It is well known that the sample resistance and thus the response to an injected cur-rent depends on the magnetization configuration. This is caused by various mag-netoresistance effects such as the anisotropic magneto resistance (AMR) [Tho57]
or the giant magneto resistance effect (GMR) [BBF+88, BGSZ89], which will not be discussed here in detail.
But, we consider the reciprocal effect: the influence of an injected current on the local magnetization also occurs. In 1984, Berger suggested that the spin-polarized conduction s-electrons can interact with the magnetic moments of the
16 Theory
more localized d-electrons and transport spin and angular momentum to the mag-netic system [Ber84].
This so-called spin-torque interaction was integrated into the LLG-equation for a macro spin model [Ber96, Slo96, BJZ98]. In these early works, two coupled ferromagnetic layers with current flowing from one layer to the other were studied.
The first layer polarizes the conduction electrons and if the second layer is not oriented parallel to the first one, the electron spin exerts a torque on the second layer magnetization trying to align it to the first layer.
Since the study of the spin-torques is a important issue of this work the fol-lowing subsections will deal with this interaction in more detail.
1.7.1 Adiabatic Spin-Torque
If current is injected into a magnetic structure, the spins of the conduction elec-tronss(x)align to the local magnetizationM(x). It is assumed that this happens adiabatically, so that the electron spins adapt without delay to the varying local magnetization as indicated in Fig. 1.5. When the electrons pass a region where the magnetization direction changes, their spins follow this direction change and there will be a change of their spin direction (ds in Fig. 1.5) and hence of their angular momentum. Due to the conservation of angular momentum, this involves a trans-fer of angular momentum to the local magnetization that hence has to change its orientation in return (−dsin Fig. 1.5).
A currentIinjected into a wire (along the x-direction) with cross sectionA car-ries during the timedta spin fraction ofds= ~IPe dt. Here we also considered that the conduction electrons might not be fully spin polarized and the effective spin po-larizationP ≤1is taken into account. When the local magnetizationMchanges, the electron spin will also change by the fractiondM/Ms=∇xM(x)dx/Ms. Hence
Figure 1.5: The spin of the injected electrons (green arrows) adiabatically aligns to the local magnetization (blue arrows). When the electron spin schanges by dsthis angular momentum has to be transferred to the local magnetization.
1.7 Influence of spin-polarized Electrons on the Magnetization 17
the total amount of angular momentum transferred to the local magnetization is:
dS=−~∇xMdx Ms
P Idt
e . (1.33)
Note the minus sign. It should also noted that only the perpendicular spin compo-nent can be transfered, since the saturation magnetization is constant. Hence, the above calculation is only valid in the limit of slow varying magnetizations. The angular momentum per volume (V =Adx) is directly proportional to the change of the magnetization:
dM = 1
2gµB· dS
~Adx. (1.34)
Considering that I/A equals the current density j and combining equations 1.33 and 1.34 finally yields:
dM =−gP µB 2eMs
j∇xMdt.
This motivates the definition of
u= gP µB 2eMs
j, (1.35)
that has units of a velocity. When we allow arbitrary current directions the deriva-tive of M in the direction of the current has to be used instead of ∇xM. This
yields: This equation equals the result for the spin-torque obtained by Thiaville and coworkers [TNMV04, TNMS05]. If one assumes that the magnetization has a constant value (low temperature regime) this description is also consistent with results obtained by Li and Zhang [LZ04a]:
∂M From these results one can expect that the spin-torque is largest in materials with low saturation magnetization Ms and high spin polarization P, since u will be maximized for a given current density.
18 Theory
1.7.2 Non-adiabatic Spin-Torque
The discussion above assumed that the injected electrons are always perfectly aligned with the local magnetization. However, for large magnetization gradients the injected conduction electrons might not be able to follow the local magne-tization completely and will lag behind by a certain amount. This creates an additional torque on the magnetization, since the electron spin s in this case has a small component perpendicular toM.
The precession of this component induces similar to an external magnetic field a torque on the magnetization that is proportional to M×dM. For this reason the non-adiabatic term is also called field-like term. The torque strength is again proportional to the current density summarized by the parameter u. For an arbi-trary current direction the derivative along the current direction has to be taken and one finally obtains:
∂M
∂t
nast
=βM×(u· ∇)M. (1.38) β is a measure for the mistracking of the electron spin, i. e for the degree of non-adiabaticity. For β = 0 the electron spin again perfectly follows the local magnetization and pure adiabatic spin-torque is present.
This derivation of the non-adiabatic spin-torque is quite simplified and the actual origin of the non-adiabaticity spin-torque term is still controversial and being discussed. Thiaville introduced this term phenomenologically to overcome discrepancies between experimental results and theory [TNMS05] (see also section 1.10).
Zhang and Li [ZL04] derived this term rigorously from theory by considering the exchange interaction between the local moments and the spin accumulation of the conduction electrons as spin-flip scattering. They coupled the electrons of transport via an s-d Hamiltonian to the magnetic system. Considering also spin relaxation by spin-flip scattering, they showed that the non-equilibrium spin density induced by the current and the variation of the local magnetization in turn creates an torque on the local magnetization. In this formulation, the non-adiabaticity is directly related to the spin-relaxation time τsf and the exchange energyJex of the s-d Hamiltonian:
βsf =τex/τsf ≈10−2, τex=~/Jex. (1.39)
1.7 Influence of spin-polarized Electrons on the Magnetization 19
In addition to the adiabatic and non-adiabatic spin-torque terms, they found two extra terms that depend on the time derivative of the magnetization. But these terms can be included as small corrections into the precession and the damping term. Later it was shown that not only spin relaxation due to spin-flip scattering, but also due to spin-orbit interaction induces a β term [TE08].
Tatara and Kohno found a similar non-adiabatic term based on linear momen-tum transfer [TK04]. In this context the non-adiabaticity is closely related to the DW resistance ρDW[LZ97, TF97]:
β = γe∆2DW
gµBP R0ρDW, (1.40)
with the DW width ∆DW and the Hall resistance R0.
In systems with fast varying magnetization, conduction electrons might be re-flected at these high magnetization gradients [XZS06, WV04]. This corresponds to the true meaning of adiabatic transport and gives also rise to a non-adiabaticity constant βna.
The effective β therefore is a sum of the contribution due to spin-relaxation βsf and non-adiabatic transport: βna: β =βsf +βna. A more detailed discussion of the origin of β can be found in [TKS08, BTE08].
1.7.3 Discussion of the Spin-Torques
Integrating the two spin-torque terms into the LLG-equation 1.32 yields:
dM
dt =−γ(M×Hef f) + α
MsM×∂M
∂t −(u· ∇)M+βM×(u· ∇)M. (1.41) The two spin-torque contributions are always perpendicular to each other and thus define a basis in the plane orthogonal to the local magnetizationM. Therefore any torque acting on the magnetizationMcan be decomposed into these two terms (Since Ms is assumed to be constant only torques can act on the magnetization).
The remaining challenge is to determine the size of the two contributions.
The most hotly debated issue is the relation betweenβ and the damping con-stant α, since it has been predicted that α and β depend similarly on the band structure [GGSM09] or are even equal [BM05]. Xiao et al. [XZS06] cast doubt on the existence of the non-adiabaticity parameter caused by spin-flip scattering, but find an oscillating non-adiabatic torque for very high magnetization
gradi-20 Theory
ents. Kohno et al. [KTSS06] calculated the non-adiabaticity parameter microscop-ically and found that in general β 6= α. Garate et al. relate β as well as α to the band structure and intrinsic spin-orbit interactions [GGSM09]. From their work it follows that α and β depend similarly on temperature and disorder, but the value of the ratio β/α depends on the band structure and is in general dif-ferent from unity. In contrast, Tserkovnyak et al. [TBBH05] as well as Barnes and Maekawa [BM05] claim that β should equal α. Controversy also exists con-cerning the type of damping, either Landau-Lifshitz [SSDZ07, SSDZ08, Sas09] or Gilbert [Smi08] damping, that has to be used. Whereas both formulations are identical in the absence of current, this is no longer the case when the spin trans-fer terms are included [SSDZ08, SSDZ07, SHK+09]. If a pure adiabatic spin-torque term is added to the LL-equation 1.31 with Landau-Lifshitz damping an additional termαM×(u·∇)Moccurs when transformed into the equivalent LLG-form (equa-tion 1.32). Thus the adiabatic spin-torque includes a field like term with β = α for Landau-Lifshitz type of damping. More general speaking: βG =βLL+α with βGbeing the non-adiabatic contribution in the LLG equation with Gilbert damp-ing and βLL for the Landau-Lifshitz damping. It was pointed out that the two equations are equivalent, if the adiabatic as well as the non-adiabatic spin-torque is considered and the choice of damping in this case is simply based on the ease of interpretation or convenience [SSDZ08].
However, to solve the discrepancies concerning the origin and the size of β, experiments addressing this topics are required. The measurement of β in real materials as well as its relation to α will give valuable information required to refine the theory.