Electron transport through honeycomb lattice ribbons with armchair edges Santanu K. Maiti1,2,∗ 9 1Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear Physics, 0 1/AF, Bidhannagar, Kolkata-700 064, India 0 2 2Department of Physics, Narasinha Dutt College, 129, Belilious Road, Howrah-711 101, India n Abstract a J 1 We address electron transport in honeycomb lattice ribbons with armchair edges attached to two semi- 2 infinite one-dimensionalmetallic electrodes within the tight-binding framework. Here we presentnumer- ically the conductance-energy and current-voltage characteristics as functions of the length and width ] of the ribbons. Our theoretical results predict that for a ribbon with much smaller length and width, l al so-calleda nanoribbon,a gapinthe conductancespectrum appearsacrossthe energyE =0. While, this h gap decreases gradually with the increase of the size of the ribbon, and eventually it almost vanishes. - This revealsa transformationfromthe semiconducting to the conducting material,andit becomes much s e more clearly visible from our presented current-voltagecharacteristics. m . t a PACS No.: 73.63.-b;73.63.Rt. m - d Keywords: Honeycomb lattice ribbon; Armchair edges; Conductance; I-V characteristic. n o c [ 1 v 8 3 2 3 . 1 0 9 0 : v i X r a ∗Corresponding Author: Santanu K. Maiti Electronic mail: [email protected] 1 1 Introduction presentbetweenthetheoryandexperiment,andthe complete knowledge of the conduction mechanism in this scale is not very well established even to- The electronic transport in nanoribbons of day. Severalcontrollingparametersaretherewhich graphenehasopenedupnewareasinnanoelectron- can regulate significantly the electron transport in ics. A graphene nanoribbon (GNR) is a monolayer aconductingbridge,andalltheseeffectshavetobe of carbon atoms arranged in a honeycomb lattice taken into accountproperly to revealthe transport structure [1, 2, 3, 4]. Due to the special electronic properties. For our illustrative purposes, here we and physical properties, graphene based materials describe very briefly some of these effects. exhibit severalnovelproperties like unconventional (i) The quantum interference effect [33, 34, 35, 36, quantum Hall effect [5], high carrier mobility [3] 37]ofelectronwavespassingthroughdifferentarms and many others. The high carrier mobility in of any conducting element which bridges two elec- graphene demonstrates the idea for fabrication of trodes becomes the most significant issue. highspeedswitchingdevicesthosehavewidespread (ii) The coupling of the electrodes with bridging applications in different fields. Some recent experi- materialprovidesanimportantsignatureinthede- ments [6, 7, 8] have also suggested that GNRs can termination of current amplitude acrossany bridge be used to design field-effect transistors and this system [33]. The understanding of this coupling to application provides a huge interest in the commu- the electrodes under non-equilibrium condition is a nity of nanoelectronics device research. Further- major challenge, and we should take care about it more, GNRs can be used to construct MOSFETs in fabrication of any electronic device. which perform much better than conventional Si (iii) The geometry of the conducting material be- MOSFETs. Inotherexperiment[9]ithasbeenpro- tweenthetwoelectrodesitselfisanimportantissue posed that a narrow strip of graphene with arm- to control the electron transmission. To emphasize chair edges, so-called a graphene nanoribbon, ex- it,Ernzerhofetal.[38]havepredictedseveralmodel hibits semiconducting behavior due to its edge ef- calculations and provided some significant results. fects, unlike carbonnanotubes oflargersizeswhich (iv) The dynamical fluctuation in the small-scale are mixtures of both metallic and semiconducting devices is another important factor which plays materials. This is due to the fact that in a nar- an active role and can be manifested through the row graphene sheet, a band gap appears across the energy E = 0, while the gap gradually disappears measurement of shot noise [39, 40], a direct con- sequence of the quantization of charge. It can be with the increase of the size of the ribbon. This used to obtain information on a system which is reveals a transformation from the semiconducting notavailabledirectlythroughtheconductancemea- to the metallic material, and such a phenomenon surements, and is generally more sensitive to the can be utilized for fabrication of electronic devices. effects ofelectron-electroncorrelationsthanthe av- Thismotivatesustostudytheelectrontransportin erage conductance. honeycomblatticeribbonswitharmchairedgesand to verify qualitativelyhowthe transformationfrom Furthermore, several other parameters of the the semiconducting to the conducting propertycan Hamiltonianthatdescribeasystemalsoprovidesig- be achieved simply by tuning the size of a ribbon. nificant effects in the determination of the current across a bridge system. The purpose of the present paper is to provide a qualitative study of electron transport in honey- Here we adopt a simple tight-binding model to comb lattice ribbons with armchair edges attached describethesystemandallthecalculationsareper- to two semi-infinite one-dimensional metallic elec- formed numerically. We address the conductance- trodes (see Fig. 1). The theoretical description energy and current-voltage characteristics as func- of electron transport in a bridge system has been tions of lengths and widths of ribbons. Our results followed based on the pioneering work of Aviram clearly predicts how a honeycomb lattice ribbon and Ratner [10]. Later, many excellent experi- with armchair edges transforms its behavior from ments [11, 12, 13, 14, 15] have been done in several the semiconductingto the metallicnature,andthis bridgesystemstounderstandthebasicmechanisms feature may be utilized in fabrication of nanoelec- underlying the electron transport. Though in liter- tronic devices. aturemanytheoretical[16,17,18,19,20,21,22,23, The paper is organized as follow. Following the 24, 25, 26, 27, 28, 29, 30, 31, 32] as well as experi- introduction (Section 1), in Section 2, we present mentalpapers[11,12,13,14,15]onelectrontrans- the model and the theoretical formulations for our port are available, yet lot of controversies are still calculations. Section 3 discusses the significant re- 2 sults, and finally, we summarize our results in Sec- full system i.e., the ribbon, source and drain, the tion 4. Green’s function is defined as, G=(ǫ−H)−1 (3) 2 Model and the synopsis of where ǫ = E+iδ. E is the injecting energy of the the theoretical background source electron and δ gives an infinitesimal imagi- nary part to ǫ. To Evaluate this Green’s function, Let us refer to Fig. 1, where a honeycomb lat- the inversion of an infinite matrix is needed since tice ribbon with armchair edges is attached to two the full system consists of the finite ribbon and the semi-infinite one-dimensional metallic electrodes, two semi-infinite electrodes. However, the entire viz, source and drain. It is important to note that system can be partitioned into sub-matrices cor- throughout this study we attach the electrodes at responding to the individual sub-systems and the Green’s function for the ribbon can be effectively written as, Source G =(ǫ−H −Σ −Σ )−1 (4) rib rib S D where H is the Hamiltonian of the ribbon which rib can be written in the tight-binding model within the non-interacting picture like, Honeycomb lattice ribbon Drain Hrib = ǫic†ici+ t c†icj +c†jci (5) X X (cid:16) (cid:17) i <ij> Figure 1: Schematic view of a honeycomb lattice ribbon with armchair edges attached to two semi- In the above Hamiltonian (Hrib), ǫi’s are the site infinite one-dimensional metallic electrodes, viz, energies, c†i (ci) is the creation (annihilation) oper- source and drain. Filled circles correspond to the atorofanelectronatthe sitei andt isthe nearest- position of the atomic sites (for color illustration, neighbor hopping integral. Similar kind of tight- see the web version). binding Hamiltonian is also used to describe the twosemi-infiniteone-dimensionalperfectelectrodes the two extreme ends of nanoribbons, as seen in wheretheHamiltonianisparametrizedbyconstant Fig. 1, to keep the uniformity of the quantum in- on-site potential ǫ0 and nearest-neighbor hopping terference effects. integral t0. In Eq. 4, ΣS = h†S−ribgShS−rib and To calculate the conductance g of the ribbon, we ΣD = hD−ribgDh†D−rib are the self-energy opera- usetheLandauerconductanceformula[41,42],and tors due to the two electrodes, where gS and gD at very low temperature and bias voltage it can be correspond to the Green’s functions of the source expressed in the form, and drain respectively. hS−rib and hD−rib are the coupling matrices and they will be non-zero only 2e2 for the adjacent points of the ribbon, and the elec- g = T (1) h trodes respectively. The matrices Γ and Γ can S D be calculated through the expression, where T gives the transmission probability of an electroninthe ribbon. This(T)canbe represented Γ =i Σr −Σa (6) S(D) S(D) S(D) in terms of the Green’s function of the ribbon and h i itscouplingtothetwoelectrodesbytherelation[41, where Σr and Σa are the retarded and ad- S(D) S(D) 42], vanced self-energies respectively, and they are con- T =Tr[ΓSGrribΓDGarib] (2) jugate with each other. These self-energies can be written as [43], where Gr and Ga are respectively the retarded rib rib and advanced Green’s functions of the ribbon in- Σr =Λ −i∆ (7) S(D) S(D) S(D) cluding the effects of the electrodes. The parame- ters Γ andΓ describe the coupling of the ribbon where Λ are the real parts of the self-energies S D S(D) to the source and drain respectively, and they can whichcorrespondtotheshiftoftheenergyeigenval- be defined in terms of their self-energies. For the ues ofthe ribbonandthe imaginaryparts ∆ of S(D) 3 the self-energies represent the broadening of these with N = 3 and M = 3 corresponds to three lin- energy levels. Since this broadening is much larger ear chains attached side by side (see Fig. 1) where thanthethermalbroadening,werestrictourallcal- each chain contains three hexagons. For simplic- culations only at absolute zero temperature. All ity, throughout our study we set the Fermi energy theinformationsabouttheribbon-to-electrodecou- E =0 and choose the units where c=e=h=1. F pling are included into these two self-energies. Let us first describe the variation of the conduc- The currentpassingacrossthe ribboncan be de- tancegasafunctionoftheinjectingelectronenergy picted as a single-electron scattering process be- E. InFig.2 we presentthe conductance-energy(g- tween the two reservoirs of charge carriers. The E) characteristics for some honeycomb lattice rib- current I can be computed as a function of the ap- bonswithfixedwidth(N =1)andvaryinglengths, plied bias voltage V through the relation [41], where (a) and (b) correspond to the linear chains with six (M = 6) and ten (M = 10) hexagons e EF+eV/2 respectively. The conductance spectra shows fine I(V)= T(E,V)dE (8) π¯hZ EF−eV/2 2.0 whereE is the equilibriumFermienergy. Here we F make a realistic assumption that the entire voltage HaL is dropped across the ribbon-electrode interfaces, and it is examined that under such an assumption L E 1.0 the I-V characteristicsdonotchangetheir qualita- Hg tive features. This assumption is based on the fact that, the electric field inside the ribbon especially for narrow ribbons seems to have a minimal effect 0.0 on the conductance-voltage characteristics. On the -4 -2 0 2 4 other hand, for quite larger ribbons and high bias E voltagestheelectricfieldinsidetheribbonmayplay a more significant role depending on the internal 2.0 structure and size of the ribbon [43], but the effect becomes too small. HbL L 3 Results and discussion E 1.0 H g In order to understand the dependence of electron transport on the lengths and widths of nanorib- 0.0 bons, in the present article, we concentrate only -4 -2 0 2 4 on the cleaned systems rather than any dirty one. E Accordingly, we set the site energies of the honey- comb lattice ribbons as ǫ = 0 for all i. The values i of the other parameters are assigned as follow: the Figure2: Conductancegasafunctionoftheenergy nearest-neighborhopping integraltin the ribbonis E for some lattice ribbons with fixed width N = 1 set to 2, the on-site energy ǫ0 and the hopping in- and varyinglengths where (a) M =6 and (b) M = tegral t0 for the two electrodes are fixed to 0 and 2 10 (for color illustration, see the web version). respectively. The parameters τ and τ are set as S D 1.5,wheretheycorrespondtothehoppingstrengths resonance peaks for some particular energies,while of the ribbon to the source and drain respectively. for all other values of the energy E, either it (g) In addition to these, we also introduce two other dropstozeroorgetsmuchsmallvalue. Attheseres- parametersN andM torevealthesizeofananorib- onance energies, the conductance gets the value 2, bon,wheretheycorrespondtothewidthandlength and hence, the transmission probabilityT becomes of the ribbon respectively. Thus, for example, a unity since the expression g = 2T holds from the nanoribbon with N = 1 and M = 4 represents a Landauer conductance formula (see Eq. 1). These linear chain of four hexagons. Hence the parame- resonancepeaksareassociatedwiththeenergylev- ter M determines the total number of hexagons in els of the nanoribbons and thus the conductance a single chain. Following this rule, a nanoribbon spectra, on the other hand, reveal the signature of 4 the energy spectra of the nanoribbons. The most in Figs. 2 and 3, we can emphasize that for a fixed importantissueobservedfromthesespectraisthat, width the centralenergygapalwaysdecreaseswith a central gap appears across the energy E =0 and the size of the nanoribbon. Now to reveal the de- the width of the gap becomes small for the chain pendenceoftheenergygaponthesystemsizemuch with 10 hexagons compared to the other chain i.e., more clearly, in Fig. 4 we show the variation of the the chain with 6 hexagons. It predicts that, for a central energy gap δE as a function of the length fixed width, the central energy gap decreases with M for some honeycomb lattice ribbons with differ- the increase of the length of the nanoribbon. In ent widths N. The red, green and blue lines cor- thesamefooting,tovisualizethedependenceofthe respond to the results for the ribbons with fixed widthontheconductance-energycharacteristics,in widths N = 1, 2 and 4 respectively. These results Fig. 3 we display the results for some honeycomb clearly emphasize that for the fixed width the gap lattice ribbons considering the width N =4, where gradually decreases with the increase of the length ofthenanoribbon. Itisalsoexaminedthatformuch larger lengths it (δE) almost vanishes (not shown 2.0 here in the figure). Quite similar nature is also ob- served if we plot the variation of the energy gap as HaL a function of the length N keeping the width M L as a constant, and due to the obvious reason we E 1.0 Hg do not plot these results further in the present de- 2.5 0.0 -4 -2 0 2 4 2.0 N=1 E 1.5 2.0 ∆E 1.0 N=4 HbL 0.5 N=2 L E 1.0 Hg 0.0 2 4 6 8 10 12 14 16 18 M 0.0 Figure 4: Variationofthe centralenergygapδE as -4 -2 0 2 4 a function of the length M for some lattice ribbons E withfixedwidthsN. Thered,greenandbluecurves correspondtoN =1,2and4respectively(forcolor illustration, see the web version). Figure3: Conductancegasafunctionoftheenergy E for some lattice ribbons with fixed width N = 4 and varying lengths (identical as in Fig. 2) where scription. These results provide us an important (a) M = 6 and (b) M = 10. Here the width of signature which concern with a transition from the theribbonsisincreasedcomparedtotheribbonsas semiconducting (finite energy gap) to the conduct- taken in Fig. 2 (for color illustration, see the web ing (zero energy gap) material, and this transition version). can be achieved simply by tuning the size of the nanoribbon. (a) and (b) represent the nanoribbons with identi- All these basic features of electron transfer can cal lengths as in Fig. 2. The results show that, due be quite easily explained from our study of the to the largesystemsizesthe g-E characteristicsex- current-voltage (I-V) characteristics rather than hibitalmostaquasi-continuousvariationacrossthe the conductance-energy spectra. The current I is energy E = 0. For both these two ribbons the en- determined from the integration procedure of the ergygapalsoappearsaroundtheenergyE =0,and transmission function (T) (see Eq. 8), where the the gapdecreaseswith the increase ofthe lengthof function T varies exactly similar to the conduc- the nanoribbon. Comparing the results presented tance spectra, differ only in magnitude by a fac- 5 tor 2, since the relation g = 2T holds from the strong coupling limit, described by the condition Landauer conductance formula (Eq. 1). The varia- τ ∼ t, current varies quite continuously with S(D) tion of the current-voltage characteristics for some the bias voltage V and achieves large current am- typical honeycomblattice ribbons with fixed width plitude compared to the weak-coupling limit. All N = 2 and varying lengths is presented in Fig. 5, these coupling effects have clearly been explained where (a) and (b) correspond to the ribbons with in many papers in the literature. The significant M = 3 and 5 respectively. The current exhibits feature observedfrom the figure (Fig. 5) is that for a staircase like behavior as a function of the ap- the fixed width (N = 2), the threshold bias volt- .8 1 HaL HaL .4 .5 HLI HLI ent 0 ent 0 r r r r u u C -.4 C -.5 -.8 -1 -4 -2 0 2 4 -4 -2 0 2 4 VoltageHVL VoltageHVL .8 1 HbL HbL .4 .5 HLI HLI ent 0 ent 0 r r r r u u C -.4 C -.5 -.8 -1 -4 -2 0 2 4 -4 -2 0 2 4 VoltageHVL VoltageHVL Figure5: CurrentI asafunctionofthebiasvoltage Figure6: CurrentI asafunctionofthebiasvoltage V for some lattice ribbons with fixed width N = 2 V for some lattice ribbons with fixed width N = 3 and varyinglengths where (a) M =3 and (b) M = and varyinglengths where (a) M =2 and (b) M = 5 (for color illustration, see the web version). 3 (for color illustration, see the web version). plied bias voltage V. This staircase like nature ap- age (V ) of electron conduction decreases with the th pearsduetotheexistenceoftheresonancepeaksin increase of the length of the ribbon. This reveals the conductance spectra since the current is com- a transformation towards the conducting material. puted by the integration process of the transmis- Quiteinthesamefashion,toseethevariationofthe sion function T. As we increase the bias voltage thresholdbiasvoltageV for othersystemsizes,in th V, the electrochemical potentials in the two elec- Fig.6weplottheresultsforsomenanoribbonswith trodes cross one of the energy levels of the ribbon fixed width N = 3 and varying lengths where (a) and accordingly a jump in the I-V curve appears. and (b) correspondto the ribbons with M =2 and The sharpness of the steps in the current-voltage 3 respectively. The results show that the thresh- characteristicsandthe currentamplitude solelyde- old bias voltages decrease much more compared to pend on the coupling strengths of the nanoribbon the nanoribbons of width N = 2. Thus both from to the electrodes, viz, source and drain. 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