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Demonstration of a NOR logic gate using a single molecule and two surface gold atoms to encode the logical input

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A logic gate has been implemented in a single trinaphthylene molecule. Each logical input controls the position of a surface Au atom that is brought closer or further away from the end of one of the naphthyl branch. Each Au atom carries 1 bit of information and is able to deform nonlocally and to shift in energy the molecular electronic states of the trinaphthylene. Probed at the end of the third naphthyl branch using scanning tunneling spectroscopy, the variations of the tunneling current intensity as a function of the Au atoms position measures the logical output of the gate. We demonstrate both theoretically and experimentally that these variations respect the truth table of a NOR logic gate.
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PHYSICAL REVIEW B 83, 155443 (2011)
Demonstration of a NOR logic gate using a single molecule and two surface gold atoms
to encode the logical input
W.-H. Soe,1,*C. Manzano,1A. De Sarkar,1F. Ample,1N. Chandrasekhar,1N. Renaud,2,*P. de Mendoza,3
A. M. Echavarren,3M. Hliwa,4,5and C. Joachim1,4
1IMRE, ASTAR (Agency for Science, Technology and Research), 3 Research Link, 117602 Singapore
2Department of Chemistry, Northwestern University, 2145 Sheridan Road, Evanston, Illinois 60208-3113, USA
3Institut of Chemical Research of Catalonia (ICIQ), Av. Pa¨
ısos Catalans 16, E-43007 Tarragona, Spain
4CEMES & MANA Satellite, CNRS, 29 rue J. Marvig, F-31055 Toulouse Cedex, France
5Fac u lt ´
e des Sciences Ben M’Sik, Universit´
e Hassan II Mohammedia, B.P. 7955 Sidi Othman, Casablanca, Morroco
(Received 8 October 2010; revised manuscript received 23 December 2010; published 26 April 2011)
A logic gate has been implemented in a single trinaphthylene molecule. Each logical input controls the
position of a surface Au atom that is brought closer or further away from the end of one of the naphthyl branch.
Each Au atom carries 1 bit of information and is able to deform nonlocally and to shift in energy the molecular
electronic states of the trinaphthylene. Probed at the end of the third naphthyl branch using scanning tunneling
spectroscopy, the variations of the tunneling current intensity as a function of the Au atoms position measures
the logical output of the gate. We demonstrate both theoretically and experimentally that these variations respect
the truth table of a NOR logic gate.
DOI: 10.1103/PhysRevB.83.155443 PACS number(s): 85.65.+h, 03.67.a, 07.79.Fc, 85.35.p
I. INTRODUCTION
Logic gates are the building block of modern electronic ar-
chitecture and are ubiquitously found in any electronic circuit
whose purpose is to perform a given arithmetic operation.
The computing power by surface area is therefore highly
dependent on the minimal surface required to implement a
given logic function. To improve this computing power, one
can scale down classical circuits, either improving lithography
techniques1or using tailor-made materials.2However, to over-
come the foreseeable difficulties limiting this miniaturization
down to the atomic scale, new electronic architectures have
to be developed. Among many other solutions, molecular
electronics is an interesting approach.3
In this frame, many solutions have been proposed to design
a small computing unit out of a few molecules.4Logic gates
have been constructed using the domino-like cascading of
CO molecules to perform the logic operation.5Intramolecular
proton transfer can also be used to induce a topological rotation
in the frontier orbitals of a naphthalocyanine molecule that
influences the electronic states of neighboring molecules.6,7
Molecular cellular automata is another promising solution
where the flow of information propagates along a molecular
network by changing the charge state of each molecule.8
Turning a single molecule into a logic circuit is a more
delicate problem. The obvious solution of shaping the
molecule like the corresponding circuit9suffers from an
important drawback: the decay of the current intensity passing
through the molecule with its length.10 Alternative solutions
have been proposed to turn the quantum properties of a single
molecule to our advantage. Stimulated Raman adiabatic
passage can thus be used on a few level system to implement
sequential arithmetic operations.11 Quantum interferences12
or negative differential resistance13 are other quantum effects
that can be used to implement logic devices. In this article we
follow the quantum Hamiltonian computing (QHC) approach
to design a logic gate in a model system and implement it
experimentally in a single molecule. We would like to stress
that the QHC approach is not qubit based and deals with
classical Boolean functions. It nevertheless uses quantum
properties of the system, that is, the nonlocal deformation of
its delocalized eigenstates, to implement a Boolean function.
However it does not rely explicitly on the superposition
principle to encode the logical input, as it is the case in a
standard qubit-based quantum computation approach.14
The QHC approach proposes to encode the logical inputs
in the Hamiltonian of the molecule and the logical output in
the tunneling current intensity passing through a part of the
molecule.15,16 The basis of this approach is described in Sec. II
using a model seven-state quantum system whose Hamiltonian
depends on two logical inputs. Modifying the delocalized
eigenstates of the system, each logical input acts nonlocally
on its electronic conductance. Switching one logical input at
one end of the system can induce strong modification of its
conductance at the other end. This is very different from the
very local action that a classical switch has on the conductance
of an electronic circuit. The corresponding scanning tunneling
microscopy (STM) experimental setup is presented in Sec. III.
Controlling the position of two surface Au atoms in the vicinity
of a trinaphthylene molecule physisorbed on a Au(111)
substrate, a digital logic gate implementing the logical negation
NOR operation of the OR Boolean operator is realised. Each
Au is approached or moved away from one of the three naphtyl
branches while the current going from the tip to the substrate
through the end of the third naphtyl branch is recorded.
Connecting an Au at one end of the molecule strongly perturbs
its conductance and therefore induces a modification of the
tunneling current intensity. Quantum chemistry calculations
of the valence molecular orbitals and the multiconfiguration
electronic states of the trinaphthylene molecule interacting
with either zero, one, or two Au atoms are presented in Sec. IV.
These calculations provide insight into the mechanism of this
molecular logic gate and reveal its connection with the model
system of Sec. II. These results indicate a method for the
future design of complex Boolean logic gates using a single
molecule.
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W. H. SOE et al. PHYSICAL REVIEW B 83, 155443 (2011)
II. THE QHC DESIGN OF A NOR LOGIC GATE
Many methods exist to calculate the electronic conduction
of a quantum system inside a tunneling junction.17 However,
the analytical expression provided by these methods are quite
difficult to handle even for simple quantum systems. This
complexity is a major obstacle in developing a simple theory
of the response of a quantum system to a modification of its
Hamiltonian.
Following a recently proposed approach,18 the electronic
conductance can be calculated from the temporal evolution
of the wave vector from one eigenstate of one electrode to an
eigenstate of the other electrode. The two scattering eigenstates
must have the same energy E. It has been demonstrated that
the value of the transmission coefficient at the energy E, noted
T(E), is proportional to the square of the oscillation frequency
between the two scattering eigenstates of energy E.19 In this
picture let us consider the model seven-state system whose
Hamiltonian is given by:
|φtip|φsub|m1|m2|m3|1|2
H=
E. ε....
.E ε....
εε ekk . .
.. kekα .
.. kke .β
.. .α . e .
.. ..β.e
(1)
and represented in Fig. 1(a). The zero matrix elements of the
Hamiltonian are represented by dots to emphasize the structure
of the system. The states |φtipand |φsub are the two scattering
eigenstates of the electrodes, and are labeled tip and sub, to
anticipate the STM implementation used in Sec. III where one
electrode is the tip of the STM and the other the substrate. Due
to the delocalization of |φtipand |φsub over the electrodes,
their interaction with the central system remains very weak and
is here taken as ε=102eV. The states |m1,|m2, and |m3
constitute the board of this simplified molecule. The coupling
strengths between the states |1,|2and this board are used
to encode the logical inputs. These latter are noted αand β:
αis the coupling strength between the states |m2and |1
and βis the coupling strength between the states |m3and
|2. These two input parameters can take the values of 0 or 1.
Changing either of these two coupling strengths changes all the
eigenstates and eigenenergies of the Hamiltonian. Therefore
the logical inputs control the current intensity going from |φtip
to |φsubeven if these two states are only connected to |m1.
Initially prepared on |φtip, the state vector |ψ(t)evolve in
time exploring the state space of the system following the
solution of the time-dependent Schrodinger equation (TDSE):
|ψ(t)=exp(iHt
¯h)|φtip. Due to the weak values of ε,the
population of the |φsub:Psub (t)=|φsub|ψ(t)|2, oscillates
smoothly driven by a single secular oscillatory term whose
oscillation frequency is . The Lowdin partitioning20 allows
us to derive an analytical expression for that depends on the
value of the logical inputs. Discretizing this expression over the
binary values of αand βleads to an approximate expression of
that associates Dirac distributions and Boolean operators:
(E,α)=α+βδ(43k222k3)
+αβδ[4(1 +3k2)22k3+k2]
+α·βδ[4(2 +3k2)22k3+2k2+1]
(2)
with =Ee. The Dirac distributions in Eq. (2) comes
from the perturbative nature of the Lowdin partitioning and
the pole of the corresponding effective Hamiltonian.21 Tuning
the value of allows us to cancel out one or several arguments
of the Dirac distributions in Eq. (2) and to select the associated
Boolean operators. Depending on which operators have been
selected, a given Boolean function is implemented in the
system. The case of the NOR gate is given by =2k.Forthis
value of , the argument of the first Dirac distribution cancels
out unlike the other ones. This selects the operator α+βand
leads to (α,β)α+βδ(0). The oscillation frequency is
then maximum for α+β=1, that is, if α=β=0, and very
small otherwise. To confirm this analysis the TDSE has been
solved numerically for E=−1eV,e=0eV,k=−1/2eV,
and ε=0.01 eV. The evolution of Psub(t) is represented in
Fig. 1(b) for the different values of αand β. In the case
(a) (b)
FIG. 1. (Color online) (a) Graphical representation of the system described by the Hamiltonian (1). |φtipand |φsub are eigenstates of the
two electrodes. The logical input are encoded in the value of αand β. (b) Population of |φsubfor E=−1eV,e=0eV,k=−1/2eV,and
ε=102eV. When α=β=0 an eigenstate of the system has the same energy as |φtipand |φsub. This leads to a maximum oscillation
frequency. This is not the case for the other values of the logical input and consequently the oscillation frequency is much lower. The QHC
design plays with the level-repulsion effect for some molecular states to be at or off resonance with the initial and target state.
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DEMONSTRATION OF A NOR LOGIC GATE USING A ... PHYSICAL REVIEW B 83, 155443 (2011)
(a) (b)
FIG. 2. (Color online) (a) Simplified QHC molecule inserted between two electrodes. The two electrodes are supposed to interact only
with the first state of the simplified molecule through the couplings V. These couplings are V=−0.15 eV. (b) The transmission coefficient of
the system represented in (a) calculated with the ESQC technique for different values of the logical input (solid black line). The T(E) follows
the same variation as the normalized oscillation frequency for the system represented in Fig. 1(a) (dashed blue line). The chemical potentials
of the electrode define the integration region of the T(E) (grey shaded area) that gives the output current intensity.
α=β=0, Psub(t) oscillates smoothly from |φtip to |φsub
with a frequency of 6.45 THz. In the other cases, the oscillation
frequency is much lower and drops to 0.05 and 0.07 THz
for α= βand α=β=1, respectively. The high frequency
obtained for α=β=0 is due to the presence of an eigenstate
of the central system with an energy Ethat connects |φtipand
|φsub. The energy of this resonant eigenstate is shifted out of
the resonance for nonnull values of the logical inputs and the
oscillation frequency decreases dramatically due to the weak
value of ε.
When the central system is inserted between two electrodes
[Fig. 2(a)], the transmission coefficient at the energy E
is proportional to the square of the oscillation frequency
determined above. This equivalence is illustrated in
Fig. 2(b) where these two quantities present the same line
shape with the usual resonance and interference patterns. The
T(E) spectra has been calculated with the elastic scattering
quantum chemistry (ESQC) technique.22 Each contacting
electrode was modeled by a linear chain with a zero on-site
energy and a site to site coupling of 2 eV. The coupling
Vbetween the last site of each electrode and the state
|m1has been set to 0.15 eV. This last parameter controls
the width of the resonances that are broaden when Vincreases.
The weak value used here leads to sharp resonances. The elec-
tronic conductance of the system at E=−1 eV is resonant,
that is, T(1eV)=1, for α=β=0 and nonresonant, that
is, T(1eV)0, for the other logical input values. This is
exactly what is predicted by Eq. (2). Consequently the output
status of the logic gate is given by the value of the electronic
conductance of the system at this precise energy to obtain the
NOR gate. However it is more stable to access the output status
using the current intensity going through the system rather than
its spectral conductance.
Following the Landauer-Buttiker approach, this intensity
is obtained integrating the transmission coefficient between
the chemical potentials of the two electrodes. These chemical
potentials depend mainly on the Fermi energy of the electrodes
and the applied bias voltage. The chemical potential are simply
set here at 1.1 eV for the tip and 0.5 eV for the substrate.
They define the integration region represented by a grey shaded
area in Fig. 2(b). To illustrate the mechanism underlying
our approach let us consider Fig. 3(a) where the changes
of the system’s eigenstates induced by αare represented
(βremaining equal to 0 for simplicity). The eigenstates
are represented supposing that each state correspond to a
pzorbital. When α=0, the lowest lying eigenstate (called
HOMO for simplicity) is located at E=−1 eV. Responsible
for the resonance observed at T(1eV)for α=β=0, this
state lies between the chemical potential of the two electrodes
[grey shaded area in Figs. 2(b) and 3(a)]. Due to this available
tunneling channel, the current intensity flowing through the
system [solid green line in Fig. 3(a)] is quite strong: 1.5 μA.
When αincreases the LUMO state [dashed red line in
Fig. 3(a)] pushed the HOMO down in energy. When αis strong
enough to push the HOMO out of the integration region, the
current intensity drops to almost zero. The same analysis can
be made when both αand βvary from 0 to 1 [Fig. 3(b)]. As
long as the HOMO remains in the integration region the current
intensity is strong and drops to zero otherwise. The width of
the HOMO, and consequently the value of V, controls the
steepness of the transition to go from a 0 to a 1 logical output.
For large V, the broad resonances lead to a smooth transition.
On the contrary, a small Vleads to sharp resonances and gives
a steeper transition. In this latter case the output intensity
surface presents large stability regions at its corner. This leads
to a self-correction of small deviations in the value of the
logical inputs that induce even smaller deviations in the output
status. The values of all the physical quantities relative to
the logical output status are summarized in the last part of
Fig. 3(b) for the four different values of the logical input.
III. A LT-UHV STM IMPLEMENTATION
OF THE NOR GATE
Due to its star-like topology, similar to the model system
presented in Sec. II, a trinaphthylene molecule [see Fig. 4(a)]
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(a) (b)
FIG. 3. (Color online) (a) Impact of αon the system’s eigenstates indicated by the dotted blue, dashed red, and solid black curves. When
α=0 the HOMO lies between the chemical potential of the two electrodes that defines the integration region (grey shaded area) leading to a
strong current intensity. If αis strong enough to push the HOMO out of the integration region, the current intensity drop to almost zero. (b)
Current intensity when αand βvary from 0 to 1. The smaller V, the steepest the transition goes from a 1 to a 0 output. Values of the different
physical quantities relative to the output status for the four input configurations.
is a good candidate to go from the theoretical design of
Sec. II to a realistic experiment. This STM experimental
setup is presented in Fig. 4(a). A trinaphthylene molecule is
physisorbed on an Au(111) surface to preserve the integrity
of its molecular electronic states. Besides, this physisorption
allows us to have a large enough space between the molecular
board and the surface where surface Au atoms can be slid into
contact with the πnetwork of the trinaphthylene. Each logical
input, αand β, controls the position of one Au atom around
the end of a given naphtyl branch. If α=0, the corresponding
Au atom is moved away from the molecule and does not
perturb its molecular electronic states. On the contrary, when
α=1, this Au atom is STM manipulated toward the end of
a naphtyl branch and perturbs all the electronic states of the
trinaphthylene. The value of βcontrols similarly the position of
a second Au atom at the end of a second naphtyl branch. The
logical output is recorded by the tunneling current intensity
going from the tip to the substrate through the end of the third
naphtyl branch.
As demonstrated below, the variations of the tunneling
current intensity with the position of the Au atoms respect the
truth table of a NOR logic gate. These variations are caused
by a local electronic perturbation of the molecule that occurs
far away from the region where its conductance is probed. If
they were acting classically, the gold atoms would only perturb
the electronic conduction of the molecule in their immediate
(a) (b)
FIG. 4. (Color online) (a) The STM experimental setup used in this article. The trinaphthylene is physisorbed on Au(111). Each logical
input, denoted αand β, controls the position of a given surface Au atom. When α=0 the corresponding Au atom is moved away from the
molecule. On the contrary when α=1, this atom is STM manipulated toward the end of a naphtyl branch. βcontrols the second Au atom in
the same way. The output of this molecular logic gate is measured in the tunneling current intensity going from the tip to the substrate through
the end of the third naphtyl branch. (b) Constant current STM experimental image of the trinaphthylene physisorbed on Au(111) with three Au
atoms in its surrounding (scale: 4 nm ×4 nm).
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surrounding. But due to the delocalization of the molecular
orbitals over the molecular board, each Au atom is able to act
nonlocally on the molecule and have a remote action on its
electronic conductance.23
Analytically pure trinaphthylene has been synthesized from
3-trimethylsilylnaphthyl-2- trifluoromethanesulphonate via a
palladium-catalyzed [2+2+2] trimerization.24,25 Once synthe-
sized, the molecules have been sublimed by free evaporation
in an UHV preparation chamber on a Au(111) surface kept
at room temperature. Thereafter, the Au(111) sample was
cooled down to 5 K. With the evaporation parameters used,
coverage under 0.1 monolayer was attained. STM imaging
shows molecules adsorbed on Au terraces and at step edges.
Single Au atoms were produced by a gentle crash of the W
tip apex on the Au(111) surface and manipulated with the
STM tip for contacting a single molecule.26 A typical image
of a trinaphthylene molecule with three companion single Au
atoms is presented in Fig. 4(b). For this image three Au atoms
were STM manipulated in close proximity to a molecule: two
for the logical inputs and one as a reference to verify whether
the molecule moved during the STM manipulation of a given
Au atom. Standard R=250 Mlow bias voltage tunneling
resistance was used for imaging and R=0.2Mfor single
Au atom manipulation in pulling mode. Whenever necessary,
the trinaphthylene molecule was manipulated to bring it closer
to the Au atoms, also in pulling mode, using an R=200 M
junction resistance with a large bias voltage. To reset the
two logical inputs to zero, the trinaphtylene can be easily
manipulated away from the two Au atoms in a pushing mode.
The three nonequivalent input configurations, that is, (0,0),
(1,0), and (1,1), are presented in Fig. 5with none, one, and two
Au atoms interacting with the trinaphthylene molecule. In each
FIG. 5. (Color online) Top: The three experimental images
obtained approaching successively none, one, and two Au atoms
near the end of the naphtyl branches. Middle: Molecular model of
the Au-trinaphthylene surface system when optimizing its atomic
scale surface structure for the experimental and calculated images
to converge scan by scan. This optimization demonstrates the
displacement of the H atoms when approaching a Au atom to
the molecule. Bottom: The theoretical ESQC-STM images of the
trinaphtylene interacting with none, one, and two surface atoms.
case, the final molecule conformation and the position of the
Au atoms at the end of the naphthyl branches were determined
by comparing experimental and calculated STM-elastic scat-
tering quantum chemistry images.22 The atomic superposition
and electron delocalization (ASED+)27,28 molecular mechan-
ics routine was used to extract the optimum molecule-surface
conformation so that the experimental and calculated images
converge. As it can be seen in Fig. 5, these calculations reveal
that contacting one Au atom to the end of a naphthyl branch
deforms its terminal phenyl group while slightly lifting up the
corresponding hydrogen atom.23
The tunneling current intensity through the third branch
of the molecule is highly sensitive to the energy position and
the lateral expansion of the trinaphtylene’s electronic states. To
understand how the manipulation of the Au atoms affects these
molecular electronic states, dI/dV maps of the trinaphthylene
molecule has been recorded for zero, one, and two gold atoms
interacting with the molecule. These maps, presented in Fig. 6,
allow us to visualize the lateral spatial extension of the frontier
orbitals of the trinaphthylene for the different value of the
FIG. 6. (Color online) Experimental and theoretical dI/dV map
of the molecule recorded at the energy of the HOMO-1, HOMO, and
LUMO of the system for zero, one, and two atoms interacting with the
molecule. A good agreement is seen between experiment and theory.
Experimental and theoretical images shows how approaching one
atom in the vicinity of the molecule deforms nonlocally its molecular
orbitals due to their delocalization over the molecular board.
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W. H. SOE et al. PHYSICAL REVIEW B 83, 155443 (2011)
logical input. ESQC STM image calculations clearly identify
the corresponding molecular orbitals of the trinaphthylene.
The case where no Au atoms are connected to the molecule,
that is, α=β=0, allows us to determine precisely the
optimum tip position above the third naphtyl branch. In this
(0,0) case, the HOMO presents two lobes roughly localized on
the two first carbon atoms of the third naphtyl branch. Since a
strong current is desired for the (0,0) input configuration, the
tip apex has been experimentally positioned on top of one of
these two lobes. Contacting Au atoms to the trinaphthylene
deforms its electronic states not only in the vicinity of the
Au atoms but all over the molecular board. The value of
the logical inputs are quantum mechanically distributed over
the molecule instead of being only available near its contact
points with the Au atoms.
The different dI/dV spectra of the molecule, represented
in Fig. 7, were recorded to determine the energy position
of its ground state, via its first oxidized (HOMO) state,
and first reduced (LUMO) states for the three nonequivalent
input configurations: (0,0), (1,0), and (1,1). In the HOMO-
LUMO gap voltage range small resonances resulting from the
superposition of surface states with the Au states appear as
shown in Fig. 7. Due to its close proximity to the vacuum
level, the LUMO is much broader than the HOMO.29 This
broadening obscures any molecular orbital splitting or shifting
obtained when the Au atoms are brought closer to the molecule.
On the contrary, the energy shift of the narrow HOMO is
clearly observed. As expected from the molecular design, the
first oxidation state of the molecule is shifted down when Au
atoms interact with the molecule. The output of the logic gate
FIG. 7. (Color online) dI/dV spectra for the three nonequivalent
Au-trinaphthylene molecule configurations. While a broad peak
corresponding to the LUMO appears in all configurations at the
positive bias, a distinct peak in the negative regime observed on
a bare molecule is split and shifted by Au atom inputs. The
HOMO of a molecule in gas phase, with a threefold symmetry,
and the degenerated HOMO-1 and HOMO-2 orbitals, with twofold
and mirror symmetries, are very close to the same energy level.
However once adsorbed on a substrate this degeneracy unravels due to
symmetry constrains resulting in the HOMO-2 orbital being dropped
out from this energy range. Therefore the peak at 1600 mV in the
spectrum taken from a bare molecule consists of only HOMO and
HOMO-1. The surface state of Au(111) is provided as a reference.
FIG. 8. (Color online) I-Vcharacteristics recorded by position-
ing the tip apex at the output side of the molecule exactly at the
maximum lobe position of the first resonance at negative bias voltage
(HOMO). Four I-Vcharacteristics were recorded corresponding to
0, 1, 1, and 2 Au input atoms at the in position. The absolute value of
the current intensity is larger in the (0,0) case than in the three other
case which is the requirement to obtain a NOR gate.
can then be measured by the conductance of the molecule
through the third branch for a bias voltage of 1.6 V. A more
convenient way to access the output status of the logic gate is to
record the tunneling current intensity measured at the fixed bias
voltage of 1.6 V. This is equivalent to integrating the dI/dV
spectrum from the Fermi energy down to the HOMO resonance
of the (0,0) input configuration. As presented in Fig. 8,the
current intensity going through the molecule is consequently
high (I0.8 nA) in the (0,0) configuration due to the presence
of a conducting channel (the HOMO level) in the integration
region. Approaching one or two gold atoms from the molecule
pushes the HOMO out of the integration region and the current
then drops to 0.4 nA. This is exactly the truth table of a NOR
gate. The difference between a 0 and a 1 logical output is
smaller than in the model case presented in the Fig. 3(b) table.
This is due to the fact that for simplicity the experiment was
performed directly on an Au(111) metallic surface and not on
an insulating surface. The 0.32 nm surface height of the board
is not enough to minimize the contribution of all the molecular
orbitals to the output tunneling current intensity. The output
measurement is far from being the one presented in the Fig. 2
model. Progresses in four-probe UHV measurement is opening
an experimental path to perform such QHC single molecule
logic gates on a semiconductor surface.30
IV. A QUANTUM CHEMISTRY INSIGHT
In order to gain insights into the mechanism of the
molecular NOR gate presented above, full semiempirical
PM6-CI molecular electronic states calculations31 have been
performed. These calculations allow us to follow how the
ground and first excited states of the molecule evolve as a
function of the number of Au atoms input. Although useful in
identifying a given molecule electronic state associated with
an observed electronic resonance, the STM-imaged molecular
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FIG. 9. (Color online) The molecular orbital diagrams of the
trinaphthylene molecule for the three different input configurations
calculated using the PM6 method. Connecting a Au atom with the
molecule deforms its molecular orbitals.
orbitals are only one part of the total description of the
electronic states of a molecule. The configuration interaction
(CI) calculation allow us to determine the contributions of the
different molecular orbitals in a given molecular electronic
state, as imaged by the dI/dV STM mapping technique.29
Starting from a semiempirical PM6 molecular orbital
description of the molecule (see Fig. 9), a multielectronic CI
calculation was performed to determine the energetic position
of the low-lying valence states for the (0,0), (1,0), and (1,1)
configurations. These calculations have been made assuming
that the physisorption of the molecule on Au(111) does not
shift the energy of its molecular orbitals. The minimum number
of self-consistent field (SCF) MOs used in our CI calculations
are, respectively, n=4, 5, and 6: the HOMO and LUMO for
(0,0), HOMO, SOMO, and LUMO for (1,0), and HOMO,
LUMO, LUMO+1, and LUMO+2 for (1,1). More MOs
can be added. But the valence states are already stabilized
with the ones considered here. The numbers of spin adapted
determinants generated by the electronic excitations in the
corresponding active space are 36, 100, and 400, respectively.
The CI calculations allow us to trace the electronic ground
state and the first excited state of the molecule from where
the first oxidation and first electronic reduction state will be
accessible in a tunneling experiment.
The energy correlation diagram of the lowest valence states
[i.e., 3 for (0,0), 5 for (1,0), and 13 for (1,1)], which only
involve the monoelectronic excitations belonging to the SCF
MOs localized on a trinaphthylene molecule, are presented
in Fig. 10. For the (1,0) and (1,1) input configurations, the
excitations, picking-up the Au 6satomic orbitals, generate
excited electronic states much higher in energy than the low-
lying ground and first excited state of the molecule. They
can be neglected in a first approximation. The valence states
are correlated by their spatial symmetry in the framework of
their common C2vsubgroup. The CI calculations presented in
Fig. 10 confirm that the ground state is shifted down in energy
by 0.1 to 0.2 eV when one or two Au atoms are connected
to the molecule. This ground state gives rise to the first large
electronic resonance observed in the tunneling junction below
FIG. 10. (Color online) Energy correlation diagram for the
ground (1A1) and first excited states of the molecule. Notice how
for energies up to the Fermi level the Au interaction spreads the
monoelectronic excitation states, many of them being uncoupled from
the surface and the tip in an STM junction.
the Fermi level (see Fig. 6). This is precisely the tunneling
resonance which is used to determine the output status of the
NOR gate. For the (1,1) input configuration, the triplet state
immediately above the ground state had not been observed
experimentally even if selection rules do not exist normally
for electronic states tunneling spectroscopy.
V. CONCLUSION
In a QHC designed molecule, a logic operation is performed
without clustering qubits in the molecule, and it is also not
necessary to bind chemical groups like molecular rectifiers,
transistors, or switches to construct a logic gate within one
single molecule. The local and classical presence of one or two
Au atoms on the inputs branches of a trinaphthylene molecule
is producing a transformation of its electronic structure readily
accessible to tunneling spectroscopy measurements on the
output branch. A molecule with a different molecular orbital
structure will produce a different Boolean truth table depend-
ing on the symmetry breaking of its molecular orbitals upon
interacting with the Au inputs and on how those molecular
orbitals are mixed up to shape the true molecular valence
states. This QHC design can be generalized to more complex
logic functions embedded in one molecule creating a definitive
link between molecular electronics and quantum design.
ACKNOWLEDGMENTS
We acknowledge the Agency of Science, Technology and
Research (A*STAR) for funding provided through the Visiting
Investigatorship Program (Phase II) Atomic Scale Technology
Project, the European Commission integrated project Pico-
Inside, and the MEXT NIMS MANA project. N.R. was
supported as part of the Non-Equilibrium Energy Research
Center (NERC), an Energy Frontier Research Center funded
by the US Department of Energy, Office of Science, Office of
Basic Energy Sciences under Award Number DE-SC0000989.
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