Quantum transport through graphene nanostructures

Transcription

1 Quantum transport through graphene nanostructures F. Libisch, S. Rotter, and J. Burgdörfer Institute for Theoretical Physics/E136, January 14, 2011 Graphene [1, 2], the rst 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 hyperne couplings [6] making graphene quantum dots promising candidates for future spin based quantum computation [4]. However, conning electrons in graphene is a challenge, mainly due to the gapless electronic structure and the Klein tunneling paradox [19]. This diculty has recently been overcome by structuring 2D graphene and quantum mechanical connement eects have been observed in nanoribbons [7, 18], single electron transistors [8] 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. To nevertheless calculate eigenstates of this size, we employ an Arnoldi-Lanczos iteration,[9] together with a shift-and-invert technique to quickly determine the eigenvalues of large sparse matrices. We use the PARPACK (parallel implicitlyrestarted Arnoldi-Lanczos) library in conjunction with the MUMPS Matrix inversion package[10], and home-made C++/FORTRAN code. The computing resources of the phoenix correspond very well to the numerical demands of our approach. In the following, we present results for eigenstates of graphene nanostructures in a magnetic eld, as well as rst results for transport properties of graphene nanoribbons. 1

2 Figure 1: (a) Magnetic-eld dependence of the eigenenergies of a graphene quantum dot with smooth connement which approximately preserves chiral symmetry. Landau levels for n = ±1, ±2 [dashed lines, see Eq. (1)] are inserted as guide to the eye. The four symbols (,,, ) mark parameter values for which eigenstates are shown in Fig. 2. (b) Close-up of the avoided crossing of two eigenstates in (a). Dots represent numerical data, the continuous line is a t to a perturbative ansatz). (c) Close-up of avoided crossings around the diabatic ridge formed by the rst Landau level (see text). Graphene nanostructures in a magnetic eld 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 eld B. Numerical tight-binding calculations and Coulomb-blockade measurements performed near the Dirac point show how the magnetic eld induces an adiabatic transition from the linear density of states at B = 0 to the Landau level regime at high elds. As our results demonstrate that this transition contains detailed signatures of the underlying graphene lattice structure including defects and localization eects at the edges. These signatures in the magnetic- eld 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 eld. In graphene, the linear band crossing at the so-called 2

3 Figure 2: Typical eigenstates (plotted is the absolute square of the wavefunction) of a graphene quantum dot with smooth connement at high magnetic eld (B=25 T), corresponding to the zeroth (a, b) and rst (c,d) Landau level. Symbols (,,, ) correspond to those in Fig. 1(a). Dirac points connects the dynamics of electrons with the free, ultrarelativistic Dirac equation.[14] One might therefore expect a magnetic-eld dependence of quantum dot eigenenergies in accordance to massless Dirac particles. This is true in the high-eld (Landau level) regime, where energies are dominated by the magnetic eld, and Landau Levels emerge at quantized energies E n, E n (B) = sgn(n) 2 e vf 2 n B, n Z 0. (1) Indeed, the magnetic-eld 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 elds: in nite graphene nanostructures, quantum connement and edge eects 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 inuence of (i) atomically sharp edges, (ii) edge disorder and (iii) lattice defects inside the quantum dot.[17] For the low-eld regime we nd 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 eld unique features of both 2D linear dispersion and of the edge connement of 3

4 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 dierent values of the particle energy E = 50meV (a), 100meV (b) and 200meV(c). the graphene QD. One prominent example is the development of the graphene specic E = 0 Landau level around the charge neutrality point [see Fig. 1(a)]. Another feature is the splitting associated with levels at the K and K cones of the graphene bandstructure in the low-magnetic eld (perturbative) limit [see Fig. 1(b)]. This scenario is further supported by the measurement of averaged peak to peak spacings and the position of the extracted crossover region in the center of the transport gap [15, 17]. These results open the way for more involved studies of the electron-hole transition including a better understanding of the 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-innite 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 nite 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 reection R. Due to the nite 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 4

5 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 nd 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 have found equally-spaced plateau signatures in transport measurements they attribute to quantization steps [18]. References [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, A. A. Firsov, Nature, 438, , (2005). [2] Y. Zhang, Y.-W. Tan, H. L. Stormer, P. Kim, Nature, 438, , (2005). [3] A. K. Geim and K. S. Novoselov, Nat. Mater., 6, 183 (2007) [4] B. Trauzettel, D. V. Bulaev, D. Loss, and G. Burkard, Nature Physics, 3, 192, (2007) [5] M. V. Berry and R. J. Mondragon, Proceedings of the Royal Society of London, A, 412, (1987) [6] H. Min et al., Phys. Rev. B, 74, (2006) [7] Z. Chen, Y.-M. Lin, M. Rooks and P. Avouris, Physica E, 40, 228, (2007) [8] (i) C. Stampfer, J. Güttinger, F. Molitor, D. Graf, T. Ihn, and K. Ensslin, Appl. Phys. Lett., 92, (2008), (ii) C. Stampfer, E. Schurtenberger, F. Molitor, J. Güttinger, T. Ihn, and K. Ensslin, Nano Lett., 8, 2378 (2008) [9] C. Lanczos, Journal of Research of the National Bureau of Standards 45, 255, (1950) [10] P. R. Amestoy, I. S. Du, J. Koster, and J.-Y. L'Excellent, SIAM Journal of Matrix Analysis and Applications 23, (2001). P. R. Amestoy, A. Guermouche, J.-Y. L'Excellent and S. Pralet, Parallel Computing 32, (2006). [11] L. A. Ponomarenko, F. Schedin, M. I. Katsnelson, R. Yang, E. H. Hill, K. S. Novoselov, A. K. Geim, Science, 320, 356 (2008) 5

6 [12] S. Rotter, J.-Z. Tang, L. Wirtz, J. Trost, and J. Burgdörfer, Phys. Rev. B 62, 1950 (2000); S. Rotter, B. Weingartner, N. Rohringer, and J. Burgdörfer, Phys. Rev. B 68, (2003). [13] D. K. Ferry and S. M. Goodnick, Transport in Nanostructures (Cambridge University Press, 1997) and references therein. [14] G. W. Semeno, Phys. Rev. Lett. 53, 2449 (1984) [15] J. Güttinger, C. Stampfer, F. Libisch, T. Frey, T. Ihn, J. Burgdörfer and K. Ensslin, Phys. Rev. Lett. 103, (2009). [16] S. Rotter, P. Ambichl, and F. Libisch, arxiv: , submitted to PRL [17] F. Libisch, S. Rotter, J. Güttinger, C. Stampfer, and J. Burgdörfer, Phys. Rev. B 81, [18] Y.-M. Lin, V. Perebeinos, Z. Chen, and P. Avouris, cond-mat/0805,0035, (2008) [19] N. Dombay, and A. Calogeracos, Phys. Rep., 315, 41<96>58 (1999) 6