Mechanism of Electron Spin Relaxation in Spiral Magnetic Structures
aa r X i v : . [ c ond - m a t . m e s - h a ll ] A p r Mechanism of Electron Spin Relaxation in Spiral Magnetic Structures
I.I. Lyapilin ∗ Institute of Metal Physics of Ural Branch of Russian Academy of Sciences, 620990 Yekaterinburg, RussiaDepartment of Theoretical Physics and Applied Mathematics,Ural Federal University, Mira St. 19, 620002 Yekaterinburg, Russia (Dated: April 2, 2019)Spin dynamics in spiral magnetic structures has been investigated. It has been shown that theinternal spatially dependent magnetic field in such structures produces a new mechanism of spinrelaxation.
I. INTRODUCTION
One of the central problems facing spintronics, such asthe development of methods of injection, generation, anddetection of spin-polarized charge carriers, is to analyzeand study the mechanisms of spin relaxation. Spin-orbitinteraction (SOI) is well-known to have a significant influ-ence on the mechanisms of electron spin relaxation. SOIcan impact on spin degrees of freedom through transla-tional ones . SOI itself does not cause spin relaxation butin combination with the scattering of the electron mo-mentum leads to its relaxation. Elliott examined the pro-cess of electron spin relaxation through the momentumscattering at impurity centers, conditional upon induc-ing the spin-orbit interaction by lattice ions . Phononscan also participate in spin relaxation. The nature ofspin relaxation involving phonons is to modulate time-dependent spin-orbit relaxation by lattice oscillations .As a result, spin relaxation occurs. The Elliott-Jafetmechanism offers for the spin relaxation frequency ω s to be proportional to the electron momentum relaxationfrequency ω p . In systems without a center of inversion,a fundamentally different mechanism of spin relaxationcan be realized. In such systems, the spin states ↑ and ↓ are not degenerate E k ↑ = E k ↓ , but the condition issatisfied E k ↑ = E − k ↓ . The lack of a center of symmetrycontributes to splitting the states subjected to SOI. Thesplitting can be described by entering an intrinsic mag-netic field B i ( k ) around which the electron spins precessat a Larmor frequency. The precession of the electronspin in the effective magnetic field together with the elec-tron momentum scattering leads to spin relaxation, and ω s ∼ ω − p . Below we consider the spin relaxation mech-anism in magnetically ordered structures, which are char-acterized by a spiral arrangement of magnetic momentsrelative to some crystal axes . The simplest case of suchstructures is an antiferromagnetic spiral or a helicoid;they can be found in rare earth metals ( Eu, Tb, Dy ) andsome oxide compounds. The structure of this type can berepresented as a sequence of atomic planes perpendicularto the axis of a helicoid. In this case, atoms in each ofthe planes have ferromagnetically ordered magnetic mo-ments. However, the magnetic moments in neighboringplanes turn at some angle θ depending on the ratio of ex-change interactions. This is because of the coexistence ofpositive exchange interaction between the nearest atomic neighbors and negative exchange interaction between theneighbors following the closest ones. The components ofthe magnetic moments of atoms oscillate in the planeof the magnetic layer S x = S x sinkz, S y = S y coskz .If, in this case, S z = 0, we have a ferromagnetic spiralwith a resulting moment. If also oscillates S z by a har-monic law, a complex magnetic structure emerges. Insuch structures, the exchange interaction between zonecharge carriers and localized moments is described by thewell-known expression H ex = − J P i s · S where j is theexchange constant. In the mean-field approximation, thisinteraction can be represented as H ex = − m ( r ) H ef ( r ),where m ( r ) = gµ B S ( r ) , g – is the spectroscopic split-ting factor, µ B is the Bohr magneton, and H ef ( r ) is theinternal effective magnetic field.It is obvious that the spatial variation of the internalfield in spiral magnetic structures should affect the spindynamics of conduction electrons. Indeed, the spin of anelectron in a state with quasi-momentum ( k ) precesses inthe effective magnetic field H ef ( r ) only for a time of theorder of the elastic scattering time τ p . After scattering,the electron goes into a state ( k ′ ) where the effectivemagnetic field has a different direction. Consequently,the evolution of the spin dynamics under such conditionsturns out to be associated with the electron momentumrelaxation.Let us look into a simple spiral when the effec-tive magnetic field rotates in a plane ( xy ), where H ef ( H cos ( Qz ) , H sin ( Qz ) ,
0) , Q = 2 π/ Λ, Λ is theperiod of the spiral structure. To gain insight into thespin dynamics, we estimate the spin relaxation mecha-nism realized under above conditions. The system athand is assumed to expose to an electric field directedalong the axis z ( E = (0 , E ) . The Hamiltonian of thesystem can be written in the form H = H k + H s + H eE + H v + H ev , , where H k , H s are the operators of the kineticand spin energy of electrons interacting with the lattice H ev and the external electric field H eE = − e P i E r i and H v is the Hamiltonian of the lattice. H k = X j p i / m,H s = − gµ B X j s i H ef = − ¯ hω sf X j ( s + e iQz j + s − j e − iQz j ) , (1)where ω sf = geH / m o c is the electron precession fre-quency in the effective field, s ± = s x ± is y . The spindynamics is determined by macroscopic equations of mo-tion for the spin subsystem of electrons. The macroscopicequations of motion can be deduced by averaging micro-scopic equations of motion over the non-equilibrium sta-tistical operator ρ ( t ) . To begin with, we write downthe microscopic equations of motion for the longitudinalcomponents of the spin and momentum of electrons. Wehave ˙ s z = iω sf { s + e iQz − s − e − iQz } + ˙ s zsv , (2)˙ p z = − e E N + iQ ¯ hω sf { s + e iQz − s − e − iQz } + ˙ p zev ( r ) , . (3)˙ A = ( i ¯ h ) − [ A, H ] , ˙ A ij = ( i ¯ h ) − [ A, H ij ] . Averagingthe microscopic equations of motion over the operator ρ ( t ), we arrive at < A i > t = Sp { A i ρ ( t ) } . The explicitform of the operator ρ ( t ) can be found within the NSOmethod . Suppose that we have carried out this proce-dure. As a result, we come to the following system ofmacroscopic equations. < ˙ s z > t = iω sf { < s + e iQz > t − < s − e − iQz > t } + < ˙ s zev > t , (4)and < ˙ p z > = iQ ¯ hω sf { < s + e iQz > t − < s − e − iQz > t } −− e E n + < ˙ p zev > t . (5)where < N > t = n is the electron concentration. Thecollisional summands in the balance equations (4) and(5) can be represented as follows : < ˙ s zev > t ∼ ( s z , s z ) ω s , < ˙ p zev > t ∼ ( p z , p z ) ω p , (6)where ( A, B ) = Z dτ Sp { Aρ τ ∆ B ρ − τ } , (7)where ∆ A = A − < A > , < · · · > = Sp {· · · ρ } , ρ isthe equilibrium Gibbs distribution. ω s is the relaxation frequency of the longitudinal spin components, and ω p is the momentum relaxation frequency; both latter arecalculated in the Born approximation over the scatter-electron interaction. ω s = gµ B χ Z −∞ dt e ǫt ( ˙ s zev , ˙ s zev ( t )) ,ω p = 1 mnT Z −∞ dt e ǫt ( ˙ p zev , ˙ p zev ( t )) , ǫ → +0 . (8)Here χ is the paramagnetic susceptibility of an electrongas χ = ( gµ B ) T · ( s + , s − ) = gµ B H · < s z > , (9) T ≡ k B T is the temperature in energy units. Takinga stationary case into account, we finally obtain the ex-pression for the spin relaxation frequency in a helicoidalmagnet: ω s ∼ mnT ¯ h Q ( s z , s z ) · ω p (10)Thus, in spiral magnets as well as in the Elliott-Jafetmechanism, the spiral relaxation is related to the electronmomentum relaxation and ω s ∼ ω p . However, againstthe Elliott-Jafet mechanism, the spin relaxation mech-anism in spiral magnetic structures is due to the pres-ence of the internal spatially-dependent magnetic field inthem. The evolution of the spin dynamics is determinedby both the period of the magnetic structure and themomentum relaxation frequency.ACKNOWLEDGMENTS The research was carriedout within the state assignment of Minobrnauki of Rus-sia (theme ”Spin No AAAA-A18-118020290104-2), sup-ported in part by RFBR (project No. 19-02-00038/19). ∗ [email protected] E.I. Rashba, Electron spin operation by electric elds: spindynamics and spin injection, Physica E , 189 (2004). R. J. Elliott, Theory of the effect of spin-orbit coupling onmagnetic rresonance in some semiconductors, Phys. Rev.,
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