Effects of metallic contacts on electron transport through graphene

0.48 MB·English

by  Salvador Barraza-Lopez

Effects of metallic contacts on electron transport through graphene
Salvador Barraza-Lopez,1 Mihajlo Vanevi´c,1,2 Markus Kindermann,1 and M. Y. Chou1
1School of Physics, Georgia Institute of Technology, Atlanta, Georgia 30332, USA
2Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands
(Dated: January 29, 2010)
Wereportonafirst-principlesstudyoftheconductancethroughgraphenesuspendedbetweenAl
contacts as a function of junction length, width, and orientation. The charge transfer at the leads
and into the freestanding section gives rise to an electron-hole asymmetry in the conductance and
0 insufficientlylongjunctionsinducestwoconductanceminimaattheenergiesoftheDiracpointsfor
1 suspendedandclampedregions,respectively. Weobtainthepotentialprofilealongajunctioncaused
0
bydopingandprovideparametersforeffectivemodelcalculations ofthejunctionconductancewith
2
weakly interacting metallic leads.

n
a PACSnumbers: 72.80.Vp,73.40.Ns,81.05.ue,05.60.Gg
J

9
2 Introduction. Grapheneisatwo-dimensionalallotropeof 13.6 nm long with various W/L ratios.
carbonwithatomsarrangedinahoneycomblattice. Suc- The difference in the work functions of the metal and

] cessful fabrication of graphene monolayers by means of graphene leads to the charge transfer and doping of the
i
c mechanicalexfoliationofgraphite[1]orepitaxialgrowth graphene layer [16]. We obtain the corresponding po-
s
- on silicon carbide [2] has ignited tremendous interest in tential profile generated by the doping of the graphene
rl this material [3]. Its favorable characteristics such as layer along the junction. Not surprisingly, the potential
mt chemical inertness, low dimensionality, extremely high intheregioncontactedbythemetalstartsdeviatingfrom
mobility, and easy control of carriers by applied gate its bulk value before reaching the geometrical edge. Fi-
.
t voltages, along with patterning using nanolithography, nite doping results in specific transport features for long
a
m open possibilities for further miniaturization of devices junctions: two conductance minima appear at the ener-
andthe emergence ofa carbon-based“post-silicon”elec- gies of the Dirac points of graphene in the clamped and
-
d tronics [2, 4]. suspended regions, respectively. For shorter junctions,
n Theelectrontransportatthenanometerscaleissignifi- where the two minima cannot be resolved, an electron-
o
cantlyaffectedbythecontacts. Theroleofmetallicleads hole asymmetry in the conductance is still appreciable.
c
[ in determining the transport properties of graphene- The impact on transport of the orientation of graphene
based junctions has been addressed by several theoret- in the junctions is found to be negligible away from the
2
ical [5–10] and experimental [11–15] studies. Yet, a Dirac point. Using the potential profile obtained from
v
parameter-free description of transport in these systems ourfirst-principlescalculationsandasmallenergybroad-
7
5 is still lacking. Previously, the leads were modeled as ening in the self-energies at the leads, we demonstrate
2 infinitely doped graphene regions that support a large that a π-electron tight-binding model can accurately re-
5 number ofpropagatingmodes [5]. Inthis model, the im- produce our first-principles transport results. This en-
.
1 portant parameter is the ratio W/L between the width ables us to predict the conductance for graphene junc-
0 (W)andlength(L)ofajunction: attheDiracpoint,the tions with leads made of other metals, and we present
0 universal, diffusive-like regime is reached for W/L ≫ 1. results for the conductance of junctions with Au leads.
1
Additional studies within a tight-binding approach have We perform transport calculations at zero source-
:
v been reported, in which either the leads form a square drain bias using the nonequilibrium Green’s function
i
X lattice [6, 7], or the coupling to the leads is modeled via (NEGF) SMEAGOL code [17], which is interfaced with
energy-level broadening [8]. the density-functional-theory (DFT) SIESTA package
r
a In this Letter, we report on large-scale first-principles [18]. We employ norm-conserving pseudopotentials [19]
calculations ofelectron transportin suspended graphene in the local-density approximation (LDA) [20], and a
takingintoaccounttheeffectsofmetallicleads. Westudy real-space grid equivalent to an energy cutoff of 310 Ry.
anon-magneticjunctionmadeofgraphenecontactedun- We explicitly construct fine-tuned basis sets for C and
derneath by two aluminum (Al) leads with a small in- Al atoms following the prescription of Junquera et al.
plane mismatch of less than 1%. This junction is a pro- [21]. The largest number of atoms (numerical orbitals)
totype system with weak interaction between the metal included in the present calculations is 464 (5600). We
contact and graphene, with no covalent bonds formed. made modifications to the transport code to improve
Our work represents the first quantitative study of elec- memoryallocationandparallelization,inordertohandle
tron transport through metal-graphene junctions to ex- calculations of this scale.
amine in detail previous models [5, 8] with dimensions Graphene junctions without metal contacts. Firstwedis-
relevant to experiment [11–14], up to 100 nm wide and cuss pristine graphene with no metallic leads attached.
2
(a) Graphene
6 layers of
Side
Aluminum
x L
zTop w 1
0
x
w 1'
z 0'
0.0
(b)
−0.2
V]
e−0.4
Vz ()[−0.6 1
2
3
−0.8
FIG. 1: (a) Band structure and (b) density of states (DOS) −8 −6 −4 −2 0 2 4 6 8
for graphene near the Dirac point (ED = 0). The two van z [nm]
Hovesingularitiesaremarked. (c)Conductanceofagraphene
ribbon98.1nmwidewithoutmetallicleads. Inset: Theresis- FIG. 2: (Color online) (a) Schematics of Al-graphene junc-
tance R diverges at the Dirac point. Dashed lines in (b) and tions. The semi-infinite leads, shown in the shaded regions,
(c) are results of tight-binding calculations. (d) Band struc- includebothgrapheneandAl. Theedgesoftheribbonareof
tureand(e)projectedDOSoncarbonatomsforgrapheneon either armchair (1) or zigzag (1’). The unit cell sizes along
Al. The highlighted lines in (d) are graphene bands. the x-direction are w0 =0.49 nm and w0′ =0.85 nm, respec-
tively. ListhedistancebetweenAlatomsatoppositesidesof
thejunction. (b) Energy location of theDirac point with re-
spect tothe Fermilevel across the junctions. Junctions 1, 2,
The band structure and density of states (DOS) in the and3haveafixedwidthW =98.1nm,andlengthsL=3.40
vicinity of the Dirac point (ED) are shown in Figs. 1(a) nm, 6.80 nm, and 13.60 nm, respectively. The DFT results
and(b), respectively. The DOSaroundED canbe fitted (dots) are shown along with fitted curves(see text).
to g(E)=D|E−ED| with D =0.11/(eV2 unit cell). In
all transport calculations described in this paper, an ef-
fective ribbon width is determined by W =nkw0, where sition of the Dirac point ED with respect to the Fermi
nk is the number of k-points used in the calculation and levelEF atthe lead,∆=ED−EF ≃−0.6eV,graphene
w0 is the size of the unit cell in the transverse (x) direc- becomes n-doped in the contact region. The DOS pro-
tion. We show in Fig. 1(c) the conductance G as a func- jectedonCatomsattheleadsgC isnonzeroattheDirac
tionofelectronenergyforagrapheneribbonofW ≈100 point with a finite value of gC(ED) ∼ 2 × 10−2 elec-
nm. Atthiswidth,thequantizedplateausintheconduc- trons/eV per graphene unit cell [Fig. 1(e)]. The fluctua-
tance become unresolvable within the energy resolution tions of gC above EF are due to hybridization.
employed(0.01eV),andGresemblesthelinearbehavior The junctions studied are schematically shown in
of the DOS around ED. The boundary conditions and Fig. 2. The leads are displayed within the shaded ar-
type of edge – zigzag or armchair – have an impact on eas. The Hamiltonian, overlap, and density matrices of
the conductance only at the vicinity of the Dirac point, the leads are obtained self-consistently from DFT bulk-
in a narrow energy range inversely proportional to the calculations and used to set up the electron density,
ribbon width. Away from ED, the conductance remains chemicalpotential,andself-energiesforthetransportcal-
unaffected by this choice with W being the same as the culations. Since the leads include graphene, the contact
geometric width in the widest ribbons studied. Tight- area between metal and graphene is essentially infinite.
binding results with a nearest-neighbor hopping param- Thisenhancesthetransparency[9],eliminatingthelarge
eter γ =−2.65eV determined fromthe LDA bandstruc- contactresistancethatwouldariseifchargecarrierswere
ture are shown by dashed lines in Figs. 1(b) and (c) for to tunnel from Al to graphene. An additionalnumber of
comparison. TheagreementwiththeDFTresultsisquite Al atoms between the leads and the freestanding section
good within ±1 eV of the Dirac point. are added so that the self-consistent electrostatic poten-
Junctions with Al-graphene contacts. The electronic tial across the junction develops smoothly. We have in-
structure of graphene in contact with Al (henceforth re- vestigated various configurations to understand the ef-
ferred to as the lead) shows a certain amount of hy- fects of the length (L), width (W), and orientation of
bridizationarisingfromtheinteractionbetweengraphene the graphene ribbon on electron transport. Junctions 1,
and the metal [Fig. 1(d)]. The leads are modeled by 1’, 2 and 3 have lengths L = 3.40 nm, 3.44 nm, 6.80
six layers of Al in the (111) orientation with a graphene nm, and 13.60 nm, respectively. The effect of varying
monolayer placed at a distance of 0.34 nm away, in the L is studied through junctions 1, 2, and 3 with a fixed
minimal-energyconfiguration[16]. Giventherelativepo- width of W =98.1 nm [see Fig. 2(b) for geometries]; the
3
ima were also observed in junctions with gate-tunable
160 2e2/h]234 barriers [24]. The inset in Fig. 3(a) shows the minimal
2e/h]2 12800 G [Min010L=40 5nLm [nm]10 15 ctibholenedofurodcrteatrhnoecfet2wneeo2a/srhh.EoArDtesfrmorjuatlnhlcepteivoaanklsuae1tsEaonfDdL(2−s,t0ua.d6nideedVb;)eictisoimvsieoss-f
G [ 40 Model L [nm] unnoticeableforthelongerjunction3. Thissmallpeakis
13.6
(a) 6.8 duetostatesthatexistonlyinthecontactareasthrough
3.4
−02.0 −1.6 −1.2 −0.8 −0.4 0.0 0.4 0.8 1.2 hybridization with Al atoms and decreases rapidly with
E [eV] increasing junction length.
102
InFig.3(b)theconductanceisplottedfortheshortest
101 junction (L= 3.4 nm) with four values of width W: 9.8
G [2e/h]2 1100−10 W [9n8m.1] nsdcmeanl,cee2o4of.f5thtnehmeco,cno4dn9ud.1cutcantnmacn,eceahneoldnps9W8ex.1afomnrmine.enetTrhgehieelsinloseugaffiarrdcitieehpnmetnliyc-
49.1
10−2 (b) 24.5 away from ED. As mentioned above, the peak at −0.6
9.8
10−3 eV only appears in very short junctions. This is yet to
−1.5 −1.0 −0.5 0.0 0.5
beobservedinexperiment,wherelongerribbonsarecus-
E [eV]
tomarily employed [25]. Set aside this peak, the open-
ing of a conduction gap with decreasing W is evident,
FIG. 3: (Color online) (a) Conductance of junctions with a
widthofW =98.1nmanddifferentlengthsL. Thecurvesare as well as the significant and nonlinear drop in conduc-
offset by 20×2e2/h for clarity. Two prominent conductance tance for the narrowest junction with W = 9.8 nm, in
minima at ED (−0.6 eV) and EF (set to zero) develop as L full accordance with experiment [25]. To investigate the
increases. The symbols represent first-principles results, and effect of the edge orientation, we study the conductance
the solid lines are from model calculations (see text). Inset: of armchair (1) and zigzag (1’) junctions of comparable
minimum conductance versus L (the dashed line is a guide
dimensions [Fig. 4(a)]. Only small deviations are found
to the eye). (b) Conductance of junctions with a length of
resultingfromthe anisotropyofthe bandstructure. The
L=3.4nmanddifferentwidthsW,highlightingtheopening
ofaconductiongapasW decreases andasharppeakpinned effect of orientation is quite small in the vicinity of ED.
at ED =−0.6 eV. Tight-binding model. We next present a tight-binding
model that can be used to study junctions bigger in size
and with arbitrary nonbonding metal contacts. To that
conductanceGasafunctionofW isstudiedforjunction end,weestimatetheeffectivepotentialprofileV(z)atthe
1 (L = 3.40 nm); and the effect of ribbon orientation atomic scale for π-electrons along the junction, directly
(armchair versus zigzag edges) is studied for junctions 1 related to the spatial dependence of doping created by
and 1’ by choosing a similar width for them: W = 98.1 themetallicleads. Thisprofilecanbeobtainedsemiclas-
nm (1) and W′ = 98.6 nm (1’). (The potential profiles sically from DFT as the value of the energy at the Dirac
in Fig. 2(b) will be discussed later.) point with respect to the Fermi level, as a function of z:
Theconductanceforjunctions1–3(withafixedwidth V(z)= ED(z)−EF. We extract this quantity from the
of W = 98.1 nm) is shown in Fig. 3(a). One notices the position-dependentenergyshiftofthevanHovesingular-
lack of symmetry with respect to both the Fermi level ities closest to the Dirac point [denoted by vH1 and vH2
(energy zero) and ED in the lead (−0.6 eV) in all cases inFig.1(b)]alongthejunction. Theresultsforjunctions
regardless of the particular values of L, in contrast to 1–3 are shown in Fig. 2(b). Significant doping occurs at
the symmetric curve for pristine graphene in Fig. 1(c) distances up to only a few nanometers from the edge of
and those displayed in Ref. [5]. Similar asymmetries metallic leads. For the two longer junctions, 2 (L=6.80
in the conductance have been seen experimentally, e.g. nm)and3 (L=13.60nm), ED is veryclosetoEF inthe
Refs. [13] and [22]. In addition to the conductance min- middle ofthe freestandingsection(z ∼0). We fitted our
imum at ED, the emergence of a second one at EF is DFT results for V(z) by the following expression [solid
apparent, and it becomes more prominent as the length lines in Fig. 2(b)]:
increases. This is because, as L→∞, the electron DOS
∆ cosh(z/λ)/cosh(L /2λ) for |z|<L /2,
approaches g(E) at the freestanding part of the junc- V(z)= eff eff
tion and the conductance becomes that of two resistors (∆ otherwise.
inseries[22]: G(E)∝g(E)gC(E)/[g(E)+gC(E)], where (1)
gC(E)∝D|E−∆| is the projected DOS of the clamped The parameters used are ∆ = −0.6 eV, λ = 1.05 nm
region. Based on the same underlying physics, a double and L = L+5a , where a = 0.142 nm is the C-C
eff 0 0
peak in resistance was experimentally demonstrated for interatomic distance. We emphasize that on the sub-
graphenedevicescomprisingtwo regionswithnoticeably nanometer scale, V(z) starts to deviate from its asymp-
different doping levels [23]. The two conductance min- totic value inside the region of intermediate Al atoms,
4
160 40 asanisolatedgraphenesheetwithaspace-dependentpo-
Orientation Au, tight-binding
tential [5]. However, we have demonstrated that to ob-
120 Armchair 30
h] Zigzag h] L [nm] tain accurate results, one needs to take into account a
e/2 80 e/220 13.6 small energy broadening [8] and allow for a finite poten-
G [2 40 G [210 40.0 tial at the contacts.
(a) (b) Conclusion. We have performed transport calcula-
0 0
−1.6 −0.8 0.0 0.8 −0.25 0.00 0.25 0.50 tions for graphene junctions attached to Al leads using
E [eV] E [eV] first-principlesnonequilibriumGreen’sfunctionmethods.
Theconductancefeaturesvarywiththelengthandwidth
FIG.4: (Coloronline)(a)Conductanceforjunctions1(arm- ofthe junction,butarelesssensitivetotheribbonorien-
chair edges, L = 3.40 nm, W = 98.1 nm, light line) and 1’ tation. Weshowthatnonbondingmetallicleadsinducea
(zigzag edges, L = 3.44 nm, W = 98.6 nm, dark dots). (b)
noticeable electron-hole asymmetry in the conductance.
Conductance for junctions consisting of graphene with Au
The opening of a conduction gap with decreasing W is
leads for two values of L from model calculations (see text).
also captured. Two minima in the conductance emerge
The junction width is W =98.1 nm.
for large enough junctions. In addition, our calculations
yield reliable information on the doping variation along
the junction for metallic leads interacting weakly with
before reaching the atomic boundary of the contact in-
graphene, and we find accurate potential profiles at the
dicated by the vertical dashed line in Fig. 2(b). In our
vicinityofthemetalboundary. Wedemonstratethatthe
calculations,sufficiently manyAl atomsare includedbe-
dominant features of our first-principles results can be
tween the bulk (shaded) leads and the freestanding sec-
reproduced by an analytically tractable effective model.
tion in order to map the transitional behavior of V(z).
The parameters of the effective model are derived from
This profileprovidesa refinementofmodels wheresharp
first-principles. As an application we use the effective
steps at the metal edge are assumed [5, 12] and will be-
model to predict the conductance in junctions with Au
come relevant as experimental junctions shrink in size.
as the supporting metal at the leads.
We have performed conductance calculations within a
We thank L. Xian, K. Park, and E. Yepez for help-
tight-binding approach using the potential profile V(z)
ful discussions. This work is supported by the Depart-
and a smearing δ =8 meV introduced through the leads
ment of Energy (Grant No. DE-FG02-97ER45632). We
self-energies [8] (to account for the nonzero conductance
acknowledge interaction with the Georgia Tech MRSEC
at ED and EF and the small peak at ED in Fig. 3). funded by the National Science Foundation (Grant No.
We obtain excellent agreement with our full-scale calcu-
DMR-02-05328) and computer support from Teragrid
lations [as indicated by the solid lines in Fig. 3(a)], with
(TG-PHY090002)and NERSC.
theexceptionofsharpdipsintheconductance[indicated
by arrowsin Fig. 3(a)] which are not reproduced. These
dipsoccuratthe anticrossingsenergiesdue tohybridiza-
tionwithaluminum[cf.bandstructureinFig.1(d)]. We
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