Information from magnetisation curvesmagnetism.eu/esm/2007-cluj/slides/pop-slides.pdf · 1 P.R....
Transcript of Information from magnetisation curvesmagnetism.eu/esm/2007-cluj/slides/pop-slides.pdf · 1 P.R....
Information from magnetisation curves
Viorel PopFaculty of Physics, Babes-Bolyai University, 4000480 Cluj-Napoca, Romania
Fundamental properties of matter and Applications
Physics is DIFFICULT and HARD to understand
keep away of physic
Magnetic moments of the electrons
Magnetism in mater
Magnetic moment of the atoms
Magnetic study of the mater
Fundamental properties and Applications
Magnetic moment of the atoms
nucleusLr
lµr
vr -e
Sr
Kinetic moment of a charge
Magnetic moment: Jgm B rr µ=
Lr
Sr
Orbital kinetic moment
Spin kinetic moment
)1(2 )1()1()1(1+
+−++++= JJ LLSSJJgLandé factor
( )
( )CGScme SIme eB eB 22 hh=
=
µ
µ
μμμμB ≡≡≡≡ spinspinspinspin magnetic momentul of free electronμB = 9,2742⋅10-24 A⋅m2
2g-g sss == ;Smem e r
r
2
1g-g == ll ;Lmem el r
r
2
V∑=
mM
r
r
magnetisation M magnetic susceptibility χχχχmagnetic permeability μμμμ
( )MHB 0 rrr
+= µ
μμμμ0 = 4π⋅π⋅π⋅π⋅10-7 H/m
HM=χ
B=µ0(H+χχχχH)= µ0(1+χχχχ)H= µH HB=µ
( )χ1µ µ 0 +=
C, Cu, Pb, H2O, NaCl, SiO 2
M Tχ
χ H(a) (b)Diamagnetic
Paramagnetic
Magnetic ordered
0m =r
χχχχ < 0
0m ≠r
J ij = 0χχχχ > 0
0m ≠r
J ij ≠ 0χχχχ >> 0
χ ≠ f(H)
CBeff ⋅⋅⋅⋅==== 8)(µµµµµµµµ CBeff ⋅⋅⋅⋅==== 4664,)(µµµµµµµµ)( 1+= JJg Beff µµ
Na, Al, CuCl 2
χ
M
H
1/C
1/χT
lawCurie;1 CT=−χ
JCN kBeff 03
µµ
⋅=
if χ(emu/mole) if χ(µB/T∙f.u)
a) ferromagneticjijJH SSi rr⋅−= 2 J ij > 0
Fe, Co, Ni, Gd…
Ms ≠ 0
1/χχχχM
TTc θ
θχ
−=
T
C
T1<T2<Tc<T3
T1
T2
T3
M
H
Ms(0) = gJ µB J
Molecular field approximation
Curie – Weiss law
θθθθ = Tc
MNH iim ====
b) antiferromagnetic
Ms=0
χχχχ
T
χχχχχχχχ⊥⊥⊥⊥
TN
1/χ
TTNθθθθ
θθθθ < 0
θχ
+=+= TCHMM BA
Η=0Η Η ΗΗ
J ij < 0BA MM rr=
MnO, Mn, Cr…
T>Tc
1/χ
T0 Tcθ
'θθθθσσσσ
χχχχχχχχ −−−−−−−−++++==== TCT011
M=MA-MB
M=MA-MB
M=MA-MB
M=MA-MB0
T0
T0
TMB MB
MB
MA MAMA
M0
M0M0
Tc
Tc
Tc
Tcomp
M MMNAA≈NBB NAA<NBBNAA>NBB
T<Tc
(a) (b) (c)
c) ferrimagnetism
Ms≠≠≠≠0
J ij < 0
Fe3O4, ferrites, GdCo5,…
MA ≠ MB
M
H
HM=χχχχ
M
Hχiχm
Hsat
MstM0 χ0
Msat
Ms
0== Hi dHdMχ
χ
H
T<Tcχi
Mst = saturation magnetisationMs = spontaneous magnetisationχm
1/χχχχMs
TTc θ
θχ
−=
T
C
Ms(0)
1/χχχχMs
TTc θ
θχ
−=
T
C
Ms(0)
M
H
Ms(T1)
Ms(T2)
Ms(T3)
χχχχp
T1
T2
T3
T1 < T2 < T3
HMM ps χχχχ++++====
01234560 100 200 300 400 500
Al5Mn3Ni2Ms (µB/f.u.) T (K)Ms = 5.03 µB/f.u.Tc = 401 K 0123450 2 4 6 8 10
Al5Mn3Ni2T = 10 K M(µB/f.u.) µ0H (T)3.63.844.24.44.64.855.25 6 7 8 9 10T = 10 K
T = 100 KT = 200 KT = 300 K
y = 5.0293 + 0.00043122x R= 0.10622
y = 4.7774 + 0.009374x R= 0.96475
y = 4.218 + 0.023918x R= 0.99702
y = 3.535 + 0.022673x R= 0.99868
M(µB/f.u.) µ0H (T)
HHaMM ps χχχχ++++ −−−−==== 1
0
0,5
1
1,5
2
2,5
3
0 2 4 6 8 10
GdCo4SiT = 4 KM(µB/f.u.)
µ0H (T)2,22,32,42,52,62,72,82,932 3 4 5 6 7 8 9 10M(µB/f.u.) µ0*H (T)
Ms(T)T→0KFor the rare earth (Gd for example)M0 = NgµB J
Ms(0) ≡ M0For 3d transition metals (Fe, Co, Ni…), the orbital moment is blocked by crystalline field:M0 = NgµB S; g ≈ 2
1/χχχχM
TTc
θχ
−=
T
C
?BoBiic k JJNgNT 3 122 )( ++++
====µµµµµµµµ
Curie temperature evaluation
T →→→→ Tc; T < Tc −−−−++++++++
++++⋅⋅⋅⋅==== cTTJJ JM TM 1113100 22 22 )( )()( )(
T2sM
Tc
Curie temperature evaluation
T →→→→ Tc; T < Tc −−−−++++++++
++++⋅⋅⋅⋅==== cTTJJ JM TM 1113100 22 22 )( )()( )(
05101520
012345
300 320 340 360 380 400LuFe4B
M2 (a.
u.) M (a.u.)
T (oC) Tc010203040506070
0246810
0 200 400 600 800M2 (a.
u.) M (a.u.)T(oC) T
2sMTc
Curie temperature evaluation
T →→→→ Tc; T < Tc −−−−++++++++
++++⋅⋅⋅⋅==== cTTJJ JM TM 1113100 22 22 )( )()( )(
0246810
200 400 600 800 1000 1200
ThFe11C1.5M(a.u.)
T(K)01020304050300 350 400 450 500 550 600 650 700M2 (
a.u.
)
T(K)O. Isnard, V. Pop, K.H.J. Buschow, J. Magn. Magn. Mat. 256 (2003) 133
234567890 200 400 600 800 1000M (a.u.) T (oC)
0246810
200 400 600 800 1000 1200SmCo5+20Fe_8hMMM(a.u.)T(K)
Phase analysis and thermal evolution
MHMbMaMFm 042 42 µµµµ−−−−⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅⋅++++++++====)(In the low magnetisation region - for example T →→→→ Tc; T < Tc
0====dMdFm HbMaM 03 µµµµ====++++ or babHMM −−−−==== 02 µµµµ
(((( ))))(((( ))))
HMCJM TJJMT TTN c cii 03220200 110 1223 µµµµµµµµµµµµ ====++++
++++++++++++−−−− )(molecular field approximations:
Nii = Tc/C
Hm = Nii M
(((( ))))c ciiT TTNa −−−−==== 0µµµµ
(((( )))) CJM TJJb 22020 110 1223++++
++++++++====)(µµµµ
T <<<< TcT = Tc T >>>> Tc.
a < 0a = 0a >>>> 0
020406080100120
0 0.2 0.4 0.6 0.8 1ThFe11C1.5
M2 (µB/f.u.)2µ0H/M (T*f.u./µB)
400 K 440 K420 KO. Isnard, V. Pop, K.H.J. Buschow, J. Magn. Magn. Mat. 256 (2003) 133
M2 T2 T3Tc T4T5T1H/M1/χ
Ms2 H/M3 H/M4H/Mc02sMT1 T2 T3
TcT4
T5
T1 <<<< T2 <<<< T3 <<<< Tc <<<< T4 <<<< T5.
M2
−−−−
−−−−++++ −−−−==== 34 4334 MHMH MHMHTMHMHTT ccc
Arrott plotA. Arrott, Phys. Rev. 108, 1394 (1957)
1/χχχχM
TTc θ
θχ
−=
T
C
Ms(0)
M
H
HM====χχχχ
Paramagnetic sampleor
Ferromagnetic sample at T>Tc
If there are some ferromagnetic impurityscMHM ++++==== χχχχ HMcHM s++++==== χχχχ
HMH1
χχχχ
CBeff ⋅⋅⋅⋅==== 8)(µµµµµµµµ CBeff ⋅⋅⋅⋅==== 4664,)(µµµµµµµµ)( 1+= JJg Beff µµ JCNkBeff 03
µµ
⋅=
if χχχχ(emu/mole) if χχχχ(µµµµB/T∙∙∙∙f.u) TTN Tc Tc
θθθθχχχχ
++++==== TC
θθθθχχχχ
−−−−==== TC
'θθθθσσσσ
χχχχχχχχ −−−−−−−−++++==== TCT011
TC====χχχχ
θθθθ θθθθθθθθ
1/χ
Ferro
para
Antiferro
Ferri
Ms(0) = gJ µB J0)( 1++++==== ppBeff JJgµµµµµµµµ
T→0K T > Tc
Ms(0) ≈ 2 µB S0 )1( +≈ ppBeff SSgµµT→0K T > Tc10 >>>>==== SSr pFor 3d transition metals (Fe, Co, Ni…), the
orbital moment is blocked by crystalline field:
For the rare earth (Gd for example): J0=Jp
1/χχχχM
TTc θ
θχ
−=
T
C
Ms(0)
1 P.R. Rhodes, E.P. Wolfarth, Proc. R. Soc. 273 (1963 ) 347.2 O. Isnard, V. Pop, K.H.J. Buschow, J. Magn. Magn. M at. 256 (2003) 1333 D. Bonnenberg, K.A. Hempel, H.P.J. Wijn, Landolt-B. orsntein new series, Vol. III, 19a,
Springer, Berlin, 1986, p. 142.4 N. Coroian, PhD thesis5 R. Ballou, E . Burzo, and V. Pop, J. Magn. Magn. Ma t. 140-144 (1995) 945.
r = 1 local moment limitr →∞∞∞∞ total delocalisation limit →∞∞∞∞2.031.691.51.321.011.00r
YCo3B25HoCo 4Si4Fe3C3ThFe11C1.5
2Co1Fe1Gd1
HH || easy magnetisation direction
HaM
∆MH ⊥⊥⊥⊥ easy magnetisation directionH
H || easy magnetisation directionHa
M∆MH ⊥⊥⊥⊥ easy magnetisation direction
low anisotropy energyspin – flop transition
H(z)(z) H(z)H MbMa
MaMaMb Mb
M
H0 (a) (b)
T < TN, antiferromagnetic materials, χχχχ⊥⊥⊥⊥ >>>> χχχχ
E. Du Trémolet de Lacheisserie (editor), Magnetisme, Presses Universitaires de Grenoble, 199 9
exac HHH ====
Density of energy in magnetic field H,
E = -χµχµχµχµ0H2/2
M
H0(a)(b)(z) H MbMa
H(z)Ma MbH(z)Ma Mb
High anisotropy energyspin – flip transition
spin–flip
andspin–flop
also in ferrimagnetic materials
metamagnetic transition
exc HH ≈≈≈≈
( ) ( )( )( )
( )χµµµχµµ
µχµχµµ
+=⋅=+⋅=
⋅=+⋅=+=+=
1
1
1
00
0
00
r
r
HH
HHMHB
µµµµi µµµµµµµµm
B
H
H
µiµmµ
M
H-Hc
HcH
Mr-Mr
B
B’A
CD
E F
O
reversible
µ >> 0magnetic coresmagnetic circuitssoft
Hc – small~0,001÷10 A/m
permanentmagnetshard
Hc – BIG ~102÷106 A/m
MH-Hc Hc
HM r-M r
BB ’ A CDE FO
( )MHBrrr
+= 0µ
Curie temperature, Tc
Fundamental research M
Application research B
Hard magnetic materials
-HB
BH(BH)max0P(BH)=const
MH
MH
Rotation mechanism Pinning mechanism
Exchange bias field
E. Girgis et al, J. Appl. Phys. 97 (2005) 103911
M HM H
Spring magnets
-100-50050100
-4 -3 -2 -1 0 1 2 3 4SmCo5 + 20Fe/8h MM
as milled
450oC 0.5h
500oC 1.5h
550oC 1.5h
600oC 0.5h
650oC 0.5hSmCo
5/2h MM
M (emu/g) H (T)SmCo 5
Fe
Hr0H ====r
SmCo 5
Fe
θ
-50050
-2 -1.5 -1 -0.5 06h+450C/0.5h2h+450C0.5hM (emu/g)
H (T)T = 4 K
050100150200250-8 -6 -4 -2 0
SmCo5_20 wt%FeT = 300K 2h/450C30'4h/450C30'6h/450C30'8h/450C30'dM/dH
H (T)
The reversibility curvesand the dM/dH variationvs H are very fine instruments in qualitative evaluation of the inter-phase hard/soft exchange coupling.
Ha
+ ++
++++
- -- - - - -
Ms Hd
+ ++ + + ++
- -- - - - -
Ms HdMH rr dd N−−−−====
Ha
sd MNH II==== sd MNH ⊥⊥⊥⊥====
dai HHHH rrrr++++======== Ha = applied field
(((( )))) (((( ))))ad dadaN NH1M MHHHHMχχχχ
χχχχχχχχχχχχχχχχ
++++====
−−−−====++++========
M
H
M
Ha
1/Nd
MHrr dd N====
The influence of the demagnetising field on the mag netisation curves
Ndx ==== Ndy ==== Ndz ====1/3.
(((( )))) MHd rr⋅⋅⋅⋅−−−−−−−−==== θθθθcos1l
θO
Mrdsphere d << l Nd = 0d >> l Nd = -1
H II easy magnetisation direction
Nd = 0
M
H
Ms(T1)
Ms(T2)
Ms(T3)
T1
T2
T3
T1 < T2 < T3
HMM ps χχχχ++++====
Hd ≠ 0
M
HHsat = Ms
H
H
NO MAGNETOCRYSTALLINE ANISOTROPY
Magnetic measurements give magnetisation (A/m)
magnetic measurements on plate shape samples
Hi = H -NMs
M
HHsat = Ha
H
M
HHsat = MsH
PERPENDICULAR ANISOTROPY
Magnetic measurements give magnetisation (A/m)
Magnetic measurements give magnetocrystalline anisotropy
-50050
-2 -1.5 -1 -0.5 0
6h+450C/0.5h2h+450C0.5h
M (emu/g)H (T)
T = 4 K
χχχχ
M H T1T2T3χχχχ
M H M HT1<T2<Tc<T3(a) (b) (c)MHχ i χm Hst
MstM0 χ0
HM=χ 0====
==== HdHdMMs
MHχ i χm Hst
MstM0 χ0
HM=χ 0====
==== HdHdMMs
M H-Hc HcMr-Mr
BB’’ A CDE FO
HH || easy magnetisation directionHaM
∆MH ⊥⊥⊥⊥ easy magnetisationdirection M HH
MHa
M 1/NdHM
HaM
χχχχ/(1+Ndχχχχ)χχχχ
MHHsat= Ms
H || thin layerH ⊥ thin layer MHHsat= Ha
H || thin layer MHHsat= Ms
H ⊥ thin layerHM H1χχχχ
scMHM += χ HMcHM s+= χ2sM T1 T2 T3Tc T4T5T1 <<<< T2 <<<< T3 <<<< Tc <<<< T4 <<<< T5.M2 T2 T3Tc T4T5T1
H/M1/χ
Ms2 H/M3 H/M4H/Mc0
-50050
-2 -1.5 -1 -0.5 06h+450C/0.5h2h+450C0.5hM (emu/g)
H (T)T = 4 K
050100150200250-8 -6 -4 -2 0
SmCo5_20 wt%FeT = 300K 2h/450C30'4h/450C30'6h/450C30'8h/450C30'dM/dH
H (T)
The reversibility curvesand the dM/dH variationvs H are very fine instruments in qualitative evaluation of the inter-phase hard/soft exchange coupling.