Abstract
In this work, we performed extensive firstprinciples simulations of highharmonic generation in the topological Diract semimetal Na_{3}Bi using a firstprinciples timedependent density functional theory framework, focusing on the effect of spinorbit coupling (SOC) on the harmonic response. We also derived an analytical model describing the microscopic mechanism of strongfield dynamics in presence of spinorbit coupling, starting from a locally U(1) × SU(2) gaugeinvariant Hamiltonian. Our results reveal that SOC: (i) affects the strongfield excitation of carriers to the conduction bands by modifying the bandstructure of Na_{3}Bi, (ii) makes each spin channel reacts differently to the driven laser by modifying the electron velocity (iii) changes the emission timing of the emitted harmonics. Moreover, we show that the SOC affects the harmonic emission by directly coupling the charge current to the spin currents, paving the way to the highharmonic spectroscopy of spin currents in solids.
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Introduction
The swift development of strongfield attoscience in solids has recently allowed for the development of many novel techniques, such as laser picoscopy of valence electrons^{1} or attosecond metrology in solids^{2}, tailored toward controlling electron dynamics in solids on unprecedented timescales^{3,4,5}. Among the possible applications of controlled strongfield dynamics in solids, one finds the possibility to perform highharmonic spectroscopy of various fundamental phenomena on femto to attosecond timescales, such as bandstructure dynamics^{6,7,8,9,10,11}, dynamical correlation effects^{12,13,14}, or the study of structural and topological phase transitions^{15,16}. Given the wealth of physical phenomena occurring in solids, much more exciting results are expected to emerge in the coming years.
Recently, topological effects, and the related Berry curvature effects, have attracted a lot of attention in the context of strongfield dynamics^{17,18,19,20,21}. In particular, several studies investigated how highharmonic generation (HHG) is affected by a topological phase transition using the Haldane model^{15,16,22} and related Haldane nanoribons models^{23}, and a more recent work investigated HHG in threedimensional topological insulators^{24}. In order to demonstrate the possibility to probe topological phase transitions using highharmonic generation in a experiment, one needs to find a material that can host these different phases, and for which one can, by tuning an external parameter, reach different parts of its phase diagram. In this respect, Na_{3}Bi was shown to be a very promising material, as its phase diagram is extremely rich and displays different topologically nontrivial phases, such as topological Dirac and Weyl semimetal phases, topological and trivial insulating phases, or a phase with nontrivial Fermi surface states with nonzero topological charges^{25}. Without breaking any system symmetry, Na_{3}Bi is a threedimensional (3D) topological Dirac semimetal^{26} where two overlapping Weyl fermions form a 3D Dirac point. It was shown that a compression of 1% along the y axis, which breaks inversion symmetry, can turn Na_{3}Bi into a topological insulator. Moreover, circularlypolarized light can be employed to break timereversal symmetry. Ab initio calculations for Na_{3}Bi driven by such circularlypolarized light showed that it induces a FloquetWeyl semimetal where the two Weyl points are separated in momentum space^{27}. Importantly, because in this case the crystal symmetries of Na_{3}Bi are still preserved, these Weyl points remain topologically protected^{25}. The capability to selectively break some of the symmetries of Na_{3}Bi either using pressure, or ultrafast circularlypolarized light pulses, allows to explore its phase diagram. However, before exploring the complex phase diagram of Na_{3}Bi, one needs to get a deeper understanding of the strongfield response in Na_{3}Bi in its pristine phase. This is the main scope of the present work.
At the hearth of many topological properties and Berrycurvaturerelated phenomena in quantum materials lies the spinorbit coupling (SOC) interaction. While SOC does not affect strongly HHG in atomic systems, mostly affecting the harmonic yield^{28}, it is known that the SOC can strongly modify the bandstructure of bulk materials, and is responsible for various phenomena such as spinHall currents^{29}, the anomalous Hall effect^{30}, and spintronics^{31}, to cite a few. Understanding how the SOC affects the strongfield dynamics in solids is therefore of paramount importance in order to later understand related phenomena such as topology. This is also why Na_{3}Bi is a very attractive choice, as it allows us to investigate SOC effects, without any Berrycurvaturerelated phenomena.
It is clear that SOC can modify the bandstructure of materials, and can, for instance, lead to bandinversion and thus induce topological insulators. SOC can also split energy bands and lift spin degeneracy, and renormalize electronic bands. Besides these modifications of the bandstructure, one might wonder if or how the SOC affects the dynamics of the excited carriers within these modified bands. Already, in the absence of any external laser field, by considering the lowest relativistic correction to the Pauli Hamiltonian containing the spinorbit interaction \({H}_{{{{\rm{SO}}}}}=\frac{\hslash e}{4{m}^{2}{c}^{2}}{{{\bf{p}}}}.({{{\boldsymbol{\sigma }}}}\times {{{\boldsymbol{\nabla }}}}V)\), where V is the potential due to the ions acting on the electrons and p their momentum, one finds that the velocity operator is modified such that \(\hat{{{{\bf{v}}}}}=\frac{i}{\hslash }[\hat{{{{\bf{r}}}}},\hat{H}]=\frac{1}{m}[{{{\bf{p}}}}+\frac{e}{c}{{{\mathcal{A}}}}]\), where \({{{\mathcal{A}}}}=\frac{\hslash }{4mc}{{{\boldsymbol{\sigma }}}}\times {{{\boldsymbol{\nabla }}}}V\) plays the role of a SU(2) gauge vector potential^{32}. This can be seen as leading to an effective magnetic field \({{{\mathcal{B}}}}={{{\boldsymbol{\nabla }}}}\times {{{\mathcal{A}}}}=\hslash [{{{\boldsymbol{\sigma }}}}.({\nabla }^{2}V)({{{\boldsymbol{\sigma }}}}.{{{\boldsymbol{\nabla }}}}){{{\boldsymbol{\nabla }}}}V]/(4mc)\). How the change of the electrons’ velocity induced by the SOC reflects in the strongfield response of solids is one of the main focus of this work. As we will see later, this change in the electrons’ velocity leads to a coupling between the charge current and the microscopic fluctuations of the spin current. Furthermore, because of minimal coupling, the spinorbit interaction is itself modified by light. However, this effect is usually negligible, as discussed below.
The relation between spincurrent and nonlinear response of bulk material has already been partly investigated in the context of perturbative nonlinear optics. It was shown that a pure spincurrent can induce a nonlinear response in GaAs in the form of a nonzero perturbative secondharmonic generation along specific directions ^{33,34}. It is therefore interesting to ask if HHG in solids could be used to measure spin currents. In ref. ^{35}, authors investigated theoretically the optical and spin harmonics emitted from iron monolayers, showing that the spin current displays a similar harmonic structure as the HHG obtained from the charge current. Whereas they showed that the spin current also exhibit nonperturbative harmonics that depends on the specific symmetries of their system, a discussion on how the HHG is directly influenced by the spin current remains elusive. To answer this question, we derived here the exact equation of motion of the physical charge current starting from a locally U(1) × SU(2) gauge invariant Hamiltonian,^{36,37} including spinorbit coupling.
The outline of this paper is as follows. In Results, we discuss some electronic properties of Na_{3}Bi, and we study its strongfield electron dynamics, focusing on the effect of spinorbit coupling on its strongfield response. In Equation of motion of charge current, we present the derivation of the equation of motion of the conserved charge current for a U(1) × SU(2) locally gaugeinvariant Hamiltonian, including necessary relativistic corrections. In particular, we obtain, in the lowest order in 1/c a formula for HHG in presence of spinorbit coupling for nonmagnetic materials. Finally, we draw our conclusions in Discussion. The details of the ab initio method used to model the electron dynamics in Na_{3}Bi are given in Methods.
Results
Using our ab initio TDDFT framework, we study HHG in bulk Dirac semimetals, taking Na_{3}Bi as a prototypical material. Its crystalline structure is shown in Fig. 1 a). A key feature to describe the electronic structure of Na_{3}Bi is the SOC. The effect of SOC on the bandstructure of Na_{3}Bi is also shown in Fig. 1. Without SOC, Na_{3}Bi is metallic, but does not display Dirac points at the Fermi energy, whereas SOC leads to the appearance of two Dirac cones at the Fermi energy, located at (0, 0, ± 0.28 π/c), in good agreement with the measured position of the Dirac cones in pristine Na_{3}Bi^{26}.
The modifications of the bandstructure induced by the SOC should affect the strongfield dynamics. Indeed, the curvature of the bands is modified, which will modify any intraband motion and the corresponding harmonic emission mechanism. The energy separation between valence and conduction bands is modified, which will affect both the excitation of carriers to the conduction bands and the interband harmonic emission by recombination of electronhole pairs. In the following, we will show that besides these expected changes, solely due to the change of the material’s bandstructure, we observe clear effects due to the change of the electron’s velocity due to the SU(2) gauge vector potential.
Highharmonic generation spectra
We start by analyzing the harmonic response of the Na_{3}Bi when excited by linearlypolarized and circularlypolarized laser fields. To be general, we consider here both y − z and x − y polarization planes, as shown in Fig. 2. Clear oddstructure harmonics are obtained in both cases, even if no clear plateau are observed in these spectra. Few points can be noticed directly: i) the intensity of the harmonics emitted from a circularlypolarized driver is not really much smaller than the one obtained for a linearlypolarized driver. In fact, for a laser polarized in the x − y plane (Fig. 2b)) the harmonic emission is even stronger for the circularlypolarized driver. An intense harmonic response to circularly polarized light is interesting as it can be used to produce highenergy photons which are circularly polarized^{38}, but it also can lead to a nonperturbative circular photogalvanic effect in this material. Latter, we will exploit the similarity of the linearly and circularly polarized light to unveil the role of the SOC on the excitation of carriers to the conduction bands. ii) The highest observed harmonics seems to be the same in all the cases, except for the linearly polarized driver in the x − y polarization plane. As a consequence, we will focus on the following on the y − z polarization plane for which the harmonic emission is stronger. iii) the two polarization planes lead to qualitatively different results for circularly polarized drivers. Indeed for the y − z plane, all odd harmonics are obtained for a circularlypolarized driver whereas for the x − y case, we observe two consecutive odd harmonics over three, the last one been suppressed, which means we observe harmonics 5, 7 but not 9, and so on. While this result looks surprising at first glance, it is only the result of selection rules^{39} for different symmetries of the different polarization planes, as already discussed and confirmed experimentally by some of us ^{11,38} and others^{40}. Indeed, in the case of the x − y polarization plane, the sixfold lattice symmetry leads to the suppression of the third harmonics, whereas the fifth and the seventh are not forbidden^{39}, as observed for instance in quartz^{38}. In the case of the y − z polarization, the fourfold symmetry leads to odd harmonics with alternating ellipticity, as observed in silicon or bulk MgO^{11,38}.
Electron dynamics and SOC
We now turn our attention to the number of electron excited by the laser pulse. In a carrier excitation process, the bandgap plays a key role in the excitation, and the excitation rate rate depends exponentially on it. We therefore expect that the number of excited electrons strongly depends on the details of the solid bandstructure in the case of interband excitation (Zener tunneling). In order to reveal the role of SOC on the excitation of carriers to conduction bands, we computed the number of excited carriers in Na_{3}Bi for a linearlypolarized laser, as well as a lefthandside (LHS) circularly polarized laser field, as shown in Fig. 3 for the same intensity. This plot shows clearly that more electrons are excited to the conduction bands when the SOC is included, irrespective of what is the polarization state of the light (see bottom panel). In the linearly polarized case, we observe clear virtual excitations at twice the frequency of the laser field, whereas in the circularlypolarized case, these oscillations are not present. This can be understood as follows: in the circularlypolarized case, electrons are ionized twice more often than in the linearly polarized case during the laser pulse, every time each components of the field reaches an extrema. Moreover, because the field strength along each direction is reduced by a factor \(1/\sqrt{2}\) for a circularlypolarized laser compared to the linearlypolarized case, excitation is roughly reduced by the same amount. This reduction of ionization for lower field strength is what we expect from singlephoton excitation through interband excitation. After the end of the pulse, the two pulses have still excited similar amount of electrons. However, the contribution from the SOC (see Fig. 3c) does not show a dependence on the polarization state nor the strength of the driver during the laser pulse, as it is almost identical for both linearly and circularlypolarized laser pulses, whose field strength differ by a factor \(1/\sqrt{2}\). We interpret the fact that the carrier excitation induced by SOC does not depend on the field strength as originating not from interband transitions, but from an intraband acceleration of carriers, through the Dirac points that appear when SOC is included. Finally, we note that the number of electrons is obtained here by a projection on the fieldfree states, which yield a gaugedependent number of electrons during the laser pulse. However, we are interested here not in the quantitative values during the laser field, but by how this quantity depends on the polarization state of the driver. Moreover at the end of the laser pulse the results are again gauge invariant, which makes our above analysis robust.
Effect of SOC on the charge and spin currents
Spinorbit coupling is a key physical ingredient needed to properly describe the topological nature of Na_{3}Bi. It is therefore interesting to investigate how much the strongfield electron dynamics is affected by the presence of SOC. As discussed in the introduction, the spinorbit coupling introduces a spindependent gauge vector field \({{{\mathcal{A}}}}=\frac{\hslash }{4mc}{{{\boldsymbol{\sigma }}}}\times {{{\boldsymbol{\nabla }}}}V\). This term leads to a different velocity of the electrons, depending on their spin, which directly implies that alongside with a charge current, there will be spin currents generated by the intense driving field when SOC is included.
We therefore computed both the charge current and the spincurrent induced by the laser field, defined in second quantization as
where \({\hat{\sigma }}_{i}\) are the Pauli matrices. Similar to Eq. (9), there is a contribution from the nonlocal operators that is omitted here for conciseness.
In absence of SOC, no spin currents are induced in the material, as Na_{3}Bi is nonmagnetic. This is clearly shown in Fig. 4 where we show the calculated charge current (panel a)) alongside with some components of the spin current (panels b)c)). Including the SOC, the electrons will behave differently depending on their spin, because of the presence of the spindependent gauge vector potential that modifies their velocity. Now we address the importance of these spin currents, compared to the magnitude of the charge current, especially for a nonmagnetic material like Na_{3}Bi. We found that the nonzero components of the macroscopic spin currents (J_{xx}, J_{zx}, J_{yy}, and J_{yz} for a laser circularly polarized in the y − z plane, J_{zx}, J_{zy}, J_{xz}, J_{yz} for a laser circularly polarized in the x − y plane) all have very similar magnitude, and we found that this magnitude is only half of the magnitude of the charge current, as shown in Fig. 4 for the case of a circularlypolarized driver. Our simulations thus reveal that strong driving fields can generate very efficiently spin currents, even when the electrons with different spin “feel” the same bandstructure. This shows that the modification of the electron’s velocity due to the SOC is not negligible. The spin currents, similar to the charge current, exhibit fast oscillations. We found (not shown here) that the harmonic spectra of the different components of the spin currents display an oddharmonic structure, and exhibit the same energy cutoff as the charge current, in agreement with the results of ref. ^{41} for Rashba and Dresselhaus models of SOC. The direct connection between spin currents and charge current, and thus HHG, is derived analytically and discussed in Equation of motion of charge current. Therefore, we focus in the following of this section on the effect of the SOC on harmonic response of the charge current.
As shown in Fig. 4a), the SOC also modifies the dynamics of the charge current. The charge current obtained in presence of SOC (black curve) is smoother than without SOC (green curve), and resembles more the time profile of the exciting laser pulse. This indicates that highfrequency components are reduced in the timedependent current, i.e., weaker harmonics are present in its power spectrum. We can directly see that the SOC reduces the harmonic response of the material by looking at the HHG spectra, see Fig. 5. For both linear and circularly polarized laser pulses, we obtain that SOC increases the linear response of the material while decreasing the yield of the harmonics. This result is counterintuitive, as we showed before that the SOC leads to more electrons excited in the conduction bands. It thus appears that more excited carriers do not directly reflect into more intense harmonic emission, which implies the carrier dynamics in presence of SOC, including their trajectories and recombination, is modified. We cannot simply understand the effect of SOC on the basis of a modified band structure, but we need to take into account how SOC reshape the electrons’ dynamics within these bands.
To further understand how the SOC can lead to less harmonic emission while more carriers are emitted, we analyzed the individual contribution of each spin channel to the charge current (j(r, t) = j^{↑↑}(r, t) + j^{↓↓}(r, t)). As shown in Fig. 6, the spindependent effective magnetic field induced by the SOC leads to a transverse motion of each individual spins, thus leading to an elliptically polarized emission from the individual spin channels, even for a linearlypolarized electric field. Again, we obtain that the spindependent gauge vector field has a important effect on the electron^’s velocity. As the effective magnetic field flips sign for each spin channel, the current associated with each of the two spin channels are of opposite ellipticity. However, because spins are degenerate in Na_{3}Bi, the transverse currents associated with each spin channel cancel each other, and no net transverse current is obtained, as shown in Fig. 6 for a linearly polarized driving field. This cancellation of the individual contributions of the transverse current leads to a smaller magnitude for the total electronic current, which reflects directly as a lower harmonic yield.
The component of the spinresolved current along the direction of the laser is also modified, as shown in Fig. 4. However, this change cannot be easily distinguished from the change in the electronic band curvature, also induced by the SOC. Still, the modification of electrons’ velocity leads to modified motion of the carriers in the bands, and we therefore expect a timedelay in the harmonic emission compared to the case in absence of SOC, due to the transverse motion of the carriers. We have performed a timefrequency analysis of the harmonic emission with and without SOC, as shown in Fig. 7, to analyze if the timing of the harmonic emission is modified in presence of SOC, as we expect from the result of Fig. 6. From the result of our timefrequency analysis, we found that the main emission event occurs at a later time when SOC is included (with a time delay of roughly 500 attoseconds), as expected from the motion of electron in presence of an effective magnetic field. The side structures are also shifted in a similar way. We also note a slightly more pronounce chirp of the emission in presence of the SOC. Finally, we found (not shown here) that a similar time delay occurs for circularly polarized laser fields, confirming that the modification of the temporal shape of the harmonic emission is present irrespective of the polarization state of the driving pulse. Overall, the modifications of the time of harmonic emission in presence of SOC demonstrate that the SOC modifies significantly the velocity of the carriers, and affects their motion in the bands. This clearly demonstrate that the modifications of the bandstructure alone account for all the changes in the HHG induce by SOC.
Equation of motion of the charge current
The equation of motion of the charge current, in absence of SOC, has been shown to provide valuable insights, for instance in the context of HHG in solids^{10}. Therefore, in this section, we aim at deriving the equation of motion for the physical charge current, in presence of spinorbit coupling.
We first consider the singleparticle Pauli Hamiltonian describing a particle of mass m and charge e in an external electromagnetic field,
where the canonical momentum is given by \(\hat{{{{\mathbf{\Pi }}}}}(t)=\frac{\hslash }{i}{{{\boldsymbol{\nabla }}}}+\frac{e}{c}{{{\bf{A}}}}(t)\), the magnetic field B(t) is given by B(t) = ∇ × A(t), where A(t) is the vector potential of the external field, and the electric field is given by \({{{\bf{E}}}}(t)=\nabla V(t)\frac{1}{c}{\partial }_{t}{{{\bf{A}}}}(t)\), with V the external scalar potential acting on the electrons. The first two terms describe respectively the kinetic energy and the potential energy of the particle. The third one is the Zeeman term, and the last one corresponds to the spinorbit coupling term. Importantly, the SOC contained in Eq. (2) allow for describing the part of the spinorbit coupling induced by the light, which is responsible for important physical effects, such as the inverse Faraday effect^{42}.
While the previous Hamiltonian has been considered in many theoretical work, it is possible to obtain a local U(1) × SU(2) symmetry by including a term of the higher order O(1/m^{3}) as explained in ref. ^{36}. The motivation for this choice lies in the fact that this will allow us later to define a U(1) × SU(2) gaugeinvariant spincurrent, which we defined in this work as the physical spin current, see Supplementary Information.
From this Hamiltonian, the equation of motion of the physical charge current is obtained, see the Supplementary Information. It is clear that this equation of motion contains many different contributions, which are not easy to analyze. In fact, it is easier to analyze the equation of motion of the macroscopic charge current, which is the source term of Maxwell equation, and therefore represent the emitted electric field. In a periodic system, the interaction term, as well as the divergences vanish, and therefore we obtain for the macroscopic average of the physical charge current, denoted \(\left\langle {\hat{j}}_{{{{\rm{phys}}}},k}(t)\right\rangle\),
Equation (3) is nothing but a forcebalance equation, and the righthandside terms are the external forces acting of the electrons. We directly identify the first term as the Lorentz force. In absence of spinorbit coupling, this is the only force term present in the equation of motion of the macroscopic charge current. The third term appears only when an electric field and a magnetic field coexist (see ref. ^{32} and references therein). The fourth term is a spin force that appears in presence of a nonuniform magnetic field, and was first evidenced by the Stern and Gerlach experiment. We note that the second term, which relates the magnetization of the electron to the external electric field, is also related to nonuniform magnetic field, if we assume that the external fields fulfill Maxwell equations. Finally, we also get a spin transverse force given by the term \(\left[{{{\bf{E}}}}\circ \left\langle {\hat{{{{\bf{J}}}}}}_{phys}\right\rangle \right]\times {{{\bf{E}}}}\) (here “∘” means a dot product on the spin index), which was discussed in ref. ^{32}. This force is proportional to the square of the electric field, and the spin current whose polarization is projected along the electric field. It is the counterpart of the force acting on a charged particle in a magnetic field j × B, in which the spin current replaces the charge current, and the electric field replaces the magnetic field
In order to proceed with the analysis of the equation of motion, we split the electricfield E into a contribution from the electronion potential v_{ext}, which we assume to be time independent (ions are clamped), and a part originating from the laser field E_{l}, assumed to be uniform (dipole approximation). Moreover, we assume that no external magnetic field is present. This leads to three contributions:
i) a part coming from the laser field
ii) an external part, from the electronion potential
and iii) a part containing cross terms coupling the electronion potential to the laser field
Let us analyze the terms we are getting. The first term in Eq. (4) of the laser contribution is the remaining of the Lorentz force. It is present without spinorbit coupling but is not responsible for any harmonic, as it only contains the frequency components of the external laser field. The second term in Eq. (4) is related to the macroscopic magnetization of the system. In a nonmagnetic material, such as Na_{3}Bi, this term vanishes. Finally, we get in Eq. (4) an highorder term coming from the spin transverse force. We implemented this term, as it is not taken care by the pseudopotentials as this is the case for the usual SOC, and checked that it is indeed numerically negligible. The cross terms in Eq. (6) contain highorder contributions (O(1/c^{4})), and are therefore negligible here. Finally, we turn our attention to the external contributions, given by Eq. (5). As a first term, we obtain the part coming from the Lorentz force, plus a highorder derivative of the electronion potential, that can be interpreted as a mass renormalization term. Given that this is also a highorder term, we also neglect it. In fact, among all the SOCrelated terms, only the second part of Eq. (5) contains a term that does not scale as 1/c^{4} but as 1/c^{2} for nonmagnetic materials. We therefore expect that the term scaling as 1/c^{2} will be the dominant contribution when SOC is included.
Retaining only this term, we obtain to the lowest order in 1/c and for a nonmagnetic materials such as Na_{3}Bi, a modified expression for HHG in presence of SOC only expressed in terms of gaugeinvariant quantities
Equation (7) is the main result of this section. It provides important physical insights into the effect of the spinorbit coupling on HHG in nonmagnetic materials. Indeed, this shows that the spin current is a source term for the equation of motion of the charge current, and therefore directly imprints into the emitted harmonic light. More precisely, we note that only the microscopic fluctuations of the spin current contribute to this extra term, and hence the HHG cannot be directly used as a probe of macroscopic spin currents. This formula offers an alternative perspective to the effect of the SOC on the strongfield response of bulk materials. The SOC does not simply modify the HHG by affecting the bandstructure of the materials, it also couples the charge current dynamics to the one of the spin current. This is the direct consequence of the change in velocity of the electrons, induced by the effective SU(2) gauge vector potential. Finally, let us comment on the fact that we considered here a nonmagnetic material. If the material is magnetic, our equation of motion Eq. (3) contains another contribution to the lowest order in the relativistic corrections, which deserves to be investigated, in particular in presence of demagnetization. Some of the higherorder contributions are also leading to harmonic emission along different directions, where there should not be any emission if SOC is omitted; and terms containing an electric field will lead to evenorder harmonic emission. Therefore one might be able to measure these terms independently of the rest of the systems’ response. A similar idea was already used to measure an effective Berry curvature in quartz from the measured HHG spectra^{43}, and our equation of motion offers many directions to use a similar logic to probe microscopic details of spin and charge current. We plan to investigate the precise role of these different terms in future works.
Discussion
In conclusion, we investigated the effect of the SOC on the strongfield electron dynamics in the topological Dirac semimetal Na_{3}Bi. We showed that the SOC affects the strongfield response by modifying the electronic bandstructure of the material, which controls the injection of carriers to the conduction bands. Beside, we showed that the SOC modifies the velocity of the electrons in the bands by acting as a spindependent effective magnetic field. As a consequence, the spinresolved electronic current becomes elliptically polarized, even for a linearlypolarized pump laser field. This modified trajectories modify the motion of the carriers, and lead to a time delay in the harmonic emission. We then derived the equation of motion of the total charge current from a locally gaugeinvariant U(1) × SU(2) Hamiltonian properly describing the spinorbit coupling in presence of a light field. From this, we showed that the SOC couples the charge current to the spin current, and we derived a formula for the HHG including to the lowest order relativistic corrections. This relation between charge and spin currents opens interesting perspectives for the optical spectroscopy of spin currents. Moreover, our equation of motion Eq. (3) can be applied to other materials, such as magnetic materials, for which other relativistic corrections might play a dominant role.
Finally, let us make some connection to some recent results obtained for the topological phase transition in the Haldane model. The increase of the number of excited electrons, as well as a time delay in the emission time of the harmonics has been identified as signatures of nontrivial topological phase^{16}, and the origin of these changes was associated to the Berry curvature of the material. Our results show that similar effects are obtained if we just include SOC, in a material which has no Berry curvature. The reason is that the Berry curvature can be seen as an effective magnetic field in momentum space^{44}, which also affects carriers trajectories^{45}. How electrons evolve in materials having both SOC and Berry curvature (topological ground state), and which effect dominates will of course require more detailed analysis, and should be the subject of future works.
Methods
Calculations were performed in the framework of realtime timedependent densityfunctional theory (TDDFT), which has been shown to yield valuable information both the microscopic^{10,12,38,46} as well as macroscopic ^{47,48} mechanisms leading to the HHG in solids and lowdimensional materials^{49,50,51}. In this work, we use an inplane lattice parameter of a = 5.448 Å and an outofplane lattice parameter c = 9.655 Å taking as the structure the one of ref. ^{25}, a realspace spacing of Δr = 0.36 Bohr, and a 28 × 28 × 15 kpoint grid to sample the Brillouin zone. We employ normconserving fully relativistic HartwigsenGoedeckerHutter (HGH) pseudopotentials, and we included spinorbit coupling unless stated differently. Because we are interested in shedding light on the role of SOC on the microscopic response to strong fields, we are not considering macroscopic effects like propagation effects due to the coupling with Maxwell equations in the present calculations. These propagation effects are important for thick samples (as shown in ref. ^{47}), but are expected to only play a minor role in thin samples and in reflection geometry. Moreover, in all the calculations, the ions are kept fixed and the coupled electronion dynamics is not considered in what follows. This is justified as the fastest optical phonon in Na_{3}Bi has a periodic of 166 fs^{52}, which is much longer that the duration of our laser pulse. For such fewcycle driver pulses, the HHG spectra from solids have been shown to be quite insensitive to the carrierenvelope phase (CEP), which is therefore taken to be zero here. We consider a laser pulse of 25 fs duration (FWHM), with a sinsquare envelope for the vector potential. The carrier wavelength λ is 2100 nm, corresponding to a carrier photon energy of 0.59 eV and the intensity of the laser field is taken to be I = 4 × 10^{11} W.cm^{−2} in matter. The timedependent wavefunctions and current are computed by propagating KohnSham equations within TDDFT, as provided by the Octopus package.^{53} The timedependent KohnSham equation within the adiabatic approximation reads
where \(\left{\psi }_{n,{{{\bf{k}}}}}\right\rangle\) is a Pauli spinor representing the Bloch state with band index n, at the point k in the Brillouin zone, \({\hat{v}}_{{{{\rm{ext}}}}}\) is the electronion potential, A(t) is the external vector potential describing the laser field, \({\hat{v}}_{{{{\rm{H}}}}}\) is the Hartree potential, \({\hat{v}}_{{{{\rm{xc}}}}}\) is the exchangecorrelation potential, and \({\hat{v}}_{{{{\rm{NL}}}}}\) is the nonlocal pseudopotential, also describing the spinorbit coupling as commonly done. As usual, the nonlocal pseudopotential contribution is made U(1) gauge invariant by properly including vectorpotential phases^{54}. However, as we will show later, this term is not enough to describe the U(1) × SU(2) gaugeinvariant spinorbit coupling. This is why we also included in \({\hat{v}}_{{{{\rm{NL}}}}}\) the term corresponding to the lightinduced modification of the spinorbit coupling, as derived and discussed in Equation of motion of charge current. In all our calculations, we employed the adiabatic local density approximation functional for describing the exchangecorrelation potential. From the timeevolved wavefunctions, we computed the total electronic current j(r, t),
where the nonlocality of the pseudopotential contributes to the charge current and leads to a spinorbit coupling contribution to it. From the knowledge of the total electronic current, the HHG spectrum is directly obtained as
where FT denotes the Fourier transform.
The total number of electron excited to the conduction bands (n_{ex}(t)) can be obtained by projecting the timeevolved wavefunctions \(\left(\left{\psi }_{n}(t)\right\rangle\right)\) on the basis of the groundstate wavefunctions \(\left(\left{\psi }_{{n}^{\prime}}^{{{{\rm{GS}}}}}\right\rangle\right)\)
where N_{e} is the total number of electrons in the system, and N_{k} is the total number of kpoints used to sample the BZ. The sum over the band indices n and \({n}^{\prime}\) run over all occupied states.
Data availability
The data that support the findings of this study are available from the corresponding authors upon request, and will be deposited on the NoMaD repository.
Code availability
The OCTOPUS code is available from http://www.octopuscode.org.
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Acknowledgements
This work was supported by the European Research Council (ERC2015AdG694097), the Cluster of Excellence ‘Advanced Imaging of Matter’ (AIM), Grupos Consolidados (IT124919) and SFB925. The Flatiron Institute is a division of the Simons Foundation.
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N.T.D. performed all the calculations and code implementation. N.T.D. and F.G.E. developed the theoretical model. All authors discussed the results and contributed to the final paper.
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TancogneDejean, N., Eich, F.G. & Rubio, A. Effect of spinorbit coupling on the high harmonics from the topological Dirac semimetal Na_{3}Bi. npj Comput Mater 8, 145 (2022). https://doi.org/10.1038/s41524022008316
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DOI: https://doi.org/10.1038/s41524022008316
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