Ultrafast optical control over spin and momentum in solids
Q.Z. Li, S. Shallcross, J.K. Dewhurst, S. Sharma, P. Elliott
UUltrafast optical control over spin andmomentum in solids
Q. Z. Li, † S. Shallcross, † J. K. Dewhurst, ‡ S. Sharma, ∗ , † and P. Elliott † † Max-Born-Institut f¨ur Nichtlineare Optik und Kurzzeitspektroskopie, Max-Born-Strasse2A, 12489 Berlin, Germany. ‡ Max-Planck-Institut f¨ur Mikrostrukturphysik, Weinberg 2, D-06120 Halle, Germany.
E-mail: [email protected]
Introduction
The coupling of laser light to matter can exert sub-cycle coherent control overmaterial properties, with optically induced currents and magnetism shown to becontrollable on ultrafast femtosecond time scales. Here, by employing laser lightconsisting of both linear and circular pulses, we show that charge of specifiedspin and crystal momentum can be created with precision throughout the firstBrillouin zone. Our hybrid pulses induce in a controlled way both adiabaticintraband motion as well as vertical interband excitation between valence andconduction bands, and require only a gapped spin split valley structure for theirimplementation. This scenario is commonly found in the 2d semi-conductors,and we demonstrate our approach with monolayer WSe . We thus establish aroute from laser light to local control over excitations in reciprocal space, openingthe way to the preparation of momenta specified excited states at ultrafast timescales. a r X i v : . [ phy s i c s . op ti c s ] D ec ub-cycle control of electrons in matter has already led to the experimental observationof femtosecond control over optically induced current , valley polarization in transitionmetal dichalcogenides (TMDC) , and the prediction from first-principles of control overmagnetic order faster than the exchange and spin-orbit times , subsequently confirmedby several experiments . The crucial limiting factor in fully implementing a coherentlightwave electronics lies in the excited spin and valley state lifetimes, in turn determined bythe fundamental scattering processes in materials: the electron-electron scattering, electron-phonon scattering, and spin scattering due to spin-orbit interaction.The decisive role in such scattering processes is played by crystal momentum, the quan-tum number associated with periodicity in solids. This determines “hot spots” in the bandstructure at which the amplitudes associated with electron-phonon interaction or spin re-laxation are large. Pre-selecting the crystal momentum of excited states by designed laserpulses thus represents a key step towards designing long-lived valley and spin excitations.However, to date precise lightwave control over the crystal momentum by light has not beendemonstrated.Laser pulses generate two distinct types of excitation in solids: intraband adiabatictransitions in which the crystal momentum k evolves according to the vector potentialas k → k + A ( t ), and diabatic transitions in which charge is excited between bands. InTMDC semi-conductors with broken inversion symmetry the diabatic transitions inducedby circularly polarized light are foundational for valleytronics. However, such excitation isby construction restricted to the high symmetry points (e.g. K and K’ points in TMDC),precluding control over crystal momentum. Circumventing this requires a new approach. Inthis work we demonstrate that hybrid laser pulses combining single cycle terahertz linearlight with circularly polarized light allows control over both adiabatic and diabatic motion,facilitating a full control over the crystal momentum. This unprecedented control over elec-trons in solids by lightwaves opens new possibilities both for the creation of spin and valleyexcited states, as well as probing the fundamental scattering processes of solids.2 a) (c) (e) (g) (i)Circular pulses Linear pulses(b) (d) (f ) (h) (j) Figure 1:
Charge excitation due to circularly and linearly polarized light.
The upper (lower) panels displaythe charge (spin) excitation in momentum space. A 1.6 eV pulse of circularly polarized light generates purespin excitation at the verticies of the Billouin zone: (a,b) σ + polarized circular light excites spin up statesat the K valley, while (d,e) σ − polarized light excites spin down states at the K’ valley. (e,f) A higherfrequency 2.0 eV pulse is in resonance both the spin split valence bands, resulting in a mixed spin but valleyspecific excitation consisting of (i) spin down excitation at K along with (ii) a surrounding halo of spin upcharge due to transitions from the higher energy spin up manifold. In panels (g-j) a single pulse of linearlight sufficient to excite interband charge results in asymmetric but a net spin unpolarized excitation, within each panel the arrows illustrating the intraband motion induced due to: (g,h) − x polarized light, (i,j) + y polarized light. Results
Due to broken inversion symmetry and strong spin-orbit coupling WSe presents an idealcase for the implementation of this idea. At the K/K’ valleys this material has significantspin spitting (a maximum of 450 meV at these K/K’ points) in addition to a gap of 1.6 eV.This allows for σ + / σ − circularly polarized light in the optical range to excite spin up/downpolarized charge at the K/K’ valleys, so called valley-spin locking. This is demonstrated, bythe means of a state-of-the-art first principles calculation performed using time-dependent-density functional theory (TDDFT), in Figs. 1a-d (for details of the computational techniquesee Methods section). A pump laser pulse with frequency equal to the gap was used (forother pulse parameters see Ref. ). Increasing the pulse frequency brings both bands intoresonance but at different crystal momenta, generating a mixed spin excited charge densityconsisting of the lower band spin at the valley centre surrounded by a halo of the higher bandspin, see Fig. 1e,f. For sufficiently strong linear light fields, the intraband motion induced3 a) (d) (g) (j) (b) (e) (h) (k)(c) (f ) (i) (l) Figure 2:
Laser control over both crystal momentum and spin polarization in WSe . A combination of aweak THz linearly polarized pulse with, at half cycle, a circularly polarized optical pulse generates chargeexcitation at pre-selected crystal momenta. With − x polarized linear light and σ − polarized circular lightthis hybrid pulse generates spin up excitation displaced from the K valley in the k x -direction by an amountdepending on the amplitude of the linear pulse, see panels (a-c) and (d-f). A − y, σ − hybrid pulse generatesspin up polarization in the vicinity of the M point, panels (g-i), while a + y, σ + hybrid pulse results in spindown polarization in the vicinity of the M point, panels (j-l). In each panel the blue arrows denote the typeof hybrid pulse with the pulse vector and scalar potentials displayed in the topmost panels. .by the pulse is accompanied by interband excitation via Landau-Zener (LZ) tunneling. Thisis shown in Fig. 1g-j for two different polarizations of linear light. LZ tunneling is effectiveat all valleys, and leads to a somewhat asymmetric valley excitation, that depends on thetrajectory in reciprocal space induced by the linear light. The excited states generated bylinear and circular pulses are thus restricted to the high symmetry points of the Brillouinzone.On the other hand, the presence of the gap allows for sufficiently weak linear light toadiabatically evolve Bloch states according to the acceleration theorem k ( t ) = k ( t = 0)+ A ( t )without exciting interband Landau-Zener transitions. Thus, by employing a single cycle4ulse we can propagate a state of arbitrary crystal momentum k ( t = 0) on the valence bandmanifold to one of the high symmetry K/K’ points and back to k ( t = 0). If, however, atexactly the centre of this pulse (i.e., at half-cycle where the A field is maximal) we applycircularly polarised light, this state will be excited to the conduction band, before returningunder the action of the linear light to k ( t = 0). Thus, charge will have effectively beenexcited vertically k ( t = 0) via a path through the “hot spots” at K/K’. Our hybrid pulsetherefore consists of a infra-red linear pulse in combination with optical circularly polarizedlight.The IR component of this hybrid pulse can control the displacement from the valley centreof the excited charge. To demonstrate this we first consider a IR Terahertz pulse centeredon frequency 0 . . (see Fig. 2a), generating the localizedcharge excitation shown in Figs. 2b,c. If we now consider a second pulse with double thevector amplitude with, in order to prevent LZ transitions, the frequency is reduced to 0 . . , the excited charge is approximatelydouble the distance from the K’ point. Note that the peak intensity is almost the same forboth pulses and both are linear polarized in − x direction. Thus by tailoring the frequencyand polarization of the IR pulse, we can excite spin at a pre-determined point in the 1BZ inprinciple arbitrarily far from the high symmetry K points.The circular component of the pulse, on the other hand, offers control over the spin-polarization of the excited charge. To demonstrate this we look at the region around an Mpoint and excite both up and down spin. In Fig. 2g-i, we use the same IR pulse as Fig. 2a-fbut instead oriented with − y polarization. This takes the M point electrons down to the Kpoint where a σ + pulse excites spin-up polarization, seen in Fig 2e,f. Conversely in Figs. 2j-l, we now change the linear polarization of the IR pulse to the + y direction and deploy a σ − pulse at half-cycle. This brings the M point up to the K’ point and excites spin-downelectrons. Thus, while both choices excite the M-point region, by tailoring the pulses we candecide whether spin-up or spin-down electrons are excited.5 a) (b) (c) Figure 3:
Impact of the width of the circular pulse on crystal momentum and spin polarization controlledcharge excitation in monolayer WSe . By increasing the duration of the circular pulse charge excitation atthe high symmetry K point occurs for a increased proportion of the intra-band trajectory, resulting in acomet like charge excitation. This should be compared with Fig. 2 in which a tighter circlar pulse results ina more focused charge excitation .Crucial to the controlled excitation of momentum in k -space described thus far is thatthe circular pulse occurs at half cycle of the weak IR linear pulse. Evidently, to create such ahybrid pulse the width of the circular pulse must be narrow as compared to that of the linearpulse. The circular pulses considered thus far have a FWHM of 7.5 fs, an experimentallychallenging laser field although one within reach of present day pump laser capability. Wethus now consider how increasing the width of the circular pulse to a standard 20fs durationimpacts the charge excitation (this laser pulse is shown in Fig. 3a). We would expect thata longer duration of inter-band excitation in relation to the intraband motion to result inthe smearing out of the focused charge excitation into a continuous scar, and indeed as seenin Fig. 3b-c we see that the intense localized charge excitations seen in Fig. 2 takes on a“comet” like appearance. Discussion
Future ultrafast nano-technology relies upon precise control over the electronic degrees offreedom of materials through designed laser pulses. In the present work we demonstratesuch a control; carefully sculpted light pulses, combining weak linearly polarized pulses with6ircularly polarized pulses, offers full control over excitations with specified spin and crystalmomentum. The materials which offer such a control require a gapped spin-split valley-typeband structure which is a common feature amongst 2d semiconductors. As the magnitudeof this gap is generally controllable via tuning by a layer perpendicular electric field (the gi-ant stark effect ) the full control over spin and crystal momentum we describe here shouldtherefore be possible in many 2d materials. Furthermore, we show that this physics oc-curs at ultrafast time scales, fully controlled by the pulse, and before electron-electron andelectron-phonon scattering processes. Such full control, in addition to representing a hith-erto unexplored richness of ultrafast phenomena in 2d materials, will be useful in probingmomentum and spin dependent scattering processes that will occur post charge excitation,such as electron-phonon coupling, and will further also be useful in tailoring the momentumcharacter of injected charge in an interface geometry. Methods
The Runge-Gross theorem establishes the time-dependent external potential as a uniquefunctional of the time dependent density, given the initial state. Based on this theorem, asystem of non-interacting particles can be chosen such that the density of this non-interactingsystem is equal to that of the interacting system for all times , with the wave function ofthis non-interacting system represented by a Slater determinant of single-particle orbitals. Ina fully non-collinear spin-dependent version of these theorems time-dependent Kohn-Sham(KS) orbitals are Pauli spinors governed by the Schr¨odinger equation: i ∂ψ j ( r , t ) ∂t = (cid:34) (cid:18) − i ∇ + 1 c A ext ( t ) (cid:19) + v s ( r , t ) + 12 c σ · B s ( r , t ) +14 c σ · ( ∇ v s ( r , t ) × − i ∇ ) (cid:35) ψ j ( r , t ) (1)7here A ext ( t ) is a vector potential representing the applied laser field, and σ are the Paulimatrices. The KS effective potential v s ( r , t ) = v ext ( r , t ) + v H ( r , t ) + v xc ( r , t ) is decom-posed into the external potential v ext , the classical electrostatic Hartree potential v H andthe exchange-correlation (XC) potential v xc . Similarly the KS magnetic field is written as B s ( r , t ) = B ext ( t ) + B xc ( r , t ) where B ext ( t ) is the magnetic field of the applied laser pulseplus possibly an additional magnetic field and B xc ( r , t ) is the exchange-correlation (XC)magnetic field. The final term of Eq. (1) is the spin-orbit coupling term. It is assumed thatthe wavelength of the applied laser is much greater than the size of a unit cell and the dipoleapproximation can be used i.e. the spatial dependence of the vector potential is disregarded.All the implementations are performed using the state-of-the art full potential linearizedaugmented plane wave (LAPW) method. Within this method the core electrons are treatedfully relativistically by solving the radial Dirac equation while higher lying electrons aretreated using the scalar relativistic Hamiltonian in the presence of the spin-orbit coupling.To obtain the 2-component Pauli spinor states, the Hamiltonian containing only the scalarpotential is diagonalized in the LAPW basis: this is the first variational step. The scalarstates thus obtained are then used as a basis to set up a second-variational Hamiltonianwith spinor degrees of freedom . This is more efficient than simply using spinor LAPWfunctions, however care must be taken to ensure that a sufficient number of first-variationaleigenstates for convergence of the second-variational problem are used. A fully non-collinearversion of TDDFT as implemented within the Elk code is used for all calculations.To calculate the crystal momentum, k , resolved excitation we use the expression N ex ( k ) = occ (cid:88) a unocc (cid:88) b |(cid:104) ψ a k ( t ) | ψ b k ( t = 0) (cid:105)| (2)Here the time-dependent KS orbitals at a given time t are projected on to the ground-state orbitals to calculate the change in occupation of the KS orbitals. Formally, withinTDDFT, the transient occupation of the excited-states does not necessarily follow that ofthe KS system. For weakly excited systems, however, the difference is expected to be small.8t should be noted that for high fluence pulses where the band renormalization effects arelarge, not applicable to this work, such an approximation would fail. Acknowledgements
QZL would like to thank DFG for funding through TRR227 (project A04). SS would liketo thank DFG for funding through SH498/4-1 and PE acknowledges funding from DFGEigene Stelle project 2059421. The authors acknowledge the North-German SupercomputingAlliance (HLRN) for providing HPC resources that have contributed to the research resultsreported in this paper.
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