Download The gating efficiency of single

The gating efficiency of single-molecule transistors
Chao-Cheng Kaun and Tamar Seideman∗
Department of Chemistry, Northwestern University,
2145 Sheridan Road, Evanston, IL. 60208-3113 USA
Abstract
Using self-consistent first principles calculations, we study the gating efficiency of fullerene
based single-molecule transistors. We find that the efficiency depends sensitively on the geometry
of the gate electrode and on the contact coupling between the molecule and the source and
drain electrodes.
In particular, a 4-rectangles gate electrode that surrounds the junction
is substantially more effective than a conventional single-rectangle gate electrode.
As the
coupling strength of the molecule to the source-drain electrode is reduced, the underlying molecular orbitals localize in space and are easier to be shifted, giving rise to enhanced gating efficiency.
Keywords:
Single-molecule Transistor; Gating Efficiency; Nonequilibrium Green function
approach; Molecular Electronics; Fullerene junction.
1
I.
INTRODUCTION
The design and fabrication of nanoscale transistors has been a long standing goal of molecular electronics1 and the focus of significant experimental and theoretical effort, inspired in
part by the important role of transistors in macro-electronics. An illuminating discussion is
given in Ref. 2. Following a proposal put forth in the theoretical study of Ref. 3, several
experimental studies4–6 have investigated the possibility of using 1,4-benzenedithiol as the
molecular moiety of a single-molecule transistor. To date, however, no pronounced gate effect has been found in such devices. By contrast, robust gate effects have been observed in a
various other molecular junctions, including C60 -,7–9 styrene-,10 and perylene tetracarboxylic
diimide (PTCDI)-11 derived molecular junctions, as well various complex structures.12
It is conceivable that the qualitative difference in gate effects of the two classes of junctions
results from their different coupling to the Fermi level. A molecular transistor typically
consists of a short molecule bridging two (source and drain) electrodes, with a third electrode
connected capacitatively and serving as a gate. Application of a gate voltage shifts the
molecular levels into alignment with the Fermi level and thus induces charge transfer between
the molecule and the source and drain electrodes. Thus, pronounced gate effects are expected
for a junction with sharp and readily-shifted resonance. The former property provides large
on/off ratios and the latter provides high gating efficiencies.
The transferred charges, however, may screen the gate potential and thus inhibit the
molecular level shift: in the extreme limit where a gate voltage is applied to a perfect
electrode, rather than to a molecular junction, the gating efficiency is very low and thus no
gate effect is expected. Therefore, if the junction has a resonant state strongly coupled to
the Fermi level (such as Au–1,4-benzenedithiol–Au4,13 ), one might expect efficient screening
of an applied external gate potential and thus limited influence of the potential on the
molecular level structure. Such transistors are expected to exhibit low gating efficiency
and hence poor gate control of current flow. Molecular junctions with weak coupling, by
contrast, are expected to better lend themselves to gate control. It is of both fundamental
and potential technological interest to quantify the above qualitative arguments, given the
potential of transistor operation as a probe of molecular junctions on the one hand and as
a potential element in future devices on the other.
The role played by the geometry of the gate electrode2 is comparably interesting to that
2
played by the level structure of the molecular moiety. On the one hand the impressive
progress of the experimental technology suggests the possibility of going beyond the simple
single-rectangle gate electrode constructs considered so far. On the other, the level shifting picture discussed above suggests the possibility of enhancing the gating efficiency of
molecular transistors through control of the gate geometry.
The Au–C60 –Au molecular junction serves as a useful model to meet these goals. As
illustrated in Refs. 8, 14, and 15, the contact couplings and the transport properties of
the Au–C60 –Au junction can be varied over a large range by variation of the inter-electrode
distance, denoted L in 14, 15 and below. For small inter-electrode distances, L ≤ 23.42
a.u., the equilibrium configuration of the junction is the symmetric structure, where the van
der Waals interaction with both electrodes is equal. This case produces a polar symmetric
current in the µA regime (I = 8.03 µA at a bias voltage of Vb = 0.19 V for L = 23.42
a.u.). When L exceeds the value that allows the fullerene to couple equally strongly to
both electrodes, the junction is asymmetric, with the C60 attached through van der Waals
interaction to one electrode. In the situation of L = 26.42 a.u., the current drops by slightly
over an order of magnitude, and its polar symmetry is lost. Further increasing L, drives
the system into the weak coupling limit, corresponding to the Coulomb blockade regime,
and the current drops to the nA range.7–9 A gate electrode, however, may stabilize the
fullerene in the symmetric configuration (the junction center), even for large values of L, by
attachment to the dielectric layer above the gate electrode. Gating effects on the Au–C60 –
Au junction were studied experimentally in the weak coupling, Coulomb blockade limit,7–9
and numerically in the strong coupling limit.16,17 In the former regime the gate effects are
pronounced, i.e., variation of the gate voltage by a few Volts can switch the system from
a zero-conductance state to a several nS7 or µS8,9 conductance state. In the latter regime,
although the gate voltage substantially modifies the transmission peak line-shapes,16,17 its
efficiency in controlling the transport properties is low.
In the present study, we investigate gate control of current flow by calculating the charge
transport from first principles using a technique18 which combines the Keldysh nonequilibrium Green’s functions (NEGF)19,20 with density functional theory (DFT)21,22 . We focus
on the question whether an efficient transistor can be produced based on a gate effect on a
molecular junction, and if so under what conditions and how. Our goals are both to investigate the potential of gate voltage as a tool for better understanding the properties of the
3
underlying junction and to investigate the single-molecule transistor as a potential element
in future molecular electronics. In addition to the molecular properties and the contact coupling, we find that the structure of the gate electrode is an important factor in the search for
a widely tunable source-drain current. We revisit the Au–C60 –Au molecular junction model
as a convenient and experimentally testable case study, but our goal is general conclusions.
For this system, a single rectangle gate is found to have only a minor effect, but extensive
gate control can be achieved by surrounding the molecule by the gate metal. Our study
provides some insight into how to generate large gate effects more generally. We find that
current modulation is due to a transmission resonance peak, which can be shifted by the
applied gate voltage. As the gate potential detunes the resonance peak from the Fermi level
of the electrodes, switching of current occurs. When the contact coupling is weakened, the
resonance peak becomes more readily shifted, and the gating efficiency improves.
II.
THEORY AND NUMERICAL METHODS
Our study is based on self-consistent NEGF-DFT formalism,18,23,24 where the Hamiltonian corresponding to the source–molecule–drain construct is expanded in a real space
linear combination of atomic orbital basis set.18,25 The atomic cores are defined by standard,
nonlocal, and norm-conserving pseudo-potentials.26 The local density approximation is used
for the description of electron exchange and correlation. The density matrix is calculated
from the non-equilibrium Green function G< (E), as
Z
i
dEG< (E),
ρ=−
2π
(1)
and G< is defined by the Keldysh equation.20
Numerically, we solve the Hartree and Poisson equations simultaneously and self consistently on a three-dimensional real space grid, including the external bias and gate potentials
as the electrostatic boundary conditions. Once the density matrix has been determined, the
Kohn-Sham effective potential is calculated. This process is iterated until numerical convergence of the self-consistent density matrix is achieved. Transport properties, such as the
transmission coefficient and the current, are then calculated as a function of the scattering
electron energy, bias voltage and gate voltage.
4
III.
RESULTS AND DISCUSSION
Figure 1 shows the structure of the C60 molecular transistor, which consists of a molecule
in contact with two (source and drain) atomic scale Au electrodes extending to reservoirs
at z = ±∞. The drain and source electrodes are composed of unit cells with 9 Au atoms
oriented in the (100) direction. Different gate geometries are investigated in the calculations
described below, one of which is depicted schematically in Fig. 1 and described below. We
consider 20 a.u. long, 32 a.u. wide gate electrodes, located at a distance of 16 a.u. (9.42
a.u.) from the axis of the junction (the C60 edge), and investigate the conductance behavior
of the device under a range of gate potential values. As discussed in Ref. 14, the fullerene
is attached to one electrode through van der Waals interaction with an equilibrium distance
of 5.13 a.u. from the C60 edge to the nearest (the source in Fig. 1) electrode. For an
inter-electrode distance of L = 26.42 a.u., the equilibrium configuration corresponds to a
distance of 8.13 a.u. of the C60 edge from the far (here drain) electrode.
Figure 2a shows the conductance (measured in the standard conductance units G0 =
2e2 /h) versus the gate voltage Vg for different geometries of the gate. The empty circles
correspond to a conventional, single-rectangle gate. We have found that a negative gate
voltage is more efficient in tuning the conductance than a positive one. For the considered
gate geometry, however, the gate has a minor effect – for as small a molecule-gate distance
as 9.42 a.u., the conductance on/off ratio at |Vg |= 5 V is merely 2.59. Shown as a solid
curve with superimposed triangles in Fig. 2a is a calculation where the single-rectangle gate
is replaced by a pair of adjacent rectangle electrodes. Here the gating efficiency increases by
a factor of two in the negative voltage region and by a factor of ca. 4 in the positive voltage
region. With this configuration the conductance on/off ratio at |Vg |= 5 V is 26.10. When
the gate potential is applied by an electrode which consists of four rectangles (surrounding
the fullerene on the four sides of the simulation box, as illustrated in Fig. 1), the gating
efficiency increases by a factor of ca. 4 in the negative gate voltage region and by a factor
of ca. 8 in the positive bias case. This situation is shown in Fig. 2a as a solid curve with
superimposed circles, and is seen to lead to a conductance on/off ratio of 1105.7 at |Vg |= 5
V. The surrounding gate is more efficient because it produces a stronger electric field at the
contact between the molecule and the source-drain electrode, similar to the electric field in
a coaxial cylindrical capacitor. Whereas the additive effect of the gate electrodes at Vg < 0
5
is, perhaps, not surprising, the superlinear behavior at Vg > 0 is more interesting. The
analysis below explains both behaviors through more subtle features of resonances under
gate potentials. We note here that, from a practical perspective, the figure of relevance
is the conductance on/off ratio. This ratio is seen to be enhanced by several orders of
magnitude through choice of the gate geometry.
Figure 2b provides a more detailed study of the surrounding gate effect, illustrating the
current-voltage characteristics for several values of the gate voltage. The current increases
with bias voltage monotonically but in a Vg -dependent fashion. As the gate voltage increases from -2 V to 2 V in 1 V increment, the current (under a bias voltage Vb = 0.19
V) increases nonlinearly from 0.062 µA to 0.873 µA. The I-V curves undergo substantial
structural modifications with changing gate voltage. A negative gate voltage is seen to be
much more effective in modulating the current than a positive gate voltage, especially for
Vb > 0.1 V.
In what follows we explore the physics underlying the current modulation by the gate
voltage. Figure 3a shows the transmission spectra for the molecular transistor of Fig. 2b
subject to different applied gate voltages. The Fermi energy of leads is indicated by the
vertical dashed line at E = 0. The five transmission spectra from low to high energy
correspond to applied gate voltages ranging from 2 V to -2 V in 1 V decrement. As the gate
voltage decreases, the transmission spectrum deviates from the Fermi level towards higher
energies, and hence the current decreases. This effect is the origin of conductance switching
in the transistor.
The shift of the transmission features is strongly dependent on their energy. The further
a transmission feature from the Fermi level the larger the spectral shift for a given value
of Vg . One consequence is the observation (Fig. 2) that the current modulation in the
negative gate voltage region is more effective than in the positive one. Another consequence
is the deformation of the I-V curve in a Vg -dependent fashion. The energy eigenvalue of
the resonance is indicated as a solid circle above each of the transmission spectra in Fig.
3a. We note that only a single molecular orbital is considered here. In reality 3 molecular
orbitals contribute to different extents to the transmission of the Au–C60 –Au junction (see
Refs. 14, 15 for detailed analysis of their properties and consequences), but for the focus of
the present work, discussion of one, specifically – the lowest energy component, suffices.
Figure 3b investigates the origin of the observation, discussed above, that the transmission
6
resonances are more readily shifted by application of a gate voltage the further the energy
eigenvalue from the Fermi level. The solid curve with superposed triangles shows the number
of excess electrons on the molecule vs. the gate voltage. For Vg > 0, the slope of this curve
is larger, indicating that the charge is more readily transferred. The curve with superposed
circles shows the resonance orbital energy vs. Vg . For Vg > 0 the slope of this curve (the
rate of change of the resonance energy) is smaller, indicating that the resonance becomes
increasingly resistive to shift by the gate voltage. Together Figs. 3a and b clarify the results
of Fig. 2 and provide more general insight into the molecular properties that determine the
gating efficiency. In the case at hand the efficiency is largely determined by screening effects.
As the resonance approaches the Fermi level, the charge transfer from the electrode to the
molecule is increasingly facilitated, and the transferred charges screen the gate potential
more efficiently. We thus expect that gate tuning would be generally more difficult for
junctions that conduct well, because such junctions are typically characterized by strong
contact coupling, which enhances electron screening effects.
Although electron screening and proximity to the Fermi level considerations suggest that
in general efficient conduction would be associated with poor gating efficiency and vice versa,
exceptions are readily imagined. To clarify this relation, and generalize the above discussion,
we consider in Fig. 4 three related constructs, differing in contact coupling due to various
structural differences. Figure 4a considers a shorter gap device, L = 24.42 a.u., where the
drain–C60 distance is reduced to 6.13 a.u. Fig. 4b returns to the long gap, L = 26.42
a.u., of Figs. 2, 3 but places the fullerene at the center of the gap, to make a symmetric
junction. Finally, in Fig. 4c, the molecule is placed at the center of a short gap, L = 24.42
a.u., junction. For the symmetric configuration junction of Figs. 4b and 4c, the center
geometry can be stabilized through dispersive interaction with a dielectric layer above the
gate electrode. For the purpose of the present analysis, the different constructs serve as
model studies.
The solid curves in Fig. 4 correspond to the Vg = 0 case and thus serve to contrast the
conductivities of the four constructs Figs. 3a, 4a, 4b and 4c. Comparison of Fig. 3a with
Fig. 4a and of Fig. 4b with Fig. 4c shows that the conductance of the asymmetric junction
increases with decreasing L faster than that of the symmetric junction. The resonance of
both the asymmetric and the symmetric devices shifts to a higher energy as L decreases.
These results are consistent with the experimental observations of Ref. 8. Comparing devices
7
with equal gaps but different fullerene location, (Fig. 3a with Fig. 4b and Fig. 4a with Fig.
4c) we find that the asymmetric device offers smaller conductance than the symmetric one,
in agreement with Ref. 14, 15.
Considering next the transport properties of these junctions subject to gate voltage, we
find that the negative gate voltage remains more efficient than the positive in shifting the
resonance. The relative gating efficiency of the different constructs, however, does not follow
their relative conductance, but rather their relative contact coupling. For instance, with the
same L, the charge transport through the symmetric device will always be better than that
through the asymmetric one for the simple reason that the latter involves a large barrier
to tunneling between the fullerene and the drain electrode (more generally, between the
molecule and the further electrode). Nonetheless, the stronger coupling of the molecule to
the nearer of the two electrodes in the asymmetric device enhances the electron screening and
gives rise to the observed less efficient gating. Thus, comparison of Figs. 3a and 4b shows
that at 26.42 a.u., the resonance shift (and hence the gating efficiency) of the asymmetric
device is smaller than that of the symmetric device.
Figure 4d shows the conductance vs. the gate voltage for the four devices. As the gate
voltage increases from -3 V to 3 V, the conductance increases in different fashions and
rates. The conductance on/off ratio at |Vg |= 3 V is 56.7 for the asymmetric long device
(solid-circle), 294.2 for the symmetric long device (empty-circle), 23.1 for the asymmetric
short device (solid-triangle), and is 28.3 for the symmetric short device (empty-triangle).
Therefore, for |Vg |= 3 V, the gate effect is most pronounced for the symmetric long device
(empty-circle), a nontrivial result although consistent with the above argumentation. In
addition, we observe the expected correlation between the degree of charge transfer to the
molecule and the gating inefficiency. The excess electrons on the molecule is 0.46 for the
asymmetric long device, 0.43 for the symmetric long device, 0.48 for the asymmetric short
device, and 0.47 for the symmetric short device. We find that the gating efficiency provides
an interesting probe of the junction properties, reflecting aspects that the I-V curves do not
reveal.
8
IV.
CONCLUDING REMARKS
Our goal in the preceding sections has been to investigate the origin of gate effects in
single molecule junctions and to develop approaches for enhancing the gating efficiency of
molecular transistors. The Au–C60 –Au junction was used as a simple but readily tunable
model system. We first illustrated the possibility of enhancing the on/off ratio by three
orders of magnitude by proper choice of the gate geometry. In particular, we showed that a
surrounding gate, consisting of four rectangles on the four sides of the junction, is considerably more efficient than a conventional gate. Next we investigated the molecular junction
properties that determine the gating efficiency. We illustrated and explained the role of
electron screening effects and its relation to the contact coupling. In particular, we pointed
out that poor contact coupling is generally associated with efficient gating. This suggests
the important role of the linking groups that connect the molecule and the electrodes, not
only for determining the conductivity but also for determining the gate effects. Finally, we
noted the sensitivity of the gating efficiency to interesting junction properties that are not
obvious from other observables.
Acknowledgments: We are grateful to the NSF for generous support within grant
number CHEM/MRD-0313638 and within the MRSEC program (DMR-0520513) at the
Materials Research Center of Northwestern University.
∗
Electronic address: [email protected]
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10
Fig. 1
Source
Vg
Drain
FIG. 1: Schematic illustration of the molecular transistor, which consists of a C60 molecule in
contact with two infinitely long Au electrodes. The gate electrode in this example surrounds the
junction axis. The distance of the C60 left (right) edge to the source (drain) electrode is 5.13 a.u.
(8.13 a.u). The inter-electrode distance is L = 26.42 a.u.
11
Fig. 2
(a)
G
0.04
0.02
0
-4
0
Vg (V)
0.05
0.1
Vb (V)
2
4
(b)
0.8
I (µ Α)
-2
0.4
0
0
0.15
FIG. 2: (a) Conductance (G0 = 2e2 /h) versus gate voltage Vg for three geometries of the gate
electrode distinguished by symbols: a standard rectangular gate electrode (empty circles); a gate
electrode consisting of two adjacent rectangles (solid triangles); a gate electrode consisting of four
rectangles that surround the molecule (solid circles). (b) I-V characteristics of a C60 transistor
with a surrounding gate electrode at a number of applied gate voltages. The five curves from low
to high current correspond to gate voltages ranging from -2 V to 2 V in 1 V increment.
12
Fig. 3
(a)
T
0.08
0.04
0
0.2
E (eV)
0.4
2
1
1
0.5
0
E (eV)
Extra electrons
(b)
0
0
-4
0
Vg (V)
4
FIG. 3: (a) Transmission spectra for the Au–C60 –Au transistor with a surrounding gate at different
gate voltages Vg . The five transmission spectra from high to low energy correspond to gate voltages
ranging from -2 V to 2 V in 1 V increment. The resonance orbital energy is indicated by solid
circles. (b) The resonance orbital energy (circles, referred to the right axis) and the extra electrons
on the molecule (triangles, referred to the left axis) vs the gate voltage Vg .
13
Fig. 4
(a)
T
1.6
0.8
0
(b)
T
1.6
0.8
0
2.4
T
(c)
1.2
0
0
0.2
E (eV)
0.4
G
(d) 0.6
0.3
0
-2
0
Vg (V)
2
FIG. 4: Transmission spectra for different Au–C60 –Au transistors with a surrounding gate: (a)
a short-gap device, L = 24.42 a.u. (the molecule-drain distance is reduced to 6.13 a.u.); (b)
a symmetric device, where the molecule is at the center of the gap (L = 26.42 a.u.); and (c)
a symmetric, short-gap device, L = 24.42 a.u. (d) Conductance vs gate voltage for the four
devices considered: asymmetric long-gap (Fig. 3a, solid-circles); asymmetric short-gap (Fig. 4a,
solid-triangles); symmetric long-gap (Fig. 4b, empty-circles); and symmetric short-gap (Fig. 4c,
empty-triangles).
14