Magnetic Anisotropy. Anisotropy. Günther Bayreuther. University of Regensburg,
93040 Regensburg, Germany. Max Planck Institute of Microstructure Physics, ...
Magnetic Anisotropy Günther Bayreuther University of Regensburg, 93040 Regensburg, Germany Max Planck Institute of Microstructure Physics, 06120 Halle, Germany
1. relevance and definition of Magnetic Anisotropy ─ MA 2. phenomenological description of MA 3. microscopic origin of MA 4. theoretical description of MA 5. how to measure the strength of MA 6. symmetry and MA 7. tailoring and manipulating MA 8. summary
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1. Relevance and definition of Magnetic Anisotropy early application of magnetic materials: the magnetic compass
Model of a Han Dynasty (206 BC–220 AD) south-indicating ladle or sinan.
Pivoting compass needle in a 14th century copy of Epistola de magnete of Peter Peregrinus (1269)
How a magnetic compass works (Wikipedia): A compass functions as a pointer to "magnetic north" because the magnetized needle at its heart aligns itself with the lines of the Earth's magnetic field. The magnetic field exerts a torque on the needle, pulling one end or pole of the needle toward the Earth's North magnetic pole, and the other toward the South magnetic pole
2
Origin of magnetic torque? magnetic dipole in a uniform magnetic field:
r r Em = −V ⋅ M ⋅ B = −V ⋅ M ⋅ B ⋅ cos ϕ Zeeman energy ϕ = ∠(M,B) dEm torque exerted on the magnetization T= = V ⋅ M ⋅ sin ϕ dϕ
r r r T =V ⋅M ×B
⇒ stable equilibrium for ϕ = 0 i.e. M aligned with B
torque on the needle?
T'=
dEa dφ
φ = ∠(M, needle)
⇒ T‘≠ 0 only if Ea = Ea(φ) , i.e. if magnetic anisotropy is present !!
Ea (φi ) = V ⋅ ε a (φi )
anisotropy energy
r r φi = ∠( M , ei ) 3
other applications of MA strong/large MA (hard magnetic materials) ▶ permanent magnets ▶ electric motors ▶ loudspeakers, microphones ▶ magnetic memories ▶ ...
weak/small MA (soft magnetic materials) ▶ transformers ▶ electromagnets ▶ electric motors - magnetic field sensors - ...
4
2. phenomenological description of MA required:
series expansion ↔
Ea (φi ) = V ⋅ ε a (φi )
r r φi = ∠( M , ei )
ε a (φi ) = K 0 + K1 ⋅ F1 (φi ) + K 2 ⋅ F2 (φi ) + ....
set of functions according to symmetries
e.g. even functions of φi for inversion symmetry Ki …. anisotropy constants
Remarks i) concept of MA inplies uniform M ! ii) MA different from anisotropy of MS:
↔ morb, mspin , SOC 0.01 - 0.1% - effect in bulk cubic materials % - effect at surfaces / in ML films
iii) anisotropic r r interaction r exchange
H = D12 ⋅ ( S1 × S 2 )
Dzyaloshinski-Moriya interaction ⇒ non-collinear spin structures 5
2. phenomenological description of MA required:
series expansion ↔
Ea (φi ) = V ⋅ ε a (φi ) ε a (φi ) = K 0 + K1 ⋅ F1 (φi ) + K 2 ⋅ F2 (φi ) + ....
set of functions according to symmetries
e.g. even functions of φi for inversion symmetry Ki …. anisotropy constants
example: (uni)axial symmetry hexagonal crystals, …
ε a = K 0 + K1 ⋅ sin 2 ϕ + K 2 ⋅ sin 4 ϕ + ... ϕ = ∠(M,symmetry axis)
uniaxial magnetic anisotropy (UMA) with
ε a = KU ⋅ sin 2 ϕ
KU>0: easy axis, hard plane KU 0, K2 = 0
10
εa surfaces for different UMA cases:
d) for K1 < 0, K2 > -K1/2
εa(ϕi)
from: R. Skomski, "Simple Models of Magnetism " Oxford University Press, 2008
11
HK HK sometimes treated as a vector field:
2 KU = MS
HK // e.a. in FMR experiments ⇒ Heff = H + HK for H // e.a.
caution !! → tensor quantity
Heff = H ─ HK
for H // h.a.
e.g. 2 x UMA with orthogonal e.a.
?
12
HK sometimes treated as a vector field: HK // e.a. in FMR experiments ⇒ Heff = H + HK
caution !! → tensor quantity e.g. 2 x UMA with orthogonal e.a.
?
13
HK sometimes treated as a vector field: HK // e.a. in FMR experiments ⇒ Heff = H + HK
caution !! → tensor quantity e.g. 2 x UMA with orthogonal e.a.
?
ε a = KU ⋅ sin 2 ϕ + KU ⋅ sin 2 ( π2 − ϕ ) = KU ⋅ (sin 2 ϕ + cos 2 ϕ ) = KU
⇔
isotropic behavior !
general case: 2 x UMA → UMA biaxial anisotropy not by superposition of 2 x UMA !! 14
cubic symmetry
bcc-Fe, fcc-Ni, fcc-Co, alloys …
ε a = K 0 + K1 ⋅ (α12α 22 + α 22α 32 + α 32α12 ) + K 2 ⋅ α12α 22α 32 + ....
αi = cos ∠(M,ei)
K1 > 0 (Fe), K2 = 0
K1 < 0 (Ni), K2 = 0
15
cubic symmetry
bcc-Fe, fcc-Ni, fcc-Co, alloys …
ε a = K 0 + K1 ⋅ (α12α 22 + α 22α 32 + α 32α12 ) + K 2 ⋅ α12α 22α 32 + .... αi = cos ∠(M,ei) a) K1 > 0 (Fe), K2 = 0 b) K1 < 0 (Ni), K2 = 0 c) K1 ≠ 0, K2 > 0
from: R. Skomski, "Simple Models of Magnetism " 16
Further remarks: i)
Ki vary with temperature
Fe
temperature [oC]
17
Further remarks: i) Ki vary with temperature ii) Ki values not unique, e.g. K2 and K3 for Fe; why? - methods! - slow convergence; Ki not independent ↔ series expansion not orthogonal,not normalized iii) Ki not “material property“; e.g. Ki vary with film thickness
18
3. microscopic origin of MA magnetic order due to exchange interaction; Heisenberg-type exchange interaction is isotropic
MA results from two fundamental interactions: dipolar (dipole▶ dipole▶dipole) dipole) interaction spin▶ spin▶orbit interaction/coupling (SOC)
19
3.1 dipolar interaction field of a point dipole
r m
r r r 2r r r µ 3(m ⋅ r )r − r m Bd (r ) = 0 4π r5 potential energy of two dipoles r r r r r r 2 3 m ⋅ R m ⋅ R − m i ij j ij i ⋅ m j Rij r r 1 Edd (ri , rj ) = − 4πµ0 Rij5
r r r Rij = ri − rj
dipolar energy of a lattice of point dipoles
r r Ed = ∑ Edd (ri , rj ) i, j
⇒
Ed ≠ 0 for non-cubic lattices Ed = 0 for an infinite cubic lattice
dipolar interaction irrelevant for Fe / Ni ? 20
why MR < MS along e.a.?
bulk (spherical) samples
50 nm Fe(001) on GaAs(001)
thin film, M // film plane [100] [110]
difference: dipolar / demagnetizing field 21
r M
ferromagnetic body ─ ellipsoid, sphere homogeneously magnetized
rr ∇B = 0 r r r B = µ0 ( H + M )
r r ∇H = C ⋅ ρ m ⇒
r r r r ⇒ ∇M = −∇H r r dM x dM y dM z ∇M = + + dx dy dz magnetic pole /charge r r density
ρ m≠ 0
if ∇M ≠ 0
sources of magnetic stray field / demagnetizing field
22
demagnetizing field Hd ↔ shape anisotropy
r r H d = − D⋅⋅ M
D … demagnetizing tensor
Hd homogeneous only for ellipsoidal bodies ! after diagonalization:
Tr D = Dx + Dy + Dz = for non-ellipsoidal objects:
{
r r r H d = H d (r )
1 4π
SI units cgs units position-dependent
to be numerically calculated with micromagnetic codes
23
sphere
cylindrical wire (∞)
thin plate/film (∞)
isotropic
UMA e.a./hard plane
UMA easy plane/h.a.
24
KU = ½ µ 0MS2
2 KU BK = MS HK =
⇒ 2KU MS
HK = MS BK = µ0MS HK = 4πMS
SI
cgs 25
thin film (L >> t) Bsat⊥ = µ0HK = µ0MS
Fe: µ0MS = 2.1 T @ 300K
VSM: M⊥(H⊥) for 50 nm Fe(001)/GaAs(001)
Problems:
determination of MS ? MOKE, AHE, …
- other anisotropies (surface, strain) - Dz < 1 with structural defects (surface roughness) 26
Remark: shape anisotropy, D
↔
continuum model, not strictly valid / fails for ultrathin films
also: atomistic model of point dipoles at lattice points incorrect ! better:
r σ (r )
from DFT
27
3.2 spin-orbit interaction (SOC) MA by SOC + crystal field interaction free atom: charge distribution related to orbital angular momentum; s-, p-, d-wave functions; ml ↔ orientation in space lattice: crystal field
→
orientation of orbital angular momentum due to Coulomb interaction
3d metals: orbital momentum is quenched by CF, but not completely due to SOC lower energy
from "Magnetism in Condensed Matter", S. Blundell, 2001 28
rotation of S by Bext
⇒ SOC: rotation of q(r) ⇒ ∆ECoul ≠ 0 ↔ MA !
magnetic moment m: orbital moment - preferential orientation via crystal field (3d) spin moment - preferential orientation via SOC crystal field ↔ crucial influence of local symmetry on MA! bulk / surface cubic / distorted lattice perfect lattice / defects: steps, …
29
4. theoretical description of MA ideally: ab initio theory for many-body problem 4.1 Density Functional Theory - DFT
rigorous in principle but: MA requires SOC, SOC ↔ relativistic effect ⇒ Dirac equation, fully relativistic treatment ! instead: Pauli addition to Hamiltonian
H SO
r r r = σ ⋅ (∇V (r ) × p ) r
successful, predictive power! e.g. spin reorientation in 1 ML Fe Gay and Richter PMA in Co1Ni2 multilayers Daalderop et al., PRL 68,682 (1992) problems: difficult/tedious for large sytems, defects; structure models (lattice distortions, strain, relaxations), disordered/ordered alloys, … 30
4.2 phenomenological theories - single ion anisotropy: energy of electric multipoles with crystal field → MA of 4f magnets - strong SOC; less appropriate for 3d metals - pair energy model L. Néel 1953: 3d metals with direct exchange interaction
εa = ∑εi; i
ε i = ∑ wik k
r µi φ
r µk r r ⋅ρ
wik = g 2 ( rik ) ⋅ (cos 2 φ − 13 ) + g 4 (rik ) ⋅ (cos 4 φ − 76 cos 2 φ + 353 ) + ... r r r r M cos φ = ρ ⋅ m = ρ x mx + ρ y m y + ρ z mz ; m= ; M g 2 ( r ) = l + m∂r ; g 4 (r ) = q + s∂r [pseudo-dipolar term + pseudo-quadrupolar term + …] εa by summation over suitable atomic pairs
⇒
ratios between Ki (absolute values → ab initio theory), Ki and λ; accounts for: different lattice symmetries, strain - magnetostriction, surfaces; e.g. theory of PMA, induced UMA in Permalloy – Ni81Fe19 , ... 31
5. how to measure the strength of MA magnetizing energy Wmag
a)
wmag =
MS
∫ HdM
εa
0
… MA energy density
Wmag (φ) - Wmin= εa (φ) M(H) loops by 1.0
5 ML Fe (001) on GaAs(001)
VSM, AGM, SQUID, MOKE + MS, …
M / MS
0.5 0.0 -0.5
[110] [1-10]
integration of anhysteretic M(H) loops
Wmag
-1.0 -1.0
-0.5
0.0
0.5
1.0
H [kOe]
32
Fe (001) on GaAs(001) 90
0 330
120
30
75ML Fe 300
38ML Fe 150
60
270
90
240
180
210
[110]
150
330
240
0
90
18 ML Fe 300
[1-10]
270
120
210
150 180
∫ HdM 0
as a function of field orientation
60
150
90
240
120
7 ML Fe
60
MS
300 270
30
magnetizing energy 0
180
330
30
wmag =
120
210
60
30
180
0
210
330
240
300 270
33
b)
fitting h.a. M(H) loops: faster alternative method if symmetry of anisotropy is known from
εa (φ)
1 4
ε m (φ , H , α ) = − K1eff sin 2 (2φ ) + K u eff sin 2 (φ ) − HM s cos(φ − α ) reversible magnetization loops along hard axis ↔ equilibrium states
dε m =0 dφ
⇒
H (m) = 2 K1eff (2m3 − m) / M s + 2 Ku eff m / M s 2 nm Fe (14 ML), T=300K 1,0 1,0 0,5
⇒ K1eff, KUeff
0,5
0,0 -0,5 -1,0
M / MS
fit to experimental h.a. M(H) / H(m) loops (here: [1-10])
0,0
-20 -15 -10 -5
0
5
10 15 20
-0,5
[110] [1-10]
-1,0 -2000 -1500 -1000
-500
0
500
1000
1500
2000
H [Oe] 34
c) Ki from torque:
dEm T= dφ
• static: torque magnetometer • dynamic: FMR
(Hres= Heff= Hext+ HK for Hext//e.a. etc.)
r M
main source of error: non-uniform , domains, incomplete saturation
⇒
measurement at high field, e.g. FMR at highest frequency best: lim Ki(1/H) 1/H → 0
35
6. Symmetry and MA example: single crystalline FM materials
6.1
(e.g.epitaxially grown Fe films)
3D → 2D, i.e. reduced extension in 1 dimension / bulk → ultrathin film ⇔ enhanced fraction of surface moments - lower coordination - lower symmetry, e.g. broken cubic symmetry
basic idea:
MAE of surface atoms different from volume atoms;
current practice: magnetic surface anisotropy = real MAE – ideal volume MAE
E
eff a
=E
vol a
+E
surf / int a
Eaeff (1) ( 2) = V ⋅ K eff = V ⋅ KV + A ⋅ ( K int + K int ) f (φi )
K eff
1 (1) ( 2) = KV + ( K int + K int ) t
test: Keff vs. 1/t or Keff ⋅ t vs. t 36
6.1.1 Perpendicular MA - PMA
predicted by L. Néel in 1953 ; exp. confirmed by Gradmann et al. 1968
(1) ( 2) film normal = symmetry axis K eff = 1 µ 0 M S2 + KV' + 1 ( K int + K int ) t 2 Fe(110) on Au(111) 1 (1) eff vol ( 2) (cgs) H = 4 π M + H + ( H + H K S K S S ) 1 ML ↔ 0.2 nm t 5 ML 2 ML
1 ML
.5 ML
HS = BS =
40 kOe 4T
treor ≈ 3 ML
G. Lugert and G. Bayreuther Thin Solid Films 175 (1989) 311 37
other examples with perpendicular e.a.: Co/Au, Co/Pt, Co/Pd, Fe/Pt !
Co/Pd polycrystalline multilayer
← Ni/Cu: surface anisotropy with perpendicular hard axis ! easy plane PMA!
general mechanism: SOC + crystal field
applications: magnetic recording media, spin torque oscillators, … 38
6.1.2 In-plane interface anisotropy - fourth order B. Heinrich et al., 1989 M. Brockmann et al., 1997
Fe(001) epitaxial films
ε a = K 0 + K1 ⋅ (α12α 22 + α 22α 32 + α 32α12 ) + K 2 ⋅ α12α 22α 32 + .... 25 ML Au/ t Fe/GaAs(001); t = 5 ML …. 75 ML
K1eff from M(H) loops Keff = Kvol + 1/t·Kint ⇒ K vol and 1
|
K1int of opposite sign !
strong thickness dependence, change of sign of K1: e.a. [100] → [110]
origin ? G. Bayreuther et al., J. Appl. Phys. 93, 8230 (2003) 39
single crystalline Fe1-xCox(001) films, 0 ≤ x ≤ 0.7 MBE-grown on GaAs(001), Ag(001), Au(001)
stable bcc structure, a0 ≈ const.
25 ML Au / X ML Fe XCo1-X / GaAs (001)
K1
eff
5
3
[x10 erg/cm ]
4
20.0 ML 10.0 ML 6.7 ML 5.0 ML 4.0 ML Fe Fe78Co22
3
Fe67Co33 Fe34Co66
2 1
Keff = Kvol + 1/t·Kint
0 -1 ⇒ K vol and
K1int always of opposite sign !
-2
1
-3 0.00 0.00
0.05 0.05
0.10 0.10
0.15 0.15
0.20 0.20
0.25 0.25
INVERSE THICKNESS [1/ML]
G. Bayreuther et al., J. Appl. Phys. 93 (2003) 8230 40
fourth-order volume and interface anisotropy constants of epitaxial Fe1-xCox (001) films on GaAs(001)
Keff = Kvol + 1/t·Kint 4 66
2 33 0
-2
K1
int
22
0 -4 -4
-2
0
K1
vol
2 5
at% Co
universal critical thickness
-2
2
[ 10 erg/cm ]
Fe1-xCox(001)
tcrit for K1eff=0; universal proportionality between K1vol and K1int
tcrit = - K1int/K1vol
4
3
[ 10 erg/cm ]
how to explain?
41
L. Nèel's pair energy model of magnetic anisotropy εa = ∑εi; i
ε i = ∑ wik
r µi φ
k
r µk
r M uniform
r r ⋅ρ
wik = g 2 ( rik ) ⋅ (cos 2 φ − 13 ) + g 4 (rik ) ⋅ (cos 4 φ − 76 cos 2 φ + 353 ) + ... r r r r M cos φ = ρ ⋅ m = ρ x mx + ρ y m y + ρ z mz ; m= ; M g 2 ( r ) = l + m∂r ; g 4 (r ) = q + s∂r Ea by summation over suitable atomic pairs
here: summation over nn and nnn pairs
⇒
2nd order term = 0 4th order term ≠ 0
42
fourth-order in-plane anisotropy in unstrained (001) bcc film K1vol =
2 16 nn nnn − q 2 q 3 a 9
K1int = −
8 nn q 2 9a
K 1int 2qnn a = − t crit = − nn vol nnn K1 8q − 9q
pure Fe: K1int < 0 ⇒ qFeFe > 0 Fe1-xCox: qnn = average of qFeFe, qFeCo and qCoCo ↔ chemical order? qFeCo, qCoCo ↔ 〈qnn 〉 < 0 for x > 0.3 random alloy: qnn/qnnn independent of x
⇒
universal
tcrit
prediction: no effect of substrate (Au, Ag, GaAs) or covering material (vacuum, Au) is experimentally observed! 43
6.2 2D → 1D: FM chains / wires
Co chains on vicinal Pt(9 9 7) XMCD data from Co L3 edge; θ = ∠(k,[111]); lines: UMA
partially explained by DFT calc. see Handbook of Magnetism and Advanced Magnetic Materials, H. Kronmüller and S. Parkin, edts. Vol. 1, p. 575 ff
44
6.2 further symmetry breaking mechanisms - orientational (chemical) order: induced UMA in Permalloy - Ni81Fe19 by oriented Ni-Fe, Ni-Ni, Fe-Fe pairs L. Néel 1953
- strained lattice: magneto-elastic anisotropy ↔ magnetostriction e.g. cubic → tetragonal
f ME (ε , α ) = B1 (α12ε 1 + α 22ε 2 + α 32ε 3 ) + B2 (α1α 2ε 6 + α1α 3ε 5 + α 2α 3ε 4 ) + ... 100 ML Fe32Co68 /GaAs(001)
curvature measured by double laser beam
D. Sander, Rep. Prog. Phys. 62, 809 (1999) 45
6.2 further symmetry breaking mechanisms - orientational (chemical) order: induced UMA in Permalloy - Ni81Fe19 by oriented Ni-Fe, Ni-Ni, Fe-Fe pairs L. Néel 1953
- strained lattice: magneto-elastic anisotropy ↔ magnetostriction e.g. cubic → tetragonal - atomic steps - …..
46
7. Tailoring and manipulating MA monoatomic steps
7.1 controlled local symmetry:
T. Leeb et al., J. Appl. Phys. 85, 4964 (1999)
nm
Fe / Ag/GaAs(1 0 29)
1 ,4 1 ,2 1 ,0 0 ,8 0 ,6 0 ,4 0 ,2 0 ,0 0
10
20
30
40
50
nm NFe [ML] 20
50
16
12
7
9
450
K1 Ku
3
eff
K 1 bzw. K u
uniaxial e.a. ⊥ step edges
eff
3
superposition of in-plane 4-fold and UMA
[ 10 erg/cm ]
500
400 350 300 250 200
6,2 ML
150 100
28,9 ML
50 0 -50 0,00
0,02
0,04
0,06
0,08
0,10
1 / NFe [1/ML]
0,12
0,14
0,16
47
7.2 oriented chemical bonds
-
Fe(001)/GaAs(001)
50 nm Fe(001) on GaAs(001)
3
KU (10 erg/cm )
10
[100]
Fe34Co66 (001)
5
eff
5
[110]
Fe (001)
1.0
0
5 ML Fe (001) on GaAs(001)
0,05
M / MS
0.5
0,10
0,15
0,20
0,25
1/t [1/ML]
0.0 -0.5
[110] [1-10]
uniaxial interface MA -1.0 -1.0
-0.5
0.0
H [kOe]
0.5
1.0
by oriented Fe-As bonds; KU dominating for ultrathin films 48
7.3 MA modified by electric field
Nature Nanotechnology 4, 158 (2009)
49
electric field induced / modified MA - strong current activities - FE/FM hybrid structures ↔ magneto-elastic coupling - multiferroic materials - …..
50
8. Summary - MA is essential for most applications of FM materials - MA originates from dipolar and S-O interaction - any symmetry breaking process creates additional MA - MAE: .01 MJ/m3 . . . . > 20 MJ/m3 1 µeV/atom . . . . > 10 meV/atom
1 ML Co clusters
Gambardella et al.
51
8. Summary - MA is essential for most applications of FM materials - MA originates from dipolar and S-O interaction - any symmetry breaking process creates additional MA - MAE: .01 MJ/m3 . . . . > 20 MJ/m3 1 µeV/atom . . . . > 10 meV/atom - theory: (relativistic) DFT, but useful phenomen. theories - many ways to manipulate MA (strength and orientation) - many opportunities for future research and applications
52
Further reading Handbook of Magnetism and Advanced Magnetic Materials, H. Kronmüller and S. Parkin, eds., Vol. 1, different chapters Ruqian Wu (p. 423 ff), A. Enders et al.(p. 575 ff), Blügel and Bihlmayer (p. 598 ff)
Ultrathin Magnetic Structures J.A.C. Bland and B. Heinrich, eds., Vol. I, ch. 2
Simple Models of Magnetism R. Skomski, Oxford University Press, 2008; ch. 3
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