# Spin Polarized Calculation and Spintronics

We have already touched upon systems that are spin polarized, for instance in the Fe example of Section Basic input file. In this Section we discuss in more detail about issues of spin polarized calculation.

In a non-spin-polarized calculation, the Hamiltonian of the system is a matrix in orbital space, $$\bf{H} = [H_{\mu\nu}]$$, where subscripts $$\mu,\nu$$ are orbital indices of the LCAO basis set. In a spin resolved calculation, the size of the Hamiltonian matrix is doubled, namely each matrix element $$H_{\mu\nu}$$ becomes a $$2\times 2$$ matrix in spin space,

(7)$\begin{split}H_{\mu\nu} \rightarrow \left[ \begin{array}{cc} H_{\mu\nu, \uparrow \uparrow} & H_{\mu\nu, \uparrow \downarrow} \\ H_{\mu\nu, \downarrow \uparrow} & H_{\mu\nu, \downarrow \downarrow} \\ \end{array} \right]\end{split}$

Similarly, the size of the density matrix $${\hat {\rho}}$$ is doubled in spin polarized calculations:

(8)$\begin{split}{\hat {\rho}}_{\mu\nu} \rightarrow \left[ \begin{array}{cc} {\hat \rho}_{\mu\nu, \uparrow \uparrow} & {\hat \rho}_{\mu\nu, \uparrow \downarrow} \\ {\hat \rho}_{\mu\nu, \downarrow \uparrow} & {\hat \rho}_{\mu\nu, \downarrow \downarrow} \\ \end{array} \right]\end{split}$

As discussed in Section Basic standard output, there are several spin types in nanodcal and the default is ‘NoSpin’. For collinear spin type CollinearSpin, the spin space is diagonal, namely $$H_{\mu\nu, \uparrow \downarrow}=H_{\mu\nu, \downarrow \uparrow}=0$$, and $${\hat \rho}_{\uparrow \downarrow}={\hat \rho}_{\downarrow \uparrow}=0$$. This is a common spin polarized situation. For collinear spin computation, the spin-up channel and spin-down channel are not directly coupled by off diagonal terms $$H_{\uparrow \downarrow}, H_{\downarrow \uparrow}$$. However, they are indirectly coupled through the exchange-correlation functional which is a functional of both $${\hat \rho}_{\uparrow \uparrow}$$ and $${\hat \rho}_{\downarrow \downarrow}$$. In spin polarized calculations, NanoDCAL uses standard LSDA or GGA XC potentials. Because the Hamiltonian and density matrices are diagonal in spin space, a collinear spin calculation takes roughly twice as long to run as a NoSpin calculation of the same system size.

Other spin types, CoplanarSpin and GeneralSpin, require non-collinear spin computations where $$H_{\mu\nu, \uparrow \downarrow}, H_{\mu\nu, \downarrow \uparrow}\neq 0$$, and $${\hat \rho}_{\uparrow \downarrow}, {\hat \rho}_{\downarrow \uparrow}\neq 0$$. In CoplanarSpin, NanoDCAL assumes spins to lie in the $$(x,z)$$ plane in spin space. In GeneralSpin, real space spins may point to any direction. Non-collinear spin computation is necessary when there is spin-orbit interaction and/or non-collinear magnetism.

## Spintronics

NanoDCAL is a powerful tool for spintronics research. There are many varieties of spintronic devices and one usually wishes to calculate spin polarized quantum transport properties such as spin current and spin resolved conductance. For theoretical formulation please refer to the :ref:theory section.

Magnetic tunnel junctions (MTJ) are important and popular spintronic devices. A MTJ consists of two ferromagnetic leads sandwiching a non-magnetic insulating tunnel barrier. The magnetic moment of the leads can be in parallel configuration (P) or anti-parallel configuration (AP). Transport properties in P and AP configurations are quite different and therefore must be calculated separately. In the /example/Group10/, you can find an example of Fe/MgO/Fe MTJ, which is perhaps the most important candidate for magnetic memory applications. For AP, transport in Fe/MgO/Fe MTJ is dominated by interface resonances and large number of transverse k-sampling is therefore necessary to resolve it. Fig. 6 has shown the transmission channels in a Fe lead versus the transverse momentum, and it gives a picture of the transmission hot spots that can only be captured by a fine k-mesh. This is quite common in MTJ calculations and special attention should be paid to it.

It is possible that the metal leads are antiferromagnetic (AF) materials. Even though the AF material has no magnetic moment, theoretical analysis predicted [NDH06] that when AF leads sandwich a non-magnetic space layer, a spin polarized transport can occur due to the bonding change at the junction interfaces. This prediction was confirmed by ab initio calculations [HWD07]. An example of calculating AF devices can be found in the /example/Group10/ directory, where the device is made Cr leads sandwiching Au space layer.

There are many spintronic systems that involve non-magnetic leads but magnetic scattering region. In the /example/Group10/, you can find such an example of Fe doped carbon nanotube (CNT). In this problem, the leads are CNT and the scattering region is also CNT but doped with Fe atoms. The scattering region is spin polarized and acts as a spin filter.

Another example in /example/Group10/ is the system where one lead is Co and the second lead is Cu, while the scattering region consists of alternating layers of Co and Cu. Such a system was used to investigate the phenomenon of spin transfer torque where the injected spin current from the Co lead transfers angular momentum to a free Co layer in the scattering region.

## Spin-orbit interaction

Spin-orbit interaction (SOI) plays important roles in magnetism. More recently SO effects were found to induce spin current in pure semiconductors without any magnetic impurity. In order to carry out SOI analysis, non-collinear spin calculation is needed. Both these capabilities have been included in NanoDCAL. An application to the three dimensional topological insulator material Bi$$_2$$Se$$_3$$ can be found in Ref. [ZHL11].

## LDA+U calculation

Calculations involving d and f electrons are more difficult due to the localized nature of these orbitals. This is however rather common in magnetism. In well known oxides such as FeO, the localization of the d electrons of Fe by the oxygen atoms opens a gap at the Fermi level, resulting to an insulating behavior for the FeO crystal. DFT with LDA misses such correlated physics. A popular and relatively simple method to study the atomic limit is to apply the LDA+U functional. NanoDCAL has implemented this method.

## Summary

Spintronic systems and magnetic devices are a main focus of NanoDCAL. Usually, spin polarized transport involve larger matrices and longer calculations. A parallel cluster computer will help substantially in the computation.

[NDH06]

Núñez, A. S., Duine, R. A., Haney, P., & MacDonald, A. H. (2006). Theory of spin torques and giant magnetoresistance in antiferromagnetic metals. Physical Review B - Condensed Matter and Materials Physics, 73(21), 214426.

[HWD07]

Haney, P. M., Waldron, D., Duine, R. A., Núñez, A. S., Guo, H., & MacDonald, A. H. (2007). Ab initio giant magnetoresistance and current-induced torques in Cr Au Cr multilayers. Physical Review B - Condensed Matter and Materials Physics, 75(17), 174428.

[ZHL11]

Zhao, Y., Hu, Y., Liu, L., Zhu, Y., & Guo, H. (2011). Helical states of topological insulator Bi2Se3. Nano Letters, 11(5), 2088–2091.