Abstract
We propose an ultrafast alloptical anomalous Hall effect in twodimensional (2D) semiconductors of hexagonal symmetry such as gapped graphene (GG), transition metal dichalcogenides (TMDCs), and hexagonal boron nitride (hBN). To induce such an effect, the material is subjected to a sequence of two strongfield singleopticalcycle pulses: A chiral pump pulse followed within a few femtoseconds by a probe pulse linearly polarized in the armchair direction of the 2D lattice. Due to the effect of topological resonance, the first (pump) pulse induces a large chirality (valley polarization) in the system, while the second pulse generates a femtosecond pulse of the anomalous Hall current. The proposed effect is fundamentally the fastest alloptical anomalous Hall effect possible in nature. It can be applied to ultrafast alloptical storage and processing of information, both classical and quantum.
1 Introduction
Twodimensional (2D) materials with honeycomb crystal structure [1], such as graphene, silicene, transition metal dichalcogenides (TMDCs), and hexagonal boron nitride (hBN), possess nontrivial topological properties in the reciprocal space [2]. Such properties are determined by the Berry curvature, which is concentrated at the K and K′ points of the Brillouin zone. While for graphene, which is a semimetal, the Berry curvature is singular at the K and K′ points and zero elsewhere, in the gapped graphene [3] (GG) and semiconductor TMDCs the Berry curvature is regular in the entire Brillouin zone with extrema at the K and K′ points. Consequently, the ultrafast electron dynamics produced by the strong optical pulses is fundamentally different in these materials [4, 5].
In graphene, for a singleoscillation chiral (“circularly polarized”) pulse, the residual (left after the pulse) population of the conduction band (CB) is almost the same for the K and K′ valleys (i.e., the induced valley polarization is very weak). For a longer pulse (with two or more optical oscillations), the valley polarization is larger; there are also pronounced fringes in the CB electron population, which form an electron interferogram caused by the accumulation of the Berry phase along the Bloch trajectories of electrons in the reciprocal space [4]. These interferograms possess characteristic forks manifesting the presence of a quantized Berry flux of ±π. The electron CB population distribution in the reciprocal space for both linearly [6] and circularlypolarized pulses [4] are asymmetric, which causes electric currents that have been recently observed experimentally [7].
In stark contrast, the twodimensional semiconductors (GG and TMDCs) placed in the field of chiral pulse, behave quite differently from graphene. Namely, there is a strong valley polarization induced by a circularlypolarized CW radiation of relatively low intensity [8–13]. A strong valley polarization can be introduced even by a singleoscillation ultrashort intense optical pulse [5]. The reason for a strong residual valley polarization in the twodimensional semiconductors is that they have broken inversion symmetry and, consequently, a finite bandgap. As a result, for a chiral pulse that breaks the timereversal (
Ultrafast generation of a large valley polarization in the GG and TMDCs by a singleoscillation chiral pulse opens up a possibility to observe an ultrafast anomalous alloptical Hall effect. The Hall current [14] can be realized only in systems with broken time reversal symmetry. For the conventional Hall effect, the time reversal symmetry is broken by an external magnetic field. In the systems, which show the anomalous Hall effect [15], the time reversal symmetry is broken even without an external magnetic field. Such anomalous Hall effect can be realized, for example, in TMDC monolayers with unbalanced population of photoexcited valleys [16, 17] or in semiconductor systems with photoinduced spin polarization and strong spin–orbit interaction [18]. In these systems, even without magnetic field, the time reversal symmetry is broken and under an applied DC electric field, the Hall current is generated. Such current is called an anomalous Hall current and it is usually probed by DC electric field. Here we propose to probe the broken time reversal symmetry of the system by an ultrashort optical pulse. Consider a second singlecycle optical pulse that is linearly polarized in the armchair direction incident normally on the already valleypolarized solid, where the valley polarization is generated by a singleoscillation chiral pulse. It is predicted to produce both a normal current in the direction of the electric field and the Hall current in the perpendicular (zigzag) direction. The latter is due to the net effect of the Berry curvature in the valleypolarized system. It changes sign for the chiral pulse of the opposite handedness. This normal current is the manifestation of the anomalous (without a magnetic field) alloptical Hall effect. The proposed alloptical anomalous Hall effect is the fundamentally fastest such an effect in nature: It takes just a single optical cycle to induce the large valley polarization and another single cycle pulse to read it out.
In this article, we consider GG, which is experimentally obtained by growing the graphene on a different substrate, i.e., on SiC [19, 20]. The GG can also serve as a generic model of TMDCs. We predict the generation of an anomalous Hall current by a combination of a strong chiral pulse, which breaks the timereversal symmetry thus playing the role of an effective magnetic field, followed by a linearly polarized probe pulse. Using the model of GG allows one to model materials with different bandgaps and to study how the anomalous Hall effect depends on the magnitude of the bandgap.
2 Model and main equations
2.1 Timedependent Schrödinger equation and its solution
A gap in graphene can be opened by breaking the inversion symmetry (
Below we consider the interaction of the GG with ultrashort optical pulses for a few femtosecond duration. The electron scattering times in graphene and other 2D materials are on the order of or significantly longer than 10 fs – see Refs. [21–26]. For ultrashort optical pulses with the duration less than 10 fs we disregard the electron scattering processes and consider the electron dynamics in the field of the pulse as coherent and collisionless. Consequently, it can be described by a timedependent Schrödinger equation (TDSE),
where H _{ k }(t) is the Hamiltonian of an electron system, which consists of the fieldfree Hamiltonian, H _{ k0}, and the interaction Hamiltonian with the field of the pulse, −e F(t)r. Here, F(t) is the pulse’s electric field, e is electron charge, k is the electron crystal wave vector. We set H _{ k0} as the nearestneighbor tight binding Hamiltonian for the GG [3, 27, 28]
where γ = −3.03 eV is the hopping integral,
and
The energies of CB and valence band (VB) are eigenvalues of H _{ k0},
where signs ± are for the CB (α = c) and the VB (α = v), respectively. The energy dispersion (4) is shown in Figure 1(c). Below we assume that initially (before the pulse) the VB is fully occupied and the CB is empty.
In solids, the applied electric field generates both the intraband (adiabatic) and interband (nonadiabatic) electron dynamics. The intraband dynamics is determined by the Bloch acceleration theorem [29], which describes the time evolution of the wave vector, k(t), in the timedependent electric field, F(t),
where q is the initial wave vector, q = k(q, −∞).
The Bloch electron trajectories of Eq. (5) determine the separatrix, which is defined as a set of initial points q in the reciprocal space for which the electron trajectories pass precisely through the corresponding K or K′ points [4]. It is a continuous line whose parametric equation is
where t ∈ (−∞, ∞) is a parameter. When the initial lattice momentum q is inside the separatrix, the corresponding Bloch trajectory, k(q, t), encircles the K or K′ point, otherwise it leaves the K or K′ point outside.
The adiabatic solutions of Schrödinger Eq. (1), which means solutions within a single band α (without an interband coupling), are the wellknown Houston functions [30],
where α = v, c for the VB and CB, respectively, and
where
The interband electron dynamics is determined by solutions of TDSE (1). Such solutions are parameterized by initial wave vector q and can be expanded in the basis of Houston functions
where β _{ α q }(t) are expansion coefficients.
It is convenient to introduce the following notations
where α and α
_{1} ≠ α are v or c,
With these notations, the Schrödinger equation in the adiabatic basis of the Houston functions (interaction representation) takes the following form
where wave function (vector of state) B _{ q }(t) and Hamiltonian H′(q, t) are defined as
Note that the interaction Hamiltonian, H′(q, t), does not have the diagonal matrix elements, which is characteristic of the interaction representation.
We express a formal general solution of this equation in terms of the evolution operator,
where
2.2 Current
The 4vector electric current density is defined as
This can also be identically written as
The bandnondiagonal (α ≠ α′) matrix elements of the velocity can be found from Eq. (21) as
The banddiagonal matrix element of velocity can also be obtained from Eq. (21) taking into account an identity
where
The 2D current density in a crystal, J (called below current for brevity), is related to the electron velocity, v as
where a factor of 2 takes into the account spin degeneracy in our model where the spin–orbit interaction is not included. Similarly, in accord with Eq. (22), the interband current is given by
Note that the current is observable and, consequently, gaugeinvariant despite the Berry connection being not gaugeinvariant. This can be verified by using an explicit gauge transformation.
3 Results and discussion
3.1 Circularly polarized pulse
We apply an ultrafast chiral (“circularlypolarized”) optical pulse, F = (F
_{
x
}, F
_{
y
}) whose waveform is symmetric with respect to a mirror reflection in the xz plane,
Here, F
_{0} is the amplitude of the pulse, u = t/τ, where τ is a characteristic halflength of the pulse (in calculations, we choose τ = 1 fs), and ± determines the handedness: + is for the righthanded and − is for the lefthanded chiral (circularly polarized) pulses. In this definition, the righthand and lefthand pulses are
We solve TDSE (1) numerically with initial conditions β
_{
c
q
} = 0 and β
_{
v
q
} = 1, i.e., the full VB and the empty CB. An optical pulse causes interband transitions and populates the CB. After the pulse, there is a stationary residual CB population remaining,
For singlecycle lefthanded and righthanded chiral pulses with the amplitude of 0.5
The different populations of the K and K′ valley can be also understood from the properties of the interband coupling at two valleys. Namely, the fundamental evolution operator (19) can be rewritten in the form
where the longitudinal component of the non Abelian Berry connection is defined as
where
and the total phase,
Here,
As we see from Eq. (30), the interband electron dynamics is determined by the total phase
As one can see in Figure 2(a), for the lefthanded chiral pulse, the topological resonance occurs in the K′ valley. In contrast, for the righthanded chiral pulse, it takes place in the K valley (Figure 2(b)). Note that the conventional resonance can also be described as a cancellation of the dynamic phase
The excitation pulses generate electric currents (see Eqs. (24) and (25)), which are experimentally observable – cf. Ref. [7]. In Figure 3, we show the x (longitudinal, i.e., along the maximum electric field of the pulse) and y (transverse) components of the current for the lefthanded and righthanded chiral pulses with an amplitude of
3.2 Linearly polarized probe and anomalous Hall effect
As described above in Section 3.1, a strong singleoscillation chiral pulse creates a large valley polarization in the GG , where the carriers predominantly occupy either K or K′ valley as determined by the pulse’s handedness. The resulting state has a broken
The anomalous Hall effect can be probed not only with a dc electric field but also with a linearlypolarized optical pulse applied after the strong (“pump”) chiral pulse. However, in this case to have a finite transferred charge, the linearlypolarized pulse must be strong: cf.: For a weak pulse, the total transferred charge will be zero due to the temporal averaging. The anomalous Hall current generated by ultrafast linearlypolarized pulse is due to two factors: (i) Finite valley polarization of the GG and (ii) anisotropic CB population distribution around the K points. This is different from a dc electric field, for which the Hall current is generated because of the finite valley polarization only.
Correspondingly, we apply a nonlinear probe: A strong linearlypolarized pulse whose field is comparable to that of the chiral pulse, i.e.,
where F
_{1} is the amplitude of the pulse. Note that for such a pulse in the absence of the valley polarization, there is only a longitudinal current J
_{
y
}: A transverse current J
_{
x
} is forbidden by the
We apply such a linearlypolarized pulse after the chiral pulse ends (i.e., at t ≥ 6 fs). The resulting currents, which are calculated from Eqs. (24) and (25), are shown in Figure 4 where the strong probe pulse is applied with its center at t = 8 fs. Note that both the longitudinal current, J _{ y } and the transverse (anomalous Hall) current J _{ x } are present in the response.
As we have already pointed out, the anomalous Hall current directly probes the valley polarization of the system. As one can see in Figure 4(a) and (c), for the pristine graphene (Δ_{
g
} = 0), the Hall current, J
_{
x
}, is precisely zero due to the absence of the valley polarization (the corresponding lines on the graphs do not change in response to the probe pulse whatsoever). With the bandgap increasing, the Hall current during the linearly polarized pulse, as expected, monotonically increases (Figure 4(a) and (c)) because the induced valley polarization increases with the bandgap. The anomalous Hall current, J
_{
x
}, changes its sign with the chirality of the pump pulse as protected by the
A remarkable property of the anomalous Hall current is that it has a very small ballistic component (that is the J
_{
x
} current after the end of the probe pulse (Figure 4(a) and (c)), so it can be considered instantaneous (inertialess). To explain this, we consider symmetry of the optical waveforms applied to the system (both chiral and linearly polarized): It is
As one can see in Figure 4(b) and (c), the longitudinal current, J _{ y } in response to the probe pulse monotonically decreases with the bandgap in accord with the decreasing CB population. Note that the J _{ y } current exists even for pristine graphene. Both components J _{ x } and J _{ y } generated in response to the strong probe pulse increase with its amplitude – see Figure 5.
We estimate an effective Hall conductivity as σ
_{
xy
} = ΔJ
_{
x
}/ΔF
_{1}, where
The classical Hall conductivity is σ _{ xy } = enc/B, where n is the 2D electron density, c is speed of light, and B is the magnetic field. We may express it in terms of an effective magnetic field, B _{eff}, which yields the same magnitude of σ _{ xy } as the anomalous Hall conductance, ∼0.2G _{0}. An estimate is B _{eff} = enc/σ _{ xy } ∼ 10^{9} G = 10^{5} T, which is a gigantic magnetic field. Consequently, the predicted anomalous alloptical Hall effect is extraordinarily strong. It can serve as an efficient source of ultrafast currents providing a direct access to the ultrafast topological charges induced in the system.
4 Conclusion
A gigantic ultrafast alloptical anomalous Hall effect occurs when two strong singleoscillation optical pulses are applied to the GG or similar hexagonalsymmetry semiconductor materials such as TMDCs or hBN. These materials possess a broken inversion symmetry and a finite direct bandgap. The two pulses, which generate the anomalous ultrafast Hall effect, are a sequence of a singlecycle chiral pulse followed by a singlecycle linearlypolarized pulse. The chiral pulse breaks down the
The subsequent application of a strong singleoscillation probe pulse that is linearlypolarized along the armchair edge (y axis) to such a system, which acquired chirality (a large valley polarization), produces a Hall current in the zigzag direction (x axis) transverse to the polarization of the probe pulse.
In our approach we used a single oscillation linearly polarized pulse to probe the valley polarization of the system. The valley degree of freedom can be also probed by a long laser pulse [39]. In this case the response of the electron system and the corresponding anomalous Hall current depend on the relation between the carrier relaxation time and the period of the pulse.
The fundamental distinction and advantage of this proposed alloptical anomalous Hall effect in 2D hexagonal semiconductors from the recent proposal [40] and observation [38] of a lightinduced anomalous Hall effect in graphene is that ours is the fundamentally fastest anomalous effect possible in nature: It takes just a single optical period to induce the strong valley polarization and just one other optical period to read it out. Such a read out can fundamentally be done either by recording the charge transferred after the probe pulse or by observing a THz radiation emitted by the Hall current that is polarized in the x direction. In sharp contrast, in Ref. [38] the chiral excitation pulse was orders of magnitude less intense and longer: Its duration was ≈500 fs, i.e., in the picosecond range vs. our pulse of just ≲5 fs duration; the read out was electrical.
There is another fundamental distinction of our predicted effect from Refs. [38, 40]. Namely, a possibility to induce the strong valley polarization by a pulse with just a single optical cycle is due to the effect of topological resonance that exists only in gapped materials such as GG and 2D semiconductors (TMDCs and hBN) but not in graphene. Therefore use of a much longer picosecond (quasiCW) pulses in Refs. [38, 40] is necessary; graphene cannot possess a singlecycle anomalous alloptical Hall effect.
The predicted ultrafast anomalous alloptical Hall effect has a potential to have applications in ultrafast memory and information processing, both classical and quantum.
Award Identifier / Grant number: DEFG0201ER15213
Award Identifier / Grant number: DESC0007043
Award Identifier / Grant number: Unassigned
Acknowledgments
We acknowledge fruitful discussions with Prof. M. I. Stockman.

Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

Research funding: Major funding was provided by Grant No. DESC0007043 from the Materials Sciences and Engineering Division of the Office of the Basic Energy Sciences, Office of Science, US Department of Energy. Numerical simulations were performed using support by Grant No. DEFG0201ER15213 from the Chemical Sciences, Biosciences and Geosciences Division, Office of Basic Energy Sciences, Office of Science, US Department of Energy. The work of V.A. was supported by NSF EFRI NewLAW Grant No. EFMA1741691.

Conflict of interest statement: The authors declare no conflicts of interest regarding this article.
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