On the theory of solid-state harmonic generation governed by crystal symmetry

One of the most widely studied and robust laws in SSHG is its selection rule, which mainly depends on the crystal symmetry and can be divided into two categories. The first type appears in the typical Floquet systems and results from periodic oscillations of laser fields. Photoexcited carriers can display dynamical symmetry and coherently generate harmonic radiation Neufeld et al. (2019a). The other is induced directly by the crystal symmetry that is not broken by applied fields. The photocurrents cancel each other out, leading to no harmonic in some particular directions.

In the strong-laser region, external electric field can be compared with the coulomb field in crystals, it cannot be regarded as a perturbation. In this case, we need to consider the influence of the time-dependent population of charge density on the nonlinear process. Therefore, compared with the perturbation approximation, a broader theory is expected to treat systems under strong laser fields.

The crystal symmetry we considered here includes the point-group symmetry and time-reversal symmetry. Due to the periodic translational symmetry, crystals are limited to 32 point groups and 122 magnetic point groups. In addition, the laser fields could also contain abundant time-space symmetry. By combining the symmetries of crystals and light fields, we can obtain a wide variety of selection rules for SSHG. In this paper, we further explore more fundamental microscopic dynamics underlying these rules.

Consider a point-group symmetry operation $\hat{G}$ on the transition dipole matrix element (see Appendix B for derivations), we have

in which $a,a^{\prime}=G^{-1}a$ denote the directions of the transition dipole moments, $G_{a^{\prime}a}=e^{i\left[\phi_{nm}^{a^{\prime}}(\mathbf{k}(t))-\phi_{nm}^{a}(\mathbf{k}(t))\right]}$ is just the $\Delta\mathrm{TDP}$ between $a$ and $a^{\prime}$ direction under $\hat{G}$. Moreover, the scalar quantities such as the dynamical phase and shift phase are invariant under $\hat{G}$.

With the addition of the light field, let’s combine $\hat{G}$ and order-$N$ time translation operator ($N\in\mathbb{N}$, $\mathbb{N}$ denotes set of natural numbers):

with $\hat{\tau}_{N}t\equiv t\pm\frac{T_{0}}{N},T_{0}=\frac{2\pi}{\omega_{0}}$ is the period of the laser field of frequency $\omega_{0}$. We apply the dynamical symmetry operation to the photocurrent,

Every wave vector $\mathbf{K}$ in the lattice is a candidate for harmonic peaks unless symmetry forbids it. For the time-dependent quasi momentum $\mathbf{k}(t)\equiv\mathbf{K}+A(t)$,

The action of the laser field may disrupt initial symmetries of crystals, but new dynamical symmetry can be induced. Based on above transformation rules, we can derive that the interband current with dynamical symmetry can transform as

The electron density $\rho_{nm}$ is approximatively invarible under the dynamical symmetry operation. Same transformation rule is followed for the intraband counterpart.

We find that all the point-group dynamical symmetry operations only induce a change of photocurrent phase, which is the $\Delta\mathrm{TDP}$. That is, the transformation occurs only at the argument or the phase of the transition dipole, but the band dispersion, transition dipole amplitude, dynamical phase and shift phase are invariant. The coherence of harmonics stem from the interference between transition dipoles with different arguments associated by the dynamic symmetry (Fig. 1(a) and 1(b) show schematics of coherent dipoles under the mirror symmetry and rotational symmetry, respectively). More detailed derivation for typical cases of basic symmetry operations can be found in Appendix C. In Table I and Table II, $\Delta\mathrm{TDP}$-determined selection rules of SSHG are shown for the mirror symmetry and rotational symmetry.

In previous works of strong-field physics, similar to the expression of gas-state harmonics under strong-field approximation, only the dynamical phase induced by coulomb barriers has been considered. However, the role of shift phase induced by applied electric barriers in SSHG has not been clarified so far. To demonstrate indispensable role of shift phase in the process of photocurrent coherence, we compare order-2 dynamical coherence processes formed by spatial-inversion symmetry with time-reversal symmetry.

The second harmonic generation is one of the most common methods to determine the inversion symmetry of crystals. We know that when we apply a monochromatic light to a centrosymmetric crystal, even-order harmonics can be canceled by destructive interference, while the odd-order harmonics show constructive interference.

Let $\hat{P}=\hat{I}\cdot\hat{\tau}_{2},\hat{I}$ is the spatial-inversion operation. Using Eq. (10), the interband current under $\hat{P}$ transforms as

Integrating over the entire $\mathrm{BZ}$, we then have $\mathbf{J}_{nm}^{I^{-1}a}\left(\hat{\tau}_{2}t\right)=-\mathbf{J}_{nm}^{a}(t)$. The reversal of the current comes from the inversed dipole moment. After applying Fourier transform, we know that the radiated photon frequency is $\omega=(2l+1)\omega_{0},l\in\mathbb{N}$. The same conclusion can be reached by analyzing the intraband current.

Similarly, we define an operator $\hat{U}$ that performs time-reversal operation $\hat{T}$ on the crystals and order-2 temporal operator $\hat{\tau}_{2}$ on the time. The time-dependent quasi momentum has $\hat{U}\mathbf{k}(t)=-\mathbf{k}(t)$. The interband current under $\hat{U}$ takes (see Appendix D for details)

By comparing Eqs. (11) and (12), it can be found that the impact of $\hat{P}$ and $\hat{U}$ on the photocurrent differs by the shift phase and $\Delta\mathrm{TDP}$. The shift phase vanish in crystals with both time-reversal symmetry and inversion symmetry (see Appendix D for derivations). However, by time-reversal symmetry alone, a phase mismatch between the dynamical phase and shift phase as well as between dynamical phase and $\Delta\mathrm{TDP}$ will be caused, and completely destructive interference cannot be formed. Since the inversion symmetry results in pure odd-order harmonic generation, the shift phase and $\Delta\mathrm{TDP}$ should be crucial factors for even-order harmonic generation in crystals with time-reversal symmetry.

A channel of the three-step model in strong-field physics includes excitation, acceleration and re-collision processes of an electron-hole pair. Let’s further consider two interference channels driven by monochromatic laser field as Fig. 1(c) shows. When inversion symmetry exists, the two channels are identical with an interval of half an optical cycle (black arrows indicate), their interference leads to pure odd-order harmonics. If the system only possesses time-reversal symmetry, then these two channels are going to be different (magenta arrows indicate). The band dispersion and transition dipole amplitude remain symmetric in $k$-space due to the protection of time-reversal symmetry. However, due to the existence of the shift vector in non-centrosymmetric crystals, electrons need to do extra work in the photoelectric field when they take interband transitions Qian et al. (2022). Thus, the external light field equivalently modulate the energy curve of electrons like coulomb field, and forming the laser-dressed effective bands with symmetry breaking in $k$-space (magenta dashed curves in Fig. 1(c)). Therefore, extra shift phase accumulates in addition to the dynamical phase when pairs of dipoles perform intraband motions. The interference condition of pure odd-order harmonics is broken and even-order harmonics can be produced. The movement of electron-hole pairs on the effective bands simultaneously accumulates the dynamical phase and shift phase, which have fully equivalent effects on the photocurrent and together constitute its phase:

This fundamental image of interference involving two channels can be easily generalized to multiple channels. Therefore, the coherence process of SSHG can be clearly described by coherent channels of dipoles on the laser-dressed effective bands.

Based on above discussions, we continue to search for rules of SSHG that could be caused by time-reversal symmetry. The inversion symmetry breaking of crystals can be judged by even-order harmonic generation, which arises from interference of non-equivalent currents between two adjacent half cycles. Similarly, we can utilize laser fields with opposite helicities to find evidence of time-reversal symmetry breaking. The helicity of elliptically polarized light can be flipped by $\hat{T}$ without considering the Poynting vector of lights. It is found that the transport processes of charge carriers are different in magnetic materials driven by lasers with different helicity. In experiments, the magnetic circular dichroism of nonlinear optical response has been used to record the magnetic switching of materials Willems et al. (2015); Fan et al. (2015). The circular dichroism caused by laser fields also have access to selective excitation of spin, valley and chirality of electron states Xiao et al. (2012); Neufeld et al. (2019b); Yu et al. (2016); Ma et al. (2017).

In the following, we will demonstrate that the circular dichroism of SSHG is directly related to the time-reversal symmetry breaking of crystal by our theoretical method. Under the time-reversal transformation, the initial right-hand helically polarized laser ($\sigma_{+}$) is changed to left-hand helically polarized laser ($\sigma_{-}$), which have $\mathbf{E}_{\sigma_{+}}(t)=\mathbf{E}_{\sigma_{-}}(-t)$, and $\mathbf{A}_{\sigma_{+}}(t)=-\mathbf{A}_{\sigma_{-}}(-t)$. Each component of the photocurrent phase has the following transformation relation,

The dynamical phase, shift phase and $\Delta\mathrm{TDP}$ all reverse signs under $\hat{T}$, but this cannot be achieved by any pure point-group symmetry. Accordingly, the interband current generated by helically polarized lasers transforms as (see Appendix D for derivations)

If we assume that the difference of Fermi-Dirac distribution is time-reversal invariant (i.e., $f_{nm}(-\mathbf{K},-t)=f_{nm}(\mathbf{K},t)$), the interband current has $\hat{T}\mathbf{J}_{nm,\sigma_{+}}^{a}(t)\hat{T}^{\dagger}=$ $-\mathbf{J}_{nm,\sigma_{+}}^{a,*}(t)$. This assumption can work in the perturbation region with low-order changing rate of electron distribution. For the intraband current, the same conclusion can be derived. Then we have $\hat{T}\mathbf{J}_{\sigma_{+}}^{a}(t)\hat{T}^{\dagger}=\mathbf{J}_{\sigma_{-}}^{a}(-t)=-\mathbf{J}_{\sigma_{+}}^{a}(t)$, so lasers with opposite helicity can produce SSHG with the same intensity. We deduce that, under the protection of the time-reversal symmetry of crystals, the intensity of low-order nonlinear optical response does not show the circular dichroism. In contrast, the circular dichroism emerges when magnetic materials are considered. It allows us to use helically polarized laser fields to detect the magnetism of crystals.

Our discussion focuses on tight-binding models with honeycomb lattice due to its universality. Considering the hopping to the nearest-neighbor sites, the Hamiltonian is

where $i$, $j$ denote different sublattices. This is the simplest two-band Hamiltonian used to describe the graphene, which is subject to $D_{6h}$ point-group and time-reversal symmetries. For calculations, we set the lattice constant to 2.5 $\mathrm{\AA}$ and the nearestneighbor hopping $t_{1}$ to 2.33 $\mathrm{eV}$. The peak intensity of the driving laser we selected is $1.2\times 10^{12}\mathrm{~{}W}/\mathrm{cm}^{2}$, wavelength is 1.9 $\mu\mathrm{m}$, and full width at half maximum is 55 $\mathrm{fs}$ in a Gaussian envelope.

Figure 2 shows the polarization characteristics of harmonics parallel and perpendicular to linearly polarized laser field as a function of crystal orientation. The orientation angle $\theta$ is set to 0° when the laser is along $\Gamma-K$ direction. Pure odd-order harmonics are generated due to the inversion symmetry of graphene [see Figs. 2(a) and 2(b)]. The $C_{6}$ axis out of plane leads to the orientation periodicity of 60° for all harmonics.

Let’s break the inversion symmetry by introducing different on-site energy for adjacent atoms, the Hamiltonian is

in which $\epsilon_{i}$= ±1 for different atoms. It can be used to describe hexagonal boron nitride (h-BN), its point-group symmetry is reduced to $D_{3h}$. Different from the graphene, even harmonics are generated due to the inversion symmetry breaking (see Figs. 2(c) and 2(d)). The inversion symmetry breaking term $M_{0}$ is set to 1.96 eV. The interference processes in time domain can be reflected by time-frequency analysis spectra. Combining Figs. 3(b) and 3(d), we now know that even-order harmonics result from the difference of harmonic radiations between adjacent half cycles. By analyzing Eq. (13), the inverted laser field every half optical cycle cannot change the dynamical phase. However, the shift phase as well as $\Delta\mathrm{TDP}$ formed by the two unequal interference channels are different, which are responsible for the different temporal harmonic radiations and even-order harmonic generation in non-centrosymmetric crystals (see channel 1 and channel 2 in Figs. 3(c) and 3(e), which are same for graphene but different for h-BN). In Fig. 3, only parallel component of photocurents is considered, thus the $\Delta\mathrm{TDP}$ is vanish here but exist for other polarization directions.

In addition, the mirror symmetry can also induce destructive interference of harmonics as we derived by Eq. (10). When the driving laser is oriented parallel to the mirror plane $\left(\theta=30^{\circ}\pm 60^{\circ}l,l\in\mathbb{N}\right)$, there are no harmonics of perpendicular polarization because the currents cancel each other out. When the laser field is perpendicular to the mirror plane $\left(\theta=60^{\circ}\pm 60^{\circ}l,l\in\mathbb{N}\right)$, it strictly follows that only odd-order harmonics are generated for the parallel polarization and even-order harmonics for the perpendicular polarization. In addition, notice that although the h-BN only has an in-plane $C_{3}$ symmetry, the harmonics can form 6-fold orientation periodicity. This can be completely attributed to the multi-cycle driving field usually used. The reversal of a multi-cycle laser field, equivalent to its carrier envelope phase shifts $\pi$, does not affect the overall intensity of harmonics. If a few-cycle driving field is used, such extra 2-fold orientation periodicity of harmonics cannot be observed. Therefore, a few-cycle driving probe is needed to determine the rotation axis of crystals.

To further discuss the effect of the mirror symmetry and time-reversal symmetry breaking, we consider the Haldane model with magnetic phase on the next-nearest neighbor sites. The Hamiltonian is expanded as

where $v_{ij}$=±1 depending on the kind of atoms that the hopping takes place between. Here, we do not care about the optical behavior of its topological properties, the complex hopping strength is considered only to break the mirror and time-reversal symmetries. The basic point-group symmetry of this Hamiltonian is reduced to $C_{3h}$. The complex hopping strength $t_{2}$=0.63 eV, its phase $\varphi=\frac{\pi}{4}$. Since the mirrors are broken, the stable destructive and constructive interference that occur in the h-BN case vanishes in the Haldane model (see $\theta=\pm 30^{\circ}l,l\in\mathbb{N}$ in Figs. 2(e) and 2(f)). The spectra keep the orientation periodicity of 60°, which arises from the $C_{3}$ axis and the multi-cycle driving field.

The helicity dependence of SSHG can be used to probe molecular chirality, which is attributed to the circular dichroism of chiral molecules Xiao et al. (2012). The similar thing happens in crystals, the mirror symmetry can protect the harmonics from the circular dichroism. Let’s apply a helically polarized laser to a crystal with mirror symmetry. Under the mirror reflection, the applied right-hand helically polarized ($\sigma_{+}$) laser is changed to left-hand helically polarized ($\sigma_{-}$) laser. The photocurrent has

where $c$ is the normal direction of mirror plane, and $M_{ca}=e^{i\left(\phi_{nm}^{M_{c}^{-1}a}-\phi_{nm}^{a}\right)}$. In other words, the mirror reflection of the laser-crystal system directly causes the reflection of photocurrent. Therefore, there is no circular dichroism in SSHG from crystals with mirror symmetry. As Fig. 4(a) shows, the harmonic spectrum of h-BN driven by circularly polarized lights with inverse helicities have the same intensity. Due to the $C_{3}$ symmetry of the h-BN and the circularly polarized driving laser we used here, only $3l\pm 1(l\in\mathbb{N})$ harmonic orders are allowed.

However, an asymmetric profile of elliptically dependent SSHG from cubic crystals has been demonstrated in Refs. Tancogne-Dejean et al. (2017); You et al. (2017); Klemke et al. (2019), which was explained by the coupled intraband and interband dynamics. Here, we attribute this asymmetry to the mirror reflection mismatch between light field and crystal. The circular dichroism may be induced if only the helicity of elliptically polarized light is reversed while its orientation angle remains (see the difference in high-order region between the blue area and red solid line in Fig. 4(b)). When we reflect the laser-crystal system simultaneously, the circular dichroism will vanish (see the red solid line and green dashed line in Fig. 4(b)).

The time-reversal symmetry can also reverse the helicity of in-plane laser fields. We can use the circular dichroism of harmonics to identify the time-reversal symmetry of crystals. We note the area shaded yellow in Fig. 4(b) that the low-order harmonics, nearly below bandgap, never exhibit circular dichroism in h-BN. However, a completely different phenomenon emerges in the Haldane model with breaking time-reversal symmetry.

We deduce from Eq. (17) that, the time-reversal symmetry of crystals prevents the low-order harmonics from circular dichroism. In order to exclude the effect of mirror symmetry, we use an elliptically polarized laser and set the orientation angle to 15°. Expectedly, the low-order harmonics of h-BN keep perfect ellipticity-dependent symmetry [see Fig. 5(a)]. However, due to the absence of the time-reversal symmetry, asymmetric ellipticity dependence can be clearly seen in the Haldane model [see Fig. 5(b)]. Thus, the circular dichroism of harmonics can be used to identify magnetic materials.

Then we consider the time-reversal enantiomer by flipping the magnetic flux of the Haldane model. As Figs. 5(b) and 5(c) show, when the phase of the flux is inverted, the ellipticity dependence is reversed exactly as well. This is equivalent to performing $\hat{T}$ on the laser-crystal system, but the intensity of low-order harmonics is unaltered. Combined with ultrafast time-resolved spectra, it is promising that the low-order harmonics can be used to observe the magnetization degree of materials or ultrafast spin dynamics Beaurepaire et al. (1996); Gong et al. (2017).