Temperature Simulations of Magnetism in Iron

1
Temperature Simulations of Magnetism in Iron
R.E. Cohen and S. Pella
Carnegie Institution of Washington
Goal To understand and predict effects of
magnetism on equation of state, elasticity, and
phase stability of iron for input into materials
models.
Non-collinear magnetic tight-binding model
For the orthogonal case

• The model is based on an accurate non-magnetic
  tight-binding model fit to LAPW (Wasserman,
  Stixrude and Cohen, PRB 53, 8296, 1996 Cohen,
  Stixrude, and Wasserman, PRB 56, 8575, 1997 58,
  5873).
• Magnetism is added via an exchange interaction
  parameterized by a single tensor, the Stoner I.
• Problems were found with the original
  implementation of this method (Mukherjee and
  Cohen, 2001).

Magnetism in iron

• bcc-Fe is stable only because of ferromagnetism
• fcc-Fe has no ordered magnetic moments,
• but has local disordered or incommensurate
  moments
• leads to anti-Invar effect (huge thermal
  expansivity)
• would not appear in phase diagram if not for
  local moments (Wasserman, Stixrude and Cohen,
  1996)
• huge effect on bulk modulus
• what about shear modulus and plasticity?

Where h is the Hamiltonian, t is the non-magnetic
part, j labels each atom, L labels each
orbital, mjL is the moment from orbital L on atom
j, IjLjL is the exchange interaction of orbital
L on orbital L On atom j, s is the Pauli spin
tensor.
For the non-orthogonal case
Methods
Comparison of LAPW and TB total energies (above)
and moments (below) for bcc Fe.

• Multiscale method using a variety of methods.
• F(V,T,?,?)FstaticFelFphononFmag
• For the free energy F, where V is volume, T,
  temperature,
• structure, and ? strain.
• The static free energy Fstatic is obtained by
  accurate
• Linearized Augmented Plane computations within
  the GGA.
• The thermal electronic free energy Fstatic is
  also obtained
• using LAPW.
• The phonon free energy Fphonon is obtained using
  molecular
• Dynamics or the particle in a cell model with a
  tight-binding
• (TB) model fit to LAPW, or first-principles
  lattice dynamics
• linear response within the quasiharmonic
  approximation.
• The magnetic free energy Fmag is obtained using
  Monte Carlo on

The non-orthogonal case is complicated by the
non-diagonal overlap matrix. All of our
computations are non-orthogonal. Our present
results use a Stoner tensor such that 3d states
polarize only d states, and there is no exchange
interaction for s and p. The Is were fit to give
the same magnetization energies as LAPW at each
volume for bcc, and then used as a function of V
for other structures. The code operates in 3
modes (1) Find self-consistent moments and
moment directions, (2) Constrain moment
directions, find self-consistent moments for
those directions, (3) Constrain moment magnitudes
and directions. The latter 2 modes are
implemented by finding staggered fields b that
give the required constraints. A self-consistent
penalty function method was used. It was also
necessary in some cases to constrain the atomic
charge so that charge does not unphysically flow
during spin polarization.
Energy vs moment for fm bcc Fe (left).
50 au
60 au
Effective Hamiltonian
70 au
80 au
Although the TB model is much faster than
self-consistent electronic structure
calculations, it is still too slow for Monte
Carlo simulations. We fit the TB results to an
effective Hamiltonian similar to that of
Rosengaard and Johansson (PRB 55, 14975, 1997).
90 au
Myrasov et al. (1992) LMTO
TB
Comparison of results for spin-wave moments
(right) and energies (below) in fcc Fe. There is
good qualitative agreement.
Sjostedt and Nordstrom (2002) LAPW
For initial tests we used only first neighbors
and no K terms. The parameters Jk and Ak were
determined from The TB model for FM and AFM bcc
energies as functions of moment at each volume.
Monte Carlo Simulations
Magnetism in Fe Summary
We used the Effective Hamiltonian in Monte Carlo
simulations for 128 atom supercells for 800,000
sweeps through all the degrees of freedom. The
results are shown below. The error bars show
the 1 std. dev. fluctuations in the momentthey
do not represent uncertainty in the mean. The
experimental Tc is 1043 K at V79.5 au.

• The non-collinear magnetic tight-binding model
  is in good agreement with most self-consistent
  calculations for Fe.
• There is no empirical input.
• Results are sensitive to I, which is obtained by
  fitting to first-principles results
• An effective Hamiltonian was fit to the TB
  results.
• Tc for bcc iron is too high. This may be due to
  the current simple nearest neighbor model.