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The recursive Green's function method for graphene

Caio H. Lewenkopf, Eduardo R. Mucciolo

arXiv:1304.3934v1cond-mat.mes-hall

TL;DR

Graphene transport calculations must account for disorder across varied geometries and scattering mechanisms. This paper provides a self-contained recursive Green’s function framework and implementation guidance for computing transport-related quantities in graphene samples, while noting specific scope limitations.

  • Problem

    Disorder appears in multiple local and long-range forms, while conductance depends strongly on device geometry, motivating reliable transport calculations for graphene.

  • Method

    The paper develops a self-contained recursive Green’s function approach using lead surface Green’s functions and efficient recursive discretization for graphene sheets and nanoribbons.

  • Results

    The paper presents procedures for evaluating graphene transport quantities including conductance, local density of states, and local current densities.

  • Takeaways & Limitations

    The framework is intended to provide the basic material needed to implement graphene transport calculations while accommodating arbitrary geometries and varied scattering processes.

  • Takeaways & Limitations

    The analytical square-lattice lead treatment omits evanescent-mode contributions, which can matter for very short systems or measurements close to contacts.

Abstract

from arXiv · show

We describe how to apply the recursive Green's function method to the computation of electronic transport properties of graphene sheets and nanoribbons in the linear response regime. This method allows for an amenable inclusion of several disorder mechanisms at the microscopic level, as well as inhomogeneous gating, finite temperature, and, to some extend, dephasing. We present algorithms for computing the conductance, density of states, and current densities for armchair and zigzag atomic edge alignments. Several numerical results are presented to illustrate the usefulness of the method.

1 Introduction

The paper motivates a tight-binding recursive Green’s function treatment of graphene transport because disorder and charge-neutrality physics challenge perturbative approaches. It presents a self-contained implementation framework for graphene sheets and nanoribbons with varied geometries and scattering mechanisms.

  • Motivation: Disorder in graphene includes local defects and edge irregularities, as well as long-range charge impurities and substrate-induced ripples.These mechanisms remain central because experimentally produced samples are still far from the perfect ballistic regime.
  • Motivation: Perturbative Dirac-Hamiltonian methods use (kFℓ)−1 as a small parameter and become limited near charge neutrality, where kFℓ≪1.Numerical methods are therefore required to investigate that regime.
  • Contribution: The recursive Green’s function method computes two- or multiple-probe conductance for arbitrary geometries while efficiently treating diverse scattering processes.The paper targets graphene transport within the tight-binding approximation using an efficient single-particle Green’s-function algorithm.
  • Related work: RGF is presented as a standard, reliable, computationally efficient, and parallelizable method for finite-size transport problems with quantum coherence.The introduction also situates it among wave-packet, kernel-polynomial, continued-fraction, and specialized ballistic-graphene approaches.
  • Contribution: The paper provides a self-contained description of the basic material needed to implement RGF calculations, alongside advanced issues and graphene-specific developments.Its scope is instructional and implementation-oriented rather than a comprehensive review.
  • Organization: The workflow covers lead surface Green’s functions, graphene discretization, local observables, phenomenological dephasing, and disorder effects.The paper organizes these topics into successive methodological sections and concludes with numerical results.

2 Elements of Linear Mesoscopic Transport

The paper formulates linear-response transport through exact single-particle Green’s functions connecting leads, using Landauer or Caroli conductance expressions. It then recasts the calculation into a slice-based recursive scheme with lead self-energies and level-width matrices.

  • Transport formulation: Linear dc conductance is computed from the exact retarded Green’s function connecting source and drain leads, using either the Landauer or Caroli formula.The two formulations provide alternative descriptions of lead coupling in the transport calculation.
  • Landauer transport: The Landauer formulation uses propagating lead modes, velocities, transverse wave functions, and transmission matrices to express conductance.The transmission matrix connects left-to-right or right-to-left propagating channels, while spin degeneracy contributes a factor of 2.
  • Graphene model: The tight-binding model represents graphene with local potentials for gating or disorder and typically nearest-neighbor hopping on a honeycomb lattice.The method can also incorporate next-to-nearest hopping and magnetic fields, although the presented treatment assumes orthogonal orbitals and nearest-neighbor hopping.
  • Computational considerations: The RGF computational cost scales as N × M3, where N is the number of slices and M is the typical number of sites per slice.The implementation is particularly recommended when disorder or irregular geometry breaks translational invariance.
  • Caroli transport: The Caroli formulation uses retarded and advanced Green’s functions across the sliced system together with lead level-width matrices derived from surface self-energies.The resulting transmission probability is real and positive because the relevant Green’s-function and coupling matrices have the required Hermitian and positivity properties.
  • Slicing scheme: The sample is divided into thin slices, with neighboring slices connected by tight-binding coupling matrices and the outer slices coupled to semi-infinite leads.The slice construction turns the system into a quasi-one-dimensional recursion problem.

3 Recursive Green’s Functions

The recursive Green’s function method builds the exact Green’s function of a two-probe graphene system by recursively connecting leads and thin slices. Left and right sweeps, followed by their combination, provide the Green’s functions needed for transport and local observables, with cost scaling as N × M3.

  • Recursive construction: Dyson formulas recursively incorporate the hopping connections between otherwise independent leads and slices to construct the full Green’s function.The perturbation selectively adds the connecting matrix elements, allowing the system to be assembled from unperturbed components.
  • Recursive sweeps: A left-to-right sweep generates GL functions, a right-to-left sweep generates GR functions, and joining the two families yields the exact Green’s function.The procedure is specialized to the two-probe conductance setup and requires intermediate recurrence relations before the full Green’s function is obtained.
  • System partitioning: The method partitions the sample into N thin slices, each containing at most M sites or cells, and couples them to left and right leads.Lead surface Green’s functions are computed before the recurrence, while inter-slice and lead-sample couplings are represented by hopping matrices.
  • Scope and extensions: The basic algorithm assumes couplings between nearest-neighbor slices, but next-nearest hopping can be included by widening slices or modifying the recursion.Doubling the unit-slice width slows computation by a factor 23, while a proper modification can incur a smaller slowdown.
  • Computational cost: N inversions, each requiring O(M3) operations, give an overall computational complexity of N × M3 for the recurrence.The right-sweep construction independently reports the same overall cost, O(N × M3).
  • Observable-specific calculations: The full Green’s functions provide the quantities needed for local current distributions, while a reduced sweep can compute transmission and reflection matrices.Local observables require G0,N+1 and Gn,n for every system slice, whereas the alternative construction is useful when only scattering matrices are required.

1. For the left sweep, we use Eq. (42) to write

The scattering matrix can be assembled using one or both directional sweeps, depending on symmetry conditions. Local observables require additional Green’s-function blocks, and the recurrence depends on finite-imaginary-part lead Green’s functions plus slice and coupling inputs.

  • Symmetry conditions: For inversion-symmetric systems, equal boundary Green’s functions and a single sweep can suffice to evaluate the whole scattering matrix.The stated relations are G00 = GN+1,N+1 and G0,N+1 = GN+1,0.
  • Symmetry conditions: With symmetric leads and no external magnetic field, time-reversal symmetry permits one sweep; otherwise both directional sweeps are needed.The two sweeps are combined to assemble the scattering matrix when these conditions are not met.
  • Local observables: Local density of states and local current flux require G0,N+1 together with Gn,n for every slice n = 1,...,N.These Green’s-function blocks are required for local observables even when a single sweep suffices for scattering quantities.
  • Required inputs: The calculation requires lead Green’s functions gL and gR, isolated-slice Hamiltonians hn, and inter-slice hopping matrices U before recursion begins.The lead Green’s functions must have finite imaginary parts to ensure convergence through lead coupling.

4 Lead Green’s Functions

The paper develops lead Green’s functions using eigenchannel decompositions for square lattices and decimation for arbitrary translation-invariant lattices. It then relates the resulting surface Green’s function to graphene leads, while noting limitations for edge orientations and evanescent modes.

  • Physical lead modeling: Lead models must account for metallic source and drain contacts and minimize contact resistance from band-structure mismatch.The chemical potential is adjusted to maximize lead density of states, and the lead lattice model is chosen to represent ohmic graphene-metal contacts.
  • Square-lattice eigenchannels: Square-lattice lead Green’s functions can be constructed in an eigenchannel basis and transformed into the site representation.The transverse modes form the channel basis, while a unitary transformation converts the channel-representation Green’s function into site-space matrix elements.
  • Square-lattice eigenchannels: The analytical square-lattice construction uses transverse hard-wall modes, longitudinal wave numbers, dispersion relations, and propagating-mode velocities.The longitudinal wave vector is identified with the continuum momentum, while propagating and evanescent modes correspond to real and complex longitudinal wave numbers, respectively.
  • Decimation method: The eigenmode expressions do not directly apply to zigzag or armchair leads because the interslice coupling is neither proportional to the identity nor commuting with the slice Hamiltonian.The paper therefore introduces a general decimation approach for semi-infinite lattices with arbitrary translation-invariant structures.
  • Decimation method: Decimation repeatedly eliminates intermediate slices, generating recursions on sites separated by 2^k until the coupling matrices are sufficiently small.The resulting surface Green’s function is then related to the lead Green’s functions; a positive η produces retarded Green’s functions and affects convergence and accuracy.

5 Device Green’s Function

This section develops device Green’s-function calculations and efficient graphene slicing schemes for armchair and zigzag geometries. The armchair “pine tree” configuration reduces slice size, while zigzag ribbons lack an equivalent alternative without next-to-nearest-neighbor connectivity.

  • Device Green’s function: The device Green’s function supports transport calculations for multilayer graphene with zigzag, armchair, or chiral edges and tight-binding disorder or gating.The RGF method computes these quantities across a broad range of device settings.
  • Slicing schemes: Efficient slicing schemes are presented for graphene monolayers with armchair and zigzag edges.The schemes are designed to support recursive calculations for both principal edge orientations.
  • Slicing armchair lattices: Armchair slices can be mounted in a “pine tree” configuration that reduces sites per slice without introducing next-to-nearest-neighbor hopping.Reducing slice size targets the matrix inversion bottleneck in rectangular geometries.
  • Lattice dimensions: Armchair and zigzag lattice sizes are related to physical graphene dimensions through separate geometric relations.The section gives dedicated size relations for each slicing scheme.
  • Slicing zigzag lattices: Zigzag ribbons can be represented by a brick-wall lattice with missing vertical bonds, but no alternative slicing reduces sites without creating next-to-nearest-neighbor connectivity.The geometry and slicing structure are illustrated with sublattice labels, vertical slices, and a highlighted dimer.

6 Evaluating Local Quantities

The RGF method provides transport-related local quantities beyond transmission, including LDOS, charge density, and current density. These calculations require exact local or lesser Green’s functions and can be more computationally demanding than conductance evaluation.

  • Local quantities: The RGF method computes local density of states and local current density in addition to transmission.These quantities extend the method from conductance calculations to spatially resolved observables.
  • Local density of states: Conductance needs retarded Green’s functions at contact slices, whereas LDOS evaluation across the device scales as O(N^2M^3).The additional factor linear in N arises because retarded Green’s functions must be evaluated at all slices of interest.
  • Local density of states: LDOS supports analyses of local magnetic-moment formation near zigzag edges or vacancies through the Stoner mechanism.The mechanism requires enhanced LDOS when strong electron-electron interaction is present.
  • Charge density: The tight-binding Hamiltonian includes local potentials for gating or disorder and can be extended through suitable hopping choices to model disorder.The paper uses orthogonal-orbital tight-binding models and discusses self-consistent occupation numbers for interacting extensions.
  • Current density: Bond currents are obtained from nonequilibrium Green’s functions, with intra-slice and inter-slice cases determined by the neighboring slice and site coordinates.In equilibrium, the lesser Green’s function symmetry yields zero bond current.
  • Current density: Current-density calculations may use recursive lesser Green’s functions or a linear-regime elimination technique requiring only retarded and advanced Green’s functions.The elimination approach avoids energy integration but scales as O(N^2M^3).
  • Current conservation: The calculated current is conserved between consecutive slices and the total current leaving any site is zero.The paper derives I(n−1)→n = In→(n+1) from the bond-current expression.

7 Dephasing

Dephasing is incorporated phenomenologically by attaching voltage probes whose chemical potentials are adjusted to produce zero net probe current. The resulting calculations require cross-conductances and, generally, self-consistent Green’s functions, while stored sweep results enable efficient inter-slice recursion.

  • Dephasing model: The voltage-probe approach models dephasing by coupling selected sites to probes that drain electrons without injecting net current.The drained electrons lose phase coherence, implementing the Büttiker and D’Amato–Pastawski phenomenology.
  • Probe conditions: Probe chemical potentials are determined by imposing Ii = 0 for every voltage probe in the linear-response equations.The left, right, and probe currents are defined as positive when flowing into the system.
  • Transport coefficients: Cross-conductance matrices are calculated from Caroli formulas and are real, positive, and symmetric without magnetic fields.Symmetry follows from the Hermitian positive coupling matrices in the absence of magnetic fields.
  • Probe implementation: Sparse probe attachment keeps Nϕ much smaller than NM, while Nϕ and the probe couplings determine the coherence length ℓϕ.The sparse arrangement limits the computational cost of cross-conductance evaluation.
  • Self-consistency: The exact Green’s functions depend on probe chemical potentials, so the dephasing calculation generally must be implemented self-consistently.The probe potentials depend on lead potentials and cross-conductance matrices.
  • Dephasing Green’s functions: Exact Green’s functions between probe-bearing slices can be obtained by recursively propagating stored left-to-right or right-to-left sweep results.The procedure starts from local Green’s functions at the relevant slices and advances through intermediate slices.
  • Dephasing Green’s functions: O(|n1 − n2|M) is the cost of the inter-slice recursion because no matrix inversions are required in these steps.This provides an efficient route for obtaining Green’s functions between separated probe locations.

8 Disorder

The disorder section shows how local potentials, strained bonds, ripples, and hopping distortions are represented in the tight-binding and RGF frameworks. It also connects off-diagonal hopping disorder to a random vector potential while favoring direct hopping renormalization for numerical modeling.

  • Disorder mechanisms: Graphene disorder includes charge inhomogeneities, substrate irregularities, ripples, strain, molecular adsorption, vacancies, and irregular edges.These mechanisms motivate microscopic disorder modeling in transport calculations.
  • Diagonal disorder: Diagonal disorder is modeled with Gaussian scatterers centered at random lattice sites, random amplitudes uniformly drawn from [−δV,δV], and concentration nimp = Nimp/A.The resulting disorder strength depends on fluctuation magnitude, scatterer range, and concentration.
  • Diagonal disorder: The dimensionless parameter K0 characterizes disorder fluctuations through the impurity-potential correlation function.K0 contains information about δV/t as well as scatterer range and concentration.
  • Diagonal disorder: The transport mean free path away from the Dirac point is obtained in the continuum limit using the Born approximation.The paper derives this relation from the disorder model and its continuum description.
  • Strain and ripples: Strain-induced bond-length changes are translated into modified hopping integrals using semi-empirical Slater–Koster parameterizations.The RGF method directly accounts for the resulting hopping changes.
  • Strain and ripples: Ripple fields alter atomic orbital overlaps and therefore modify nearest-neighbor and next-to-nearest-neighbor hopping matrix elements.The ripple profile is generated from a substrate-correlation model under the stated low-temperature assumption.
  • Vector-potential disorder: Off-diagonal hopping disorder can be mapped to a random vector potential in the continuum description.The paper derives the mapping from hopping distortions to a corresponding vector field.
  • Vector-potential disorder: For tight-binding and RGF calculations, modeling strain with renormalized hoppings is simpler than projecting onto the K and K′ cones.This choice avoids issues associated with preserving time-reversal symmetry in the continuum projection.

9 Some Numerical Results

The RGF calculations benchmark clean graphene transport, resolve current distributions in armchair and zigzag flakes, and examine disorder effects in sheets and nanoribbons. Results reproduce analytical behavior in sufficiently large ballistic systems while revealing crossover and localization phenomena under disorder.

  • 9.1 Ballistic transport in clean samples: Clean-sheet conductivities and Fano factors agree well with analytical results for large systems, but finite-size oscillations and asymmetry appear when systems are too small.The deviations are associated with insufficient system size and, in particular, transport dominated by evanescent modes.
  • 9.1 Ballistic transport in clean samples: Different aspect ratios confirm that deviations from the analytical curve occur only when ribbons are too short and evanescent modes dominate transport.
  • 9.1 Ballistic transport in clean samples: At E = 0, zigzag flakes carry current primarily through edge states, whereas armchair flakes show a nearly uniform current distribution.Slightly away from E = 0, the zigzag current distribution changes drastically, with nearly no current at the edges.
  • 9.1 Ballistic transport in clean samples: Edge-current densities in zigzag nanoribbons change quantitatively and qualitatively when next-nearest-neighbor hopping replaces the nearest-neighbor model.The appropriate tight-binding model is tied to fitting DFT calculations or explaining experimental manipulation and characterization of nanoribbon edges.
  • 9.2 Disordered graphene sheets: Long-range disorder produces a ballistic-to-diffusive crossover: σ0 approaches the pure-ballistic prediction for L/ℓ < 1, while diffusive conductivity scales with ln(L/ℓ) for L/ℓ ≫ 1.In the coherent diffusive regime, the conductivity minimum has substantial sample-to-sample fluctuations and weak logarithmic dependence on mean free path.
  • 9.3 Nanoribbons: Even small random edge-site removal rapidly smears nanoribbon conductance steps, while edge disorder produces localized states visible in the local density of states.These results demonstrate the difficulty of observing conductance quantization experimentally in disordered nanoribbons.

C Random Flux Estimate

This section estimates the root-mean-square value of the random magnetic field generated by a random vector potential.

  • C Random Flux Estimate: The calculation concerns a random magnetic field generated by a random vector potential.
  • C Random Flux Estimate: The target quantity is the root-mean-square value of that random magnetic field.
  • C Random Flux Estimate: The section introduces an estimate rather than reporting a transport result.
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