Transcript Document

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.
Magnetism in iron
For the orthogonal case:
1
h jLjL  t jLjL   0   mijL' I jLjL '  i  bij   i
2 L'


1
E  Ebs   m jL I jLjL ' m jL'   b j  m j
4 jLL '
j
Non-collinear magnetic tightbinding model
•The model is based on an accurate nonmagnetic 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).
Magnetization Energy (mRy)
•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, σ is the Pauli spin tensor.
For the non-orthogonal case:
1
1
h jLj 'L '  t jLj 'L '   0   s jLjL ' mijL' I jLjL '  mij 'L ' I jLjL '  i   s jLjL ' bij  bij '  i
4 j 'L '
2 j 'L '


1
E  Ebs   m jL I jLjL ' m jL'   b j  m j
4 jLL '
j
60
50
40
30
20
10
0
40
50
60
70
80
90
100
mijL 
Volume (Bohr3)
Methods
F(V,T,,)=Fstatic+Fel+Fphonon+Fmag
1.25
1.2
2
1.15
1.5
I(eV)
Magnetic Moment (  B )
C
The code operates in 3 modes: (1) Find self-consistent moments and moment directions, (2) Constrain moment directions, fin
2.5
The static free energy Fstatic is obtained by accurate
Linearized Augmented Plane computations within the GGA.
s
The non-orthogonal case is complicated by the non-diagonal overlap matrix. All of our computations are non-orthogonal. Our
no exchange interaction for s and p. The I’s were fit to give the same magnetization energies as LAPW at each volume for bcc
3
For the free energy F, where V is volume, T, temperature,
 structure, and  strain.
C
kn
*
knj ' L '
knj ' L '
Comparison of LAPW and TB total energies
(above) and moments (below) for bcc Fe.
Multiscale method using a variety of methods.
w
*
knjL jLj ' L '
1.1
1
1.05
0.5
1
0
40
50
60
70
80
90
100
Volume (Bohr 3)
3
Moment (B)
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.
2
2
1
Myrasov et al. (1992) LMTO
V=78.84
V=81.54
V=84.29
V=90.0
1
E (V ,  'ij )  cijkl (V ) 'ij  'kl
2
Accurate magnetic tight-binding model
fit to LAPW
E (mRyd)
5
0
0
-5
-10
0.00
V=78.84
V=81.54
V=84.29
V=90.00
0.25
0.50
0.75
10
-5
-10
0.00
0.50
0.75
Effective Hamiltonian
0.1
70 au
80 au
90 au 4
0.05
0
-0.05
0
1
2
-0.1
3
Magnetic Moment ( B)
1.00
E   Ak m j
kj
0.25
0.50
2k
  1
  2
1
  J jj 'e j  e j '   K jj ' e j  e j ' 
2 j j'
2 j j'
J jj '   J m j m j '
k
k
0.75
1.00
Although the TB model is much faster than selfconsistent 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).
k
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.
k
q
K jj '   K m j m j '
k
k
k
k
Monte Carlo Simulations
TB
0.50
0.75
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 moment—they do not represent
uncertainty in the mean. The experimental Tc is 1043 K at V=79.5 au.
1.00
Sjostedt and Nordstrom
(2002) LAPW
0.25
0.75
60 au
0.15
1
V=71.874
V=73.852
V=75.191
V=76.545
V=77.916
V=79.304
0
0.50
0.2
q
q
5
0.25
2
0
0.00
V=71.87
V=78.84
V=81.54
V=84.29
V=90.0
0.25
TB
V=71.874
V=73.852
V=75.191
V=76.545
V=77.916
V=79.304
-5
-10
1.00
0.00
0
1.00 0.00
0.75
V=71.87
V=78.84
V=81.54
V=84.29
V=90.0
Energy vs moment for fm bcc Fe (left).
50 au
1.00
q
2.5
1600
1400
2
Temperature (K)
 Elasticity from strain energy density
(isochoric strain)
Myrasov et al. (1992) LMTO
1
Sjostedt and Nordstrom
(2002) LAPW
10
5
0.50
Magnetic Moment (  B)
 E 
P  

 V 
10
0.25
(b)
Comparison of results for
spin-wave moments
(right) and energies
(below) in fcc Fe. There
is good qualitative
agreement.
E (mRyd)
 GGA (PBE) and LDA
 k=12x12x12 (hcp), k= 6x12x12
(orthorhombic)
 RMTKmax=9.0 with RMT=2.0
 3s, 3p, 3d, 4s, and 4p as valence
electrons
 Equation of State
(c)
3
Moment (B)
LAPW:
 Spin polarized DFT (collinear)
3
(a)
0
0.00
The magnetic free energy Fmag is obtained using Monte Carlo on
an effective Hamiltonian fit to TB and LAPW.
Magnetization Energy (Ry)
The thermal electronic free energy Fstatic is also obtained
using LAPW.
0.25
1.5
1
0.5
1200
1000
800
600
400
200
0
0
40
0
500
1000
Temperature (K)
1500
50
60
70
80
2000
Volume (Bohr3)
90
100
• The non-collinear magnetic tightbinding 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.