Time-domain remark of ballistic orbital-angular-momentum currents with big leisure size in tungsten

Present extraction

To extract the in-plane sheet present I_C flowing contained in the pattern from the measured terahertz sign S_THz, we first measure our set-up response perform H_SE by having a reference electro-optic emitter (50 μm GaP on a 500 μm glass substrate) on the identical place because the pattern, which yields a reference terahertz sign (S_{mathrm{THz}}^{mathrm{ref}}). By calculating the emitted terahertz electrical area from that reference emitter E^ref, H_SE is set by fixing the convolution ({{{S}}}_{{rm{THz}}}^{{rm{ref}}}left({{t}}proper)=left({{{H}}}_{{{SE}}} * {{{E}}}^{{rm{ref}}}proper)left({{t}}proper)) (equation (2)) for H_SE (ref. ⁴³). Additional measured inputs for this calculation are the excitation spot measurement with a full-width at half-maximum of twenty-two μm, the excitation pulse vitality of 1.9 nJ and a Fourier-transform-limited pump pulse with a spectrum centred at 800 nm and a full-width at half-maximum of 110 nm. We carry out the deconvolution instantly within the time area by recasting it as a matrix equation¹⁶.

Subsequent, the electrical area E instantly behind the pattern is obtained from the recorded terahertz sign S_THz with the assistance of the derived perform H_SE by fixing the analogous equation S_THz(t) = (H_SE * E)(t) for E. Lastly, the sheet cost present (Supplementary Desk 1) as proven in Fig. 3 is derived from a generalized Ohm’s regulation²⁸, which within the frequency area at frequency ω/2π reads

Right here, −e is the electron cost and the pattern impedance Z(ω) is given by Z₀/[1 + n_sub + Z₀dσ(ω)] with the free-space impedance Z₀, the substrate refractive index n_sub ≈ 2 (ref. ⁵⁶) and the metal-stack thickness d = d_FM + d_PM. The measured imply pattern conductivity σ (Supplementary Desk 1) is roughly frequency-independent because of the giant Drude scattering charge (Supplementary Fig. 2). To allow a comparability of terahertz currents from totally different samples, we normalize I_C by the absorbed fluence within the FM. The info proven in Fig. 3 have been obtained in a dry-air ambiance.

Pattern preparation

The FM|PM samples (FM = Ni and Py, PM = Pt, Ti, Cu and W) are fabricated on glass substrates (thickness 500 μm) or thermally oxidized Si substrates (625 μm) by radio-frequency magnetron sputtering beneath an Ar ambiance of 6N purity. The pattern construction and thickness are described in Supplementary Desk 1. For the sputtering, the bottom stress within the chamber is decrease than 5 × 10⁻⁷ Pa. To keep away from oxidation, SiO₂ (thickness 4 nm) is sputtered on the floor of the movies. All sputtering processes are carried out at room temperature. The W movies are predominantly within the β-phase for d_W < 10 nm, with an α-phase content material that grows with d_W and dominates for d_W > 10 nm (ref. ⁹).

Estimate of digital temperatures

We calculate the digital temperature enhance ΔT_e0 upon pump-pulse absorption by

Right here, T₀ = 300 Okay is the ambient temperature, F_l is the absorbed fluence within the respective layer l = FM or PM (Supplementary Desk 1), d is the layer thickness and γT_e is the particular digital warmth capability at digital temperature T_e with γ = 300 J m⁻³ Okay⁻² for W, 320 J m⁻³ Okay⁻² for Ni, 330 J m⁻³ Okay⁻² for Ti and 90 J m⁻³ Okay⁻² for Pt (ref. ⁵⁷).

To acquire the absorbed fluences in every layer, we be aware that the pump electrical area is sort of fixed all through the pattern (Supplementary Fig. 9). Due to this fact, the native pump absorption scales solely with the imaginary half Imε of the dielectric perform ε at a wavelength of 800 nm, which equals 22.07 for Ni, 9.31 for Pt, 19.41 for Ti and 19.71 for W (ref. ⁵⁸). Consequently, the absorbed fluence is set by

with the whole absorbed fluence F_tot that’s obtained from the absorbed pump energy (Supplementary Desk 1) and the beam measurement on the pattern (as described beforehand).

Experimental error estimation

The error bars for the utmost place t_max of the transient cost present I_C(t) (Fig. 4b) are estimated as ±20% of the read-off delay worth (round markers in Fig. 4a), however at least 5 fs. The uncertainty within the relative amplitude of I_C(t_max) (Fig. 4c) and the relative space of I_C(t) (Fig. 4e) is, respectively, estimated as ±20% and ±10%, each reflecting the standard signal-to-noise ratio of the extracted present traces (Fig. 4a). The error bars for the full-width of I_C(t) at half-maximum (Fig. 4d) are obtained from the uncertainty of the delay (Fig. 4b) with subsequent multiplication by (sqrt{2}), which accounts for the error propagation of a distinction of two portions.

Mannequin of L transport

To mannequin the ballistic present within the PM, we assume {that a} δ(t)-like transient L accumulation within the FM generates an digital wave packet, which has orbital angular momentum ΔL_okay0 alongside the route of the FM magnetization M and imply wave vector okay within the PM proper behind the FM/PM interface, that’s, at z = 0⁺ (Fig. 5a).

Within the case of purely ballistic transport, this wave packet propagates into the PM bulk in line with ΔL_okay(z, t) = ΔL_okay0 δ(z − v_okayzt), the place v_okayz is the z element of the wave-packet group velocity. Be aware that we prohibit ourselves to okay values with non-negative v_okayz values. The overall pump-induced L present density flowing into the depth of the PM is for z > 0 given by the sum

Assuming that ΔL_okay0 arises from states not too removed from the Fermi vitality, the summation of equation (6) is roughly proportional to an integration over the Fermi floor components with v_okayz ≥ 0. One obtains

the place z > 0, and

is the L weight of the z-axis group velocity v_z. In equation (7), we phenomenologically account for the comfort of the ballistic present with time fixed τ by introducing the issue e^−t/τ. Performing the combination of equation (7) yields

To find out a believable form of w(v), we be aware that ΔL_okay0 is non-zero inside a number of 0.1 eV across the Fermi vitality E_F owing to the width of the photoexcited and quickly enjoyable electron distribution³³. Assuming a spherical Fermi floor and isotropic ΔL_okay0, we’ve ΔL_okay0 ∝ δ(E_okay − E_F) and v_okayz = v_Fcosθ, the place E_okay is the band construction, v_F is the Fermi velocity and θ is the angle between okay and the z axis. Due to this fact, after turning equation (8) into an integral, the integrand (Delta {L}_{{bf{okay}}0}updeltaleft({v}_{{rm{F}}}{{cos }}theta -{v}_{z}proper)) ({{rm{d}}^{3}{bf{okay}}}) turns into proportional to (updeltaleft({v}_{{rm{F}}}{{cos }}theta -{v}_{z}proper)) ({rm{d}}{{cos }}theta) in spherical coordinates, resulting in

the place Θ is the Heaviside step perform. In different phrases, all velocities ({v}_{z}={v}_{{rm{F}}}{{cos }}theta) from 0 to v_F have equal weight.

Within the case of purely diffusive transport, we use the L diffusion equation for μ_L (ref. ²³). With a localized accumulation μ_L(z, t) ∝ δ(z) at time t ≈ 0, the buildup disperses in line with the well-known resolution

for z > 0 and t > 0. Right here, D is the diffusion coefficient that equals ({v}_{L}^{2}tau /3) within the case of a spherical okay-space floor carrying the L wave packets, v_L is their group velocity and τ is their velocity leisure time. To find out the present density, we apply Fick’s regulation²³, j_L = −D∂μ_L/∂z to equation (11) and acquire

Ab initio estimate of the L velocity

The orbital velocity is estimated for the majority W in a body-centred cubic construction. The ab initio self-consistent calculation of the digital states is carried out inside density purposeful idea through the use of the FLEUR code², which implements the full-potential linearly augmented plane-wave (FLAPW) technique⁵⁹. The trade–correlation impact is included within the scheme of the generalized gradient approximation through the use of the Perdew–Burke–Ernzerhof purposeful⁶⁰. The lattice parameter of the cubic unit cell is about to five.96a₀, the place a₀ is the Bohr radius. For the muffin-tin potential, we set R_MT = 2.5a₀ for the radius and l_max = 12 for the utmost of the harmonic growth. Additional, we set the plane-wave cut-offs for the interstitial area to (4.0{a}_{0}^{-1}), (10.1{a}_{0}^{-1}) and (12.2{a}_{0}^{-1}) for the idea set, the trade–correlation purposeful and the cost density, respectively. For the okay-points, a 16 × 16 × 16 Monkhorst–Pack mesh is outlined.

From the converged digital construction, we acquire the maximally localized Wannier features through the use of the WANNIER90 code⁶¹. We use 18 Wannier states with (s,{p}_{x},{p}_{y},{p}_{z},{d}_{{z}^{2}},{d}_{{x}^{2}-{y}^{2}},{d}_{xy},{d}_{yz},{d}_{zx}) symmetries for spin up and down because the preliminary guess. The utmost of the inside (frozen) vitality window is about 5 eV above the Fermi vitality for the disentanglement, and the outer vitality window is outlined by the minimal and most energies of the 36 valence states obtained from the FLAPW calculation. The Hamiltonian, place, orbital-angular-momentum operators and spin-angular-momentum operators are remodeled from the FLAPW foundation into the maximally localized Wannier perform foundation. The ensuing digital band construction and texture of the orbital-angular-momentum operator L are displayed in Supplementary Fig. 9.

From this life like tight-binding mannequin, the orbital-momentum-weighted velocity averaged over the Fermi floor (FS) is calculated by

the place v_α and L_β are the α element of the rate and the β element of the orbital-angular-momentum operators, respectively. The |nokay〉 is the eigenstate of the Hamiltonian with vitality E_nokay and band index n, and ({f}_{n{bf{okay}}}^{,{prime} }) is the vitality by-product of the Fermi–Dirac distribution perform. To polarize L_β, we add a small orbital Zeeman coupling alongside the β route to the naked Hamiltonian. We affirm that the results of equation (13) modifications by lower than 1% when the orbital Zeeman splitting is elevated from 10 meV to 30 meV. The okay-space integrals in equation (13) are carried out on a 256 × 256 × 256 mesh.

The orbital velocity is estimated by

the place ({langle {L}_{beta }^{2}rangle }_{{rm{FS}}}) is obtained by equation (13), however with v_α changed by L_β. The result’s proven in Supplementary Fig. 10.

Particulars of ab initio LCC calculations

A skinny W stack of 19 body-centred cubic (110) atomic layers is calculated by the self-consistent ab initio technique utilizing the identical FLAPW parameters as for the majority calculation given beforehand, apart from the mesh of okay-points for which we use a 24 × 24 Monkhorst–Pack mesh. For Wannierization, we acquire 342 maximally localized Wannier features, ranging from Wannier states with (s,{p}_{x},{p}_{y},{p}_{z},{d}_{{z}^{2}},{d}_{{x}^{2}-{y}^{2}},{d}_{xy},{d}_{yz},{d}_{zx}) symmetries for spin up and down because the preliminary guess. We outline the utmost of the inside window at 2 eV above the Fermi vitality.

From the Hamiltonian of the W skinny movie, the okay-space orbital-angular-momentum texture on the Fermi floor is obtained by

the place ({langle {{bf{L}}}_{{rm{prime}}}rangle }_{n{bf{okay}}}=langle n{bf{okay}}|{{bf{L}}}_{{rm{prime}}}|n{bf{okay}}rangle) is the expectation worth of the orbital angular momentum for the 2 atoms on the highest floor, T = 300 Okay is temperature and okay_B is the Boltzmann fixed.

To calculate the cost present as a consequence of LCC, we contemplate the orbital-dependent chemical potential

as a perturbation. Right here, L_γ is the γ element of the orbital-angular-momentum operator, and r_β is the β element of the place operator, which is well-defined alongside the z axis (Fig. 1), and ({varepsilon }_{beta gamma }) could be interpreted as an orbital-dependent electrical area. The charge-current density alongside the α route is given by the Kubo method

the place V is the quantity of the system and Γ = 25 meV is a phenomenological broadening parameter. The LCC response is characterised by the tensor

which relates (langle {j}_{a}rangle) and ε_βγ by (langle {j}_{alpha }rangle ={sigma }_{{LCmathrm{C}},alpha beta }^{{L}_{gamma }} epsilon_{beta gamma}). The okay-space integral is carried out on a 400 ×  400 mesh. The z-resolved LCC response is proven in Supplementary Fig. 11.

Reporting abstract

Additional data on analysis design is offered within the Nature Portfolio Reporting Abstract linked to this text.