Abstract
The generation of highorder harmonics in gases enabled to probe the attosecond electron dynamics in atoms and molecules with unprecedented resolution. Extending these techniques to solids, which were originally developed for atomic and molecular gases, requires a fundamental understanding of the physics that has been partially addressed theoretically. Here, we employ timedependent densityfunctional theory to investigate how the electron dynamics resulting in highharmonic emission in monolayer hexagonal boron nitride is affected by the presence of vacancies. We show how these realistic spinpolarised defects modify the harmonic emission and demonstrate that important differences exist between harmonics from a pristine solid and a defected solid. In particular, we found that the different spin channels are affected differently by the presence of the spinpolarised point defect. Moreover, the localisation of the wavefunction, the geometry of the defect, and the electron–electron interaction are all crucial ingredients to describe highharmonic generation in defected solids.
Introduction
Recent advances in midinfrared and terahertz laser sources have demonstrated the generation of nonperturbative highorder harmonics from solids, including semiconductors, dielectrics, and nanostructures below their damage threshold.^{1,2,3,4,5,6,7,8,9,10,11,12,13,14} With the pioneering work of Ghimire et al.,^{2} highharmonic generation (HHG) in solids offers fascinating avenues to probe lightdriven electron dynamics in solids on attosecond timescale,^{15,16,17,18,19,20} and to image energy banddispersion of solids.^{6,10,21} Moreover, due to the high electron density in solids in comparison to gases, HHG from solids may be superior for higher harmonic yield. Finally, HHG in solids represents an attractive route towards compact tabletop lightsource for coherent and bright attosecond pulses in the extreme ultraviolet and soft xray energy regime.^{1,9,22,23}
Due to the growth processes, defects are inevitable in real solids.^{24} Defects in materials can appear in the form of vacancies, impurities, interstitials (all of these can be neutral as well as charged), dislocations, etc. Defectinduced microscopic modifications in a material significantly affect on its macroscopic properties.^{25} Electronic, optical, vibrational, structural, and diffusion properties of solids with defects have been thoroughly reviewed over the past century.^{26,27,28,29,30,31,32} Defect engineering is used to achieve desirable characteristics for materials, e.g., doping has revolutionised the field of electronics.^{33} Defects can also be highly controlled, and it is possible to create isolated defects such as nitrogenvacancy defects in diamond^{34,35} or single photon emitters in twodimensional materials.^{36}
The influence of defects in solids is not well explored in strongfield physics. In this work, we aim to address the following questions: Is it possible to observe defectspecific fingerprints in strongfield driven electron dynamics? Or does the electron–electron interaction play a different role for defects and bulk materials? Moreover, some defects are also spin polarised in nature. This raises the question if it is possible to control the electron dynamics for different spin channels independently in a nonmagnetic host material. As we will show below, HHG is a unique probe that helps us to explore these interesting questions.
We aim to model strongfield driven electron dynamics in a defectedsolid with the least approximations. In this work, we employ ab initio timedependent density functional theory (TDDFT) to simulate the strongfield driven electron dynamics in defectedsolid.^{37} This allows us to come up with theoretical predictions relevant for real experiments on defectedsolid without empirical input. There are various theoretical predictions for the HHG from doped semiconductors by using simple onedimensional model Hamiltonians.^{38,39,40} In ref., ^{38} by using TDDFT, it is predicted that there is an enhancement of several orders of magnitude in the efficiency of HHG in a donordoped semiconductor. Using an independent particle model, Huang et al. contrastingly found that the efficiency of the second plateau from the doped semiconductor is enhanced.^{39} Similar calculations within a tightbinding Anderson model indicates that the disorder may lead to wellresolved peaks in HHG.^{41} The conceptual idea for the tomographic imaging of shallow impurities in solids, within a onedimensional hydrogenic model, has been developed by Corkum and coworkers.^{42} Even though these pioneering works have shown that defects can influence HHG, many points remain elusive. So far, only model systems in one dimension have been considered, and no investigation of realistic defects (through geometry optimisation and relaxation of atomic forces) has been carried out. Beyond the structural aspect, several other essential aspects need to be investigated in order to obtain a better understanding of HHG in defectedsolids such as the importance of electron–electron interaction (that goes beyond singleactive electron and independentparticle approximations), the role of the electron’s spin, the effect of the symmetry breaking due to the defects, etc. Our present work aims to shed some light on some of these crucial questions. For that, we need to go beyond the onedimensional model Hamiltonians.
In order to investigate how the presence of defects modifies HHG in periodic materials, we need to select some systems of interest. Nowadays, twodimensional (2D) materials are at the centre of tremendous research activities as they reveal different electronic and optical properties compared to the bulk solids. Monolayer 2D materials such as transitionmetal dichalcogenides,^{14,43} graphene^{44,45} and hexagonal boron nitride (hBN)^{46,47,48} among others have been used to generate strongfield driven highorder harmonics. Many studies have examined HHG in hBN when the polarisation of the laser pulse is either inplane^{47,48} or outofplane^{46} of the material. Using an outofplane driving laser pulse, TancogneDejean et al. have shown that atomiclike harmonics can be generated from hBN.^{46} Also, hBN is used to explore the competition between atomiclike and bulklike characteristics of HHG.^{46,47} In MoS_{2}, it has been demonstrated experimentally that the generation of highorder harmonics is more efficient in a monolayer in comparison to its bulk counterpart.^{43} Moreover, HHG from graphene exhibits an unusual dependence on the laser ellipticity.^{44} Lightdriven control over the valley pseudospin in WSe_{2} is demonstrated by Langer et al.^{14} These works have shed light on the fact that 2D materials are promising for studying lightdriven electron dynamics and for more technological applications in petahertz electronics^{49} and valleytronics.^{50}
Monolayer hBN is an interesting material for the study of electronic and optical properties. hBN is a promising candidate for lightemitting devices in the far UV region due to the strong exciton emission.^{51,52} Due to this technological importance, several experimental and theoretical studies have been carried out for hBN with defects.^{36,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68} Different kinds of defects in hBN can be classified as vacancy (monovacancies to cluster of vacancies), antisite, and impurities. In particular, defects like monovacancies of boron and nitrogen atoms in hBN are among the most commonly observed defects. Recently, Zettili and coworkers have shown the possibility of engineering a cluster of vacancies and characterising them using ultrahighresolution transmission electron microscopy.^{56,61,63} Signatures of defects in hBN are identified by analysing cathodoluminescence and photoluminescence spectra.^{51,64} It is shown that the emission band around 4 eV originates from the transitions involving deep defect levels.^{65} Ultrabright singlephoton emission from a single layer of hBN with nitrogen vacancy is achieved experimentally, for example in largescale nanophotonics and quantum information processing.^{36,53} In refs., ^{54,65,66,67,69} different kinds of defects in hBN are modelled, and their effects on the electronic and optical properties are thoroughly investigated.
In the present work, using the wellestablished supercell approach to model the defects in hBN, we analyse their influence on HHG. Due to the partial ionic nature of its bonds, hBN is a wide bandgap semiconductor with an experimental bandgap of 6 eV. This makes hBN an interesting candidate for generating highorder harmonics without damaging the material. Moreover, 2D material like hBN enable us to easily visualise the induced electron density and localised defect states easily. The presence of boron or nitrogen vacancies in hBN acts as a spinpolarised defect. These factors make hBN an ideal candidate for exploring spinresolved HHG in nonmagnetic defected solids. Below, we will discuss HHG in hBN with and without spinpolarised monovacancies.
Results
Computational approach
hBN has a twoatoms primitive cell. We model the vacancy in a 5 × 5 (7 × 7) supercell with 50 atoms (98 atoms) (See methods section for the details of structural optimisation) following the methods in ref. ^{54} The size of the 5 × 5 supercell is large enough to separate the nearest defects with a distance greater than 12 Å. This minimises the interaction between the nearby defects,^{54,69} and the defect wavefunctions are found to be welllocalised within the supercell.^{66} In the present work, we are not considering more than one point defect. Both of our vacancy configurations (single boron as well as single nitrogen vacancy) have a total magnetic moment of +1 μ_{B}, which is consistent with the earlier reported results.^{54,66,69} For the low defectconcentration considered here (2 and ~1%), the strength of the total magnetic moment is independent of the defect concentrations.^{69} We have found that the 7 × 7 and 5 × 5 supercells converged to similar groundstates.^{54}
In all the present calculations, we consider a laser pulse of 15femtosecond duration at fullwidth halfmaximum with sinesquared envelope and a peak intensity of 13.25 TW/cm^{2} in matter (for an experimental inplane optical index n of ~2.65^{70}). The carrier wavelength of the pulse is 1600 nm, which corresponds to a photon energy of 0.77 eV. The polarisation of the laser is linear and its direction is normal to the mirror plane of hBN. The symmetry of the pristine hBN permits the observation of the harmonics in the parallel and perpendicular directions to the laser polarisation. The direction resolved analysis of the HHG spectrum is shown in the Supplementary Information. For pristine hBN, the band gap is found to be 4.73 eV within DFTPBE level. The energy bandgap falls near the sixth harmonic of the incident photon energy and laser parameters are wellbelow the damage threshold of the pristine.
Highharmonic generation in hexagonal boron nitride with a boron vacancy
We start our analysis by comparing the HHG from pristine hBN and from hBN with a boron vacancy. Removal of a boron atom makes the system spinpolarised.^{54,69} The highharmonic spectrum of hBN with a boron vacancy (V_{B}) and its comparison with the spectrum of pristine hBN is presented in Fig. 1a. The spectrum of V_{B} is different from the pristine hBN as evident from the figure. There are two distinct differences: first, the below bandgap harmonics corresponding to V_{B} are significantly enhanced (see Fig. 1c). Second, the harmonics, closer to the energy cutoff, have a much lower yield for V_{B} in comparison to the pristine material.
An interesting aspect for the spinpolarised defects in a nonmagnetic host is that the effect of the defect states, compared to bulk states, can be identified clearly by examining the spinresolved spectrum. The harmonics corresponding to spinup and spindown channels in V_{B} are shown in Fig. 1b. As reflected in the figure, the below bandgap harmonics are different for both the channels. The strength of the third harmonic corresponding to the spindown channel is much stronger (by an order of magnitude) in comparison to the associated spinup channel as evident from Fig. 1d. This indicates that the increase in the yield observed in Fig. 1a–c predominantly originates from the spindown channel. At variance, the decrease in harmonic yield in higher energies matches well for both the spinchannels. This is a strong indication that the defect states do not play a direct role in this part of the HHG spectrum of V_{B}, but instead that bulk bands predominantly affect this spectral region, as discussed below.
Gap states and electron dynamics
To understand the difference in the harmonics yield associated with spinup and spindown channels, let us analyse the groundstate energy bandstructure. The unfolded bandstructures of V_{B} for spinup and spindown channels are shown in Fig. 2a, d, respectively. The bandstructure within the bandgap region is zoomed for both the channels and shown in Fig. 2b, c. As visible from the figure, there is one spinup (labelled as 1 in Fig. 2b) and two spindown (labelled as 2 and 3 in Fig. 2c) defect levels within the bandgap of pristine hBN. One defect state corresponding to the spinup channel is pushed within the valence bands (see Fig. 2a). All three defect states within the bandgap are found to be unoccupied. These factors make the V_{B} a triple acceptor. The corresponding wavefunctions for these defect states are presented in Fig. 2e–g. The wavefunctions are found to be localised around the vacancy as expected for dispersionless states. The p_{x} and p_{y} states of the nitrogen atoms in the vicinity of a boron vacancy contribute to these defect states, giving a σcharacter to the vacancy wavefunctions.^{69}
Unlike the pristine hBN, spinup and spindown electrons in V_{B} see a different bandstructure near the bandgap, since the spinresolved ingap states are different. Therefore, spinup and spindown electrons evolve differently in the presence of the laser pulse. It means that interband transitions and ionisation involving the defect states will contribute differently to the spectrum. Hence, the spectral enhancement of the third harmonic can be understood as follows: as visible from the spindown bandstructure, there is an additional defect state near the valence band, which allows spindown electrons to be ionised or to recombine through multiple channels and contribute more to the third harmonic (see Fig. 1).
It is evident from the band structure that additional defectstates effectively reduce the minimal band gap needed to reach the conduction bands. Due to the relaxation of atoms next to the vacancy, the pristine bands are also slightly modified. However, this modification is found to be negligible compared to the photon energy of the laser and is not further discussed. In order to understand how the presence of the defect states influences the interbandtunnelling, the number of excited electrons during the laser pulse is evaluated (see Fig. 3a). In the presence of defect states, there are mostly two possible ways in which ionisation can be modified compared to the bulk material. A first possibility is the direct ionisation of the defect states if they are occupied, or filling them if they are originally unoccupied. Another possibility is a double sequential ionisation, in which the defect states play an intermediate role in easing the ionisation to the conduction bands. In V_{B}, there is a finite probability of finding the electrons in conduction bands even after the laser pulse is over (see the red curve in Fig. 3a). In contrast to this, the pulse is not able to promote a significant portion of the valence electrons to the conduction bands permanently in the case of pristine hBN as the bandgap is significantly large (see the black curve in Fig. 3a). More precisely, for the 5 × 5 supercell, we found that 1.6 electrons are ionised, compared to 0.25 in the case of the same cell in the pristine material.
To achieve a better understanding of the possible ionisation mechanisms, we also consider the induced electron density (n_{ind}) at two different times near the peak of the vector potential (marked as 1 and 2 in Fig. 3b) and at the end of the laser field (Fig. 3f, g). As reflected in Fig. 3d, e, the spinpolarised induced densities of V_{B} have a pronounced localised component near the defects and resemble the spatial structure of the initially unoccupied defect wavefunctions (see Fig. 2e–g). The induced densities at the end of the vector potential show that electrons remain in the defect states even after the laser pulse is over (see Fig. 3f, g). Considering that the three defect states are originally unoccupied, we cannot conclude that more electrons are ionised to the conduction bands. It is most likely that ingap defect states are filled during the laser excitation as 1.6 electrons are excited and V_{B} is a triple acceptor.
Overall, these results show that the electron dynamics in acceptordoped solids imply a net transfer of population to the originally unoccupied gap states, but for a wide bandgap host no more electrons are promoted to conduction bands. This explains why only the loworder third harmonic is directly affected by the presence of spinpolarised defect states (as evidenced by the spin dependence of the spectrum). The low density of these defect states means that fewer photons are absorbed. We note that the irreversible population change, assisted by the defect states, ultimately implies that more energy is absorbed by the defected solid, which leads to a lower damage threshold in comparison to the pristine. However, the intensity considered here is low enough to see such an effect.
Effect of electron–electron interaction
We have established so far that the increase of the loworder harmonic yield is compatible with the presence of the defect states in the band gap of hBN. This is an independentparticle vision, in which we used the groundstate bandstructure of V_{B} to explain the observed effect on the HHG spectrum. We now turn our attention to the higher energy harmonics, for which the harmonic yield is decreased. This seems not to be compatible with a simple vision in terms of the singleparticle band structure, especially in view of the fact that more electrons are excited by the laser pulse, as shown in Fig. 3a.
To understand this, let us investigate the effect of the electron–electron interaction on the electron dynamics in V_{B} and pristine hBN. Within the dipole approximation, the HHG spectrum can be expressed as^{37,71}
Here, n_{ind}(r, t) is the induced electron density, v_{0} is the electronnuclei interaction potential, N_{e} is the total number of electrons, and E is the applied electric field. n_{ind}(r, t) is the difference of the total timedependent electron density n(r, t) and the groundstate electron density n_{0}(r), i.e., n_{ind}(r, t) = n(r, t) − n_{0}(r). Also, n(r, t) is decomposed in spinpolarised fashion as n(r, t) = n_{↑}(r, t) + n_{↓}(r, t). If one analyses this expression, it is straightforward to understand how the introduction of vacancies can change the harmonic spectrum, through a change in the local potential structure near the defect. This results in the change in gradient of the electron–nuclei interaction potential v_{0}, which is independent of electrons interacting with each other or not. Apart from this explicit source of change, it is clear that the dynamics of the induced density, evolving from a different groundstate also leads to the modifications in the harmonic spectra. The fact that the ground states are different, and hence n_{0}(r) is different, does not affect the harmonic spectrum because of the absence of time dependence in both n_{0} and v_{0}. The possible difference between the HHG spectra can, therefore, be understood in terms of independent particle effects (originating from the structural change of the nuclear potential v_{0}) and interaction effects from the induced density n_{ind}.
In order to disentangle these two sources of differences between pristine and defect hBN, we simulate the harmonic spectra within an independent particle (IP) model by freezing the Hartree and exchangecorrelation potentials to the groundstate value. The harmonic spectra for pristine hBN and V_{B} within TDDFT and IP are compared in Fig. 4. In the case of pristine hBN, the HHGspectra obtained by the IP model and TDDFT are similar. Hence, there is no significant manybody effect in HHG from pristine hBN with an inplane laser polarisation, at least as described by the PBE functional used here. A similar finding has been reported for Si^{37} and MgO,^{72} within the localdensity approximation. Only the highorder harmonics display small differences and there the electron–electron interaction reduces the harmonic yield, as found in ref. ^{38} In contrast, the HHGspectra obtained by IP and TDDFT are significantly different for V_{B} as reflected in Fig. 4b. This indicates that the electronelectron interaction is essential for HHG in defected solids. The total currents corresponding to the harmonic spectra for the pristine hBN and V_{B} are shown in Supplementary Fig. 2.
For the case of V_{B}, the induced electron density is displayed in Fig. 3. This helps us to understand how the spatial structure of the defect wavefunctions influences the spectrum. Similarly, Fig. 3c indicates that the spatial density oscillations are responsible for HHG in pristine hBN. It is clear that the induced density is different in the two systems. The substantial difference in the harmonic spectra of V_{B} obtained by two approaches, TDDFT and IP, is due to the socalled local field effects. This is explored in detail in the following paragraphs. It will be shown that this difference is responsible for the decrease of the harmonic yield for V_{B}.
In the presence of an external electricfield, the localised inducedcharge acts as an oscillating dipole near the vacancy. The dipole induces a local electric field, which screens the effect of the external electric field. This is usually referred to as local field effects. The same mechanism is responsible for the appearance of a depolarisation field at the surface of a material driven by an outofplane electric field.^{46,47} It is important to stress that this induced dipole is expected to play a significant role here, due to the σcharacter of the vacancy wavefunctions. The induced electric field is directly related to the electron–electron interaction term as clearly shown in ref. ^{47} As shown in Fig. 4b, the harmonic yield at higher energies is increased if we neglect local field effects, i.e., we treat electrons at the IP level. Moreover, the electron–electron interaction also affects the third harmonic, as shown in Fig. 4, but leads to an increase of the harmonic yield. We, therefore, attribute this effect not to local field effects but to correlation effects. It is important to stress that in Maxwell equations, the source term of the induced electric field is the induced density (summed over spin). Therefore, the induced electric field acts equally on the HHG from both spin channels, which is why the decrease of the yield occurs equally on both spin channels, see Fig. 1b.
We conclude that the modifications in the HHG spectrum due to a point defect originate from a complex interplay of two important factors: the electronic transitions including the ingap defect states, and the electron–electron interaction. This indicates that HHG in defectedsolids cannot be fully addressed via IP approximation as this can lead to wrong qualitative predictions in certain cases.
Finally, we discuss the dependence of the defect concentration on the HHG spectrum by computing the HHG for a 7 × 7 supercell with a boron vacancy, which corresponds to a ~1% doping concentration. The harmonic spectrum is presented in Supplementary Fig. 4. The third harmonic enhancement persists with comparable intensity even where there is a lower defect concentration, whereas the higher energy region of the harmonic spectrum matches well with the pristine spectrum. This is consistent with the observation that the higher energy spectrum is dominated by the bulk bands. This indicates that some of the present findings depend on the defect concentration and may not be observed below a certain concentration threshold.
Highharmonic generation in hexagonal boron nitride with a nitrogen vacancy
Until now, we have discussed HHG in hBN with a boron vacancy. Now we will explore HHG in hBN with a nitrogen vacancy. Similar to the V_{B} case, hBN with a nitrogen vacancy (V_{N}) is also spin polarised. Figure 5a presents the highorder harmonic spectrum of V_{N} and its comparison with the spectrum of pristine hBN. As apparent from the figure, the spectrum of V_{N} resembles the pristine spectrum more closely, except in the below bandgap regime and for an increase of the yield between 30 and 35 eV. All the laser parameters are identical to the case of V_{B}. The spectra of V_{N} and V_{B} could be expected to resemble each other as one atom from the pristine hBN has been removed in both of them. However, this is not the case, as evident from Fig. 5b and we note that the highenergy part of the spectrum of V_{N} is much closer to the one of pristine hBN than V_{B}, except above 30 eV. The spectra of V_{N} and V_{B} are also fairly different. On close inspection of the below bandgap spectrum, one finds that the third harmonic in V_{N} is significantly enhanced with respect to the pristine case (see Fig. 5c). The same observation was made in the case of HHG in V_{B}. To understand whether the reason behind this enhancement is identical to V_{B} or not, we examined the spinresolved harmonics in V_{N}. The below bandgap spinresolved spectrum reveals that the third harmonic has a greater contribution from the spinup electron than the spindown electron (see Fig. 5d; also see Supplementary Fig. 5 for the corresponding spectra in the full energy range). This contrasts the findings in V_{B} where the major contribution originated from the spindown electron. To understand this behaviour, let us explore the groundstate bandstructure of V_{N}.
The unfolded bandstructure of V_{N} is presented in Fig. 6. Each of the three boron atoms has an unpaired electron following the removal of a nitrogen atom from pristine hBN. One spinup and one spindown vacancy states emerge within the bandgap and are located more closely to the conduction band (see Fig. 6a–d). One additional defect state is pushed further towards the conduction band in each case. The spinup defect state is found to be occupied. This makes V_{N} a single donor.^{54} The p_{z}states of the boron atoms, which are in the vicinity of a nitrogen vacancy, contribute to the defect states. This gives a πcharacter to the defect wavefunctions (Fig. 6e, f).^{54,66} Similar to V_{B}, only gap states are analysed. The spinup defect level is occupied and close to the conduction band, which explains the major contribution of the spinup electron to the third harmonic in V_{N}. Electrons in this state can easily get ionised to the conduction bands and add more spectral weight to the third harmonic. Note that, unlike V_{B}, the local symmetry in V_{N} is preserved, which also explains why the HHG spectrum from V_{N} is close to the HHG from pristine hBN.
The unfolded bandstructures of V_{N} and V_{B} mainly explain the significant enhancement of the third harmonic and its different spinpolarised nature. To explain the overall difference in the harmonic spectrum, the snapshots of n_{ind} in V_{N} at different times along the vector potential are presented in Fig. 6g–j. In the realspace picture, defect contribution arises from the localised induced density, which can be seen from the integral in Eq. (1). The depletion of the spinup defect state as well as induced electron density in the spindown defect state can be observed at the end of the pulse (see Fig. 6h, j).
In comparison to V_{B}, the effect of screening due to the local field effects is weaker in the case of V_{N}. In comparison to V_{B}, the weaker impact of the local field effects on V_{N} can be attributed to the following two reasons: (1) the induced charge density has a pronounced localised component around the nitrogen vacancy. As evident from Fig. 6e, f, the wavefunctions of spinup and spindown electrons have similar spatial structure, whereas the corresponding induced charge densities have opposite sign (see Fig. 6g–j). This partial cancellation of the spinresolved induced charge density lowers the total induced charge and results in weaker local field effects. (2) For the inplane laser polarisation, the induced dipole due to the πlike defect states gets much less polarised than for the σlike defect states. This results in weaker screening in V_{N} than V_{B}.
The weaker local field effects in V_{N} are fully consistent with the HHG spectra of V_{N}, obtained through TDDFT and IP models (see Supplementary Fig. 3). Here, the third harmonic enhancement is well captured within the IP model, but the increase in the yield between 30 and 35 eV is found to originate from the electron–electron interaction (see Supplementary Fig. 3). Finally, in the HHG spectrum of V_{N}, the defect state plays a role even at higher orders (see the spinresolved spectra in Supplementary Fig. 5), though these effects are feeble.
In essence, the total HHG spectra corresponding to V_{B} and V_{N} include the contributions from the energybands of pristine hBN as well as from the transitions including gap states. Moreover, electron–electron interaction plays significant and different roles in both cases. The harmonic spectrum of the defected solid preserves some piece of information of the pristine structure in the higher energy regime along with the characteristic signatures of the defect in the near bandgap regime, or close to the cutoff energy for V_{N}.
Discussion
In summary, we have investigated the role of vacancydefect in solidstate HHG. For this purpose, we considered hBN with a boron or a nitrogen atom vacancy. In simple terms, one may assume that hBN with a boron atom vacancy or with a nitrogen atom vacancy would exhibit similar HHG spectra since a single atom from hBN has been removed. However, this is not the case as boron and nitrogen vacancies lead to qualitatively different electronic structures, and this is apparent from their corresponding gap states. It has been found that once an atom is removed from hBN, either boron or nitrogen, the system becomes spinpolarised with a nonzero magnetic moment near the vacancy. As a consequence, the defectinduced gap states are found to be different for each spin channel and for each vacancy, which we found to be strongly reflected in the loworder harmonics. These contributions are strongly spindependent, according to the ordering and occupancy of the defect states. Altogether, the role of the defect states can be understood by analysing the spinpolarised spectra, and the findings are in accordance with the spin polarised bandstructure. This establishes one aspect of the role of defect states in strongfield dynamics in solids.
In addition, the vacancy wavefunctions of V_{B} and V_{N} show σ and πcharacters, respectively, which lead to different qualitative changes in the harmonic spectra of vacancies, due to the localfield effects and electron correlations. These different behaviours are caused by the creation or absence of an induced dipole, which may counteract the driving electric field, and directly depends on the spatial shape of the defect state wavefunctions. Moreover, the electron–electron interaction also manifests itself in the decrease of the harmonic yield close to the energy cutoff in the V_{B} case, whereas this effect is completely absent in the V_{N} or pristine cases. This implies that the nature of the vacancies in V_{B} and V_{N} is entirely different, as reflected in their HHG spectra, even at a defect concentration as low as 2%. The HHG spectrum of V_{N} is similar to the pristine hBN, whereas the spectrum of V_{B} differs significantly. These effects essentially imply that some defects are more suited to the modification of the HHG spectra of the bulk materials. This in turn opens the door for tuning HHG by engineering defects in solids.
From our work, we can also estimate about other known defects in hBN. If one considers the doping impurity instead of vacancy, e.g., carbon impurity, the bandstructure of hBN remains spinpolarised in nature near the band gap.^{54,65} If a boron atom is replaced by a carbon atom, one occupied spinup and one unoccupied spindown defect levels appear. Both the defect levels are near the conduction band with the wavefunctions contributed from the p_{z} orbitals of carbon and nearby nitrogen atoms. So, the carbon doping defect is expected to show qualitatively similar behaviour in the HHG spectra as that observed in the case of V_{N}. On the other hand, one occupied spinup and one unoccupied spindown states appear near the conduction bands when nitrogen is replaced with a carbon atom. The wavefunction in this case is contributed mostly by the p_{z} orbitals of the carbon as well as nearby boron atoms. In this case, the enhancement in the below bandgap harmonics is expected due to the defect states near the valence bands similar to V_{B}. However, the effect of screening is expected to be lower compared to V_{B} as the nature of wavefunctions here is similar to that of V_{N}.
In the case of bivacancy in hBN, one occupied defect state exists near the valence band and two unoccupied defect states near the conduction band (see ref. ^{65}). In this situation, if the separation between defect states and the nearby bands is small compared to the photon energy, no significant changes in the below bandgap harmonics are expected.
Let us finally comment on the appealing possibility of performing imaging of spinpolarised defects in solids using HHG. As spin channels are not equivalent in the studied defects, one might think about using circularly polarised pulses to probe each spin channel independently. However, in the context of dilute magnetic impurities, as studied here, a crystal will host as many defects with positive magnetic moments as the negative ones, and the signals for up or down spin channel will appear as identical after macroscopic averaging.
Our work opens up interesting perspectives for further studies on strongfield electron dynamics in twodimensional and extended systems, especially involving isolated defects. Further work may address the possibility of monitoring electronimpurity scattering using HHG, more complex defects such as bivacancies, and a practical scheme for imaging buried defects in solids.
Methods
Geometry relaxation
Geometry optimisation was performed using the DFT code Quantum ESPRESSO.^{73,74} Both the atomic coordinate and the lattice constant relaxation were allowed. Forces were optimised to be below 10^{−3} eV/Å. We used an energy cutoff of 150 Ry, and a kpoint grid of 10 × 10 × 1kpoints. We used a vacuum region of more than 20 Å to isolate the monolayer from its periodic copies. The relaxed lattice constants of 12.60 and 12.57 Å for V_{B}, and 12.48 Å for V_{N} were obtained while for pristine hBN, it was found to be 12.56 Å. The structure of hBN with a boron vacancy was relaxed by lowering the local threefold symmetry.^{54} The lowering of the symmetry with the boron vacancy is attributed to the JahnTeller distortion, which is found to be independent of the defect concentration.^{54,65,69}
TDDFT simulations
By propagating the KohnSham equations within TDDFT, the evolution of the timedependent current is computed by using the Octopus package.^{75,76} TDDFT within generalised gradient approximation (GGA) with exchange and correlations of PerdewBurkeErnzerhof (PBE)^{77} is used for all the simulations presented here. The adiabatic approximation is used for all the timedependent simulations. We used normconserving pseudopotentials. The realspace cell was sampled with a grid spacing of 0.18 Å, and a 6 × 6 kpoints grid was used to sample the 2D Brillouin zone. The semiperiodic boundary conditions are employed. A simulation box of 74.08 Å along the nonperiodic dimension, which includes 21.17 Å of absorbing regions on each side of the monolayer, is used. The absorbing boundaries are treated using the complex absorbing potential method and the cap height h is taken as h = −1 atomic units (a.u.) to avoid the reflection error in the spectral region of interest.^{78}
Effective singleparticle bandstructure or spectral function for the vacancy structures is visualised by unfolding the bandstructure of 5 × 5 supercell with 50 atoms.^{79,80,81} The spinpolarised calculations are used to address the spinpolarised vacancies.
The Fouriertransform of the total spinpolarised timedependent electronic current j_{σ}(r, t) is used to simulate the HHG spectrum as
where \({\mathcal{FT}}\) and σ stand for the Fouriertransform and the spinindex, respectively.
The total number of excited electrons is obtained by projecting the timeevolved wavefunctions (\(\left{\psi }_{n}(t)\right\rangle\)) on the basis of the groundstate wavefunctions (\(\left{\psi }_{n^{\prime} }^{{\rm{GS}}}\right\rangle\))
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 (AIM), Grupos Consolidados (IT124919), SFB925, the Flatiron Institute (a division of the Simons Foundation), and Ramanujan fellowship (SB/S2/ RJN152/2015).
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N.T.D., A.R., and G.D. conceived the idea, designed the research and supervise the work. M.M.S. performed all the calculations. All authors discussed the results and contributed to the final manuscript.
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Mrudul, M.S., TancogneDejean, N., Rubio, A. et al. Highharmonic generation from spinpolarised defects in solids. npj Comput Mater 6, 10 (2020). https://doi.org/10.1038/s415240200275z
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