Table Of ContentElectron transport through honeycomb lattice ribbons with
armchair edges
Santanu K. Maiti1,2,∗
9 1Theoretical Condensed Matter Physics Division, Saha Institute of Nuclear Physics,
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1/AF, Bidhannagar, Kolkata-700 064, India
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2 2Department of Physics, Narasinha Dutt College, 129, Belilious Road, Howrah-711 101, India
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Abstract
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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
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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.
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PACS No.: 73.63.-b;73.63.Rt.
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d Keywords: Honeycomb lattice ribbon; Armchair edges; Conductance; I-V characteristic.
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∗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. It is ob- Figs.5and6weclearlyobservethatV canbereg-
th
served that, in the limit of weak coupling, defined ulated very nicely by tuning the size (both length
by the condition τ << t, current shows stair- and width) of the nanoribbon. For quite largerrib-
S(D)
case like structure with sharp steps. While, in the bons the threshold bias voltage eventually reduces
6
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8
