Unidirectional magnetoresistance and spin-orbit torque in NiMnSb
J. Železný, Z. Fang, K. Olejník, J. Patchett, F. Gerhard, C. Gould, L. W. Molenkamp, C. Gomez-Olivella, J. Zemen, T. Tichý, T. Jungwirth, C. Ciccarelli
UUnidirectional magnetoresistance and spin-orbit torque inNiMnSb
J. ˇZelezn´y, Z. Fang, K. Olejn´ık, J. Patchett, F. Gerhard, C. Gould, L. W.Molenkamp, C. Gomez-Olivella, J. Zemen, T. Tich´y, T. Jungwirth,
1, 5 and C. Ciccarelli Institute of Physics, Czech Academy of Sciences,Cukrovarnick´a 10, 162 00, Praha 6, Czech Republic Cavendish Laboratory, University of Cambridge, CB3 0HE, United Kingdom Physikalisches Institut (EP3), Universit¨at W¨urzburg,Am Hubland, D-97074 W¨urzburg, Germany Faculty of Electrical Engineering, Czech Technical University in Prague,Technick´a 2, Prague 166 27, Czech Republic School of Physics and Astronomy, University of Nottingham,Nottingham NG7 2RD, United Kingdom a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b bstract Spin-dependent transport phenomena due to relativistic spin-orbit coupling and broken space-inversion symmetry are often difficult to interpret microscopically, in particular when occurring atsurfaces or interfaces. Here we present a theoretical and experimental study of spin-orbit torqueand unidirectional magnetoresistance in a model room-temperature ferromagnet NiMnSb with in-version asymmetry in the bulk of this half-heusler crystal. Besides the angular dependence onmagnetization, the competition of Rashba and Dresselhaus-like spin-orbit couplings results in thedependence of these effects on the crystal direction of the applied electric field. The phenomenol-ogy that we observe highlights potential inapplicability of commonly considered approaches forinterpreting experiments. We point out that, in general, there is no direct link between the current-induced non-equilibrium spin polarization inferred from the measured spin-orbit torque and theunidirectional magnetiresistance. We also emphasize that the unidirectional magnetoresistancehas not only longitudinal but also transverse components in the electric field – current indiceswhich complicates its separation from the thermoelectric contributions to the detected signals incommon experimental techniques. We use the theoretical results to analyze our measurements ofthe on-resonance and off-resonance mixing signals in microbar devices fabricated from an epitaxialNiMnSb film along different crystal directions. Based on the analysis we extract an experimentalestimate of the unidirectional magnetoresistance in NiMnSb.
I. INTRODUCTION
Anisotropic magnetoresistance (AMR) is an example of a relativistic transport phe-nomenon in ferromagnets with a long history. McGuire and Potter provided a phenomeno-logical model that fully explained the angular dependence of AMR. The model was based ona general argument that the magnetisation dependent conductivity tensor σ ij ( m ), like anyother observable, reflects the symmetry of the ferromagnetic crystal. This means that ifa symmetry operation belongs to the crystallographic point group, the conductivity tensoris left invariant under the same symmetry operation. As an example, when an electricalcurrent flows in an isotropic (polycrystalline) ferromagnet, the magnetisation direction de-fines the only axis of rotational symmetry, which results in a cos(2 θ ) angular dependence ofAMR, with θ being the angle between the magnetisation and the current.2hile AMR can be observed in any magnetic system, ferromagnetic conductors withan inversion asymmetric crystal structure have been more recently identified as a fruitfulplatform for discovering and utilizing a range of new relativistic spintronics phenomena. Apart from the exchange field B ex , which splits the electronic structure in majority andminority spin bands, the inversion symmetry breaking in the lattice together with spin-orbit coupling introduces an additional splitting ∆ H SO = B SO ( k ) · s . Here B SO ( k ) is aneffective magnetic field that depends on the crystal momentum k , while s represents the spin-polarisation. The key property of B SO ( k ) is that it is odd in k and thus it results in oppositespin splitting for opposite k . In inversion asymmetric strained zinc-blende semiconductorslike GaAs, B SO ( k ) is a combination of Rashba and Dresselhaus symmetry terms, B RSO ( k ) ∝ ( k y , − k x ) and B DSO ( k ) ∝ ( k x , − k y ). A direct manifestation of this splitting is found in thespin-orbit torque (SOT) in, e.g., (Ga,Mn)As that emerge when an electrical current isapplied to this inversion-asymmetric ferromagnetic semiconductor. In metallic systemssuch as half-heusler NiMnSb studied here, B SO ( k ) is not described by the simple linear-in- k Rashba and Dresselhaus form. SOT, nevertheless, contains Rashba and Dreselhaus-liketerms analogous to those observed in the zinc-blende semiconductors as those reflect thecrystal symmetry which is the same in zinc-blende and half-heusler crystals. SOT in bulk non-centrosymmetric crystals is associated with a current-induced non-equilibrium spin polarisation δ s SO , an effect often referred to as the Edelstein effect or theinverse spin-galvanic effect (iSGE). When δ s SO is perpendicular to the magnetization,it will exert a torque on it via the exchange coupling. The component of δ s SO that isparallel to B ex does not lead to a magnetization torque and is, therefore, transparent toany experimental method that relies on driving the magnetisation out of equilibrium bySOT. Besides SOT, B SO ( k ) together with B ex can also lead to magneto-transport terms that aresecond order in the applied electric field and, unlike the first-order AMR, are odd under themagnetization. The unidirectional magnetoresistance (UMR) is an example of such a second-order magneto-transport effect that was previously reported in non-centrosymmetric ferro-magnet/paramagnet bilayers or bulk ferromagnets. The origin of UMR was consideredto be closely connected to the phenomenology of the giant magnetoresistance (GMR).
While in the GMR multilayer, the fixed reference ferromagnetic layer acts as an externalsource of spin δ s , in UMR this is replaced with δ s SO generated internally by the spin-3rbit coupling. As for GMR, spin-dependent scattering within the ferromagnet results inthis scenario in a different resistance depending on the orientation of δ s SO relative to themagnetization of the probed ferromagnet. Moreover, the accumulated spin can introduce aproportional change in the exchange splitting of the bands, further affecting the conductivityby influencing spin transmission of minority and majority spins. Another UMR mechanismconsiders that the magnons’ population of the ferromagnet is increased or decreased de-pending on the orientation of δ s SO relative to the magnetisation. This leads to a changein the average magnetisation, which also results in a net change of the magnetoresistance.Although it is possible to experimentally distinguish these different contributions for theirparticular dependence on current density and magnetic field, they all share a commonorigin in the component of δ s SO collinear with the ferromagnet’s magnetisation.In Sec. II we report our symmetry analysis and ab initio calculations of current inducedspin polarization and UMR in NiMnSb. We use this model system to highlight potentialmisconceptions when inferring these quantities from experiment. In Sec. III we then discussour measurements in NiMnSb microbars patterned along different crystal directions, and inSec. IV we summarize our main results. II. THEORETICAL RESULTS AND GENERAL IMPLICATIONS FOR THE ANAL-YSIS OF EXPERIMENTSA. Current-induced spin polarization and spin-orbit torque
The tetragonal distortion of the non-centrosymmetric cubic unit cell of NiMnSb, inducedby the lattice mismatch with the substrate, lowers its symmetry to a − m symmetry pointgroup and results in a Dresselhaus-like k -linear term of the spin-orbit coupling. Experimentsin NiMnSb epilayers show an additional Rashba-like k -linear term of the spin-orbit couplingwhich we model by introducing a shear strain. This lowers the symmetry further to a pointgroup mm
2. When an electrical current is passed in the plane perpendicular to the growthdirection, carriers acquire a non-equilibrium spin polarization δ s SO , which in general can bedecomposed into a component that is parallel to the in-plane magnetisation of the NiMnSbfilm, δ s (cid:107) SO , and a component that is perpendicular to it, δ s ⊥ SO . This includes both intrinsicand extrinsic contributions. 4e use ab-initio calculations to evaluate the current-induced non-equilibrium spin polar-ization as a function of the directions of magnetization and applied electric field (see Ap-pendix C for the description of the numerical method). We consider here only the in-planecomponents of current-induced spin-polarization since this is the component relevant for theUMR and the field-like torque. For these calculations we used the relaxation time τ chosenso that the theoretical conductivity matches the experimental value of 3 . × Scm − . Thetetragonal and shear strains are chosen to make the Rashba and Dresselhaus contributions ofcomparable strength, resulting in a non-trivial dependence of δ s SO on the crystal directionof the applied electric field. In Fig. 1 we plot the perpendicular ( δ s ⊥ SO ) and parallel ( δ s (cid:107) SO )components of the current-induced non-equilibrium spin polarization. The dependence of δ s ⊥ SO ∼ cos( θ − θ SO ) on the magnetization angle θ corresponds to what we would expect tofind for a field-like SOT generated by a magnetisation-independent effective spin-orbit field, h SO ≈ − J ex δ s SO /m , where J ex is the exchange constant between carrier spins and the in-plane magnetisation m . θ SO would then correspond to the angle of δ s SO for the given crystaldirection of the applied electric field. In our case, θ SO = 0 for the electric field along [110] or[1¯10] axes since the Rashba and Dresselhaus-like spin-orbit fields are (anti)parallel for thesetwo crystal directions. For [100] or [010] axes, the two spin-orbit fields are orthogonal toeach other and their vector sum results in θ SO (cid:54) = 0.Remarkably, when we include δ s (cid:107) SO , total δ s SO becomes magnetization-dependent (seeFigs. 1). We emphasize that this dependence of δ s SO ( h SO ) on the magnetization anglewould be invisible when measuring SOT, which only depends on δ s ⊥ SO . SOT can be reliablyobtained from on-resonance mixing-signal measurements since the detected signal can bedirectly attributed to the precessing magnetization driven by the SOT. Our calculations inFig. 1 demonstrate, however, that a simple harmonic dependence of δ s ⊥ SO on the magne-tization angle, when inferred from the SOT measurement, does not imply that the total δ s SO is constant and that its parallel component δ s (cid:107) SO is a 90 ◦ -phase shifted replica of theperpendicular component. Therefore, if considering δ s (cid:107) SO as the driving mechanism behindUMR, SOT measurements cannot be, in general, used to quantify experimentally δ s (cid:107) SO in agiven structure. 5 . Unidirectional magnetoresistance When writing the total current density j up to the second order in the applied electricfiled E as, j i = j (1) i + j (2) i = σ ij E j + ξ ijk E j E k , (1)UMR has been associated with the longitudinal component of the ξ ijk transport coefficient. Formally, ξ ijk is described by the second order quantum mechanical Kubo formula. However,finding accurate solutions of the formula is a major challenge, in particular in the presenceof electron scattering. Here we analyze ξ ijk using a semiclassical Boltzmann approximation,where ξ ijk = − e τ (cid:88) n (cid:90) d k (2 π ) v in v jn v kn ∂ f ∂(cid:15) n ( k ) , (2)and v n ( k ) = (cid:126) ∂(cid:15) n ∂ k , f is the Fermi-Dirac equilibrium distribution function, and (cid:15) n ( k ) is theband energy (see Appendix A for the derivation of this formula).Since the group velocity v ( k ) is odd under space inversion, the second-order term willvanish in crystals that have inversion symmetry. Moreover, it will also vanish in nonmag-netic crystals since v ( k ) is odd under time-reversal. Furthermore, similarly to the anomalousHall effect in coplanar magnetic systems, the second-order term will vanish in the absence ofspin-orbit coupling, as the system will then be invariant under combined spin rotation andtime-reversal symmetry. ξ ijk will thus be present in magnetic crystals with broken inver-sion symmetry as a consequence of the spin-orbit coupling. These are the same symmetryrequirements as for the existence of SOT.In Fig. 2 we plot the calculations of the longitudinal and transverse components of thesecond-order current obtained from the Boltzmann equation (2), normalized to the first-order current as a function of the magnetization angle for different directions of the electricfield and for a current density of 10 Am − . The relative amplitudes of the longitudinalcomponent for different directions of the applied electric field are similar to the relativeamplitudes of δ s ⊥ SO (cf. Figs. 1 and 2). In both cases, the amplitudes are comparable forthe electric field along [100] and [010] axes, while the largest/smallest amplitude is obtainedfor fields along [110] / [1¯10] axes. From these results we can expect similar amplitude ratiosalso in the measured UMR and SOT. We point out, however, that this is not necessarilya consequence of a common proportionality of UMR and SOT to δ s SO . The Boltzmann6pproximation formula (2) gives an explicit example of a contribution to UMR with no directrelationship to δ s SO . The similar amplitude ratios in the two cases are merely a reflection ofa common Rashba-Dresselhaus-like symmetry of the underlying spin-orbit coupled electronicstructure.Our calculations in Fig. 2 also illustrate that the second-order current can have a sizabletransverse component. Specifically in NiMnSb, the transverse component has a comparableamplitude to the longitudinal component for [100] and [010] crystal directions. Here we recallthat in earlier experimental studies, the separation of UMR from competing thermo-electriccontributions was based on the assumption that UMR had only a longitudinal componentwhile the thermal effects contributed to both longitudinal and transverse signals. Ourresults show that using the transverse signal for experimentally calibrating the magnitudeof the thermal contribution is not, in general, reliable because the transverse component canalso contain a contribution from UMR.
III. MEASURED DATA AND DISCUSSION
In our experiments, we pattern all our bard from the same 34 nm thick film of ferromag-netic NiMnSb epitaxially grown on a 200 nm thick In . Ga . As buffer layer on an Fe:InPinsulating substrate and capped with a 5 nm MgO layer. The vertical lattice constant of5.97 ˚A indicates that the film is under compressive strain and is close to a stoichiometriccomposition. Fig. 3a illustrates our measurement set-up. The NiMnSb film is patternedinto 4 × µ m bars along different crystal directions and mounted on the rotational stageof an electromagnet. A microwave current I cos( ωt ) is passed in the bars and the polariz-ing action of the spin-orbit coupling induces an oscillating non-equilibrium spin population δ s SO ( ω ) which scales linearly with the current. The transverse component δ s ⊥ SO is responsi-ble for generating torques on the magnetisation via the effective field h ⊥ SO ≈ − J ex δ s ⊥ SO /m .As described in the previous paragraph, the longitudinal component δS || is responsible forgenerating both longitudinal and transverse components of the unidirectional magnetoresis-tance. Here we focus on the longitudinal components only and measure the dc longitudinalvoltage V dc via a bias tee.When ferromagnetic resonance is excited, rectification between the microwave currentand the oscillating AMR results in a resonance in V dc . The resonance is clearly visible in7he 2D plots in Fig. 3b. It shows V dc as a function of the external magnetic field and itsdirection θ with respect to the current direction, for bars parallel to the [¯110] and [100]axes. From the analysis of its line-shape (see Ref. 6 for details) we are able to quantify h ⊥ SO .In agreement with previous studies on different systems, we experimentally identifythe θ -dependence of h ⊥ SO for the different crystal directions in which the microbars arepatterned, as shown in Fig. 4. As in the theoretical calculations of δ s ⊥ SO , the amplitudes ofmeasured h ⊥ SO for bars along the [100] and [010] axes are similar, while the largest/smallestamplitudes are obtained for the [110] / [1¯10] directions, consistent with the combined Rashba-Dresselhaus-like symmetry of the NiMnSb electronic struture. The theoretical magnitudeof | h ⊥ SO | ≈ | J ex δ s ⊥ SO /m | ∼ µ T at 10 Am − current density was reported earlier in Ref. 8and is of the same order of magnitude as in the experiment.In Fig. 3b we notice that the resonance is sitting on a sinusoidally varying background,which we refer to as V BG . This is also shown in Fig. 5, where the θ -dependence of thebackground voltage at a saturating magnetic field is plotted for bars patterned along thefour different crystal directions [100], [010], [110] and [¯110]. The striking feature is that themagnitude of V BG is again strongly dependent on the bar direction, however, the directionscorresponding to the largest and smallest V BG switched place compared to h SO (or δ s SO ).We note that these amplitude ratios of V BG , as well as the crystal direction dependent phaseshifts, are reproducible in different physical sets of samples patterned along the four crystaldirections and do not depend on applied power, as shown in Appendix D.To interpret the measured V BG we now consider UMR and the thermoelectric contribu-tion, namely the anomalous Nernst effect (ANE). When exciting the system by the appliedac current, an out-of-plane temperature gradient due to Joule heating can result in an electri-cal signal detected in the sample plane due to ANE. Since the heat deposited by the currentscales with the square of the current density, ANE is a second-order effect in the electricfield, just like the second order term of the conductivity. Based on a recent experimentalmeasurement of ANE in NiMnSb and our numerical simulation of the heat gradient, weestimate that the contribution to V BG due to ANE is ∼ . − . µ V per current densityof 10 Am − (see Appendix E). While the magnitude is similar to that of the measuredsignals in Fig.5, we do not expect a strong dependence of the ANE contribution to V BG onthe crystal direction of the applied current. This is because ANE requires only the time-reversal symmetry breaking by the magnetization while broken spatial symmetries of the8rystal only lead to additional, higher order corrections.UMR, on the other hand, is generated by the inversion symmetry breaking which isof the combined Rashba-Dresselhaus-like form in our NiMnSb samples. As discussed inthe theory section, this leads to a strong crystal direction dependence of the UMR. Sincetheory suggests that the amplitude ratios for the different crystal directions of SOT andUMR are similar, we can use this as a constraint when fitting the measured V BG data. Theresults of the fitting shown in Fig. 5 were obtained by assuming fixed amplitude ratiosof UMR, corresponding to the measured amplitude ratios of the SOT fields h ⊥ SO , plus anANE contribution with an amplitude which is independent of the crystal direction. Theextracted experimental ANE component is of the same order of magnitude as the aboveestimate. The fitted UMR contribution is an order of magnitude larger than our theoreticalvalue which we attribute to the crude semiclassical Boltzmann approximation used in theUMR calculations. In the future, more elaborate Kubo formula calculations seem necessaryto capture UMR in NiMnSb on the quantitative level. They will also allow for verifyingthe correspondence between the SOT and UMR amplitude ratios, and by this for morefirmly establishing the fitting method we used to separate the UMR and thermoelectriccontributions in the measured data. IV. SUMMARY
Based on our study of ferromagnetic NiMnSb with non-centrosymmetric bulk crystalstructure we make the following observations regarding the explored non-equilibrium spin-orbit coupling effects: (i) A harmonic dependence on the magnetization angle of the com-ponent of the current-induced spin polarization transverse to the magnetization does notimply, in general, that the in-plane component is its 90 ◦ phase shifted replica with the sameamplitude. Despite the harmonic dependence of the transverse component, the total spinpolarization vector, and the corresponding total current-induced spin-orbit field vector, isnot necessarily independent of the magnetization angle. As a result, a measurement of SOT,driven by the transverse component of the current-induced spin-orbit field, should not beused, in general, for extracting the total non-equilibrium spin-orbit field (spin polarization)vector. (ii) The approximate Boltzmann theory of UMR, together with the approximate as-sumption of SOT being proportional to the current-induced spin polarization, suggest that9he amplitude ratios of UMR and SOT for electric fields applied along different crystal di-rections are similar. This can be used for separating experimental UMR and thermoelectric(e.g. ANE) contributions, by employing independently measured SOT. On the other hand,measurements of the diagonal and off-diagonal components in the electric field – current in-dices is not, in general, a reliable tool for separating UMR and thermoelectric contributionsbecause both can have sizable diagonal and off-diagonal components. (iii) Finally, the Boltz-mann theory also illustrates that UMR can have microscopic contributions which are notdirectly related to the current-induced spin polarization vector. Similar phenomenologiesobserved in UMR and SOT can be a mere reflection of the common underlying relativisticelectronic structure with broken time and space inversion symmetries. ACKNOWLEDGMENTS
We acknowledge the Grant Agency of the Czech Republic Grant No. 19-18623Y, Ministryof Education of the Czech Republic Grants LM2018110, LNSM-LNSpin and e-InfrastructureCZ – LM2018140, EU FET Open RIA Grant No. 766566 and support from the Instituteof Physics of the Czech Academy of Sciences and the Max Planck Society through theMax Planck Partner Group Programme. CC acknowledges support from the Royal Soci-ety. JPP acknowledges support from the EPSRC. The work of JZ was supported by theMinistry of Education, Youth and Sports of the Czech Republic from the OP RDE pro-gramme under the project International Mobility of Researchers MSCA-IF at CTU No.CZ.02.2.69/0.0/0.0/18 070/0010457. T. R. Mcguire and R. I. Potter, IEEE Transactions on Magnetics , 1018 (1975). F. E. Neumann, in
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Calculations of current-induced spin polarization for different directions ofelectric field.
The plots show the total spin polarization, as well as its components that arelongitudinal and transverse to the magnetization, as a function of the magnetization orientationwithin the (001)-plane. In each plot, θ denotes the angle of the magnetization measured from theelectric field. Dashed lines represent harmonic fits to the numerical points. IG. 2.
Second-order current calculations for different directions of the electric field.
The plots show the second order current separated into parallel and perpendicular components tothe electric field as a function of the magnetization orientation within the (001)-plane. In eachplot, θ denotes the angle of the magnetization measured from the electric field. The fits were doneusing the lowest order expansion of the second-order conductivity tensor given in Eq. (B1). Thistensor does not depend on E and thus the fitting is done together for all directions of E . Thefitting is done for both the transverse and longitudinal components together. We plot the resultsas a ratio of the second-order current to the first order longitudinal current. This ratio dependslinearly on the electric field, and the values are given for an electric field that corresponds to alongitudinal current density of 10 Am − . IG. 3.
Experimental set-up and on and off-resonance measurements of the mixing dcsignal. (a)
Design of the experimental layout. (b)
Longitudinal dc voltage measured as a functionof the external field B and its direction θ with respect to the bar for a bar patterned along the[¯110] and one patterned along the [100] direction. A current density of 5.7x10 Am − was usedfor the [¯110] bar while a current density of 1.3x10 Am − was used for the [100] bar. IG. 4.
Experimental SOT measurements for different bar orientations.
The plots showthe measured transverse components of the effective field B so as a function of the magnetizationorientation within the (001)-plane. In each plot θ denotes the angle of the magnetization measuredfrom the electric field, as illustrated in Fig. 3(a). Dashed lines are the harmonic fits. IG. 5.
Extraction of the UMR and ANE contributions from the measured backgroundvoltage.
The plots show the measured background voltage as a function of the magnetizationorientation within the (001)-plane. The UMR and ANE fits are done assuming that the ANEis isotropic and UMR is described by the lowest order second-order conductivity tensor given inEq. (B1) with the additional assumption that the relative amplitudes of the UMR for different bardirections are the same as the relative magnitudes of the SOT. This leaves three fitting parameters,which can be understood as the overall UMR magnitude, UMR phase shift for the [100] and [010]directions and the ANE magnitude. ppendix A: Second-order Boltzmann formula derivation Here we derive the second-order Boltzmann formula (2). The general form of the Boltz-mann formula for a distribution function g ( t, r , k ) under the assumptions of a stationaryand spatially homogeneous g and no magnetic field can be expressed as: − e (cid:126) E · ∇ k g = (cid:18) dgdt (cid:19) col , (A1)where e is the (positive) elementary charge, E is the electric field and (cid:0) dgdt (cid:1) col is the changeof the distribution function due to scattering.We will assume the constant relaxation-time approximation: (cid:18) dgdt (cid:19) col = − g − g τ , (A2)where τ is the relaxation time and g is the equilibrium distribution function, which forelectrons is the Fermi-Dirac distribution function: g ( r , k ) = f FD ( (cid:15) ( k )) = 1 e (cid:15) ( k ) − µkBT + 1 . (A3)Within the constant relaxation-time approximation the Boltzmann formula has the followingform: − e (cid:126) E · ∇ k g = − g − f FD τ . (A4)To find a solution for g up to second order in E we expand g in powers of E : g = g + g i E i + g ij E i E j , (A5)and insert it into the Boltzmanm formula (A4). Keeping only the terms up to E we find: − e (cid:126) E i (cid:18) ∂g ∂k i + ∂∂k i ( g j E j ) (cid:19) a = − g i E i + g ij E i E j τ . (A6)Since this equation must hold for all E , the coefficients for the E and E terms on bothsides of the equation must be the same. Therefore g i = eτ (cid:126) ∂g ∂k i , (A7) g ij = eτ (cid:126) ∂g j ∂k i = e τ (cid:126) ∂ g ∂k i ∂k j . (A8)18onsidering that the dependence of g on k is only through (cid:15) we find: g i = eτ (cid:126) ∂g ∂(cid:15) ∂(cid:15)k i , (A9) g ij = e τ (cid:126) (cid:18) ∂ g ∂(cid:15) ∂(cid:15)∂k i ∂(cid:15)∂k j + ∂g ∂(cid:15) ∂ E ∂k i ∂k j (cid:19) . (A10)Taking into account the relation v ( k ) = 1 (cid:126) ∂(cid:15)∂ k (A11)we have g i = eτ v i ∂g ∂(cid:15) , (A12) g ij = e τ (cid:18) ∂ g ∂(cid:15) v i v j + ∂g ∂(cid:15) (cid:126) ∂v j ∂k i (cid:19) . (A13)Electrical current is then given by: J = − e (cid:90) d k (2 π ) v ( k ) g. (A14)We note that this integral is done over the first Brillouin zone or any other unit cell in thereciprocal space. The first order contribution to the current is given by J = − e τ E i (cid:90) d k (2 π ) v v i ∂g ∂(cid:15) , (A15)and the second order contribution: J = − e τ E i E j (cid:90) d k (2 π ) v (cid:18) v i v j ∂ g ∂(cid:15) + 1 (cid:126) ∂v j ∂k i ∂g ∂(cid:15) (cid:19) . (A16)Note that in a multi-band system, these expression give a contribution from each individualband and the total current will be a sum over all bands.The expression for the second order current can be further simplified. We first define asecond order conductivity tensor ξ ijl : J i = ξ ijl E j E l . (A17)From Eq. (A18) we have 19 ijl = ξ aijl + ξ bijl , (A18)where ξ aijl = − e τ (cid:90) d k (2 π ) v i v j v l ∂ g ∂(cid:15) (A19) ξ bijl = − e τ (cid:90) d k (2 π ) v i (cid:126) ∂v l ∂k j ∂g ∂(cid:15) . (A20)Alternatively, using Eqs. (A7) and (A8), ξ can be written in the form ξ ijl = − e τ (cid:126) (cid:90) d k (2 π ) v i ∂∂k j (cid:18) v l ∂g ∂(cid:15) (cid:19) , (A21)Using integration by parts ξ ijl = − e τ (cid:126) (cid:90) Γ dk (2 π ) v i v l ν j ∂g ∂(cid:15) + e τ (cid:126) (cid:90) d k (2 π ) ∂v i ∂k j v l ∂g ∂(cid:15) , (A22)here Γ signifies integral over the unit cell boundary and ν is the outward unit normal vectorto the boundary. The first term in this relation in fact vanishes. To see that this is thecase it is useful to use for the integration a unit cell of the reciprocal space spanned by thereciprocal lattice vectors b , b , b (see Fig. (6)), instead of the first Brillouin zone. Thefirst term in (A22) is then given by sum over 6 surfaces. Since g is a periodic function of k , also ∂g ∂(cid:15) must be periodic. This means that at the opposite boundaries of the reciprocalunit cell ∂g ∂(cid:15) is the same. However, since the outward unit normal vector ν is opposite forthe opposite boundaries the whole term vanishes.Combining Eqs. (A21) and (A22) we have ξ ijl = ξ aijl + ξ bijl = − ξ blji (A23)To simplify this further we need to show that ξ ijl is symmetric under interchanging any twoindices. This is clearly satisfied for ξ aijl , however, it is less obvious for ξ bijl . From Eq. (A17),we see that ξ ijl = ξ ilj must hold and thus also ξ bijl = ξ ,bilj should be satisfied. This can beexplicitly verified from (cid:126) ∂v l ∂k j = ∂ (cid:15)∂k l ∂k j , which is symmetric as long as (cid:15) is sufficiently smoothfunction. To show that the tensors are also symmetric under interchanging the other indices,we consider: ξ jil − ξ ijl = − ξ blij + ξ blji = 0 . (A24)20 IG. 6. (a)
Unit cell of the reciprocal space defined by the reciprocal lattice vectors b , b , b .Image adapted from https://commons.wikimedia.org/w/index.php?curid=29922624 under CC BY-SA 3.0. (b) Coordinate system used for the symmetry analysis. The vectors x , y , z define thecartesian coordinate system used for the symmetry analysis. a , a , a are the lattice vectors ofthe NiMnSb in presence of shear strain. Without the strain, the lattice vectors would correspondto the conventional lattice of the cubic NiMnSb lattice and would be oriented along the cartesiancoordinate system. Then it must also hold that the ξ ijl and ξ bijl tensors are symmetric under interchangingindices i and l : ξ ijl = ξ jil = ξ jli = ξ lji . (A25)We can thus rewrite Eq. (A23) as ξ ijl = ξ aijl + ξ bijl = − ξ bijl . (A26)Therefore, ξ bijl = − ξ bijl / ξ ijl = ξ aijl − e τ (cid:90) d k (2 π ) v i v j v l ∂ g ∂(cid:15) . (A27) Appendix B: Symmetry of second order currents
Here we study the symmetry of the second-order currents. Similarly to other responsephenomena, the second order currents will in general contain both time-reversal even ( T -even) and time-reversal odd ( T -odd) components. Here we consider only the time-reversalodd component since this component corresponds to the UMR. In the following ξ ijk willdenote the T -odd second-order conductivity tensor. Furthermore, we assume the ξ ijk tensoris symmetric under interchanging any two indices. As shown in Appendix A, this holds for21he Boltzmann formula that we use in our calculations. It is not clear whether this holdsfor the ξ ijk in general, thus the symmetry analysis here should be taken to refer specificallyto the Boltzmann contribution. The method for the symmetry analysis is analogous to theone used in Ref. 26. We have implemented the second order symmetry analysis in the opensource code Symmetr. All the results shown here are given in a cartesian coordinate systemdescribed in Fig. 6(b). Since in the experiment the magnetization always lies in the [001]plane, we consider here only magnetization in this plane. In Table I we give the generalshape of the ξ ijk for general direction of magnetization within this plane as well as for the[110] and [1-10] directions where the symmetry is higher. ξ xjl ξ yjl ξ zjl M ⊥ [001] ξ xxx ξ yxx ξ yxx ξ yyx
00 0 ξ zzx ξ yxx ξ yyx ξ yyx ξ yyy
00 0 ξ zzy ξ zzx ξ zzy ξ zzx ξ zzy M || [110] or [1 − ξ xxx ξ yxx ξ yxx − ξ yxx
00 0 ξ zzx ξ yxx − ξ yxx − ξ yxx − ξ xxx
00 0 − ξ zzx ξ zzx − ξ zzx ξ zzx − ξ zzx TABLE I. Symmetry of second-order conductivity tensor in NiMnSb for different directions of themagnetization. Note that the ξ ijk coefficients between different directions of M are not in generalrelated. To describe the dependence of the second-order currents on magnetization, it is useful toexpand the ξ ijk tensor in powers of the magnetization. We consider only the lowest order22erm since it describes well both the calculations and the experiment: ξ xjl = M x x − M y x M x x − M y x M z x M x x − M y x M x x − M y x M z x M x x − M y x ,ξ yjl = M x x − M y x M x x − M y x M x x − M y x M x x − M y x − M z x − M z x M x x − M y x , (B1) ξ zjl = M z x M x x − M y x − M z x M x x − M y x M x x − M y x M x x − M y x . Here x i denotes free parameters of the expansion. Appendix C: Calculation description
The calculations utilize the FPLO density-functional theory (DFT) code for descrip-tion of the electronic structure. This DFT uses a local orbitals basis set for solving theKohn-Sham equations. This makes it easy to transform the DFT Kohn-Sham Hamiltonianinto a Wannier form, which is needed for the transport calculations. We use the full set ofbasis orbitals for this transformation, which makes the Wannier Hamiltonian very accurate.This is crucial for the second-order calculations since the second order contributions are verysmall and very sensitive to small symmetry violations that often exist in Wannier Hamilto-nians generated by the more conventional approach based on maximally localized Wannierfunctions. The transport calculations utilize Wannier interpolation to evaluate the responseformula on a tight grid in the reciprocal space. We have implemented the second-orderBoltzmann calculation in the freely accessible Linres code. We have also utilized this codefor the calculations of the current-induced spin polarization.The spin-polarization calculations use the following Kubo formula for the esponse tensor(i.e., tensor such that δs SO ,i = χ ij E j ): χ ij = − e (cid:126) V π Re (cid:88) k ,m,n (cid:104) u n ( k ) | ˆ S i | u m ( k ) (cid:105) (cid:104) u m ( k ) | ˆ v j | u n ( k ) (cid:105) Γ (( E F − E n ( k )) + Γ )( E F − E m ( k )) + Γ ) . S is the spin operator, u n ( k ) are the Bloch functions of a band n , k is the Blochwave vector, ε n ( k ) is the band energy, E F is the Fermi energy, ˆ v is the velocity operatorand Γ is a quasiparticle broadening parameter that describes the disorder strength. Thisformula becomes equivalent to the first-order constant relaxation time Boltzmann formulafor small Γ (with τ = (cid:126) / T -even part of the Kubo formula. We do not consider here the T -odd componentsince experimentally it is seen that the in-plane SOT has a field-like character and thecorresponding current-induced spin-polarization is thus T -even. Our test calculations ofthe T -odd component also suggest that it is much smaller than the T -even component forrealistic values of Γ.The DFT calculations utilized 12x12x12 k-points and the GGA-PBE potential. Thecurrent-induced spin-polarization and the second-order conductivity response calculationsuse a 400x400x400 k-mesh, which we have confirmed to be sufficient for good convergence.To estimate the value of Γ we calculate the first order conductivity using a conductivityformula analogous to Eq. (C1) and choose the Γ so that the conductivity matches theexperimental conductivity. This corresponds to Γ ≈ τ ≈ .
036 eV was used, because the samplesused in those experiments had somewhat larger conductivity. We note that our calculationsof the current-induced spin-polarization were done for the Γ = 0 .
036 eV value and havebeen afterwords rescaled to Γ = 0 .
05 eV, assuming 1 / Γ scaling. This is quite accurate sincefor theses value of Γ the formula is very close to the Boltzmann formula and thus has 1 / Γscaling.
Appendix D: Power dependence and reproducibility
In Fig. 7 we show that, as expected, the background voltage V BG scales linearly withpower and that the angular dependence of V BG is independent of the power. In Fig. 8 weshow that the results are reproducible between different bars patterned on the same chip.24 P (mW) q S O ( deg ) a. b. V B G ( V ) q (deg) V B G ( V ) P (mW)
FIG. 7. (a)
Top: Phase shift θ SO of V BG , fitted by V sin( θ + θ SO ), for a bar along [100] at differentvalues of the microwave power. Bottom: angular dependence of V BG for different microwave powerspassed in the bar, showing no phase shift. (b) Power dependence of V for the same bar.FIG. 8. Angular dependence of V BG , normalized to a current density of 10 Am − , measuredfor two different bars patterned on the same chip along each of the four crystal directions. Top:measurements for the first set of bars. Bottom: measurement for the second set of bars. Appendix E: Modeling temperature gradient using Finite Element Method
The system is excited by an ac current passing mainly through the NiMnSb layer whichresults in Joule heating. The heat scales with the square of the current density and dis-sipates into surrounding material giving rise to temperature gradients. The out-of-planetemperature gradient drives ANE which contributes to the electrical signal detected in thesample plane. We perform a simulation of heat transfer in the cross-section of our deviceutilizing the finite element method (FEM) as implemented in COMSOL Multiphysics . Thesoftware solves the heat equation numerically using an automatically generated triangularmesh with density adjusted to the size of individual domains of the device. The geometry25s shown in Figure 9 and consists of the NiMnSb wire (thickness = 37 nm, width = 1 µ m)deposited on a (In,Ga)As mesa (thickness = 200 nm, width = 1 µ m). There is also an InPsubstrate (thickness = 1 µ m, width= 2 µ m), a thin MgO capping layer (thickness = 5 nm,width = 1 µ m), and a He atmosphere included in the simulation. FIG. 9. Simulated geometry - cross-section of the device, NiMnSn conducting the ac current ishighlighted in blue, thin MgO capping layer (poorly visible in this plot) is included in the simulation.
The reference temperature, T ref , is set to 300 K (before the Joule heating takes place).The current density applied to NiMnSb is 10 or 10 Am − . At the boundary of thesimulated area the temperature is fixed to T ref or a thermal insulation is assumed (except thebottom boundary again fixed to T ref ). These two types of boundary condition correspondto very efficient cooling (transfer of heat to the surroundings) or to extremely poor cooling,respectively. The real system would fall between these two limiting cases. Figures 10 and 11show the simulated steady-state temperature distribution for the case with boundaries fixedto T ref and with insulating boundaries, respectively.The former case (Fig. 10) is simulated with applied current density of 10 and 10 Am − but we show the temperature profile only for 10 Am − . The 10 times larger current densityresults visually in the same temperature profile but the maximum temperature increase, T-T ref , is 100 times larger.The latter case (Fig. 11) is simulated only with applied current density of 10 Am − andall the heat is dissipated only via the bottom boundary – through the substrate. The out-of-plane temperature gradient is evaluated along a vertical cut-line (along the z-coordinate)26 IG. 10. Temperature distribution in case the applied current density is 10 Am − and allboundaries are set to T ref = 300 K – the most efficient cooling.FIG. 11. Temperature distribution in case the applied current density is 10 Am − and the top,left and right boundary of the simulated domain are insulating and the bottom boundary is set toT ref = 300 K - the least efficient cooling. running through the middle of the mesa. Figure 12 shows the temperature increase, T-T ref , along the cut-line within the NiMnSb layer (1200 nm to 1237 nm) for the three casessimulated.As expected, the case with insulating boundaries shows a larger increase in temperaturebut the dependence on the z-coordinate is the same. The case with 10 times larger appliedcurrent shows a 100 times larger increase of temperature which is due to the Joule heatingscaling with the square of the applied current density.27 IG. 12. Temperature increase T-T ref along vertical cut-line withing the NiMnSb layer.FIG. 13. Out-of-plane temperature gradient along vertical cut-line within the NiMnSb layer.
Finally, Figure 13 shows the out-of-plane temperature gradient generating ANE in theNiMnSb wire. It is evaluated simply as a numerical derivative of the temperature givenin Figure 12 for the three cases. Note that the gradient decreases linearly in the NiMnSblayer towards the top surface. The efficiency of the cooling (insulating boundaries or fixedtemperature) does not affect the gradient so the approximation of the cooling mechanismassumed in our model should not significantly compromise the validity of our numericalresults. The main result of the FEM simulation relevant to the experimental device is thatthe average out-of-plane temperature gradient is of the order of 10 Km − for current densityof 10 Am − . 28 E [Sm − ] σ T [Wm − K − ] c p [Jkg − K − ] ρ [kgm − ]MgO 0 200 σ E , thermalconductivity σ T , heat capacity c p , and mass density ρ . Our FEM results depend on the material parameters used. We have measured the elec-trical conductivity of the individual layers at room temperature. The room temperaturethermal conductivity and heat capacity parameters of our films are estimated based onliterature as listed in Table II. In case of NiMnSb they are estimated based on relatedmaterials.35,36