Graphene Stacks Amplify Light Interactions, Boosting Data Processing Potential

A new study details how light interacts with rhombohedral graphene to generate high-harmonics, a process with potential for future optical technologies. Jessica O. de Almeida and colleagues at University of Stuttgart and Los Alamos National Laboratory demonstrate that the unique chiral Bloch states within this multilayer graphene system sharply influence high-harmonic generation. The research reveals a relationship between the number of graphene layers and both the strength and order of the generated harmonics, with the quantum geometry of the material imprinted on the resulting light. Furthermore, the team’s analysis of valley splitting and doping effects highlights the potential of rhombohedral graphene as a platform for exploring complex nonlinear optical behaviour and controlling the circular dichroism of emitted light.

Scientists are unlocking the potential of rhombohedral graphene for manipulating light, building on recent advances in engineering multilayer materials to enhance their electronic properties. High-harmonic generation, a process where intense laser light creates new frequencies, offers a route towards compact optical devices and probing quantum materials. Achieving practical optical information processing demands scalable and efficient systems. Rhombohedral graphene’s layered structure provides a novel means of controlling light emission during this process.

The arrangement of graphene layers directly influences the circular dichroism of the generated harmonics, which is the difference in absorption of left- and right-handed circularly polarised light. Quantum geometry, specifically the chiral Bloch states describing electron movement, imprints itself on the emitted light’s characteristics. The number of graphene layers dictates the strength and order of these harmonics, revealing a scalable method for manipulating light polarisation.

Valley states within this system effect high harmonic generation. The “winding” of the Bloch states scales linearly with n, mirroring the dominant harmonic order. The location of the strongest quantum geometry in momentum space, on a ring of finite radius, is imprinted on the time-dependent momentum distribution at the beginning of the strong laser pulse. Interaction-induced splitting of the two valleys leads to a complex interaction of opposite chiralities, directly visible in the dependence of circular dichroism on the harmonic order.

Analysis of doping identifies a quantity that tracks the net chirality of the occupied states. Rhombohedral graphene constitutes a promising platform for exploring rich nonlinear optical phenomena. Systematic study of effects rooted in the non-trivial momentum dependencies of multi-component Bloch states, often collectively referred to as quantum geometry, has become a very active field of study in condensed matter physics, relevant to a broad variety of phenomena.

This is particularly true in systems with low symmetries and, consequently, fewer constraints on the associated matrix elements. If all mirror symmetries and time-reversal symmetry are broken, the Bloch states are chiral and exhibit a non-zero net Berry curvature Ω. Rhombohedral multi-layer graphene (RMG), where graphene layers are stacked such that the B sublattice of one layer is stacked directly below the A sublattice of the subsequent layer, provides a particularly exciting and recent example. Without two-fold rotational symmetry C2z or inversion, the Bloch states of RMG are chiral with Ω= Interestingly, Ω(k) and quantum geometric effects in general are concentrated on a ring of finite radius in momentum (k) space encircling the K and K′ points and the magnitude is tunable by the number n of rhombohedrally stacked layers.

In combination with the fact that this system exhibits strongly correlated physics, including the spontaneous emergence of an imbalance between the two valleys, which breaks time-reversal symmetry, RMG has attracted a lot of interest in the context of quantum geometry. Harmonic generation, a non-linear optical phenomenon where driving a system by a high-intensity laser at a fundamental frequency ω0 leads to the emission at frequencies ω(l) = lω0, with integer l > 1, has also attracted significant attention in recent years. Beyond its technological relevance, one of the central reasons is its relation to the aforementioned quantum geometry and topology of Bloch bands.

This phenomenon, including with large l (HHG), is also very actively studied in two-dimensional systems, such as graphene, both experimentally and theoretically. Consequently, the physics of harmonic generation in RMG is being studied for different numbers n of graphene layers. The low symmetries of the system, particularly the lack of C2z symmetry, is a key difference compared to the related flat-band system twisted bilayer graphene. The role of the quantum geometry of the Bloch states for harmonic generation is demonstrated.

Investigation of the HHG reveals the consequences of an interaction-induced valley imbalance and doping. Circular dichroism (CD) of the harmonic generation spectrum is employed as a sensitive measure of the valley-resolved physics, widely used as a probe of chirality and proposed as a way of accessing Chern numbers. A sign change of the dominant circular dichroism signature as a function of the number n of graphene layers is found due to the competition of the two valleys with different interaction-induced gaps.

The remainder of this paper is organized as follows. Section II introduces the basic model and formalism used. To illustrate the key physics in a minimal setting, analysis of HHG spectra begins by considering only one of the two valleys of RMG, as detailed in Section III. Section IV incorporates both valleys and their interaction. Section V summarizes the results. Multiple appendices provide additional data and analytical details.

This section introduces the effective two-band model used to describe the low-energy physics of RMG, which can be more generally thought of as a minimal prototype of a chiral continuum model. This model is coupled to an external electromagnetic field associated with the applied strong pulsed laser and the method of computing the resultant HHG is outlined. Section A details the Bloch Hamiltonian of RMG. Rhombohedral stacks of n graphene layers are considered, where honeycomb lattices with two sublattices, Aj and Bj in the jth graphene layer, are stacked such that Bj and Aj+1 are vertically on top of each other.

An important experimental tuning parameter of the system is an external displacement field, denoted by D, which effectively creates a potential offset between neighboring layers. A full tight-binding model on the honeycomb lattice would give rise to a total of 2n bands (per spin). However, near charge neutrality, only two low-energy bands exist, one just above and one below the Fermi level in the vicinity of the K (ξ = +) and K′ (ξ = −). The weight of their Bloch states is localized on the non-dimerized sites, i.e., the sublattices A1 and Bn. This allows construction of a minimal two-band model (per spin) for each of the two valleys ξ = ±, with the Bloch Hamiltonian h(n) ξ (k) = −μ + w −γ1(−k∗ ξ/kc)n −γ1(−kξ/kc)n −μ −w. Here, k = (kx, ky) is the momentum deviation from the respective K or K′ point and the complex momentum coordinate kξ = kx + iξky is introduced. Besides, μ is the chemical potential, w is controlled by the displacement field, γ1 is the interlayer nearest neighbour hopping amplitude between superposed sites Bj-Aj+1, and the momentum scale is kc = 2γ1/( √ 3aγ0). In the latter, an is the intralayer distance between sublattices and γ0 the intralayer nearest neighbour hoppings of the original tight-binding model.

An important advantage of this effective two-band model is that the eigenstates and the associated matrix elements needed are readily derived analytically (see Appendix A). The spectrum reads as εp(k) = −μ + p q w2 + γ2 1(k/kc)2n, where p = ± labels the two bands and k = q k2x + k2y = |kξ|. With increasing n, the bands become flatter in a range of momenta of order kc. The associated increase in the density of states is also expected to be the reason why RMG displays multiple correlated phases, as revealed by recent experiments. Note that this minimal model leads to a dispersion that is even in momentum, εp(k) = εp(−k). This is why the band energies are the same in both valleys ξ = ±. The Bloch states encode the non-trivial quantum geometry of the bands and, in particular, that each valley is chiral, with opposite orientations in the two valleys, as required by time-reversal symmetry. This chirality can be illustrated and quantified using the Berry curvature, given by Ω(ξ) p (k) = h ∇k × A(ξ) p (k) i z, where A(ξ) p (k) = i ⟨φ(ξ) k,p|∇kφ(ξ) k,p⟩is the Berry connection.

As will become important later, Ω(ξ) p (k) = Ω(ξ) p (−k) for this model. The Berry curvature is sizable only in a ring-shaped region in k-space, as can be seen in Fig0.1(c). The resulting valley-resolved Chern number is C(ξ) p = 1 2π Z d2k Ω(ξ) ± (k) = pξ n 2, showing that it is proportional to the number of layers n. The half-integer quantization for odd n is a consequence of the continuum approximation. The main goal is to describe the higher harmonics generated by the coupling of the system to a strong pulsed laser with an electric field profile E(t) = −∂tA(t). Within the dipole approximation and using the length gauge, this leads to the additional time-dependent term Vint(t) = −eE(t) · r in the Hamiltonian.

Importantly, the position operator r acts non-trivially in the Bloch basis, with matrix elements given by ⟨Ψ(ξ) k,p|r|Ψ(ξ′) k′,p′⟩= δξ,ξ′ −iδp,p′(∇k′δ(k −k′)) + δ(k −k′)d(ξ) pp′(k), where the dipole matrix elements with respect to the periodic part of the Bloch states are given by d(ξ) pp′(k) = i ⟨φ(ξ) k,p|∇kφ(ξ) k,p′⟩ and it is used that this minimal model is block diagonal in the valley basis (no intervalley coherence). The diagonal matrix elements are equal to the Berry connection in the respective band and valley, d(ξ) pp (k) = A(ξ) p (k), probing the chirality of the bands. Importantly, also the off-diagonal components are non-zero which will drive transitions across the band gap. The explicit analytical form of these matrix elements is provided in Appendix A. To solve the time-dependent Schrödinger equation i | Ψ(t)⟩= H(t) |Ψ(t)⟩, where H(t) is the full Hamiltonian including the band structure and the coupling to E(t), the electronic wave function is expanded in the Bloch basis, |Ψ(t)⟩= X p=± Z d2k αp,ξ(k, t) |Ψ(ξ) k,p⟩. As a result of the term in the first line of Eq., the time-dependent Schrödinger equation expressed in terms of the expansion coefficients αp,ξ(k, t) ∈C involves momentum derivatives. Translating these coefficients to a “moving frame” via αp,ξ(k, t) =: eA(t)·∇Kβp,ξ(K, t) with K = k −A(t), eliminates the momentum derivatives. The resultant “semi-conductor Bloch equations” (SBEs) then read as βp,ξ(K, t) = −i h εp(k) + E(t) · A(ξ) p (k) i βp,ξ(K, t) −iE(t) · X p′=p d(ξ) pp′(k)βp′,ξ(K, t). Here and in the following equations, k should be understood as k(t) = K + A(t).

Researchers demonstrated high-harmonic generation in rhombohedral stacks containing n layers of graphene. The study revealed that the chiral Bloch states within these graphene systems significantly influence this process, with the winding of these states scaling linearly with the number of layers. This finding indicates that rhombohedral graphene provides a viable platform for investigating nonlinear optical phenomena. Furthermore, the research identified a connection between valley splitting and circular dichroism, and detailed how the net chirality of occupied states can be tracked; the authors suggest further analysis of these effects is warranted.