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Atomistic Model of Ferrimagnetic GdFeCo

Thomas Ostler, Richard Evans and Roy Chantrell October 2, 2009

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

OUTLINE

Motivation for developing the model Benefits of atomistic modelling Model details Mean Field results Static properties of Ferrimagnetic materials Dynamic properties Summary and conclusion Magneto-Optical Reversal

ℏω2 ℏω2 ℏ(ω1 − Ωm) ℏω1 L=1 L=0 ℏΩm

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

OUTLINE

Motivation for developing the model Benefits of atomistic modelling Model details Mean Field results Static properties of Ferrimagnetic materials Dynamic properties Summary and conclusion

1 2 3 4 5 6 100 200 300 400 500 600 Hc [T] (Sweep Rate=108T/s) T [K] 32% 36%

Efficient calculation of static magnetic properties

0.0 0.5 1.0 1.5 2.0 2.5 200 400 600 800 1000 1200 Ms [μB/atom] T [K]

0% RE 12% RE 18% RE 24% RE 30% RE 36% RE

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

OUTLINE

Motivation for developing the model Benefits of atomistic modelling Model details Mean Field results Static properties of Ferrimagnetic materials Dynamic properties Summary and conclusion Atomistic level magnetisation dynamics

• 0.40
• 0.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

• 1.0

0.0 10.0 20.0 30.0 40.0 50.0 Mz/M0 Time [ps]

790K 910K 970K 1030K 1090K 1210K 1270K

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

OUTLINE

Motivation for developing the model Benefits of atomistic modelling Model details Mean Field results Static properties of Ferrimagnetic materials Dynamic properties Summary and conclusion

H S S x H S x S x H

LLG, Hamiltonian, lattice structures....

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

OUTLINE

Motivation for developing the model Benefits of atomistic modelling Model details Mean Field results Static properties of Ferrimagnetic materials Dynamic properties Summary and conclusion

0.0 0.5 1.0 1.5 2.0 1050 T[K] 0.0 0.5 1.0 1.5 2.0 Ms (LLG) [μb per atom] 0% 1% 2% 3% 0% 1% 2% 3%

M(T), χ, χ⊥

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 100 200 300 400 500 600 χ,⊥/μb [1/T Per Atom] Time [s] χ χ⊥

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

OUTLINE

Motivation for developing the model Benefits of atomistic modelling Model details Mean Field results Static properties of Ferrimagnetic materials Dynamic properties Summary and conclusion

0.0 0.5 1.0 1.5 2.0 100 200 300 400 500 600 αeƒƒ T [K] Ferrimagnet Ferromagnet

M(T), Hc(T), τ, τ⊥ and αeff

0.0 0.5 1.0 1.5 2.0 2.5 200 400 600 800 1000 1200 Ms [μB/atom] T [K]

0% RE 12% RE 18% RE 24% RE 30% RE 36% RE

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

MOTIVATION

Potential new mechanism for HDD writing of magnetic bits Magneto-Optical reversal seen in Amorphous GdFeCo Quantum or Thermodynamic process? Or both?

Quantum: Stimulated Raman emission Thermodynamics: Combination of heat and Inverse Faraday Effect (IFE)

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

Motivation

Potential new mechanism for HDD writing of magnetic bits Magneto-Optical reversal seen in Amorphous GdFeCo Quantum or Thermodynamic process? Or both?

Quantum: Stimulated Raman emission Thermodynamics: Combination of heat and Inverse Faraday Effect (IFE)

ℏω2 ℏω2 ℏ(ω1 − Ωm) ℏω1 L=1 L=0 ℏΩm

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

Motivation

Potential new mechanism for HDD writing of magnetic bits Magneto-Optical reversal seen in Amorphous GdFeCo Quantum or Thermodynamic process? Or both?

Quantum: Stimulated Raman emission Thermodynamics: Combination of heat and Inverse Faraday Effect (IFE)

200 400 600 800 1000 0.0 0.5 1.0 1.5 2.0 4 8 12 16 20 Te/ [K] B [T] Time [ps]

Te T B

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

STORY SO FAR

40fs laser pulses cause reversal depending on chirality of light Induced IFE leads to effective field (∝ E × E∗) Raman type scattering effect causes mixing of L=1 wavefunction with L=0, causing ultrafast reversal with energy of a magnon Insufficient numbers of photons at frequencies ω1 and ω2

ℏω2 ℏω2 ℏ(ω1 − Ωm) ℏω1 L=1 L=0 ℏΩm

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

STORY SO FAR

40fs laser pulses cause reversal depending on chirality of light Induced IFE leads to effective field (∝ E × E∗) Raman type scattering effect causes mixing of L=1 wavefunction with L=0, causing ultrafast reversal with energy of a magnon Insufficient numbers of photons at frequencies ω1 and ω2

ℏω2 ℏω2 ℏ(ω1 − Ωm) ℏω1 L=1 L=0 ℏΩm

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

BENEFITS OF ATOMISTIC MODELLING

With the use of the Landau-Lifshitz-Gilbert equation we have the benefit of time resolved magnetisation dynamics

• n the very short time scale

dSi dt = − γi (1 + λ2

i )µsi

[Si(t)×Hi(t)]− λiγi (1 + λ2

i )µsi

{Si(t)×[Si(t)×Hi(t)]} On-site damping, gyromagnetic ratio and magnetic moment is very important for Ferrimagnets

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

BENEFITS OF ATOMISTIC MODELLING

In theory any lattice structure is possible Fitting of Hamiltonian to ab-initio data is possible to provide accurate physics for atomistic model Atomistic calculations can be parameterized for use with a macro-spin model namely the Landau-Lifshitz-Bloch (LLB) equation

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

BENEFITS OF ATOMISTIC MODELLING

In theory any lattice structure is possible Fitting of Hamiltonian to ab-initio data is possible to provide accurate physics for atomistic model Atomistic calculations can be parameterized for use with a macro-spin model namely the Landau-Lifshitz-Bloch (LLB) equation

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

BENEFITS OF ATOMISTIC MODELLING

In theory any lattice structure is possible Fitting of Hamiltonian to ab-initio data is possible to provide accurate physics for atomistic model Atomistic calculations can be parameterized for use with a macro-spin model namely the Landau-Lifshitz-Bloch (LLB) equation

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

OUTLINE

Motivation for developing the model Benefits of atomistic modelling Model details Mean Field results Static properties of Ferrimagnetic materials Dynamic properties Summary and conclusion

H S S x H S x S x H

LLG, Hamiltonian, lattice structures....

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

HAMILTONIAN

Ab-initio level work is underway to fit a Hamiltonian to magneto-optically active GdFeCo composition Currently use a generic Hamiltonian

Uni-axial anisotropy, nearest neighbour Heisenberg exchange and Zeeman term

Exchange: TM-TM = FM, RE-RE = FM, TM-RE = AFM H = −1 2

N

• i=1

N

• j=1

JijSi · Sj −

N

• i=1

D(Si · ni)2 −

N

• i=1

µiB · Si

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

HAMILTONIAN

Ab-initio level work is underway to fit a Hamiltonian to magneto-optically active GdFeCo composition Currently use a generic Hamiltonian

Uni-axial anisotropy, nearest neighbour Heisenberg exchange and Zeeman term

Exchange: TM-TM = FM, RE-RE = FM, TM-RE = AFM H = −1 2

N

• i=1

N

• j=1

JijSi·Sj−

N

• i=1

D(Si·ni)2−

N

• i=1

µiB·Si

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

HAMILTONIAN

Ab-initio level work is underway to fit a Hamiltonian to magneto-optically active GdFeCo composition Currently use a generic Hamiltonian

Uni-axial anisotropy, nearest neighbour Heisenberg exchange and Zeeman term

Exchange: TM-TM = FM, RE-RE = FM, TM-RE = AFM H = −1 2

N

• i=1

N

• j=1

JijSi · Sj −

N

• i=1

D(Si · ni)2 −

N

• i=1

µiB · Si (1)

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

INTRODUCING THERMAL NOISE

We use a stochastic thermal noise term which obeys the conditions ζi(t) = 0; ζa

i(t)ζb j(t′) = 2δijδabδ(t − t′)λikBT

µiγi White noise term presumes phonon and electron system act on timescale much faster than spins Question: Is this assumption valid for ultra-fast laser experiments?

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

MEAN FIELD IMPURITY MODEL

Use Langevin equation mTM = L µTMρM0mTM kBT

• ; H = ρM

Assume as we add RE impurity Tc = (1 − x)T 0

c and mean

field form of exchange mTM = L 3T 0

c (1 − x)mTM

T

• ; JTM-TM =

3kBTc 2zs(s + 1) RE feels exchange field from TM, JTM-RE = κJTM-TM HRE = −JTM-REmTM · mREz

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

RESULT

Once we know the field on the RE by solving Langevin function for TM self-consistently we can calculate magnetisation of RE

0.0 0.5 1.0 1.5 2.0 1050 T[K] 0.0 0.5 1.0 1.5 2.0 Ms (LLG) [μb per atom] 0% 1% 2% 3% 0% 1% 2% 3%

Good agreement with Langevin dynamics for low impurity levels Poor agreement for high impurity levels (≈ 6%)

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

TWO SUB-LATTICE MEAN FIELD MODEL

Motivation: Can we parameterise a atomistic model for LLB model? Can χ(T), χ⊥(T) and A(T) contain enough information, e.g. excitation of exchange mode and high coercivity? Coupled integral equation of each sub-lattice m1 = S1 = 2π π

0 S1exp−βH sin(θ1)dθ1dφ1

2π π

0 exp−βH sin(θ1)dθ1dφ1

m2 = S2 = 2π π

0 S2exp−βH sin(θ2)dθ2dφ2

2π π

0 exp−βH sin(θ2)dθ2dφ2

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

TWO SUB-LATTICE MEAN FIELD MODEL

Hamiltonian H = − JRE-RES1 · m1(1 − p)2Z − JRE-TMS1 · m2p(1 − p)Z − JTM-TMS2 · m2p2Z − JTM-RES2 · m1p(1 − p)Z − d(S1z

2(1 − p) + S2z 2(1 − p))

− B(µ1S1(1 − p) + µ2S2p) (2) Essentially same a Langevin Model, but with p representing the proportion of TM

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

RESULT

Some interesting results

• f susceptibility as a

function of temperature at compensation point Two maxima for χ Interesting minima for χ⊥ We know τ⊥ ∝ χ⊥. Results from Langevin dynamics model show a decrease in relaxation time at this point.

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 100 200 300 400 500 600 χ,⊥/μb [1/T Per Atom] Time [s] χ χ⊥

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

ANALYSIS AND FURTHER WORK

ANALYSIS Does not work too well for low percentages as we do not have two sub-lattices From what we have seen during Langevin dynamics simulations we seem to have an agreement A single lattice LLB model could be enough to describe dynamics seen for LLG model FURTHER WORK Parameterising LLG model for inputs into LLB model Micromagnetic (LLB) simulations of µm films Comparison with LLG model will show us if we have captured the physics

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

ANALYSIS AND FURTHER WORK

ANALYSIS Does not work too well for low percentages as we do not have two sub-lattices From what we have seen during Langevin dynamics simulations we seem to have an agreement A single lattice LLB model could be enough to describe dynamics seen for LLG model FURTHER WORK Parameterising LLG model for inputs into LLB model Micromagnetic (LLB) simulations of µm films Comparison with LLG model will show us if we have captured the physics

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

OUTLINE

Motivation for developing the model Benefits of atomistic modelling Model details Mean Field results Static properties of Ferrimagnetic materials Dynamic properties Summary and conclusion

0.0 0.5 1.0 1.5 2.0 100 200 300 400 500 600 αeƒƒ T [K] Ferrimagnet Ferromagnet

M(T), Hc(T), τ, τ⊥ and αeff

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

STATIC PROPERTIES, M(T)

Two sub-lattices antiferromagnetically exchange coupled, with different degrees of criticality leads to magnetic compensation point

0.0 0.5 1.0 1.5 2.0 2.5 200 400 600 800 1000 1200 Ms [μB/atom] T [K]

0% RE 12% RE 18% RE 24% RE 30% RE 36% RE

• 1.0
• 0.5

0.0 0.5 1.0 1.5 100 200 300 400 500 600 700 M/M0 T [K]

TM RE JTM-RE = −2.0 JTM-RE = −1.0 JTM-RE = −0.5 JTM-RE = −0.1 JTM-RE = 0.0

As JTM-RE is increased we get polarisation of the two sub-lattices

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

COERCIVITY

Experimentally coercivity diverges This is due to magnetic compensation point Net Zeeman energy is low

1 2 3 4 5 6 100 200 300 400 500 600 Hc [T] (Sweep Rate=108T/s) T [K] 32% 36%

Reproduced very well in simulations Eventually reverses because high field increases magnetisation

• f parallel sub-lattice, so

net Zeeman energy is no longer zero. Needs high fields to do so

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

ANGULAR MOMENTUM COMPENSATION POINT

Due to inhomogeneities in lattice field potential different species have different angular momentum contributions Results in different gyromagnetic ratios According to theory for high exchange between sub-lattices, we have an angular momentum compensation point (TA) γeff = M1 − M2 M1/γ1 − M2/γ2 When denominator goes to zero this point is TA Results in divergence in damping αeff = α1M1/γ1 + α2M2/γ2 M1/γ1 − M2/γ2

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

EXPERIMENT AND THEORY

At TA (50K above TM) they measure a large increase in damping Began by setting TA=TM Calculated τ⊥ by fitting thermally equilibrated macro-spin coherently rotated to 30 degrees to cos(ωt) exp(−t/τ) Ferrimagnet shows intrinsically different behaviour

0.0 0.5 1.0 1.5 2.0 100 200 300 400 500 600 αeƒƒ T [K] Ferrimagnet Ferromagnet Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

ANGULAR MOMENTUM COMPENSATION POINT

We can move TA without moving TM by changing γi for each species Carried out same simulation as previous slide but with different γTM

0.0 0.5 1.0 1.5 2.0 100 200 300 400 500 αeƒƒ T [K] γTM = γRE γTM > γRE γTM < γRE

Far away from TA analytic expression does not fit because exchange between sub-lattices is low Near compensation point we have excitement of an exchange mode, which acts like high exchange

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

DYNAMICS

We began by looking at the longitudinal relaxation time as a function of TM-RE composition Compared the Ferrimagnet to a Ferromagnet At Curie point we saw increase in the longitudinal relaxation time This may help to explain the speed of magneto-optical reversal

5e-13 1e-12 1.5e-12 2e-12 2.5e-12 3e-12 3.5e-12 4e-12 100 200 300 400 500 600 700 τ T [K] Ferrimagnet Ferromagnet

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

MAGNETO-OPTICAL REVERSAL

Can simulate magneto-optical reversal by utilising the two temperature model for electron and lattice system Ce ∂Te ∂t = −∇ · ¯ Qe − G(Te − Tl) + S(¯ r, t) τe ∂ ¯ Qe ∂t + ¯ Qe = −Ke∇Te Cl ∂Tl ∂t = −∇ · ¯ Ql + G(Te − Tl) τl ∂ ¯ Ql ∂t + ¯ Ql = −K1∇Tl Presumption: have a thin enough film to produce uniform heating Laser fluency presumed Gaussian temporally and uniform spatially

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

MAGNETO-OPTICAL REVERSAL

Experiments show IFE produces field ∼ 20T Presume Gaussian field Field and temperature profile looks like

200 400 600 800 1000 0.0 0.5 1.0 1.5 2.0 4 8 12 16 20 Te/ [K] B [T] Time [ps]

Te T B

We couple spins to electron temperature as we do not know exact energy transfer from electron system to spins There seems to be a way to get heat into 4f spin system very quickly in RE’s from conduction electrons

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

MAGNETO-OPTICAL REVERSAL

The combination of the field and thermal effects results in a “window” for reversal

• 0.40
• 0.20

0.00 0.20 0.40 0.60 0.80 1.00 1.20

• 1.0

0.0 10.0 20.0 30.0 40.0 50.0 Mz/M0 Time [ps]

790K 910K 970K 1030K 1090K 1210K 1270K

For a particular field width (in time) as we increase the maximum electron temperature we see an opening of the window During this window we see a “linear” reversal of the macro-spin At the upper end of the window we put too much heat into the system, which disorders the system closing the window

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

EFFECT OF FIELD PULSE

As we increase the field pulse width we see a widening of the window

• 1.00
• 0.50

0.00 0.50 1.00 700 900 1100 1300 1500 σ Tm [K]

Δt = 0.4ps Δt = 0.6ps Δt = 0.8ps

σ =

• +1

if mz (50ps) > 0 −1

• therwise

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

ROLE OF RARE EARTH

There is a large question as to whether the role of the RE is important for these processes? Does the RE provide a means of channelling energy quickly into the 4f moments? Does the RE broaden the frequencies available for Raman activation? Calculations of a single spin LLB Ferromagnet suggest the RE is not necessary

∆t[ps] T max

e

[K] 1.4 1.2 1 0.8 0.6 0.4 0.2 1300 1100 900 700

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

SUMMARY: MEAN FIELD

Impurity model provides some validation of computational model for low concentrations A two sublets mean field model can be used as input for the LLB model Using this LLB model we can compare see if we can describe the physics of a Ferrimagnet accurately If not then we may need to revisit the LLB equation and derive it for a more general case

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

SUMMARY: STATICS USING ATOMISTIC MODEL

Magnetic compensation point Coercivity as a function of temperature Angular momentum compensation point Comparison with experiment is qualitatively very good

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

SUMMARY: DYNAMICS

We have shown that the longitudinal relaxation time of the Ferrimagnet is faster than that of the Ferromagnet at the Curie point This is important for the explanation of the fast reversal times involved in magneto-optical reversal The presence of a reversal window supports the work done experimentally Changing the width of the field pulse, which is equivalent to increasing the laser pulse time increases/decreases the size of the window We have good agreement with micromagnetic (LLB) simulations

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo

OUTLOOK

Using DFT calculations we hope understand some more of the fundamental physics of the system We hope to continue work on the LLB equation to see if the LLB equation can used for Ferrimagnets Magneto-optical calculations will continue in the hope of explaining the role of the RE Calculation of spin wave spectra to see excitation of exchange mode and its role in magneto-optical reversal

Thank you for listening

Thomas Ostler, Richard Evans and Roy Chantrell Atomistic Model of Ferrimagnetic GdFeCo