Noncollinear calculations for metallic nanowires

Version: 2016.3

In the tutorial Introduction to noncollinear spin you learned how to perform a simple noncollinear calculation. In this tutorial you will apply the same procedure to more advanced systems. In a first example, you will consider a metal nanowire connecting two electrodes of the same material. In the second example, you will consider an infinite wire and study the effect of spin-orbit coupling and spin-orientation on the electronic structure. The calculations will be very similar to the two works published by Czerner et al.: [CYM08][CYM10].

Building the device

Set up the Ni(111) surface

Open the builder_icon Builder, click on Add ‣ From Database, search for the Nickel fcc bulk structure and plus_icon add it to the Stash. Open Builders ‣ Surface (Cleave), enter Miller indices (111) as shown in the figure below, and click Next.

../../_images/cleave_Ni111.png

In the next window, define a \(2 \times 2\) surface supercell as shown below, then click Next.

../../_images/cleave_Ni111_2x2.png

Set the out-of-plane direction to Periodic and normal (electrode), set the thickness to 6 layers (see image below) and click Finish.

../../_images/cleave_Ni111_electrode.png

Finally, add some vacuum above the (111) surface. Go to Bulk Tools ‣ Lattice Parameters and set the C vector length to 40 Å. Be careful to select to “Keep Cartesian coordinates constant when changing the lattice”.

../../_images/lattice_parameters2.png

Note

In principle, you can also create the surface by adding the vacuum in the Surface (Cleave) tool by choosing a slab configuration. The procedure followed here will make the visualization of the next steps easier.

Set up the Ni wire between two Ni(111) surfaces

Select three Ni atoms at the surface and click the centroid plugin on the toolbar at the top of the builder_icon Builder to add a Ni atom in the geometric center of the selected atoms.

../../_images/centroid.png

Then, go to Coordinate Tools ‣ Translate tool to move this atom by 2.035 Å above the surface. This distance corresponds approximately to the Ni(111) interlayer distance.

../../_images/move_out.png

To create the wire, use again Coordinate Tools ‣ Translate but this time select the Copy option and translate by 2.49248 Å, which corresponds to the Ni-Ni distance. Do it twice and you are halfway to creating a 5-atom wire embedded between two Ni electrodes.

../../_images/move_out_copy.png

Take note of the Z coordinate of the last Ni atom added to the chain (19.2306 Å) and select all the Ni atoms in your system except this last Ni atom. Then, go to Coordinate Tools ‣ Mirror, select the predefined xy mirror axes and enter 19.2306 Å as the Z coordinate of the mirror point P. Remember to check the Copy box to make a copy of the mirrored object, and press Apply and you will obtain the structure shown below.

../../_images/mirror_ni5.png

Create the device

Click on Device Tools ‣ Device From Bulk and keep the predefined electrode lengths corresponding to three Ni layers.

../../_images/device_from_bulk_ni5.png

Setting up the collinear calculation and analyzing the results

In this step you will run a collinear calculation and save the corresponding state in a NetCDF file. In the next step, you will run a noncollinear calculation by using the collinear state as an initial guess. In this way, you will save a lot of computational time, since the convergence of a noncollinear calculation can be hard to achieve otherwise.

Send the device structure you have previously created to the script_generator_icon Script Generator and add the following blocks:

  • Add a calculator_icon New Calculator with the following parameters:

    • LDA exchange-correlation functional
    • Spin-polarized calculation
    • \(4 \times 4 \times 100\) grid for k-points sampling
    • SingleZetaPolarized basis set
    • Electron temperature set to 2400 K
  • Add an initial_state_icon Initial State with the following options:

    • Select User spin and check that Spin for Nickel is set to 1.
  • Add an analysis_icon Analysis > MullikenPopulation

  • Change the output file name to ‘Ni5_collinear.nc‘.

Note

The high electron temperature considerably improves convergence.

Send the script to the job_manager_icon Job Manager and run the calculation. Once the job is done (in serial it can take up to three hours), the LabFloor will be populated by the twoprobe_icon DeviceConfiguration and labfloor_mullikenpopulation_icon MullikenPopulation objects.

Select the labfloor_mullikenpopulation_icon MullikenPopulation and drag and drop it on the viewer_icon Viewer to visualize the converged collinear spin state.

../../_images/spin_visualizer_para.png

The directions of the arrows is not so interesting or surprising - this is a collinear calculation with the electrodes polarized parallel to each other. We do however see a strong net spin polarization in the atomic wire compared to the surface layers.

Setting up the noncollinear calculation

In the tutorial Introduction to noncollinear spin you have learned how to set up a noncollinear calculation. Here you will learn how to set up the initial spins of a noncollinear calculation based on the collinear converged result.

In order to set the initial spin state you need to specify the spin direction in physical spherical coordinates (r, θ, φ), with the following important definitions (see also the figure below):

  • θ is the angle with the z axis
  • φ the polar angle in the xy plane relative to the x-axis
../../_images/polar_coordinates3.png

Warning

If you start your job by reading a previously converged spin state, as described below in this tutorial, the reference axis (Z in the picture above) for each spin of each atom is the axis of the corresponding converged spin state.

Drag and drop the twoprobe_icon DeviceConfiguration from the LabFloor to the script_generator_icon Script Generator and modify the following parameters:

  • In the existing calculator_icon New Calculator, set the spin option to ‘Noncollinear

  • Add an initial_state_icon Initial State with the following options:

    • Select User spin and check that Spin for Nickel is set to 1
    • Check the option Use old calculation and use as filename ‘ni5_collinear.nc
  • Add analysis_icon Analysis > MullikenPopulation

Send the script to the editor_icon Editor and modify the initial_state_icon Initial State block as follows:

# -------------------------------------------------------------
# Initial State
# -------------------------------------------------------------

# Define the spin rotation
theta = 180*Degrees
left_spins = [(i, 1, 90*Degrees, 0*Degrees) for i in range(24)]
center_spins = [(i+24, 1, 90*Degrees-theta*(i+1)/6, 0*Degrees) for i in range(5)]
right_spins = [(i+29, 1, 90*Degrees-theta, 0*Degrees) for i in range(24)]
spin_list = left_spins+center_spins+right_spins
initial_spin = InitialSpin(scaled_spins=spin_list)

old_calculation = nlread('ni5_collinear.nc', DeviceConfiguration)[0]

device_configuration.setCalculator(
    calculator,
    initial_spin=initial_spin,
    initial_state=old_calculation,
)
device_configuration.update()
nlsave('ni5_noncollinear_out-plane.nc', device_configuration)
nlprint(device_configuration)

The setup above corresponds to an out-of-plane spin rotation between two anti-ferromagnetic electrodes, as reported in Refs. [CYM08] and [CYM10] (cf. Figure 1 in Ref. [CYM08]). In a later section you will also consider an in-plane rotation.

Note

The scaled_spin argument follows the format (atom index, initial scaled spin, θ, φ) as documented in the InitialSpin entry in the Reference Manual.

Remember that the spin orientation you enter here is relative to the collinear spin state read before. In this case, the collinear state is ferromagnetic, which corresponds to θ=0 and φ=0.

Save the script in your project directory, check that the ‘ni5_collinear.nc‘ file is present in the same directory and run the script.

Analyzing the results

Antiparallel configuration, out-of-plane rotation

From the LabFloor drag and drop the labfloor_mullikenpopulation_icon MullikenPopulation object contained in ‘Ni5_noncollinear_out-plane.nc‘ file into the viewer_icon Viewer to visualize the converged noncollinear spin state.

../../_images/spin_visualizer_out-plane.png

For a more detailed analysis, open the text representation of labfloor_mullikenpopulation_icon MullikenPopulation. Here you can read the spin-up, spin-down, θ and φ components for each atom.

The figure below shows the direction and size of the spin state of the atoms in the wire (red arrows) and also of two atoms in the electrode extension regions (black arrows) for comparison. You can compare this plot to the results reported in Refs. [CYM08] and [CYM10].

../../_images/2D_plot_Ni5.png

Antiparallel configuration, in-plane rotation

Another possibility is to rotate the spins in th xy plane by changing the φ angle. To achieve this, modify the script above as indicated below:

# Define the spin rotation
phi = 180*Degrees
left_spins = [(i, 1, 90*Degrees, 0*Degrees) for i in range(24)]
center_spins = [(i+24, 1, 90*Degrees, 0*Degrees+phi*(i+1)/6) for i in range(5)]
right_spins = [(i+29, 1, 90*Degrees, 0*Degrees-phi) for i in range(24)]
spin_list = left_spins+center_spins+right_spins
initial_spin = InitialSpin(scaled_spins=spin_list)

Run the calculation for this initial spin configuration and display the corresponding labfloor_mullikenpopulation_icon MullikenPopulation in the viewer_icon Viewer:

../../_images/spin_visualizer_in-plane.png

Including spin-orbit coupling in noncollinear calculations

ATK also allows you to perform noncollinear calculations including spin-orbit coupling (SOC). In this example, you will investigate the effects of noncollinear spin and spin-orbit coupling to the simple case of an infinite Ni chain [CYM10]. In order to improve the convergence of the calculation, you will first set up a spin polarized calculation and only afterwards perform a noncollinear and finally a calculation with SOC, starting from the previously converged state.

Building the Ni wire

Open the builder_icon Builder and create a bulk configuration corresponding to an infinite chain of Ni atoms with an interatomic distance of 2.49248 Å (Ni-Ni distance).

../../_images/Ni_wire_builder.png

Collinear calculation

Send the structure to the script_generator_icon Script Generator, and setup the calculation as follows:

  • Add a calculator_icon New Calculator with the following parameters

    • PBE exchange-correlation functional
    • Spin-polarized calculation
    • \(1 \times 1 \times 13\) k-points sampling
    • SG15-Medium basis set
  • initial_state_icon Initial State

    • Select User spin and check that Spin for Nickel is set to 1.
  • Change the output file name to ‘ni_collinear.nc‘.

Note

In order to run a calculation including spin-orbit later on you need fully relativistic pseudopotentials. In this case you will use the SG15 Pseudopotentials and basis sets .

Once you are ready, run the calculation.

Noncollinear calculation with spin-orbit coupling

Drag and drop the twoprobe_icon DeviceConfiguration contained in the ‘ni_collinear.nc‘ file from the LabFloor to the script_generator_icon Script Generator and modify the following parameters:

  • In the existing calculator_icon New Calculator, set the spin option to ‘Noncollinear Spin-Orbit

  • Add an initial_state_icon Initial State

    • Select User spin and check that Spin for Nickel is set to 1.
    • Check the option Use old calculation and use as filename ‘ni_collinear.nc‘’
  • Add analysis_icon Analysis > BandStructure

    • 200 points
    • G,Z path

Send the script to the editor_icon Editor and modify the initial_state_icon Initial State block as follows:

# -------------------------------------------------------------
# Initial State
# -------------------------------------------------------------

# Define the spin rotation
theta = 0*Degrees
left_spins = [(0, 1, theta, 0*Degrees) ]
initial_spin = InitialSpin(scaled_spins=left_spins)

old_calculation = nlread('Ni_collinear.nc', BulkConfiguration)[0]

bulk_configuration.setCalculator(
    calculator,
    initial_spin=initial_spin,
    initial_state=old_calculation,
)
bulk_configuration.update()
nlsave('Ni_noncollinear_phi0.nc', bulk_configuration)
nlprint(bulk_configuration)

Then run the calculation. You can notice that in the calculation the direction of the spin is parallel to the C axis (theta=0*Degrees). Once the calculation is done, repeat the calculation by setting the spin direction perpendicular to C (theta=90*Degrees), and compare the two resulting band structures (see figure below).

../../_images/bands_phi0.png ../../_images/bands_phi90.png

The results are in good agreement with the observations reported in [CYM10]. In particular, the effect of the SOC is clearly visible from the band splitting when the spin is oriented parallel to the wire axis. For both spin orientations you can also observe anticrossing of bands.

References

[CYM08](1, 2, 3, 4) Michael Czerner, Bogdan Yu. Yavorsky, and Ingrid Mertig. Fully relaxed magnetic structure of transition metal nanowires: First-principles calculations. Phys. Rev. B, 77:104411, 2008. doi:10.1103/PhysRevB.77.104411.
[CYM10](1, 2, 3, 4, 5) Michael Czerner, Bogdan Yu. Yavorsky, and Ingrid Mertig. The role of noncollinear magnetic order and magnetic anisotropy for the transport properties through nanowires. Phys. Stat. Sol. B, 247:2594, 2010. doi:10.1002/pssb.201046190.