eld theory has yielded excitonic antifer- romagnetism in a two-dimensional model [20]. A material which has been a focal point of discussions in this context is Ca 2RuO 4. In neutron scattering ex- periments an in-plane AFM ordering has been measured below the N eel temperature TN≈110Kand neutron- scattering spectra can only be explained by taking into account substantial SOC [21{24]. Accordingly, excitonic magnetism, where the magnetic moment arises from ad- mixture of J=1 component into the ionic J=0 state, has been argued to describe this compound [17, 21]. However, a strong crystal eld (CF), favoring doubly occupied xy orbitals, is also clearly present in Ca 2RuO 4and compli- cates the analysis, because it would favor a description in terms of a spin-one system. This is backed by a structural phase transition accompanying the metal-insulator tran- sition. SOC would in this picture be only a correction aecting excitations [24, 25]. In a previous publication, some of us have used the variational cluster approach (VCA) based on ab initio pa- rameters to show that excitonic antiferromagnetism can coexist with substantial CF's and that Ca 2RuO 4falls into this regime [26, 27] of orbitally polarized excitonic antiferromagnetism. In the present paper, we study the competition of CF
and SOC int4 2gsystems in more depth and for a wider parameter space. We investigate an eective spin-orbit model obtained in second-order per- turbation theory, as also used for Ca 2RuO 4[26]. This extends the comparison of CF and SOC acting on the itinerant regime (without magnetic ordering) [10] to mag- netic Mott insulators. Our work is also complementary to a very recent study using the Hartree-Fock approach to investigate the dependence of magnetic ordering on SOC, CF, and tilting of octahedra, which focused on pat- terns with smaller unit cells of one or two Ca ions [28]. We obtain
- phase diagrams using both Monte-Carlo (MC) simulations for the semiclassical limit of the model and exact diagonalization (ED) for the quantum systemarXiv:2103.06033v1 [cond-mat.str-el] 10 Mar 20212 and provide excitation spectra for the various magnetic phases. As expected [26], stripy magnetism is found when both SOC and CF are weak, and checkerboard order (as seen in Ca 2RuO 4) takes over when either becomes strong enough to suciently lift orbital degeneracy. For nega- tive CF, i.e., disfavoring doubly occupied xyorbitals, we nd an additional intermediate phase with rather com- plex magnetic order. Overall, we nd the agreement between the semiclassical and quantum models to be quite good, with phase boundaries between the magnetic phases only moderately dierent. Similarly, the transi- tion to a paramagnetic (PM) state at strong SOC in the full quantum-mechanical model is compared to an eec- tive triplon model [14], valid at strong SOC, and found to agree. Finally, we present the dynamic spin structure factor of the spin-orbital model to discuss signatures of the various magnetic phases accessible to neutron scat- tering experiments. In Sec. II, we introduce models, i.e., the full spin- orbital superexchange model as well as the triplon model valid for strong SOC, and methods. In Sec. III A, we rst go over the limiting cases of the spin-orbital system at dominant CF, the triplon scenario, discuss the intri- cate interplay of spin and orbital order for small CF and SOC, and nally give the phase diagram for intermedi- ate values in Sec. III B. The phase diagram is compared to results of semiclassical MC calculations for the same model in Sec. III C. Section III D presents the dynamic- spin-structure-factor data corresponding to neutron scat- tering experiments for the various phases. Finally Sec. IV gives a summary of the results found in this paper. II. MODEL AND METHODS A. Spin-orbit model Ca2RuO 4has ad4conguration, meaning that four electrons reside in three t2gorbitals, from now on referred to asxy-,zx-, andyz-orbital. The kinetic part of this Hamiltonian can be written as Hkin=3 /summation.disp m=1/summation.disp /uni27E8i;j/uni27E9m/summation.disp ;(t;mc† i;;cj;;+h:c:); (1) wheremare the three dierent bond types introduced in Fig. 1 and t;mis the hopping amplitude depending on the orbital avor and the bond type m. Tab. I gives the amplitudes for all possible t;mfor a square lattice geom- etry.c† i;;(ci;;) is creating (annihilating) an electron in orbitalat siteiwith spin. The possible hopping paths for Ca 2RuO 4[26, 29] are shown in Fig. 1. On near- est neighbor bonds (NN) only two orbitals are active (e.g. xyandzxforx-bonds), while for next-nearest neighbor bonds (NNN) only the xyorbital has a nonzero hopping amplitude (see Tab. I). 11 xyxy xy xyyz yz zx z xzx zx yz yz32 2FIG. 1. Possible hopping pro- cesses in Ca 2RuO 4(based on [26]). The xyorbital can hop inx- (bond 1) and y- direction (bond 2) and has a nonzero hopping amplitude for next-nearest neighbors (bond 3). Thezxandyzorbital can hop only on bond 1 and 2 re- spectively.t;mAmplitude txy;1txy txy;2txy txy;3tNNN tzx;1tzx tzx;2 0 tzx;3 0 tyz;1 0 tyz;2tyz tyz;3 0 TABLE I. Possible hop- ping parameters t;mfrom equations (1) and (4)-(6) as well as their amplitudes for a square lattice geom- etry. The parameter m here indicates the bond type introduced in Fig.1 whileare thet2gor- bitals. (a ) (b ) (c) (d ) ≙ ≙ JH FIG. 2. Displayed are the dierent possible hopping processes from (5) and (4). In (a) and (b) virtual hoppings where the orbital conguration is preserved are shown. In (a) the double occupancy is at the same orbital, while in (b) the double occu- pancy resides at dierent orbitals. (c) and (d) display second order hoppings where the orbital congurations change. In the \pair- ip" process (c) the change arises due to the last term in (2), while the \swap" process (d) arises due to a dif- ferent orbital hopping back then forth. The onsite interaction has the form of a Kanamori- Hamiltonian [30] Hint=U/summation.disp i;ni↑ni↓+U′/summation.disp i;/summation.disp <nini− +(U′−JH)/summation.disp i;/summation.disp <nini −JH/summation.disp i;≠(c† i↑ci↓c† i↓ci↑−c† i↑c† i↓ci↓ci↑);(2) with intraorbital Hubbard interaction U, interorbital U′=U−2JHand Hund's coupling JH. Since the computational cost to calculate this Hamil- tonian via ED is very high we only consider a low energy sector of our Hilbert space. We focus here on the Mott-3 insulating regime with large UandJH. The low-energy sector is then given by states where each site contains ex- actly four electrons (two holes), as Usuppresses charge uctuations. Hund's-rule coupling JHmoreover ensures that exactly one orbital per site is doubly occupied and that the electrons in the remaining two half-lled orbitals form a total spin S=1. This means we have three dier- ent orbital congurations and a S=1 spin state, leading to a subspace of nine states. The orbital congurations are labeled with the orbital which is doubly occupied from here on. It turns out (see [31]) that this orbital degree of freedom can be mapped to an eective angular momentum with Lx=Lyz=−i(/divides.alt0xy/uni27E9/uni27E8zx/divides.alt0−/divides.alt0zx/uni27E9/uni27E8xy/divides.alt0) Ly=Lxz=−i(/divides.alt0yz/uni27E9/uni27E8xy/divides.alt0−/divides.alt0xy/uni27E9/uni27E8yz/divides.alt0) Lz=Lxy=−i(/divides.alt0zx/uni27E9/uni27E8yz/divides.alt0−/divides.alt0yz/uni27E9/uni27E8zx/divides.alt0); (3) where the notation Lwith an orbital index is intro- duced to make the expression of equations (4)-(6) more straightforward and can be easily translated into the x-, y- andz-component of the angular momentum L. The eective spin-orbital Hamiltonian is then obtained by treating the hopping term in second order perturba- tion theory. This gives a Kugel-Khomskii type Hamilto- nian [32, 33], where only virtual hopping processes of the formd4d4→d5d3→d4d4take place. The eective spin- orbital superexchange Hamiltonian includes both orbital as well as spin-orbital interactions. Spin-orbital superex- change terms that preserve orbital occupations of the two sites are HOP=3 /summation.disp m=1/summation.disp /uni27E8i;j/uni27E9m/summation.disp ≠/bracketleft.alt4t2 ;mU+JH U(U+2JH) ×(SiSj−1)(1−L2 )i(1−L2 )j +/parenleft.alt4t2 ≠(;);m(U+JH) U(U+2JH)−(t2 ;m+t2 ;m)JH U(U−3JH)/parenright.alt4 ×(SiSj−1)(1−L2 )i(1−L2 )j/bracketright.alt4: (4) Here we used the aforementioned mapping from orbitals to eective angular momentum L. Having two orbitals of the same avor means only the electrons in the other two orbitals are allowed to perform a virtual hopping (Fig.2 (a)), while for dierent avors each orbital can be involved in such a hopping process (Fig.2 (b)). Furthermore, there are spin-orbital couplings that change orbital congurations. These can be separated in so called \pair- ip" (Fig.2 (c)) processes where two orbitals of the same avor ip their avor to another one and \swap" processes (Fig.2 (d)) where two orbitals ofdierent avor exchange their avor HOF=3 /summation.disp m=1/summation.disp /uni27E8i;j/uni27E9m/summation.disp ≠/bracketleft.alt4−t;mt;mJH U(U+2JH) ×(SiSj−1)(LL)i(LL)j +/parenleft.alt4t;mt;m(U−JH) U(U−3JH)/parenright.alt4 ×(SiSj+1)(LL)i(LL)j/bracketright.alt4: (5) Finally, additional orbital terms aect sites iandjwith dierent orbital occupation: HL⋅L=3 /summation.disp m=1/summation.disp /uni27E8i;j/uni27E9m/summation.disp ≠/bracketleft.alt4t;mt;m2JH U(U−3JH) ×(LL)i(LL)j −(t2 ;m+t2 ;m)1 (U−3JH) ×(1−L2 )i(1−L2 )j/bracketright.alt4: (6) The full superexchange interaction of two sites can be summarized as H=HOF+HOP+HL⋅L (7) Using the hoppings symmetry allowed on a square lat- tice up to second neighbors (see Fig. 1 and Tab. I), one obtains the eective spin-orbital model that can, e.g., be applied to Ca 2RuO 4[26]. In addition to these intersite interactions we also in- clude SOC and the CF splitting
. The SOC terms can be written in the form HSOC=/summation.disp iSi⋅Li=i/summation.disp i/summation.disp ;; ;′  ′c† i;;ci; ;′;(8) where denotes the Levi-Civita symbol and are Pauli matrices [10, 34]. SOC favors the total angular momentum to be J=0, while the CF favors a double occupancy of the xyorbital. Projected onto the low- energy Hilbert space spanned by S=1 and L=1, they can be written as HIon=HSOC+HCF=/summation.disp iSiLi+
/summation.disp i(Lz i)2:(9) Going beyond previous eective models [14, 35, 36], our model thus fully captures the in uence of the Hund's couplingJHand gives the possibility to investigate anisotropic hoppings as well as the ;
→0 limits. The competition between the last two terms, CF
and SOC, is one of the main topics of this paper. We thus x the remaining parameters to values appropriate for Ca 2RuO 4[37]. Hopping processes between NN sites and NNN sites were included with hopping parameters4 / FIG. 3. /uni27E8J2/uni27E9of the spin-orbit model (red) and triplon number nbof the triplon model (blue) are plotted in dependency of SOC. The dashed blue line denotes the phase transition to the PM phase in the triplon model, which is determined via d2nb d2=0. The parameters are chosen to be txy=0:2 eV;tyz= tzx=0:137 eV;tNNN=0:1 eV,U=2 eV,JH=0:34 eV, and
=0:1 eV. set totxy=0:2 eV,tyz=tzx=0:137 eV,tNNN=0:1 eV, and
=0:25 eV via density-functional theory [37]. How- ever, we found that results only dier in details when more symmetric NN hoppings txy=tyz=tzxare used or when NNN hopping is left o. Substantial onsite Coulomb repulsion and Hund's-rule coupling U=2 eV andJH=0:34 eV, as can be inferred from x-ray stud- ies [38], stabilize a Mott insulator with robust onsite spinS=1. Previous calculations using the VCA have shown [26] that most of the weight of the ground state is indeed captured by states that minimize Coulomb inter- actions (2), so that a super-exchange treatment and the resulting spin-orbital model can be justied. B. PM phase and triplon model For strong SOC, we expect our system to be in a PM phase where each ion is in the J=0 state [14, 26]. Transi- tion into magnetically ordered states occurs then via con- densation of triplons. We are going to compare the large- SOC limit of the full spin-orbit superexchange model to a triplon model appropriate for signicant SOC. We take an approach like in Ref. [36] and project (4)-(6) onto the low energy subspace of the SOC Hamiltonian, i.e., ontotheJ=0 andJ=1 states /divides.alt0J=0;MJ=0/uni27E9=1√ 3(/divides.alt0MS=1;ML=−1/uni27E9+/divides.alt0−1;1/uni27E9−/divides.alt00;0/uni27E9) /divides.alt0J=1;MJ=1/uni27E9=1√ 2(/divides.alt01;0/uni27E9−/divides.alt00;1/uni27E9) /divides.alt0J=1;MJ=0/uni27E9=1√ 2(/divides.alt01;−1/uni27E9−/divides.alt0−1;1/uni27E9) /divides.alt0J=1;MJ=−1/uni27E9=1√ 2(/divides.alt0−1;0/uni27E9−/divides.alt00;−1/uni27E9) (10) and projecting out the J=2 levels. We then can dene triplon operators T† 1/slash.left0/slash.left−1(T1/slash.left0/slash.left−1) which create (annihilate) the respective J=1 triplet state and annihilate (create) the J=0 singlet. These operators can then be rewritten to Tx/slash.lefty/slash.leftz(for further details see [14]). C. Methods To investigate these models we use ED on an eight site cluster with√ 8×√ 8 geometry to determine a
− phase diagram as well as the dynamical spin structure factor (DSSF) for specic
and . This is done for both the full spin-orbital model (Sec. II A) as well as the triplon model introduced in Sec. II B. While the spin-orbital model is capable of cap- turing the physics at weak SOC, for strong SOC the triplon model is numerically more accessible due to the reduction of the Hilbert space. To conrm the results of ED and get a better under- standing of the phases identied, we additionally per- formed semiclassical parallel tempering MC calculations with the full spin-orbital model for a 4 ×4 cluster. The easier approach of a fully classical treatment, meaning a parametrization of SiandLias three dimensional real vectors, is not sucient here. A simple example can be found in the (LL)iterms in the Hamiltonian. There is a clear dierence between calculating this expression with scalar components of a three dimensional vector and representing the angular momenta as non-commutative matrices. We accomplish the latter by instead consider- ing trial wave functions of direct-product form /divides.alt0 /uni27E9=/uni2297.big.disp i(/divides.alt0Si/uni27E9⊗/divides.alt0Li/uni27E9); (11) where the rst product runs over all sites i. We allow all complex linear combinations of the Lzeigenvalues /divides.alt0Li/uni27E9=1;i/divides.alt0ML=−1/uni27E9+2;i/divides.alt0ML=0/uni27E9+3;i/divides.alt0ML=+1/uni27E9with T i∗ i=1, and analogously for the spin /divides.alt0Si/uni27E9. These trial wave functions are used to calculate the energy, i.e., the energy becomes a (real-valued) function of classical com- plex vectors i. Classical Markov-chain Monte Carlo is then based on this energy function. A similar approach has been used for quadrupole cor- relations in a spin-1 model with biquadratic interaction [39]. In this context one might refer to our method as5 00.020.040.060.080.10 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 Δ/eVλ/eVPM ? FIG. 3 FIG. 4. Phase diagram for large SOC in the triplon model introduced in Sec. II B. The parameters are chosen to be txy= 0:2 eV;tyz=tzx=0:137 eV;tNNN=0:1 eV,U=2 eV, and JH=0:34 eV. For large SOC, the J=0 phases arises where no triplons are prevalent, while for
<0 the z-AFM and for
>0 thexy-AFM phase is favored. aSU(3)⊗SU(3)semiclassical Monte-Carlo simulation. Compared to ED the numerical expenses of this method are minute. A big drawback of the product state nature of the basis is its inability to accurately represent the singlet and hence nd the paramagnetic phase. How- ever, we have the triplon model to conrm ED data in this parameter range. The Monte Carlo code is used as a counterpart of the triplon model for low spin-orbit cou- pling. Finally we point out that all terms in the Hamiltonian are represented as matrices in the chosen basis and the scalar denitions of spin components or other observables are recovered by simply constructing the expectation val- ues regarding /divides.alt0 /uni27E9. III. RESULTS A. Limiting regimes 1.
/uni226BLimit Presumably the most straightforward limit of the t4 2g model is the case of dominant CF
/uni226Bfavoring the xy orbital to be fully occupied. The two remaining orbitals zxandyzare then half lled and form a spin one. Mag- netic ordering within the plane is then determined by the ratio of NNN and NN superexchange, with the latter usually dominating and favoring a checkerboard pattern. 2./uni226B
Limit For dominant SOC /uni226B
, the ground state becomes theJ=0 state without magnetic moment and there- fore leads to a PM phase. Decreasing SOC leads to the possibility of an admixture of the J=1 states to the k k k k k (d) (a) (b) (c)λ λ Δ Δ FIG. 5. In-plane- ( x-y) and out-of-plane ( z) SSF S(k;;
). is varied in (a) with
=0 eV and (b) with
=0:25 eV.
is varied in (c) with =0 and (d) with =0:06 eV. The momenta kaccessible on an√ 8×√ 8 cluster are k=(0;0), (;0),(0;),(± 2;± 2)and(;). Remaining parameters are given in Sec. II A. ground state, since the energy gap between the Jstates is decreasing and superexchange is driving the transition between the J=0 and theJ=1 states [14, 15]. This triplon-condensation transition leads to a nite magneti- zation and magnetic ordering can be described with the triplon model introduced in Sec. II B. The transition from a magnetically ordered state to the PM state is seen in Fig. 3, which shows the triplon num- ber obtained using ED for the triplon model on a√ 8×√ 8 cluster. CF is here set to
=0:1 eV, where the magnetic order has a checkerboard pattern. The in ection point of the triplon number vs. SOC (at≈0:07 eV) was taken as the phase boundary to the PM phase. Figure 3 also shows the expectation value /uni27E8J2/uni27E9obtained for the full spin-orbital superexchange model. While there is no ob- vious signal, like, e.g., an in ection point, for the phase boundary, increasing leads to a decrease of /uni27E8J2/uni27E9, in agreement with the triplon number. Figure 4 gives the
-phase diagram for the triplon model at intermediate to large, where it can be assumed to be valid. Magnetic order switches from in-plane to out-of-plane at
≈0, and the phase diagram is in qualitative agreement with [35] for
≥0, whereJH=0 and isotropic hopping were used. The triplon model is naturally not able to capture the physical behavior for small SOC. From here on we will therefore focus on performing our calculations with the full spin-orbital model. 3.=0Limit The opposite limit of =0 has been investigated for varied Coulomb repulsion Uand Hund's coupling JH[40]. The calculations in [40] were done with a full nonpertur- bative treatment of the Hubbard-like Hamiltonian, which limited the cluster size to 2 ×2. In agreement with our6 results, obtained with the full spin-orbital model, large orbital degeneracy at small CF 0 /uni2272
/uni22720:24 eV leads to a complex stripy spin-orbital pattern [26, 40]. For larger positive
/uni22730:24 eV, CF dominates and double occupancy is uniformly in the xyorbital. Therefore the Hamiltonian reduces to orbital-preserving terms which yield a simple Heisenberg spin Hamiltonian, while NNN interactions are frustrated. These eects cause a phase transition from the stripy phase to a checkerboard pat- tern. The magnetic ordering can be inferred from the spin structure factors (SSF) for =0 and variable CF that are summarized in Fig. 5 (c). In addition to the stripy andxy-polarized checkerboard patterns seen for
/uni22730, we nd checkerboard order again for
/uni226A0. In this negative-
regime, the xyorbital is half lled to gain in- plane kinetic (resp. superexchange) energy, while double occupation of xzandyzorbitals alternate in a checker- board pattern as well. The overall ordering is thus remi- niscent of that obtained for vanadates with two t2gelec- trons [41]. In the regime −0:12 eV<
/uni22720, an additional phase is nally seen that has nite SSF's for several momenta: (;0),(;), and(0;0). To clarify the nature of this phase, we performed MC simulations on a 4 ×4 cluster, where we include weak SOC =0:01 eV for numerical reasons. In Fig. 6 (a)-(d), snapshots of the four phases appearing in the MC results are shown for the whole
range discussed above. For
=0 the pattern be- comes an alternation of AFM and FM stripes [Fig. 6 (b)], which leads to maxima at Sz(;0);Sz(0;);Sz(;), andSz(0;0)in the momentum space comparable to the signatures in the SSF of the ED. This phase is from here on referred as \3-up-1-down". Overall, the phases seen in the semiclassical model are in good agreement with the characterization based on ED results. For
=−0:2 eV [Fig. 6 (a)], the out-of-plane checkerboard AFM pattern is the ground state with a maximum at Sz(;)and a clear checkerboard pattern inz-direction in position space. After the novel \3-up- 1-down" phase at
≈0, positive
≈0:125 eV leads to a stripy pattern with a maximum at Sz(;0)[Fig. 6 (c)] and larger
=0:2 eV to the in-plane AFM order with maxima at Sx(;)andSy(;), both in accordance with the ED results. Reference [28], which restricts itself to FM and N eel AFM phases, reports an FM phase with some AFM correlations at weak CF,i.e., also sees compe- tition of FM and AFM tendencies roughly where we nd the stripy and \3-up-1-down" patterns. B. Phase diagram of the full spin-orbital model After discussing the limiting cases of small and large CF and SOC, we now investigate the
- plane. We rst study the static SSF for xed and
[Fig. 5 (a)- (d)]. Since we perform ED on an eight site cluster, the SSF is only obtainable for eight dierent kvalues, from FIG. 6. Spin components Sz(a)-(c) and Sx(d) per site as well as for all relevant wavevectors kfor a 4 ×4 square lattice. Calculations were performed with semiclassic parallel temper- ing MC. In (a)-(d) snapshots of the dierent phases arising in the parameter range −0:2<
<0:2 and 0:01 eV<<0:08 eV are shown. These can be directly compared to the ED results of Fig. 9. which only four are unique. These are k=(0;0)resp. FM ordering, (;0)resp. stripy ordering, (;)resp. AFM ordering and (/slash.left2;/slash.left2). In Fig. 5 (a)-(d) only the SSF's with appreciable weight are displayed. Our goal is an understanding of the impact of and
on the spin- orbital state. Hopping parameters txy,tyz,tzx, Coulomb repulsionU, and Hund's coupling JHwhere chosen as introduced in Sec. II A. In Fig. 5 (a), CF is xed to
=0 eV and one sees three phases depending on the strength of SOC. For small SOC (<0:02 eV), one nds the stripy phase, where (;0)- SSF's have similar in-plane and out-of-plane components. This is in concordance with the results of VCA calcula- tions of [26] as well as ED calculations on a 2 ×2 cluster [40, 42]. Increasing the SOC to 0 :02 eV<<0:04 eV gives rise to a phase with contributions from in- and out-of-plane (;0) as well as ( ;) structure factors and additionally7 00.010.020.030.040.050.060.07 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4PM Δ/eVλ/eV 9(d)9(c)9(b)9(a) Ca2RuO4 FIG. 7.
−phase diagram obtained by ED calculations on√ 8×√ 8 cluster. The PM phase (dark grey) was identied via the triplon model of Sec. II B. Sketches show the spin ordering for the respective phase, calculated via MC on a 4 ×4 cluster. The white dots denote the snapshots of the taken in Fig. 9, to investigate the dynamical SSF (further information see Sec. III D), including the parameter setting for Ca 2RuO 4. calculations. the (0,0) out-of-plane contribution. This phase is the \3- up-1-down" state already discussed in the limit =0 (see Sec. III A). Increasing SOC further ( >0:04 eV) leads to an out-of-plane AFM phase. This phase is identical with the out-of-plane AFM phase arising in the triplon model (orange phase in Fig. 4). This phase was also found at
=0 eV and substantial SOC in the VCA calculations of [26]. Further increase of SOC starts to reduce the SSF at(;)again, and nally suppresses all AFM order, see the discussion of the triplon model and Fig. 4. The results for a large xed CF at
=0:25 eV are displayed in Fig. 5 (b). Starting from SOC =0 eV, the ground state is an isotropic AFM phase. SOC induces a smooth transition to an in-plane AFM order. This is due to the fact that positive
favors the double occupancy of thexy-orbital and therefore Lz=0, and ascouples spin and orbital momenta, this also leads to a decrease of theSzcomponent. Lastly in Fig. 5 (d) SOC is set to the value =0:06 eV. As already mentioned earlier
>0:04 eV stabilizes an in-plane AFM pattern, due to the preference of Lz=0 which results in a preference of Sz=0 due to SOC. If the CF is small or has a negative sign, out-of-plane AFM ordering is favored since the xyorbital is mostly singly occupied. This transition is well captured by the triplon model discussed in Sec. III A. The information gained from the ED SSF's, supported by semiclassical MC in case of the \3-up-1-down" pat- tern, as well as the information on the transition to the PM phase inferred from the triplon model is summarized in the
−phase diagram in Fig. 7. To obtain this phase diagram we performed multiple sweeps by varying
for xed (and vice versa ), like in Fig. 5, and included the PM phase from the triplon model. We did this be- cause the transition is better identiable than with /uni27E8J2/uni27E9 (see Fig. 3). If both CF and SOC are weak, the inter- ? 6(a) 6(b) 6(c) 6(d)FIG. 8. Phase diagram depending on and
obtained via MC for remaining parameters as given in Sec. II A. White dots denote the snapshots of Fig. 6. action terms introduced in (4)-(6) dominate, which favor a stripy alignment of spins (light blue) together with a complex orbital pattern [26, 34]. For large CF, the double occupation locates either at the xy(
>0) or alternates betweenzx- andyz-orbitals (
<0), which results in an x-y-AFM (light orange) or z-AFM order (dark orange) respectively. These phases are both very robust against SOC, which favors a J=0 PM phase (dark grey). The competition between the z-AFM and the stripy phase at small negative CF and small SOC, causes the \3-up-1- down" phase to arise (dark blue). Locating Ca 2RuO 4in this phase diagram (correspond- ing white dot in Fig. 7), puts it solidly within the in-plane AFM phase, as expected from experiment [17, 24, 37, 38]. Also, Ca 2RuO 4does not appear to be very close to any CF-driven phase transition and its AFM order can thus be expected to be rather robust. C. Comparison of semiclassical and quantum models Having made use of a semiclassical Markov chain MC to identify the \3-up-1-down" phase, we now compare the semiclassical and quantum phase diagrams more gener- ally. Several snapshots for weak SOC were already dis- cussed in Sec. III A to clarify the regime of weak SOC and
/uni22720. To obtain a full semiclassical phase diagram, sev- eral CF sweeps for dierent strengths of SOC between −0:2 eV<
<0:2 eV were performed and the SSF for (;),(;0)and(0;0)are calculated. The results yield the phase diagram shown in Fig. 8 [white points denote the locations of the snapshots of Fig. 6 (a)-(d)]. For dominant SOC >0:04 eV and
<0, an out-of-plane AFM phase arises (dark orange in Fig. 8), while posi- tive CF
>0 eV gives rise to an in-plane AFM phase (light orange). For strong CF /divides.alt0
/divides.alt0>0 both phases stay stable up to =0. For weak SOC <0:04 eV and CF 0:05 eV<
<0:15 eV, the interaction part in the Hamil- tonian becomes dominant. This is similar to Sec. III B,8 Sx(k,ω) Sy(k,ω) Sz(k,ω) (0,π)(0,0)(π,π)(π,0)k (,)π 2π 2(,)π 2π 2 0 0.02 0.04 0.06 0.08 0.1 ω/ eV0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1 ω/ eV(0,π)(0,0)(π,π)(π,0)k (,)π 2π 2(,)π 2π 2(b) (c) (d)TxyTz(a) FIG. 9. Dynamical spin structure factor S(k;!)for (a)
= 0:25 eV;=0:065 eV, (b)
=0:0 eV;=0:06 eV, (c)
= 0:15 eV;=0 eV, and (d)
=0:0 eV;=0:03 eV. Parameters in (a) are the ones used to describe Ca 2RuO 4[26] and capture the characteristics of INS experiments. which leads to an out-of-plane stripy phase (light blue). The competition between the stripy and the out-of-plane AFM phase leads to the \3-up-1-down" phase (dark blue) already discussed in Sec. III A at −0:05 eV<
<0:05 eV for<0:04 eV. This phase diagram is in good qualitative agreement with the spin-orbit model (Fig. 4). The exact location of the phase transitions dier somewhat between semiclassi- cal and quantum models. In comparison to the ED simu- lations, in the semiclassical calculations the AFM phases (both in- and out-of-plane) are more dominant. While ED predicts the z-AFM phase to end at
≈−0:1 eV for=0 eV, in the semiclassical simulations the z-AFM phase stays robust until
≈−0:05 eV (same for the x−y- AFM ordering see Fig. 7 and Fig. 8). While the origin for the the dierence might lie in the small clusters used (especially for ED), it is quite plausible that quantum uctuations have the strongest impact near orbital de- generacy. The fact that semiclassical MC captures the same phases as ED, gives a promising pathway that ef- fective spin-orbital models can also be studied on signif- icant larger cluster size with semiclassical MC while still giving reasonable results. D. Dynamic spin-structure factor Experimentally, the various phases might be distin- guished via magnetic excitations. Therefore we dis- cuss here the signatures expected for the dynamic spin- structure factor S(k;!)=−1 Im/uni27E80/divides.alt0S(−k)1 !−H+i0+S(k)/divides.alt00/uni27E9; (12) / / (a) (b)FIG. 10. Excitation energy !maxatk=(0;0)(blue) and hole density nh xyin thexyorbital (red). (a) Depending on SOC for CF
=0:25 eV and (b) depending on CF for SOC =0:065 eV. which gives an !resolution of the phases introduced in Fig. 5. This can then be compared to inelastic neutron scattering [17, 24]. In Fig. 9 the DSSF's of the four dis- tinct phases are shown. The locations of these snap- shots in the phase diagram are denoted with white dots in Fig.7. 1. Excitations of the in-plane AFM regime For
=0:25 eV and=0:065 eV [Fig. 9 (a)], the the Goldstone mode at (;)allows us to identify the in- plane AFM phase found above in Fig. 5 (b) and (d). The spectrum of Fig. 9(a) was already presented in Ref. [26] as the parameters closely t Ca 2RuO 4. As already dis- cussed in [26] the in-plane (red guideline) and out-of- plane (blue guideline) transverse modes can be identi- ed. Especially the in-plane transverse mode shows an excellent agreement to [17] reproducing the maximum at k=(0;0)and!=0:54 eV. This maximum, a characteristic signature of the x-y symmetry of the magnetic moments, strongly depends on the hole density nh xyin thexy-orbital, which is nh xy≈0:25 in Fig. 9(a). Figure 10(a) shows the dependence of nh xy and of the excitation energy !Max(0;0)on SOC. The excitation energy at k=(0;0)increases steadily from a minimum at !≈0:02 eV to the maximum !=0:54 eV for the Ca 2RuO 4parameters in Fig. 9 (a). Having a maximum at k=(0;0)is thus closely connected to nite - but not necessarily large - hole density in the xy-orbital. Without SOC, strong CF
=0:25 eV localizes the two holes in the zx- andyz-orbital, with nh xy≈0:05, in agree- ment with ab-initio calculations for Ca 2RuO 4performed without SOC [43, 44]. Increasing SOC softens this po- larization because it couples SandLand thus competes with
. SOC increases the hole density at xyso that it reachesnh xy=0:25 at=0:065 eV. On one hand, this implies that the xy-orbital continues to be rather close to fully occupied and justies the picture of Ca 2RuO 4 as orbitally ordered [25]. On the other hand, Figs. 10(a) and 9(a) reveal that the relatively few holes in the xy- orbital have a decisive impact on magnetic excitations. Vice versa , if SOC is xed and the CF is increased [Fig. 10 (b)] the maximum at k=(0;0)vanishes. Starting at=0:065 eV and
=0:25 eV the maximum is, as9 Mx(k,ω) My(k,ω) Mz(k,ω) (b) 0 ω/ eV0 0.02 0.04 0.06 0.08 0.1 ω/ eV(0,π)(0,0)(π,π)(π,0)k (,)π 2π 2(,)π 2π 2 ωGap ωGap(a) 0.02 0.04 0.06 0.08 0.1 0.12 FIG. 11. Dynamical magnetic structure factor M(k;!)for (a)
=0:0 eV and (b)
=0:25 eV with substantial SOC =0:12 eV.!Gapmarks the energy gap between the ground state and the lowest lying excitation. already discussed, at !=54 meV. Increasing
up to
=0:6 eV strongly suppresses the hole density in the xy-orbital and at the same time leads to a minimum in the excitation spectrum at k=(0;0)and!=36 meV. It is of note that while the hole density appears to be linked to!atk=(0;0), it is not the only in uence. This can be concluded by the fact that for the parameter settings
=0:25 eV;=0:015 eV and
=0:6 eV;= 0:065 eV the hole densities are very similar ( nh xy≈0:05) while!Max(0;0)of the excitation diers by a factor of 1 :8 between strong and weak values of SOC and CF. This means that SOC and CF also have direct in uence to the excitation at k=(0;0)in addition to the indirect in uence via the hole density of nh xy. Taken together, the extensive study of the excitation at k=(0;0)has shown that excitation spectra already dier from the one measured in [17] for relatively weak changes inand
, even though the ground state of Ca 2RuO 4is quite robust against such perturbations. It is therefore remarkable that the DSSF in Fig. 9(a) of the eective model is in such close agreement with the experimental data. 2. Excitations of the PM and various out-of-plane AFM phases Decreasing CF to
=0:0 eV and leaving =0:06 eV, the lowest excitation only has out-of-plane contributions [Fig. 9(b)]. This indicates z-AFM ordering, cf. Fig. 5 (a) and (d), although the system is here close to the PM state, see Fig. 7. Choosing a large value for SOC =0:12 eV rmly puts the system into the PM state, and the excitation minimum at (;)moves to higher !. This can be seen in Fig. 11(a), with the magnetization M=2S−Land the dynamical magnetic structure factor obtained analogue to (12). The excitation gap is !Gap= 0:046 eV [Fig. 11(a)] meaning there is a signicant energy cost for the system to create a triplon. Increasing the CF to
=0:25 eV [Fig. 11 (b)] one can see that (i) the lowest-energy triplon has now x- ycharacter and (ii) its energy is decreased signicantly to!Gap=0:027 eV. The nite CF thus reduces triplon (0,π)(0,0)(π,π)(π,0)k (,)π 2π 2(,)π 2π 2 (0,π)(0,0)(π,π)(π,0)k (,)π 2π 2(,)π 2π 2 0 0.02 0.04 0.06 0.08 0.1 ω/ eV0 0.02 0.04 0.06 0.08 0.1 ω/ eV0 0.02 0.04 0.06 0.08 0.1 ω/ eV0 0.02 0.04 0.06 0.08 0.1 ω/ eVLx(k,ω) Ly(k,ω) Lz(k,ω) Sx(k,ω) Sy(k,ω) Sz(k,ω) (a) (b) (c) (d)FIG. 12. Dynamical Spin and Orbital structure factors for crystal eld
=−0:3 eV and weak SOC. (a) and (c) show the DSSF (12) while (b) and (d) give the orbital analogue based on (3). In (a) and (b), =0:002 eV and in (c) and (d), =0:01 eV. energy so that they can eventually condense into mag- netic order. This can also be seen nicely in Fig. 7 where Ca2RuO 4(corresponding white dot in Fig. 7) would be in the PM phase if it had no signicant CF splitting. Spectra for the stripy and and \3-up-1-down" phases realized near orbital degeneracy are shown in Fig. 9 (c) resp. (d). The stripy phase (
=0:15 eV;=0 eV) in Fig. 9 (c) not only shows spin isotropy but also a degen- eracy between x- (;0) andy-stripy (0;) order. Finally, the DSSF of the \3-up-1-down" phase from Fig. 5 (a) is displayed in Fig. 9 (d) and shows the many ordering vectors contributing for !→0. The last phase to be discussed in detail is the checker- board AFM order with out-of-plane anisotropy at
/uni22720. For=0, moderate CF
≈−0:3 eV is enough to x thexyorbital to half lling, so that either xzoryz orbitals are double occupied. These two states alter- nate in a checkerboard pattern with the same unit cell as a Heisenberg-symmetric AFM. A corresponding mag- netic excitation spectrum is shown in Fig. 12(a), where weak=0:0002 eV induces slight Ising anisotropy into a nearly isotropic spectrum. For the orbital analogue to the DSSF, the spin operator Sin (12) is replaced by angular-momentum operators (3). The resulting spec- trum shown in Fig. 12(b) is, however, featureless, be- cause alternating order in real orbitals is quadrupolar and would show up in the (Lx)2−(Ly)2∝nxz−nyzchannel. Already for rather small =0:01 eV, however, Ising anisotropy in spin excitations is very pronounced with an ordered moment along zand a substantial excitation gap, see Fig. 12(c). At the same time, orbital order is now also clearly dipolar and peaked at (;), see Fig. 12(d). SOC has thus coupled spin and orbital ordering into a checkerboard pattern with Lz=1,Sz=−1 in one sub- lattice and Lz=−1,Sz=1 on the other. In contrast10 to
>0 discussed above, where SOC induces a gradual crossover from a Heisenberg spin-one system to an ex- citonic AFM state, the transition between the isotropic and Ising states is here much more abrupt. IV. DISCUSSION AND CONCLUSIONS In this paper we investigate an eective low-energy spin-orbital Hamiltonian for spin-orbit coupled Mott in- sulators like Ca 2RuO 4. This model interpolates from the strong-SOC regime, where a description in terms of triplons is applicable, to vanishing SOC and moreover includes Hund's coupling and anisotropic hopping. For this model, we performed ED calculations on a√ 8×√ 8 square lattice to obtain both static and dynamic SSF's for varying CF
and SOC . The results for the static SSF indicated the existence of four distinct phases. Namely a z-AFM and xy-AFM with checkerboard pattern, stripy- AFM and a \3-up-1-down" phase at small CF and SOC /uni22730. The stripy and \3-up-1-down" arise near orbital degeneracy, i.e., when neither SOC nor CF dominate, out of the competition and partial frustration of various superexchange terms. The two checkerboard phases, in contrast, extend to large CF's and include excitonic vari- ants at moderate SOC, whereas strong SOC nally drives a transition to a PM state. We supplemented the ED analysis of the full quantum model with MC calculations for a semiclassical variant of the same spin-orbital model on a 4 ×4 cluster. Over- all agreement between the semiclassical and quantum- mechanical models was quite good, with the largest dif- ferences found around orbital degeneracy
;≈0. The transition to the PM J=0 phase at strong SOC cou- pling was claried with the help of an eective triplon model comparable to [14]. Combining these results gave us a complete
−phase diagram that establishes the competition of CF and SOC for strongly correlated t4 2gsystems. We also investigate the DSSF and show that there is a remarkable correspondence [26] between calculations based on ab initio parameters and neutron-scattering re- sults, despite the fact that the calculations appear to strongly depend on the hole density in the xy-orbital. Pa- rameter dependence is also quite sensitive, which makes this a stringent test of the model that allows a distinction between orbital degeneracy lifted by a CF or by SOC. We further give spectra expected for the other phases found with the model. In contrast to the gradual impact of SOC on the exci- tations of the orbitally polarized regime
>0, a much clearer transition is revealed at
<0. Relatively small SOC is enough to switch from alternating orbital order and Heisenberg AFM to order involving complex orbitals. However, coupling to further lattice distortions, not dis- cussed here, would be expected to push this transition to stronger SOC. This physics might also be relevant to t2 2g systems, i.e., with two electrons as in vanadates, where similar alternating or orbital order arises [41]. Although SOC for two electrons has opposite sign than the two- hole case discussed here, this does not qualitatively aect results in the parameter regime with eective Ising sym- metry, i.e. when
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1903.03360v1.Intra__and_inter_orbital_correlated_electron_spin_dynamics_in___rm_Sr_2_Ir_O_4___spin_wave_gap_and_spin_orbit_exciton.pdf

arXiv:1903.03360v1 [cond-mat.str-el] 8 Mar 2019Intra- and inter-orbital correlated electron spin dynamic s in Sr2IrO4: spin-wave gap and spin-orbit exciton Shubhajyoti Mohapatra1and Avinash Singh1 1Department of Physics, Indian Institute of Technology, Kanpu r - 208016, India (Dated: March 11, 2019) Transformation of Coulomb interaction terms to the pseudo- orbital basis consti- tuted by J= 1/2 and 3/2 states arising from spin-orbit coupling provides a ver- satile tool. This formalism is applied to investigate magne tic anisotropy effects on low-energy spin-wave excitations as well as high-energy sp in-orbit exciton modes in Sr2IrO4. The Hund’s coupling term explictly yields easy-plane anis otropy, resulting in gapless (in-plane) and gapped (out-of-plane) modes, in a greement with recent res- onant inelastic x-ray scattering (RIXS) measurements. The collective mode of inter- orbital, spin-flip, particle-hole excitations with approp riate interaction strengths and renormalized spin-orbit gap yields two well-defined propag ating spin-orbit exciton modes, with energy scale and dispersion in good agreement wi th RIXS studies. PACS numbers: 75.30.Ds, 71.27.+a, 75.10.Lp, 71.10.Fd2 I. INTRODUCTION The iridium based transition-metal oxides exhibiting novel J=1/2 Mott insulating states have attracted considerable interest in recent years in view of the ir potential for host- ing collective quantum states such as quantum spin liquids, topologica l orders, and high- temperature superconductors.1The effective J=1/2 antiferromagnetic (AFM) insulating state in iridates arises from a novel interplay between crystal field , spin-orbit coupling and intermediate Coulomb correlations. Exploration of the emerging qua ntum states in the iridate compounds therefore involves investigation of the correlat ed spin-orbital entangled electronic states and related magnetic properties. Amongtheiridiumcompounds, thequasi-two-dimensional (2D)squa re-latticeperovskite- structured iridate Sr 2IrO4is of special interest as the first spin-orbit Mott insulator to be identified and because of its structural and physical similarity with L a2CuO4.2,3It exhibits canted AFM ordering of the pseudospins below N´ eel temperature TN≈240 K. The canting of the in-plane magnetic moments tracks the staggered IrO 6octahedral rotations about the caxis. The effectively single (pseudo) orbital ( J=1/2) nature of this Mott insulator has motivated intensive finite doping studies aimed at inducing the superc onducting state as in the cuprates.4–10 Recent technological advancements in resonant inelastic X-ray sc attering (RIXS) have been instrumental in the elucidation of the pseudospin dynamics in Sr 2IrO4. In the first published data,11spectra along high-symmetry directions in the Brillouin zone reveal a sin- gle gapless spin-wave mode with a dispersion of ∼200 meV, indicating isotropic nature of pseudo-spin interactions. In subsequent investigations of both p arent and electron-doped compounds, the limited energy resolution of RIXS could not also reso lve any spin-wave gap.8,12However, recent measurements conducted with improved energy resolution point to apartiallyresolved ∼30meVspin-wave gapat theΓ point,13which hasbeen furtherresolved via high-resolution RIXS and inelastic neutron scattering (INS), bo th of which indicate an- other spin-wave gap between 2 to 3 meV at ( π,π).14These low-energy features correspond to different spin-wave modes associated with basal-plane and out-o f-plane fluctuations, in- dicating the presence of anisotropic spin interactions. In addition to spin-wave modes, RIXS experiments have also reveale d a high-energy dis- persive feature in the energy range 0.4-0.8 eV. Attributed to elect ron-hole pair excitations3 across the spin-orbit gap between the J=1/2 and 3/2 bands, this distinctive mode is re- ferred to as the spin-orbit exciton.11,15–18Unusual magnetism has been predicted in recent theoretical investigations for 5 d4and 5d5systems arising from the condensation of spin-orbit excitons.19–22 The anisotropic magnetic interactions such as the pseudo-dipolar ( PD) and Dzyaloshinskii-Moriya (DM) terms within the effective J=1/2 spin model for Sr 2IrO4ac- count for the canted AF state, but do not yield true magnetic aniso tropy and spin-wave gap as the relevant spin-dependent hopping term can be gauged aw ay.4,23,24The easy basal- plane anisotropy has been proposed to arise from the Hund’s couplin g term when virtual excitations to J=3/2 states are included.23,25–27Recently, the pseudo-spin-lattice coupling has been proposed to account for the structural orthorhombic ity, the easy-axis anisotropy within the basal plane, and alignment of moments along the crystallog raphic direction.28 However, electron itineracy, finite mixing between J=1/2 and 3/2 sectors, and weak corre- lation effects play key roles in explaining the magnetic properties of Sr 2IrO4, and thus put limitations on these phenomenological spin models. In terms of multi-orbital itinerant-electron approaches, collectiv e magnetic excitations werestudied withintheHartree-Fock(HF)andtherandomphase a pproximation(RPA).24,29 In ref. [29], the spin-wave mode was shown to be split into two branch es - one gapless and the other gapped - and the spin-wave gap was explained in terms of t he anisotropic exchange coupling attributed to the interplay between Hund’s coupling and spin -orbit coupling. How- ever, the crucial role of the finite magnetic moment in the nominally fille dJ=3/2 sector on the expression of the magnetic anisotropy was not studied. In ref . [24], the focus was on understanding the strong zone-boundary spin-wave dispersion, which was demonstrated as arising from finite- Uand finite-SOC effects. Appreciable mixing between J=1/2and3/2 sectors, especially near the Fermi energy, has beenshownininvestigationsofthepseudo-orbital-resolvedelectr onicbandsusingthedensity functional theory (DFT) approach30–32and realistic three-orbital models.24,33–35Significant deviation from ideal fillings ( n1/2=1 andn3/2=4) for the two sectors in Sr 2IrO4,30and small magnetic moment in the J=3/2 sector have also been reported,24,34implying break-down of the one-band ( J=1/2) picture in real systems. Within a minimal extension of this pictu re which can provide a unified description of the observed high-energy features as discussed above, investigation of the coupling and excitations between the J=1/2 andJ=3/2 sectors4 is therefore of particular interest. In this paper, we therefore plan to investigate intra- and inter-or bital correlated-electron spin dynamics in Sr 2IrO4. Detailed comparison with RIXS data can provide experimental evidence of the several distinctive features associated with the r ich interplay of spin-orbit coupling, Coulomb interaction, and realistic multi-orbital electronic b and structure. These key features include: (i) dispersion of spin-wave and spin-orbit exc iton modes, (ii) finite- U and finite-SOC effects, (iii) mixing between J=1/2 and 3/2 sectors, (iv) Hund’s-coupling- induced spin-rotation-symmetry-breaking and spin-wave gap, (v ) correlation-induced spin- orbit gap renormalization, (vi) coupling between collective and single- particle excitations, and (vii) coupling between magnetic moments in the J=1/2 and 3/2 sectors. The structure of the paper is as follows. After a brief description of the three-orbital model and the pseu do-spin-orbital basis in Sec. II, the transformation of various Coulomb interaction terms from the original three- orbital basis to the pseudo-orbital basis is presented in Sec. III, explicitly showing easy x-y plane anisotropy resulting from the Hund’s coupling term. Represen tation of the AFM state in the pseudo-orbital basis is discussed in Sec. IV. Formulation of th e spin-wave propagator and the calculated dispersion showing the spin-wave gap are presen ted in Secs. V and VI. The spin-orbit gap renormalization due to the relative energy shift b etween the J=1/2 and 3/2 states arising from the density interaction terms is discussed in Sec. VII. The spin-orbit exciton as a resonant state formed by the correlated propagatio n of the inter-orbital, spin- flip, particle-hole excitation across the renormalized spin-orbit gap is investigated in Sec. VIII. Finally, conclusions are presented in Sec. IX. II. THREE-ORBITAL MODEL: PSEUDO-ORBITAL BASIS Due to large crystal-field splitting ( ∼3 eV) in the IrO 6octahedra, the low-energy physics in thed5iridates is effectively described by projecting out the empty e glevels which are well above the t2glevels. Spin-orbit coupling (SOC) further splits the t 2gstates into (up- per)J=1/2 doublet and (lower) J=3/2 quartet with an energy gap of 3 λ/2. Four of the five electrons fill the J=3/2 states, leaving one electron for the J=1/2 sector, rendering it magnetically active in the ground state. Corresponding to the three Kramers pairs above, we introduce th reepseudo orbitals5 FIG. 1: The ‘pseudo-spin-orbital’ energy level scheme for t he three Kramers pairs along with their orbital shapes. The colors represent the weights of real spi n↑(red) and ↓(blue) in each pair. (l= 1,2,3) withpseudo spins (τ=↑,↓) each. The |J,mj/angbracketrightand the corresponding |l,τ/angbracketrightstates have the form: |l= 1,τ=σ/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingle1 2,±1 2/angbracketrightbigg = [|yz,¯σ/angbracketright±i|xz,¯σ/angbracketright±|xy,σ/angbracketright]/√ 3 |l= 2,τ=σ/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingle3 2,±1 2/angbracketrightbigg = [|yz,¯σ/angbracketright±i|xz,¯σ/angbracketright∓2|xy,σ/angbracketright]/√ 6 |l= 3,τ= ¯σ/angbracketright=/vextendsingle/vextendsingle/vextendsingle/vextendsingle3 2,±3 2/angbracketrightbigg = [|yz,σ/angbracketright±i|xz,σ/angbracketright]/√ 2 (1) where|yz,σ/angbracketright,|xz,σ/angbracketright,|xy,σ/angbracketrightare the t 2gstates and the signs ±correspond to spins σ=↑/↓. The coherent superposition of different-symmetry t2gorbitals, with opposite spin polariza- tion between xz/yzandxylevels implies spin-orbital entanglement, and also imparts unique extended 3D shape to the pseudo-orbitals l= 1,2,3, as shown in Fig 1. Inverting theabovetransformation, weobtaintherepresentat ion ofthethree-orbitalbasis states in terms of the pseudo-orbital basis states, given below in t erms of the corresponding creation operators: a† yzσ a† xzσ a† xyσ = 1√ 31√ 61√ 2 iσ√ 3iσ√ 6−iσ√ 2 −σ√ 3√ 2σ√ 30 a† 1τ a† 2τ a† 3τ (2) where,σ=↑/↓andτ=σ. Now, we consider the free part of the three-orbital model Hamilto nian including the SOC6 and band terms represented in the basis ( yzσ,xzσ,xy ¯σ): HSO+Hband=/summationdisplay kµσψ† kµσ Eyz kiσλ 2+Eyz|xz k−σλ 2 −iσλ 2+Eyz|xz k Exz kiλ 2 −σλ 2−iλ 2Exy k ψkµσ (3) whereψ† kµσ=/parenleftig a† kyzσa† kxzσa† kxy¯σ/parenrightig ,Eµ kare the band energies for the three orbitals µ, andλ is the SOC constant. The orbital mixing hopping term Eyz|xz karises from the staggered IrO 6 octahedral rotations in Sr 2IrO4. Applying the transformation given in Eq. (2), the above Hamiltonian is transformed to the pseudo-orbital basis |1,τ=↑,↓/angbracketright,|2,τ=↑,↓/angbracketright, and|3,τ=↑,↓/angbracketright. The orbital mixing hopping term Eyz|xz kleads to pseudo-spin-dependent terms in this basis, which breaks s pin- rotationsymmetry. However, these spin-dependent terms canb egaugedaway by a spin- and site-dependent unitary transformation,24leaving the spin-independent form: HSO+Hband= /summationtext klmElm kψ† kl1 1ψkm, which isinvariant under theSU(2)transformation ψkm→ψ′ km= [U]ψkm in pseudo-spin space. In the above discussion, the two magnetic sublattices correspond ing to the staggered magnetic order have not been included for compactness. The band termEkincludes nearest- neighbor (NN) and next-nearest-neighbor (NNN) hopping terms e tc., which therefore con- nect different or same magnetic sublattice(s), as will be discussed in Sec. IV. III. COULOMB INTERACTION TERMS IN PSEUDO-ORBITAL BASIS We consider the on-site Coulomb interaction terms: Hint=U/summationdisplay i,µniµ↑niµ↓+U′/summationdisplay i,µ<ν,σniµσniνσ+(U′−JH)/summationdisplay i,µ<ν,σniµσniνσ +JH/summationdisplay i,µ/negationslash=ν(a† iµ↑a† iν↓aiµ↓aiν↑+a† iµ↑a† iµ↓aiν↓aiν↑) (4) in the three-orbital basis ( µ,ν=yz,xz,xy ), including the intra-orbital ( U) and inter-orbital (U′) density interaction terms, the Hund’s coupling term ( JH), and the pair hopping term (JH). Herea† iµσandaiµσare the creation and annihilation operators for site i, orbitalµ, spinσ=↑,↓, and the density operator niµσ=a† iµσaiµσ. Usingthetransformationfromthethree-orbitalbasistothepse udo-orbitalbasis( m,m′= 1,2,3) described earlier, and keeping density as well as spin-flip interact ion terms which are7 relevant for the present study, we obtain (for site i): Hint(i) =1 2/summationdisplay m,m′,τ,τ′Uττ′ mm′nmτnm′τ′+/parenleftbiggU−U′ 3/parenrightbigg/summationdisplay τa† 1τa† 2τa1τa2τ +/parenleftbiggU−2JH−U′ 6/parenrightbigg/summationdisplay τ/parenleftig a† 2τa† 3τa2τa3τ+2a† 3τa† 1τa3τa1τ/parenrightig (5) where the transformed interaction matrices Uττ′ mm′in the new basis have the following form: Uττ mm′= 0U′U′−2 3JH U′0U′−1 3JH U′−2 3JHU′−1 3JH0 , Uττ mm′= 1 3(U+2U′)1 3(U+2U′−3JH)1 3(U+2U′−JH) 1 3(U+2U′−3JH)1 2(U+U′)1 6(U+5U′−4JH) 1 3(U+2U′−JH)1 6(U+5U′−4JH)1 2(U+U′) (6) for pseudo-spins τ′=τandτ′=τ, whereτ=↑,↓. Similar transformation to the Jbasis has been discussed recently, focussing only on the density interac tion terms.36 Using the spherical symmetry condition ( U′=U-2JH), the transformed interaction Hamil- tonian can be written in terms of the local density and spin operator s as: Hint(i) =/parenleftbigg U−4 3JH/parenrightbigg n1↑n1↓+(U−JH)[n2↑n2↓+n3↑n3↓] −4 3JHS1.S2+2JH[Sz 1Sz 2−Sz 1Sz 3] +/parenleftbigg U−13 6JH/parenrightbigg [n1n2+n1n3]+/parenleftbigg U−7 3JH/parenrightbigg n2n3 (7) where the spin operator Sm=ψ† mτ 2ψmand the density operator nm=ψ† m1ψm=nm↑+nm↓ in terms of the local pseudo-spin-orbital field operator ψ† m= (a† m↑a† m↓). TheHubbard-like interactionterms Umnm↑nm↓∼ −UmSm.Smareinvariant under pseudo- spin rotation, as is the Hund’s-coupling-like term S1.S2. Furthermore, under the corre- sponding SU(2) transformation ψm→ψ′ m= [U]ψm, the total density terms nmare in- variant. Therefore, the only interaction terms which break spin ro tation symmetry and are thus responsible for magneto-crystalline anisotropy in Sr 2IrO4are theSz 1Sz 2andSz 1Sz 3 terms. As discussed earlier, the magnetically active sector is the no minally half-filled m= 1 pseudo-orbital. Magnetic moments in the nominally doubly occupied m= 2,3 orbitals are8 very small. As Sz 2andSz 3are proportional to Sz 1within a classical spin picture, the mag- netic anisotropy terms can be written as D(Sz 1)2, corresponding to an effective single-ion anisotropy. As shown below, we will find that D>0, indicating easy x-yplane anisotropy. IV. AF STATE: STAGGERED FIELD TERM We consider the ( π,π) ordered AF state on the square lattice, focussing on the stagge red field terms within the pseudo-orbital basis arising from the Hartree -Fock (HF) approxima- tion of the various interaction terms in Eq. (7). The charge terms c orresponding to density condensates in this approximation will be discussed in Sec. VII. For g eneral ordering direc- tion with components ∆l= (∆x l,∆y l,∆z l), the staggered field term for sector lis given by: Hsf(l) =/summationdisplay ksψ† kls/parenleftig −sτ.∆l/parenrightig ψkls=/summationdisplay ks−sψ† kls ∆z l∆x l−i∆y l ∆x l+i∆y l−∆z l ψkls (8) whereψ† kls= (a† kls↑a† kls↓),s=±1 for the two sublattices A/B, and the staggered field components ∆x,y,z l=1,2,3are self-consistently determined from: 2∆z 1=U1mz 1+2JH 3mz 2+JH(mz 3−mz 2) 2∆x,y 1=U1mx,y 1+2JH 3mx,y 2 2∆z 2=U2mz 2+2JH 3mz 1−JHmz 1 2∆x,y 2=U2mx,y 2+2JH 3mx,y 1 2∆z 3=U3mz 3+JHmz 1 2∆x,y 3=U3mx,y 3 (9) in terms of the staggered pseudo-spin magnetizations ml=mx l,my l,mz lfor the three pseudo- orbitalsl= 1,2,3. In practice, it is easier to choose set of ∆l=1,2,3and self-consistently determine the Hubbard-like interaction strengths Ul=1,2,3such that U1=U−4 3JHandU2= U3=U−JHusing Eq. (9). The interaction strengths are related by Ul=2,3=Ul=1+JH/3. Transforming back to the three-orbital basis ( yzσ,xzσ,xy ¯σ), the staggered-field contri-9 bution for the l= 1 sector is illustrated below: Hl=1 sf=/summationdisplay kσssσψ† kσs ∆z 1 3 1iσ−σ −iσ1i −σ−i1 δσσ′+/parenleftbigg−∆x 1+i∆y 1 3/parenrightbigg 1iσ−σ −iσ−1i −σ−i−1 δ¯σσ′ ψkσ′s (10) which has similar structure as the spin-orbit coupling term. Including the SOC and band terms, thefullHFHamiltonianconsideredinourbandstructureand spinfluctuationanalysis is given by HHF=HSO+Hband+Hsf, where, HSO+Hband=/summationdisplay kσsψ† kσs ǫyz k′iσλ 2−σλ 2 −iσλ 2ǫxz k′iλ 2 −σλ 2−iλ 2ǫxy k′ δss′+ ǫyz kǫyz|xz k0 −ǫyz|xz kǫxz k0 0 0 ǫxy k δ¯ss′ ψkσs′ (11) in the composite three-orbital, two-sublattice basis, showing the d ifferent hopping terms connecting the same and opposite sublattice(s). Corresponding to the hopping terms in the tight-binding model, the v arious band disper- sion terms in Eq. (11) are given by: ǫxy k=−2t1(coskx+cosky) ǫxy k′=−4t2coskxcosky−2t3(cos2kx+cos2ky)+µxy ǫyz k=−2t5coskx−2t4cosky ǫxz k=−2t4coskx−2t5cosky ǫyz|xz k=−2tm(coskx+cosky). (12) Heret1,t2,t3are respectively the first, second, and third neighbor hopping ter ms for the xyorbital, which has energy offset µxyfrom the degenerate yz/xzorbitals induced by the tetragonal splitting. For the yz(xz) orbital,t4andt5are the NN hopping terms in y(x) andx(y) directions, respectively. Mixing between xzandyzorbitals is represented by the NN hopping term tm. We have taken values of the tight-binding parameters ( t1,t2,t3,t4, t5,tm,µxy,λ) = (1.0, 0.5, 0.25, 1.028, 0.167, 0.2, -0.7, 1.35) in units of t1, where the energy scalet1= 280 meV. Using above parameters, the calculated electronic band structure shows AFM insulating state and mixing between pseudo-orbital sectors.24,3310 -0.8585-0.858-0.8575-0.857-0.8565-0.856-0.8555-0.855 0 0.1 0.2 0.3 0.4 0.5 0.6EHFg Canting angle ( φ)x-y plane z-direction (a)tm= -0.15 JH= 0 -0.8542-0.854-0.8538-0.8536-0.8534-0.8532-0.853-0.8528-0.8526 0 0.2 0.4 0.6 0.8 1EHFg θ/(π/2)easy x-y planefinite JH tm= 0 (b) FIG. 2: Variation of AFM state energy per state (a) for x-yplane ordering with canting angle φ including finite yz−xzorbital mixing hopping showing degeneracy at the optimal ca nting angle with the z-ordered AFM state, and (b) with the staggered field polar ang leθshowing easy x-y plane anisotropy for finite Hund’s coupling. Canted AFM state and JHinduced easy-plane anisotropy The octahedral-rotation-induced orbital mixing hopping term ( tm) betweenyzandxz orbitals generates PD ( Sz iSz j) and DM [( /vectorSi×/vectorSj).ˆz] anisotropic interactions in the strong coupling limit.24However, the AFM-state energy is invariant with respect to chang e of ordering direction from zaxis tox-yplane provided spins are canted at the optimal canting angle, thus preserving the gapless Goldstone mode. Fig. 2(a) show s the variation of AFM- state energy with canting angle ( φ) for ordering in the x-yplane. The energy minimum at the optimal canting angle is exactly degenerate with the energy for z-direction ordering. The Hund’s-coupling-induced easy-plane magnetic anisotropy is exp licitly shown in Fig.2(b) by the variation of AFM-state energy with polar angle θcorresponding to stag- gered field orientation in the x-zplane, with ∆z 1= (∆+∆ ani)cosθand ∆x 1= ∆sinθ. Here ∆ represents the spin-rotationally-symmetric part [( U1m1+2JH 3m2)/2] of the staggered field term forl=1 orbital. The symmetry-breaking term ∆ anicorresponds to the additional con- tribution1 2JH(mz 3−mz 2), as seen from Eq. (9). Here ∆=0.9, ∆ ani=−0.01, and the orbital mixing hopping term tmhas been set to zero for simplicity. The simplified analysis presented in this subsection, with staggered field only for the l=1 orbital, serves to explicitly illustrate the magnetic anisotropy features within our band picture.11 V. MAGNETIC ANISOTROPY AND GAPPED SPIN WAVE In view of the Hund’s-coupling-induced easy-plane anisotropy as dis cussed above, we consider the x-ordered AFM state. The spin-wave propagator corresponding t o transverse spin fluctuations should therefore yield one gapless mode ( ydirection) and one gapped mode (zdirection). Accordingly, we consider the time-ordered spin-wave p ropagator: χ(q,ω) =/integraldisplay dt/summationdisplay ieiω(t−t′)e−iq.(ri−rj)/angbracketleftΨ0|T[Sα im(t)Sβ jn(t′)]|Ψ0/angbracketright (13) involving the transverse α,β=y,zcomponents of the pseudo-spin operators Sα imandSβ jn for pseudo orbitals mandnat lattice sites iandj. In the random phase approximation (RPA), the spin-wave propaga tor is obtained as: [χ(q,ω)] =[χ0(q,ω)] 1−2[U][χ0(q,ω)](14) where the bare particle-hole propagator: [χ0(q,ω)]αβ ab=1 4/summationdisplay k/bracketleftigg /angbracketleftϕk−q|τα|ϕk/angbracketrighta/angbracketleftϕk|τβ|ϕk−q/angbracketrightb E+ k−q−E− k+ω−iη+/angbracketleftϕk−q|τα|ϕk/angbracketrighta/angbracketleftϕk|τβ|ϕk−q/angbracketrightb E+ k−E− k−q−ω−iη/bracketrightigg (15) was evaluated in the composite spin-orbital-sublattice basis (2 spin c omponents α,β=y,z ⊗3 pseudo orbitals m= 1,2,3⊗2 sublattices s,s′= A,B) by integrating out the fermions in the (π,π) ordered state. Here Ekandϕkare the eigenvalues and eigenvectors of the Hamiltonian matrix in the pseudo-orbital basis, the indices a,b= 1,6 correspond to the orbital-sublattice subspace, and the superscript +( −) refers to particle (hole) energies above (below) the Fermi energy. The amplitudes ϕm kτwere obtained by projecting the kstates in the three-orbital basis on to the pseudo-orbital basis states |m,τ=↑,↓/angbracketrightcorresponding to theJ= 1/2 and 3/2 sector states, as given below: ϕ1 k↑=1√ 3/parenleftbig φyz k↓−iφxz k↓+φxy k↑/parenrightbig ϕ1 k↓=1√ 3/parenleftbig φyz k↑+iφxz k↑−φxy k↓/parenrightbig ϕ2 k↑=1√ 6/parenleftbig φyz k↓−iφxz k↓−2φxy k↑/parenrightbig ϕ2 k↓=1√ 6/parenleftbig φyz k↑+iφxz k↑+2φxy k↓/parenrightbig ϕ3 k↑=1√ 2/parenleftbig φyz k↓+iφxz k↓/parenrightbig ϕ3 k↓=1√ 2/parenleftbig φyz k↑−iφxz k↑/parenrightbig (16) in terms of the amplitudes φµ kσin the three-orbital basis ( µ=yz,xz,xy ). The rotationally invariant Hubbard- and Hund’s coupling-like terms ha ving the form Sα imSβ inδαβare diagonal in spin components ( α=β). The on-site Coulomb interaction terms12 0 50 100 150 200 250ωq (meV) (π,0) ( π,π) (π/2,π/2) (0,0) ( π,0)yz FIG. 3: The calculated spin-wave dispersion in the three-or bital model with staggered field in the xdirection. The easy x-yplane anisotropy arising from Hund’s coupling results in on e gapless mode and one gapped mode corresponding to transverse fluctua tions in the yandzdirections, respectively. are also diagonal in the sublattice basis ( s=s′). The interaction matrix [ U] in Eq. (14) is therefore obtained as: [U] = U12 3JH0 2 3JHU20 0 0U3 δαβδss′+ 0−JHJH −JH0 0 JH0 0 δαzδβzδss′ (17) in the pseudo-orbital basis. While the first interaction term above p reserves spin rotation symmetry, the second interaction term (corresponding to the Sz imSz interms in Eq. 7) breaks rotation symmetry and is responsible for easy x-yplane anisotropy. The spin wave energies are calculated from the poles of Eq. 14. The 12 ×12 [χ0(q,ω)] matrix was evaluated by performing the ksum over the 2D Brillouin zone divided into a 300 ×300 mesh. VI. SPIN-WAVE DISPERSION The calculated spin-wave energies in the x-ordered AFM state are shown in Fig. 3. Here we have taken staggered field values ∆x l=1,2,3= (0.92,0.08,−0.06) in units of t1, which ensures self-consistency for all three orbitals, with the given rela tionsU2=U3=U1+JH/3. Using the calculated sublattice magnetization values mx l=1,2,3=(0.65,0.005,-0.038), we obtain Ul=1,2,3=(0.80,0.83,0.83) eV, which finally yields U=U1+4 3JH=0.93 eV for JH=0.1 eV.13 25 30 35 40 45 50 0.01 0.02 0.03 0.04 0.05 0.2 0.3 0.4 0.5 0.6 0.7 0.8spin-wave gap (meV) λ (eV) magnetic moment |mx 3| FIG. 4: Variation of spin-wave gap with magnetic moment |mx 3|in thel=3 orbital (blue curve). The magnitude of |mx 3|decreases with SOC strength (red curve) due to suppression o f mixing between J=1/2 and 3/2 sectors. Here ( U,JH)=(0.93 eV, 0.1 eV). The spin-wave dispersion clearly shows the Goldstone mode and the g apped mode, corre- sponding to transverse spin fluctuations in the yandzdirections, respectively. The easy x-y plane anisotropy arising from Hund’s coupling results in energy gap ≈40 meV for the out- of-plane (z) mode. The two modes are degenerate at ( π,0) and (π/2,π/2). The excitation energy at ( π,0) is approximately twice that at ( π/2,π/2), and the strong zone-boundary dispersion in this iridate compound was ascribed to finite- Uand finite-SOC effects.24The calculated spin-wave dispersion and energy gap are in very good agr eement with RIXS measurements.11,13–15 The electron fillings in the different pseudo orbitals are obtained as nl=1,2,3≈ (1.064,1.99,1.946). Finite mixing between the J=1/2 and 3/2 sectors is reflected in the small deviations from ideal fillings and also in the very small magnetic mo ment values for l= 2,3 as given above, which play a crucial role in the expression of magnet ic anisotropy and spin-wave gap in view of the anisotropic interaction terms in Eq. ( 7) involving the Hund’s coupling JH. The values λ=0.38 eV,U=0.93 eV, and JH=0.1 eV taken above lie well within the estimated parameter range for Sr 2IrO4.18,35 We have investigated the crucial role of the small J=3/2-sector magnetic moment by studying the variation of the spin-wave gap with SOC strength which effectively controls the mixing between J=1/2 and 3/2 sectors. Fig. 4 shows that the spin-wave gap sharply increases with magnetic moment |mx 3|in thel=3 orbital (the dominant moment), indicating14 a finite-SOC effect on the experimentally observed out-of-plane sp in-wave gap in Sr 2IrO4. The opposite sign of the magnetic moment mx 3as compared to mx 1(due to spin-orbital entanglement) plays a vital role in the easy-plane anisotropy. VII. RENORMALIZED SPIN-ORBIT GAP As another application of the transformation described in Sec. III , we now consider the relative energy shift between the J=1/2 and 3/2 states arising from the density interaction terms in Eq. (7). This relative shift effectively renormalizes the spin- orbit gap and plays an important role in determining the energy scale of the spin-orbit excit on, as discussed in the next section. Corresponding to the total density condensate /angbracketleftnl↑+nl↓/angbracketrightin the HF approxi- mation of the density interaction terms, the spin-independent self -energy contributions for the three orbitals are obtained as: Σl=1 dens=U/angbracketleftbigg1 2n1+n2+n3/angbracketrightbigg −JH/angbracketleftbigg2 3n1+13 6n2+13 6n3/angbracketrightbigg Σl=2 dens=U/angbracketleftbigg n1+1 2n2+n3/angbracketrightbigg −JH/angbracketleftbigg13 6n1+1 2n2+7 3n3/angbracketrightbigg Σl=3 dens=U/angbracketleftbigg n1+n2+1 2n3/angbracketrightbigg −JH/angbracketleftbigg13 6n1+7 3n2+1 2n3/angbracketrightbigg (18) The formally unequal contributions will result in relative energy shift s between the three orbitals depending on the electron filling. With /angbracketleftn1/angbracketright=1 and/angbracketleftn2/angbracketright=/angbracketleftn3/angbracketright=2 for thed5system having nominally half-filled and filled orbitals, the relative energy shift: ∆dens= Σl=1 dens−Σl=2,3 dens=U−3JH 2(19) betweenl=1 and (degenerate) l=2,3 orbitals. ForU >3JH, the relative energy shift enhances the energy gap between J=1/2 and 3 /2 sectors, effectively resulting in a correlation-induced renormalizat ion of the spin-orbit gap and the spin-orbit coupling. For d4systems with nominally /angbracketleftn1/angbracketright=0, the relative energy shift increases to U−3JH. This enhancement of the spin-orbit gap renormalization is seen in recent DFT study of the hexagonal iridates Sr 3LiIrO6and Sr 4IrO6with Ir5+(5d4) and Ir4+ (5d5) ions, respectively.37 The SOC strength is renormalized as ˜λ=λ+2∆dens/3 by the relative energy shift. With ∆dens= (U−3JH)/2≈0.3 eV for the parameter values considered earlier, we obtain ˜λ≈15 (π,0) ( π,π) (π/2,π/2) (0,0) 0 100 200 300 400 500ω (meV) 0.1 1 10 100 (π,0) (a)(π,0) ( π,π) (π/2,π/2) (0,0) 0 100 200 300 400 500ω (meV) 0.1 1 10 100 (π,0) (b) FIG. 5: (a) Calculated spin-wave spectral function with (a) the renormalized SOC and (b) the bare SOC, showing the squeezing of the spin-wave mode by the parti cle-hole excitations near ( π/2,π/2). 0.6 eV, which is in agreement with the correlation-enhanced SOC stre ngth obtained in a recent DFT study of Sr 2IrO4.35The SOC renormalization also improves the comparison of spin-wave dispersion with experiment near ( π/2,π/2) as shown in Fig. 5(a). With the bare SOC strength, the collective spin-wave mode is squeezed by the par ticle-hole excitation, as seen in Fig. 5(b). The renormalized spin-orbit gap increases the par ticle-hole excitation energy and thereby removes the flattening. By effectively suppre ssing the mixing between J=1/2 and 3/2 sectors, the SOC renormalization also strengthens t he AFM state. VIII. SPIN-ORBIT EXCITON The low-energy collective (spin-wave ) modes investigated in Secs. V and VI essentially involve intra-orbital spin-flip excitations within the magnetically activ eJ=1/2 sector. In this section, we will investigate inter-orbital, spin-flip, particle-hole excitations across the spin-orbit gap between the nominally filled J=3/2 sector and the half-filled J=1/2 sector. For thez-ordered AFM state, we consider the composite pseudo-spin-orb ital fluctuation propagator: χ−+ so(q,ω) =/integraldisplay dt/summationdisplay ieiω(t−t′)e−iq.(ri−rj)/angbracketleftΨ0|T[S− i,m,n(t)S+ j,m,n(t′)]|Ψ0/angbracketright (20)16 involving the inter-orbital spin-lowering and -raising operators S− i,m,n=a† in↓aim↑and S+ j,m,n=a† jm↑ajn↓at lattice sites iandj, describing the propagation of a spin-flip particle-hole excitation between different pseudo orbitals mandn. Althoughthe most general propagator would involve S− i,m,nandS+ j,m′,n′, the above simplified propagator is a good approximation in view of the orbital restrictions on the particle-hole states as discu ssed below. Also, we have considered the z-ordered AFM state as the weak easy-plane anisotropy has negligib le effect on the spin-orbit exciton. In the ladder-sum approximation, the spin-orbital propagator is o btained as: [χ−+ so(q,ω)] =[χ0 so(q,ω)] 1−U[χ0so(q,ω)](21) where the relevant interactions U=Uττ mnfor the spin-flip particle-hole pair are given in Eq. (6), and the bare particle-hole propagator: [χ0 so(q,ω)]mn ss′=/summationdisplay k/bracketleftigg /angbracketleftϕn k−q|τ−|ϕm k/angbracketrights/angbracketleftϕm k|τ+|ϕn k−q/angbracketrights′ E+ k−q−E− k+ω−iη+/angbracketleftϕn k−q|τ−|ϕm k/angbracketrights/angbracketleftϕm k|τ+|ϕn k−q/angbracketrights′ E+ k−E− k−q−ω−iη/bracketrightigg (22) was evaluated using the projected amplitudes given in Eq. 16. The lad der-sum approxima- tionwithrepeated(attractive)interactionsrepresents resona ntscatteringoftheparticle-hole pair, resulting in a resonant state split-off from the particle-hole co ntinuum, which we iden- tify as the spin-orbit exciton mode. The dominant contribution to the bare particle-hole propagator ab ove will correspond to particle (+) states in the nominally half-filled pseudo-orbital m=1 (J=1/2 sector) and hole (−) states in the nominally filled pseudo-orbitals n=2,3 (J=3/2 sector). Due to these restrictions, the bare propagator essentially becomes diagonal in the composite particle-hole orbital basis ( m′=m,n′=n), which justifies the simplified propagator considered above. In order to focus exclusively on the high-energy spin-orbit exciton mo de, particle-hole exci- tations within the J=1/2 sector (which yield the low-energy spin-wave mode) have been excluded. Fig. 6 shows the spin-orbit exciton spectral function: Aq(ω) =1 πIm Tr/bracketleftbig χ−+ so(q,ω)/bracketrightbig (23) as an intensity plot for qalong the high symmetry directions of the BZ. For clarity, we have considered here the particle-hole propagator for m=1 andn=3,2 separately in Eq. (22), for which the relevant interaction terms are: Uττ 13=U-5JH/3 andUττ 12=U-7JH/3. Here, we have17 (π/2,π/2)(π,0) (π,π) (π/2,π/2) (0,0) ( π,0) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9ω (eV) 0.001 0.01 0.1 1 10 n = 3 (a) (π/2,π/2)(π,0) (π,π) (π/2,π/2) (0,0) ( π,0) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9ω (eV) 0.001 0.01 0.1 1 10 n = 2 (b) FIG. 6: The spin-orbit exciton spectral function Aq(ω) for the two cases: (a) n=3 and (b) n=2, showingwell defineddispersivemodesnearthelower edge oft hecontinuum. Theexciton represents collective spin-orbital excitations across the renormali zed spin-orbit gap. takenU=0.93 eV and JH=0.1 eV as in Sec. VI, the three-orbital model parameters are sam e as in Sec. IV, and the renormalized spin-orbit gap has been incorpor ated. Thespin-orbitexcitonspectralfunctioninFig. 6(a)clearlyshowsa welldefinedpropagat- ing mode near the lower edge of the continuum with significantly higher intensity compared to the continuum background. With increasing interaction strengt h, this mode progressively shifts to lower energy further away from the continuum, and beco mes more prominent in intensity, confirming its distinct identity from the continuum backgr ound. Fig. 6(b) shows a similar exciton mode for the other case ( m=1,n=2), with slightly higher energy and reduced dispersion as well as significant damping. The relatively reduced interactionstrength Uττ 12forthismodeaccountsfortheslightlyhigher energy. Thecalculate d dispersion and energy scale of the two spin-orbit exciton modes are in excellent agreement withthetwoexcitonmodesreportedinRIXSinvestigations11,13ofSr2IrO4aswellasprevious theoretical studies.18 IX. CONCLUSIONS Transformation of the various Coulomb interaction terms to the ps eudo-orbital basis formed by the J=1/2 and 3/2 states was shown to provide a versatile tool for inves tigating magnetic anisotropy effects as well as the spin-orbit exciton modes in the strongly spin-orbit coupled compound Sr 2IrO4. Explicitly pseudo-spin-symmetry-breaking terms were obtained18 (dominantly ∼JHSz 1Sz 3), resulting in easy x-yplane anisotropy and gap for the out-of-plane spin-wave mode, reflecting the importance of mixing with the J=3/2 sector in determining the magnetic properties of this compound. Well-defined propagating spin-orbit exciton modes were obtained re presenting collective modes of inter-orbital, spin-flip, particle-hole excitations, with bot h dispersion and energy scale in excellent agreement with RIXS studies. 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2003.00582v2.Spin_orbit_interaction_and_spin_selectivity_for_tunneling_electron_transfer_in_DNA.pdf

Spin-orbit interaction and spin selectivity for tunneling electron transfer in DNA Solmar Varela,1,Iskra Zambrano,2Bertrand Berche,3Vladimiro Mujica,4and Ernesto Medina2, 5,y 1Yachay Tech University, School of Chemical Sciences & Engineering, 100119-Urcuqu , Ecuador 2Yachay Tech University, School of Physical Sciences & Nanotechnology, 100119-Urcuqu , Ecuador 3Laboratoire de Physique et Chimie Th eoriques, UMR Universit e de Lorraine-CNRS 7019 54506 Vanduvre les Nancy, France 4School of Molecular Sciences, Arizona State University, Tempe, Arizona 85287-1604, USA 5Simon A. Levin Mathematical, Computational and Modeling Sciences Center, Arizona State University, Tempe, Arizona 85287-1604, USA (Dated: March 18, 2020) Electron transfer (ET) in biological molecules such as peptides and proteins consists of electrons moving between well dened localized states (donors to acceptors) through a tunneling process. Here we present an analytical model for ET by tunneling in DNA, in the presence of Spin-Orbit (SO) interaction, to produce a strong spin asymmetry with the intrinsic atomic SO strength in meV range. We obtain a Hamiltonian consistent with charge transport through orbitals on the DNA bases and derive the behavior of ET as a function of the injection state momentum, the spin-orbit coupling and barrier length and strength. A highly consistent scenario arises where two concomitant mechanisms for spin selection arises; spin interference and dierential spin amplitude decay. High spin ltering can take place at the cost of reduced amplitude transmission assuming realistic values for the SO coupling. The spin ltering scenario is completed by addressing the spin dependent torque under the barrier, with a consistent conserved denition for the spin current. I. INTRODUCTION Extensive studies show that electronic transfer in biological systems (for example, photosynthesis and respiration1) is fast and ecient, which can be ex- plained by means of tunneling processes through organic molecules1,2. Hopeld3was one of the rst who devel- oped a theory in terms of electron tunneling through a square potential barrier to analyze the electronic transfer in biological systems, nding the expected exponential decrease with the spatial separation of localized states and providing a mechanism to understand the function of the structural characteristics on electron transport molecules. Beratan et al.4,5obtained similar results by showing that the transfer of long-distance electrons in proteins decreases with distance, reinforcing the electron tunneling process as a transport mechanism in these sys- tems. In addition, they showed that the dependence on distance in proteins is related to the structure and that tunneling is mediated by consecutive electronic in- teractions between connecting donor with acceptor sites. Electron transfer by pure quantum tunneling have been shown to occur over distances between 20-40 A6,7in bi- ological molecules such as proteins and DNA or con- jugated structures. Such processes are temperature in- sensitive indicating that they are not activated and are partially coherent8. Spin active tunneling in chiral molecules has not re- ceived considerable attention in spite of its relevance. Recently, Michaeli and Naaman9considered tunneling through the dipole potential produced by hydrogen bond- ing in a helical geometry. This is a very important model since it is akin to both DNA, oligopeptides with an - helix, strong spin polarizers due to the Chiral-Induced Spin Selectivity (CISS) eect10{13. Nevertheless, their model does not discuss the details of tunneling coupled tothe SO interaction so each component propagates equally through the dipole barrier. Tunneling processes coupled to spin-activity has been modelled previously for the case of time reversal sym- metry breaking. Buttiker14proposed a model for a spin active barrier that considers a magnetic eld under the barrier to study the polarization of the transmitted waves and the characteristic dwell times for each spin compo- nent. Spin polarization is obtained because the decay strength is spin dependent due to the Zeeman energy contrast in the magnetic eld. This mechanism is quite articial for real molecules and it is in any case very weak but it suggests a similar mechanism using the SO split- ting energy basis of the CISS eect. Here, we propose to extend the Buttiker model to the spin-orbit Hamiltonian previously concocted for DNA15, including realistic as- sumption of a small doping of either electrons or holes by the surface-molecule contact. We nd that, analogously to the mechanism found by Buttiker, the energy splitting associate with the spin-orbit term generates dierent de- cay rates for each spin species. The dierent rates pro- duce an exponentially large polarization eect albeit the proven value of the coupling in the meV range16. This work is organized as follows: In section II we depart from the Hamiltonian model of reference15de- scribingcoupling between neighboring DNA bases including intrinsic SO pathways. A small doping is as- sumed to tune linear and quadratic terms in the Hamil- tonian in reciprocal space. In sections III and IV we solve for the tunneling problem with the derived Hamil- tonian and discuss the spin-polarization as a function of the tunneling length and SO coupling strength. We also discuss how the torque term in the spin continuity equa- tion that accounts for spin polarization for a time rever- sal symmetric potential. We end with the Summary and Conclusions.arXiv:2003.00582v2 [cond-mat.mes-hall] 17 Mar 20202 II. MOLECULAR HAMILTONIAN The full model Hamiltonian for DNA, incorporating the Stark eect for electric elds along the axis of the molecule and atomic Spin-orbit coupling, has been de- rived recently by Varela et al15. The model involves the orbital basisfpx;py;pz;sgon each base on a single he- lix, assuming weak coupling to the partner strand. The Fermi level for one orbital per base would be at half ll- ing, while light doping of the molecule by electrons or holes e.g. from contact with a substrate, determines the dispersion relation around the Fermi energy.We clarify that mobile electrons in the bases come from orbitals17. While these orbitals maybe thought of as fully lled, in- teractions with neighboring bases and surrounding envi- ronment will transfer electrons, process that we model as a change in the lling of these orbitals. The same model will result, perturbatively if we assume half lling and dope with electrons or if we assume fully lled orbitals and dope with holes. Figure 1 shows the -stacking model18for the dou- ble helix, showing only single pzorbital standing from the basis pairs. The wavefunction overlaps and Spin- Orbit (SO) couplings are derived from a tight-binding Slater-Koster analytical approach with lowest order per- turbation theory15. The helical/chiral structure results in a rst order SO coupling akin to that of carbon nanotubes19. The dependence on the chirality and pitch of the helix is built into the SO coupling parameter in the Bloch Hamiltonian. FIG. 1. Orbital model for transport in DNA. The gure de- picts the electron carrying orbitals ( pzorbital perpendicular to the base planes) coupled by VppSlater-Koster matrix ele- ments. It is well known that any transport mechanism occurs by electron transfer through these orbitals17.The largest contributions to the Hamiltonian, consid- ering only the intrinsic spin-orbit coupling of atoms in- volved inorbitals (N, C, O), comprises two terms H= (" 2p+ 2tf(k))1s2g(k)SOsy; (1) where 1srepresents the unit matrix in spin space and syis the Pauli matrix representing the spin degree of freedom in the local coordinate system of the molecule. The rst term in the Hamiltonian (1), involves the base 2porbital energy ( " 2p) and the kinetic energy with t, diagonal in spin space, depending on explicit structural parameters of the molecule t=V pp+b2
2 V ppV pp
(82r2(1cos
) +b2
2); (2) where
is the twist angle per base, randbare the radius and pitch of the helix, respectively and V; ppare the Slater-Koster overlaps between consecutive bases. The second term in (1), is the spin active term SO=4prb
(1cos
) V ppV pp
(" 2p" 2p)(82r2(1cos
) +b2
2);(3) wherepis the atomic SO coupling of double bonded atoms in the bases (either C, O or N) and "; 2pare the bare energies if the 2 pvalence orbital is either (per- pendicular to the base) or bonded. The SOpa- rameter, like the t, also includes all geometrical char- acteristic of the helix. Finally, f(k) = cos( k
R) and g(k) = sin( k
R) are the functions of reciprocal space withRthe lattice parameter and kthe wave vector in the local system of the helix. This Hamiltonian only includes the dominant spin active terms derived from the geome- try dependent spin-orbit coupling. Additional spin active terms are three to six orders of magnitude smaller15. The previous simplied model is based on the ba- sis setfpx;py;pzgorbitals per DNA base so half ll- ing is assumed ( px;pyare-bonded while pzis single lled)15. Charge transfer/doping by the environment of the molecule or by the substrate on which the molecule is attached, can add or subtract charge shifting the dis- persion from the in ection point Kfor the purely ki- netic Hamiltonian. Thus, the Fermi energy corresponds toK= 0, so that k=K+qdescribes a perturbative doping in the vicinity of the Fermi level. Expanding to lowest order in qand assuming that k
R1 we have f(k) = 1q2R2 2+O(q4)::: ; g(k) =qR+O(q3):::; (4) and the resulting Hamiltonian is: H= " 2p+ 2t 1q2R2 2 1s2qRSOsy: (5) Note that to lowest order in qwe have a quadratic dis- persion for the kinetic term and a linear dispersion for the spin-orbit term.3 In the sense of k
ptheory20we can requantize this Hamiltonian to treat the tunneling problem in the vicin- ity of the Fermi level: q!i@xand~2=2R2jtj!m. Eliminating constant energy terms we arrive at H=1 2m(i~@x)21+y(i~@x); (6) where=2R ~SOandyis the Pauli matrix. This derivation results in the same Hamiltonian surmised in reference21and leads to the detailed physics of the CISS eect in the absence of tunneling. The Hamiltonian in reference15can then be considered as a microscopic derivation of the continuum description. Note also that in the model recently proposed for Helicene22. III. POTENTIAL BARRIER We now introduce the previous model under a poten- tial barrier assuming, as shown in Figure 2, that electrons are injected from (and partially re ected back to) a donor localized state and received at an acceptor site. One might also consider dipole barriers as expected from hy- drogen bond generated potential identied in reference9. We consider an incident state of momentum px, where xis along the helix tangent. Electrons interact with a potential barrier of height V0and widtha. In the barrier region the SO interaction is active (see Fig.2 and refer- ence23). The scattering problem is then dened by H=( p2 x 2m+Vo 1+ypx; 06x6a donor=accept:states ; outside(7) The parameters used for the injected momentum, bar- rier height and the range of spin-orbit values are se- lected as follows: by electron transfer from experimen- tal techniques based on coupling articial donor and acceptor sites has been tested using a series of well- conjugated molecules including metallo-intercalators, or- ganic intercalators, organic end-cappers24. Measure- ments, using DNA as a bridge, report tunneling between 10-40 A6,7,25,26. On the other hand the barrier heights reported are in the range of 0.5-2.5 eV27,28, either by the potential dierence between the metal intercalators as donors/acceptors or the substrate in an STM setup, and the HOMO state of Guanine29. The donor connement potential can give an idea of the approximate kvector values being injected into the barrier, assuming carriers are in the ground state. In in- tercalators such as those in reference26or STM setups8 report connement over one or two base pairs. We can estimatek= 0:4 nm1, corresponding to the incident energy ofE= 0:24 eV, and V0= 2 eV. With these ex- perimentally derived parameters, we can see the conse- quences of dierential spin tunneling with the derived Hamiltonian. FIG. 2. Scattering potential barrier model with SO interac- tion (red hatch). The label for the incident ( A) and scattered (BandF) wave functions amplitudes are indicated. The well parameters are estimated in the text on the basis of polaron transport. IV. TUNNELING PROBLEM Once we have estimated the barrier parameters and the polaron well parameters, we can fully solve the 1D scattering problem by assuming an initial pure spin state. To determine the scattering properties we can solve the problem with simple plane wave injection conditions. The Hamiltonian Hacts on spinors with the form (x) = "(x) #(x) ; (8) where the arrows indicate the spin components. If the incident beam is given by in(x) = A" A# eikx; (9) and the spinor for the scattered beam is out(x) = F" F# eikx; (10) then, the spin asymmetry of the scattered beam cam be written in the form Pz=jF"j2jF#j2 jF"j2+jF#j2: (11)4 Now, Considering an incident electron with energy E and wave vector k, the general solutions are 1= A" A# eik1x+ B" B# eik1x;x60;(12) 3= F" F# eik3x;x>a; (13) and in the region of the barrier, the solution when E >V 0 is 2= C"eiq"x C#eiq#x + D"eiq"x D#eiq#x ; 06x6a;(14) and, forE <V 0 2= C"eq"x C#eq#x + D"eq"x D#eq#x ; 06x6a; (15) whereCandDare the amplitudes inside barrier region. Solving the eigenvalue problem H =E for each of the regions, we have that wave vectors for the electron in 1 and 3 are k1=k3=p 2mE= ~and for region 2, the wave vector qdepends on the spin orientation and, if E >V 0is given by qs=r k2q2 0+m ~2 +sm ~ ; (16) and ifE <V 0, then qs=r jq2 0k2jm ~2 ism ~ ; (17) whereq2 0= 2mV0=~2,k2= 2mE= ~2.sis the label asso- ciated with the spin up(down) such that s= +(). One can see the explicit dependence of qwith the spin s,V0 and with the SO magnitude, . Note that if E > V 0 then wave vector qsin the barrier region is real and the amplitudes will oscillate due to standing wave patterns between the edges of the barrier and the spin precession (relative changes in the spinor amplitudes) due to the SO coupling. The coecients are determined by the requirement of the continuity of the wave function at x= 0 and x=afollowing reference30: 1;s(0) = 2;s(0), 2;s(a) = 3;s(a) and ^vx;1 1(0) = ^vx;2 2(0);^vx;2 2(a) = ^vx;3 3(a) where the velocity ^ vx=@H=@pxin regions 1 and 3 have the form ^vx;1= ^vx;3= px=m 0 0px=m ; (18) and in region 2 ^vx;2= px=mi i px=m : (19)A. Energies below the barrier E <V 0 Below the barrier transmission will be the most com- mon physical scenario where we have an interplay be- tween three energies: i) the incoming energy of the elec- tron estimated by the quantum well that precedes the barrier, ii) the barrier height V0and iii) the SO energy that has been estimated to be in the meV range16. It is useful to consider some possible values of the wavevector inside barrier qs(Eq.17): = 0,qs=p jq2 0k2jand no spin activity is expected. Simple wave function decay is expected. jq2 0k2j>(m=~)2, thenqswill be a complex number (6= 0). Then we have an underdamped decay of the barrier wavefunction. Ifjq2 0k2j<(m=~)2, thenqsis purely imaginary number and the wave function is a plane wave. When the spin-orbit energy ESO, approachesj~2k2=2m V0j, a transition is expected between the two previous regimes. FIG. 3. Spin asymmetry Pzas a function of ain nm and the energy of the SO interaction in meV. The values for the incident wave function of electron k= 0:44 nm1and the barrier height q0= 1:2 nm1are xed. All these regimes are depicted in gures 3 and 4 for the polarization as a function of the barrier length and the SO energy equivalent m2=2. The range chosen of the SO energy is in agreement with the values computed in ref.16due to hydrogen bonding. Figure 3 shows the situation deep below the barrier where the wavefunction oscillates and decays (see Fig. 5) in an under-damped sit- uation because of the SO coupling. At zero SO coupling no spin polarization is observed. Once we have a nite5 the polarization is exponentially enhanced but there are also interference eects due to dierent oscillation frequencies of the j";#ispin components. This gives a reentrant eect where polarization can increase and then decrease as a function of the barrier width. Note the po- larization can increase a factor of three for a change in between 0:1 and 1 nm in barrier length. At 1 nm barrier length and 40 meV Rashba coupling (not capped by the atomic SO coupling because it is a combination of SO and Stark interactions16for DNA and Oligopeptides) we nd a polarization of 30%. FIG. 4. Spin asymmetry Pzas a function of ain nm and the energy of the SO interaction in meV. The values for the incident wave function k= 0:44 nm1and the barrier height q0= 0:46 nm1are xed. Spin ltering by tunneling in spin active media, gen- erates a high polarization with the expected molecular SO coupling, the amplitude is also exponentially small. Experimental accounts for the polarization rates should be able to check for this feature in time resolved experi- ments or essays that can change the tunneling length by e.g. mechanical stretching16,31. Figure 4 depicts a dierent regime where one has an input energy close to the barrier height. There we see a stronger re-entrant eect that extends for even lower values of the SO energy while increasing the needed bar- rier lengths for the same polarization enhancement as in Fig.3. The gure also shows the expected transition to plane wave behavior at jq2 0k2j(m=~)2under the barrier, because of the SO energy scale. FIG. 5. Spin component probabilities for energies in the vicin- ity of the barrier height. The observe oscillation accounts for the reentrant behavior predicted for the polarization. We usedk= 0:440 nm1,q0= 0:446 nm1,m2=2 = 80 meV anda= 5 nm. FIG. 6. Spin asymmetry Pzversus the barrier length aand the input momentum k, where the barrier height q0= 1:2 nm1, and the SO coupling energy m2=2 = 30 meV are xed. The dotted line represents a reasonable value for k argued in the text. Note the possibility of tuning between the barrier length and the input momentum determined by the pre-barrier well. Finally Fig.6 shows the sensitivity of the barrier polar- izing strength as a function of the input momentum (de- termined by the input well states). The gure also shows the possibility of tuning the well associated momentum and the barrier length to achieve large ltering ecien- cies. The existence of this mechanism for ltering could be evidenced by stretching/compressing the molecule in6 order to modify the tuning parameters and thus the l- tering power of the system. B. Above barrier energies E >V 0 The range of energies above the potential barrier are dominated by "interference" polarization as shown by the reentrant plot in Fig.7. Here there is no exponential de- cay and polarization is produced by the relative oscilla- tions of the two spin amplitudes. We believe this is not a generic situation for electron transfer in molecules where tunneling is predominant. If the energy is close to the barrier height one spin component can have energies be- low the barrier while the other is above the barrier and polarization can be enhanced by the same mechanism as in Fig 4. From the gure we can also see that the in- terference mechanism is less eective in producing high polarization values (up to 20%). FIG. 7. Spin asymmetry Pzas a function of ain nm andin meV. The values for the incident energy of electron k= 0:44 nm1and the barrier height q0= 0:30 nm1are xed. C. Spin currents and torque dipoles It has been shown that in the presence of SO cou- pling the conventional denition of spin current as a matrix element of ^Js= (1=i~)fv;szgis incomplete and unphysical32. The consistent spin current density should be written in the form: Is= Ren y(~ r)^Is (~ r)o ; (20) where ^Is=d(^r^sz)=dtis the eective spin current oper- ator, and ( ~ r) is the spatially dependent wave function.Developing the denition of the conserved spin current we have ^Is=d^ r dt^sz+^ rd^sz dt; =1 i~ [^ r;^H]^sz+^ r[^sz;^H] ; =^Js+^P; (21) where ^His the Hamiltonian of the system, ^ szis the spin operator for the zcomponent, ^Jsis the conventional spin current operator and the extra term ^Pis the torque dipole density from the corresponding torque density  due to the presence of the SO coupling. Considering our Hamiltonian (7), the two terms in Eq.21 are ^Js=i~2 2m @xm=~ m=~@x ; (22) and ^P=ix~ 0@x @x0 : (23) The torque density can be then computed by the relation Ts= Re ydsz dt  = Re y1 i~[^sz;^H]  =r
Ps; (24) Figure 8 shows the torque density integrated over the bar- rier length as a function of physical values for the SO en- ergy. The gure shows the range where there is a torque dierential between spin species producing net spin po- larization seen previously. The sharp dip indicated the SO coupling that produces pure wave behavior under the barrier (qspurely imaginary, see Eq.16). It is curious to note also (see inset), there is no linear regime for small that shows spin polarization. Figure 9 shows similar behavior as a function of the barrier length. Again there is no linear regime for polarized currents. One can think of torques taking away angular momentum depending on the spin species as the mechanism for generating spin po- larization under the barrier. This is a very clear insight derived from the consistent formulation of the conserved spin current denition32. As a concrete estimate of the change in angular mo- mentum produced by the torque density: Using the inputkvector range in Fig.6 to estimate the barrier dwell time14which fork= 0:44 nm1is 1014(see reference33). From this estimate we can compute, from Fig.9, the total change in angular momentum is
L 0:1~=2. This is a polarization that is comparable to that reported in Fig.6. V. SUMMARY AND CONCLUSIONS We have derived a Hamiltonian for a model of doped DNA that includes a SO coupling term that depends7 FIG. 8. Torque density 2in the region 2 for the two spin- components as a function of the SO coupling energy with k= 0:440 nm1,q0= 0:446 nm1, anda= 5 nm. Note there is no linear regime (see inset) for spin ltering. FIG. 9. Torque density Tin the barrier region for the two spin-components, as a function of the width barrier with k= 0:440 nm1,q0= 0:446 nm1andm2=2 = 14 meV.linearly on crystal momentum. We assume that elec- trons tunnel under a barrier of length abetween conned electron-phonon/polaron states. The SO couples dier- ently to each component of the spinor yielding a net spin- polarized output. The output polarization can be very large, e.g. 60% for realistic values of the SO coupling16, depending on the relation between the barrier length and the inputkvector of the electron. This is of course at the cost of a small spin current amplitude. We have also discussed the source of spin polarization as due to the existence of a torque density that dierentiates between up and down spin, using a consistent formulation of the spin current32. This mechanism is checked with an esti- mate of the change in angular momentum of the electron this torque density produces. Thus there is no need to invoke large unphysical SO strengths to achieve large po- larization values, as measured in the experiments. A nal feature that bears out of the model is that spin ltering has no linear regime as a function of the SO strength and the barrier length. These results seem to oer an alter- native interpretation to models that require time reversal symmetry breaking e.g. wave function leakage to explain spin polarization in the context of the CISS eect34. One important conclusion related to the generality of the model is its validity for very general sequences of DNA and Oligopeptides as long as transport the mech- anism involves short range tunneling35. The tunneling mechanism for transport is present both in uniform and heterogeneous sequences that have been studied36. Given the latter the physically relevant ingredients in the min- imal model are: a linear in kSO coupling with meV strength due to C/N atoms and a consistent conserved spin-current denition providing an angular momentum changing torque density. ACKNOWLEDGMENTS This work was supported by CEPRA VIII Grant XII-2108-06 Mechanical Spectroscopy funded by CEDIA, Ecuador. We acknowledge useful discussions with Ji r  Svozil k. svarela@yachaytech.edu.ec yemedina@yachaytech.edu.ec 1J. Winkler, A. Di Biblio, N. Farrow, J. Richards, and H. Gray, P. Appl. Chem. 71, 1753 (1999). 2J. Blumberger, Chem. Rev. 115, 11191 (2015). 3J. J. Hopeld, Proc. Nat. Acad. Sci. 71, 3640 (1974). 4D. Beratan, J. N. Onuchic, and J. Hopeld , J. Chem. Phys. 86, 4488 (1987). 5D. Beratan, J. N. Betts, and J. N. Onuchic, Science 252, 1285 (1991). 6J. R. Winkler, A. R. Dunn, C. R. Hess, and H. B. Gray ,Electron tunneling through iron and copper proteins . Bioinorganic Electrochemistry (Netherlands: Springer,2008). 7C. J. Murphy, M. R. Arkin, Y. Jenkins, N. D. Ghatlia, S. H. Bossmann, N. J. Turro, and J. K. Barton, Science 262, 1025 (1993). 8L. Xiang, J. L. Palma, C. Bruot, V. Mujica, M. Ratner, and N. Tao, Nat. Chem. 7, 221 (2015). 9K. Michaeli and R. Naaman, J. Phys. Chem. C 123, 17043 (2019). 10I. Carmeli, V. Skakalova, R. Naaman, and Z. Vager, Sci- ence41, 761 (2002). 11I. Carmeli, K. S. Kumar, O. Hei er, C. Carmeli and R. Naaman, Angew. Chem. Int. Ed. 53, 8953 (2014).8 12Z. Xie, T. Z. Markus, S. Cohen, Z. Vager, R. Gutierrez, and R. Naaman, Nano Lett. 11, 4652 (2011). 13D. Mishra, T. Z. Markus, R. Naaman, M. Kettner, and B. Gohler, Proc. Natl. Acad. Sci. 110, 14872 (2013). 14M. Buttiker , Phys. Rev. B 27, 6178 (1983). 15S. Varela, V. Mujica, and E. Medina, Phys. Rev. B 93, 155436 (2016). 16S. Varela, B. Monta~ nes, F. Lopez, B. Berche, B. Guillot, V. Mujica, and E. Medina , J. Chem. Phys. 151, 125102 (2019). 17L. G. D. Hawke, G. Kalosakas, and C. Simserides , Mol. Phys. 107, 1755 (2009). 18J. C. Genereux and J. K. Barton, Chem. Rev. 110, 1642 (2010). 19T. Ando, J. Phys. Soc. Jap. 69, 1757 (2000). 20R. Winkler, Spin-Orbit Coupling Eect in Two- Dimensional Electron and Hole Systems . Springer-Verlag Berlin Heidelberg (2003). 21E. Medina, L. A. Gonzalez-Arraga, D. Finkelstein-Shapiro, B. Berche, and V. Mujica , Science 142, 194308 (2015). 22M. Geyer, R. Gutierrez, V. Mujica, and G. Cuniberti, J. Phys. Chem. C 123, 27230 (2019). 23P. H. Cribb, S. Nordholm, and N.S. Hush, Chem. Phys. 44, 315 (1979). 24G. B. Schuster, Springer: New York, 236and237, (2004). 25S. Risser, D. Beratan and T. Meade, J. Am. Chem. Soc. 115, 2508 (1993).26A. R. Arnold, M. A. Grodick, and J. K. Barton, Cell Chem- ical Biology 23, 183 (2016). 27E. Wierzbinski, R. Venkatramani, K. L. Davis, S. Bezer, J. Kong, Y. Xing, E. Borguet, C. Achim, D. N. Beratan, and D. H. Waldeck, ACS Nano 7, 5391 (2013). 28N. V. Grib, D. A. Ryndyk, R. Gutierrez and G. Cuniberti, J. Biophys. Chem. 1, 77 (2010). 29E. Macia and F. Triozon and S. Roche, Phys. Rev. B. 71, 113 (2005). 30L. W. Molenkamp, G. Schmidt, and G. E. W. Bauer, Phys. Rev. B 64, 121202 (2001). 31S. Varela, V. Mujica and E. Medina , Chimia 72, 411 (2018). 32J. Shi, P. Zhang, D. Xiao and Q. Niu, Phys. Rev. Lett. 96, 076604 (2006). 33R. J. Behm, N. Garcia, and H. Rohrer, Scanning Tunnel- ing Microscopy and Related Methods ,NATO ASI Series E, Applied Science, 184 (1990). 34S. Matiyahu, Y. Utsumi, A. Aharony, O. Entin-Wohlman, and C. A. Balseiro, Phys. Rev. B 93, 075407 (2016). 35B. Giese, J. Amaudrut, A.-K. Kohler, M. Spormann, and S. Wessely, Nature 412, 318 (2001); L.M. Xiang, J. L. Palma, V. Mujica, M.A. Ratner, and N.J. Tao, Nature Chem, 7, 221 (2015); A. Shah, S. Shahzad, K. Ahmad, and H.-B. Kraatz, Chem. Rev. DOI: 10.1039/c4cs00297k (2015). 36T. Aqua, R. Naaman, and S. S. Daube, Langmuir 19, 10573 (2003); Angew. Chem. Int. Ed. 53, 1 (2014).

2401.15424v1.Interplay_of_altermagnetism_and_weak_ferromagnetism_in_two_dimensional_RuF__4_.pdf

Interplay of altermagnetism and weak ferromagnetism in two-dimensional RuF 4 Marko Milivojevi´ c ,1, 2Marko Orozovi´ c ,3Silvia Picozzi,4Martin Gmitra ,5, 6and Srdjan Stavri´ c3, 4,∗ 1Institute of Informatics, Slovak Academy of Sciences, 84507 Bratislava, Slovakia 2Faculty of Physics, University of Belgrade, 11001 Belgrade, Serbia 3Vinˇ ca Institute of Nuclear Sciences - National Institute of the Republic of Serbia, University of Belgrade, P. O. Box 522, RS-11001 Belgrade, Serbia 4Consiglio Nazionale delle Ricerche CNR-SPIN, c/o Universit´ a degli Studi ”G. D’Annunzio”, 66100 Chieti, Italy 5Institute of Physics, Pavol Jozef ˇSaf´ arik University in Koˇ sice, 04001 Koˇ sice, Slovakia 6Institute of Experimental Physics, Slovak Academy of Sciences, 04001 Koˇ sice, Slovakia (Dated: January 30, 2024) Abstract Gaining growing attention in spintronics is a class of magnets displaying zero net mag- netization and spin-split electronic bands called altermagnets. Here, by combining density functional theory and symmetry analysis, we show that RuF 4monolayer is a two-dimensional d-wave altermagnet. Spin-orbit coupling leads to pronounced spin splitting of the electronic bands at the Γ point by ∼100 meV and turns the RuF 4into a weak ferromagnet due to non trivial spin-momentum locking that cants the Ru magnetic moments. The net magnetic moment scales linearly with the spin-orbit coupling strength. Using group theory we derive an effective spin Hamiltonian capturing the spin-splitting and spin-momentum locking of the electronic bands. Disentanglement of the altermagnetic and spin-orbit coupling induced spin splitting uncovers to which extent the altermagnetic properties are affected by the spin-orbit coupling. Our results move the spotlight to the non trivial spin-momentum locking and weak ferromagnetism in the two-dimensional altermagnets relevant for novel venues in this emerging field of material science research. 1arXiv:2401.15424v1 [cond-mat.mtrl-sci] 27 Jan 2024I. INTRODUCTION Recent discovery of non-relativistic spin splitting in electronic bands of materials with compensated collinear magnetic order opened new venue in antiferromagnetic spintronics1–3. These intriguing materials constitute a separate magnetic phase, be- sides ferromagnets (FM) and antiferromagnets (AF), dubbed the name altermagnets (AM)4,5or, alternatively, antiferromagnets with non-interconvertible spin-structure mo- tif pair6–8. Altermagnets display a profusion of diverse physical phenomena, including anomalous Hall effect9, piezoelectricity10, and chiral magnons11; they can be used for efficient spin-to-charge conversion12, to generate orientation-dependent spin-pumping currents13and to proximitize superconductor materials14–16, just to scratch the mani- fold of their possible applications. Many altermagnetic compounds have been classified so far17,18, including MnTe19,20, MnF 26, RuO 24,9,21and CrSb22. In MnTe the spin splitting of 380 meV is experi- mentally determined by angle-resolved photoemission spectroscopy (ARPES) measure- ments, thus confirming its altermagnetic status previously predicted by density func- tional theory (DFT) calculations. For possible spin-transport applications, apart from the magnitude of the splitting energy, the positioning of the spin-split bands is highly important. In this respect CrSb may be a promising candidate for producing spin- polarized currents as a giant spin splitting of ∼600 meV occurs just below the Fermi energy22. Regarding their dimensionality, all the previously mentioned altermagnets are three- dimensional (3D) compounds, and as of now, reports on two-dimensional (2D) alter- magnets are limited. In the work of Brekke et al23it is proposed that magnon-mediated superconductivity in a 2D altermagnet could result with a superconductor with substan- tially enhanced critical temperature. However, this is accomplished using a microscopic model, and no specific material has been suggested to test this result. Very recently, monolayer Cr 2Te2O is predicted as a promising platform for practical realization of the spin Seebeck and spin Nernst effects of magnons24. Further, it is shown that monolayer MnPS 3, which is a conventional antiferromagnet, can be converted into altermagnet if one of its S layers is replaced by Se25. However, in thus obtained MnP(S,Se) 3Janus monolayer, the spin splitting of the relevant top valence band is modest, not exceed- ing≈10 meV in the best case. Considering this, the endeavor to fabricate the Janus 2monolayer might be deemed excessively demanding in comparison to the anticipated benefits. Hence, for prospective applications in miniature and highly efficient spintronic devices, it would be beneficial to have 2D materials that inherently exhibit altermag- netic properties. There is yet an important peculiarity concerning 2D magnets in general. Namely, in the absence of an external magnetic field the long-range order of spins on a 2D lattice can sustain above T= 0 K temperature only if the system possesses the magnetic anisotropy energy (MAE)26. MAE is crucial here as it gaps the magnon spectra. Otherwise, the low-energy magnon excitations would destroy the long-range order at arbitrary non-zero temperature. In this respect, altermagnets are not any different from ferromagnets and antiferromagnets. MAE, in turn, requires the presence of a sizable spin-orbit coupling (SOC) in the system. Therefore, for a realistic description of a 2D altermagnet the SOC must be accounted for. This raises a question – to which extent its non-relativistic properties, e.g. the hallmark spin splitting of bands, are affected by SOC? Here, we show that in the non-relativistic limit monolayer RuF 4is a 2D altermagnet and we describe the spin splitting of band structure and spin textures of the alter- magnetic phase. Subsequently, we analyze the influence of SOC on the altermagnetic properties. The structure of monolayer RuF 4, its crystallographic and magnetic sym- metries, are described in Subsection II A. The altermagnetic phase is analyzed in Sub- section II B. Weak ferromagnetism and SOC-induced spin splitting of bands are exposed in Subsections II C and II D. Finally, we summarize the paper in Section III with an outlook on the link between altermagnetism and weak ferromagnetism. II. RESULTS A. The structure and crystal symmetries of RuF 4monolayer Monolayer RuF 4is composed of ruthenium atoms sitting in centers of fluorine octahe- dra. It is a van der Waals material that can be exfoliated from the bulk compound which was experimentally synthesized in monoclinic P21/ccrystallographic space group (group No. 14)27. We will refer to an Ru atom with its surrounding F ligands as the Ru cluster . In each Ru cluster two out of six F atoms belong only to that cluster, whereas four of them are shared with four nearest neighbors. Crystal lattice of monolayer RuF 4 3has two sublattices with symmetrically distinct Ru clusters, RuAand RuB, that are arranged as depicted in Fig. 1. Figure 1: Structure of monolayer RuF 4and a schematic depiction of transformations g1=I andg2= (Ux|t1 2) that are generators of its crystallographic group. Distinct Ru clusters are depicted by red (RuA) and blue (RuB) octahedra. Two perpendicular lattice vectors that define the RuF 4plane, a1anda2, have dif- ferent norm, a1= 5.131˚A and a2= 5.477˚A, as obtained from our DFT calculations. Therefore, we will refer to the two directions defined by these vectors as the shorter and the longer in-plane directions. The structure of monolayer RuF 4belongs to the p21/b11 layer group (layer group No. 17)28which has two generators: the space inversion g1=I and the non-symmorphic group element g2= (Ux|t1 2) that is the combination of two op- erations each transforming RuAinto RuBand vice versa: a rotation Uxby 180◦around the shorter in-plane direction a1and a fractional lattice translation t1 2=1 2(a1+a2). Therefore, the group symmetry consists of four elements, G={e, g1, g2, g1g2}, where e is the identity element and g1g2represents a joint action of the group generators. Im- portantly, the crystal structure is centrosymmetric , but the Ru-Ru bond does not have the point of inversion, meaning that the Dzyaloshinskii-Moriya interaction can occur between the Ru magnetic moments29. We will discuss this in detail in the remainder of the manuscript. 4B. Non-relativistic description of magnetic properties: 2D altermagnet RuF 4has a compensated net magnetic moment due to the opposite magnetic moments on the two Ru sublattices. Octahedral crystal field of F−ions acts on Ru4+ion, leading to the t2g−egsplitting of its 4 dorbitals and leaving two spin-unpaired electrons in thet2gmanifold. Therefore, from the crystal field theory, magnetic moments of 2 µB are expected on Ru atoms. Our DFT calculations give 1 .464µB, in agreement with the value reported in Ref.30. The discrepancy between the expected and the DFT- calculated magnetic moments is due to the hybridization between the Ru 4 dand the F 2porbitals. Concretely, the F atoms that belong to a single Ru cluster exhibit moderate magnetic moments of 0.214 µB, induced by the nearest Ru magnetic moment, that are coupled ferromagnetically to that moment. On the other hand, the F atoms shared between two clusters are bonded to the two opposite-spin Ru atoms and are thus non- magnetic. In summary, the magnitude of the magnetic moment on each Ru cluster is |m|=|m(Ru) + 2 m(F)|= 1.9µB. In the absence of an external magnetic field and without SOC, the direction of magnetic moments and thus of the N´ eel vector ( N) is arbitrary. To perform symmetry analysis, we assume that Npoints along the longer in-plane lattice direction a2, which coincides with the y-axis in our choice of the coordinate frame (see Fig. 1). Space inversion g1has a trivial effect on magnetic moments as they are pseudovectors, whereas g2leaves the magnetic order invariant because the rotation Uxand the fractional lattice translation t1 2each reverse the magnetic order. Therefore, all the symmetry operations from the p21/b11 layer group leave the magnetic order unaltered and are thus the symmetries of the magnetic system as well, meaning that the magnetic space group is equal to its parent crystallographic group (type-I). Given that (1) the magnetic space group is of type-I31and (2) the combination θIof time reversal θand space inversion symmetry Iis broken due to the magnetic order, the non-relativistic spin splitting in monolayer RuF 4is allowed7,17. In Fig. 2a we plot the band structure of monolayer RuF 4calculated with spin- polarized DFT calculations. The system is a semiconductor with an indirect band gap of 0 .83 eV between the valence band maximum, located on the XY path, and the conduction band minimum located at S. Along the chosen k-path in the Brillouin zone (BZ), the spin-up and spin-down bands are degenerate along the ΓX, XS, and ΓY lines, 5Figure 2: Non-relativistic RuF 4band structure calculated with spin-polarized DFT calcula- tions. In (a), the ΓXSΓYX path is assumed and the spin splitting energies of the top valence (V1/2) and bottom conduction (C 1/2) bands are given in the inset (yellow rectangle). In (b) the top valence band V 1/2is illustrated once again along the S′ΓS path, and the d-wave nature of its spin-momentum locking in the full Brillouin zone (green rectangle) at −0.15 eV below the Fermi energy is illustrated in (c). whereas a sizable spin splitting ∆ En(k) =E↑ n(k)−E↓ n(k) (nis the band index) is observable along the ΓS and XY lines. Notably, of all the bands in the energy window from−1.5 eV below to 1 .5 eV above EF, the spin splitting is the largest for the top valence (V 1/2shown in Fig. 2b) and the bottom conduction bands (C 1/2), reaching 163 meV and 117 meV, respectively (yellow rectangle in Fig. 2a). The spin-momentum locking of the bands V 1/2presented in Fig. 2c clearly shows that monolayer RuF 4is a d-wave altermagnet4. 6To model the spin splitting of these bands around the Γ point we employ a simple time-reversal breaking Hamiltonian, compatible with the system’s symmetry, Hss=1 2αkxkyσy. (1) Here σy=±1 and αis the band-dependent parameter determined by fitting the model to the DFT bands. For the V 1/2and C 1/2, up to the 25% of the ΓX line, the spin splitting in an arbitrary k-direction can be described using αV= 3450 meV ˚A2and αC= 2368 meV ˚A2. As DFT calculations pointed out, the spin degeneracy of bands is not lifted along the ΓX and ΓY lines. This can be explained by analyzing how the group elements connect different points of the energy surface En(kx, ky, s). Keeping the assumption that magnetic moments are in the y-direction, the action of group elements reflects on the electronic bands as g1En(kx, ky, s) = En(−kx,−ky, s), g2En(kx, ky, s) = En(kx,−ky,−s), g1g2En(kx, ky, s) = En(−kx, ky,−s). (2) Based on the equations (2), one concludes that along ΓX, where k= (kx,0), and ΓY, where k= (0, ky), the relations g2En(kx,0, s) =En(kx,0,−s), g1g2En(0, ky, s) =En(0, ky,−s) (3) express the spin degeneracy. On the other hand, these constraints are not present along an arbitrary k-path and the spin splitting of bands is not forbidden. This is exactly the case of the ΓS and XY lines, where we observe a noticeable non-relativistic spin splitting, which is the hallmark of the altermagnetic phase. C. SOC turning altermagnet into a weak ferromagnet Once the SOC is included, the spin space couples to the real space and the N´ eel vector Nobtains a defined crystallographic direction. Our DFT calculations reveal that the shorter in-plane lattice direction (the y-axis) is the preferential direction for 7N, with MAE separating it from the longer in-plane direction ( x-axis) and from the out-of-plane direction ( z-axis) by 1 .12 meV /Ru and 7 .04 meV /Ru, respectively. Due to SOC, the magnetic moments are getting canted from the y-axis towards the x-axis by 3 .2◦, as depicted in Fig. 3a. Canting lowers the total energy only slightly (0.08 meV /Ru), but it induces a net magnetic moment of |M|=|mA+mB|= 0.22µB which points in the + xdirection. Therefore, upon the inclusion of SOC, the monolayer RuF 4changes its magnetic phase from the altermagnetic to the weak ferromagnetic (WF) phase. Figure 3: (a) SOC-induced canting of magnetic moments on two sublattices of RuF 4. (b) Spin textures of the top valence bands V 1and V 2at energy -0.15 eV below the Fermi level. (c) Dependence of induced net magnetic moment on the SOC strength. The direction of the induced magnetic moment in RuF 4can be inferred from the group theory analysis. In general, the magnetic moment M= (Mx, My, Mz) of any system must stay invariant under the action of all the group symmetry elements. Since the inversion g1has a trivial action on the pseudovector Mthrough the Drepresenta- tion41,D(g1)M=M, whereas the group elements g2andg1g2change the sign of the 8yandzcomponent of M,D(g2)M=D(g1g2)M= (Mx,−My,−Mz)≡(Mx, My, Mz), both the MyandMzmust be zero. Therefore, the net magnetic moment of RuF 4must point along the x-axis, in accordance with our DFT results. The transition from the altermagnetic to the weak ferromagnetic phase leaves its fingerprints on the spin texture . Spin texture is a vector field in the reciprocal space defined through the expectation value of the spin operator s= (sx, sy, sz) in a given Bloch state |ψn kx,ky⟩, ⟨si⟩n kx,ky=⟨ψn kx,ky|si|ψn kx,ky⟩, i=x, y, z. (4) The spin texture, like the net magnetic moment, complies with the symmetry of the system. We provide detailed derivation in Appendix A and arrive here straight at the conclusion by expressing the action of the group elements on the spin texture, g1:⟨si⟩n kx,ky=⟨si⟩n −kx,−ky, g2:⟨si⟩n kx,ky= (−1)1−δi,x⟨si⟩n kx,−ky, g1g2:⟨si⟩n kx,ky= (−1)1−δi,x⟨si⟩n −kx,ky, (5) where δi,xis the Kronecker delta, equal to 1 if i=xand 0 otherwise. The Eq. 5 shows that the spin texture in the I quadrant of the BZ ( kx>0, ky>0) is related to the spin texture in the II, III, and IV through the action of the group elements g1g2,g1, and g2, respectively. We can illustrate these relations with a concrete example. In Fig. 3c we show the spin texture of the valence bands V 1/2at the constant energy surface −0.15 eV below the Fermi level. Now, it is clear that the spin expectation values ⟨sy/z⟩kx,kyfrom the II and IV quadrants cancel out the spin expectation values in the I and III quadrants (actually ⟨sz⟩is zero everywhere, which is not shown in the plot). The cancellation of the spin expectation values from different quadrants of the BZ leads to the vanishing of MyandMzin the real space. Indeed, each component of the net magnetic moment can be expressed as an integral of the corresponding spin expectation values over the BZ, Mi=Pocc nR BZ⟨si⟩n kx,kyd2k, where the ”occ” denotes that the sum runs over the occupied states. On the other hand, ⟨sx⟩kx,ky>0 in all the four quadrants, yielding non-zero Mx. In the following, we question the microscopic mechanism responsible for canting the magnetic moments. In the work of Wang et al30it is shown that the Dzyaloshinskii- Moriya interaction (DMI) acting between the Ru moments is mainly responsible for 9their canting, with some contribution from the single-ion anisotropy (SIA). The DMI between the opposite-spin Ru atoms is allowed because the Ru-Ru bond does not display inversion29, as we mentioned in the Subsection II A. Given that SOC is the common origin of both the DMI and SIA, the magnitude of the induced net magnetic moment must depend on the SOC strength. To shed light on this relation, we scale the SOC strength in DFT calculations by using the modified SOC constant eλ=ξλ, where the dimensionless parameter ξranges from 0 (no SOC) to 1 (realistic SOC). Then, we constrain the magnetic moments in the directions that are canted by angle αfrom the y-axis. For a set of directions αthe equilibrium canting angle αξis found by minimizing the auxiliary function Eξ(α) =c0+c2(α−αξ)2, where Eξ(α) is the total energy of the system for a particular canting angle αand SOC strength eλ=ξλ. Finally, for each ξwe perform DFT calculations for magnetic moments canted by αξand we calculate the net magnetic moment. The plot in Fig. 3c shows that the net magnetic moment is linear in SOC strength. The linearity of the dependence M(˜λ) can be explained by turning to the microscopic expression for the Dzyaloshinskii-Moriya interaction. First, we noticed that the mag- nitude of the Ru cluster’s magnetic moment m=m(Ru) + 2 m(F) is almost unaffected by SOC, as the magnitude of the magnetic moment on Ru atom changes only slightly from 1 .464µBforξ= 0 to 1 .453µBforξ= 1 while |m(F)|doesn’t change at all, staying 0.214µB. Second, the magnitude of the net magnetic moment M=mA+mBand the canting angle αξare related by M= 2msinαξ, where m=|mA/B|. Now, if we assume that canted magnetic moments stay in the xy-plane, the DMI energy can be expressed as EDMI=D·(mA×mB) =DcosθDsin(2αξ), (6) where θD=∡(D, z). As pointed out by Moriya29the magnitude of the Dzyaloshinskii vector is linear in SOC constant, D∼λ(here D∼˜λas we are using modified SOC constant), whereas its direction depends on the symmetry of the structure and on the atomic positioning. Therefore, the angle θDdoes not depend on ˜λand the DMI energy can be expressed as EDMI∼˜λsin(2αξ). As long as the canting angle is small the approximate relation EDMI∼2˜λαξholds. Noticing that the DMI energy is quadratic in Dzyaloshinskii vector magnitude42,EDMI∼D2, the last relation shows that the canting angle is linear in the modified SOC constant, αξ∼˜λ. Finally, for small canting 10angles, the linearity of the M(˜λ) dependence stems from the relation M∼2mαξ∼m˜λ. D. SOC-induced spin splitting of bands Spin-orbit coupling, besides inducing net magnetic moment in the 2D altermagnet RuF 4, is responsible for significant spin splitting of the electronic bands. To shed light on this SOC effect, in Fig. 4(a)-(b) we plot the band structure calculated with SOC along the same k-path and in the same energy window as in the Fig. 2. In addition, we project the spin expectation values ⟨sx⟩kx,kyand⟨sy⟩kx,kyover each band to reveal how their spin polarization evolves along different k-directions. Due to SOC, the bands V 1/2and C 1/2, which are degenerate in the AM phase along some special directions, are now split in almost the entire BZ, displaying highly nonuni- form splitting (Fig. 4c). Intriguingly, the non-relativistic (AM-induced) and relativistic (SOC-induced) spin splitting reach their respective maxima in different regions of the BZ: while SOC-induced splitting is maximal at Γ and S points, where the bands were previously degenerate in the AM phase, the AM-induced splitting in the middle of the ΓS and XY lines is barely affected by SOC. Apart from the splitting energies, the polarization of bands displays non-uniform k-dependence. For example, the V 1/2bands are polarized in the x-direction along the ΓX and ΓY lines, with ⟨sx⟩<0 for V 1and⟨sx⟩>0 for V 2(Fig. 4a-b). The polarization in the x-direction along these k-lines is the most evident for the V 7/8bands. On the other hand, both ⟨sx⟩and⟨sy⟩are nonzero and vary substantially along the ΓS and XY lines, meaning that V 1/2have mixed polarization along these directions. Also, there are k-directions along which the bands are almost unpolarized, such as the XS line where ⟨sx⟩ ≈ ⟨sy⟩ ≈0. The point that deserves special attention is the lifting of spin degeneracy along the XΓY path. What is particularly striking is a large spin splitting at Γ, which is a time- reversal invariant momenta (TRIM) point, of ∆V1/2 Γ= 86 meV and ∆C1/2 Γ= 79 meV for the V 1/2and C 1/2bands, respectively. Moreover, ∆ Γscales linearly with the SOC strength, as shown in Fig. 4d. At first glance, this linear splitting seems to occur due to a nonzero net magnetic moment pointing in the x-direction, but our additional DFT calculations reveal that even when the Ru magnetic moments are strictly constrained in the y-direction (so that M= 0), the spin splitting of bands along XΓY persists. 11Figure 4: Non-collinear relativistic DFT calculations of the band structure for RuF 4along high symmetry lines. (a) spin expectation values ⟨sx(k)⟩and (b) ⟨sy(k)⟩plotted as a band color. (c) spin splitting of the top valence V 1/2and bottom conduction bands C 1/2is along the same k-path given, with the AM-induced spin splitting plotted for comparison with dashed lines. (d) spin splitting of V 1/2and C 1/2at the Γ point calculated for the different SOC strength ξ. (e) the difference between spin splitting energies with and without SOC of V 1/2 and C 1/2bands along the ΓS path for different SOC strengths ξ. To shed light on this, let us consider a non-relativistic Hamiltonian of the AM phase Hnrel AMand include SOC as a perturbation. The complete derivation is exposed 12in Appendix B but here we discuss the most important conclusions. Assuming that two Bloch wavefunctions corresponding to the same k= (kx, ky) point, |Ψ1⟩kx,ky= |ψ1⟩ ⊗ |↑⟩ and|Ψ2⟩kx,ky=|ψ2⟩ ⊗ |↓⟩ , are with the same energy E, the perturbation HSOC= (∇V×p)·σ=λL·σin the {|Ψ1⟩kx,ky,|Ψ2⟩kx,ky}basis has the matrix elements of the form ⟨HSOC⟩=λ ℓy i(ℓx−ℓz) −i(ℓx−ℓz) ℓy , (7) where ℓi=ℓi(kx, ky),i=x, y, z , are real numbers that can be connected to nonzero matrix elements of the pseudovector operator L, see Appendix B. From the Eq. 7, the SOC contribution to the level Eisλ ℓy±(ℓx−ℓz) , meaning that SOC lifts the degeneracy of |Ψ1⟩kx,kyand|Ψ2⟩kx,kyinducing the spin splitting of ∆ k= 2λ(ℓx−ℓz). The derivation exposed in Appendix B applies to any two degenerate Bloch wavefunctions as it only exploits the action of the group element g2and the assumption that Npoints in the y-direction. Therefore, the conclusions of Appendix B apply to any kpoint, showing that the linear-in-SOC spin splitting is a cooperative effect of breaking the time-reversal symmetry due to the magnetic order and of the spin-orbit coupling. The spin degeneracy along the ΓS and XY lines is lifted already in the AM phase, but the inclusion of SOC changes this spin splitting drastically. To show how the SOC- induced splitting gradually develops over the AM-induced one, we plot the difference of the SOC-induced and AM-induced splitting energies for V 1/2and C 1/2bands along the ΓS path for different SOC strengths ˜λ=ξλ( Fig. 4e). As ξvaries from 0 to 1 the SOC- induced splitting increases, but the fraction of the ΓS path where it dominates over the AM-induced splitting never surpasses 40%. Therefore, the AM-induced splitting is still dominant along the ΓS line despite the sizable SOC-induced spin splitting. III. DISCUSSION AND CONCLUSION In this work we have shown that in the non-relativistic limit, the monolayer RuF 4 represents a two-dimensional d-wave altermagnet. Using the type-I magnetic space group, we have revealed the symmetry properties of the non-relativistic spin splitting in the altermagnetic phase. By applying the theory of invariants, we have derived a simple model Hamiltonian that captures the spin-momentum locking of the top valence 13and the bottom conduction bands, both of which can be activated for spin transport by doping. In the relativistic limit, we have shown that spin-orbit coupling induces weak ferromagnetism and significantly changes the spin splitting and the spin texture of bands. Using the perturbation theory, we explained the appearance of a sizable SOC-induced spin splitting at the time-reversal invariant Γ point, which is reminiscent of Zeeman splitting but occurs even in the limit of vanishing net magnetic moment. We have shown that the magnetic moment induced due to canting of Ru spins is linear in SOC strength. This implies that in altermagnetic compounds containing heavy atoms, which exhibit very strong SOC, the altermagnetic phase could be significantly altered by weak ferromagnetism. Finally, it’s worth noting that Autieri et al.32recently highlighted that the emergence of weak ferromagnetism in metallic altermagnets is a property unparalleled in conven- tional antiferromagnets. As such, the appearance of weak magnetization in metallic compounds can be used to demarcate the conventional antiferromagnets from alter- magnets. Yet, as we have shown in this work, RuF 4is a semiconductor and we argue that the appearance of SOC-induced weak ferromagnetism may be a general property of all the altermagnetic compounds, not just metals. This becomes especially encour- aging once one realizes that the small component of the magnetic moment, induced by SOC and perpendicular to the N´ eel vector, could be employed as a knob to control the direction of dominant magnetic arrangement in altermagnets. Indeed, the N´ eel vector in weak ferromagnets can be dragged by a small external magnetic field using the ex- perimental technique exposed in the work of Dmitrienko et al.33. In this way one could hopefully control the direction of spin-polarized currents in altermagnets. Therefore, we strongly believe that the relation between weak ferromagnetism and altermagnetism warrants greater attention, and would stimulate interest to delve into this intriguing topic. IV. METHODS DFT calculations were performed using the VASP code34. The effects of electronic exchange and correlation were described at the Generalized Gradient Approximation (GGA) level using the Perdew–Burke–Ernzerhof (PBE) functional35. The Kohn-Sham wavefunctions were expanded on a plane wave basis set with a cutoff of 500 eV. We 14used the pseudopotentials with Ru 4 p,5s,4dand F 2 s,2pstates as valence states. The lattice constants a1= 5.131˚A and a2= 5.477˚A were obtained from spin-polarized non-relativistic DFT calculations using the Birch-Murnaghan equation of state and assuming the antiparallel arrangement of Ru magnetic moments. In the out-of-plane direction, we used the lattice constant of a3= 20 ˚A to ensure sufficient separation between the periodic replicas. The atomic positions were relaxed until all the forces’ components dropped below 0 .002 eV /˚A2. The Brillouin zone was sampled with 10 ×9×1 kpoints mesh during the relaxation and with a finer 15 ×14×1 mesh during the self- consistent field calculations. To find the canting angle of the Ru magnetic moment we performed non-collinear DFT calculations with SOC and for a set of directions we constrained the magnetic moments using the penalty functional36. For producing the bandstructure and spin texture plots we used the Pyprocar package37. For setting up DFT calculations and for structural visualization we used the Atomic Simulation Environment38,XCrySden39, and Vesta40packages. Acknowledgments We acknowledge support by the Italian Ministry for Research and Education through the Nanoscience Foundries and Fine Analysis (NFFA-Trieste, Italy) project and through the PRIN-2017 project ”TWEET: Towards ferroelectricity in two dimensions” (IT- MIUR grant No. 2017YCTB59). M.M, M.O., and S.S. acknowledge the financial sup- port provided by the Ministry of Education, Science, and Technological Development of the Republic of Serbia. This project has received funding from the European Union’s Horizon 2020 Research and Innovation Programme under the Programme SASPRO 2 COFUND Marie Sklodowska-Curie grant agreement No. 945478. M.G. acknowledges financial support provided by Slovak Research and Development Agency provided under Contract No. APVV-SK-CZ-RD-21-0114 and by the Ministry of Education, Science, Research and Sport of the Slovak Republic provided under Grant No. VEGA 1/0695/23 and Slovak Academy of Sciences project IMPULZ IM-2021-42 and project FLAG ERA JTC 2021 2DSOTECH. 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Crystallogr. 44, 1272 (2011), ISSN 0021-8898. 41D(g1) =1 0 0 0 1 0 0 0 1 ,D(g2) =1 0 0 0−1 0 0 0−1 ,D(g1g2) =1 0 0 0−1 0 0 0−1 42For a quantum spin-1 system described by Hamiltonian H=JS1·S2+D·(S1×S2), D=Dez, the DMI correction of the eigenvalues is equal to ±√ J2+D2∼ ±J 1 +D2 2J2
andJ 2(1±3q 1 +8D2 9J2)∼J 2(1±3(1 +4D2 9J2)), where it is assumed that the DMI is much smaller than the Heisenberg exchange. Therefore, the DMI correction to the energy is EDMI∼D2/J. Appendix A: General constraints on spin expectation values in the relativistic case Let|ψn kx,ky⟩represents an eigenstate of the altermagnetic Hamiltonian in a relativistic phase Hrel AM Hrel AM|ψn kx,ky⟩=En(kx, ky)|ψn kx,ky⟩, (A1) with group symmetry G=p21/b11 of both the relativistic and/or altermagnetic phase, since the magnetic space group is equal to the crystallographic group, see Section II B. The commutation relation [ d(g),Hrel AM] = 0 holds for any element g∈G, where dis the representation of the group acting in the Hilbert space of the Hamiltonian Hrel AM. For a given state |ψn kx,ky⟩, the spin expectation values are defined as ⟨si⟩n kx,ky=⟨ψn kx,ky|si|ψn kx,ky⟩, i=x, y, z. (A2) 18Using the action of the group element g∈Gin the Hilbert space in which the Hamil- tonian Hacts, the previous equation can be transformed to ⟨si⟩n kx,ky=⟨ψn kx,ky|d†(g)d(g)sid†(g)d(g)|ψn kx,ky⟩ (A3) By analyzing the action of nontrivial group elements g1=I,g2= (Ux|t1 2) and g1g2, one obtains general constraints on the spin expectation values g1:⟨si⟩n kx,ky=⟨si⟩n −kx,−ky, g2:⟨si⟩n kx,ky= (−1)1−δi,x⟨si⟩n kx,−ky, g1g2:⟨si⟩n kx,ky= (−1)1−δi,x⟨si⟩n −kx,ky, (A4) where δi,xis the Kronecker delta, equal to 1 if i=xand 0 otherwise. To obtain the previous relations, we have used the following identities g1:d(g1)sid†(g1) =si, d (g1)|ψn kx,ky⟩=|ψn −kx,−ky⟩, g2:d(g2)sid†(g2) = (−1)1−δi,xsi, d (g2)|ψn kx,ky⟩=|ψn kx,−ky⟩ g1g2:d(g1g2)sid†(g1g2) = (−1)1−δi,xsi, d(g1g2)|ψn kx,ky⟩=|ψn −kx,ky⟩. (A5) In short, the set of equations A4 show that in a system with p21/b11 symmetry the spin texture ⟨sx⟩can accumulate, yielding a finite net magnetic moment in the x-direction, whereas the ⟨sy⟩and⟨sz⟩from different quadrants of the BZ cancel out, as explained in the main text and illustrated in Fig. 3b. Furthermore, the equations A4 in the special cases of the Γ point and along the ΓX and ΓY lines, suggest that spin expectation values are nonzero in the x-direction solely since for ky= 0 (ΓX line) we have ⟨sx⟩n kx,0=⟨sx⟩n kx,0, ⟨sy⟩n kx,0=−⟨sy⟩n kx,0= 0, ⟨sz⟩n kx,0=−⟨sz⟩n kx,0= 0, (A6) while along the ΓY line ( ky= 0) one gets ⟨sx⟩n 0,ky=⟨sx⟩n 0,ky, ⟨sy⟩n 0,ky=−⟨sy⟩n 0,ky= 0, ⟨sz⟩n 0,ky=−⟨sz⟩n 0,ky= 0. (A7) 19This conclusion is in agreement with the band structure plotted in Fig. 4 which shows that along the XΓY path the x-component of spin is nonzero (blue and red bands Fig. 4a) while the y-component equals to zero (gray bands Fig. 4b). Also, it should be mentioned that the conclusions agree with the ones given in Section B, where we have shown that the SOC is solely responsible for the observed splitting along the high-symmetry lines ΓX and ΓY, as well as in the Γ point. Appendix B: Perturbation theory of SOC-induced removal of spin degeneracy at arbitrary kpoint The spin splitting at the Γ point and along the ΓX and ΓY lines can be explained using the perturbation theory, where the SOC Hamiltonian is introduced as a perturba- tion. To do this, we apply a first-order degenerate perturbation theory, valid for bands not only along the XΓY path, but also at other kpoints of the Brillouin zone which host the degenerate bands in the altermagnetic phase. We describe the two degenerate states as |Ψ1⟩=|ψ1⟩ ⊗ |↑⟩ and|Ψ2⟩=|ψ2⟩ ⊗ |↓⟩ , where |ψ1/2⟩corresponds to the orbital part of the wave function, while |↑/↓⟩are eigenstates of the Pauli σyoperator for the eigenvalues 1 and −1, respectively. We chose σybecause we assumed that the spin quantization axis is along the y-direction. Now, if theHnrel AMis the non-relativistic one-electron Hamiltonian (AM emphasizes here that this is the non-relativistic Hamiltonian of the altermagnetic phase), then the following relation holds Hnrel AM|Ψ1/2⟩=E|Ψ1/2⟩. (B1) In the orbital space the action of the group element g2is through the representation dO(g2), while in the magnetic space acts as a representation dM,dM(g2) =0 1 1 0
, which flips the magnetic moments dM(g2)|↑/↓⟩=|↓/↑⟩. Using the commutation relation between the Hamiltonian and the elements of the group, [ dO(g2)⊗dM(g2),Hnrel AM] = 0, one can say that  dO(g2)⊗dM(g2) Hnrel AM|ψ1⟩ ⊗ |↑⟩ = dO(g2)⊗dM(g2) E|ψ1⟩ ⊗ |↑⟩ Hnrel AM dO(g2)|ψ1⟩ ⊗ |↓⟩ =E dO(g2)|ψ1⟩ ⊗ |↓⟩ . (B2) The last relation is equal to Hnrel AM|ψ2⟩ ⊗ |↓⟩ =E|ψ2⟩ ⊗ |↓⟩ , which leads us to con- clude that the relation dO(g2)|ψ1⟩=|ψ2⟩must hold. Furthermore, using the fact that 20dO(g2)dO(g2) =dO(g2 2) =I, where Iis the identity matrix, one can conclude that dO(g2 2)|ψ1⟩=|ψ1⟩, which suggests that the relation dO(g2)|ψ2⟩=|ψ1⟩holds. Therefore, the group element g2interchanges the orbital parts of two degenerate states. This will be useful for calculating the matrix elements of the SOC operator in the {|Ψ1⟩,|Ψ2⟩} basis. To account for the SOC effects we use the first-order perturbation theory and cal- culate the allowed matrix elements of the operator HSOC= (∇V×p)·σ, where∇V represents the gradient of the crystal potential, pthe momentum operator, and σthe Pauli operator. Now, the fact that ( ∇V×p) transforms as a pseudovector allows us to rewrite the SOC Hamiltonian as HSOC=λL·σ, where λdescribes the SOC strength. Note that Lis not the angular momentum but it transforms like one. Now, the matrix elements of HSOCin the {|Ψ1⟩,|Ψ2⟩}basis are ⟨HSOC⟩=λ ⟨ψ1|Ly|ψ1⟩ i⟨ψ1|Lx|ψ2⟩ − ⟨ψ1|Lz|ψ2⟩ −i⟨ψ2|Lx|ψ1⟩ − ⟨ψ2|Lz|ψ1⟩ −⟨ ψ2|Ly|ψ2⟩ . (B3) Here we have used that the Pauli operators in a given spin basis {|↑⟩=1√ 2 −i|+⟩+ |−⟩
,|↓⟩=1√ 2 i|+⟩+|−⟩
}(|±⟩are eigenvectors of the σzoperator, σz|±⟩=±|±⟩ ) are equal to σx=0 i −i 0
,σy=1 0 0−1
,σz=0−1 −1 0
. The symmetry argument in this case gives us ⟨ψ1|Ly|ψ1⟩=⟨ψ1|d† O(g2)dO(g2)Lyd† O(g2)dO(g2)|ψ1⟩=−⟨ψ2|Ly|ψ2⟩=ℓy,(B4) where we have used that dO(g2)Lyd† O(g2) =−Lyand that ℓyis the real number. Also, we calculate the matrix elements ⟨ψ1|Lx|ψ2⟩and⟨ψ1|Lz|ψ2⟩, ℓ12 x=⟨ψ1|Lx|ψ2⟩=⟨ψ1|d† O(g2)dO(g2)Lxd† O(g2)dO(g2)|ψ2⟩=⟨ψ2|Lx|ψ1⟩= (ℓ12 x)∗, ℓ12 z=⟨ψ1|Lz|ψ2⟩=⟨ψ1|d† O(g2)dO(g2)Lzd† O(g2)dO(g2)|ψ2⟩=−⟨ψ2|Lz|ψ1⟩=−(ℓ12 z)∗, (B5) and use the relations dO(g2)Lxd† O(g2) =LxanddO(g2)Lzd† O(g2) =−Lz. The conditions obtained in (B5) imply that ℓ12 x=ℓx∈ R andℓ12 z= iℓz, where ℓz∈ R, allowing us to simplify ⟨HSOC⟩to ⟨HSOC⟩=λ ℓy i(ℓx−ℓz) −i(ℓx−ℓz) ℓy . (B6) 21Using the final form of ⟨HSOC⟩given by Eq. (B6), the SOC-induced energy contribu- tion to the degenerate level in the altermagnetic case is equal λ ℓy±(ℓx−ℓz) . The spin splitting due to the SOC is equal to 2 λ(ℓx−ℓz). Furthermore, by calculating the eigenvectors of E0I2+⟨HSOC⟩, where E0represents the energy of the double degenerate bands, while I2is the 2 ×2 identity matrix in the eigenbasis {|↑⟩,|↓⟩}of the altermag- netic phase, we find that the spin expectation values have the nonzero component in thex-direction only, consistent with the general conclusion from Appendix A. However, the first-order perturbation theory predicts ⟨sx⟩=±1 (in the units of ℏ/2) at the Γ point, which is not consistent with the data obtained from first-principles calculations, where ⟨sx⟩<1. This implies that the higher-order perturbation theory needs to be applied to account for the interaction with the bands higher in energy, responsible for the reduction of the intensity of the spin expectation value. 22

2206.10222v1.Faraday_patterns_in_spin_orbit_coupled_Bose_Einstein_condensates.pdf

arXiv:2206.10222v1 [cond-mat.quant-gas] 21 Jun 2022Faraday patterns in spin-orbit coupled Bose-Einstein cond ensates Huan Zhang,1,2,3Sheng Liu,1,2,3,∗and Yong-Sheng Zhang1,2,3,† 1CAS Key Laboratory of Quantum Information,University of Sc ience and Technology of China, Hefei, 230026, China 2CAS Center for Excellence in Quantum Information and Quantu m Physics, Hefei, 230026, China 3Hefei National Laboratory, University of Science and Techn ology of China, Hefei, 230088, China (Dated: June 22, 2022) We study the Faraday patterns generated by spin-orbit coupl ing induced parametric resonance in a spinor Bose-Einstein condensate with repulsive interact ion. The collective elementary excitations of the Bose-Einstein condensate, including density waves a nd spin waves, are coupled as the result of the Raman-induced spin-orbit coupling and a quench of the relative phase of two Raman lasers without the modulation of any of the system’s parameters. We observed several higher paramet- ric resonance tongues at integer multiples of the driving fr equency and investigated the interplay between Faraday instabilities and modulation instabiliti es when we quench the spin-orbit coupled Bose-Einstein condensate from zero-momentum phase to plan e-wave phase. If the detuning is equal to zero, the wave number of combination resonance barely cha nges as the strength of spin-orbit cou- pling increases. If the detuning is not equal to zero after a q uench, a single combination resonance tongue will split into two parts. I. INTRODUCTION Spontaneous pattern formation is an important uni- versal phenomenon in chemistry, biology, and physics [ 1]. The pattern formation in a classical driving system was studied by Faraday in 1831 [ 2] using different kinds of liquid. He found that when the driving frequency ex- ceeds the critical value, the surface of the liquid becomes spatially modulated, which is related to the parametric instability. Since therealizationofBose-Einsteinconden- sate (BEC), one can study hydrodynamics of quantum fluid in experiment. This new type of nonlinear quan- tum fluid shows some different features from the classical counterpart. A BEC of the cold atoms manifests much more interesting nonlinear phenomena due to its quan- tum nature, such as solitons [ 3,4], vortices [ 5–9] and Faraday waves [ 10–12]. The tunable cold atom system provides a promising platform to investigate these fas- cinating phenomena. The nonlinear dynamics of BECs stem from the collision interaction between the atoms, which can be adjusted by exploiting the Feshbach reso- nances [13]. The Faraday patterns of a BEC will emerge if one periodically drives the strength of nonlinearity [10,12,14–17], or if the trap confinement of the BEC is periodically modulated [ 11,18–20], in which the non- linear interaction is effectively oscillated. Realization of the synthetic spin-orbit (SO) coupled BEC [21,22] enables study of some interesting phenom- enaincondensedmattersystems,suchasspin-Halleffect, supersolidity and topological insulators [ 23]. Raman- induced SO coupling can be generated by two counter propagating beams of laser light along the xdirection of an initially confined BEC in a typical experiment. By tuning the system’s parameter, the ground state of ∗shengliu@ustc.edu.cn †yshzhang@ustc.edu.cnthe SO-coupled BEC will be in different phases, such as plane-wave phase, zero-momentum phase, or stripe phase. The spontaneous pattern formation of the SO- coupledBEChasbeenstudiedinRefs. [ 24–27], wherethe patterns are induced by modulation instabilities, a mov- ing barrier, or the Kibble-Zurek mechanism, etc. How- ever, investigation of spontaneous pattern formation in- duced by Faraday instabilities in the SO-coupled BEC is still lacking. In a recentpaper [ 28], the authorsfound that the Fara- day patterns can also be excited from a quench of the phase of the Rabi coupling without any modulation of the system’s parameters. The effective modulation of interaction can be realized when the interaction coeffi- cients satisfy g2 12/negationslash=g11g22and the average populations of two hyperfine states of the spinor BEC experience Rabi oscillation. However, according to the simulation in Ref. [ 28], it takes a long time for the Faraday pat- terns to emerge and in a practical experiment, the tun- ing range of |g2 12−g11g22|is limited. Motivated by Ref. [28], here we show that the Faraday patterns in a SO- coupled BEC ( gij[i,j= 1,2] =g) can be induced by the Raman-induced SO coupling and a quench of rela- tive phase of two Raman lasers without modulating any of the system’s parameters such as the external poten- tial, effective interaction, etc. In a typical experiment, the strength of SO coupling is highly tunable and can be adjusted by changing the angle between the two incident Raman lasers. One can also effectively tune the strength of SO coupling by periodic modulation of the power of the Raman lasers [ 29,30]. The spontaneous formation of the Faraday patterns is the direct consequence of the coupling of two types of collective elementary excitations (spin waves and density waves) due to the SO coupling and the Rabi oscillation induced by the quench. In our numerical simulation, we can observe the Faraday pat- terns emerge in a much shorter time. In the resonance diagram,weobservedseveralhigherresonancetonguesat integer multiples of the driving frequency. If the detun-2 ing is not equal to zero after the quench, the resonance tongues will split into two parts as the strength of SO couplingincreases. Afterquenchingthe SO-coupledBEC from zero-momentum phase to plane-wave phase, we can observe the coexistence of modulation instabilities and Faraday instabilities. Our paper is organized as follows. In Sec. IIwe derive the coupled Gross-Pitaevskii (GP) equations of the SO- coupled BEC and its analytical solutions. In Sec. IIIwe derivethe resonanceconditionsofoursystem. Section IV presentsthe instabilityanalysisandthe numericalresults of integrating the GP equations. In Sec. Vwe discuss more general cases. Section VIis our conclusion. II. SYSTEM MODEL We consider a two-dimensional homogeneous BEC with Raman-induced SO coupling along the xdirection at zero temperature under the mean-field description. This two-dimensional geometry can be realized by ap- plying a strong harmonic trapping potential of frequency ωzin thezdirection. As we focus on the homoge- neous case, the external potential in two dimensions is V(x,y) = 0. The two incident counter propagating lasers act on the atoms at some angle with the xdi- rection to synthesize the Rashba [ 31] and the Dressel- haus [32] SO-coupling interaction with equal contribu- tions. Using the rotation approximation, the single par- ticle Hamiltonian of interacting SO coupled BEC can be written as [ 21]Hsp=ˆp2/(2m)+δEσz/2+/planckover2pi1Ωcos(2krx− δωt+φ)σx/2−/planckover2pi1Ωsin(2krx−δωt+φ)σy/2, where ˆp is the two-dimensional momentum operator, mis the mass of the cold atoms, δEis the energy level splitting, Ω is the strength of Raman coupling, δωis the detun- ing of two Raman lasers, kris the projected wavenum- ber of Raman lasers in the xdirection and φis the relative phase of two Raman lasers. In the corotating frame of the effective field, under the unitary rotation U= exp[i(krx−δωt/2)σz],Hr sp=U†HspU−i/planckover2pi1U†∂U/∂t [29]. The total mean field Hamiltonian becomes H=Hr sp+Hint, (1a) Hr sp= (ˆp+/planckover2pi1krˆex)2 2m+/planckover2pi1∆ 2/planckover2pi1Ω 2eiφ /planckover2pi1Ω 2e−iφ(ˆp−/planckover2pi1krˆex)2 2m−/planckover2pi1∆ 2 , (1b) Hint=/parenleftbigg g11|Ψ1|2+g12|Ψ2|20 0 g21|Ψ1|2+g22|Ψ2|2/parenrightbigg , (1c) where ∆ = δE//planckover2pi1−δωis the two-photon detuning, gij(i,j= 1,2) is the interaction coefficients, Ψ1=Ψ1(r) andΨ2=Ψ2(r) represent the macroscopic condensate’s wave functions where r= (x,y) is the two-dimensional spatial vector and Ω is the strength of Raman coupling.-4 -2 0 2 4-2-1012345 -4 -2 0 2 4(a) (b) FIG. 1. Dimensionless dispersion relation ˜E(∆ = 0) −= k2 x/2−/radicalbig γ2k2x+Ω2/4 with Ω = Ω i(solid line) and Ω = Ω f (dash line) corresponding to situations of before and after a quench, respetively. Circles in the figure mean the initial states before a quench located at kx= 0 before the quench, which indicates the SO-coupled BEC is in zero-momentum phase. (a) An example of both Ω iand Ω fare larger than Ω c. (b) An example of Ω i>Ωcand Ω f<Ωc. The strength of SO coupling is /planckover2pi1kr/m. Diagonalizing the single particle Hamiltonian (1b), we can obtain the dispersion relationship E±=p2 2m+/planckover2pi12k2 r 2m±/radicalbigg (/planckover2pi1krpx m+/planckover2pi1∆ 2)2+/planckover2pi12Ω2 4,(2) whereE−is the ground state energy of the SO-coupled BEC. The corresponding eigenfunctions are φ+=eip+·r//planckover2pi1/parenleftbigg cos(θ/2) sin(θ/2)/parenrightbigg ,φ−=eip−·r//planckover2pi1/parenleftbigg sin(θ/2) −cos(θ/2)/parenrightbigg , (3) where sinθ=/planckover2pi1Ω//radicalbig 4(/planckover2pi1pxkr/m−/planckover2pi1∆/2)2+/planckover2pi12Ω2. The time evolution of the spinor BEC is governed by the Schr¨ odinger equation i/planckover2pi1∂ ∂t/parenleftbigg Ψ1(r,t) Ψ2(r,t)/parenrightbigg =H/parenleftbigg Ψ1(r,t) Ψ2(r,t)/parenrightbigg . (4) Substituting Eqs. ( 1) into Eq. ( 4), using the length unit x0=/radicalbig /planckover2pi1/mωz, the time unit t0= 1/ωz, and the energy unite0=/planckover2pi1ωz, we can obtain the coupled dimensionless3 GP equations, i∂˜Ψ1(˜r,˜t) ∂˜t=−˜∇2˜Ψ1(˜r,˜t) 2−iγ∂˜Ψ1(˜r,˜t) ∂˜x +˜g11|˜Ψ1(˜r,˜t)|2˜Ψ1(˜r,˜t)+˜g12|˜Ψ2(˜r,˜t)|2˜Ψ1(˜r,˜t) +˜∆ 2˜Ψ1(˜r,˜t)+˜Ω 2eiφ˜Ψ2(˜r,˜t), (5a) i∂˜Ψ2(˜r,˜t) ∂˜t=−˜∇2˜Ψ2(˜r,˜t) 2+iγ∂˜Ψ2(˜r,˜t) ∂˜x +˜g22|˜Ψ2(˜r,˜t)|2˜Ψ2(˜r,˜t)+˜g21|˜Ψ1(˜r,˜t)|2˜Ψ2(˜r,˜t) −˜∆ 2˜Ψ2(˜r,˜t)+˜Ω 2e−iφ˜Ψ1(˜r,˜t). (5b) Here˜∇denotesdimensionlesstwo-dimensionalderivative andγ=krx0is the dimensionless strength of SO cou- pling; other variables and parameters with tilde are also dimensionless. We will omit all tildes in the equations (5) for convenience in our following discussions. After above treatment, the dimensionless dispersion relation ˜E−(∆ = 0) =k2 x/2−/radicalbig γ2k2x+Ω2/4 is shown in Fig. 1. In the case of ∆ = 0, when Ω >Ωc= 2γ2, the ground stateofthe SO-coupledBECisinzero-momentumphase; whileifΩ<2γ2, thegroundstateoftheSOcoupledBEC is in plane-wave phase, which corresponds to the dressed spin states |↓′/angb∇acket∇ight= cos(θ/2)|↑/angb∇acket∇ight −sin(θ/2))|↓/angb∇acket∇ightand|↑′/angb∇acket∇ight= sin(θ/2)|↑/angb∇acket∇ight−cos(θ/2)|↓/angb∇acket∇ight, wheresinθ= Ω//radicalbig 4γ2k2x+Ω2, |↑/angb∇acket∇ightand|↓/angb∇acket∇ightare the bare spin states of SO coupled BEC. We consider two kinds of quenching scenarios in the following: (1) When t= 0, we prepare the ground state of the SO coupled BEC in the zero-momentum phase (Ωi>Ωc) and the relative phase of two lasers φ=π. Whent>0, we suddenly quench the system by shifting the relative phase of two lasers to φ/negationslash=πand keep the strength of Raman coupling unchanged (Ω f= Ωi). (2) We can quench both the relative phase of two lasers to φ/negationslash=πand the strength of Raman coupling to Ω f<Ωc simultaneously, after which the SO-coupled BEC will en- ter the plane-wave phase. After the quench, the initial state will not be in the ground state of the new Hamil- tonian, it will start to evolve. Figure 1shows two kinds of scenarios before and after the quench. In the present paper, we set gij(i,j= 1,2) =g= 1.0. In the following sections, we set the detuning ∆ to zero andΨ1(r,0) =Ψ2(r,0) = const but quench the relative phase of two lasers from φi=πtoφf=−π/2 att >0 for convenience in our discussions. We can obtain the following uniform solution from the coupled equations (5) Ψ1(t) =e−igtψ1(t), (6a) Ψ2(t) =e−igtψ2(t), (6b)where ψ1(t) =1√ 2(cos(Ω ft/2)−sin(Ωft/2),(7a) ψ2(t) =1√ 2(cos(Ω ft/2)+sin(Ω ft/2).(7b) More general cases will be discussed in Sec. V. III. PARAMETRIC RESONANCE INDUCED BY SPIN-ORBIT COUPLING In order to investigate the dynamics of excitations, we can add small fluctuations δΨ1andδΨ2to the uniform solution ( 6) and obtain Ψf 1(t) =e−igt(ψ1(t)+δΨ1), (8a) Ψf 2(t) =e−igt(ψ2(t)+δΨ2). (8b) Here we use the transformation in Ref. [ 28] and define δΨd δΨs = ψ1(t)ψ2(t) −ψ2(t)ψ1(t) δΨ1 δΨ2 ,(9) whereδΨdandδΨsare the density and the spin fluctu- ations, respectively. Substituting Eqs. ( 8) and (9) into the coupled GP equations ( 5) and neglecting the second- orderandthethird-ordertermsof δΨdandδΨs, weobtain i∂(δΨd) ∂t=−∇2 2δΨd+iγsin(Ωft)∂(δΨd) ∂x +iγcos(Ωft)∂(δΨs) ∂x+g(δΨd+δΨ∗ d), (10a) i∂(δΨs) ∂t=−∇2 2δΨs+iγcos(Ωft)∂(δΨd) ∂x −iγsin(Ωft)∂(δΨs) ∂x. (10b) It is now clear that the equations of fluctuations of den- sity waves and spin waves become coupled with each otherdue to the SO couplingeven if gij(i,j= 1,2) =g> 0 with driving frequency Ω f. If the strength of SO cou- pling is zero, Eqs. ( 10) become decoupled, which is the sameastheresultinRef. [ 28], wheretheFaradaypattern is absent although the relativephase oftwo Raman lasers is not equal to zero. In Ref. [ 28], the interatomic interac- tion effectively oscillates in time when the population of two states |↑/angb∇acket∇ightand|↓/angb∇acket∇ightoscillates since g2 12/negationslash=g11g22. In our case, if the system is in the zero-momentum phase, the interatomic interaction need not oscillate with time and the Faraday patterns will be introduced by SO coupling.4 For convenience, in the following discussion we set ky= 0 andγ= 0. The Bogoliubov-de Gennes (BdG) Hamiltonian in momentum space becomes ˆHBdG= k2 x/2+g g 0 0 −g−k2 x/2−g0 0 0 0 k2 x/2 0 0 0 0 −k2 x/2 .(11) The positive eigenvalues of the matrix are ωd=/radicalbig ǫk(ǫk+2g),ωs=ǫk, whereǫk=k2 x/2. Assume X to be the fundamental solution of the differential equa- tioni˙x=ˆHBdGx, where x= (δΨd,δΨ∗ d,δΨs,δΨ∗ s)T. ThenX= exp(−iˆHBdGt). The monodromy matrix F0=X(T) = exp( −iHBdGT), where T = 2 π/Ωfis the period of driving. The multipliers ρhare the eigenvalues of the monodromy matrix F0, ρh(h=d,s) = exp(±iωhT) = exp( ±i2πωh/Ωf).(12) The parametric resonance conditions are 2ωd=nΩf, n= 1,2,3,..., (13a) 2ωs=nΩf, n= 1,2,3,..., (13b) ωd±ωs=nΩf, n= 1,2,3,.... (13c) The cases of ( 13a) and (13b) are called fundamental res- onance while the cases of ( 13c) are called combination resonance. The solutions of Eqs. ( 13) are kd=±/radicalbigg/radicalig 4g2+n2Ω2 f−2, n= 1,2,3,...(14a) ks=±/radicalbig nΩf, n= 1,2,3,... (14b) kd+s=±nΩf/radicalbig nΩf+g, n= 1,2,3,... (14c) Specifically, thecaseof ωd−ωs=nΩfhasnorealsolution ofkx. When 2ωh/negationslash=nΩforωd±ωs/negationslash=nΩf(n= 1,2,...), the multipliers are not double and are complex quantities lying on the complex unit circle, i.e., |ρh|= 1. Therefore, the system is stable when γ= 0. When ky= ∆ = 0 and for smallγ >0, we can expand the monodromy matrix nearγ= 0 [37] (see the Appendix). If at least one of the modules of the multipliers |ρh|>1, the system is dynamically unstable. If both modules of the multipliers |ρd,s|<1, the system will be asymptotically stable. The instability will occur in the vicinity of resonance points (14a), (14b), and ( 14c), at which the multipliers are dou- ble. IV. INSTABILITY ANALYSIS AND THE NUMERICAL RESULTS ( ∆ = 0) We investigate the formations of Faraday patterns of the SO coupled BEC numerically by integrating the two dimensional GP Eqs. ( 5) using a pseudospectral method [33]. The spatial size of our system is 409.6 ×409.6. The 01234 0 1 2 3 401234 0 1 2 3 4 -505 0 1 2 3 4 00.10.20.3 (f)(c) (b) (e) (d)(a) FIG. 2. In the first column, t= 52.5 andγ= 0.8. In the second column, t= 37.5 andγ= 1.2. In the third column, t= 22.5 andγ= 1.5. (a)-(c) display the density distribution log(|Ψ1(k)|2) in the momentum space. (d)-(f) show the real part of Floquet exponent max[Re( λ)] calculated by the time dependent BdG equations. The plots are symmetric about kx andkyaxis, so we only show the results in the first quadrant. Here, Ω f= 2. wavefunction is discretized into a 2048 ×2048 mesh grid andthe periodicboundaryconditionisadopted[ 28]. The resolution in real space is ∆ x= ∆y= 0.2 and in momen- tum space it is ∆ k= 2π/409.6≈0.015 to ensure that thetypicalwavelengthandwavenumberofdynamicsare larger than ∆ xand ∆k, respectively. The initial states areΨ1(t= 0) =Ψ2(t= 0) = 1/√ 2 plus small Gaussian noise. The resonance peaks can be recognized by the Fourier transformation of spatial wave functions. The resonanceregions can also be calculated through Floquet analysis, by which we can calculate the unstable regions. Following the same procedure in Ref. [ 28], we assume the uniform solution Ψj(t) =e−iµtfj(t)(j= 1,2), (15) whereµis a constant and f(t) is the complex periodic function with Rabi oscillation period T. We add small excitations to the solution ( 15) and obtain Ψj(r,t) =e−iµt[fj(t)+δΨj(r,t)],(16) δΨj(r,t) =uj(t)e−ik·r+v∗ j(t)eik·r.(17) Herekis the wave vector of the excitations. Substituting Eqs. (16) and (17) into GP equations ( 5), we can obtain the coupled time dependent BdG equations i ˙u1 ˙v1 ˙u2 ˙v2 =L(t) u1 v1 u2 v2 . (18)5 -20 0 20-20 0 20 0.96 0.98 1 -20 0 20 0.66 0.68 0.7 0.72 -20 0 20 0.06 0.08 0.1 (c) (b) (a) FIG. 3. (a)-(c) are density distributions of |Ψ1|2in real space att= 52.5,37.5, and 22 .5 corresponding to γ= 0.8, 1.2, and 1.5, respectively. Here Ω f= 2.0. Here L(t) = A1+γkxB1C1−iΩf 2C2 −B∗ 1−A1+γkx−C∗ 2−C∗ 1−iΩf 2 C∗ 1+iΩf 2C2A2−γkxB2 −C∗ 2−C1+iΩf 2−B∗ 2−A2−γkx , (19a) A1=k2 x/2+k2 y/2−µ+2g|f1(t)|2+g|f2(t)|2,(19b) A2=k2 x/2+k2 y/2−µ+2g|f2(t)|2+g|f1(t)|2,(19c) B1=gf2 1(t), (19d) B2=gf2 2(t), (19e) C1=gf1(t)f∗ 2(t), (19f) C2=gf1(t)f2(t). (19g) The monodromy matrix F=Texp[−i/integraltextT 0L(t)dt], whereTis the time-ordered operator. We calculate the monodromy matrix Fnumerically using the fourth or- der Runge-Kutta algorithm. The multipliers ρ=eλT are obtained by diagonalizing the matrix F. The real part of the Floquet exponent is defined as max[Re( λ)]. If max[Re(λ)]>0, the system is unstable. When ∆ = 0 and Ω f= 2.0 the results of time evolu- tion of the GP equations are presented in Figs. 2(a)-2(c), which display the density distribution of log( |Ψ1(k)|2) in the first quadrant of the momentum space, and in Fig. 3, which displays the real space density distribution of |Ψ1|2. The unstable regions identified by the value of max[Re(λ)] are displayed in Fig. 2(d)-2(f). We observe that the Faraday patterns appear earlier ( t<60t0) than that in Ref. [ 28] (t >100t0). In Figs. 3(a)-3(c), the re- sulting spatial modulations of density in the xdirection are denser than that in the ydirection due to the larger wavenumber in the kxdirection of the momentum space. We canalsoseethatthe resonanceregionsin the firstrow 0.5 1 1.5 2 2.5 3 3.5 400.511.522.53 00.050.10.150.20.250.3 0.5 1 1.5 2 2.5 300.010.02 FIG. 4. The diagram of the real part of Floquet expo- nent max[Re( λ)] withkxandγwhen ∆ = ky= 0. The dashed line represents combination resonance ( n= 1) at kx= 2/√ 3≈1.15. The solid line represents combination res- onance ( n= 2) atkx= 4/√ 5≈1.79. The dash dot line rep- resents combination resonance ( n= 3) atkx= 6/√ 7≈2.27. Inset: The real part of Floquet exponent max[Re( λ)] as a function of kxwhenγ= 0.1. Here, Ω f= 2.0. of Fig.2match very well with the unstable regions in the second row of Fig. 2. We can clearly see four arcs in Figs.2(a) and2(b), where γ= 0.8 andγ= 1.2, respec- tively. The Raman-induced SO coupling is set to be in thexdirection so the resonance peaks are diminished in thekydirection. We can also understand this from the BdG Hamiltonian ( A.1), in which the driving terms are all multiplied by γandkx. In thekydirection,kx= 0 and the driving terms become to zero, which indicates there is no coupling between the two types of excita- tions. Therefore, there is no resonance in this situation. The leftmost arcs in Figs. 2(a) and2(b) correspond to the combination resonance conditions ωd+ωs= Ωfand kd+s(n= 1) = 2/√ 3≈1.15. The other two fundamental resonance arcs ( n= 1) are too faint to be seen. This can be understood by calculating max[Re( λ)]. The re- sults are shown in Figs. 2(d)-2(e), in which the other two fundamental resonance arcs ( n= 1) are also ex- tremely faint where max[Re( λ)]≈10−4∼10−2at the resonance peaks. The rest three arcs correspond to the resonance conditions ( 13a)-(13c) (n= 2). If we look at the rightmost arcs in Figs. 2(d)-Fig. 2(f) more closely, we can find another very thin and dim resonance arcs intercepting with the rightmost arcs. These thin arcs correspond to the higher combination parametric reso- nance, i.e., ωd+ωs= 3Ωf. In Figs. 2(b) and 2(c), the strength of SO coupling γ >1, and the system is in plane-wave phase. The modulation unstable region will start to grow and merge with the Faraday unstable re- gion. In this case the SO-coupled BEC is in the dressed spin state |↑′/angb∇acket∇ightand|↓′/angb∇acket∇ight. The effective interspecies g1′2′ and intraspecies g1′1′,g2′2′coefficients can be expressed in terms of bare interaction coefficient g[34], i.e.,6 0 1 2 3 4 5 600.511.522.53 00.10.20.30.4 FIG. 5. The diagram of |Im(ω±)|maxaboutkxandγwhen ky= 0,g= 1, and Ω f→0. g1′1′=g, (20a) g2′2′=g, (20b) g1′2′= 2g−cos2θ, (20c) where cosθ=γkx//radicalig γ2k2x+Ω2 f. Now the miscibility condition becomes η=(g2 1′2′−g1′1′g2′2′) g2 1′1′=(2g−cos2θ)2−g2 g2.(21) We setg= 1 in the present paper so ηis always larger than zero, which indicates the dressed spin states are immiscible (modulation unstable) when the system en- ters into the plane-wave phase. We also calculated the diagram of Floquet exponent Re( λ)maxaboutγandkx (ky= 0). The result is shown in Fig. 4. The dashed, solid, and dash-dot vertical lines represent three com- bination resonances ( n= 1,2,3), respectively. We can see the three combination resonances coincide with these vertical lines respectively, which indicates the combina- tion resonance wave numbers kxbarely change when the strength of SO coupling increases. The three resonance regions merge with the modulation unstable region when γisaround1.5. Thehigh resonancetongue( n= 3)inter- cepts with the spin-wave unstable region ( n= 2), other much higher resonancetongues ( n>3) are too dim to be seen. When the value of γis close to zero, the system is asymptotically stable except near the first combination resonance ( n= 1) (see the Appendix), so the rightmost four resonance peaks (including the n= 3 resonance) in Fig.4disappear (see the inset in Fig. 4). In the limit of Ω f→0, the dispersion relation of exci- tations can be described by [ 35] ω2 ±=1 2(Λ1±/radicalig Λ2 1+4Λ2), (22)4 4.5 5 5.500.10.20.30.40.5 FIG. 6. The value of max[Re( λ)] as function of kxwhen γ= 2.5 andky= 0 with different values of Ω f. The solid line represents the analytical result of |Im(ω±)|maxwhen Ω f→0 . wherek2=k2 x+k2 yand Λ1=1 2k2(k2+2g)+2k2 xγ2, (23a) Λ2=−k2/bracketleftig γ2/parenleftig γ2k2 x−1 2k2(k2+2g)/parenrightig +k2 16(k2+2g)2−k2g2 4/bracketrightig . (23b) If the solutions of Eq. ( 23) are complex numbers, the amplitude of excitation grows exponentially with time. We can define |Im(ω±)|maxas the growing rate. We cal- culate|Im(ω±)|maxas a function of kx(ky= 0) andγ numerically; the result is displayed in Fig. 5. The loca- tion of modulation unstable regionsmoves in the positive kxdirection as the strength of SO coupling γincreases, similar to the behavior with the unstable region in Fig. 4whenγ >1. The rightmost region in Fig. 4repre- sents the spin wave excitation; it will merge with the modulation unstable region when the strength of Raman coupling Ω f→0, as we can see in Fig. 6. V. DISCUSSIONS OF GENERAL CASES A.∆/negationslash= 0 When ∆ /negationslash= 0, the ground state of the SO coupled BEC will be in plane-wave phase. [In the present paper, we setgij(i,j= 1,2) =g= 1, so the stripe phase is absent in our system.] We assume Ψ1(r,0) =Ψ2(r,0) = const. We not only follow the quenching scenarios in Sec. II but also quench the detuning from ∆ = 0 to ∆ /negationslash= 0 si- multaneously. Then we can obtain the following uniform solution of the coupled equations ( 5)7 0.5 1 1.5 2 2.5 3 3.50.511.522.5 00.050.10.150.20.25 FIG. 7. The diagram of real part of Floquet exponent max[Re( λ)] withkxandγwhen ∆ = 0 .4 andky= 0. The dashed, solid and dash-dot lines represent the resonance co n- ditionωd+ωs=/radicalBig Ω2 f+∆2,ωd+ωs= 2/radicalBig Ω2 f+∆2and ωd+ωs= 3/radicalBig Ω2 f+∆2, respectively. Here, Ω f= 2.0. Ψ1(t) =e−igtψ1(t), (24a) Ψ2(t) =e−igtψ2(t), (24b) where ψ1(t) =1√ 2[cos(/radicalig Ω2 f+∆2t/2)−eiξsin(/radicalig Ω2 f+∆2t/2)], (25a) ψ2(t) =1√ 2[cos(/radicalig Ω2 f+∆2t/2)+eiξsin(/radicalig Ω2 f+∆2t/2)], (25b) sinξ=∆/radicalig Ω2 f+∆2. (25c) Following the standard procedure in Sec. III, we gener- alize the transformation in Eq. ( 9) and define δΨd δΨs = ψ∗ 1(t)ψ∗ 2(t) −ψ2(t)ψ1(t) δΨ1 δΨ2 ,(26) whereδΨdandδΨsare the density and the spin fluctua- tions, respectively. Substituting Eqs. ( 25) and (26) into the coupled GP equations ( 5) and neglecting the second- order and third-order terms of δΨdandδΨs, we obtain 0 0.5 1 1.5 2 2.5 302 00.050.10.15 FIG. 8. The diagram of real part of Floquet exponent max[Re( λ)] withkxandφwhen ∆ = ky= 0 and γ= 0.4. The dashed line represents combination resonance ( n= 1) at kx= 2/√ 3≈1.15. The dotted line represents combination resonance ( n= 2) atkx= 4/√ 5≈1.79. Here, Ω f= 2.0. i∂(δΨd) ∂t=−∇2 2δΨd+iγcosξsin(/radicalig Ω2 f+∆2t)∂(δΨd) ∂x +iγe−iξ/braceleftig isinξ+cosξcos(/radicalig Ω2 f+∆2t)/bracerightig ×∂(δΨs) ∂x+g(δΨd+δΨ∗ d), (27a) i∂(δΨs) ∂t=−∇2 2δΨs+iγe−iξ/braceleftig isinφ +cosξcos(/radicalig Ω2 f+∆2t)/bracerightig∂(δΨd) ∂x −iγcosξsin(/radicalig Ω2 f+∆2t)∂(δΨs) ∂x. (27b) The resonance conditions near γ= 0 are the same as Eqs. (13) if we replace Ω fwith/radicalig Ω2 f+∆2. When the detuning∆ isnotequaltozero, the systemisin theplane wavephase. Itsunstablebehaviorhassomenew features. We plot the diagram of the real part of Floquet exponent max[Re(λ)] withkxandγin Fig.7. We observe that the combination resonance tongues will split into two parts as the strength of SO coupling γincreases. The first (n= 1) two fundamental resonancepeaks reappear while they are absent when ∆ = 0. B. Arbitrary relative phase φf We assume ∆ = 0, Ψ1(r,0) =Ψ2(r,0) = const and quench the relative phase from φi=πto an arbitrary φf/negationslash=π. Then we can obtain the uniform solution of the coupled equations ( 5) Ψ1(t) =e−igtψ1(t), (28a) Ψ2(t) =e−igtψ2(t), (28b)8 where ψ1(t) =1√ 2[cos(Ωf 2t)−ieiφfsin(Ωf 2t)],(29a) ψ2(t) =1√ 2[cos(Ωf 2t)−ie−iφfsin(Ωf 2t)].(29b) Using the generalized transformation ( 26), following the same procedures in Sec. VA, we can obtain the following coupled equations i∂(δΨd) ∂t=−∇2 2δΨd−iγsinφfsin(Ωft)∂(δΨd) ∂x +iγ[cos(Ω ft)+icosφfsin(Ωft)]∂(δΨs) ∂x +g(δΨd+δΨ∗ d), (30a) i∂(δΨs) ∂t=−∇2 2δΨs+iγ[cos(Ω ft)−icosφfsin(Ωft)] ×∂(δΨd) ∂x+iγsinφfsin(Ωft)∂(δΨs) ∂x. (30b) Whenγisclosetozero, thecorrespondingresonancecon- ditions are the same as Eqs. ( 13). However, we should bear in mind that the initial state is the ground state of H[φf= (1 +2m)π],m= 1,2,3,.... Ifφf= (1 +2m)π, m= 1,2,3,..., there will be no Faraday patterns. We plot the diagram of real part of the Floquet exponent of max[Re( λ)] withkxandφfwhen ∆ = ky= 0 and γ= 0.4 in Fig. 8. The dashed line represents combi- nation resonance ( n= 1) atkx= 2/√ 3≈1.15. The dotted line represents combination resonance ( n= 2) at kx= 4/√ 5≈1.79. Two fundamental resonances ( n= 1) and other higher resonance tongues ( n>2) are too faint to be seen. We can see that the resonance regions are symmetric about φ=π. When the relative phase φ=π, there is no resonance region in the diagram, which means there is no pattern formation. VI. CONCLUSION We have investigated the Faraday instability of homo- geneous spinor BEC with SO coupling by a quench. We found that in the zero-momentum phase, the spatial pat- terns will emerge even if the interspecies and intraspecies interactionsarethe same. This isduetothefact thattwo fundamental excitations (excitations of spin waves and density waves) will be coupled with each other because of the SO coupling and the Rabi oscillations of the two components. We observe higher parametric resonance tonguesatintegermultiplesofthedrivingfrequency. The system is asymptotically stable except at the nearby of the first combination resonance. When we quench the SO coupled BEC from zero-momentum phase to plane- wave phase, the modulation instability begins to play0 0.5 1 1.5 2 2.5 30.99940.99960.99981 = 0 =0.005 = 0.01 = 0.02 FIG. 9. The maximum module of multipliers of the mon- odromy matrix Fas the function of wave number kxwith different strengths of SO coupling γ. Here, Ω f= 2.0. the role of exponentially growing excitations. When the detuning is equal to zero, the wave number of the com- bination resonance barely changes as the strength of the SO coupling increases and for changes of relative phase of the two lasers. If the detuning is not equal to zero af- ter a quench, a single combination resonance tongue will split into two parts as the strength of the SO coupling increases. Recently, the BEC in an optical box poten- tial was realized [ 36]; we hope our theoretical calculation will inspire the observation of pattern formation of SO coupling BEC in the box potential. ACKNOWLEDGMENTS This workhas been supported by the National Natural Science Foundation of China (Grants No. 92065113)and the National Key R&D Program. Appendix: stability of the system near γ= 0 Whenky= ∆ = 0 and γ/negationslash= 0, the Bogoliubov-de Gennes Hamiltonian H BdG(γ,t) in momentum space be- comes k2 x 2+S+g g C 0 −g−k2 x 2+S−g0 C C 0k2 x 2−S 0 0 C 0 −k2 x 2−S ,(A.1) where S =−γkxsin(Ωft), (A.2a) C =−γkxcos(Ωft). (A.2b) If 0< γ≪1, we can expand the monodromy matrix F=Texp[−i/integraltextT 0HBdG(γ,t)dt] nearγ= 0 and keep the terms of the first order of γ[37], F=F0(I+Aγ), (A.3)9 whereIis the identity matrix and F0= P(T) Q(T) 0 0 Q∗(T) P∗(T) 0 0 0 0 e−iωsT0 0 0 0 eiωsT ,(A.4) A=/integraldisplayT 0X−1∂HBdG(γ,t) ∂γXdt. (A.5) After some calculation, we can reduce Eq. ( A.5) to a more simple form A= 0 0M N 1 0 0N∗ 1M∗ M∗N∗ 20 0 N2M0 0 , (A.6) where M=−/integraldisplayT 0kxP(t)cos(Ω ft)e−iωstdt, (A.7a) N1=−/integraldisplayT 0kxQ(t)cos(Ω ft)eiωstdt, (A.7b) N2=−/integraldisplayT 0kxQ(t)cos(Ω ft)e−iωstdt, (A.7c) P(t) = cos(ωdt)+isin(ωdt)(ǫk+g)/ωd,(A.7d) Q(t) =igsin(ωdt)/ωd. (A.7e)The multipliers ρare the eigenvalues of the matrix F. We numerically calculate the maximum module of eigen- values|ρ|maxwithg= Ωf/2 = 1.0 and different values ofγ. The result is presented in Fig. 9.kx= 2/√ 3 corresponds to the wave number of combination reso- nance (n= 1).kx=/radicalbig 2√ 5−2 corresponds to the density waves fundamental resonance ( n= 2) . In Eqs. (14),kd< kd+s< ksifg= Ωf= 1.0. Thus when kx>/radicalbig 2√ 5−2, the multipliers are always smaller than 1 and the system is asymptotically stable. [1] M. C. Cross and P. C. Hohenberg, Pattern formation outside of equilibrium, Rev. Mod. Phys. 65, 851 (1993). [2] M. 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2012.06933v1.Pure_Bulk_Orbital_and_Spin_Photocurrent_in_Two_Dimensional_Ferroelectric_Materials.pdf

1Pure B ulk Orbital and Spin P hoto current in Two-Dimensional Ferroelectric Materials Xingchi Mu, Jian Zhou* Center for Advancing Materials Performance from the Nanoscale, State Key Laboratory for Mechanical Behavior of Materials, Xi ’an Jiaotong University, Xi ’an 710049, China Abstract We elucidate light-induced orbital and spin current through nonlinear response theory, which generalizes the well -known bulk photovoltaic effect in centrosymmetric broken materials from charge to the spin and orbital degrees of freedom. We use two - dimensional nonmagnetic fer roelectric materials (such as GeS and its analogues) to illustrate th is bulk orbital/spin photovoltaic effect, through first -principles calculations. These materials possess a vertical mirror symmetry and time -reversal symmetry but lack of inversion symmet ry. We reveal that in addition to the conventional photocurrent that propagates parallel to the mirror plane (under linearly polarized light) , the symmetric forbidden current perpendicular to the mirror actually contains electron flows , which carry angular momentum information and move oppositely . One could observe a pure orbital moment current with zero electric charge current . This hidden photo -induced orbital current leads to a pure spin current via spin -orbit coupling interactions. Therefore, a four -terminal device can be designed to detect and measure photo -induced charge, orbital, and spin currents simultaneously. All these currents couple with electric polarization ܲ , hence their amplitude and direction can be manipulated through ferroelectric phase transition. Our work provides a route to generalizing nanoscale devices from their photo -induced electronic s to orbitronic s and spintronic s. Keywords: bulk photovoltaic effect; spintronics; orbitronics; two-dimensional ferroelectrics; symmetry analysis; first -principles calculation 2Introduction. Bulk photovoltaic (BPV) effect ,1 which converts incident alternating optical field into direct electric current in centrosymmetric broken materials, has attracted tremendous attention during the past few decade s for its easy manipulation and low ener gy cost . Comparing with conventional light -to-current conversion in a p -n junction between two semiconductors, BPV effect produces electric current everywhere light shines onto the material , which could significantly enhance the conversion efficiency and d ensity. From physic s point of view, BPV effect is a second order nonlinear optical effect, which includes two photons (with frequency ߱ and −߱ ; absorption and emission) and an electron (with moving velocity ݒ.)2 The BPV effect uses electron charge degree of freedom (DOF) to generate a biased electric potential in semico nductor s,3-6 which are serving as promising electronic device s. In order to further increase the information read/write kinetics and storage density, one may resort to new DOF s of electron, such as its spin angular momentum . The study of the int rinsic spin and its induced magnetic moment is thus referred to as “spintronics ”,7-9 which has been shown to hold unprecedented potential in the future miniaturized devices, especially in the field of quantum computing and neuromorphic computing . Roughly speaking, when the velocities of electrons in the spin up and spin down channels are different ( ݒ↑−ݒ↓≠0), there is a collection motion of electron sp in and leads to a nonzero spin current. In addition to spin, another DOF that could produce angul ar momentum and magnetic moment is the electron orbital which describes the electron travel ling around one or a few nuclei . It is a n overlooked DOF because in most conventional materials, the orbital moment is significantly or completely quenched under str ong and symmetric crystal field. However, when the symmetry and strength of crystal field are reduced , especially in low -dimensional materials, orbital DOF may play an important role in their magnetic properties, topological behaviors, and valleytronic features.10-13 Similar as spintronics, this novel field is thus termed as “orbitronics ”,14,15 which is predicted to further enhance information read/write speed significantly. If the electron velocities carrying different orbital magnetic moment s are different ( e.g., ݒ−ݒି≠0), then 3one could also expect an orbital current , analogous to spin current. Note that such an orbital current has been predicted in the linear response Hall effect picture ,16-18 in addition to spin Hall effect19,20 and valley Hall effect.11,21 -23 In the current work, we predict that in addition to nonlinear BPV effect, there exists a hidden orbital current which carries colossal orbital moment when light shines onto two-dimensional (2D) nonmagnetic ferroelectric materials. We refer to this effect as bulk orbital photovoltaic (BOPV) effect, which is described by a second order nonlinear optic al process. We use 2D ferroelectric group -IV monochalcogenide monolayers (GeS, SnS, GeSe, SnSe , GeTe, and SnTe )24-28 and group -V single elemental monolayer Bi(110)29,30 to illustrate our theory. Th is family of materials have been experimentally fabricated and proved to possess flexible and robust in-plane ferroelectrics.31,32 They all belong to ݊݉ܲ 21ത space group , which contains a vertical mirror symmetry ( ℳ௫), and the electric polarization is along ݕ .When linearly polarized light (LPL) (with their polarization direction along ݔ or ݕ )is irradiated onto them, conventional nonlinear BPV current is parallel to the polarization direction ݕ ,while the net BPV along ݔ is zero, according to symmetry considerations . However, we apply first -principles calculations and reveal that there are unexpected hidden electron movements along the +ݔ and −ݔ ,which carries different orbital moments . Hence, one could expect a finite and pure BOPV current to be measured in the ݔ-direction. Here pure orbital current means no electric charge current is mixed. We also show that the spin-orbit coupling (SOC) interaction could convert such BOPV current into spin DOF, namely, bulk spin photovoltaic (BSPV) current (also along ݔ.)33-36 When circularly polarized light (CPL) is used, the conventional charge BPV current will be along ݔ ,while the BOPV and BSPV current s flow along ݕ. Results and Discussion . When light (propagating along out -of-plane direction ݖ ) irradiates onto the 2D materials, one could adopt closed circuit boundary condition37 and use electric field ( )as the natural variable. The BPV , BOPV , and BSPV effect can be eval uated by (Einstein summation convention adopted ) ࣤ(߱=0)=ߪ(0;߱,−߱)ܧ(߱)ܧ(−߱), 4ࣤ;(߱=0)=ߪ;(0;߱,−߱)ܧ(߱)ܧ(−߱), ࣤ;ௌ(߱=0)=ߪ;ௌ(0;߱,−߱)ܧ(߱)ܧ(−߱) . (1) Here ࣤ , ࣤ; , ࣤ;ௌ are electric charge current, orbital current, and spin current density that propagat e along the ܿ-direction , respectively, and ܧ (ܧ) is electric field component ( ܽ,ܾ,ܿ=ݔ,ݕ.) ܮ and ܵ are orbital and spin angular momentum components, respe ctively. Here, we focus on their ݖ-component ( ݅=ݖ )which is most frequently measured and observed experimentally. Actually we have demonstrated that under glide plane symmetry ቄℳ௭|ቀଵ ଶଵ ଶ0ቁቅ , the in -plane spin angular momentum component is suppressed in most k-points of the Brillouin zone (BZ) .30 Now we focus on the ℳ௫ mirror plane effects . One can apply a simple symmetry analysis to examine the non -dissipation response features of these coefficients. Under LPL ( ܽ=ܾ=ݔ,ݕ,) which does not break ℳ௫ symmetry, the charge current would also obey ℳ௫. Hence, ࣤ௫=ℳ௫ࣤ௫=−ࣤ௫=0, so that both ߪ௫௫௫ and ߪ௬௬௫ are zero (symmetr y forbidden). On the other hand, the vertical direction current ࣤ௬ can be nonzero, hence the ߪ௫௫௬ and ߪ௬௬௬ are finite , indicating that BPV electric charge current only flows along ݕ . This is consistent with previous works.38,39 However, since both ܮ௭ and ܵ௭ transform as pseudovector s – they flip their sign under ℳ௫ . One thus expects that the ߪ௫௫௫; , ߪ௫௫௫;ௌ , ߪ௬௬௫; and ߪ௬௬௫;ௌ would become nonzero, while the ߪ௫௫௬; , ߪ௫௫௬;ௌ , ߪ௬௬௬; and ߪ௬௬௬;ௌ all vanish. These suggest that the zero valued BPV current ࣤ௫ actually do not indicate the electron motions are completely frozen in the ݔ- direction. There exists hidden electron flows , which carries orbital and spin angular momentum instead of charge DOF . We illustrate this hidden spin/orbital photocurrent in Figure 1(a). Since the charge photocurrent and spin/orbital angular momentum (or magnetic moment) photocurrent are perpendicular to each other, one could apply a four -terminal device to measure and observe them simultaneously . 5 Figure 1. (a) Schematic plot of bulk angular momentum photo -conductivity (BOPV and BSP V) effect. Unlike BPV that propagates along the mirror plane of monolayer GeS, the BOPV and BSPV current s carrying orbital and spin DOF information are travel ling normal to the mirror plane (ܯ௫) under light . A four -terminal device can be used to detect all these photocurrents. The yellow electrode measures charge DOF, while the blue electrodes are magnetic and could measure angular moment um DOF. (b) Interband transition (pink arrows) picture , where black arrows indicate angular momentum or magnetic momen t from orbital and spin DOF . Solid and dashed curves represent occupied (valence) and unoccupied (conduction) bands of the semiconductor , respectively . We calculate the nonlinear photo -conductivity coefficients explicitly. According to the second order Kubo response theory and within the independent particle approximation framework ,5,40 one can compute the complex nonlinear photo - conductivity from its band structure via ߯;ࣩ(0;߱,−߱)=మ ℏమఠమ∫ௗయ (ଶగ)య∑ ݂௩ೌ௩್;ࣩ (ఠି୧/ఛ)(ఠାஐି୧/ఛ)ஐୀ±ఠ . (2) This is based on a three band model that includes band -݉ , ݊ , and ݈. The phenomenological carrier lifetime ߬ is taken to be 0.2 ps , and i=√−1. The direction scripts ܽ,ܾ,ܿ=ݔ,ݕ. ݂=݂−݂ and ߱=߱−߱ are the occupation and frequency difference between band -݈ and band -݊ ,respectively . Velocity parameter is defined as ݒ=⟨݈|ݒො|݉⟩ and the orbital ( and spin) current operator is adopted to be ݆;ࣩ=⟨݉|{ݒ,ࣩ௭}|݊⟩=ଵ ଶ⟨݉|ݒࣩ௭+ࣩ௭ݒ|݊⟩ with ࣩ௭=ܮ௭ (or ܵ௭). For the charge current, it can be replaced by ݆=⟨݉|݁ݒ|݊⟩. The explicit k-dependence on these 6quantities are omitted. If during interband transition the ݆; or ݆;ௌ rises, it could lead to finite BOPV or BSPV current . We schematically plot this physical process in Figure 1b. Note that this is different from the intra -band optical responses in doped semiconductors .41 Under LPL, the nonlinear photo -conductivity in Eq. (1) can be evaluated as ߪ;ࣩ=ℜ߯;ࣩ . For CPL irradiation (along ݖ) , since the field takes the form i[(߱)×∗(߱)]௭, the response can be evaluated by ߪ;ࣩ=ଵ ଶℑ൫߯;ࣩ−߯;ࣩ൯. We will focus on LPL photo -conductivity in the main text, and plot and discuss the CPL respo nses in Supporting Information. We can denote t he numerator of Eq. (2 in the format ܰ ;௭=ݒݒݒࣩ௭ , which can be used to determine the (dissipationless) symmetry allowed and forbidden responses. The time -reversal symmetry gives ࣮ݒ()=−ݒ,∗(−) and ࣮ࣩ௭()=−ࣩ௭∗(−) , hence, one could have ࣮ܰ ;௭()=ܰ ;௭∗(−) for a Kramers pair (here ⋅∗ indicates complex conjugate of quantity ⋅). Integration over the first Brillouin zone ( BZ) yields a real number. As for mirror symmetry, since ℳ௫ݒ௫()=−ݒ௫൫෩൯ , ℳ௫ݒ௬()=ݒ௬൫෩൯ (෩ is mirror symmetr ic image of , ݇෨௫=−݇௫,݇෨௬=݇௬ ), and ℳ௫ࣩ௭()=−ࣩ௭൫෩൯ , we have ℳ௫ܰ௫ ;௭()=ܰ௫ ;௭൫෩൯ and ℳ௫ܰ௬ ;௭()=−ܰ௬ ;௭൫෩൯. The latter is odd under mirror operation. We thus prove that BOPV and BSPV currents only occur along the ݔ- direction under ݔ or ݕ- LPL, consistent with previous analysis . In the long relaxation time approximation, one could demonstrate that LPL irradiation yields the BPV shift current and the CPL illumination gives an injection current for time -reversal symme tric systems. However, for the BSPV and BOPV , we find that the LPL induced photocurrent is injection -like, which is proportional to the relaxation time ߬ .The CPL, on the other hand, induces shift -like current (see Supporting Information for detailed discussions).42 Now we apply Eq. (2) to compute nonlinear photo -conductivity in 2D nonmagnetic ferroelectric materials. Taking monolayer GeS as an example (Fig ure 2a), we calculate 7its LPL induced BOPV conductivity. In practice, the BZ integration in Eq. (2) is ∫ௗయ (ଶగ)య=ଵ ∑ݓ ,where ܸ is the total volume of simulation supercell and ݓ is the weight of each k-point . In the 3D periodic boundary condition, the supercell of 2D materials contains artificial vacuum space along ݖ , whose contribution needs to be eliminated. According to previou s works, we rescale this result by using an effective thickness of 2D materials ݀ (taken to be 0.6 nm) , which is estimated by the layer -to- layer distance when these 2D materials are van der Waals stacked into bulk. Thus, we can rescale the photo -conductivity by ߪଶୈ=ߪୗେℎ/݀ , where ߪୗେ and ℎ are the supercell calculated conductivity and the supercell lattice constant along ݖ , respectively.43,44 This makes the second order conductivity of 2D materials consistent with conventional quantities of 3D bulk materials. In the following, we w ill report the ߪଶୈ values. As shown in Figure 2b, one sees that consistent with our previous symmetry analysis, ߪ௫௫௬; and ߪ௬௬௬; are exactly zero through all optical frequency, while ߪ௫௫௫; and ߪ௬௬௫; are finite . This numerical results demonstrate that the nonlinear BOPV current flows along the ݔ-direction, which is normal to the mirror plane. Our BPV conductivity calculation confirms that the ݔ-direction charge current is zero (see Supporting Information), whic h also agrees with previous works. Thus, it suggests that the same amount of electrons carrying opposite ݖ- component angular momentum are transporting along ݔ and −ݔ directions, which can be called as pure orbital current (charge current ࣤ௫=݁(ݒ௫;+ݒ௫;ି)=0 , orbital current ࣤ௫;= ݈௭ݒ௫;−݈௭ݒ௫;ି≠0 , where ݈௭ is the averaged ݖ- component angular momentum expectation value , similar as spin up and spin down in the spin DOF ). As for the ݕ- direction electron current, it is a pure charge current with ࣤ௬=݁(ݒ௬;+ݒ௬;ି)≠0 and orbital current ࣤ௬;=݈௭ݒ௬;−݈௭ݒ௬;ି=0 , indicating that all electrons are moving along the same direction ( ݕ or −ݕ )and the amount of electrons carrying ݈௭ and −݈௭ are the same . This is consistent with illustrations in Figure 1a. 8 Figure 2. (a) Atomic geometry of monolayer GeS. (b) Calculated BOPV conductivity as a function of incident linearly polarized light frequency. (c) jDOS ߩ(߱,) distribution in the first BZ at ߱=2.83 eV and ߱=2.17 eV . (d) k- resolved BOPV conductivity ߫௫௫௫; (at ߱=2.83 eV) and ߫௬௬௫; (at ߱=2.17 eV). When the incident light energy is b elow the direct bandgap of monolayer GeS (1.91 eV), all photo -conductivities are zero since no interband transition occurs. At incident optical energy of ℏ߱=2.83 eV, the nonlinear BOPV conductivity ߪ௫௫௫; reaches a negative peak of −636 .2 ఓ మℏ ଶ . Here the unit ఓ మ is that of nonlinear charge B PV conductivity, and ℏ ଶ converts it to angular momentum, similar as that from Hall conductance to spin Hall conductance. If one measures magnetic moment, this unit becomes ݃ఓ మఓಳ ଶ, where ݃ is Landé g-factor (݃≃−1, ݃ௌ≃−2) and ߤ is Bohr magneton. In order to further examine its momentum space contribution, we first plot the k-resolved joint density of states (jDOS) ߩ(߱,) at ߱=2.83 eV (Figure 2c, left panel) . The jDOS reads ߩ௩(߱)=ଵ (ଶగ)మ∫ߩ(߱,)݀ଶ=ଵ (ଶగ)మ∫∑ ߜ(ߝ−ߝ௩−ℏ߱),௩ ݀ଶ, (3) where ߝ is eigenvalue of band -݊ at momentum , and ܿ and ݒ represent the 9conduction and valence bands, respectively. The integral is taking in the first BZ. According to Sokhotski -Plemelj formula, jDOS represent s the resonant band transition between band -݈ and band -݊ in Eq. (2). One could see that ߩ௩(߱=2.83 eV,) is mainly contributed around the ±ܺ and ±ܻ points in the BZ. We next plot the real part of the integrand of Eq. (2) , ߫௫௫௫;(߱,)=ℜ∑ ݂௩ೣ௩ೣೣ;ಽ (ఠି୧/ఛ)(ఠାஐି୧/ఛ)ஐୀ±ఠ (Figure 2d, left panel) , integrating which over the first BZ yields ߯௫௫௫;=ߪ௫௫௫;. One clearly sees that in the momentum space, ߫௫௫௫;(߱,) keeps the ℳ௫ symmetry, and is mainly contributed around ±ܺ and ±ܻ. The jDOS around Γ point does not contribute any significant photo -conductivities. When the ݕ-polarized LPL is shined, it reaches a peak of 913 .95 ఓ మℏ ଶ at ߱=2.17 eV . The k-resolved jDOS and ߫௬௬௫;(߱,) are shown in the right panels of Figures 2c and 2d. Again, the distribution shows ℳ௫ symmetry, which locates around the valleys at (0,±0.54,0) Å–1, but not around Γ. The SOC interaction that breaks the spin rotational symmetry usually splits the spin up and spin down degeneracy in the centrosymmetric broken systems (such as Rashba and Dresselhaus splitting). Here we show that such spin polarization in band dispersion also produces finite BSPV effect . In Figure 3a we plot the calculated BSPV conduc tivity of monolayer GeS. Analogues to the BOPV , under LPL illumination, BSPV also flows along the ݔ-direction, giving finite ߪ௫௫௫;ௌ and ߪ௬௬௫;ௌ, while ߪ௫௫௬;ௌ= ߪ௬௬௬;ௌ=0. However, we find that the magnitude of BSPV conductivities a re generally much smaller than that of the BOPV . Comparing Figure 2b and 3a, one could see that the magnitude of BOPV conductivity is about one order of magnitude larger than the BSPV conductivity. For example, the ߪ௫௫௫;ௌ(߱=2.83 eV)=113 .37 ఓ మℏ ଶ and ߪ௬௬௫;ௌ(߱=2.17 eV)=3.60 ఓ మℏ ଶ , much smaller than the corresponding BOPV magnitudes at the same frequency . We also plot their momentum space contribution s (߫௫௫௫;ௌ and ߫௬௬௫;ௌ, Figure 3b) , which show that they locate similarly as in the BOPV conductivities , and the ℳ௫ symmetry still retains. Note that these 2D ferroelectric 10monolayers possess four electron valleys in the first BZ (near ±ܺ and ±ܻ) .Hence, we show that the photocurrent is mainly contributed from these valleys, which may provide promising physical properties among orbitronics, spintronics, and valleytronics. Figure 3. (a) BSPV photo conductivity component of monolayer GeS under LPL . (b) Momentum space distribution s of the integrand ߫௫௫௫;ௌ(߱=2.83 eV,) and ߫௬௬௫;ௌ(߱=2.17 eV,). According to solid state physics theory , orbital moment in a bulk material is usually strongly quenched by the symmetric crystal field, so that it is the spin polarization that mainly contributes to the total magnetic moment. Hence, the orbital moment contribution is omitted in most cases. However, here we find that BOPV conductivity is generally much larger than that of BSPV . According to Eq . (2), the dominate interband contribution is a two -band transition, namely, |݉〉=|݊〉, and the |݈〉 band lies on the other side of the Fermi level (hence ݂≠0). We will limit our discussion on this two -band model. Thus, the difference between BOPV and BSPV conductivity can be understood by comparing ⟨{ݒ௫,ܮ௭}⟩ and ⟨{ݒ௫,ܵ௭}〉 for the low energy bands (near Fermi level) , which is determined by the velocity and orbital/spin texture . In order to illustrate it more cle arly, we plot the k-space distribution of orbital and spin angular momentum (⟨ܮ௭⟩ and ⟨ܵ௭⟩) of the highest valence band (VB) and the second highest valence band (VB −1), as shown in Figure 4a and Figure 4b. Here VB and VB −1 are actually Rashba splitting bands . One clearly observes that the ⟨ܮ௭⟩ distribution on the VB and VB −1 are similar, while the ⟨ܵ௭⟩ distribution on them show opposite values. 11 This is because that the orbital texture is determined by the crystal field once the material forms , and changes marginally under SOC. On the other hand, the Rashba - type spin splitting yields that ⟨ܵ௭⟩ flips its sign between the VB and VB −1 at each k. We also plot the spin and orbital angular momentum distributions of the lowest two conduction bands i n Supporting Information, and similar results can be seen. The velocity texture distributions on VB and VB −1 are also similar (but not identical) (Figure 4c). We plot ⟨{ݒ௫,ܮ௭}⟩ and ⟨{ݒ௫,ܵ௭}⟩ of bands near the Fermi level in Supporting Information. From all these evidences, we show that the crystal field determined orbital responses are similar at the Rashba splitting bands , while their contributions to the spin responses are opposite (but not completely cancelled due to small velocity distribution difference ). Therefore, the BSPV conductivity is usually much smaller than that of BOPV . 12 Figure 4. Momentum space distribution of (a) ⟨ܮ௭⟩ , (b) ⟨ܵ௭⟩ and (c) velocity texture ൫⟨ݒ௫⟩,⟨ݒ௬⟩൯ of VB and VB –1 in the first BZ . (d) BSPV response function under different SOC strength parameter ߣ .Inset: The peak magnitude change of BSPV ߪ௫௫௫;ௌ(߱=2.83 eV) as a function of ߣ. Linear relation can be clearly seen. One has to note that the band dispersion will be significantly ch anged under very strong SOC, 13so that such linearity may not hold when ߣ is very big . In order to further understand the mechanism of BOPV and BSPV photo - conductivit ies, we artificially tune the SOC interaction ܪୗ=ߙୗ⋅/ℏଶ strength by multiply ing a pre -factor ߣ∈[0,1] . Here ߣ=0 turns off the SOC, and ߣ=1 indicates full SOC. We find that the BOPV (and BPV) conductivity marginally changes under different ߣ (see Supporting Information). However, the BSPV conductivity linearly reduces to zero from ߣ=1 (full SOC) to ߣ=0 (no SOC), as shown in Figure 4d. This clearly demonstrates that the BOPV effect is ubiquitous even without SOC since orbital texture originates from crystal field, while SOC is crucial for BSPV effect in these nonmagnetic systems. Therefore, we could conclude that the BOPV effect arises when crystal is formed, and then it leads to BSPV effect through a finite SOC interaction ( ܪୗ∝⋅ .)Similar relation can also be seen in the orbital and spin Hall effects .16 Note that very strong SOC may not necessarily imply further enhanced BSPV , as the band dispersion would be significantly affected. For the ferroelectric materials, one could easily apply external (electrical, mechanical, and optical) fields to modulate its polarization ( for example, from ܲ to −ܲ). The transition barrier between different ferroic orders is usually a high symmetric geometry, which is centrosymmetric and not electrically polarized (ܲ=0). We now examine the BOPV and BSPV photo -conductivity under different electric polarizations. In Figure 5 we plot the polarization dependent BOPV and BSPV photo -conductivit ies. One clearly observes that all these conductivities diminishes at ܲ=0 state. This is consistent with symmetry analysis, ܫመܰ ;௭()=−ܰ ;௭(−) ,where ܫመ is inversion symmetry operator (angular moment a ܮ and ܵ are invariant under ܫመ). We also note that when the polarization flips (corresponding to a 180° -rotation from ܲ to −ܲ), the photo -conductivities reverse their flowing direction while keeping same magnitudes. If a 90° -rotation occurs, these conductivities also rotate 90°, flowing along the ±ݕ- direction. Hence, one could control the ferroic polarization or der to manipulate the BOPV and BSPV photocurrents , as well as the BPV effect . 14 Figure 5. Polarization dependent (a) BOPV and (b) BSPV conductivity of monolayer GeS. The reversal of polarization ܲ flips the photocurrent, while the high symmetric structure ( ܲ=0) forbids any photocurrents. We now calculate the BOPV and BSPV conductivities for other analogues, namely, monolayers GeSe, SnS, SnSe, GeTe, SnTe, and Bi. Note that even though the monolaye r Bi is a single elemental material, Peierls instability occurs due to strong s and p orbital hybridization, which leads to charge transfer within each atomic layer. Thus, the monolayer Bi also shows in -plane ferroelectricity and fascinating optical proper ties. All these BOPV and BSPV photo -conductivity results are shown in Figure 6. For the BOPV conductivity, we observe clear similarities for all these systems , because their electronic band structure can be described by the same low energy model .30 By comparing the main peaks in Figure 6a and 2b, we find that (for the group IV-VI systems) when the system is composed by small cation and large anion, the BOPV photo -conductivity shows larger peak values (over 10,000 ఓ మℏ ଶ). Hence, the 15monolayer GeTe shows largest photocurrent responses, while the orbital photo - conductivity of monolayer SnS is smallest. However, the BSPV does not have such similarity as the S OC interaction strength (proportional to Z4) determines its responses. Figure 6. (a) BOPV and (b) BSPV conductivity coefficient s of other similar 2D ferroelectric monolayers. Table 1. Polarization dependent bulk ( charge, spin, and orbital) photovoltaic conductivities along ݔ .Here ߟ represents left - or right -handed CPL. Polarization of LPL is along ݔ or ݕ. All eight different types of photocurrents can be realized under light and ferroicity. Symbols SC and IC indicate shift and injection current, respectively. For the ݕ-directional currents, one can apply a 90° -rotation to yield similar results. Polarization ܲା௬ ܲି௬ ܲା௫ ܲି௫ Mirror ℳ௫ ℳ௫ ℳ௬ ℳ௬ LPL ߪ௫;ࣩ (IC) −ߪ௫;ࣩ (IC) ߪ௫ (SC) −ߪ௫ (SC) CPL ߪఎ௫=−ߪିఎ௫ (IC) −ߪఎ௫=ߪିఎ௫ (IC) ߪఎ௫;ࣩ=−ߪିఎ௫;ࣩ (SC) −ߪఎ௫;ࣩ=ߪିఎ௫;ࣩ (SC) 16Conclusion. We predict robust and pure bulk photovoltaic currents in the carrier orbital and spin degree s of freedom. Using nonmagnetic 2D ferroelectric materials (GeS and its analogues) as exemplary materials, we show that the mirror symmetry forbidden BPV conductivity actually contains hidden electron motions , which carries orbital moment flow with zero net electric charge current . Under SOC interaction, the photo -induced orbital current could convert into spin current. Both of these currents are perpendicular to conventional BPV electric current , so that they can be purely and exclusively detected and used . When ferroi c order switches , the photo -conductivities rotate their directions accordingly . We summarize such polarization and light dependent photovoltaic effects in Table 1. Our prediction of pure BOPV and BSPV effects can be easily detected and observed in experiments, and may provide potential ultrafast spintronic and orbitronic applications of 2D in -plane ferroelectric materials, in addition to their electronic features, especially when a four -terminal device is applied . Methods. We use first -principle s density functional theory to calculate the geometric, electronic, and optical properties of 2D monolayer GeS and analogous systems, as implemented in the Vienna ab initio simulation package (V ASP).45 The generalized gradient approximation (GGA) in the Perdew -Burke -Ernzerhof (PBE) form46 is adopted to treat the exchange correlation functional in the Kohn -Sham equation. A vacuum space of 12 Å along the out -of-plane ݖ- direction is used, to eliminate the interactions between different periodic images. Projector -augmented wave (PAW)47 method is used to treat the core electrons, and the valence electrons are represented by planewave basis set, with a kinetic cutoff energy chosen to be 350 eV . The first Brillouin zone is represented by the Monkhorst -Pack k-mesh scheme48 with a (9×9×1) grid for geometric and electronic structure calculations. Convergence criteria of total energy and force component on each ion are set as 1×10-7 eV and 0.01 eV/Å, respectively. Spin -orbit coupling interactions are self-consistently included in all calculations , unless otherwise noted . In order to evaluate the nonlinear optical conductivities, we fit the electronic structure by atomic orbital tight-binding model in atomic orbital basis set ( s and p orbitals), as implemented in the Wannier90 package,49,50 17and the optical conductivities are integrated on a denser k-mesh of ( 901×901×1) grid. The convergence of k-grid density is carefully tested. As for the estimate of orbital angular momentum contributed intra -atomically , we use |ݏ〉=ܻ, |௫〉=ଵ √ଶ(ܻଵିଵ− ܻଵଵ) , |௬〉= √ଶ(ܻଵିଵ+ܻଵଵ) , and |௭⟩=ܻଵ as basis set and calculate the ir matrix components ⟨݉|ܮ௭|݊⟩ , with the intra-atomic orbital angular momentum ܮ௭= ݅ℏቌ0 −ଵ ଶߪି ଵ ଶߪା 0ቍ , where ߪ±=ߪ௫±݅ߪ௬ . The spin operators are proportional to conventional Pauli matrix, ܵ=ℏ ଶߪ , where ߪ (݅=ݔ,ݕ,ݖ) are Pauli matrices . We test and verify our calculation procedure by comparing with previous orbital momentum calculations and BPV effect computat ions. Acknowledgments. This work was supported by the National Natural Science Foundation of China under Grant Nos. 21903063 and 11974270 , and the Startup Funding Program of Xi ’an Jiaotong University . J.Z. acknowledges helpful discussions with Haowei Xu and Dr. Hua Wang, Dr. R uixiang Fei, and P rof. Ya ng Gao on the nonlinear optical effect and theory, and discussions with Prof. Yi Pan for potential observation s. Supplementary Information. Electronic band dispersion of monolayer GeS and its tight binding fitting , BPV and BOPV photo -conductivities of monolayer GeS under different SOC strengths , k-space distribution of spin 〈ܵ௭〉 and orbital angular momentum 〈ܮ௭〉 of the CB and CB+1 , momentum space resolved 〈{ݒ௫,ܵ௭}〉 and 〈{ݒ௫,ܮ௭}〉 for the bands near Fermi level, circularly polarized light induced BPV , BOPV , and BSPV conductivities, and BPV photo -conductivity coefficient of monolayer GeS . Corresponding authors : *J.Z.: jianz hou@xjtu.edu.cn 18Competing Interests. The authors declare no competing interests. Author Contribution. J.Z. conceived the concept and wrote the code . X.M. performed calculations. X.M. and J.Z. analyzed data and wrote the manuscript . Data Availability. All data generated or analyzed during this study are included in this published article (and its Supplementary In formation files), and are available from the authors upon reasonable request . Code Availability. 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2012.10647v5.Non_orthogonal_Spin_Momentum_Locking.pdf

Non-orthogonal spin-momentum locking Tugrul Hakiogluy,1,Wei-Chi Chiu,2,yRobert S. Markiewicz,2Bahadur Singh,3and Arun Bansil2 1Energy Institute and Department of Physics, Istanbul Technical University Maslak 34469, Istanbul, Turkey 2Department of Physics, Northeastern University, Boston, MA 02115, USA 3Department of Condensed Matter Physics and Materials Science, Tata Institute of Fundamental Research, Colaba, Mumbai 400005, India Spin-momentum locking is a unique intrinsic feature of strongly spin-orbit coupled materials and a key to their promise of applications in spintronics and quantum computation. Much of the existing work, in topological and non-topological pure materials, has been focused on the orthogonal locking in the vicinity of the point where the directions of spin and momentum vectors are locked perpendicularly. With the orthogonal case, enforced by the symmetry in pure systems, mechanisms responsible for non-orthogonal spin-momentum locking (NOSML) have drawn little attention, although it has been reported on the topological surface of -Sn. Here, we demonstrate that, the presence of the spin-orbit scattering from dilute spinless impurities can produce the NOSML state in the presence of a strong intrinsic spin-orbit coupling in the pristine material. We also observe an interesting coupling threshold for the NOSML state to occur. The relevant parameter in our analysis is the de ection angle from orthogonality which can be extracted experimentally from the spin-and-angle-resolved photoemission (S-ARPES) spectra. Our formalism is applicable to all strongly spin-orbit coupled systems with impurities and not limited to topological ones. The understanding of NOSML bears on spin-orbit dependent phenomena, including issues of spin-to-charge conversion and the interpretation of quasiparticle interference (QPI) patterns as well as scanning-tunneling spectra (STS) in general spin-orbit coupled materials. I. INTRODUCTION Spin-momentum locking (SML) occurs commonly in spin-orbit coupled low dimensional materials with or without topological bands[1{4]. Its telltale signatures in- volve forbidden backscattering [5, 6] from non-magnetic impurities (no `U-turn') and enhancement of weak an- tilocalization eects [7]. SML enables electrical control of spin polarization in nonequilibrium transport and thus plays a key role in spintronics and spin-based quantum information sciences applications [2, 8] in the capability to drive a spin-polarized current with polarization per- pendicular to the current density [9{11]. The orthogonal SML (OSML){see Fig.1.a, is common in materials exhibiting SML[1{3]. The OSML state is the result of in-plane Rashba spin-orbit coupling (SOC) ob- served rst time on the Au (111) surface long before the topological materials were discovered[12]. In topological insulators, OSML with a -Berry phase is an essential feature of the surface electron bands[13]. It has been utilized in the electrical detection of magnon decay[14]. Despite its broad presence in strongly spin-orbit cou- pled materials, violations of the SML has been seen in real materials. In such cases, spin and momentum are weakly unlocked within a narrow range of angular de- viations constrained by the crystal symmetries. An S- ARPES study of the Au/Ge (111) surface revealed such examples[15] and similar eects have been reported in high-temperature superconductors[16]. In certain topo- hakioglu@itu.edu.tr yThese authors contributed equally.logical insulators the spin wiggles around the Fermi sur- face due to the hexagonally warped Fermi surface but respects the OSML[17]. Deviations from the orthog- onal picture were also observed experimentally in the Bi2ySbyTexSe3xfamily[18] as shown in the Fig.1.b. We can call these weak violations from the orthogonally locked state as type-I violations. It has been shown that high-order corrections to the theoretical k:pHamiltonian can induce deviations from the orthogonal picture[19]. Many body interactions also cause similar eects, as the electron-phonon interaction in this material[20{23] was recently studied in this context[24, 25]. The triple and septuple windings of the spin vector have also been stud- ied theoretically as violations of the OSML[26, 27]. Another type of deviation from the perfect OSML state is not in the locking of the spin and momentum but in their orthogonality, i.e. the non-orthogonal spin- momentum locking (NOSML) as illustrated in Figs.1.c and d (called as the type-II violations of the OSML). Such a state has been reported on the topological sur- face of strained -Sn[28, 29]. Here, S-ARPES and Mott polarimetry reveal the presence of a radial component of the spin (Fig1.c) with a signicant inward deviation of  090'20on a circular Fermi surface[30]. The out-of-plane spin Szis observed to vanish in conformity with the absence of the out-of-plane SOC. Note that - Snis inversion symmetric in unstrained and strained phases [29, 31{37]. The authors of Ref.[28] also point at the presence of electron-impurity interaction through an analysis of the electronic self-energy. This last observation is of key importance in our the- ory of the NOSML. Our approach is not limited to topo- logical surface states but addresses NOSML as a generalarXiv:2012.10647v5 [cond-mat.mes-hall] 15 May 20232 phenomenon in materials with strong SOC. The presence of inversion and time-reversal symmetries substantially constrains the Hamiltonian for treating non-interacting bands. Origin of the NOSML lies beyond the realm of warped electronic bands and details of the lattice struc- ture are not important for generating this eect. Indeed, the OSML state is strictly enforced in pure materials due to theC1vsymmetry. We therefore study impurity ef- fects in this article as the source of NOSML. II. THE THEORY OF INTERACTING SPIN Our starting point is the time-reversal invariant Hamil- tonian in the pseudo-spin jkibasis ( ~= 1)[1, 38, 39]: H0= (k)0+gk: (1) where= (x;y;z) is the pseudospin representing the spin-orbit coupled total angular momentum states[1], k= (kx;ky) =k(cos;sin) is the electron wavevector relative to the Dirac point at k= 0 andkandare the spin-independent and isotropic bare electron band and the chemical potential, respectively. The Hamilto- nian in Eq.(1) is the most basic Hamiltonian in spin- tronics as well as topological surfaces. A pair of such Hamiltonians can be used to model states in Dirac and Weyl semimetals as well as Rashba type interface states. The spin-orbit vector gkis normally composed of an in- plane component gk=g0^zkwithg0as the Rashba type in-plane SOC and ^zas the surface unit normal vec- tor to the xyplane dened by the kvector, and an anisotropic out-of-plane component g?k. We also rep- resent the in-plane and out-of-plane components of the spin as well as the self-energy vectors below using the same notation. The g?kas well as the out-of-plane component of the spin are zero in our case due to the azymuthal rotational symmetry. The eigenstates jki of Eq.(1), where =is the spin-orbit band index, include the chiral spin-1/2 state not only attached to the dominantjpziorbitals as considered conventionally, but also the in-plane orbitals jpxiandjpyi. The role played by the in-plane orbitals is strongly material depen- dent, which has been demonstrated experimentally[40] and theoretically[41]. It is known that these eects do not violate the OSML[13, 17]. For our purposes in this work we ignore these in-plane orbitals and consider that the orbital texture is solely determined by the out-of- planejpziorbitals. Further discussion on this point is made in Section.V. With this considered, the pseudospin is given byhJi=hkjjki= (=2)^gk, where ^gkis the unit spin-orbit vector, which coincides with the actual spinS(k) = (=2)^gk. Since the in-plane component gk of the spin-orbit vector gkis perpendicular to k, the spin S(k) is locked orthogonally to the electron momentum kthroughout, yielding the OSML. The Eq.(1) is clearly insucient to describe all strongly spin-orbit coupled surfaces and additional terms may be present due to the symmetries. For instance, the cubic (a) (c) (d)(b)vvvvFIG. 1. A schematic of various planar spin-momentum lock- ing cases for a chiral band. (a) The OSML. (b) Weakly un- locked case of Bi2Se3andBi2Te3. The weak out-of-plane component is not shown. (c, d) NOSML with  <0 (c) and >0 (d).is dened in Eq.(15). Dresselhaus SOC is present in the absence of inversion symmetry in ordinary semiconductors[1]. Its realization inBi2X3type strong topological insulators results in the hexagonal warped Fermi surfaces but the OSML is still respected[17]. In few other cases, the anisotropy may persist down to the point[42]. Note that, NOSML is ideally an isotropic eect and anisotropic warping in the band structure may hinder its observation. Therefore, we ignore the anisotropy and examine rotational symmetry allowed Hamiltonians in the study of the NOSML state. The minimal Hamiltonian which can yield a NOSML state is[1, 38, 39] H1= k:: (2) This Hamiltonian is known to be present in the Kane model between the 7cand 6cbands of zinc-blende structures[1, 43, 44]. The total Hamiltonian H0+H1 is equivalent toH0by a complex rotation of the spin- orbit constant, and eective spin-orbit coupling is de- ned asgk=g0^zk+ k. The energy spectrum is lin- ear and spin-momentum pair is locked non-orthogonally at  0=(=2 +tan1 =g0) with thedescribing the upper and the lower Dirac cones. While, Eq.(2) can eas- ily accommodate a non-orthogonal state of the spin and momentum in zinc-blende structures[44], it is not appli- cable when the inversion symmetry holds since that re- quires = 0. This prompts us to think that the NOSML in inversion symmetric systems may have its origin fun- damentally beyond the class of symmetry allowed single particle Hamiltonians. Here we demonstrate that, the impurities in the real materials may provide a striking clue for the general source of the NOSML. It is known that the electron-impurity interaction, combined with the strong SOC, gives rise to the spin-orbit scattering in ad-3 dition to the scalar scattering channels, which then leads to a number of observable transport phenom- ena. These are linearly dependent on the spin-orbit scattering strength[45] such as corrections in the mo- mentum and spin relaxation, spin-dependent diusion, weak localization/antilocalization[2] and anomalous spin- texture[46]. The spin-orbit scattering between the impu- rity and the electron bands also provides a platform for NOSML and this is the main focus of this work. In this work, we use the interacting Green's function formalism for the renormalized spin[25]. In this approach the spin is given by S(k) = 2^G(K) (3) Here, theindicates that S(k) is calculated at the phys- ical energy pole position E=Ekof the full Green's function[25]. G(K) =gk+(K), withK= (k;iE) in the Matsubara Green's function formalism, is the renormalized spin-orbit vector where ^G=G=jGjis its unit vector. Here it is crucial to note that, Genters as a simple sum of the spin-orbit vector gkof the non- interacting structure and the interactions represented by the spin-dependent self-energy (SDSE). The gkis aligned perpendicularly to the momentum. The renor- malized spin-orbit vector Ghowever, develops a non- orthogonal component as a result of the interactions. We recently demonstrated this theoretically using the electron-phonon interaction and the Fermi surface warp- ing yielding six-fold symmetric type-I violations of the OSML in the topological insulator Bi2Se3[24, 25] [see Fig1.(b)]. Here, we demonstrate that, the electron- impurity interaction can have a similar consequence with- out the need of a warped Fermi-surface yielding a non- orthogonally locked conguration of the spin and mo- mentum [Fig1.(c) and (d)]. The spin-dependent-self-energy (SDSE) vector (K) represents the impurity average of the microscopic scat- tering events between the electron and the impurity (see Appendix A). The full spin-neutral and the spin- dependent parts of the self energies can be combined in the pseudospin matrix form as, (K) =  0(K)0+(K): (4) where  0is the spin neutral self-energy (SNSE) and = (x;y;z) is the SDSE as introduced before. The total change in the spin between this interacting model and the non-interacting one is expectedly decided by the SDSE which is given by
S(k) = (=2) [^G(k)^gk]. Here we took the dierence of two cases with and with- out interactions using Eq.(3). In the weak interaction limit[25] this takes an elegant form with the leading term
S(k)' 2 k jgkj^k (5) where  k=(K):^kis the component of the SDSE along the momentum. The Eq.(5) states that the inter- actions can cause both type-I and type-II violations of theOSML. Since there are strong symmetry considerations in the pure crystal structure, the generation of a nite  k is not trivial. In this work, we study the electron-non- magnetic impurity scattering as a new mechanism for the non-orthogonally locked type-II state as shown in Fig.1 (c) and (d). III. THE SPIN-ORBIT IMPURITY SCATTERING Whatever the mechanism is, the formalism in Eqs. (3- 5) hinges upon an accurate model for the self-energy in Eq.(4). The electron-impurity scattering is represented as a scattering potential V(j) ei=V(j) 0+V(j) sothe spin independent and spin-orbit scattering parts are given byV(j) 0andV(j) sorespectively. In the notation, the su- perscript refers to the j'th impurity (see Appendix B). The spin-orbit coupling in the pristine sample is as- sumed to be suciently strong compared to the electron- impurity interaction. By this approach a simple pic- ture can be obtained where the leading order contri- bution to NOSML can be isolated from the other sec- ondary eects. The scattering matrix between the ini- tialjii=jkiand the naljfi=jk00istates is T0(k;k0) =P jhk00jV(j) eijkigiven in the Born ap- proximation by[2, 45, 47{57] and Appendix B by, T0(k;k0) =X jei(kk0):Rjt(j) 0(k;k0) (6) where the exponential phase factor accounts for the im- purity scattering phase shifts occuring at random cen- tersRjandt(j) 0(k;k0) is the scattering amplitude of the electron o the j'th impurity from the initial to the - nal state which can be derived microscopically once the impurity-electron scattering potential is known. We will assume that there is only one kind of impurity and drop thejindex int(j) 0. The scattering of an external spin- less impurity with the electron under the in uence of the spin-orbit coupling is an old textbook problem which has been studied before[52]. The eective interaction is ba- sically a superposition of two independent parts. The rst part is a spin independent channel contributing to the momentum distribution and relaxation. The second part has been shown to arise as a result of the interac- tion between spin-orbit coupled electrons and the spinless impurity. The scattering matrix is then given by[55] (see Appendix B), t(k;k0) =a00+c0^k^k0: (7) where thet0in Eq.(6) corresponds to the matrix el- ement of the Eq.(7) with the spin indices ;0. The rst term describes the spinless scattering with a0as the scattering strength and the second term is the spin-orbit scattering with the strength c0. These coecients are generally functions of k,k0as well as the details of the4 microscopic electron-impurity interaction [2, 51{54] (see Appendix B). We further assume dilute impurity limit ni3 F whereniis the impurity concentration and Fis the Fermi wavelength of the scattered electrons. In this limit, we neglect the interference between multiple scattering events. IV. THE SELF-ENERGY DUE TO THE SPIN-ORBIT SCATTERING A. The spin-independent self-energy The OSML is strictly enforced by the C1vsymmetry near the point in pure crystals[58, 59]. In strongly spin- orbit coupled materials, deviations from this orthogonal picture requires a sucient impurity coupling and the renormalization of the Bloch states. Including the impu- rity scattering perturbatively, the eect vanishes in the rst order of the perturbation since at this level the elec- tron self energy averages out to zero over the impurities (see Appendix A). Here, the NOSML emerges beyond the second order in the electron self energy and this includes the renormalization of the Bloch states. The Feynman diagrams of the Green's functions and the self energies are summarized in Fig.(4) of the Appendix A. Another point to stress is that, the NOSML can be con- cealed by warping or other anisotropy eects. To keep the formulation at a fundamental level, we limit ourselves to the case when such phenomena are absent or suciently weak and consider an isotropic band k=k2=(2m) near the point. The full impurity averaged self-energy in Eq.(4) is dened as [47, 48, 60], (K) =ni 2Zdk0 (2)2 X t(k;k0) [1 +^G(k0;E):]G(k0;E)t(k0;k) (8) TheG(k;E) = 1=(EEk) is the Green's function of the eigenband with index andEk=~k+jG(k;E)j as the renormalized energy band with ~k=k+ Ref0g andG=gk+. Thenidependence in Eq.(8) comes from the averaging over the random impurity positions Rias given in Eq.(6) and shown in the Appendix A. We note that, the dependence of the self energy on the impu- rity concentration in Eq.(8) is not linear due to the non- linear dependence in the renormalized spin-orbit vector Gon the self-energy. Furthermore, these equations can be obtained from our more general theory of the surface electrons interacting with the lattice excitations studied in Ref.[25] when the phonon excitation energy vanishes in the static limit. The spin-independent and spin-dependent parts of Eq.(8) are given by, 0= Trfg=2;= Trfg=2 (9)We assume that the spin-orbit scattering is weak com- pared to the spin-independent one. We also neglect the overall phase of the t(k;k0) and assume that a0is real. The latter can then be directly related to the spin- independent self-energy  0. Using the Eq's.(9) we have, Imf0(E)g'm 4nia2 0 1g0q g2 0+2 mE! :(10) which is related to the life-time = 1=Imf0gof the electron momentum due to its scattering with the impu- rities. Another importance of this equation is the con- nection with the experiment, i.e. Im f0(E)gcan be di- rectly extracted from the experimental quasiparticle mo- mentum distribution[28]. We will use the Im f0gas a phenomenological parameter replacing nidependence throughout. It will be shown in the next section that the SDSE as found by the second equation in (9) has a dierent dependence on ni. This is brought by the renormalized spin-orbit vector on the right hand side in Eq.(8) which may lead to a critical boundary separating the OSML and the NOSML phases as discussed below. B. The spin-dependent self-energy and the NOSML We now turn to the spin-dependent component in Eq.(8), which can be extracted by using the second of the Eqs.(9). Since the out-of-plane component of the gk is absent due to the rotational symmetry, gk=gkand zis absent. We start by writing = (x;y;0) in the polar form using the radial ^kand the azymuthal ^gkunit vectors as = g^gk+ k^k (11) where g=:^gkand k=:^kare the components of along the ^gkand ^kdirections respectively and they are scalar functions independent from the direction of ^k. Using these scalar components is particularly useful in the impurity averaging since  gand kare not aected by the scattering directions of the kvector, a crucial factor in the impurity averaging considering the random orientations in each scattering event (see Appendix A). The Eq.(11) is equivalently written as xiy=ei(+=2)Ck (12) which is a quite convenient way of writing the SDSE since Ckis represented in terms of the scalar components of the SDSE elegantly as Ck= g+ik. The real part renor- malizes the spin-orbit strength since g0k!g0k+ g. This renormalization can be ignored since g0kis su- ciently strong. The imaginary part  k, on the other hand, is an emerging component which is the main cause of the deviation in the spin-momentum locking angle5 FIG. 2. Spin deviation angle kis illustrated for two dierent casesk<0 andk>0. Thekhas the same sign in the upper and lower Dirac cones which is shown for the k>0 case in the inset. from the orthogonality as shown in Eq.(5). Using Eq.(12) in Eq.'s (9) and (8) we nd Ck=Zk0dk0 2z0(k;k0)F(k0) (13) where F(k0) =X g0k0+Ck0 jG(k0;E)j EEk0(14) which numerically couples  gand k. Herez0(k;k0) is a complex scalar function depending on the scattering strengths in the Eq.(7). In the simplest case of constant scattering strengths, z0is just a complex number. In order to make a connection with the spin-texture mea- surements and obtain some quantitative estimates, we now dene a microscopic spin-deviation angle kas il- lustrated in Fig.(2). From the geometry and using Eq.(3) we nd[18, 24, 25] sink=S:^k jSj!k jGj(15) We further identify two cases in Fig.2 as k<0 and k>0. The Eq.(15) also yields that the kin the upper and the lower Dirac cones have opposite signs as required by the time reversal symmetry and shown by the inlet in Fig.(2). We now shift our attention to the numerical solution of the Eq.(13) which reveals the dependence of the spin- deviation angle kon the impurity scattering as well as the spin-orbit coupling strengths. We now dene a small dimensionless quantity = Im  0=EFwhich is linearly dependent on ni. Concerning the solution for the k, we concentrate on the upper Dirac band = + and solve the Eq.(13). It is easy to see that +kvanishes when z0 is purely real and varies linearly with  c= Imz0with a steep behavior near  c= 0. The calculated +kat the Fermi level is shown in Fig.3 as  candare varied. The inset therein refers to the behaviour when z0is purely imaginary. FIG. 3. The k(in degrees) at the Fermi surface for the = + band in Eq.(15) using the Eq.(13) as the spin-orbit scat- tering amplitude  cand theare varied at a xed spin-orbit coupling strength corresponding to  g0=g0kF=EF'0:4. The inset at the top right illustrates a sharp boundary between the OSML (= 0) and the NOSML ( 6= 0) phases determined by the critical values of the  g0and= Im  0=EF. The color scale is for the jkjand applies to both plots. We used kF= 0:035A1andEF= 150meV for the normalization[28]. V. DISCUSSION AND CONCLUSION Due to the conservation of the total angular momen- tumJ=L+S, the orbital congurations can aect the spin texture[40, 41] and, since NOSML is a weak eect due to the small scattering strength  c, it is important to understand whether the in-plane orbitals jpxiandjpyi can change the observed picture in the Section IV.B. In- cluding the contribution of these in-plane orbitals, the spin-orbital state is given up to the linear order in k by[40, 41, 61], jjki= (u0v1k) (jpzi ji) ip 2(v0u1kw1k) (jpri ji) (16) +1p 2(v0u1k+w1k)jpti ji whereu0;1;v0;1;w1are material dependent coecients, ji= (1=p 2)[j"iieij#i] is the chiral spin-1 =2 vor- tex state,ji=j()iandpr(pt) are the radial (tan- gential) in-plane combinations of the px;py-orbitals given by thejpr(pt)i= cos(sin)jpxi+ sin(cos)jpyi. One may consider that jjkishould have been used in this work instead of jki. Although this is principally correct, is has been studied before that the jjkidoes not change the spin or the spin-momentum orthogonal- ity at the single-particle Hamiltonian level[13, 17]. Determination of the constants a0(k;k0) andc0(k;k0) in Eq.(7) with their full momentum dependence is a fundamentally important problem. Experimentally, the6 quasiparticle interference (QPI) with spectroscopic STM can be a promising probe of spin-orbit scattering[55, 57]. With this technique the authors in Ref.[55] estimated c=k2 F'80A2for the polar semiconductor BiTeI . Here, the relation between electron-impurity scattering and the spin texture provides an alternative method of extracting cwhen the warping anisotropy is absent. For a system with inversion symmetry,  ccan be found once the kof the spin texture could be measured by using S-ARPES. We know that +k'20in the case of -Sn[30] and the warping is nearly absent in the surface bands. Using Fig.3 and this +k, we nd that  c=k2 F'40A2putting this material as a strong topological spin-orbit impurity scatterer. In summary, we showed that the presence of the spin- orbit scatterings from non-magnetic impurities, an eect which is expected to be nite when the impurities are present in strongly spin-orbit coupled realistic materi- als, can provide a mechanism for the deviations from the well-established phenomenon of OSML to the one with a non-orthogonal locking. The NOSML angle which can be measured experimentally, is a non-linear function of the impurity concentration and we nd that its appear- ance requires a critical spin-orbit strength. It will be interesting to explore this new state experimentally in more general topological/non-topological systems at var- ious spin-orbit coupling strengths and impurity concen- trations. Our theory should pave the road for the full investigation of the eect of the impurities on the spin- momentum locking also including the magnetic impuri- ties. We end with a nal remark that, our study high- lights additional richnesses of spin textures brought by the impurity eects in strongly spin-orbit coupled mate- rials. VI. ACKNOWLEDGEMENT T.H.'s research is supported by the ITU-BAP project TDK-2018-41181. He dedicates this work to the 250th birthday of the Istanbul Technical University (est. 1773) where a major part of this work was done. T.H. also thanks Northeastern University for support during his visit. The work at Northeastern University was sup- ported by the US Department of Energy (DOE), Of- ce of Science, Basic Energy Sciences Grant No. DE- SC0022216 (accurate modeling of complex magnetic states) and beneted from Northeastern University's Ad- vanced Scientic Computation Center and the Discov- ery Cluster and the National Energy Research Scientic Computing Center through DOE Grant No. DE-AC02- 05CH11231. Appendix A: IMPURITY AVERAGING Here we discuss details of the impurity averaging of the electron self-energy and the Green's function. Diagram- FIG. 4. Feynman diagrams corresponding to the rst order electron-impurity vertex in Eq.(7) (a), the Green's function in the matrix form (b), the electron self-energy in the matrix form (c). matically the electron-impurity interaction is described by the Feynman diagrams as shown in Fig.(4). We consider that the impurity at the random posi- tionRjis scattered by electrons with initial and nal momentak;k0. By the impurity averaging we mean a two-step process. The rst is that the kinetic phase ei(kk0):Rjof the electron wavefunction acquired at the j'th scattering is randomized by the random position Rj of thej'th impurity. This leads to the average over the impurity positions as described in a separate section be- low. The second crucial factor is that the random impu- rity positions also lead to randomized incidence direction of the electron between two scattering events. In order to avoid averaging over the random initial-nal momen- tum orientations at each scattering, we must form scalar quantities of the self-energy vector as  k=:^kand g=:^gkas the component of the self-energy along the momentum and along the spin-orbit vector. The  kand gare these scalar quantities which are not aected by the random directions of the initial state vector kbefore each scattering. We dene the average over the impurity positions by hOiimp=Z dRO(R)P(R) (A1) HereP(R) is the classical distribution of the impurity positions and O(R) is a generic quantity to be averaged. In our case the impurity positions are completely random withP(R) = 1= with being the area in which the impurities are randomly scattered. Considering that the impurity-electron interaction is weak, we use a perturbative expansion of the electron7 Green's function including the rst and second order terms in the impurity-electron scattering matrix elements T0(k;k0). In this section we derive the impurity aver- aged full self-energy given by the Eq.(8) of the main text. The latter is given by, h(K)iimp=hT(k;k)iimp +Zdk0 (2)2hT(k;k0)G(k0;E)T(k0;k)iimp (A2) whereT(k;k0) is the same as the Eq.(6) in the ma- trix form. The Feynman diagrams corresponding to theT0(k;k0) are shown in Fig.(4.a). In Eq.(A2) the G0(k0;E) is the interacting electron Green's function of the0band in terms of the 2 2-matrix form in the electron-pseudospin space. In order to nd this quantity, we rst start with the matrix Dyson equation 1 G(k;E)=1 G0(k;E)(k;E) (A3) with G0(k;E) =1 Ekgk:(A4) representing the non-interacting Green's function and the (k;E) =  0(k;E)0+(k;E):the full electron self- energy. TheG(k;E) in the Eq.(A3) can then be com- pactly written as G(k;E) =X G(k;E) (A5) where G(k;E) =1 2[1 +^G(k;E):]G(k;E) (A6) with G(k;E) =1 EEk(A7) as the exact Green's function of the quasiparticles in the eigenband of the Hamiltonian in Eq.(1). The G(k;E) = gk+(k;E) is the renormalized spin-orbit vector and ^Gis the unit vector of G. Eq.(A6) is the di- rect sum of the contributions from each spin-orbit band singled out by the physical pole-position of the G(k;E) atE=Ek. We now leave the Green's functions aside and examine the full self-energy in Eq.(A2) diagrammatically in order to derive the dependence of Eq.(8) on the impurity con- centration. The rst term hT0(k;k)iimpis the impurity average of the Eq.(6) for which we use: NimpX j=1hei(kk0):Rjiimp=1 Z d3RNimpX j=1ei(kk0):R =nik;k0 (A8)where the average impurity concentration is given by ni=Nimp= . We therefore have that hT0(k;k0)iimp= nik;k0t0(k;k) which can be ignored since it implies the absence of scattering on the average. We now shift our attention to the second term in Eq.(A2). This requires the knowledge of the full Green's function. The result is (temporarily omitting some indices for simplicity), hTG(k0;E)Tiimp'1 X i;jhei(kk0):(RiRj)iimp t(k;k0)G(k0;E)t(k0;k) (A9) where we used t(k;k0) =ty(k0;k) for the unitarity of the scattering matrix. We now work on the relevant part in Eq.(A9) which depends on the impurity average. By denition X i;jhei(kk0):(RiRj)iimp=1 X i=j1 +X i6=jhei(kk0):(RiRj)iimp (A10) The impurity averaging over a totally random impurity distribution yields random interference between dierent impurities when Ri6=Rjyielding a vanishing contribu- tion fork06=k. This term is therefore (with 1 Nimp) X i6=jhei(kk0):(RiRj)iimp=n2 impk;k0(A11) Hence it averages out to zero when k6=k0like the rst order impurity average in Eq.(A8). The net eect of this term is therefore essentially the same as the rst order impurity-vertex. The net eect of the impurity averaging in Eq.(A10) is therefore provided by the rst term on the right hand side as (1 = )P i1 =Ni= =ni. Eq.(A9) is therefore given by hTGTiimp=nit(k;k0)G(k0;E)t(k0;k) (A12) Using this result in Eq.(A2) we nd h(K)iimp=niZdk0 (2)2t(k;k0)G(k0;E)ty(k;k0) (A13) The Eq.(A13) is the full electron self energy correspond- ing to the Eq.(4) in the manuscript. Using Eq.(9), the Eq.(A13) yields the Eq.(8) in the manuscript where we dropped the explicit impurity averaging symbol h:::iimp. Next we consider the second type of average which is due to the random orientations of the initial/nal mo- menta. The  k=h:^kiimpand g=h:^gkiimpare meaningful quantities for impurity averaging since both are scalars and unaected by the random directions of8 FIG. 5. Second interference diagrams for the electron self- energy contributing to the NOSML which have linear depen- dence on the spin-orbit scattering strength c0. The solid line represents the bare electron propagator, and the dashed line represents the two scattering events with the impurity Rj. the scattered electron momenta. It can be explicity seen that, the transformation in Eq.(12) separates the ran- dom orientation of the ^kand^k0by separating out 0 in the angular average, and indeed, what remains is the Ck= g+ikwhich is perfectly a scalar complex func- tion ofk. In order to obtain a self consistent expression for Ck= gik, we apply the Eq.'s(9), (11) and (12) in Eq.(A13). The real and imaginary parts of Ckdene a coupled set of equations given by ( = xiy; = xiy) (k;E) =ni 2Zdk0 (2)2X G(k0;E)t(k;k0)t(k0;k) 1 2Trf(1 +G(k0;E):)g(A14) where; = 0;x;y;z andt(k;k0) refers to the scat- tering matrix t(k;k0) =t(k;k0). We further as- sume that the coecients a0andc0int(k;k0) explic- itly depend on the scattering angle  = 0[52]. We then use the Eq.(12) on both sides of this expresion and carry out the angular integrations for  = 0to obtain an expression for Ck. Since ^k^k0= sin  ^z, the scattering matrix is conned to those terms with ;= 0;zin Eq.(7). Applying this in Eq.(A14), with G=GxiGy=ei(+=2)g0k+ and the Eq.(12) for , Ck=Zk0dk0 2z0(k;k0)F(k0): (A15) Here, z0=Im  0 4mZd 2eih (t0t0 0tzt0 z) + (t0 0tzt0 zt0)i (A16) andF(k) is given by the Eq.(14) in the manuscript. In Eq.(A16) the short notation t0 0andt0 zimply that t0 0(k;k0) =t0(k0;k) andt0 z(k;k0) =tz(k0;k). We now use the fact that t0(k;k0) andtz(k;k0) in the nota- tion of Eq.(A14) are respectively given by a0(k;k0) andc0(k;k0) sin  in the Eq.(7). These coecients have been calculated for the problem at hand as a0(k;k0) = A0+B0cos  andc0(k;k0) =iC0+ 4D0sin  where A0;B0;C0;D0are real constants. It can be shown that the rst paranthesis on the right hand side in Eq.(A16) contributes to the real part of the Ck, whereas the sec- ond one is imaginary and contributes to its imaginary part krendering the NOSML as an interference eect between the scalar and the spin-orbit impurity scatter- ing. The angular integration in Eq.(A16) can be done immediately yielding the complex number, z0=Im  0 4m(A0+iD0)B0 (A17) which can then be used in the Eq.(13). Appendix B: THE SCATTERING MATRIX t(k;k0) In order to construct the t(k;k0) we start from a gen- eral electron-spinless impurity scattering potential Vei(r) as the sum of individually localized electron-impurity po- tentials at each impurity position Rjas Vei(r) =NimpX j=1v(j) ei(rRj): (B1) Thev(j) ei(r) is a sum of the spin independent and the spin- orbit scattering potentials v(j) 0andv(j) so, respectively as v(j) ei(r) =v(j) 0(r) +v(j) so(r); where (B2) v(j) so(r) =:h rv(j) 0(r)pi We consider only one type of impurity and assume that v(j) eiis the same for all impurities. The general quantum state of the Bloch electrons is given by k(r) =eik:ruk(r) (B3) withuk(r) carrying information about the orbital sym- metries. The scattering amplitude is given by T0(k;k0) =hk00jVeijki =Z dr  k00(r)Vei(r) k(r) (B4) Inserting Eq.(B1) into Eq.(B4), we nd, T0(k;k0) =NimpX jei(kk0):Rjt0(k;k0) (B5) where t0(k;k0) = [~v0(k;k0)]0+ [~vso(k;k0)]0(B6) with [~vX(k;k0)]0=Z dr  k00(r)(r)vX(r) k(r) (B7)9 whereX= 0 for the spinless and X=sofor the spin-orbit scatterings as in Eq.(B2). Due to the spin- less character of the v0(r) in Eq.(B7), [~ v0(k;k0)]0= a0(k;k0)0. From the Eq.(B7) it is easily observed that, fork;k0'0 near the center of the linear dispersion, a0(k;k0) andc0(k;k0) are proportional to the Fouriertransform of the spinless and the spin-orbit potentials in Eq.(B2). The Eqs.(B4)-(B7) comprise a generic deriva- tion of Eq.(7) in the main text. This general formulation can be applied to more specic cases only when the Bloch state in Eq.(B3) and the electron-impurity potentials in the Eq.(B2) are given. [1] R. 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1507.07103v1.Quantum_entanglement_in_the_one_dimensional_spin_orbital_SU_2___otimes_XXZ__model.pdf

arXiv:1507.07103v1 [cond-mat.str-el] 25 Jul 2015Quantum entanglement in the one-dimensional spin-orbital SU(2)⊗XXZmodel Wen-Long You,1,2Peter Horsch,1and Andrzej M. Ole´ s1,3 1Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany 2College of Physics, Optoelectronics and Energy, Soochow Un iversity, Suzhou, Jiangsu 215006, People’s Republic of China 3Marian Smoluchowski Institute of Physics, Jagiellonian Un iversity, prof. S. /suppress Lojasiewicza 11, PL-30348 Krak´ ow, Poland (Dated: August 22, 2018) We investigate the phase diagram and the spin-orbital entan glement of a one-dimensional SU(2)⊗XXZ model with SU(2) spin exchange and anisotropic XXZ orbital exchange interac- tions and negative exchange coupling. As a unique feature, t he spin-orbital entanglement entropy in the entangled ground states increases here linearly with system size. In the case of Ising or- bital interactions we identify an emergent phase with long- range spin-singlet dimer correlations triggered by a quadrupling of correlations in the orbital se ctor. The peculiar translational invariant spin-singlet dimer phase has finite von Neumann entanglemen t entropy and survives when orbital quantum fluctuations are included. It even persists in the is otropic SU(2) ⊗SU(2) limit. Surpris- ingly, for finite transverse orbital coupling the long-rang e spin singlet correlations also coexist in the antiferromagnetic spin and alternating orbital phase m aking this phase also unconventional. Moreover we also find a complementary orbital singlet phase t hat exists in the isotropic case but does not extend to the Ising limit. The nature of entanglemen t appears essentially different from that found in the frequently discussed model with positive c oupling. Furthermore we investigate the collective spin and orbital wave excitations of the dise ntangled ferromagnetic-spin/ferro-orbital ground state and explore the continuum of spin-orbital exci tations. Interestingly one finds among the latter excitations two modes of exciton bound states. Th eir spin-orbital correlations differ from the remaining continuum states and exhibit logarithmic sca ling of the von Neumann entropy with increasing system size. We demonstrate that spin-orbital e xcitons can be experimentally explored using resonant inelastic x-ray scattering, where the stron gly entangled exciton states can be easily distinguished from the spin-orbital continuum. PACS numbers: 75.25.Dk, 03.67.Mn, 05.30.Rt, 75.10.Jm I. INTRODUCTION Spin-orbital coupling phenomena are ubiquitous in solids and have been known to exist since the early days of quantum mechanics and band theory, but only re- cently it was realized that the quantum nature of or- bital degrees of freedom plays a crucial role in the fields of strongly correlated electrons [1–7] and cold atoms [8– 12]. The growing evidence of spin-orbital entanglement (SOE)accumulatedduetonovelexperimentaltechniques whichprobeavarietyofunderlyingelectronicstates. The strong Coulomb interactions and the relativistic spin- orbit interaction entangle locally the spin and orbital de- grees of freedom [13] which display an amazing variety of fundamentally new and fascinating phenomena, rang- ing from topologically nontrivial states [14], relativistic Mott-insulating behavior in 5 d[15, 16] and 4 d[17, 18] transition-metal oxides and entanglement on superex- change bonds in spin-orbital models [6, 19]. Other more recent developments include entangled spin-orbital exci- tations[20,21], dopedspin-orbitalsystems[22], skyrmion lattices in the chiral metal MnSi [23], multiferroics, spin- Hall effects [24], Majorana and Weyl fermions [25], topo- logicalsurfacestates[26], Kondosystems[27], exoticspin textures in disordered systems, to name just a few. To date, experimental observation of a dynamic spin- orbital state has been a challenge. Apart from the in-trinsic anisotropy and the relative complexity of the or- bital couplings, it has been shown that the interplay be- tween the two frustrated degrees of freedom may lead to exotic states of matter. An x-ray scattering study of a dynamic spin-orbital state in the frustrated magnet Ba3CuSb2O9supports spin liquid state [28, 29], while FeSc2S4[30–32] and the d1effective models on the trian- gular lattice [33] and on the honeycomb lattice [34, 35] are found to be candidates for spin-orbital liquids in the theory. Recently remarkable progress was achieved due to rapidly developed resonant inelastic x-ray scattering (RIXS) techniques [36] which helped to explore the el- ementary excitations in Sr 2CuO3[37, 38] and Sr 2IrO4 [39], with antiferromagnetic (AF) and ferro-orbital (FO) order in ground states. Orbital order in the spin-gapped dimerised system Sr 3Cr2O8below the Jahn-Teller tran- sition was also identified [40]. However, it remains chal- lenging experimentally and theoretically, mainly owing to the lack of an ultimate understanding of spin-orbital correlations. In the Mott insulators with an idealized perovskite structure, the low-energy physics is described by spin- orbital models, similar to the Kugel-Khomskii model [2], where the spin and orbital are considered on equal foot- ing as dynamic quantum variables [4]. Spin interaction possesses SU(2) symmetry, which will be broken however by the relativistic spin-orbit coupling. It couples spins2 to the orbitals, that are in general non-SU(2)-symmetric in a solid. However, this coupling can frequently be ne- glected in realistic 3 dsystems and one is left in general with entangled spin-orbital superexchange problem [6], that is the eigenstates cannot be written as products of spin and orbital wave functions. One immediate conse- quence of entanglement is that spin and orbital terms cannot be factorized in the mean-field approach. Or- bitals arespatiallyanisotropicandthus their interactions have lower symmetry than the spin ones which reflects the directional dependence of the orbital wave functions. For the fixed occupation of orbitals, the magnitude and sign of the spin-orbital superexchange interactions follow the classical Goodenough-Kanamorirules [41], but quan- tum fluctuations change them and make it necessary to consider spin-orbital interplay in entangled states on ex- change bonds [19]. Therefore, it is important to measure whether eigenstates are entangled or not. AnaturalmeasureofSOEisthevonNeumann entropy (vNE) which we write first for the nondegenerate ground state|Ψ0∝an}bracketri}ht, S0 vN≡ −TrA{ρ(0) Alog2ρ(0) A}. (1.1) Here we consider a system Ω composed of two non- overlapping subsystems [42], i.e., Ω = A∪B,A∩B=∅, andρ(0) Ais the reduced density matrix. It is obtained by integrating the density matrix over subsystem B, i.e., ρ(0) A= TrB|Ψ0∝an}bracketri}ht∝an}bracketle{tΨ0|. However, one has to realize that information contained in entanglement entropy depends crucially on how one partitions the Hilbert space of the system. To investigate SOE we use here as two subsys- temsAandBthe spin and orbital degrees of freedom in the entire chain. Standard spin-orbital phases may have entanglement in only one sector and here we concentrate on joint SOE [43]. This choice is distinct from the one conventionally made when the system is separated into two spatially complementary parts [44], for instance in frustrated spin chains [45] or in the periodic 1D Ander- son model [46]. Though much attention was devoted to the ground state in the past [47], it has been noticed only recently that the entanglement entropy of low-energy excitations may provide even more valuable insights [43, 48, 49] which are of crucial importance to understand the origin ofquantum phasetransitionsin spin-orbitalsystems[50]. The well known area law of the bipartite entanglement entropy restricts the Hilbert space accessible to a ground state of gapped systems [51, 52], while the area law is violated by a leading logarithmic correction in critical systems, whose prefactor is determined by the number of chiral modes and precisely given by Widom conjecture [53]. In this respect, the application of the entanglement entropy in describing quantum criticality in many-body Hamiltonian merits a lot of studies [42, 54]. On the other hand, the excited states have the mix- ture of logarithmic and extensive entanglement entropy, and the logarithmic states are expected to be negligiblein number compared to all the others. The entanglement in excited state is proven always larger than that of the ground state of a spin chain [55]. For a spin-orbital cou- pled system, the division of spin and orbital operators retains the real-space symmetries, which is beneficial to the calculation of mutual entanglement. In two-particle states, the SOE is determined by the inter-component coherence length [43], as though the state has sufficient decay of correlations [45]. The aim of this paper is to use the entanglement en- tropy to investigate the full phase diagram of the one- dimensional (1D) anisotropic spin-orbital SU(2) ⊗XXZ model. The main motivation for considering the Ising asymmetry in the orbital sector comes from the obser- vation that spin-orbital entanglement is large when both subsystems, i.e., spin and orbital sectors, reveal strong quantum fluctuations. Thus the Ising anisotropy which is present in many physical systems introduces addi- tional control of orbital fluctuations and thereby pro- vides an important control parameter for SOE. Here we focus on the model with negative exchange inter- action. This choice of the exchange coupling restricts somewhat joint spin-orbital fluctuations being particu- larly large near the SU(4) symmetric point in the 1D spin-orbital model with positivecoupling constant [56], but opens novel possibilities for entangled states, as we show below [50]. An interesting phase with entangled ground state, consisting of alternating spin singlets along the spin-orbital ring, is found for Ising orbital interac- tions when the dimerization in the spin channel induces the change from FO to alternating orbital (AO) correla- tions. Here we report the complete phase diagram of the anisotropic SU(2) ⊗XXZspin-orbital model, with two phases of similar nature which gain energy from singlet correlations leading to dimerization, either in spin or in orbital sector. These phases were overlooked before in the fully symmetric case, i.e., in the phase diagram of the isotropic SU(2) ⊗SU(2) model [43]. We also analyze the nature of spin-orbital excited states, particularly in the case of the disentangled ferro- magnetic (FM) and FO ground state, labeled as FM/FO order. We also analyze entanglement entropy in the ex- cited states for the FM/FO phase and show that spin- orbital excitations form a continuum, supplemented by collective bound states. The latter states are character- ized by a logarithmic scaling behavior, and as we show could be detected by properly designed RIXS experi- ments [57–59]. The paper is organized as follows. The model is intro- duced in Sec. II. In Sec. III we present an analytic solu- tion for the ground state in the Ising limit of the orbital interactions. A more general situation with anisotropic XXZorbital interaction is analyzed in Sec. IVA, and the phase diagram for the isotropic SU(2) ⊗SU(2) model is reported in Sec. IVB. This model and the obtained SOE are different from the AF case, as shown in Sec. IVC. Next we determine the elementary excitations in the FM/FO phase in Sec. V and show that they are en-3 tangled although the ground state is disentangled. The vNE spectral function is presented in Sec. VIA, includ- ing the scaling behavior of the bound states which is contrasted with that in the AF/AO ground state. In Sec. VIB we explore the possibilities of investigating entanglement in the present 1D spin-orbital model by RIXS. The paper is concluded by a discussion and brief summary in Sec. VII. Some additional technical insights which are accessible by an exact solution of the two-site model are presented in the Appendix. II. THE 1D SPIN-ORBITAL SU(2) ⊗XXZ We consider the 1D spin-orbital Hamiltonian which couplesS= 1/2 spins and T= 1/2 orbital (pseudospin) operators, H=−J/summationdisplay jHS j(x)⊗HT j(∆,y),(2.1) with SU(2) spin Heisenberg interaction HS j(x), orbital anisotropic XXZinteraction HT j(∆,y), HS j(x) =/vectorSj·/vectorSj+1+x, (2.2) HT j(∆,y) = ∆/parenleftbig Tx jTx j+1+Ty jTy j+1/parenrightbig +Tz jTz j+1+y.(2.3) We take below J= 1 as the energy unit. The model Eq. (2.1) has the following parameters: (i) xandy which determine the amplitudes oforbital and spin ferro- exchange interactions, −Jxand−Jy, respectively, and (ii)∆whichinterpolatesbetweenthe Heisenberg(∆ = 1) and Ising (∆ = 0) limit for orbital interactions. When ∆ = 1, the spin and orbital interactions are on equal footing and the symmetry of the Hamiltonian (2.1) is en- hancedtoSU(2) ⊗SU(2) — thismodel describesageneric competition between FM and AF spin, and between FO and AO bond correlations [43]. We emphasize that the coupling constant −Jisneg- ative, so at large x >0 andy >0 it gives a disentan- gled FM/FO ground state, see below — therefore the model may be called in short FM. This choice of the exchange coupling restricts somewhat joint spin-orbital fluctuations being large near the SU(4) symmetric point, (x,y) = (0.25,0.25), in the 1D spin-orbital model with positive, i.e., AF coupling constant [56], but opens other interesting possibilities for entangled states, as we have shown recently [50]. Both total spin magnetization Sz andorbitalpolarization Tzareconserved,andtimerever- sal symmetry leads to the total momentum either k= 0 ork=π. Before analyzing the spin-orbital model of Eq. (1) in more detail, let us summarize briefly the properties of the well known AF model, with positive coupling con- stantJ. The SU(4) symmetric Hamiltonian found at (x,y) = (0.25,0.25) is an integrable model which can be solved in terms of the Bethe Ansatz[56, 60]. Away from the SU(4) symmetric points this choice of the cou- pling constant favors the phases with spin-orbital orderdepending on the actual values of xandy, and the phase diagram obtained by numerical methods includes in gen- eral phases with all types of coupled spin-orbital order, i.e., FM/FO, AF/FO, AF/AO, and FM/AO, as well as the gaplessspin-orbitalliquid phase nearthe SU(4) point [61, 62]. In addition, Schwinger boson analysis gives phases with spin-orbital valence-bond correlations and also spin valence bond and orbital valence-bond phases [63]. The latter two show a tendency towards dimerised spin or orbital correlations which occur here in the prox- imity of the SU(4) point. For some special choice of pa- rametersthemodelcanbesolvedexactly: (i)when∆ = 1 andx=y= 3/4, the exact ground state is doubly de- generate with the spins and the orbitals forming singlets on alternate bonds, while (ii) when ∆ = 0, x= 3/4 and y= 1/2, the non-Haldane spin-liquid ground state can be analytically obtained [64, 65], and (iii) several inte- grable cases were presented for interactions with special symmetries [66, 67], or (iv) with XYorbital interactions (∆ =∞) [20]. The form of Eq. (2.1) is not the most general one but is representative for real spin-orbital systems with anisotropic orbital interactions. In real systems the or- bital part contributes by additional superexchange terms which are not coupled to SU(2) spin interaction [4]. For instance, in the case of t2gorbital degrees of freedom as in the perovskite titanates or vanadates, the interactions along theccubic axis involve the doublet of two orbitals active along it, i.e., the yzandzxorbitals [68]; a similar situation is encountered in a tetragonal crystal field of a quasi-1D Mott insulator [69], or for pxandpyorbitals of a 1D fermionic optical lattice [8–10]. A priori, due to the quartic spin-orbital joint term, ∝(/vectorSj·/vectorSj+1)[∆(Tx jTx j+1+Ty jTy j+1)+Tz jTz j+1]intheHamil- tonian Eq. (2.1) the spin-orbital interactions are entan- gled, and the spin and orbital operators cannot be sepa- rated from each other in the correlation function, except for some ground or excited states in which the SOE van- ishes. The spin-orbital bond correlations (2.4) Ctot 1≡/angbracketleftBig (/vectorSj·/vectorSj+1)[∆(Tx jTx j+1+Ty jTy j+1)+Tz jTz j+1]/angbracketrightBig , (2.4) are uniform in the considered system and Ctot 1does not depend on the site index j. We investigate below these composite quartic correlations and show that they could also be surprisingly large. As an additional criterion of setting up the phase diagram, we use below the fidelity susceptibility which elucidates the change rate of ground states in the parameter space [70]. It serves as an or- der parameter to characterize the phase diagram of the anisotropic(∆ <1) spin-orbitalmodel (2.1). The fidelity susceptibility is defined as follows, χF(λ)≡ −2 lim δλ→0lnF(λ,δλ) (δλ)2, (2.5) where the fidelity F(λ,δλ) =|∝an}bracketle{tΨ0(λ)|Ψ0(λ+δλ)∝an}bracketri}ht|,(2.6)4 is taken along a certain path in the parameter space in the vicinity of the point λ≡λ(∆,x,y). III. ISING ORBITAL INTERACTIONS ( ∆ = 0) In the Ising limit of orbital interactions (∆ = 0) the Hamiltonian (2.1) simplifies and has SU(2) ⊗Z2symme- try — it is a prototype model for the directional orbital interactions with quenched quantum fluctuations in t2g systems. This may happen in real compounds in two ways: (i) either only one of the two active orbitals is oc- cupied by one electron and contributes in the hopping processes along the 180◦bonds [71] or 90◦bonds [72], or (ii) the orbital degrees of freedom are quenched in the presence of strong crystal field. In both these cases the orbital exchange (orbital-flip) processes are blocked and orbital interaction are of a classical Ising-like form. Such Ising interactions are frustrated when they emerge in higher dimension, as in the well-studied orbital com- pass model [73–75] and in Kitaev model [76], see also a recent review on the compass model [77]. It is now in- triguing to ask what happens to the SOE in this case. It may be still triggered by spin fluctuations while the model with Ising spin interactions (A.2) is classical. The phase diagram of the model Eq. (2.1) at ∆ = 0, i.e., in the absence of orbital fluctuations, which follows from fidelity susceptibility (2.5) is displayed in Fig. 1. As expected, one finds four trivial combinations of spin- orbital order: FM/FO (phase I), AF/FO (phase II), AF/AO (phase III), and FM/AO (phase IV). All these phases have the entanglement entropy (1.1) S0 vN= 0 and spins and orbitals disentangle. Transitions between pairs of them are given by straight lines and may be also obtained rigorously by the mean-field approach. The ground state of a L-site chain stays in the subspace Sz= 0,Tz= 0, momentum k= 0 (always degenerate withSz= 0,Tz= 0,k=πfor all parameters) in phases III (AF/AO), IV (FM/AO) and V, while it is found in the subspaces Sz= 0,Tz=±L/2,k= 0 in phases I (FM/FO) and II (AF/FO) (of course, in phases I and IV also other values of Sz∝ne}ationslash= 0, with −L/2≥Sz≥L/2, are allowed and the ground states have the respective degen- eracy). The ground states with energy E0= 0 are highly degeneratewhen x<−1/4along the critical line y= 1/4 between phases III and IV, suggesting that antiparallel orbitals erase the spin dynamics. Along the critical line y=−1/4 between phases I and II, the ground states are also highly degenerate when x≥3/4, and parallel or- bitals on the bonds (in FO order) quench again the spin fluctuations. Although the orbital interactions are Ising-like, entan- gled spin-orbital ground state occurs in phase V. In order to understand better emergent phase V, we introduce the longitudinal equal-time spin/orbital structure factor, de- fined for a ring of length L(with a lattice constant a= 1; FIG. 1. (Color online) Spin-orbital entanglement entropy S0 vN Eq. (1.1) and the phase diagram in the ( x,y) plane of the SU(2)⊗Z2spin-orbital model (2.1) with ∆ = 0 as obtained for the system size of L= 8 sites. The critical lines are discerned by both fidelity susceptibility and analytical method. Phas es I-IV are disentangled ( S0 vN= 0) with order defined as follows: FM/FO (phase I), AF/FO (phase II), AF/AO (phase III), and FM/AO (phase IV). The spin and orbital textures in phase V with finite entropy S0 vN>0 are explained in the text. we use periodic boundary conditions) by Szz(k) =1 LL/summationdisplay j,j′=1e−ik(j−j′)∝an}bracketle{tSz jSz j′∝an}bracketri}ht,(3.1) Tzz(k) =1 LL/summationdisplay j,j′=1e−ik(j−j′)∝an}bracketle{tTz jTz j′∝an}bracketri}ht.(3.2) The calculation of the equal-time structure factor Szz(k) for a model of uncorrelated nearest neighbor dimers was compared with the one for the kagome lattice ZnCu3(OD)6Cl2[78]. One finds analytically that in the case ∆ = 0, a cosine-like spin structure factor, i.e., Szz(k)∝(1−cosk), is revealed in phase V for y=−1/4, implying that only nearest neighbor spins are correlated. This finding is essential as the short-range spin correla- tion indicates here a translationinvariantdimerised spin- singlet state which has the same spin structure as the Majumdar-Ghosh (MG) spin state [79]. However, this state is not triggered here by frustrated interactions J1 andJ2, but is evidently induced by the correlations in the orbital sector. In the Ising limit we obtain the analytic ground state for phase V as described below. The essential feature is that the energy is gained by spin singlets occupying the bonds with AO states, while the bonds connecting two spin singlets have FO order, see Fig. 2(a). To construct the ground state, we introduce the corresponding four5 FIG. 2. (Color online) (a) One of four translational equiva- lently spin and orbital configurations in the Ising limit of t he spin-orbital model (2.1) at ∆ = 0 and y=−0.25. The spins form isolated dimers (shaded ovals). (b) A single orbital ex - citation and induced spin configuration. (c) A single spin fli p makes a singlet-triplet spin excitation, but does not induc e any change in orbital correlations. configurations in the orbital sector: |φ1∝an}bracketri}ht=|++−−++−−···∝an}bracketri}ht, |φ2∝an}bracketri}ht=|−++−−++−···∝an}bracketri}ht, |φ3∝an}bracketri}ht=|−−++−−++···∝an}bracketri}ht, |φ4∝an}bracketri}ht=|+−−++−−+···∝an}bracketri}ht. (3.3) The solutions are classified by the momenta correspond- ing to the translational symmetry of the system. The orbital wave functions in the ground state for the mo- mentak= 0,π/2,3π/2,πcorrespond to: |φk∝an}bracketri}ht=1 2/parenleftbig |φ1∝an}bracketri}ht+eik|φ2∝an}bracketri}ht+e2ik|φ3∝an}bracketri}ht+e3ik|φ4∝an}bracketri}ht/parenrightbig .(3.4) In spin subspace there are two distinct (but nonorthogo- nal) states: |ψD 1∝an}bracketri}ht= [1,2][3,4]···[N−1,N], |ψD 2∝an}bracketri}ht= [2,3][4,5]···[N,1], (3.5) where the singlets are located on odd (even) bonds. Here a singlet is defined by [ l,l+1] = (| ↑↓∝an}bracketri}ht−| ↓↑∝an}bracketri}ht )/√ 2. Onerepresentativecomponentofthegroundstatewith the orbital part |φ4∝an}bracketri}htaccompanied by the spin state |ψD 1∝an}bracketri}ht is shown in Fig. 2(a). The ground state in the k= 0 subspace is given by the superposition |Φk=0∝an}bracketri}ht= 1 2/parenleftbig |φ1∝an}bracketri}ht⊗|ψD 2∝an}bracketri}ht+|φ2∝an}bracketri}ht⊗|ψD 1∝an}bracketri}ht+|φ3∝an}bracketri}ht⊗|ψD 2∝an}bracketri}ht+|φ4∝an}bracketri}ht⊗|ψD 1∝an}bracketri}ht/parenrightbig =1√ 2/parenleftbigg|φ1∝an}bracketri}ht+|φ3∝an}bracketri}ht√ 2⊗|ψD 2∝an}bracketri}ht+|φ2∝an}bracketri}ht+|φ4∝an}bracketri}ht√ 2⊗|ψD 1∝an}bracketri}ht/parenrightbigg .(3.6)The state |Φk=0∝an}bracketri}htis entangled both in individual spin and orbitalsubspaces, and alsois characterizedby SOEalong thechain. Suchamany-bodystate,andsimilarstatesob- tained for other momenta, k=±π/2 andk=π, give an exact value of the vNE, S0 vN= 1. The resulting fluctu- ations between these states suppress conventional order and the system features finite entropy even at zero tem- perature, in contrast to the naive expectation from the third law of thermodynamics. The emergent excitations are also entangled and fundamentally different from the individual spin or orbital ones, see Figs. 2(b) and 2(c). As the orbital correlations are classical in the Ising limit, we can determine all the phase boundaries analyt- ically by considering the spin interactions for various or- bital configurations. The lower boundary between phase III (AF/AO) and V at y=−1/4 can be determined by comparing the uniform state with energies of AO corre- lation on a bond, i.e., ∝an}bracketle{tTz jTz j+1∝an}bracketri}ht=−1/4, with the al- ternating state of pairs of the same orbitals shown in Fig. 2(a), i.e., ∝an}bracketle{tTz jTz j+1∝an}bracketri}ht= (−1)j/4, which coexists with spin dimer order (spin interactions vanish for a pair of identical orbitals). One finds the following effective spin Hamiltonian in this case: HDIM=1 2/summationdisplay j∈odd/parenleftBig /vectorSj·/vectorSj+1+x/parenrightBig ,(3.7) and the correspondinggroundstate energy per site in the thermodynamic limit is E0 DIM=1 4(−0.75+x). (3.8) The dimerised phase competes with the AO order coex- isting with the 1D resonating valence-bond spin state, with energy E0 AO=1 2(−0.4431+x). (3.9) Hence, one finds that E0 DIM< E0 AOforx >0.136. The quadrupling due to spin-orbital interplay in phase V is well seen by the calculation of the four-spin correlation function which we define following Refs. [80, 81], D(r) =1 L/summationdisplay i/bracketleftBig/angbracketleftBig (/vectorSi·/vectorSi+1)(/vectorSi+r·/vectorSi+r+1)/angbracketrightBig −/angbracketleftBig /vectorSi·/vectorSi+1/angbracketrightBig/angbracketleftBig /vectorSi+r·/vectorSi+r+1/angbracketrightBig/bracketrightBig .(3.10) Ify=−1/4, spin dimer correlations alternate and D(r) = (−1)r/parenleftbigg3 8/parenrightbigg2 , (3.11) which follows from Eq. (3.10) for the alternating spin singlets, ∝an}bracketle{t/vectorSi·/vectorSi+r∝an}bracketri}ht=−3[1−(−1)r]/8. Indeed, one finds this value (3.11) for x∈[0.2,0.7] and the result is robust and the same for systems sizes L= 12 andL= 16, see Fig. 3. On the contrary, for x <0.2 the values of D(r)6 FIG. 3. Dimer correlation function D(r) (3.10) obtained for the anisotropic SU(2) ⊗Z2spin-orbital model for different val- ues ofx∈[0,0.7] in phases V and III and for the ring of length: (a)L= 12, and (b) L= 16 sites. Parameters: y=−0.25 and ∆ = 0. decrease with increasing distance r, and would vanish in the thermodynamic limit of L→ ∞. Wheny <−1/4, there are three competing phases with predetermined orbital configurations (AO, DIM, or FO) and the corresponding spin interactions given by ef- fective spin Hamiltonians: HAO=/parenleftbigg1 4−y/parenrightbigg/summationdisplay j(/vectorSj·/vectorSj+1+x),(3.12) HDIM=/parenleftbigg1 4−y/parenrightbigg/summationdisplay j∈odd(/vectorSj·/vectorSj+1+x) −/parenleftbigg1 4+y/parenrightbigg/summationdisplay j∈even(/vectorSj·/vectorSj+1+x),(3.13) HFO=−/parenleftbigg1 4+y/parenrightbigg/summationdisplay j(/vectorSj·/vectorSj+1+x).(3.14) Inthiscasealsoeveninter-singletbondscontributetothe energy in the |DIM∝an}bracketri}htstate, but the spin correlations van- ish, i.e.,∝an}bracketle{t/vectorSj·/vectorSj+1∝an}bracketri}ht= 0. It is obvious that HAOandHFO standforthesame(translationalinvariant)spinHamilto- nian, andHFOwill have lower ground state energy when x>−∝an}bracketle{t/vectorSj·/vectorSj+1∝an}bracketri}htAF≃0.4431. The dimerised AF Heisen- berg chain (3.13) related to spin-Peierls state cannot be solved trivially, with the exception of the free-dimer limit (y=−0.25) and the uniform Heisenberg limit ( y=−∞) [82], One finds the ground state energy per site ε∞(δ) of a pure dimerised spin chain [83], ε∞(δ) =3 41 1+α/parenleftbigg 1+α2 8+α3 32+···/parenrightbigg ,(3.15) withα= (1−δ)/(1 +δ) andδ≡1/|4y|/greaterorsimilar0.4. For δ/lessorsimilar0.4, ε∞(δ) = ln2−(ln2−1)|δ|4/3.(3.16)In such a case, E0 DIM=y[ε∞(δ)−x]. The overwhelming dimerised phase will persist in a range of negative values ofy, and the boundaries close at y=−∞, as is indicated by structure factors and fidelity susceptibility. The phase transitions in the phase diagram of Fig. 1 imply the discontinuous changes of order parameters in first-order quantum phase transitions. The orbital order changes from phase II (AF/FO) to phase III (AF/AO), as shown in Ref. [50], but the N´ eel order persists in both of them and manifests itself in the two-spin correlation, ∝an}bracketle{tSz iSz i+r∝an}bracketri}ht. For translational invariant and orthonormal linear combinations of the symmetry-broken N´ eel (AF) states, |ΦAF 1∝an}bracketri}ht=| ↑↓↑↓ ··· ↑↓∝an}bracketri}ht , |ΦAF 2∝an}bracketri}ht=| ↓↑↓↑ ··· ↓↑∝an}bracketri}ht , (3.17) there are spin ∝an}bracketle{tSz iSz i+r∝an}bracketri}ht= (−1)r/4 and dimer D(r) = 0 correlations (for r∝ne}ationslash= 0), while for dimer states |ΦDIM 1∝an}bracketri}ht and|ΦDIM 2∝an}bracketri}ht,∝an}bracketle{tSz iSz i+r∝an}bracketri}ht=0 (forr∝ne}ationslash=±1) andD(r)∝ne}ationslash= 0 (for r∝ne}ationslash= 0), see Fig. 3, respectively. These results reflect the long-range nature of the two types of order. The AF classical spin correlations (3.17) are replaced by a power law for the AF spin S= 1/2 chain in the thermodynamic limit, /angbracketleftBig /vectorSi·/vectorSi+r/angbracketrightBig ∼(−1)r/radicalbig ln|r| |r|, (3.18) which is equivalently revealed by the structure factors Szz(k) andTzz(k) defined by Eqs. (3.1) and (3.2). IV. ENTANGLEMENT IN THE GROUND STATES A. The anisotropic orbital interactions ( 0<∆≤1) When ∆ = 0, there is no dynamics in the orbital sec- tor, and the orbital structure factor is dominated by a single mode which follows from the Z2symmetry [50]. This changes when 0 <∆≤1 and the quantum fluctu- ations in the orbital sector contribute. In order to un- derstand the modifications of the phase diagram in the entire interval 0 <∆≤1, we select ∆ = 0 .5 and study the longitudinal equal-time spin/orbital structure factor, defined for a ring of length Lin Eqs. (3.1) and (3.2). The most important change at finite ∆ occurs for the phase transition between phases V and III (AF/AO) which becomes continuous for fixed yand decreasing x, with a gradual change of spin correlations from the al- ternating singlets to an AF order along the chain [50]. Here we discuss in more detail the intermediate case of ∆ = 0.5. First we address the phases with uniform spin- orbital order. The spin structure factors has distinct peaks atk= 0 for FM order and at k=πfor AF order. Similarly, one finds a maximum of the orbital structure factorTzz(k) atk= 0 for FO order and at k=πfor AO order. These structure factors complement one another7 and one finds that the spin correlations are somewhat weaker due to stronger spin fluctuations, while the or- bital fluctuations are moderate at this value of ∆ = 0 .5. The dimerised phase V is characterized by a remark- ably different behavior, see Figs. 4(c) and 4(d). Phase V, found at the Epoint in Fig. 5, has spin dimers ac- companied by an orbital pattern with the periodicity of four sites, see Fig. 2(a). Spin correlations give a sharp maximum of Szz(k) atk=πas for AF states, while two symmetric peaks of Tzz(k) atk=π/2 andk= 3π/2 in- dicate quadrupling of the unit cell in the orbital channel. When the model evolves towards the SU(2) ⊗SU(2) limit with increasing ∆, one expects also a similar phase VI with interchanged role of spin and orbital correlations. Indeed, this complementary phase emerges already at small ∆>0 and is identified by the respective structure factors shown also in Figs. 4(c) and 4(d). The above analysis of the structure factors demon- strates that the phase diagram found for ∆ = 0 .5 con- tains six distinct phases, see Fig. 5. The quantum phase transitions at the boarder lines I-V and II-V are of first order. The phase transitionbetween phases III (AF/AO) and V is a first order transition only for ∆ = 0, and here this transition is continuous [84]. As described above, phase VI emerges at finite ∆ but is still quite narrow in the phase diagram of Fig. 5. Also the phase transition FIG. 4. (Color online) The spin Szz(k) (a,c), and orbital Tzz(k) (b,d) structure factors obtained for the selected points shown in the phase diagram of the anisotropic spin-orbital SU(2)⊗XXZ model, see Fig. 5: (a,b) A,B,CandDin phases I-IV, and (c,d) EandFin phases V and VI. Param- eters: ∆ = 0 .5 andL= 8 sites./s65 /s66/s67 /s68 /s69/s70 /s45/s49 /s48 /s49/s45/s49/s48/s49 /s83 /s118/s78 /s86/s73 /s86 /s32/s32 /s120/s121/s73/s86 /s73 /s73/s73/s73 /s73/s73/s48/s46/s48 /s48/s46/s56 /s49/s46/s54 /s50/s46/s52 /s51/s46/s50 /s52/s46/s48 FIG. 5. (Color online) Phase diagram in the ( x,y) plane and spin-orbital entanglement S0 vN(1.1) (right scale) in different phases (indicated by color intensity) of the anisotropic sp in- orbital SU(2) ⊗XXZ model (2.1) with ∆ = 0 .5, obtained for a ring of L= 8 sites. Phases I-IV correspond to FM/FO, AF/FO, AF/AO, FF/AO order in spin-orbital sectors. Or- bitals (spins) follow a quadrupled pattern accompanied by spin (orbital) dimer correlations in phase V (VI). Six label ed points are: A= (−1,1),B= (−1,−0.5),C= (0.5,1), D= (0.5,−0.5),E= (1,−0.5), andF= (−1,0.38) — they are used to investigate spin and orbital structure factors i n different phases, see Fig. 4. The phase boundaries, deter- mined by the dominant modes of structure factors are shown by solid (dashed) lines for the first (second) order quantum phase transitions. from phase III to phase VI is continuous. We have ver- ified that due to the short-range nature of spin-orbital correlations, the size L= 12 is sufficient as the phase boundaries are here almost the same as for the ring of L= 8 sites. TABLE I. The spin-orbital configurations, momenta k, the total spin Szand orbital Tzquantum numbers of the ground states I-VI found for the anisotropic SU(2) ⊗XXZ model at ∆<1. All states in these subspaces are nondegenerate ( d= 1), but in case of Sz=L/2 there are L+ 1 degenerate states for total S=L/2, and for Tz=L/2 and ∆ <1 an equivalent state for −orbitals has Tz=−L/2. At ∆ = 0 phase VI is absent and the ground state degeneracy of phase V changes tod= 4 corresponding to momenta k= 0,±π/2,π. phase spin state orbital state k SzTz I ↑↑↑↑↑↑↑↑ + + + + + + ++ 0 L/2L/2 II ↑↓↑↓↑↓↑↓ + + + + + + ++ 0 0 L/2 III ↑↓↑↓↑↓↑↓ +−+−+−+−0 0 0 IV ↑↑↑↑↑↑↑↑ +−+−+−+−0L/2 0 V (S= 0) singlets + −−+ +−+−0 0 0 VI ↑↓↓↑↑↓↓↑ (T= 0) singlets 0 0 08 The spin-orbital phases I-VI found at ∆ >0 are sum- marized in Table I. Phases I-IV have polarized or alter- nating spin and orbital components, combined in all pos- sible ways into phases: FM/FO, AF/FO, AF/AO, and FM/AO. Phases with either Sz=L/2 orTz=L/2 (FM or FO) have of course also degeneracy with respect to other possible values of SzorTz(the latter only at ∆ = 1 when Tis also a good quantum number). In addi- tion, there are two phases with dimer orbital (phase V) or dimer spin (phase VI) correlations. It is remarkable that these two phases survive in the isotropic model at ∆ = 1, see Sec. IVB. For ∆>0 also phase III is characterized by finite SOE, and it expands to higher values of yalong vertical lines for fixed x<−0.25. Indeed, the onset of entangled region in the phase diagram of Fig. 5 moves to higher valuesofywith increasing∆, seeFig. 6. At ∆ = 0 .25the entanglement entropy S0 vNdevelops a narrow peak with a maximum at y≃0.15. This maximum broadens up and moves somewhat to the right (to higher y) when the orbital fluctuations increase with increasing ∆ towards the isotropic SU(2) ⊗SU(2) model. A sharp increase of the entropy to S0 vN= 1, visible in all the curves shown in Fig. 6, signals the SOE in phase VI which increases with increasing ∆. This large SOE can develop because phase VI is similar to phase V — it does not break trans- lation invariance of the model and different spin-orbital configurations contribute simultaneously. To detect SOE we employ here not only the vNE, S0 vN Eq. (1.1), but also a direct measure by the spin-orbital /s45/s49 /s48 /s49/s48/s49/s50 /s32/s32/s83 /s118/s78 /s121/s32 /s61/s48/s46/s48 /s32 /s61/s48/s46/s50/s53 /s32 /s61/s48/s46/s53 /s32 /s61/s49/s46/s48 FIG. 6. (Color online) Spin-orbital entanglement entropy S0 vN (2.1) in the ground state of the spin-orbital model (2.1) as a function of yfor selected values of ∆. The onset of phase VI is detected at ∆ >0 by a step-like increase of entropy to S0 vN= 1. Parameters: x=−0.5 andL= 8./s45/s49 /s48 /s49/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53 /s45/s49 /s48 /s49/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53 /s40/s97/s41 /s32/s32 /s120/s32/s83 /s118/s78 /s32/s56/s32/s67 /s49/s40/s98/s41 /s32/s32 /s120/s32/s83 /s49 /s32/s84 /s49 FIG. 7. (Color online) The onset of phase V and its gradual change into phase III with decreasing xin the ground state of the spin-orbital model (2.1): (a) spin-orbital entangle ment entropy S0 vN(1.1) and the spin-orbital correlation function C1(4.1); (b) spin S1(4.2) and orbital T1(4.3) correlation functions. The onset of phase V is signaled by a step-like increase of the entropy to S0 vN= 1. Parameters: ∆ = 0 .5, y=−0.5 andL= 8. correlation function on a bond [19], C1≡1 LL/summationdisplay i=1/bracketleftBig/angbracketleftBig (/vectorSi·/vectorSi+1)(/vectorTi·/vectorTi+1)/angbracketrightBig −/angbracketleftBig /vectorSi·/vectorSi+1/angbracketrightBig/angbracketleftBig /vectorTi·/vectorTi+1/angbracketrightBig/bracketrightBig , (4.1) and we compare it with the conventional intersite spin- and orbital correlation functions: S1≡1 LL/summationdisplay i=1/angbracketleftBig /vectorSi·/vectorSi+1/angbracketrightBig , (4.2) T1≡1 LL/summationdisplay i=1/angbracketleftBig /vectorTi·/vectorTi+1/angbracketrightBig . (4.3) The above general expressions imply averaging over the exact (translation invariant) ground state found from Lanczos diagonalization of a ring of length L. While S1(4.2) andT1(4.3) correlations indicate the tendency towards particular spin and orbital order, C1(4.1) quan- tifies the SOE — if C1∝ne}ationslash= 0 spin and orbital degrees of freedomareentangledandthemean-fielddecouplingcan- not be applied in Eq. (2.1) as it generates systematic errors. To gain a better insight into the nature of a phase transition between phases II (AF/FO) and V and be- tween V and III (AF/AO) which occur for decreasing x at a fixedy<−1/4, we study SOE vNE S0 vN, joint spin- orbitalC1(4.1), and individual spin S1(4.2) and orbital T1(4.3) correlations in Fig. 7. Both S0 vNandC1show a very similar behavior with a maximum within phase V, see Fig. 7(a). The SOE is lower in phase III than in phase V, i.e., below x≃ −0.25, and decreases further with decreasing x. These two phases have rather similar9 spin correlations S1, but orbital correlations T1are sim- ilar to spin ones only within phase III (AF/AO); above x≃ −0.25 they vary fast within phase V and almost disappear ( T1≃0) near the transition point to phase II (AF/FO), see Fig. 7(b). B. Isotropic spin-orbital SU(2) ⊗SU(2) model The phase diagram of the isotropic SU(2) ⊗SU(2) model (Fig. 8) includes the same six phases as the one obtained at ∆ = 0 .5 (Fig. 5), with spin-orbital corre- lations explained in Table I. The main differences to the phasediagramat∆ = 0 .5isasomewhatreducedstability rangeof phase IV (FM/AO), and also phase II (AF/FO), bothdestabilizedbyenhancedspin-orbitalfluctuationsin and around phase III (AF/AO). The range of entangled ground states is broad and includes phase III as well as phases V and VI on its both sides. The phase transitions from phase I where all quantum fluctuations are absent to either phase II or IV are given by straight lines and may be determined using mean-field approach. It is quite unexpected that the dimerised phases V and VI survive in the phase diagram of the isotropic SU(2)⊗SU(2) model at ∆ = 1 .0, see Fig. 8. These two phases emerge in between a disentangled phase II and III in the case of phase V, and similarly between phases IV and III in the case of phase VI, and are stabilized by robust quadrupling of orbital or spin correlations which was overlooked before [43]. This is not so surprising as one expects that isotropic spin-orbital interactions would lead instead to uniform phases only. In each case the ef- fective exchange interaction changes sign in one (either spin or orbital) channel which resembles the mechanism of exotic magnetic order found in the Kugel-Khomskii model [85]. The phases V and VI emerge by the same mechanism as phase V for the Ising orbital interactions, see Sec. III. In the isotropic model this phase and phase VI with com- plementaryspin-orbitalcorrelationsoccurinasymmetric way with respect to the x=yline, see Fig. 8. Orbital correlations in the case of phase V (spin correlations in the case of phase VI) change gradually towards FO (FM) order in phase II (IV) with increasing x(y). The qua- drupling of the unit cell seen in both phases in such cor- relations, shown in Fig. 9, may be seen as a precursor of this transition. Both phases are stable only in a rather narrow range and disappear for y→ −∞orx→ −∞, respectively, as presented in the inset of Fig. 8 for phase VI. The spin-orbital interactions and the mechanism sta- bilizing these phases are different from spin-Peierls and orbital-Peierls mechanisms in the 1D SU(2) ⊗SU(2) spin- orbital model with positive exchange ( J=−1) [63]. We emphasize that the present mechanism of dimerization is effective only in one (spin or orbital) channel and thus it is also distinct from the spin-orbital dimerization in a FM chain found at finite temperature [86]. A characteristic feature of SOE in phase VI (and simi-/s45/s49 /s48 /s49/s45/s49/s48/s49 /s73/s73/s73 /s120/s121/s48/s46/s48 /s49/s46/s48 /s50/s46/s48 /s51/s46/s48 /s52/s46/s48 /s53/s46/s48 /s54/s46/s48/s73/s73/s73/s86/s73/s73/s86 /s86/s83 /s118/s78 /s45/s49/s48 /s45/s53 /s48/s48/s46/s52/s48/s46/s54/s48/s46/s56 /s73/s73/s73/s86/s73/s73/s86/s73 /s86 /s121 /s120 FIG. 8. (Color online) Phase diagram in the ( x,y) plane and spin-orbital entanglement S0 vN(1.1) (right scale) in the ground state of the isotropic spin-orbital SU(2) ⊗SU(2) model (2.1) with ∆ = 1 .0 obtained for a ring of L= 12 sites. Phases I-IV correspond to FM/FO, AF/FO, AF/AO, FM/AO order in spin-orbital sectors. Orbitals (spins) follow a quadrup led pattern accompanied by spin (orbital) dimer correlations i n phase V (VI). The phase boundaries determined by dominant modes of structure factors, shown by solid (dashed) lines, a re of first (second) order. Inset shows the extended range of phase VI which separates phases IV and III for −10< x < 0. larinphaseV)isacompetitionbetweenthespin(orbital) quadrupling correlations along the chain which support orbital (spin) singlets, with the AF/AO order character- /s48 /s50/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s32/s40/s120/s44/s121/s41/s61/s40/s48/s46/s54/s44/s45/s49/s46/s48/s41 /s32/s40/s120/s44/s121/s41/s61/s40/s45/s49/s46/s48/s44/s48/s46/s54/s41/s40/s97/s41 /s32/s32/s83/s122/s122 /s40/s107/s41 /s107/s50/s48/s46/s48/s48/s46/s53/s49/s46/s48 /s32/s40/s120/s44/s121/s41/s61/s40/s48/s46/s54/s44/s45/s49/s46/s48/s41 /s32/s40/s120/s44/s121/s41/s61/s40/s45/s49/s46/s48/s44/s48/s46/s54/s41/s40/s98/s41/s32/s84/s122/s122 /s40/s107/s41 /s32 /s107 FIG. 9. (Color online) The structure factors obtained for a spin-orbital ring Eq. (2.1) of L= 8 sites in the full Hilbert space at ∆ = 1 .0: (a) spin Szz(k), and (b) orbital Tzz(k). The points (0 .6,−1.0) (filled squares) and ( −1.0,0.6) (filled circles) correspond to phase V and VI, see Fig. 8.10 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s52 /s32/s32/s83 /s118/s78 /s121/s32/s120/s61/s45/s48/s46/s51 /s32/s120/s61/s45/s48/s46/s53 /s32/s120/s61/s45/s49/s46/s48 /s32/s120/s61/s45/s49/s46/s53 /s32/s120/s61/s45/s50/s46/s48 FIG. 10. (Color online) Spin-orbital entanglement entropy S0 vNin the ground state of the SU(2) ⊗SU(2) spin-orbital model (2.1) as a function of yfor representative values of x∈[−2.0,−0.3]. The onset of phase VI at decreasing yis detected by a step-like increase of entropy at the IV-VI phas e transition. istic of phase III (AF/AO). For x∈[−1.0,−0.3] the vNE S0 vNincreases discontinuously at the phase transition IV- VI, next drops somewhat and next increases further, see Fig. 10. It exhibits a broad maximum, moving to lower values ofywith decreasing x. This behavior shows that two phases (VI and III) compete in this regime. For still lower values of xthe SOE entropy is smaller and almost constant when ydecreases deeply into phase III. C. Entanglement in the SU(2) ⊗SU(2) models The ground state obtained in the present spin-orbital SU(2)⊗SU(2) model for phase III (AF/AO) is distinct from the one found for positive coupling constant, i.e., J=−1 in Eq. (2.1). We elucidate this difference by studying both models along the symmetry line x=y in the phase diagram. Phase I (FM/FO) is found in the present case for x=y >1/4, while in the case of positive coupling constant it becomes the ground state forx=y<−1/4 [61]. First we consider SOE detected by the vNE S0 vNEq. (1.1), and by the joint spin-orbital correlation function C1(4.1), see Fig. 11. To understand better the tran- sition from phase I to phase III, we consider the corre- lation functions along the symmetry line x=y. Phase I with FM/FO order is disentangled in both cases. At the quantum phase transition to phase III, signalled by a rapid increase of both S0 vNand 8|C1|, we observethat the entropy reaches the highest value at the onset of phase III, and then decreases when xdecreases and one moves deeper into the entangled phase III. In the present model (2.1) one finds C1>0 which is imposed by the negative/s45/s49 /s48 /s49/s48/s50/s52 /s45/s49 /s48 /s49/s48/s50/s52/s32/s32 /s120/s32/s83 /s118/s78 /s32/s56/s32/s67 /s49/s40/s97/s41 /s74/s61/s49 /s74/s61/s45/s49 /s32/s32 /s120/s32/s83 /s118/s78 /s32/s56/s32/s124/s67 /s49/s124/s40/s98/s41 FIG. 11. (Color online) von Neumann entropy S0 vN(1.1) and joint spin-orbital bond correlation C1(4.1) as obtained in the SU(2)⊗SU(2) model (2.1) along the symmetry x=yline in the phase diagram with the ring of L= 8 sites for: (a) the model with negative coupling −J=−1 [43], and (b) the model with positive coupling −J= 1 constant [56]. coupling constant. The entropy maximum and also the maximum of 8 C1are sharp indeed and signal the onset of phase III, see Fig. 11(a). The model with a pos- itive coupling constant behaves differently — here the joint spin-orbital correlations are negative C1<0, and bothS0 vNand 8|C1|have flat maxima in a range of x and only at x≃0.5 both drop rapidly. This large SOE forx∈[−0.25,0.5] indicates a spin-orbital liquid phase which forms near the SU(4) point x=y= 1/4[61]. Only atx>0.5 the strong spin-orbital fluctuations are weak- ened when phase III is approached and SOE decreases. A special feature of the present SU(2) ⊗SU(2) spin- orbital model is a very distinct behavior along the I- III phase transition line x+y= 1/2, see Fig. 8. The ground state energy of phase I (FM/FO) in which quan- tum fluctuations are absent, E0=−J(1/4+x)2, is found by taking exact classical values of spin (4.2) and or- bital (4.3) correlations on the bonds, S1=T1= 1/4. The energy decreases when the transition at x= 1/4 is approached. On the contrary, coming from the other side it is not allowed to assume classical correlations, S1=T1=−1/4, as then the Hamiltonian would van- ish at the transition. In fact the energy E0=−J/4 can be also obtained mainly from enhanced joint spin-orbital correlations, ∝an}bracketle{t(/vectorSj·/vectorSj+1)(/vectorTj·/vectorTj+1)∝an}bracketri}ht= 5/16andC1= 1/4, see Fig. 11. At the phase transition the spin and orbital correlations are very weak , i.e.,S1=T1≃ −1/8. This is a very peculiar situation as joint spin-orbital correla- tions cannot be factorized and damp to a large extent individual spin and orbital correlations. A qualitatively different nature of the SOE in both SU(2)⊗SU(2) models is also captured by its effect on the individual spin (orbital) correlations, see Fig. 12. In the present model (2.1) with J= 1,C1>0 damps individ-11 /s45/s49 /s48 /s49/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53 /s45/s49 /s48 /s49/s45/s48/s46/s53/s48/s46/s48/s48/s46/s53 /s74/s61/s49/s74/s61/s45/s49/s32/s32/s83 /s49 /s120/s40/s97/s41 /s32/s32/s83 /s49 /s120/s40/s98/s41 FIG. 12. (Color online) Spin S1(4.2) and orbital T1(4.3) (T1=S1) bond correlations as obtained in the SU(2) ⊗SU(2) model (2.1) along the symmetry line x=yin the phase di- agram with the ring of L= 8 sites for: (a) the model with negative exchange J= 1 [43], and (b) the model with positive exchange J=−1 [56]. ual spin and orbital fluctuations near the quantum phase transition stronger than in the case of positive coupling (J=−1), cf. Figs. 12(a) and 12(b). However, the joint fluctuations C1decrease fast when J= 1, while they are robust in the spin liquid phase for J=−1 when x∈[0.25,0.50). Atx= 0 one finds S1=T1≃ −0.37, already much below the value of S1=T1≃ −0.22 found forJ=−1. The spin S1and orbital T1correlations be- come similar in both phases deeply within phase III, as found by comparing these values at x=−1 forJ= 1 /s52 /s54 /s56 /s49/s48 /s49/s50/s48/s50/s52/s54/s56 /s32/s74/s61/s49 /s32/s74/s61/s45/s49/s32/s83 /s118/s78 /s32/s32 /s76 FIG. 13. (Color online) von Neumann entropy S0 vN(1.1) ob- tained in the isotropic SU(2) ⊗SU(2) model (2.1) at the max- imum seen in Fig. 11(a) at x=y= 0.249 (squares) and at J=−1 atx=y=−0.249 (circles), i.e., at the onset of phase III, both for rings of L= 4,6,···,12 sites and the linear fits to the data (dashed lines).and atx= 1 forJ=−1. Here SOE is weak because spins and orbitals fluctuate almost independently. The sharp peak of S0 vNfound near the phase transi- tion forJ= 1 is rather unusual, see Fig. 11(a). In this regime of parameters the ground state energy E0 is lowered by positive joint spin-orbital correlations C1, while negative S1=T1≃ −1/8 [Fig. 12(a)] increase it somewhat. Large vNE S0 vNis found near the phase tran- sition for rings of even length, starting from S0 vN≃1 for L= 4. It is remarkable that the vNE at the maximum scales with system size L, see Fig. 13. This behavior is unique and proves that SOE which takes place at every bond is extensive and extends here over the entire ring. The model with positive coupling constant, J=−1, has also a similar linear scaling, S0 vN∝L, but the entropy is smaller and systematic fluctuations between the rings of length of 4nand 4n+2sites, seen in Fig. 13. are distinct and indicate a crucial role played here by global SU(4) singlets. V. ENTANGLED ELEMENTARY EXCITATIONS: FM/FO ORDER A. Analytic approach When the ground state is disentangled, SOE is gen- erated locally in excited states [87] and would not scale linearly with system size. Indeed, we analyzed the low- energy excitations of the disentangled FM/FO phase in the 1D SU(2) ⊗SU(2) spin-orbital model in Ref. [43] and found a much weaker dependence on system size. We investigated the SOE for spin-orbital bound states (BSs) and spin-orbital exciton (SOEX) state and found a log- arithmic scaling, while the entropy saturates for other separable (trivial) spin-orbital excitations. One finds that the vNE is controlled by the spin-orbital correla- tion length ξand decays logarithmically with ring length L. Here we consider again the FM/FO disentangled groundstate |0∝an}bracketri}ht, obtained for the anisotropicspin-orbital SU(2)⊗XXZmodelwiththeexactlyknowngroundstate energyE0, H|0∝an}bracketri}ht=E0|0∝an}bracketri}ht. (5.1) Using equation of motion method one finds spin (magnon) excitations with dispersion ωS(Q) =/parenleftbigg1 4+y/parenrightbigg (1−cosQ), (5.2) and orbital (orbiton) excitations [88], ωT(Q) =/parenleftbigg1 4+x/parenrightbigg (1−∆cosQ).(5.3) The spin-orbital continuum is given by Ω(Q,q) =ωS/parenleftbiggQ 2−q/parenrightbigg +ωT/parenleftbiggQ 2+q/parenrightbigg .(5.4)12 Next, weconsiderthepropagationofamagnon-orbiton pair excitation along the FM/FO chain, by exciting si- multaneously a single spin and a single orbital. The translationsymmetryimposesthattotalmomentum Q= 2mπ/L(form= 0,···,L−1) is conserved during scat- tering. The scatteringofmagnon and orbitonwith initial (final) momenta {Q 2−q,Q 2+q}({Q 2−q′,Q 2+q′}) and the totalmomentum QisrepresentedbytheGreen’sfunction [89], G(Q,ω) =1 L/summationdisplay q,q′∝an}bracketle{t∝an}bracketle{tS+ Q 2−q′T+ Q 2+q′|S− Q 2−qT− Q 2+q∝an}bracketri}ht∝an}bracketri}ht,(5.5) for a combined spin ( S− Q 2−q) and orbital ( T− Q 2+q) excita- tion. The analytical form reads G(Q,ω) =G0(Q,ω)+Π(Q,ω), (5.6) Π(Q,ω) =−2(1+∆)+(1 −∆2)Fss(Q,ω) 4[1+Λ(Q,ω)]H2 cc(Q,ω) −2(1−∆)+(1−∆2)Fcc(Q,ω) 4[1+Λ(Q,ω)]H2 ss(Q,ω) +(1−∆2)Fsc(Q,ω) 2[1+Λ(Q,ω)]Hcc(Q,ω)Hss(Q,ω),(5.7) where the noninteracting Green’s function is given by G0(Q,ω) =1 L/summationdisplay qG0 qq(Q,ω), (5.8) G0 qq(Q,ω) =1 ω−Ω(Q,q), (5.9) and Hcc(Q,ω) =1 L/summationdisplay q/parenleftbigg cosQ 2−cosq/parenrightbigg G0 qq(Q,ω),(5.10) Hss(Q,ω) =1 L/summationdisplay q/parenleftbigg sinQ 2−sinq/parenrightbigg G0 qq(Q,ω).(5.11) One finds the denominator in Π( Q,ω) (5.7), 1+Λ(Q,ω) =/bracketleftbigg 1+1 2(1+∆)Fcc(Q,ω)/bracketrightbigg/bracketleftbigg 1+1 2(1−∆)Fss(Q,ω)/bracketrightbigg −1 4(1−∆2)F2 sc(Q,ω), (5.12) with Fcc(Q,ω) =1 L/summationdisplay q(cosQ 2−cosq)2 ω−Ω(Q,q), (5.13) Fss(Q,ω) =1 L/summationdisplay q(sinQ 2−sinq)2 ω−Ω(Q,q), (5.14) Fsc(Q,ω) =1 L/summationdisplay q(sinQ 2−sinq)(cosQ 2−cosq) ω−Ω(Q,q). (5.15)Here we define a phase δ∈[0,2π] which quantifies the difference of dynamic properties of magnon and orbiton excitations throughout the Brillouin zone in the form tanδ=(4y+1)−(4x+1)∆ (4y+1)+(4x+1)∆tan/parenleftbiggQ 2/parenrightbigg .(5.16) For the symmetry line x=yEq. (5.16) greatly simplifies for the isotropic model at ∆ = 1 and gives δ(Q) = 0, which has been studied in Ref. [43]. Also, a quantity describes the position relative to the continuum is given by a(Q,ω) =ω−(x+y+1 2) b(Q), (5.17) with b(Q) =/bracketleftBigg/parenleftbigg x+1 4/parenrightbigg2 ∆2+/parenleftbigg y+1 4/parenrightbigg2 + 2∆/parenleftbigg x+1 4/parenrightbigg/parenleftbigg y+1 4/parenrightbigg cosQ/bracketrightbigg1/2 .(5.18) The excitations at the top and at the bottom of the con- tinuum correspond to a(Q,ω) = 1 and −1, respectively. In the noninteracting case, we have the imaginary and real parts of Eq. (5.6), ℑG0(Q,ω) =−θ(1−|a(Q,ω)|) b(Q)/radicalbig 1−a2(Q,ω),(5.19) ℜG0(Q,ω) =θ(a(Q,ω)−1) b(Q)/radicalbig a2(Q,ω)−1 −θ(−1−a(Q,ω)) b(Q)/radicalbig a2(Q,ω)−1,(5.20) whereθ(x) is Heaviside step function whose value is zero for negative argument and 1 for nonnegative argument. The frequency dependence of the imaginarypart exhibits square-root singularities in Eq. (5.19) at the bottom and at the top of the continuum [90]. B. Numerical studies We note that the inclusion of the spin-orbital attrac- tion will smear out the singularities by Eq. (5.6) since a more pronounced divergence of the numerator than the denominatoroccurswhen a(Q,ω)→ ±1, and they cancel each other, i.e., ℑG(Q,ω) = 0. Furthermore, the poles ofG(Q,ω) are determined by 1+Λ(Q,ω) = 0. (5.21) Our analysis shows that for given Qmost the real so- lutions of Eq. (5.21) are interspersed within the contin- uum, but these modes are unstable to two free waves. However, a small number of solutions may lie well below the continuum. To investigate the spectra we begin with13 the asymmetric SU(2) ⊗XXZmodel. As it is shown in Fig. 14, the attractive interactions shift spin-orbital BSs outside the continuum [91–93]. The binding energy ap- proaches zero for the isotropic SU(2) ⊗SU(2) model, but is finite for anisotropic SU(2) ⊗XXZmodel due to a gap in the orbital excitation spectrum, and the small- qbe- havior of the binding energy reveals that the BSs appear for arbitrarily small wave number. The BSs can be also obtained by the equation of motion method for a spin-orbital joint excitation, S− mT− m+l|0∝an}bracketri}ht. The collective mode follows from Eq. (5.21). Such a collective spin-orbital excitation (bound state) in- volves spin and orbital flips at many sites and can be written as follows, |Ψ(Q)∝an}bracketri}ht=1√ L/summationdisplay m,lal(Q)eiQmS− mT− m+l|0∝an}bracketri}ht =/summationdisplay qaqS− Q 2−qT− Q 2+q|0∝an}bracketri}ht, (5.22) with the coefficients aq=1√ L/summationdisplay lal(Q)e−i(Q 2−q)l.(5.23) The correlation length ξ≡/summationtext ll|al|2defines the average size of spin-orbital BSs or excitons and is much smaller than the system size, i.e., 0 < ξ≪L. The correlation length becomesextensivefora trivialcontinuumstate, as shown in Fig. 15. The analytic solution of this equation is tedious but straightforward. The dispersion of the col- lective excitation, ωBS(Q), can be analyzed in a simple way at some special points, including Q= 0 andQ=π. In the isotropic model (at ∆ = 1), Eq. (5.21) reduces to 1 +Fcc(Q,ω) = 0. In this case there is at most one solution for every Q[43]. Nevertheless, the anisotropic orbitalcouplingwill inducemorebranchesinpartofBril- louin zone (see Fig. 14). When ∆ <1 andQ= 0, Fsc(0,ω)=0 and then Eq. (5.21) reduces to 1+1 2(1+∆)Fcc(0,ω) = 0, (5.24) or 1+1 2(1−∆)Fss(0,ω) = 0. (5.25) The solution that follows from Eq. (5.24) is given by ωBS,1(0) =(α+3β)2−/radicalbig (α−17β)(α−β)3 8β +x(1−∆)−3β+1 2, (5.26) whereα= (∆x+y),β= (1+∆)/4. The instability of such a mode given by ωBS,1(0) = 0 sets up the threshold of the FM/FO state separating it from the AF/AO state (phase III) [43, 92]. Moreover,the solution for Eq. (5.25) is given by ωBS,2(0) =x+y−(α+β)2 (1−∆)+β.(5.27)/s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s52/s54/s56 /s40/s97/s41 /s32/s32/s40/s81 /s41 /s81/s47 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s52/s54/s56 /s40/s98/s41 /s32/s32/s40/s81 /s41 /s81/s47 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48/s48/s50/s52/s54/s56 /s40/s99/s41 /s32/s32/s40/s81 /s41 /s81/s47 FIG. 14. (Color online)(a) Excitation spectra of a ring of L= 40 sites for the spin-orbital SU(2) ⊗XXZ model with ∆ = 0 .5 as function of momentum Qat: (a)x=y= 1/4, (b)x= 0.375,y= 0.125, and (c) x= 0.27,y= 0.23. The dotted (blue), dashed (green) lines inside the spin-orbital conti nuum Ω(Q,q) denote the orbital and SOEX excitations, i.e., ωT(Q) andωSOEX (Q), respectively, that are degenerate. The (red) solid lines show spin excitations. We find that when α <(1−3∆)/4, bothωBS,1(0) and14 /s48 /s49 /s50 /s51/s48/s53/s49/s48/s49/s53/s50/s48 /s32/s32 /s81/s32 /s66/s83/s44/s49/s40/s120/s61/s48/s46/s53/s44/s121/s61/s48/s46/s53/s41 /s32 /s66/s83/s44/s49/s40/s120/s61/s48/s46/s51/s55/s53/s44/s121/s61/s48/s46/s49/s50/s53/s41 /s32 /s66/s83/s44/s50/s40/s120/s61/s48/s46/s51/s55/s53/s44/s121/s61/s48/s46/s49/s50/s53/s41 FIG. 15. (Color online) The correlation length ξfor increasing momentum Qas obtained for a ring of L= 80 sites in the anisotropic SU(2) ⊗XXZ model with ∆ = 0 .5. ωBS,2(0) exist, while only ωBS,1(0) survives when (1 + ∆)/4> α >(1−3∆)/4. Finally, for α >(1 +∆)/4 no bound states exist. When ∆<1 andQ=π, one finds that Fsc(π,ω)=0. Analogously, Eq. (5.21) has two solutions, ωBS,1(π) and ωBS,2(π) when−(3+∆)/4<y−∆x<(1−∆)/4, with explicit expressions for their energies: ωBS,1(π) =x+y+1 2−1+∆ 4 −[/parenleftbig y+1 4/parenrightbig −∆/parenleftbig x+1 4/parenrightbig ]2 1+∆, (5.28) ωBS,2(π) =x+y+1 2 +ζ−(2γ−1−∆)3 2√2γ−9+9∆ 8(∆−1),(5.29) with γ= (y+1/4)−∆(x+1/4), ζ= 3+4γ−4γ2−6∆−4γ∆+3∆2.(5.30) Wheny−∆x<−(3+∆)/4 and (1 −∆)/4<y−∆x< (3∆+1)/4, only one ωBS,1(π) solution exists. In case of y−∆x= (1−∆)/4,ωBS,2(π) mergeswithlowerboundary of the continuum. C. Propagating spin-orbital exciton states Especially at the SU(4) symmetric point, i.e., at x= y= 1/4, spinon and orbiton are strongly coupled to form a joint SOEX state inside the spin-orbital contin- uum across the whole Brillouin zone. Usually such an elementary spin-orbital excitation in the continuum is unstable and decays into a spinon and an orbiton [38]. However, it is surprising that such a SOEX state propa- gates here as a undamped on-site spin-orbital excitation/s48 /s49 /s50 /s51/s48/s50/s52/s54 /s32/s32/s40/s81 /s41 /s81 FIG. 16. (Color online) Excitation spectrum for the spin- orbital SU(2) ⊗Z2model (2.1) on a ring of L= 40 sites. The dotted (blue) and dashed-dotted (gray) lines indicate the o r- bital excitation ωT(Q) and the lower boundary of the con- tinuum Ω( Q,q),ωc(Q) — both are degenerate. The dashed (green) and solid (red) lines denote the SOEX excitations ωSOEX (Q), and spin excitations ωS(Q). Parameters: ∆ = 0 .0, x= 0.375, and y= 0.125. [88], e.g.S− lT− l|0∝an}bracketri}htat sitel, within the spin-orbital con- tinuum with ξ= 0 (see Eq. (5.23)). It is straightforward to derive, [H,S− lT− l]|0∝an}bracketri}ht= [Cl(x,y)+Dl(x,y)]|0∝an}bracketri}ht,(5.31) where Cl(x,y) = (x+y)S− lT− l−∆ 4/parenleftbig S− l−1T− l−1+S− l+1T− l+1/parenrightbig , Dl(x,y) =−1 2/bracketleftbigg ∆/parenleftbigg x−1 4/parenrightbigg/parenleftbig S− lT− l−1+S− lT− l+1/parenrightbig +/parenleftbigg y−1 4/parenrightbigg/parenleftbig S− l−1T− l+S− l+1T− l/parenrightbig/bracketrightbigg .(5.32) Thedissipativeterm Dl(x,y)vanisheswhen x=y= 1/4. Consequently, the dispersion of the SOEX state is given by ωSOEX(Q) =1 2(1−∆cosQ).(5.33) The SOEX state for x= 1/4 is degenerate with the orbital wave excitation, see Eq. (5.3) and Fig. 14(a). When ∆ = 1 and x=y >0.25, there is a quasi-SOEX state inside the spin-orbital continuum for Q < π, with ξ<1. Whenx∝ne}ationslash=y, theresidualsignaloftheSOEXstate denotedbyfinite ξvanishesandthe BS ωBS,2(π) appears. The smaller the ∆ is, the more values of QgiveωBS,2(Q). There is no BS at Q=πfory−∆x >(3∆ + 1)/4. Furthermore, away from the symmetric point, the SOEX will acquire a finite linewidth due to residual interactions into magnon-orbiton pairs, Γ =ℑ{G−1(Q,ω)}. (5.34)15 The exciton spectral weight can be calculated from the self-energy Σ( Q,ω), zQ=/bracketleftbigg 1−/parenleftbigg∂Σ(Q,ω) ∂ω/parenrightbigg/bracketrightbigg−1/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle ωSOEX(Q),(5.35) where the SOEX energy is given by by the pole, ωSOEX(Q) =ℜ{G−1(Q,ω)}. (5.36) If Γ/[ω−ωSOEX(Q)]2→0, the exciton is stable. The decay rate of the SOEX increases with growing x>1/4 and also for decreasing momenta Q, and they coincide at ∆ = 0. In the limit of Ising orbital interactions (∆ = 0), the orbital part becomes classical and orbitons are disper- sionless, indicating localized orbital excitations, see Fig. 16. We find that two BSs, ωBS,1(0) andωBS,2(0), exist wheny <1/4, and there are no BSs otherwise. In con- trast, atQ=πone finds two solutions, ωBS,1(π) and ωBS,2(π), when −1/4< y <1/4 and no BS is found fory >1/4. In Fig. 16, both BSs, ωBS,1andωBS,2, are undamped. In this case, ωT(Q) = 1/4 +xand is degenerate with the lower boundary of the continuum ωc(Q) = 1/4 +x. Moreover, ωSOEX(Q) =x+y, and especially when x=y= 1/4, these excitations coincide. In this case all excitations are dispersionless, see Fig. 16. The spin-orbital BS in Fig. 16 appears below the bot- tom of the spin-orbital continuum and is stabilized by its binding energy. VI. VON NEUMANN ENTROPY SPECTRA A. The spectra in the FM/FO phase To investigate the degree of entanglement of spin- orbital excited states, we introduce the vNE spectral function in the Lehmann representation, SvN(Q,ω) =−/summationdisplay nTr{ρ(µ) Slog2ρ(µ) S}δ{ω−ωn(Q)}, (6.1) where we use a short-hand notation ( µ) = (Q,ωn) for momentum Qand excitation energy ω, and ρ(n) S= TrT|Ψn(Q)∝an}bracketri}ht∝an}bracketle{tΨn(Q)| (6.2) is the spin ( S) density matrix obtained by tracing over the orbital ( T) degrees of freedom. In Fig. 17 we present the analytic results for the vNE spectral function when ∆ = 0.5,x= 0.375, andy= 0.125. The parity symme- try is broken at x∝ne}ationslash=yand ∆∝ne}ationslash= 1. The even and odd excitations show diverse behavior of their entanglement, as is displayed in Fig. 17. Inspection of the vNE spectra shows that the entanglement reaches a local maximum at the BSs and SOEX states, and all these states have short-range correlation length ξ./s48 /s50 /s52 /s54/s81/s61/s48/s46/s48 /s81/s61/s48/s46/s50 /s81/s61/s48/s46/s52 /s81/s61/s48/s46/s54 /s81/s61/s48/s46/s56 /s48/s50/s52/s54/s40/s97/s41 /s83 /s118/s78 /s48 /s50 /s52 /s54/s81/s61/s48/s46/s48 /s81/s61/s48/s46/s50 /s81/s61/s48/s46/s52 /s81/s61/s48/s46/s54 /s81/s61/s48/s46/s56 /s48/s50/s52/s54 /s83 /s118/s78/s40/s98/s41 FIG. 17. (Color online) The vNE spectral function SvN(Q,ω) (6.1) for the anisotropic spin-orbital SU(2) ⊗XXZ model (2.1) with ∆ = 0 .5 on a ring of L= 160 sites, see Fig. 14(b), obtained for different momenta Q≤0.8π, and for: (a) even excitations, and (b) odd excitations. Isolated vertical li nes below the continuum indicate the BS, with dispersion given by the dashed (red) line. Parameters: x= 0.375,y= 0.125. We have derived an asymptotic form of the vNE as a function of ξ[43], SvN≃log2/braceleftbiggL (1+ξ)/bracerightbigg . (6.3) One finds the asymptotic logarithmic scaling of vNE of spin-orbitalBSsand SOEXstatewhosemagnon-orbitons correlationlength is short-range,and the vNE is givenby SvN= log2L+c0. (6.4) In particular, c0= 0 for the SOEX state and c0<0 otherwise. Such relation is displayed in Fig. 18. Fig. 15 implies that ωBS,1are always stable for all momenta whileωBS,2are undamped for large momenta. When both xandyareawayfrom1 /4,the SOEXstate is unstable and decays into a spinon and an orbiton. In this case the correlation length ξbecomes extensive. We have verified that the scaling is entirely different in such16 /s54 /s55 /s56 /s57 /s49/s48/s54/s56/s49/s48 /s32/s32/s83 /s118/s78 /s108/s111/s103 /s50/s76/s32 /s66/s83/s44/s49/s40/s48/s46/s56 /s41 /s66/s83/s44/s50/s40/s48/s46/s56 /s41/s40/s97/s41 /s53 /s54 /s55 /s56 /s57 /s49/s48/s52/s54/s56/s49/s48/s32 /s32/s40/s98/s41/s83 /s118/s78 /s108/s111/s103 /s50/s76/s32 /s66/s83/s44/s49/s40/s48/s46/s56 /s41 /s32 /s66/s83/s44/s50/s40/s48/s46/s56 /s41 FIG. 18. (Color online) Scaling behavior of the entanglemen t entropy SvNof the spin-orbital BSs at Q= 0.8π(points) forx= 0.375,y= 0.125, and for: (a) ∆ = 0 .5 and (b) ∆ = 0. Lines represent logarithmic fits to Eq. (6.4) with: (a) c0=−0.421 and −0.842; (b) c0=−0.368 and −1.122. a case and the entropy of the SOEX scales instead as a power law, SvN=c1 L+c0. (6.5) The vNE saturates in the thermodynamic limit. B. RIXS spectral functions in the FM/FO state The entanglement spectral function SvN(Q,ω) has a similar form as any other dynamical spin or charge cor- relation function. There is, however, an important dif- ference — as there is no direct probe for the vNE of an arbitrary state, the SOE spectra can be calculated but cannot be measured directly. On the other hand, we have shown before [43] that the intensity distribution of certain RIXS spectra of spin-orbital excitations in fact probe qualitatively SOE.We introduce the spectral function of the coupled spin- orbital excitations at distance l, Al(Q,ω) =1 πlim η→0Im∝an}bracketle{t0|Γ(l)† Q1 ω+E0−H−iηΓ(l) Q|0∝an}bracketri}ht. (6.6) Here Γ(0) Q=1√ L/summationdisplay jeiQjS− jT− j (6.7) is the local operator for an on-site joint spin-orbital exci- tation measured in RIXS [57–59]. We employ as well the even and odd parity operators, Γ(1±) Q=1√ 2L/summationdisplay jeiQj/parenleftbig S− j+1±S− j−1/parenrightbig T− j,(6.8) which serve to probe the nearest neighbor spin-orbital excitations. Intuitively, theon-sitespectralfunction A0(Q,ω) high- lights the SOEX state. It is found that both ωSOEX(Q) andωBS(Q) are solutions of Eq. (5.21) when x=y= 1/4. However, the weight of the BS is here zero. Now the spectral function is given by A0(Q,ω) =δ{ω−ωSOEX(Q)},(6.9) which is confirmed in Fig. 19(a). As the point ( x,y) moves away from the symmetric point, i.e., x=y= 1/4, the BSs gain spectral weight which decreases with mo- mentumQ. The spectral weight vanishes at Q=πac- companying a square-root singularity at the lower bound of the continuum, see Fig. 19(b). The δpeak turns into /s48 /s49/s48/s50/s52/s54 /s48 /s49 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s32/s32/s65 /s48/s40/s81/s44 /s41/s40/s97/s41 /s32/s40/s98/s41 /s32/s32 /s40/s99/s41/s32 /s32 FIG. 19. (Color online) The spectral function of the on-site spin-orbital excitation A0(Q,ω) for the anisotropic SU(2)⊗XXZ model with ∆ = 0 .5, and for: (a) x= 0.25, y= 0.25, (b)x= 0.5,y= 0.5, and (c) x= 0.375,y= 0.125. In each panel the momenta Qrange from π/10 (bottom) to 9π/10 (top); the peak broadening is η= 0.001. The red dot- ted (green dashed) lines mark the positions of BSs (SOEX) states. Gray dash-dot line signals the onset of the continuu m.17 /s48 /s49/s48/s50/s52/s54 /s48 /s49 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s32/s32/s65 /s49/s43/s40/s81/s44 /s41/s40/s97/s41 /s32/s40/s98/s41 /s32/s32 /s40/s99/s41/s32 /s32 FIG. 20. (Color online) The spectral function of the spin- orbital excitation at nearest neighbors A1+(Q,ω) with even parity for the anisotropic SU(2) ⊗XXZ model with ∆ = 0 .5, and for: (a) x= 0.25,y= 0.25, (b)x= 0.5,y= 0.5, and (c) x= 0.375,y= 0.125. In each panel the momenta Qrange fromπ/10 (bottom) to 9 π/10 (top); the peak broadening is η= 0.001. The red dotted (green dashed) lines mark the positions of BSs (SOEX) states. Gray dash-dot line signals the onset of the continuum. a broad peak. A difference between xandyinduces a second branch of BSs and it gains larger spectral weight at large momenta than the first BS, see Fig. 19(c). Alto- gether, the evolution of spectral weight is similar to that of the correlation length in Fig. 15. The BSs can be captured also by the spectral func- /s48 /s49/s48/s50/s52/s54 /s48 /s49 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s32/s32/s65 /s49/s45/s40/s81/s44 /s41/s40/s97/s41 /s32/s40/s98/s41 /s32/s32 /s40/s99/s41/s32 /s32 FIG. 21. (Color online) The spectral function of the spin- orbital excitation at nearest neighbors A1+(Q,ω) with odd parity for the anisotropic SU(2) ⊗XXZ model with ∆ = 0 .5, and for: (a) x= 0.25,y= 0.25, (b)x= 0.5,y= 0.5, and (c) x= 0.375,y= 0.125. In each panel the momenta Qrange fromπ/10 (bottom) to 9 π/10 (top); the peak broadening is η= 0.001. The red dotted (green dashed) lines mark the positions of BSs (SOEX) states. Gray dash-dot line signals the onset of the continuum.tions for the nearest neighbor excitations, see Figs. 20 and 21. With increasing momentum Q, the spectral weight of even-parity excitation A1+(Q,ω) at the first BS decreases (see Fig. 20), while the spectral weight of odd-parity excitation A1−(Q,ω) rises (see Fig. 21). A1+(Q,ω) reaches its valley for quasi-SOEX states, but no such feature could be found in A1−(Q,ω). VII. DISCUSSION AND CONCLUSIONS Motivated by the discovery of new Majumdar-Ghosh- like valence-bond spin singlet phases triggered by or- bital correlations, we havestudied the spin-orbitalentan- glement (SOE) in the one-dimensional (1D) anisotropic SU(2)⊗XXZspin-orbital model with the negative ex- change interaction. The asymmetry between spin and orbital degrees of freedom yields a better insight into the phase diagram and the mechanisms responsible for the different types of order observed for this system. In ad- dition to the four uniform phases I-IV, our study demon- strates that a gapful phase V exists in case of classical Ising orbital interactions, i.e., in the SU(2) ⊗Z2model. It is characterized by quadrupling of the unit cell seen as a maximum of the orbital structure factor at k=π/2. Fory=−1/4 this provides a perfect dimer structure of spin singlets in the whole region of stability of this phase, where the dimer spin correlations D(r) develop and uncover long-range dimer order. The dimer phase V is quite robust and survives when the orbital quantum fluctuations at ∆ >0 are taken into account. The phase diagram is still richer at finite ∆ >0, when quantum orbital fluctuations develop and induce an or- bital dimer phase VI, with a complementary role of spin and orbital correlations to phase V. The emergence of the nonuniform phase V is a result of the joint interac- tion between spin fluctuation and orbital degree of free- dom, and thus phase V carries finite SOE. The orbital fluctuations enhance the SOE in ground state near the III/V phase transition and lead to phase VI when the {x,y}parameters are interchanged. We also anticipate that these dimer phases may survive in higher dimen- sions. In fact, Lieb-Schultz-Mattis theorem is applicable tothepresentmodelforarbitrary xandyandnonzero∆, where a finite gap exists above these degenerate ground states. Both phases V and VI are gapped phases with alternating spin and orbital singlets, respectively. As we have shown, the phase boundaries can be captured by SOE and fidelity susceptibility. The phase transition be- tween phases III and V is a first-order transition in the ∆ = 0 case, and the transition changes to continuous when ∆>0. An important consequence of finite SOE in the ground state is that it invalidates mean-field decoupling of spin and orbital degrees of freedom, as this would imply a spin-orbital product ground state. A similar restriction applies to the entangled elementary excitations in the disentangled ferromagnetic phase with ferro-orbital or-18 der in spin-orbital systems which were analyzed here with the help of the von Neumann entropy spectral func- tion. Spin-orbital excitations are highlighted by nontriv- ial SOE, especially by logarithmic scaling of SOE in this phase. A priori, theSOEmakesitnecessarytotreattheeigen- statesofagivenmodel exactly. In fact, sinceamean-field decoupling shields the SOE, it fails to describe elemen- tary excitations even qualitatively correctly in a num- ber of spin-orbital models. In antiferromagnetic ground states with ferro-orbital order it was demonstrated both in theory [37] and experiment [38] that the spin-orbital excitation fractionalizes into freely propagating spinon and orbiton, giving rise to spin-orbital separation under specific condition. The SOE in the spin-orbital separa- tion remainsunclear. The low-lyingexcitations in phases II(AF/FO)andIII (AF/AO)arespinwaveswith vanish- ing SOE, corresponding to a two-spinon continuum of an antiferromagneticspin chain. The low-lying excitation in phases V and VI corresponds to spin-orbital excitation, as shown in Figs. 2(b) and 2(c). The problem of the SOE in elementary excitations in other phases remains a challenge for future studies. Summarizing, we have shown that the anisotropic SU(2)⊗XXZspin-orbital model with negative exchange coupling has remarkably different behavior and phase di- agrams from the well known SU(2) ⊗SU(2) model with positive exchange coupling. While the spin-orbital liq- uid phase is absent in the former case, we have found that the joint ferromagnetic/ferro-orbital fluctuations are surprisingly strong at the quantum phase transi- tion to the antiferromagnetic spin order which gives even stronger SOE than that established for the 1D isotropic SU(2)⊗SU(2)model with positive exchange coupling. ACKNOWLEDGMENTS We thank Krzysztof Wohlfeld for insightful discus- sions. W.-L.Y. acknowledges support by the Natural Science Foundation of Jiangsu Province of China un- der Grant No. BK20141190 and the NSFC under Grant No. 11474211. A.M.O. kindly acknowledges support by Narodowe Centrum Nauki (NCN, National Science Cen- ter) under Project No. 2012/04/A/ST3/00331. Appendix: Phase diagrams for the two-site model To understand better the phase boundaries in the anisotropic SU(2) ⊗XXZspin-orbital model with neg- ative exchange coupling (2.1), we present here an exact solution for the system of L= 2 sites [91], H12=−J/parenleftBig /vectorS1·/vectorS2+x/parenrightBig ×/bracketleftbigg∆ 2/parenleftbig T+ 1T− 2+T− 1T+ 2/parenrightbig +Tz 1Tz 2+y/bracketrightbigg ,(A.1)−1 0 1−101 xyI IV II III −1 0 1−101 xyI IV II III FIG. 22. (Color online)(a) The phase diagram of 2-site model when ∆ = 0. (b) ∆ = 0 .5. Phases I-IV correspond to FS/FO,AS/FO, AS/AO, FS/AO configurations in spin- orbital sectors. Again,{x,y}are the parameters, and 0 ≤∆≤1 inter- polates between the Ising Z2(∆ = 0) and Heisenberg SU(2) (∆ = 1) symmetry. The orbital interaction with XXZsymmetry can be exactly diagonalized and one finds 4 eigenstates: |++∝an}bracketri}ht,|−−∝an}bracketri}ht, (|+−∝an}bracketri}ht+|−+∝an}bracketri}ht)/√ 2, (|+−∝an}bracketri}ht−|− +∝an}bracketri}ht)/√ 2, correspondingtoeigenvalues: 1 /4, 1/4, ∆/2−1/4,−∆/2−1/4, respectively. At ∆ = 0 we recover doubly degenerate configurations from the latter two, while at ∆ = 1 we recover a triplet T= 1 from the first three. In any case, the third component of the triplet, (|+−∝an}bracketri}ht+|−+∝an}bracketri}ht)/√ 2, is always an entangled ex- cited state with the present choice of parameters, while the orbital singlet, ( |+−∝an}bracketri}ht−|− +∝an}bracketri}ht)/√ 2, is an entangled ground state for some parameters. The spin part is classified as a triplet S= 1 or a singlet S= 0, and these states are accompanied by the orbital statesdescribedabove. ThisgivesthestatesI-IVinTable19 TABLE II. The spin-orbital configuration for the model (A.1) , with phases I-IV defined by distinct spin Sand orbital state as as obtained at 0 <∆<1. The lowest energy E0/Jhas degeneracy dand becomes the ground state at the respective values of {x,y}parameters, Fig. 22. At ∆ = 0 the degenera- cies of phases III and IV change to 2 and 6 for the Z2⊗Z2 symmetry, while at ∆ = 1 the degeneracies for the ground states I-IV are 9, 3, 1, 3 and follow from the SU(2) ⊗SU(2) symmetry. phase S orbital states E0/J d I 1 |+ +/an}bracketri}ht,|−−/an}bracketri}ht −/parenleftbig1 4+x/parenrightbig/parenleftbig1 4+y/parenrightbig 6 II 0 |+ +/an}bracketri}ht,|−−/an}bracketri}ht −/parenleftbig −3 4+x/parenrightbig/parenleftbig1 4+y/parenrightbig 2 III 01√ 2(|+−/an}bracketri}ht−|− +/an}bracketri}ht)/parenleftbig −3 4+x/parenrightbig/parenleftbig1 4+∆ 2−y/parenrightbig 1 IV 11√ 2(|+−/an}bracketri}ht−|− +/an}bracketri}ht)/parenleftbig1 4+x/parenrightbig/parenleftbig1 4+∆ 2−y/parenrightbig 3 II,andoneofthemisthegroundstateforanypointinthe (x,y) plane. All phase transitions are first order, with a change of spin or orbital state. The phases II and IV are symmetricfor∆ = 1,andthetransitionbetweenIandIII occurs alongthe x+y= 1/2 line [43]. At ∆ = 0 phase IV existsfory>1/4whilephaseII onlyfor x>3/4,seeFig.22(a). This reflects the essential difference between the orbital configurations in the Ising limit and the quantum spin states. Note that spin singlet S= 0 is an entangled state, but the larger Hilbert space gives no spin-orbital entanglement in any phase. Thus, the total energies and the phase diagram are easily deduced, see Fig. 22. From the comparison of the energies E0of phases III and IV (Table II) one can see that the boundary between III and IV, given by the straight line y= 1/4 + ∆/2, moves upwards with increasing ∆, see Fig. 22(b). Ac- cordingly, the phase boundary between phases I and III is also modified. The interplay between spins and or- bitals develops and leads to interesting consequences of entanglement for rings with L≥4. 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1607.05782v1.Emergent_spin_valley_orbital_physics_by_spontaneous_parity_breaking.pdf

Emergent spin-valley-orbital physics by spontaneous parity breaking Satoru Hayami1, Hiroaki Kusunose2, and Yukitoshi Motome3 1Department of Physics, Hokkaido University, Sapporo 060-0810, Japan 2Department of Physics, Meiji University, Kawasaki 214-8571, Japan 3Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan E-mail: hayami@phys.sci.hokudai.ac.jp Abstract. The spin-orbit coupling in the absence of spatial inversion symmetry plays an important role in realizing intriguing electronic states in solids, such as topological insulators and unconventional superconductivity. Usually, the inversion symmetry breaking is inherent in the lattice structures, and hence, it is not easy to control these interesting properties by external parameters. We here theoretically investigate the possibility of generating the spin-orbital entanglement by spontaneous electronic ordering caused by electron correlations. In particular, we focus on the centrosymmetric lattices with local asymmetry at the lattice sites, e.g., zig-zag, honeycomb, and diamond structures. In such systems, conventional staggered orders, such as charge order and antiferromagnetic order, break the inversion symmetry and activate the antisymmetric spin-orbit coupling, which is hidden in a sublattice- dependent form in the paramagnetic state. Considering a minimal two-orbital model on a honeycomb lattice, we scrutinize the explicit form of the antisymmetric spin- orbit coupling for all the possible staggered charge, spin, orbital, and spin-orbital orders. We show that the complete table is useful for understanding of spin-valley- orbital physics, such as spin and valley splitting in the electronic band structure and generalized magnetoelectric responses in not only spin but also orbital and spin-orbital channels, re ecting in peculiar magnetic, elastic, and optical properties in solids. PACS numbers: 72.25.-b, 71.10.Fd, 71.70.Ej, 75.85.+t Keywords : spatial inversion symmetry, spin-orbit coupling, electron correlation, odd- parity multipole, spin-valley physics, topological insulator, magnetoelectric eect Submitted to: J. Phys.: Condens. Matter 1. Introduction The spin-orbit coupling (SOC), which originates from the relativistic motion of electrons, is a source of interesting electronic states in solids. In particular, in the systems without spatial inversion symmetry, the SOC acquires an antisymmetric component with respectarXiv:1607.05782v1 [cond-mat.str-el] 19 Jul 2016Emergent spin-valley-orbital physics by spontaneous parity breaking 2 to the wave vector [1, 2]. This is called the antisymmetric SOC (ASOC), typied by the Rashba SOC near a surface and interface [3, 4, 5], and the Dresselhaus SOC in the cubic systems lacking inversion symmetry [6, 7]. The ASOC has been extensively studied as it results in intriguing physics, such as Dirac electrons at the surface of topological insulators [8, 9], the spin Hall eect [10, 11], multiferroics [12, 13], and the noncentrosymmetric superconductivity [14]. noncentrosymmetric latticecentrosymmetric lattice(a) (b) with local asymmetry Figure 1. Schematic pictures of (a) a noncentrosymmetric system and (b) a centrosymmetric system with local asymmetry. In the latter, spatial inversion symmetry is broken at the lattice sites (vertices on each triangle) despite the presence of inversion symmetry in the whole system. The red spheres represent the inversion centers. The ASOC is in general described by the simple Hamiltonian in the form of HASOC(k) =g(k)
(krV)
; (1.1) whereg(k) is the axial vector, which is asymmetric with respect to k,is the vector of Pauli matrices describing the spin degree of freedom in electrons, and rVis an asymmetric potential gradient. The kdependence of g(k) is determined by the potential gradient rV, which depends on the symmetry of the system. The magnitude of ASOC is predominantly determined by three microscopic parameters [15, 16]: (I) the atomic SOC, (II) transfer integrals between orbitals with dierent parity, and (III) local hybridizations between orbitals with dierent parity. Note that, while (I) and (II) exist even in the centrosymmetric systems, (III) vanishes when the atomic site is an inversion center. In order to realize a gigantic ASOC for applications to electronics and spintronics, it is desired to enhance these three parameters. However, it is dicult to control them exibly because such microscopic parameters are intrinsically determined by the lattice structures and the atomic orbitals. In the present study, we theoretically explore the possibility of controlling the ASOC by spontaneous symmetry breaking in the electronic degrees of freedom. There are two key ingredients, in addition to (I)-(III) above: one is a centrosymmetric lattice structureEmergent spin-valley-orbital physics by spontaneous parity breaking 3 with local asymmetry, and the other is an electronic order which breaks the inversion symmetry spontaneously, as we detail below. In the rst place, we consider a class of centrosymmetric lattices whose inversion centers are located not at the lattice sites but at osite positions [17, 18, 16]. A schematic example is shown in gure 1, in comparison with the noncentrosymmetric case. In the gures, the upward and downward triangles, whose vertices represent the lattice sites, are placed periodically in a dierent manner. Figure 1(a) represents an example of noncentrosymmetric lattice structures; there is no inversion center in the system. On the other hand, in gure 1(b), while the system possesses the inversion symmetry at some intersite positions as displayed by the red spheres, there are no inversion center at every lattice site. This type of local asymmetry is found in many lattice structures with the sublattice degree of freedom, for instance, zigzag chain, honeycomb, and diamond lattices. Interestingly, in these lattices, the local asymmetry generates a potential gradient at each lattice site, rVs, which depends on the sublattice s. This leads to an ASOC in the sublattice-dependent form, whose Hamiltonian is given by HASOC(k) =X sgs(k)
X s(krVs)
: (1.2) This is unchanged for the spatial inversion operation with respect to the intersite inversion centers, k$kandrVs$rVs0=rVs(s6=s0). Therefore, in contrast to the ASOC in noncentrosymmetric systems, this type of sublattice-dependent ASOC does not contribute to bulk properties, such as spin splitting in the band structure and the magnetoelectric eect. Thus, equation (1.2) can be regarded as the \hidden" ASOC: the system retains ASOC but the net component is zero because of the cancellation between the sublattices. The second key ingredient, the spontaneous inversion symmetry breaking by electronic orders, plays a crucial role in activating the \hidden" ASOC. When the electronic order occurs with breaking the inversion symmetry, it induces a net component of ASOC. Such an emergent ASOC by spontaneous parity breaking has recently been attracted interest as it brings new aspects in the physics of SOC [19, 18, 16, 20, 21]. This type of ASOC has exible controllability through the phase transition in the electronic degree of freedom. In other words, the ASOC can be varied, or even switched on and o, by external parameters, such as pressure, electric and magnetic elds. In addition, the form of the ASOC, i.e., the kdependence and amplitude, which is usually determined by the lattice and band structures, is also controllable through the spontaneous electronic orders. Furthermore, the emergent ASOC gives rise to intriguing electronic structures and transport properties. This is due to the fact that the spontaneous electronic orders in the presence of the \hidden" ASOC simultaneously accompany multipoles with an odd parity, such as magnetic quadrupole, electric octupole [22], and magnetic toroidal moments [18, 16, 20, 23, 24, 25]. Such odd-parity multipoles bring about a peculiar modulation of the electronic structures and unconventional o-diagonal responses [26, 27, 28]. For example, a staggered antiferromagnetic (AFM) order onEmergent spin-valley-orbital physics by spontaneous parity breaking 4 the zigzag chain, which accompanies a ferroic toroidal order (see section 2), modies the electronic band structure in an asymmetric way in the momentum space, and results in unusual o-diagonal responses including the magnetoelectric eect [18, 16, 24] and asymmetric modulation of collective spin-wave excitations [29]. This indicates that the odd-parity multipoles open the further possibility of enriching the spin-orbital entangled phenomena. In the previous work [20], the authors addressed the issue of emergent ASOC by analyzing a minimal microscopic model on a honeycomb lattice. The eects of various charge, spin, orbital, and spin-orbital orders that break spatial inversion symmetry spontaneously were studied by the symmetry analysis as well as the mean-eld approximations. In this paper, we push forward this issue in a more general framework. Specically, we derive the explicit form of the ASOC for all the possible symmetry breakings including charge, spin, orbital, and spin-orbital channels in the same minimal model. We also provide a comprehensive survey of o-diagonal responses as well as the modulations of electronic structures. These analyses will clarify how the emergent ASOC gives rise to fascinating properties, such as the spin and valley splitting in the band structure, asymmetric band modulation with a band bottom shift, and peculiar o-diagonal responses, e.g., the spin and valley Hall eects and magnetoelectric eects. The results provide a comprehensive reference for further exploration of the ASOC physics, as our method presented here is straightforwardly applicable to other systems with more realistic electronic and lattice structures. The rest of the paper is organized as follows. In section 2, we describe how the ASOC is hidden in centrosymmetric systems with local asymmetry and how it is activated by conventional electronic orderings. We present several examples of odd- parity multipoles induced by the electronic orders, including the magnetic toroidal multipoles. In section 3, we introduce a minimal tight-binding model on the honeycomb lattice including both the atomic SOC and electron correlations. We present all the possible staggered orders that break spatial inversion symmetry, and categorize them into seven classes from the viewpoint of the symmetry [20]. In section 4, we investigate the explicit form of the eective ASOC resulting from the spontaneous electronic ordering. We provide the complete table of the emergent ASOC for all seven classes. Then, we examine the nature of the ordered states in seven classes as well as the paramagnetic state in detail, focusing on the spin and valley splitting in the band structure, band deformation with a band bottom shift, spin and valley Hall eects in section 5, and their o-diagonal responses to an electric current which include magnetoelectric eects in section 6. Finally, section 7 is devoted to summary and perspectives for future study. In Appendix A, we discuss the derivation of eective ASOC. We also describe the eect of ASOC on the band structure from the viewpoint of the eigenvalue analysis in Appendix B.Emergent spin-valley-orbital physics by spontaneous parity breaking 5 local asymmetry × charge order → ferroelectric order (a) local asymmetry × antiferromagenetic order → toroidal and quadrupole orders (b)y x zA B TQ T P y x zy x z Figure 2. Schematic pictures for (a) charge ordering on the zigzag chain, which leads to a ferroelectric order (uniform alignment of electric dipoles Pin theydirection), and (b) antiferromagnetic ordering with aligning the magnetic moments along the zdirection, which induces a ferroic order of toroidal moments Tin thexdirection and magnetic quadrupoles Q; see the bottom panel. The red and blue spheres in (a) represent the local charge densities, while the arrows in (b) the local magnetic moments. 2. Hidden antisymmetric spin-orbit coupling and odd-parity multipoles In this section, we describe how odd-parity multipoles are induced by the spontaneous electronic ordering on centrosymmetric lattices with local asymmetry. Let us consider an example of the one-dimensional zigzag chain, as shown in gure 2. By taking the x axis in the chain direction and the zaxis in the perpendicular direction to the plane on which the zigzag chain lies, the potential gradient (local electric eld) appears in the ydirection with alternating sign between the sublattices A and B due to the lack of the inversion symmetry at each lattice site. Therefore, taking kkxandrVskyin equation (1.2), we obtain the ASOC with gs(k)/(0;0;kx@Vs=@y). Thus,gs(k) has theEmergent spin-valley-orbital physics by spontaneous parity breaking 6 opposite sign for A and B sublattices and it is canceled out in the whole system. This is an example of the \hidden" ASOC mentioned in the previous section. In the presence of such hidden ASOC, spontaneous symmetry breaking by electron correlations can activate the odd-parity multipoles, such as magnetic quadrupole, electric dipole, and toroidal dipole. For example, when a charge ordering occurs on the zigzag chain as nA6=nB[nA(B)is the electronic density in the A (B) sublattice], a uniform electric polarization is induced in the ydirection,Py/(nAnB)j@Vs=@yj[gure 2(a)]. This is regarded as the ferroelectric ordering resulting from the uniform alignment of electric dipoles. Meanwhile, when we consider a staggered collinear magnetic ordering in thezdirection given by mz A=mz B[mz A(B)is the magnetic moment in the zdirection in the A (B) sublattice], the magnetic toroidal moment is induced, together with the magnetic quadrupole moment [gure 2(b)]. Here, the toroidal moment tis dened by t/P i(riSi), whereriis the position vector from the inversion center to the lattice siteiandSiis the magnetic moment at site i[27, 28]. By using this expression, we nd that the toroidal moment is induced in the xdirection:Tx/(mz Amz B)j@Vs=@yj. As easily imagined from the above examples, a variety of multipoles can be realized according to the lattice structures and types of electronic orders. We show representative examples in gure 3. The second and third columns represent the electric and magnetic multipoles with the rank lin the rst column, respectively. The last column shows the magnetic toroidal multipoles, which appear in the multipole expansion of a vector potential [27, 28, 16]. In the table, the blue boxes represent the multipoles with even parity. These are conventional multipoles, which appear even in the presence of spatial inversion symmetry at the lattice sites. The well-known examples of the even-parity multipoles are the electric charge (monopole) ( l= 0), magnetic dipole ( l= 1), and electric quadrupole (l= 2). In these examples, the multipoles are dened at each lattice site, and they are aligned in a staggered way, as shown in gure 3. In other words, the minimal unit of multipoles is a single lattice site for these cases. On the other hand, the multipoles can be dened as spatially extended objects over several lattice sites, as described above. The toroidal quadrupole ( l= 2) and magnetic octupole ( l= 3) in gure 3 are such examples of even-parity multipoles. In the former, the toroidal dipole moments Tare aligned in the lattice plane to form quadrupoles, as shown in the gure. Meanwhile, in the latter, the all-in/all-out ordering of magnetic moments on the pyrochlore lattice is regarded as a magnetic octupole order from the symmetry point of view [30, 31, 32]. We note that this is also interpreted as an antiferroic monopole ordering on the cubic lattice (the unit is a pair of upward and downward tetrahedrons), as shown in the inset z. On the other hand, the red boxes represent rather unconventional odd-parity multipoles, which appear only when the spatial inversion symmetry is broken at the lattice sites. It is worth noting that the odd-parity multipoles are realized by conventional charge and magnetic orderings on the centrosymmetric lattices with local zIn the similar way, the staggered charge ordering on the square lattice (as shown in l= 0 electric monopole) is regarded as the electric quadrupole ordering in the unit of four sublattices.Emergent spin-valley-orbital physics by spontaneous parity breaking 7 electric magnetic toroidal in spin ice magnetic excitationsin spin ice (quadrupole) (octupole)(monopole) (dipole)rank CO on square z-AFM on square CO on zigzag z-AFM on zigzag z-FM on theAFOO on square CO on honeycomb z-AFM on honeycomb all-in/all-out orderz-AFM on zigzag + ʵ ʵʵ ʵ++ + P T P T T in spin all P Shastry-Sutherland lattice Figure 3. Examples of the electric, magnetic, and toroidal magnetic multipoles induced by spontaneous electronic ordering, up to the rank l= 3. The red (blue) boxes stand for multipoles with odd (even) parity. CO, FM, AFM, and AFOO represents the charge ordered, ferromagnetic, antiferromagnetic, and antiferroorbital ordered states, respectively. The prex zindicates that the magnetic moments are along the z direction (perpendicular to the lattice plane). The size of the spheres re ects the magnitude of local charge densities, and the arrows represent local magnetic moments. The spheres in the tetrahedron in the magnetic multipoles with l= 0 andl= 3 represent the magnetic monopoles and antimonopoles. The toroidal dipoles Tand electric dipoles Pare also shown in the gure. The cubic in the box of the l= 3 magnetic multipole represents the schematic arrangement of the magnetic monopoles (+) and antimonopoles ( ) in a staggered way. See the main text for details.Emergent spin-valley-orbital physics by spontaneous parity breaking 8 asymmetry, except for the magnetic monopole ( l= 0)x. We have already described the cases in the electric dipole ( l= 1), toroidal dipole ( l= 1), and magnetic quadrupole (l= 2) by taking the example of the zigzag chain in gure 2. The odd-parity multipoles on the zigzag chain are naturally extended to those on the honeycomb lattice, since the honeycomb lattice is constructed from the superposition of the zigzag chains connected by threefold rotational symmetry; for instance, the superposition of the toroidal (electric) dipoles connected by threefold rotational symmetry results in the the toroidal (electric) octupole ( l= 3), as shown in gure 3. Thus, the spontaneous symmetry breaking by conventional electronic ordering can induce unconventional odd-parity multipoles. This is unique nature of the centrosymmetric lattice systems with local asymmetry. Once these odd-parity multipoles are activated, new quantum states accompanying such as an antisymmetric spin splitting and o-diagonal responses, e.g., a magnetoelectric response, are expected. Furthermore, such o-diagonal responses are controllable through the phase transition in the electronic degrees of freedom. In the following sections, taking a two-orbital model on the honeycomb lattice as a fundamental example, we examine how the ASOC is generated by spontaneous inversion symmetry breaking, what types of odd-parity multipoles are activated, and how they in uence the electronic and transport properties. 3. Two-orbital model on a honeycomb lattice In this section, we present a model to investigate spin-orbital coupled phenomena induced by spontaneous electronic ordering. In section 3.1, we introduce the model Hamiltonian including charge, spin, and orbital degrees of freedom on a honeycomb lattice. We show how staggered-type electronic orders break symmetries in the honeycomb-lattice model in section 3.2. 3.1. Model Hamiltonian For studying the spontaneous breaking of spatial inversion symmetry by electron correlations, we consider a minimal model Hamiltonian on the honeycomb lattice [gure 4(a)]. We include the orbital degree of freedom for d-electron systems under a crystalline electric eld, such as the trigonal, trigonal prismatic, and square antiprismatic conguration of the ligands, which splits the atomic energy levels by the magnetic quantum number m=2 (dx2y2anddxyorbitals),m=1 (dzxanddyz), andm= 0 (dz2), as schematically shown in gure 4(b). Among them, we take into account only two orbitals with m=1k. The Hamiltonian H, which we consider in the present study, consists of the one- body partH0and the two-body part H1: H=H0+H1; (3.1) xThe magnetic monopole corresponds to a magnetic excitation in spin ice [33, 30, 31]. kThe following results are straightforwardly generalized for the m=2 case.Emergent spin-valley-orbital physics by spontaneous parity breaking 9 AB xy(a)(b)CEF SOCz Figure 4. (a) Schematic picture of a honeycomb lattice; the primitive translational vectors are taken as a1= (p 3=2;1=2)aanda2= (p 3=2;1=2)a, where the lattice spacing is given by a=p 3 and we set a= 1 throughout this paper. A and B represent the sublattices. (b) Schematic picture of the atomic energy levels of dorbitals. We focus on four orbitals ( m;) =f(+1;");(1;#);(+1;#);(1;")gafter the level splitting by the trigonal crystalline electric eld (CEF) and the atomic SOC. H0=t0X km( 0;kcy AkmcBkm+ H:c:)t1X km( m;kcy AkmcBkm+ H:c:) + 2X imcy im(m)cim; (3.2) H1=X i0X mnm0n0Umnm0n0 2cy imcy in0cin00cim0+X hi;ji0X mm0Vnimnjm00;(3.3) wherecy skm(cskm) is the creation (annihilation) operator for sublattice s= A or B, wave vectork, orbitalm=1 ( m=m), and spin ="or#;cy im(cim) is the real-space representation. The rst and second terms of H0in equation (3.2) represent the intra- and inter- orbital hoppings between nearest-neighbor sites, respectively. The sum of kis taken over the folded Brillouin zone throughout this paper. The kdependence in the hopping terms is given by n;k=3X j=1!(j1)neik
j=  n;k(n= 0;1); (3.4) where!=e2i=3;1= (a1a2)=3,2= (a1+ 2a2)=3, and3=(2a1+a2)=3 [a1anda2are primitive translational vectors as shown in gure 4(a)]. The additional phase factors !nin equation (3.4) come from the angular-momentum transfers between dierent orbitals, which brings about the asymmetry with respect to kand results in peculiar o-diagonal responses in Sec. 6. The third term in equation (3.2) represents the atomic SOC. This term has a nonzero matrix element only for the diagonal component with respect to orbitals (the Ising type) as we consider only m=1. The SOC splits the atomic energy levels represented by ( m;) = (+1;"), (+1;#), (1;"), and (1;#) into two groups,Emergent spin-valley-orbital physics by spontaneous parity breaking 10 f(+1;");(1;#)gandf(+1;#);(1;")g, as shown in the rightmost panel in gure 4(b). Meanwhile, the rst term of H1in equation (3.3) stands for the on-site Coulomb interaction. We take Ummmm =U, andUmnmn =U0=U2JH, andUmnnm = Ummnn =JH(m6=n), whereU,U0, andJHare the intra-orbital repulsion, inter-orbital repulsion, and Hund's-rule coupling, respectively. The second term in equation (3.3) is the Coulomb repulsion between nearest-neighbor sites, which is introduced to discuss a charge order; here, nim=cy imcim. In the following sections, we set 2 t0= 1 as the energy unit. We dene the electron density asne=P skmhcy skmcskmi=(NsNk), whereNsis the number of sublattices and Nkis the number of grid points in the folded Brillouin zone. 3.2. Eect of spontaneous parity breaking We focus on four symmetries in the noninteracting Hamiltonian H0in equation (3.2) [20]: spatial inversion (P), time-reversal (T), 2=3 rotation around the zaxis (R), and mirror for thexzplane (M). These symmetry operations are represented by using three Pauli matrices,for sublattice, for spin, and for orbital indices, as P:x;T: iyxK;R:Oe2iz=3;M: iz; (3.5) whereKis a complex conjugation operator and Ois the cyclic permutation operator of the site indices around the rotation center. We note that Otransforms as O n;k= !n n;k. The orbital operators, x,y, andz, correspond to the electric quadrupoles, l2 xl2 y,lxly+lylx, and the magnetic dipole lz, respectively, in the m=1 subspace. Thus, the former two are time-reversal even, and the latter one is time-reversal odd. The transformation properties of relevant operators and quantities are summarized in table 1. These symmetries are spontaneously broken once a phase transition is caused by electron correlations. We here consider only the staggered electronic orders with an alternative sign between the A and B sublattices on the bipartite honeycomb lattice, as they are the simplest realizations of the breaking of Psymmetry (spontaneous parity breaking). The staggered orders commonly include the component of zwith the ordering wave vector Q= (0;0) (for simplicity, we omit the orderings represented by xandy). There are sixteen candidates for such staggered orders, which are denoted as(; = 0;x;y;z ): a charge order 00(CO), three spin orders 0(-SO), three orbital orders 0(-OO), and nine spin-orbital orders (-SOO). Here, ;=x;y;z , and0and0are 22 unit matrices. The symmetry-breaking elds corresponding to these orders are obtained in a general form through the mean-eld decoupling of two-body interactions in equation (3.3): ~H1=hX sk0X mm0cy skm(s)]0 mm0cskm00; (3.6) wherehis the magnitude of the symmetry-breaking elds and p(s) = +1 (1) fors= A (B). In the following sections, we deal with symmetry-breaking elds in equation (3.6) instead of the two-body Hamiltonian in equation (3.3).Emergent spin-valley-orbital physics by spontaneous parity breaking 11 Table 1. The transformation of relevant operators, matrix elements, and irreducible functions appearing in the eective ASOC in the following sections. Rtransforms the two-dimensional representation like r0=Rz(2=3)r, whereRz(2=3) is 2=3 rotation matrix around the zaxis and two-dimensional vector r= (x;y).fE(k) = fE1(k)ifE2(k). P(x)T(iyxK)R(Oe2iz=3)M(iz) note x xx xx y yy yy z zz z z x xx 0 x x l2 xl2 y y yy 0 y y lxly+lylx z zz z z lz   !1 =xiy x xx x x yyy y y zzz z z     =xiy n;k  n;k n;k!n n;k n;k n;k=  n;k fA(k)fA(k)fA(k)fA(k)fA(k)1 16ky(3k2 xk2 y) fork!0 fE(k)fE(k)fE(k)!1fE(k)fE(k)3i 2(kx+ iky) fork!0 Table 2. Eight symmetry classes of the paramagnetic state and sixteen staggered ordered states categorized in terms of the presence ( ) or absence () of the four symmetries of the system: spatial inversion ( P), time-reversal ( T), 2=3 rotation around the zaxis (R), and mirror symmetry for the xzplane (M) [20]. PM stands for the paramagnetic state. CO, SO, OO, and SOO represent charge, spin, orbital, and spin-orbital orders, respectively, and the prexes like zdenote the type of orders. In the columns for SS, VS, and BD, the checkmark ( X) shows that the spin splitting, valley splitting, and band deformation with a band bottom shift in the electronic structure takes place under the corresponding order, respectively. The columns for sME(u), sME(s), oME(u), and oME(s) indicate the magnetoelectric responses; the prex s and o represent spin and orbital, respectively, and the u and s in the parentheses denote the uniform and staggered component, respectively. See the main text for details. # P T R M SS VS BD sME(u) sME(s) oME(u) oME(s) 0 PM { { { { { { { 1 CO, zz-SOO X { { { { { { 2x=y-OO  X { { { X X X 3xz=yz -SOO  X { { { { { { 4z-SO,z-OO  {X { { { { { 5zx=zy -SOO   { { X { X X X 6x=y-SO   { { { { { { { 7xx=yy=xy=yx -SOO    { { { X X { X The sixteen staggered orders break the symmetries T,R, andMin a dierent way. We categorize them into seven classes with respect to the symmetries, as summarized in table 2 [20]. This symmetry analysis will provide a useful reference for discussing theEmergent spin-valley-orbital physics by spontaneous parity breaking 12 ASOC induced by each electronic ordering, as described in section 4. It is also helpful for understanding of microscopic and macroscopic physical properties, such as the electronic band structure (section 5) and o-diagonal responses including magnetoelectric eects (section 6). 4. Antisymmetric Spin-Orbit Coupling Induced by Electronic Ordering In this section, analyzing the Hamiltonian H0+~H1, we show how the eective ASOC is activated by spontaneous symmetry breaking. In section 4.1, we discuss the eective ASOC already existing in the paramagnetic state, which is hidden in the sublattice- dependent form. In section 4.2, we show the explicit form of the ASOC induced by symmetry breaking for several examples of the electronic orders. We summarize the ASOCs for all the sixteen possible orders in section 4.3. 4.1. Paramagnetic state First, we consider the ASOC in the paramagnetic state by taking into account only H0 in equation (3.2). In the paramagnetic state, the ASOC is hidden in the sublattice- dependent form, as discussed in section 2. One way to obtain the explicit form of this hidden ASOC is to treat the electron transfers between dierent sublattices as the perturbation. The details of the perturbative calculations are described in Appendix A, and we quote the general result of the eective ASOC in equation (A.6): He(k) =X [gu (k)0+gs (k)z]; where the term proportional to 0(z) represents the uniform (staggered) eective ASOC. As shown in Appendix A, we nd that there are three types of the ASOC in the paramagnetic state, which are represented by gs z0(k)zz04p 3t2 1 fA(k)zz0; (4.1) gs zx(k)zzx4p 3t0t1 fE1(k)zzx; (4.2) gs zy(k)zzy4p 3t0t1 fE2(k)zzy; (4.3) wherefA(k),fE1(k), andfE2(k) are given by fA(k) =" cos p 3kx 2! cosky 2# sinky 2 ; (4.4a) fE1(k) =" cos p 3kx 2! + 2 cosky 2# sinky 2 ; (4.4b) fE2(k) =p 3 sin p 3kx 2! cosky 2 ; (4.4c)Emergent spin-valley-orbital physics by spontaneous parity breaking 13 respectively. Note that fA(k) [fE1(k) andfE2(k)] and0(xandy) belongs to the A (E) irreducible representation of C3group, andffE1(k)g2+ffE2(k)g2has sixfold rotational symmetry. Equations (4.1)-(4.3) indicate that these ASOCs are hidden in the sublattice-dependent form and canceled out within the unit cell [ fA;E1;E2(k)zis invariant underP]. The ASOC in equation (4.1) has the similar form to the conventional one which has been discussed in topological insulators [34]. The threefold rotational symmetry in thekdependence in equation (4.4 a) re ects the symmetry of the honeycomb lattice. In fact, the asymptotic form of fA(k) in the limit of k!0 is obtained as fA(k)!1 16ky(3k2 xk2 y); (4.5) which is compatible with the threefold rotational symmetry. Owing to the ASOC, the paramagnetic state in our two-orbital Hubbard model shows the quantum spin Hall eect, as discussed in [35]. The form of ASOC in equation (4.1) is similar to the eective single-orbital Hubbard model discussed in [34]. However, our two-orbital model exhibits richer behavior in the quantum spin Hall eect than the single-orbital one: our model exhibits several quantized values of the spin Hall conductivity depending on t1and, at not only 1/2 lling but also 1/4 lling [35]. On the other hand, equations (4.2) and (4.3) represent dierent types of the ASOC: they include not only the spin component but also the orbital component, xory. As described in section 6, they give rise to the coupling between the order parameters and an external electric current. The ASOCs in equations (4.2) and (4.3) have k-linear contributions in the k!0 limit, as fE1;E2(k)!3 2ky;x: (4.6) However, this does not mean that these ASOCs break the threefold rotational symmetry because the net ASOC is proportional to the linear combination of the two contributions as [fE1(k)x+fE2(k)y]zz, which commutes with the threefold rotational operation, R. Once the threefold rotational symmetry is broken, fE1(k) andfE2(k) become unbalanced, giving rise to linear magnetoelectric couplings, as will be discussed in later sections. We summarize the result of the hidden ASOC in the paramagnetic state in table 3 for comparison with the symmetry broken cases discussed in the following sections. Table 3. Staggered ASOCs, gs (k) (; = 0;x;y;z ), in the paramagnetic state. In the table, A, E 1, and E 2represent the coecient of the ASOC as fA(k),fE1(k), andfE2(k), respectively. In the presence of threefold rotational symmetry, fE1andfE2 appear as the same weight in the ASOC, as they constitute two-dimensional irreducible representation of C3group. See also table 4 for the ordered states. #gs 00gs x0gs y0gs z0gs 0xgs 0ygs 0zgs xxgs xygs xzgs yxgs yygs yzgs zxgs zygs zz 0 PM { { { A { { { { { { { { { E 1E2{Emergent spin-valley-orbital physics by spontaneous parity breaking 14 4.2. Ordered states Next, we discuss the ASOC additionally induced by spontaneous electronic orders, taking into account the symmetry-breaking term ~H1in equation (3.6) in addition to H0in equation (3.2). When an electronic order breaks spatial inversion symmetry, the ASOC acquires a spatially uniform component. In this section, we discuss the explicit form of such emergent ASOC by taking three typical examples out of the sixteen ordered states: CO (section 4.2.1), zx-SOO (section 4.2.2), and xx-SOO (section 4.2.3). The detailed derivation is given in Appendix A. We note that staggered components are also induced, but they have the same functional forms as those in the paramagnetic state listed in table 3; hence, we do not discuss them here. The summary including all the other cases will be presented in section 4.3. 4.2.1. CO (class #1) In the CO state belonging to the class #1, ~H1is proportional to z00. Under this parity breaking order, an additional ASOC is induced in the form of gu 0z(k)00z/ht2 1 2fA(k)00z: (4.7) See equation (A.8) for the derivation. The ASOC is spatially uniform, as it is proportional to 0. Furthermore, the uniform ASOC is proportional to h, which indicates that it is induced by the spontaneous electronic ordering. The kdependence of the ASOC preserves the threefold rotational symmetry, as discussed in equation (4.5). This is consistent with the fact that the CO state does not break the threefold rotational symmetry, as shown in table 2. From the viewpoint of symmetry, the eective ASOC in equation (4.7) preserves time-reversal symmetry as fA(k)zis invariant under the time-reversal operation, while it breaks inversion symmetry. This is similar to the Rashba and Dresselhaus SOCs [3, 4, 6, 7]. Indeed, it leads to the spin splitting in the band structure, which is similar to the systems with these SOCs, as discussed in section 5.2. 4.2.2.zx-SOO (class #5) Next, we discuss the case of the zx-SOO state in the class #5 ( ~H1/zzx). In this case, a uniform ASOC is induced in the form of gu zz(k)0zz/ht0t1 2fE1(k)0zz: (4.8) Thekdependence is dierent from that in the CO state in equation (4.7): the ASOC in thezx-SOO state has a linear contribution with respect to kasfE1(k)/kyin the limit ofk!0 [see equation (4.6)]. This is because the threefold rotational symmetry is broken by the zx-SOO, as shown in table 2. Such a linear term in the ASOC leads to the asymmetric band deformation with respect to k, as discussed in section 5.4. It also gives rise to the linear magnetoelectric eect, as discussed in section 6.2. From the symmetry point of view, the eective ASOC in equation (4.8) breaks time-reversal symmetry [ fE1(k)zzis time-reversal odd] as well as inversion symmetry. This indicates that the ASOC induced by zx-SOO is regarded as an eective \toroidal"Emergent spin-valley-orbital physics by spontaneous parity breaking 15 eld along the kydirection as the k-linear contribution of fE1(k),ky, is perpendicular to the spin moment, z. In other words, the zx-SOO accompanies a ferroic toroidal order in the kydirection. In this situation, there is no guarantee that kyandkyare equivalent, and hence, the asymmetric band deformation is allowed (see section 5.4). 4.2.3.xx-SOO (class #7) The last example discussed here is the xx-SOO state in the class #7 ( ~H1/zxx). The ordered state in this category breaks all four symmetries, as shown in table 2. The emergent ASOC in the xx-SOO state is also obtained through equation (A.8), whose explicit form is given by gu xz(k)0xz/ht0t1 2fE1(k)0xz: (4.9) Here, thekdependence of the ASOC is linear in kyin thek!0 limit, similar to that in the zx-SOO order, consistent with the breaking of threefold rotational symmetry in the xx-SOO. Furthermore, the ASOC breaks the mirror symmetry because equation (4.9) includes x, which does not commute with the mirror operator M= iz in equation (3.5); this is also consistent with the symmetry of the order parameter. This form of ASOC induces the uniform longitudinal magnetoelectric eect but no asymmetric band deformation since the k-linear contribution of fE1(k),ky, and the spin component, x, are in the xyplane, as discussed in section 6.1. Similarly, in the xy-SOO state, the transverse uniform magnetoelectric eect occurs owing to gu xz(k)/fE2(k) with an eective \toroidal" eld along the kzdirection. 4.3. Summary of Antisymmetric Spin-Orbit Coupling Table 4. Emergent uniform ASOCs, gu (k) (;= 0;x;y;z ), in the sixteen ordered states with parity breaking. The notations are common to table 3. # gu 00gu x0gu y0gu z0gu 0xgu 0ygu 0zgu xxgu xygu xzgu yxgu yygu yzgu zxgu zygu zz 1CO { { { { { { A { { { { { { { { { zz-SOO { { { A { { { { { { { { { E 1E2{ 2x-OO { { { { { { E 1{ { { { { { { { { y-OO { { { { { { E 2{ { { { { { { { { 3xz-SOO { A { { { { { E 1E2{ { { { { { { yz-SOO { { A { { { { { { { E 1E2{ { { { 4z-SO { { { { { { { { { { { { { { { A z-OO A { { { E 1E2{ { { { { { { { { { 5zx-SOO { { { { { { { { { { { { { { { E 1 zy-SOO { { { { { { { { { { { { { { { E 2 6x-SO { { { { { { { { { A { { { { { { y-SO { { { { { { { { { { { { A { { { 7xx-SOO { { { { { { { { { E 1{ { { { { { yy-SOO { { { { { { { { { { { { E 2{ { { xy-SOO { { { { { { { { { E 2{ { { { { { yx-SOO { { { { { { { { { { { { E 1{ { {Emergent spin-valley-orbital physics by spontaneous parity breaking 16 Performing similar analysis for other ordered states, we obtain the uniform ASOCs induced by each order. Table 4 summarizes the results, in which A, E 1, and E 2represent that the induced uniform ASOC has the kdependence of fA(k),fE1(k), andfE2(k), respectively. For example, in the case of CO in the class #1 described in section 4.2.1, the system exhibits the uniform ASOC given by gu 0z(k)00z/fA(k)00z, while the zx-SOO in the class #5 in section 4.2.2 induces gu zz(k)0zz/fE1(k)0zz. Table 4 shows that a variety of uniform ASOCs are induced by spontaneous parity breaking. They have dierent spin and orbital dependences according to the types of ordered phases, which are useful for understanding of peculiar electronic structures under each electronic order. Moreover, table 4 is also useful for understanding of the physical properties in each phase, such as the magnetoelectric eects, discussed in section 6. 5. Electronic Structure In this section, we show the electronic band structures of the paramagnetic and ordered states classied into eight classes #0-7 in tables 2, 3, and 4. We discuss the relationship between the ASOCs induced by electronic ordering and the band structures. After brie y introducing the typical band structure in the paramagnetic state (class #0) in section 5.1, we show the spin splitting in the classes #1, #2, and #3 (section 5.2), valley splitting in the class #4 (section 5.3), and asymmetric band deformation with a band bottom shift in the class #5 (section 5.4) in the electronic structures. We also present the complementary understanding of the peculiar band modulations from the eigenvalues of the Hamiltonian in Appendix B. Hereafter, we take t0= 0:5,t1= 0:5, and= 0:5 if not explicitly stated. 5.1. Band structure in the paramagnetic state (class #0) 0 -22 K’ K MK’K M(a) 0 kx0 π 2π -π -2π-π -2ππ2π ky (b)Energy Figure 5. (a) Electronic band structure for the paramagnetic state. (b) The energy contours slightly below the Fermi level at half lling ( E=0:35) corresponding to the dashed line in (a). The hexagon represents the Brillouin zone. Figure 5 shows the band structure in the paramagnetic state, calculated from theEmergent spin-valley-orbital physics by spontaneous parity breaking 17 noninteracting Hamiltonian H0in equation (3.2). In the paramagnetic state, there are four bands separated by energy gaps due to both the atomic SOC and inter-orbital hoppingt1. Each band is doubly degenerate as both spatial inversion and time-reversal symmetries are preserved. At the commensurate llings, ne= 1, 2, and 3, the system becomes insulating. These insulating states are topological insulators; the spin Hall conductivity [see equation (5.1)] is quantized at a nonzero integer value, as mentioned above [35]. This is due to the presence of the staggered ASOCs, gs z0(k),gs zx(k) and gs zy(k), in table 3. The energy contours slightly below the Fermi level at half lling is also shown in gure 5(b). Re ecting the presence of spatial inversion and threefold rotational symmetries, the six regions near the K and K' points in the Brillouin zone are all equivalent. Such degeneracy is lifted once the electronic ordering breaks the symmetries, as discussed in the following sections. 5.2. Spin splitting (class #1, #2, #3) In this section, we show that particular parity breaking orders split the electronic bands depending on the spin degree of freedom, called spin splitting. The spin splitting occurs in the classes #1, #2, and #3 (SS in table 2), but in a dierent form in each class. Note that the spin splitting occurs only in the presence of time-reversal symmetry. 5.2.1. Class #1 First, we discuss the electronic structure in the CO state classied into the class #1. Figure 6(a) shows a schematic picture of CO. The order parameter in the CO state breaks only spatial inversion symmetry, as shown in table 2. Figure 6(b) shows the band structure in the CO state calculated from the Hamiltonian H0+~H1at h= 0:05. The red (blue) curves denote the up(down)-spin component, which indicates that the spin splitting occurs in the CO state. The spin splitting is antisymmetric with respect to the wave vector k: the low energy states in both valence and conduction bands at half lling are predominantly spin-up polarized at the K' point, while they are spin-down polarized at the K point. This is also seen in the energy contours slightly below the Fermi level at half lling, as shown in gure 6(c). The antisymmetric spin splitting is large around the K and K' points, as shown in gure 6(b). This is consistent with the fact that the ASOC in equation (4.7) does not include the linear term with respect to the wave vector, but the third-order term [see also equation (4.5)]. It is also interesting to examine the electronic structure from the topological aspect. For this purpose, we compute the spin Hall conductivity by using the linear response theory: SH xy=e 21 iV0X mnkf("nk)f("mk) "nk"mkJ(s)nm x;kJmn y;k "nk"mk+ i; (5.1) whereV0is the system volume, f(") is the Fermi distribution function, and "mkand jmkiare the eigenvalue and eigenstate of H0+~H1.J(s)mn ;k =hmkjJ(s) jnkiis theEmergent spin-valley-orbital physics by spontaneous parity breaking 18 0 -22Energy(b) K’ K M 0 -2 2 Energy ) (c) K’K M 0 -22Energy(e) K’ K M 0 -2 2 Energy(f) 0 -22Energy(h) K’ K M0 kx0 π 2π -π -2π-π -2ππ2π ky 0 kx0 π 2π -π -2π-π -2ππ2π ky 0 kx0 π 2π -π -2π-π -2ππ2π ky (i) 2Class #1: CO Class #2: x-OO Class #3: xz-SOO (a) (d) (g) Figure 6. The schematic pictures of the charge, spin, and orbital order in (a) CO, (d)x-OO, and (g) xz-SOO phases are shown. In the picture for CO in (a), the size of the spheres re ects the magnitude of local charge densities. For x-OO in (d), the lobes represent the dx2y2orbitals. For xz-SOO in (g), the arrows on the spheres show the local magnetic moments along the xdirection and the circular arrows around the spheres represent the orbital currents. The xz-SOO state is given by the superposition of the two congurations. Electronic band structures of the mean-eld Hamiltonian H0+~H1for the (b) CO, (e) x-OO, and (h) xz-SOO states. The energy contours slightly below the Fermi level at half lling ( E=0:35) corresponding to the dashed lines in (b), (e), and (h) are shown in (c), (f), and (i), respectively. We take h= 0:05 (0:2) for (b) and (c) [(e), (f), (h), and (i)]. In (b), (c), (e), and (f), the red (blue) curves show the bands with up(down)-spin polarization in the zdirection, while those are for up(down)-spin polarization in the xdirection in (h) and (i). matrix element of the spin current operator, which is dened by J(s) =1 2fz;Jgin thedirection (Jis the current operator and f


g is an anticommutator). We set e=4= 1 (eis the elementary charge). Thus, SH xyrepresents the coecient for theEmergent spin-valley-orbital physics by spontaneous parity breaking 19 spin current in the xdirection induced by the electric current in the ydirection{. We take temperature T= 0:001 and the damping factor = 0:001. 0 1 2 0.00 0.25 0.50(a) -3-2-1 0 1 2 3 0.00 0.25 0.50(b)gap 0.00.20.40.6 Figure 7. (a)hdependence of the spin Hall conductivity at half lling in the CO state (left axis). The slight deviations from integer values are due to the nite-size eects. The energy gap is also shown (the right axis). (b) hdependences of the Hall conductivities at the K and K' points. Figure 7(a) shows the hdependence of the spin Hall conductivity in the CO state. Forh < = 2 = 0:25, the spin Hall conductivity SH xyis quantized at 2, indicating that the system is a topological insulator [35]. With increasing h, the band gap shrinks as shown in gures 7(a) [see also gure B1(a) in Appendix B], and it closes at the K and K' points at h==2, whereSH xychanges discontinuously from 2 to 0. For further increasingh, the gap opens again, but SH xyremains at 0, as shown in gure 7(a): the system is a trivial band insulator for h>= 2. The result indicates the transition from the topological insulator to a trivial band insulator at h==2. On the other hand, the CO state also exhibits the valley Hall eect, re ecting the emergence of the valley degree of freedom (inequivalence between the K and K' points) [36]. Figure 7(b) shows hdependences of the Hall coecients xyat the K and K' points. Here, xyis calculated from the correlation between the electric currents Jx andJywithin the linear response theory similar to equation (5.1): xy() =e2 ~1 iX mnf("n)f("m) "n"mJnm x;Jmn y; "n"m+ i; (5.2) where ~is the Dirac constant and stands for either the K or K' points. As shown in gure 7(b), the Hall conductivities at the K and K' points become nonzero with opposite signs due to the presence of time-reversal symmetry. They are critically enhanced with the inverse square of the energy gap while approaching the gap closing point at h==2 from both sides. The result indicates that we can obtain gigantic valley Hall responses by controlling the order parameter in the CO phase. {In the case of insulators, the electric current should be replaced with the electric eld. This is also the case for the o-diagonal responses in equation (6.1).Emergent spin-valley-orbital physics by spontaneous parity breaking 20 The antisymmetric spin splitting also occurs for the zz-SOO phase in the same class #1, but in a dierent form: the energy levels just above and below the Fermi level at half lling at the K point have opposite spin polarizations. The dierence is explained by the spin dependence of the emergent ASOCs for the CO and zz-SOO states; the former is proportional to 0zwhich aects the spin sector via the spin-orbit coupling /zz, whereas the latter is directly proportional to z0(gu z0), as shown in table 4. 5.2.2. Class #2 Next, we turn to the class #2. The x=y-OO states in this class are characterized by the simultaneous breaking of spatial inversion and rotational symmetries, as shown in table 2. In this situation, a uniform ASOC with k-linear contribution appears, as shown in table 4. In the x-OO case, whose schematic picture is shown in gure 6(d), we obtain the ASOC from equation (A.8) in the form of gu 0z(k)00z/ht0t1 2fE1(k)00z; (5.3) which is proportional to kyin the limit of k!0, as shown in equation (4.6). On the other hand, the ASOC for y-OO is given as gu 0z(k)00z/ht0t1 2fE2(k)00z; (5.4) which is proportional to kxin the limit of k!0. Thek-linear dependence of the emergent ASOC in the class #2 leads to a dierent type of spin splitting from the class #1. For instance, in the x-OO phase, the ky- linear contribution brings about an antisymmetric spin splitting occurs in a way that the bands with up(down)-spin polarization are shifted to the + ky(ky) direction. The typical band structure is shown in gure 6(e). Note that the band dispersion still satises the relation "(k) ="(k) due to the presence of time-reversal symmetry. As seen in the energy contours plotted in gure 6(f), however, the band structure is no longer symmetric with respect to the threefold rotation. Another distinct feature from the class #1 is the kdependence of the magnitude of the antisymmetric spin splitting. The spin splitting for the present x-OO order takes place predominantly around the point rather than the K and K' points as shown in gure 6(e), whereas that for the CO appears conspicuously around the K and K' points as shown in gure 6(b). This is because the lowest-order ASOC is linear in kin the class #2. The situation is similar to the other state in this class #2, the y-OO phase, while the bands are split in the kx direction re ecting the kx-linear contribution in the ASOC in equation (5.4). 5.2.3. Class #3 Finally, let us discuss the class #3. The order parameter in this class breaks spatial inversion and mirror symmetries but it retains rotational symmetry, as shown in table 2. In this case also, the band structure exhibits spin splitting: gures 6(h) and 6(i) show the typical examples under xz-SOO whose schematic picture is shown in gure 6(g). However, the spin polarization is not along the zaxis but in the xyplane. The spin splitting is understood from the form of the induced ASOC. For instance, inEmergent spin-valley-orbital physics by spontaneous parity breaking 21 thexz-SOO state, according to table 4, the induced ASOCs are summarized into the form, [c1fA(k)0+c2ffE1(k)x+fE2(k)yg]x0: (5.5) The form of emergent ASOC is similar to that in the zz-SOO in the class #1, while the spin component is xinstead ofz. Thus, the ASOC induced by the xz-SOO state leads to the spin splitting with the spin quantization axis along the xdirection, as shown in gures 6(h) and 6(i). Similarly, the spin splitting along the ydirection is obtained for theyz-SOO phase in the same class. 5.3. Valley splitting (class #4) Energy0 -22 K’ K MK’K M0 kx0 π 2π -π -2π-π -2ππ2π ky (b) (c) Class #4: z-SO (a) K’ K M 0 2 2 Figure 8. (a) The schematic picture for z-SO is shown. (b) Electronic band structure of the mean-eld Hamiltonian H0+~H1for thez-SO state at h= 0:05. (c) The energy contours slightly below the Fermi level at half lling ( E=0:35) corresponding to the dashed line in (b). In this section, we show that the parity breaking order in the class #4 leads to a dierent type of modulation of the band structure. In this case, the spin splitting is absent, but the bands are modulated in a dierent way between the K and K' points. This is called valley splitting (VS in table 2). Thez-SO andz-OO belonging to the class #4 both break spatial inversion and time- reversal symmetries simultaneously, as shown in table 2. Hence, there is no guarantee that the energy eigenvalue at kis degenerate with that at k, as in the toroidal ordered state [18, 16]. In fact, these orders can be regarded as toroidal octupole orders (see section 2), and the lowest-order contribution of the emergent ASOC is of third order with respect to k(see table 4)+. Specically, in the z-SO phase in gure 8(a), the band structure is modulated as shown in gure 8(b): at half lling, the gap at the K' point becomes smaller than that at the K point. Correspondingly, the hole pockets at the K' point are larger than those at the K points, as shown in the energy contours in gure 8(c). Thus, the z-SO leads +The ASOCs with k-linear component do not break the rotational symmetry of the system because the net component given by their linear combination preserves the rotational symmetry, similar to the discussion in Sec. 4.1.Emergent spin-valley-orbital physics by spontaneous parity breaking 22 to valley splitting in the band structure, as pointed out in [37]. This valley splitting is induced by the ASOC generated by the z-SO, which is represented by gu zz(k)0zz/ht2 1 2fA(k)0zz: (5.6) The form of the ASOC is similar to that in the CO state in equation (4.7); the dierence is in the spin component, 0!z, re ecting the time-reversal symmetry breaking. The valley splitting is understood by the appearance of fA(k) due to the rotational symmetry and the fact that fA(k)zzis invariant under simultaneous transformations ofPT(see also Appendix B). Furthermore, the spin Hall conductivity also shows the similar behavior to that in the class #1, as shown in gure 7(a): the similar topological transition takes place at h==2. The valley splitting is also seen in the z-OO phase in the same class, but in a dierent way from the z-SO case. In the z-OO phase, for example, the energy levels just above and below the Fermi level at half lling are shifted downward (upward) at the K' (K) point. This is due to the spin and orbital dependence of the emergent ASOC, gu 00(k)000, as listed in table 4. 5.4. Asymmetric band deformation (class #5) 0 -22Energy K’ K MK’K M0 kx0 π 2π -π -2π-π -2ππ2π ky (b) (c) Class #5: zx-SOO (a) K’ K M 0 2 2 Figure 9. (a) The schematic picture for zx-SOO is shown. (b) Electronic band structure of the mean-eld Hamiltonian H0+~H1for thezx-SOO state at h= 0:3. (c) The energy contours slightly below the Fermi level at half lling ( E=0:35) corresponding to the dashed line in (b). Finally, we discuss the electronic band structure in the presence of parity breaking orders classied in the class #5, which break spatial inversion, time-reversal, and rotational symmetries, as shown in table 2. In this case, for instance, in the zx-SOO state schematically shown in gure 9(a), the emergent ASOC includes the ky-linear contribution as inferred by equation (4.8), similar to the case of the x-OO state in the class #2 [equation (5.3)]. Thus, we expect the asymmetric band deformation in the ky direction. The dierence between the two classes is the spin dependence: the former includesz, while the latter 0. This leads to the dierent type of the asymmetric band deformation in the class #5, as demonstrated in gures 9(b) and 9(c): the bands areEmergent spin-valley-orbital physics by spontaneous parity breaking 23 modulated with a band bottom shift, with retaining the spin degeneracy as fE1(k)zz is invariant under PT. The band deformation is similar to the ferroic toroidal ordered cases discussed in [18, 16, 24]. In the case of the zy-SOO phase in the same class #5, a similar band deformation takes place with the band bottom shift to the kxdirection because of the induced ASOC, gu zz(k)/kx. 6. O-diagonal Responses In this section, we discuss the o-diagonal responses of the paramagnetic and ordered states classied into eight classes. Specically, we show the spatially uniform and staggered moments induced by an electric current in sections 6.1 and 6.2, respectively. We use the linear response theory by computing the tensors between order parameters and electric current in each ordered state: KT =2 iV0X mnkf("nk)f("mk) "nk"mkmnm T;kJmn ;k "nk"mk+ i; (6.1) wheremnm T;k=hnkjT jmki[T = u (u=0) or s (s=z)]. Thus,Ku (Ks ) represents the coecient for the uniform (staggered) order parameter /induced by the electric current in the direction. In equation (6.1), we set gBe=(2h) = 1 (gis theg-factor and Bthe Bohr magneton). 6.1. Uniform response to electric current 0 4 8 12 0 1 2 3 4 0 1 2 3 0.0 0.5 1.0 1.5 2.0 0 1 2 3 0.0 0.5 1.0 1.5 2.0(b) (c) Figure 10. (a) The coecients of magnetization-current correlations, Ku x0x,Ku y0y, andKs z0y, as functions of the electron density nein thexx-SOO (class #7) phases at h= 0:2, temperature T= 0:01, and the damping factor = 0:01. (b)hdependences ofKu x0x,Ku y0y, andKs z0yin thexx-SOO phase at ne= 0:1. (c)dependences of Ku x0x, Ku y0y, andKs z0yin thexx-SOO phase at ne= 0:1 andh= 0:2. Let us rst discuss the uniform magnetoelectric responses, i.e., the uniform magnetic moments induced by the electric current, which are described by Ku 0: we consider mnm u0;kwith=x;y;z in equation (6.1). For instance, Ku x0xis the coecient for the uniform magnetization in the xdirection induced by the electric current in the xEmergent spin-valley-orbital physics by spontaneous parity breaking 24 Table 5. Uniform o-diagonal responses in paramagnetic and sixteen staggered ordered states. C, S , O, and SO (; =x;y;z ) represent the charge, spin, orbital, and spin-orbital orders, corresponding to the coecients Ku 00,Ku 0,Ku 0, andKu , respectively. =x;yin the table represents that the coecient becomes nonzero for the electric current along the direction. The superscriptindicates the nonzero response even in the absence of SOC, = 0. # C S xSySzOxOyOzSOxxSOxySOxzSOyxSOyySOyzSOzxSOzySOzz 0 PM { { { { { { { { { { { { { { { { 1CO { { { { xy{ { { { { { { y x { zz-SOO { { { { x y { { { { { { { yx{ 2x-OO { { { { xyy{ { { { { { y x x y-OO { { { { yxx{ { { { { { x y y 3xz-SOO { { { { { { { yx{x y { { { { yz-SOO { { { { { { { x y {yx{ { { { 4z-SO { { { { y x { { { { { { { xy{ z-OO { { { { yx{ { { { { { { x y { 5zx-SOO { { { { y x x { { { { { { xyy zy-SOO { { { { x y y { { { { { { yxx 6x-SO { { { { { { { xy{y x { { { { y-SO { { { { { { { y x {xy{ { { { 7xx-SOO { x y { { { { xyyy x x { { { yy-SOO { x y { { { { x y y yxx{ { { xy-SOO { y x { { { { yxxx y y { { { yx-SOO { y x { { { { y x x xyy{ { { direction. This is a longitudinal magnetoelectric eect. Remarkably, among sixteen possible electronic orders, only the SOO states belonging to the class #7 exhibit the uniform magnetoelectric eects [20] [sME(u) in table 2]. Each phase in the class #7 shows two components of Ku 0:Ku x0xandKu y0yforxx- andyy-SOO and Ku x0yand Ku y0xforxy- andyx-SOO, respectively. This result indicates that, in the present honeycomb-lattice model, the uniform magnetoelectric eect is observed when both threefold rotational and mirror symmetries are broken in addition to time-reversal symmetry. As a typical example, we show the results for the xx-SOO state. In this case, Ku y0y becomes nonzero in the entire region of ne, whileKu x0xbecomes zero only for ne= 2, as shown in gure 10(a). The result indicates that a uniform magnetization in the x (y) direction can be induced by an electric current in the x(y) direction under the xx-SOO.Ku x0xandKu y0ybecome nonzero even in the insulating cases at commensurate llingsne= 1, 2, and 3 (except for Ku x0xatne= 2), since the dominant contribution comes from the inter-band components in equation (6.1). Other phases in the class #7 also show similar behavior in the dierent components mentioned above. The staggered o-diagonal responses such as Ks z0yin gure 10 will be discussed in the next subsection. We also show handdependences of the uniform magnetic response to the electric current at ne= 0:1 in gures 10(b) and 10(c), respectively. Both curves show qualitatively similar behavior: for small hand, the uniform response increases linearlyEmergent spin-valley-orbital physics by spontaneous parity breaking 25 tohand, while it decreases for large handafter showing a broad peak. The decrease to zero ash!1 (!1 ) is explained by the asymptotic form of the emergent ASOC, which is approximated as h=(2+h2)/1=h(/h=2) [see also equation (4.9)]. Next, we discuss uniform responses in the orbital channel. The xandyorbital components, Ku 0xandKu 0y, represent the electric quadrupole responses to the electric current. We nd that the ordered states in the classes #1, #2, #4, and #5 show nonzero values of Ku 0xandKu 0y. Comparing with the emergent ASOC in table 4, we note that nonzero responses with Ku 0xandKu 0yare obtained in the presence of gu (k) (;= 0;z). On the other hand, we obtain nonzero Ku 0zin the classes #2 and #5 [oME(u) in table 2]. The zorbital component, Ku 0z, represents the magnetic dipole response, asz/lz. In this case, we nd that the breaking of threefold rotational symmetry is necessary for nonzero Ku 0z, in addition to the presence of the emergent ASOC /gu (k) (; = 0;z) (see tables 4 and 5). Interestingly, there are nonzero orbital-current responses in the ordered states belonging to the class #2 despite the presence of time- reversal symmetry. This is understood by decomposing the electric current operator J into two parts as J=@H @k=J(0) +J(1) ; (6.2) where J(0) =t0X km@ 0;k @kcy AkmcBkm+ H:c: ; (6.3) J(1) =t1X km@ m;k @kcy AkmcBkm+ H:c: : (6.4) The former J(0) represents the current originating from the intra-orbital hopping t0, while the latter J(1) from the inter-orbital t1. The latter is the orbital o-diagonal current carrying the orbital angular momentum z, which is proportional to xand y. Combining J(1) with the order parameters xandyin the class #2, we obtain a response in the zcomponent, i.e., the nonzero Ku 0z. In addition to the magnetic and orbital responses, we also have uniform responses in spin-orbital channels. All such uniform o-diagonal responses in the paramagnetic and sixteen staggered ordered states are summarized in table 5. We note that the responses become nonzero only when t16= 0, while some of them remain nonzero even for = 0 (indicated by the superscriptsin table 5). In the paramagnetic state, no response is induced by the electric current because there is no uniform ASOC. Meanwhile, each ordered phase shows some o-diagonal responses because of the uniform ASOC induced by the spontaneous parity breaking in table 4. For instance, in the x-OO phase in the class #2, when the electric current is applied in the xdirection, the uniform hxi,hzyi, andhzzimoments are induced, while hyi,hzi, andhzxiare induced when the electric current is applied in the ydirection. This indicates that the electric quadrupolesEmergent spin-valley-orbital physics by spontaneous parity breaking 26 l2 xl2 y(lxly+lylx) are detected in the electric current applied in the x(y) direction in thex-OO phase. 6.2. Staggered response to electric current Table 6. Table of staggered o-diagonal responses in paramagnetic and sixteen staggered ordered states. The notations are common to table 5. # C S xSySzOxOyOzSOxxSOxySOxzSOyxSOyySOyzSOzxSOzySOzz 0 PM { { { { xy{ { { { { { { y x { 1CO { { { { xy{ { { { { { { y x { zz-SOO { { { { xy{ { { { { { { y x { 2x-OOx{ {y xyy{ { { { { { y x x y-OOx{ {y xyy{ { { { { { y x x 3xz-SOO { { { { xy{ { { { { { { y x { yz-SOO { { { { xy{ { { { { { { y x { 4z-SO { { { { xy{ { { { { { { y x { z-OO { { { { xy{ { { { { { { y x { 5zx-SOOx{ {y xyy{ { { { { { y x x zy-SOOx{ {y xyy{ { { { { { y x x 6x-SO { { { { xy{ { { { { { { y x { y-SO { { { { xy{ { { { { { { y x { 7xx-SOOx{ {y xyy{ { { { { { y x x yy-SOOx{ {y xyy{ { { { { { y x x xy-SOOx{ {y xyy{ { { { { { y x x yx-SOOx{ {y xyy{ { { { { { y x x -2 0 2 4 0 1 2 3 4(a) 0 1 2 0.00 0.04 0.08 0.12 0.16 0.0 0.1 0.2 0.3 0.4 0.5 0.0 0.1 0.2(b) (c) Figure 11. (a) The coecient of magnetization-current correlation, Ks z0y, as function of the electron density nein thex-OO and (class #2) at h= 0:1, temperature T= 0:01, and the damping factor = 0:01. (b)hdependence of Ks 00xin thex-OO phase at ne= 0:1 and= 0:5. (c)dependence of Ks 00xin thex-OO phase at ne= 0:1 and h= 0:1. We turn to the staggered magnetoelectric eects. The staggered magnetoelectric responses were discussed for the centrosymmetric systems with local asymmetry, even in the paramagnetic state [18, 16]. In the present honeycomb-lattice model, however, the linear magnetoelectric eect does not appear as long as the threefold rotationalEmergent spin-valley-orbital physics by spontaneous parity breaking 27 symmetryRis preserved; once Ris broken by spontaneous electronic ordering, the ASOC is induced with k-linear contributions (see table 4), which gives rise to staggered linear magnetoelectric responses, as indicated by sME(s) in table 2. As a typical example, we examine the x-OO in the class #2. Figure 11(a) shows Ks z0yas a function of the electron density ne.Ks z0ybecomes nonzero in the entire region of ne, except when the system is insulating at integer ne. The result indicates that the staggered magnetic moment in the zdirection is induced by the electric current in the ydirection for thex-OO order (other staggered magnetic responses are all zero). This is because thex-OO breaks the rotational symmetry R, which activates the ASOC contributing to the linear magnetoelectric term, as discussed in section 3.2. Similar staggered magnetoelectric eects are obtained in the class #7. Figure 10(a) shows a staggered response, Ks z0y, as a function of ne. In this case also, the staggered magnetic response is induced, consistent with the ASOC in the xx-SOO phase in table 4. We also show handdependences of Ks z0yatne= 0:1 in gures 10(b) and 10(c), respectively. Both curves show similar behavior to the uniform magnetic responses discussed in section 6.1: for small h(), the staggered responses increase with increasing h() due to the emergence of the ASOC, while they gradually decrease and approach to zero for large h(). However, in contrast to the uniform ones, the same component Ks z0y becomes nonzero in the staggered response for all the ordered states in the class #7. Note that the staggered magnetoelectric responses predominantly come from the intra- orbital components in equation (6.1), leading to a strong dependence on the damping factor. Table 6 summarizes the complete table of the staggered o-diagonal responses in the paramagnetic and sixteen staggered ordered states. Similar to the uniform ones in table 5, all the nonzero responses appear when t16= 0, while some of them remain nonzero even for = 0, as shown in table 6. As a typical example of the o-diagonal responses except for the magnetoelectric eects, we discuss the staggered CO response in thex-OO phase. Figures 11(b) and 11(c) show handdependences of Ks 00xat ne= 0:1, respectively. The behaviors are dierent from those in the magnetoelectric eects in gures 10(b) and 10(c). In the hdependence in gure 11(b), the staggered CO response, Ks 00x, continues to increase as increasing h, which indicates that a large response persists in the region where the order parameter in the x-OO develops well and almost saturates. On the other hand, the dependence in gure 11(c) decreases asincreases. The nonzero value of Ks 00xat= 0 originates in the eective ASOC induced by the order parameters and inter-orbital hopping t1, which indicates that the atomic SOC does not play a role in this staggered CO response. In fact, the eective ASOC in the x-OO phase has the form of gu 0z(k)00z/(t0t1=h)fE1(k)00zin the regionh[cf. equation (5.3)]. Atne1:6 and 2:4,Ks z0ybecomes zero in changing the sign although the system is metallic.Emergent spin-valley-orbital physics by spontaneous parity breaking 28 7. Summary and Concluding Remarks To summarize, we have investigated the eect of electronic orders which break the spatial inversion symmetry spontaneously in the spin-orbital coupled systems on the centrosymmetric lattices with local asymmetry. We have claried how the ASOC is generated by the staggered charge, spin, orbital, and spin-orbital orders, taking a minimal two-orbital model on a honeycomb lattice. We derived the explicit form of the eective ASOC for all the cases and analyzed them in detail from the symmetry point of view. On the basis of the analysis of the ASOC, we have discussed the nature of each electronic orders as well as the paramagnetic state. The results are summarized in table 2. In the following, let us brie y review the main results. In the paramagnetic state, the ASOC is hidden in the sublattice-dependent form (table 3). The hidden ASOC is activated by spontaneous parity breaking, in a dierent way depending on the symmetry of the electronic order parameters. We classied all the possible staggered orders into the seven classes #1-#7 by symmetry (table 2) and derived the eective ASOCs for each case (table 4). Using the comprehensive table of the ASOC, we have examined the electronic properties, such as the spin and valley splitting of the band structure, and the o-diagonal responses to an external electric current. In the classes #1, #2, and #3, we showed that the emergent ASOC gives rise to the antisymmetric spin splitting of the band structure. Interestingly, the spin splitting appears in a dierent manner in dierent classes, which is understood from the form of the ASOC in each class. The topological phase transition was discussed for the CO state in the class #1. We also discussed that the ASOC in the class #2 leads to peculiar o-diagonal responses in both spin and orbital channels, which originates from the breaking of threefold rotational symmetry. On the other hand, in the classes #4 and #5, the band structures exhibit peculiar deformation, which is ascribed to the violation of time-reversal symmetry in addition to the spatial inversion symmetry. In the class #4, we showed that the emergent ASOC leads to the valley splitting of the band structure. We nd a topological transition also in this case, similar to the class #1. Meanwhile, in the class #5, the band bottom shift from the point is induced by thek-linear ASOC, similar to the toroidal ordered cases. In this case also, we claried that the system exhibits the o-diagonal responses in the spin and orbital channels, similar to the class #2. In the class #7, in which the spin-orbital orders break all the four symmetries considered here, we pointed out that the system exhibits both uniform and staggered magnetoelectric responses, in addition to the staggered orbital response. Summarizing the results, we have completed the tables for the uniform and staggered responses of the ordered parameters to an electric current (tables 5 and 6). Our results provide a comprehensive reference for further exploration of emergent physical properties induced by the spontaneously-generated ASOC. Our minimal model includes the essential ingredients for such emergent physics: the atomic SOC, electron hopping between orbitals with dierent angular momentum, electron correlations, and local asymmetry of the lattice structure. The obtained ASOC includes charge, spin,Emergent spin-valley-orbital physics by spontaneous parity breaking 29 and orbital degrees of freedom, which is the generalization of the conventional ASOC studied in the eld of semiconductors and topological insulators. In other words, we have extended the ASOC physics to multiband correlated electron systems. Our present analysis paves the way for investigating new noncentrosymmetric physics by spontaneous parity breaking, such as new types of electromagnetic and transport properties induced by the emergent ASOC. Let us conclude by making several remarks on the future problems. One of the intriguing problems is the extension of the present analysis to other degrees of freedom. In the present study, we elucidated the eect of the ASOC on the electronic structures and o-diagonal responses by using the spin-charge-orbital coupled model. Further peculiar o-diagonal responses can be expected by including the coupling to collective modes, such as magnons from magnetic excitations and phonons from the lattice distortion. For example, recent experiments showed that the asymmetric magnon excitations in chiral ferromagnets lead to nonreciprocal magnon propagations [38, 39]. Although such studies were limited to the noncentrosymmetric systems thus far, similar results will be obtained in a more controllable way for centrosymmetric systems with local asymmetry. In fact, the authors claried that the AFM orders on the zigzag chain and honeycomb lattice accompany asymmetric magnon dispersions with respect to the wave vector [29]. Such extensions may result in controlling the multiferroic o-diagonal phenomena, e.g., the magnetostriction and piezoelectric eect [40]. Another interesting problem is the physics related with domain and interface in the systems with the spontaneous ASOC [41]. For instance, in the pyrochlore lattice systems, the surfaces and domains can induce characteristic electronic states and o- diagonal responses [32, 42]. Moreover, a gapless domain state appears on the honeycomb lattice in the presence of the staggered potential [43]. Our results will serve as a good reference for comprehensive understanding of such peculiar surface/domains states. Finally, the experimental exploration of the physics of emergent ASOC is an important future problem. There are good candidate materials, e.g., trichalcogenides MX0X3(M: transition metal, X: chalcogen, X0= P, Si, Ge) [44, 45]. We have predicted several new phenomena related with the staggered electronic ordering, such as the spin- orbital orders. Although our model is a skeleton model and further sophistication is necessary to compare with the experiments, we believe that our analyses dig out the new essential features of the spin-charge-orbital coupled systems, which are possibly explored in monolayer MXX0 3. Moreover, similar interesting physics is expected also in other centrosymmetric lattices with local asymmetry like the zig-zag chain and diamond lattice, which are found in several f-electron compounds, such as UGe 2[46, 47, 48], URhGe [49, 50, 51], UCoGe [52, 53, 54], LnM 2Al10(Ln= Ce, Nd, Gd, Dy, Ho, Er, M= Fe, Ru, Os) [55, 56, 57, 58, 59, 60, 61, 62], RT2X20(R= Pr, La, Yb, U, T= Fe, Co, Ti, V, Nb, Ru, Rh, Ir, X= Al, Zn) [63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74], and-YbAlB 4[75, 76, 77, 78]. Further eorts from both theoretical and experimental sides are highly desired for such exploration. Our results will also be useful for extending the spin-orbit physics inEmergent spin-valley-orbital physics by spontaneous parity breaking 30 noncetrosymmetric systems. For example, in the monolayer transition metal dichalcogenides MX 2with 2H structure [79, 80, 81, 82], the transition metal Mand the chalcogenXcomprise the honeycomb lattice by aligning in a staggered way. Compared to the honeycomb-lattice model in the present study, this corresponds to the CO state. Indeed, the spin splitting in the band structure is observed in the monolayer 2H-MX 2[36], as predicted for the class #1 in our model. Then, once, e.g., z-SO is realized in the monolayer 2H- MX 2, a valley splitting is expected in addition to the spin splitting, as discussed in [37]. Such a possibility can be examined by introducing a staggered potential into our model. Thus, our results predict the emergent properties by spontaneous electronic orders also in noncentrosymmetric systems in a comprehensive way. Similar arguments can be applied to materials with a zincblende-type lattice structure, which can be regarded as the CO state on the diamond lattice. Acknowledgments This work was supported by JSPS KAKENHI Grant Numbers 24340076, 15K05176, 15H05882 (J-Physics), and 15H05885 (J-Physics), the Strategic Programs for Innovative Research (SPIRE), MEXT, and the Computational Materials Science Initiative (CMSI), Japan. Appendix A. Derivation of Eective Antisymmetric Spin-Orbit Coupling In this appendix, we derive the eective ASOC in the model given by equations (3.2) and (3.6). To this end, we perform the canonical transformation to obtain the eective ASOC at one of two sublattices by eliminating other sublattice [83, 84]. For this purpose, let us divide our mean-eld Hamiltonian into the model space and that connecting to out of the model space as H0+~H1=X kss00mm0hmjHss0(k)jm00icy skmcs0km00; H(k) = HA(k) 0 0HB(k)! + 0H0(k) H0y(k) 0! ; (A.1) where the 44 matrices are given by HA;B(k) = 2zzh; (A.2) H0(k) =[t0 0;k0+t1 2( +1;k++ 1;k)]0: (A.3) In the lowest order with respect to H0(k), the eective Hamiltonian for the A sublattice is given by HA e(k) =HAH0H1 BH0y: (A.4) Similarly, the eective Hamiltonian for the B sublattice is given by HB e(k) =HBH0yH1 AH0: (A.5)Emergent spin-valley-orbital physics by spontaneous parity breaking 31 In general, the 8 8 eective Hamiltonian He(k) can be expanded in terms of in addition to 0andzas He(k) =X  gu (k)0+gs (k)z : (A.6) Considering equation (A.6) at the A and B sublattices, we have the relations, HA;B e(k) =X [gu (k)gs (k)]: (A.7) Solving these relations, we obtain the coecients in the form gu;s (k) =1 8Tr (HA eHB e) ; (A.8) whereHA;B e(k) are given by equations (A.4) and (A.5). To demonstrate the usage of equations (A.6) and (A.8), we consider the paramagnetic state ( h= 0). By the straightforward calculation of equations (A.4) and (A.5), we have HA;B e=HA;B4p 3t2 1 fA(k)z04p 3t0t1 [fE1(k)x+fE2(k)y]z +t2 1 (j +1;kj2+j 1;kj2)zz2t2 0 j 0;kj2zz; (A.9) where fA(k) =1 4p 3(j +1;kj2j 1;kj2) =" cos p 3kx 2! cosky 2# sinky 2 ; (A.10) fE1(k) = RefE(k) =" cos p 3kx 2! + 2 cosky 2# sinky 2 ; (A.11) fE2(k) =ImfE(k) =p 3 sin p 3kx 2! cosky 2 ; (A.12) with fE(k) =1 2p 3( +1;k  0;k  1;k 0;k): (A.13) Here, the functions, fA(k),fE1(k), andfE2(k), are all antisymmetric with respect to k, as shown in gure A1. Thus, the uniform components of the eective Hamiltonian are symmetric with respect to k. The staggered components are antisymmetric and they are given by HASOC e =4p 3t2 1 fA(k)z04p 3t0t1 [fE1(k)x+fE2(k)y]z:(A.14)Emergent spin-valley-orbital physics by spontaneous parity breaking 32 (a) 0 K’K M0 kx0 π 2π -π -2π-π -2ππ2π ky 1 -1(b) 0 K’K M0 kx0 π 2π -π -2π-π -2ππ2π ky 1 -12 -2(c) 0 K’K M0 kx0 π 2π -π -2π-π -2ππ2π ky 1 -12 -2 Figure A1. (Color online) The contours of (a) fA(k), (b)fE1(k), and (c)fE2(k). The hexagons represent the Brillouin zone. CO(a) z-SO(b) Figure B1. Schematic diagram of the energy eigenvalues at the K' point when introducing (a) CO (class #1) and (b) z-SO (class #4). The red (blue) levels show the bands with up(down)-spin polarization. Appendix B. Eigenvalue analysis In this appendix, we reexamine the eect of ASOC from the viewpoint of eigenvalues of the Hamiltonian. We discuss the correspondence between the ASOC derived from equation (A.8) and the energy eigenvalues by direct diagonalization of the mean-eld Hamiltonian,H0+~H1. We show that the peculiar electronic structures, such as the antisymmetric spin splitting and band deformation, are understood from the eigenvalue analysis.Emergent spin-valley-orbital physics by spontaneous parity breaking 33 First, we consider the CO state in section 5.2.1 (class #1). The antisymmetric spin splitting is derived by directly calculating the energy eigenvalues at the K and K' points in the CO state. At the K' point, the eigenvalues are easily obtained by the diagonalization of H0+~H1as "" CO(K0) =r 2 + 4+ (3t1)2; 2; "# CO(K0) =r 2 4+ (3t1)2;+ 2; (B.1) where=2h. The energy levels are schematically displayed in gure B1(a). The eigenvalues clearly show that the spin splitting takes place under the CO at the K' point, which is consistent with the argument related to the eective ASOC in equation (4.7). The opposite spin splitting occurs at the K point because hchanges intohat the K point. Similarly, the valley splitting in the z-SO state discussed in section 5.3 (class #4) is understood from the energy eigenvalues at the K and K' points. The eigenvalues in thez-SO case are obtained by changing the sign of the hterms for down spins in those for the CO case above, namely, " zSO(K0) =r 2 + 4+ (3t1)2; 2; (B.2) " zSO(K) =r 2 4+ (3t1)2;+ 2; (B.3) whererepresents +1 (-1) for up (down) spin. Thus, each eigenvalue is doubly degenerate in terms of spin, as schematically shown in gure B1(b). The result explains the valley splitting; for example, at half lling, the gap at the K' point is 2h, while that at the K point is + 2h. Next, we discuss the x-OO phase (class #2). The k-linear contribution in the emergent ASOC in this state discussed in section 5.2.2 is also explained by directly examining the eigenvalues of the Hamiltonian. For the x-OO phase ( ~H1/z0x), we can expand the eigenvalues with respect to handt1up to the rst order:  xOO(k) = D kp 3ht1 D kj 0;kjfE1(k)! ;  D+ k+p 3ht1 D+ kj 0;kjfE1(k)! ; (B.4) whererepresents +1 (-1) for up (down) spin and Dkis dened by D k= 2t0j 0;kj: (B.5) Therefore, the change of the eigenvalues by introducing his represented by
 xOO(k)ht 1fE1(k): (B.6) In the limit of k!0, we obtain
 xOO(k)ht 1ky: (B.7)Emergent spin-valley-orbital physics by spontaneous parity breaking 34 The result in equation (B.7) gives the consistent dependence on h,t1, andkywith equation (5.3). Similarly, the eigenvalues in the limit of k!0 for they-OO state is obtained in the consistent form with equation (5.4) as
" yOO(k)ht 1kx: (B.8) Now, we turn to the class #5. The eective ASOC in this case is derived from the diagonalization by using the fact that the Hamiltonian in the zx-SOO phase has a similar form to that in the x-OO phase in the class #2, with a dierence in the presence ofzin~H1. Hence, the eigenvalues in the zx-SOO state are obtained by replacing h withhin the results of the class #2. Then, the change of the eigenvalues by the zx-SOO in the limit of k!0 reads
" zxSOO(k)ht1ky: (B.9) Similar to the class #2, the eigenvalues are proportional to ky, but without any spin dependence. This explains the asymmetric band modulation with a band bottom shift in thekydirection discussed in section 5.4. So far, we have shown that the spin splitting, valley splitting, and the band deformation in the band structure are well understood by performing the direct diagonalization. In some cases, however, the eigenvalue analysis does not work out properly. We here consider the band structure in the xz-SOO case in section 5.2.3 (class #3). By the similar procedure of the previous examples, we can derive the asymptotic form of the eigenvalue in the limit of k!0 by diagonalizing the Hamiltonian:
 xzSOO(k)/ht1q k2 x+k2 y: (B.10) The isotropic kdependence in equation (B.10) indicates that the band structures are modulated in the symmetric way in this case. Hence, we can extract no information about the ASOC in the xz-SOO state, in contrast to the discussion given in section 5.2.3. Furthermore, any band modulations do not seem to occur by equation (B.10), although thexz-SOO shows the spin splitting, as shown in gure 6(h). This example signals the failure of the eigenvalue analysis. Indeed, the eigenvalues of the equation (5.5) are isotropic, and only the eigenvectors have the information about their spin dependences. Meanwhile, the derivation of the eective ASOC by equation (A.8) is straightforward and gives comprehensive understanding of the modulations of the band structure and the o-diagonal responses to an electric current. 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1403.4195v1.Spin_Hall_and_Edelstein_effects_in_metallic_films__from_2D_to_3D.pdf

Spin Hall and Edelstein eects in metallic lms: from 2D to 3D J.Borge1, C. Gorini2, G. Vignale3, and R. Raimondi1 1Dipartimento di Matematica e Fisica, Universit a Roma Tre, Via della Vasca Navale 84, Rome, Italy 2Service de Physique de l' Etat Condens e, CNRS URA 2464, CEA Saclay, F-91191 Gif-sur-Yvette, France and 3Department of Physics and Astronomy, University of Missouri, Columbia MO 65211, USA Abstract A normal metallic lm sandwiched between two insulators may have strong spin-orbit coupling near the metal-insulator interfaces, even if spin-orbit coupling is negligible in the bulk of the lm. In this paper we study two technologically important and deeply interconnected eects that arise from interfacial spin-orbit coupling in metallic lms. The rst is the spin Hall eect, whereby a charge current in the plane of the lm is partially converted into an orthogonal spin current in the same plane. The second is the Edelstein eect, in which a charge current produces an in-plane, transverse spin polarization. At variance with strictly two-dimensional Rashba systems, we nd that the spin Hall conductivity has a nite value even if spin-orbit interaction with impurities is neglected and \vertex corrections" are properly taken into account. Even more remarkably, such nite value becomes \universal" in a certain conguration. This is a direct consequence of the spatial dependence of spin-orbit coupling on the third dimension, perpendicular to the lm plane. The non-vanishing spin Hall conductivity has a profound in uence on the Edelstein eect, which we show to consist of two terms, the rst with the standard form valid in a strictly two-dimensional Rashba system, and a second arising from the presence of the third dimension. Whereas the standard term is proportional to the momentum relaxation time, the new one scales with the spin relaxation time. Our results, although derived in a specic model, should be valid rather generally, whenever a spatially dependent Rashba spin-orbit coupling is present and the electron motion is not strictly two-dimensional. 1arXiv:1403.4195v1 [cond-mat.mes-hall] 17 Mar 2014I. INTRODUCTION Spin-orbit coupling gives rise to several interesting transport phenomena arising from the induced correlation between charge and spin degrees of freedom. In particular, it allows one to manipulate spins without using magnetic electrodes, having as such become one of the most studied topics within the eld of spintronics.1{14Among the many interesting eects that arise from spin-orbit coupling, two stand out for their potential technological impor- tance: the spin Hall eect15and the Edelstein eect16,17. The spin Hall eect consists in the appearance of a z-polarized spin current owing in the y-direction produced by an electric eld in the x-direction.18{22The generation of a perpendicular electric eld by an injected spin current, i.e. the inverse spin Hall eect, has been observed in numerous settings and presently provides the basis for one of the most eective methods to detect spin currents.23{25 The Edelstein eect16,17consists instead in the appearance of a y-spin polarization in re- sponse to an applied electric eld in the x-direction. It has been proposed as a promising way of achieving all-electrical control of magnetic properties in electronic circuits.18,19,26{31 The two eects are deeply connected,32{34as we will see momentarily. There are, in principle, several possible mechanisms for the spin Hall eect, and it is useful to divide them in two classes. We call them either extrinsic or intrinsic, depending on whether their origin is the spin-orbit interaction with impurities or with the regular lattice structure. In this work we will focus exclusively on intrinsic eects. This means that the impurities (while, of course, needed to give the system a nite electrical conductivity) do not couple to the electron spin. Bychkov and Rashba devised an extremely simple and yet powerful model35describing the intrinsic spin-orbit coupling of the electrons in a 2-Dimensional Electron Gas (2DEG) in a quantum well in the presence of an electric eld perpendicular to the plane in which the electrons move. In spite of its apparent simplicity, this analytically solvable model has several subtle features, which arise from the interplay of spin-orbit coupling and impurity scattering. The best-known feature is the vanishing of the Spin Hall Conductivity (SHC) for a uniform and constant in-plane electric eld.36{38This would leave spin-orbit coupling with impurities (not included in the original Bychkov-Rashba model) as the only plausible mechanism for the experimentally observed spin Hall eect in semiconductor-based 2DEGs18,19.39 However it has been recently pointed out that the vanishing of the SHC need not occur in 2V−V+ Interfacial spin-orbit (Rash ba) z x yFIG. 1. (color online) Schematic representation of a thin metal lm sandwiched between insulators with asymmetric interfacial spin-orbit couplings. V+andVare the heights of the two interfacial potential barriers. These potentials generate interfacial spin-orbit interactions of the Rashba type, whose strength is controlled by the eective Compton wavelengths +andrespectively. systems which are not strictly two-dimensional, as explicitly shown in a model schematically describing the interface of the two insulating oxides LaAlO 3and SrTiO 3(LAO/STO)40. Even more recently41, it has been suggested that a large SHC could be realized in a thin metal (Cu) lm that is sandwiched between two dierent insulators, such as oxides or the vacuum.42 Such a system is shown schematically in Fig. 1. The inversion symmetry breaking across the interfaces produces interfacial Rashba-like spin-orbit couplings, thus allowing metals without substantial intrinsic bulk spin-orbit to host a non-vanishing SHC. The spin-orbit coupling asymmetry { or, more generally, the fact that the spin-orbit interaction is not homogeneous across the thickness of the lm { is the core issue in this novel approach. In this paper we will study the in uence of the interfacial spin-orbit couplings on the Edelstein and spin Hall eects in this class of heterostructures. Before proceeding to a detailed study of the model depicted in Fig. 1, it is useful to recall the deep connection32{34that exists between the spin Hall and Edelstein eects in the Bychkov-Rashba model, described by the Hamiltonian H=p2 2m+(xpyypx); (1) wheremis the eective electron mass and is the Bychkov-Rashba spin-orbit coupling constant given by =2eEz=~, withthe materials' eective Compton wavelength, Ez 3the electric eld perpendicular to the electron layer, and ethe absolute value of the electron charge. It is convenient to describe spin-orbit coupling in terms of a non-Abelian gauge eldA=Aaa=2, withAx y= 2mandAy x=2m.43{45If not otherwise specied, superscripts indicate spin components, while subscripts stand for spatial components. The rst consequence of resorting to this language is the appearance of an SU(2) magnetic eld Bz z=(2m)2, which arises from the non-commuting components of the Bychkov-Rashba vector potential. Such a spin-magnetic eld couples the charge current driven by an electric eld, say along x, to thez-polarized spin current owing along y. This is very much similar to the standard Hall eect, where two charge currents owing perpendicular to each other are coupled by a magnetic eld. The drift component of the spin current can thus be described by a Hall-like term [Jz y]drift=SHE driftEx: (2) It is however important to appreciate that this is not yet the full spin Hall current, i.e. SHE drift is not the full SHC. In the diusive regime SHE drift is given by the classic formula SHE drift = (!c)D=e, where!c=B=m~is the \cyclotron frequency" associated with the SU(2) magnetic eld, is the elastic momentum scattering time, and Dis the Drude conductivity. For a more general formula see Eq. (6) below. In addition to the drift current, there is also a \diusion current" due to spin precession around the Bychkov-Rashba eective spin-orbit eld. Within the SU(2) formalism this current arises from the replacement of the ordinary derivative with the SU(2) covariant derivative in the expression for the diusion current. The SU(2) covariant derivative, due to the gauge eld, is rjO=@jO+i[Aj;O]; (3) withOa given quantity being acted upon. The normal derivative, @j, along a given axis j is shifted by the commutator with the gauge eld component along that same axis. As a result of the replacement @!r diusion-like terms, normally proportional to spin density gradients, arise even in uniform conditions and the diusion contribution to the spin current turns out to be [Jz y]diff=2m ~Dsy; (4) whereD=v2 F=2 is the diusion coecient, vFbeing the Fermi velocity. In the diusive 4regime the full spin current Jz ycan thus be expressed as Jz y=2m ~Dsy+SHE driftEx: (5) For a detailed justication of Eq. (5) we refer the reader to Refs. 44 and 46. The factor in front of the spin density in the rst term of Eq.(5) can also be written as an eective velocity Lso=s. HereLso=~(2m)1is the typical spin length due to the dierent Fermi momenta in the two spin-orbit split bands, whereas s=~2(4m22D)1is the Dyakonov-Perel spin relaxation time. In terms of andsone has SHE drift=e 8~2 s; (6) which is indeed equivalent to the classical surmise given after Eq. (2). If we introduce the total SHC and the Edelstein Conductivity (EC) dened by Jz y=SHEEx; sy=EEEx (7) we may rewrite Eq.(5) as EE=s Lso SHESHE drift
: (8) In the standard Bychkov-Rashba model a general constraint from the equation of motion dictates that under steady and uniform conditions Jz y= 0. Therefore the EC reads EE=s LsoSHE drift=em 2~2=eN0; (9) which is easily obtained by using the expressions given above and the single particle density of states in two dimensions, N0=m=2~2. The remarkable thing is that this expression remains unchanged for arbitrary ratios between the spin splitting energy and the disorder broadening of the levels. However, in a more general situation with a non-zero SHC the EC would consist of the two terms appearing in Eq. (8). The latter equation is the \deep connection" mentioned earlier between the Edelstein and the spin Hall eect. The rst term on the r.h.s. is the \regular" contribution to the EC, the only surviving one in the Bychkov- Rashba model where the full SHC vanishes. The second term is \anomalous" in the sense that it does not appear in the standard Bychkov-Rashba model, but it does appear in more general models such as the one we discuss in this paper. Notice that the \regular" term is proportional to (see Eq. (6)), while the \anomalous" term, being proportional to the 5Dyakonov-Perel relaxation time sand, in the diusive regime, is inversely proportional to the momentum relaxation time. At variance with the Bychkov-Rashba model, the one we choose for our system is not strictly two-dimensional, and we take into account several states of quantized motion in the direction perpendicular to the interface ( z). Another crucial feature of this model is the occurrence of two dierent spin-orbit couplings at the two interfaces. The dierence arises because (i) the interfacial potential barriers V+andVare generally dierent, and (ii) the eective Compton wavelengths +and, characterizing the spin-orbit coupling strength at the two interfaces, are dierent. Our central results for the generic asymmetric model are SHE=ncX n=1e 4~
E(3) nkFn
EnkFn; (10) and EE=ncX n=1eN0 kFn~h
EnkFn+
E(3) nkFn(n) DPi ; (11) the sums running over the nclledz-subbands of the thin lm. To each subband there correspond a Fermi wavevector (without spin-orbit) kFn, an intraband spin-orbit energy splitting with a linear- and a cubic-in- kpart
EnkFn= (2E0n2=d)[kFn(2 +2 ) +2mk3 Fn ~2(6 +V+6 V)] (12) 
E(1) nkFn+
E(3) nkFn(13) and a Dyakonov-Perel spin relaxation time (n) DP = 21 + (2
EnkFn=~)2 (2
EnkFn=~)2 : (14) In the above formulas dis the lm thickness and E0=~2=2md2. Two particularly interesting regimes are apparent. First, a \quasi-symmetric" conguration, dened by equal spin-orbit strengths,+=, but dierent barrier heights, V+6=V. In this case
E(1) nk= 0 (due to Ehrenfest's theorem47) and a most striking result is obtained: the SHC has a maximal value ofe 4~(independent of !) times the number of occupied bands SHE=ncX n=1e 4~: (15) 6At the same time the \anomalous" EC is at its largest. A second very interesting cong- uration is a strongly asymmetric insulator-metal-vacuum junction, += 0;V+!1 and ;VV. In this case the SHC becomes directly proportional to the gap V SHE=ncX n=1e 4~32mk2 FnV4: (16) Notice however that the SHC cannot be made arbitrarily large simply by engineering a large V, since the above result holds provided 2 mk2 FnV4=~2<1. The paper is organized as follows. In Sec. II we introduce and discuss the model. In Secs. III and IV we calculate the SHC and the EC, respectively. Both Sections are technically heavy and can be skipped at a rst reading, leading straight to Sec. V where the physical consequences of our results are discussed and special regimes are analyzed. Sec. VI presents our summary and conclusions. II. THE MODEL AND ITS SOLUTION Following Ref. 41, we model the normal metallic thin lm via the following Hamiltonian H=p2 2m+VC(z) +HR+U(r); (17) where the rst term represents the kinetic energy associated to the unconstrained motion in thexyplane and p= (px;py) is the standard two-dimensional momentum operator. The nite thickness dof the metallic lm is taken into account by a conning potential VC=V+(zz+) +V(zz); (18) whereVis the height of the potential barrier at z=d=2 and(z) is the Heaviside function. The third term in Eq.(17) describes the Rashba interfacial spin-orbit interaction in thexyplane located at z=d=2 HR=2 V(zz)2 +V+(zz+) ~(pyxpxy); (19) whereare the eective Compton wavelengths for the two interfaces, x;y;zare the Pauli matrices. The last term in Eq.(17) represents the scattering from impurities aecting the motion in the xyplane and r= (x;y) is the coordinate operator. The impurity potential is taken in a standard way as a white-noise disorder with variance hU(r)U(r0)i= 7(2N0)1(rr0), whereN0is the two-dimensional density of states previously introduced. We will assume throughout that the Fermi energy EFnin each subband is much larger than the level broadening ~=and use the self-consistent Born approximation. The eigenfunctions of the Hamiltonian (17) have the form nks(r;z) =eik
r p A1p 20 @1 iseik1 Afnks(z); (20) whereAis the area of the interface, k= (kx;ky) is the in-plane wave vector, ris the position in the interfacial plane and zis the coordinate perpendicular to the plane. kis the angle between kand thexaxis. These states are classied by a subband index n= 1;2::, which plays the role of a principal quantum number, an in-plane wave vector k, and an helicity index ,s= +1 or1 which determines the form of the spin-dependent part of the wave function. By inserting the wave function (20) into the Schr odinger equation for the Hamiltonian (17) we nd the following equation for the functions fnks(z) describing the motion along the z-axis ~2 2mf00 nks(z) + VC(z)ks 2 V(z+d=2)2 +V+(zd=2) fnks(z) =nksfnks(z); (21) where the full energy eigenvalues are Enks=~2k2 2m+nks: (22) By taking into account the continuity of the wave function fnks(z) atz=d=2 and the discontinuities of its derivatives we obtain for the eigenvalue nksthe following transcendental equation arctan0 BB@pr d2 d2  d dsk1 CCA+ arctan0 BB@pr d2 d2 + +d d++sk1 CCA+p=n; (23) where the energy is measured in units of E0=~2=(2md2) set by the thickness of the lm. In the absence of spin-orbit coupling ( = 0) and for innite heights of the potential (V!1 ), the solution reduces to the well-known energy levels nks=E0n2. In the general case with both andVnite we use perturbation theory by assuming dlarge. There are 8two natural length scales associated with the conning potential d=~=p2mVso that we expand in the small parameters d=d. Since all the energy scales are set by E0, we nd useful to describe the spin-orbit coupling in terms of the parameters =2 =din such a way that the product E0=~has the dimensions of a velocity, just as the typical Rashba coupling parameter. In the following we make an expansion to rst order in d=dand up to third order in k. For the eigenvalues of (21) we nd nks=E0n2 12d+d+ d+se1k+e2k2+se3k3 (24) and the eigenfunctions fnks(z) =cnkssin" n d+d 1ks+d+ 1++ksd 2+z+d 1ks# ; (25) where cnks=s 4 de[2(se1k+e2k2+se3k3)]; de=d+d++d; e1= 2d+ d+d d ;e2=2d+ d2 ++d d2  ;e3= 2d+ d3 +d d3  :(26) Notice that the sign of the coecients e1ande3depends on the relative strength of the spin-orbit coupling and barrier heights V. To avoid troubles with minus signs in the following calculations, we assume that the couplings are labeled in such a way that +>, andV+>Vso thate1;e3>0. In the next Section we evaluate the SHC assuming that n=ncis the topmost occupied subband. In the following we use units such that ~=c= 1. III. SPIN HALL CONDUCTIVITY The SHC is dened as the non-equilibrium spin density response to an applied electric eld. By using a vector gauge with the electric eld given by E=@tA, the Kubo formula, corresponding to the bubble diagram of Fig.2, reads SHE= lim !!0Imhhjz y;jxii !; (27) 9FIG. 2. Feynman bubble diagram for the EC(a+b) or SHC(c). The empty right dot indicates the spin density (EC) or the spin current density (SHC) bare vertex, the left empty one indicates the normal velocity operator, and the full dot is the dressed charge current density vertex. where we have introduced the spin current operator jz y=zky=2mand the charge current operatorjx=e^vx. The number current operator, besides the standard velocity component, includes a spin-orbit induced anomalous contribution ^ vx=kx=m+^x. Without vertex corrections, the anomalous contribution reads ^x=^vx= 2 +V+(zz+)2 V(zz) y: (28) This expression can be written in terms of the exact Green functions and vertices as SHE=lim !!0Ime !X nn0kk0ss0hn0k0s0j^vxjnksihnksjjz yjn0k0s0iZ1 1d 2Gns(+;k)Gn0s0(;k0): (29) wheree>0 is the unit charge, =!=2 andGns(;k) = (Enks+ isgn=2)1is the Green function averaged over disorder in the self-consistent Born approximation with self energy ns(r;r0;) =(rr0) 2N0Gns(r;r;): (30) After performing the integral over the frequency we obtain SHE=e 2X nn0kss0hn0ks0j^vxjnksihnksjjz yjn0ks0iGR nksGA n0ks0; (31) where we have introduced the retarded and advanced zero-energy Green functions at the Fermi level GR;A nks=1 Enks+i=2(32) 10and exploited the fact that plane waves at dierent momentum kare orthogonal. To proceed further we need the expression for the vertices. It is easy to recognize that the standard part of the velocity operator kx=mdoes not contribute since it requires s=s0, whereas the matrix elements of jz ydier from zero only for s6=s0. Explicitly we have hn0ks0jkxjnksi=kxhfn0ks0jfnksis0s=hfn0ks0jfnksikcosks0s (33) hnks0j^vxjnksi= (coskz;s0s+ sinky;s0s)
Enk khfnks0jfnksi (34) hnksjjz yjn0ks0i=hfnksjfn0ks0ik 2msinkx;ss0; (35) where
Enk= (Enk+Enk)=2 =E0n2(e1k+e3k3) is half the spin-splitting energy in the n-th band. Eq.(34) is straightforwardly obtained from the eigenvalue equation (21) for the functionsfnks(z). Let us now discuss the overlaps between the wave functions hfnksjfn0k0s0i. Ifn=n0we have hfnksjfnk0s0i=de 2cnkscnk0s0 1e1(ks+k0s0) +e2(k2+k02) +e3(k3s+k03s0) 4 ; (36) which is unity plus corrections of order ( d=d) whens;k6=s0;k0. Ifn6=n0hfnksjfn0k0s0iis at least of order ( d=d). Before continuing our calculation we observe that it is important to distinguish between the intra-band ( n=n0) and the inter-band ( n6=n0) contributions. The inter-band contributions are of second order in d=d, because they are proportional tohfnksjfn0ks0i2. Since we limit our expansion to the rst order in d=dwe will from now on neglect these contributions. Notice, however, that this approximation is no longer valid when the intra-band splitting controlled by e1ande3vanishes. In this case one cannot avoid taking into account the inter-band contributions. In the same spirit, we also approximate the intra-band overlap hfnksjfnk0s0i'1, because all of our results are at least linear in ( d=d) and we neglect higher order terms. The anomalous contribution to the velocity vertex, ^x, can be computed following the procedure described in Ref. 37 according to the equations (see Fig.3) ^x= ~ x+1 2N0X k0GR k0^xGA k0; ~ x=^vx+1 2N0X k0GR k0k0 x mGA k0~ (1)+ ~ (2)(37) 11FIG. 3. Ladder resummation for the spin-dependent part of the dressed charge current density vertex. The dashed line represents the correlation between propagators scattering o the same impurity site. To extend the treatment to the present case, the projection must be made over the states jnksi. Assuming that the impurity potential does not depend on z, the matrix elements of the eective vertex ~ (2)are: (2)nn ss0(k)hnksj~ (2)jnks0i=1 2N0X n1k0s1hnksjn1k0s1iGR n1k0s1k0 x mGA n1k0s1hn1k0s1jnks0i; (38) and (1)nn ss0(k)hnksj~ (1)jnks0iis given by Eq.(34). The matrix elements hnksjn1k0s1iand hn1k0s1jnks0iare those of the impurity potential: hnksjn1k0s1i=1 2hfnksjfn1k0s1i 1 +ss1ei(k0k) (39) hn1k01jnks0i=1 2hfn1k0s1jfnks0i 1 +s0s1ei(k0k) : (40) By observing that k0 x=k0cosk0, one can perform the integration over the direction of k0in 12the expression of (2)nn ss0(k) 1 4Z2 0dk0 2 1 +ss1ei(k0k) cosk0 1 +s0s1ei(k0k) =s1 8 seik+s0eik ; (41) to get (2)nn ss0(k) =(coskz;ss0+ sinky;ss0) 16N0X n1k0s1s1hfnksjfn1k0s1ihfn1k0s1jfnks0iGR n1k0s1k0 mGA n1k0s1: (42) Approximatinghfnksjfn1k0s1inn1, summing over s1, and integrating over kwith the technique shown in the Appendix yields (2)nn ss0(k) =(coskz;ss0+ sinky;ss0)E0n2(e1+ 2e3k2 Fn); (43) where we have introduced the spin-averaged Fermi momentum in the n-th subband k2 Fn 2m=E0n2: (44) On the other hand (1)nn ss0(k) is given by (1)nn ss0(k) = (coskz;ss0+ sinky;ss0)E0n2(e1+e3k2 Fn) (45) wherekhas been replaced by kFnat the required level of accuracy. Combining (1)nn ss0(k) and (2)nn ss0(k) as mandated by Eq. (37) we nally obtain nn x;ss0(k) =(coskz;ss0+ sinky;ss0)E0n2e3k2 Fn: (46) Next we project the equation for the vertex corrections in the basis of the eigenstates and get the following integral equation: nn x;ss0(k) = nn x;ss0(k) +1 2N0X n1n2k0s1s2hnksjn1k0s1iGR n1k0s1n1n2 x;s1s2(k0)GA n2k0s2hn2k0s2jnks0i; (47) which, by conning to intra-band processes only, can be solved with the ansatz nn x;ss0(k) = n(kFn)(cos(k)(z)ss0+ sin(k)(y)ss0) yielding nn x;ss0(k) = nn x;ss0(k)(n) DP : (48) By performing the integral over momentum and summing over the spin indices in Eq.(31), one obtains the SHC as SHE=ncX n=1e 82 (n) DPn(kFn)
EnkFn=kFn; (49) 13wherencis the number of occupied bands. If vertex corrections are ignored, i.e., if we approximate n(kFn) =
EnkFn=kFn(cf. Eq.(34)), Eq.(49) gives us SHE drift=ncX n=1e 82 (n) DP; (50) which, in the weak disorder limit ( !1 ), reproduces the result of Ref. 41, i.e.SHE drift = (e=8)nc. If instead the renormalized vertex (48) is properly taken into account, we obtain SHE=ncX ne 4e3k2 Fn e1+e3k2 Fn: (51) Notice that, being proportional to 4 (e1/2 ,e3/6 ), this result is consistent with the result obtained in Ref. 40 for a dierent but related model. Making use of the explicit expressions for e1ande3we nally get the previously reported result of Eq.(10). IV. EDELSTEIN CONDUCTIVITY In the d.c. limit, i.e., for !!0, the Edelstein conductivity (EC) is dened by EE= lim !!0Imhhsy;jxii !: (52) That can be written as: EE=lim !!0Ime !X nn0kk0ss0hn0k0s0j^vxjnksihnksjsyjn0k0s0iZ1 1d 2Gns(+;k)Gn0s0(;k0); (53) After performing the integral over frequency we get EE=e 2X nn0kss0hn0ks0j^vxjnksihnksjsyjn0ks0iGR nksGA n0ks0; (54) where we have used again the orthogonality of the eigenvectors with dierent momentum. As shown in Fig.2, we consider the bare vertex for the spin density sy=y=2 and the two vertices for the number current density ^ vx=^x+kx=m,37{^xbeing the renormalized spin-dependent part of the vertex. Clearly, the two parts of the number current vertex yield two separate contributions to the EC and we are now going to evaluate them separately. We 14then evaluate the (a) diagram in Fig.2 as: EE;(a)=e 4mX nn0kss0hn0ks0jkxjnksihnksjyjn0ks0iGR nksGA n0ks0; (55) where the matrix elements of the spin vertex is hnksjyjn0ks0i=hfnksjfn0ks0i(coskz;ss0sinky;ss0): (56) Settingn0=nand using Eq.(24) for the energy eigenvalues, we can perform the integra- tion over the momentum in Eq.(55) obtaining for EE;(a)the expression EE;(a)=ncX n=1eN0E0n2 e1+ 2e3k2 Fn
; (57) Next we evaluate the (b) diagram in Fig.2 as: EE;(b)=e 4X nn0kss0hn0ks0j^xjnksihnksjyjn0ks0iGR nksGA n0ks0; (58) We setn=n0and insert the result obtained in Eq.(48) for hnks0j^xjnksi. Since both the matrix elements of ^xandycontain terms proportional to cos( k) and sin(k), we must distinguish between s=s0(rst term in Eq.(46)) and s6=s0(second term in Eq.(46)). If s=s0we have EE;(b) 1 =e 4X nkshnsj~xjnksihnksjyjnksiGR nksGA nks (59) The integral over the momentum can be done with the technique shown in the Appendix to yield EE;(b) 1 =ncX neN0E0n2e3k2 Fn(n) DP 2: (60) Ifs6=s0we have instead EE;(b) 2 =e 4X nkshnksj~xjnksihnksjyjnk0siGR nksGA nks: (61) So we can conclude that EE;(b) 2 =ncX n=1eN0E0n2e3k2 Fn (2
EnkFn)2(62) with
EnkFndened in Eq.(12). Combining the (a) and (b) contributions, the nal result for the Edelstein conductivity is found to be: EE=ncX n=1eN0E0n2 e1+ 3e3k2 Fn+2e3k2 Fn (2
EnkFn)2 ; (63) which is easily seen to be equivalent to Eq. (11). 15V. DISCUSSION The two central results (63) and (51) may be interpreted along the lines outlined in the introduction. We begin by noticing that both conductivities are expressed as simple sums of independent subband contributions, hence the relation (8) is valid separately within each subband. The second step is the identication of the quantity s=Lsofor a given subband. Clearlysmust be identied with the Dyakonov-Perel relaxation time (n) DPdened in (14). For the spin-orbit length Lsoone notices that the quantity 2 pFin the Rashba model corresponds to the band splitting, and hence must here be replaced by 2
EnkFn. This yields, after restoring ~in the following, L(n) so=~vFn 2
EnkFn; (64) i.e.s=Lso!(n) DP=L(n) so. With this prescription one can apply Eq. (8) subband-by-subband and obtain EE;(n)=(n) DP L(n) soh SHE; (n)SHE; (n) drifti ; (65) whereSHE; (n);SHE; (n) drift stand for the n-th band contribution to Eqs. (51) and (50), respec- tively. It is now immediate to see that a sum over the subbands leads to the EC of Eq. (63). We may thus conclude the following: a non vanishing SHC in the presence of Rashba spin- orbit coupling gives rises to an anomalous EC scaling with the inverse scattering time; conversely, an anomalous EC yields a non-vanishing SHC. We now consider two physically interesting limiting cases of the general solution: 1. the insulator-metal-vacuum junction, += 0V+!1 ,= V=V; 2. lms with the same spin orbit constant coupling at the two interfaces, =+=. In the rst case we get EE=ncX n2eN0E0n22 d~ 1 +~22V 82E3 0n4 ; (66) SHE=ncX ne 4~32mk2 FnV4: (67) There are some experimental studies of metal-metal-vacuum junctions that shows giant spin- orbit coupling48and where one could test the prediction of Eqs.(66-67). Though Eq. (67) is 16obtained for small values of the parameter 2 mk2 FnV4=~21, the structure of the result is quite interesting: it suggests that this kind of device, the insulator-metal-vacuum junction, could be an ecient spintronic device, its transport properties being proportional to the barrier height V. In the second case let us rst assume a \quasi-symmetric" conguration, i.e. though +=, the barrier heights are dierent, V+6=V. We then obtain that the spin splitting of the bands vanishes to linear order in k(e1= 0) (see footnote 48) so that SHE=ncX ne 4~; (68) and EE=ncX neN0
EnkFn kFn~ 3 +~2 2(
EnkFn)2 : (69) The SHC in this limit is independent of . This very striking result is reminiscent of the universal resulte 8~obtained for a single Bychkov-Rashba band when vertex corrections are ignored.4However vertex corrections are now fully included, yet the SHC is not only nite, but independent ofand equal to the single band universal result multiplied by a factor2! We emphasize that this result has nothing to do with the non-vanishing intrinsic SHC that arises in certain generalized models of spin-orbit coupling with winding number higher than 1.49Rather, it has everything to do with the k-dependence of the transverse subbands describing the electron wave function in the z- direction. We also nd that the anomalous part of the Edelstein eect becomes large, as it is proportional to 1 =
EnkFn, and the splitting vanishes with the third power of kat smallk. Let us nally discuss the fully inversion-symmetric limit of the model, +=and V+=V. We notice that in this case the limit of Eq. (51) does not exist, because both e1 ande3vanish (the spin splitting is identically zero!) while the value of Eq. (51) depends on the order in which e1ande3tend to zero, in particular on whether they tend to zero simultaneously, or e1tend to zero before e3, as in the \quasi-symmetric" case above. The origin of this apparently unphysical non-analytic behavior can be traced back to the singular character of the vertex (48) for vanishing spin splitting. Under these circumstances, the Dyakonov-Perel spin relaxation time (14) diverges, apparently implying spin conservation. However, even in the inversion-symmetric limit, interband eects provide spin relaxation processes which regularize the vertex. Such eects are typically negligible away from the 17inversion-symmetric limit, since they are proportional to the square of the wave-function overlap between dierent bands and therefore scale as ( d=d)2. However, in the inversion- symmetric limit they cannot be neglected. A full analysis of interband eects is beyond the scope of the present paper, and we limit ourselves to a heuristic discussion of the physical origin of the spin relaxation mechanism due to interband virtual transitions. In the inversion-symmetric limit, the Hamiltonian is invariant upon the simultaneous operations of space inversion along the z-direction (z!z) and helicity ipping ( s!s), i.e., a full mirror re ection in the xyplane. Hence the eigenfunctions can be classied as even or odd under such a re ection: fnks(z) =Pnfnks(z): (70) wherePn=1. Furthermore the parity eigenvalue Pnis the same as in the absence of spin-orbit interaction, because the re ection commutes with the spin-orbit interaction. Since states of opposite helicity are degenerate, one can construct, in each band n, states that are linear combinations of the helicity eigenstates ji nk"=1 2(fnk+(z)j+i+fnk(z)ji) (71) nk#=1 2(fnk+(z)j+ifnk(z)ji): (72) These can be rewritten in terms of the eigenstates j"iandj#iofzand, after using (70), one obtains nk"=fnk+(z) +Pnfnk+(z) 2j"i+ ieikfnk+(z)Pnfnk+(z) 2j#i (73) nk#=fnk+(z)Pnfnk+(z) 2j"i+ ieikfnk+(z) +Pnfnk+(z) 2j#i: (74) One sees immediately that, within the rst Born approximation, impurity scattering cannot produce spin ipping within a band because the matrix element of the z-independent disorder potential between nk" nk0#vanishes by symmetry. On the other hand, spin ipping may occur in the second Born approximation by going through an intermediate state in a band of opposite parity. For example, an electron may rst jump, under the action of the disorder potential, to a state of opposite spin in an unoccupied band of opposite parity; then in a second step it may return to the original band without ipping its spin. Alternatively the spin may remain unchanged in the transition to the unoccupied band, and ip on the way back to the original band. As a result of such 18second-order processes, a new mechanism of spin relaxation arises, which we call inter-band spin relaxation , with rate 1 IB. When this additional relaxation mechanism is taken into account, the diverging DP relaxation time in Eq. (48) for the vertex is replaced by the nite total spin relaxation time ( 1 DP+1 IB)1. Thus, the non-analyticity is cured. The regime analyzed in this paper corresponds to the situation in which 1 DP1 IB, and inter-band spin relaxation can be neglected. Clearly, when looking at the fully symmetric limit, with vanishing spin splitting, inter-band relaxation must be taken into account, to- gether with inter-band contributions to the SHC and EC. Once more, a full- edged treatment of this regime is beyond the scope of the present work. VI. CONCLUSIONS We have developed a simple model for describing spin transport eects and spin-charge conversion in heterostructures consisting of a metallic lm sandwiched between two dier- ent insulators. All the eects we have considered depend crucially on the three-dimensional nature of the system { in particular, the fact that the transverse wave functions depend on the in-plane momentum { and on the lack of inversion symmetry caused by the dierent properties of the top and bottom metal-insulator interfaces, each characterized by a dierent barrier height (gap) and spin-orbit coupling strength. After a careful consideration of vertex corrections we nd that the model supports a non-zero intrinsic SHC, in sharp contrast to the 2DEG Rashba case. Strikingly, in a \quasi-symmetric" junction the SHC reaches a maximal and universal value. We have also calculated the Edelstein eect for the same model and found that the induced spin polarization is the sum of two dierent contribu- tions. The rst one is analogous to the term found in the 2DEG Rashba case, whereas the second \anomalous" one has a completely dierent nature. Namely, it is inversely propor- tional to the scattering time, indicating that it is caused by the combined action of multiple electron-impurity scattering and spin-orbit coupling. We have also discussed the general connection between the non-vanishing SHC and the anomalous term in the EC. Further- more, by Onsager's reciprocity relations, our results are immediately relevant to the inverse Edelstein eect34,50,51, in which a non-equilibrium spin density induces a charge current. The above features, although discussed here for a specic model, are expected to be gen- eral, proper to any non-strictly two-dimensional system in which the spin-orbit interaction 19is non-homogeneous across the conning direction. Technical applications of this idea could lead to a new class of spin-orbit-coupling-based devices. ACKNOWLEDGMENTS CG acknowledges support by CEA through the DSM-Energy Program (project E112-7- Meso-Therm-DSM). GV acknowledges support from NSF Grant No. DMR-1104788. Appendix A: Integrals of Green functions To perform the integral of Eq.(55) we exploit the poles with the Cauchy theorem of residues. We use the formulae37 X kGR nksGA nksf(k) = 2Nnsf(kFns); (A1) X kGR nkGA nk+f(k) =2N0 1i2
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0905.1818v1.Pauli_Spin_Blockade_in_the_Presence_of_Strong_Spin_Orbit_Coupling.pdf

arXiv:0905.1818v1 [cond-mat.mes-hall] 12 May 2009Pauli Spin Blockade in the Presence of Strong Spin-Orbit Cou pling J. Danon and Yu. V. Nazarov Kavli Institute of NanoScience, Delft University of Techno logy, 2628 CJ Delft, The Netherlands (Dated: October 25, 2018) We study electron transport in a double quantum dot in the Pau li spin blockade regime, in the presence of strong spin-orbit coupling. The effect of spi n-orbit coupling is incorporated into a modified interdot tunnel coupling. We elucidate the role of t he external magnetic field, the nuclear fieldsinthedots, andspinrelaxation. Wefindqualitativeag reement withexperimentalobservations, and we propose a way to extend the range of magnetic fields in wh ich blockade can be observed. Blockade phenomena, whereby strong interactions be- tween single particles affect the global transport or exci- tation properties of a system, are widely used to control and detect quantum states of single particles. In single electron transistors, the electrostatic interaction between electrons can block the current flow [1], thereby enabling precise control over the number of charges on the tran- sistor [2]. In semiconductor quantum dots, the Pauli ex- clusion principle can lead to a spin-selective blockade [3], which has provento be apowerful tool forread-outofthe spin degree of freedom of single electrons [4, 5, 6, 7, 8]. In this spin blockade regime, a double quantum dot is tuned such that current involves the transport cycle (0,1)→(1,1)→(0,2)→(0,1), (n,m)denotingacharge statewith n(m) excesselectronsintheleft(right) dot(see Fig. 1(a)). Since the only accessible (0 ,2) state is a spin singlet,thecurrentisblockedassoonasthesystementers a (1,1) triplet state (Fig. 1(b)): transport is then due to spin relaxation processes, possibly including interaction with thenuclearfields[9]. Thisblockadehasbeenusedin GaAs quantum dots to detect coherentrotationsofsingle electron spins [4, 5], coherent rotations of two-electron spin states [6], and mixing of two-electron spin states due to hyperfine interaction with nuclear spins [7, 8]. Motivated by a possibly large increase of efficiency of magnetic andelectric controloverthe spin states [10, 11], also quantum dots in host materials with a relatively largeg-factor and strong spin-orbit interaction are being investigated. Very recently, Pauli spin blockade has been demonstrated in a double quantum dot defined by top gates along an InAs nanowire [12, 13]. However, as com- pared to GaAs, spin blockade in InAs nanowire quantum dots seems to be destroyed by the strong spin-orbit cou- pling: significant spin blockade has been only observed at very small external magnetic fields ( <∼10 mT [12]). An important question is whether there exists a way to extend this interval of magnetic fields. To answer that question, one first has to understand the physical mech- anism behind the lifting of the blockade. In this work we study Pauli spin blockade in the pres- ence of strong spin-orbit mixing. We show that the only way spin-orbit coupling interferes with electron trans- port through a double dot is by introducing non-spin- conserving tunneling elements between the dots. Thisyields coupling of the (1 ,1) triplet states to the outgoing (0,2) singlet, thereby lifting the spin blockade. How- ever, for sufficiently small external magnetic fields this does not happen. If the (1 ,1) states are not split apart by a large Zeeman energy, they will rearrange to one coupled, decaying state and three blocked states. When the external field B0is increased, it couples the blocked states to the decaying state. As soon as this field in- duced decay grows larger than the other escape rates (i.e.B2 0Γ/t2>Γrel, where Γ is the decay rate of the (0,2) singlet, tthe strength of the tunnel coupling, and Γrelthe spin relaxation rate [14]) the blockade is lifted. Therefore, the current exhibits a dip at small fields. The presence of two random nuclear fields in the dots (of typical magnitude K∼1 mT) complicates matters since it adds another dimension to the parameter space. We distinguish two cases: If the nuclear fields are small compared to t2/Γ, they just provide an alternative way Lead Dot µL µRΓ t ∆(a) (b) ↑ ↑↑/xmarkbld Γ|T+〉|T–〉 |α〉 |β〉 |S02 〉Γso – Γso +Γso βΓrel ∆ (1,1) (0,2) (0,1) B0(c) FIG. 1: Double quantum dot in the Pauli spin blockade regime. (a) The double dot is coupled to two leads. Due to a voltage bias, electrons can only run from the left to the righ t lead. (b) Energy diagram assuming spin-conserving interdot coupling. The only accessible (0 ,2) state is a spin singlet: all (1,1) triplet states are not coupled to the (0 ,2) state and the current is blocked. (c) Energy levels and transition rates a s- sumingnon-spin-conserving interdot coupling. We consider the ‘high’-field limit and neglect the effects of the nuclear fields. Then three of the four (1 ,1) states can decay, leaving only one spin blockaded state |α/angbracketright. Isotropic spin relaxation ∼Γrelcauses transitions between all (1 ,1) states.2 to escape spin blockade, which may compete with spin relaxation. There is still a dip at small magnetic fields, and the current and width of the dip are determined by the maximum of Γ relandK2Γ/t2. In the second case, K≫t2/Γ, the current may exhibit either a peak or a dip, depending on the strength and orientation of the spin-orbit mixing. If there is a peak in this regime, the cross-over from dip to peak takes place at K∼t2/Γ. Let us now turn to our model. We describe the relative detuning of the (1 ,1) states and the (0 ,2) states by the Hamiltonian ˆHe=−∆|S02∝an}bracketri}ht∝an}bracketle{tS02|, where|S02∝an}bracketri}htdenotes the (0,2)spinsingletstate. The energiesofthe four(1 ,1) states are further split by the magnetic fields acting on the electronspins, ˆHm=B0(ˆSz L+ˆSz R)+/vectorKL·ˆ/vectorSL+/vectorKR·ˆ/vectorSR, whereˆ/vectorSL(R)istheelectronspinoperatorintheleft(right) dot (for InAs nanostructures g∼7 [16]). We chose the z-axis along /vectorB0, and included two randomly oriented ef- fective nuclear fields /vectorKL,Rresulting from the hyperfine coupling of the electron spin in each dot to Nnuclear spins (in InAs quantum dots N∼105[16], yielding a typical magnitude K∝1/√ N∼0.6µeV). We treat the nuclear fields classically, disregarding feedback of the electron spin dynamics which could lead to dynamical nuclear spin polarization [17]. Let us now analyze the possible effects of spin-orbit coupling. (i) It can mix up the spin and orbital structure of the electron states. The resulting states will remain Kramers doublets, thus giving no qualitative difference with respect to the common spin up and down doublets. (ii) The mixing also renormalizes the g-factor that de- fines the splitting of the doublets in a magnetic field. This, however, is not seen provided we measure B0in units of energy. (iii) The coupling also can facilitate spin relaxation [18], but this is no qualitative change either. Some of these aspects have been investigated in [19]. The only place where strong spin-orbit interaction leads to a qualitative change is the tunnel coupling be- tween the dots. It provides a finite overlap of states dif- fering in index of the Kramers doublet (in further dis- cussion we refer to this index as to ‘spin’), this allow- ing for a very compact model incorporating the inter- action. The most general non-‘spin’-conserving tunnel- ing Hamiltonian for two doublet electrons in a left and rightstate reads ˆHt=/summationtext α,β/braceleftBig tL αβˆa† LαˆaRβ+tR αβˆa† RαˆaLβ/bracerightBig , α,βbeing the spin indices, ˆ a† L(R)and ˆaL(R)the elec- tron creation and annihilation operatorsin the left(right) state, and tL,Rcoupling matrices. We impose condi- tions of hermiticity and time-reversibility on ˆHtand concentrate on the matrix elements between the (1 ,1) states and |S02∝an}bracketri}htin our double dot setup. In the con- venient basis of orthonormal unpolarized triplet states |Tx,y∝an}bracketri}ht ≡i1/2∓1/2{|T−∝an}bracketri}ht∓|T+∝an}bracketri}ht}/√ 2,|Tz∝an}bracketri}ht ≡ |T0∝an}bracketri}ht, and the (1,1) singlet |S∝an}bracketri}ht, this Hamiltonian reads ˆHt=i/vectort·/vector|T∝an}bracketri}ht∝an}bracketle{tS02|+t0|S∝an}bracketri}ht∝an}bracketle{tS02|+h.c.,(1)with/vector|T∝an}bracketri}ht ≡ {|Tx∝an}bracketri}ht,|Ty∝an}bracketri}ht,|Tz∝an}bracketri}ht}. The model therefore adds a 3-vector of new coupling parameters, /vectort={tx,ty,tz}, to the usual spin-conserving t0, the vector being a ‘real’ vector with respect to coordinate transformations. If the energy scale of spin-orbit interaction is larger or compa- rable to the energy distance between the levels in the dot (which is believed to be the case in InAs structures), the mixing of the doublet components is of the order of 1. Therefore we assume that all four coupling parameters are generally of the same order of magnitude t0,x,y,z∼t. As the structure of the localized electron wave functions is very much dependent on the nanostructure design and its inevitable imperfections, the direction of /vectortis hard to predict: therefore we consider arbitrary directions. We describe the electron dynamics with an evolution equation for the density matrix [9]. Next to the Hamil- tonian terms, we complement the equation with (i) the rates∼Γ describing the decay of |S02∝an}bracketri}htand the refill to a (1,1) state, and (ii) a small electron spin relaxation rate Γ rel≪Γ. The full evolution of the electron density matrix then can be written as dˆρ dt=−i[ˆHe+ˆHm+ˆHt,ˆρ]+Γˆρ+Γrelˆρ.(2) Experimentally, the temperature exceeds the Zeeman en- ergy [12], allowing us to assume isotropic spin relaxation: each (1,1) state will transit to any of the other (1 ,1) states with a rate Γ rel/3. Explicitly, we use Γrelˆρ= −Γrelˆρ+1 6Γrel/summationtext α,dˆσα dˆρˆσα d, ˆσα L(R)being the Pauli ma- trices in the left(right) dot. Motivated by experimental work, we assume that the decay rate Γ of |S02∝an}bracketri}htis by far the largest frequency scale in(2), i.e.Γ ≫B0,K,t,Γrel(inprincipleΓcanbecompa- rable with the detuning ∆). Under this assumption, we separatethe time scalesand derivethe effective evolution equation for the density matrix in the (1 ,1) subspace dˆρ dt=−i[ˆHm+ˆH′ t,ˆρ]−Goutˆρ+Ginˆρ+Γrelˆρ.(3) The decay and refill terms are now incorporated into Gout kl,mn= 2{δkmTn2T2l+δlnTk2T2m}Γ/(Γ2+4∆2) Gin kl,mn=δklTn2T2mΓ/(Γ2+4∆2),(4) whereTa2≡ ∝an}bracketle{ta|ˆHt|S02∝an}bracketri}ht. The coupling between the dots gives also rise to an exchange Hamiltonian ( H′ t)ij= 4∆/(Γ2+4∆2)Ti2T2j, withH′ t∼Goutprovided Γ ∼∆. The diagonal elements of Goutgive us the decay rates: If we consider |T±∝an}bracketri}htand|T0∝an}bracketri}ht, the three triplet states split by an external magnetic field, we find Γso ±≡Gout ±±,±±= 2Γ(t2 x+t2 y)/(Γ2+ 4∆2) and Γso 0≡Gout 00,00= 4Γt2 z/(Γ2+ 4∆2), all of which are ∼Γso∼t2/Γ. Let us neglect for a moment the nuclear fields, and focus on zero detuning, ∆ = 0. This allows us to grasp qualitatively the peculiarities of the spin blockade lifting,3 determined by competition between the Hamiltonian ( ∼ B0) and dissipative terms ( ∼t2/Γ,Γrel) in Eq. 3. At sufficiently large fields, the basis states |T0∝an}bracketri}htand |S∝an}bracketri}htare aligned in energy. The spin-orbit modulated tunnel coupling then sets the difference between these states, which is best seen in a basis that mixes the states,|α∝an}bracketri}ht ≡ {t0|T0∝an}bracketri}ht+itz|S∝an}bracketri}ht}//radicalbig t2 0+t2zand|β∝an}bracketri}ht ≡ {itz|T0∝an}bracketri}ht+t0|S∝an}bracketri}ht}//radicalbig t2 0+t2z. Now|α∝an}bracketri}htis a blocked state, i.e.Gout αα,αα= 0, while |β∝an}bracketri}htdecays with an effective rate Γso β≡Gout ββ,ββ= 4Γ(t2 0+t2 z)/(Γ2+4∆2). In Fig. 1(c) we give the energy levels of the five states and all transition rates in the limit of ‘large’ external fields. It is clear that the system will spend most time in the state |α∝an}bracketri}ht. The current is determined by the spin-relaxation decay rate of this state to any unblocked state, 3Γ rel/3 = Γrel. Let us note that if nbstates out of nstates are blocked, such a decay produces on average n/nbelectrons tunneling to the outgoing lead before the system is recaptured in a blocked state. Therefore, the current is I/e= 4Γrel. This picture holds until the decay rates of the three non-blocked states become comparable with Γ rel, which takes place at B0∼√ΓsoΓrel. To understand this, let us startwith consideringtheoppositelimit, B0≪√ΓsoΓrel. In this case all four (1 ,1) states are almost aligned in energy, and the instructive basis to work in is the one spanned by a single decaying state |m∝an}bracketri}ht ≡ {i/vectort·/vector|T∝an}bracketri}ht+ t0|S∝an}bracketri}ht}//radicalBig |/vectort|2+t2 0, and three orthonormal states |1∝an}bracketri}ht,|2∝an}bracketri}ht and|3∝an}bracketri}htthat are not coupled to |S02∝an}bracketri}ht. AtB0= 0 three of the four states are blocked, and spin relaxation to the unblocked state proceedswith a rate Γ rel/3. A relaxation process produces on average n/nb= 4/3 electron trans- fers, so that the total current is reduced by a factor of 9 in comparison with the ‘high’-field case, I/e=4 9Γrel. Thisfactorof9agreesremarkablywellwith experimental observations (see Fig. 2b in Ref. [12]). We now add a finite external field B0to this picture. Since/vectortis generally not parallel to B0, the external field will split the states |1∝an}bracketri}ht,|2∝an}bracketri}htand|3∝an}bracketri}htin energy and mix two of them with the decaying state |m∝an}bracketri}ht. This mixing results in an effective decay rate ∼B2 0/Γso, which may compete with the spin relaxation rate Γ rel. AtB0∼√ΓsoΓrel, we cross over to the ‘high’-field regime described above, where only one blocked state is left. Therefore, the cur- rent exhibits a dip (suppression by a factor 9) around zero field with a width estimated as√ΓsoΓrel(Fig. 2). Letusnowincludetheeffectsofthenuclearfields /vectorKL,R on a qualitative level. If the fields are small compared to the scale t2/Γ, their only relevant effect is to mix the states described above. This mixing creates a new pos- sibility for decay of the blocked states, characterized by a rate Γ N∼K2/Γso. This rate may compete with spin relaxation ∼Γrel, and could cause the current to scale with Γ Nand the width of the dip with K. In the oppo- site limit, K≫t2/Γ, the nuclear fields dominate the en- ergy scales and separation of the (1 ,1) states at B0<∼K.Then, generally all four states are coupled to |S02∝an}bracketri}hton equal footing and the spin blockade is lifted. Qualita- tively, this situation is similar to that without spin-orbit interaction (see Eqs 10-12 in [9]). Without spin-orbit in- teraction, an increase of magnetic field leads to blocking of two triplet states, resulting in a current peak at zero field.Withspin-orbit interaction, tx,ystill couple the split-off triplets to the decaying state. Depending on the strength and orientation of /vectort, the current in the limit of ‘high’ fields can be either smaller or larger than that at B0= 0, so we expect either peak or dip. If it is a peak, the transition from peak to dip is expected at K∼Γso, that is, at t∼√ KΓ. Indeed, such a transition has been observed upon varying the magnitude of the tunnel cou- pling (Fig. 2 in Ref. [12]). If we assume that K∼1.5 mT and associate the level broadening observed ( ∼100µeV) with Γ, we estimate t∼8µeV, which agrees with the range of coupling energies mentioned in [12]. Let us now support the qualitative arguments given above with explicit analytical and numerical solutions. The current through the double dot is evaluated as I/e=ρ22Γ,ρ22being the steady-state probability to be in|S02∝an}bracketri}ht, as obtained from solving Eq. 2. We give an an- alytical solution for ∆ = 0, neglecting the nuclear fields, and expressing the answer in terms of the dimensionless parameter /vectort/t0=/vector η. Under these assumptions, we find I=Imax/parenleftbigg 1−8 9B2 c B2+B2c/parenrightbigg , (5) withBc= 2√ 2(1 +|/vector η|2)(η2 x+η2 y)−1/2t0/radicalbig Γrel/Γ and Imax= 4eΓrel. The current exhibits a Lorentzian-shaped dip (see Fig. 2, compare with Fig. 2b in Ref. [12]). The widthBcand the limits at low and ‘high’ fields agree with the qualitative estimations given above. To include the effect of the two nuclear fields, we com- pute steady-state solutions of (2) and average over many configurationsof /vectorKL,R[9]. InFig.3wepresenttheresult- ing current versus magnetic field and detuning for three different regimes. To produce the plots we turned to concrete values of the parameters, setting Γ = 0.1 meV, Γrel= 1 MHz, /vector η= 0.25× {1,1,1}. We averaged over 5000 configurations of /vectorKL,R, randomly sampled from a -40 -20 0 20 40 01234I [e Γrel ] B0 [t 0(Γrel /Γ)½]η = {.5,.5,.5} → η = {.1,.1,.5} → η = {0,0,.5} → FIG. 2: Current as a function of B0, at ∆ = 0, and neglecting the nuclear fields. Around zero field a dip is observed, its width depends on the magnitude and orientation of /vector η.4 010 20 ∆ = 0 04812 I [pA] I [pA] B0 [mT] ∆ [ µeV] (a) (b) (c) (d) t20/Γ = 0.2 µeV t20/Γ = 6 µeV ∆ = 0 0 400 800 0404812 -40 0 40 B0 = 0 mT B0 = 2 mT 0.1 0.2 0.3 0I [pA] t20/Γ = 2 neV B0 = 80 mT ∆ = 0 0.1 0.2 0.3 0B0 = 0 mT B0 = 2 mT B0 = 80 mT (e) (f) 30 812 16 B0 = 0 mT B0 = 80 mT B0 = 10 mT B0 = 5 mT FIG. 3: The current I=eρ22Γ for (a,b) large, (c,d) intermedi- ate, and (e,f) small tunnel coupling. The dip observed aroun d zero field (a) disappears when t2 0/Γ∼K(c) and evolves into a peak for even smaller tunnel coupling (e). normal distribution with a r.m.s. of 0.4 µeV. In Fig. 3(a) and (b) we assumed large tunnel coupling, t2 0/Γ = 6µeV so that KΓ/t2 0= 0.07 is small. In (a) we plot the current at ∆ = 0, while in (b) we plot it ver- sus detuning for different fixed B0. We observe in (a) a Lorentzian-like dip in the current at B0= 0. While it looks similar to the plots in Fig. 2, the width is de- termined by the nuclear fields since K≫Γrel. The curvecanbeaccuratelyfitwiththeLorentzian(5), giving Bc= 7.4KandImax= 0.62K2Γ/t0. Fig.3(b)illustrates the unusual broadeningofthe resonantpeakwith respect to its natural width determined by Γ. The width in this case scales as ∼t2 0/Kand is determined by competition of Γsoand ΓN. These plots qualitatively agree with data presented in Fig. 2b in Ref. [12]. In Fig. 3(c) and (d) we present the same plots, for smaller tunnel coupling, t2 0/Γ = 0.2µeV = 0.5K. We included in plot (c) the curves for two random nuclear field configurations: It is clear that the current strongly depends on /vectorKL,R, which agrees with our expectation that in the regime Γ rel<ΓN the current I∝ΓN∝K2. Remarkably, averaging over many configurations smooths the sharp features at small B0(c.f. [9]). Plots (d) exhibit no broadening with re- spect to Γ, in correspondence with Fig. 2a of Ref. [12]. In Fig. 3(e) and (f) we again made the same plots for yet smaller tunnel coupling, t2 0/Γ = 2 neV ≪K. Since the nuclear fields now dominate the splitting of the (1 ,1) states, we see a peak comparable to the one in Fig. 4 of Ref. [9] surmounting a finite background current due to spin-orbit decay of the split-off triplets. We expect our results to hold for any quantum dot system with strong spin-orbit interaction. Indeed, recent experiments on quantum dots in carbon nanotubes in the spin blockade regime [15] display the very same specificfeatures, as e.g. a zero-field dip in the current. Nowthat we understand the originofthe lifting ofspin blockade, we also propose a way to extend the blockade region. If one would have a freely rotatable magnet as source of the field B0, one would observe a large increase in width of the blockade region as soon as /vectorB0and/vectortare parallel. One can understand this as follows. If /vectorteffec- tively points along the z-direction, txandtyand thus Γso ±are zero: the states |T±∝an}bracketri}htare blocked (see Fig. 2). As |T±∝an}bracketri}htare eigenstates of the field B0, this blockade could persist up to arbitrarily high fields. Since |T0∝an}bracketri}htand|S∝an}bracketri}ht are rotated into |α∝an}bracketri}htand|β∝an}bracketri}ht, current will then scale in general with the anti-parallel component of spin instead of only the spin singlet. To conclude, we presented a model to study electron transport in the Pauli spin blockade regime in the pres- ence of strong spin-orbit interaction. It reproduces all features observed in experiment, such as lifting of the spin blockade at high external fields or at low interdot tunnel coupling. We explain the mechanisms involved and identify all relevant energy scales. We also propose a simple way to extend the region of spin blockade. We acknowledge fruitful discussions with A. Pfund, S. Nadj-Perge, S. Frolov, and K. Ensslin. This work is part of the research program of the Stichting FOM. [1] T. A. Fulton and G. J. Dolan, Phys. Rev. Lett. 59, 109 (1987); D. V. Averin and K. K. Likharev, J. Low Temp. Phys.62, 345 (1986). [2] R. C. Ashoori, et al., Phys. Rev. Lett. 68, 3088 (1992). [3] K. Ono, et al., Science 297, 1313 (2002); K. Ono and S. Tarucha, Phys. Rev. Lett. 92, 256803 (2004). [4] F. H. L. Koppens, et al., Nature 442, 766 (2006). [5] K. C. Nowack, et al., Science 318, 1430 (2007). [6] J. R. Petta, et al., Science 309, 2180 (2005). [7] F. H. L. Koppens, et al., Science 309, 1346 (2005). [8] A. C. Johnson, et al., Nature 435, 925 (2005). [9] O. N. Jouravlev and Yu. V. Nazarov, Phys. Rev. Lett. 96, 176804 (2006). [10] C. Flindt, et al., Phys. Rev. Lett. 97, 240501 (2006). [11] V. N. Golovach, et al., Phys. Rev. B 74, 165319 (2006). [12] A. Pfund, et al., Phys. Rev. Lett. 99, 036801 (2007). [13] A. Pfund, et al., Physica E 40, 1279 (2008). [14] Throughout the paper we present energies and magnetic fields in terms of frequencies. This corresponds to setting ¯h=gµB= 1. [15] H. O. H. Churchill, et al., Phys. Rev. Lett. 102, 166802 (2009). [16] A. Pfund, et al., Phys. Rev. B 76, 161308 (2007). [17] J. Danon, et al., arXiv:0902.2653 (2009); M. S. Rudner and L. S. Levitov, Phys. Rev. Lett. 99, 036602 (2007). [18] A. V. Khaetskii and Yu. V. Nazarov, Phys. Rev. B 64, 125316 (2001). [19] C. L. Romano et al, arXiv:0801.1808 (2008); H.-Y. Chen et al., J. Phys.: Cond. Matt. 20, 135221 (2008).

1110.0114v1.Spin_orbital_phase_synchronization_in_the_magnetic_field_driven_electron_dynamics_in_a_double_quantum_dot.pdf

arXiv:1110.0114v1 [nlin.CD] 1 Oct 2011Spin-orbital phase synchronization in the magnetic field-d riven electron dynamics in a double quantum dot. L. Chotorlishvili,1,4E.Ya. Sherman,2,3Z. Toklikishvili,5, A. Komnik,4and J. Berakdar1 1Institut f¨ ur Physik, Martin-Luther Universit¨ at Halle-W ittenberg, Heinrich-Damerow-Str.4, 06120 Halle, Germany 2Departamento de Quimica F´ ısica, Universidad del Pa´ ıs Vasco UPV/EHU, 48080 Bilbao, Spain 3IKERBASQUE, Basque Foundation for Science, E-48011 Bilbao , Spain 4Institut f¨ ur Theoretische Physik, Universit¨ at Heidelbe rg, Philosophenweg 19, D-69120 Heidelberg,Germany 5Physics Department of the Tbilisi State University, Chavchavadze av.3, 0128, Tbilisi, Georgia Abstract We study the dynamics of an electron confined in a one-dimensi onal double quantum dot in the presenceofdrivingexternalmagneticfields. Theorbitalmo tionoftheelectroniscoupledtothespin dynamics by spin orbit interaction of the Dresselhaus type. We derive an effective time-dependent Hamiltonian model for the orbital motion of the electron and obtain a synchronization condition between the orbital and the spin dynamics. From this model we deduce an analytical expression for the Arnold tongue and propose an experimental scheme for realizing the synchronization of the orbital and spin dynamics. 1I. INTRODUCTION Phase synchronization and related phenomena are among the most fascinating effects of nonlinear dynamics. Besides the deep fundamental interest[1–7], p hase synchronization has a broad range of applications in chemistry [8], ecology [9], astronomy [10], in the field of informationtransferusingchaoticsignals[11], andforthecontrol ofhighfrequencyelectronic devices [12]. In nonlinear dissipative systems, phase synchronizatio n occurs if the frequency of the external driving field is close to the eigenfrequency of the sy stem. In this case, for a certain frequency interval of the driving field the oscillations of th e nonlinear dissipative system can be synchronized with the perturbing force. Usually, fo r stronger driving fields the frequency interval for the synchronization becomes broade r and the synchronization protocol is more efficient. The broad and growing interest in the pha se synchronization calls for the analysis of new possible realizations of this phenomenon. A pa rticularly interesting issue is the application of the synchronization protocols for magnet ic nanostructures, which have rich applications [11–16] and exhibit interesting nonlinear dyna mical properties that can be exploited as a testing ground for dynamical systems [17, 18]. A principal challenge in nanoscience is to find an efficient procedure fo r the manipulation of the systems states. A high level of accuracy on the state cont rol is required especially in such applications as quantum computing, where a precise tailoring of the entangled states is highly desirable [19, 20]. With this in mind several physical systems we re considered up to now, e. g. Josephson junction qubits and Rydberg atoms in a qua ntum cavities [19, 20], ion traps [21], single molecular nanomagnets [22], and nanoelectrome chanical resonators [23, 24]. Among others, one of the most promising systems are elect ron spins confined in two-dimensional quantum dots [25–27] and in one-dimensional nan owires and nanowire- based quantum dots [28–31]. The key element of the corresponding models is the spin orbit (SO) coupling term, which is linear in the electron momentum. Such mom entum-dependent coupling offers a new way of manipulating the spin by changing the elect ron momentum via a periodic electric field. This is the idea of the electric-dipole spin reson ance proposed by Rashba and Efros for the electrons confined in nanostructures o n the scale of 10 nm [27]. However, the external electric field can strongly affect the orbita l dynamics and thus the system is driven out of the linear regime [32]. The nonlinearities usually r esult in a complex behavior of the affected systems and their dynamics might become c omplicated and even 2unpredictable. On the other hand, the nonlinearity may also lead to a number of interesting phenomena. In spite of the huge interest in the systems with SO cou pling, the influence of the spin dynamics on the orbital motion was not addressed yet in f ull detail. With the present work we would like to bridge this gap. Our goal is to investigat e the possibilities of controlling the orbital motion of the electron via external magnetic fields acting on its spin and via the spin-orbit coupling. That can be considered as opposite t o the electric-dipole spin resonance protocol proposed by Rashba and Efros [27]. We will demonstrate that: (i) by using an external driving field one can achieve a sufficient degree o f control over the orbital motion, and (ii) as a result, one can devise a very efficient syn chronization protocol of the orbital motion and the spin dynamics based on the application o f a pulsed external magnetic field. II. THEORETICAL MODEL We consider a model system of a single electron confined in a double qu antum dot de- scribed by a potential of the form U(x) =U0/bracketleftbig −2(x/d)2+(x/d)4/bracketrightbig . HereU0is the energy barrier separating two minima with 2 dbeing the distance between them. We assume the system is dissipative, and the dissipation is a thermal effect appearin g due to a coupling to environment. The dissipation, which impacts mainly on the orbital m otion, is essen- tial for the synchronization processes we are going to discuss late r in the text. For strong driving magnetic fields, the influence of the dissipation on the spin dyn amics is negligibly small and can be ignored. In addition, we assume that the temperat ure is low enough to prevent the activated over-the-barrier motion. For the particu lar value of U0∼20 meV the low-temperature regime means T <100 K. For the GaAs-based structure with the electron effective mass mbeing 0.067 of the free electron mass and d∼100 nm, the tunneling prob- ability is small and a classical consideration is justified. To quantify th e SO interaction we use a coupling term of the Dresselhaus type Hso=αPxσx, wherePxis the momentum of the electron and σxis the Pauli matrix. Therefore, the Hamiltonian of the one dimensiona l system reads: H=P2 x 2m+U(x)+αPxσx+µBgBz(t)σz 2+µBgBx(t)σx 2, (1) 3 FIG. 1. Schematic illustration of the infinite series of exte rnal magnetic field pulses applied to the system. A series of the short pulses Bz(t) =B0∞/summationtext t=0δτ(t−nT) with the pulse width τand time interval between pulses T, is applied along the zaxis. Series of the pulses with the larger width T and a shorter interval between the pulses τ,Bx(t) =B0∞/summationtext t=0∆T(t−τn) is applied along the xaxis. The amplitude of the pulses B0is the same in both cases. whereµBistheBohrmagnetonand gistheelectronLand´ efactor. Here Bz(t) =B0∞/summationtext t=0δτ(t− nT) is an infinite series of external magnetic field pulses with the pulse st rengthB0, which is applied along the zaxis. The temporal width of the pulses applied along the z-axis is smaller than the interval between the pulses τ≪T(in what follows we set T= 1). On the other hand, for the pulses along the x-axisBx(t) =B0∞/summationtext t=0∆T(t−τn) the pulse duration is larger than interval between pulses T≪τ(cf. Fig. 1). A different route to the control of the spin-dynamics in double quantum dots via electric field pulses is out lined in [33]. Introducing the characteristic maximum momentum of the electron Pmax x=√2mU0 we can estimate the maximal precession rate of the spin due to the S O coupling Ωmax so= (2α//planckover2pi1)√2mU0, while the magnetic field pulse of the amplitude B=B0induces a spin precession with the rate Ω B=µB|g|B0//planckover2pi1. Therefore, if Ω B>Ωmax sowe can during the pulse neglect the spin rotation produced by the SO coupling and the spin is c ompletely controlled by the external driving fields. We need a protocol with two driving fie lds in order to fulfill the synchronization requirements as discussed later in the text. N amely, for the control of the spin dynamics via the external driving fields, the amplitudes of th e fields should belarge, B0>(2α/µB|g|)√2mU0. On the other hand, a strong constant magnetic field produces a high frequency precession of the spin Ω B=µB|g|B0//planckover2pi1, while for the synchronization we need to tune the precession frequency up or down keeping fixed th e strong driving field 4amplitude. Below we will show that the optimal conditions for the sync hronization are realized using two types of the driving pulses. Applying short pulses a long thez-axisτ≪T, Bz(t) =B0∞/summationtext t=0δτ(t−nT) and long pulses along the x-axisBx(t) =B0∞/summationtext t=0∆T(t−τn), we can realize a spin precession with a frequency, which is inversely prop ortional to the time interval between the short pulses Ω ∼1/Tindependently from the driving field strength B0. In what follows, for convenience we use dimensionless units via the t ransformations E→E/4U0,x→x/d,t→t/radicalbig 4U0/m,Px→Px/√2mU0,ε→α/4U0. III. DISSIPATIVE SYSTEM AND THE PROBLEM OF PHASE SYNCHRONIZA- TION BETWEEN ORBITAL AND SPIN MOTION A. Spin dynamics in pulsed magnetic fields As was stated above the synchronization can occur if the frequen cy of the driving field is close to the eigenfrequency of the nonlinear dissipative system. I f this is the case, in the particular frequency interval, the oscillations of the nonlinear dissip ative system and the field can be synchronized. With the increase in the driving field amplitud e, the synchroniza- tion can occur in a broader frequency interval, and the synchroniz ation protocol becomes more efficient. Our aim is to develop a method for the synchronization of the dynamics of the electron spin and the orbital motion, using an external drivin g magnetic field and SO coupling. Although the magnetic field is not coupled to the orbital m otion directly, a sufficiently strong field influences the orbital motion through the sp in dynamics if the SO coupling is present. If SO term is relatively small Ωmax so<ΩB, that is ΩB=µB|g|B0 /planckover2pi1>Ωmax so=2α /planckover2pi1/radicalbig 2mU0, (2) the spin and, correspondingly, the orbital motion can be controlled externally. From Eq. (1) it is easy to see, that in between the short pulses the electron spin r otates around the x-axis and the equations of motion for the electron spin in this case read ˙σx= 0,˙σy=−ΩBσz,˙σz= ΩBσy. (3) On the other hand, during the short pulses we have ˙σx=−ΩBσy,˙σy= ΩBσx,˙σz= 0. (4) 5Considering the dynamics due to the pulse acting on the spin at the mo ment of time t=t0, we can split the evolution operator ˆTevdefined as σ/parenleftBig t[−] 0+T/parenrightBig =ˆTevσ/parenleftBig t[−] 0/parenrightBig (5) into two parts: ˆTev=ˆTR׈Tδ, where σ/parenleftBig t[+] 0/parenrightBig =ˆTδσ/parenleftBig t[−] 0/parenrightBig , σ/parenleftBig t[−] 0+T/parenrightBig =ˆTRσ/parenleftBig t[+] 0/parenrightBig . (6) Here we introduced the notations t[+] 0≡t0+0 andt[−] 0≡t0−0. The operator ˆTRdescribes the rotationof the electron spin around the x-axis produced by the long pulse of the external driving magnetic field Bx(t) =B0∞/summationtext t=0∆T(t−τn), which is applied along the x-axis and ˆTσ corresponds to the evolution produced by the short pulses Bz(t) =B0∞/summationtext t=0δτ(t−nT) applied along the z-axis. Integrating Eq. (4) for a short time interval t∈/parenleftBig t[−] 0,t[+] 0/parenrightBig we obtain ˆTδ(σx) =σx(t[−] 0)cos(Ω Bτ)−σy(t[−] 0)sin(Ω Bτ), (7) ˆTδ(σy) =σx(t[−] 0)sin(Ω Bτ)+σy(t[−] 0)cos(Ω Bτ). (8) Integrating Eq. (3) during the time interval t∈/parenleftBig t[+] 0,t[−] 0+T/parenrightBig of long applied pulse we find ˆTR(σy) =/radicalbigg 1−/parenleftBig σx/parenleftBig t[+] 0/parenrightBig/parenrightBig2 cos(ΩBT), (9) ˆTR(σz) =/radicalbigg 1−/parenleftBig σx/parenleftBig t[+] 0/parenrightBig/parenrightBig2 sin(ΩBT). (10) Combining Eq. (7) with Eq. (9) we finally can reconstruct the complet e picture of the full time evolution of the electron spin: σy n+1=/radicalbig 1−(σx n+1)2cos((n+1)Ω BT), σz n+1=/radicalbig 1−(σx n+1)2sin((n+1)Ω BT), σx n+1=σx ncos(ΩBτ)−/radicalbig 1−(σxn)2sin(ΩBτ)cos(nΩBT).(11) The recurrent relations Eq. (11) describe the spin dynamics. Accu racy of the employed approximations may be checked by the validity of the normalization co nditionσ2= 1.In order to identify, whether the nonlinear map (11) is chaotic or regu lar, we evaluate the Lyapunov exponent for the spin system [34]. Taking into account th e peculiarity of the 6 FIG. 2. Lyapunov exponent as a function of the initial spin co mponent σx 0. After N= 1000 iterations theLyapunov exponent is negative λ(σx 0)<0, meaningthat the spindynamics is regular. T= 1, Ω Bτ= 1, Ω BT= 20. At these conditions, the dependence is weak. system (11), which is the fact, that the equation for the xcomponent σx n+1is self-consistent σx n+1=f(σx n), we deduce for the Lyapunov exponent λ(σx 0) = lim N→∞ δσ→01 Nln/vextendsingle/vextendsingle/vextendsingle/vextendsinglefN(σx 0+δσ)−fN(σx 0) δσ/vextendsingle/vextendsingle/vextendsingle/vextendsingle= lim N→∞1 Nln/vextendsingle/vextendsingle/vextendsingle/vextendsingledfN(σx 0) dσx 0/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(12) Here,σx 0is the initial value of the spin projection and the small increment of th e initial valuesσx 0+δσquantifies the sensitivity of the recurrence relations (11) with res pect to the slight change in the initial conditions. After some algebra from Eqs. ( 11) and (12) we finally obtain: λ(σx 0) = lim N→∞1 NN−1/summationdisplay n=0ln/vextendsingle/vextendsingle/vextendsingle/vextendsingledf(σx n) dσx n/vextendsingle/vextendsingle/vextendsingle/vextendsingle=1 NN−1/summationdisplay n=0ln/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsinglecos(ΩB)+σx n/radicalbig 1−(σx n)2sin(ΩB)cos(nΩB)/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle. (13) The results of the numerical calculations are presented on Figs. 2 a nd 3. From Fig. 2 we see that the Lyapunov exponent is negative, λ(σx 0)<0 and therefore the spin dynamics is regular, since the initial distance between two neighboring trajec tories starting from the initial points σx 0andσx 0+δσis not increasing asymptotically after an infinite number of iterations δσeNλ(σ0). Therefore, from Figs. 2 and 3 we conclude, that the dynamics of the electron spin is controlled by the magnetic field pulses, thus follow ing equation σx(t) = σx 0cos(Ωt) we achieve the spin manipulation by magnetic fields. The spin rotation frequency is determined by the time interval between the short pulses Ω ≈1/T. 7 FIG. 3. Time dependence of the spin projection σx n≡σx(n).T= 1, Ω Bτ= 1, Ω BT= 20. From Fig. 3 we see that σx n=σx(t) is periodic in time and the frequency of oscillations Ω is in versely proportional to the time lapse between pulses Ω ≈1/T. B. Synchronization of the spin and the orbital motion. With the spin dynamics discussed in the previous subsection, for the orbital motion of electron we can write the following effective Hamiltonian assuming that σx(t)≈cos(Ωt): H=P2 x 2m+U(x)+αPxcos(Ωt). (14) The equation of motion for the system (14) has the form: ¨x+γ˙x−x+x3=−βsin(Ωt), (15) where two new dimensionless quantities are introduced: β=αΩm/4U0andγ→γ/4U0m. We seek a solution of Eq. (15) using the following ansatz x(t) =1 2A(t)eiΩt+1 2A∗(t)e−iΩt. (16) Assuming that the amplitude A(t) in Eq. (16) is a slow variable the following condition applies ˙A(t)eiΩt+˙A∗(t)e−iΩt= 0. (17) 8 FIG. 4. Arnold tongue diagram in terms of the decay constant γand field frequency Ω, plotted using the conditions (25). Shadowed regions define paramete r values for which the synchronization is possible. Taking into account Eqs. (16) and (17) from Eq. (15) we deduce: iΩ˙A(t)eiΩt−/parenleftbiggΩ2 2A(t)eiΩt+c.c./parenrightbigg + +γ/parenleftbiggiΩ 2A(t)eiΩt+c.c./parenrightbigg −/parenleftbigg1 2A(t)eiΩt++c.c./parenrightbigg + (18) +1 8/parenleftbig A3(t)e3iΩt+3|A(t)|2A(t)eiΩt+c.c./parenrightbig =β 2i/parenleftbig eiΩt−c.c./parenrightbig . Multiplying Eq. (18) by the exponent e−iΩtand averaging it over the fast phases we find: ˙A(t)+(Ω2)+1)i 2ΩA(t)+γ 2A(t)−3i 8Ω|A(t)|2A(t) =−β 2Ω. (19) Introducing the notations A(t) = 2√γz(t), τ=tγ 2,∆ =Ω2+1 γΩ, ε=β 2Ωγ3/2, (20) we can rewrite Eq. (19) in a more compact form ˙z(τ)+i∆z(τ) =−z(τ)+3i Ω|z(τ)|2z(τ)+ε. (21) Inserting z(τ) =R(τ)eiϕ(τ)into Eq. (21), for the real and imaginary parts we obtain ˙R(τ) =−R+εcosϕ(τ), ˙ϕ(τ)+∆ =3 ΩR2(τ)−ε R(τ)sinϕ(τ).(22) 9 FIG. 5. Arnold tongue in terms of the parameters ∆ =/parenleftbig Ω2+1/parenrightbig /γΩ, (∆,ε) plotted using Eq. (26). Pointbbelongs to the domain where a synchronization is not possibl e, while abelongs to the synchronization domain. Using Eq. (22) and setting ˙R= 0, ˙ϕ= 0 for the stationary solutions we obtain: f(ξ) =ξ+ξ/parenleftbigg3 Ωξ−∆/parenrightbigg2 =ε2, (23) ξ1,2=Ω 9/parenleftBig 2∆±√ ∆2−3/parenrightBig , (24) whereR2=ξandξ1,2are rootsof the equation df(ξ)/dξ= 0. In order to identify the Arnold tongue [35], which shows the regions in which a synchronization is poss ible, we utilize the standard condition df(ξ)/dξ= 0. From Eq. (24) it is not difficult to see that the roots of the equation df(ξ)/dξ= 0 are real if ∆ >√ 3. Taking into account that ∆ = (Ω2+1)/γΩ we can rewrite the inequality in the form Ω2+ 1> γΩ√ 3. Consequently we obtain the following criteria for the synchronization 0<Ω<1 2(γ√ 3−/radicalbig 3γ2−4),Ω>1 2(γ√ 3+/radicalbig 3γ2−4), γ >2√ 3(25) Graphical representation of the conditions (25) is shown in Fig. 4. Eq. (25) defines the synchronization area in terms of the externa l field frequency Ω and the dissipation constant γ. The minima points of the function f(ξ), (23) does not depend on the SO coupling constant α. Therefore criteria (25) is independent of the values 10 FIG. 6. Orbital and spin dynamics plotted using the exact num erical integration of the Eq. (15). Panel a) corresponds to the values of parameters for which a s ynchronization is possible (see Fig. 5, point a). In particular/parenleftBig ε/√ Ω = 4.41,∆ = 1.88/parenrightBig . The right panel b) corresponds to the point b on Fig.5,/parenleftBig ε/√ Ω = 1.05,∆ = 3.03/parenrightBig , i.e. the dynamics occurs outside of the synchronization ar ea. We see that in the former case the orbital and the spin dynamic s are in phase and in the latter case the phase difference is about π/2 . of the SO coupling strength as well. Nevertheless, inserting the roo tsξ1,2of the equation df(ξ)/dξ= 0 into Eq. (23) one can derive more illustrative and precise criteria in the form of the parametrical curve: 2 81(±3√ ∆2−3+9∆+∆3∓∆2√ ∆2−3)−/parenleftbiggε√ Ω/parenrightbigg2 = 0. (26) The parametrical curve defined by Eq. (26) represents the bord er of the synchronization domain, see Fig. 5. Taking into account that β=αΩm/4U0,ε=β/2Ωγ3/2we obtain ε√ Ω=α√ Ωm 8U0γ3/2. (27) From Eq. (27) we see that the parameters of Fig. 5 depend on the o scillation frequency that can be easily controlled by tuning the time interval between puls es Ω≈1/T. All other parameters in Eq. (27), such as the SO coupling constant α, barrier height U0, and the electron effective mass mare internal characteristics of the system whereas the decay constant γis related to the thermal effects. Using Eq. (26) and Fig. 5 one can s ynchronize the electron orbital motion with its spin dynamics. 11IV. CONCLUSIONS. We have investigated the classical electron dynamics in a double dot p otential, with the spin of electron being controlled by external magnetic fields. We hav e shown that the orbital electrondynamicscanbecontrolledveryeffectively bythefieldinthe presenceofaspin-orbit coupling. Using the proposed protocol of magnetic field pulses of diff erent duration we have shown that it is possible to synchronize the spin and the orbital motio n of the electron. In particular, if the driving field amplitude is large enough, B0>(2α/µB|g|)√2mU0, the spin dynamics is periodical in time. Then σx(t) =σx 0cos(Ωt), where the oscillation frequency is inversely proportional to the time interval between pulses Ω ≈1/Tand can be tuned independently from the amplitude of the pulses B0. As a consequence the orbital dynamics canbestudiedwithreduced effective, time-dependent, one-dimen sional model. Byusing this model we found the synchronization condition between the orbital and the spin dynamics. Furthermore, we derived an analytical expression for the Arnold t ongue that defines the values of the parameters for which a synchronization is possible. Sin ce in the designed protocol the spin precession rate is determined by the interval be tween the applied pulses we believe that it can be realized in future experiments on semiconduc tor quantum dot devices. Acknowledgments The financial support by the Deutsche Forschungsgemeinschaft (DFG) through SFB 762, and contract BE 2161/5-1, Grant No. KO -2235/3, and STCU Grant No. 5053 is gratefully acknowledged. 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1106.5193v1.Intrinsic_spin_orbit_coupling_in_superconducting_δ_doped_SrTiO3_heterostructures.pdf

Intrinsic spin-orbit coupling in superconducting -doped SrTiO 3heterostructures M. Kim,1Y. Kozuka,1C. Bell,1, 2Y. Hikita,1and H. Y. Hwang1, 2, 3 1Department of Advanced Materials Science, University of Tokyo, Kashiwa, Chiba 277-8651, Japan 2Department of Applied Physics and Stanford Institute for Materials and Energy Science, Stanford University, Stanford, California 94305, USA 3Japan Science and Technology Agency, Kawaguchi, 332-0012, Japan (Dated: June 8, 2021) We report the violation of the Pauli limit due to intrinsic spin-orbit coupling in SrTiO 3het- erostructures. Via selective doping down to a few nanometers, a two-dimensional superconductor is formed, geometrically suppressing orbital pair-breaking. The spin-orbit scattering is exposed by the robust in-plane superconducting upper critical eld, exceeding the Pauli limit by a factor of 4. Transport scattering times several orders of magnitude higher than for conventional thin lm superconductors enables a new regime to be entered, where spin-orbit coupling eects arise non- perturbatively. Unconventional superconductivity is a subject of great theoretical and experimental interest [1{3]. A central is- sue in this eld is the discovery and understanding of non-trivial pairing mechanisms, such as the spin-triplet Cooper pair, which has been explicitly investigated in heavy fermions [1], Sr 2RuO 4[4], and crystals with broken inversion symmetry [5]. Recently, novel pairing has also been predicted in two-dimensional systems breaking in- version symmetry [6, 7]. Experimentally, measurements of the superconducting upper critical eld Hc2give vital information. In particular, violations of the Pauli para- magnetic limit [8, 9] can be used to unravel the nature of the electron spins in the superconducting state. Notably, the presence of spin-orbit coupling (SOC) can be quan- tied [10], as demonstrated by the Hc2studies of metal thin-lm superconductors [11] and bilayer systems where interface SOC drastically enhances Hc2[12]. Electron-doped SrTiO 3(STO) has attracted much at- tention as the lowest-density superconductor [13] with high-mobility [14]. These characteristics enable the cre- ation of novel low dimensional systems [15], and are vital to shed light on the rich physics present at the LaAlO 3/SrTiO 3(LAO/STO) interface, where the pres- ence of the Rashba spin-orbit interaction has been dis- cussed, aecting both the normal and superconducting state transport properties [16, 17]. However, despite the fact that the conduction band structure of STO is sim- ilar top-type GaAs [18, 19], the latter a model system for spintronics, the role of possible intrinsic SOC in the transport properties of doped STO is still unclear. In this Letter, we study the violation of the supercon- ducting Pauli limit due to intrinsic SOC in a systematic series of symmetric, doped STO heterostructures. Using the-doping technique, we selectively add Nb dopants in a narrow region inside an otherwise continuous undoped STO host crystal. As the thickness of the dopant layer is reduced, the destruction of superconductivity by orbital pair-breaking is geometrically suppressed, and the super- conducting Hc2is enhanced for magnetic elds applied parallel to the dopant plane. In the thin regime, whenthe dopant layer is just a few nanometers thick, the su- perconductivity is robust beyond the conventional Pauli limit, demonstrating the presence of spin-orbit scatter- ing (SOS) in the STO. Moreover, due to the absence of a surface or interface close to the dopant plane, and the spreading of the electron wavefunctions into the undoped STO, the electronic mean free path does not collapse as the dopant thickness decreases. Thus we preserve trans- port scattering times several orders of magnitude higher than for conventional thin lm superconductors. In this regime, the intrinsic band SOC eects arise as a non- perturbative correction to the transport, despite the rel- atively long absolute SOS times. The samples were fabricated with various thicknesses of 1 at:% doped Nb:SrTiO 3(NSTO) lms embedded between cap and buer layers of undoped STO, using pulsed layer deposition. High-temperature growth, above 1050C, in a low oxygen partial pressure of less than 107Torr was chosen to achieve high-quality STO lms, by managing the defect chemistry of the strontium and oxygen vacancies [20]. On a TiO 2terminated STO (100) 0.5 0.4 0.3 0.2 0.1 0.0Tc (K) 1 10 100 1000 d (nm)(a) (b) 1.0 0.5 0.0R/Rn 0.45 0.25 0.05 T (K)457 99 8.849 20 5.5 6.94.53.9 FIG. 1. (color online) (a) Sheet resistance R, normalized by the normal-state value Rnas a function of temperature T. Numbers refer to the -doped layer thickness din nm. (b) Superconducting transition temperature Tcversusd.Tcis dened by the temperature at the half value of the normal- state resistance; 10 %-90 % width of resistance is shown as an error bar.arXiv:1106.5193v1 [cond-mat.supr-con] 26 Jun 20112 1.0 0.8 0.6 0.4 0.2 0.0Hc2/Hc2( =0 °) 100 50 0 (°)Subs. 457 99 49 5.5(a)1.0 0.8 0.6 0.4 0.2 0.0Hc2/Hc2(0 K) 1.0 0.5 0.0 t 5.5 8.8 20 49 99 457 Subs.(1 - t)(b)1.0 0.8 0.6 0.4 0.2 0.0Hc2///Hc2//(0 K) 1.0 0.5 0.0 t 5.5 8.8 20 49 99 457 Subs. (1 - t)(1 - t)1/2(c) 1101001000 Length (nm) 1 10 100 1000 d (nm)d = dTinkhamGL dTinkham(d) FIG. 2. (color online) (a) Angular dependence of the upper critical eld Hc2at 50 mK, normalized by the value at = 0 . Dashed curves are tting results obtained to Tinkham's model. Results for representative samples (numbers refer to din nm) and a bulk 1 at :% NSTO substrate (Subs.) are shown. (b) Normalized perpendicular upper critical eld H? c2=H? c2(0 K) plotted as a function of the reduced temperature t=T=T c.H? c2(0 K) was obtained by extrapolation to T= 0 K from a tting to the data in (b) over the range of 0 :7t1. (c) Normalized parallel upper critical eld with same tting procedure. (d) Ginzburg-Landau coherence length GL(open rectangles) and dTinkham (closed diamonds) versus d. Dashed line is d= dTinkham . substrate, a 100 nm undoped STO buer layer was rst grown, followed by the 1 at :% NSTO layer with various thicknesses in the range 3.9 nm d457 nm. A 100 nm undoped STO cap layer was grown above the doped layer, to prevent surface depletion [21]. Post-annealing in a moderate oxidizing condition was used to ll oxygen va- cancies formed during growth. Transport measurements were made using a standard four-probe method with sam- ple cooling achieved using a dilution refrigerator with an in-situ rotator. For zero eld measurements, the residual magnetic eld was reduced below an absolute value of 0H= 0:1 mT, where 0is the vacuum permeability. All samples were superconducting at low temperatures, as shown in Fig :1 (a). The transition temperatures Tc dened by the temperature below which the resistance was 50 % of the normal-state value, were in the range 253 mKTc374 mK, as shown in Fig :1 (b). All samples, except for the two thinnest, showed sharp 10 %- 90 % transition widths ( 10 mK). While samples with thicknessd8.8 nm showed relatively constant Tc( 260 mK), several thinner samples showed a higher Tc while maintaining a sharp transition, suggesting possible changes to the superconducting properties close to the two-dimensional (2D) limit. We note that although the transition broadening in some of the thinner samples may relate to inhomogeneities, this is also reminiscent of the suggested Bose metal phase between the superconducting and insulating states [22]. Firstly, the anisotropy of Hc2was used to measure the dimensionality of the superconductivity. We investigated the variation of Hc2by rotating the sample with respect to the magnetic eld, as shown in Fig :2 (a). A bulk 1 at : % NSTO substrate was also measured as a reference. As ddecreased, a clear modulation of Hc2as a function of the anglebetween the magnetic eld and the sample plane was found. Here Hc2was dened as the eld at which the resistance was half that of the normal state. For sampleswithd99 nm, excellent ts to these data could be made using Tinkham's model [23], which is valid when the superconducting thickness is less than the Ginzburg- Landau coherence length dTinkham<GL(0) [24]. These ts are shown in Fig :2 (a). The dimensional crossover of superconductivity is more clearly demonstrated by the temperature dependence of Hc2, therefore we next measured H? c2(t) andHk c2(t), the out-of-plane ( = 90) and in-plane ( = 0) upper criti- cal elds respectively (here t=T=Tc), as shown in Figs :2 (b) and (c). In the perpendicular eld geometry, all sam- ples showed a linear temperature dependence. In the parallel eld geometry, for d99 nm, however, Hk c2(t) showed a clear square root form, which is characteris- tic of the 2D superconducting state. These data clearly demonstrate a three-dimensional (3D) to 2D crossover of the superconducting character as a function of d. By estimating dTinkham andGL(0) from the H? c2andHk c2 data, we nd that dTinkham decreases in proportion to 2.52.01.51.00.50.0µ0H (T)0.30.20.10.01/d (nm-1)Hc2//Hcp Hc2⊥ (x5) FIG. 3. (color online) Hk c2(circles),H? c2(squares, scaled by factor of ve), and the Pauli paramagnetic limit Hp c(trian- gles) plotted by 1 =d.Hk c2andH? c2are 50 mK data. Dashed lines are guides to the eye.3 0.012460.124 0.1 1 10 so-2-1012logd = 5.5 nm Hc2 ()2.0 1.5 1.0 0.5 0.0μ0Hc2 (T) 100806040200 (°)2.0 1.5 1.0 0.5 0.0μ0Hc2 (T) 0.25 0.00 T (K)Hc2// Hc2(x10)(a) (b) 10-1410-1310-1210-11 (s) 3 4 5 678910 d (nm)0.0010.010.11tr/sotr/so trso(c) FIG. 4. (color online) (a) Contour plot showing the deviation between the WHH simulation and the experiment by using Hc2() data ford= 5.5 nm. The minimum point is located at = 6.4,= 0.14. (b) Hc2() of 5.5 nm thick sample at 50 mK. Dotted line is the best t obtained from the WHH simulation. Inset: H? c2(T) andHk c2(T) data and the WHH theory t (dotted line). (c) Variation of soandtrwithdfor the ve thinnest samples, obtained from best ts to the WHH theory. An error bar of so is given by assuming 10 % thickness variation of the superconducting layer. The ratio tr=sois also shown on the right axis. the growth thickness d, and in the thinnest sample is much smaller than GL(0)100 nm, as plotted in Fig :2 (d), conrming the 2D nature of the superconductivity. A crucial and intriguing aspect of the Hk c2(t) data is the violation of the Pauli paramagnetic limit. The Pauli paramagnetic limiting eld [8, 9] is given by Hp c=
0=p 2B, whereBis the Bohr magneton (with a g- factor of two), and
0= 1:76kBTcis the BCS supercon- ducting gap for a weak-coupling superconductor, where kBis Boltzmann's constant. This limit is appropriate, since via tunneling bulk doped STO is known to be in the weak-coupling regime [25]. The variation of H? c2,Hk c2, andHp cas a function of 1 =dis shown in Fig :3.Hk c2ex- ceeds the Pauli limiting eld Hp cby a factor of more than four in the thinnest sample, while H? c2andHp cremained essentially constant. In the case of a 2D superconductor in a parallel mag- netic eld, if the sample is thin enough that orbital de- pairing is suppressed, spin paramagnetism is the domi- nant mechanism for destroying superconductivity [11]. In the presence of SOS, however, Hk c2can be robust beyond the Pauli limit. It should be noted that the renormaliza- tion of normal-state properties by many-body eects [26] can also enhance the Pauli limit. However, Shubnikov-de Haas oscillations in these -doped samples showed that the electron mass is consistent with the band structure at low temperature [15]. Given the low electron-density, strong correlation eects near half-lling are also absent. To investigate the eects of SOS in more detail, we performed a numerical t of the Hc2(;t) data to the Werthamer-Helfand-Hohenberg (WHH) theory [10] tak- ing into account corrections for the thin lm case [24]. Within this theory, the two crucial tting parameters, the orbital depairing parameter and the SOS rate so, are given by =~ 2mD; (1)so=2~ 3kBTcso; (2) where ~is the Plank constant divided by 2 ,mis the eective electron mass, D=v2 Ftr=3 is the diusion con- stant, and vFis the Fermi velocity. trandsoare, re- spectively, the transport and SOS times. For various  andso, we calculated the sum of the squares of the dif- ferences between the WHH model and the Hc2() data (we denote this sum as ), for the case d= 5:5 nm, as shown in Fig :4 (a). A unique minimum value of  was found, giving an excellent t to the experimental data, as shown in Fig :4 (b). We obtain tr= 6:21014s andso= 6:81013s, fromandso, respectively. This clearly indicates that the SOS is a highly signicant factor aecting the normal and superconducting trans- port properties, in spite of its relatively small absolute value. This is due to an appealing point of the -doped STO structures, where tris several orders of magnitudes higher than for conventional metal superconductors [27]. By performing similar analysis on data for the other samples where Hk c2exceededHp c, we obtain the thickness dependences of soandtr, as shown in Fig :4 (c). A clear decrease of sowith decreasing dis found, while tris rel- atively unchanged. This dependence suggests that the SOS is not dominated by either the Elliott-Yafet mecha- nism where so/tr, or the D'yakonov-Perel' mechanism whereso/1=tr[28]. This rather unexpected result sug- gests that the SOS observed has a dierent origin. We next clarify this point by comparison with other systems. We emphasize that this combination of SOC with high mobility conduction electrons places our system is in a dierent regime compared to other thin lm structures that violate the Pauli limit. For example, the use of heavy atoms to induce SOC in superconducting bilayers has been studied [12]. However, in this case, as is usual for conventional superconducting thin lms, the mean free path collapses in the thin limit. A similar collapse occurs with substrate gating at the LAO/STO heteroin- terface [29] where an asymmetric conning potential and4 Rashba SOC is expected [16, 17]. -doping is a crucial determinant for this dierence: since there is no obvious surface or interface surrounding the conducting layer, the scattering length due to disorder is unchanged (and even increased) with decreasing d[30]. Additionally, the sym- metry of the structure, giving rise to zero net eective electric eld, means that Rashba SOC is absent. We can thus interpret this as intrinsic SOC of STO, due to the d-orbitals of the Ti atoms [18]. Indeed the bulk conduction bands of STO have a similar structure to the valence bands of GaAs, where non-perturbatively large SOC has been demonstrated [19]. However in the case of STO there are few studies of electron SOS. It should be noted that calculations indicate that these con- ned-doped samples have a multiple subband structure [15], therefore the change of soobserved may relate to intersubband-induced spin-orbit interaction [31], or in- tersubband scattering [32], which in turn are in uenced by changes of the band structure as dchanges. The fact that the energy scale of the observed SOC ( 2 meV) is bigger than the superconducting gap ( 40eV) suggests the possibility of mixed spin-triplet and singlet states, giving rise to novel superconducting states in these low dimensional layers [6, 7, 33]. Moreover, in the normal state, the combination of high mobility conduction and SOC imply that STO can be usefully employed in a wide range of controllable spintronic architectures, including the spin Hall eect, which have until now been domi- nated by metals and traditional semiconductors. Thus this system is ideal for future measurements, which can be made over a range of densities both via growth con- trol [34], and eld-eect gating, the latter simultaneously introducing Rashba contributions to the SOC [35]. The authors thank M. Lippmaa for experimental as- sistance, M :R:Beasley, A:Bhattacharya, A :Kapitulnik, K:H:Kim, A:H:MacDonald, A :F:Morpurgo, B :Spi- vak, and H:Takagi for discussions. C :B:and H:Y:H:ac- knowledge support by the Department of Energy, Oce of Basic Energy Sciences, Division of Materials Sciences and Engineering, under contract DE-AC02-76SF00515. M:K:acknowledges support from the Japanese Govern- ment Scholarship Program of the Ministry of Education, Culture, Sport, Science and Technology, Japan. [1] M. Sigrist and K. Ueda, Rev. Mod. Phys. 63, 239 (1991). [2] P. Fulde and R. A. Ferrell, Phys. Rev. 135, A550 (1964). [3] A. I. Larkin and Y. N. Ovchinnikov, Zh. Eksp. Teor. Fiz. 47, 1136 (1964). [4] A. P. Mackenzie and Y. Maeno, Rev. Mod. Phys. 75, 657 (2003). [5] E. Bauer et al. , Phys. Rev. Lett. 92, 027003 (2004). [6] L. P. Gor'kov and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001). [7] S. K. Yip, Phys. Rev. B 65, 144508 (2002).[8] B. S. Chandrasekhar, Appl. Phys. Lett. 1, 7 (1962). [9] A. M. Clogston, Phys. Rev. Lett. 9, 266 (1962). [10] N. R. Werthamer, E. Helfand, and P. C. Hohenberg, Phys. Rev. 147, 295 (1966). [11] P. M. Tedrow and R. Meservey, Phys. Rev. B 8, 5098 (1973). [12] X. S. Wu, P. W. Adams, Y. Yang, and R. L. McCarley, Phys. Rev. Lett. 96, 127002 (2006). [13] C. S. Koonce, M. L. Cohen, J. F. Schooley, W. R. Hosler, and E. R. Pfeier, Phys. Rev. 163, 380 (1967). [14] O. N. Tufte and P. W. Chapman, Phys. Rev. 155, 796 (1967). [15] Y. Kozuka et al. , Nature 462, 487 (2009). [16] A. D. Caviglia et al. , Phys. Rev. Lett. 104, 126803 (2010). [17] M. Ben Shalom, M. Sachs, D. Rakhmilevitch, A. Palevski, and Y. Dagan, Phys. Rev. Lett. 104, 126802 (2010). [18] L. F. Mattheis, Phys. Rev. B 6, 4718 (1972). [19] B. Grbi c et al. , Phys. Rev. B 77, 125312 (2008). [20] Y. Kozuka, Y. Hikita, C. Bell, and H. Y. Hwang, Appl. Phys. Lett. 97, 012107 (2010). [21] A. Ohtomo and H. Y. Hwang, Appl. Phys. Lett. 84, 1716 (2004). [22] N. Mason and A. Kapitulnik, Phys. Rev. Lett. 82, 5341 (1999). [23] M. Tinkham, Phys. Rev. 129, 2413 (1963). [24] See supplemental material for detailed descriptions of Tinkham's and the WHH model. [25] G. Binnig and H. E. Hoenig, Solid State Commun. 14, 597 (1974). [26] T. P. Orlando, E. J. McNi, S. Foner, and M. R. Beasley, Phys. Rev. B 19, 4545 (1979). [27] R. R. Hake, Appl. Phys. Lett. 10, 189 (1967). [28] I. Zuti c, J. Fabian, and S. Das Sarma, Rev. Mod. Phys. 76, 323 (2004). [29] C. Bell et al. , Phys. Rev. Lett. 103, 226802 (2009). [30] Y. Kozuka et al. , Appl. Phys. Lett. 97, 222115 (2010). [31] E. Bernardes, J. Schliemann, M. Lee, J. C. Egues, and D. Loss, Phys. Rev. Lett. 99, 076603 (2007). [32] J. E. Hansen, R. Taboryski, and P. E. Lindelof, Phys. Rev. B 47, 16040 (1993). [33] V. L. Ginzburg, Phys. Lett. 13, 101 (1964). [34] M. Kim et al. , cond-mat/1104.3388. [35] T. Koga, J. Nitta, T. Akazaki, and H. Takayanagi, Phys. Rev. Lett. 89, 046801 (2002). Supplemental Material - Intrinsic Spin-Orbit Coupling in Superconducting -doped SrTiO 3 Heterostructures Tinkham's model The magnetic-eld response of a two-dimensional su- perconductor can be described by Tinkham's model [1], which assumes that the thickness of the superconductor is thinner than the Ginzburg-Landau coherence length, dTinkham<GL. It should be noted that the model does not include the eects of spin-orbit scattering or the Pauli paramagnetic limit, and assumes an isotropic supercon- ducting wavefunction. According to the model, the an-5 gular dependence of the upper critical eld can be shown to be Hc2() sin H? c2+ Hc2() cos Hk c2!2 = 1; (1) whereis the angle between the magnetic eld and the sample plane. The temperature dependence of the upper critical eld in perpendicular and parallel eld geometry is given by H? c2(t) =0 2GL(0)2(1t); (2) Hk c2(t) =0p 12 2GL(0)dTinkham(1t)1 2; (3) wheret=T=T cis the reduced temperature,  0=h=2e= 2:071015Wb is the ux quantum, and GL(0) is the Ginzburg-Landau coherence length extrapolated to T= 0 K. From Eqs :2 and 3,dTinkham andGL(0) can be found using dTinkham =s 60H? c2 (Hk c2)2; (4) GL(0) =s 0 2H? c2: (5) Thus,dTinkham can be calculated by measurement of H? c2 andHk c2of a sample experimentally. As noted by Ben Shalomet al: [2], in the case of the LaAlO 3/SrTiO 3in- terface, the value of dTinkham is an upper bound on the thickness. In our case we nd good agreement between the grown dopant layer thickness danddTinkham in thick samples, but dTinkham deviates slightly from dasHk c2 exceeds the Pauli limiting eld in the thinnest samples (d8:8 nm), indicating a limit of the model. WHH theory The Werthamer-Helfand-Hohenberg (WHH) theory [3] was used to more quantitatively t the superconducting upper critical eld, Hc2, data in the main text, in order to determine the spin-orbit scattering time in the system. Within this theory Hc2is the implicit solution of the equation lnt+ 1 2+iso 4   1 2+h+1 2so+i 2t + 1 2iso 4   1 2+h+1 2soi 2t 1 2
= 0; where is the digamma function. With a slight cor- rection [4] from the original WHH paper, the terms are dened as h=DeH c2 kBTc; (6)so=2~ 3kBTcso; (7) =r (h)21 42so; (8) =~ 2mD; (9) whereDis the diusion constant, sothe spin-orbit scat- tering time, and mthe electron mass. To include the eect of the nite thickness of a thin lm, the term hin Eq:6 should be replaced by hang(), which is given by [5] hang() =D 2kBTc 2eHc2jsin()j+1 3~(deH c2cos())2 ; (10) wheredis thickness of superconducting layer. In the case of =so= 0 (no spin-orbit coupling, but also no Pauli paramagnetic limit), the above formula is reduced to the orbital term only, where Hc2is the solution of lnt+ 1 2+h 2t 1 2 = 0: (11) In tting the data, the orbital only case (Eq. 11) could not accurately t the data for samples with Hk c2larger than the Pauli paramagnetic limit. However we could obtain a very successful t with the full WHH theory (Eq. 6), as discussed and shown in the main text (Fig. 4). Several sources of error should be considered in this t. Firstly, it has been argued that cooling supercon- ducting ultra-thin lms below 60 mK is extremely dif- cult [6], thus the Hc2(t) data may articially saturate at low temperatures due to a lack of cooling. However, 1284λso7.06.05.04.0d (nm) FIG. S1. Optimal value of sofrom the WHH tting for the d= 5:5 nm sample, depending on the thickness dused in the model.6 3.02.52.01.51.0µ0Hc2// (T)12840λsoWHHSchopohl et al.Experiment FIG. S2. Variation of Hk c2withsousing a non-perturbative spin-orbit coupling model, compared to the original WHH model, adapted from Schopohl et al: [8]. by tting to data at temperatures only above 75 mK, we nd an increase of the value of soof only 2 % compared to the full tting curve. Secondly, the in uence of error in the value of dused in Eq:10 can also be considered. This eect is demonstrated in Fig : S1, where the value ofsoobtained from the tting is plotted against dfor thed= 5.5 nm sample. As is clear, sensitivity of the t to variation in dgives rise to variation in so; we obtain so= (6:81:0)1013s ford= 5:50:5 nm. We note that the WHH theory makes various simpli- fying assumptions: the superconductor should be in the dirty limit, where the electron mean free path is shorter than the BCS coherence length `BCS. Secondly, the spin-orbit scattering time is less than the total scatter- ing timesotr. In order to estimate these parame- ters, we assume a single band approximation with spher-ical Fermi surface and used an electron eective mass m= 1:24m0, wherem0is the bare electron mass, ex- tracted from Shubnikov-de Haas oscillations [7]. In the d= 5.5 nm sample, we found `100 nm,BCS470 nm,tr6:21014s. Therefore, we conclude that our system is in the dirty limit, and spin-orbit coupling can be treated as a perturbation. This however becomes less clear with decreasing thickness, for which tr=so0:1. As a further check, we used a non-perturbative theory proposed by Schopohl et al: [8]. The value of Hk c2for varioussousing this theory is shown in Fig : S2, along with the original WHH model (neglecting the nite size corrections of Aoi etal: [5]). Since in absolute terms the observed values of Hk c2are not large due to the relatively lowTc, we estimate an error of only 20 % in the de- termination of soby using the original WHH model, as shown, which does not signicantly aect the result of the original tting. [1] M. Tinkham, Phys. Rev. 129, 2413 (1963). [2] M. Ben Shalom, M. Sachs, D. Rakhmilevitch, A. Palevski, and Y. Dagan, Phys. Rev. Lett. 104, 126802 (2010). [3] N. R. Werthamer, E. Helfand, and P. C. Hohenberg, Phys. Rev.147, 295 (1966). [4] A. L. Fetter and P. C. Hohenberg in R. D. Parks, ed., Superconductivity (Marcel Dekker Inc., 1969). [5] K. Aoi, R. Meservey, and P. M. Tedrow, Phys. Rev. B 7, 554 (1973). [6] K. A. Parendo, B. Tan, K. H. Sarwa, and A. M. Goldman, Phys. Rev. B 73, 174527 (2006). [7] Y. Kozuka et al. , Nature 462, 487 (2009). [8] N. Schopohl and K. Scharnberg, Physica B & C 107, 293 (1981).

2006.13651v1.Landau_levels_in_spin_orbit_coupling_proximitized_graphene__bulk_states.pdf

Landau levels in spin-orbit coupling proximitized graphene: bulk states Tobias Frankand Jaroslav Fabian Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany (Dated: June 25, 2020) We study the magnetic-eld dependence of Landau levels in graphene proximitized by large spin- orbit coupling materials, such as transition-metal dichalcogenides or topological insulators. In ad- dition to the Rashba coupling, two types of intrinsic spin-orbit interactions, uniform (Kane-Mele type) and staggered (valley Zeeman type), are included, to resolve their interplay with magnetic orbital eects. Employing a continuum model approach, we derive analytic expressions for low- energy Landau levels, which can be used to extract local orbital and spin-orbit coupling parameters from scanning probe spectroscopy experiments. We compare dierent parameter regimes to iden- tify ngerprints of relative and absolute magnitudes of intrinsic spin-orbit coupling in the spectra. The inverted band structure of graphene proximitized by WSe 2leads to an interesting crossing of Landau states across the bulk gap at a crossover eld, providing insights into the size of Rashba spin-orbit coupling. Landau level spectroscopy can help to resolve the type and signs of the intrinsic spin-orbit coupling by analyzing the symmetry in energy and number of crossings in the Landau fan chart. Finally, our results suggest that the strong response to the magnetic eld of Dirac electrons in proximitized graphene can be associated with extremely large self-rotating magnetic moments. I. INTRODUCTION The modication of band structures of two- dimensional materials by proximity eect oers a way to design systems with novel properties. In particular for spintronics in graphene1this turns out to be a very promising way to enhance spin-orbit coupling (SOC) or introducing exchange interaction by proximity with other materials2{4. In this way we can harvest the best from both materials, on the one hand the excellent electronic transport properties of graphene, paired with, e.g., the special optical properties of transition metal dichalcogenides (TMDs)5. TMDs are two-dimensional van der Waals insulators and are a perfect t to graphene, because the Dirac cone is situated inside the TMD band gap and leaves the Dirac cone mainly intact. Due to the strong native SOC in TMDs on the order of hundreds of meV, via hybridization processes, the SOC can transfer to graphene. By now these spin-orbit coupling proximity eects are no longer abstract theoretical concepts, as they have been demonstrated in many experiments6. Due to the break- ing of spatial symmetries by the substrate, the funda- mental properties and spin degeneracy of the graphene Hamiltonian can be altered and a gap be introduced in the originally massless dispersion. Interpretation of ex- perimental data7{9hints towards predominant induction of sublattice resolved intrinsic spin-orbit coupling, so- called valley Zeeman intrinsic spin-orbit coupling, where intrinsic spin-orbit coupling is of same magnitude but of dierent signs in the two sublattices. Other types of eects, such as the breaking of the sublattice symme- try via a staggered potential or enhancement of Rashba spin-orbit coupling are also possible10. Determination of model parameters for symmetry- broken graphene by means of density functional the- ory (DFT) calculations was only carried out for commen-surate lattice arrangements, but in real systems, lattice mismatch eects could play an important role. For ex- ample, the staggered sublattice potential may eectively average out, but could be sizeable locally due to the for- mation of moir e patterns11,12. It is thus important to de- vise more direct methods than weak antilocalization and spin precession measurements to explore the local elec- tronic processes and energy scales. This could be, for example, achieved by scanning tunneling spectroscopy supplemented with a transverse magnetic eld13,14. It was demonstrated earlier15that this method can be used to extract local Rashba parameters in two-dimensional electron gases by comparison with Landau level models. In this spirit, we focus on single-particle Landau level spectra (with approximatep Bmagnetic eld behavior) rather than on transport signatures (with approximate linear-Bbehavior)16. The physics of symmetry-broken honeycomb lattices has been extensively studied theoretically16{20. However, the interplay between orbital magnetic elds and prox- imity SOC is a relatively recent subject; see, for example, Ref. 16 for monolayer and Ref. 21 for bilayer graphene. Due to graphene's linear energy dispersion, the Lan- dau levels of Dirac electrons follow a dierent magnetic eld (B) dependence than in two-dimensional electron gases22{24. The graphene Landau levels are given by "n(B) = sgn(n)p 2v2 Fe~jnjB(n= 0;1;2;:::), where vFis the Fermi velocity. In the absence of Zeeman cou- pling, the levels are fourfold degenerate due to the spin and valley degrees of freedom. The Landau level with orbital index n= 0 is special, being located solely in one sublattice, depending on the valley degree of freedom25. When symmetries in a hexagonal lattice become broken as in, e.g., buckled silicene, the fourfold Landau level de- generacy is lifted19and gaps can be introduced. While spin-orbit coupling proximitized graphene has many sim- ilarities to silicene, in proximitized graphene intrinsicarXiv:2006.13651v1 [cond-mat.mes-hall] 24 Jun 20202 SOC is sublattice resolved and Rashba SOC is momen- tum independent4,17,20. The interplay of uniform intrin- sic and Rashba SOC with orbital magnetic elds was studied in Refs. 17 and 18. Rashba17nds analytic so- lutions for the Landau states which have in-plane spin- orbit texture and whose hallmark is a twofold degener- ate zero energy Landau level. De Martino et al. conrm this result and provide additional analytic results for uni- form intrinsic SOC and study magnetic eld edge state physics, where they observe phase transitions between dierent spin-ltered edge regimes. In Ref. 26 it was shown that using Landau level spectroscopy the magni- tude of induced intrinsic SOC can be extracted in prox- imitized bilayer graphene systems employing capacitance measurements. In a recent work16properties of proximitized graphene under the in uence of orbital magnetic elds was stud- ied, where emphasis is put on calculations of transport fan diagrams and Hall conductivities27. This work nds very rich fan diagrams with characteristic level splittings, allowing one to infer the relative sizes of Rashba and in- trinsic SOC from scaling observations. The focus is on the behavior of higher-lying Landau levels. Our work addresses specically low-energy and low- magnetic eld states which are accessible to scanning tunneling experiments. Additionally, we provide analytic expressions for all four low-energy Landau levels in high magnetic elds. Finally, we also consider the interplay of the valley Zeeman spin-orbit interaction with a staggered potential, which is important to distinguish non-inverted and inverted spectral regimes. We use a Hamiltonian for the bulk Landau level spec- trum of SOC proximitized graphene, including Zeeman interaction, with parameters from DFT predictions4. We show that spectral features of this Hamiltonian may be used to extract important model parameters from exper- iment by applying magnetic elds up to 15 Tesla. From the model Hamiltonian we extract high- Beld formulas for the energetic behavior of low-energy Landau levels, comparing them with numerical data. Excellent corre- spondence is found even for smaller values of magnetic elds. These formulas could provide a direct way to measure local intrinsic sublattice-resolved spin-orbit cou- pling as well as the local sublattice potential by means of Landau level scanning tunneling spectroscopy13{15. Important, this would also provide a direct way to deter- mine whether the intrinsic spin-orbit coupling in prox- imitized heterostructures is of uniform28or valley Zee- man type29,30, dierentiating between topologically triv- ial and nontrivial regimes. These regimes can also be identied by the number of crossings of the low-energy Landau levels in the fan diagram as well as from their electron-hole symmetry. In the quantum valley spin Hall (QVSH) regime31, in which the graphene spin-orbit coupling gap is inverted, we nd unique signatures of the gap inversion in the Lan- dau level spectrum. In general, strong orbital magnetic response at lowB-elds is accompanied by a crossing ofspecic Landau levels from the electron to the hole sec- tors and vice versa. The crossing of states happens at a critical magnetic eld, lying in the mT range, for which we provide a formula. The formula oers a method to extract the Rashba spin-orbit coupling strength. The strong magnetic eld dependence of those crossing states can be interpreted in terms of magnetic moments, which as we show, are on the order of 1000 Bfor realistic parameters. This nding is further corroborated by ex- plicit calculation of the self-rotating magnetic moments for proximitized graphene structures. The paper is organized as follows. In Sec. II we in- troduce the proximity Hamiltonian and show its form in a magnetic eld. Low-energy levels for large mag- netic elds are derived and compared to realistic proxim- ity scenarios. Orbital magnetic moments induced by the band structure of proximitized graphene are discussed in Sec. III before concluding with Sec. IV. II. CONTINUUM: SOC PROXIMITY HAMILTONIAN IN MAGNETIC FIELD When graphene is brought into contact with other sur- faces, internal symmetries of graphene, such as the sixfold rotational symmetry or the horizontal mirror symmetries are broken. This is re ected in the low-energy electronic structure, where the spin degeneracy of the Dirac bands is lifted and a gap can be introduced due to chiral symme- try breaking. On a microscopic level, symmetry reduc- tion manifests in terms of extra electron hopping pro- cesses, which are forbidden in pristine graphene. We follow the conventions introduced in Ref. 10, den- ing aC3v-symmetric, linearized eective Hamiltonian H=H k+H
+H R+H I (1) H k=~vF(kxxkyy) s0; (2) H
=
z s0; (3) H R=R(x syy sx); (4) H I=1 2 A I(z+0) +B I(z0) sz:(5) The spin-orbit coupling proximity Hamiltonian (1) is given for a specic valley =1 (representing the K/K0points) and separates into the bare graphene part, Eq. (2), the staggered potential part, Eq. (3), the Rashba spin-orbit coupling, Eq. (4), and the sublattice-resolved intrinsic spin-orbit coupling part, Eq. (5). The bare graphene part involves the Fermi velocity vF=p 3at=2~, which depends on the nearest-neighbor hopping strength tand lattice constant a. For details of the underlying microscopic hopping processes see Refs. 4, 10, and 31. The symbols andsstand for unit and Pauli matrices describing sublattice and real spin degrees of freedom, respectively.3 A. SOC proximity Hamiltonian in orbital magnetic eld To study the orbital eects of a transverse magnetic eld on the proximitized graphene system, we apply the continuum approximation and minimal coupling substi- tution kx!(px+eAx)=~!kxeBy ~: (6) The Landau gauge with vector potential A(r) = (By;0;0) is applied to produce a magnetic eld per- pendicular to graphene. In our notation Bis always the modulus of theB-eld and=indicates its sign. Im- plicitly, we assume a plane wave ansatz in the xdirection. Following the approach of Ref. 18, we dene bosonic Lan- dau ladder operators a;aythat lead to a replacement rule of the original matrix elements of ~vF kx+eBy ~ +iky !~!Ba; (7) ~vF kx+eBy ~ iky !~!Bay: (8) We dene the cyclotron frequency !B=p 2vF=lBwith the magnetic length as lB=p ~=eB. The operators are the standard textbook ladder operators with ajni=pnjn1i,ayjni=pn+ 1jn+ 1i, acting on harmonic oscillator eigenfunctions jni;withn2N0. Substituting Eqs. 7 and 8 into Eq. (1) leads to the following positive B-eld andK-valley Hamiltonian H=+ =+B= (9) 0 BB@
+A I 0~!Bay2iR 0
A I 0~!Bay ~!Ba 0
B I 0 2iR~!Ba 0
+B I1 CCA; given in the basis ( A",A#,B",B#). To solve the Schr odinger equation, we employ the spinor ansatz =+ =+B;n;m=0 BB@cA" n;mjni cA# n;mjn+ 1i cB" n;mjn1i cB# n;mjni1 CCA: (10) This ansatz represents a combination of dierent har- monic oscillator eigenfunctions for dierent spinor com- ponents, which are consistent with the matrix of Eq. (9) acting on it. The quantum number mlabels the subset of solutions for orbital index n, with linear combination coecients cn;m. After acting on the ansatz, we obtain the matrix H=+ =+B;n= (11) 0 BB@
+A I 0~!Bpn 2iR 0
A I 0~!Bpn+ 1 ~!Bpn 0
B I 0 2iR~!Bpn+ 1 0
+B I1 CCA:A more general expression for the other valley K0and opposite magnetic elds can be found in the appendix. B. Low-energy Landau levels Hamiltonian of Eq. (11) for a general n1 can be solved by numerical diagonalization. This section deals with the discussion of low orbital indices nand their as- sociated eigenstates. Harmonic oscillator eigenfunctions are only dened for quantum numbers n0. Here, in principle also negative orbital indices are allowed, as long as they do not produce trivial solutions, e.g., for n=1, we obtain =+ =+B;n=1=0 B@0 j0i 0 01 CA; (12) eectively projecting Hamiltonian of Eq. (11) to the A# subspace. This leads to the Hamiltonian and eigenvalue of H=+ =+B;n=1=
A I; (13) for a state that is completely localized in the Asublattice and which is independent of magnetic eld. For the next higher orbital index n= 0 atK, one has the ansatz =+ =+B;n=0;m=0 BB@cA" 0;mj0i cA# 0;mj1i 0 cB# 0;mj0i1 CCA; (14) leading to an eective 3 3 Hamiltonian H=+ =+B;n=0=0 @
+A I 0 2iR 0
A I~!B 2iR~!B
+B I1 A:(15) Analytical diagonalization of Hamiltonian of Eq. (15) is possible, but solutions are too lengthy to be given here. In the following we also add a Zeeman term HZ=gsBB0 sz; (16) withB=e~=2me, the Bohr magneton and electron spin g-factorgs. Pure graphene under the action of an orbital magnetic eld has four degenerate zero-energy states, two for each spin and two for each valley. If the proximity parameters of graphene tend to zero, these zero energy modes should be reproduced. So far we found one of these states with Eq. (12). Another one turns out to be included in the n= 0 space, by analyzing the dependence of eigenenergies of Hamiltonian of Eq. (15) with respect to large magnetic elds to orderO(1=B). Including also Zeeman energy4 and carrying out the analysis for the other valley, this leads to the four low-energy states Es  E+A" +=
+A I+gsBB+O1 B ; (17) E+A# +=
A IgsBB; (18) EB" +=
+B I+gsBB+O1 B ; (19) EB# +=
B IgsBB: (20) Interestingly, the B-eld selects states in the Asub- lattice in valley = + and from Bsublattice in valley =, as indicated by the superscripts. Even without the Zeeman term, an orbital magnetic eld is able to break the Kramers degeneracy of levels by coupling to SOC hopping processes, which is not possible, for exam- ple, in pure graphene (without SOC). Equations (17){ (20) could be used in experiment to extract
, A I, and B I. For negative magnetic elds ( =), creation and annihilation operators are interchanged ( a$ ay) in Eq. (9), leading to a dierent n-dependent Hamiltonian H =B;nwith accordingly adapted wave function ansatz (see also appendix). We nd the symmetry relation H B;n= (0 sx)yH B;n(0 sx); (21) which leads to a connection between levels at opposite magnetic elds Es B=Es B; (22) i.e. for each positive magnetic eld B, there is a cor- responding level at Bwith the same energy, but with ipped valley and spin quantum numbers, in line with the operation of time reversal. C. Realistic parameter values for graphene on TMDs A general motivation for this study is to develop tech- niques to extract parameters in conjunction with local magneto-spectroscopy. In experiment, spin-orbit cou- pling and orbital model parameters may vary spatially due to formation of moir e patterns originating from a twist angle between graphene and the substrate or due to lattice mismatch. In this section we show which types of local fan charts can be expected and whether nger- prints of specic parameter regimes can be recognized therein. To this end, we take ab-initio extracted parameters for commensurate system combinations4and compare spin- resolved eigenvalues of Hamiltonian of Eq. (11) at = in Figs. 1 and 2 versus increasing magnetic eld [includ- ing the Zeeman term of Eq. (16)] with the bulk band structure.In the case of graphene on WS 2, whereA IB I, A I<
,R< A I, i.e. the non-inverted regime, we see that for small magnetic eld values, the Landau levels ap- proach exactly the eigenspectrum at the Kpoint of the bulk spectrum. Further, for small magnetic eld values, Zeeman energy is not important and we nd two con- stant eigenvalues of spin-down polarization, as indicated by formulas (18) and (20). The other two eigenvalues (17) and (19) are represented already at very small magnetic elds of about 0.03 T. Zeeman energy and the presence of the linear-in-Blow-energy Landau levels become evi- dent for large magnetic elds. It is surprising that spin degeneracy is almost immediately broken by the mag- netic eld, where it selects only the innermost Landau levels to have spin-down polarization of dierent valleys. This is inverted by application of negative magnetic elds (not shown). The second higher Landau level shows an anticrossing, which mixes spin-up and down species and relates to the Rashba spin-orbit interaction. For high magnetic elds (above 20 T), we nd the low-energy Lan- dau levels to cross, which could be used to extract model parameters with the help of Eqs. (17){(20). In the inverted case of graphene on WSe 2, for which A IB I,A I>
,R< A I, i.e., intrinsic spin-orbit coupling dominates all energy scales, the Landau level structure is very similar to the previous case except for very small magnetic elds in the mT regime. We observe a very strong response to the magnetic eld where two levels are crossing at a certain critical eld and the system isno longer gapped for all magnetic elds. This will be discussed in more detail in the next section. Moreover, we apply a set of parameters, which are the same as for WSe 2, butB I!B I, resulting in about equal sized intrinsic SOC parameters. The spectra for this situation are shown in Fig. 3. This large uniform intrinsic SOC results in a quantum spin Hall topologi- cal insulator32. Even for large magnetic elds, we see no crossing of energy levels at zero magnetic elds in Fig. 3. Conversely, having a negative sign for the intrin- sic spin-orbit couplings results in two crossings at about zero energy, see Fig. 4. In literature commonly only the case of negative intrinsic spin-orbit couplings (in our con- vention) is studied18. The number of crossings and the symmetry with respect to energy could be used to qual- itatively determine the relative and absolute signs of the intrinsic SOC parameters and therefore provide informa- tion about topologically trivial and nontrivial regimes. Electron-hole symmetry in the fan chart is a ngerprint of uniform, Kane-Mele SOC, absence of electron-hole sym- metry points to staggered, valley Zeeman type of intrinsic SOC. D. Discussion of gap closing Gap inversion in graphene on WSe 2leads to very cu- rious eects even without magnetic elds. Pseudohelical edge states can appear which are as robust as topolog-5 FIG. 1. Bulk and Landau level spectrum for graphene/WS 2for small and large values of the magnetic eld B. The bulk spectrum is from the Kpoint. Straight lines are plots of the low-energy Landau level formulas. Color code corresponds to szexpectation values (red is spin up, blue is spin down). Parameters: t=2:657 eV,
= 1 :31 meV,R= 0:36 meV, A I= 1:02 meV,B I=1:21 meV. FIG. 2. Bulk and Landau level spectrum for graphene/WSe 2for small and large values of the magnetic eld B. The bulk spectrum is from the Kpoint. Straight lines are plots of the low-energy Landau level formulas. Color code corresponds to szexpectation values (red is spin up, blue is spin down). Parameters: t=2:507 eV,
= 0 :54 meV,R= 0:56 meV, A I= 1:22 meV,B I=1:16 meV. ical states from a topological insulator31. The inverted regime behaves peculiar with magnetic eld as well; we observe a gap closing in Fig. 2 as compared to the pre- served gap in WS 2. This gap closing is not related to the Zeeman energy but is a purely orbital eect. In Fig. 5 we zoom into the low- Bphysics of Fig. 2. From our analysis we can see that only exactly one state from each valley is able to cross without interaction close to zero energy. One of the states stems from the 3 3 Hamiltonian of Eq. (15), the other from the other valley counterpart. The states in question are the A#from = + andB#from=. By L owdin-downfolding the 33 Hamiltonians for n= 0 at energy E= 0 to these subspaces, an approximation to the eigenvalue canbe obtained in both cases, leading to expressions H=+ e=
A I+(
+A I)2 vB (
+A I)(
B I) + 42 R;(23) H= e=
B I(
B I)2 vB (
+A I)(
B I) + 42 R;(24) wherev=~!B=p B=p 2e~vF. These expressions only contain linear terms in B. This is due to the downfolding procedure, which neglects higher order terms. Equations (23) and (24) are plot- ted in Fig. 5. Starting at the exact eigenvalues of the bulk spectrum at the K/K0points, they cross at a cer- tain magnetic eld, at about the same point where the numerical data intersects. We tested this to hold for dif- ferent combinations of parameters in the inverted regime. Thus, we can combine Eqs. (23) and (24) to estimate the6 FIG. 3. Bulk and Landau level spectrum for graphene/Kane-Mele-WSe 2withA I=B Ifor small and large values of the magnetic eldB. The bulk spectrum is from the Kpoint. Straight lines are plots of the low-energy Landau level formulas. Color code corresponds to szexpectation values (red is spin up, blue is spin down). Parameters: t=2:507 eV,
= 0 :54 meV, R= 0:56 meV,A I= 1:22 meV,B I= 1:16 meV. FIG. 4. Bulk and Landau level spectrum for graphene/Kane-Mele-WSe 2withA I=B Ifor small and large values of the magnetic eldB. The bulk spectrum is from the Kpoint. Straight lines are plots of the low-energy Landau level formulas. Color code corresponds to szexpectation values (red is spin up, blue is spin down). Parameters: t=2:507 eV,
= 0 :54 meV, R= 0:56 meV,A I=1:22 meV,B I=1:16 meV. critical magnetic eld Bcrit=(A IB I2
)((
+A I)(
B I) + 42 R) 2v(2
+A IB I); (25) for which we expect the inner states to cross. Insert- ing model values for WSe 2, we obtain a critical eld of 1.8 mT. While being very small, the critical eld scales quadratically with the intrinsic SOC. For tenfold in- creased intrinsic SOC, predicted by some experiments6, one expects critical eld values on the order of 100 mT. Formula (25) depends on the Rashba parameter, which oers a way to extract the parameter by rst determining the other parameters with the low-energy Landau level expressions of Eqs. (17){(20) and then plugging in the critical magnetic eld. Further, Eqs. (23) and (24) can be interpreted in termsof a Zeeman-like (linear-in- B) response. Magnetic mo- ments atKandK0can be identied as mK=K0=2 v(
A=B I) (
+A I)(
B I) + 42 R; (26) which for staggered A I=B I=Isimplies to mK=K0=2 v(
+I) (
+I)2+ 42 R: (27) Inserting WSe 2parameters, we obtain m0:4 eV=T = 7256B. Interestingly, for I=R= 0, this formula re- duces to four times the magnetic moment found by Xiao et al.33and thus can be brought into connection with the self-rotating magnetic moment discussed in Sec. III. The magnetic moment of Eq. (27) is only an approximation as it follows from a linearized Hamiltonian, but has the7 FIG. 5. Zoom into the low B-eld and low-energy Landau level states of graphene on WSe 2of Fig. 2. Dashed lines are eigenvalues for downfolded Hamiltonians onto subspaces of then= 0 Landau Hamiltonian. correct qualitative behavior, namely it scales quadrati- cally with the velocity and is inversely proportional to the energy gap; compare to formula (28) in Sec. III. III. MAGNETIC MOMENTS The response to orbital magnetic elds in this system is very strong, particularly in the inverted-band case. Here, we want to pick up the idea of magnetic moments appearing in Eq. (26). It is known that inversion sym- metry breaking in graphene, as experienced in proximi- tized graphene, can have consequences for the semiclas- sical motion of electrons33. Specically, it manifests in a valley Hall eect, which generates a transversal val- ley current, i.e., local imbalance of valley population, a consequence of valley-dependent Berry curvature. This eect is very similar to the intrinsic spin Hall eect which originates from a spin-dependent Berry curvature. Berry curvature around graphene valleys can have very peculiar shapes in these systems and determines their topological properties31. Inversion symmetry allows also for nite magnetic mo- ments in reciprocal space mnk, dependent on the band and k vector of the Bloch state. This magnetic moment contribution plays an important role in the formation of the macroscopic orbital magnetization of solids34and should be detectable in experiment33. It also has implica- tions for the semiclassical single-particle band energy of electrons, which gets modied by an additional Zeeman- like term"B nk="nkmnkB35. The often called self-rotating magnetic moment, can be expressed in terms of perturbation theory36 mz nk=ie 2~h@kxunkjHkEnkj@kyunkix$y =ie 2~X l6=nhunkj@H @kxjulkihulkj@H @kyjunki EnkElkx$y:(28) −0.01 0.00 0.01 k[1/nm]−6−4−20246E [meV] −0.01 0.00 0.01 k[1/nm]0.51.01.52.02.53.0m [1000µB](a) WS 2 −0.01 0.00 0.01 k[1/nm]−6−4−20246E [meV] −0.01 0.00 0.01 k[1/nm]−3−2−1012345m [1000µB] (b) WSe 2 FIG. 6. Low-energy band structures (left) around the Kpoint of proximitized graphene (on WS 2and WSe 2) with their as- sociated magnetic moments (right). Colors of the bands cor- respond to band indices which are in correspondence in both gures. Without time-reversal symmetry breaking, mz nktrans- forms like angular momentum or Berry curvature and is odd in k. This is in nice agreement with formulas (23) (24), and Eq. (27), re ecting the dierent signs in the dierent valleys. We evaluate Eq. (28) for DFT-extracted model param- eters4in Fig. 6 next to their bulk band structures for the non-inverted and inverted systems graphene on WS 2 and WSe 2, respectively. For the non-inverted case, see Fig. 6(a), we nd same signs of magnetic moments for all the bands in one valley. The magnitudes of the magnetic moments are exceptionally large compared to the Bohr magnetonB, enhanced by a factor of up to 3000. The large size of the magnetic moment can be made plau- sible by estimating m=Bmev2 F=
E104, using vF106m/s and inserting a typical gap energy scale of
E= 1 meV. The asymmetry between the four bands stems from the slight deviations in magnitude between the intrinsic spin-orbit coupling parameters. The inverted case, Fig. 6(b) is dierent. Directly at the Kpoint, we again see the behavior of only positive mag- netic moments, which could have been guessed from the8 previous result, the bands are just inverted with respect to each other. Interactions among the bands are slightly altered compared to WS 2, changing the overall magni- tude. We see magnetic moments of almost 5000 B, which is in the same order of magnitude as the results from the approximate formula (27). However, directly at the anticrossings, which are generated by Rashba spin- orbit coupling, negative magnetic moments are formed. These negative magnetic moments could be a ngerprint of the inverted gap regime in experiments. IV. CONCLUSION In this work we show that Landau level spectroscopy in proximitized graphene can be an alternative way to directly measure the local magnitude of orbital and spin- orbit coupling parameters. From the Landau level fan diagrams it is possible to distinguish the nature (relative and absolute signs) of intrinsic spin-orbit couplings by counting the crossings of low-energy Landau levels and by analysis of their electron-hole symmetry. Further, the inverted band structure in graphene on WSe 2opens up a new regime, where the bulk gap is not preserved for all magnetic eld values. This could have potential con- sequences for the edge state physics in these systems, which remains to be studied. Interestingly, the response of the crossing Landau levels to magnetic elds is in the same order of magnitude as the self-rotating mag- netic moments generated in the valleys of proximitized graphene. The magnetic moments can be giant, about several 1000 Bcompared to the electron spin g-factor. The direct relation between the response of Landau levels and the magnetic moments is subject to future work.ACKNOWLEDGMENTS This work was funded by the Deutsche Forschungsge- meinschaft (DFG, German Research Foundation) SFB 1277 (project-id 314695032). The authors gratefully ac- knowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by pro- viding computing time on the GCS Supercomputer SuperMUC at Leibniz Supercomputing Centre (LRZ, www.lrz.de). Appendix A: General magnetic eld Hamiltonian for eld directions and valleys For a general valley =1 and sign of magnetic eld =1, the eective Hamiltonian matrix for orbital num- bernis given by H B;n=H
+H R+H I+gsBB0 sz (A1) ~!Bx p n+ (1)=2 0 0p n+ (1 +)=2 ; TABLE I. Wave function spinor component ansatzes for the general Hamiltonian of Eq. (A1) for valleyand sign of mag- netic eldin the basis ( A";A#;B";B#).   spinor components constant solution + +jni;jn+ 1i;jn1i;jni A# + jn1i;jni;jni;jn+ 1i B# +jni;jn1i;jn+ 1i;jni B" j n+ 1i;jni;jni;jn1i A" following the same derivation as in Sec. II. The Hamilto- nian of Eq. (A1) is only well dened in combination with the wave function ansatz it acts uppon, which are listed in Tab. I. Emails to: tobias.frank@physik.uni-regensburg.de 1W. Han, R. K. Kawakami, M. Gmitra, and J. Fabian, Nat. Nano 9, 794 (2014). 2T. Frank, M. Gmitra, and J. Fabian, Phys. Rev. B 93, 155142 (2016). 3K. Zollner, M. Gmitra, T. Frank, and J. Fabian, Phys. Rev. B 94, 155441 (2016). 4M. Gmitra, D. Kochan, P. H ogl, and J. Fabian, Phys. Rev. B93, 155104 (2016). 5M. Gmitra and J. Fabian, Phys. Rev. B 92, 155403 (2015). 6J. H. Garcia, M. Vila, A. W. Cummings, and S. Roche, Chem. Soc. Rev. 47, 3359 (2018). 7A. W. Cummings, J. H. Garcia, J. Fabian, and S. Roche, Phys. Rev. Lett. 119, 206601 (2017). 8T. S. Ghiasi, J. Ingla-Ayn es, A. A. Kaverzin, and B. J. van Wees, Nano Lett. 17, 7528 (2017). 9L. A. Ben tez, J. F. Sierra, W. Savero Torres, A. Arrighi, F. Bonell, M. V. Costache, and S. O. Valenzuela, Nat.Phys. 14, 303 (2018). 10D. Kochan, S. Irmer, and J. Fabian, Phys. Rev. B 95, 165415 (2017). 11Y. Li and M. Koshino, Phys. Rev. B 99, 075438 (2019). 12A. David, P. Rakyta, A. Korm anyos, and G. Burkard, Phys. Rev. B 100, 085412 (2019). 13G. Li and E. Y. Andrei, Nat. Phys. 3, 623 (2007). 14L.-J. Yin, S.-Y. Li, J.-B. Qiao, J.-C. Nie, and L. He, Phys. Rev. B 91, 115405 (2015). 15J. R. Bindel, M. Pezzotta, J. Ulrich, M. Liebmann, E. Y. Sherman, and M. Morgenstern, Nat. Phys. 12, 920 (2016). 16T. P. Cysne, J. H. Garcia, A. R. Rocha, and T. G. Rap- poport, Phys. Rev. B 97, 085413 (2018). 17E. I. Rashba, Phys. Rev. B 79, 161409(R) (2009). 18A. De Martino, A. H utten, and R. Egger, Phys. Rev. B 84, 155420 (2011). 19C. J. Tabert and E. J. Nicol, Phys. Rev. Lett. 110, 197402 (2013).9 20V. Y. Tsaran and S. G. Sharapov, Phys. Rev. B 90, 205417 (2014). 21J. Y. Khoo and L. Levitov, Phys. Rev. B 98, 115307 (2018). 22K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov, Nature 438, 197 (2005). 23Y. Zheng and T. Ando, Phys. Rev. B 65, 245420 (2002). 24L. Brey and H. A. Fertig, Phys. Rev. B 73, 195408 (2006). 25C. J. Tabert, Electronic Phenomena in 2D Dirac-like Sys- tems: Silicene and Topological Insulator Surface States , Ph.D. thesis, University of Guelph, Ontario, Canada (2015). 26J. O. Island, X. Cui, C. Lewandowski, J. Y. Khoo, E. M. Spanton, H. Zhou, D. Rhodes, J. C. Hone, T. Taniguchi, K. Watanabe, L. S. Levitov, M. P. Zaletel, and A. F. Young, Nature 571, 85 (2019). 27A graphene proximity model very similar to ours is em- ployed, where we use another convention10,K!K0, A=B I!A=B I.28T. Wakamura, F. Reale, P. Palczynski, M. Q. Zhao, A. T. C. Johnson, S. Gu eron, C. Mattevi, A. Ouerghi, and H. Bouchiat, Phys. Rev. B 99, 245402 (2019). 29Z. Wang, D.-K. Ki, H. Chen, H. Berger, A. H. MacDonald, and A. F. Morpurgo, Nat. Commun. 6, 8339 (2015). 30B. Yang, M.-F. Tu, J. Kim, Y. Wu, H. Wang, J. Alicea, R. Wu, M. Bockrath, and J. Shi, 2D Mater. 3, 031012 (2016). 31T. Frank, P. H ogl, M. Gmitra, D. Kochan, and J. Fabian, Phys. Rev. Lett. 120, 156402 (2018). 32C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005). 33D. Xiao, W. Yao, and Q. Niu, Phys. Rev. Lett. 99, 236809 (2007). 34T. Thonhauser, D. Ceresoli, D. Vanderbilt, and R. Resta, Phys. Rev. Lett. 95, 137205 (2005). 35D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010). 36M.-C. Chang and Q. Niu, Phys. Rev. B 53, 7010 (1996).

1511.03004v2.Boundary_conditions_and_Green_function_approach_of_the_spin_orbit_interaction_in_the_graphitic_nanocone.pdf

arXiv:1511.03004v2 [cond-mat.mes-hall] 11 May 2017Boundary conditions and Green function approach of the spin -orbit interaction in the graphitic nanocone J. Smotlacha1,2,∗and R. Pincak3,1,† 1Bogoliubov Laboratory of Theoretical Physics, Joint Insti tute for Nuclear Research, 141980 Dubna, Moscow region, Rus sia 2Faculty of Nuclear Sciences and Physical Engineering, Czech Technical University, Brehova 7, 110 00 Prague, Czech Republic 3Institute of Experimental Physics, Slovak Academy of Scien ces, Watsonova 47,043 53 Kosice, Slovak Republic (Dated: September 6, 2021) The boundary effects affecting the Hamiltonian for the nanoco ne with curvature–induced spin– orbit coupling were considered and the corresponding elect ronic structure was calculated. These boundary effects include the spin–orbit coupling, the elect ron mass acquisition and the Coulomb interaction. Different numbers of the pentagonal defects in the tip were considered. The matrix and analytical form of the Green function approach was used for t he verification of our results and the increase of their precision in the case of the spin–orbit cou pling. PACS numbers: 73.22.Pr, 81.05.ue, 71.70.Ej, 72.25.-b. Keywords: Tight-binding method, Graphitic nanocone, Spin –orbit coupling, Boundary conditions, Coulomb interaction, Green function approach I. INTRODUCTION The electronic structure of the carbon nanoparticles is of great in terest in the todays physical research. These nanostructures have a large potential use as electronic nanodev ices in the computer science. The main structure of this kind is the plain graphene, but other kinds of the nanostructur es like fullerene, nanotubes etc. were already largely investigated. The main characteristics of the electronic properties is the local de nsity of states ( LDoS). It can be calculated using the different methods; here we use the continuum gauge field- theory approach [1, 2] which leads to the solution of a 2-component Dirac-like equation. The result we get is the sum of the squares of the components of the normalized solution. This method was used in [3–10] for the calculation of the ele ctronic structure in the close surrounding of different kinds of the defects and of fullerene, nanocone, nanotu bes and wormhole. One phenomenon has not been yet considered enough in the graphe ne nanostructures: the effect of the spin-orbit coupling (SOC). The reason is that this effect is very weak in the case of the plain gr aphene, but it can grow considerably in other kinds of the carbon nanostructures becaus e of the curvature and the orbital overlap of the next-nearest neighbouring atomic sites. Furthermore, the influe nce of the impurities could play a crucial rule. Then, we distinguish the intrinsic and the extrinsic (Rashba) term of the SOC. A lot of theoretical work concerning this effect was performed in [6, 11–18]. There are more ways how to incorporate these terms into the Hamilt onian for the ground state of the given nanos- tructure. For the case of the graphitic nanocone [7], different met hods are described in [4] and [19]: in these papers, the terms corresponding to the SOCare added into different blocks of the corresponding matrix and, co nsequently, it is mixed with different terms of the remaining part of the Hamiltonian. Here, we will be more interested in the method used in [4], where the used formalism is an analog of the proced ure used for the case of the nanotubes [6]. In [4], the boundary effects were not considered. Their significance is (among others) connected with the needs of the quadratic integrability of the resulting wave function. They com e from the rapid change of the geometry close to the conical tip and from the finite size of the conical nanostructur e. The change of the geometry could give rise to the mass of the fermion [10], but it could be also simulated by an addition of a fictious charge into the conical tip [20]. Next, the influence of the pseudopotential is also significant [21 ]. Finding of the analytical solution of the resulting Hamiltonians is a very difficult task. In most of our calculations, we use the numerical methods. To verify them and achieve a higher p recision, the Green function approach is also used. ∗Electronic address: smota@centrum.cz †Electronic address: pincak@saske.sk2 In this paper, we review the results of the calculations concerning t he influence of the SOCon the electronic structure of the graphitic nanocone in [4], discuss the influence of t he boundary effects and incorporate the Green function formalism. In section II, we remind the main results from [4]. In section III, we present the influence of the possible boundary effects connected with the topology of the molec ular surface: the relativistic acquisition of the mass connected with the curvature and the Coulomb interaction. F inally, in section IV, we verify some our numerical results using the Green function method and we derive the perturb ation scheme to the second order for the purpose of getting more precise results of the calculation of the LDoS. II. LOCAL DENSITY OF STATES OF THE NANOCONE INFLUENCED BY THE SPIN–ORBIT COUPLING In [4], we derived the Hamiltonian for the curvature–induced SOCin the graphitic nanocone using the method of Ando [6]. We started with the case of the Hamiltonian without the SOC[7] and after incorporating the appropriate SOCterms, we finally got ˆHs=/planckover2pi1v/parenleftigg 0 ∂r−i1 rξxσx(/vector r)−is∂ϕ (1−η)r−Ay rσy−3η 2(1−η)r+1 2r −∂r+i1 rξxσx(/vector r)−is∂ϕ (1−η)r−Ay rσy−3η 2(1−η)r−1 2r0/parenrightigg , (1) where all the parameters connected with SOCand with the geometry of the nanocone are described in the mention ed paper. To calculate the LDoS, we solve the equation ˆHsψ(r,ϕ) =Eψ(r,ϕ) (2) and finally get the matrix equation (9) in [4]. There, this equation is solv ed in a numerical way and the resulting LDoSforn= 0 is sketched in Fig. 3 of that paper. The numerical method of finding the solution used in [4] can be used fo r other modes. In Fig. 1, we see the LDoSfor the same numbers of the defects, but here it corresponds to the sum of the solutions corresponding to n=−1,0,1,2,3. We can see that while for the case of 1 and 2 defects in the tip, the LDoSshould grow to infinity forr= 0 for an arbitrary value of energy, this effect is restricted only to zero energy in the case of 3 defects in the tip. FIG. 1: 3D graphs of the LDoSfor the graphitic nanocone influenced by the SOC. Here, the LDoScorresponds to the sum of the solutions corresponding to n=−1,0,1,2,3. The number of the defects in the tip in the particular cases :N= 1 (left), N= 2 (middle) and N= 3 (right). From these results follows that there could be a strong localization o f the electrons in the tip. But we did not considerdifferentboundaryeffectswhichcouldsignificantlyinfluenc ethisbehavior. Oureffortistofindaquadratically integrable solution which preserves the main features of the found approach. This means that for 1 and 2 defects the rise of theLDoSclose tor= 0 will be suppressed or restricted to a much smaller area around r= 0, and the peak forr= 0 andE= 0 in the case of 3 defects will be preserved. In the last case of 3 de fects, the calculations show that the main contribution to the revealed peak is the mode n=−1 and that considering the next modes the resulting character of the LDoSforr= 0 will approach the behavior of the case of 1 and 2 defects (Fig. 2) .3 FIG. 2: 3D graphs of the LDoSfor the graphitic nanocone with 3 defects in the tip influence d by the SOC:n=−1 (left) and the case corresponding to the sum with −5≤n≤4 (right). The Hamiltonian (1) is not the only possibility how to express the Hamilto nian for the SOC: we can also use the form of the Hamiltonian where the spin–orbit coefficients and the diffe rential operators occupy different blocks of the corresponding matrix and find the analytical solution. This possibility is outlined in [19], where the Rashba term is excluded and the intrinsic terms are moved to the diagonal positions of the Hamiltonian. So the resulting Hamiltonian (Eq. (10) in [19]) has the form ˆHs=/planckover2pi1v/parenleftbigg △soσ′ x−i∂r−i 2r−iνs r −i∂r−i 2r+iνs r−△soσ′ x(/vector r)/parenrightbigg (3) withσ′ x=σxcosα+σzsinαand sinα= 1−N 6. Let us stress that here the intrinsic term does not depend on r and theSOCdepends on the vortex angle only. The solution for this Hamiltonian is g iven by the modified Bessel function of the second kind. On the contrary, the analytical solut ion of (1) was not found; we can only say that the found numerical solution [4] is similar to the Bessel function of the fir st kind. Working on the Hamiltonian (1) investigated in [4], let us investigate th e boundary effects to get more precise information about the electronic structure close to r= 0. III. INFLUENCE OF BOUNDARY EFFECTS: SMOOTH GEOMETRY AND COU LOMB INTERACTION In the real graphitic nanocone, the tip has not a sharp form as one would expect for the cone geometry, but the geometry becomes smooth in this region (Fig. 3); it means, there is s uch a value of r0that our predictions fail for r<r0. From the possible ways how to remove this discrepancy, we outline th ese 2 methods: having to find a way to simulate the smooth geometry in the conical tip, we either suppose t hat due to a big curvature and the connected relativistic effects, the massless fermion, for which we solve equatio n (2), acquires a non-negligible mass m0[10] or a charge is considered in the conical tip from which the Coulomb interac tion comes influencing the results. A. Method of a massive fermion4 FIG. 3: The deviation of the geometry of the graphitic nanoco ne from the geometry of the real nanocone. Forr<r0, we can write the corrected version of the matrix equation (9) fro m [4]: m0 0∂r+F r−i rC 0 m0−i rD ∂r+F r −∂r+F−1 ri rD−m00 i rC−∂r+F−1 r0−m0 fn↑(r) fn↓(r) gn↑(r) gn↓(r) =E fn↑(r) fn↓(r) gn↑(r) gn↓(r) . (4) For higher values of r, where the curvature is considerably smaller, the mass is considere d to be zero and the form of the matrix equation is changed into the mentioned form. We can solve this equation numerically and analytically, as well. In the ca se of the numerical solution, especially the case of 3 defects would cause a shift and decoupling of the peak s in the corresponding graph. However, the main purpose of using the boundary conditions - getting the solution whic h is quadratically integrable - is not achieved here. For the purpose of getting the analytical solution, we suppose in (4 ) that the mass m0acquired by the fermion is considerably larger than other effects contained in the Hamiltonian, i.e.ξx= 0 andm0≫r,ξy,F,fn↑(and the same up to the 4-th order of the powers and the differentiations). Then , the solution can be very roughly approximated as fn↑(r) =rζ1exp/parenleftbig ζ2r2/parenrightbig ,ζ1=1−6F+4F2 1+4F,ζ2=E2−m2 0 2+8F, (5) fn↓(r) = irζ1 ξyexp/parenleftbig ζ2r2/parenrightbig ·/parenleftbigg −ζ1+F−1−2ζ2r2−E2−m2 0 2F−1r2/parenrightbigg , (6) gn↑(r) =E−m0 2F−1rζ1+1exp/parenleftbig ζ2r2/parenrightbig , (7) gn↓(r) = irζ1+1 ξyexp/parenleftbig ζ2r2/parenrightbig ·/bracketleftbiggE−m0 2F−1(ζ1+2−F+2ζ2r2)/bracketrightbigg . (8) The graphs of the LDoScorresponding to this solution coincide with the graphs for the nume rical solution with m0≫E, so the potential peaks for an arbitrary number of the defects d isappear here. B. Method of a charge simulation5 The influence of the charge considered in the conical tip is expresse d in the Hamiltonian by the presence of the diagonal term −κ r, whereκ= 1/137 is the fine structure constant. So this term substitutes the m ass term in (4): −κ r0∂r+F r−i rC 0 −κ r−i rD ∂r+F r −∂r+F−1 ri rD −κ r0 i rC−∂r+F−1 r0−κ r fn↑,C(r) fn↓,C(r) gn↑,C(r) gn↓,C(r) =E fn↑,C(r) fn↓,C(r) gn↑,C(r) gn↓,C(r) . (9) Here, the coefficients C,Dcan be considered nonzero, as well as zero, depending on whether we consider the parallel influence of both the Coulomb interaction and the SOCor the Coulomb interaction only. The analytical solution for the second case is given in [20] and we outline the corresponding LDoSin Fig. 4. (It is possible to consider nonzero mass in this case but here we suppose, in agreement with (9), m0= 0). In the expression for the analytical solution, nhas half-integer values, here we considered −2.5≤n≤2.5. The outlined solution, similarly as the solution for the case of the SOConly in Fig. (1), signalizes the localization of the electrons for r= 0, but on the contrary to that figure, for higher rand energies close to zero, the LDoSdecreases to zero. For the first case, we use the analog of the numerical method used in [4]: we express the solution in the form of an infinite series (Eqs. (A1), (A2) in [4]) and substitute it into the sy stem (9). By the addition of the requirement ξ1=ξ+ 2, we get a system of recurrence equations for the coefficients ak,bk,ck,dk,k≥0. From the boundary conditions follows α= 0 and the same for the coefficients a0,b0,a1,b1. Then, the value of the parameter ξis determined from the requirement of the nonzero solution of the sy stem for the coefficients a2,b2,c0,d0, i. e. the determinant of the corresponding matrix should be nonzero. In th is way, we get ξ=1 2(−5±/radicalig 1−4CD−4F+4F2±/radicalbig −CD(1−2F)2+κ2(C2−D2+κ2)). (10) We have to choose one of the 4 possible values of ξ; here we chose the value with plus signs on the critical places. Then, the coefficients a2,b2,c0,d0create the components of the corresponding eigenvector. Now, fork≥2, we can easily calculate the coefficients ak,bk,ck−2,dk−2in the chosen limit. The coefficient βcan be chosen arbitrarily; we choseβ= 0. The resulting graphs of the LDoSare in Fig. 5. Here, similarly as in Fig. 1, −1≤n≤3. Comparing Figs. 4 and 5 we see that the same problem appears in both cases: the electrons arelocalized in the tip for an arbitraryenergyfor an arbitrarynumberofthedefectsandtheuniquenessofthepeakf orzeroenergy,whichcorrespondedtothemode n=−1 and 3 defects in the case of the SOConly (Fig. 1), is distorted by the divergence of the LDoSat all energies for r= 0. FIG. 4: Graphs of the LDoSfor the graphitic nanocone influenced by the Coulomb interac tion (with the exclusion of the SOC) for different distances rfrom the tip, −2.5≤n≤2.5 and for 1, 2 and 3 defects. We see that similarly as in the case of the SOConly, we get the localization of the electrons in the tip for 1 and 2 defects at all energies but, furthermore, there is no energy restriction for the case of 3 defects as well. C. Comparison of the results6 FIG. 5: Graphs of the LDoSfor the graphitic nanocone influenced by the Coulomb interac tion (including the influence of the SOC) for different distances rfrom the tip, −1≤n≤3 and for different numbers of the defects. Let us investigate the character of the divergence of the LDoSforr= 0 closer. In Fig. 6, we see the dependence of theLDoSonrvariable for zero energy in the case of the influence of the SOConly and of the simultaneous influence of the SOCand the Coulomb interaction. We see here that in comparison with the first case, in the second case the decrease of the LDoSclose tor= 0 is much faster and one could suppose that the quadratic integra bility of the acquired solution is achieved here. To verify this hypothesis, we have to do the integration of the LDoSclose tor= 0 in all of the outlined cases. This task is still in progress. FIG. 6: Behaviour of the LDoSfor zero energy close to r= 0 for different numbers of defects in the conical tip: influen ce of theSOConly (left) and the simultaneous influence of the SOCand the Coulomb interaction (right). In Figs. 7 and 8, we see the comparison of all the investigated effect s which can influence the electronic structure of the graphitic nanocone. From the graphs follows that at higher d istances, the biggest significance has the effect of theSOCand its influence strongly depends on the number of the defects, w hile the effect of the Coulomb interaction (with or without the SOC) nearly does not depend on the number of the defects. On the other hand, at the lower distance, just after the fast dec rease of the LDoSclose tor= 0, the effect of theSOCdecreases significantly and this decrease is stronger for the highe r number of the defects. The effect of the Coulomb interaction in the presence of the SOCdecreases as well for the higher number of the defects and it approaches zero for E= 0. If the Coulomb interaction occurs without the SOC, its influence, thanks to the decrease of other effects, is getting more significant, but significant change s of its magnitude are appreciable only close to zero.7 FIG. 7: Comparison of the influence of the SOCand the Coulomb interaction at the distance r= 1 from the tip for different numbers of the defects: simultaneous influence of both effect s (denoted by SC), the Coulomb interaction only (C) and the SOC only (S). FIG. 8: The same at the distance r= 5 from the tip. IV. GREEN FUNCTION APPROACH The results described in the previous sections were acquired by add ing different influences to the basic Hamiltonian from [7], but the presented solutions are distorted by mistakes com ing from the used methods of finding the numerical solution. We can verify their validity using different kinds of the Green function method. The first kind follows from the chemical structure of the molecule an d estimated energy of the spin–orbit coupling corresponding to the given atomic site. On this base, we compose th e matrix H(E) which describes the energy of the atomic bonds and of the corresponding spin–orbit couplings. Its inv erse matrix corresponds to the Green function matrix Gfrom which the LDoSfor different atomic sites will be calculated. The second kind follows from the analytical expression for the Gree n function corresponding to the pure graphitic nanocone without any other influences [7] from which the Green fun ction corresponding to the spin–orbit coupling will be calculated as a perturbation. A. Matrix method Using this method, we verify especially the presence of the peak for zero energy close to the conical tip found in section II for the case of 3 defects and mode n=−1. It was found using the numerical method of the solution of the Dirac-like equation. The principle of this method consists in the const ruction of the matrix of the type ( N,N), where Nis the number of the atoms in the given molecule. In this matrix, the at omic sites are characterized by the diagonal elements and each interatomic bond is characterized by its energy in the non-diagonal element at the appropriate position. For example, for the 2-atomic molecule, the appropriate m atrix has the form H(E) =/parenleftbigg E t t E/parenrightbigg , (11) wheretis the energy corresponding to the interatomic bond. In the case o f the consideration of the spin–orbit8 interaction, we perform this substitution for each matrix element: E→/parenleftbigg E+△(r) 0 0E−△(r)/parenrightbigg (12) for the diagonal elements and t→/parenleftbigg t0 0t/parenrightbigg (13) for the non-diagonal elements. Here, ±△(r) denotes the increase or the decrease, resp. of the energy by t he energy corresponding to the spin–orbit interaction depending on the dista nce from the tip. In the above mentioned case of the 2-atomic molecule it means the transformation of the correspo nding matrix in the following way: /parenleftbigg E t t E/parenrightbigg → E+△1(r) 0 t 0 0E−△1(r) 0 t t 0E+△2(r) 0 0 t 0E−△2(r) (14) For theN-atomic molecule we create in this way the matrix of the type (2 N,2N). Then, the local density of states corresponding to the m-th atomic position in this molecule we calculate from the expression LDoS(E) =1 πIm(G2m−1,2m−1(E−i0)+G2m,2m(E−i0)), (15) where it holds for the Green function matrix G G(E) =H−1(E). (16) In this way, we create the corresponding matrices H(E) and we calculate the local density of states for the different atomic positions in the case of the nanocones with different numbers of the defects in the tip. The placement of the defects can differ and in this way, the general result could be influen ced. FIG. 9: Local density of states for nanocones with different n umbers and configurations of pentagonal defects in the tip: 1 defect (left), different configurations of 2 defects (middle , right). In Figs. 9 and 10 we see the local density of states calculated for diff erent configurations of 1, 2, 3 and 4 defects in the tip. In details, the results differ from the results in section II, where all the defects are concentrated in the9 FIG. 10: Local density of states for nanocones with different numbers and configurations of defects in the tip: triangular defect (left), 3 defects (middle), 4 defects (right). As shown in [2 2], the cases of npentagonal defects and 1 defect with 6 −nedges are equivalent, so, the case of triangular defect can be cons idered as 1 of possible configurations of 3 pentagonal defect s. conical tip, while here they are placed in a limited area around the tip. B ut the main feature remains the same: In the case of 3 defects, a significant peak around the tip close to zer o energy appears. This feature is not so significant for the case of other numbers of the defects - other peaks appe ar in other distances from the tip and this reduces the significance of possible peaks close to the tip. Moreover, the case o f 3 defects is the only case for which we can get an approximate value of LDoSin the tip (r= 0) - only in this case a configuration with an atom in the tip exists. But still, for other numbers of the defects, we see some of the previou sly predicted phenomena - for example in Fig. 9, the difference of the character of LDoSfor different configurations of 2 defects is in agreement with the pr edictions from [23]: the configuration in the right part shows much more signific ant metallic properties than the configuration in the middle part. In accordance with the results presented in section II for the cas e of the solution of the Dirac-like equation, in all the results presented in this section the bonds between the car bon atoms are considered the same. In the real nanostructure, the strength of the bonds can differ and we can p redict it through the geometric optimization. The dependence △(r) is considered to be linear, so, △(r) =△0·r, (17) where△0= 0.8meV [24]. B. Analytical method In this method, we calculate the LDoSusing the expression LDoS(E) =1 πImTrG(E−i0), (18) whereGis the Green function corresponding to the Hamiltonian (1). This exp ression is formally identical to (15), but here the calculations follow from the analytical expression for t he Green function which we can get with the help of the perturbation calculation. It starts from the Green functio nG0which corresponds to the Hamiltonian without the influence of the SOC[7], where the analytical solution is known. In this calculation, we divide the Hamiltonian (1) as ˆHs=ˆH0s+ˆVSOs (19)10 with ˆH0s=/planckover2pi1v 0/parenleftig ∂r−is1 r(1−η)∂ϕ+1 2r−3 2η (1−η)r/parenrightig ⊗1/parenleftig −∂r−is1 r(1−η)∂ϕ−1 2r−3 2η (1−η)r/parenrightig ⊗1 0 ,(20) and ˆVSOs=/planckover2pi1v/parenleftigg 0 −iδγ′ 4γRσx(/vector r)−Ayσy r iδγ′ 4γRσx(/vector r)−Ayσy r0/parenrightigg , (21) here we denote 1=/parenleftbigg 1 0 0 1/parenrightbigg . (22) In the perturbation scheme holds G(r′′,ϕ′′,r′,ϕ′;E) =G0(r′′,ϕ′′,r′,ϕ′;E)+G1(r′′,ϕ′′,r′,ϕ′;E)+G2(r′′,ϕ′′,r′,ϕ′;E)+...= =G0(r′′,ϕ′′,r′,ϕ′;E)+/integraldisplay G0(r′′,ϕ′′,r,ϕ;E)VSOs(r,ϕ)G0(r,ϕ,r′,ϕ′;E)r(1−η)drdϕ+ +/integraldisplay G0(r′′,ϕ′′,r1,ϕ1;E)VSOs(r1,ϕ1)G0(r1,ϕ1,r2,ϕ2;E)VSOs(r2,ϕ2)G0(r2,ϕ2,r′,ϕ′;E)r1r2(1−η)2dr1dr2dϕ1dϕ2+...etc. (23) Here,G0(r′′,ϕ′′,r′,ϕ′;E) =/angbracketleftr′′,ϕ′′|(H0s−E)−1|r′,ϕ′/angbracketrightis specified in [7]. There could be a mismatch thanks to different size of G0andVSOs: the matrix G0as defined in [7] is of the kind 2 ×2 while the matrix VSOsis of the kind 4×4. This discrepancy is solved in such a way that G0in the calculations is considered as 1⊗G0. We will calculate the approximation of the Green function to the first and to the second order. 1. First order approximation In the first order approximation, the Green function has the form (G1s)k,l=/planckover2pi1υδ 8πγ′ 4γ/radicalbig η(2−η) (1−η)2/parenleftbigg −1 0 0 1/parenrightbigg/summationdisplay n∈Z+∞/integraldisplay 0(al1 n(r,r′)a2k n+1(r′,r)−al2 n(r,r′)a1k n+1(r′,r))dr+ +i/planckover2pi1υδps 4π/radicalbig η(2−η) (1−η)2/parenleftbigg 0 1 −1 0/parenrightbigg ·/summationdisplay n∈Z∞/integraldisplay 0/parenleftbig al2 n(r,r′)a1k n(r′,r)+al1 n(r,r′)a2k n(r′,r)/parenrightbig dr. (24) For the purpose of the calculation of the LDoS, we are interested in the trace of the corresponding matrix. Since this tracehasthe zerovalue, the firstorderapproximationtothe LDoSiszero. It meansthat the first orderapproximation is not sufficient to calculate the corrections to the LDoSfollowing from the influence of the SOC. The particular elements of the corresponding matrix are nonzero, but it is irreleva nt for us for this moment. So we will try to find the result from the approximation to the second order. 2. Second order approximation11 In the second order approximation, the trace of the matrix corre sponding to the Green function (which is needed for the definition of the LDoSin (18) ) has the form TrG2,s=−/planckover2pi12v2 16π2δ2η(2−η) (1−η)3 /parenleftbiggγ′ 4γ/parenrightbigg2/summationdisplay |n01−n12|=1∞/integraldisplay 0∞/integraldisplay 0I1(r′,r1,r2)dr1dr2−8p2/summationdisplay n∈Z∞/integraldisplay 0∞/integraldisplay 0I2(r′,r1,r2)dr1dr2 ,(25) where I1(r′,r1,r2) =a12 n01(r2,r′)/parenleftBig a12 n12(r1,r2)a11 n01(r′,r1)−a11 n12(r1,r2)a21 n01(r′,r1)/parenrightBig +a11 n01(r2,r′)/parenleftBig a21 n12(r1,r2)a21 n01(r′,r1)−a22 n12(r1,r2)a11 n01(r′,r1)/parenrightBig + +a22 n01(r2,r′)/parenleftBig a12 n12(r1,r2)a12 n01(r′,r1)−a11 n12(r1,r2)a22 n01(r′,r1)/parenrightBig +a21 n01(r2,r′)/parenleftBig a21 n12(r1,r2)a22 n01(r′,r1)−a22 n12(r1,r2)a12 n01(r′,r1)/parenrightBig ,(26) I2(r′,r1,r2) =a12 n(r2,r′)/parenleftBig a12 n(r1,r2)a11 n(r′,r1) +a11 n(r1,r2)a21 n(r′,r1)/parenrightBig +a11 n(r2,r′)/parenleftBig a22 n(r1,r2)a11 n(r′,r1) +a21 n(r1,r2)a21 n(r′,r1)/parenrightBig + +a22 n(r2,r′)/parenleftBig a12 n(r1,r2)a12 n(r′,r1) +a11 n(r1,r2)a22 n(r′,r1)/parenrightBig +a21 n(r2,r′)/parenleftBig a22 n(r1,r2)a12 n(r′,r1) +a21 n(r1,r2)a22 n(r′,r1)/parenrightBig . (27) Now, we will make the substitution with the help of [7] and we will integra te overr1,r2with the help of [25]. However, thanks to the appearance of the multiples of 3 coefficient sakl n(ri,rj), with the respect to their definition in [7], the next integration over 3 variables p1,p2,p3is needed. It will be performed numerically and the energy variable will be considered complex in the result. We compare the results acquired by using the Green function appro ach and with the help of the numerical method used in [4], where the solution for the mode n= 0 is presented: from (18) follows LDoS 2(E) =1 πImTr [G0(E−i0)+G1(E−i0)+G2(E−i0)], (28) where the lower index 2 in the LDoS 2means the mentioned precision. To calculate G0, we exploit the expression (A.6) in [7] for the calculation of Tr G0: E π/integraldisplay∞ 0dpp /planckover2pi12v2p2−E2/summationdisplay n∈Z[J2sn−1(pr)J2sn−1(pr′)+J2sn(pr)J2sn(pr′)], (29) wherer→r′. In the sum, we use only the term J2sn(pr)J2sn(pr′) forn= 0. TrG1is zero for arbitrary n, as follows from our calculations for the first order approximation. We calculat e TrG2using (25). FIG. 11: LDoScalculated using the numerical method in [4] (left - see also the right part of Fig. 3 in [4]), zeroth order approximation calculated using the relations in [7] (middl e) and second order approximation calculated using (25) (ri ght), performed for 3 defects and the values are n= 0,r′= 5. If the second order approximation is sufficient, then the graph in th e left part of Fig. 11 should be given by the sum of the graphs in the middle and in the right part. But at first glance, t he graphs in the middle and in the right part have different scales, so their sum cannot approach the graph in th e left part. The reason is that no normalization was considered during the calculation of the Green function approach. So, using (28), we try to evaluate the expression 1 πImTr [G0(E−i0)+NG2(E−i0)], (30)12 /s45/s49 /s48 /s49/s48/s44/s48/s48/s44/s53/s49/s44/s48/s76/s68/s79/s83 /s50 /s32/s32 /s69/s45/s49 /s48 /s49/s48/s44/s48/s48/s48/s44/s48/s50/s48/s44/s48/s52/s48/s44/s48/s54 /s32/s32/s76/s68/s79/s83 /s50 /s69 FIG. 12: Expression (30) evaluated using 2 different values o fN:N= 1 (left) and N= 1/20 (right). whereNis the normalization constant. In Fig. 12, we see that (30) will coincide with the left part of Fig. 11 mu ch better for the case N= 1/20. But still it is not complete coincidence, especially for the energy values close t o zero. So it means that the next order of the Green approach is needed. 3. Case of the finite size In the case of the finite size of the conical structure, the energy levels would be quantized and the Green function approach would be modified for the case of the discrete spectrum. The discrete energy levels would be calculated on the base of an additional requirement on the wave function which sh ould have the zero value on the border. We use the simplest case of the nanocone without the SOCpresented in [7]. This case provides the solution of the form (see (49) in [7]) ψ(r) =1 2/radicalbig π(1−η)/parenleftigg Jν(|E| /planckover2pi1vr) signEJν+1(|E| /planckover2pi1vr)/parenrightigg . (31) Particular parameters have this form: ν=sn−η 1−η,/planckover2pi1v=3 2t (32) wheretis the energy of the C-C bond (about 3 eV). So we require that one o f the components of the wave function has the zero value on the border: Jν(rmax·2 3|E′|) = 0,resp.Jν+1(rmax·2 3|E′|) = 0, (33) whereE′andrmaxare measured in the units of tand the length of C-C bond, respectively. To demonstrate the character of the resulting energy levels, we will investigate the cas es of 1 and 3 defects, i.e. η= 1/6 andη= 1/2 and make the choice of the parameters s= 1,n= 2. Next, we take the value rmax= 20 as the border value of r. Then, for the particular cases, the low-energy spectrum contains thes e values: -1 defect:ν= 2.5 andE′= 0,±0.432259,±0.682126,±0.924221,±1.1636,±1.40168,±1.63904,±1.87596 or ν= 3.5 andE′= 0,±0.524095,±0.781284,±1.02735,±1.26927,±1.50914,±1.74782,±1.98576 -3 defects:ν= 3 andE′= 0,±0.478512,±0.732077,±0.97614,±1.21676,±1.45571,±1.6937,±1.93111 or ν= 4 andE′= 0,±0.569126,±0.829853,±1.07794,±1.3212,±1.56202,±1.80143 We see that the dependence of the energy levels on the number of t he defects in the tip is not very strong. To calculate the resulting LDoS, the Green function approach for the discrete spectrum would be performed with the help of the summation over other modes n.13 V. CONCLUSION The boundary effects influencing the electronic structure of the g raphitic nanocone were investigated and we used mostly numerical methods for the corresponding calculations. We w anted to find a quadratically integrable solution which would preserve the main features of the solution found in [4], e specially the isolated peak for r= 0 andE= 0 in the case of 3 defects which could have a potential use for the con struction of the probing tip in the atomic force microscopy. We achieved this goal only partially - the found solution f or the case of the simultaneous influence of the Coulomb interaction and the SOCin Fig. 5 (as well as the analytical solution for the case of the Coulomb interaction only from [20] sketched in Fig. 4) seems to be quadratically integrable , but the mentioned isolated peak for the case of 3 defects disappeared. We also tried to find the possible correct ions caused by the considerable rise of the mass of the electron bounded on the molecular surface close to r= 0 (coming from the rapid curvature), but the resulting changesof the LDoSconnected with this effect seem to be negligible (regardlesswe use th e numerical or the analytical solution). For the case of the spin–orbit interaction, the numerical results w ere verified using the matrix form of Green function method for the different numbers and configurations of t he defects. It was shown that although some small disagreements in the results were presented, the main character of theLDoSremained the same, especially for the case of 3 defects and the zero energy peak close to the tip. We can also do a comparison with the results in [26], where the calculations were performed for the case of the nanocone with 0 (nanodisk), 1 and 2 pentagonal defects in the tip. The investigated nanostructures contain very high number of car bon atoms - the nanostructures with the number of atoms about 250 and 5000 were chosen. The results also show an ap pearance of a zero energy peak in LDoSclose to the tip. For the case of 5000 atoms, the height of the peak does not depend on the number of the defects, for the case of 250, the height of this peak increases a little with the number of the defects. So, we can suppose that for our case of tens of atoms, the results would confirm our calculations - t he peak would be much more significant for higher number of defects, especially the case of 3 defects. The results coming from the analytical form of the Green function a pproach for the continuous spectrum (infinite size of the nanocone) don’t correlate sufficiently with the results of the numerical approach in [4]. This is given by more reasons: first, we performed the calculations of the difficult in tegrals numerically with the help of the program Mathematica, so the precision of the acquired results is limited. Seco ndly, we did the calculations to the second order only. Maybe we should use also the third order approach. In further calculations, we could also consider the electronic struc ture of the double graphitic nanocone. The corresponding Hamiltonian would differ by the presence of a new cont ribution which would be constant for all range ofrexcluding a small interval close to the tip, where a strong influence o f the interaction with the neighboring atomic orbitals caused by the curvature would increase the resulting effec t. Although up to now, no effective method for the fabrication of the g raphene nanocones was found, an evidence of occurrence of these materials was described in [27]. Moreover, nex t theoretical calculations and molecular simulations were performed [28, 29]. These results promise practical applicatio ns of the graphitic nanocones in close future. New calculations related to the spin–orbital coupling in graphene [30] as well as the theoretical and experimental investigations of graphene monolayer and bilayer from the last days [31, 32] contribute to this goal. ACKNOWLEDGEMENTS — The work was supported by the Science and T echnology Assistance Agency under Contract No. APVV-0171-10, VEGA Grant No. 2/0037/13 and Minis try of Education Agency for Structural Funds of EU in the frame of project 26220120021, 26220120033 and 261 10230061. R. Pincak would like to thank the TH division at CERN for hospitality. [1] D. P. DiVincenzo and E.J. Mele, Phys. Rev. B 29, 1685 (1984). [2] J. Callaway, Quantum Theory of the Solid State (Academic, New York, 1974). [3] E. A. Kochetov, V. A. Osipov and R. Pincak, J. Phys.: Conde ns. Matter 22, 395502 (2010). [4] R. Pincak, J. Smotlacha and M. Pudlak, Eur. Phys. J. B 881, 17 (2015). [5] J. Smotlacha, R. Pincak and M. Pudlak, Eur. Phys. J. B 84, 255 (2011). [6] T. Ando, J. Phys. Soc. Jpn. 69, 1757 (2000). [7] Yu. A. Sitenko, N. D. Vlasii, Nucl. Phys. B 787, 241 (2007). [8] J. Gonzalez, F. Guinea and J. Herrero, Phys. Rev. B 79, 165434 (2009). [9] J. Gonzalez and J. Herrero, Nucl. Phys. B 825, 426443 (2010). [10] J. Smotlacha and R. Pincak, Quantum Matter 4, (2015). [11] C. L. Kane and E. J. Mele, Phys. Rev. Lett. 95, 226801 (2005).14 [12] F. Kuemmeth, S. Ilani, D. C. Ralph, P. L. McEuen, Nature 452, 448 (2008). [13] T. Fang, W. Zuo and H. Luo, Phys. Rev. Lett. 101, 246805 (2008). [14] G. A. Steele et al., Nature Communications DOI:10.1038 /ncomms2584. [15] D. Huertas–Hernando, F. Guinea and A. Brataas, Phys. Re v. B74, 155426 (2006). [16] J. S. Jeong, J. Shin and H. W. Lee, Phys. Rev. B 84, 195457 (2011). [17] M. del Valle, M. Marganska and M. Grifoni, Phys. Rev. B 84, 165427 (2011). [18] K. N. Pichugin, M.Pudlak and R.G. Nazmitdinov, Eur. Phy s. J. B87, 124 (2014). [19] T. Choudhari, N. Deo, EPL 108, 57006 (2014). [20] B. Chakraborty, K. S. Gupta, S. Sen, J. Phys. A 46, 055303 (2013). [21] R. Pincak, J. Smotlacha and M. Pudlak, Physica B 441, 58 (2014). [22] R. Pincak, M. Pudlak, chapter in Progress in Fullerene Research , ed. F. Columbus, Nova Science Publishers, New York (2007). [23] P. E. Lammert and V. H. Crespi, Phys. Rev. B 69035406 (2004). [24] G. A. Steele, F. Pei, E. A. Laird, J. M. Jol, H. B. Meerwald t, L. P. Kouwenhoven, Nature Communications 4, 1573 (2013). [25] M. Abramowitz and I.A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1972). [26] P. Ulloa, A. Latg, L. E. Oliveira and M. Pacheco, Nanosca le Res. Lett. 8, 384 (2013). [27] S. N. Naess, A. Elgsaeter, G. Helgesen and K. D. Knudsen, Sci. Technol. Adv. Mater. 10, 6 (2009). [28] D. Ma, H. Ding, X. Wang, N. Yang, X. Zhang, Int. J. Heat. Ma ss. Tran. A 108, 940 (May 2017). [29] S. Conti, H. Olbermann and I. Tobasco, Math. Models Meth ods Appl. Sci. 0, 1 (2016). [30] R. Mohammadkhani, B. Abdollahipour and M. Alidoust, EP L111, 6 (2015). [31] D. D. Borges, C. F. Woellner, P. A. S. Autreto, D. S. Galva o,Water Permeation through Layered Graphene-based Mem- branes: A Fully Atomistic Molecular Dynamics Investigatio n, arXiv:1702.00250 (2017). [32] E. del Corro, M. Pea-Alvarez, K. Sato, A. Morales-Garci a, M. Bousa, M. Mracko, R. Kolman, B. Pacakova, L. Kavan, M. Kalbac, and O. Frank, Fine tuning of optical transition energy of twisted bilayer graphene via interlayer distance modulation , accepted to Phys. Rev. B (2017).

1012.3575v1.Spin_Relaxation_in_Quantum_Wires.pdf

arXiv:1012.3575v1 [cond-mat.mes-hall] 16 Dec 2010Spin Relaxation in Quantum Wires P. Wenk∗ School of Engineering and Science, Jacobs University Breme n, Bremen 28759, Germany S. Kettemann† School of Engineering and Science, Jacobs University Breme n, Bremen 28759, Germany, and Asia Pacific Center for Theoretical Physics and Division of Advanced Materials Science Pohang University of Science and Technol ogy (POSTECH) San31, Hyoja-dong, Nam-gu, Pohang 790-784, South Korea The spin dynamics and spin relaxation of itinerant electron s in quantum wires with spin-orbit coupling is reviewed. We give an introduction to spin dynami cs, and review spin-orbit coupling mechanisms in semiconductors. The spin diffusion equation w ith spin-orbit coupling is derived, using only intuitive, classical random walk arguments. We g ive an overview of all spin relaxation mechanisms, with particular emphasis on the motional narro wing mechanism in disordered conduc- tors, the D’yakonov-Perel’-Spin relaxation (DPS). Here, w e discuss in particular, the existence of persistent spin helix solutions of the spin diffusion equati on, with vanishing spin relaxation rates. We then, derive solutions of the spin diffusion equation in qu antum wires, and show that there is an effective alignment of the spin-orbit field in wires whose wid th is smaller than the spin precession lengthLSO. We show that the resulting reduction in the spin relaxation rate results in a change in the sign of the quantum corrections to the conductivity. F inally, we present recent experimental results which confirm the decrease of the spin relaxation rat e in wires whose width is smaller than LSO: the direct optical measurement of the spin relaxation rate , as well as transport measurements, which show a dimensional crossover from weak antilocalizat ion to weak localization as the wire width is reduced. Open problems remain, in particular in nar rower, ballistic wires, were optical and transport measurements seem to find opposite behavior of the spin relaxation rate: enhancement, suppression, respectively. We conclude with a review of the se and other open problems which still challenge the theoretical understanding and modeling of th e experimental results.2 Contents I. Introduction 3 II. Spin Dynamics 3 A. Dynamics of a Localized Spin 3 B. Spin Dynamics of Itinerant Electrons 4 1. Ballistic Spin Dynamics 4 2. Spin Diffusion Equation 4 3. Spin-Orbit Interaction in Semiconductors 5 4. Spin Diffusion in the Presence of Spin-Orbit Interaction 7 III. Spin Relaxation Mechanisms 9 A. D’yakonov-Perel’ Spin Relaxation 10 B. DP Spin Relaxation with Electron-Electron and Electron- Phonon Scattering 11 C. Elliott-Yafet Spin Relaxation 12 D. Spin Relaxation due to Spin-Orbit Interaction with Impur ities 12 E. Bir-Aronov-Pikus Spin Relaxation 13 F. Magnetic Impurities 13 G. Nuclear Spins 13 H. Magnetic Field Dependence of Spin Relaxation 13 I. Dimensional Reduction of Spin Relaxation 14 IV. Spin-Dynamics in Quantum Wires 14 A. One-Dimensional Wires 14 B. Spin-Diffusion in Quantum Wires 14 C. Weak Localization Corrections 17 V. Experimental Results on Spin Relaxation Rate in Semicond uctor Quantum Wires 19 A. Optical Measurements 19 B. Transport Measurements 20 VI. Critical Discussion and Future Perspective 20 VII. Summary 21 Symbols 22 Acknowledgments 22 References 223 I. INTRODUCTION The emerging technology of spintronics intends to use the ma nipulation of the spin degree of freedom of individual electrons for energy efficient storage and transport of infor mation.1In contrast to classical electronics, which relies on the steering of charge carriers through semiconductors, sp intronics uses the spin carried by electrons, resembling ti ny spinning tops. The difference to a classical top is that its an gular momentum is quantized, it can only take two discrete values, up or down. To control the spin of electrons, a detail ed understanding of the interaction between the spin and orbital degrees of freedom of electrons and other mechanism s which do not conserve its spin, is necessary. These are typically weak perturbations, compared to the kinetic ener gy of conduction electrons, so that their spin relaxes slowl y to the advantage of spintronic applications. The relaxatio n, or depolarization of the electron spin can occur due to the randomization of the electron momentum by scattering fr om impurities, and dislocations in the material, and due to scattering with elementary excitations of the solid such as phonons and other electrons, when it is transferred to the randomization of the electron spin due to the spin-orbit interaction. In addition, scattering from localized spins , such as nuclear spins and magnetic impurities are sources of electron spin relaxation. The electron spin relaxation can be reduced by constraining the electrons in low dimensional structures, quantum wells (confined in one direction, free in two dimensions), quantum wires ( confined in two direction s, free in one direction), or quantum dots ( confined in all three directions). Although spin relaxation is typical ly smallest in quantum dots due to their discrete energy leve l spectrum, the necessity to transfer the spin in spintronic d evices, recently lead to intense research efforts to reduce t he spin relaxation in quantum wires, where the energy spectrum is continuous. In the following we will review the theory of spin dynamics and relaxation in quantum wires, and compar e it with recent experimental results. After a general introduction to spin dynamics in Section II, we discuss all relevant spin relaxation mechanisms and how they depend on dimension, temperature, mobility, charge carrier densi ty and magnetic field in Section III. In particular, we review recent results on spin relaxation in semiconducting quantu m wires, and its influence on the quantum corrections to their conductance in Section IV. These weak localization corrections are thereby a very sen sitive measure of spin relaxation in quantum wires, in addition to optical methods as we review in Section V. We set /planckover2pi1= 1in the following. II. SPIN DYNAMICS Before we review the spin dynamics of conduction electrons a nd holes in semiconductors and metals, let us first reconsider the spin dynamics of a localized spin, as governe d by the Bloch equations. A. Dynamics of a Localized Spin A localized spin ˆ s, like a nuclear spin, or the spin of a magnetic impurity in a so lid, precesses in an external magnetic field Bdue to the Zeeman interaction with Hamiltonian HZ=−γgˆ sB, whereγgis the corresponding gyromagnetic ratio of the nuclear spin or magnetic impurity spin, respectively, which we will set equal to one, unless needed explicitly. This spin dynamics is governed by the Blo ch equation of a localized spin, ∂tˆ s=γgˆ s×B. (1) This equation is identical to the Heisenberg equation ∂tˆ s=−i[ˆ s,HZ]for the quantum mechanical spin operator ˆ sof anS= 1/2-spin, interacting with the external magnetic field Bdue to the Zeeman interaction with Hamiltonian HZ. The solution of the Bloch equation for a magnetic field pointi ng in the z-direction is ˆsz(t) = ˆsz(0), while the x- and y- components of the spin are precessing with frequency ω0=γgBaround the z-axis, ˆsx(t) = ˆsx(0)cosω0t+ˆsy(0)sinω0t, ˆsy(t) =−ˆsx(0)sinω0t+ ˆsy(0)cosω0t. Since a localized spin interacts with its environment by ex change interaction and magnetic dipole interaction, the precession will depha se after a time τ2, and the z-component of the spin relaxes to its equilibrium value sz0within a relaxation time τ1. This modifies the Bloch equations to the phenomenological equations, ∂tˆsx=γg(ˆsyBz−ˆszBy)−1 τ2ˆsx ∂tˆsy=γg(ˆszBx−ˆsxBz)−1 τ2ˆsy ∂tˆsz=γg(ˆsxBy−ˆsyBx)−1 τ1(ˆsz−sz0). (2)4 B. Spin Dynamics of Itinerant Electrons 1. Ballistic Spin Dynamics The intrinsic degree of freedom spin is a direct consequence of the Lorentz invariant formulation of quantum mechanics. Expanding the relativistic Dirac equation in th e ratio of the electron velocity and the constant velocity of lightc, one obtains in addition to the Zeeman term, a term which coup les the spin swith the momentum pof the electrons, the spin-orbit coupling HSO=−µB 2mc2ˆ s p×E=−ˆ sBSO(p), (3) where we set the gyromagnetic ratio γg= 1.E=−∇V, is an electrical field, and BSO(p) =µB/(2mc2)p×E. Substitution into the Heisenberg equation yields the Bloch equation in the presence of spin-orbit interaction: ∂tˆ s=ˆ s×BSO(p), (4) so that the spin performs a precession around the momentum de pendent spin-orbit field BSO(p). It is important to note, that the spin-orbit field does not break the invariance under time reversal ( ˆ s→ −ˆ s,p→ −p), in contrast to an external magnetic field B. Therefore, averaging over all directions of momentum, the re is no spin polarization of the conduction electrons. However, injecting a spin-polar ized electron with given momentum pinto a translationally invariant wire, its spin precesses in the spin-orbit field as the electron moves through the wire. The spin will be oriented again in the initial direction after it moved a leng thLSO, the spin precession length. The precise magnitude ofLSOdoes not only depend on the strength of the spin-orbit intera ction but may also depend on the direction of its movement in the crystal, as we will discuss below. 2. Spin Diffusion Equation Translational invariance is broken by the presence of disor der due to impurities and lattice imperfections in the conductor. As the electrons scatter from the disorder poten tial elastically, their momentum changes in a stochastic way, resulting in diffusive motion. That results in a change o f the the local electron density ρ(r,t) =/summationtext α=±|ψα(r,t)|2, whereα=±denotes the orientation of the electron spin, and ψα(r,t)is the position and time dependent electron wave function amplitude. On length scales exceeding the ela stic mean free path le, that density is governed by the diffusion equation ∂ρ ∂t=De∇2ρ, (5) where the diffusion constant Deis related to the elastic scattering time τbyDe=v2 Fτ/dD, wherevFis the Fermi velocity, and dDthe Diffusion dimension of the electron system. That diffusio n constant is related to the mobility of the electrons, µe=eτ/m∗by the Einstein relation µeρ=e2νDe, whereνis the density of states per spin at the Fermi energy EF. Injecting an electron at position r0into a conductor with previously constant electron densityρ0, the solution of the diffusion equation yields that the elect ron density spreads in space according to ρ(r,t) =ρ0+exp(−(r−r0)2/4Det)/(4πDet)dD/2, wheredDis the dimension of diffusion. That dimension is equal to the kinetic dimension d,dD=d, if the elastic mean free path leis smaller than the size of the sample in all directions. If the elastic mean free path is larger than the sample size in one direction the diffusion dimension reduces by one, accordingly. Thus, on average the variance of the distance t he electron moves after time tis/angb∇acketleft(r−r0)2/angb∇acket∇ight= 2dDDet. This introduces a new length scale, the diffusion length LD(t) =√Det. We can rewrite the density as ρ=/angb∇acketleftψ†(r,t)ψ(r,t)/angb∇acket∇ight, whereψ†= (ψ† +,ψ† −)is the two-component vector of the up (+), and down (-) spin fe rmionic creation operators, and ψthe 2-component vector of annihilation operators, respect ively,/angb∇acketleft.../angb∇acket∇ightdenotes the expectation value. Accordingly, the spin density s(r,t)is expected to satisfy a diffusion equation, as well. The spin density is defined by s(r,t) =1 2/angb∇acketleftψ†(r,t)σψ(r,t)/angb∇acket∇ight, (6) whereσis the vector of Pauli matrices, σx=/parenleftbigg 0 1 1 0/parenrightbigg ,σy=/parenleftbigg 0−i i0/parenrightbigg ,andσz=/parenleftbigg 1 0 0−1/parenrightbigg .5 Thus the z-component of the spin density is half the differenc e between the density of spin up and down electrons, sz= (ρ+−ρ−)/2, which is the local spin polarization of the electron system . Thus, we can directly infer the diffusion equation for sz, and, similarly, for the other components of the spin densit y, yielding, without magnetic field and spin-orbit interaction,2 ∂s ∂t=De∇2s−s ˆτs. (7) Here, in the spin relaxation term we introduced the tensor ˆτs, which can have non-diagonal matrix elements. In the case of a diagonal matrix, τsxx=τsyy=τ2, is the spin dephasing time, and τszz=τ1the spin relaxation time. The spin diffusion equation can be written as a continuity equ ation for the spin density vector, by defining the spin diffusion current of the spin components si, Jsi=−De∇si. (8) Thus, we get the continuity equation for the spin density com ponentssi, ∂si ∂t+∇Jsi=−/summationdisplay jsj τsij. (9) 3. Spin-Orbit Interaction in Semiconductors While silicon and germanium have in their diamond structure an inversion symmetry around every midpoint on each line connecting nearest neighbor atoms, this is not the case for III-V-semiconductors like GaAs, InAs, InSb, or ZnS. These have a zinc-blende structure which can be obtained fro m a diamond structure with neighbored sites occupied by the two different elements. Therefore the inversion symmetr y is broken, which results in spin-orbit coupling. Similarl y, that symmetry is broken in II-VI-semiconductors. This bulk inversion asymmetry (BIA) coupling, or often so called Dresselhaus-coupling, is anisotropic, as given by3 HD=γD/bracketleftbig σxkx(k2 y−k2 z)+σyky(k2 z−k2 x)+σzkz(k2 x−k2 y)/bracketrightbig , (10) whereγDis the Dresselhaus-spin-orbit coefficient. Confinement in qu antum wells with width aon the order of the Fermi wave length λFyields accordingly a spin-orbit interaction where the mome ntum in growth direction is of the order of 1/a. Because of the anisotropy of the Dresselhaus term, the spin -orbit interaction depends strongly on the growth direction of the quantum well. Grown in [001]direction, one gets, taking the expectation value of Eq. ( 10) in the direction normal to the plane, noting that /angb∇acketleftkz/angb∇acket∇ight=/angb∇acketleftk3 z/angb∇acket∇ight= 0,3 HD[001]=α1(−σxkx+σyky)+γD(σxkxk2 y−σykyk2 x). (11) whereα1=γD/angb∇acketleftk2 z/angb∇acket∇ightis the linear Dresselhaus parameter. Thus, inserting an ele ctron with momentum along the x- direction, with its spin initially polarized in z-directio n, it will precess around the x-axis as it moves along. For nar row quantum wells, where /angb∇acketleftk2 z/angb∇acket∇ight ∼1/a2≥k2 Fthe linear term exceeds the cubic Dresselhaus terms. A speci al situation arises for quantum wells grown in the [110]- direction, where it turns out that the spin-orbit field is po inting normal to the quantum well, as shown in Fig. 1, so that an electron whose spin is initially polarized along the normal of the plane, remains polarized as it moves in the quantum well. In q uantum wells with asymmetric electrical confinement the inversion symmetry is broken as well. This structural in version asymmetry (SIA) can be deliberately modified by changing the confinement potential by application of a gat e voltage. The resulting spin-orbit coupling, the SIA coupling, also called Rashba-spin-orbit interaction4is given by HR=α2(σxky−σykx), (12) whereα2depends on the asymmetry of the confinement potential V(z)in the direction z, the growth direction of the quantum well, and can thus be deliberately changed by applic ation of a gate potential. At first sight it looks as if the expectation value of the electrical field Ec=−∂zV(z)in the conduction band state vanishes, since the ground stat e of the quantum well must be symmetric in z. Taking into account the coupling to the valence band,5,6the discontinuities in the effective mass,7and corrections due to the coupling to odd excited states,8yields a sizable coupling parameter depending on the asymmetry of the confinement potential6,9. This dependence allows one, in principle, to control the ele ctron spin with a gate potential, which can therefore be used as the basis of a spin transistor.106 SIA 00 BIA/LBracket1111/RBracket1 00 BIA/LBracket1001/RBracket1 00 /LBracket1100/RBracket1/LBracket1010/RBracket1 BIA/LBracket1110/RBracket1 0 /LBracket11/OverBar/OverBar10/RBracket10/LBracket1001/RBracket10/LBracket1110/RBracket1 Figure 1: The spin-orbit vector fields for linear structure i nversion asymmetry (Rashba) coupling, and for linear bulk i nversion asymmetry (BIA) spin orbit coupling for quantum wells grown in [111], [001] and [110] direction, respectively. We can combine all spin-orbit couplings by introducing the s pin-orbit field such that the Hamiltonian has the form of a Zeeman term: HSO=−sBSO(k), (13) where the spin vector is s=σ/2. But we stress again that since BSO(k)→BSO(−k) =−BSO(k)under the time reversal operation, spin-orbit coupling does not break tim e reversal symmetry, since the time reversal operation also changes the sign of the spin, s→ −s. Only an external magnetic field Bbreaks the time reversal symmetry. Thus, the electron spin operator ˆ sis for fixed electron momentum kgoverned by the Bloch equations with the spin-orbit field, ∂ˆ s ∂t=ˆ s×(B+BSO(k))−1 ˆτsˆ s. (14) The spin relaxation tensor is no longer necessarily diagona l in the presence of spin-orbit interaction. In narrow quantum wells where the cubic Dresselhaus couplin g is weak compared to the linear Dresselhaus and Rashba couplings, the spin-orbit field is given by BSO(k) =−2 −α1kx+α2ky α1ky−α2kx 0 , (15) which changes both its direction and its amplitude |BSO(k)|= 2/radicalbig (α2 1+α2 2)k2−4α1α2kxky, as the direction of the momentum kis changed. Accordingly, the electron energy dispersion cl ose to the Fermi energy is in general7 anisotropic as given by E±=1 2m∗k2±αk/radicalbigg 1−4α1α2 α2cosθsinθ, (16) wherek=|k|,α=/radicalbig α2 1+α2 2, andkx=kcosθ. Thus, when an electron is injected with energy E, with momentum kalong the [100]-direction, kx=k,ky= 0, its wave function is a superposition of plain waves with the positive momentak±=∓αm∗+m∗(α2+2E/m∗)1/2. The momentum difference k−−k+= 2m∗αcauses a rotation of the electron eigenstate in the spin subspace. When at x= 0the electron spin was polarized up spin, with the Eigenvecto r ψ(x= 0) =/parenleftbigg 1 0/parenrightbigg , then, when its momentum points in x-direction, at a distance x, it will have rotated the spin as described by the Eigenvector ψ(x) =1 2/parenleftbigg1 α1+iα2 α/parenrightbigg eik+x+1 2/parenleftbigg1 −α1+iα2 α/parenrightbigg eik−x. (17) In Fig. 2we plot the corresponding spin density as defined in Eq. ( 6) for pure Rashba coupling, α1= 0. The spin 0 LSO/Slash12 LSO/Minus101 xsz Figure 2: Precession of a spin injected at x= 0, polarized in z-direction, as it moves by one spin precessio n length LSO=π/m∗α through the wire with linear Rashba spin orbit coupling α2. will point again in the initial direction, when the phase diff erence between the two plain waves is 2π, which gives the condition for spin precession length as 2π= (k−−k+)LSO, yielding for linear Rashba and Dresselhaus coupling, and the electron moving in [100]- direction, LSO=π/m∗α. (18) We note that the period of spin precession changes with the di rection of the electron momentum since the spin-orbit field, Eq. ( 15), is anisotropic. 4. Spin Diffusion in the Presence of Spin-Orbit Interaction As the electrons are scattered by imperfections like impuri ties and dislocations, their momentum is changed ran- domly. Accordingly, the direction of the spin-orbit field BSO(k)changes randomly as the electron moves through the sample. This has two consequences: the electron spin direct ion becomes randomized, dephasing the spin precession and relaxing the spin polarization. In addition, the spin pr ecession term is modified, as the momentum kchanges randomly, and has no longer the form given in the ballistic Bl och-like equation, Eq. ( 14). One can derive the diffu- sion equation for the expectation value of the spin, the spin density Eq. ( 6) semiclassically,11,12or by diagrammatic8 expansion.13In order to get a better understanding on the meaning of this e quation, we will give a simplified classical derivation, in the following. The spin density at time t+∆tcan be related to the one at the earlier time t. Note that for ballistic times ∆t≤τ, the distance the electron has moved with a probability p∆x,∆x, is related to that time by the ballistic equation, ∆x=k(t)∆t/mwhen the electron moves with the momentum k(t). On this time scale the spin evolution is still governed by the ballistic Bloch equa tion Eq. ( 14). Thus, we can relate the spin density at the position xat the time t+∆t, to the one at the earlier time tat position x−∆x: s(x,t+∆t) =/summationdisplay ∆xp∆x/parenleftbigg/parenleftbigg 1−1 ˆτs∆t/parenrightbigg s(x−∆x,t)−∆t[B+BSO(k(t))]×s(x−∆x,t)/parenrightbigg . (19) Now, we can expand in ∆tto first order and in ∆xto second order. Next, we average over the disorder potentia l, assuming that the electrons are scattered isotropically, a nd substitute/summationtext ∆xp∆x...=/integraltext (dΩ/Ω)...whereΩis the total angle, and/integraltextdΩdenotes the integral over all angles with/integraltext(dΩ/Ω) = 1 . Also, we get (s(x,t+∆t)−s(x,t))/∆t→ ∂ts(x,t)for∆t→0, and/angb∇acketleft∆x2 i/angb∇acket∇ight= 2De∆t, whereDeis the diffusion constant. While the disorder average yields /angb∇acketleft∆x/angb∇acket∇ight= 0, and/angb∇acketleftBSO(k(t))/angb∇acket∇ight= 0, separately, for isotropic impurity scattering, averagin g their product yields a finite value, since ∆xdepends on the momentum at time t,k(t), yielding /angb∇acketleft∆xBSOi(k(t))/angb∇acket∇ight= 2∆t/angb∇acketleftvFBSOi(k(t))/angb∇acket∇ight, where /angb∇acketleft.../angb∇acket∇ightdenotes the average over the Fermi surface. This way, we can a lso evaluate the average of the spin-orbit term in Eq. (19), expanded to first order in ∆x, and get, substituting ∆t→τthe spin diffusion equation, ∂s ∂t=−B×s+De∇2s+2τ/angb∇acketleft(∇vF)BSO(p)/angb∇acket∇ight×s−1 ˆτss, (20) where/angb∇acketleft.../angb∇acket∇ightdenotes the average over the Fermi surface. Spin polarized e lectrons injected into the sample spread diffusively, and their spin polarization, while spreading d iffusively as well, decays in amplitude exponentially in tim e. Since, between scattering events the spins precess around t he spin-orbit fields, one expects also an oscillation of the polarization amplitude in space. One can find the spatial dis tribution of the spin density which is the solution of Eq. (20) with the smallest decay rate Γs. As an example, the solution for linear Rashba coupling is,12 s(x,t) = (ˆeqcosqx+Aˆezsinqx)e−t/τs, (21) with1/τs= 7/16τs0where1/τs0= 2τk2 Fα2 2and where the amplitude of the momentum qis determined by Deq2= 15/16τs0, andA= 3/√ 15, andˆeq=q/q. This solution is plotted in Fig. 3forˆeq= (1,1,0)/√ 2. In Fig. 4we plot the linearly independent solution obtained by interchanging cosandsinin Eq. ( 21), with the spin pointing in z-direction, initially. We choose ˆeq= ˆex. Comparison with the ballistic precession of the spin, Fig 4shows that the period of precession is enhanced by the factor 4/√ 15in the diffusive wire, and that the amplitude of the spin densi ty is modulated, changing from 1toA= 3/√ 15. 0 /LParen115/Slash14/RParen1LSO/Slash12 /LParen115/Slash14/RParen1LSOx/Minus0.500.5Sy /Minus0.500.5 Sz Figure 3: The spin density for linear Rashba coupling which i s a solution of the spin diffusion equation with the relaxatio n rate7/16τs. The spin points initially in the x−y-plane in the direction (1,1,0).9 0 Lso/Slash12 Lso/Minus/FractionBarExt1 20/FractionBarExt1 2 x/LessS/Greaterz 0.770.891/VertBar1S/VertBar1 Figure 4: The spin density for linear Rashba coupling which i s a solution of the spin diffusion equation with the relaxatio n rate 1/τs= 7/16τs0. Note that, compared to the ballistic spin density, Fig. 2, the period is slightly enhanced by a factor 4/√ 15. Also, the amplitude of the spin density changes with the posi tionx, in contrast to the ballistic case. The color is changing in proportion to the spin density amplitude. Injecting a spin-polarized electron at one point, say x= 0, its density spreads the same way it does without spin- orbit interaction, ρ(r,t) = exp(−r2/4Det)/(4πDet)dD/2, whereris the distance to the injection point. However, the decay of the spin density is periodically modulated as a func tion of2π/radicalbig 15/16r/LSO.14The spin-orbit interaction together with the scattering from impurities is also a sourc e of spin relaxation, as we discuss in the next Section together with other mechanisms of spin relaxation. We can fin d the classical spin diffusion current in the presence of spin-orbit interaction, in a similar way as one can derive th e classical diffusion current: The current at the position ris a sum over all currents in its vicinity which are directed t owards that position. Thus, j(r,t) =/angb∇acketleftvρ(r−∆x)/angb∇acket∇ight where an angular average over all possible directions of the velocity vis taken. Expanding in ∆x=lev/v, and noting that /angb∇acketleftvρ(r)/angb∇acket∇ight= 0, one gets j(r,t) =/angb∇acketleftv(−∆x)∇ρ(r)/angb∇acket∇ight=−(vFle/2)∇ρ(r) =−De∇ρ(r). For the classical spin diffusion current of spin component Si, as defined by jSi(r,t) =vSi(r,t), there is the complication that the spin keeps precessing as it moves from r−∆xtor, and that the spin-orbit field changes its direction with the direction of the electron velocity v. Therefore, the 0-th order term in the expansion in ∆xdoes not vanish, rather, we get jSi(r,t) =/angb∇acketleftvSk i(r,t)/angb∇acket∇ight −De∇Si(r,t), whereSk iis the part of the spin density which evolved from the spin den sity atr−∆xmoving with velocity vand momentum k. Noting that the spin precession on ballistic scales t≤τis governed by the Bloch equation, Eq. ( 14), we find by integration of Eq. ( 14), thatSk i=−τ(BSO(k)×S)iso that we can rewrite the first term yielding the total spin diffusion cu rrent as jSi=−τ/angb∇acketleftvF(BSO(k)×S)i/angb∇acket∇ight−De∇Si. (22) Thus, we can rewrite the spin diffusion equation in terms of th is spin diffusion current and get the continuity equation ∂si ∂t=−De∇jSi+τ/angb∇acketleft∇vF(BSO(k)×S)i/angb∇acket∇ight−1 ˆτsijsj. (23) It is important to note that in contrast to the continuity equ ation for the density, there are two additional terms, due to the spin orbit interaction. The last one is the spin relaxa tion tensor which will be considered in detail in the next section. The other term arises due to the fact that Eq. ( 20) contains a factor 2in front of the spin-orbit precession term, while the spin diffusion current Eq. ( 22) does not contain that factor. This has important physical c onsequences, resulting in the suppression of the spin relaxation rate in q uantum wires and quantum dots as soon as their lateral extension is smaller than the spin precession length LSO, as we will see in the subsequent Sections. III. SPIN RELAXATION MECHANISMS The intrinsic spin-orbit interaction itself causes the spi n of the electrons to precess coherently, as the electrons move through a conductor, defining the spin precession lengt hLSO, Eq. ( 18). Since impurities and dislocations in the conductor randomize the electron momentum, the impurit y scattering is transferred into a randomization of the10 electron spin by the spin-orbit interaction, which thereby results in spin dephasing and spin relaxation. This results in a new length scale, the spin relaxation length, Ls, which is related to the spin relaxation rate 1/τsby Ls=/radicalbig Deτs. (24) A. D’yakonov-Perel’ Spin Relaxation D’yakonov-Perel’ spin relaxation (DPS) can be understood q ualitatively in the following way: The spin-orbit field BSO(k)changes its direction randomly after each elastic scatteri ng event from an impurity, that is, after a time of the order of the elastic scattering time τ, when the momentum is changed randomly as sketched in Fig. 5. Thus, the Figure 5: Elastic scattering from impurities changes the di rection of the spin-orbit field around which the electron spi n is precessing. spin has the time τto perform a precession around the present direction of the s pin-orbit field, and can thus change its direction only by an angle of the order of BSOτby precession. After a time twithNt=t/τscattering events, the direction of the spin will therefore have changed by an an gle of the order of |BSO|τ√Nt=|BSO|√ τt. Defining the spin relaxation time τsas the time by which the spin direction has changed by an angle of order one, we thus find that 1/τs∼τ/angb∇acketleftBSO(k)2/angb∇acket∇ight, where the angular brackets denote integration over all ang les. Remarkably, this spin relaxation rate becomes smaller, the more scattering event s take place, because the smaller the elastic scattering tim e τis, the less time the spin has to change its direction by prece ssion. Such a behavior is also well known as motional , ordynamic narrowing of magnetic resonance lines.15A more rigorous derivation for the kinetic equation of the sp in density matrix yields additional interference terms, not t aken into account in the above argument. It can be obtained by iterating the expansion of the spin density Eq. ( 19) once in the spin precession term, which yields the term /angbracketleftBigg s(x,t)×/integraldisplay∆t 0dt′BSO(k(t′))×/integraldisplay∆t 0dt′′BSO(k(t′′))/angbracketrightBigg , (25) where/angb∇acketleft.../angb∇acket∇ightdenotes the average over all angles due to the scattering fro m impurities. Since the electrons move ballistically at times smaller than the elastic scattering time, the momenta are correlated only on time scales smaller thanτ, yielding /angb∇acketleftki(t′)kj(t′′)/angb∇acket∇ight= (1/2)k2δijτδ(t′−t′′). Noting that (A×B×C)m=ǫijkǫklmAiBjCland/summationtextǫijkǫklm=δilδjm−δimδjlwe find that Eq. ( 25) simplifies to −/summationtext i(1/τsij)Sj, where the matrix elements of the spin relaxation terms are g iven by,16 1 τsij=τ/parenleftbig /angb∇acketleftBSO(k)2/angb∇acket∇ightδij−/angb∇acketleftBSO(k)iBSO(k)j/angb∇acket∇ight/parenrightbig , (26) where/angb∇acketleft.../angb∇acket∇ightdenotes the average over the direction of the momentum k. These non-diagonal terms can diminish the spin relaxation and even result in vanishing spin relaxatio n. As an example, we consider a quantum well where the linear Dresselhaus coupling for quantum wells grown in [001]direction, Eq. ( 11), and linear Rashba-coupling, Eq. ( 12), are the dominant spin-orbit couplings. The energy dispersi on is anisotropic, as given by Eq. ( 16), and the spin-orbit fieldBSO(k)changes its direction and its amplitude with the direction o f the momentum k: BSO(k) =−2 −α1kx+α2ky α1ky−α2kx 0 , (27)11 with|BSO(k)|= 2/radicalbig (α2 1+α2 2)k2−4α1α2kxky. Thus we find the spin relaxation tensor as, 1 ˆτs(k) = 4τk2 1 2α2−α1α20 −α1α21 2α20 0 0 α2 . (28) Diagonalizing this matrix, one finds the three eigenvalues (1/τs)(α1±α2)2/α2and2/τswhereα2=α2 1+α2 2, and 1/τs= 2k2τα2. Note, that one of these eigenvalues of the spin relaxation t ensor vanishes when α1=α2=α0. In fact, this is a special case, when the spin-orbit field does no t change its direction with the momentum: BSO(k)|α1=α2=α0=2α0(kx−ky) 1 1 0 . (29) In this case the constant spin density given by S=S0 1 1 0 , (30) does not decay in time, since the spin density vector is paral lel to the spin orbit field BSO(k), Eq. ( 29), and cannot precess, as has been noted in Ref. [ 17]. It turns out, however, that there are two more modes which d o not decay in time, whose spin relaxation rate vanishes for α1=α2. These modes are not homogeneous in space, and correspond to precessing spin densities. They were found previously in a n umerical Monte Carlo simulation and found not to decay in time, being called therefore persistent spin helix .18,19Recently, a long living inhomogeneous spin density distrib ution has been detected experimentally in Ref. [ 20]. We can now get these persistent spin helix modes analytically, by solving the full spin diffusion equation Eq. ( 20) with the spin relaxation tensor given by Eq. ( 28). We can diagonalize that equation, noting that its eigenfunctions are plain waves S(x)∼exp(iQx−Et). Thereby one finds, first of all, the mode with Eigenvalue E1=DeQ2, with the spin density S=S0 1 1 0 exp(iQx−DeQ2t). (31) Indeed for Q= 0, the homogeneous solution, it does not decay in time, in agre ement with the solution we found above, Eq. ( 31). There are, however, two more modes with the eigenvalues E±=1 τs(˜Q2+2±2|˜Qx−˜Qy|), (32) where˜Q=LSOQ/2π. At˜Qx=−˜Qy=±1, these modes do not decay in time. These two stationary solut ions, are S=S0 1 −1 0 sin/parenleftbigg2π LSO(x−y)/parenrightbigg +S0√ 2 0 0 1 cos/parenleftbigg2π LSO(x−y)/parenrightbigg , (33) and the linearly independent solution, obtained by interch angingcosandsinin Eq. ( 33). The spin precesses as the electrons diffuse along the quantum wire with the period LSO, the spin precession length, forming a persistent spin helix, as shown in Fig. 6. B. DP Spin Relaxation with Electron-Electron and Electron- Phonon Scattering It has been noted, that the momentum scattering which limits the D’yakonov-Perel’ mechanism of spin relaxation is not restricted to impurity scattering, but can also be due to electron-phonon or electron-electron interactions.21–24 Thus the scattering time, τis the total scattering time as defined by,21,221/τ= 1/τ0+1/τee+1/τep, where1/τ0is the elastic scattering rate due to scattering from impurities w ith potential V, given by 1/τ0= 2πνni/integraltext (dθ/2π)(1−cosθ)| V(k,k′)|2, whereνis the density of states per spin at the Fermi energy, niis the concentration of impurities with potentialV, andkk′=kk′cos(θ). In degenerate semiconductors and in metals, the electron- electron scattering rate is given by the Fermi liquid inelastic electron scattering r ate1/τee∼T2/ǫF. The electron-phonon scattering time 1/τep∼T5decays faster with temperature. Thus, at low temperatures t he DP spin relaxation is dominated by elastic impurity scattering τ0.12 0 LSO/Slash12 LSOx /MinusS00S0 /LessS/Greatery/Minus/Radical1/Radical1Extens/Radical1Extens/Radical1Extens/Radical1Extens/Radical1Extens2S00/Radical1/Radical1Extens/Radical1Extens/Radical1Extens/Radical1Extens/Radical1Extens2S0 /LessS/Greaterz Figure 6: Persistent spin helix solution of the spin diffusio n equation for equal magnitude of linear Rashba and linear Dr esselhaus coupling, Eq.( 33). C. Elliott-Yafet Spin Relaxation Because of the spin-orbit interaction the conduction elect ron wave functions are not Eigenstates of the electron spin, but have an admixture of both spin up and spin down wave f unctions. Thus, a nonmagnetic impurity potential Vcan change the electron spin, by changing their momentum due to the spin-orbit coupling. This results in another source of spin relaxation which is stronger, the more often t he electrons are scattered, and is thus proportional to the momentum scattering rate 1/τ.25,26For degenerate III-V semiconductors one finds27,28 1 τs∼∆2 SO (EG+∆SO)2E2 k E2 G1 τ(k), (34) whereEGis the gap between the valence and the conduction band of the s emiconductor, Ekthe energy of the conduction electron, and ∆SOis the spin-orbit splitting of the valence band. Thus, the El liott-Yafet spin relaxation (EYS) can be distinguished, being proportional to 1/τ, and thereby to the resistivity, in contrast to the DP spin scattering rate, Eq. ( 26), which is proportional to the conductivity. Since the EYS d ecays in proportion to the inverse of the band gap, it is negligible in large band gap semiconduc tors likeSiandGaAs . The scattering rate 1/τis again the sum of the impurity scattering rate,25the electron-phonon scattering rate,26,29and electron-electron interaction,30 so that all these scattering processes result in EYS. In non- degenerate semiconductors, where the Fermi energy is below the conduction band edge, 1/τs∼τT3/EGattains a stronger temperature dependence. D. Spin Relaxation due to Spin-Orbit Interaction with Impur ities The spin-orbit interaction, as defined in Eq. ( 3), arises whenever there is a gradient in an electrostatic po tential. Thus, the impurity potential gives rise to the spin-orbit in teraction VSO=1 2m2c2∇V×k s. (35) Perturbation theory yields then directly the correspondin g spin relaxation rate 1 τs=πνni/summationdisplay α,β/integraldisplaydθ 2π(1−cosθ)|VSO(k,k′)αβ|2, (36) proportional to the concentration of impurities ni. Hereα,β=±denotes the spin indices. Since the spin-orbit interaction increases with the atomic number Zof the impurity element, this spin relaxation increases as Z2, being stronger for heavier element impurities.13 E. Bir-Aronov-Pikus Spin Relaxation The exchange interaction Jbetween electrons and holes in p-doped semiconductors resu lts in spin relaxation, as well.31,32Its strength is proportional to the density of holes pand depends on their itinerancy. If the holes are localized they act like magnetic impurities. If they are iti nerant, the spin of the conduction electrons is transferred by the exchange interaction to the holes, where the spin-orbit splitting of the valence bands results in fast spin relaxati on of the hole spin due to the Elliott-Yafet, or the D’yakonov-P erel’ mechanism. F. Magnetic Impurities Magnetic impurities have a spin Swhich interacts with the spin of the conduction electrons by the exchange interaction J, resulting in a spatially and temporarily fluctuating local magnetic field BMI(r) =−/summationdisplay iJδ(r−Ri)S, (37) where the sum is over the position of the magnetic impurities Ri. This gives rise to spin relaxation of the conduction electrons, with a rate given by 1 τMs= 2πnMνJ2S(S+1), (38) wherenMis the density of magnetic impurities, and νis the density of states at the Fermi energy. Here, Sis the spin quantum number of the magnetic impurity, which can take the valuesS= 1/2,1,3/2,2.... Antiferromagnetic exchange interaction between the magnetic impurity spin an d the conduction electrons results in a competition between the conduction electrons to form a singlet with the i mpurity spin, which results in enhanced nonmagnetic and magnetic scattering. At low temperatures the magnetic i mpurity spin is screened by the conduction electrons resulting in a vanishing of the magnetic scattering rate. Th us, the spin scattering from magnetic impurities has a maximum at a temperature of the order of the Kondo temperatur eTK∼EFexp(−1/νJ), whereνis the density of states at the Fermi energy.33–35In semiconductors TKis exponentially small due to the small effective mass and the resulting small density of states ν. Therefore, the magnetic moments remain free at the experim entally achievable temperatures. At large concentration of magnetic impuriti es, the RKKY-exchange interaction between the magnetic impurities quenches however the spin quantum dynamics, so t hatS(S+1)is replaced by its classical value S2. In Mn-p-doped GaAs, the exchange interaction between the Mn do pants and the holes can result in compensation of the hole spins and therefore a suppression of the Bir-Aronov-Pi kus (BAP) spin relaxation.36 G. Nuclear Spins Nuclear spins interact by the hyperfine interaction with con duction electrons. The hyperfine interaction between nuclear spins ˆIand the conduction electron spin, ˆs, results in a local Zeeman field given by37 ˆBN(r) =−8π 3g0µB γg/summationdisplay nγnˆI,δ(r−Rn), (39) whereγnis the gyromagnetic ratio of the nuclear spin. The spatial an d temporal fluctuations of this hyperfine interaction field result in spin relaxation proportional to its variance, similar to the spin relaxation by magnetic impurities. H. Magnetic Field Dependence of Spin Relaxation The magnetic field changes the electron momentum due to the Lo rentz force, resulting in a continuous change of the spin-orbit field, which similar to the momentum scatteri ng results in motional narrowing and thereby a reduction of DPS:28,38–40 1 τs∼τ 1+ω2cτ2. (40)14 Another source of a magnetic field dependence is the precessi on around the external magnetic field. In bulk semi- conductors and for magnetic fields perpendicular to a quantu m well, the orbital mechanism is dominating, however. This magnetic field dependence can be used to identify the spi n relaxation mechanism, since the EYS does have only a weak magnetic field dependence due to the weak Pauli-parama gnetism. I. Dimensional Reduction of Spin Relaxation Electrostatic confinement of conduction electrons can redu ce the effective dimension of their motion. In quantum dots, the electrons are confined in all three directions, and the e nergy spectrum consists of discrete levels like in atoms. Therefore, the energy conservation restricts relax ation processes severely, resulting in strongly enhanced s pin relaxation times in quantum dots.41,42Then, spin relaxation can only occur due to absorption or emi ssion of phonons, yielding spin relaxation rates proportional to the inelast ic electron-phonon scattering rate.41Quantitative comparison of the various spin relaxation mechanisms in GaAs quantum do ts resulted in the conclusion that the spin relaxation is dominated by the hyperfine interaction.43–45A similar conclusion can be drawn from experiments on low tem perature spin relaxation in low density n-type GaAs, where the locali zation of the electrons in the impurity band results in spin relaxation dominated by hyperfine interaction as well.46,47For linear Rashba and linear Dresselhaus spin-orbit coupling we can see from the spin diffusion equation Eq. ( 20) with the DP spin relaxation tensor Eq. ( 28) that the spin relaxation vanishes, when the spin current Eq. ( 22) vanishes, in which case the last two terms of Eq. ( 20) cancel exactly. The vanishing of the spin current is imposed by hard wall boundary condition for which the spin diffusion current vanishes at the boundaries of the sample, jSin|Boundary= 0, wherenis the normal to the boundary. When the quantum dot is smaller than the spin precession leng thLSOthe lowest energy mode thus corresponds to a homogeneous solution with vanishing spin relaxation rate. Cubic spin-orbit coupling does not yield such a vanishing of the DP spin relaxation rate. Only in quantum dots whose siz e does not exceed the elastic mean free path lethe DP spin relaxation from cubic spin relaxation becomes dimin ished. In quantum wires , the electrons have a continuous spectrum of delocalized states. Still, transverse confinem ent can reduce the DP spin relaxation as we review in the next section. IV. SPIN-DYNAMICS IN QUANTUM WIRES A. One-Dimensional Wires In one dimensional wires, whose width Wis of the order of the Fermi wave length λF, impurities can only reverse the momentum p→ −p. Therefore, the spin-orbit field can only change its sign, wh en a scattering from impurities occurs. BSO(p)→BSO(−p) =−BSO(p). Therefore, the precession axis and the amplitude of the spi n orbit field does not change, reversing only the spin precession, so that the D’ya konov-Perel’-spin relaxation is absent in one dimensional wires.48In an external magnetic field, the precession around the magn etic field axis, due to the Zeeman-interaction is competing with the spin-orbit field, however. Then, as the el ectrons are scattered from impurities, both the precession axis and the amplitude of the total precession field is changi ng, since |B+BSO(−p)|=|B−BSO(p)|/negationslash=|B+BSO(p)|, resulting in spin dephasing and relaxation, as the sign of th e momentum changes randomly. B. Spin-Diffusion in Quantum Wires How does the spin relaxation rate depend on the wire width Wwhen the quantum wire has more than one channel occupied,W >λ F? Clearly, for large wire widths, the spin relaxation rate sh ould converge to a finite value, while it vanishes for W→λF. It is both of practical importance for spintronic applicat ions and of fundamental interest to know on which length scales this crossover occurs. Basicall y, there are three intrinsic length scales characterizing t he quantum wire relative to its width W. The Fermi wave length λF, the elastic mean free path leand the spin precession lengthLSO, Eq. ( 18). Suppression of spin relaxation for wire widths not exceed ing the elastic mean free path le, has been predicted and obtained numerically in Refs. [ 11,49–53]. Is the spin relaxation rate also suppressed in diffusive wires in which the elastic mean free path is smaller than the wire wi dth as in the wire shown schematically in Fig. 7? We will answer this question by means of an analytical deriva tion in the following. The transversal confinement imposes that the spin current vanishes normal to the boundar y,jSin|Boundary= 0. For a wire grown along the [010]15 Figure 7: Elastic scatterings from impurities and from the b oundary of the wire change the direction of the spin-orbit fie ld around which the electron spin is precessing. direction, n= ˆexis the unit vector in the x-direction. For wire widths Wsmaller than the spin precession length LSO, the solutions with the lowest energy have thus a vanishing tr ansverse spin current, and the spin diffusion equation Eq. (20) becomes ∂si ∂t=−De∂yjSiy+τ/angb∇acketleft∇vF(BSO(k)×S)i/angb∇acket∇ight−/summationdisplay j1 ˆτsijsj. (41) with jSix|x=±W/2= (−τ/angb∇acketleftvx(BSO(k)×S)i/angb∇acket∇ight−De∂xSi)|x=±W/2= 0, (42) whereWis the width of the wire. One sees that this equation has a pers istent solution, which does not decay in time and is homogeneous along the wire, ∂yS= 0. In this special case the spin diffusion equation simplifies t o12 ∂tS=−1 τsα2 α2 1−α1α20 −α1α2α2 20 0 0 α2 S. (43) Indeed this has one persistent solution given by S=S0 α2 α1 0 , (44) Thus, we can conclude that the boundary conditions impose an effective alignment of all spin-orbit fields, in a direction identical to the one it would attain in a one-dimensional wir e, along the [010]-direction, setting kx= 0in Eq. ( 27), BSO(k) =−2ky α2 α1 0 , (45) which therefore does not change its direction when the elect rons are scattered. This is remarkable, since this alignmen t already occurs in wires with many channels, where the impuri ty scattering is two-dimensional, and the transverse momentum kxactually can be finite. Rather, the alignment of the spin-orb it field, accompanied by a suppression of the DP spin relaxation rate occurs due to the constraint on th e spin-dynamics imposed by the boundary conditions as soon as the wire width Wis smaller than the length scale which governs the spin dynam ics, namely, the spin precession length LSO. It turns out that the spin diffusion equation Eq. ( 41) has also two long persisting spin helix solutions in narrow wires13,54which oscillate periodically with the period LSO=π/m∗α. In contrast to the situation in 2D systems we reviewed in the previous Section, in quantum wires of width W < L SOthese solutions are long persisting even for α1/negationslash=α2. These two stationary solutions, are S=S0 α1 α −α2 α 0 sin/parenleftbigg2π LSOy/parenrightbigg +S0 0 0 1 cos/parenleftbigg2π LSOy/parenrightbigg , (46)16 0 /FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExt/FractionBarExtLSO 2 LSOy /Minus101 Sy/Slash1S0/Minus101 Sz/Slash1S0 0/FractionBarExt/FractionBarExt/FractionBarExtΠ 4/FractionBarExt/FractionBarExt/FractionBarExtΠ 2/CurlyPhi Figure 8: Persistent spin helix solution of the spin diffusio n equation in a quantum wire whose width Wis smaller than the spin precession length LSOfor varying ratio of linear Rashba α2=αsinϕand linear Dresselhaus coupling, α1=αcosϕ, Eq.( 46), for fixed αandLSO=π/m∗α. and the linearly independent solution, obtained by interch angingcosandsinin Eq. ( 46). The spin precesses as the electrons diffuse along the quantum wire with the period LSO, the spin precession length, forming a persistent spin helix, whose x-component is proportional to the linear Dresselha us-coupling α1while its y-component is proportional to the Rashba-coupling α2as seen in Fig. 8. A similar reduction of the spin relaxation rate is not effect ive for cubic spin-orbit coupling for wire widths exceeding the elastic m ean free path le. One can derive the spin relaxation rate as function of the wire width for diffusive wires le<W <L SO. The total spin relaxation rate, in the presence of both linear Rashba spin-orbit coupling α2and linear and cubic Dresselhaus coupling α1, andγD, is as function of wire widthWgiven by,54 1 τs(W) =1 12/parenleftbiggW LSO/parenrightbigg2 δ2 SO1 τs+De(m∗2ǫFγD)2, (47) where1/τs= 2p2 F(α2 2+(α1−m∗γDǫF/2)2)τ. We introduced the dimensionless factor, δSO= (Q2 R−Q2 D)/Q2 SOwith Q2 SO=Q2 D+Q2 RwhereQDdepends on Dresselhaus spin-orbit coupling, QD=m∗(2α1−m∗ǫFγ).QRdepends on Rashba coupling: QR= 2m∗α2. Thus, for negligible cubic Dresselhaus spin-orbit coupli ng the the spin relaxation length increases when decreasing the wire width Was, Ls(W) =/radicalbig Deτs(W)∼L2 SO W. (48) This can be understood as follows:54–56In a wire whose width exceeds the spin precession length LSO, the area an electron covers by diffusion in time τsisWLs. To achieve spin relaxation, this area should be equal to the corresponding 2D spin relaxation area Ls(2D)2, whereLs(2D) =LSO/(2π). Thus, the smaller the wire width, the larger the spin relaxation length becomes, Ls∼(LSO)2/Win agreement with Eq. ( 48). For larger wire widths, the spin diffusion equation can be solved as well, and one finds that the spin relaxation rate does not increase monotonously to the 2D limiting value but shows oscillations on the scale LSO, which can be understood in analogy to Fabry-Pérot resonances.54For pure linear Rashba coupling that behavior can be derived analytically, in the approximation of a homogeneous spin density in transverse direction, yieldin g a relaxation rate given by 1 τs(W) =De 2Q2 SO/parenleftbigg 1−sin(QSOW) QSOW/parenrightbigg , (49) whereQSO= 2π/LSO. Furthermore, taking into account the transverse modulati on of the spin density by performing an exact diagonalization of the spin diffusion equation with the transverse boundary conditions, Eq. ( 42), one finds17 forW >L SOmodes which are localized at the boundaries and have a lower r elaxation rate than the bulk modes.12,13 For pure Rashba spin relaxation we find that there is a spin-he lix solution located at the edge whose relaxation rate 1/τs=.31/τs0is smaller than the spin relaxation rate of bulk modes 1/τs= 7/16τs0. C. Weak Localization Corrections Quantum interference of electrons in low-dimensional, dis ordered conductors results in corrections to the electrica l conductivity ∆σ. This quantum correction, the weak localization effect, is k nown to be a very sensitive tool to study dephasing and symmetry breaking mechanisms in conductors.57–59The entanglement of spin and charge by spin-orbit interaction reverses the effect of weak localization and the reby enhances the conductivity, the weak antilocalization effect. The quantum correction to the conductivity ∆σarises from the fact, that the quantum return probability to a given point x0after a time t,P(t), differs from the classical return probability, due to quant um interference. As the electrons scatter from impurities, there is a finite pr obability that they diffuse on closed paths, which does increase the lower the dimension of the conductor. Since an e lectron can move on such a closed orbit clockwise or anticlockwise as shown in light and dark blue in Fig. 9, with equal probability, the probability amplitudes of bot h paths add coherently, if their length is smaller than the dep hasing length Lϕ. In a magnetic field, indicated by the Figure 9: Electrons can diffuse on closed paths, orbit clockw ise or anticlockwise as indicated by the light and dark blue a rrows, respectively. Middle figure: Closed electron paths enclose a magnetic flux from an external magnetic field, indicated as t he red arrow, breaking time reversal symmetry. Right figure: The sc attering from a magnetic impurity spin, breaks the time reve rsal symmetry between the clock- and anticlockwise electron pat hs. red arrow in the middle Fig. 9, the electrons acquire a magnetic flux phase. This phase depe nds on the direction in which the electron moves on the closed path. Thus, the quantu m interference is diminished in an external magnetic field since the area of closed paths and thereby the flux phases are randomly distributed in a disordered wire, even though the magnetic field can be constant. Similarly, the sca ttering from magnetic impurities breaks the time reversal invariance between the two directions in which the closed pa th can be transversed. Therefore magnetic impurities diminish the quantum corrections in proportion to the rate w ith which the electron spins scatter from them due to the exchange interaction, 1/τMs, Eq. ( 38). Thus, the quantum correction to the conductivity, ∆σis proportional to the integral over all times smaller than t he dephasing time τϕof the quantum mechanical return probability P(t) =λd Fρ(x,t), wheredis dimension of diffusion, andρis the electron density. In the presence of spin-orbit scatt ering, the sign of the quantum correction changes to weak antilocalization as was as predicted by Hikami, Larkin , and Nagaoka60for conductors with impurities of heavy elements. As conduction electrons scatter from such impuri ties, the spin-orbit interaction randomizes their spin, Fig.10. The resulting spin relaxation suppresses interference of time reversed paths in spin triplet configurations, while interference in singlet configuration remains unaffec ted as indicated in Fig. 10. Since singlet interference reduces the electron’s return probability it enhances the conducti vity, the weak antilocalization effect. Weak magnetic fields suppress also these singlet contributions, reducing the co nductivity and resulting in negative magnetoconductivity . If the host lattice of the electrons provides spin-orbit int eraction, the spin relaxation of DP or EY type does have the same effect of diminishing the quantum corrections in the triplet configuration. When the dephasing length Lϕis smaller than the wire width W, the quantum corrections are determined by the interferenc e of 2-dimensional closed diffusion paths, and as a result, the conductivity inc reases logarithmically with Lϕwhich increases itself as the temperature is lowered. At low temperatures, the electr on-electron scattering is the dominating mechanism of spin dephasing, yielding Lϕ∼T−1/2. One can derive the magnetic field dependence of that quantum correction18 Figure 10: As electrons diffuse, their spin precesses around the spin-orbit field, which changes its orientation, when th e electron is scattered. Electrons which enter closed paths with the sa me spin leave it therefore with a different spin if they choose the path in the opposite sense, as indicated by the light and dark blue arrows. However, electrons which enter the closed path with opposite spin, and move through the closed path in opposite s ense, attain the same quantum phase. This is a consequence of time reversal invariance. nonperturbatively.42,60–64An approximate expression showing the logarithmic depende nce explicitly is given by ∆σ=−1 2πlnB+4 3HMs+Hϕ Hτ+1 2πlnB+Hϕ+Hs+2 3HMs Hτ+1 πlnB+Hϕ+cHs+2 3HMs Hτ, (50) in units of e2/h. All parameters are rescaled to dimensions of magnetic field s:Hϕ= 1/(4eDeτϕ) = 1/(4eL2 ϕ), Hτ=/planckover2pi1/(4eDeτ), the spin relaxation field due to spin orbit relaxation, Hs=/planckover2pi1/(4eDeτs),61and the spin relaxation field due to magnetic impurities HMs=/planckover2pi1/(4eDeτMs). Here1/τsis the DP relaxation rate in the 2D limit derived in the previous section.61,65The prefactor cdepends on the particular spin-orbit interaction. For line ar Rashba-coupling, c= 7/16. Note that 7/16τsis the smallest spin relaxation rate of an inhomogeneous spi n density distribution13as derived in the section II B 4.1/τMsis the magnetic scattering rate from magnetic impurities, E q. (38). Indeed we see that the first term does not depend on the DP spin relaxatio n rate. This term originates from the interference of time reversed paths, indicated in Fig. 10, which contributes to the quantum conductance in the single t state, |S= 0;m= 0/angb∇acket∇ight= (| ↑↓/angb∇acket∇ight− | ↓↑/angb∇acket∇ight )/√ 2, the minus sign in front of the second term is the origin of the change in sign in the weak localization correction. The other three terms a re suppressed by the spin relaxation rate, since they originate from interference in triplet states, |S= 1;m= 0/angb∇acket∇ight= (| ↑↓/angb∇acket∇ight+| ↓↑/angb∇acket∇ight)/√ 2,|S= 1;m= 1/angb∇acket∇ight,|S= 1;m=−1/angb∇acket∇ight which do not conserve the spin symmetry. Thus, at strong spin -orbit induced spin relaxation the last three terms are suppressed and the sign of the quantum correction switches t o weak antilocalization. In quasi-1-dimensional quantum wires which are coherent in transverse direction, W < L ϕthe weak localization correction is further enhanced, and increases linearly with the dephasing length Lϕ. Thus, for WQSO≪1the weak localization correction is54 ∆σ=√HW/radicalBig Hϕ+1 4B∗(W)+2 3HMs−√HW/radicalBig Hϕ+1 4B∗(W)+Hs(W)+2 3HMs −2√HW/radicalBig Hϕ+1 4B∗(W)+1 2Hs(W)+4 3HMs, (51) in units ofe2/h. We defined HW=/planckover2pi1/(4eW2), and the effective external magnetic field, B∗(W) =/parenleftbigg 1−1//parenleftbigg 1+W2 3l2 B/parenrightbigg/parenrightbigg B. (52)19 The spin relaxation field Hs(W)is forW <L SO, Hs(W) =1 12/parenleftbiggW LSO/parenrightbigg2 δ2 SOHs, (53) suppressed in proportion to (W/LSO)2. Taking one transverse mode into account the quantum conduc tivity correction is plotted in Fig. 11for different wire withs for pure Rashba SOC, showing the cros sover from weak localization (positive magnetoconductivity) to weak antilocalization (negative magnetoconductivity). In analogy to the effective magnetic Figure 11: The quantum conductivity correction in units of 2e2/has function of magnetic field B(scaled with bulk relaxation fieldHs), and the wire width W(scaled with LSO/2π), for pure Rashba coupling, δSO= 1. field, Eq. ( 52), the spin orbit coupling acts in quantum wires like an effect ive magnetic vector potential.55One can expect that in ballistic wires, le>W, the spin relaxation rate is suppressed in analogy to the flux cancellation effect, which yields the weaker rate, 1/τs= (W/Cle)(DeW2/12L4 SO), whereC= 10.8.66–68A dimensional crossover from weak antilocalization to weak localization and a reduction of spin relaxation has recently been observed experimental ly in quantum wires as we will review in the next Section. V. EXPERIMENTAL RESULTS ON SPIN RELAXATION RATE IN SEMICOND UCTOR QUANTUM WIRES A. Optical Measurements Optical time-resolved Faraday rotation (TRFR) spectrosco py69has been used to probe the spin dynamics in an array of n-doped InGaAs wires by Holleitner et al. in Ref. [ 70,71]. The wires were dry etched from a quantum well grown in the [001]-direction with a distance of 1 µm between the wires. Spin aligned charge carriers were creat ed by absorption of circularly-polarized light. For normal in cidence, the spins point then perpendicular to the quantum well plane, in the growth direction [001]. The time evolutio n of the spin polarization was then measured with a linearly polarized pulse, see inset of Fig. 1c of Ref. [ 70]. The time dependence fits well with an exponential decay ∼exp(−∆t/τs). As seen in Fig. 2a of Ref. [ 70], the thus measured lifetime τsat fixed temperature T= 5K of the spin polarization is enhanced when the wire width Wis reduced70: While for W >15µm it isτs= (12±1)ps, it increases for channels grown along the [100]- direction t o almostτs= 30 ps, and in the [110]- direction to about τs= 20ps. Thus, the experimental results show that the spin relaxa tion depends on the patterning direction of the wires: wires aligned along [100] and [010] show equivalent s pin relaxation times, which are generally longer than the spin relaxation times of wires patterned along [110] and [ 110]. The dimensional reduction could be seen already for wire widths as wide as 10µm, which is much wider than both the Fermi wave length and the e lastic mean free path lein the wires. This agrees well with the predicted reduction o f the DP scattering rate, Eq. ( 47) for wire widths smaller than the spin precession length LSO. From the measured 2D spin diffusion length Ls(2D) = (0.9−1.1)µm,20 and its relation to the spin precession length Eq. ( 18),LSO= 2πLs(2D), we expect the crossover to occur on a scale ofLSO= (5.7−6.9)µm as observed in Fig. 2a of Ref. [ 70]. FromLSO=π/m∗αwe get with m∗= 0.064mea spin-orbit coupling α= (5−6)meVÅ. According to Ls=√Deτs, the spin relaxation length increases by a factor of/radicalbig 30/12 = 1.6in the [100]-, and by/radicalbig 20/12= 1.3in the [110]- direction. The spin relaxation time has been found to attain a maximum, h owever, at about W= 1µm≈Ls(2D), decaying appreciably for smaller widths. While a saturation of τscould be expected according to Eq. ( 47) for diffusive wires, due to cubic Dresselhaus-coupling, a decrease is unexpecte d. Schwab et al., Ref. [ 12], noted that with wire boundary conditions which do not conserve the spin of the conduction e lectrons one can obtain such a reduction. A mechanism for such spin-flip processes at the edges of the wire has not ye t been identified, however. The magnetic field dependence of the spin relaxation rate yields further confirmation that the dominant spin relaxation mechanism in these wires is DPS: It follows the predicted behavior Eq. ( 40), as seen in Fig. 3a of Ref. [ 70], and the spin relaxation rate is enhanced toτs(B= 1T) = 100 ps for all wire growth directions, at T= 5K and wire widths of W= 1.25µm. B. Transport Measurements A dimensional crossover from weak antilocalization to weak localization and a reduction of spin relaxation has recently been observed experimentally in n-doped InGaAs qu antum wires,72,73in GaAs wires,74as well as in Al- GaN/GaN wires.75The crossover indeed occurred in all experiments on the leng th scale of the spin precession length LSO. We summarize in the following the main results of these expe riments. Wirthmann et al., Ref. [ 72], measured the magnetoconductivity of inversion-doped In As quantum wells with a density ofn= 9.7×1011/cm2, and a measured effective mass of m∗= 0.04me. In the wide wires the magnetoconductivity showed a pronounced weak antilocalization peak, which agre ed well with the 2D theory,61,65with a spin-orbit-coupling parameter of α= 9.3meVÅ. They observed a diminishment of the antilocalization peak which occurred for wire widths W <0.6µm, atT= 2K, indicating a dimensional reduction of the DP spin relaxat ion rate. Schäpers et al. observed in Ga xIn1−xAs/InP quantum wires a complete crossover from weak antiloc alization to weak localization for wire widths below W= 500 nm. Such a crossover has also been observed in GaAs-quantum w ires by Dinter et al., Ref. [ 74]. Very recently, Kunihashi et al., Ref. [ 76] observed the crossover from weak antilocalization to weak localization in gate controlled InGaAs quantum wires. The asymmetric poten tial normal to the quantum well could be enhanced by application of a negative gate voltage, yielding an increas e of the SIA-coupling parameter α, with decreasing carrier density, as was obtained by fitting the magnetoconductivity of the quantum wells to the theory of 2D weak localization corrections of Iordanskii et al., Ref. [ 65]. Thereby, the spin relaxation length Ls=LSO/2πwas found to decrease from 0.5µm to0.15µm, which according to LSO=π/m∗αcorresponds to an increase of αfrom(20±1)meVÅ at electron concentrations of n= 1.4×1012/cm2toα= (60±1)meVÅ at electron concentrations of n= 0.3×1012/cm2. The magnetoconductivity of a sample with 95 quantum wires in par allel showed a clear crossover from weak antilocaliza- tion to localization. Fitting the data to Eq. ( 51) a corresponding decrease of the spin relaxation rate was ob tained, which was observable already at large widths of the order of t he spin precession length LSOin agreement with the theory Eq. ( 47). However, a saturation as obtained theoretically in diffus ive wires, due to cubic BIA-coupling was not observed. This might be due to the limitation of Eq. ( 47), to diffusive wire widths, le< W, while in ballistic wires a suppression also of the spin relaxation due to cubic BIA-co upling can be expected, since it vanishes identically in 1-D wires, see section IVA. Also, an increase of the spin scattering rate in narrower wi res,W < L s(2D), was not observed in contrast to the results of the optical experimen ts, Ref. [ 70], reviewed above. The dimensional crossover has also been observed in the hete rostructures of the wide gap semiconductor GaN.75The magnetoconductivity of 160 AlGaN/GaN- quantum wires were m easured. The effective mass is m∗= 0.22me, all wires were diffusive with le< W. For electron densities of n≈5×1012/cm2an increase from Ls(2D)≈550nm toLs(W≈130nm)>1.8µm, and for densities n≈2×1012/cm2an increase from Ls(2D)≈500nm to Ls(W≈120nm)>1. µm was observed. Using Ls(2D) = 1/2m∗α, one obtains for both densities n, the spin- orbit coupling α≈5.8meVÅ. A saturation of the spin relaxation rate could not be ob served, suggesting that the cubic BIA-coupling is negligible in these structures. We note, that an enhancement of the spin relaxation rate as in the optical experiments of narrow InGaAs quantum wires, Ref. [ 70], was not observed in these AlGaN/GaN-wires. VI. CRITICAL DISCUSSION AND FUTURE PERSPECTIVE The fact that optical and transport measurements seem to find opposite behavior, enhancement and suppression of the spin relaxation rate, respectively, in narrow wires, calls for an extension of the theory to describe the crossove r21 to ballistic quantum wires. This can be done, using the kinet ic equation approach to the spin-diffusion equation,12a semiclassical approach,77,78or an extension of the diagrammatic approach.13In particular, the dimensional crossover of DPS due to cubic Dresselhaus coupling, which we found not t o be suppressed in diffusive wires, needs to be studied for ballistic wires, le> W, as many of the experimentally studied quantum wires are in t his regime. Furthermore, using the spin diffusion equation, one can study the dependen ce on the growth direction of quantum wires, and find more information on the magnitude of the various spin-orbit coupling parameters, α1,α2,γD, by comparison with the directional dependence found in both the optical measureme nts70of the spin relaxation rate, as well as in recent gate controlled transport experiments.76 In narrow wires, corrections due to electron-electron inte raction can become more important and influence especially the temperature dependence. Ref. [ 71] reports a strong temperature dependence of the spin relaxa tion rate in narrow quantum wires. As shown in Ref. [ 23], the spin relaxation rates obtained from the spin diffusion equation and the quantum corrections to the magnetoconductivity can be diffe rent, when corrections due to electron-electron interacti on become important. As the DPS becomes suppressed in quantum w ires other spin relaxation mechanisms like the EYS may become dominant, since it is expected that the dimension al dependence of EYS is less strong. In more narrow wires, disorder can also result in Anderson localization. S imilar as in quantum dots,41,45this can yield enhanced spin relaxation due to hyperfine coupling, Eq. ( 39). The spin relaxation in metal wires is believed to be domina ted by the EYS mechanism, which is not expected to show such a strong wire width dependence, although this needs to be explored in more detail. Even dilute concentrations of magn etic impurities of less than 1 ppm, do yield measurable spin relaxation rates in metals and allow the study of the Kon do effect with unprecedented accuracy.34,35 VII. SUMMARY The spin dynamics and spin relaxation of itinerant electron s in disordered quantum wires with spin-orbit coupling is governed by the spin diffusion equation Eq. ( 20). We have shown that it can be derived by using classical rand om walk arguments, in agreement with more elaborate derivatio ns.12,13In semiconductor quantum wires all available experiments show that the motional narrowing mechanism of s pin relaxation, the D’yakonov-Perel’-Spin relaxation (DPS) is the dominant mechanism in quantum wires whose width exceeds the spin precession length LSO. The solution of the spin diffusion equation reveals existence of persiste nt spin helix modes when the linear BIA- and the SIA-spin- orbit coupling are of equal magnitude. In quantum wires whic h are more narrow than the spin precession length LSO there is an effective alignment of the spin-orbit fields givin g rise to long living spin density modes for arbitrary ratio of the linear BIA- and the SIA-spin-orbit coupling. The resu lting reduction in the spin relaxation rate results in a change in the sign of the quantum corrections to the conducti vity. Recent experimental results confirm the increase of the spin relaxation rate in wires whose width is smaller th anLSO, both the direct optical measurement of the spin relaxation rate, as well as transport measurements. These s how a dimensional crossover from weak antilocalization to weak localization as the wire width is reduced. Open probl ems remain, in particular in narrower, ballistic wires, were optical and transport measurements seem to find opposit e behavior of the spin relaxation rate: enhancement, suppression, respectively. The experimentally observed r eduction of spin relaxation in quantum wires opens new perspectives for spintronic applications, since the spin- orbit coupling and therefore the spin precession length rem ains unaffected, allowing a better control of the itinerant elect ron spin. The observed directional dependence moreover can yield more detailed information about the spin-orbit co upling, enhancing the spin control for future spintronic devices further.22 Symbols τ0elastic scattering time τeescattering time due to electron-electron interaction τepscattering time due to electron-phonon interaction τtotal scattering time 1/τ= 1/τ0+1/τee+1/τep. ˆτsspin relaxation tensor Dediffusion constant, De=v2 Fτ/dD, wheredDis the dimension of diffusion. leelastic mean free path LSOspin precession length in 2D. The spin will be oriented again in the initial direction after it moved ballistically the lengthLSO. 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2011.00882v2.Spin_orbital_magnetic_response_of_relativistic_fermions_with_band_hybridization.pdf

RIKEN-iTHEMS-Report-20 Spin-orbital magnetic response of relativistic fermions with band hybridization Yasufumi Araki,1Daiki Suenaga,2Kei Suzuki,1and Shigehiro Yasui3, 4 1Advanced Science Research Center, Japan Atomic Energy Agency (JAEA), Tokai 319-1195, Japan 2Research Center for Nuclear Physics (RCNP), Osaka University, Osaka 567-0047, Japan 3Research and Education Center for Natural Sciences, Keio University, Hiyoshi 4-1-1, Yokohama, Kanagawa 223-8521, Japan 4RIKEN iTHEMS, RIKEN, Wako 351-0198, Japan Spins of relativistic fermions are related to their orbital degrees of freedom. In order to quantify the eect of hybridization between relativistic and nonrelativistic degrees of freedom on spin-orbit coupling, we focus on the spin-orbital (SO) crossed susceptibility arising from spin-orbit coupling. The SO crossed susceptibility is dened as the response function of their spin polarization to the \orbital" magnetic eld, namely the eect of magnetic eld on the orbital motion of particles as the vector potential. Once relativistic and nonrelativistic fermions are hybridized, their SO crossed susceptibility gets modied at the Fermi energy around the band hybridization point, leading to spin polarization of nonrelativistic fermions as well. These eects are enhanced under a dynamical magnetic eld that violates thermal equilibrium, arising from the interband process permitted by the band hybridization. Its experimental realization is discussed for Dirac electrons in solids with slight breaking of crystalline symmetry or doping, and also for quark matter including dilute heavy quarks strongly hybridized with light quarks, arising in a relativistic heavy-ion collision process. I. INTRODUCTION Relativistic fermions arise at various energy scales. While relativistic dynamics of fermions is generally de- scribed by the Dirac equation, with four-component spinor eld [1], massless relativistic fermions can be described by the Weyl equation, with two-component spinor eld [2]. Originally, those equations were invented to describe elementary particles obeying Lorentz sym- metry at high energy. Recently, they are also applied to electrons in some crystalline materials, classied as Dirac and Weyl semimetals, which are intensely studied over the past decade [3{9]. In those semimetals, energy bands of electrons exhibit pointlike crossing structures in momentum space, called Dirac or Weyl points, around which the low-energy excitations of electrons can be ef- fectively described as massless Dirac or Weyl fermions. While the characteristics of relativistic fermions them- selves have been broadly studied, we note here that rela- tivistic fermions coexist with nonrelativistic fermions in some cases. It is found in some crystalline materials that Dirac or Weyl points coexist with other bands irrelevant to those point-node structures at the same energy [10{ 16]. Slight breaking of crystalline symmetries by lattice deformations or disorders may lead to hybridization be- tween the Dirac or Weyl bands and the irrelevant bands [17]. For example, the magnetic alloy Co 3Sn2Se2, which is a sibling of the magnetic Weyl semimetal Co 3Sn2S2, is found to exhibit an anticrossing structure between the Weyl cones and the band irrelevant to them, due to the strong band inversion by spin-orbit coupling (SOC) on Se [18{20]. Interband hybridization can occur in quark matter as well: light quarks, with avors u;d; ors, are usually treated as massless Dirac fermions, in compari- son with heavy quarks ( cand sometimes b). If the heavy quarks are dilute enough, they form bound states with the light quarks at low momentum due to color exchange,proposed as the QCD Kondo eect [21{40]. Such a sit- uation is proposed to occur at a short timescale after a relativistic heavy-ion collision process. Once nonrelativistic fermions are mixed and hy- bridized with relativistic fermions, the relativistic eect, including SOC, may get modied. In order to quan- tify the eect of the hybridization between relativistic and nonrelativistic degrees of freedom on SOC, we fo- cus on the spin-orbital (SO) crossed susceptibility, which constitutes a part of the magnetic susceptibility [41{48]. The SO crossed susceptibility, or the SO susceptibility in short, is dened as the response function of spin polariza- tion (spin magnetization) to the orbital magnetic eld, namely, the eect of magnetic eld on the orbital motion of particles via the vector potential. The SO suscepti- bility arises from SOC, namely, the correlation of spin and orbital degrees of freedom, which is the major conse- quence of the relativistic eect [49]. In connection with measurable transport properties, the SO susceptibility is related to the spin Hall conductivity [50, 51], which is one of the typical transport properties arising from SOC [52{54]. In particular, the characteristics of the SO suscep- tibility for relativistic (Dirac and Weyl) fermions have been intensely studied over the past few years, mainly in connection with topological insulators and semimetals [41{43, 46{48]. Since the spin of a massless relativistic fermion is locked to its momentum [55], known as spin- momentum locking, it is proposed that the SO suscepti- bility of massless relativistic fermions shows a universal behavior, depending linearly on the Fermi energy (chemi- cal potential) [42, 56]. However, for multiband systems including nonrelativistic dispersions, general idea on the SO susceptibility has not been well established, despite its rising importance in both solid states and quark mat- ter. Such a lack of general idea on the SO susceptibility is in a clear contrast to the situations in the spin-spin andarXiv:2011.00882v2 [cond-mat.mes-hall] 11 May 20212 orbital-orbital susceptibilities, which were generally for- mulated and studied for various kinds of materials from the mid-20th century [57{67]. Based on the above background, here we study the eect of band hybridization on the SO crossed suscepti- bility. First, we derive a formula for the SO susceptibility applicable to general multiband systems. Based on the obtained formula, we evaluate the eect of hybridization. In order to evaluate the dierence in the SO susceptibil- ity related to the presence or absence of the hybridiza- tion, we use a simple model composed of massless Dirac fermions obeying spin-momentum locking and nonrela- tivistic fermions free from SOC. We nd that, if the magnetic eld is suddenly switched on and violates the thermal equilibrium of the fermions, which we call the dy- namical process, the susceptibility gets strongly reduced at the Fermi level in the vicinity of the band hybridiza- tion point. Owing to the band hybridization, the nonrel- ativistic fermions acquire spin polarization as well, which also becomes signicant in the dynamical process. We give a qualitative understanding of these modications of the susceptibilities using the perturbation theory with a simple quantum mechanics, which we show in a similar manner with the well-established Van Vleck paramag- netism, namely the interband modication of spin-spin and orbital-orbital susceptibilities assisted by SOC [58]. This article is organized as follows. In Sec. II, we de- rive a general formula for the SO crossed susceptibility by the linear response theory using the Matsubara for- malism. In Sec. III, we apply the obtained formula to Weyl fermions as a test case, to see the consistency with the previous literatures [42, 56]. Section IV is the main study in this article, where we introduce a minimal hy- brid model with Dirac and nonrelativistic degrees of free- dom, and evaluate the SO crossed susceptibility using the obtained formula. We discuss how the spin polarization in each sector, namely Dirac or nonrelativistic, gets mod- ied by the hybridization. In Sec. V, we give some dis- cussion on possible experimental methods to capture the obtained behavior of the susceptibilities, both in solids and quark matter. Finally, we summarize our analysis in Sec. VI. Detailed denitions of the susceptibilities and calculation processes are shown in the appendixes. Throughout this article, we take the natural unit with ~= 1, and the speed of light cand the charge of particle e(<0) are left as constants. II. GENERAL ANALYSIS In this section, we derive a general formula for the SO crossed susceptibility by the linear response theory. We start with the denition of the SO crossed suscepti- bility, and evaluate it perturbatively by using the Mat- subara Green's functions. After rearranging the obtained terms with the momentum-space (Bloch) eigenstates, the SO susceptibility is expressed in terms of the geometric quantities related to the band eigenstates, namely, theBerry connection, the Berry curvature, and the orbital magnetic moment. A. Linear response theory The SO crossed susceptibility is dened as the response function of spin magnetization Msto the orbital mag- netic eldBo[42]. We here give a brief discussion how the spin and orbital degrees of freedom are distin- guished, by considering both electrons in solid states and elementary particles in high-energy physics, and show the denition of the SO crossed susceptibility. For detailed discussion about magnetic susceptibility among the spin and orbital degrees of freedom, see Appendix A. The spin magnetization is composed of the spin polar- ization of fermions, Ms= hSi; (1) where the coecient =gBis the gyromagnetic ratio, withgtheg-factor for the fermions and Bthe Bohr mag- neton, andSis the spin operator of the fermions. (Note that the spin magnetic moment of a negative-charge par- ticle is antiparallel to the spin polarization.) The orbital magnetic eld Bois dened with the U(1) vector po- tentialAsatisfyingBo=rA, which couples to the particles in terms of the covariant derivative in contin- uum [2], r7!rieA; (2) and the Peierls phase on lattice [68], tij7!tijexp ieZrj ridr
A ; (3) for the hopping amplitude between lattice sites riand rj. We should be careful about the role of Bo. In the context of relativistic quantum electrodynamics (QED), where the charged particles with Lorentz symmetry are coupled to the electromagnetic elds, the eect of mag- netic eld is fully described by the vector potential, namely,Boin our denition. Bocouples to both the spin and orbital degrees of freedom in this framework. On the other hand, in the low-energy eective theory for nonrelativistic fermions, which is derived from the low- momentum expansion of massive Dirac fermions, Bodoes not fully describe the eect of the magnetic eld. The Zeeman coupling, namely the direct coupling between the magnetic eld and the spin angular momentum, is given separately from Bocoupled as the vector potential. This framework applies to electrons in solid states, including those with Dirac or Weyl dispersion at low energy in their momentum-space band structures. In this frame- work, the eect of magnetic eld on the magnetization via the Zeeman coupling, which is rather straightforward and has been well studied in the context of magnetism, is excluded from our analysis in response to Bo.3 Al(q, iωm) vl Si 〈Si(q, iωm)〉k-q, iωn-iωm k, iωnΠ(q, iωm) AlSi FIG. 1. The loop diagram corresponding to the response func- tion Si Algiven by Eqs. (9) and (11), with the Matsubara for- malism. The solid lines represent the fermion propagators, and the wavy line corresponds to the external eld. With the above notations, the SO crossed susceptibil- ity is dened as a tensor so ij(i;j=x;y;z ) satisfying the relation Ms i(q; ) =so ij(q; )Bo j(q; ) (4) between the spin magnetization Msdened in Eq. (1) and the orbital magnetic eld Bo, whereqand are the wave number (momentum) and the frequency (energy) of the applied magnetic eld Bo. As long as the spin polar- ization is well dened, so ijis uniquely dened in both the relativistic and nonrelativistic regimes. Below we derive the response function to the orbital magnetic eld Boby the perturbative expansion with respect to the vector po- tentialA, in a way similar to the perturbative derivation process of the orbital-orbital susceptibility [62{64]. We start with the translationally invariant system de- scribed by the momentum-space Hamiltonian H0=X k y(k)H(k) (k); (5) with the fermionic eld operator (k) and the kernel ma- trix of Hamiltonian H(k) acting on the components of the eld operator. This assumption applies to both contin- uum with continuous translational symmetry and crys- tals with discrete translational symmetries. By diagonal- izing the matrix H(k), we obtain the energy-momentum dispersiona(k), corresponding to the band dispersion in crystals, and the eigenstate jua(k)i, which are related by H(k)jua(k)i=a(k)jua(k)i (6) withathe label for the eigenstate. We are here interested in the expectation value of the spin polarization hS(r;t)iunder the vector poten- tialA(r;t). The local spin operator S(r;t) is dened with the fermionic eld operators ( y; ) as S(r;t) = y(r;t)S (r;t); (7) where Sis the matrix acting on the spin subspace of the fermionic elds, usually related to the Pauli matrices Si=i=2. As the linear response of spin polarizationhSi=x;y;zito the vector potential Al=x;y;z, we focus on the response function Si Aldened by hSi(r;t)i=Z dr0dt0Si Al(rr0;tt0)Al(r0;t0);(8) or its Fourier transform hSi(q; )i= Si Al(q; )Al(q; ) (9) at arbitrary frequency and momentum q. In order to evaluate the response function Si Al(q; ), we rst note that the coupling to the vector potential with an arbitrary momentum A(q) is given as the per- turbation term Ho(k;k0) =e 2 v(k) +v(k0)
A(kk0); (10) with the velocity matrix v(k) =@H(k)=@k[see Eq. (A3)]. Based on this coupling, the response of the spin polarization hSiiis given with the Matsubara for- malism at one loop, as shown in Fig. 1, hSi(q;i!m)i=eAl(q;i!m) 2VX i!n;kTr SiG(k;i!n) (11) [vl(k) +vl(kq)]G(kq;i!ni!m) ; using the unperturbed Green's function G(i!n;k) = [i!n+H(k)]1. Here= 1=Tis the inverse temper- ature,Vis the volume of the system, is the chemical potential (Fermi energy) of the fermions, and  !mand !nare bosonic and fermionic Matsubara frequencies, re- spectively. We do not consider vertex correction to the velocity vertex, as long as we omit interaction among fermions or impurity scattering. By evaluating the Mat- subara sum over i!nand performing the analytical con- tinuationi!m! +i0 (see Appendix B for detailed derivation process), we obtain Si Al(q; ) =e VX kX abFab(k;q; )Mil ab(k;q):(12) The factor Mil ab(k;q) =1 2hub(kq)jSijua(k)i (13) hua(k)jvl(k) +vl(kq)jub(kq)i measures the correlation between spin and orbital degrees of freedom mediated by the single-particle states jua(k)i andjub(kq)i, and the factor Fab(k;q; ) =f(a(k))f(b(kq)) a(k)b(kq) i0(14) species the spectral weight from the above two states, withfthe Fermi distribution function. Since the orbital magnetic eld Boand the vector po- tentialAare related by Bo=rA, or Bo j(q; ) =ijlhqhAl(q; ) (15)4 in momentum space, the susceptibility tensor so ij(q; ) dened by Eq. (4) can be derived from Si Al(q; ) satis- fying Eq. (9), so ij(q; ) =i 2 jlh@Si Al(q; ) @qh: (16) Owing to the antisymmetrization by jlh,so ij(q; ) be- comes gauge independent. B. Static and dynamical susceptibilities We are mostly interested in the behavior of so ij(q; ) in the low-frequency ( !0) and long-wavelength ( q!0) limits. There are two ways in taking the low-frequency limit, which correspond to dierent physical situations as follows [69{72]: •Static limit | If one takes = 0 (or i!m= 0) from the beginning, the response function gives the be- havior of the system in a thermal equilibrium that is reconstructed under the external eld, such as the Landau levels under Bo. This picture is valid if the system thermalizes to the new equilibrium as quickly as the external eld is applied, corre- sponding to the case 1& , whereis the relax- ation time that phenomenologically characterizes the timescale of thermalization process. •Dynamical limit | If one keeps 6= 0 at rst step and takes the limit !0 after evaluating q!0, the response function gives the response arising from the nonequilibrium modulation of the parti- cle distribution, driven by the introduction of the external eld. This picture is valid if the thermal- ization process is slow enough so that the distribu- tion function of particles cannot follow the applied external eld, corresponding to the case 1. . In addition to the above conditions, in order to apply the low-frequency limit !0, the frequency should be smaller than the energy scale of band splitting (level repulsion), such as the bandgap. In the presence of inter- band hybridization (characterized by the parameter hin Sec. IV), it leads to level repulsion at the band crossing point, providing one characteristic energy scale for this condition. While the static limit is mainly considered for the SO crossed susceptibility in previous work [41{48], the dy- namical limit is important as well, since it is related to experimental measurements with a magnetic eld applied in a short time scale, such as a pulse magnetic eld. Es- pecially, in relativistic heavy-ion collision processes, the magnetic eld is generated soon after the two nuclei col- lide peripherally, which is more likely to be described by the dynamical limit. We therefore consider the SO crossed susceptibilities in both the static and dynamicallimits, which we distinguish by so(sta) ij andso(dyn) ij , and compare them in the following discussions. A dierence between the static and dynamical lim- its emerges in the limiting behavior of the weight factor Fab(k;q; ) given in Eq. (14): •Interband eect | If two bands aandbare dierent (a(k)6=b(k), ora6bfor simplicity of notation), both the static and dynamical limits give the same factor Fab(k;q; )ja6b!f(a(k))f(b(k)) a(k)b(k):(17) •Intraband eect | Ifaandbcorrespond to a same band or degenerate bands ( a(k) =b(k), ora b), there arises a dierence between the static and dynamical limits. By taking !0 rst, the static limit gives Fab(k;q; )jab!f0(a(k)); (18) withf0() =@f=@ , since the numerator and the denominator inFabsimultaneously approach zero underq!0. On the other hand, in the dynamical limit, withq!0 taken rst, only the numerator approaches zero and this factor vanishes. The dierence in the limiting behavior of the intraband eect results in the dierence in the susceptibility, as we demonstrate in the following discussions. C. Identication with geometric quantities The SO crossed susceptibility obtained by Eq. (16) can be further evaluated by expanding the energies and the eigenfunctions by qup to its rst order. The q-expansion yields thek-space gradient of the energy rka(k) = va(k), namely the group velocity, and the gradient of the eigenfunction jrkua(k)i. In order to rearrange the obtained terms, it is instructive to introduce the multi- band expressions of the geometrical quantities character- izing thek-space structure of the wave functions [73{78]. (Note that these multiband expressions are introduced to simplify the obtained formulas, and hence are rigor- ously dierent from the precise multiband denitions in- troduced in Ref. 77.) The physical meanings of the ge- ometric quantities are given in terms of the wave-packet picture [79], where a wave packet localized in real space and momentum space is constructed as linear combina- tion of the momentum-space wave functions jua(k)i. (For simplicity of notations, we do not explicitly denote the argumentkbelow.) We here introduce the Berry connection Aab, the or- bital magnetic moment mab, the Berry curvature ab, and the spin Berry curvature (Si) ab. Below we give their denitions and their physical meanings on the basis of the wave-packet picture. The Berry connection Aab=ihuajrkubi; (19)5 namely the matrix element of the position operator r= irk, is related to the shift of the wave-packet center in real space due to the quantum interference. The orbital magnetic moment is dened as mab=ie 2hrkuaj(abH)jrkubi; (20) with ab= (a+b)=2. (The cross product acts on the Cartesian components arising from the momentum gra- dientrk=P j=x;y;zej@kj, whereex;y;zare the unit vec- tors in the Cartesian coordinate.) mabis related to the orbital angular momentum intrinsic to the wave packet, which arises from geometrical structure of the wave func- tions. It is in analogy with the \self-rotation" of a clas- sical rigid body, and is distinct from motion of the wave- packet center [74{76, 78]. The Berry curvature ab=ihrkuajjrkubi (21) and the \spin Berry curvature" (Si) ab=ihrkuajSijrkubi (22) roughly correspond to the circulating current and spin current, respectively, arising from the geometrical struc- ture of the wave functions. Since all these eects couple to the vector potential or the orbital magnetic eld in real space, these geometric quantities appear in the response functions to the orbital magnetic eld. Using the expressions of the geometric quantities intro- duced above, we can classify the SO crossed susceptibility into three terms, so ij=(A) ij+(m) ij+( ) ij; (23) where the rst term picks up the contribution from the Berry connection, the second term from the orbital mag- netic moment, and the third term from the Berry cur- vature and the spin Berry curvature. (The detailed cal- culation process is shown in Appendix B 4.) Here we introduce the shorthand notations for the spin matrix el- ementSi ab=huajSijubi, the Fermi distribution function fa=f(a(k)), and the weight factor Fab=8 < :f0 a (ab) fafb ab(a6b)(24) arising fromFabin Eq. (14). With these notations, the Berry connection term is given as so(sta:A) ij =e VX kX a6b(f0 aFab)Re[(vaAab)jSi ba] (25) so(dyn:A) ij =e VX kX a6b(1 2f0 aFab)Re[(vaAab)jSi ba]; (26)and the orbital magnetic moment term as so(sta:m) ij = VX kX abFabReh mj abSi bai (27) so(dyn:m) ij = VX kX a6bFabReh mj abSi bai ; (28) in the static and dynamical limits, respectively. The Berry curvature term so( ) ij =e 2VX kX abfaRe j abSi ba +X afa (Si)j aa ; (29) arising from the Berry curvature and the spin Berry cur- vature, takes the same form for the static and dynamical limits, since this term is originally proportional to fafb and the intraband eect abgives no contribution to this term. The static SO susceptibility, namely the response of the spin magnetization to the orbital magnetic eld in equilibrium, is equivalent to its counterpart in terms of the Onsager's reciprocity theorem [80]: the response of the orbital magnetization in equilibrium [78, 81{83] Mo=ie 2VX a;kfahrkuaj(a+H2)jrkuai(30) to the spin magnetic eld (Zeeman splitting) Hs= Z drBs
S; (31) consistently reproduces the static susceptibility obtained above [see Eq. (A12)]. Since the formula Eq. (30) is valid only in equilibrium, we cannot rederive the dynamical SO crossed susceptibility, which is based on the nonequilib- rium distribution disturbed by the magnetic eld, from this reciprocity. The Berry-curvature term, arising from all the occu- pied states in the Fermi sea, contributes to the static and dynamical susceptibilities in the same manner. Since it counts up the entire contribution from the Fermi sea, the susceptibility depends on the momentum-space cut- o, which corresponds to the structure of the Brillouin zone in crystals. In order to extract the universal behav- ior arising from the relativistic dispersion and the band hybridization in later sections, we do not concentrate on the value of so ijitself, but discuss its dependence on the Fermi level throughout this article. We evaluate its deviation from the value at = 0,
so ij() =so ij()so ij(= 0); (32) which is the quantity considered in Ref. 42 for Dirac and Weyl fermions.6 III. SINGLE WEYL NODE Before going on to detailed analysis with interband hy- bridization eect, let us check how the above formula works by taking a single Weyl node as a simplest test case, which is in parallel with the analyses in Refs. 42 and 56. We shall see that, due to spin-momentum locking, the SO crossed response of Dirac and Weyl fermions is closely related to the chiral magnetic eect (CME) and the chi- ral separation eect (CSE), which are the response phe- nomena with respect to the orbital magnetic eld, well studied in the context of relativistic eld theory [84{89]. In lattice systems, the Nielsen{Ninomiya theorem [90, 91] requires that Weyl nodes with opposite chiral- ities should appear in pairs. We can still rely on the single Weyl-node picture, as long as we neglect the con- tribution from kaway from the Weyl points and extract the-dependence. This picture is valid if the Weyl points are well separated in momentum space. At momenta kaway from the Weyl points, the corresponding band energya(k) should be located far above or below the Fermi level , so that the region around the Weyl points will give the dominant contribution to the response phe- nomenon. Under such conditions, we can treat the quasi- particle excitations around each Weyl point separately. A. SO crossed susceptibility If we assume spherical symmetry around the Weyl points, we can use the momentum-space Hamiltonian as a 22-matrix, H(k) =vFk
; (33) where the momentum kis dened as the relative mo- mentum from the Weyl point, with the spherical coor- dinatek=k(sincos;sinsin;cos). This matrix acts on the spin-1 =2 space with spin-up and down states, where the Pauli matrix corresponds to the spin oper- atorSbyS==2. The chirality (right/left) for each Weyl node is identied by =, andvFdenotes the Fermi velocity around the Weyl point. This Hamilto- nian yields the conventional linear dispersion, with the positive-energy branch (k) =vFjkjand negative-energy branch(k) =vFjkj, corresponding to the eigenfunc- tions ju+(k)i= ei=2cos 2 ei=2sin 2 ;ju(k)i= ei=2sin 2 ei=2cos 2 : (34) For the chirality = +, the positive-energy branch cor- responds toju+iand the negative-energy branch to jui, and vice versa for =. Taking the eigenstates juias the basis, the intraband and interband geometrical quan-tities within the single Weyl node are given as A;=1 2kcote;A;=1 2k(ie+e);(35) m;=evF 2kek;m;= 0; (36) ;=1 2k2ek; ;=1 2k2cotek; (37) (Si) ;=1 4k2(ei kei cot)ek; (38) with the unit vectors ek=k=jkj,e=ezek=sin, e=eek. The matrix elements of the spin operator S==2 read S;=1 2ek;S;=1 2(iee): (39) Using the above geometrical quantities and matrix el- ements, the SO crossed susceptibility tensor for a single Weyl node, at Fermi level , can be straightforwardly obtained,
so(sta) ij () =e  82vFij; (40)
so(dyn) ij () =e  242vFij; (41) at zero temperature. The static susceptibility, given by Eq. (40), correctly reproduces the result in Ref. 42 ob- tained by explicitly counting up the contributions from the Landau levels under the magnetic eld. On the other hand, the magnitude of the dynamical suscepti- bility given by Eq. (41) is one third that of the static susceptibility, which has not been explicitly mentioned in the context of the SO crossed susceptibility. Such a dierence between static and dynamical limits is also seen in the CME and the CSE, which we shall discuss in de- tail below. The -dependence in the SO crossed suscep- tibility of Weyl fermions implies that, under an orbital magnetic eld Bo, spin polarization of the Weyl fermions can be induced by varying the chemical potential . This electron spin polarization will exert a spin torque on mag- netization if the system has a ferromagnetic order, which is proposed as the charge- or voltage-induced torque in the context of magnetic Weyl semimetals [56]. B. Chiral magnetic/separation eects The SO crossed susceptibility of Dirac and Weyl fermions is closely related to the CME, namely the cur- rent response against an orbital magnetic eld [84{89]. For a single Weyl node with chirality , the current op- eratorjand the spin operator Sare related as j=evF y =2evFS: (42) Therefore, when the spin polarization hSi(sta) =e 82vFBo;hSi(dyn)  =e 242vFBo(43)7 is induced by the magnetic eld Bo, the chirality- dependent current hji(sta) =e2 42Bo;hji(dyn)  =e2 122Bo(44) is induced accordingly. For a pair of Weyl nodes, or a single Dirac node, the net current vanishes once it is summed over the chirality =. In case the chemi- cal potentials of the two chiralities =are dierent, which is characterized by the chiral chemical potential 5= (+)=2, the net current does not fully cancel. The currenthji=hji++hjiarises in response to Bo, hji(sta)=e25 22Bo;hji(dyn)=e25 62Bo; (45) which is consistent with the static and dynamical CME obtained by the eld-theoretical approach [92, 93] and the semiclassical approach [94{96]. We should be careful that the CME in equilibrium is unrealistic in lattice systems, once one takes into account all the occupied states below the Fermi level, including the states away from the Weyl points [97{100]. On the other hand, the dynamical CME, which is explicitly referred to as the gyrotropic magnetic eect (GME) [96] or the natural optical activity [93] as well, is still present in crystals, arising from the eld- induced modulation of the density of states at the Fermi surfaces. In the absence of 5, the charge current hjivan- ishes in total, whereas the chiral current hj5i=hji+ hji, corresponding to the currents of right-handed and left-handed fermions owing in opposite directions, is present. The chiral current arises in response to the or- bital magnetic eld Bo, hj5i(sta)=e2 22Bo;hj5i(dyn)=e2 62Bo; (46) which is known as the chiral separation eect (CSE) in the context of the relativistic eld theory [101, 102]. The dierence between the static and dynamical limit is present in the CSE as well [103]. Note that the chiral currenthj5iis proportional to the net spin polarization hSi=hSi++hSi, hj5i=X hji=X (2evF)hSi=2evFhSi; (47) due to spin-momentum locking. While the denition of the SO susceptibility is valid as long as the particles have spin degrees of freedom, the CSE is well dened only if the chirality is dened as a good quantum number, and hence we can regard the CSE as the typical example of the SO crossed response arising exclusively for chiral fermions. IV. BAND HYBRIDIZATION EFFECT Based on the general formula obtained above, we now discuss the behavior of the SO crossed susceptibility, inthe hybrid system of Dirac and nonrelativistic fermions. Using a minimal model Hamiltonian including Dirac and nonrelativistic fermions, we discuss the eect of band hy- bridization on the SO susceptibility. We evaluate both the static and dynamical susceptibilities
so(sta=dyn)as functions of the Fermi energy , and discuss how they get modied from those for Dirac and Weyl fermions men- tioned in the previous section. We shall also separate the induced spin polarization into that from the Dirac bands and that from the nonrelativistic bands, to see the hybridization-induced eect in more detail. We mainly consider their behavior at zero temperature. A. Minimal model Let us take into account a single species of Dirac fermions and a single species of nonrelativistic fermions, both with spin 1 =2, in three dimensions. Here the eld operators consist of six components in total: the four- component Dirac sector is labeled by chirality (R/L) and spin ("=#) in the Weyl representation, Dirac = ( R"; R#; L"; L#)T; (48) while the two-component nonrelativistic (NR) sector is labeled by spin ("=#), NR= ( NR"; NR#)T: (49) We dene the model Hamiltonian for each sector as HDirac =Z d3r y Dirac(r)(ivFr
) Dirac(r) (50) HNR=Z d3r y NR(r)r2 2m+0 NR(r): (51) The-matrices for the Dirac sector are dened with the Weyl representation, = diag(;). Here we assume that the momentum is locked not to the pseudospin, such as atomic orbital or sublattice degrees of freedom in crys- tals, but to the real spin, so that acts on the real spin degrees of freedom [104]. For simplicity of discus- sion, we set the Fermi velocity vFfor the Dirac sector isotropic around the Dirac point k= 0, and we take the free-particle dispersion for the nonrelativistic sector. mdenotes the eective mass at band bottom, and 0is the energy dierence (oset) of the band from the Dirac point. We now take into account hybridization between the Dirac and nonrelativistic bands, and consider its eect on the SO crossed susceptibility. The hybridization arises if there is a slight violation of crystalline symmetries that protect the Dirac-node structure, or an interaction be- tween the Dirac and nonrelativistic sectors. Whereas its detailed structure depends on the microscopic proper- ties, namely the crystal structure, angular momenta of the constituent atomic orbitals, etc., our main interest is rather conceptual, to see the behavior of the SO crossed8 0246 -2 -4 02402468 -2 -4 -6 0246 vFk / |ε0| vFk / |ε0|ε(k) / |ε0|(a) ε0 > 0 (b) ε0 < 0 ε1ε2 FIG. 2. Band structure of the minimal hybridized model de- ned by Eqs. (53) and (54), for the energy oset (a) 0>0 and (b)0<0. The parameters are taken as m= 3 j0jand h= 0:2j0j. The dashed lines show the bands without the hybridization h.1and2in (a) are the energies of the band crossing points in the absence of hybridization, which we shall use in later calculations. susceptibility in the vicinity of the band hybridization point. We therefore set up a simple structure of hy- bridization, which satises spherical symmetry, conserves spin, and acts on the right-handed and left-handed com- ponents with the same weights. The hybridization term is then parametrized by a single real value h, with Hhyb=hZ d3rX s=";#h y R;s NR;s+ y L;s NR;s+ H:c:i : (52) With this hybridization term, the total Hamiltonian H= HDirac +HNR+Hhybcan be written as a 6 6-matrix in momentum-space representation as follows: H=X k y(k)H(k) (k); (53) H(k) =0 @vFk
 0h 0vFk
h h hk2 2m+01 A;(54) with the eld operator = ( R"; R#; L"; L#; NR"; NR#)T: (55) Since this model Hamiltonian keeps the right-handed and left-handed components in the Dirac sector equiva- lent, it yields three bands, each of which is twofold de- generate and spherically symmetric around k= 0. Here we note that H(k) commutes with the operator ek
, corresponding to the helicity of a particle. Therefore, H(k) can be separated into two helicity subspaces char-acterized by the eigenvalue =, with the 33-matrix H(k) =0 @vFk 0h 0vFk h h hk2 2m+01 A (56) for each subspace. The typical band structure based on this Hamiltonian is shown by Fig. 2. In the present model, the quadratic dispersion from the nonrelativistic sector coexists with the linear dispersion from the Dirac sector. Therefore, if the energy oset 0is positive, as shown in Fig. 2(a), the nonrelativistic band intersects the particle (electron) branch of the Dirac band twice, whose energy levels are labeled as 1and2in the fol- lowing discussions. At these points, the bands develop anticrossing with the amplitude h. If0is negative, as shown in Fig. 2(b), the nonrelativistic band intersects the antiparticle (hole) and particle branches of the Dirac band once for each, yielding a gap at the crossing point with the antiparticle branch. In the present calculation, we take0positive, and investigate the behavior of the SO crossed susceptibility mainly around the hybridiza- tion points 1;2. Our model dened here is composed of minimal num- ber of degrees of freedom, in order to extract the com- mon feature in the SO susceptibility caused solely by the presence or absence of the band hybridization. While realistic systems including relativistic fermions generally have richer internal degrees of freedom, such as orbital and sublattice degrees of freedom for electrons in solid states and color and avor degrees of freedom for quarks in high-energy physics, dependence on such detailed in- ternal structures for each system is beyond our interest in this article. B. Static and Dynamical suceptibilities With the model Hamiltonian dened in the previous subsection, we now evaluate the SO crossed susceptibil- ity
so() in both static and dynamical limits. We rst consider the response of the net spin polarization, by tak- ing the spin operator Sas a matrix diag( ;;)=2 act- ing on the 6-component eld operator in Eq. (55). Here we x the band parameters 0>0 andm= 30, and vary the band hybridization parameter hand the Fermi energy to capture typical structures in
so(), arising from the band hybridization. We evaluate the formula ob- tained in Sec. II C numerically, based on the band eigen- states of the model Hamiltonian. The quantities with energy dimensions are rescaled by 0in the present cal- culations. Since the system is assumed to satisfy spher- ical symmetry, the susceptibility tensor possesses only the diagonal part
so ij() =
so()ij, which we shall evaluate in the following discussion. We rst compare the static susceptibility
so(sta) and the dynamical susceptibility
so(dyn)under the band hybridization, with those estimated with the Dirac fermions without hybridization, which we call the \pure9 -2.00.02.04.06.00.02.04.06.0 -2.0 μ / ε0Δχso (μ)/e2γ 4π2vF ε1 ε2(sta)(dyn) h = 0 h = 0.2ε0 h = 0.5ε0 h = 1.0ε0 FIG. 3. Behavior of the SO crossed susceptibility
so(), as functions of the Fermi energy . The solid and dashed lines are the static and dynamical susceptibility, respectively, with the hybridization parameter hvaried as shown in the inset table. The vertical dashed lines ( 1;2) correspond to the band hybridization points, which are identical to those shown in Fig. 2(a). The parameters are taken as 0>0 andm= 30. Dirac" case [Eqs. (40) and (41)]. The results are shown in Fig. 3 as functions of . In the vicinity of the band hybridization points 1;2, both the static susceptibility
so(sta)and the dynamical susceptibility
so(dyn)de- viate from those in the pure Dirac case. They asymp- totically reach the pure Dirac behavior at the energies away from1;2, since the hybridization eect on the band eigenstates is signicant only around these points. For the static susceptibility
so(eq)(), we nd three nonanalytic cusps for each value of h. These cusps cor- responds to the band edges, namely the minima and maxima of the bands under the hybridization. The origin of such a nonanalytic behavior can be traced back to the density of states, which becomes nonana- lytic at each band edge. We can see that it comes from the intraband part of the magnetic-moment contribution so(sta:m) ij given by Eq. (27), since it is accompanied with the factorf0(a) that gives the density of states at zero- temperature limit. Nonanalyticity in the static suscepti- bility is also found for massive Dirac fermions [42], arising at the edges of the mass gap, which can also be attributed to the above mechanism. In contrast, the dynamical susceptibility
so(dyn)() is obtained as a smooth function in , since it does not contain the intraband Fermi-surface contribution. Al- though it still contains the Fermi-surface eect f0(a) in so(dyn:A) ij , the velocity vain the same term reaches zero at the band edge, canceling nonanalyticity from the den- sity of states. Aside from the nonanalyticity, we should note that the dynamical susceptibility
so(dyn)() shows a rel- atively large deviation from the pure Dirac case around the hybridization points 1;2. In particular, around 2, 0.02.04.0 -2.0Δχso(sta) (μ)/e2γ 4π2vF ε1 ε2 -2.00.02.04.06.0 μ / ε00.0e2γ 4π2vF 1.02.0 -1.0Δχso(dyn) (μ)/(b)Dynamical Total Dirac NR6.0(a)Static Total Dirac NRFIG. 4. The SO crossed susceptibilities separated into the Dirac sector so Dirac and the nonrelativistic (NR) sector so NR, and their sum (Total, so). Panel (a) shows the static sus- ceptibilities, while Panel (b) shows the dynamical suscepti- bilities. Both results are obtained with the oset of the NR band0>0, the eective mass for the NR band m= 30, and the hybridization parameter h= 0:20. the static susceptibility appears almost insensitive to the hybridization eect, whereas the dynamical sucep- tibility gets suppressed by the hybridization. This is because the dynamical susceptibility is dominated by the interband processes: the contribution from the in- terband processes, accompanied with the weight factor Fab= [f(a)f(b)]=(ab), becomes signif- icant atkaround the hybridization point, as the two bandsaandbget close to one another. As a result, the dynamical SO crossed susceptibility acquires a large modication from the band hybridization, in comparison with the static susceptibility. C. Response of Dirac and nonrelativistic sectors In order to understand the hybridization-induced mod- ication in so() in more detail, we separate it into the contributions from the Dirac and nonrelativistic sectors. The spin magnetization for the Dirac sector Ms Dirac and that for the nonrelativistic sector Ms NRcan be evaluated10 separately, with the spin operators SDirac =1 20 @0 0 00 0 0 01 A;SNR=1 20 @0 0 0 0 0 0 0 01 A:(57) As the response functions of these sector-resolved spin magnetizations to the orbital magnetic eld Bo, we de- ne the SO crossed susceptibilities for the Dirac and non- relativistic sectors, Ms Dirac;i= hSDirac;iiso Dirac;ijBo j (58) Ms NR;i= hSNR;iiso NR;ijBo j; (59) which are obtained by using SDirac andSNRinstead of Sin the formulas shown in Sec. II C. Based on the above denition, the sector-resolved sus- ceptibilities as functions of the Fermi energy are ob- tained as shown in Fig. 4, (a) in the static limit and (b) in the dynamical limit. We here emphasize that so NR becomes nonzero around the hybridization points 1;2 both in the static and dynamical limits, among which the eect in the dynamical limit appears rather signi- cant. This indicates that the nonrelativistic sector shows a nite spin polarization in response to the orbital mag- netic eld, even though the nonrelativistic fermions are originally not subject to SOC. Let us here give a qualitative discussion about the mechanism how the dynamical susceptibility is strongly in uenced by the band hybridization, by using a pertur- bation theory with a simple quantum mechanics. What we need to evaluate is the spin magnetic moment of the hybridized states around the hybridization points. Since the orbital magnetic moment maa(k) of a massless Dirac (or Weyl) fermion is parallel to its spin magnetic moment aa(k) = Saa(k), as seen from Eqs. (36) and (39), here we substitute the orbital magnetic eld with an ef- fective spin magnetic eld acting selectively on the Dirac sector, coupled to the spins as He= Bs Dirac
SDirac. Note that this substitution of eective eld is valid in describing only the direction of the induced spin polar- ization, not its magnitude including parameter depen- dences. We then take into account the hybridization between the Dirac and nonrelativistic sectors, and con- sider the response of spin magnetic moments to this ef- fective magnetic eld. For the sake of clarity, we take the direction of Bs Dirac asz-axis, which yields He= Bs DiracSz Dirac, and focus on the magnetic moments in this direction, as we are here interested in the responses in spatially isotropic systems. At the momentum kcwhere the Dirac and nonrela- tivistic bands cross each other, two states juDirac(kc)i andjuNR(kc)iget hybridized. The hybridized states are given as linear combinations of the two states, jui=1p 2[juDiracijuNRi]; (60) where the eigenenergies satisfy+= 2h. If the hybridization does not mix spins, as we have assumedin the present model, juDiraciandjuNRiparticipating to the hybridization have the same spin polarizations. If the Fermi level lies between +and, only the occupied statejuicontributes to the spin polarization. Therefore, we see here how the spin magnetic moment of the statejuigets perturbed by the eective magnetic eldBs Dirac. When the eective magnetic eld Bs Dirac is applied, the statejuiis perturbed by He, jui=ju+ihu+jHejui += Bs Dirac 2hju+ihu+jSz Diracjui (61) at rst order in Bs Dirac. By this perturbation, the spin magnetic moments of juiprojected onto the Dirac and nonrelativistic sectors, which we denote as = hujSjui( = Dirac;NR), get modied as z = hujSz jui=2 RehujSz jui = 2Bs Dirac hRe [hu+jSz juihujSz Diracju+i]:(62) For the Dirac sector, the modication z Dirac = 2Bs Dirac hjhu+jSz Diracjuij2(63) is parallel to Bs Dirac, from which we can qualitatively understand the enhancement of the SO response in the Dirac sector. We note that this mechanism is similar to the Van Vleck paramagnetism, where the paramagnetic susceptibility is enhanced by the interband eect that is allowed by SOC [58]. On the other hand, for z NR, the matrix elements in Eq. (62) become hu+jSz NRjui=1 2huNRjSz NRjuNRi (64) hujSz Diracju+i=1 2huDiracjSz DiracjuDiraci; (65) which yields z NR= 2Bs Dirac 4hhuNRjSz NRjuNRihuDiracjSz DiracjuDiraci: (66) Since we have assumed that juDiraciandjuNRihave the same spin direction, huDiracjSz DiracjuDiraciand huNRjSi NRjuNRihave the same signs, which is the case with the present model Hamiltonian in Eq. (54). There- fore, the product of the two matrix elements in Eq. (62) becomes negative, yielding z NRantiparallel to Bs Dirac. This discussion provides qualitative interpretation about the negative SO response induced in the nonrelativistic sector seen in Fig. 4(b), which is due to the structure of the hybridized states. Our calculation results of the SO crossed susceptibility using the minimal model in Eqs. (53) and (54) are well described by the above discussion with a simple quan- tum mechanics. This discussion is valid no matter what kind of internal degrees of freedom is present, such as11 orbital, sublattice, avor, or color. Therefore, we can understand that the modications in the SO susceptibil- ities found in our calculation are not the artifact from the minimal model employed here. It is the common feature available in any relativistic fermion systems hybridized with nonrelativistic fermion degrees of freedom, as long as the hybridization mixes two states with the same spin direction. V. IMPLICATION ON EXPERIMENTS Finally, we give some discussions about the implica- tions of our ndings on experiments. In order to realize our idea in experimental measurements, we rst need to note the hierarchy of energy (time) scales, among the relaxation rate (inverse relaxation time) 1, the hy- bridization energy h, and the frequency of the external magnetic eld . As mentioned in Sec. II B, the static limit is valid for < 1< h, and the dynamical limit applies to the case 1< < h. For example, the transport calculations in graphene with charged impu- rities give the relaxation time around 1ps, corre- sponding to the frequency 11THz and the energy 4meV [105]. This can be regarded as the typical scale of relaxation for Dirac electrons at low carrier density, with the Fermi velocity comparable to that of graphene (vF= 3106m=s). With this relaxation timescale, tran- sition between the static and dynamical behaviors in the susceptibility can be achieved by varying the frequency of the external magnetic eld around the terahertz regime. The hybridization energy hshould be larger than 1, so that the spectral broadening by the imaginary part of the fermion self energy may not obscure the band splitting ofharising from the hybridization eect. A. Solid states For electrons in solid states, the response to a magnetic eld measured in experiments contains both the response to the orbital magnetic eld discussed throughout this article and the response to the spin magnetic eld via the Zeeman eect. In order to extract the orbital eect, one may excite the orbital degrees of freedom selectively by a circularly polarized light, and observe the magnetic circular dichroism, namely the dierence in the light ab- sorption depending on the polarization of the light [106]. Another way to identify the orbital eect is to subtract the spin eect from the full response to the magnetic eld. In order to extract the spin eect, one may rely on the exchange coupling between the spins of localized electrons in magnetic elements and the spins of itinerant electrons, which takes the same form with the Zeeman coupling. By introducing magnetic dopants in bulk sam- ple, or the magnetic proximity eect in thin-lm geom- etry attached with a magnetic material, we can mimic the spin magnetic eld for the itinerant electrons, fromwhich we may extract the response to the spin magnetic eld. Aside from the total spin polarization, we are also in- terested in the spin polarization separated into the Dirac and nonrelativistic sectors, as discussed in Sec. IV C. In order to distinguish the spin polarization by the sectors, the nuclear magnetic resonance (NMR) spectroscopy will be helpful in some materials [107]. We may rely on the Knight shift in the NMR spectrum, which arises from the hyperne coupling bewteen the electron spin and the nuclear magnetic moment [108]. The Knight shift pro- vides information about the electron spin polarization belonging to each constituent element in the compound. Therefore, if the Dirac and nonrelativistic bands in the material come from dierent elements, such as in a Dirac semimetal with impurity dopants, the Knight shift may provide information about the sector-resolved spin polar- ization mentioned in our discussion. B. Quark matter Our discussion can also be applied to quark matter. In particular, we may consider mixture of heavy quarks, corresponding to the avor cand sometimes b, and light quarksu,d, ands. Light quarks, having Dirac masses relatively smaller than heavy quarks, can be treated as Dirac fermions, while heavy quarks behave as nonrela- tivistic fermions at low momentum. Throughout a rela- tivistic heavy-ion collision process, the generated quark matter will be subject to a magnetic eld, if the colli- sion of two heavy nuclei is noncentral [88, 109{112]. In a manner similar to the CME and the CSE proposed in quark matter, this magnetic eld will give rise to the spin polarization of both the light and heavy quarks. Since light quarks are described as relativistic Dirac fermions, the magnetic eld couples to them only via the vector potential. While heavy quarks behave as nonrelativistic fermions, the Zeeman eect on them is almost negligible due to their large Dirac masses. Therefore, the eect of the magnetic eld on the spin polarization of quarks can be dominantly described by the SO crossed susceptibility so. Hybridization of light and heavy quarks is possible, if heavy quarks are dilute enough in comparison with light quarks. Heavy quarks form bound states with light quarks by the strong interaction, which is proposed as the QCD Kondo eect [21{40]. Under such a hybridiza- tion, our analysis on the SO susceptibility implies that heavy quarks, corresponding to the nonrelativistic sec- tor in our analysis, develop spin polarization in response to a magnetic eld, although the Zeeman coupling for heavy quarks is weak. While the spin polarization of heavy quarks cannot be measured directly, it may be captured as spin polarization of heavy hadrons includ- ing heavy quarks ( corb) after hadronization, where the quarks are cooled down and conned in hadrons. In the hadronization process, spin polarization of heavy quark12 can be transferred to spin polarization of a  cor b baryon. Therefore, measurement of the spin polariza- tion of the  cor bbaryon is one of the promising ways to observe the hybridization induced by the QCD Kondo eect which is not experimentally veried so far. In order to understand such an eect in quark matter precisely, one needs to determine the microscopic structure of in- teraction and parameters specic to the system, which is left for further analysis [113]. VI. CONCLUSION In the present article, we have focused on the SO crossed susceptibility, namely the response function of the spin magnetization Mscomposed of the spin po- larization of fermions, to the orbital magnetic eld Bo described by the U(1) vector potential. The SO crossed susceptibility quanties the relativistic eect acting on fermions, since it arises as a consequence of SOC, which is the relativistic eect. The idea of SO susceptibility is applicable to any kind of fermion system, not limited to solid states but also to quark matter. One of the main issues discussed in this work is the comparison of the SO crossed susceptibilities in two lim- its, namely the static and dynamical limits. While the behavior of the SO crossed susceptibility in the static limit, induced by a slowly introduced magnetic eld keep- ing the equilibrium, is broadly discussed in the context of topological materials, its dynamical-limit behavior, un- der an abruptly introduced magnetic eld that drives the distribution out of equilibrium, is discussed systemati- cally for the rst time. As a result of our analysis, we have found that the dierence between the static and dynamical SO sus- ceptibilities becomes signicant in the presence of band hybridization. We have seen this tendency by using the hybridized model of Dirac fermions obeying spin- momentum locking and nonrelativistic fermions free from SOC. In the dynamical limit, the SO susceptibility gets strongly modied by the hybridization, and the spins of the nonrelativistic fermions also respond to the orbital magnetic eld, even though they are not originally sub- ject to SOC. These modication eects can be under- stood as the outcome of interband perturbation eect allowed by the band hybridization, which is in a mecha- nism similar to the Van Vleck paramagnetism. The framework of our discussion applies at various en- ergy scales, such as electrons in solids and quark matter after heavy-ion collision in accelerators. In the present article, we have taken a simple model with minimal num- ber of degrees of freedom, with which we are successful in extracting a common feature in the SO susceptibility triggered by the hybridization of relativistic and nonrel- ativistic degrees of freedom. Of course, in realistic sys- tems, there may be more diverse internal degrees of free- dom, depending on the atomic orbitals and crystalline symmetries for electrons in solid states and color and a-vor degrees of freedom for quarks in high-energy physics. Detailed treatment of the crossed response phenomena, based on the microscopic model for each setup, will be left for future analysis. ACKNOWLEDGMENTS Y. A. is supported by the Leading Initiative for Ex- cellent Young Researchers (LEADER). D. S. wishes to thank Keio University and Japan Atomic Energy Agency (JAEA) for their hospitalities during his stay there. K. S. is supported by Japan Society for the Promotion of Science (JSPS) KAKENHI (Grants No. JP17K14277 and JP20K14476). S. Y. is supported by JSPS KAK- ENHI (Grant No. JP17K05435) and by the Interdisci- plinary Theoretical and Mathematical Sciences Program (iTHEMS) at RIKEN. Appendix A: Orbital and spin magnetizations In this part of the appendix, we review how a magnetic eld couples to the orbital and spin degrees of freedom, and distinguish the SO crossed susceptibility from the other types of magnetic susceptibilities. For relativistic fermions under Lorentz symmetry, their coupling to a magnetic eld Bis given in terms of co- variant derivative, with the vector potential A(r) corre- sponding to the magnetic eld B=rA. On the other hand, for nonrelativistic fermions, such as elec- trons trapped in crystals (including electrons in topologi- cal semimetals with the Dirac- or Weyl-type dispersion at low energy), their coupling to a magnetic eld is classied into (a) the orbital eect and (b) the spin eect, which can be derived by downfolding the relativistic theory to the nonrelativistic limit. (a) The orbital eect is given in terms of covariant derivative, which is similar to the gauge coupling in the relativistic theory. If the dynamics of fermions is de- scribed by the continuum Hamiltonian H0=Z dr y(r)H(p) (r); (A1) where (r) is the (multi-component) eld operator of the fermions and H(p) is dened by substituting the momen- tum operator p=irto the momentum-space Hamil- tonianH(k), the vector potential shifts the Hamiltonian as H=Z dr y(r)H(peA(r)) (r); (A2) withethe electric charge of the fermion. Therefore, the perturbation by the magnetic eld is extracted as Ho=e 2Z dr y(r)fv(p);A(r)g (r); (A3)13 with the velocity operator v(p) =@H(p)=@p. (b) The spin eect is the so-called Zeeman splitting, namely the coupling between the spin angular momen- tum and the magnetic eld. If the spin density operator of the fermions is given as S= yS , with Sthe ma- trix acting on the components of the eld operator, the spin eect of the magnetic eld is given in terms of the Zeeman term, Hs= Z drB
S= Z dr y(B
S) : (A4) Here =gBis the parameter called gyromagnetic ratio, withBthe Bohr magneton and gis theg-factor for the fermions. Note that gmay depend on internal degrees of freedom, such as the species of fermions, the atomic orbitals that the electrons in the crystal belong to, etc., whereas we here neglect such detailed structures. Although the origins of the above two eects are the same magnetic eld B, one may formally distinguish them by using dierent labels for the magnetic eld, Bo andBs. Under the orbital and spin magnetic elds, the partition function Z[Bo;Bs] is given from the perturbed HamiltonianH0+Ho[Bo] +Hs[Bs] by tracing out the fermionic degrees of freedom ( y; ). From this parti- tion function, the magnetization can also be dened sepa- rately for the orbital and spin sectors, as thermodynamic variables conjugate to BoandBs: Mo=lnZ Bo Bo=Bs=0;Ms=lnZ Bs Bo=Bs=0: The spin magnetization Msis composed of the spins of the fermions, related to the expectation value of spin po- larizationhSias Ms= hSi= h yS i: (A5) On the other hand, the orbital magnetization Mocomes from the orbital angular momenta of the fermions, cor- responding to the circulating electric current carried by the fermions. Since the position operator ris ill-dened in unbounded systems, the momentum-space formalism of orbital magnetization Mocannot be given so simply as the spin magnetization Ms[see Eq. (30)]. The magnetic susceptibility ijis dened as the tensor characterizing the response of magnetization Mito the magnetic eld Bj. As the magnetic eld and the mag- netization are separated into the orbital and spin parts dened above, the magnetic susceptibility can be sepa- rated into the four parts: [Spin-spin] Ms i=ss ijBs j; (A6) [Spin-orbital] Ms i=so ijBo j; (A7) [Orbital-spin] Mo i=os ijBs j; (A8) [Orbital-orbital] Mo i=oo ijBo j: (A9) The spin-spin response is known as the Pauli paramag- netism, namely, the spin polarization induced by the Zee- man splitting, and the orbital-orbital part often gives riseto the Landau diamagnetism, due to the orbital magnetic moment of the quantum Hall states under the Landau quantization. The spin-orbital crossed parts soandos are not classied with either of them, which require the correlation between the spin and orbital degrees of free- dom. The above four susceptibilities are given in terms of the partition function Z[Bo;Bs], ss ij=2lnZ Bs iBs j; so ij=2lnZ Bs iBo j; (A10) os ij=2lnZ Bo iBs j; oo ij=2lnZ Bo iBo j: (A11) The spin-orbital crossed parts soandossatisfy the relation so ij=os ji; (A12) which is the outcome of the Onsager's reciprocity theo- rem [80]. Although magnetic eld and magnetization in the rel- ativistic regime cannot be separated into the orbital and spin parts, the idea of the SO crossed susceptibility so still applies. The spin magnetization can be dened from the spin polarization, Ms= hSi, and the magnetic eldBcouples to the particles only via the vector poten- tial. Therefore, in the relativistic regime, the response of the spin magnetization to the magnetic eld is described in terms of sodened above. Appendix B: Detailed derivation process of the SO susceptibility tensor In this appendix, we show details of the derivation pro- cess toward the formula for so, whose nal form is given in Sec. II C. 1. Perturbation by vector potential The starting point is the perturbation by the coupling to the vector potential A. With the perturbation Hamil- tonianHogiven by Eq. (10), the linear perturbation of the Green's function becomes G(k;i!n;k0;i!0 n) (B1) =G(k;i!n)Ho(k;i!n;k0;i!0 n)G(k0;i!0 n); which is nondiagonal in momentum and Matsubara fre- quency. Using this perturbation of Green's function, the expectation value of the spin polarization hS(q;i!m)iin-14 duced by the vector potential A(q;i!m) reads hSi(q;i!m)i =Z drd eiq
r+i!mhSi(r;)i (B2) =Z drd eiq
r+i!mh y(r;)Si (r;)i (B3) =Zdrd (V)2X i!n;i!0nX k;k0ei(q+k0k)
r+i(!m+!0 n!n) h y(k0;i!0 n)Si (k;i!n)i (B4) =1 VX i!n;k y(kq;i!ni!m)Si (k;i!n) (B5) =1 VX i!n;kTr [SiG(k;i!n;kq;i!ni!m)] (B6) =1 VX i!n;kTr SiG(k;i!n)H(k;kq) (B7) G(kq;i!ni!m) =eAl(q;i!m) 2VX i!n;kTr SiG(k;i!n) (B8) [vl(k) +vl(kq)]G(kq;i!ni!m) Si Al(q;i!m)Al(q;i!m); (B9) wherei!m=i2 mis the bosonic Matsubara frequency corresponding to the frequency of the vector potential. This form corresponds to Eq. (11). By the analytical continuation i!m! +i0, we obtain the linear response to the vector potential, hSi(q; )i= Si Al(q; )Al(q; ); (B10) for nite momentum qand frequency . 2. Description with band eigenstates We need to evaluate the response tensor Si Al(q; ) up toO(q), to apply Eq. (16). By decomposing the Green's function as G(k;i!n) =X ajua(k)ihua(k)j i!+na(k); (B11) whereadenotes the band index satisfying H(k)jua(k)i=a(k)jua(k)i (B12)andi!+ n=i!n+, the response tensor is given in terms of matrix elements as Si Al(q;i!m) =e 2VX i!n;kTrn SiG(k;i!n) (B13) [vl(k) +vl(kq)]G(kq;i!ni!m)o =e 2VX i!n;kX abhub(kq)jSijua(k)i i!+na(k) i!+ni!mb(kq) hua(k)jvl(k) +vl(kq)jub(kq)i (B14) e VX i!n;kX abMil ab(k;q) i!+na(k) i!+ni!mb(kq): (B15) HereMil ab(k;q) dened in the last line corresponds to Eq. (13). The Matsubara summation is evaluated as 1 X i!n1 i!+na(k) i!+ni!mb(kq) =f(a(k))f(b(kq)+i!m) i!m+a(k)b(kq)(B16) =f(a(k))f(b(kq)) i!m+a(k)b(kq)(B17) i!m! +i0!f(a(k))f(b(kq)) a(k)b(kq) i0Fab(k;q; ); which corresponds to Eq. (14). We thus obtain Eq. (12), Si Al(q; ) =e VX kX abFab(k;q; )Mil ab(k;q):(B18) 3. Expansion by q We need to expand O Aj(q; ) up to O(q) to de- rive the response to the magnetic eld. By expanding Fab(k;q; ) andMil ab(k;q) as Fab(k;q; ) =F(0) ab(k; )qhF(1)h ab(k; ) +O(q2); (B19) Mil ab(k;q) =M(0)il ab(k)qhM(1)ilh ab(k) +O(q2); (B20) so ij(q= 0; ) can be obtained from Eq. (16) as so ij(q= 0; ) =i 2 jlh@Si Al(q; ) @qh q=0(B21) =ie 2VjlhX kX abh F(1)h abM(0)il ab+F(0) abM(1)ilh abi :15 The expansion of the matrix elements Mil ab(k;kq) is given by M(0)il ab=huajvljubihubjSijuai; (B22) M(1)ilh ab=1 2huaj@khvljubihubjSijuai (B23) +huajvlj@khubihubjSijuai +huajvljubih@khubjSijuai; where the dependence on kis not explicitly written for simplicity. vh ais the group velocity, dened by vh a= @kha. Since the rst term in M(1)ilh abcontains@khvl= @kh@klH(k), which is symmetric in h$l, it does not contribute to so ij(q; ) after the antisymmetrization by jlh. On evaluating the weight factor Fab(k;q; ), we should be careful about the dierence between the static and dy- namical limits, which comes from the absence or presence of the frequency in the denominator. •For the interband processa6=b, the numera- tor and the denominator remain nite in the limit q!0 and !0 irrespective of the order of tak- ing these two limits, and hence the q-expansion of Fab(k;q; !0) is straightforwardly given as F(0) ab=fafb ab; (B24) F(1)h ab=vh afafb (ab)2f0 a ab ; in both the static and dynamical limits. •For the intraband processa=b, the denominator approaches zero in the limit q!0 and !0, and hence we need to care about the order of tak- ing these two limits. In the static limit, where the limit = 0 is taken rst, both the numerator and the denominator approach zero in the limit q!0, yielding Fab(k;q; = 0)ja=b (B25) =f(a(k))f(a(kq)) a(k)a(kq)q!0!f0(a(k)) =f0 a; which is of O(q0). In the dynamical limit, where is kept nite rst, the denominator remains nite in the limit q!0. Therefore, the q-expansion becomes Fab(k;q; )ja=b (B26) =f(a(k))f(a(kq)) a(k)a(kq) =qhvh af0 a +O(q2); which is ofO(q1). Although it appears to diverge in the limit !0, it does not contribute to so(dyn) after the antisymmetrization, as we shall see below.4. Rearrangement with geometric quantities From theq-expansion given by Eq. (B21), we are now ready to rearrange the obtained terms into the geometric quantities, Aab=ihuajrkubi; (B27) mab=ie 2hrkuaj(abH)jrkubi; (B28) ab=ihrkuajjrkubi; (B29) (Si) ab=ihrkuajSijrkubi; (B30) whose physical meanings are brie y explained in Sec. II C. We also use the shorthand notations ab=1 2(a+b); (B31) Si ba=hubjSijuai: (B32) The matrix element of the velocity operator can be trans- formed as huajvjubi=huajrkHjubi (B33) =rkhuajHjubihrkuajHjubihuajHjrkubi =rkaabbhrkuajubiahuajrkubi =vaab+ (ba)huajrkubi: Therefore, we obtain the relation vjuai=vajubi+ (aH)jrkuai: (B34) a. Intraband contribution The intraband contribution diers in the static and dynamical limits. In the static limit, the weight factor is F(0) abja=b=f0 a; (B35) and hence the intraband contribution to the susceptibil- ity becomes so(sta:intra) ij =ie 2VjlhX kX abf0 aM(1)ilh ab: (B36) HereM(1)ilh abfora=breads (by omitting the term symmetric in l$h) huajvlj@khubihubjSijuai+huajvljubih@khubjSijuai(B37) = vl ahuaj@khubi+h@kluajaHj@khubi Si ba (B38) +vl aabX b0h@khuajub0ihub0jSijuai:16 Therefore, we obtain so(sta:intra) ij =1 VX kX abf0 ahe 2(vaAab)j+mj abi Si ba +e 2VX kX ab0f0 a(vaAab0)jSi b0a(B39) =e 2VX kX a6bf0 a(vaAab)jSi ba (B40) 1 VX kX abf0 amj abSi ba: In the dynamical limit, the weight factor starts from O(q1), F(1)h abja=b=vh af0 a ; (B41) and hence the intraband contribution to the susceptibil- ity becomes so(dyn:intra) ij =ie 2VjlhX kX abvh af0 a M(0)il ab:(B42) HereM(0)il abfora=breads huajvljubihubjSijuai=vl aabSi aa: (B43) Using this form, the right-hand side of Eq. (B42) contains the factorvh avl a, which is symmetric in l$hand vanishes under the antisymmetrization. Therefore, the intraband part for the dynamical susceptibility vanishes, so(dyn:intra) ij = 0: (B44) b. Interband contribution The dierence in the static and dynamical limits does not appear in the interband contribution. The expansion ofMreads M(0)il ab=huajvljubihubjSijuai (B45) =(ab)huaj@klubiSi ba; (B46) M(1)ilh ab=1 2huaj@khvljubihubjSijuai (B47) +huajvlj@khubihubjSijuai +huajvljubih@khubjSijuai =1 2huaj@kl@khHjubiSi ba (B48) +vl ahuaj@khubiSi ba +h@kluajaHj@khubiSi ba +h@kluajabjubiX ch@khubjuciSi ca:By multiplying the weight factor, we are left with F(1)h abM(0)il ab=vh b f0 bfafb ab huaj@klubiSi ba;(B49) F(0) abM(1)ilh abfafb abSi ba (B50)  vl ahuaj@khubi+h@kluajaHj@khubi + (fafb)h@kluajubiX ch@khubjuciSi ca: Since the rst term in Eq. (B48) is symmetric in l$ hand does not contribute to the susceptibility, we have omitted its contribution to the right-hand side of Eq. (B50). By identifying them with the geometric quan- tities term by term, the interband contribution to the susceptibility reads so(inter) ij =ie 2VjlhX kX a6bh F(1)h abM(0)il ab+F(0) abM(1)ilh abi (B51) =e 2VjlhX kX a6b f0 bfafb ab vh bAl abSi ba (B52) e 2VjlhX kX a6bfafb abvl aAh abSi ba 1 VX kX a6bfafb abh mj ab+e 4(ab) j abi Si ba ie 2VjlhX kX abc(fafb)h@kluajubih@khubjuciSi ca; where the condition a6bin the last line is omitted due to the factor ( fafb), which vanishes for ab. By further processing these terms, we obtain so(inter) ij =e 2VX kX a6bf0 a(vaAba)jSi ab (B53) e 2VX kX a6bfafb ab (vaAba)jSi ab+ (vaAab)jSi ba 1 VX kX a6bfafb abh mj ab+e 4(ab) j abi Si ba +ie 2VjlhX kX abcfah@kluajubihubj@khuciSi ca ie 2VjlhX kX abcfbh@khubjucihucjSijuaihuaj@klubi17 =e 2VX kX a6bf0 a(vaAba)jSi ab (B54) e VX kX a6bfafb abRe (vaAab)jSi ba 1 VX kX a6bfafb abmj abSi ba +e 2VX kX abfa+fb 2 j abSi ba+e 2VX bfb (Si)j bb:By adding the intraband contribution obtained in Eq. 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1604.05810v2.Second_post_Newtonian_Lagrangian_dynamics_of_spinning_compact_binaries.pdf

arXiv:1604.05810v2 [gr-qc] 22 Apr 2016Second post-Newtonian Lagrangian dynamics of spinning com pact binaries Li Huang and Xin Wu∗ Department of Physics &Institute of Astronomy, Nanchang University, Nanchang 330 031, China The leading-order spin-orbit coupling is included in a post -Newtonian Lagrangian formulation of spinning compact binaries, which consists of the Newtonian term, first post-Newtonian (1PN) and 2PN non-spin terms and 2PN spin-spin coupling. This makes a 3 PN spin-spin coupling occur in the derived Hamiltonian. The spin-spin couplings are mainly re sponsible for chaos in the Hamiltonians. However, the 3PN spin-spin Hamiltonian is small and has diffe rent signs, compared with the 2PN spin-spin Hamiltonian equivalent to the 2PN spin-spin Lagr angian. As a result, the probability of the occurrence of chaos in the Lagrangian formulation witho ut the spin-orbit coupling is larger than that in the Lagrangian formulation with the spin-orbit coup ling. Numerical evidences support the claim. PACS numbers: 04.25.Nx, 04.25.-g, 05.45.-a, 45.20.Jj I. INTRODUCTION On February 11, 2016, the LIGO Scientific Collabora- tion and Virgo Collaboration announced the detection of gravitational wave signals (GW150914), sent out from the inspiral and merger of a pair of black holes with masses 36 M⊙and 29M⊙[1]. The gravitational wave dis- covery directly confirmed a major prediction of Albert Einstein’s 1915 general theory of relativity. The LIGO projectwasoriginallyproposedinthe1980sanditsinitial funding was approved in 1992. Since then, the dynamics ofspinningcompactbinarieshasreceivedmoreattention. A precise theoretical waveform template is necessary to matchwithgravitationalwavedata. Asakindofdescrip- tionofthewaveformsanddynamicalevolutionequations, the post-Newtonian (PN) approximation to general rel- ativity was used. Up to now, the PN expansion of the relativistic spinning two-body problem has provided the dynamicalnon-spinevolutionequationsandthe spinevo- lution equations to fourth post-Newtonian (4PN) order [2-7]. A key feature of the gravitational waveforms from a chaotic system is the extreme sensitivity to initial condi- tions [8-10], and therefore the chaos is a possible obstacle to the method of matched filtering. For this reason, sev- eral authors were interested in the presence or absence of chaotic behavior in the orbital dynamics of spinning black hole pairs [11-17]. There were three debates about this topic. Sixteen years ago fractal methods were used to show that the 2PN harmonic-coordinates Lagrangian formula- tion of two spinning black holes admits chaotic behavior [11]. Here the Lagrangian includes contributions from the Newtonian, 1PN and 2PN non-spin terms and the ef- fects of spin-orbit and spin-spin couplings.[41] However, the authors of [12] made an opposite claim on ruling out chaos in compact binary systems by finding no positive Lyapunov exponents along the fractal of [11]. The au- ∗Electronic address: xwu@ncu.edu.cnthors of [18, 19] refuted this claim by finding positive Lyapunovexponents, and pointed out that the reasonfor the false Lyapunov exponents obtained in Ref. [12] lies in using the Cartesian distance between two nearby tra- jectories by continually rescaling the shadow trajectory. This is a debate with respect to Lyapunov exponents re- sulting in two different claims on the chaotic behavior of comparable mass spinning binaries. In fact, the true reason for the discrepancy was found in Ref. [20] and should be that different treatments of the Lyapunov ex- ponents were given in the three papers [12, 18, 19]. The authors of [12] used the stabilizing limit values as the values of Lyapunov exponents, whereas those of [18, 19] used the slopes of the fit lines. It is clear that obtaining the limit values requires more CPU times than obtaining the slopes. Second debate is different descriptions of chaotic re- gions and parameter spaces. It was reported in Ref. [21] that increasing the spin magnitudes and misalign- ments leads to the transition to chaos, and the strength of chaos is the largest for the spins perpendicular to the orbital angular momentum. However, an entirely differ- ent description from Ref. [22] is that chaosoccursmainly when the initial spins are nearly antialigned with the or- bital angular momentum for the binary configuration of masses 10 M⊙and 10M⊙. These descriptions seem to be apparently conflicting, but they can all be correct, as mentioned in Ref. [23]. This is because a compli- catedcombinationofallparametersandinitialconditions rather than single physical parameter or initial condition is responsible for yielding chaos. No universal rule can be given to dependence of chaos on each parameter or initial condition. Third debate relates to different dynamical behav- iors of PN Lagrangian and Hamiltonian conservative systems at the same order. In other words, the de- bate corresponds to a question whether the two formu- lations are equivalent. The equivalence of Arnowitt- Deser-Misner (ADM) Hamiltonian and harmonic coor- dinate Lagrangian approaches at 3PN order were proved by two groups [24, 25]. This result is still correct for ap- proximation accuracy to next-to-next-to-leading (4PN)2 order spin1-spin2 couplings [4]. Recently, the authors of [26] resurveyed this question. In their opinion, this equivalence means that all the known results of the ADM Hamiltonian approach can be transferred to the harmonic-coordinates Lagrangian one, or those of the harmonic-coordinatesLagrangianapproachcanbe trans- ferred to the ADM Hamiltonian one. In fact, the two approaches are not exactly equal but are approximately related. In this case, dynamical differences between the twoapproachesarenotavoided. Foratwo-blackholesys- tem with twobodies spinning, the PN Lagrangianformu- lation of the Newtonian term and the 1.5PN spin-orbit coupling is non-integrable and possibly chaotic, whereas the PN Hamiltonian formulation of the Newtonian term and the 1.5PN spin-orbit coupling is integrable and non- chaotic [26]. This is due to the difference between the two formulations, 3PN spin-spin coupling. That is, the Lagrangian is exactly equivalent to the Hamiltonian plus the 3PN spin-spin term. As a result, the 3PN spin-spin effect leads to the non-integrability of the Hamiltonian equivalent to the Lagrangian. Ten years ago, similar claims were given to the two PN formulations of the two-black hole system with one bodies spinning [21, 27- 29]. An acute debate has arisen as to whether the com- pact binaries of one bodies spinning exhibit chaotic be- haviour. A key to this question in Ref. [30] is as follows. The construction of canonical, conjugate spin variables in Ref. [31] showed directly that four integrals of the total energy and total angular momentum in an eight- dimensional phase space of a conservative Hamiltonian equivalent to the Lagrangian determine the integrability of the Lagrangian. Precisely speaking, no chaos occurs in the PN conservative Lagrangian and Hamiltonian for- mulations of comparable mass compact binaries with one body spinning. One of the main results of [26, 32] is that a PN La- grangian approach at a certain order always exists a for- mal equivalent PN Hamiltonian. This is helpful for us to study the Lagrangian dynamics using the Hamilto- nian dynamics. Following this direction, we shall re- visit the 2PN ADM Lagrangian dynamics of two spin- ning black holes, in which the Newtonian, 1PN and 2PN non-spin terms and the 1.5PN spin-orbit and 2PN spin- spin contributions are included. A comparison between the Lagrangian and related Hamiltonian dynamics will be made, and a question of how the orbit-spin coupling exerts an influence on chaos resulted from the spin-spin coupling in the Lagrangianwill be particularlydiscussed. The present investigation is unlike the work [11], where the onset of chaos in the 2PN Lagrangian formulation was mainly shown. It is also unlike the paper [33], where the effect of the orbit-spin coupling on the strength of chaos caused by the spin-spin coupling was considered but the 1PN and 2PN non-spin terms were not included in the Lagrangian. In our numerical computations, the velocity of light cand the constant of gravitation Gare taken as ge- ometrized units, c=G= 1.II. POST-NEWTONIAN APPROACHES Suppose that compact binaries have masses M1and M2, and then their total mass is M=M1+M2. Other parameters are mass ratio β=M1/M2, reduced mass µ=M1M2/Mand mass parameter η=µ/M=β/(1+ β)2. In ADM center-of-mass coordinate system, ris a relative position of body 1 to body 2, and vis a veloc- ity of body 1 relative to the center of mass. Let unit radial vector be n=r/rwith radius r=|r|. The evolu- tion of spinless compact binaries can be described by the following PN Lagrangian formulation L0=LN+1 c2L1PN+1 c4L2PN. (1) The above three parts are the Newtonian term LN, first order PN contribution L1PNand second order PN con- tribution L2PN. They are expressed in Ref. [34] as LN=1 r+v2 2, (2) L1PN=1 8(1−3η)v4+1 2[(3+η)v2+η(n·v)2]1 r −1 2r2, (3) L2PN=1 16(1−7η+13η2)v6+1 8[(7−12η−9η2)v4 +(4−10η)η(n·v)2v2+3η2(n·v)4]1 r +1 2[(4−2η+η2)v2+3η(1+η)(n·v)2]1 r2 +1 4(1+3η)1 r3. (4) In fact, these dimensionless equations deal with the use of scale transformation: r→GMr,t→GMtandL0→ µL0. In terms of Legendre transformation H0=p·v− L0with momenta p=∂L/∂v, we have the following Hamiltonian H0=HN+1 c2H1PN+1 c4H2PN, (5) where these sub-Hamiltonians are HN=p2 2−1 r, (6) H1PN=1 8(3η−1)p4−1 2[(3+η)p2+η(n·p)2]1 r +1 2r2, (7) H2PN=1 16(1−5η+5η2)p6+1 8[(5−20η−3η2)p4 −2η2(n·p)2p2−3η2(n·p)4]1 r+1 2[(5+8η) p2+3η(n·p)2]1 r2−1 4(1+3η)1 r3.(8)3 The two Hamiltonians H1PNandH2PNare the results of [34]. Besides them, other higher-order PN terms can be derived from the Lagrangian L0. For example, a third order PN sub-Hamiltonian was given in [26] by H3PN=3 16(−η+7η2−12η3)p8+1 8r[(2−7η+3η2 +30η3)p6+(4η−11η2+36η3)(n·p)2p4 +6(η2−3η3)(n·p)4p2]+1 4r2[(5+18η +21η2−9η3)p4+4(5η2+2η3)(n·p)4 +(14η−7η2−27η3)(n·p)2p2] +1 2r3[(−3−31η−7η2+η3)p2 +(−η−η2+7η3)(n·p)2]. (9) It is obtained from coupling of the 1PN term L1PNand the 2PN term L2PN. Since the difference between the 2PN Lagrangian L0and the 2PN Hamiltonian H0is at least 3PN order, the two PN approaches are not exactly equivalent. Clearly, a Hamiltonian that is equivalent to the 2PN Lagrangian[42] cannot be at second order but should be at a higher enough order or an infinite order. This is one of the main results of [26]. When the two bodies spin, some spin effects should be considered. Now, the leading-orderspin-spincouplingin- teraction L2ss[35], as one kind of spin effect, is included in the Lagrangian L0. In this sense, the obtained La- grangian becomes L1=L0+L2ss, (10) where L2ss=−1 2r3[3 r2(S0·r)2−S02],(11) S= (2+3 2β)S1+(2+3β 2)S2, S0= (1+1 β)S1+(1+β)S2. Note that each spin variable is dimensionless, namely, Si=SiˆSiwith spin magnitude Si=χiMi2/M2(0≤ χi≤1)and3-dimensionalunitvector ˆSi. Because L2ssis independent of velocity, it does not couple the 1PN term L1PNor the 2PN term L2PNvia the Legendre trans- formation. It is only converted to a spin-spin coupling Hamiltonian H2ss=−L2ss. (12) Adding this term to the Hamiltonian H0, we have the following Hamiltonian H1=H0+H2ss. (13) When another kind of spin effect, the leading-order spin-orbit coupling L1.5so[35], is further included in theabove-mentionedLagrangian L1, we obtain a Lagrangian formulation as follows: L2=L1+L1.5so, (14) L1.5so=−1 r3S·(r×v). (15) UnlikeL2ss,L1.5sodepends on the velocity. Therefore, the Legendre transformation results in the appearance of the leading-orderspin-orbit Hamiltonian H1.5soobtained from the coupling of the Newtonian term LNand the spin-orbit term L1.5so, the next-order spin-orbit Hamil- tonianH2.5soobtainedfromthecouplingofthe1PNterm L1PNand the spin-orbit term L1.5soand the next-order spin-spinHamiltonian H3ssobtainedfromthecouplingof the leading-order spin-orbit term L1.5soand itself. These Hamiltonians are written in [26] as H1.5so=−L1.5so|v→p, (16) H2.5so=1 r3(3η−1 2p2−3+η r)S·(r×p),(17) H3ss=1 2r6[r2S2−(S×r)2]. (18) We take three Hamiltonians: H2=H1+H1.5so, (19) H3=H2+H2.5so+H3PN, (20) H4=H3+H3ss. (21) ForthethreeHamiltonians, H4isthe bestapproximation to the Lagrangian L2although the former is not exactly equivalent to the latter. As an important point to note, L2exhibits chaos when the two objects spin and the spin effects are L1.5sobut are not L2ss. This is because many higher-order spin-spin couplings (such as H3ss), included in a higher enough order Hamiltonian equivalent to L2 (with its equations of motion to another higher enough order), make this equivalent Hamiltonian non-integrable [26]. However,anyPNconservativeLagrangianapproach is always integrable and non-chaotic when only one body spins and the spin effects are not restricted to the spin- orbit couplings. This is due to integrability of the equiv- alent Hamiltonian [30, 31]. Clearly, the above-mentioned same PN order La- grangian and Hamiltonian approaches (e.g. L2andH2) aretypically nonequivalent, and thereforethey both have different dynamics to a large extent. Now, we are mainly interested in knowing how the spin-orbit coupling L1.5so affects the chaotic behavior yielded by the spin-spin cou- plingL2ssin the above Lagrangians. In other words, we should compare differences between the L1andL2 dynamics. Considering that the 2PN spin-spin coupling H2ssequivalent to L2ssand the 3PN spin-spin coupling H3ssassociated to L1.5soplay an important role in the onset of chaos in the Hamiltonians, we should also focus on differences between the H3andH4dynamics. These discussions await the following numerical simulations.4 III. NUMERICAL COMPARISONS Let us take initial conditions r(0) = (17 .04,0,0), v(0) = (0,0.094,0), dynamical parameters χ1=χ2= 1, β= 0.79, and initial unit spin vectors ˆS1= (0.1,0.3,0.8)//radicalbig 0.12+0.32+0.82, ˆS2= (0.7,0.3,0.1)//radicalbig 0.72+0.32+0.12. An eighth- and ninth-order Runge-Kutta-Fehlberg algo- rithm of variable step sizes [RKF8(9)] is used to solve the related Lagrangian and Hamiltonian systems. This algo- rithm is highly precise. In fact, it gives an order of 10−11 to the accuracy of the Lagrangian L2and an order of 10−12to the accuracy of the Hamiltonian H2, as shown in Fig. 1. Here, the errors of L2andH2were estimated according to two different integration paths. The error ofL2, i.e., the error of H2, was obtained by applying the RKF8(9) to solve the 2PN Lagrangian equations of L2, while the errorof H2was given by applying the RKF8(9) to solvethe 2PNHamiltonian equations of H2. Although theerrorshavesecularchanges,theyaresosmallthatthe obtained numerical results should be reliable. The evolution of the orbit in Fig. 2 demonstrates that thesameorderLagrangianandHamiltonianformulations L2andH2diverge quickly from the same starting point. This supports again the general result of [26] on the non- equivalence of the PN Lagrangian and Hamiltonian ap- proaches at the same order. For the given orbit, we investigate dynamical differ- ences among some Lagrangian and Hamiltonian formu- lations using several methods to find chaos. A. Chaos indicators A power spectral analysis method is based on Fourier transformation and gives the distribution of frequencies to a certain time series. It can roughly detect chaos from order. Discrete frequencies are usually regardedas power spectra of regular orbits, whereas continuous frequencies are generally referred to as power spectra of chaotic or- bits. In light of this criterion, distributions of continuous frequency spectra in Figs. 3(a) and (d)-(f) seem to show the chaoticity of the Lagrangian L1and the Hamiltoni- ansH2,H3andH4. On the other hand, distributions of discrete frequency spectra in Figs. 3(b) and (c) describe theregularityoftheHamiltonians H1andtheLagrangian L2. It is sufficiently proved that the same PN order La- grangian and Hamiltonian approaches ( L1andH1,L2 andH2)havedifferentdynamics. Itisworthemphasizing that the method of power spectra is ambiguous to differ- entiate among complicated periodic orbits, quasiperiodic orbitsandweaklychaoticorbits. Therefore,morereliable qualitative methods are necessarily used. As a common method to distinguish chaos from or- der, a Lyapunov exponent characterizes the average ex- ponential deviation of two nearby orbits. The varia-tional method and the two-particle one are two algo- rithms for calculating the Lyapunov exponent [36, 37]. Although the latter method is less rigorous than the for- mer method, its application to a complicated system is more convenient. For the use of the two-particle method, the principal Lyapunov exponent is defined as λ= lim t→∞1 tlnd(t) d(0), (22) whered(0) andd(t) are separations between two neigh- boring orbits at times 0 and t, respectively. The best choice of the initial separation d(0) is an order of 10−8 under the circumstance of double precision [36]. In addi- tion, avoiding saturation of orbits needs renormalization. A global stable system[43] is chaotic if λreaches a stabi- lizing positive value, whereas it is ordered when λtends to zero. In terms of this, it can be seen clearly from Fig. 4 that the four approaches L1,H2,H3andH4with pos- itive Lyapunov exponents are chaotic, and the two for- mulations H1andL2with zero Lyapunov exponents are regular. NotethateachofthesevaluesofLyapunovexpo- nents is given after integration time 2 ×105. Obtaining a reliable stabilizing limit value of Lyapunov exponent usually costs an extremely expensive computation [20]. A quicker method to find chaos is a fast Lyapunov indicator (FLI) [38, 39]. It was originally calculated us- ing the length of a tangential vector and renormalization is unnecessary. Similar to the Lyapunov exponent, this indicator can be further developed with the separation between two nearby trajectories [40]. The modified indi- cator is of the form FLI = log10d(t) d(0). (23) An appropriate choice for renormalization within a rea- sonable amount of time span is important. See Ref. [40] formoredetailsofthisindicator. TheFLIincreasesexpo- nentially with time for a chaotic orbit, but algebraically for a regular orbit. The completely different time rates are used to distinguish between the two cases. Based on this point, the dynamical behaviors of the six approaches L1,L2,H1,H2,H3andH4can be described by the FLIs in Fig. 5. These results are the same as those given by the Lyapunov exponents in Fig. 4. Here each FLI was obtainedafter t= 5×104. In this sense, the FLI isindeed a faster method to identify chaos than the Lyapunov ex- ponent. Because of this advantage, the method of FLIs is widely used to sketch out the global structure of phase spaceorto providesomeinsight into dependence ofchaos on single physical parameter or initial condition [23]. B. Effects of varying the mass ratio on chaos Fixing the above initial conditions, initial spin config- urations and spin parameters ( χ1andχ2), we let the mass ratio βrun from 0 to 1 in increments of 0.01. For5 TABLE I: Chaotic parameter spaces of each approach Approach Mass Ratio β L1 0.05, 0.06, 0.08, [0.11,0.16], [0.18,0.83], 0.86, 0.86, [0.89,0.92], 0.94 H1 [0.12,0.19], [0.23,0.27], 0.30, [0.33,0.34], [0.40,0.42], [0.62,0.64] L2 0.03, 0.05, [0.09,0.41], [0.43,0.45], 0.47, 0.54 H2 0.09, [0.17,0.22], [0.24,0.99] H3 [0.09,0.22], [0.24,0.65], [0.67,0.97] H4 [0.09,0.10], [0.15,0.17], 0.19, 0.48, 0.62, 0.65, 0.68, 0.75, 0.79, 0.82, [0.84,0.91], 0.93 each value of β, FLI is obtained after integration time t= 5×104. In this way, we plot Fig. 6 in which de- pendence of FLI for each PN approach on βis described. It is found through a number of numerical tests that 7.5 is a threshold of FLI to distinguish between regular and chaoticorbits. A globalstable orbit is chaoticif its FLI is largerthanthe threshold, but orderedifits FLI issmaller thanthethreshold. Inthissense, this figureshowsclearly the correspondence between the mass ratio and the or- bital dynamics. It is shown sufficiently that the PN Lagrangian and Hamiltonian approximations at the same order, ( L1,H1) and (L2,H2), have different dynamical behaviors in sig- nificant measure. More details on the related differ- ences are listed in Table 1. It can also be seen that the chaotic parameter space of L1is larger than that ofL2. That means that the spin-orbit coupling L1.5so makes many chaotic orbits in the system L1evolve into ordered orbits.[44] In other words, the probability of the occurrence of chaos in the system L1without L1.5sois large, but that in the system L2withL1.5sois small. Thus,L1.5soplays an important role in weakening or suppressing the chaotic behaviors yielded by L2ss. Here, the chaotic behaviors weakened (or suppressed) do not meanthatanindividualorbitmustbecomefromstrongly chaotic to weakly chaotic (or from chaotic to nonchaotic) whenL1.5sois included in L1. This orbit may become more strongly chaotic, or may vary from order to chaos. For example, β= 0.03 in Table 1 corresponds to the regularity of L1but the chaoticity of L2. In fact, the chaotic behaviors weakened or suppressed mean decreas- ing the chaotic parameter space. The result obtained in the generalcasewith the PNterms L1PNandL2PNis an extension to the special case without the PN terms L1PN andL2PNin Ref. [33]. As the authors of [33] claimed, this result is due to different signs of H2ss(equivalent to L2ss) andH3ss(associated to L1.5so). The two spin-spin terms are responsible for causing chaos in the Hamilto- nianH4, andH2sshas a more primary contribution tochaos. It is further shown in Fig. 6 and Table 1 that H4 with the inclusion of H3sshas weak chaos and a small chaotic parameter space, compared with H3with the ab- sence ofH3ss. This is helpful for us to explain why L1.5so can somewhat weaken or suppress the chaoticity caused byL2ssin the PN Lagrangian system L2. IV. CONCLUSIONS When a Lagrangian function at a certain PN order is transformed into a Hamiltonian function, many addi- tionalhigher-orderPNtermsusuallyoccur. Inthis sense, the PN Lagrangian and Hamiltonian approaches at the same order are generally nonequivalent. The equivalence between the Lagrangian formulation and a Hamiltonian system often requires that the Euler-Lagrangian equa- tions and the Hamiltonian should be up to higher enough orders or an infinite order. For the Lagrangian formulation of spinning compact binaries, which includes the Newtonian term, 1PN and 2PNnon-spintermsand2PNspin-spincoupling,theLeg- endre transformation gives not only the same order PN Hamiltonian but also many additional higher-order PN terms, such as the 3PN non-spin term. Therefore, the sameorderLagrangianandHamiltonianapproacheshave some different dynamics. This result is confirmed by nu- merical simulations. This Lagrangian is non-integrable and can be chaotic under an appropriate circumstance due to the absence of a fifth integral of motion in the equivalent Hamiltonian. When the 1.5PNspin-orbit cou- pling is added to the Lagrangian, the 3PN spin-spin cou- pling appear in the derived Hamiltonian. The 3PN spin- spin Hamiltonian is small and has different signs com- pared with the 2PNspin-spin Hamiltonian. In this sense, the probability of the occurrence of chaos in the La- grangian formulation without the spin-orbit coupling is large, whereas that in the Lagrangian formulation with the spin-orbit coupling is small. That means that the leading-order spin-orbit coupling can somewhat weaken or suppress the chaos yielded by the leading-order spin- spin coupling in the PNLagrangianformulation. Numer- ical results also support this fact. Acknowledgments This research has been supported by the Natural Sci- ence Foundation of Jiangxi Province under Grant No. [2015] 75, the National Natural Science Foundation of China under Grant No. 11533004 and the Postgradu- ate Innovation Special Fund Project of Jiangxi Province under Grant No. YC2015-S017. [1] B.P. Abbott et al. Phys. Rev. Lett. 116, 061102 (2016) [2] M. Levi, Phys. Rev. D 85, 064043 (2012)[3] T. Damour, P. Jaranowski, G. Sch¨ afer, Phys. Rev. D 89, 064058 (2014)6 [4] M. Levi, J. Steinhoff, JCAP 1412, 003 (2014) [5] M. Levi, J. Steinhoff, JHEP 1509, 219 (2015) [6] M. Levi, J. Steinhoff, JHEP 1506, 059 (2015) [7] M. Levi, J. Steinhoff, JCAP 1601, 008 (2016) [8] K. Kiuchi, H. Koyama, K.I. Maeda, Phys. Rev. D 76, 024018 (2007) [9] Y.Wang, X.Wu, Commun. Theor. Phys. 56, 1045 (2011) [10] Y. Z. Wang, X. Wu, S. Y. Zhong, Acta Phys. Sin. 61, 160401 (2012) [11] J. Levin, Phys. Rev. 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D 91, 024042, 2015 [27] C. K¨ onigsd¨ orffer, A. Gopakumar, Phys. Rev. D 71, 024039 (2005)[28] A. Gopakumar, C. K¨ onigsd¨ orffer, Phys. Rev. D 72, 121501 (2005) [29] J. Levin, Phys. Rev. D 74, 124027 (2006) [30] X. Wu, G. Huang, Mon. Not. R. Astron. Soc. 452, 3617 (2015) [31] X. Wu, Y. Xie, Phys. Rev. D 81, 084045 (2010) [32] R. C. Chen, X. Wu, Commun. Theor. Phys. 65, 321 (2016) [33] H. Wang, G. Q. Huang, Commun. Theor. Phys. 64, 159 (2015) [34] L. Blanchet, B. R. Iyer, Classical Quantum Gravity 20, 755 (2003) [35] A. Buonanno, Y. Chen, T. Damour, Phys. Rev. D 74, 104005 (2006) [36] G. Tancredi, A. Sa nchez, F. Roig, Astron. J. 121, 1171 (2001) [37] X. Wu, T.Y. Huang, Phys. Lett. A 313, 77 (2003) [38] C. Froeschl´ e, E. Lega, R. Gonczi, Celest. Mech. Dyn. Astron.67, 41 (1997) [39] C. Froeschl´ e, E. Lega, Celest. Mech. Dyn. Astron. 78, 167 (2000) [40] X. Wu, T.Y. Huang, H. Zhang, Phys. Rev. D 74, 083001 (2006) [41] The leading-order spin-orbit coupling is at 1.5PN orde r and the leading-order spin-spin coupling is at 2PN order. [42] Here, the equations of motion derived from this La- grangian are required to arrive at a higher enough order or an infinite order. [43] As an example, a global stable binary system in Ref. [30] means that the two objects do not run to infinity, and do not merge, either. In fact, the binaries move in a bounded region. [44] The chaoticity of the system L1is due to the spin-spin coupling L2ss.L2minusL1isL1.5so.7 /s48/s46/s48 /s49/s46/s48 /s50/s46/s48 /s51/s46/s48 /s52/s46/s48 /s53/s46/s48/s48/s46/s48/s49/s46/s48/s50/s46/s48/s51/s46/s48/s52/s46/s48/s53/s46/s48/s54/s46/s48 /s40/s97/s41/s32/s32/s32/s32/s32/s76/s50 /s32/s32/s72/s32/s40/s49/s48/s45/s49/s49 /s41 /s116/s32/s40/s49/s48/s52 /s77/s41/s48 /s49 /s50 /s51 /s52 /s53/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48 /s40/s98/s41/s32/s32/s32/s32/s32/s32/s72/s50 /s32/s32/s72/s32/s40/s49/s48/s45/s49/s50 /s41 /s116/s32/s40/s49/s48/s52 /s77/s41 FIG. 1: (color online) Hamiltonian errors of the 2PN Lagrang ian and Hamiltonian approaches, L2andH2. (a) The RKF8(9) is used to solve the Euler-Lagrangian equations of L2so as to obtain the difference ∆ Hbetween the Hamiltonian H2(t) at time tand the initial Hamiltonian H2(0). (b) The integrator directly solves the Hamiltonian H2and obtain the difference ∆ H. /s45/s50/s48 /s45/s49/s53 /s45/s49/s48 /s45/s53 /s48 /s53 /s49/s48 /s49/s53 /s50/s48/s45/s50/s48/s45/s49/s53/s45/s49/s48/s45/s53/s48/s53/s49/s48/s49/s53/s50/s48 /s32/s76/s50 /s32/s72/s50 /s32/s32/s89 /s88/s40/s97/s41 −10010 −10010−505 XY ZL2 H2(b) FIG. 2: (color online) Comparison of orbits in the 2PN Lagran gian and Hamiltonian approaches, L2andH2. (a) The orbits projected onto the X-Yplane, and (b) the orbits in the three-dimensional Euclidea n space.8 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55/s48/s46/s48/s48/s46/s53/s49/s46/s48/s49/s46/s53/s50/s46/s48 /s40/s97/s41/s32/s32/s32/s32/s32/s76/s49 /s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50 /s41/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56/s48/s50/s52/s54/s56 /s40/s98/s41/s32/s32/s32/s32/s72/s49 /s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50 /s41/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52 /s40/s99/s41/s32/s32/s32/s32/s76/s50 /s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50 /s41 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50/s48/s49/s50/s51/s52/s53 /s40/s100/s41/s32/s32/s32/s32/s32/s32/s32/s72/s50 /s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50 /s41/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56/s48/s49/s50/s51/s52/s53 /s72/s51/s40/s101/s41 /s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50 /s41/s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48 /s49/s46/s50 /s49/s46/s52 /s49/s46/s54/s48/s49/s50/s51/s52/s53 /s40/s102/s41/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s32/s72/s52 /s32/s32/s80/s111/s119/s101/s114/s32/s115/s112/s101/s99/s116/s114/s117/s109 /s70/s114/s101/s113/s117/s101/s110/s99/s121/s32/s40/s49/s48/s45/s50 /s41 FIG. 3: Power spectra of six PN approaches. /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s45/s52/s46/s53/s45/s52/s46/s48/s45/s51/s46/s53/s45/s51/s46/s48/s45/s50/s46/s53/s45/s50/s46/s48/s45/s49/s46/s53/s45/s49/s46/s48/s45/s48/s46/s53 /s40/s97/s41 /s72/s49/s72/s51/s76/s49 /s32/s32/s108/s111/s103 /s49/s48/s32/s124 /s124 /s108/s111/s103 /s49/s48/s32/s116/s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53 /s53/s46/s48/s45/s54/s45/s53/s45/s52/s45/s51/s45/s50/s45/s49 /s40/s98/s41 /s72/s50 /s72/s52 /s76/s50 /s32/s32/s108/s111/s103 /s49/s48/s32/s124 /s124 /s108/s111/s103 /s49/s48/s32/s116 FIG. 4: (color online) Lyapunov exponents λof six PN approaches.9 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52 /s40/s97/s41 /s72/s51 /s72/s49/s76/s49 /s32/s32/s70/s76/s73 /s108/s111/s103 /s49/s48/s32/s116/s49/s46/s48 /s49/s46/s53 /s50/s46/s48 /s50/s46/s53 /s51/s46/s48 /s51/s46/s53 /s52/s46/s48 /s52/s46/s53/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48 /s40/s98/s41 /s76/s50/s72/s52/s72/s50 /s32/s32/s70/s76/s73 /s108/s111/s103 /s49/s48/s32/s116 FIG. 5: (color online) Fast Lyapunov indicators (FLIs) of si x PN approaches. /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53 /s76/s49/s40/s97/s41 /s32/s32/s70/s76/s73 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s51/s54/s57/s49/s50/s49/s53/s49/s56 /s72/s49/s40/s98/s41 /s32/s32/s70/s76/s73 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53/s51/s48/s51/s53/s52/s48/s52/s53 /s40/s99/s41/s32/s32/s32/s32/s32/s32/s76/s50 /s32/s32/s70/s76/s73 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s49/s48/s50/s48/s51/s48/s52/s48/s53/s48/s54/s48/s55/s48 /s40/s100/s41 /s72/s50 /s32/s32/s70/s76/s73 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s53/s49/s48/s49/s53/s50/s48/s50/s53 /s40/s101/s41/s32/s32/s32/s32/s72/s51 /s32/s32/s70/s76/s73 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s50/s52/s54/s56/s49/s48/s49/s50/s49/s52 /s40/s102/s41/s32/s32/s32/s32/s32/s72/s52 /s32/s32/s70/s76/s73 FIG. 6: (color online) Dependence of FLI on the mass ratio βfor each approach.

2106.02286v2.Efficient_conversion_of_orbital_Hall_current_to_spin_current_for_spin_orbit_torque_switching.pdf

1 Efficient conversion of orbital Hall current to spin current fo r spin-orbit torque switching Soogil Lee1,†, Min-Gu Kang1,†, Dongwook Go2,3, Dohyoung Kim1, Jun-Ho Kang4, Taekhyeon Lee4, Geun-Hee Lee4, Nyun Jong Lee5, Sanghoon Kim5, Kab-Jin Kim4, Kyung-Jin Lee4, and Byong-Guk Park1,* 1 Department of Materials Scien ce and Engineering and KI for Nanocentury, KAIST, Daejeon 34141, Korea 2Peter Grünberg Institut and Institute for Advanc ed Simulation, Forschungszentrum Jülich and JARA, 52425 Jülich, Germany 3Institute of Physics, Johannes Gutenber g University Mainz, 55099 Mainz, Germany 4Department of Physics, KAIST, Daejeon 34141, Korea 5Department of Physics, Unive rsity of Ulsan, Ulsan 44610, Korea †These two authors equally c ontributed to this work. *Correspondence to: bgpark @kaist.ac.kr (B.-G.P.) 2 Abstract Spin Hall effect, an electric generation of spin current, allow s for efficient control of magnetization. Recent theory revea led that orbital Hall effect creates orbital current, which can be much larger than sp in Hall-induced spin current. H owever, orbital current cannot directly exert a torque on a ferromagnet, requiring a co nversion process from orbital current to spin current. Here, we report two effective methods of the conversion through spin-orbit coupling engineering, which allows us to una mbiguously demonstrate orbital-current-induced spin torque, or orbital Hall torque. We find that orbital Hall torque is greatly enhanced by introducing either a rare-earth f erromagnet Gd or a Pt interfacial layer with strong sp in-orbit coupling in Cr/ferroma gnet structures, indicating that the orbital current generated in Cr is efficiently convert ed into spin current in the G d o r P t l a y e r . F u r t h e r m o r e , w e s h o w t h a t t h e o r b i t a l H a l l t o r q ue can facilitate the reduction of switching current of perpendicular magnetization i n spin-orbit-torque-based spintronic devices. Introduction Spin Hall effect (SHE) that creates a transverse spin current b y a charge current in a non- magnet (NM) with strong sp in-orbit coupling (SOC)1 has received much attention because the resulting spin-orbit torque (SOT) offers efficient control of m agnetization in NM/ferromagnet (FM) heterostructures of various spintronic devices2–9. Similar to the SHE, the orbital Hall effect (OHE) generates an orbital current, a flow of orbital an gular momentum10–13. The OHE has distinctive features compared to the SHE; first, the OHE or iginates from momentum-space orbital textures, so it universally occurs in multi-orbital sys tems regardless of the magnitude of 3 SOC12. Thus, non-trivial orbital current can be generated even in 3 d transition metals with weak SOC13. Second, theoretical calculatio ns show that orbital Hall condu ctivity is much larger than spin Hall conductivity in many materials, including those commo nly used for SOT such as Ta and W10,11,13. This suggests that the spin torque caused by the OHE, or orbi tal Hall torque (OHT)14,15, can be larger than the SHE-induc ed spin torque, enhancing the spin-torque efficiency in spintronic devices. Recently, several experimenta l reports have claimed that significant SOT observed in FM/Cu/oxide structures, in which th e SHE is known to be negligible, is of orbital current origin16–18. However, there seems to be no consensus on whether those results provide evidence of OHT. One issue is that there is no exchange coupling between orbital angular momentum ( L) and local magnetic moment, and thus the orbital current canno t directly give a torque on magnetization. To make the OHT exert on the local magnetic moment of the FM, the L must be converted to the spin angular momentum ( S)14,15. Therefore, finding an efficient method of “ L-S conversion” is crucial to utilizing the OHT for the manipulati on of the magnetization direction. In this Article, we experimentally demonstrate two effective L-S conversion techniques of engineering SOC of either an FM o r an NM/FM interface. We emplo y Cr as an orbital current source material because it has been theoretically predicted tha t C r h a s a l a r g e o r b i t a l H a l l conductivity 𝜎ୌେ୰ of ~8,200 ( ħ/e)(ꞏcm)-1 while having a relatively small spin Hall conductivity 𝜎ୗୌେ୰ o f 130 (ħ/e)(ꞏcm)-1 with the opposite sign13. Here, ħ is the reduced Planck constant and e is the electron charge. For the Cr/FM bilayers, overall charge -to-spin conversion efficiency, referred as effective spin Hall angle 𝜃ୗୌୣ, is expressed as14 𝜃ୗୌୣൌ ሺ2𝑒/ℏሻሺ𝜎ୗୌେ୰𝜎ୌେ୰𝜂ିௌሻ/𝜎 ௫௫େ୰, (1) where 𝜎௫௫େ୰ is the electrical conductivity of Cr and 𝜂ିௌ i s t h e L-S conversion coefficient. 4 Here, we assume perfect transmission ( 𝑇୧୬୲=1) of both spin and orb ital currents through the Cr/FM interface. Note that the second term on the right side of Eq. (1) corresponds to the OHE contribution to 𝜃ୗୌୣ, which depends on the magnitude and sign of 𝜂ିௌ. We demonstrate how to achieve a large 𝜂ିௌ by engineering SOC of an FM or an NM interfacial layer. First, we employ a rare-earth FM of Gd with strong SOC, which increases t he OHT in Cr/Gd heterostructures by ten times com pared to that in Cr/Co heteros tructures, indicating that the orbital current generated in Cr is efficiently converted to spi n current in the FM Gd layer. Second, we modify the Cr/Co 32Fe48B20 (CoFeB) interface by inse rting a 1 nm Pt layer to facilitate L-S conversion. This leads to an enhancement in OHT, allowing us t o demonstrate OHT-induced magnetization switching of perpendicular magnetizat ion in Cr/Pt/CoFeB heterostructures. Since the OHE is expected to occur generally in various materials, our results demonstrating the significant OHT generated through the L-S conversion techniques broaden the scope of material engineerin g to improve spin-torque switch ing efficiency for the development of low-pow er spintronic devices. Results and Discussion Orbital Hall torque generated by orbital current in Cr To demonstrate the OHE in Cr and associated OHT, we investigate the current-induced spin- torque in Cr/FM heterostructures for two different FMs of Co an d Ni. Figures 1a,b illustrate the role of 𝜂ିௌ i n 𝜃ୗୌୣ of the Cr/FM samples, where 𝑆ୗୌ is the spin angular momentum generated by SHE and 𝑆ୌ is the spin angular momentum co nverted from orbital angular momentum due to OHE ( 𝐿ୌ). Note that we assume that Ni has a greater 𝜂ିௌ than that of Co ( 𝜂ିௌ୧𝜂ିௌେ୭ሻ because 𝜎ୗୌ୧ is an order of magnitude larger than 𝜎ୗୌେ୭ 19,20. This is also 5 supported by a recent first principle calculation demonstrating that the W/Ni bilayer exhibits a positive 𝜃ୗୌୣ despite the negative 𝜎ୗୌ o f W15. This is attributed to the increased orbital current contribution to 𝜃ୗୌୣ by the large and positive 𝜂ିௌ୧. Figure 1a shows the case of a Cr/Co bilayer with 𝜂ିௌେ୭~0, where 𝑆ୗୌ is dominant and thus 𝜃ୗୌୣ is mainly determined by 𝜎ୗୌେ୰ of negative sign. On the othe r hand, for the Cr/Ni bilayer ha ving sizable 𝜂ିௌ୧ , non- negligible 𝑆ୌ caused by the conversion of 𝜎ୌେ୰ contributes to 𝜃ୗୌୣ (Fig. 1b). Since 𝑆ୌ i s a positive value ( 𝜂ିௌ୧0 & 𝜎ୌେ୰0 ሻ , opposite to 𝑆ୗୌ, 𝜃ୗୌୣ of the Cr/Ni heterostructures becomes positive when the magnitude of 𝑆ୌ is larger than that of 𝑆ୗୌ. To test whether the orbital current generated in Cr gives rise to OHT, we perform i n-plane harmonic Hall measurements of Co (3.0 nm)/Cr (7.5 nm) and Ni (2.0 nm)/Cr (7.5 nm) Hall-bar patterned samples (Fig. 1c). Figures 1d,e show representative 2nd harmonic Hall resistance ( 𝑅௫௬ଶன) versus azimuthal angle ( 𝜑 ) curves under different external magnetic fields ( 𝐵ୣ୶୲ ). 𝑅௫௬ଶனሺ𝜑ሻ is expressed as21, 𝑅௫௬ଶனሺ𝜑ሻൌሼ ሾ 𝑅ୌଵனሺ𝐵ୈ/𝐵ୣሻ𝑅 ∇்ሿcos𝜑 ሾ2𝑅ୌଵனሺ𝐵𝐵 ୣሻ/𝐵 ୣ୶୲ሿሺcosଷ𝜑െ cos𝜑ሻሽ , ( 2 ) where 𝑅ୌଵன and 𝑅ୌଵன are the 1st harmonic anomalous Hall and planar Hall resistances, respectively; 𝐵ୈ ሺ 𝐵 ሻ is the damping-like (field-like) effective field; 𝐵ୣ is the effective magnetic field, including the demagnetization field a nd anisotropy field of FM; 𝑅∇் is the thermal contributions, and 𝐵ୣ i s t h e c u r r e n t - i n d u c e d O e r s t e d f i e l d . T h e 𝑅ୌଵன and 𝑅ୌଵன data are shown in Supplementary Note 1. Figure 1f shows the cos𝜑 component of 𝑅௫௬ଶன divided by 𝑅ୌଵன [𝑅ୡ୭ୱఝଶன/𝑅ୌଵன] as a function of 1/𝐵 ୣ for the two FM/Cr samples, of which the slope represents BDLT and associated 𝜃ୗୌୣ. We find that the Co/Cr sample shows a negative 6 slope, and thus a negative 𝜃ୗୌୣ . This is consistent with the negative 𝜎ୗୌେ୰ reported both theoretically and experimentally22–24. In contrast, the Ni/Cr sample exhibits a positive slope, indicating a positive 𝜃ୗୌୣ. The sign reversal of 𝜃ୗୌୣ in the Ni/Cr sample is attributed to the increased contribution of the o rbital current in Cr by the L-S conversion in Ni ( 𝜎ୌେ୰𝜂ିௌ୧0). Note that the ሺcosଷ𝜑െc o s 𝜑 ሻ component of 𝑅௫௬ଶன divided by 𝑅ୌଵன , representing 𝐵 𝐵ୣ, of the Ni/Cr sample is larger than that of the Co/Cr sample ( Supplementary Note 2). This might also be related to the incr eased orbital current in the N i/Cr sample, which, however, should be clarified through f urther investigation. To verify whether the orbital current in Cr is the main cause o f the measured torque, we perform two control experiments. First, we investigate the role of FM in determining 𝜃ୗୌୣ b y measuring the 𝑅ୡ୭ୱఝଶன/𝑅ୌଵன of the Co (3 nm)/Pt (5 nm) and Ni (2 nm)/Pt (5 nm) structures, in which Cr is replaced by Pt, which has positive 𝜎ୗୌ୲ and 𝜎ୌ୲10,11,13,25. Figure 1f shows positive slopes and corresponding positive 𝜃ୗୌୣ’s for both the FM/Pt samples, indicating that the sign change of 𝜃ୗୌୣ in the FM/Cr samples is not due to the FM layer itself20. Second, we examine the interfacial contributions26–31 to 𝜃ୗୌୣ by measuring the Cr thickness ( tCr) dependence of the damping-like torque efficiency, 𝜉ୈ ൌ ሺ2𝑒/ℏሻሺ𝑀 ୗ𝑡𝐵ୈ/𝐽େ୰ሻ , f o r t h e F M / C r s a m p l e s . Here, MS is the saturation magnetization, tFM is the FM thickness, and JCr is the current density flowing in Cr (Supplementary Note 3). If the positive 𝜃ୗୌୣ of the Ni/Cr samples is due to the interfacial effect, DLT decreases with increasing tCr and eventually changes its sign to negative for thicker tCr’s where bulk Cr with negative 𝜎ୗୌେ୰ dominates. However, this is not the case, as shown in Fig. 1g; for both FM/Cr samples, the magnitude of DLT increases with tCr, while maintaining its sign unchanged, which demonstrates that there i s no significant interfacial contribution to 𝜃ୗୌୣ in the FM/Cr samples. These re sults corroborate that the OHE in Cr 7 primarily governs the 𝜃ୗୌୣ of the FM/Cr samples, providing an excellent pla tform to study L- S conversion engineering. Efficient L-S conversion through rare-earth ferromagnet Gd We now present two techniques to enhance the 𝜂ିௌ of the Cr/FM structures. First, we introduce a rare-earth FM Gd, wh ich is expected to have a large 𝜂ିௌ due to its strong SOC32,33. Figure 2a illustrates the L-S conversion process in Cr/Gd heterostructures, where 𝜂ିௌୋୢ i s negative because of its negative spin Hall angle34. In this case, 𝑆ୌ due to the orbital current (𝜂ିௌୋୢ𝜎ୌେ୰<0) is in the same direction as 𝑆ୗୌ (𝜎ୗୌେ୰<0), so they add up constructively with each other. This would result in enhanced 𝜃ୗୌୣ in the Cr/Gd heterostructure compared to the Cr/Co heterostructure. To verify this idea, we prepare Hall-bar patterned samples of G d (10 nm)/Cr (7.5 nm) and Co (10 nm)/Cr (7.5 nm) structures and conduct in-plane harmonic Hall measurements. Note that the measurements are performed at 10 K to avoid any side e ffects due to the large difference in Curie temperatures between Gd (~293 K) and Co (~1 ,400 K). Figures 2b and 2c show the 𝑅௫௬ଶனሺ𝜑ሻ data measured under different Bext’s of the Gd/Cr and Co/Cr samples, respectively, which are well described by Eq. (2) and are repre sented by solid curves. The 𝑅ୌଵன and 𝑅ୌଵன data are shown in Supplementary Note 1. Figure 2d shows 𝑅ୡ୭ୱఝଶன /𝑅ୌଵன versus 1/𝐵 ୣ for the Gd/Cr and Co/Cr samples. We find two interesting point s; first, both samples exhibit negative slopes, indicating 𝜃ୗୌୣ൏0. Second, the Gd/Cr sample has a much larger slope or 𝐵ୈ than that of the Co/Cr sample. The estimated DLT of the Gd/Cr sample is 0.21±0.01, which is about ten times greater than that of the Co /Cr sample ( 0.018±0.002) (Supplementary Note 3). Note that DLT of the Gd/Cr samples increases with the tCr 8 (Supplementary Note 4), indicating that DLT originates from the bulk Cr bulk, which is the orbital current in Cr. The large enhancements of DLT o r 𝜃ୗୌୣ demonstrate that the OHT contribution can be increase d by introducing FMs with large 𝜂ିௌ. Magnetization switching by efficient L-S conversion through Pt interfacial layer We next demonstrate another L-S conversion technique that modi fies the NM/FM interface by inserting a Pt layer. This method has the advantage that it can be easily incorporated into perpendicularly magnetized CoFeB/MgO structures, which is a bas ic component of various spintronic devices35–37. Figure 3a illustrates the conversion process in a Cr/Pt/CoFeB structure, where the 𝐿ୌ originating from Cr is converted to 𝑆ୌ in the Pt layer. Since Pt has a positive 𝜂ିௌ୲ due to positive 𝜎ୗୌ୲ , 𝑆ୌ would be positive ( 𝜎ୌେ୰𝜂ିௌ୲0 ) , w h i l e 𝑆ୗୌ is negative (𝜎ୗୌେ୰൏0). Thus, the OHT due to 𝑆ୌ is the opposite of the spin Hall torque due to 𝑆ୗୌ. To examine the effect of Pt insertion on OHT, we perform current-i nduced magnetization switching experiments as schema tically illustrated in Fig. 3b. Figure 3c shows switching curves as a function of pulse current density ( Jpulse) for Cr (10.0 nm)/Pt (0 or 1.0 nm)/CoFeB (0.9 nm)/MgO (1.6 nm) Hall- bar patterned sample s. Note that an in-pl ane magnetic field Bx of +20 mT is applied along the current direction for deterministic swi tching of the perpendicular magnetization2,4,38. The Cr/CoFeB sample shows a c ounterclockwise switching curve consistent with negative 𝜃ୗୌୣ, caused primarily by the SHE in Cr. Interestingly, the switchi ng polarity is reversed by introduci ng a Pt (1 nm) insertion layer . The clockwise switching curve of the Cr/Pt/CoFeB sampl e corresponds to positive 𝜃ୗୌୣ, which is the expected sign in the OHT scenario (Fig. 3a). Note that the sign of BDLT and associated 𝜃ୗୌୣ for Cr (5.0 nm)/(Pt 0 or 1 nm)/CoFeB (0.9 nm)/MgO (1.6 nm) samples, obtained from the perp endicular harmonic 9 measurement (Supplementary Note 5), is also consistent with is the switching result. The sign change in 𝜃ୗୌୣ can be caused by the inserted Pt itself with positive 𝜎ୗୌ୲. To rule out this possibility, we investigate the tCr dependence of the curre nt-induced magnetization switching for the samples, where tCr ranges from 2.0 nm to 12.5 nm. Figure 3d shows the switching efficiency39,40 𝜉ୗ ሾൌ ሺ2𝑒/ℏሻሺ𝑀 ୗ𝑡େ୭ୣ 𝐵/𝐽ୗሻሿ as a function of tCr. Here, 𝑡େ୭ୣ is the CoFeB thickness, 𝐵 is the domain wall propagation field, and 𝐽ୗ is the switching current dens ity (Supplementary Note 6). Interestingly , we find that the magnitude of 𝜉ୗ for both samples increases with increasing tCr, while its sign remains unchanged for all tCr’s used in this study. Since the contribution of the spin curre nt generated from Pt to 𝜃ୗୌୣ i n the Cr/Pt/CoFeB structures w ill decrease with increasing tCr, the similar thickness dependence of 𝜉ୗ indicates that the 𝜃ୗୌୣ of both samples predominantly originates from the Cr layer, not from the Pt interfacial layer ; the SHE and OHE in Cr are th e main sources of 𝜃ୗୌୣ for the Cr/CoFeB and Cr/Pt/CoFeB samples, respectively. These results demonstrate that the OHT can be effectively modified by interface SOC engineering and is cap able of switching the perpendicular magnetization. In conclusion, we experimentally demonstrate non-trivial OHT, s pin torques originating from the orbital current in Cr, by introducing two effective wa ys of orbital-to-spin ( L-S) conversion, which is a key ingr edient of OHT generation. First, we employ a rare-earth FM of Gd having a larger L-S conversion efficiency than that of conventional 3 d FMs. This greatly improves the SOT efficiency of the Cr/Gd bilayers compared to t hat of the Cr/Co bilayers. Second, we introduce a Pt interfacial layer in the Cr/CoFeB bil ayers to facilitate L-S conversion. This allows the OHT to control the perpendicular magnetization in the Cr/Pt/CoFeB heterostructures. Since orbital currents can occur in various m aterials regardless of the SOC 10 strength, our results provide a unique strategy based on orbita l currents to develop material systems with enhanced SOT efficiency. 11 Methods Film preparation and Hall-bar fabrication. Bilayers of FM (Co, Ni)/Cr, FM (Co, Ni)/Pt, Gd /Cr, and Co/Cr for harmonic measurements were deposited on Si/S iO2 or Si/Si 3N4 substrates using DC magnetron sputtering und er a base pressure of <2.6×10-5 Pa, while Cr/CoFeB and Cr/Pt/CoFeB structures for switc hing experiments were deposited on a highly resistive Si substrate using DC and RF magnetron sputtering under a base pre ssure of <4.0 ൈ10-6 Pa. An underlayer of Ta (1 nm)/AlO x (2 nm), or Ta (1.5 nm) layers were used to obtain smooth roughness; a capping layer of Ta (2~3 nm) was used to prevent f urther oxidation. All metallic layers and the MgO layer were g rown with a working pressure of 0.4 Pa and a power of 30 W at room temperature. The AlO x layer was formed by deposition of an Al layer and subsequent plasma oxidation with an O 2 pressure of 4.0 Pa and a power of 30 W for 75 s. Hall-bar-patt erned devices with widths of 5, 10, or 15 μm were defined using photo lithography and Ar ion-milling. Spin-orbit torque characterization. In-plane harmonic measurement with AC current (frequency of 11 Hz) was performe d to evaluate the spin-orbit t orque of the heterostructures. Both 𝑅௫௬ଵன and 𝑅௫௬ଶன were recorded by two lock-in amplifiers at the same time while varying the azimuthal angle ( 𝜑) under a constant external field Bext and a current density Jx of 1×1011 A/m2. Current-induced magnetization switching measurements. Magnetization switching experiments were conducted by appl ying a current pulse (pulse w idth of 30 μs) with a constant external magnetic field ( Bx) of +20 mT. The magnetization st ate was checked by anomalous Hall resistance ( RAHE) after applying the current pulse. 12 References 1. Sinova, J., Valenzuela, S. O ., Wunderlich, J., Back, C. H. & Jungwirth, T. Spin Hall effects. Rev. Mod. Phys. 87, 1213–1260 (2015). 2. Miron, I. M. et al. Perpendicular switching of a si ngle ferromagnetic layer induce d by in-plane current injection. Nature 476, 189–193 (2011). 3. 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We also thank Byoung Kook Kim at the KAIST Analysis Center for Research Advancement (KARA) for his support on the magnetic pro perties measurement. This work was supported by the National Research Foundation of Korea (2015M3D1A1070465, 2020R1A2C2010309, 2020R1A2C3013302). Author contributions The study was performed under the supervision of B.-G.P. S.L. a nd M.-G.K. fabricated samples and conducted the in-plane harmoni c measurements and spin-orbit torque switching experiments with the help of D.K., J.-H. K., T.L., G.-H.L., N.J .L., and S.K. S.L., M.-G.K., and B.-G.P. performed data analysis w ith the help of D.G., K.-J.K., and K.-J.L. S.L ., M.-G.K., and B.-G.P. wrote the manuscript with the help of all authors. Competing interests Authors declare no co mpeting interests. Data availability The data that support the findings of this study are available from the corresponding author upon reasonable request. 17 Figure 1. Orbital-current-induced spin torque in Cr/ferromagnet heterostructures. a,b, Schematic illustrations of the angular momentum transfer by spin current (red arrows) and orbital current (blue arrows) for Co ( a) and Ni ( b) FMs. The spin (orbital) angular momentum is represented by S (L). The source of S and L is marked by the subscript of SH (OH) for SHE (OHE). 𝜼𝑳ି𝑺𝐂𝐨ሺ𝐍𝐢ሻ is the orbital-to-spin conversion efficiency of Co (Ni). c, The 𝑹𝒙𝒚𝟐𝛚ሺ𝝋ሻ measurement geometry for a Hall-bar sample. 𝝋 i s the azimuthal angle of the external magnetic field ( Bext) with respect to the current direction. d , e , Azimuthal angle ( ) dependent 2nd harmonic Hall resistance, 𝑹𝒙𝒚𝟐𝛚ሺ𝝋ሻ, under different Bext of Co (3.0 nm)/Cr (7.5 nm) ( d) and Ni (2.0 nm)/Cr (7.5 nm) ( e) samples. The solid lines are fitting curves using Eq. 2. Each 𝑹𝒙𝒚𝟐𝛚ሺ𝝋ሻ curve of Co/Cr ( d) and Ni/Cr ( e) is shifted by a y-axis offset to clearly show the Bext dependence. f, 𝑹𝐂𝐎𝐒𝝋𝟐𝛚/𝑹𝐀𝐇𝐄𝟏𝛚 as function of 1/ Beff of Ni/Cr (red), Co/Cr (li ght-green), Ni/Pt (magenta), and Co/Pt (green) samples. Each solid line is a linear fitting line. g, Cr thickness ( tCr) dependent damping-like torque efficiency ( DLT) of Ni/Cr (red) and Co/Cr (light-green) samples, where tCr ranges from 2 to 12.5 nm. The DLT’s of the reference Ni/Pt (magenta) and Co/Pt (green) samples a re included for comparison. Lines are guide to eyes. All measurements are c onducted at 300 K. 18 Figure 2. Efficient L-S conversion by rare-earth Gd with strong SOC. a, Schematic illustration of the orbital-to-spin conversion by 𝜼𝑳ି𝑺𝐆𝐝 in the Cr/Gd heterostructure. b,c, 𝑹𝒙𝒚𝟐𝛚ሺ𝝋ሻ under different Bext of Gd (10 nm)/Cr (5 nm) ( b) and Co (10 nm)/Cr (5 nm) ( c) samples. Each 𝑹𝒙𝒚𝟐𝛚ሺ𝝋ሻ of Gd/Cr ( b) and Co/Cr (c) is shifted by a y-axis offset to clearly show Bext dependence. The solid lines are fitting curves using Eq. (2). d, 𝑹𝐂𝐎𝐒𝝋𝟐𝛚/𝑹𝐀𝐇𝐄𝟏𝛚 v s . 1 / Beff of Gd/Cr (brown) and Co/Cr (blue-green) samples. Each solid li ne is a linear fitting line. All measurements are conducted at 10 K. 19 Figure 3. OHT-driven magnetization switching by in Cr/Pt/CoFeB heterostructures. a, Schematic illustration of the orbital-to-spin conversion by 𝜼𝑳ି𝑺𝐏𝐭 in the Cr/Pt/CoFeB heterostructure. b, The magnetization switching measurem ent geometry for a Hall-bar sam ple, in which an in-plane magnetic field Bx is applied along the pulsed current Jpulse. c, Magnetization switching curves of Cr (10.0 nm)/ CoFeB (0.9 nm) and Cr (10.0 nm)/Pt (1.0 nm)/CoFeB (0.9 nm) samp les under a Bx of +20 mT. The green and magenta symbols represent to the samples without and with the Pt insertion layer, respectively. Open (closed) symbols indicate magnetization swit ching from down-to-up (up-to-down) directions. The switching polarity is indicated by an arrow in the center of curve. d, tCr-dependent switching efficiency ( 𝝃𝐒𝐖) of the Cr/CoFeB (green) and Cr/Pt/CoFeB (magenta) samples. Li nes are guide to eyes. All measurements are conducted at 300 K.

1403.2759v2.Renormalization_of_anticrossings_in_interacting_quantum_wires_with_Rashba_and_Dresselhaus_spin_orbit_couplings.pdf

arXiv:1403.2759v2 [cond-mat.mes-hall] 31 May 2014Renormalization of anticrossings in interacting quantum w ires with Rashba and Dresselhaus spin-orbit couplings Tobias Meng,1Jelena Klinovaja,2and Daniel Loss1 1Department of Physics, University of Basel, Klingelbergst rasse 82, CH-4056 Basel, Switzerland 2Department of Physics, Harvard University, Cambridge, Mas sachusetts 02138, USA (Dated: June 3, 2021) We discuss how electron-electron interactions renormaliz e the spin-orbit induced anticrossings between different subbands in ballistic quantum wires. Depe nding on the ratio of spin-orbit cou- pling and subband spacing, electron-electron interaction s can either increase or decrease anticrossing gaps. When the anticrossings are closing due to a special com bination of Rashba and Dresselhaus spin-orbit couplings, their gap approaches zero as an inter action dependent power law of the spin- orbit couplings, which is a consequence of Luttinger liquid physics. Monitoring the closing of the anticrossings allows to directly measure the related renor malization group scaling dimension in an experiment. If a magnetic field is applied parallel to the spi n-orbit coupling direction, the anticross- ings experience different renormalizations. Since this diff erence is entirely rooted in electron-electron interactions, unequally large anticrossings also serve as a direct signature of Luttinger liquid physics. Electron-electron interactions furthermore increase the sensitivity of conductance measurements to the presence of anticrossing. PACS numbers: 73.21.Hb,71.70.Ej, 71.10.Pm I. INTRODUCTION The presence of the electronic spin degree of free- dom in nanoscale and mesoscale semiconductor systems has driven a large amount of research during the past decades. Prominent examples range from spintronics,1 over spin qubits,2,3to topological insulators, and the quantum spin Hall effect.4–7One important effect is thereby the coupling between the electron spin and its motion. In semiconductor systems, this spin-orbit cou- pling is usually distinguished into the Rashba coupling,8 generated by structural inversion symmetry breaking, and the Dresselhaus coupling9due to bulk inversion symmetry breaking. In some one-dimensional systems, such as carbon nanotubes, spin-orbit coupling can lead to helical modes, modes in which opposite spins prop- agate in opposite directions.10In other systems, such as quantum wires and nanoribbons, only the combina- tion of spin-orbit coupling and an externally applied magnetic field allows for the generation of helical11–15 and fractional helical16,17phases. In conjunction with proximity-induced superconductivity, these systems have been proposed to host Majorana zero modes18,19and parafermionic bound states,16,20respectively. In quantum wires with multiple subbands, the spin- orbit coupling gives rise to both intra- and inter-subband couplings. While the intra-subband spin-orbit coupling results in the usual relative shift of the dispersions for spin up and spin down in momentum space, the inter- subband spin-orbit coupling lifts crossings of different bands of opposite spin,21–27much like the magnetic field does in the single subband case.11The lifting of the inter-subband crossings furthermore leads to a regime of (partial) spin polarization of the current inside the wire,23,26–28which resembles the quasi-helical regime in single subband wires. In a bosonized language, thepartially gapped quasi-helical regime in single subband Rashba nanowires can be understood as a consequence of the magnetic field being a relevant perturbation in the renormalization group (RG) sense with respect to the gapless quantum wire fixed point. During the RG flow, electron-electron interactions renormalize the magnetic field to stronger values.29In the present work, we ana- lyze to what extent electron-electron interactions renor- malize the analogous inter-subband anticrossing gaps in multi-subband quantum wires with spin-orbit couplings. We find that depending on system parameters, the inter- subband anticrossing gaps either grow or shrink in the presence of electron-electron interactions. The paper is organized as follows. First, we define a two subband model in Sec. II, and perform an RG anal- ysis in Sec. III. Section IV discusses how the monitoring of anticrossings as a function of an applied electric field allowsto measure Luttinger liquid power laws. In Sec. V, we give an outlook on anticrossings occurring at the bot- tom of the second subbands, which are not captured by our Luttinger liquid approach. We close by commenting on the effects of an external magnetic field, including its interplay with the anticrossings, in Sec. VI, and discuss the sensitivity oftransportmeasurementsto the presence of the anticrossings in an interacting wire in Sec. VII. II. EXPERIMENTAL SETUP AND MODEL In this work we focus on ballistic quantum wires defined in two-dimensional electron gases (2DEGs) by means of electrostatic gates since these systems offer a large in situ tunability (see Fig. 1). Our analysis is, however, equally applicable to other types of quantum wires with spin-orbit couplings. Together with a gate modulating the electron density in the wire, the setup2 FIG. 1: The proposed experimental setup consists of a two- dimensional electron gas (2DEG), in which the electrons are confined to a quasi one-dimensional wire region (red). The confinement is due to electrostatic gates placed atop the 2DEG (green), and depleting it outside the wire region (ligh t yellow). By modulating the electrostatic potential of the gates, the width of the wire can be changed in-situ. The elec- tron density inside the wire can be controlled byan addition al gate (not shown). depicted in Fig. 1 allows one to control the number of occupied subbands, their fillings, and the energy differ- ence between the band bottoms. Concretely, we consider 2DEGs defined in InAs heterostructures because of their relativelylargespin-orbitcoupling. Ofparticularinterest are samples exhibiting both sizable Rashba and Dressel- haus spin-orbit couplings. While the latter is fixed for a given sample, the Rashba spin-orbit coupling can be controlled by changing an applied electric field.30,31In particular, this allows us to access the regime of partial compensation between the two couplings32,33discussed in Sec. IV. Using units of /planckover2pi1= 1, we model this experimental setup by the Hamiltonian H=H2D+Hint.+HR+HD+Hconf., (1) H2D=/integraldisplay d2rΨ†(r)/parenleftBigg −∂2 x−∂2 y 2m−µ/parenrightBigg 12×2Ψ(r), Hint.=/integraldisplay d2r/integraldisplay d2r′U(r−r′)ρ(r)ρ(r′), HR=α/integraldisplay d2rΨ†(r) (iσx∂y−iσy∂x) Ψ(r), HD=β/integraldisplay d2rΨ†(r) (iσx∂x−iσy∂y) Ψ(r), where Ψ( r) = (c↑(x,y),c↓(x,y))Tis the vector of an- nihilation operators cν(x,y) for electrons of spin νat position r= (x,y)T. The electrons have an effective massm, and their chemical potential is denoted by µ. The total charge densities ρ(r) =/summationtext νc† ν(r)cν(r) inter- act via the (screened) Coulomb repulsion U(r−r′),α denotes the Rashba and βthe Dresselhaus spin-orbit coupling strength, and σiare the Pauli matrices. For a heterostructre that has been grown along the crystal-lographic [001] direction, the xandydirections corre- spond to the crystallographic [100] and [010] directions, respectively.34 To make the possible partial compensation of Rashba and Dresselhaus spin-orbit couplings more apparent, it is useful to perform a change of coordinates both in real space and spin space by introducing x′=−(x−y)/√ 2, y′=−(y+x)/√ 2,σx′= (σx−σy)/√ 2, andσy′= (σy+ σx)/√ 2. This transformation yields HSOI=HR+HD =−(α+β)/integraldisplay d2r′Ψ†(r′)iσx′∂y′Ψ(r′) (2) +(α−β)/integraldisplay d2r′Ψ†(r′)iσy′∂x′Ψ(r′). The first term in Eq. (2), proportional to ( α+β), cou- ples different subbands due to its derivative in the y′- direction, while the second term, proportional to ( α−β), corresponds to a spin-orbit coupling within a given sub- band. Thislattertermcouldbe removedfromtheHamil- tonian by virtue of the gauge transformation cσ(x′,y′) = eiσzx′kSOc′ σ(x′,y′) at the expense of introducing an addi- tional phase factor for the inter-subband term.29In spite of simplicity of this gauge transformation, we do not use itinthepresentwork. Thetransversalconfinementofthe electrons due to electrostatic gates, finally, is modeled by a harmonicpotential. We focus on a confinement parallel to thex′-axis, which is described by the Hamiltonian Hconf.=/integraldisplay d2r′Ψ†(r′)1 2mΩ2y′212×2Ψ(r′).(3) This potential gives rise to electronic subbands whose transversal wave functions are harmonic oscillator eigen- states. The bottoms of the subbands are offset by an energy difference of δ12= Ω. A. Luttinger liquid description For simplicity, we focus on the case in which only the lowest two subbands are occupied, and neglect all higher subbands. The effects of higher subbands are similar to the mixing observed in the two subband model,21–27,35 and their neglect restricts our model to sufficiently low (Fermi) energies, such that the coupling to the third sub- band is not yet important. In order to analyze the effects of Coulomb repulsion, we derive a Luttinger liquid description of the system by first decomposing the two-dimensional fermionic opera- tors into operators acting within the different subbands n= 1,2;cσ(x′,y′)→cn,σ(x′). After a rotation in spin space,whichbringstheintra-subbandspin-orbitcoupling to a diagonal form, the kinetic energy and intra-subband3 spin-orbit couplings are described by the Hamiltonian H0=/integraldisplay dx′/summationdisplay σc† 1,σ(x′)/parenleftbigg−∂2 x′ 2m−µ/parenrightbigg c1,σ(x′) +/integraldisplay dx′/summationdisplay σc† 2,σ(x′)/parenleftbigg−∂2 x′ 2m+δ12−µ/parenrightbigg c2,σ(x′) (4) +/integraldisplay dx′/summationdisplay n,σ(α−β)σc† n,σ(x′)i∂x′cn,σ(x′), whereσ=↑,↓≡+1,−1. Next, we restrict the Hamil- tonian to low energy excitations close to the Fermi mo- menta±kFnin the lowest two subbands, which gives rise to left-moving and right-moving modes, cn,σ(x′)≈ eix′kFnRn,σ(x′) +e−ixkFnLn,σ(x′), and linearize the ki- netic energy around these momenta. Here, kFndenotes the Fermi momentum in the nth subband. In this ef- fective low energy description, the screened Coulomb in- teraction yields a number of matrix elements connect- ing the low energy modes. For our purpose, however, only the leading order terms corresponding to (approxi- mately) zero momentum transfer need to be kept track of. In particular, we have checked that additional sine- Gordon terms corresponding to, for instance, backscat- tering processes, are RG irrelevant, or less relevant than and competing with the sine-Gordon terms correspond- ing to inter-subband spin-orbit coupling. The Coulomb repulsion can thus be described by renormalized values of the velocities and Luttinger liquid parameters in the charge sectors of each of the bands, along with a term couplingthechargedensitiesinthetwobands.36Bosoniz- ing the effective low energy Hamiltonian,37we obtain the Luttinger liquid description H=H1+H2+HCoulomb 12+HSOI 12,(5) where Hi=/integraldisplaydx′ 2π/summationdisplay j=c,s/bracketleftbigguij Kij(∂x′φij)2+uijKij(∂x′θij)2/bracketrightbigg (6) describes the bosonic charge cand spin sexcitations in band one and two, propagating with effective velocities uij, and characterized by Luttinger liquid parameters Kij. In our approximation, the velocities and Luttinger liquid parameters are given by uij=vFi/Kij,Kic= [1+2U/(πvFi)]−1/2, andKis= 1 (Uis the zero momen- tum transfer Coulomb matrix element, and vFidenotes the Fermi velocity in band i). The fields φijrelate to the integrated charge and spin densities in band j, while the fields θijare proportional to the integrated charge and spin currents.37They obey the canonical commuta- tionrelations[ φij(x),θi′j′(x′)] =δii′δjj′(iπ/2)sgn(x′−x). TheLuttingerliquidtheoryisvalidatlengthscaleslarger than a short-distance cutoff a. We choose the bare value of this cutoff to be given by the lattice constant a0. The Coulomb repulsion between the bands yields the term HCoulomb 12=/integraldisplaydx′ 2π2U12(∂x′φ1c)(∂x′φ2c),(7)-0.5 0 0.5 1 1.5 -2 -1 0 1 2Energy [meV] Wave vector [107 m-1]1, ↑2, ↑ 1, ↓2, ↓FS (a) -0.5 0 0.5 1 1.5 -2 -1 0 1 2Energy [meV] Wave vector [107 m-1]1, ↑2, ↑ 1, ↓2, ↓(b) -0.5 0 0.5 1 1.5 -2 -1 0 1 2Energy [meV] Wave vector [107 m-1]1, ↑2, ↑ 1, ↓2, ↓BS (c) FIG. 2: Bandstructure of the two-subband quantum wire in the non-interactingcase and without interband couplings ( the labelsi,σindicate subband index and spin polarization). The spectra are calculated using Eqs. (1) and (2) after setting U= 0,α+β= 0, and α−β=α0. The red circles highlight the inter-subband crossings that are lifted in the presence of inter-subband spin-orbit coupling. ( a) Band structure calcu- lated for a large energy difference of δ12= 0.8meV between the bottoms of the first and second subbands, corresponding to a situation in which the intersubband spin-orbit couplin g induces forward scattering (FS). ( b) The case where the cross- ings occur at the bottoms of the second bands (with an offset ofδ12≈0.4meV). Finally, ( c) indicates that mall band off- sets, here δ12= 0.2meV, lead to interband backscattering (BS). Higher unoccupied bands are neglected. withU12= 2U/π. The inter-subband spin-orbit coupling HSOI 12, finally, lifts the crossings between subbands which would be present in the bandstructure otherwise, see Fig. 2. For this plot and in the remainder, we use the the effec- tive mass m= 0.0229mein terms of the electron rest massme, the InAs lattice constant a0= 6.0583˚A, and we consider a bare intra-subband spin-orbit coupling4 α−β=α0=kSO/mcorrespondingtoaspin-orbitlength ofk−1 SO= 127nm.34,38The value of the Coulomb matrix element Udepends on microscopic details, and in par- ticular on the screening length. We choose it such that the Luttinger liquid parameter in the first subband takes the experimentally realistic value39–41K1c= 0.65 when the chemical potential matches the energy of the crossing between the lower subbands 1 ,↑and 1,↓(in Fig. 2, this corresponds to µ= 0). In the Luttinger liquid picture, the inter-subband spin- orbit coupling corresponds to a sine-Gordon term. This term, in general, is rapidly oscillating in the space due to the finite momentum transfer in the scattering pro- cess and can, thus, be neglected.37Only when the chem- ical potential is tuned to (close to) the subband crossing points, as shown in Fig. 2, the sine-Gordon terms are non-oscillating (very slowly oscillating), and need to be kept. In this case, the inter-subband spin-orbit coupling corresponds to HSOI 12=/integraldisplaydx′ 2πa/parenleftBig α(1) 12cos/parenleftBig Ψf,b 1/parenrightBig +α(2) 12cos/parenleftBig Ψf,b 2/parenrightBig/parenrightBig , (8) where we have dropped the Klein factors37, and where α(1) 12=α(2) 12= (α+β)/radicalbig mδ12/2. The arguments of the sine-Gordon term HSOI 12depend on whether the (even- tually lifted) band crossing points occur between modes moving in the same or in opposite directions. In the for- mer case, HSOI 12encodesinter-subbandforwardscattering (indicated in Eq. (8) by the superscript f), see Fig. 2( a), while it correspondsto inter-subbandbackscatteringoth- erwise (superscript b), see Fig. 2( c). The interesting in- termediate case, in which the crossing occurs at the bot- tomoftheupperbandsasshowninFig.2( b), isnotacces- sible in our bosonized calculation. If the chemical poten- tial is close to the band bottom, the state of the system is sensitive to the band curvature which we neglect, how- ever. The electrons are furthermore strongly influenced by electron-electron interactions and interband pair tun- neling processes,42which may induce a partial gap in the system.43We therefore expect our analysis to hold only as long as the Fermi velocities in the second band aresuf- ficiently large. For concreteness, we restrict our analysis to the regime of Luttinger liquid parameters Kc2/greaterorsimilar0.5, and comment on the opposite regime in Sec. V. 1. Forward scattering In the case of forward scattering, the arguments in the sine-Gordon term in Eq. (8) read Ψf 1=φ1c−φ1s+θ1c−θ1s−φ2c−φ2s−θ2c−θ2s√ 2, (9a) Ψf 2=φ1c+φ1s−θ1c−θ1s−φ2c+φ2s+θ2c−θ2s√ 2. (9b)While Ψf 1and Ψf 2commute with each other, they do not commute with themselves, [Ψf k(x),Ψf l(x′)] = 2iπδklsgn(x′−x). This non-trivial commutation rela- tion encodes that although forward scattering lifts the inter-subband crossings, it cannot open up a gap. 2. Backscattering For backscattering, which occurs in the situation de- picted in Fig. 2( c), the arguments of the sine-Gordon term in Eq. (8) are given by Ψb 1=φ1c−φ1s+θ1c−θ1s+φ2c+φ2s−θ2c−θ2s√ 2, (10a) Ψb 2=φ1c+φ1s−θ1c−θ1s+φ2c−φ2s+θ2c−θ2s√ 2. (10b) Since backscattering does open up a (partial) gap in the spectrum, the fields Ψb 1and Ψb 2commute both with themselves and with each other. III. INTER-SUBBAND ANTICROSSINGS AND THEIR RENORMALIZATIONS The inter-subband spin-orbit coupling giving rise to the sine-Gordon terms in Eq. (8) lifts the band crossings shown in Fig. 2. In Fig. 3, we plot the bandstructures corresponding to α−β=α0, and an interband coupling α+β= 0.5α0, whileallotherparametersarechosenasin Fig. 2. In particular, we stick to the non-interacting case U= 0forthisfigure,suchthatthefermionicHamiltonian given in Eq. (1) can be diagonalized. In the remainder, we discuss how the spectra shown in Fig. 3 are renor- malized by electron-electron interactions as a function of the inter-subband spacing δ12, or the intra-subband spin- orbit coupling α−β, which are experimentally tunable by electrostatic gates. A. Perturbative RG analysis The renormalization of the inter-subband spin-orbit couplingsdue to the Coulomb repulsion can be addressed with a Luttinger liquid renormalization group (RG) analysis37of the sine-Gordonterms given in Eq. (8). The strength of the renormalizations depends on the ratio of kinetic energy and Coulomb repulsion. As can be in- ferred from Eq. (4), a modification of either the band off- setδ12or the intra-subband spin-orbit coupling ( α−β) not only allows to change between forward and backscat- tering, but also leads to modified values of the Fermi5 -0.5 0 0.5 1 1.5 -2 -1 0 1 2Energy [meV] Wave vector [107 m-1]FS (a) -0.5 0 0.5 1 1.5 -2 -1 0 1 2Energy [meV] Wave vector [107 m-1](b) -0.5 0 0.5 1 1.5 -2 -1 0 1 2Energy [meV] Wave vector [107 m-1]BS (c) FIG. 3: Bandstructure of the two-subband quantum wire in the non-interacting case for relatively large interband co u- plings (all parameters are chosen as in Fig. 2, except for α+β= 0.5α0). The band offsets are again given by δ12= 0.8meV in panel ( a),δ12≈0.4meV in panel ( b), and δ12= 0.2meV in panel ( c). velocities at the crossing point, vF2 vF1=/vextendsingle/vextendsingle/vextendsingle1−δ12 2m(α−β)2/vextendsingle/vextendsingle/vextendsingle 1+δ12 2m(α−β)2, (11) which is depicted in Fig.4. The modified Fermi velocities in turn affect the Luttinger liquid parameters Kic= [1+ 2U/(πvFi)]−1/2as depicted in Fig. 5, and thus the RG flow. The RG equations for the inter-subband spin-orbit couplings α(i) 12arederived in a real space RG scheme with a running short-distance cutoff a(b) =a0bby expanding the action corresponding to Eq. (6) to first order in α(i) 12, see Appendix A. Taking into account that the argument of the sine-Gordon term changes between forward scat- 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1vF2 / vF1 δ12 [meV]BS FS FIG. 4: Ratio of the Fermi velocities vF2/vF1at the crossing points (see Fig. 2). The dashed vertical line indicates the i n- terbandoffsetatwhichthebandcrossings occuratthebottom of the second band. For smaller δ12, the inter-subband spin- orbit coupling gives rise to backscattering (BS), while lar ger values of δ12lead to forward scattering (FS). The dotted lines delimit the excluded regime in which strong interactions in the second subband eventually modify the physics (we choose K2c= 0.5 as the criterion to distinguish weak from strong interactions). 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1Kic δ12 [meV]BS FS K1cK2c FIG. 5: Luttinger liquid parameters Kicin the charge sector of bandi= 1,2 if the chemical potential intersects the cross- ing points of the spectra shown in Fig. 2, as a function of the band offset δ12. For different band offsets, the crossing point occurs at different points of the bandstructure, which alter s the Fermi velocities. This in turn modifies the value of the Luttinger liquid parameters. Like in Fig. 4, the dotted vert i- cal lines (and the dotted horizontal line) indicate K2c= 0.5, the dashed vertical line indicates the transition point bet ween forward (FS) and backscattering (BS). tering and backscattering, we obtain the RG equation dα(i) 12 dln(b)= (1−g12)α(i) 12, (12) where the RG scaling dimension g12depends on the Lut- tinger liquid parameters set by the Coulomb interaction strength, and the velocities. Its calculation requires the diagonalization of the quadratic part of the bosonized Hamiltonian, which is detailed in Appendix A. Fig. 6 shows (1 −g12) as a function of the band offset δ12. We find that (1 −g12) is always negative for forward6 -0.4-0.2 0 0.2 0.4 0 0.2 0.4 0.6 0.8 1g12 [meV] δ12 [meV]BS FS FIG. 6: The parameter (1 −g12) as a function of the band off- setδ12(g12is the RG scaling dimension of the inter-subband spin-orbit couplings). As in Fig. 4, the vertical dashed lin e marksthetransitionbetweenforward (FS)andbackscatteri ng (BS), thevertical dottedlines indicate theregime inwhich the band crossing occurs close to the bottom of the upper bands. scattering, resulting in an effectively reduced interband spin-orbit coupling. Backscattering, on the other hand, is generally enhanced under RG. The scaling dimension g12, finally, grows (and, extrapolating our theory, is even pushed to RG irrelevant values g12>2) when the band crossings approach the bottom of the upper bands. This trend can be traced back to the ratio of Fermi velocities shown in Fig. 4. The more these Fermi velocities dif- fer, the harder it is to scatter, say, a fast electron from band one into the slowly moving modes of band two. This effect is known from Coulomb-drag setups,44–46and has also been analyzed for the RKKY interaction in two- subband quantum wires.36 B. Renormalized anticrossing gaps in the backscattering regime We now turn to the case where inter-subband spin- orbit coupling allows for backscattering, such that it re- sults in a partial gap. Assuming both finite size effects and finite temperature effects to be small, the RG flow is integrated until the running gap associated with the inter-subband coupling reaches the running band width (at high temperatures or in short wires, the RG flow is cut off by either the temperature or the wire length, re- spectively, and renormalization effects due to Coulomb interactions are less pronounced). The gap can be de- fined by the expansion of the cosine to second order.37At the end of the RG flow, the renormalized inter-subband spin-orbit coupling α(i) 12∗reads α(i) 12∗=α(i) 12b∗1−g12=vF2 a/parenleftBigg aα(i) 12 vF2/parenrightBigg1/(2−g12) ,(13) whereb∗2−g12=vF2/(aα(i) 12) is the RG scale at which the flow stops. The gap therefore scales as an interaction- dependent powerlawofthebareinter-subbandspin-orbit-0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 -1 -0.5 0 0.5 1Energy [meV] Wave vector [107 m-1](a) -0.15-0.1-0.05 0 0.05 0.1 0.15 0.2 0.25 -1 -0.5 0 0.5 1Energy [meV] Wave vector [107 m-1](b) FIG. 7: Bandstructureof the twosubbandquantumwire for a band offset of δ12= 0.1meV,α−β=α0, andα+β= 0.1α0. Panel (a) shows the unrenormalized bandstructure, panel ( b) the one renormalized by electron-electron interactions. coupling, α(i) 121/(2−g12), which only reduces to a linear de- pendence in the non-interacting case. We illustrate the renormalization for a band offset of δ12= 0.2meV, for a bare spin-orbit coupling of α+β= 0.1α0in Fig. 7, by feeding the renormalized values α(1) 12∗andα(2) 12∗back into the non-interacting theory, and find that the gap of the anticrossingsisrenormalizedtoroughlytwiceits bare value for the depicted set of parameters. The fact that interactions do not affect the anticrossings more strongly has two reasons. The value of 1 −g12is relatively small, and the bare couplings α(i) 12are large (the relative shift of the dispersions for spin up and spin down electrons in momentum space is of the order of the Fermi momen- tum). The RG flow is thus rapidly cut off by the gap. IV. CLOSING OF THE ANTICROSSING BY ELECTRIC FIELDS The renormalizationofthe inter-subbandanticrossings becomes especially important when the system is tuned to the point of equally strong Rashba and Dresselhaus spin orbit interactions, α→ −β, where the anticrossing splittings close. At this special point, the system ex- hibits a number of unusual properties, such as a negative magnetoresistance (weak localization),47and giant spin relaxation anisotropies.48It furthermore allows for the construction of nonballistic spin-field-effect transistors,32 and for a persistent spin helix,32,33,49,50, the control of7 spin decoherence in quantum dots,51–53and it guaran- tees the absence of spontaneous magnetic order.54These effects are intimately related to the conservation of the spin along the axis set by the intersubband spin-orbit coupling for α→ −β.32,33Since the intra-subband spin- orbit coupling can furthermore be removed by a gauge transformation (in the absence of a Zeeman term),29the special point α=−βcan also be understood as a wire without spin-orbit coupling. The size of the renormalized inter-subband anticross- ingsduringthetuningof α+βtozerocanbemonitoredin situ, forinstancebyvirtueoftunnelingspectroscopy,39–41 or by optical techniques.55–57Its non-linear dependence on the bare parameter α+βas a function of the electric field is a direct measure of Luttinger liquid physics, and more precisely of the RG scaling dimension g12. Figure 8 displays the nonlinear scaling of the interband anticross- ings when α→ −βfor fixed β=−α0/2 andδ12= 0.2, that is in the backscattering regime, where the inter- subband spin-orbit coupling is enhanced by electron- electron interactions. We find that the Luttinger liquid power law can be measured over several decades (espe- cially since the bare value of α+βcan be tuned to exceed α0), even when taking into account realistic energy res- olutions of 0 .005meV, or equivalently temperatures up to∼50mK, or finite size effect for wire lengths of a few microns. While the variation of αat fixed βim- plies a variation of the intra-subband spin-orbit coupling α−β, Fig. 8 shows that this does not affect the power law of the anticrossinggap in a noticeable way. The mea- surement range is, in our model, limited by the subband spacing. When the size of the anticrossing splitting is of the order of the spacing to the next higher subbands, the two-subband model used here becomes inaccurate. To remedy this shortcoming, one may increase the subband spacingδ12, and the intra-subband spin-orbit coupling α−βin such a way that the anticrossings still occur between modes moving in opposite directions. This al- lows to increase the maximal value of the inter-subband spin-orbit coupling α+βwhilst keeping the two-subband approximation justified, and to thereby extend the mea- surement range described by the present theory. V. BAND CROSSINGS AT THE BOTTOM OF THE UPPER BANDS As mentioned above, the regimeofanticrossingsoccur- ring at the bottom of the upper bands is not captured by our theory: not only the ratio of Coulomb repulsion energy to kinetic energy becomes very large, but also the curvature of the upper bands becomes eventually im- portant. These effects are beyond our theory based on a Luttinger liquid approach for weakly interacting elec- trons. We can, however, expect some of the features of our calculation to remain valid even for Kc2<0.5, a regime which can only be reached with long-range in- teractions. Most importantly, we expect the RG scaling0.0010.0050.010.050.10.5 0.001 0.01 0.1 1Anticrossing energy gap [meV] (α + β)/α0renormalized unrenormalized FIG. 8: Scaling of the inter-subband anticrossings as a func - tion of the bare inter-subband interaction α+βfor fixed β=−α0/2. The unrenormalized gap depends linearly on α+β, while the renomormalized one obeys the power law defined in Eq. (13). As an example, this graphs contrasts the renormalized and unrenormalized anticrossing gaps for δ12= 0.2, in which case the anticrossing gap scales with the exponent 1 /(2−g12)≈0.91 as function of the bare inter- subbandspin-orbitcoupling. Thedottedlinemarks anenerg y of 0.005meV ≡50mK, which constitutes a realistic experi- mental measurement resolution. dimension g12to grow when band bottoms of the upper bands are approached. One can consequently expect the size of the inter-subband anticrossing gap to be largely reduced in the strongly interacting regime close to the band bottoms. It would thus be worthwhile to also mon- itor the size of the inter-subband anticrossing gaps as the chemical potential approaches the bottom of the upper bands in an experiment. VI. EFFECTS OF AN APPLIED MAGNETIC FIELD So far, we have not included the effects of a possibly applied magnetic field. Depending on its direction, such a field affects the properties of the wire in a number of ways. We discuss the two limiting cases of a field aligned either parallely or perpendicularly to the spin quantiza- tion axis set by the intra-subband spin-orbit coupling. For a general direction, the magnetic field exhibits all of the effects discussed in Secs. VIA and VIB. A. Magnetic field parallel to the spin-orbit direction If the magnetic field is applied along the spin quanti- zation axis set by the intra-subband spin-orbit coupling, its Zeeman effect shifts spin up and down relative to each other in energy. The initially degenerate anticross- ings between the bands 1 ,↓and 2,↑, and 1,↑and 2,↓ then occur at different energies, see Fig. 9. The stronger the magnetic field becomes, the more the inter-subband crossings (eventually lifted by the spin-orbit coupling)8 -0.4-0.2 0 0.2 0.4 0.6 0.8 1 -2 -1 0 1 2Energy [meV] Wave vector [107 m-1]1, ↑2, ↑ 1, ↓2, ↓ FIG. 9: Band structure of the two subband quantum wire in the non-interacting case, without interband couplings, and with an applied magnetic field parallel to the spin quantiza- tion axis set by the intra-subband spin-orbit coupling (we u se U= 0,α+β= 0, and α−β=α0forδ12= 0.2meV and a Zeeman energy of ±0.2meV for spin up and spin down). For this value of the magnetic field, crossing between the bands 1,↓and 2,↑corresponds to forward scattering, while the one between 1 ,↑and 2,↓corresponds to backscattering. are pushed towards the forward scattering regime. Im- portantly,sincethecrossingsarenotdegenerateinenergy anymore, they are also not equally affected by the mag- netic field. We find that if without magnetic field, the crossings occur in the backscattering regime, an interme- diate magnetic field can create the interesting situation that one of the (eventually lifted) crossings corresponds to forward scattering, while the other one corresponds to backscattering. This is precisely the case in the sit- uation depicted in Fig. 9. As can be inferred from the discussion of Sec. III, electron-electron interactions then result in differently large renormalizations for the two anticrossings. Since, furthermore, the bare energy gaps of the anticrossings are identical, α(1) 12=α(2) 12, we thus find that any difference in gaps of the two anticrossings is a direct signature of electron-electroninteractions, and their interplay with the magnetic field. B. Magnetic field perpendicular to the spin-orbit direction As is well-known from the single subband case, a mag- netic field applied perpendicular to the direction set by the intra-subband spin-orbit coupling opens partial gaps at the crossings of the bands i,↑andi,↓(with 1 = 1 ,2), see Fig. 10. At the inter-subband crossing points, the distinct inversion symmetries of the two transversalwave functions prevents the opening of a gap in the absence of spin-orbit coupling. If the chemical potential is tuned to either ofthe crossingpoints between the subband i,↑and i,↓, the magnetic field is described by the Hamiltonian H(i) B=/integraldisplay dx∆B πacos/parenleftBig√ 2(φic+θis)/parenrightBig ,(14)where ∆ B=gµBB/2 (gis theg-factor,µBdenotes Bohr’s magneton, while Bis amplitude of the ap- plied magnetic field). Away from these crossing points, the Hamiltonian describing the magnetic field contains rapidly oscillating factors, and can thus be dropped. In complete analogy to the single subband case,29electron- electron interactions renormalize the partial gaps opened by the magnetic field to the values ∆∗ i= ∆B/parenleftbigg/planckover2pi1vF,i a0∆B/parenrightbigg(1−g∆,i)/(2−g∆,i) ,(15) wherethe scalingdimensions g∆,idepend onthe strength of electron-electron interactions, and the Fermi velocities within the bands, as well as on the subband mixing due to both inter-subband spin-orbit coupling, and Coulomb repulsion. If the subband offset δ12is larger than the en- ergy scale associated with the intra-subband spin-orbit coupling, such that the second subband lives at ener- gies higher than the energy of the crossing between the bands 1,↑and 1,↓, we find that g∆,1= (K1c+1/K1s)/2. Considering concretely the Luttinger liquid parameters Kic= 0.65 andKis= 1,39–41this exponent becomes g∆,1≈0.85, and thus significantly different from its non- interacting value is g(0) ∆,1= 1. It is, however, of the same order as the scaling exponent g12of the inter-subband anticrossings in the backscattering regime, as can be in- ferred from Fig. 6. The calculation of the scaling dimen- siong∆,2(and similarly of g∆,1for small band offsets δ12) requiresthediagonalizationoftheHamiltonianaccording to the discussion of Appendix A. Because the power law behavior of ∆ B,1and ∆ B,2can be measured over several decades by simply tuning the external magnetic field, we conclude in analogy to Sec. IV that the monitoring of the gaps as a function of the applied field Bconstitutes an additional signature of Luttinger liquid physics. As a further effect of the magnetic field, the distortion of the bandstructure due to the presence of additional gaps as compared to the case without field modifies the Fermi velocities at the eventually lifted inter-subband crossing points, and therefore also modifies the renormalizationof the inter-subband anticrossings. VII. TRANSPORT SIGNATURES In electronic transport, the opening of partial gaps is heralded by a reduction of the conductance. This re- duction can be calculated in an inhomogenous Luttinger model, which also takes into account the leads.17,58–60 A universally quantized conductance is then obtained in thescalinglimit ofanRGrelevantsine-Gordonpotential, such as the inter-subband spin-orbit coupling, or a mag- netic field perpendicular to the spin-quantization axis set by the intra-subband spin-orbit coupling. Electron- electron interactions are a crucial ingredient in observing the conduction reduction in an experiment,17,29,61since9 -0.2-0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 -2 -1 0 1 2Energy [meV] Wave vector [107 m-1] FIG. 10: Band structure of the two subband quantum wire in the non-interacting case, without interband couplings, and withanapplied magnetic fieldperpendicular tothespinquan - tization axis set by the intra-subband spin-orbit coupling (we useU= 0,α+β= 0, and α−β=α0forδ12= 0.2meV and ∆ B= 0.05meV). The dotted lines indicate the bands without magnetic field. they drive the system towardsthe universal scaling limit. Put differently, interactions boost the magnitude of the gap as compared to, for instance, temperature, and finite size gaps, which reduces non-universal corrections to the conductance present for small gaps. In a quantum wire with two subbands, and both Rashba and Dresselhaus spin-orbit couplings, we thus find that the conductance is very sensitive to the mag- nitude, and orientation of an external electric field. If the field is close to the point of partial compensation between Rashba and Dresselhaus couplings, α≈ −β, the conductance through the ballistic wire amounts to G= 4e2/h. If the electric field deviates from this spe- cial point, the anticrossing gaps open, and are quickly renormalized to stronger values by electron-electron in- teractions. This causes the conductance to drop to half its value, i.e. G→2e2/h. VIII. SUMMARY In this work, we have analyzed the effect of electron- electron interactions on spin-orbit couplings in multi- subband quantum wires. Depending on the ratio of the energy spacing between the subbands and the intra- subband spin-orbit coupling, electron-electron interac- tions can either increase or decrease the inter-subband spin-orbit coupling. For large spacings or small Rashba interactions, when the inter-subband spin-orbit interac- tion couples electrons moving into the same direction, theinter-subbandspin-orbitcouplingliftsthedegeneracy between the bands, but does not open up a gap. If spin- orbit interaction couples electrons moving into opposite directions, a gap opens. Unless the chemical potential is close to the bottom of the upper bands, this gap is en- larged by the presence of electron-electron interactions. Analyzing this renormalization with a renormalizationgroup approach, we showed that the effective spin-orbit couplings are interaction-dependent power laws of the bare spin-orbit parameters, and thus of an applied elec- tric field. We have discussed how the scaling dimension of the inter-subband spin-orbit coupling can be measured by monitoring the closing of the anticrossings as one fine- tunes the bare parameters to the point of partial com- pensation between Rashba and Dresselhaus couplings. We then commented on the strongly interacting regime occurringwhen the chemical potential is close to the bot- toms of the upper bands, where our calculation hints at a strong reduction of the anticrossing splitting. Finally, we discussed the effects of an external mag- netic field. If the latter is applied parallel to the spin quantizationaxissetbytheintra-subbandspin-orbitcou- pling, a suitably chosen field can shift the inter-subband crossings such that one of them results in forward scat- tering, while the other one corresponds to backscatter- ing. In this case, electron-electron interactions enlarge the anticrossing that results from backscattering, while they decrease the one due to forward scattering. The dif- ference in the gap size of the anticrossings is thus a con- venient signature of electron-electron interactions. If the magnetic field is applied perpendicular to the direction set by the intra-subband spin-orbit coupling, it opens additional partial gaps. Similar to the inter-subband an- ticrossings, these gaps experience a substantial renormal- ization in the presence of electron-electron interactions. Monitoring the partial gaps opened by the magnetic field as a function of the strength of this field thus again al- lowsone toobservethe scalingdimensionofthe magnetic field in an experiment, which constitutes an further di- rect measure of Luttinger liquid physics. The distortions of the bandstructure resulting from the gaps opened by the magnetic field furthermore affect the Fermi velocities at the (lifted) crossing points, and therefore changes the renormalizationoftheanticrossings. Finally, wehavedis- cussed that the conductance through the wire is reduced by a factor of two when the inter-subband anticrossings open, and get renormalized due to electron-electron in- teractions. In a future work, it would not only be interesting to investigate the physics close to the bottoms of the up- per bands in more detail, but also to analyze the ef- fects of disorder, which we have neglected in the present work. In general, a quantum wire with scalar impurities is known to be susceptible to Anderson localization. One can speculate, however, that localization is suppressed in the quasi-helical regime just below the bottom of the sec- ond band, similarto the single subband casewith applied magnetic field.62 Acknowledgments This workhas been supported by the Swiss NF, NCCR QSIT, and the Harvard Quantum Optical Center (JK).10 Appendix A: Diagonalization of the quadratic part of the Hamiltonian The diagonal electronic Hamiltonian is obtained from H1+H2[see Eq. (6)] by the canonical transformation φ1c=/radicalBigg u1cK1c uc+(1+A2c)φc++/radicalBigg A2cu1cK1c uc−(1+A2c)φc−, (A1a) φ2c=/radicalBigg A2cu2cK2c uc+(1+A2c)φc+−/radicalBigg u2cK2c uc−(1+A2c)φc−, (A1b) θ1c=/radicalbigguc+ u1cK1c(1+A2c)θc++/radicalBigg A2cuc− u1cK1c(1+A2c)θc−, (A1c) θ2c=/radicalBigg A2cuc+ u2cK2c(1+A2c)θc+−/radicalbigguc− u2cK2c(1+A2c)θc−, (A1d) with the velocities uc±= (A2) /radicaltp/radicalvertex/radicalvertex/radicalbtu2 1c+u2 2c 2±/radicalBigg/parenleftbiggu2 1c−u2 2c 2/parenrightbigg2 +U2 12u1cK1cu2cK2c, and with Ac=2U12√u1cK1cu2cK2c/radicalBig (u2 1c−u2 2c)2+4U2 12u1cK1cu2cK2c+u2 1c−u2 2c. (A3) UsingKis= 1, the quadratic part of the electronic Hamiltonian can then be written as He=/summationdisplay k=±uck 2π/integraldisplay dx′/parenleftBig (∂zφck)2+(∂zθck)2/parenrightBig (A4) +/summationdisplay i=1,2vFi 2π/integraldisplay dx′/parenleftBig (∂zφis)2+(∂zθis)2/parenrightBig . A generalized form of the transformation given in Eq. (A1) would allow one to take into account the spin density-density interaction, charge current-current in- teraction and spin current-current interaction neglected here. In order to calculate the scaling dimension of the sine-Gordon terms given in Eq. (8), one first needs to decompose the fields Ψf,b 1,2into the new fields φ±andθ±. Considering first forward scattering, we find Ψf 1=/summationdisplay i=±(ciφi+diθi)−/summationdisplay j=1,2φjs+θjs√ 2 Ψf 2=/summationdisplay i=±(ciφi−diθi)+/summationdisplay j=1,2φjs−θjs√ 2,where the constants cianddifollow from Eqs. (A1), and read c+=/radicalBigg u1cK1c 2uc+(1+A2c)−/radicalBigg A2cu2cK2c 2uc+(1+A2c),(A5a) d+=/radicalbigguc+ 2u1cK1c(1+A2c)−/radicalBigg A2cuc+ 2u2cK2c(1+A2c), (A5b) c−=/radicalBigg A2cu1cK1c 2uc−(1+A2c)+/radicalBigg u2cK2c 2uc−(1+A2c),(A5c) d−=/radicalBigg A2cuc− 2u1cK1c(1+A2c)+/radicalbigguc− 2u2cK2c(1+A2c). (A5d) Since the quadratic part of the Hamiltonian is now diagonal, the scaling dimensions can be obtained from a standard RG analysis of the sine-Gordon potential.37We find that α(1) 12andα(2) 12obey the same RG equation, dα(i) 12 dln(b)= (1−g12)α(i) 12, (A6) with g12=/summationdisplay i=±c2 i+d2 i 4+K1s+1/K1s+K2s+1/K2s 8(A7) for forward scattering. In the case of backscattering, on the other hand, we obtain Ψb 1=/summationdisplay i=±(˜ciφi+˜diθi)−φ1s+θ1s−φ2s+θ2s√ 2, Ψb 2=/summationdisplay i=±(˜ciφi−˜diθi)−φ2s+θ2s−φ1s+θ1s√ 2, with ˜c+=/radicalBigg u1cK1c 2uc+(1+A2c)+/radicalBigg A2cu2cK2c 2uc+(1+A2c),(A8a) ˜d+=/radicalbigguc+ 2u1cK1c(1+A2c)−/radicalBigg A2cuc+ 2u2cK2c(1+A2c), (A8b) ˜c−=/radicalBigg A2cu1cK1c 2uc−(1+A2c)−/radicalBigg u2cK2c 2uc−(1+A2c),(A8c) ˜d−=/radicalBigg A2cuc− 2u1cK1c(1+A2c)+/radicalbigguc− 2u2cK2c(1+A2c). 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2101.11438v1.Spin_memory_loss_induced_by_bulk_spin_orbit_coupling_at_ferromagnet_heavy_metal_interfaces.pdf

Spin-memory loss induced by bulk spin-orbit coupling at ferromagnet/heavy-metal interfaces Mijin Lim1and Hyun-Woo Lee1,a) Department of Physics, Pohang University of Science and Technology, Pohang, 37673, Republic of Korea (Dated: 28 January 2021) A spin current through a ferromagnet/heavy-metal interface may shrink due to the spin-flip at the interface, resulting in the spin-memory loss. Here we propose a mechanism of the spin-memory loss. In contrast to other mechanisms based on interfacial spin-orbit coupling, our mechanism is based on the bulk spin-orbit coupling in a heavy-metal. We demonstrate that the bulk spin-orbit coupling induces the entanglement between the spin and orbital degrees of freedom and this spin-orbital entanglement can give rise to sizable spin-flip at the interface even when the interfacial spin-orbit coupling is weak. Our mechanism emphasizes crucial roles of the atomic orbital degree of freedom and induces the strong spin-memory loss near band crossing points between bands of different orbital characters. A spin current can be both generated1,2and relaxed3by the spin-orbit coupling (SOC). Accurate determination of spin current characteristics such as the spin generation efficiency (spin Hall angle) and the relaxation rate (or spin diffusion length) is difficult, however. Reported experimental values of these characteristic parameters are spread sometimes over one order of magnitude even for seemingly same systems4. An important source of this spread is ferromagnet (FM)/heavy- metal (HM) interfaces3,5,6. Experimental schemes to quantify a spin current, such as spin-orbit torque measurements1,2and spin pumping measurements7,8, utilize a FM/HM bilayer and the spin polarization can be reduced at the FM/HM interface as a spin current passes through it. It is thus important to take account of this so-called spin-memory loss3,5,9,10for quanti- tative analysis of a spin current11–15. There are ongoing effors to clarify mechanisms of the spin-memory loss16–25. Many theoretical16–19and experimen- tal24,25works focused on effects of the interfacial SOC such as the Rashba spin-momentum coupling. However much less attention has been paid to roles of the bulk SOC (within a HM) despite the fact that the bulk SOC in a HM is strong in most experimental situations with the strong spin-memory loss. The bulk SOC effects are taken into account in a few first-principles calculations21–23but are not analyzed explic- itly. In this study, we investigate effects of the bulk SOC S
L [Eq. (2)] on the spin memory loss at a FM/HM interface. In centrosymmetric systems such as fcc Pt bulk, the bulk SOC does not induce any spin splitting. Thus in order to investigate effects of the bulk SOC, it is crucial to take into account the atomic orbital degree of freedom. We use a model Hamilto- nian [Eq. (1)] that takes into account the atomic orbital degree of freedom explicitly and demonstrate that the bulk SOC can induce sizable spin-memory loss at a FM/HM interface. A re- mark is in order. Our mechanism based on the bulk SOC may look similar to other mechanisms based on the interfacial SOC in the sense that the bulk SOC can give rise to an interfacial SOC when it is combined with the broken inversion symme- try (HISB[Eq. (4)]) near the FM/HM interface. However as a)Electronic mail: hwl@postech.ac.krdemonstrated below, the spin-memory loss by the bulk SOC can be significant even when the interfacial SOC arising from the bulk SOC and the inversion symmetry breaking is weak, 0.05 eV
Å. This is in contrast to the interfacial SOC based mechanisms of the spin memory loss, which require stronger interfacial SOC (1 eV
Å) for the significant spin memory loss. A key element of the bulk SOC based mechanism is the spin-orbital entanglement. Since an electron incident on a FM/HM interface from the HM has its spin entangled to its orbital degree of freedom due to the bulk SOC, even "boring" orbital scattering at the interface can flip the electron spin and result in the spin-memory loss (see Fig. 3 for illustration). A FM/HM bilayer is modeled as in Fig. 1(a), where both FM ( z=1;2;


) and HM ( z=0;1;2;


) have the sim- ple cubic lattice structure. Each lattice site can host t2gd- orbitals ( dxy,dyz, and dzx). The bilayer is described by the tight-binding Hamiltonian, Htot=Ht2g+HSOCQ(z+0:5)+HexQ(z0:5)+HISB;(1) where Ht2gis the kinetic energy term that describes the nearest-neighbor hoppings between the t2gorbitals. Since nearest-neighbor inter-orbital hoppings violate the inversion symmetry of the simple cubic lattice, Ht2gcontains only intra- orbital hoppings (for details, see supplementary material S1). We adopt the same intra-orbital hopping parameters for both FM and HM so that the inversion symmetry breaking is min- imized at the FM/HM interface (thereby suppressing the in- terfacial SOC unless HISBis turned on). The FM and HM are distinguished only by the bulk SOC HSOCfor the HM and the exchange coupling Hexfor the FM. Q(z)in Eq. (1) is the Heaviside step function. HSOCfor the HM ( z0)reads HSOC=2l ¯h2å i;s;s0C+ i;sL
SCi;s0; (2) andHexfor the FM ( z<0) reads Hex=Jå i;s;s0C+ i;sS
MCi;s0; (3) where C+ i;s= (c+ i;xy;s;c+ i;yz;s;c+ i;zx;s)is the three-component electron creation operator at the lattice site iwith spin s.LarXiv:2101.11438v1 [cond-mat.mes-hall] 27 Jan 20212 kzπ0−π kx0π−π (b) (a)zSOC strength [eV]00.50zExchange interaction[eV]00.50(a)(b)(c)-orbital size is reduced. 20181101FMHM -0kz-5-4-3-2-1012Energy-0kz-5-4-3-2-1012Energy(d)(e)zSOC strength [eV]00.50zExchange interaction[eV]00.50(a)(b)(c)-orbital size is reduced. 20181101FMHM -0kz-5-4-3-2-1012Energy-0kz-5-4-3-2-1012Energy(d)(e)20-2-4−π0π (c)20-2-4−π0π kzkzFMHM↑↓dxydyzdzx(d)(e) −πkyπ0 kzπ0−πkx0π−π−πkyπ0 FIG. 1. (a) Schematic figure of a FM/HM bilayer with both FM and HM in the simple cubic lattice. Each lattice site can host t2g d-orbitals, dyz,dzxanddxy. (b),(c) Energy band structures of a FM (b) and a HM (c) for (kx;ky) = (p 3;0). The solid and dotted lines in (b) denote the spin up and down bands, respectively. In (b) and (c), thedxy,dyzanddzxcharacter bands are colored red, green, and blue, respectively. The gray dashed region in (c) denotes the avoided band crossing, near which the orbital character of the bands varies rapidly with k. (d),(e) Three-dimensional plots of the Fermi surfaces in a HM at the Fermi energy EF=3:0 eV . In (d), the colors are chosen to visualize the inner (blue) and outer Fermi surfaces. In (e), the colors represent the orbital character as in (c). Note that the orbital character varies rapidly with knear the avoided band crossing. The following parameter values are used throughout this paper; ( ts,tp,J, l) = (1, 0.2, 0.5, 0.5)[eV]. andSare the orbital and spin angular momenta of the t2gd- orbitals, and Mis a three-dimensional unit vector of magne- tization. Here landJare the atomic SOC strength and the exchange interaction energy, respectively. To assess effects of the inversion symmetry breaking, we introduce HISB, HISB=gå i;s[C† i;xy;sCi+ˆx;yz;s+C† i;xy;sCi+ˆy;zx;s+H.c.] gå i;s[C† i;xy;sCiˆx;yz;s+C† i;xy;sCiˆy;zx;s+H.c.];(4) where the sum over the lattice index iruns only within the z=0 (interfacial layer of the HM) and the z=1 (interfacial layer of the FM) layers since the inversion symmetry break- ing is strongly localized in the two layers. HISBdescribes thenearest-neighbor inter-orbital hoppings between the dxyor- bital and the dyz(dzx) orbital along the x(y) direction. These inter-orbital hoppings can exist only when the inversion sym- metry is broken26. Thus the inter-orbital hopping strength gin HISBcan be regarded as the strength of the inversion symme- try breaking. As demonstrated in Ref.27,HISBintroduces the orbital-momentum coupling (see supplementary material S1), which, near the Gpoint, is proportional to L
(kˆz). Com- bined with HSOC,HISBgenerates the Rashba spin-momentum coupling26,27. Eigenstates of the HM are two-fold degenerate [Fig. 1(c)] due to the inversion symmetry whereas the degeneracy is lifted in the FM [Fig. 1(b)] due to Hex. Scattering eigenstates (Fig. 3) of the FM/HM bilayer are calculated by matching the incident, transmitted, and reflected waves through the scat- tering boundary conditions (see supplementary material S2 for details). The matching also takes into account evanescent waves localized near the FM/HM interface. For concreteness, M=ˆzis assumed, which motivates the zaxis to be used as the spin quantization axis, although spin-up ( s=") states are inevitably superposed with spin-down ( s=#) states due to HSOCin the HM. Figure 3 illustrates the nature of the super- position between states with different spins. To examine the spin-flip at the FM/HM interface, we calcu- late the spin currents carried by incident, reflected, and trans- mitted waves, separately. The conventional definition of the spin current is used, isz;X=hyXj1 2fsz;vzgjyXi, where szand vzare the z-directional spin and velocity. Here jyXidenotes incident ( X=I), reflected ( X=R), or transmitted ( X=T) wave. The z-polarized spin current is calculated since the spin zcomponent flip is entirely due to the spin-memory loss whereas the spin xorycomponents can be flipped through Hexeven without the spin-memory loss. We also calculate the charge current ich;Xcarried by incident ( X=I), reflected (X=R), and transmitted ( X=T) waves, separately. For each scattering state, the spin-memory retention rate DR(T)during the reflection ( R) and transmission ( T) is defined by DR(T)=isz;R(T)=isz;I ich;R(T)=ich;I: (5) Note that Eq. (5) is defined so that pure charge scattering does not suppress DR(T)(see supplementary material S3). Thus if DR(T)is smaller than 1, it implies the spin-memory loss. We also define the spin-flip probability PR(T)for the reflection and transmission, PR(T)= (1DR(T) 2)100[%]: (6) Figure 2 shows DR(T)andPR(T)at the Fermi energy EF= 3:0 eV as a function of kxwhile kyis fixed to zero. For the reflection [Fig. 2(a)], PRvanishes when the strength gofHISB is zero, and increases as gincreases. But the increase of PR is not uniform over the Fermi surface but instead strongly lo- calized near the gray dashed region, where the avoided cross- ing between energy bands of different orbital character occurs [Figs. 1(c), 1(d)] and thus orbital character of the Fermi sur- face varies rapidly with k[Fig. 1(e)]. Similarly, the spin-flip3 (a)(b) 10.50-0.5-1 kx0 π/2 πkx0 π/2 π0255075100 10.50-0.5-1⏤⏤"= 0.1⏤⏤"= 0.08⏤⏤"= 0.06⏤⏤"= 0.04⏤⏤"= 0.02⏤⏤"= 0× ⏤⏤"= 0.1⏤⏤"= 0.08⏤⏤"= 0.06⏤⏤"= 0.04⏤⏤"= 0.02⏤⏤"= 0× 0255075100210106 FIG. 2. The spin-memory retention rate DR(T)and the spin-flip probability PR(T)for the reflection (a) and the transmission (b) at the Fermi energy EF=3:0 eV as a function of the in-plane momen- tumkx. The other component kyof the in-plane momentum is set to zero. Results for different strength gofHISBare denoted by differ- ent colors. The gray dashed region corresponds to the avoided band crossing between the orange- and blue-colored bands in Fig. 1(d). probability PTfor the transmission [Fig. 2(b)] also shows large values only near the gray dashed region. Interestingly, PTde- pends on gweakly and can have sizable values even when g vanishes. Reasons for the enhanced PR(T)are discussed be- low. As a side remark, we mention that PRmay go above 50% near the gray dashed region for g=0:1 eV
Å , implying that the spin-memory is not merely lost but instead reversed . To understand the origin of the spin-memory loss, we first examine the effect of HSOCon the wave function structure of eigenstates in the HM. HSOCis proportional to L
S=LzSz+1 2(LS++L+S); (7) where LzandSzare the z-directional orbital and spin angular momenta. The two last terms LS+andL+Sconvertjdzx;si andjdyz;sitojdxy;si, where sdenotes"or#. Thus both of the two-fold degenerate eigenstates, jfHM;1iandjfHM;2i, are superpositions of spin up and down components, jfHM,1i=cu yzdyz;" +cu zxjdzx;"i+cd xydxy;# ; jfHM,2i=cd yzdyz;# +cd zxjdzx;#i+cu xydxy;" :(8) Note that for each of jfHM;1iandjfHM;2i, the spin up and down components have different orbital characters. Thus jfHM;1iandjfHM;2iare spin-orbital entangled states. The de- gree of the entanglement is weak far away from the avoided crossing [Fig. 1(d)] but the entanglement becomes strong near the avoided crossing. The spin-orbital entanglement plays an important role for the spin-memory loss. Figure 3 shows schematically the scat- tering process for an incident wave jyIiwith wave function characterjfHM,1i. Suppose cu yzorcu zxis larger than cd xy, so thatjyIicarries a spin current with the spin polarization up. When HISBis absent ( g=0), the spin polarization of the trans- mitted wave arising from the incident wave is determined by how easily the three componentsdyz;" ,jdzx;"i,dxy;# in the incident wave match the wave function character in the FM. Thus if the wave function in the FM has strongdxy;# character at EF, the spin polarization of the transmitted wave HMFM ++ ++ ... ++ yzzxxy v2FIG. 3. Schematic figure of the scattering process in a FM/HM heterostructure. An incident wave (black solid arrow) is scattered at the FM/HM interface to generate outgoing waves (dashed black arrows). The gray vertical arrow at the interface denotes the evanes- cent wave. The incident and reflected waves in the HM are inevitably superpositions of spin up (red up arrows) and down (blue down ar- rows) components due to the spin-orbital entanglement by HSOC. Or- bital characters of the incident, reflected, and transmitted waves are schematically depicted. jyTihas sizable spin down component, implying that mostly spin up nature of the incident wave is suppressed. Figure 1(b) indeed shows sizabledxy;# character at EF=3:0 eV . This explains the spin-memory loss in PTeven for g=0 [Fig. 2(b)]. Next, we turn on gand investigate the effect of HISBon the spin-memory loss during the reflection ( PR). For small g, the effect of HISBonPRcan be analyzed perturbatively throughhyRjHISBjyIiwithjyIiofjfHM;1icharacter andjyRi ofjfHM;2icharacter. HerejfHM;1iandjfHM;2icarry the the spin currents with opposite spin polarization, and thus the magnitude ofhyRjHISBjyIican be used as a measure of the spin-flip through HISB. According to Eq. (4), HISBconverts jdzx;sianddyz;s todxy;s (or vice versa). This change in the orbital character makes hyRjHISBjyIinonvanishing, re- sulting in the spin-flip. Here we emphasize that HISB[Eq. (4)] does not modify the spin degree of freedom at all but the spin is effectively flipped nevertheless since the spin and or- bital degrees of freedom are entangle in jfHM;1iandjfHM;2i [Eq. (8)]. Together with the fact that the spin-orbital entangle- ment becomes strong near the avoided crossing, this explains why PRincreases with gespecially near the avoided crossing [Fig. 2(a)]. It is worth comparing the present mechanism based on the bulk SOC with those16–19based on the interfacial Rashba spin-momentum coupling. There are connections and also differences. The bulk SOC mechanism is connected to the Rashba spin-momentum coupling in the sense that HISBand HSOC, which are the core elements of the spin-memory loss for the reflection PR, give rise to the Rashba spin-momentum coupling as well26,27. This connection can be understood by recalling the recognition27thatHISBamounts to the orbital- momentum coupling (see supplementary material S1), which has a particularly simple form Lk
ˆznear the Gpoint. When combined with HSOCS
L,HISBgenerates the Rashba spin-momentum coupling aRSk
ˆz. It has been argued284 that the combination of HISBandHSOCis an important source of the Rashba spin-momentum coupling in many systems. However the bulk SOC mechanism differs from the Rashba spin-momentum coupling mechanisms in that the latter mech- anisms16–19neglect the atomic orbital degree of freedom com- pletely and thus the spin-orbital entanglement cannot play any role. We demonstrated above that the spin-orbital entan- glement is responsible for the spin-memory loss during the transmission ( PT) even when HISB=0 (thus Rashba spin- momentum coupling is absent). Also for the spin-memory loss during the reflection ( PR), for which HISBis essential (thus Rashba spin-momentum coupling is present), the spin- orbital entanglement allows for the sizable spin-memory loss even from weak HISBwith g0.1 eV
Å, which amounts to the Rashba spin-momentum coupling strength aRof0.05 eV
Å (see supplementary material S4 for the evaluation of aR). This value is comparable to aR=0:03 eV
Å for the weak Rashba system, Ag(111) surface29, but much smaller than the value1 eV
Å assumed in the theoretical studies16,17of the spin-memory loss by the Rashba spin-momentum coupling. An important implication of the bulk SOC mechanism is that the degree of the spin-memory loss is not uniform in the momentum space but is strong near the band crossing (Fig. 2) because the spin-orbital entanglement is strong there. It is in- teresting that the first-principles calculations of spin-flip scat- tering probabilities for Cu/Pd interfaces22also find large spin- flip probabilities near band crossing points, where the orbital hybridization occurs and the spin-orbital entanglement be- comes strong. Although our result (Fig. 2) based on the sim- ple Hamiltonian Htothas clear limitations in terms of quan- titative predictions, this qualitative agreement with the first- principles calculation result suggests that the bulk SOC mech- anism may be relevant in real materials. Finally we mention that the intrinsic spin Hall effect in HMs, which is an impor- tant mechanism of the spin current generation in FM/HM bi- layers, arises mainly in the momentum space regions close to the band crossing where the orbital hybridization occurs30–32. Thus the bulk SOC mechanism may be especially relevant for the spin-Hall-effect-induced spin transport. To conclude, we investigated the spin-flip scattering at a FM/HM interface induced by the bulk SOC ( HSOC) with ex- plicit account of the atomic orbital degree of freedom. Even when the inversion symmetry breaking ( HISB) at the FM/HM interface is strongly suppressed, the spin-flip can still occur at the interface due to the spin-orbital entanglement caused by the bulk SOC. Additional spin-flip arises when the inversion symmetry breaking ( HISB) at the FM/HM interface is turned on. Our work proposes another spin-memory loss mechanism, in which the spin-orbital entanglement by the bulk SOC plays key roles. 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1301.6355v1.Kondo_Effect_in_the_Presence_of_Spin_Orbit_Coupling.pdf

arXiv:1301.6355v1 [cond-mat.str-el] 27 Jan 2013Kondo Effect in the Presence of Spin-Orbit Coupling Takashi Yanagisawa Electronics and Photonics Research Institute, National In stitute of Advanced Industrial Science and Technology (AIST) Tsukuba Central 2, 1-1-1 Umezono, Tsu kuba 305-8568, Japan (Dated: Received April 8, 2012; published online September 4, 2012) Recently, a series of noncentrosymmetric superconductors has been a subject of considerable interest since the discovery of superconductivity in CePt 3Si. In noncentrosymmetric materials, the degeneracy of bands is lifted in the presence of spin-orbit c oupling. This will bring about new effects in the Kondo effect since the band degeneracy plays an importa nt role in the scattering of electrons by localized spins. We investigate the single-impurity Kon do problem in the presence of spin-orbit coupling. We examine the effect of spin-orbit coupling on the scattering of conduction electrons, by using the Green’s function method, for the s-d Hamiltonian, with employing a decoupling procedure. As a result, we obtain a closed system of equations of Green’s functions, from which we can calculate physical quantities. The Kondotemperature TKis estimated from a singularity of Green’s functions. We show that TKis reduced as the spin-orbit coupling constant αis increased. When 2 αkFis comparable to or greater than kBTK(α= 0),TKshows an abrupt decrease as a result of the band splitting. This suggests a Kondo collapse accompanied with a sharp decrease of TK. The logT-dependence of the resistivity will be concealed by the spin -orbit interaction. I. INTRODUCTION The Kondo effect has attracted many researchers since thediscoveryofthe solutionoftheresistanceminimum[1, 2]. The effect arises from the interactions between a sin- gle magnetic atom and the many electrons in a metal. Metals, when magnetic atoms are added, and rare earth compounds exhibit many interesting phenomena that are related to the Kondo effect. The spin-flip scattering of a conduction-electron spin by a localized impurity spin gives rise to a term proportional to ln Tin the resistivity. Superconductors without inversion symmetry have at- tracted much attention since the discovery of supercon- ductivity in CePt 3Si[3]. A group of noncentrosymmet- ric rare-earth compounds has been reported to exhibit superconductivity: for example, Li 2Pt3B[4, 5], CeIrSi 3, CeCoGe 3, CeIrGe 3[6–8], and LaNiC 2[9]. The absence of spatial inversion yields the splitting of bands due to a spin-orbit interaction[10, 11]. The influence of the spin-orbit interaction was dis- cussed very recently in two-dimensional systems starting from the single-impurity Anderson model[12–14]. In the conventional Kondo problem, the conduction-electron states with spin up and down are degenerate. We expect that the band splitting has a large effect on the Kondo effect, and is closely related to a multi-channel Kondo problem. The purpose of this paper is to investigate this subject on the basis ofthe s-d Hamiltonian with the spin- orbit interaction of Rashba type in three dimensions at finite temperature. We calculate Green’s functions and evaluate the Kondo temperature TKfrom a singularity of them. We show that TKis reduced as a result of the band splitting and shows a abrupt decrease when αkFis comparable to kBTK. The paper is organized as follows. In Section II we show the Hamiltonian, and in Section III we derive equa- tions for Green’s functions. We obtain an approximate solution in Section IV. The Kondo temperature and cor-rection to resistivity are discussed in subsequent Sections V and VI. In Section VII we examine the strong limit of the spin-orbit interaction where the details ofcalcuations are shown in Appexdix. II. MODEL HAMILTONIAN The Hamiltonian is H=H0+Hsd=HK+Hso+Hsd where HK=/summationdisplay kξk(c† k↑ck↑+c† k↓ck↓), (1) Hso=/summationdisplay k[α(ikx+ky)c† k↑ck↓+α(−ikx+ky)c† k↓ck↑], (2) Hsd=−J 21 N/summationdisplay kk′[Sz(c† k↑ck′↑−c† k↓ck′↓)+S+c† k↓ck′↑ +S−c† k↑ck′↓]. (3) ξkis defined by ξk=ǫk−µwhereǫkis the dispersion relation of the conduction electrons and µis the chemi- cal potential. ckσandc† kσare annihilation and creation operators, respectively. We set H0=HK+Hso.S+, S−andSzdenote the operators of the localized spin. We consider the spin-orbit interaction of Rashba type in Hso.αindicates the coupling constant of the spin-orbit interaction. The term Hsdindicates the s-d interaction between the conduction electrons and the localized spin, with the coupling constant J.Jis negative for the anti- ferromagnetic interaction.2 III. GREEN’S FUNCTIONS First, we define Green’s functions of the conduction electrons Gkk′σ(τ) =−∝an}bracketle{tTτckσ(τ)c† k′σ(0)∝an}bracketri}ht, (4) Fkk′(τ) =−∝an}bracketle{tTτck↓(τ)c† k′↑(0)∝an}bracketri}ht, (5) whereTτis the time ordering operator. We note that the spin operators satisfy the following relations: S±Sz=∓1 2Sz, SzS±=±1 2S±,(6) S+S−=3 4+Sz−S2 z, (7) S−S+=3 4−Sz−S2 z. (8) We also define Green’s functions which include the lo- calized spins as well as the conduction electron opera- tors. They are for example, following the notation of Zubarev[15], ∝an}bracketle{t∝an}bracketle{tSzck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htτ=−∝an}bracketle{tTτSzck↑(τ)c† k′↑(0)∝an}bracketri}ht,(9) ∝an}bracketle{t∝an}bracketle{tS−ck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htτ=−∝an}bracketle{tTτS−ck↓(τ)c† k′↑(0)∝an}bracketri}ht,(10) ∝an}bracketle{t∝an}bracketle{tSzck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htτ=−∝an}bracketle{tTτSzck↓(τ)c† k′↑(0)∝an}bracketri}ht,(11) ∝an}bracketle{t∝an}bracketle{tS−ck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htτ=−∝an}bracketle{tTτS−ck↑(τ)c† k′↑(0)∝an}bracketri}ht.(12) The Fourier transforms are defined as usual: Gkk′σ(τ) =1 β/summationdisplay ne−iωnτGkk′σ(iωn),(13) Fkk′(τ) =1 β/summationdisplay ne−iωnτFkk′(iωn),(14) ∝an}bracketle{t∝an}bracketle{tSzck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htτ=1 β/summationdisplay ne−iωnτ∝an}bracketle{t∝an}bracketle{tSzck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiωn,(15) ······. (16) From the commutation relations [H0,ck↑] =−ξkck↑−α(ikx+ky)ck↓,(17) [Hsd,ck↑] =−J 2N/summationdisplay k′(−Szck′↑−S−ck′↓),(18) the equation of motion for Gkk′↑(τ) reads ∂ ∂τGkk′↑(τ) =−δ(τ)δkk′−ξkGkk′↑(τ) −α(ikx+ky)Fkk′(τ) +J 2N/summationdisplay q[∝an}bracketle{t∝an}bracketle{tSzcq↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htτ+∝an}bracketle{t∝an}bracketle{tS−cq↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htτ]. (19)Similarly, the equation of motion for Fkk′is ∂ ∂τFkk′(τ) =−ξkFkk′−α(−ikx+ky)Gkk′↑(τ) −J 2N/summationdisplay q[∝an}bracketle{t∝an}bracketle{tSzcq↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htτ−∝an}bracketle{t∝an}bracketle{tS+cq↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htτ]. (20) We define Γkk′(τ) =1 β/summationdisplay ne−iωnΓkk′(iωn) =∝an}bracketle{t∝an}bracketle{tSzck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htτ+∝an}bracketle{t∝an}bracketle{tS−ck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htτ,(21) Φqk(τ) =1 β/summationdisplay ne−iωnτΦqk(iωn) =∝an}bracketle{t∝an}bracketle{tSzcq↓−S+cq↑;c† k↑∝an}bracketri}ht∝an}bracketri}ht, (22) then we obtain (iωn−ξk)Gkk′↑(iωn) =δkk′+α(ikx+ky)Fkk′(iωn) −J 2N/summationdisplay qk′Γqk′(iωn),(23) (iωn−ξk)Fkk′(iωn) =α(−ikx+ky)Gkk′↑(iωn) +J 2N/summationdisplay qΦqk′(iωn).(24) To obtain the solution to the equations above, we need Green’s functions in eqs.(9)-(12). The equations of mo- tion for∝an}bracketle{t∝an}bracketle{tSzck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htand∝an}bracketle{t∝an}bracketle{tS−ck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htare (iω−ξk)∝an}bracketle{t∝an}bracketle{tSzck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω =∝an}bracketle{tSz∝an}bracketri}htδkk′+α(ikx+ky)∝an}bracketle{t∝an}bracketle{tSzck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω −J 2N/summationdisplay q/bracketleftBig ∝an}bracketle{t∝an}bracketle{tS2 zcq↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω+1 2∝an}bracketle{t∝an}bracketle{tS−cq↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω/bracketrightBig −J 2N/summationdisplay qq′/bracketleftBig ∝an}bracketle{t∝an}bracketle{tS+ck↑c† q↓cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω − ∝an}bracketle{t∝an}bracketle{tS−ck↑c† q↑cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω/bracketrightBig , (25) (iω−ξk)∝an}bracketle{t∝an}bracketle{tS−ck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω =α(−ikx+ky)∝an}bracketle{t∝an}bracketle{tS−ck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω −J 4N/summationdisplay qq′∝an}bracketle{t∝an}bracketle{tS−cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω +J 2N/summationdisplay qq′/bracketleftBig ∝an}bracketle{t∝an}bracketle{tS−ck↓c† q↑cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω −∝an}bracketle{t∝an}bracketle{tS−ck↓c† q↓cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω−2∝an}bracketle{t∝an}bracketle{tSzck↓c† q↓cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω/bracketrightBig −J 2N/summationdisplay q′∝an}bracketle{t∝an}bracketle{tS+S−cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω. (26)3 We use the commutation relation S+S−= 3/4+Sz−S2 z to obtain (iω−ξk)Γkk′(iω) =δkk′∝an}bracketle{tSz∝an}bracketri}ht+α(ikx+ky)∝an}bracketle{t∝an}bracketle{tSzck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω +α(−ikx+ky)∝an}bracketle{t∝an}bracketle{tS−ck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω −J 2N/summationdisplay q/bracketleftBig3 4∝an}bracketle{t∝an}bracketle{tcq↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω+Γqk′(iω)/bracketrightBig −J 2N/summationdisplay qq′/bracketleftBig ∝an}bracketle{t∝an}bracketle{tS+ck↑c† q↓cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω − ∝an}bracketle{t∝an}bracketle{tS−ck↑c† q↑cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω−∝an}bracketle{t∝an}bracketle{tS−ck↓c† q↑cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω +∝an}bracketle{t∝an}bracketle{tS−ck↓c† q↓cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω + 2∝an}bracketle{t∝an}bracketle{tSzck↓c† q↓cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω/bracketrightBig . (27) Here we assume that ∝an}bracketle{tSz∝an}bracketri}ht= 0. Now we need ∝an}bracketle{t∝an}bracketle{tSzck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htand∝an}bracketle{t∝an}bracketle{tS−ck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htto obtain a solu- tion for Γ kk′. The equations for ∝an}bracketle{t∝an}bracketle{tSzck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htand ∝an}bracketle{t∝an}bracketle{tS−ck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htread (iω−ξk)∝an}bracketle{t∝an}bracketle{tSzck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω =α(−ikx+ky)∝an}bracketle{t∝an}bracketle{tSzck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω +J 2N/summationdisplay q/bracketleftBig ∝an}bracketle{t∝an}bracketle{tS2 zcq↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω−1 2∝an}bracketle{t∝an}bracketle{tS+cq↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω/bracketrightBig −J 2N/summationdisplay qq′/bracketleftBig ∝an}bracketle{t∝an}bracketle{tS+c† q↓cq′↓ck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω − ∝an}bracketle{t∝an}bracketle{tS−c† q↑cq′↓ck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω/bracketrightBig , (28) (iω−ξk)∝an}bracketle{t∝an}bracketle{tS−ck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω =−δkk′∝an}bracketle{tS−∝an}bracketri}ht+α(ikx+ky)∝an}bracketle{t∝an}bracketle{tS−ck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω +J 4N/summationdisplay q′∝an}bracketle{t∝an}bracketle{tS−cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω +J 2N/summationdisplay qq′/bracketleftBig ∝an}bracketle{t∝an}bracketle{tS−ck↑c† q↓cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω − ∝an}bracketle{t∝an}bracketle{tS−ck↑c† q↑cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω+2∝an}bracketle{t∝an}bracketle{tSzc† q↓cq′↓ck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω/bracketrightBig . (29) IV. APPROXIMATE SOLUTION We assume that the spin-orbit coupling αis small and we keep terms up to the order of α. We adopt the ap-proximation that ∝an}bracketle{t∝an}bracketle{tSzck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω=α(−ikx+ky) iω−ξk∝an}bracketle{t∝an}bracketle{tSzck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω, (30) ∝an}bracketle{t∝an}bracketle{tS−ck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω=α(ikx+ky) iω−ξk∝an}bracketle{t∝an}bracketle{tS−ck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω.(31) This means that we have neglected the terms of the order ofJαin the right-hand side. Then we obtain /parenleftBigg iω−ξk−α2(k2 x+k2 y) iω−ξk/parenrightBigg Γkk′(iω) =−J 2N/summationdisplay q/bracketleftBig3 4∝an}bracketle{t∝an}bracketle{tcq↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω+Γqk′(iω)/bracketrightBig −J 2N/summationdisplay qq′/bracketleftBig ∝an}bracketle{t∝an}bracketle{tS+ck↑c† q↓cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω − ∝an}bracketle{t∝an}bracketle{tS−ck↑c† q↓cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω−∝an}bracketle{t∝an}bracketle{tS−ck↓c† q↑cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω +∝an}bracketle{t∝an}bracketle{tS−ck↓c† q↓cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω+2∝an}bracketle{t∝an}bracketle{tSzck↓c† q↓cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω/bracketrightBig (32) We use the same approximation in the right-hand side of eq.(23), that is, ( iω−ξk)Fkk′(iω) =α(−ikx+ ky)Gkk′↑(iω), and we have (iω−ξk)Gkk′↑(iω) =δkk′−J 2N/summationdisplay pΓpk′(iω) +α2(k2 x+k2 y) iω−ξkGkk′↑(iω) (33) Here, we employ the decoupling approximation proce- dure for Green’s functions[16, 17]. Many-body Green’s functions are approximated as follows. ∝an}bracketle{t∝an}bracketle{tS−ck↑c† q↑cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω≈ ∝an}bracketle{tck↑c† q↑∝an}bracketri}ht∝an}bracketle{t∝an}bracketle{tS−cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω +∝an}bracketle{tS−c† q↑cq′↓∝an}bracketri}ht∝an}bracketle{t∝an}bracketle{tck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω, (34) ∝an}bracketle{t∝an}bracketle{tS+ck↑c† q↓cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω≈ ∝an}bracketle{tS+c† q↓cq′↑∝an}bracketri}ht∝an}bracketle{t∝an}bracketle{tck↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω − ∝an}bracketle{tS+c† q↓ck′↑∝an}bracketri}ht∝an}bracketle{t∝an}bracketle{tcq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω, (35) ∝an}bracketle{t∝an}bracketle{tSzck↓c† q↓cq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω≈ ∝an}bracketle{tck↓c† q↓∝an}bracketri}ht∝an}bracketle{t∝an}bracketle{tSzcq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω − ∝an}bracketle{tSzc† q↓ck↓∝an}bracketri}ht∝an}bracketle{t∝an}bracketle{tcq′↑;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω, (36) ∝an}bracketle{t∝an}bracketle{tS−ck↓c† q↓cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω≈ ∝an}bracketle{tcq↓c† q′↓∝an}bracketri}ht∝an}bracketle{t∝an}bracketle{tS−ck↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω +∝an}bracketle{tck↓c† q↓∝an}bracketri}ht∝an}bracketle{t∝an}bracketle{tS−cq′↓;c† k′↑∝an}bracketri}ht∝an}bracketri}htiω. (37)4 We define nk=/summationdisplay q∝an}bracketle{tc† q↑ck↑∝an}bracketri}ht=/summationdisplay q∝an}bracketle{tc† q↓ck↓∝an}bracketri}ht, (38) mk= 3/summationdisplay q∝an}bracketle{tS−c† q↑ck↓∝an}bracketri}ht= 2/summationdisplay q(∝an}bracketle{tSzc† q↑ck↑∝an}bracketri}ht+∝an}bracketle{tS−c† q↑ck↓∝an}bracketri}ht). (39) We used the relation obtained from the rotational sym- metry in the spin space, ∝an}bracketle{tS−c† q↑cq′↓∝an}bracketri}ht=∝an}bracketle{tS+c† q↓cq′↑∝an}bracketri}ht= 2∝an}bracketle{tSzc† q↑cq′↑∝an}bracketri}ht=−2∝an}bracketle{tSzc† q↓cq′↓∝an}bracketri}ht. (40) Then, after the analytic continuation iω→ω+iδ, we have /parenleftbigg ω−ξk−α2k2 ⊥ ω−ξk/parenrightbigg Γkk′(ω)+/parenleftbigg3 4−mk/parenrightbigg ×J 2N/summationdisplay qGqk′(ω)+/parenleftbigg nk−1 2/parenrightbiggJ N/summationdisplay qΓqk′(ω) = 0, (41) /parenleftbigg ω−ξk−α2k2 ⊥ ω−ξk/parenrightbigg Gkk′(ω)+J 2N/summationdisplay qΓqk′(ω) =δkk′, (42) whereweset k2 ⊥=k2 x+k2 y. Then, weobtainfromeqs.(41) and (42) that Γkk′(ω) =G0 k(ω)/parenleftbigg mk−3 4/parenrightbiggJ 2NG0 k′(ω) −G0 k(ω)/bracketleftBig/parenleftbigg nk−1 2/parenrightbigg J+/parenleftbigg mk−3 4/parenrightbigg/parenleftbiggJ 2/parenrightbigg2 F(ω)/bracketrightBig ×1 N/summationdisplay qΓqk(ω), (43) where G0 k(ω) =1 2/parenleftbigg1 ω−ξk+αk⊥+1 ω−ξk−αk⊥/parenrightbigg (44) F(ω) =1 N/summationdisplay kG0 k(ω), (45) G(ω) =1 N/summationdisplay k/parenleftbigg nk−1 2/parenrightbigg G0 k(ω), (46) Γ(ω) =1 N/summationdisplay k/parenleftbigg mk−3 4/parenrightbigg G0 k(ω). (47) Because of 1 N/summationdisplay qΓqk(ω) =J 2NΓ(ω)G0 k(ω)1 1+JG(ω)+(J/2)2Γ(ω)F(ω), (48)we obtain Γkk′(ω) =J 2NG0 k(ω)G0 k′(ω)/bracketleftBig/parenleftbigg mk−3 4/parenrightbigg (1+JG(ω)) −/parenleftbigg nk−1 2/parenrightbigg JΓ(ω)/bracketrightBig ×1 1+JG(ω)+(J/2)2Γ(ω)F(ω), (49) Gkk′(ω) =δkk′G0 k(ω)−J2 4NΓ(ω)G0 k(ω)G0 k′(ω) ×1 1+JG(ω)+(J/2)2Γ(ω)F(ω) =δkk′G0 k(ω)+J NG0 k(ω)G0 k′(ω)t(ω),(50) where we defined t(ω) =−J 4Γ(ω) 1+JG(ω)+(J/2)2Γ(ω)F(ω).(51) mkis given by m∗ k= 2/summationdisplay q(∝an}bracketle{tSzc† k↑cq↑∝an}bracketri}ht∝an}bracketle{tS−c† k↑cq↓∝an}bracketri}ht) = 2/summationdisplay qΓqk(τ=−0) =2 β/summationdisplay qωneiωnδΓqk(iωn), (52) whereδis an infinitesimal constant. Because mkis real, we obtain mk=−41 β/summationdisplay ωneiωnδG0 k(iωn)t(iωn).(53) Similarly we have nk=1 β/summationdisplay ωnG0 k(iωn)(1+JF(iωn)t(iωn)).(54) V. KONDO TEMPERATURE A singularity of t(ω) determines the characteristic temperature of the system. We investigate the high- temperature region where mk= 0. Then, Γ(ω) =−3 4F(ω). (55) We obtain for k=k′ Gkk(ω)−1=G0 k(ω)−1−3J2 16NF(ω) 1+JG(ω)+O(J4). (56) The Kondo temperature TKis determined by the van- ishing of the denominator in this equation: 1−J1 2N/summationdisplay k/parenleftbigg1 ω−ξk+αk⊥+1 ω−ξk−αk⊥/parenrightbigg ×1 4/parenleftbigg tanh/parenleftbiggξk−αk⊥ 2TK/parenrightbigg +tanh/parenleftbiggξk+αk⊥ 2TK/parenrightbigg/parenrightbigg = 0, (57)5 where ξk=1 2m(k2 ⊥+k2 z)−µ. (58) We have used nk= (f(ξk−αk⊥) +f(ξk+αk⊥))/2 by neglecting the term of the order of J2. By using the expansion, tanh/parenleftBigz 2/parenrightBig =∞/summationdisplay n=−∞1 z−iπ(2n+1),(59) the equation for TKis 1 =J∞/summationdisplay n=−∞1 8(2π)2/radicalbigg2m µ/integraldisplayK 0dk⊥k⊥/bracketleftBig 2iπTKsign(2n+1) ω−iπ(2n+1)TK +iπTKsign(2n+1) ω+2αk⊥−iπ(2n+1)TK +iπTKsign(2n+1) ω−2αk⊥−iπ(2n+1)TK/bracketrightBig , (60) whereKisacutoffandweuseanapproximateexpression /integraldisplayK −Kdkz1 ω−k2 ⊥/(2m)+αk⊥−k2z/(2m)+µ ×1 k2 ⊥/(2m)+k2z/(2m)+αk⊥−µ−iπ(2n+1)TK ≈/radicalbigg2m µiπsign(2n+1)TK1 ω+2αk⊥−iπ(2n+1)TK. (61) We set an cutoff n0≡D/(2πTK) in the summation with respect to n. By using the formula for the digamma function, n0/summationdisplay n=01 n+1 2+x=ψ/parenleftbigg1 2+x+n0/parenrightbigg −ψ/parenleftbigg1 2+x/parenrightbigg ,(62) we obtain 1 =|J|1 32π2/radicalbigg 2m µ/integraldisplayK 0dk⊥k⊥/bracketleftBig 4log/parenleftbigg2eγD πTK/parenrightbigg +2ψ/parenleftbigg1 2/parenrightbigg −1 2ψ/parenleftbigg1 2−ω+2αk⊥ i2πTK/parenrightbigg −1 2ψ/parenleftbigg1 2+ω+2αk⊥ i2πTK/parenrightbigg −1 2ψ/parenleftbigg1 2−ω−2αk⊥ i2πTK/parenrightbigg −1 2ψ/parenleftbigg1 2+ω−2αk⊥ i2πTK/parenrightbigg/bracketrightBig . (63) We setµ=k2 F/(2m) andK= 2kF, and expand the digamma function in terms of αk⊥/(2πTK). Forω= 0, we have 1 =|J|mkF 2π2/bracketleftBig log/parenleftbigg2eγD πTK/parenrightbigg −7ζ(3) 2π2/parenleftbigg2αkF TK/parenrightbigg2 +31ζ(5) 12π4/parenleftbigg2αkF TK/parenrightbigg4 −.../bracketrightBig . (64)This yields the temperature TKas kBTK=2eγD πexp/bracketleftBig −1 ρF|J|−7ζ(3) 2π2/parenleftbigg2αkF kBTK/parenrightbigg2 +31ζ(5) 12π4/parenleftbigg2αkF kBTK/parenrightbigg4 −···/bracketrightBig , (65) where we introduced the Boltzmann constant kBand the density of states ρF=mkF/(2π2). This is a self- consistency equation for TK, and yields x= exp/bracketleftBig −0.21314/parenleftBigαr x/parenrightBig2 +0.0550/parenleftBigαr x/parenrightBig4 −0.01655/parenleftBigαr x/parenrightBig6 +0.005396/parenleftBigαr x/parenrightBig8 −0.001822/parenleftBigαr x/parenrightBig10 +···/bracketrightBig ≡g(x), (66) with variables x=TK/T0 K, αr= 2αkF/kBT0 K,(67) where kBT0 K=2eγD πexp/parenleftbigg −1 ρF|J|/parenrightbigg . (68) We expanded g(x) in powers of αr/xup to the tenth order. The function g(x) is shown in Fig.1, where higher- order terms are small and negligible except near x∼ 0. The equation x=g(x) has no finite solution when αr>1.045. This indicates that TKvanishes when the spin-orbit coupling αkFis greater than 1 .045kBT0 K, and indicates a Kondo collapse with a sharp decrease of TK. This may overestimate the reduction of TK. Whenαis very large, if we use the asymptotic relation ψ(1/2+z)∼ log(z), we obtain 1≃ρF|J|/bracketleftBig log/parenleftbigg2eγD πkBT/parenrightbigg −1 2log/parenleftbigg2αkF πkBT/parenrightbigg +1 4/bracketrightBig .(69) This yields kBTK≃√e 2αkF(2eγD)2 πexp/parenleftbigg −2 ρF|J|/parenrightbigg =π√e αrkBT0 K, (70) forαr≫1. We show TKas a function of αrin Fig.2. We expect that,in the strong coupling limit αr≫1, TKshould approach that of single-band model: kBTα K=2eγD πexp/parenleftbigg −2 ρF|J|/parenrightbigg . (71) We will show this in the section 7. This agrees with eq.(70) for αkF∼D. This is very small compared to the original TKbecauseTα K/T0 K≃kBT0 K/D. Therefore TKdecreases as αris increased and shows up a sharp decrease at αr∼1.6 00.511.5 0 0.5 1 1.5 2g xαr = 0.5 1 1.5 FIG. 1: g(x) = exp( −(7ζ(3)/2π2)(αr/x)2+···) up to the tenth order of αr/xas a function of xforαr= 0.5, 1 and 1.5. The straight line is a linear function x. 00.51 0 0.4 0.8 1.2 1.6TK/TK0 αr FIG. 2: TKas a function of αr.x=g(x) has no solution for αr>1.045 (Kondo collapse) as indicated by dotted line. The dashed line is an expected line. VI. CORRECTION TO RESISTIVITY The imaginary part of Gkk(ω)−1gives the scattering rate of conduction electrons due to the localized spin.The inverse of the life time τk(ω) is 1 τk(ω)=niNImGkk(ω)−1 =3niJ2 16πρα(ω)1+JK(ω) (1+JK(ω))2+(JL(ω))2, (72) whereniis the concentration of magnetic impurities. We havedefined K(ω) = ReG(ω+iδ),L(ω) =−ImG(ω+iδ), andρα(ω) =−(1/π)ImF(ω+iδ). Because we obtain K(0)≃ρF/bracketleftBig log/parenleftbigg2eγD πkBT/parenrightbigg −7ζ(3) 4π2/parenleftbigg2αkF kBT/parenrightbigg2/bracketrightBig ,(73) the formula of the conductivity yields σ=−2e2 3/integraldisplay τk(ξk)v2 k∂f ∂ξkρ(ξk)dξk ≃2e2 3v2 Fρ(0)16 3πniJ2ρα(0)(1−|J|K(0)) ≃2e2 3v2 FρF16 3πni|J|ρF ρα(0)/bracketleftBig log/parenleftbiggT T0 K/parenrightbigg +7ζ(3) 4π2/parenleftbigg2αkF kBT/parenrightbigg2/bracketrightBig . (74) We have the term ( α/T)2that comes from the spin-orbit interaction, and this term will conceal the logarithmic dependence of the resistivity. Then, the electrical resis- tivityRin the high temperature region T≫T0 Kis R=R0 ×/bracketleftBigg 1−|J|ρFlog/parenleftbigg2eγD πkBT/parenrightbigg +7ζ(3) 4π2|J|ρF/parenleftbigg2αkF kBT/parenrightbigg2/bracketrightBigg−1 (75) whereR0is a constant. If the term ( α/kBT)2is larger than the logarithmic term, the resistivity even shows R∼T2. Hence, the spin-orbit coupling may change the temperature dependence of the resistivity drastically. VII. STRONG SPIN-ORBIT COUPLING CASE In this section let us consider the case with strong spin-orbit coupling. For this purpose, we diagonalize the Hamiltonian H0: H0=/summationdisplay k(c† k↑c† k↓)/parenleftbigg ξkα(ikx+ky) α(−ikx+ky)ξk/parenrightbigg/parenleftbigg ck↑ ck↓/parenrightbigg =/summationdisplay k/bracketleftBig (ξk−αk⊥)α† kαk+(ξk+αk⊥)β† kβk/bracketrightBig ,(76) wherek⊥=/radicalBig k2x+k2y, andαkandβkare defined by αk=ukck↑+vkck↓, (77) βk=−v∗ kck↑+ukck↓. (78)7 The coefficients ukandvkare uk=1 2, vk=−ikx+ky√ 2k⊥, (79) satisfyingu2 k+|vk|2= 1. We consider the case where the band split is so large that we can neglect the upper band. This means that we keep terms that contain α-operators only. In this approximation the s-d interaction term is Hα sd=−J 21 N/summationdisplay kk′[Sz{(ukuk′−vkv∗ k′)α† kαk′ +S+vkuk′α† kαk′+S−ukv∗ k′α† kαk′].(80) This is the model of one-channel conduction-electron band that interacts with the localized spin. Let us consider the following Green’s function: Gα kk′(τ) =−∝an}bracketle{tTταk(τ)α† k′(0)∝an}bracketri}ht. (81) By using the same method in previous sections, Gα kk′is shown to be Gα kk′(z) =δkk′ z−ξkα+J 2N1 2+vkv∗ k′ (z−ξkα)(z−ξk′α)t(z), (82) for arbitrary complex number zwhere we defined t(z) =3J 16Fα(z) 1+J 2Gα(z)−3 16/parenleftbigJ 2/parenrightbig2Fα(z)2,(83) Fα(z) =1 N/summationdisplay k1 z−ξkα, (84) Gα(z) =1 N/summationdisplay k¯nkα−1/2 z−ξkα. (85) We derive this formula in Appendix. The Kondo tem- peratureTα Kis determined from a singularity of t(z) in the same way as previous sections. We obtain kBTα K=2eγD πexp/parenleftbigg −2 |J|ρF/parenrightbigg . (86) The characteristic energy Tα Kis reduced significantly comparedtothe conventionalKondotemperatureby fac- tor 2 in the exponential function. This factor appearsbe- cause the number of channel of the conduction electrons in this case is just half of the normal Kondo system. The resistivity is also calculated as R=R0/bracketleftBig 1+ρF|J| 2log/parenleftbigg2eγD πkBT/parenrightbigg +···/bracketrightBig ,(87) with a factor 1 /2.VIII. DISCUSSION We investigated the Kondo effect in the presence of spin-orbit coupling. The influence of band splitting was examinedbyusingtheGreen’sfunctionmethodwherewe adopted the decoupling schemeto obtain an approximate solution. The Kondo temperature is reduced by the spin- orbit interaction, and shows asudden decreasewhen αkF is of the order of kBT0 K. We call this the Kondo collapse due to the spin-orbit coupling. The Kondo effect is sup- pressed and the log T-dependence of the resistivity will be weakened and concealed. The reduction of TKas a re- sultofthespin-orbitcouplingisconsistentwiththeresult for the single-impurity Anderson model using the numer- ical renormalization group technique[13]. In their work the Kondo temperature is a decreasing function of the Rashba energy ER∝α2when the level of the localized electrons is lowered, that is, in the Kondo region, while it is a increasing function when the localized level is not deep. The variation of the Kondo temperature is approx- imately linear as a function of ER, namely, quadratic in α. This is consistent with our result which shows a small variation of the Kondo temperature with the quadratic correction when αis small. In a recent work[12], the Kondo temperature is increased in the presence of the Dzyaloshinski-Moriya interaction. The Dzyaloshinsky- Moriya interaction, however, vanishes in the Kondo re- gion with particle-hole symmetry ǫd=−U/2. Hence the result in ref.[12] seems consistent with the result for the s-d model. Asalimitofstrongspin-orbitinteraction,wecaninves- tigate a crossover to a one-channel Kondo problem. The Kondo problem with the spin-orbit coupling is closely related to a multi-channel Kondo problem. The Kondo temperature is reduced considerably because the degen- eracy of the conducting electrons becomes half of the conventional Kondo system in this limit. The specific heat also exhibits a log T-term in the present model with one-channel conduction band, and this term appears in the fifth-orderof ρJ. Thisagreeswith the originalKondo problem. The author expresses his sincere thanks to K. Yamaji, I. Hase and J. Kondo for helpful discussion. IX. APPENDIX In this appendix we derive the equation of motion for Green’s functions for the single-band s-d model and dis- cuss its physical properties. A. Green’s functions H0was diagonalized by αkandβk: αk=ukck↑+vkck↓, (88) βk=−v∗ kck↑+ukck↓. (89)8 The coefficients ukandvkare uk=1 2, vk=−ikx+ky√ 2k⊥, (90) satisfyingu2 k+|vk|2= 1. The s-d interaction part be- comes Hsd=−J 21 N/summationdisplay kk′[Sz{(ukuk′−vkv∗ k′)α† kαk′ −(ukuk′−v∗ kvk′)β† kβk′ −(ukvk′+vkuk′)α† kβk′−(ukv∗ k′+v∗ kuk′)β† kαk′} +S+(vkuk′α† kαk′−ukvk′β† kβk′−vkvk′α† kβk′ +ukuk′β† kαk′) +S−(ukv∗ k′α† kαk′−v∗ kuk′β† kβk′+ukuk′α† kβk′ −v∗ kvk′β† kαk′)]. (91) The single-bands-d modelcontainsonlythe followings-d interaction, Hα sd=−J 21 N/summationdisplay kk′[Sz{(ukuk′−vkv∗ k′)α† kαk′ +S+vkuk′α† kαk′+S−ukv∗ k′α† kαk′].(92) We consider the following Green’s functions: Gα kk′(τ) =−∝an}bracketle{tTταk(τ)α† k′(0)∝an}bracketri}ht,(93) ∝an}bracketle{t∝an}bracketle{tSzαk;α† k′∝an}bracketri}ht∝an}bracketri}htτ=−∝an}bracketle{tTτ(Szαk)(τ)α† k′(0)∝an}bracketri}ht,(94) ∝an}bracketle{t∝an}bracketle{tS+αk;α† k′∝an}bracketri}ht∝an}bracketri}htτ=−∝an}bracketle{tTτ(S+αk)(τ)α† k′(0)∝an}bracketri}ht.(95) The Fourier transforms are defined similarly: Gα kk′(τ) =1 β/summationdisplay ωneiωnτGα kk′(iωn),(96) ∝an}bracketle{t∝an}bracketle{tSzαk;α† k′∝an}bracketri}ht∝an}bracketri}htτ=1 β/summationdisplay ωneiωnτ∝an}bracketle{t∝an}bracketle{tSzαk;α† k′∝an}bracketri}ht∝an}bracketri}htω,(97) ∝an}bracketle{t∝an}bracketle{tS+αk;α† k′∝an}bracketri}ht∝an}bracketri}htτ=1 β/summationdisplay ωneiωnτ∝an}bracketle{t∝an}bracketle{tS+αk;α† k′∝an}bracketri}ht∝an}bracketri}htω.(98) The equations of motion for these Green’s functions are derived as follows. iωnGα kk′(iωn) =δkk′+ξkαGα kk′(iωn) −J 2N/summationdisplay q[(ukuq−vkv∗ q)∝an}bracketle{t∝an}bracketle{tSzαq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn +vkuq∝an}bracketle{t∝an}bracketle{tS+αq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn +ukv∗ q∝an}bracketle{t∝an}bracketle{tS−αq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn], (99)iωn∝an}bracketle{t∝an}bracketle{tSzαk;α† k′∝an}bracketri}ht∝an}bracketri}htiωn=ξkα∝an}bracketle{t∝an}bracketle{tSzαk;α† k′∝an}bracketri}ht∝an}bracketri}htiωn +J 2N/summationdisplay q[−(ukuq−vkv∗ q)∝an}bracketle{t∝an}bracketle{tS2 zαq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn +vku∗ q∝an}bracketle{t∝an}bracketle{tS+(nkα−1 2)αq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn −ukv∗ q∝an}bracketle{t∝an}bracketle{tS−(nkα−1 2)αq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn], (100) iωn∝an}bracketle{t∝an}bracketle{tS+αk;α† k′∝an}bracketri}ht∝an}bracketri}htiωn=ξkα∝an}bracketle{t∝an}bracketle{tS+αk;α† k′∝an}bracketri}ht∝an}bracketri}htiωn +J 2N/summationdisplay q[−(ukuq−vkv∗ q)∝an}bracketle{t∝an}bracketle{tS+(nkα−1 2)αq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn + 2ukv∗ q∝an}bracketle{t∝an}bracketle{tSznkααq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn−ukv∗ q∝an}bracketle{t∝an}bracketle{tS+S−αq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn], (101) iωn∝an}bracketle{t∝an}bracketle{tS−αk;α† k′∝an}bracketri}ht∝an}bracketri}htiωn=ξkα∝an}bracketle{t∝an}bracketle{tS+αk;α† k′∝an}bracketri}ht∝an}bracketri}htiωn +J 2N/summationdisplay q[(ukuq−vkv∗ q)∝an}bracketle{t∝an}bracketle{tS−(nkα−1 2)αq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn −2vkuq∝an}bracketle{t∝an}bracketle{tSznkααq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn−vkuq∝an}bracketle{t∝an}bracketle{tS−S+αq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn], (102) wherenkα=α† kαk. We have unknown functions ∝an}bracketle{t∝an}bracketle{tSankααq;α† k′∝an}bracketri}ht∝an}bracketri}htfora=z, + and−. B. Approximate Solution To obtain a consistent solution, we adopt the following approximation: ∝an}bracketle{t∝an}bracketle{tSankααq;α† k′∝an}bracketri}ht∝an}bracketri}ht=∝an}bracketle{tnkα∝an}bracketri}ht∝an}bracketle{t∝an}bracketle{tSaαq;α† k′∝an}bracketri}ht∝an}bracketri}ht.(103) Using this approximationand the relation S+S−= 3/4+ Sz−S2 z, we obtain (iωn−ξkα)Gα kk′(iωn) =δkk′ +/parenleftbiggJ 2N/parenrightbigg2/summationdisplay q¯nq−1/2 iωn−ξq/summationdisplay q′/bracketleftBig vkuq′∝an}bracketle{t∝an}bracketle{tS+αq′;α† k′∝an}bracketri}ht∝an}bracketri}htiωn +ukv∗ q′∝an}bracketle{t∝an}bracketle{tS−αq′;α† k′∝an}bracketri}ht∝an}bracketri}htiωn + (ukuq′−vkv∗ q′)∝an}bracketle{t∝an}bracketle{tSzαq′;α† k′∝an}bracketri}ht∝an}bracketri}htiωn/bracketrightBig +3 8/parenleftbiggJ 2N/parenrightbigg2/summationdisplay q1 iωn−ξqα/summationdisplay q′(ukuq′+vkv∗ q′)Gα q′k′(iωn), (104)9 where ¯nq=∝an}bracketle{tnqα∝an}bracketri}ht. Here we define Γkk′(iωn) =/summationdisplay q/bracketleftBig (ukuq−vkv∗ q)∝an}bracketle{t∝an}bracketle{tSzαq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn +vkuq∝an}bracketle{t∝an}bracketle{tS+αq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn+ukv∗ q∝an}bracketle{t∝an}bracketle{tS−αq;α† k′∝an}bracketri}ht∝an}bracketri}htiωn/bracketrightBig . (105) This quantity reads after substituting the equations for ∝an}bracketle{t∝an}bracketle{tSaαq;α† k′∝an}bracketri}ht∝an}bracketri}ht Γkk′=J 2N/summationdisplay q1 iωn−xikα/summationdisplay q′/bracketleftBig −vkuq′/parenleftbigg ¯nq−1 2/parenrightbigg ∝an}bracketle{t∝an}bracketle{tS+αq′;α† k′∝an}bracketri}ht∝an}bracketri}htiωn −ukv∗ q′/parenleftbigg ¯nq−1 2/parenrightbigg ∝an}bracketle{t∝an}bracketle{tS−αq′;α† k′∝an}bracketri}ht∝an}bracketri}htiωn −(ukuq′−vkv∗ q′)/parenleftbigg ¯nq−1 2/parenrightbigg ∝an}bracketle{t∝an}bracketle{tSzαq′;α† k′∝an}bracketri}ht∝an}bracketri}htiωn −3 8(ukuq′+vkv∗ q′)Gα q′k′(iωn)/bracketrightBig =−J 2N3 8/summationdisplay q¯nk−1/2 iωn−ξqαΓkk′−J 2N3 8/summationdisplay q1 iωn−ξqα ×/summationdisplay q′(ukuq′+vkv∗ q′)Gα q′k′(iωn). (106) Then we obtain Gα kk′(iωn) =δkk′ iωn−ξkα+3 8/parenleftbiggJ 2N/parenrightbigg21 iωn−ξkα ×/summationdisplay q′1 iωn−ξq′α1 1+J 2N/summationtext p¯np−1/2 iωn−ξpα ×/summationdisplay q(ukuq+vkv∗ q)Gα qk′(iωn).(107) We have set uk= 1/√ 2. Because vksatisfiesvk=−v−k and|vk|2= 1/2, we have /summationdisplay kv∗ kGα kk′(iωn) =v∗ k′ iωn−ξk′α/bracketleftBig 1−3 8/parenleftbiggJ 2/parenrightbigg21 2Fα(iωn)2 ×1 1+(J/2)Gα(iωn)/bracketrightBig−1 ,(108) where we set Fα(z) =1 N/summationdisplay k1 z−ξkα, (109) Gα(z) =1 N/summationdisplay k¯nkα−1/2 z−ξkα. (110) We define t(z) =3J 16Fα(z) 1+J 2Gα(z)−3 16/parenleftbigJ 2/parenrightbig2Fα(z)2.(111)ThenGα kk′and Γkk′read Gα kk′(z) =δkk′ z−ξkα+J 2N1 2+vkv∗ k′ (z−ξkα)(z−ξk′α)t(z), (112) Γkk′(z) =−1 2+vkv∗ k′ z−ξk′αt(z), (113) forarbitrarycomplexnumber z. TheKondotemperature Tα Kis determined from a singularity of t(z) in the same way as previous sections. We obtain Tα K=2eγD πexp/parenleftbigg −2 |J|ρF/parenrightbigg . (114) The characteristic energy Tα Kis reduced significantly comparedto the conventionalKondotemperatureby fac- tor 2 in the exponential function: Tα K∼/parenleftbiggT0 K D/parenrightbigg T0 K. (115) This factor appearsbecause the number of channel of the conduction electronsin this caseis just halfofthe normal Kondo system. The resistivity is also calculated as R=R0/bracketleftBig 1+ρF|J| 2log/parenleftbigg2eγD πkBT/parenrightbigg +···/bracketrightBig ,(116) with a factor 1 /2. C. Entropy and Specific Heat The energy expectation value E=∝an}bracketle{tH∝an}bracketri}htis given by E=/summationdisplay kξk∝an}bracketle{tα† kαk∝an}bracketri}ht−J 2N/summationdisplay kk′∝an}bracketle{t{Sz(ukuk′−vkv∗ k′) +S+vkuk′+S−ukv∗ k′}α† kαk′∝an}bracketri}ht =1 β/summationdisplay kωnξkαG† kk(iωn)−J 21 βN/summationdisplay kΓkk(iωn) =1 β/summationdisplay kωnξkα iωn−ξkα+J 21 βN/summationdisplay kωniωnt(iωn) (iωn−ξkα)2. (117) The expectation value of the interaction Hamiltonian is denoted as V.Vis given by V=−J 2N/summationdisplay kk′∝an}bracketle{t{Sz(ukuk′−vkv∗ k′)+S+vkuk′ +S−ukv∗ k′}α† kαk′∝an}bracketri}ht =−J 21 βN/summationdisplay kΓkk(iωn) =J 21 βN/summationdisplay kωnt(iωn) iωn−ξkα. (118)10 This is written as V=J 2ρ(0)Re/integraldisplayD −Ddωf(ω)t(ω−iδ),(119) where we adopted the approximation Fα(ω±iδ) =∓πρ(0)i. (120) ρ(ω) is the density of states of conduction electrons. We needt(z) to estimate V.Gα(z), which appears in the denominator of t(z), contains a singularity. Gα(z) is written as Gα(z) =Rα(z)+J 2N1 βN/summationdisplay kωn1 z−ξkαt(iωn) (iωn−ξkα)2, (121) where Rα(z) =1 β/summationdisplay ωnFα(iωn)−Fα(z) z−iωn−1 2Fα(z).(122) Rα(z) is evaluated as[18] Rα(ω−iδ)≈ρ(0)/bracketleftbigg ψ/parenleftbigg1 2+βD 2π/parenrightbigg −ψ/parenleftbigg1 2+iβz 2π/parenrightbigg/bracketrightbigg , (123) whereψis the digamma function and Dis the cutoff energy. We use the following relation, 1+ρ(0)J 2/bracketleftBig log/parenleftbiggD 2πkBT/parenrightbigg −ψ/parenleftbigg1 2+iβω 2π/parenrightbigg/bracketrightBig =ρ(0)J 2/bracketleftBig logTα K T−g(βω)/bracketrightBig , (124) where g(x) =ψ/parenleftbigg1 2+ix 2π/parenrightbigg −ψ/parenleftbigg1 2/parenrightbigg .(125) Then the interaction energy is V=−3π 16ρ(0)JIm/integraldisplayD −Ddωf(ω)1 log(Tα K/T)−g(βω), (126) where we neglected the term of the order of ( ρ(0)J)2in the denominator of t(z).Vhas a logarithmic temper- ature dependence. Because of the relation between the free energy and V, V=J∂F ∂J, (127) the additional entropy ∆ S(T) is ∆S(T) =−∂ ∂T(F−F0) =−/integraldisplayJ 0dJ′ J′∂ ∂TV(J′,T).(128)To estimate V, we use the expansion formula for the Fermi distribution function f(ω): /integraldisplayD −Ddωf(ω)h(ω) =/integraldisplay0 −Ddωh(ω)+π2 6(kBT)2h′(0), (129) for a differentiable function h(ω). Using this, we obtain V=−3π 16ρ(0)JIm/integraldisplay0 −Ddω1 log(Tα K/T)−g(βω) −3π 16π2 6(kBT)2ρ(0)JIm∂ ∂ω1 log(Tα K/T)−g(βω)/vextendsingle/vextendsingle/vextendsingle ω=0. (130) We are interested in logarithmic terms log( D/kBT), log(D/kBT)2and so on in the region |log(Tα K/T)| ≫1. The second term is written as V2=−π4 128kBTρ(0)J1 (log(Tα K/T))2.(131) This is expanded as in terms of ρ(0)J: V2=π4 32/parenleftbiggρ(0)J 2/parenrightbigg4 kBTlog/parenleftbiggD kBT/parenrightbigg −3π4 64/parenleftbiggρ(0)J 2/parenrightbigg5 kBTlog/parenleftbiggD kBT/parenrightbigg2 .(132) In the first term of V, the logarithmic corrections never emerge from the region where βωis large because we haveg(βω)∼log(βω) for largeωand theT-dependence is canceled with log( Tα K/T). Whenβωis small,g(βω) is expressed in a power series of βω. A dominant contri- bution is of the order of (log( Tα K/T))−2. The integral is restricted on the interval ( −kBT,0) and the first term V1 is estimated as V1≃ −3π 16ρ(0)J1 (log(Tα K/T))2/integraldisplay0 −kBTdωImψ/parenleftbigg1 2+iβω 2π/parenrightbigg =−3π 16kBTρ(0)J1 (log(Tα K/T))2π/bracketleftBig −1 8+0.0052 −0.00738+0.000026···/bracketrightBig . (133) As a result, Vis given as V=−A 2kBTρ(0)J1 (log(Tα K/T))2,(134) for a constant A>0. From the relation Tα K=Dexp(2/(ρ(0)J)), we have dρ(0)J ρ(0)J=−1 log(Tα K/D)dlogTα K.(135) Using this formula, the entropy obtained from the inter- action energy Vis ∆S=−∂∆F ∂T, (136)11 where ∆F=−kBA/bracketleftBig T1 (log(D/T))2/parenleftBigρ(0)J 2+1 log(Tα K/T) −2 log(D/T)log/vextendsingle/vextendsingle/vextendsingleρ(0)J 2log/parenleftbiggTα K T/parenrightBig/vextendsingle/vextendsingle/vextendsingle/parenrightbigg/bracketrightBig ,(137) is the free energy. Because of the relation log/parenleftbiggTα K T/parenrightbigg =2 ρ(0)J+log/parenleftbigg2eγ πD kBT/parenrightbigg ,(138) up to the fifth order of ρ(0)J, ∆Sis given as ∆S=kBA/bracketleftBig1 3/parenleftbiggρ(0)J 2/parenrightbigg3 +1 2/parenleftbiggρ(0)J 2/parenrightbigg4 −1 2/parenleftbiggρ(0)J 2/parenrightbigg4 log/parenleftbiggD kBT/parenrightbigg +3 5/parenleftbiggρ(0)J 2/parenrightbigg5/parenleftbigg logD kBT/parenrightbigg2 −6 5/parenleftbiggρ(0)J 2/parenrightbigg5 log/parenleftbiggD kBT/parenrightbigg/bracketrightBig . (139) The logarithmic term first appears in the fourth order of ρ(0)J. Then the correction to the specific heat ∆ C= T∂∆S/∂Tis ∆C kB≃A 2/parenleftbiggρ(0)J 2/parenrightbigg4/bracketleftBig 1−12 5/parenleftbiggρ(0)J 2/parenrightbigg log/parenleftbiggD kBT/parenrightbigg/bracketrightBig . (140) Hence the specific heatexhibits alogarithmicbehaviorat lowtemperatures. Alog T-termappearsinthefifth order ofρ(0)J; this agrees with the original Kondo problem[2]. In the original Kondo problem, the entropy and the spe- cific heat were evaluated as[2, 19] ∆Ssd≃kBπ2 4(ρJ)3/bracketleftBig 1−3ρJlog/parenleftbiggD kBT/parenrightbigg/bracketrightBig ,(141) ∆Csd≃kB3π3 4(ρJ)4/bracketleftBig 1−4ρJlog/parenleftbiggD kBT/parenrightbigg/bracketrightBig .(142)This suggests that[18] ∆Csd≃kB3π3 4(ρJ)4 1 (1+ρJlog(D/kBT)4 ≃kB3π3 41 (log(TK/T))4, (143) as an expansion in terms of 1 /log(TK/T). In the present model, the coefficients are reduced, where 4 is reduced to 12/5 in front of ρJlog(D/kBT) in the specific heat com- pared to the usual s-d model, and the divergence near the Kondo temperature is moderated. Because the for- mation of a local singlet by the the conduction electrons is weakened in a one-channel case, the entropy decreases more slowly as the temperature is decreased. In the region |log(D/kBT)| ≫1 and|log(Tα K/T)| ≫ 1, ∆Sis obtained as a double-power series of 1/log(D/kBT) and 1/log(Tα K/T): ∆S≃kBA/bracketleftBig1 (log(D/kBT))2/parenleftbiggρ(0)J 2+1 log(Tα K/T)/parenrightbigg/bracketrightBig . (144) Then we obtain ∆C≃kBA1 (log(D/kBT))21 (log(Tα K/T))2.(145) [1] J. Kondo: Prog. Theor. Phys. 32 (1964) 37. [2] J. Kondo: Solid State Physics 23 (1969) 183. [3] E.Bauer, G. Hilshcer, H.Michor, Ch.Paus, E.W.Sceidt, A. Gribanov, Yu. Seropegin, H. Noel, M. Sigrist, and P. Rogl: Phys. Rev. Lett. 92 (2004) 027003. [4] H. Q. Yuan, D. F. Agterberg, N. Hayashi, P. Badica, D. Vandervelde, K. Togano, M. Sigrist and M. B. Salamon: Phys. Rev. Lett. 97 (2006) 017006. [5] M. Nishiyama, Y. Inada and G. Q. Zheng: Phys. Rev. Lett. 98 (2007) 047002. [6] N. Kimura, K. Ito, H. Aoki, S. Uji and T. Terashima: Phys. Rev. Lett. 98 (2007) 197001. [7] M. A. Measson, H. Muranaka, T. Kawai, Y. Ota, K. Sugiyama, M. Hagiwara, K. Kindo, T. Takeuchi, K. Shimizu, F. Honda, R. Settai and Y. Onuki: J. Phys. Soc. Jpn. 78 (2009) 124713. [8] F. Honda, I. Bonalde, K. Shimizu, S. Yoshiuchi, Y. Hi-rose, T. Nakamura, R. Settai and Y. Onuki: Phys. Rev. B81 (2010) 140507. [9] A. D. Hillier, J. Quintanilla and R. Cywinski: Phys. Rev. Lett. 102 (2009) 117007. [10] K. Samokhin, E. S. Zijlstra and S. K. Bose: Phys. Rev. B69 (2004) 094514. [11] I. Hase and T. Yanagisawa: J. Phys. Soc. Jpn. 78 (2009) 084724. [12] M. Zarea, S. E. Ulloa and N. Sandler: Phys. Rev. Lett. 108 (2012) 046601. [13] R. Zitko and J. Bonca: Phys. Rev B84 (2011) 193411. [14] X. Y. Feng and F.-C. Zhang: J. Phys. Condens. Matter 23 (2011) 105602. [15] D. Zubarev: Sov. Phys. Uspekhi 3 (1960) 320; Nonequi- librium Statistical Thermodynamics , Plenum Pub. Corp. (1974). [16] Y. Nagaoka: Phys. Rev. 138 (1965) A1112.12 [17] D. R. Hamann: Phys. Rev. D158 (1967) 570. [18] J. Zittartz and E. M¨ uller-Hartmann: Z. Phys. 212 (1968 ) 380.[19] K. Yosida and H. Miwa: Prog. Theor. Phys. 41 (1969) 1416.

2306.02965v1.Accurate_and_efficient_treatment_of_spin_orbit_coupling_via_second_variation_employing_local_orbitals.pdf

Accurate and efficient treatment of spin-orbit coupling via second variation employing local orbitals Cecilia Vona,1,∗Sven Lubeck,1,∗Hannah Kleine,1Andris Gulans,2and Claudia Draxl1, 3 1Institut f¨ ur Physik and IRIS Adlershof, Humboldt-Universit¨ at zu Berlin, 12489 Berlin, Germany 2Department of Physics, University of Latvia, LV-1004 Riga, Latvia 3European Theoretical Spectroscopic Facility (ETSF) (Dated: June 6, 2023) A new method is presented that allows for efficient evaluation of spin-orbit coupling (SOC) in density-functional theory calculations. In the so-called second-variational scheme, where Kohn- Sham functions obtained in a scalar-relativistic calculation are employed as a new basis for the spin-orbit-coupled problem, we introduce a rich set of local orbitals as additional basis functions. Also relativistic local orbitals can be used. The method is implemented in the all-electron full- potential code exciting . We show that, for materials with strong SOC effects, this approach can reduce the overall basis-set size and thus computational costs tremendously. I. INTRODUCTION Spin-orbit coupling (SOC) is crucially important for accurate electronic-structure calculations of many mate- rials. To illustrate, SOC is responsible for lifting the degeneracy of low-energy excitons in transition-metal dichalcogenides (TMDCs) [1–3], opening a tiny gap in graphene [4–6], and dramatically lowering the fundamen- tal band gap in halide perovskites [7]. However, the im- pact of SOC is not limited to changing features of the electronic bands. It affects bond lengths [8–10], phonon energies [8, 11], and even turns deep defects into shallow ones [12]. In density-functional-theory (DFT) computations, SOC is treated differently in the various methods and codes. In this context, the family of full-potential lin- earized augmented planewaves (LAPW) methods is com- monly used as the reference, e.g., for new implementa- tions [13] or for assessing pseudopotentials [14]. Com- monly used LAPW codes [15–18] employ similar strate- gies to account for SOC. For the low-lying core orbitals, the standard approach is to solve the radial 4-component Dirac equation, assuming a spherically symmetric poten- tial. For the semi-core and valence electrons, the common strategy is to employ a two-step procedure [19]. First, the Kohn-Sham (KS) problem is solved within the scalar- relativistic approximation (first variation, FV). Then, the solutions of the full problem including SOC are con- structed using the FV wavefunctions as the basis. This step is known as the second variation (SV). The strategy relies on an assumption that SOC introduces a small per- turbation, and, indeed, this scheme is appropriate and efficient for many materials, since all the occupied and only a handful of unoccupied bands are sufficient. Un- der these circumstances, the two-step procedure offers a clear computational advantage over methods where SOC is treated on the same footing with other terms of the Hamiltonian [10, 20, 21]. ∗These two authors contributed equally.Some materials, however, require more involved calcu- lations than others. For example, it was argued by Schei- demantel and coworkers [22] that Bi 2Te3requires the consideration of unoccupied bands of at least 8 Ry above the Fermi level to give reliable results. Even more strik- ing, in the halide perovskites, the full set of KS orbitals is needed for convergence [23]. These cases illustrate that for some materials SOC cannot be considered as a small perturbation. Moreover, it is known that scalar- and fully-relativistic orbitals have different asymptotic behavior at small distances from the nuclei. Most no- tably, SOC introduces a splitting within the p-orbitals into spinors, where the radial part of the p3/2solution goes to zero, while the p1/2one diverges. This behavior cannot be recovered in terms of scalar-relativistic (SR) functions. Therefore, in Refs. 19 and 24, the SV basis was extended by additional basis functions, local orbitals (LOs), that recover the correct asymptotic behavior of thep1/2orbitals. This approach is a step forward com- pared to conventional SV calculations. By introducing, however, exactly one shell of p1/2LOs per atom, it does not offer the possibility of systematic improvement to- ward the complete-basis-set limit. There are examples of SR calculations in literature where an extensive use of LOs is required to reach precision targets [25–27]. Fur- thermore, Ref. 21 demonstrated this point also in the con- text of fully-relativistic calculations. We therefore con- clude that the state-of-the-art SV approaches, be it with or without p1/2LOs, are not sufficient for a systematic description of SOC in condensed-matter systems. In this work, we introduce a new approach, termed sec- ond variation with local orbitals (SVLO), which makes use of the fact that relativistic effects are strongest around the atomic nuclei. Therefore, in comparison to the standard SV approach, it is important to increase the flexibility of the basis specifically in these regions. To sat- isfy this need, we express the solution of the full problem in terms of FV wavefunctions and rich sets of LOs. In contrast to the usual approach, all LOs are treated as ex- plicit basis functions, also on the SV level. In addition, we include LOs obtained from solving the Dirac equa-arXiv:2306.02965v1 [cond-mat.mtrl-sci] 5 Jun 20232 tion (termed Dirac-type LOs in the following) beyond p1/2functions. Based on the implementation in the all- electron full-potential package exciting [15], we demon- strate and validate our method in band-structure and total-energy calculations of Xe, MoS 2, PbI 2,γ-CsPbI 3, and Bi 2Te3. II. METHOD A. Conventional second variation We consider the two-component KS equations X σ′=↑,↓ˆHσσ′Ψikσ′(r) =εikΨikσ(r) (1) for the spin components σ=↑,↓. The resulting single- particle spinorsP σΨikσ(r)|σ⟩have eigenenergies εik, where iis the band index and kthe Bloch wave vector. The Hamiltonian ˆHσσ′consists of a SR part and a spin- orbit part that couples the two spin components: ˆHσσ′=δσσ′ˆHSR σ+ˆHSOC σσ′. (2) As described in Refs. 10, 20, and 21, Eq. 1 can be solved directly, i.e., non-perturbatively (NP), requiring a sig- nificantly larger computational effort compared to a SR calculation. Given that ˆHSOC σσ′typically leads to a small correction, it is unsatisfactory to pay the full price for the NP solution. For this reason, often the conventional SV method is employed. In this approach, first, one solves the scalar-relativistic problem in FV, ˆHSR σΨFV jkσ(r) =εFV jkσΨFV jkσ(r), (3) and subsequently uses the resulting FV eigenstates (which are the FV KS wavefunctions) as a basis for the SV eigenstates ΨSV ikσ(r) =X jCSV kσjiΨFV jkσ(r). (4) Here, jruns over all Noccoccupied and a limited number Nunocc of unoccupied FV KS states. Approximating the exact solution Ψ ikσ(r) by ΨSV ikσ(r), one obtains the SV eigenequation for the expansion coefficients CSV kσji X σ′j′Hkσσ′jj′CSV kσ′j′i=εSV ikCSV kσji, (5) where Hkσσ′jj′are the matrix elements of ˆHσσ′, as defined in Eq. 2, with respect to the basis functions ΨFV jkσ(r), Hkσσ′jj′= ΨFV jkσˆHσσ′ΨFV j′kσ′ =δσσ′δjj′εFV jkσ+ ΨFV jkσˆHSOC σσ′ΨFV j′kσ′ .(6)B. Second variation with local orbitals The SVLO approach makes use of the underlying LAPW+LO method that is utilized to solve the FV prob- lem in Eq. 3. Within the LAPW+LO method, KS or- bitals are represented by two distinct types of basis func- tions, namely LAPWs, ϕGk(r), and LOs, ϕµ(r), which are indexed by reciprocal lattice vectors Gand LO in- dices µ, respectively, ΨFV jkσ(r) =X GCkσGjϕGk(r) +X µCkσµjϕµ(r).(7) In order to avoid linear dependency issues between FV eigenfunctions and LOs in our new approach, we modify these FV eigenfunctions such that LO contributions are neglected, and only the first sum in Eq. 7 is further considered: ¯ΨFV jkσ(r) =X GCkσGjϕGk(r). (8) We combine these modified FV functions with the origi- nal set of LOs to form the SVLO basis ΨSVLO ikσ(r) =X jCSVLO kσji¯ΨFV jkσ(r) +X µCSVLO kσµiϕµ(r). (9) The total basis-set size in the SVLO method includes the number of these LO basis functions, NLO. To summarize, the total number of basis functions in the two methods is NSV(LO) b=( Nocc+Nunocc SV Nocc+Nunocc +NLO SVLO .(10) In both cases, Nunocc is a computational parameter, and the results need to be converged with respect to it. We note in passing that the SVLO basis is not orthogonal and thus carries a slight computational overhead com- pared to the conventional SV method, since it leads to a generalized eigenvalue problem. The SVLO method is implemented in exciting . How the different types of LOs are constructed will be de- scribed in the next section. Unlike Ref. 19 and 24, our approach uses the entire set of LOs from the FV basis (including Dirac-type LOs if necessary) as the basis in the SV step. C. Local orbitals LOs are basis functions with the characteristic of be- ing non-zero only in a sphere centered at a specific nu- cleus α[34]. They take the form of atomic-like orbitals which read ϕµ(r) =δα,αµδl,lµδm,m µUµ(rα)Ylm(ˆrα), (11)3 where Ylm(ˆrα) are spherical harmonics and Uµ(rα) are linear combinations of two or more radial functions USR µ(rα) =X ξaµξuαξl(rα;εαξl). (12) The index ξsums over different radial functions. These radial functions uαξl(rα;εαξl) are the solutions of the SR Schr¨ odinger equation and/or their energy derivatives (of any order), evaluated at predefined energy parameters εαξl. Depending on their purpose all radial functions have the same energy parameter or one corresponding to a different state. To account for the asymptotic behavior of relativistic orbitals at the atomic nuclei, we build LOs in which the radial functions are solutions of the Dirac equation UDirac µ(rα) =X ξ,JaµξJuαξJl(rα;εαξJl). (13) Here, the radial functions and the energy parameters are characterized by the additional quantum number J. We sum over the index Jto show that it is possible to com- bine radial functions with different total angular momen- tum (but the same angular momentum l). It is also pos- sible to combine J-resolved radial functions with SR ra- dial functions. In the following we call any LOs including at least one J-resolved radial function, Dirac-type LOs. With this, we can add one or more LOs with any rela- tivistic quantum number, going beyond what has been suggested by Singh [19]. This approach, also used in Ref. 10, is convenient since the general form of the LOs of Eq. 12 is kept. III. COMPUTATIONAL DETAILS We consider a set of five materials, including 3D and 2D semiconductors and a topological insulator, with dif- TABLE I: Structural information and convergence parameters used in the calculations of the considered materials. Rmin MTGmaxis the product of the largest reciprocal lattice vector, Gmax, considered in the LAPW basis and the (smallest) MT radius, Rmin MT. For MoS 2, the latter refers to the S sphere ( RMT=2.05 a0). For more detailed information, we refer to the input files provided at NOMAD. Material Xe MoS 2 PbI2CsPbI 3Bi2Te3 a[˚A] 6.20 3.16 4.56 8.86 10.44 b[˚A] 6.20 3.16 4.56 8.57 10.44 c[˚A] 6.20 15.88 6.99 12.47 10.44 Space group Fm3-m P-6m2 P-3m1 Pnam R-3m Ref. 28, 29 30 31 32 33 RMT[a0] 3.00 2.30/2.05 2.90 2.90 2.80 Rmin MTGmax 8 8 8 9 10 k-mesh 4×4×46×6×16×6×43×3×26×6×6ferent atomic species, stoichiometry, and degree of SOC. For all of them, we employ experimental atomic struc- tures. All calculations are performed with the package exciting [15] where the new method is implemented. Exchange and correlation effects are treated by the PBE parametrization of the generalized gradient approxima- tion [35, 36]. Core electrons are described by means of the 4-component Dirac equation considering only the spher- ically symmetric part of the KS potential. For semicore and valence electrons, the zero-order regular approxima- tion [37, 38] is used to obtain the SR and SOC contribu- tions to the kinetic energy operator. The SOC term is applied only within the muffin-tin region and is evaluated by the following expression: ˆHSOC=c2 (2c2−V)21 rdV drσL, (14) where σandVare the vector of Pauli matrices and the spherically symmetric component of the KS potential, respectively. The structural and computational parameters are dis- played in Table I. The respective k-mesh and the dimen- sionless LAPW cutoff Rmin MTGmaxare chosen such that total energies per atom and band gaps are within a nu- merical precision of 10−2eV/atom and 10−2eV, respec- tively. The actual LAPW basis cutoff Gmaxis determined by dividing Rmin MTGmaxby the smallest MT radius Rmin MT of the considered system. SR calculations serve for comparison with the other methods to investigate the magnitude of SOC effects. To determine the advantages of the SVLO over the SV method, we have carefully monitored the convergence of all considered quantities with respect to the number of SV(LO) basis functions. The NP method, as described in Ref. 10, is used as a reference for this assessment. For SR and SV calculations, we employ SR LOs. As we mainly address pstates in our examples, we label this case as p. The LO set including Dirac-type LOs, referred to as p1/2, is constructed by adding to the SR LOs two p1/2-type LOs for each pstate. Due to their pcharacter, each LO gives rise to three degenerate basis functions. The method is, however, fully general such to include relativistic LOs of other characters. For instance, we ex- plore the effect of d3/2LOs in MoS 2since its valence band maximum (VBM) and conduction band minimum (CBm) exhibit predominant d-character [39]. As the impact on the total energy and the electronic structure turns out to be negligible, however, we do not include this case in the following analysis. In the other materials, we addi- tionally investigate the effects of p3/2LOs, by replacing thepLOs. Due to their similar behavior near the nuclei, their impact is, however, only of the order of 10−2eV or smaller, which is within our convergence criteria. For this reason, we do not consider them further. p1/2LOs are used in SVLO and in the corresponding NP reference when specified. The number of LOs used in the different systems is displayed in Table II together with the size of the LAPW basis and the number of occupied valence4 states. To obtain the band-gap position for Bi 2Te3, for the different sets of LOs ( p-type and the p1/2-type), we perform on top of the self-consistent NP calculation an additional iteration with a 48 ×48×48k-mesh. The so determined respective k-points are then included in the band-structure path, from which we extract the final val- ues of the energy gaps in the SV and SVLO calculations. All input and output files are available at NOMAD [40]. TABLE II: Basis functions considered in the calculations of the five studied systems. NLAPW is the number of LAPWs, NLO(N1/2 LO) the number of LO basis functions for calculations without (with) Dirac-type LOs. The last column shows the number of occupied valence states, Nocc. Material NLAPW NLON1/2 LONocc Xe 138 26 38 13 MoS 2 939 35 - 13 PbI2 318 73 109 33 CsPbI 33236 496 736 228 Bi2Te3 895 141 201 57 IV. RESULTS A. Xe Our analysis starts with solid Xe. Fig. 1 shows the convergence behavior of the total energy with respect to the number of basis functions, taking the NP calculation as a reference. For comparison, we also show the results of the conventional SV method. Since we employ the same number of occupied states in the SV and the SVLO method, and for the pandp1/2sets of LOs (Table II), the number of basis functions on the x-axis does not include the occupied states, i.e.,˜NSV(LO) b=NSV(LO) b−Nocc. NSV(LO) bis defined in Eq. 10 that applies to both meth- ods. The number of LO basis functions in the equation is also predetermined, therefore, the increase in ˜NSV(LO) b reflects only the number of unoccupied bands. We will always refer to ˜NSV(LO) bwhen discussing the basis-set size. In the case of Xe, we consider 26 SR LO basis func- tions (see Table II). Strikingly, the total-energy differ- ences obtained by the SVLO method stay within 2 ×10−3 eV/atom when employing a total number of basis func- tions comparable with the number of LO basis functions, while the SV method requires all available FV states to reach values even one order of magnitude larger (7 ×10−2 eV/atom). To visualize this behavior better, Fig. 2 de- picts the convergence of both methods on a logarithmic scale. We can observe that the SVLO method reaches convergence within 10−6eV/atom with ∼80 basis func- tions. In this figure, we also analyze the convergence FIG. 1: Convergence behavior of the total energy with respect to the number of basis functions ˜NSV(LO) b (excluding occupied states). The energy of the NP calculation is taken as a reference. Blue circles indicate the SVLO scheme, green diamonds the conventional SV treatment. The right panel zooms into the gray region where the SVLO method converges. of the energy gap, Eg, and the SOC splitting at the Γ point, δSOC. When SOC is considered, Egdecreases by 0.43 eV due to the splitting of the (disregarding spin) three-fold degenerate VBM into a single state and a double-degenerate state by about 1.30 eV (see also Table III and Fig. 3). For both quantities, we observe that the SVLO method reaches a precision of the order of 10−4 eV already with a number of basis functions comparable with the number of LO functions; with approximately 80 basis functions even two orders of magnitude better. In contrast, the SV treatment, employing all available FV KS eigenstates, only converges within 10−3eV and 10−2 eV for the energy gap and the SOC splitting, respectively. If we consider a target precision often used for production calculations such as 10−2eV/atom for the total energy and 10−2eV for energy gaps and SOC splittings, the advantage of the SVLO method is particularly consider- able for the total energy. In contrast, the SV energy gap reaches the target precision at a number of empty states smaller than the number of LO basis functions, and the corresponding SOC splitting requires approximately 75 empty states. Dirac-type LOs turn out to be significant for the SOC splitting which increases by 0 .1 eV (Table III) upon adding four p1/2-type LOs, each of them contributing three degenerate basis functions (Table II). Their effect on the energy gap is negligible. The convergence behav- ior of the energy gap and the SOC splitting with respect to the number of basis functions is comparable to that of the SVLO method with SR LOs (Fig. 2). Contrarily, the total energy converges to a worse precision (within 10−4eV/atom). Also with Dirac-type LOs the analyzed quantities reach the targeted precision with a few empty5 FIG. 2: Convergence behavior of total energy, energy gap, and SOC splitting in Xe, MoS 2, PbI 2, and CsPbI 3, with respect to the number of basis functions used in the SV(LO) methods. Note the logarithmic scale on the y-axes. For the energy differences, the NP results serve as a reference. In the NP reference calculations, we employ sets of por p1/2LOs, depending on the method to compare with. Green diamonds stand for the SV method with SR LOs. All other results are obtained with the SVLO method, using different types of LOs: those obtained using SR (Dirac-type) orbitals are indicated by blue circles (red triangles). The vertical lines mark the respective number of LO basis functions. For PbI 2, we display δ2 SOCand the energy difference δ1 SOC(both indicated in Fig. 3). The gray shaded areas are guides to the eye for highlighting the points which are within the target precision (10−2eV/atom for the total energy and 10−2eV for the other quantities). states in addition to the LO basis functions. In the Ap- pendix, we explain why the SV and the SVLO method do not converge to the same precision.B. MoS 2 The transition-metal dichalcogenide MoS 2is among the most studied 2D materials, and a candidate for many6 TABLE III: Energy gaps, Eg, and SOC splittings, δSOC, of the considered materials computed with the SVLO method for different sets of LOs. For comparison, scalar-relativistic (SR) results are shown. For Bi 2Te3, that does not exhibit any SOC splitting, we show the energy difference between the highest valence band (VB) and the lowest conduction band (CB) at Γ, EΓ→Γ. Note that in this material, SOC not only changes the magnitude of the gap but also the position of the VBM and the CBm. Both are again altered when Dirac-type LOs are considered. MethodEg[eV] δSOC[eV] EΓ→Γ[eV] Xe MoS 2PbI2CsPbI 3 Bi2Te3 Xe MoS 2PbI2(δ1 SOC) PbI 2(δ2 SOC) CsPbI 3Bi2Te3 SR 6.22 1.78 2.20 1.64 0.25 (Γ →Γ) - - 0.94 - - 0.25 SVLO ( p)5.79 1.71 1.66 0.82 0.10 (B →B)1.30 0.15 1.25 0.63 0.71 0.58 SVLO ( p1/2)5.79 1.40 0.55 0.03 (D →C)1.40 1.45 0.68 0.76 0.69 FIG. 3: Band structures of Xe (upper-left panel), Mo 2(upper-right panel), PbI 2(lower-left panel), and CsPbI 3 (lower-right panel) computed with different methods and types of local orbitals. Black lines correspond to SR calculations and blue (red) lines to the SVLO method without (with) Dirac-type orbitals. The VBM is set to 0. At the right of each panel, we zoom into the corresponding region indicated by a gray box. applications in optoelectronics. SOC reduces the energy gap by 0.07 eV only (Table III), caused by a splitting of the VBM. Although this splitting is rather small, i.e., 0.15 eV, it is fundamental as, being not considered, could lead to the unphysical prediction of an indirect band gap [41]. Moreover, the splitting at the K-point of the Brillouin zone (BZ) is essential for the accurate descrip- tion of the optical spectra [1, 2]. Regarding the con-vergence behavior (second column of Fig. 2), we observe a small improvement of the SVLO method over the SV method for the total energy: With around 40 basis func- tions, the SVLO (SV) method reaches a precision of the order of 10−3eV/atom (10−2eV/atom). For the energy gap and the SOC splitting, both methods reach conver- gence with a few basis functions and reproduce the NP treatment with a precision of the order of 10−4eV.7 C. PbI 2 Lead iodide, PbI 2, is a semiconductor used for detec- tors, and it is also a precursor for the heavily inves- tigated solar-cell materials, the lead-based halide per- ovskites. Like in the latter, in PbI 2, the SOC effects are massive. The band gap reduces by 0.54 eV (Table III, Fig. 3) and (disregarding spin) the two-fold degenerate second conduction band (CB) experiences a splitting of δ2 SOC= 0.63 eV. There is also an increase in the energy distance between the CBm and the second CB which is 0.31 eV (Table III). For convenience, we label this en- ergy difference as δ1 SOC. For PbI 2, the advantages of the SVLO method over SV are considerable. With a num- ber of basis functions comparable to the number of LO functions (here 73, see Table II), the SVLO method has reached the target precision for all considered quantities (Fig. 2). Contrarily, the SV method, requires basically all empty states ( ∼375 basis functions) for reaching the target precision for the total energy; ∼225 empty bands are needed for the energy gap and ∼150 for δ1 SOC, while only∼10 empty states for δ2 SOC. Except for the total en- ergy, for which the SV method converges to a precision of the order of 10−2eV/atom and the SVLO method of the order of 10−5eV/atom, both approaches converge to comparable precision. For an accurate prediction of the electronic structure, p1/2-type LOs are crucial (Fig. 3). We add 4 for each species, with a total of 36 LOs basis functions (see Ta- ble II). They reduce the energy gap further by 0.26 eV and increase δ1 SOCby additional 0.20 eV. δ2 SOCincreases by 0.05 eV only (Table III). The convergence behavior with p1/2-type LOs is overall comparable to that with SR LOs. Note that the two curves appear shifted by these 36 additional basis functions. Although this number of LO basis functions is considerable for such a system (see Table II), the speed-up with respect to the SV method is significant also when Dirac-type LOs are employed. D. CsPbI 3 CsPbI 3is among the most studied inorganic metal halide perovskites [23, 42]. We consider it in the or- thorhombic γ-phase that contains 20 atoms. Being com- posed by three heavy elements, SOC effects are enor- mous. The band gap decreases from 1.64 eV with SR to 0.82 eV when SOC is considered (Table III). This is caused by a 0.71 eV splitting of the (disregarding spin) two-fold degenerate CBm (Fig. 3). When Dirac-type LOs are added, the gap further reduces by 0.27 eV, while the splitting increases by only 0.05 eV (Table III). Although SV and SVLO( p) converge to the same re- sults within the target precision, the computational effort required for the two approaches is noticeably different. In the limit of large unit cells –CsPbI 3is the largest one con- sidered here– the dominant contribution to the run time comes from the tasks that scale cubically with respectto the system size. These tasks include the construction of the Hamiltonian matrices and the diagonalization. As shown in Fig. 2, a converged SV calculation requires that essentially all unoccupied bands are included for solving the full problem. In this light, SV does not offer any advantage over the NP approach. In contrast, to con- verge the SVLO( p) calculation, it is sufficient to use a significantly smaller basis with Nocc= 228, NLO= 496, andNunocc∼0 (see Table II). Taking into account the spin degrees of freedom, the size of the Hamiltonian ma- trix in the SV step is ∼1500. As discussed above, di- agonalization is also required in the FV step, where the dimension of the SR Hamiltonian is ∼3800. This step is therefore the most computationally intensive in this ex- ample. Compared to the NP calculation, we find that total time spent on the FV and SV steps is reduced by a factor of 3.6. Finally, the inclusion of p1/2-type LOs increases NLOto 736 and thus also slightly increases the size of the SV diagonalization problem. E. Bi 2Te3 Bi2Te3is a topological insulator with a single Dirac cone at Γ [43, 44]. It is characterized by strong SOC ef- fects, shifting the fundamental band gap from Γ to an off- symmetry point in the mirror plane of the first Brillouin zone that is displayed in the bottom panel of Fig. 4. The positions of the VBM and CBm are highly sensitive to the structure and the choice of the exchange-correlation functional, thus there are controversial results present in the literature. Ref. 45 presents an overview of this diver- sity that increases when more accurate methods, such as theGW approximation, are applied [46, 47]. All these aspects together make Bi 2Te3computationally challeng- ing. Bi2Te3crystallizes in a rhombohedral structure with R-3m symmetry, shown in the top panel of Fig. 4. It consists of five layers, with alternating Te and Bi sheets, repeated along the z-direction. There are two chemically inequivalent Te sites. Including SOC, the band structure undergoes signifi- cant changes that are further enhanced when Dirac-type LOs are added [48, 49] (top and middle panels of Fig. 5). A relevant difference is observed at Γ where the valence band (VB) and the CB obtained from SOC calculations show a hump as a consequence of the band-inversion char- acteristic of this material [43, 45]. Differently from sim- ilar topological insulators, the hump is well preserved in spinor GW calculations, which include the off-diagonal elements of the self-energy, even though the band disper- sion is strongly altered [47]. SR calculations lead to a direct band gap of 0.25 eV at Γ (Table III). By adding SOC –but no Dirac-type LOs– it reduces to 0.10 eV and is located at point B=(0.67, 0.58, 0.58), which appears sixfold in the BZ. Our results are comparable with those of Ref. 49 and Ref. 50. In the former, a direct band gap of 0.13 eV at (0.667,0.571, 0.571) was measured, while in8 Quintuple layerabc BiTe2Te1 xyzkxkykz FIG. 4: Top: Crystal structure of Bi 2Te3, built by Bi (pink) and two chemically inequivalent Te atoms (Te 1 orange, Te 2gold). Bottom: Corresponding Brillouin zone. The mirror plane containing the points depicted in the band structure in Fig. 5 is indicated in red. the latter, a value of 0.11 eV was computed, but different from our result, it was reported to be indirect. However, VBM and CBm are very close to each other being lo- cated at (0.652, 0.579, 0.579) and (0.663, 0.568, 0.568), respectively. One may assign these differences to the use of different k-grids and crystal structures (here we use a= 10 .44˚A and θ= 24 .27◦[33], while Refs. [49, 50] use an experimental structure with a= 10 .48˚A and θ= 24.16◦). By adding 4 p1/2-type LOs for each species, i.e., a total of 60 basis functions (Table II), the gap reduces to 0.03 eV and becomes indirect. In the bottom panel of Fig. 5, we observe that the VB is lowered at B and raised at D=(0.52, 0.35, 0.35) where the VBM is now located. The CB is not altered at B but lowered at C=(0.65, 0.54, 0.54) which is the approximate location of the CBm (the reso- lution being limited by the 48 ×48×48k-mesh). Points C an D are six-fold degenerate. D is located between Γ and A=(0.64, 0.43, 0.43), which lies on the path between ZandU. C and B are close to Z→F. Larson [48] and Huang and coworkers [49] obtained gaps of 0.05 eV FIG. 5: Band structure of Bi 2Te3, computed without (top panel) and with SOC (other panels). The coordinates of the high-symmetry points are U (0.823,0.339, 0.339), Z (0.5,0.5,0.5), F (0.5,0.5,0.0), L (0.5,0.0,0.0); those of points A, B, C, and D are given in the text. The dashed vertical lines in the two top panels indicate the position of point D. The bottom panel zooms into the region of the band edges, showing the direct (indirect) band gap computed with p(p1/2) LOs. Note that, differently from Fig. 3, the energy zero is not at the VBM but in the middle of the band gap. and 0.07 eV, respectively, with p1/2-type LOs. The lo- cations of the band extrema slightly differ between the three works where ours is in better agreement with that of Larson [48]. As evident from Fig. 6, for Bi 2Te3, our new method has clear advantages over the conventional SV method, which reaches the target precision for the total energy only with basically all available FV KS states (about9 ∼1000, see Table II), while the SVLO method requires a basis-set size comparable with the number of LO basis functions (141 for the p-set and 201 for the p1/2-set). The SVLO method converges in either case within a precision of 10−4eV/atom. Like for the other materials, the elec- tronic structure obtained by SV, converges faster than the total energy, but the convergence is still not compa- rable with that of the SVLO method. To obtain an en- ergy gap within a precision of 10−2eV, the SV method requires about twice the number of basis functions (with- out including the occupied states); to obtain EΓ→Γwith a precision of ∼10−4eV, the basis size needs to be fur- ther doubled. The band gap converges to a precision of ∼10−5eV, while the SV method cannot go lower than 10−4eV. V. DISCUSSION AND CONCLUSIONS In this work, we have introduced a novel approach – the SVLO method– to treat spin-orbit coupling in DFT calculations efficiently. It allows us to obtain rapid con- vergence and highly precise results, e.g., band energies within the order of 10−4eV or even better. SOC split- tings and total energies within a precision of 10−2eV and 10−2eV/atom, respectively, can actually be obtained with a number of basis functions that is comparable to the number of occupied states plus a set of LOs. Its effi- ciency is owing to the fact that SOC effects mainly come from regions around the atomic nuclei where atomic-like functions play the major role in describing them. We have demonstrated this method with examples of very different materials. The use of the SVLO method is most efficient when SOC effects are strong. In the cases, we also observe significant contributions of p1/2LOs. Ob- viously, the overall gain of our method is getting more pronounced the bigger the system is. In summary, by providing a method that allows for reliable and efficientcalculations of SOC, our work contributes to obtaining highly-accurate electronic properties at the DFT level. ACKNOWLEDGMENTS This work was supported by the German Research Foundation within the priority program SPP2196, Perovskite Semiconductors (project 424709454) and the CRC HIOS (project 182087777, B11). A.G. ac- knowledges funding provided by European Regional Development Fund via the Central Finance and Con- tracting Agency of Republic of Latvia under the grant agreement 1.1.1.5/21/A/004. Partial support from the European Union’s Horizon 2020 research and innovation program under the grant agreement N °951786 (NOMAD CoE) is appreciated. APPENDIX In the examples discussed above, the SV method of- ten converges to worse precision than the SVLO method. This may appear counter intuitive since the two methods should be equivalent if the SV basis includes all available FV KS states. The reason for the seeming discrepancy comes from the fact that the LAPW basis-set size may be different at different k-points, depending on their symme- try. In contrast, in the SV method, the size of the basis is controlled by an input parameter and limited by the number of available FV KS orbitals. In our implementa- tion, the same number is considered for all k-points. 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B 68, 125210 (2003).12 FIG. 6: Convergence behavior of total energy, energy gap, and energy difference between the highest VB and the lowest CB at Γ with respect to the number of second-variational basis functions, ˜NSV(LO) b, for Bi 2Te3. FIG. 7: Same as Fig. 2 for Xe, but for one k-point only.

0808.0069v1.Electric_field_driven_long_lived_spin_excitations_on_a_cylindrical_surface_with_spin_orbit_interaction.pdf

arXiv:0808.0069v1 [cond-mat.mtrl-sci] 1 Aug 2008Electric-field driven long-lived spin excitations on a cyli ndrical surface with spin-orbit interaction P. Kleinert∗ Paul-Drude-Intitut f¨ ur Festk¨ orperelektronik, Hausvogteiplatz 5-7, 10117 Berlin, Germany (Dated: November 22, 2018) Abstract Based on quantum-kinetic equations, coupled spin-charge d rift-diffusion equations are derived for a two-dimensional electron gas on a cylindrical surface . Besides the Rashba and Dresselhaus spin-orbit interaction, the elastic scattering on impurit ies, and a constant electric field are taken into account. From the solution of the drift-diffusion equati ons, a long-lived spin excitation is identified for spins coupled to the Rashba term on a cylinder w ith a given radius. The electric-field driven weakly damped spin waves are manifest in the componen ts of the magnetization and have the potential for non-ballistic spin-device applications . PACS numbers: 72.25.Dc,72.20.My,72.10.Bg 1I. INTRODUCTION In the emerging field of spintronics, a main issue is the avoidance of sp in randomization while stimulating controlled rotations of spins in a spin field effect trans istor. Therefore, the recent proposal1for a spintronic device, which operates in the non-ballistic regime, re ceived considerable interest. According to this design, spin relaxation is su ppressed, when the Rashba and Dresselhaus spin-orbit interaction (SOI) constants a re tuned by an external gate voltage so that their couplings become equal. In fact, the sup pression is a consequence of an exact spin-rotation symmetry,2,3which leads to a strong anisotropy of the in-plane spin-dephasing time. At the presence of an in-plane electric field, th e persistent spin helix is converted into a field-dependent internal eigenmode.4In close analogy to space-charge waves in crystals, these field mediated spin excitations can be probe d by optical grating techniques.3Both spin and charge pattern, which are generated by polarized las er beams, provide the required wave vector for the excitation of internal eig enmodes. Suppressed spin- relaxation occurs not only in (001) GaAs/Al xGa1−xAs quantum wells with balanced Rashba and Dresselhaus SOI strengths but also in (110) quantum wells with D resselhaus coupling.5 Most activities in the field of spintronics that are based on the Rashb a and Dresselhaus SOI refer to a plane two-dimensional electron gas (2DEG), which is c onfined by a semicon- ductor quantum well. However, the diversity of spin-related pheno mena markedly increases for different geometries of the 2DEG. In dependence on the curva ture of the surface, in which the 2DEG resides, additional contributions to the SOI appear that may lead to new spin effects. Examples of current interest provide microtubes fab ricated by exploiting the self-rolling mechanism of strained bilayers. These rolled-up structu res exhibit pronounced optical resonances6arising from micron-sized cylindrical resonators or give rise to nove l magnetoresistance oscillations, which were observed in the ballistic t ransport of electrons on cylindrical surfaces.7For non-ballistic spintronic device applications, the prediction of a conserved spin component, which arises when the Rashba coupling constant γ1equals the quantity /planckover2pi1/2m∗R(withRbeing the radius of the cylinder and m∗the effective mass of the 2DEG) is most interesting.8The identification of this novel long-lived spin mode by fabricated curved samples seems to be feasible with the present -day technology.9,10,11,12 It is the aim of this paper to study the field dependence of the predic ted spin helix by a systematic consideration of electric-field mediated eigenmodes of a spin-charge coupled 22DEG that is confined to a circular cylinder. Both Rashba and Dresse lhaus SOIs as well as spin-independent elastic impurity scattering are taken into accoun t. II. BASIC THEORY While in classical mechanics the restricted motion of particles on curv ed surfaces is un- ambiguously described by equations of motion, the quantum-mecha nical study of curved systems starts from two different perspectives. In the first, wid ely used method, the three- dimensional Schr¨ odinger equation is converted to its two-dimensio nal counterpart by an appropriate confining procedure.13This approach naturally accounts for the fact that the curved structures are embedded in a three-dimensional space, in which electric and mag- netic fields could be present. The second alternative description of the carrier dynamics on curved samples completely rests on a two-dimensional model.14In our study of SOI on the surface of a cylinder, we apply the widely accepted first approach, which was already used for studying spin effects on curved surfaces.15,16,17,18The second-quantized version of the Hamiltonian has the form16 H0=∞/integraldisplay 0dϕ 2π/summationdisplay kz/braceleftbigg/summationdisplay sa† kzs(ϕ)/bracketleftbigg/planckover2pi12k2 z 2m∗+/hatwidep2 ϕ 2m∗/bracketrightbigg akzs(ϕ) (1) +γ1/summationdisplay s,s′a† kzs(ϕ)[σz ss′/hatwidepϕ−/planckover2pi1kzΣss′]akzs′(ϕ) +γ2/summationdisplay s,s′a† kzs(ϕ)/bracketleftbigg1 2(Σss′/hatwidepϕ+/hatwidepϕΣss′)−/planckover2pi1kzσz ss′/bracketrightbigg akzs′(ϕ)/bracerightbigg , in which Rashba and Dresselhaus contributions appear with the coup ling constants γ1and γ2, respectively. The creation [ a† kzs(ϕ)] and annihilation [ akzs(ϕ)] operators depend on the spin index s, the wave vector component kzalong the axis of the cylinder, and the angle ϕ. The SOI terms include the Pauli matrices σ, the transverse momentum operator /hatwidepϕ, and a matrix /hatwideΣ that introduces off-diagonal elements with respect to the spin ind ex. These quantities are defined by /hatwidepϕ=−i/planckover2pi1 R∂ ∂ϕ,/hatwideΣ = 0−ie−iϕ ieiϕ0 . (2) 3The periodic boundary conditions on the cylinder surface are accou nted for by a discrete Fourier transformation, which is applied in the form akz↑(ϕ) =∞/summationdisplay m=−∞eimϕakzm↑, akz↓(ϕ) =eiϕ∞/summationdisplay m=−∞eimϕakzm↓. (3) By this transformation, the projection of the total angular mome ntum on the cylinder axis appears and the Hamiltonian simplifies considerably. In addition to the SOI, both elastic scattering on impurities with the short-range coupling strength Uand an external electric fieldE(applied along the cylinder axis) are taken into account. As it is assum ed throughout the paper that the radius Rof the cylinder is much larger than the lattice constant, we introduce the electron momentum vector k= (kϕ,kz,0), kϕ=/parenleftbigg m+1 2/parenrightbigg /R, (4) in order to express the Hamiltonian in a form that is very similar to the c ase of planar geometry. We obtain H=/summationdisplay k,sε(k)a† ksaks+/summationdisplay k/summationdisplay s,s′(/planckover2pi1ω1(k)·σss′)a† ksaks′ (5) +U/summationdisplay k,k′/summationdisplay sa† ksak′s−ieE·/summationdisplay k,s∇κa† k−κ 2sak+κ 2s/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle κ=0, with the parabolic dispersion relation ε(k) =/planckover2pi12k2 2m∗−/planckover2pi1 2R/parenleftbigg γ1−/planckover2pi1 4m∗R/parenrightbigg . (6) The main part of the SOI is included in the vector ω1(k) =/parenleftbigg 0,−(γ1kz−γ2kϕ),kϕ(γ1−/planckover2pi1 2m∗R)−γ2kz/parenrightbigg . (7) The model description of spin-independent scattering onimpurities inEq. (5) hasthe advan- tage of simplicity, permitting us an exact treatment of scattering in the Born approximation via the scattering time τdefined by 1 τ=2πU2 /planckover2pi1/summationdisplay k′δ(ε(k)−ε(k′)). (8) All information about the spin-orbit coupled electron ensemble is pro vided by the spin- density matrix fs s′(k,k′|t) =/angb∇acketlefta† ksak′s′/angb∇acket∇ightt, (9) 4which is calculated from quantum-kinetic equations. A transparent physical interpretation of final results is facilitated by considering the projected spin vect or on a local trihedron. This transformation is achieved by f=/summationdisplay s,s′fs s′Sss′, (10) with the following transformation matrices Sϕ=1 2 0ie2iϕ −ie−2iϕ0 , Sz=1 2 1 0 0−1 , Sr=1 2 0e2iϕ e−2iϕ0 ,(11) which project to the cylinder axis ( Sz), as well as to the tangential ( Sϕ) and normal ( Sr) directions. To proceed, the wave vectors are shifted according t ok→k+κ/2 andk′→ k−κ/2 withkϕ= (m+m′+1)/2Randκϕ= (m−m′)/R. The derivation of spin-charge coupled kinetic equations is carried out by applying the same steps as in our previous study of the planar geometry.19The final result is expressed by the kinetic equations ∂ ∂tf(k,κ|t)−i/planckover2pi1 m∗k·κf+iω(κ)·f+e /planckover2pi1E·∇kf=1 τ(f−f), (12) ∂ ∂tf(k,κ|t)−i/planckover2pi1 m∗(k·κ)f−2ω(k)×f+iω(κ)f+e /planckover2pi1(E·∇k)f =1 τ(f−f)−/planckover2pi1ω(k) τ∂ ∂ε(k)f+1 τ∂ ∂ε(k)/planckover2pi1ω(k)f, (13) in which a new SOI vector appears given by ω(κ) = (ω1y(κ)sin(2ϕ),ω1y(κ)cos(2ϕ),−ω1z(κ)). (14) On the right-hand side of Eq. (13), there remain scattering contr ibutions that are propor- tional to the SOI. These terms are necessary for a consistent tr eatment of a homogeneous 2DEG. The bar over quantities in Eqs. (12) and (13) indicates an inte gration over the polar angleαof the vector k=k(cosα,sinα,0). The kinetic Eqs. (12) and (13) serve as a starting point for various studies of spin effects on cylindrical surfaces. We mention only the field- induced spin accumulation and the influence of the spin degree of fre edom on the charge current (as well as the appearance of a ”spin current” on the cylin der).19In this paper, we do not follow this interesting line of reasoning but look for a solution of Eqs. (12) and (13) in the long-wavelength and low-frequency drift-diffusion regime. 5The envisaged macroscopic behavior of the coupled spin-charge sy stem is established during an evolution period, in which a nonequilibrium spin polarization and charge density still exist, whereas the energy of particles already thermalized.20In this transport regime, the following separation ansatz for the mean components fandfis justified f(k,κ|t) =−F(κ|t)n′(ε(k)) dn/dε F,f(k,κ|t) =−F(κ|t)n′(ε(k)) dn/dε F, (15) wheren(ε(k)) denotes the Fermi distribution function and n=/integraltext dερ(ε)n(ε) is the equilib- rium carrier density (with ρ(ε) being the density of states). n′(ε(k)) is a short-hand notation fordn/dε(k). Adopting this approximation also for the field contributions on the left-hand side of Eqs. (12) and (13), we obtain a set of linear equations for th e components of the spin-density matrix. The solution is expanded with respect to κand integrated over the angleα. This procedure leads to coupled equations for the charge Fand spin Fdistribu- tion functions, the solution of which is easily integrated over the ene rgyε(k). The resulting coupled spin-charge drift-diffusion equations take the form /bracketleftbigg∂ ∂t−iµE·κ+Dκ2/bracketrightbigg F+i µBω(κ)·M+2im∗τ /planckover2pi1µB([Λ×ω(κ)]·M) = 0,(16) /bracketleftbigg∂ ∂t−iµE·κ+Dκ2+/hatwideΓ/bracketrightbigg M+e m∗cM×Heff −χ(/hatwideΓHeff)F n−2im∗µ c[Λ×ω(κ)]F=G, (17) for the charge density Fand the magnetization M=µBFwithµB=e/planckover2pi1/2m∗cbeing the Bohr magneton. The vector Gon the right-hand side of Eq. (17) accounts for the source of an external spin generation. Furthermore, Dandµdenote the diffusion coefficient and the mobility that are related to each other via the Einstein relation µ=eDn′/n. The Pauli susceptibility is given by χ=µBn′. Scattering times that refer to various spin components are collected by the symmetric matrix /hatwideΓ, which is given by /hatwideΓ =4Dm∗2 /planckover2pi12 a2 11+a2 12+a2 31+a2 32−(a22a32+a21a31)−(a11a21+a22a12) −(a22a32+a21a31)a2 11+a2 12+a2 21+a2 22−(a12a32+a11a31) −(a11a21+a22a12)−(a12a32+a11a31)a2 21+a2 22+a2 31+a2 32 ,(18) where the quantities aijare expressed by the spin-orbit coupling constants a11=γ2sin(2ϕ), a21=γ2cos(2ϕ), a12=−γ1sin(2ϕ), a22=−γ1cos(2ϕ),(19) 6a31=−/parenleftbigg γ1−/planckover2pi1 2m∗R/parenrightbigg , a32=γ2. (20) The electric field is accounted for by the vector Λ= (a21µEϕ+a22µEz,a31µEϕ+a32µEz,a11µEϕ+a12µEz), (21) from which an effective magnetic field Heff=2m∗2c e/planckover2pi1(Λ+2iDω(κ)), (22) is derived that enters Eq. (17) for the field-induced magnetization . The appearance of a magnetic field Heff, which is solely due to the electric field, illustrates why there is a perfect electric-field analog of the Hanle effect.21 The drift-diffusion Eqs. (16) and (17) for the charge density F(κ,t) and magnetization F(κ,t) provide the basis for the study of many spin-related phenomena o f a curved 2DEG in the drift-diffusion regime. Here, we shall focus on spin-related eig enmodes of the system. III. LONG-LIVED SPIN WAVES A solution of Eq. (17) is searched for under the condition that the r etroaction of spin on the charge density can be neglected so that the carrier density is g iven by its equilibrium value (F=n). Performing a Laplace transformation with respect to the time va riabletand introducing the abbreviations M′=M−χHeff,Σ =s−iµE·κ+Dκ2, (23) withsbeing the Laplace variable, Eq. (17) is converted into the linear equa tions, ΣM′+/hatwideΓM′+e m∗cM′×Heff (24) =M(0)+G/s−2im∗µ cΛ×ω(κ)n−χΣHeff, which are symbolically written as /hatwideTM′=Q. Eigenmodes of the spin subsystem are calcu- lated from the zeros of the determinant of the matrix /hatwideT. A simple but cumbersome algebra leads to the result det/hatwideT= Σ(σ2+ω2 H)+g2/parenleftbigg σ+(µE)2 D/parenrightbigg , (25) 7inwhich theshort-handnotations ωH= (e/m∗c)Heffandσ= Σ+g1areused. Thecoupling constants g1andg2are given by g1= 24Dm∗2 /planckover2pi12/bracketleftbigg γ2 1+γ2 2−/planckover2pi1 2m∗R/parenleftbigg γ1−/planckover2pi1 4m∗R/parenrightbigg/bracketrightbigg , (26) g2=/parenleftbigg4Dm∗2 /planckover2pi12/parenrightbigg2/bracketleftbigg γ2 2−γ1/parenleftbigg γ1−/planckover2pi1 2m∗R/parenrightbigg/bracketrightbigg2 . (27) The cubic equation det /hatwideT= 0 with respect to the Laplace variable shas three solutions, whichgivethedispersionrelationsofspinexcitations. Mosteigenmod eshaveafinitelifetime. However, there is one long-lived spin excitation, whose damping comp letely disappears for a given wave number κz. This mode appears for a model without any Dresselhaus SOI (γ2= 0), when the coupling constant γ1matches the quantity /planckover2pi1/2m∗R. In this case, we obtain (s→iω) ω1,2=−µEz(κz±K)−iD(κz±K)2, (28) withK= 2m∗γ1//planckover2pi1being a wave number that is built from the Rashba spin-orbit coupling constant γ1. This soft mode becomes increasingly undamped in the limit κz→K. The persistent spin mode of this kind, which is a consequence of a new spin -rotation symmetry, has no counterpart in the planar Rashba model and is a distinct feat ure that solely appears on a cylinder surface. In order to excite the persistent spin wave, a regular lattice of spin polarization Qr perpendicular to the cylinder surface is provided by laser pulses. Fo r simplicity, the spin generation is assumed to have the form Qr=Qr0 2[δ(κz−κ0)+δ(κz+κ0)]. (29) Under the condition Qϕ=Qz= 0, the solution M=χHeff+/hatwideT−1Qof Eq. (17) is expressed by Mz=µEzK σ2+ω2 HQr, Mr=σ σ2+ω2 HQr. (30) Inthederivationoftheseequations, itwasconsideredthattheinv erse Fouriertransformation with respect to κϕleads toϕ= 0. The inverse Laplace transformation and the integration overkzgive for the non-vanishing components of the field-mediated magne tization the final 8results Mr(z,t) =Qr0 2/braceleftbigg e−D(κ0+K)2tcos[κ0z+µEz(κ0+K)t] +e−D(κ0−K)2tcos[κ0z+µEz(κ0−K)t]/bracerightbigg , (31) Mz(z,t) =M(−) z(z,t)−M(+) z(z,t), (32) with M± z(z,t) =µEz (µEz)2+(2Dκ0)2Qr0 2/braceleftbigg 2Dκ0cos[κ0z+µEz(κ0±K)t] −µEzsin[κ0z+µEz(κ0±K)t]/bracerightbigg e−D(κ0±K)2t. (33) Both components MzandMrconsist of a strongly and weakly damped oscillating term. Under the resonance condition κ0=K, the first mode quickly disappears, whereas the second mode becomes completely undamped. A smooth dependence on the electric field Ez persists in the magnetization Mzalong the cylinder axis. A slight detuning of the resonance, however, leads to the appearance of an electric-field driven spin wa ve, the damping of which is extremely weak. The frequency of this long-lived spin excitation is d irectly controlled by the applied electric field. The situation is similar to the persistent spin h elix of a planar 2DEG. Therefore, it is supposed that the robust spin wave on a cylin der and its direct manipulation by an electric field has the potential to be utilized in futur e spintronic device applications. IV. SUMMARY Nanostructures with a great variety of novel geometries like curv ed graphene systems and rolled-up 2DEG are now experimentally available. Hence, the rigorous theoretical study of the dynamics on such curved surfaces became a subject of rec ent interest. Especially spin effects have been treated because the curvature of the sur face gives rise to additional contributions to the SOI. Consequently, the number of possible sp in effects considerably increases in nanostructures with curved geometries. This observ ation further stimulates activities in the field of spintronics. A key point regarding spin-field-e ffect transistors is the exclusive manipulation of spin by means of an electric field. Particu larly attractive is the proposal for a device working in the non-ballistic regime, where s pin scattering in a 9planar 2DEG is suppressed due to a spin-rotation symmetry.1A similar effect that occurs on the surface of a cylinder was studied in the present paper. Base d on quantum-kinetic equations for the spin-density matrix, rigorous coupled spin-char ge drift-diffusion equations were systematically derived for a cylinder, whose radius is much large r than the lattice constant. From the solution of these equations, the dispersion re lation of field-dependent spin eigenmodes are identified. In general, there are three damped spin excitations, the character of which is determined by the coupling constants γ1andγ2of the Rashba and Dresselhaus SOI. For the pure Rashba model ( γ2= 0), a long-lived spin wave exists, when theradius Rofthecylinder matchesthecondition R=/planckover2pi1/2m∗γ1. Thisfindingisofparticular interest as an applied electric field stimulates a nearly undamped spin w ave. The excitation mechanism of spin waves has the same character as the excitation o f space-charge waves, which are normally strongly damped. Nevertheless, space-charge waves in crystals were clearly demonstrated in experiment. To my knowledge, completely un damped space-charge waves do not exist. Their damping is reduced, however, in the regime of negative differential conductivity due to a negative Maxwellian relaxation time.22The complete disappearance of the damping of an excitation is a novelty that occurs in special spin subsystems with a k-linear SOI. The above mentioned peculiarity of the Rashba model on a cylindrical surface has no counterpart in a planar 2DEG. 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2305.05830v1.Inverse_orbital_Hall_effect_and_orbitronic_terahertz_emission_observed_in_the_materials_with_weak_spin_orbit_coupling.pdf

1 Inverse orbital Hall effect and orbitronic terahertz emission observed in the materials with weak spin -orbit coupling Ping Wang,1 Zheng Feng,2 Yuhe Yang,1 Delin Zhang,1,* Quancheng Liu ,3 Zedong Xu,1 Zhiyan Jia,1 Yong Wu,4 Guoqiang Yu,5 Xiaoguang Xu,4 Yong Jiang1,* 1 Institute of Quantum Materials and Devices, School of Electronic and Information Engineering ; State Key Laboratory of Separation Membrane and Membrane Processes, Tiangong University, Tianjin, 300387, China. 2 Microsystem & T erahertz Research Center, C AEP , Chengdu, 610200, China. 3 School of Information Engineering, Southwest University of Science and Technology, Mianyang, 621010 , China. 4 School of Materials Science and Eng ineering, University of Science and Technology Beijing, Beijing, 100083 , China . 5 Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing, 100190, China. These authors contributed equally: Ping Wang, Zheng Feng, Yuhe Yang, Delin Zhang . *Corresponding authors. Email: zhangdelin@tiangong.edu.cn (D.L.Z.), and yjiang@tiangong.edu.cn (Y.J.) 2 Abstract : The Orbital Hall effect , which originat es from materials with weak spin -orbit coupling , has attracted considerable interest for spin-orbitronic applications . Here, we demonstrate the inverse effect of the orbital Hall effect and observe orbitronic terahertz emission in the Ti and Mn materials . Through spin -orbit transition in the ferromagnetic layer , the generated orbital current can be convert ed to charge current in the Ti and Mn layers via the inverse orbital Hall effect . Furthermore , the inserted W layer provides an additional conversion of the orbital -charge current in the Ti and Mn layers, significantly enhanc ing the orbitronic terahertz emission . Moreover , the orbitronic terahertz emission can be manipulated by cooperati ng with the inverse orbital Hall effect and the inverse spin Hall effect in the different sample configurations . Our results not only discover the physical mechanism of condensed matter physics but also pave the way for designing promising spin-orbit ronic devices and terahertz emitters . 3 INTRODUCTION The materials with spin -orbit coupling (SOC) possess two distinct channels of angular momentum, namely spin angular momentum (S) and orbital angular momentum ( L), which generate spin current and orbit current in a transverse direction with an applied electric field1-8. Recently, charge -spin and spin -charge conversions have been demonstrated in the heavy metals (HM) (e.g. Ta, W) and quantum topological materials (e.g. topological insulators and semimetals) with strong SOC through spin Hall effect (SHE) or Rashba -Edelstein effect and inverse spin Hall effect (ISHE) , respectively1-4,9-12. These effects have opened up potential technical applications, including spin-orbit torque (SOT) devices and spintronic terahertz (THz) emitters11-18. However, due to the domination of the spin contribution in the nonmagnetic materials (NM) with strong SOC , the orbital contribut ion has not been given much attention . Recently, the orbital Hall effect (OHE) has been theoretically and experimental ly observed in NM with weak SOC, where the efficient orbital current is obtain ed with an applied electric field7,8,19. Through OHE, the charge current ( JC) can be converted to orbital current ( JL), and then to spin current ( JS), which can exert a n orbital torque to switch the magnetization of the ferromagnetic layer, thereby operating the orbitronic devices20-24. However, the partner of OHE, inverse orbital Hall effect (IOHE), as well as the orbitronic THz emitters exhibiting advantages of high emission efficiency, ultra - broad bandwidth, excellent performance, and flexible tunability remains elusive9,16-18,25,26. To remedy this, we choos e Ti and Mn as the OHE source to study IOHE and the efficiency of orbitronic THz emission originat ing from IOHE. RESULTS AND DISCUSSION Inverse orbital Hall effect 4 The SHE and I SHE phenomena involve the conversion of JC→JS and JS→JC in heavy metals or quantum material s with strong SOC3,10, where a transverse flow of spin angular momentum and voltage are generated , respectively , as shown in Fig. 1a. The direction of JS, JC and the conversion efficiency of JC→JS, JS→JC depend on the sign and value of spin Hall angle (θSH), as given by the equations JS ~ θSH σ×JC and JC ~ θSH JS×σ, where σ is the spin polarization2,4,16. However, t he OHE and IOHE phenomena refer to the conversion of JC→JL and JL→JC, respectively, in NM with weak SOC , resulting in a transverse flow of orbital angular momentum and voltage , as depicted in Fig. 1b. Through OHE, JC is convert ed to JL in the NM layer without relying on the strong SOC. JL is then transfer red to JS, which generate s orbital torque in the FM layer due to its SOC, as widely reported20-24. IOHE is the inverse process of OHE , where JS generated in the FM layer first converts to JL, then flows into the NM layer and convert s to JC. The direction of JL, JC and the conversion efficiency of JC→JL, JL→JC depend on the sign and value of orbital Hall angle ( θOH), which is given by the equations JL ~ θOH σOH×JC and JC ~ θOH JL×σOH, where σOH is the orbital polarization in OHE . θOH = σL (2e/ћ) is similar to θSH = σS (2e/ћ), where σS, σL, and are spin and orbital Hall conductivity and resistivit y, respectively8. To investigate IOHE as proposed above, we designed and fabricated the bilayered structure s of Co (2 nm) /X (4-60 nm) (X = Ti, Mn) and carr ied out the experiment s using THz emission spectroscopy18. The Co layer used here can effectively achieve JS→JL conversion due to its relatively large spin-orbital conversion efficiency (CCo), where the sign and value of CCo depend on the spin -orbit correlation R = <L·S> of the Co layer20,21 ,27. Meanwhile, Ti and Mn possess large OHE as demonstrated in theory a nd experiment7,8,19,21,28,29. Figure 2 a illustrates the physical mechanism of the JS→JL→JC conversion in the Co/X (X = Ti, Mn) bilayered structures . When a 5 femtosecond (fs) laser is pump ed into the Co layer, both JS and JL are simultaneously generated due to SOC of the Co layer (The possible contribution of JL can be directly generated in the Co layer by the laser ). JL will flow into the Ti or Mn layers and convert to JC,IOHE due to IOHE (indicated by the blue arrow) , while JS will directly convert to JC,ISHE due to ISHE in the Ti or Mn layers (label ed by the red arrow) . The phase of JC,ISHE and the conversion efficiency of JS→JC,ISHE depend on the sign and value of θSH of the Ti or Mn layers (θSH,Ti/Mn < 0)8,30, which can be identified from the polarity and amplitude of the THz signal16. Compar ed to ISHE , the phase and amplitude of JC,IOHE depend on the sign of CCoθOH,Ti/Mn, as it arises from the conversion of JS→JL→JC,IOHE (where CCo > 0 and θOH,Ti/Mn > 0)8,19,20 ,31. In the bilayered structures of Co/ X (where X = Ti, Mn), t he phase of JC,ISHE and JC,IOHE is opposite , and this can be expressed in the polarity of the THz signal. Although JC,ISHE arises from X (where X = Ti, Mn), it is negligible owing to the very small θSH,Ti/Mn of X (X = Ti, Mn)30. Therefore , JC,IOHE originat ing from Ti or Mn dominates the THz signal . Figure 2 b and Supplementary Figure 1 show the normalized THz signals of the Co (2 nm) /Ti (4-60 nm) and Co (2 nm) /Mn (4-60 nm) structures with different thickness es (d) of Ti and Mn (the normalized THz signals are obtained by normalizing the original THz signals to the laser absorbance of FM layer and the THz radiation impendence , which represents the spin/orbit -charge conversion efficiency9,32, see details in Methods ). The o bvious THz signals can be observed in the Co (2 nm) /Ti (4-60 nm) and Co (2 nm) /Mn (4-20 nm) structures , suggesting that the JS→JC conversion occurs . The polarit y of the THz signals is opposite when the samples were flipped due to the opposite direction of JL during the measurement (see Supplementary Fig. 2), further confirming the JL→JC conversion in the Co/Ti and Co/Mn structures . To eliminate the contribution of the substrate, Co, Mn , Ti, or MgO layers, we performed THz experiments on these layers , and the results are presented in 6 Supplementary Fig. 3. We found that the contribution of the THz signals from these layers is negligible . Moreover , to confirm the JS→JC conversion in the Co/Ti and Co/Mn structures due to IOHE, we measured the THz signal s of the Co (2 nm) /W (2 nm) and Co (2 nm) /Pt (2 nm) bilayered structure s, where JC,ISHE mainly originat es from ISHE in the W and Pt layer s16. If the THz signals of the Co (2 nm) /Ti (4 nm) and Co (2 nm) /Mn (4 nm) bilayered structures come from ISHE, th e polarit y of the THz signal s will be the same as that of the Co (2 nm) /W (2 nm) bilayered structure but opposite to that of the Co (2 nm) /Pt (2 nm) bilayered structure . This is because Ti, Mn and W have the same negative sign of θSH, while Pt has a positive sign of θSH (θSH,Ti = -3.6× 10-4, θSH,Mn = -1.9× 10-3, θSH,W = -3.3× 10-1, θSH,Pt = 1.2× 10-1)30,33,34. However, the polarit y of the THz signal s in the Co/Ti and Co/Mn bilayered structures is opposite to that of the Co/W bilayered structure but the same as that of the Co/Pt bilayered structure , as shown in Fig. 2c. The polarit y of the THz signal s for the Co/Ti and Co/Mn bilayered structures is consistent with the sign of CCo θOH,Ti /Mn, indicating that the THz signals in these structures are induced by IOHE. In addition, the absolute values of the normalized THz peak -to-peak signals (THz ΔPK , which represents the difference between the first peak and second peak ) of the Co (2 nm) /Ti (4- 60 nm) bilayered structures first increase up to the maximum at dTi = 40 nm and then decrease , persisting even at dTi = 60 nm, as depicted in Fig. 2d. This behavior is in agreement with the long orbital diffusion length of Ti ( λOD,Ti)19, as reported previously . In contrast to Ti, Mn has a relatively short orbital diffusion length (λOD,Mn). Enhanced orbitronic THz emission assisted by efficient spin -orbital current conversion To gain a deeper understand ing of the crucial JS→JL→JC conversion and the orbitronic THz emission based on IOHE , as well as the additional contribution from ISHE, we introduced inserted layers (W, Ti, Mn) with different SOC between the Co and X (X= Ti, Mn) layers. The 7 normalized THz signals of the Co/M/X (M=W, Ti, Mn; X= Ti, Mn) structures are p resented in Fig. 3 and Supplementary Fig. 4. When 2 -nm W with a strong SOC ( RW < 0, CW < 0) is inserted20,33, the amplitude of the THz signals for the Co (2 nm) /W (2 nm) /X (4 nm) (X = Ti, Mn) structures can be significantly enhanced compared to the Co (2 nm) /X (4 nm) (X = Ti, Mn) and Co (2 nm) /W (2 nm) bilayered structures, as shown Fig. 3a. Furthermore, Fig. 3a displays the corresponding difference of absolute value (|ΔPK (Co/W/ Ti)-(Co/W)| and |ΔPK (Co/W/ Mn)-(Co/W)|) of the THz ΔPK between the Co (2 nm) /W (2 nm) /X (4 nm) (X = Ti, Mn) and Co (2 nm) /W (2 nm) structures , as well as the THz |ΔPK | of the Co (2 nm) /X (4 nm) (X = Ti, Mn) bilayered structures. We have observed that |ΔPK (Co/W/ Ti)-(Co/W)| and |ΔPK (Co/W/ Mn)-(Co/W)| exhibit similar magnitudes, which are more than one order of the magnitude larger than that of the Co (2 nm) /X (4 nm) (X = Ti, Mn) bilayered structure s. These resu lts indicate that the insert ed W layer not only generates JC,ISHE (as indicated by the red arrow in Fig. 3b), but also contributes to JC,IOHE (as indicated by the blue arrow in Fig. 3b) in the Co (2 nm) /W (2 nm) /X (4 nm) (X= Ti, Mn) structures. The insert ed W layer provide s an additional large JL,W resulting from the coupling between L and S due to its large SOC (JL from the Co layer is small er compared to the W layer and is not labeled in the figure ). JL,W enter s the Ti or Mn layer s and convert s to JC,IOHE by IOHE ( JC,IOHE ~ CW θOH,T i/Mn JS). JC,IOHE and JC,ISHE in the Co (2 nm) /W (2 nm) /X (4 nm) (X = Ti, Mn) structures have the same phase. Thus , the orbitronic THz emission of the Co (2 nm) /W (2 nm) /X (4 nm) (X = Ti, Mn) structures primarily originates from JC ~ θSH,W JS + CW θOH,T i/Mn JS. However, when the Ti (4 nm) and M n (4 nm) layer s with weak SOC ( RTi,Mn < 0) are inserted, the |ΔPK (Co/Mn/Ti)- (Co/Mn)| and |ΔPK (Co/Ti/Mn)-(Co/Ti)| values for the Co (2 nm) /Ti (4 nm) /Mn (4 nm) and Co (2 nm) /Mn (4 nm) /Ti (4 nm) structures , respectively , are smaller than those of the Co (2 nm) /X (4 nm) (X = Ti, Mn) structures , as depicted in Fig. 3c. In the Co/Ti/M n and Co/Mn/T i structures, the 8 conversion of JS→JL predominantly occurs in the Co layer , and JL further converts to JC in the Ti and M n layers without an additional JL like in the W layer . Therefore, the orbitronic THz emission of the Co/Ti/Mn and Co/ Mn/Ti structures still primarily originates from the conversion of JS→JL by the SOC of the Co layer and JL→JC by IOHE of the Ti and Mn layers , as shown in Fig. 3d. Collaboration between inverse orbital and spin Hall effects Inspired by the physical mechanism of the Co/W/ X (X = Ti, Mn) structures, we investigate d the orbitronic THz emission of the different structures : Co (2 nm) /Ti (4-100 nm)/W (2 nm) and Ti (4-100 nm)/Co (2 nm) /W (2 nm) , derived from the combined contribution of IOHE and ISHE . The conversion process is illustrated in Figs. 4a and 4c. When the Ti layer is inserted between the Co and W layers, JS from the Co layer will be divided into two parts, one part will transfer into the W layer through the Ti layer and convert to JC,ISHE due to ISHE , the other part will convert to JL and flow into the Ti layer . JL will then convert to JC,IOHE due to IOHE. JC,ISHE and JC,IOHE have a 180-degree phase shift, as shown in Fig. 4a. During this process, JL from the W layer is not induced. Therefore, we did not observe the enhance d amplitude of the THz signals in the Co/Ti/W structure, as we did in the Co/W /X (X = Ti, Mn) structures. Additionally , we found that the polarity and amplitude of the THz signals can change in the Co/Ti/W structure with an increas e in dTi, as presented in Fig. 4b. The reason is that the Co (2 nm)/Ti (10 nm) structure has a larger THz |ΔPK | than that of the Co (2 nm)/Ti (4 nm) structure (see Fig. 2b), and JC,ISHE and JC,IOHE have a 180 degree phase shift . Thus the THz |ΔPK | of the Co (2 nm)/Ti (4 nm)/W (2 nm) structure is larger tha n that of the Co (2 nm)/Ti (10 nm)/W (2 nm) structure. However, w hen dTi > λSD,Ti, JS from the Co layer is blocked by the Ti layer and cannot convert to JC,ISHE in the W laye r. In this case, JC,IOHE from the Ti layer dominate s the THz signals , resulting in small THz |ΔPK | 9 with the opposite polarity compared to the Co (2 nm)/Ti (4 nm)/W (2 nm) and Co (2 nm)/Ti (10 nm)/W (2 nm) structures in the Co (2 nm)/Ti (40 -100 nm)/W (2 nm) structure . In the Ti/Co/W structure, the THz polarity is identical to that of the Co/W structure. The amplitude of the THz emission first increase s and then decrease s with the increasing dTi, reaching maximum amplitude (much larger than that of the Co/W structure ) at dTi = 10 nm , as shown in Fig. 4d. The THz signals primarily origin ate from JS→JC,W by ISHE of the W layer and JS→JL→JC,Ti by IOHE of the Ti layer . JC,ISHE from the W layer and JC,IOHE from the Ti layer have no phase shift as plotted in Fig . 4c. Therefore, the THz |ΔPK | increases first and then decreases with increas ing dTi. Addition ally, we found that the amplitude of the THz emission also depend s on the thickness of the nonmagnetic and ferromagnetic layers9, besides the cooperation or competition between ISHE and IOHE . We have demonstrated the occurrence of IOHE in materials with weak spin -orbit coupling such as Ti and Mn , and observed the orbitronic terahertz emi ssion in the Co/Ti and Co /Mn structures. T he introduction of a W layer has been found to significantly enhance the orbitronic terahertz emi ssion by inducing an additional conversion of the orbital -charge current. Furthermore, the manipulation of orbitronic THz emission has been achieved by designing specific structural configurations that cooperat e with IOHE and ISHE. This study not only helps to explore the physical mechanism of IOHE , but also provides guidance for the design and fabrication of spin-orbit ronic devices and THz emitters. 10 METHODS Sample preparation All the samples in this study were prepared on the Al2O3 (0001) single crystal substrates using an ultrahigh vacuum magnetron sputtering system at room temperature . To prevent oxidation , a 5-nm MgO capping layer was deposited on the samples. During the deposition process, the working gas Ar was set to 2.5 mTorr. Co (2 nm) /X (4-60 nm) (X = Ti, Mn) structures were used to investigate IOHE. Co (2 nm) /W (2 nm)/ X (4-100 nm) ( X = Ti, Mn) structures were utilized to study the enhancement of IOHE by inserting materials with strong SOC. Co/Ti (4-100 nm)/W (2 nm) and Ti (4 -100 nm)/ Co (2 nm) /W (2 nm) structures were employed to understand the collaboration and competition between IOHE and ISHE. The Co (2 nm) /W (2 nm) , Co (2 nm)/Pt (2 nm), Co (2 nm)/MgO (5 nm), Ti (4 nm)/MgO (5 nm), Mn (4 nm)/MgO (5 nm) , MgO (5 nm) structure s were prepared as the reference sample s. THz emission measurement The THz emission measurements were conducted using a home -made THz emission spectroscopy setup that utilized a Ti: sapphire laser oscillator with a center wavelength of 800 nm, a pulse duration of 100 fs, an average power of 2 W, and repetition rate of 80 MHz. The fs laser beam was split into a pump and a probe beam . The pump beam was directed onto the sample under normal incidence to excite it , and the THz waves generated were detected using the electro -optic sampling technique with the probe beam. For detecti on, a 2-mm thick ZnTe (110) electro -optic crystal was used. The samples were subjected to a n in-plane magnetic field of 50 mT. All measurements were conducted at room temperature in a dry-air environment. Normalization of the THz signals 11 The normalized THz signals can be obtained by normalizing the original THz signals (ETHz) to the laser absorbance (AFM) of the FM layer and the THz radiation impendence (Z). The laser absorbance ( AFM) of the FM layer is defined by the equation32: FM FM total totaltAAt (1) where Atotal and ttotal are the total laser absorbance and total thickness of the FM/NM bilayer or FM/NM 1/NM 2(NM 1/FM/NM 2) trilayer, tFM is the thickness of the FM layer. By measuring the total laser absorbance, one can obtain AFM of the FM layer. For the FM/NM bilayer : 0 0 FM FM NM NM 1ZZn Z t t (2) For the FM/NM 1/NM 2(NM 1/FM/NM 2) trilayer : 0 0 FM FM NM1 NM1 NM2 NM2 1ZZn Z t t t (3) The THz radiation impedance , Z, can be obtained by measuring the THz transmission of the sample and the bare substrate. Normaliz ation of the THz signals can be performed using the equation: THz FMEEAZ (4) Furthermore , the relationship between the THz signal ( ETHz) and the efficiency of the spin (orbit) -charge conversion c an be described by adding the orbit part using the equation9: THz SH S OH L FMEE J JAZ (5) where θSH and θOH are the spin Hall angle and the orbit Hall angle that characterize the efficiency of spin -charge and orbit -charge conversions, respectively. By using the above 12 equation, the normalized THz signal can be shown to be proportional to the spin/orbit -charge conversion efficiency. Therefore , the spin/orbit -charge conversion efficiency of the NM layers can be estimated through the normalized THz signals. DATA AVAILABILITY The data that support the findings of this study are available from the corresponding author upon reasonable request. ACKNOWLEDGMENTS This work was partially supported by the National Natural Science Foundation of China (Grant Nos. 52271240 , 51731003, 52061135205, 51971023, 51971024, 51927802 , 52271186 , 52201292 ) and Beijing Natural Science Foundation Key Program (Grant No. Z190007). Z.F . acknowledges the National Natural Science Foundation of China (NSFC) (62027807) and the National Key R&D Program of China (2021YFA1401400) . We would like to thank the Analytical & Testing Center of Tiangong University. COMPETING INTERESTS The authors declare no competing interests. AUTHOR CONTRIBUTIONS P.W., Z.F., Y.H.Y., and D.L.Z. contributed equally to this work. D.L.Z initialized and conceived this work. Y.J. coordinated and supervise d the project. D.L.Z. and P.W. conceived the experiments and designed all the samples. Y.H.Y. prepared the samples , Z.F. and Q.C.L. 13 performed the THz characterization. Z.D.X., Z.Y.J., Y.W., G.Q.Y. and X.G.X. characterized the basic properties of the samples . 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Engineering the spin -orbit -torque efficiency and magnetic properties of Tb/Co ferrimagnetic multilayers by stacking order. Phys. Rev. Appl. 17, 044056 (2022). 28. Hayashi, H. et al. Observation of long -range orbital transport and giant orbital torque. Commun. Phys. 6, 32 (2023). 29. Seifert, T. S. et al. Time -domain observation of ballistic orbital -angular -momentum currents with giant relaxation length in tungsten. http s://doi.org/10.48550/arXiv.2301.00747 (2023) . 30. Du, C., Wang, H., Yang, F. & Hammel, P. C., Systematic variation of spin -orbit coupling with d-orbital filling: large inverse spin Hall effect in 3 d transition metals. Phys. Rev. B 90, 140407(R) (2014). 31. Xu, Y. et al. Inverse orbital hall effect discovered from light -induced terahertz emission. https://doi.org/10.48550/arXiv.2208.01866 (2022) . 32. Zhang, H., et al., Laser pulse induced efficient terahertz emission from Co/Al heterostructures. Phys. Rev. B 102, 024435 (2020). 33. Pai, C. F. et al. Spin transfer torque devices utilizing the giant spin Hall effect of tungsten. Appl. Phys. Lett. 101, 122404 (2012). 34. Obstbaum, M. et al. Inverse spin Hall effect in Ni 81Fe19/normal -metal bilayers. Phys. Rev. B 89, 060407 (R) (2014). 16 Fig. 1 Physical mechanism of SHE /ISHE and OHE/IOHE . a SHE and ISHE refer to the conversion s of JC→JS and JS→JC in the heavy metals or quantum material s with strong spin- orbit coupling ( SOC ), where a transverse flow of spin angular momentum and voltage are generated , respectively . b OHE and IOHE are the conversion of JC→JL and JL→JC in the materials with weak SOC, where a transverse flow of orbital angular momentum and voltage are induced, respectively . 17 Fig. 2 Orbitronic THz emission originated from IOHE . a Conversion path of JS→JC in the structure with ferromagnetic (FM ) and nonmagnetic (NM) materials with weak spin -orbit coupling (SOC) . When a femtosecond laser pumps into the FM layer, JS and JL are simultaneously generated because of SOC of the Co layer. JL will flow into the NM layer and convert to JC due to IOHE (labeled by the blue arrow ); JS will directly convert to JC,ISHE via ISHE (labeled by the red arrow) . b The normalized THz signals of the Co (2 nm)/Ti (4-60 nm) structures . The obvious THz signals can be observed in the Co/Ti structures , indicating the conversion of JS→JC. c Comparison of the THz polarit y of the Co/Ti, Co/Mn , Co/W and C o/Pt -1600160 -1600160 -1600160 -1600160 1 2 3-1600160-1600160 -1600160 -4500450 1 2 3-8000800 0 20 40 60-2000Co(2)/Ti(4) Co(2)/Ti(10)Normalized THz signals (a.u.)Co(2)/Ti(20) Co(2)/Ti(40) t (ps)Co(2)/Ti(60) JLJCNM with weak SOC RNM<0 JSJLFM CFM>0 Hext× - SOC - JSJCJC,ISHE JC,IOHE LS Co(2)/Ti(4) Normalized THz signals (a.u.) THz ΔPK (a.u.)a b c Co(2)/Mn(4) Co(2)/W(2) Co(2)/Pt(2) t (ps) d (nm)d18 structures . The Co/Ti and Co/Mn structures have the opposite and same TH z polarit y to those of the Co/W and Co/Pt structure s, respectively . d The THz ΔPK of the Co/Ti (red color ) and Co/Mn (blue color ) structures as a function of the thickness of the Ti and Mn layers , indicating the long orbital diffusion length of Ti. Fig. 3 Enhanced orbitronic THz emission assisted by efficient spin -orbital current conversion . a The normalized THz signals of the Co/W and Co/W/X (X = Ti, Mn) structures and the corresponding extracted difference of THz |ΔPK|. 2-nm W inserted with strong SOC can enhance the THz amplitudes of the Co/W/ X (X = Ti, Mn) structures compared to the Co/W and Co/ X (X = Ti, Mn) structures . b The W layer provides an additional large JL, then enter s into the Ti or Mn layers and convert s to JC,IOHE by IOHE . c The normalized THz signals of the Co/X (X = Ti, Mn) , Co/Ti/Mn , and Co/Mn/Ti structures and the corresponding extracted difference of THz |ΔPK|. The difference of THz |ΔPK| of the Co/Ti/Mn , Co/Mn/Ti , and Co/ X (X = Ti, Mn) structures are smaller than those of the Co/W/X (X = Ti, Mn) structure s. d The routine conversion of JS→JL by SOC of the Co layer and JL→JC by IOHE of the Ti and Mn layers. The error bar s refer to the noise of the THz signals . -8000800 -8000800 1 2 3-8000800-50050 -50050 -50050 1 2 3-50050 JLJCTi,Mn RTi,Mn <0 Co CCo>0 Hext - JC,IOHE JLJCTi,Mn RTi,Mn<0 - JC,IOHE JSJLSOCS L Co(2)/W(2) Co(2)/W(2)/Ti(4)Normalized THz signals (a.u.)Co(2)/W(2)/Mn(4) t (ps)050010001500 THz |ΔPK| (a.u.)a b dc Co(2)/Ti(4) JSJLW RW<0 Co CCo>0 Hext SOC - JSJC JC,ISHE JLJCTi,Mn RTi,Mn<0 - JC,IOHE LS× ×CW<0 Co(2)/Mn(4)/Ti(4) t (ps)Normalized THz signals (a.u.)Co(2)/Mn(4) Co(2)/Ti(4)/Mn(4) 0153045 THz |ΔPK| (a.u.) Co/W/Ti-Co/WCo/MnCo/Ti Co/W/Mn-Co/W Co/Mn/Ti-Co/MnCo/Ti Co/Ti/Mn-Co/TiCo/Mn19 Fig. 4 Collaboration between IOHE and ISHE. a The JS→JL→JC and JS→JC conversion of the Co/Ti/W structure . JS from the Co layer will be separated into two parts, one part will transfer into the W layer through the Ti layer and convert to JC,ISHE due to ISHE ; the other part will convert to JL and flow into the Ti layer, then convert to JC,IOHE due to IOHE. JL from the W layer is not induced . b The normalized THz signals of the Co (2 nm) /Ti (4-100 nm) /W (2 nm) structures and the corresponding extracted THz ΔPK. c The JS→JL→JC and JS→JC conversion of the Ti/Co/W structure . The THz signals originate from JS→JC,W by ISHE of the W layer and JS→JL→JC,Ti by IOHE of the Ti layer . d The normalized THz signals of the Ti (4-100 nm) /Co (2 nm) /W (2 nm) structures and the corresponding extracted THz ΔPK. The error bar s refer to the noise of the THz signals . -9000900 -9000900 -9000900-100001000 -100001000 -100001000 -9000900 1 2 3-9000900-100001000 1 2 3-100001000 JCJLTi RTi<0 Co RCo>0 Hext - JC,IOHEW RW<0 JSJLSOCS L - JSJCJC,ISHE × × 4 nmCo(2)/Ti(4-100)/W(2) 60 nm100 nm4 nm10 nm40 nm4 nm10 nm40 nm60 nm100 nm 10 nm 40 nmNormalized THz signals (a.u.) 4 nmTi(4-100)/Co(2)/W(2)a bc d 10 nm 40 nmNormalized THz signals (a.u.) 60 nm 100 nm t (ps)-50005001000 THz ΔPK (a.u.) 60 nm JLJCTi RTi<0 Co RCo>0 Hext - JC,IOHEW RW<0 JSJLSOCS L - JSJCJC,ISHE × 100 nm t (ps)010002000 THz ΔPK (a.u.)

1102.2406v1.Chiral_spin_states_in_polarized_kagome_spin_systems_with_spin_orbit_coupling.pdf

Chiral spin states in polarized kagome spin systems with spin-orbit coupling Jia-Wei Mei,1, 2Evelyn Tang,2and Xiao-Gang Wen2, 1 1Institute for Advanced Study, Tsinghua University, Beijing, 100084, P. R. China 2Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA (Dated: Nov, 2010) We study quantum spin systems with a proper combination of geometric frustration, spin-orbit coupling and ferromagnetism. We argue that such a system is likely to be in a chiral spin state, a fractional quantum Hall (FQH) state for bosonic spin degrees of freedom. The energy scale of the bosonic FQH state is of the same order as the spin-orbit coupling and ferromagnetism | overall much higher than the energy scale of FQH states in semiconductors. I. INTRODUCTION Landau symmetry breaking1,2has been the standard theoretical concept in the classication of phases and transitions between them. However, this theory turned out insucient when the fractional quantum Hall (FQH) state3,4was discovered. These states (FQH states and spin liquids) are not distinguished by their symmetries; instead they have new topological quantum numbers such as robust ground state degeneracy5,6and robust non- Abelian Berry's phases7. The topological order8,9asso- ciated with topological quantum numbers has been pro- posed for the classication of these states. Recently, it was realized that topological order can be interpreted as patterns of long range quantum entanglement10{12. This long range entanglement has important applications for topological quantum computation: the robust ground state degeneracy can be used as quantum memory13; fractional defects from the entangled states which carry fractional charges4and fractional statistics14{16(or non- Abelian statistics17,18) can perform fault tolerant quan- tum computation19,20. Although it has attractive concepts and applications, topological order is only realized at very low tempera- tures in FQH systems3,4. In this paper we present a proposal to realize highly entangled topological states at higher temperatures. The ideal is to combine geomet- ric frustration, spin-orbit coupling and ferromagnetism in quantum spin systems. Both spin-orbit coupling and ferromagnetism can have high energy scales and appear at room temperature. Their combination breaks time- reversal symmetry which leads to rich and complicated interference from quantum spin uctuations. In this pa- per we show that they can lead to highly entangled topo- logical states at high temperatures. Quantum spins on the kagome lattice are ge- ometrically frustrated systems. They appear in the following compounds: Herbertsmithite Zn Cu 3(OH)6Cl2,21{23Kapellasite Cu 3Zn(OH)6Cl2,24 Y0:5Ca0:5BaCo 4O7,25MgxCu4x(OH)6Cl2,26 CaBaCo 4O7,27Pr3Ga5SiO 14,28Nd3Ga5SiO 14,29 BaCu 3V2O8(OH)2,30Cu(1,3-benzenedicarboxylate),31 KFe 3(OH)6(SO4)2,32YBaCo 4O7,33,34YBaCo 3AlO 7, YBaCo 3FeO 7,33,35 -Cu2(OD)3Cl,36Ni5(TeO 3)4Br2, Ni5(TeO 3)4Cl2,37Cu3V2O7(OH)2=2H2O,38,39Cs2Cu3CeF 12,40Cs2Cu3SnF 12,Rb2Cu3SnF 12,41 Cu2(OD)3Cl,42Cs2Cu3ZrF 12,Cs2Cu3HfF 1243and Co3V2O8.44Motivated by these materials, in this paper we study the Heisenberg model on the kagome lattice with additional spin-orbit interaction and Zeeman couplingP iBzSz i. Some related theoretical work can be found in Ref. 45,46. In Ref. 45 a model with spin-orbit interaction but no Zeeman coupling is studied; some mean-eld spin liquid states are found. In Ref. 46 a model with Zeeman coupling but no spin-orbit interaction is studied via numerical calculations. Two magnetization steps are found at M=M max = 1=3 (stronger) and 2 =3 (weaker) for a 36 spin cluster. In this paper, we study the state with magnetization hSz ii= 1=3. In section II, we write down the quantum spin model with spin-orbit coupling on the Kagome lat- tice. In section III, we map the spin model to the hard- core bosonic model in III A , construct three trial wave- functions for the polarized spin system hSz ii= 1=3 in III B and then evaluate the energy expectation for these three states in III C. We nd that the bosonic quantum Hall state has the lowest energy. Lastly, we discuss the materials realization in III D. In the Appendix A, we also discuss spin-orbit coupling in the transition metal oxide materials. II. QUANTUM SPINS ON THE KAGOME LATTICE WITH SPIN-ORBIT COUPLING The kagome lattice has 3 sites (labelled 1, 2 and 3) within every unit cell with the primitive vectors a1= 2a^x anda2=a(^x+p 3^y) (ais the lattice constant), see Fig. 1(a). The unit cell contains one hexagon and two triangles so it is geometrically frustrated. As shown in Fig. 1 (b), the triangle 4123on the kagome plane in Herbertsmithite ZnCu 3(OH)6Cl221{23 contains three copper cations surrounded by distorted octahedrons sharing one chlorine corner while each pair shares an oxygen corner. Mediated by this oxygen, the Cu 3d9electron hops from site r1tor2, e.g. see Fig. 1(c). Inversion symmetry for Herbertsmithite breaks down explicitly, leading to a non-uniform charge distribution in the kagome lattice. For convenience, we model the chargearXiv:1102.2406v1 [cond-mat.str-el] 11 Feb 20112 (a) 1 23 3'1' 2' E: effective electric fieldcharge center (b) (c) 1 2XYXY 1 2 3 θθ FIG. 1: (color online) (a) The kagome lattice with three dif- ferent sites l= 1;2;3 within the unit cell. Inversion symmetry breaking via a charge center in the hexagon leads to the ef- fective electric eld Eijon the bond rij, represented by green arrows which point from the middle of the bond to the cen- ter of every triangle on the kagome lattice. (b) The triangle 4123in the kagome lattice for Herbertsmithite21{23. (c) The nearest neighbor bond r12: the electron hops from site r1to r2mediated by the oxygen atom. center as being in the hexagon, see Fig.1(a). When hop- ping from r1tor2, the electron sees the electric eld E12 (labelled by the green arrow in Fig. 1(a)). The eective electric eld couples to the electron through the spin- orbit coupling vector D12=E12r12in the Rashba manner, where r12=r1r247 t12=tX 0 exp(i~ 
D12)0cy 1c20+ h.c. (1) Here~ = (x;y;z) are the Pauli matrices. The coe- cientshould be chosen to make the spin-orbit coupling vector D12dimensionless. Note that D12=D12. Including on-site interactions we obtain the Hubbard model with spin-orbit coupling for S= 1=2 electrons on the kagome lattice H=tX 0 (ei~ 
Dij)0cy icj0+ h.c. +UX ini"ni#(2) whereiandjdenote nearest neighbors. For a specied bond rij, we can make a gauge transformation48 ci!~ci=X 0(ei(D=2)~ 
nij)0ci0 cj!~cj=X 0(ei(D=2)~ 
nij)0cj0 (3)where Dij=nijD. Then H=tX (~cy i~cj+ h.c.) +UX i~ni"~ni# (4) Using standard second-order perturbation theory, we ob- tain the exchange term Jij=J~Si
~Sj (5) HereJ= 4t2=Uis the exchange coupling for the rotated spin operator ~Si=P 0~ci~ 0~cj0. On the kagome lattice, we cannot nd a gauge trans- formation as in Eq. (3) that would be compatible for each site. So we have to write the Hamiltonian in terms of the original spin operators. On every bond, the ro- tated spin operators are related to the original ones as follows: ~Si= (1cos(D))(^nij
Si)^nij+ cos(D)Si sin(D)Si^nij (6) ~Sj= (1cos(D))(^nij
Sj)^nij+ cos(D)Sj + sin(D)Sj^nij (7) Thus we obtain the quantum spin model on the kagome lattice including spin-orbit coupling48 H=JX hiji(cos(2D)Si
Sj+ sin(2D)(SiSj)
^nij +2 sin2(D)(Si
^nij)(Sj
^nij)
: (8) III. POLARIZED SPIN STATE WITH TOPOLOGICAL ORDER A. Hardcore bosonic model For simplicity we choose the spin-orbit coupling vectors Dijperpendicular to the kagome plane: D12=D23= D31=D1020=D2030=D3010=D^z(only in-plane ef- fective electric elds Eijare considered). We use the Holstein-Primako transformation: S+ i=by i;S i=bi;Sz i=1 2by ibi (9) wherebiis the hardcore bosonic operator [ bi;by j] =ij, ni=by ibi1. This maps the spin model (8) onto a hardcore bosonic model49,50 H=J 2X hiji exp[( ^nij
^z)i2D]by ibj+ h.c. +JX hijininj (10) which describes interacting hardcore systems with hop- ping under eective uxes as shown in Fig. 2: within the triangles4123andr102030, there are uxes 1= 6D; in3 1 233'1' 2' 1φ 1φ2φ FIG. 2: (color online) Flux distribution for the hardcore bosons. the hexagon, there is ux 2=21. When16= 0; mod 2, these eective uxes break time-reversal sym- metry for this model. Now let us consider just one boson described by only the hopping term in the above Hamiltonian. The hopping Hamiltonian has three bands. We calculate the Berry curvatures over the Brillouin zone for the lowest band in the presence of the ux for dierent spin-orbit couplings: D= 0:025, 0:1 and=8 . The Berry curvature is dened as follows Fn(k) =ij@kiA(n) j(k); A(n) i(k) =ihunkj@kijunki(11) whereunkis the Bloch wave packet in the n-th band of the hopping Hamiltonian Ht=J 2X hiji exp[( ^nij
^z)i2D]by ibj+ h.c. Htjunki=nkjunki (12) In this paper, instead of using ^kxand ^kyas the axes ink-space, we use ^k1=kxand ^k2= (^kx+p 3^ky)=2 for convenience. The dispersion nkhas three bands (la- belledn=bfor the bottom band, n=mfor the middle band andn=tfor the top band) as shown in Fig. 3 (a), (c) and (e). In all cases ( D= 0:025,D= 0:1 and D==8) the three bands have nonzero Chern numbers Cb= 1,Cm= 0 andCt=1, where the Chern number C1 2R BZd2kFn(k). We plot the Berry curvature of the bottom band for D= 0:025, 0:1 and=8 in Fig. 3 (b), (d) and (f). We see that when D= 0:1, the lowest band is separated from the other bands by an energy gap and the lowest band is quite at. Since the lowest band has a non-zero Chern numberCb= 1, it simulates the rst Landau level in free space. By analogy to the quantum Hall eect in high magnetic eld, the hardcore bosons are likely to form a = 1=2 bosonic quantum Hall state when there is half a boson per unit cell. The boson lling number is f= 1=6 per site which corresponds to the spin polarization hSz ii= 1=2f= 1=3 (13) In other words, the polarized spin state hSz ii= 1=3 is likely to be a chiral spin liquid51| a topologically or- dered state. −pi0pi −pi0pi−1.5−1−0.500.511.52 k1(a). Band dispersions (D=0.025) k2εk/J k1k2(b). Bottom band curvature (D=0.025) −pi −pi/2 0 pi/2 pi−pi−pi/20pi/2pi 0123x 10−3 −pi0pi −pi0pi−1012 k1(c). Band dispersions (D=0.1) k2εk/J k1k2(d). Bottom band curvature (D=0.1) −pi −pi/2 0 pi/2 pi−pi−pi/20pi/2pi 0510x 10−4 −pi0pi −pi0pi−2−101 k1(e). Band dispersions (D= π/8) k2εk/J (f). Bottom band curvature (D= π/8) k1k2 −pi −pi/2 0 pi/2 pi−pi−pi/20pi/2pi 246810x 10−3FIG. 3: (color online) (a), (c) and (e): Band dispersions of the hardcore boson in the presence of the ux as shown in Fig. 2 forD= 0:025,D= 0:1 andD==8, respectively; (b), (d) and (f) are the corresponding bottom band curvatures for (a), (c) and (e). B. Fermionic constructions for the bosonic wavefunctions To study the topologically ordered chiral spin state, we will employ the fermionic approach to construct trial bosonic wavefunctions . The many-body bosonic wave- function can be represented as follows ji=X fx1;


;xNbg(x1;


;xNb)jfxigi (14) Here the sum is over all possible boson congura- tionsjfxigi=jfx1;


;xNbgi=ay x1


ay xNbj0iand (x1;


;xNb) is the symmetric wavefunction. In this paper, we are only concerned with translationally invari- ant ground states. The many-body bosonic wave function for a product4 state (PS) has the form (x1;


;xNb) =NY i=1l(xi) (15) wherel(xi+aj) =l(xi) to maintain translational in- variance.l= 1;2;3 denotes sites within the unit cell and the vector aj,j= 1;2, is a Bravais vector for the kagome lattice. So the many-body wave function is labelled by three complex parameters 1,2, and3corresponding to the spin orientation on the three sites in each unit cell. The mean eld ground state of this type is obtained by minimizing the average energy by varying 1,2, and 3. This type of spin ordered states without topological order is the main competing state for the ground state of our model. To construct the bosonic ground state with topological order, we split the the hardcore boson into two species of fermions bi=ii (16) whereiandiare fermion operators which satisfy the constraint on every site: ni=ni=nib. The congu- ration becomes jfxigi=y x1y x1


y xNby xNbj0i (17) and the symmetric wave function factorizes as follows (fxig) = (fxig) (fxig) (18) where (fxig) and (fxig) are the antisymmetric fermionic wavefunctions for iandi. Using this frac- tionalization we construct two ansatz wavefunctions: the bosonic quantum Hall state (QHS) and the spin Hall state (SHS). The fermionic wavefunctions (fxig) and (fxig) can be constructed from the mean eld tight binding Hamiltonian H=teX hiji y ijexp(iA ij) + h.c. H=teX hiji y ijexp(iA ij) + h.c. (19) The lling factors for the fermions i,iaref= 1=6 per site, namely half per unit cell. For the QHS and SHS, we need the lling factor corresponding to one particle per unit cell which can be realized by inserting half a ux quantum (=) in the original unit cell to double the unit cell: ! 2+ 2! 1=; ! =; (20) where! 1and! 2are uxes in the kagome unit cell, see Fig. 2. In the presence of these uxes, we specify a gauge forA! ijin this tight-binding model (19) to obtain single particle wavefunctions in the bottom band: !(ki;xj)(!=;), where kis the Bloch momentum vector for the doubled unit cell. Thus the fermionic wavefunctions are the determinants of these single particle wave func- tions: !(fxig) = det[ !(ki;xj)] (21) where!=; andi;j= 1;2;


;Nb. For the QHS state, we set  1= 1and 2= 2; for the SHS state, we set 1= 1and 2= 2. For the QHS, each fermion has the same Chern number C!= 1; for the SHS,C= 1 andC=1. We now consider the eective theory for the QHS and SHS. For the QHS and SHS, fermionic excitations are gapped out. There is a gauge freedom for the frac- tionalization in Eq. (16): the gauge transformation i!ieii,i!ieiidoes not change the bosonic operatorbi. The Hamiltonian with the gauge uctuations is given by H=teX hiji y ijexp(i~Aij+iaij) + h.c. H=teX hiji y ijexp(i~Aijiaij) + h.c. (22) The non-zero Chern number for each fermion species im- plies that the low energy eective action for the gauge elds is given by L=i 4X !C!a@a+::: (23) where:::represents higher order terms. For the QHS, C=C= 1 so we obtain the low-energy eective theory LQHS=i 2a@a+1 42g(@a)2(24) This describes the = 1=2 FQH state for bosons corre- sponding to the chiral spin state rst introduced in Ref. 51. We note that although the andfermions have the same Chern number, the sign of the coupling of each fermion to the U(1) gauge eld ais opposite. Thus a 2 ux ofacreates anfermion and annihilates a  fermion. For the SHS, C= 1,C=1 and we obtain the low-energy eective theory LSHS=1 42g(@a)2(25) Here theandfermions in the SHS have opposite Chern number and the sign of the coupling of the two fermions to the U(1) gauge eld ais also opposite. Thus a 2 ux ofacreates anfermion and a fermion which corresponds to a bboson, i.e. it describes a spin ip. So the magnetic eld of the U(1) gauge eld acorresponds to the spin Szdensity. Since spin Szis conserved, the U(1) instanton is forbidden. Thus the 2+1D U(1) gauge5 D=0.025 D=0.1 D=pi/8−0.25−0.2−0.15−0.1−0.050Energy per siteUnit: J PS QHS SHS FIG. 4: (color online) Energy per site for D= 0:025,D= 0:1 andD==8 respectively. theory above is not conned and the U(1) gauge eld a remains gapless. As this gapless U(1) gauge eld cor- responds to the spin Szdensity, the gapless spin den- sity uctuations imply that eiSzspin rotation is spon- taneously broken. Thus the SHS is a spin XY ordered state. C. Numerical results For the three dierent states (PS, QHS and SHS), we can evaluate the expected energies for the bosonic model (10): E() =hjHji hji=X fxigeL(fxig)jhfxigjij2 hji(26) where we dene the local energy eL(fxig) =hjHjfxigi hjfxigi. We evaluate the energy in (26) by appropriately averag- ing the local energy eL(fxig) over a set of congurations jfxigidistributed according to the square of the wave functionjhfxigjij2, generated with a standard varia- tional Monte Carlo method. Then we use a minimization function to optimize the expectation values on an 8 8 lattice. In Fig. 4, we plot the energy per site of the three states for D= 0:025, D= 0:1 andD==8. ForD= 0:025, the PS and SHS have energies close to each other. The QHS has a better energy. As the spin-orbit coupling is increased, the SHS becomes worse in energy. Both the PS and QHS gain in energy and the PS gains much more. When D==8, the PS gives results close to the QHS. With small spin- orbit coupling ( D= 0:025 andD= 0:1), the bottom band of the hopping Hamiltonian in (10) is at and has a smooth curvature over the Brillouin zone. The classic PS cannot gain much energy through condensation of the lowest states. When Dincreases, the bottom band becomes more convex and the PS will gain a lot of energy through condensation. Our numerical results indicate that the topologically ordered QHS (the chiral spin state) is a serious candidate for a kagome spin system with spin-orbit coupling and FIG. 5: (color online) A scheme to tune the lling num- ber of hardcore bosons bi: the kagome lattice couples to a ferromagnetic substrate by the exchange interaction Hint= JexP iSm i
Si; we can tune the substrate magnetization hSm ii by an applied magnetic eld. spin polarization. This may be a realistic route for the discovery of new topologically ordered states in quantum spin systems. D. Practical realization Here we explore the possibility of obtaining a po- larized state ( Sz i6= 0) experimentally. This can be achieved by applying a magnetic eld, which adds a term Hh=BP inito the Hamiltonian (10). However, the exchange energy J100meV is usually very large, so experimentally accessible magnetic elds cannot polarize the spin to Sz= 1=3. Hence we should nd other ways of obtaining a large eective magnetic eld. One way is to place the kagome lattice on a ferromag- netic substrate, see Fig. 5. The exchange interaction is Hint=JexP iSm i
Si, hereJexis the exchange coupling between spins Sm ion the substrate and spins Sion the kagome plane. The exchange coupling Jecan be very large when the kagome plane structure matches that of the substrate perfectly. Such a large eective magnetic eld can polarize the spin on the kagome lattice. A third way is to insert ferromagnetic atoms in the kagome system. If these ferromagnetic atoms form a fer- romagnetic state, the exchange interaction can also in- duce spin polarization on the kagome lattice. IV. SUMMARY In this paper, we study quantum spin systems on the kagome lattice with spin-orbit coupling and and spin polarization. We argue that such a system can be in a topologically ordered chiral spin state, a FQH state for bosonic spin degrees of freedom. The energy scale of the bosonic FQH state is of the same order as the spin-orbit coupling and ferromagnetism | overall much higher than the energy scale of FQH states in semicon- ductors. This result suggests exploration of topologically ordered states in quantum spin systems with a proper combination of geometric frustration, spin-orbital cou- pling and ferromagnetism. This research is supported by NSF Grant No. DMR- 1005541 and NSFC 11074140.6 Appendix A: Discussion of spin-orbit coupling The Rashba spin-orbit coupling is weak, around jDj= 0:025 for Herbertsimthsite Zn Cu 3(OH)6Cl2. This small value prompts us to nd other mechanisms to increase the strength of the spin-orbit coupling. In the literature, the spin-orbit coupling is also dis- cussed on the atomic level and with a large strength of coupling, e.g. around 0.2 eV and 0.4 eV for the 4 dand 5delectrons, respectively. We hope to relate the atomic spin-orbit coupling to the form represented in Eq. 1. This has been achieved for the 5 d5electron in Sr 2IrO452 and we discuss the general case below. AS= 1=2 electron can be found in d1,d5andd9 orbtitals in transition metal cations, e.g. in Mo5+, Ir4+ and Cu2+respectively. Due to the crystal eld , the vefold degenerate dstate is split into a doublet egand a triplett2g.d1andd5with a ligand octahedron and d9 with a ligand tetrahedron belong to t2g. The triplet t2g has strong spin-orbit coupling Hi=li
si: (A1) Here siis the spin operator and l= 1 is the eec- tive angular momentum with jlz i= 0i  jXYiiand jlz i=1i  1p 2(ijXZiijYZii) (X,YandZare local axes supporting by the local octahedron or tetrahe- dron , e.g. see Fig. 1(b)). The strong spin-orbit coupling (A1) splitst2ginto two groups with eective angular mo- mentumJz e= 1=2 andJz e= 3=2, respectively. The Jz e= 1=2 singlet contains a Kramers doublet: j~"ii=p 2 3j1;"ii+1 3j0;#iiandj~#ii=p 2 3j+ 1;#ii+1 3j0;"ii. Here we are concerned with the Kramers doublet labelled by the pseudospin ~si= ~i=2. There is a pbond between the cation and mediating oxygen atom. When the two cations and oxygen lie on a straight line along the bond, the values of the overlap for hlz= 0jpiOandhlz=1jpiOare the same. In Herbert- smithite, they are not along a straight line and form a triangle, see Fig. 1(c). As a result, dierent orbits have dierent overlaps with the porbital in the oxygen h~jpiOwhen the electron hops on the bond r12fromr1tor2: jhlz=1jpiOj>jhlz= 0jpiOj (A2) For the sake of simplicity, we neglect the overlap jhlz= 0jpiOjhere and are only concerned with jhlz=1jpiOj. Then we can write the spin-orbit in terms of the pseu- dospin in this manner Hil(ri)
~s(ri) (A3) Here the orbital angular momentum l(ri) can be regarded as the eective magnetic eld on the pseudospin ~s(ri). It is very interesting that the orientation of the magnetic eldl(ri) varies from each site ri. When the hopping encloses a loop, e.g. 4123in Fig. 1 (b), an electron obtains a non-zero Berry phase Berry related to the spin- orbit coupling vector Din Eq. (1) as follows Berry =I dr
D(r) (A4) where is the closed loop. To rotate the Kramers doublet from the local axes ( X, YandZ) to the global axes ( x,yandz) we use j~;fxgi= (ei~ 
ni=2)j;fXgi (A5) wherefxg=eili
nifXg. The hopping process on the bond r12is given as t12=X 0 t0(r1;r2)cy 1c20+ h.c. (A6) with the hopping parameter t0(r1;r2)(ei~ 
n12)01h1jOpihOpj1i2  t(ei~ 
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1109.4745v1.Intrinsic_coupling_of_orbital_excitations_to_spin_fluctuations_in_Mott_insulators.pdf

arXiv:1109.4745v1 [cond-mat.str-el] 22 Sep 2011Intrinsic coupling of orbital excitations to spin fluctuati ons in Mott insulators Krzysztof Wohlfeld,1Maria Daghofer,1Satoshi Nishimoto,1Giniyat Khaliullin,2and Jeroen van den Brink1 1IFW Dresden, P. O. Box 27 01 16, D-01171 Dresden, Germany 2Max-Planck-Institut f¨ ur Festk¨ orperforschung, Heisenb ergstrasse 1, D-70569 Stuttgart, Germany (Dated: June 9, 2022) We show how the general and basic asymmetry between two funda mental degrees of freedom present in strongly correlated oxides, spin and orbital, ha s very profound repercussions on the elementary spin and orbital excitations. Whereas the magno ns remain largely unaffected, orbitons become inherently coupled with spin fluctuations in spin-or bital models with antiferromagnetic and ferroorbital ordered ground states. The composite orbiton -magnon modes that emerge fractionalize again in one dimension, giving rise to spin-orbital separat ion in the peculiar regime where spinons are faster than orbitons. PACS numbers: 75.25.Dk, 75.30.Ds, 71.10.Fd, 74.72.Cj In transition metal oxides, different 3 dorbitals near the Fermi level can have similar energy and thereby con- tribute to the low-energy physics. In presence of strong correlationschargefluctuations become suppressedand a Mott insulator is realized when there is a commensurate number of electrons per unit cell. The effective Hamilto- nian that emerges, often referred to as Kugel-Khomskii (KK) Hamiltonian, can be expressed in terms ofspin and orbital operators [1]. From a formal viewpoint the spin and orbital operators are very similar because they form identical algebras, but the way in which these operators enter into realistic KK Hamiltonians is very different. While the spin wavefunction is in essence rotationally invariant and the coupling between spin operators there- foreSU(2) symmetric, the symmetry in orbital space is much lower due to the ubiquitous crystal field act- ing on the orbital wavefunctions in crystals. This re- duced symmetry causes the orbital-orbital interaction to be very anisotropic in space and often inherently frus- trated, which causes exotic effects such as macroscopic degeneracy of the groundstate or the emergence of non- Abeliantopologicalexcitationsin thecaseofthe compass or Kitaev models, respectively [2], which may be relevant for quantum computation [3, 4]. We will show that this asymmetry between spin and orbital degrees of freedom has fundamental repercussions on the coupling of the elementary spin and orbital excitations – the magnons and orbitons, respectively: whereas the magnons remain largely unaffected, orbitons become inherently coupled with spin fluctuations in spin-orbital models with anti- ferromagnetic(AF)andferroorbital(FO)orderedground states. This is relevant in the experimental context as substantial progress is being made in measuring or- bitons[5,6]andtheirdispersion,inparticularinresonant inelastic x-ray scattering (RIXS) [7, 8], where we predict the coupling of the orbiton to magnetic fluctuations to be clearly discernible. In thestandard approach the complex problem of in- tertwined spin-orbital excitations is solved using a mean- field decoupling, i.e. considering magnons in a fixed or-bital background or orbitons in a fixed spin background. While such an approach can work well to obtain the correct spin and orbital orderings consistent with the Goodenough-Kanamori rules [1] and in some cases with ferromagnetic (FM) order [9], we show here that it fails todescribeorbitalexcitationsevenqualitativelycorrectly foranumberofspin-orbitalmodels. We avoidthisdecou- pling by mapping the coupled orbiton-magnon dynamics onto the well-controlled problem of a hole propagating in a magnetic background: the extensively studied sin- gle holet–Jmodel. In particular, we find that in one dimension (1D) the orbital excitation fractionalizes into freelypropagatingspinonandorbiton, givingrisetospin- orbital separation in the peculiar regime where spinons are faster than orbitons. Model and problem statement.— The generic form of the KK Hamiltonian [1] in the Mott-insulating limit is H= 2J/parenleftBig/summationdisplay /angbracketlefti,j/angbracketrightHS i,jHT ij+/summationdisplay iHT i/parenrightBig , (1) whereiandjare lattice sites and on each bond ∝angbracketlefti,j∝angbracketrightthe spin-spin interaction is HS ijwhereas the orbital-orbital one isHT ij. To capture the genericdifferences between orbitonsandmagnons,itisenoughtobreaktheSU(2)ro- tation symmetry in orbital space, which is here achieved by the large local crystal field breaking the degeneracy between the orbitals, expressed as HT i=Ez 2JTz i. We keep for simplicity the rotational symmetry in the in- teractions so that for the spins HS ij=Si·Sj+1 4and for the orbitals HT ij=Ti·Tj+1 4, whereS(T) are the spin (orbital) operators that fulfill the SU(2) algebra for S= 1/2 (T= 1/2) spins (pseudospins). The constant J >0 gives the energy scale of the spin-orbital superex- change and the symmetry breaking field for the orbitals isEz. Note that for the case of Ez= 0 (not considered here), this model has an SU(4) symmetry even higher than the combined SU(2)×SU(2) symmetries, which re- sults in the ground state given by the Bethe Ansatz and composite spin-orbital gapless excitations in addition to the separate spin and orbital ones (see, e.g., Ref. [10]).2 This model describes the low-energy physics, determined by singly occupied sites, of a two-orbital Hubbard model in the limit of a large onsite Coulomb repulsion Uand vanishing Hund’s exchange JH, cf. Eqs. (1-5) and (29) in Ref. [9]. Here we are interested in the orbital excitations of the model Eq. (1) when Ez≫Ji.e., the orbitalsplittings are larger than magnetic coupling energy. This is a realistic regime for many strongly correlated compounds such as 1D or two-dimensional (2D) cuprates [11]. As one degree of freedom is completely polarized, the ground state is easily found and given by all electrons occupying a single orbital in an AF state. But while decoupling spin and or- bital degrees of freedom works here for the ground state, it is not at all appropriate for orbital excitations – this is the problem investigated below. Decoupling of spin and orbital sector.— The ground state|ψ∝angbracketright=|ψS∝angbracketright ⊗ |ψO∝angbracketrightof Eq. (1) is described by the ground state |ψS∝angbracketright=|AF∝angbracketrightof an AF Heisenberg system formed by spins in the lower-energy orbital, i.e., an FO ordered state |ψO∝angbracketright=|FO∝angbracketright. The orbital excitations are reached by flipping an orbital, i.e., by promoting an elec- tronatsite jfromtheoccupiedlowerorbitaltotheempty higher band at the same site, expressed by the orbital raising operator T+ j. The momentum-dependent orbital excitation is given by T+ k=/summationtext jeikjT+ j[T− k= (T+ k)†], and the spectral function describing its dynamics is O(k,ω) =1 πlim η→0ℑ∝angbracketleftψ|T− k1 ω+Eψ−H−iηT+ k|ψ∝angbracketright.(2) Firstwediscussthe orbitonspectralfunction bytaking the orbital excitation to be independent of the magnetic excitation. One can then rewrite the orbital operators by use of Holstein-Primakoff bosons (see, e.g., Ref. [12]), keep only quadratic terms in the expansion (orbital-wave theory)and, noticingthat the groundstate |ψO∝angbracketrightdoesnot contain bosons, obtain O(k,ω) =δ[ω−ωOW(k)], (3) with a mean-field orbital-wave dispersion ωOW(k) = Ez−1 2zJOW(1−γk). Here,zis the coordination num- ber,γkis the lattice structure factor, and the effective orbital exchange constant JOW= 2J∝angbracketleftψS|Si·Sj+1 4|ψS∝angbracketright. The orbital excitation on the mean-field level is thus a quasiparticle with a cosine-like dispersion with period 2π: for example in 1D we obtain an effective reduced JOW≃ −0.4J≪J, cf. the thick line in Fig. 2. An anal- ogousprocedurefor magnons inFM planeswithalternat- ingorbitals(AO)hasbeenappliedtoLaMnO 3orKCuF 3, and similarly yields magnons with a reduced bandwidth, but without any other trace of the AO order, in agree- ment with experiment [13]. We will see, however, that for orbitons this framework of mean-field decoupling is greatly oversimplified. Mapping onto an effective t–Jmodel.— The orbital- wave approximation puts all the impact of the AF orderj j+ 1 j j+ 1j+ 1 j+ 1j+ 1j+ 1(a) 12 (b) 12j j jjVIRTUAL STATE VIRTUAL STATE FIG. 1: (color online) Two superexchange processes moving an electron in an excited orbital (indicated by oval) from si te jto its neighbor j+ 1: (a) and (b) describe orbiton motion when spins along the bond are antiparallel or parallel, re- spectively, see text. The states in the grey middle panels ar e not part of the low-energy Hilbert space corresponding to (1 ). These virtual excitations within the full two-orbital Hubbard model illustrate theorigin ofthose superexchangeinterac tions of Hamiltonian (1) that propagate the orbiton; note that the spin of the excited electron is conserved. intoJOW, which is a mean-field average of the sum of twodistinct superexchangeprocessesin which the orbital excitation may propagate through the lattice. The first one, which correspondsto( T+ jT− j+1+h.c.)(S+ jS− j+1+h.c.) processes in Hamiltonian (1), allows for orbiton prop- agation when the spins on the bond are antiparallel, see Fig. 1(a). The second one, which corresponds to (T+ jT− j+1+h.c.)(Sz jSz j+1+1/4) processes in Hamiltonian (1), allows for orbiton propagation when the spins on the bond are parallel, see Fig. 1(b). Crucially (see next para- graph), this figure illustrates that in both cases the spin of the electron in the upper orbital 2 is conserved dur- ing the orbiton propagation. This is because spin-orbital Hamiltonian (1) is, as mentioned above, a low energy limit of the two-orbital Hubbard model with the Hund’s exchangeJH= 0 and the spins of individual electrons in the superexchange process cannot be flipped (see middle panels of Fig. 1). In more realistic spin-orbital models the Hund’s exchange is finite [1], but the processes which would not conserve the electron’s spin in the excited or- bital are small ( ∝JH/U) and thus could be neglected. To be explicit, we now focus on the 1D case and em- ploya Jordan-Wignertransformation[14]. 2D and three- dimensional (3D) cases are discussed afterwards. We thus introduce S+ j=βjeiπQ, S− j= e−iπQβ† j, Sz j= 1 2−njβ,whereQ=/summationtextj−1 l=1nlβandβ† jcreate spinons whileT+ j= e−iπ¯Qα† j, T− j=αjeiπ¯Q, Tz j=njα−1 2, where¯Q=/summationtextj−1 l=1nlαandα† jcreates a pseudospinon. Since the spin of the propagating electron in the up- per orbital 2 is conserved (see above), one may cal- culate the spectral function O(k,ω) for, e.g., spin-up in the upper orbital. We are then allowed to replace T† k→/summationtext jeikjT+ j(1 2+Sz j) = e−iπ¯Qα† j(1−njβ) in Eq. (2) and thus terms in the Hamiltonian that create or annihi-3 late both a spinon and a pseudospinon at the same site lead to a vanishing contribution to the spectral function. Phase factors cancel in one dimension and we obtain O(k,ω) =1 πlim η→0ℑ∝angbracketleft¯ψ|αk1 ω+E¯ψ−¯H−Ez−iηα+ k|¯ψ∝angbracketright,(4) with the effective fermionic Hamiltonian ¯H=−1 2J/summationdisplay /angbracketlefti,j/angbracketright/parenleftbig β† iαiα† jβj+αiα† j+h.c./parenrightbig +J/summationdisplay /angbracketlefti,j/angbracketright/bracketleftBig1 2(β† iβj+h.c.)−1 2(niβ+njβ)+niβnjβ/bracketrightBig ,(5) and an implicit constraint ∀jβ† jβj+α† jαj≤1. Here|¯ψ∝angbracketright is a tensor product of the magnetic ground state |ψS∝angbracketright= |AF∝angbracketrightexpressed in terms of spinons and a vacuum state for pseudospinons [recall that we consider here a single orbiton only; this also allowed us to skip quartic terms in pseudospinons in Eq. (15)]. At this point, we observe that the resulting effective Hamiltonian is in fact a Hamiltonian for the t–Jmodel written in terms of the Jordan-Wigner fermions with the above constraint [15]. By introducing the electron oper- atorspj↑=α† j,pj↓=α† jβjeiπQacting in the restricted Hilbert space without double occupancies we obtain O(k,ω)=1 πlim η→0ℑ∝angbracketleft˜ψ|p† k↑1 ω+E˜ψ−˜H−Ez−iηpk↑|˜ψ∝angbracketright,(6) with thet–JHamiltonian ˜H=−t/summationdisplay /angbracketlefti,j/angbracketright,σ(p† iσpjσ+h.c.)+J/summationdisplay /angbracketlefti,j/angbracketright(Si·Sj+1 4ninj),(7) wherenj=/summationtext σnjpσand the hopping parameter tis defined ast=J/2 [16]. The ground state |˜ψ∝angbracketrightis now the tensor product of a vacuum state for holes and the |ψS∝angbracketright state. We have thus mapped the single orbiton in the FO and AF chain, with dynamics governed by Hamiltonian (1), onto a single hole doped into the undoped AF chain with its dynamics governed by Hamiltonian (17). Numerical results for the t–Jmodel.— To flesh out the resulting coupling between orbitons and spin fluc- tuations we use Lanczos exact diagonalization to evalu- ate Eq. (16) on a finite chain (28 sites). The spectral function is shown in Fig. 2: the spectrum differs qualita- tively from the orbital-wave result shown as a thick line in Fig. 2. It now consists of multiple peaks (expected to merge into incoherent spectrum in the thermodynamic limit) instead of one single excitation. There is a dom- inant feature at the lower edge of the spectrum, but its periodicity is πreflecting the doubled unit cell of the AF order. Thet–Jmodel with J > t[16] is not easily accessi- ble in the Hubbard type models, as it would formally FIG. 2: (color online) Spectral function O(k,ω) of orbital excitation obtained via the mapping onto the t–Jmodel, Eq. (16), evaluated using Lanczos exact diagonalization on a 28 site chain. A broadening η= 0.03JandEz= 10J. The thick line shows orbital excitation in a mean-field (orbital-wave ) approach, Eq. (3). (c)(b)(a) FIG. 3: (color online) Schematic representation of the orbi tal motion and the induced spin fluctuations giving rise to spin- orbital separation in 1D. The first hop of the excited state (a→b) creates a spinon (wavy line) that moves via spin exchange ∝J. Next hop (b →c) does not produce any extra spinons: an ‘orbiton’ freely propagating as a ‘holon’ with a n effective hopping t∼J/2 is created. correspond to small onsite interaction U, where the t– Jmodel is no longer valid. In this regime, the spinon moves faster than the holon, and the entire lower edge of the spectrum is thus given by ‘holon’ states [17]. If the orbiton takes the place of the holon as argued here, this exotic behavior should be observable in RIXS exper- iments and one would expect a dominant excitation with orbiton-character with dispersion ω≈Ez−2tsin|k|at the bottom of the spectrum. The latter would be ex- pected to extend up to ω≈Ez+√ J2+4t2−4tJcosk and to contain an intermediate feature still well-visible within the continuum with the dispersion of the purely4 orbiton-character scaling as ω≈Ez+2tsin|k|. Figure 3 illustrates how electron exchange processes can let an orbital excitation propagate through the sys- tem after creating a spinon in the first step. The spinon itself moves via spin flips ∝J > t, faster than the or- biton, and the two get well separated. The orbital-wave picture, on the other hand, would require the orbital ex- citation to move without creating the spinon in the first step. As can be inferred from Fig. 3, this is only possible for imperfect N´ eel AF spin order so that the averages of processes shown in Fig. 1(a) and (b) are finite. 2D and 3D.— Remarkably, the standard OW picture becomes even worsein higher dimensions: in 2D (3D) cu- bic lattices, the mean-field orbital coupling JOWalmost vanishes due to ∝angbracketleftψS|Si·Sj+1 4|ψS∝angbracketright ≃ −0.08 (−0.05); the orbital dynamics is then entirely governed by coupling to the spin fluctuations [18]. The t–Jmodel description of asingleorbiton Eqs. (16-17) is general and valid for any dimension: the mapping rests entirely on the fact that spin on the excited orbital is conserved. In particu- lar, the Jordan-Wigner fermionization applied to the 2D case gives the t–Jmodel expressed in terms of Jordan- Wigner fermions [15] and consequently the 2D version of Eqs. (16-17), see [19] for details. A very good approximate solution for the higher- dimensional t–Jmodel with J > tcan be obtained by perturbation theory, because J > tcorresponds to weak coupling, see Eq. (7) of Ref. [20]. The solu- tion shows that in 2D or 3D a single orbiton in the undoped AF system described by the t–Jmodel can- not fractionalize due the magnetic string effect. Still, the orbiton dressed with spin fluctuations is mobile on a renormalized scale [20]. For example the 2D case di- rectly corresponds to the result in Ref. [21] which gives ω≈Ez−x1+x2(coskx+cosky)2+x3(cos2kx+cos2ky) for the orbiton dispersion ( xiare positive parameters ∝t2/JwhenJ > t, cf. Ref. [21] for exact values). We thus expect to observe, e.g. in high resolution RIXS experiment on 2D cuprates, a spectrum similar to the one-particle Greens function of the higher-dimensional t–Jmodel, with a low-energy quasi-particle orbiton peak ∝ωand additional incoherent part [21]. Crucially, while the parameters xidepend onJ/tand would vary in more realistic spin-orbital models [16], the general shape of the orbiton dispersion ω– the minimum at ( π/2,π/2), sad- dle point at ( π,0), maxima at (0 ,0) and (π,π) – is robust reflecting the fact that the coherent motion of orbiton is possible only within a given spin-sublattice. No analogue in spin sector.— Let us now revisit the analoguewherethespinispolarized(FM) andtheorbital sector shows AO order due to, e.g., a large Jahn-Teller effect. An example would be the FM and AO planes in LaMnO 3or KCuF 3[13]. Many studies have shown that the orbital degrees of freedom merely renormalize the spin excitation in this case and do not change the peri- odicity of magnons (see, e.g., Ref. [22]). This qualitativedifference is caused by the fact that in realistic cases the Jahn-Teller stabilized AO order is much more classical and robust, thus suppressing the creation of orbital ex- citations by the excited spin and giving larger weight to the pure spin excitation. In other words, spin waves are typically below the orbital gap and well protected by the underlying SU(2) symmetry and Goldstone theorem. In conclusion, we have shown that orbitons in realistic spin-orbitalmodelswith AF-spin andferroorbitalground state are so strongly coupled to the spin excitations that the usual mean-field decoupling of two sectors breaks down. In fact, we have presented an exact mapping of the problem onto an effective t–Jmodel and have shown that the orbiton in such models behaves like a single hole in undoped AF Mott insulator. However, since the typi- calsuperexchangeparametersforKKspin-orbitalmodels lead toJ >t, the study of orbiton problem provides ac- cess to a regime of the t–Jmodel which has been thought to be ‘unphysical’ in terms of single-band Hubbard mod- els. Finally, in 1D signatures of spin-orbital separation are expected again in the peculiar regime where spinons are faster than orbitons. We thank G. Jackeli, M.W. Haverkort and A.M. Ole´ s for discussions. Support from the Alexander von Hum- boldt Foundation (K.W.) and the DFG Emmy Noether Program (M.D.) is acknowledged. [1] K.I. Kugel and D.I. Khomskii, Sov. Phys. Usp. 25, 231 (1982). [2] G. Jackeli and G. Khaliullin, Phys. Rev. Lett. 102, 017205 (2009). [3] A.Y. Kitaev, Annals of Physics 303, 2 (2003). [4] B. Dou¸ cot, M. Feigel’man, L. Ioffe, and A. Ioselevich, Phys. Rev. B 71, 024505 (2005). [5] C. Ulrich et al., Phys. Rev. Lett. 97, 157401 (2006). [6] C. Ulrich et al., Phys. Rev. Lett. 103, 107205 (2009). [7] A. Kotani and S. Shin, Rev. Mod. Phys. 73, 203 (2001). [8] L.J.P. Ament et al., Rev. Mod. Phys. 83, 705 (2011). [9] J. van den Brink et al., Phys. Rev. B 58, 10276 (1998). [10] Y.-Q. Li, M. Ma, D.-N. Shi, and F.-C. Zhang, Phys. Rev. B60, 12781 (1999). [11] It turns out that the results below are valid also when th e symmetry of the orbital superexchange term is further broken (as it may be the case in some compounds). [12] K. Wohlfeld, A.M. Ole´ s, and P. Horsch, Phys. Rev. B 79, 224433 (2009). [13] F. Moussa et al., Phys. Rev. B 54, 15149 (1996); B. Lake, D.A. Tennant, and S.E. Nagler, Phys. Rev. Lett. 85, 832 (2000). [14] P. Jordan and E. Wigner, Z. Physik 47, 631 (1928). [15] S. Barnes and S. Maekawa, Journal of Physics: Con- densed Matter 14, L19 (2002). [16] Note that inequivalent hoppings of orbitals [not inclu ded in model Eq. (1)] would lead to t≡2t1t2/U,J= 4t2 1/U andt∝negationslash=J/2, where t1(t2) is the hopping of electrons in the lower (upper) orbital. Still J > tholds since t1≥t2. [17] M. Brunner, F. Assaad, and A. Muramatsu,5 Eur. Phys. J. B 16, 209 (2000); H. Suzuura and N. Nagaosa, Phys. Rev. B 56, 3548 (1997). [18] G. Khaliullin and S. Maekawa, Phys. Rev. Lett. 85, 3950 (2000). [19] See supplementary material below for details of the derivation. [20] S. Schmitt-Rink, C.M. Varma, and A.E. Ruckenstein, Phys. Rev. Lett. 60, 2793 (1988). [21] G. Martinez and P. Horsch, Phys. Rev. B 44, 317 (1991). [22] G. Khaliullin and S. Okamoto, Phys. Rev. B 68, 205109 (2003). SUPPLEMENTARY MATERIAL In what follows we show that the t–Jmodel descrip- tion of the orbiton problem [Eqs. (6-7) in the main text] is not only valid in the 1D case but also in higher di- mensions. To be explicit we concentrate now on the 2D case (from which the 3D case follows in a straightforward way) and introduce the Jordan-Wigner fermions αandβ for pseudospins and spins: S+ j=βjeiπQj, (8) S− j= e−iπQjβ† j, (9) Sz j=1 2−njβ, (10) whereQj=/summationtextj−1 l=1nlβandβ† jcreate spinons while T+ j= e−iπ¯Qjα† j, (11) T− j=αjeiπ¯Qj, (12) Tz j=njα−1 2, (13) where¯Qj=/summationtextj−1 l=1nlα. Note that this is the same trans- formation as in the main text but, to keep track of the phase factors, we explicitly wrote the site index jof the phase factors Qjand¯Qj. Next, similarly as in 1D, the spin of the propagating electron in the upper orbital 2 is conserved, and one may calculate the spectral function O(k,ω) for, e.g., spin-up in the upper orbital. We are then allowed to replace T† k→/summationtext jeikjT+ j(1 2+Sz j) = e−iπ¯Qjα† j(1−njβ) in Eq. (2) in the main text and thus terms in the Hamiltonian that create or annihilate both a spinon and a pseudospinon at the same site also lead to a vanishing contribution to the spectral function. In 2D one has to take care of the phase factors. How- ever,thecrucialobservationisthatforthecaseofthe sin- gleorbiton the phase factors ¯Qjassociated with a pseu- dospinon either do not contribute at all [Eq. (14) below]or cancel for the nearest neighbor bonds [Eq. (15) below] – similarly to 1D. Thus, these are only the spin phase factorsQjwhich are in the end present in the spectral function and Hamiltonian written in terms of the spinons and pseudospinons: O(k,ω) =1 πlim η→0ℑ∝angbracketleft¯ψ|αk1 ω+E¯ψ−¯H−Ez−iηα+ k|¯ψ∝angbracketright, (14) with the effective fermionic Hamiltonian ¯H=−1 2J/summationdisplay /angbracketlefti,j/angbracketright/parenleftbig e−iπQiβ† iαiα† jβjeiπQj+αiα† j+h.c./parenrightbig +J/summationdisplay /angbracketlefti,j/angbracketright/bracketleftBig1 2(e−iπQiβ† iβjeiπQj+h.c.) −1 2(niβ+njβ)+niβnjβ/bracketrightBig . (15) As stated above (and mentioned for the 1D case in the main text), here we have an implicit constraint ∀jβ† jβj+ α† jαj≤1 while|¯ψ∝angbracketrightis a tensor product of the magnetic ground state |ψS∝angbracketright=|AF∝angbracketrightexpressed in terms of spinons and a vacuum state for pseudospinons. At this point, we observe that the resulting effective Hamiltonian is in fact a Hamiltonian for the t–Jmodel written in terms of the Jordan-Wigner fermions with the above constraint. By introducing the electron opera- torspj↑=α† j,pj↓=α† jβjeiπQjacting in the restricted Hilbert space without double occupancies we obtain O(k,ω)=1 πlim η→0ℑ∝angbracketleft˜ψ|p† k↑1 ω+E˜ψ−˜H−Ez−iηpk↑|˜ψ∝angbracketright, (16) with thet–JHamiltonian ˜H=−t/summationdisplay /angbracketlefti,j/angbracketright,σ(p† iσpjσ+h.c.)+J/summationdisplay /angbracketlefti,j/angbracketright(Si·Sj+1 4ninj), (17) wherenj=/summationtext σnjpσand the hopping parameter tis defined as t=J/2. The ground state |˜ψ∝angbracketrightis now the tensor product of a vacuum state for holes and the |ψS∝angbracketright state. We have thus mapped the problem of a single or- bital excitation in the 2D FO and AF ground state, with dynamics governed by Hamiltonian Eq. (1) in the main text, onto a single hole doped into the undoped 2D AF ground state with its dynamics governed by Hamiltonian (17).

2102.05489v1.Suppression_of_effective_spin_orbit_coupling_by_thermal_fluctuations_in_spin_orbit_coupled_antiferromagnets.pdf

Suppression of eective spin-orbit coupling by thermal uctuations in spin-orbit coupled antiferromagnets Jan Lotze1and Maria Daghofer1, 2 1Institut f ur Funktionelle Materie und Quantentechnologien, Universit at Stuttgart, 70550 Stuttgart, Germany 2Center for Integrated Quantum Science and Technology, University of Stuttgart, Pfaenwaldring 57, 70550 Stuttgart, Germany (Dated: February 11, 2021) We apply the nite-temperature variational cluster approach to a strongly correlated and spin- orbit coupled model for four electrons (i.e. two holes) in the t2gsubshell. We focus on parameters suitable for antiferromagnetic Mott insulators, in particular Ca 2RuO 4, and identify a crossover from the low-temperature regime, where spin-orbit coupling is essential, to the high-temperature regime where it leaves few signatures. The crossover is seen in one-particle spectra, where xzandyzspectra are almost one dimensional (as expected for weak spin-orbit coupling) at high temperature. At lower temperature, where spin-orbit coupling mixes all three orbitals, they become more two dimensional. However, stronger eects are seen in two-particle observables like the weight in states with denite onsite angular momentum. We thus identify the enigmatic intermediate-temperature 'orbital-order phase transition', which has been reported in various X-ray diraction and absorption experiments atT≈260K, as the signature of the onset of spin-orbital correlations. I. INTRODUCTION Ruthenium oxides have for decades attracted consid- erable attention, rst for their complex phase diagrams that bear some similarities to those high- TNcuprate superconductors, with superconducting and Mott insu- lating phases [1]. More recently, the interplay of spin- orbit coupling (SOC), electron itineracy, electronic cor- relations, and lattice degrees of freedom, which are all present, has attracted attention. The exotic behavior emerging on this stage includes a potential spin liq- uid in-RuCl 3[2, 3], enigmatic superconductivity in Sr2RuO 4[4] and a non-equilibrium Weyl semi metal in Ca2RuO 4[5]. This last compound, Ca 2RuO 4, had already been dis- cussed as a Mott insulating end member of the fam- ily of compounds including enigmatic superconductors. Its high-temperature metal-insulator transition has been well described by a combination of density-functional the- ory and dynamical mean-eld theory [6]. The emerg- ing picture is that of a lattice-supported Mott transi- tion, where the xyorbital is lowered in energy and be- comes nearly doubly occupied, while a Mott gap opens in the approximately half lled xzandyzorbitals. SOC, which had alternatively been argued to drive the metal- insulator transition[7], was later shown to have only a weak impact on the gap [6]. However, this changes decisively when it comes to the magnetic properties of the antiferromagnetic state ob- served at even lower temperatures. For weak SOC and dominant crystal eld (CF), one would expect the half- lledxzandyzorbitals to form a spin one, while the doubly occupied xyorbital would be magnetically in- ert. However, magnetic excitations turn out to show pronounced X-Y-symmetry as well as Higgs modes [8, 9], which can more naturally be explained in terms of 'ex- citonic' antiferromagnetism, which fundamentally relieson SOC. Orbital angular momentum of two t2gholes can be modeled as an eective L=1. In the idealized picture of an undistorted Ru-O octahedron (i.e. with equivalent xy,yz, andxzorbitals) SOC would couple total spin S=1 withL=1 into a singlet ground state with to- tal angular momentum J=0 [10]. When superexchange connects ions, however, higher-energy triplets gain ki- netic energy and may condense into a magnetically or- dered state. In one dimension, the resulting ground-state phase diagram has been established by use of the density- matrix renormalization group and includes a parameter regime supporting excitonic magnetism [11, 12]. Recent numerical work using a combination of density-functional theory and variational cluster approach (VCA) has fur- ther indicated that the excitonic scenario with SOC as a decisive player indeed applies to the antiferromagnetic low-temperature state of Ca 2RuO 4[13]. It would thus be highly desirable to investigate tem- peratures between the ground state with a large role for SOC and the high-temperature state, where it only yields small corrections, also with a view towards other ruthenates with similar energy scales. While the metal- insulator transition and the interplay of lattice and corre- lations is accessible to dynamical mean-eld theory with a Monte-Carlo impurity solver, the fermionic minus-sign problem is present at lower temperatures [14]. Adjusted one-particle states based on total angular momentum can reduce the minus-sign problem [15], however, such an op- timal basis cannot easily be identied in realistic models, where CFs or anisotropic hoppings compete with SOC. In order to address low temperature scales of spin- orbit coupled t2gorbitals, described by the three-orbital Hubbard model of Sec. II, we thus implement a nite- temperature variant of the VCA, see Sec. II A. Exact diagonalization is used to solve a small cluster and to extract its self energy, which is then used to evaluate the Green's function of the thermodynamic limit. This al-arXiv:2102.05489v1 [cond-mat.str-el] 10 Feb 20212 lows us to treat the antiferromagnetic order, and since we focus here on the Mott insulating regime, bath sites are less necessary. Based on the results presented in Sec. III, we iden- tify a temperature range above the N eel temperature, but in the Mott insulating regime, where the spin-orbital character strongly changes. While the onsite-singlet and triplet states describe the ionic state at low tempera- tures very well, as expected for the excitonic scenario, they become less useful at higher temperatures. Here, the original orbitals provide a clearer picture, especially in the presence of a CF, as will be seen in the one-particle spectra discussed in Sec. III B. Int2gmodels with SOC of a magnitude suitable for excitonic magnetism, there is thus a third temperature scale intermediate between the metal-insulator transition related to charge uctuations and lattice distortions and the N eel temperature related to magnetic degrees of free- dom. We discuss in Sec. IV how this ties in with the enig- matic 'orbital ordering' transition that has been debated at intermediate temperatures in Ca 2RuO 4[16, 17]. II. MODEL AND METHODS We study here a three-orbital Hubbard model for t2g electrons on a square lattice, where we focus on nearest- neighbor (NN) hopping and tetragonal symmetry. The kinetic energy is then diagonal in orbital indices and takes the form Hkin=−t/summation.disp /uni27E8i;j/uni27E9;c† i;xy;cj;xy;(1) −t/summation.disp /uni27E8i;j/uni27E9∥x;c† i;xz;cj;xz;−t/summation.disp /uni27E8i;j/uni27E9∥y;c† i;yz;cj;yz;+H.c. whereci;;(c† i;;) annihilates (creates) an electron with spin=↑;↓in orbital=xy;xz;yz at sitei. Nearest- neighbor bonds /uni27E8i;j/uni27E9along the two direction xandyare considered and we use tas our unit of energy. Tetragonal CF splitting HCF=−
/summation.disp i;ni;xy; (2) withni;xy;=c† i;xy;ci;xy;and
>0 is motivated by the shortened octahedra of the low-temperature phase of Ca2RuO 4. For a lling of four electrons (two holes), it favors a doubly occupied xyorbital with half lled xz andyzorbitals. We will tune it to interpolate between an orbitally polarized spin-one system at large
and a more equal interplay of degenerate orbitals at small
. In the present paper, the impact of SOC is particularly important, which takes the form HSOC=/summation.disp ili⋅si=i 2/summation.disp i/summation.disp ;; ;′"  ′c† i;;ci; ;′(3)fort2gorbitals, with the totally antisymmetric Levi- Civita tensor " and Pauli matrices ,=x;y;z [18, 19]. We focus here on intermediate SOC that is not strong enough to suppress magnetic ordering. Finally, there are eective onsite Coulomb interac- tions [20] Hint=U/summation.disp i;ni↑ni↓+U′ 2/summation.disp i;/summation.disp ≠nini (4) +1 2(U′−JH)/summation.disp i;/summation.disp ≠nini −JH/summation.disp i;≠(c† i↑ci↓c† i;↓ci↑−c† i↑c† i↓ci↓ci↑) with Coulomb interaction U,U′and Hund's coupling JH connected via U′=U−2JH. We are here less interested in varying interactions and rather focus on the Mott in- sulating regime with a Hund's-rule coupling larger than SOC, so that L-Scoupling provides a clearer descrip- tion thanj-jcoupling. We thus choose U=12:5tand JH=2:5t, which is consistent with their order of magni- tude in Ca 2RuO 4[21]. A. Finite-temperature variational cluster approach We use the nite-temperature [22] VCA [23, 24], where the grand potential of the system is approximated in terms of a 'reference system' that consists of small dis- connected clusters, but has the same electron-electron interactions as the Hamiltonian of interest [25]: (cl)= cl+Tr ln(−G−1 cl)−Tr ln(−G−1 CPT) (5) with the grand potential cland Green's function Gcl obtained from the cluster. The CPT-Green's Function G−1 CPT=(G−1 cl−G−1 cl;0+G−1 0)−1replaces the non-interacting cluster Green's function Gcl;0by the non-interacting Green's function G0of the full system. The approxi- mation thus consists in replacing the self energy of the physical system by that of the small cluster. In order to improve the approximation, the self-energy{functional approach [24] allows us to optimize the cluster self energy clby varying one-particle parameters of the reference Hamiltonian. The best approximation to the system's grand potential is a stationary point of w.r.t. the vari- ational one-particle parameters. Note that this variation aects only the small cluster, the non-interacting Green's function given by the one-particle part of the physical Hamiltonian remains xed. We numerically evaluate the cluster grand potential cl=−−1ln()=−−1ln/summation.disp me−"m(6) with partition function  and cluster-energies "n, as well3 as the cluster Green's function, whose electron part reads [G(+) cl](z)=/summation.disp m;ne−"m /uni23A1/uni23A2/uni23A2/uni23A2/uni23A2/uni23A2/uni23A3/uni27E8.alt2 m/divides.alt2c/divides.alt2 + n/uni27E9.alt2/uni27E8.alt2 + n/divides.alt2c† /divides.alt2 m/uni27E9.alt2 z−E+nm+i 0+/uni23A4/uni23A5/uni23A5/uni23A5/uni23A5/uni23A5/uni23A6 (7) with energy dierence E± nm="± n−"m. We largely follow Seki et al. [22] and obtain the spectrum and eigenstates with band Lanczos to resolve (approximately) degenerate eigenenergies. We use eight starting vectors and converge 120 eigenvectors, which are used to evaluate the Green's functions in a second Lanczos run. In this second step, band Lanczos did not turn out to be advantageous, and we thus use the conventional algorithm. Assembling the cluster Green's function is accelerated with the help of a high-frequency expansion for frequency arguments with absolute values larger than that of the largest pole [22]. [Other frequencies are obtained via (7).] Following Seki et al. , the high-energy Green's function is expanded to 15-th order as G(z)=∞ /summation.disp k=0M(k)  zk+1(8) with moments M(0) (z)=and M(k>0) =/summation.disp m;n(E+ nm)ke−"m /uni27E8.alt2 m/divides.alt2c/divides.alt2 + n/uni27E9.alt2/uni27E8.alt2 + n/divides.alt2c† /divides.alt2 m/uni27E9.alt2 +/summation.disp m;n(−E− nm)ke−"m /uni27E8.alt2 m/divides.alt2c† /divides.alt2 − n/uni27E9.alt2/uni27E8.alt2 − n/divides.alt2c/divides.alt2 m/uni27E9.alt2: (9) In order to x the density to N=4 electrons (i.e. two holes), the grand potential is transformed to the free en- ergyF(N;V;T)= (;V;T)+Nby means of a Legen- dre transform[26]. There are thus at least two variational parameters, the chemical potential to x the density and the cluster chemical potential ′to ensure thermody- namic consistency[27, 28]. Additionally, we consider an- tiferromagnetic order parameters with ordered moment within the a-b-plane or along the c-axis. Previous work forT=0 has shown that the zand in-plane components lead to quite dierent grand potentials (as expected for nite SOC), but that the grand potential is very similar for operators like spin, magnetization, or total angular momentum [13]. For this reason and in order to easily compare to the spin-one antiferromagnet, we use here the spin as the order parameter. It has also been shown that a sizable CF as well as SOC both favor 'checkerboard' magnetic patterns with ordering vector Q=(;), so that we use the ctitious Weiss eld HWeiss=h/summation.disp ieQriSx/slash.leftz i(10) withilabeling the site at riandSx i=∑c† i;;↑ci;;↓+ H.c. andSz i=∑(c† i;;↑ci;;↑−c† i;;↓ci;;↓)are thexandz component of the total spin, respectively.For variational parameters ,′andhgiving station- ary grand potentials, we evaluate one-body expectation values (like magnetization or orbital densities) from the Green's function, as is done for T=0. The entropy Sis determined as the derivative of the grand potential via contour integrals: S=Scl+SCPT+Scl−( CPT− cl)/slash.leftT (11) with the contributions Scl=−( cl−/uni27E8H/uni27E9cl)/slash.leftT; (12) SCPT=/uni222E.dispCdzf(z)tr[(GclG−1 CPTGcl)−1Gmod]/slash.leftT2;(13) Scl=/uni222E.dispCdzf(z)tr[(−G−1 cl)Gmod]/slash.leftT2(14) and the abbreviations /uni27E8H/uni27E9cl=−1/summation.disp m"mexp(−"m); (15) Gmod=/summation.disp m;n(/uni27E8H/uni27E9cl−"m)−1exp(−m)[G+ cl+G− cl] −/summation.disp m;n(Tz)−1exp(−m)/bracketleft.alt4@G+ cl @z+@G− cl @z/bracketright.alt4:(16) The specic heat C(T)is then obtained as the numerical derivativeC(T)=T(
S/slash.left
T)of the entropy. Since the ordered moment in an excitonic magnet arises through a superposition of the J=0 state preferred by onsite SOC and J=1 states [10], it is helpful to con- sider weights /uni27E8J/uni27E9found in eigenstates of total onsite an- gular momentum J. Unfortunately, this is a two-particle quantity and thus not readily available from the VCA. We approximate it as the exact-diagonalization expecta- tion value obtained for the Nsitessites of the reference cluster with optimized parameters: /uni27E8J/uni27E9∶=1 Nsites/summation.disp i;Jz/summation.disp me−"m/uni27E8m/divides.alt0Ji;Jz i/uni27E9/uni27E8Ji;Jz i/divides.alt0m/uni27E9:(17) Eigenstates /divides.alt0Ji;Jz i/uni27E9denote here the state at site idened by angular momentum J=L+S. III. RESULTS A. Temperature dependence of the onsite angular momentum AtT=0, VCA for the t2gmodel with four electrons and CF
=0 has revealed two dierent ordering pat- terns depending on [13]: at small /uni22720:4t, orbitals and spins order in a stripy pattern with orthogonal order- ing momenta (0;)for spins and (;0)for orbitals. For larger, excitonic antiferromagnetic (AFM) order with momentum (;)takes over, where the out-of-plane z component is favored over in-plane directions. We are here interested in the latter regime and thus focus on >0:4t.4 0.00.10.20.30.40.00.20.40.60.81.01.21.4 (a) T/tmλ/t= 0.6 λ/t= 0.8 λ/t= 1.0 0.00.10.20.30.40.00.51.01.52.02.53.0 (b) T/tC FIG. 1. Thermodynamics of the excitonic antiferromagnet without CF splitting. (a) gives the magnetic order parameter, i.e. staggered out-of-plane spin and (b) the specic heat for SOC=0:6t;0:8t;t. (We do not intend to discuss the compli- cated spin and orbital stripe pattern found at =
=0 [13], and accordingly leave out the regime of small /uni22720:4t.) 0.00.10.20.30.40.00.20.40.60.81.0 TN(a) T/t/angbracketleftJ/angbracketright 0.00.10.20.30.4TN(b) T/t0.00.10.20.30.4TN(c) T/t/angbracketleftJ= 0 /angbracketright /angbracketleftJ= 1 /angbracketright /angbracketleftJ= 2 /angbracketright FIG. 2. Temperature evolution of the average weight found in eigenstates of the total angular momentum, see Eq. (17). (a)=0:6t, (b)=0:8t, and (c) for =t. Figure 1(a) shows the ordered spin moment depending on temperature. As expected, the value at T=0 is re- duced when larger increases the energy gap between the ionic singlet and triplet states, which in turn reduces the triplet admixture into the ground state. Somewhat sur- prisingly, the N eel temperature is not monotonic. While we can certainly not exclude strong nite-size eects due to the 2×1-site cluster, an alternative explanation may be that system at smallest =0:6tis aected by its close- ness to the competing stripy phase. The corresponding specic heat is given in Fig. 1(b) and has a second broad hump at higher temperature T>TNin addition to the expected peak at the magnetic ordering transition. This feature exists for all three values of and shifts to slightly higher temperatures for =t. Figure 2 shows the average weight Eq. (17) found in eigenstates with J=0;1;2 of the total onsite angular momentum. Weights are constant in the regime of con- stant magnetization, and the J=0 (J=1) state looses (gains) weight when magnetic order is lost. This is in clear contrast to a (somewhat articial) transition to a paramagnet at constant temperature: reducing the or- dering eld hatT=0 pushes weight from the J=1 states into theJ=0 state [13]. At TN, the curves get abruptly steeper and weights in J=0 andJ=1 states change substantially at higher temperatures T>TN. Weight in theJ=2 states is completely negligible below TN, but 0.0 0.1 0.2 0.30.00.51.01.52.0 (a) T/tmλ/t= 0.1 λ/t= 0.8 0.0 0.1 0.2 0.30.00.51.01.52.02.5(b) T/tC 0.0 0.1 0.2 0.30.00.20.40.6 TN (c) T/t/angbracketleftJ/angbracketright 0.0 0.1 0.2 0.30.00.20.40.6 TN (d) T/t/angbracketleftJ/angbracketright/angbracketleftJ= 0 /angbracketright /angbracketleftJ= 1 /angbracketright /angbracketleftJ= 2 /angbracketrightFIG. 3. Thermodynamics of spin antiferromagnet at large CF
=5t. (a) gives the magnetic order parameter, i.e. staggered magnetization and (b) the specic heat for SOC =0:1tand =0:8t. (=0 was numerically less stable.) (c) and (d) give the average overlaps of Eq. (17) for =0:1tand=0:8t, resp. similarly begins to grow at T>TN. We are going to ar- gue that this spin-orbital rearrangement is the origin of the second hump in the specic heat. For comparison, Fig. 3(a) gives the magnetization and specic heat for CF
=5tthat is large enough to enforce complete orbital polarization with a doubly occupied xy orbital at all temperatures shown. The two holes are then found in xzandyzorbitals and form a conventional spin one, with an ordered moment close to two in the AFM state. The specic heat shown in Fig. 3(b) has here only the peak at the N eel temperature and no further features. The expected weights in eigenstates with total onsite angular momentum J=0;1;2 are given in Fig. 3(c) and (d) and present a quite dierent picture from the excitonic case discussed in Fig. 2: while the weights in J=0 andJ=1 states change appreciably below TN, only little variation is seen above TN. Finally, Figs. 4 and 5 discuss intermediate
=1:5t, an order of magnitude appropriate to describe Ca 2RuO 4. Ground-state VCA calculations have here shown in- plane magnetic moments to be favored over out-of-plane moments [13], in agreement with the AFM state of Ca2RuO 4. Again, the N eel temperature is not very sensi- tive to SOC while the ordered moment is substantially reduced by it. The system without SOC has the largest ordered moment close to two, see Fig. 4(a). As will be dis- cussed below, its xyorbital is completely lled below the N eel temperature, see Fig. 5(a), so that it comes close to a spin-one scenario. Larger ≥0:6treduce the ordered moment, which indicates that orbital polarization is not strong enough to quench SOC. The specic heat shown5 0.00.10.20.30.40.00.51.01.52.0 (a) T/tmλ/t= 0.0 λ/t= 0.6 λ/t= 0.8 λ/t= 1.0 0.00.10.20.30.40.00.51.01.52.02.53.0 (b) T/tC FIG. 4. Transition from spin to excitonic antiferromagnet at
=1:5t. (a) gives the magnetic order parameter, i.e. stag- gered in-plane spin magnetization and (b) the specic heat for SOC=0;0:6t;0:8t;t. 0.00.51.01.52.0 TN (a)/angbracketleftnρ/angbracketright ρ=xy ρ=yz ρ=xzTN (b)TN (c)TN (d) 0.00.10.20.30.40.00.20.40.6 TN(e) T/t/angbracketleftJ/angbracketright 0.00.10.20.30.4TN(f) T/t/angbracketleftJ= 0 /angbracketright /angbracketleftJ= 1 /angbracketright /angbracketleftJ= 2 /angbracketright 0.00.10.20.30.4TN(g) T/t0.00.10.20.30.4TN(h) T/t FIG. 5. Spin-orbital onsite wave function for
=1:5t. (a-d) show the orbital-resolved densities for =0;0:6t;0:8t;tand (e-h) the weights in Jstates according to Eq. (17). in Fig. 4(b) looks qualitatively much more similar to the results for
=0 than to those for
=5t, as a second hump atT>TNis clearly seen. While the transition from spin-one to excitonic antifer- romagnetism is a gradual crossover, it was estimated to occur at≈0:7tin the ground state [13]. The J-weights qualitatively agree, with Fig. 5(e) for =0 being simi- lar to the spin-one scenario of Fig. 3(c,d), while Figs. 5 (g,h) for=0:8tand=1 resemble more the excitonic case of Fig. 2. Figure 5(f) for =0:6tlies somewhere in between, again in line with the previous estimate. The second hump in the specic heat for =0 can be understood by noting that the orbital densities in Fig. 5(a) do not remain constant above TN. Since the CF is here just strong enough to ll the xyorbital at T=0, nite temperature can induce xy-holes and these orbital uctuations are re ected in the specic heat. The weights found in states J=0;1;2, in contrast do here not change above TN, see Fig. 5(e), when there is no SOC. In the opposite limit =t, the orbital densities are nearly constant, a small dierence between xzandyz belowTNbeing due to magnetic symmetry breaking. Weights inJ=0 andJ=1 states depend here strongly on temperature at T>TN, see Fig. 5(h). While low T/uni2272TN strongly suppressed any weight in J=2 states for
=0, -20-15-10-50510(!)=t(a) 00.6 1.2 (b) 00.6 1.2 (c) 00.6 1.2 -20-15-10-50510(!)=t(d) (e) (f) (0;0)(;0)(;)(0;0)-20-15-10-50510 k(!)=t(g) (0;0)(;0)(;)(0;0) k(h) (0;0)(;0)(;)(0;0) k(i)FIG. 6. Orbital-resolved one-particle spectral density for
= 0,=0:8tand temperatures (a-c) T=0, (d-f)T=0:14t≈TN, and (g-i)T=0:35t. Orbital character is xyin (a), (d), and (g),yzin (b), (e), and (h), and xzin (c), (f), and (i). see Fig. 2, it is here nearly constant, because it is con- nected to the clear orbital polarization nxy>nxz/slash.leftyz[13]. B. Signatures of SOC in one-particle spectra Figure 6 shows the VCA one-particle spectral density for
=0 and temperatures T=0,T/uni2273TNandT/uni226BTN. At all temperatures, the occupied states are split into three subbands at energies !/uni22725t,!≈10t, and!/uni227315t (with the last having lower weight), which can be related to Hund's-rule coupling [21]. Below TN, some signatures of the doubling of the unit cell are visible in the form of shadow bands around (0;0)and(;). Apart from this feature, the predominant eect of temperature is making the spectra less coherent. Overall, temperature eects are here rather subtle. Temperature-driven orbital reconstruction reveals it- self slightly more when CF and SOC compete, see the one-particle spectra shown in Fig. 7 for
=1:5tand =t. The ground-state spectrum Fig. 7(a-c) shows again a slight shadow band due to the doubling of the unit cell and both the empty band (of predominantly xz/yzchar- acter) and the highest occupied band (of predominantly xycharacter) have a two-dimensional dispersion, simi- lar to Fig. 6(a-c). In the spectra taken around TN, see Fig. 7(d-f), the shadow band has vanished. The occu- piedxz/yzstates have become more coherent than in the ground state. This rather unconventional behavior may be related to the ladder-like features that were re-6 -20-15-10-50510(!)=t(a) 00.6 1.2 (b) 00.6 1.2 (c) 00.6 1.2 -20-15-10-50510(!)=t(d) (e) (f) (0;0)(;0)(;)(0;0)-20-15-10-50510 k(!)=t(g) (0;0)(;0)(;)(0;0) k(h) (0;0)(;0)(;)(0;0) k(i) FIG. 7. Orbital-resolved one-particle spectral density for
= 1:5t,=tand temperatures (a-c) T=0, (d-f)T=0:16t≈TN, and (g-i)T=0:35t. Orbital character is xyin (a), (d), and (g),yzin (b), (e), and (h), and xzin (c), (f), and (i). cently found in a strong-coupling t-J-like model with- out SOC, where they arise in the AFM state due to the anisotropic hoppings of these orbitals [29]: when mag- netic order is lost, the ladder features become weaker and the underlying dispersion is seen more easily. It is rather one-dimensional, as expected for xz/yzorbitals without SOC. Such a weaker impact of SOC at higher binding en- ergies is somewhat reminiscent of the correlation-induced energy dependence of SOC previously reported for metal- lic Sr 2RuO 4[30]. In the unoccupied xzandyzstates, on the other hand, incoherent features have gained weight in addition to the coherent band dominating the T=0 spectrum. They do not follow the two-dimensional dispersion of the coher- ent band, but are more one dimensional. At high tem- peratureT=0:35t, nally, the unoccupied bands show mostly the one-dimensional dispersion characteristic of xz/yzorbitals in the absence of SOC, see Fig. 7(h-i). In the presence of a CF, SOC thus only couples the three orbitals into a 2D dispersion at lower temperatures and lower excitation energies, while spectra at higher tem- peratures and energies look similar to the case without SOC. IV. DISCUSSION AND CONCLUSIONS We have shown that temperature strongly aects the spin-orbital onsite state in the PM Mott insulating state of spin-orbit coupled t4 2gsystems. We have investigatedparameter sets supporting excitonic AFM order at low temperatures, with and without a crystal eld. As long as the CF is not strong enough to completely quench the orbital degree of freedom, we consistently nd a second broad hump in the specic heat, in addition to the peak atTN. In the same temperature range, onsite total an- gular momentum changes substantially. In one-particle spectra, low-energy excitations stem- ming fromxzandyzorbitals are two-dimensional in the ground state, but become more one-dimensional at higher temperatures. This can also be interpreted as SOC being most eective at low temperatures. Overall, signatures of SOC and of the temperature-driven spin-orbital rear- rangement are rather subtle in one-particle spectra. Even at low temperatures, where SOC is essential do reproduce the dispersion of magnetic excitations [8, 9], one-particle spectra have thus been reasonably well described already without taking SOC into account [21, 29]. However, we argue that X-ray diraction and absorp- tion experiments performed on Ca 2RuO 4show signatures of the spin-orbital rearrangement found here. Parame- ters for this compound correspond roughly to those of Figs. 4(c,d) and 7, i.e.
≈1:5tand≈0:8t−1t[13]. At temperatures of ≈260 K, i.e. between the metal-insulator transition (which goes together with a structural phase transition) and the N eel transition, signatures of another phase transition were reported early on and interpreted in terms of orbital order [16, 17]. Since this additional transition does not break any spa- tial symmetries, one can rule out orbital stripe [31] or checkerboard [32] patterns theoretically predicted for ab- sent (or weak) SOC. More recent work established that the transition cannot be related to a change in orbital densities, leaving only the phase in a complex orbital su- perposition as a possibility [33]. This would t with our ndings of an SOC-driven spin-orbital rearrangement. When SOC prefers the J=0 state at low temperatures, this implies for each spin projection a specic phase rela- tion between the orbitals. In contrast, no denite phases are expected at higher temperatures where SOC is less active. We have thus identied the enigmatic orbital-order transition in Ca 2RuO 4as a transition to a spin-orbit cou- pled onsite wave function. This implies, e.g., that a spin up (down) prefers the complex /divides.alt0lz=±1/uni27E9orbital over the opposite state. This is somewhat reminiscent of ferro- orbital order into complex orbitals, which was early on proposed as a scenario for Ca 2RuO 4[17]. More generally, complex-orbital order has been suggested to play a role in doped manganites [34] and the Vervey transition of mag- netite [35]. Spontaneous complex-orbital order is, how- ever, rare, because lattice distortions like the Jahn-Teller eect favor real orbitals. 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0907.2092v2.A_virtual_intersubband_spin_flip_spin_orbit_coupling_induced_spin_relaxation_in_GaAs__110__quantum_wells.pdf

arXiv:0907.2092v2 [cond-mat.mes-hall] 7 Aug 2009A virtual intersubband spin-flip spin-orbit coupling induc ed spin relaxation in GaAs (110) quantum wells Y. Zhou1and M. W. Wu1,2,∗ 1Hefei National Laboratory for Physical Sciences at Microsc ale, University of Science and Technology of China, Hefei, Anhui , 230026, China 2Department of Physics, University of Science and Technolog y of China, Hefei, Anhui, 230026, China (Dated: November 8, 2018) A spin relaxation mechanism is proposed based on asecond-or der spin-flip intersubbandspin-orbit coupling together with the spin-conserving scattering. Th e corresponding spin relaxation time is calculated via the Fermi golden rule. It is shown that this me chanism is important in symmetric GaAs (110) quantum wells with high impurity density. The dep endences of the spin relaxation time on electron density, temperature and well width are studied with the underlying physics analyzed. PACS numbers: 72.25.Rb; 71.70.Ej; 73.21.Fg In recent years, semiconductor spintronics has be- come a focus of intense experimental and theoretical re- search [1, 2, 3, 4, 5, 6]. One of the key factors for the design of the spin-based device is to understand the spin relaxation such that the information is well pre- served before required operations are completed. In n- type zinc-blende semiconductors, like GaAs, the lead- ing spin relaxation mechanism in most situation is the D’yakonov-Perel’ (DP) mechanism [7, 8, 9, 10], which is from the joint effects of the momentum scattering and the momentum-dependent effective magnetic field (inhomogenous broadening [11, 12, 13]) induced by the Dresselhaus [15] and/or Rashba [16, 17] spin-orbit cou- pling(SOC).However,insymmetric(110)-orientedGaAs quantum wells (QWs), when only the lowest subband is occupied, the in-plane components of the spin-orbit field vanish and the effective magnetic field only exists along the growth direction [18]. As a result, electrons with spin polarization along the growth direction can not precess around the effective field, and therefore the DP mecha- nism is absent. It is also noted that in the presence of an in-plane external magnetic field, the in-plane and out- of-plane spin relaxations are mixed, thus the DP mecha- nism still leads to the spin relaxation/dephasing in this system, as point out by Wu and Kuwata-Gonokami [19]. Experimentally Ohno et al.[20] first observed very long spin relaxation time (SRT) in GaAs (110) QWs, which exceeds the SRT in (100) QWs by more than one orderof magnitude. Later the spin dynamics in this system was studied by many works [21, 22, 23, 24, 25, 26, 27, 28, 29]. In most of these works, the main reason limiting the SRT is thought to be the Bir-Aronov-Pikus (BAP) mecha- nism [30, 31], which is due to the electron-hole exchange interaction (holes are from the optical excitation in n- type samples). In the spin noise spectroscopy measure- ments by M¨ uller et al.[28], the excitation of semicon- ductor is negligible, and hence the BAP mechanism is avoided. They reportedevenlongerSRT about 24ns and attributed it to the DP mechanism due to the random Rashba fields caused by fluctuations in the donor den-sity [32, 33]. In this note, we propose another spin relax- ation mechanism to understand the slow spin relaxation in theabsenceof the DP and BAP mechanisms, which is based on a second-order spin-flip process of the intersub- band SOC together with the spin-conserving scattering. We will show that this mechanism is important in sym- metric GaAs (110) QWs with high impurity density. We start our investigation from symmetrically doped GaAs (110) QWs with the growth direction along the z- axis. The well width is assumed to be small enough so that only the lowest subband is occupied for the tem- perature and electron density we discuss. The envelope functions of the relevant subband are calculated under the finite-well-depth assumption. The barrier layer is chosen to be Al 0.39Ga0.61As where the barrier height is 328meV[34]. TheHamiltoniancanbe writtenas( /planckover2pi1≡1) H=/summationdisplay nn′kσσ′/bracketleftBig (ǫk+En)δnn′δσσ′+hnn′(k)·σσσ′ 2/bracketrightBig ×c† nkσcn′kσ′+HI. (1) Hereǫk=k2/2m∗is the energy spectrum of the elec- tron with two-dimensional momentum k= (kx,ky);En represents the quantized energy of the electron in the n- th subband; σis the Pauli matrix. In symmetric QWs without external gate voltage, the Rashba SOC can be neglected [35], and the spin-obit field h(k) is only from the Dresselhaus term. In the (110) coordinate system, h(k) can be written as hnn′ x(k) =γD/bracketleftbig −(k2 x+2k2 y)∝angbracketleftn|kz|n′∝angbracketright+∝angbracketleftn|k3 z|n′∝angbracketright/bracketrightbig , hnn′ y(k) = 4γDkxky∝angbracketleftn|kz|n′∝angbracketright, hnn′ z(k) =γDkx/parenleftbig k2 x−2k2 y−∝angbracketleftn|k2 z|n′∝angbracketright/parenrightbig δnn′,(2) whereγD= 30 eV·˚A3denotes the Dresselhaus SOC coef- ficient [36] and ∝angbracketleftn|km z|n′∝angbracketright=/integraltext dzφ∗ n(z)(−i∂/∂z)mφn′(z) withφn(z) representing the envelope function of the electron in n-th subbands. Since ∝angbracketleftn|kz|n∝angbracketright= ∝angbracketleftn|k3 z|n∝angbracketright= 0, all the in-plane components of the in- tersubband spin-orbit field vanish as mentioned above.2 However, ∝angbracketleft1|kz|2∝angbracketright ∝negationslash= 0, thus the in-plane compo- nents of the intersubband spin-orbit field are still present. The interaction Hamiltonian HIis composed of the electron-ionized-impurity, electron-longitudinal- optical(LO)-phonon, electron-acoustic(AC)-phononand electron-electron Coulomb interactions. Their expres- sions can be found in textbooks [37, 38]. FIG. 1: (Color online) Schematic representation of a second - order spin-flip process based on the intersubband spin-orbi t fieldHsoand the spin-conserving scattering Hsc. Asecond-orderspin-flip processcan be constructed via the intersubband spin-orbit field and the spin-conserving scattering as depicted in Fig. 1. The matrix element of the spin-flip transition from the electron-impurity scat- tering (which is referred to as the spin-flip electron- impurity scattering in the following) is given by U↑↓(k,k′,qz) =∝angbracketleft1k↑ |Hso|2k↓∝angbracketright ∝angbracketleft2k↓ |Hsc ei|1k′↓∝angbracketright ǫk′−ǫk−∆E21 +∝angbracketleft1k↑ |Hsc ei|2k′↑∝angbracketright ∝angbracketleft2k′↑ |Hso|1k′↓∝angbracketright −∆E21.(3) Here ∆E21=E2−E1stands for the energy splitting between the first and second subbands of electrons. Al- though the above process is similar to the short-range Elliott-Yefet spin relaxation caused by virtual scatter- ing [1, 39], the intermediate virtual states are chosen to be the states in the high conduction subband in stead of the states in the valence band in the previous work [39]. From the energy conservation ǫk=ǫk′, the symmetry of the form factor I12(qz) =I21(qz) =∝angbracketleft1|eiqzz|2∝angbracketrightand the symmetry of the spin-orbit field h21 /bardbl(k) =−h12 /bardbl(k), U↑↓(k,k′,qz) can be rewritten as U↑↓(k,k′,qz) =Uk′−kI12(qz) −∆E21[h12(k)−h12(k′)]·σ↑↓.(4) Similarly, one can obtain the matrix element of the spin- flip electron-phonon scattering Dη ↑↓(k,k′,qz) =DQηI12(qz) −∆E21[h12(k)−h12(k′)]·σ↑↓,(5) whereQ= (k′−k,qz) is the three-dimensional momen- tum.Uk′−kandDQηrepresent the matrix elements of the electron-impurity and electron-phonon interactions respectively, whose expressions are given in detail inRef. 40. From Eqs. (4) and (5), it is seen that the matrix elements of the spin-flip electron-impurity and electron-phononscatteringsareproportionalto |h12(k)− h12(k′)|/∆E21. Thus the term containing k3 zinh(k), which is independent of two-dimensional momentum k, has no contribution to the spin-flip scattering. By using the Fermi golden rule, the SRT is given by T1−1=Tei 1−1+Tep 1−1with [41] Tei/ep 1−1 =2/summationtext kk′qzΓei/ep ↑↓(k,k′,qz)fk(1−fk′) /summationtext kk′fk(1−fk′).(6) Herefkis the equilibrium electron distribution func- tion. Γei/ep ↑↓(k,k′,qz) represents the transition rate of the spin-flip electron-impurity or electron-phononscattering, which can be written as Γei ↑↓(k,k′,qz) = 2π|U↑↓(k,k′,qz)|2δ(ǫk−ǫk′),(7) Γep ↑↓(k,k′,qz) = 2π/summationdisplay η|Dη ↑↓(k,k′,qz)|2[NQηδ(ǫk−ǫk′ +ωQη)+(NQη+1)δ(ǫk−ǫk′−ωQη)].(8) HereNQηandωQηare the distribution function and en- ergy spectrum of the phonon with mode ηand momen- tumQ. It is noted that the spin relaxation process we discuss is only from the virtual intersubband scattering. The spin relaxation due to the real intersubband scatter- ing [14, 24, 25] is negligible due to the small well width in the present investigation. Our main results are summarized in Figs. 2–4. In Fig. 2, we plot the SRTs due to the spin-flip electron- impurity, electron-AC-phonon and electron-LO-phonon scatterings together with the total SRT with all the spin- flip scatterings included as function of temperature for Ni=Ne(solid curves) and Ni= 0.01Ne(dashed curves). Ne= 1.8×1011cm−2anda= 16.8 nm. In the low impurity density case with Ni= 0.01Ne, it is seen that the spin relaxation due to the spin-flip electron-AC-phononand spin-flip electron-impurity scat- terings are comparable at low temperature and the con- tribution from the spin-flip electron-LO-phonon scatter- ing becomes more important and even dominates at high temperature. In the high impurity density case with Ni=Ne, thespin relaxationdue tothespin-flipelectron- impurity scattering is dominant at most temperatures we discuss, and the contribution from the spin-flip electron- LO-phonon scattering becomes comparable to that from thespin-flip electron-impurityscatteringwhen T >80K. We also compare our results with the experiment by M¨ ulleret al.[28] (square) in Fig. 2. By fitting the mea- suredmobilitygiveninRef.28, oneobtains Ni∼0.01Ne. In this case, it is seen that the SRT obtained from our model is two orders of magnitude larger than the experi- mental data at T= 20 K. However, in the high impurity case with Ni=Ne, it is shown that the spin relaxation3 101102103104105106107 10 100 T ( K )T1 ( ns )total EI LO AC exp FIG. 2: (Color online) SRTs due to the spin-flip electron- impurity, electron-AC-phonon and electron-LO-phonon sca t- terings as well as the total SRT with all the spin-flip scatter - ings included vs.temperature Twith well width a= 16.8 nm and electron density Ne= 1.8×1011cm−2. The impurity densities are Ni=Ne(solid curves) and 0 .01Ne(dashed curves), respectively. The square represents the experime ntal result in Ref. 28. The cross represents the SRT due to the RRDP mechanism for Ni=Ne, calculated via the relation TRRDP 1∝1/τp. 101102103104 10 100 T ( K )T1 ( ns )Ne = 0.5 × 1011 cm-2 2 × 1011 cm-2 FIG. 3: (Color online) SRTs with all the spin-flip scattering s included vs.temperature TforNi=Ne= 0.5×1011cm−2 and 2×1011cm−2, respectively. The well widths are a= 10 nm (•) and 16 .8 nm (/square). The black dashed vertical lines indicate the Fermi temperatures Te Ffor both electron densi- ties. rate due to our mechanism is enhanced by more than one order of magnitude. Meanwhile, the spin relaxation rate due to the random Rashba field induced DP (RRDP) mechanism [32, 33] is suppressed significantly due to the increase of the electron-impurity scattering. Here we es- timate the SRT due to the RRDP mechanism at Ni=Ne (cross) by using the relation TRRDP 1∝1/τp[1]. It is seen that the mechanism from our model becomes more im-101102103104 0.1 1 Ne ( 1011 cm-2 )T1 ( ns )T = 20 K 50 K FIG. 4: (Color online) SRTs with all the spin-flip scattering s included vs.electron density Nefor temperatures T= 20 K and 50 K, with impurity densities Ni=Ne(•) andNi= 1011cm−2(/square). The black dashed vertical lines indicate the electron densities satisfying Ee F=kBTfor both temperatures. portantcomparedwiththeRRDP mechanisminthehigh impurity case. Now we discuss the temperature dependence of the SRT. In Fig. 3 we plot the SRT as function of temper- ature for different electron densities and well widths. It is shown that the SRT first decreases slowly and then rapidly with T. The turning point is roughly around the electron Fermi temperature Te F=Ee F/kB. Since the electron-impurity scattering has a weak temperature de- pendence, the temperature dependence of the SRT is mainly from the k-dependent spin-orbit field h(k). From Eqs. (4) and (5), it is seen that only the quadratic terms inh(k) contribute to the spin-flip scattering. When tem- peratureincreases,moreelectronsaredistributedatlager momentum states, and then the contribution from h(k) becomes larger. This leads to a larger spin-flip scatter- ing rate and hence reduces the SRT. It is also noted the average momentum of electrons is not sensitive to tem- perature in the degenerate regime ( T < Te F). Thus the SRT decreases slowly at low temperature. From Fig. 3, it is also seen that the SRT becomes shorter for wider well width. It can be understood as follows. Since the form factor is weakly dependent on well width, one ob- tains that T1∝(∆E21/∝angbracketleft1|kz|2∝angbracketright)2from Eqs. (4)–(8). It is known that ∆ E21∝a−2and∝angbracketleft1|kz|2∝angbracketright ∝a−1under the infinite well-depth assumption. Thus the SRT decreases with an increase of well width. The electron density dependence of the SRT is also in- vestigated. In Fig. 4, the SRT is plotted as function of electron density for different temperatures and impurity densities. We first discuss the case with fixed impurity densityNi= 1011cm−2. It is seen that the SRT first in- creases and then decreases with Newith a peak around the electron density satisfying Ee F=kBT. This peak originates from the competition of two effects. On one4 hand, similar to the discussion in temperature depen- dence, with the increase of electron density, the average kincreases and thus the contribution from h(k) is en- hanced. This effect leads to a decrease of the SRT. On the other hand, the screening of electrons also increases with electron density, which suppresses the spin-flip scat- tering and increases the SRT. When the electron density is low and Ee F< kBT, i.e., in the nondegenerate regime, the average kchanges little with Ne. Thus the effect of the increase in the screening is dominant and the SRT increases with increasing density. However, when the electron density is high enough so that Ee F> kBT, i.e., in the degenerate regime, the effect of the increase of the contribution from h(k) becomes more important. Conse- quentlytheSRTdecreases. Thenweturntothecasewith Ni=Ne. ItisseenthattheSRTdecreasesmonotonically with electron density. This is because the effect of the in- crease of the impurity scattering surpasses the one of the increase of screening, and makes the SRT decrease even in low electron density (nondegenerate) regime. Never- theless, one still can see that the SRT decreases more rapidly in the degenerate regime. In conclusion, we have proposed a spin relaxation mechanism based on a second-order spin-flip process of the intersubband spin-flip SOC and the spin-conserving scattering, and calculated the corresponding SRT via the Fermi golden rule. We show that this mechanism is im- portant in symmetric GaAs (110) QWs with high impu- rity density. The temperature, well width and electron density dependences of the SRT are also investigated. This work was supported by the National Natural Sci- ence Foundation of China under Grant No. 10725417, the National Basic Research Program of China under Grant No. 2006CB922005and the Knowledge Innovation Project of Chinese Academy of Sciences. We thank M. Q. Weng for valuable discussions. ∗Author to whom correspondence should be addressed; Electronic address: mwwu@ustc.edu.cn. [1] F. Meier, B.P. Zakharchenya(Eds.), Optical Orientatio n, North-Holland, Amsterdam, 1984. [2] S.A. Wolf, D.D. Awschalom, R.A. Buhrman, J.M. Daughton, S. von Moln´ ar, M.L. Roukes, A.Y. Chtchelka- nova, D.M. Treger, Science 294, (2001) 1488. [3] D.D. Awschalom, D. Loss, N. Samarth (Eds.), Semi- conductor Spintronics and Quantum Computation, Springer-Verlag, Berlin, 2002. [4] I.ˇZuti´ c, J. Fabian, S. Das Sarma, Rev. Mod. Phys. 76, (2004) 323. [5] J. Fabian, A. Matos-Abiague, C. Ertler, P. 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Bel’kov, S.A. Tarasenk o, D. Schuh, W. Wegscheider, T. Korn, C. Sch¨ uller, D. Weiss, S.D. Ganichev, Phys. Rev. B 79, (2009) 245329. [30] G.L. Bir, A.G. Aronv, G.E. Pikus, Zh. Eksp. Teor. Fiz. 69, (1975) 1382. [31] G.L. Bir, A.G. Aronv, G.E. Pikus, Sov. Phys. JETP 42, (1976) 705. [32] E.Y. Sherman, Appl. Phys. Lett. 82, (2003) 209. [33] E.Y. Sherman, Phys. Rev. B 67, (2003) 161303(R). [34] E.T. Yu, J.O. McCaldin, T.C. McGill, Solid State Phys. 46, (1992) 1. [35] Bernardes et al. [E. Bernardes, J. Schliemann, M. Lee, J.C. Egues, D. Loss, Phys. Rev. Lett. 99, (2007) 076603; see also R. S. Calsaverini, E. Bernardes, J.C. Egues, D. Loss, Phys. Rev. B 78, (2008) 155313] showed that the inter-subband Rashba SOC is nonzero even in sym- metric QWs due to the distinct parities of the enve- lope functions in different subbands. 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[41] The matrix elements of the spin-flip electron-electron Coulomb scattering are proportional to [ |h12(k)−h12(k′)|/∆E21]2, while the matrix elements of the spin- flip electron-impurity and electron-phonon scatterings are proportional to |h12(k)−h12(k′)|/∆E21. In the pa- rameter regime we discuss, |h12(k)−h12(k′)|/∆E21< 0.01. Consequently, the contribution from the spin-flip electron-electron Coulomb scattering is negligible.

1907.00319v2.Intrinsic_Spin_Hall_Effect_with_Spin_Tensor_Momentum_Coupling.pdf

arXiv:1907.00319v2 [cond-mat.mes-hall] 3 Feb 2020Intrinsic Spin Hall Effect with Spin-Tensor-Momentum Coupling Yong-Ping Fu1,∗, Fei-Jie Huang2and Qi-Hui Chen3,∗∗ 1Department of Physics, West Yunnan University, Lincang 677 000, China 2Department of Physics, Kunming University, Kunming 650214 , China 3School of Physical Science and Technology, Southwest Jiaot ong University, Chengdu 610031, China Corresponding author:∗Email: ynufyp@sina.cn;∗∗Email: qhchen@swjtu.edu.cn Abstract We derive the spin continuity equation by using the Noether’s theore m. A new type of spin-tensor Hall current is found in the continuity equa tion. The spin-tensor Hall current is originating from the coupling of the spin- tensor and the momentum. The intrinsic spin Hall effect in the two-dimensiona l fermion model with the spin-orbit coupling and spin-tensor-moment um cou- pling is studied. The total spin Hall conductivity with the presence of the spin-tensor Hall current is calculated. The numerical results indica te that the total spin Hall conductivity is enhanced by the contribution of t he spin- tensor-momentum coupling. The spin-tensor-momentum coupling m ay in- crease the spin transport in the intrinsic spin Hall effect. Keywords: spin-orbit coupling, spin-tensor-momentum coupling, spin Hall effect 1. Introduction The spin-orbit coupling (SOC) plays an important role in many quantum phenomena of solid and ultracold atomic systems. One important exa mple is the spin Hall effect (SHE) [1]. The SHE is studied in the semiconductor heterostructures. The spin Hall current is independent of the sk ew scattering ofthemovingmagneticmomentsandthemagneticimpurities[2]. Inth eSHE a dissipationless spin Hall current can be induced by an electric field in h ole- doped semiconductors [3, 4, 5]. The SHE has been observed experim entally Preprint submitted to Physica B February 4, 2020in GaAs [6, 7, 8]. The effect does not depends on the electron-impur ity scattering [9, 10, 11, 12, 13], but is related to the substantial SOC . In the momentum ( p) direction perpendicular to the electric current, the effective torque from the Rashba SOC tilts the spins up for p>0 and down for p<0. The effect is intrinsic in electron systems, and is called as intrinsic spin H all effect (ISHE). The spin Hall conductance is found to be a universal value, and is independent of the Rashba coupling strength [14, 15, 16, 17, 18]. Recently, a new type of spin-tensor-momentum coupling (STMC) is p ro- posed in the Bose-Einstein condensation [19]. It is found that the c oupling between the two components of particle’s spin and momentum (e.g., o rbit) leads to new types of stripe superfluid phase and multicritical points for phase transitions. The STMC may open a door for exploring many oth er interesting physics [20, 21, 22, 23, 24, 25]. For example, the new pr oper- ties of the SHE in the STMC system. If a physical system depends on two spin degrees of freedom, the concept of spin-tensor can be used to study this physical problem. The electric current depends only on the velocity degree of freedom, while the spin current depends on the velocity and spin d egrees of freedom. The spin-tensor current depend on velocity and spin- tensor de- grees of freedom, the STMC will inevitably lead to more abundant phy sical phenomena with the increasing of the degree of freedom. This paper is organized as follows. We discuss the spin continuity equa - tion by using the Noether’s theorem, and define the new spin-tenso r Hall current in Sec.2. In Sec.3 we study the ISHE in a model with the Rashb a SOC and STMC. The conductivities of the spin Hall and spin-tensor Ha ll currents are calculated. Finally the conclusion is presented in Sec.4. 2. Spin Continuity Equation and Spin-Tensor Hall Current Noether’s theorem indicates that the invariant of the system unde r a con- tinuous transformation will lead to the corresponding conservatio n current [26]. One can rigorously derive the total-angular momentum conserv ation equation by using the Noether’s theorem. The spin current of the D irac fermion coupled with the external electromagnetic field can be extr acted from total-angular momentum current (Appendix A). The non-rela tivistic approximation of the spin current can be obtained by using the stan dard Foldy-Wouthuysen (FW) transformation [27, 28]. The spin continuit y equa- 2tion up to the order of 1 /c2reads ∂ ∂tρi s+∇jJji s=Ti s, (1) the spin density is ρi s=ψ+siψ+ψ+1 4c2/braceleftBig [v×(v×s)]i−[(v×s)×v]i/bracerightBig ψ, (2) wheres=/planckover2pi1σ/2 is the spin operator, v=π/mis the velocity operator, π= p−eA/cis the momentum operator. The second term of the spin density is therelativisticcorrectionupto1 /c2orderandisvery small. Thecorrectionis thecontributionfromthecarrier’sspincouplingtoitsmomentum. It includes the spin-vector potential interaction A×sand the spin-orbit coupling p×s. The spin current can be derived as Jji s=ψ+1 4/parenleftbig sivj−sjvi/parenrightbig ψ+ψ+i 2/bracketleftbig si(τ×v)j−sj(τ×v)i/bracketrightbig ψ +ψ+1 2δijs·vψ−ψ+1 12c2/parenleftBig Jji s(0)v2+σ·vJji s(0)σ·v/parenrightBig ψ +H.c., (3) herewedefine τ=σ/2asthepseudospinand Jji s(0)=(sivj−sjvi)/4+i[si(τ×v)j−sj(τ×v)i]/2. The first term of Eq.(3) is the spin Hall current Jji SH=ψ+1 4/parenleftbig sivj−sjvi/parenrightbig ψ+H.c.. (4) Under the time reversal transformation t→ −t, we have s→ −s,π→ −π, the spin Hall current satisfies the time-reversal-symmetry. The r esponse equation of the spin Hall current Jij SH=σSHεijkEk[2, 3] requires the anti- symmetry of iandjcomponents, here σSHis the spin Hall conductivity. The same antisymmetry structure appears in the Eq.(4). We define the second term of Eq. (3) as the spin-tensor Hall curre nt Jji STH=ψ+i 2/bracketleftbig si(τ×v)j−sj(τ×v)i/bracketrightbig ψ+H.c.. (5) Mathematically, the rank-2 spin-tensor is defined as Nij=Pij+Dij, where Pij= (SiSj+SjSi)/2 is the symmetry tensor and Dij= (SiSj−SjSi)/2 is the antisymmetry tensor. Sidenotes the fermion spin sior pseudospin τi. In 3Eq. (5)si(τ×v)j=siτkvlεkljdenotes the coupling of the spin-tensor and the momentum, where siτk(i/ne}ationslash=j/ne}ationslash=k) is the antisymmetry spin-tensor. The spin-tensor introduced in Ref. [19] applies to systems with spin s=1. The spin operator of the boson is not satisfied with the commutation rela tion. Therefore the spin-tensor of Ref. [19] is a kind of symmetry tenso r. The spin-tensor Hall current is originating from the coupling of the s pin- tensor and the momentum. We note that the spin-tensor Hall curr ent is not a higher order correction of the non-relativistic approximation, it is in the same order as the spin Hall current. The imaginary number iin Eq.(5) is to ensure that the spin-tensor current operator [ isi(τ×v)j] is the Hermite current operator. In the time-reversal transformation s→ −sandπ→ −π the spin-tensor current is also time-reversal symmetry protect ed. Because of the pseudospin τdoes not contain the constant /planckover2pi1, where the dimension of /planckover2pi1 is [J·s], the pseudospin is dimensionless, and therefore in the time-rever sal transformation we have τ→τ. It is also easy to proved that the spin-tensor current is antisymmetric current for iandjcomponent. These properties indicate that the second term of Eq.(3) is a kind of spin-tensor Hall c urrent. A new velocity operators can be obtained by analogy with the conven - tional spin Hall current. We define the velocity of the spin-tensor c urrent as vST= 2(τ×v), and therefore the total spin Hall current can be defined as Jji tot−SH=ψ+1 4/parenleftbig sivj tot−sjvi tot/parenrightbig ψ+H.c., (6) where the total velocity is vtot=v+ivST. The third term of Eq.(3) is the scalar current, and δijψ+(s·v)ψ= 0 wheni/ne}ationslash=j. The fourth term of Eq.(3) includes the operators of spin Hall current and spin-tensor Hall current. This term is a kind of relativis tic correction from the SOC and STMC. The relativistic correction is of t he 1/c2order and is small enough to be neglected. In the external electromagnetic field, the total spin current is no t con- served because of the coupling of the angular momentum and exter nal field. The non-relativistic limit of the spin torque (up to 1 /c2order) can be derived in the form Ti s=ψ+e mc(s×B)ψ−ψ+e/planckover2pi12 4m2c2(∇×E)ψ +ψ+e 2m2c2/braceleftBig [π×(E×s)]i+[(s×E)×π]i/bracerightBig ψ, (7) 4whereBisthemagneticfieldand Eistheelectricfield. Wefindthatthespin torque is the result of the relativistic corrections. The first term o f Eq.(7) is the coupling of the magnetic moment µ(=es/mc) and the external field B. The second term of the torque is the contribution from curl E. The second term equals zero when the external electric field Eis constant in the SHE. The third term in Ti sis the contribution of π×v′and is finite for the electric orbital motion in the presence of SOC [29], where v′= (e/2m2c2)(s×E) is the relativistic correction of the velocity. If the external field B= 0, the spin torque is of the order 1 /c2, the spin current is approximately conserved. Because of the coupling of the spin and orbit, the contribution of sp in angular momentum and the orbit angular momentum to the torque ca n not be directly divided. Both the spin and orbit angular momentums contr ibute to the spin torque. In Ref. [29] the authors ignore the total cont ribution of the orbit angular momentum to the spin torque. The authors in Ref. [30] use the Gordon decomposition to divide the spin current into the con vective and internal parts, and derive the continuity equation for the con vective spin current with the absence of the external electromagnetic fie ld. They consider influence of the external electromagnetic field by using th e simple transformation p→πandi/planckover2pi1∂/∂t→i/planckover2pi1∂/∂t−eA0in the final results. In our derivation the external field Aµis included in the total-angular momentum conservation equation and the FW transformation. Thus the resu lts of the continuity equation of the spin current is more valid in this paper. 3. Spin Hall and Spin-Tensor Hall Conductivities in ISHE 3.1. Model Hamiltonian In the two-dimensional fermion system (2DFS) the substantial Ra shba SOC and the external electric field Elead to the spin Hall current [14, 15, 16, 18, 31]. The Hamiltonian including the SOC and STMC is given by H0=p2 2m+HR+HST, (8) whereHR=−2λ /planckover2pi1(sypx−sxpy) is the Rashba SOC Hamiltonian, λis the Rashba coupling constant [32]. In the x−yplane the Rashba coupling of the spinsandthemomentum ( px,py) is(s×p)z=sxpy−sypx, where theRashba SOC termsxpydescribes the coupling between the linear momentum pyand the spinsxin thepymomentum direction. In the pxmomentum direction, 5/s48/s50/s48/s52/s48/s69 /s70/s69 /s69 /s112 /s70/s45/s112 /s70/s43 /s46/s46/s69 /s70/s112 /s70/s45/s112 /s70/s43 /s46 /s46 /s69/s32 /s47 /s32/s112 /s43/s112 /s45 /s46/s46/s69 /s67 /s112 /s112 /s40/s97 /s41/s40/s98 /s41 Figure 1: Schematic of the splitting of energy states due to the Ras hba SOC (a) and SOC+STMC (b). Solid and dash arrow stand for spin and pseudos pin, respec- tively. SOC+STMC leads to four eigenstates of the 2DFS. We take th e eigenvalues as /an}b∇acketle{ts/an}b∇acket∇i}ht=/planckover2pi1/2(up arrow), −/planckover2pi1/2(down arrow), and /an}b∇acketle{tτz/an}b∇acket∇i}ht= 1/2(up arrow), −1/2(down arrow). the Rashba SOC can be described as sypx. We define the Rashba-like STMC asTzxpyandTzypxin the (px,py) momentum space, where Tij=τisjis the spin-tensor. Thespin-tensorHallcurrent willnotappearinthe2DFSwiththeabs ence of the STMC. In order to induce the spin-tensor Hall current in 2DF S, we construct a Hamiltonian including the SOC and STMC HST=4ζ m/planckover2pi12[(Tzypx)(sypx)−(Tzxpy)(sxpy)], (9) and we define ζas the dimensionless coupling constant of the STMC, and 0< ζ <1. The physical origin of the pseudospin τican be the double sub- lattices structure of the 2DFS with SOC. After simple algebraic oper ations, the energy states of Eq. (9) can be written as EST=±ζcos2θp2/2m, where θ=arctan(py/px). The pseudospin /an}b∇acketle{tτz/an}b∇acket∇i}ht=±1/2 corresponds to the two sublattice energy states. The Rashba SOC splits energy states into two branches (Fig.1(a)) Eη=p2 2m−ηλp, (10) 6whereη= +1 for /an}b∇acketle{ts/an}b∇acket∇i}ht=/planckover2pi1/2 andη=−1 for/an}b∇acketle{ts/an}b∇acket∇i}ht=−/planckover2pi1/2. The SOC and STMC lead to four eigenstates (Fig.1(b)) Eη,η′=p2 2m(1+η′ζcos2θ)−ηλp, (11) whereη′= +1 for /an}b∇acketle{tτz/an}b∇acket∇i}ht= 1/2,η′=−1 for/an}b∇acketle{tτz/an}b∇acket∇i}ht=−1/2, and tan θ= py/pxwith the momentums px=pcosθandpy=psinθ. The SOC+STMC Hamiltonian has four eigenstates, the fermions occupy the lower en ergy band E+,−in the spin Hall effect. The fermi energy EFis much larger than the slitting energy ∆ [33], we assume EF>Ec/greaterorsimilar∆, whereEcis the cross point oftheenergybands E+,+andE−,−. The differenceoftheFermiradii pF+and pF−of the majority ( E+,−) and minority ( E−,−) spin bands can be calculated by the eigenvalues EF=p2 F+ 2m(1−ζcos2θ)−λpF+ =p2 F− 2m(1−ζcos2θ)+λpF−, (12) and we have △pF=pF+−pF−=2mλ 1−ζcos2θ. (13) If theFermi energy issmaller thantheenergy Ec, such asE′ F=E′, theFermi space is between the energy bands E′ +,+(p′ F−) andE′ +,−(p′ F+). The difference △p′ Fof the Fermi radii p′ F+andp′ F−depends on the Fermi energy E′ F. 3.2. Spin Hall and Spin-Tensor Hall Conductivities In this section we discuss the spin Hall current which is polarized in the ˆz-direction and flows in the ˆ y-direction, and is perpendicular to the charge current ˆx-direction. Based on Eq.(6) the total spin Hall current operator is defined as ˆJyz tot−SH=1 4(szvtoty−syvtotz) + H.c.=1 4({sz,vtoty} − {sy,vtotz}), herevtotz= 0. We define the velocities according to the Heisenberg equations vy=1 i/planckover2pi1[y,p2 2m+HR], (14) and vSTy=1 i/planckover2pi1[y,HST]. (15) 7The velocity operators of the spin current and the spin-tensor cu rrent can be calculated as vy=py m+2λ /planckover2pi1sxandvSTy=−2ζτzpy m. In the SOC+STMC 2DFS the pseudospin and spin have the same index z, we redefine the total velocity operator as vtoty=vy+vSTyto ensure the Hermiticity of the spin current. The stable spin Hall effect requires that the spin is along a certain di- rection. However the spin is precessing with time due to the SOC, the pre- cession equation of the spin can be obtained by the Heisenberg equa tions (dsi(t)/dt= [si,H]/i/planckover2pi1) as ds1(t) dt=−2λ /planckover2pi1sz(t)p1, (16) ds2(t) dt=−2λ /planckover2pi1sz(t)p2, (17) dsz(t) dt=2λ /planckover2pi1[s1(t)p1+s2(t)p2], (18) where ˆx1and ˆx2denote the azimuthal and radial direction in momentum space, respectively. We consider the presence of the external electric field, the syste m Hamil- tonian is given by H=H0+eExx, whereExis the external electric field in the ˆx-direction. Using the Heisenberg equation, we have px=px0−eExt wherepx0is the initial momentum at t= 0. The fermi surface is displaced an amounteExtdue to the presence of the external electric field. Applying the adiabatic spin dynamics [14], the ˆ x2component of the spin can be approx- imated tos2(t) =ssinθp1(t)/p2for a weak field Exand a short instant t. Substituting the above approximation in Eq.(17) we have the ˆ zcomponent of the spin sas sz=−/planckover2pi1 2λp2ds2(t) dt. (19) In the weak field and short instant approximation, ˙ p1(t)≈˙px(t) =−eExand p2≈p, the electric field induces a linear response of spin sz: sz=e/planckover2pi1ssinθ 2λp2Ex. (20) 8The SOC+STMC Hamiltonian has four eigenstate, just the lower ener gy bandE+,−(majority spin band) contributes to the total spin current, here the values of the spin and the pseudospin are /an}b∇acketle{ts/an}b∇acket∇i}ht=/planckover2pi1/2 and/an}b∇acketle{tτz/an}b∇acket∇i}ht=−1/2. From the definition of spin Hall and spin-tensor Hall current, the co r- responding current operators are given by ˆJyz SH=1 4({sz,vy}−{sy,vz}) and ˆJyz STH=1 4({sz,vSTy}−{sy,vSTz}), wherevz=vSTz= 0 in this 2DFS. The definition of the spin Hall current is different from the conventional definition of the spin current ( ˆJyz spin=1 2{sz,vy}), because the conventional spin current operator can be divided into two parts ˆJyz spin=ˆJyz spin(1)+ˆJyz spin(2), where ˆJyz spin(1)=1 4({sz,vy}−{sy,vz}), (21) and ˆJyz spin(2)=1 4({sz,vy}+{sy,vz}). (22) The antisymmetry part ˆJyz spin(1)is the spin Hall current operator. The oper- atorˆJyz spin(2)does not satisfy the antisymmetry structure of indexes yandz, and therefore ˆJyz spin(2)is a kind of non-spin-Hall current operator. Only the antisymmetry part of the spin current contributes to the SHE [3, 2 9]. It is inappropriate to define ˆJyz spin(1)as a spin Hall current operator in the SHE. The spin Hall and spin-tensor Hall currents in the SOC+STMC system can be expressed as JSH=/summationtext k/an}b∇acketle{tˆJSH/an}b∇acket∇i}htfDandJSTH=/summationtext k/an}b∇acketle{tˆJSTH/an}b∇acket∇i}htfD, where k=p//planckover2pi1isthewavevectorand fDistheequilibriumFermi-Diracdistribution. At zero temperature the spin current is given by Jyz SH=/integraldisplaypF+ pF−d2p (2π/planckover2pi1)2εxyze/planckover2pi12Exsinθpy 8λmp2 =εxyzσSHEx, (23) where the spin Hall conductivity is σSH=e 16π2/integraldisplay2π 0dθsin2θ 1−ζcos2θ. (24) The spin-tensor Hall current is calculated by Jyz STH=/integraldisplaypF+ pF−d2p (2π/planckover2pi1)2εxyze/planckover2pi12ζExsinθpy 8λmp2 =εxyzσSTHEx, (25) 9/s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51 /s48/s46/s52 /s48/s46/s53 /s48/s46/s54 /s48/s46/s55 /s48/s46/s56 /s48/s46/s57 /s49/s46/s48/s49/s46/s48/s49/s46/s53/s50/s46/s48/s50/s46/s53/s51/s46/s48/s51/s46/s53/s52/s46/s48 /s32 /s83/s72/s40/s119/s105/s116/s104/s32/s83/s84/s77/s67/s41 /s32 /s83/s72/s40/s119/s105/s116/s104/s111/s117/s116/s32/s83/s84/s77/s67/s41 /s32/s32/s47/s101 /s32/s40 /s49/s48/s45/s50 /s41 Figure 2: The numerical results of the spin Hall conductivity vs the c oupling strength ζ. We have used the dimensionless quantity σ/e. The triangle symbol stands for the spin Hall conductivity of the conventional spin Hall current without the STMC. The diamond symbol stands for the total spin Hall conductivity of the total Ha ll current ( JSH+JSTH) with the STMC. where the spin-tensor Hall conductivity is σSTH(ζ) =ζσSH(ζ). (26) Eq.(24) and (26) indicates that the spin Hall and spin-tensor Hall co nduc- tivities are not universal values, they depend on the STMC paramet erζ. If ζ= 0, the STMC becomes to zero ( σSTH= 0) and the spin Hall conductiv- ity can be calculated as σSH=e/16π. The numerical results indicate that σSTH≈e/16πatζ= 1. Sinova et al.found that the spin Hall conductivity in a system with Rashba SOC remains the universal value e/8π[14, 16, 18]. The two results of the spin Hall conductivities differ by only a factor 1 /2. This difference comes from the different definitions of the spin Hall cu rrent. In Fig.2 we plot the spin Hall conductivities as a function of STMC strength in the condition of EF> Ec. The spin Hall conductivity with the absence of the STMC is not a universal value, and we find 0 .8e/16π<σSH< e/16πin the region of 0 < ζ <1. The total spin Hall conductivity of the total Hall current ( JSH+JSTH) is larger than the conventional spin Hall conductivity. The spin Hall conductivity is enhanced by the STMC effe ct. 10The enhancement to the value of the spin Hall conductivity is most ob vious atζ≈1. By comparing with the conventional value( e/16π) of the spin Hall conductivity, the total spin Hall conductivity is enhanced by a fact or of 2 at ζ≈1. It is difficult to distinguish the intrinsic spin Hall effect (ISHE) from the extrinsic spin Hall effect (ESHE) in the SOC system experimentally [1]. However the numerical results show that the effect of STMC enhan ces the total spin Hall conductivity. The enhancement phenomenon of the spin Hall conductivity may provide an experimental method to distinguish the ISHE from the ESHE. For example, the 2D free fermi gas in the double sub lattices structure is a good candidate to verify the effect of the enhancem ent in the experiments. 4. Conclusion The new spin-tensor Hall current due to the coupling of the spin-te nsor and the momentum is found in the spin continuity equation by using the Noether’s theorem. We propose a model of 2DFS with the SOC and ST MC. The spin Hall conductivity and the spin-tensor Hall conductivity of t he ISHE are studied in the model. We find that the STMC effect enhances the t otal spin Hall transport of the ISHE. The enhancement is more evident w ith the increasing of the STMC strength. Our results may motivate fur ther theoretical studies of ISHE. Acknowledgements Y. P. Fu acknowledges the support of National Natural Science Fo un- dation of China (No. 11805029) and the Yunnan Applied Fundamenta l Research Projects (No. 2017FD250). F. J. Huang acknowledges the sup- port from Yunnan Local Colleges Applied Basic Research Projects ( No. 2017FH001-112). Q. H. Chen acknowledges the support from Nat ional Nat- ural Science Foundation of China (No. 11947404) and the Fundame ntal Research Funds for the Central Universities (No. 2682016CX078 ). Appendix A. Relativistic Total-Angular Momentum Conserva tion and Spin Continuity Equation According to the Noether’s theorem [26], the conservation of the t otal- angularmomentumofafermiondescribedbyaLagrangian L=L(xµ,θσ,∂µθσ) 11is given by ∂αMαµν= 0, (A.1) where∂α=/parenleftbig∂ c∂t,∇/parenrightbig . The total-angular momentum tensor contains the spin angular momentum tensor (SAMT) and orbit angular momentum tens or (OAMT),Mαµν=Sαµν+Lαµν. The SAMT is defined as Sαµν=∂L ∂(∂αθσ)Iµν σρθρ, (A.2) where the coefficient Iµν σρ=(1/4)[γµ,γν]σρ(for the fermion) and gµ σgν ρ−gµ ρgν σ (for electromagnetic field) [28, 34]. The γ-matrices is represented as γµ=/parenleftbigg/parenleftbigg1 0 0−1/parenrightbigg ,/parenleftbigg0σ −σ0/parenrightbigg/parenrightbigg , (A.3) σis the Pauli matrices. gµν=diag(1,−1,−1,−1) is the Minkowski metric tensor (µ,ν= 0,1,2,3). The Greek letters denote the Lorentz indices, the Latin letters denote three-vector indices. The OAMT can be written as Lαµν=xµTαν−xνTαµ, (A.4) whereTµνis the energy-momentum tensor Tµν=∂L ∂(∂µθσ)∂νθσ−gµνL. (A.5) The fieldθdenotes the fermion or electromagnetic field ( θ= (Ψ,Aµ)). Thesystem ofDiracfermions(Ψ)coupledwiththeelectromagneticfi led(Aµ) is described by the Lagrangian L=¯Ψ(i/planckover2pi1cγµ∂µ−mc2)Ψ−e¯ΨγµAµΨ−1 4FµνFµν, (A.6) whereFµν=∂µAν−∂νAµis the electromagnetic tensor, and ¯Ψ = Ψ+γ0. After the partial derivative calculations we have ∂αSαµν=i/planckover2pi1c 4∂α/parenleftbig¯Ψγα[γµ,γν]Ψ/parenrightbig +JνAµ−JµAν +(∂αAν)(∂αAµ)−(∂νAα)(∂αAµ) −(∂αAµ)(∂αAν)+(∂µAα)(∂αAν), (A.7) 12and ∂αLαµν=¯Ψ(i/planckover2pi1cγµ∂ν−i/planckover2pi1cγν∂µ)Ψ −(∂µAα)(∂νAα)+(∂αAµ)(∂νAα) +(∂νAα)(∂µAα)−(∂αAν)(∂µAα), (A.8) where the electronic current is Jµ=∂νFνµ=e¯ΨγµΨ, and the energy- momentum tensor is Tµν=¯Ψi/planckover2pi1cγµ∂νΨ−Fµρ∂νAρ−gµνL. The energy- momentum is conserved ( ∂µTµν= 0). By substituting (A.7) and (A.8) into the conservation equation of total-angular momentum, Eq. (A.1) c an be rewritten as ∂α/parenleftbiggi/planckover2pi1c 4¯Ψγα[γµ,γν]Ψ/parenrightbigg =¯Ψc(γνπµ−γµπν)Ψ, (A.9) here we define the momentum operator as πµ=i/planckover2pi1∂µ−e cAµ. If we only consider the vector indices of µandν, one can easily derive ∂ ∂t/parenleftbigg Ψ+εijk/planckover2pi1 2ΣkΨ/parenrightbigg +∇l/parenleftbigg Ψ+cαlεijm/planckover2pi1 2ΣmΨ/parenrightbigg = Ψ+c/parenleftbig πiαj−πjαi/parenrightbig Ψ, (A.10) whereεijkis the Levi-Civita symbol. Here the Σ and αmatrices are Σ=/parenleftbiggσ0 0σ/parenrightbigg , (A.11) and α=/parenleftbigg 0σ σ0/parenrightbigg . (A.12) Sinceεijmεijn= 2δm nandεijkaibj= (a×b)k, the relativistic continuity equation of the spin can be derived as ∂ ∂tρi D+∇jJji D=Ti D, (A.13) whereρi D= Ψ+si DΨ is the spin density of the DiracFermion, Jji D= Ψ+vj Dsi DΨ is the relativistic spin current originating from the contribution of th e spin angular momentum, and Ti D= Ψ+(π×vD)iΨ is the spin torque originating from the contribution of the orbit angular momentum. Here we defin e the velocity operatoras vD=cαandthe spinoperator as sD=/planckover2pi1Σ/2. 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[13] W. K. Tse, J. Fabian, I. Zutic, S. D. Sarma. Phys. Rev. B 72(2005) 241303. [14] J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, A. H. M ac- Donald. Phys. Rev. Lett. 92(2004) 126603. [15] D. Culcer, J. Sinova, N. A. Sinitsyn, A. H. MacDonald, Q. Niu. Phy s. Rev. Lett. 93(2004) 046602. [16] S. Q. Shen. Phys. Rev. B 70(2004) 081311. [17] J. Schliemann, D, Loss. Phys. Rev. B 69(2004) 165315. 14[18] S. Zhang, Z. Yang. Phys. Rev. Lett. 94(2005) 066602. [19] X. W. Luo, K. Sun, C. W. Zhang. Phys. Rev. Lett. 119(2017) 193001. [20] H. P. Hu, J. P. Hon, F. Zhang, C. W. Zhang. Phys. Rev. Lett. 120 (2018) 240401. [21] R. Masrour, A. Jabar. Physica A 539(2020) 122878; Physica A 497 (2018) 211; Journal of Magnetism and Magnetic Materials 426(2017) 225; Superlattices and Microstructures 98(2016) 78. [22] A. Jabar, R. Masrour. Physica B 539(2018) 21; Physica A 538(2020) 122959; Physica A (2019) doi:https://doi.org/10.1016/j.physa.2019.123377; Physica A 514 (2019) 974; Physica A 515(2019) 270; Chemical Physics Letters 700 (2018) 130. [23] R. Masrour, E. K. Hlil. Physica A 456(2016) 215. [24] R. Masrour, A. Jabar, E.K. Hlil. Journal of Magnetism and Magne tic Materials 453(2018) 220; Intermetallics 91(2017) 120. [25] G. Kadim, R. Masrour, A. Jabar. Journal of Magnetism and Mag netic Materials 499(2020) 166263. [26] E. Noether. Nachr. Ges. Wiss. Goettingen Math.-Phys. Kl. 1918(1918) 235. [27] L. L. Foldy, S. A. Wouthuysen. Phys. Rev. 78(1950) 29. [28] W. Greiner. Relativistic Quantum Mechanics: Wave Equations (Springer, Berlin) (2000). [29] Y. Wang, K. Xia, Z. B. Su, Z. S. Ma. Phys. Rev. Lett. 96(2006) 066601. [30] R. Shen, Y. Chen, Z. D. Wang, D. Y. Xing. Phys. Rev. B 74(2006) 125313. [31] S. Q. Shen. Phys. Rev. Lett. 95(2005) 187203. [32] Y. A. Bychkov, E. I. Rashba. J. Phys. C 17(1984) 6039. 15[33] J. Nitta, T. Akazaki, H. Takayanagi, T. Enoki. Phys. Rev. Lett .78 (1997) 1335. [34] W. Greiner, J. Reinhardt. Field Quantization ( Springer, Berlin) ( 1996). [35] M. Milletari, M. Offidani, A. Ferreira, R. Raimondi. Phys. Rev. Lett . 119(2017) 246801. 16

1105.4215v1.Quantum_states_and_linear_response_in_dc_and_electromagnetic_fields_for_charge_current_and_spin_polarization_of_electrons_at_Bi_Si_interface_with_giant_spin_orbit_coupling.pdf

arXiv:1105.4215v1 [cond-mat.mes-hall] 21 May 2011Quantum states and linear response in dc and electromagneti c fields for charge current and spin polarization of electrons at Bi/Si interface with g iant spin-orbit coupling D.V. Khomitsky∗ Department of Physics, University of Nizhny Novgorod, 23 Gagarin Avenue, 603950 Nizhny Novgorod, Russian Federat ion (Dated: November 9, 2018) An expansion of the nearly free-electron model constructed by Frantzeskakis, Pons and Grioni [Phys. Rev. B 82, 085440 (2010)] describing quantum states at Bi/Si(111) in terface with giant spin- orbit coupling is developed and applied for the band structu re and spin polarization calculation, as well as for the linear response analysis for charge curren t and induced spin caused by dc field and by electromagnetic radiation. It is found that the large spin-orbit coupling in this system may allow resolving the spin-dependent properties even at room temperature and at realistic collision rate. The geometry of the atomic lattice combined with spin- orbit coupling leads to an anisotropic response both for current and spin components related to the orientation of the external field. The in-plane dc electric field produces only the in-plane compon ents of spin in the sample while both the in-plane and out-of-plane spin components can be excited by normally propagating electromagnetic wave with different polarizations. PACS numbers: 73.20.At, 71.70.Ej, 72.25.-b, 78.67.-n I. INTRODUCTION It is well-known that the knowledge of a material with high values of spin-orbit (SO) coupling parameters is a goal for many theoretical, experimental, and device research groups in condensed matter physics and spin- tronics due to its fascinating spin-related properties and possible applications in the information processing and storage. Among the candidates which attract consid- erable attention throughout the last decade there are Bi/Si(111) surface alloys which band structure has been experimentally studied for several years and recently modeled in a paper1by Frantzeskakis, Pons and Gri- oni. This material, in line with the other examples of ”metal-on-semiconductor” systems with large SO cou- pling, has been a subject of many both experimental and theoretical papers since it seems very promising to make use of a material combining the large SO split- ting of Bi and conventional semiconductor technology of Si which is one of the main goals of spintronics.2–4 Here we shall mention only some of the numerous re- sults of research on the Bi-covered Si surface proper- ties with various crystal orientation of Si substrate. In particular, the scanning tunneling microscopy has been used to determine the surface structure of Bi/Si some 18 years ago,5and the analysis of the atomic geome- try and electronic structure continued further6,7focus- ing throughout the recent years mainly on the atomic surface geometry and spin-resolved band structure re- construction where the methods of angle-resolved pho- toemission spectroscopy have been applied1,8–15. Other methods included the low-energyelectron diffraction and atomic force microscopy16,17, and besides pure Si, the ∗E-mail: khomitsky@phys.unn.ruSi-Ge superlattices have been used as a substrate for Bi coverage,18and later the lateral Ge-Si nanostructure prepared on the Si/Bi surface have been studied by the scanning tunneling microscopy.19Also, for the Bi/Si sys- tem there were studies of energetic stability and equi- librium geometry20and the possibility of designing iron silicide wires along Bi nanolines on the hydrogenated Si surface,21, and of the thermal response upon the fem- tosecond laser excitation.22It is well-known that Bi is a material with very big SO splitting, and thus it at- tracts a steady interest in its potential applications in spintronics where various schemes of combining it with semiconductors are suggested, one of the most recent be- ing an investigation of BiTeI bulk material where the SO splitting reaches a very high value of 0 .4 eV.23 From the list of papers mentioned above it is evident that the geometric properties of atom arrangement and the resulting band structure have already been studied for Bi/Si systems by many experimental and theoreti- cal groups. However, much lower attention have been given so far to the prediction and observation of different effects caused by the electron system response to an ex- ternal excitation, including such basic properties as the charge current and spin polarization in the dc field which are often are considered as the starting point of the re- sponse calculations, especially for systems with impor- tant role of SO coupling.24–27Besides the response to the dc electric field, the optical properties of SO-split band spectrum always attracted significant attention starting from the conventional semiconductor structures with big SO coupling.28–32In our previous papers we have ob- served an important role of SO coupling in conventional InGaAs-based semiconductor superlattices on the energy bandformation33whichdirectlyaffectedbothchargeand spin response for the excitation by the electromagnetic radiation34,35and by the dc electric field.36It is known that the spin polarization configurations in semiconduc-2 tors may have a rather long relaxation time4,37,38which makes them as important as the conventional chargecur- rent setups for their applications in the nanoelectronics and spintronics. While there is no doubt that the electron properties in Bi-covered Si interface differ from the ones in conven- tionalsemiconductorstructureswith strongSO coupling, the questions regarding their SO-dependent response to the external dc and electromagnetic fields remain to be very important since we are still in the beginning of our way towards understanding and utilizing such novel ma- terials with giant SO coupling as Bi/Si. The goal of the present paper is to apply a modified and expanded version of simple but adequate nearly free electron model1for the band structure of electrons in Bi covering the Si(111) surface which allows us to cal- culate various physical characteristics of this material in the linear response regime, including the charge current and spin polarization caused by dc electric field with dif- ferent orientations, and also to obtain the response of non-equilibrium spin polarization excited by the electro- magnetic field with various polarizations. Indeed, such physical quantities can be among the first ones measured in the nearest experiments on Bi/Si, and hence it is of interest to calculate them beforehand both qualitatively and, when possible, quantitatively. Since we do not know exactly for the present moment many material parame- ters of the electron system in Bi-covered Si interface, in- cluding such parameters as the electron surface concen- tration and mobility, the dielectric tensor, the relaxation rates for charge and spin, etc, we sometimes cannot yet calculate the effects in the absolute measurable units and use the standard label ”arbitrary units” instead. Still, we believe that the comparison of the output results for the same physical parameter calculated at different con- ditions always has a value since it allows to predict the relative significance of them when the conditions are var- ied. We find both common and distinct features of the charge and spin system response in Bi/Si compared to well-known GaAs or InGaAs semiconductor structures. Thus, we believe that our findings can be a good starting point for further theoretical and experimental studies of charge and spin response in Bi/Si system at various ex- ternal fields. The paper is organized as follows: in Sec.II we in- troduce the expanded version of the nearly free electron HamiltonianforBi/Sianddiscussthe bandstructureand spin polarization in the energy bands, in Sec.III we solve the kinetic equation and calculate the chargecurrent and spin polarization in the dc electric field, in Sec.IV we study the excited spin polarization in the framework of the linear response theory for different polarizations and frequencies of the incident electromagnetic wave, and we give our conclusions in Sec.V.II. HAMILTONIAN AND QUANTUM STATES Recent proposal of several theoretical models for band structurecalculationofBi/Sielectronsurfacestates1pro- vided a variety of choices for studies of the correspond- ing Hamiltonian and quantum states. Here we shall start withthe simplestnearlyfreeelectron(NFE) modelwhich was proposed initially for the description of the band structure mainly in the vicinity of the Mpoint at the surface Brillouin zone (SBZ) of Bi/Si having a hexago- nal shape shown by dashed contour in Fig.1. We shall only briefly describe it here since the detailed derivation and discussion is available in the original paper.1The choice of the reciprocal lattice vectors initially has been restricted to three vectors G1,G2andG3connecting four equivalent Gamma-points Γ(0),Γ(1),Γ(2)andΓ(3). In the framework of the NFE approach for each Gamma- point the standard 2 ×2 Rashba Hamiltonian of a free electron in the ˆ σzbasis has been written with the center of the quasimomentum at the corresponding Γ(n)point. As a result, an 8 ×8 matrix is derived giving the en- ergy bands and two-component eigenvectors(the Rashba spinors) describing the spin polarization in the reciprocal space. We are going to use an expanded version of this model by including the remaining reciprocal vectors G4,G5 andG6into our basis of nearest neighbor sites connect- ing the center Gamma-point Γ(0)with all surrounding pointsΓ(1),...,Γ(6), as it is shown by hollow vectors G1,...,G6in Fig.1 where several hexagonal SBZ-s are shown by solid contours, thus creating a 14 ×14 Hamil- tonian matrix. We assume the previously determined1 values of geometrical parameters ΓM= 0.54˚A−1and ΓK= 0.62˚A−1. Such an expansion allows us to treat a much wider area of the SBZ compared to the region near theM-point1and to keep the symmetry of the non- trivial hexagonal Bi/Si(111) trimer structure with one monolayer of Bi atoms.1,6,7,13 Our Hamiltonian may thus be described via its matrix elements in the following form: Hnn′=HR(k+Gn)δnn′+Vnn′, (1) and the electron spinor wavefunction is constructed as Ψk(r) =/summationdisplay nankψnk(r) (2) where the conventional form of Rashba Hamiltonian is used, HR(k) =/parenleftbigg ¯h2k2/2m α R(ky+ikx) αR(ky−ikx) ¯h2k2/2m/parenrightbigg ,(3) and the matrix elements of the periodic potential cou- pling the free electron states are3 FIG. 1: Surface Brillouin zones (SBZ) of Bi/Si with a hexago- nal shape shown by solid lines and the reciprocal lattice (ho l- low) vectors G1,...,G6connecting the equivalent Gamma- pointsΓ(0),...,Γ(6)of the nearest neighbor approximation for the nearly free electron model. The spin-split Rashba paraboloids are centered in each of the Gamma-points. Vnn′=/angb∇acketleftψn|/summationdisplay mV0exp(iGmr)|ψn′/angb∇acket∇ight.(4) The basis functions ψnk(r) in Eq.(2) are the well- known Rashba spinors ψnk=ψk+Gnwhere ψk=eikr √ 2/parenleftbigg1 ±eiArg(ky−ikx)/parenrightbigg , (5) and the ( ±) sign corresponds to two eigenvalues for Rashba energy spectrum E(k) = ¯h2k2/2m±αRk. We continue to adopt here the known values of material pa- rameters for Eqs (1)-(4) and constants used during the initial construction1of the NFE model for Bi/Si, namely, we putm= 0.8m0,αR= 1.1 eV·˚A andV0= 0.3 eV. After diagonalization of the Hamiltonian (1) we ar- rive to the energy band spectrum E=Es(kx,ky) where s= 1,2,...labelsthe energybandsoftheelectronsinthe Bi/Si(111) system. The three-dimensional plot of the en- ergyband structure is presented in Fig.2 for the four low- est bands labeled from 1 to 4. These lowest bands seem to be of the primaryimportance for the electronresponse analysissincethe Fermilevelis reportedlylocated1in the middle of them at EF= 1.6 eV, i.e. between the band No.2andbandNo.3, asitcanbeseeninFig.2. Oneofthe most important features of the spectrum in Fig.2 stem- ming from the lattice geometry is the hexagonal symme- try of the energy bands in k-space which implies, among FIG. 2: (Color online) Energy band structure of the electron s onBi-covered Si(111) surface for thefour lowest bandslabe led from 1 to 4. The Fermi level is located at EF= 1.6 eV between the band No.2 and band No.3 where a small global energy gap of around 0 .1 eV is formed. The cross-sections of our 3D band plot shown here accurately repeat the 2D plots for the energy dispersion lines along various directions in the SBZ which were studied earlier in the framework of the NFE model.1 other things, the absence of the kx↔kysymmetry, lead- ing to rich properties of the spin response phenomena as weshallseebelow. Itshouldbementionedthatthecross- sectionsofour3Dbandplotshownhereaccuratelyrepeat the 2D plots for the energy dispersion lines along vari- ous directions in the SBZ which were studied earlier in the framework of the NFE model.1We would like to add here just some new quantitative data: the cross-sections of the energy band surfaces along the Γ−Mdirection reported previously1might created an impression of a large energy gap in the whole spectrum between bands 2 and 3 while the complete 3D presentation of these bands in Fig.2 indicates that the global structure of the energy bands in the whole 2D SBZ leaves this gap opened but with a much smaller width of around 0 .1 eV. Of course, the precise values of energy gaps may vary from model to model and can be specified more precisely during the future experimental and theoretical analysis. Another important characteristic of quantum states in a system with significant SO coupling is the spin polar- ization of the eigenstates ψkin the Brillouin zone which can be defined as the vector field in the reciprocal space with the components ( m=x,y,z) Sm(k) =/angb∇acketleftψk|ˆσm|ψk/angb∇acket∇ight. (6) As usual for the Hamiltonian with a pure Rashba SO coupling term, the out-of-plane component Szof the spin4 FIG. 3: Two-dimensional spin polarization distributions (Sx(k),Sy(k)) for the lowest energy bands No.1 (a) and No.2 (b) from the band spectrum shown in Fig.2, with the hexag- onal SBZ marked by a solid contour. The initial Rashba counter clock-wise and clock-wise patterns are present in a wide area surrounding the SBZ center, but more complicated vector field structure near the SBZ edge. fieldvanishes. Theremainingcomponentsforma2Dspin polarization distribution in the SBZ which forms a spe- cific vector field picture for each of the energy bands. In Fig.3 and Fig.4 we show the 2D spin polarization distri- butions (Sx(k),Sy(k)) for the lowest energy bands No.1 (Fig.3a), No.2 (Fig.3b), No.3 (Fig.4a) and No.4 (Fig.4b) from the band spectrum shown in Fig.2, with the hexag- onal SBZ marked by a solid contour. As for the spins in two lowest subbands shown in Fig.3, one can see here that the initial Rashba counter clock-wise and clock-wise FIG. 4: Spin polarization distributions ( Sx(k),Sy(k)) for the higher energy bands No.3 (a) and No.4 (b) from the band spectrum shown in Fig.2, with the hexagonal SBZ marked by a solid contour. The spin polarization in band No.3 (a) and especially in bandNo.4 (b) demonstrates more newpropertie s compared to the free Rashba states, including both the shape of the spin vector field which captures now more features of the hexagonal geometry of the SBZ, and the arising of new lo- cal vortices at various points of symmetry of the SBZ, mainly near its corners. patterns of spin directions are present in a rather wide area surrounding the SBZ center, but more complicated vector field structure arises near the SBZ edge. The spin polarization in higher band No.3 and especially in band No.4 shown in Fig.4 demonstrates more new properties compared to the free Rashba states, including both the shape of the spin vector field which captures now more features of the hexagonal geometry of the SBZ, and the5 arising of new local vortices at various points of symme- try of the SBZ, mainly near its corners. It can be concluded fromthe analysisofthe spin polar- ization in the energy bands of Bi/Si system that certain propertiesofthe initial basisofRashbastates remainvis- ible. However, the change of the space symmetry to the hexagonal type without the element of axial symmetry and without the x↔ysymmetry may probably lead to both common and distinct features in the current and spin response to the application of various external fields compared to the well-known properties of 2DEG with Rashba SO coupling. This assumption will be confirmed and illustrated below. III. CHARGE CURRENT AND SPIN POLARIZATION RESPONSE FOR DC FIELD Itisknownthattheresponseofatwo-dimensionalelec- tron gas with SO coupling to a constant electric field may be accompanied not only by the charge current but also by the spin polarization.26,27,36The most significant propertiesofsuchresponseforapureRashbaSOterm(3) in the Hamiltonian is the arising of the in-plane trans- verse polarization, i.e. the Sy(x)spin component when the electric field is applied along the x(y) direction while the out-of-plane Szcomponent is absent in case of the accurately included relaxation processes which is some- timesrelatedalsotothe absenceofthe spinHalleffect for ak-linearRashbamodelinthepresenceofthedisorder.39 So, it is natural to start the analysis of the electron sys- tem response with the calculation of the charge current and field-induced spin. We shall start with the calculation of the non- equilibrium electric field-affected distribution function ˜fm(k,Ei) in the m-th energy band. If the system is sub- jected to a constant and uniform electric field Eiparallel to the i-th axis, then in the collision frequency approxi- mation the kinetic equation for ˜fm(k,Ei) can be written as36 eEi∂˜fm(k,Ei) ∂ki=−ν[˜fm(k,Ei)−fm(k)],(7) whereνis the collision rate and fm(k) = 1/(1 + exp[(Em(k)−µ)/kBT])is the Fermi equilibrium distribu- tion function in the m-th band. Since the Bi/Si energy spectrum is characterized by a very large SO splitting and the band widths of the order of 1 eV, it may be a promising candidate for spin-dependent phenomena visi- ble at room temperature. Thus, in the following we shall assume that T= 293 K and consider a value of ν= 1012 s−1. As we have said before, the estimate for the col- lision rate as well for many other material parameters for the Bi/Si system is presently based on the assump- tions rather than of the solid experimental facts since we are still in the beginning of the investigation for this new material. Still, we believe that our qualitative andsometimes quantitative results can be useful for predict- ing some novel properties of the electron and spin system response. The mean charge current density ji(Ei) measured for 2D system in units of current divided by the unit of transverse system size and the mean spin polarization Sj,j=x,y,z, can be found as36 ji(Ei) =en/summationdisplay m,k/angb∇acketleftψmk|vi|ψmk/angb∇acket∇ight˜fm(k,Ei),(8) Sj(Ei) =/summationdisplay m,k/angb∇acketleftψmk|σj|ψmk/angb∇acket∇ight˜fm(k,Ei),(9) wherenis the surface concentration of the electrons on the Bi-covered Si surface, σi(i=x,y,z) are the Pauli matrices, and the velocityoperator vi=∂H/∂k iincludes the SO part proportional to the Rashba parameter αR and acts on the spinor wavefunctions (2) via the matrices vx=/parenleftbigg −i¯h∇x/m iα/ ¯h −iα/¯h−i¯h∇x/m/parenrightbigg , (10) vy=/parenleftbigg −i¯h∇y/m α/ ¯h α/¯h−i¯h∇y/m/parenrightbigg .(11) In order to take a closer look for onto the expectations of the charge current density, we shall calculate jx(Ex) andjy(Ey) as well as Si(Ex) andSi(Ey) (i=x,y,z) by applying Eq.(8) for the electron surface concentra- tionn= 1014cm−2which may be reasonable since the surface density of atoms on Bi-covered Si(111) surface is estimated on the order of 1015cm−2according to the experimental observations.9,10 The results for the charge current density (8) and spin polarization (9) are shown in Fig.5(a) and Fig.5(b) re- spectively. Onemayseeaconventionallineardependence ofthechargecurrentontheappliedelectricfieldthrough- out the whole range of fields up to 2 kV/cm, and almost linear dependence for the significant non-zero induced spin components Sy(Ex) andSx(Ey). Other in-plane components Sx(Ex) andSy(Ey) marked by arrow are also present in Fig.5 but their magnitude is much lower compared to Sy(Ex) andSx(Ey), and the out-of plane Szcomponent is negligibly small. It is evident that the lattice asymmetry with respect to the x↔yinterchange transform has lead to a slight but distinct asymmetry in the current amplitude of around 12%, and the dominat- ingSy(Ex) andSx(Ey) induced spin components demon- stratethe well-knowntransversein-planecharacterofthe induced spin for linear Rashba SO coupling. It should be noted that the local probe measurements of induced spin polarizations(ormagnetization) may detect the non-zero static and dynamic local magnetization40in the spot un- der the probe even in case of total zero mean spin value6 FIG. 5: (a) Charge current density (8) and (b) spin polar- ization (9) induced by the dc electric field applied along x (solid curves) and y(dashed curves). A conventional linear dependence of the charge current on the applied electric fiel d throughout the whole range of fields and almost linear depen- dence for the significant non-zero induced spin components Sy(Ex) andSx(Ey) are visible. Other in-plane components Sx(Ex) andSy(Ey) marked by arrow also present but their magnitude is much lower, and the out-of plane Szcomponent is negligibly small. (9). The examples of such systems with zero total spin polarization but non-zero spin spatial distribution (spin textures) can be found among the models of semicon- ductor superlattices with SO coupling both with35and without34,36externalmagneticfield,buttheirexperimen- tal observation and device application are currently lim- ited bytheprobeandmanipulationtechnologyofthe size of artificial superlattices and quantum wells rather then probing and utilizing the spatial magnetic configurations on the scale of individual atoms in the lattice. As to the predicted non-zero mean values of the induced spin such as those predicted here for Bi/Si, they are related to the whole sample and thus should be detectable. We be- lieve that the predictions of the charge current and spin polarization generation made in this Sec. can be useful in designing novel spintronic devices where the induced spin components are coupled in a well-defined manner to the direction of the applied electric field, and this effect survives at room temperature and finite collision rate.IV. SPIN POLARIZATION EXCITED BY ELECTROMAGNETIC FIELD The response of the spin system in materials with sig- nificant SO coupling on the application of an external electromagnetic radiation is among the most important and straightforwardly obtained characteristics since the optical manipulation of spins is one of the main goals of spintronics in general, and the linear response theory for the electromagnetic radiation effects is well-established and easily applied. It was found in various papers that the spins with different polarizations can be excited, de- pending on the symmetry of the electron Hamiltonian, the type and strength of the SO terms, and on the po- larization of the incident radiation.2,3,28–31,34,35As in the previous Sec., we shall calculate the response functions for the room temperature and for a realistic collision broadeningsincetherelativelylargescaleofenergybands and SO splitting in the Bi/Si electron system compared to the conventional GaAs, InGaAs or pure Si semicon- ductor structures can make Bi/Si being a promising can- didate for the observation and control of the predicted radiation-induced effects in the devices operating even at room temperatures, as we hope. The electromagnetic wave is considered to be propa- gating normally to the Bi/Si(111) interface along the z axis, and is characterized by the polarization of the elec- tricfieldvector E=E0exp(i(k·r−ωt)) inthe (xy)plane, E0= (E0x,E0y,0). In the dipole approximation the in- teraction of the electromagnetic field with the electrons is described via the velocity operators(10),(10) which in- clude the SO part. We start with the calculation of the absorption coefficient α(ω) =4π2e2 m2ωc√εV/summationdisplay n,n′,k|(e·v)nn′k|2×(12) ×δ(En′k−Enk−¯hω)(fnk−fn′k) (13) wheree= (ex,ey,0) is the polarization vector for the incident wave, v= (vx,vy,0) is given by the velocity operators (10),(11), fnkandfn′kare Fermi equilibrium distribution functions, Vis the sample volume, and the summation is taken over all energy bands n,n′and the SBZ points k. As we have already said, we don’t know yet the exact values for many of the material param- eters for Bi/Si including the dielectric constant ε, and thus we shall focus mainly on the dependence of (13) on the incident wave frequency and will scale the absolute value of absorption in arbitrary units which can always be rescaled when the values material parameters will be clarified in future experiments. The second quantity which frequency dependence we shall present together with the absorption is the induced spin polarization Sm(ω) which can be derived by apply- ing the Kubo linear response theory35,41,42and has the following form:7 Sm(ω) =−ieE0l 8πm¯h/summationdisplay n,n′,kfnk−fn′k ωnn′(k)× (14) ×S(m) n′n(k)v(l) nn′(k) ω−ωnn′(k)+iν.(15) Here the interband matrix elements of the spin m-th component operator S(m) n′n(k) =/angb∇acketleftψn′k|ˆσm|ψnk/angb∇acket∇ightas well as the matrix elements for the l-th component of the velocity operator (10),(11) enter depending on the incident wave polarization and on the desirable output for the spin component, ¯ hωnn′(k) =Enk−En′k, the parameterνis level broadening which we take as being equal to the collision rate introduced in the previous Sec, andVcis the unit cell volume. As before, we shallassume thatT= 293 K and ν= 1012s−1which should provide us with realistic expectations regarding the absorption and induced spin dependence on the incoming photon energy. The results for the photon energy dependence of the absorption coefficient (13) and the induced spin polariza- tion(15)areshowninFig.6throughFig.9fortheincident wave linearly polarized along x(Fig.6), along y(Fig.7) directions, and for circular σ+(Fig.8) and σ+(Fig.9) po- larizations. The photon energyinterval is chosento cover the whole energy band range of the four lowest bands shown in Fig.2 where the most effective transitions oc- cur between the states below and above the Fermi level. One can see that both the in-plane spin components Sx, Syand the out-of plane component Szcan be excited on a comparable scale which is a distinguishable feature of thelowerhexagonalsymmetrycombinedwithRashbaSO coupling compared with one-dimensional26,34,36or two- dimensional square35lattices with Rashba SO coupling. If one compares the results for x- andy-polarized radi- ationinFig.6andFig.7, itcanbeseenthat, similartothe dc current properties discussed in the previous Sec., the x↔ylattice and energy band asymmetry in the hexago- nalgeometryofthe wholeproblemis reflectedherein dif- ferentshapeandamplitude fortheabsorptioncoefficients in Fig.6(a) and 7(a). Again, by looking onto the relative magnitudeofthe excitedspin componentsin Fig.6(b)-(d) and in Fig.7(b)-(d) one can see that the pure Rashba SO couplingis reflected in the dominatingexcited Sxcompo- nent in Fig.6 and correspondingly in the dominating Sy component in Fig.6, i.e., in the in-plane and transverse direction relative to the electric field vector of the inci- dent wave. As to the results for the circular σ±-polarized radiation presented in Fig.8 and Fig.9, on can see that the common features of the response to both x-polarized andy-polarized incoming wave from Fig.6 and Fig.7 can be seen on the spin component figures since both vxand vyoperators here enter the expressions (13) and (15) for the response. The absorption coefficient is totally insen- sible to the direction of rotation for the incoming wave as it can be seen by comparing Fig.8(a) and Fig.9(a). The shape of the photon energy dependence for the excited FIG. 6: (a) Absorption coefficient and (b) - (d) components of the induced spin density in the Bi/Si surface electron gas shown as a function of the incoming photon energy for the linearx-polarized incident radiation propagating normally to the interface plane. The greatest absorption and most of the spin polarization peaks are induced in the photon energy nea r ¯hω∼0.5 eV corresponding to the transitions between the highest occupied band 2 and the lowest unoccupied band 3 of the electron energy spectrum shown in Fog.2. Both the in- planespincomponents Sx,Syandtheout-ofplanecomponent Szcan be excited.8 FIG. 7: Photon energy dependence of (a) absorption coeffi- cient and (b) - (d) components of the induced spin density for the same parameters as in Fig.6 but for the y-polarized incident radiation. The x↔ylattice and energy band asym- metryinthehexagonal geometryis reflectedindifferentshap e and amplitude for the absorption coefficients in this Fig. and in Fig.6. The Rashba term in SO coupling is reflected in the dominating excited Sxcomponent in this Fig. and the domi- natingSycomponent in Fig.6. FIG. 8: Photon energy dependence of (a) absorption coeffi- cient and (b) - (d) components of the induced spin density for the same parameters as in Fig.6 but for the circular σ+- polarized incident radiation. The common features of the response to both x-polarized and y-polarized incoming wave from Fig.6 and Fig.7 can be seen on the spin component fig- ures.9 FIG. 9: Photon energy dependence of (a) absorption coef- ficient and (b) - (d) components of the induced spin den- sity for the same parameters as in Fig.6 but for the circular σ−-polarized incident radiation. The absorption coefficient is unchanged from the σ+case in Fig.8(a) while the excited spin components demonstrate quantitative differences whil e maintaining the same type of shape for the photon energy dependence.spin components in Fig.8(b)-(d) and Fig.9(b)-(d) is dif- ferent forσ+andσ−polarizations, but these differences have a quantitative rather than a qualitative character since the hexagonal symmetry of the lattice and the en- ergy bands does not make any of these two polarizations preferable from the point of view of the response quanti- ties (13) and (15). In conclusion, the calculation and analysis of the ab- sorption and spin polarizationresponse to the monochro- matic electromagnetic radiation with normal incidence and having different polarizations demonstrates that this radiation is most effectively absorbed in the photon en- ergy range of around 0 .5 eV corresponding to the pho- ton wavelength λ= 2.47µmwhere both in-plane and out-of-plane spin components can be excited at realistic temperature and collision broadening on a comparable scale with relative amplitudes depending on the precise value of frequency and the polarization of the incident radiation. These properties can be useful for designing new optical and spintronic devices coupling the electron spin with light and operating at room temperature. V. CONCLUSIONS We have developed an expansion of the nearly free- electron model1describing the energy bands and spin polarization for the electron states at Bi/Si(111) inter- face with giant spin-orbit coupling, and applied it for the linear response analysis for charge current and induced spin caused by dc field and by electromagnetic radiation. Itwasfoundthat the largespin-orbitcouplingin thissys- tem may allow resolving the spin-dependent properties even at room temperature and at realistic collision rate. The geometry of the atomic lattice combined with spin- orbit coupling leads to an anisotropic response both for currentand spin components related to the orientationof the external field. The in-plane dc electric field produces only the in-plane components of spin in the sample while both the in-plane and out-of-plane spin components can beexcitedbynormallypropagatingelectromagneticwave with different polarizations. The qualitative predictions of the charge and spin response in a novel and promising Bi/Si system may be useful for the forthcoming detailed theoretical and experimental studies which may lead to the development of principally new electronic, optical and spintronic devices operating at room temperature. Further theoretical and especially experimental studies of this promising system with big SO coupling allow- ing the survivability of the spin-related effects at room temperature are expected bringing us new and fascinat- ingphenomenawithbothfundamental,experimentaland device-related results.10 Acknowledgments The author is grateful to V.Ya. Demikhovskii, A.M. Satanin, A.A. Perov for helpful discussions, and to A.A.Chubanov for technical assistance. The work was sup- ported by the RFBR Grants 11-02-00960a and 11-02- 97039/Regional, and by the RNP Program of Ministry of Education and Science RF. 1E. Frantzeskakis, S. Pons, and M. Grioni, Phys. Rev. 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0707.3908v1.Quasiclassical_approach_and_spin_orbit_coupling.pdf

arXiv:0707.3908v1 [cond-mat.mes-hall] 26 Jul 2007Quasiclassicalapproachandspin-orbitcoupling CosimoGoriniaPeter SchwabaMichaelDzierzawaaRobertoRaimondib aInstitut f¨ ur Physik, Universit¨ at Augsburg, 86135 Augsbu rg, Germany bDipartimento di Fisica ”E. Amaldi”, Universit` a di Roma Tre , Via della Vasca Navale 84, 00146 Roma, Italy Abstract We discuss the quasiclassical Green function method for a tw o-dimensional electron gas in the presence of spin-orbitco upling, with emphasis on the meaning of the ξ-integration procedure. As an application of our approach, we demonstrate how the spin-Hallconductivity, inthe presence ofspin-flipscatte ring,can beeasilyobtained fromthespin-densitycontinui ty equation. Key words: EP2DS-17, manuscript, LaTeX-2e, style files PACS:72.25.Ba, 72.25.Dc 1. Introduction The quasiclassical technique is one of the most pow- erful methods to tackle transport problems. Its main virtue relies in the fact that starting from a micro- scopic quantum formulation of the problem at hand it aims at deriving a simpler kinetic equation resembling the semiclassical Boltzmann one. In deriving such an equation some of the information at microscopic level is suitably incorporated in a set of parameters charac- terizingthephysicalsystematmacroscopic level.Since the first application to superconductivity, this equa- tion is known as the Eilenberger equation (for a review see for instance [1]). We have recently derived[2] such an equation for a two-dimensional electron gas in the presence of spin orbit coupling with Hamiltonian H=p2 2m+b·σ, (1) whereb(p) is a momentum dependent internal mag- netic field. In the case of Rashba spin-orbit coupling 1Corresponding author. E-mail:raimondi@fis.uniroma3.itb=αp∧ˆ ez. In Ref. [2] we adopted the standard ξ- integrationproceduretoarriveattheEilenbergerequa- tion, and, though this leads to correct results, we feel theneed for a deeper understanding, which we provide in the present paper. In so doing we follow an analysis carried out by Shelankov[3]. Finally, we use the Eilen- berger equation to study the response to an external electric field in the presence of magnetic impurities. 2. The quasiclassical approach In deriving the Eilenberger equation a key observa- tion is that, by subtracting from the Dyson equation its hermitian conjugate, one eliminates the singular- ity for equal space-time arguments and gets a simpler equation for the ξ-integrated Green function ˇg(ˆp,x) =i πZ dξˇG(p,x), ξ=ǫ(p)−µ.(2) HereˇG(p,x) is the Green function in Wigner space, i.e. the Fourier transform of ˇG(x1,x2) with respect to Preprint submitted to Physica E 30 October 2018the relative coordinate r=x1−x2. The “check” indi- cates that the Green function is a 2 by 2 matrix in the Keldysh space [1]. To shed some light on the meaning of theξ-integration, let us consider first the space de- pendence of the two-point retarded Green function for free electrons in the absence of spin-orbit coupling GR(x1,x2) =X peip·r ω−ξ+i0+,r=x1−x2.(3) Atlarge distances, theintegral is dominatedbytheex- trema of the exponential under the condition of con- stant energy. This forces the velocity to be parallel or antiparallel to the line connecting the two space argu- ments,∂pǫ(p)∝r. Itis thenusefultoconsider themo- mentum components parallel ( p/bardbl) and perpendicular (p⊥) tor. Given the presence of the pole, one can ex- pand the energy in powers of the two momentum com- ponents ξ(p/bardbl,p⊥) =vF(p/bardbl−pF)+p2 ⊥/2m. In the case of the retarded Green function, the important region is that with velocity parallel to r. We then get GR(x1,x2)=Zdp⊥dp/bardbl (2π)2eip/bardblr ω−vF(p/bardbl−pF)−p2 ⊥ 2m+i0+ =−iei(pF+ω/vF)r vFZdp⊥ 2πe−ip2 ⊥r/2pF =−r 2πi pFrN0ei(pF+ω/vF)r, N0=m 2π.(4) One sees how the Green function is factorized in a rapidly varying term ∼eipFr/√pFr, and a slow one, ei(ω/vF)r. This suggests to write quite generally GR(x1,x2)=−r 2πi pFrN0eipFrgR(x1,x2) =GR 0(r,ω= 0)gR(x1,x2) (5) wheregR(x1,x2) is slowly varying and GR 0indicates thefree Green function. Explicitly, in the presentequi- librium case gR(x1,x2) =i 2πZ dξeiξr/vF ω−ξ+i0+=eiωr/v F.(6) For the advanced Green function one can go through the same steps with the difference that the integral is dominated by the extremum corresponding to a veloc- ity antiparallel to r, so that one has the ingoing wave replacingtheoutgoingone.Inthenon-equilibriumcase Shelankov has shown that gR(x1,x2) =i 2πZ dξeiξr/vFGR(p,x),p=pˆr(7)andfurthermorethatthequasiclassical Greenfunction corresponds to the symmetrized expression gR(ˆ p;x) = lim r→0i πZ dξcos„ξr vF« GR(p,x) (8) when sending to zero the relative coordinate r. When the spin-orbit coupling is present the Green function becomes amatrix in spin space and the Fermi surface splits into two branches ǫ±(p) =p2 2m±|b|. We always assume this splitting to be small compared to the Fermi energy, i.e. |b|/ǫF≪1. In the case of the Rashba interaction we write GR(x1,x2) =−X ±r 2πi p±rN±eip±r1 2n P±,˜gR(x1,x2)o (9) whereP±=|±∝an}bracketri}ht∝an}bracketle{t±|is the projector relative to the ±energy branch and the curly brackets denote the anticommutator. This ansatz allows us to proceed in Wignerspaceasbefore,whileretainingtheinformation onthecouplingandcoherenceofthetwobands.Eq.(9) is the equivalent in real space of the ansatz for the Green function G(p,x) used in Ref.[2]. With such an ansatz, Eq.(9), we obtain from Eq.(7) gR(x1,x2) =X ±1 2N0n N±P±,˜gR(x1,x2)o .(10) What we have explicitly shown for the retarded com- ponent of the Green function can be extended to the advanced and Keldysh components too. Notice that gRand ˜gRcoincide in the absence of spin-orbit cou- pling, since in that case N±=N0. The derivation of theEilenbergerequationcannowbedonefollowingthe steps detailed in Ref.[2]. We do not repeat them here and give just the final result X ν=±` ∂tˇgν+1 2npν m+∂p(bν·σ),∂xˇgνo +i[bν·σ,ˇgν]´ =−iˆˇΣ,ˇg˜ , (11) where ˇgν= (1/2){Pν,ˇg}, ˇg= ˇg++ ˇg−and both the momentum pνand the internal field bνare evaluated at theν-branch of the Fermi surface. Finally, ˇΣ is the self-energy. It is often convenient to expand ˇ gin terms of Pauli matrices, ˇ g= ˇg0+ˇg·σ, to explicitly separate charge and spin components. Physical quantities like chargeandspindensitiesandcurrentsarerelatedtothe Keldysh component of ˇ g. For example the spin current forsl,l=x,y,zis 2jl s(x,t) =−1 2πN0Zdǫ 2πJK l s(ǫ;x,t),(12) where ˇJl s=X ν=±∝an}bracketle{t1 2npν m+∂p(bν·σ),ˇgνo ∝an}bracketri}htl(13) and∝an}bracketle{t...∝an}bracketri}htis the angle average over the directions of p. 3. Magnetic impurities and spin currents Focusing on the Rashba interaction, we study the effects of magnetic impurities on spin currents. In [4] and [5] the problem has been recently tackled via dia- grammatic techniques.Weshowhowanalogous results can be obtained in a simple and rather elegant way relying on eq.(11). As it is well known, spin currents arising from the spin Hall effect in such a system are completelysuppressedbythepresenceofnon-magnetic scatterers. By taking the angular average of eq.(11), one obtains a set of continuity equations for the vari- ous spin components which let one easily understand the origin of this cancellation. Explicitly, by assum- ings-wave and non-magnetic impurities randomly dis- tributed in the system V1(x) =X iU δ(x−Ri), (14) the self-energy in the Born approximation turns out to beˇΣ1=−i∝an}bracketle{tˇg∝an}bracketri}ht/2τ, 1/τbeing the momentum scatter- ing rate. The continuity equations for the l=x,y,z spin components then read ∂t∝an}bracketle{tˇgl∝an}bracketri}ht+∂x·ˇJl s= 2∝an}bracketle{tb0∧ˇg∝an}bracketri}htl. (15) A rather important peculiarity of the Rashba Hamil- tonian is that it lets one write the vector product ap- pearing above in terms of the various spin currents, so that, by choosing for example l=y, we are left with ∂t∝an}bracketle{tˇgy∝an}bracketri}ht+∂x·ˇJy s=−2mαˇJz s,y. (16) Understationaryandhomogeneous conditionsthisim- plies the vanishing of the ˇJz s,yspin current. As soon as magnetic impurities are introduced in the system, their presence changes the self-energy and leads to the appearance of additional terms in Eq.(16). We assume the magnetic scatterers to be also isotropic and ran- domly distributed V2(x) =X iB·σδ(x−Ri), (17)and, proceeding again in the Born approximation, we obtain the self-energy ˇΣ =ˇΣ1+ˇΣ2=−i 2τ∝an}bracketle{tˇg∝an}bracketri}ht−i 6τsf3X l=1σl∝an}bracketle{tˇg∝an}bracketri}htσl.(18) Here 1/τsfis the spin-flip rate. With this, and by con- sideringagainstationaryandhomogeneousconditions, Eq.(16) becomes 2mαJKz s,y+4 3τsf∝an}bracketle{tgK y∝an}bracketri}ht= 0, (19) whichintermsoftherealspincurrentandpolarization means jsz y=−2 3mατsfsy. (20) By assuming a low concentration of magnetic impuri- ties, wecan useineq.(20) thevalueofthe y-spinpolar- ization valid in their absence, sy=−|e|EατN 0[6],E beingtheexternal,homogeneouselectricfield.Wethen get the spin Hall conductivity to first order in τ/τsf σsH=|e| 3πτ τsf, (21) aresultsthatdiffersfromthoseonRefs.[4,5].Thisisnot surprising for Ref.[5] , which neglects normal impurity scattering and then considers the opposite limit. The reasonwhyourresultdoesnotagreewiththelowmag- netic impurity-concentration limit of eq.(20) of Ref.[4] is not clear to us and deserves further investigation. ThisworkwassupportedbytheDeutscheForschungs- gemeinschaft through SFB 484 and SPP 1285 and by CNISM under Progetto d’Innesco 2006. References [1] J. Rammer and H. Smith, Rev. Mod. Phys. 58, 323 (1986). [2] R. Raimondi, C. Gorini, P. Schwab, and M. Dzierzawa, Phys. Rev. B 74, 035340 (2006). [3] A. I. Shelankov, J. Low Temp. Phys. 60, 29 (1985). [4] J.Inoue,T.Kato,Y.Ishikawa,H.Itoh,G.E.W.Bauer, L. W. Molenkamp, Phys. Rev. Lett. 97,046604 (2006). [5] P. Wang, Y. Li, X. Zhao, Phys. Rev. B 75, 075326 (2007). [6] V. M. Edelstein, Solid state Commun. 73, 233 (1990); J. Phys.: Condens. Matter 5, 2603 (1993). 3

1110.0558v3.Spin_orbit_coupled_dipolar_Bose_Einstein_condensates.pdf

arXiv:1110.0558v3 [cond-mat.quant-gas] 8 Jun 2012Spin-orbit-coupled dipolar Bose-Einstein condensates Y. Deng1,2, J. Cheng3, H. Jing2, C.-P. Sun1, and S. Yi1 1State Key Laboratory of Theoretical Physics, Institute of T heoretical Physics, Chinese Academy of Sciences, P.O. Box 2735, Beijing 100190, China 2Department of Physics, Henan Normal University, Xinxiang 4 53007, China and 3Department of Physics, South China University of Technolog y, Guangzhou 510640, China (Dated: October 22, 2018) We propose an experimental scheme to create spin-orbit coup ling in spin-3 Cr atoms using Raman processes. By employing the linear Zeeman effect and optical Stark shift, two spin states within the ground electronic manifold are selected, which results in a pseudo-spin-1 /2 model. We further study the ground state structures of a spin-orbit-coupled C r condensate. We show that, in addition to the stripe structures induced by the spin-orbit coupling , the magnetic dipole-dipole interaction gives rise to the vortex phase, in which a spontaneous spin vo rtex is formed. PACS numbers: 37.10.Vz, 03.75.Mn Over the past few years, there has been rapidly grow- ing interest in engineering Abelian and non-Abelian arti- ficial gauge fields in ultracold atomic gases [1–6]. Partic- ularly, the non-Abelian gauge field, or more specifically the spin-orbit (SO) coupling, is of fundamental impor- tance in many branches of physics. Fascinating examples include the quantum spin-Hall effect and the topologi- cal insulators in condensed matter physics [7]. With the enormous tunability of the interaction and geometry, ul- tracold atomic gases may offer a tremendous opportunity for studying exotic quantum phenomena in many-body systems with SO coupling [8–16]. In their pioneer experiments, the NIST group have re- alized the light-induced vector potentials [17], the syn- thetic magnetic fields [18], and the electric forces [19] in ultracold Rb gases through Raman processes [4], which differs from most dark-state based theoretical propos- als [20] in that the linear Zeeman shift is compensated by the two-photon detuning. More remarkably, they also created a two-component SO-coupled condensate of Rb atoms and observed the phase transition from spatially mixed to separated states [21]. An important ingredient in this experiment is that the quadratic Zeeman shift is employed to separate two desired spin states from the re- maining one. Hence, this scheme is inapplicable to atoms without nuclear spin, such as certain isotopes of Cr and Dy, in which the quadratic Zeeman effect is absent. In this Letter, we propose an experimental scheme to create SO coupling in spin-352Cr atoms by selecting two internal states from the J= 3 ground electronic mani- fold. Similar to the NIST group’s scheme, ours also re- lies on Raman processes. However, we utilize the optical Stark shift to compensate the linear Zeeman shift so that the lowest two levels are near degenerate and well sepa- rated from other levels, which leads to a pseudo spin-1 /2 model. The proposed scheme has the advantages that only a moderate magnetic field strength is required and it also applies to atoms without nuclear spin. An interesting feature of the Cr atom is that it pos- sesses a large magnetic dipole moment, which makes the scalar Cr condensate an important platform for demon-(a) (b) FIG. 1: (color online). (a) Scheme for creating SO coupling i n a Cr atom. Two Raman beams, propagating along ˆ x+ ˆyand −ˆx+ ˆywith frequency difference ∆ ωL, are linearly polarized along ˆzand ˆx+ ˆy, respectively. A bias field B0is applied along the negative zaxis, which generates a Zeeman shift ωZin the ground state manifold. (b) Level diagram for the Raman coupling within the |J= 3/angbracketrightground state manifold by utilizing the |J′= 2/angbracketrightexcited state. strating the dipolar effects [22]. Moreover, when an atom’s spin degree of freedom becomes available, mag- netic dipole-dipole interaction (MDDI) also couples the spin and orbital angular momenta, which is responsible for the Einstein-de Haas effects [23, 24], the spontaneous demagnetization [25] of the Cr condensate, and the spon- taneous spin vortices [26–28] in spinor condensates. Un- fortunately, in spin-3 Cr condensates, contact interaction also contains spin-exchange terms which are much larger than the strength of the MDDI [29]. Therefore, the spin vortex phases are yet to be observed. In the pseudo spin- 1/2 Cr condensate, we show that only the MDDI con- tains spin-exchange terms and a spontaneous spin vortex is readily observable.2 We consider a condensate of52Cr atoms subjected to a bias magnetic field B0along the negative z-axis. The Zeeman shift within the ground state manifold is /planckover2pi1ωZ= gsµB|B0|withgs= 2 being the electron spin g-factor andµBthe Bohr magneton. Here, the quadratic Zeeman shift is zero because of the absence of the nuclear spin. As shown in Fig. 1, atoms are illuminated by a pair of linearly polarized Raman beams which propagate along ˆx+ ˆyand−ˆx+ ˆywith frequencies ωL+ ∆ωLandωL, respectively. The ground- (7S3) to excited-state (7P2)transitions are coupled by the Rabi frequencies Ω 1eik1·r and Ω 2eik2·r, wherek1=kL(ˆx+ˆy) andk2=kL(−ˆx+ˆy) are the wave vectors of the Raman beams with kL=√ 2π/λandλbeing the wave length of the lasers. For simplicity, Ω 1,2areassumedtobereal. Ifthefrequencyof thelasersisfardetuned fromthe ground-toexcited-state transition, i.e., |Ω1,2/∆| ≪1 with ∆ being the detuning, the excitedstatescanbe adiabaticallyeliminated toyield the atom-light interaction Hamiltonian /planckover2pi1 U2 ΩRX U 2T ΩRX∗∆c+U1+U2ΩR(X+X∗T)U2T U2T∗ΩR(X∗+XT∗) 2∆c+U1+2U2ΩR(X+X∗T)U2T U2T∗ΩR(X∗+XT∗) 3∆c+U1+2U2ΩR(X+X∗T)U2T U2T∗ΩR(X∗+XT∗) 4∆c+U1+2U2ΩR(X+X∗T)U2T U2T∗ΩR(X∗+XT∗) 5∆c+U1+U2ΩRX∗T U2T∗ΩRXT∗6∆c+U2 ,(1) where ∆ c=ωZ+ ∆ωLis the two-photon detuning, ΩR=−Ω1Ω2/∆ is the Rabi frequency for the Raman coupling,U1,2=−Ω2 1,2/∆ are the optical Stark shifts induced by the laser fields Ω 1and Ω 2, respectively, and T(t)≡e2i∆ωLtandX(x)≡e2ikLxare introduced for short-hand notation. The physical significance of Eq. (1) canbereadilyunderstood [30]byusingtheleveldiagram [Fig. 1(b)]. From Hamiltonian (1), it is apparent that, under the conditions ∆ c+U1≈0 and|U2|,|ΩR| ≪ |∆c|, the en- ergy levels mJ= 3 and 2 can be separated from other levels due to the large Zeeman shift. These conditions can be satisfied by choosing ωZ= Ω2 1/∆ and assuming that|∆ωL/ωZ| ≪1 and|Ω2/Ω1| ≪1, which eventually leads to an effective two-level Hamiltonian: ˆh=p2 2MˆI+/planckover2pi1/parenleftbigg −∆ωL/2 ΩRe2ikLx ΩRe−2ikLx∆ωL/2/parenrightbigg ,(2) for pseudo spin-up | ↑/angbracketright=|mJ= 3/angbracketrightand -down | ↓/angbracketright= |2/angbracketright, whereˆIis the identity matrix and a constant term, −(U2+∆ωL/2)ˆI, has been added to obtain Eq. (2). We note that the atom-light interaction term in ˆhcan be intuitively treated as an effective magnetic field, Beff=/planckover2pi1(gsµB)−1(2ΩRcos2kLx,−2ΩRsin2kLx,−∆ωL). Unlike the NIST group’s scheme [21], here, an optical Stark shift −Ω2 1/∆ is used to compensate the linear Zee- man shift, so that only the levels mJ= 3 and 2 are Raman coupled near resonance (∆ ωL≈0). To proceed further, let us focus on the motion of an atom along the xaxis by freezing its yandzdegrees of freedom. By applying a simple gauge transform [12], thesingle-particle Hamiltonian can be recast into ˆh′ x=/parenleftbigg/planckover2pi12q2 2M+EL/parenrightbigg ˆI+2κqˆσz+/planckover2pi1ΩRˆσx−/planckover2pi1∆ωL 2ˆσz,(3) whereq=px//planckover2pi1is the quasimomentum, EL= /planckover2pi12k2 L/(2M) is the single-photon recoil energy, ˆ σx,y,zare the Pauli matrices, and κ=EL/kLis the SO coupling strength. Even though κis independent of Raman cou- pling strength, SO coupling strength is still tunable by varying the relative angle of the Raman beams [21]. It can be readily shown that, after dropping the constant ELterm, the eigenenergies of Eq. (3) are E±(q) =/planckover2pi12q2 2M±/radicaligg /planckover2pi12Ω2 R+/parenleftbigg 2κq−/planckover2pi1∆ωL 2/parenrightbigg2 ,(4) in analogy to those in the spin-1 Rb condensate. In par- ticular, on the lower branch E−(q), there exist two local minima at q±≃ ±kL/radicalbig 1−/planckover2pi12Ω2 R/(4E2 L) when/planckover2pi1ΩR/lessorsimilar 2ELand/planckover2pi1∆ωL/lessorsimilarEL. The corresponding energies are E−(q±)≃ −EL−/planckover2pi12Ω2 R/(4EL)±/planckover2pi1∆ωL/2. The states with quasimomenta /planckover2pi1q−and/planckover2pi1q+(labeled as | ↑′/angbracketrightand | ↓′/angbracketright, respectively) represent the dressed spin states in which atoms condense in the absence of the interactions. Here, we would like to discuss the experimental feasi- bility of our scheme. The transition wavelength from the ground to excited state is 429 .1nm, which corresponds to a recoil energy EL//planckover2pi1≃(2π)10kHz. Other laser param- eters can be set up as follows. Since the linear Zeeman shiftωZin our proposal plays the role of the quadratic Zeeman shift in the NIST experiment [21], we may set /planckover2pi1ωZ= 3.8EL, which implies that the laser intensity |Ω1|2= 3.8EL|∆|//planckover2pi1is about the same order of magni- tude as that used in the experiment. To allow the Ra- man coupling Ω Rto vary from 0 to EL, which covers the3 most interesting parameter region in the experiment, the maximum value of |Ω2|can be chosen as 0 .26|Ω1|. Con- sequently, the maximum value of U2is less than 0 .26EL, whichjustifiestheneglectingof U2inEq. (1). Finally, we point out that the SO coupling strength κin our scheme is 3.14 times larger than that in the Rb experiment due tothe smallermassandthe shortertransitionwavelength of Cr atom. Now we turn to study the many-body effect in a SO- coupled Cr condensate. To this end, we first write down the single-particle Hamiltonian, which, in the second quantized, takes the form ˆH0=/integraldisplay drˆΨ†(r)/bracketleftig ˆh+V(r)−µ/bracketrightig ˆΨ(r),(5) whereV(r) =Mω2 ⊥(x2+y2+γ2z2)/2 is an axially symmetric harmonic trap with ω⊥being the radial trap frequency and γthe trap aspect ratio, µis the chem- ical potential, and ˆΨ(r) = [ˆψ↑(r),ˆψ↓(r)]Tis the field operator for the bare spin states. We note that ˆH0 can also be expressed in terms of dressed spin states by using the transform ˆψ↑(r)≃ˆψ↑′(r)−εe2ikLxˆψ↓′(r) andˆψ↓(r)≃ −ˆψ↓′(r) +εe−2ikLxˆψ↑′(r), whereε≃ /planckover2pi1ΩR/(4EL+/planckover2pi1∆ωL)≪1 in the weak Raman coupling limit/planckover2pi1ΩR/EL≪1. In terms of the bare spin states, the collisional inter- action takes the form ˆHc=1 2/integraldisplay dr/parenleftbigg g6ˆψ† ↑ˆψ† ↑ˆψ↑ˆψ↑+5g4+6g6 11ˆψ† ↓ˆψ† ↓ˆψ↓ˆψ↓ +2g6ˆψ† ↑ˆψ† ↓ˆψ↓ˆψ↑/parenrightig , (6) whereg4,6= 4π/planckover2pi12a4,6/Mwitha4= 58aBanda6= 112aBbeing thes-wave scattering lengths for the colli- sional channel with total spin angular momentum j= 4 and 6, respectively [31]. This result can be understood as follows. The collision between two mJ= 3 atoms can happen only in the total spin j= 6 channel; con- sequently, it has a scattering length a6. For collisions betweenmJ= 3 and 2 atoms, the projection of the to- tal spin along the z-axis ismj= 5, which is conserved during collision. Therefore, this collision also happens in thej= 6 channel. But when two mJ= 2 atoms collide with each other, both j= 6 and 4 channels will con- tribute. Apparently, the spin-up and -down states are immiscible. The MDDI for the pseudo spin-1 /2 system can be de- composed into ˆHd=ˆH(1) d+ˆH(2) dwith ˆH(1) d=gd/radicalbigg 4π 5/integraldisplaydrdr′ |r−r′|3Y2,0(ˆe)/bracketleftig −9ˆψ† ↑ˆψ′† ↑ˆψ′ ↑ˆψ↑ −4ˆψ† ↓ˆψ′† ↓ˆψ′ ↓ˆψ↓−12ˆψ† ↑ˆψ′† ↓ˆψ′ ↓ˆψ↑+3ˆψ† ↑ˆψ′† ↓ˆψ′ ↑ˆψ↓/bracketrightig ,(7) ˆH(2) d=−gd/radicalbigg 9π 5/integraldisplaydrdr′ |r−r′|3/bracketleftig 2Y2,−1(ˆe)/parenleftig 3ˆψ† ↑ˆψ′† ↑ˆψ′ ↑ˆψ↓ +2ˆψ† ↑ˆψ′† ↓ˆψ′ ↓ˆψ↓/parenrightig +√ 6Y2,−2(ˆe)ˆψ† ↑ˆψ′† ↑ˆψ′ ↓ˆψ↓+h.c./bracketrightig ,(8) x(µm)y(µm) −24 024−24024 FIG. 2: Integrated densities (columns 1 and 3 for spin-up and -down, respectively) and phases of the condensate wave functions on the z= 0 plane (columns 2 and 4 for spin-up and -down, respectively). From the first to the fourth rows, the frequency differences are /planckover2pi1∆ωL/EL= 0.01, 0.04, 0.0875, and 0.1, respectively. here,gd=µ0g2 sµ2 B/(4π) withµ0being the vacuum per- meability and µBthe Bohrmagneton, ˆ e= (r−r′)/|r−r′| is an unit vector, and we have adopted the notations ˆψα≡ˆψα(r) andˆψ′ α≡ˆψα(r′) withα=↑and↓. The first three terms of ˆH(1) drepresent the intra- and inter- species dipolar interactions in a mixture of mJ= 3 and 2 atoms, and the last term is the exchange dipolar interac- tion.ˆH(2) dis of particular interest. It represents the SO coupling containing in the MDDI and does not conserve the atom number in the individual spin state. However, the total angular momentum is conserved by ˆH(2) d. In this spin-1 /2 model, interactions have a much sim- pler form compared to those in the spin-3 system. In particular, here, only the MDDI contains spin-exchange terms. As will be shown, even though gdis much smaller thang4,6, the spin associateddipolareffect canbe readily detected in pseudo spin-1 /2 Cr condensates. We now investigate the ground state structures of the SO-coupled dipolar condensate using the mean-field the- ory. To this end, the field operators ˆψαare replaced by the condensate wave function ψα=/angbracketleftˆψα/angbracketright, which can be obtained by numerically minimizing the free energy functional F[ψ↑,ψ↓] =/angbracketleftˆH0+ˆHs+ˆHd/angbracketright. Specifically, we consider a Cr condensate with N= 106atoms. The pa- rameters for the trapping potential are chosen as ω⊥= (2π)100Hz and γ= 6, representing a three-dimensional pancake-shaped trap. Furthermore, the Rabi frequency for Raman coupling is fixed at /planckover2pi1ΩR=−0.01EL. Since4 x(µm)y(µm) −8−4048−8−4048 FIG. 3: (color online). Vector plot of the transverse compo- nents of s(r) on the z= 0 plane for /planckover2pi1∆ωL= 0.0875EL. The grayscale indicates the integrated density ¯ n↑(x,y). 00.030.060.090.1200.20.40.60.81 ¯h∆ωL/EL/tildewideNα(a) 00.030.060.090.12−1−0.8−0.6−0.4−0.20 ¯h∆ωL/EL¯Lz/¯h(b) PP VP PP↑ ↓ FIG. 4: (color online). Reduced atom number /tildewideNα(a) and av- erage orbital angular momentum ¯Lz(b) as functions of ∆ ωL. ε≪1 is satisfied, we shall discuss only the ground state in terms of the bare spin states. In Fig. 2, we plot the integrated density, ¯ nα(x,y) = N−1/integraltext dz|ψα(r)|2, and the phases of the condensate wave functions for various ∆ ωL’s. When ∆ ωLis small (the first row in Fig. 2), the single-particle energies of the two pseudo spin states are nearly degenerate such that the ground state structure is mainly determined by the in- teractions. Apparently, both ˆHcandˆHd(in the pancake- shaped trap) favor the spin-down state. Therefore, | ↓ /angbracketright becomes dominantly populated, which we refer to as the polarized phase (PP). As shown in the fourth row of Fig. 2, the PP also occurs when the frequency difference ∆ωL(or, equivalently, the zcomponent of the effective magnetic field Beff) is sufficient large, under which | ↑/angbracketright becomes dominantly occupied. The common feature of the PPs is that the wave func- tion of the highly populated state is structureless, as any structure developed in the high density spin state would cost too much kinetic energy. On the other hand, a striped structure forms in both the density and phase of the less populated spin state. The phase stripe can be intuitively understood as follows. To lower the energy, the pseudo spin density, s(r) =/summationtext αβψ∗ αˆσαβψβ, has to be antiparallel to the local effectively magnetic field, which requires the relative phase of the condensate wave func- tion to takethe formarg( ψ↑)−arg(ψ↓)∼π+2kLx. Since the phase of the highly populated state is a constant, thephase of the other spin state is then periodically modu- lated along the xdirection. The density stripe in the less populated spin state is caused by the immiscible nature of the two-component condensate. More remarkably, we observe a vortex phase (VP) for intermediate ∆ ωLvalues. As shown in the second and third rows of Fig. 2, a singly quantized vortex appears in the spin-up statedue to the SO couplingin the ˆH(2) dterm of the MDDI. In the VP, the atom numbers in spin-up and -down states become comparable [Fig. 4(a)], which allows the spin s(r) of the atom to form significant trans- verse components. As shown in Fig. 3, since the MDDI is minimized with a head-to-tail spin configuration, the transverse components of s(r) are forced to form a spin vortex. Consequently,thewavefunction ψ↑developsa2 π phase winding, representing a vortex state. The reason that the vortex state appears only on the spin-up com- ponent is due to the immiscibility of our two-component system, which results a density depletion at the center ofψ↑. Therefore, forming a vortex in the spin-up state costs less kinetic energy. Moreover, in the VP, the phase stripes alsoappear in the low density regionsofboth spin states, which is the manifestation of the SO coupling in- duced by the light fields. To determine the phase boundaries, we plot the ∆ωLdependences of the reduced atom number, /tildewideNα= N−1/integraltext dr|ψα|2, and the average orbital angular momen- tum,¯Lz=N−1/summationtext α/integraltext drψ∗ αˆLzψα, in Fig. 4, where ˆLz= −i/planckover2pi1(x∂ ∂y−y∂ ∂x) is thez-component of the orbital angu- larmomentum. As can be seen, the phase boundariesare marked by two critical ∆ ωLvalues, ∆ω∗ L= 0.031EL//planckover2pi1 and ∆ω∗∗ L= 0.088EL//planckover2pi1. For ∆ω∗ L<∆ωL<∆ω∗∗ L, the condensate lies in the VP; otherwise, it is in the PP. Within the VP, atom numbers and orbital angular mo- mentum change dramatically as one varies ∆ ωL. In conclusion, we have proposed an experimental scheme to generate SO coupling in spin-3 Cr condensates via Raman processes. Optical Stark shift is employed to selecting two spin states from the atom’s ground elec- tronic manifold. The proposed scheme should be read- ily realizable experimentally. Subsequently, the ground- statestructuresofaSO-coupledCrcondensatehavebeen investigated. We show that the interplay between the light fields and the MDDI gives rise to the polarized and vortex phases. In particular, a spontaneous spin vortex is formed in VP. The spin vortex state can be experimen- tally identified if it is observed that the spin-up conden- sate is a vortex state and the spin-down condensate is a vortex-free one. Finally, we point out that our scheme should also apply to the Dy atom [32], which has an even larger dipole moment. We thank Ruquan Wang for helpful discussions. 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1512.02398v1.Multi_orbital_quantum_antiferromagnetism_in_iron_pnictides_____effective_spin_couplings_and_quantum_corrections_to_sublattice_magnetization.pdf

arXiv:1512.02398v1 [cond-mat.str-el] 8 Dec 2015Multi-orbital quantum antiferromagnetism in iron pnictid es — effective spin couplings and quantum corrections to sublatt ice magnetization Sayandip Ghosh, Nimisha Raghuvanshi, Shubhajyoti Mohapatra, Ashish Kumar, and Avinash Singh Department of Physics, Indian Institute of Technology Kanpur 208016, India∗ (Dated: July 26, 2018) Towards understanding the multi-orbital quantum antiferr omagnetism in iron pnictides, effective spin couplings and spin fluctuation indu ced quantum corrections to sublattice magnetization are obtained in the ( π,0) AF state of a realistic three band interacting electron model involving xz,yz, andxyFe 3d orbitals. The xy orbital is found to be mainly responsible for the generation of strong ferromagnetic spin coupling in the bdirection, which is critically important to fully account f or the spin wave dispersion as measured in inelastic neutron sc attering experiments. The ferromagnetic spin coupling is strongly suppressed as t hexyband approaches half filling, and is ascribed to particle-hole exchange in th e partially filled xyband. The strongest AF spin coupling in the adirection is found to be in the orbital off diagonal sector involving the xzandxyorbitals. First order quantum corrections to sublattice magnetization are evaluated for the three orbit als, and yield a significant 37% average reduction from the Hartree-Fock value. PACS numbers: 75.30.Ds, 71.27.+a, 75.10.Lp, 71.10.Fd2 I. INTRODUCTION The rich phase diagram exhibited by iron pnictides1,2including magnetic, structural, and superconducting phase transitions3have stimulated intensive investigations aimed at detailed understanding of their macroscopic physical behavior in te rms of their complex multi-orbital electronic structure as revealed by first-principle ca lculations4–11and angle re- solved photoemission spectroscopy (ARPES) experiments12–16. The magnetic state exhibits (π,0) magnetic ordering of Fe moments in the a−bplane, with a concomitant structural distortion a > b, possibly correlated with the ferro orbital order nxz> nyzas seen in ARPES studies14. Inelastic neutron scattering studies of magnetic excitations in iron pnictides have been carried out extensively17–23, and clearly reveal well defined spin wave excitations with en- ergy scale ∼200 meV, persisting even above the N´ eel temperature23, indicating that short range antiferromagnetic (AF) and ferromagnetic (F) order rema in in the aandbdirections, respectively, even above the disordering temperature for long-r ange magnetic order. This persistence of short range anisotropic magnetic order may accou nt for the narrow nematic phase24,25above the N´ eel temperature where the ferro orbital order14and structural distor- tion survive, as well as the temperature dependence of the measu red anisotropies in aand bdirections of magnetic excitations and resistivity14,23,26. Spin wave excitations in iron pnictides have been theoretically studied for multi-band models27–31in the random phase approximation (RPA), including the orbital matr ix com- ponents of the spin wave spectral weight30. The magnitude of the intra-orbital Coulomb interaction term considered in these investigations typically lie in the in termediate coupling range (U∼1−2 eV), resulting in moderately well developed local moments in the ( π,0) magnetic state, which has interesting implications on the multi-orbita l quantum antiferro- magnetism in these compounds. In this paper we will study the effect ive spin couplings generated by particle-hole exchange, their orbital contributions , and quantum corrections to sublattice magnetization in the ( π,0) AF state of a realistic three-orbital model which yields Fermi surface structure, spin wave excitations, and ferro orbital ordering in quantita- tive agreement with experiments. One important feature of the measured spin wave dispersion is that the spin wave en- ergy is maximum at the ferromagnetic zone boundary (FZB), slightly higher than at the3 antiferromagnetic zone boundary (AFZB). This feature is quite sig nificant as the FZB spin wave energy provides a sensitive measure of the effective ferroma gnetic (F) spin coupling in the (π,0) state31. It is only when F spin coupling is included that the FZB spin wave energy becomes finite, and even maximum over the entire Brillouin zon e when it exceeds the AF spin coupling. Understanding the microscopic mechanism behin d the origin of this strong F spin coupling and its interplay with the complex multi-orbital e lectronic structure as observed in iron pnictides should provide significant insight toward understanding the multi-orbital antiferromagnetism in these compounds. The structure of this paper is as below. Following a brief account in Se ction II of the realistic three bandmodel for iron pnictides interms of the xz,yz, andxyFe3d orbitals, the effective spin couplings are introduced in Section III in terms of the p article-hole propagator [χ0] evaluated in the ( π,0) magnetic state. Evaluation of quantum corrections to sublattic e magnetization is discussed in Section IV in terms of transverse spin c orrelations /angbracketleftS− iµS+ iµ/angbracketright and/angbracketleftS+ iµS− iµ/angbracketright. Results for the calculated effective spin couplings, quantum corre ctions, and discussion oftheir variationwith xyorbital energy offset arepresented in Section V, followed by conclusions in Section VI. II. THREE-ORBITAL MODEL AND ( π,0) MAGNETIC STATE We consider a minimal three-orbital model31involving dxz,dyzanddxyFe 3dorbitals. The tight binding Hamiltonian in the plane-wave basis is defined as: H0=/summationdisplay k,σ,µ,νTµν(k)a† kµσakνσ (1) where T11=−2t1coskx−2t2cosky−4t3coskxcosky T22=−2t2coskx−2t1cosky−4t3coskxcosky T33=−2t5(coskx+cosky)−4t6coskxcosky+εxy T12=T21=−4t4sinkxsinky T13=¯T31=−2it7sinkx−4it8sinkxcosky T23=¯T32=−2it7sinky−4it8sinkycoskx (2)4-1 1 -1 1ky/π kx/π (0,0) ( π,0) ( π,π)'sus.ot' q 0 100 200 300 ω (meV) 0 5 FIG. 1: (a) Fermi surfacein theunfolded BZ for thethree-orb ital model31with hoppingparameters as given in Table I. The main orbital contributions are shown as:dxz(red),dyz(green), and dxy (blue). (b) The spin wave spectral function in the ( π,0) SDW state for the three-orbital model at half filling. TABLE I: Values of the hopping parameters in the three-orbit al model (in eV) t1t2t3t4t5t6t7t8 0.1 0.32−0.29−0.06−0.3−0.16−0.15−0.02 are the tight-binding matrix elements in the unfolded Brillouin Zone ( −π≤kx,ky≤π). Here,t1andt2are the intra-orbital hoppings for xz(yz) alongx(y) andy(x) directions, respectively, t3andt4are the intra and inter-orbital hoppings along diagonal direction fo r xzandyz,t5andt6are intra-orbital NN and NNN hoppings for xy, whilet7andt8the NN and NNN hybridization between xyandxz/yz. Finally, εxyis the energy difference between thexyand degenerate xz/yzorbitals. The Fermi surface for the three-orbital Hamiltonian (1) for hopp ing parameter values given in Table 1 is shown in Figure 1(a) corresponding to near half filling. There are two circular hole pockets around the center and elliptical electron pock ets around ( ±π,0) and (0,±π) in the unfolded BZ. The two hole pockets involve primarily the xzandyzorbitals,5 while the electron pockets centered at ( ±π,0) [(0,±π)] arise mainly from the hybridization of thexyandyz[xz] orbitals. All of these features are in good agreement with results from DFT calculations and ARPES experiments. We now consider the ( π,0) ordered magnetic (SDW) state of this model. The various electron-electron interaction terms included are: HI=U/summationdisplay i,µniµ↑niµ↓+(U′−J/2)µ<ν/summationdisplay i,µ,νniµniν−2Jµ<ν/summationdisplay i,µ,νSiµ·Siν +J′µ<ν/summationdisplay i,µ,ν(a† iµ↑a† iµ↓aiν↓aiν↑+H.c.), (3) whereSiµ(niµ) refer to the local spin (charge) density operators for orbital µ. The first and second terms are the intra-orbital and inter-orbital Coulomb inte ractions respectively, the third term is the Hund’s rule coupling and the fourth term the “pair-h opping” term. In the following, we will consider U= 1.2 eV and J≈U/4. This interaction strength corresponds to the intermediate coupling regime, which is in accord with recent DFT + DMFT study of magnetism in iron pnictides32. Extending the two-sublattice basis approach for the SDW state in a single-band model28 to a composite three-orbital, two-sublattice basis, the Hartree- Fock (HF) level Hamiltonian matrix in this composite basis (A xzAyzAxyBxzByzBxy) is obtained as: Hσ HF(k) = −σ∆xz+ε2y k0 0 ε1x k+ε3 kε4 k ε7x k+ε8,1 k 0−σ∆yz+ε1y kε7y kε4 kε2x k+ε3 k ε8,2 k 0 −ε7y k−σ∆xy+ε5y k+εxy−ε7x k−ε8,1 k−ε8,2 kε5x k+ε6 k ε1x k+ε3 k ε4 k ε7x k+ε8,1 kσ∆xz+ε2y k0 0 ε4 k ε2x k+ε3 k ε8,2 k0σ∆yz+ε1y kε7y k −ε7x k−ε8,1 k−ε8,2 kε5x k+ε6 k 0 −ε7y kσ∆xy+ε5y k+εxy (4) for spinσ, in terms of the band energies corresponding to hopping terms alon g different directions: ε1x k=−2t1coskxε1y k=−2t1cosky6 ε2x k=−2t2coskxε2y k=−2t2cosky ε5x k=−2t5coskxε5y k=−2t5cosky ε3 k=−4t3coskxcoskyε4 k=−4t4sinkxsinky ε6 k=−4t6coskxcosky ε7x k=−2it7sinkxε7y k=−2it7sinky ε8,1 k=−4it8sinkxcoskyε8,2 k=−4it8coskxsinky (5) and the self-consistent exchange fields defined as 2∆ µ=Umµ+J/summationtext ν/negationslash=µmνin terms of sublattice magnetization mµfor orbital µ. The calculated spin wave spectral function31in the SDW state for the three-orbital model is shown in Fig. 1(b). Evidently, spin wave excitations are highly disper sive, and do not decay into the particle-hole continuum. The energy scale of spin exc itations is ∼200 meV with a well-defined maximum at the ferromagnetic zone boundary [ q= (0,π)]. These fea- tures of spin wave excitations are in excellent agreement with result s from inelastic neutron scattering measurements, confirming the realistic nature of the t hree-orbital model. III. EFFECTIVE SPIN COUPLINGS In the (π,0) magnetic state of the multi-band interacting electron model, the transverse spin fluctuation (spin wave) propagator in the random phase appro ximation (RPA) can be expressed in a symmetrised form: [χ−+(q,ω)] =[χ0(q,ω)] 1−[U][χ0(q,ω)] =1 [U]−[U][χ0(q,ω)][U]−1 [U](6) where [χ0] is the bare particle-hole propagator in the orbital-sublattice basis evaluated by integrating out the fermionic degrees of freedom, [ U] is the local (on-site) interaction matrix with intra-orbital terms Uand inter-orbital (Hund’s rule coupling) terms J, and the matrix [U][χ0(q,ω)][U] is Hermitian. Broadly, the important features of spin wave excitations in the spo ntaneously-broken- symmetry ( π,0) magnetic state of the three-band model are: (i) presence of z ero-energy Goldstone modes related to continuous spin rotation symmetry of t he interacting electron7 Hamiltonian and (ii) coupling between the three orbitals due to orbital hybridization in the tight-binding model. We therefore consider mapping to an effective s pin model: H=/summationdisplay /angbracketleftij/angbracketright,µνJµν ijSiµ.Sjν (7) in terms of spin-1/2 operators Siµcorresponding to the three electronic orbitals µ= xz,yz,xy which allows for effective spin interactions (couplings) between differ ent orbitals while preserving the spin rotation symmetry. Since positive spin wave energies are associated with increase in spin in teraction energy corresponding to specific spin twisting modes, the effective spin cou plings are inherently present in the RPA level spin propagator of Eq. (6). Based on earlie r spin wave studies in interacting electron models, as briefly discussed below, we will consid er the correspondence: −Jµν ij 2=/summationdisplay q/summationdisplay µ′ν′[U]µµ′[χ0(q)]µ′ν′ ss′[U]ν′νeiq.(ri−rj)(8) where indices s,s′correspond to the sublattices (A/B) which sites iandjbelong to, and we have set ω= 0 in the bare particle-hole propagator [ χ0(q,ω)]. The above form clearly suggests exchange of the particle-hole propagator [ χ0] as the origin of the effective inter- site spin couplings in interacting electron models with purely local inter actions of the form −Uµµ′Siµ.Siµ′. In the following, we will focus on the first neighbor spin couplings in th ea (AF) and b(F) directions (1a and 1b) and the second neighbor coupling along th e diagonal direction (2), evaluated by considering the corresponding lattice c onnectivity vectors δ= ri−rjin Eq. (8). In the (π,π) AF state of the single-band Hubbard model at half filling, extensive ly stud- ied in the context of quantum antiferromagnetism in cuprates, the above prescription yields AF spin coupling 4 t2/Uin the strong coupling limit33. In the ferromagnetic state of the Ferromagnetic Kondo Lattice Model (FKLM), which has been exten sively studied for under- standing metallic ferromagnetism in doped manganites, the calculate d spin wave dispersion in the strong coupling (double exchange) limit ( JH/t→ ∞) corresponds exactly to that for the Quantum Heisenberg Ferromagnet with NN spin coupling given by J2 Hχ0(q)34. Perhaps most relevant is that in the ( π,0) magnetic state of the half filled t−t′−U model, this correspondence exactly yields the AF first and second n eighbor spin couplings J1= 4t2/UandJ2= 4t′2/Uin the strong coupling limit, and also the effective F spin8 couplings of opposite sign for the doped t−t′−Umodel where the F spin couplings are gen- erateddueto particle-holeexchange inthepartiallyfilled bandasinme tallic ferromagnets31. Finally, in the limit of vanishing electron interaction strength (free ele ctron limit), the above prescription reduces exactly to the RKKY interaction, and thus allo ws for continuous inter- polation between the weak and strong coupling limits. It should be not ed that dynamical effects on effective spin couplings (through ωdependence in [ χ(0)(q,ω)] have been neglected, but should not affect the results qualitatively in the intermediate cou pling regime. IV. QUANTUM CORRECTIONS TO SUBLATTICE MAGNETIZATION Towards understanding the quantum antiferromagnetism in cupra te antiferromagnets, quantum spin fluctuations have been studied intensively in view of the ir important role in diverse macroscopic properties such as existence of long-range o rder, quantum corrections to sublattice magnetization, perpendicular susceptibility, spin wave velocity, ground state energy, and spin correlations36,37. In terms of the half-filled Hubbard model representation for the AF ground state, quantum spin fluctuations reduce the su blattice magnetization to nearly 60% of the classical (HF) value in two dimensions in the stron g-coupling limit (U/t→ ∞). The sublattice magnetization quantum corrections for the individ ual orbitals in the (π,0) state of the three band model are therefore of interest. We w ill use the approach in terms of transverse spin correlations which is valid in the full range of interaction strength for the Hubbard model and interpolates properly to the strong co upling limit38. Extending this approach to the multi-orbital antiferromagnet, the correct ed sublattice magnetization for orbital µis obtained from: mµ=mHF µ−δmSF µ (9) where the first-order, quantum spin-fluctuation corrections: δmSF µ=/angbracketleftS+ iµS− iµ/angbracketright+/angbracketleftS− iµS+ iµ/angbracketright /angbracketleftS+ iµS− iµ/angbracketright−/angbracketleftS− iµS+ iµ/angbracketright−1. (10) The transverse spin correlations above (equal-time, same-site) a re evaluated from the retarded part of the RPA level transverse spin fluctuation propa gator: /angbracketleftS− iµ(t)S+ iµ(t′→t−)/angbracketright=/summationdisplay q/integraldisplay∞ 0−dω πIm[χ−+(q,ω)]AA µµ9 /angbracketleftS+ iµ(t)S− iµ(t′→t−)/angbracketright=/summationdisplay q/integraldisplay∞ 0−dω πIm[χ+−(q,ω)]AA µµ =/summationdisplay q/integraldisplay∞ 0−dω πIm[χ−+(q,ω)]BB µµ (11) for lattice site ion A sublattice, using the spin-sublattice symmetry in the AF state w hich relates correlations on A and B sublattices via /angbracketleftS+ iµS− iµ/angbracketrightA=/angbracketleftS− iµS+ iµ/angbracketrightB. V. RESULTS AND DISCUSSION A. Effective spin couplings The effective spin couplings Jµν δwere calculated from Eq. (8), with the qsummation over the two dimensional Brillouin Zone performed for grids upto 60 ×60 to ensure no significant variation. The results are shown in Table 2 for the first and second n eighbors indicated byδ≡1a,1b,2, and for the reference case ǫxy=0.8 eV as considered for the spin wave plot of Fig. 1(b). Among the diagonal terms, the F (negative) spin coup ling (1b) is maximum for thexyorbital, whereas AF spin couplings are strong for xyandyzorbitals (1 a) and forxzandyzorbitals (2). The dominant off-diagonal term is the AF spin coupling (1 a) involving the xzandxyorbitals. Interestingly, this off-diagonal term is the maximum AF spin coupling, highlighting the multi-orbital character of quantum an tiferromagnetism and the role of strong orbital hybridization in the three band model. As all three orbitals follow identical ( π,0) magnetic ordering, it is useful to consider the total (orbital summed) spin couplings Jδ=/summationdisplay µ,νJµν δ (12) which yieldtheeffective couplingsasconsidered inphenomenological s pin modelstodescribe the spin wave dispersion in iron pnictides19. The most significant feature of the total spin couplings, given in Table 3, is the ferromagnetic (negative) first neig hbor spin coupling (1 b), which is in agreement with Ref.19where it was shown that spin wave dispersion throughout the BZ and the maximum at ( π,π) can be explained by a suitably parameterized Heisenberg Hamiltonianwithaneffectiveferromagneticexchangeinteraction( J1b<0)inthe bdirection. The present work provides the microscopic origin of this ferromagn etic interaction as due to10 Jµν 1a(meV) xz yz xy xz6.22 15.54 40.33 yz15.54 32.32 21.06 xy40.33 21.06 34.09 Jµν 1b(meV) xz yz xy xz17.06 0.53 -5.75 yz0.53 -17.17 -2.62 xy-5.75 -2.62 -48.74 Jµν 2(meV) xz yz xy xz28.86 15.38 -5.16 yz15.38 31.66 0.67 xy-5.16 0.67 -13.44 TABLE II: Effective spin couplings Jµν δfor first and second neighbors, evaluated from Eq. (8) for the reference case ǫxy= 0.8 eV. δ1a1b2 Jδ(meV)226.48 -64.52 68.86 TABLE III: The total (orbital summed) effective spin coupling s for first and second neighbors. the usual particle-hole exchange process in an itinerant-electron model, with the xyorbital being mainly responsible for the strong F spin coupling generated. We now investigate the effect of orbital order nxz−nxy(controlled by the offset energy εxy) on spin wave energies and effective spin couplings. Fig. 2 shows the v ariation of (a) the electronic densities ( nµ) for the three orbitals and (b) the spin wave energies at the ferromagnetic q= (0,π) and antiferromagnetic ( π/2,0) zone boundaries. While nyz remains fixed at 1 (half filling), there is a significant transfer of elect ronic density from xytoxzorbital with increasing xyorbital energy offset. The strong enhancement in the FZB spin wave energy with the depletion in xyorbital electronic density from 1 is due to the corresponding enhancement in the F spin coupling contribution o f thexyorbital, as discussed below. Fig. 3(a)shows thevariation(againwith εxy)ofthetotal(orbitalsummed) spincouplings11 0 0.5 1 1.5 2 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Electronic density nµ εxy (eV)µ=xz yz xy 0 50 100 150 200 250 300 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9ω (meV) εxy (eV)AFZB FZB FIG. 2: Variation of (a) electronic density nµfor the three orbitals and (b) spin wave energy at the F and AF zone boundaries with the xyorbital energy offset εxy. -100 0 100 200 300 400 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Jδ (meV) εxy (eV) 1a 1b 2 -80-40 0 40 80 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9Jδµ (meV) εxy (eV)xz (1b) xy (1b) yz (1a) FIG. 3: Variation of the (a) total (orbital summed) spin coup lingsJδand (b) the individual orbital contributions Jµ δto the effective spin couplings for the first and second neighbo rs with the xy orbital energy offset εxy. for the first neighbors (1 a,1b) in theaandbdirections and the second neighbor (2) along the diagonal direction. For higher values of the xyorbital energy offset, the effective spin coupling J1bis strongly negative, clearly highlighting the strong F spin generated by the particle-hole process mainly in the partially filled xyband. With decreasing xyorbital offset energy, as the xyband approaches half filling, the F component of the spin coupling generatedbyparticle-holeexchangedecreasessubstantially, re sultinginthenetspincoupling changing sign from negative to positive. Due to orbital hybridization in the three band tight binding model (1) , the effective12 spin couplings considered earlier include contributions from the orbit al off-diagonal parts [χ0]µ/negationslash=νof the particle-hole propagator as well. We therefore consider the individual orbital contributions to the effective spin couplings: −Jµ δ 2=U2/summationdisplay q[χ0(q)]µµ ss′eiq.(ri−rj)(13) which are shown in Fig. 3(b) for the three orbitals µ=xz,yz,xy . Thexyorbital yields strong F spin coupling (1 b) for higher values of xyorbital offset energy, which sharply decreases in magnitude, even changing sign to AF spin coupling, with d ecreasing xyorbital offset energy. This reduction is evidently related to the xyorbital approaching half filling, which strongly suppresses the F spin coupling generated by the par ticle-hole exchange. We now address the origin of the strong F spin coupling generated in t hexyorbital sector as seen in Fig. 3(b). Ferro spin couplings are generated gen erically through exchange of the particle-hole propagator, as in metallic ferromagnets with pa rtially filled bands, and are especially strong when the Fermi energy lies near a van Hove pea k in the DOS, which enhances the favorable delocalization contribution and suppresse s the unfavorable exchange contribution35. At first sight, it may seem that the electronic structure of the ga pped (π,0) magnetic state has little in common with metallic ferromagnets where t he Fermi energy lies within the band. However, the xyorbital sector is indeed only partially filled [Fig. 2(a)] in the gapped ( π,0) state due to the strong hybridization with xz/yzorbitals, and we attribute the generation of F spin coupling in the bdirection to particle-hole processes in this partially filledxysector. Indeed, the F (negative) spin coupling is maximum at the upper end of the scale [Fig. 3(b)] where the xyelectronic density is maximally depleted ( nxy≈0.75) and the energy gap for particle-hole excitations is also minimum, which together result in s trong enhancement of the F spin coupling generated by particle-hole exchange. With dec reasingxyorbital energy offset, as the xyorbital approaches half filling ( nxy≈1), the particle-hole-exchange mediated F component of the spin coupling decreases substantially, resulting in the net spin coupling changing sign from negative to positive. As also seen from Fig. 3(b), the AF spin coupling (1 a) for the half-filled yzorbital shows no variation with xyorbital energy offset. For the xzorbital, the strongly AF spin coupling (1b) near half filling (due to large hopping term t2) decreases substantially with xyorbital offset energy as xzband filling increases beyond 1 [Fig. 2(a)], indicating F contribution13 0 50 100 150 200 250 300 1b 1a 2Jδ (meV)J J=0 FIG. 4: Comparison of the effective spin couplings Jδwith and without the Hund’s rule coupling termJ, highlighting the significant role of Jin stabilizing the ( π,0) state. from the additional states transferred from the xyband. This reduced AF spin coupling in the ferromagnetic ( b) direction also contributes to the stabilization of the ( π,0) state by suppressing magnetic frustration. Each spin coupling term (8) involves altogether nine terms correspo nding to µ′ν′= xz,yz,xy , out of which eight terms involve the Hund’s rule coupling term J, indicating the importance of Jin the effective spin couplings. Fig. 4 shows the total (orbital summe d) spin couplings Jδ=/summationtext µνJµν δevaluated with and without Jfor the three neighbors. The significant reduction (to nearly half in all three cases) in the total s pin couplings when Jis set to zero highlights the importance of the Hund’s rule coupling term in the effective spin couplings and therefore on the overall stabilization of the ( π,0) magnetic state. B. Quantum corrections to sublattice magnetization For evaluating the quantum corrections to sublattice magnetizatio n, theqsummation over the BZ in Eq. (11) was performed over a 60 ×60 grid as earlier. The ωintegral was performed by evaluating the transverse spin fluctuation spectra l function for 1000 ωpoints extending upto 4 eV to ensure that optical modes and particle-hole excitations up to highest energies are included. Besides the Goldstone mode, optical modes a re also generally present formulti-orbitalmagneticsystems. The calculatedtransverse sp incorrelationsandquantum corrections to sublattice magnetization are given in Table 4 for the t hree orbitals, showing maximum and minimum quantum corrections for the xzandyzorbitals, respectively.14 µ/angbracketleftS+ iµS− iµ/angbracketright/angbracketleftS− iµS+ iµ/angbracketright/angbracketleft[S+ iµ,S− iµ]/angbracketright/angbracketleft2Sz iµ/angbracketrightδmiµ xz0.640.088 0.55 0.570.32 yz0.790.054 0.73 0.750.15 xy0.700.081 0.62 0.630.26 TABLE IV: Transverse spin correlations and sublattice magn etization quantum corrections for the three orbitals. It should be noted that the spin identities following from the spin comm utation relations: /angbracketleft[S+ iµ,S− iµ]/angbracketright=/angbracketleft2Sz iµ/angbracketright (14) between the RPA level transverse spin correlations and the HF leve l magnetizations provide a stringent check on the numerical accuracy, and are obeyed to h igh degree as seen from Table 4. With the total (orbital summed) quantum correction to sublattice magnetization /summationtext µδmµ= 0.73, and the total HF level sublattice magnetization/summationtext µmHF µ=/summationtext µ/angbracketleft2Sz µ/angbracketright= 1.95, the orbital-averaged reduction in sublattice magnetization due to quantum spin fluc- tuation induced quantum corrections in the ( π,0) AF state of the three band model is quite significant at about 37%. VI. CONCLUSIONS Originating from particle-hole exchange, effective spin couplings Jµν δfor first and second neighbors were evaluated in the ( π,0) magnetic state of a realistic three band interacting electron model for iron pnictides involving xz,yz,xy Fe 3d orbitals. Variation of these spin couplings with the xyorbital energy offset provides valuable insight into the multi- orbital quantum antiferromagnetism in these compounds. The xyorbital was found to be mainly responsible for the generation of strong F spin coupling betwe en first neighbors in thebdirection, which is critically required to fully account for the spin wave dispersion measured from inelastic neutron scattering experiments. The F sp in coupling is strongly suppressed with decreasing orbital order as the xyband approaches half filling, and is ascribed, asinmetallicferromagnets, toparticle-holeexchangeint hepartiallyfilled xyband, which provides the microscopic basis for the negative (ferromagne tic) exchange interaction15 (J1b<0) as considered in phenomenological spin models. Significantly, the strongest AF spin coupling between first neighbor s intheadirection lies in the orbital off diagonal sector involving the xzandxyorbitals, highlighting the important role of strong orbital hybridization on effective spin couplings. While t he AF spin coupling inadirection forthe half-filled yzorbital was foundto beconstant with increasing xyorbital energy offset, the frustrating AF spin coupling in bdirection for the xzorbital was found to decrease, thus reducing the magnetic frustration substantia lly. The Hund’s rule coupling term was found to contribute significantly to the effective spin coup lings and therefore to the overall stabilization of the ( π,0) magnetic state. The first-order spin-fluctuation induced quantum corrections to sublattice magnetization were evaluated from the transverse spin correlations, and yield ma ximum reduction for the overfilled xzorbital and minimum reduction for the half filled yzorbital. The orbital- averaged reduction in sublattice magnetization due to quantum spin fluctuations in the (π,0) AF state of the three band model was found to be significant at a bout 37%. Acknowledgements SG and NR acknowledge financial support from the Council of Scient ific and Industrial Research, India. References ∗Electronic address: avinas@iitk.ac.in 1J. Zhao, Q. Huang, C. de la Cruz, S. Li, J. W. Lynn, Y. Chen, M. A. Green, G. F. Chen, G. Li, Z. Li, J. L. Luo, N. L. Wang, and P. Dai, Nature Mat. 7, 953 (2008). 2S. Nandi, M. G. Kim, A. Kreyssig, R. M. Fernandes, D. 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2002.01151v1.Isotropic_All_electric_Spin_analyzer_based_on_a_quantum_ring_with_spin_orbit_coupling.pdf

arXiv:2002.01151v1 [cond-mat.mes-hall] 4 Feb 2020Isotropic All-electric Spin analyzer based on a quantum rin g with spin-orbit couplings Shenglin Peng,1,2Wenchen Luo,2,∗Jian Sun,2Ai-Min Guo,2Fangping Ouyang,1,2,3,†and Tapash Chakraborty4,‡ 1State Key Laboratory of Powder Metallurgy, and Powder Metal lurgy Research Institute, Central South University, Changsha, P. R. China 410083 2School of Physics and Electronics, Central South Universit y, Changsha, P. R. China 410083 3School of Physics and Technology, Xinjiang University, Uru mqi, P. R. China 830046 4Department of Physics and Astronomy, University of Manitob a, Winnipeg, Canada R3T 2N2 (Dated: February 5, 2020) Here we propose an isotropic all electrical spin analyzer in a quantum ring with spin-orbit cou- pling by analytically and numerically modeling how the char ge transmission rates depend on the polarization of the incident spin. The formalism of spin tra nsmission and polarization rates in an arbitrary direction is also developed by analyzing the Ahar onov-Bohm and the Aharonov-Casher effects. The topological spin texture induced by the spin-or bit couplings essentially contributes to the dynamic phase and plays an important role in spin transpo rt. The spin transport features de- rived analytically has been confirmed numerically. This int eresting two-dimensional electron system can be designed as a spin filter, spin polarizer and general an alyzer by simply tuning the spin-orbit couplings, which paves the way for realizing the tunable and integrable spintronics device. I. INTRODUCTION Manipulation of the spin degrees of freedom and the conduction charges in low-dimensional quantum struc- tures has been attracting considerable interest, due to wide range of potential applications in semiconductor spintronics and quantum computation. How to control, modulate, or detect the spin degree of freedom at the mesoscopic scale is a key step for the application of the spin coherence in electronic devices. The quantum ring [1, 2] is an ideal platform to take into consideration the Aharonov-Bohm (AB) and the Aharonov-Casher (AC) effects to show the nature of the quantum interference in conductance. The transport properties of similar nano- devices have received considerable attention, especially in the spin transport device subject to the Rashba spin- orbit coupling (SOC) [3–8], but the presence of Dressel- haus SOC or combination of both SOCs[9, 10] have not been investigated sufficiently as yet. The interplay of the Rashba SOC and the quantum interference has been widely reported in the literature. No spin is being polarized [11–13] in the transmission in the two-lead rings with equal arm length and without a magnetic flux or an impurity. This is because in this case the interference phase of the two different eigentransport channels is entirely due to the AC effect. The signs of phases are opposite but the absolute values are equal resulting in equal transmission rate for opposite spins. Topolarizethe spin, weneed to introducemagneticfields [14–17], use unequal length arms [11, 18, 19], doping [20– 22], or contact three or more leads [23, 24]. Quantum interference between the two arms of the ring provides suitable means for controlling the spin in ∗Electronic address: luo.wenchen@csu.edu.cn †Electronic address: ouyangfp@csu.edu.cn ‡Electronic address: Tapash.Chakraborty@umanitoba.cathe nano-scale, which has been proven by the Green’s function method [18, 25] or Griffith’s boundary condi- tions [15, 19–21, 23, 24, 26]. The first order linear ap- proximation with full transparent contacts was also re- ported [11, 14, 16, 27], albeit without the backscattering effect. The S-matrix method [28, 29] presents a rough assessment of the backscattering by fixing the energy- dependentcouplingparameterbetweentheleadsandring as constant. We note that previous works on spin trans- port properties in the quantum ring were not compre- hensive. For spin-unpolarized input current these works often only focused on spin polarization in the zdirection or the direction of the eigenstates of the ring. The total polarizability, polarization direction, and spin polariza- tion in arbitrary directions were rarely discussed. Work in the case of the arbitrarily spin-polarized incident are difficult to find in the literature. In this work, we present an analytical model for one- dimensional (1D) rings and numerical studies of real- istic two-dimensional (2D) quantum rings in the non- equilibrium Green’s function (NEGF) method [6] where both the Rashba and Dresselhaus SOCs are present. We derivethe formulaforthe transmissionratesforarbitrary spin polarization and generalize them to the cases of the polarized incident spin. A density matrix describing the spin-polarized (in arbitrary direction) input current is alsointroducedintothe Green’sfunction equation, which results in the same results obtained by the analytical 1D model. The transmission rate Tcan be up to unity with the fully polarized output in a proper magnetic field and with a proper Rashba SOC. When the input current is spin-polarized the transmis- sion rate depends on the direction of the input polariza- tion and the output current is still spin polarized. So the quantum ring is also acting as a spin torque which may be useful in spintronics. This property also guides us finding the way to design an omnidirectional spin an- alyzer. In contrast, the optical polarization analyzer is simpler since the polarization is perpendicular to the di-2 rection of the light. However, the spin polarization can be along an arbitrary direction on the Bloch sphere. The spin analyzer in a particular direction can be achieved in the ferromagnetism systems [30]. The arbitrary spin analyzer needs the light involved [31, 32], which is diffi- cult to be integrated. Here, we just need to measure the conductances in different strengthes of the SOC to ob- tain the polarization of the incident spin, which is easier to integrate on the chip. It is interesting that in such a simple system, the spin filter, spin polarizer and spin an- alyzer can be achieved by just tuning the magnetic field or the Rashba SOC via the gate [33–36]. II. THE TRANSPORT PROPERTIES IN THE ONE-DIMENSIONAL MODEL To understand the transport properties in a quantum ring, the one-dimensional (1D) model is usually applied. The ring is contacted with the left and the right leads at ϕ=πand 0, respectively. In this work, we suppose that the electron is injected from the left lead, then it travels through the ring in two different paths, one from ϕ=π to 0 clockwise (the upper arm) and the other from ϕ=π to 2πcounterclockwise (the lower arm), as shown in Fig. 1(a). As discussed in the previous work [37], the 1D model works very well when the radius is not too large. The 1D model here, at least, is a good approximation which results in the correct physical pictures. Another approx- imation of neglecting the Zeeman effect is also adopted. In the relatively low magnetic field ( B <3T), the Zee- man coupling is weak and could be neglected. We can also numerically verify that this approximation is appro- priate in low magnetic fields. For simplicity, we first consider only the Rashba SOC being present. If the Zeeman coupling is neglected the energyspectrum of the 1D ring is given by [12, 15, 16, 38, 39]Eµ n=τ/parenleftBig nµ j−ΦAB 2π−Φµ AC 2π/parenrightBig2 wherenµ jis the orbital quantum number, and the index µ= 1,2 represents the spin eigenstates |↑∝an}bracketri}htand|↓∝an}bracketri}ht, andj=±represents the clockwise and counterclockwise electron motions, respec- tively. Also, τ=/planckover2pi12 2m∗r2 0is the energy unit, Φ AB= 2πNis the AB phase with the relative magnetic flux N=eBr2 0 2/planckover2pi1, and Φµ AC= (−1)µ/parenleftBig/radicalbig 1+4β2 1−1/parenrightBig πis the AC phase withβ1=g1m∗r0//planckover2pi1. The corresponding eigenstates are given by Ψµ j(ϕ) = 1√ 2πe−inµ jϕχµ(ϕ), whereχ1(ϕ) =/parenleftbig cosθ1 2,−eiϕsinθ1 2/parenrightbigT andχ2(ϕ) =/parenleftbig sinθ1 2,eiϕcosθ1 2/parenrightbigT, with tanθ1= 2β1[39]. It is clear that the directions of the spin polarization are along (θ1,ϕ) and (π−θ1,π+ϕ) for the two eigenstates respectively. The schematic diagram of the total transport is explic- itly drawn in Fig. 1(a). The incident current can be de- composed into the two eigenstates χ1,2, and the electronis transported by these two channels. The transmission rate is given by (see the Method), Tµ=KΦµ KK′+/bracketleftbig 4k2 0(Φµ−K′)+k2sin2(πk0r0)/bracketrightbig2,(1) whereK= 16k2k2 0sin2(πk0r0), Φµ= cos2ΦAB+Φµ AC 2and K′= cos2(πk0r0). For vanishing magnetic field the AB phase vanishes and the SOC induced energy shift U0is neglected, then Eq. ( 1) agrees with the results obtained in Ref. [12, 15]. If there is a constant potential Uadded at the contact then the magnetic field for Tµ= 0 is not changed while the magnetic field for Tµ= 1 is slightly shifted. Hence, for the spin filter the contact defect is not very important. The numerator of Eq. ( 1) indicates that the trans- mission rate oscillates with the incident energy E, and cos2ΦAB+Φµ AC 2means that Tµoscillates with the increase of the magnetic field Bor the coupling strength of the SOCg1. When the magnetic field vanishes, Φ AB= 0 and T1=T2, resulting in a completely unpolarized transport if the incident spin is unpolarized. However, if both of the AB and the AC phases are taken into consideration in a proper magnetic field the spin can be fully polarized after traversing the ring. If we want an 100% polarized spin current output then the phases must satisfy Φ AB+ Φµ AC=πso that the eigenstate χµis completed filtered out, and only the other eigenstate is left. For a given SOC differ- ent AB phases (different magnetic flux) lead to differ- ent eigenstate filtering. The magnetic flux difference of the two nearest eigenstates filtering is then given by ∆N=1 2(/radicalbig 1+4β2 1−1). This result of constructing a perfect spin filter is consistent with the results of the S- matrix method [28]. If the incident spin is unpolarized, then the spin can be composed of an arbitrary direction ( θ′,ϕ′) and its opposite direction ( π−θ′,π+ϕ′) independently. The two transport channels do not interfere with each other, and the transmission rates can be obtained by projecting the two eigen transmission rates onto the two directions, T(θ′,ϕ′)+=/summationtext µ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftBig χ(θ′,ϕ′)/bracketrightBig† χµ(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Tµ, andT(θ′,ϕ′)−=/summationtext µ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/bracketleftBig χ(π−θ′,π+ϕ′)/bracketrightBig† χµ(0)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Tµ,where χ(θ′,ϕ′)≡/parenleftBig cosθ′ 2,eiϕ′sinθ′ 2/parenrightBigT . The upper index of χ stands for the direction of the spin of the state. The spin polarization of the outcoming current in an arbitrary di- rection can be found to be P(θ′,ϕ′)=/bracketleftbig χ1(0)/bracketrightbig†σ(θ′,ϕ′)χ1(0)Pχ,(2) where the spin matrix along the direction of ( θ′,ϕ′) is σ(θ′,ϕ′)= (σxcosϕ′+σysinϕ′)sinθ′+σzcosθ′,andPχ= (T1−T2)/(T1+T2)isthespinpolarizationinthedirection of the two eigenstates at the contact, ( θ1/2,0). Since3 |P(θ′,ϕ′)| ≤ |Pχ|, the outcoming polarization is always along the direction of the eigenstate χ1orχ2. The transmission rates when the incident spin is un- polarized are well studied. Next we consider the case where the incident spin is polarized in an arbitrary di- rection along ( θ,ϕ). Irrespective of the incident electron is a pure or a mixed state, the transmission rate is always obtained by T(θ,ϕ)=/summationdisplay µ/vextendsingle/vextendsingle/vextendsingle/vextendsingle/parenleftBig χ(θ,ϕ)/parenrightBig† χµ(π)/vextendsingle/vextendsingle/vextendsingle/vextendsingle2 Tµ, =T1cos2/parenleftbiggθin ∆ 2/parenrightbigg +T2sin2/parenleftbiggθin ∆ 2/parenrightbigg ,(3) whereθin ∆is the angle between the direction ( θ,ϕ) and the direction of the spin polarization of χ1(π) which is (θ1,0). It means that the arbitrarily polarized spin is projected to the two conjugate eigensatesofthe ring, and the transmission rate of the spin is the sum of the two eigen channels. Moreover, for the unpolarized incident current, we can decompose it into two conjugate parts, and we get T(θ,ϕ)+T(π−θ,π+ϕ)=T1+T2. In fact, we can define the transmission rate T(θ,ϕ) (θ′,ϕ′)± where the upper index is the polarization of the inci- dent spin and the lower index represents the transmis- sion rate along the direction ( θ′,ϕ′) (for (θ′,ϕ′)+) or (π−θ′,π+ϕ′) (for (θ′,ϕ′)−). If the incident electrons arebeinginjectedonebyoneandissupposedtobeapure state, then the outcoming wave function at the right lead canbefoundas χ(θ,ϕ) out=/summationtext µ[χµ(π)]†χ(θ,ϕ)tµχµ(0).The transmission and the polarization rates are then given by T(θ,ϕ) (θ′,ϕ′)±=/bracketleftBig χ(θ,ϕ) out/bracketrightBig† σ(θ,ϕ)±χ(θ,ϕ) out (4) P(θ,ϕ) (θ′,ϕ′)=/bracketleftBig χ(θ,ϕ) out/bracketrightBig† σ(θ,ϕ)χ(θ,ϕ) out, (5) whereσ(θ,ϕ)±=|(θ,ϕ)±∝an}bracketri}ht∝an}bracketle{t(θ,ϕ)±|is the density matrix of the eigenstate of the matrix σ(θ,ϕ). By analyzing Eq. ( 3), it is easy to obtain that max(T(θ,ϕ)) = max( T1,T2) and min( T(θ,ϕ)) = min(T1,T2). Therefore, the incident spin having maxi- mum and the minimum transmission rates must be par- allel to the polarization directions of the two eigenstates, respectively. The generic spin torquing is given by Eq. ( 5), but the presence of a magnetic field makes the analytical result a bit complicated. Forsimplicity, weconsiderthe magnetic field approaching zero, so that Φ AB→0 andT1=T2. The spin polarizations for an arbitrarily polarized inci- dent current are P(θ,ϕ) x=−sin2θ1cosθ+cos2θ1sinθcosϕ, P(θ,ϕ) y= sinϕsinθ, P(θ,ϕ) z= cos2θ1cosθ+sin2θ1sinθcosϕ.(6) Whenϕ= 0, then P(θ,ϕ) x=−sin(2θ1−θ),P(θ,ϕ) y= 0,P(θ,ϕ) z= cos(2θ1−θ). It means that the incident andoutcoming spins are all in the xOzplane, the spin passes the ring and is torqued a fixed angle in the xOzplane, (θout,ϕout) = (θ−2θ1,0). It can be intuitively under- stood by the spin textures of the eigenstates that there is noycomponent spin at ϕ= 0,πin the ring [37]. The torqued angle is only related to the strength of the SOC. This special case goes back to the result obtained in Ref. [40], and the more special case, P(0,0) z= cos(2θ1) was obtained in the path-integral approach [17]. A se- ries of the ring may be able to tune the spin polarization arbitrarily. If only the Dresselhaus SOC is present, the anal- ysis above is still valid, but some terms need to be changed. The AC phase needs to be replaced by Φµ AC= −(−1)µ/parenleftBig/radicalbig 1+4β2 2/parenrightBig πwhereβ2=g2m∗r0//planckover2pi1. Theeigen- states of the ring also need to be changed to χ1(ϕ) =/parenleftbig cosθ2 2,ie−iϕsinθ2 2/parenrightbigT, χ2(ϕ) =/parenleftbig sinθ2 2,−e−iϕcosθ2 2/parenrightbigT, with tanθ2= 2β2, and the additional potential is U0= −β2 2. All other calculations remain unchanged. When both the SOCs are present then it would be difficult to have analytical results for the transport prob- lem. WethenseekthesolutionsnumericallyintheNEGF method. III. NUMERICAL RESULTS OF THE SPIN AND CHARGE TRANSPORT PROPERTIES IN TWO-DIMENSIONAL MODELS The spin transmission rates Tαand the spin polar- ization rates Pαare important variables characterizing the transport properties. The spin polarization rate Pα is the probability of the spin of the outcoming electron projected to the αaxis.P0is the total polarizationof the outcoming spin. If P0= 1, then the spin of the current at the drain is fully polarized along a certain direction, otherwise the outcoming current contains different com- ponents of the spin at the same time and it is not fully polarized. We here numerically calculate the two rates to explore the transport properties of a more realistic two- dimensional quantum ring contacted by the source and drain on the two ends of a diameter. We then discuss how the spin of the current is polarized and filtered by the quantum ring with the SOCs when the incident elec- tron is spin unpolarized, and compare the 2D numerical results with the analysis in the 1D model. For simplicity and without loss of generality we con- sider the ring on the surface of the InAs semiconductor. We adopt the tight-binding Hamiltonian (details shown in the appendix) to perform the numerical calculations by applying the Green’s functions [6, 41]. The device is indicated in Fig. 1(b), in which the lattice constant is 1 nm [42]. The energy spectrum of the ring without the source and the drain in such a tight-binding model is similar to that of the ring in the parabolic potential cal- culated in Fock-Darwin basis [43], as shown in Fig. 1(c). So the tight-binding model itself is reliable and is a very4 /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013 /uni000000ed/uni00000014/uni00000011/uni00000018 /uni000000ed/uni00000014/uni00000011/uni00000013 /uni000000ed/uni00000013/uni00000011/uni00000018 /uni00000013/uni00000011/uni00000013 /uni00000013/uni00000011/uni00000018 /uni00000014/uni00000011/uni00000013 /uni00000014/uni00000011/uni00000018/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013 Lwr0rw rw Ψµ in Ψµ outΨµ 1 Ψµ 2 /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000018 /uni00000013/uni00000011/uni00000014/uni00000013 /uni00000013/uni00000011/uni00000014/uni00000018 /uni00000013/uni00000011/uni00000015/uni00000013/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013 /uni00000015/uni00000011/uni0000001b /uni00000016/uni00000011/uni00000013 /uni00000016/uni00000011/uni00000015 /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000018 /uni00000013/uni00000011/uni00000018/uni00000013 /uni00000013/uni00000011/uni0000001a/uni00000018 /uni00000014/uni00000011/uni00000013/uni00000013¯hg2= 0nm·meV¯hg1= 20nm ·meVB= 3.0TTz↓Tz↑ /uni00000013/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000011/uni00000015/uni00000018 /uni00000013/uni00000011/uni00000018/uni00000013 /uni00000013/uni00000011/uni0000001a/uni00000018 /uni00000014/uni00000011/uni00000013/uni00000013/uni00000014/uni00000013/uni00000013/uni00000013/uni00000015/uni00000013/uni00000013/uni00000013/uni00000016/uni00000013/uni00000013/uni00000013/uni00000017/uni00000013/uni00000013/uni00000013/uni00000018/uni00000013/uni00000013/uni00000013/uni00000019/uni00000013/uni00000013/uni00000013/uni0000001a/uni00000013/uni00000013/uni00000013(a) ( b) ( c) (d) ( e)B[T] x/r0 T B[T] k[π/a] k[π/a] energy[meV]y/r0energy[meV]energy[meV] FIG. 1: (color online). (a) The schematic transport in a 1D ri ng. (b) The device with r0=15 nm, rw=5 nm, lead (red) width Lw=10 nm and lattice constant a=1 nm. (c) The energy spectrum of tight-biding ring (black) w ith/planckover2pi1g1= 20.0 nm ·meV and g2= 0. (d) The energy spectrum of the lead without the Zeeman cou pling or SOCs. (e) Three figures from left to right: The energy of the input electrons; Around B= 3T, the energy spectrum of the ring within the energy of the i nput electrons, i.e. E <250 meV; The transmission versus the energy of the input elec trons. accurate approximation for the real physical system. Fig.1shows the two-lead transport device and an ex- ample of the transport property of the ring. The lead is 10 nm wide in the ydirection and is semi-infinite along thexaxis. We consider low-energy transport only, and the input electronsare on the lowestenergyband allowed in the lead as shown in Fig. 1(d). In Fig. 1(e), we find that the transmissions are only allowed when the energy of the incident electron is close to the energy levels of the ring. Previous studies have offered the possibilities that the quantumringwith SOCscanactasthespinfilterandpo- larizer. We apply the NEGF method in a full 2D model quantum ring with SOCs, as shown in Fig. 1. More- over, we take all the realistic conditions, Zeeman cou- pling, finite width of the ring and the rotational symme- try breaking, into consideration. In fact, the 1D model still works qualitatively. The spin filtering can be basi- cally explained by the transmission rates Tµin Eq. (1) varying with the competition of the AB and the AC phases in different magnetic fields. In a proper magnetic field and with a proper SOC, one channel can be shielded and the other one is fully survived, so that both T0and P0can be up to 1. Details can be found in the appendix. As discussed in the 1D model, if only the Rashba SOC is present then TyandPywill be suppressed. The direc- tion of the spin polarizer can be tuned by the strength of the Rashba SOC in the plane xOz. If only the Dressel-haus SOC is present, then TxandPxwill be suppressed. The direction of the spin polarizer is then in the plane yOz. If both of the SOCs are present then the situation becomes complex and the spin polarizer can be controled morewidely. However, we find that if the outcoming spin needs to be polarized well, then it is better to keep one SOC dominating the system. The competition of the two SOCs makes the spin more difficult to be polarized, as shown in the appendix. We note that in such a simple device the spin filter- ing and spin polarizer can be realized. The unpolarized spinistransportedthroughthesimplequantumringwith Rashba SOC, and then the outcoming spin is polarized. The directions other than the outcoming polarization are filtered, and the current is spin polarized. Moreover, the RashbaSOCcan be easilytuned, sothat the polarization of the outcoming spin can be easily tuned by a gate. IV. ISOTROPIC ALL-ELECTRIC SPIN ANALYZER Now we would like to consider the case when the in- cident electrons are already fully polarized. Similar to the light polarizer, the ring with the SOCs in fact can be acted as a spin analyzer. If the incident electron is already spin polarized in the direction of ( θin,ϕin) in the spherical coordinate of the spin space, then the transmis-5 sion rate is given by Tα(E) = Tr/bracketleftbig σαΓ(E)G(E)σ(θin,ϕin)+Γ(E)G†(E)/bracketrightbig ,(7) whereσ(θin,ϕin)+is the density matrix of the polarized state,the Green’sfunction Gandthe broadeningfunction Γ can be found in Ref. [41]. We note that the outcoming spin is still spin polarized, but is torqued by an angle given by Eq. ( 6). Using Eq. ( 7), we can clearly decompose the unpo- larized incident ψinin the basis of σz. In the density matrix form, |ψin∝an}bracketri}ht∝an}bracketle{tψin|=/parenleftBig |ψin z↑∝an}bracketri}ht∝an}bracketle{tψin z↑|+|ψin z↓∝an}bracketri}ht∝an}bracketle{tψin z↓|/parenrightBig /2. The incident wave function can be divided into two parts with opposite spin polarization, and each part provides a transport channel. The total transmission rate is the sum of the transmission rates of the two channels, since there is no coherence between the two channels. In the appendix, we can clearly see how the spin textures and the current evolve in the ring for different channels in which the spin is decomposed along + zor−z. A. Transmission rates for the polarized incident spin current WesupposethattheringisonlycoupledbytheRashba spin-orbitinteraction /planckover2pi1g1= 20nm·meVandtheincident current is already spin polarized. The polarization direc- tion of the incident spin ( θin,ϕin) varies and the charge transmission rate is indicated in Fig. 2. Fig.2(a) shows the case when T1= 1,T2= 0 andP0= 1. The one- dimensional analytical model predicts that the outcom- ingpolarizationisalongthe eigenstate χ1(0), (0.161π,π), and theχ2channel is closed T2= 0. In the two- dimensional model, it indicates that the outcoming po- larization is along ( θout,ϕout) = (0.214π,0.984π) always, where the channel of χ1is free to transport and the other channel (χ2) is completely closed. It means that the po- larization angle of the eigenstate of the 2D ring at ϕ= 0 is (0.214π,0.984π). This difference comes from the Zee- man effect and the finite width. It implies that these effects can also generate a finite Pyin the ring with the Rashba SOC only, which is significantly differenti from the 1D model. Moreover,the incident polarizationwith the maximum transmission rate among all the directions in the spin spaceisalsoalongthe eigenstate χ1 2D(π), (θmax in,ϕmax in) = (0.214π,0.016π). We note that ( θmax in,ϕmax in) and (θout,ϕout) are mirror symmetry to the zaxis. The transmission of the spin-polarized input current in arbitrary direction is determined by the projection of (θin,ϕin) to (θmax in,ϕmax in), since the channel of χ2 2Dis closed. For a more general case, both transport channels of the eigensates allow electrons to pass ( Tmax 0,Tmin 0> 0), as shown in Figs. 2(b) and (c), the maximum trans- mission rate Tmax 0= 0.962 corresponds to the incident polarization ( θmax in,ϕmax in) = (0.792π,0.963π), and its output polarization is ( θmax out,ϕmax out) = (0.792π,0.037π).For the minimum transmission rate, we have Tmin 0= 0.591, (θmin in,ϕmin in) = (0.208π,1.963π), (θmin out,ϕmin out) = (0.792π,1.037π). In this case, ( θout,ϕout) is no longer a fixed angle along χ1 2D, but changes with the angle of incidence (θin,ϕin). Interestingly, we also find numerically that the gen- eral relation between the incident angle and the charge transmission rates T0is given by T0=Tmax 0cos2/parenleftbiggθ∆ 2/parenrightbigg +Tmin 0sin2/parenleftbiggθ∆ 2/parenrightbigg ,(8) whereθ∆is the angle between the incident spin po- larization ( θin,ϕin) and the special angle ( θmax in,ϕmax in), whether the outcoming spin is polarized or not. This equation is exactly the same as Eq. ( 3) that we found for the 1D model. The only difference is that in Eq. ( 3),T1,2 correspond to the transmission rate of the eigenstates of theringχ1,2. However,inthe2Dringthe Tmax 0andTmin 0 correspond to the eigenstates of the 2D ring which are a little different from those of the 1D ring. The arbitrary spin is projected to the angles of the eigenstates of the ring, (θmax in,ϕmax in) and(π−θmax in,ϕmax in+π). This projec- tion then gives directly the transmission rate in Eqs. ( 3) and (8). The Zeeman coupling, circle symmetry break- ing, and finite width only change the spin-polarization direction of eigenstates χµ, the properties predicted by the 1D analytical model are retained, which implies that we could use the quantum ring to design the integrable spin devices. B. Design of a spin analyzer The ring acts as a spin torque: it allows the electron to pass but the spin polarization must be torqued. If the ring is coupled by the Dresselhaus spin-orbit interaction only, the similar spin torque occurs. The only difference is the outcoming angle of the spin, which is the mirror symmetry of the incident angle (for the maximum trans- mission rate only) to the plane xOz. The direction de- pendent transmission rate is also given by Eqs. ( 3) and (8). According to the property of the angle dependent transmission rate in Eq. ( 8), we can realize a spin ana- lyzerintheringdevice. Beforethemeasurement,weneed to knowTmax 0andTmin 0in a given magnetic field. They can be determined by the measurement of the transmis- sion rates of the known spin polarized incidents. We use three spin polarized incident with Px,y,z= 1, re- spectively, and one spin unpolarized incident to identify the following parameters: Tmax 0,Tmin 0,θmax in,ϕmax in. The transmission rate for the unpolarized incident is marked asT, and we have already known T=Tmax 0+Tmin 0. The three transmission ratesfor different spin polarizationin- cident are marked as T(x),T(y) andT(z). Applying Eq. (8) toT(x),T(y) andT(z), we find another three equa- tions. So four equations in all can be solved and the four variablesTmax 0,Tmin 0,θmax in,ϕmax inare found.6 /uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c /uni00000010/uni00000014/uni00000011/uni00000013/uni00000013/uni0000028c/uni00000010/uni00000013/uni00000011/uni00000019/uni0000001a/uni0000028c/uni00000010/uni00000013/uni00000011/uni00000016/uni00000016/uni0000028c/uni00000013/uni00000011/uni00000013/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000016/uni00000016/uni0000028c/uni00000013/uni00000011/uni00000019/uni0000001a/uni0000028c/uni00000014/uni00000011/uni00000013/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000013/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000014/uni0000001a/uni0000028c/uni00000013/uni00000011/uni00000016/uni00000016/uni0000028c/uni00000013/uni00000011/uni00000018/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000019/uni0000001a/uni0000028c/uni00000013/uni00000011/uni0000001b/uni00000016/uni0000028c/uni00000014/uni00000011/uni00000013/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000001c/uni00000013/uni00000011/uni00000019/uni00000018/uni00000013/uni00000011/uni0000001a/uni00000014/uni00000013/uni00000011/uni0000001a/uni0000001b/uni00000013/uni00000011/uni0000001b/uni00000017/uni00000013/uni00000011/uni0000001c/uni00000013/uni00000013/uni00000011/uni0000001c/uni00000019 θin θin θinϕout θout T0 (d) (c) (b)B= 1.500T ¯hg1= 33.5nm·meV ¯hg2= 0nm ·meV /uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000018/uni0000028c/uni00000014/uni00000011/uni00000013/uni0000028c/uni00000014/uni00000011/uni00000018/uni0000028c/uni00000015/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000014/uni0000001a/uni00000013/uni00000011/uni00000016/uni00000016/uni00000013/uni00000011/uni00000018/uni00000013/uni00000013/uni00000011/uni00000019/uni0000001a/uni00000013/uni00000011/uni0000001b/uni00000016/uni00000014/uni00000011/uni00000013/uni00000013 θinT0ϕin(a)B= 2.408T ¯hg1= 33.5nm·meV ¯hg2= 0nm ·meV FIG. 2: (color online). The energy of the incident electron i sEin= 198.5 meV. (a) The total transmission rate T0for different angles of the polarization of the incident electron ( θin,ϕin) when/planckover2pi1g1= 33.5 nm·meV and g2= 0 atB= 2.408 T. (b) The total transmission rate T0and (c)-(d) the angles of the polarization of the outgoing el ectron (θout,ϕout) for different polarization of the incident electron ( θin,ϕin) atB= 1.5 T. The scenario to analyze the spin polarization by de- tecting the charge transmission rates for different SOCs then can be established. The scheme is described as fol- lows: First, the ring is coupled by the two spin-orbit interac- tions (g1,g2). The transmission rates are shown in colors in Fig.3(a).Tmax 0and the corresponding incident po- larization ( θmax in,ϕmax in)1is already known, as the blue vector in Fig. 3(d). Once we measure the transmission rateT0, we can find the angle θ∆1between the incident polarization angle ( θin,ϕin) and (θmax in,ϕmax in)1by apply- ing Eq. ( 8). However, the possible polarization direction in the three-dimensional space of the spin can be along any element of the cone, shown as the blue circle in Fig. 3(d). we project the angles of the elements of the cone onto the (θ,ϕ) plane to obtain the solid line in Fig. 3(a). Second, we tune the Rashba SOC and the transmis- sion rates are shown in colors in Figs. 3(b). The angle of the maximum transmission rate, ( θmax in,ϕmax in)2, is repre- sented by the green vector in Fig. 3(d). Then measure the transmission rate to obtain the angle θ∆2to find the second cone. The spin polarization is possibly located in the solid green line in Fig. 3(b), where the dashed line represents the first measurement. So the incident polar- ization must be at one of the intersection points of the two lines. Thirdly, we tune the Rashba SOC again to find the third line which is shown in Fig. 3(c). The three lines must intersect at the same point which is the unique di- rection of the polarization of the incident spin. The in- tersection point can also be seen in the spin space in Fig. 3(d). Here the external magnetic field is fixed and can be integrated on the chip. In fact, the three curves in the (θ,ϕ) plane always intersect at the same point for anymagnetic field. A proper magnetic field results in better discrimination. We note that the spin analyzer could be also achieved by a single SOC. In the 1D model, a single SOC only twists the spin in one direction ( xory). The incident angle can not be uniquely determined, there are always two intersection points, no matter how many times we tune the strength of the SOC. However, in the real 2D ring, the spin can be twisted more widely. The unique intersection can appear. We show the numerical results in Fig.3(e) where only the Rashba SOC exists and the Dresselhaus SOC is absent. It is clear that after three measurements with different strengthes of the Rashba SOC, all the cones intersect at the unique intersection and the other intersection has been lifted. So the spin polarization can also be identified more easily. V. CONCLUSION In summary, we present a detailed study of the trans- port properties of the device in which a quantum ring is in contact with two leads at the ends of one diameter. When the SOC is introduced, different phases are added in the matter wave of the electron with different spins. So that the transmission rates for different spins are no longer degenerate. By detailed analytical and numerical studies, we find that in a simple quantum ring device, the spin unpolarized current can be spin polarized parallel to the eigenstates of the ring for appropriate SOC and the magnetic field. The direction of the polarization can be tuned easily by the SOC and the magnetic field as well. This simple device is therefore proposed to be a spin po- larizer. Moreover, similar to the light polarizer/analyzer, it can also be designed as an omnidirectional all-electric7 (d) xyz /uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000013/uni0000028c/uni00000014/uni00000011/uni00000013/uni0000028c/uni00000015/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni0000001a/uni00000015/uni00000013/uni00000011/uni0000001a/uni00000018/uni00000013/uni00000011/uni0000001a/uni0000001c/uni00000013/uni00000011/uni0000001b/uni00000016/uni00000013/uni00000011/uni0000001b/uni00000019/uni00000013/uni00000011/uni0000001c/uni00000013/uni00000013/uni00000011/uni0000001c/uni00000017 ¯hg1= 40nm ·meVT0 θinϕin(c)B= 1.8T ¯hg2= 20nm ·meV /uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000013/uni0000028c/uni00000014/uni00000011/uni00000013/uni0000028c/uni00000015/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni0000001b/uni00000016/uni00000013/uni00000011/uni0000001b/uni00000019/uni00000013/uni00000011/uni0000001b/uni0000001b/uni00000013/uni00000011/uni0000001c/uni00000014/uni00000013/uni00000011/uni0000001c/uni00000016/uni00000013/uni00000011/uni0000001c/uni00000019/uni00000013/uni00000011/uni0000001c/uni0000001b ¯hg1= 10nm ·meVT0 θinϕin(b)B= 1.8T ¯hg2= 20nm ·meV /uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000013/uni0000028c/uni00000014/uni00000011/uni00000013/uni0000028c/uni00000015/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni0000001b/uni00000014/uni00000013/uni00000011/uni0000001b/uni00000017/uni00000013/uni00000011/uni0000001b/uni0000001a/uni00000013/uni00000011/uni0000001c/uni00000013/uni00000013/uni00000011/uni0000001c/uni00000016/uni00000013/uni00000011/uni0000001c/uni00000019/uni00000013/uni00000011/uni0000001c/uni0000001c ¯hg1= 0nm·meVT0 θinϕin(a)B= 1.8T ¯hg2= 20nm ·meV (f) xyz /uni00000013/uni00000011/uni00000013/uni0000028c /uni00000013/uni00000011/uni00000018/uni0000028c /uni00000014/uni00000011/uni00000013/uni0000028c/uni00000013/uni00000011/uni00000013/uni0000028c/uni00000014/uni00000011/uni00000013/uni0000028c/uni00000015/uni00000011/uni00000013/uni0000028c ¯hg1= 10nm ·meV ¯hg1= 25nm ·meV ¯hg1= 35nm ·meV θinϕin(e)B= 1.8T ¯hg2= 0nm·meV FIG. 3: (color online). The energy of the incident electron i sEin= 198.5 meV, and the background magnetic field B= 1.8 T. (a)-(d) The colors represent the transmission rates for t he SOCs /planckover2pi1(g1,g2) = (0,20),(10,20),(40,20) nm·meV, respectively. The solid curve in (a) - (d) represents the possible angles of the incident polarization after the first, second and the thi rd measurements, respectively. The dash lines in (b) and (c) re presents the previous measurements. (d) The possible polar ization is a cone for each measurement in the spin space. The blue, gre en and red arrows stand for the vector ( θmax in,ϕmax in)1,2,3, respectively. The intersection of the three cones is the pur ple vector representing the incident polarization. (e)-(f ) The intersections of the possible angles in three Rashba SOCs /planckover2pi1g1= 10,25,35 nm·meV, respectively, when the Dresselhaus is absent. spin analyzer by simply measuring the transmission rate of the polarized incident via Eq. ( 8). These findings pave the way to control the system in spintronics and may be useful in quantum computation. It also contributes an easy and controllable proposal to the design of the high- performance all-electric transport device. VI. ACKNOWLEDGEMENT This workis supported by the NSF-China under Grant No. 11804396. J.S. is supported by the NSF-China un- der Grant No. 11804397. A.G. acknowledges financial support by the NSF-China under Grant No. 11874428, 11504066, and the Innovation-Driven Project of Central South University (CSU) (Grant No. 2018CX044). F.O. acknowledges financial support by the NSF-China under Grant No. 51272291, the Distinguished Young Scholar Foundation of Hunan Province (Grant No. 2015JJ1020), and the CSU Research Fund for Sheng-hua scholars (Grant No. 502033019). T.C. would like to thank Jun- saku Nitta for helpful discussion, in particular, for point- ing out Ref. [28].Appendix A: Method and Formalism Here we explicitly derive the transmission rate in Eq. (1). We consider electrons as plane waves in the two leads with momentum /planckover2pi1kand energy E=/planckover2pi12k2 2m∗. Note that there is an additional potential U0=−β2 1in- duced by the Rashba SOC [38], so that in the two arms, E=/planckover2pi12k2 0 2m∗+U0. The wave vector in the two arms are [12] kµ j=k0+j/parenleftBig ΦAB 2πr0+Φµ AC 2πr0/parenrightBig .The incident current can be decomposed into the two eigenstates χ1,2, and the elec- tron is transported by this two channels. We note that in this case the two channels are independent and there is no interference between the two eigenstates. If the incident spin is polarized along χµthen the outcoming polarization is still along χµ. If the incident current is decomposed into other two orthogonal states, then the interference is difficult to deal with. The wave function at the left lead contains the inci- dent and the reflection, Ψµ in. The wave function of the upper arm also contains two parts, the clockwise and the anticlockwise movements, Ψµ 1. In the same manner, the wave function of the lower arm is Ψµ 2. The output wave function is marked Ψµ out. All these wave functions are8 given by Ψµ in=/parenleftbig eikx+rµe−ikx/parenrightbig χµ(π),(S1) Ψµ 1=/summationdisplay jCjejikµ jxχµ(φ), (S2) Ψµ 2=/summationdisplay jDjejikµ −jxχµ(φ), (S3) Ψµ out=tµeikxχµ(0), (S4) whererµis the reflection rate, C,Dare the parameters which can be determined by the continuous condition, andtµis the variable characterizing the transport prop- erties of the device. The transmission rate is thus given byTµ=|tµ|2. By applying the Griffith boundary condi- tions [15, 19–21, 23, 24, 26], the wave functions and the currents must be continuous at the two leads ( x=±r0 orθ= 0,π), we obtain six equations to solve the six vari- ables. Among them, the most wanted transmission rate can be solved, Tµ=KΦµ KK′+/bracketleftbig 4k2 0(Φµ−K′)+k2sin2(πk0r0)/bracketrightbig2,(S5) which is shown as Eq. (1). In order to calculate the transport properties numeri- cally, it is convenient to discretize the continuous Hamil- tonian. We discretize Hon the sites of a square lattice with the lattice constant ato obtain the tight binding Hamiltonian. It is obtained by calculating the matrix el- ements in the basis of position. The tight binding Hamil- tonian is given by H=/summationdisplay i/parenleftbigg Vi+4t+∆ 2σz/parenrightbigg c† ici−/summationdisplay /angbracketlefti,j/angbracketright(t+sij)c† icjeiθij, (S6) t=/planckover2pi12 2m∗a2, (S7) sij=−i/planckover2pi1g1 2a2/parenleftbig σx∆y−σy∆x/parenrightbig −i/planckover2pi1g2 2a2/parenleftbig σy∆y−σx∆x/parenrightbig , (S8) θij=e /planckover2pi1(Axi∆x+Ayi∆y), (S9) whereiruns over all sites, ∝an}bracketle{ti,j∝an}bracketri}htrepresents the nearest neighbouring hopping only, xiandyiare thexandy coordinates of site i, and ∆xij=xj−xi, ∆yij=yj−yi. For convenience, we apply a hard-wall potential instead of the parabolic potential, Vi=/braceleftBigg 0|ri−r0|/lessorequalslantrw ∞ |ri−r0|>rw, (S10) whereri=/radicalbig x2 i+y2 iand the width of the ring is rw. We connect two parallel leads to the ring, then the transmis- sion properties can be obtained by using the nonequilib- rium Green’s function (NEGF).It is worthwhile to note that the continuous model and the tight binding model are compatible and all the ob- servable quantities in these two models are almost equal (the small errors vanish when a→0). Moreover, the en- ergy spectrum has no essential difference in a parabolic potentialfromthatinahard-wallpotential, if rwmatches the confinement /planckover2pi1ω. Then we consider the transport properties in such a lattice model with tight-binding Hamiltonian. The spin transmission rate Tof the elec- tron transporting from the left lead to the right lead is defined by using the NEGF method [6, 41], Tα(E) = Tr{σα[ΓR(E)GRL(E)ΓL(E)G† LR(E)]},(S11) whereα∈ {x,y,z},σx,y,zare the Pauli matrices and σ0 is the unit matrix. The Green’s function is defined by the projection of the full Green’s function [41], GRL=PRGPL, (S12) G(E) = (E−H−ΣR−ΣL)−1,(S13) wherePR,PLare the projection operators to the right and the left leads, Σ R,ΣLare the self-energy of the right and the left leads, respectively. The broadening function is defined by Γ j=i/bracketleftBig Σj−Σ† j/bracketrightBig . Tα(E) is the transmission rate of the ( α∈ {x,y,z}) component of the spin or the total charge transmission (α= 0) while the energy of the incident electron is E. Then the spin polarization rate Pis defined as: Pα=Tα T0×100%,α∈ {x,y,z} (S14) andP0=/radicalBig T2x+T2y+T2z/T0=/radicalBig P2x+P2y+P2z.P0 represents the spin polarization of the outcoming elec- tron. IfP0= 1, then the spin is fully polarized. If P0= 0, the spin is fully unpolarized. By diagonalizing the tight-binding Hamiltonian in Eq. (S6), we can have the value of the wave functions at each site,ψ(ri), which is a two component spinor. The phys- ical quantities can then be obtained. The spin fields are calculated by σα(ri) =ψ†(ri)σαψ(ri), (S15) and the density is given by n(ri) =ψ†(ri)ψ(ri). The average value of the observable quantity is thus given by ∝an}bracketle{tA∝an}bracketri}ht=/summationtext iψ†(ri)Aψ†(ri)∆x∆y. The in-plane field can be described by the vector field σ(r) = (σx(r),σy(r)). The current operators can be derived by jµ=−δH δA, so that the on-site current densities are given by jx(ri) =e 2m∗/bracketleftBig ψ†(ri)Pxψ(ri)+(Pxψ(ri))†ψ(ri)/bracketrightBig −eψ†(ri)(g1σy+g2σx)ψ(ri), (S16) jy(ri) =e 2m∗/bracketleftBig ψ†(ri)Pyψ(ri)+(Pyψ(ri))†ψ(ri)/bracketrightBig +eψ†(ri)(g1σx+g2σy)ψ(ri). (S17)9 /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001b/uni00000014/uni00000011/uni00000013(b) ¯hg1= 20nm ·meV ¯hg2= 0nm·meV E= 198.5meVBmaxT B′ maxT BminT B′ minT /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001b/uni00000014/uni00000011/uni00000013(a) ¯hg1= 0nm·meV ¯hg2= 0nm·meV E= 198.5meV Tz↑Tz↓BmaxT BminT /uni00000013 /uni00000015 /uni00000017 /uni00000019 /uni0000001b /uni00000014/uni00000013/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000015/uni00000013/uni00000011/uni00000017/uni00000013/uni00000011/uni00000019/uni00000013/uni00000011/uni0000001b/uni00000014/uni00000011/uni00000013(c) ¯hg1= 0nm·meV ¯hg2= 20nm ·meV E= 198.5meV /uni00000014/uni00000013/uni00000013 /uni00000014/uni00000018/uni00000013 /uni00000015/uni00000013/uni00000013 /uni00000015/uni00000018/uni00000013/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013(e) ¯hg1= 0nm·meV ¯hg2= 20nm ·meV B= 2.67T /uni00000014/uni00000013/uni00000013 /uni00000014/uni00000018/uni00000013 /uni00000015/uni00000013/uni00000013 /uni00000015/uni00000018/uni00000013/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013TxTyTzT0 (d) ¯hg1= 20nm ·meV ¯hg2= 0nm·meV B= 2.67T∆BmaxT=B′ maxT−BmaxT ∆BminT=BminT−B′ minT ∆BT=B′ minT−B′ maxTB[T] B[T] B[T] E[meV]TT E[meV]T FIG. S1: (color online). Tz↑andTz↓(a) without SOC, (b) with the Rashba SOC only, and (c) with the Dresselhauss SOC only. Tαα∈(x,y,z) (d) with the Rashba SOC only and (e) with the Dresselhauss SOC only. The current is contributed by three parts, jα(ri)≡jz↑,α(r)+jz↓,α(r)+jSOC,α(r),(S18) where jz↑,α(ri) =e 2m∗/bracketleftbig ψ∗ ↑Pαψ↑+(Pαψ↑)∗ψ↑/bracketrightbig ,(S19) jz↓,α(ri) =e 2m∗/bracketleftbig ψ∗ ↓Pαψ↓+(Pαψ↓)∗ψ↓/bracketrightbig ,(S20) jSOC,x(ri) =−eψ†(g1σy+g2σx)ψ, (S21) jSOC,y(ri) =eψ†(g1σx+g2σy)ψ. (S22) The on-site wave function spinor is ψ=/parenleftbigψ↑ψ↓/parenrightbigT, and ↑,↓are related to the eigenstates of the spin operator σz. Appendix B: Spin transmissions in different SOCs – Spin filtering and spin polarizer Westudyindetailshowthetransmissionrateisrelated to the magnetic field and the SOCs. We suppose that the input electrons are spin unpolarized. If there is no SOC, the transported electrons are spin unpolarized as well, i.e.,Tz↑=Tz↓forg= 0. However, the Zeeman coupling makes the transmission rate different, especially in a strong magnetic field, as shown in Fig. S1(a). Since the minimum transmission rates for spin up and down are all located at the same magnetic field, it would bedifficult to suppress one spin to zero and keep the other spin finite. If the SOCs are introduced into the system, we find that the transmission rate curves for spin down and spin up arewell separated. Ifonly the RashbaSOC is present, the curve of Tz↓is shifted left and the curve of Tz↑is shifted right as shown in Fig. S1(b). If only the Dressel- haus SOC exists, the shift of the curves is just opposite to that in a Rashba ring, as shown in Fig. S1(c). If there is no magnetic field, the electron transports through the upper arm and the lower arm with the same phase added ( T1=T2), so that the transmission rates for different spins depend on the ring itself and are equal, as showninFigs. S1(b)and(c). However,infinitemagnetic fields the time-reversal symmetry is broken, the spin de- generacy in the ring will be lifted more in the presence of either Rashba or Dresselhas SOC strongly. Such a com- bination of the magnetic field and the SOCs can lead to significant spin filtering effect. In the numerical curves (Figs. S1(b) and (c)), the spin filtering appears periodically in a magnetic field, since the term Φ AB+ΦACin transmission rate Eq. (1) only depends on the magnetic field. For instance, if only the Rashba SOC is present and the energy of the incident electron is Ein= 198.5 meV in Fig. S1(b), the lowest magnetic field where the spin down is suppressed is at B= 2.67T, which can be further lowered by increasing the radius of the ring (at the same magnetic flux). In this case,Tz↓→0 andTz↑is finite, so that the output electrons are almost polarized to spin up, Pz→1. For different energies of the input electrons, the transmission rates are shown in Figs. S1(d) and (e). The negative Ti representsthe icomponentofthe outputspinispolarized in the negative direction of the iaxis. In fact, according toFig.S1(d)andtheanalysisofthe1Dmodel, theoutput spin is polarized along χ1,2between the zand−xaxis, sinceTy≈0 andTx,zare finite. We now compare the transmission curves of the ring without SOC(Fig. S1(a)) andthe ringwith RashbaSOC (Fig.S1(b)). Thefirstmaximumratefor Tz↑isatBmaxT in the ring without SOC. After the Rashba SOC is set in, both of the Tz↑andTz↓transmission rates are shifted. Wesupposethatthe firstmaximumvalueof Tz↑isshifted toB′ maxT. In the same manner, the first minimum rate in the ring without SOC is at BminT= 2.83T, while the first minimum rate Tz↓is shifted to B′ minT. We define the parameters ∆ BmaxT=B′ maxT−BmaxT, ∆BminT= BminT−B′ minT, and ∆BT=B′ minT−B′ maxTto study how the Rashba SOC shifts the transmission rate curve and changes the polarization of the spin. We show that in Fig. S2(a) ∆BTdecreases with the increase of the Rashba SOC g1, due to the change of the AC phase Φµ AC, just as predicted in the 1D model. When /planckover2pi1g1= 33.5 nm·meV, ∆BT= 0, which means that the transmission rate of spin down is suppressed to minimum and the transmission rate of spin up is maximum. Inter- estingly, at this point the total charge transmission rate is exactly 1 as shown in Fig. S2(b). Hence, χ1is com-10 /uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013(b) Tx Ty Tz T0 /uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013(a) ∆BminT ∆BmaxT ∆BT /uni00000014/uni0000001a/uni00000018 /uni00000015/uni00000013/uni00000013 /uni00000015/uni00000015/uni00000018/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013(e) Px Py Pz P0 /uni00000014/uni0000001a/uni00000018 /uni00000015/uni00000013/uni00000013 /uni00000015/uni00000015/uni00000018/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013(d) Tx TyTz T0 /uni00000013 /uni00000015/uni00000013 /uni00000017/uni00000013 /uni00000019/uni00000013/uni000000ed/uni00000014/uni00000011/uni00000013/uni000000ed/uni00000013/uni00000011/uni00000018/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013 Px Py Pz P0(c)¯hg1[nm·meV] ¯hg1[nm·meV] E[meV] E[meV]∆B[T] ¯hg1[nm·meV] T TorPP FIG. S2: (color online). (a) Magnetic field shifts ∆ Binduced by the Rashba SOC. (b) Spin transmission rates Tand (c) spin polarization Pin different g1with incident electron en- ergy 198.5 meV, in the magnetic field where the minimum Tz↓ is, so that P0= 1 always. (d) The spin transmission rate T and (e) the spin polarization Pat/planckover2pi1g1= 33.5 nm ·meV and B=2.41 T (corresponding to ∆ BT= 0) with energy. pletely suppressed, and χ2passes the ring freely. Mean- while, theycomponent of the spin is almost suppressed, Ty≈0, and both |Tx|and|Px|increase, as shown in Figs.S2(b) and (c). We are then able to control the direction of the spin polarization by tuning the Rashba SOC. The tunable spin polarizer is thereby established. In Fig.S2(d), we show how the spin polarization and the transmission rate vary with the energy of the incident electron. Basically, Tyis close to 0 and Pyis also always very small. It is nonzero comparing with the 1D model, due to the Zeeman coupling and the width of the ring. In Figs.S2(d) and (e) we show that if the energies of the incident electrons are in the region [185 ,205] meV, the spin transmission and polarization are stable. So that the outcoming current which is obtained by integral of the transmission rate Tover this region is almost fully spin polarized. In order to exclude the unwanted trans- mission below 185meV, we can apply a gate to lift the whole energy band of the lead. Inourringdevice, theRashbaSOCtilts the spintothe xaxis, while the Dresselhaus SOC flip the spin towards theydirection. It can also be understood simply as fol- lows. When the magnetic field is absent, the effective vector potential induced by the SOCs is ASOC x=−m e/planckover2pi1(g1σy+g2σx), (S1) ASOC y=m e/planckover2pi1(g1σx+g2σy). (S2) Suppose the incident wave function is spin polarized, ψin += (1 0)T, then the outcoming wave function influ- enced by the SOC is given by ψout∝e−iAx·2(r0+rw)ψin,since the coordinate difference in the ydirection is zero. If there is only Rashba existing, ψout R∝(1+iγg1σy)/parenleftbigg 1 0/parenrightbigg =/parenleftbigg 1 −γ/parenrightbigg ,(S3) whereγ= 2(r0+rw)m e/planckover2pi1g1>0. So we have ∝an}bracketle{tσx∝an}bracketri}ht= −2γ <0 and∝an}bracketle{tσy∝an}bracketri}ht= 0. The spin is torqued from the +zdirection to −x. If the incident electron is spin down,ψin −= (0 1)T, thenψout R= (γ1)T, and then ∝an}bracketle{tσx∝an}bracketri}ht= 2γ >0 and∝an}bracketle{tσy∝an}bracketri}ht= 0. In Fig. S2(d), however spin down is suppressed in the transport, and the spin upψin +is flipped to the −xaxis. On the other hand, if only the Dresselhaus SOC is present, we can do the same calculation. For ψin +,ψout D= (1iγ)T, so that ∝an}bracketle{tσx∝an}bracketri}ht= 0 and∝an}bracketle{tσy∝an}bracketri}ht= 2γ >0. Forψin −,ψout D= (iγ1)T, so that ∝an}bracketle{tσx∝an}bracketri}ht= 0 and ∝an}bracketle{tσy∝an}bracketri}ht=−2γ <0. In Fig. S2(e), the spin up is suppressed, while the spin down ψin −is flipped to the −yaxis in the transport. The analysis agrees with the numerical results perfectly. As drawn in Fig. S3, we show the relation between the spin transmission rates and the spin polarizations for different SOCs. If only the Rashba SOC is existing, thenTyandPywill be suppressed shown in Fig. S3(a). The direction of the spin polarizer can be tuned by the strength of the Rashba SOC in the plane xOz. If only the Dresselhaus SOC is present, then TxandPxwill be suppressedshowninFig. S3(b). Thedirectionofthespin polarizer is then in the plane yOz. If both of the SOCs are present, then the situation becomes complicated and spin polarizer can be controled more widely, as shown in Fig.S3(c). However, we find that if the outcoming spin needs to be polarized well, then it is better to keep one SOC dominating the system. The competition of the two SOCs makes the spin more difficult to be polarized. Appendix C: Spin textures and current in the transport The incident spin is supposed to be unpolarized, so that the wave function of the incident electrons ψin can be decomposed to two parts in any direction of the spin polarization. Without the loss of generality, we decompose the incident electron in the basis of σz, |ψin z↑|2=|ψin z↓|2. The spins of the two parts are indepen- dently polarized along zor−zdirection, respectively. For each part of the incident electron, it contributes one transmission channel in the transport. Then we can fig- ure out which channel plays more important role in the transport. The wave function of the incident electron is supposed to be the wave function of the lowest band of the lead. By employing Eq. (7) we can obtain the wave function in the ring by the Green’s function method, ψring=GτLψin z↑(↓), (S1) whereτLis the coupling matrix between the incident (left) lead and the ring[41]. We againemploy the current11 /uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013 /uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019 /uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019 /uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019 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 /uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000016/uni00000016/uni00000013/uni00000011/uni00000019/uni0000001a/uni00000014/uni00000011/uni00000013/uni00000013/uni00000014/uni00000011/uni00000016/uni00000016/uni00000014/uni00000011/uni00000019/uni0000001a/uni00000015/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013 /uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019 /uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019 /uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019 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 /uni00000013/uni00000014/uni00000015/uni00000016/uni00000017/uni00000018/uni00000019/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000016/uni00000016/uni00000013/uni00000011/uni00000019/uni0000001a/uni00000014/uni00000011/uni00000013/uni00000013/uni00000014/uni00000011/uni00000016/uni00000016/uni00000014/uni00000011/uni00000019/uni0000001a/uni00000015/uni00000011/uni00000013/uni00000013 /uni00000013/uni00000017/uni0000001b/uni00000014/uni00000015/uni00000014/uni00000019/uni00000015/uni00000013/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013 /uni00000013/uni00000017/uni0000001b/uni00000014/uni00000015/uni00000014/uni00000019/uni00000015/uni00000013 /uni00000013/uni00000017/uni0000001b/uni00000014/uni00000015/uni00000014/uni00000019/uni00000015/uni00000013 /uni00000013/uni00000017/uni0000001b/uni00000014/uni00000015/uni00000014/uni00000019/uni00000015/uni00000013 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 /uni00000013 /uni00000017 /uni0000001b/uni00000014/uni00000015 /uni00000014/uni00000019 /uni00000015/uni00000013/uni0000001a/uni00000018/uni00000014/uni00000013/uni00000013/uni00000014/uni00000015/uni00000018/uni00000014/uni00000018/uni00000013/uni00000014/uni0000001a/uni00000018/uni00000015/uni00000013/uni00000013/uni00000015/uni00000015/uni00000018/uni00000015/uni00000018/uni00000013 /uni00000013/uni00000011/uni00000013/uni00000013/uni00000013/uni00000011/uni00000014/uni00000019/uni00000013/uni00000011/uni00000016/uni00000015/uni00000013/uni00000011/uni00000017/uni0000001a/uni00000013/uni00000011/uni00000019/uni00000016/uni00000013/uni00000011/uni0000001a/uni0000001c/uni00000013/uni00000011/uni0000001c/uni00000018¯hg1= 20nm ·meV,¯hg2= 0nm·meV ¯hg1= 0nm·meV,¯hg2= 20nm ·meV ¯hg2= 5nm·meV,B= 3.0T(a) (b) (c)P0 Pz Py Px T0 P0 Pz Py Px T0B[T] B[T] B[T] B[T] B[T0] ¯hg1[nm·meV] ¯hg1[nm·meV] ¯hg1[nm·meV] ¯hg1[nm·meV] ¯hg1[nm·meV]energy[meV] energy[meV] energy[meV] FIG. S3: (color online). The transmission Tand spin polarizability Pwith (a) Rashba SOC only, (b) Dresselhauss SOC only, and (d) fixed Dresselhauss SOC but tunable Rashba SOC. densities jz↑,jz↓andjSOCdefined in Eqs. ( S19) to (S22), whereψneeds to be replacedby ψring. We can define the transmission density tα(r) proportional to the current, tα(r) =−/planckover2pi1 2e·meVjα(r). (S2) The transmission rate is thus obtained by Tα=/summationtext itα(ri), whereiincludes all the sites between the lead and the ring. For simplicity, we consider only the Rashba SOC in two different cases: (i) /planckover2pi1g1= 20 nm ·meV atB= 2.76T, the electron of the incident energy Ein= 198.5 meV has the transmission rate T0= 0.372; (ii) /planckover2pi1g1= 20 nm ·meV atB= 0.1T, the transmission rate of the electron with Ein= 172 meV is T0= 1.998. In Figs. S4(a) and (b), we show how the incident wave functions ψin z↑= (1 0)T andψin z↓= (0 1)Tare transported through the ring, respectively, where the outcoming spin is polarized and the transport rate is relatively low. In Figs. S4(c) and (d), we show how the incident wave functions transport in the ring when the magnetic field is B= 0.1T, where the electrons pass through the ring freely but the spin is not polarized at all. When the transportreachesthe equilibriumstatus, the spin and charge densities and the current densities areplotted in Fig. S4. Both the charge densities and the spin textures shown in the first two columns (from left to right) of Fig. S4are periodically distributed in the ring as a stationary wave, due to the interference of the mat- ter wave of the electron. The spin textures also support the analysis of the outcoming spins derived in Eq. ( S3), i.e. at the right lead ∝an}bracketle{tσy∝an}bracketri}ht= 0, thexcomponent spin is generated in the transport by the SOC and the direction ofσx(r) depends on the polarization of the incident spin. Comparing with the case without the magnetic field [3, 4], here the vector potential of the external magnetic field and the effective vector potential induced by the SOC give different phases to the upper and the lower arms, respectively. This phase difference leads to differ- ent transmission for different spins and can be observed by the transport experiment. In the spin up channel in case (i), electron is mostly transported by the current jz↑, which means the SOC does not contribute a lot in the transmission. In the spin down channel, the SOC flips spin and induces stronger transmission. However, the transmission of this channel isstillweak, onlycontributes1 /20ofthe spinup channel. In this case, the spin is thus strongly polarized. In the case (ii), both of the two channels have high transmission12 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni00000013/uni00000014/uni00000015×/uni00000014/uni00000013/uni000000ed/uni00000015 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni00000013/uni00000014/uni00000015/uni00000016×/uni00000014/uni00000013/uni000000ed/uni00000016 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018×/uni00000014/uni00000013/uni000000ed/uni00000015 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018×/uni00000014/uni00000013/uni000000ed/uni00000015 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni000000ed/uni00000014/uni00000013/uni00000014×/uni00000014/uni00000013/uni000000ed/uni00000018tz↑ tz↓ tSOC ttotal˚A−2 ˚A−10.354 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018×/uni00000014/uni00000013/uni000000ed/uni00000018¯hg1= 20nm ·meV,¯hg2= 0nm·meV,B= 2.67T,E= 198.5meVy/r0˚A−2(a) /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni000000ed/uni00000014/uni00000013/uni00000014×/uni00000014/uni00000013/uni000000ed/uni00000018˚A−2 ˚A−10.017 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni00000013/uni00000011/uni00000013/uni00000013/uni00000011/uni00000018/uni00000014/uni00000011/uni00000013/uni00000014/uni00000011/uni00000018×/uni00000014/uni00000013/uni000000ed/uni00000018y/r0˚A−2(b) /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni000000ed/uni00000015/uni00000013/uni00000015×/uni00000014/uni00000013/uni000000ed/uni00000019˚A−2 ˚A−11.000 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni00000013/uni00000014/uni00000015/uni00000016×/uni00000014/uni00000013/uni000000ed/uni00000019¯hg1= 20nm ·meV,¯hg2= 0nm·meV,B= 0.10T,E= 172.0meVy/r0˚A−2(c) /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni000000ed/uni00000015/uni00000013/uni00000015×/uni00000014/uni00000013/uni000000ed/uni00000019 x/r0 x/r0 x/r0 x/r0 x/r0˚A−2 ˚A−1 σz 0.998 /uni000000ed/uni00000014 /uni00000013 /uni00000014/uni000000ed/uni00000014/uni00000013/uni00000014 /uni00000013/uni00000015/uni00000017×/uni00000014/uni00000013/uni000000ed/uni00000019 x/r0y/r0˚A−2 ρ(d) FIG. S4: (color online). Transport status and transmission flow density for electrons with fixed incident energy and magn etic fields in a Rashba ring. The first two columns (from left to righ t) show the charge and spin fields when the transport reaches the equilibrium status, the colors represent the density an dσz(r), respectively. In (a) and (b), strong spin filtering is foun d, i.e. the transmission rate of spin up is much higher than that of sp in down. In (c) and (d) the outcoming spin is fully unpolarize d although the total transmission rate is almost 1. For (a) and (b),/planckover2pi1g1= 20 nm ·meV,B= 2.67 T and the incident energy is 198.5 meV. For (c) and (d), /planckover2pi1g1= 20 nm ·meV,B= 0.1 T and the incident energy is 172 meV. rate, close to 1. The current in the spin down channel is obviously imbalanced in the upper and lower arms. However the outcoming spin has half in spin up and half in spin down, which means that the ring in this case is good in transport but fails to polarize the spin. There are circular currents in the ring, which do not contribute to total transmission, when the transmissionrate is low. It keeps the current conserved. If the trans- mission is high, the internal circling is weak, but the im- balance between the currents of the upper and the lower arms is explicit. From the detailed transport pictures shown in Fig. S4, we can clearly see how the electron passes through the ring. This method is general and can be applied to other systems as well. [1]T. Chakraborty, A. Manaselyan, and M. Berseghyan, inPhysics of Quantum Rings (Springer, Berlin 2018), edited by V.M. Fomin. [2]T. Chakraborty, and P. Pietil¨ ainen, Phys. Rev. B 52, 1932 (1995); Hong-Yi Chen, P. Pietil¨ ainen, and Tapash Chakraborty, Phys. Rev. B 78, 073407 (2008). [3]S. Datta and B. Das, Appl. Phys. Lett. 56, 665 (1990). 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[14]S. Gumber, A B. Bhattacherjee, P K. Jha, Phys. Rev. B 98, 205408 (2018). [15]B. Moln´ ar, F. M. Peeters, and P. Vasilopoulos, Phys. Rev. B69, 155335 (2004). [16]D. Frustaglia and K. Richter, Phys. Rev. B 69, 235310 (2004). [17]P. Lucignano, D. Giuliano, and A. Tagliacozzo, Phys. Rev. B76, 045324 (2007). [18]J. Wang, and K. S. Chan, J. Phys.: Condens. Matter 21, 245701 (2009). [19]H.Z.Tang, L.X.Zhai, M.Shen, J.J.Liu, PhysicsLetters A378, 2790-2794 (2014). [20]S. Bellucci and P. Onorato, J. Phys.: Condens. Matter 19, 395020 (2007); Eur. Phys. J. B 73, 215 (2010). [21]R. Citro, F. Romeo, M. Marinaro, Phys. Rev. B 74, 115329 (2006). [22]V. M. Kovalev, A. V. Chaplik, Journal of Experimental and Theoretical Physics, 103, 781 (2006). [23]S. Saeedia and E. Faizabadi, Eur. Phys. J. B 89, 118 (2016). [24]L. X. Zhai, Y. Wang, Z. An, AIP Advances 8, 055120 (2018). [25]Qing-feng Sun, Jian Wang and Hong Guo, Phys. Rev. B 71, 165310 (2005). [26]S. Griffith, Trans. Faraday Soc. 49, 345 (1953). [27]C. Li, Y. J. Li, D. X. Yu, C. L. Jia, New J. Phys. 20, 093023 (2018). [28]N. Hatano, R. Shirasaki, and H. Nakamura, Phys. Rev. A75, 032107 (2007). [29]A S. Naeimi, Eslami, M. Esmaeilzadeh, J. Appl. Phys. 113, 044316 (2013). [30]Evgeny Y. Tsymbal and Igor ˘Zuti´ c,Handbook of spin transport and magnetism , CRC Press, Boca Raton (2012). [31]Di-Jing Huang, Jae-Yong Lee, Jih-Shih Suen, G. A. Mul- hollan, A. B. Andrews, and J. L. Erskine, Rev. Sci. In- strum.64, 3474 (1993). [32]C. Jozwiak, J. Graf, G. Lebedev, N. Andresen, A. K. Schmid, A. V. Fedorov, F. El Gabaly, W. Wan, A. Lan- zara, and Z. Hussain, Rev. Sci. Instrum. 81, 053904 (2010). [33]J. Nitta, T. Akazaki, H. Takayanagi, and T. Enoki, Phys. Rev. Lett. 78, 1335 (1997). [34]C. R. Ast, D. Pacil´ e, L. Moreschini, M. C. Falub, M. Papagno, K. Kern, M. Grioni, J. Henk, A. Ernst, S. Os- tanin, and P. Bruno, Phys. Rev. B 77, 081407(R) (2008). [35]Y. Kanai, R. S. Deacon, S. Takahashi, A. Oiwa, K. Yoshida, K. Shibata, K. Hirakawa, Y. 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1211.2018v1.Interaction_induced_staggered_spin_orbit_order_in_two_dimensional_electron_gas.pdf

Interaction induced staggered spin-orbit order in two-dimensional electron gas Tanmoy Das Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. We propose and formulate an interaction induced staggered spin-orbit order as a new emergent phase of two-dimensional Fermi gases. We show that when some form of inherent spin-splitting via Rashba-type spin- orbit coupling renders two helical Fermi surfaces to become significantly ‘nested’, a Fermi surface instability arises. To lift this degeneracy, a spontaneous symmetry breaking spin-orbit density wave develops, causing a surprisingly large quasiparticle gapping with chiral electronic states. Since the staggered spin-orbit order is associated with a condensation energy, quantified by the gap value, destroying such spin-orbit interaction costs sufficiently large perturbation field or temperature or de-phasing time. BiAg 2surface state is shown to be a representative system for realizing such novel spin-orbit interaction with tunable and large strength, and the spin-splitting is decoupled from charge excitations. These functional properties are relevant for spin-electronics, spin-caloritronics, and spin-Hall effect applications. PACS numbers: 71.70.Ej, 73.20.-r, 75.25.Dk, 79.60.Bm In semiconductor heterostructures, a charge particle mov- ing on a symmetry-breaking electric field experiences an effective ‘anisotropic’ magnetic field due to relativistic ef- fect, which couples to its spin. Such spin-orbit coupling (SOC), known as Rashba-[1, 2] or Dresselhaus-type[3] SOC, has proven to be a useful ingredient for realizing many physical concepts such as spin Hall effect,[4] spin torque current[5, 6], spin domain reversal,[7] and exotic emer- gent superconducting,[8] and magnetic phases,[9] which are relevant for the applications of spin-electronics,[10] spin- caloritronics[11] and quantum information processing. Along with long spin lifetime, spin relaxation, and immunity to perturbations or disorders, a highly desired recipe for these applications is decoupling the spin current from associated charge flow. This is a difficult task as the electric field gen- erated either by the broken inversion symmetry and/ or by the magnetic field itself naturally generates charge current per- pendicular to the spin current, thus greatly hinders the spin transport.[12, 13] Despite the discovery of giant SOC in a large class of condensed matter systems,[2, 14–19] practical realization and device implementation of these concepts have thus posed challenging.[10, 11] In this work, we introduce a theoretical proposal for gener- ating and manipulating staggered spin-orbit entangled helical state via electronic interaction. A key ingredient for realiz- ing such state is to have some form of intrinsic SOC, whose strength is not important, prior to the inclusion of interac- tion. For such systems, the ground state is defined by more exotic quantum numbers such as total angular momentum ( J forjj-type SOC) or helical quantum number ( =for Rashba- or Dresselhaus-type SOC), rather than typical spin- , orbital-, or momentum alone. Therefore, if the interac- tion strength is less than the SOC strength, a typical spin- or orbital-density wave alone is prohibited to form. On the other hand, we demonstrate here that novel emergent phases of mat- ter can arise in the spin-orbit channel even without necessarily breaking the time-reversal symmetry. We formulate the aforementioned postulates on the basis of a Rashba-type SOC ground state; however the idea is generaland can be extended to other systems. Rashba SOC splits the non-interacting FS into two concentric helical Fermi pockets, and thereby a FS ‘hot-spot’ Qdevelops where degeneracy in- duces FS nesting between the opposite helical states. As a result of such instability, a translational symmetry breaking spontaneous collective ordering of helical degree of freedom develops, causing a giant SODW. The resulting SODW spa- tially modulates with a periodicity determined by the ‘hot- spot’ wavevector Q. The SODW renders gapping in the quasi- particle states, and the corresponding gap energy
thermo- dynamically represents its strength. Unlike in topological in- sulators or in other SOC systems where a spin-degenerate point exists with zero gapping, here the finite gap
protects the SODW phase from external perturbations and spin de- phasing. This means the spin-splitting survives up to a finite strength of Zeemen energy EZ(comparable to
), determin- ing the critical value of a time-reversal breaking perturbation such as magnetic field, Bc, and spin de-phasing time s. As a proposal, we show that a feedback effect of this order is the presence of a spin-orbit collective mode, which physically represents the interaction between two electrons with opposite spins and orbitals, or spin-orbit entangled quantum numbers, and that it can be detected via two-particle probes such as po- larized inelastic neutron scattering. Lattice model for Rashba coupling:- We start with a system of two-component Fermi gas in the presence of Rashba-type SOC. In the single-particle description of the system, the non- interacting FS is spin-polarized on the two-dimensional (2D) momentum space. For some systems, as in the surface state of BiAg 2deposited on various substrates,[2, 19] the FSs can yield a shape to generate dominant FS nesting, and hence an unstable one-body ground state. If two momenta across the nesting ‘hot-spot’ is connected by spin flips, it cannot sup- port a charge density wave scenario. Furthermore, if the nest- ing commences between different segments of the same band, a spin-density wave may arise if the interaction strength can overcome the SOC strength. On the other hand, in such case a SODW can arise via inter-helical FS nesting instability even if the interaction strength is lower than the SOC strength.arXiv:1211.2018v1 [cond-mat.str-el] 9 Nov 20122 -0.25 0.25 -0.25 0.25 -0.25 0.25 (kx,0) [π/a] (kx,ky) [π/a] (-kx,ky) [π/a] -0.25 0.25 kx [π/a] 0 ky [π/a] -0.25 0 0.25 -1 0 1 Energy (eV) 0 0.04 LOW HIGH Re χ0 (1/eV) |↓> |↑> (a) (b) (c) (d) (e) (100) (110) (1ī0) FIG. 1. (Color online) (a) Non-interacting bands split by single- fermion Rashba-type SOC plotted along (100)-direction. The blue solid line is the coherent eigenstate, while the background is the associated single-particle spin-resolved spectral weight. The spin- polarization is depicted by a red (down-spin) to black (up-spin) gra- dient colormap. Along (100), the Rashba term vanishes, and thus the spin-resolved spectral weight gradient is absent here. The counter polarization of the spin texture along (110)- and (1 10)-directions is demonstrated in (b) and (c), respectively. (d) Corresponding FS map is plotted in the same color scale. (e) Real part of non- interacting susceptibility at zero excitation energy reveals the devel- opment of paramount inter-helical state nesting at the ‘hot-spot’ vec- torQ0:115(;). The non-interacting Hamiltonian in the two component fermion fields k= [ k;"; k;#]TasH0(k) = y k[k1 iR(xsinkyysinkx)] k. Herekis the free-fermion dispersion term, modeled by nearest-neighbor electronic hop- pingtask=2t[coskx+ cosky]EF, whereEF is the chemical potential. The second term is the 2D lat- tice generalization of the standard Rashba SOC term Hso= iR(^^k)z, with ^being the Pauli matrices and Rbeing the Rashba SOC strength. The helical dispersion spectrum ofH0isE k;0=kR[sin2kx+ sin2ky]1=2. The pa- rameters are obtained by fitting to the experimental dispersion from Ref. 2, as listed in Ref. 21 and the computed dispersion is plotted in Figs. 1(a), 1(b), 1(c) along different high-symmetry lines, and the corresponding FS is given in Fig. 1(d). FS instability:- We now investigate the FS instabilities of the system by evaluating the bare susceptibility in the particle- hole channel. is computed by convolving the single-particle Green’s function G(k;i!n) = (i!nH0(k))1, yielding (q;ipm) =P k;nG(k;i!n)G(k+q;i!n+ipm), wherei!nandipmare the fermionic and bosonic Matsubara frequencies, respectively. The result plotted in the 2D qspace at zero en- ergy (the real frequency is obtained by taking analytical con- tinuation from the Matsubara frequency) in Fig. 1(e) exposes that the nesting at Q= 0:115(;)is dominant. Qcon- nects two momenta lying on different helical bands as shown in Fig. 1(d). Due to the definite chirality of the FS, different orientations of Q= 0:115(;)vector are decoupled, and thus are exclusively included in the Hamiltonian. Spin-orbit density wave:- Based on these results, we now desire to write down a two-body interacting Hamiltonian in the reduced Brillouin zone in which the Nambu-Gor’kov spinor becomes k= [ k"; k#; k+Q"; k+Q;#]T. In this notation the interaction in the singlet-channel can be characterized by a contact interaction parameter gand cor- respondingly, HI(k) =g y k;" k;# y k+Q;" k+Q;#. In or- der to reduce the two-body problem into ordered Fermionic ground state, we decouple the interaction term HIby intro- ducing a auxiliary spin-orbit field
(k) =g y k+Q;[z x]0 k;0[note that k;and k+Q;0belong to differ- ent helical states, having two Pauli matrices zandxfor spin and momentum, and represents a tensor product be- tween them]. Employing mean-field approximation to the spin-orbit field
(k), we obtain the total Hamiltonian as H= y k[H0(k) +
(k)z x+h:c:] kwhich leads to the excitation spectrum as E; k=S ++[(S )2+
2]1=2; withS (k) = (E k;0E k;0)=2; (1) where ==is the helical index due to SOC, and =is the split band index due to translational sym- metry breaking. We evaluate the order parameter
self- consistency as a function of temperature to a given contact potential. See supplementary material (SM)[22] for techni- cal details. The obtained temperature dependence and critical temperature demonstrate the spontaneous behavior of mean- field gap opening. Quasiparticle gapping:- Fig. 2 gives our main result in which the nature of the quasiparticle gap opening due to SODW is demonstrated and compared with angle-resolved photoemission spectroscopy (ARPES) data.[2] The blue lines are the coherent quasiparticle bands, plotted on top of the spin-resolved spectral weight [red (spin down) to black (spin up) colormap]. In the SODW state, two helical states split into several sub-bands, and a gap opens at the energy where two opposite helical states are connected by the Qvector. Along (100)-direction, the gap opening occurs in the filled state. The nature of gap opening and multiple number of shadow bands are in detailed agreement with recent ARPES data on the sur- face state of BiAg 2alloy deposited on the monolayres (MLs) of Ag/Au(111) heterostructure.[2, 19] Similarly, along the di- agonal direction,
appears slightly above the Fermi level, and tiny hole-like pockets develop, as visible on the FS map in Fig. 2(e). The constant energy surface map at an energy E=- 110 meV illustrates how the main band and the folded band3 -0.5 0.5 0 kx [π/a] (1/Ǻ) kx [π/a] (1/Ǻ) kx = ky [π/a] (1/Ǻ) -0.5 0.5 0 0 0 -0.3 -0.6 -0.5 0.5 -0.5 0.5 -0.5 0.5 -0.5 0.5 |↓> |↑> (a) (b) (c) (d) (e) (f) d e f Energy (eV) FIG. 2. (Color online) Quasiparticle gapping due to SODW. (a) Electronic structure and associated spin-texture are plotted along (100)- direction. (b) Same as (a) but along (110) axis. The gap opening occurs below and above the Fermi level in both cases, while the spin- degeneracy at -point is intact. Three horizontal arrows dictate three energy scales where the constant energy maps are presented in (d), (e), and (f). (c) The ARPES result of the gap opening in the BiAg 2surface state (obtained via deposition of the sample on varying number of monolayers of Ag/Au heterostructure), taken from Ref. [2]. The present theory, replotted from (a) on top of the experimental data with red dashed line, is in detailed agreement with the ARPES data. The number of sub-bands developed in the SODW is clearly evident in the experimental data as well, demonstrating further the interaction origin of the gap. Results are presented in the unfolded Brillouin zone to facilitate comparison with experimental data. are nested along (100) -direction (in the particle-hole channel) where the quasiparticle gapping occurs, see Fig. 2(d). Sim- ilarly, at an energy E=62 meV above the Fermi level, the quasiparticle state is fully gapped along the diagonal direc- tion; Fig. 2(f). Tunable spin-orbit order:- Interestingly, the ARPES data[2] also reveals that the measured gap value
varies upon chang- ing the thickness of the Ag film, with a clear evidence of Kramers degenerate point at -momentum for all cases. These observations indicate that the systems undergoes quantum phase transition with a spontaneous symmetry breaking in the spin-orbit channel, whereas the time-reversal symmetry is not explicitly broken. It is interesting to note that a similar broken symmetry state other than time-reversal symmetry is obtained on the surface state of topological insulators,[23, 24] and also proposed to be responsible for the enigmatic ‘hidden-order’ state in heavy fermion URu 2Si2,[25] despite having different forms of spin-orbit coupling quantum state in these systems. ARPES data plotted in Figs. 3(c), 3(f), 3(i) show that the gap decreases from
= 120 meV for ML2 to
= 80 meV for ML4 to
= 60 meV for ML16. To theoretically explain this gap variation, we self-consistently tune the contact po- tentialg, while keeping the Rashba term Rfixed. We find that the gap closing is associated with chemical potential vari- ation (beyond rigid band shift approximation), and the gappedregion moves towards the Fermi level. We also find that the critical temperature below which the SODW sets in decreases with decreasing g; see SM[22]. Subsequently, the hole pock- ets along the diagonal directions grow in size. The large he- lical FSs with tunable area will be of considerable value for generating and detecting spin-resolved transports. In what fol- lows, the new interaction induced spin-orbit effect is tunable and large, even when the intrinsic SOC is fixed and weak. Spin-orbit correlation function and emergent collective mode:- Finally, to unravel the mechanism of SODW order, we deduce the emergence of associated collective excitations in the spin-orbit channel. The general form of the spin-orbit po- larization vector is defined as J(q;) =P k y k;()[z x]0 k+q;0(), and correspondingly its bare correlator 0(q;) =hTJ(q;)J(q;0)i, whereis the imaginary time and the operator Tdenotes standard time-ordering be- tween the fermionic fields. The result is presented as a func- tion of excitation energy after performing Fourier transforma- tion, and including many-body correction within the treatment of random-phase approximation (RPA): =0=(1U0), whereUis the interaction matrix defined in the SM[22]. The corresponding result of the imaginary part of RPA sus- ceptibilityis given in Fig. 4. Along the diagonal direction, a robust collective mode is visible at Q0:115(;)with its energy proportional to the gap value. The result is presented4 -0.5 0.5 0 kx [π/a] (1/Ǻ) 0 -0.25 0 -0.5 0.5 0 -0.5 0.5 0 -0.5 0.5 |↓> |↑> ML2 ML4 ML16 Energy (eV) (a) (b) (c) (d) (e) (f) (g) (h) (i) FIG. 3. (Color online) Tunable spin-orbit order. (a) Quasiparti- cle dispersion along (100)-direction for self-consistently evaluated gap
=120 meV . (b) Corresponding FS map. (c) Comparison with ARPES data for the case of ML2.[2] (d)-(f) Same as (a)-(c) but for a smaller gap of
= 80 meV and comparison with data for ML4 (reproduced from Fig. 2). The result for even a smaller gap of
= 60 meV is compared with experimental data for ML16 in (g)- (i). As the gap gradually decreases with increasing number of mono- layers of Ag/Au heterostructure, the location of the gap approaches the Fermi level, suggesting that the interaction induced spin-order ef- fect is tunable, irrespective of a constant value of the non-interacting Rashba-coupling strength. for the case of ML2, however, the !res
relation is re- produced for other cases (in the bare susceptibility level, the peak appears at the gap energy, however, the many body RPA correction shifts the mode energy to a slightly lower value). A downward dispersion of the mode is also visible centering Q, and extending to q!0andq!0:124(;)with vanish- ing intensity. A second collective mode appears at an energy which is about half of the first mode energy due to multi-band effect. The first mode is a robust feature tied to the emergent spontaneous symmetry breaking SODW, whereas the inten- sity and energy of the second mode varies and its appearance is subject to the strength of interaction. The constant energy profiles at the first and second mode energies are depicted in -0.125 0.125 0 qx=qy [π/a] (1/Ǻ) -0.125 0.125 qx [π/a] (1/Ǻ) 0.1 0 0.2 LOW HIGH Excitation Energy (eV) Im χ (a) (b) (c) FIG. 4. (Color online) Collective spin-orbit mode. (a) Computed spectrum of imaginary part of RPA susceptibility in the spin-orbit channel in the SODW state plotted along (110) direction. The param- eters for this calculation is same as deduced for ML2 configuration before. A sharp peak in intensity at Q0:115(;)is clearly visi- ble around!
0:12eV . A weak downward dispersion branch away from this mode can be marked. The second mode around
=2 is a multiband feature. (b)-(c) Constant energy profiles of the same susceptibility at first and second mode energies, respectively. Figs. 4(b),(c). The bosonic spin-orbit excitation represents electron-electron interaction with simultaneous spin and or- bital flips, or an entangled helical index flip.[22] A polar- ized inelastic neutron scattering measurement, which directly probes the imaginary part of the susceptibility, will be able to detect this mode. We note that no such collective mode de- velops in the charge susceptibility, and thus confirming that the present spin-orbit interaction does not activate any charge excitation. Outlook and discussion:- The proposed spin-orbit order is a novel quantum phase of matter which can also arise in va- riety of other materials in which the non-interacting wave- function is defined by exotic quantum number such as heli- cal index, pseudospin, total angular momentum (typically in heavy-fermions and actinides) owing to SOC of various na- tures. Depending on the nature of the interaction and broken symmetry, the spin-orbit order can as well emerge as short range order in which other interacting phases such as quantum spin-Hall effect, spin-orbit nematic phase may exist. Here, we take a 2D electron gas as a representative example of how a SODW can arise due to the FS instability in a Rashba-type SOC background. In this context, it is interesting to point out the recent experimental findings of quasiparticle gapping in the surface state of topological insulator due to quantum phase transition. Such gap opening in the absence of time-reversal symmetry breaking and without the destruction of bulk topo- logical properties,[23, 24] violates the conventional topologi-5 cal paradigm and criterion.[26] We envisage that it is sugges- tive of an emergent time-reversal invariant spin-orbit order, whose detail is required to be formulated in future study. Some important advantages of the present spin-orbit or- der than the typical single-particle SOC can be noted. (1) In topological insulators or in Rashba systems, the functional use of spin-orbit effect is subject to counter propagating he- lical spin state which is topologically protected by Kramers spin-degeneracy with zero gap at the -point.[26] Therefore the presence of any type of time-reversal breaking impurity or defects, with strength as small as infinitesimal value will destroy the protection, and thus barring the exciting useful- ness of topologically protected transport properties. On the other hand, the present spin-orbit order is not only protected by symmetry, but most importantly it is thermodynamically shielded with a condensation energy equal to the tunable and large gap. Therefore, it would require a sufficiently large value of magnetic field such that its associated Zeeman en- ergyEZgBB(gis ‘g’-factor, Bis Bohr magneton)] can overcome the condensation gap energy. (2) Spin de-phasing timeswhich is an important ingredient for the spintronics and quantum computing applications is considerably larger here (determined by EZenergy), and is tunable. (3) Another crucial benefit of the interaction induced spin-orbit effect is that, unlike in typical electric field or magnetic field induced SOC, it does not necessarily activate a charge flow, and thus will be highly valuable for solely generating spin-transport. Another experimental verification of the broken symme- try spin-orbit order is detection of spin Nernst effect. Since Nernst effect is sensitive to reconstructed FS, a manifestation of broken symmetry phase,[27, 28] spin-orbit order will gen- erate a spin-resolved thermal current which are detectable in recent days laboratory facilities.[29] The author is grateful to M. J. Graf, A. V . Balatsky, J.-X. Zhu, S. Raghu, P. W ¨olfle for many fruitful discussions. The work is supported by the U.S. DOE through the Office of Sci- ence (BES) and the LDRD Program and faciliated by NERSC computing allocation. [1] E. I. Rashba, Sov. Phys. Solid State 2, 1109-1122 (1960). [2] Y . A. Bychkov, and E. I. Rashba, JETP Lett. 39, 78-81 (1984). [3] G. Dresselhaus, Phys. Rev. 100, 580-586 (1955). [4] J. E. Hirschm, Phys. Rev. Lett. 83, 1834-1837 (1999). [5] A. Manchon, and S. Zhang, Phys. Rev. B 78, 212405 (2008). [6] I. M. Miron, et al. Nat. Mater. 9, 230-234 (2010). [7] A. Chernyshov, et al. Nat. Phys. 5, 656 - 659 (2009). [8] L. P. Gor’kov, and E. I. Rashba, Phys. Rev. Lett. 87, 037004 (2001). [9] C. Wu, J. Zaanen, and S.-C. Zhang, Phys. Rev. Lett. 95, 247007 (2005). [10] F. Pulizzi, Nat. Mater. 11, 367 (2012) [11] G. E. W. Bauer, E. Saitoh, and B. J. va Wees, Nature Mater. 11, 391-399 (2002). [12] S. D. Ganichev, et al. Nat. Phys. 2, 609 - 613 (2006). [13] A. Hoffmann, Phys. Stat. Sol. (c) 4, 4236 (2007).[14] S. LaShell, B. A. McDougall, and E. Jensen, Phys. Rev. Lett. 77, 3419-3422 (1996). [15] T. Nakagawa, et al. Phys. Rev. B 75, 155409 (2007). [16] K. Ishizaka, et al. Nature 10, 521-526 (2011). [17] Y . J. Lin, K. Jim ´enez-Garcia, I. B. Spielman, Nature 471, 83-88 (2011). [18] Y . Xia, et al. Nat. Phys. 5, 398-402 (2009). [19] A. Crepaldi, et al. Phys. Rev. B 85, 075411 (2012). [20] H. Bentmann, et al. Phys. Rev. Lett. 108, 196801 (2012). [21] The values of all parameters are obtained by fitting the quasiparticle-spectrum and gap with the experimental value from Ref. 2. We find t=1:216eV , andR= 1:25=jtj. The inter- action potential is self-consistently evaluated to be g=jtj=1, 0.84, and 0.73 which gives the experimental gap values of
= 0.12, 0.08 and 0.06 in eV unit for ML2, ML4 and ML16, respectively. [22] See supplementary materials for details calculations. [23] E. Sato, et al. Nat. Phys. 7, 840 (2011). [24] S. Y . Xu, et al. arXiv:1204.6518. [25] T. Das, Sci. Rep. 2, 596 (2012). [26] L. Fu, C. L. Kane, and E. J. Mele, Phys. Rev. Lett. 98, 106803 (2007). [27] V . Oganesyan, and I. Ussishkin, Phys. Rev. B 70, 054503 (2004). [28] T. Das, Phys. Rev. B 86, 064527 (2012). [29] S.-G. Cheng et al. Phys. Rev. B 78, 045302 (2008). SUPPLEMENTARY MATERIAL In the supplementary mayerial we provide the detailed derivation of the spin-orbit order parameter. Due to the definite chirality of the FS, different orientations of Qi= 0:115(;)are decoupled and are exclusively included in the Hamiltonian. In this spirit we define the Nambu- Gor’kov spinor as k= [ k;"; k;#; k+Qi;"; k+Qi;#;:::]T, wherei= 14. Since macroscopically the gap value is same for all four Qivectors, the interaction potential is con- stant which gives the two-body spin-orbit term as HI(k) = gP i y k;" k;# y k+Qi;" k+Qi;#. In the functional path in- tegral formalism, the partition function of the Fermi gas is Z=R D yD exp (S[ y; ])(~=kB= 1throughout), where the action is S[ y; ] =R 0dP0 k[ y@ +H0+ HI], with= 1=Tand the prime over the summation repre- sents that the summation is performed in the reduced Brillouin zone. In order to reduce the above-derived action into an or- dered Fermionic problem, we decouple the four-field in- teraction term by introducing the auxiliary spin-orbit cou- pling field
(k) =g y k+Q;[z x]0 k;0. Em- ploying the Hubbard-Stratonovich transformation[1] and integrating out the fermionic fields, we obtain Z=R D
yD
exp (Seff[
y;
]), where the effective spin- orbit order reads Seff[
y;
] =1 2Tr ln [G1(k;)]R 0P kdj
(k)j2=g. The inverse single-particle Green’s6 FIG. 5. Self-consistent gap values as a function of temperature for
(0)=jtj0.1 withg=jtj=1. The critical temperature for the spin- orbit density wave, Tso, is estimated for this case to be around 800 K. The line depicts a fit of the function
(T)=
(0) = (1T=Tso)0:35, which is close to the BCS critical exponent of 0.5. Inset: The critical temperatureTso(blue squares) and gap amplitude
(0) (red circles) varies linearly with the interaction potential strength g. The data are presented with respect to jtj. function in the fermionic Matsubara frequency is obtained as G1=0 B@i!n1H0(k)x
::: x
yi!n1H0(k+Q1)::: .........1 CA:(2) The other terms belong to the three values of Qi. At the mean- field level,
(k)represents the gap parameter which is same for all values of Qi. In this case, the effective action fur- ther simplifies to Seff[
y;
] =1 2ln [detG1(k;i!n)] Nj
(k)j2=g, whereNis the size of the system. From det[G1(k;E)] = 0 , we derive the excitation spectrum as given in Eq. 1 in the main text. Finally, by using the relation for the thermodynamic potential =1=lnZ, we obtain the self-consistent condition for the number operator and thegap function as: n"=#=0X k;Z1 1d! 2ImG11=22(k;;! +i)f(!);(3)
=g0X k;Z1 1d! 2ImG12(k;;! +i)f(!);(4) whereis infinitesimally small broadening parameter as f(!) is the Fermi-Dirac distribution function for the excitation en- ergy of!. We solve Eqs. 3 and 4, self-consistently to obtain the value of quasiparticle gap to match the experiment. The values of all parameters are obtained by fitting the quasiparticle-spectrum and gap with the experimental value from Ref. [2]. We find t=1:216eV , andR= 1:25=jtj. The interaction potential is self-consistently evaluated to be g=jtj=1, 0.84, and 0.73 which gives the experimental gap val- ues of
= 0.12, 0.08 and 0.06 eV for ML2, ML4 and ML16, respectively. We also deduce the temperature dependence of the gap parameter as given in the supplementary Fig. 5. The result reproduces a mean-field like behavior with the critical exponent 0:35, which can be compared with the typical BCS value of 0:5. The critical temperature below which a spin- orbit order sets in relies on the contact potential (see inset to Fig. 5). SPIN-ORBIT EXCITATION Finally, we expand on the deduction of the spin-orbit sus- ceptibility. The spin-orbit operator is defined in the main text as J(q;) =X k y k;()[z x]0 k+q;0(): (5) Substituting the expression for spin-orbit polarization in the spin-orbit susceptibility formula, we obtain 0(q;) =1 NhTJ(q;)J(q;0)i=1 NX k;k0hT y k;() k+q;0() y k0;(0) k0q;0(0)i (6) Rewriting the above susceptibility in the reduced Brillouin zone (denoted by prime over the summation), we get 0(q;) =1 2N0X k;k0" hT y k;() k+q;0() y k0;(0) k0q;0(0)i+X i hT y k;() k+q;0() y k0+Qi;(0) k0q+Qi;0(0)i +hT y k+Qi;() k+q+Qi;0() y k0;(0) k0q;0(0)i+hT y k+Qi;() k+q+Qi;0() y k0+Qi;(0) k0q+Qi;0(0)ii (7)7 k, ω, ν k+q, ω+Ω, ν′ Γνν′ Γνν′ (a) (b) ν ν′ FIG. 6. (a) Real-space view of spin-orbit density wave having two opposite spin-orbit states sitting in different sublattices. Arrows stand for spin while circles with different colors give different or- bitals. We chose a commensurate modulation here along the diagonal for illustration convenience. (b) Spin-orbit bubble which is responsi- ble for electron-electron interactions having opposite spin and orbital quantum numbers. The interaction vertex 0= [z x]0. Expanding the above four-fermionic fields into two ordered fields in the spin-orbit channel and employing the Wick’s theorem,[3] we can write down the above equation in terms of Green’s functions as 0(q;) =1 2N0X kX ijGij(k;;)Gij(k+q;0;); (8)whereGijare the components of the Green’s function given in Eq. 1 above. Here we employ the momentum conserva- tion relationk0=k+qork0=k+q+Qi. Each term in0represent a polarization bubble as represented in supple- mentary Fig. 6(b) which is responsible for interaction between two electrons with opposite orbitals and spins, or with oppo- site helical index . The explicit form of the susceptibility ten- sor isijkl 0(q;ipm) =1=NP k;n;;0iy (k)j (k)ky 0(k+ q)l 0(k+q)(f(E k)f(E0 k+q)=(i!nE kE0 k+q):Here i andEare the eigenvector and eigenvalue of the Hamil- tonian deduced in Eq. 2, with ;0being the corresponding band indices, and i;j;k;l their orbital counterparts. Within RPA framework, the perturbation interaction term for the sus- ceptibility is HRPA =UP nn+U0P nn, where ==is the helical index and U, andU0are the in- teraction strength between same and opposite helical states. Since the interaction here is Coulomb type which acts on charge particles, therefore, without any loss of generality, we takeU=U0g= 0:35eV for ML2 configuration. It is in- teresting to note here that despite the same value of interaction between both helical states, the long-range order is prohib- ited along the spin-channel due to strong spin-orbit coupling (U <<R), but from symmetry argument, an order parame- ter in the spin-orbit channel ( U0-term) is allowed. [1] R. L. Stratonovich, Soviet Physics Doklady 2, 416 (1957). [2] H. Bentmann, et al. , Phys. Rev. Lett. 108, 196801 (2012). [3] G. Mahan, Plenum Publishers, New York, USA (third edition) (2000).

2210.00600v1.Spin_orbit_coupling_and_Kondo_resonance_in_Co_adatom_on_Cu_100__surface__DFT_ED_study.pdf

Spin-orbit coupling and Kondo resonance in Co adatom on Cu(100) surface: DFT+ED study A. B. Shick and M. Tchaplianka Institute of Physics, Czech Academy of Science, Na Slovance 2, CZ-18221 Prague, Czech Republic A. I. Lichtenstein Institute of Theoretical Physics, University of Hamburg, 20355 Hamburg, Germany and European X-Ray Free-Electron Laser Facility, Holzkoppel 4, 22869 Schenefeld, Germany (Dated: October 4, 2022) We report density functional theory plus exact diagonalization of the multi-orbital Anderson impurity model calculations for the Co adatom on the top of Cu(001) surface. For the Co atom d-shell occupation nd8, a singlet many-body ground state and Kondo resonance are found, when the spin-orbit coupling is included in the calculations. The dierential conductance is evaluated in a good agreement with the scanning tuneling microscopy measurements. The results illustrate the essential role which the spin-orbit coupling is playing in a formation of Kondo singlet for the multi-orbital impurity in low dimensions. I. INTRODUCTION The electronic nanometer scaled devices require the atomistic control of their behaviour governed by the elec- tron correlation eects. One of the most famous corre- lation phenomena is the Kondo eect originating from screening of the local magnetic moment by the Fermi sea of conduction electrons, and resulting in a formation of a singlet ground state [1]. Historically the Kondo screening was detected as a resistance increase below a characteris- tic Kondo temperature TKin dilute magnetic alloys [2]. Recent advances in scanning tuneling microscopy (STM) allowed observation of the Kondo phenomenon on the atomic scale, for atoms and molecules at surfaces [3, 4]. In these experiments, an enhanced conductance near the Fermi level ( EF) is found due to the formation of a sharp Abrikosov-Suhl-Kondo [5{7] resonance in the electronic density of states (DOS). One case of the Kondo eect the most studied exper- imentally and theoretically is that of a Co adatom on the metallic Cu substrate [3, 8{10]. The experimental STM spectra display sharp peaks at zero bias, or so called "zero-bias" anomalies, similar to the Fano-resonance [11] found in the atomic physics, which are associated with the Kondo resonance. The theoretical description of the Kondo screening in multiorbital dmanifold is dicult since the whole dshell is likely to play a role. Very re- cently, theoretical electronic structure of the Co atom on the top of Cu(100) was considered [10] using nu- merically exact continuous-time quantum Monte-Carlo (CTQMC) method [12] to solve the multiorbital sin- gle impurity Anderson model [13] (SIAM) together with the density-functional theory [14] as implemented in the W2DYNAMICS package [15, 16]. However, the spin-orbit coupling (SOC) was neglected. The peak in the DOS at Electronic address: shick@fzu.czEFwas obtained in these calculations, and was inter- preted as a signature of the Kondo resonance. Alternative interpretation was proposed [17] which is based on the spin-polarized time-dependent DFT in con- junction with many-body perturbation theory. These au- thors claim that the "zero-bias" anomalies are not neces- sarily related to the Kondo resonance, and are connected to interplay between the inelastic spin excitations and the magnetic anisotropy. Thus the controversy exists con- cerning the details of the physical processes underlying the Kondo screening in Co@Cu(100). In this work, we revisit Co@Cu(100) case making use of the combination of DFT with the exact diagonalization of multiorbital SIAM (DFT+ED) including SOC. We demonstrate that SOC plays crucial role in formation of the singlet ground state (GS) and the Kondo resonance. II. METHODOLOGY: DFT + EXACT DIAGONALIZATION The exact diagonalization (ED) method is based on a numerical solution of the multi-orbital Anderson im- purity model (AIM) [13]. The continuum of the bath states is discretized. The ve d-orbitals AIM with the full spherically symmetric Coulomb interaction, a crystal eld (CF), and SOC is written as, H=X kmkmby kmbkm+X mddy mdm +X mm00 l
s+
CF
 0 mm0dy mdm00 +X km Vkmdy mbkm+h:c: (1) +1 2X mm0m00m0000Umm0m00m000dy mdy m00dm0000dm00: The impurity-level position dwhich yield the desired hndi, and the bath energies kmare measured from thearXiv:2210.00600v1 [cond-mat.str-el] 2 Oct 20222 FIG. 1: The ball model (top view) for Co@4Cu 8] supercell. The specic choice of the Cartesian reference frame is show. With this choice, the local Green's function without SOC becomes diagonal in the basis of cubic harmonics m=fxz;yz;xy;x2 y2;3z2r2g(a); orbitally resolved DOS (b); orbitally resolved hybridization Im
(c) for the Co adatom on Cu(001), chemical potential , that was set to zero. The SOC pa- rameter species the strength of the spin-orbit coupling, whereas
CFmatrix describes CF acting on the impu- rity. The hybridization Vmkparameters describe the cou- pling of substrate to the impurity orbitals. These param- eters are determined from DFT calculations, and their particular choice will be described below. The last term in Eq.(1) represents the Coulomb in-teraction. The Slater integrals F0= 4.00 eV, F2=7.75 eV, andF4= 4.85 eV are used for the Coulomb inter- action [9, 10]. They correspond to the values for the CoulombU= 4 eV and exchange J= 0.9 eV for Co which are in the ballpark of commonly accepted Uand Jfor transitional 3 d-metals. The DFT calculation were performed on a supercell of four Cu(100) layers, and the Co adatom followed by four3 empty Cu layers modeling the vacuum. Fig. 1A shows ball model of the Co@[4Cu 8] supercell employed for the adsorbate atop of Cu . The structure relaxation was per- formed employing the VASP method [18] together with the generalized gradient approximation (GGA) to spin- polarized DFT without SOC. The adatom-substrate dis- tance as well as the atomic positions within two Cu(100) layers underneath were allowed to relax. The relaxed dis- tance between the Co adatom in a fourfold hollow po- sition and the rst Cu substrate layer of 2.91 a:u:is in a good agreement with previously reported value of 2.87 a:u[10]. In order to obtain the bath parameters in the AIM Hamiltonian Eq.( 1) we make use of the recipes of the dynamical mean-eld theory (DMFT) [19, 20], and em- ploy the DFT(LDA) local Green's function G0(z) [G0(z)] 1 2=1 VBZZ BZd3k z+EFHDFT(k)1 1 2;(2) calculated with help of the full-potential linearized aug- mented plane wave method (FLAPW) [21, 22], in order to dene the parameters for the Eq.(1) . Here, the en- ergyzis counted from the Fermi energy EF, and the index lm marks thed-orbitals in the MT-sphere of the Co adatom. Note that the non-spin-polarised LDA is used to extract the hybridization function
( z). The orbitally resolved density of states (DOS) together with the hybridization function Im
are shown in Fig. 1B,C. They are compatible with the results of Ref. [23]. Further details of constructing the discrete bath model are given in Appendix A. The tted bath parameters are shown in Table IV. These parameters are used to build the AIM Hamiltonian Eq.( 1). The SOC parameter = 0.079 eV is taken from LDA calculations in a standard way, =ZRMT 0drr1 2(Mc)2dV(r) dr(ul(r))2; making use of the radial solutions ulof the Kohn- Sham-Dirac scalar-relativistic equations [24], the rela- tivistic mass M=m+ (ElV(r))=2c2at an appro- priate energy El, and the radial derivative of spherically- symmetric part of the LDA potential. III. RESULTS AND DISCUSSION The total number of electrons N, and thed-shell oc- cupation are controlled by the dparameter. It has a meaning of the chemical potential =din Eq. (1). In DMFT it is quite common to use =Vdc, the spherically- symmetric double-counting which has a meaning of the mean-eld Coulomb energy of the d-shell, and to use standard (AMF) Vdc= (U=2nd+2l 2(2l+1)(UJ)nd) [27] form, or the fully localized limit (FLL) Vdc= (U J)=2 (nd1) [28]. Since precise denition of nddependson the choice of the localized basis, we adopt a strategy of Ref. [9], and consider a value of as a parameter. A. Co in the bulk Cu At rst, we consider the Co impurity in the bulk Cu making use of the CoCu 15supercell model. DFT+ED cal- culations for dierent values of in a comparison with previous DFT+CTQMC relsults [9] are described in de- tails in Ref. [25]. Here, we adjust the value of in order to have the Co atom d-shell occupation nd8. This valence of Co in the bulk Co follows from DFT calcula- tions [9, 25]. Without SOC we found that the value of = 27:4 corresponds to the nd8 occupation. The GS solution without SOC (see Table I) is the j iN=30singlet, and the exited triplet is0.4 eV higher in the energy. Note that each eigenstatej iNof Eq.( 1) corresponds to an integer Noccupation ( d-shell + bath) since ^Ncommutes with Hamiltonian Eq.( 1). For each j i, the probabilities to nd the atomic eigenstates jniwith integer occupation dn,Pn=hnj ih jni, and thed-shell occupation nd=P nPnndn. The corresponding density of d-states (DOS) [1]: A() =1 ZImX ;;h jc j ih jcy j i +i+EE[eE+eE](3) where the; run over the eigenstates of Hamiltonian Eq.( 1), fm;gmarks the single particle spin-orbital, is shown in Fig. 2a, with the peak in DOS very near EF. The expectation values of the total h jJzj i, orbital h jLzj i, and spinh jSzj iangular momenta for the j iN=30singlet GS and the exited triplet are shown in Table I. They correspond to a solution of the Kondo model for localized S=1 2anti-ferromagnetically coupled to a single band of conduction electrons [26]. Together with the Kondo peak in DOS (cf. Fig. 2a) our DFT+ED solution corresponds to the Kondo singlet state. When SOC is included, and the spin is not a good quantum number, there are a minor changes in the char- acter for= 27:5 (nd8), the GS solution j iN=30: GS is a singlet, and the exited triplet consists of an ef- fectivejJ= 1;Jz=1;0;1idegenerate states which are 0.5 eV higher in the energy. The DOS has a peak in DOS very near EF. It is seen that weak 3 d-shell SOC plays no essential role for the Co impurity in the Cu host. These calculations show that our DFT+ED approach is capable to reproduce the Kondo singlet for Co in the bulk Cu for nd= 8, in agreement with conclusions of DFT+CTQMC [9]. Also, in agreement with commonly accepted point of view [32], we show that the presence of SOC does not lead to essential modication of a Kondo model.4 TABLE I: The total number of particles ( d-shell + bath) N, the expectation values h jJzj i,h jLzj i,h jSzj iangu- lar momenta, non-zero probabilities Pdnto nd the atomic eigenstatesjniwith integer occupation dnfor GS and low- energy excitation energies for dierent values of . without SOC Energy (eV) JzLzSzPd7Pd8Pd9 =27.4 eV,nd= 8.05 N=30 -148.5822 0. 0. 0. 0.20 0.51 0.26 -148.1014 0.53 0 0.53 0.22 0.55 0.20 -148.1014 0 0 0 0.22 0.55 0.20 -148.1014 -0.53 0 -0.53 0.22 0.55 0.20 with SOC Energy (eV) JzLzSzPd7Pd8Pd9 =27.5 eV,nd= 7.99 N=30 -149.4028 0. 0. 0. 0.19 0.51 0.27 -148.9296 0.94 0.49 0.45 0.21 0.55 0.21 -148.9296 0. 0. 0. 0.21 0.55 0.21 -148.9296 -0.94 -0.49 -0.45 0.21 0.55 0.21 FIG. 2: DOS for the Co in the bulk Cu without SOC for = 27.4 eV (a), and with SOC for = 27.5 eV (b). B. Co on Cu(001) Now we turn to a salient aspect of our investigation, the Co adatom on Cu(001) surface. Considering a value ofas a parameter, we analyse the ground state (GS) ofEq.( 1) with and without SOC for dierent values of . Making use of grand-canonical averages at low temper- aturekBT=1= (1=500) eV (20K) we calculate the expectation values of total number of electrons ( d-shell + bath)hNi, the charge uctuation ( hN2ihNi2)1 2near the GS, the expectation values of spin ( S), orbital (L) and total spin-orbital ( J) moments, and show them in Table II together with the d-shell occupation ndfor the GS, and corresponding Pnprobabilities, with and without SOC. For the values of = 26 eV and 27 eV, the GS is the eigenstatej iN=26, and is a combination of d7(Pd7 0:3) andd8(Pd80:6). These state have a non-integer ndoccupation due to hybridization of the atomic d-states with the substrate. Nevertheless, the ( hN2ihNi2)1 2 0 pointing on the absence of charge uctuations. The Svalues lie between of S= 3=2 (the atomic d7,4F), andS= 1 (the atomic d8,3F), while the Lis close to the atomic L= 3. The expectation values of the z-axis projections of the total h jJzj i, orbitalh jLzj i, and spinh jSzj iangular momenta for GS and low-energy excitation energies for = 27.0 eV are shown in Tab. III. It is seen that without SOC the GS can be interpreted as S= 1-like triplet. For = 28 eV, the GS is the eigenstate j iN=27, and the contributions of d7(Pd70:1) andd8 (Pd80:5) are reduced while d9,2D(Pd90:3) is increased. Again, there are no charge uctuations near the GS. This GS looks similar to S= 1=2 doublet (see Tab. III). When the SOC is included, for the values of = 26 eV, 27 eV the eigenstate j iN=26is split to the lowest energy singlet plus excited doublet (see Tab. III). These states approximately correspond to jJ= 1;Jzieigenstates of the eective Hamiltonian [29], ^HMA=D^J2 z+E(^J2 x^J2 y); (4) with the uniaxial magnetic anisotropy D4:5 meV, and E= 0. For= 28 eV, the GS remains j iN=27doublet. The corresponding densities of d-states (DOS) for the values of= 26 eV, 27 eV, 28 eV are shown in Ap- pendix B Fig. 5. There is are similarities in the DOS with and without SOC: no peak in DOS in a close vicin- ity ofEF. For these values of and without SOC there are no singlet GS, and no Kondo resonances in the DOS. In a presence of SOC, even their GS become singlets for = 26;27 eV, no Kondo peaks are formed. For = 28 eV the GS solution remains a doublet without Kondo resonance in the DOS. Since the change in the GS with the variation of  between 27 eV and 28 eV is observed, we further adjust the values of in order to keep the same nd8 without and with the SOC. In case of =27.4 eV and without the SOC, we obtain a non-integer hNi=26.55, non-zero (hN2ihNi2)1 20:5 charge uctuations, and nd=7.93. This solution is formally close to \ d8" state but actually a combination of of d7(Pd70:21),d8(Pd80:58), and d9(Pd90:18) atomic states (see Tab. II).5 TABLE II: The chemical potential (eV), the occupation hNi, uctuation (hN2ihNi2)1 2,ndoccupation, non-zero probabil- itiesPdnto nd the atomic eigenstates jniwith integer occupation dn, spin, orbital and total moments of the impurity d-shell for dierent values of . Grand-canonical averages are at low temperature kBT=1= (1=500) eV. without SOC (eV)hNi(hN2ihNi2)1 2ndPd6Pd7Pd8Pd9S L J 26 26.00 0.00 7.57 0.05 0.34 0.56 0.03 1.10 3.07 3.40 27 26.00 0.01 7.74 0.03 0.27 0.62 0.08 1.03 3.01 3.32 27.4 26.55 0.50 7.93 0.02 0.21 0.58 0.18 0.94 2.87 3.15 28 27.00 0.00 8.17 0.01 0.14 0.51 0.33 0.82 2.68 2.91 with SOC (eV)hNi(hN2ihNi2)1 2ndPd6Pd7Pd8Pd9S L J 26 26.00 0.00 7.58 0.05 0.34 0.57 0.04 1.09 3.07 3.89 27 26.00 0.00 7.75 0.03 0.26 0.62 0.08 1.03 3.01 3.82 27.6 26.38 0.48 7.96 0.02 0.20 0.58 0.19 0.93 2.86 3.51 28 27.00 0.00 8.17 0.01 0.14 0.51 0.33 0.82 2.68 3.16 TABLE III: The total number of particles ( d-shell + bath) N, the expectation values h jJzj i,h jLzj i,h jSzj ian- gular momenta, non-zero probabilities Pdnto nd the atomic eigenstatesjniwith integer occupation dnfor GS and low- energy excitation energies for dierent values of . without SOC Energy (eV) JzLzSzPd7Pd8Pd9 =27.0 eV N=26 -142.2319 0.0 0.0 0.0 0.27 0.62 0.08 -142.2319 0.90 0.0 0.90 0.27 0.62 0.08 -142.2319 -0.90 0.0 -0.90 0.27 0.62 0.08 =27.4 eV N=26 -145.3478 0.00 0.0 0.00 0.23 0.61 0.13 -145.3478 0.81 0.0 0.81 0.23 0.61 0.13 -145.3478 -0.81 0.0 -0.81 0.23 0.61 0.13 N=27 -145.3490 0.57 0.0 0.57 0.19 0.55 0.23 -145.3490 -0.57 0.0 -0.57 0.19 0.55 0.23 =28.0 eV N=27 -150.1992 0.53 0.0 0.53 0.14 0.51 0.33 -150.1992 -0.53 0.0 -0.53 0.14 0.51 0.33 with SOC Energy (eV) JzLzSzPd7Pd8Pd9 =27.0 eV N=26 -142.3054 0.00 0.0 0.00 0.26 0.62 0.08 -142.3009 1.48 0.91 0.57 0.26 0.62 0.08 -142.3009 -1.48 -0.91 -0.57 0.26 0.62 0.08 =27.6 eV N=26 -146.9950 0.00 0.0 0.00 0.21 0.61 0.16 -146.9912 1.10 0.70 0.40 0.21 0.61 0.16 -146.9912 -1.10 -0.70 -0.40 0.21 0.61 0.16 N=27 -146.9931 1.43 0.95 0.48 0.18 0.54 0.26 -146.9931 -1.43 -0.95 -0.48 0.18 0.54 0.26 =28.0 eV N=27 -150.2373 1.37 0.91 0.45 0.14 0.51 0.33 -150.2373 -1.37 -0.91 -0.45 0.14 0.51 0.33 There is a peak near EFin the DOS shown in Fig. 3(a). Note that similar peak in DOS was obtained in CTQMC calculations [10] without SOC with the same choice of the Coulomb-Uand the exchange- J, andnd=8 very close to our calculations. In Ref. [10] it is interpreted as a spectral FIG. 3: DOS for the Co@Cu(001) as a function of = 27.5 eV without SOC (a), and with SOC (b), = 27.4 eV without SOC (c), and 27.6 eV with SOC (d). signature of the Kondo eect. As follows from Eq.( 3) the presence of such a peak signals the (quasi)-degeneracy of the eigenvalues EN, andEN1. These are thej iN=27 doublet andj iN=26triplet states which dier in the energy by 1.2 meV (see Tab. III), with the doublet GS j iN=27. Since there is no singlet GS, the DOS peak at6 EFis not a Kondo resonance, and signals the presence of valence uctuations [29]. When the SOC is included, and with =27.6 eV, there is a non-integerhNi=26.38, with non-zero charge uctu- ations (hN2ihNi2)1 20:5, andnd=7.96 (see Tab. II). Again, the DOS has a peak at EFwhich is shown in Fig. 3(d). In this case, the the (quasi)-degeneracy occurs between the singlet j iN=26state being 1.9 meV lower in the energy than the j iN=27doublet (see Tab. III). The DOS peak at EFdue toj iN=26-to-j iN=27transition can be interpreted as a Kondo resonance. For the singlet GS we can use the renormalized per- turbation theory [13] in order to esimate the Kondo tem- perature, TK= 4ZIm[
(EF)]; (5) where ^ZTr[(^IdRe[()])=d(EF)]1A(EF)] Tr[A(EF)] is a quasiparticle weight, and A(EF) is the DOS ma- trix from Eq.(3). We obtain Z=0.097, and corresponding TK= 0:019 eV (220 K). It exceeds the experimen- tal estimate TK= 88 K [3] of the Kondo scale. Indeed, Eq. (5) serves as an order of magnitude estimate of TK. The scanning tunnelling spectroscopy measures the dif- ferential conductance G(V) through the adatom, and al- lows to probe the DOS. Comparison between the ex- perimental and theoretical G(V) is the most direct way to distinguish between dierent theoretical approxima- tions and to identify the most appropriate theoretical ap- proach. Experimentally G(V) of Co@Cu(100) was studied in Ref. [3]. Observed step-like behaviour was interpreted in terms of interference between two tunnelling channels: (i) tunnelling to the d-DOS shown in Fig. 5, and (ii) tun- nelling into the conduction electrons of the Cu substrate modied by the presence of the Co adatom. At the low bias, the dierential conductance is then expressed [30] in the basis of cubic harmonics as, G(!)X m 1 + m((1qm2) Im[Gm(!)] + 2qmRe[Gm(!)]
); (6) whereGm(Gmm) is a Green's function of the Hamil- tonian Eq.( 1), m Im[
m(EF)] is a hybridization between the d-levelmand the substrate shown in Fig. 1C, andqmis a Fano parameter. For the strongly localized Co adatom d-orbitals [31], qmRe[G0;m(EF)]=Im[G0;m(EF)]: The calculatedG(V) is in a fair quantitative agreement with the experimental data [3]. Note that our results seem to agree with the experiments better than those of Ref. [17]. Contrary to proposal of the Ref. [17], attempt- ing to explain the zero-bias anomaly in Co@Cu(100) as the results of inelastic spin excitations, our theory demonstrates that they can be better explained from the point of view of the "Kondo" physics. FIG. 4: Dierential conductance Gcalculated making use of the Eq.( 6). IV. SUMMARY The many-body calculations within the multi-orbital SIAM for the Co adatom on the Cu(100) surface are performed. DFT calculations were used to dene the in- put for the discrete bath model of forty bath orbitals, and the SOC included. We found that the peak in the DOS atEFcan occur for the Co atom d-shell occu- pationnd8, and is connected to quasi-degenerate ground state of the SIAM. Without SOC, the lowest energy state is an eective S= 1=2-like doublet, and next to it there is an eective S= 1-like triplet, so the resonance in the DOS( EF) does not represent a Kondo resonance. When SOC is included, the triplet states are split like jJ= 1;Jzieigenstates in a presence of the magnetic anisotropy ^HMA=D^J2 z, so that the jJ= 1;Jz= 0isinglet becomes a ground state. The cor- responding DOS( EF) peak corresponds to the Kondo res- onance. This solution is veried by comparison with ex- perimentally observed zero-bias anomaly in the dieren- tial conductance. Our calculations illustrate the essential role which the SOC, and corresponding uniaxial magnetic anisotropy, is playing in a formation of Kondo singlet in the multi-orbital low-dimensional systems. V. ACKNOWLEDGMENTS Financial support was provided by Operational Pro- gramme Research, Development and Education nanced by European Structural and Investment Funds and the Czech Ministry of Education, Youth and Sports (Project No. SOLID21 - CZ.02.1.01/0.0/0.0/16 019/0000760), and by the Czech Science Foundation (GACR) grant No. 21-09766S. The work of A.I.L. is supported by European Research Council via Synergy Grant 854843 - FAST- CORR.7 Appendix A: Fitting the bath hybridization With the specic choice of the Cartesian reference frame (see Fig. 1), the local Green's function G0(z) be- comes diagonal in the basis of cubic harmonics m= fxz;yz;xy;x2y2;3z2r2g. Moreover, it is convenient to use the imaginary energy axis over the Matsubara fre- quenciesi!n. The corresponding non-interacting Green's function of the Eq.(1) will then become G0;m(i!n) =1 i!nm
m(i!); with the hybridization function
m(i!n) =i!nmG1 0;m(i!n): (A1) Thus, the hybridization function Eq. (A1) can be eval- uated making use of the local Green's function G0(z). The discrete bath model is built by nding bath ener- gies and amplitudes which reproduce the continuous hy- bridization function as closely as possible. ~
m(i!n) =KX k=1V2 km i!nkm: (A2) The tting is done by minimizing the residual function, fm(fkm;Vkmg) =PN! n=11 ! n~
m(i!n)
m(i!n)2 ; (A3) using the limited-memory, bounded Broyden{Fletcher{ Goldfarb{Shanno method [33, 34], with the parameters kmandVkmas variables. The factor1 ! nwith = 0:5 is used to attenuate the signicance of the higher frequen- cies. TABLE IV: Values of the dshell
CF(eV), the bath energies k m(eV), and hybridisation parameters Vk m(eV) evaluted from LDA . m xz yz xy x2y23z2r2
CF -0.043 -0.043 0.117 0.053 -0.082 k=1;m-2.16 -2.16 -1.99 -2.01 -2.57 Vk=1;m0.85 0.85 0.65 0.65 0.72 k=2;m-0.08 -0.08 0.001 -0.02 -0.05 Vk=2;m0.18 0.18 0.08 0.10 0.13 k=3;m0.51 0.51 1.45 0.53 0.43 Vk=3;m0.36 0.36 0.55 0.34 0.32 k=4;m7.56 7.56 7.80 8.16 7.72 Vk=4;m2.08 2.08 2.12 1.78 1.70Appendix B: DOS as a function of for Co on Cu(001) FIG. 5: DOS for the Co@Cu(001) with and without SOC as a function of = 26 eV (a), 27 eV (b), and 28 eV (c)8 [1] G. D. Mahan, Many-particle physics ( Springer Science & Business Media, Boston, MA, 2000). [2] P. Monod, Phys. Rev. Lett. 19, 1113 (1967). [3] N. Knorr, M. A. Schneider, L. Diekh oner, P. Wahl and K. Kern , Phys. Rev. Lett. 88, 096804 (2002). [4] A. Zhao, Q. Li, L. Chen, H. Xiang, W. Wang, S. Pan, B. Wang, X. Xiao, J. Yang , J. Hou et al. ,Science 309, 1542 (2005). [5] A. Abrikosov, Phys. Phys. Fiz. 2, 5 (1965). [6] H. Suhl, Phys. Rev. 138, A515 (1965). [7] Y. Nagaoka, Phys. Rev. 138, A1112 (1965). [8] P. Wahl, L. Diekh oner, M. A. Schneider, L. Vitali, G. Wittich and K. Kern, Phys. Rev. Lett. 93, 176603 (2004). [9] B. Surer, M. Troyer, P. Werner, T. O. Wehling, A. M. L auchli , A. Wilhelm and A. I. Lichtenstein, Phys. Rev. B85, 085114 (2012). [10] A. Valli, M. P. Bahlke, A. Kowalski, M. Karolak, C. Her- rmann and G. Sangiovanni, Phys. Rev. Res. 2, 033432 (2020). [11] U. Fano, Phys. Rev. 1241866 (1961). [12] A. N. Rubtsov , V. V. Savkin and A. I. Lichtenstein, Phys. Rev. B 72, 035122 (2005). [13] A. C. Hewson The Kondo Problem to Heavy Fermions , ( Cambridge University Press, Cambridge, 1993) [14] P. Hohenberg and W. Kohn, Phys. Rev. 136, B864 (1964). [15] N. Parragh, A. Toschi, K. Held and G. Sangiovanni, Phys. Rev. B 86, 155158 (2012). [16] M. Wallerberger, A. Hausoel, P. Gunacker, A. Kowalski, N. Parragh, F. Goth, K. Held and G. Sangiovanni, Com- puter Physics Communications 235, 388 (2019). [17] J. Bouaziz, F. S. M. Guimar~ aes and S. Lounis, Nat. Com- mun. 11, 1 (2020).[18] G. Kresse and J. Furthm uller , Comput. Mater. Sci. 6, 15 (1996). [19] A. Georges, G. Kotliar, W. Krauth and M. J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). [20] A. I. Lichtenstein and M. I. Katsnelson, Phys. Rev. B 57, 6884 (1998). [21] E. Wimmer, H. Krakauer, M. Weinert and A. J. Freeman, Phys. Rev. B 24, 864 (1981). [22] W. Mannstadt and A. J. Freeman, Phys. Rev. B 55, 13298 (1997). [23] D. Jacob, J. Phys.: Condens. Matter 27, 245606 (2015). [24] A. MacDonald, W. Pickett and D. Koelling , J. Phys. C: Solid State Phys. 13, 2675 (1980). [25] M. Tchaplianka, A. B. Shick, J. Kolorenc, J. Phys.: Conf. Ser.2164 , 012045 (2022). [26] K. Yosida, Phys. Rev. 147, 223 (1966). [27] V. I. Anisimov, J. Zaanen and O. K. Andersen, Phys. Rev. B 44, 943 (1991). [28] I. V. Solovyev, P. H. Dederichs and V. I. Anisimov, Phys. Rev. B 50, 16861 (1994). [29] M. Tchaplianka, A. Shick and A. Lichtenstein, New J. Phys. 23, 103037 (2021). [30] K. R. Patton, S. Kettemann, A. Zhuravlev and A. Licht- enstein Phys. Rev. B 76, 100408(R) (2007). [31] T. O. Wehling, H. P. Dahal, A. I. Lichtenstein, M. I. Katsnelson, H. C. Manoharan and A. V. Balatsky , Phys. Rev. B 81, 085413 (2010). [32] G. Bergmann, Phys. Rev. Lett. 57, 1460 (1986). [33] C. Zhu, R. H. Byrd , P. Lu and J. Nocedal, ACM Trans. Math. Software 23, 550 (1997). [34] J. L. Morales and J. Nocedal, ACM Trans. Math. Soft- ware 38, 1 (2011).

1611.01442v2.Limits_on_dynamically_generated_spin_orbit_coupling__Absence_of__l_1__Pomeranchuk_instabilities_in_metals.pdf

Limits on dynamically generated spin-orbit coupling: Absence of l= 1Pomeranchuk instabilities in metals Egor I. Kiselev,1Mathias S. Scheurer,1Peter Wölfle,1, 2and Jörg Schmalian1, 3 1Institut für Theorie der Kondensierten Materie, Karlsruher Institut für Technologie, 76131 Karlsruhe, Germany 2Institut für Nanotechnologie, Karlsruher Institut für Technologie, 76344 Karlsruhe, Germany 3Institut für Festkörperphysik, Karlsruher Institut für Technologie, 76344 Karlsruhe, Germany An ordered state in the spin sector that breaks parity without breaking time-reversal symmetry, i.e., that can be considered as dynamically generated spin-orbit coupling, was proposed to explain puzzling observations in a range of different systems. Here we derive severe restrictions for such a state that follow from a Ward identity related to spin conservation. It is shown that l= 1 spin-Pomeranchuk instabilities are not possible in non-relativistic systems since the response of spin-current fluctuations is entirely incoherent and non-singular. This rules out relativistic spin- orbit coupling as an emergent low-energy phenomenon. We illustrate the exotic physical properties of the remaining higher angular momentum analogues of spin-orbit coupling and derive a geometric constraint for spin-orbit vectors in lattice systems. I. INTRODUCTION Fermi liquid (FL) theory [1–4] forms the intellectual foundation of our understanding of quantum fluids such as3He, fermionic atomic gases, as well as simple met- als and numerous strongly-correlated systems. The in- herent instability of the FL state was analyzed early-on by Pomeranchuk [5], who derived the threshold values (2l+ 1)for the phenomenological Landau parameters Fs;a lbeyond which newly ordered states are expected to emerge. Here, lcorresponds to the angular momentum channel under consideration and s(a) refers to order in the charge (spin) sector. These criteria were derived from an analysis of the quasiparticle contribution to the energy of a FL. Famous examples of Pomeranchuk in- stabilities are phase separation for Fs 0! 1, ferromag- netism forFa 0! 1, or charge- and spin nematic order forFs;a 2!5. Our main interest here will be the behav- ior forFa 1!3, which was proposed by Wu and Zhang [6] and which would lead to a state with order in the spin sector that breaks parity while time-reversal sym- metry remains intact. Such order has been invoked to explain the behavior in systems as diverse as chromium [7, 8], Sr 3Ru2O7[9], the hidden order in URu 2Si2[10], or the physics in the vicinity of a ferromagnetic quantum phase transition [11]. The associated dynamic generation of spin-orbit coupling of Ref. 6 was analyzed in great de- tail in Ref. 12. Even3He was argued to be in the vicinity of a corresponding instability [6]. In this paper, we demonstrate that l= 1Pomeranchuk instabilities in the spin channel are not possible. Specif- ically, the divergence in the quasiparticle susceptibility is canceled by a vanishing vertex that connects the re- sponse of quasiparticles and bare fermions. As a conse- quence, the response of the system is entirely incoherent and the analysis of the energy balance of Ref. 5 turns out to be incomplete. The identical result is obtained within FL theory if the proper form of the spin current intermsofthequasiparticledistributionfunctionisused.The origin of this peculiar behavior is that the spin cur- rent, which is closely related to the instability, is itself not a conserved quantity, yet it is the current of a con- served density. This allows to draw rigorous conclusions from associated Ward identities that exclude second or- der phase transitions. At the level of the Hartree-Fock approximation, this has been found earlier in Ref. 13. Our arguments imply that the absence of a second-oder instability for l= 1is, in fact, exact. Following an argu- ment by Bloch (see Refs. 14 and 15), we also show that a first order transition is not allowed either. While some of those conclusions for Pomeranchuk instabilities could have been drawn from the vast literature on the FL the- ory and its microscopic foundations (see in particular the workbyLeggettinRef.16), theongoingdiscussionofthis instability in Refs. 6–12 seems to warrant a detailed anal- ysis of this issue. In addition, it is shown that relativistic spin-orbit coupling cannot emerge due to spontaneous symmetry breaking of the electron liquid at low energies. Spontaneously generated spin-orbit interactions are only expected in higher angular momentum channels with un- conventional residual symmetry groups exhibiting exotic physical behavior such as enhanced anomalous Hall con- ductivities. Finally, the implications of our findings for lattice systems are discussed. II. POMERANCHUK INSTABILITY The phenomenological formulation of FL theory is based on the celebrated parametrization of the energy change due to quasiparticle excitations [1, 2] Eqp=P k" knqp k, with single-particle energy " k=vF(jkjkF) ++1 FX k0;0F;0 k;k0nqp k00;(1) where,kF,vF=m mv0 F,F=m m0 F, andm=m denote, respectively, the Fermi energy, momentum, ve-arXiv:1611.01442v2 [cond-mat.str-el] 1 Apr 20172 Figure 1. Distortion of the spin-up (green) and spin-down (blue) Fermi surface for the l= 1Pomeranchuk instability in the (a) charge and (b) spin channel. locity, the density of states, and the mass renormaliza- tion. For the interaction function we use the usual ex- pansion for isotropic Fermi surfaces in the absence of spin-orbit coupling, F;0 k;k0=Fs k;k0+0Fa k;k0,Fr() =P1 l=0Fr lPl(cos), with Legendre polynomials Pland cos=ek
ek0. Herer=sandr=afor the symmet- ric (charge) and anti-symmetric (spin) channel, respec- tively. Pomeranchuk concluded that an instability with a spontaneous deformation of the Fermi surface (see, e.g., Fig. 1) kF!kF+ks F;l('kF) +ka F;l('kF)(2) occurs, when Fs;a l! (2l+ 1). Here'kFis the angle of the Fermi wave vector relative to some arbitrarily chosen axis. Specifically, the energy change due to quasiparticle excitations was found to be [5] Eqp=1 0 Fm mX l;r=fs;agjnqp;r lj2 1 +Fr l 2l+ 1 +O (nqp)4 : (3) Here,nqp;r lis the change in the quasiparticle occupa- tion in momentum space caused by kr F;l('kF)and acts as order parameter of the new phase. If m=mremains finite and the coefficient of jnqp;r lj2becomes negative, an instability to a state with nqp;r l6= 0is energetically fa- vored. Thishappensonce Fr lreachestheabovethreshold value. The response of quasiparticles can also be charac- terized by the quasiparticle susceptibility [6] r qp;l=F 1 +Fr l 2l+1: (4) Forl= 0this corresponds to the well known expressions of the charge and spin susceptibilities for the symmetric (s) and antisymmetric ( a) channel, respectively. r qp;l diverges as Fr l! (2l+ 1), which coincides with the Pomeranchuk instability criterion. Below we demonstrate that the leading order expan- sion of the full energy with respect to the electron mo- mentum distribution nr 1is rigorously given by Er l=1=1 0 Fjnr 1j2+O (nr 1)4 ; (5)ruling out the corresponding Pomeranchuk instability and demonstrating that the analysis of the energy (3) of quasiparticles is not sufficient. This is closely related to the fact that the susceptibility of the system is not adequately expressed in terms of its quasiparticle contri- bution in Eq. (4). In the microscopic foundation of FL theory [17], one dividesthesingle-particleGreen’sfunction Gk(!)inaco- herentquasiparticlecontributionandanincoherentback- ground, Gk(!) =Z !+i0+" k+Ginc: k(!): (6) Zis the spectral weight and " kthe single-particle energy in Eq. (1). A detailed analysis of the susceptibilities of a many-body system was given by Leggett [16], who finds r l= ( r lZ)2r qp;l+r inc;l: (7) Here,r qp;lis the quasiparticle response of Eq. (4) and r lthe vertex that connects the response of quasiparti- cles and bare fermions. Finally, r inc;lis the incoherent response of the system, a contribution that is directly re- lated to the incoherent part Ginc: k(!)of the single-particle Green’s function (6). The incoherent response of the sys- tem can now be expressed as r inc;l=r(!) llim !!0lim q!0r l(q;!); (8) taking into account that r qp;lvanishes in this limit. Next we discuss implications for susceptibilities that are caused by conservation laws, i.e., associated with a Hermitian operator that commutes with the Hamilto- nianHof the system, [;H] = 0:An example is charge or particle-number conservation (  =N). In a non- relativistic system with Hamiltonian H=Z r y (r) ~2r2 2m+U(r) (r) +1 2Z r;r0 y (r) y (r0)V(rr0) (r0) (r);(9) the conservation of the components of the total spin  =Sjis another example. Here we use ( y ) to represent the annihilation (creation) of a bare fermion of spinand apply the conventionR r:::=R ddrP :::. Let us first focus on the case without crystal potential U(r), which yields the bare dispersion "k=~2k2 2m, and comment on the implications of a finite crystal po- tential at the end. We write  =R ddr(r)with density  q= 1 VP k; y k+q 2 k kq 2in momentum space and form factor k. The charge density corresponds to  k= ,andaspindensityamountsto  k=j ,j= 1;2;3, with Pauli matrices j. Let us assume that  qcom- mutes with the interacting (non-quadratic) part Hintof the Hamiltonian,   q;Hint = 0: (10)3 This is the case, e.g., for interactions of the form Hint= f[f qg]such asHint=P qVq q q. Most importantly, this also holds for both the spin and charge density in the case of the non-relativistic solid-state Hamiltonian (9) with electron-electron Coulomb interaction. If Eq. (10) applies, we can derive a Ward identity (see the Appendix A) that implies (q=0;!6= 0) = 0 (11) for the susceptibility of the conserved density  qand (q)ij Jij J(q!0;0) =X k;    k2@nk @ki@"k @kj;(12) forthesusceptibility Jofthecurrent Jassociatedwith  q. In Eq. (12), nkdenotes the momentum occupation and the limit q!0has to be performed along the ith direction after the limit !!0, which is indicated by the superscript (q). Note, Eq. (12) is valid for an arbitrary dispersion"kand not limited to Galilean invariant sys- tems. The fact that the susceptibility of Eq. (12) is finite also excludes critical phases that might exist without a finite order parameter. The implication of Eq. (11) for susceptibilities of con- served quantities is obvious and well established. Let the form factor be  k=j ,j= 0;3. Then follows from Eq.(11)that s;a inc;0=s;a(!) 0 = 0, i.e., theentireresponse of the system is coherent. The same Ward identity can alsobeusedtoshowthat s;a 0=Z1andleadstothewell known relation r 0=r qp;0with quasiparticle susceptibil- ity given in Eq. (4). This means that the susceptibility of a conserved density is fully determined by the quasipar- ticle response. Using the formalism of Ref. 16, one can also determine the next order corrections in q=!, r 0(q;!) =q2 !2m m 1 +Fr 1 3 +Oq !4 :(13) Let us next exploit the Ward identity (12) for the cur- rent. If the Fermi energy is the largest energy scale, we can replace@"k @kibykF mcos'kin the current operator J showing that the charge ( k=0) and spin ( k=3) current susceptibilities correspond to l= 1instabilities withr=sandr=a, respectively. Analyzing Eq. (12) yields our key result s;a 1=0 F, which, via Legendre transformation, leads to Eq. (5). A Pomeranchuk in- stability in the l= 1channel is therefore not possible. If one further uses the continuity equation !2 q2(q;!) =J(q;!)J(q;0)one can identify the vertex and incoherent contribution for the spin- current susceptibility [16] r 1=Z1m m 1 +1 3Fr 1 ; r inc;1=0 F 1m m 1 +1 3Fr 1 ;(14)where we used Eq. (13). Let us analyze Eq. (14) for the charge and spin response separately. For a Galilean- invariant system, the charge current is itself a conserved quantity and Eq. (11) implies that s inc;1= 0. Thus, we recover the celebrated resultm m= 1 +1 3Fs 1. In addi- tion, it follows s 1=Z1ands 1=s qp;1=0 F. While no Pomeranchuk instability will take place in the l= 1 charge channel, these results are fully consistent with the analysis of Ref. 5 as the entire response in the l= 1 charge channel of Galilean-invariant systems is coherent and captured by quasiparticle excitations. The situation is different for currents that are, them- selves, not conserved quantities, such as the spin current. If one approaches the Pomeranchuk threshold value, the vertex a 1vanishes and there is no contribution of the susceptibility due to quasiparticles. The divergence of the quasiparticle contribution of the susceptibility is sup- pressed by the vanishing vertex. As a consequence, the entire response becomes incoherent and the energetic analysis that led to Eq. (3) is not applicable, leading to Eq. (5) instead. It is interesting to note, however, that the correct re- sult for the physical spin current susceptibility may be obtained within FL theory if the proper form of the spin current is used (see the Appendix B). There we also de- rive the response of the spin (charge) momentum current density to an external field of l= 2symmetry, which is again very different from the l= 2quasiparticle suscep- tibilities of Eq. (4), i.e., 26=qp;2. The arguments given so far exclude critical behavior in thel= 1channel with diverging susceptibility. To ex- clude a state with finite order parameter nr 1that might be reached via a first order transition, we first note that na 1(ns 1) is proportional to the expectation value of J with=3(=0). We can then apply the arguments of Refs. 14 and 15 to show that, given any state with fi- nite expectation value of J, there is always a state with lower energy. Consequently, there is also no first order transition to an l= 1Pomeranchuk phase. III. CONSTRAINTS ON SPONTANEOUS GENERATION OF SPIN-ORBIT COUPLING Theresultspresentedaboveyieldstrongrestrictionson the interaction-induced generation of spin-orbit coupling
H=Z k; y kg(k)
 k;g(k) =g(k);(15) with order parameter g(k). Assume that the Hamilto- nian (such as the non-relativistic Hamiltonian in Eq. (9) withU= 0) has a symmetry group that is a direct product SO (d)L SO(3)Sin orbital and spin space with d= 3(d= 2) for three-dimensional (two-dimensional) systems. Wecanthenexpand g(k)intermsofbasisfunc- tions (spherical harmonics Yl;mford= 3andeil'kfor d= 2)transformingundertheirreduciblerepresentations4 Figure 2. Part (b) and (d) show two examples of complex tra- jectoriesz(t)that are allowed by Eq. (17) with corresponding spin-textures (for = +1) given in (a) and (c), respectively. In the limit of a spherical Fermi surface, the texture of (a) coincides with the l= 3Dresselhaus coupling in Eq. (16). ofSO (d)L. Notingthatthebasisfunctionsof l= 1aresu- perpositions offkjg, we conclude that the l= 1channel can be discarded. Since l 1 = (l+1)l(l1), Eq. (15) cannot contain a term that is invariant under SO (d)L+S, the set of simultaneous spin and orbital rotations, which is the point group of a system with relativistic spin-orbit coupling. In this sense, relativistic spin-orbit coupling cannot occur as an emergent low-energy phenomenon. However,spin-orbitcouplinginhigherangularmomen- tumchannels( l3)canbegeneratedspontaneouslyand if this does occur, some rather exotic behavior follows as we discuss next. Focusing for simplicity on d= 2, the remaining possible spin-orbit vectors are of the form g(k) =g0(cos(l'k);sin(l'k);0)T; l= 3;5;:::;(16) withresidualsymmetrygroupSO (2)L3=lS3generatedby thecombination L3=lS3oftheout-of-planecomponents L3andS3of the orbital and spin angular momentum operators. For the upper (lower) sign, Eq. (16) can be seen as generalizations of the Rashba (Dresselhaus) spin- orbit term with gwindingw=l(w=l) times on the Fermi surface. The generalized Dresselhaus term with l= 3is illustrated in Fig. 2(a). The unconventional form (16) of the spin-orbit cou- pling has interesting physical consequences: The Berry curvature (finite in the presence of a Zeeman termP k y kh3 k) is enhanced by a factor of lcompared to the usual Rashba-Dresselhaus scenario which affects many electronic properties [18]. E.g., the anomalous Hall conductivity xy[22], relating an applied electricfield to a perpendicular electric current for h6= 0, is enhanced by a factor of l,xy=l
xyjl=1. To illus- trate another consequence of a spin-orbit vector of the form (16), let us assume that, in analogy to the pro- posal of Ref. 19, the system shows superconductivity in thes-wave channel. Describing the latter on the mean- field level by
2P k y k" y k#+H.c. and focusing on the relevant regime [19] where the chemical potential lies in the Zeeman-induced gap at k= 0, we follow Ref. 20 and project the theory onto the lower, effectively spin- less, band. We find that this low-energy model exhibits (k1ik2)l-wave pairing. Recalling that l3, this not only corresponds to very exotic pairing but also leads to a topological class-D invariant =land, thus,lchiral Majorana modes at the edge of the system. IV. LATTICE EFFECTS Finally, let us discuss the modifications in the pres- ence of a lattice, i.e., when the periodic potential U(r) in Eq. (9) is finite. Since this additional term again commutes with the spin and charge density, we can still exclude all phases with order parameter of the form Ojj0=P k y kjkj0 k,j= 0;1;2;3,j0= 1;:::;d. Note that this even holds when ions are taken into account as dynamical degrees of freedom since all interactions are functions of the electron and ion density alone. The main difference is that we cannot rule out any of the ir- reducible representations of the lattice point group Gp sincekjandkjf('k)transform identically under Gpfor any function fthat is invariant under Gp. Nonetheless, our results still lead to significant restrictions. To see this, let us first introduce fermionic operators cnq, with band index nand crystal momentum q, diagonalizing the non-interacting part H0of the Hamiltonian with result- ing bandstructure Enq. Assuming that only one band n=n0is relevant, it holds Ojj0=P qcy n0qjvj0(q)cn0q, where the summation is restricted to the first Brillouin zoneandvj(q)@qjEn0q. Toproceed, wefocuson d= 2 and treat the interaction-induced spin-orbit coupling on the mean-field level, i.e., addP qcy n0qg(q)
cn0qwith g(q) = (R1(q);R 2(q);0)Tto the Hamiltonian where R1 (R2) transforms as x(y) underGpand=1. Evalu- ating the constraint hOjj0i= 0in the regime where the Fermi energy is the largest relevant energy scale of the system, we obtain the equivalent condition Z'ir(Gp) 0d'w(')jR(')jei(')= 0;(17) where'is the polar angle parameterizing the Fermi sur- face (restricted to the irreducible part of the Brillouin zone, 0< ' < ' ir(Gp)),w(') =k2 F(')=j^nF(')kF(')j is a Fermi surface weight function, depending on the Fermi momentum kFand the Fermi-surface nor- mal ^nF, and(')denotes the angle between R(') and ^nF(')[see Fig. 2(a)]. Upon defining z(t) :=5 Rt 0d'w(')jR(')jei('), we see that any spontaneously generated spin-orbit texture must lead to a closed tra- jectoryfz(t)j0< t < ' ir(Gp)g. This restriction is il- lustrated in Fig. 2(b-d) for the point group Gp=C4 where'ir= 2and the boundary condition (0) = ('ir)mod2is dictated by rotational symmetry. Most importantly, we see in Fig. 2(c-d) that, as opposed to the continuum limit Gp=SO(2)L, spin-orbit vectors with net winding w=1are possible albeit with much more complex structure than the conventional Rashba or Dres- selhaus spin-orbit coupling as dictated by Eq. (17). V. CONCLUSION In summary, we have shown that neither a charge- nor a spin-Pomeranchuk instability with l= 1can occur in a non-relativistic metallic solid-state system. The diver- gence of the quasiparticle susceptibility in the l= 1spin channel and even the complete quasiparticle contribution is in fact removed by the vanishing of the vertex coupling quasiparticles and real electrons. The actual response may be calculated exactly and is found to be completely non-singular. The identical result follows within FL the- ory, if care is taken that the quasiparticle spin current receives a correction term induced by the quasiparticle energy change. Our findings imply that relativistic spin- orbit coupling with residual symmetry group SO (3)L+S cannot be spontaneously generated. Furthermore, any realistic lattice model involving spontaneously generated spin-orbit vectors g(q) = (R1(q);R2(q);0)Tmust sat- isfy the severe constraint in Eq. (17) which is illustrated geometrically in Fig. 2. ACKNOWLEDGMENTS We acknowledge fruitful discussions with B. Jeevanesan and S. Sachdev. Appendix A: Ward identity and its relation to susceptibilities In this appendix, we discuss how the Ward identity leading to Eqs. (11) and (12) of the main text is derived. WefirstnotethatifEq.(10)holds, thedynamicsofthe density will be governed by the noninteracting part H0of the Hamiltonian, @t q=i H0; q , and leads to a conti- nuity equation with current J=P k@"k @k y k k k. In order to determine the associated susceptibilities  andJ, we analyze the correlator L  k;k0;q(;0) =D T y k+q 2() kq 2() y k0q 2 (0) k0+q 2(0)E ; (A1)wheredenotesimaginarytimeand Tthetime-ordering operator. Following [21], one obtains from the Heisen- berg equation of motion of  qthe Ward identity for L  k;k0;q(;0). After Fourier transformation to frequen- cies, it reads X k; i  "k+q 2"kq 2  kL  k;k0;q(i ;i 0) =  k0 Gk0+q 2(i 0)Gk0q 2 (i 0+i ) ;(A2) whereGk(i )istheexactsingle-particleGreen’sfunction on the imaginary axis. Its retarded analytic continuation has been introduced in Eq. (6). The Ward identity (A2) allows to draw conclusions for the two susceptibilities (q;i ) =TX k;k0; 0X   k  k0L  k;k0;q(i ;i 0); (A3a) ij J(q;i ) =TX k;k0; 0X   k@"k @ki  k0@"k0 @k0 jL  k;k0;q(i ;i 0): (A3b) Multiplying both sides of Eq. (A2) by   k0and  k0@"k0 @k0 j, summation over k0, 0as well as ,readily leads (after analytic continuation i !!+i0+) to Eqs. (11) and (12), respectively. Appendix B: Response functions within FL theory In this section, we first show that the exact result for the spin-current susceptibility resulting from the Ward identity (A2) can alternatively be obtained within FL theory and then apply this approach to the l= 2spin and charge channel. 1. Spin-current susceptibility. To obtain the correct form for the spin current, we use that the distribution function nk(r;t)obeys the Landau-Boltzmann equation. Linearized in the external field and Fourier transformed, nk(r;t)nk(r;t) n0 k(r;t) =P q;!nk(q;!)eiqri!t, the latter reads (!q
vk)nk+q
vk@n0 k @"k"k=I(A1) where"k="ext k+1 FP k0;0F;0 k;k0nk00andvk= rk"k=k=mis the quasiparticle velocity, Iis the col- lision integral, and n0 kis the equilibrium (Fermi) distri- bution function. The applied external field leads to a quasiparticle energy contribution "ext k. In the case of interest to us (spin-current response) "ext k=1 mk
A, whereAis a spin-dependent vector potential.6 The spin conservation law !nSq
jS= 0follows by multiplying Eq. (A1) by and summing over k;, where the collision integral drops out on account of the spin conservation in two-particle collision processes. The spin density nSand the spin current density are defined as nS=X k;nk; jS=X k;vk2 4nk@n0 k @"k1 FX k0;0F;0 k;k0nk003 5: (A2) Weobservethatthespincurrentconsistsoftwocontribu- tions: a “direct” term and a “backflow” term. Using the usual expansion of the Landau interaction function F;0 k;k0 (given in the main text) and assuming nk/k
A, we have X k0;0F;0 k;k0nk00'Fa 1m kF2 vkX k0;00vk0nk00: (A3) In the limit of !;q!0, it holds nk=1 exp("k+"ext k) + 1 = "k1 mk
A@n0 k @"k(A4) with solution nk=@n0 k @"k1 m(k
A)1 1 +Fa 1=3:(A5)Substituting into the expression (A2) for the spin current density, we find jS=1 1 +Fa 1=3X k;2vkk
A m @n0 k @"k [1 +Fa 1=3] =1 3k2 F m2m mFA=n mA (A6) in agreement with the microscopic result following from the Ward identity. 2. The case l= 2. The phenomenological derivation using the kinetic equation may be extended to higher angular momentum channels, employing some additional assumptions. We consider the case l= 2. To begin with the spin chan- nel, the change in the quasiparticle distribution function caused by an external field in the spin channel of l= 2 symmetry, "ext k=1 mkkD; 6=;(A7) is given by nk=@n0 k @"k1 mkkD1 1 +Fa 2=5:(A8) The quasiparticle distribution is not an observable quan- tity. A possible observable is the momentum (charge or spin) current density. Although the (charge or spin) cur- rent is not a conserved quantity, we may still derive an expression for it from the kinetic equation (A1), !jS=X k;vk[!nk@n0 k @"k1 FX k0;0F;0 k;k0!nk00] =X k;vk2 4q
vkn0 k+Ik@n0 k @"k1 FX k0;0F;0 k;k0(q
vk0n0 k00+Ik00)3 5 (A9) where we defined n0 k=nk@n0 k @"k"k. The equation describes the change of the local current density by flow out of or into the volume element in terms of the diver- gence of the momentum spin current density and by relaxation processes, !jS;=X qijS;;(A10) where the momentum spin current tensor is defined by =X k;vk0 @vkn0 k@n0 k @"k1 FX k0;0F;0 k;k0vk0n0 k001 A: (A11) A possible instability of the system with respect to a deformation of the Fermi surface of d-wave type, as ex- pressed by the external field "ext kdefined above, should show up as a divergence of the susceptibility a l=2= D7 =X k;vk0 @vkn0 k D@n0 k @"k1 FX k0;0F;0 k;k0vk0n0 k00 D1 A: (A12) We may calculate a l=2by substituting n0 k=@n0 k @"k1 mkkD (A13) where the factor 1=(1 +Fa 2=5)present in nkhas dropped out. The resulting expression a l=2=1 5kF mn m(1 +Fa 1=3) (A14)does not diverge when 1 +Fa 2=5!0and is actually independent of Fa 2. The susceptibility does, however, diverge when m= 1 +Fs 1=3!0or vanishes at 1 + Fa 1=3 = 0, both signaling an instability of the system. The analogous derivation for the l= 2charge suscep- tibility gives s l=2=1 5kF mn m: (A15) [1] L. Landau, Sov. Phys. JETP 3, 920 (1957). [2] L. Landau, Sov. Phys. JETP 5, 101 (1957). [3] G. Baym and C. Pethick, Landau Fermi-Liquid Theory: Concepts and Applications (Wiley-VCH, 1991). [4] D. Vollhardt and P. Wölfle, The Superfluid Phases of Helium 3 (Dover Publications, 2012). [5] I. Pomeranchuk, Sov. Phys. JETP 8, 361 (1958). [6] C. Wu and S.-C. Zhang, Phys. Rev. Lett. 93, 036403 (2004). [7] J.E. Hirsch, Phys. Rev. B 41, 6820 (1990). [8] J.E. Hirsch, Phys. Rev. B 41, 6828 (1990). [9] Y. Yoshioka and K. Miyake, Journal of the Physical So- ciety of Japan 81, 023707 (2012). [10] C.M. Varma and L. Zhu, Phys. Rev. Lett. 96, 036405 (2006). [11] A. V. Chubukov and D. L. Maslov, Phys. Rev. Lett. 103, 216401 (2009). [12] C. Wu, K. Sun, E. Fradkin, and S.-C. Zhang, Phys. Rev.B75, 115103 (2007). [13] J. Quintanilla and A.J. Schofield, Phys. Rev. B 74, 115126 (2006). [14] D. Bohm, Phys. Rev. 75, 502 (1949). [15] N. Bray-Ali and Z. Nussinov, Phys. Rev. B 80, 012401 (2009). [16] A. Leggett, Physical Review 140, A1869 (1965). [17] A. A. Abrikosov, L. P. Gorkov, and I. E. Dzyaloshinskii, Methods of Quantum Field Theory in Statistical Physics (Dover Books on Physics) (Dover Publications, 1975), revised english ed., ISBN 9780486632285. [18] D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. 82, 1959 (2010). [19] J. D. Sau, R. M. Lutchyn, S. Tewari, and S. Das Sarma, Phys. Rev. Lett. 104, 040502 (2010). [20] J. Alicea, Phys. Rev. B 81, 125318 (2010). [21] U. Behn, Phys. Status Solidi 88, 699 (1978). [22] We here focus on the intrinsic, i.e., disorder-independent, contribution to the anomalous Hall conductivity.

1603.07544v3.Interplay_of_Coulomb_interaction_and_spin_orbit_coupling.pdf

arXiv:1603.07544v3 [cond-mat.str-el] 13 Jul 2016Interplay of Coulomb interaction and spin-orbit coupling J¨ org B¨ unemann,1,2Thorben Linneweber,3Ute L¨ ow,3Frithjof B. Anders,3and Florian Gebhard1 1Fachbereich Physik, Philipps-Universit¨ at Marburg, D-35 032 Marburg, Germany 2Institut f¨ ur Physik, BTU Cottbus-Senftenberg, D-03013 Co ttbus, Germany 3Lehrstuhl f¨ ur Theoretische Physik II, Technische Univers it¨ at Dortmund, D-44227 Dortmund, Germany (Dated: Version of June 15, 2016, approved by JB, FG) We employ the Gutzwiller variational approach to investiga te the interplay of Coulomb interaction and spin-orbit coupling in a three-orbital Hubbard model. A lready in the paramagnetic phase we find a substantial renormalization of the spin-orbit coupli ng that enters the effective single-par- ticle Hamiltonian for the quasi-particles. Only close to ha lf band-filling and for sizable Coulomb interaction we observe clear signatures of Hund’s atomic ru les for spin, orbital, and total angular momentum. For a finite local Hund’s-rule exchange interacti on we find a ferromagnetically ordered state. The spin-orbit coupling considerably reduces the si ze of the ordered moment, it generates a small ordered orbital moment, and it induces a magnetic anis otropy. To investigate the magnetic anisotropy energy, we use an external magnetic field that til ts the magnetic moment away from the easy axis (1 ,1,1). PACS numbers: 71.10.Fd,71.27.+a,71.70.Ej,75.10.Lp I. INTRODUCTION In atomic physics, the spin-orbit coupling (SOC) plays an important role because it determines the value of the total angular momentum in the ground state according to Hund’s third rule. After maximizing the total spin s (first rule) and the total orbital moment l(second rule), the quantum number for the total angular momentum is j=|l−s|(j=l+s) below (above) half filling (third rule).1,2ThethirdruleappliesinthelimitwheretheSOC is small compared to the average Coulomb interaction of the electrons, i.e., for all ‘light atoms’, including transi- tion metals. Note that the quantum numbers sandlare, in fact, well defined only in the limit of a vanishing SOC. For atoms in a solid, the situation is obviously much more complicated because neither of the three quantum numbers s,l, orjis well defined due to a breaking of the rotational symmetry. Yet, we know that some of the basic mechanisms of Hund’s rules are still relevant. For example, the maximization of the spin is a direct consequence of intra-atomic exchange correlations that are caused by the electronic Coulomb interactions. The verysameCoulombinteractionisthereasonformagnetic order in solids, e.g., in ferromagnets. The SOC is only a small perturbation to the dominant Coulomb interaction in transition metals and their compounds. Nevertheless, it can have profound consequences, e.g., for the direction of the magnetic moment, the so-called ‘easy axis’. From a theoretician’s point of view, the analysis of the interplay and/or competition of a strong local Coulomb interaction and a (comparatively small) SOC in a solid is rather demanding. Even the study of simplifying models for the Coulomb interaction, such as multi-orbital Hub- bard models, poses a tremendously difficult task. Any study of such models is possible with a limited numeri- cal accuracy only, e.g., in determining the ground-state energy. Giventhe fundamentaluncertaintiesin thetreat- ment of the sizable Coulomb correlations it is non-trivialto come to firm conclusions on the effects of the SOC. Therefore, most theoretical studies on the interplay of Coulomb interaction and SOC focused on insulating or spin states and/or assumed a rather large SOC.3–6For the study of (itinerant) 4 d, 5dorfelectron systems, the dynamical mean field theory has been used frequently in recent years, see, e.g., Refs. [7–10]. In such systems, however, the SOC tends to be significantly larger than in transition metals and their compounds that we have primarily in mind in our present model study. In this work, we employ the Gutzwiller approach11 to investigate approximate variational ground states for multi-orbital Hubbard models. The analytical evalua- tion of expectation values for Gutzwiller wave functions poses a difficult many-body problem that requires addi- tional approximations. Most often applied in the context of multi-band models is the ‘Gutzwiller approximation’ which becomes exactfor the Gutzwiller wavefunctions in thelimitofinfinitespatialdimensions.11–14Itcanbeused to evaluate expectation values for a large set of model parameters, see Sect. IIB. This allows us to study sys- tematically the subtle interplay of Coulomb correlations and spin-orbit coupling. We consider a Hubbard model with three degenerate t2gorbitals on a three-dimensional cubic lattice. In the first part of our investigation we concentrate on the in- terplay of Coulomb interaction and spin-orbit coupling for paramagnetic metallic ground states. We find that the Coulomb interaction enhances the effective SOC be- tween the quasi-particles. In addition, we investigate the significance of Hund’s rules. Only Hund’s first rule ap- proximately applies in strongly correlated paramagnetic metallic systems. It is well known that for a finite (local) exchange in- teraction, multi-orbital Hubbard models tend to favor ferromagnetic states for sufficiently large Coulomb inter- actions. In the second part of our investigation we in- vestigate if and to what extent the ferromagnetic states2 are modified by the spin-orbit coupling. We find that the SOC opposes the formation of ferromagnetic order in metals. While, in the absence of SOC, the ordered moment has no preferred direction, the SOC aligns it along the ‘easy-axis’, and induces a small ordered orbital moment. Recently, the Gutzwiller method and the density func- tional theory (DFT) were combined in a self-consistent manner;15,16aformalderivationcanbefoundinRef.[17]. The Gutzwiller-DFT was applied to a number of materi- als, for example to nickel and iron, see Refs. [17,18], and references therein. From a methodological point of view, our model study in this work provides a first step to- wards a self-consistent treatment of the SOC within the Gutzwiller-DFT scheme. This work is organized as follows. In Sect. II we intro- duceourmodelandsummarizetheGutzwillervariational approach. In Sect. III we discuss our results for param- agnetic and ferromagnetic ground states. Summary and conclusions, Sect. IV, close our presentation. Technical details are deferred to two appendices. II. MODELS AND METHOD In this section, we introduce our model and explain the Gutzwiller variational approach that we use for its investigation. A. Hamiltonian We study a Hubbard model with three t2gorbitals per site on a simple-cubic lattice in three dimensions. The Hamiltonian of this system has the form ˆH=ˆH0+ˆHC+ˆHso, (1) whereˆH0denotes the kinetic energy of the electrons, ˆHC describes their Coulomb interaction, and ˆHsomodels the spin-orbit coupling. 1. Kinetic energy and density of states We consider electrons that move between t2gorbitalsb andb′on sitesiandjof our simple-cubic lattice with L sites. In second quantization the single-particle Hamilto- nian reads ˆH0=/summationdisplay i/\e}atio\slash=j/summationdisplay σ,σ′tσ,σ′ i,jˆc† i,σˆcj,σ′, (2) where we introduce the combined spin-orbital index σ≡(b,s), b∈ {1,2,3}, s∈ {↑,↓}.(3) The crystal-field energies are set to zero, tσ,σ′ i,i= 0.-1.5-1-0.5 00.511.50 01 12 23 34 45 56 67 70 0.2 0.4 0.6 0.8 1DOS( ) EFnσσn EFDOS( ) FIG. 1: Density of states at the Fermi energy EFas a function ofEF(blue) and the orbital occupation nσ(black). We use the standard parameterization for the hopping amplitudes in (2) with some generic Slater-Koster pa- rameters19 t(1),(2),(3) π = 0.3t,−0.1t,0.025t, (4) t(2),(3) σ= 0.1t,0.01t, (5) t(1),(2),(3) δ= 0.1t,−0.025t,0.02t (6) for the electron transfers up to 3rd nearest neighbors. By including hoppings beyond the nearest neighbors we makesure that there are no artificial features in ourband structure, suchasnestingvectorsorparticle-holesymme- try. In transition metal compounds, the value of tis of the order of 1eV. In our pure model study in this work, we will simply set t= 1 as our energy unit. The single-particle Hamiltonian (2) can be readily di- agonalized in momentum space, ˆH0=/summationdisplay k/summationdisplay σ,σ′εk;σ,σ′ˆc† k,σˆck,σ′ (7) with the bare dispersion εk;σ,σ′≡1 L/summationdisplay i/\e}atio\slash=jtσ,σ′ i,jeik(Ri−Rj), (8) andkfrom the first Brillouin zone. The remaining task isthe diagonalizationofthe 6 ×6matrix εk;σ,σ′foreachk to obtain the (bare) band structure. For non-interacting electrons, all energy levels up to the Fermi energy EF are filled in the ground state. The corresponding density of states at the Fermi-energy EFis shown in Fig. 1 as a functionofboth EFandoftheaverageorbitaloccupation 0≤nσ≤1. The total bandwidth is W≈3.4. Apparently, the Hamiltonian for the kinetic energy is not particle-holesymmetric, as can be seen from the den- sity of states at the Fermi energy. Fig. 1 clearly shows3 that DOS( nσ)/\e}atio\slash= DOS(1 −nσ). To study the influence of the spin-orbit coupling, we shall later investigate a particle-hole symmetric kinetic energy. For this case, we use the somewhat artificial Slater-Koster parameters for electron transfers between nearest neighbors only, t′(1) π= 0.2, t′(1) δ= 0.1, (9) which lead to a symmetric density of states of band- widthW′= 2. Inourferromagneticcalculationswefocusonthefilling nσ≈0.4 where the (paramagnetic) density of states has a maximum at the Fermi energy. At such a maximum we canexpect a strongertendency towardsferromagnetic order according to the Stoner criterion.20 2. Local interactions The Coulomb and spin-orbit interaction are assumed to be purely local, ˆHC=/summationdisplay iˆHi;C,ˆHso=/summationdisplay iˆHi;so.(10) The local Coulomb interaction for a model with three degenerate t2gorbitals reads21 2ˆHi;C=U/summationdisplay b,sˆni,b,sˆni,b,¯s+/summationdisplay b(/\e}atio\slash=)b′ s,s′(U′−δs,s′J)ˆni,b,sˆni,b′,s′ +J/summationdisplay b(/\e}atio\slash=)b′/bracketleftbigg/parenleftig ˆc† i,b,↑ˆc† i,b,↓ˆci,b′,↓ˆci,b′,↑+h.c./parenrightig +/summationdisplay sˆc† i,b,sˆc† i,b′,¯sˆci,b,¯sˆci,b′,s/bracketrightbigg ,(11) where we use the convention ¯↑=↓,¯↓=↑, and ˆni,b,s= ˆc† i,b,sˆci,b,scounts the electrons with spin sin orbital b on sitei. Note that for t2g-orbitals the three parame- ters in (11) are not independent because they obey the symmetry relation U′=U−2J.21 For the SOC we use ˆHi;so=/summationdisplay σ,σ′ǫso i;σ,σ′ˆc† i,σˆci,σ′. (12) When weworkwith the followingorderforourlocalbasis |σ/a\}bracketri}ht, |1/a\}bracketri}ht=|yz,↑/a\}bracketri}ht,|2/a\}bracketri}ht=|yz,↓/a\}bracketri}ht,|3/a\}bracketri}ht=|xz,↑/a\}bracketri}ht,...,|6/a\}bracketri}ht=|xy,↓/a\}bracketri}ht, (13) the six-dimensional SOC matrix ˜ ǫsoin (12) has the well- known form ˜ǫso=−iζ 2 0−˜σ3˜σ2 ˜σ30−˜σ1 ˜σ2˜σ10 ≡ζ˜Σ (14)with the standardtwo-dimensionalPaulimatrices ˜ σ1, ˜σ2, ˜σ3, and the SOC constant ζ. The local Hamiltonian ˆHi;loc=ˆHi;C+ˆHi;so (15) in the 64-dimensional local Hilbert space is readily diag- onalized, ˆHi;loc|Γ/a\}bracketri}hti=EΓ|Γ/a\}bracketri}hti. (16) For parameter values that are typical for transition met- als,ζ/J= 0.2...1.0 andJ/U= 0.2, the atomic spec- trum has a generic form. In table I we list the degener- ate eigenspacesof ˆHi;loc, orderedby increasingenergy for given particle number 0 ≤nloc≤6. We give the degen- eracygof each level, its total spin s, orbital moment l, and total ‘angular momentum’ j. #nlocgslj 101000 1141/213/2 2121/211/2 125112 223111 321110 425022 521000#nlocgslj 161000 1521/211/2 2541/213/2 141110 243111 345112 445022 541000 #nlocgslj 1343/203/2 2341/223/2 3361/225/2 4321/211/2 5341/213/2 TABLE I: Degenerate eigenspaces of ˆHi;loc, ordered by energy for a given particle number 0 ≤nloc≤6 with a specification of the degeneracy g, total spin s, orbital moment l, and total ‘angular momentum’ j. Since the rotational symmetry is broken in our cubic environment, the quantum numbers landjdo, in fact, not label eigenstates of the total ‘angular momentum’ operator. It is well known, however, that in the t2gsub- space we have ˜l2=/summationdisplay i∈{x,y,z}˜l2 i= 211 (17)4 for the vector ˜lof the three matrices ˜lx= 0 0 0 0 0 i 0−i 0 , ˜ly= 0 0−i 0 0 0 i 0 0 , ˜lz= 0 i 0 −i 0 0 0 0 0 . (18) Hence, the orbital moment behaves like that of l= 1 states (‘T-P equivalence’),21because/a\}bracketle{tˆl2/a\}bracketri}ht= 1(1+1) = 2. To be more precise, one finds ˜li=−˜l(l=1) i (19) where on the r.h.s. we introduced the representation of the orbital momentum for ( l= 1)p-orbitals. Due to the T-P equivalence we can label the multiplet states |Γ/a\}bracketri}htby a quantum number jthat formally corresponds to a to- tal angular momentum of l= 1 orbitals. Table I shows that Hund’s rules are still valid for the ground states of all particle numbers if we make the replacement l→ −l in Hund’s third rule, as a consequence of eq. (19). In particular, as seen from table I, the local spectrum is not particle-hole symmetric. As we will show in Sect. IIIA1, the particle-hole asymmetry induced by the SOC is vis- ible in our itinerant three-band lattice model even when we work with a symmetric density of states. B. Gutzwiller wave functions and energy functional 1. Wave functions Forthe variationalinvestigationofthe Hamiltonian(1) we use the Gutzwiller wave functions |ΨG/a\}bracketri}ht=/productdisplay iˆPi|Ψ0/a\}bracketri}ht, (20) where|Ψ0/a\}bracketri}htis a normalized single-particle product state (Slater determinant) and the local Gutzwiller correlator is defined as ˆPi=/summationdisplay Γ,Γ′λi;Γ,Γ′|Γ/a\}bracketri}htii/a\}bracketle{tΓ′| ≡/summationdisplay Γdλi;Γd|Γd/a\}bracketri}htii/a\}bracketle{tΓd|.(21) Here, weintroducethe matrix ˜λiof(complex) variational parameters λi;Γ,Γ′which allows us to optimize the occu- pation and the form of the eigenstates |Γd/a\}bracketri}htiofˆPi. We assume that the matrix ˜λiis Hermitian which en- sures that the eigenstates |Γd/a\}bracketri}htiexist and form a basis of the local Hilbert space. Without SOC it is usually a sen- sible approximation to work with a diagonal (and hencereal) matrix ˜λi. For a finite SOC, however, it is essen- tial to include at least some non-diagonal elements in ˜λi. In this work, we will take into account all non-diagonal parameters in λi;Γ,Γ′with states |Γ/a\}bracketri}htiand|Γ′/a\}bracketri}htithat have the same particle number. The evaluation of expectations values with respect to the wave function (20) poses a difficult many-particle problem that cannot be solved in general. As shown in Refs.[11,22], itispossibletoderiveanalyticalexpressions forthe variationalground-stateenergyin thelimit ofinfi- nite spatial dimensions ( D→ ∞). An application of this energy functional to finite-dimensional systems is usually termed ‘Gutzwiller approximation’. It will also be used in this work. One should keep in mind, however, that the Gutzwiller approximationhas its limitations, and the study of some phenomena requires an evaluation of ex- pectation values in finite dimensions.23,24 Since the energy functional of the Gutzwiller approx- imation has been derived in detail in previous work, we will only summarize the main results in this section. In the following we are only interested in systems and wave functions that are translationally invariant. Hence, we shall drop lattice-site indices whenever this does not lead to ambiguities. 2. Constraints As shown in Refs. [11,22] it is most convenient for the evaluation of Gutzwiller wave functions in infinite spatial dimensions to impose the following (local) constraints /a\}bracketle{tˆP†ˆP/a\}bracketri}htΨ0−1≡gc 1(˜λ,|Ψ0/a\}bracketri}ht) = 0,(22) /a\}bracketle{tˆc† σˆP†ˆPˆcσ′/a\}bracketri}htΨ0−Cσ′,σ≡gc σ,σ′(˜λ,|Ψ0/a\}bracketri}ht) = 0 (23) for the local correlation operators ˆP≡ˆPi. Here, we introduce the local density matrix ˜C≡˜Ciwith the ele- ments Ci;σ,σ′=/a\}bracketle{tˆc† i,σ′ˆci,σ/a\}bracketri}htΨ0. (24) Note that the order of indices in (24) has been chosen deliberately because it slightly simplifies the analytical results in Sect. IIB5. The constraints can be evaluated by means of Wick’s theorem; explicit expressions are given in Appendix A. In systems with a high symmetry, the matrix ˜Cis often diagonal, e.g., for dorbitals in a cubic environment. In such a case, one usually has to take into account only the diagonal constraints (23), because the l.h.s. of (23) for σ/\e}atio\slash=σ′is automatically zero for all values of ˜λithat are included in the variational Ansatz. Here, the matrix ˜Cis non-diagonalin oursystemwith afinite SOC.Evenifone introducesalocalbasiswhichhasadiagonallocaldensity matrixwith respectto |Ψ0/a\}bracketri}ht, seeAppendix A,onestill has to take into account some non-diagonal constraints.5 3. Expectation values Each local operator ˆOi, e.g., the operator ˆHi;so, can be written as ˆOi=/summationdisplay Γ,Γ′OΓ,Γ′ˆmi;Γ,Γ′, (25) ˆmi;Γ,Γ′≡ |Γ/a\}bracketri}htii/a\}bracketle{tΓ′|. (26) In infinite dimensions the expectation value of ˆOihas the form /a\}bracketle{tˆOi/a\}bracketri}htΨG=/summationdisplay Γ1,Γ2,Γ3,Γ4OΓ2,Γ3λ∗ Γ2,Γ1λΓ3,Γ4/a\}bracketle{tˆmi;Γ1,Γ4/a\}bracketri}htΨ0, (27) where the remaining expectation values m0 i;Γ,Γ′≡ /a\}bracketle{tˆmi;Γ,Γ′/a\}bracketri}htΨ0 (28) can readily be evaluated using Wicks theorem, see Ap- pendix A. The expectation value of a hopping operator in infinite dimensions reads ( i/\e}atio\slash=j) /angbracketleftbig ˆc† i,σ1ˆcj,σ2/angbracketrightbig ΨG=/summationdisplay σ′ 1,σ′ 2qσ′ 1σ1/parenleftig qσ′ 2σ2/parenrightig∗/angbracketleftbig ˆc† i,σ′ 1ˆcj,σ′ 2/angbracketrightbig Ψ0,(29) where an analytical expression for the (local) renormal- ization matrix qσ′ σis also given in Appendix A. Note that the matrix qσ′ σis, in general, neither real nor Hermi- tian. Any symmetries among its elements are caused by those of the orbital basis states |σ/a\}bracketri}htand the form of the Gutzwiller wave function. For example, if we have no SOC and no magnetic or orbital order in our degenerate three-band system, the renormalization matrix has the simple form qσ′ σ=δσ,σ′√qwith only one renormalization factor for all orbitals. 4. Structure of the energy functional In a translationally invariant system, the expectation values that we introduced in the previous section lead to the following variational energy functional (per lattice site) EG/parenleftbig˜λ,|Ψ0/a\}bracketri}ht/parenrightbig =/summationdisplay σ1,σ2 σ′ 1,σ′ 2qσ′ 1σ1/parenleftig qσ′ 2σ2/parenrightig∗ Eσ1,σ2,σ′ 1,σ′ 2 +/summationdisplay Γ,Γ1,Γ2EΓλ∗ Γ,Γ1λΓ,Γ2m0 Γ1,Γ2.(30) Here, we introduce the tensor Eσ1,σ2,σ′ 1,σ′ 2≡1 L/summationdisplay i/\e}atio\slash=jtσ1,σ2 i,j/a\}bracketle{tˆc† i,σ′ 1ˆcj,σ′ 2/angbracketrightbig Ψ0 =1 L/summationdisplay kεk;σ1,σ2/angbracketleftbig ˆc† k,σ′ 1ˆck,σ′ 2/angbracketrightbig Ψ0(31)with the bare dispersion εk;σ,σ′from eq. (8). The energy (30) is a function of λΓ,Γ′and|Ψ0/a\}bracketri}htwhere |Ψ0/a\}bracketri}htenters(30), (31)solelythroughthe(non-interacting) density matrix ˜ ρwith the elements ρ(iσ),(jσ′)≡ /a\}bracketle{tˆc† j,σ′ˆci,σ/a\}bracketri}htΨ0. (32) Note that the non-local elements of ˜ ρ(i/\e}atio\slash=j) determine the tensor (31) while its local elements ρ(iσ),(iσ′)=Ci;σ,σ′, (33) as introduced in eq. (24), enter the elements qσ′ σof the renormalization matrix, the expectation value (28), and the constraints (22), (23). The energy EG=EG(˜λ,˜ρ,˜C) (34) has to be minimized with respect to the variational pa- rameters λΓ,Γ′and the density matrices ˜ ρand˜Cobeying the constraints (22), (23), (33), and ˜ρ2= ˜ρ . (35) Thisadditionalconstraintensuresthat ˜ ρcorrespondstoa Slater determinant |Ψ0/a\}bracketri}ht. Note that introducing the local density matrix ˜Cas an independent variational object in (34), at the expense of the additional constraint (33), is actually not necessary. Instead one could consider the energysolelyasafunction of ˜λ,˜ρ. Ourformofthe energy functional, however, will turn out to be slightly more convenient because, in the Gutzwiller approximation, ˜ ρ enters the energy in a non-linear way only through its local elements. 5. Minimization of the energy functional We introduce the real and the imaginary parts of the variational parameters λΓ,Γ′=λ(r) Γ,Γ′+iλ(i) Γ,Γ′. (36) Due to the Hermiticity of ˜λwe have λ(r) Γ,Γ′=λ(r) Γ′,Γ, (37) λ(i) Γ,Γ′=−λ(i) Γ,Γ′→λ(i) Γ,Γ= 0, (38) which leads to a number nvof independent (and real) variational parameters λ(r/i) Γ′,Γ. They will be considered as the components vzof thenv-dimensional vector v= (v1,...,v nv)T. (39) The (in general) complex constraints (22), (23) are not all independent, e.g., because of the Hermiticity of ˜ gc.6 We denote the set of all independent real and imaginary parts of (22), (23) by the ncreal constraints gl(v,˜C) = 0 ( l= 1,...,n c). (40) The constraints (33), (35), and (40) are implemented via Lagrangeparameters ησ,σ′, Ω(iσ),(jσ′), and Λ l. This leads to the Lagrange functional LG≡EG(v,˜ρ,˜C)−/summationdisplay lΛlgl(v,˜C) −/summationdisplay σ,σ′ησ,σ′/summationdisplay i(Cσ′,σ−ρ(iσ′),(iσ)) −/summationdisplay i,j/summationdisplay σ,σ′Ω(iσ),(jσ′)[˜ρ2−˜ρ](jσ′),(iσ)(41) which provides the basis of our minimization. As shown, e.g., in Ref. [25], the minimization of (41) with respect to ˜ ρleads to the effective single-particle Hamiltonian ˆHeff 0=/summationdisplay i,j/summationdisplay σ,σ′(¯tσ,σ′ i,j+δi,jησ,σ′)ˆc† i,σˆcj,σ′(42) with the renormalized hopping parameters ¯tσ1,σ2 i,j(v,˜C) =/summationdisplay σ′ 1,σ′ 2qσ1 σ′ 1(v,˜C)/parenleftig qσ2 σ′ 2(v,˜C)/parenrightig∗ tσ′ 1,σ′ 2 i,j.(43) The optimal Slater determinant |Ψ0/a\}bracketri}htis the ground state ofˆHeff 0, ˆHeff 0|Ψ0/a\}bracketri}ht=Eeff 0|Ψ0/a\}bracketri}ht. (44) From the minimization of (41) with respect to ˜Cwe ob- tain an equation for ησ,σ′in (42), ησ,σ′=∂ ∂Cσ,σ′EG−/summationdisplay lΛl∂ ∂Cσ,σ′gl.(45) Finally, the minimization with respect to v ∂ ∂vZEG−/summationdisplay lΛl∂ ∂vZgl= 0 (46) determines the Lagrange parameters Λ land the optimal value of v. Equations (42)–(46) need to be solved self- consistently. In Appendix B we explain in more detail how we solve this problem numerically. Note that our minimization algorithm does not require the constraints gl(v,˜C) to be independent. This is a major advantage over the method that we had proposed in the earlier work [25]. III. RESULTS In the following we discuss the paramagnetic and the ferromagnetic cases separately.A. Paramagnetic ground states 1. Effective spin-orbit coupling Without any breaking of spin or orbital symmetries, the minimization of the Gutzwiller energy functional leads to effective on-site energies (45) that have the same form as the SOC (14) but with the coupling constant ζ replacedby ζeff. Thischangefromthebaretoaneffective coupling constant also changes the quasi-particle disper- sion ofˆHeff 0. Therefore, the energy splittings at certain high-symmetrypoints as seen in ARPES experiments are a measure for the effective, not the bare spin-orbit cou- pling. Note that extracting the quasi-particle dispersion from our Gutzwiller approach relies on a Fermi-liquid interpretation.26However, all changes of the effective single-particle Hamiltonian, e.g., energy shifts, are re- lated to changes of certain ground-state expectation val- ues. Sincethelatterarevariationallycontrolled,itisvery likely that the exact single-particle spectrum reflects the same trends. 0.2 0.4 0.6 0.811.52 nσrSOC FIG. 2: SOC-renormalization rSOCas a function of the orbital occupation nσforJ= 0,ζ= 0.05 (red), ζ= 0.1 (blue), ζ= 0.2 (black) and U= 2,3,4 (in ascending order). In Fig. 2 we show the renormalization of ζ, rSOC≡ζeff/ζ (47) as a function of the orbital occupation nσfor the three bare values ζ= 0.05,0.1,0.2 and interaction parameters U= 2,3,4 andJ= 0. Apparently, the effective spin- orbit coupling increases as a function of U, apart from a small region of an almost filled shell where ζeff(U)< ζ. ForU= 4, which is approximately equal to the band width, the spin-orbit couping can be renormalized by a factor two or more, rSOC(U= 4,nσ= 2/3,ζ= 0.2)≈ 2.3. This substantial increase is clearly visible in the quasi-particle band structure, see Fig. 3, where we show the bare ( U=J= 0) and renormalized band structures (U= 4,J/U= 0.2) fornσ= 2/3 andζ= 0.2. For7✲ ✁ ✂ ✲ ✄✁ ☎ ✲ ✄✁ ✂ ✲ ✂✁ ☎ ✂✁ ✂ ✂✁ ☎ ✄✁ ✂✥ ● ✆ ✥ ✝ ✆ ● ✝ ❊✞❊ ❋ FIG. 3: Quasi-particle bands along high-symmetry lines for ζ= 0.2,nσ= 2/3,U= 0,J= 0 (red), and U= 4,J/U = 0.2 (black). example, the splitting of the bands at the Γ-point and the R-point is noticeably enhanced in presence of the Coulomb interaction. Therenormalization rSOCisnotmonotonousasafunc- tion of the bare coupling ζ. Moreover, it is notparticle- hole symmetric, i.e., it is not invariant under the trans- formation nσ→1−nσ. This is only partly caused by the particle-hole asymmetry of the bare density of states in Fig. 1. As discussed already in Sect. IIA2, the SOC in- herently breaks the particle-hole symmetry. To illustrate this point, we show the results for a symmetric density of states that results from the nearest-neighbor electron transfers(9), displayedintheinsetofFig.4. Asseenfrom the figure, the SOC alone induces a particle-hole asym- metry in the renormalization of the effective spin-orbit 0.2 0.4 0.6 0.80.811.21.41.61.8 0.2 0.4 0.6 0.80.811.21.41.61.8 -1-0.5 00.5 10123456 nσEFDOSrSOC FIG. 4: SOC-renormalization rSOCfor the symmetric density of states from the nearest-neighbor electron transfers (9) , as a function of the orbital occupation nσforζ= 0.1,J= 0, U= 3 (black) and U= 4 (blue); inset: density of states at the Fermi energy.0.2 0.4 0.6 0.80.911.11.21.3 nσrSOC FIG. 5: SOC-renormalization rSOCas a function of the orbital occupation nσforJ/U = 0.2,ζ= 0.05 (red), ζ= 0.1 (blue), ζ= 0.2 (black), and U= 1 (circles), U= 2 (squares). 0.2 0.4 0.6 0.80.40.60.811.2 nσrSOC FIG. 6: SOC-renormalization rSOCas a function of the orbital occupation nσforJ/U = 0.2,ζ= 0.05 (red), ζ= 0.1 (blue), ζ= 0.2 (black), and U= 2.5. coupling. We not in passing, that band structures with a fairly similar density of states may, nevertheless, display a very different nσdependence of rSOC. Apparently, the full momentum dependence of the band structure deter- mines the details of the rSOC-curves. For finite values of the exchange interaction J, the ef- fective coupling constants are smaller than for J= 0, in general. This can be seen in Figs. 5 and 6 where we show the renormalization for J/U= 0.2, andU= 1,2 (Fig. 5), U= 2.5 (Fig. 6). Note that for U=Uc<∼2.5 there appears a Brinkmann-Rice type of insulating phase27at half filling where the renormalization matrix ˜ qis zero. Therefore we could perform our calculations shown in Fig. 6 only away from half filling. The dependence of the renormalization on the band- fillingnσappears to be even more complicated for fi-8 niteJ, in particular in the region around half filling. One must keep in mind, however, that there is a ‘triv- ial’ contribution to the renormalization of ζwhich sim- ply stems from the band-width renormalization induced by the renormalization matrix qσ′ σ. To understand this effect, we consider, for the sake of argument, a renormal- ization matrix of the simplest form qσ′ σ=δσ,σ′√q. In that case, the effective hopping parameters in (42) are given by ¯tσ,σ′ i,j=qtσ,σ′ i,j. Hence, in order to obtain the same expectation values of |Ψ0/a\}bracketri}htas in the non-interacting limit, we must introducea scaling ζ→qζ < ζ. The effect of the enhancement of ζeffis therefore amplified by the renormalization of the hopping parameters. For a more quantitative analysis, we define an average value ¯qof the bandwidth renormalization through ¯q=/a\}bracketle{tˆH0/a\}bracketri}htG//a\}bracketle{tˆH0/a\}bracketri}ht0, (48) i.e., ¯qquantifies the reduction of the average kinetic en- ergy in presence of the Coulomb interaction. The rel- ative SOC-renormalization is then plotted in Fig. 7 for the same parameters as in Fig. 5. It shows that the non- trivial renormalization is, in fact, largest in the region around half filling. Moreover, it is actually fairly inde- pendent of the bare SOC, a feature that cannot be seen in the original representation of the data in Fig. 5. 0.2 0.4 0.6 0.81234 nσrSOC/q- FIG. 7: Relative SOC-renormalization rSOC/¯qas a function of the orbital occupation nσforJ/U = 0.2,ζ= 0.05 (red), ζ= 0.1 (blue), ζ= 0.2 (black) and ζ= 0.1 (blue), ζ= 0.2 (black), andU= 1 (circles), U= 2 (squares), U= 2.5 (diamonds). 2. Hund’s rules in a solid? In the introduction we raised the question if, and to what extent, Hund’s rules are still discernible in a solid. To clarify this issue, we define the three ‘quantum num-bers’s,l,jvia the local expectation values /a\}bracketle{tˆS2 i/a\}bracketri}htG=s(s+1), /a\}bracketle{tˆL2 i/a\}bracketri}htG=l(l+1), /a\}bracketle{t(ˆSi+ˆLi)2/a\}bracketri}htG=j(j+1). (49) 0.2 0.4 0.6 0.800.511.52 nσj,l,s FIG. 8: Quantum numbers j(black), l(blue),s(red) as a function of the orbital occupation nσforJ/U = 0.2,ζ= 0.05, andU= 1 (solid), U= 2 (dashed), U= 2.5 (dotted). Figure 8 shows these three numbers for ζ= 0.05, J/U= 0.2 andU= 1,2,2.5. The bars give the values in the atomic limit, as extracted from the ground states in table I. As expected, all quantum numbers move towards their atomic values when we increase the Coulomb inter- action parameters. This is best visible near half-filling when the system is close to the metal-insulator transi- tion that appears at half filling. 0.1 0.2 0.300.511.52 0.7 0.8 0.900.511.52j,l,s j,l,s nσ σn FIG. 9: Quantum numbers j(black), l(blue),s(red) as a function of the orbital occupation nσforJ/U = 0.2,ζ= 0.05, andU= 6 (solid), U= 9 (dashed).9 As shown in previous work,11,28this transition is of first-order where in the Gutzwiller insulating state all atomsareintheirgroundstate. Thismeansthatat Ucall three quantum numbers will jump to their atomic values athalffilling. Forallother(integer)fillings, thesystem is still rather itinerant and some of the quantum numbers, in particular j, deviate significantly from their atomic values. This is best visible at a filling of nσ= 2/3 where the value of jis far off its atomic value jatomic= 0. The results change only slightly when we increase the values ofU(andJ) as can be seen from Fig. 9 where we display j,l, andsfor larger values of Uaway from half filling. The difference between the behavior close to half fill- ing and the other integer fillings can be understood from the atomic spectra. The high-spin ground state at half filling is only slightly changed by a small SOC and, most importantly, its degeneracy is not lifted. Hence, the en- ergydifference between the Hund’s-rule ground state and the first excited state is of the order of J. In contrast, at all other integer fillings, the ground states are cre- ated by a splitting of the (degenerate) ground states at ζ= 0, caused by the SOC. Therefore, the energy differ- ence between the Hund’s-rule ground state and the first excited states is much smaller away from half filling. As a consequence, it is energetically not favorable to lose a lot of kinetic energy by only occupying the Hund’s-rule ground state. Unlike in the half-filled case, the Hund’s- rule ground state does not dominate the quantum num- bers in the metallic phase at or around other integer fill- ings. As seen from Figs. 8 and 9, only Hund’s first rule is seen to be obeyed in strongly correlated paramagnetic metals close to integer fillings. B. Ferromagnetic ground states Withoutthespin-orbitcoupling,theHamiltoniancom- mutes with the total spin operator. Hence, the energy of a ferromagnetic ground state cannot depend on the di- rection of the magnetic moment. For finite SOC, there is a preferred direction of the moment, the so-called ‘easy axis’. In order to find this axis, we minimize the energy functional with respect to |Ψ0/a\}bracketri}htwithout any bias on the magnetic-moment direction using a completely general matrixησ,σ′. It turns out that in our system and for the parameters considered in this section, the magnetic moment always points into the (1 ,1,1)-direction. 1. Ordered moment In Fig. 10 we display the total spin S≡ |/a\}bracketle{tˆSi/a\}bracketri}ht|for seven different values of ζ(0≤ζ≤0.3) as a function of Ufor J/U= 0.2. As seen from the figure, the SOC destabilizes the ferromagnetic order, i.e., the value Ucfor noticeable ferromagnetic order ( S >0.1) substantially increases as a function of ζ. Concomitantly, the ordered magnetic moment m= 2Sstrongly depends on the SOC as long as1 1.5 200.51S U FIG. 10: Spin Sin (1,1,1)-direction as a function of Uwith J/U = 0.2 fornσ= 0.4 andζ= 0.05 (blue), 0 .1 (red), 0 .15 (green), 0 .2 (maroon), 0 .255 (violet), 0 .3 (orange). the magnetic order is weak, S <1/2. The SOC becomes asmallperturbation onlyin the saturationregion, S >1. The SOC not only reduces the ordered spin moment, it also induces an orbital moment, i.e., L≡ |/a\}bracketle{tˆLi/a\}bracketri}ht|is non- zero. This is shown in Fig. 11 where we display Lfor the same parameters as in Fig. 10, apart from ζ= 0 where L= 0. Theorbitalcontributiontothemagneticmoment, however, remains rather small, of the order of 10% of the spin moment, especially for values of ζ <0.1 that are realistic for transition metals. Therefore, the gain in orbital moment does not compensate the loss in the ordered spin moment induced by the spin-orbit coupling. 1 1.5 200.050.1L U FIG. 11: Orbital moment Lin (1,1,1)-direction as a function ofUwithJ/U = 0.2 fornσ= 0.4 andζ= 0.05 (blue), 0 .1 (red), 0.15 (green), 0 .2 (maroon), 0 .255 (violet), 0 .3 (orange).10 0 0.5 101e-052e-053e-054e-055e-056e-05 1.3 1.4 1.501e-052e-053e-054e-055e-056e-05∆E α/α-1,2 1,2 FIG. 12: Anisotropy energy ∆ E1,2as a function of the magnetic-moment direction that is rotat ed, (1), from (1 ,0,0) to (1,1,1) (solid lines) and, (2), from (1 ,0,0) to (1,1,0) (dashed lines) with maximum rotation angles ¯ α1= arccos 1 /√ 3 and ¯α2=π/4; parameters: U= 1.2 (black), U= 1.25 (blue), U= 1.3 (red), U= 1.35 (green), U= 1.4 (orange), U= 1.45 (violet), U= 1.5 (maroon), J/U = 0.2,ζ= 0.1; inset: maximal anisotropy energy as a function of UforJ/U = 0.2,ζ= 0.1. 2. Anisotropy energy Finally, we take a look at the ‘anisotropy energy’, i.e., the dependence of the energy on the magnetic-moment direction. Tothisend, wecouldintroduceadditionalcon- straints that fix the moment direction during the mini- mization. However, this would require additional pro- gramming work that we prefer to avoid. Therefore, we apply an external magnetic field that allows us to change the magnetic-moment direction. In fact, this is how the anisotropy energy would actually be measured. Since our field is just a technical tool, we couple it to the spin only, i.e., we add ˆHB=−B/summationdisplay ieB·Si (50) to the Hamiltonian of our system. Here, eBis the direc- tion of the magnetic field that we adjust in our calcula- tions. The size of the field amplitude Bmust be chosen with care to obtain meaningful results. On the one hand, it must be large enough to force the magnetic moment into all directions that we aim to investigate, i.e., in theground state we must approximately find /a\}bracketle{tSi/a\}bracketri}htG||eB. On the other hand, the variation in the field contribution to the energy must be small compared to the variation of the system’s energy that we actually want to determine. Meeting these criteria becomes difficult, in particular, in the regionof small magneticmoments. In all calculations that we are going to present below, we found that a field amplitude of B= 0.002 leads to meaningful results for the anisotropy energy. In the following we consider rotations of the mag- netic moment from the (1 ,0,0) direction, (1), into the (1,1,1) direction, and, (2), into the (1 ,1,0) direction. The corresponding maximal rotation angles are ¯ α1= arccos(1/√ 3) and ¯α2=π/4, respectively. From our minimization we obtain the two energies E1,2(α) as a function of the angle α. Since, as mentioned before, the easy axis always points into the (1 ,1,1)-direction, we de- fine the anisotropy energy as ∆ Ei(α)≡Ei(α)−E1(¯α1). This quantity is displayed in Fig. 12 for several values of U(and consequently also different values of the magnetic moment) for J/U= 0.2 andζ= 0.1. The figure shows that, although the anisotropy energy is quite small, of the order of several ten µeV per site, our approach is11 1.2 1.4 1.6 1.800.00010.00020.00030.00040.0005∆Emax U FIG. 13: Maximal anisotropy energy as a function of Ufor J/U = 0.2,ζ= 0.2 (black), ζ= 0.15 (blue), ζ= 0.1 (red). perfectly capable to resolve it. The maximal anisotropy energy ∆ Emax, i.e., its value for α= 0 is a non-trivial function of U. This can be seen from the inset of Fig. 12 where we display ∆ Emax. When we increase the SOC, the anisotropy energies change significantly, see Fig. 13 where we show ∆ Emaxas a function of Uforζ= 0.1,0.15,0.2. The non-monotonic behavior of ∆ Emaxhas its cause in the band structure. For example, the maxima in the red and blue curves and thecorrespondingstructureintheblackcurvecorrespond to almost the same magnetization, cf. Fig. 10. To extract the genuine ζdependence of ∆ Emax, it is best to consider states with the same moment. This is done in Fig. 14 where we display ∆ Emaxas a function ofζfor values of Uwhich lead to the same ordered spin moments. These curves reveal that the anisotropy de- pends very sensitively on ζfor small values of ζwhereas it becomes linear for sizable ζ. IV. SUMMARY AND CONCLUSIONS In this work we investigated the interplay of local Coulomb interactions and the spin-orbit coupling in a three-orbital Hubbard model in three dimensions. Based on the Gutzwiller approximation to general multi-band Gutzwiller wave functions, we find that the Coulomb in- teraction leads to a considerable renormalization of the effectiveSOCinparamagneticmetals; the spin-orbitcou- plings can be enhanced over their atomic values by a fac- tor of more than two. This effect could be seen in exper- iment as enhanced band splittings in the quasi-particle dispersion. Hund’s rulesdetermine spin and orbitalmoments ofan atom. In metallic systems, signatures of Hund’s rules are visible only close to half band-filling. For all other (in- teger) fillings, the local Hund’-rule ground states cannot dominate over states with other quantum numbers be-0.1 0.200.00010.00020.00030.00040.00050.0006∆Emax ζ FIG. 14: Maximal anisotropy energy as a function of ζfor values of Uwith the same ordered spin moment of S= 0.4 (black), S= 0.45 (blue), S= 0.5 (red), S= 0.6 (green), and J/U = 0.2. causethis wouldbe veryunfavorableforthe electrons’ki- neticenergy. At best, Hund’sfirstruleappliesin strongly correlated metallic systems close to integer fillings. For ferromagnetic ground states, we find magnetiza- tion curves that are significantly influenced by the spin- orbit coupling. Overall, the SOC tends to destabilize the ferromagnetic order. For example, it shifts the onset of ferromagnetism to higher values of the Coulomb param- eters. In the presence of an ordered spin moment, the SOC has two main effects: (i), the magnetic spin mo- ment points into a preferred direction (easy axis), and, (ii), it generates a small but finite orbital moment in the same direction as the spin moment. We analyzed the magnetic anisotropy by applying an external magnetic field with constant strength and vary- ing direction. Our method is capable to resolve the anisotropy energy which can be rather small for spin- orbit couplings that are realistic for transition metals. As a function of the Coulomb interaction, the anisotropy energy shows a non-monotonic behavior which we could trace back to details of the electronic band structure. In this study we worked with the most general Ansatz for a Gutzwiller wave function. For the calculation of anisotropy energies, it is mandatory to avoid the often used approximation of a diagonal variational-parameter matrix because this approximation results in anisotropy energies that can be off by several orders of magnitude. For our three-band model, it is possible to include all elements of the variational-parameter matrix. Of course, this cannot be done for five d-bands. Therefore, strate- gies must be developed to include only the most signifi- cantmatrixelements. Inaseparate,moretechnicalwork, we analyze in detail the importance of non-diagonal vari- ational parameters, and show how to obtain accurate re- sults with a properly chosen subset of such parameters.29 Our method can directly be applied to materials that12 can be described by effective three-band models, e,g,, Sr2RuO4. It will be interesting to see the consequences ofthe substantialspin-orbit coupling onthe ground-state phase diagram and other electronic properties of these systems.8,30 Acknowledgements We thank R. Schade for valuable discussions on opti- mization algorithms. This work was supported in part by the Priority Pro- gramme 1458 of the Deutsche Forschungsgemeinschaft (DFG) under GE 746/10-1. T.L., U.L., and F.B.A. ac- knowledge the financial support by the Deutsche For- schungsgemeinschaft and the Russian Foundation of Ba- sic Research through the Transregio TRR 160. The authors gratefully acknowledge the computing time granted by the John-von-Neumann Institute for Computing (NIC), and provided on the supercomputer JURECA at J¨ ulich Supercomputing Centre (JSC) under project no. HDO08. Appendix A: Energy functional and its derivatives 1. Local basis The local density matrix (24) is non-diagonal when we include the spin-orbit coupling. For a fixed state |Ψ0/a\}bracketri}ht, however, we can always find a local basis, described by operators ˆd† i,γ=/summationdisplay σui;σ,γˆc† i,σ, ,ˆdi,γ=/summationdisplay σu∗ i;σ,γˆci,σ(A1) and a unitary matrix ˜ ui, so that the local density matrix ˜Diis diagonal, Di;γ′,γ≡ /a\}bracketle{tˆd† i,γˆdi,γ′/a\}bracketri}htΨ0=δγ,γ′ni,γ.(A2) Working with this new orbital basis |γ/a\}bracketri}htis quite use- ful because the energy functional (30) as well as the constraints (22), (23) have a much simpler form, see Sect. A3. In general, the basis |γ/a\}bracketri}htis not uniquely defined. For instance, in our three-band model without any charge or magnetic order, our local density matrix has the form ˜C=n011−∆nso 0˜Σ (A3) with˜Σ as defined in (14). The diagonalization of (A3) leadstoatwo-foldandafour-folddegenerateset ofstates |γ/a\}bracketri}htwith the occupation numbers n0−2∆nso 0andn0+ ∆nso 0, respectively. Therefore, the states |γ/a\}bracketri}htare defined only up to an arbitrary unitary transformation within these two degenerate sub-spaces. Even for a system with three non-degenerateorbitalsthere would be a remaining two-fold degeneracy in the spectrum of ˜C.2. Atomic spectrum We introduce the configuration basis |I/a\}bracketri}htof the local Hilbert space, |I/a\}bracketri}ht ≡/productdisplay σ∈Iˆc† σ|0/a\}bracketri}ht ≡ˆc† σ1...ˆc† σ|I||0/a\}bracketri}ht,(A4) where the operators ˆ c† σare in ascending order, i.e., we haveσ1< σ2... < σ |I|where|I|is the number of par- ticles in state |I/a\}bracketri}ht. Using the standard mathematical no- tations for set operators, we frequently encounter the states|I∪σ/a\}bracketri}htor|I\σ/a\}bracketri}htwhich result from the local cre- ation/annihilation of an electron. Since we work with fermions, we define the minus-sign function fsgn(σ,I)≡ /a\}bracketle{tI∪σ|ˆc† σ|I/a\}bracketri}ht. (A5) With the basis (A4), we can readily set up the local Hamilton matrix Hloc I,I′=/a\}bracketle{tI|ˆHloc|I′/a\}bracketri}ht (A6) and determine its eigenstates |Γ/a\}bracketri}ht=/summationdisplay ITI,Γ|I/a\}bracketri}ht (A7) by standard numerical techniques. For the numerical minimization of the Gutzwiller energy functional, how- ever, we prefer to work with the orbital states |γ/a\}bracketri}htand its corresponding configuration basis |J/a\}bracketri}ht ≡/productdisplay γ∈Jˆd† γ|0/a\}bracketri}ht ≡ˆd† γ1...ˆd† γ|I||0/a\}bracketri}ht.(A8) One way to determine the expansion of |Γ/a\}bracketri}htwith respect to this basis, |Γ/a\}bracketri}ht=/summationdisplay JAJ,Γ|J/a\}bracketri}ht, (A9) would be to transform the local Hamiltonian ˆHlocto the basis|γ/a\}bracketri}htand to set up and diagonalize the Hamilton ma- trixHloc J,J′. Alternatively, one may determine the eigen- states (A7) and calculate the coefficients AJ,Γin (A9) from the formula AJ,Γ=/summationdisplay ITI,Γ/a\}bracketle{tJ|I/a\}bracketri}ht, /a\}bracketle{tJ|I/a\}bracketri}ht= Det(u∗ σi,γj),(σi∈I,γj∈J).(A10) 3. Energy functional For a (still general) orbital basis |γ/a\}bracketri}ht, we find the fol- lowing expression for the constraints (22), (23), /summationdisplay Γ,Γ1,Γ2λ∗ Γ,Γ1λΓ,Γ2m0 Γ1,Γ2= 1,(A11) /summationdisplay Γ,Γ1,Γ2λ∗ Γ,Γ1λΓ,Γ2m0 Γ1∪γ,Γ2∪γ′=δγ,γ′nγ,(A12)13 where |Γ∪γ/a\}bracketri}ht ≡ˆd† γ|Γ/a\}bracketri}ht=/summationdisplay J(γ/∈J)fsgn(γ,J)AJ,Γ|J∪γ/a\}bracketri}ht, (A13) m0 Γ,Γ′≡ /a\}bracketle{tˆmΓ,Γ′/a\}bracketri}htΨ0. (A14) Since|J/a\}bracketri}htis a basis of the local Hilbert space, all expec- tation values of the form (A14) are determined by the determinants m0 J,J′≡ /a\}bracketle{tˆmJ,J′/a\}bracketri}htΨ0=/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingleΩJ,J′−ΩJ,¯J Ω¯J,J′¯Ω¯J,¯J/vextendsingle/vextendsingle/vextendsingle/vextendsingle/vextendsingle.(A15) Here, Ω J,J′are the matrices ΩJ,J′= Dγ′ 1,γ1Dγ′ 2,γ1... D γ′ |J′|,γ1 Dγ′ 1,γ2Dγ′ 2,γ2... D γ′ |J′|,γ2 ............ Dγ′ 1,γ|J|Dγ′ 2,γ|J|... Dγ′ |J′|,γ|J| ,(A16) in which the entries are the elements of the uncorrelated local density matrix (A2) that belong to the configura- tionsJ= (γ1,...,γ |J|) andJ′= (γ′ 1,...,γ′ |J′|). The matrix¯Ω¯J,¯Jin (A15) is defined by ¯Ω¯J,¯J= 1−Dγ1,γ1−Dγ1,γ2...−Dγ|¯J|,γ1 −Dσ2,σ11−Dσ2,σ2...−Dγ|¯J|,γ2 ............ −Dγ1,γ|¯J|−Dγ2,γ|¯J|...1−Dγ|¯J|,γ|¯J| , (A17) withγi∈¯J≡(1,...,N)\(J∪J′). So far we havenot used yet the defining condition (A2) of the|γ/a\}bracketri}ht-basis. By applying it, the expectation val- ues (A15) have the much simpler form m0 J,J′=δJ,J′m0 J, m0 J=/productdisplay γ∈Jnγ/productdisplay γ/∈J(1−nγ).(A18) It is this simplification that makesthe use ofthe |γ/a\}bracketri}ht-basis particularly convenient in the evaluation of ground-state expectation values. For the calculation of derivatives with respect to Dγ,γ′, however, we have to start from the general expression (A15), see Sect. A4. With the above results, eqs. (A14)–(A17) , we can cal- culate the local energy as Eloc=/summationdisplay Γ,Γ1,Γ2EΓλ∗ Γ,Γ1λΓ,Γ2m0 Γ1,Γ2.(A19) In the|γ/a\}bracketri}ht-basis we have explicitly m0 Γ1,Γ2=/summationdisplay J1,J2AJ1,Γ1A∗ J2,Γ2m0 J1,J2.(A20)For a ground-state calculation, this expression can be simplified further using eq. (A18). Finally, the renormalization matrix has the form qγ′ γ=/summationdisplay Γ1,...,Γ4λ∗ Γ2,Γ1λΓ3,Γ4/a\}bracketle{tΓ2|ˆd† γ|Γ3/a\}bracketri}ht ×/summationdisplay J1,J4AJ1,Γ1A∗ J4,Γ4Hγ′ J1,J4(A21) with Hγ′ J1,J4≡(1−fγ′,J1)/a\}bracketle{tJ4|ˆdγ′|J4∪γ′/a\}bracketri}htm0 J1,J4∪γ′ +/parenleftig fγ′,J4m0 J1\γ′,J4+(1−fγ′,J4)m0;γ′ J1\γ′,J4/parenrightig ×/a\}bracketle{tJ1\σ′|ˆdγ′|I1/a\}bracketri}ht, (A22) and fγ,J≡ /a\}bracketle{tJ|ˆd† γˆdγ|J/a\}bracketri}ht (A23) is either zero or unity. Here, the expectation value m0;γ′ J1\γ′,J4has the same form as the one in (A15), except that the index ¯Jhas to be replaced by ¯J\γ′. We need the general result (A21) for the renormalization matrix for the calculation of derivatives with respect to non- diagonal elements of Dγ,γ′, see Sect. A4. For a ground- state calculation one can use eq. (A2) and obtain the simpler expression qγ′ γ=1 nσ′/summationdisplay Γ1...Γ4λ∗ Γ2,Γ1λΓ3,Γ4/a\}bracketle{tΓ2|ˆd† γ|Γ3/a\}bracketri}htm0 Γ1,Γ4∪γ′(A24) which may also be written in the form31 qγ′ γ=1 nσ′/a\}bracketle{tˆP†ˆd† γˆPˆdγ′/a\}bracketri}htΨ0. (A25) In summary, the Gutzwiller energy functional in the |γ/a\}bracketri}ht-basis is given as EG(v,˜ρ,˜D) =/summationdisplay γ1,γ2 γ′ 1,γ′ 2qγ′ 1γ1/parenleftig qγ′ 2γ2/parenrightig∗ Eγ1,γ2,γ′ 1,γ′ 2 +/summationdisplay Γ,Γ1,Γ2EΓλ∗ Γ,Γ1λΓ,Γ2m0 Γ1,Γ2. (A26) Here, we applied the transformation to the |γ/a\}bracketri}ht-basis Eγ1,γ2,γ′ 1,γ′ 2=/summationdisplay σ1,σ2 σ′ 1,σ′ 2u∗ σ1,γ1uσ2,γ2uσ′ 1,γ′ 1u∗ σ′ 2,γ′ 2Eσ1,σ2,σ′ 1,σ′ 2 (A27) and qγ′ γ=/summationdisplay σ,σ′uσ,γu∗ σ′,γ′qσ′ σ. (A28)14 4. Derivatives The minimization algorithm which we explain in this section requires the calculation of derivatives of the en- ergyandoftheconstraintswith respecttothe variational parameters vzand the local density matrix ˜Cor˜D. a. Derivatives with respect to vZ The constraints, the local energy, and the renormaliza- tion factors are all quadratic functions of the variational parameters vz, i.e., they are of the form f(v) =/summationdisplay Z,Z′fZ,Z′vZ′vZ. (A29) The fast calculation of derivatives ∂vZf(v) =/summationdisplay Z′(fZ,Z′+fZ,Z′)vZ′(A30) is then possible if all coefficients fZ,Z′are stored in the main memory. In our calculations we observe that the number of contributing coefficients fZ,Z′in the expan- sion is particularly large in the renormalization factors when we include non-diagonal elements in the varia- tional parameter matrix λΓ,Γ′. Hence, our minimiza- tion for the three-orbital model that includes all nv= 924 non-diagonal variational parameters is numerically much more demanding than the minimization, e.g., for a five-orbital model with only diagonal parameters ( nv= 1024). b. Derivatives with respect to Cσ,σ′ Forthecalculationoftheeffectiveon-siteenergies(45), we need to determine the derivatives of the energy and of the constraints with respect to Cσ,σ′. Again, it is easier to calculate the derivatives first in the γ-basis and then transform them via ∂ ∂Cσ,σ′=/summationdisplay γ,γ′u∗ σ,γuσ′,γ′∂ ∂Dγ,γ′.(A31) For the derivatives of the constraints and of the local energy, we just need to determinethe derivativeof(A15). This gives ∂ ∂Dγ,γm0 J,J′=δJ,J′m0 J,J/braceleftigg 1/nγforγ∈J −1/(1−nγ) forγ /∈J (A32) forγ=γ′, and ∂ ∂Dγ′,γm0 J,J′=δ¯I,I\γδ¯I,I′\γ′m0¯I,¯I (1−nγ)(1−nγ′)(A33)forγ/\e}atio\slash=γ′, where γ∈Jandγ′∈J′. The only re- maining problem is to calculate derivatives of the object m0;¯γ J,J′that appears in the definition of the renormaliza- tion matrix, eqs. (A21), (A22) with respect to Dγ′,γ. It contributes only when γ/\e}atio\slash= ¯γandγ′/\e}atio\slash= ¯γ. Then we can use the simple relationship ∂ ∂Dγ′,γm0;¯γ J,J′=1 1−n¯γ∂ ∂Dγ′,γm0 J,J′.(A34) Appendix B: Minimization algorithm 1. Inner minimization For a given single-particle state |Ψ0/a\}bracketri}ht, or, equivalently, a given single-particle density matrix ˜ ρ, we have to minimize the energy functional (34) obeying the con- straints (40). In Ref. [25] we introduced a very efficient method for this minimization which was used in a num- ber of previous studies, for example on elementary iron and nickel.17,18This method, however, is only applicable if the gradients Fl≡∂vgl(v) (B1) of the constraints (40) are linearly independent because it requires a matrix Wl,l′≡Fl·Fl′to be regular. In principle, this problem can be overcome by a group- theoretical analysis that identifies the maximum set of independent constraints. Such a solution, however, is rather cumbersome and it runs into difficulties if one aims to study the transition between minima with differ- ent point-group symmetries. Even if we ensure that the gradients Flare linearly independent, however, we ob- serve that the algorithm introduced in Ref. [25] becomes prohibitively slow when we aim to minimize the energy functional for a general (complex) variational parameter matrixλΓ,Γ′. For this reason we tested a couple of alternative min- imization algorithms that are discussed in textbooks on numerical optimization.32We found the ‘Penalty and Augmented Lagrangian Method’ (PALM) to be most useful in our context when combined with an un- constrained Broyden-Fletcher-Goldfarb-Shanno (BFGS) minimization. We shall briefly summarize these methods in the following. a. PALM In the PALM one studies the functional LPALM G(v,{Λl},µ)≡EG(v)−/summationdisplay lΛlgl(v)+µ 2/summationdisplay l[gl(v)]2, (B2) which contains Lagrange parameter terms ( ∼Λl) and penalty terms ( ∼µ). In a pure ‘penalty method’ one would set Λ l= 0 and minimize (B2) for a given value of15 µ >0. If, in the minimum v=v0, the constraints are sufficiently well fulfilled, i.e., /summationdisplay lgl(v0)2< g2 c (B3) with some properly chosen value of gc, we may con- siderE0=EG(v0) as a decent approximation for the Gutzwiller ground-state energy. Otherwise, we increase µand start another minimization. For our Gutzwiller energy functional it turns out that the convergence to the minimum is much faster when we use a full PALM algorithm with Lagrange parameters Λl/\e}atio\slash= 0. This method works as follows.32 (i) StartfromsomeinitialvaluesΛ l= Λl;0andµ=µ0, e.g., Λ l;0= 1 and µ0= 50|EG(vnc)|wherevnc lare the variational parameters in the non-interacting limit, i.e., with λΓ,Γ′=δΓ,Γ′. (ii) Minimize LPALM G;0(v)≡LPALM G(v,{Λl;0},µ0) (B4) with respect to v. For this step we use the method of steepest descent combined with the BFGS method, see Sect. B1b. We denote the min- imum found in step (ii) by v0. (iii) Set Λl;k+1= Λl;k−µkgl(v0), (B5) µk+1=βµk (B6) with some properly chosen number β >1. In our calculations we worked with β= 2. (iv) Go back to step (ii) until eq. (B3) is satisfied. b. Steepest decent and BFGS method We still haveto choose a method for the unconstrained minimization in step (ii) in the PALM. It is a major ad- vantage of our Gutzwiller minimization that calculating gradients of the energy or of the constraints works just as fast as the calculation of these objects themselves. Of course, this is only the case when we use eq. (A30) and do not try to calculate the gradients numerically from the difference quotient. LetE(v) be our functional and F0=∂vE(v)|v=v0(B7) its gradient at the point v0. Then the simplest way of minimizing E(v) is the ‘method of steepest descent’ where the one-dimensional function ∆E(α) =E(v0+αF0) (B8) is minimized with respect to α. Instead of the optimal valueα=α0, in practical numerics we use a value ˜ α0that reduces the value of our functional E(v). We cal- culate a new point v0→v0+ ˜α0F0and reiterate the procedure until |F0|is below a pre-determined thresh- old. It is the decisive advantage of this method that it always converges towards a (potentially local) minimum as long as the functional is well-behaved, which we can take for granted in physics. The main disadvantage of the method is its rather slow convergence. Therefore, we found it necessary to combine it with a faster algorithm, the BFGS method, which, however, works reliably only in the vicinity of the minimum. The starting point of the BFGS method is a second- order expansion of the functional E(v0+δv)≈E(v0)+F0·δv+1 2δvT·˜H0·δv,(B9) where˜H0is the Hessian matrix of second derivatives at the point v0. Provided that ˜H0is positive definite, the right-hand site is minimized for δv=−˜B0·F0, (B10) where˜B0=˜H−1 0. Making iterative steps in the varia- tional parameter space by means of eq. (B10) is a multi- dimensional version of the Newton method. The main obstacle of the Newton method is the nu- merical calculation of ˜H0and the solution of eq. (B10). Therefore, it is better to use a so-called ‘quasi Newton method’ of which BFGS is one example. This method employs eq. (B10) without calculating ˜B0(or˜H0) ex- actly. It works as follows.32 (i) Start at some point vkand calculate the gradient Fkand the inverse ˜Bkof the Hessian matrix. Due to the benign structure of our functional we can afford this initial calculation of ˜Bkbecause it is done only once. (ii) Calculate the new point vk+1=vk−˜Bk·Fk. (B11) (iii) Calculate Fk+1from Fk+1=∂vE(v)|v=vk+1(B12) and an approximate update of ˜Bkfrom ˜Bk+1= (˜1−αkskyT k)˜Bk+1(˜1−αkyksT k)+αksksT k, (B13) where sk≡vk+1−vk, yk≡Fk+1−Fk, αk≡yT ksk. (B14) (iv) Go back to step (ii) until |Fk|is below some pre- defined threshold. Within the BFGS method it is not ensured that go- ing from vktovk+1always leads to a decrease of our functional. Therefore, we need the method of steepest decent as a backup to reach a region in the variational parameter space where the BFGS method converges.16 2. Outer minimization Given the optimum variationalparameters v0from the inner minimization we need to determine a new single- particle state by means of eqs. (42)–(46). All derivatives ineqs.(45)–(46)arecalculatedwiththeformulaegivenin eq. (B7). Then, the remaining problem is the calculation of the Lagrange parameters Λ lfrom eqs. (46). The num- bernvof these linear equations is usually much larger than the number of Lagrange parameters nc. Due to a possible inter-dependence of the constraints, the solution of the equations may not be unique. Hence, we cannot use the trick of Ref. [25] (see Sec. 4.2.1 of that work), which led to a number of nclinear equations. Here, wechoosetodetermineoneoftheinfinitelymanypossible sets of Lagrange parameters by minimizing the functional Y({Λl}) =/summationdisplay Z/parenleftigg ∂EG ∂vZ/vextendsingle/vextendsingle/vextendsingle/vextendsingle v=v0−/summationdisplay lΛl∂gl ∂vZ/vextendsingle/vextendsingle/vextendsingle/vextendsingle v=v0/parenrightigg2 (B15) with respect to Λ l. Note that the lack of uniqueness for the Lagrange parameters Λ lhas no consequences for the fields (45). The latter are always uniquely defined, apart from a total energy shift that can be absorbed in the chemical potential. With the fields (45) and the renormalization matrix determined, we diagonalize (42) and determine |Ψ0/a\}bracketri}htby means of the standard tetrahedron method. 1F. Hund. Zeitschrift f¨ ur Physik , 40:742, 1927. 2F. Hund. Zeitschrift f¨ ur Physik , 42:93, 1927. 3J.-P. Julien, J.-X. Zhu, and R. C. Albers. Phys. Rev. B , 77:195123, 2008. 4L. Du, L. Huang, and X. Dai. Eur. Phys. J. B , 86:94, 2013. 5L. Du, X. Sheng, H. Weng, and X. Dai. Europhys. Lett. , 101:27003, 2013. 6A. Farrell and T. Pereg-Barnea. Phys. Rev. B , 89:035112, 2014. 7H. Shinaoka, S. Hoshino, M. Troyer, and P. Werner. Phys. Rev. Lett. , 115:156401, 2015. 8G. Zhang, E. Gorelov, E. Sarvestani, and E. Pavarini. Phys. Rev. Lett. , 116:106402, 2016. 9Z. Y. Meng, Y. B. Kim, and H.-Y. Kee. Phys. Rev. Lett. , 113:177003, 2014. 10A. O. Shorikov and V. I. Anisimov. arxiv:1412.0140. 11J. B¨ unemann, W. Weber, and F. Gebhard. Phys. Rev. B , 57:6896, 1998. 12W. Metzner and D. Vollhardt. Phys. Rev. Lett. , 59:121, 1987. 13F. Gebhard. Phys. Rev. B , 41:9452, 1990. 14K. zu M¨ unster and J. B¨ unemann. Phys. Rev. B , 2016. 15K. M. Ho, J. Schmalian, and C. Z. Wang. Phys. Rev. B , 77:073101, 2008. 16X. Deng, X. Dai, and Z. Fang. Europhys. Lett. , 83:37008, 2008. 17T. Schickling, J. B¨ unemann, F. Gebhard, and W. Weber. New Journal of Physics , 16:93034, 2014. 18T. Schickling, J. B¨ unemann, L. Boeri, and F. Gebhard.Phys. Rev. B , 2016. 19J. C. Slater and G. F. Koster. Phys. Rev. , 94:1498, 1954. 20E. C. Stoner. Proc. of the Royal Soc. London A , 165:372, 1938. 21S. Sugano, Y. Tanabe, and H. Kamimura. Multiplets of Transition-Metal Ions in Crystals . Pure and Applied Physics 33, Academic Press, New York, 1970. 22J. B¨ unemann, F. Gebhard, and W. Weber. In A. Narlikar, editor,Frontiers in Magnetic Materials . Springer, Berlin, 2005. 23J. B¨ uneman, T. Schickling, and F. Gebhard. Europhys. Lett., 98:27006, 2012. 24J. Kaczmarczyk, J. Spa/suppress lek, T. Schickling, and J. B¨ une- mann.Phys. Rev. B , 88:115127, 2013. 25J. B¨ unemann, F. Gebhard, T. Schickling, and W. Weber. physica status solidi (b) , 249:1282, 2012. 26J. B¨ unemann, F. Gebhard, and R. Thul. Phys. Rev. B , 67:75103, 2003. 27W. F. Brinkman and T. M. Rice. Phys. Rev. B , 2:4302, 1970. 28J. B¨ unemann and W. Weber. Phys. Rev. B , 55:4011, 1997. 29J. B¨ unemann, T. Linneweber, and F. Gebhard. physica status solidi , 2016. 30S. Hoshino and P. Werner. Phys. Rev. Lett. , 115:247001, 2015. 31M. Fabrizio. Phys. Rev. B , 76:165110, 2007. 32J. Nocedal and S. J. Wright. Numerical Optimization . Springer, New York, 2nd edition, 2006.

1511.03454v1.Spin_Swapping_Transport_and_Torques_in_Ultrathin_Magnetic_Bilayers.pdf

Spin Swapping Transport and Torques in Ultrathin Magnetic Bilayers Hamed Ben Mohamed Saidaoui1and A. Manchon1 1Physical Science and Engineering Division, King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Kingdom of Saudi Arabia Planar spin transport in disordered ultrathin magnetic bilayers comprising a ferromagnet and a normal metal (typically used for spin pumping, spin Seebeck and spin-orbit torque experiments) is investigated theoretically. Using a tight-binding model that treats extrinsic spin Hall eect, spin swapping and spin relaxation on equal footing, we show that the nature of spin-orbit coupled transport dramatically depends on ratio between the layers thickness dand the mean free path . While spin Hall eect dominates in the diusive limit ( d), spin swapping dominates in Knudsen regime (d.). A remarkable consequence is that the symmetry of the spin-orbit torque exerted on the ferromagnet is entirely dierent in these two regimes. Introduction Spin-orbit coupling is responsible for a wide variety of phenomena that have attracted a huge amount of attention recently [1, 2]. Among the most prominent phenomena, one can mention the inverse spin galvanic eect, i.e. the electrical generation of a nonequi- librium magnetization [3] and the spin Hall eect [4, 5], i.e. the conversion of an unpolarized charge current into a pure spin current. The nature spin Hall eect has been scrutinized intensively lately due to its central role in spintronics. While the original theory was based on carrier scattering against extrinsic spin-orbit coupled impurities [5], the importance of the band structure's Berry curvature has been recently unveiled, producing large dissipationless (i.e. scattering independent) spin Hall eects [6]. Although the proper theoretical treat- ment of this eect in the diusive limit continues rais- ing debates [7], experiments tend to conrm the impor- tance of intrinsic spin Hall eect in 4 dand 5dtransi- tion metals [8, 9]. In the opposite limit, i.e. when the system size becomes comparable to the mean free path (so-called Knudsen regime), the nature of spin Hall eect changes subtly as quantum and semiclassical size eects emerge [10]. An accurate description of spin-orbit cou- pled transport in this regime is crucial as ultrathin nor- mal metal/ferromagnet bilayers (e.g. Pt/NiFe etc.) are now commonly used in spin pumping [11], spin Seebeck eect [12] and spin-orbit torque [13, 14] measurements. The limitations of the current models of spin Hall ef- fect are best illustrated by the puzzles raised by spin- orbit torque experiments in such ultrathin multilayers. In magnetic systems lacking inversion symmetry, spin-orbit coupling enables the electrical control of the magnetic or- der parameter [15{18]. This spin-orbit mediated torque has been observed in a various materials combinations in- volving heavy metals [13, 14], oxides [19] and topological insulators [20]. Experimentally, the torque possesses two components referred to as damping torque (even in mag- netization direction) and eld-like torque (odd in magne- tization direction). Consensually, the damping torque is associated with the spin Hall eect occurring in the bulk of the heavy metal [21] [see Fig. 1(a)], while the eld-like torque arises from the inverse spin galvanic eect induced FIG. 1. (Color online) Schematics of (a) spin Hall and (b) spin swapping eects in a bilayer composed of a normal metal (blue) and a ferromagnet (yellow) with magnetization min the diusive and Knudsen regimes, respectively. The charge current jeis injected in the plane of the layers and results in a spin current Js owing perpendicular to the interface. by spin-orbit coupling present at the interface between the heavy metal and the ferromagnet [15, 16]. However, experiments have reported complex material dependence parameters that are not accounted for by these models [22, 23]. In particular, sizable eld-like torques have been observed in systems where interfacial spin-orbit coupling is expected to be small [24]. In spite of intense theoreti- cal eorts [25, 26], no ecient mechanisms related to spin Hall eect and able to generate sizable eld-like torque have been identied [21]. In this Letter, we theoretically demonstrate that the nature of extrinsic spin-orbit coupled transport in disor- dered ultrathin magnetic bilayers dramatically depends on the transport regime. When disorder is strong and transport is diusive, spin Hall eect dominates leading to a damping torque in agreement with the widely ac- cepted physical picture [21]. In contrast, when disorder is weak and the system size is of the order of the carrier mean free path, spin swapping [27] becomes increasingly important, leading to a dominant eld-like torque. General principles Consider a metallic bilayer com- posed of a spin-orbit coupled normal metal and a fer- romagnet without spin-orbit coupling (see Fig. 1). A current jeis injected in the plane of the bilayer and ex- ert a torque on the ferromagnet. Disregarding interfa-arXiv:1511.03454v1 [cond-mat.mes-hall] 11 Nov 20152 cial spin-orbit coupling, two spin-orbit coupled transport phenomena are present in this system. First, due to in- trinsic and/or extrinsic spin Hall eect in the normal metal, a spin current ows along the normal to the inter- facezwith a spin polarization along ( zje). This results in a spin torque on the form m[(zje)m], which is of damping-like form [21] (even in magnetization), see Fig. 1(a). In addition, electrons owing in the ferromag- net acquire a spin polarization along mand may scatter towards the normal metal. Once in the normal metal, these electrons experience spin swapping: upon scatter- ing on spin-orbit coupled impurities, they experience a spin-orbit eld oriented normal to the scattering plane [i.e. along ( zje)] and about which their spin precess [27, 28]. Upon this reorientation, a spin current polarized along m(zje) is injected into the ferromagnet and in- duces a eld-like torque (odd in magnetization), see Fig. 1(b), even in the absence of interfacial spin galvanic ef- fect. Since these two eects operate in distinct disorder regimes, namely spin Hall eect necessitates strong disor- der while spin swapping survives even for weak disorder [28], the nature of the torque should dramatically change from one regime to the other. Numerical results To investigate these eects quanti- tatively, we computed the spin transport in a magnetic bilayer using a tight-binding model [28, 29]. The system is a two-dimensional square lattice connected laterally to external leads. The full Hamiltonian of the central sys- tem reads ^H=X i;j;;0f(ij0+
ij 2m
^0)^c+ i;j;^ci;j;0+h:c:g X i;j;tN(^c+ i+1;j;^ci;j;+ ^c+ i;j+1;^ci;j;+h:c:) X i;jti1;j i;j1(^c+ i;j;"^ci1;j1;"^c+ i;j;#^ci1;j1;#) X i;jti;j i1;j1(^c+ i;j1;"^ci1;j;"^c+ i;j1;#^ci1;j;#):(1) Here the rst term at the right-hand side of Eq. (1) is the spin-independent onsite energy in which ij=0+ ij, 0being the onsite energy and ij2[=2;=2] a ran- dom potential of strength that introduces disorder in the system. The second term is the exchange interac- tion (
ij) between the spin of the carriers and the local magnetic moment of direction mon site (i;j). The third term in the Hamiltonian corresponds to the nearest neighbor hopping energy ( tN). The last two terms are the next-nearest neighbor hopping parameters that ac- count for the disorder-driven spin-orbit coupled scatter- ing. The next-nearest neighbor hopping parameter reads ti;j i0;j0=itN(i;ji0;j0), whereis the dimensionless spin-orbit coupling strength. The operator ^ c+ i;j;(^ci;j;) creates (annihilates) a particle with spin at position (i;j). This approach models spin Hall eect, spin swap- ping and spin relaxation on equal footing [28] and onecan tune the relative strength between spin Hall eect and spin swapping by changing the disorder strength (and thereby the mean free path) and the spin-orbit cou- pling strength . 0102000.010.020.03<δSy>(ℏ/2) α = 0.2 α = 0.4 α = 0.6 0.5 100.20.40.60.8τDL τFL 102030−0.0500.050.10.15 Width (a)α = 0.5 α = 0.2 0.5 100.10.20.30.4 Spin orbit strength (α)τDL τFL01020−0.00200.0020.004<δSx>(ℏ/2) 01020−0.0500.05<δSx>(10-3 ℏ/2) <δSy>(10-3 ℏ/2) Torkance ( 10-2 h/e) Torkance (10-4 h/e)Width (a) Spin orbit strength (α) (a) Γ = 2.2 eV (c) Γ = 0.1 eV(b) (d) FIG. 2. (Color online) (a,c) Spin density prole along the magnetic bilayer width for strong (=2.2 eV) and weak dis- order ( = 0 :1 eV) regimes. The vertical dashed line separates the normal metal (left) from the ferromagnetic layer (right). The main panels (inset) represent the largest (smallest) spin density component for various . (b,d) Corresponding spin torkance components as a function of . The parameters are tN=0=
= 1 eV and m=z. Let us now consider the current-driven spin density in a two-dimensional bilayer in ( x,z) plane (see Fig. 1). The width of the ferromagnet (normal metal) is 10 a(20 a), and the length of the bilayer is 30 a, whereais the square lattice parameter. Figure 2(a) shows the nonequi- librium spin density prole along the bilayer width ob- tained for a strongly disordered system ( = 2 :2 eV) and various spin-orbit coupling strengths, . Figure 3(c,d) displays the corresponding two dimensional map of the spin density components, SxandSy, respectively. The spin density is mainly aligned along Sy(main panel) and has a small Sxcontribution (inset). Remarkably, Sysmoothly accumulates over the layer width, as ex- pected from spin Hall eect in the diusive regime. No- tice though that oscillations stemming from quantum co- herence survive even for this amount of disorder, as no extrinsic quantum dephasing is introduced. The small Sxcomponent is conned at the interface, which illus- trates its spin swapping origin: it only survives within a distance of the order of the mean free path, as illustrated in Fig. 1(b). As a consequence, in diusive regime, the ef- ciency of the torque (torkance) exerted on the magnetic layer =G=R d Sm( is the volume of the magnet, Gis the conductance of the bilayer) is dominated by a3 FIG. 3. (Color online) Two-dimensional mapping of the spin density components, (a,c) Sxand (b,d)Sy, in (a,b) weak and (c,d) strong disordered regimes. The dashed line rep- resents the interface between the normal metal and the fer- romagnet and the white arrow indicates the direction of the magnetization. The parameters are the same as in Fig. 2. damping-like component DL, i.e.m[(zje)m]. Nonetheless, when reducing the spin-orbit strength the spin Hall eect decreases and one observes a transition between damping-dominated torque ( DL) to eld-like- dominated torque ( FL), as illustrated in Fig. 2(b). This crossover occurs because spin relaxation, which is detri- mental to spin swapping [28], decreases with thereby enhancing spin swapping. This region can be widened by decreasing the disorder strength, as shown in Fig. 4. The case of weak disorder is even more remarkable, as shown in Fig. 2(c) [two dimensional mapping is given in Fig. 3(a,b)]. Phase coherence results in quantum oscillations of both Sx(main panel) and Sy(inset). Nevertheless, while the oscillations of Syin the normal metal are symmetric with respect to the center of the layer (a reminiscence of the standing nature of the wave functions), thereby resulting in a vanishing spin current injection, the oscillations of Sxare distorted and result in an eective spin current injection into the adjacent fer- romagnet. As a consequence, the torkance is dominated by the eld-like component, FL, i.e.m(zje) [right panel of Fig. 2(d)] for all , in agreement with the phenomenological discussion provided above. In Fig. 4, the ratio FL=DLis displayed as a func- tion of disorder and spin-orbit coupling strengths. In- terestingly, we nd that the torque is dominated by the eld-like component in the weak disorder/weak spin-orbit coupling regime, while it is dominated by the damping like component in the strong disorder and/or strong spin- orbit coupling regime. These dierent phases can be di- rectly attributed to the spatial dependences shown in Fig. 2. When spin Hall eect dominates (diusive regime), the torque is mostly damping-like, and when spin swap- ping dominates (Knudsen regime), the torque is mostly FIG. 4. (Color online) Ratio between the magnitude of the eld-like torque and damping-like torque, FL=DL, as a func- tion of and . The ratio is given in logarithmic scale and the dashed line indicates FL=DL=1. The parameters are the same as in Fig. 2. eld-like. This behavior has been reproduced by varying the thickness of the normal metal while keeping the dis- order xed (not shown). These simulations demonstrate that in ultrathin bilayers eld-like torques do not neces- sarily arise from interfacial spin galvanic eect, but can emerge due to spin dependent scattering in the normal metal. A necessary condition is that the thickness of the normal metal ought to be of the order of the mean free path. Drift-diusion model Let us now address the nature of the spin swapping torque in the diusive regime. In- deed, it may look surprising that spin swapping dramat- ically decreases when disorder is strong. The spin-orbit coupled spin transport in the normal metal can be mod- eled using the spin diusion equation developed in Ref. 30 in the 1stBorn approximation eje=N=rc+sh 2r; (2) e2Ji s=N=ri 2sheirc+sw 2ei(r);(3) whereNis the bulk conductivity, sh==k Fis the Hall angle from side jump scattering (within 1stBorn ap- proximation, skew scattering is absent) and sw= 2=3 is the spin swapping coecient. andkFare the mean free path and Fermi wavevector, respectively. cand are the spin-independent and spin-dependent electro- chemical potentials, related to the charge and spin ac- cumulation by c=n=eNand=S=eN, andNis the density of state. Note that jeis the current density vector whereasJsis the spin density tensor and Ji sis the i-th spin component of the spin current. This set of equa- tions is combined with the spin and charge accumulation continuity equations r
jc= 0 and r
Js==sf wheresfis the spin relaxation time. The spin transport4 in the ferromagnet is modeled by similar drift-diusion equations [21]. To model the torque exerted on the ferromagnet, we assume that the spin dephasing in the magnetic layer is so short that the incoming spin current is entirely ab- sorbed within a few monolayers from the interface. The boundary conditions are then written [31] je=g
c+ g
k;Jk s;z= g
c+g
k;(4) J? s;z=2(g"# r
op+g"# i
ip)my 2(g"# r
ip+g"# i
op)m(ym); (5) whereJk s;z=Js;z
mandJ? s;z=Js;zJk s;z
mis the spin current transverse to the magnetization m. We de- neg= (g"+g#)=2 and = (g"g#)=2g,gsbeing the interfacial conductance for spin sandg"#=g"# r+ig"# iis the (complex) mixing conductance. The algebra to ob- tain the interfacial spin current is cumbersome but does not present technical diculties. We nd that the torque possesses two contributions, =sh+sw, associated with spin Hall and spin swapping respectively, and sh=~shjN D0h (~g"# rj~g"#j2)m(ym)~g"# imyi ; (6) sw=sw~shjN Dh ~g"# rNmzmx+ ~g"# iNmzm(xm) +j~g"#j2mxmz ; (7) D0(1~g"# r)2+sw(N+ ~g"# r)(1~g"# r) sin2: In order to keep the notation compact, we dened the eective spin Hall angle ~ sh=sh(1cosh1dN=N sf) and normalized mixing conductances ~ g"# j= 4~N sfg"# j=N, where ~N sf=N sf=tanh(dN=N sf) is the eective spin dif- fusion length of the normal metal. Finally, N= 4~N sf=N 4~F sf=F+1=gand0= 1 +N. HereFandF sfare the conductivity and spin diusion length of the ferromag- netic layer, and jNis the charge current density owing in the normal metal. The spin Hall torque, sh[Eq. (6)], solely arises from spin Hall eect (/sh) and produces the regular damp- ing torque m(ym), with a small contribution to the eld-like torque my[21]. In the presence of spin swapping, these two torques are renormalized by the de- nominator Dthat depends on sin2through the spin swapping coecient sw(where cos=m
z). More in- terestingly, the spin swapping torque, sw[Eq. (7)], arises from the interplay between spin swapping and spin Hall eect (/swsh) and generates three additional torque components. Assuming ~ g"# i~g"# r1, to the leading order in ~g"# r, the torques reduce to sh~shjN D0~g"# rm(ym); (8) swsw~shjN D~g"# rNmzmx; (9)and the ratio between these two contributions is given by swN 1+N. Therefore, in the diusive regime, since most of the current ows in the bulk of the normal metal the contribution of the spin swapping close to the interface is vanishingly small [as shown in the inset of Fig. 2(b)] and the only manner spin swapping contributes to the torque is through its interplay with spin Hall eect, which is second order in spin-orbit coupling and reasonably much smaller than the spin Hall eect in agreement with our tight-binding calculations. Discussion and perspectives These results are of di- rect relevance for experiments on current-driven spin- orbit torque [13, 14], but also spin pumping [11] and spin Seebeck [12] measurements in ultrathin magnetic multilayers. As a matter of fact, most of these investi- gations are conducted on multilayers comprising metals with thicknesses from 10 nm down to less than 1 nm (see, e.g. Refs. [22, 23]). In sputtered thin lms, the grain size ranges from 5 to 10 nm, which suggests that the trans- port is not diusive and that extrinsic spin swapping can lead to sizable eld-like torque even in the absence of interfacial inverse spin galvanic eect. In addition, in the numerical calculations reported here the eect of in- trinsic (Berry phase-induced) spin Hall eect, dominant in 4dand 5dtransition metals, was disregarded. Then, one can reasonably expect that the spin-orbit coupling in the band structure should also induce spin swapping [32]. This eect is well-known in semiconductors where the coherent precession about the local spin-orbit eld induces, e.g., D'yakonov-Perel spin relaxation. Intrinsic spin swapping can be estimated using ab initio calcula- tions but it may be dicult to disentangle this eect from interfacial inverse spin galvanic eect [25]. We conclude this work by commenting on the impact of spin swapping on spin pumping, the Onsager recipro- cal of spin transfer torque. When excited by a radio- frequency eld (or by thermal magnons), the precess- ing magnetization pumps a spin current, polarized along m@tm, into the normal metal [33]. Such a spin current can be converted into a charge current through inverse spin Hall eect [11], but it also enhances the mag- netic damping of the ferromagnet [33]. Upon spin swap- ping this pumped spin current is converted into another spin current polarized along y(m@tm) (ybeing the direction of the spin-orbit eld perpendicular to the scattering plane). While this new spin current does not contribute to additional electric signal, it should produce a corrective damping torque on the form my@tm, i.e. an anisotropic magnetic damping. This eect vanishes by symmetry in homogeneous ferromagnets, but is ex- pected to survive in magnetic domain walls resulting in unconventional magnetic damping. Further theoretical investigations and experimental explorations are neces- sary to uncover the full implications of this eect. A.M. acknowledges inspiring discussions with T. Valet and H.B.M.S. thanks S. Feki for his valuable technical5 support. This work was supported by the King Abdullah University of Science and Technology (KAUST). aurelien.manchon@kaust.edu.sa [1] T. Jungwirth, J. Wunderlich, and K. Olejnk, Nat. Mate- rials11, 382 (2012). [2] A. Manchon, H.C. Koo, J. Nitta, S. Frolov and R.A. Duine, Nat. Materials 14, 871 (2015). [3] E.L. Ivchenko, G.E. Pikus, Pisma Z. Eksp. Teor. Fiz. 27, 640 (1978); JETP Lett. 27, 604 (1978). [4] G. Vignale, J. Supercond. Nov. Magn. 23, 3 (2010). [5] M. I. Dyakonov and V. I. Perel, Phys. Lett. A 35, 459, (1971). [6] S. Murakami, N. Nagaosa and S.-C. Zhang, Science 301, 1348 (2003); J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004). [7] I. A. Ado, I. A. Dmitriev, P. M. Ostrovsky, and M. Titov, EuroPhys. Lett. 111, 37004 (2015). [8] L. Vila, T. Kimura, and Y. C. Otani, Phys. Rev. Lett. 99, 226604 (2007). [9] C. Du, H. Wang, F. Yang, and P. C. Hammel, Phys. Rev. B90, 140407 (2014). [10] X. Wang, J. Xiao, A. Manchon, S. Maekawa, arXiv:1407.8278 (2014). [11] E. Saitoh, M. Ueda, H. Miyajima, and G. Tatara, Appl. Phys. Lett., 88, 182509 (2006). [12] K. Uchida et al., Nat. Mater. 9, 894 (2010). [13] I. M. Miron et al., Nature 476, 189-193 (2011). [14] L. Liu, C.-F. Pai, Y. Li, H. W. Tseng, D. C. Ralph, and R. A. Buhrman, Science 336, 555 (2012). [15] B. A. Bernevig and S.C. Zhang, Phys. Rev. Lett. 95, 016801 (2005); I. Garate and A.H. MacDonald, Phys. Rev. B. 80, 134403 (2009).[16] A. Manchon and S. Zhang, Phys. Rev. B 78, 212405 (2008); Phys. Rev. B 79, 094422 (2009). [17] J. Zelezn y et al., Phys. Rev. Lett. 113, 157201 (2014). [18] A. Brataas and K. Hals, Nature Nanotechnology 9, 86 (2014). [19] X. Qiu et al., Nat. Nanotech. 10, 333 (2015). [20] A. R. Mellnik et al., Nature 511, 449 (2014); Y. Fan et al., Nature Mater. 13, 699 (2014). [21] P. M. Haney, H. -W. Lee, K. -J. Lee, A. Manchon, and M. D. Stiles, Phys. Rev. B 87, 174411 (2013). [22] Avci et al., Phys. Rev. B 89, 214419 (2014); Pai et al., Phys. Rev. B 92, 064426 (2015). [23] Kim et al., Nat. Materials 12, 240 (2012); Phys. Rev. B 89, 174424 (2014). [24] Fan et al., Nat. Comm. 4:1799 (2012); Pai et al., Appl. Phys. Lett. 104, 082407 (2014). [25] F. Freimuth, S. Bl ugel, and Y. Mokrousov, Phys. Rev. B 90, 174423 (2014); P. M. Haney, H. -W. Lee, K. -J. Lee, A. Manchon, and M. D. Stiles, Phys. Rev. B 88, 214417 (2013). [26] X. Wang and A. Manchon, Phys. Rev. Lett. 108, 117201 (2012); Li et al., Phys. Rev. B 91, 134402 (2015); Lee et al., Phys. Rev. B 91, 144401 (2015). [27] M. B. Lifshits and M. I. Dyakonov, Phys. Rev. Lett. 103, 186601, (2009). [28] H. B. M. Saidaoui, Y. Otani and A. Manchon, Phys. Rev. B92, 024417 (2015). [29] C. W. Groth, M. Wimmer, A. R. Akhmerov, X. Waintal, New J. Phys. 16, 063065 (2014). [30] R. V. Shchelushkin and Arne Brataas, Phys. Rev. B 71, 045123 (2005). [31] A. Brataas, G. E.W. Bauer, P. J. Kelly, Phys. Rep. 427, 157 (2006). [32] S. Sadjina, A. Brataas, and A. G. Malshukov, Phys. Rev. B85, 115306 (2012). [33] Y. Tserkovnyak, A. Brataas, G. E.W. Bauer, Phys. Rev. Lett. 88, 117601 (2002).

1310.8016v1.Spin_orbital_state_induced_by_strong_spin_orbit_coupling.pdf

arXiv:1310.8016v1 [cond-mat.str-el] 30 Oct 2013Spin-orbital state induced by strong spin-orbit coupling Hiroaki Onishi Advanced Science Research Center, Japan Atomic Energy Agen cy, Tokai, Ibaraki 319-1195, Japan Abstract. To clarify a crucial role of a spin-orbit coupling in the emer gence of novel spin-orbital states in 5d-electron compounds such as Sr 2IrO4, we investigate ground state properties of a t2g-orbital Hubbard model on a square lattice by Lanczos diagonalizatio n. In the absence of the spin-orbit coupling, the ground state is spin singlet. When the spin-or bit coupling is strong enough, the ground state turns into a weak ferromagnetic state. The weak ferromagnetic state is a singlet state in terms of an effective total angular momentum. Regard ing the orbital state, we find the so-called complex orbital state, in which real xy,yz, andzxorbital states are mixed with complex coefficients. 1. Introduction Novel quantum phenomenon driven by the interplay of strong s pin-orbit coupling and Coulomb interaction has been a subject of significant study in the fiel ds of condensed matter physics and materials science. Since spin and orbital degrees of freedo m are entangled at a local level by the spin-orbit coupling, the total angular momentum provides a good description of ordered phases and fluctuation properties. On the other hand, we would envis age possible realization of devices with exotic functionalities that utilize cross correlatio n involving magnetic and electric degrees of freedom, in which magnetic behavior can be manipulated by an applied electric field, while electric behavior can be controlled by a magnetic field. Recently, a 5 dtransition metal oxide Sr 2IrO4, in which Ir4+ions have five electrons with a low-spin (t2g)5state, has attracted growing attention as a candidate for a n ovel Mott insulator characterized by an effective total angular momentum Jeff=1 2, whereJeff=−L+S, rather than spin commonly observed in 3 dMott insulators [1, 2]. The spin-orbit coupling in the t2gmanifold stabilizesthe Jeff=1 2state. Theemergentbehaviorofthe Jeff=1 2Mottinsulatorhasbeenexplored in experiments such as angle resolved photoemission spectr oscopy [1], optical conductivity [1, 3], resonant x-ray scattering [2], and resonant inelastic x-ra y scattering [4]. An isostructural iridate Ba2IrO4has also been reported as the Jeff=1 2Mott insulator [5]. Theoretical efforts have been devoted to clarify the characteristics of the Jeff=1 2Mott insulator on the basis of first-principles calculations [1, 6] and model calculations [7, 8, 9, 10]. The purpose of this paper is to gain an insight into the spin-o rbital state under the spin-orbit coupling from a microscopic viewpoint. We investigate grou nd-state properties of a t2g-orbital Hubbardmodelincludingthespin-orbitcouplingby exploit ing numerical techniques. We discuss a spin-orbit-induced ground-state transition from a spin s inglet state to a singlet state in terms of the effective total angular momentum.(xy, yz, zx) ⊗ ( ↑, ↓)jeff =1/2 jeff =3/2 3λ/2 spin-orbit coupling α↑ α↓ α↑ α↓ β↑ β↓ β↓ γ↑ γ↑ spin down spin up Figure 1. Local electron configuration andwavefunction ofthe t2gorbitals underthespin-orbit coupling. To visualize the wavefunction, we draw the surfac e defined by r=/radicalbig/summationtext σ|ψ(θ,φ,σ)|2in the polar coordinate with σbeing real spin, while the color denotes the weight of spin up and down states. 2. Model and numerical method We consider triply degenerate t2gorbitals on a square lattice in the xyplane, and the electron number per site is five, correspondingto the low-spin state o f Ir4+ions. Thet2g-orbital Hubbard model is given by H=/summationdisplay i,a,τ,τ′,σta ττ′(d† iτσdi+aτ′σ+h.c.)+λ/summationdisplay iLi·Si+U/summationdisplay i,τρiτ↑ρiτ↓+U′ 2/summationdisplay i,σ,σ′,τ/negationslash=τ′ρiτσρiτ′σ′ +J 2/summationdisplay i,σ,σ′,τ/negationslash=τ′d† iτσd† iτ′σ′diτσ′diτ′σ+J′ 2/summationdisplay i,σ/negationslash=σ′,τ/negationslash=τ′d† iτσd† iτσ′diτ′σ′diτ′σ, (1) wherediτσisanannihilationoperatorforanelectron withspin σ(=↑,↓)inorbitalτ(=xy,yz,zx ) at sitei,ρiτσ=d† iτσdiτσ, andSiandLirepresent spin and orbital angular momentum operators, respectively. The hopping amplitude is given by tx xy,xy=tx zx,zx=ty xy,xy=ty yz,yz=t, and zero for other combinations of orbitals. Hereafter, tis taken as the energy unit. λdenotes the spin-orbit coupling.U,U′,J, andJ’ are intra-orbital Coulomb, inter-orbital Coulomb, excha nge (Hund’s rule coupling), and pair-hopping interactions, respectiv ely. We assume that U=U′+J+J′due to the rotational symmetry in the local orbital space, and J=J′due to the reality of the xy,yz, andzxorbital functions. We numerically investigate ground-state properties of the 2×2 four-site system by Lanczos diagonalization. Because of the three orbitals, the number of bases per site is 43=64, and the matrix dimension of the Hamiltonian becomes huge as the syst em size increases. In general, the matrix dimension is reduced by decomposing the Hilbert spac e into a block-diagonal form by usingsymmetries of the Hamiltonian. Inthe present case, th e total numberof electrons is a good quantum number. We cannot use the zcomponent of the total spin as a good quantum number, since the spin SU(2) symmetry is broken by the spin-orbit cou pling. We have not utilized the lattice symmetry such as the translational symmetry. Thus t he matrix dimension is 10 ,626 for four sites, while it grows to 377 ,348,994 for eight sites. 3. Results First we briefly discuss the local electron configuration in t he atomic limit. As shown in Fig. 1, thet2glevel is split by the spin-orbit coupling into a jeff=1 2doublet and a jeff=3 2quartet, in which the eigenstates are characterized by the effective to tal angular momentum jeff. The0123 0 0.5 1spin orbitalangular momentum λ(a) 0123 0 0.5 1total (=L+S) effective total (= −L+S)angular momentum λ(b) Figure 2. The magnitude of the sum total of angular momenta in the whole system atU′=10 andJ=2. (a) Spin and orbital angular momenta. (b) Total and effecti ve total angular momenta. eigenstates are |α±/angbracketright=1√ 3|xy±/angbracketright±1√ 3|yz∓/angbracketright+i√ 3|zx∓/angbracketrightwith eigenenergy λfor thejeff=1 2doublet, and|β±/angbracketright=1√ 2|yz∓/angbracketright∓i√ 2|zx∓/angbracketrightand|γ±/angbracketright=/radicalBig 2 3|xy±/angbracketright∓1√ 6|yz∓/angbracketright−i√ 6|zx∓/angbracketrightwith eigenenergy −λ 2 for thejeff=3 2quartet, where we introduce βandγorbitals to distinguish two Kramers doublets in thejeff=3 2quartet, and pseudospins to label two states in each Kramers doublet. Note that due to the entanglement of spin and orbital states, the wavef unction exhibits anisotropic charge and spin distributions. Since we have five electrons for an Ir4+ion, the lower jeff=3 2quartet is fully occupied, while the upper jeff=1 2doublet is half-filled. Now we move on to the Lanczos results. In Fig. 2(a), we present the magnitude of sum total of spin and orbital angular momenta in the whole system, defin ed by/angbracketleft(/summationtext iSi)2/angbracketright=Stot(Stot+1) and/angbracketleft(/summationtext iLi)2/angbracketright=Ltot(Ltot+1), respectively. At λ=0,Stotis found to be zero, indicating a spin singlet ground state. As λincreases,Stotgradually increases and a sudden change occurs at a transition point, above which the ground state turns to a wea k ferromagnetic state. Note that the induced spin moment is reduced from the maximum value1 2×4=2. We also find an abrupt change for Ltot, implying that the orbital configuration is reorganized. Fi gure 2(b) represents the magnitude of sum total of total and effective total angular momenta in the whole system, defined by /angbracketleft(/summationtext iJi)2/angbracketright=Jtot(Jtot+ 1) and /angbracketleft(/summationtext iJeff,i)2/angbracketright=Jeff,tot(Jeff,tot+ 1), respectively. Jtotis enhanced in the weak ferromagnetic state. In contrast, Jeff,totdecreases and approaches zero in the limit of large λ. Thus the weak ferromagnetic state is regarded as a singlet s tate in terms of the effective total angular momentum. To clarify how the orbital state changes as λvaries, we measure the charge density in each of thet2gorbitals,nτ=/summationtext σ/angbracketleftρiτσ/angbracketright, in different two basis sets. Note that due to the translationa l invariance, the charge density is equivalent in all sites. I n Fig. 3(a), we plot the charge density in the real xy,yz, andzxorbitals. At λ=0, we find that nxy=1.5 andnyz=nzx=1.75, since the itinerancy of orbitals depends on the hopping direction and holes preferably occupy the xyorbitals rather than the yzandzxorbitals. As λincreases,nxy,nyz, andnzxapproach5 3, indicating that xy,yz, andzxorbital states are mixed with equal weight. Note that nyzand nzxare equivalent irrespective of λ. Transforming the basis set into the complex α,β, andγ orbitals, we can see the characteristics of the orbital stat e from the viewpoint of the effective total angular momentum. As shown in Fig. 3(b), we observe a sh arp change at the transition point, implying again the reorganization of the orbital con figuration. For large λ, theαorbitals are singly occupied and relevant to the low-energy property , while theβandγorbitals are fully11.52 0 0.5 1xy yz zx λequivalent yz and zxcharge density (a) 11.52 0 0.5 1α β γ λcharge density (b) Figure 3. The charge density in each of the t2gorbitals for different basis sets at U′=10 and J=2. (a) Real xy,yz, andzxorbitals. (b) Complex α,β, andγorbitals. occupied, consistent with the local electron configuration in the atomic limit, as shown in Fig. 1. This clearly indicates the emergence of a complex orbital st ate, in which spin and orbital states are entangled with complex number coefficients. 4. Summary We have studied ground-state properties of the t2g-orbital Hubbard model with the spin-orbit coupling by Lanczos diagonalization. We have found that due to the spin-orbit coupling, the ground state changes from the spin singlet to the singlet in t erms of the effective total angular momentum. Thecomplex orbitalstatecharacterized by Jeff=1 2emerges inthespin-orbit-induced state. To clarify novel magnetism in 5 dtransision metal oxides such as Sr 2IrO4, an important issue is to understand the excitation dynamics under the str ong spin-orbit coupling, which we will discuss elsewhere in future. Acknowledgments The author thanks G. Khaliullin, S. Maekawa, M. Mori, T. Shir akawa, H. Watanabe, and S. Yunoki for discussions. Part of numerical calculations wer e performed on the supercomputer at Japan Atomic Energy Agency. This work was supported by Grant -in-Aid for Scientific Research of Ministry of Education, Culture, Sports, Science, and Tec hnology of Japan. References [1] Kim B J, Jun H, Moon S J, Kim J Y, Park B G, Leem C S, Yu J, Noh T W, Kim C, Oh S J, Park J H, Durairaj V, Cao G and Rotenberg E 2008 Phys. Rev. Lett. 101076402 [2] Kim B J, Ohsumi H, Komesu T, Sakai S, Morita T, Takagi H and A rima T 2009 Science3231329 [3] Moon S J, Jin H, Choi W S, Lee J S, Seo S S A, Yu J, Cao G, Noh T W an d Lee Y S 2009 Phys. Rev. B80 195110 [4] Ishii K, Jarrige I, Yoshida M, Ikeuchi K, Mizuki J, Ohashi K, Takayama T, Matsuno J and Takagi H 2011 Phys. Rev. B83115121 [5] Okabe H, Isobe M, Takayama-Muromachi E, Koga A, Takeshit a S, Hiraishi M, Miyazaki M, Kadono R, Miyake Y and Akimitsu J 2011 Phys. Rev. B83155118 [6] Jin H, Jeong H, Ozaki T and Yu J 2009 Phys. Rev. B80075112 [7] Jackeli G and Khaliullin G 2009 Phys. Rev. Lett. 102017205 [8] Watanabe H, Shirakawa T and Yunoki S 2010 Phys. Rev. Lett. 105216410 [9] Onishi H 2011 J. Phys. Soc. Jpn. 80SA141 [10] Wang F and Senthil T 2011 Phys. Rev. Lett 106136402

1305.3810v2.Interplay_between_spin_orbit_interactions_and_a_time_dependent_electromagnetic_field_in_monolayer_graphene.pdf

Interplay between spin-orbit interactions and a time-dependent electromagnetic eld in monolayer graphene Andreas Scholz,Alexander L opez, and John Schliemann Institute for Theoretical Physics, University of Regensburg, D-93040 Regensburg, Germany (Dated: May 25, 2022) We apply a circularly and linearly polarized terahertz eld on a monolayer of graphene taking into account spin-orbit interactions of the intrinsic and Rashba types. It turns out that the eld can be used not only to induce a gap in the energy spectrum, but also to close an existing gap due to the dierent reaction of the spin components with circularly polarized light. Signatures of spin-orbit coupling in the density of states of the driven system can be observed even for energies where the static density of states is independent of spin-orbit interactions. Furthermore it is shown that the time evolution of the spin polarization and the orbital dynamics of an initial wave packet can be modulated by varying the ratio of the spin-orbit coupling parameters. Assuming that the system acquires a quasi stationary state, the optical conductivity of the irradiated sample is calculated. Our results conrm the multi step nature of the conductivity obtained recently, where the number of intermediate steps can be changed by adjusting the spin-orbit coupling parameters and the orientation of the eld. PACS numbers: 71.70.Ej, 73.22.Pr, 78.67.Wj I. INTRODUCTION Since a monolayer of graphite was isolated and detected for the rst time,1many theoretical and experimental studies on this remarkable and surprising material have been published.2Even several years after its discovery graphene remains one of the most intense research topics in solid state physics. This expresses the high expectations and hopes physicists have for graphene as being a building block for novel electronic devices. While in the beginning the focus of graphene research was mainly set on spin-independent phenomena, as it was claimed that spin-orbit interactions (SOIs) are virtually unimportant in graphene,3,4recent experimental and theoretical works have demonstrated that spin-orbit coupling (SOC) eects might be important as the characteristic parameters can be enlarged signicantly.5{10This, in principle, opens up the possibility of using spin-related phenomena in this outstanding material with exceptional electronic properties. Moreover, and indeed fortunately, graphene is not the only promising two-dimensional hexagonal system and thus many of the ndings of spin-related research in graphene can also be applied to other systems. As an example we mention a monolayer of MoS 211{13 which can at low energies eectively be described as two uncoupled gapped graphene systems, where both the band gap and the SOIs turn out to be large,14{16 and silicene, a two-dimensional honeycomb lattice made of silicon atoms.17{19Furthermore, restricting ourselves to nearest-neighbor and interlayer hopping, the Hamiltonian of bilayer graphene is formally equivalent to that of monolayer graphene with a purely Rashba SOC, where the Rashba coecient is substituted by the interlayer hopping constant. In recent works the eects of an external time- dependent eld on two-dimensional materials such asa monolayer20{27or bilayer28of graphene, HgTe/CdTe quantum wells,29,30orn- and p-doped electron gases31{34 have been discussed. It was shown that an electromagnetic eld can induce gaps in the energy spectrum of graphene20{23and even move and merge the Dirac points.27Both aspects might be interesting for future applications such as transistors. Furthermore, in Ref.29the possibility of changing the topology of a HgTe/CdTe quantum well by applying linearly polarized light leading to so-called Floquet topological insulators has been reported. In this work we study how SOIs of the intrinsic and Rashba types manifest themselves in a monolayer of graphene under the in uence of a time-dependent eld whose energy is on the terahertz (THz) regime. The SOC coupling constants are chosen to be of the order of the photon energy. This work is organized as follows. In Sec. II, we introduce the model Hamiltonian and brie y summarize the main properties of the solution of Schr odinger's equation according to Floquet's theorem. In Sec. III, the energy spectrum and the density of states (DOS) of the driven system is discussed and signatures arising from the interplay of SOC and the THz eld are pointed out. In Sec. IV, the dynamics of physical observables such as the spin polarization and the position operators are studied. In Sec. V, the optical conductivity of the irradiated sample is calculated for various combinations of the SOC parameters. Finally, Sec. VI summarizes the main results of this work. II. THE MODEL We use the Kane-Mele model3(setting ~= 1 throughout this work) ^H0=vFk
s0+Izsz+R(s)ez (1)arXiv:1305.3810v2 [cond-mat.mes-hall] 28 Aug 20132 to describe a monolayer of graphene including SOIs of the intrinsic (I) and Rashba ( R) types at one Kpoint. The Pauli matrices (s) and the unit matrix 0(s0) act on the pseudospin (real spin) space. The other Kpoint can be described by the above Hamiltonian with x!x andz!z. The eect of the electromagnetic eld can be incorporated by the minimal coupling scheme k! k+eA(t). As the vector potential does not depend on the position operators, the Hamiltonian remains diagonal in momentum space and we can treat kas a number instead of a dierential operator. The time-dependent contribution to the Hamiltonian, ^H1(t) =evFA(t)
s0; (2) is assumed to be periodic in time, i.e., ^H1(t+T) =^H1(t), whereT= 2= and is the frequency of the radiation eld. The vector potential describing a monochromatic wave propagating perpendicular to the graphene plane can be assumed to be either classical, A(t) =p 2E0 [cospcos tex+ sinpsin tey];(3) or quantized, A(t) =A cosp ^aei t+ ^ayei t
ex +isinp ^aei t^ayei t
ey ; (4) where, among obvious notation, the parameter A contains geometric information about the cavity surrounding the system. The eld is either circularly (p= 45) or linearly polarized (e.g., along the x direction for p= 0). Quantizing the electromagnetic eld adds a degree of freedom described by the bosonic operators ^a(y), which comes along with a new conserved quantity given by the helicity ^h=^J+ ^ay^a, where the angular momentum ^J=xkyykx+z=2 generates rotations of the carrier degrees of freedom in real and pseudospin space. To treat the electromagnetic eld as a quantized operator is important in situations where the charge carriers have a signicant back-action on the eld which in turn can alter the particle dynamics itself. To analyze this aspect further, let us consider the case of a vanishing eld and neglect SOIs for the moment. Now assuming a wave packet with initial momentum along theyaxis and the pseudospin initially in the x direction, the dynamics of the system in the Heisenberg representation is given by35{41 d2 dt2xH(t) =2v2 Fksin (2vFkt) (5) andd2yH(t)=dt2= 0. From the classical expression for the radiative power of dipolar radiation,42P= (er)2=60c3, we nd the time-averaged energy loss per time as P=e2v4 Fk2 30c37:12102k2 nm2meV ps; (6)i.e., for a wave vector of k= 0:1 nm1the radiative power is of order 104meV/ps. Due to the very large Fermi velocity of vF= 106m/s in graphene, a time scale of 1 ps corresponds, for appropriate initial conditions, to a distance of 1 m traveled by the wave packet. Therefore, the above loss rate should be seen as a small eect.43Hence the energy loss due to dipolar radiation induced by Zitterbewegung can be neglected compared to other energy scales in typical experimental situations. Accordingly, in what follows we will treat the electromagnetic eld as a classical quantity and not as an operator. For convenience we introduce the dimensionless quantity =vFeE0= 2. Due to the periodicity of ^H(t) = ^H0+^H1(t), the solution of Schr odinger's equationh i@=@t^Hi j k;i= 0 obeys Floquet's theorem44,45and thus is of the form j k;i=ei"k;tj k;(t)i; (7) where;=1 are band indices. The Floquet states j k;(t)ihave the same periodicity as the Hamiltonian and can be expanded in a Fourier series:46 j k;(t)i=1X n=1ein tn k; : (8) The original problem can now be reduced to the diagonalization of the time-independent Floquet Hamiltonian whose components are dened by  ^HF nm=1 TZT 0dt^H(t)ei(nm) tn nm:(9) The time evolution of an arbitrary state with respect to an initial time t0is captured by the operator ^Uk(t;t0) =X 0;0ei"k;00(tt0)j k;00(t)ih k;00(t0)j: (10) Notice that the energies and wave functions entering Eq. (7) are not uniquely dened as n k;E = ein tj k;i(withn2Z) is a solution of Schr odinger's equation as well. The corresponding quasienergy "n k;= "k;+n diers only by a multiple of the THz energy. Hence the choice of the eigenenergies is ambiguous as they describe the same physical situation. In order to get a well-dened quantity that is the same for all "n k;, we furthermore introduce the time-averaged (or quasi stationary) energy23,47{49 "k;=1 TZT 0dtD k;(t)j^H(t)j k;(t)E : (11) In general, there is a non trivial relation between the quasienergies and the mean energies. Notice that in the absence of the driving Eq. (11) reproduces the energies of the unperturbed system (see below).3 FIG. 1: (Color online) Quasienergy spectrum under circularly polarized light ( p= 45) for various combinations of the SOC parameters: ( R= ;I= ) = (a) (0;0), (b) (0;0:1), (c) (0:1;0) (c), and (d) (0 :1;0:1). The eld strength was set to = 0:3. III. ENERGY SPECTRUM AND DENSITY OF STATES A. Energy bands The energy bands of the static problem ( = 0) can readily be obtained: E(k) =R+q v2 Fk2+ (RI)2: (12) For a nite driving, the eigensystem is calculated numerically by diagonalization of the Floquet Hamiltonian in Eq. (9). As mentioned above, this leads to an innite number of eigenenergies46where only four of them are physically independent (corresponding to the dimension of the problem), while all others can be obtained by adding or subtracting a multiple of the energy of the electromagnetic eld Eem= . In Figs. 1 and 2 the quasienergies within the rst and second Brillouin zones (BZs) are shown as red lines for dierent combinations of the SOC parameters for a xed eld strength of = 0:3. The black dashed line in Fig. 1 shows, for comparison, Eq. (12) projected to the BZ. The SOC parameters are chosen to be of the order of the THz energy, e.g., in the present case R=I0:1 . As mentioned in the introduction, it has been demonstrated thatRandIcan be enlarged by several orders of magnitude by choosing proper adatoms5{7or a suitable environment8{10which allows values of RandIin the THz (meV) range. Our results depend only on the ratio R=I= and on the coupling strength . Hence they may also be applied to elds with larger frequencies (such as the mid-infrared) provided the SOC parameters are large enough. The advantage of a THz eld, however, is that the eld energies are far below the energies of opticalphonons (of about 200 meV),50such that excitations of optical phonons are suppressed. Circular polarization. The unperturbed energy spectrum of Eq. (12) consists of twofold spin-degenerate bands ifR= 0, while for a nite Rashba coecient structure inversion symmetry is broken and the bands split up; see the dashed lines in Fig. 1. Once ^H1(t) is turned on, in Figs. 1(a), 1(c) and 1(d) a gap opens up right the Dirac point, separating the valence and conduction bands. Here the bands are parabolic around the Kpoint but closely follow the linear behavior of the unperturbed result for vFk& . Similarly, a nite gap also appears in the mean energies lifting the Kpoint degeneracy, e.g., in Fig. 3(a) with a gap of 0= 4 2=p 1 + 42. For nite SOIs the bands react dierently on the THz eld and hence the degeneracy present in the static case of Fig. 1(b), where I= 0:1 andR= 0, disappears. Right at the Dirac point the quasienergy gap vanishes, while a new gap opens up between the conduction (or valence) band states with dierent spin orientations. Two of the four bands are now linear and not parabolic as in the case of = 0. Similarly, the gap in the time-averaged energies shown in Fig. 3(b) is closed. For larger momenta, vFk >0:7 , the spin splitting in Fig. 3(b) eventually becomes so small that the bands are virtually degenerate again. From Fig. 1 we can see that besides the gap at the Dirac point, additional gaps appear at vFkn =2 (n2Z). While these gaps are quite large for vFk =2 and , its value strongly decreases for larger momenta and seems to vanish for vFk&1:5 . The reason for these gaps is the existence of photon resonances,23i.e., the absorption and emission of photons, similar to the ac Stark eect in semiconductors.44Here transitions might occur at the resonant points EE00nEem. In the vicinity of the resonances, vFk0:5n , the average energies drop to zero. For large enough momenta the dips eventually become so narrow that they seem to disappear. In case spin degeneracy is broken (i.e., E+6= E), the above resonant condition can be fullled for multiple values of kand hence we observe not one but several nearby dips in the average energy spectrum, as shown in Figs. 3(c) and 3(d). Linear polarization. If the eld is linearly polarized, in the following along the xdirection, the energy spectrum is expected to be strongly anisotropic. In contrast to the circular case spin degeneracy is broken only ifR6= 0. From Fig. 2(a) we can see that for R= 0 and I= 0 the quasienergy spectrum for an in-plane angle ofk= 0, where tan k=ky=kx, exactly follows the unperturbed spectrum, i.e., the eld has no in uence.23 However, if at least one of the SOC parameters is nite, the valence and conduction bands no longer touch at vFk0:5 , where deviations from the static results are largest, and the THz eld induces a gap as shown in Fig. 2(b). The corresponding time-averaged energies, shown in Figs. 4(b)-4(d), exhibit characteristic dips at4 FIG. 2: (Color online) Quasienergy spectrum under linearly polarized light ( p= 0) for various combinations of the SOC parameters and momentum in plane orientations (tan k=ky=kx): (R= ;I= ) = (0;0) (left column), (0 ;0:1) (second from left), (0:1;0) (second from right), and (0 :1;0:1) (right). The eld strength was set to = 0:3. vFk0:5n , as for circularly polarized light. However, from Figs. 4(a) and 4(d) we can see that only those bands are aected by the THz eld that are (in the static limit) not linear but parabolic in momentum while the linear FIG. 3: (Color online) Mean energies derived from Eq. (11) under circularly polarized light ( p= 45) for various combinations of the SOC parameters: ( R= ;I= ) = (a) (0;0), (b) (0;0:1), (c) (0:1;0), and (d) (0 :1;0:1). The eld strength was set to = 0:3.bands remain unchanged and in particular ungapped. Notice that contrary to the above case where p= 45 the positions of the dips in the average energies are nearly the same for both spin orientations. Fork= 45(see the middle row of Fig. 2), we observe remarkable gaps in all quasienergy spectra at vFk0:5 andvFk . In addition, for nite SOIs an additional small gap opens up at the Kpoint separating the valence and conduction bands; see Figs. 2(g) and 2(h). The time- averaged energies as shown in the middle row of Fig. 4 resemble the circular result of Fig. 3. The important dierences, however, are the absence [Figs. 4(e) and 4(h)] or reduction [Figs. 4(f) and 4(g)] of the gap at the Dirac point and the fact that the THz does not cause an additional spin splitting of the bands; compare Fig. 1(b) and Fig. 2(f). In contrast to the case of k= 0the positions of the resonant dips clearly split up for R6= 0. Finally, for an in-plane angle perpendicular to the polarization direction, i.e., k= 90, again in all four cases a distinct gap opens up at vFk0:5 . While for R= 0 the Kpoint energies do not change, a small gap opens up in the quasienergies in Figs. 2(k) and 2(l) where R6= 0. Furthermore, the dips in the mean energies of cases Figs. 4(i) and 4(j) are suppressed for vFk= but they are clearly present in Figs. 4(k) and 4(l).5 FIG. 4: (Color online) Mean energies derived from Eq. (11) under linearly polarized light ( p= 0) for various combinations of the SOC parameters and momentum in plane orientations (tan k=ky=kx): (R= ;I= ) = (0;0) (left column), (0 ;0:1) (second from left), (0 :1;0) (second from right), and (0 :1;0:1) (right). The eld strength was set to = 0:3. B. Density of states In Figs. 5 and 6 the time-averaged DOS,23 D(E) =gvX k;1X n=1 n k;jn k; [E"k;+n ]; (13) is shown for various combinations of the SOC parameters with (red solid line) and without (black dashed) electromagnetic eld for circularly and linearly polarized light, respectively. The eld amplitude was set to = 0:3. The prefactor gv= 2 is due to the valley degeneracy. The static DOS for zero energy, shown as the dashed lines in Figs. 5 and 6, is zero in (a) and (b) and nite in (c) and (d). In the latter case ( R=I) the charge neutrality point is shifted to R=I. The electromagnetic eld yields a nite weightD n k;jn k;E to the subbands in the rst BZ even for momenta vFk > 0:5 . This leads to a distinct increase of the DOS for small energies compared to the eld-free situation.20In Fig. 6(b), for example, the DOS is greatly enhanced for jEj<I, while in the static case D(E) = 0 in this regime. As the quasienergies "k;have several extrema located atvFk  =2 and (see Figs. 1 and 2), the DOS exhibits pronounced Van Hove singularities.20,23 While due to the isotropy of the quasienergy spectrumin the case of circularly polarized light these singularities occur for arbitrary angles of k, for a linearly polarized eld not all angles lead to Van Hove singularities. As a consequence, the associated peaks rise much more strongly for p= 45compared to p= 0. In the former, the DOS drops down almost vertically and remains roughly constant around vFk0:5 and . This is in clear contrast to the linearly polarized case, where the decrease of the DOS is much smoother and the DOS becomes peaked, with D(E) being roughly linear around vFk0:5 and .28Moreover, if spin degeneracy is lifted, the DOS shows additional dips in between neighboring Van Hove singularities. This is also true in Fig. 5(b), where the splitting is caused by the THz eld and not by the Rashba term. In the static limit signatures of SOIs in the DOS can be seen only in a narrow region with E.0:25 , while for larger energies it is virtually the same in all cases (see the dashed lines in Figs. 5 and 6). This changes once the eld is switched on. Here SOC manifests itself even for larger energies. Comparing, e.g., Figs. 5(a) and 5(c), we see a remarkable dierence even for energies E due to the additional dips and peaks in the DOS. This can be understood from the quasienergy spectrum, e.g., in Fig. 1(c), where due to the breaking of spin degeneracy several nearby points with a horizontal dispersion exist. For circularly polarized light qualitatively the same happens also for the case of a purely intrinsic coupling6 (R= 0) as the bands split up for 6= 0. However, this splitting is signicant only for small momenta and hence the multiple dips in Fig. 5(b) can be seen only for energies around E0:5 . IV. SPIN POLARIZATION AND WAVE PACKET DYNAMICS We now discuss the dynamics of the real spin expressed by the operator ^SH;j(t) =H;0sH;j(t) (j2 fx;y;zg). We restrict ourselves to an initial state described by a Gaussian wave packet for a single momentum,40which is appropriate for a suciently broad initial wave packet:37 rin(t0) =1pder2 2d20 B@1 2 3 41 CA: (14) In the following the spinor components in Eq. (14) are chosen as1=i2=i3=4= 0:5, i.e., the initial state is in general a linear combination of the static eigenvectors. Because of d dt^SH;z(t) =2R[H;xsH;x(t) +H;ysH;y(t)]; changes in the initial out of plane spin polarization (SP) hSz(t0)i=j1j2+j2j2j3j2j4j2; whereh:i:=hinj:jini, can be induced only if the Rashba contribution is nite. Similarly, for the other two spin directions, hSx(t0)i= 2 Ref13+ 24g FIG. 5: (Color online) Time-averaged density of states calculated from Eq.(13) under circularly polarized light (p= 45) for various combinations of the SOC parameters: (R= ;I= ) = (a) (0 ;0), (b) (0;0:1), (c) (0:1;0), and (d) (0:1;0:1). The eld strength was set to = 0:3. FIG. 6: (Color online) Time-averaged density of states calculated from Eq. (13) under linearly polarized light (p= 0) for various combinations of the SOC parameters: (R= ;I= ) = (a) (0 ;0), (b) (0;0:1), (c) (0:1;0), and (d) (0:1;0:1). The eld strength was set to = 0:3. and hSy(t0)i= 2 Imf13+ 24g; whose dynamics are described by d dt^SH;x=y (t) = 2 RH;x=ysH;z(t)IH;zsH;y=x (t) ; at least one of the SOC coecients has to be nonzero in order to get a nontrivial time evolution. In the following we set without loss of generality t0= 0. In Fig. 7 we x the intrinsic parameter I= 0:25 and vary the Rashba constant at the Dirac point. The eld strength is set to = 0:5. While for R6=I the in plane SP of the static system, exemplarily shown for thexcomponent in Fig. 7(a), shows fast oscillations around zero; right at the point R=Ithe expectation valueshSx(t)iandhSy(t)ioscillate around a nite value. Subsequently the mean polarization Sx=yshown as black and red lines in Fig. 8(a), calculated for a total simulation time of t= 10 000, vanishes or is very small for R6=I, while Sx=y=0:5 forR=I. If we now turn on the THz eld, the time evolution of the spin operators clearly becomes more complicated; see Figs. 7(c) and 7(e). If the eld is linearly polarized along the xdirection, Sy is nite only for R=I, with Sy=0:5 at that point, as in the static case. However, this is no longer true for Sxas can be seen from Fig. 8(b) where the peak for the xcomponent disappears. For circularly polarized light both the peaks for Sxand SyatR=I vanish and a signicantly reduced dip at R0:34 appears. The out-of-plane SP ( Sz) of the static system oscillates around zero, where the period of the oscillations increases for larger R; see Fig. 7(b). Hence in contrast toSx=ythe mean polarization Szvanishes for arbitrary7 FIG. 7: (Color online) Time evolution of the xandz components of the spin polarization without electric eld (top row), under a linearly polarized eld ( p= 0,= 0:5) (middle row), and for circular polarization ( p= 45,= 0:5) (bottom row), as a function of the Rashba coecient. Parameters: I= 0:25 ,k= 0. R, as can be seen from the green line in Fig. 8(a). This remains true for linearly polarized light where Sz0 in all cases. Compared to that the situation for circularly polarized light is quite dierent. Here the zcomponent of the averaged spin oscillates as a function of Rand, depending on the magnitude of the Rashba parameter, Szcan be either positive or negative or zero for R= 0 andR0:33 . Notice that even though the intrinsic parameter has been xed to I= 0:25 in the above discussion, our ndings remain qualitatively the same for other values of I, and in particular the peak, e.g., in Fig. 8(a) always appears right at the point where R=I. A possible way to detect the SP has been described in Ref.51. Here the sample is scanned by a cantilever in magnetic resonance force microscopy, where the detected shift in frequency turns out to be related to the SP. The time evolution of the position operators in Heisenberg representation is given by d dt^rH(t) =ih ^H;^rHi =vFHsH;0(t): (15) Note that contrary to electron and hole gas systems35{37 the dissipative term proportional to momentum ismissing in Eq. (15) due to the Dirac-like nature of the charge carriers in graphene. By calculating the usual velocity operator ^vH(t) =vFHsH;0(t) it is thus possible to extract the orbital dynamics of the system, hr(t)i:=hinj^rH(t)jini; with respect to an initial wave packet given in Eq. (14). In Fig. 9 this is shown for circularly (a) and linearly (b) polarized light of strength = 0:5 for xed I= 0:5 and two dierent values of the Rashba SOC parameters for a total simulation time of t= 1000. While for p= 45andR6= 0 the trajectory resembles an ellipse, and hence the particle becomes localized, as exemplarily shown in the red curve in Fig. 9(a) for R= 0:5 , the basic propagation is along the ydirection if the Rashba contribution vanishes and, compared to hyi, only moderate deviations from the initial position in the x direction can be seen. Forp= 0andR= 0:5 [see the red curve in Fig. 9(b)], the main dynamics is along the xaxis with small oscillations around hyi= 0, while in the other case ofR= 0:6 (green line) the trajectory is again bounded in a nite region around hxi=5vF= and hyi=25vF= , respectively. V. OPTICAL CONDUCTIVITY In this section the optical conductivity of irradiated graphene is calculated. As we are not interested in processes that appear right after or before the THz eld is turned on and o, we consider the system in a quasi stationary state and assume the probability distribution to be of the form P/e"k;, where FIG. 8: (Color online) Mean spin polarization as a function of the Rashba parameter (a) without electric eld, (b) under a linearly polarized eld ( p= 0,= 0:5), and (c) for a circular polarization ( p= 45,= 0:5). The total simulation time is t= 10 000. Parameters: I= 0:25 , k= 0.8 FIG. 9: (Color online) Orbital dynamics hr(t)icalculated for a total simulation time of t= 1000 for (a) circularly ( p= 45) and (b) linearly ( p= 0) polarized light for various Rashba SOC coecients: R= 0 (black line), 0 :5 (red), and 0:6 (green). Parameters: I= 0:5 ,= 0:5,k= 0. "k;are the average energies introduced in Eq. (11) and = 1=Tthe inverse temperature. The quasi equilibrium density matrix in the basis of the Floquet states then reads23,47{49hk;j^qejk;00i=;0;0f[";(k)]. In the following, we restrict ourselves to zero temperature such that the Fermi distribution function reads f[E] = [EFE], withEFbeing the chemical potential. The expression for the dissipative part of the time- averaged longitudinal optical conductivity, obtained from the nonequilibrium Green's function method derived in Ref.23, then reads Refxx(!)g=gve2 !X k;m;jX ;;0;0D nj k;00^vxn k;E2 (f["k;]f["k;00])[!+"k;"k;00j ]: (16) The quasienergies and states entering Eq. (16) are chosen to be in the rst BZ, although any other choice is possible as well. From the function in Eq. (16) we can see that, in principle, transitions between all kinds of subbands are possible. In the static limit only those subbands that correspond to the energies of Eq. (12) have a nonzero weight and Eq. (16) reproduces previous results.52{54 For a nite driving the weight of the other subbands becomes nonzero, whereas it increases for larger driving amplitudes, and hence additional transitions become possible. In Figs. 10 and 11 we show the optical conductivity calculated for a xed Fermi energy of EF= 3 under the in uence of circularly and linearly polarized light, respectively. The eld strength is = 0, 0:5, and 1:0. The main feature of the static conductivity, as shown, e.g., in the dashed curve in Fig. 10(a), is its steplike behavior at != 2EF, where transitions from the valence to the conduction band become possible. Switching on the time-dependent eld leads to several additional steps in xx,23due to photon-assisted processes. By comparing e.g., Figs. 10(a) and 10(c), it becomes clear that the number of steps increases for larger coupling strengths as the weight is distributed over a broader range of subbands. The eect of the Rashba term, which leadsto a distinct breaking of the spin degeneracy of each subband, furthermore induces several intermediate steps as the number of possible transitions in the function of Eq. (16) becomes much larger. From Figs. 10(b) and 11(b) we can see that the basic structure of  xxis the same for p= 45andp= 0, but in the latter the conductivity turns out to be slightly smoother. By increasing the eld strength to = 1:0 we observe dips in the conductivity at !=n ,23where the eect is clearly larger forp= 45thanp= 0; see, e.g., Figs. 10(c) and 11(c). The inclusion of the Rashba term creates further dips for slightly smaller and larger frequencies, respectively. These dips are due to the appearance of gaps in the quasienergy spectrum (see the discussion in Sec. III), as some transitions are no longer possible. From Figs. 10(c) and 10(d) one can see that while the static conductivities (dashed curves) are quite similar in both cases, i.e., the eect of Ris only slight, remarkable dierences occur in the driven case, and hence SOC eects are greatly enhanced. VI. CONCLUSIONS In this work the eect of a time-dependent electric eld on a monolayer of graphene including SOIs of the intrinsic and Rashba types has been studied. We have demonstrated that a circularly polarized THz eld can be used not only to induce a gap at the Dirac point, which transforms graphene from a semimetal to an insulator, but also to close an existing gap in the quasienergies. In the opposite case of a linear polarization the spectrum turned out to be highly anisotropic and, depending on the strength of the SOC parameters and on the orientation of the eld, gaps in the spectrum might appear at the Kpoint and at the FIG. 10: (Color online) Optical conductivity under circularly polarized light ( p= 45) for various eld strengths = 0 (black dashed curves), 0 :5 [red lines in (a) and (b)], and 1 :0 [(c) and (d)], and SOC parameters in units of 0=e2=4.9 FIG. 11: (Color online) Optical conductivity under linearly polarized light ( p= 0) for various eld strengths = 0 (black dashed curves), 0 :5 [red lines in (a) and (b)], and 1 :0 [(c) and (d)], and SOC parameters in units of 0=e2=4. photon resonances vFk0:5n , or become suppressed. While the eect of SOIs on the DOS of the static sample could be seen only for energies E.0:25 , due to the existence of a multiple number of dips, signatures of SOC in the DOS of irradiated graphene appear even at much larger energies. By introducing a time-dependent eld it turned out to be possible to induce a nite net spin polarization in the sample. The sign and magnitude, e.g., of the out-of-planepolarization, can be modulated by changing the ratio of the SOC parameters, which can be done experimentally by adjusting the Rashba coecient via an electric gate. In the last part of this work the longitudinal optical conductivity was calculated. As reported already in Ref.23, the conductivity of irradiated graphene exhibits a multi step structure as transitions between a variety of subbands become possible. The number of steps depends not only on the coupling strength, but also on the magnitude of the Rashba parameter and on the polarization direction. Furthermore, for large enough coupling strengths the conductivity drops down for frequencies around the photon energy.23As for the DOS, compared to the static result the eect of SOIs on the optical conductivity is greatly enhanced for 6= 0, which is mainly caused by the Rashba contribution. Finally, let us point out that even though the SOC parameters within this work have been chosen to be smaller than (but comparable to) the energy of the eld, our ndings [such as the appearance of gaps in the quasienergy spectrum or the oscillatory behavior of the out-of-plane spin polarization in Fig. 8(c)] are not limited to this case, but can also be observed in the opposite case ofR=I& . Acknowledgments We thank M. Busl, M. Grifoni, S. Kohler, and M. W. Wu, for useful discussions. This work was supported by Deutsche Forschungsgemeinschaft via Grant No. GRK 1570. To whom correspondence should be addressed. Electronic address:andreas.scholz@physik.uni-regensburg.de 1K. S. Novoselov, A. K. 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1004.1352v2.Spin_charge_and_spin_orbital_coupling_effects_on_spin_dynamics_in_ferromagnetic_manganites.pdf

arXiv:1004.1352v2 [cond-mat.str-el] 12 May 2010Spin-charge and spin-orbital coupling effects on spin dynamics in ferromagnetic manganites Dheeraj Kumar Singh, Bhaskar Kamble, and Avinash Singh∗ Department of Physics, Indian Institute of Technology Kanpur Correlation-induced spin-charge and spin-orbital coupli ng effects on spin dynam- ics in ferromagnetic manganites are calculated with realis tic parameters in order to provide a quantitative comparison with experimental res ults for spin stiffness, magnon dispersion, magnon damping, anomalous zone-bounda ry magnon softening, and Curie temperature. The role of orbital degeneracy, orbi tal ordering, and orbital correlations on spin dynamics in different doping regimes is h ighlighted. PACS numbers: 75.30.Ds,71.27.+a,75.10.Lp,71.10.Fd2 I. INTRODUCTION The role of charge and orbital fluctuations on magnetic couplings an d excitations is of strong current interest in view of the several zone-boundary anomalies observed in spin-wave excitation measurements in the metallic ferromagnetic ph ase of colossal mag- netoresistive manganites.1–7The presence of short-range dynamical orbital fluctuations has been suggested in neutron scattering studies of ferromagne tic metallic manganite La1−x(Ca1−ySry)xMnO3.7These observations areof crucial importancefor a quantitative u n- derstanding of the carrier-induced spin-spin interactions, magno n excitations, and magnon damping, and have highlighted possible limitations of existing theoretic al approaches. For example, the observed magnon dispersion in the Γ-X direction sh ows significant softening near the zone boundary, indicating non-Heisenberg beh aviour usually modeled by including a fourth neighbour interaction term J4, and highlighting the limitation of the double-exchange model. Similarly, the prediction of magnon-phonon coupling as the origin of magnon damping2and of disorder as the origin of zone-boundary anomalous softenin g8 have been questioned in recent experiments.4–7Furthermore, the dramatic difference in the sensitivity of long-wavelength and zone-boundary magnon modes o n the density of mobile charge carriers has emerged as one of the most puzzling feature. Observed for a finite range of carrier concentrations, while the spin stiffness remains almost co nstant, the anomalous softening and broadening of the zone-boundary modes show subs tantial enhancement with increasing hole concentration.4,5 The role of orbital degeneracy, orbital ordering and orbital corr elations on spins dynam- ics in ferromagnetic manganites in the entire hole-doping range 0 < x<∼0.5 is still not fully understood due to lack of quantitative comparisons of theore tical calculations with experimental results in manganites, as highlighted in the recent rev iew.5 Most of the earlier investigations including correlation-induced O(1 /S) quantum cor- rections to the magnon spectrum were carried out in the strong-c oupling (double-exchange) limitandcouldnotsatisfactorilyaccountfortheobservedanomalou szone-boundarymagnon softening.9Whereas, withtypicalparametervalues t∼0.2−0.5eVforthemobile egelectrons and 2J∼2 eV for their exchange coupling to the localized t2gcore spins, the intermediate- coupling regime J∼Wappears to be more appropriate for the ferromagnetic mangantie s.10 Recent theoreticalstudies intheintermediate-coupling regimeand including theCoulomb3 repulsion between band electrons have demonstrated some realist ic features such as doping dependent asymmetry of the ferromagnetic phase and enhanced magnon softening and non- Heisenberg behaviour, thereby highlighting the importance of corr elated motion of electrons on spin dynamics.11,12However, these investigations do not include orbital degeneracy a nd inter-orbital Coulomb interaction, and do not provide any quantita tive comparisons with experimental results on manganites. Earlier theoretical investiga tions of the role of orbital- lattice fluctuations and correlations on magnetic couplings and excit ations have mostly been limited to ferro orbital correlations,13and have not addressed the physically relevant stag- gered orbital correlations. A self-consistent investigation of the interplay between spin and or bital orderings has recently carried out using a two-orbital ferromagnetic Kondo latt ice model (FKLM) includ- ing the Jahn-Teller coupling,14and doping dependence of the calculated Curie temperature was compared with experiments for different ferromagnetic manga nites. Finite Jahn-Teller distortion and orbital ordering were shown to be self-consistently generated at low doping. However, again only ferro orbital ordering was included in this analys is. Recent investigations of the correlated motion of electrons using a non-perturbative, inverse-degeneracy-expansion based, Goldstone-mode-prese rving approach have highlighted theroleofspin-chargeandspin-orbitalcouplingeffectsonmagnet iccouplingsandexcitations in orbitally degenerate metallic ferromagnets.15–18While spin-charge coupling was indeed found to yield strong magnon softening, damping, and non-Heisenb erg behaviour, it was only on including orbital degeneracy, inter-orbital interaction, an d a new class of spin- orbital coupling diagrams that low-energy staggered orbital fluct uations, particularly with momentum near ( π/2,π/2,0)corresponding to CE-type orbital correlations, were found18to genericallyyieldstrongintrinsicallynon-Heisenberg(1 −cosq)2magnonselfenergycorrection in three dimensions, resulting in no spin stiffness reduction, but stro ngly suppressed zone- boundary magnon energies in the Γ-X direction. Inthis paper we will apply these new results specifically to thecase of ferromagnetic man- ganites. We will consider a two-orbital FKLM including intra- and inter -orbital Coulomb interactions, with realistic values of hole bandwidth, lattice paramet er, and Hund’s coupling etc. Such a quantitative comparison should provide, within a physica lly transparent the- oretical approach, insight into the role of orbital ordering, orbita l degeneracy, and orbital correlations on spin dynamics in ferromagnetic manganites, seamles sly covering the entire4 hole-doping range 0 < x<∼0.5 within a single theoretical framework. The outline of the paper is as follows. Spin-charge coupling effects on spin dynamics in a single-band FKLM are presented in Section II, highlighting the stro ng differences from the canonical double-exchange behaviour. Spin stiffness and spin- wave energies are then obtained in the orbitally degenerate state of a two-orbital FKLM in S ection III, and com- pared with neutron-scattering results for the optimally ferromag netic manganites ( x≈0.3). Inter-orbital interaction Vis included next andSection IV describes spin dynamics in theor- bitally ordered ferromagnetic state. The hole-doping ( x) dependence of the estimated Curie temperature is compared with experimental results for two differe nt cases corresponding to wide- and intermediate-band manganites. Finally, the role of ( π/2,π/2,0)-type orbital cor- relations on zone-boundary magnon softening and instability of the FM state near x= 0.5 are discussed in section V. II. SPIN-CHARGE COUPLING MAGNON SELF ENERGY The interplay between itinerant carriers in a partially filled band and loc alized magnetic moments is conventionally studied within the ferromagnetic Kondo lat tice model (FKLM): H=−t/summationdisplay /angbracketleftij/angbracketrightσa† iσajσ−J/summationdisplay iSi.σi (1) involving a local exchange interaction between the localized spins Siand itinerant electron spinsσi. Due to crystal-field splitting of the degenerate Mn 3 dorbitals into egandt2glevels, the partially filled band in ferromagnetic manganites corresponds to the mobile egelectrons whereas the three Hund’s-coupled localized t2gelectrons yield the localized magnetic mo- ments with spin quantum number S= 3/2. We will first consider the single-band FKLM, which is appropriate for t he orbitally de- generate state of the optimally doped ferromagnetic manganites n ear hole doping x≈0.3. As the doped holes are shared equally by the two degenerate orbita ls, this hole doping cor- responds to band filling n= (1−x)/2≈0.35 in each band. Orbital ordering and orbital correlations will be incorporated later within a two-orbital FKLM inclu ding intra-orbital and inter-orbital Coulomb interactions. Throughout, we will consid er a simple cubic lattice with lattice parameter a.5 β αα (b) α βα (c)ββ αβ (a)α FIG. 1: The first-order quantum corrections to the irreducib le particle-hole propagator φ(q,ω). The first-order magnon self energy for the single-band FKLM, res ulting from quantum corrections to the irreducible particle-hole propagator φ(q,ω) shown in Fig. 1, was recently obtained as:17 Σmagnon(q,ω) =J2(2S)/summationdisplay Q/integraldisplaydΩ 2πi/parenleftBigg 1 Ω+ω0 Q−iη/parenrightBigg/summationdisplay k/bracketleftBigg/parenleftBigg 1 ǫ↑+ k−q+Q−ǫ↑− k+ω−Ω−iη/parenrightBigg ×1 2S/parenleftBigg 1−2JS ǫ↓+ k−q−ǫ↑− k+ω−iη/parenrightBigg2 (2) whereω0 Qrefers to the bare magnon energies, ǫσ k=ǫk−σJSare the exchange-split band energies, and the superscripts + /−indicate particle ( ǫσ k> ǫF) and hole ( ǫσ k< ǫF) energies. Incorporating correlation-induced self-energy and vertex corr ections, the above magnon self energy represents a spin-charge coupling between magnons and c harge excitations in the partially-filled majority-spin band, with the coupling vertex explicitly v anishing for q= 0, thus ensuring that the Goldstone mode is explicitly preserved. This r esult was obtained by applying the systematic inverse-degeneracy (1 /N) expansion scheme to a purely fermionic representation for the FKLM which allows conventional many-body diagrammatic expan- sion. The imaginary part of the magnon self energy in the strong-coupling limit: 1 πImΣmagnon(q,ω) =/parenleftbigg1 2S/parenrightbigg2/summationdisplay Q/summationdisplay k(ǫk−q−ǫk+ω)2δ(ǫ↑+ k−q+Q−ǫ↑− k+ω+ω0 Q) (3) yields finite magnondamping and linewidth at zero temperature, arisin g frommagnondecay into intermediate magnon states accompanied with majority-spin ch arge excitations. On the other hand, at the classical level (random phase approximatio n), magnon damping is6 0 0.01 0.02 0.03 0.04 0.05 0 0.2 0.4 0.6 0.8 1D (0) nS=3/2W=12t(a)J/W=1/6 1/4 1/2 1 0 0.01 0.02 0.03 0.04 0.05 0 0.2 0.4 0.6 0.8 1D n(b) FIG. 2: Comparison of the bare (a) and renormalized (b) spin s tiffness for the single-band FKLM, showing the substantial spin stiffness reduction due to the sp in-charge coupling effect. possible only due to decay into the Stoner continuum, and is therefo re completely absent in the intermediate and strong coupling regimes where the magnon spe ctrum lies well within the Stoner gap. The small- qbehaviour of the magnon self energy in Eq. (2) yields the first-orde r quantum correction to spin-stiffness in ddimensions: D(1)= Σ(1)(q)/q2=1 d(2S)2/summationdisplay Q,k(∇ǫk)2 ǫ↑+ k−q+Q−ǫ↑− k+ω0 Q(4) which is down by a factor (1 /2S) relative to the bare (classical) spin stiffness: D(0)=ω(0) q/q2=1 d(2S)/bracketleftbigg1 2/angbracketleft∇2ǫk/angbracketright−/angbracketleft(∇ǫk)2/angbracketright 2JS/bracketrightbigg . (5) The two competing terms above of order tandt2/Jcorrespond to delocalization energy loss and exchange energy gain upon spin twisting, and determine the overall stability of the ferromagnetic state against long wavelength fluctuations. Th e spin stiffness quantum correction D(1)is only weakly dependent on J(through ω0 Q), and involves only an exchange contribution, which is suppressed for electronic spectral distribu tions characterized by dom- inant saddle-point behaviour ( ∇ǫk= 0), resulting in enhanced spin stiffness D=D(0)−D(1) and ferromagnetic stability. In the double-exchange limit ( J→ ∞), only the delocalization part of the classical spin stiffness survives. This contribution has a particle-hole symmetric p arabolic band-filling dependence. Its behaviour with electron filling nis identical to that with hole doping x, vanishing at both ends n= 0 and n= 1 and increasing symmetrically with added electrons or holes, leading to the conventional understanding with in the DE model that7 0 0.1 0.2 0.3 0.4 0.5Magnon Energy and Damping qJ =0.7W S = 3/2 Χ Μ R Γn = 0.35 Γω0 q ωq Γqx10 FIG. 3: The bare and renormalized magnon energies and magnon damping for the single-band FKLM. The magnon damping linewidths are nearly one-tenth of the renormalized magnon energies in the Γ-X direction. ferromagnetism increases with added carriers. However, both th e classical and quantum exchange contributions involving the ( ∇ǫk)2terms in Eqs. (4) and (5) break this particle- hole symmetry, and the stiffness actually vanishes at n <1 before the band is completely filled. Since the spin stiffness quantum correction involves only an exchang e contribution, in- cludingquantumcorrectionisthereforeeffectively equivalent toen hancing thebareexchange contribution by decreasing J. Fig. 2 shows the renormalized spin stiffness D=D(0)−D(1) in units of ta2, evaluated from Eqs. (4) and (5) for different values of J. For band fillings belown= 0.5, the renormalized spin stiffness for J∼Wis indeed seen to be close to the bare spin stiffness for J/t= 4. Figure 3 shows a comparison of the bare and renormalized magnon dis persions in units of t, andalso themagnondamping obtained forthe single-band FKLM. Be sides the substantial reductionoftherenormalizedmagnonenergies, wealsofindthatint heintermediatecoupling regime (J∼W), themagnon self energy becomes increasingly non-Heisenberg like , resulting inωX< ωR/3, as indeed observed. The magnon linewidth Γ qobtained from the imaginary part of the magnon self energy increases with momentum q, but the ratio Γ q/ωqis found to remain nearly constant at ∼1/10upto the zone boundaryin theΓ-Xdirection, inagreement with magnon linewidth measurements.7In the X-M, M-R, and R-Γ directions, we find the ratio Γ q/ωqto be smaller than 0.1.8 0 50 100 150 200 0 0.2 0.4 0.6 0.8 1D (meV Å2) nJAF=2.8meV S=3/2t=240meV J/W=1/2 1 2 FIG. 4: The renormalized spin stiffness calculated for a two-o rbital model, including an AF contri- bution due to the Mn superexchange. For J≈W, the calculated values are close to the measured stiffness ∼170 meV ˚A2for the wide-band compound La 1−xSrxMnO3. Thespin-chargecouplingeffect onmagnonexcitationscanbereadily extended tothetwo- band FKLM involving interaction term −J/summationtext iSi.(σiα+σiβ) in Eq. (1). Fig. 4 shows the renormalized spin stiffness for this two-band model obtained by dou bling the spin stiffness resultD=D(0)−D(1)from Eqs. (4,5), corresponding to the independent contributions of the two degenerate egorbitals. Here we have taken t= 240meV and lattice parameter a= 3.87˚A for manganites. Corresponding to the AF superexchange intera ction between Mn spins, an AF contribution to spin stiffness D(0) AF=zJAFSa2/6 was also subtracted from the above result with JAF= 2.8meV and lattice coordination z= 6. Near optimal band filling n≈0.35 (hole doping x= 1−2n≈0.3) for ferromagnetic man- ganites, the calculated spin stiffness for J≈Wis in close agreement with the experimentally measured values ∼170 meV ˚A2for the wide-band compound La 1−xSrxMnO3. The strong suppression near n=0.5 (x=0) seen above with decreasing Jcould also be a contributing factor towards destabilizing ferromagnetism in manganites as xapproaches zero, in addition to orbital correlations. Fig. 5 shows a comparison of the calculated magnon dispersion for th e two-band FKLM along different symmetry directions in the Brillouin zone with experimen tal data for the narrow-band compound La 0.7Ca0.3MnO3obtained from neutron scattering measurements.5 The renormalized magnon energy was obtained using ωq= 2(ω0 q−Σq)−zJAFS(1−γq). A relatively smaller hopping term t= 180meV (bandwidth W= 12t≈2eV) was taken in the calculation for this narrow-band compound. Ab-initio calculations als o yield an estimated9 0 20 40 60 80 100 120ωq (meV) qJ =0.7W S = 3/2 Χ Μ R Γ Γn = 0.35 JAF = 2.1meVt=180meV experimental calculated FIG. 5: Comparison of calculated renormalized magnon energ y dispersion with experimental data for the narrow-band compound La 0.7Ca0.3MnO3from neutron scattering measurements.5 bandwidth W= 2eV for La 1−xCaxMnO3.19The renormalized magnon energies are broadly in good agreement with the neutron scattering measurements. In cluding the zone-boundary magnon softening resulting from the spin-orbital coupling effect dis cussed later will improve the agreement near the X point. 0 0.5 1 1.5 2 0 50 100 150 200 250 300 350 400〈Sz〉 T (K)JAF=2.1meVS=3/2J=0.7W t=180meVn=0.35 FIG. 6: Temperature dependence of magnetization calculate d using the self-consistent Callen scheme with the same set of parameters as above. The calculat ed Curie temperature is close to the measured values for narrow-band manganites. From the renormalized magnon energies obtained as above, the finit e-temperature spin dynamics was investigated using the self-consistent Callen scheme in which the magnetiza- tion for a quantum spin- Sferromagnet:20 /angbracketleftSz/angbracketright=(S−Φ)(1+Φ)2S+1+(S+1+Φ)Φ2S+1 (1+Φ)2S+1−Φ2S+1, (6)10 where the boson (magnon) occupation number: Φ =1 N/summationdisplay q1 eβ˜ωq−1(7) in terms of the thermally renormalized magnon energies: ˜ωq=ωq/angbracketleftSz/angbracketright/S (8) Figure6showsthetemperaturedependence ofmagnetizationobt ainedbyselfconsistently solving the coupled set of equations (6)-(8). With the same set of p arameters as in Fig. 5 for the magnon dispersion fit, the calculated Curie temperature ( ∼280K) is close to the measured value ∼250K for LCMO. Asorbitalcorrelationsarerelativelyunimportantatthesefillings, t heorbitallydegenerate FKLM calculation presented here provides a good description of fer romagnetic properties. Orbital ordering and fluctuations due to inter-orbital interaction and Jahn-Teller distortion become important near quarter filling ( x∼0) and near x∼0.5, and therefore must be included as both strongly suppress ferromagnetism. Effects of or bital ordering and orbital fluctuations on spin dynamics are discussed below. III. ORBITALLY ORDERED FERROMAGNETIC STATE Weakly doped manganites such as La 1−xSrxMnO3and La 1−xCaxMnO3exhibit an or- bitally ordered state for x<∼0.2, as inferred from x-ray diffraction and neutron scattering experiments.7,21In this section, we will therefore consider spin dynamics in an orbitally or- dered ferromagnet with staggered orbital ordering, and examine the behaviour of the Curie temperature with the onset of orbital ordering in the low-doping re gime. We therefore consider a two-orbital correlated FKLM: H=−t/summationdisplay /angbracketleftij/angbracketrightσµa† iσµajσµ−J/summationdisplay iµSiµ.σiµ+U/summationdisplay iµniµ↑niµ↓+V/summationdisplay iniαniβ (9) correspondingtothetwo egorbitalsµ=α,βpersite, andincludeintra-orbital( U)andinter- orbital(V)Coulombinteractions. Withincreasing V, theorbitallydegenerate ferromagnetic state becomes unstable towards an orbitally ordered ferromagne tic state with staggered orbital ordering near half-filling (in the majority-spin band). In the pseudo-spin space of the two orbitals, the orbitally-ordered state is exactly analagous to th e antiferromagnetic state11 of the Hubbard model with staggered spin ordering.22For simplicity, we consider staggered orbital ordering in all three directions, although the end compound LaMnO 3exhibits only planar staggered orbital ordering. The Jahn-Teller-phononic term is also considered to be important in m anganites, espe- cially in the low and intermediate doping range.23However, the inter-orbital Coulomb inter- action hasbeen suggested to bemuch stronger thanthe electron -phonon coupling in order to account for the observed insulating behaviour in undoped manganit es above the Jahn-Teller transition and the bond length changes below it.24,25Generally, Coulombic and Jahn-Teller- phononic approaches for manganites have been shown to be qualita tively similar.26Indeed, a mean-field treatment of the Jahn-Teller term14yields an electronic exchange-field term in orbital space proportional to the orbital magnetization /angbracketleftniσα−niσβ/angbracketright, exactly as would be obtained from the inter-orbital interaction term. We consider an orbitally-ordered ferromagnetic state with stagge red orbital ordering: /angbracketleftniα↑/angbracketrightA=/angbracketleftniβ↑/angbracketrightB=n+M/2 /angbracketleftniβ↑/angbracketrightA=/angbracketleftniα↑/angbracketrightB=n−M/2 (10) where the staggered orbital order Mcharacterizes the density modulation on the two sub- lattices A and B. For simplicity, we consider a saturated (half-metallic ) ferromagnetic state with empty spin- ↓bands:/angbracketleftniα↓/angbracketright=/angbracketleftniβ↓/angbracketright= 0. The ferromagnetic ordering is chosen to be in the ˆzdirection. The effective spin couplings and magnon energies are again determine d from the particle- hole propagator, now evaluated in the orbitally ordered state. The magnon energies are obtained as: ωq=J2(2S)[λ(0)−λ(q)] (11) whereλ(q) is the maximum eigenvalue of the transverse spin propagator: χ−+(q,ω) =/summationdisplay µ=α,β[χ0 µ(q,ω)] 1−U[χ0 µ(q,ω)](12) in terms of the bare particle-hole propagators [ χ0 µ(q,ω)] for the two orbitals µ, which are 2×2 matrices in the two-sublattice basis, with [ χ0 β]AA/BB= [χ0 α]BB/AAfollowing from the orbital-sublattice symmetry in the orbitally-ordered state. The ba re particle-hole propaga-12 tors are obtained by integrating out the fermions in the orbitally ord ered ferromagnetic state described by the effective single-particle Hamiltonian matrix: H0 µσ(k) = Γσ 1 0 0 1 + −µσ∆σǫk ǫkµσ∆σ (13) withintheself-consistent field(Hartree-Fock)approximation. He reΓ↑=−JS+VnandΓ↓= JS+(U+V)narethe orbitallyindependent fields with energy difference Γ ↓−Γ↑=Un+2JS correspondingtotheusualexchangebandsplitting, and∆ ↑=VM/2and∆ ↓= (U−V)M/2 aretheself-consistently determined exchange fields correspond ing to theorbital ordering M. The spin ( σ) and orbital ( µ) indices in the matrix are + and −for spins ↑and↓, and orbitals αandβ, respectively. The eigenvalues and eigenvectors of the above Hamiltonian matrix yie ld the bare-level band-electron energies and amplitudes for orbital µand spin σ: E0 kµσ= Γσ±/radicalBig ∆2σ+ǫ2 k a2 kµσ⊖=1 2/parenleftBigg 1+µσ∆σ/radicalbig ∆2σ+ǫ2 k/parenrightBigg =b2 kµσ⊕ b2 kµσ⊖=1 2/parenleftBigg 1−µσ∆σ/radicalbig ∆2σ+ǫ2 k/parenrightBigg =a2 kµσ⊕ (14) where⊕and⊖refer to the two eigenvalue branches ( ±). Orbital ordering splits the electron bands with energy gaps 2∆ ↑=VMand 2∆ ↓= (U−V)Mfor the two spins. At quarter filling (n= 1/2), the spin- ↑lower band ⊖iscompletely filled andtheupper band ⊕isempty, yielding a ferromagnetic insulator, and a metal-insulator transition o ccurs when orbital ordering melts, either driven by band overlap with decreasing Vor when the temperature exceedstheorbitalmeltingtemperature. Inoursaturatedferr omagneticstatewith /angbracketleftni↓/angbracketright= 0, both branches of the spin- ↓electron band are above the Fermi energy. The similarity of the orbitally-ordered state with the antiferromagn etic state of the Hub- bard model results not only in similar expressions for quasiparticle en ergies and amplitudes as above, but also in an identical self-consistency condition: 1 V=/summationdisplay k↑⊖1 2/radicalBig ∆2 ↑+ǫ2 k(15) which implicitly determines the magnitude of the orbital order m= 2∆↑/Vas a function of the effective interaction strength V. When only nearest-neighbor hopping is present, orbital13 0 50 100 150 200 250 300 350 400 0 0.1 0.2 0.3 0.4 0.5Tc (K) doping (x)(a) orbitally orderedt=180 meV orbitally degenerateV=6 V=0 expt 0 50 100 150 200 250 300 350 400 0 0.1 0.2 0.3 0.4 0.5Tc (K) doping (x)(b)orbitally orderedt=240 meV orbitally degenerateV=5 V=0 expt FIG. 7: Comparison of the doping ( x) dependence of calculated Tcwith experiments for the two compounds (a) La 1−xCaxMnO3and (b) La 1−xSrxMnO3. Staggered orbital ordering sharply suppress ferromagnetism due to band narrowing and reduced e lectronic delocalization. ordering at quarter filling sets in for any positive Vdue to Fermi-surface nesting, whereas a finitecriticalinteractionstrengthisrequiredwhenfrustratingne xt-nearest-neighbor hopping terms are included. For finite doping xaway from quarter filling, the staggered orbital order parameter mde- creases rapidly and vanishes at a critical doping value xc. Although long-range orbital order may be susceptible to orbital fluctuations at finite doping in the same way as the staggered spin ordering is in the doped antiferromagnet,31this HF result does provide a measure of the local orbital order. The pseudogap structure associated with sh ort-range orbital order in the metallic phase has been observed in recent high-resolution scanning tunneling microscopy and spectroscopy measurements of La 0.7Sr0.3MnO3and La 0.625Ca0.375MnO3thin films.27 ForJ∼WandU∼W, the Hund’s coupling contribution 2 JSto the exchange band splitting typically dominates over the intra-orbital (Hubbard) inter action contribution Um. Consequently, the calculated magnon energies depend rather wea kly onU, and we have therefore dropped it for simplicity in the calculations presented belo w. Figure 7 shows a comparison of the doping dependence of the calcula ted tran- sition temperature Tcwith experiments for the two compounds La 1−xCaxMnO3and La1−xSrxMnO3.28,29Assuming a Heisenberg form ωq=zJS(1−γq) for the magnon spec- trum, the transition temperature Tc= (1/3)zJS(S+ 1)//summationtext q(1−γq)−1= (5/6)ωXwas calculated approximately from the magnon energy at the X point ( π,0,0) obtained from14 Eq. (11). An AF superexchange interaction JAFbetween the neighboring Mn core spins was also included which correspondingly reduces the magnon energy. St aggered orbital ordering was found to persist up to x∼0.25, as indicated by the dashed vertical lines, and is clearly seen to sharply suppress ferromagnetism due to band narrowing a nd reduced electronic de- localization. In the orbitally degenerate state obtained beyond this critical doping value, the calculated Curie temperature is indicated by V= 0. As earlier, we have taken hop- ping energies t= 180meV and t= 240meV for the narrow-band and wide-band compounds La1−xCaxMnO3and La 1−xSrxMnO3, corresponding to bandwidths ∼2eV and ∼3eV, re- spectively. Other parameters are V/t= 6 and 5, J AF= 2.8meV and 2.1meV for the two compounds. Also, J/t= 4 in both cases, as the bare magnon energy in the orbitally degen- erate case then approximately corresponds to the renormalized m agnon energy (Fig. 2) for J∼Wonincluding thespin-charge coupling effect asdiscussed insection II .The anomalous zone-boundary magnon softening observed in narrow-band comp ounds will sharply reduce Tcasxapproaches 0.5. The above analogy with the antiferromagnet can be readily extende d to inter-orbital fluctuations and orbital waves (orbiton) in analogy with transvers e-spin fluctuations and spinwaves.22Thisanalogythendirectly yieldstheorbitonexcitationspectrum zJS/radicalbig1−γ2q in terms of the orbital exchange coupling J ≡4t2/Vin the strong-coupling limit, quantum reduction in the orbital order Mdue to quantum orbital fluctuations, orbital order-disorder transition at Tc∼(1/3)zJS(S+1)//summationtext q(1−γ2 q)−1/2in the strong-coupling limit ( V≫t) and orbital-order melting induced metal-insulator transition in the we ak-coupling limit. Similarly, hole motion in an orbitally ordered ferromagnet will lead to scr ambling of the local orbital order, string of broken orbital bonds, incoherent h ole spectral function and quasiparticle and band-gap renormalizations due to multiple orbiton e mission-absorption processes, in analogy with corresponding results for the AF.30Insights into the stability of theorbitally-orderedstateatfinitedopingawayfromquarterfilling cansimilarlybeobtained fromcorrespondingresults forthedopedantiferromagnet.31However, ifrealisticinter-orbital hoppingterms tαβareincluded, thecontinuousorbital-rotationsymmetryisbroken, resulting in a gapped orbiton spectrum, making the orbitally ordered state int rinsically different.15 IV. CE-TYPE ORBITAL CORRELATIONS AND ANOMALOUS MAGNON SOFTENING The orbitally degenerate ferromagnetic metallic state of manganite s also typically be- comes unstable near hole doping x= 0.5 due to onset of CE-type orbital correlations. The role of such planar staggered orbital correlations with momentum n ear (π/2,π/2,0) on spin dynamics was recently investigated for a two-orbital Hubbard mod el: H=−t/summationdisplay /angbracketleftij/angbracketrightσ(a† iασajασ+a† iβσajβσ)+U/summationdisplay i(niα↑niα↓+niβ↑niβ↓)+/summationdisplay iVniαniβ (16) including intra and inter orbital Coulomb interactions UandV. In the saturated ferromag- netic state (band filling nequal to magnetization m), electron correlation effects were incor- porated within the framework of a non-perturbative Goldstone-m ode-preserving approach.18 The magnon self energy due to coupling between spin and orbital fluc tuations [Fig. 8] was shown to generically yield strong intrinsically non-Heisenberg (1 −cosq)2momentum depen- dence, implying no spin stiffness reduction but strongly suppressed zone-boundary magnon energies in the Γ-X direction. In this section we will extend these res ults to the two-orbital correlated FKLM as in Eq. (9) with similar manganite parameters as us ed in previous sections. Thecorrelation-inducedspin-orbital coupling magnonself energy f orthetwo-orbitalHub- bard model was obtained as: Σsp−orb(q)≈m2/angbracketleft[Γsp−orb(q)]2/angbracketrightQ′,Ω′ Ωspin+Ωorb−iη(17) where Ω orband Ω spinrepresent characteristic orbital and spin fluctuation energy sca les, the angular brackets /angbracketleft /angbracketrightrefer to averaging over the orbital fluctuation modes Q′. The spin- orbital interaction vertex [Γ sp−orb(q)], represented diagrammatically in Fig. 8, was shown to explicitly vanish at momentum q= 0 in accordance with the Goldstone mode requirement, and yields the dominant qdependence of the magnon self energy. Although the same overall structure is obtained for the correlate d FKLM, there are two essential differences. First, the magnon self energy correct ion for the FKLM is given by Σ(n) magnon=J2(2S)φ(n)intermsofthecorrepondingquantumcorrection φ(n)totheirreducible particle-hole propagator (instead of U2mφ(n)as for the Hubbard model). Second, since the bare particle-hole propagator χ0for the mobile electrons in the FKLM involves the total16 U U α α↑ ↓V Vβαβ + + ≡↑ ↑ ↑ ↑ ↓ ↓ ↓ ↓ ↓V V VU(a)↑ ↑↑ (b)↑Γ Γ Γ FIG. 8: The spin-orbital coupling diagrams for the irreduci ble particle-hole propagator (a) can be represented in terms of a spin-orbital interaction vertex Γ sp−orb, the three diagrammatic contri- butions to which are shown in (b) involving three-fermion ve rtices. The missing fourth diagram vanishes because of the assumption of complete polarizatio n. exchange splitting Um+2JS, the spin-fluctuation propagator χ0/(1−Uχ0) corresponding to theUladders as in Fig. 8 yields a gapped spectrum with characteristic gap e nergy∼2JS (instead of the O( t) magnon energy scale for the Hubbard model). Consequently, thefirst-orderspin-orbitalcouplingmagnonselfe nergiesforthetwomodels are approximately related by: ΣFKLM sp−orb(q) = ΣHM sp−orb(q)×/parenleftbiggJ22S U2m/parenrightbigg/parenleftbiggt 2JS/parenrightbigg . (18) ForU= 20t,m≈0.3(astakenintheHubbardModelcalculation18)andJ≈W, therelative factor is about 1/10. For simplicity, we have therefore obtained th e spin-orbital coupling magnon self energy for the FKLM from the earlier Hubbard model re sult by multiplying by this relative factor. Doubling the magnon self energy to account for the independent co ntributions of the two degenerate egorbitals, the renormalized magnon energy ωq=ω0 q−Σsp−orb(q) for the two- orbital correlated FKLM is shown in Fig. 9 for different hole dopings. H ereω0 qis the bare magnon energy for the two-band FKLM including the negative contr ibutionzJAFS(1−γq) of the superexchange interaction JAF= 2.1meV between Mn core spins. This bare magnon energy was evaluated for J= 4 as it then approximately corresponds to the renormalized17 0 5 10 15 20 25 30 35 0 0.5 1 1.5 2 2.5 3ωq (meV) qV/t = 4 J ≈ W t = 180 meVωq 0 x=0.0 0.2 0.3 0.4 0.5 FIG. 9: Renormalized magnon energies ωq=ω0 q−Σsp−orb(q) including the spin-orbital coupling magnon self energy, plotted along the Γ-X direction for differ ent hole dopings, showing the anoma- lous zone-boundary magnon softening due to coupling of spin excitations with CE-type, period 4 a, planar orbital correlations with momentum modes near ( π/2,π/2,0). magnon energy (Fig. 2) for J≈Won including the spin-charge coupling effect as discussed in section II. The above analysis clearly shows the importance of CE-type stagge red orbital correla- tions on the observed anomalous zone-boundary magnon softenin g. Indeed, a sharp onset of CE-type orbital correlations at momentum ( π/2,π/2,0) has been observed in neutron scattering studies near the Curie temperature,7which gradually diminishes in intensity and sharpness with decreasing Ca concentration in La 1−x(CaySr1−y)xMnO3crystals. Interest- ingly, this behaviour is exactly similar to the resistivity temperature p rofile of these crystals with varying Ca concentration.32Taken in conjunction with our spin-orbital coupling theo- retical result of strong zone-boundary magnon softening produ ced by such CE-type orbital correlations,18this sharp onset of orbital correlations near Tcwould imply a sudden magnon softening and collapse of the ferromagnetic state. As the spin-or bital coupling magnon self energy∼(1−cosq)2does not renormalize the spin stiffness, this behaviour could also ac- count for the finite spin stiffness observed just below Tcin the low-bandwidth materials, which has so far been a puzzling feature. In the earlier Hubbard model calculation, the spin-orbital interact ion term [Γ sp−orb]2was calculated for the orbital fluctuation mode Q′= (π/2,π/2,0) and an averaging factor of18 1/10 was included, as obtained on averaging over all orbital fluctua tion modes assuming a flat distribution. As fluctuation modes near ( π/2,π/2,0) become increasingly prominent due to onset of CE type combined charge-orbital correlations nea r half doping (stabilized by an inter-site Coulomb interaction V′ninj), the enhanced weight of such staggered orbital correlations will substantially increase the magnon self energy and h ence the anomalous magnon softening as xapproaches 0.5. V. CONCLUSIONS Various aspects of spin dynamics in ferromagnetic manganites were theoretically investi- gated in terms of a realistic two-orbital FKLM including intra- and inte r-orbital Coulomb interactions, staggered orbital ordering, and orbital correlatio ns. For the same set of man- ganite parameters, the calculations are in close agreement with rec ent neutron scattering experiments onspinstiffness, magnondispersion, magnonlinewidth, Curietemperature, and anomalous magnon softening. The set of manganite parameters ta ken were lattice parame- tera= 3.87˚A, hopping terms t= 240meV and t= 180meV (corresponding to bandwidths W≈3eV and ≈2eV, respectively) for the wide-band (LSMO) and narrow-band (L CMO) compounds, Hund’s coupling J≈W, inter-orbital Coulomb repulsion V/t≈5, and Mn spin superexchange interaction JAF≈2.5meV. In the orbitally degenerate ground state appropriate for the opt imally doped ferromag- netic manganites with x∼0.3, the spin-charge coupling magnon self energy for the FKLM was shown to yield substantial suppression of ferromagnetism in th e physically relevant fi- niteJregime. Incorporating this magnon energy renormalization, we obt ained spin stiffness D∼150meV˚A2, Curie temperatures Tc∼300K, and the ratio of the magnon linewidth to magnon energy Γ q/ωq∼1/10 for magnon modes upto the zone boundary in the Γ-X direction, and the calculated magnon dispersion fitted well with rece nt neutron scattering data for La 1−xCaxMnO3. The onset of orbital ordering at low doping ( x<∼0.25) due to the inter-orbital interaction Vwas found to sharply suppress Tcdue to band narrowing effects associated with the staggered orbital order. The doping dependence of the calculate d Curie temperature for two bandwidth cases corresponding to narrow-band and wide-ban d compounds was in close agreement with the observed behaviour of Tcfor LCMO and LSMO.19 Finally, the role of CE-type planar orbital fluctuations with momentu m near (π/2,π/2,0) was investigated on spin dynamics in the orbitally degenerate ferrom agnet near half doping (x∼0.5). The magnon self energy due to correlation-induced spin-orbita l coupling arising from inter-orbital interaction and fluctuation was evaluated for t he correlated FKLM from theearlier Hubbardmodel result. Forthesamesetofmanganitepa rameters, thespin-orbital coupling magnon self energy with U∼Wwas found to yield anomalous zone-boundary magnon softening of the same magnitude as observed in neutron sc attering experiments. Onset of the CE-type charge-orbital correlations and the spin-o rbital coupling are therefore evidently responsible for the instability of the ferromagnetic state near half doping. ∗Electronic address: avinas@iitk.ac.in 1H. Y. Hwang, P. Dai, S-W. Cheong, G. Aeppli, D. A. Tennant, and H. A. Mook, Phys. Rev. Lett.80, 1316 (1998). 2P. Dai, H. Y. Hwang, J. Zhang, J. A. Fernandez-Baca, S.-W. Che ong, C. Kloc, Y. Tomioka, and Y. Tokura, Phys. Rev. B 61, 9553 (2000). 3T. Chatterji, L. P. Regnault, and W. Schmidt, Phys. Rev. B 66, 214408 (2002). 4F. Ye, Pengcheng Dai, J. A. Fernandez-Baca, Hao Sha, J. W. Lyn n, H. Kawano-Furukawa, Y. Tomioka, Y. Tokura, and Jiandi Zhang, Phys. Rev. Lett. 96, 047204 (2006). 5F. Ye, P. Dai, J. A. Fernandez-Baca, D. T. Adroja, T. G. Perrin g, Y. Tomioka, and Y. Tokura, Phys. Rev. B 75, 144408 (2007). 6J. Zhang, F. Ye, H. Sha, P. Dai, J. A. Fernandez-Baca, and E. W. Plummer, J. Phys.: Condens. Matter19, 315204 (2007). 7F. Moussa, M. Hennion, P. Kober Lehouelleur, D. Reznik, S. Pe tit, H. Moudden, A. Ivanov, Ya. M. Mukovskii, R. Privezentsev, and F. Albenque-Rullier , Phys. Rev. B 76, 064403 (2007). 8Y. Motome and N. Furukawa, Phys. Rev. B 71, 014446 (2005). 9For a recent review see, for example, M. D. Kapetanakis, A. Ma nousaki, and I. E. Perakis, Phys. Rev. B 73, 174424 (2006). 10E. Dagotto, T. Hotta, and A. Moreo, Phys. Rep. 344, 1 (2001). 11D. I. Golosov, Phys. Rev. B 71, 014428 (2005). 12M. D. Kapetanakis and I. E. Perakis, Phys. Rev. B 75, 140401 (2007).20 13G. Khaliullin and R. Kilian, Phys. Rev. B 61, 3494 (2000). 14M. Stier and W. Nolting, Phys. Rev. B 75, 144409 (2007). 15A. Singh, Phys. Rev. B 74, 224437 (2006). 16S. Pandey and A. Singh, Phys. Rev. B 75, 064412 (2007); ibid.76, 104437 (2007); ibid.78, 014414 (2008). 17S. Pandey, S. K. Das, B. Kamble, S. Ghosh, D. K. Singh, R. Ray, a nd A. Singh, Phys. Rev. B 77, 134447 (2008). 18D. K. Singh, B. Kamble, and A. Singh, Phys. Rev. B 81, 064430 (2010). 19G. Trimarchi and N. Binggeli, Phys. Rev. B 71, 035101 (2005). 20H. B. Callen, Phys. Rev. 130, 890 (1963). 21M. Pissas, I. Margiolaki, G. Papavassiliou, D. Stamopoulos , and D. Argyriou, Phys. Rev. B 72, 064425 (2005). 22A. Singh and Z. Teˇ sanovi´ c, Phys. Rev. B 41, 614 (1990); A. Singh, Phys. Rev. B 43, 3617 (1991); A. Singh, cond-mat/9802047 (1998). 23A. J. Millis, P. B. Littlewood, and B. I. Shraiman, Phys. Rev. Lett.74, 5144 (1995). 24P. Benedetti and R. Zeyher, Phys. Rev. B 59, 9923 (1999). 25S. Okamoto, S. Ishihara, and S. Maekawa, Phys. Rev. B 65, 144403 (2002). 26T. Hotta, A. L. Malvezzi, and E. Dagotto, Phys. Rev. B 62, 9432 (2000). 27U. R. Singh, A. K. Gupta, G. Sheet, V. Chandrasekhar, H. W. Jan g, and C.-B Eom, Appl. Phys. Lett. 93, 212503 (2008); U. R. Singh, Ph.D. Thesis, IIT Kanpur (2009) . 28Y. Tokura and Y. Tomioka, J. Magn. Magn. Mater. 200, 1 (1999). 29S.-W. Cheong and H. Hwang, Ferromagnetism vs. Charge/Orbital Ordering in Mixed-Valen t Manganites , chapter in “Colossal Magnetoresistive Oxides”, edited by Y. Tokura (Gordon and Breach, Monographs in Condensed Matter Science, 1998). 30P. Srivastava and A. Singh, Phys. Rev. B 70, 115103 (2004). 31A. Singh and H. Ghosh, Phys. Rev. B 65, 134414 (2002). 32Y. Tomioka, A. Asamitsu, and Y. Tokura, Phys. Rev. B 63, 024421 (2000).

1203.1159v1.Short_range_correlations_in_dilute_atomic_Fermi_gases_with_spin_orbit_coupling.pdf

arXiv:1203.1159v1 [cond-mat.quant-gas] 6 Mar 2012Short-range correlations in dilute atomic Fermi gases with spin-orbit coupling Zhenhua Yu Institute for Advanced Study, Tsinghua University, Beijin g 100084, China (Dated: March 22, 2021) We study the short-range correlation strength of three dime nsional spin half dilute atomic Fermi gases with spin-orbit coupling. The interatomic interacti on is modeled by the contact pseudopoten- tial. In the high temperature limit, we derive the expressio n for the second order virial expansion of the thermodynamic potential via the ladder diagrams. We fur ther evaluate the second order virial expansion in the limit that the spin-orbit coupling constan ts are small, and find that the correlation strength between the fermions increases as the forth power o f the spin-orbit coupling constants. At zero temperature, we consider the cases in which there are sy mmetric spin-orbit couplings in two or three directions. In such cases, there is always a two-body b ound state of zero net momentum. In the limit that the average interparticle distance is much la rger than the dimension of the two-body bound state, the system primarily consists of condensed bos onic molecules that fermions pair to form; we find that the correlation strength also becomes bigg er compared to that in the absence of spin-orbit coupling. Our results indicate that generic spi n-orbit coupling enhances the short-range correlations of the Fermi gases. Measurement of such enhanc ement by photoassociation experiment is also discussed. PACS numbers: 05.30.Fk, 05.70.Ce, 67.10.-j, 67.85.Lm I. INTRODUCTION Recent experimental advances in generating synthetic gauge fields are motivated by simulating charged parti- cles in solid state systems by neutral atoms [1, 2]. In the presence of external magnetic fields, the degener- acy of a manifold of the atom’s hyperfine spin states is lifted. The coupling between the manifold of the hy- perfine spin states and external laser fields gives rise to dressed states. The adiabatic elimination of the high en- ergydressedstatesresultsinalowenergyeffectiveHamil- tonian in which synthetic gauge fields emerge. By such schemes, uniform vector potentials [3], synthetic mag- netic [4] and electric fields [5] are realized in condensates of87Rb atoms. With the magnetic field’s strength and the laserfrequencyfine tuned, aspin-orbitcouplingbilin- ear in momentum and pseudo-spin operator components in one direction is engineered as well [6]. The possibil- ity of inducing spin-orbit couplings in two [7] and three directions [8] is further discussed theoretically. Similar attempts to synthesize gauge fields in atomic Fermi gases are under active experimental exploration [9]. It is an interesting question how the introduction of spin-orbit coupling would affect the correlations of di- lute atomic gases. In the BEC-BCS crossover problem, Tan noticed that the correlations in a homogeneous di- lutetwo-componentatomicFermigashaveanasymptotic form [10] ∝angb∇acketleftψ† ↑(r)ψ† ↓(0)ψ↓(0)ψ↑(r)∝angb∇acket∇ight=C/parenleftbigg1 r−1 as/parenrightbigg2 ,(1) in the regime where ris much less than dthe mean dis- tance between the particles and bigger than r0the range of the interatomic interaction potential U(r). Hereψσ are the field operators for fermions and asis the s-wave scattering length. The correlation (or contact) strengthat short distance Chas been shown to be linked with thermodynamic quantities through a seriesof remarkable relations [10–12], named as Tan’s relations, one of which is C=−m 4π∂f ∂a−1s, (2) wheremis the mass of particles and the free energy den- sity isf=−Tlog(Tre−H/T)/VwithTthe temperature andVthe volume of the system. In the presence of spin-orbit coupling, Eqs. (1) and (2) should hold when r0is much smaller than the length scale corresponding to the spin-orbit coupling strength. For specific, let us consider the spin-orbit coupling of the formhso=/summationtext i=x,y,zκipiσi, whereσiare the Pauli matrices and κiare the spin-orbit coupling constants. Originally, in the absence of hso, Eq. (1) can be de- rived from the observation [12] that given the scale sep- arationr0≪din dilute Fermi gases, when one writes ∝angb∇acketleftψ† ↑(r)ψ† ↓(0)ψ↓(0)ψ↑(r)∝angb∇acket∇ight=Cχ2(r), in the regime r/lessorsimilarr0, χ(r) satisfiesthe Schr¨ odingerequationofthe relativemo- tion of two interacting fermions with zero energy /bracketleftbigg −∇2 m+U(r)/bracketrightbigg χ(r) = 0. (3) The asymptotic form χ(r)∼(1/r−1/as) forr > r 0 is required to connect with the behavior of χ(r) in the regimer<r0which is solely determined by U(r);aspa- rameterizes the effects of U(r) onχ(r). Note that since we assume that U(r) is not fine tuned close to any reso- nance other than in the s-wave channel, the non s-wave parts of ∝angb∇acketleftψ† ↑(r)ψ† ↓(0)ψ↓(0)ψ↑(r)∝angb∇acket∇ightis neglected for r/lessorsimilarr0 due to the strong suppressionby the centrifugalpotential barrier. The introduction of hsowould modify Eq. (3). Since experimental values of κiare about 1 /d[6, 9], the inverseof the mean interparticle distance, we expect that2 the correction to χdue tohsois of orderκir0, which is negligible in dilute gases. A two-body calculation us- ing a square well model potential for equal spin-orbit couplings in three directions agrees with our expectation [13]. Given χunchanged in the lowest order of κir0, in the same way as used in Refs. [12, 14], one can show that Eq. (2) stands. However, how spin-orbit coupling would affect the magnitude of the correlation strength at short distance is the problem that we are going to study below. In this paper, we consider three dimensional spin half Fermi gases with spin-orbit coupling hso=/summationtext i=x,y,zκipiσi. The Hamiltonian of the system is H=H0+Hint+Hso, H0=/integraldisplay d3r∇Ψ†(r)∇Ψ(r)/2m, Hint=¯g/integraldisplay d3rψ† ↑(r)ψ† ↓(r)ψ↓(r)ψ↑(r), Hso=/integraldisplay d3rΨ†(r)hsoΨ(r). (4) HereΨ = (ψ↑,ψ↓)T. Thebarecouplingconstant ¯ gforthe contact pseudopotential U(r) = ¯gδ(r) defined with the momentum cutoff Λ is related to the s-wave scattering lengthasvia the renormalization m 4πas=1 ¯g+mΛ 2π2. (5) We take /planckover2pi1= 1 throughout. In the high temperature limit, we derive the second order virial expansion of the thermodynamic potential of the system by the ladder di- agrams. We further evaluate the second order virial ex- pansion perturbatively for small κiand find that the cor- relation strength Cincreases as the forth power of κi. At zero temperature, we consider two cases, equal spin-orbit couplings in two or three dimensions, in which there al- ways exits a two-body bound state with zero net momen- tum no matter the value of as[15]. In the limit that the mean interparticle distance is much larger than the size of the two-body bound state, by the fact that the leading contribution to the ground state energy comes from the binding energy of the bound state which fermions pair into, we show that Cbecomes bigger compared to that intheabsenceofspin-orbitcoupling. Ourresultsindicate that nonzerospin-orbitcoupling genericallyenhances the correlation strength of the Fermi gases. Such enhance- ment can be detected in photoassociation experiment. II. VIRIAL EXPANSION AND LADDER DIAGRAMS When the temperature Tis high, the grand canoni- cal partition function, Z= Tre−(H−µN)/Twithµthe chemical potential, can be approximated by a virial ex- pansionintermsofthe fugacity η=eµ/Twhichis asmall number. The effects of pairwise interactions first appearin the second order virial expansion. For two component Fermi gases interacting through a short range central po- tential without spin-orbital coupling, the second virial coefficient due to interactions has been derived [16] b2=/summationdisplay ne|En|/T+/summationdisplay ℓ/integraldisplay∞ 0dk π(2ℓ+1)dδℓ(k) dke−k2/mT, (6) whereEnare the binding energies of two-body bound states andδℓare the phase shifts in the ℓth partial waves. In the presence of spin-orbital coupling, hsocouples scat- terings in different partial waves to each other; this cou- pling renders classification in terms of angular momen- tum as in Eq. (6) impossible. However, since the ladder diagrams exhaust the two-body scattering processes, we use them to calculate the second order virial expansion [17, 18]. It is instructive to demonstrate how the ladder dia- grams reproduce a result agreeable with Eq. (6) for a Fermi gas whose Hamiltonian is H0+Hint. The varia- tion of the thermodynamic potential Ω = −TlogZdue to the ladder diagrams is [18] δΩ =/summationdisplay P/integraldisplay Cdζ 2πifB(ζ)log/parenleftbiggm 4πas−Π(P,ζ)/parenrightbigg ,(7) with Π(P,ζ) =/integraldisplayd3q (2π)3/braceleftbigg1−f(ξP/2+q)−f(ξP/2−q) ζ−ξP/2+q−ξP/2−q+1 ǫq/bracerightbigg . (8) Hereξq=ǫq−µ,ǫq=q2/2m,fBandfare the Bose and Fermi distribution functions respectively. The renor- malization Eq. (5) has been used to obtain Eq. (7). The branchcut ofthe logarithmicfunction lies on the positive real axis of its argument. The contour Cwraps the real axis of the integral variable ζ. To order of η2, we neglect the Fermi distribution func- tions in Π, since they contribute at least an extra fac- tor ofη; the argument of the logarithmic function in Eq. (7) becomes the inverse of the T-matrix in vacuum t−1(ζ−P2/4m+2µ), where t(ζ) =/bracketleftbiggm 4πas+im3/2 4πζ1/2/bracketrightbigg−1 . (9) After changingthe variable ζ′=ζ−P2/4m+2µ, we have δΩ =/summationdisplay P/integraldisplay Cdζ′ 2πifB(ζ′+P2/4m+2µ)log/parenleftbig t−1(ζ′)/parenrightbig . (10) It is generically true that the singularities of log(t−1(P,ζ′)) on the real axis of ζ′are left bounded. We deform the contour Cto wrap the part of the real axis ofζ′right to the most left singularity. On this contour3 C, we expand fB(ζ′+P2/4m+ 2µ) in the integrand to the lowest order of η, which is of order η2, as δΩ≈η2/summationdisplay Pe−P2/4mT/integraldisplay Cdζ′ 2πie−ζ′/Tlog/parenleftbig t−1(ζ′)/parenrightbig .(11) Direct evaluation of the above equation gives δΩ(0)=−23/2η2TV λ3˜b2(λ/as), (12) with ˜b2(λ/as) =θ(as)eλ2/2πa2 s+/integraldisplay∞ 0dk π(−as) 1+(kas)2e−k2λ2/2π =1 2[1+Erf(λ/√ 2πas)]eλ2/2πa2 s. (13) Here the thermal wavelength is λ≡/radicalbig 2π/mT. Equation (13) has been obtained by calculating the partition func- tion from the two-body eigenenergies in Refs. [14, 19]. Given that the contact pseudopotential ¯ gδ(r) scatters only the s-wave, and cot δs(k) =−1/kas, and−1/ma2 sis the binding energy for the only bound state when as>0, Eq. (13) agrees with Eq. (6). For fixed density n=N/V withNthe total number of fermions, Eq. (12) is δΩ(0)=−2−1/2n2TVλ3˜b2(λ/as) (14) sincen= 2η/λ3to order of η, In the presence of Hso,δΩ retains the form of Eq. (7) with Π replaced by Πso(P,ζ) =1 4/integraldisplayd3q (2π)3 /summationdisplay α,α′=±11−αα′φP,q ζ−Eα P/2+q−Eα′ P/2−q+4 ǫq , (15) withEα q=ξq+α∆q, ∆q=/radicalbig/summationtext iκ2 iq2 i, andφP,q= [/summationtext iκ2 i(P2 i/4−q2 i)]/∆P/2+q∆P/2−q. The index α=±1 picks up different helicity branches ofnoninteracting par- ticles [2]. Note that Fermi distribution functions have been neglected in Π sofor the same reason as stated be- fore. III. PERTURBATION IN HIGH TEMPERATURE LIMIT While to obtain the behavior of the second order virial expansionδΩ as a function of arbitrary values of κire- quires a full evaluation of the multi-dimensional integral (cf. Eqs. (7) and (15)), in the following, we calculate δΩ perturbatively in terms of mλκiin the high Tlimit in whichλ→0. Since the integrand of Π sois invariant un- der the transformation κi→ −κi, the perturbation seriesofδΩ consists of only even orders of mκiλ. To the forth order ofκi m 4πas−Πso(P,ζ′+P2/4m−2µ) =t−1(ζ′)−4/integraldisplayd3q (2π)3 ×/bracketleftBigg/summationtext iκ2 iq2 i (ζ′−q2/m)3+/summationtext i,jκ2 iκ2 j(4q2 iq2 j+p2 iq2 j−pipjqiqj) (ζ′−q2/m)5/bracketrightBigg . (16) For the second order virial expansion δΩ to second or- der ofκi, δΩ(2) =η2/parenleftBigg/summationdisplay iκ2 i/parenrightBigg/summationdisplay P/integraldisplay Cdζ′ 2πie−(ζ′+P2/4m)/Tim3 8π√mζ′t(ζ′) =−m/parenleftBigg/summationdisplay iκ2 i/parenrightBigg 23/2η2V λ3˜b2(λ/as) =/bracketleftBigg/summationdisplay i(mλκi)2/2π/bracketrightBigg δΩ(0). (17) Equation (17) can be reproduced from the diagrams shown in Fig. (1). The perturbation δΩ byHsocomes from the ladder diagrams [20] with the vertex Hsoat- tached. Since the system in the absence of Hsoare in- variant under the reflection of momentum p→ −p, the diagrams with a single vertex/summationtext iκipiσiattached must be identically zero. Of second order of κi, similarly, the diagrams proportional to κiκjvanish ifi∝negationslash=j; fori=j, twoκipiσimust attach to the same pair of free particle propagatorsas shown in Fig. (1). To see this point, given that the three directions are equivalent, let us consider attachingκxpxσxto the ladder diagrams. We choose the free particle propagators diagonalizing the zcompo- nent of the spin operator σz. Since the interactions have SU(2) symmetry and κxpxσxflips the spin by unity, dia- grams in Fig. (1) are the only nonzero ones contributing to ∆Ω(2)within the ladder diagrams. The class of the diagrams represented by the left one in Fig. (1) gives δΩ(2) l=T2 V/summationdisplay P,q,ζ,z,i/bracketleftbiggκ2 i(Pi/2+qi)2 (ζ−z−ξP−q)(z−ξP+q)3 +κ2 i(Pi/2−qi)2 (ζ−z−ξP+q)(z−ξP−q)3/bracketrightbigg t(ζ−P2/4m+2µ), (18) and the class by the right gives δΩ(2) r=−T2 V/summationdisplay P,q,ζ,z,iκ2 i(P2 i/4−q2 i) (ζ−z−ξP−q)2(z−ξP+q)2 ×t(ζ−P2/4m+2µ). (19) Hereζandzare the bosonic and fermionic Matsubara frequenciesrespectively. Oforder η2, thesumofEqs.(18) and (19) equals Eq. (17).4 g− Hsog− Hso FIG. 1: The Ladder diagrams contributing to the variation of the thermodynamical potential of the oder of the square of the spin-orbit coupling. The point vertices are ¯ g, which are connected by a pair of free particle propagators of H0. The wiggling vertices correspond to Hso. For fixed density n, since n= 2η[1 +/summationtext i(mλκi)2/4π]/λ3to first order of ηand second order ofκi,δΩ(0)+δΩ(2)=−2−1/2TVλ3n2˜b2(λ/as); the second virial coefficient is still ˜b2(λ/as), not changed to the order of κ2 i. Consequently, the correlationstrength C does not change as well. This absence of modification of Ccan be understood in the following way. The contribu- tions toδΩ proportional to κ2 ifrom the three directions are additive and of the same form. We can reproduce the coefficient of κ2 zin Eq. (17) by introducing the perturbation Hamiltonian Hp=/integraltextd3rΨ(r)κzpzσzΨ(r) toH0+Hint. Note that the unitary tranformation U= exp[imκz/integraltext d3rz(ψ† ↑(r)ψ↑(r)−ψ† ↓(r)ψ↓(r))] transforms K=H0+Hint+Hp−µNas UKU†=K−(mκ2 z/2)N. We have Ω(T,µ,κ z) =−Tlog(Tre−K/T) = Ω(T,µ+mκ2 z/2,0). (20) For the second order virial expansion δΩ of orderκ2 z, δΩ(T,µ,κ z) =δΩ(T,µ,0)+∂δΩ(T,µ,0) ∂µ(mκ2 z/2) =δΩ(0)[1+(mλκz)2/2π], (21) which agrees with Eq. (17). For the correlation strength C, let us write its correction of second order κias A/summationtext i(mλκi)2. In the case of with spin-orbit coupling only inzdirection,δC=A(mλκz)2. However, accord- ing to Eq. (20), the effect of the spin-orbit coupling only in onedirection is equivalent to shifting µ. Physically, for fixed density, a chemical potential shift should not affect Cat all; one concludes A= 0. The effects of the spin-orbit coupling on Ccan be re- vealed by calculating δΩ to the forth power of κi. FromEq. (16), the non-cross terms ∝κ4 igive δΩ(4) nc=−η2/parenleftBigg/summationdisplay iκ4 i/parenrightBigg 23/2V λ3/integraldisplay Cdζ′ 2πie−ζ′/T ×/bracketleftbiggim7/2 32πζ′3/2t(ζ′)−m5 128π2ζ′t2(ζ′)/bracketrightbigg =/summationdisplay i(mλκ2 i)4δΩ(0)/8π2, (22) a result expected from expanding Eq. (20) in terms of κ2 z. The cross terms ∝κ2 iκ2 jfori∝negationslash=jare δΩ(4) c=η2 /summationdisplay i<jκ2 iκ2 j 23/2V λ3m5 16π/integraldisplay Cdζ′ 2πie−ζ′/T ×/braceleftbigg it(ζ′)/bracketleftbiggmT 21 (mζ′)5/2−1 31 (mζ′)3/2/bracketrightbigg +1 4πζ′t2(ζ′)/bracerightbigg . (23) After changing the integral variable z= 2πζ′/Tand in- tegrating by part, we have δΩ(4) c=η2 /summationdisplay i<jκ2 iκ2 j 23/2V λ3m3λ2 4/integraldisplay Cdz 2πie−z/2π ×/bracketleftbigg −π z1 (α+i√z)4+i 31√z1 (α+i√z)3 +1 6z1 (α+i√z)2+2i 3π1√z1 (α+i√z)/bracketrightbigg ,(24) withα=λ/as. Direct evaluation of the contour integral yields δΩ(4) c=−23/2η2TV λ3F(λ/as) 8π /summationdisplay i<jm4λ4κ2 iκ2 j (25) where F(α) =eα2/2π/parenleftbiggπ α4−2 3α2+1 π/parenrightbigg/bracketleftBig 1+Erf(α/√ 2π)/bracketrightBig −π α4+1 6α2−√ 2/parenleftbigg1 α3−1 3πα/parenrightbigg . (26) Figure (2) shows F(λ/as) as a function of λ/as, positive and well-behaved everywhere. To the same order of κi n=2η λ3/braceleftBigg 1+m2λ2 4π/parenleftBigg/summationdisplay iκ2 i/parenrightBigg +m4λ4 16π2/bracketleftBigg 1 2/summationdisplay iκ4 i +1 3 /summationdisplay i<jκ2 iκ2 j , (27)5       )¬D V ¬D V FIG. 2: (Color online) F(λ/as) vsλ/ascalculted from Eq. (26). the thermodynamical potential variation in terms of nis δΩ =−2−1/2TVλ3n2/bracketleftbigg ˜b2(λ/as)+/parenleftbiggF(λ/as) 8π −˜b2(λ/as) 6π2/parenrightBigg /summationdisplay i<jm4λ4κ2 iκ2 j .(28) By Eq. (2) the variation of Cis δC= 2−3/2λ2n2 /summationdisplay i<jm4λ4κ2 iκ2 j Γ(λ/as) (29) with Γ(λ/as) =∂ ∂(λ/as)/parenleftBigg F(λ/as) 8π−˜b2(λ/as) 6π2/parenrightBigg .(30) Figure (3) shows that Γ is always positive; the correla- tion strength increases as the forth power of κiin the limitmλκi→0. Note that since δC∝/summationtext i<jκ2 iκ2 j, the correlation strength changes only if spin-orbit coupling constants are nonzero at least in two directions. IV. ZERO TEMPERATURE At zero temperature, we consider that there is a sym- metric spin-orbit coupling in either two or three direc- tions, i.e. , κx=κy=κandκz= 0, orκx=κy= κz=κ. It has been shown that for the two cases there is always a two-body bound state with zero center of mass momentum for all as[15]. The eigenenergy of this bound stateǫis given by m 4πas−Πso(0,ǫ) = 0, (31) where in the expression of Π sothe chemical potential is set zero. In the limit that the mean distance between particlesd∼n−1/3is much larger than the size of the     ¬D V¬D V  FIG. 3: (Color online) Γ( λ/as) vsλ/ascalculated from Eq. (30). bound state ℓb, the system can be considered as primar- ily consisting of bosonic molecules which two fermions pair into. These molecules condense into the zero mo- mentum state. Thus we expand in terms of the small numberℓbn1/3and have the leading contribution to the energy of the system coming from the binding energy of the molecules as E=Nǫ/2. Forκx=κy=κandκz= 0, Eq. (31) reduces to [15] 1 mκas=/radicalbigg 1+Eb mκ2−log/parenleftBigg 1+/radicalbigg 1+Eb mκ2/parenrightBigg −1 2log/parenleftbiggmκ2 Eb/parenrightbigg . (32) The binding energy Ebis defined with respect to the scattering threshold energy Eth=−mκ2asEb≡ |ǫ− Eth|. The wavefunction of the bound state is [15] ψb(r) =1√ N(ψs(r)| ↑↓ − ↓↑∝angb∇acket∇ight +ψa(r)| ↑↑∝angb∇acket∇ight+ψ∗ a(r)| ↓↓∝angb∇acket∇ight), (33) whereψs(r) =−/summationtext k,αcos(k·r)/[2(ǫk+α∆k)−ǫ],ψa(r) = i/summationtext k,ααe−iφksin(k·r)/[2(ǫk+α∆k)−ǫ] withφkthe azimuthal angle of k, and the normalization factor N=/summationtext k,α2V/[2(ǫk+α∆k)−ǫ]2. It is instructive to check that Cextracted from the correlation at short distance through Eq. (1) satisfies Eq. (2). By the wavefunction ψb, C= lim r→0r2∝angb∇acketleftψ† ↑(r)ψ† ↓(0)ψ↓(0)ψ↑(r)∝angb∇acket∇ight = (N/V) lim r→0r2N−1|ψs(r)|2 = (4π2)−1N−1m2VN. (34) On the other hand, from Eq. (31) −m 4π∂(E/V) ∂(1/as)=m2N (4π)2V 1 2V/summationdisplay k,α1 [ǫ−2(ǫk+α∆k)]2 −1 , (35)6 which matches Eq. (34). In the case of κx=κy=κz=κ, from Eq. (31) the binding energy Ebis given by [15] mEb=1 4/parenleftBigg 1 as+/radicalBigg 1 a2s+4m2κ2/parenrightBigg2 ; (36) the bound state wavefunction is ψb(r) =1√ N/bracketleftBigg e−√mEbr r/parenleftBigg κ/radicalbig Eb/msin(mκr)+cos(mκr)/parenrightBigg ×| ↑↓ − ↓↑∝angb∇acket∇ight +i/parenleftBig (/radicalbig mEb+1/r)sin(mκr) −mκcos(mκr))e−√mEbr √mEbr| ↑↓+↓↑∝angb∇acket∇ightˆr/bracketrightBigg ,(37) where the subscript means that ˆris the spin quan- tization axis. The normalization factor is N= 2π[m2κ2/(mEb)3/2+ 1/(mEb)1/2]. It is also straight- forward to show that Cextracted from the correlation function maintains Eq. (2). In the absence of spin-orbit couplings, C=n/4πasin the BEC limit n1/3as→0+. The correlation strength decreases as asincreases and acquires a universal value C≈2.7×(3π2n)4/3/36π4at unitarity 1 /as= 0. In the BCS limitn1/3as→0−,C=a2 sn2/4. For the two cases considered above, spin-orbit coupling ensures the exis- tenceofthetwo-bodyboundstateofzeronetmomentum. In the limit ℓbn1/3→0,Cis contributed primarily from the bound state that the fermions pair into, and thus is proportional to the density n. In the limit κas→0+, for symmetric 2D couplings, Eb= 1/ma2 s+mκ2−m3κ4a2 s/3 andC= (n/4πas)(1 +m4κ4a4 s/6); for symmetric 3D couplings, Eb= 1/ma2 s+ 2mκ2−m3κ4a2 sandC= (n/4πas)(1+m4κ4a4 s/2). Thevariationof Cproportional toκ4and ratio between the coefficients of the κ4terms is expectedfromperturbativecalculationsasdonebeforeat high temperatures. Generally, combining Eqs. (2), (32) and (36), we have lim n→0C(κx=κy=κz= 0) C(κx=κy=κz=κ) =/braceleftBigg4 2+√ 1+4m2κ2a2s+1/√ 1+4m2κ2a2s<1,foras>0; 0, foras<0. (38) Figure (4) plots γ= lim n→0C(κx=κy=κz= 0)/C(κx=κy=κ,κz= 0) versus 1 /κas. ThusCin- creases in the presence of symmetric spin-orbit coupling in two or three directions. V. DISCUSSION Our results at high temperature and at zero tempera- ture indicate that generic spin-orbit couplings enhances«D V¤      FIG. 4: (Color online) The ratio γvs 1/κas. the correlation strength C. This enhancement is the net result of the two-fold effects brought about by hso: the change of the density of states of noninteracting particles ρ(ω) and the mixing of scattering in different channels. The former effect is clearly reflected in determining the zero net momentum two-body bound state with symmet- ric spin-orbit coupling in two or three directions. For the two cases, Eq. (31) involves only the scattering between two particles of the same helicity and can be written as m 4πas−/integraldisplay∞ 0dω/parenleftbiggρ(ω) ǫ+mκ2−2ω+m 2π2/radicalbiggm 2ω/parenrightbigg = 0. (39) The change of ρ(ω), especially in the limit ω→0 from ∼√ωto∼const.and∼1/√ωrespectively, gives rise to the two-body bound state no matter the value of as. The existence of this bound state guarantees the increase of Cin the low density limit at zero temperature. In the situation where the scattering between two particles with nonzeronet momentum shall be taken into account, scat- terings in intra- and inter-helicity channels are coupled together. This mixture can be seen by expressing Hint in terms of the fermion operators diagonalizing H0+Hso [21]. Terms in Π sowhich determines the second order virial expansion correspond to different channels. Re- cently Ref. [22] employ the BCS mean field theory to study the ground state of attractive spin half fermions with spin-orbit coupling, and find that the BCS pairing gap ∆ increases with spin-orbit coupling constants. This finding agrees with ours since within the BCS theory, C=−(m/4π)[∂(E/V)/∂a−1 s]∼∆2. The enhancement of Cin Fermi gases with spin-orbit coupling can be measured by photoassociation experi- ment. Previous experiment [23], across the Feshbach res- onance between the lowest two hyperfine states of6Li at magnetic field 834G, associates a pair of Fermi atoms in the closed channel of the Feshbach resonance into a molecular state. The resultant molecules lose from the trap confining the Fermi gas. The loss rate of atoms R has been shown to be proportional to C[12, 14, 24]. In the presence of spin-orbit coupling, the relation between7 the loss rate Rand the correlation strength Cshould remain intact since the photoassociation only involves physical processes at a distance ∼r0much shorter than the scaleintroduced by the spin-orbitcouplingconstants. The changeof Cshall be clearlyreflectedin the atomloss rate. Acknowledgement We thank Shizhong Zhang for bringing Ref. [17] to our attention. We are grateful to Ran Qi and Zeng-Qiang Yufor extensive and instructive discussions. We achnowl- edge Tin-Lun Ho, Hui Zhai and Xiaoling Cui for criti- cal reading of the manuscript. This work is supported in partbyTsinghuaUniversityInitiativeScientificResearch Program, and NSFC under Grant No. 11104157. [1] J. Dalibard, F. Gerbier, G. Juzeli¯ unas and P. ¨Ohberg, Rev. Mod. Phys. 83, 1523 (2011). [2] H. Zhai, arXiv:1110.6798. [3] Y.-J. Lin, R.L. Compton, A.R. Perry, W.D. Phillips, J.V. Porto, and I.B. Spielman, Phys. Rev. Lett. 102, 130401 (2009). [4] Y.-J. Lin, R.L. Compton, K. Jim´ enez-Garc´ ıa, J.V. Port o, and I.B. Spielman, Nature 462, 628 (2009). [5] Y.-J. Lin, R. L. Compton, K. Jim´ enez-Garc´ ıa, W.D. Phillips, J.V. Porto, and I.B. Spielman, Nat. Phys. 7, 531 (2011). [6] Y.-J. Lin, K. Jim´ enez-Garc´ ıa, and I.B. Spielman, Na- ture471, 83 (2011). [7] D.L. Campbell, G. Juzeli¯ unas, and I.B. Spielman, Phys. Rev. A. 84, 025602 (2011). [8] B.M. Anderson, G. Juzeli¯ unas, I.B. Spielman, and V.M. Galitski, arXiv:1112.6022. [9] P. Wang, Z.-Q. Yu, Z. Fu, J. Miao, L. Huang, S. Chai, H. Zhai, and J. Zhang, submitted for publication. [10] S. Tan, Ann. Phys. 323, 2952 (2008); 323, 971 (2008). [11] E. Braaten and L. Platter, Phys. Rev. Lett. 100, 205301 (2008). [12] S. Zhang and A.J. Leggett, Phys. Rev. A 79, 023601 (2009). [13] X. Cui, Phys. Rev. A 85, 022705 (2012). [14] Z. Yu, G.M. Bruun, and G. Baym, Phys. Rev. A 80, 023615 (2009).[15] J.P. Vyasanakere, and V.B. Shenoy, arXiv:1108.4872. [16] G. E. Uhlenbeck and E. Beth, Physica 3, 729 (1936); E. Beth and G. E. Uhlenbeck, Physica 4, 915 (1937). [17] M. Randeria, Bose-Einstein condensation , ed. by A. Griffin, D.W. Snoke, and S. Stringari, (Cambridge Univ. Press, Cambridge, 1995), P. 355. [18] C. J. Pethick and H. Smith, Bose-Einstein condensa- tion in dilute gases , 2nd edition, (Cambridge Univ. Press, Cambridge, 2008), Sec. 17.4. [19] T.-L. Ho, and E.J. Mueller, Phys. Rev. Lett. 92, 160404 (2004). [20] P. Nozi` eres, and S. Schmitt-Rink, J. of Low Temp. Phys. 59, 195 (1985). [21] T. Ozawa, andG.Baym, Phys.Rev.A 84, 043622(2011); Phys. Rev. A 85, 013612 (2012). [22] J.P. 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2109.02871v1.Creating_moving_gap_solitons_in_spin_orbit_coupled_Bose_Einstein_condensates.pdf

Creating moving gap solitons in spin-orbit-coupled Bose-Einstein condensates Jin Su,1, 2Hao Lyu,1Yuanyuan Chen,1and Yongping Zhang1, 1International Center of Quantum Articial Intelligence for Science and Technology (QuArtist) and Department of Physics, Shanghai University, Shanghai 200444, China 2Department of Basic Medicine, Changzhi Medical College, Changzhi 046000, China A simple and ecient method to create gap solitons is proposed in a spin-orbit-coupled spin-1 Bose-Einstein condensate. We nd that a free expansion along the spin-orbit coupling dimension can generate two moving gap solitons, which are identied from a generalized massive Thirring model. The dynamics of gap solitons can be controlled by adjusting spin-orbit coupling parameters. I. INTRODUCTION It is known that the balance between dispersion and nonlinearity may generate solitons [1, 2]. Engineering dispersion and nonlinearity becomes a merited means to control over the existence of solitons and their dynamics. There is a particular kind of engineered dispersion, which is in the form of an avoided energy level crossing with an energy gap that forbids propagation of linear waves. The balance between such dispersion and nonlinearity places solitons inside the energy gap. Due to their specic loca- tion, they are named as gap solitons [3, 4]. Generalized massive Thirring models, possessing this dispersion as well as nonlinearity, become an ideal platform to theoret- ically stimulate gap solitons [5{11]. In experiments, peri- odic potentials can carry out energy gap opening in dis- persion. The observation of gap solitons has been imple- mented experimentally in various nonlinear periodic op- tical systems, including ber Bragg gratings [12], waveg- uide arrays [13], optically induced photonic lattices [14{ 16], temporal lattices [17], and microcavities with peri- odic modulations [18]. In atomic Bose-Einstein condensates (BECs), optical lattices and spin-orbit coupling are responsible for dis- persion engineering [19, 20]. Energy gaps can be opened up around Brillouin zone edges by optical lattices. Gap solitons are predicted theoretically [21, 22] and observed experimentally [23] inside these gaps. The study on matter-wave gap solitons in optical lattices soon repre- sents an active research eld [1, 2, 24{27]. Spin-orbit cou- pling, which can be implemented experimentally by Ra- man dressing of atomic hyperne states [28], can modify dispersion in an interesting way that is dierent from op- tical lattices. The spin-orbit-coupled dispersion features many degenerated energy minima and has local avoided crossings [29]. It was predicted long time ago that the lo- cal spin-orbit-coupled energy gap can mimic generalized massive Thirring model and can support the existence of gap solitonlike solutions [30{34]. However, until now their experimental observations are not yet achieved. A possible experimental challenge may lay in prepa- ration. In order to decorate BECs with spin-orbit cou- pling, Raman lasers are always ramped up adiabatically yongping11@t.shu.edu.cnin experiments [28, 29]. Such loading procedure prepares initial spin-orbit-coupled BECs that are far away from the local energy gaps. It is a challenge for experiments to push initial states into the local energy gaps with high precision. A similar need to transport initial BECs into energy gaps also exists for optical lattices experiments of gap solitons. Precise controllability of optical lattices makes the transport possible. In the experiment, it has been realized by manipulating optical lattices in the way that rst accelerating and then keeping optical lattices to move at a constant velocity [23]. In spin-orbit-coupled experiments, it is hard to manipulate Raman lasers. This is because that two pseudo-spin states are coupled reso- nantly by Raman lasers via a two-photon transition. The adjusting of Raman laser frequency gives rise to an ex- tra time-dependent detuning for the resonant transition. The detuning will destroy eects of spin-orbit coupling. In the present paper, we propose a simple and ecient approach to create gap solitons in spin-orbit-coupled BECs. This method does not need external operations for pushing initial states. We nd that a free expan- sion triggered by suddenly switching o the harmonic trap along the spin-orbit coupling direction can transfer atoms into the regime of spin-orbit-coupled energy gaps. Atoms inside the regime reach the balance between the dispersion and nonlinearity to form gap solitons. Our study is in times of rapid advance in the eld of spin-orbit-coupled BECs [35{38]. Various novel phenom- ena have been revealed theoretically and experimentally in these systems. The study of solitons in spin-orbit- coupled BECs is attracting much attention. Gap soli- tons in the local energy gaps can exist for both repul- sive and attractive interactions [32, 33]. Besides gap soli- tons, spin-orbit coupling can also support another kind of bright solitons [39, 40]. These solitons are only pro- duced by attractive interactions and exist in the semi- innite gap of the spin-orbit-coupled dispersion. Vari- ous features of these bright solitons have been addressed theoretically [41{48]. It is interesting to mention that spin-orbit coupling can stabilize bright solitons in high dimensions [49{56], which are always unstable without spin-orbit coupling. On the other hand, optical lattices can open global energy gaps in spin-orbit-coupled dis- persion [57]. Therefore, spin-orbit-coupled lattices can accommodate gap solitons and the symmetries of spin- orbit coupling bring them new characters [58{60].arXiv:2109.02871v1 [cond-mat.quant-gas] 7 Sep 20212 We present a straightforward expansion method to cre- ate gap solitons in spin-orbit-coupled spin-1 BECs. The spin-1 spin-orbit coupling has been experimentally im- plemented in a87Rb BEC [61]. In the spin-orbit-coupled dispersion, there are two independent local avoided cross- ings sitting at opposite sides in momentum space. Free expansion can push some atoms into both avoided cross- ings. Two moving gap solitons can be created and move in opposite direction. The spin-orbit-coupled spin- 1 BECs provide versatile and tunable systems to con- trol over dynamics of gap solitons. We adjust spin-orbit- coupled dispersion to slow down one of them and to make one disappear. The scheme of the paper is organized as follows. In Sec. II, we describe the idea of existence of gap solitons in a spin-orbit-coupled spin-1 BEC. We show the spin- orbit-coupled dispersion has two avoided crossings. An eective model including two bands is built up to analyze dynamics around the avoided crossings. From the eec- tive model, we can get analytical gap soliton solutions. In Sec. III, we demonstrate that free expansion can dynam- ically generate two moving bright solitons. The solitons are identied to be gap solitons by tting soliton pro- les with analytical gap soliton solutions. In Sec. IV, we describe that dynamics of moving gap solitons can be manipulated, and the conclusion follows in Sec. V. II. GAP SOLITONS IN A SPIN-ORBIT-COUPLED SPIN-1 BEC Three hyperne states of87Rb can be coupled to- gether by three Raman lasers [61]. Along the Raman lasers propagation direction, there are momentum ex- changes between lasers and atoms. The proper manipula- tion of the momentum exchange generates spin-orbit cou- pling [38]. Such Raman coupling leads to single-particle spin-orbit-coupled Hamiltonian Hsocas [61] Hsoc=1 2 i@ @x+ 4Fz2 +p 2 Fx+Fz+F2 z;(1) with (Fx;Fy;Fz) being spin-1 Pauli matrices. The Rabi frequency is proportional to the intensity of Raman lasers. The detuning can be adjusted by changing the frequency of Raman lasers. represents the quadratic Zeeman shift and its value depends on the magnitude of the bias magnetic eld. The spin-orbit coupling is de- noted by the momentum displacement along the xdirec- tion (where Raman lasers propagate) for three hyperne states. Dynamics of a spin-1 BEC with the Raman-induced spin-orbit coupling is described by the Gross-Pitaevskii equation (GPE), i@ @t= Hsoc+V(x) +g0 j 1j2+j 2j2+j 3j2
 : (2) - 8 - 4 0 4 8 0 4 8 - 3 . 0 - 2 . 5 - 2 . 0 - 1 . 5 0 5 1 . 5 2 . 0 2 . 5 3 . 0 0 5 L i n e a r S p e c t r u m Quasimomentum( a ) ( b ) ( c ) ( c ) Q u a s i m o m e n t u m - 1 . 0 - 0 . 5 0 . 0 0 . 5 1 . 0 ( b ) Q u a s i m o m e n t u m L i n e a r S p e c t r u m L i n e a r S p e c t r u mFIG. 1. The dispersion relation of the spin-orbit-coupled Hamiltonian Hsocin the quasimomentum space. The param- eter are = 1, = 1, and= 0. (a) The full three bands. Around the smallest separation between the two lower bands, the dispersion appears as two avoided energy crossings, which are shown by the shadow areas. (b) and (c) are zoom-in of the shadow areas labeled in (a) respectively. Color scales repre- sent the polarization dened as ( j 2j2j 1j2)=(j 1j2+j 2j2) in (b) and (j 2j2j 3j2)=(j 2j2+j 3j2) in (c). The wave functions = ( 1; 2; 3)Tdescribe occu- pations of three hyperne states. The GPE is quasi- one-dimensional, which can be achieved by loading the BEC into an one-dimensional optical dipole waveguide so that motion along the transverse direction ( y;z) are frozen. For the convenience of numerical calculations, the above GPE is dimensionless. The units of length, energy and time are chosen as 2 =kRam,~2k2 Ram=4mand 4m=(~k2 Ram), respectively. Here mis the atom mass and kRam= 2=Ramis the wavevector with Ram= 790nm the wavelength of Raman lasers [61]. The harmonic trap isV(x) =f2 xx2=2 with dimensionless fxrelating to the trap frequency !x,fx= 2p 2m!x=(~k2 Ram). The interac- tion coecient in Eq. (2) is g0= 4Na0m!r=(~kRam) with Nbeing the atom number, a0being the spin-independent scattering length and !rbeing the trap frequency along the transverse direction. For a87Rb BEC,a0= 100aB withaBthe Bohr radius. We neglect the spin-dependent scattering length a2since it is very small in experi- ments [61]. The wave functions satisfy normalization condition,R dx(j 1j2+j 2j2+j 3j2) = 1. The dispersion relation of the spin-orbit-coupled Hamiltonian Hsochas interesting features. A typical dis- persion is shown in Fig. 1. In quasimomentum kspace, there are three parabola locating at k=4 and 0 respec- tively. Local gaps are opened by the Rabi frequency at crossings between parabola. At last, the dispersion rela- tion has three bands. The lowest band possesses three en- ergy local minima. The bias between them can be tuned by changing the detuning and quadratic term. Around the smallest separations between the two lower bands, the dispersion appears as two local avoided energy crossings. In the regimes of avoided crossings [shown by the shadow areas in Fig. 1(a)], the lowest band possesses negative ef- fective mass. The left avoided crossing is dominated by the components 1and 2, and the involvement of the3 -1500-1000-5000500100015000.0000.0040.0080.012- 1500-1000-5000500100015000.0000.0040.0080.012- 6-4-202460123- 6-4-2024601234Densityx D ensityx DispersionDensityQ uasimomentumD ensityQ uasimomentum0369 02 505 007 501 000-1610-80508051610 xt t=600t =1000(b)( c)( d)( e)(a) FIG. 2. Symmetric expansion triggered by suddenly switching o the longitudinal harmonic trap. The initial state is the ground state with the trap parameter fx= 0:6. (a) The evolution of total density n(x) =j 1j2+j 2j2+j 3j2as a function oft. Two sharp density peaks are formed and propagate oppositely with a very long lifetime. Snapshots taken at t= 600 andt= 1000 are shown in (b) and (c) respectively. In (d) and (e) the normalized quasimomentum-space densities (red lines) dened in Eq. (6) at t= 600 and 1000 respectively are shown. Black lines are the dispersion relation of spin-orbit-coupled Hamiltonian Hsoc. Shadowed areas indicate avoided energy crossings featuring negative eective mass in the lowest band. In (d), circles are tting to the right avoided crossing. The tting is 2 :5p  2(k2:25)2+ 2with  = 1:517 and  = 1. The other parameters are = 1, = 1,= 0 andg0= 10. component 3is very small. This is can be seen from the polarization ( j 2j2j 1j2)=(j 1j2+j 2j2) demon- strated by color scales in Fig. 1(b) for the left avoided crossing. While the right avoided crossing is mainly oc- cupied by the components 2and 3, and the popula- tion of the component 1can be neglected. The po- larization for the right avoided crossing is dened as, (j 2j2j 3j2)=(j 2j2+j 3j2), which is demonstrated in Fig. 1(c). Since the local avoided crossings are far from the higher band and are dominated by two components, if physics is only relevant to the avoided crossings, they can be eectively described by the Hamiltonian, He=  z(i@ @xkcen) + x; (3) with eigenstates = (a;b)Tbeing two components. z andxare spin-1/2 Pauli matrices. The dispersion re- lation ofHeisp  2(kkcen)2+ 2, having the form of avoided crossing in the quasimomentum space.  de- scribes the gap size of the avoided crossing,  depicts its steepness, and kcenis the location where the avoided crossing is centered. The avoided crossings, existing in the spin-orbit- coupled Hamiltonian Hsoc, provide a versatile bed for the investigation of gap solitons [30{32]. Assuming atoms are conned around an avoided crossing, their dynamics may be described by the eective model as, i@ @t=He+g0(jaj2+jbj2): (4)In above equation, interactions between atoms are con- sidered by the last term. The eective model, in lit- erature, is known as a generalized massive Thirring model [5{11]. The massive Thirring model, sharing the sameHebut having specic nonlinearity, is completely integrable [62]. The eective model in Eq. (4) includes a generalized nonlinearity. It is known that the outstand- ing feature of the model is to admit analytical moving gap soliton solutions [7, 8, 30{32, 63]. They are,  a b =Aeixkcen1
sech i 2

sech +i 2
 ; (5) with A=s  (1v2) 2g e2+ei e2+eiv eisin ; = (x= +vt) sin p 1v2; = (vx= +t) cos p 1v2; and
=1v 1 +v1 4 : Two parameters v(1< v < 1) and (0 << ) determine the velocity and amplitude of gap solitons. These moving gap solitons are supported by the avoided crossing dispersion. For the spin-orbit-coupled4 -200-195-190-185-180-175-1700.0000.0040.0080.0120.0000.0020.0040.006 Total densityx (b) Density distribution( a) FIG. 3. Matching of density distributions between the negative-direction moving peak taken at t= 600 in Fig. 2 and an analytical soliton solution from Eq. (5). All param- eters are same as in Fig. 2. In (a), each component density is presented. Black solid lines from top to bottom are j 3j2, j 2j2andj 1j2. Red open-circle lines are gap soliton solution. Upper and lower ones correspond to jbj2andjaj2respec- tively. In (b), total density distribution j 1j2+j 2j2+j 3j2 (black solid line) and jaj2+jbj2(red open-circle line) are shown. The parameters of the avoided crossing are  = 1:517 and  = 1. The parameters of analytical gap solitons are v=0:306 and = 0 :342. spin-1 system, two avoided crossings sit at opposite sides withkcen6= 0 and are far from k= 0 [as shown in Fig. 1(a)]. In experiments, to articially introduce spin- orbit coupling into the BEC, Raman lasers should be ramped up adiabatically, in this way, the BEC is always prepared in spin-orbit-coupled ground state which locates at the global minimum of lowest band in the dispersion relation of Hsoc(i.e.,k= 0). In order to create mov- ing gap solitons, the transport of initially prepared BEC fromk= 0 tokcenis prerequisite. However, the precise manipulation of the transport in experiments may be a challenge. In the following, we reveal that a free expan- sion could be an experimentally accessible way to create moving gap solitons. III. CREATING MOVING GAP SOLITONS BY EXPANSION We start from a spin-orbit-coupled ground state ob- tained by numerically solving Eq. (2) using the imaginary time evolution. The harmonic trap V(x) is then suddenly switched o. The quench allows the BEC to expand along the longitudinal direction xin the presence of spin-orbit FIG. 4. Asymmetric expansion with a nonzero detuning = 0:3. The detuning gives rise to the asymmetry of two avoided crossing with respect to k= 0. (a) Total density evolution. (b) The normalized quasimomentum-space denisty (red line) att= 900 and the dispersion relation of Hsoc(black lines). The shadow areas indicate avoided crossings, the ttings of which are represented by circles. We use the tting as 2 :28p  2(k2:2)2+ 2for the right avoided crossing and 2 :7p  2(k+ 2:5)2+ 2for the left one with  = 1:517 and  = 1. (c,d) Matching of expansion result at t= 900 (black solid lines) and analytical gap soliton solutions in Eq. (5) (red open- circle lines). (c) Soliton moves in negative direction with v= 0:4 and = 0 :27. Black solid lines from top to bottom correspond toj 3j2,j 2j2andj 1j2. Red open-circle lines from upper to lower are jbj2andjaj2. (d) Soliton moves in positive direction with v= 0:3 and = 0 :335. Black solid lines from top to bottom correspond to j 1j2,j 2j2andj 3j2. Red open-circle lines from upper to lower are jaj2andjbj2. The other parameters are = 1, = 1 andg0= 10. coupling. The interactions between atoms are converted into driving forces to move atoms. If quench-induced ex- citations are weak comparing with the band separation between the two lower bands in the dispersion relation, the expansion mainly involves the lowest band [20]. The expansion of total density, n(x) =j 1(x)j2+ j 2(x)j2+j 3(x)j2, is presented in Fig. 2(a). In coor- dinate space, the expansion of the total density is sym- metric with respect to x= 0. Around t= 150, two sharp5 symmetric density peaks appear. They move oppositely at a constant velocity for a very long time and keep their height unchanged. Fig. 2(b) and 2(c) demonstrate two snapshots of the evolution, which are taken at t= 600 andt= 1000 respectively. Beside the density peaks, there are other excited modes. As the other modes prop- agate, the density peaks become more obvious. The cho- sen parameters result in a symmetric dispersion relation with respect to k= 0 [shown in Fig. 2(d)]. Initial state is centered around k= 0. During expansion, quasimomen- tum is symmetrically extended in opposite directions. As quasimomentum increases, some atoms enter into neg- ative eective mass regimes of lowest band [indicated by shadowed areas in Fig. 2(d) and 2(e)]. When atom number inside these regimes is apparently larger than other occupation of quasimomentum, the density peaks are emergent in coordinate space. Fig. 2(d) and 2(e) show two snapshots of normalized quasimomentum-space density distributions. The normalized quasimomentum- space density is dened as, j 1(k)j2+j 2(k)j2+j 3(k)j2 R dk[j 1(k)j2+j 2(k)j2+j 3(k)j2]; (6) withj i(k)j2being the density of ith component in quasi- momentum space. From these plots, it is clear that the occupation of negative eective mass regimes completely dominates. The appearance of the sharp density peaks in coordinate space always accompanies the dominant oc- cupation of negative eective mass regimes. It looks like that some atoms are \trapped" by negative eective mass regimes. Such trapped states can sustain for a long time, this requires a proper range of interaction coecient: if the coecient is small, there will be not enough driv- ing resource for expansion to push atoms into negative eective mass regimes; if it is large, atoms in negative eective mass regimes shall be continuously moved out by further expansion. The longevous sharp density peaks in coordinate space in Fig. 2(a) are identied as moving gap solitons. Nearby the negative eective mass regimes, the two lower bands are close to each other and form avoided crossings. The eective model in Eq. (4) provides a simple way to de- scribe atoms inside the regimes. To precisely mimic the avoided crossings in the dispersion of Hsoc, we rst x the parameters  ; andkceninHeby tting the dis- persion ofHeto the avoided crossings. The tting of the right avoided crossing as an example is demonstrated in Fig. 2(d). We shall reveal that the sharp density peaks are moving gap solitons in Eq. (5). For this purpose, we extract information of the velocity and amplitude of the density peaks. Then we apply them to the analytical gap soliton solutions to x the free parameters vand in Eq. (5). In Fig. 3, we show the negative-direction mov- ing density peak together with the proles of gap solitons. The density peak is taken at t= 600 in Fig. 2(a). On ac- count of the mismatch of time origin for the density peak and gap solitons, we choose a tthat is dierent from 600 for gap solitons. With this t, the center of gap soliton -6- 4- 20 2 4 6 012- 500- 495- 490- 485- 4800.0030.0090 .0000.00602 004 006 008 0010001200-750750- 150001500xt t = 900(a)( b)(a)( c) DensityQ uasimomentum048D ispersion x DensityFIG. 5. Asymmetric expansion with = 1. All panels show same quantities as in Fig. 4. But here g0= 8. The tting in (b) is 2p  2(k2)2+ 2with  = 1:517 and  = 1. In (c), parameters of the gap soliton solution are v= 0:54 and = 0:36. Black solid lines from top to bottom correspond toj 3j2,j 2j2andj 1j2. Red open-circle lines from upper to lower arejbj2andjaj2. coincides with the density peak. It is shown that proles of gap solitons match with the density peak very well. Therefore, the density peak is a gap soliton and originates from the occupation of the right avoided crossing. As the right avoided crossing mainly comes from the coupling between 2and 3, the occupation of the rst compo- nent 1[the bottom solid line in Fig. 3(a)] is very small. We notice that spatial modulation exists in the density peak, which is more evident in the second component 2 [the middle solid line in Fig. 3(a)] . This is because be- side the dominant occupation of avoided crossing regimes the expansion also produces some modes around k= 0 [see Fig. 2(d) and 2(e)]. These modes are primarily com- posed of the second component 2, and are occupied by just a few atoms. The interference between them and the density peaks contributes to the modulation on the top of the density peaks. The total density n(x) also shows modulation, and gap soliton solution can t it very well,6 which is demonstrated in Fig. 3(b). We conclude that the density peaks are moving gap solitons, which can be created through a free expansion. An initial spin-orbit-coupled BEC can be pushed to oc- cupy the regime of avoided crossings in dispersion rela- tion, where atoms form gap solitons. IV. MANIPULATING MOVING GAP SOLITONS In this section, we demonstrate that the spin-orbit- coupled spin-1 system is very versatile for controlling dy- namics of gap solitons. It can be achieved by adjusting the detuning inHsoc. In the previous section, two sym- metric gap solitons can be created with a zero detuning = 0. We reveal that in the presence of the detuning two gap solitons become asymmetric: for a small detun- ing, two gap solitons have dierent velocity; for a large detuning one of them disappears. Expansion of the total density with = 0:3 is presented in Fig. 4(a). The expansion can still generate two mov- ing gap solitons. However, the negative-direction moving one has a larger velocity. In the presence of the detuning, the dispersion relation of Hsocbecomes asymmetric with respect tok= 0. The right avoided crossing is energeti- cally lower than the left one, and during the increase of quasimomentum in the expansion more atoms are sent into the right regime as shown in Fig. 4(b). In Fig. 4(c) and 4(d), we match expansion results with analytical gap soliton solutions in Eq. (5). The good match in Fig. 4(c) and 4(d) veries that the expansion with a small detun- ing induces asymmetric gap solitons. Oscillations in 2 still exist due to the excited modes around k= 0. Expansion of the total density with a larger detuning (= 1) is shown in Fig. 5(a). There is only one solitonproduced from background and it moves in the negative direction. With this detuning, the right avoided crossing in the dispersion relation of Hsocbecomes more obviously lower than the left one [see Fig. 5(b)]. The expansion can easily delivers atoms into the right regime. This causes that there are no enough atoms in the left regime to form gap solitons. Therefore, only atoms in the right regime can form soliton. The density of soliton in Fig. 5(c) shows that populations of 2and 3dominate, which conrms that the soliton comes from the occupation of the right avoided crossing. The match presented in Fig. 5(c) proves that the soliton is a gap soliton. It should be noted that the interaction coecient g0is smaller than that in Fig. 4. We numerically nd that to excite only one soliton the range ofg0would be 7:0< g0<9:3, which corresponds to the total atom number 2000 <N < 2660 if the typical transverse tap !r= 2150Hz is considered. V. CONCLUSION We present the straightforward free expansion method to create gap solitons in an experimentally realizable spin-orbit-coupled spin-1 BEC. Via expansion, some atoms can be delivered into the regime of spin-orbit- coupled energy gaps to form gap solitons. With this method, the created gap solitons are movable. 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1605.04716v2.Characteristics_of_persistent_spin_current_components_in_a_quasi_periodic_Fibonacci_ring_with_spin_orbit_interactions__Prediction_of_spin_orbit_coupling_and_on_site_energy.pdf

arXiv:1605.04716v2 [cond-mat.mes-hall] 17 Oct 2016Characteristics of persistent spin current components in a quasi-periodic Fibonacci ring with spin-orbit interactions: Prediction of spin-orb it coupling and on-site energy Moumita Patra1and Santanu K. Maiti1,∗ 1Physics and Applied Mathematics Unit, Indian Statistical I nstitute, 203 Barrackpore Trunk Road, Kolkata-700 108, India In the present work we investigate the behavior of all three c omponents of persistent spin current in a quasi-periodic Fibonacci ring subjected to Rashba and D resselhaus spin-orbit interactions. Analogous to persistent charge current in a conducting ring where electrons gain a Berry phase in presence of magnetic flux, spin Berry phase is associated dur ing the motion of electrons in presence of a spin-orbit field which is responsible for the generation of spin current. The interplay between two spin-orbit fields along with quasi-periodic Fibonacci s equence on persistent spin current is described elaborately, and from our analysis, we can estima te the strength of any one of two spin- orbit couplings together with on-site energy, provided the other is known. PACS numbers: 73.23.Ra, 71.23.Ft, 71.70.Ej, 73.23.-b I. INTRODUCTION The study of spin dependent transport in low- dimensional quantum systems, particularly ring-like ge- ometries, has always been intriguing due to their strange behavior and possible potential applications in designing spintronic devices. To do this proper spin regulation is highly important. Several attempts1–11have been made in the last couple of decades, and undoubtedly, a wealth of literature knowledge has been established towards this direction. Mostly external magnetic fields or ferromag- netic leads were used12,13to control electron spin but none of these are quite suitable from experimental per- spective. This is because confining a large magnetic field in a small region e.g., ring-like geometry is always a diffi- A BN2 1Y X FIG. 1: (Color online). Schematic view of a 8-site meso- scopic Fibonacci ring (5th generation) subjected to Rashba and Dresselhaus spin-orbit interactions. The ring is con- structed with two basic atomic units AandB, and they are described by two colored filled circles. cult task, and at the same footing major problem arises during spin injection from ferromagnetic leads through a conducting junction due to large inconsistency in re- sistivity. Certainly it demands new methodologies and attention has been paid towards the intrinsic proper- ties14–25like spin-orbit (SO) coupling of the materials. Usually two types of SO fields are encountered in study- ing spin dependent transport phenomena. One is known asRashbaSOcoupling26whichisgeneratedduetobreak-ing of symmetry in confining potential, and thus, can be regulated externally27by gate electrodes. While the other, defined as Dresselhaus SO coupling28, is observed due to the breakingofstructural symmetry. The SO field plays an essential role in spintronic applications as it di- rectly couples to the electron’s spin degree of freedom. In presence of such SO field a net circulating spin cur- rentis established29in a conducting ring, analogous to magnetic flux driven persistent charge current30–32. The magnetic flux introduces a phase, called Berry phase, to moving electrons which produces net charge current by breaking time reversal symmetry between clockwise and anti-clockwisemovingelectrons, while aspin Berryphase is associated in presence of SO coupling which generates spin current. The works involving persistent spin current in ring- like geometries studied so far are mostly confined to the perfect periodic lattices or completely random ones33–36. But, to the best of our knowledge, no one has addressed the behavior of SO-interaction induced spin current in quasi-periodic lattices which can bridge the gap between an ordered lattice and a fully random one. In addition, the earlier studies essentially focused on only one com- ponent (viz, Z-component), though it is extremely inter- esting and important too to study all three components of spin current to analyze spin dynamics of moving elec- tronsin presenceofSO fields. Motivated with this, in the present work we explore the behavior of persistent spin current in a one-dimensional (1D) quasi-periodic ring ge- ometry where lattice sites are arrangedin a Fibonacci se- quence37–39, thesimplest exampleofaquasi-periodicsys- tem. A Fibonacci chain is constructed by two basic units AandBfollowing the specific rule A→ABandB→A. Thus, applying successively this substitutional rule, star- ing fromAlattice orBlattice we can construct the full lattice for any particular generation, say p-th generation, obeying the prescription Fp=Fp−1⊗Fp−2, and connect- ing its two ends we get the required ring model. Thus, if we start with Alattice then A,AB,ABA,ABAAB , ABAABABA ,..., etc., are the first few generations,2 and the series is characterized by the ratio of total num- ber ofAandBatoms which is called as Golden mean τ (= 1+√ 5/2). Now, instead of considering AandBas lattice points if we assign them as bond variables then bond model of Fibonacci generation is established40,41, and when both these lattice and bond models are taken intoaccountitbecomesa mixed model38. Inourmodelwe consider only site representation of Fibonacci sequence startingwith lattice A, for the sakeofsimplification. The responseofotherwillbediscussedelsewhereinourfuture work. In this work, we address the behavior of all three com- ponentsofpersistentspincurrentinaFibonacciringsub- jected to Rashba and Dresselhaus SO couplings. Within a tight-binding framework we calculate the current com- ponents using second-quantized approach which is the most convenient tool for such calculations. The inter- play between Rashba and Dresselhaus SO fields on the current components exhibits several interesting patterns that can be utilized to estimate any oneofthe SO fields if we know the other one, and also we can estimate the site energy ofA- orB-type site provided any one of these two is given. Our analysis can be utilized to explore spin de- pendent phenomena in any correlated lattices subjected to such kind of SO fields. This is the first step towards this direction. The work is arranged as follows. In Sec. II we present the model and theoretical framework for calculations. The results are discussed in Sec. III, and finally we con- clude in Sec. IV. II. MODEL AND THEORETICAL FORMULATION The mesoscopic Fibonacci ring subjected to Rashba and Dresselhaus SO couplings is schematically depicted in Fig. 1, and the Hamiltonian of such a ring reads as, H=H0+HR+HD (1) which includes the SO-coupled free term ( H0), Rashba Hamiltonian ( HR) and the Dresselhaus Hamiltonian (HD). Using tight-binding (TB) framework we can ex- press these Hamiltonians39for aN-site ring as follows: H0=/summationdisplay nc† nǫncn+/summationdisplay n/parenleftBig c† n+1tcn+c† ntcn+1/parenrightBig (2a) HR=−/summationdisplay n/parenleftBig c† n+1(iσx)αcosφn,n+1cn+h.c./parenrightBig −/summationdisplay n/parenleftBig c† n+1(iσy)αsinφn,n+1cn+h.c./parenrightBig (2b) HD=/summationdisplay n/parenleftBig c† n+1(iσy)βcosφn,n+1cn+h.c./parenrightBig +/summationdisplay n/parenleftBig c† n+1(iσx)βsinφn,n+1cn+h.c./parenrightBig (2c)where the site index nruns from 1 to N, and we use the conditionN+1 = 1. The other factors are: ǫn=/parenleftbigg ǫn↑0 0ǫn↓/parenrightbigg ,t=/parenleftbigg t0 0t/parenrightbigg ,cn=/parenleftbigg cn↑ cn↓/parenrightbigg , α=/parenleftbigg α0 0α/parenrightbigg ,β=/parenleftbigg β0 0β/parenrightbigg wheretis the nearest-neighbor hopping integral and ǫn↑(ǫn↓) represents the on-site energy of an up (down) spin electron sitting at the nth site. As we are not considering any magnetic type interaction the site en- ergies for up and down spin electrons become equal i.e., ǫn↑=ǫn↓=ǫn(say). For the A-type sites we refer ǫn=ǫA, and similarly, ǫn=ǫBforB-type atomic sites. When these two site energies are identical (viz, ǫA=ǫB) the Fibonacci ring becomes a perfect one, and in that case we set them to zero without loss of any generality. αandβare the Rashba and Dresselhaus SO coupling strengths, respectively, and φn,n+1= (φn+φn+1)/2, whereφn= 2π(n−1)/N.σi’s (i=x,y,z) are the Pauli spin matrices in σzdiagonal representation. Looking carefully the Rashba and Dresselhaus Hamil- tonians (Eqs. 2(b) and 2(c)) one can find that these two Hamiltonians are connected by a unitary transformation HD=UH RU†, whereU=σz(σx+σy)/√ 2. This hid- den transformation relation leads to several interesting results, in particular when the strengths of these two SO couplings are equal, which will be available in our next section (Sec. III). To calculate spin current components we define the operator33Ik=1 2N(σk˙x+˙xσk), wherek=x,y,zde- pending on the specific component. In this expression ˙x is obtained by taking the commutation of position oper- atorx(=/summationtext nC† nnCn) with the Hamiltonian H. After simplification the current operator gets the form: Ik=iπ N/summationdisplay n/parenleftBig c† nσkt†n,n+1 φcn+1−h.c./parenrightBig +iπ N/summationdisplay n/parenleftBig c† nt†n,n+1 φσkcn+1−h.c./parenrightBig (3) where,tn,n+1 φis a (2×2) matrix whose elements are tn,n+1 φ1,1=t tn,n+1 φ2,2=t tn,n+1 φ1,2=−iαe−iφn,n+1+βeiφn,n+1 tn,n+1 φ2,1=−iαeiφn,n+1−βeiφn,n+1. Once the current operator is established (Eq. 3), the in- dividual current components carried by each eigenstate, say|ψm/an}bracketri}ht, can be easily found from the operation Ik,m= /an}bracketleftψm|Ik|ψm/an}bracketri}ht, where |ψm/an}bracketri}ht=/summationtext n/parenleftBig am n↑|n↑/an}bracketri}ht+am n↓|n↓/an}bracketri}ht/parenrightBig .3 am nσ’sarethecoefficients. Doingaquitelongandstraight- forward calculation we eventually get the following cur- rent expressions for three different directions ( X,Yand Z) as follows: Iz,m=2πit N/summationdisplay n/bracketleftbig/braceleftbig am∗ n,↑am n+1,↑−h.c./bracerightbig −/braceleftbig am∗ n,↓am n+1,↓− h.c.}] (4a) Ix,m=2πit N/summationdisplay n/braceleftbig/parenleftbig am∗ n,↑am n+1,↓+am∗ n,↓am n+1,↑/parenrightbig −h.c./bracerightbig +2π N/summationdisplay n/braceleftbig/parenleftbig am∗ n+1,↑am n,↑+am∗ n+1,↓am n,↓/parenrightbig +h.c./bracerightbig × (βsinφn,n+1−αcosφn,n+1) (4b) Iy,m=2πt N/summationdisplay n/braceleftbig/parenleftbig am∗ n+1,↓am n,↑−am∗ n+1,↑am n,↓/parenrightbig +h.c./bracerightbig +2π N/summationdisplay n/braceleftbig/parenleftbig am∗ n+1,↑am n,↑+am∗ n+1,↓am n,↓/parenrightbig +h.c./bracerightbig × (βcosφn,n+1−αsinφn,n+1) (4c) Thus, at absolute zero temperature, the net current for aNeelectron system becomes Ik=Ne/summationdisplay m=1Ik,m. (5) III. RESULTS Based on the above theoretical framework now we present our numerical results which include three dif- ferent current components carried by individual energy eigenstates, net currents of all three components for a particular electron filling Ne, and the possibilities of de- termining any one of the SO fields as well as on-site ener- gies provided the other is known. As we are not focusing on quantitative analysis considering a particular mate- rial, we choose c=e=h= 1 for the sake of simplifica- tion and fix the nearest-neighborhopping integral t= 1.5 throughout the analysis. The values of other parameters are given in subsequent figures. A. Current components carried by distinct energy levels Before analyzing net current components for a specific filling factor, let us first focus on the behavior of indi- vidual state currents as they give more clear conducting signature of separate energy levels which essentially gov- ern the net response of any system.In Fig. 2 we present the variation of Z-component of spincurrentcarriedby individualenergylevelsforacom- pletely perfect (left column) and a Fibonacci ring (right column) in presenceofdifferent SOcouplings. Severalin- /LParen1a/RParen1 Α/EquΑl0.5,Β/EquΑl0 /Minus3.25 0 3.25/Minus0.16500.165 EmIz,m/LParen1d/RParen1Α/EquΑl0.5,Β/EquΑl0 /Minus3.25 0.2 3.65/Minus0.07200.072 EmIz,m /LParen1b/RParen1 Α/EquΑl0,Β/EquΑl0.5 /Minus3.25 0 3.25/Minus0.16500.165 EmIz,m/LParen1e/RParen1Α/EquΑl0,Β/EquΑl0.5 /Minus3.25 0.2 3.65/Minus0.07200.072 EmIz,m /LParen1c/RParen1 Α/EquΑlΒ/EquΑl0.5 /Minus3.6 0 3.6/Minus0.16500.165 EmIz,m/LParen1f/RParen1Α/EquΑlΒ/EquΑl0.5 /Minus3.6 0.2 4/Minus0.0720.00.072 EmIz,m FIG. 2: (Color online). Z-component of spin current ( Iz,m) carried by individual energy levels (levels are indexed by t he parameter mandEmbeing the energy eigenvalue of mth level) for 34-site ring (8th generation) considering differ ent values of αandβ, where the first and second columns corre- spond to ǫA=ǫB= 0 and ǫA=−ǫB= 1, respectively. teresting featuresare observed. (i) In presence ofany one of the two SO fields individual state currents exhibit a nice pattern for the perfect case where the current starts increasing (each vertical line represents the current am- plitude for each state) from the energy band edge and reaches to a maximum at the band centre. While, for the Fibonacci ring sub-spectra with finite gaps, associ- ated with energy sub-bands, are obtained which is the generic feature of any Fibonacci lattice, like other quasi- crystals42,43. In each sub-band higher currents are ob- tained from central energy levels while edge states pro- vide lesser currents, similar to a perfect ring. Thus, set- ting the Fermi energy, associated with electron filling Ne, at the centre or towards the edge of each sub-band one can regulate current amplitude and this phenomenon can be visualized at multiple energies due to the appearance of multiple energy sub-bands. Here it is also crucial to note that no such phenomenon will be observed in a completely random disordered ring as it exhibits only localized states for the entire energy band region. (ii) Successive energy levels carry currents in opposite direc- tions which gives an important conclusion that the net Z-component of current is controlled basically by the top most filled energy level, similar to that what we get in the case of conventional magnetic flux induced persistent charge current in a conducting mesoscopic ring. (iii) Fi-4 nally, it is important to see that the direction of individ- ual state currents in a ring subjected to only Rashba SO coupling gets exactly reversed when the ring is described with only Dresselhaus SO interaction, without changing any magnitude. Thus, if both these two SO fields are present in a particular sample and if they are equal in magnitude then current carried by different eigenstates drops exactly to zero due to mutual cancellation of cur- rent caused by these fields. This vanishing nature of spin currentcanbeprovedasfollows. Itisalreadypointedout thatHRandHDare connected by a unitary transforma- tionHD=UH RU†. Therefore, if |ψm/an}bracketri}htbe an eigenstate ofHRthenU|ψm/an}bracketri}ht(=|ψ′ m/an}bracketri}ht) will be the eigenstate of the Hamiltonian HD. This immediately gives us the follow- ing relation: Iz,m|D=/an}bracketleftψ′ m|Iz,m|ψ′ m/an}bracketri}ht =/an}bracketleftψm|U†Iz,mU|ψm/an}bracketri}ht =/an}bracketleftψm|U†1 2N(σz˙x+˙xσz)U|ψm/an}bracketri}ht =/an}bracketleftψm|1 2N(−σz˙x−˙xσz)|ψm/an}bracketri}ht =−Iz,m|R (6) From the above mathematical argument the sign rever- sal ofIz,munder interchanging the roles played by α andβcan be easily understood. Certainly this vanishing behavior can be exploited to determine any one among these two SO fields if the other is given. In particular the determination of Dresselhaus strength will be much eas- ier as for a specific material it is constant, while Rashba strength can be tuned with the help of external gate po- tential. Here, it is worthy to note that the interplay of Rashba and Dresselhaus SO couplings has also been re- ported in several other contexts, and particularly when these two strengths are equal, persistent spin helix44–47 hasobservedwhichisofcourseoneofthe mostimportant and attractive areas of spintronics. In Figs. 3 and 4 we present the characteristics of X andYcomponents of spin current, respectively, carried by individual energy levels for the identical ring size and parameter values as taken in Fig. 2 for finer comparison of all three components. The observations are notewor- thy. (i) For the perfect ring (viz, ǫA=ǫB= 0), both XandYcomponents of current are zero for each en- ergy eigenstates, while non-zero contribution comes from the Fibonacci ring. In order to explain this behavior let us focus on Fig. 5, where the velocity direction of a moving electron is schematically shown at different lat- tice points of a ring placed in the X-Yplane. Now, consider the Rashba and Dresselhaus Hamiltonians in a continuum representation where they get the forms: HR=α/parenleftbig σypx−σxpy/parenrightbig andHD=β/parenleftbig σypy−σxpx/parenrightbig wherepxandpyare the components of palongXand Ydirections, respectively. Thus, at the point A of a pure Rashba ring only pywill contribute (since here px= 0) toHR, while it is −pyat the point C (Fig. 5). Simi- larly, for the points B and D the contributing terms are/LParen1a/RParen1 Α/EquΑl0.5,Β/EquΑl0 /Minus3.25 0 3.25/Minus0.16500.165 EmIx,m/LParen1d/RParen1 Α/EquΑl0.5,Β/EquΑl0 /Minus3.25 0.2 3.65/Minus0.002800.0028 EmIx,m /LParen1b/RParen1 Α/EquΑl0,Β/EquΑl0.5 /Minus3.25 0 3.25/Minus0.16500.165 EmIx,m/LParen1e/RParen1 Α/EquΑl0,Β/EquΑl0.5 /Minus3.25 0.2 3.65/Minus0.0300.03 EmIx,m /LParen1c/RParen1 Α/EquΑlΒ/EquΑl0.5 /Minus3.6 0 3.6/Minus0.16500.165 EmIx,m/LParen1f/RParen1 Α/EquΑlΒ/EquΑl0.5 /Minus3.6 0.2 4/Minus0.100.1 EmIx,m FIG. 3: (Color online). X-component of spin current ( Ix,m) carried by distinct energy levels of a mesoscopic ring where different spectra correspond to the identical meanings as de - scribed in Fig. 2. The ring size and other physical parameter s are also same as taken in Fig. 2. pxand−px, respectively. As a result of this the net contribution to HRbecomes zero, and this is equally true for any other diagonally opposite points (though in these cases both pxandpycontribute) which leads to a vanishing spin current along XandYdirections for a perfect Rashba ring. The same argument is also valid in the case of a pure Dresselhaus ring. But, as long as the symmetry between the diagonally opposite points is brokenthe mutual cancellationdoes not takeplace which results a finite non-zero spin current along these two di- rections. This is exactly what we get in a Fibonacci ring, and in the same footing, wecan expect non-zerospin cur- rent for any other disordered rings. (ii) In the Fibonacci ring theX-component ( Y-component) of spin current in presence of αmaps exactly in the opposite sense (i.e., identical magnitude but opposite in direction) to the Y- component ( X-component) of current under swapping the roles played by αandβ(right columns of Figs. 3 and 4). This phenomenon can be explained from the following mathematical analysis. Iy,m|D=/an}bracketleftψ′ m|Iy,m|ψ′ m/an}bracketri}ht =/an}bracketleftψm|U†Iy,mU|ψm/an}bracketri}ht =/an}bracketleftψm|U†1 2N(σy˙x+˙xσy)U|ψm/an}bracketri}ht =/an}bracketleftψm|1 2N(−σx˙x−˙xσx)|ψm/an}bracketri}ht =−Ix,m|R (7)5 /LParen1a/RParen1 Α/EquΑl0.5,Β/EquΑl0 /Minus3.25 0 3.25/Minus0.16500.165 EmIy,m/LParen1d/RParen1 Α/EquΑl0.5,Β/EquΑl0 /Minus3.25 0.2 3.65/Minus0.0300.03 EmIy,m /LParen1b/RParen1 Α/EquΑl0,Β/EquΑl0.5 /Minus3.25 0 3.25/Minus0.16500.165 EmIy,m/LParen1e/RParen1 Α/EquΑl0,Β/EquΑl0.5 /Minus3.25 0.2 3.65/Minus0.002800.0028 EmIy,m /LParen1c/RParen1 Α/EquΑlΒ/EquΑl0.5 /Minus3.6 0 3.6/Minus0.16500.165 EmIy,m/LParen1f/RParen1 Α/EquΑlΒ/EquΑl0.5 /Minus3.6 0.2 4/Minus0.100.1 EmIy,m FIG. 4: (Color online). Y-component of persistent spin cur- rent (Iy,m) for separate energy levels of a conducting ring in presence of αandβ, where the different spectra represent the similar meanings as described in Fig. 2. The physical param- eters remain unchanged as taken in Fig. 2. Y XAB C D FIG. 5: (Color online). Velocity direction (green arrow) of a moving electron at some typical points (filled colored circl es) of a ring placed in the X-Yplane. and Ix,m|D=/an}bracketleftψ′ m|Ix,m|ψ′ m/an}bracketri}ht =/an}bracketleftψm|U†Ix,mU|ψm/an}bracketri}ht =/an}bracketleftψp|U†1 2N(σx˙x+˙xσx)U|ψm/an}bracketri}ht =/an}bracketleftψm|1 2N(−σy˙x−˙xσy)|ψm/an}bracketri}ht =−Iy,m|R (8) Equations 7 and 8 clearly describe the interchange of X andYcomponentsofcurrentunderthereciprocationof α andβ. (iii) Quite interestingly we see that both for these two components ( XandY) the states lying towards theenergy band edge carry higher current compared to the innerstates,unlikethe Z-componentofcurrentwhereop- posite signature is noticed. This feature is also observed in other quasi-periodic rings as well as in a fully random one. In addition, a significant change in current ampli- tude takes place between the two current components when the ring is subjected to either αorβ, even if these strengthsareidentical(Figs.3(d) and(e); (Figs.4(d)and (e)), though its proper physical explanation is not clear to us. (iv) Finally, it is important to note that at α=β none of these XandYcomponents of current vanishes (see Figs. 3(f) and 4(f)) since Iy,m|D/ne}ationslash=−Iy,m|Rand also Ix,m|D/ne}ationslash=−Ix,m|R. B. Components of net current for a particular electron filling Now we focus on the behavior of all three components of spin current for a particular electron filling and the total spin current taking the contributions from these individual components. In Fig. 6 we present the variation of Z-component of spincurrentasafunctionofSOcouplingfora89-sitering /LParen1a/RParen1 Β/EquΑl0 0.00.51.0/Minus0.120.110.34 ΑIz/LParen1c/RParen1 Β/EquΑl0 0.00.51.00.054 0.014 /Minus0.026 ΑIz /LParen1b/RParen1Α/EquΑl0 0.00.51.00.12 /Minus0.11 /Minus0.34 ΒIz /LParen1d/RParen1Α/EquΑl0 0.00.51.0/Minus0.054/Minus0.0140.026 ΒIz FIG. 6: (Color online). Z-component of spin current ( Iz) as a function of any one of two SO fields (keeping the other at zero) for a 89-site ring (10th generation) considering Ne= 50, where the left and right columns correspond to ǫA=ǫB= 0 andǫA=−ǫB= 1, respectively. considering Ne= 50. In the first column the results are shownforaperfectring, whileforthe Fibonacciringthey are presented in the other column. The current exhibits an anomalous oscillation with SO coupling and its am- plitude gradually decreases with increasing the coupling strength. This oscillation is characterized by the cross- ing of different distinct energy levels (viz, degeneracy) of the system, and also observed in other context35,39. The other feature i.e., the phase reversal of Izby interchang-6 ing the parameters αandβcan be well understood from our previous analysis, and thus, the net Z-component /LParen1a/RParen1 Β/EquΑl0 0.0 0.5 1.0012 ΑIx/Multiply10/Minus3 /LParen1b/RParen1 Α/EquΑl0 0.0 0.5 1.00.03.87.6 ΒIx/Multiply10/Minus3 FIG. 7: (Color online). X-component of spin current ( Ix) as a function of any one of the two SO interactions for a 89-site (10th generation) Fibonacci ring ( ǫA=−ǫB= 1) considering Ne= 50. /LParen1a/RParen1Β/EquΑl0 0.0 0.5 1.00.0 /Minus3.8 /Minus7.6 ΑIy/Multiply10/Minus3 /LParen1b/RParen1Α/EquΑl0 0.0 0.5 1.00 /Minus1 /Minus2 ΒIy/Multiply10/Minus3 FIG. 8: (Color online). Y-component of spin current ( Iy) as a function of any one of the two SO interactions for a 89-site (10th generation) Fibonacci ring ( ǫA=−ǫB= 1) considering Ne= 50. of spin current should vanish under the situation α=β, which is not shown here to save space. In addition, it is also observed that the net current amplitude in the Fi- bonacci ring (right column of Fig. 6) for any αorβis much smaller than the perfect one (left column ofFig. 6), and it is solely associated with the conducting nature of different energy levels those are contributing to the cur- rent. The nature of current carryingstates can be clearlynoticed from the spectra given in Fig. 2, where the cur- rents carried by distinct energy levels of the Fibonacci ring are much smaller compared to the perfect one. The behaviors of other two current components (viz, XandY) are shown in Figs. 7 and 8. Since both these two components are zero for the perfect ring, here we present the results only for a Fibonacci ring consider- ing the identical parameter values and electron filling as taken in Fig. 6. The current exhibits an oscillation, and unlikeZ-component, the oscillating peak increases with increasing SO interaction. Finally, focus on the spectra shown in Fig. 9 where we present the variation of net spin current Istaking the individualcontributionsfromthreedifferentcomponents. It is defined as Is=/radicalBig I2x+I2y+I2z. Both the perfect and Β/EquΑl0/LParen1a/RParen1 0 1 20.00.10.2 ΑIs Β/EquΑl0/LParen1c/RParen1 0 1 20.00.0240.048 ΑIs Α/EquΑl0/LParen1b/RParen1 0120.00.10.2 ΒIs Α/EquΑl0/LParen1d/RParen1 0120.00.0240.048 ΒIs FIG. 9: (Color online). Net spin current Is(considering the contributions from all three components) as a function of an y one of two SO interactions (setting the other at zero) for a 89-site (10th generation) ring with 50 electrons, where the first and second columns correspond to ǫA=ǫB= 0 and ǫA=−ǫB= 1, respectively. Fibonacci rings are taken into account those are placed in the first and second columns of Fig. 9, respectively. In both these two cases we find oscillating behavior of current following the current components as discussed in Figs. 6-8. For the ordered ring since the contribution comes only from Iz, the oscillation of Isgradually dies out with SO coupling. While for the Fibonacci ring as IxandIyalong with Izcontribute to Is, a finite but small oscillation still persists even for higher values ofSO coupling. The another feature obtained from the spectra i.e., lesserIsin Fibonacci ring compared to the perfect one for any non-zero SO coupling is quite obvious. C. Prediction of on-site energy In this sub-section we discuss the possibilities of esti- mating any one of the two on-site potentials ( ǫAandǫB)7 of a Fibonacci ring if we known the other one. This can be done quite easily by analyzing the behav- ior of current amplitude of individual components as a function of either ǫAorǫB, setting the other constant, keeping in mind that a distinct feature may appear when these two site energies become identical since the current components are significantly influenced by the disorder- ness. In Fig. 10 we show the variation of Izas a function of ǫAfor a 55-site Fibonacci ring with 55 electrons consid- 0.0 0.5 1.0 1.5 2.06.4911.6 ΕAIz/Multiply10/Minus3 FIG. 10: (Color online). Izvs.ǫAfor a 55-site (9th gener- ation) Fibonacci ring for three distinct values of ǫB, where the red, navy and orange curves correspond to ǫB= 0.5, 1 and 1.5, respectively. The other physical parameters are: Ne= 55,α= 1 and β= 0. 0.0 0.5 1.0 1.5 2.00.04.59 ΕAIx/Multiply10/Minus5 FIG. 11: (Color online). Same as Fig. 10 where the variation ofIxwithǫAis shown. ering three distinct values of ǫBthose are represented by three different colored curves. Here we take α= 1 and fixβto zero. Interestingly we see that Izreaches to the maximum (shown by the dotted line) when the two site energies are equal. While, the current amplitude gets re- duced with increasing the deviation of site energies i.e., |ǫA−ǫB|. This is solely associated with localizing behav- ior of electronic waves and directly linked with previous analysis. Thus, for aparticularmaterialcomposedoftwo different lattices one can determine the site energy of any one by varying the other and observing the maximum of Iz. It takes place only when ǫA=ǫB. In the same footing, we can also find a definite con- dition from the other two current components throughwhich site energy is predicted. The results are presented in Figs. 11 and 12 for the identical ring and parameter 0.0 0.5 1.0 1.5 2.00.0 /Minus22.5 /Minus45 ΕAIy/Multiply10/Minus5 FIG. 12: (Color online). Same as Fig. 10 where the variation ofIywithǫAis shown. values astaken in Fig. 10. From these spectra it is clearly seen that when ǫAbecomes equal to ǫB, bothXandY components of spin current drop exactly zero, and this vanishing behavior leads to the possibility ofdetermining site energy. Before we end this sub-section it is important to note that from the practical point of view one may think how site energies of a large number of A-type orB-type sites can be tuned for large sized ring. This is of course a diffi- cult task. But, through the present analysis we intend to establish that if we take a ring geometry with few foreign atoms, then this prescription will be useful since tuning the site energies of only these few atoms by means of ex- ternal gate potential site energies of parent atoms can be estimated. IV. SUMMARY AND CONCLUSIONS In summary, we have made a comprehensive analysis of all three components of persistent spin current in a quasi-periodic Fibonacci ring with Rashba and Dressel- haus SO interactions. Within a tight-binding framework we compute all the current components based on second- quantized approach. Several distinct features have been observed those can be utilized to determine any one of the two SO fields as well as the site energies when the other is known. Theresultsstudiedinthisworkareforagenericmodel, not related to any specific material, and thus, can be extended to any such correlated as well as uncorrelated lattice models and can providesome basicinputs towards spin dependent transport phenomena. V. ACKNOWLEDGMENT MP is thankful to University Grants Commission (UGC), India for research fellowship.8 ∗Electronic address: santanu.maiti@isical.ac.in 1S. A. Wolf, D. D. Awschalom, R. A. Buhrman, J. M. Daughton, S. von Moln´ ar, M. L. Roukes, A. Y. 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1604.08782v1.Tunable_spin_charge_conversion_through_topological_phase_transitions_in_zigzag_nanoribbons.pdf

Tunable spin-charge conversion through topological phase transitions in zigzag nanoribbons Hang Li1and Aur elien Manchon1 1King Abdullah University of Science and Technology (KAUST), Physical Science and Engineering Division (PSE). Thuwal, 23955-6900, Saudi Arabia. (Dated: June 9, 2021) We study spin-orbit torques and charge pumping in magnetic quasi-one dimensional zigzag nanoribbons with hexagonal lattice, in the presence of large intrinsic spin-orbit coupling. Such a system experiences topological phase transition from a trivial band insulator to a quantum spin Hall insulator either by tuning the magnetization direction or the intrinsic spin-orbit coupling. We nd that spin-charge conversion eciency (i.e. spin-orbit torque and charge pumping) is dramati- cally enhanced at the topological transition, displaying a substantial angular anisotropy. PACS numbers: 72.25.Dc,72.20.My,75.50.Pp Introduction - Topological insulators, a new phase of matter, have attracted intense research interest due to their nontrivial physical properties and potential appli- cations in spintronics1. Similarly to conventional or band insulators they possess a band gap in the bulk. Yet, dierently from conventional insulators they support time-reversal-symmetry-protected spin-polarized surface or edge states in the bulk band gap. These materials may experience a topological phase transition from topologi- cal to band insulators by structural design and manipula- tions such as doping with impurities2,3, applying a strain or pressure4,5, inducing lattice distortion or enhancing spin-orbit coupling via nonmagnetic substrates6. Inter- estingly, even without structural manipulation, topologi- cal phase transition can also be driven by coupling topo- logical insulators with magnetic substrates. For exam- ple, a transition from a band insulator to a quantum anomalous Hall insulator can be achieved by inducing magnetic exchange in silicene via a proximate magnetic layer7,8. Among these studies, the in uence of topologi- cal phase transitions on Hall conductivities and spin tex- tures in momentum space has been conrmed3,9. From a topological standpoint, a charge or spin current in a topological insulator is also a topological current. Hence, unlike semiconductors and metals, charge conductivities and spin-polarized edge states in topological insulators can be controlled not only by an electric eld but also by topological phase transitions. Besides topological phase transitions, charges owing at the surface or edge of topological insulators are accom- panied by a non-equilibrium spin polarization due to the large spin-momentum locking of surface states10. Such a magneto-electric eect can be used to excite and switch the magnetization of a ferromagnet deposited on the surface, as studied theoretically11,12and demonstrated experimentally13{15. This spin-orbit torque displays a larger electrical eciency compared with spin torque in bilayers involving heavy metals16,17. Alternatively, the spin-to-charge conversion present at the surface of topo- logical insulators can be probed through charge pumping, i.e. the Onsager reciprocal of spin-orbit torques24,25. In fact, while a charge current creates a torque on the mag-netization, a precessing magnetization induces a charge current along the interface. This eect was originally ob- served in magnetic bilayers involving heavy metals and attributed to the inverse spin Hall eect present in the bulk of the heavy metal18. This observation has been recently extended to two dimensional systems such as hexagonal lattices19, semimetal surfaces20and more re- cently to the surface of topological insulators21{23. In these systems, the spin-charge conversion is attributed to the spin-momentum locking induced by interfacial (Rashba or Dirac) spin-orbit coupling. While magneto- electric eects have been studied in topological insulators in the metallic regime11,12, the in uence of topological phase transitions on these mechanisms has been essen- tially overlooked. In particular, besides the emergence of quantized magneto-electric eect10, it is not clear how the topologically non-trivial edge states contribute to spin-orbit torque and charge pumping. In this paper, we theoretically investigate both charge pumping and spin-orbit torque in quasi-one-dimensional zigzag nanoribbons with a hexagonal lattice in the presence of intrinsic spin-orbit coupling and mag- netic exchange. Depending on the spin-orbit coupling strength, this system displays topological phase transi- tions between trivial (metallic) and non-trivial (quan- tum spin/anomalous Hall) phases7,33. We demonstrate that spin-charge conversion eciency is dramatically en- hanced at the topological transition, resulting in large damping-like spin-orbit torque and DC charge pumping. Spin-orbit torque and charge pumping - Let us rst formulate the reciprocity relationship between spin-orbit torques and charge pumping (see also Ref. 24 and 25). We start from the denition of magnetization dynamics and charge current @tm= m@mF+ ^
E; Jc=^
E+^
@mF; (1) where@mF=@m =Msis the eective eld that drives the dynamics of the magnetization in the absence of charge ow. is the magnetic energy density and Msis the saturation magnetization. Eis the electricarXiv:1604.08782v1 [cond-mat.mes-hall] 29 Apr 20162 eld that drives the charge current through the con- ductivity tensor ^ in the absence of magnetization dy- namics. ^and^are the tensors accounting for current- driven torques and charge pumping, respectively. We can rewrite these two equations in a more compact form  @tni @tmi =Lni;fj eLni;fj m Lmi;fj eLmi;fj m fj e fj m (2) where we dene the particle current @tni=SJc;i=e , the electric and magnetic forces fj e=deEj,fj m=B@mjF. Onsager coecients are then Lni;fj eLni;fj m Lmi;fj eLmi;fj m = Wij=e2ij=B ij=d( =B)(eiej)
m : (3) Here, we consider a magnetic volume of width W, thick- nessdand section normal to the current ow S=Wd. Applying Onsager reciprocity principle24,26 Lni;fj m(m) =Lmj;fie(m); (4) and we obtain ij(m)=B=ji(m)=d. In two- dimensional magnets with interfacial inversion asymme- try, the spin-orbit torque T= ^
Ecan be parsed into two components (see e.g. Refs. 12 and 36) T=DLm((zE)m) +FLm(zE);(5) referred to as the damping-like ( DL) and eld-like torque (FL). Hence, by denition ij(m) =DL[m((zej)m)]
ei +FL[m(zej)]
ei: (6) Then, applying Onsager reciprocity, we obtain the charge pumping coecient ij(m) =(B=d)DL(m)[m((zei)m)]
ej + (B=d)FL(m)[m(zei)]
ej: (7) And nally, the charge current induced by the magneti- zation dynamics reads Jc=B d DL(m)z(m@tm) +B d FL(m)z@tm: (8) This equation establishes the correspondance between the current-driven spin-orbit torque and the charge cur- rent pumped by a time-varying magnetization. In the fol- lowing, we will compute the current-driven spin density Sfrom Kubo formula [Eq. (10)]. The torque is simply T= (2Jex=~)mS, so that that the conclusions drawn for spin-orbit torques equally apply to charge pumping. Model and method - Let us now consider a single- layered zigzag nanoribbon with a hexagonal lattice (e.g. silicene, germanene, stanene etc.) deposited on top of a ferromagnetic layer. The ferromagnetic layer may be 1 2 3 4 . . . . . . . N y x z (a) (b) (c) (d) FIG. 1. (Color online) (a) Top view of zigzag silicene-like nanoribbons switched by a nonmagnetic topological insulator and a magnetic topological insulator. The super unit cell as indicated by the red rectangle. (b) Band structure for dier- ent magnetization direction m. (c) Phase diagram for various Rashba and magnetization. (d) Density of states for dierent magnetization direction. The current is directed along the x axis. The parameters are tso= 36 meV and Jex= 10 meV. chosen as EuO27or YIG19,28, and induces a weak ex- change coupling on the spin-polarized carriers as well as Rashba spin-orbit coupling. In a tight-binding representation, Hamiltonian for silicene-like material can be described by29 H0=X hi;jit^c+ i;^cj;+itso 3p 3X hhi;jiic+ i;vijsz cj; +i2tR 3X hi;jic+ i;^z
(sdij)cj;+JexX i;c+ i;s
Mci;: (9) where ^c+ i;(^ci;) creates (annihilates) an electron with spinon sitei.hi;ji(hhi;jii) runs over all the possible nearest-neighbor (next-nearest-neighbor) hopping sites. tR(tso) is the Rashba (intrinsic) spin-orbit coupling con- stant.vij=1 when the trajectory of electron hopping from the site jto the siteiis anti-clockwise (clockwise). Jexis the ferromagnetic coupling constant. The rst term denotes the nearest-neighbor hopping, the second term denotes the intrinsic spin-orbit coupling and the third one represents the extrinsic Rashba spin-orbit coupling. The fourth term is the exchange interaction between the spin of the carrier and the local moment of the ferromag- net. We assume that the nanoribbon is uniform and peri- odic along the transport direction. A super unit cell is chosen as shown in the red rectangle in Fig. 1(a). To3 compute the spin torques and charge pumping, we rst evaluate the nonequilibrium spin density Susing Kubo formula,30 S=e~ 2AReX k;a;bh kbj^sj kaih kajE
^vj kbi [GR kbGA kaGR kbGR ka]; (10) where Eis the electric eld, ^v=1 ~@H @kis the velocity operator,GR ka= (GA ka)= 1=(EFEka+i). is the energy spectral broadening, and Ais the unit cell area. EFis the Fermi energy, Ekais the energy of electrons in banda. The eigenvector j k;aiin bandacan be found by diagonalizing Eq. (9). Equation (10) contains both intraband ( a=b) and interband ( a6=b) contributions to the nonequilibrium spin density (see the discussion in Ref. 36). The former is related to impurity scattering and the latter only includes intrinsic contributions re- lated to Berry curvature at = 0. We ignore the vertex corrections as they only result in a renormalization fac- tor of the order of unity in two dimensional hexagonal lattices31. For a nanoribbon in the absence of spin-orbit coup- ing, the eigenvalues and eigenvectors around the Dirac point are independent on the magnetization direction. However, when intrinsic spin-orbit couping is present, it acts as a valley-dependent antiferromagnetic eective eld along the z direction. In the low energy limit, it readsso^z ^sz. When the magnetization is di- rected along the x axis, the cooperation of magnetic ex- change and Rashba spin-orbit coupling can open up a band gap turning the system into a (trivial) band insula- tor, as shown in the left panel of Fig. 1(b) (see also Ref. 32). The corresponding density of states in the left panel of Fig. 1(d) displays an evident gap. In contrast, when the magnetization is directed along the z axis, the system evolves towards the quantum spin Hall regime (insulat- ing bulk and conducting spin-polarized edges) as shown in the right panel of Fig. 1(b). It is related to the fact that the magnetic eld couples with the intrinsic spin- orbit coupling and leads to the redistribution of ground states33. Unlike the band insulator, the corresponding density of states show a parabolic dependence on energy as shown in the right panel of Fig. 1(d). For silicene-like materials, the exchange coupling is about 30 meV34. In this parametric range, there are only two dierent topo- logical phases: trivial band insulator and quantum spin Hall insulator as shown in Fig. 1(c). The others topo- logical phases such as quantum anomalous Hall insulator stand beyond this parametric range. Non-equilibrium spin density and torques - In order to understand the in uence of topological phase transition on spin-orbit torque, we rst investigate the in uence of intrinsic spin-orbit coupling on the nonequilibrium spin density in a non magnetic nanoribbon. In this system, Rashba spin-orbit coupling enables the electrical gener- ation of a non-equilibrium spin density, Sy, an eect known as the inverse spin galvanic eect and studied in /s48/s53/s48/s48/s49/s48/s48/s48 /s45/s52/s45/s50/s48/s50/s52 /s48/s46/s48/s48/s48 /s48/s46/s48/s49/s53/s48/s49/s48/s50/s48/s51/s48/s52/s48 /s45/s48/s46/s48/s49/s53 /s48/s46/s48/s48/s48 /s48/s46/s48/s49/s53/s45/s48/s46/s50/s45/s48/s46/s49/s48/s46/s48/s48/s46/s49/s48/s46/s50/s45/s48/s46/s48/s48/s50 /s48/s46/s48/s48/s48 /s48/s46/s48/s48/s50/s53/s48/s53/s53/s54/s48/s54/s53/s55/s48 /s40/s99/s41/s83/s73/s110/s116/s114/s97 /s121 /s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41/s83/s73/s110/s116/s114/s97 /s121 /s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41 /s32/s32/s116/s115/s111/s32/s51/s32/s109/s101/s86 /s32/s32/s116/s115/s111/s32/s54/s32/s109/s101/s86 /s32/s32/s116/s115/s111/s32/s57/s32/s109/s101/s86 /s32/s32/s116/s115/s111/s32/s51/s48/s32/s109/s101/s86 /s32/s32/s116/s115/s111/s32/s51/s51/s32/s109/s101/s86 /s32/s32/s116/s115/s111/s32/s51/s54/s109/s101/s86/s40/s97/s41 /s40/s100/s41/s83/s73/s110/s116/s101/s114 /s121 /s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41/s83/s73/s110/s116/s101/s114 /s121 /s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41 /s69 /s102/s32/s40/s101/s86/s41/s40/s98/s41 /s69 /s102/s40/s101/s86/s41FIG. 2. (Color online) Intraband and interband components of spin density as a function of Fermi energy in a non-magnetic nanoribbon without (a)-(b) and with intrinsic spin-orbit cou- pling (c)-(d). The electric eld is directed along the x axis. The parameters are tso= 0, Jex= 0, tR= 20 meV and = 0 :3meV . details in bulk two dimensional hexagonal crystals31,37. In Fig. 2 we present the intraband (a,c) and interband contributions (b,d) to the non-equilibrium spin density in a nanoribbon as a function of Fermi energy without (a,b) and with (c,d) intrinsic spin-orbit coupling. When the in- trinsic spin-orbit coupling is absent [Fig. 2(a,b)], the sys- tem is metallic and the intraband component dominates the spin density, indicating that carriers at the Fermi sur- face dominate the transport. The intraband component [Fig. 2(a)] is one order of magnitude larger than the in- terband component [Fig. 2(b)], in agreement with the results obtained for two-dimensional graphene-like mate- rials, or two-dimensional electron gases35{37. When the intrinsic spin-orbit coupling is turned on [Fig. 2(c,d)], it opens up a bulk band gap and induces spin-polarized edge states. In the quantum spin Hall regime (small Fermi energy, no bulk transport), the intraband and in- terband contributions are of the same order of magni- tude, while beyond the quantum spin Hall regime (large Fermi energy, both edge and bulk transport coexist), the intraband contribution dominates the spin density. Let us now turn our attention towards the case of a magnetic nanoribbon. In our conguration, E=Ex, and the non-equilibrium spin density can be parsed into two components, S=SDLym+SFLy; (11) referred to as damping-like ( SDL) and eld-like ( SFL). We plot the eld-like and the damping-like spin densities with and without intrinsic spin-orbit coupling in Fig. 3.4 /s45/s49/s48/s48/s48/s49/s48/s48/s50/s48/s48 /s45/s48/s46/s48/s48/s53 /s48/s46/s48/s48/s48 /s48/s46/s48/s48/s53/s48/s50/s52/s48/s45/s50/s48/s45/s52/s48/s45/s54/s48/s45/s56/s48 /s45/s48/s46/s48/s48/s53 /s48/s46/s48/s48/s48 /s48/s46/s48/s48/s53/s48/s50/s48/s52/s48/s32/s83/s73/s110/s116/s114/s97 /s70/s76/s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41 /s32/s116 /s115/s111 /s32/s32/s61/s48 /s32/s116 /s115/s111 /s32/s32/s61/s51/s54/s32/s109/s101/s86/s40/s97/s41/s83/s73/s110/s116/s101/s114 /s70/s76/s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41 /s40/s98/s41 /s69 /s102/s32/s40/s101/s86/s41/s83/s73/s110/s116/s101/s114 /s68/s76/s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41/s40/s99/s41 /s40/s100/s41/s120/s120/s40/s101/s50 /s47 /s32/s41/s32 /s32/s53 /s69 /s102/s32/s40/s101/s86/s41 FIG. 3. (Color online)(a) Intraband, (b)-(c) interband spin density and (d) conductance as a function of Fermi energy without and with intrinsic spin-orbit coupling. The magneti- zation is directed along the z axis.The exchange coupling is xed to Jex= 10 meV and other parameters are the same as in Fig. 2. When the intrinsic spin-orbit coupling is absent and the exchange interaction is present, the intraband compo- nent dominates the eld-like spin density in Fig. 3(a) and (b) similar to the case without exchange interac- tion displayed in Fig. 2(a,b). Moreover, the damping- like spin density [Fig. 3(b)] is smaller than the eld-like spin density [Fig. 3(a)] because the former is a correc- tion arising from the precession of non-equilibrium spin density around the magnetization caused by the accel- eration of carriers in the electric eld30,36,38. When the intrinsic spin-orbit coupling is turned on, the nanorib- bon enters the quantum spin Hall regime: transport only occurs through spin-polarized edge states, resulting in quantized conductance [Fig. 3(d)]. The interband and interband eld-like spin densities [Fig. 3(a,c)] becomes of comparable magnitude but with opposite sign, while the damping-like spin density is signicantly enhanced [Fig. 3(b)]. As a result, the damping-like spin density dominates over the eld-like spin density . Furthermore, since the conductance is only due to edge states, the over- all electrical eciency of the torque (= torque magnitude / conductance) is dramatically enhanced in the quantum spin Hall regime. The topological phase transition can be induced not only by tuning the intrinsic spin-orbit coupling but also by rotating the magnetization as shown in Fig. 1(b,c). In Fig. 4, we plot the intraband and interband contribu- tions to spin density as a function of the magnetization angle for dierent Fermi energies in the absence (a,b,c) or presence (d,e,f) of intrinsic spin-orbit coupling. Dra- matic features can be observed depending on whether the nanoribbon experiences a phase transition or not. When intrinsic spin-orbit coupling is absent [Fig. /s45/s54/s48/s48/s45/s52/s48/s48/s45/s50/s48/s48 /s48/s50/s48/s52/s48/s54/s48 /s45/s48/s46/s51/s48/s46/s48/s48/s46/s51/s48/s46/s54 /s48/s49/s50/s51 /s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48/s45/s50/s48/s45/s49/s48/s48/s49/s48/s50/s48 /s48 /s51/s48 /s54/s48 /s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48/s48/s49/s53/s51/s48/s52/s53/s54/s48/s40/s99/s41/s83/s73/s110/s116/s114/s97 /s70/s76/s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41/s83/s73/s110/s116/s114/s97 /s70/s76/s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41 /s40/s97/s41/s40/s100/s41 /s40/s98/s41/s83/s73/s110/s116/s101/s114 /s70/s76/s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41 /s40/s101/s41/s83/s73/s110/s116/s101/s114 /s70/s76/s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41/s32/s69 /s102/s32/s32/s51/s109/s101/s86 /s32/s69 /s102/s32/s32/s54/s109/s101/s86 /s32/s69 /s102/s32/s32/s57/s109/s101/s86 /s32/s69 /s102/s32/s32/s49/s50/s109/s101/s86 /s32/s69 /s102/s32/s32/s49/s53/s109/s101/s86/s83/s73/s110/s116/s101/s114 /s68/s76/s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41/s83/s73/s110/s116/s101/s114 /s68/s76/s47/s101/s69/s40/s101/s86/s45/s49 /s110/s109/s45/s49 /s41 /s40/s100/s101/s103/s114/s101/s101/s41/s40/s102/s41/s116 /s83/s79/s32/s61/s32/s51/s54/s32/s109/s101/s86 /s40/s100/s101/s103/s114/s101/s101/s41/s116 /s83/s79/s32/s61/s32/s48FIG. 4. (Color online) Intraband and interband spin density as a function of magnetization angle for dierent Fermi en- ergy. (a)-(c) without intrinsic spin-orbit coupling and (d)-(f) with intrinsic spin-orbit coupling. 4(a,b,c)], or when intrinsic spin-orbit coupling is present and the Fermi energy large enough [ >10 meV in Fig. 4(d,e,f)], the nanoribbon remains metallic independently on the magnetization direction. The spin density adopts the form given in Eq. (11) and commonly observed in two dimensional Rashba gases39. Minor angular depen- dence is observable due to the small distortion of the Fermi surface (see also Ref. 40). In contrast, when in- trinsic spin-orbit coupling is turned on and the Fermi energy is small enough [ <10 meV in Fig. 4(d,e,f)], the nanoribbon experiences a topological phase transi- tion from the metallic ( 0;) to the quantum spin Hall regime ( =2). This transition is clearly seen in Fig. 4(d), where the intraband eld-like spin density decreases dramatically (but does not vanish) upon set- ting the magnetization away from 0;. Correspond- ingly, the interband damping-like and eld-like contribu- tions display an abrupt and dramatic enhancement when the magnetization angle is varied through the topological phase transition. Charge pumping - By the virtue of Onsager reciprocity, the results obtained above for the current-driven spin densities apply straightforwardly to the charge pump- ing through Eq. (8). From the denition of the torque, DL= 2JexSDL=EandFL= 2JexSFL=E, and hence- forth the charge current pumped by a precessing magne-5 tization reads Jc=2JexB d SDL Ez(m@tm) +2JexB d SFL Ez@tm: (12) The rst component gives both AC and DC signals41, while the second term is purely AC. The study of non- equilibrium spin density reported above indicates that the second component z@tmdominates in the metal- lic regime (since SFL> S DL), while the rst compo- nentz(m@tm) can be dramatically enhanced in the quantum spin Hall regime ( SDL> S FL). Fur- thermore, because changing the magnetization direction can induce topological phase transitions, one expects that charge pumping with the magnetization lying out of the plane of the two dimensional nanoribbon is much more ecient than when the magnetization lies in the plane. A large charge pumping eciency is expected at the topo- logical phase transition. Notice though that the DC charge pumping vanishes when the magnetization pre- cesses around the normal to the plane as hm@tmiz. Discussion and conclusion - In summary, we have in- vestigated the impact of topological phase transition on the nature of spin-orbit torque and charge pumping in quasi-one dimensional hexagonal nanoribbons. By tun- ing the magnetization angle or the intrinsic spin-orbitcoupling, the system can change from a band insulator to a quantum spin Hall insulator. We nd that spin- charge conversion eciencies (i.e. damping torque and charge pumping) are signicantly enhanced in the quan- tum spin Hall regime. Recently, a gigantic damping torque has been reported at the surface of topological insulators, with electrical ef- ciencies about two orders of magnitude larger than in transition metal bilayers14. To the best of our knowledge, no theory is currently able to explain this observation (see discussion in Ref. 12). Although the present model does not precisely apply to the experimental case, it empha- sizes that close or in the quantum spin Hall regime, (i) the electrical eciency of the spin-orbit torque is dramat- ically enhanced due to the reduction of the conductance and, most remarkably, (ii) the competition between in- terband and intraband contributions reduce the eld-like torque, resulting in a dominating damping-like torque. Such an eect, properly adapted to the case of topologi- cal insulators, could open interesting perspectives for the smart design of ecient spin-orbit interfaces through the manipulation of topological phase transition. ACKNOWLEDGMENTS The research reported in this publication was sup- ported by King Abdullah University of Science and Tech- nology (KAUST). Aurelien.Manchon@kaust.edu.sa 1F. Ortmann, S. Roche and S. Valenzuela, Topological insu- lators: Fundamentals and Perspectives (Wiley-VCH Ver- lag, Weinheim, Germany, 2015). 2Y. L. 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0706.0370v2.Kinetic_investigation_on_extrinsic_spin_Hall_effect_induced_by_skew_scattering.pdf

arXiv:0706.0370v2 [cond-mat.mes-hall] 10 Jan 2008Kinetic investigation on extrinsic spin Hall effect induced by skew scattering J. L. Cheng and M. W. Wu∗ Hefei National Laboratory for Physical Sciences at Microsc ale and Department of Physics, University of Science and Technology of China, Hefei, Anhui , 230026, China (Dated: October 30, 2018) The kinetics of the extrinsic spin Hall conductivity induce d by the skew scattering is performed from the fully microscopic kinetic spin Bloch equation appr oach in (001) GaAs symmetric quantum well. In the steady state, the extrinsic spin Hall current/c onductivity vanishes for the linear- k dependent spin-orbit coupling and is very small for the cubi c-kdependent spin-orbit coupling. The spin precession induced by the Dresselhaus/Rashba spin-or bit coupling plays a very important role in the vanishment of the extrinsic spin Hall conductivity in the steady state. An in-plane spin polarization is induced by the skew scattering, with the hel p of the spin-orbit coupling. This spin polarization is very different from the current-induced spi n polarization. PACS numbers: 72.25.Pn, 72.25.Rb, 71.70.Ej, 71.10.-w Generating and manipulating the spin polarization in semiconductors is one of the important prerequisites for the realization of the new spintronic device.1Spin Hall effect (SHE) is considered as the convenient method to generatespin polarization in additional to the traditional methods such as the external magnetic field, the cir- cular/linear polarized laser,2,3the spin-galvanic effect4 and the spin injection from the ferromagnetic metal to semiconductor.5The SHE is induced by intrinsic or ex- trinsic spin-orbit coupling (SOC)6which gives rise to the spin current vertical to the charge current without ap- plying external magnetic field and/or spin accumulation at sample edges.7,8,9Experimentally, the spin Hall con- ductivity (SHC) is estimated indirectly by the spin accu- mulation at the sample edges,10,11,12,13,14or the charge current with a transverse magnetic field applied in a gyrotropic systems.15,16,17Recently, a direct electronic measurementsoftheSHEisgivenbyValenzuelaandTin- kham in the metallic conductor.18All these effects are explained as the intrinsic6,19,20,21,22,23,24,25,26,27and/or the extrinsic SHE6,7,28,29,30,31,32,33,34,35,36,37,38theoreti- cally by using the Kubo formula19,20,30,31or the Boltz- mann equation6,7,23,34,36with only the carrier-impurity scattering included. The intrinsic SHE is induced by the intrinsic SOC (i.e., the Dresselhaus39and/or the Rashba40SOCs) with the applied external electric field and the resulting SHC was at first thought as dissipationless one in perfect crystals.19,20Later investigations proved that this kind of SHC disappears even for the infinitesimal impurity density with the vertex correction22,23,41,42,43,44in the Rashba model or the linear Dresselhaus model in quan- tum wells,45but remains a finite value for the cubic SOC.46However, the spin current is not an observable quantity. The Laughlin’s gauge gedanken experiment in- dicates that the intrinsic SHE cannot lead to anyspin ac- cumulation at sample edges,24,25unless in a mesoscopic system. In additional to the Dresselhaus and the Rashba SOC, the mixing between the valance and the conduc- tion bands gives rise to two corrections: One is theadditional spin-dependent electron-impurity2,48or the electron-phonon16,17skew scattering. The extrinsic SHE induced by the skew scattering alone has been widely studied by using both the Kubo formula and the Boltz- mannequationmethod,6,7,28,29,30,34,35,36andthe nonzero extrinsic SHC is obtained. Lately, Tse and Das Sarma31 have proved the vanishment of the extrinsic SHC by considering the vortex correction of the linear SOC in the Kubo formula. However, a fully microscopic cal- culation of the extrinsic SHC from the kinetic equation approach is still missing and the vanishment of the ex- trinsic SHC by the vortex correction from the Kubo ap- proach needs to be verified from the kinetic approach. The other is the additional spin-dependent position and velocity operators which bring the correction to the defi- nition of the spin current, and are referred to as the side- jump mechanism.6Here two corrections on the definition should be specified. The first comes from the electrical potential which gives an intrinsic-like contribution and againcannot contribute to the spin accumulationaccord- ing to Refs. 24,25,47. The second comes from the spin dependent scattering in which high order correlationsbe- tween different wave vectors need to be considered.6,34It is hard to include the second correctionin the Boltzmann equation approach,6though it also gives rise to the ex- trinsic SHE, together with the skew scattering. In the following, we concentrate on the extrinsic SHE induced by the skew scattering. The spin polarization is not only accumulated at the sample edges due to the extrinsic SHE, but also observed simultaneously inside the samples which is induced by the chargecurrent.10,11,13,49,50,51,52Bycomparingthe ex- periments in Refs. 10,13,51, it is easy to find that both spin polarizations are in the same order. In theory, En- gelet al.49and Trushin and Schliemann52attributed the current-induced spin polarization (CISP) in the homoge- neous system as the results of the current induced effec- tive magnetic field (EMF) from the SOC.51Tarasenko showed that the spin-flip phonon scattering in asym- metric quantum wells can also induce this kind of spin2 polarization.50However, the spin-conserving skew scat- tering in two dimensional GaAs semiconductor can also induce spin polarization inside the sample due to the ex- trinsic SHE. This effect has not been studied in the liter- ature. Moreover,a fully microscopickinetic investigation on the extrinsic SHE is also missing in the literature. In this paper, we focus on the kinetic process of the extrinsic SHE induced by the skew scattering in symmet- ric GaAs (001) quantum well from the kinetic spin Bloch equation (KSBE) approach.53,54We demonstrate the im- portant role ofthe spin precessioninduced by the (intrin- sic) Dresselhaus/Rashba SOC to the SHE and show that itisinadequatetostudytheextrinsicSHEfromtheKubo formalism without considering the Dresselhaus/Rashba SOC in the literature. We further show that the extrin- sic SHE can generate spin polarizations in homogeneous system. By using the non-equilibrium Green function method and the generalized Kadanoff-Baym Ansatz,55we con- struct the KSBE53,54for electrons as follows ∂ρk(t) ∂t−eE∂ρk ∂kx+∂ρk ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle coh+∂ρk ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle scat+∂ρk ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle ss= 0.(1) Here,ρk(t) =/parenleftbigg fk↑ρk↑↓ ρk↓↑fk↓/parenrightbigg is electron density ma- trix with wave vector kat time t. The applied elec- tric field Eis assumed along the x-axis and the mag- netic field Bis along the x-y(well) plane. The coher- ent term describes the spin precession along the mag- netic/effective magnetic field and is given by∂ρk ∂t/vextendsingle/vextendsingle/vextendsingle coh= i[1 2(ΩD(k)+gµBB)·σ,ρk(t)], where ΩD(k) =γ(kx(k2 y− (π a)2),ky((π a)2−k2 x),0) represents the EMF from the Dresselhaus SOC39withγstanding for the material- determinedSOCstrength56,57andabeingthewellwidth. Infinite-well-depth assumption is adopted here and only the lowest subband is taken into account due to small well width. The spin conserving scattering∂ρk ∂t/vextendsingle/vextendsingle/vextendsingle scatis givenbythespinconservingelectron-impurityscattering, the electron-phonon scattering and the electron-electron Coulomb scattering which are given in detail in Ref. 58. The skew scattering is given by the third order expan- sion of the electron-impurity scattering and reads:6,30,31 ∂ρk ∂t/vextendsingle/vextendsingle/vextendsingle/vextendsingle ss=−2π2Niλγ2/summationdisplay k1k2;q1q2δ(εk2−εk)δ(εk1−εk) ×V(k−k1,q1)V(k1−k2,q2)V(k2−k,−q1−q2) ×{(k−k1)×(k2−k)·σ,ρk2}. (2) Hereλγ2=η(2−η) 2m∗Eg(3−η)is the strength of the skew scat- tering with η=∆ ∆+Eg.2Egand ∆ are the band gap and the spin-orbit splitting of the valance band separately. V(q,qz) =−e2 q2+q2 z+κ2I(qza 2π) withI(x) =eiπxsin(πx) πx(1−x2) being the form factor and κdenoting the Debye-H¨ ucke screening constant. This term produces an asymmetric spin-conserving scattering of electrons so that the spin- up/down electrons prefer to be scattered to the left/rightE kxky FIG. 1: (Color online) The schematic for the skew scatter- ing and the intrinsic spin orbit coupling in the k-space. ⊙ (⊗) stands for the spin-up (-down) polarization. The blue ar- rows give the direction of the EMF from the Dresselhaus SOC which rotates the spin polarization to the direction indica ted by the red dashed arrows. side. A schematic for the skew scattering is given in Fig. 1 for a left moving electron. We first analysis the KSBE for some simple cases. Af- ter carrying out the summation over k, one gets the equation of continuity for the spin density S=/summationtext ksk=/summationtext kTr[ρkσ] as ∂S(t) ∂t−/summationdisplay kΩD(k)×sk(t)−gµBB×S(t) = 0.(3) The summation of the spin-conservingscattering is natu- rally zero. The summation of the skew-scattering in Eq. (2) for the two-dimension system is also zero because it can onlyskew the spin-up and -down electronsseparately instead of flip them. Equation (3) is consistent with the result in Ref. 44 which is obtained by using the general operator commutation relations. When only the linear- kterms in ΩD(k) is retained, for the steady state one obtains Jz y=egµBBy γm∗((π/a)2−∝an}bracketle{tk2x∝an}bracketri}ht)Sz. (4) HereJz=−e/summationtext k/planckover2pi1k/m∗(fk↑−fk↓) is the spin cur- rent which is the simplest but the most widely-used definition6without considering contribution from the side-jump effect. The SHC is given as σz y=Jz y/E. In the derivation, we also take the approximation/summationtext kkxk2 ysz k= −m∗ e∝an}bracketle{tk2 y∝an}bracketri}htJz xand/summationtext kkyk2 xsz k=−m∗ e∝an}bracketle{tk2 x∝an}bracketri}htJz ywith∝an}bracketle{tk2 x/y∝an}bracketri}ht, the average value of k2 x/y. If the strain-induced SOC Hs=α(kyσx−kxσy) (with the same expression of the Rashba SOC) is taken into account, Eq. (4) changes into Jz y=γ((π a)2−∝an}bracketle{tk2 y∝an}bracketri}ht)By−2αBx γ2((π a)2−∝an}bracketle{tk2x∝an}bracketri}ht)((π a)2−∝an}bracketle{tk2y∝an}bracketri}ht)−4α2egµB m∗Sz.(5) Withouttheexternalmagneticfield, Eqs.(4)and(5)give Jz y= 0, which verifies the zero extrinsic SHC given by3 Ref. [31] from the Kubo approach. However, the skew scattering does generate the spin currents when the elec- tric field is applied (in Fig. 1), but it is just the dynamic one. To make clear how the spin current disappears, we rewrite the x-component of Eq. (3) as ∂Sx/∂t=m∗γ[(π/a)2−∝an}bracketle{tk2 z∝an}bracketri}ht]Jz y/e . (6) It is hence easy to find that the spin-Hall current is con- verted to the spin polarization along the x-axis and it tends to zero in the steady state. This can further be understood from Fig. 1: after the spin current is excited by the skew scattering, the spin polarization distributes (⊙and⊗) anti-symmetrically at ±ky. However the y components of the EMF due to the SOC (the blue ar- rows) have the same symmetry and rotate the spin po- larizations to the −xdirection (the red dashed arrows). Therefore, the spin Hall current is converted to the in- plane spin polarization. Now we show the time evolution of the SHC σz yand the spin polarization Px=Sx/Necalculated by numeri- cally solving the KSBE with all the scattering explicitly included at T= 200 K in Fig. 2. The parameters in the calculation are taken as following: the electron and im- purity densities Ne=Ni= 4×1011cm−2;a= 7.5 nm; E= 0.1 kV/cm; γ= 11.4 eV·˚A3; and√ λγ= 2.07˚A. From the figure, the SHC increases with time first from zero value to a maximum one in nearly 1 ps, then de- creases slowly to a very small value (in stead of zero due to the inclusion of the Dresselhaus SOC with the cubic kterms) in a characteristic time scale about 50 ps. The spin polarization is along the −x-axis and increases from zero to its steady state Px= 1.2×10−4. The above evolution is easy to understand with the help of the schematic in Fig. 1. When the positive elec- tric field is applied, the skew scattering scatters the spin- down electrons to the upper panel of the k-space and the spin-up ones to the lower panel. This leads to the spin currents flowing along the y-direction. The strength of the skew scattering is determined by the shift of the elec- tron distribution, so the SHC increases fast to its max- imum value σz y∼2.5×10−3e2 hat the time scale of the charge current (see the dashed curve in the inset of Fig. 2). Then the spin polarization precesses along the y- components of the EMF from the SOC to the −xdirec- tion to generate the in-plane spin polarization. Due to the symmetry, the spin polarizations along the yandz axes are zero. It is interesting to see that the obtained SHC in the steady state is orders of magnitude smaller than the one obtainedfromtheKuboformulawidelyusedinthelitera- ture where only the lowest orderdiagram is considered.30 With only the impurity scattering included at zero tem- perature, the Kube formula gives the extrinsic SHC30as 2πm∗λγ2εF /planckover2pi12σx∼3.37×10−3σxwith the chargeconductiv- ityσx=ne2τ m. This value is orders of magnitude larger thanourresult σz y∼4.1×10−8σx, obtainedfromthefully microscopic KSBE. The difference is caused by the inho- mogeneous broadening59in spin precession due to thet (ps) σx(102e2/h)σzy(10−2e2/h)0.1 04 2 00.3 0.2 0.1 0 t(ps) −10−4Pxσzy(10−2e2/h)1.2 0.8 0.4 010008006004002000100 10−1 10−2 10−3 10−4 10−5 FIG.2: Time evolutionoftheSHCwith(solid curve)/without (chain curve) coherent terms and the time evolution of the spin polarization (dashed curve) along the x-axis atT= 200 K.Ni=Ne= 4×1011cm−2and the electric field E= 0.1 kV/cm. The corresponding time evolutions of the SHC of the first 4 ps are shown in the inset, together with the charge conductivity σx. It is noted that the scales of the spin polarization/charge conductivity are on the right hand sid e of the frames. Dresselhaus/Rashba SOC. To show this effect, we drop the coherent term and plot the time evolutionofthe SHC by chain curves in Fig. 2. We find that the steady SHC σx y∼0.4×10−3σxis much closer to the one above given by the Kubo formula at 0 K. Therefore, we conclude that it is not suitable to calculate the extrinsic SHC without considering the spin precession. Furthermore, the side jump mechanism gives an time-independent contribution asσz y=−2λγ2e2Ne∼ −2×10−3e2/hwith the opposite sign of the one from the skew scattering. It is further noted that althoughthe spin precessionin- duced by the EMF leadsto the vanishment ofthe SHC, it plays a very important role in generating the spin polar- ization. From the discussion above, it is easy to see that the skew scatteringalonecannot generateanyspin polar- ization due to the anti-symmetrical spin polarization at ±ky. Differing fromthe CISP49wherethe SOCis usedto provide a current induced EMF,10,58here the SOC acts asan anti-symmetricalprecessionfield, which is the same as the spin polarization induced by the spin-dependent phonon scattering.3We further note that the spin polar- ization induced by the skew scattering is very different from the CISP: (i) The spin polarization induced by cur- rent induced EMF prefers the small well width which gives a large EMF,49whereas the one induced by the skew scattering and the spin precession prefers a large well width. The later can be seen as following: from Fig. 2, the evolution of the extrinsic spin Hall concurrent can be written as Jz y(t) =Jz y,0e−t/τswith the relaxation timeτs.Jz y,0approximates to the maximum value which4 is only determined by the skew scattering. Therefore Sx=m∗τsγ((π/a)2−∝an}bracketle{tk2 x∝an}bracketri}ht)Jz y,0/e. For the D’yakonov- Perel’ mechanism,60τs∝[γ((π/a)2−∝an}bracketle{tk2 x∝an}bracketri}ht)]−2. Hence Sx∝γ((π/a)2−∝an}bracketle{tk2 x∝an}bracketri}ht)−1. (ii) When the electric field is along the x-axis, for the Dresselhaus SOC, the CISP is along the xdirection, while the polarization induced by the skew scattering is along the −xdirection. How- ever, for the strain-induced or the Rashba SOC, the spin polarizations from both mechanisms are along the same direction. (iii) The CISP decreases with the impurity density49becausehighimpuritydensityreducestheEMF effectively. However, the skew-scattering-induced spin polarization increases with impurity density as the skew scattering is proportional to the impurity density. In summary, we investigate the SHC and the spin po- larization induced by the k-asymmetric spin-conserving skew scattering in symmetrical (001) GaAs quantum well from the fully microscopic KSBE approach at hightemperature with all the scattering explicitly included. We find the spin precession induced by the Dressel- haus/Rashba SOC has a very important effect on the extrinsic SHC and verify the vanishment of the SHC for lineark-dependent SOC. We also show that the SHC in- duced by the skew scattering calculated from the Kubo formula in the literature is inadequate without consider- ing the spin precession. Finally we show that with the joint effects from the skew scattering and the spin pre- cession, an in-plane spin polarization can be generated which can be further rotated to the z-direction by apply- ing an external in-plane magnetic field. The authors acknowledge valuable discussions with Z. Y. Weng. 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2103.16721v1.Numerical_analysis_of_the_spin_orbit_coupling_parameters_in_III_V_quantum_wells_using_8_band_Kane_model_and_finite_difference_method.pdf

IOP Publishing Journal Title Journal XX (XXXX) XXXXXX https://doi.org/XXXX/XXXX xxxx- xxxx/xx/xxxxxx 1 © xxxx IOP Publishing Ltd Numerical analysis of the spin -orbit coupling parameters in III- V quantum wells using 8- band Kane model and finite -difference method V E Degtyarev, S V Khazanova, A A Konakova) and Yu A Danilov Faculty of Physics , National Research Lobachevsky State University of Nizhny Novgorod, Nizhny Novgorod, Russia a) E-mail: anton.a.konakov @gmail .com Received xxxxxx Accepted for publication xxxxxx Published xxxxxx Abstract By means the envelope fu nction approximation, 8- band Kane model and a finite -difference scheme with the coordinate space discretization, we numerically performed calculations of the spin- orbit coupling parameters for 2D electron gas confined in both symmetric and asymmetric [0 0 1] quantum wells based on zinc -blende III -V semiconductors. Influence of the quantum well band parameters and width as well as the magnitude of the external electric field applied along the growth direction on the Dresselhaus and Rashba spin -orbit coupling parameters is investigated. It has been found that in the symmetric InGaAs/GaAs quantum wells linear -in-momentum spin -orbit coupling disappears for the third electron subband at certain values of well width and the indium content. It is also shown that in asymmetric InGaAs/GaAs structures the spin- orbit coupling parameters can be equal at a certain electric field that is the condition for the realization of the SU(2) spin symmetry and formation of persistent spin helices. Besides, we calculated the spin -orbit coupling in the persistent spin helix regime as a function of the well width, indium content and external field. The proposed approach for the calculation of the spin -orbit coupling parameters can be applied to other 2D structures with the spin -orbit c oupling. Keywords: spin-orbit coupling, envelope function approximation, 8 -band Kane model, finite difference method, persistent spin helix 1. Introduction Due to effects of quantum confinement a magnitude of spin-dependent phenomena sufficiently increases in low - dimensional semiconductor structures [1]. Thus, two - dimensional (2D) electron systems are actively investigated for design of current and future spintronic devices [2]. A principal intere st to 2D structures in the spintronics applications is determined by the spin-orbit coupling (SOC) that leads to the spin splitting and subsequent reconstruction of their energy spectrum [3] and causes a number of physical phenomena, such as spin -galvanic effect [4], spin Hall effect [5] and spin- dependent tunneling [6, 7], which are key for creation of devices based on spin -dependent phenomena. Presence of heavy elements (indium, antimony, bismuth) in heterostructures and influence of external potential can lead to even more noticeable spin -related phenomena and, consequently, need for detailed study of SOC. In 2D crystalline systems there are two main linear -in- wavevector (linear -in-k) contributions to SOC: the first one is the Dresselhaus SOC due to asymm etry of a host crystal unit cell [ 8] and the second one is the Rashba interaction Journal XX (XXXX) XXXXXX V E Degtyarev et al 2 caused by asymmetry of a heterostructure [ 9]. The latter one can be effectively controlled by an applied bias [ 10–12]: an external electric field modifies the Rashba SOC para meter and, therefore, significantly affects kinetic [1 3, 14] and dynamical [1 5, 16] spin characteristics of a 2D electron gas. Moreover, Rashba and Dresselhaus terms can interfere resulting in the spin splitting anisotropy. In particular, for electron stat es in III- V quantum wells (QWs) with the [0 0 1] growth direction the SU(2) spin symmetry and corresponding persistent spin helix (PSH) state can be realized at the certain SOC parameters relations [1 7, 18]. Currently physics and applications of the PSH re gime in 2D systems are of a great interest [ 19, 20, 21] due to the suppression of the D’yakonov -Perel’ electron spin relaxation mechanism [ 22] and reduction of the spin relaxation rate [2 3]. Therefore, manipulation of the SOC parameters and PSH state by means an external electric field extends design possibilities of spintronic devices. It should be noted that experimental determination of the SOC parameters remains a difficult task [ 24]. In this regard, their preliminary calculation allows one to predict s pin- related properties of low -dimensional structures, compare them with experimental data and accordingly design QWs with required features. Earlier the spin splitting of electron states in direct- band III-V QWs were calculated in works of Pfeffer and Zawadzki [25, 26] and Pfeffer [2 7, 28]. In Refs. [25, 27, 28] the envelope function approximation with the 14- band Kane model was used in order to receive analytical expressions for Rashba and Dresselhaus spin splitting, while in [ 26] the SOC parameters were calculated with the " conventional " 8-band Kane model. All calculations were performed taking into account the discontinuity of band parameters and external electric field, generating the asymmetry of the QW. The same multiband envelope function approximati on based on both 88× and 1414× Hamiltonians was used in [ 29]. One of the main conclusions of this work that application of 14 - band Kane model to calculations of the spin splitting gives the minor improvements over results obtained with the use of "conventional " 88× model, while the number of parameters, equations and, in this way, complexity of calculations rises sufficiently. Thus, the use of 8 -band Kane model adopted for calculations of spin -dependent properties is more preferable. While above mentioned works concentrate on the SOC parameters calculation for the lowest lying electron subband, some experimental data show, that in the conduction band of III-V QWs there can be occupied more, than on e subband [30–33]. For such systems it is important to know the SOC parameters for several quantum subbands. Theoretical description of the SOC in QWs with two subbands was first proposed in [ 34] and developed then in [ 35, 36]. In these works authors using 8-band Kane model received the effective 2 -subband Hamiltonian in the envelope’s subspace containing both intra - and intersubband linear -in-k SOC. Note that for low energies and long wavelengths the correction to the spectra from conventional intrasubband SOC is linear -in-k, while contribution to the dispersion curves from intersubband SOC starts from the second order in the wave vector [ 35]. In all of these studies [2 5–28, 34–36] there were used, in fact, the averaging the additional SOC (Rashba or Rashba and Dresselhaus) over the states, obtained in the envelope function approximation, that was first proposed by D’yakonov and Kachorovsky in 1986 [37 ]. The main advantage of this approach is that it is semi -analytical way to obtain the expressions for the SOC parameters: it is necessary to calculate the envelope function and then determine essential matrix elements. However, here we show that it is only the first approximation to the SOC parameters, because some information about spectra and wave functions is lost through the averaging procedure. In this work, we use 8 -band Kane model and direct numerical approach to simulate the SOC parameters that enables increasing calculation accuracy and taking into account band structure peculiarities. Computer simulation and numerical calculations allow one to reveal the correlation between the technological parameters of growth and the SOC magnitudes in heterostructures with an arbitrary com position and geometry. Our aim is to define the spin splitting for different energy subbands and to investigate the possibility of controlling the SOC parameters for various QWs. In particular, we propose a technique to extract the Dresselhaus parameter fo r symmetric heterostructures and quantitatively determine the Rashba interaction in the presence of the structure asymmetry caused by the electric bias applied along the growth direction. Moreover, we search for a tendency, how the wave function penetratio n in the barrier region of QW changes the SOC parameters. We also reveal factors that affect the ratio between Rashba and Dresselhaus contributions, which is important for realization of the PSH regime. All our calculations are carried out for In xGa 1-xAs/G aAs QWs. This paper is organized as following. In Sec. 2 we present theoretical background of the SOC parameters definition in zinc-blende semiconductor 2D systems and propose new numerical approach for their calculation. In Sec. 3 we demonstrate and discu ss calculated Rashba and Dresselhaus parameters for In xGa 1-xAs/GaAs QWs as functions of QW width, indium content and external bias. Peculiar attention is devoted to parameters that allow realization of the PSH state. Finally, Sec. 4 conclude the work. 2. Theoretical background and method of calculations 2.1 The spin- orbit coupling in zinc -blende type quantum wells In this work we theoretically investigated the electronic states in In xGa 1–xAs/GaAs QWs with account of SOC . As Journal XX (XXXX) XXXXXX V E Degtyarev et al 3 known, the effective Hamiltonian near the bottom of the conduction band of a bulk crystal with a zinc blende structure up to terms of the third order in the wave vector has the form [8]: ( )zz yy xx DmH σκσκσκγσ ˆ ˆ ˆ ˆ 2ˆ 022 ++ + =k, (1) where m is electron effective mass, ()2 2 z y x x kkk−=κ , ()2 2 x z y y kkk−=κ and ()2 2 y xz z kkk−=κ are combination s of wave vector components due to effective magnetic field, caused by SOC, xk, yk and zk are three components of the wave vecto r, xσˆ, yσˆ, zσˆ and 0ˆσ are Pauli matrices and unit matrix respectively, Dγ is bulk Dresselhaus constant. In other words, the second term of (1) is bulk Dresselhaus SOC caused by the absence of an inversion centr e in the crystal unit cell. Furthermore in the work, for clarity , the term " SOC parameter " refers to 2D system, while " SOC constant " corresponds to bulk Dresselhaus SOC. The SOC 3D Hamiltonian can be rewriten in explicit form as ()() ( ).ˆ ˆˆ ˆ ˆ ˆ 2 22 2 2 3 yxy xyx Dyy xx zD y x zzD D kk kkk kk kk k H σσγσσγ σγ − ++− −− = (2) Confinement of carriers in 2D structures results in to modification of SOC. Assuming that spectrum is quantized in the QW growth direction, for small 2D wave vectors electron states are described by two -component ( taking spin into account) envelope function sn,ψ, where n is QW subband index and s characterizes the spin projection. We suppose that QW s are grown in []001 crystallographic direction, which corresponds to the z-axis in Hamiltonian (2). Difference between material parameters in QW structure leads to need of the symmetrization procedure for k- dependent terms in the growth direction [ 38]: ()() ().,2, 2 zzzkzz zzikzi k D zDD D zDz ∂∂ ∂∂→− ∂∂+∂∂−→∂∂−→ γ γγγ γ (3) After a veraging over the nth QW subband with envelope function ()zsn,ψ the Hamiltonian (2) takes the form [26, 37]: ()()()( )yxy xyxn D yy xx nn D kk kk k k H σσγσσβ ˆ ˆ ˆ ˆ ˆ 2 2 2 − +−= , (4) where () nD nzzz∂∂ ∂∂=γβ (5) is the Dresselhaus parameter in the nth subband , ()()n Dn D zγγ= , and brackets n... stands for averaging over envelope function in the nth subband. Asymmetry of the potential profile results in addition of linear in wave vector spin -orbit contribution to the Hamiltonian (the so -called Rashba term) [ 9]: ()()yx xy nn R k k H σσα ˆ ˆ ˆ − = . (5) where nα is Rashba parameter in the nth subband . The asymmetry can be implemented with external electric field applied along the strcuture growth direction. In this case the Rashba parameter for each subband is proportional to the magnitude of applied field [ 9]. In QW structure s the SOC contribution int o electron Hamiltonian should contain both linear and cubic in wave vector terms. Excluding third -order wave vector terms in Hamiltonian (4) [ 26] and the interface contributions [ 39, 40], the effective 2D Hamiltonian for electrons in nth subband in zinc bl ende semiconductor asymmetric QW can be written as () ()().ˆ ˆ ˆ ˆˆ 2ˆ 02 2 2 2 yy xx n yx xy ny x D k k k kmkkH σσβσσασ −+− +++= (6) It should be noted th at the Hamiltonian (6) is applicable only for small wave vectors near the QW conduction band bottom, therefore it describes the dispersion only for the first electron subband [41]. Th us, more accurate calculation of the QW energy spectra and SOC parameters requires usage of multi -band k·p-models [42, 43], specifically, the 8-band Kane model [44, 45]. Within the framework of this approach, the k·p-interaction of conduction and valence band s is taken into account exactly, while influence of all other band s is treated as a perturbation . Using this method the heterostructure energy spectrum and corresponding 8 - component envelope function ()zkky xsn ,,,Ψ are calculated from Schrödinger -like equation ()() ()(),,, ,,, ˆ ,, ˆ , ,, 88 88 zkk kkEzkk zVIzikk H y x sn y x sny x sn y x Ψ == Ψ + ∂∂−× × (7) Journal XX (XXXX) XXXXXX V E Degtyarev et al 4 where ()z y x kkk H ,, ˆ 88× is the Kane Hamiltonian [ 44], ()zV contains the confining potential including the asymmetry induced by external gate, ()y xsn kkE ,, is the energy of the electronic state in nth subband with 2D wave vector k and the spin projection s. 2.2. Numerical method of calculations In general, the analytical solution of equation (7) is difficult and the most universal approach is based on the numerical methods. For numerical solution of equation (7) we used the finite difference method (FDM). In the framework of this method Hamiltonian parameters (let us denote them as ()zC ) and envelope functions are determined as spatial functions on the nodes of a one -dimensional uniform grid in the QW growth direction: ()() ()( ) , ,, ,,, 00 j y x y xj jdzzkk zkkC jdzzC zC Ψ=+ Ψ→ Ψ≡+→ (8) where dz is the grid spacing. In order to get the solution we convert the differential operators (3) in to matrix operators [46]. Thus, substituting the transformed operators to equation (7), we come to Hermitian eigenvalue problem parameterized with two -dimensional wave vector. However, the additional difficulties of this computational scheme may be associated with full or partial mixing between computed eigenspectrum and the so -called "spurious " solutions [47] described below. 2.3. The problem of " spurious" states Spurious solutions exhibiting unphysical behavior can be found in different parts of the simulated energy spectrum including the band gap. These states are characterized by incorrect subband bending in k-space, as well as high - frequency spatial oscillations. On the other hand, presence and features of these solutions strongly depend on the choice of numerical procedure [48]. In order to describe the reason for the appearance of such unphysical states let us consider a " toy" model 22× k·p- Hamiltonian describing the interaction of nondegenerate conduction and valence band s: + −+=× 22 22 ~ ~~ ~ ˆ kL E kPikPi kA EH VC, (9) where CE, VE are the positions of the conduction and valence band edges respectively, P~ is the momentum matrix element. The 8 -band Kane Hamiltonian used in this work has blocks equavalent to (9), so the following discussion is applicable for the c onsidered Hamiltonian . The numerical solution of energy spectrum problem for an infinite discrete mesh with FDM or finite -elements -based scheme results in the eigenvalue problem: i jjD ji E H Ψ=Ψ∑)( , , (10) where i, j are the mesh node indices, and the discrete Hamiltonian operator )( ,D jiH has the three -diagonal form [49]: = εεε ˆˆ0ˆˆˆ0ˆˆ )( , tt tt HD ji , (11) where εˆ and tˆ are the matrices describing the discrete form of the Hamiltonian. For the Hamiltonian (9) they are written as: ++ = 22 ~200~2 ˆ dzLEdzAE CC ε , −−− = 22 ~ 2~2~ ~ ˆ dzL dzPdzP dzA t . (12) By applying the Bloch theorem to (10) one yields the dispersion equation: ()() ( )()Ψ=Ψ−+ ++kE ikdz t ikdz t expˆ expˆˆε . (13) The solution E (k) of the dispersion relation (13) has at least two extrem a: the first one corresponding to the actual physical solution of the Hamiltonian (9) at 0=k and the second one at dz kπ= (unphysical valley) responsible for the presence of " spurious " states [50]. Then, s ubstituting dz kπ= to the dispersion equation (13) for FDM version of the equation (7) we obtain the energy position UVE of the unphysical valley which can be written as : + += Am dzE EC UV 02 224, (14) where 0m is the free electron mass, and A describes quadratic -in-k interaction of the conduction band with the remote bands. Journal XX (XXXX) XXXXXX V E Degtyarev et al 5 Decreasing of grid spacing results in shift of unphysical valley with respect to conduction band minimum. The direction of the shift is determined by the sign of A m+022 in (14). Neglecting the interaction with bands not included in the 8 -band k·p-Hamiltonian, we can make the shift to be positive by setting 0=A according to Bastard [51]. In other words, to eliminate the problem of spurious states one has to fix the conduction band term of the Hamiltonian to be 0222m k and recalculate the interband momentum matrix term P by means the electron effective mass value. Thus , adjusting the grid spacing we shift up the energy values of spurious solutions eliminating their mixing with physical states. 2.4. Extracting the Dresselhaus and Rashba linear -in-k parameters from numerical calculations During numerical solution of equati on (7) one get the electron energy dispersion: () ()k kn sn Es mkE ∆+= 2 222 ,, (15) where ()knE∆ is the spin splitting of nth electron subband. Angle dependence of this energy splitting can be written as () ϕβαβα 2sin 2 22 2 nn n n n k E ++=∆ k , (16) where ()x ykk arctan=ϕ . For realization of the PSH state it is important to know the ratio between Rashba and Dresselhaus parameters. O ur aim is to reveal a tendency in the behavior of the ir values in the single QWs with variable parameters. For direct extr acting of the individual contribution nα and nβ values we input the linear -in-k coefficients of the spin splitting ±nb along corresponding direction ()1 ,1±=Σ± in the momentum space: ( ) kbk k k k kEn n n y x n ±=±=±= =∆ βα22 ,2 . (17) By numerically determining coefficients ±nb we get nα and nβ values from the equation (17) for each subband. The proposed approach is similar to one used earlier in [52 ] for experimental extraction of Rashba and Dresselhaus SOC parameters. It should be noted th at Rashba and Dresselhaus contributions depend on a number of macroscopic characteristics, such as parameters of materials forming a QW, its width, doping profile and growth temperature, carrier density. Thus, in this work we theoretically investigate the SOC in InGaAs/GaAs QW for different width d with variable indium content using the 8 -band envelope function approximation and FDM . We show that the behavior of the Dresselhaus and Rashba p arameters varies greatly with a QW subband index. Besides we defined the QW parameters required for the PSH state formation. The suggested method to calculate the spin splitting and the SOC parameters is self-sufficient and can be viewed as one of the main results of the work. 3. Calculation results and discussion 3.1. Dresselhaus parameter in the symmetric quantum well of different width Let us consider first a single symmetric In xGa 1–xAs/GaAs QW grown along the direction []001 . Kane Hamiltonian parameters for In xGa 1-xAs semiconductor alloys were taken from Ref. [45]. Because in symmetric structures, due to parity , the intrasubband SO interaction contains only the Dresselhaus cont ribution , therefore we calculate this parameter for each subband with different QW width d. In the simplest approximation of infinitely deep well the Dresselhaus parameter has following form (see, e.g. , [53]): 222 dnD nγπβ= . (18) According to this approximation the Dresselhaus term monotonically decreases with QW width for all quantum subbands . If we solve this problem numerically , by means of method described in 2.4, we get a different result for the Dresselhaus parameter with increasing width QW. Fig ure 1 demonstrates the dependence of the Dresselhaus parameter absolute value on QW width calculated for three lower QW subbands. It is evident that only for the first subband this parameter behaves according to the e quation (18), i.e. monotonically drops with increasing QW width. As the subband index growth , the dependence ()dnβ becomes complicated, moreover, for the third subband w e observe a change in the sign of the Dresselhaus parameter at a certain value of the well width . Journal XX (XXXX) XXXXXX V E Degtyarev et al 6 5 10 15 20 25 30 350.00.10.20.30.40.5 |β| [meV⋅nm] QW width [nm] |β1| |β2| |β3| Figure 1. Absolute value of the Dresselhaus parameter for InxGa 1–xAs/GaAs ( 4.0=x ) QWs , calculated for three electron subbands as a function of QW width. One reason of such a behaviour is incorrectness of equation (18) for finite -depth QWs . In reality the electron wave function is not completely localized inside the QW but penetrates into barriers layers [53]. Moreover, t he fraction of this penetration depends significa ntly on the heterostructure materials parameters and system’s geometry. 3.2. Dresselhaus parameter in the symmetric quantum well with variable In content In this section we suggest an explanation of t he non-trivial behavior of the Dresselhaus parameter )(dnβ for different subband index, QW width and i ndium content. According to equation (5) )(dnβ depends on Dresselhaus constants both in well and barrier regions. Difference in the well and barrier materials leads to the electron density redistribution between them and, as a consequence, changing of contribution s to the Dresselhaus parameter. It is evident that the resultant Dresselhaus parameter should be proportional to the wave function fraction multiplied by corresponding bulk Dresselhaus constant )(zDγ both of the well and the barrier materia l. In other words, when calculating parameter nβ, one need to take into account the real value )(zDγ for each coordinate point of the structure . As known for InxGa 1-xAs solid solutions , the bulk Dresselhaus constant Dγ varies drastically with indium content, hence it can differently contribute to resultant value of the Dresselhaus parameter and lead not only to the value chan ging but also to the sign inver sion [54]. Consequently, the Dresselhaus par ameter can unusually behave with simultaneo us electron subband index and QW width increasing [55]. The indium content varying results in changing of band offset s at the interface, therefore the effective depth of the well changes for all subbands. However, behavior of the Dresselhaus parameter for the third subband looks the most intriguing (figure 1), so let us consider it in more detail. At figure 2 there are shown the dependence of the Dresselhaus parameter modulus 3β on the QW width for several values of In content. The dependence is nonmonotonic with maximum for QW width 12~ nm. Moreover, the Dresselhaus parameter substantially decreases with increase of In content x. Such a behaviour i s due to change of Dγ in the well region, including its sign. B esides, the linear -in-k Dress elhaus parameter equals zero in QWs with 36.0>x at some certain QW width . 10 15 20 250,00,10,20,30,40,5 x=0.4 x=0.38 |β3| [meV⋅nm] QW width [nm] x=0.36 x=0.34 x=0.32 KP approximation Figure 2. Absolute value of the Dresselhaus parameter calculated for third electron subband (3β) as a function of QW width with variable In content x. a. b. Figure 3. (a) The well region component ()A 3β of the Dresselhaus parameter for third electron subband calculated using single -band effective mass aprroximation. (b) Total (sum of the well and barrier regions ) value 3β as a function of QW width for third electron subband . x stands for In content in well layer . In order to estimate the Dresselhaus contribution for each regions (well and barrier), we split the envelope function into two fractions: corresponding to the well layer region (A) and barrier region (B). For symmetric QW it is sufficient to make 10 12 14 16 18 20-0,04-0,020,000,020,040,06 x(In)=0.34 x(In)=0.35 x(In)=0.32 x(In)=0.33 QW width [nm] x(In)=0.36 x(In)=0.37 x(In)=0.38β3(A) [meV⋅nm] 10 12 14 16 18 20-0,04-0,020,000,020,040,060,080,10 β3 [meV⋅nm] QW width [nm] x=0.32 x=0.34 x=0.36 x=0.38 x=0.4EM approximationJournal XX (XXXX) XXXXXX V E Degtyarev et al 7 calculati ons in the framework of the effective mass approximation [56 ]: such an approach gives qualitative description. The Dresselhaus parameter ()dnβ for the nth subband can be presented as a sum of two contributions: ()() ()()()(),, ,22 , ,∫∂∂=+= + BAn nBA DBA nB nA n n zzz dz ψψ γββββ (19) where ()A nβ and ()B nβ correspond to In xGa 1-xAs well and GaAs barrier regions, respectively, ()znψ is the envelope function obtained in the framework of the single -band effective mass method . According to our estimates, the quantitative error of the effective mass method in comparison with the Kane model (or, in other words, the error of the averaging procedure based on the effective mass approximation described in detail in Sec. 2 ) for In0.4Ga 0.6As/GaAs QW is about 1.5 times for the first electron subband and about 3 times for the third subband. Figure 3(a) shows the Dresselhaus parameter ()()dA 3β of the A region and figure 3(b) demonstrates the total value ()d3β as function of QW width for the third subband with different In content. Here we are taking into account that the penetration of the wave function into the barrier regions varies both with the well width and with subband index. Obviously the third subband is characterized by more significant electron density redistribution and wave function penetration into the barrier. It is seen that A -region contribution is nonmonotonic with QW width and increasing for any indium content. We believe that it occurs due to the simultaneous superposition of two physical effects: the barrier wave function part (B) is increasing but the quantum state average kinetic energy is decreasing. In other words, as the subband index increases, the kinetic energy becomes smaller, but th e penetration into the barrier increases . Summing the two separately calculated well/barrier components according to ( 19) we obtain )(3dβ dependence, which is also nonmonotonic. Moreover , in InGaAs the bulk Dresselhaus constant changes its sign from some indium content. It is seen that this dependence ( figure 3(b)) qualitatively coincides with the behavior of the parameter 3β (at the corresponding In content) calculated in the framework of the 8 -band k·p-model (figure 2). Thus, the results of multi -band calculations and effective mass method show, that for higher QW subbands the averaged Dresselhaus parameter can vary within wide range, turning to zero at the definite com binations of QW parameters. It should be noted that obtained result is specific for peculiar QW materials InxGa 1-xAs/GaAs because bulk Dresselhaus constant Dγ strongly varies from well to barrier. In particular, the same Dresselhaus parameter behavior is unreachable for the heterostructures based on GaAs/GaAsSb materials because bulk Dresselhaus constant Dγ has the same sign at any solid solution contents. 3.3. PSH at the asymmet ric QW potential, caused by the transverse electric field Further we consider asymmetric In xGa 1–xAs/GaAs QW potential profile with different width, in which both Dresselhaus and Rashba SOC are present. Asymmetric potential is obtained by means of an external electric field ( 4.00÷=E mV/nm) applied in the structure growth direction. Solving the equation (7), we calculate the energy spectrum taking into account SOC, described by superposition of Rashba and Dresselhaus effects. In this Section our aim is to reveal the electric filed range that satisfy the conditions for the PSH regime realization [ 17, 18] with different QW width and In content. It is known the PSH state corresponds to the numerical equality of Rashba and Dresselhaus constant s (n nβα= ) (see f igure 4). 0 2 4 60.00.51.01.52.02.5 α3α2 β3β2 SOC parameters [meV ⋅nm] Electric field strength [mV/nm] 1st subband 2nd subband 3rd subband β1α1 Figure 4. Rashba and Dresselhaus parameters (nα and nβ) dependence on the electric fiel d strength for thr ee lower electron subbands in In xGa 1–xAs/GaAs QW with the In content 4.0=x and well width 10=d nm. By means the method, described above, we calculate SOC parameters in In xGa 1–xAs/GaAs QW structure for all subbands as a function of applied transverse electric field (figure 4). It is obvious that PSH state for the first subband can be realized with a small In content in the QW ( x is about 0.2). At the same time it is not difficult to sho w that increasing of indium content in QW should lead to possibility of PSH formation for excited subbands. Journal XX (XXXX) XXXXXX V E Degtyarev et al 8 0.10 0.15 0.20 0.25 0.30 0.35 0.401234567 PSH field [mV/nm] In content d=5 nm d=7 nm d=10 nm d=15 nm Figure 5. PSH field values for the first electron subband ( 1 1βα= ) as a function of indium content in In xGa 1– xAs/GaAs QW with different QW width 15,10,7,5=d nm. Thus, for QW with a large In content ( x is about 0.4), applying an electric field one can obtain a PSH regime with various combinations of heterostructure geometry and In content. At figure 4 it is depicted the Rashba and Dresselhaus parameters crossing points n nβα= , corresponding to PSH states for three electron subbands. One can see from this figure, that the parameter nα induc ed by asymmetry structure is almost linear function of the external field and is almost independent on the width well [5 7]. The Dresselhaus parameter , in its turn, is not vary with the increasing field and is determined, in principal, by subband index, In content and QW width (Sec. 3.1). Since the Rashba parameter is well controlled by the electrical field ( figure 4), we can draw a conclusion, that for PSH realization one can adjust it by gate voltage for any Dresselhaus parameter determined by the structure itself . Also, we identified that the Dresselhaus parameter nonmonotonic dependence on the QW width with the subband index increasing results in a nontrivial behavior of the PSH field (the electric field that allows us to realize the PSH condition ). From figure 4 it is seen, that the PSH field required for the third subband (red crossing point) has lesser value than it for second subband (green crossing point) at a given quantum well width. Alternating Dresselhaus parameter behavior for the third subband is very interesting result from fundamental point of view. But, senso stricto , experimentally observed QW electronic properties are mostly determined by the first subband, which is typically populated with majority of carriers. Therefore, we limit f urther our considerations to the simulation of QW electronic properties for only one subband. In particular, one of the aims of this research is to show that even for the first subband we can create the PSH regime in the wide range of QW material and geometric parameters at the real electric field strength. Smoothly varying the indium content and QW width let us observe how electric field the required for PSH symmetry point 1 1βα= is changing (see figure 5). Fig ure 5 shows that the PSH field almost linearly decreases with In content increasing (the 10% shift of In content leads to the change of PSH field around 0.5 mV/nm). Note also, that for all considered QWs the relation between the PSH field and In content depends weakly on the QW width. It is also shown that for fixed indium content, PSH field decreases rapidly with QW width growth. 7.07.58.08.59.0 9.5 10.02345 0.100.150.200.250.30α 1=β1 [meV ⋅nm] PSH field [mV/nm]QW width [nm] Figure 6. PSH SOC parameters, calculated for the first electron subband (1 1βα= ) as a function of electric field strength in In xGa 1–xAs/GaAs QW with variable indium content ( 4.0...1.0=x ) and QW width 10,5.8,7=d nm. Figure 6 demonstrates a summary of our calculations for asymmetric QWs. On this figure one can see the first subband SOC parameter (the Rashba parameter equals the Dresselhaus one) as a function of the PSH field for different In contents. Thus, by combining the QW width and In content varying we can realize the PSH regime with different precession frequency. It is necessary to emphasize that calculated PSH fields can be easily obtained experimentally, that is important for possible applications. 4. Conclusion By direct numerical calculations, a technique has been proposed that makes it possible to extract the individual contribution of the Dresselhaus and Rashba SOC parameters for the QWs conduction band of based on III -V semiconductors with a zinc blende structure in the presence of a transverse uniform electric field. The calculations carried out in this paper show that the asymmetry of the potential contributes to an increase in the Journal XX (XXXX) XXXXXX V E Degtyarev et al 9 spin splitting for all electron subbands in the QW . The Rashba parameter calculated in the framework of the 8 -band Kane model depends linearly on the applied electric field, remaining practically insensitive to the width of the QW and the index of the size -quantized subband. In turn, the Dresselhaus parameter is practically independent on the external applied field, but it varies significantly with the subband index and with the well width. Taking into account the redistribution of the electron wave function between the well and the barrier regions in the symmetric finite -depth QW, we explain the nonmonotonic Dresselhaus parameter dependence on the electron subband index and, in particular, its alternating behavior with QW width for third subband. Moreover , as a specific result we obtain the possibility to eliminate the linear -in-k Dresselhaus term in QW at the certain parameters. We have also demonstrated that applying of external transverse electric field to In xGa 1–xAs/GaAs QW allows one to effectively control the ratio between the Rashba and Dresselhaus parameters. The requirement of n nβα= can be fulfilled by the variation of both Rashba and Dresselhaus terms, which in turn d epend on a variety of parameters, such as subband index, electric field strength (external gate voltage), QW materials and structural design. Thus, by combining the configuration of the structure and the external electric field, one can realize the PSH states. It should be noted both SOC parameters are highly sensitive to the heterostructure material kind. We showed earlier [5 8] that for another pair of heteromaterials (for example InGaSb/GaSb), the equality of the Rashba and Dresselhaus constants is extrem ely difficult with available electric fields. Our results can be used for design of semiconductor spintronics structures with required characteristics. 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1505.02530v2.Current_induced_spin_polarization_and_spin_orbit_torque_in_graphene.pdf

arXiv:1505.02530v2 [cond-mat.mes-hall] 11 Oct 2015Current-induced spinpolarizationandspin-orbit torque i ngraphene A. Dyrdał1and J. Barna´ s1,2 1Faculty of Physics, Adam Mickiewicz University, ul. Umulto wska 85, 61-614 Pozna´ n, Poland 2Institute of Molecular Physics, Polish Academy of Sciences , ul. M. Smoluchowskiego 17, 60-179 Pozna´ n, Poland (Dated: October 13, 2015) Using the Green function formalism we calculate a current-i nduced spin polarization of weakly magnetized graphene with Rashba spin-orbit interaction. In a general c ase, all components of the current-induced spin polarization are nonzero, contrary to the nonmagnetic limi t, where the only nonvanishing component of spin polarization is that in the graphene plane and normal to elec tric field. When the induced spin polarization is exchange-coupled to the magnetization, it exerts a spin-or bit torque on the latter. Using the Green function methodwehavederivedsomeanalyticalformulasforthespin polarizationandalsodeterminedthecorrespond- ing spin-orbit torque components. The analytical results a re compared with those obtained numerically. Vertex correctionsduetoscatteringonrandomlydistributedimpu ritiesisalsocalculatedandshowntoenhancethespin polarizationcalculated inthe bare bubble approximation. PACS numbers: 72.20.My, 72.80.Vp,72.25.-b I. INTRODUCTION One of the main issues of the present-dayspin electronics, that is of great importance for further development of high- density memory devices and magnetic random access mem- ories, is effective manipulation of magnetization by a spin - polarized current. The general idea of switching and contro l- ling orientation of a magnetic moment with electric current flowing through a system is based on coupling between the electron spin and magnetic moments. Two such interactions turned out to be especially useful – exchange interaction an d spin-orbitcoupling. In a magnetically nonuniform system, the spin-polarized current generates a torque that is a consequence of: (i) ex- change coupling between the conduction electrons and mag- netization, and (ii) conservationof angular momentum in th e system. The torque appears then as a result of the spin an- gular momentum transfer from a spin-polarized current (or pure spin current) to magnetic moments. Therefore, this torque is called spin-transfer torque.1,2Such a torque leads, among others, to magnetic switching in spin vales and to do- mainwalldisplacements,asobservedrecentlyinmanyexper - iments. Moreover,thesephenomenagiveapossibilitytocon - struct low-power non-volatile memory cells (STT-MRAM, racetrack memory), integrated circuits employing a logic- in- memory architecture as well as logic schemes processing in- formationwith spins.3 Another possibility to control orientation of magnetic mo- mentsis based on a spin torquethat appearsdue to spin-orbit interaction in the system. The corresponding torque exerte d on the magnetization is usually referred to as the spin-orbi t torque, and appears also in a magnetically uniform system, like a single uniform layer. Physical mechanism of the spin- orbit torque is based on a nonequilibrium spin polarization of the system, which is induced by an external electric field (current)inthepresenceofspin-orbitinteraction. Sucha spin polarizationwaspredictedlongtimeagoinnonmagneticsys - tems, where an electric current flowing through the system with spin-orbit interaction was shown to induce not only the transversespin-current4,5(so-calledspinHall effect),but alsoa spin-polarizationof conductionelectrons.6–11In the case of two-dimensionalelectron gas with Rashba spin-orbit inter ac- tion, the induced spin polarization is in the plane of the ele c- tron gas and normal to the electric field. Such a nonequilib- riumspin-polarizationmaybetreatedasaneffectivemagne tic field, which may lead to reorientationof a magneticmoment, and also can modifyor induce magneticdynamics. The spin- orbittorquewasanalyzedinrecentfewyearsinmanypapers, mainly in metallic and semiconductor heterostructures.12–17 While the current-induced spin polarization, known also as the inverse spin-galvanic effect,18is well known and was in- vestigatedtheoreticallyaswellasexperimentallyinther ecent three decades, the role of geometric phase in this effect, an d consequently in the spin-orbit torque, was invoked only ver y recently.19–21 Inthispaperweconsiderthecurrent-inducedspinpolariza - tion and spin-orbit torque in graphene, which is assumed to be deposited on a substrate that ensures the presence of spin - orbit interaction of Rashba type22. We also assume that the grapheneismagnetized,whichmaybeeitherduetothe mag- netic proximity effect to a ferromagnetic substrate (or cov er layer), or due to magnetic atoms (nanoparticles) on its sur- face.23–27CoexistenceoftheRashbaspin-orbitinteractionand proximity-inducedmagnetismingraphenewaspredictedthe - oretically and also observed experimentally.27–31As the spin transfer torque in ferromagnetic graphene junctions was al - ready considered theoretically (see e.g. Yokoyama and Lin- der32), the problem of spin torques induced by spin-orbit in- teractioningrapheneis ratherunexplored. It has been shown that the current-induced spin polariza- tioninadefect-freenonmagneticgraphenewithRashbaspin - orbit interaction is oriented in the graphene plane and is al so normalto the currentorientation. Moreover,sign of the spi n- polarization depends on the chemical potential and also on thesignoftheRashbaspin-orbitcouplingparameter.33When the Fermi level passes through the Dirac points, the spin po- larization becomes reversed. In this paper we show that the current-induced spin polarization in magnetic graphene ha s generally all three components. In the approximation linea r withrespecttothemagnetization,oneofthesecomponentsi s2 equal to that in the case of a nonmagnetic graphene, i.e. it is proportional to the relaxation time. The leading terms in th e othertwo componentsareindependentofthe relaxationtime . The paper is organized as follows. In Sec. 2 we describe the model and present a general formula describing current- induced spin polarization. Analytical formulasas well as n u- merical results for the current-induced spin polarization are presentedin Sec. 3. Vertexcorrectionis calculated in sect ion 4, while the spin-orbit torque is described and discussed in Sec. 5. SummaryandfinalconclusionsareinSec. 6. II. MODELANDMETHOD Transport properties of graphene close to the charge neu- tralitypointaredeterminedmainlybyelectronsinthevici nity of Dirac points. The corresponding effective-mass Hamilto - nian,H0 K,whichdescribesthelow-energyelectronicstatesin graphene around the Kpoint of the Brillouin zone, can be writtenasasum ofthreeterms,34 H0 K=H0+HR+HM. (1) The first term, H0, describes the low energy electronic states of pristine graphene, and can be written as a matrix in the pseudospin(sublattice)space, H0=v/parenleftbigg 0 ( kx−iky)σ0 (kx+iky)σ00/parenrightbigg ,(2) wherev=/planckover2pi1vF, withvFdenoting the electron velocity in graphene, which is constant. The second term in Eq. (1) de- scribesthe Rashbaspin-orbitinteractionduetoa substrat e, HR=λ/parenleftbigg 0σy+iσx σy−iσx0/parenrightbigg , (3) withλbeing the Rashba spin-orbit coupling parameter. The last term of the Hamiltonian (1) represents the influence of an effective exchange field ˜Mcreated by a nonzero magne- tization. Such a magnetization can appear in graphene, for instance, due to the proximity effect to a magnetic substrat e. Thistermcanbewrittenin theform, HM=−˜M·/parenleftbigg σ0 0σ/parenrightbigg , (4) wheretheexchangefield ˜Mismeasuredinenergyunits. This field can be related to the magnetization Mand the local ex- changeinteractionbetweentheconductionelectronsandma g- netization in the two-dimensional graphene, Jex(r−r′) = Jexδ(r−r′),viathe formula ˜M= (Jex/2gµB)M, Here,g is the Lande factor ( g= 2),µBis the Bohr magneton, while positive and negative Jex(measured in the units of Jm2) cor- respondto antiferromagneticand ferromagneticcoupling, re- spectively. In the above equations, σis the vector of Pauli matrices, σ= (σx,σy,σz), while the matrix σ0denotes the unit matrix in the spin space. Note that the so-called intrin sic spin-orbit interaction in graphene is very small and theref ore itisneglectedinourconsideration. Inageneralcase,them ag- netization vector Mmay be oriented arbitrarilyin space, and FIG.1. (Color online) Schematic of the system under conside ration. Graphene is on a substrate which assures a nonzero magnetiza tion and also a spin-orbit interaction of Rashba type. Orientati on of the magneticmoment Misdescribedbytheangles θandξ. Anexternal electricfieldisoriented along the axis y. itsorientationwillbedescribedbytwosphericalangles, θand ξ,asindicatedinFig.1. Moreover,theabsolutemagnitudeof Mis assumed to be constant, |M| ≡M= const. Hamil- tonian for the second non-equivalentDirac point, K′, can be obtained from HKby reversing sign of the wavevector com- ponentkxandsubstitution σy→ −σyinHR. In the lowest order with respect to the exchange field ˜M, the casual Green function corresponding to the Hamiltonian (1),G0 k={[ε+µ+iδsign(ε)]−H0 K}−1has poles at ε= En−µ−iδsign(ε),whereEn(n= 1−4)areeigenvaluesof theHamiltonian(1)withoutthe term HM. These eigenvalues havethe followingform: E1,2=∓λ−/radicalbig k2v2+λ2 (5) E3,4=∓λ+/radicalbig k2v2+λ2, (6) whereE1,2correspond to the valance bands, while E3,4de- scribetheconductionbands. Note,thebandscorresponding to n= 2andn= 3toucheach otherat the Dirac point( k= 0), whileagapequalto 4λappearsbetweenthebands n= 1and n= 4. Inthepresenceofadynamicalexternalelectricfieldapplie d along the axis y, the total Hamiltonian for electrons near the Kpointtakestheform H=H0 K+HA K, (7) wherethe secondterm, HA K=−eˆvyAy(t) =−iev /planckover2pi1/parenleftbigg 0−σ0 σ00/parenrightbigg Ay(t),(8) is the perturbationdue to interactionwith the time-depend ent electromagnetic field represented by the vector potential Ay(t) =Aye−iωt. Here,eis the electroncharge, ˆvyis they- component of the electron velocity operator, ˆv=∂H0 K/∂k, whereasωis the frequency of the dynamical field (later we will takethelimit of ω→0). Whenanelectriccurrentflowsinthesystemduetotheelec- tricfield,electronspinsbecomepolarizedasaresultofthe co- operationofthe currentand Rashbaspin-orbitcoupling. Th is nonequilibriumspin polarization of conductionelectrons can becalculated(inthezero-temperaturelimit)usingthefol low- ingformula: Sα(t) =−iTr/integraldisplayd2k (2π)2ˆSαGk(t,t′)|t′=t+0,(9)3 whereGk(t,t′)isthezero-temperaturecausalGreenfunction correspondingtothetotal Hamiltonian H(see Eq.7),and ˆSα isthespinvertexfunctiondefinedas ˆSα=/planckover2pi1 2/parenleftbigg σα0 0σα/parenrightbigg . (10) UponFouriertransformationwithrespecttothetimevariab les and expansion in a series with respect to the vector potentia l Ay=−iEy/ω, the expression (9) for the induced nonequi- libriumspindensitytakestheform Sα(ω) =eEy ωTr/integraldisplayd2k (2π)2/integraldisplaydε 2πˆSαG0 k(ε+/planckover2pi1ω)ˆvyG0 k(ε).(11) Inthedc limit, ω→0,theaboveformulaleadsto thefollow- ingexpressionforthespinpolarization: Sα=e 2πEy/planckover2pi1Tr/integraldisplayd2k (2π)2ˆSαG0R kˆvyG0A k,(12) whereG0R(A) kis the retarded(advanced)Green functioncor- responding to the unperturbed Hamiltonian (1), taken at the Fermi level ( ε= 0). Upon taking into account Eqs (10) and (12), and also includingthe contributionfromthe secon d Dirac point, the expression for the induced spin polarizati on acquiresthe form Sα=e/planckover2pi12 2πEyTr/integraldisplayd2k (2π)2/parenleftbigg σα0 0σα/parenrightbigg G0R kˆvyG0A k.(13) Basedonthisformulawecalculateanalyticallyaswellasnu - merically the current-induced spin polarization, as descr ibed anddiscussedinthesubsequentsection. III. CURRENT-INDUCEDSPINPOLARIZATION From the general formula (13) one finds the following ex- pressionforthe α-th componentofthespin polarization: Sα=e/planckover2pi1 2πEy/integraldisplaydkk (2π)2Tα Π4 n=1(µ−En+iΓ)(µ−En−iΓ), (14) whereTαisdefinedas Tα=/planckover2pi1/integraldisplay2π 0dφTr/bracketleftbigg/parenleftbigg σα0 0σα/parenrightbigg g0R kˆvyg0A k/bracketrightbigg .(15) Here,g0R(A) kis the nominator of the retarded (advanced) Greenfunction, φstandsfortheanglebetweentheaxis xand thewavevector k,whileΓ =/planckover2pi1/2τ,whereτisthemomentum relaxation time. The parameter Γ(or equivalently relaxation timeτ)willbetreatedhereasaphenomenologicalparameter, whicheffectivelyincludescontributionsduetomomentumr e- laxation from various scattering processes (scattering on im- purities,otherstructuraldefects,phonons,orelectron- electron scattering). Note that Γdepends in general on the chemical potential µand may be also different in the two Rashba sub- bands. However,whentheFermilevelisinthetwosubbands, we assumeforsimplicitythesame Γforbothofthem.Up to the terms linear in the exchange field ˜M, the func- tionsTα(α=x,y,z),seeEq.(15),canbewrittenasfollows: Tx= 16λπvµ(k4v4−µ4+4λ2µ2),(16a) Ty= 64πvλ˜Mµ(k2v2−2λ2)Γsinθ,(16b) Tz=−64πλ˜Mv3k2µΓcosθsinξ.(16c) Note, the dependenceon the orientationof Mis containedin theaboveexpressionsfor TyandTz, whileTxisindependent ofM. Equations(14)and(16)allowfindingspinpolarization in a general case, i.e. for an arbitrary relaxation time. How - ever, some analytical expressions for all components of the spin polarizationcan be obtained in the limit of low impurit y concentration,i.e. forlongrelaxationtimes( τ→ ∞). Consider first the x-component of the spin polarization. CombiningEq.(16a)withEq.(14)andmakingthesubstitutio n√ k2v2+λ2=γ,oneobtains Sx= 8e/planckover2pi1Eyλµ ×/integraldisplay∞ λdγγ v(2π)2(γ2−λ2)2−µ4+4λ2µ2 [(µ+λ+γ)2+Γ2][(µ−λ+γ)2+Γ2] ×1 [(µ+λ−γ)2+Γ2][(µ−λ−γ)2+Γ2].(17) Fromthisformulafollowsthat Sxisindependentof ˜Min the linear approximation with respect to the exchange field. For long relaxation times we get the same analytical formulas as thoseinthecase ofnonmagneticgraphene,33i.e., Sx=e 4π2λ±µ v(λ±µ)µEyτ (18) forthe Fermilevellyingintherange −2λ < µ < 2λ,and Sx=±e 4π2λ v(µ2−λ2)µ2Eyτ (19) for|µ|>2λ. In both above equations (as well as below), the upper and lower signs correspond to µ >0andµ <0, respectively. ThespinpolarizationgivenbyEqs(18)and(19) ispropor- tional toτ. However, one should bear in mind that these for- mulas were derived on the assumption of long τ. Therefore, one may expectsome deviationsfrom this formulawhen τis finite and not too long. In Fig. 2(a) we show variation of the Sxcomponentofspinpolarizationwiththechemicalpotential µand relaxationtime τ, obtainedby numericalintegrationof the formula (17). Figures 2b and 2c, in turn, present cross- sections of the density plots shown in Fig. 2(a) for constant values of τandµ, respectively. The results obtained from the analytical formulas are compared in Figs 2(b) with those obtained by numerical integration of the formula (17). From this comparison follows that for τof the order of 10−11s or smaller, there are some deviations from the results given by theanalyticalformulas,thoughthese deviationsare notla rge. Forτoftheorderof 10−10sorlonger,numericalresultsmatch quite well those obtainedfrom the analytical formulas. Sin ce4 FIG. 2. (Color online) Spin polarization components induce d by a current: (a,b,c) show the x-component, (d,e,f) show the y-component, whereas (g,h,i) show the z-component. The top panel (a,d,g) shows the spin polarizati on components as a function of chemical potential µ and relaxation time τ. The medium panel (b,e,h) shows the spin polarization compo nents as a function of chemical potential µfor indicated values of τ, while the bottom panel (c,f,i) shows the polarization comp onents as a function of τfor indicated values of µ. The right parts of (d,e,f)present the corresponding shaded regions inthe lef tparts. The solidanddashed lines in(b) represent the resul tsbased onthe analytical formulas andnumerical integration,respectively. The cur ves forτ→ ∞in(e,h)correspond toanalyticalsolutions. Theother para meters are: λ= 2meV,Ey= 1V/cm,˜M= 0.1meV,θ=π/3, andξ=π/2. theSxcomponent is the same in magnetic and nonmagnetic limits (within the approximationsused here), and in the non - magneticlimititwasconsideredandanalyzedinRef.[33],w e will notdiscussthiscomponentinmoredetail. From Eqs (14) and (16b) one finds the y-componentof the spinpolarizationin thefollowingform: Sy= 32eEy/planckover2pi1λµ˜M ×/integraldisplay∞ λdγγ v(2π)2(γ2−3λ2)Γsinθ [(µ+λ+γ)2+Γ2][(µ−λ+γ)2+Γ2] ×1 [(µ+λ−γ)2+Γ2][(µ−λ−γ)2+Γ2].(20) In the limit of slow relaxation, Γ→0, the above formula leadsto thefollowinganalyticalresults: Sy=±e/planckover2pi1 4π˜M λsinθµ(µ±2λ)−2λ2 2vµ(µ±λ)Ey(21) for|µ|<2λ, and Sy=±e/planckover2pi1 4π˜M λsinθµ2−4λ2 v(µ2−λ2)Ey (22)for|µ|>2λ. Numerical results for the y-component of the current- induced spin polarization, obtained by numerical integrat ion of the formula (20) are shown in Fig.2(d) as a function of chemical potential µand relaxation time τ. Figures 2(e,f) present cross-sections of Fig.2(d). Figure 2(e) additiona lly shows the results obtained from analytical formulas, see th e curvesfor τ→ ∞. RightpartsofFig.2(d,e,f)presentinmore detail the corresponding shaded regions. Similarly as the x- component, Syis antisymmetric with respect to reversal of the sign of Fermi energy, and its dependence on µalso re- veals some steps at µ=±2λ. These steps are associated with the edges of the bands E1andE4. Moreover, when the Fermi level is at the Dirac point ( µ= 0), the analytical so- lution (21) for Sybecomes divergent. To understand origin of the divergency in the analytical solution for τ→ ∞, one shouldnotethat the solutionforthe x-componentis also infi- nite forτ→ ∞, independentlyof µ. This clearly shows that thelimitof τ→ ∞isnotphysicalasthedissipationprocesses are necessary in order to stabilize a finite current-induced de- viation of the system from equilibrium, and thus also a finite5 current density and spin polarization. Therefore, in Fig.2 (e) wecomparethenumericalresultsbasedonthecorresponding analyticalformulaswiththoseobtainedbynumericalinteg ra- tion. This comparisonclearly shows that the results obtain ed from the analytical formulas are roughly in agreement with thoseobtainedfromnumericalintegration,exceptthevici nity ofµ= 0, where the analytical solution diverges for µ→0, while the numerical results based on Eq.(20) are then finite. Moreover, some discrepancy also occurs around µ=±2λ, but now the difference is finite and rather small. Thus, one should bear in mind that the analytical results (21) and (22) forthey-componenthave limited applicabilityrange,and are notapplicablefor µin thevicinityoftheDiracpoints. TheSzcomponent can by found from Eqs (14) and (16c) andacquirestheform Sz=−32eEy/planckover2pi1λµ˜Mcosθsinξ ×/integraldisplay∞ λdγγ v(2π)2(γ2−λ2)Γ [(µ+λ+γ)2+Γ2][(µ−λ+γ)2+Γ2] ×1 [(µ+λ−γ)2+Γ2][(µ−λ−γ)2+Γ2].(23) Similar calculations as those done for the y-component lead to the followinganalytical expressionsin the limit of long re- laxationtime: Sz=∓e/planckover2pi1 4π˜Mλcosθsinξµ±2λ 2v(µ±λ)Ey(24) forµ <2λ, and Sz=∓e/planckover2pi1 4π˜M λcosθsinξµ2−2λ2 v(µ2−λ2)Ey(25) forµ >2λ. In Fig.2(g) we present the z-component of the current- induced spin polarization, calculated as a function of the chemical potential and relaxation time by numerical integr a- tion of the formula (25). In turn, Figs.2(h,i) show cross- section of Fig.2(g). In Fig.2(h) we additionally compare th e numerical results with those obtained from analytical solu - tion. Now, the analytical solution is not divergent, see the curve for τ→ ∞. When the relaxation time is sufficiently small, the numerical results obtained from Eq. (23) deviate fromtheresultsobtainedonthebasisoftheanalyticalform u- las. These deviationsare rather small for τ/greaterorsimilar10−11s, except theregionnearthe zerochemicalpotential. However,thedi f- ference between the analytical and numerical results aroun d µ= 0isnowmuchless pronouncedthanit wasin thecaseof they-component(compareFig.2(e) and Fig.2(h). In turn, for τ/lessorsimilar10−11s the deviations become remarkable in the whole rangeofthechemicalpotentialsshowninFig.2(h). All the components of the spin polarization ( Sx,Syand Sz) vanish at µ= 0and are antisymmetric with respect to the sign reversal of the chemical potential. Numerical resu lts presented above show that the spin polarization strongly de - pends on the Fermi level position. In the close vicinity of theDiracpoints,the y-componentofthespinpolarizationhas pronouncedpeaks(positiveaboveandnegativebelow µ= 0).The other two components behave more regularly in this re- gion. Allthreecomponentsexhibitsomecusps(ordips)when µisinthevicinityof µ=±2λ,i.e.,whentheFermilevelap- proaches the top edge of the band E1or bottom edge of the bandE4. The spin polarization also remarkably depends on the Rashba parameter λ. This dependence reveals peculiari- tiesofthecorrespondingelectronicstructure,andremark ably depends on the Rashba parameter. In numerical calculations we assumed the Rashba spin-orbit couplingparameter λ= 2 meV. Generally, this parameter depends on the substrate (or cover layer), and in real systems varies from a few to a few tensofmeV,see eg. Refs[35–40]. IV. VERTEXCORRECTION In the preceding section we have calculated spin polariza- tion induced by electric field assuming effective relaxatio n timeτ(or equivalentlyrelaxation rate Γ). Both,τand chem- ical potential were treated there as independent parameter s. When consideringa specific relaxation mechanism, these pa- rameters usually are not independent. Since the dominant scattering processes are on impurities, we consider now thi s problem in more details. Assume the scattering potential created by randomly distributed weak short-range scattere rs, whichmaybewrittenas V(r)s0σ0withGaussiancorrelations /an}bracketle{tV(r)V(r′))/an}bracketri}ht=niV2δ(r−r′)(wheres0andσ0and de- noteunitmatrixinthepseudo-spinandspin subspacerespec - tively). Detailed calculation of the self energydue to scattering on the point-like impurities gives Γ1,4=niV2 2v2(|µ| −λ)and Γ2,3=niV2 2v2(|µ|+λ), whereniistheimpurityconcentration whileVis the impurity scattering potential. When |µ| ≫λ, then indeed Γ1,4≃Γ2,3≡Γ. Otherwise, we take Γas the averageof Γ1,4andΓ2,3,i.e.Γ =niV2 2v2|µ|. When calculating the impurity averaged conductivity, it is wellknownthatnon-crossingdiagramsgiveanimportantcon - tributionandrenormalizetheresultsobtainedinthe barebub- bleapproximation. Such a vertex renormalization is known to have a significant influence on the spin current induced viathe spin Hall effect. In the case of two-dimensional electron gas with Rashba spin-orbit interaction it totally can- cels the spin Hall conductivity obtained in the bare bubble approximation.41–44However, this is not a general property and in other systems the vertex corrections can only reduce partlythespin Halleffect.45–47 Theproblemofdisorderingraphenewasdiscussedinmany papers.48–50However,thereisstillalackofinformationonthe influenceof disorderand impuritieson spin-orbit drivenph e- nomena in graphene. This problem was raised by Sinitsyn et al.51and Gusynin et al.52in the contextof spin Hall and spin Nernsteffectinthepresenceofintrinsicspin-orbitinter action in graphene and in the case of spin-independent random po- tential. In this case problem becomes simpler because one can reduce the model to 2×2space. Such a simplification, however, is not possible in the presence of Rashba spin-orbi t interaction.6 Intheweakscatteringlimit,thelocalizationcorrections are vanishingly small and therefore only noncrossing ladder di a- gramsareimportant. Thesummationovertheladderdiagrams can be represented by the vertex corrections to the current- inducedspinpolarization. Therenormalizedspinvertexfu nc- tionisthengivenbythe followingequation:53 ˜Sα=ˆSα+niV2/integraldisplayd2k (2π)2GA k˜SαGR k,(26) whereˆSαis definedby Eq. (10). For the point-likescattering potentialonecanpostulatethe vertexfunction ˜Sαintheform ˜Sα=aα/planckover2pi1 2/parenleftbigg σx0 0σx/parenrightbigg +bα/planckover2pi1 2/parenleftbigg σy0 0σy/parenrightbigg +cα/planckover2pi1 2/parenleftbigg σz0 0σz/parenrightbigg +dα/planckover2pi1 2/parenleftbigg σ00 0σ0/parenrightbigg ,(27) forα=x,y,z, whereaα,bα,cα, anddαare certain param- eters to be determined. To find these parameters we multiply Eq.(26)bythematrixasspecifiedbelowandtake thetrace, Tr/braceleftbigg/parenleftbigg σi0 0σi/parenrightbigg ˜Sα/bracerightbigg = Tr/braceleftbigg/parenleftbigg σi0 0σi/parenrightbigg ˆSα/bracerightbigg +niV2/integraldisplayd2k (2π)2Tr/braceleftbigg/parenleftbigg σi0 0σi/parenrightbigg GA k˜SαGR k/bracerightbigg ,(28) fori= 0,x,y,z. TakingintoaccountEq.(27),onefindsthen a set ofequationsforthecoefficients aα,bα,cα,dα. We recall that in this paper the exchangefield due to prox- imity effect is assumed to be small, so the current-induced spinpolarizationislimitedtothetermslinearintheexcha nge field. Consequently,thevertexcorrectionisalsocalculat edin thelowestorderappropriatetohavespinpolarizationline arin M. Forα=xwe findthat: bx=cx=dx= 0, (29) ax=1 1−niV2Ix, (30) where Ix=/integraldisplaydkk 2πχx(µ,Γ)/producttext4 n=1(µ−En+iΓ)(µ−En−iΓ)(31) and χx(µ,Γ) =k6v6+k4v4(3Γ2−µ2)+(Γ2+µ2)3 +4(Γ4−µ4)λ2+k2v2(Γ2+µ2)(3Γ2−µ2+4λ2) ≈(k2v2−µ2)/parenleftbig k4v4−µ4+4µ2λ2/parenrightbig +/parenleftbig 3k4v4+3µ4+2k2v2/parenleftbig µ2+2λ2/parenrightbig/parenrightbig Γ2.(32) In the above equation, only terms up to the second order in Γhave been retained, while terms of higher order have been omitted. Forα=ywe findthefollowingcoefficients: ay=cy=dy= 0, (33) by=ax=η. (34)Inturn,for α=zwefind az=bz=dz= 0, (35) cz=1 1−niV2Iz=ζ, (36) where Iz=/integraldisplaydkk 2πχz(µ,Γ)/producttext4 n=1(µ−En+iΓ)(µ−En−iΓ)(37) and χz(µ,Γ) =k6v6+k2v2/parenleftbig 3Γ2−µ2/parenrightbig/parenleftbig Γ2+µ2/parenrightbig +/parenleftbig Γ2+µ2/parenrightbig3+k4v4/parenleftbig 3Γ2−µ2−2λ2/parenrightbig +/parenleftbig Γ2+µ2/parenrightbig/parenleftbig 2/parenleftbig Γ2−3µ2/parenrightbig λ2+8λ4/parenrightbig ≈/parenleftbig (k2v2−µ2)2−4µ2λ2)/parenrightbig/parenleftbig k2v2+µ2−2λ2/parenrightbig +/parenleftbig 3k4v4+2k2v2µ2+3µ4−4µ2λ2+8λ4/parenrightbig Γ2.(38) Finallytherenormalizedspin-vertexfunctionsare: ˜Sx=/planckover2pi1 2η/parenleftbigg σx0 0σx/parenrightbigg (39) ˜Sy=/planckover2pi1 2η/parenleftbigg σy0 0σy/parenrightbigg (40) ˜Sz=/planckover2pi1 2ζ/parenleftbigg σz0 0σz/parenrightbigg (41) This means that the results obtainedin the bare bubble ap- proximation should be multiplied only by a numerical factor to take into account the vertex corrections due to disorder. More specifically, the results for SxandSyshould be mul- tipliedbythefactor ηwhilethosefor Szshouldbemultiplied byζ. Thesituationissignificantlydifferentfromthatfoundin thecaseofspinHalleffect. Thisisbecausetransportpheno m- ena and spin polarization are affected by scattering on impu - ritiesinremarkablydifferentways. In Fig.3(a) we show the renormalization parameter ηas a function of chemical potential and relaxation time. Now the relaxation time is connected with the chemical potentia l through the relation/planckover2pi1 τ=niV2 v2|µ|. A single point in the τ,µ space corresponds to a well defined value of niV2. How- ever, possible values of niV2have been limited in Fig.3 to niV2<(niV2)max, where(niV2)maxisa certainmaximum value which is physically reasonable. The central white re- gion is bounded by the condition /planckover2pi1/τ= (niV2)max|µ|/v2 and is excluded for the considered parameters. In Fig.3c, in turn, we show the parameter ηas a function of the relaxation time and the ratio niV2/(niV2)max. As one might expect, thisfigureshowsthat the normalizationparameter ηbecomes reducedwith decreasing niV2. Figures 3(b) and 3(d) present cross-sections of Fig.3(a) and 3(c), respectively. The abo ve described results for ηshow that the SxandSycomponents are remarkablyrenormalizedby the vertexcorrectionand ar e enhanced by a factor of the order of 2 (between 1 and 3).7 FIG.3. (Color online) The parameter ηas afunction of the chemical potential and relaxationtime ( a) andas a functionof relaxation timeand niV2/(niV2)max(c). Figures (b) and (d) show ηas a function of chemical potential µ(b) andniV2/(niV2)max(d) for indicated values the relaxation time. Figures (e)-(h) show the same variatio ns as figures (a)-(d), but for the parameter ζ. The white regions in (a) and (e) are excluded for the assumed value of (niV2)max= 0.4×10−2(eV·nm)2. The other parameters are as inFig.2. This enhancement of the spin polarization is comparable to that found in the case of two-dimensional electron gas with Rashbe interaction.8The parameter ζ, in turn, is shown in Fig.3e-f. Itisofthesameorderofmagnitudeastheparamete r ηand depends on the chemical potential and relaxation time ina similarway,so wewill notdiscussit in moredetail. V. SPIN-ORBITTORQUE The current-inducedspin polarizationis exchange-couple d to the local magnetization Mand thus exerts a torque on M. According to Eq.(4), energy of this interaction per unit are a canbewrittenas Eex=−(2//planckover2pi1)˜M·S,whereSistheinduced spinpolarization. Takingintoaccounttherelationbetwee n˜M andM,onefindsthespin-orbittorqueperunitarea, τ,exerted on the magnetization (more precisely on the corresponding equilibriumspinpolarizationofthesystem)intheform τ=2 /planckover2pi1˜M×S=Jex gµB/planckover2pi1M×S. (42) Let us consider in more detail some specific situations as concerns relative orientation of the magnetization and ele c- tric field (current). Let us start with the situation when the magnetization Misintheplaneofthesystemandperpendic- ular to the current. This corresponds to θ= 0andξ= 0 (Mx=M/ne}ationslash= 0andMy=Mz= 0). From the abovegeneral equationfollowsthatthespin-orbittorquecanbethenwrit tenina generalformas τ=A(−ˆjMxSz+ˆkMxSy), (43) whereˆi,ˆj, andˆkare unit vectors along the axes x,yand z,respectively,andweintroducedthefollowingabbreviati on: A=Jex/gµB/planckover2pi1. TakingintoaccountEqs(21), (22), (24) and (25), one finds immediately that the spin-orbit torque in thi s geometry disappears because both SyandSzcomponent of thespinpolarizationvanish. Consider now the situation corresponding to θ= 0and ξ=π/2(My=M/ne}ationslash= 0andMx=Mz= 0), i.e. the casewhenthemagnetizationisparalleltotheelectriccurr ent. From Eq.(42) follows that the spin-orbit torque has the fol- lowinggeneralform: τ=A(ˆiMySz−ˆkMySx). (44) TheSzcomponentis now nonzero,andthus both, SzandSx contributetothetorqueinthisgeometry. Whenθ=π/2,namelyMz=M/ne}ationslash= 0andMx=My= 0, themagnetizationisperpendiculartothegrapheneplane. T he spin-orbittorquetakesthenthegeneralform, τ=A(−ˆiMzSy+ˆjMzSx). (45) Similarly as in the preceding situation, both SyandSxare nonzeroanddeterminethe torque. Inthelasttwocasesthespin-orbittorquecontainstwocom- ponents: linear term with respect to Jex(proportional to Sx) and quadratic term in Jex(proportional to SzandSy). The8 spin orbit torque contains one component proportionalto th e relaxation time and another component whose the dominant partisindependentonthe relaxationtime. In a general case of arbitrary orientation of the magnetic moment, magnitude and character of the spin-orbit torque varies with the orientation of the magnetic moment. This is because two components of the current-induced spin polar- ization depend on the magnetization, while the third one is independentof M. Asaresultthespintorquemayhavefield- likeand(anti)dampingterms. VI. SUMMARY We have calculated current-induced spin polarization in graphenedepositedonaferromagneticsubstrate,thatensu res not only Rashba spin-orbit interaction but also a ferromag- netic moment in the graphene layer. To describe electronic spectrum of graphene we have used Kane Hamiltonian that describes low-energy states around the Dirac points. Using the zero-temperature Green functions formalism and linear response theory, we have derived analytical formulas for th e spin polarization, up to the terms linear in M. Numerical re- sults based on the analytical formulas have been compared withthoseobtainedbynumericalintegrationprocedure. Fr om this comparison we have formulated applicability conditio nsof the analytical results. Significant deviations of the ana lyt- ical results from those based on numerical integration have beenfoundforrelaxationtimessmallerthan 10−10s. Thenonequilibrium(current-induced)spinpolarizatione x- erts a torque on the magnetization viathe exchange interac- tion. This torque containsa term which is proportionalto th e x-component of the induced spin polarization and therefore is proportionalto the momentumrelaxation time. The torque alsoincludesa componentwhosemainpartisindependentof therelaxationtime. The spin-orbit torque due to the interplay of external electric field and Rashba coupling at the interface between grapheneandamagneticlayercanbeusedforinstancetotrig - ger magnetic dynamics and/or magnetic switching. Indeed, such a switchingwas observedexperimentallyin a recent pa- per by Wang et al.54However, instead of graphene they used MoS2– anothertwo-dimensionalhoneycombcrystal. 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1201.4306v1.Josephson_dynamics_of_a_spin_orbit_coupled_Bose_Einstein_condensate_in_a_double_well_potential.pdf

arXiv:1201.4306v1 [cond-mat.quant-gas] 20 Jan 2012Josephson dynamics of a spin-orbit coupled Bose-Einstein c ondensate in a double well potential Dan-Wei Zhang,1Li-Bin Fu,2Z. D. Wang,3and Shi-Liang Zhu1,4,∗ 1Laboratory of Quantum Information Technology and SPTE, South China Normal University, Guangzhou, China 2Science and Technology Computation Physics Laboratory, Institute of Applied Physics and Computational Mathematic s, Beijing 100088, China 3Department of Physics and Center of Theoretical and Computa tional Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China 4Center for Quantum Information, IIIS, Tsinghua University We investigate the quantum dynamics of an experimentally re alized spin-orbit coupled Bose- Einstein condensate in a double well potential. The spin-or bit coupling can significantly enhance the atomic inter-well tunneling. We find the coexistence of i nternal and external Josephson effects in the system, which are moreover inherently coupled in a com plicated form even in the absence of interatomic interactions. Moreover, we show that the spin- dependent tunneling between two wells can induce a net atomic spin current referred as spin Josephs on effects. Such novel spin Josephson effects can be observable for realistically experimental co nditions. PACS numbers: 03.75.Lm, 67.85.Hj I. INTRODUCTION Based on the Berry phase effect [1, 2] and its non- Abeliangeneralization[3], thecreationofsyntheticgauge fields in neutral atoms by controlling atom-light inter- action has attracted great interest in recent theoretical studies [4–18], and has been realized in spinor Bose- Einstein condensates (BECs) in the pioneering experi- ments of the NIST group [19, 20] and also in several sub- sequent experiments of other groups [21–23]. The neu- tral atoms in the generated effective Abelian and non- Abelian gauge fields behave like electrons in an electro- magnetic field [19, 21, 22] or electrons with spin-orbit (SO) coupling [20]. Different from electrons that are fermions, the atoms with the synthetic SO coupling can be bosons and typically BECs. This bosonic counterpart of the SO coupled materials has no direct analogin solid- state systems and thus has received increasing attention [24–44] for different types of SO coupling, different inter- nal atomic structures (pseudospin-1/2, spin-1 and spin-2 bosons, etc.), and different external conditions (homoge- nous, trapped and rotated). These theoretical investiga- tions focus mainly on the static properties of SO coupled BECsand haverevealrichphase diagramsofthe ground- states [25–28, 30, 31] and exotic vortex structures [33– 35, 39–41]. However, to our knowledge, their dynamics has been less studied [24, 42–44], where the SO coupled BECs are demonstrated to exhibit interesting relativistic dynamics, such as analogs of self-localization [42], Zit- terbewegung [43] and Klein tunneling [44] under certain conditions. On the other hand, quantum dynamics of a BEC in a double well potential has been widely investigated. In ∗Electronic address: slzhu@scnu.edu.cn ) (xV x 0 b bL R pL nRnLpR 2: 2:ppJ nnJ(a)(b) npJ pnJ FIG. 1: (Color online) A schematic representation of (a) a SO coupled BEC in a double well trap. (b) The dynamic process of the system, where the blue solid and dashed lines repre- sent the inter-well tunneling without and with spin-flippin g, respectively, the green circles represent the Raman coupli ng. The inter-well spin-flipping tunneling induced by the Raman coupling is negligible under current experiment condition s [20, 50]. The atomic interaction terms are not shown in the figure. particular, the coherent atomic tunneling between two wells results in oscillatory exchange of the BEC, which is analogous to the Josephson effects (JEs) for neutral atoms [45–48]. The weakly interacting BECs provide a further context [46–48] for JEs in superconductor sys- tems because they display a nonlinear generalization of typical d.c. and a.c. JEs and macroscopic quantum self- trapping, all of which have been observed in experiments [49–51]. Apartfromtheconventionalsingle-speciesBECs [45–48], theJosephsondynamicsoftwo-speciesBECs[52] andspinorBECswithout SOcoupling[53] havealsobeen studied [54–57]; however, the dynamics of SO coupled BECs is yet to be explored. In this paper, we investigate the dynamics of a specific SO coupled BEC, which was realized in the experiment of the NIST group, in a double well trapping potential.2 We find that the SO coupling in the system contributes to and increases the atomic tunneling to a large extent, which can significantly enhance the atomic JEs. The full dynamics of the system contains both internal and exter- nal JEs, which are moreoverinherently coupled in a com- plicated form even in the absence of interatomic interac- tions. We further demonstrate that the spin-dependent Josephsontunnelingcanleadtoanetatomicspincurrent by varying conditions, which we refer to as spin Joseph- son effects. The predicted spin-Josephson currents are robustagainsttheparameteradjustmentandvaryingini- tial conditions, and can be observable in the SO coupled BECs under realistic experimental conditions. The paper is organized as follows. In Sec. II we con- struct a model that can be used to study the quantum dynamics of a SO coupled BEC in a double well poten- tial. Then, in Sec. III, the Josephson dynamics of the constructed system is investigated, with the complicat- edly coupled internal and external JEs being addressed. In Sec. IV we demonstrate that the spin JEs exhibit in the system under realistic conditions. Brief discussions and a short conclusion are given in Sec. V. II. MODEL In a very recent experiment, the NIST group realized a synthetic SO coupling in the87Rb BEC, in which a pairofRamanlasersgenerateamomentum-sensitivecou- pling between two internal atomic states [20]. In the bare pseudo-spin basis | ↑/an}bracketri}htb=|mF= 0/an}bracketri}htand| ↓/an}bracketri}htb=|mF= −1/an}bracketri}ht, the SO coupling is described by the single particle Hamiltonian given by [20] ˆh=p2 2mˆI+1 2/parenleftbigg δΩe2ikLx Ωe−2ikLx−δ/parenrightbigg ,(1) wherepis the atomic momentum in the xyplane,mis the atomic mass, δis the tunable detuning behaved as a Zeeman filed, kLis the wave number of the Raman laser, and Ω is the Raman coupling strength. Such kind of SO coupling is equivalent to that of an electronic system withequalcontributionfromRashbaandDresselhausSO coupling, and thus it is effective just in one-dimension (1D). So we restrict our discussions in 1D and focus on the motion of atoms along xaxis by freezing their yand zdegrees of freedom. To proceed further, we introduce the dressed pseudo- spins| ↑/an}bracketri}ht=e−ikLx| ↑/an}bracketri}htband| ↓/an}bracketri}ht=eikLx| ↓/an}bracketri}htb[20, 41], then the single-particle Hamiltonian in 1D (along xaxis) can be written as ˆh0=¯h2ˆk2 x 2m+2αˆkxσz+Ω 2σx+δ 2σz,(2) whereˆkxis the atomic wave vector operator, and α= Er/kLis the SO coupling strength with Er= ¯h2k2 L/2m being the single-photon recoil energy. The dispersion re- lation of the single particle Hamiltonian (2) with δ= 0isE±(kx) =¯h2k2 x 2m±/radicalbig 4α2k2x+Ω2/4, which exhibits a structure of two branches. We are interested in the lower energy one E−(kx). There is only one minimum inkx= 0 for large Raman coupling Ω >4Er, where the atoms of both atomic levels condense. However, the lower branch for Ω <4Erpresents two minima for condensation of dressed pseudo-spin-up (left one) atoms and dressed pseudo-spin-down (right one) atoms, respec- tively. The Raman coupling and a small δmodulate the population of atoms in these two states [20]. Here we fo- cus on the later regime, i.e. Ω <4Er, because such BEC with spin-separated and non-zero central momentum is more interesting in contrast to a regular BEC with zero central momentum. To be clearer, we can rewrite the Hamiltonian (2) as ˆh0=/parenleftbigg H↑Ω/2 Ω/2H↓/parenrightbigg , (3) whereH↑=¯h2 2m(ˆk2 x+ 2kLˆkx) +δ 2andH↓=¯h2 2m(ˆk2 x− 2kLˆkx)−δ 2. Since it is more straightforward to describe the system in terms of dressed pseudo-spin states com- pared with using bare ones, we will work in the dressed pseudo-spin space and simply refer to dressed pseudo- spin as spin for convenience hereafter. We also note that the parameters kL, Ω andδin the single-particle Hamil- tonian can be tuned independently in a wide range [20], making the SO coupled BEC a suitable platform for in- vestigatingthe Josephsondynamicsin thepresenceofSO coupling. Now we turn to consider such a SO coupled BEC in a double well potential denoted by V(x) as shown in Fig. 1(a). Note that the double well potential here is assumed to be spin-independent. To investigate the dynamics of the system, we adopt the two-mode approximation [45– 48] with the field operator ˆΨσ(x)≃ˆaLσψLσ(x)+ˆaRσψRσ(x), (4) whereψjσ(x) is the ground state wave function of the j well (j=L,R) with spin σ(σ=↑,↓), and ˆajσis the an- nihilation operator for spin σin thejwell, satisfying the bosonic commutation relationship [ˆ ajσ,ˆa† kσ′] =δjkδσσ′. The validity of the two-mode approximation holds under twoconditions: the weakatomicinteractionandsmallef- fectiveZeemansplitting, astheatomscannotbepumped outoftheloweststateofeachwellinthis case. Inthe sec- ond quantization formalism, the total Hamiltonian reads H=/integraldisplay dxˆΨ†(x)/bracketleftig ˆh0+V(x)+ˆhint/bracketrightig ˆΨ(x),(5) where the two-component field operator ˆΨ = (ˆΨ↑,ˆΨ↓)T, and the interaction Hamiltonian ˆhintwill be specified below. Substituting Eq. (4) into Eq. (5), one can rewrite3 the total Hamiltonian as H=/summationdisplay j,σεjσˆa† jσˆajσ+/summationdisplay σσ′/parenleftig Jσσ′ˆa† LσˆaRσ′+h.c./parenrightig +Ω 2/summationdisplay j/parenleftig ˆa† j↑ˆaj↓+h.c./parenrightig +δ 2/summationdisplay j/parenleftig ˆa† j↑ˆaj↑−ˆa† j↓ˆaj↓/parenrightig +Hint,(6) whereεjσ=/integraltext dxψ∗ jσ(x)[Hσ+V(x)]ψjσ(x)≈1 2¯hωj− Eris the single-particle ground state energy in the jwell withωjbeing the harmonic frequency of this well,Jσσ=/integraltextdxψ∗ Lσ(x)[Hσ+V(x)]ψRσ(x) andJσ¯σ=/integraltextdxψ∗ Lσ(x)Ω 2ψR¯σ(x) (withσand ¯σreferring to different spins) are the tunneling terms shown in Fig. 1(b). In addition, the interaction Hamiltonian is given by Hint=1 2/summationdisplay j/parenleftig g(j) ↑↑ˆa† j↑ˆa† j↑ˆaj↑ˆaj↑+g(j) ↓↓ˆa† j↓ˆa† j↓ˆaj↓ˆaj↓ +2g(j) ↑↓ˆa† j↑ˆa† j↑ˆaj↓ˆaj↓/parenrightig , (7) whereg(j) σσ′=2¯h2aσσ′ ml2 ⊥/integraltext dx|ψjσ(x)|2|ψjσ′(x)|2is the effec- tive 1D interacting strength with aσσ′being thes-wave scattering length between spin σandσ′andl⊥being the oscillator length associated to a harmonic vertical con- finement. Note that here we have ignored the inter-well atomic interactions because the s-wave scattering length (which is on the order of nanometers) is much smaller than the inter-well distance (which is on the order of micrometers). We have also dropped the inter-well cou- pling since its strength is exponentially smaller than the intra-well counterpart. The Hamiltonian (6) describes the dynamic process of the system schematically shown in Fig. 1(b). For simplicity, we assume the double well potential to be symmetric as shown in Fig. 1(a), with each well hav- ing the same harmonic trapping frequency ω. Thus we haveεL=εRandg(L) σσ′=g(R) σσ′. Such kind of double well potential can be generated in experiments [51] with the form V(x) =a(x2−b2)2, (8) where the parameters aandbare both tunable in the experiments [51]. Expanding V(x) nearx=±b, one ob- tains its harmonic form as V(2)(x).=1 2mω2(x±b)2and thusa=mω2/8b2. The ground state wavefunctions of the BEC in each well potential with each spin can be ap- proximately represented by its corresponding lowest en- ergy single-particle wavefunction, which can be worked out by solving the equations [ Hσ+1 2mω2(x±b)2]ψjσ= εjσψjσ(here±are forj=L,R, respectively). The re- sults are [24] ψL↑=ϕ(L) 0(x)e−ikLx, ψL↓=ϕ(L) 0(x)eikLx, ψR↑=ϕ(R) 0(x)e−ikLx, ψR↓=ϕ(R) 0(x)eikLx,(9)1 2 3 4−0.4−0.200.20.40.60.8J(η) / Er 1 2 3 4−0.200.20.40.60.8 1 2 3 4−0.200.20.40.60.8 b / µmJ(η) / Er 1 2 3 4−0.200.20.40.60.8 b / µm 100J(T) 100J(V) |J(SO)| 100J(R) (c) I0 = 3 µm(a) I0 = 1 µm (b) I0 = 2 µm (d) I0 = 4 µm FIG. 2: (Color online) The energy scales of tunneling terms J(η)as a function of bfor (a)l0= 1µm, (b)l0= 2µm, (c) l0= 3µm, and (d) l0= 4µm, respectively. In (a)-(d) we set Ω/¯h=Er/¯h= 22.5 kHz. whereϕ(L) 0(x) =1√ l0√πe−(x+d)2/2l2 0, andϕ(R) 0(x) = 1√ l0√πe−(x−d)2/2l2 0withl0=/radicalbig ¯h/mωbeing the oscillator length. Substituting Eq. (9) into the expressions of Jσσ′, one can obtain J↑↑=J(T)+J(SO)+J(V)+J(Z), J↓↓=J(T)+J(SO)+J(V)−J(Z), J↑↓=J↓↑=J(R),(10) where the terms J(T)=−¯h2 2m/integraltext dxϕ(L) 0ϕ′′(R) 0,J(SO)= −Er/integraltext dxϕ(L) 0ϕ(R) 0,J(V)=/integraltext dxϕ(L) 0V(x)ϕ(R) 0,J(Z)= δ 2/integraltext dxϕ(L) 0ϕ(R) 0, andJ(R)=Ω 2/integraltext dxϕ(L) 0e−2ikLxϕ(R) 0. Comparedwith theatomictunnelingofaregularBEC, the SO coupled BEC in this system exhibits two addi- tional tunneling channels, the SO coupling induced tun- neling term J(SO)and the Raman coupling induced one J(R). Toclarifythe effectsofthese termsinthe tunneling processes,weneedtoworkoutandtocomparetheenergy scales of all the terms J(η), whereη={T,V,Z,SO,R }. Substituting Eqs. (9) and (8) into Eq. (10), we can ob- tain the following analytical solutions J(η)=ξηe−b2/l2 0, (11) whereξT=¯h2 2ml2 0(1 2−b2 l2 0),ξV=¯h2 8mb2l4 0(3 4l4 0−b2l2 0+b4), ξZ=δ/2,ξSO=−Er, andξR= Ωe−k2 Ll2 0. Since the Zee- man filedδis independently tunable to the double-well structure and should be small, we here further assume δ≪Erand thus we focus on the comparison among J(SO,R,T,V ). TheeffectsofZeeman-splittinginducedtun- neling will be specified in the Sec. IV. Forl0∼√ 2b, we haveξT∼0 andξV∼¯h2 4mb2, and thus|ξSO| |ξV|= 8π2(b λL)2>∼100. Here we have assumed4 the same wavelength λL= 2π/kL= 0.8µm and recoil frequencyEr/¯h= 22.5 kHz as those in the experiments [20],bandl0to be on the order of micrometers [50, 51]. In fact, in the regime of b2/l2 0∼[0.5,2], we find that |ξSO|>∼100max{|ξT|,|ξV|}. (12) Besides, one can check that ξR∼Ωe−64forl0∼1µm andλL= 0.8µm, and thus the Raman-coupling induced tunneling is negligible in this system. The comparisons amongJ(η)forsometypicalparametersareshownin Fig. 2. In otherwords,wefind that underrealisticexperiment conditions [20, 50], the spin-flipping tunneling induced by Raman coupling is negligible but the SO coupling in- duced tunneling term J(SO)dominates and moreover it greatly enhances atomic tunneling in this system. Thus we may rewrite the tunneling terms as J↑↓=J↓↑≈0, J↑↑≈J↓↓≈J(SO)=−γEr,(13) whereγ= exp(−b2/l2 0)∼[0.1,0.6]. It is worthwhile to note that the new tunneling terms J(SO)andJ(R)in this SO coupled system are both tunable, enabling us to study the interesting effects of SO coupling in the atomic inter-well tunneling. For instance, one can decrease the effective wave number in xaxis to the scale kL∼1/l0so thatJ(R)∼0.37Ωand then the Raman-couplinginduced tunneling can revive. This can be achieved by adjusting the angle between the applying Raman lasers and the trapping potential or alternatively by using lasers with larger wavelength. In addition, in the same way one can tune the recoil energy Erto identify the enhancement of atomic tunneling due to the SO coupling (i.e. the effect ofJ(SO)) in experiments. As a first step to inves- tigate the system under current experiment conditions, we here concentrate on the tunneling regime governed by Eq. (13). III. FULL DYNAMICS OF THE SYSTEM We are now in the position to investigate the quan- tum dynamics of the system constructed in the previous section. We first address the non-interacting case, i.e.Hint= 0 in Eq. (6), in which the single-particle Hamil- tonian is given by H0≃J↑↑/parenleftig ˆa† L↑ˆaR↑+ˆaL↑ˆa† R↑/parenrightig +J↓↓/parenleftig ˆa† L↓ˆaR↓+ˆaL↓ˆa† R↓/parenrightig +Ω 2/parenleftig ˆa† L↑ˆaL↓+ˆaL↑ˆa† L↓+ˆa† R↑ˆaR↓+ˆaR↑ˆa† R↓/parenrightig +δ 2/parenleftig ˆa† L↑ˆaL↑−ˆa† L↓ˆaL↓+ˆa† R↑ˆaR↑−ˆa† R↓ˆaR↓/parenrightig . (14) Here we have dropped the tunneling terms Jσ¯σsince these spin-flipping tunneling progresses can be negligible in the current experiment conditions [20, 50]. In order to study the dynamic properties of the system, we need to work with the equation of motion. The corresponding Heisenberg equations read i¯hd dtˆajσ= [ˆajσ,H0] =Jσσˆa¯jσ+Ω 2ˆaj¯σ+(−1)pδ 2ˆajσ, (15) whereσand ¯σrefer to different spin, while jand¯j to different wells, and p= 0,1 are forσ=↑,↓, re- spectively. Using the mean-field approximation, one has ˆajσ≃ /an}bracketle{tˆajσ/an}bracketri}ht ≡ajσwithajσbeingcnumbers. Thus we can rewrite the equations of motion as i¯h˙ajσ=Jσσa¯jσ+Ω 2aj¯σ+(−1)pδ 2ajσ.(16) By defining a four-component wavefunction Φ = (aL↑,aL↓,aR↑,aR↓)T, Eq. (16) is rewritten as i¯hd dtΦ = HMΦ, where the Hamiltonian of the system is given by HM= δ 2Ω 2J↑↑0 Ω 2−δ 20J↓↓ J↑↑0δ 2Ω 2 0J↓↓Ω 2−δ 2 . (17) We now look into the JEs in this system. Let us fur- ther express ajσasajσ=/radicalbig Njσeiθjσ, where the particle numbersNjσand phases θjσare all time-dependent in general. According to Eq. (16), we can obtain i¯h˙Njσ 2−¯hNjσ˙θjσ=Jσσ/radicalbigNjσN¯jσei(θ¯jσ−θjσ) +Ω 2/radicalbig NjσNj¯σei(θj¯σ−θjσ)+(−1)pδ 2Njσ.(18) Separating the image and real parts of Eq. (18) yields two groups of equations as ˙Njσ=2Jσσ ¯h/radicalbigNjσN¯jσsin(θ¯jσ−θjσ)+Ω ¯h/radicalbig NjσNj¯σsin(θj¯σ−θjσ), ˙θjσ=Jσσ ¯h/radicalig N¯jσ Njσcos(θ¯jσ−θjσ)+Ω 2¯h/radicalig Nj¯σ Njσcos(θj¯σ−θjσ)+(−1)pδ 2¯h.(19) Eq. (19) consists actually of eight coupled equations. To simplify these equations, we introduce φσ=θRσ−θLσ andρσ=NRσ−NLσfor the phase and particle number differences between two wells with the same spin σ, andφj=θj↓−θj↑andρj=Nj↓−Nj↑for the phase and particlenumberdifferencesbetweentwospinsinthesame5 wellj, respectively. Thus we can obtain ˙ρσ=L1sinφσ+1 2/summationdisplay j(−1)qL2sinφj, ˙ρj=1 2/summationdisplay σ(−1)pL1sinφσ+L2sinφj,(20) whereL1=−4Jσσ ¯h√NRσNLσ,L2=−2Ω ¯h/radicalbigNj↑Nj↓, and q= 0,1 forj=R,L, respectively ( p= 0,1 forσ=↑,↓, respectively). From the above Eq. (20), we find the coexistence of internal JE related to ˙ ρj(φj) and external JErelatedto ˙ ρσ(φσ). Moreover,theinternalandexternal JEs are inherently coupled in a more complicated form. Before ending this section, we briefly discuss the weakly interacting cases, which have been assumed to meet the requirement of two-mode approximation. In this regime, the mean-field analysis still works well, and the dropped term Hintcan be taken into count within the previous discussions. This leads to two additional terms related to interactions into Eq. (16), and now the equations of motion are given by i¯h˙ajσ=Jσσa¯jσ+Ω 2aj¯σ+(−1)pδ 2ajσ +gσσ|ajσ|2ajσ+gσ¯σ|aj¯σ|2ajσ,(21) where the interacting strength gσσ′can be found as gσσ′=√ 2¯h2aσσ′√πml2 ⊥l0. The estimation of the interaction en-ergyand the Josephsondynamics in the presence ofweak interactions will be presented in the next section. IV. JOSEPHSON EFFECTS IN WEAK RAMAN COUPLING REGIMES In the preceding section, we have shown that the SO coupled BEC in a double well potential exhibits the com- plicated coupled external and internal Josephson dynam- ics. We, in this section, consider a specific dynamic pro- cess of the system in the weak Raman coupling regime (i.e. Ω/Er≪1), where the external Josephson dynamic dominates. Infact, themanipulationanddetectionofthe SO coupled BECs in this regime have been performed in experiments [20]. For the weak Raman coupling, we find that the ra- tioν≡ |Jσσ|/Ω can reach several hundreds from Eq. (13). Thus within the time scale τ∼¯h/Ω≃45 ms for Ω = 0.001Er, one can ignore the effects of the spin- flipping tunneling, which leads to two external Josephson tunneling processes for different spins. The spins in this regime are conserved and then the total particle number of spinσ Nσt=NLσ+NRσare time-independent con- stants. We assume N↑t=N↓t=Ntfor simplicity. The equations of motion (21) in this case can be rewritten as i¯hd dt/parenleftbigg aLσ aRσ/parenrightbigg =/parenleftbigg (−1)pδ 2+gσσ|aLσ|2+gσ¯σ|aL¯σ|2Jσσ Jσσ (−1)pδ 2+gσσ|aRσ|2+gσ¯σ|aR¯σ|2/parenrightbigg/parenleftbigg aLσ aRσ/parenrightbigg .(22) Bydefiningthenormalizedinter-wellparticlenumberdif- ference for spin σasZσ= [NRσ−NLσ]/Nt(−1≤ Zσ≤ 1), the equations of motion (19) become rather simple in this case (similar to those for the regular two species BECs [54]), which are given by ˙Zσ=−2Jσσ ¯h√1−Zσsinφσ, ˙φσ=Jσσ ¯hZσ√ 1−Z2σcosφσ+Uσσ ¯hZσ+Uσ¯σ ¯hZ¯σ+(−1)pδ 2¯h. (23) The spin-dependent atomic density current is given by Iσ=Nt·˙Zσ. (24) From Eq.(24), we can define the net spin current as Is=I↑−I↓, (25) and the total atomic current as Ia=I↑+I↓. (26) We first considerthe JEs ofthe system in the noninter- acting limit, i.e. Uσσ=Uσ¯σ= 0 in Eq. (23), which canbe realizedbyFeshbachresonance[58]. Under this condi- tion, the two external Josephson tunneling processes for different spins are decoupled. We numerically calculate Eqs. (23), with some typical results of the time evolu- tion ofZσfor different initial conditions being shown in Fig. 3. In the calculations, we have assumed the zero Zeeman filed δ= 0 in Fig. 3(a) and (b), and small Zee- man fieldδ= 0.01Erin (c) and (d). Compared with zero Zeeman field cases, a small Zeeman field results in a deviation in Josephson tunneling strengths Jσσand in time-cumulative phases δ/2¯hfor different spins. Here Z↑(t) andZ↓(t) demonstrate the oscillatory Josephson tunnelings which are similar to the early results in Ref. [47]. As shown in Fig. 3(a-d), they are spin-dependent and the dynamic evolution of each one depends on its own tunneling strength, phase and initial conditions. We also calculate Iσ,IsandIain this regime with typical results being shown in Fig. 4. It is interesting to see that the spin-dependent atomic density currents due to the spin-related Josephson tunnelings give rise to a net spin current (cf. Fig. 4), and moreover in some certain6 0246810−1−0.500.51Zσ 0246810−1−0.500.51 0246810−1−0.500.51 t / msZσ 0246810−1−0.500.51 t / msZ↑ Z↓ (d)(b) (a) (c) FIG. 3: (Color online) The time evolution of Zσin nonin- teracting limitation. In (a) and (b) we have δ= 0 and J↑↑=J↓↓=−0.1Er; In (c) and (d) we have δ= 0.01Er, J↑↑=−0.905Er, andJ↓↓=−0.105Er. The initial conditions areZ↑(0) =−Z↓(0) = 0.3,φ↑(0) = 0.5φ↓(0) =π/4 in (a) and (c); and Z↑(0) =Z↓(0) = 0, φ↑(0) =φ↓(0) =π/4 in (b) and (d). 0 2 4 6 8 10−4000−2000020004000(Iσ, Is, Ia) / Nt 0 2 4 6 8 10−4000−2000020004000 t / ms(Iσ, Is, Ia) / Nt I↑ I↓ Is Ia(b)(a) FIG. 4: (color online) Josephson currents in the noninter- acting limit. The time evolution of spin-dependent atomic currents Iσ, a net spin current Isand a total atomic current Iain (a) for the same conditions in Fig. 3(a); and in (b) for the some conditions in Fig. 3(b). initial conditions the total atomic current can be zero, which leads to a new interesting pure spin currents (cf. Fig. 4(b)). We call such new JEs as spin Josephson effects, which can be observable in experiments by mea- suring the time-evolution of spin-dependent population imbalance of the atomic gas [59]. For weakly interacting cases, we have to estimate0246810−1−0.500.51Zσ 0246810−1−0.500.51 0246810−1−0.500.51 t / msZσ 0246810−1−0.500.51 t / msZ↑ Z↓(b)(a) (c) (d) FIG. 5: (Color online) The time evolution of Zσin the weakly interacting regime with Uσσ= 0.01ErandUσσ′= 0.011Er. Other parameters and initial conditions in (a)-(d) are the same with those in Fig. 3(a)-(d), respectively. the interaction energy UσσandUσ¯σ, which should be Uσσ,Uσ¯σ≪¯hωdue to the two-mode approximation. This requirement results in Uσσ≈Uσ¯σ≪0.1Er. In this regime, the two spin-Josephson tunneling processes are coupled via atomic interactions. To understand the effects of the interaction, we show in Fig. 5 some typical results of the time evolution of Zσfor the same initial conditions and parameters in Fig. 3. It clearly demon- strate that the modification of Zσ(t) due to atomic in- teractions is not significant and even very minor in some cases (such as the case for Zσ(0) = 0 and δ= 0 in Fig. 3(b), 5(b)) since the interactionenergyis smallcompared with the tunneling energy. Therefore the spin Josephson dynamics still exhibit a similar oscillatory feature in this regime. To see more clearly the oscillatory properties of the spin JEs, we have numerically calculated the frequency spectra of the net spin currents Isfor various conditions (such as those in Fig. 3 and Fig. 5). We find that the spectra for different cases exhibit a single peak centered at the slightly shifted frequency, as seen in Fig. 6. The single-peak feature shown in Fig. 6 implies that the spin currentIs(t) can well be described by a sin-function, while the weak interatomic interactions or the small Zee- man field can merely modify the period and amplitude of the current slightly. Thus we conclude that the spin JEs in this system are robust against the parameter ad- justment and initial conditions.7 0.0 0.2 0.4 0.6 0.8 1.03(a) 3(c) 5(a) 5(c)Spectrum of Is Frequency (kHz) FIG. 6: (Color online) Spectra of the net spin currents Isfor the cases in Fig. 3(a,c) and in Fig. 5 (a,c). A single peak in the spectrum of each case implies that the spin current Is(t) is well described by a sin-function. V. DISCUSSION AND CONCLUSION Before concluding, we briefly discuss another specific dynamic process of the system in the relatively strong Raman coupling regime, Ω ≫ |Jσσ|, which can be real- izedsuchasbytuning Jσσ∼ −0.1ErandΩ∼Er. Inthis regime, within the time scale ¯ h/|Jσσ|one can ignore theatomicinter-welltunneling andconsideronlythe internal dynamics in each single well. The atomic tunneling be- tween two spins refers to spin-flipping is induced by the Raman coupling, and thus such Josephson tunneling is in the spin space. 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1712.07908v1.Cooper_Pairing_in_A_Doped_2D_Antiferromagnet_with_Spin_Orbit_Coupling.pdf

arXiv:1712.07908v1 [cond-mat.supr-con] 21 Dec 2017Cooper Pairing in A Doped 2D Antiferromagnet with Spin-Orbi t Coupling Jingxiang Zhao1, Qiang Gu1,2∗ 1Department of Physics, University of Science and Technolog y Beijing, Beijing 100083, P. R. China 2Beijing Key Laboratory for Magneto-Photoelectrical Compo site and Interface Science, School of Mathematics and Physics, University of Science an d Technology Beijing, Beijing 100083, P. R. China (Dated: April 29, 2021) We study the two-dimensional Hubbard model with the Rashba t ype spin-orbit coupling within and beyond the mean-field theory. The antiferromagnetic gro und state for the model at half-filling and the Cooper pairing induced by antiferromagnetic spin flu ctuations near half-filling are examined based on the random-phase approximation. We show that the an tiferromagnetic order is suppressed and the magnetic susceptibility turns out to be anisotropic in the presence of the spin-orbit coupling. Energy spectrums of transverse spin fluctuations are obtain ed and the effective interactions between holes mediated by antiferromagnetic spin fluctuations are d educed in the case of low hole doping. It seems that the spin-orbit coupling tends to form s+p-wave Cooper pairs, while the s+d-wave pairing is dominant when the spin-orbit coupling is absent. I. INTRODUCTION Spin fluctuations may result in effective attractive interactions between fermions, and this mechanism plays an important role in understanding unconventional superconductivity1. And it attracts much attention till today2,3. For ferromagnetic or nearly ferromagnetic Fermi systems, spin fluctuations are favorable to spin- triplet Cooper pairs. For example, it was suggested that the p-wave triplet Cooper pairing in superfluid3He, a nearly ferromagnetic Fermi liquid, should be induced by spin fluctuations4. Furthermore, Fay and Appel pointed out that the longitudinal ferromagnetic spin fluctuations could cause p-wave effective attraction within the ferro- magnetic state5. This theory provides an candidate ex- planation on the superconductivity in the ferromagnetic superconductors, such as UGe 26and UCoGe7, whose su- perconductivity (SC) state coexists with the itinerant- electron ferromagnetic order. Similarly, antiferromagnetic (AFM) spin fluctuations can also give rise to Cooper pairing. Schrieffer, Wen and Zhang have proposed an AFM spin fluctuation mech- anism, the spin-bag model8, to explain the high-T c superconductivity9. This model is based on the half- filled Hubbard model on the square-lattice, which favors the AFM ground state in the large-U limit and thus cor- responds to the AFM order of the parent materials of cuprate superconductors. Schrieffer et al.suggested that the AFM spin fluctuation should induce d-wave Cooper pairs between holes in the weak hole-doping case. The AFM order is also present in many other super- conductors or their parent materials, including heavy- fermion and iron-based superconductors. For instance, the SC phase of the heavy fermion material CePt 3Si has been proved to coexist with antiferromagnetism10, and so has the low-pressure SC phase of CeCu 2Si211. Vari- ous AFM orders appear in iron-based superconductors, such as the collinear AFM state in LaO 1−xFxFeAs12, the bi-collinear state in Fe 1+ySexTe1−x13, and the blocked checkerboard AFM order in K 0.8Fe1.6Se214. Recently, some groups report the microscopic coexistence of anti-ferromagnetism and superconductivity in the iron-based materials15,16. It is naturally supposed that AFM spin fluctuations might play an important role in iron- based superconductors. For example, some reports sug- gest that the anti-ferromagnetic spin fluctuations of LaFeAsO 1−xFx17and Fe 1−xCoxSe18should mediate the s±wave superconducting state. One more important issue is that the spin-orbit cou- pling(SOC)maybepresentinsomeofforgoingsupercon- ductors. Particularly,theSOCisinevitablyresultedfrom the lack of structure inversion symmetry and therefore it must be considered in the non-centrosymmetric (NCS) superconductor19. It is known that the SOC can cause the admixture of spin-up and spin-down20,21, which es- sentially influences the spin degree of freedom. Resulting from the spin-mixing, SOC could lead to the mixture of spin-singlet and spin-triplet pairing symmetry22. Actu- ally, SOC has stimulated much researchinterest in recent years since it plays an important role in other condensed matter systems, e.g. Quantum spin Hall insulator23and atomic Fermi superfluid24. Due to the effect of SOC on spin states, it is infered that SOC could affect the spin fluctuations and influence the orbital symmetry of Cooper pairs mediated by spin fluctuations. Various models were studied to discuss the role of SOC in the orbital symmetry. Some papers em- ployedatwo-bandmodelandstatedthat SOCcouldsplit the degeneracy of p-wave states25. This model combined theitinerantelectronswith localmomentsandcouldhelp understand some unconventional superconductivity, e.g., Sr2RuO4. But it was not suitable for some superconduc- tors whose SC and AFM order originated from the same band electrons, for example, some NCS superconductors which also exhibited the AFM order. Therefore, a single- band model was necessary. Some papers studied a single- band Hubbard model with SOC26,27and reported that SOC might be in favor of the d+f-wave pairing states or p+d-wave states. However, these papers neglected the AFM fluctuation. In some intensive studies on NCS superconductivity, the AFM fluctuation was introduced in different ways. Some people selected parameters to2 fit spin susceptibility into experimental results manifest- ing anti-ferromagnetism28, while some people employed a staggered field to describe the AFM order29. All of them reportedthatthe mixtureofspin-singletandtriplet Cooper pairs was resulted from SOC and the orbital symmetry was obtained, e.g., s+p-wave or p+d+f-wave states. But the AFM order could not consistently ob- tained in these papers. Thus a better single-band model is necessary for consistently studying the effect of SOC on the AFM order and fluctuations. In this paper, we investigate the half-filled Hubbard model with the Rashba SOC in a two-dimensional (2D) square lattice. The central issue of this paper is to exam- inetheinfluenceofSOContheCooperpairingintermedi- atedbytheAFMspinfluctuations. Ourmodelisasingle- band model, which suggests both AFM order and super- conductivity are originated from one-band electrons. It is might help for understanding the magnetic properties andpairingsymmetryofsomequasitwo-dimensionallay- ered superconductors with SOC, for instance, iron-based superconductors30,31and NCS32. The paper is organized as follows. The model is de- scribed in Section II. Ground state properties for the model at half-filled are studied based on the mean-field approximation and the RPA. The sublattice magnetiza- tion, the spectrum of transverse spin excitation and the ratio of transverse versus longitudinal spin susceptibil- ity at (π,π) are calculated. Section IV discusses effec- tive interactions between holes induced by the AFM spin fluctuations in the case of weak hole doping, with the emphasis on pairing effects in the s,p, anddchannels. The conclusions are given in the last section. II. THE MEAN-FIELD MODEL We start from a single-band half-filled Hubbard model with SOC in a two-dimensional square lattice. H=/summationdisplay k,σεkc† kσckσ+α/summationdisplay kg(/vectork)·s(/vectork)+U/summationdisplay ini↑ni↓, whereεk=−2t(coskxa+ coskya) is the kinetic energy arising from electron hoping between the nearest neighbours with abeing the lattice con- stant. The second term of Hamiltonian describes the spin-orbit coupling with αbeing the coupling strength. Here the type of SOC takes the form as26,29: s(/vectork)=/summationtext σ,σ′σσ,σ′c† k,σck,σ′andg(/vectork)=(−vy(/vectork),vx(/vectork),0), wherevx,y(/vectork)=∂εk/∂kx,y. With the definition, the g vector, which is ( −2tsinkya,2tsinkxa,0), protects the symmetry and periodicity of the Brillouin zone. In the following, we assume a= 1 for simplicity. By introducing v(/vectork)=−2t/radicalBig sin2ky+sin2kx, the term of SOC, −2tα/summationtext k(sinky±isinkx) can be denoted as α/summationtext kv(ˆk)exp(±iφk), where φk=arctan(sin kx/sinky).In this case, the Hamiltonian has the form, H=/summationdisplay k,σεkc† kσckσ+U 2N/summationdisplay k,k′,q/summationdisplay σσ′c† k−qσc† k′+qσ′ck′σ′ckσ +α/summationdisplay kv(ˆk)eiφkc† k↑ck↓+α/summationdisplay kv(ˆk)e−iφkc† k↓ck↑.(1) In the case of AFM, by using the mean-field ap- proach (MFA), the interaction term can be written as −US/summationtext k,σ,σ′c† k+Q,σσz σσ′ck,σ′3,8, where Q= (π,π), the nesting vector of Fermi surface as shown in Fig. 1, and S=/angbracketleftbig G/vextendsingle/vextendsingleSz Q/vextendsingle/vextendsingleG/angbracketrightbig /N, whereSz Q=/summationtext kc† k+Q,σσz σσ′ck,σ′and |G/angbracketrightisthe groundstateofthemodel. Therefore,the AFM order can be studied consistently. When the spin-orbit coupling is ignored, |G/angbracketrightis the same as the ground state of the antiferromagnetism defined as in Ref.[8]. Through introducing new fermion-operators fk,η(η= 1,2,3,4), the Hamiltonian can be diagonalized via the Bogliubov transformation. In the process of diagonalization, some equations between kandk+Qare:i) the nesting Fermi surface results in εk+Q=−εk;ii) the principal value ofφkis confined in ( −π,π]. Accordingly φk+Q=φk+π is defined to keep v(k+Q)e−iφk+Q=−v(k)e−ikφkand φ−k=φk+πto preserve g(−k) =−g(k). The relation- ship between electron operators and the quasi-particles operators is expressed as, ck↑=eiφk 2/bracketleftbig uk+fc k,++uk−fc k,−+νk+fv k,++νk−fv k,−/bracketrightbig , ck↓=e−iφk 2/bracketleftbig −uk+fc k,++uk−fc k,−−νk+fv k,++νk−fv k,−/bracketrightbig , ck+Q↑=eiφk 2/bracketleftbig νk+fc k,++νk−fc k,−−uk+fv k,+−uk−fv k,−/bracketrightbig , ck+Q↓=e−iφk 2/bracketleftbig νk+fc k,+−νk−fc k,−−uk+fv k,++uk−fv k,−/bracketrightbig , (2) where,u±=1 2/radicalBig 1+ξk,± Ek,±,ν±=1 2/radicalBig 1−ξk,± Ek,±,Ek,±= /radicalBig ξ2 k,±+∆2is the eigenvalue with ξk,±=εk±αvkand ∆ =−US/2. The diagonalized Hamiltonian can be formed as below, H=/summationdisplay k′Ek,+/parenleftBig fc† k,+fc k,+−fv† k,+fv k,+/parenrightBig +/summationdisplay k′Ek,−/parenleftBig fc† k,−fc k,−−fv† k,−fv k,−/parenrightBig .(3) where/summationtext′represents the summation extending over the magnetic zone without SOC displayed in Fig. 1. The Fermi surface and the energy spectrum of electrons are split by SOC as shown in Fig. 1 and Fig. 2, respectively. The nesting Fermi surface which stands for the AFM or- der is broken by SOC, which suggests that SOC should suppress AFM order. Moreover, the system still retains the periodicity as shown in Fig. 2 so the first BZ can still represent the symmetry of the model. So the numeri- cal analysis will be reduced in the first BZ rather than magnetic BZ in the presence of SOC.3 /s107 /s107/s81/s121 /s120 FIG. 1. (Color online) The schematic of the first Brillouin zone; the solid line represents the Fermi surface at half-fil ling without SOC. The dashed lines represent the split Fermi sur- faces resulted from SOC. /s45/s54 /s45/s51 /s48 /s51 /s54/s45/s52/s45/s50/s48/s50/s52 /s32/s32/s69 /s107 /s107 /s120 FIG. 2. (Color online) The schematic diagram of the eigenval - ues. It is shown that the period of energy bands are protected by Rashba SOC. The dotted line locating in the middle of fig- ure represents the chemical potential. In the half-filled case, the conductive bands, fc k,+and fc k,−, are empty for particles and the other bands, fv k,+ andfv k,−are valence bands which are full filled by parti- cles, so the ground state can be defined as: fc k,+|G/angbracketright=fc k,−|G/angbracketright= 0 ;fv† k,+|G/angbracketright=fv† k,−|G/angbracketright= 0.(4) III. GROUND STATE PROPERTIES ON THE RANDOM-PHASE APPROXIMATION To quantitatively study the effect exerted by SOC on the AFM order, we employ the foregoing definitions of ground state to obtain the self-consistent equation of the/s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48 /s32/s32/s83 /s85/s47/s116/s32 /s61/s48 /s32 /s61/s48/s46/s53 /s32 /s61/s49/s46/s48 /s32 /s61/s49/s46/s53 /s32 /s61/s50/s46/s48 FIG. 3. (Color online) The sublattice magnetization S of the system with SOC obtained by the mean-field approach. αis the strength of the reduced spin-orbit coupling. sublattice magnetization S: S=1 N/angbracketleftbig G/vextendsingle/vextendsingleSz Q/vextendsingle/vextendsingleG/angbracketrightbig =−1 N/summationdisplay k′/parenleftbigg∆ Ek,++∆ Ek,−/parenrightbigg =−2∆ U.(5) We show the relationship between Sand the Hubbard interaction U/tin Fig. 3. It must be pointed out that the MFA is more applicable when the Hubbard interac- tion is strong, so U/t >1 is shown. It does not imply that a critical value of Hubbard interaction is defined. As shown, Sincreases as the Hubbard interaction Uis enhanced. To study the role of SOC theoretically, the strength of SOC is selected from 0 to 2. The results with α= 0 correspond to the absence of SOC8. IfUis fixed, we can find that Sdecreases with αincreased. Fig. 4 exhibits that the weaker the interaction Uis, the smaller SOC suppressing the magnetization to zero is. These re- sults suggest that the Hubbard interaction be beneficial to AFM order, while AFM order should be suppressed by SOC. The suppression of Sby SOC might be due to the width-broadening of the Hubbard bands by SOC. It is similar to the decreasing of Hubbard interaction. The system might be in favor of paramagnetic metal when U is small33. As shown in Fig. 3, the weaker Uis, the smallerSis. It is known that the MF approach can give a quali- tative description. To quantitatively study the effect of sublattice on the Hubbard interaction U, We have to dis- cuss the effect of fluctuations on the ground state. Based onthegroundstateofthemodel, thedefinitionsofcharge and spin correlation functions of electrons are Ref. [8], ¯χ00(q,q′,t) =i 2N/angbracketleftG|Tρq(t)ρ−q′(0)|G/angbracketright ¯χij(q,q′,t) =i 2N/angbracketleftG|TSi q(t)Sj −q′(0)|G/angbracketright,(6) whereρq=/summationtext k,αc† k+q,αck,αis the charge density oper- ator, and Si q=/summationtext k,α,βc† k+q,ασi αβck,βis the spin density4 /s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56/s49/s46/s48 /s32/s85/s47/s116 /s61 /s50/s46/s48 /s32/s85/s47/s116 /s61 /s53/s46/s48 /s32/s85/s47/s116 /s61 /s57/s46/s48 /s32/s32/s83 FIG. 4. (Color online) The sublattice magnetization S v.s. the strength of SOC αfor different Hubbard interaction U, which is obtained by mean-field approach.operator. |G/angbracketrightis the ground state defined by Eq. (4). In the case of half-filling, the non-vanishing terms of Eq. (6) are /angbracketleftBig fv† pλ(t1)fc kλ′(t1)fc† kλ′(t)fv pλ(t)/angbracketrightBig , (7) where,t < t1. The reason lies in that a particle only annihilatesfirstly, thencreatesinthefulledvalencebands and the process is just the opposite one in the empty conduction bands. Based on the transformation, Eq. (2), the correlation functions with SOC can be obtained as below, ¯χ00 0(q,ω) =/summationdisplay k,σG0 σσ(k+q,ω)G0 σσ(k+q,ω)+/summationdisplay kG0 ↑↓(k+q,ω)G0 ↓↑(k+q,ω)+/summationdisplay kG0 ↓↑(k+q,ω)G0 ↑↓(k+q,ω) =−1 2N/summationdisplay k′/parenleftBig 1+e−i(φk+q−φk)+ei(φk+q−φk)/parenrightBig/bracketleftbig ¯χ00 0f1(q,ω)+ ¯χ00 0f2(q,ω)+ ¯χ00 0f3(q,ω)+ ¯χ00 0f4(q,ω)/bracketrightbig .(8) ¯χ00 0f1(q,ω) =1 4/bracketleftbigg/parenleftbigg 1−ξk,+ξk+q,++∆2 Ek,+Ek+q,+/parenrightbigg/parenleftbigg1 ω−Ek,+−Ek+q,++iδ+1 −ω−Ek,+−Ek+q,++iδ/parenrightbigg/bracketrightbigg ¯χ00 0f2(q,ω) =1 4/bracketleftbigg/parenleftbigg 1−ξk,+ξk+q,−+∆2 Ek,+Ek+q,−/parenrightbigg/parenleftbigg1 ω−Ek,+−Ek+q,−+iδ+1 −ω−Ek,+−Ek+q,−+iδ/parenrightbigg/bracketrightbigg ¯χ00 0f3(q,ω) =1 4/bracketleftbigg/parenleftbigg 1−ξk,−ξk+q,++∆2 Ek,−Ek+q,+/parenrightbigg/parenleftbigg1 ω−Ek,−−Ek+q,++iδ+1 −ω−Ek,−−Ek+q,++iδ/parenrightbigg/bracketrightbigg ¯χ00 0f4(q,ω) =1 4/bracketleftbigg/parenleftbigg 1−ξk,−ξk+q,−+∆2 Ek,−Ek+q,−/parenrightbigg/parenleftbigg1 ω−Ek,−−Ek+q,−+iδ+1 −ω−Ek,−−Ek+q,−+iδ/parenrightbigg/bracketrightbigg .(9) The singularity of ¯ χ00 0fiis in the lower half plane when t >0. When t <0, the singularity is in the upper half plane. The longitudinal spin correlation function, ¯χzz 0(q,ω) =/summationdisplay σG0 σσ(k+q,ω)G0 σσ(k+q,ω)−/summationdisplay kG0 ↑↓(k+q,ω)G0 ↓↑(k+q,ω)−/summationdisplay kG0 ↓↑(k+q,ω)G0 ↑↓(k+q,ω) =−1 2N/summationdisplay k′/parenleftBig 1−e−i(φk+q−φk)−ei(φk+q−φk)/parenrightBig/bracketleftbig ¯χ00 0f1(q,ω)+ ¯χ00 0f2(q,ω)+ ¯χ00 0f3(q,ω)+ ¯χ00 0f4(q,ω)/bracketrightbig .(10) We can find that ¯ χzz 0(q,ω) is different from ¯ χ00 0(q,ω), other than the case without SOC, where ¯ χzz 0(q,ω)=¯χ00 0(q,ω). This is because G0 ↑↓andG0 ↓↑are non-zero as the spins mixed by SOC. The transverse spin corre lation function are presented as follows, ¯χ+− o(q,ω) =−1 2N/summationdisplay k′/bracketleftBig ¯χ+− of1(q,ω)+ ¯χ+− of2(q,ω)+ ¯χ+− of3(q,ω)+ ¯χ+− of4(q,ω)/bracketrightBig (11) ¯χ+− Q(q,ω) =−1 2N/summationdisplay k′/bracketleftBig ¯χ+− Qf1(q,ω)+ ¯χ+− Qf2(q,ω)+ ¯χ+− Qf3(q,ω)+ ¯χ+− Qf4(q,ω)/bracketrightBig , (12)5 ¯χ+− of1(q,ω) =1 4/bracketleftbigg/parenleftbigg 1−ξk,+ξk+q,+−∆2 Ek,+Ek+q,+/parenrightbigg/parenleftbigg1 ω−Ek,+−Ek+q,++iδ+1 −ω−Ek,+−Ek+q,++iδ/parenrightbigg/bracketrightbigg ¯χ+− of2(q,ω) =1 4/bracketleftbigg/parenleftbigg 1−ξk,+ξk+q,−−∆2 Ek,+Ek+q,−/parenrightbigg/parenleftbigg1 ω−Ek,+−Ek+q,−+iδ+1 −ω−Ek,+−Ek+q,−+iδ/parenrightbigg/bracketrightbigg ¯χ+− of3(q,ω) =1 4/bracketleftbigg/parenleftbigg 1−ξk,−ξk+q,+−∆2 Ek,−Ek+q,+/parenrightbigg/parenleftbigg1 ω−Ek,−−Ek+q,++iδ+1 −ω−Ek,−−Ek+q,++iδ/parenrightbigg/bracketrightbigg ¯χ+− of4(q,ω) =1 4/bracketleftbigg/parenleftbigg 1−ξk,−ξk+q,−−∆2 Ek,−Ek+q,−/parenrightbigg/parenleftbigg1 ω−Ek,−−Ek+q,−+iδ+1 −ω−Ek,−−Ek+q,−+iδ/parenrightbigg/bracketrightbigg ,(13) ¯χ+− Qf1(q,ω) =∆ Ek,+/parenleftbigg1 ω−Ek,+−Ek+q,++iδ−1 −ω−Ek,+−Ek+q,++iδ/parenrightbigg ¯χ+− Qf2(q,ω) =∆ Ek,+/parenleftbigg1 ω−Ek,+−Ek+q,−+iδ−1 −ω−Ek,+−Ek+q,−+iδ/parenrightbigg ¯χ+− Qf3(q,ω) =∆ Ek,−/parenleftbigg1 ω−Ek,−−Ek+q,++iδ−1 −ω−Ek,−−Ek+q,++iδ/parenrightbigg ¯χ+− Qf4(q,ω) =∆ Ek,−/parenleftbigg1 ω−Ek,−−Ek+q,−+iδ−1 −ω−Ek,−−Ek+q,−+iδ/parenrightbigg . (14) In the limit of α= 0, all the correlation functions are the same as Ref. (8). The subscript oandQin Eq. (13) and (14) are q′=qandq′=q+Qrespectively. χ+− 0(q,q′;ω) =χ+− o(q;ω)δ(q′−q) +χ+− Q(q,ω)δ(q′−q+Q),(15) is a 2×2 matrix. With RPA, the susceptibility is obtained by the follow- ing formulas, ¯χ00 RPA(q,ω) =/summationdisplay i¯χ00 0fi(q,ω) 1+U¯χ00 0fi(q,ω), (16) ¯χzz RPA(q,ω) =/summationdisplay i¯χzz 0(q,ω) 1−U¯χzz 0(q,ω), (17) ¯χ+− RPA(q,ω) =/summationdisplay i/summationdisplay q′χ+− ofi(q,q′,ω)(1−Uχ+− 0fi(q,q′,ω))−1.(18) To investigate the effect of SOC on the spin fluctuations, the energy spectrum of transverse spin fluctuations is de- duced by calculating the pole of transverse dynamical spin susceptibility, as shown in Fig. 5. In the absence of SOC, there is a gapless point at q=( π,π), which is con- sistent with the Goldstone theorem34. However, a gap opens in the presence of SOC, even though the AFM or- der remains. The similar phenomenon is also reported in Ref. [35] which have discussed the magnetic excitation in Sr2IrO4. A gap will open at the (0, 0) point, which is due to the spin-orbit coupling. That the gap opens in the presence of SOC might result from that SOC breaks the continuous symmetry36, so Goldstone theory is not applicable./s48/s49/s50 /s32 /s61 /s48 /s32 /s61 /s48/s46/s53 /s32 /s61 /s49/s46/s48 /s32 /s61 /s49/s46/s53/s32 /s40/s48/s44/s32/s48/s41/s40 /s44/s32 /s41/s40/s48/s44/s32 /s41/s40/s48/s44/s32/s48/s41/s32/s32 /s113 /s32/s32/s32 FIG. 5. (Color online)The energy spectrum of transverse spi n fluctuation with different strength of SOC, which is along the symmetry route (0 ,0) →(0,π)→(π,π)→(0, 0). To study the effect of the fluctuations on the sublattice magnetization, we use the definition of sublattice mag- netization according to Ref. [8], S=−i N/summationdisplay k′/integraldisplaydω 2πTr/bracketleftbig σzG0(k,k+Q;ω)/bracketrightbig (19) With ignoring fluctuations, we define the single Green function with respect to the ground state, |G/angbracketright, G0(k,k+Q;ω) =∆σz ω2−E2 k++iδ+∆σz ω2−E2 k−+iδ.(20) In this case, S=−1 N/summationdisplay k′/parenleftbigg∆ Ek++∆ Ek−/parenrightbigg =−2∆ U, (21)6 which is the same as the result of the MFA, Eq. (5). When the fluctuations are considered in our model, the fullGreen’sfunctioncanbeobtainedbyDyson’sequation with the self-energy which is established by Eq. (16-18). 1 Gαβ(k,k′;ω)=1 G0 αβ(k,k′;ω)−Σαβ(k,k′;ω),(22) wherek′=k+Q. Replacing G0by the full Green func- FIG. 6. The Feynman diagram of Green function mod- ified by longitudinal spin-fluctuations and transverse spin - fluctuations. The double lines are the single-particle Gree n function of the quasi-particles tionGin Eq.(21), the numerical results of the sublattice magnetization is shown in Fig. 7. The magnetization is suppressed by the SOC, which agreeswith the results ob- tained by MFA. Comparing with the MFA, the value of Sis smaller with the same strength of SOC. /s48 /s50 /s52 /s54 /s56 /s49/s48/s48/s46/s48/s48/s46/s50/s48/s46/s52/s48/s46/s54/s48/s46/s56 /s32 /s61/s48 /s32 /s61/s49/s46/s48 /s32 /s61/s49/s46/s53 /s32 /s61/s50/s46/s48 /s32/s32/s83 /s85/s47/s116 FIG. 7. (Color online) The sublattice magnetization S is af- fected by fluctuations for different SOC, which are obtained by RPA. The ratio of the transverse susceptibility to the longi- tudinal one with q= (π,π) is calculated. The depen- dence of ratio on the SOC is exhibited by Fig. 8. The Hubbard interaction U= 9.7tis selected to correspond with Ref. [37], which suggeststhat the anisotropyofanti- ferromagnetic fluctuations of Ba 0.68K0.32Fe2As2may be/s48/s46/s48 /s48/s46/s53 /s49/s46/s48 /s49/s46/s53 /s50/s46/s48/s49/s46/s48/s49/s46/s50/s49/s46/s52/s49/s46/s54 /s32/s32/s32/s43/s45 /s40/s81 /s41/s47/s122/s122 /s40/s81 /s41/s32/s85/s61/s51/s46/s48/s116 /s32/s85/s61/s53/s46/s48/s116 /s32/s85/s61/s57/s46/s55/s116 FIG. 8. (Color online) The ratio of the transverse and longi- tudinal susceptibility for different Hubbard interactions . The triangle, circle and rhombus are together on ”1” when the strength of SOC is zero. due to the spin-orbit coupling. The parent material of Ba0.68K0.32Fe2As2emerges AFM order and can be re- graded as quasi two-dimensional square lattice31, which can be illuminated via our model. The role of SOC in the anisotropy of susceptibility for Sr2RuO4 was also re- portedbyEremin et al.38, whichisagreedwith ourstudy. Our calculation shows that 1) the ratio is 1 when α=0, that is the spin fluctuations are isotropic in the absence of the spin-orbit coupling; 2) the ratio is larger than 1 with the increasing of the coupling, which indicates that the spin-orbit coupling can result in an anisotropy of the spin-fluctuations which agrees with Ref. [37]; 3) ratio de- creases with the Hubbard interaction decreased when we fix the strength of coupling. IV. COOPER PAIRING NEAR HALF-FILLING In the absence of SOC, the anti-ferromagnetic fluc- tuations of the weak holes doped Hubbard model fa- vors d-wave paring, which has been used to give an in- terpretation on the cuprates superconductors8. In this section, the effective interaction intermediated by anti- ferromagnetic spin fluctuations with SOC is studied and the symmetry of Cooper pairs is discussed. We now assume that the system is weak holes doped. At half- filling, the valence band is full and the conduction band is empty. The electronsofthe top ofthe valence band are removed in the case of weak hole doping, and the system canemergea metallic behaviorand superconductivityre- sulted from the holes. First, the BCS type Hamiltonian7 is, Hint=1 4N/summationdisplay k,k′/summationdisplay αα′ ββ′Vc(k′,k)δα′αδβ′βc† k′α′c† −k′β′c−kβckα −1 4N/summationdisplay k,k′/summationdisplay αα′ ββ′Vz(k′,k)σz α′ασz β′βc† k′α′c† −k′β′c−kβckα −1 4N/summationdisplay k,k′/summationdisplay αα′ ββ′V+−(k′,k)σ+ α′ασ− β′βc† k′α′c† −k′β′c−kβckα. (23) whereVc(k′,k) is induced by charge-fluctuations and Vz(k′,k) andV+−(k′,k) are caused by longitudinal and transverse fluctuations, respectively. Vc(k′,k) = 2U−¯χ00 RPA(k′,k), Vz(k′,k) = ¯χzz RPA(k′,k), V+−(k′,k) = ¯χ+− RPA(k′,k). (24) It iswellknownthat the symmetryofpairscanbeshown by the energy gap of superconductors, ∆ SC, defined by BCS theory as, ∆SC(k) =/summationdisplay k′V(k′,k)∆SC(k′)/radicalbig Ek+∆SC(k′).(25) Based on Eq. (25), we can find that the symmetry of ∆SC(k) is the same as the interaction on V(k). In our model, the function ∆ SC(k) cannot be known directly, so the self-consistent equation is too difficult to solve. And thus we can discuss the symmetry of gap according to the symmetry of the interaction. To find out domi- nant channels of interaction, the Vzis expanded in par- tial waves5,39. For three dimensions, the interaction can be expressed in terms of spherical harmonics function Ym l(x), V(k,k′) =/summationdisplay l,m(2l+1)Vm lYm l(θk,φk)Ym l(θk′,φk′), Vm l=/integraldisplay dθdφV(k,k′)Ym l(θk,φk)Ym l(θk′,φk′).(26) where,θandφare solid angles of− →k. Our model is twodimensional, sowe haveto use LegendrePolynomials Pl(x) to express the interaction. V(k,k′) =/summationdisplay l(2l+1)VlPl(cosθk)Pl(cosθk′), Vl=/integraldisplay1 −1dcosθkdcosθk′V(k,k′)Pl(cosθk)Pl(cosθk′). (27)where, cos θ=kx//radicalBig k2x+k2y. Accordingto the BCS-type Hamiltonian, the interactions can be obtained as follow, Hint=/summationdisplay k,k′VS(k,k′)c† k′,↑c† −k′,↓c−k,↓ck,↑ +/summationdisplay k,k′,σVT(k,k′)c† k′,σc† −k′,σc−k,σck,σ.(28) The spin-singlet interaction VS land triplet VT l, which consist of the interactions arising from the charge, longi- tudinal and transverse spin fluctuations, can be obtained by, VS(k,k′) =Vc(k,k′)+Vz(k,k′)−2V+−(k,k′), VT(k,k′) =Vc(k,k′)−Vz(k,k′). (29) Tostudy the effect ofSOC on Vl, wenumericallycalcu- lated the strength of l= 0,1,2, that is s-, p- and d-wave channel of interaction, with the strength of SOC α=0.01, 0.1, 0.2, 0.3, 0.5, 0.8, 1.0. The relation of the strength of partial wave for interaction to SOC has been shown in Fig. 9. We can find that the values of l= 0 and l= 1 are negative and l= 2 is positive. In order to facilitate comparison, the results without SOC are also calculated. The strength of s-wave potential, l= 0, is about -0.17, and p-wave, l= 1, is position and very small, 0.05. For l= 2, d-wave, the strength is about -0.14. It means that SOC is in favor of s- and p-wave attractive interactions mediated by AFM fluctuations other than d-wave, which is different from the case without SOC3,8, where d-wave pairing is dominant. Meanwhile, the strength of p-wave potential is as strong as s-wave. So the spin-orbit cou- plingcouldbringoutthemixtureofspin-singletandspin- triplet Cooper pairs and the orbital degree of freedom is an admixture of s+p-wave. As mentioned in the intro- duction, SOC could lead to the mixture of spin-singlet and spin-triplet Cooper pairs22. For spin-singlet Cooper pairs, the space wave function should be symmetric, for example, s-wave or d-wave, and for triplet, the space wave function should be antisymmetric, e.g. p-wave or f-wave. Our calculations suggest that s+p-wave Cooper pair be favorable with respect to the model considered in our works.It may be helpful in understanding the pairing symmetry ofNCS superconductors. Many workson NCS superconductors have reported that the spin degree of freedom is the mixture of spin-singlet and triplet27,40,41, however a consensus on the symmetry of orbital degree of freedom has not been reached. According to our cal- culations, SOC tends to form the s+p-wave Cooper pairs which is mediated by AFM fluctuations when the Hub- bard model is adopted, which is agreed with Ref. [28] and Ref. [29]. In contrast to the two papers, we cal- culate the partial waves of the effective interaction to study the pairing symmetry, which is more direct than them. Ref. [26] and [41] suggest that SOC should induce d+f or s+f pairing states, but their models are different from ours. For the two-band models, some papers also indicate that SOC could play an important role in the8 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s50/s50/s45/s48/s46/s50/s48/s45/s48/s46/s49/s56/s45/s48/s46/s49/s54/s45/s48/s46/s49/s52/s45/s48/s46/s49/s50/s45/s48/s46/s49/s48/s86/s83 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s45/s48/s46/s51/s48/s45/s48/s46/s50/s53/s45/s48/s46/s50/s48/s45/s48/s46/s49/s53/s45/s48/s46/s49/s48/s45/s48/s46/s48/s53/s48/s46/s48/s48/s86/s80 /s48/s46/s48 /s48/s46/s50 /s48/s46/s52 /s48/s46/s54 /s48/s46/s56 /s49/s46/s48/s48/s46/s48/s54/s48/s46/s48/s56/s48/s46/s49/s48/s48/s46/s49/s50/s48/s46/s49/s52/s48/s46/s49/s54/s86/s68 FIG. 9. (Color online) The strength of interactions for l=0, 1, 2, which corresponds to s-, p-, and d-wave pairing, respec tively. /s48/s51/s48/s54/s48/s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48 /s50/s49/s48 /s50/s52/s48 /s50/s55/s48/s51/s48/s48/s51/s51/s48/s48/s46/s49/s48/s48/s46/s49/s53/s48/s46/s50/s48 /s48/s46/s49/s48 /s48/s46/s49/s53 /s48/s46/s50/s48/s86/s83/s32 /s61/s48/s46/s48 /s32 /s61/s48/s46/s49 /s32 /s61/s48/s46/s50 /s32 /s61/s48/s46/s51 /s32 /s61/s48/s46/s53 /s32 /s61/s48/s46/s56 /s32 /s61/s49/s46/s48 /s48/s51/s48/s54/s48/s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48 /s50/s49/s48 /s50/s52/s48 /s50/s55/s48/s51/s48/s48/s51/s51/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50/s48/s46/s51 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50 /s48/s46/s51/s43 /s45/s32 /s61/s48/s46/s48 /s32 /s61/s48/s46/s49 /s32 /s61/s48/s46/s50 /s32 /s61/s48/s46/s51 /s32 /s61/s48/s46/s53 /s32 /s61/s48/s46/s56 /s32 /s61/s49/s46/s48/s86/s80/s48/s51/s48/s54/s48/s57/s48 /s49/s50/s48 /s49/s53/s48 /s49/s56/s48 /s50/s49/s48 /s50/s52/s48 /s50/s55/s48/s51/s48/s48/s51/s51/s48/s48/s46/s48/s48/s46/s49/s48/s46/s50 /s48/s46/s48 /s48/s46/s49 /s48/s46/s50/s32 /s61/s48/s46/s48 /s32 /s61/s48/s46/s49 /s32 /s61/s48/s46/s50 /s32 /s61/s48/s46/s51 /s32 /s61/s48/s46/s53 /s32 /s61/s48/s46/s55 /s32 /s61/s48/s46/s56 /s32 /s61/s49/s46/s48 /s45/s45 /s43/s43/s86/s68 FIG. 10. (Color online) The dependence of interactions on th e angle of momentum for l=0, 1, 2, which corresponds to VS, VP, and VD, respectively. p-wave pairing state. Sigrist et al.25and Annet et al.42 havestudiedt-JmodelandanattractiveHubbardmodel, respectively. Both of them stated that the chiral p-wave stateofSr 2RuO4shouldbeduetoSOC.It seemsthatthe effect of SOC on pairing symmetry might be dependent on the individual models. For the s- and p-wave, the strength of interaction de- creases with the increasing of SOC when the strength is strong. It implies that the large SOC might be bad for the superconductivity. However, the interaction in- creases initially with SOC and decreases afterwards. It seems that the SOC suppresses the magnetization, which possibly enhance the spin fluctuations. Furthermore, the interaction induced by spin fluctuations is promoted. When the strength of SOC is very large, all the pair- ing potentials are suppressed by SOC. It indicates that large SOC is bad for superconductivity which is agreed with the dependence of critical temperature on SOC43. Fig. 10 illustrate the dependence of interactions on the momentum. According to Eq. (25), it could describe the symmetry of the gap of superconductivity. Obviously, the s-wave is angular-isotropy, and it may be a conven- tional s-wavestate. For p-wave, the state should be l= 1 andm= 0 in terms of the Legendre function of two di- mension. And d-wave potential is also m=0, which is the same as p-wave.V. CONCLUSION In summary, we have studied the ground state of the two-dimensional Hubbard model with Rashba SOC on a square lattice. Both the results obtained by MFA and RPA show that the sublattice magnetization decreases with the increasing of SOC for a fixed Hubbard inter- action. Moreover, the magnetization for RPA is smaller than MFA with the same U and α. The suppression of AFM order caused by SOC might be resulted from that SOC broadens the sub-Hubbard bands. Besides, a gapped energy spectrum of transverse spin fluctua- tions and an anisotropy of spin susceptibility, which are brought about by SOC, are present. Furthermore, we have discussed the effective pairing interactions between electrons mediated by AFM spin fluctuations in the case of weak hole doping. The calcu- lations about the partial waves of interactions indicate that p-wave potential can be induced by SOC. The d- wave potential which is dominant without SOC is sup- pressed by SOC. Moreover, the strength of s-wavealways exists whether SOC is present or not. It seems that the SOC tends to form s+p paring rather than s+d pairing. Note added .We are just aware of a numerical work studying the mechanism of p-wave Cooper pairs44. They state that the degeneracy of various p-wave states is split by the magnetic anisotropy. The anisotropy might result from SOC. This paper indicates that the symmetry of Cooper pairs mediated by spin-fluctuations is still a hot topic.9 ACKNOWLEDGMENTS This work is supported by the National Natural Sci- ence Foundation of China (Grant No. 11274039)and theFundamentalResearchFundsfortheCentralUniversities of China. ∗Corresponding author: qgu@ustb.edu.cn 1D. J. Scalapino, Rev. Mod. Phys. 84, 1383 (2012). 2F. Essenberger, A. Sanna, A. Linscheid, F. Tandetzky, G. Profeta, P. Cudazzo, and E. K. U. Gross, Phys. Rev. B 90, 214504 (2014). 3A. T. Rφmer, I. Eremin, P. J. Hirschfeld, and B. M. An- dersen, Phys. Rev. B 93174519 (2016). 4D. D. Osheroff, Rev. Mod. Phys. 69, 667 (1997). 5D. Fay and J. Appel, Phys. Rev. B 22, 3173 (1980). 6S. S. Saxena, P. Agarwal, K. Ahilan, F. M. Grosche, R. K. W. Haselwimmer, M. J. Steiner, E. Pugh, I. R. Walker, S. R. Julian, P. Monthoux, G. G. Lonzarich, A. Huxley, I. Sheikin, D. Braithwaite and J. Flouquet, Nature 406, 587 (2000). 7T. Hattori, Y. Ihara, Y. Nakai, K. Ishida, Y. Tada, S. 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1711.03255v2.Relativistic_Spin_Orbit_Interactions_of_Photons_and_Electrons.pdf

Relativistic spin-orbit interactions of photons and electrons D. A. Smirnova,1,V. M. Travin,2, 3,K. Y. Bliokh,1, 4,and F. Nori4, 5 1Nonlinear Physics Centre, RSPE, The Australian National University, Canberra, ACT 0200, Australia 2V. N. Karazin Kharkov National University, 4 Svobody Sq., Kharkov 61022, Ukraine 3Institute for Low Temperature and Structure Research, Polish Academy of Sciences, POB 1410, 50-950 Wroclaw, Poland 4CEMS, RIKEN, Wako-shi, Saitama 351-0198, Japan 5Physics Department, University of Michigan, Ann Arbor, Michigan 48109-1040, USA Laboratory optics, typically dealing with monochromatic light beams in a single reference frame, exhibits numerous spin-orbit interaction phenomena due to the coupling between the spin and orbital degrees of freedom of light. Similar phenomena appear for electrons and other spinning particles. Here we examine transformations of paraxial photon and relativistic-electron states carrying the spin and orbital angular momenta (AM) under the Lorentz boosts between dierent reference frames. We show that transverse boosts inevitably produce a rather nontrivial conversion from spin to orbital AM. The converted part is then separated between the intrinsic (vortex) and extrinsic (transverse shift or Hall eect) contributions. Although the spin, intrinsic-orbital, and extrinsic-orbital parts all point in dierent directions, such complex behavior is necessary for the proper Lorentz trans- formation of the total AM of the particle. Relativistic spin-orbit interactions can be important in scattering processes involving photons, electrons, and other relativistic spinning particles, as well as when studying light emitted by fast-moving bodies. I. INTRODUCTION In the past decade, the spin-orbit interactions (SOIs) of light | including spin Hall eects, spin-to-orbital an- gular momentum (AM) conversions, etc. | have be- come an inherent part of modern optics, see [1{5] for reviews. The vast majority of known SOI eects origi- nate from the fundamental polarization and AM prop- erties of monochromatic Maxwell elds in a single lab- oratory reference frame [1, 6]. Similar phenomena have also been described for relativistic electrons and other spinning particles [7{12]. Electron SOIs play an impor- tant role in atomic physics, condensed matter, and could also aect the dynamics of relatvistic free-electron states carrying intrinsic AM [13{15]. At the same time, there is considerable recent interest onrelativistic transformations of photons and other par- ticles carrying intrinsic AM [16{20]. Transverse Lorentz boosts of wave beams break down their monochromatic- ity and induce a number of nontrivial relativistic AM- dependent phenomena. In particular, the Lorentz trans- formations of the intrinsic and extrinsic AM dier sig- nicantly from each other. Requiring their consistency brings about the relativistic Hall eect (i.e., the boost- induced transverse position shift) related to the delocal- ized nature of the wave AM [16{20]. Relativistic properties and transformations of AM- carrying waves are important from both the fundamen- tal and practical viewpoints. These are involved in the \proton spin puzzle" in QCD [21, 22], studies of \chi- ral fermions" [17, 18], and collisions of spinning parti- cles [17, 19]. Moreover, there is a rapidly growing in- These authors contributed equally to this work.terest in scattering of photons, electrons, and other high- energy particles carrying intrinsic orbital AM [14, 23{25]. Naturally, the Lorentz transformations of wavepackets or beams carrying spin and orbital AM are of great impor- tance for these topics. Importantly, most of the recent studies of Lorentz transformations of the wave AM considered the intrin- sic spin (polarization) and orbital (vortex) AM on equal footing. For example, the Hall-eect shift of the energy centroid is largely independent of the spin or orbital na- ture of the intrinsic AM [16, 17, 19, 20, 26]. However, in this paper, we show that the spin and orbital AM of photons and relativistic electrons are transformed quite dierently under Lorentz boosts. To illustrate this crucial dierence, we put forward the following paradox about the transformation of the spin of a photon. The spin AM of a paraxial photon can be well approx- imated by the plane-wave expression S=~P=P[27{ 29], where 2(1;1) is the polarization helicity and Pis the photon momentum. Assuming P=Pz(over- Figure 1. Transverse Lorentz transformation of the photon spin. The spin of an electromagnetic wave, S, is rotated by the angle= sin1u(a), which is in contrast to the Lorentz transformation of a relativistic AM J(provided the boost momentum N=0):J0= J(b) [16, 30].arXiv:1711.03255v2 [physics.optics] 28 Jan 20182 bars denote the unit vectors of the corresponding axes), we perform the transverse Lorentz boost characterized by the velocity v=vxand the corresponding Lorentz factor = 1=p 1u2,uv=c. This transformation preserves the helicity (which is Lorentz-invariant for massless particles) and rotates the propagation direction by the angle= sin1u= cos1 1. As a result, the photon spin in the boosted reference frame (indicated by primes) becomes S0=~[ 1z0ux0], Fig. 1(a). However, this contradicts the Lorentz transformations of the rel- ativistic AM tensor, which consists of the the AM Jand the \boost momentum" N[30]. Indeed, assuming that the photon is represented by a large paraxial wavepacket (close enough to the plane wave) with energy W=~! (!=kcis the frequency, kis the wavevector), momen- tumP=~kz, and position of the centroid R=ctz, the boost momentum vanishes in the original reference frame: N=ctPRW=c =0. Then, the Lorentz trans- formations of the AM J=Jzyields J0= J[16, 30, 31], Fig. 1(b). Obviously, the photon spin AM Scannot fol- low this rule because >1, while the spin is restricted to the (~;~) range. In a more fundamental context, the dierence between the spin and AM transformations come from the fact that the spin of a relativistic particle follows the Pauli-Lubanski four-vector rather than the rank-2 AM tensor [32{34]. In this paper, we resolve the above controversy by con- sidering trasversely-localized optical beams carrying both spin and orbital AM. We derive quite nontrivial Lorentz transformations of the spin and orbital AM carryied by paraxial photons, as well as the relativistic Hall-eect shifts caused by photon's spin and orbital AM. We nd that Lorentz boosts inevitably produce spin-to-orbital AM conversion as well as nontrivial spin and orbital Hall- eect shifts, i.e., relativistic SOIs . We also perform anal- ogous Lorentz-boost calculations for the Dirac-electron beams, and show that most of their AM transformation features are similar to the photon case, albeit modied by the nite electron mass. II. RELATIVISTIC TRANSFORMATIONS OF OPTICAL BEAMS A. General formalism We rst introduce the general formalism for calcula- tions of dynamical properties (energy, momentum, AM, etc.) of generic free-space Maxwell elds. This is mosty based on the results of works [6, 35, 36]. The real electric and magnetic elds E(t;r) andH(t;r) are represented via their complex Fourier (plane-wave) components:  E(t;r) H(t;r) =2ReZd3k (2)3=2 E(k) H(k) ei!t+ik
r;(1) where!(k) =kc. Due to Maxwell's equations, the Fourier components are orthogonal to the wavevector:E
k=H
k= 0, and it is instructive to make a transfor- mation to the local k-space coordinates with the longi- tudinal axis attached to the wavevector. The elds have only two transverse components in these coordinates, and using the basis of circular polarizations corresponds to thehelicity representation of Maxwell elds. The transition to this basis is realized by the unitary transformation [6] n ~E(k);~H(k)o =^V^U(k)fE(k);H(k)g: (2) Here ^U(k) =^Rz(')^Ry(#)^Rz(') =0 @12 sin2# 2cos2'sin2# 2sin 2'sin#cos' sin2# 2sin 2'12 sin2# 2sin2'sin#cos' sin#cos' sin#sin' cos#1 A is the rotational matrix superimposing the longitudinal axis with the wavevector (( #;') are the spherical angles of the k-vector and ^Ry;zare the corresponding rotational matrices), whereas ^V=1p 20 @1i0 1i0 0 0p 21 A: is the constant matrix of the transition to the circular- polarization basis. Omitting the vanishing longitudinal component of the elds (2), we end up with the two-component electric eld ~E= ~E+;~ET and the corresponding magnetic eld ~H=i^~Efollowing from Maxwell's equations. Here ^= diag(1;1) is the helicity operator and throughout the paper we use Gaussian-like units with "0=0= 1. We now dene the \ photon wavefunction " [35, 36] in the helicity representation as (k) =1p 2Nh ~E(k) +i^~H(k)i =r 2 N~E(k);(3) where the normalization factor Nis the number of pho- tons dened below. Then, the expectation value of an operator ^Ocan be calculated as [35, 36] O=h j^Oj iZd3k ~! y(k)
(^O) (k): (4) Note that the factor !1(k) in Eq. (4) is crucial for non- monochromatic elds. Assuming the one-photon normal- izationh j i= 1, the number of photons in Eq. (3) is N= 2Rd3k ~!j~E(k)j2. For further calculations, we need operators of the en- ergyW, momentum P, position R, spin AM S, orbital AML, and boost momentum N(see [16, 30, 37{39] for the latter quantity). According to the works [6, 35, 36], in the helicity representation, projected on the 2D subspace3 of Maxwell elds ~E(k), these operators read ^W=~!;^P=~k;^R=irk^AB(k); ^S=~^k k;^L=^R^P;^N=ct^P^R^W=c: (5) Here ^AB(k) = ^k1[(1cos#)=sin#]'('is the unit vector of the azimuthal coordinate ') is the Berry con- nection , which determines the covariant derivative and parallel transport of E(k)?kon the sphere S2=fk=kg (the electric eld E(k) belongs to the vector ber bundle over this sphere) [1, 3, 6, 36, 40]. The operators ^W,^P, the total AM ^J=^S+^L, and ^Nprovide 10 generators of the Poincar e group , and their expectation values are conserved in free space [35, 36, 38, 39]. Note that the expectation value of the position op- erator (5), R, describes the \ photon centroid ", while theenergy centroid of the eld is determined as RE= c(ctPN)=W. These two positions can dier from each other in non-monochromatic elds; they play a crucial role in the Lorentz transformations of the AM and rela- tivistic Hall eects [16, 17, 19, 26, 30]. We also note that the same expectation values (4) can be obtained without transition to the helicity represen- tation (2). In the canonical momentum representation, the 6-component \photon wavefunction" is given by the Fourier components (1): can(k) =1p NfE(k);H(k)g; (6) whereN=Rd3k ~! jE(k)j2+jH(k)j2
. In this represen- tation, the operators (5) have canonical form without the Berry connection: ^Rcan=irk;^Lcan=^Rcan^P; ^Ncan=ct^P^Rcan^W=c; (7) whereas the spin operator ^Scan is given by the momentum-independent spin-1 3 3 matrices [6, 29, 35]. Although the canonical operators have simpler form, the canonical photon wavefunction (6) is considerably com- plicated, having six components instead of two. There- fore, below we employ the helicity representation (2){(5) for photonic calculations, but use the canonical represen- tation analogous to Eqs. (7) for the Dirac-electron calcu- lations in Section III. In addition to the expectation values of operators (5) in the momentum representation, we will use the spatial energy and Poynting-momentum densities in the coordi- nate representation of real elds (1) [30]: w= E2+H2
=2;p=c1(EH): (8) The integral energy, momentum, total AM, and boost momentum of a localized eld are then determined as W=R wd3r,P=R pd3r,J=R (rp)d3r, and N=R (ctprw=c)d3r. For a one-photon eld, these values are equivalent to the corresponding expectation values calculated using Eqs. (3){(5).We nally describe the Lorentz boosts of a generic elec- tromagnetic eld. The real elds fE(t;r);H(t;r)gare transformed as components of the antisymmetric rank- 2 eld tensor, together with the Lorentz transformation of the four-coordinates ( ct;r) [30]. The Fourier com- ponentsfE(k);H(k)gacquire the extra factor 1, be- cause the dierential in the integrals (1) is transformed asd3k0= d3kdue to the Lorentz contraction. Consid- ering the boost with the velocity v=vx, this yields: E0 x= 1Ex; H0 x= 1Hx; E0 y=EyuHz; H0 y=Hy+uEz; E0 z=Ez+uHy; H0 z=HzuEy: (9) This eld transformation is accompanied by the Lorentz boost of the four-wavevector ( !=c;k): !0= (!vkx); k0 x=  kxv! c2 ; k0 y=ky; k0 z=kz: (10) The boosted elds in the helicity representation,n ~E0(k0);~H0(k0)o , are obtained from the elds (9) via the unitary transformation (2) involving the boosted wavevectors k0(10) and the corresponding spherical an- gles (#0;'0). B. The Lorentz boost of a Bessel beam We are now in the position to consider a photon state carrying spin and orbital AM, the simplest model of which being provided by monochromatic Bessel beams [6, 14, 41]. The Fourier spectrum of the z-propagating Bessel beam is a circle lying on the sphere of radius k=k0=!0=cat the polar angle #=#0, see Fig. 2(a). Assuming well-dened helicity =1 (i.e., the same right-hand or left-hand circular polarizations of all plane waves in the beam spectrum), the electric eld of the Bessel beam can be written as [6]: ~E(k) =A 2 1 + 1 (kk0)(##0)ei`';(11) whereAis the eld amplitude, is the delta-function, and exp(i`) indicates a vortex with the integer topolog- ical charge`, which is responsible for the intrinsic orbital AM carried by the beam [27{29]. Using Eq. (2), we obtain the Bessel-beam eld compo- nents in the Cartesian coordinates: E(k)=Ap 20 @abe2i' i a+be2i'
2p abei'1 A(kk0)(##0)ei`'; (12) wherea= cos2(#0=2) andb= sin2(#0=2). Since we are dealing with the helicity eigenstate, ^ ~E=~E, the corre- sponding magnetic eld is H(k) =iE(k).4 Figure 2. Monochromatic z-propagating Bessel beam (11) and (12) (a,b) and the same beam in the reference frame moving with velocity v=vx(c,d). The Fourier spectra (i.e., the wavevector distributions with color-coded vortex phases exp(i`')) (a,c) and the real-space distributions of the energy and Poynting-momentum densities (8) (b,d) are shown. One can see the non-monochromatic character ( !0=k0c6= const) of the boosted beam, its elliptic Lorentz-contraction deforma- tion, and the relativistic Hall-eect shift of the energy cen- troid:Y0 E= (v=!0)(`+). For better visibility, we used nonparaxial beams with the following parameters: = 1, `= 2, sin#0= 0:4 (a,c), sin#0= 0:7 (b,d),u= 0:8 (c,d), and =k0sin#0(b,d). Evaluating the Fourier integrals (1), we nd the real Bessel-beam elds E(t;r) and H(t;r), and plot the trasverse real-space distributions of the energy and Poynting-momentum densities (8) in Fig. 2(b). Parax- iality implies #01, but the Bessel beams are exact solutions of Maxwell's equations for any values of #0. Calculating the expectation values (3){(5) of the en- ergy, momentum, spin and orbital AM, etc., for the Bessel-beam eld (11), we obtain [6]: W=~!0;P=~k0cos#0z'~k0z; L=~[`+(1cos#0)]z'~`z;S=~cos#0z'~z; J=~(+`)z;R?=N?=0; (13) where we used the paraxial approximation #01, and the subscript?indicates the transverse ( x;y) compo- nents. Note that the Bessel beams are delocalized in the longitudinal z-direction and are not square-integrable in the transverse plane. Therefore all the integrals of squared elds and the normalization factor Ndiverge but their ratios (13) are nite [6, 16]. The longitudinal photon and energy centroid coordinates ZandZEare ill-dened in the beam, but if we were to consider a long z-localized wavepacket, we would approximately obtain Z=ZE= (c2P=W)t=ct. This corresponds to the van- ishing boost momentum Nz= 0. Equations (13) presentthe expected picture of a paraxial photon carrying intrin- sic spin () and orbital ( `) AM [29]. We now perform the Lorentz boost (10) and (9) of the Bessel beam (12). This brings about cumbersome but exact expressions for the boosted Bessel-beam elds fE0(k0);H0(k0)g, E0(t0;r0);H0(t0;r0) , and the corre- sponding helicity-representation eld ~E0(k0). Figures 2(c,d) show the Fourier spectrum and the real-space transverse distributions of the energy and Poynting mo- mentum densities (8) for these elds (cf. [16, 19, 42]). One can see that the boosted eld is not monochromatic anymore (!0=k0c6= const), it is elliptically deformed due to the Lorentz contraction, and its energy centroid is shifted in the transverse direction, Y0 E6= 0, which is a manifestation of the relativistic Hall eect [16, 17, 19, 26]. At the same time, since the helicity is Lorentz-invariant, the boosted eld is still the helicity eigenstate: ^ ~E0(k0) = ~E0(k0),=1, and H0(k0) =iE0(k0). Most importantly, we can now calculate the expecta- tion values (3){(5) for the boosted Bessel beam. In the #01 approximation, this yields: W0=~ !0;P0=~k0(z0 ux0); L0=~[` +( 1)]z0+~ux0; S0=~ 1z0ux0 ;J0=~ (`+)z0; (14) R0 ?=v 2!0(`+ 2)y0vtx0;N0 ?=~ u(`+)y0: These equations contain the central results of this work, which are also illustrated in Fig. 3. The energy and momentum (14) present the standard Lorentz trans- formation of the quantities (13): W0= W,P0= P Wv=c2. The boost momentum also agrees with the Lorentz transformation of the relativistic AM ten- sor [16, 30]: N0= Jv=c. This corresponds to the transverse Hall-eect shift of the energy centroid R0 E?+vt=Jv=W = (v=! 0)(`+)y0[Fig. 2(d)], in agreement with recent results [16] (for = 0) and [17{19] (for `= 0). At the same time, the AM parts and the \photon centroid" in Eqs. (14) exhibit several unusual features. First, the spin AM is indeed transformed as expected for a polarized plane wave: S0=~P0=P0, Fig. 1(a), and in contrast to the relativistic AM transformation, Fig. 1(b). Second, this paradox is resolved by the non- trivial transformation of the orbital AM L0, which ac- quires unexpected helicity-dependent terms , both longi- tudinal and transverse. This signals the relativistic spin- orbit interactions of light , cf. [1]. As a result, the to- tal AM J0=L0+S0is transformed exactly as expected for the relativistic AM with N=0:J0= J. Third, the photon centroid R0 ?exhibits the natural drift vt in the moving frame and the transverse Hall-eect shift Y0= (v=2!0)(`+ 2). This diers from the previously analyzed spinless and massive-particle cases [16, 26] by thefactor of 2 before the helicity [43]. This unexpected5 Figure 3. Transformations of the spin and orbital AM in a paraxial vortex beam under a transverse Lorentz boost (u= 0:6 here). (a) The original monochromatic beam car- ries the spin AM S=P=Pdue to the circular polarization (helicity) (= 1 here), as well as the intrinsic orbital AM Lint=`P=Pdue to the vortex ( `= 2 here). (b) The boosted beam carries spin AM tilted together with the beam momen- tum: S0=P0=P0[Fig. 1(a)], the intrinsic orbital AM Lint0, Eq. (15), due to the elliptically deformed and tilted vortex, and the extrinsic orbital AM caused by the transverse shift (Hall eect) of the beam centroid: Lext0=R0P0. Although all these contributions point in dierent directions, the total AM is transformed according to the Lorentz transformation: J0= J[Fig. 1(b)]. factor plays an important role in the Lorentz transforma- tions (14) of the photon AM. Indeed, the photon centroid R0allows us to sepa- rate the intrinsic (vortex-related) and extrinsic (shift- induced) contributions to the orbital AM [1, 6, 16, 29]: Lext0=R0P0=~(`+ 2) 1 2z0+u 2x0 ; Lint0=L0Lext0=~` + 1 2z0u 2x0 :(15) Here we used the longitudinal photon position Z0= 1ctbecause of the oblique propagation at the angle = cos1 1. Remarkably, the form of the intrinsic or- bital AM Lint0can be clearly explained by the geometric deformations of the vortex phase in the beam. Namely, the vortex is elliptically deformed due to the Lorentz con- traction with the factor of and also tilted by the angle , as shown in Fig. 3(b) (because the phase fronts in the boosted beam are near-perpendicular to the momen- tumP0). It is easy to show that these deformations, x!x= ,kx! kx, andz=xtan=u x!ux, result in the intrinsic orbital AM (15) for a vortex wavefunc- tion /(x+iy)`. Importantly, the x0-directed term inLint0, related to the tilt of the vortex, was missed in previous studies [16, 42] only focused on the longitudi- nalz0-component of the AM. The set of equations (14) and (15) show that both this new term and the factor of 2 before the helicity in the centroid shift R0ensure theproper Lorentz transformation of the total AM J. In addition to the analytical k-space calculations of the expectation values (14), we numerical calculated the val- uesW0,P0,J0, and N0using the r-space integration of the energy and Poynting-momentum densities (8) in the transformed Bessel beam. The results were in agreement with Eqs. (14). Here we should make two important re- marks. First, since Bessel beams are delocalized along the longitudinal z-axis, the integration should be per- formed over a 2D cross-section of the beam. In doing so, the result depends on the choice of the cross-section, sim- ilar to the \geometric spin Hall eect of light" [45, 46]. We found that the proper Lorentz transformation of the AM is obtained using the integration in the tilted plane z0=u x0parallel to the phase fronts (i.e., orthogonal to the momentum) of the boosted beam, Fig. 3(b). This is in agreement with the Wigner-translation approach used in [19]. In the k-space calculations of Eqs. (14) the tilted-cross-section condition was also used as @=@k0 z= u @=@k0 x. Second, we note that the Berry connection ^AB(k0) in the operators (5) played a crucial role in ob- taining the transformed quantities (14). In the paraxial limit#0!0, it is determined by the mean momentum P0and equals ^AB=^k1 0[(1 1)=( u)]y. This illu- minates the geometric SOI origin of the nontrivial trans- formations (14) [1, 6]. III. RELATIVISTIC TRANSFORMATIONS OF DIRAC-ELECTRON BEAMS A. General formalism It is interesting to check if the nontrivial transforma- tions (14) and (15) of the spin and orbital AM quantities are specic to photons (i.e., massless spin-1 particles) or these have a universal character. For this purpose, we consider a similar Lorentz-transformation problem for a Bessel-beam state of the Dirac electron [12], i.e., a mas- sive spin-1/2 particle. We rst recall the Dirac equation in the standard rep- resentation [32]: i~@ @t= ^
^Pc+^mc2 ; (16) where (r;t) is the four-component bi-spinor wavefunc- tion, ^P=i~ris the momentum operator in the coor- dinate representation, mis the electron mass, and ^= 0^ ^0 ; ^= 1 0 01 are the 44 Dirac matrices with ^being the vector of the 22 Pauli matrices. The wavefunction can be represented as the Fourier integral, i.e., as a superposition of Dirac plane waves: (r;t) =Zd3k (2)3=2~ (k)ei!t+ik
r: (17)6 Here,!(k) =p k2c2+2,=mc2=~, and the Fourier amplitudes can be factorized as: ~ (k)=f(k)(k);(k)=1p 2!p!+p!^
k
 ; (18) wheref(k) is the scalar Fourier amplitude, (k) is the normalized polarization bi-spinor ( y= 1), k=k=k, and= a b is the two-component polarization spinor (y= 1) describing the spin state of the plane-wave electron in its rest frame [12, 14, 32]. The Fourier amplitudes ~ (k) can be regarded as the (non-normalized) Dirac wavefunction in the canonical momentum representation. In this representation, the operators of the energy, momentum, position, spin, or- bital angular momentum, and boost momentum have a canonical form similar to Eq. (7): ^W=~!;^P=~k;^R=irk; (19) ^S=~ 2 0^ ^0 ;^L=^R^P;^N=ct^P^R^W=c: The normalized (one-electron) expectation values are cal- culated similar to Eq. (4): O=1 Nh~ j^Oj~ i=1 NZ d3k~ y(k)
(^O)~ (k);(20) with the number of electrons N=R d3kj~ (k)j2. Note that, in contrast to photons, the inner product for elec- trons does not involve the !1(k) factor. This is because the squared wavefunction amplitudes correspond to the particle and energy densities for electrons and photons, respectively. It is worth remarking that one can alternatively use the Foldy-Wouthuysen momentum representation for the calculation of the expectation values for Dirac electrons. This representation, diagonalizing the Dirac Hamilto- nian, allows one to reduce the wavefunction to the two components(k), but complicates the operators with the Berry-connection terms, similar to the helicity represen- tation (2) and (5) for photons [7, 8, 12, 34]. We nally introduce the Lorentz transformation (with the velocity v=vx) of the Dirac wavefunction ~ (k). Akin to the transformation of the Fourier components of Maxwell elds, Eq. (9), it acquires an extra 1factor and reads [32]: ~ 0=1p 2 p + 1p 1 ^x ~ : (21) The Lorentz transformation of the electron four- wavevector ( !=c;k) is still given by Eq. (10). B. The Lorentz boost of a Dirac-Bessel beam The Bessel-beam state of the Dirac electron (i.e., the Dirac-Bessel beam) is constructed similar to the opticalbeam (11) and (12), Fig. 2(a). The scalar part and the polarization spinors of the two spin states of the electron are given by [12, 14]: f(k)=A(kk0)(0)ei`;+= 1 0 ;= 0 1 : (22) These states correspond to the well-dened z-components of the electron spin, sz=1=2, in its rest frame. Al- ternatively, one can choose two states with well-dened helicity [14, 32], but these states reduce to the same  states in the paraxial approximation 01. Substsituting the wavefunction (18) and (22) into Eqs. (19) and (20), we obtain the expectation values of the energy, momentum, AM, etc. for the Bessel-Dirac electron in the paraxial limit: W=~!0;P=~k0z;R?=N?=0; L=~`z;S=~szz;J=~(sz+`)z: (23) This coinsides with Eqs. (13) with the only dierence that now!0=p k2 0c2+2, and the spin quantum number sz=1=2 substitutes the helicity =1. Note that the longitudinal boost momentum also vanishes, Nz= 0, when we assume the relativistic equation of motion Z= ZE= (c2P=W)t. Now, performing the Lorentz transformation (21) and (10) of the Dirac-Bessel wavefunction (18) and (22), we calculate the expectation values (19) and (20) in the boosted reference frame. Remarkably, this results in for- mulae very similar to photonic Eqs. (14): W0=~ !0;P0=~k0 z0 u!0 k0cx0 ; L0=~[` +sz( 1)]z0+~szuk0c !0x0; S0=~sz 1z0uk0c !0x0 ;J0=~ (`+sz)z0;(24) R0 ?=v 2!0(`+ 2sz)y0vtx0;N0 ?=~ u(`+sz)y0: The main dierence from the photonic case is that !0=k0c6= 1, and these factors modify the directions of the boosted momentum, spin, and orbital AM. One can see that the modied transformations (24) exactly correspond to the fact that the Pauli-Lubanski four-pseudovector (0;) = (S
P;SW=c) [32], orthog- onal to the electron four-momentum ( W=c; P), is trans- formed as a four-vector under Lorentz boosts. This has several consequences. First, the direction of the spin AM does not follow the momentum of the electron: S0,P0. Second, the absolute value of the spin AM diminishes: S0= (~=2)p 1u22=!2 0< S , which can be inter- preted as partial depolarization of the boosted electron. Finally, the transformation of the Pauli-Lubaski vector also describes the transformation of the electron helic- ity~=  0=P. In the original and boosted frames, the7 helicity becomes: =sz; 0=szq 1 +u22 k2 0c2; (25) which clearly indicates that the helicity is Lorentz- invariant only for massless particles. Akin to the photonic case, Eqs. (15), we separate the intrinsic (vortex-related) and extrinsic (shift-related) parts of the electron orbital AM: Lext0=R0P0=~(`+ 2sz) 1 2z0+u 2k0c !0x0 ; Lint0=L0Lext0=~` + 1 2z0u 2k0c !0x0 ;(26) where we used Z0= (c2P0 z=W0)t. Thus, the intrinsic and extrinsic orbital AM of electrons are also analogous to those of photons (up to modication by the k0c=!0 factors). In particular, as it should be, the vortex-related intrinsic orbital AM depends only on the vortex quantum number`and is independent of the spin sz. Similar to the case of optical beams, the expectation values of the ^R-dependent operators for electrons de- pend on the choice of the beam cross-section used in the integration. We found that Eqs. (24), consistent with the Lorentz transformations of the relativistic AM, are obtained only when choosing the tilted cross sec- tionz0=x0tanS, i.e.@=@k0 z= tanS@=@k0 x, where S= tan1(u k 0c=!0). Notably, this angle corresponds to the direction of the electron spin AM S0rather than momentum P0(for photons these directions coincide). Understanding this peculiarity requires further investiga- tions of properly 3D localized Dirac wavepackets, which is beyond the scope of this study. IV. DISCUSSION We have considered relativistic transformations of the spin and orbital AM of paraxial photons and Dirac elec- trons under the transverse Lorentz boost. The main re- sults are summarized in Eqs. (14), (15), (24), (26), and in Fig. 3. We have found that the Lorentz transformations of these quantities, as well as of other beam character- istics, exhibit quite nontrivial forms, which together en- sure the proper Lorentz transformation of the total AM and resolve the paradox with the transformation of the photon spin, Fig. 1. Most importantly, the transverse Lorentz boost inevitably produces the spin-to-orbital AM conversion (i.e., helicity-depend terms in the orbital AM) and nontrivial redistribution between the intrinsic (vor- tex) and extrinsic (shift) parts of the orbital AM. These eects have the geometric origin and evidence the rela- tivistic SOIs of light . Although we considered the particular case of Bessel beams (allowing analytical calculations), the results are generic for paraxial azimuthally-symmetric beams orwavepackets. This is because all derived transformations have very clear geometric/relativistic explanations, inde- pendent of the particular type of the beam. Note also that we considered only transverse Lorentz boosts. It is easy to see that a longitudinal z-boost does not break the monchromaticity of the beam and can only modify its parameters (13). Until this breaks the paraxiality of the beam (i.e., for #1 0), the spin and orbital AM (13) remain practically unchanged. We also note that the general formalism developed in this work alows one to perform the Lorentz transformations of arbitrary Maxwell and Dirac elds and to determine their prop- erties in any reference frame. These results can play an important role in scattering processes involving relativis- tic particles carrying intrinsic AM, as well as in studies of light emitted by fast-moving bodies. It should be emphasized that the nontrivial transfor- mations found in this work are actually xed by funda- mental reasons. Namely, the transformation of the to- tal AM Jand the boost momentum Nare determined by the Lorentz boost of the rank-2 AM tensor , while the spin AM Sfollows the boost of the Pauli-Lubanski four-vector (this is applied to both electrons and photons [33, 34]). Hence, the total AM and spin AM inevitably obey dierent transformations. The dierence between these two determines the nontrivial form of the orbital AML. Moreover, the orbital AM can be split into the extrinsic part Lext(determined by the position of the particle) and the intrinsic one Lint(related to the vortex phase structure of the wavefunction). If we adopt the fact that the intrinsic contribution must depend only on the vortex quantum number `(but not on the spin state  orsz), this unambiguously determines the position shift R0proportional to ( `+ 2) or (`+ 2sz). Interestingly, such dependence was previously known only for the mag- netic moment of the Dirac electron [12, 32, 44], and was directly associated with the g= 2 gyromagnetic factor for the electron spin. Our calculations show that this combination is universal for the relativistic Hall eect, independently of the spin and mass of the particle. The dierence between relativistic transformations of the spin and orbital AM can also be compared with the dierence in the commutation relations of the quantum-mechanical versions of these quantities [6, 34, 47]. Neither spin nor orbital AM operators (assuming their second-quantization or covariant Berry-connection forms) obey the canonical SO(3) commutation rules, while the total AM does. 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1711.02460v1.Spin_Orbit_Coupling_and_Magnetic_Anisotropy_in_Iron_Based_Superconductors.pdf

Spin-Orbit Coupling and Magnetic Anisotropy in Iron-Based Superconductors Daniel D. Scherer1and Brian M. Andersen1 1Niels Bohr Institute, University of Copenhagen, Juliane Maries Vej 30, DK-2100 Copenhagen, Denmark We determine theoretically the effect of spin-orbit coupling on the magnetic excitation spectrum of itinerant multi-orbital systems, with specific application to iron-based superconductors. Our microscopic model includes a realistic ten-band kinetic Hamiltonian, atomic spin-orbit coupling, and multi-orbital Hubbard interactions. Our results highlight the remarkable variability of the resulting magnetic anisotropy despite constant spin-orbit coupling. At the same time, the magnetic anisotropy exhibits robust universal behavior upon changes in the bandstructure corresponding to different materials of iron-based superconductors. A natural explanation of the observed universality emerges when considering optimal nesting as a resonance phenomenon. Our theory is also of relevance to other itinerant system with spin-orbit coupling and nesting tendencies in the bandstructure. Introduction. The investigation of magnetism in Fe-based superconducting materials (FeSCs) has proven to be a very rich avenue of research [1]. Symmetry-distinct magnetic phases have been experimentally identified, both colinear and coplanar [2–9], in agreement with theoretical models [10– 14]. Recently, it was discovered that distinct colinear phases exhibit completely different orientations of the ordered mo- ments [15], pointing to effects from spin-orbit coupling (SOC). The SOC is typically considered weak in the FeSCs, and hence neglected in many theoretical studies. However, recent focus on details of magnetic anisotropies as seen by polarized neutron scattering [16–18], including sizable spin gaps in the ordered states 15 meV [1], and considerable SOC-induced band splittings of 10-40 meV [19–21], have reinvigorated the interest in a detailed understanding of SOC and its role in magnetism and superconductivity of these ma- terials. In addition, obtaining a quantitative description of the magnetic anisotropy has important implications for the gen- eral understanding of the magnetism in terms of mainly local- ized or itinerant electrons [1, 16]. Finally, we note that the im- portance of SOC has recently been highlighted through the ex- perimental report of topological states and Majorana fermions in a certain class of FeSCs [22, 23]. Experimentally, spin-polarized neutron scattering measure- ments have mapped out the energy ( !) and temperature ( T) dependence of the magnetic anisotropy. Below, we denote byMa,Mb, andMcthe magnetic scattering polarized along the orthorhombic a,b, andcaxes, respectively. Focusing first on undoped BaFe 2As2, in the magnetic state below TN the scattering fulfills the hierarchy Mc> Mb> Ma. This is in agreement with QAF= (;0;)ordered moments aligned antiferromagnetically in the ab-plane along the longer aaxis, and implies that transverse out-of-plane fluctuations alongcare cheaper than in-plane transverse fluctuations in theb-direction [1, 17, 24, 25]. The results in the param- agnetic (PM) state at T > T NatQAFcan be summarized by the following points: 1) The low-energy magnetic re- sponse is isotropic McMbMaat highTbut be- comes increasingly anisotropic with Ma> Mc&Mbas TapproachesTN[17, 24, 26]. The fact that Mais largest agrees with the condensation of moments along the aaxis belowTN. 2) This PM magnetic anisotropy close to TNisobserved only at !.6meV [17]. The doping-dependence of the magnetic anisotropy obtained from electron- and hole- doped BaFe 2As2[18, 26–29], NaFeAs [30], and FeSe [16] has given rise to the following additional points: 3) Doping of BaFe 2As2tends to enhance the c-axis polarized low-energy magnetic fluctuations in the PM phase such that a range ex- ists whereMc&Ma> Mb. The enhanced susceptibility alongcis consistent with the out-of-plane moment orientation of theC4-symmetric magnetic phase observed in Na-doped BaFe 2As2[15]. In the nematic PM phase of FeSe, Mcalso dominates the inelastic response [16]. 4) At sufficiently large doping (e.g. 15% Ni in BaFe 2As2), the magnetic anisotropy vanishes [31]. The hierarchy of the magnetic susceptibilities, their !- andT-dependence, and their switching as a function of dop- ing has remained an outstanding puzzle, and may naively seem at odds with an atomically defined single-ion spin-orbit- generated magnetic anisotropy. For example, it has been sug- gested that intervening effects of orbital fluctuations may be at play [17]. Clearly, it is desirable to acquire a microscopic understanding of the interplay between SOC and electronic interactions in the magnetism of FeSCs. Here, within a realistic ten-band description that properly incorporates atomic SOC, we provide a theoretical explana- tion for the above points 1)-4). We classify the spin-resolved contributions to the particle-hole propagator into different types of excitations. By virtue of SOC, the spin-dependent particle-hole excitations generate a hierarchy in the energy gaps for spin excitations. We propose a general mecha- nism for the doping-dependence of the resulting magnetic anisotropy that turns out to be determined by the position of the optimal nesting of the band on the energy axis and the dominant orbital content of the participating single-particle states. From that perspective, our study is relevant not just to FeSCs, but any itinerant system with SOC and nested bands. Both theT- and!-dependence of the anisotropy follow es- sentially from the smallness of the SOC energy-scale together with the enhancement of magnetic scattering close to T- or interaction-driven SDW-instabilities. Model. Upon inclusion of atomic SOC, the itinerant elec- tron system of the FeSC materials is described by a multior- bital Hubbard Hamiltonian H=H0+HSOC+Hintfor thearXiv:1711.02460v1 [cond-mat.supr-con] 7 Nov 20172 electronic degrees of freedom of the 3dshell of iron. The non- interacting part describing the electronic structure consists of a hopping Hamiltonian H0and an atomic SOC HSOC. We de- fine the fermionic operators cy li,clito create and destroy, respectively, an electron on sublattice lat siteiin orbital with spin polarization .H0is written as H0=X X l;l0;i;jX ;cy li t li;l0j0ll0ij cl0j;(1) where hopping matrix elements t li;l0jare material specific and the electronic filling is fixed by the chemical potential 0. The indices l;l02fA;Bgdenote the 2-Fe sublattices, cor- responding to the two inequivalent Fe-sites in the 2-Fe unit cell due to the pnictogen(Pn)/chalcogen(Ch) staggering about the FePn/FeCh plane. The orbital indices ; label the five 3d-orbitals at a given Fe-site. The orbitals of xzandyzsym- metry transform to xzandyzunder a glide-plane transfor- mation [32]. Invariance under the glide-plane transformation thus requires a phase difference of between certain inter- orbital hopping-matrix elements. It is convenient to work in a basis where this phase difference is absorbed in the defi- nition of the local basis for the xz;yz -orbitals on AandB sublattices, respectively. For the A-sublattice let therefore ;2fxz;yz;x2y2;xy;3z2r2g, while for the Bsub- lattice we take ;2f~xz;~yz;x2y2;xy;3z2r2g, where ~xz=xzand~yz=yz. In this ‘phase-staggered’ basis, the atomic SOC Hamiltonian becomes HSOC= 2X l;iX ;X ;0cy lil]
0cli0;(2) with coupling strength and the angular momentum operator in vector notation [Ll] with components [Lx l];[Ly l];[Lz l]in the phase-staggered basis of 3d-orbitals, and is the vector of Pauli matrices. The phase-staggering results in different matrix representations on theA- andB-sublattice for the angular momentum operator. One obtains Lx A=Lx B,Ly A=Ly BandLz A=Lz B. While thez-component remains unaffected, the couplings of x- and y-components change sign between A- andB-sublattice. We note that while in the absence of SOC the ten-orbital model is unitarily equivalent to a five-orbital model formulated in the 1-Fe Brillouin zone, the breakdown of this equivalence in the presence of SOC can be understood as due to the two different matrix representations of the angular momentum operator in the phase-staggered basis. Electronic interactions of the 3 d states are modeled by a local Hubbard-Hund interaction term Hint=UX l;i;nli"nli#+ U0J 2X l;i;<;;0nlinli0 2JX l;i;<Sli
Sli+J0X l;i;<;cy licy liclicli;(3) parametrized by an intraorbital Hubbard- U, an interorbital couplingU0, Hund’s coupling Jand pair hopping J0, satis- fyingU0=U2J,J=J0. The operators for local charge FIG. 1. Fermi surfaces in the 1-Fe BZ ( (Kx;Ky)denotes momenta in the 1-Fe BZ coordinate system) extracted from the electronic spec- tral function with 0= 0 eV and= 0:025eV for (a) LaFeAsO, (b) BaFe 2As2and (c) FeSe. The dashed square denotes the 2-Fe BZ. and spin are nli=nli"+nli#withnli=cy licliand Sli= 1=2P 0cy li0cli0, respectively. Below, we will consider three sets of hopping parameters t li;l0jfor different FeSC parent materials: LaFeAsO [33], BaFe 2As2[34], and FeSe [35], see Fig. 1 for the correspond- ing Fermi surfaces. The effect of hole- or electron-doping is obtained by a rigid shift in the chemical potential 0. For fur- ther details on the bandstructures and the effects of SOC, we refer to the Supplementary Material (SM) [41]. Spin susceptibility. To make connection to neutron scat- tering, we compute the imaginary-time spin-spin correlation function (here i;jrefer to the spatial directions x;y;z ) ij(i!n;q) =g2 2Z 0dei!nhTSi q()Sj q(0)i;(4) withg= 2and the Fourier transformed electron spin operator for the 2-Fe unit cell given as Si q() =1p NX k;l;;;0cy kql()i 0 2ckl0():(5) To account for interaction effects in the weak-coupling regime, we evaluate the correlation functions in the random- phase approximation (RPA) in the absence of spin-rotation invariance, see SM [41]. Performing analytic continuation i!n!!+ i, with > 0a small smearing parameter, we gain access to the momentum- and frequency-resolved spec- tral density of magnetic excitations with different spatial po- larizations probed by polarized neutron scattering. We have Mi(!)Im[ii(!+ i;QAF)]; (6) in a coordinate system x=a,y=b,z=caligned with the orthorhombic crystal axes and QAF=Q1;2with the nesting vectors Q1= (;0),Q2= (0;), where Q2is related to Q1 by aC4rotation in the abplane. The cross-terms with i6=j vanish for the commensurate wavevector QAF. Since the interaction term Hintis rotationally symmetric, it cannot create anisotropy in the magnetic response. Hence, all SOC-driven anisotropy is contained purely in the particle-hole propagator, and therefore the origin of anisotropy is found in the structure of the non-interacting susceptibility. In terms of3 FIG. 2. (a),(b),(c) Chemical potential dependence of the total and orbitally ( =only) resolved isotropic contribution to the static non- interacting susceptibility with SOC = 0:025 eV atkBT= 0:01eV for the (a) LaFeAsO and (b) BaFe 2As2and (c) FeSe model with fixed wavevector Q1. (d),(e),(f) Corresponding anisotropic contributions and (g),(h),(i) summed particle-hole amplitudes contributing to the anisotropic magnetic response for = 0:015eV (dashed), = 0:025eV (solid) and = 0:035eV (dot-dashed). the sublattice-, orbital-, and spin-resolved electronic Greens function, the non-interacting susceptibility reads ij 0(q) =1 4X 1:::4i 12j 34G23G41; (7) where for compact notation we defined G23G41g2 4NX Gl2;l03(k)Gl04;l1(kq); withq= (i!n;q)andk= (ip;k),i!n;ipbeing bosonic and fermionic Matsubara frequencies, respectively, and the shorthandP(:::) =P kP l;l0P ;(:::). Perform- ing the Matsubara sum yields a Lindhard-factor dressed by wavevector-dependent matrix elements, see SM [41]. We can then extract the isotropic contribution to the susceptibility as 0=1 4X G+GG]: (8) The anisotropic contributions,
ii 0=ii 00, can be ex- pressed in terms of three particle-hole amplitudes
xx 0= ++;
yy 0= ++; (9)
zz 0= ++; (10) where we have defined the summed amplitudes ++=1 2X GG; +=1 2X GG;(11) =1 4X GGG]: (12)In the non-nematic PM state, the anisotropic response at Q2 is related to that at Q1by aC4transformation about the c-axis:
xx=yy 0 (Q2) =
yy=xx 0 (Q1)and
zz 0(Q2) =
zz 0(Q1). The amplitude , measuring the difference of equal- and opposite-spin (w.r.t. to the z-axis pointing out-of- plane) particle-hole propagation is insensitive to a C4rota- tion. Likewise, the amplitude +corresponds to processes that are possible due to SOC, but do not change the total spin along the z-direction. In contrast, the spin-flip amplitude ++, where both electron and hole with a fixed initial spin propagate to the opposite spin state by virtue of SOC, reacts by a sign change. A commonality between the bands is the sublattice structure of the anisotropy-generating particle-hole amplitudes. While ++receives only inter-sublattice contri- butions, +andonly come form intra-sublattice terms. The physical interpretation of the particle-hole bubble di- agrams can be made more transparent by considering SOC within perturbation theory. We find that the leading contri- bution to the anisotropy at QAFemerges at order 2(see SM [41] for details). This is in contrast to previous work [36], where the leading anisotropy was found to be of the form J2 and depended crucially on a finite Hund’s coupling. We ad- ditionally investigated the importance of the sign of , see SM [41], and found that results for the magnetic anisotropy are only weakly affected. Anisotropy without interactions. Our findings for the doping-dependence of the magnetic anisotropy for the non- interacting LaFeAsO, BaFe 2As2, and FeSe models at kBT=4 0:01eV are shown in Fig. 2 for several values of . For the 1111 and 122 bands, there exists a clear correlation be- tween the position of the optimal nesting condition on the energy axis (that is only weakly dependent on small ), see Fig. 2(a),(b), and the central peak in the static anisotropic re- sponse as a function of 0, seen in Fig. 2(d),(e). Indeed, the characteristic 0-dependence of the anisotropy can be qual- itatively reproduced in a simple level model, see SM [41], where the optimal nesting condition is replaced by isolated levels withxyandyzorbital content, coupled by SOC. This simple model also provides the same type of spin-dependent particle-hole amplitudes as seen in the tight-binding models, cf. Fig. 2(g),(h), pointing to a universal mechanism behind the doping-dependence of the magnetic anisotropy across the FeSC materials. In this picture the behavior of
ii 0with dop- ing is determined largely by the position of the optimal nesting condition on the energy axis and the symmetry properties of the participating orbitals. For all three tight-binding models, the hierarchy in the mag- netic anisotropy changes with 0. While the different realiza- tions of the hierarchy are already apparent at = 0:015eV , in- creasingenlarges the doping range with a particular form of the hierarchy. For LaFeAsO and BaFe 2As2we obtain a dom- inatingxx 0in the undoped case, while on the hole-(electron- )doped side, an extended region with dominating zz 0(yy 0) exists. Sufficiently far away from the nesting resonance, the magnetic anisotropy drops rapidly. These findings are in ex- cellent agreement with properties 3) and 4) highlighted in the introduction. The most prominent difference in the doping- dependence occurs on the hole-doped side, where
xx 0and
yy 0in the LaFeAsO and FeSe models do not display zero crossings, as opposed to the BaFe 2As2case. In addition, the FeSe model, where optimal nesting for xyandyzorbitals is weakened and occurs in different places on the 0-axis, see Fig. 2(c), displays a dominating zz 0in the undoped case for sufficiently large . This agrees with the recent findings in Ref.16, see Fig. 2(f). Both weak hole-doping or increasing  enhance the dominance of out-of-plane spin-fluctuations com- pared to in-plane fluctuations. The anisotropy is driven by the same type of particle-hole excitations in all models, cf. Fig. 2(g),(h),(i). Only ++and +yield sizable contribu- tions in the LaFeAsO and BaFe 2As2bands, with basically vanishing. For FeSe the -amplitude is stronger compared to the 1111 and 122 cases. Anisotropy with interactions. When including interactions, additional (inter-sublattice and inter-orbital) contributions of the particle-hole propagator enter, that are not included in the susceptibility of the non-interacting system. The properties of the electronic particle-hole propagator together with the inter- action vertex, however, fully determine the gap-structure of magnetic excitations with different polarization. While inter- orbital contributions can in principle be enhanced by Hund’s coupling, we did not observe a modification of the hierarchy in magnetic anisotropy between the bare and RPA results. In this respect, the static bare susceptibility provides a measure of the gap-sizes of spin excitations with different polarization. FIG. 3. (a) Imaginary part of the interacting susceptibilities as a function of !at wavevector Q1for the BaFe 2As2model with = 0:025eV atkBT= 0:01eV close to the interaction driven SDW instability (with J=U=4) for different chemical potentials: 0= 0eV ,U= 0:815eV (solid),0=0:05eV ,U= 0:898eV (dotted) and 0= 0:05eV ,U= 1:030eV (dashed). (b) T- dependence of the static part of the RPA susceptibility for = 0eV with= 0:025eV andU= 0:816eV ,J=U=4. The dashed vertical line marks the SDW transition temperature TN. We can thus connect the results in Fig. 2 to the doping dependence of the magnetic scattering amplitudes Mi. Fo- cusing on BaFe 2As2, cf. Fig. 2(e), our weak-coupling ap- proach yields Ma>Mc>Mbin an extended region around 0= 0eV , consistent with a stripe SDW state with ordered moments along a. The formation of a finite SDW order below TNresults in the gapping of excitations parallel to the mo- ment direction. For sufficiently low Tin the stripe magnetic state, we can thus expect Mc> Mb> Ma. Returning to the discussion of the PM state, for sufficiently strong SOC, hole-doping first leads to a regime with Mc> Ma> Mb, with a subsequent crossover to Mc>Mb>Maupon further hole-doping, all consistent with the observed reorientation of magnetic moments in a C4-symmetric magnetic phase [15]. We show the !-dependent RPA results for the imaginary part of the susceptibility in the various regimes in Fig. 3(a) for interaction parameters UandJclose to the interaction driven SDW-instability with fixed wavevector. For the un- doped case ( 0= 0 eV) the!-dependent anisotropy in the magnetic scattering is clearly visible and diminishes quickly for!&67meV . In the hole- ( =0:05eV) and electron- doped (= 0:05eV) cases, the changes in the hierarchy of magnetic scattering can be observed with an overall de- crease of the magnetic scattering, while at the same time the anisotropy appears over a larger energy range. These differ- ences to the undoped case are simply due to the increasing de- gree of incommensurability of the wavevector associated with the leading SDW-instability, while we observe the magnetic scattering at the commensurate wavevector QAF. Thus, the5 magnetic excitations at QAFobtain larger gaps for the doped cases than for the undoped case shown in Fig. 3(a). The T- dependence of Re[ii(0 + i;QAF)]is shown in Fig. 3(b), wherexxdiverges as T!TN. The anisotropy increases strongly in the proximity to the SDW transition, while it re- mains small for elevated T. The results shown in Fig. 3 are in excellent agreement with the points 1) and 2) discussed in the introduction. 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Schmalzl, N. Qureshi, and M. Braden, Scientific Reports 7, 10307 (2017). [41] Supplementary material.1 Supplementary Material: “Spin-Orbit Coupling and Magnetic Anisotropy in Multiband Metals” EFFECT OF sign()ON ELECTRONIC BANDSTRUCTURE AND ANISOTROPY The spin-orbit coupling (SOC) leads to a non-equivalence of 2-Fe and 1-Fe unit-cell descriptions of the iron-based supercon- ductors (FeSCs). Within the 2-Fe description, a finite SOC splits those states that are degenerate at the boundary of the 2-Fe Brillouin zone due to glide-plane symmetry. The electronic states are, however, still 2-fold degenerate in the paramagnetic state due to time-reversal and inversion symmetry. In Fig. S1 we demonstrate the effect of the spin-orbit coupling on the electronic (a)λ=0.050eV (f)λ=0.050eV (k)λ=0.050eV (b)λ=0.025eV (g)λ=0.025eV (l)λ=0.025eV (c)λ=0.0eV (h)λ=0.0eV (m)λ=0.0eV (d)λ=−0.025eV (i)λ=−0.025eV (n)λ=−0.025eV (e)λ=−0.050eV (j)λ=−0.050eV (o)λ=−0.050eV FIG. S1. High-symmetry cuts through bandstructure for (a)-(e) LaFeAsO model [S1], (f)-(j) BaAs 2Fe2model [S2] and (k)-(o) FeSe model [S3] with SOC strength of varying sign and magnitude. The bandstructures were obtained from the electronic spectral function. The path through momentum space goes from toMoverXand back to , where momenta are specified with respect to the 2-Fe BZ. bandstructure for the 2D tight-binding models for LaFeAsO [S1], BaFe 2As2[S2] and FeSe [S3]. The FeSe model was obtained from performing a self-consistent mean-field calculation, yielding a sizable nearest-neighbor hopping renormalization. Within the notation of Ref. S3, the parameters for the mean-field calculation were ~V= 0:74eV and ~V0= 0. As can be seen from the bandstructures, SOC leads to splittings and shifts at both center and boundary of the Brillouin zone. We show bandstructures for both>0and<0where the effects of SOC are slightly different as to which type of splittings occur and in which direction states at the Brillouin zone center are shifted to. We additionally explore the effect of a negative SOC, <0, on the ansiotropy
ii 0, where we restrict ourselves to the same jj-values as in the main text, see Fig. S2. In the LaFeAsO model, we observe a suppression of the hole-doped 0-region with dominant zz 0compared to the >0case, which can be traced back to an increase of the ++amplitude in the corresponding doping regime. In the BaFe 2As2model, additional zero-crossing appear in
xx 0 and
yy 0on the hole-doped side, while for =0:035eV the zero-crossings are removed. Both LaFeAsO and BaFe 2As22 (a) (d) (g) (b) (e) (h) (c) (f) (i) FIG. S2. (a),(b),(c) Chemical potential dependence of the total and orbitally ( =only) resolved isotropic contribution to the static non- interacting susceptibility with SOC = 0:025 eV atkBT= 0:01eV for the (a) LaFeAsO and (b) BaFe 2As2and (c) FeSe model with fixed wavevector Q1. (d),(e),(f) Corresponding anisotropic contributions and (g),(h),(i) summed particle-hole amplitudes contributing to the anisotropic magnetic response for = 0:015eV (dashed), = 0:025eV (solid) and = 0:035eV (dot-dashed). models show the same qualitative behavior on the electron-doped side, as they do for >0. In the case of FeSe, the magnetic anisotropy shows the same qualitative behavior as for positive for both hole- and electron-doping. We conclude that sign() can have a qualitative influence on the 0-dependence of the anisotropy, but the changes in the hierarchy of ii 0strongly depend on quantitative differences in the doping dependence of particle-hole amplitudes ++, +and. RPA CORRELATION FUNCTIONS Following Ref. S4, we here describe the RPA formalism we employ to analyze the collective excitations of FeSCs. As appropriate for the presence of a general SOC term, we assume the absence of spin-rotation symmetry. While in a paramagnetic state without SOC the conservation of electronic spin facilitates a decoupling of the RPA equations for transverse and longitudinal fluctuations, this is in general no longer the case in the presence of SOC. Since SOC also generates a coupling between charge- and spin-fluctuations already at the Gaussian level (in the language of effective actions for collective excitations), the RPA equations need to be extended to account for the mixing of charge- and spin-excitations for general transfer momenta. For high-symmetry momenta, like the stripe wave-vectors Q1= (;0)andQ2= (0;), the coupling between charge and spin sector vanishes. The formalism we present below is, however, general and not restricted to specific momenta. We compute the imaginary-time spin-spin correlation function (where i;jrefer to the spatial directions x;y;z ) ij(i!n;q) =g2 2Z 0dei!nhTSi q()Sj q(0)i; (S1) with the Fourier transformed electron spin operator for the 2-Fe unit cell given as Si q() =1p NX k;l;;;0cy kql()i 0 2ckl0(): (S2)3 (a) (b) k−q kl3µ3σ3l4µ4σ4 l1µ1σ1 l2µ2σ2 = l3µ3σ3l4µ4σ4 l1µ1σ1 l2µ2σ2RPA RPA + l3µ3σ3l4µ4σ4 l1µ1σ1 l2µ2σ2 l3µ3σ3l4µ4σ4 l1µ1σ1 l2µ2σ2 FIG. S3. (a) Bubble diagram for the non-interacting generalized correlation function Eq. (S6). The labels at the vertices denote the incoming and outgoing quantum numbers l;; denoting sublattice, orbital and spin. The fermionic propagator, represented by full lines with arrows, includes the effects of SOC to infinite order in the SOC strength . The fermionic propagator also carries a frequency-momentum quantum numberk= (ip;k)andq= (i!n;q)denotes a bosonic transfer frequency/momentum. (b) Diagrammatic representation of the RPA equa- tion Eq. (S13) to compute Eq. (S6) within the RPA approximation. The internal quantum numbers that are summed over are not specified. The dashed horizontal line denotes the interaction vertex [U]defined in Eq. (S14)-(S17). We note that the interaction vertex, although represented by a horizontal line, contains both direct and exchange contributions in terms of the microscopic electronic interaction. We note that we typically specify the transfer momentum qwith respect to the coordinate system of the 1-Fe Brillouin zone. It is then understood that ‘ kq’ refers to subtraction of the two vectors in a common coordinate system. Here Tdenotes the time-ordering operator with respect to the imaginary-time variable 2[0;), withthe inverse temperature and i 0thei-th Pauli matrix. From the imaginary part of ij(i!n;q), we can extract the spectrum of spin-excitations that are probed by neutron scattering. The density susceptibility is defined as 00(i!n;q) =1 2Z 0dei!nhTNq()Nq(0)i; (S3) with the density operator Nq() =1p NX k;l;;cy kql()ckl(): (S4) To derive RPA expressions for the above quantities in the absence of spin-rotation symmetry, it proves useful to introduce the generalized correlation function []l111;l222 l333;l444(i!n;q) =1 NZ 0dei!nX k;k0hTcy kql111()ckl222()cy k0+q0l333(0)ck0l444(0)i: (S5) To ease notation we introduce a combined index X(l;; )by collecting sublattice, orbital and spin indices. In the absence of interactions, the correlation function []X1;X2 X3;X4(i!n;q)[]l111;l222 l333;l444(i!n;q)reduces to [0]X1;X2 X3;X4(i!n;q) =1 NZ 0dei!nX k;k0hTcy kqX1()ckX2()cy k0+q0X3(0)ck0X4(0)i0 (S6) =1 NZ 0dei!nX X1;:::;X 4X kGX2;X3(;k)GX4;X1(;kq) (S7) =1 NX p;kGX2;X3(ip;k)GX4;X1(ipi!n;kq) (S8) =1 NX k;n1;n2[Mn1;n2(k;q)]X1;X2 X3;X4f(En1(kq))f(En2(k)) i!n+En1(kq)En2(k); (S9) with the eigenenergies En(k)of the Hamiltonian H0+HSOC andf() = [exp((0)) + 1]1the Fermi-Dirac distribution. Here we defined the imaginary-time Greens function GX1;X2(;k) =hTckX1()cy kX2(0)i0=1 X neinGX1;X2(in;k); (S10) with GX1;X2(in;k) =X nUX1;n(k)U X2;n(k) inn(k); (S11)4 withn(k) =En(k)0. The orbital-dressing factors entering the components of the bare correlation function read [Mn1;n2(k;q)]X1;X2 X3;X4=U X1;n1(kq)UX2;n2(k)U X3;n2(k)UX4;n1(kq): (S12) The unitary matrix Ul;n(k)diagonalizes the quadratic Hamiltonian H0+HSOC. The RPA equation for the generalized correlation function reads as []X1;X2 X3;X4(i!n;q) = [0]X1;X2 X3;X4(i!n;q) + [0]X1;X2 Y1;Y2(i!n;q)[U]Y1;Y2 Y3;Y4[]Y3;Y4 X3;X4(i!n;q): (S13) Repeated indices are summed over in Eq. (S13). The bare fluctuation vertex [U]X1;X2 X3;X4[U]l111;l222 l333;l444originates from the Hubbard-Hund interaction and describes how electrons scatter off a collective excitation in the particle-hole channel. Since we employ the Hubbard-Hund interaction with interaction parameters preserving spin-rotational symmetry, it is still possible to classify the scattering of collective excitations according to their total spin. Accordingly, the vertex can be split into three different contributions as [U]X1;X2 X3;X4= [U1]X1;X2 X3;X4+ [U2]X1;X2 X3;X4+ [U3]X1;X2 X3;X4; (S14) whereU1andU3describe the scattering of opposite spin and equal spin fluctuations in the longitudinal channel, respectively, whileU2describes the scattering of transverse spin fluctuations. The vertex contribution U1is defined as [U1]l;l l;l=U; [U1]l;l l;l=U0;[U1]l;l l;l=J;[U1]l;l l;l=J0;with6= (S15) where denotes the opposite spin polarization to . TheU1contribution is zero for all other sublattice, orbital or spin index combinations. For the equal spin fluctuation vertex, we find the non-zero elements [U3]l; l;l=(U0J);[U3]l; l;l= (U0J);with6=: (S16) For the transverse channel, we obtain [U2]l; l;l=U; [U2]l; l;l=U0;[U2]l; l;l=J;[U2]l; l;l=J0;with6=; (S17) and zero else. For (residual) continuous spin-rotational symmetry, the transverse and longitudinal channels decouple and can be treated independently. We computed the non-interacting bubble with a 50 50 discretization for the electronic momenta in the 2D 2-Fe BZ and then solved the linear matrix equation Eq. (S13) for the RPA correlation function. The exploration of 3D bandstructure effects on the magnetic anisotropy is beyond the scope of the present investigation. The RPA approximation to the spin susceptibilities ij(i!n;q)and the density susceptibility 00(i!n;q)can be recovered by forming the appropriate linear combinations of correlation functions: 00(i!n;q) =X l;l0X ;1 2X 1;:::; 41234[]l1;l2 l03;l04(i!n;q); (S18) ij(i!n;q) =X l;l0X ;g2 2X 1;:::; 4i 12 2j 34 2[]l1;l2 l03;l04(i!n;q): (S19) The susceptibilities of the non-interacting model can be obtained in the same way by simply replacing the RPA approximation by the bare correlation function. For the sake of completeness, we specify the spin-configurations [1234]that are summed in Eqs. (S18),(S19) to arrive at the physical susceptibilities: 00: (+[""""] + [""##] + [##""] + [####]); xx: (+["#"#] + ["##"] + [#""#] + [#"#"]) yy: (["#"#] + ["##"] + [#""#][#"#"]); zz: (+[""""][""##][##""] + [####]); xy: i (["#"#] + ["##"][#""#] + [#"#"]); xz: (+["#""]["###] + [#"""][#"##]); yx: i (["#"#]["##"] + [#""#] + [#"#"]); yz: i (["#""] + ["###] + [#"""][#"##]); zx: (+["""#] + [""#"][##"#][###"]); zy: i (["""#] + [""#"] + [##"#][###"]): The susceptibilities 0iandi0that describe a response of charge (spin) due to an external field coupling linearly to spin (charge) can be obtained in a completely analogous fashion, but are not considered here.5 SECOND-ORDER PERTURBATION THEORY IN SPIN-ORBIT COUPLING (a) λLiσi σ2σ3 kσ4 σ4λLiσi σ1σ2 k−qk −q σ1σ1σ2σ2σ3 σ3 (b) λL+σ− λL+σ−k−qk −q kk↑↑ ↓ ↓↓ ↓ ↑ ↑ (c) λL−σ+ λL−σ+k−qk −q kk↓↓ ↑ ↑↑ ↑ ↓ ↓ (d) λL−σ+ λL+σ−k−qk −q kk↓↓ ↓ ↓↑ ↑ ↑ ↑ (e) λL+σ− λL−σ+k−qk −q kk↑↑ ↑ ↑↓ ↓ ↓ ↓ (f) λLzσz σσ λLzσz σ′σ′k−qk −q kkσσ σ′ σ′σ σ σ′ σ′ FIG. S4. Feynman diagrams to second order in for the non-interacting susceptibility. Solid lines denote the non-interacting Greens function without SOC. While this Greens function is diagonal in the spin quantum number and both ""- and##-components are identical in the para- magnetic state without external fields, we indicate the spin indices on the Greens function lines to show the flow of the spin quantum number through the particle-hole bubble diagrams. (a) Contribution to the isotropic part of the susceptibility, cf. Eq. (S37). An equivalent diagram with the SOC insertions on the lower fermion line contributes as well. (b),(c) Contributions to the in-plane anisotropy, cf. Eqs. (S26),(S27). (d),(e) Contributions to the out-of-plane anisotropy, cf. Eqs. (S28),(S30). (f) Feynman diagram measuring the difference in the particle-hole amplitudes with parallel and antiparallel spin, cf. Eq. (S30). As another route to gain insight into the mechanism behind SOC-driven anisotropy, we now consider second-order perturba- tion theory for the non-interacting susceptibility in the SOC strength . We arrive at the following results for the components of the non-interacting Greens functions Gl";l00"(k) = [G0]l;l00(k) + 2[Cz 1]l;l00(k) +2 40 @X j[Cjj 2]l;l00(k) +X jki"zjk[Cjk 2]l;l00(k)1 A+O(3);(S20) and Gl#;l00#(k) = [G0]l;l00(k) 2[Cz 1]l;l00(k) +2 40 @X j[Cjj 2]l;l00(k)X jki"zjk[Cjk 2]l;l00(k)1 A+O(3);(S21) as well as Gl#;l00"(k) =[C+ 1]l;l00(k) + i2 2X jk "xjk+ i"yjk
[Cjk 2]l;l00(k) +O(3); (S22) and Gl";l00#(k) =[C 1]l;l00(k) + i2 2X jk "xjki"yjk
[Cjk 2]l;l00(k) +O(3); (S23) with [Ci 1]l;l00(k) =X sX 1;2[G0]l;s1(k)[Li s]12[G0]s2;l00(k); (S24) [Cij 2]l;l00(k) =X s;s0X 1:::4[G0]l;s1(k)[Li s]12[G0]s2;s03(k)[Lj s0]34[G0]s03;l00(k); (S25) where we defined [C 1]l;l00(k) =1 2([Cx 1]l;l00(k)i[Cy 1]l;l00(k))andG0denotes the non-interacting Greens function with- out spin-orbit coupling. Defining the shorthand notation G12G34g2 4NX kX l;l0X ;0Gl1;l02(k)Gl03;l4(kq);6 we then arrive at G"#G"#=g2 4NX kX l;l0X ;02[C 1]l;l00(k)[C 1]l00;l(kq) +O(3); (S26) G#"G#"=g2 4NX kX l;l0X ;02[C+ 1]l;l00(k)[C+ 1]l00;l(kq) +O(3); (S27) G"#G#"=g2 4NX kX l;l0X ;02[C 1]l;l00(k)[C+ 1]l00;l(kq) +O(3); (S28) G#"G"#=g2 4NX kX l;l0X ;02[C+ 1]l;l00(k)[C 1]l00;l(kq) +O(3): (S29) Thus, anisotropy in the susceptibility emerges at second order in . We further obtain 1 4X GGG] =g2 4NX kX l;l0X ;02 4[Cz 1]l;l00(k)[Cz 1]l00;l(kq) +O(3): (S30) Plugging in the decomposition of the Greens function over Eigenstates of H0, we find (where aandbare the appropriate labels corresponding to the spin-combinations 11and22) G11G22=g2 4NX kX l;l0X ;02[Ca 1]l;l00(k)[Cb 1]l00;l(kq) +O(3) (S31) =g2 4N2X kX n1:::n4Mn1;n2;n3;n4(k;q)[La]n1n2(k)[Lb]n3n4(kq) L(2) ani(i!n;n1(k);n2(k);n3(kq);n4(kq)) +O(3); and 1 4X GGG] =g2 4NX kX l;l0X ;02 4[Cz 1]l;l00(k)[Cz 1]l00;l(kq) +O(3) (S32) =g2 4N2 4X kX n1:::n4Mn1;n2;n3;n4(k;q)[Lz]n1n2(k)[Lz]n3n4(kq) L(2) ani(i!n;n1(k);n2(k);n3(kq);n4(kq)) +O(3): Here we defined Mn1;n2;n3;n4(k;q)X l;l0X ;0[Mn1;n2;n3;n4(k;q)]l;l l00;l00; (S33) with the generalized product of orbital-dressing factors [Mn1;n2;n3;n4(k;q)]l11;l22 l33;l44U l11;n4(kq)Ul22;n1(k)U l33;n2(k)Ul44;n3(kq); (S34) as well as the band-space matrix-elements of the angular momentum operator [La]nn0(k) =X sX 1;2U s1;n(k)[La s]12Us2;n0(k): (S35) We further defined the Lindhard-type factor L(2) ani(i!n;1;2;3;4) =1 X pY j=1;21 ipjY j0=3;41 ipi!nj0: (S36)7 (a) (c) (b) (d) FIG. S5. Perturbative results (solid curves) for
ii 0and ++, +andto order2as a function of chemical potential 0for (a),(c) LaFeAsO and (b),(d) BaFe 2As2models with = 0:025eV andT= 0:01eV compared to the exact numerical result (dashed curves). For the isotropic contribution, we find 1 4X G+GG] =g2 4NX kX l;l0X ;0[G0]l;l00(k)[G0]l00;l(kq) (S37) g2 4NX kX l;l0X ;02 4X j [Cjj 2]l;l00(k)[G0]l00;l(kq) + [G0]l;l00(k)[Cjj 2]l00;l(kq) +O(3) =g2 4NX kX n1;n2Mn1;n2(k;q)L(0)(i!n;n1(k);n2(kq)) +g2 4N2 4X kX n1:::n4Mn1;n3;n4;n4(k;q)X j[Lj]n1n2(k)[Lj]n2n3(k) L(2) iso(i!n;n1(k);n2(k);n3(k);n4(kq)); +g2 4N2 4X kX n1:::n4Mn4;n4;n1;n3(k;q)X j[Lj]n1n2(kq)[Lj]n2n3(kq) L0(2) iso(i!n;n1(kq);n2(kq);n3(kq);n4(k)) +O(3); with Mn1;n2(k;q)X l;l0X ;0[Mn1;n2(k;q)]l;l l00;l00; (S38) and the Lindhard-type factors L(2) iso(i!n;1;2;3;4) =1 X pY j=1;2;31 ipj1 ipi!n4; (S39) L0(2) iso(i!n;1;2;3;4) =1 X pY j=1;2;31 ipi!nj1 ip4: (S40) From these results we can read off that two SOC-insertions on one fermion line in the bubble contribute to the isotropic part, while one SOC-insertion on each fermion line generates anisotropy. The perturbative evaluation of the non-interacting susceptibility thus reveals that the leading contributions to the anisotropy in the diagonal components of the susceptibility tensor come at second order with respect to the SOC strength . AtO(3), the sign of the coupling can enter. As long as SOC is a perturbative scale,8 one can generally expect that the difference between the >0and<0cases atO(3)are most prominent, where the O(2) results indicate a change in the hierarchy of anisotropies. In Fig. S5 we compare second-order perturbation theory in SOC to the exact numerical evaluation of the non-interacting susceptibility for the chemical potential dependence of the anisotropy. As is clear from Fig. S5(a),(c) the perturbation theory yields a faithful representation of the main trends observed in the full numerical result for the LaFeAsO model, while it tends to overestimate the magnitude of the anisotropic contributions. For even smaller than 0:025eV , the overall quantitative agreement tends to become better. For the BaFe 2As2model, on the other hand, qualitative agreement between full numerical evaluation and second-order perturbation theory is only found in the vicinity of the optimal nesting condition, see Fig. S5(b),(d). We additionally note that the failure of perturbation theory comes mostly from the large deviations of the ++-type amplitude, while bothand +seem to be captured rather well by the perturbative result. FERMIONIC LEVEL MODEL Here, we define a simple (non-interacting) model with fermionic levels having a well-defined orbital character in the 3 d- manifold that are coupled by SOC. We then evaluate the susceptibility for this simple system and find that the anisotropy shows a behavior that is qualitatively very similar to the variation of the spin anisotropy in the tight-binding models as a function of the chemical potential 0. To keep a certain degree of generality, we keep the five 3 dorbitals but do not include a sublattice structure. It is important to note that the level structure in this simple Hamiltonian has nothing to do with the crystal field in the tight-binding models. The level structure rather reflects the k-space nesting (‘resonance’ between single-particle states at different kwith different orbital character). The Hamiltonian reads H=H0+HSOC; (S41) with H0=X X cy (0)c (S42) and HSOC= 2X ;X ;0cy ]
0c0: (S43) We now compute ij 0(i!n) =g2 2X ;X 1;:::; 4i 12 2j 34 2[0]1;2 3;4(i!n) (S44) =g2 2X ;X n1;n2Si n1;n2Sj n2;n1L(0)(i!n;n1;n2); (S45) with the matrix elements Si 1n1;2n21 2X 12U 11;n1i 12U22;n2; (S46) Sj 3n3;4n41 2X 12U 33;n3j 34U44;n4; (S47) and the Lindhard-factor L(0)(i!n;1;2)1 X p1 ip11 ipi!n2=f(1)f(2) i!n+12: (S48) The eigenstates of the Hamiltonian Hobtain a non-trivial orbital structure by virtue of the SOC that ultimately entangles the spin and orbital degree of freedom. We also note that the only chemical potential dependence comes from the Lindhard-factor, while the orbital and spin structure of the eigenstates depends on the chosen level structure and the SOC strength . We also note, that while the SOC will shift the levels, it will not lead to a splitting of the originally spin-degenerate levels. So each level retains its two-fold degeneracy.9 We investigate the model numerically (where we always consider the static limit i!n!i0+) and focus on two levels with xy andyzorbital character, respectively (the remaining levels are shifted to large negative energies). We then plot i) the isotropic contribution ii) the anisotropic contributions iii) the functions ++, +andthat measure different types of particle-hole excitations contributing to the anisotropy in Fig. S6, where +++1 2[G"#G"#+G#"G#"]; (S49) ++1 2[G"#G#"+G#"G"#]; (S50) 1 4[G""G##+G##G""G##G##G""G""]; (S51) with the same summation conventions as defined above. For the chosen orbitals, only the y-component of the angular-momentum operator has non-vanishing matrix elements. We note that while plotting all quantities in absolute units, we do not intend to relate the ‘physics’ of this simple model to a tight-binding model. Rather, it serves as an analogy, that demonstrates that the qualitative (a) (b) (c) (d) (e) (f) (g) (h) (i) FIG. S6. Isotropic and anisotropic magnetic response of the simple level model with varying SOC strength, where for (a)-(c) = 0:015eV , (d)-(f)= 0:025eV , (g)-(i)= 0:035eV . (a),(d),(g) Isotropic contribution to the susceptibility. A resonance (understood as a peak in the isotropic part of the susceptibility) occurs as the chemical potential sweeps across the position of levels with xyandyzorbitals. The level corresponding to the remaining orbitals are shifted to large negative energies and do not contribute here. Both xzy andyzorbitals have the same contribution to the total susceptibility. Increasing SOC leads to a broadening of the resonance. (b),(e),(h) Chemical potential dependence of the anisotropy. The degeneracy xx 0=zz 0occurs due to the simplicity of the level model. (c),(f),(i) Particle-hole amplitudes for the level model. Also here, the relation ++= +is due to the simplicity of the level model. behavior of the chemical potential dependence of the anisotropic response is determined to a large extent by i) the orbitals involved in the resonance and correspondingly ii) the subspace of orbitals the angular-momentum operator is acting on, as well as iii) the position of the resonance on the energy axis. It is clear from Fig. S6 that including xyandyztype orbitals leads to the same type of particle-hole excitations as in the tight-binding models, where -type excitations vanish and ++and + determine the anisotropy. In contrast to the tight-binding models, the level model produces amplitudes ++and +that obey ++= +. Aside from asymmetry around the position of the resonance, the qualitative 0-dependence of the anisotropy response in the full model could be obtained from the level model when globally shifting the +amplitude to lower energies on the energy axis. While the property ++= +is robust in the level model to both level splitting and/or hybridization,10 bringing additional levels of different orbital character closer to the xyandyzlevels can lift the degeneracy between
xx 0and
zz 0, essentially by producing a finite amplitude. As can be seen from Fig. S6(b),(e),(h) an increase of SOC widens the region where a particular hierarchy in the magnetic anisotropy is realized, i.e., the zero-crossings of
xx 0,
yy 0and
zz 0 move further away from the position of the peak in the isotropic part of the susceptibility. In the same way as the nesting condition in the full tight-binding model has most orbital contribution from xyandyzatQ1, while at Q2the dominant contributions come fromxyandxz, replacingyzbyxzchanges the sign of ++and yields ++= +. In that case,
yy 0and
zz 0become degenerate. Aside from the degeneracy, this behavior of the anisotropy is fully in line (at least on a qualitative level) with the behavior observed in the tight-binding models, again demonstrating both the presence of a resonance and the symmetry of the involved orbitals as the deciding, universal factors. Albeit its simplicity, gaining a detailed understanding of the interplay of matrix elements and the Lindhard-factor seems to be involved, at least in the sense that it is an intricate interplay of inter- and intra-orbital contributions that give rise to the observed chemical potential dependence in the magnetic anisotropy. Since we do not expect these details to carry over to the tight-binding models, we do not delve into a discussion here. While the level model clearly demonstrates universality in the chemical potential dependence of the anisotropy response, it cannot capture all the effects influencing the anisotropy in the actual tight-binding models. [S1] H. Ikeda, R. Arita, and J. Kune  s, Phys. Rev. B 81, 054502 (2010). [S2] H. Eschrig and K. Koepernik, Phys. Rev. B 80, 104503 (2009). [S3] D. D. Scherer, A. Jacko, C. Friedrich, E. ¸ Sa¸ sio ˘glu, S. Blügel, R. Valentí, B. M. Andersen, Phys. Rev. B 95, 094504 (2017). [S4] D. D. Scherer, I. Eremin, and B. M. Andersen, Phys. Rev. B 94, 180405(R) (2016).