1
Magnetism
Magnetism
Quantummechanics
Electron-Electron
Interactions
2
Outline
• Magnetism is a purely quantum phenomenon!
• Diamagnetism
• Paramagnetism
• Effects of electron-electron interactions
Hund’s rules for atoms
Atoms in a magnetic field – Curie law
Magnetism in transition metals, rare earths
• Magnetic order and cooperative effects in solids
Transition temperature TC
Curie-Weiss law
• Magnetism: example of an “order parameter”
3
Definitions
Magnetization M = the magnetic moment per unit volume.
• Diamagnetic material: < 0
• Paramagnetic material: > 0
• Ferromagnetic material: M 0 even without external field
Magnetic susceptibility per unit volume = 0M/B
B: macroscopic magnetic field intensity,
0 : permeability of free space
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Introduction to Solid State Physics by Kittel (2005)
5
Hund’s rules and electron interactions
• Hund’s rules:
1st rule: maximum spin for electrons in a given shell
=> Electron-electron interactions
Reason – parallel-spin electrons are kept apart because they
must obey the exclusion principle –thus the repulsive
interaction between electrons is reduced for parallel spins!
2nd rule: maximum angular momentum possible for the given
spin orientation
Reason – maximum angular momentum means
electrons are going the same direction around the nucleus –
stay apart – lower energy!
6
Hund’s rules and electron interactions
3rd rule: the total angular J=| L±S | with the sign being
determined by whether the shell of orbitals is more than
half filled (+) or less than half filled (-).
Reason
i i
i
H l L S = =
always favoring L counter-aligned with S
When the shell is more than half filled, the additional spin
still want to counter-align with its own angular momentum,
but then L would be aligned with the net spin.
7
External field2
0 ( )2
pH V r
m= +
2( )
( )2
B
eAp
cH g B V rm
+
= + +
adding an external magnetic field:
1
2A B r=
22
2
2
1( ) ( )
2 2 4 2
B
p e eH V r B r p B r
m mc mc
g B
= + + +
+
8
External field2
22
2
1( ) ( )
2 2 4 2B
p e eH V r B r p B r g B
m mc mc = + + + +
( ) ( )2 2 2
B
e e ep B r B r p B l B l
mc mc mc = = =
22
0 2
1( )
2 4B
eH H B l g B r
mc = + + +
paramagnetic term
diamagnetic term
9
Diamagnetism
• Consider a single “closed shell” atom in a magnetic field (In a
closed shell atom, spins are paired and the electrons are
distributed spherically around the atom, i.e., no orbital angular
momentum → there is no total angular momentum.)
• Diamagnetism results from an electric current being set up in
atoms due to an external magnetic field
• Lenz’s law – when the magnetic flux is changed, an induced
current is set up in such a direction as to oppose the flux change.
10
Larmor Diamagnetism2
2
2
1
2 4
eH B r
mc=
• Susceptibility 2 2
0
26
e r
mc
−=
2 2 2 2 22
2 2 2
2 2 28 8 12
e e B e BE B r x y r
mc mc mc = = + =
22
26
E emoment r B
B mc
= − = −
For large conductive molecules, this term would be very large.
Introduction to Solid State Physics by Kittel (2005)
11
Larmor Diamagnetism
https://www.youtube.com/watch?v=KlJsVqc0ywM
BH g B =
Introduction to Solid State Physics by Kittel (2005)
12
Free spin ½ (Curie or Langevin) paramagnetism
exp( ) exp( )B B
B B
B BZ
k T k T
= + −
lnBF k T Z= −
tanh( )BB
B
BFM
B k T
= − =
13
Free spin ½ (Curie or Langevin) paramagnetism
2
0
0lim B
HB
nM
H k T
→
= =
Curie law
exp( ) exp( )
tanh( )
exp( ) exp( )
B B
B B BB B
B B B
B B
B B
k T k T BM
B B k T
k T k T
− −
= =
+ −
Introduction to Solid State Physics by Kittel (2005)
1 2( ) BM N N = −
14
Free spin J (Curie or Langevin)
paramagnetism
( )BH B l g = +
2 2( ) [ ]
L J S JB L gS B J g
J J+ = +
2 22 2 2 2
2 2[ ]
2 2
J L J L J S J SB J g
J J
+ − − + − −= +
2 2 2 2 2 2
2 2[ ]
2 2
J L S J S LB J g
J J
+ − + −= +
' Bg B J=
1 1 ( 1) ( 1)' ( 1) ( 1)[ ]
2 2 ( 1)
S S L Lg g g
J J
+ − += + + −
+Landau g-value
15
Free spin J (Curie or Langevin)
paramagnetism
'exp( )
Z
JB Z
J J B
g BJZ
k T
=−
= −
exp( )
exp( )
J
J
J
J J
m J
J J
J
m J
m m x
m
m x
=−
=−
−
=
−
' B
B
g Bx
k T
=
1J
Zm
Z x
= −
' B JM ng m=
16
Free spin J (Curie or Langevin)
paramagnetism
exp( )J
J
J
m J
Z m x=−
= −
exp( )(1 exp[ (2 1) ]) sinh[(2 1) / 2]
1 exp( ) sinh[ / 2]
Jx J x J xZ
x x
− − + += =
− −
2 00
(1 )(1 ...)
1
na ya y y
y
−+ + + =
−
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Free spin J (Curie or Langevin)
paramagnetism
y xJ=
' ( )B JM ng JB y=
2 1 2 1 1 1( ) coth( ) coth( )
2 2 2 2J
J JB y y y
J J J J
+ += −
Brillouin function
1/21/ 2, tanh( )J B y= =
sinh[(2 1) / 2]
sinh[ / 2]
J xZ
x
+=
1J
Zm
Z x
= −
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Free spin J (Curie or Langevin)
paramagnetism
2
0 0 ( ' ) ( 1)
3
B
B
M n gM J J C
H B k T T
+= = =
2 1 2 1 1 1( ) coth( ) coth( )
2 2 2 2J
J JB y y y
J J J J
+ += −
y is around 2*10-3 at RT.
31( )
3
Jy y
J
+ +
' ( 1) '
3
B B
B
ng J g JBM
k T
+=
C: Curie constant
19
Introduction to Solid State Physics by Kittel (2005)
20
Introduction to Solid State Physics by Kittel (2005)
21
Introduction to Solid State Physics by Kittel (2005)
Quenched
orbital
moment due
to crystal field
J. Stöhr and H. C. Siegmann, Magnetism, Springer (2006)
23
zL x yi i y x
= = −
should be real.
zn L n must be purely imaginary…
Since the wave functions under crystal field are real functions,
zn L n
L is purely imaginary, but is Hermitian.
0zn L n =All components of the orbital angularmomentum of a non-degenerate state arequenched.
Quenched orbital moment due to crystal field
24
Van Vleck paramagnetism
For J=0 in the ground state (the filled shell), there is no
paramagnetic effect for the first-order perturbation.
2
0
0
0 ( )B
n n
B l g nE
E E
+ =
−
0 ( ) 0 0B B l g + =
However, there would be finite contribution from the second-order
perturbation.
It is positive and temperature-independent.
22
0
0 ( )2 z zB
n n
L gS nN
V E E
+=
−
Van Vleck
paramagnetism
25
Pauli paramagnetism for free electron gas (1925)
• Spin up electrons (parallel to field) are shifted opposite to spin
down electrons (antiparallel to field).
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Pauli paramagnetism for free electron gas (1925)
Density of states for both spins
One spin orientation
BE B =
• Energies shift by
2
( )
(1 / 2) ( )2 ( )
B
B F B B F
M N N
D E B D E B
= −
= =
• Magnetization
27
Pauli paramagnetism for free electron gas (1925)
2
( )
(1 / 2) ( )2 ( )
B
B F B B F
M N N
D E B D E B
= −
= =
2(3 / 2) / ( )B B FM N B k T= Independent of T!
• Free electron gas, D(EF)=3N/2kBTF
• This is a way to measure the density of states!
(Note: There are corrections from the electron-electroninteractions. )
• Unlike the paramagnetism of magnetic ions, here the magnitudeof ~ diamagnetism’s (suppressed by a factor of kBT/EF (Pauliexclusion principle))
• The first method for attaining temperatures much below 1K.
28
Adiabatic demagnetization (proposed by Debye, 1926)
Introduction to Solid State Physics by Kittel (2005)
Thermal contact at T1 is provided by He gas
(a→b), and the thermal contact is broken by
pumping the gas (b→c).
• Freezing is effective only if entropy contribution from spin is
dominant (usually need T << TD, i.e., entropy contribution from
lattice vibrations is negligible.)
29
Introduction to Solid State Physics by Kittel (2005)
2 1( / )T T B B=
Can reach < 10-3 K
Adiabatic demagnetization
30
Ferromagnetism and
Antiferromagnetism
31
Introduction to Solid State Physics by Kittel (2005)
32
Magnetic materials
• What causes some materials (e.g. Fe) to be ferromagnetic?
• Others like Cr are antiferromagnetic (what is this?)
• Magnetic materials tend to be in particular places in the
periodic table: transition metal, rare earths.
• Starting point for understanding: the fact that open-shell atoms
have moments.
33
Modern Physics for Scientists and Engineers by Thornton and Rex (2013).
34
Understanding magnetic materials
• In most magnetic materials (Fe, Ni, ….) the first step in
understanding magnetism is to consider the material as a collection
of atoms where each atom has a magnetic moment
• Of course the atoms change in the solid, but this gives a good
starting point – qualitatively correct
Small!
• Ferromagnetism is not from magnetic dipole-dipole interaction,
nor the spin-orbit interaction. It is a result of electrostatic
interaction!
Dipole-dipole interaction:
35
When are atoms magnetic?
• An atom MUST have a magnetic moment if there is an odd
number of electrons – spin ½ (at least)
• “Open shell” atoms have moments – Hund’s rules
1st rule: maximum spin for electrons in a given shell
2nd rule: maximum angular momentum for the given spin
orientation
Mn2+:3d5
Fe2+:3d6
36
Exchange interactions
1 1 2 2 1 2 1 2( , , , ) ( , ) ( , )orbit spinr s r s r r s s =
1 2
1( , ) : , , ( ),
2spin s s +
1( )
2 −
1 2
1 1( , ) : ( 12 21 ), ( 12 21 )
2 2orbit r r + −
e-
e-
37
Exchange interactions
sin 0
12 [( 12 21 ) ( 12 21 )]
2gletE V= + + +
0
12 [( 12 21 ) ( 12 21 )]
2tripletE V= + − −
sin 02gletE K J= + +
02tripletE K J= + −
12 21J V=
12 12K V=
exchange integral
direct integral
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Exchange interactions2 2 2
1 2 1 21 2
( )
2
S S S SS S
+ − − =
1 1 2 21 2
( 1) ( 1) ( 1)
2
S S S S S SS S
+ − + − + =triplet
1 3 1 31 2
12 2 2 2
2 4
− −
= =
1 2
1 3 1 30 1
32 2 2 2
2 4S S
− −
= = −singlet
sin 02gletE K J= + +
02tripletE K J= + −0 1 22 2
2
JH K J S S= + − −
39
Spontaneous Magnetic Order
,
2 ij i j B i
i j i
H J S S g B S= − +
• In an insulator (electrons could not hop from atom to atom)
• Exchange energy J
0ijJ Spins are anti-aligned.
Spins are aligned.0ijJ
The couplings between spins would drop rapidly as the distance
between spins increases.
Only consider nearest-neighbors.
40
Spontaneous Magnetic Order
,
2 i j
i j
H J S S
= −
For an uniform system without any applied field
,
2 ij i j B i
i j i
H J S S g B S
= − +
Heisenberg Hamiltonian
In the case of a ferromagnet, there could ordering of magnetic
moments even in the absence of any applied magnetic field:
spontaneous magnetic order.
41
Mean field theory (Weiss 1906)
For the ith spin of all the other spin in the solid, taken as the mean
effect field of the FM system
,
2 ij i j B i
i j i
H J S S g B S
= − +
i B i effH g S B= −
2 ij j
j
eff mf
B
J S
B B B Bg
= − = +
mean filed or
molecular field
mf mfB M=
42
Mean field theory (Weiss 1906)
Ferromagnetism
' ( ') ( ')B J S JM ng JB y M B y= =
' ( )'
B mf
B
g B My J
k T
+=
' ( )B JM ng JB y=
2 1 2 1 1 1( ) coth( ) coth( )
2 2 2 2J
J JB y y y
J J J J
+ += −
Paramagnetism
' B
B
g By J
k T
=
43
Mean field theory (Weiss 1906)
Solid State Physics by Schmool (2017)
44
Curie temperature
' 1y
31( ') ' ( ' )
3J
JB y y y
J
+ +
1' '
3 '
B CS
B mf
k TJM M y y
J g J
+= =
Without external magnetic field:
''
B mf
B
g My J
k T
= '
'
B
B mf
k TM y
g J =
' ( 1)
3
B mf S
C
B
g J MT
k
+=
45
Mean field theory (Weiss 1906)
Solid State Physics by Schmool (2017)
46
Mean field theory (Weiss 1906)
' ( )1
3
B mf mfC
S B mf S
g B M B MTM JJ
M J k T T M
+ ++=
CT T
( )mf C CM T T BT − =
/C mf
C
TM
B T T
= =
− Curie-Weiss law
47
Example of a phase transition to a state of new order
• At high temperature, the material is paramagnetic. Magnetic
moments on each atom are disordered.
• At a critical temperature Tc the moments order. Total
magnetization M is an “order parameter”
• Transition temperatures: 1043 K in Fe, 627 K in Ni, 292 K in Gd
Introduction to Solid State Physics by Kittel (2005)
48
Symmetry Breaking
,
2 i j
i j
H J S S
= −
It is rotational symmetric: the magnetization could point in any
direction and the energy would be the same.
Anisotropy energy
If the anisotropy energy is extremely large, it would force spin to
be either SZ=S or –S.
2
,
2 ( )z
i j i
i j i
H J S S S
= − −
Ising model
In some materials, the spin are lying in the xy plane.
XY model
49
Easy axis
Introduction to Solid State Physics by Kittel (2005)
50
Spin wave
Introduction to Solid State Physics by Kittel (2005)
Introduction to Solid State Physics by Kittel (2005)
51
Ferrimagnets
52
Antiferromagnetism (predicted by Neel, 1936)
• Magnetic moments can also order to give no net moment –antiferromagnetism
• Transition temperature Ttransition = TNeel
MnO (O2+ not shown)
Introduction to Solid State Physics by Kittel (2005)
53
Antiferromagnetism (predicted by Neel, 1936)
Phys. Rev. B 58, 11583 (1998)
54
Antiferromagnetism (AFM)
' ( )1
3
B mf mfN
S B mf S
g B M B MTM JJ
M J k T T M
− −+=
( )mf N NM T T BT + =
/N mf
N
TM
B T T
= =
+Curie-Weiss law for AFM
ij j
j
eff mf
B
J S
B B B Bg
= − = +
mf mfB M= −0ijJ
Magnetization as a function of field in AFM
materials
Introduction to Solid State Physics by Kittel (2005)
Magnetization of CoV2O6 as a function of field
Zhangzhen He et. al., J. Am. Chem. Soc. 131, 7554 (2009).
57
Introduction to Solid State Physics by Kittel (2005)
58
Reciprocal susceptibility of magnetite, Fe3O4
Introduction to Solid State Physics by Kittel (2005)
Direct exchange
59
Direct exchange
If the electrons on neighboring magnetic atoms interact via an
exchange interaction, this is known as direct exchange.
Usually direct exchange cannot be an important mechanism in
controlling the magnetic properties because there is no sufficient
overlap between neighboring orbitals.
60
Indirect exchange
indirect exchange: super exchange, double exchange, RKKY
The exchange interaction is normally very short-ranged, and thus the
longer-ranged interactions is called super-exchange interaction.
Superexchange can be defined as an indirect exchange interaction
between non-neighboring magnetic ions which is mediated by a non-
magnetic in-between ion.
61
Superexchange
Because superexchange involves the oxygen orbital as well as
metal ion, it is a second-order process and is derived from second-
order perturbation theory.
The matrix element is controlled by a parameter called the
hopping integral t, which is proportional to the band width of the
conduction band or the bandwidth in a simple tight-binding model.
Goodenough-Kanamori rule
The size of the superexchange depends on the magnitude of the
magnetic moments on the metal atoms and the metal–oxygen (M–
O) orbital overlap and the M–O–M bond angle.
AFM
FM
J. Stöhr and H. Siegmann, Magnetism (2006).
, ,0 ,
, 0 0
,0 0
, 0 0 0
t
t U
−
−
2
1
2
2
3
0
tE
U
E
tE U
U
= −
=
= +
1
2
3
, ,0
,
,0 ,
t
U
t
U
= +
=
= −
The antiferromagnetic coupling lowers the energy of the system
by allowing these electrons to become delocalized over the whole
structure, thus lowering the kinetic energy.
64J. Stöhr and H. Siegmann, Magnetism (2006).
65
Double exchange (Zener, 1951)Clarence Zener introduced double exchange to explain the
magneto-conductive properties of mixed-valence solids, notably
doped Mn perovskites.
A. Urushibara et al., Phys. Rev. B (1995).
66
Phase diagram of La1-xSrxMnO3
A. Urushibara et al., Phys. Rev. B (1995).
67
Double exchange (Zener, 1951)
Because in O2− the p-orbitals are fully occupied, the process has to
proceed in two steps by “double exchange”.
The movement of an electron from O to one ion followed by a
transfer of a second electron from the other ion into the vacated O
orbital.
Zener proposed a mechanism for hopping of an electron from one
Mn to another through an intervening O2−.
68
Double exchange (Zener, 1951)
J. Stöhr and H. C. Siegmann, Magnetism (2006)
, , ,
, 0
, 0
, 0 0
U t
t U
U
−
−
1
2
3
0
E U t
E
E U t
= −
=
= +
1
2
3
, ,
,
, ,
= −
=
= +
The double exchange interaction favors a ferromagnetic alignment.
69
Double exchange (Zener, 1951)
J. Stöhr and H. C. Siegmann, Magnetism (2006)
The ability to hop reduces the
overall energy and thus LSMO
ferromagnetically aligns to save
energy.
The FM alignment then allows
the eg electrons to hop through
the crystal and thus the material
becomes metallic.
Fe3O4
Fe3+
Fe3+
Fe2+
double-exchange
Super-exchange
71
Double exchange (Zener, 1951)
In Fe3O4, a double exchange interaction ferromagnetically aligns
the octahedral Fe2+ and Fe3+ ions, while the superexchange
between tetrahedral and octahedral Fe3+ is antiferromagnetic.
Therefore, the two sets of Fe3+ ions cancel out, leaving a net
moment due to the Fe2+ ions alone. So, Fe3O4 is a ferrimagnetic
system.
In general, two metal atoms which are bonded through O may have
a valency that differs by one, such as for the cases Mn3+ (3d4) and
Mn4+ (3d3) in La1−xSrxMnO3 (LSMO) and for Fe2+ (3d6) and Fe3+
(3d5) in Fe3O4.
The electron is thus delocalized over the entire M–O–M group and
the metal atoms are said to be of mixed valency.
72
Solid-State Physics by James Patterson and
Bernard Bailey (2010)
Ruderman-Kittel-Kasuya-Yosida (RKKY)
interactionThe spin polarization of the conduction electrons oscillates in sign
as a function of distance from the localized moment and that the
spin information was carried over relatively large distances.
73
Screening can exist for either spin or charge scattering and results
in oscillations of the charge or spin density around the scattering
center
The oscillations in the charge density around a point-charge
impurity were first derived in 1958 by Friedel and hence go by the
name Friedel oscillations.
When conductive electrons are scattered by an atom, they will
rearrange themselves in order to minimize the disturbance. This
process is called screening.
Ruderman-Kittel-Kasuya-Yosida (RKKY)
interaction
74
Friedel oscillations
dI/dV as a function of distance
from a monatomic step.
M. F. Crommie et. al., Nature 363, 524–527(1993)
RKKY exchange
In the 4f metals the indirect mechanism involves the outer 5d
electrons which partly overlap with the 4f shell.
In contrast to the case of super-exchange, the indirect coupling
between two atoms thus proceeds through the outer electronic
states of the atoms themselves rather than through the electronic
states of a third atom.
J. Stöhr and H. C. Siegmann, Magnetism, Springer (2006)
RKKY exchange
The RKKY exchange coefficient J(R) is found to be oscillatory with
distance R according to
2 4
3 2 3 4
16 cos(2 ) sin(2 )( )
(2 ) (2 ) (2 )
e F F F
F F
A m k k R k RJ R
k R k R
= −
It makes a damped oscillation with distance from positive to negative
values.
2
3 2 3
2 cos(2 )1, ( )
(2 )
e F FA m k k R
R J RR
=
Therefore, depending upon the separation between a pair of ions their
magnetic coupling can be ferromagnetic or antiferromagnetic.
RKKY exchange
J. Stöhr and H. C. Siegmann, Magnetism, Springer (2006)
Ni80Co20 layers across a Ru spacer layer
Exchange interactions
Basic Aspects of the Quantum Theory of Solids by Khomskii (2010)
Exchange interactions
Basic Aspects of the Quantum Theory of Solids by Khomskii (2010)
80
Magnetic Domains and Hysteresis
81
Magnetic Domains and Hysteresis
In an actual ferromagnet, the material does not really break apart,
but different regions would have magnetization in different
directions to minimize energy.
The regions where the moments are aligned in one given direction is
called as a domain. The boundary of a domain is known as a domain
wall.
82
Hysteresis in Ferromagnet
Solid-State Physics by Ibachand Lueth (2009)
83
Hysteresis in Ferromagnet
Introduction to Solid State Physics by Kittel (2005)
X-ray microscopy
J. Stöhr and H. C. Siegmann, Magnetism, Springer (2006)
X-rays offer capabilities, such as elemental and chemical state
specificity, variable sampling depth, and the capability to follow
ultrafast processes on the picosecond scale.
One tunes the photon energy to a resonance and fix the photon
polarization and the magnetic contrast depends on the orientation of
the photon polarization relative to the magnetic orientation.
If now the sample contains microscopic regions with different
magnetic orientations, the signal from these regions will vary because
of the dichroic absorption effect.
Scanning Transmission X-ray Microscopy
(STXM)
J. Stöhr and H. C. Siegmann, Magnetism, Springer (2006)
In this approach the energy resolution is
given by the monochromator in the
beam line and the spatial resolution is
determined by the size of the X-ray spot.
The resolution is typically about 30 nm
with resolutions down to 10 nm or less.
They are “bulk” sensitive, in the sense
that the transmitted intensity is
determined by the entire thickness of the
sample.
Transmission Imaging X-ray Microscopy
(TIXM)
J. Stöhr and H. C. Siegmann, Magnetism, Springer (2006)
The spatial resolution of 15 nm has
been obtained.
Modern microscopes use a
monochromatic incident beam with
ΔE/E ~ 1/5,000 which also allows
spectroscopic studies of the detailed
near-edge fine structure.
X-ray Photoemission Electron Microscopy (X-
PEEM)
J. Stöhr and H. C. Siegmann, Magnetism, Springer (2006)
The third imaging method is based on X-
rays-in/electrons-out.
The sample is illuminated by a
monochromatic X-ray beam that is only
moderately focused, typically to tens of
micrometers, so that it matches the
maximum field of view of a photoelectron
microscope.
Most PEEM microscopes do not incorporate
an energy analyzer or filter and thus the
secondary electrons provides a suitably large
signal.
TIXM images of Fe/Gd multilayer
J. Stöhr and H. C. Siegmann, Magnetism, Springer (2006)
Exchange BiasExchange bias, arises if a thin film of a ferromagnet (FM), such as Co,
has a common interface with an antiferromagnet (AFM) such as CoO.
The size of the effect could only
be explained by assuming an
AFM–FM exchange interaction.
Many AFMs are best described by
two identical sublattices. In each
sublattice, the spins are parallel
generating a magnetization just
like in a FM.
MnO (O2+ not shown)Introduction to Solid State Physics by Kittel (2005)
The occurrence of exchange bias due to a “bias field” arising from the
antiferromagnet.
Exchange Bias
J. Stöhr and H. C. Siegmann, Magnetism, Springer (2006)
If a FM is deposited on an AFM in the absence of an external field, the
magnetization loop will still be symmetric and exhibit uniaxial
anisotropy.
Cooling FM-AFM systems across the TN, the magnetization loop may
be shifted horizontally in either the negative or positive field direction.
Exchange BiasThis case corresponds to a unidirectional magnetic anisotropy, since
the positive and negative external field directions are no longer
equivalent.
The field HB is called the transferred exchange field or the bias field.
J. Stöhr and H. C. Siegmann, Magnetism, Springer (2006)