Spin transport in Magnetic Tunnel Junctions
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- Laureen Kennedy
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1 Spin transport in Magnetic Tunnel Junctions This tutorial shows how to simulate and analyze the electronic transport properties of magnetic tunnel junctions (MTJs). You will study the collinear and non-collinear spin-dependent transport properties of the Fe-MgO-Fe magnetic tunnel junction. In particular you will study the electronic transmission, the tunnel magnetoresistance and the spin-transfer torque.
2 Setup the MTJ device Builder Download the already relaxed Fe-MgO-Fe geometry and save it in the Project Folder. Select the bulk configuration and send it to the Builder. Use Device Tools Device From Bulk to generate the device structure. Send the structure to the Script Generator.
3 Setup the script for transmission calculation: spin parallel Script Generator New Calculator Spin: Polarized Exchange-correlation: SGGA.PBE Electron temperature: 1200 Kelvin. K-point sampling: (7,7,100) Initial State. Initial State: User spin Relative spin: 0 for Mg and O and 1 for Fe. MullikenPopulation Forces
4 Setup the script for transmission calculation: spin parallel Script Generator TransmissionSpectrum E0=E1= 0 ev Energy Points=1 K-points grid sampling: (151x151)
5 Inspecting the forces From the LabFloor, select the Forces object. Next, click Text Representation... button. The forces on the equivalent electrode atoms inside the central cell are not calculated (zero). For the other atoms, the forces are small: relaxing the geometry is a good strategy for this system.
6 Inspecting the Mulliken Population From the LabFloor, select the MullikenPopulation object. Click Text Representation... button. Check that each atom is correctly spin polarized: Inspect the leftmost column, this shows the total Mulliken population per spin channel for each atom. All the Fe atoms polarized with their majority spin pointing up. The Mg atoms close to the interface couple antiparallel with Fe. The O atoms are unpolarized.
7 Analyzing the k-parallel transmission It is clear from the plots that the down spin (minority) transmission is strongly supressed in magnitude and exhibits peaks at certain points in the Brillouin zone.
8 Setup the script for Transmission calculation: spin anti-parallel Script Generator New Calculator Spin: Polarized Exchange-correlation: SGGA.PBE Electron temperature: 1200 Kelvin. K-point sampling: (7,7,100) Initial State. Initial State: User spin Relative spin: 0 for Mg and O and 1/-1 for Fe. MullikenPopulation Forces TransmissionSpectrum E0=E1= 0 ev Energy Points=1 K-points grid sampling: (151x151)
9 Analysis of k anti-parallel transmission In this case the transmission for the up and down will be identical due to the mirror symmetry of the system. Note: due to the time inversion symmetry, propagation from left to right is always identical to propagation from right to left. The up components of the transmission spectrum corresponds to up electrons from the left transmission propagating into the right electrode. The down component of the transmission spectrum corresponds to the down electrons in the right propagating into the left electrode.
10 Tunneling magnetoresistance The tunneling magneto resistance (TMR) is defined as: GP is the conductance through the junction with parallel spin alignment GAP is the conductance for the anti-parallel spin alignment Script provided (tmr.py) to calculate the conductance from the transmission spectra of the junction with parallel and anti-parallel spin aligment: Execute the script and find the TMR:
11 Adaptive k-grid Adaptive k-grid Object for defining the k-point sampling used for the calculation of transmission and spin-transfer torque. Application: automatically zoom in on the significant features of the electron transmission spectrum. Useful when the k-dependent transmission is dominated by localized peaks and the total computed transmission may depend critically on resolving those peaks. Adaptive algorithm: divides the Brillouin zone into triangles and integrates over each triangle. Each triangle becomes increasingly small in an iterative manner until convergence is reaced, but stops if the number of refinement levels reaches the max_number_of_levels (20 by default). The integral over each triangle is estimated as: In order to estimate whether or not the Io integral is converged, two types of errors available: Absolute: Absolute converged when the absolute value of the Brillouin zone integral in all triangles is lower than the chosen tolerance Relative: Relative converged when the relative value of the Brillouin zone integral in all triangles is lower than the chosen tolerance. Absolute or Relative error? If one have knowledge about the function values, the use of Absolute error may be advantageous, otherwise the use of Relative error will be more useful.
12 Adaptive grid vs. MonkhorstPack grid: total transmission convergence at 0 ev Transmission computed for Parallel and antiparallel configurations Basis set: DoubleZetaPolarized for Mg and O. SingleZetaPolarized for Fe. Exchange and correlation: SGGA.PBE K-point sampling: 6,6,100 Electron temperature: 1200 Kelvin Adaptive k-grid Relative Tolerance: 10, 1, 0.1, 0.01, 0.001, and Monkhorst-Pack grid K-pont sampling: (5x5), (9x9), (17x17), (33x33), (65x65), (129x129), (257x257), (513x513), (1025x1025) Well converged calculations of the total transmission through the MTJ obtained with no less than 103 Monkhorst-Pack k-points and with a relative tolerance of 10-2 using the adaptive k-point grid.
13 Parallel transmission Provide the adaptive_grid.py script. This script calculates the transmission for the Fe MgO Fe device with parallel spin orientation using both a dense MP grid and the AdaptiveGrid method. Monkhorst-Pack grid: 500x500 k-points Adaptive grid: relative tolerance of Download and run the script.
14 Spin-up transmission Monkhorst-Pack grid Transmission spectrum obtained using a MP grid of (500x500) kpoint sampling. The transmission spectrum obtained with the dense MP grid is different from the one obtained before using (151x151) k-point sampling. Now circular feature in the middle of the spectrum with zero transmission immediately around the gamma point. The improved k-point sampling has only increased the total k-point averaged spin-up transmission by a factor of 4.
15 Spin-up transmission Adaptive grid Spin-up transmission spectrum evaluated using the Adaptive grid with a 10-2 relative tolerance. Black dots indicate the sampled k-points. The adaptive grid has zoomed in on the circular feature of the k-dependent transmission, causing a larger k-point density around the circular feature.
16 Spin-down transmission Above, transmission obtained using Monkhorst-Pack grid. The total kpoint averaged spin-down transmission is Below, transmission obtained using Adaptive grid. The total k-point averaged spin-down transmission is The adaptive grid has zoomed in on the transmission peaks. The total k-point averaged spindown transmission has changed 5%.
17 Restricting the grid range Provide the zoom.py script. This script shows how the adaptive grid class can be used to resolve only a specific features of the spin-up and spin-down k-dependent transmission. A relative tolerance of 10-3 is used to fully resolve the transmission features. The below figures show the spin-up and spin-down transmissions.
18 Computing the non-collinear device configuration Editor Requires non-collinear definition of the electron spin. Device configuration: read the collinear configuration mgo_para.nc New Calculator : replace SGGA.PBE by NCGGA.PBE exchange and correlation. Define the spin rotation: theta= 120 degrees Setup the initial state: new calculator and spin Update and save the device configuration.
19 Mulliken population and Spin Transfer Torque Scripter Analysis from file Select the mgo_nonco_dc.nc file MullikenPopulation SpinTransferTorque. Energy: 0eV Contributions: Left to calculate left right linear response current. Send the script to the Editor and replace the Monkhorst-Pack grid by the Adaptive grid.
20 Convergence of the total spin-transfer torque We are using a relative tolerance of 10-2 Adaptive grid for the adaptive grid. The spin-transfer torque must be computed using the relative tolerance at which the total transferred torque at the right side is converged. We have calculated the total spin-transfer torque in the right side for different relative tolerances. Use of 10-2 relative tolerance seems to be sufficient. Using a Monkhorst-Pack grid the total spintransfer torque does not reach the convergence. Monkhorst-Pack grid
21 Analysis of the results Mulliken Population Inspect the Mulliken charge and the components of the spin polarization vector using the text representation. The sum of up and down populations corresponds to the Mulliken charge (number of electrons). The difference of these populations combined with the two angles (Theta and Phi) forms the spin polarization vector. This vector can be visualized using the Viewer.
22 Analysis of the results Spin-transfer torque The spatial components of the spin-transfer torque can be plotted using the 1D Projector. Visualize the spatial components of the Spintransfer torque using the Viewer. The below figure shows the isosurface of the y component.
23 Optimizing the junction geometry Builder 1.- Use Add From Plugin Magnetic Tunnel Junction (FeMgO-style). Change the number the barriers to 6 and number to surface layers to 2 on both left and right side and leave the other parameters as they are by default. 2.- Convert the device geometry to bulk send the structure to the Script Generator to optimize it by keeping Rigid the last two layers of Fe.
24 Optimizing the junction geometry Script Generator New Calculator Spin: Polarized Exchange-correlation: SGGA.PBE Electron temperature: 1200 Kelvin. K-point sampling: (7,7,100) Initial State. Initial State: User spin Relative spin: 0 for Mg and O and 1 for Fe.
25 Optimizing the junction geometry Script Generator OptimizeGeometry Keep the first and last four atoms rigid. Forces
26 Analysis of the results Forces Zero in x-y direction. Less that the optimization criteria (0.05 ev/ang) for atoms which have been relaxed. For the electrode extension the forces are slightly larger and the force vectors point out of the cell (cell under compressive strain). Adding more surface to increase the cell length in z direction would lower the strain but would have very little effect on the results in the present case.
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