Gradient expansion formalism for generic spin torques

Gradient expansion formalism for generic spin torques Atsuo Shitade RIKEN Center for Emergent Matter Science Atsuo Shitade, arxiv:1708.03424.

Outline 1. Spintronics a. Magnetoresistance and spin torques b. Two microscopic formalisms for spin torques 2. Gradient expansion a. Wilson line b. Moyal product 3. Application to spin torques a. 1st. order spin renormalization and Gilbert damping b. 2nd. order spin transfer torque and ββ-term

Magnetic systems nonvolatile memories Reading = magnetization measurement Giant MagnetoResistance Baibich et al., PRL 61, 2472 (1988). Binasch et al., PRB 38, 4828 (1989). Tunnel MagnetoResistance Julliere, Phys. Lett. 54A, 225 (1975). Miyazaki and Tezuka, J. Magn. Magn. Mater. 139, L231 (1995). Moodera et al., PRL 74, 3273 (1995). Fert Grünberg Nobel Prize 2007 antiparallel parallel Fe Cr Fe Cr Fe Cr Fe Cr Fe Cr Fe Cr

Magnetic systems nonvolatile memories Writing = magnetization reversal Magnetic field Magnetoresistive Random Access Memory Spin Transfer Torque STT-MRAM Berger, J. Appl. Phys. 55, 1954 (1984). Berger, PRB 54, 9353 (1996). Slonczewski, J. Magn. Magn. Mater. 159, L1 (1996). ħnn = nn HH sp nn + ττ ττ = ħssnn ħααnn nn ȷȷ s nn ββnn ȷȷ s nn Spin renormalization Gilbert damping STT ββ-term ββ-term Due to spin relaxation Zhang and Li, PRL 93, 127204 (2004). ββ = 0, ββ αα, or ββ = αα Thiaville et al., Europhys. Lett. 69, 990 (2005).

Real materials are more complicated Textured ferromagnets w/o Spin-Orbit Interactions STT and ββ-term ττ = ȷȷ s nn ββnn ȷȷ s nn Uniform ferromagnets w/ SOIs Spin-Orbit Torque ττ = 2mmmm/ħ zz ȷȷ ss nn for Rashba SOI Manchon and Zhang, PRB 78, 212405 (2008); PRB 79, 094422 (2009). Textured ferromagnets w/ SOIs Topological insulator surface Yokoyama et al., PRB 81, 241410 (2010); Sakai and Kohno, PRB 89, 165307 (2014). different forms of STT, ββ-term, and SOT Noncentrosymmetric Weyl ferromagnet Kurebayashi and Nomura, arxiv:1702.04918. nn is assumed to have small fluctuations from zz. small-amplitude formalism SOT only, no STT or ββ-term

Small-amplitude formalism Assume small transverse fluctuations around a uniform state nn = zz + uu, uu XX = uu QQ ee iiqq μμxx μμ = uu xx, uu yy, 0, uu 2 1 Pick up the first order w.r.t. QQ, uu, EE four-point vertices Kohno et al., JPSJ 75, 113706 (2006). vv ii --EE ii σσ ββ --uu ββ vv jj --QQ jj impurities Cannot extrapolate the small-amplitude dynamics to the finiteamplitude one except for some simple cases

Gauge transformation formalism Apply a gauge transformation ψψ XX UU XX ψψ XX, nn XX σσ σσ zz Magnetization texture is described by an emergent gauge field AA μμ XX iiuu XX XX μμuu XX = uu XX XX μμuu XX σσ. Pick up the first order w.r.t. AA μμ, EE ii Kohno and Shibata, JPSJ 76, 063710 (2007). vv ii --EE ii σσ ββ vv jj ββ --AA jj impurities Quenched magnetic impurities are transformed to dynamical ones, which yield an additional AA μμ.

Motivations nn can be controlled by EE. In real materials, we have (non)magnetic impurities, SOIs, sublattice degree of freedom (in antiferromagnets), etc. Small-amplitude formalism Cannot be applied to the finite-amplitude dynamics. Gauge transformation formalism Needs careful calculation not to miss AA μμ. Particularly in the presence of magnetic impurities; otherwise αα = 0. Can we derive a first-principles formula (into which we only have to input our Hamiltonian)? Gradient expansion

Gradient expansion Green s function GG xx 1, xx 2 which satisfies the Dyson equation, L xx 1 GG xx 1, xx 2 Σ GG xx 1, xx 2 = δδ(xx 1 xx 2 ) Wigner representation to deal with convolution, GG XX 12, pp 12 = dd DD xx 12 ee iipp μμ 12μμxx 12/ħ GG (xx 1, xx 2 ) Gauge covariance Wilson line GG (xx 1, xx 2 ) transforms as GG xx 1, xx 2 = VV xx 1 GG xx 1, xx 2 VV xx 2. GG xx 1, xx 2 should transform as GG xx 1, xx 2 = VV XX 12 GG xx 1, xx 2 VV (XX 12 ). Wigner representation of convolution Moyal product Levanda and Fleurov, J. Phys.:Condens. Matter 6, 7889 (1994); Ann. Phys. 292, 199 (2001). Ordinary product if translationally invariant, but not in general.

Wilson line Gauge covariance Wilson line GG (xx 1, xx 2 ) transforms as GG xx 1, xx 2 = VV xx 1 GG xx 1, xx 2 VV xx 2. GG xx 1, xx 2 should transform as GG xx 1, xx 2 = VV XX 12 GG xx 1, xx 2 VV (XX 12 ). Wilson line which satisfies WW xx 1, xx 2 = VV xx 1 WW xx 1, xx 2 VV (xx 2 ), WW xx 1, xx 2 PP exp ii ħ xx 1 ddyyμμ AA μμ yy xx 2 Locally gauge-covariant Green s function GG xx 1, xx 2 WW XX 12, xx 1 GG xx 1, xx 2 WW xx 2, XX 12 GG XX 12, pp 12 = dd DD xx 12 ee iipp μμ 12μμxx 12/ħ GG (xx 1, xx 2 )

Moyal product Wigner representation of convolution Moyal product Ordinary product if translationally invariant, but not in general. Moyal product Locally gauge-covariant convolution AA XX 12, pp 12 BB XX 12, pp 12 dd DD xx 12 ee iipp μμ 12μμxx 12/ħ WW XX 12, xx 1 dd DD xx 3 AA xx 1, xx 3 BB xx 3, xx 2 WW xx 2, XX 12 = dd DD xx 12 dd DD xx 3 ee iipp μμ 12μμxx 12/ħ ee iipp μμ 13μμxx 13/ħ ee iipp μμ 32μμxx 32/ħ WW XX 12, xx 1 WW xx 1, XX 13 AA XX 13, pp 13 WW XX 13, xx 3 WW xx 3, XX 32 BB XX 32, pp 32 WW XX 32, xx 2 WW xx 2, XX 12 Integrand is expanded w.r.t. xx 13, xx 32 around XX 12. Wilson loop is evaluated by the Stokes theorem. Nonabelian Stokes theorem for a nonabelian gauge field Aref eva, Theor. Math. Phys. 43, 353 (1980); Bralić, PRD 22, 3090 (1980). Dyson equation L Σ GG = 1

Moyal product Minimum for calculating spin torques AA BB = AA BB + iii/2 PP DD AA, BB + iii/2 PP F AA, BB + iii/2 2 PP DD F AA, BB PP DD AA, BB PP F AA, BB = XX λλaa ppλλ BB ppλλ AA XX λλbb = F μμμμ ppμμ AA ppνν BB U(1) field strength PP DD F AA, BB = F μμμμ XX λλ ppμμ AA ppλλ ppνν BB ppλλ ppμμ AA XX λλ ppνν BB More for something AA BB = + 1/2! iii/2 2 PP DD 2 AA, BB + 1/2! iii/2 2 PP F 2 AA, BB PP DD 2 AA, BB = XX λλ XX κκaa ppλλ ppκκ BB 2 XX λλ ppκκ AA ppλλ XX κκbb + ppλλ ppκκ AA XX λλ XX κκbb PP F 2 AA, BB = F μμμμ F ρρρρ ppμμ ppρρ AA ppνν ppσσ BB more terms up to 4th. order w.r.t. xx 13, xx 32 for a nonabelian field strength

Perturbation theory of Green s function Dyson equation L Σ GG = 1 AA BB = AA BB + iii/2 PP DD AA, BB + iii/2 PP F AA, BB + iii/2 2 PP DD F AA, BB GG = GG 0 + ħ/2 GG DD + ħ/2 GG F + ħ/2 2 GG DD F Σ = Σ 0 + ħ/2 Σ DD + ħ/2 Σ F + ħ/2 2 Σ DD F First-principles formula (PP = DD, F) GG 0 1 GG PP = Σ PP GG 0 iipp PP GG 0 1, GG 0 STT and ββ-term SOT Spin renormalization Gilbert damping GG 0 1 GG PP QQ = Σ PP QQ GG 0 + Σ QQ GG PP + iipp PP Σ QQ, GG 0 iipp PP GG 0 1, GG QQ + PP QQ ii 2 PP PP QQ GG 0 1, GG 0 Advantages Deals with spacetime gradient and field strength on an equal footing Guarantees gauge covariance Contains retarded, advanced and lesser Green s functions Spin expectation value is available from the above formula.

Benchmark: textured ferromagnet w/o SOIs Hamiltonian H XX, pp = pp 2 /2mm JJ nn XX σσ Impurities by the self-consistent Born approximation Nonmagnetic vv i δδ XX XX ijj Magnetic vv s mm jj σσ δδ XX XX sjj ; otherwise αα = ββ = 0. What we calculate Spin torques ττ = nn JJ σσ Spin expectation value σσ XX = ±iii dddd 2πππ dd3 pp 2πππ 3 tr σσ GG< XX, ξξ, pp Momentum integral and spin trace are carried out during the self-consistent calculation of the self-energy.

Spin renormalization and Gilbert damping 1st. order w.r.t. time gradient XX 0 GG 0 1 GG PP = Σ PP GG 0 iiηη PP IIII II GG 0 1 JJ GG 0 PP IIII PP AA, BB ηη IIII PP II AA JJ BB XX ηη μμ pp νν pp DD = ηη νν XX μμ μμ DD = δδ νν pp ηη μμ pp νν F = F μμμμ Lesser Green s function GG PP < = ± GG PP R GG PP A ff pp 0 + GG PP < 1 ff pp 0 GG 0 R 1 GG PP R = Σ PP R GG 0 R iiηη PP IIII II GG 0 R 1 JJ GG 0 R Fermi-surface terms only for XX 0 or F jjj GG 0 R 1 GG PP < 1 = Σ PP < 1 GG 0 A + iiηη PP IIpp 0 II GG 0 R 1 GG 0 R GG 0 A GG 0 R 1 GG 0 A 1 II GG 0 A Example of calculation gg R DD XX, ξξ = gg R DDD gg R DDD Σ R DDD Momentum integral ξξ = AA ξξ Σ R DDD ξξ nn XX nn XX σσ ξξ + BB ξξ ξξ = (γγ i γγ s /3)gg R DD2 ξξ Contribution to spin renormalization γγ i nn i vv i 2, γγ s nn s vv s 2

STT and ββ-term 2nd. order w.r.t. space gradient XX ii and electric field F jjj Lesser Green s function GG < R PP QQ = ± GG PP QQ A < GG PP QQ ff pp 0 + GG 1 PP QQ ff < pp 0 + GG 2 PP QQ ff pp 0 R GG 1 R 0 GG PP QQ = Σ R PP QQ GG RR 0 + Σ R QQ GG R PP + iiηη IIII PP II Σ R QQ JJ GG R 0 iiηη IIII R PP II GG 1 0 JJ GG R QQ + (PP QQ) + ηη PP IIII ηη QQ KKKK II KK GG 0 R 1 JJ LL GG 0 R R GG 1 < 0 GG 1 < PP QQ = Σ 1 PP QQ GG A 0 + Σ R < QQ GG 1 < PP + (Σ 1 QQ GG A PP + iiηη IIII < PP II Σ 1 QQ JJ GG A IIpp 0 iiηη 0 PP II Σ R QQ GG R 0 GG A 0 Σ R QQ Σ A A QQ II GG 0 iiηη PP IIII II GG 0 R 1 JJ GG QQ < 1 + iiηη PP IIpp 0 II GG 0 R 1 GG QQ R GG QQ A GG 0 R 1 GG 0 A 1 II GG QQ A ηη PP IIpp 0 ηη QQ KKKK II KK GG 0 R 1 LL GG 0 R GG 0 A + LL GG 0 R 1 GG 0 A 1 II KK GG 0 A + PP QQ ) Example of calculation R R gg DD F XX, ξξ = gg DD F1 Momentum integral ξξ F iii XX iinn XX σσ /mm Contribution to ββ-term R gg DD F1 R Σ DD F1 R ξξ = CC(ξξ)Σ DD F1 ξξ + DD ξξ R ξξ = (γγ i γγ s /3)gg DD F1 ξξ

Chemical-potential dependence ττ = nn JJ σσ ττ = ħ 4ππ 2 2mmmm ħ 2 3/2 ττ rrrrrr = JJ 1/2 dddd ff ξξ ττ ren + ττ αα nn nn 1 4ππ 2 2mmmm ħ 2 1/2 Igg RR DDD ξξ ff < ξξ iigg 1 DDD /2 F iii ττ STT + ττ ββ nn XX iinn Fermi-sea term of GG DD < ττ αα = JJ 1/2 dddd ff ξξ < iigg 1 DDD ξξ /2 Fermi-surface term of GG DD < ττ STT = JJ 1/2 dddd ff ξξ < 1 gg DD F2 ξξ /2ii Fermi-surface term < of GG DD F ττ ββ = JJ 1/2 dddd ff ξξ RR Igg DD F1 ξξ + ff < ξξ gg 1 DD F1 /2ii Fermi-sea term < of GG DD F

Summary Gradient expansion: Wilson line Wigner representation Perturbation theory w.r.t. spacetime gradient and field strength Explicit gauge covariance Application to spin torques Spin renormalization and Gilbert damping 1st. order w.r.t time gradient SOT 1st. order w.r.t. electric field STT and ββ-term 2nd. order w.r.t. space gradient and electric field