NANO MAGNETIC MATERIALS AND APPLICATIONS
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Sep 09, 2024
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About This Presentation
NANOMAGNETIC MATERIALS
Slide Content
1
A. K. Majumdar
Indian Institute of Science Education & Research
Kolkata, India
Indian Institute of Technology, Kanpur, India
(1972-2006)
University of Florida, Gainesville, July 11-14, 201 1
Magnetism
A. K. Majumdar
Indian Institute of Science Education & Research
Kolkata, India
Indian Institute of Technology, Kanpur, India
(1972-2006)
University of Florida, Gainesville, July 11-14, 201 1
Magnetism
2
Plan of the Talk
History of Magnetism
Diamagnetism
Paramagnetism
Ferromagnetism
Weiss molecular field theory
Heisenbergs theory
Blochs spin-wave theory
Band theory of ferromagnetism
Spin glass (1972)
R & D in Magnetic Materials after 1973
Magnetoresistance (MR)
Ordinary or normal magnetoresistance (OMR)
Anisotropic magnetoresistance (AMR) in a ferromagnet
Giant magnetoresistance (GMR)
Hall effect in ferromagnets Fe-Cr multilayers: GMR and Hall effect Applications Nano-magnetism
Superparamagnetism
Examples: Ni nano/TiN
Exchange bias
Tunnel magnetoresistance (TMR)
Colossal magnetoresistance (CMR)
Plan of the lectures
Plan of the Talk
History of Magnetism
Diamagnetism
Paramagnetism
Ferromagnetism
Weiss molecular field theory
Heisenbergs theory
Blochs spin-wave theory
Band theory of ferromagnetism
Spin glass (1972)
R & D in Magnetic Materials after 1973
Magnetoresistance (MR)
Ordinary or normal magnetoresistance (OMR)
Anisotropic magnetoresistance (AMR) in a ferromagnet
Giant magnetoresistance (GMR)
Hall effect in ferromagnets Fe-Cr multilayers: GMR and Hall effect Applications Nano-magnetism
Superparamagnetism
Examples: Ni nano/TiN
Exchange bias
Tunnel magnetoresistance (TMR)
Colossal magnetoresistance (CMR)
Plan of the lectures
3
The study of magnetism started with the discovery o f the Lodestone (or Magnetite Fe
3
O
4
)
around 500-800 B.C. in
Greece & China.
Lodestones attract pieces of iron and the attraction can only be
stopped by placing between them an iron plate, whi ch acts as
a
Magnetic Shield.
The directional property of the Lodestone was utilized to design Compass,
which was invented around 100 A.D. in China.
Theoretical understanding of magnetism came
only
in the 19
th
century along with some basic applications .
History of Magnetism
The study of magnetism started with the discovery o f the Lodestone (or Magnetite Fe
3
O
4
)
around 500-800 B.C. in
Greece & China.
Lodestones attract pieces of iron and the attraction can only be
stopped by placing between them an iron plate, whi ch acts as
a
Magnetic Shield.
The directional property of the Lodestone was utilized to design Compass,
which was invented around 100 A.D. in China.
Theoretical understanding of magnetism came
only
in the 19
th
century along with some basic applications .
History of Magnetism
4
Diamagnetism ⇒
The weakest manifestation of magnetism is
Diamagnetism
⇒⇒⇒⇒
Change of orbital moment of electrons due to applied
magnetic field.
⇒
The relevant parameter which quantifies the strengt h of
magnetism is the
Magnetic Susceptibility, cccc,
which is defined as
c= Lt
H®0
(dM/dH).
⇒
Diamagnetism arises from two basic laws of Physics, viz., Faradays law & Lenzs law: An electron moves around a nucleus in a circle of r adius, r.
A magnetic field
H
is applied. The induced electric field,
E(r),
generated during the change is
E(r) 2πr = -d/dt (H πr
2
)/c
or, E(r) = - (r/2c) dH/dt.
Diamagnetism
Diamagnetism ⇒
The weakest manifestation of magnetism is
Diamagnetism
⇒⇒⇒⇒
Change of orbital moment of electrons due to applied
magnetic field.
⇒
The relevant parameter which quantifies the strengt h of
magnetism is the
Magnetic Susceptibility, cccc,
which is defined as
c= Lt
H®0
(dM/dH).
⇒
Diamagnetism arises from two basic laws of Physics, viz., Faradays law & Lenzs law: An electron moves around a nucleus in a circle of r adius, r.
A magnetic field
H
is applied. The induced electric field,
E(r),
generated during the change is
E(r) 2πr = -d/dt (H πr
2
)/c
or, E(r) = - (r/2c) dH/dt.
Diamagnetism
5
This E(r) produces a force & hence a torque = e E(r) r
= dL/dt = (e r
2
/2c) dH/dt, (e > 0).
The extra ∆L(L was already there for the orbital motion)
due to the turning of the field = (e r
2
/2c) H.
The corresponding moment
m
= - (e/2mc) ∆L = - (e
2
r
2
/4mc
2
) H, r
2
= x
2
+ y
2
.
For N atoms per unit volume with atomic no. Z, the Magnetic Susceptibility
cccc
= - (Ne
2
Z<r
2
>
av ) /6m
c
2
.
QM treatment also produces the same answer!!!
π
cis negative»- 10
-6
in cgs units in the case of typical diamagnetic materials. In
SI units it is
- 4πx 10
-6
.
Diamagnetism
This E(r) produces a force & hence a torque = e E(r) r
= dL/dt = (e r
2
/2c) dH/dt, (e > 0).
The extra ∆L(L was already there for the orbital motion)
due to the turning of the field = (e r
2
/2c) H.
The corresponding moment
m
= - (e/2mc) ∆L = - (e
2
r
2
/4mc
2
) H, r
2
= x
2
+ y
2
.
For N atoms per unit volume with atomic no. Z, the Magnetic Susceptibility
cccc
= - (Ne
2
Z<r
2
>
av ) /6m
c
2
.
QM treatment also produces the same answer!!!
π
cis negative»- 10
-6
in cgs units in the case of typical diamagnetic materials. In
SI units it is
- 4πx 10
-6
.
Diamagnetism
6
Diamagnetism QM treatment
The Hamiltonian of a charged particle in a magnetic field B is
H = (p eA/c)
2
/2m +eφ, A = vector pot, φ= scalar pot, valid for both
classical & QM.
K.E. is not dependent on B, so it is unlikely that Aenters H. But
p = p
kin
+ p
field
= mV + eA/c (Kittel, App. G, P: 654)
& K.E. = (mV) 2
/2m = (p eA/c)
2
/2m,whereB =ÑÑÑÑX A.
So,B-field-dependent part of H is ieh/4 pmc[ÑÑÑÑ. A+A. ÑÑÑÑ] + e
2
A
2
/2mc
2
.(1)
Diamagnetism QM treatment
The Hamiltonian of a charged particle in a magnetic field B is
H = (p eA/c)
2
/2m +eφ, A = vector pot, φ= scalar pot, valid for both
classical & QM.
K.E. is not dependent on B, so it is unlikely that Aenters H. But
p = p
kin
+ p
field
= mV + eA/c (Kittel, App. G, P: 654)
& K.E. = (mV) 2
/2m = (p eA/c)
2
/2m,whereB =ÑÑÑÑX A.
So,B-field-dependent part of H is ieh/4 pmc[ÑÑÑÑ. A+A. ÑÑÑÑ] + e
2
A
2
/2mc
2
.(1)
7
Diamagnetism QM treatment
c= Lt
H®0
(dM/dH) = - (Ne
2
Z<r
2
>av )/6mc
2
independent of T.
M = -¶E/¶B = - (Ne
2
Z<r
2
>
av
)/6mc
2
B for N atoms/volume with atomic no. Z.
Since B = H for non-magnetic materials
If B is uniform and II z-axis, A
X
= - ½ y B, A
Y
= ½ x B, and A
Z
= 0.
So, (1) becomes H =(e/2mc) B [ih/2p(x ¶/¶y - y ¶/¶x)] + e
2
B
2
/8mc
2
(x
2
+ y
2
).
¯ ¯
L
Z
⇒Orbital PM; E = e
2
B
2
/12mc
2
<r
2
>
by 1st order
perturbation theory.
Diamagnetism QM treatment
c= Lt
H®0
(dM/dH) = - (Ne
2
Z<r
2
>av )/6mc
2
independent of T.
M = -¶E/¶B = - (Ne
2
Z<r
2
>
av
)/6mc
2
B for N atoms/volume with atomic no. Z.
Since B = H for non-magnetic materials
If B is uniform and II z-axis, A
X
= - ½ y B, A
Y
= ½ x B, and A
Z
= 0.
So, (1) becomes H =(e/2mc) B [ih/2p(x ¶/¶y - y ¶/¶x)] + e
2
B
2
/8mc
2
(x
2
+ y
2
).
¯ ¯
L
Z
⇒Orbital PM; E = e
2
B
2
/12mc
2
<r
2
>
by 1st order
perturbation theory.
8
Some applications:
Meissner effect
in a superconductor:
cas high as 1 below T
C
perfect diamagnetism to make B = 0
inside
(flux expulsion) below H
C1
.
Substrates
of present day magnetic sensors are
mostly diamagnetic like Si, sapphire, etc.
Diamagnetism
M(T) shows
negative and temperature
independent
magnetization.
Some applications:
Meissner effect
in a superconductor:
cas high as 1 below T
C
perfect diamagnetism to make B = 0
inside
(flux expulsion) below H
C1
.
Substrates
of present day magnetic sensors are
mostly diamagnetic like Si, sapphire, etc.
Diamagnetism
M(T) shows
negative and temperature
independent
magnetization.
9
Paramagnetism
Atoms/molecules in solids/liquids with odd no of el ectrons(S¹0):
free Na atoms with partly filled inner shells, comp ounds like
molecular oxygen and organic biradicals, metals (P auli), etc.
contribute to electron paramagnetism.
Free atoms/ions with partly filled inner shells, e. g., Mn
2+
, Gd
3+
, U
4+
show ionic paramagnetism.
A collection of magnetic moments,
m
, interact with external magnetic
field
H
:
Interaction energy
U = -m . H.
Magnetization results from the orientation of the m agnetic moments but
thermal disorders disturb this orientation.
Paramagnetism
Paramagnetism
Atoms/molecules in solids/liquids with odd no of el ectrons(S¹0):
free Na atoms with partly filled inner shells, comp ounds like
molecular oxygen and organic biradicals, metals (P auli), etc.
contribute to electron paramagnetism.
Free atoms/ions with partly filled inner shells, e. g., Mn
2+
, Gd
3+
, U
4+
show ionic paramagnetism.
A collection of magnetic moments,
m
, interact with external magnetic
field
H
:
Interaction energy
U = -m . H.
Magnetization results from the orientation of the m agnetic moments but
thermal disorders disturb this orientation.
Paramagnetism
10
Paramagnetism
The energy levels, according to quantum mechanics, is given by
Paramagnetism
The energy levels, according to quantum mechanics, is given by
11
⇒
In this
<m> vs. H/T
plot
paramagnetic saturation is
observed only at very high H & low
T, i.e.,
x >>1 when coth x®®®®1 and
< m > ®®®®
Ng
B
S
⇒
At 4 K & 1tesla,
<m> ~ 14 %
of its
saturation value.
⇒
For ordinary temperatures like 300
K & 1 T field,
x <<1.
Then
coth x
®®®®1/x + x/3
and so
Ngμ
B
.
Paramagnetism
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
S=3/2 FOR Cr3++
Magnetic Moments
Magnetic Field/Temperature(B/T)
H
Tk
SSg
m
B
B
+
>® <
3
)1 (
m
Thus
<m>
varies linearly with field.
< m > = Ng
mmmm
BS [coth X- 1/X] = L(X) = Langevin function when S
®®®® ¥¥¥¥
with X = Sx.
⇒
In this
<m> vs. H/T
plot
paramagnetic saturation is
observed only at very high H & low
T, i.e.,
x >>1 when coth x®®®®1 and
< m > ®®®®
Ng
B
S
⇒
At 4 K & 1tesla,
<m> ~ 14 %
of its
saturation value.
⇒
For ordinary temperatures like 300
K & 1 T field,
x <<1.
Then
coth x
®®®®1/x + x/3
and so
Ngμ
B
.
Paramagnetism
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
S=3/2 FOR Cr3++
Magnetic Moments
Magnetic Field/Temperature(B/T)
H
Tk
SSg
m
B
B
+
>® <
3
)1 (
m
Thus
<m>
varies linearly with field.
< m > = Ng
mmmm
BS [coth X- 1/X] = L(X) = Langevin function when S
®®®® ¥¥¥¥
with X = Sx.
12
So the
Magnetic Susceptibility,
cccc
at temperature T is
cccc
= ( N g
2
mmmm
B
2
S(S+1)/3k
B
)*1/T
= C/T
,
where
N
= No. of ions/vol,
g
is the Lande factor
,
mmmm
B
is the Bohr
magneton and
C
is the
Curie constant.
This is the famous
Curie law for paramagnets.
Some applications:
To obtain temperatures lower than 4.2 K, paramagnetic salts like,
CeMgNO
3
, CrKSO
4
, etc.
,
are kept in an isothermal 4.2 K bath
& a field of, say, 1 T applied.
>>>
Entropy of the system decreases & heat goes out. Fi nally the
magnetic field is removed adiabatically.
>>> Lattice temperature drops to mK range.
Similarly microkelvin is obtained using nuclear demagnetization.
Paramagnetism
So the
Magnetic Susceptibility,
cccc
at temperature T is
cccc
= ( N g
2
mmmm
B
2
S(S+1)/3k
B
)*1/T
= C/T
,
where
N
= No. of ions/vol,
g
is the Lande factor
,
mmmm
B
is the Bohr
magneton and
C
is the
Curie constant.
This is the famous
Curie law for paramagnets.
Some applications:
To obtain temperatures lower than 4.2 K, paramagnetic salts like,
CeMgNO
3
, CrKSO
4
, etc.
,
are kept in an isothermal 4.2 K bath
& a field of, say, 1 T applied.
>>>
Entropy of the system decreases & heat goes out. Fi nally the
magnetic field is removed adiabatically.
>>> Lattice temperature drops to mK range.
Similarly microkelvin is obtained using nuclear demagnetization.
Paramagnetism
13
Paramagnetism
Fascinating magnetic properties, also quite complex in na ture:
Ce58: [Xe] 4f
2
5d
0
6s
2
, Ce
3+
= [Xe] 4f
1
5d
0
since 6s
2
and 4f
1
removed
Yb70: [Xe] 4f
14
5d
0
6s
2
, Yb
3+
=
[Xe] 4f
13
5d
0
since 6s
2
and 4f
1
removed
Note: [Xe] = [Kr] 4d
10
5s
2
5p
6
.
In trivalent ions the outermost shells are identical 5s
2
5p
6
like neutral Xe.
In La (just before RE) 4f is empty, Ce
+++
has one 4f electron, this
number increases to 13 for Yb and 4f
14
at Lu, the radii contracting from 1.11
Å (Ce) to 0.94 Å (Yb) →
Lanthanide Contraction.
The number of 4f
electrons compacted in the inner shell is what determines their magnetic
properties. The atoms have a (2J+1)-fold degenerate gro und state which is
lifted by a magnetic field.
Rareearth ions
In a Curie PM
=
(
)
T
Np
T
C
T
g J NJ
B B
B
k
m
k
m
c
b
3 3
1
2 2 2 2
= =
+
=
, p = effective Bohr Magnetron number
.
(
)
[
]
1
+
JJg
)1 (2
)1 ( )1 ( )1 (
1
+
+
-
+
+
+
+=
JJ
LL SS JJ
, g = Lande factor
p calculated from above for the
+3
RE
ions using Hunds rule agrees
very well with experimental values except for
&
3+
Eu
+3
Sm
.
Paramagnetism
Fascinating magnetic properties, also quite complex in na ture:
Ce58: [Xe] 4f
2
5d
0
6s
2
, Ce
3+
= [Xe] 4f
1
5d
0
since 6s
2
and 4f
1
removed
Yb70: [Xe] 4f
14
5d
0
6s
2
, Yb
3+
=
[Xe] 4f
13
5d
0
since 6s
2
and 4f
1
removed
Note: [Xe] = [Kr] 4d
10
5s
2
5p
6
.
In trivalent ions the outermost shells are identical 5s
2
5p
6
like neutral Xe.
In La (just before RE) 4f is empty, Ce
+++
has one 4f electron, this
number increases to 13 for Yb and 4f
14
at Lu, the radii contracting from 1.11
Å (Ce) to 0.94 Å (Yb) →
Lanthanide Contraction.
The number of 4f
electrons compacted in the inner shell is what determines their magnetic
properties. The atoms have a (2J+1)-fold degenerate gro und state which is
lifted by a magnetic field.
Rareearth ions
In a Curie PM
=
(
)
T
Np
T
C
T
g J NJ
B B
B
k
m
k
m
c
b
3 3
1
2 2 2 2
= =
+
=
, p = effective Bohr Magnetron number
.
(
)
[
]
1
+
JJg
)1 (2
)1 ( )1 ( )1 (
1
+
+
-
+
+
+
+=
JJ
LL SS JJ
, g = Lande factor
p calculated from above for the
+3
RE
ions using Hunds rule agrees
very well with experimental values except for
&
3+
Eu
+3
Sm
.
14
Paramagnetism
Hunds rules
SL-
SL+
Formulated by Friedrich Hund, a German Physicist around 1927.
The ground state of an ion is characterized by:
1. Maximum value of the total spinS allowed by Paulis exclusion principle.
2. Maximum value of the total orbital angular momentum L consistent with
the total value of S, hence PEP.
3. The value of the total angular momentum J is equal to
when the shell is less than half full & when the shell is
more than half full.
When it is just half full, the first rule gives L= 0 and so J = S.
Paramagnetism
Hunds rules
SL-
SL+
Formulated by Friedrich Hund, a German Physicist around 1927.
The ground state of an ion is characterized by:
1. Maximum value of the total spinS allowed by Paulis exclusion principle.
2. Maximum value of the total orbital angular momentum L consistent with
the total value of S, hence PEP.
3. The value of the total angular momentum J is equal to
when the shell is less than half full & when the shell is
more than half full.
When it is just half full, the first rule gives L= 0 and so J = S.
15
Paramagnetism
Explanation of 2
nd
rule:
Best approached by model calculations.
electrons.Therefore,all electrons tend to become
are separated more and hence have less positive P.E. than for
Explanation 1
st
rule:
Its origin is Paulis exclusion principle and
Coulomb repulsion between electrons. ¯
giving maximum S.
+2
Mn
Mn:
2 5
4 3s d
+2
Mn
5
3d
Example of rules 1 & 2:
,(half- filled 3d sub shell).
All 5 spins can be║to each other if each electron occupies a different
orbital and there are exactly 5 orbitals characterized by or bital angular
quantum nos. m
l
= -2, -1, 0, 1, 2. One expects S = 5/2 and so ∑m
l
= 0.
ThereforeLis 0 as observed.
Paramagnetism
Explanation of 2
nd
rule:
Best approached by model calculations.
electrons.Therefore,all electrons tend to become
are separated more and hence have less positive P.E. than for
Explanation 1
st
rule:
Its origin is Paulis exclusion principle and
Coulomb repulsion between electrons. ¯
giving maximum S.
+2
Mn
Mn:
2 5
4 3s d
+2
Mn
5
3d
Example of rules 1 & 2:
,(half- filled 3d sub shell).
All 5 spins can be║to each other if each electron occupies a different
orbital and there are exactly 5 orbitals characterized by or bital angular
quantum nos. m
l
= -2, -1, 0, 1, 2. One expects S = 5/2 and so ∑m
l
= 0.
ThereforeLis 0 as observed.
16
Paramagnetism
Explanation of 3
rd
rule:
Consequence of the sign of the spin-orbit interaction. For asingle
electron, energy is lowest whenSis antiparallel toL(L.S= - ve). But
the low energy pairs m
l
and m
s
are progressively used up as one adds
electrons to the shell. By PEP, when the shell is more than hal f full the
state of lowest energy necessarily has the S║L.
Examples of rule 3: Ce
3+
= [Xe] 4f
1
5s
2
5p
6
since 6s
2
and 4f
1
removed. Similarly Pr
3+
= [Xe]
4f
2
5s
2
5p
6
. Nd
3+
, Pm
3+
, Sm
3+
,Eu
3+
,Gd
3+
,Tb
3+
, Dy
3+
, Ho
3+
,Er
3+
,Tm
3+
,Yb
3+
have 4f
3
to 4f
13
. Take Ce
3+
: It has one 4f electron, an f electron has l = 3,
s = ½,
4f shell is less than half full
(full shell has 14 electrons), by third
rule =J=½LS½= L- 1/2 = 5/2.
J
s
L
1 2+
2/5
2
F
+3
PrSpectroscopic notation
⇒⇒⇒⇒
[L = 0, 1, 2, 3, 4, 5, 6 are S, P, D, F, G, H, I].
has 2 4felectrons: s = 1, l = 3. Both cannot have m
l
= 3 (PEP),
so max. Lis not 6 but 5.J=½LS½= 5 -1 = 4.
⇒⇒⇒⇒
3
H
4
Paramagnetism
Explanation of 3
rd
rule:
Consequence of the sign of the spin-orbit interaction. For asingle
electron, energy is lowest whenSis antiparallel toL(L.S= - ve). But
the low energy pairs m
l
and m
s
are progressively used up as one adds
electrons to the shell. By PEP, when the shell is more than hal f full the
state of lowest energy necessarily has the S║L.
Examples of rule 3: Ce
3+
= [Xe] 4f
1
5s
2
5p
6
since 6s
2
and 4f
1
removed. Similarly Pr
3+
= [Xe]
4f
2
5s
2
5p
6
. Nd
3+
, Pm
3+
, Sm
3+
,Eu
3+
,Gd
3+
,Tb
3+
, Dy
3+
, Ho
3+
,Er
3+
,Tm
3+
,Yb
3+
have 4f
3
to 4f
13
. Take Ce
3+
: It has one 4f electron, an f electron has l = 3,
s = ½,
4f shell is less than half full
(full shell has 14 electrons), by third
rule =J=½LS½= L- 1/2 = 5/2.
J
s
L
1 2+
2/5
2
F
+3
PrSpectroscopic notation
⇒⇒⇒⇒
[L = 0, 1, 2, 3, 4, 5, 6 are S, P, D, F, G, H, I].
has 2 4felectrons: s = 1, l = 3. Both cannot have m
l
= 3 (PEP),
so max. Lis not 6 but 5.J=½LS½= 5 -1 = 4.
⇒⇒⇒⇒
3
H
4
17
Paramagnetism
Exactly ½ full 4f shell:
: has 7 4f electrons; s = 7/2, L = 0, J = 7/2
+3
Nd
2/9
4
I
+3
Pm
4
5
I
has 3 4felectrons: s = 3/2, L =6, J=6-3/2=9/2
⇒
has 4 4felectrons: s = 2, L= 6, J = L- S = 4
⇒
+3
Gd
2/7
8
S
⇒
+3
Ho
¯
8
5
I
4f shell is more than half full:
has 10 4felectrons: 7 will be ,
3 will be . S = 2, L= 6 [3 2 1 0 -1 -2 -3] ; J = 6 + 2 = 8
⇒
and so on.
Note: 4f shell is well within the inner core (localized) surrounde d by 5s
2
, 5p
6
, and
6s
2
& so almost unaffected by crystal field (CF).
3d transition element ions
3d transition element ions, being in the outermost shel l, are affected by strongly
inhomogeneous electric field, called the crystal field (CF) of neighboring ions in
a real crystal. L-Scoupling breaks, so states are not specified by J. (2L + 1)
degenerate levels for the free ions may split by the CF andLis often quenched.
p calculated from J gives total disagreement with experiments.
Paramagnetism
Exactly ½ full 4f shell:
: has 7 4f electrons; s = 7/2, L = 0, J = 7/2
+3
Nd
2/9
4
I
+3
Pm
4
5
I
has 3 4felectrons: s = 3/2, L =6, J=6-3/2=9/2
⇒
has 4 4felectrons: s = 2, L= 6, J = L- S = 4
⇒
+3
Gd
2/7
8
S
⇒
+3
Ho
¯
8
5
I
4f shell is more than half full:
has 10 4felectrons: 7 will be ,
3 will be . S = 2, L= 6 [3 2 1 0 -1 -2 -3] ; J = 6 + 2 = 8
⇒
and so on.
Note: 4f shell is well within the inner core (localized) surrounde d by 5s
2
, 5p
6
, and
6s
2
& so almost unaffected by crystal field (CF).
3d transition element ions
3d transition element ions, being in the outermost shel l, are affected by strongly
inhomogeneous electric field, called the crystal field (CF) of neighboring ions in
a real crystal. L-Scoupling breaks, so states are not specified by J. (2L + 1)
degenerate levels for the free ions may split by the CF andLis often quenched.
p calculated from J gives total disagreement with experiments.
18
Paramagnetism
Details: If E is towards a fixed nucleus, classically all L
X
, L
Y
, L
Z
are constants (fixed plane for a central force). In QM L
Z
& L
2
are
constants of motion but in a non-central field (as in a crystal) the orbital
plane is not fixed & the components of Lare not constants & may be
even zero on the average. If <L
Z
> = 0, L is said to be quenched.
3d transition element ions (Contd.)
In an orthorhombic crystal, say, the neighboring cha rges produce
about the nucleus a potential V = Ax
2
+ By
2
(A + B)z
2
satisfying Laplace
equation & the crystal symmetry. For L =1, the orbi tal moment of all 3
energy levels have <L
Z
> = 0. The CF splits the degenerate level with
separation >> what the B-field does. For cubic symm etry, there is no
quadratic term in V & so p electron levels will remai n triply degenerate
unless there is a
spontaneous
displacement of the magnetic ion, called
Jahn-Teller effect
, which lifts the degeneracy and lowers its energy.
Mn
3+
has a large JT effect in manganites which produces Colossal
Magnetoresistance (CMR).
For details see C. Kittel, ISSP (7
th
Ed.) P. 425-429; Ashcroft & Mermin, P. 655-659.
Paramagnetism
Details: If E is towards a fixed nucleus, classically all L
X
, L
Y
, L
Z
are constants (fixed plane for a central force). In QM L
Z
& L
2
are
constants of motion but in a non-central field (as in a crystal) the orbital
plane is not fixed & the components of Lare not constants & may be
even zero on the average. If <L
Z
> = 0, L is said to be quenched.
3d transition element ions (Contd.)
In an orthorhombic crystal, say, the neighboring cha rges produce
about the nucleus a potential V = Ax
2
+ By
2
(A + B)z
2
satisfying Laplace
equation & the crystal symmetry. For L =1, the orbi tal moment of all 3
energy levels have <L
Z
> = 0. The CF splits the degenerate level with
separation >> what the B-field does. For cubic symm etry, there is no
quadratic term in V & so p electron levels will remai n triply degenerate
unless there is a
spontaneous
displacement of the magnetic ion, called
Jahn-Teller effect
, which lifts the degeneracy and lowers its energy.
Mn
3+
has a large JT effect in manganites which produces Colossal
Magnetoresistance (CMR).
For details see C. Kittel, ISSP (7
th
Ed.) P. 425-429; Ashcroft & Mermin, P. 655-659.
19
Paramagnetism
Pauli paramagnetism in metals
Paramagnetic susceptibility of a free electron gas: There is a
change in the occupation number of up & down spin electrons even
in a non-magnetic metal when a magnetic field B is applied.
The magnetization is given by (
T << T
F
)
M = (N
up
- N
down
) =
2
D ( ε
F
) B, D ( ε
F
) = 3N/2ε
F
.
\\\\
χ= 3 N ∆
2
/ 2ε
F ,independent of temperature,
where D ( ε
F
) = Density of states at the Fermi level ε
F
.
Paramagnetism
Pauli paramagnetism in metals
Paramagnetic susceptibility of a free electron gas: There is a
change in the occupation number of up & down spin electrons even
in a non-magnetic metal when a magnetic field B is applied.
The magnetization is given by (
T << T
F
)
M = (N
up
- N
down
) =
2
D ( ε
F
) B, D ( ε
F
) = 3N/2ε
F
.
\\\\
χ= 3 N ∆
2
/ 2ε
F ,independent of temperature,
where D ( ε
F
) = Density of states at the Fermi level ε
F
.
20
Any theory of ferromagnetism has to explain:
i) Existence of spontaneous magnetization
M
below
T
C
(Paramagnetic to ferromagnetic transition temperatu re).
ii) Below T
C
, a small
H
0
produces
M
S
from M ~ 0 at H = 0.
There are three major theories on ferromagnetism: 1. Weiss molecular field theory
2. Heisenbergs theory
3. Blochs spin-wave theory (T << Tc).
Ferromagnetism
Any theory of ferromagnetism has to explain:
i) Existence of spontaneous magnetization
M
below
T
C
(Paramagnetic to ferromagnetic transition temperatu re).
ii) Below T
C
, a small
H
0
produces
M
S
from M ~ 0 at H = 0.
There are three major theories on ferromagnetism: 1. Weiss molecular field theory
2. Heisenbergs theory
3. Blochs spin-wave theory (T << Tc).
Ferromagnetism
21
π
Weiss proposed that:
i) Below
T
C
spontaneously magnetized domains, randomly oriented
give M ~ 0 at H=0. A small
H
0
produces domain growth with
M|| H
0
.
ii) A very strong
molecular field, H
E
of unknown origin aligns the
atomic moments within a domain.
π
Taking alignment energy ~ thermal energy below Tc,
For Fe:
H
E
= k
B
T
C
/ m~ 10
7
gauss ~ 10
3
T !!!
E
d-d ~ [
1.
2 - (
1.r
12) (
2.r
12)]/4πε
0r
3
.
Classical dipole-dipole interaction gives a field of ~ 0. 1 T only
& is anisotropic but ferromagnetic anisotropy is only a se cond
order effect.
So ???
π
Weiss postulated that
H
E
= lM,
where
l
is the
molecular field
parameter
and
M
is the
saturation magnetization.
Ferromagnetism
Weiss molecular field theory
π
Weiss proposed that:
i) Below
T
C
spontaneously magnetized domains, randomly oriented
give M ~ 0 at H=0. A small
H
0
produces domain growth with
M|| H
0
.
ii) A very strong
molecular field, H
E
of unknown origin aligns the
atomic moments within a domain.
π
Taking alignment energy ~ thermal energy below Tc,
For Fe:
H
E
= k
B
T
C
/ m~ 10
7
gauss ~ 10
3
T !!!
E
d-d ~ [
1.
2 - (
1.r
12) (
2.r
12)]/4πε
0r
3
.
Classical dipole-dipole interaction gives a field of ~ 0. 1 T only
& is anisotropic but ferromagnetic anisotropy is only a se cond
order effect.
So ???
π
Weiss postulated that
H
E
= lM,
where
l
is the
molecular field
parameter
and
M
is the
saturation magnetization.
Ferromagnetism
Weiss molecular field theory
22
Curie-Weiss law
Curie theory of paramagnetism gave M = [( N g
2
m
B
2
S(S+1)/3k
B
)*1/T]*H
= (C/T)*H.
Replacing H by H
0
+ lM we get
M = CH
0
/(T-Cl)
\\\\
cccc
= C/(T-T*
C
),
where
T*
C
= Cl
= paramagnetic Curie
temperature.
Putting g ~ 2, S = 1, M = 1700 emu/g
one gets l= 5000 & H
E
= 10
3
T for Fe.
This theory fails to explain
c(T) for
T <T*
C
.
Ferromagnetism
Curie-Weiss law
Curie theory of paramagnetism gave M = [( N g
2
m
B
2
S(S+1)/3k
B
)*1/T]*H
= (C/T)*H.
Replacing H by H
0
+ lM we get
M = CH
0
/(T-Cl)
\\\\
cccc
= C/(T-T*
C
),
where
T*
C
= Cl
= paramagnetic Curie
temperature.
Putting g ~ 2, S = 1, M = 1700 emu/g
one gets l= 5000 & H
E
= 10
3
T for Fe.
This theory fails to explain
c(T) for
T <T*
C
.
Ferromagnetism
23
Ferromagnetism
Ferromagnetism
24
Ferromagnetism
Ferromagnetism
25
Ferromagnetism
Ferromagnetism
26
Heisenbergs Theory (Exchange effect)
2 1 2 1
2
2
*
1
*
)()( )()(rd rd ruru
r r
e
ruru J
i j
j i
j i ij
rrr r
r r
r r
∫
-
=
Ferromagnetism
⇒
Heisenberg found the origin of Weiss molecular field in t he quantum
mechanical exchange effect, which is basically electrical in nature.
Electron spins on the same or neighboring atoms tend to be coupled by
the exchange effect a consequence of Paulis Exclusion Pri nciple. If u
i
and u
jare the two wave-functions into which we put 2 electrons, t here
are two types of states that we can construct according to the ir having
antiparallel or parallel spins. These are
Ψ(r
1,r
2) =1/Ö2 [u
i(r
1) u
j(r
2)±u
j(r
1) u
i(r
2)] where
±correspond to space symmetric/antisymmetric (spin singlet, S = 0/spin
triplet, S = 1) states.
Singlet: spin anisymmetric: S = 0: (↑↓-↓↑)/Ö2 , m = 0.
Triplet: spin symmetric: S = 1:↑↑m = 1;↓↓m = -1;
(↑↓+↓↑)/Ö2 , m = 0.
Also, if you exchange the electrons between the two states, i.e.,
interchange r
1and r
2,Ψ(-) changes sign butΨ(+) remains the same. If
the two electrons have the same spin(║), they cannot occupy the same
r. ThusΨ(-) = 0 ifr
1=r
2. Now if you calculate the average of the
Coulomb energy e
2
/│r
12│in these two states we find them different by
which is the exchange integral.
Heisenbergs Theory (Exchange effect)
2 1 2 1
2
2
*
1
*
)()( )()(rd rd ruru
r r
e
ruru J
i j
j i
j i ij
rrr r
r r
r r
∫
-
=
Ferromagnetism
⇒
Heisenberg found the origin of Weiss molecular field in t he quantum
mechanical exchange effect, which is basically electrical in nature.
Electron spins on the same or neighboring atoms tend to be coupled by
the exchange effect a consequence of Paulis Exclusion Pri nciple. If u
i
and u
jare the two wave-functions into which we put 2 electrons, t here
are two types of states that we can construct according to the ir having
antiparallel or parallel spins. These are
Ψ(r
1,r
2) =1/Ö2 [u
i(r
1) u
j(r
2)±u
j(r
1) u
i(r
2)] where
±correspond to space symmetric/antisymmetric (spin singlet, S = 0/spin
triplet, S = 1) states.
Singlet: spin anisymmetric: S = 0: (↑↓-↓↑)/Ö2 , m = 0.
Triplet: spin symmetric: S = 1:↑↑m = 1;↓↓m = -1;
(↑↓+↓↑)/Ö2 , m = 0.
Also, if you exchange the electrons between the two states, i.e.,
interchange r
1and r
2,Ψ(-) changes sign butΨ(+) remains the same. If
the two electrons have the same spin(║), they cannot occupy the same
r. ThusΨ(-) = 0 ifr
1=r
2. Now if you calculate the average of the
Coulomb energy e
2
/│r
12│in these two states we find them different by
which is the exchange integral.
27
Ferromagnetism
Ferromagnetism
28
Ferromagnetism
Ferromagnetism
29
Ferromagnetism
Ferromagnetism
30
Ferromagnetism
Assuming nearest neighbour exchange interaction, J
ij= J & S
i S
we get the Hamiltonian
H = - gm
B
H
0
NS - ½ {2JNZ(S)
2
}
,
where Z = no. of N.N.s; N = no. of atoms/vol.
Simple algebra gives, replacing
H
0
by
H
0
+ H
E
(= lM),
using
T
C= C l,
C= N g
2
m
B
2
S(S+1)/3k
B
&
M = Ngm
B
S
in
H = -M(H
0
+ ½ H
E
),
llll= (2JZ/Ng
2
mmmm
B
2
)
&
J = (3k
B
T
C
/2ZS(S+1))
~ 1.2 x 10
-2
eV
for Fe with S = 1.
Taking S = ½ , one gets
(k
B
T
C
/ ZJ) = 0.5
from mean-field theory
= 0.325
from Rushbrooke & Wood theory.
j
ji
ji
i ij
i
Z
i B
SSJ S H g Hr
r
∑
∑
¹
- -=
,
0
. 2
m
j i ij e
SSJ Hr
r
. 2-=
In the presence of a field H
o, the Hamiltonian becomes
Heisenberg gave the exchange Hamiltonian the form
which is isotropic.
Ferromagnetism
Assuming nearest neighbour exchange interaction, J
ij= J & S
i S
we get the Hamiltonian
H = - gm
B
H
0
NS - ½ {2JNZ(S)
2
}
,
where Z = no. of N.N.s; N = no. of atoms/vol.
Simple algebra gives, replacing
H
0
by
H
0
+ H
E
(= lM),
using
T
C= C l,
C= N g
2
m
B
2
S(S+1)/3k
B
&
M = Ngm
B
S
in
H = -M(H
0
+ ½ H
E
),
llll= (2JZ/Ng
2
mmmm
B
2
)
&
J = (3k
B
T
C
/2ZS(S+1))
~ 1.2 x 10
-2
eV
for Fe with S = 1.
Taking S = ½ , one gets
(k
B
T
C
/ ZJ) = 0.5
from mean-field theory
= 0.325
from Rushbrooke & Wood theory.
j
ji
ji
i ij
i
Z
i B
SSJ S H g Hr
r
∑
∑
¹
- -=
,
0
. 2
m
j i ij e
SSJ Hr
r
. 2-=
In the presence of a field H
o, the Hamiltonian becomes
Heisenberg gave the exchange Hamiltonian the form
which is isotropic.
31
Blochs Spin-wave Theory (T <<Tc)
In the ground state of a FM (at 0 K) the atoms at d ifferent lattice
sites have their maximum z-component of spin,
S
j
z
= S
.
As T increases the system is excited out of its gro und state
giving rise to sinusoidal
Spin Waves,
as shown below.
Ferromagnetism
The amplitude of this wave at j
th
site is proportional to
S-S
j
z.
The energy of these spin wavesis quantized and the energy quantum
is called a
magnon.
Blochs Spin-wave Theory (T <<Tc)
In the ground state of a FM (at 0 K) the atoms at d ifferent lattice
sites have their maximum z-component of spin,
S
j
z
= S
.
As T increases the system is excited out of its gro und state
giving rise to sinusoidal
Spin Waves,
as shown below.
Ferromagnetism
The amplitude of this wave at j
th
site is proportional to
S-S
j
z.
The energy of these spin wavesis quantized and the energy quantum
is called a
magnon.
32
Ferromagnetism
Ferromagnetism
33
Ferromagnetism
, satisfying H y= E y,
↑
Ferromagnetism
, satisfying H y= E y,
↑
34
Ferromagnetism
The next 9 slides could be summarized as follows:
The spin raising and lowering operators S
j
±
= S
j
x
± i S
j
y
operate on the
eigenstates of the spin deviation operator h
j. The spin raising operator raises the z-
component of the spin and hence lowers the spin deviation and vice-versa. Then one
defines Boson creation (a
j
+
) and annihilation (a
j) operators as well as the number
operator (a
j
+
a
j) in terms of S
j
±
and S
j
z
. These relations are called Holstein-Primakoff
transformation. Then one makes a transformation of S
j
±
and S
j
z
in terms of magnon
creation and annihilation operators, b
k
+
and b
k
. Kis found from the periodic boundary
condition. b
k
+
b
k
is the magnon number operator with eigenvalue n
k
for the magnon state
K.
Finally, the energy needed to excite a magnon in the state K, in the case of a
simple cubic crystal of nearest neighbor distance a is
(h/2p)ω
k
= g m
B
H
0
+ 2 J S a
2
k
2
.
This is the magnon dispersion relation where E ~ k
2
like that of free electrons.
Ferromagnetism
The next 9 slides could be summarized as follows:
The spin raising and lowering operators S
j
±
= S
j
x
± i S
j
y
operate on the
eigenstates of the spin deviation operator h
j. The spin raising operator raises the z-
component of the spin and hence lowers the spin deviation and vice-versa. Then one
defines Boson creation (a
j
+
) and annihilation (a
j) operators as well as the number
operator (a
j
+
a
j) in terms of S
j
±
and S
j
z
. These relations are called Holstein-Primakoff
transformation. Then one makes a transformation of S
j
±
and S
j
z
in terms of magnon
creation and annihilation operators, b
k
+
and b
k
. Kis found from the periodic boundary
condition. b
k
+
b
k
is the magnon number operator with eigenvalue n
k
for the magnon state
K.
Finally, the energy needed to excite a magnon in the state K, in the case of a
simple cubic crystal of nearest neighbor distance a is
(h/2p)ω
k
= g m
B
H
0
+ 2 J S a
2
k
2
.
This is the magnon dispersion relation where E ~ k
2
like that of free electrons.
35
Ferromagnetism
Ferromagnetism
36
Ferromagnetism
Ferromagnetism
37
Ferromagnetism
Ferromagnetism
38
Ferromagnetism
Ferromagnetism
39
Ferromagnetism
Ferromagnetism
40
Ferromagnetism
Ferromagnetism
41
Ferromagnetism
Ferromagnetism
42
Ferromagnetism
Ferromagnetism
43
Ferromagnetism
Ferromagnetism
44
Ferromagnetism
Ferromagnetism
45
Ferromagnetism
D(ω) d ω= g(k) d
3
k
Ferromagnetism
D(ω) d ω= g(k) d
3
k
46
Ferromagnetism
Ferromagnetism
47
Ferromagnetism
Ferromagnetism
48
Ferromagnetism
Ferromagnetism
49
Ferromagnetism
Ferromagnetism
50
Ferromagnetism
Ferromagnetism
51
Using the spin-wave dispersion relation for a cubic
system
(h/2p)w
k
= gm
B
H
0
+ 2Jsa
2
k
2
,
one calculates the number of
magnons, n
in thermal
equilibrium at temperature T to be proportional to T
3/2
and hence
M(T) = M(0) [1+ A
Z
{3/2, T
g
/T} T
3/2
+ B
Z
{5/2,T
g
/T}T
5/2
]. This is the famous Blochs T
3/2
.
Similarly specific heat contribution of
magnons
is also
~
T
3/2
.
Both are in very good agreement with experiments.
Ferromagnetism
Spontaneous magnetization, M
S
of a ferromagnet as a
function of reduced
temperature
T/T
C.
Blochs T
3/2
law
holds only
for
T/T
C << 1.
Ferromagnetism Ferromagnetism
Summary of spin-wave theory
Using the spin-wave dispersion relation for a cubic
system
(h/2p)w
k
= gm
B
H
0
+ 2Jsa
2
k
2
,
one calculates the number of
magnons, n
in thermal
equilibrium at temperature T to be proportional to T
3/2
and hence
M(T) = M(0) [1+ A
Z
{3/2, T
g
/T} T
3/2
+ B
Z
{5/2,T
g
/T}T
5/2
]. This is the famous Blochs T
3/2
.
Similarly specific heat contribution of
magnons
is also
~
T
3/2
.
Both are in very good agreement with experiments.
Ferromagnetism
Spontaneous magnetization, M
S
of a ferromagnet as a
function of reduced
temperature
T/T
C.
Blochs T
3/2
law
holds only
for
T/T
C << 1.
Ferromagnetism Ferromagnetism
Summary of spin-wave theory
52
Ferromagnetism
Hysteresis curve M(H)
M
S M
r H
C
Permanent magnets:
Ferrites: low cost, classical industrial needs,
H
C = 0.4 T.
NdFeB: miniaturization, actuators for read/
write heads, H
C = 1.5 T, very
high energy product = 300 kJ/m
3
.
AlNiCo & SmCo: Higher price/energy, used
if irreplaceable.
Ferromagnetism
Hysteresis curve M(H)
M
S M
r H
C
Permanent magnets:
Ferrites: low cost, classical industrial needs,
H
C = 0.4 T.
NdFeB: miniaturization, actuators for read/
write heads, H
C = 1.5 T, very
high energy product = 300 kJ/m
3
.
AlNiCo & SmCo: Higher price/energy, used
if irreplaceable.
53
Ferromagnetism
Soft magnetic materials
Used in channeling magnetic induction flux: Large M
S & initial permeabilityμ
Minimum energy loss (area of M-H loop).
Examples: Fe-Si alloys for transformers, FeB amorphous alloys for tiny transformer
cores, Ferrites for high frequency & low power applicati ons, YIG
in microwave range, Permalloys in AMR & many multilayer devices.
Stainless steel (non-magnetic ???): Very interesting magnetic phases
from competing FM/AFM interactions.
Ni Fe Cr Alloys
Ferromagnetism
Soft magnetic materials
Used in channeling magnetic induction flux: Large M
S & initial permeabilityμ
Minimum energy loss (area of M-H loop).
Examples: Fe-Si alloys for transformers, FeB amorphous alloys for tiny transformer
cores, Ferrites for high frequency & low power applicati ons, YIG
in microwave range, Permalloys in AMR & many multilayer devices.
Stainless steel (non-magnetic ???): Very interesting magnetic phases
from competing FM/AFM interactions.
Ni Fe Cr Alloys
54
Collective-electron or band theory of ferromagnetism
The localized-moment theory breaks down in two
aspects. The magnetic moment on each atom or ion
should be the same in both the ferromagnetic and
paramagnetic phases & its value should correspond to
an integral number of m
B.
None are observed
experimentally. Hence the need of a band theory or
collective-electron theory. In Fe, Ni, and Co, the Fermi
energy lies in a region of overlapping 3d and 4s ba nds
as shown in Fig. 1. The rigid-band modelassumes that
the structures of the 3d and 4s bands do not change
markedly across the 3d series and so any difference s in
electronic structure are caused entirely by changes in
the Fermi energy (E
F
). This is supported by detailed
band structure calculations.
Fig. 1: Schematic 3d and 4s densities
of states in transition metals. The
positions of the Fermi levels (E
F
) in
Zn, Cu, Ni, Co, Fe, and Mn are shown.
Collective-electron or band theory of ferromagnetism
The localized-moment theory breaks down in two
aspects. The magnetic moment on each atom or ion
should be the same in both the ferromagnetic and
paramagnetic phases & its value should correspond to
an integral number of m
B.
None are observed
experimentally. Hence the need of a band theory or
collective-electron theory. In Fe, Ni, and Co, the Fermi
energy lies in a region of overlapping 3d and 4s ba nds
as shown in Fig. 1. The rigid-band modelassumes that
the structures of the 3d and 4s bands do not change
markedly across the 3d series and so any difference s in
electronic structure are caused entirely by changes in
the Fermi energy (E
F
). This is supported by detailed
band structure calculations.
Fig. 1: Schematic 3d and 4s densities
of states in transition metals. The
positions of the Fermi levels (E
F
) in
Zn, Cu, Ni, Co, Fe, and Mn are shown.
55
Collective-electron or band theory of ferromagnetism
The exchange interaction shifts the energy of the 3d band
for electrons with one spin direction relative to that with
opposite spin. If E
F
lies within the 3d band, then the
displacement will lead to more electrons of the lower-energy
spin direction and hence a spontaneous magnetic moment in the
ground state (Fig. 2). The exchange splitting is negligible for the
4s electrons, but significant for 3d electrons.
⇒
In Ni, the displacement is so strong that one 3d sub-band is
filled with 5 electrons, and the other contains all 0.54 holes. So
the saturation magnetization of Ni is M
S= 0.54Nm
B, where N is
the total number of Ni atoms.
⇒⇒⇒⇒the magnetic moments of
the 3d metals are not integral number of mmmm
B!
⇒
In Cu, E
F
lies above the 3d band. Since both the 3d sub-
bands are filled, and the 4s band has no exchange-splitting, the
numbers of up- and down-spin electrons are equal. Hence no
magnetic moment.
Fig. 2: Schematic 3d and 4s up-
and down-spin densities of states
in a transition metal with
exchange interaction included.
⇒
In Zn, both the 3d and 4s bands are filled and hence no magnetic moment.
⇒
In lighter transition metals, Mn, Cr, etc., the exchange interac tion is much
weaker & ferromagnetism is not observed.
Collective-electron or band theory of ferromagnetism
The exchange interaction shifts the energy of the 3d band
for electrons with one spin direction relative to that with
opposite spin. If E
F
lies within the 3d band, then the
displacement will lead to more electrons of the lower-energy
spin direction and hence a spontaneous magnetic moment in the
ground state (Fig. 2). The exchange splitting is negligible for the
4s electrons, but significant for 3d electrons.
⇒
In Ni, the displacement is so strong that one 3d sub-band is
filled with 5 electrons, and the other contains all 0.54 holes. So
the saturation magnetization of Ni is M
S= 0.54Nm
B, where N is
the total number of Ni atoms.
⇒⇒⇒⇒the magnetic moments of
the 3d metals are not integral number of mmmm
B!
⇒
In Cu, E
F
lies above the 3d band. Since both the 3d sub-
bands are filled, and the 4s band has no exchange-splitting, the
numbers of up- and down-spin electrons are equal. Hence no
magnetic moment.
Fig. 2: Schematic 3d and 4s up-
and down-spin densities of states
in a transition metal with
exchange interaction included.
⇒
In Zn, both the 3d and 4s bands are filled and hence no magnetic moment.
⇒
In lighter transition metals, Mn, Cr, etc., the exchange interac tion is much
weaker & ferromagnetism is not observed.
56
Band structure of 3d magnetic metals & Cu
M
S= 2.2 mmmm
BM
S= 1.7 mmmm
B
M
S= 0.6 mmmm
BM
S= 0.0 mmmm
B
Band structure of 3d magnetic metals & Cu
M
S= 2.2 mmmm
BM
S= 1.7 mmmm
B
M
S= 0.6 mmmm
BM
S= 0.0 mmmm
B
57
The Slater-Pauling curve(1930)
The collective-electron and rigid-band models are f urther supported by the well-known plot
known as the Slater-Pauling curve. They calculated the saturation magnetization as a
continuous function of the number of 3d and 4s vale nce electrons per atom across the 3d
series. They used the rigid-band model, and obtaine d a linear increase in saturation
magnetization from Cr to Fe, then a linear decrease , reaching zero magnetization at an
electron density between Ni and Cu. Their calculat ed values agree well with those
measured for Fe, Co, and Ni, as well as Fe-Co, Co- Ni, and Ni-Cu alloys. The alloys on the
right-hand side are strong ferromagnets. The slope of the branch on the right is θ1 when the
charge difference of the constituent atoms is small , Z ~ < 2. The multiple branches (on left)
with slope ≈ 1, as expected for rigid bands, are for alloys of late 3d elements with early 3d
elements for which the 3d-states lie well above the Fermi level of the ferromagnetic host &
we have to invoke Friedels virtual bound states.
The average atomic moment is plotted against the nu mber of valence (3d + 4s) electrons.
The Slater-Pauling curve(1930)
The collective-electron and rigid-band models are f urther supported by the well-known plot
known as the Slater-Pauling curve. They calculated the saturation magnetization as a
continuous function of the number of 3d and 4s vale nce electrons per atom across the 3d
series. They used the rigid-band model, and obtaine d a linear increase in saturation
magnetization from Cr to Fe, then a linear decrease , reaching zero magnetization at an
electron density between Ni and Cu. Their calculat ed values agree well with those
measured for Fe, Co, and Ni, as well as Fe-Co, Co- Ni, and Ni-Cu alloys. The alloys on the
right-hand side are strong ferromagnets. The slope of the branch on the right is θ1 when the
charge difference of the constituent atoms is small , Z ~ < 2. The multiple branches (on left)
with slope ≈ 1, as expected for rigid bands, are for alloys of late 3d elements with early 3d
elements for which the 3d-states lie well above the Fermi level of the ferromagnetic host &
we have to invoke Friedels virtual bound states.
The average atomic moment is plotted against the nu mber of valence (3d + 4s) electrons.
58
Magnetism
FERRIMAGNET ANTIFERROMAGNET FERROMAGNET
Magnetism
FERRIMAGNET ANTIFERROMAGNET FERROMAGNET
59
Spin glass
ω
To distinguish between an antiferromagnet, a paramagnet and a spin glass.
t = 0
t = ∞
Neutron scattering gives m = 0 but Q ¹¹¹¹0 for T < T
g
.
Spin glass
ω
To distinguish between an antiferromagnet, a paramagnet and a spin glass.
t = 0
t = ∞
Neutron scattering gives m = 0 but Q ¹¹¹¹0 for T < T
g
.
60
The research on magnetism till late 20th century fo cused on its basic
understanding and also on applications as soft and hard magnetic materials.
In 1980s it was observed that surface and interfac e properties deviate
considerably from those of the bulk. With the adven t of novel techniques
e-beam evaporation
Ion-beam sputtering
Magnetron sputtering
Molecular beam epitaxy (MBE)
excellent
magnetic thin films
could be prepared with tailor-made properties which
can not be obtained from bulk materials alone. Magn etic recording industry was
also going for miniaturization and thin film techno logy fitted their requirement.
R & D in Magnetic Materials after 1973
Then came, for the first time, the industrial appli cation of electronic properties
which depend on the spin of the electrons giving ri se to the so-called
Giant
Magnetoresistance(GMR),
discovered in 1986.
Since early 80s the
Anisotropic Magnetoresistance effect (AMR)
has been used
in a variety of applications, especially in read he ads in magnetic tapes and later on
in hard-disk systems, gradually replacing the older inductive thin film heads. Now,
GMR recording heads offer a stiff competition to th ose using AMR effects.
So, what is Magnetoresistance ???
The research on magnetism till late 20th century fo cused on its basic
understanding and also on applications as soft and hard magnetic materials.
In 1980s it was observed that surface and interfac e properties deviate
considerably from those of the bulk. With the adven t of novel techniques
e-beam evaporation
Ion-beam sputtering
Magnetron sputtering
Molecular beam epitaxy (MBE)
excellent
magnetic thin films
could be prepared with tailor-made properties which
can not be obtained from bulk materials alone. Magn etic recording industry was
also going for miniaturization and thin film techno logy fitted their requirement.
R & D in Magnetic Materials after 1973
Then came, for the first time, the industrial appli cation of electronic properties
which depend on the spin of the electrons giving ri se to the so-called
Giant
Magnetoresistance(GMR),
discovered in 1986.
Since early 80s the
Anisotropic Magnetoresistance effect (AMR)
has been used
in a variety of applications, especially in read he ads in magnetic tapes and later on
in hard-disk systems, gradually replacing the older inductive thin film heads. Now,
GMR recording heads offer a stiff competition to th ose using AMR effects.
So, what is Magnetoresistance ???
61
I. Ordinary or normal magnetoresistance (OMR)
Due to the Lorentz force acting on the electron tra jectories in a
magnetic field. MR ~ B
2
at low fields. MR is significant only at low
temperatures for pure materials at high filds.
II. Anisotropic magnetoresistance (AMR) in a ferrom agnet
In low field LMR is positive and TMR is
negative.
Negative MR after saturation due to quenching
of spin-waves by magnetic field.
FAR is an inherent property of FM
materials originating from spin-orbit
interaction of conduction electrons with
localized spins.
FAR(ferromagnetic anisotropy of resistivity )
= (Dr
//s
-Dr
^s
)/ r
//s
Magnetoresistance
I. Ordinary or normal magnetoresistance (OMR)
Due to the Lorentz force acting on the electron tra jectories in a
magnetic field. MR ~ B
2
at low fields. MR is significant only at low
temperatures for pure materials at high filds.
II. Anisotropic magnetoresistance (AMR) in a ferrom agnet
In low field LMR is positive and TMR is
negative.
Negative MR after saturation due to quenching
of spin-waves by magnetic field.
FAR is an inherent property of FM
materials originating from spin-orbit
interaction of conduction electrons with
localized spins.
FAR(ferromagnetic anisotropy of resistivity )
= (Dr
//s
-Dr
^s
)/ r
//s
Magnetoresistance
62
RKKYinteraction~ cos(2k
f
r)/(2k
f
r)
3 .
Establishedbyexperimentsonlightscatteringbyspinwaves.
[ P. Grünberg et al., Phys. Rev. Lett. 57, 2442(1986).]
At high fields the spins align
with the field (saturating at H
sat)
and the resistance is reduced.
Magnetoresistance is negative!
At low fields the interlayer antiferromgnetic
coupling causes the spins in adjacent layers
to be antiparallel and the resistance is high
III. Giant Magnetoresistance (GMR)
Fe-Cr is a lattice matched pair : Exchange coupling of ferromagnetic Fe
layers through Cr spacers gives rise to a negative giant magnetoresistance
(GMR) with the application of a magnetic field.
Introduction
Magnetoresistance
RKKYinteraction~ cos(2k
f
r)/(2k
f
r)
3 .
Establishedbyexperimentsonlightscatteringbyspinwaves.
[ P. Grünberg et al., Phys. Rev. Lett. 57, 2442(1986).]
At high fields the spins align
with the field (saturating at H
sat)
and the resistance is reduced.
Magnetoresistance is negative!
At low fields the interlayer antiferromgnetic
coupling causes the spins in adjacent layers
to be antiparallel and the resistance is high
III. Giant Magnetoresistance (GMR)
Fe-Cr is a lattice matched pair : Exchange coupling of ferromagnetic Fe
layers through Cr spacers gives rise to a negative giant magnetoresistance
(GMR) with the application of a magnetic field.
Introduction
Magnetoresistance
63
Fe-Cr
Magnetoresistance is defined by
.%100
),0(
),0( ),(
´
-
=
T
T TH
MR
r
r
r
(1)
Giant Magnetoresistance (GMR)
Giant Magnetoresistance (GMR)
Bulk scatteringInterface scattering
[ Magnetic Multilayers and Giant Magnetoresistance, Ed. by Uwe Hartmann,
Springer Series in Surface Sciences, Vol. 37, Berlin (1999).]
Fe-Cr
Magnetoresistance is defined by
.%100
),0(
),0( ),(
´
-
=
T
T TH
MR
r
r
r
(1)
Giant Magnetoresistance (GMR)
Giant Magnetoresistance (GMR)
Bulk scatteringInterface scattering
[ Magnetic Multilayers and Giant Magnetoresistance, Ed. by Uwe Hartmann,
Springer Series in Surface Sciences, Vol. 37, Berlin (1999).]
64
Sample structure
Cr(t Å)
30 bi-layers
of Fe/Cr
Sample details
Si/Cr(50Å)/[Fe(20Å)/Cr(tÅ)]´30/Cr((50-t)Å) Varying Cr thickness t= 6, 8, 10, 12, and 14 Å
Cr ( 50-t )Å
Cr 50 Å
Si Substrate
Fe/Cr multilayers prepared by
ion-beam sputtering technique
.
Ar and Xe ions were used.
Beam current 20 mA /30 mA and energy 900eV/1100eV.
Fe(20 Å)
bi-layer
GMR in Fe-Cr multilayers
Sample structure
Cr(t Å)
30 bi-layers
of Fe/Cr
Sample details
Si/Cr(50Å)/[Fe(20Å)/Cr(tÅ)]´30/Cr((50-t)Å) Varying Cr thickness t= 6, 8, 10, 12, and 14 Å
Cr ( 50-t )Å
Cr 50 Å
Si Substrate
Fe/Cr multilayers prepared by
ion-beam sputtering technique
.
Ar and Xe ions were used.
Beam current 20 mA /30 mA and energy 900eV/1100eV.
Fe(20 Å)
bi-layer
GMR in Fe-Cr multilayers
65
GMR vs. H for 2 samples.
⇒
Negative GMR of 21 % at 10 K and 8 % at 300 K with H
sat»13 kOe.
⇒
Hardly any hysteresis
⇒
strong AF coupling.
GMR in Fe-Cr multilayers
GMR vs. H for 2 samples.
⇒
Negative GMR of 21 % at 10 K and 8 % at 300 K with H
sat»13 kOe.
⇒
Hardly any hysteresis
⇒
strong AF coupling.
GMR in Fe-Cr multilayers
66
ρ
H
= E
y
/ j
x
= R
0
B
z
+
0
R
S
M ,
where R
0
= ordinary Hall constant (OHC),
R
S
= extra-ordinary or spontaneous Hall
constant (EHC), B = magnetic induction
and M= magnetization.
In ferromagnetic metals and alloys
ω
0 M
s
ω
0R
s
M
s
Slope=
R
0
B
ρ
H
ρ
H
vs. B of a ferromagnet.
Hall Effect
R
0
( Ordinary Hall constant ) : In a 2-band model (as in Fe ) consisting of electrons and holes
R
0
= (s
e
2
/ n
e
e
e
+ s
h
2
/ n
h
e
h
) / (s
e
+ s
h
)
2
,
where s= conductivity, ne = charge density, e
e
<0. ω
Sign of R
0
determines relative conductivity.R
0
>0 for Fe at 300 K.
ω
Eq.(4) reduces to R
0
= 1/ne for a single band.
Eq.(4)
ρ
H
= E
y
/ j
x
= R
0
B
z
+
0
R
S
M ,
where R
0
= ordinary Hall constant (OHC),
R
S
= extra-ordinary or spontaneous Hall
constant (EHC), B = magnetic induction
and M= magnetization.
In ferromagnetic metals and alloys
ω
0 M
s
ω
0R
s
M
s
Slope=
R
0
B
ρ
H
ρ
H
vs. B of a ferromagnet.
Hall Effect
R
0
( Ordinary Hall constant ) : In a 2-band model (as in Fe ) consisting of electrons and holes
R
0
= (s
e
2
/ n
e
e
e
+ s
h
2
/ n
h
e
h
) / (s
e
+ s
h
)
2
,
where s= conductivity, ne = charge density, e
e
<0. ω
Sign of R
0
determines relative conductivity.R
0
>0 for Fe at 300 K.
ω
Eq.(4) reduces to R
0
= 1/ne for a single band.
Eq.(4)
67
R
S
( Extra-ordinary or spontaneous Hall constant ):
a) Classical Smit asymmetric scattering(AS).
b) Non-classical transport (side-jump).
ω
Boltzmann Eq. is correct to the lowest order in ( «1) (t
= relaxation time),
true for pure metals and dilute alloys at low temperatures.
ω
R
S
causedby AS of electrons by impurities in the presence of spin-orbit
interaction in a ferromagnet .
ω
Boltzmann Eq.
a) Classical Smit asymmetric scattering (AS)
,
where [s] = relaxation frequency tensor ~ impurity concentration.
ρ
Off-diagonal elements describe AS proportional to M.
ρ
Diagonal elements give Ohmic r.
ω
Boundary condition: j
y= 0
⇒⇒⇒⇒
ρ
H
and ρ.
ω
R
S
= aρ.
[L. Berger, Phys. Rev. 177, 790(1969).]
F
E
t
h
[ ]
js
e
m
Bj Een
dt
pdr rr r
r
- ´+ =
Introduction (theory)
Hall Effect
R
S
( Extra-ordinary or spontaneous Hall constant ):
a) Classical Smit asymmetric scattering(AS).
b) Non-classical transport (side-jump).
ω
Boltzmann Eq. is correct to the lowest order in ( «1) (t
= relaxation time),
true for pure metals and dilute alloys at low temperatures.
ω
R
S
causedby AS of electrons by impurities in the presence of spin-orbit
interaction in a ferromagnet .
ω
Boltzmann Eq.
a) Classical Smit asymmetric scattering (AS)
,
where [s] = relaxation frequency tensor ~ impurity concentration.
ρ
Off-diagonal elements describe AS proportional to M.
ρ
Diagonal elements give Ohmic r.
ω
Boundary condition: j
y= 0
⇒⇒⇒⇒
ρ
H
and ρ.
ω
R
S
= aρ.
[L. Berger, Phys. Rev. 177, 790(1969).]
F
E
t
h
[ ]
js
e
m
Bj Een
dt
pdr rr r
r
- ´+ =
Introduction (theory)
Hall Effect
68
b) Non-classicaltransport (side-jump mechanism) ω
is not small
⇒
Concentrated and disordered alloys, high temperatures.
[R. Karplus and J. M. Luttinger, Phys. Rev. 95, 115 4 (1954).]
ω
Calculation: Free electron plane wave (e
i kx
) is scattered by a short-range
square well impurity potential
with V(r)= 0 for r > R & V(r)= V
0for r < R. Using Born approx. one finds
a side-wise displacement of the wave-packet y ~ 0.1-0.2 nm
(side-jump).
F
E
t
h
Z Z
SL
r
V
r cm
rV
m
H
¶
¶
+ + Ñ -=
1
2
1
)(
2
2 2
2
2
h
ρTransport theory for arbitrary w
ct= e B t/ m, w
c= cyclotron frequency gives
ρ
ρ
H
= R
0
B
z+
0
R
S
M
,
where R
S
= b r
2
.
Combining Eqs.(3) & (4) for the most general case one gets
R
S
= a r+ b r
2
.
[L. Berger and G. Bergmann , in the Hall effect and its applications, edited by C. L.
Chien and C. R. Westgate (Plenum, New York , 1980), p.55 and references therein.]
Introduction (theory)
Hall Effect
b) Non-classicaltransport (side-jump mechanism) ω
is not small
⇒
Concentrated and disordered alloys, high temperatures.
[R. Karplus and J. M. Luttinger, Phys. Rev. 95, 115 4 (1954).]
ω
Calculation: Free electron plane wave (e
i kx
) is scattered by a short-range
square well impurity potential
with V(r)= 0 for r > R & V(r)= V
0for r < R. Using Born approx. one finds
a side-wise displacement of the wave-packet y ~ 0.1-0.2 nm
(side-jump).
F
E
t
h
Z Z
SL
r
V
r cm
rV
m
H
¶
¶
+ + Ñ -=
1
2
1
)(
2
2 2
2
2
h
ρTransport theory for arbitrary w
ct= e B t/ m, w
c= cyclotron frequency gives
ρ
ρ
H
= R
0
B
z+
0
R
S
M
,
where R
S
= b r
2
.
Combining Eqs.(3) & (4) for the most general case one gets
R
S
= a r+ b r
2
.
[L. Berger and G. Bergmann , in the Hall effect and its applications, edited by C. L.
Chien and C. R. Westgate (Plenum, New York , 1980), p.55 and references therein.]
Introduction (theory)
Hall Effect
t = 0t = 0
t«0
t »0
t«0
Y Y
t »0
d
y
S
a)b)
Skew scattering Side jump scatterin g
[L. Berger and G. Bergmann , in the Hall effect and its applications, edited by C. L. Chien
and C. R. Westgate (Plenum, New York , 1980), p.55 and references therein.]
Hall Effect
69
t«0
t »0
t«0
Y Y
t »0
d
y
S
a)b)
Skew scattering Side jump scatterin g
[L. Berger and G. Bergmann , in the Hall effect and its applications, edited by C. L. Chien
and C. R. Westgate (Plenum, New York , 1980), p.55 and references therein.]
Hall Effect
69
Hall effect in GMR multilayers
ω
All the theories discussed earlier are valid for ho mogeneous ferromagnets.
ω
Scaling law is valid only in the local limit; the m ean free path λ« d, the
layer thickness. It is invalid in the long mean fre e path limit; λ»d. ω
Zhang had shown the failure of scaling law in com posite magnetic-
nonmagnetic systems: ωThe standard boundary condition used for calculatin g ρ
H
, j
y
(z) = 0, is not
valid here for all z, but j
y
(z), integrated over z, is zero.
ωThe two-point local Hall conductivity is given by
σ
yx
(z, z') µ(m λ
SO
M
Z
σ
CIP
(z, z'))/τ(z), (3)
where σ
CIP
(z, z') is theCIP two-point Ohmic conductivity.
ωTo get σ
yx
, integrate σ
yx
(z, z') over z and z' and sum over the
spin variables.
[S. Zhang , Phys. Rev. B 51 , 3632(1995).]
Theory of Hall effect in inhomogeneous ferromagnets Theory of Hall effect in inhomogeneous ferromagnets
70
ω
All the theories discussed earlier are valid for ho mogeneous ferromagnets.
ω
Scaling law is valid only in the local limit; the m ean free path λ« d, the
layer thickness. It is invalid in the long mean fre e path limit; λ»d. ω
Zhang had shown the failure of scaling law in com posite magnetic-
nonmagnetic systems: ωThe standard boundary condition used for calculatin g ρ
H
, j
y
(z) = 0, is not
valid here for all z, but j
y
(z), integrated over z, is zero.
ωThe two-point local Hall conductivity is given by
σ
yx
(z, z') µ(m λ
SO
M
Z
σ
CIP
(z, z'))/τ(z), (3)
where σ
CIP
(z, z') is theCIP two-point Ohmic conductivity.
ωTo get σ
yx
, integrate σ
yx
(z, z') over z and z' and sum over the
spin variables.
[S. Zhang , Phys. Rev. B 51 , 3632(1995).]
Theory of Hall effect in inhomogeneous ferromagnets Theory of Hall effect in inhomogeneous ferromagnets
70
ωFor
homogeneous magnetic materials
,σ
CIP
(z, z'), is proportional to τ(z)
and ρ
yx
is simply proportional to the square of the ordinar y resistivity ρ
2
.
ωFor inhomogeneous magnetic systems:
σ
yx
is found in terms of average relaxation times ( τ) and thickness (t)
of the components
σ
yx
= λ
SO
M
Z
A t
m
∑
s
(t
m
+ t
nm
τ
m
s
/ τ
nm
)
-1
. (4)
ω
Thus Hall conductivity depends on the ratio of rel axation times &
ρ
xy
~ ρ
2
is no more valid. Experiments give n between ~ 1 to ~ 4.
Finally, Zhang had shown that n could be < 2, > 2, or = 2 if MFP
of only the magnetic layer or the non-magnetic laye r or
both but in
a fixed ratio, is temperature dependent.
Theory of Hall effect in inhomogeneous ferromagnets
71
homogeneous magnetic materials
,σ
CIP
(z, z'), is proportional to τ(z)
and ρ
yx
is simply proportional to the square of the ordinar y resistivity ρ
2
.
ωFor inhomogeneous magnetic systems:
σ
yx
is found in terms of average relaxation times ( τ) and thickness (t)
of the components
σ
yx
= λ
SO
M
Z
A t
m
∑
s
(t
m
+ t
nm
τ
m
s
/ τ
nm
)
-1
. (4)
ω
Thus Hall conductivity depends on the ratio of rel axation times &
ρ
xy
~ ρ
2
is no more valid. Experiments give n between ~ 1 to ~ 4.
Finally, Zhang had shown that n could be < 2, > 2, or = 2 if MFP
of only the magnetic layer or the non-magnetic laye r or
both but in
a fixed ratio, is temperature dependent.
Theory of Hall effect in inhomogeneous ferromagnets
71
Motivation
An interplay between magnetic
state and electrical transport
GMR !
To investigate Hall effect in inhomogeneous magnetic systems
like Fe/Cr
multilayers for better understanding of GMR
Both magnetism &
electrical transport are involved
.
Hall effect !
Q1
.Why is Hall effect important in GMR systems ?
Q2.
Does any scaling relation exist for GMR systems?
Is it the same or is it different from that of homo geneous ferromagnets ?
Hall effect
72
An interplay between magnetic
state and electrical transport
GMR !
To investigate Hall effect in inhomogeneous magnetic systems
like Fe/Cr
multilayers for better understanding of GMR
Both magnetism &
electrical transport are involved
.
Hall effect !
Q1
.Why is Hall effect important in GMR systems ?
Q2.
Does any scaling relation exist for GMR systems?
Is it the same or is it different from that of homo geneous ferromagnets ?
Hall effect
72
Hall effect and Transverse/Longitudinal Magnetoresistance measured
by 5-probe and 4-probe dc methods, respectively using
a home-made variable temperature cryostat and a 7 T
superconducting magnet
Accuracy of V
H»1 in 3000
,, ,, V
R»1 in 10
5
.
Magnetization measured with a Quantum Design SQUID
Magnetometer.
Experimental Techniques
73
by 5-probe and 4-probe dc methods, respectively using
a home-made variable temperature cryostat and a 7 T
superconducting magnet
Accuracy of V
H»1 in 3000
,, ,, V
R»1 in 10
5
.
Magnetization measured with a Quantum Design SQUID
Magnetometer.
Experimental Techniques
73
0 1 2 3 4 5 6 7
0.0
1.5
3.0
4.5
6.0
7.5
9.0
Peak position
T = 300K
T = 4.2K
F8 (30L)
Hall Resistivity (10
-9
Wm)
Field (tesla)
Hall resistivity ( r
H
) vs. applied magnetic field ( m
0
H
applied
).
Q 3
. Why do the humps appear in the EHE just before th e
saturation field in these GMR systems?
Hall Effect data Hall Effect data
In Hall geometry B= m
0
[H
applied
+ (1-N) M], N=demagnetization factor.
74
0.0
1.5
3.0
4.5
6.0
7.5
9.0
Peak position
T = 300K
T = 4.2K
F8 (30L)
Hall Resistivity (10
-9
Wm)
Field (tesla)
Hall resistivity ( r
H
) vs. applied magnetic field ( m
0
H
applied
).
Q 3
. Why do the humps appear in the EHE just before th e
saturation field in these GMR systems?
Hall Effect data Hall Effect data
In Hall geometry B= m
0
[H
applied
+ (1-N) M], N=demagnetization factor.
74
0 1 2 3 4 5 6 7
-0.32
-0.24
-0.16
-0.08
0.00
T = 300K
T = 4.2K
F8 (30L)
Magnetoresistance Ratio
Field (tesla)
MR ratio vs. applied magnetic field ( m
0
H
applied
).
Magnetoresistance Magnetoresistancedata
75
-0.32
-0.24
-0.16
-0.08
0.00
T = 300K
T = 4.2K
F8 (30L)
Magnetoresistance Ratio
Field (tesla)
MR ratio vs. applied magnetic field ( m
0
H
applied
).
Magnetoresistance Magnetoresistancedata
75
Explanation of anomalous humps in EHE lies in itscorrelation with GMR
Plots of R
s
M, M and ragainstapplied magnetic field ( m
0
H
applied
).
All show saturation around 3 tesla.
Hall Effect Hall Effect
76
Plots of R
s
M, M and ragainstapplied magnetic field ( m
0
H
applied
).
All show saturation around 3 tesla.
Hall Effect Hall Effect
76
5.6 5.8 6.0 6.2
0.0
0.3
0.6
0.9
1.2
1.5F10(30L)
n = 3.45 ± 0.11
F10(10L)
n = 3.07 ± 0.04
R
s~
r
n
ln R
s
ln
r
(at B = 0)
Plot of ln R
s
vs. ln r(rbeing the resistivity at B = 0).
Q 4.
Why has the scaling law failed ?
Scaling relation Scaling relation
Fits to R
S
= a r+ b r
n
with a = 0 yield nmuch larger than 2 as found earli er in
Fe-Cr, Co-Cu, and Co-Ag films.
[Y. Aoki et al., J. Magn. Magn. Mater. 126 , 448
(1993); T. Lucin´ski et al., J. Magn. Magn. Mater. 160 , 347 (1996); V. Korenivski et
al., Phys. Rev. B 53 , R 11 938 (1996).]
The exponent is larger for samples having higher GM R !!!
77
0.0
0.3
0.6
0.9
1.2
1.5F10(30L)
n = 3.45 ± 0.11
F10(10L)
n = 3.07 ± 0.04
R
s~
r
n
ln R
s
ln
r
(at B = 0)
Plot of ln R
s
vs. ln r(rbeing the resistivity at B = 0).
Q 4.
Why has the scaling law failed ?
Scaling relation Scaling relation
Fits to R
S
= a r+ b r
n
with a = 0 yield nmuch larger than 2 as found earli er in
Fe-Cr, Co-Cu, and Co-Ag films.
[Y. Aoki et al., J. Magn. Magn. Mater. 126 , 448
(1993); T. Lucin´ski et al., J. Magn. Magn. Mater. 160 , 347 (1996); V. Korenivski et
al., Phys. Rev. B 53 , R 11 938 (1996).]
The exponent is larger for samples having higher GM R !!!
77
Problems of Hall effect analysis in composite systems
like Fe/Cr multilayers Homogeneous ferromagnets
like Bulk Fe
Inhomogeneous ferromagnets
like Fe/Cr (GMR)
R
sand Ohmic resistivity (
r
)
have hardly any field dependence
Only Fe causes AHE, not Cr
Both R
sand Ohmic resistivity
(
r
) are field dependent
GMR systems and homogeneous bulk systems should not be treated on equal footings !!!
In ln R
svs. ln rplot, it does not matter
whether one uses the resistivity( r) in zero
field or saturation field (the scaling law
remains unaltered).
The antiferromagnetic coupling plays a
crucial role in making both Extraordinary
Hall effect & GMR field dependent.
Since Rs should not have any field dependence (Eq. (2)), one could ar gue that the above
field dependence is mainly due to the reduction in the current density through the FM
layers while B-field switches the system gradually from an tiparallel to parallel state.
↓↓ ↓
Hall effect Hall effect
↓Extraordinary Hall voltage at a given field depends
on the current density passing through the FM layers.
78
like Fe/Cr multilayers Homogeneous ferromagnets
like Bulk Fe
Inhomogeneous ferromagnets
like Fe/Cr (GMR)
R
sand Ohmic resistivity (
r
)
have hardly any field dependence
Only Fe causes AHE, not Cr
Both R
sand Ohmic resistivity
(
r
) are field dependent
GMR systems and homogeneous bulk systems should not be treated on equal footings !!!
In ln R
svs. ln rplot, it does not matter
whether one uses the resistivity( r) in zero
field or saturation field (the scaling law
remains unaltered).
The antiferromagnetic coupling plays a
crucial role in making both Extraordinary
Hall effect & GMR field dependent.
Since Rs should not have any field dependence (Eq. (2)), one could ar gue that the above
field dependence is mainly due to the reduction in the current density through the FM
layers while B-field switches the system gradually from an tiparallel to parallel state.
↓↓ ↓
Hall effect Hall effect
↓Extraordinary Hall voltage at a given field depends
on the current density passing through the FM layers.
78
5.4 5.6 5.8 6.0
0.0
0.4
0.8
1.2
1.6
R
s~
r
n
n = 1.95 ± 0.03
Master plot for (Fe/Cr)
10
ln R
s
ln
r
F8 F10 F12 F14
5.4 5.7 6.0 6.3
0.0
0.5
1.0
1.5
2.0
R
s~
r
n
Master plot for (Fe/Cr)
30
n = 2.05 ± 0.03
F8 F10 F12 F14
ln R
s
ln
r
Plot of ln R
s (B) vs. ln r(B).
Scaling exponent n =
1.95 for 10 bi-layer &
2.05 for 30 bi-layer
series. Each plot has
4 samples x 4 fields x
12 temperatures =
192 points.
Scaling relation Scaling relation
A realization of the charge confinement in a Quantum Well
Spin-polarized quantum well states are
formed in the non-magnetic Cr spacer layer
due to multiple reflections of electrons from
the interfaces of adjacent magnetic Fe
layers. They mediate the exchange coupling
between the FM layers.
Majority electrons get gradually confined
to the Cr spacer layer as H increases. This
reduces the current in the Fe layers which,
in turn, causes the decrease of both R
s& r.
79
0.0
0.4
0.8
1.2
1.6
R
s~
r
n
n = 1.95 ± 0.03
Master plot for (Fe/Cr)
10
ln R
s
ln
r
F8 F10 F12 F14
5.4 5.7 6.0 6.3
0.0
0.5
1.0
1.5
2.0
R
s~
r
n
Master plot for (Fe/Cr)
30
n = 2.05 ± 0.03
F8 F10 F12 F14
ln R
s
ln
r
Plot of ln R
s (B) vs. ln r(B).
Scaling exponent n =
1.95 for 10 bi-layer &
2.05 for 30 bi-layer
series. Each plot has
4 samples x 4 fields x
12 temperatures =
192 points.
Scaling relation Scaling relation
A realization of the charge confinement in a Quantum Well
Spin-polarized quantum well states are
formed in the non-magnetic Cr spacer layer
due to multiple reflections of electrons from
the interfaces of adjacent magnetic Fe
layers. They mediate the exchange coupling
between the FM layers.
Majority electrons get gradually confined
to the Cr spacer layer as H increases. This
reduces the current in the Fe layers which,
in turn, causes the decrease of both R
s& r.
79
80
Robotics and automotive sensors (e.g. in car).
Measuring electrical current in cables.
Pressure sensors (GMR in conjunction with
magnetostrictive materials), Microphone.
Solid-state Compass systems.
Sensitive detection of magnetic field.
Magnetic recording and detection of landmines.
1973: Rare earth transition metal film in magneto- optic recording.
1979: Thin film technology for heads in hard disks (both read and write
processes) (IBM).
1991: AMR effect using permalloy films for sensors in HDD by IBM.
1997: GMR sensors in HDD by IBM.
Currently both
GMR and TMR
are used for application in sensors and MRAMs .
Comparison between AMR / GMR effect:
In contrast to AMR, GMR is isotropic & GMR effect i s more robust than AMR.
Giant Magnetoresistance (GMR)
Applications of thin-film technology
Robotics and automotive sensors (e.g. in car).
Measuring electrical current in cables.
Pressure sensors (GMR in conjunction with
magnetostrictive materials), Microphone.
Solid-state Compass systems.
Sensitive detection of magnetic field.
Magnetic recording and detection of landmines.
1973: Rare earth transition metal film in magneto- optic recording.
1979: Thin film technology for heads in hard disks (both read and write
processes) (IBM).
1991: AMR effect using permalloy films for sensors in HDD by IBM.
1997: GMR sensors in HDD by IBM.
Currently both
GMR and TMR
are used for application in sensors and MRAMs .
Comparison between AMR / GMR effect:
In contrast to AMR, GMR is isotropic & GMR effect i s more robust than AMR.
Giant Magnetoresistance (GMR)
Applications of thin-film technology
81
Nano-scale magnetism:
Magnetic properties like T
c
(Curie
temperature), M
s
(Saturation magnetization), and H
c
(Coercive
field) change as the size reduces to< 100 nm, due t o higher
surface /volume ratio.
(A) (B) (C)
(A) Specific magnetization vs. temperature at 1.2 t esla.
(B) Specific magnetization vs. average diameter of Ni particles.
(C) Coercivity vs. average diameter of Ni particles .
[You-wei Du, Ming-xiang Xu, Jian Wu, Ying-bing Shi, and Huai-xian Lu,
J. Appl. Phys. 70, 5903 (1991).]
Electrical transport
properties are seriously affected by grain
boundary scattering.
Nano-magnetism
Nano-scale magnetism:
Magnetic properties like T
c
(Curie
temperature), M
s
(Saturation magnetization), and H
c
(Coercive
field) change as the size reduces to< 100 nm, due t o higher
surface /volume ratio.
(A) (B) (C)
(A) Specific magnetization vs. temperature at 1.2 t esla.
(B) Specific magnetization vs. average diameter of Ni particles.
(C) Coercivity vs. average diameter of Ni particles .
[You-wei Du, Ming-xiang Xu, Jian Wu, Ying-bing Shi, and Huai-xian Lu,
J. Appl. Phys. 70, 5903 (1991).]
Electrical transport
properties are seriously affected by grain
boundary scattering.
Nano-magnetism
82
Recording Media
A good medium must have high M
r
and H
C
.
In the year 2000 areal storage density of 65 Gbit/s q. inch was obtained in
CoCrPtTa deposited on Cr thin films/Cr
80
Mo
20
alloy.
To achieve still higher density like 400Gbit/sq. in ch it is necessary to use
patterned medium instead of a continuous one. This is an assembly of
nano-scale magnetically independent dots, each dot representing one bit
of information.
Some techniques are self-organization of nano-parti cles, nano-imprints,
or local ion irradiation.
Problem of reducing magnetic particle size is the s o-called
Superparamagnetic limit.
What is Superparamagnetism ???
Nano-magnetism
Recording Media
A good medium must have high M
r
and H
C
.
In the year 2000 areal storage density of 65 Gbit/s q. inch was obtained in
CoCrPtTa deposited on Cr thin films/Cr
80
Mo
20
alloy.
To achieve still higher density like 400Gbit/sq. in ch it is necessary to use
patterned medium instead of a continuous one. This is an assembly of
nano-scale magnetically independent dots, each dot representing one bit
of information.
Some techniques are self-organization of nano-parti cles, nano-imprints,
or local ion irradiation.
Problem of reducing magnetic particle size is the s o-called
Superparamagnetic limit.
What is Superparamagnetism ???
Nano-magnetism
83
Superparamagnetism
There are several types of anisotropies, the most c ommon is the magnetocrystalline
anisotropy caused by the
spin-orbit interaction
in a ferromagnet. It is easier to magnetize along c ertain
crystallographic directions. This energy term is di rection dependent
although its magnitude is much less than that of th e exchange term. It
only dictates the direction of
M.
Thus the axis of quantization z is
always in a direction for which the anisotropy ener gy is minimum.
The energy of a ferromagnetic particle (many atoms ) with uniaxial
anisotropy constant K
1
(energy/vol.) and magnetic moment
mmmm
making
an angle
θ
with
H
|| z-axis is
(1)
Superparamagnetism
There are several types of anisotropies, the most c ommon is the magnetocrystalline
anisotropy caused by the
spin-orbit interaction
in a ferromagnet. It is easier to magnetize along c ertain
crystallographic directions. This energy term is di rection dependent
although its magnitude is much less than that of th e exchange term. It
only dictates the direction of
M.
Thus the axis of quantization z is
always in a direction for which the anisotropy ener gy is minimum.
The energy of a ferromagnetic particle (many atoms ) with uniaxial
anisotropy constant K
1
(energy/vol.) and magnetic moment
mmmm
making
an angle
θ
with
H
|| z-axis is
(1)
84
Superparamagnetism
where V is the volume of the particle. Eq. (1) has two minima at
θ = 0 and
p
whose energies are
and an energy barrier in between.
Assuming that
M
of the particles spend almost all their time
along one of the minima, then the no. of particles jumping from min. 1 to
min. 2 is a function of the barrier height
ε
m
ε
1
, where
ε
m
= energy of the
maximum. To get
ε
m
ω
Put
∂ε/ ∂θ = 0 = sinθ (2K
1
V cosθ + mH)
Superparamagnetism
where V is the volume of the particle. Eq. (1) has two minima at
θ = 0 and
p
whose energies are
and an energy barrier in between.
Assuming that
M
of the particles spend almost all their time
along one of the minima, then the no. of particles jumping from min. 1 to
min. 2 is a function of the barrier height
ε
m
ε
1
, where
ε
m
= energy of the
maximum. To get
ε
m
ω
Put
∂ε/ ∂θ = 0 = sinθ (2K
1
V cosθ + mH)
85
Superparamagnetism
ωsinθ = 0
gives two minima at
θ = 0 and p.
ω
The other solution gives the m aximum.
ω
Substituting
cosθ
in Eq. (1) one gets
where
mmmm= VM
and
|M| =M
S.
Superparamagnetism
ωsinθ = 0
gives two minima at
θ = 0 and p.
ω
The other solution gives the m aximum.
ω
Substituting
cosθ
in Eq. (1) one gets
where
mmmm= VM
and
|M| =M
S.
86
The frequency of jump from min. 1 to min.2 is
Converting to relaxation time for H = 0 gives
Superparamagnetism
Neels estimate of f
0 was 10
9
s
-1
. Current values are ~ 10
10
s
-1
.
& K
1V = Energy barrier.
Similarly,
where
The frequency of jump from min. 1 to min.2 is
Converting to relaxation time for H = 0 gives
Superparamagnetism
Neels estimate of f
0 was 10
9
s
-1
. Current values are ~ 10
10
s
-1
.
& K
1V = Energy barrier.
Similarly,
where
87
π
This table shows prominently
the exponential behaviour of τ
(R). For, say, Ni τincreases by 5
orders of magnitude as R changes
from 75 to 85 Å.
π
If τ>> τ
exp
, experimental time
scale, no change of
M
could be
observed during τ
exp
→ Stable
ferromagnetism.
π
If τ
exp
~ 100 s (SQUID), 75 Å Ni
particle will notshow stable FM
but if the probe is Mossbauer (10
-8
s) it is indeeda FM.
Superparamagnetism
π
If t<< t
exp,
M
will flip many many times during t
exp& average
M
will be 0.
Thus there is a loss of Stable ferromagnetism since the relaxa tion time, t, is too small.
π
This table shows prominently
the exponential behaviour of τ
(R). For, say, Ni τincreases by 5
orders of magnitude as R changes
from 75 to 85 Å.
π
If τ>> τ
exp
, experimental time
scale, no change of
M
could be
observed during τ
exp
→ Stable
ferromagnetism.
π
If τ
exp
~ 100 s (SQUID), 75 Å Ni
particle will notshow stable FM
but if the probe is Mossbauer (10
-8
s) it is indeeda FM.
Superparamagnetism
π
If t<< t
exp,
M
will flip many many times during t
exp& average
M
will be 0.
Thus there is a loss of Stable ferromagnetism since the relaxa tion time, t, is too small.
88
⇒
The final form of m along the direction of H comes out to be
where B
S
(x) is the
Brillouin functionParamagnetism
>= <
Tk
SH g
SB g m
B
B
S B
m
m
-
+ +
=
S
x
S S
x S
S
S
2
coth
2
1
2
)1 2(
coth
2
1 2
where
S is the spin of the ion.
< m > =g
mmmm
B
[coth x-1/x] = L(x) = Langevin function when S
®®®® ¥¥¥¥
.
⇒
The final form of m along the direction of H comes out to be
where B
S
(x) is the
Brillouin functionParamagnetism
>= <
Tk
SH g
SB g m
B
B
S B
m
m
-
+ +
=
S
x
S S
x S
S
S
2
coth
2
1
2
)1 2(
coth
2
1 2
where
S is the spin of the ion.
< m > =g
mmmm
B
[coth x-1/x] = L(x) = Langevin function when S
®®®® ¥¥¥¥
.
89
⇒
In this
<m> vs. H/T
plot
paramagnetic saturation is
observed only at very high
H & low T.
⇒
At 4 K & 1tesla,
<m> ~ 14%
of its saturation value.
⇒
For ordinary temperature like
300 K & 1 T field,
x <<1
and
.
So
<m>
varies linearly with field.
Paramagnetism
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
S=3/2 FOR Cr3++
Magnetic Moments
Magnetic Field/Temperature(B/T)
H
Tk
SSg
m
B
B
+
>® <
3
)1 (
m
⇒
In this
<m> vs. H/T
plot
paramagnetic saturation is
observed only at very high
H & low T.
⇒
At 4 K & 1tesla,
<m> ~ 14%
of its saturation value.
⇒
For ordinary temperature like
300 K & 1 T field,
x <<1
and
.
So
<m>
varies linearly with field.
Paramagnetism
0 2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
S=3/2 FOR Cr3++
Magnetic Moments
Magnetic Field/Temperature(B/T)
H
Tk
SSg
m
B
B
+
>® <
3
)1 (
m
90
π
But in a field
H
,
M
will behave as a paramagnet as shown in <m> vs.
H/T plot of last page with easy saturation when all the particles align
at a much lower field & higher temperature since S in Brillouin function
(~ SH/T) is now 10
3
- 10
4
. This isSuperparamagnetism(SPM) ---
super meaning very high as in Superconductor.
π
Transition from stable FM to SPM shifts to smaller parti cle size when
T is decreased since τis a function of V/T. The temperature, T
B at
which τ~ τ
exp, is called the Blocking temperature. Above T
B, SPM
with all <m> vs. H/T curves coalesceto one but no hysteresis. Below
T
B, it is in a blocked(FM) state with hysteretic M
r and H
C . T
B
moves to lower temperatures at higher fields due to the lowering of
barrier height.
π
A single particle of only several Å dimension cannot be h andled easily.
So, experiments are carried out with an ensemble of ma gnetic particles
with a size distribution.
Superparamagnetism
π
But in a field
H
,
M
will behave as a paramagnet as shown in <m> vs.
H/T plot of last page with easy saturation when all the particles align
at a much lower field & higher temperature since S in Brillouin function
(~ SH/T) is now 10
3
- 10
4
. This isSuperparamagnetism(SPM) ---
super meaning very high as in Superconductor.
π
Transition from stable FM to SPM shifts to smaller parti cle size when
T is decreased since τis a function of V/T. The temperature, T
B at
which τ~ τ
exp, is called the Blocking temperature. Above T
B, SPM
with all <m> vs. H/T curves coalesceto one but no hysteresis. Below
T
B, it is in a blocked(FM) state with hysteretic M
r and H
C . T
B
moves to lower temperatures at higher fields due to the lowering of
barrier height.
π
A single particle of only several Å dimension cannot be h andled easily.
So, experiments are carried out with an ensemble of ma gnetic particles
with a size distribution.
Superparamagnetism
91
Superparamagnetism
0 50 100 150 200 250 300
0.0
5.0x10
-7
1.0x10
-6
1.5x10
-6
2.0x10
-6
2.5x10
-6
3.0x10
-6
3.5x10
-6
4.0x10
-6
M (emu)
T (K)
M (T) at H = 200 Oe for 6 nm Ni
sample. Langevin/Brillouin
function M= (cothx-1/x)well with
= 2700
B
where x = H/k
B
T.
Samples are single layers of Ni nanoparticles with non-conducting Al
2
O
3
on
both sides deposited on both Si and Sapphire substr ates using PLD technique.
M(H,T)measured using Quantum Design MPMS (SQUID magnetometer).
Diamagnetic contribution of substrate subtracted ( typically χ = -2 x 10
-4
emu/tesla).
Superparamagnetism
0 50 100 150 200 250 300
0.0
5.0x10
-7
1.0x10
-6
1.5x10
-6
2.0x10
-6
2.5x10
-6
3.0x10
-6
3.5x10
-6
4.0x10
-6
M (emu)
T (K)
M (T) at H = 200 Oe for 6 nm Ni
sample. Langevin/Brillouin
function M= (cothx-1/x)well with
= 2700
B
where x = H/k
B
T.
Samples are single layers of Ni nanoparticles with non-conducting Al
2
O
3
on
both sides deposited on both Si and Sapphire substr ates using PLD technique.
M(H,T)measured using Quantum Design MPMS (SQUID magnetometer).
Diamagnetic contribution of substrate subtracted ( typically χ = -2 x 10
-4
emu/tesla).
92
Sample details
D. Kumar, H. Zhou, T. K. Nath, Alex V. Kvit, and J. Narayan, Appl. Phys. Lett. 79, 2817 (2001).
Base pressure = 5´10
-7
Torr
Substrate temperature = 600 C
Energy density and repeatition rate of
the laser beam are 2J/cm
2
and 10Hz.
Five alternating layers of Ni (
nano dots
) and TiN (
metallic matrix
)
were deposited on Si using PLD method:
TiN
Ni
STEM-Z image of Ni nanoparticles
embedded in TiN metal matrix.
Si
Why TiN?
Chemical stability, hardness, acts as
diffusion barrier for both Ni and Si, high
electrical conductivity, grows as a buffer
layer epitaxially on Si.
Nano-magnetism
Sample details
D. Kumar, H. Zhou, T. K. Nath, Alex V. Kvit, and J. Narayan, Appl. Phys. Lett. 79, 2817 (2001).
Base pressure = 5´10
-7
Torr
Substrate temperature = 600 C
Energy density and repeatition rate of
the laser beam are 2J/cm
2
and 10Hz.
Five alternating layers of Ni (
nano dots
) and TiN (
metallic matrix
)
were deposited on Si using PLD method:
TiN
Ni
STEM-Z image of Ni nanoparticles
embedded in TiN metal matrix.
Si
Why TiN?
Chemical stability, hardness, acts as
diffusion barrier for both Ni and Si, high
electrical conductivity, grows as a buffer
layer epitaxially on Si.
Nano-magnetism
93
A cross-sectional STEM-Z image
Ni [220] || TiN[002]
Ni[110] || TiN[110]
Ni [002] || TiN[002]
Ni[110] || TiN[110]
Single layer epitaxial Ni nano dots on TiN/Si(100) template
ω
Ni separation ≈ 10 nm
ω
Triangular morphology
with 17 nm base and
9 nm height
ω
Some have rectangular
morphology. ω
Thickness of
TiN = 6´32 nm.
ω
Ni particles grow epitaxially on TiN acting as a te mplate which also
grows epitaxially on Si. ω
Ni-nano/TiN is an ideal system for studying electrical transport in magnetic
nano particles due to epitaxial growth and conducting nature of TiN .
Nano-magnetism
A cross-sectional STEM-Z image
Ni [220] || TiN[002]
Ni[110] || TiN[110]
Ni [002] || TiN[002]
Ni[110] || TiN[110]
Single layer epitaxial Ni nano dots on TiN/Si(100) template
ω
Ni separation ≈ 10 nm
ω
Triangular morphology
with 17 nm base and
9 nm height
ω
Some have rectangular
morphology. ω
Thickness of
TiN = 6´32 nm.
ω
Ni particles grow epitaxially on TiN acting as a te mplate which also
grows epitaxially on Si. ω
Ni-nano/TiN is an ideal system for studying electrical transport in magnetic
nano particles due to epitaxial growth and conducting nature of TiN .
Nano-magnetism
94
HallEffect
0 1 2 3 4 5
0
1x10
-10
2x10
-10
3x10
-10
4x10
-10
5x10
-10
6x10
-10
T=25K
T = 300K
Hall effect
Ni-Nano/TiN
Hall effect
Pure Ni-bulk
11K 35K 60K 85K 105K 115K 135K 160K 185K 210K 235K 260K 290K
T=290K T=11K
Hall Resistivity (- r
H
)(ohm-m)
Field (tesla)
Hall resistivity ( r
H) vs. applied magnetic field ( m
0H
applied).
ω
ρ
His negative at all temperatures from 11 to 290 K.
ω
In Hall geometry B =
0
[H
applied
+ (1-N) M], N=demagnetization factor.
So, ρ
H= R
o
0H
applied+ R
S
0M,
Nano-magnetism
HallEffect
0 1 2 3 4 5
0
1x10
-10
2x10
-10
3x10
-10
4x10
-10
5x10
-10
6x10
-10
T=25K
T = 300K
Hall effect
Ni-Nano/TiN
Hall effect
Pure Ni-bulk
11K 35K 60K 85K 105K 115K 135K 160K 185K 210K 235K 260K 290K
T=290K T=11K
Hall Resistivity (- r
H
)(ohm-m)
Field (tesla)
Hall resistivity ( r
H) vs. applied magnetic field ( m
0H
applied).
ω
ρ
His negative at all temperatures from 11 to 290 K.
ω
In Hall geometry B =
0
[H
applied
+ (1-N) M], N=demagnetization factor.
So, ρ
H= R
o
0H
applied+ R
S
0M,
Nano-magnetism
95
-3000 -2000 -1000 0 1000 2000 3000
-1.8
-1.2
-0.6
0.0
0.6
1.2
1.8
0.20
0.25
0.30
0.35
0 100 200 300
50
100
150
200
250
Hc
H
C
(Oe)
T (K)
M
r
(10
-4
emu)
Mr
M (10
- 4
emu)
Field (tesla)
(10 K) (100 K) (300 K)
Magnetization vs. applied field at 10, 100 and 300 K.
Inset: Coercivity & remanent magnetization vs. T.
Magnetization
Nano-magnetism
[P. Khatua, T. K. Nath, and A. K. Majumdar, Phys. R ev. B
73, 064408
(2006).]
-3000 -2000 -1000 0 1000 2000 3000
-1.8
-1.2
-0.6
0.0
0.6
1.2
1.8
0.20
0.25
0.30
0.35
0 100 200 300
50
100
150
200
250
Hc
H
C
(Oe)
T (K)
M
r
(10
-4
emu)
Mr
M (10
- 4
emu)
Field (tesla)
(10 K) (100 K) (300 K)
Magnetization vs. applied field at 10, 100 and 300 K.
Inset: Coercivity & remanent magnetization vs. T.
Magnetization
Nano-magnetism
[P. Khatua, T. K. Nath, and A. K. Majumdar, Phys. R ev. B
73, 064408
(2006).]
96
FeMn providing local magnetic field to Co layer, an d NiFe is magnetically soft.
GMR > 20 %.
Read heads for HDD detect fields ~ 10 Oe.
Ferromagnetic
Substrate
Ta
FeMn
Co
Cu
NiFe
Ta
}
Antiferromagnetic
Exchange-biased Spin-valve structure
FeMn providing local magnetic field to Co layer, an d NiFe is magnetically soft.
GMR > 20 %.
Read heads for HDD detect fields ~ 10 Oe.
Ferromagnetic
Substrate
Ta
FeMn
Co
Cu
NiFe
Ta
}
Antiferromagnetic
Exchange-biased Spin-valve structure
97
Tunnel Magnetoresistance (TMR)
Discovery of GMR effect in Fe/Cr superlattice in 1986 and giant tunne l magnetoresistance
(TMR) effect at room temperature (RT) in 1995 opened up a new field of science and technology
called spintronics. The latter provides better understanding on the physic s of spin-dependent
transport in heterogeneous systems. Perhaps more significantly, such studies have contributed to
new generations of magnetic devices for information storage and magnetic sensors.
A magnetic tunnel junction (MTJ), which consists of a thin
insulating layer (a tunnel barrier) sandwiched between two
ferromagnetic electrode layers, exhibits TMR due to spin-
dependent electron tunneling. MTJs with an amorphous
aluminium oxide (AlO) tunnel barrier have shown
magnetoresistance (MR) ratios up to about 70 % at RT and
are currently used in magnetoresistive random access memory
(MRAM) and the read heads of hard disk drives. In 2004 MR
ratios of about 200 % were obtained in MTJs with a single-
crystal MgO(001) barrier. Later CoFeB/MgO/CoFeB MTJs
were also developed having MR ratios up to 500 % at RT.
Tunnel Magnetoresistance (TMR)
Discovery of GMR effect in Fe/Cr superlattice in 1986 and giant tunne l magnetoresistance
(TMR) effect at room temperature (RT) in 1995 opened up a new field of science and technology
called spintronics. The latter provides better understanding on the physic s of spin-dependent
transport in heterogeneous systems. Perhaps more significantly, such studies have contributed to
new generations of magnetic devices for information storage and magnetic sensors.
A magnetic tunnel junction (MTJ), which consists of a thin
insulating layer (a tunnel barrier) sandwiched between two
ferromagnetic electrode layers, exhibits TMR due to spin-
dependent electron tunneling. MTJs with an amorphous
aluminium oxide (AlO) tunnel barrier have shown
magnetoresistance (MR) ratios up to about 70 % at RT and
are currently used in magnetoresistive random access memory
(MRAM) and the read heads of hard disk drives. In 2004 MR
ratios of about 200 % were obtained in MTJs with a single-
crystal MgO(001) barrier. Later CoFeB/MgO/CoFeB MTJs
were also developed having MR ratios up to 500 % at RT.
98
Tunnel Magnetoresistance (TMR)
(a) Magnetizations in the two electrodes are
aligned parallel (P).
(b) Magnetizations are aligned antiparallel (AP).
D
2↓D
2↑
¯ ¯
SPIN-POLARIZED TUNNELING
In an MTJ, the resistance of the junction
depends on the relative orientation of the
magnetization vectors Min the two electrodes.
When the Ms are parallel, tunneling probability is maximum b ecause electrons
from those states with a large density of states c an tunnel into the same states in
the other electrode. When the magnetization vectors are antiparallel, there will
be a mismatch between the tunneling states on each side of the junction. This
leads to a diminished tunneling probability, hence, a larger resistance.
D
1↑ and D
1↓,respectively, denote the density of states
at E
Ffor the majority-spin and minority-spin bands
in electrode 1, and D
2↑ and D
2↓ are respectively
those for electrode 2.
|D
↑- D
¯| >> than what is shown for
high polarization materials!!!
⇒⇒⇒⇒
Tunnel Magnetoresistance (TMR)
(a) Magnetizations in the two electrodes are
aligned parallel (P).
(b) Magnetizations are aligned antiparallel (AP).
D
2↓D
2↑
¯ ¯
SPIN-POLARIZED TUNNELING
In an MTJ, the resistance of the junction
depends on the relative orientation of the
magnetization vectors Min the two electrodes.
When the Ms are parallel, tunneling probability is maximum b ecause electrons
from those states with a large density of states c an tunnel into the same states in
the other electrode. When the magnetization vectors are antiparallel, there will
be a mismatch between the tunneling states on each side of the junction. This
leads to a diminished tunneling probability, hence, a larger resistance.
D
1↑ and D
1↓,respectively, denote the density of states
at E
Ffor the majority-spin and minority-spin bands
in electrode 1, and D
2↑ and D
2↓ are respectively
those for electrode 2.
|D
↑- D
¯| >> than what is shown for
high polarization materials!!!
⇒⇒⇒⇒
99
Tunnel Magnetoresistance (TMR)
Gang Mao, Gupta, el al. at Brown & IBM obtained be low 200 Oe a
GMR of 250 % in MTJs with electrodes made of epita xial filmsof
doped half-metallicmanganite La
0.67
Sr
033
MnO
3
(LSMO) and insulating
barrier of SrTiO
3
using self-aligned lithographic process to pattern the
junctions to micron size. They confirmed the spin-p olarized tunneling as
the active mechanism with P ~ 0.75. The low saturat ion field comes from
the fact that manganites are magnetically soft, hav ing a coercive field as
small as 10 Oe. They have also made polycrystalline films with a large
number of the grain boundaries and observed large M R at low fields.
Here the mechanism has been attributed to the spin- dependent scattering
across the grain boundaries that serves to pin the magnetic domain walls.
Assuming no spin-flip scattering, the MR ratio betw een the two
configurations is given by,
(DR/R
P
) = (R
AP
-R
P
)/R
P
= 2P
2
/(1-P
2
), where R
P
/R
AP
is the resistance for
the P/AP configuration & P= (D
↑
D
¯
)/ (D
↑
+ D
¯
) = spin-polarization
parameter of the magnetic electrodes. A half-metal corresponds to P = 1.
Tunnel Magnetoresistance (TMR)
Gang Mao, Gupta, el al. at Brown & IBM obtained be low 200 Oe a
GMR of 250 % in MTJs with electrodes made of epita xial filmsof
doped half-metallicmanganite La
0.67
Sr
033
MnO
3
(LSMO) and insulating
barrier of SrTiO
3
using self-aligned lithographic process to pattern the
junctions to micron size. They confirmed the spin-p olarized tunneling as
the active mechanism with P ~ 0.75. The low saturat ion field comes from
the fact that manganites are magnetically soft, hav ing a coercive field as
small as 10 Oe. They have also made polycrystalline films with a large
number of the grain boundaries and observed large M R at low fields.
Here the mechanism has been attributed to the spin- dependent scattering
across the grain boundaries that serves to pin the magnetic domain walls.
Assuming no spin-flip scattering, the MR ratio betw een the two
configurations is given by,
(DR/R
P
) = (R
AP
-R
P
)/R
P
= 2P
2
/(1-P
2
), where R
P
/R
AP
is the resistance for
the P/AP configuration & P= (D
↑
D
¯
)/ (D
↑
+ D
¯
) = spin-polarization
parameter of the magnetic electrodes. A half-metal corresponds to P = 1.
100
Another class of materials, the rare-earth
manganiteoxides, La
1-x
D
x
MnO
3
(D = Sr, Ca,
etc.),show Colossal Magnetoresistance effect
(CMR) with a very large negative MRnear their
magnetic phase transition temperatures (T
C
)
when subjected to a tesla-scale magnetic field.
The double-exchangemodel of Zener and a
strong e-ph interaction from the Jahn-Teller
splitting of Mn d levels explain most of their
magnetotransport properties.
Though fundamentally interesting, the CMR
effect, achieved only at large fields and below
300 K, poses severe technological challenges to
potential applications in magnetoelectronic
devices, where low field sensitivity is crucial.
Colossal Magnetoresistance (CMR)
Another class of materials, the rare-earth
manganiteoxides, La
1-x
D
x
MnO
3
(D = Sr, Ca,
etc.),show Colossal Magnetoresistance effect
(CMR) with a very large negative MRnear their
magnetic phase transition temperatures (T
C
)
when subjected to a tesla-scale magnetic field.
The double-exchangemodel of Zener and a
strong e-ph interaction from the Jahn-Teller
splitting of Mn d levels explain most of their
magnetotransport properties.
Though fundamentally interesting, the CMR
effect, achieved only at large fields and below
300 K, poses severe technological challenges to
potential applications in magnetoelectronic
devices, where low field sensitivity is crucial.
Colossal Magnetoresistance (CMR)
101
Colossal Magnetoresistance (CMR)
Colossal Magnetoresistance (CMR)
102
Colossal Magnetoresistance (CMR)
the e
g
orbital, which can hop into the neighboring Mn
4+
sites (Double
Exchange).The spin of this conducting electron is aligned wit h the local spin
(S = 3/2) in the t
2g
3 orbitals of Mn
3+
due to strong Hund's coupling. When the
manganite becomes ferromagnetic, the electrons in t he e
g
orbitals are fully
spin-polarized. The band structure is such that all the conduction electrons are
in the majority band. This kind of metal with empty minority band is
generally called a half-metal and so manganites hav e naturally become a good
candidate for the study of spin-polarized transport .
The origin of CMR stems from the strong
interplay among the electronic structure,
magnetism, and lattice dynamics in manganites.
Doping of divalent Ca or Sr impurities into
trivalent La sites create mixed valence states of
Mn
3+
(fraction: l-x) and Mn
4+
(fraction: x).
Mn
4+
(3d
3
) has a localized spin of S = 3/2 from
the low-lying t
2g
3orbitals, whereas the e
g
obitals
are empty. Mn
3+
(3d
4
) has an extra electron in
t
2g
e
g
t
2g
e
g
Colossal Magnetoresistance (CMR)
the e
g
orbital, which can hop into the neighboring Mn
4+
sites (Double
Exchange).The spin of this conducting electron is aligned wit h the local spin
(S = 3/2) in the t
2g
3 orbitals of Mn
3+
due to strong Hund's coupling. When the
manganite becomes ferromagnetic, the electrons in t he e
g
orbitals are fully
spin-polarized. The band structure is such that all the conduction electrons are
in the majority band. This kind of metal with empty minority band is
generally called a half-metal and so manganites hav e naturally become a good
candidate for the study of spin-polarized transport .
The origin of CMR stems from the strong
interplay among the electronic structure,
magnetism, and lattice dynamics in manganites.
Doping of divalent Ca or Sr impurities into
trivalent La sites create mixed valence states of
Mn
3+
(fraction: l-x) and Mn
4+
(fraction: x).
Mn
4+
(3d
3
) has a localized spin of S = 3/2 from
the low-lying t
2g
3orbitals, whereas the e
g
obitals
are empty. Mn
3+
(3d
4
) has an extra electron in
t
2g
e
g
t
2g
e
g
103
Colossal Magnetoresistance (CMR)
Superexchange favors antiferromagnetism
Double exchange
Zener (1951) offered an explanation that remains at the core of our unders tanding of magnetic
oxides. In doped manganese oxides, the two configurations
y
1: Mn
3+
O
2
-2
Mn
4+
and y
2: Mn
4+
O
2
-2
Mn
3+
are degenerate and are connected by the so-called double-exchange matrix element. This
matrix element arises via the transfer of an electron from Mn
3+
to the central O
2
-2
simultaneous
with transfer from O
2
-2
to Mn
4+
. The degeneracy of y
1and y
2, a consequence of the two
valencies of the Mn ions, makes this process fundamentally different from the above
conventional superexchange. Because of strong Hunds coupling, the transfer-ma trix element
has finite value only when the core spins of the Mn ions are aligned fe rromagnetically. The
coupling of degenerate states lifts the degeneracy, and the system resonates between y
1and y
2
if the core spins are parallel, leading to a ferromagnetic, conduct ing ground state. The splitting
of the degenerate levels is k
BT
Cand, using classical arguments, predicts the electrical
conductivity to be s x e
2
a h T
C/T where a is the Mn-Mn distance and x, the Mn
4+
fraction.
Colossal Magnetoresistance (CMR)
Superexchange favors antiferromagnetism
Double exchange
Zener (1951) offered an explanation that remains at the core of our unders tanding of magnetic
oxides. In doped manganese oxides, the two configurations
y
1: Mn
3+
O
2
-2
Mn
4+
and y
2: Mn
4+
O
2
-2
Mn
3+
are degenerate and are connected by the so-called double-exchange matrix element. This
matrix element arises via the transfer of an electron from Mn
3+
to the central O
2
-2
simultaneous
with transfer from O
2
-2
to Mn
4+
. The degeneracy of y
1and y
2, a consequence of the two
valencies of the Mn ions, makes this process fundamentally different from the above
conventional superexchange. Because of strong Hunds coupling, the transfer-ma trix element
has finite value only when the core spins of the Mn ions are aligned fe rromagnetically. The
coupling of degenerate states lifts the degeneracy, and the system resonates between y
1and y
2
if the core spins are parallel, leading to a ferromagnetic, conduct ing ground state. The splitting
of the degenerate levels is k
BT
Cand, using classical arguments, predicts the electrical
conductivity to be s x e
2
a h T
C/T where a is the Mn-Mn distance and x, the Mn
4+
fraction.
Thanks
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