Quantum transport through graphene nanostructures

Transcription

1 Quantum transport through graphene nanostructures S. Rotter, F. Libisch, L. Wirtz, C. Stampfer, F. Aigner, I. Březinová, and J. Burgdörfer Institute for Theoretical Physics/E136 December 9, 2009 Graphene [1,2], the first true two-dimensional (2D) solid, is attracting considerable attention, mostly due to unique dynamics of electrons near the Fermi energy which closely mimics that of a massless Dirac Hamiltonian. Moreover, the double cone structure near the K and K points of the sub-lattices in reciprocal space gives rise to a near pseudo-spin degeneracy, suggesting an analog of Dirac four spinors. Envisioned applications range from high-mobility nanoelectronics [3], spin-qubits in graphene quantum dots [4] and the creation of neutrino billiards [5]. Spin coherence times in graphene are expected to be very long due to potentially weak spin-orbit and hyperfine couplings [6] making graphene quantum dots promising candidates for future spin based quantum computation [4]. However, confining electrons in graphene is a challenge, mainly due to the gapless electronic structure and the Klein tunneling paradox [7]. This difficulty has recently been overcome by structuring 2D graphene and quantum mechanical confinement effects have been observed in nanoribbons [8,9], single electron transistors [10] and graphene quantum billiards [11]. However, to simulate realistic graphene quantum nanodevices, the numerical solution of the Schrödinger equation has remained a computational challenge. This is partly due to the fact, that experimentally realizable structures feature typical sizes of 50nm, containing well over a million carbon atoms. At the Institute for Theoretical Physics an extension of the widely used Recursive Green s Function Method (RGM) [12] was developed which can bypass several of the limitations of conventional techniques. Key ingredient of this approach is the decomposition of the scattering geometry into separable substructures ( modules ) for which all the numerical procedures can be performed very effectively. All the modules are eventually connected with each other by means of matrix Dyson equations such that they span the entire scattering region (see Fig. 1). In this way we reach a high degree of computational efficiency. Adapting our Modular Recursive Green s Function Method (MRGM) [13] to different scatter- 1

2 (a) (b) (c) (d) Figure 1: (a) Single graphene block described by the matrix H 0. (b) A half-infinite waveguide is assembled by periodic repetition of (a) (see shaded area). By cutting off a finite-length module from a half-infinite waveguide (b), a rectangular module is calculated (c), which is then used to assemble a large-scale scattering structure (d). ing scenarios we are able to study a variety of different transport phenomena in previously unexplored parameter regimes. In 2009, we have adapted the MRGM method to the simulation of graphene nanostructures (See Fig. 1). In the following, we present results for eigenstates of graphene nanostructures in a magnetic field, as well as predictions for transport properties of graphene nanoribbons. Graphene nanostructures in a magnetic field We investigate the electronic eigenstates of graphene quantum dots with realistic size (e.g. up to 50 nm diameter) in the presence of a perpendicular magnetic field B. Numerical tight-binding calculations and Coulomb-blockade measurements performed near the Dirac point show how the magnetic field induces an adiabatic transition from the linear density of states at B = 0 to the Landau level regime at high fields. As our results demonstrate that this transition contains detailed signatures of the underlying graphene lattice structure including defects and localization effects at the edges. These signatures in the magneticfield dependence of graphene energy levels thus serve as sensitive indicators for the quality of graphene quantum dots and, in further consequence, for the validity of the Dirac-picture in describing the experiment. We focus on the eigenenergies of graphene-quantum dots as a function of a perpendicular magnetic field. In graphene, the linear band crossing at the so-called Dirac points connects the dynamics of electrons with the free, ultrarelativistic Dirac equation [14]. One might therefore expect a magnetic-field dependence of quantum dot eigenenergies in accordance to massless Dirac particles. This is true in the high-field (Landau level) regime, where energies are dominated by the 2

3 Figure 2: (a) Energy levels of a graphene-based quantum dot (diameter d = 50nm) as a function of a perpendicular magnetic field. (b) Same as (a), with a charging energy of 18meV and Zeeman splitting included. (c) Experimental data exstracted from Coulomb blockade measurements for a 50 nm graphene quantum dot. The kink pattern due to crossings with the first exited Landau level can be identified both in experiment and theory. (d) The mean level spacing as a function of magnetic field. magnetic field. Indeed, the magnetic-field dependence of the addition spectrum has been exploited in recent work to (approximately) pin down the electron-hole crossover point [15]. However, the analogy to Dirac particles does not hold in the perturbative regime of very small magnetic fields: in finite graphene nanostructures, quantum confinement and edge effects as well as lattice defects increase the complexity of this problem. To assess the role of edges and disorder, we simulate numerically an idealized graphene structure with smooth edges and consider separately the influence of (i) atomically sharp edges, (ii) edge disorder and (iii) lattice defects inside the quantum dot. For the low-field regime we find significant deviations from expectations based on the Dirac equation. This is due to the strong dependence on conserved sublattice symmetry (or, equivalently, chiral symmetry), which is broken in the case of lattice defects or edges. We observe in the evolution of the spectrum as a function of the magnetic field unique features of both 2D linear dispersion and of the edge confinement of the graphene QD. One prominent example is the development of the graphene specific E = 0 Landau level around the charge neutrality point [see Fig. 2(a)]. The overall tendency of states converging towards the zero-energy Landau level is accompanied by kinks which we attribute to crossings with higher Landau levels, which can be observed both in theory [see Fig. 2(b)] as well as experiment [see Fig. 2(c)]. This scenario is further supported by the analysis of averaged peak to peak spacings and the position of the extracted crossover region in the center of the transport gap [see Fig. 2(d)]. These results open the way for more involved studies of the electron-hole transition including a better understanding of the 3

4 (a) W 20nm (b) 100nm (c) (d) Figure 3: (a) Graphene nanoconstriction assembled out of 19 rectangular building blocks. The width W of the most narrow rectangle is 20 nm, corresponding to 106 unit cells. (b) Scattering wave function for different values of the particle energy E = 50meV (a), 100meV (b) and 200meV(c). addition spectra and spin states in graphene quantum dots. Transport through graphene nanoribbons We consider scattering through a graphene nanoconstriction of width W connected to two semi-infinite waveguides [i.e. we do not model the contact region, see Fig. 3(a)]. The edge of the ribbon is cut along a zigzag line of the hexagonal lattice. In a finite region of length L, we introduce edge roughness or defects into the ribbon [see shaded area Fig. 3(a)] and calculate the resulting transmission T and reflection R. Due to the finite width of the nanoribbon, the transverse component of the wavevector becomes quantized. As a consequence, the cone-like dispersion relation of graphene is reduced to a discrete set of curves. The distance in energy between adjacent curves is proportional to W 1, i.e. discrete modes (and thus quantization steps) are more widely spaced for narrow ribbons. As a result, the conductance G of an ideal graphene quantum point contact features quantization steps with the height of two conductance quanta, 2e 2 /h (neglecting spin), due to the two contributions of the K and K cone. We find that the quantization plateaus in the conductance of graphene nanoribbons are very sensitive to disorder. As a consequence, the experimental demonstration of size quantization peaks in graphene remains elusive, although recent publications claim to 4

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