Quantum Transport in Nanostructured Graphene Antti-Pekka Jauho

Transcription

1 Quantum Transport in Nanostructured Graphene Antti-Pekka Jauho ICSNN, July 23 rd 2018, Madrid

2 CNG Group Photo

3 Three stories 1. Conductance quantization suppression in the Quantum Hall Regime, Caridad et al., Nature Commun. 9, 659 (2018) 2. Ballistic tracks in graphene nanoribbons, Aprojanz et al, submitted 3. Ballistic quantum transport and tunable band gaps in Dirac Ring systems, Jessen et al., to be submitted

4 Common themes In nanostructured graphene transport is often dominated by edge effects It is very difficult to fabricate ideal edges: the edges are often irregular, disordered, illcharacterized. Many exoctic theory predictions are not robust against disorder effects The three examples illustrate various methods to produce well-defined edges, and how experiment and theory can be made to match actually, if you are lucky - extremely well.

5 Story 1: Conductance quantization suppression in the QHE regime Similarities and differences between 1D channels and QHE 25 nm ZGNR 65 nm ZGNR, B=20 T

6 Caridad et al, Nature Commun. 9, 659 (2018) Sample 1: low disorder; Sample 2: normal disorder

7 Conundrum Why do extremely smooth edges show conductance plateaux which exceed the values expected from simple size quantization Why does the increased disorder occurring in standard samples restore the expected plateaux?

8 Charge distribution nonuniformity due to gate Comsol simulations of carrier density (green: nanoribbon, blue: narrow constriction, red: wide constriction)

9 Consequences of charge nonuniformity Red dos: bulk channels not present in uniform case Solid: actual band structure; dashed: uniform gating

10 Tight-binding simulations (including nonuniform gating)

11 Conclusion Story 1 Careful etching (RIE instead of oxygen plasma ashing) allows one to fabricate low-disorder nanowires (mobilities comparable to unzipped CNTs) A QHE regime with suppressed quantization can be studied with these samples A consideration of nonuniform charge density allows quantitative modeling of the conductance features Electron-electron interactions play an important role in shaping the actual potential landscape

12 Story 2: Ballistic tracks in graphene nanoribbons Baringhaus et al. (Tegenkamp group) observed exceptional ballistic transport in epitaxial graphene nanoribbons in dual probe STM measurements (Nature 506, 349 (2014)): The conductance maintains its quantized value for probe separations of several microns in these experiments. However, the value of conductance could not be explained (looks as if only one spin-component is involved??)

13 Cut a wedge in the GNR In subsequent experiments a wedge was cut in the GNR:

14 Excellent agreement with measurement and quantum transport simulations Baringhaus et al., PRL 116, (2016) The peaks exceeding G0 are due to quantum interference (the small area, with area W x L, under the wedge acts as a quantum cavity with a few standing waves) However, the single-channel transport i.e. G0 - was put in by hand

15 New refinement: measurements with a movable injector Note the sharp, and the blunt tip. The sharp tip can be moved across the ribbon, as indicated by the red arrow Whatever is measured, is very robust: panel (d) shows back and forth scans of the sharp tip, which retrace the conductance values. The measurements pose a challenge: explain the values of the conductance plateaux!

16 Details of spatially resolved conductance Sharp tip in red region: G = G0 Sharp tip in blue region: G= 4 G0 Sharp tip in purple region: G= 8 G0

17 Theoretical model (1) The following features are included in the model: The lower edge has a zig-zag structure and its magnetism is modeled by a mean-field Hubbard term. Upper edge has no Hubbard term An edge sublattice mass term (comes from bonding of the lower edge) A transverse field term, created due to charge transfer at the upper edge. (All these terms have been (separately) discussed in the literature and are reasonably well-understood (but not put together, as we do))

18 Theoretical model (2) Tight-binding model: The Green s functions are calculated for an infinite ribbon while the Gammas describe the couplings to the sharp tip, and the broad tip.

19 Theoretical results The exceptional channel (red) is clearly at the lower end of the NR consistent with earlier experiments. Experiment:

20 Conclusion Story 2 A movable tip is able to resolve spatially separated channels: this gives a concrete meaning to the abstract channel This work could be the first where the spatially separated channels emerge from a microscopic model, and are experimentally detected This would could have far reaching implications for many other systems where edge and bulk transport compete.

21 Story 3: Ballistic quantum transport and tunable band gaps in Dirac Ring systems A main hindrance for many applications of graphene is the lack of the band gap Many theoretical methods exist to create a gap: nanoribbons, periodic gates, periodic adsorbates, bilayers in transverse fields. We took inspiration from photonic crystals: a periodic modification of the dielectric environment can (experimentally) deflect a light beam consisting of photons (perhaps not totally unlike of massless Dirac Fermions in graphene )

22 Graphene antidot lattices GAL

23 Miniband structure for GAL Tight-binding description, allowing for a modified hopping integral at edges (obtained from DFT), according to Son et al., PRL 97, (2006) For a {7,3} structure with triangular lattice symmetry one finds a band gap of 0.73 ev

24 Fabrication (large area GAL) Important technological breakthrough (Spring 2010):

25 Typical results

26 A show stopper The GALs fabricated in this way are large and seemingly quite regular, but: It is very difficult to transfer them to an insulating surface All transfer methods (at least the ones we tried) leave remnants on the GALs, with worsened electronic properties In fact, the remaining graphene between the antidots is severly damaged, with a very low mobility (An unexpected side benefit: these structures may turn out to be useful in sensor or optical applications)

27 A new beginning. Use the Columbus egg (a trivial trick solves a huge problem in one sweep): encapsulate graphene between BN, and do the e-beam etching through the protecting BN layer. This technique was pioneered by the Regensburg group. They have demonstrated clear commensurability oscillations in nanostructured graphene samples.

28 State-of-the-art (compilation from unpublished CNG work) Bjarke Jessen, Lene Gammelgaard et al, unpublished

29 Magnetotransport in GALs In recent work (Power et al, PRB (2017) we showed that an extraordinary agreement with conductance experiments (Regensburg, Sandner et al) on holey graphene can be obtained if the full power of simulations is used. Below: examples of computed current distirubtions in a GAL with applied magnetic field. Also semiclassical trajectories are shown.

30 Comparison with experiments Expt results from Regensburg S.R. Power et al, PRB Simulations are an avergae over five different realizations of mild disorder but no temperature smoothing is included. (M signifies the beginning of a (modified) QHE regime.)

31 CNG sample & measurements Pristine weak T-dep GAL insulating T-dep Jessen & Gammelgaard et al (unpublished)

32 Pristine vs nanostructured Note the difference between Landau fans: pristine material has linear LL s whicle nanostructured sample has nonlinear fan shape, and a finite gap, which closes when B is increased. The pristine material shows both primary and secondary Moiré features, while only the primary is visible for the GAL. It is remarkable that Moiré survives nanopatterning!

33 Comparison with experiment and TBsimulations Experiment and TB simulations join seamlessly for positive gate voltages For negative VG, the Moiré feature (not included in the simulations) obscures part of the data

34 A detail of the Dirac Ring energy spectrum The bands actually consist of many Dirac ring energy levels with a complicated field dependence (one example is highlighted as the dashed line in the inset).

35 zooming in..the dashed line highlights the behavior of a typical level. See also da Costa et al., PRB (2014)

36 Look closer at the Moiré feature at negative gate voltage The Moiré structure at VG=- 7.5V beautifully clones the GAL gap around VG=0, as well as its fine structure. is this generic?

37 Conclusion Story 3 By using BN protective layers it is possible to nanostructure graphene so gently, that disorder effects are minimized, and the good conductive properties of the pristine material preserved, so that quantum ballistic effects become visible A band gap is formed, in analog with photonic crystals, and in agreement with theoretical predictions. This is a true band gap, not the mobility gap seen in many previous experiments The underlying Moiré pattern due to mismatch of the lattice constants is preserved, remarkably well The Moiré pattern seems to clone the fine structure around the GAL band gap There are many other applications of GALs which we believe have now become within experimental reach eg. qubits based on defect states as visualized in early theoretical research. Design your own structure on hbn, etch through it and voilá! The method should be applicable to other 2D materials as well.