Band Theory of Ferromagnets

7/29/2019 Band Theory of Ferromagnets

1/12

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


2/12

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)


3/12

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


s e ectrons are assume to ma e no contr ut ons to sp n-

unbalance in 3d-elements


4/12

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 =



5/12

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



6/12

770=C CT

At RT M M 029.0=

C

RT

TPartial

Complete

alignment Complete

randomness

=

Normalized saturated magnetization (for Fe) vs. reduced

ex


empera ure Cw ere C s e ur e empera ure .


7/12

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



8/12

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.



9/12

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.


grows y o


10/12

Magnetization growth (2-domain model) as H is

increased starting from H=0 to H=Hsat

Movement2 domains ofdomain-

wall

DomainSin le

rotationdomain



11/12

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



12/12

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)

Documents

Band Theory of Ferromagnets