Anomalous ultrafast all-optical Hall effect in gapped graphene

2.1 Time-dependent Schrödinger equation and its solution

A gap in graphene can be opened by breaking the inversion symmetry ( P ), i.e., the symmetry between two sublattices [3], A and B. To describe the GG, we consider two-band tight-binding Hamiltonian, which includes an extra diagonal term with the on-site energies Δ _g /2 and −Δ _g /2 at two sublattices A and B, respectively – see Figure 1(a). This difference in the on-site energies breaks down the P -symmetry causing the bandgaps of Δ _g to open up at the K- and K′-points – see the schematics of the Brillouin zone in Figure 1(b). Note that the electron spectra in the K and K′ valleys are identical as protected by the time-reversal symmetry – see Figure 1(c), while the Berry curvatures are opposite.

Figure 1:

(a) Honeycomb crystal structure of graphene with sublattices A and B. (b) The first Brillouin zone of graphene with two valleys K and K′. (c) Energy dispersion a function of crystal wave vector for GG with the band gap of 1 eV.

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 time-dependent Schrödinger equation (TDSE),

where H _k (t) is the Hamiltonian of an electron system, which consists of the field-free 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 nearest-neighbor tight binding Hamiltonian for the GG [3, 27, 28]

where γ = −3.03 eV is the hopping integral,

and a = 2.46 A ̊ is the lattice constant.

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 time-dependent 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 well-known Houston functions [30],

where α = v, c for the VB and CB, respectively, and Ψ k ( α ) are the lattice-periodic Bloch functions in the absence of the pulse field. Here the dynamic phase, ϕ α ( D ) , and the geometric phase, ϕ α ( B ) , are defined as

where A α α = Ψ q ( α ) | i ∂ ∂ q | Ψ q ( α ) is the intraband Berry connection.

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 Φ α q ( H ) ( r , t ) as

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, A α α 1 ( q ) is the interband (non Abelian) Berry connection [2, 31, 32], ϕ α α 1 ( D ) ( q , t ) is the transition dynamic phase, and ϕ α α 1 ( B ) ( q , t ) is the transition Berry phase. Note that the interband dipole matrix element, D α α 1 ( q ) , which determines the optical transitions between the VB and CB at a wave vector q, is related to the transition Berry connection as D α α 1 ( q ) = e A α α 1 ( q ) .

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, S ̂ ( q , t ) , as follows

where T ̂ is the well-known time-ordering operator [33], and the integral is affected along the Bloch trajectory (Eq. (5)): d k ( t ) = e ℏ F ( t ) d t . We solve Eq. (15) numerically for each value of the initial reciprocal wave vector, q. From this solution we can find the electric current, J ( t ) = J x ( t ) , J y ( t ) , generated during the pulse.

2.2 Current

The 4-vector electric current density is defined as j ̂ = ( e ρ ̂ , e ρ ̂ v ̂ ) , where ρ ̂ is the operator of charge density, and v ̂ is the operator of velocity. The latter can be defined for a given lattice momentum k as

This can also be identically written as

The band-nondiagonal (α ≠ α′) matrix elements of the velocity can be found from Eq. (21) as

The band-diagonal matrix element of velocity can also be obtained from Eq. (21) taking into account an identity ∂ ∂ k , H k 0 = ∂ ∂ k H k 0 as

where v α , k ( g ) = ∂ ∂ k E α ( k ) is the group velocity in a band α at a lattice momentum k.

The 2D current density in a crystal, J (called below current for brevity), is related to the electron velocity, v as J = e a 2 v , where a is the lattice constant (see Eq. (3)). This current, J, is a sum of the interband and intraband contributions, J(t) = J ^(intra)(t) + J ^(inter)(t). In accord with Eq. (23), the intraband current can be expressed 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, gauge-invariant despite the Berry connection being not gauge-invariant. This can be verified by using an explicit gauge transformation.

3.1 Circularly polarized pulse

We apply an ultrafast chiral (“circularly-polarized”) optical pulse, F = (F _x , F _y ) whose waveform is symmetric with respect to a mirror reflection in the xz plane, P x z , as defined by the following parametrization

Here, F ₀ is the amplitude of the pulse, u = t/τ, where τ is a characteristic half-length of the pulse (in calculations, we choose τ = 1 fs), and ± determines the handedness: + is for the right-handed and − is for the left-handed chiral (circularly polarized) pulses. In this definition, the right-hand and left-hand pulses are T -reversed with respect to each other. The waveforms of a right-hand pulse and a left-hand pulse are depicted in the insets in Figure 2(a) and (b), respectively.

Figure 2:

Residual CB population N CB ( r e s ) ( k ) for GG with the band gap of 2 eV in the extended zone picture after a chiral single-cycle excitation pulse. (a) The excitation optical pulse is left-handed with the amplitude of F 0 = 0.5 V A ̊ − 1 . Inset: Waveform of the pulse F(t) = {F _x (t), F _y (t)} as a function of time t. (b) The excitation optical pulse is right-handed with the amplitude of F 0 = 0.5 V A ̊ − 1 . Inset: Waveform of the pulse as a function of time t. The solid white lines show the boundary of the first Brillouin zone with the K, K′ points indicated. The separatrix (Eq. (6)) is shown in panels (a) and (b) by red solid lines.

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, N CB ( r e s ) ( q ) = | β c q ( t = ∞ ) | 2 .

For single-cycle left-handed and right-handed chiral pulses with the amplitude of 0.5 V A ̊ − 1 the distributions of the residual CB population for a GG with a bandgap of Δ _g = 2 eV are shown in Figure 2(a) and (b), respectively. In a pristine graphene, Δ _g = 0, a chiral pulse with a waveform symmetric with respect to the P x z mirror reflection produces a strictly zero valley polarization [34], i.e., the K and K′ valleys are populated equally. In sharp contrast, for a GG, there is a large valley polarization. This is because that for GG, the P x z symmetry is broken. As a result, for a chiral pulse, which breaks down the time-reversal symmetry, the response of the GG in the K and K′ valleys is different.

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 A ̂ ‖ ( q , t ) = A ̂ ( q , t ) F ( t ) / F ( t ) , and d k ( t ) = e ℏ F ( t ) d t . Explicitly, matrix A ̂ ‖ ( q , t ) has the form

where

and the total phase, ϕ cv ( t o t ) , is defined as

Here, ϕ cv ( T ) is the topological phase, and ϕ cv ( A ) ( q , t ) = arg A ‖ ( q , t ) is the phase of the interband coupling amplitude.

As we see from Eq. (30), the interband electron dynamics is determined by the total phase ϕ cv ( t o t ) , which is a sum of the dynamic phase, ϕ cv ( D ) , and topological phase, ϕ cv ( T ) . The symmetry of the dynamic and topological phases with respect to the valley index (pseudospin) is opposite: The dynamic phase is even while the topological phase is odd. Assume that in one valley, say K, at an initial lattice momentum q, the dynamic and topological phases have opposite signs and cancel one another. This is accord with Eq. (28) will lead to a coherent accumulation of transition amplitude and, consequently, a large population of the CB. At the same time, because of the valley antisymmetry of the topological phases, the dynamic and topological phases in valley K′ will add to each other causing rapid oscillation of the integrand and mutual compensation of contributions over time in Eq. (28), leading to a low CB population. This is an effect of the topological resonance [34].

As one can see in Figure 2(a), for the left-handed chiral pulse, the topological resonance occurs in the K′ valley. In contrast, for the right-handed 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 ϕ cv ( D ) ≈ Δ g t / ℏ (where Δ _g is the bandgap) and the field phase −ωt, which occurs for ω ≈ Δ _g /ℏ. In sharp contrast to the topological resonance, the conventional resonance is symmetric with respect to the valley index.

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 left-handed and right-handed chiral pulses with an amplitude of 0.5 V A ̊ − 1 calculated for different values of the bandgap, Δ _g . As one can see, both the longitudinal and transverse currents are generated. The magnitude of these currents decreases with the bandgap along with the corresponding reduction in the CB population. The longitudinal current, J _x , for graphene (Δ _g = 0) does not have a ballistic (dc) component: After the pulse ends, only decaying oscillations due to interband contribution are present – see Figure 3(a) and (c). This is because the dc current is purely intraband (cf. Eqs. (24) and (25)) and, therefore, it is completely determined by the residual CB populations, which, for pristine graphene, are P y z -symmetric due to its inherent P x z symmetry [34]. This results in a complete vanishing of the ballistic J _x current for pristine graphene. With the opening of the bandgap, the P x z symmetry is broken, and there is a non-zero but still small ballistic current.

Figure 3:

Currents J _x [panel (a)] and J _y [panel (b)] excited by a left-hand circularly polarized pulse with the amplitude of F 0 = 0.5 V A ̊ − 1 . The corresponding band gaps are marked in panel (a). In panels (c) and (d) the GG is excited by a right-hand circularly polarized pulse.

3.2 Linearly polarized probe and anomalous Hall effect

As described above in Section 3.1, a strong single-oscillation 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 T symmetry. A probe dc electric field applied to such a system will cause a Hall effect in the absence of any external or internal magnetic field, which is the anomalous Hall effect [35–38].

The anomalous Hall effect can be probed not only with a dc electric field but also with a linearly-polarized optical pulse applied after the strong (“pump”) chiral pulse. However, in this case to have a finite transferred charge, the linearly-polarized 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 linearly-polarized 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 linearly-polarized pulse whose field is comparable to that of the chiral pulse, i.e., ∼ 0.1 − 0.5 V A ̊ − 1 . For such a pulse, the optical nonlinearity (rectification) would define a predominant direction of the charge transfer both for longitudinal current (in the direction of the linear polarization) and for the transverse current (the anomalous Hall current). We consider a pulse linearly polarized along the y axis with the following waveform

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 P y z symmetry of the system.

We apply such a linearly-polarized 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.

Figure 4:

Electric currents J _x [panel (a)] and J _y [panel (b)] generated by a left-hand circularly polarized pulse (2 fs ≥ t ≥ −2 fs) followed by a linearly polarized pulse (10 fs ≥ t ≥ 6 fs). The amplitudes of circularly and linearly polarized pulses are the same, F 0 = F 1 = 0.5 V A ̊ − 1 . The corresponding band gaps are marked in panel (a). Panels (c) and (d) are the same as panels (a) and (b) but excited by a combination of right-hand circularly polarized pulse and a linear polarized pulse.

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 T -reversal symmetry. This anomalous Hall current causes a net charge transfer in the x direction, which can be measured experimentally.

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 TP x z . For graphene, it is also the symmetry of the system. Thus for the graphene, the ballistic current is twice forbidden: The TP x z symmetry forbids the valley polarization by the applied chiral pulse, and it also directly forbids the J _x current because under it J _x transforms to −J _x . For the gapped (semiconductor) materials, the P x z symmetry is not exact. Nevertheless, the ballistic anomalous Hall current, J _x , is still very small as our computations show. Concluding, the anomalous Hall current excited by a strong linearly-polarized probe pulse acting after the strong chiral pump pulse is ultrafast (existing predominantly within the duration of the probe pulse; it is odd (changes its sign) with respect to the pump chirality.

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.

Figure 5:

Electric currents J _x [panel (a)] and J _y [panel (b)] generated by a left-hand circularly polarized pulse (2 fs ≥ t ≥ − 2 fs) followed by a linearly polarized pulse (10 fs ≥ t ≥ 6 fs). The amplitude of the circularly polarized pulse is F 0 = 0.5 V A ̊ − 1 , while the corresponding amplitudes of the linearly polarized pulse are marked in panel (a). The band gap is 2 eV.

We estimate an effective Hall conductivity as σ _xy = ΔJ _x /ΔF ₁, where Δ J x ∼ 0.3 A µ m , and Δ F 1 ∼ 2 V A ̊ − 1 , where both ΔF ₁ and ΔJ _x are obtained from Figure 5 as the full range of the change of the corresponding quantity for field F 1 = 1 V A . Using these values, we estimate the effective Hall conductivity as σ _xy ∼ 0.2G ₀, where G 0 = e 2 π ℏ is the conductance quantum.

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 ∼ 10⁹ G = 10⁵ T, which is a gigantic magnetic field. Consequently, the predicted anomalous all-optical 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.