NON-EQUILIBRIUM CURRENT FLUCTUATIONS IN GRAPHENE AThesis Presented to The Academic Faculty by Alexander David Wiener In Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the School of Physics Georgia Institute of Technology May 2013 Copyright c 2013 by Alexander David Wiener NON-EQUILIBRIUM CURRENT FLUCTUATIONS IN GRAPHENE Approved by: Professor Markus Kindermann, Professor Carlos Sa de Melo Advisor School of Physics School of Physics Georgia Institute of Technology Georgia Institute of Technology Professor Walter de Heer Professor Angelo Bongiorno School of Physics School of Chemistry Georgia Institute of Technology Georgia Institute of Technology Professor Phillip N. First Date Approved: 5 December 2012 School of Physics Georgia Institute of Technology For my teachers “Bitter are the roots of study, but how sweet their fruit.” Aristotle iii ACKNOWLEDGEMENTS More people deserve acknowledgement for their assistance in this work than I could recognize in one thesis. Besides my parents, who were an endless source of support emotionally, intellectually and occasionally financially, I wish to thank my friends Brett Ashenfelter and Lillian MacLeod for their love and encouragement throughout this long process. I also owe a debt to the sta↵and to all of my friends in the School of Physics. In particular, I thank Ed Greco, Daniel Borrero, Danny Caballero, Adam Perkins, Lee Miller, Chris Malec, Jeremy Hicks and anyone else I’ve forgotten for their companionship and for thoughtful discussions on many topics. I am especially thankful for the collaboration and commiseration of Seungjoo Nah, who’s influence throughout this process has been invaluable. This work would not have been com- pleted without the intellectual and emotional support of all of these people, and I am grateful for the role they have played in this period of my life. Finally, I thank my teachers, past and present, for sharing their unique understanding of the world and for encouraging me to form my own. In particular, this work would not have been possible without the guidance of my advisor Markus Kindermann, or the committee members who generously o↵ered their time in my assistance. iv TABLE OF CONTENTS DEDICATION .................................. iii ACKNOWLEDGEMENTS .......................... iv LIST OF TABLES ............................... vii LIST OF FIGURES .............................. viii SUMMARY .................................... xii IINTRODUCTION............................. 1 II THE ELECTRONIC STRUCTURE OF GRAPHENE ...... 8 2.1 Lattice structure and dispersion relation . 8 2.2 MasslessDiracfermions. 12 2.3 Cyclotron mass, wave functions and helicity . 14 2.4 Klein tunneling . 16 III NOISE IN MESOSCOPIC CONDUCTORS ............. 21 3.1 Thermal noise . 21 3.2 Shot noise . 23 3.2.1 Single particle shot noise . 24 3.2.2 The two-terminal Landauer-B¨uttiker formula . 27 3.3 Shot noise on graphene . 33 3.3.1 Theory . 33 3.3.2 Experiment . 37 IV UNAMBIGUOUS SIGNATURES OF EVANESCENT MODE TRANS- PORT ON GRAPHENE ......................... 39 4.1 Evanescent mode transport through two evanescent strips in series . 40 4.1.1 Wave functions and the transmission probability . 41 4.1.2 The universal limit . 43 4.1.3 Digression on the probability distribution . 48 4.1.4 Voltage induced signatures of evanescent mode transport . 50 v 4.2 Evanescent mode transport in a graphene superlattice . 53 4.2.1 Transmission through a cascade of evanescent strips in series 54 4.2.2 The long superlattice limit . 57 4.2.3 Analyticresults ......................... 58 4.3 Discussion and Conclusions . 62 VDEPENDENCEOFTHEFANOFACTORONTHECHEMICAL POTENTIAL IN GRAPHENE NANO-RIBBONS ......... 64 5.1 Calculation of the potential . 65 5.1.1 Model . 67 5.1.2 The conformal map . 68 5.1.3 Corrections due to screening by the graphene sheet . 71 5.2 Transport calculations . 72 5.2.1 Analytic calculation of transmission probability . 72 5.2.2 Analyticresults ......................... 77 5.3 Numericalcalculation . 79 5.3.1 Digression on an alternate extraction method . 81 5.4 Discussion and Conclusions . 82 VI CONCLUSION ............................... 83 APPENDIX A — GRAPHENE BOUNDARY CONDITIONS .. 86 REFERENCES .................................. 89 VITA ........................................ 103 vi LIST OF TABLES 1Formofthec functions appearing in the transmission probability Eq. (90) for real ( )andcomplex( )valuesofk . When k is com- < = R R plex, it is assumed to be of the form ikR where kR is a real number. Hence, all references to kR in the table assume a real number, and the complex nature of the wave number is already reflected in the form of the coefficients as they appear here. 51 vii LIST OF FIGURES 1Carbonallotropesindi↵erentdimensions.Graphene(2D)canbecut and folded into fullerenes (0D) and carbon nanotubes (1D). Graphite (3D) consists of vertically stacked layers of 2D graphene. (Adapted from Geim et al. [68]) . 2 2Thehexagonalgraphenelattice(a)withtwocarbonatomsperunit cell, labelled A and B.Thelatticevectorsare ~a and the nearest { i} neighbor vectors are ~δ .ThecorrespondingBrillouinzone(b)isalso { i} hexagonal, with reciprocal lattice vectors ~b and Dirac cones located { i} at the K and K0 points. (Adapted from Castro Neto et al. [42]) . 9 3ElectronicdispersionrelationE~k for monolayer graphene, plotted in units of t with t =2.7eVandt0 = 0.2t eV. The spectrum is linear in the neighborhoods of the six corners− of the hexagonal Brillouin zone. One such region is magnified to the right of the plot. All but two of these corners, known as the K and K0 points, are equivalent and can be connected by reciprocal lattice vectors. (Adapted from Castro Neto et al. [42]) . 11 4(a)AparticlewithenergyE is incident on a potential barrier of height V0 and width L.Classicaltrajectoriesareconfinedtotheregionto the left of the potential, marked with a dark gray rectangle. Quantum mechanical particles can tunnel through the potential barrier and be transmitted, but the transmission is exponentially damped with in- creasing potential height or width. Particles obeying a Dirac equation tunnel easily through such barriers, with a probability increasing to unity for normal incidence. (b) Overhead view of a massless Dirac fermion incident on a potential barrier in graphene. The particle has the wave function I left of the potential and an incident angle of φ with respect to the horizontal. The wave function is II within the potential barrier region, where the trajectory is at angle ✓.Tothe right of the barrier, the wave function is given by III with angle φ. (Adapted from Castro Neto et al. [42]) . 18 5Atwo-terminalscatteringsysteminwhichamesoscopicsampleiscon- nected to two reservoirs labelled L and R.Bothreservoirsarein thermal equilibrium, and reservoir i = L, R ,attemperatureT with { } i chemical potential µi,hasincoming(outgoing)statewavefunctionsIi (Oi). (Adapted from Blanter and B¨uttiker [25]) . 28 viii 6 Agraphenefielde↵ect transistor geometry, as considered by Tworzyd lo et al.,[167]consistsofasheetofgraphenewithlengthL and width W is contacted at either end by ideal leads (gray rectangles). A gate electrode (not shown) controls the potential in the central graphene region. (Adapted from Tworzyd lo et al. [167]) . 34 7SchematicofagraphenestripofwidthW consisting of a highly doped region (gray rectangle) of length λ at voltage Vd separated by weakly doped regions of length L and R.Separategateleads(notshown) control the voltages VL and VR in the weakly doped regions. The system is contacted at either end by ideal leads (not shown) at the lead potential V`.Theelectrostaticpotentialinthestripisplotted above the schematic as a function of x.................. 41 8FanofactorF for two graphene strips as in Fig. 7 for L = R and chemical potential at the Dirac point, as a function of the length of 2 the highly doped region in units of d L .Thecurveiscalculatedfrom Eq. (53) in the limit N and| is| shown magnified in the inset for small values of the lead!1 thickness. The Fano factor approaches 0.33... as λ 0, in agreement with the calculations of Tworzyd lo et al. [167] For large! values of the length of the highly doped region, the Fano factor approaches 0.25...,inagreementwiththeanalyticcalculations presented in the text. 44 9LengthratiodependenceoftheFanofactorfortwographenestrips as in Fig. 7 with chemical potential at the Dirac point. The curve is calculated from the integrals given in Eq. (74) and Eq. (76) in the 2 continuum limit with λ d L . The Fano factor shows a minimum of 0.25 ... for the symmetric| case| (R/L =1)andtendstoward0.33 ... as R/L 0orR/L ,inagreementwiththeresultsofTworzyd lo et al. [167]! . .!1 . 47 10 The Fano factor for two graphene strips as in Fig. 7 for L = R and λ L2 as a function of gate voltage V in the right graphene | d| R region, measured in units of vF/L.Gateleadsfixthechemicalpotential in the left evanescent region to the Dirac point (VL =0).TheFano factor is 0.25 ...when VR is at the Dirac point, as previously calculated, and it approaches 0.33 ... as V v /L................ 52 R F 11 Schematic of a segment of a graphene superlattice of width W ,con- sisting of evanescent graphene regions of length L separated by doped regions (gray rectangles) of length λ at the potential Vd.Separategate leads (not shown) fix the voltages in the graphene regions to the Dirac point, V =0.Thesuperlatticeiscontactedateitherendbyleadsat potential V` (not shown). The superlattice potential is plotted above the schematic as a function of x...................... 53 ix