Transcript 投影片 1 - National Cheng Kung University

12. Ferromagnetism and AntiFerromagnetism
Ferromagnetic Order
Curie Point and the Exchange Integral
Temperature Dependence of the Saturation
Magnetization
Saturation Magnetization at Absolute Zero
Magnons
Quantization of Spin Waves
Thermal Excitation of Magnons
Neutron Magnetic Scattering
Ferrimagnetic Order
Curie Temperature and Susceptibility of
Ferrimagnets
Iron Garnets
Antiferromagnetic Order
Susceptibility Below the Neel Temperature
Antiferromagnetic Magnons
Ferromagnetic Domains
Anisotropy Energy
Transition Region Between Domains
Solitons
Origin of Domains
Coercivity and Hysteresis
Single Domain Particles
Geomagnetism and Biomagnetism
Magnetic Force Microscopy
Magnetic Bubble Domain
Ferromagnetic Order
Spontaneous moment: M  0 when Hext = 0.
M =saturation moment 0
M =saturation moment 0
M = 0
M =saturation moment 0
M =saturation moment 0
Curie Point and the Exchange Integral
Paramagnet
N ions with spin S
Exchange (internal aligning) Field BE
(Large)
Thermal Agitation
(Small)
M = Magnetization = average magnetic moment per unit moment.
Mean field approximation:
BE   M
Ferromagnet
System in thermal
equilibriun at T
λ indep of T.
Curie temperature TC :
critical temperature for transitions between para- & ferro-magnetic phases
Paramagnetic phase under applied field Ba :
p 
C
T
Curie law
C
C
C

Ba

B
M   p  Ba  BE    B a   M 
a
C
T

C



T
T 1   
 T 
C
C
→


Curie-Weiss law
TC  C
T  C
T  TC
Using ( see Quantum Theory of
Paramagnetism, Chap.11 ),
Ni,
TC = 358C
N J  J  1 g 2 B2
C
3kB
we have

TC
3kBTC

C N J  J  1 g 2 B2
For iron, TC  1000K, g  2, J = S = 1 (Fe ~3d8), N = 1023 cm−3 so that λ  7300.
With M  1700, we have BE  107 G = 103 T .
Whereas a magnetic ion produces a field at its neighbor  μB / a3  103 G = 0.1 T.
Paul exclusion principle → 2 e’s with parallel spins have lower electrostatic energy.
This energy difference is called the exchange energy.
Heisenberg model :
Mean field with z n.n. → J 
U  2 J Si  S j
3kBTC
2 zS  S  1
For iron with S = 1, J = 8.3 meV
J = exchange integral
S=½
sc
bcc
kB T / z J 0.28 0.325
Better statistics
fcc
0.346
Temperature Dependence of the Saturation Magnetization
Mean field theory for T < TC , S = 1/2 , and replacing the Curie law with Brillouin expression
 M 
M  N  tanh 

k
T
 B

→
m
m  tanh  
t 
where
→ t=1
T << TC :
m
M
N
t
kB T
N 2 
N 2 
TC 
kB
tanh   1  2 e 2 
 2 2  N 
M  M  0   M T   2 N  exp  

k
T

B

M of simple (single domain) magnet
= saturation magnetization
Ni
 2 2  N 
 2TC 
M  2 N  exp  

2
N

ex
p



k
T
 T 

B

Experiment:
M
 A T 3/2
M  0
( compared well with spin wave theory)
A  7.5  10–6 deg –3/2 for Ni
A  3.4  10–6 deg –3/2 for Fe
Saturation Magnetization at Absolute Zero
M s  0  nB N B
3d 6 band
3d 7 ferromag
3d 8
4f
7
simple
ferromag
Cause for non-integral nB :
• spin-orbit interaction.
• local magnetization of conduction e.
• ferrimagnetism
Known simple ferromagnetic insulators
(spin all lined up in ground state):
CrBr3 , EuO, EuS
Ferromagnetism that can be described by band electron model:
transition metals Fe, Co, Ni
Cu: 3d10 4s1
paramagnetic
T > TC
Ni: 3d8 4s2
ferromagnetic
nB = 0.61
n –n = 0.54 e/atom
Problem 11.6
T < TC
Magnons
Magnon = Quantum of spin wave
Nearest neighbor Heisenberg model for atomic chain of N atoms:
U  2 J
N
S
p 1
p
 S p 1
S=½
G.S.: U0 = –2 N J S2
1st E.S.: U1 = U0 + 8J S2
U0 < U < U1
Magnon:
Total Δ Sz =1
Ref: Ziman, “Principles of the theory of solids”, 2 nd ed., §10.11.
N
S
U  2 J
p 1
p
 S p 1
Terms involving pth spin :
 2J

2 J S p   S p 1  S p 1   μ p   
S p 1  S p 1 
 g B



Bp  
 μ p  B p
Equation of motion:
d Sp
dt
  g B S p  B p
 μp Bp
Linearization for
S S
x
p
S ,S
2J
S p 1  S p 1 

g B
Exchange / effective field
Cartesian coordinates:
z
p
μ p  g B S p
y
p
d S pi
dt

2J
d S px
dt
d S py
dt
d S pz
dt
 2 J S p   S p 1  S p 1 
 i j k S pj  S pk 1  S pk 1 

2J S

0
 2S
2J S
y
p
 S py1  S py1 
 2S
x
p
 S px1  S px1 
d S px
dt
d S py
dt
d S pz
dt

2J S

 2S
2J S
x
p
 S px1  S px1 
0
S py  v ei  k p a   t 

 S py1  S py1 
 2S
S px  u ei  k p a   t 
4J S
y
p
→
1  cos ka 

i


  2 J S 2  e  i k a  ei k a




2J S

i


  4 J S 1  cos ka 


4J S
u iv
real solution:
2  e
i k a
i

 ei k a  
u 
   0
 v 


1  cos ka    u 
i
   0
 v 


S px  u cos  k p a   t 
S py  u sin  k p a   t 

4J S
1  cos ka 
Long wavelength limit:


n.n. interaction only:
cubic lattices:
2J S
a2k 2

  2 J S  z   cos k  δ 


δ
2 2
2
 

2
J
S
a
k

D
k
ka 1
Neutron scattering:
D at 295K
[ meV A2]
Fe
Co
Ni
281
500
364
a = lattice constant
Quantization of Spin Waves
Magnon is a boson :


1


1
 k  k  nk    4 J S 1  cos ka   nk  
2
2


Thermal Excitation of Magnons
nk 
Bose-Einstein distribution:
1
k / k B T
e
Average number of excited magnons at T :

k
ka << 1:
D   

k
nk

V
 2 
3
2J S
2
ak
→
2
k 
1
nk   d  D   n  
4 J S a2
k
V
k
1
V

V 
2

d


4

k

 k  2 3
4 J S a 2 k 8 2 J S a 2 4 2  2J S a 2 
V

4 2
V

4 2



2 
2
J
S
a


3/2 
 kB T 

2 
 2J S a 
 d
0
3/2

e
 / kB T
V

1
4 2

 kB T 

2 
2
J
S
a


3/2 
 dx
0
 k T 
 0.0587 V  B
2 
 2J S a 
3
 
2 2
3/2
x
ex 1
3/2
Let N = Q/a3 = numbers of atoms per unit volume, then Q = 1,2,4 for sc, bcc, fcc lattices.
 nk g B
M
 k
M  0
 N S g B
3/2
0.0587  kB T 



QS  2J S 
Bloch T 3/2 law

Neutron Magnetic Scattering
Fe
bcc,
h+k+l = even
Mn Pt3
Neutrons interacts with magnetic moments (p, n & e)
k n  kn  k  G
2 2
kn2
kn

 k
2M n 2M n
2
Ferrimagnetic Order
Magnetite ( Fe3O4 = FeO  Fe2 O3 ):
Ferric ( Fe3+ : d 5 ) ion: S = 5/2.
If all spins are aligned, then
Ferrous ( Fe2+ : d 6 ) ion: S = 2.
5 5

nB  0   2    2   14
2 2

per Fe3O4
Observed values is 4.1 (see Table 1) → Ferrimagnetism
spinel
MgAl2O4
cancelled
Ferrimagnetism is found in ferrites ( MO  Fe2 O3 ),
M = Zn, Cd, Fe, Ni, Co, Mg.
Most are poor conductors (good as rf transformer cores).
a ~ 8A
All J < 0 (antiferromagnetic)
|JAB | > |JAA | , |JBB |
→ AA, BB //, AB anti //.
Mg: 8 tetrahedral sites
Al: 16 octahedral sites
O: 32 vertices
B A   M A   MB
U 
BB   M A  MB
 , ,   0
1
1

B

M

B

M
 A A B B  2   M 2A  2 M A  M B  M 2B 
2
→ U is lowest when
For M A  MB  0
M A  MB  0
lowest U is 0
( ferrimagnetism )
 MA  MB  0 
 Ferrimagnetism is favorable if U < 0, i.e.,
2 M AM B   M A2  M B2
This condition doesn’t rule out other special configurations that may have lower energies.
Curie Temperature and Susceptibility of Ferrimagnets
B A   M B
Let λ = ν = 0 →
MA 
Mean field approx.:
B B   M A
CA
 Ba   M B 
T
MB 
U   M A  MB
CB
 Ba   M A 
T
C A   M A   C A 
 T


    Ba
C

T
M
 B
 B   CB 
→
For Ba = 0, a non-zero solution is possible only if
T
CB 
CA 
0
T
Fe3O4
→
TC  
CACB
For Ba  0, M A  T CA   C ACB Ba
T 2  TC2
MB 
→ 
T CB   C ACB
Ba
T 2  TC2
M A  M B T  CA  CB   2 CACB

T 2  TC2
Ba
Iron Garnets
Iron Garnets: cubic ferrimagnetic insulators M3Fe5O12 , e.g., YIG ( Y3Fe5O12 ).
Ionic states: M+3, Fe+3 (ferric, S = 3/2, L = 0 )
In YIG (Y3Fe5O12), Y+3 is diamagnetic.
At T = 0, each Fe+3 contributes  5B .
However, 5 Fe+3 → 3  (d-sites) + 2  (a sites) → net 1 
→ nB(0) = 5 per formula unit.
Mean field at a site (due to d site ions)  −1.5104 Md .
TC = 559K is due to a−d interaction.
Fe+3 are in L = 0 state → interaction with phonons is weak
→ sharp resonance lines
Antiferromagnetic Order
MnO
(NaCl lattice, Mn on fcc)
~ sc,
a = 8.85A
X-ray:
a = 4.43A for both.
~ fcc,
a = 4.43A
neutron
Spins on single (111) plane are //.
Spins on adjacent (111) planes are anti-//.
For an anti-ferromagnet in paramagnetic phase (at T > TN ),
 T C   M A  C 

  M     Ba
C

T

 B  C 

2C T   C 
2C

T 2   2C 2
T  C
→

2C
T  TN
T C   C2
MA  MB  2
Ba
T  2 C2
2C
experiment T  

TN   C
See Table 2 for values of TN / θ.
Discrepancy due mostly to 2nd neighbor interactions & sublattice structures.
  

Adding constant −ε to mean field gives
TN
 
Prob 12.3
Susceptibility Below the Neel Temperature
For T > TN , χ is independent of direction of Ba .
For T <TN , χ depends on whether Ba is // or  to Sz .
U   MA  MB  Ba   MA  MB 
Case  :
MA  MB  M
MA  MB  M 2 cos 2
B a  M A  B a  M B   Ba M cos
→
1
  2   Ba M sin 
2
1
2
2



M
1

2



U   M cos 2  2Ba M sin 

  2 Ba M 
 2

2
Ground state:

→
dU
 4 M 2  2 Ba M  0
d
→

ˆ  2M B
ˆ  Ba Bˆ
MA  MB  2M sin  B
a
a
 a
1
 
at T = 0

Ba
2 M
φ→0
U   MA  MB  Ba   MA  MB 
Case // :
MA  MB  M
MA  MB  M 2
Ba  M A  Ba  MB  Ba M
→
U   M 2
independent of Ba →
// T  0  0
// T  TN    T  TN 
MnF2
In strong fields, spins jump from
// to , which has lower U.
See D.Wagner,”Introduction to the
Theory of Magnetism”, p.167-70.
Antiferromagnetic Magnons
S2zp  S  S2zp1
Setting
d S pi
dt
2J

the linearized version of the equations of motion
 i j k S pj  S pk 1  S pk 1 
becomes
d S2xp
dt
d S2yp
dt


dt
S
S

2p
 2S
2J S
y
2p
 2S
S
x
2p
y
2 p 1
S

2iJ S
 2S

2p
S

2 p 1
ve
S

x
2 p 1
d S2xp 1
dt

d S2yp 1
dt
S

2 p 1
d S2p1

dt
u
i 2 p k a  t 
i  2 p  1 k a
y
2 p 1
 t 

2J S

 2S
2J S
y
2 p 1
 2S
 S2yp  S2yp 2 
x
2 p 1
 S2xp  S2xp 2 
we have
ue

2 p 1
S
x
2 p 1
S  Sx  i S y
Setting
d S2p
2J S
2J S
→
v
2J S

2iJ S
 2S
 2u  v  e  i k a  ei k a  


 2v  u  e  i k a  ei k a  



2 p 1
 S2p  S2p2 
u
v

2J S
2J S
 2u  v  e  i k a  ei k a  


 2v  u  e  i k a  ei k a  


   ex

 cos ka
4J S
→
 v  ex  v  u cos ka 
 cos ka  u 
   0
  ex   v 
Nonzero solution only if
→
ex  
 u  ex u  v cos ka 
  ex ex cos ka
2
2
2
2
 0    ex  ex cos ka
ex cos ka
  ex
 2  ex2 sin 2 ka
  ex sin ka

 ex a k
ka 0
Magnons were observed
up to 0.93 TN in MnF3
RbMnF3
Ferromagnetic Domains
For T < TC , observed M of macroscopic specimen are often much less than expected.
Reason:
Full spin alignment occurs only in small regions called domains.
Direction of magnetization in different domains are uncorrelated so that M tends to be 0
when Ha = 0.
M=0
Domains also form in antiferromagnets, ferroelastics, superconductors, and
sometimes metals undergoing strong dHvA effect.
For weak Ha , volume of domains with M // Ha increases → total M increases.
As Ha increases, domains begin to merge or disappear,
the process becomes irreversible (hysteresis).
In strong Ha , M in all domains begin to align with Ha.
Hysteresis loop
Hc = coercivity
Br = Remanence
Bs = Saturation induction
Anisotropy Energy
In ferromagnetic crystals, M tends to align with certain directions (easy axes).
This preference is due to the anistropy (magnetocrystalline) energy.
bcc
fcc
hexagonal
spin-orbit coupling
makes L , S anti-parallel.
For Co
(hcp):
For Fe
(bcc):
U K  K1 sin 2   K2 sin 4 
K1  4.1106 erg / cm3
θ = angle from hexagonal axis.
K2  1.0 106 erg / cm3
U K  K1 12 22   22 32   3212   K 2 12 22 32
K1  4.2 105 erg / cm3
αj = direction cosine wrt cubic axes.
K2  1.5 105 erg / cm3
Transition Region between Domains
Bloch wall
U  2 J Si  S j  2J S 2 cos 

 0
Let
 1 
 2J S 2 1   2 
 2 
wex  U   U  0
 2J S 2 1  cos  
For a spin-flip (φ = π) between adjacent sites :
For small φ,
wex  4J S 2
wex  J S 2 2
2 2
J
S

2  
Doing a spin-flip in N steps : wex  NJ S   
N
N
2

wex
However, larger N increases wall thickness & hence the anisotropy energy.
0
N 
Exchange energy for line of N atoms thru wall
Let σ denote energy per unit area of wall, then
J S 2 2
wex 
N
 wall   ex   anis
2 2
J
S

For a cubic crystal with lattice constant a,  ex 
N a2
J S 2 2
 wall 
 K Na
N a2
 wall
J S 2 2
  2 2  Ka  0
N
N a
→
N
 anis  K  wall thickness 
 K Na
J S 2 2
K a3
 wall  2
 300 in Fe
KJ S 2
a
Accurate calculation for a 180 wall in a (100) plane gives
 1 erg /cm2 in Fe
 wall  2
2K1J S 2
a
Origin of Domains
domians of
closure
U = U0
U =U0 / 2
U =U0 / 4
U=0
U=0
Coercivity and Hysteresis
Coercivity HC = field requires to reverse B to 0.
Supermalloy
Fe-Si
Alnico V
SmCo5
HC [G]
0.002
0.5
600
10,000
Application
pulse
transformer
commercial
transformer
loud speaker
magnet
Soft
less impurity / strain
e.g., Fe-Si, Fe-Co-Mn, Ni78Fe22, NiZn, MnZn,
metallic glass Fe79B13Si9, amorphous alloys
Si: ρ , εanis .
Ni78Fe22 : εanis  0, κm  0.
High HC of small grains or powder ( d < 10−5 cm ):
single domains, ΔM by rotation.
Rare earth metals in alloy with Mn, Fe, Co, Ni have
high K & εanis ~ 2K/M.
E.g., Fe (Hc ~500G), SmCo5 (Hc ~ 2K/M ~ 290kG ),
Nd2 Fe14 B ( B H ~50MGOe )
Hard
higher strain
e.g. precipitated phase
Alnico V
Single Domain Particles
Dominant application of ferromagnetism:
Magnetic recording devices, e.g., hard disks, audio & video tapes.
Recording medium are composed of single domain particles / regions.
Ideal single domain particle = elongated (acicular) particle with M // long axis.
For digital behavior, particle size ~ 10−100 nm.
First successful recording material is acicular τ-Fe2O3 with
long/short axes ratio ~5:1, HC ~ 200 Oe, length < 1 m.
Better material is CrO2 , with axes ratio ~20:1, HC ~ 500 Oe.
Superparamagnetism: paramagnetism below TC or TN .
Single domain consists of chain of spheres with fixed µ.
M follows Curie-Brillouin-Langevin law if particles are in liquid & free to rotate.
If liquid is then frozen, a permanent M retains after removal of Ha
Geomagnetism and Biomagnetism
Sedimentary rocks containing single domain particles can carry remanent M that gives a
record of earth’s H at the time of deposition.
These can be 500 million yrs old.
They provide the strongest support to continental drift theory ( plate tectonics).
Brunhes (1906) discovered earth’s H can reverse direction
(explained by standard dynamo theory).
Such reversal took place abruptly every 104 to 107 yrs.
Single domain particles of Fe2O3 are behind the directionseeking (magnetotaxis) behavior of biological entities such as
migrating birds, bees, and bacteria.
Magnetotaxis bacteria with chain
of 50 nm particles of Fe2O3
Magnetic Force Microscopy (M F M)
Resolution ~ 10-100 nm