Ultrafast laser-induced demagnetization of solids: Understanding the mechanism with real-time TDDFT simulations

Max-Planck Institute ofMicrostructure Physics

Halle (Saale)

E.K.U. Gross

Beaurepaire et al, PRL 76, 4250 (1996)

First experiment on ultrafast laser induced demagnetization

Possible mechanisms for demagnetization

Possible mechanisms for demagnetisation

• Direct interaction of spins with the magnetic component of the laserZhang, Huebner, PRL 85, 3025 (2000)

Possible mechanisms for demagnetisation

• Direct interaction of spins with the magnetic component of the laserZhang, Huebner, PRL 85, 3025 (2000)

• Spin-flip electron-phonon scattering(ultimately leading to transfer of spin angular momentum to the lattice)Koopmans et al, PRL 95, 267207 (2005)

Possible mechanisms for demagnetisation

• Direct interaction of spins with the magnetic component of the laserZhang, Huebner, PRL 85, 3025 (2000)

• Spin-flip electron-phonon scattering(ultimately leading to transfer of spin angular momentum to the lattice)Koopmans et al, PRL 95, 267207 (2005)

• Super-diffusive spin transportBattiato, Carva, Oppeneer, PRL 105, 027203 (2010)

Possible mechanisms for demagnetisation

• Direct interaction of spins with the magnetic component of the laserZhang, Huebner, PRL 85, 3025 (2000)

• Spin-flip electron-phonon scattering(ultimately leading to transfer of spin angular momentum to the lattice)Koopmans et al, PRL 95, 267207 (2005)

• Super-diffusive spin transportBattiato, Carva, Oppeneer, PRL 105, 027203 (2010)

• Our proposal for the first 50 fs: Laser-induced charge excitation followed by spin-orbit-driven demagnetization of the remaining d-electrons

Quantity of prime interest: vector field of spin magnetization

Quantity of prime interest: vector field of spin magnetization

Cr monolayer in ground state

Basic 1-1 correspondence:The time-dependent density determines uniquelythe time-dependent external potential and hence allphysical observables for fixed initial state.

( ) ( )v rt rt1-1←→ ρ

Time-dependent density-functional theory

KS theorem:The time-dependent density of the interacting system of interest canbe calculated as density

of an auxiliary non-interacting (KS) system

with the local potential

( ) ( )2N

j = 1 ρ rt = rtjϕ∑

( ) ( ) ( ) ( )3S

r ' tv r ' t ' rt v rt d r '

r r 'ρ

ρ = + + −∫

( ) [ ]( ) ( )2 2

j S ji rt v rt rtt 2m

∂ ∇ϕ = − + ρ ϕ ∂

( ) ( )xcv r ' t ' rt ρ

(E. Runge, E.K.U.G., PRL 52, 997 (1984))

Theoretical approach:

( ) ( )( ) [ ]( ) [ ]( )

[ ]( )( ) ( ) ( )

21, , , , ,2

, , ,2

k laser S B S

BS k

i r t i A t v r t B r tt

v r t i r tc

ϕ ρ µ σ ρ

µ σ ρ ϕ

∂ = − ∇− + − ⋅∂ + ⋅ ∇ × − ∇

m m

m

Generalization: Real-time TDDFT with SOC

[ ]( ) ( ) ( ) [ ]( )3,, , , ,S lattice xc

r tv r t v r d r v r t

r rρ

ρ ρ′

′= + +′−∫m m

[ ]( ) ( ) [ ]( ), , , , ,S external xcB r t B r t B r tρ ρ= +m m

( ),k r tϕwhere are Pauli spinors

K. Krieger, K. Dewhurst, P. Elliott, S. Sharma, E.K.U.G., JCTC 11, 4870 (2015)

• Wave length of laser in the visible regime(very large compared to unit cell)

Dipole approximation is made(i.e. electric field of laser is assumed to be spatially constant)

Laser can be described by a purely time-dependent vector potential

• Periodicity of the TDKS Hamiltonian is preserved!

• Implementation in ELK code (FLAPW) (http://elk.sourceforge.net/)

Aspects of the implementation

• Wave length of laser in the visible regime(very large compared to unit cell)

Dipole approximation is made(i.e. electric field of laser is assumed to be spatially constant)

Laser can be described by a purely time-dependent vector potential

• Periodicity of the TDKS Hamiltonian is preserved!

• Implementation in ELK code (FLAPW) (http://elk.sourceforge.net/)

Aspects of the implementation

ELK = Electrons in K-SpaceorElectrons in Kay's Space

Kay Dewhurst Sangeeta Sharma

• Wave length of laser in the visible regime(very large compared to unit cell)

Dipole approximation is made(i.e. electric field of laser is assumed to be spatially constant)

Laser can be described by a purely time-dependent vector potential

• Periodicity of the TDKS Hamiltonian is preserved!

• Implementation in ELK code (FLAPW) (http://elk.sourceforge.net/)

Aspects of the implementation

( ) ( ) ( )j ij ii1. Set , t c tψ = χ∑r r

( ) ( )2. Compute , t and , tρ r m r

( ) ( ) ( ) ( )s s s KSˆ3. Compute v , t , , t , , t to give H tr B r A r

( )ij i KS jˆ4. Compute H H t≡ χ χ

ik kj j ij5. Solve H d d for d and =∈ ∈

( ) ( )ki t*ij jk lk ilkl

6. Compute c t t d d e c t− ∈ ∆+ ∆ =∑7. Goto 1

Algorithm for time propagation

K. Krieger, K. Dewhurst, P. Elliott, S. Sharma, E.K.U.G., JCTC 11, 4870 (2015)

Analysis of the results

components of spin moment

Calculation without spin-orbit coupling

( )z KS zi ˆ ˆM t H ,

t∂ = σ ∂

{ }( ) ( ) ( ) ( ){ }

3x KS,y y KS,x

3s y s z2

d r M (r, t)B (rt) M (r, t)B (rt)

1 ˆ ˆd r x v r, t j r, t y v r, t j r, t2c

= −

+ ⋅ × − ⋅ ×

∫

∫ ∇ ∇

KS ext XCB (rt) B (rt) B (rt)= +

ˆˆj(r, t) p= σ⊗

spin current tensor

Exact equation of motion

( )z KS zi ˆ ˆM t H ,

t∂ = σ ∂

{ }( ) ( ) ( ) ( ){ }

3x KS,y y KS,x

3s y s z2

d r M (r, t)B (rt) M (r, t)B (rt)

1 ˆ ˆd r x v r, t j r, t y v r, t j r, t2c

= −

+ ⋅ × − ⋅ ×

∫

∫ ∇ ∇

KS ext XCB (rt) B (rt) B (rt)= +

ˆˆj(r, t) p= σ⊗

Global torqueexerted by BKS

spin current tensor

Exact equation of motion

( )z KS zi ˆ ˆM t H ,

t∂ = σ ∂

{ }( ) ( ) ( ) ( ){ }

3x KS,y y KS,x

3s y s z2

d r M (r, t)B (rt) M (r, t)B (rt)

1 ˆ ˆd r x v r, t j r, t y v r, t j r, t2c

= −

+ ⋅ × − ⋅ ×

∫

∫ ∇ ∇

KS ext XCB (rt) B (rt) B (rt)= +

ˆˆj(r, t) p= σ⊗

Global torqueexerted by BKS

Global torque = 0, if Bext = 0 (due to zero-torque theorem for Bxc )

spin current tensor

Exact equation of motion

Note: Ground state of bulk Fe, Co, Ni is collinear

Demagnetization occurs in two steps:

- Initial excitation by laser moves magnetization from atomic regioninto interstitial region. Total Moment is basically conservedduring this phase.

- Spin-Orbit term drives demagnetization of the more localized electrons until stabilization at lower moment is achieved

Playing with laser parameters

Influence of approximation for xc functional

The four steps of any functional theory

Step 1: Basic Theorems (Hohenberg-Kohn-Sham/ Runge-Gross)

Step 2: Find approximate functionals for

Step 3: Write code that solves the KS equations efficiently

Step 4: Run code for interesting systems/questions

( ) ( )xcv r ' t ' rt ρ

Problem: In all standard approximations of Exc (LSDA, GGAs)m(r) and Bxc(r) are locally parallel

S. Sharma, J.K. Dewhurst, C. Ambrosch-Draxl, S. Kurth, N. Helbig, S. Pittalis, S. Shallcross, L. Nordstroem E.K.U.G., Phys. Rev. Lett. 98, 196405 (2007)

Why is that important?

Ab-initio description of spin dynamics:

microscopic equation of motion (following from TDSDFT)

XC Sm(r, t) m(r, t) B (r, t) J (r, t) SOC= × − ∇⋅ +

in absence of external magnetic field

Consequence of local collinearity: m×Bxc = 0: → possibly wrong spin dynamics→ how important is this term in real-time dynamics?

Construction of a novel GGA-type functional

Traditional LSDA: Start from uniform electron gas in collinear magnetic state. Determine from QMC or MBPT and parametrize touse in LSDA.

New non-collinear functional: Start from spin-spiralphase of e-gas. Determine from MBPT andparametrize to use as non-collinear GGA.

XCe [n,m]

XCe [n,m]

XCe [n, m]XCe [n, m]

F.G. Eich and E.K.U. Gross, Phys. Rev. Lett. 111, 156401 (2013)

( )( )( )

2

scosm m ssin

1 s

⋅

= ⋅ −

q rr q r

Magnetisation of a spin-spiral state in the uniform electron gas

SSW SSWxc xc (n,m,q,s)ε = ε

Illustration of spin spiral waves along one spatial coordinate for two different choices of wavevectorq=k1/2.

( ) ( )( ) ( ) ( )

2T2

2 4T T

Ds

D m d=

+r

rr r r ( ) ( ) ( ) ( )

( ) ( )

2 4T T2

4T

D m dq

m D+

=r r r

rr r

[ ] ( ) ( ) ( ) ( ) ( )( )GGA 3 SSWxc xcE n,m d r n n ,m ,q ,s= ε∫ r r r r r

( ) ( ) ( )( ) 22Td m m= × ∇r r r ( ) ( ) ( )( ) 2

TD m m= × ∇⊗r r r

F.G. Eich and E.K.U. Gross, Phys. Rev. Lett. 111, 156401 (2013)

Beyond 3D bulk

Cr monolayer

0 50 100 150 200 (fs)

Streamlines for Jx, the spin-current vector field of the x component of spin, around a Ni atom in bulk (left) and for the outermost Ni atom in the slab (right).

0 50 100 150 200 (fs)

Heusler compounds

GaMn

Mn3Ga

Laser parameters: ω=2.72eV Ipeak= 1x1015 W/cm2 J = 935 mJ/cm2 FWHM = 2.42 fs

Global moment |M(t)| preservedLocal moments around each atom change

Mn3Ga

P. Elliott, T. Mueller, K. Dewhurst, S. Sharma, E.K.U.G., arXiv 1603.05603

Ga 0.02 μBMn -3.14 μBNi -0.37 μB

Ni2MnGa

See loss in global moment

Laser parameters: ω=2.72eV Ipeak= 1x1015 W/cm2 J = 935 mJ/cm2 FWHM = 2.42 fs

Ni2MnGa

Also change in local momentsTransfer of moment from Mn to Ni (does not require SOC)Followed by spin-orbit mediated demagnetization on Ni

Ni2MnGa

P. Elliott, T. Mueller, K. Dewhurst, S. Sharma, E.K.U.G., arXiv 1603.05603

Summary

• No demagnetization without Spin-Orbit coupling

• Demagnetization in first fs is a two-step process:1. Initial excitation of electrons into highly excited states

(without much of a change in the total magnetization)2. Spin-orbit coupling drives demagnetization of localized

electrons (mainly d electrons)

• No significant change in Mx and My

• New xc functional derived from spin-spiral phase of uniform e-gasyields results very similar to non-collinear LSDA

• Ultrafast transfer of spin moment between sublattices of Heuslercompounds: Easily understood on the basis of the ground-state DOS

Florian Eich

Kay Dewhurst

Sangeeta SharmaKevin Krieger

Peter Elliott