Spin-Polarized NEGF Convergence Guide

Version: O-2018.06

QuantumATK delivers a market-leading and highly optimized implementation of the non-equilibrium Green’s function (NEGF) method for spin-polarized electron transport calculations. Achieving fast and reliable convergence of the NEGF calculation to obtain the electronic ground state can sometimes be challenging as discussed in NEGF Convergence Guide for spin-unpolarized devices. This guide provides additional guidance when dealing with spin-polarized devices, e.g., magnetic tunnel junctions (MTJ), see also Spin transport in magnetic tunnel junctions.

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Systems investigated

  • Fe(001)|MgO reference structure
../../_images/fe-mgo-reference-structure.png
  • Fe(001)|MgO|Fe(001) magnetic tunnel junction (MTJ)
../../_images/mtj-structure.png

Computational parameters for the reference Fe|MgO structure

  • \(1 \times 1\) lateral unit cell Fe(001)|MgO.
  • Dielectric layer: 13 MgO monolayers.
  • Ferromagnetic metal: 7 Fe monolayers.
  • Norm-conserving pseudopotential: SG15.
  • LCAO basis set: Medium.
  • K-points mesh: \(21 \times 21 \times 1\).
  • Mesh cutoff: \(150\) Hartree.
  • SCF threshold: \(10^{−5}\) Hartree
  • Forces threshold: \(0.05\) eV/Å.
  • Geometry optimization: full relaxation of Fe and MgO structure and lattice parameters.

Computational parameters for the Fe|MgO|Fe device

  • Basic parameters
    • \(2 \times 2\) lateral unit cell Fe(001)|MgO|Fe(001) junction.
    • Dielectric spacer: 13 MgO monolayers.
    • Norm-conserving pseudopotential: SG15.
    • LCAO basis set: Medium.
    • Mesh cutoff: \(150\) Hartree.
    • SCF threshold: \(10^{−5}\) Hartree.
  • Structural parameters taken from a reference Fe(001)|MgO structure
    • Bulk Fe lattice parameter: \(a_{\rm Fe} = 3.00\) Å.
    • Fe-O separation distance: \(d_{\rm Fe-O} = 2.11\) Å.
    • Mg-O separation distance: \(d_{\rm Mg-O} = 2.18\) Å.
  • Parameters that have been varied
    • Electrodes
      • Length: 4 or 6 Fe monolayers
    • Bias voltage
    • K-point sampling
      • Device calculations: \(11\times 11\) and \(15\times 15\).
      • Electrode calculations: \(11\times 11\times 151\) and \(15\times 15\times 151\).
    • Mixing parameters
      • Damping factor: 0.1 and 0.05.
      • Number of history steps: 20 and 12.

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Calculating the self-energy matrix

  • Green’s-function matrix
../../_images/self-energy.png

\(\Sigma^{\rm L}\) (\(\Sigma^{\rm R}\)) is the self-energy matrix of the left (right) electrode.

1. Method overview

2. Practical methods

  • In practice, RecursionSelfEnergy and SparseRecursionSelfEnergy are the methods of choice for both equilibrium and non-equilibrium calculations.
  • The best method for equilibrium and non-equilibrium calculations in terms of speed and memory footprint can be chosen by using DevicePerformanceProfile to analyze the performance of the device calculation.
../../_images/performance-profile.png

Zero and finite-bias convergence: mixing parameters

  • General considerations
    • All calculations converge, independently on the parameters used.
    • The convergence is almost unaffected by the variation of the computational parameters.
    • Convergence at finite bias voltages is much faster than at zero bias, because the density matrix from the most close bias voltage is used as the initial state.
    ../../_images/convergence-1.png
  • If slightly less aggressive mixing parameters are used compared to the default ones (lower damping_factor and number_of_history_steps), the convergence at zero (finite) bias voltage is slightly slower (faster).

    ../../_images/convergence-2.png
  • Increasing the k_point_sampling along A and B from \(11\times 11\) to \(15\times 15\) results in a slightly slower convergence at zero bias voltage, but in a faster convergence at finite bias voltages.

    ../../_images/convergence-3.png

Zero and finite-bias convergence: electrode-length and k-point sampling

  • Increasing the electrode length from 4-layers to 6-layers results in a slightly slower convergence at zero-bias voltage. Convergence at finite-bias voltages is very similar.

  • The 6-layers electrode calculations are more memory intensive than the 4-layers electrode calculations. In case of insufficient memory, it is possible to reduce the memory footprint by re-evaluating the self-energies at each SCF step storage_strategy=NoStorage().

    ../../_images/convergence-4.png

Electrode validator

  • The electrode validator allows one to check if the electrode repetition is sufficiently long in the C direction. The length of the electrode ultimately depends on the basis set used in the calculations.

  • In the present calculations, the four-layers electrode is slightly too short and some matrix elements are discarded. For production calculations with Fe electrodes and the SG15-Medium pseudopotential/basis set, a 6-layers electrode should be used.

    ../../_images/convergence-5.png

Contact support

If you failed to converge your device calculation even after following our guidelines, write a post on the online QuantumWise Forum (https://quantumwise.com/forum) or contact QuantumATK support (quantumatk-support@synopsys.com). In both cases, please attach your QuantumATK script and log file.