Controlling the carriers in graphene via antidot lattices. Project description Page 1 of 6

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1 Controlling the carriers in graphene via antidot lattices. Project description Page 1 of 6 Controlling the carriers in graphene via antidot lattices The discovery in 2004 of graphene [1], an atomically thin layer of carbon, ignited a flurry of research and earned the discoverers last year s Nobel Prize in physics. Graphene possesses several unique and intriguing physical properties [2] that make it an attractive material for fundamental research as well as applied science, as evident from the interest of companies such as IBM [3] and Samsung [4]. Due to its extraordinary electronic quality [5], graphene is expected to play a major role in future electronic devices, possibly replacing silicon as the material of choice. However, pristine graphene lacks an electronic band gap; the electrons in graphene are difficult to control and graphene is not immediately applicable for semiconductor devices. Along with co-workers, I have previously suggested so-called graphene antidot lattices (GALs) as a way of turning graphene into a semiconductor [6, 7]. In the simplest form, a GAL is a periodically perforated graphene sheet. While several aspects of GALs have already been studied, focus has so far largely been on the properties of isolated sheets of GALs. The purpose of this project is to take the next step towards GAL-based electronics, by conducting theoretical research on the fundamental building blocks of future GAL-based devices. These studies will both increase our understanding of these unique structures and serve as guidelines towards experimental realizations of GAL-based electronic devices. The first year of the project will be carried out in collaboration with Prof. Thomas Garm Pedersen at Aalborg University (AAU), while the second year will be carried out in collaboration with Prof. Antti-Pekka Jauho at the Technical University of Denmark (DTU), to ensure an ongoing collaboration between these universities on graphene antidot lattices. An external stay at Aalto University in Finland is planned in the second year of the project. Background Graphene is a truly remarkable material. A single atomic layer of carbon atoms, it has a mean free path at room temperature of the order of millimeters and extremely high electron mobility [5]. In pristine graphene, carriers behave as massless particles with a linear dispersion relation consisting of two linear bands intersecting at the Fermi energy. These so-called Dirac fermions display several peculiar properties, such as an unconventional integer quantum Hall effect [8] and Klein tunneling [9], the latter manifesting itself as perfect normal incidence transmission through potential barriers. This, as well as the lack of a band gap, presents a great challenge for utilizing graphene in semiconductor devices, where tight control of the flow of carriers is required. During the last few years, GALs have emerged as a way of introducing a band gap in graphene in a controlled manner, by periodically modulating the graphene sheet [6, 7]. The modulation opens up a band gap in a manner that is closely analogous to providing mass to the otherwise massless Dirac fermions. The appearance of a controllable mass greatly extends the possible applications of graphene and opens up for a wealth of new, intriguing phenomena of this unique material. The

2 Controlling the carriers in graphene via antidot lattices. Project description Page 2 of 6 Figure 1. (left) Pristine graphene yields massless Dirac fermions with a vanishing band gap. Adding a periodic modulation, here in the form of perforations of the sheet, provides a mass to the carriers, rendering graphene semiconducting. Here, the lines indicate bonds between carbon atoms sitting at each vertex. (right) GAL waveguides, one possible application of GALs that will be investigated. Here, blue circles indicate antidots, while the background is graphene. Sandwiching a region of pristine graphene between GALs is expected to result in a quasi-one dimensional electron waveguide, in analogy with photonic crystal waveguide structures. magnitude of the mass, and the resulting band gap, can be effectively controlled via the shape and size of the holes. However, the source of the modulation need not be actual holes in graphene, but can also, e.g., be in the shape of hydrogen adsorbed on graphene as in a recent experimental realization of GALs [10]. Our original proposal has already sparked a very active sub-field of graphene research, with several theoretical and experimental groups working on the subject, and with fabrication of GALs already achieved by different methods [10, 11, 12]. In particular, the field has matured to a stage where mass-production of GALs seems feasible. Project objectives By combining GALs with regions of pristine graphene, systems can be realized in which the relativistic Dirac particles of graphene have spatially varying mass. In this project I plan to explore theoretically the properties of such unique systems, which are expected to yield both fascinating physics as well as functioning as the fundamental building blocks of future GAL-based devices. The hypothesis of the project is that such combined structures of pristine graphene and GALs will possess properties useful for semiconductor devices, while maintaining the main features that make graphene attractive for electronics. The objective is to theoretically determine the properties of the structures, prove their applicability for devices, and provide general guidelines for optimizing their properties for GAL-based electronics. Specifically, the following lines of research will be pursued in the project: A. GAL waveguides. By sandwiching a region of pristine graphene between GALs, carriers may be confined within the massless graphene region, thereby creating quasi-one dimensional graphene waveguides in a manner analogous to photonic crystal waveguides.

3 Controlling the carriers in graphene via antidot lattices. Project description Page 3 of 6 Extended defects in graphene have recently received attention as a means of generating metallic wires in graphene [13]. Also, very recently, electrical gates have been used to achieve electron guiding in graphene [14]. However, contrary to such proposals, GAL waveguides may offer several distinct advantages. The carriers are expected to be confined largely to a region of pristine graphene, and the waveguides should thus inherit the exceptional electronic properties of graphene. This will result in, e.g., suppressed backscattering in GAL waveguides due to Klein tunneling. While the proposal in [14] also allows guiding in pristine graphene, this comes at the cost of requiring very precise electrical contact with the graphene sheet. Furthermore, the exact shape of the antidots contributes a further parameter for tuning the transport properties of the waveguides. B. GAL mass barriers. In general, the interface between regions of different mass, particularly between pristine graphene and GALs, is expected to reveal interesting properties. A GAL region between two graphene sheets creates a barrier in which the electron acquires a nonzero mass. The transport properties of electrons launched from one graphene sheet towards the barrier will be studied. Such mass barriers represent the smallest device component one can imagine for GAL-based electronics. C. GALs in magnetic fields. The interplay between high magnetic fields and a periodic lattice leads to very rich physics [15]. -The larger unit cell of GALs, along with the favorable energy scaling of graphene, may allow such phenomena to be observed at significantly lower magnetic fields. Such studies are interesting for bulk GALs as well as in relation to the GAL waveguide proposal, where a magnetic field is expected to strongly influence the transport properties. In particular, GAL waveguides in magnetic fields may prove useful for spintronic applications, in which the spin rather than the charge of the electron serves as the fundamental unit of information. Scientific significance Several scientific challenges must be met in order for the project to be successful. The size of the envisioned structures presents a challenge in its own, because computational cost is such that several assumptions must first be made in order to simplify the modeling. Such assumptions require a thorough understanding of the physics of the problem and must be validated. Furthermore, in order to provide general guidelines for optimization of the structures for devices, simpler analytical expressions are in some cases preferable to purely numerical results. Obtaining such analytical results is a significant challenge for these relatively complicated structures. When these challenges are met, I expect that on a short term, the project will increase our fundamental understanding of the interfaces between GALs and pristine graphene, serving as guidelines towards GAL devices. Furthermore, specific structures useful for GAL-based devices will be

4 Controlling the carriers in graphene via antidot lattices. Project description Page 4 of 6 studied, which should be of great benefit for experimental research in the field of GALs. On a longer term, this is expected to help pave the way for graphene-based electronics. Research plan The three research lines A-C share many similarities as far as methodologies go. The low-energy electronic structure of graphene is quite well described by the Dirac equation (DE), which emerges as a linearization of a tight-binding (TB) approach to the problem. By introducing a mass term in the DE, yielding so-called gapped graphene, this can be used to model the low-energy properties of GALs [16]. The advantage of the DE approach is that it may lead to closed-form analytical expressions, serving as useful guidelines to the overall dependence of the electronic properties on the various parameters, e.g., antidot size and lattice periodicity. I plan to employ the DE approach for initial studies of B, for which analytical solutions for the transmission coefficients should be obtainable. Also, C will be analyzed using both a DE approach and a TB model, where the magnetic field is introduced via a Peierls substitution. To investigate the transport properties of the proposed structures, I plan to employ standard Green s function (GF) methods [17], in which the device region is coupled to infinite leads modeled via self-energy terms in the Hamiltonian. Time schedule The schedule is outlined in the chart below, with activities highlighted for each quarter. Each activity terminates in a milestone described below. - Realization of electron guiding in graphene is starting to receive significant attention, so I plan to start the project by investigating research line A using a TB model. Milestone 1 (M1) consists of confirmation of the expected localization properties of the structure, and the calculation and analysis of the waveguide dispersion relations. Afterwards, some time will be needed in order to implement the GF methods required in order to investigate in more detail the transport properties of the proposed structures. Once the methods are implemented (M2), I will use them to investigate the transport properties of the GAL waveguides. M3 is thus a confirmation of the wave guiding properties using GF methods, and determination of a figure of merit characterizing the guiding efficiency and its dependence on the parameters of the waveguide. Completion of M3 coincides with my relocation to DTU, where experimental work on graphene antidot lattices is taking place. I thus plan to arrive with results on the waveguide structures that will hopefully be useful for the experimental activities taking place there. I will then move on to study B using the DE approach in order to obtain analytical expressions for the transport coefficients through a mass barrier (M4). The GF methods will then be adapted to the mass barrier and numerical calculations of a more realistic model of the mass barrier will be carried out. This will terminate in M5, an evaluation of the range of validity of the results of M4 as well as an analysis of the usefulness of the mass barrier for transistor devices. I plan to visit Aalto University as a visiting researcher during this activity, to benefit from their expertise on electron

5 Controlling the carriers in graphene via antidot lattices. Project description Page 5 of 6 transport in graphene. I expect further visits later to be arranged if a more permanent collaboration emerges. Upon completion of M5, I will move on to research line C. I plan to start by studying the general properties of gapped graphene in magnetic fields, using the methods outlined above, and then move on to studying the effects of a magnetic field on the transport in GAL waveguides. I expect to obtain analytical results for the magneto-optical properties of gapped graphene and obtain preliminary results on the use of GAL waveguides for spintronics (M6). Due to the overlap between methods for A-C the exact time schedule is of course highly flexible and will be adapted to take into account results from other researchers as well as potential unexpected results from our own studies. Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Research A using TB (M1) AAU AAU Implement GF methods (M2) AAU Research A using GF (M3) AAU Research B using DE (M4) DTU Research B using GF (M5) Aalto Research C (M6) DTU DTU My qualifications and scientific contributions I have a strong background in theoretical solid state physics, with a Master Thesis on the topic of solid state quantum computing and a Ph.D. thesis covering diverse aspects of solid state physics, such as slow light in photonic crystals (4 publications), - spin qubits in nanostructures (5 publications), and graphene antidot lattices (2 publications). As one of the authors of the original GAL proposal I have had the opportunity of being part of this sub-field of graphene research right from the beginning. With regards to the methods outlined in the research plan below, I have extensive experience with tight-binding models (TB), both in the field of GALs but also as part of research on the piezoresistivity of silicon (2 publications), where development of the TB model was a major cornerstone of the project. My background in photonic crystal structures provides me with knowledge that I expect will be very useful for studies of the proposed waveguide structures. Publication of results I expect each of the milestones M1 and M3-M6 to result in at least one journal publication in highprofile journals such as the Physical Review series. Furthermore, several annual conferences feature graphene as a prominent topic, so I plan to submit conference contributions to two conferences per year. This will ensure that our results are presented to a wide audience, while also serving as an opportunity for initiating international collaborations. Graphene has received a lot of attention in the popular press lately, so I will pursuit the possibility of popular accounts in, e.g., Ingeniøren of our research as well. Ethical aspects I have not identified any ethical aspects of the research.

6 Controlling the carriers in graphene via antidot lattices. Project description Page 6 of 6 Bibliography [1] Electric field effect in atomically thin carbon films, K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva and A.A. Firsov, Science 308, 666 (2004). [2] The electronic properties of graphene, A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov and A.K. Geim, Reviews of Modern Physics 81, 109 (2009). [3] 100-GHz transistors from wafer-scale epitaxial graphene, Y.-M. Lin, C. Dimitrakopoulos, K.A. Jenkins, D.B. Farmer, H.-Y. Chiu, A. Grill and Ph. Avouris, Science 327, 662 (2010). [4] Roll-to-roll production of 30-inch graphene films for transparent electrodes, S. Bae, H. Kim, Y. Lee, X. Xu, J.- S. Park, Y. Zheng, J. Balakrishnan, T. Lei, H.R. Kim, Y. I. Song, Y.-J. Kim, K.S. Kim, B. Özyilmaz, J.-H. Ahn, B.H. Hong and S. Iijima, Nature Nanotechnology 5, 574 (2010). [5] Giant intrinsic carrier mobilities in graphene and its bilayer, S.V. Morozov, K.S. Novoselov, M.I. Katsnelson, F. Schedin, D.C. Elias, J.A. Jaszczak and A.K. Geim, Physical Review Letters 100, (2008). [6] Graphene Antidot Lattices Designed Defects and Spin Qubits, T.G. Pedersen, C. Flindt, J. Pedersen, N.A. Mortensen and A.-P. Jauho, Physical Review Letters 100, (2008). [7] Electronic properties of graphene antidot lattices, J.A. Fürst, J.G. Pedersen, C. Flindt, N.A. Mortensen, M. Brandbyge, T.G. Pedersen and A.-P. Jauho, New Journal of Physics 11, (2009). [8] Unconventional Integer Quantum Hall Effect in Graphene, V.P. Gusynin and S.G. Sharapov, Physical Review Letters 95, (2005). [9] Chiral tunneling and the Klein paradox in graphene, M.I. Katsnelson, K.S. Novoselov and A.K. Geim, Nature Physics 2, 620 (2006). [10] Bandgap opening in graphene induced by patterened hydrogen - adsorption, R. Balog et al., Nature Materials 9, 315 (2010). [11] Graphene nanomesh, J. Bai, X. Zhong, S. Jiang, Y. Huang and X. Duan, Nature Nanotechnology 5, 190 (2010). [12] Fabrication and characterization of large-area, semiconducting nanoperforated graphene materials, M. Kim, N.S. Safron, E. Han, M.S. Arnold and P. Gopalan, Nano Letters 10, 1125 (2010). [13] An extended defect in graphene as a metallic wire, J. Lahiri, Y. Lin, P. Bozkurt, I.I. Oleynik and M. Batzill, Nature Nanotechnology 5, 326 (2010). [14] Gate-controlled guiding of electrons in graphene, J.R. Williams, T. Low, M.S. Lundstrom and C.M. Marcus, Nature Nanotechnology, advanced online publication 13 February 2011, doi: /nnano [15] Energy levels and wave functions of Bloch electrons in rational and irrational magnetic fields, D.R. Hofstadter, Physical Review B 14, 2239 (1976); Butterfly-like spectra and collective modes of antidot superlattices in magnetic fields, E. Anisimovas and P. Johansson, Physical Review B 60, 7744 (1999). [16] Optical response and excitons in gapped graphene, T.G. Pedersen, A.-P. Jauho and K. Pedersen, Physical Review B 79, (2009). [17] Electronic Transport in Mesoscopic Systems, S. Datta, Cambridge University Press (1995).