Microscopic theory for the light-induced anomalous Hall effect in graphene

TL;DR

This paper models the light-induced anomalous Hall effect in graphene using a quantum Liouville approach, revealing different mechanisms in weak and strong laser fields and linking them to Floquet band topology.

Contribution

It introduces a comprehensive quantum model explaining the anomalous Hall effect in graphene under ultrafast light, highlighting the role of Floquet-Bloch bands and population imbalances.

Findings

Hall effect from asymmetric photocarrier populations in weak fields

Steady state driven by topologically non-trivial Floquet bands in strong fields

Potential for designing ultrafast quantum devices with Floquet engineering

Abstract

We employ a quantum Liouville equation with relaxation to model the recently observed anomalous Hall effect in graphene irradiated by an ultrafast pulse of circularly polarized light. In the weak-field regime, we demonstrate that the Hall effect originates from an asymmetric population of photocarriers in the Dirac bands. By contrast, in the strong-field regime, the system is driven into a non-equilibrium steady state that is well-described by topologically non-trivial Floquet-Bloch bands. Here, the anomalous Hall current originates from the combination of a population imbalance in these dressed bands together with a smaller anomalous velocity contribution arising from their Berry curvature. This robust and general finding enables the simulation of electrical transport from light-induced Floquet-Bloch bands in an experimentally relevant parameter regime and creates a pathway to designing ultrafast quantum devices with Floquet-engineered transport properties.

Equations

$$\frac{d}{dt}{\rho }_{\mathbf{K}(t)}(t)$$

$$J_H^{POP}(t)=-\frac{2e}{\hslash (2\pi {)}^2}\sum _{b=v,c}\int d\mathbf{k}\frac{\partial {ϵ}_{b\mathbf{K}(t)}}{\hslash \partial k_x}{\rho }_{bb,\mathbf{K}(t)}.$$

$$J_H^{D-POP}=-\frac{2e}{\hslash (2\pi {)}^2}\sum _{b=v,c}\int d\mathbf{k}{\tilde{v}}_{b\mathbf{k}}^{NO}{\tilde{n}}_{b\mathbf{k}}^{NO},$$

$${\sigma }_{xy}^T=\frac{2e^2}{\hslash }\int \frac{d\mathbf{k}}{(2\pi {)}^2}\sum _b{n}_{b\mathbf{k}}^{NO}{\Omega }_{b\mathbf{k}}^{NO}.$$

Full text

Optical control of functional materials has emerged as an important research front bridging condensed matter physics Basov et al. (2017) and ultrafast spectroscopy Krausz and Stockman (2014). Many noteworthy phenomena have been discovered in optically-driven quantum solids, including light-induced superconductivity Fausti et al. (2011); Mitrano et al. (2016), various types of photo-initiated insulator-metal transitions Fiebig et al. (1998); Cavalleri et al. (2001); Rini et al. (2005); Liu et al. (2012), light control of microscopic interactions like electron-phonon coupling Pomarico et al. (2017); Kennes et al. (2017); Sentef (2017), and theoretically predicted Floquet-topological phases of matter Oka and Aoki (2009); Kitagawa et al. (2011); Lindner et al. (2011); Sentef et al. (2015); Hübener et al. (2017). Floquet-topological phases in particular have stimulated much interest but direct evidence of electron-photon Floquet-dressed states in solids is scarce to this date Wang et al. (2013); Mahmood et al. (2016), in contrast to the field of artificial lattices Struck et al. (2011); Aidelsburger et al. (2011); Miyake et al. (2013); Jotzu et al. (2014); Fläschner et al. (2016); Näger et al. (2018); Madison et al. (1998); Longhi et al. (2006); Lignier et al. (2007); Rechtsman et al. (2013); Aidelsburger et al. (2018); Jotzu et al. (2015); Parker et al. (2013); Kennedy et al. (2015); Aidelsburger et al. (2014); Ozawa et al. (2018); Asteria et al. (2018).

Recently a light-induced anomalous Hall effect was observed in graphene using ultrafast transport techniques McIver et al. (2018). A key challenge for the interpretation of the reported effects lies in the competition between Floquet engineering of Hamiltonians versus the role of electronic population effects. For the case of laser-driven graphene, the latter are particularly important as the pump laser is generically resonant with electronic excitations. Here we provide a theoretical framework within which this class of experiments McIver et al. (2018) can be interpreted.

A graphene lattice subjected to circular driving has been studied theoretically in a variety of frameworks Oka and Aoki (2009); Kitagawa et al. (2011); Lindner et al. (2011); Bukov et al. (2015); Foa Torres et al. (2014); Jotzu et al. (2014); Usaj et al. (2014); Mikami et al. (2016); Dehghani et al. (2014, 2015); Sentef et al. (2015); Kundu et al. (2014). We focus here on the low-frequency driving regime and find that the driven-dissipative dynamics together with the applied bias field plays a crucial role in understanding the transport properties of the Floquet-engineered state. Our real-time simulations contain both the population imbalance of excited photocarriers in the Dirac cone of graphene as well as the Floquet-topological Berry curvature of photon-dressed bands. We find that population effects play an important role under the low-frequency driving used in the experiments in both weak and strong driving limits. In the weak-driving regime, light-induced Hall transport originates mainly from population imbalance of photocarriers in the bare bands. By contrast, in the strong field regime clear topological-Floquet states are formed, and the light-induced Hall effect is characterized by a population imbalance of these Floquet bands that outweighs, however, the Floquet-Berry curvature contribution which is predicted to dominate in the high-frequency regime Oka and Aoki (2009); Foa Torres et al. (2014); Dehghani et al. (2014). Our results demonstrate that Floquet-engineering in solids is a reality, even with significant dissipation. These findings are in good agreement with the experimental results McIver et al. (2018) and provide a microscopic interpretation of the observed light-induced anomalous Hall effect.

To model the electron dynamics in graphene under electromagnetic fields, we employ a quantum Liouville equation for the one-particle reduced density matrix $\rho_{\boldsymbol{K}(t)}$,

[TABLE]

with phenomenological relaxation $\hat{D}_{\boldsymbol{K}(t)}$ and the Dirac Hamiltonian, $H_{\boldsymbol{K}(t)}=\hbar v_{F}\tau_{z}\sigma_{x}K_{x}(t)+\hbar v_{F}\sigma_{y}K_{y}(t)$, where $v_{F}$ is the Fermi velocity, $\sigma_{x/y}$ are the Pauli matrices, and $\boldsymbol{K}(t)=\boldsymbol{k}+e\boldsymbol{A}(t)/\hbar c$ is the lattice momentum coupled to an external vector potential $\boldsymbol{A}(t)$. The linear Dirac-cone approximation is justified in the present context of low-frequency driving and moderate deviations from half-filled bands with a chemical potential shift $\mu$ McIver et al. (2018). The different chirality of the Dirac fermions at the $K$ and $K^{\prime}$ points is given by $\tau_{z}=\pm 1$. The phenomenological dissipation $\hat{D}_{\boldsymbol{K}(t)}$ based on the relaxation time approximation is added to account for relaxation and dephasing effects Meier et al. (1994); 2018satoSI (201). We note that effects of dissipatively coupled Floquet-topological systems have been discussed previously Dehghani et al. (2014, 2015); Seetharam et al. (2015); Mikami et al. (2016), but not in the concrete context of the present work. Here we set the relaxation time $T_{1}$ to $100$ fs, and the dephasing time $T_{2}$ to $20$ fs. However, the qualitative behavior of the light-induced Hall effect does not strongly depend on the choice of $T_{1}$ and $T_{2}$.

To evaluate the Hall conductivity $\sigma_{xy}$, we apply a weak static source-drain electric field along the $y$-axis of the model and compute the transverse current along the $x$-axis under the presence of a circular laser pulse. Both the static electric field and the laser pulse are included in the model via the vector potential $\boldsymbol{A}(t)$. Following the same analysis as in Ref. McIver et al., 2018, we define the Hall current as the difference of the transverse current induced by right- and left-handed circular laser fields (pump dichroism) 2018satoSI (201). To match experimental values we set the wavelength of the circular laser pulse to $\lambda=6.5$ $\mu$m, which corresponds to the mean photon energy of $\hbar\omega_{\text{MIR}}\approx 190$ meV. The pulse duration is $1$ ps (FWHM). Note that the present model leads to the identical Hall current around $K$ and $K^{\prime}$ with $\tau_{z}=\pm 1$.

First, we investigate the light-induced Hall effect in the weak-driving regime, $E_{\text{MIR}}$ $=$ 1 MV/m. Fig. 1 (a) shows the computed Hall conductivity $\sigma_{xy}$ as a function of $\mu$, while Fig. 1 (b) shows the experimental Hall conductivity with the weakest experimental fluence $0.01$ mJ McIver et al. (2018). Our results confirm that the Hall conductivity is proportional to the laser intensity in this regime. Therefore, the single-photon absorption process is expected to play a dominant role in the light-induced Hall effect. Indeed, both the theoretical and experimental results consistently show a strong suppression of the Hall effect once $\mu$ reaches $\pm\hbar\omega_{\text{MIR}}/2$, which is when single-photon absorption becomes suppressed. The experimental result shows a sign change of the Hall conductivity in the higher doping regime while the theoretical result shows the same sign in the whole chemical potential range. The sign-change feature might be understood by a negative offset due to the inverse Faraday effect from a substrate Hertel (2006). However, since a detailed analysis of the substrate effect is beyond the scope of the present work, this aspect will be investigated in the future.

To clarify the microscopic origin of the Hall current in the weak field regime, we perform a perturbative analysis which shows that the Hall current originates from population imbalance of photocarriers in the Brillouin zone (BZ) 2018satoSI (201). An excess of photocarriers is generated on one side of the Dirac cone compared to the other, and the non-symmetric photocarrier-distribution results in a net Hall current. Furthermore, the perturbative analysis reveals that the population imbalance is induced by the interference of two excitation paths: one of them is the single-photon absorption process in the bare Dirac band (black-dashed line in Fig. 1 (c)), while the other one is the single-photon absorption in the tilted Dirac band (blue-solid line in Fig. 1 (c)), where the tilt is induced by the static source drain field. To confirm this conclusion we further compute the conduction band population $\rho_{cc,\boldsymbol{K}(t)}(t)=\mathrm{Tr}\left[|u^{s}_{c\boldsymbol{K}(t)}\rangle\langle u^{s}_{c\boldsymbol{K}(t)}|\rho_{\boldsymbol{K}(t)}(t)\right]$ using instantaneous eigenstates of the Hamiltonian; $\hat{H}_{\boldsymbol{K}(t)}|u^{s}_{b\boldsymbol{K}(t)}\rangle=\epsilon_{b\boldsymbol{K}(t)}|u^{s}_{b\boldsymbol{K}(t)}\rangle$, where $b$ denotes the band index, $v$ and $c$ for valence and conduction bands, respectively. Figure 1 (d) shows the cycle-averaged conduction-band population pump dichroism. A population imbalance along the $x$-direction is clearly observed under the source-drain field along the $y$-direction. From this, we compute the population contribution to the Hall current, $J^{POP}_{H}(t)$, by multiplying populations with corresponding band velocities,

[TABLE]

The result shown in Fig. 1 (a) accounts for most of the total Hall conductivity. Overall our analysis demonstrates that in the weak-field regime the Hall current mainly originates from a population imbalance of photocarriers in the bare Dirac band. Importantly, the population imbalance mechanism does not rely on the specific properties of the Dirac Hamiltonian. Therefore, the mechanism is general and can induce anomalous Hall currents even in conventional semiconductors 2018satoSI (201).

Having established an asymmetric population imbalance in the Dirac bands as the source of light-induced anomalous Hall currents in the weak-field regime, we move on to explore the strong optical driving regime. We set $E_{\text{MIR}}$ to $20$ MV/m, which is the peak field intensity for the largest reported experimental fluence, $0.23~{}mJ$ McIver et al. (2018). Fig. 2 (a) shows the computed Hall conductivity $\sigma_{xy}$ as a function of $\mu$, while Fig. 2 (b) shows the corresponding experimental result McIver et al. (2018). The theoretical and experimental data are in good agreement, with both exhibiting conductivity maxima around $\mu=0$ as well as $\pm\hbar\omega_{\text{MIR}}/2$. Fig. 2 (c) shows the calculated Floquet band structure for the same driving parameters. The Floquet bands show characteristic gap openings at $\epsilon_{b\boldsymbol{k}}=0$, and $\pm\hbar\omega_{\text{MIR}}/2$. Strikingly, the peaks in $\sigma_{xy}$ near $\pm\hbar\omega_{\text{MIR}}/2$ coincide with the outer edges of the Floquet bands (blue arrows in Fig. 2 (c)), and the width of the central plateau of $\sigma_{xy}$ is approximately the width of the gap at the Dirac point. This already indicates a close relation between the formation of Floquet-Bloch bands and the anomalous Hall effect for strong laser driving.

In order to clarify the role of Floquet-Bloch states in the generation of anomalous Hall currents, we analyze to what extent they represent the full dynamics of the system under the driving conditions. Since the complete dynamics is described by the time-dependent density matrix $\rho_{\boldsymbol{K}(t)}$, we analyze the time evolution in terms of its instantaneous eigenstates, the time-dependent natural orbitals $|u^{NO}_{b\boldsymbol{k}}(t)\rangle$ Löwdin (1955), giving $\rho_{\boldsymbol{K}(t)}=\sum_{b=v,c}n^{NO}_{b\boldsymbol{k}}(t)|u^{NO}_{b\boldsymbol{k}}(t)\rangle\langle u^{NO}_{b\boldsymbol{k}}(t)|$. This orbital basis for the dynamics gives the full dressed states of the system without the assumptions required by Floquet theory and allows us to make a detailed assessment of the Floquet picture.

The cycle-averaged population of the natural orbitals is computed as $\tilde{n}^{NO}_{b\boldsymbol{k}}=\int^{T_{\text{MIR}}}_{0}dtn^{NO}_{b\boldsymbol{k}}(t)/T_{\text{MIR}}$ and Figure 2 (d) shows the pump dichroism of the natural-orbital population $\tilde{n}^{NO}_{b\boldsymbol{k}}$. Consistent with the population imbalance in the weak-field regime (Fig. 1 (d)), the natural-orbital population also shows the imbalance along the $x$-direction. From the population of the natural orbitals, we can compute the dressed-band population contribution to the Hall current as

[TABLE]

where $\tilde{v}^{NO}_{b\boldsymbol{k}}$ is an effective dressed-band natural-orbital velocity, $\tilde{v}^{NO}_{b\boldsymbol{k}}=v_{F}\int^{T_{\text{MIR}}}_{0}dt\langle u^{NO}_{b\boldsymbol{k}}(t)|\sigma_{y}|u^{NO}_{b\boldsymbol{k}}(t)\rangle/T_{\text{MIR}}$. The blue curve in Fig. 2 (a) shows the contribution of the natural-orbital population computed by Eq. (3). The contribution of the natural-orbital population reproduces the full signal very well in the whole investigated chemical-potential range, indicating that a dressed state picture gives a valid description of the light-induced Hall current. As a reference, we also evaluated the contribution from the population imbalance in the bare Dirac band with Eq. (2) and show the result as the green curve in Fig. 2 (a). From the results it is clear that the population imbalance in the bare Dirac band is not sufficient to describe the full signal. Thus we confirmed that the dressed states describe a significantly different electronic structure and that the light dressing enhances the light-induced Hall effect.

Having established the role of dressed bands in the strong-field regime, we turn to the relation between Floquet states and natural orbitals under continuous circular laser fields without the source-drain field. For this purpose, we introduce the Floquet fidelity, $S_{\boldsymbol{k}}$, as a measure of similarity between Floquet states $|u^{F}_{b\boldsymbol{k}}(t)\rangle$ and natural orbitals $|u^{NO}_{b\boldsymbol{k}}(t)\rangle$. The Floquet fidelity is defined as the absolute value of the determinant of the fidelity matrix, $S_{\boldsymbol{k}}=|\mathrm{det}F_{\boldsymbol{k}}|$, where each element of the matrix $F_{\boldsymbol{k}}$ is the cycle-averaged overlap between natural orbitals and Floquet states, $F_{\boldsymbol{k},ij}=\int^{T_{\text{MIR}}}_{0}dt\left|\langle u^{NO}_{ik}(t)|u^{F}_{jk}(t)\rangle\right|^{2}/T_{\text{MIR}}$ 2018satoSI (201). The Floquet fidelity $S_{\boldsymbol{k}}$ can take a maximum value of one only if the natural orbitals and the Floquet states are identical.

Fig. 3 (a) shows the Floquet fidelity $S_{\boldsymbol{k}}$ at the Dirac point as a function of the driving field strength. The system is far from the Floquet limit in the weak-field regime ($S_{\boldsymbol{k}}\approx 0$), while it approaches the Floquet limit in the strong-field regime ($S_{\boldsymbol{k}}\approx 1$). This behavior is consistent with the above findings: the population imbalance in the bare Dirac band dominates the Hall current in the weak-field regime, while the dressed-state picture is more appropriate in the strong-field regime.

A map of the Floquet fidelity $S_{\boldsymbol{k}}$ in the BZ in the strong field regime is shown in the inset of Fig. 3 (a), where the single-photon resonance, $\hbar v_{F}|\boldsymbol{k}|=\hbar\omega_{\text{MIR}}/2$, is indicated as a red circle. This result shows that Floquet states are established throughout a large portion of the BZ including the Dirac point, except for a ring close to the the single-photon resonance ($S_{\boldsymbol{k}}\approx 1$), where the steady state appears to be strongly disturbed ($S_{\boldsymbol{k}}\approx 0$). In relation to the Floquet-bandstructure in Fig. 2 (c), this indicates that the circular laser field is able to establish the outer edges of the Floquet states (the blue arrows in Fig. 2 (c)), while the inner edges (red arrows) do not form. Moreover, the realization of the outer Floquet edges is supported by the appearance of the peaks in Fig. 2 (a) that are positioned at the resonance energies.

Having demonstrated the relevance of a Floquet dressed-state picture under strong-field driving, we now discuss the role of Berry curvature of the dressed states for the Hall current. We compute the cycle-averaged Berry curvature of the natural orbitals, $\Omega^{NO}_{B}(\boldsymbol{k})=-i\int^{T_{\text{MIR}}}_{0}dt\left[\nabla_{\boldsymbol{k}}\times\langle u^{NO}_{b\boldsymbol{k}}(t)|\nabla_{\boldsymbol{k}}|u^{NO}_{b\boldsymbol{k}}(t)\rangle\right]_{z}/T_{\text{MIR}}$ shown in Fig. 3 (c), while Fig. 3 (d) shows the Berry curvature of the corresponding Floquet state at $E_{\text{\text{MIR}}}=20$ MV/m. Both the natural orbitals and the Floquet state consistently show a positive Berry curvature at the Dirac point ($\boldsymbol{k}=0$) and at the first resonance ($v_{F}|\boldsymbol{k}|=\hbar\omega_{\text{MIR}}/2$). Indeed the natural-orbital Berry curvature integrated over the BZ is found to be $\int d\boldsymbol{k}\Omega^{NO}_{B}(\boldsymbol{k})=\pm\pi$ for each Dirac cone, which is consistent with the topological Floquet-Chern insulator picture Oka and Aoki (2009).

Finally we quantify the Hall currents expected solely from the time-averaged Berry curvature of the bands by computing the Hall conductivity Oka and Aoki (2009)

[TABLE]

The result shown in Fig. 3 (b) clearly demonstrates that light-induced Berry-curvature has a non-zero contribution to the Hall conductivity. However, this contribution is much smaller. Furthermore, by computing the dependence of this effect on the field strength, we find that it changes sign 2018satoSI (201).

Our modelling suggests that the recently observed light-induced anomalous Hall effect in graphene results mainly from the asymmetric distribution of photocarriers in the topologically non-trivial Floquet-Bloch states while their Berry curvature contribution is clearly smaller. Nevertheless, it is striking that the theoretical Hall conductivity saturates on the order of $\sim 2e^{2}/h$, which is consistent with the experimental observation McIver et al. (2018), despite the strong population imbalance contribution 2018satoSI (201). This may just be a coincidence, but the possibility of hidden connections to topological invariants should be investigated further. The anomalous velocity contribution from the Berry curvature of the Floquet-Bloch bands could become larger at higher laser frequencies or for more off-resonant driving, which might be realized in other material platforms Claassen et al. (2016); Shin et al. (2018). The population imbalance effect can also be expected to play a role in quantum simulation experiments studying Floquet effects, such as of ultracold fermions in driven optical lattices Jotzu et al. (2014); Fläschner et al. (2016); Aidelsburger et al. (2014), as well as in other driven quantum materials.

More broadly, these results highlight the importance of adopting a dressed-state picture in understanding the non-equilibrium electrical transport properties of optically-driven solids. They also demonstrate the viability of Floquet-engineering under experimentally realistic conditions, which opens the doors to exploring a host of other exciting non-equilibrium quantum transport phenomena. For example, Floquet-engineering of effective couplings in solids, such as dynamical Hubbard U, has recently been proposed as a means to significantly modify electronic states in correlated insulators Tancogne-Dejean et al. (2018); Gong et al. (2009); Singla et al. (2015) which can also induce nontrivial topology Topp et al. (2018). Future work could for instance explore the possibilities for light-controlled topological edge states Usaj et al. (2014); Claassen et al. (2018) that are opened up by the present ultrafast-transport setup and complementary ultrafast real-space imaging techniques Cocker et al. (2016); Liu et al. (2015).

This work was supported by the European Research Council (ERC-2015-AdG694097) and the Deutsche Forschungsgemeinschaft through the SFB 925. The Flatiron Institute is a division of the Simons Foundation. S.A.S. gratefully acknowledges the fellowship from the Alexander von Humboldt Foundation. M.A.S. acknowledges financial support by the DFG through the Emmy Noether programme (SE 2558/2-1). P.T. acknowledges the received funding from the European Unions Horizon 2020 research and innovation programme under the Marie Sklodowska-Curie grant agreement No 793609. M.N. acknowledges support from Stiftung der Deutschen Wirtschaft. L.M., A.R. and A.C. acknowledge support from the Cluster of Excellence ’Advanced Imaging of Matter’ (AIM)