Table Of ContentTwin lead ballistic conductor based on nanoribbon edge
transport
Martin Konˆopka and Peter Dieˇska
SLOVAK UNIVERSITY OF TECHNOLOGY in Bratislava,
Faculty of Electrical Engineering and Information Technology,
Institute of Nuclear and Physical Engineering,
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Department of Physics,
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Ilkoviˇcova 3, 812 19 Bratislava, Slovak Republic
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M E-mail: [email protected]
1 31st March 2017
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]
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a Abstract
h
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If a device like a graphene nanoribbon (GNR) has all its four corners attached to electric
s
e
current leads, the device becomes a quantum junction through which two electrical circuits
m
. can interact. We study such system theoretically for stationary currents. The 4-point energy-
t
a
dependent conductance matrix of the nanostructure and the classical resistors in the circuits
m
- are parameters of the model. The two bias voltages in the circuits are the control variables
d
n of the studied system while the electrochemical potentials at the device’s terminals are non-
o
triviallydependentonthevoltages. Forthespecialcaseofthelinear-responseregimeanalytical
c
[ formulae for the operation of the coupled quantum-classical device are derived and applied.
2 For higher bias voltages numerical solutions are obtained. The developed approach allows to
v
5 study scenarios ranging from independent circuits to strongly coupled ones. The analytical
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formulae transparently show, among other quantities, how a difference between quasi-Fermi
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3
levels of a particular pair of leads can be continuously tuned to a target value including zero.
0
. ForthechosenmodeloftheGNRwithhighlyconductivezigzagedgeswedeterminetheregime
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0 in which the single device carries two almost independent currents.
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: Keywords: multiterminal,conductance,transport,edge,corner,graphene,nanoribbon,
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i chemical potential
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1 Introduction
Electronic transport through graphene nanoribbons has attracted attention in particular
because their zigzag edges exhibit magnetic properties and support specific modes which
1
may influence the charge transport [1, 2, 3]. Edges of larger-scale graphene ribbons have
been proposed to serve similarly or analogously to optical fibres or waveguides [4]. In
experimental works studying electronic transport through a nanoribbon, the flake is typ-
ically contacted to electrodes across its whole width, see for instance Ref. [5]. However,
if the edges are to be used for the transport, it may be appropriate to make contacts
only to the corners of such ribbon. This poses a question if a single graphene nanorib-
bon or an alternative planar atomistic structure could serve as two wires carrying two
independent electric currents. Obviously, there would be some interaction between the
currents flowing along the two parallel edges and they would not be fully independent.
Still, if the energy ranges of the high density of states, resulting from the zigzag (ZZ)
edge modes, are used for the transport, the conductance along the ZZ direction may
be significantly larger than the conductance along the perpendicular (armchair, AC)
direction. In such case the two ZZ edges might be considered as two relatively indepen-
dent spatially separated conductors. More generally, such scenario would be the case if
the 4×4 conductance matrix of a ribbon preferred one of the directions over the per-
pendicular and diagonal directions. On the other hand, a nanoribbon with significant
conductance along several directions would allow to study the intriguing regime in which
the two classical circuits are coupled through the quantum ballistic device. Regardless
of the conductance characteristics, such setups would naturally require four electrodes
contacted to the corners of the nanoribbon.
Electronic transport through graphene flakes with electrodes contacted at their corners
has been studied computationally in the work [6] with focus on non-rectangular flakes
such as trapezoids or triangles. The work has addressed the question how the two-
contact conductance qualitatively depends on the magnitude of the angles in the two
contacted corner areas and on the types of the edges forming the corner. It was in
addition found that the usual nearest-neighbor (NN) tight-binding (TB) model was
insufficient for proper description of the edge-induced transport properties (provided
that the electrodes were attached to the corners).
Four-terminal phase-coherent conductance has been considered by Bu¨ttiker [7] in order
to clarify the occurrence of asymmetric magnetoresistances. The author considers a
sample contacted to two circuits which is a setup similar to the one considered in the
present work. On the contrary, the model of Bu¨ttiker does not include any classical
resistorsandislimitedtothelinearregime. Wewillreturntoitinthepresentationofour
results. Four-terminal schemes have been studied both theoretically and experimentally
in setups with two voltage probes implying that two of the four terminals carry zero
net currents. Analysis of this specific model can be found as early as in the above
cited work by Bu¨ttiker. In Ref. [8] an experimental realisation has been described
using a one-dimensional single-mode quantum wire. The authors observed vanishing
2
four-terminal resistance in the wire, in accordance with the theoretical considerations
that the resistance of a perfect quantum wire should be zero. A different type of a four-
terminal quantum transport device was considered in Ref. [9], with the proposal of using
such a device for electron wave computing. Coherent electronic circuits characterised by
capacitances in addition to resistances have been studied in the literature. In Ref. [10]
violation of Kirchhoff’s laws has been experimentally confirmed for a fully coherent
quantum resistance-capacitance circuit with the capacitor formed by a quantum dot.
A newer review of the coherent RC circuit experimental probing with focus on time-
dependent transport can be found in Ref. [11].
In this paper we study a coupled quantum-classical device consisting of a single GNR
and of two classical circuits with the four leads contacted to the corners of the GNR.
The 4-terminal quantum subsystem is assumed to function in the ballistic regime. The
electric currents are driven by DC bias voltage sources in the circuits which include
resistors. Transparent analytical formulae displaying the interplay of the quantum and
classical elements of the system are derived for both the electric currents as well as
the electrochemical or electrostatic potentials in the device’s leads. We show that under
certain conditions the two electrical circuits of the scheme operate almost independently.
Hence such a GNR could serve as two relatively independent conductors which might
be found useful in experimental setups and practical applications as it would allow to
shrink the device size. The regime in which the circuits are significantly coupled through
the ballistic device presents an interesting physical model and is cover by our study as
well.
In Sect. 2 we define the system under study. In Sect. 3 we derive main results of our
work both for the linear and non-linear regimes. We also provide several details on
the quantum-mechanical description of the studied system. In Sect. 4 we apply the
theoretical modelling to study the setup defined in Sect. 2. We conclude our results in
Sect. 5.
2 The model
The physical model. An atomistic model of one of the considered GNRs is shown in
Fig. 1. The red-marked carbon atoms in the corners form the contact areas where the
four electrodes (not shown) are attached. If this is accomplished, the whole system in a
simple electrical setup can be described by the scheme drawn in Fig. 2. The resistances
R and R of the two classical resistors are known parameters. The control variables
A B
are the static bias voltages V and V on the two DC sources, not the electrochemical
A B
potentials in the leads. The potentials as well as the two currents are unknown non-
trivialfunctionsofthebiasvoltagesandhavetobedeterminedasitisexplainedindetail
3
Figure 1: The graphene nanoribbon (GNR) with the four electrodes attached at its corners. Each
electrode is modelled by a bunch of 54 identical mono-atomically thin wires [6] (not shown) coupled
to the red-marked carbon atoms. Precise GNR dimensions are L = 71b (along the ZZ edges) and
W = 62a (along the AC edges), with a = 0.142nm being the nearest-neighbor distance in graphene
√
and b = a 3 the lattice parameter. The GNR is composed of 6006 atoms. In an alternative adapted
convention [12] the GNR’s dimensions are (143,42).
below in the text. The stationary regime is assumed for all elements of the scheme. We
consider a low-temperature regime in which inelastic scattering of the electrons in the
GNR and in its contacts can be neglected. Hence the GNR with its contacts represents
a quantum subsystem of the entire scheme. The electronic transport properties of such
a contacted device can therefore be described by a 4×4 conductance matrix G. This
matrix is computed using a microscopic model described below. For sufficiently low bias
voltages the linear regime with I ,I ∝ V ,V can be assumed and the conductance
A B A B
matrix is then considered as energy-independent. This will allow us to derive analytical
formulae for currents I (V ,V ) and I (V ,V ) in terms of the fixed system parameters
A A B B A B
G, R , and R .
A B
We recall that in the scheme under study (Fig. 2) not only the currents but also the
electrochemical potentials µ , µ and µ are unknown non-trivial functions, µ (V ,V ),
2 3 4 α A B
which have to be determined and must strictly be distinguished from the bias voltages.
Thisrequirementisdictatedbytheschemeitselfinwhicheventhedifferencesµ −µ and
2 1
µ −µ are a priori unknown because the currents through the resistors are unknown.
4 3
Thechemicalpotentialattheterminal1isfixedbytheground-levelcondition. Westudy
also the non-linear regime in which the conductance matrix G is energy dependent.
The microscopic description. We consider GNRs with the electrodes attached to
their corners according to Fig. 1. The model of the electrodes and the electronic struc-
ture description are similar to those used in Ref. [6] with a notable exception of the
more general model of the electrodes in the present work. We use the independent-
electron approximation and an extended tight-binding (TB) hamiltonian for the GNR
with interactions up to the 3rd NN included. The hopping parameters t = −2.97eV,
B
t(cid:48) = −0.073eV and t(cid:48)(cid:48) = −0.33eV are taken from Ref. [13]. The orbital overlaps are
B B
neglected. The extended interaction range within the GNR was found important to
4
Figure 2: The electric scheme of the considered device. Given control variables are the two bias
voltages V a V . Additional parameters are the classical resistances R and R and the 4 × 4,
A B A B
possibly energy-dependent, conductance matrix G of the central ballistic device. Stationary regime
is assumed for each element of the scheme. The arrows on the leads mark the conventionally positive
directions of the currents I and I in the circuits.
A B
account for the contributions of the ZZ edge state to the electronic transport if the elec-
trodes are attached to corners of graphene flakes [6]. We do not consider the spin-related
effects; the focus of the present work is the description of a four-terminal nanoribbon de-
vice coupled to two classical branches of the circuits. A generalisation of our formalism
including the spin effects at least on the level of the mean-field approximation to Hub-
bard hamiltonian would be relatively straightforward although computationally much
more expensive [14]. Quantitatively accurate modelling of GNR’s electronic structure
especially for narrow ribbons would also require an inclusion of the electron-electron
interactions into the model [12].
Eachelectrodeinourmodelisformedbyabunchofmutuallynon-interactingmonoatom-
ically thin wires [15, 6] described within the 2nd NN approximation with the couplings
t = t and t = t(cid:48) . The particular model drawn in Fig. 1 and used for our quantitative
1 B 2 B
analyses employs 54 wires per electrode. Using just a single wire would allow us to
study very small currents only. The introduction of the 2nd NN couplings t for the
2
atoms of the wires is the only important modification of the used microscopic theory in
comparison to work [6]. The equilibrium on-site (atomic orbital) energies are denoted
as (cid:15) and are assumed to be the same for the entire system, including the electrodes.
The extended TB model of the monoatomically thin infinite periodic wires yields the
one-electron dispersion relation
N
(cid:88)far
E(K) = (cid:15)+ 2t cos(nK) (1)
n
n=1
5
with N being farthest nearest neighbor considered in the wires; here N = 2. The
far far
dispersion relation can be applied also to the electrons in the semi-infinite wires which
the electrodes of our model are composed of. Infinite as well as semi-infinite wire’s
eigenstates start at the energy E = E(0) = (cid:15) + 2t + 2t . For typical numerical
min 1 2
parameters of the extended TB model the dispersion relation supports eigenstates up to
E = E(π) = (cid:15)−2t +2t . The Fermi energy of the wires is
max 1 2
E = E(π/2) = (cid:15)−2t (2)
F 2
Introduction of the extended TB model of the electrodes thus causes the positive shift
−2t of the Fermi energy compared to the basic TB model. We note that if the model of
2
the wires was extended up to the 3rd NN, this would have no effect on Fermi energy (2)
and overall would have a negligible impact on our results as we have verified in a few
calculations. The above defined on-site energy (cid:15) is the same for all atoms of the system.
Its value therefore can be chosen arbitrarily. We use (cid:15) = 2t = 2t(cid:48) which results in
2 B
having E = 0.
F
The couplings between the wires and the GNR again extend up to the 2nd NN from the
vertex atom of the GNR (the atom of the GNR, to which a given wire is attached). In
other words, the vertex atom is coupled to two closest atoms of the semi-infinite wire.
Our model assumes the same coupling parameters t and t as defined above for the
1 2
intra-wire interactions.
3 Theory
3.1 Quantum-mechanical calculations
Standard Green’s function formalism (described for instance in Ref. [16]) is used to
calculate eigenfunctions of the entire system. Knowledge of the eigenfunctions allows us
to compute the transmission and reflection coefficients for electrons incoming from the
wires. With the exception of the extended TB model of the wires the calculations have
been accomplished according to the route described in Ref. [6]. The Green’s function
formalism requires knowledge of the semi-infinite wire’s eigenstates which are provided
in appendix A.
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3.2 Landauer-Bu¨ttiker formula
We will employ the well-known Landauer-Bu¨ttiker (LB) type formula for stationary
currents through the leads of a multi-terminal device:
2e (cid:88)NT (cid:90) µα
I = − T (E)dE , α ∈ {1,2,...,N } (3)
α α←β T
h
µ
β=1 β
β(cid:54)=α
N is the number of the terminals (also the electrodes), T (E) is the transmission
T α←β
coefficientfromtheterminalβ totheterminalαandµ aretheelectrochemicalpotentials
γ
associated with the individual wires. Formula (3) assumes (i) zero temperature, (ii) the
time-reversal symmetry (TRS) and (iii) the trivial description of the spin degrees of
freedom (spin degeneracy). The TRS implies that
(cid:88) (cid:88)
T (E) = T (E) (4)
α←β β←α
β((cid:54)=α) β((cid:54)=α)
We note that in our model each electrode (attached at its corresponding terminal) is
assumed to be composed of a number of the mono-atomically thin wires. We assume
that α and β label the electrodes, not the wires. The TRS formula (4) is equally exactly
valid for these composite electrodes as it would be valid if the indices α and β labelled
the individual wires attached on the central device.
The charge conservation for the central device implies the Kirchhoff’s first law:
(cid:88)NT
I = 0 (5)
α
α=1
The electrochemical potential µ at the terminal α is conveniently expressed in terms
α
of the electrostatic potential U associated with the terminal (the electrode):
α
µ ≡ −eU (6)
α α
We recall that in our model (see its description in Sect. 2 and in Fig. 2) the electrochem-
ical potentials themselves as well as the currents are non-trivial unknown functions of
the two bias voltages V and V as will be further explained below.
A B
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3.3 Inclusion of the circuits for a 4-terminal device
3.3.1 Formulation of the problem
Consider a device with 4 terminals, i.e. N = 4 like that drawn in Fig. 2. Each terminal
T
has an electrode connected to it. In our basic statement of the LB formula (3) above
the (stationary) currents I obey to Kirchhoff’s first law (5) and no other constraints
α
were supposed. Now we put in consequences the assumption that the four leads form
the two circuits and there are also the two resistors included so that the entire system
can be schematically depicted as on Fig. 2. The lower circuit (A) has one of its branches
grounded. For such 4-terminal device we have to require the following additional con-
straining condition,
I +I = 0 (7)
3 4
which is more restrictive than just Kirchhoff’s first law (5). We assume the stationary
regime for each element of the entire scheme, i.e. also for the sources. Therefore the
condition (7) is necessary otherwise an unphysical charge imbalance would be gradually
built on the voltage source V and that would in addition contradict the assumption of
B
the stationarity. The former condition (5) together with the equation (7) then imply a
similar condition also on the currents I and I :
1 2
I +I = 0 (8)
1 2
The presence of the circuits requires that, in addition to the Eqs. (7) and (8), Kirchhoff’s
second law has to be fulfilled:
U = U +V −R I , (9a)
2 1 A A A
U = U +V −R I (9b)
4 3 B B B
We recall the relation (6): µ ≡ −eU . According to the scheme in Fig. 2, we can choose
α α
U = 0 (10)
1
without any significant loss of generality. This auxiliary setting allows us to accomplish
derivations using the potentials U , U and U instead of the differences U −U , U −U
2 3 4 2 1 3 1
and U −U . Hence the three electrostatic potentials U , U , U and the four currents
4 1 2 3 4
I are unknown quantities and have to be determined as functions of the two given bias
α
voltages V and V .
A B
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3.3.2 The linearity approximation
Under the assumption of the section’s title, the formula (3) for the electric currents takes
the frequently accounted form
(cid:88)NT
I = G .(U −U ) (11)
α αβ α β
β=1
β(cid:54)=α
We stress again that for the scheme under study (see Fig. 2) the electrostatic potentials
U can not be reduced to or confused with the bias voltages V and V . See also the
γ A B
reasoning in Sect. 2. The linear conductance matrix elements are
2e2
G ≡ T (12)
αβ α←β
h
The 4 equations (11) together with the constraints (7), (8) and with the Eqs. (9) form an
inhomogeneous set of 8 linear equations with 7 unknowns U , U , U , I , ..., I . Known
2 3 4 1 4
parameters are the two resistances R and R , the matrix elements G satisfying the
A B αβ
TRS of the form (4) and the two bias voltages V and V . I.e. the two voltages V
A B A
and V , are supposed to be the control variables in such an experiment, not the three
B
electrostatic potentials U , U , and U . There is a linear dependence among the 8
2 3 4
equations. Once the set is properly reduced to remove the dependence, we obtain an
inhomogeneous set with a unique solution. The equations are solved in appendix B.1
and resulting linear relations I = I (V ,V ) and I = I (V ,V ) are obtained:
A A A B B A A B
(cid:0) (cid:1)
α +γR V −α V
AA B A AB B
I = (13a)
A
1+α R +α R +γR R
AA A BB B A B
(cid:0) (cid:1)
α +γR V −α V
BB A B BA A
I = (13b)
B
1+α R +α R +γR R
AA A BB B A B
where
1
(cid:2) (cid:3)
α = S S −(G +G )(G +G ) (14a)
AA 1 13 14 31 41
S
1
α = (G G −G G ) (14b)
AB 13 24 14 23
S
1
α = (G G −G G ) (14c)
BA 31 42 41 32
S
1
(cid:2) (cid:3)
α = S S −(G +G )(G +G ) (14d)
BB 3 31 32 13 23
S
and
γ = α α −α α (15)
AA BB BA AB
9
with
S = G +G +G +G = G +G +G +G (16)
13 23 14 24 31 32 41 42
and
(cid:88) (cid:88) 2e2
S = G = G = (N −T ) (17)
α αβ βα α α→α
h
β(cid:54)=α β(cid:54)=α
N is the number of channels per electrode α. In our model its is the number of the
α
elementary wires composing the electrode α. The expressions α , S and S [see also
CC(cid:48) α
Eqs. (B.19) and (B.20)], which depend on the quantum conductance matrix elements G
only, have been introduced by Bu¨ttiker [7] as it will be discussed below.
As stated above, the electrostatic potentials U , α = 2,3,4, are non-trivial (although
α
linear, under the assumptions of Sect. 3.3.2) functions of the bias voltages on the batter-
ies. Using Eqs. (9), (10) and (13) we first find the potential differences (see also Fig. 2)
(1+α R )V +α R V
BB B A AB A B
U −U = (18a)
2 1
1+α R +α R +γR R
AA A BB B A B
(1+α R )V +α R V
AA A B BA B A
U −U = (18b)
4 3
1+α R +α R +γR R
AA A BB B A B
restoring in this way the reference potential U which can be set to an arbitrary value
1
and the explicit presence of which allows us to write more general and symmetrical
formulae. Next we determine the potentials differences in the perpendicular direction
and across the sample. To achieve this goal, we first use the Eq. (B.12a) which yields
U = (I −aU )/b for U = 0. U is already know from the Eq. (18a). Accomplishing
4 A 2 1 2
lengthy algebraic manipulations with multiple use of the TRS relations of the form (3)
[see also the Eq. (12)] and finally restoring the reference potential U we arrive at
1
1
(cid:2)(cid:0) (cid:1) (cid:0) (cid:1)
U −U = − G +G V + G S +G G R V
4 2 31 41 A 41 3 31 43 B A
SD(G,R)
(19a)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:3)
− G +G V − G S +G G R V
13 23 B 23 1 13 21 A B
1
(cid:2)(cid:0) (cid:1) (cid:0) (cid:1)
U −U = G +G V + G S +G G R V
3 1 42 32 A 32 4 42 34 B A
SD(G,R)
(19b)
(cid:0) (cid:1) (cid:0) (cid:1) (cid:3)
− G +G V − G S +G G R V
24 14 B 14 2 24 12 A B
10
