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Transition to Quantum Chaos in Weakly Disordered Graphene Nanoflakes

We analyze numerically ensembles of tight-binding Hamiltonians describing highly-symmetric graphene nanoflakes with weak diagonal disorder induced by random electrostatic potential landscapes. When increasing the disorder strength, statistical distribution of energy levels evolves from Poissonian to Wigner, indicating the transition to quantum chaos. Power laws with the universal exponent map the disorder strength in nanoflakes of different sizes, boundaries, and microscopic disorder types onto a single parameter in additive random-matrix model.


Introduction
Soon after the discovery of graphene-an atomically-thin monolayer of carbon atoms arranged in a honeycomb lattice [1]-it was shown experimentally that electrons in this material behaves as spin-1/2 massless Dirac particles [2], in agreement with much earlier theoretical prediction by Semenoff [3]. For this reason, the nanostructures in graphene have attracted much attention, leading physicists to reexamine classic effects of quantum transport [4] in search of novel features that arise from the unusual conical band structure, chirality, or the presence of additional quantum number (valley index) [5,6]. In particular, a Coulomb-blockade experiment on quantum dots consist of graphene nanoflakes and normal metallic leads [7] shown signatures of quantum chaos (the energy-level repulsion) for the flake size smaller then 100 nm, but without clear identification of the system symmetry class. Some more light was shed on this issue with theoretical work [8], showing that measurable quantities may indicate different symmetry class in the case of open than closed quantum dot. Later, the energy-level statistics of closed and irregular graphene flakes obtained from numerical diagonalization of tight-binding Hamiltonians [9,10] was found to coincide with those given by the Gaussian orthogonal ensemble (GOE) of random matrices [11].
In this paper, we follow the numerical approach established by Refs. [8,9,10] but focus on regular (hexagonal) graphene flakes with a weak diagonal disorder attributed to the substrate-induced random electrostatic potential landscape (see Fig. 1). The results show, that the energy-level statistics of such systems coincide with those given by additive random matrices of the form H 0 + λV [12], where H 0 is the diagonal random matrix (and thus has Poisson statistics) and V is GOE matrix. We also found, that the parameter λ is related to the extensive quantity N tot K 0 (where N tot is the total number of carbon atoms arXiv:1112.5078v1 [cond-mat.mes-hall] 21 Dec 2011 and K 0 is an intensive measure of the disorder strength) via the scaling law λ ∝ (N tot K 0 ) α , with α 0.6 regardless boundary conditions and microscopic details of the disorder model. The paper is organised as follows. In Sec. 2, we recall the basic findings on possible symmetry classes of chaotic nanosystems containing Dirac fermions, and present microscopic models of disorder in graphene nanoflakes. In Sec. 3, the random matrix model describing the transitions to quantum chaos is applied to rationalize level-spacing distributions obtained from numerical diagonalization of tight-binding Hamiltonians. The conclusions are given in Sec. 4.

Dirac fermions in disordered graphene
In this Section, we present two different microscopic models of disorder in graphene nanoflakes, representing the random electrostatic potential landscape abruptly or smoothly varying on the length-scale of the lattice spacing a = 0.246 nm. But first, let us briefly recall (after Ref. [8]) the discussion of possible symmetry classes of such nanosystems.

Symmetries of the Hamiltonian
The effective Hamiltonian for low-energy excitations of electrons in graphene in the absence of magnetic field has a form of the Dirac Hamiltonian where v F ≈ 10 6 m/s is the energy-independent Fermi velocity, σ i and τ i (i = 1, 2, 3) are the Pauli matrices acting on sublattice and valley degrees of freedom (respectively), and σ 0 (τ 0 ) denotes the unit matrix. M (x, y) and U (x, y) are the mass term and the external electrostatic potential. Symmetries of the Hamiltonian (1) are defined by the following antiunitary operations: standard time reversal T , and two "special time reversals" where C denotes complex conjugation. The mass term breaks the symplectic symmetry associated with T sl , leading to the two distinct possible scenarios: (i) In the case of weak intervalley scattering, T v commutes with H eff , so the system consists of two independent subsystems (one for each valley). Each subsystem lacks time-reversal symmetry, as T commutes only with full H eff . Because the Kramer's degeneracy (T 2 v = −I), the Hamiltonian consists of two degenerate blocks, each of which belonging to the Gaussian Unitary Ensemble (GUE). The analogous scenario was considered by Berry and Mondragon [13] for neutrino billiards, lacking the valley degrees of freedom.
(ii) In the case of strong intervalley scattering caused by irregular and abrupt system edges, or by the potential abruptly varying on the scale of atomic separation, the two sublattices are also nonequivalent, so both special time-reversal symmetries T sl and T v became irrelevant. T commutes with H eff leading to the orthogonal symmetry class.
The existing numerical studies for closed systems of irregular shapes [8,9,10] show that the typical intervalley scattering time is always shorter than the time required to resolve a level spacing (Heisenberg's time) leading to the scenario (ii). Some features of the scenario (i) were found in open systems [8], for which the intervalley scattering time needs to be compared with much shorter escape time. Such systems are, however, beyond the scope of this paper, as we focus on regular and weakly-disordered systems, for which the intervalley scattering itself may be suppressed.

Disorder in the tight-binding model of graphene
The lattice Hamiltonian for disordered graphene reads The hopping-matrix element γ ij = −γ if the orbitals |i and |j are nearest neighbors on the honeycomb lattice (with γ = 2 The electrostatic potential contains a contribution U gate from gate electrodes (slowly varying with the site position r i ) and a random contribution U imp from impurities. For small nanoflakes one can choose U gate U 0 = 0, whereas a realization of disorder potential is generated by randomly choosing N imp lattice sites R n (n = 1, . . . , N imp ) out of N tot , and by randomly choosing the amplitudes U n ∈ (−δ, δ). The potential is then smoothed over a distance ξ by convolution with a Gaussian, namely The special case of ξ a, N imp = N tot corresponds to the Anderson model on a honeycomb lattice, considered in work [14] on spectral statistics of nanotubelike structures. Earlier, the model constituted by Eqs. (3,4) with ξ a was shown to reproduce basic transport properties of disordered mesoscopic graphene samples [15,16]. It has not been considered, however, in the discussion of spectral statistics of nanoflakes so far.
We further define the Fourier transform of two-point correlation function where the system area A = 1 4 √ 3N tot a 2 , and the averaging takes place over possible realizations of the disorder (4) (so U imp = 0). For the length scales large compared to ξ, the dimensionless correlator becomes a representative measure of the disorder strength. For q = 0, we The numerical value of the ratio K q /K 0 at q = ± 2π 3a , 0 approximates the intervalley scattering rate, and is as small as 2 × 10 −6 for ξ = √ 3 a (used in the numerical simulations presented in remaining parts of the paper).

Additive matrix model for transition Poisson-GOE
Before presenting the numerical results for spectral statistics of graphene nanoflakes, let us briefly review corresponding additive random-matrix models and resulting nearest-neighbor spacings distributions [12]. When large integrable system undergoes transition to quantum chaos, its spectral properties can be modelled by the following random Hamiltonian where H 0 is diagonal random matrix, which elements follow a Gaussian distribution with zero mean and the variance (H 0 ij ) 2 = δ ij , the parameter λ ∈ [ 0, ∞ ], and V is a member of one of the Gaussian ensembles. In particular, for transition Poisson-GOE, elements of V are real numbers chosen to follow a Gaussian distribution with zero mean and the variance V 2 ij = (1 + δ ij )/N , where N is the matrix size.

Energy-level distributions for disordered graphene flakes
In this Subsection, the central question of the present work is addressed, namely: Whether the statistic interpolating between Poisson and GOE, P (λ; S) (8) is capable of describing nearest-neighbor spacings distributions P (S) for weakly-disordered graphene flakes? In other words, may the additive-random matrix model defined via Eq. (7) be applicable for such relativistic nanosystems? To answer this question, we focus on two systems of a high symmetry: hexagonal flakes with entirely armchair or zigzag edges, each of which is showing Poisson statistic in the absence of disorder (providing the level degeneracy is properly taken into account). As already mentioned in Sec. 2, two distinct models of disorder are applied to each system: Anderson model, defined by setting ξ = 0 and N imp = N tot in Eqs. (3,4), or smooth disorder, with ξ = √ 3a and N imp N tot .
To obtain the statistics P (S), we diagonalized numerically tight-binding Hamiltonians (3) for the flake containing N tot 10 4 atoms and 200 − 400 independent disorder realizations for either type of edges, disorder models, and each disorder strength quantified by the correlator K 0 (6). Some additional effort is required when unfolding the spectra: Unlike for two-dimensional gas of Schrödinger electrons, for which average density of states ρ is assumed to be energy-independent, for bulk graphene we have [13] ρ(E) A|E|/[π( v F ) 2 ] (per spin). For small systems studied here, boundary effects lead to additional states appearing near E ±γ (armchair edges) or E 0 (zigzag edges). Also, the impurity potential (4) introduces some bound states for |E| < δ. All these additional states, however, are localized on areas small in comparison to A, and thus not contribute to the spectrum obtained in a Coulomb-blockade experiment such as reported in Ref. [7]. For this reason, we limit the energy Table 1. Least-square fitted power-laws λ fit (ζ) = λ1ζ α , with ζ ≡ NtotK0 (lines in Fig. 3). Numbers in parenthesis are standard deviations for the last digit.

Disorder model
Armchair range [cf. E min and E max in Fig. 1(c),(d)] such that The constant term ρ 0 and the effective area A eff A are determined via leastsquare fitting of Eq. (9) to the actual ρ(E) obtained by numerical averaging over independent disorder realizations.
Our numerical results are presented in Figs. 2 and 3. First, we compare the statistics P (S) on two selected examples of nanosystems considered: the hexagon with armchair edges and Anderson-type disorder ( Fig. 2(a)-(c)) and the hexagon with zigzag edges and smooth disorder (Fig. 2(d)-(f)). Although some systematic deviations of P (S) from the best-fitted interpolating statistics P (λ fit ; S) (8) are visible for S > 1 due to a finite system size (notice that a better agreement is observed for N tot = 10584 than for 8322), P (λ fit ; S) reproduces the actual nearest-neighbor spacings distribution with a good accuracy for both systems and wide range of K 0 . We further notice, that similar values of λ fit are reached for the second system at K 0 typically 5 − 8 times larger than for the first system.
The dependence of λ fit on the total disorder strength N tot K 0 for all datasets available is illustrated in Fig. 3 (datapoints) in the logarithmic scale. The particular choice of the independent variable ζ ≡ N tot K 0 allows us to find the approximating relations λ fit λ fit (ζ), which still differ between the systems with different edges or disorder types, but remain unchanged when varying N tot and K 0 independently with the remaining parameters fixed. Also, for smooth disorder, we vary N imp having δ fixed at δ/γ = 0.1 or 0.5 (corresponding to the absence or presence of charge puddles in the physical system).
Least-square fitted power-laws λ fit (ζ) are listed in Table 1 and plot in Fig.  3(a),(b) (lines). The power-laws fail for λ fit 1, as P (λ fit ; S) becomes indistinguishable from GOE statistics in this range. They are, however, closely-followed by the datapoints for smaller λ fit -s. We further verify the obtained λ fit (ζ)-s for the case of smooth impurity potential, by taking the systems approximately four times larger in area (namely, N tot = 34062 for armchair edges or 42366 for zigzag edges), but generating only one disorder realization for each K 0 . Such an approach reproduces the experimental procedure of Ref. [7], where the spectrum of a single system was obtained. Additionally, the corresponding flake diameters 2R A = 87 √ 3 a 37 nm and 2R Z = 168 a 41 nm are of the same order of magnitude as diameters reported in Ref. [7]. The new datapoints (open triangles in Fig. 3) still follow the corresponding power-laws, providing that λ fit 1.
Probably, the most remarkable feature of these results is that all graphene nanoflakes considered show transition Poisson-GOE when increasing the disorder strength, with no signatures of GUE statistics. This is expected for the flakes with armchair edges which couple the valleys [5], or with zigzag edges and Anderson-type disorder [15], for which the intervalley scattering restores time-reversal symmetry. The absence of GUE statistics seems surprising in the case of zigzag edges accompanied by the smooth impurity potential. In such case, some intervalley scattering originates from six 120 • corners, a role of which may become decisive for spectral statistics of closed nanosystems.

Conclusions
We find that the additive random-matrix model, describing a transition to quantum chaos in Hamiltonian systems, is also relevant when discussing spectral statistics of highly-symmetric graphene nanoflakes with a weak diagonal disorder. The functional relation between the model parameter λ and the disorder strength N tot K 0 has a form of a power law, with the universal exponent α 0.6, which is insensitive to the boundary type or to the microscopic model of the impurity potential.
In the chaotic range, regular graphene flakes show energy-level statistics characteristic for the Gaussian Orthogonal Ensemble (GOE) of random matrices, indicating the strong scattering of Dirac fermions between the valleys. This coincides with earlier findings for irregular nanoflakes [8,9,10].