Transcript Magnetostriction in Fe-based alloys.

Theory of magnetostriction in Invar materials.
S. Khmelevskyi
Center for Computational Materials Science,
Vienna University of Technology
P. Mohn, A. V. Ruban, I. Turek
Phenomenon of magnetostriction.
Re-orientation
.
“Para-process”
Applied field reduces thermal disorder
orienting moments along field.
It is the source of the volume
magnetostriction in all ferromagnets
Applied field induces a band splitting.
Small effect.
It may be ignored for ferromagnets.
H=100T ~ 1mRy
However it is the source of para-process at
T = 0K
Even in non-magnetic materials!
Re-orientation of the magnetization along the field
spin-orbit coupling
anisotropy
Anomalous thermal expansion in Invar
Thermal expansion anomaly of Invar-type.
ωlat – thermal expansion due to
lattice vibrations, usually well
follows Gruneisen law.
ωm – magnetic contribution,
which vanishes in paramagnetic
state. It exists in all metallic
magnets
ωs0 – spontaneous volume
magnetostriction
for Fe-based Invar ωs0 ~1-2%
ωs0(Fe-Ni) ~ 2.1%
Magnetic contribution to the thermal expansion can be viewed as sort of spontaneous paraprocess.
m (T )  m (M (T ))   (Hmol . (M (T )))
h
m
Experimental examples.
“classical” Invar alloys Fe-Ni
RECo2
(Zr1-xNbx)Fe2 – Laves Phase compounds.
After M. Shiga and Y. Nakamura
J.Phys. Soc. Jap. (1979)
Modern measurements at low T: example (Er-Y)Co2
ωs0 – need not to be large
if Tc is small.
Modern measurements at low T reveals
a lot of materials with Invar type anomaly
Which has a small Tc.
R. Hauser et al. Phys. Rev. B 61, 1198 (2000)
Some Thermodynamics.
Vonsovskyi, Shur (1948)
T  M s  
  
~






H


T

 p,T

 p, H p
 ~ J 0 -effective exchange constant
Bean, Rodbell (1962) local moment model
2
F (M ,  ) 
 J 0 ( ) M s2  N kbT S mag
2
 dJ0  2
M s
 d 
s  
“Simple” example 1: RECo2 Laves Phases.
Fixed Spin Moment
calculations for YCo2
(Dy,Er)Co2
M  2B sˆz
Dy3+
Heff  Hcrit
2I0 g  1 Jˆ  sˆ 
s-d model
sinhJ  12  y 




G M ,  , T  Fd M ,  , T  k BT ln
sinh y 2
y
( g  1) I 0 M
k BT B
Fd is the Helmholtz potential
of the itinerant subsystem
Fd M ,  , 0  E(M , )   Cij M 2i j
i, j
Heff  Hcrit
Spin-fluctuations included
 = DV / V
Stoner Co band
1  10
-3
DyCo 2
HoCo 2
0
20
40
60
80
100
120
140
T(K)
C M
ij
2i
M 2 M  m
2
     Fd M ,  , T 
j
i, j
S. Khmelevskyi, P.Mohn, JMMM, 272-276 (2003)525
“Simple” example 2: hcp Gadolinium.
TB-LMTO
Disordered Local Moment. Partial
DLM Gd(4f-up)1-xGd(4f-down)x
f-electrons – open core (m = 7B).
Thermal expansion
A. Lindbaum and M. Rotter
s (exp)~ 0.5%
S.Khmelevskyi, I. Turek, P. Mohn, PRB 70, (2003)
s (cal.) 
V FM   V DLM 
 0.53%
V DLM 
Intermediate conclusions.
• Invar anomaly is magnetovolume effect related to the spontaneous
volume magnetostriction due to change of magnetization with
temperature (very trivial).
•Such effect exists in ALL magnetic materials (even in ALL nonmagnetic in applied external field).
•One need just to explain why in given Invar material such a contribution
large enough to compensate (or be comparable in size to !) a thermal
expansion due to lattice vibration in the temperature interval from 0K to
Tc.
•There is no intrinsic and unique feature of Invar materials, which is
absent in “normal” ones. The difference between them only in quantity related size of spontaneous magnetostrictions, Tc (and Gruneisen
coefficient).
Fe-based transition metal alloys.
• Classical Invar systems:
(technical Invars).
• bcc Fe-Co
• AFe2 – Laves Phases.
fcc Fe-Ni, Fe-Pt, Fe-Pd
Problems.
•One cannot separate a system into electronic subsystems, one of which is
responsible for the intrinsic temperature dependent molecular field acting on
another subsystem with anomalous magnetostrictive properties.
•We should have working approach to tackle with finite temperature magnetism of
itinerant magnets in intermediate regime between local and weak itinerant cases
(our choice is DLM – “Do it better if …”).
•Lots of additional material dependent complications: chemical disorder,
antiferromagnetic interactions on frustrated lattice etc.
•50 years of intensive theoretical development on Fe-Ni, which are very complex
alloy.
Disordered Local Moments state.
Model for paramagnetic state: disordered alloy of the same sort of atoms with spin-up and spin
down randomly distrubuted over the lattice sites.
(Cyrot, Gyorrfy)
Alloy analogy:

A1-xBx
binary metallic alloy
A()1-x A()x
partially ordered state
with local moments
Using Coherent Potential Approximation
A()0.5 A()0.5 – model paramagnetic state above Tc
Disordered Fe-Pt

 M M
M  M loc coth loc
 k BT

TB-LMTO, LDA spd-basis
Partial DLM calculations
97.0
Tc=ΓMloc2/(3kB)
Fe+50-xFe-x Pt50
Fe+70-xFe-x Pt30
Fe+74-xFe-xPt26
90.5
90.0
2/(3k )
Tc=ΓMloc
89.5
B
PM
89.0
88.5
88.0
PM
87.5
fractional length change(DL/L)
PM
96.5
volume (bohr3/atom)

k BT 
 

M

M
loc


2 10
-3
Fe70Pt30
Fe76Pt24
87.0
5
10
15
20
25
x (concentration)
s 
V FM   V DLM 
V DLM 
30
35
0.2
0.4
0.6
0.8
T/Tc
Fe74Pt26: so(exp)=1.7% ; so(calc)=1.9%
S. Khmelevskyi, I. Turek, P. Mohn, PRL, 91 037201 (2003)
1.0
fcc Fe-Pd, fcc Fe-Pt and bcc Fe-Co
2.5
Fe1-x Cox bcc
Fe1-x Pt x fcc
Fe1-x Pdx fcc
s x 10-2
2.0
s 
1.5
V FM   V DLM 
V DLM 
1.0
0.5
0.0
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
9.2
9.3
3.0
2.9
M(B/atom)
2.8
Maximum of the spontaneous
magnetostriction corresponds to the
maximum of the local Fe moment
drop in paramagnetic state.
2.7
2.6
2.5
2.4
2.3
2.2
2.1
2.0
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
9.0
9.1
valence electron concentration
9.2
9.3
Khmelevskyi and Mohn, PRB 69 (2004)
Why local Fe moment anomalously decreases in Invar alloy composition?
Fe70Pt30
25
25
20
20
e/a=8.6
15
DOS(st./Ry/spin/atom)
15
DOS(st./Ry/spin/atom)
Fe50Pt50
10
5
0
-5
-10
-15
-20
Fe70Pt30
Fe FM
Fe DLM
e/a=9
10
5
0
-5
-10
-15
-20
-25
Fe50Pt50
Fe FM
Fe DLM
-30
-25
-0.4
-0.3
-0.2
E(Ry)
-0.1
0.0
0.1
0.2
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
E(Ry)
Transition from strong ferromagnetic state at T=0K to the weak ferromagnetic state in PM region
Ordered and disordered Fe3Pt
Cu3Au - structure
Long-Range Order parameter:
where c(Fe) is total
concentration of Fe in the alloy
and cI(Fe) is a concentration of
Fe atoms on Pt sites.
Ordered and disordered Fe3Pt
s 
V FM   V DLM 
V DLM 
Bulk KKR-ASA, spdf
basis, and Muffin tin
electrostatic corrections
Spontaneous magnetostriction
moderately decreases with
increasing of ordering
Khmelevskyi, Ruban, Kakehashi, Mohn,
Johansson, PRB 72 (2005)
Nothing is new since there is exist nothing new.
Fe-Ni case.
Akai and Dederichs, PRB B 47, 8739 (1993)
consistent with TB-LMTO calculations with spd-basis
Fe65Ni35 KKR-ASA LSDA spd-basis
D. D. Johnson, F. J. Pinski, J. B. Staunton, B. L.
Gyorffy, and G.M. Stocks, in Physical
Metallurgy of Controlled ExpansionInvar-type
Alloys, 1990
Crisan et al. Phys. Rev. B 66, 014416 (2002)
Problem number 1: exchange interactions in Fe-Ni
H 

J
e
 ij i e j
i , j{ Ni }
Inter-atomic exchange interactions of
Heisenberg Hamiltonian calculated using
Lichtenstein Green function formalism
(Magnetic Force Theorem). GGA results.
INVAR alloy Fe65Ni65 become antiferromagnetic
at volumes lower then experimental ones
Antiferromagnetic scenario cannot be ruled out
What is magnetic ground state of Fe65Ni35?
Fe65Ni65 alloy
Calculations with Local self-consistent
Green Function methods (LSGF)
512 atoms super cell
Moment of Fe atoms with 11 and 12 Fe
nearest neighbors oriented anti-parallel
to the total magnetization.
Ruban, Khmelevskyi, Mohn, Johansson PRB
76 (2007)
Wang et al. JAP (1998)
Problem number 2: LSDA
Fe65Ni35 INVAR alloy.
V FM   V DLM 
s 
V DLM 
GGA, Full-Charge Density EMTO results: so(exp)=2.2% ; so(calc)=3.2%
Calculations of effective inter-atomic chemical interactions and MC simulations shows that
Fe-Ni cannot be considered as partially ordered alloy.
Short-Range order effects is also very weak.
Ruban, Khmelevskyi, Mohn, Johansson PRB 76 (2007)
antiferromagnetic scenario: contra-example.
(Zr1-xNbx)Fe2 – Laves Phase compounds.
and
YFe2 – non-Invar system.
Calculations with DLM.
KKR-ASA, spdf basis with MT-corrections, GGA
Rws(FM)
Rws(DLM)
ωs0(cal./exp.)
mFe(FM),
mFe(DLM),
mA(FM),
a.u.
a.u.
%
μB
μB
μB
ZrFe2
2.901
2.876
2.6 / 1.0
2.12
1.62
-0.84
Zr0.7Nb0.3Fe2
2.872
2.843
2.5 / 0.8
2.04
1.57
-0.71
YFe2
3.014
3.001
1.3 / ~0.0
2.24
2.06
-0.8
Zr0.7Nb0.3Fe2
Zr0.7Nb0.3Fe2
Conclusions.
• The Invar effect has common origin in all Invar-type magnetic systems.
Spontaneous volume magnetostriction, which large enough to compensate
thermal expansion. It should be only RELATIVELY large.
• In all considering cases the source of large magnetostriction is decrease of the
size of the local moments induced by the thermal disorder of magnetic moments.
The source of this decrease may be different in different Invars.
• The difference between Invar and non-Invar systems is quantitative –
not qualitative.