Properties of the short period CoRoT-planet population I Theoretical planetary mass spectra

Physical properties of the shallow sediments in late Pleistocene formations,Ursa Basin,Gulf of Mexico,and their implications for generation and preservation of shallow overpressuresN.T.T.Binh a ,*,T.Tokunaga a ,1,T.Nakamura b ,2,K.Kozumi b ,2,M.Nakajima b ,2,M.Kubota b ,2,H.Kameya c ,3,M.Taniue c ,3aDepartment of Environment Systems,School of Frontier Sciences,University of Tokyo,5-1-5Kashiwanoha,Kashiwa-shi,Chiba 277-8563,Japan bGeotechnical Department,Dia Consultants,Saitama,Japan cCore Laboratory,Oyo Corporation,Niigata,Japana r t i c l e i n f oArticle history:Received 14September 2007Received in revised form 20January 2009Accepted 23January 2009Available online 31January 2009Keywords:Basin modellingShallow overpressure Fluid flowDeepwater environmenta b s t r a c tUnderstanding the evolution of abnormally high fluid pressures within sedimentary formations is critical for analysing hydrogeological processes and assessing drilling risks.We have constructed a two-dimensional basin model and have performed numerical simulations to increase the understanding of the history of fluid flow and shallow overpressures in the Pleistocene and Holocene formations in the Ursa basin,deepwater Gulf of Mexico.We measured physical properties of sediments,such as porosity and permeability,in the laboratory and estimated in situ pore pressures from preconsolidation pressures.We obtained porosity–effective stress relationships from measurements of bulk density,grain density and preconsolidation pressures in the laboratory.Porosity–effective stress relationships were also obtained from downhole density logs and measured pore pressures.The porosity–effective stress and porosity–permeability relationships obtained were applied in two-dimensional basin simulations.Results showed that high pore pressures developed shortly after sediment deposition.Peaks in pore pressure ratios were related to high sedimentation rates of mass transport deposits and the incision of the Ursa teral flows from the area where the overburden is thick towards the area where it is thin have occurred at least since 30ka.Present pore pressure and temperature distributions suggest that lateral flows play a role in re-distributing heat in the basin.Ó2009Elsevier Ltd.All rights reserved.1.IntroductionUnderstanding pore pressure regimes and fluid flow patterns in sedimentary basins and their temporal changes is important for investigating the evolution of sedimentary basins,the stability of slopes,and related geodynamics.For example,lateral fluid flows in a sedimentary formation are controlled by pore pressure gradients,sedimentation rates,and permeability distribution (Bethke,1986).Flemings et al.(2005)speculated that focusing of fluid flows may result in slope instability on continental slopes.In 2005,the Inte-grated Ocean Drilling Program (IODP)conducted Expedition 308tostudy shallow overpressure and fluid flow in the Ursa region,continental slope of the Gulf of Mexico.Eight holes were drilled at three well sites U1324,U1323and U1322.The wells were logged and in situ measurements were made.Geopressured sediments from Pleistocene and Holocene formations were found in these wells (Expedition 308Scientists,2005;Myers et al.,2007).In this area,Byrd et al.(1996),Eaton (1999),Ostermeier et al.(2002),and Flemings et al.(2005)described the existence of shallow water flow phenomena,and discussed the problems encountered with shallow water flows.Shallow water flows are associated with a variety of drilling problems and seafloor damage,including uncontrolled flow of sands in the annulus of the borehole being drilled,borehole washouts,craters,and cracks on the seafloor (Alberty et al.,1997;Alberty,2000;Faul et al.,2000;Ostermeier et al.,2002).They are considered to be the cause of major problems for the oil and gas industry working in the Mark-Ursa region (Alberty,2000).The loss of many wells in the Ursa Development Project in the Mississippi Canyon Block 810was an extreme example of severe damage due to violent shallow water flows (Furlow,1998).*Corresponding author.Department of Earth Sciences,Durham University,Durham DH13LE,United Kingdom.Tel.:þ44(0)1913343972;fax:þ44(0)1913342301.E-mail address:binh.nguyen@ (N.T.T.Binh).1Tel.:þ81471364708;fax:þ81471364709.2Tel.:þ81486543011;fax:þ81486543833.3Tel.:þ81252745656;fax:þ81252716765.Contents lists available at ScienceDirectMarine and Petroleum Geologyjournal homepage :www.else /locate /marpetgeo0264-8172/$–see front matter Ó2009Elsevier Ltd.All rights reserved.doi:10.1016/j.marpetgeo.2009.01.018Marine and Petroleum Geology 26(2009)474–486Recently,Sawyer et al.(2007a)described the lithology of the main depositional elements in the Mars-Ursa region and interpreted the geological evolution of the area for the past70ky based on high resolution3D seismic data and well log data.Long et al.(2007)and Flemings et al.(2006,2008)presented pore pressure penetrometer measurements made during IODP Expedition308and documented vertical and lateral variation in overpressure at Sites U1322and U1324.They showed that overpressures have reached60%of the hydrostatic effective stress at Site U1322and70%at Site U1324. Sawyer et al.(2007b)and Dugan et al.(2007)integrated physical properties of mass transport complexes(MTCs)and concluded that consolidation behaviour in MTCs is different from that in the sedi-ments encasing the MTCs.Furthermore,low permeability in MTCs precludes drainage of overpressure in the Ursa region.In this study,we conducted numerical modelling,supported by data from laboratory experiments using samples obtained from IODP Expedition308,to examine lateralfluidflow and pore pressure evolution in the Pleistocene and Holocene sedimentary sections of the Ursa basin.In the geotechnical laboratory,we measured physical properties of sediments such as porosity and permeability and estimated in situ pore pressures from preconsolidation pressures. Then porosity–effective stress relationships were constructed from IODP results and the estimated in situ pore pressures.The porosity–effective stress and porosity–permeability relationships obtained were applied in two-dimensional basin simulations.Two-dimensional modelling cases presented here include the effects of both the uneven sedimentation rates along the cross-section and the ancient channel activities on the hydrogeological system.The modelling results can help to improve the under-standing of the history offluidflow and overpressure in the study area.2.Geological settingIn this section,we summarize what is known from previous research results about the lithology and deposition of sediments in the Ursa basin.This information will be used to construct the geological model for our basin modelling study which will be dis-cussed in Section4.The Ursa Basin is located about200km southeast of New Orleans on the continental slope of the Gulf of Mexico(Fig.1).It is a salt-withdrawal mini-basin with water depth in the range800–1400m,and with sediments originating from the Mississippi River system(Expedition308Scientists,2005).This study focuses on the Late Pleistocene to Holocene sedimentary section of the Ursa Basin, i.e.,from the base of the Blue Unit to the seafloor(Fig.2).According to Sawyer et al.(2007a)and Expedition308Scientists (2005),sediments deposited from the base of the Blue Unit to the seafloor can be divided intofive units from bottom to top:the Blue Unit,the Ursa Canyon channel-levee system,the Southwest Pass Canyon channel-levee system,mass transport deposits,and the distal deposits and hemipelagic drape(Fig.2).The Blue Unit is composed of interbedded sands and clays with a maximum two-way travel time of250–300ms(Sawyer et al.,2007a).In the study area,the Blue Unit thins towards west due to the incision by the Southwest Pass and Ursa Canyon systems(Fig.2)(Sawyer et al., 2007a).The Blue Unit was most likely deposited during the MIS Stage4eustatic sea level fall and at the time of an eastward shift in the drainage pattern of the Mississippi River(Piggott and Pulham, 1993;Winker and Booth,2000;Ruddiman,2001).The base of the Blue Unit was interpreted to have been deposited at around68ka based on biostratigraphic data from nannofossils assemblages (Winker and Booth,2000).Note that Urgeles et al.(2007)recently suggested the age of the base of the Blue Unit to be89ka,but without giving a reason.Thus,we assumed the age of deposition to be68ka. The Ursa Canyon runs from the northwest to the southeast(Sawyer et al.,2007a).The channel-levee system is composed of a channelfill, channel-margin slides,and levees(Sawyer et al.,2007a).The channel margin and levees consist of silty clays while the channelfill consists of sands and silty clays(Sawyer et al.,2007a).The Ursa Canyon incised the Blue Unit and channel-margin slides play as hydraulic barrier within the Blue Unit(Fig.2)(Sawyer et al.,2007a).Fig.1.Location of the study area and the cross-section used for two-dimensional basin modelling.Contours show water depth.The names of the blocks are also shown.After Sawyer et al.(2007b).N.T.T.Binh et al./Marine and Petroleum Geology26(2009)474–486475The Southwest Pass Canyon system is younger than,and lies to the west of the Ursa Canyon system (Fig.2).It eroded much of the western levee of the Ursa channel and completely buried its fill and eastern levee.It has similar characteristics as the Ursa Canyon system but is even larger (Sawyer et al.,2007a ).The Southwest Pass Canyon also contains a belt of rotated channel-margin slides (Fig.2),which is up to 5.5km wide.The canyon fill itself is approximately 1.3–1.6km wide (Sawyer et al.,2007a ).Mass transport deposits (MTDs)are situated within the mud-rich levee deposits above the Blue Unit in the studied area (Fig.2)(Sawyer et al.,2007a ).MTDs consist of silty clays (Sawyer et al.,2007b;Dugan et al.,2007).The first mass transport deposit is located on the eastern levee of the Ursa Canyon channel-levee system,and the second mass transport deposit is located within the eastern levee of the Southwest Pass Canyon channel-levee system (Fig.2b)(Sawyer et al.,2007a ).Distal deposits and hemipelagic drape overlie the MTDs.The distal deposits are composed primarily of olive-green and reddish brown clay interbedded with black clay and are capped by hemi-pelagic clay that is rich in nannofossils and foraminifera (Expedi-tion 308Scientists,2005).3.Physical properties of the soft sedimentsSoft sediment samples obtained from IODP Expedition 308were analysed,focusing mainly on porosity–permeability relationships.Index properties of the samples used for consolidation tests are shown in Table 1.Grain densities were in the range 2760–2900kg/m 3and were found to be higher than those measured onboard the ship (2667–2780kg/m 3)(Expedition 308Scientists,2005).We fol-lowed JIS A 1202-1999(Japanese Industrial Standards Committee,1999)and JGS 0111-2000(Japanese Geotechnical Society,2000a )for the measurements of the grain densities of the samples,so the reasons for the discrepancy between the density measurements are unknown.However,the effects of the differences of the grain densities on the calculated porosities were rather small,i.e.,less than 2%,and hence were not significant,at least for this study.3.1.Porosity–permeability relationships3.1.1.Evaluation of permeability from 1-D consolidation testsA total of fourteen samples were tested either by using an incremental loading oedometer (IL)or aconstant-rate-of-strainFig.2.(a)Seismic line AA 0and (b)interpreted cross-section.The dotted blue box shows the modelled area.Modified from Sawyer et al.(2007a)(For interpretation of the references to colour in this figure legend,the reader is referred to the web version of this article).N.T.T.Binh et al./Marine and Petroleum Geology 26(2009)474–486476oedometer (CRS).Both the IL and CRS oedometers had a loading capability up to 10MPa.Prior to testing,the samples were removed from the sealed core liner and trimmed to a height of 2.5cm and a diameter of 6.0cm.Once trimmed,the wet mass of the sample was recorded before the sample was transferred into the consolidation cell.The cells for both oedometers were made of stainless steel so that only vertical sediment deformation could occur.The sample was set into the cell and placed between two porous stones.Saturated filter papers were used to separate the sample from the porous stones and to prevent fine-grained sediment particles from blocking the drainage paths through the porous stones.After the cell was assembled,it was transferred to the oedometer.Side friction was minimized using silicone grease.General sample preparation and testing procedures followed the guidelines set out by JIS A 1217-2000and JIS A 1227-2000(Japanese Industrial Standards Committee,2000b,c )and JGS 0411-2000and JGS 0412-2000(Japanese Geotechnical Society,2000c,d ).In the IL tests,the cells were submerged in saline water to keep the samples fully water-saturated.The samples were allowed to settle for 24h after the application of each incremental load.Time,loading pressure,and the change in specimen height were measured.The end of primary consolidation was evaluated using the square root of time method proposed by Taylor (1948).Then,the coefficient of vertical consolidation (C v )and the coefficient of volume compressibility (m v )were indirectly evaluated based on the one-dimensional Terzaghi’s consolidation theory (Terzaghi,1943;Lambe and Whitman,1979).In the CRS tests,backpressure of 200kPa was applied to the samples for 24h to ensure they were fully water-saturated before loading with a constant axial strain rate of 0.03%/min.Backpressure was kept constant at 200kPa until the end of the tests.Time,axial load,the change in specimen height,and excess pore pressure at the base of the sample were measured.C v and m v were calculated using linear finite strain theory (Smith and Wahls,1969).Hydraulic conductivity (K )was estimated from C v ,m v and density of water (r w )byK ¼C v m v r w g(1)where g is gravitational acceleration (Lambe and Whitman,1979).In the equation (1),the unit of K is m/s,C v is m 2/s,m v is 1/Pa,g is m/s 2,r w is kg/m 3.Intrinsic permeability can be obtained from hydraulic conduc-tivity through the following relationship (Hubbert,1940)K ¼k r w gm(2)where k is intrinsic permeability (m 2)and m is viscosity of pore fluid (Pa s).The intrinsic permeability reported here is for seawater with density and viscosity to be 1025kg/m 3and 0.963Â10À3Pa s,respectively.3.1.2.ResultsThe permeability values obtained from fourteen consolidation tests ranged from 5Â10À17to 1Â10À19m 2.For comparison,the results are plotted in Figs.3and 4with data from a previous study in the Gulf of Mexico (Bryant et al.,1975).Bryant et al.(1975)classified their measurements based on the grain size of the samples.For grain size analysis,we used a hydrometer and fol-lowed the method JIS A 1204-2000(Japanese Industrial Standards Committee,2000a )and JGS 0131-2000(Japanese Geotechnical Society,2000b ).We also estimated shale volume of the sediments from gamma ray log data.The analysed samples from MTDs and silty clays have shale volumes in the range 59–78%(Table 1).Thus,the data obtained for MTDs and silty clays were compared with group 2of Bryant et al.(1975),which had clay content in the range 60–80%(Fig.3).Since the gamma ray data were not good enough at very shallow depths near the seafloor,we did not use these datatoTable 11E-191E-181E-171E-161E-15PorosityP e r m e a b i l i t y (m 2)Silty clays MTDs Group 2 (Bryant et al., 1975)Fig. 3.Porosity and permeability obtained from IL tests and CRS tests for mass transport deposits (open triangles)and silty clays (open squares)in the Ursa Basin,and the published results in the Gulf of Mexico (black circles)(Bryant et al.,1975).The dotted line and the black line were modelled porosity–vertical permeability relation-ship used for MTDs and for silty clays,respectively.N.T.T.Binh et al./Marine and Petroleum Geology 26(2009)474–486477calculate shale volumes for hemipelagic clay.Based on visual core description and smear slide analyses,hemipelagic clay samples are mainly composed of clay (Expedition 308Scientists,2005).Therefore,we assume that the data for hemipelagic clays should be comparable with the group 1samples of Bryant et al.(1975)which had clay content higher than 80%(Fig.4).Based on these compar-isons,the results from this study were confirmed to be consistent with data from the previous study.An IL test and a CRS test were conducted with specimens from the same sample U1324B-19H-CC for comparison.The intrinsic permeability values obtained from the CRS test were about 1.17–1.2times higher than those obtained from the IL test for the same porosity (Fig.5).However,the slopes of the permeability–porosity lines on the semi logarithmic plots were parallel (Fig.5),suggesting that the trend of permeability reduction by mechanical consoli-dation can be estimated from this slope.We used the equationk ¼A exp ðB f Þ(3)where A and B are lithology-dependent constants to express rela-tionships between permeability (k )and porosity (4)(Table 2).We used the porosity–permeability relationships shown in Table 2for the two-dimensional simulation described in the next section.3.2.Preconsolidation pressure measurementsPreconsolidation pressure is usually interpreted to be the maximum effective overburden stress that sediments have expe-rienced (Casagrande,1936;Lambe and Whitman,1979).Knowing the preconsolidation pressure (P c )and in situ overburden stress (S v ),we can estimate the in situ pore pressure (P p )through the following equation assuming that the sediments are normally consolidated.P p ¼S v ÀP c (4)In this study,preconsolidation pressures of the sediments were determined by using Casagrande (1936)’s method to analyse the IL and CRS test data.In situ pore pressures calculated from P c are shown in Table 3.The values obtained are consistent with in situ measurements (Flemings et al.,2008)(Fig.6)except for the point at 1566mbsl (metres below sea level)at site U1324.Also,in situ pore pressuremeasurements at 1316.5mbsl and 1366.5mbsl seemed to be lower than the expected pore pressures from preconsolidation pressures.The in situ measurements were conducted with good coupling between the measurement device and the formation at these depths (Flemings et al.,2008).These differences could be explained by the declines of pore pressures due to recent drilling activity,as suggested at the site U1322by Flemings et al.(2008)even though the precise reason is quite difficult to state.Based on the data shown in Fig.6,we consider that the sections between seafloor and 1127mbsl at site U1324and between seafloor and 1375.5mbsl at site U1322are normally consolidated.3.3.Porosity–effective stress relationshipsPorosity–effective stress relationships were constructed for normal consolidation sections based on porosities calculated from density logs at sites U1324and U1322(Expedition 308Scientists,2005)(Fig.7).Overburden stress was estimated from density logs and water depths.In hemipelagic clay sections,porosities decrease quickly from 80%to 55%(Fig.7(a)).For mass transport deposits and silty clays,porosities can be modelled by porosity–effective stress relation-ships of the Athy (1930)type,with different parameters for the silty clays and mass transport deposits (Fig.7(b),(c)).4.Two-dimensional pore pressure and fluid flow modelling Basin modelling is one of the methods which can be used to predict pore pressure as well as fluid flow in a sedimentary basin(e.g.,Welte and Yalcin,1988;Burrus et al.,1992;Dore´et al.,1993;Hermanrud,1993;Neuzil,2003).Fully coupled and integrated basin simulators provide information on pore pressures,porosities,temperatures and fluid flow patterns through time.We conducted two-dimensional simulation using SIGMA-2D (Okui et al.,1996,1998).1.E-191.E-181.E-171.E-16PorosityP e r m e a b i l i t y (m 2)Fig.5.Porosity–permeability relationships obtained from the IL test (black squares)andCRS test (open squares)on the sample 1324B-19H-CC.The lines were obtained by least square fits.Table 2Constants obtained for vertical permeability–porosity relationships.See text for 1.E-191.E-181.E-171.E-161.E-151.E-14PorosityP e r m e a b i l i t y (m 2)Fig.4.Porosity and permeability obtained from hemipelagic clays in the Ursa Basin using IL consolidation tests (open squares),and the published results in the Gulf of Mexico (black circles)(Bryant et al.,1975).The black line was used as the modelled porosity–vertical permeability relationship for hemipelagic clays.N.T.T.Binh et al./Marine and Petroleum Geology 26(2009)474–4864784.1.Model construction:geological model,lithology model,and physical propertiesThe cross-section was chosen to be parallel to the local slope direction in the study area (Fig.1).Hence,it is considered to contain the principal direction of fluid flow related to sedimentation and topography.There are three IODP Expedition 308well sites along this section,i.e.,U1324,U1323and U1322,and data from these wells can be used for calibration.A lithology model was made based on previous studies in the area (Expedition 308Scientists,2005;Sawyer et al.,2007a ).Because of the lack of detailed data,it was necessary to simplify the lithology distribution.Four types of lithology,hemipelagic clays,mass transport deposits,sands,and silty clays,were used to model the sediments in the study area (Fig.8).Since sheet sands in the Blue Unit have been considered to be the source of shallow water flows in the Ursa area (Eaton,1999;Ostermeier et al.,2002;Winker and Shipp,2002),the distribution of sheet sands in the Blue Unit and the Ursa channel fill sands were modelled in detail,whereas the other lithologies were rather simplified.The modelled cross-section was part of cross-section AA 0in Fig.2(b),and was divided into 28columns,26layers and one erosion event (Fig.8).Based on previous research results (Expedition 308Scientists,2005;Sawyer et al.,2007a ),the distribution of the sediments was set as follows (Table 4).From seafloor to seismic horizon S10,the lithology is hemipelagic clays (layers 1and 2).From seismic horizon S10to seismic horizon S20,the lithology is silty clays and MTDs (layers 3–6).From seismic horizon S20to seismic horizon S30,the lithology is MTDs (layers 7–9).From seismic horizon S30to seismic horizon S60,the lithology is silty clays and MTDs with small amounts of sand (layers 10–13).The Ursa channel (layers 14–17)was modelled as sands and silty clays with the lithology distribu-tion being based on the seismic interpretation (Fig.2)(Sawyer et al.,2007a ).The Blue Unit includes sands and interbedded silty clays (layers 18to 24).Layers 25and 26are silty clays below the Blue Unit.The simulation was run for a total period of 85ky.Erosion by the incision of the Ursa Canyon was considered (Fig.8).Based on previous studies on sea level cycles in the Mis-sissippi river depositional system and in the Gulf of Mexico (Ruddiman,2001;LoDico et al.,2006),we considered that erosion had occurred after the deposition of the Blue Unit and before the sea level rise at 58ka.The timing of erosion was set to lie between 60ka and 58ka (Table 4).The eroded thickness was modelled to be between 50and 250m,assuming that the thickness of the Blue Unit had been constant throughout this cross-section before erosion occurred.During erosion,the palaeo-water depth in the eroded area increased by an amount equals to the eroded thickness.The relationships between porosity and vertical effective stress and between porosity and permeability for hemipelagic clays,MTDs and silty clays obtained from our experimental study (Figs.3,4and 7and Table 2)were used as input data for basin simulation.We also used published relationships between porosity and vertical effective stress and between porosity and permeability for sands in the Gulf of Mexico (Aniekwena et al.,2003)(Fig.9).Anisotropy in permeability was considered.Freeze and Cherry (1979)suggested that clays and shales show horizontal toverticalTable 3aU1324U1322Pressure (MPa)D e p t h (m b s l )Overburden stressCalculated from PcPressure (MPa)Fig.6.Pore pressures calculated from preconsolidation pressures (filled squares)(a)at site U1324and (b)at site U1322.The in situ measurements (Flemings et al.,2008)are also shown by open triangles.The units mbsl are metres below sea level.N.T.T.Binh et al./Marine and Petroleum Geology 26(2009)474–486479anisotropy ratio in the range3:1–10:1.In this study,the ratios between horizontal permeability and vertical permeability were chosen to be10:1for hemipelagic clays and mass transport deposits,and to be3:1for silty clays.For sands,since the perme-ability in the horizontal direction was found to be nearly equal to,or only slightly larger than that in the vertical direction(Hatanaka et al.,1997),the horizontal to vertical anisotropy ratio was chosen to be1:1.Other physical properties for each lithology required for the two-dimensional basin modelling include grain density,matrix thermal conductivity,and heat capacity.These data were estimated from published results(Expedition308Scientists,2005;Rossane et al.,2004;Waples and Waples,2004)(Table5).4.2.Boundary and initial conditionsGeological information can be used to select the appropriate boundary conditions.For this purpose,the distribution of faults in the layers of higher hydraulic conductivity was studied.Because the majority of sediments are clays and silty clays,and fault displacements tend to producefine-grained material in general(e.g.,Ferrill and Morris,2001),all faults are assumed to act as sealing faults.Faults are located at the eastern boundary of the cross-section(Fig.2(b)).At the western boundary,the Blue Unit is completely eroded by the incision of the Southwest Pass channel (Fig.2).Thus,the body of low hydraulic conductivity in the Southwest Pass channel can be treated as a seal for westwardfluid Fig.8.The geological model used for the basin modelling.Dotted line shows erosion by the Ursa Canyon.50100150200250PorosityVerticaleffectivestress(kPa)100200300400500600700800PorosityVerticaleffectivestress(kPa)Porositya b cFig.7.Porosity–effective stress relationships for(a)hemipelagic clays,(b)MTDs,and(c)silty clays.Lines in thefigure and the equations show the modelled porosity–effective stress relationships used for two-dimensional basin modelling.N.T.T.Binh et al./Marine and Petroleum Geology26(2009)474–486480flow in the Blue Unit.Therefore,no-flow boundary conditions were assigned to both side boundaries.For the basal boundary,no fluid flow conditions were set because the base of the cross-section is composed of a mud-rich layer.The initial pore pressures at each time step were set to be the calculated pore pressures at the previous time step.For the newly added grids,hydrostatic pore pressures were applied.The initial temperatures were set to be those obtained from the previous time step for existing grids,and seawater temperature for the newly added grids.Boundary conditions for the heat flow in SIGMA-2D are specified temperatures along the upper boundary,a specified heat flux along the bottom,and no flux along side boundaries (Okui et al.,1996,1998).At the upper surface of the model,temperatureand pressure were modelled as seafloor temperature and seafloor pressure.Temperatures at the seafloor were set to be in the range 4.0–4.5 C depending on the water depth.Pressure at the seafloor was considered to be hydrostatic.4.3.Model calibrationDuring calibration,the measured data from wells,i.e.,pore pressure,temperature,and porosity,were compared with the simulated results at the well locations to optimize the parameters used in the model.Sedimentation rates were adjusted by trial-and-error to obtain reasonable agreement between simulated and observed values.The geological model used in this study was made up of 26layers (Fig.8and Table 4).In the modelled cross-section,the ages of the top of the layers 1,2,6,9,12,13,15and 19were determined based on the estimated ages of the known key horizons (Table 4)(Expedition 308Scientists,2005;Sawyer et al.,2007a ).Because other layers were situated between horizons of known ages,the ages of these layers were adjusted to lie within the constrained ranges.Through this procedure,the appropriate sedimentation rates (Table 4)were chosen to obtain good correlations between the calculated results and the measured data.Present-day heat flow at the base of the model was estimated by fitting temperatures calculated from the model and in situ measurements (Expedition 308Scientists,2005).The heat flow value obtained was 36.3mW/m 2.This value is quite low in comparison with heat flow values seen in the continental crust,which ranges from 40to 46mW/m 2(Smith and Dees,1982).Mello and Karner (1996),Nagihara and Jones (2005)and Jones et al.(2003)suggested that the rapid deposition of thick sections of young sediments in the Mississippi fan has suppressed regional isotherms,resulting in anomalously low heat flow.We assumed that the palaeo heat flows were the same as present-day heatflows.Table 4Ages and thicknesses for the modelled cross-section AA 0.S10to S80represents keyTable 5Physical properties of four types of lithology estimated from Expedition 308050010001500200025003000350040000.30.40.50.60.70.8PorosityV e r t i c a l e f f e c t i v e s t r e s s (k P a)1.0E-151.0E-141.0E-131.0E-121.0E-111.0E-100.10.20.30.40.5PorosityP e r m e a b i l i t y (m 2)abFig.9.(a)Porosity–effective stress and (b)porosity–permeability relationships for sand.From Aniekwena et al.(2003).N.T.T.Binh et al./Marine and Petroleum Geology 26(2009)474–486481。

英语语⾔学复习专题英语语⾔学复习专题A Review on English LinguisticsTiger ZhouSchool of Foreign Languages,Shanghai Dianji University⼀.Explain the following terms1.dualityDuality refers to the fact that language is a system,which consists of two sets of structures,or two levels.At the lower level,there is a structure of sounds,which are meaningless by themselves. At the higher level,there is a structure of meanings where the sounds of language can be grouped and regrouped into a large number of units of meaning./doc/6b316456657d27284b73f242336c1eb91b37336b.html plementary distributionWhen two or more than two allophones of the same phoneme do not distinguish meaning and occur in different phonetic environments,then the allophones are said to be in complementary distribution.3.categoryCategory refers to a group of linguistic items which fulfill the same of familiar functions in a particular language such as sentence,a noun phrase or a verb.4.phrase structure ruleA certain word can only co-occur with certain other words.There must be certain grammatical mechanism that ensures the appropriate positions that specifiers,heads and complements occupy in phrase structure.Such special type of grammatical mechanism that regulates the arrangement of elements that make up a phrase is called a phrase structure rule.5.homonymyHomonymy refers to the phenomenon that words having different meanings have the same form,i.e.different words are identical in sound or spelling,or in both.6.displacementDisplacement means that language can be used to refer to things which are present or not present,real or imagined matters in the past,present,or future,or in far away places.In other words,language can be used to refer to contexts removed from the immediate situations of the speaker.This property provides speakers with an opportunity to talk about a wide range of things, free from barriers caused by separation in time or place.7.minimal pairsWhen two different forms are identical in every way except for one sound segment which occurs in the same place in the strings,the two sound combinations are said to form a minimal pair.8.deep structureIt is the structure that is formed by the XP rule in accordance with the head’s sub-categorization properties and that corresponds most closely to the meaningful grouping of words.9.contextIt is generally considered as constituted by the knowledge shared by the speaker and the hearer.10.hyponymyHyponymy refers to the sense relation between a more general,more inclusive word and a more specific word.(The word which is more general in meaning is called the superordinate,and the more specific words are called its hyponyms.)⼆.Short AnswersChapter1Introduction2.What are the major branches of linguistics?What does each of them study?The major branches of linguistics are:(1)phonetics:it studies the sounds used in linguistic communication;(2)phonology:it studies how sounds are put together and used to convey meaning in communication;(3)morphology:it studies the way in which linguistic symbols representing sounds are arranged and combined to form words;(4)syntax:it studies the rules which govern how words are combined to form grammatically permissible sentences in languages;(5)semantics:it studies meaning conveyed by language;(6)pragmatics:it studies the meaning in the context of language use.3.In what basic ways does modern linguistics differ from traditional grammar?The general approach thus traditionally formed to the study of language over the years is roughly referred to as“traditional grammar.”Modern linguistics differs from traditional grammar in several basic ways.Firstly,modem linguistics is descriptive while traditional grammar is prescriptive.Second,modem linguistics regards the spoken language as primary,not the written.Traditional grammarians,on the other hand,tended to emphasize,maybe over-emphasize,the importance of the written word,partly because of its permanence.from traditional grammar also in that it does not force languages4.Is modern linguistics mainly synchronic or diachronic?Why?In modem linguistics,a synchronic approach seems to enjoy priority over a diachronic one. Because people believed that unless the various states of a language in different historical periods are successfully studied,it would be difficult to describe the changes that have taken place in its historical development.6.How is Saussure's distinction between langue and parole similar to Chomsky's distinction between competence and performance?Saussure's distinction and Chomsky's are very similar,they differ at least in that Saussure took a sociological view of language and his notion of langue is a matter of social conventions,and Chomsky looks at language from a psychological point of view and to him competence is a property of the mind of each individual.8.What are the main features of human language that have been specified by C.Hockett to show that it is essentially different from animal communication system?The main features of human language are termed design features.They include:a)ArbitrarinessLanguage is arbitrary.This means that there is no logical connection between meanings and sounds.A good example is the fact that different sounds are used to refer to the same object in different languages.b)ProductivityLanguage is productive or creative in that it makes possible the construction and interpretation of new signals by its users.This is why they can produce and understand an infinitely large number of sentences,including sentences they have never heard before.c)DualityLanguage consists of two sets of structures,or two levels.At the lower or the basic level there is a structure of sounds,which are meaningless by themselves.But the sounds of language can be grouped and regrouped into a large number of units of meaning,which are found at the higher level of the system.d)DisplacementLanguage can be used to refer to things which are present or not present,real or imagined matters in the past,present,or future,or in far-away places.In other words,language can be used to refer to contexts removed from the immediate situations of the speaker.This is what “displacement”means.e)Cultural transmissionWhile human capacity能⼒for language has a genetic basis,i.e.,we were all born with the ability to acquire language,the details of any language system are not genetically transmitted,but instead have to be taught and learned.Chapter2Phonology2.What is voicing and how is it caused?Voicing is a quality of speech sounds and a feature of all vowels and some consonants in English. It is caused by the vibration of the vocal cords.3.Explain with examples how broad transcription and narrow transcription differ?The transcription with letter-symbols only is called broad transcription.This is the transcription normally used in dictionaries and teaching textbooks for general purposes.The latter,i.e.the transcription with letter-symbols together with the diacritics is called narrow transcription.This is the transcription needed and used by the phoneticians in their study of speech sounds.With the help of the diacritics they can faithfully represent as much of the finedetails as it is necessary for their purpose.The example is the consonant[p].We all know that[p]is pronounced differently in the two words pit and spit.In the word pit,the sound[p]is pronounced with a strong puff of air,but in spit the puff of air is withheld to some extent.In the case of pit,the[p]sound is said to be aspirated and in the case ofspit,the[p]sound is unaspirated.This difference is not shown in broad transcription, but in narrow transcription,a small raised“h”is used to show aspiration,thus pit is transcribed as [p?it]and spit is transcribed as[spit].3.What criteria are used to classify the English vowels?Vowels may be distinguished as front,central,and back according to which part of the tongue is held highest.To further distinguish members of each group,we need to apply another criterion,i.e. the openness of the mouth.Accordingly,we classify the vowels into four groups:close vowels, semi-close vowels,semi-open vowels,and open vowels.A third criterion that is often used in the classification of vowels is the shape of the lips.In English,all the front vowels and the central vowels are unrounded vowels,i.e.,without rounding the lips,and all the back vowels,with the exception of[a:],are rounded.Then theEnglish vowels can also be classified according to the length of the sound.The long vowels are all tense vowels and the short vowels are lax vowels. 4. A.Give the phonetic symbol for each of the following sound descriptions:1)voiced palatal affricate[?]2)voiceless labiodental fricative[f]3)voiced alveolar stop[d]4)front,close,short[?]5)back,semi-open,long[?:]6)voiceless bilabial stop[p]B.Give the phonetic features of each of the following sounds:1)[d]voiceless alveolar stop2)[l]voiced alveolar liquid3)[?]voiceless palatal affricate4)[w]voiced bilabial glide5)[?]back,close,(rounded)short6)[?]front,open,(unrounded)short9.Explain with examples the sequential rule,the assimilation rule,and the deletion rule.Rules that govern the combination of sounds in a particular language are called sequential rules.There are many such sequential rules in English.For example,if a word begins with a[l]or a [r],then the next sound must be a vowel.That is why[lbik][lkbi]are impossible combinations in English.They have violated the restrictions on the sequencing of phonemes.The assimilation rule assimilates one sound to another by“copying”a feature of a sequential phoneme,thus making the two phones similar.Assimilation of neighboring sounds is, for the most part,caused by articulatory or physiological processes.When we speak,we tend to increase the ease of articulation.This“sloppy慵懒的;马虎的”tendency may become regularized as rules of language.Deletion rule tells us when a sound is to be deleted although it is orthographically represented.Wehave noticed that in the pronunciation of such words as sign,design,and paradigm,there is no[g]sound although it is represented in spelling by the letter g.But in their corresponding formssignature,designation,and paradigmatic,the[g]represented by the letter g is pronounced.The rule can be stated as:Delete a[g]when it occurs before a final nasal consonant.Given the rule,the phonemic representation of the stems in sign–signature,resign–resignation,paradigm–paradigmatic will include the phoneme/g/,which will be deleted according to the regular rule if no suffix is added.10.What are suprasegmental features?How do the major suprasegmental features of English function in conveying meaning?The phonemic features that occur above the level of the segments are called suprasegmental features.The main suprasegmental features include stress,intonation,and tone.The location of stress in English distinguishes meaning.There are two kinds of stress:word stress and sentence stress.For example,a shift of stress may change the part of speech of a word from a noun,to a verb although its spelling remains unchanged.Tones are pitch variations which can distinguish meaning just like phonemes.Intonation plays an important role in the conveyance of meaning in almost every language,especially in a language like English.When spoken in different tones,the same sequence of words may have different meanings.Chapter3Morphology1.Divide the following words into their separate morphemes by placing a“+”between each morpheme and the next:a.micro+filmb.be+draggle+edc.announce+mentd.pre+digest+ione.tele+communicate+ionf.fore+fatherg.psycho+physics h.mechan+ist6.The following sentences contain both derivational and inflectional affixes.Underline all of the derivational affixes and circle(blacken)the inflectional affixes.a)The farmer’s cow s escap ed.b)It was rain ing.c)Those sock s are inexpensive.d)Jim need s the new er copy.e)The strong est rower continu ed.f)She quickly clos ed the book.g)The alphabetization went well.Chapter4Syntax1.What is syntax?Syntax is a branch of linguistics that studies how words are combined to form sentences and the rules that govern the formation of sentences.2.What is phrase structure rule?The grammatical mechanism that regulates the arrangement of elements(i.e.specifiers,heads, and complements)that make up a phrase is called a phrase structure rule.The phrase structural rule for NP,VP,AP,and PP can be written as follows:NP→(Det)N(PP)...VP→(Qual)V(NP)...AP→(Deg)A(PP)...PP→(Deg)P(NP)...The general phrasal structural rule(X stands for the head N,V,A or P):The XP rule:XP→(specifier)X(complement)3.What is category?How to determine a word’s category?Category refers to a group of linguistic items which fulfill the same or similar functions in a particular language such as a sentence,a noun phrase or a verb.To determine a word’s category,three criteria are usually employed,namely meaning, inflection and distribution.A word’s distributional facts together with information about its meaning and inflectional capabilities help identify its syntactic category.4.What is coordinate structure and what properties does it have?The structure formed by joining two or more elements of the same type with the help of a conjunction is called coordinate structures.Coordinate structures exhibits four important properties:a)There is no limit on the number of coordinated categories that can appear prior to theconjunction.b)A category at any level(a head or an entire XP)can becoordinated.c)Coordinated categories must be of the same type.d)The category type of the coordinate phrase is identical to the category type of theelements being conjoined.6.What is deep structure and what is surface structure?There are two levels of syntactic structure.The first,formed by the XP rule in accordance with the head’s sub-categorization properties,is called deep structure(or D-structure).The second, corresponding to the final syntactic form of the sentence which results from appropriate transformations,is called surface structure(or S-structure).8.The following phrases include a head,a complement,and a specifier.Draw the appropriate tree structure for each.13.The derivations of the following sentences involve the inversion transformation.Give the deep structure and the surface structure of each of thesesentences.a)Would you come tomorrow?b)What did Helen bring to the party?c)Who broke the window?Chapter5Semantics1.What are the major views concerning the study of meaning?1)The naming theory was proposed by the ancient Greek scholar Plato.According to this theory,the linguistic forms or symbols,in other words,the words used in a language are simply labels of the objects they stand for.So words are just names or labels for things.2)The conceptualist view has been held by some philosophers and linguists from ancient times.This view holds that there is no direct link between a linguistic form and what it refers to(i.e.,between language and the real world);rather,in the interpretation of meaning they are linked through the mediation of concepts in the mind.3)Contextualism holds that meaning should be studied in terms of situation,use,context ––elements closely linked with language behavior.The representative of this approach was J.R. Firth,famous British linguist.4)Behaviorism attempts to define the meaning of a language form as the“situation in which the speaker utters it and the response it calls forth in the hearer.”This theory,somewhat close to contextualism,is linked with psychological interest.2.What are the major types of synonyms in English?The major types of synonyms are dialectal synonyms,stylistic synonyms,emotive or evaluative synonyms,collocational synonyms,and semantically different synonyms.3.Explain with examples“homonymy”,“polysemy”,and“hyponymy”.1)Homonymy refers to the phenomenon that words having different meanings have the same form,i.e.,different words are identical in sound or spelling,or in both.When two words are identical in sound,they are homophones.When two words are identical in spelling,they are homographs.When two words are identical in both sound and spelling,they are complete homonyms.2)While different words may have the same or similar meaning,the same one word may have more than one meaning.This is what we call polysemy,and such a word is called a polysemic word.There are many polysemic words in English,The fact is the more commonly used a word is, the more likely it has acquired more than one meaning.3)Hyponymy refers to the sense relation between a more general,more inclusive word anda more specific word.The word which is more general in meaning is called the superordinate, and the more specific words are called itshyponyms.Hyponyms of the same superordinate areco-hyponyms to each other.Hyponymy is a relation of inclusion;in terms of meaning,the superordinate includes all its hyponyms.5.Identify the relations between the following pairs of sentences:1)“T om’s wife is pregnant”presupposes“T om has a wife.”2)“My sister will soon be divorced”presupposes“My sister is a married woman.”3)“He likes seafood”is entailed by”He likes crabs.”4)“They are going to have another baby”presupposes“They have a child.”8.Try to analyze the following sentences in terms of predication analysis:1)The man sells ice-cream.MAN,ICE-CREAM(SELL)2)Is the baby sleeping?BABY(SLEEP)3)It is snowing.(SNOW)4)The tree grows well.TREE(GROW)Chapter6Pragmatics1.What does pragmatics study?How does it differ from traditional semantics?Generally speaking,pragmatics is the study of meaning in the context.It studies meaning in a dynamic way and as a process.In order to have a successful communication,the speaker and hearer must take the context into their consideration so as to effect the right meaning and intention.The development and establishment of pragmatics in the1960s and1970s resulted mainly from the expansion of the study of semantics.However,it is different from the traditional semantics.The major difference between them lies in that pragmatics studies meaning in a dynamic way,while semantics studies meaning in a static way.Pragmatics takes context into consideration while semantics does not.Pragmatics takes care of the aspect of meaning that is not accounted for by semantics.2.Why is the notion of context essential in the pragmatic study of linguistic communication?The notion of context is essential to the pragmatic study of language.It is generally considered as constituted by the knowledge shared by the speaker and the hearer.Various contents of shared knowledge have been identified,e.g.knowledge of the language they use,knowledge of what has been said before,knowledge about the world in general,knowledge about the specific situation in which linguistic communication is taking place,and knowledge about each other.Context determines the speaker’s use of language and also the heater’s interpretation of what is said to him. Without such knowledge,linguistic communication would not be possible,and without considering such knowledge,linguistic communication cannot be satisfactorily accounted for in a pragmatic sense.Look at the following sentences:(1)How did it go?(2)It is cold in here.(3)It was a hot Christmas day so we went down to the beach in the afternoon and had a good time swimming and surfing.Sentence(1)might be used in a conversation between two students talking about an examination,or two surgeons talking about an operation,or in some other contexts;(2)might be said by the speaker to ask the hearer to turn on the heater,or leave the place,or to put on more clothes,or to apologize for the poor condition of the room,depending on the situation of context;(3)makes sense only if the hearer has the knowledge that Christmas falls in summer in the southern hemisphere.3.How are sentence meaning and utterance meaning related,and how do they differ?A sentence is a grammatical concept,and the meaning of a sentence is often studied as the abstract,intrinsic property of the sentence itself in terms of predication.But if we think of a sentence as what people actually utter in the course of communication,it becomes an utterance, and it should be considered in the situation in which it is actually uttered(or used).So it is impossible to tell if“The dog is barking”is a sentence or an utterance.It can be either.It all depends on how we look at it and how we are going to analyze it.If we take it as a grammatical unit and consider it as a self-contained unit in isolation from context,then we are treating it as a sentence.If we take it as something a speaker utters in a certain situation with a certain purpose, then we are treating it as an utterance.Therefore,while the meaning of a sentence is abstract,and decontextualized,that of anutterance is concrete,and context-dependent.The meaning of an utterance is based on sentence meaning;it is the realization of the abstract meaning of a sentence in a real situation of communication,or simply in a context.Now,take the sentence“My bag is heavy”as an example. Semantic analysis of the meaning of the sentence results in the one-place predication BAG(BE HEAVY).Then a pragmatic analysis of the utterance meaning of the sentence varies with the context in which it is uttered.For example,it could be uttered by a speaker as a straightforward statement, telling the hearer that his bag is heavy.It could also be intended by the speaker as an indirect,polite request,asking the hearer to help him carry the bag.Another possibility is that the speaker is declining someone’s request for help.All these are possible interpretations of the same utterance “My bag is heavy”.How it is to be understood depends on the context in which it is uttered and the purpose for which the speaker utters it.While most utterances take the form of grammatically complete sentences,some utterances do not,and some cannot even be restored to complete sentences.5.According to Austin,what are the three acts a person is possibly performing while making an utterance.Give an example.According to Austin’s new model,a speaker might be performing three acts simultaneously when speaking:locutionary act,illocutionary act,and perlocutionary act.A locutionary act is the act of uttering words,phrases,clauses.It is the act of conveying literal meaning by means of syntax,lexicon and phonology.An illocutionary act is the act of expressing the speaker’s intention;it is the act performed in saying something.A perlocutionary act is the act performed by or resulting from saying something;it is the consequence of,or the change brought about by the utterance;it is the act performed by saying something.Let’s look at an example:You have left the door wide open.The locutionary act performed by the speaker is his utterance of the words“you”,“have”,“door”,“open”,etc.,thus expressing what the words literally mean. The illocutionary act performed by the speaker is that by making such an utterance he has expressed his intention of speaking,i.e.asking someone to close the door,or making a complaint, depending on the context.The perlocutionary act refers to the effect of the utterance.If the hearer gets the speaker’s message and sees that the speaker means to tell him to close the door,the speaker has successfully brought about the change in the real world he has intended to;then the perlocutionary act is successfully performed.7.What is indirect language use?How is it explained in the light of speech act theory?When someone is not saying in an explicit and straightforward manner what he means to say, rather he is trying to put across his message in an implicit,roundabout way,we can say he is using indirect language.8.What are the four maxims of the CP?Try to give your own examples to show how flouting these maxims gives rise to conversational implicature? Cooperative Principle,abbreviated as CP.It goes as follows:Make your conversational contribution such as required at the stage at which it occurs by the accepted purpose or direction of the talk exchange in which you are engaged.To be more specific,there are four maxims under this general principle:1)The maxim of quantitya)Make your contribution as informative as required(for the current purpose of the exchange).b)Do not make your contribution more informative than is required.2)The maxim of qualitya)Do not say what you believe to be false.b)Do not say that for which you lack adequate evidence.3)The maxim of relationa)Be relevant.4)The maxim of mannera)Avoid obscurity of expression.b)Avoid ambiguity.c)Be brief(avoid unnecessary prolixity).d)Be orderly.。

The electronic properties of grapheneA.H.Castro NetoDepartment of Physics,Boston University,590Commonwealth Avenue,Boston,Massachusetts02215,USAF.GuineaInstituto de Ciencia de Materiales de Madrid,CSIC,Cantoblanco,E-28049Madrid,SpainN.M.R.PeresCenter of Physics and Department of Physics,Universidade do Minho,P-4710-057,Braga,PortugalK.S.Novoselov and A.K.GeimDepartment of Physics and Astronomy,University of Manchester,Manchester,M139PL,United Kingdom͑Published14January2009͒This article reviews the basic theoretical aspects of graphene,a one-atom-thick allotrope of carbon, with unusual two-dimensional Dirac-like electronic excitations.The Dirac electrons can be controlled by application of external electric and magneticfields,or by altering sample geometry and/or topology.The Dirac electrons behave in unusual ways in tunneling,confinement,and the integer quantum Hall effect.The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers.Edge͑surface͒states in graphene depend on the edge termination͑zigzag or armchair͒and affect the physical properties of nanoribbons.Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties.The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented.DOI:10.1103/RevModPhys.81.109PACS number͑s͒:81.05.Uw,73.20.Ϫr,03.65.Pm,82.45.MpCONTENTSI.Introduction110II.Elementary Electronic Properties of Graphene112A.Single layer:Tight-binding approach1121.Cyclotron mass1132.Density of states114B.Dirac fermions1141.Chiral tunneling and Klein paradox1152.Confinement and Zitterbewegung117C.Bilayer graphene:Tight-binding approach118D.Epitaxial graphene119E.Graphene stacks1201.Electronic structure of bulk graphite121F.Surface states in graphene122G.Surface states in graphene stacks124H.The spectrum of graphene nanoribbons1241.Zigzag nanoribbons1252.Armchair nanoribbons126I.Dirac fermions in a magneticfield126J.The anomalous integer quantum Hall effect128 K.Tight-binding model in a magneticfield128 ndau levels in graphene stacks130 M.Diamagnetism130 N.Spin-orbit coupling131 III.Flexural Phonons,Elasticity,and Crumpling132 IV.Disorder in Graphene134A.Ripples135B.Topological lattice defects136C.Impurity states137D.Localized states near edges,cracks,and voids137E.Self-doping138F.Vector potential and gaugefield disorder1391.Gaugefield induced by curvature1402.Elastic strain1403.Random gaugefields141G.Coupling to magnetic impurities141H.Weak and strong localization142I.Transport near the Dirac point143J.Boltzmann equation description of dc transport indoped graphene144 K.Magnetotransport and universal conductivity1451.The full self-consistent Born approximation͑FSBA͒146 V.Many-Body Effects148A.Electron-phonon interactions148B.Electron-electron interactions1501.Screening in graphene stacks152C.Short-range interactions1521.Bilayer graphene:Exchange1532.Bilayer graphene:Short-range interactions154D.Interactions in high magneticfields154VI.Conclusions154 Acknowledgments155 References155REVIEWS OF MODERN PHYSICS,VOLUME81,JANUARY–MARCH20090034-6861/2009/81͑1͒/109͑54͒©2009The American Physical Society109I.INTRODUCTIONCarbon is the materia prima for life and the basis of all organic chemistry.Because of the flexibility of its bond-ing,carbon-based systems show an unlimited number of different structures with an equally large variety of physical properties.These physical properties are,in great part,the result of the dimensionality of these structures.Among systems with only carbon atoms,graphene—a two-dimensional ͑2D ͒allotrope of carbon—plays an important role since it is the basis for the understanding of the electronic properties in other allotropes.Graphene is made out of carbon atoms ar-ranged on a honeycomb structure made out of hexagons ͑see Fig.1͒,and can be thought of as composed of ben-zene rings stripped out from their hydrogen atoms ͑Pauling,1972͒.Fullerenes ͑Andreoni,2000͒are mol-ecules where carbon atoms are arranged spherically,and hence,from the physical point of view,are zero-dimensional objects with discrete energy states.Fullerenes can be obtained from graphene with the in-troduction of pentagons ͑that create positive curvature defects ͒,and hence,fullerenes can be thought as wrapped-up graphene.Carbon nanotubes ͑Saito et al.,1998;Charlier et al.,2007͒are obtained by rolling graphene along a given direction and reconnecting the carbon bonds.Hence carbon nanotubes have only hexa-gons and can be thought of as one-dimensional ͑1D ͒ob-jects.Graphite,a three dimensional ͑3D ͒allotrope of carbon,became widely known after the invention of the pencil in 1564͑Petroski,1989͒,and its usefulness as an instrument for writing comes from the fact that graphite is made out of stacks of graphene layers that are weakly coupled by van der Waals forces.Hence,when one presses a pencil against a sheet of paper,one is actually producing graphene stacks and,somewhere among them,there could be individual graphene layers.Al-though graphene is the mother for all these different allotropes and has been presumably produced every time someone writes with a pencil,it was only isolated 440years after its invention ͑Novoselov et al.,2004͒.The reason is that,first,no one actually expected graphene to exist in the free state and,second,even with the ben-efit of hindsight,no experimental tools existed to search for one-atom-thick flakes among the pencil debris cov-ering macroscopic areas ͑Geim and MacDonald,2007͒.Graphene was eventually spotted due to the subtle op-tical effect it creates on top of a chosen SiO 2substrate ͑Novoselov et al.,2004͒that allows its observation with an ordinary optical microscope ͑Abergel et al.,2007;Blake et al.,2007;Casiraghi et al.,2007͒.Hence,graphene is relatively straightforward to make,but not so easy to find.The structural flexibility of graphene is reflected in its electronic properties.The sp 2hybridization between one s orbital and two p orbitals leads to a trigonal planar structure with a formation of a ␴bond between carbon atoms that are separated by 1.42Å.The ␴band is re-sponsible for the robustness of the lattice structure in all allotropes.Due to the Pauli principle,these bands have a filled shell and,hence,form a deep valence band.The unaffected p orbital,which is perpendicular to the pla-nar structure,can bind covalently with neighboring car-bon atoms,leading to the formation of a ␲band.Since each p orbital has one extra electron,the ␲band is half filled.Half-filled bands in transition elements have played an important role in the physics of strongly correlated systems since,due to their strong tight-binding charac-ter,the Coulomb energies are large,leading to strong collective effects,magnetism,and insulating behavior due to correlation gaps or Mottness ͑Phillips,2006͒.In fact,Linus Pauling proposed in the 1950s that,on the basis of the electronic properties of benzene,graphene should be a resonant valence bond ͑RVB ͒structure ͑Pauling,1972͒.RVB states have become popular in the literature of transition-metal oxides,and particularly in studies of cuprate-oxide superconductors ͑Maple,1998͒.This point of view should be contrasted with contempo-raneous band-structure studies of graphene ͑Wallace,1947͒that found it to be a semimetal with unusual lin-early dispersing electronic excitations called Dirac elec-trons.While most current experimental data in graphene support the band structure point of view,the role of electron-electron interactions in graphene is a subject of intense research.It was P .R.Wallace in 1946who first wrote on the band structure of graphene and showed the unusual semimetallic behavior in this material ͑Wallace,1947͒.At that time,the thought of a purely 2D structure was not reality and Wallace’s studies of graphene served him as a starting point to study graphite,an important mate-rial for nuclear reactors in the post–World War II era.During the following years,the study of graphite culmi-nated with the Slonczewski-Weiss-McClure ͑SWM ͒band structure of graphite,which provided a description of the electronic properties in this material ͑McClure,1957;Slonczewski and Weiss,1958͒and was successful in de-scribing the experimental data ͑Boyle and Nozières 1958;McClure,1958;Spry and Scherer,1960;Soule et al.,1964;Williamson et al.,1965;Dillon et al.,1977͒.From 1957to 1968,the assignment of the electron and hole states within the SWM model were oppositetoFIG.1.͑Color online ͒Graphene ͑top left ͒is a honeycomb lattice of carbon atoms.Graphite ͑top right ͒can be viewed as a stack of graphene layers.Carbon nanotubes are rolled-up cylinders of graphene ͑bottom left ͒.Fullerenes ͑C 60͒are mol-ecules consisting of wrapped graphene by the introduction of pentagons on the hexagonal lattice.From Castro Neto et al.,2006a .110Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009what is accepted today.In1968,Schroeder et al.͑Schroeder et al.,1968͒established the currently ac-cepted location of electron and hole pockets͑McClure, 1971͒.The SWM model has been revisited in recent years because of its inability to describe the van der Waals–like interactions between graphene planes,a problem that requires the understanding of many-body effects that go beyond the band-structure description ͑Rydberg et al.,2003͒.These issues,however,do not arise in the context of a single graphene crystal but they show up when graphene layers are stacked on top of each other,as in the case,for instance,of the bilayer graphene.Stacking can change the electronic properties considerably and the layering structure can be used in order to control the electronic properties.One of the most interesting aspects of the graphene problem is that its low-energy excitations are massless, chiral,Dirac fermions.In neutral graphene,the chemical potential crosses exactly the Dirac point.This particular dispersion,that is only valid at low energies,mimics the physics of quantum electrodynamics͑QED͒for massless fermions except for the fact that in graphene the Dirac fermions move with a speed v F,which is300times smaller than the speed of light c.Hence,many of the unusual properties of QED can show up in graphene but at much smaller speeds͑Castro Neto et al.,2006a; Katsnelson et al.,2006;Katsnelson and Novoselov, 2007͒.Dirac fermions behave in unusual ways when compared to ordinary electrons if subjected to magnetic fields,leading to new physical phenomena͑Gusynin and Sharapov,2005;Peres,Guinea,and Castro Neto,2006a͒such as the anomalous integer quantum Hall effect ͑IQHE͒measured experimentally͑Novoselov,Geim, Morozov,et al.,2005a;Zhang et al.,2005͒.Besides being qualitatively different from the IQHE observed in Si and GaAlAs͑heterostructures͒devices͑Stone,1992͒, the IQHE in graphene can be observed at room tem-perature because of the large cyclotron energies for “relativistic”electrons͑Novoselov et al.,2007͒.In fact, the anomalous IQHE is the trademark of Dirac fermion behavior.Another interesting feature of Dirac fermions is their insensitivity to external electrostatic potentials due to the so-called Klein paradox,that is,the fact that Dirac fermions can be transmitted with probability1through a classically forbidden region͑Calogeracos and Dombey, 1999;Itzykson and Zuber,2006͒.In fact,Dirac fermions behave in an unusual way in the presence of confining potentials,leading to the phenomenon of Zitter-bewegung,or jittery motion of the wave function͑Itzyk-son and Zuber,2006͒.In graphene,these electrostatic potentials can be easily generated by disorder.Since dis-order is unavoidable in any material,there has been a great deal of interest in trying to understand how disor-der affects the physics of electrons in graphene and its transport properties.In fact,under certain conditions, Dirac fermions are immune to localization effects ob-served in ordinary electrons͑Lee and Ramakrishnan, 1985͒and it has been established experimentally that electrons can propagate without scattering over large distances of the order of micrometers in graphene͑No-voselov et al.,2004͒.The sources of disorder in graphene are many and can vary from ordinary effects commonly found in semiconductors,such as ionized impurities in the Si substrate,to adatoms and various molecules ad-sorbed in the graphene surface,to more unusual defects such as ripples associated with the soft structure of graphene͑Meyer,Geim,Katsnelson,Novoselov,Booth, et al.,2007a͒.In fact,graphene is unique in the sense that it shares properties of soft membranes͑Nelson et al.,2004͒and at the same time it behaves in a metallic way,so that the Dirac fermions propagate on a locally curved space.Here analogies with problems of quantum gravity become apparent͑Fauser et al.,2007͒.The soft-ness of graphene is related with the fact that it has out-of-plane vibrational modes͑phonons͒that cannot be found in3D solids.Theseflexural modes,responsible for the bending properties of graphene,also account for the lack of long range structural order in soft mem-branes leading to the phenomenon of crumpling͑Nelson et al.,2004͒.Nevertheless,the presence of a substrate or scaffolds that hold graphene in place can stabilize a cer-tain degree of order in graphene but leaves behind the so-called ripples͑which can be viewed as frozenflexural modes͒.It was realized early on that graphene should also present unusual mesoscopic effects͑Peres,Castro Neto, and Guinea,2006a;Katsnelson,2007a͒.These effects have their origin in the boundary conditions required for the wave functions in mesoscopic samples with various types of edges graphene can have͑Nakada et al.,1996; Wakabayashi et al.,1999;Peres,Guinea,and Castro Neto,2006a;Akhmerov and Beenakker,2008͒.The most studied edges,zigzag and armchair,have drastically different electronic properties.Zigzag edges can sustain edge͑surface͒states and resonances that are not present in the armchair case.Moreover,when coupled to con-ducting leads,the boundary conditions for a graphene ribbon strongly affect its conductance,and the chiral Dirac nature of fermions in graphene can be used for applications where one can control the valleyflavor of the electrons besides its charge,the so-called valleytron-ics͑Rycerz et al.,2007͒.Furthermore,when supercon-ducting contacts are attached to graphene,they lead to the development of supercurrentflow and Andreev pro-cesses characteristic of the superconducting proximity effect͑Heersche et al.,2007͒.The fact that Cooper pairs can propagate so well in graphene attests to the robust electronic coherence in this material.In fact,quantum interference phenomena such as weak localization,uni-versal conductancefluctuations͑Morozov et al.,2006͒, and the Aharonov-Bohm effect in graphene rings have already been observed experimentally͑Recher et al., 2007;Russo,2007͒.The ballistic electronic propagation in graphene can be used forfield-effect devices such as p-n͑Cheianov and Fal’ko,2006;Cheianov,Fal’ko,and Altshuler,2007;Huard et al.,2007;Lemme et al.,2007; Tworzydlo et al.,2007;Williams et al.,2007;Fogler, Glazman,Novikov,et al.,2008;Zhang and Fogler,2008͒and p-n-p͑Ossipov et al.,2007͒junctions,and as“neu-111Castro Neto et al.:The electronic properties of graphene Rev.Mod.Phys.,V ol.81,No.1,January–March2009trino”billiards ͑Berry and Modragon,1987;Miao et al.,2007͒.It has also been suggested that Coulomb interac-tions are considerably enhanced in smaller geometries,such as graphene quantum dots ͑Milton Pereira et al.,2007͒,leading to unusual Coulomb blockade effects ͑Geim and Novoselov,2007͒and perhaps to magnetic phenomena such as the Kondo effect.The transport properties of graphene allow for their use in a plethora of applications ranging from single molecule detection ͑Schedin et al.,2007;Wehling et al.,2008͒to spin injec-tion ͑Cho et al.,2007;Hill et al.,2007;Ohishi et al.,2007;Tombros et al.,2007͒.Because of its unusual structural and electronic flex-ibility,graphene can be tailored chemically and/or struc-turally in many different ways:deposition of metal at-oms ͑Calandra and Mauri,2007;Uchoa et al.,2008͒or molecules ͑Schedin et al.,2007;Leenaerts et al.,2008;Wehling et al.,2008͒on top;intercalation ͓as done in graphite intercalated compounds ͑Dresselhaus et al.,1983;Tanuma and Kamimura,1985;Dresselhaus and Dresselhaus,2002͔͒;incorporation of nitrogen and/or boron in its structure ͑Martins et al.,2007;Peres,Klironomos,Tsai,et al.,2007͓͒in analogy with what has been done in nanotubes ͑Stephan et al.,1994͔͒;and using different substrates that modify the electronic structure ͑Calizo et al.,2007;Giovannetti et al.,2007;Varchon et al.,2007;Zhou et al.,2007;Das et al.,2008;Faugeras et al.,2008͒.The control of graphene properties can be extended in new directions allowing for the creation of graphene-based systems with magnetic and supercon-ducting properties ͑Uchoa and Castro Neto,2007͒that are unique in their 2D properties.Although the graphene field is still in its infancy,the scientific and technological possibilities of this new material seem to be unlimited.The understanding and control of this ma-terial’s properties can open doors for a new frontier in electronics.As the current status of the experiment and potential applications have recently been reviewed ͑Geim and Novoselov,2007͒,in this paper we concen-trate on the theory and more technical aspects of elec-tronic properties with this exciting new material.II.ELEMENTARY ELECTRONIC PROPERTIES OF GRAPHENEA.Single layer:Tight-binding approachGraphene is made out of carbon atoms arranged in hexagonal structure,as shown in Fig.2.The structure can be seen as a triangular lattice with a basis of two atoms per unit cell.The lattice vectors can be written asa 1=a 2͑3,ͱ3͒,a 2=a2͑3,−ͱ3͒,͑1͒where a Ϸ1.42Åis the carbon-carbon distance.Thereciprocal-lattice vectors are given byb 1=2␲3a͑1,ͱ3͒,b 2=2␲3a͑1,−ͱ3͒.͑2͒Of particular importance for the physics of graphene are the two points K and K Јat the corners of the graphene Brillouin zone ͑BZ ͒.These are named Dirac points for reasons that will become clear later.Their positions in momentum space are given byK =ͩ2␲3a ,2␲3ͱ3aͪ,K Ј=ͩ2␲3a ,−2␲3ͱ3aͪ.͑3͒The three nearest-neighbor vectors in real space are given by␦1=a 2͑1,ͱ3͒␦2=a 2͑1,−ͱ3͒␦3=−a ͑1,0͒͑4͒while the six second-nearest neighbors are located at ␦1Ј=±a 1,␦2Ј=±a 2,␦3Ј=±͑a 2−a 1͒.The tight-binding Hamiltonian for electrons in graphene considering that electrons can hop to both nearest-and next-nearest-neighbor atoms has the form ͑we use units such that ប=1͒H =−t͚͗i ,j ͘,␴͑a ␴,i †b ␴,j +H.c.͒−t Ј͚͗͗i ,j ͘͘,␴͑a ␴,i †a ␴,j +b ␴,i †b ␴,j +H.c.͒,͑5͒where a i ,␴͑a i ,␴†͒annihilates ͑creates ͒an electron with spin ␴͑␴=↑,↓͒on site R i on sublattice A ͑an equiva-lent definition is used for sublattice B ͒,t ͑Ϸ2.8eV ͒is the nearest-neighbor hopping energy ͑hopping between dif-ferent sublattices ͒,and t Јis the next nearest-neighbor hopping energy 1͑hopping in the same sublattice ͒.The energy bands derived from this Hamiltonian have the form ͑Wallace,1947͒E ±͑k ͒=±t ͱ3+f ͑k ͒−t Јf ͑k ͒,1The value of t Јis not well known but ab initio calculations ͑Reich et al.,2002͒find 0.02t Շt ЈՇ0.2t depending on the tight-binding parametrization.These calculations also include the effect of a third-nearest-neighbors hopping,which has a value of around 0.07eV.A tight-binding fit to cyclotron resonance experiments ͑Deacon et al.,2007͒finds t ЈϷ0.1eV.FIG.2.͑Color online ͒Honeycomb lattice and its Brillouin zone.Left:lattice structure of graphene,made out of two in-terpenetrating triangular lattices ͑a 1and a 2are the lattice unit vectors,and ␦i ,i =1,2,3are the nearest-neighbor vectors ͒.Right:corresponding Brillouin zone.The Dirac cones are lo-cated at the K and K Јpoints.112Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009f ͑k ͒=2cos ͑ͱ3k y a ͒+4cosͩͱ32k y a ͪcosͩ32k x a ͪ,͑6͒where the plus sign applies to the upper ͑␲*͒and the minus sign the lower ͑␲͒band.It is clear from Eq.͑6͒that the spectrum is symmetric around zero energy if t Ј=0.For finite values of t Ј,the electron-hole symmetry is broken and the ␲and ␲*bands become asymmetric.In Fig.3,we show the full band structure of graphene with both t and t Ј.In the same figure,we also show a zoom in of the band structure close to one of the Dirac points ͑at the K or K Јpoint in the BZ ͒.This dispersion can be obtained by expanding the full band structure,Eq.͑6͒,close to the K ͑or K Ј͒vector,Eq.͑3͒,as k =K +q ,with ͉q ͉Ӷ͉K ͉͑Wallace,1947͒,E ±͑q ͒Ϸ±vF ͉q ͉+O ͓͑q /K ͒2͔,͑7͒where q is the momentum measured relatively to the Dirac points and v F is the Fermi velocity,given by v F =3ta /2,with a value v F Ӎ1ϫ106m/s.This result was first obtained by Wallace ͑1947͒.The most striking difference between this result and the usual case,⑀͑q ͒=q 2/͑2m ͒,where m is the electron mass,is that the Fermi velocity in Eq.͑7͒does not de-pend on the energy or momentum:in the usual case we have v =k /m =ͱ2E /m and hence the velocity changes substantially with energy.The expansion of the spectrum around the Dirac point including t Јup to second order in q /K is given byE ±͑q ͒Ӎ3t Ј±vF ͉q ͉−ͩ9t Јa 24±3ta 28sin ͑3␪q ͉͒ͪq ͉2,͑8͒where␪q =arctanͩq x q yͪ͑9͒is the angle in momentum space.Hence,the presence of t Јshifts in energy the position of the Dirac point and breaks electron-hole symmetry.Note that up to order ͑q /K ͒2the dispersion depends on the direction in mo-mentum space and has a threefold symmetry.This is the so-called trigonal warping of the electronic spectrum ͑Ando et al.,1998,Dresselhaus and Dresselhaus,2002͒.1.Cyclotron massThe energy dispersion ͑7͒resembles the energy of ul-trarelativistic particles;these particles are quantum me-chanically described by the massless Dirac equation ͑see Sec.II.B for more on this analogy ͒.An immediate con-sequence of this massless Dirac-like dispersion is a cy-clotron mass that depends on the electronic density as its square root ͑Novoselov,Geim,Morozov,et al.,2005;Zhang et al.,2005͒.The cyclotron mass is defined,within the semiclassical approximation ͑Ashcroft and Mermin,1976͒,asm *=12␲ͫץA ͑E ͒ץEͬE =E F,͑10͒with A ͑E ͒the area in k space enclosed by the orbit andgiven byA ͑E ͒=␲q ͑E ͒2=␲E 2v F2.͑11͒Using Eq.͑11͒in Eq.͑10͒,one obtainsm *=E Fv F2=k Fv F.͑12͒The electronic density n is related to the Fermi momen-tum k F as k F2/␲=n ͑with contributions from the two Dirac points K and K Јand spin included ͒,which leads tom *=ͱ␲v Fͱn .͑13͒Fitting Eq.͑13͒to the experimental data ͑see Fig.4͒provides an estimation for the Fermi velocity andtheFIG.3.͑Color online ͒Electronic dispersion in the honeycomb lattice.Left:energy spectrum ͑in units of t ͒for finite values of t and t Ј,with t =2.7eV and t Ј=−0.2t .Right:zoom in of the energy bands close to one of the Diracpoints.FIG.4.͑Color online ͒Cyclotron mass of charge carriers in graphene as a function of their concentration n .Positive and negative n correspond to electrons and holes,respectively.Symbols are the experimental data extracted from the tem-perature dependence of the SdH oscillations;solid curves are the best fit by Eq.͑13͒.m 0is the free-electron mass.Adapted from Novoselov,Geim,Morozov,et al.,2005.113Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009hopping parameter as v F Ϸ106ms −1and t Ϸ3eV,respec-tively.Experimental observation of the ͱn dependence on the cyclotron mass provides evidence for the exis-tence of massless Dirac quasiparticles in graphene ͑No-voselov,Geim,Morozov,et al.,2005;Zhang et al.,2005;Deacon et al.,2007;Jiang,Henriksen,Tung,et al.,2007͒—the usual parabolic ͑Schrödinger ͒dispersion im-plies a constant cyclotron mass.2.Density of statesThe density of states per unit cell,derived from Eq.͑5͒,is given in Fig.5for both t Ј=0and t Ј 0,showing in both cases semimetallic behavior ͑Wallace,1947;Bena and Kivelson,2005͒.For t Ј=0,it is possible to derive an analytical expression for the density of states per unit cell,which has the form ͑Hobson and Nierenberg,1953͒␳͑E ͒=4␲2͉E ͉t 21ͱZ 0F ͩ␲2,ͱZ 1Z 0ͪ,Z 0=Άͩ1+ͯE t ͯͪ2−͓͑E /t ͒2−1͔24,−t ഛE ഛt4ͯE t ͯ,−3t ഛE ഛ−t ∨t ഛE ഛ3t ,Z 1=Ά4ͯE t ͯ,−t ഛE ഛtͩ1+ͯE tͯͪ2−͓͑E /t ͒2−1͔24,−3t ഛE ഛ−t ∨t ഛE ഛ3t ,͑14͒where F ͑␲/2,x ͒is the complete elliptic integral of thefirst kind.Close to the Dirac point,the dispersion is ap-proximated by Eq.͑7͒and the density of states per unit cell is given by ͑with a degeneracy of 4included ͒␳͑E ͒=2A c ␲͉E ͉v F2,͑15͒where A c is the unit cell area given by A c =3ͱ3a 2/2.It is worth noting that the density of states for graphene is different from the density of states of carbon nanotubes ͑Saito et al.,1992a ,1992b ͒.The latter shows 1/ͱE singu-larities due to the 1D nature of their electronic spec-trum,which occurs due to the quantization of the mo-mentum in the direction perpendicular to the tube axis.From this perspective,graphene nanoribbons,which also have momentum quantization perpendicular to the ribbon length,have properties similar to carbon nano-tubes.B.Dirac fermionsWe consider the Hamiltonian ͑5͒with t Ј=0and theFourier transform of the electron operators,a n =1ͱN c͚ke −i k ·R na ͑k ͒,͑16͒where N c is the number of unit ing this transfor-mation,we write the field a n as a sum of two terms,coming from expanding the Fourier sum around K Јand K .This produces an approximation for the representa-tion of the field a n as a sum of two new fields,written asa n Ӎe −i K ·R n a 1,n +e −i K Ј·R n a 2,n ,b n Ӎe −i K ·R n b 1,n +e −i K Ј·R n b 2,n ,͑17͒ρ(ε)ε/tρ(ε)ε/tFIG.5.Density of states per unit cell as a function of energy ͑in units of t ͒computed from the energy dispersion ͑5͒,t Ј=0.2t ͑top ͒and t Ј=0͑bottom ͒.Also shown is a zoom-in of the density of states close to the neutrality point of one electron per site.For the case t Ј=0,the electron-hole nature of the spectrum is apparent and the density of states close to the neutrality point can be approximated by ␳͑⑀͒ϰ͉⑀͉.114Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009where the index i =1͑i =2͒refers to the K ͑K Ј͒point.These new fields,a i ,n and b i ,n ,are assumed to vary slowly over the unit cell.The procedure for deriving a theory that is valid close to the Dirac point con-sists in using this representation in the tight-binding Hamiltonian and expanding the opera-tors up to a linear order in ␦.In the derivation,one uses the fact that ͚␦e ±i K ·␦=͚␦e ±i K Ј·␦=0.After some straightforward algebra,we arrive at ͑Semenoff,1984͒H Ӎ−t͵dxdy ⌿ˆ1†͑r ͒ͫͩ3a ͑1−i ͱ3͒/4−3a ͑1+i ͱ3͒/4ͪץx +ͩ3a ͑−i −ͱ3͒/4−3a ͑i −ͱ3͒/4ͪץy ͬ⌿ˆ1͑r ͒+⌿ˆ2†͑r ͒ͫͩ3a ͑1+i ͱ3͒/4−3a ͑1−i ͱ3͒/4ͪץx +ͩ3a ͑i −ͱ3͒/4−3a ͑−i −ͱ3͒/4ͪץy ͬ⌿ˆ2͑r ͒=−i v F͵dxdy ͓⌿ˆ1†͑r ͒␴·ٌ⌿ˆ1͑r ͒+⌿ˆ2†͑r ͒␴*·ٌ⌿ˆ2͑r ͔͒,͑18͒with Pauli matrices ␴=͑␴x ,␴y ͒,␴*=͑␴x ,−␴y ͒,and ⌿ˆi†=͑a i †,b i †͒͑i =1,2͒.It is clear that the effective Hamil-tonian ͑18͒is made of two copies of the massless Dirac-like Hamiltonian,one holding for p around K and the other for p around K Ј.Note that,in first quantized lan-guage,the two-component electron wave function ␺͑r ͒,close to the K point,obeys the 2D Dirac equation,−i v F ␴·ٌ␺͑r ͒=E ␺͑r ͒.͑19͒The wave function,in momentum space,for the mo-mentum around K has the form␺±,K ͑k ͒=1ͱ2ͩe −i ␪k /2±e i ␪k /2ͪ͑20͒for H K =v F ␴·k ,where the Ϯsigns correspond to the eigenenergies E =±v F k ,that is,for the ␲*and ␲bands,respectively,and ␪k is given by Eq.͑9͒.The wave func-tion for the momentum around K Јhas the form␺±,K Ј͑k ͒=1ͱ2ͩe i ␪k /2±e −i ␪k /2ͪ͑21͒for H K Ј=v F ␴*·k .Note that the wave functions at K and K Јare related by time-reversal symmetry:if we set the origin of coordinates in momentum space in the M point of the BZ ͑see Fig.2͒,time reversal becomes equivalent to a reflection along the k x axis,that is,͑k x ,k y ͒→͑k x ,−k y ͒.Also note that if the phase ␪is rotated by 2␲,the wave function changes sign indicating a phase of ␲͑in the literature this is commonly called a Berry’s phase ͒.This change of phase by ␲under rotation is char-acteristic of spinors.In fact,the wave function is a two-component spinor.A relevant quantity used to characterize the eigen-functions is their helicity defined as the projection of the momentum operator along the ͑pseudo ͒spin direction.The quantum-mechanical operator for the helicity has the formhˆ=12␴·p ͉p ͉.͑22͒It is clear from the definition of h ˆthat the states ␺K͑r ͒and ␺K Ј͑r ͒are also eigenstates of h ˆ,h ˆ␺K ͑r ͒=±12␺K͑r ͒,͑23͒and an equivalent equation for ␺K Ј͑r ͒with inverted sign.Therefore,electrons ͑holes ͒have a positive ͑negative ͒helicity.Equation ͑23͒implies that ␴has its two eigen-values either in the direction of ͑⇑͒or against ͑⇓͒the momentum p .This property says that the states of the system close to the Dirac point have well defined chiral-ity or helicity.Note that chirality is not defined in regard to the real spin of the electron ͑that has not yet ap-peared in the problem ͒but to a pseudospin variable as-sociated with the two components of the wave function.The helicity values are good quantum numbers as long as the Hamiltonian ͑18͒is valid.Therefore,the existence of helicity quantum numbers holds only as an asymptotic property,which is well defined close to the Dirac points K and K Ј.Either at larger energies or due to the presence of a finite t Ј,the helicity stops being a good quantum number.1.Chiral tunneling and Klein paradoxIn this section,we address the scattering of chiral elec-trons in two dimensions by a square barrier ͑Katsnelson et al.,2006;Katsnelson,2007b ͒.The one-dimensional scattering of chiral electrons was discussed earlier in the context on nanotubes ͑Ando et al.,1998;McEuen et al.,1999͒.We start by noting that by a gauge transformation the wave function ͑20͒can be written as115Castro Neto et al.:The electronic properties of grapheneRev.Mod.Phys.,V ol.81,No.1,January–March 2009。