First-principles study of spin-transfer torques in layered systems with noncollinear magnetization

An efficient first-principles method was developed to calculate spin-transfer torques in layered system with noncollinear magnetization. The complete scattering wave function is determined by matching the wave function in the scattering region with the Bloch states in the leads. The spin-transfer torques are obtained with the aid of the scattering wave function. We applied our method to the ferromagnetic spin valve and found that the material (Co, Ni, and ${\text{Ni}}_{80}{\text{Fe}}_{20}$) dependence of the spin-transfer torques could be well understood by the Fermi surfaces. Ni has much longer spin injection penetration length than Co. Interfacial disorder is also considered. It is found that the spin-transfer torques could be enhanced by the interfacial disorder in some system.

Figure 1

(Color online) Sketch of the configuration used for current-induced switching. A scattering region is sandwiched by left- $\left(\mathbf{L}\right)$ and right-hand $\left(\mathbf{R}\right)$ leads that have translational symmetry and are partitioned into principle layers perpendicular to the transport direction. The scattering region contains $N$ principle layers, but the structure and chemical composition are, in principle, arbitrary. The switching layer FM can be Co, Ni, and ${\text{Ni}}_{80}{\text{Fe}}_{20}$.

Figure 2

Illustration of incoming current and outgoing current for the $\mathbf{R}\text{th}$ site. Assuming the particle current comes from $(I-1)\text{st}$ layer to $(I+1)\text{st}$ layer. The arrow lines denote the current related to $\mathbf{R}\text{th}$ site and dotted lines denote the coupling between sites irrelevant to $\mathbf{R}\text{th}$ site.

Figure 3

(Color online) (a) The angular dependence of total spin torques on free magnet Co of $\text{Co}\left(\theta \right)|\text{Cu}\left(9\text{ML}\right)\phantom{|}|\text{Co}\left(15\text{ML}\right)\phantom{|}|\text{Cu}\phantom{|}$, where the electron current flows from the fixed magnet to the free magnet. (b) The angular dependence of the total conductance of the same system.

Figure 4

(Color online) The layer dependence of STT on interfacial unit cell in the spin valve $\text{Co}\left(\theta =90\right)|\text{Cu}\phantom{|}|\text{FM}\phantom{|}|\text{Cu}\phantom{|}$, where the free magnet FMs are Co, Ni, and ${\text{Ni}}_{80}{\text{Fe}}_{20}$ respectively.

Figure 5

(Color online) The layer dependence of STT on interfacial unit cell for the spin injection setup of single interface: (a) $\text{Cu}|\text{Co}\phantom{|}$ (b) $\text{Cu}|\text{Ni}\phantom{|}$.

Figure 6

(Color online) Dependence of the total in-plane STT on the thickness of the free magnet $d$ for $\theta =\pi /2$. The self-consistent potential of the collinear system $\text{Co}\left(0°\right)|\text{Cu}\phantom{|}|\text{FM}\phantom{|}|\text{Cu}\phantom{|}$ is used and the noncollinear potentials are obtained with rigid approximation, which is justified because the Cu spacer is thick enough.

Figure 7

(Color online) (a) and (b) The layer dependence of STT on interfacial unit cell when the spin injected through a single $\text{Cu}|\text{Co}\phantom{|}$ interface for different ${\mathbf{k}}_{\parallel }$ points in 2D BZ.

Figure 8

(Color online) First row: Fermi-surface projection of the Co bulk fcc Brillouin zone on to a plane perpendicular to the [111] direction. Left-hand panel is for majority electron and right-hand panel is for minority electron with band 3, 4, and 5 Fermi surfaces. The second row is Ni bulk.

Figure 9

(Color online) (a) The concentration $x$ dependence of the conductance and (b) the total in-plane spin torques on the free magnet of spin valve $\text{Co}\left(\theta =90°\right)|\text{Cu}\phantom{|}|\text{FM}\phantom{|}|\text{Cu}\phantom{|}$ with interfacial alloy ${\text{Cu}}_x{\text{FM}}_{1-x}$, where the solid symbol for FM is Co and the open symbol for FM is Ni. The error bar gives the uncertainty due to the randomly generated disorder configurations.

Figure 10

(Color online) The layer dependence of STT on interfacial unit cell in the spin valve $\text{Co}\left(\theta =90°\right)|\text{Cu}\phantom{|}|\text{FM}\phantom{|}|\text{Cu}\phantom{|}$ with interfacial disorder: (a) FM is Co, (b) FM is Ni. The zone labeled by $\Pi$ is the layers with substitutional alloy ${\text{Cu}}_{50}{\text{FM}}_{50}$.

Figure 11

The thickness $d$ dependence of the averaged in-plane spin torques on the atom in free magnet and the total conductance of the spin valve $\text{Co}\left(\theta =90°\right)|\text{Cu}\phantom{|}|\text{Co}\phantom{|}|\text{Cu}\phantom{|}$.

Figure 12

(Color online) STT for $\text{Co}|\text{Cu}\phantom{|}|\text{Co}\phantom{|}|\text{Cu}\phantom{|}$ for in-plane and out-of-plane plotted as a function of the normalized area element $\Delta {\mathbf{k}}_{\parallel }/A_{BZ}$, which is used in the 2D BZ summation. $\Delta {\mathbf{k}}_{\parallel }/A_{BZ}=1/N$, where $N$ is the total number of $k$ points used in the whole 2D BZ.