Magnetocrystalline anisotropy of adatoms:

How to make it big?

O. Sipr1 S. Mankovsky2 S. Bornemann2 J. Vackar1

H. Ebert2 J. Minar2,3

1Institute of Physics AS CR, Prague http://www.fzu.cz/~sipr

2Department Chemie, Ludwig-Maximilians-Universitat Munchen

3New Technologies Research Center, University of West Bohemia, Pilsen

26. August 2014 / Condensed Matter in Paris 2014

1

Outline

◮ MAE: What is it about?

◮ Scaling of MAE with SOC: case of adatoms

◮ SOC-induced splitting of DOS as a marker of MAE

◮ Message to the mankind

2

Magnetic Anisotropy Energy (MAE)

◮ Difference between total energies for two orientations of themagnetization M with respect to the crystal lattice:

EMAE = E (M1) − E (M2)

3

Magnetic Anisotropy Energy (MAE)

◮ Difference between total energies for two orientations of themagnetization M with respect to the crystal lattice:

EMAE = E (M1) − E (M2)

◮ Magnetocrystalline contribution to MAE is due to the spinorbit coupling (SOC).

3

Magnetic Anisotropy Energy (MAE)

◮ Difference between total energies for two orientations of themagnetization M with respect to the crystal lattice:

EMAE = E (M1) − E (M2)

◮ Magnetocrystalline contribution to MAE is due to the spinorbit coupling (SOC).

◮ What is the specific mechanism that gives rise to the MAE fora concrete system?

◮ Can we see a signature of MAE in the electronic structure?

◮ The way the MAE scales with the SOC tells us somethingabout its mechanism.

◮ People still strive to understand how to make MAE as high aspossible [Jesche et al. Nature Comm. (2014), Rau et al.

Science (2014), Khajetoorians & Wiebe Science (2014),Antropov et al. arXiv (2014)].

3

Total energies and single-particle (band) energies

◮ Total energy within the DFT framework:

Etot =

occ∑i

Ei−1

2

∫dr

∫dr′

n(r)n(r′)

|r − r′|+Exc[n]−

∫drn(r)vxc[n](r)

◮ Kohn-Sham functional is extremal ⇒ differences in Etot canbe approximated by differences in single-particle or “band”energies:

EMAE ≈

occ∑i

Ei (M1) −

occ∑i

Ei (M2)

Issue to be addressed:How are the band energies affected by the spin-orbit coupling?

4

Change of band energy upon inclusion of SOC

◮ SOC-perturbed Hamiltonian (SOC is scaled by λ):

H = H0 + HSOC = H0 + λ ξ(r)σ · L

◮ Lowest-order non-vanishing correction to the ground state isthe second-order term:

δE (2) =∑j 6=0

|〈ψ0|HSOC |ψj〉|2

E0 − Ej

◮ δE (2) scales as λ2 ⇒ MAE scales with SOC quadratically.

5

Change of band energy upon inclusion of SOC

◮ SOC-perturbed Hamiltonian (SOC is scaled by λ):

H = H0 + HSOC = H0 + λ ξ(r)σ · L

◮ Lowest-order non-vanishing correction to the ground state isthe second-order term:

δE (2) =∑j 6=0

|〈ψ0|HSOC |ψj〉|2

E0 − Ej

◮ δE (2) scales as λ2 ⇒ MAE scales with SOC quadratically.

◮ However: There may be contributions to the MAE whichcannot be described within perturbation theory (e.g., if SOC islarge or if degenerate states are important).

5

If the perturbation theory cannot be used. . .

◮ Inspiration:

For free atoms, SOC splits the originally degenerate states by

∼ mℓ λ cos θ ,

mℓ is the (orbital) magnetic quantum number,λ is the SOC scaling factor,θ is the angle between M and the spin quantization axis.

◮ If the degenerate states are near the Fermi level, theSOC-induced splitting may push some levels above EF ,lowering thus the energy [Daalderop et al. (1990,1991,1994),Wang et al. (1993), Ravindran et al. (2001)].

◮ Contributions due this effect could scale linearly with the SOCstrength λ but the situation is more complicated. . . [Lessard et

al. (1997), Gimbert & Calmels (2012)]

6

SOC-scaling of contribution due to degenerate states

Gimbert & Calmels PRB (2012)

◮ For extended systems, thedegeneracy is limited only to asmall part of the Brillouin zone,hence it does not affect the overallquadratic scaling of the MAE withthe SOC.

7

SOC-scaling of contribution due to degenerate states

Gimbert & Calmels PRB (2012)

◮ For extended systems, thedegeneracy is limited only to asmall part of the Brillouin zone,hence it does not affect the overallquadratic scaling of the MAE withthe SOC.

◮ What if the k-dependence issuppressed (as in the case ofadatoms)?

Let us have a look. . .

7

Calculate MAE for adatoms

Fe, Co, and Ni adatomson Au(111) surface

◮ Fully relativistic Green’s-function KKR formalism [Ebert,

Kodderitzsch and Minar RPP (2011)]

◮ MAE evaluated via the torque◮ Definition: EMAE ≡ E

(x) − E(z)

EMAE > 0 ⇔ easy axis is normal to the surface

◮ Scaling of the SOC done via identifying the SOC-related term by

means of an approximate Dirac equation [Ebert et al. PRB 53, 7721

(1996) extending the scheme of Koelling & Harmon (1977) and

MacLarren & Victora (1994)]

8

When things work: Free-standing monolayers

-5

-4

-3

-2

-1

0E

MA

E=

E(x

)-

E(z

)[m

eV]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

spin-orbit coupling scaling factor

calculated EMAE

quadratic fit by -2.1 2

free-standing Co monolayer

9

MAE as function of SOC: Fe adatom

0

1

2

3

4

5

6

7

EM

AE

=E

(x)-

E(z

)[m

eV]

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

spin-orbit coupling scaling factor

calculated EMAE

quadratic fit by 3.6 2

linear fit by 5.6 - 1.9

Fe adatom on Au(111)

Fits sought within the range λ=0.8–1.2, i.e., for realistic SOC values.

10

MAE as function of SOC: Fe adatom

2

3

4

5

EM

AE

=E

(x)-

E(z

)[m

eV]

0.8 0.9 1.0 1.1 1.2

spin-orbit coupling scaling factor

calculated EMAE

quadratic fit by 3.6 2

linear fit by 5.6 - 1.9

Fe adatom on Au(111)

Fits sought within the range λ=0.8–1.2, i.e., for realistic SOC values.

10

SOC-scaling of MAE: verdict for adatoms

For 3d adatoms on Au(111), the MAE does not scale with theSOC quadratically (in the range of realistic SOC values).

Rather, this scaling is quasi-linear.

⇒ For realistic SOC values, we are in the regime where the MAEcannot be described by a perturbation theory.

11

DOS for Fe adatom

0

2

4n↑ (E

)[e

V-1

]M z

= 1

Fe adatom on Au(111)

0

2

4

n↓ (E)

[eV

-1]

-5 -4 -3 -2 -1 0 1

energy [eV]

FeAu nearest to Fe

12

DOS for Fe adatom

0

2

4n↑ (E

)[e

V-1

]M z

= 1

Fe adatom on Au(111)

0

2

4

n↓ (E)

[eV

-1]

-5 -4 -3 -2 -1 0 1

energy [eV]

FeAu nearest to Fe

12

Effect of SOC on mℓ-resolved DOS: Fe adatom

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1] M z

= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mz = -2mz = +2mz = -1mz = +1mz = 0

M z= 1

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1] M z

= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mx = -2mx = +2mx = -1mx = +1mx = 0

M z= 1

Fe adatomon Au(111)

13

Effect of SOC on mℓ-resolved DOS: Fe adatom

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1] M z

= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mz = -2mz = +2mz = -1mz = +1mz = 0

M z= 1

-2

0

2

4

6

EM

AE

[meV

]

-1.0 -0.5 0.0 0.5 1.0

Etop - EF [eV]

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1] M z

= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mx = -2mx = +2mx = -1mx = +1mx = 0

M z= 1

Fe adatomon Au(111)

MAE as functionof the top of the valence band

13

Effect of SOC on mℓ-resolved DOS: Co adatom

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1] M z

= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mz = -2mz = +2mz = -1mz = +1mz = 0

M z= 1

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1] M z

= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mx = -2mx = +2mx = -1mx = +1mx = 0

M z= 1

Co adatomon Au(111)

14

Effect of SOC on mℓ-resolved DOS: Co adatom

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1] M z

= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mz = -2mz = +2mz = -1mz = +1mz = 0

M z= 1

0

5

10

EM

AE

[meV

]

-1.0 -0.5 0.0 0.5 1.0

Etop - EF [eV]

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1] M z

= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mx = -2mx = +2mx = -1mx = +1mx = 0

M z= 1

Co adatomon Au(111)

MAE as functionof the top of the valence band

14

Effect of SOC on mℓ-resolved DOS: Ni adatom

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

M z= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mz = -2mz = +2mz = -1mz = +1mz = 0

M z= 1

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

M z= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mx = -2mx = +2mx = -1mx = +1mx = 0

M z= 1

Ni adatomon Au(111)

15

Effect of SOC on mℓ-resolved DOS: Ni adatom

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

M z= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mz = -2mz = +2mz = -1mz = +1mz = 0

M z= 1

-10

0

10

20

EM

AE

[meV

]

-1.0 -0.5 0.0 0.5 1.0

Etop - EF [eV]

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

M z= 0

0.0

0.5

1.0

1.5

n↓ (E)

[eV

-1]

-1.0 -0.5 0.0 0.5 1.0

energy [eV]

mx = -2mx = +2mx = -1mx = +1mx = 0

M z= 1

Ni adatomon Au(111)

MAE as functionof the top of the valence band

15

Conclusions

◮ For adatoms, if there is a degeneracy between states ofdifferent mℓ, it is not restricted only to a small region of thek-space and hence it affects the DOS.

◮ Because of this, a significant contribution to the MAE appearswhich cannot be described within the perturbation theory (itdoes not scale quadratically with the SOC. . . )

◮ In such a case, the mechanism behind the (large) MAE is thepushing of one of the mℓ-resolved DOS components above theFermi level.

16

Conclusions

◮ For adatoms, if there is a degeneracy between states ofdifferent mℓ, it is not restricted only to a small region of thek-space and hence it affects the DOS.

◮ Because of this, a significant contribution to the MAE appearswhich cannot be described within the perturbation theory (itdoes not scale quadratically with the SOC. . . )

◮ In such a case, the mechanism behind the (large) MAE is thepushing of one of the mℓ-resolved DOS components above theFermi level.

Thank you!

16