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Band theory of Ferromagnetism
As Fe-atoms approach to form asolid, 3d and 4s orbitals overlap.
Paulis principle applies to all of them
over app ng atoms must eato splitting of 1 E-level into N-
levels---energy band formation
1 mg Fe has 1019 atoms each
energy level in isolated atom must
split into 1019 levels
Density of levels in band N(E)
No. of levels in dE = N(E)dE;
Average E-separation = 1/N(E)
Fig 8.22
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
More splitting in 4s-levels
Ref.: Introduction to Magnetic Materials B D Cullity
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Band theory of Ferromagnetism
As there are 5 3d-levels per atom with a
Ni atom has 8d
and 2s e-s
capacity of10 electrons, DOS of d-level is
higher
Results:
9.4 d e-s,
0.6 s e-s
e extent to w c t ese eve s are
occupied (dotted line) would depend on
number of(3d+4s) electrons in atom
Ref.:Introduction to
agnet c ater a sB D Cullity
Fig 8.22From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
Exchange forces cause spin unbalance =net moment/atom
10 atoms case (1 e- per atom)
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Ferromagnetism
Water-in-a-tank analogy
EXCHANGE force is like
a DAM holding water
across it
one or more electrons to
higher E. (these levels
(a) The split d-band. (b) Thes-band is not affected.
The arrows in the bands are s in ma netic moments.
mus no e w e y
spaced, else exchange
forces will not be able to
cause electron transfer)
LOW N(E) of s-band implies widely spaced levels therein
Fig 8.22From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
s e ectrons are assume to ma e no contr ut ons to sp n-
unbalance in 3d-elements
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Maximum imbalance when one half-band is full of 5 electrons
s/atomeelectrons4ofnumber
s/atomeelectrons43ofnumber
-
-
sx
s)d(n
=
+=
s/atomeelectrons3ofnumber -dn-x =
At saturation, 5 d-electrons have spin UP & [(n x) 5] have spin DN
The magnetic moment per atom is therefore,
)](10[}]5){(5[ xnxnH == Max spin unbalance is proportional to number of unfilled e-states in d-band
Ni: Observed H=0.6. Using this and n=10 (83d+24s) x=0.60
)6.10( nH =
Fig 8.22From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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CRIETRIA FOR EXISTANCE OF FRROMAGNETISM
The electrons responsible for FM must lie in partially filled
bands in order that there are vacant levels in which electronswith unpaired spins can move in
The DOS in the bands must be high, so that the increase in
energy caused by spin alignment will be small
The atoms must be the right distance apart so that the
exchan e forces can enable the d-electron s ins in one atom
to align the spins in neighboring atoms
Fig 8.22From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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770=C CT
At RT M M 029.0=
C
RT
TPartial
Complete
alignment Complete
randomness
=
Normalized saturated magnetization (for Fe) vs. reduced
ex
Fig 8.22From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
empera ure Cw ere C s e ur e empera ure .
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For Fe, Eex=kTC 90 meV
Maximum alignment of atomic Magnets Saturation Magnetization, (Msat)
Lattice vibrations OR Thermal Energy (or kT) DISORIENT the atomic
ma nets
Fig 8.22From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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Magnetic Domains in a single Xtal
Magnetostatic
Energy density
= B2/2
0
2
2
B
Domain/Bloch wall = 0MWhen cooled
from above TC
(a) Magnetized bar of ferromagnet with only one domain and hence an external B
(b) Formation of2 domains with opposite M reduces the external B. There are,
, .
(c) Domains of closure fitting at the ends eliminates the external fields at the ends.
(d) A speciment with several domains and closure domains.
There is no external B and the s ecimen a ears un-ma netized.
Fig 8.22From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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Magnetization along EASY direction along which spinalignments are easiest (exchange Interaction is strongest/maximum)
(a) An unmagnetized crystal of iron in the absence of an applied magnetic field.
H=0
.
(b) When an external field is applied the domain wall migrates into domain B which
enlarges A and B. The result is that the specimen now acquires net magnetization.
Fig 8.23From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
grows y o
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Magnetization growth (2-domain model) as H is
increased starting from H=0 to H=Hsat
Movement2 domains ofdomain-
wall
DomainSin le
rotationdomain
Fig 8.23From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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Magnetization along
EASY direction alongw c sp n a gnmen s are
easiest (exchange Interactionis strongest/maximum)
Hsat along [100] ~40 Oe
Hsat along [111] ~400 Oe
Along [111] M grows by
OP : domain wall movement
Magnetocrystalline anisotropy in a single iron crystal.Mvs.Hdepends on the
crystal on the crystal direction and is easiest along [100] and hardest along [111]
: oma n ro a on
Fig 8.24From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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Table 8.4 : Exchange interaction, magnetocrystalline anisotropy energyK, and
saturation magnetostriction coefficient sat
Material Crystal Eex kTC(meV)
Easy Hard K
(mJ cm3)
sat
( 106 )
Fe BCC 90 ;
cube edge
; cube
diagonal
0 :
48:
20 [100]
20 [111]
Co HCP 120 // to c axis 0: // to c axis
450: to c
axis
< > < >
cube
diagonal
edge 5:
24 [111]
From Principles of Electronic Materials and Devices, Third Edition, S.O. Kasap ( McGraw-Hill, 2005)
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