Spin Currents In Non-collinear Antiferromagnets

Spin currents in non-collinearantiferromagnetsJakub Železný, Yang Zhang, Yan Sun, Claudia Felser, Binghai Yan,Rafael Hernandez, Libor Šmejkal, Joao Godhino, JoergWunderlich, Tomás Jungwirth

Antiferromagnetic spintronicsRising interest in antiferromagnets for spintronics applicationsSome advantages over ferromagnets: Fast dynamics No stray fields, insensitive to magnetic fields Wide range of antiferromagnetic materials including metals,semiconductors, insulators, multiferroics, superconductors butThe antiferromagnetic order is difficult to manipulate and detectMany spintronics effects, which exist in ferromagnets are prohibited bysymmetry in most antiferromagnets

Spin-orbit torques in antiferromagnetsElectrical control and detection of the antiferromagnetic orderhas been demonstratedA key to efficient manipulation of theantiferromagnetic order is using a staggeredmagnetic fieldAsMn BCan be generated by spin-orbit torqueMn ACuCuMnAsZelezny et al., PRL 113, 157201 (2014)Wadley et al., Science 351, 587–590 (2016)Bodnar et al., arXiv:1706.02482Meinert et al., arXiv:1706.06983

Nature physics focused issue on antiferromagnetic spintronicsOverview - T. Jungwirth et al., Nature Physics 14, 200–203 (2018)Transport - J. Železný et al., Nature Physics 14, 220-228 (2018)Opto-spintronics - P. Němec et al., Nature Physics 14, 229-241 (2018)Spin textures and dynamics - O. Gomonay et al., Nature Physics 14, 213-216 (2018)Topology - L. Šmejkal et al., Nature Physics 14, 242-251 (2018)Synthetic antiferromagnets - R. A. Duine et al., Nature Physics 14, 217-219 (2018)

Non-collinear antiferromagnetsNon-collinear order is common in antiferromagnetsTriangular antiferromagnets Mn3XMn3Rh, Mn3Pt, Mn3IrMn3Ga, Mn3Ge, Mn3SnSimplest example of a non-collinear magnetic system

Spin currentsSpin-polarized current in ferromagnetsEOdd under time-reversalNon-relativistic originSpin-Hall effectEEven under time-reversalRelativistic originSpin currents are responsible for giant and tunnelingmagnetoresistance, spin-transfer torque, spin-orbit torque,

Definition of spin currentSpin current operator is normally defined as:Issues with this definition:1. Spin is not conserved so the spin current does not satisfy thecontinuity equationShi et al., PRL 96, 076604 (2006)2. Spin current can occur in equilibriumRashba, Phys. Rev. B 68, 241315(R) 20033. Spin currents also can exist in insulators4. Spin current cannot be directly measuredBut it’s a well defined (and useful) physical quantity!For accurate estimates and comparison with experiment it’s best tocalculate directly spin accumulation, torque

Spin-polarized currentIn most antiferromagnets spin-polarized current is not allowed bysymmetryIn the Mn3X non-collinear antiferromagnets spin-polarized currentis presentEZelezny et al., PRL 119, 187204 (2017)

Origin of the odd spin currentslongitudinalSpin-polarized currentTotal spin currentsEEtransversemagnetic spin Hall effectspin current spin x velocityZelezny et al., PRL 119, 187204 (2017)

Linear responseThese two parts have precisely opposite transformation under time-reversalUnder time-reversal:We use a constant band broadening to simulate disorder

Linear response“Real” part“Imaginary” partConductivity:even - ordinary conductivityodd - Anomalous HalleffectSpin-orbit torque:Odd - field-like torqueeven – antidampinglike torqueSpin current:odd – spin-polarizedcurrenteven - spin Hall effectOdd components can only exist in magnetic materials

CalculationsMn3SnMn3IrThe odd spin currents dependstrongly on disorder unlike theintrinsic spin Hall effectSpin current angle:Both large longitudinal and transverse spincurrents present𝜶 (%)𝜶 7Mn3Ir3.51.9Mn3Pt4.21.9Spin currents are smaller thantypically in ferromagnets, butstill relatively largeIn BCC Fe 𝜶 18%𝜶 1%Zelezny et al., PRL 119, 187204 (2017)

Magnetic spin Hall effectOdd – magnetic spin Hall effectEven - spin Hall effectSymmetry and origin distinct from the conventional spin Hall effectOdd:Even:No experimental evidence so far

TSpin-polarized 310KNcurrent in a collinear systemagonalNiAs structurenThe two sublattices are not connected by T inversion or T translationeMnTemagneticMn sheetstation in theplanenotknow n!Spin current angle for spin-polarized current and the magnetic SHE 1%allyptypeconductor(undoped)

SHE calculationsSpin Hall effect exist in these materials evenwithout spin-orbit coupling!(No spin-orbit coupling no relativistic effects)Yang Zhang et al., Phys. Rev. B 95, 075128 (2017)Yang Zhang et al., New J. Phys. 20 073028 (2018)

What is the origin of the SHE?Originates from the magnetic order instead of spin-orbit couplingSimilar to AHE or orbital magnetic moment in non-coplanarmagnetic systemsTaguchi et al., Science 291, 2573 (2001)

What is the origin of the SHE?Can be understood in terms of symmetry and is not necessarily related to topologyNon-collinear magnetic order breaks symmetry similarly to spin-orbit couplingExample:In a coplanar system time-reversal 180 spin rotation is a symmetry withoutSOCtime-reversalspin rotationNo AHE in a coplanar magnetic systemBrinkman et al., Proc. Royal Soc. London A,294, 343-358 (1966)Litvin et al., Physica 76, 538-554 (1974)

SOT in bilayer systemsSHE and the magnetic SHE will generate a spin-orbit torque in FM/AFM bilayersystemsThe spin-orbit torque has been experimentally studied in a number of systemsJThe SHE and the magnetic SHE exist even without spin-orbit couplingThe spin-orbit torque does not necessarily originate from spin-orbit couplingThsitoyan et al., PRB 92, 214406 (2016)Zhang et al., Science Advances 2, (2016)Young-Wan oh et al., Nature Nanotech., (2016)

Transverse spin currentsMagnetic and conventional spin Hall effect will depend differently on presence ofmagnetic domains and have different symmetryJJoddevenSensitive to the antiferromagneticorder and presence of domainsSpin current with spin-polarization along the spin current flow is possible

Antiferromagnetic junctionsj“1”Spin-transfer torque andgiant or tunnelingmagnetoresistance“0”FM junctionSimilar effects weretheoretically predicted, butare very sensitive to disorderand have never beenobserved experimentally“1”“0”AFM junctionJ. Železný, Nature Phys. 14, 220–228 (2018)

Spin-polarized currentIn non-collinear antiferromagnets, current is spin-polarized andthus same approach can be used as in ferromagnetsRobust giant or tunneling magnetoresistance and spin-transfer torque should bepresentE

180º reversalSignificance of the AHE and the spin-polarized current is in part because theyallow to detect 180º reversal of all magnetic moments180º reversal is equivalent to time-reversalj 𝜎Ejs 𝜎𝑠E𝜎even σ M1, M2, σ M1, M2, 𝜎𝑠even σ𝑠 M1, M2, σ𝑠 M1, M2, 𝜎odd σ M1, M2, σ M1, M2, 𝜎𝑠odd σ𝑠 M1, M2, σ𝑠 M1, M2, 𝜎odd AHE𝜎sodd spin-polarized current andmagnetic SHEIn AFM with T inversion or T translation symmetry both 𝜎odd and 𝜎sod vanish

Second-order effectsIn AFM with T inversion or T translation 180º reversal cannot be detected withlinear transport effectsPossible with second order, however!𝜎(2), allowed in systems with broken inversion symmetryAsThe odd component of 𝜎(2) allowed even in AFMswith PT symmetryMn BMn ACu

180º switching in CuMnAsSwitching in CuMnAs is normally 90º measured by AMRJ. Godinho et al., arXiv:1806.02795Wadley et al., Science 351, 587–590 (2016)180º switching

Origin of the second-order signalThe original idea behind the experiment was that the second-order signal isdue to tilting of the magnetic moments by the spin-orbit torqueBut the second order signal could also simply be a second orderconductivity, unrelated to a spin-polarization or a torque

Origin of the second-order signalUsing the Boltzmann equation within the constantrelaxation time approximation:The second-order term can be understood as originating from k 𝑘 bandassymetryRequires broken inversion and time-reversal spin-orbit coupling (or non-collinearmagnetic order)

Second-order measurementsSecond harmonics commonly used to measure spin-orbit torquesUnidirectional magnetoresistanceOlejnik et al., Phys Rev B 91, 180402(R) (2015)Avci et al., Nature Physics 11, 570–575 (2015)

SummaryNoncollinear antiferromagnets are very attractive for spintronics sincethey combine advantages of ferromagnets and antiferromagnets Spin-polarized currentAHE, Kerr effectspin Hall effect (without SOC)Detecting 180 switchingpossible Fast magnetic dynamics Insensitive to externalmagnetic fields No stray fieldsSpin-polarized current not limited to ferromagnets, but can exist incollinear and non-collinear antiferromagnetsMagnetic spin Hall effect: in magnetic materials a transverse currentdistinct from the spin Hall effect can existSpin-Hall effect can exist without SOCZelezny et al., PRL 119, 187204 (2017)Yang Zhang et al., New J. Phys. 20 073028 (2018)

Spin current operator is normally defined as: Issues with this definition: 1. Spin is not conserved so the spin current does not satisfy the continuity equation 2. Spin current can occur in equilibrium 3. Spin currents also can exist in insulators 4. Spin current cannot