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Abstract
We investigate the interaction between the counter-rotating bicircular field and the trivial and topological insulator with anomalous Hall conductivity (AHC) to show the effect of the asymmetric spin band and topological invariant. We show that the reaction of the system to the counter-rotating bicircular field is classified into the high-field and low-field regimes. In the high-field regime, it is shown that the AHC of the system is controlled by the phase difference between the ω0 and 2ω0 fields. We also show that in the low-field regime, the AHC of the topological insulator is determined by the helicity of the laser, while the AHC is negligible in the trivial insulator. For the spin-orbit coupling (SOC), it is demonstrated that the high SOC increases the required field amplitude for the transition from the low-field to the high-field regime. Also, we show that strong SOC leads to an additional sign change of the AHC in the high-field regime, but with different origins in the trivial and topological insulator.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Laser-matter interaction of two-dimensional (2D) hexagonal lattice has been a hot topic in the ultrafast laser and solid state physics community. The hexagonal lattice has shown topological states such as the Chen insulator and the topological insulator, as demonstrated by the Haldane model [1] and the Kane-Mele model [2], respectively. In addition, the hexagonal lattice has a valley pseudospin, which is considered as an additional degree of freedom in addition to spin [3,4]. Research on valley polarization is even extended to the bilayer system [5–7].
Regarding selective valley excitation, a counter-rotating bicircular laser of $\omega _{0}$ and $2\omega _{0}$ has attracted interest. It has been shown that a bicircular laser in a honeycomb lattice can selectively excite a specific valley independently of the vector potential amplitude in contrast to a circularly polarized laser [8]. In addition, the bicircular laser can be used to control the valley polarization even in materials with vanishing Berry curvature, such as graphene [9–11]. However, the control of valley polarization in topological materials or strong spin-orbit coupling (SOC) material with a counter-rotating bicircular laser field has not been investigated yet.
In this study, we investigate the valley asymmetry of the trivial and topological insulator in the honeycomb lattice. We use a counter-rotating bicircular laser of $\omega _{0}$ and $2\omega _{0}$ which have a trefoil Lissajous figure of the electric field and the vector potential as shown in Fig. 2. Using semiconductor Bloch equations (SBEs), anomalous Hall conductivity (AHC) is calculated as an observable for asymmetry of the system or as an indicator of valley Hall effect. We compare the result of the trivial insulator with that of the topological insulator to show the effect of the asymmetric spin band and the topological invariant. By controlling the laser parameter and material parameter, we understand the mechanism of the valley asymmetry in the trivial and topological insulator under the counter-rotating bicircular field.
2. Theory
2.1 Kane-Mele model
The Kane-Mele model [2,12] for topological insulators in this study is given by
In the simulation, we use the lattice constant $a_{0}$ = 1 Å and the hopping energy $t_{1}$ = 0.075 a.u. ($\approx$ 2.04 eV), unless otherwise noted. We choose $t_{2}$ = 0.0 a.u. for the trivial insulator and $t_{2}$ = 0.025 a.u. ($\approx$ 0.68 eV) for the topological insulator. The corresponding energy dispersion and Berry curvature of the given parameter are shown in Fig. 1.
Fig. 1. Band structure of the trivial ($C=0$) and topological insulator ($C=\pm 1$). (a) Energy dispersion of the trivial and topological insulator. Berry curvature of the conduction band of (b) the trivial and (c) the topological insulator ($C=-1$ spin band only). $C=\pm 1$ correspond to the spin bands of the topological insulator.
2.2 Semiconductor Bloch equations
In the simulations, we use the SBEs [13–16] to calculate the time evolution of the system. We solve the equation of motion for the density matrix in a moving frame. The SBEs in the moving frame are given by
For the decay term $\Gamma$, we apply the dephasing time $T_{2}$, which is a phenomenological parameter that integrates the effect of every decay process such as electron-electron interaction, electron-phonon interaction, and electron-impurity interaction [14–16]. However, since the dephasing time $T_{2}$ is not defined from the Wannier gauge, we need to transform the density matrix from the Wannier gauge to the Hamiltonian gauge. With this transformation, the dephasing term $\Gamma$ is given by
where $\rho ^{\rm {(H)}} = U^{\dagger } \rho ^{\rm {(W)}} U$ and $U$ is the eigenvector matrix of the Hamiltonian. We use the dephasing time $T_{2}$ = 220 a.u. ($\approx$ 5.3 fs) in this study. Different dephasing times do not change the result qualitatively, as shown in Figs. 7 and 8 in the Appendix A.From the calculated density matrix, we can calculate all other observables, such as the conduction band population, the current density, and the AHC which is given by
2.3 Counter-rotating bicircular laser
We use the counter-rotating bicircular laser pulse with frequency $\omega _{0}$ and $2\omega _{0}$ in the simulations. The vector potential of the bicircular laser is given by
A counter-rotating bicircular laser from Eq. (6) gives a trefoil Lissajous figure of the electric field and the vector potential. The bicircular laser and the honeycomb lattice have 3-fold rotational symmetry and therefore, they have geometry correspondence, which is shown in Fig. 2. As shown in Fig. 2(a), the trefoil geometry of the electric field is well-matched with the honeycomb lattice. In Fig. 2(b), the vector potential is drawn in the Brillouin zone (BZ). Figures 2(c)-(e) show how the Lissajous figure of the vector potential is oriented for different phases difference between the $\omega _{0}$ and $2\omega _{0}$ fields. Changing the phase difference between two fields is equivalent to rotating the Lissajous figure of the vector potential. By examining the trefoil geometry, we easily predict that the excitation of the valley is maximized at a phase difference of 90$^\circ$ or 270$^\circ$.
Fig. 2. Trefoil geometry of the electric field and vector potential of the counter-rotating bicircular laser. (a) Honeycomb lattice in real space is drawn with the Lissajous figure of the electric field. (b) Brillouin zone of the honeycomb lattice is drawn with the Lissajous figure of the vector potential. (c,d,e) Rotation of the Lissajous figure of the vector potential for different phases between the $\omega _{0}$ and $2\omega _{0}$ fields. Field amplitude and ratio $R$ are adjusted for a visualization purpose.
3. Results and discussion
Figure 3 shows the change of AHC after the bicircular laser for various laser parameters. The first row of Fig. 3 is for a topologically trivial material and the second row is for a topological insulator. The results for the RCP $\omega _{0}$ pulse are plotted in the first column of Fig. 3 and those for the LCP $\omega _{0}$ pulse are in the second column. We note that the variation of the final AHC for different phases is different in the low-field regime and in the high-field regime. Electric field amplitude is scanned up to $E_{0}$ = 0.02 a.u. ($I_{0} \approx 1.4\times 10^{13}$ W/cm$^{2}$) to cover both low/high-field regime clearly, even though the high-field might above the damage threshold of a material in the experiment.
Fig. 3. Anomalous Hall conductivity (AHC) of the trivial and topological insulator after bicircular laser pulses for various electric field strengths $E_{0}$ and phase differences $\varphi$. (a) and (c) for the case of right-handed circularly polarized (RCP) $\omega _{0}$ pulse. (b) and (d) for the case of left-handed circularly polarized (LCP) $\omega _{0}$ pulse.
The final AHC in the low-field regime has a sign corresponding to the helicity of the $\omega _{0}$ pulse in the trivial and topological insulator. The sign of the AHC in the low-field regime is the same for both the trivial and topological insulators, but the magnitude of the AHC is much smaller for the trivial insulator. The same sign of the AHC in the low-field regime can be understood by the valley index or the Berry curvature of the valley. After the topological transition, the sign of the Berry curvature from the valley where the bandgap is located is flipped as seen in Fig. 1(b) and (c). This topological transition, or band inversion, changes the sign of the AHC from the inverted valley, but at the same time, the helicity of the field corresponding to the optical selection rule is also changed. As the net effect vanishes, the sign of the AHC in the low-field regime is the same for both the trivial and topological insulator.
When the electric field is strong, the AHC shows periodicity with respect to the phase difference. The dependence on phase difference can be understood by the relative location of the trefoil structure of the bicircular laser and the honeycomb lattice, as shown in Fig. 2. The final AHC in the high-field regime has peak value when the phase differences between the $\omega _{0}$ and $2\omega _{0}$ fields are set at 90$^\circ$ or 270$^\circ$. It is consistent with previous studies [8–10]. Interestingly, the topological insulator shows the opposite sign of the AHC compared to that of the trivial insulator in the high-field regime. The sign change of the AHC can be understood by the sign change of the Berry curvature in Fig. 1; the Berry curvature of the system changes its sign after the topological transition, and therefore, the AHC also changes its sign. It is worth noting that the conduction band population asymmetry $(n_{c}^{K} - n_{c}^{K'})/(n_{c}^{K} + n_{c}^{K'})$ shows a different behavior with the AHC in the topological insulator. As the SOC strength increases, the magnitude of Berry curvature in each valley does not match each other, and therefore, the population asymmetry of the conduction band is not a good indicator in the topological insulator [see Figs. 10 in Appendix B].
For a deeper understanding, we calculate the conduction band population for the trivial and topological insulators after the bicircular laser pulse, as shown in Fig. 4. Each column in Fig. 4 corresponds to different electric field strength; the left column is for the weakest field ($E_{0}$ = 0.002 a.u.) and the right column for the highest ($E_{0}$ = 0.020 a.u.). The first row in Fig. 4 is the conduction band population of the spinless system (duplicated bands of topologically trivial material, Chern number $C=0$). The second and third rows in Fig. 4 correspond to the electron density of the conduction bands of the topological insulator that have Chern number $C=1$ and $C=-1$, respectively.
Fig. 4. Electron population in the conduction band after the bicircular laser pulse. For each column, the peak strengths of the electric field are $E_{0}$ = 0.002, 0.008, and 0.020 a.u., respectively (corresponding peak intensities are $I_{0} \approx 1.4\times 10^{11}, 2.24\times 10^{12}, 1.4\times 10^{13}$ W/cm$^{2}$, respectively). The first row shows the electron densities of the spinless system (trivial insulator). The second and third rows show the spin bands of the topological insulator that have Chern number $C=+1$ and $C=-1$, respectively. $C=+1$ have bandgap point near the K point, and $C=-1$ have bandgap point near the K$'$ point. The boundary of the first Brillouin zone is drawn by the red line. The phase difference between the $\omega _{0}$ and $2\omega _{0}$ fields is fixed at $\varphi = 270^\circ$.
It is worth noting that the trivial insulator has the same bandgap at the K and K$'$ point, while the spin band of the topological insulator, the Chern number of which is $C=+1$ ($C=-1$), has the smallest bandgap near the K (K$'$) point. Also note that an RCP light match with a negative (positive) Berry curvature of conduction (valence) band in a two band model. Therefore, excitation from an RCP field is stronger at the K$'$ point in trivial insulator but at the K point in $C=+1$ spin band.
When the electric field is weak, the excitation from the small band gap region is strong and dominates other regions. In the topological insulator, the remaining population of the conduction band after the bicircular pulse comes mainly from the minimum bandgap region of each spin band; however, the population of the valley whose valley index matches with the helicity of the $\omega _{0}$ pulse [K point in $C=+1$ spin band, Fig. 4(d)] is larger than that of the valley whose valley index does not match with the helicity of the $\omega _{0}$ pulse [K$'$ point in $C=-1$, Fig. 4(g)]. Therefore, the ACH in the low-field regime is determined by the helicity of the $\omega _{0}$ pulse, as shown in Figs. 3(c) and (d). In the trivial insulator [Fig. 4(a)], since the bandgap at the K and K$'$ point are the same, the final population of the conduction band after the bicircular pulse depends only on the helicity of the $\omega _{0}$ pulse. Note that we choose the field amplitude of $\omega _{0}$ pulse is higher than that of $2\omega _{0}$ pulse ($R$ = 0.5), and therefore, the helicity of the $\omega _{0}$ pulse is more important than that of the $2\omega _{0}$ pulse. The counter-rotating bicircular field indicates that $\omega$ and $2\omega$ field have opposite helicity to each other, and therefore, the AHC of trivial insulator almost vanishes in the low-field regime, as shown in Fig. 3(a) and (b). It is worth noting that the field helicity corresponding to the valley changes after the topological transition; in other words, the valley corresponding to the RCP laser is K$'$ in the trivial insulator [Fig. 4(a)], but K in the topological insulator [Fig. 4(d) and (g)].
When the electric field is strong, the excitation from the small band gap region is no longer dominant. The area that corresponds to the trefoil geometry of the laser field is excited as the electric field increases, and at a certain point, this excitation becomes dominant over the excitation through the minimum bandgap [Fig. 4(f) and (i)]; for example, the excited valley changes from K to K$'$ at $C=+1$ spin band as the field amplitude increases [Fig. 4(d) and (f)]. At this point, the AHC is determined by the valley selected by the trefoil geometry of the laser field and its Berry curvature, not by the valley of minimum bandgap.
Figure 5 shows how the AHC is affected by the strength of the SOC and the electric field. We choose the phase difference of $\varphi = 90^\circ$ for the excitation at K [as in Fig. 5(a)] and of $\varphi = 270^\circ$ for excitation at K$'$ (Fig. 5(b)). Note that the topological transition does not change the sign of the AHC directly. The sign of the final AHC remains the same before and after the topological transition, especially for the small bandgap region. Also note that when the SOC strength $t_{2}$ is larger, a higher electric field strength is needed to enter the high-field regime; in Fig. 5, the boundary between low-field and high-field regime becomes higher as $t_{2}$ increases. The larger SOC means the larger difference between the energy dispersion of two spin bands; therefore, a larger electric field is needed to excite the area that does not contain the smallest bandgap. Another important factor is whether the valley selected by the trefoil geometry of the laser field and the helicity of the $\omega _{0}$ pulse are well-matched or not. For the topologically trivial case, K$'$ valley is matched with RCP $\omega _{0}$ pulse, and trefoil geometry of the laser field which excites the K$'$ valley corresponds to the phase difference $\varphi$ = 270$^\circ$ (the laser field with the phase difference of $\varphi$ = 90$^\circ$ excites K valley). As a result, the AHC for the trivial insulator is stronger at the phase difference $\varphi$ = 270$^\circ$ [Fig. 5(b)] than at the phase difference $\varphi$ = 90$^\circ$ [Fig. 5(a)]. The sign of Berry curvature is opposite in the topological insulator case. For the topological insulator, the AHC in Fig. 5(a) is stronger than in Fig. 5(b), since the valley corresponds to RCP $\omega _{0}$ pulse is changed from K$'$ to K, but laser geometry is not changed. Moreover, Fig. 5(b) shows the sign change of the AHC which comes from the mismatch of the valley corresponding to the RCP $\omega _{0}$ pulse (K) and the valley selected by the trefoil geometry of the laser field (K$'$).
Fig. 5. Anomalous Hall conductivity (AHC) with respect to spin-orbit coupling strength and field strength of a bicircular laser. The AHC is scanned for various electric field strengths $E_{0}$ and spin-orbit coupling hopping $t_{2}$. The phase difference between the $\omega _{0}$ and $2\omega _{0}$ fields is set at (a) $\varphi = 90^\circ$ and (b) $\varphi = 270^\circ$. The green line indicates the topological transition point. The region between the green line represents trivial insulators, and the other region indicates topological insulators.
To check what happens in the system between topological transition and to check whether the results from above are universal or not, we show the final AHC along with the phase diagram of the Kane-Mele model in Fig. 6. In the low-field regime [Fig. 6(a)], we observe strong AHC only near the topological transition because the effect by small bandgap dominates all other factors. In the high-field regime [Fig. 6(b)], the AHC is rather strong through the whole region of the topological insulator, except near $M_{0}=0$.
Fig. 6. Anomalous Hall conductivity (AHC) after the bicircular laser pulse in the phase diagram of the Kane-Mele model. The green line indicates the topological transition position; the region below the green line represents topological insulators, and the region above the green line indicates trivial insulators. (a) AHC in the low-field regime ($E_{0}$ = 0.004 a.u., $I_{0} \approx 5.6\times 10^{11}$ W/cm$^2$) and (b) AHC in the high-field regime ($E_{0}$ = 0.02 a.u., $I_{0} \approx 1.4\times 10^{13}$ W/cm$^2$). The phase difference between the $\omega _{0}$ and $2\omega _{0}$ fields is set at $\varphi = 270^\circ$.
At the topological transition point, the AHC vanishes; however, near the transition point, the AHC has a strong value and its sign depends on the helicity of the pulse $\omega _{0}$. In the low-field regime, the AHC quickly decreases to zero as we move away from the transition point. However, in the high-field regime, there is an oscillatory structure: as we move away from the transition point, the AHC becomes zero, and then it becomes strong again and then vanishes again. It is worth noting that the sign of the AHC changes in each oscillation cycle with high SOC strength. In the high-field regime, when the SOC is strong enough, both the trivial insulator and the topological insulator show a sign change of the AHC, but the strength of the AHC after the sign change is much weaker in the trivial insulator than in the topological insulator. This shows the limitation of the analysis of Fig. 5. With high electric field strength and strong SOC, an additional factor is set in; therefore, the sign of the AHC in the high-field regime is not determined only by the trefoil geometry of the laser field and valley index. As shown in Fig. 4(i), the bicircular laser does not exactly excite a high symmetry point with the combination of asymmetric band and high field, making a structured distribution of the population. This triangular hole of no population becomes larger as the SOC increases, and at a certain point, the triangular hole becomes too large that the excited population does not correspond to the Berry curvature of the valley. This is the reason why we also observe the change in the sign of the AHC in the trivial insulator, but its strength is much weaker than that of the topological insulator because the differences come from the different mechanisms.
4. Conclusion
We calculate the AHC of the trivial and topological insulator with the bicircular laser pulse. We show that in the low-field regime, the AHC of the topological insulator is determined by the helicity of the $\omega _{0}$ pulse, while the AHC is negligible in the trivial insulator. In the high-field regime, we show that the AHC of both the trivial and topological insulators depends on the phase difference between the $\omega _{0}$ and $2\omega _{0}$ fields, but the sign of the AHC is opposite. We also show that the matching between the valley selected by the helicity of $\omega _{0}$ pulse and the valley selected by the trefoil geometry of the laser field is important when the electric field strength is not too high. In addition, we show that the boundary between the low-field and high-field regime changes with the SOC strength. Finally, with high SOC, changing on-site energy difference or amount of broken inversion symmetry gives a change of sign of the AHC, but the strength of the AHC after the sign inversion of the AHC is much different in the trivial and topological insulators. The present work becomes a starting point to manipulate the quantum properties of quantum materials.
Appendix
A. Effect of the dephasing time to anomalous Hall conductivity
We show the effect of the dephasing time $T_{2}$ to the anomalous Hall conductivity (AHC) in Fig. 7 and 8. Figure 7 shows the AHC with the dephasing time of 55 a.u. ($\approx$ 1.3 fs), which is four times smaller from the dephasing time used in the manuscript. Figure 8 shows the AHC with the dephasing time of 1000 a.u. ($\approx$ 24 fs), which is about five times greater than the dephasing time used in the manuscript. In both cases, the dependence of AHC on the phase difference $\varphi$ and the electric field amplitude $E_{0}$ preserves a similar structure to the result in the manuscript.
Fig. 7. Anomalous Hall conductivity (AHC) of the trivial and topological insulator after bicircular laser pulses for various electric field strengths $E_0$ and phase differences $\varphi$ with a dephasing time of 55 a.u. ($\approx$ 1.3 fs).
Fig. 8. Anomalous Hall conductivity (AHC) of the trivial and topological insulator after bicircular laser pulses for various electric field strengths $E_0$ and phase differences $\varphi$ with a dephasing time of 1000 a.u. ($\approx$ 24 fs).
B. Population asymmetry after the counter-rotating bicircular laser field
Previous studies have shown the control of valley polarization with the counter-rotating bicircular laser field using the population asymmetry of the conduction band [8–10]. For comparison, the analysis using the population asymmetry is also performed. We define the population asymmetry $S_{K-K'}$ as
where $n_{c}^{K}$ and $n_{c}^{K'}$ are the populations of the conduction band at the $K$ and $K'$ valleys, respectively. We integrate the blue (red) region in Fig. 9 for the $n_{c}^{K}$ ($n_{c}^{K'}$).We plot the population asymmetry $S_{K-K'}$ of the trivial and topological insulators in Fig. 10. For topological insulator, we add two $S_{K-K'}$ from each spin band. For the trivial insulator, the result is expected from previous studies [8–10]. As noted in the main text, for topological insulator, the population asymmetry is not a good indicator of the valley polarization.
Funding
Korea Institute for Advancement of Technology (HRD Program for Industrial Innovation, P0008763); National Research Foundation of Korea (2020R1A2C2103181, 2022M3H4A1A04074153, RS-2022-00154676).
Acknowledgements
This work has been supported in part by the National Research Foundation of Korea (NRF) Grants (Grant No. 2022M3H4A1A04074153, No. 2020R1A2C2103181, and RS-2022-00154676) funded by the Ministry of Science, ICT, and by Korea Institute for Advancement of Technology(KIAT) grant funded by the Korea Government(MOTIE) (P0008763, HRD Program for Industrial Innovation)
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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