Information from magnetisation curves

Viorel PopFaculty of Physics, Babes-Bolyai University, 4000480 Cluj-Napoca, Romania

Fundamental properties of matter and Applications

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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.