BASICS AND MAGNETIC MATERIALS K.-H. Mller Institut fr Festkrper- - - PowerPoint PPT Presentation
BASICS AND MAGNETIC MATERIALS K.-H. Mller Institut fr Festkrper- und Werkstoffforschung Dresden, POB 270116, D-01171 Dresden, Germany 1 INTRODUCTION 2 MAGNETIC MOMENT AND MAGNETIZATION 3 LOCALIZED ELECTRON MAGNETISM 4
(1190): describes the compass
de Magnete
divorced physics from metaphysics
H.C. Oersted (1777-1851)
J.C. Maxwell (1831-1871)
A.M. Ampère (1755-1836)
H.A. Lorentz (1853-1928)
B & E & ) (
xB
v E v + = e m&
→ exchange interaction ←
ferrites 55% Nd-Fe-B 30% permanent magnets 20% soft magnets 27% magnetic recording 53%
Brain ; Intergalactic space 10-13 Heart 10-10 Galaxy 10-9 urbanic noise 10-6…10-8 Surface of earth 5·10-5 near power cable (in home) 10-4 Surface of sun 10-2 surface of magnetite 5·10-1 simple resistive coil 10-1 permanent magnets 1
16 superconducting coils 20 high performance resistive coils (stationary) 26 hybrid magnets (resistive + supercond.) 40 long pulse coil (100ms) 60 short pulse coil (10 ms) 80 Anisotropy fields in solids ≤ 102
3·102 Exchange fields in solids ≤ 103 explosive flux compression 3·103 Neutron stars 108
microscopic form of Maxwell's equ.
i e h + ε = ∇ & x h e & µ − = ∇ x = ∇h r
0 =
ε ∇ e r = ∇ + i &
j P e M h + + ε = − ∇ & & ) ( x ρ = + ε ∇ ) ( P e
macroscopic form of Maxwell's equ.
j D H + = ∇ & x = ∇B H E & µ − = ∇ x ρ = ∇D = ∇ + ρ j
macroscopic fields
µ ≡ / B h E e = M B H − µ ≡ / P E D + ε ≡
magnetic moment and magnetization
τ = = ∇ τ − ≡ ⇒ −∇ = ∇
V V
M d M r m M H K d
P ∇ − ρ = r P M j i & + ∇ + = x
magnetization M and polarization P
∇ τ = = ∇ τ − =
V V
) ( ) (
x x
M r M r m d 2 1 d K
L L v r m m e m e m e 2 g 2 d 2
m
= = ρ τ =
x
(Zeeman effect) spin of the electron:
scale mirror iron bar
∂B/∂z ≠ 0
Ag atoms
) (
xB
v E v + = e m&
) ) ( ) ( S L B S L ( λ − + + + − =
2 2 2
y x 4 B 2 2 2 m e m e
2
H H
⇒ m = <µ> = – ∂<H >/∂B M = N <µ> χij = µ0∂Mi/ ∂Bj = – µ0N ∂2< H >/∂Bi∂Bj diamagnetic susceptibility (in 10-6): H2O: –9, alcohol: –7.2
1 2 3 4 5 6 7 8 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Brillouin function
levitated glass cube melted by a laser beam
0 D – as in the examples above 1 D – as a train on a rail
YBCO NdFeB
problems: (i) thermodynamically metastable states (ii) the field generated by the samples own magnetization (iii) how to define a correct expression for magnetic work
U = < H > + µ0 Hm (a Legendre transformation) (iv) magnetostatic interaction is long range ⇒ ⇒ both expressions are correct: work done on different systems !
µ + δ = µ + δ =
V
M H m H d d Q d Q dU
µ − δ = µ − δ = > <
V
H M H m d d Q d Q d
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⇒ Curie's law M = χ H with χ = C/T
– large electron density ⇒ delocalized (itinerant) electrons – e.g. in Li- or Na-metal ⇒ Hund's rule magnetic moment disappears ⇒ a small, (nearly) temperature independent susceptibility – small electron densities ⇒ strongly correlated electrons ⇒ they can be localized and carry a magnetic moment ⇒ examples: MnO, FeO, CoO, CuO (antiferromagnets); EuO, CrCl3 (ferromagnets)
) * ( ) (
2 2 F 2 B
3 1 1 E 2 m m N − µ µ = χ
with
m e 2
B
h | | = µ
⇒ metals with large or moderate m* (e.g. Na: m/m* ≈ 1) are Pauli paramagnets, χ > 0 ⇒ those with small m* (e.g. Bi m/m* ≈ 102) are Landau diamagnets, χ < 0
MOLECULAR SUSCEPTIBILITY TEMPERATURE [°C]
800 600 400 200 10-3 10-4 10-5 10-6
⇒ discrete Landau levels: ⇒ de Haas-van Alphen effect, Shubnikov-de Haas effect ⇒ an oscillation of M(H)
* 2 ) k ( H 1/2) (n E
2 z n
m m e h h + µ + = * | |
with H || z
0.04 0.08 0.12 0.16 0.20
0.0 0.1 0.2 0.3 Applied field µ0H (T)
CeBiPt H II [100]
25 5
10
1/(µ0H) (1/T)
50
T = 4.2 K
] [M M c M ) (E I 1 ) (E 4 1 E
6 4 2 F F 2 B
O N N + + − µ + µ − = ) ( MH
⇒ finite value of M, even for H = 0, if I N(EF) > 1 (Stoner condition)
⇒ exchange enhancement
) (E I 1 ) (E 2
F F 2 B
N N − µ µ = χ
⇒ modified theory of spin fluctuations (T. Moriya 1978) ⇒ e.g. the afm in Cr ⇒ e.g. in Pd
Semiconductor
Ge, Si, …
Semimetal
Bi, Sb, …
valence band conduction band
Half-metal
CrO2, Mn2VAl
EF
weak ferromagnet
Fe, …
weak ferromagnetism
α-Fe2O3, La2CuO4
M1 M2
spin canting in AFM
4s 3d
strong ferromagnet
Co, Ni, …
4s 3d
in alloys and compounds ⇒
susceptibility temperature (K)
CuSO4·5H2O
in alloys and compounds
Temperature (K) Magnetization (mT)
in alloys and compounds
⇒ no rotational symmetry ⇒ Lz no good quantum number ⇒ Lz-mixing : Lz/ħ = +2, -2 ⇒ dx2-y2
number of 3d -electrons magnetic moment [ µB ]
spin-only value Hund's rule (free ion) values
h / ) 1 ( 2
B
S S µ µ + =
⇒ "High-spin–low-spin transition" or "spin quenching"
∆(CEF) < ∆(HR) ⇒ S = 2ħ (high spin) Hund's rule value ∆(CEF) > ∆(HR) ⇒ S = 0 (low spin)
⇒ no rotational symmetry ⇒ Lz no good quantum number ⇒ Lz-mixing : Lz/ħ= +2, -2 ⇒ dx2-y2
d(Cu-O) ≈ 1.92Å a-b plane P4/mbm c
⇒ S = ħ/2
Magnetic field (Oe)
200 400 600 2 4 6 8
T = 2K
Magnetization [emu/g]
⇒ zero field splitting (simplest case): , J/ħ integral ⇒ Jz = 0
) ) ( ) ( S L B S L ( λ − + + + − =
2 2 2
y x 4 B 2 2 2 m e m e
2
H H
⇒ Van Vleck paramagnetism
⇒ "Kramers ions" (e.g. Cu++, Nd3+) ⇒ "non-Kramers ions" (even numbers of electrons as in Fe++, Pr3+) ⇒ singlets possible
2 z
J D ~
CEF =
H
χ-1 [a.u.] temperature temperature χ-1 [a.u.]
10 energy levels
S = ħ, L = 5ħ , J = 4ħ χ(T→0) finite ⇒ Van Vleck paramagnetism
S = 3ħ/2, L = 6ħ, J = 9ħ/2 χ(T→0) → ∞ ⇒ Curie-Langevin paramagnetism Pr2(SO4)3·8H2O Nd2(SO4)3·8H2O
⇒ zero field splitting (simplest case): , J/ħ integral ⇒ Jz = 0
) ) ( ) ( S L B S L ( λ − + + + − =
2 2 2
y x 4 B 2 2 2 m e m e
2
H H
⇒ Van Vleck paramagnetism
⇒ "Kramers ions" (e.g. Cu++, Nd3+) ⇒ "non-Kramers ions" (even numbers of electrons as in Fe++, Pr3+) ⇒ singlets possible
2 z
J D ~
CEF =
H
5 j i 2 j i dip
r 4 3 r j) (i, E π − µ = ) ( ) ( ) ( r m r m m m
⇒ is omnipresent in magnetic materials ⇒ results in magnetic ordering temperatures of typically 1 K ⇒ is long range ⇒ anisotropic
H'(r)M(r) τ µ − =
d 2 E
self
µ =
j i, j i j i, self
M M D V 2 E
=
i i
1 D
Di ≥ 0 demagnetization factors ⇒ "demagnetizing fields" <H'i(r)>V = – Di M (bodies of arbitrary shape!) with
, ) ( ) ( ' r M r H ∇ − = ∇ = ∇ ) ( ' x r H
H' M sphere D =1/3 H' = – M/3 hollow sphere D =1/3
H'=0
M H' = 0 H' = –M <H'> = – M/3 cube D = 1/3 <H'> = – M/3 H'(r)
non-uniform
prolate spheroid M D = 0 ⇒ H' = 0 prolate spheroid M D = 1/2 ⇒ H' = –M/2 long cylinder M H' = –M/2 H' = M/2 D = 0 ⇒ <H'> = 0
5 j i 2 j i dip
r 4 3 r j) (i, E π − µ = ) ( ) ( ) ( r m r m m m
⇒ is omnipresent in magnetic materials ⇒ results in magnetic ordering temperatures of typically 1 K ⇒ is long range
H'(r)M(r) τ µ − =
d 2 E
self
µ =
j i, j i j i, self
M M D V 2 E
=
i i
1 D
Di ≥ 0 demagnetization factors ⇒ "demagnetizing fields" <H'i(r)>V = – Di M (bodies of arbitrary shape!)
with ⇒ tensor character of (Di,j) ⇒ shape anisotropy
, ) ( ) ( ' r M r H ∇ − = ∇ = ∇ ) ( ' x r H
5 ij ij j ij i 2 ij j i dip
r 4 3 r j) (i, E π − µ = ) ( ) ( ) ( r m r m m m
⇒ its effect is sensitive to to the presence of other interactions 1.) two dipoles governed by Edip only ⇒
2.) if additionally m1 || m2 required (strong exchange interaction) typical easy-axis magnetic anisotropy
) ( ϑ − π µ =
2 3 ij 2 dip
3 1 r 4 m E cos
ϑ
rij rij rij
3.) if the dipoles are confined to a certain axis (by strong CEF) antiferromagnetic alignment ferromagnetic alignment a) b)
axis axis
rij rij
wave function antisymmetric in r1 and r2 symmetric in r1 and r2 energy difference at R0 : Eant – Esymm ≡ -J
s = s1 + s2 Ψ(s=1, sz=1) = (χ1(↑) χ2(↑) si =1/2 Ψ(s=1, sz=0) = {(χ1(↑) χ2(↓) + χ1(↓) χ2(↑)}/√2
(with S = ħs etc.)
Ψ(s=1, sz=-1) = (χ1(↓) χ2(↓) ⇒ Hex(i,j)= – J sisj ⇒ description in spin space although purely electrostatic in its nature ⇒ isotropic
H2-molecule
R E(R)
spin ↑↑ spin ↑↓
⇒ high ordering temperatures (molecular field of P. Weiss: <JSi> ) ⇒ isotropic Hex(i,j)= – J si sj
> <
⇒ long range and J oscillating in magnitude and sign, e.g. in 4f-elements
mobile Mn-3d electrons mediate the exchange between neighbouring Mn magnetic ions ⇒ ferromagnetic metals
in materials with singlet CEF ground states of non-Kramers ions e.g. the ferromanget PrPtAl : Tc ≈ 6 K ∆(CEF) = 21 K the antiferromagnet PrNi2B2C : TN = 4 K
in ionic compounds (e.g. Cu++ ⇒ kBJ ≈ –2000 K) mediated by anions (e.g. O--) ⇒ mostly J < 0
105 104 103 102 Energy/kB [K]
intratomic correlation between electrons crystalline electric field SL coupling SL coupling crystal field room temperature interatomic exchange coupling Zeeman energy (external field )
2 z sia
S D ~ =
H
caused by CEF
µ − µ − − ∇ τ =
V
' ) ( ) ( ] 2 M K M A [ d F
2 s 2 2 s 2
MH MH nM M
⇒ continuum description (micromagnetism) simplest form of magnetic anisotropy in uniaxial materials
⇒ represented by the demagnetization tensor (Dij):
µ =
j i, j j i, i self
M D M V 2 E
⇒ uniaxial bodies excluded: and spheres, cubes etc.
ϑ − = ϑ − − µ =
⊥ ⊥ 2 sh 2 2 self
K const 1) (3D D [ 2 VM E cos cos ]
⇒ ⇒ K = KCEF + Ksh with Ksh = (3D⊥ – 1) µ0VM2/2 e.g. ALNICO : needles of Fe-Co (1 µm x 1 nm; µ0Ms ≈ 2.4 T ⇒ D⊥ = 0.5) embedded in low-Ms Ni-Al
= ) (D j
i, ⊥
D
⊥
D
⊥
D 2
⇒ tends to orient spins Si and Sj perpendicular to each other and to d ⇒ causes canting of the sublattice magnetizations ⇒ weak ferromagnetism for sufficiently low symmetry (e.g. α-Fe2O3)
anisotropy because Sx
2 = Sy 2 = Sz 2 = ħ2/4 are constants
⇒ anisotropy of the g-factor in
e.g. CuSO4·6H2O : gz = 2.20, gy = gz = 2.08
combination of exchange interaction with CEF and S-L-interaction
j ij i ae
D S S ˆ =
H
symmetric tensor
ij
D ˆ
⇒ "pseudodipolar interaction"
e.g. W.H. Meiklejohn and C-P. Bean 1957: exchange anisotropy
ferromagnet antiferromagnet CoO coated Co particles
H(kOe) M (a.u.) M (a.u.) H(kOe)
T = 77 K T = 1.8 K
Ag80Mn20 spin glass both curves obtained after field-cooling ⇒ exchange bias in information storage technology
"spring magnets" (e.g. Nd4Fe77B19)
µ − µ − − ∇ τ =
V
' ) ( ) ( ] 2 M K M A [ d F
2 s 2 2 s 2
MH MH nM M
simplest form of magnetic anisotropy in uniaxial materials ⇒ at H = 0 a first order phase transition : M changes from nMs to – nMs
F ϑ π
0 ≤ H < HA HA < H H < – HA – HA < H < 0
ϑ
M n, H
JHc ≥ HA = 2 K/(µ0M)
⇒ for – JHc < H < 0 the magnetized state is metastable
"Brown's paradox " ⇒ ⇒ ⇒ explained by imperfections in the material ⇒ supported by thermal fluctuations and quantum tunneling ⇒ soft magnetic materials : JHc << Ms ⇒ hard magnetic (or permanent magnet) materials : JHc ≥ Ms ⇒ need of well defined microstructure ⇒ materials sciences
H M H = 0
1.5 1.0 0.5
Magnetization [Tesla] µ0H [ T ] (BH)max ≈ 430kJ/m3
nucleation of reverse domains
0.0 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 hot pressed die upset
Polarization J [ T ] External field µ0H [ T ]
1.4 1.0 0.6 0.2 Magnetization [Tesla] µ0H [ T ]
2.0 1.0
1.0
1 2 3 4 5 6
t0 = 60 s S/Js = - 5.3 10-3 *
S/Ms = -5·10-3 t0 = 60 s
1 2 3 4 5 5 10 15 20 25
t0 = 88 s S/Js = 5 10-4
4Fe 77B 19
S/Ms = 5·10-4 t0 = 88 s
2 4
300 400 350
with Cu++ (S = ħ/2) magnetic moments plane: d = 2 chains: d = 1 isolated clusters: d = 0 dimer: d = 0 Cu O
O Cu
Cu F
they will be described by
⇒ they are magnetically isotropic ⇒ nevertheless their behaviour is very different because thermal as well as quantum fluctuations make cooperative phenomena very sensitive to d
d ground state µstagg/µB TN/| J | Tmax/| J| in χ(T) χ(T→0) t ≡ T/|J| 3 Néel AFM 0.85 0.93 – ≈ const. 2 Néel AFM 0.61 0.94 0.04·(1 + t ) 1 spin liquid – 0.64 ~ (1 + 0.5·ln(7.7/t)) dimer S = 0 singlet – 0.62 ~ e-1/t/t (spin gap)
j i s
χ J/Ng2µB
2
T/⎮J⎮ 1 2 3 0.04 Curie 0.06 0.08 0.10
d = 2 d = 3
T
χ
TN Curie
d = 1 and d = 0
χ T/⎮J⎮ 0.5 1.0
chain dimer
(C.M. Soukoulis et. al. 1991)
) 3ln(32J/J J 4 T k
N B ⊥
π =
⇒ nevertheless at high temperatures it behaves as a d = 2 system ⇒ a maximum of χ at kBTmax ≈ 0.94 J. example : La2CuO4: kBJ ≈ 1500 K , J⊥/J ≈ 10-5 , resulting TN ≈ 300 K Tmax ≈ 1400 K ⇒
⇒ also in d = 0 clusters and other low-d- systems
2
2 1 2 1 dim
/ ) (
, , , , + − − +
ψ ψ − ψ ψ = Ψ
large dimensionality, d ≥ 2 ⇒ rich variety of different afm structures
neutron scattering, x-ray magnetic scattering, Mössbauer spectroscopy, nuclear magnetic resonance (NMR) and muon spin relaxation (µSR)
⇒ no antiferromagnet but a singlet state ⇒ no magnetic moments at the sites 1 and 2
in spin-ladder compounds e.g. in SrCu2O3
⇒ much more complicated < >
alternating fm planes x = 0
fm x = 0.3
alternating fm chains x = 0.5
"simple" afm x = 1
magnetic moments mediated by itinerant electrons (RKKY)
⇒ small net magnetization (as e.g. in α-Fe2O3, La2CuO4)
inversion center
⇒ the center of inversion is not between the sites the afm sublattices ⇒ WFM
d ≠ 0
antiparallel and different in size e.g.Fe3O4, GdCo5
antiferromagnetic commensurate : incommensurate c-axis modulated : (spiral structure) q = 0.916c* Additionally there is an incommensurate a-axis modulated with q = 0.585a*
a a ≈ 105° ≈ 75° ≈ 60° ≈ 90° c 45°
are in competition ⇒ the system can form more than two magnetic sublattices
solid structures (as e.g. amorphous solids) ⇒ thermodynamic or kinetic "glass" state ? ⇒ no equilibrium net magnetization ⇒ nevertheless strong hysteresis phenomena ⇒ no long range order
frustration in a tetrahedron (as in the pyrochlore structure) with afm coupled spins i (i =1 … 4): ⇒ the afm alignment cannot be satisfied, at the same time, for all spin pairs S1 S4 S3 S2
⇒ due to the non-linearity of the problem a domain wall will be formed ⇒ wall width (Nd2Fe14B : 3 nm) ⇒ wall energy per unit area
A/K ~ δ AK ~ γ
in random anisotropy systems domain like structures ⇒ competition of exchange and random anisotropy ⇒ even without magnetostatic selfenergy
spontaneous formation of domains
2 s c
M AK D µ / ~
(Nd2Fe14B : 300 nm)
µ − − ∇ τ =
V
' ) ( ) ( ] 2 M K M A [ d F
2 s 2 2 s 2
MH nM M
δ easy axis domain domain localization
Hot-deformed melt-spun Nd-Fe-B : grain size ≈ 200 …300 nm ≤ Dc ≈ 400 nm
10 µm 10 µm 20 µm 20 µm
thermally demagnetized dc-field demagnetized
– by Kerr micoscopy –
10 µm
– by Kerr micoscopy –
2 4
0,0 0,5 1,0 1,5
polarisation J (T) applied field µ0H (T) 100µm VACODYM 722HR
Kerr microscopy
2 4
0,0 0,5 1,0 1,5
polarisationJ (T) applied field µ0H (T) 100µm VACOMAX 240HR
Kerr microscopy
⇒ change at T = Tc (or TN) ≈ J/kB from the paramagnetic state to fm or afm
by spin waves
by mean field approximations M/Ms ≈ (1 – T/Tc)1/2
by scaling and renormalization theory
M/Ms T/Tc
states of different symmetry (L.D. Landau)
M/Ms T/Tc 1 1
ferromagnet
MnAs
e.g. MnAs, RCo2 (R = Dy, Ho, Er) and GdSi2Ge2 ⇒ magnetic refrigeration
z j j i, z i s
s J∑ − =
M/Ms
J < 0 T (K) kBTN = 2.27J M/Ms ≈ ( 1 – T/TN)1/8 ⇒ strong influence of the dimensionality on the critical behaviour
z = ±1)
⇒ many different types of phase transitions
antiferromagnetically at TN[Cu] ≈ 95 K
100 200 300 Temperature (K) 15 10 5
Magnetization (a.u.)
to M in ferrimagnets at T = Tcomp
ϑ Ms c
contributions of different subsystems to magnetic anisotropy are different example Nd2Fe14B :
+ ϑ + ϑ τ =
V
] [ K
4 2 2 1 A
K K d E sin sin
i.e. K → K1 , … K1 = K1[Nd] + K1[Fe] ⇒ ⇒ ⇒ ⇒ a transition at T = TSR : easy axis → easy cone K2 K1
TSR = 135 K
tilting angle ϑt TSR temperature [ K ] ϑt
if afm exchange is relatively weak compared to the anisotropy field
interaction between the fm planes Dy Co Si
H | | [001]
µ (µB/f.u.)
relatively strong compared to the anisotropy field
2
flop transition, examples GdCo5, DyCo12B6
10 20 30 40 50 60
Hcr = 46 T
magnetization (arb. units) magnetic field (Tesla)
Co Gd H H > Hcr
Gd Co M M1 M2 Hcr ∼ |J||M1 – M2| T = 5 K
⇒ competition of magnetostatic (Zeeman) energy with the anisotropy energy of different orders
5 10 15 20 25
10 20 30 40 Nd2Fe14B H || a magnetisation [µB/f.u.] applied field µ0H [T]
130 K 30 K FOMP
⇒ anisotropy in the tetragonal basal plane
e.g. TmSb, PrNi5, Pr ⇒ from Van Vleck paramagnetism to Langevin-Curie paramagnetism
⇒ from Pauli paramagnetism to ferromagnetism, e.g. YCo2, LuCo2, La(Fe,Si)13
⇒ modification of scaling and renormalization ⇒ consequences of the quantum critical behaviour at non-zero temperature ?
µ − − =
i x i x B j i, z j z i
s H g s s J
si
z = ± 1/2 , J > 0 ⇒ Ising ferromagnet
Temperature [ K ] µ0Hx [ T ]
6 4 2
in a field applied perpendicularly to its magnetic axis it becomes a (quantum) paramagnet
quantum critical points (QCP)
crossover to 3d behaviour at low temperatures occurs only above a critical value of J⊥/J J' = J J J⊥
J⊥/J TN /J
0.2 0.4 0.6 0.8 1 0.5 1.0 1.5
QCP
the spin gap ∆ opens immediately upon the introduction of non-zero J' ⇒ no QCP
Barnes et al. 1993
0.5 1.0 2.0 J'/J ∆/J