# Controlling the entropy of a single-molecule junction

Eugenia Pyurbeeva  School of Physics and Astronomy, Queen Mary University of
London, Mile End Road, London E1 4NS, UK Chunwei Hsu  Kavli Institute of
Nanoscience, Delft University of Technology, Lorentzweg 1, Delft 2628 CJ, The
Netherlands David Vogel  Department of Chemistry, University of Basel, St.
Johanns‐Ring 19, 4056 Basel, Switzerland Christina Wegeberg  Department of
Chemistry, University of Basel, St. Johanns‐Ring 19, 4056 Basel, Switzerland
Marcel Mayor  Department of Chemistry, University of Basel, St. Johanns‐Ring
19, 4056 Basel, Switzerland Herre van der Zant  Kavli Institute of
Nanoscience, Delft University of Technology, Lorentzweg 1, Delft 2628 CJ, The
Netherlands Jan A. Mol<EMAIL_ADDRESS>School of Physics and Astronomy,
Queen Mary University of London, Mile End Road, London E1 4NS, UK Pascal
Gehring<EMAIL_ADDRESS>IMCN/NAPS, Université Catholique de
Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium

###### Abstract

Single molecules are nanoscale thermodynamic systems with few degrees of
freedom. Thus, the knowledge of their entropy can reveal the presence of
microscopic electron transfer dynamics, that are difficult to observe
otherwise. Here, we apply thermocurrent spectroscopy to directly measure the
entropy of a single free radical molecule in a magnetic field. Our results
allow us to uncover the presence of a singlet to triplet transition in one of
the redox states of the molecule, not detected by conventional charge
transport measurements. This highlights the power of thermoelectric
measurements which can be used to determine the difference in configurational
entropy between the redox states of a nanoscale system involved in conductance
without any prior assumptions about its structure or microscopic dynamics.

Institute for Nanotechnology (INT), Karlsruhe Institute of Technology (KIT),
P.O. Box 3640, 76021 Karlsruhe, Germany Lehn Institute of Functional Materials
(LIFM), School of Chemistry, Sun Yat-Sen University (SYSU), 510275 Guangzhou,
China

The thermodynamic concept of entropy is key to the understanding of many
chemical processes, including electron transfer reactions 1. Spontaneous
change only occurs when entropy increases and the universe becomes more
disordered. While the statistical interpretation of entropy first given by
Boltzmann connects entropy as a thermodynamic state function to the number of
microstates available to the system, this connection is generally not
quantifiable – macroscopic systems, containing Avogadro’s numbers ($\sim
10^{23}$) of molecules, simply have too many configurations for the individual
to be tractable. At the single-molecule level, however, electron transfer
reactions lead to significant changes in configurational entropy related to
the electronic structure that can be quantified due to the relatively small
number of states of the system 2. The ability to directly measure entropy of
individual molecules – without the need of a priori knowledge of its
electronic structure – can thus provide insights into their non-trivial
quantum states, such as (high-)spin ground states 3, 4, Kondo states 5, 6, 7,
Majorana modes 8, and non-abelian anyons 9, 10, and shed light on the
discriminating role of entropy in driving electron transfer rates.

Recent experiments have demonstrated the potential for direct entropy
measurements in mesoscopic quantum systems 11, 12, 13, 14. These studies
connect macroscopic observable quantities, including electrical current and
charge, to microscopic configuration entropy in the form of spin-degeneracy.
Here, we scale this approach down to the single-molecule level and determine
the redox-dependent entropy of an individual nitronyl nitroxide radical (shown
in Figure 1a) directly from its thermoelectric conductance. The entropy
measurements indicate that the neutral molecule holds an unpaired electron and
is therefore in a doublet ground state, while the ground state of the reduced
molecule is a singlet with all electrons paired. However, by applying a
magnetic field to the reduced molecule, a low-lying triplet excitation state
is revealed which was not observed in conventional charge transport
measurements.

Figure 1: a) Sketch of the molecule (a free radical, green, located at the a
nitronyl nitroxide side group, blue, which is connected to a backbone, black)
connected to the source and drain lead. The red arrows indicate electron
tunneling on and off the molecule. b) Bias and gate voltage dependent
differential conductance and c) thermocurrent of a single NNR molecule
junction.

To measure the entropy difference $\Delta S$ between the neutral (with $N$
electrons) and the reduced (with $N+1$ electrons) free radical molecule
directly from its thermoelectric response, we first develop the necessary
theoretical framework. To this end, we find the elements of the Onsager matrix
that are related to the charge current $I$. Starting from a thermodynamic
relation between the internal energy of the molecule, its redox state, and its
configurational entropy 15, we derive an expression for the conductance
$G=\partial I/\partial V_{sd}$ (where $V_{sd}$ is the bias voltage across the
molecule) from rate equations (see SI) that is consistent with that derived
for sequential electron tunneling through a quantum dot16. Moreover, we derive
an expression for the thermoelectric conductance $L=\partial I/\partial\Delta
T$ (where $\Delta T$ is the temperature difference across the molecule) which
takes a similar form:

$\displaystyle G\propto\dfrac{1}{T}$
$\displaystyle\left[1-f(\varepsilon)\right]f(\varepsilon-T\Delta S),$ (1)
$\displaystyle L\propto\dfrac{\varepsilon}{T^{2}}$
$\displaystyle\left[1-f(\varepsilon)\right]f(\varepsilon-T\Delta S),$ (2)

where $f$ is the Fermi-Dirac distribution, $\varepsilon$ the single-electron
energy level with respect to the electrochemical potential of the electrodes,
and $T$ the equilibrium temperature of the system. Here we recognise the free
energy of formation $\Delta F=\varepsilon-T\Delta S$ associated with the
isothermal electrode reaction as the driving force for electron transfer17. We
also observe that, as a result, the conductance is maximum when
$\varepsilon=T\Delta S/2$, as was experimentally demonstrated previously18,
14. More importantly, the thermoelectric response is asymmetric around
$\varepsilon=0$ and the degree of asymmetry scales directly with $\Delta S$.
In the remainder of this Letter, we will therefore exploit this asymmetry as a
direct measure for the entropy difference.

The conductance and thermoelectric response of the free radical are measured
simultaneously by contacting it in an electromigrated breakjunction with
integrated micro-heaters and gate electrodes using a double lock-in
technique19, 14. Here, we use an organic radical molecule (nitronyl nitroxide
radical). It consists of a 1,4-bis(4-pyridyl)benzene backbone and a nitronyl
nitroxide side group where an unpaired electron resides as shown in Fig. 1a.
This makes it a single-spin system in its neutral state. Single-molecule
junctions are formed by opening a nm-sized gap in a thin gold bridge using
electromigration followed by self-breaking19 and depositing of single-
molecules from solution after nano-gap formation. Figure 1c and d show the
conductance and thermocurrent $L\Delta T$ as a function of the applied bias
($V_{sd}$) and gate ($V_{g}$) voltage. We observed regions of low conductance
(Coulomb-blocked, black in Figure 1b), corresponding to the $N$ and $N+1$
redox states of the free radical, separated by a region of sequential electron
tunneling. Furthermore, lines inside the sequential tunneling regime which run
in parallel to the edges of the Coulomb diamond can be observed, which we
attribute to vibrational excitations of the molecule 20, 21, 22, 23, 14. The
thermocurrent (Figure 1c) is zero at the Coulomb diamond edges and reaches
minimum/maximum values close to these edges (blue and red lines), similar to
our previous findings on Gd-tpy complexes 14.

Figure 2: a) Experimental thermocurrent and b) conductance (open circles), and
theoretical fits (solid lines) using equations 1 and 2 (in the small
temperature difference assumption) at three different magnetic fields. The
yellow shaded areas indicate the minimum thermocurrent $I_{\mathrm{th,min}}$.
The vertical dotted lines indicate $I_{\mathrm{th}}=0$.

In the following, we will focus on the gate-dependent conductance and
thermocurrent traces at zero bias ($V_{sd}=0$), as shown in Figure 2. We find
a peak in conductance around a gate value of 4.8 - 4.9V, where an extra charge
is continuously added and removed to/from the molecule leading to electron
transfer from one side of the molecule to the other. For gate voltages $\gg
4.9$ V $N+1$ charges reside on the molecule and it is in its reduced state.
The gate-dependent thermocurrent changes sign around the position of the
conductance peak and possesses a pronounced asymmetry between its maximum and
minimum. As explained above, this asymmetry is a direct measure of the entropy
difference between the redox states of the free radical. We further observe
that the degree of asymmetry of the thermocurrent, as well as the shift of the
conductance peak, vary when a magnetic field is applied (see Figure 2b and c).
This is a clear indication that the entropy of the system can be controlled by
a magnetic field. To quantify this entropy change $\Delta S$ as a function of
magnetic field we fit the zero-bias conductance and thermocurrent traces shown
in Figure 2 to Equations 1 and 2 (see solid lines in Figure 2) using
$\varepsilon=\varepsilon_{0}-\alpha V_{g}$, where $\alpha$ is the lever-arm
given by the capacitance between the free radical and the electrodes 16, 24,
25.

Figure 3: a) Entropy change of the molecule obtained from fitting equation 1
and 2 to the experimental data (in the small temperature difference
assumption) as a function of magnetic field (points). Solid line represents
the best fit of the entropy change values to the difference in Gibbs entropy
between the charge states in the doublet $\rightarrow$ singlet+triplet model.
b) Energy level structure for the molecule in its neutral state (N electrons)
and reduced state (N+1 electrons) as a function of magnetic field, for the
values of $\Delta_{ST}$ and g-factor from fit in 3a. It can be seen that the
point of zero entropy change (dashed line at approximately 6T) approximately
corresponds to the same energy splitting between the two doublet states and
the two lower-lying states of the $N+1$ charge states, meaning that the
transition effectively occurs between two identical doublet states (vertical
arrows). The yellow spheres (and corresponding arrows) illustrate the spin
states of the free radical (on the nitronyl nitroxide unit) and the extra
added electron on the backbone (in the reduced case, N+1).

Figure 3a shows the resulting entropy difference between the $N$ and $N+1$
redox states as a function of magnetic field determined by fitting to Eqn 2.
At zero magnetic field $\Delta S$ approaches a value of $-k_{B}\ln 2$, where
$k_{B}$ is the Boltzmann constant. Because the Gibbs entropy is equal to
$-k_{B}\sum p\ln p$, where $p$ is the occupation probability of each
microstate of the system, we can conclude that the $N$ charge state has 2
accessible microstates with equal occupation probabilities, while the $N+1$
charge state has one. This agrees with the free radical ground state having an
unpaired electron in its ground state, and thus being a spin-doublet, and the
reduced state having all electrons paired, being a singlet.

When the magnetic field is increased, we observe an increase of $\Delta S$,
which reaches a value of zero and changes sign at $B\sim 6$ T. With the
application of the magnetic field, the degeneracy of the doublet ($N$
electrons) ground state is lifted. As the field – and with it the Zeeman
splitting – is increased, the probability of occupation of the bottom energy
level of the doublet begins to dominate. This can explain the increase of
$\Delta S$ at low magnetic fields, however this is not consistent with the
subsequent change of sign of $\Delta S$ at approximately 6T. The magnetic-
field-dependent entropy therefore points to a low-lying triplet state of the
reduced free radical where the entropy associated to the partial population in
the $N+1$ singlet and triplet states can exceed the entropy of the $N$ doublet
population. We confirm these results by comparing them to previously employed
methods 15, 14 based on the analysis of the difference in the energy levels
corresponding to zero thermocurrent point and the conductance peak, where we
obtained a similar entropy difference dependence to the thermocurrent
asymmetry analysis (see SI section 2).

To determine the singlet-triplet splitting $\Delta_{ST}$ in the $N+1$ redox
state, we find the expected entropy difference for the doublet $\rightarrow$
singlet+triplet transition, assuming a Gibbs distribution in each of the
charge states, as a function of magnetic field, singlet-triplet splitting
$\Delta_{ST}$ and the g-factor. We use this dependence to fit the
experimentally obtained $\Delta S$ dependence on magnetic field with
$\Delta_{ST}$ and the g-factor as a fitting parameters (the fitting result is
shown in Figure 3). This approach yields a $g$-factor of 2.7, deviating from
the value of 2 for a free electron. Such deviations have been found before in
nano-scale junctions and have been attributed to orbital contributions.26, 27
Furthermore, we extract a singlet-triplet splitting $\Delta_{ST}=1.7\pm 0.1$
meV. This low energy can be explained by assuming that the extra electron
(when reducing the molecule) is added to the molecular back-bone rather than
to the nitronyl nitroxide side group of the molecule. $\Delta_{ST}$ would then
correspond to the exchange interaction between the two free electrons (on the
side group and the backbone), similar to what has been found experimentally in
similar free radical all-organic systems 28.

The small singlet-triplet splitting, and the resulting entropy contribution of
the non-zero $N+1$ triplet population, is the reason for the deviation of
$\Delta S$ from $-k_{B}\ln 2$ at zero magnetic field. Moreover, at $B\sim 6$ T
the splitting between the $|T_{+}\rangle$ triplet state and the singlet is
equal to the splitting between the doublet states (see Figure 3b), and the
Gibbs entropy of the two redox states are approximately equal,
$-k_{B}\left[p_{\uparrow}\ln p_{\uparrow}+p_{\downarrow}\ln
p_{\downarrow}\right]\approx-k_{B}\left[p_{S}\ln p_{S}+p_{T_{+}}\ln
p_{T_{+}}\right]$, (where $p$s are the occupation probablities of $\uparrow$ –
the spin up doublet state, $\downarrow$ – spin down doublet state, $S$ – the
singlet state and $T_{+}$ – the lowest-lying triplet state), as the population
in $|T_{0}\rangle$ and $|T_{-}\rangle$ are negligible, where $T_{0}$ and
$T_{-}$ are the middle- and high-energy triplet states respectively. When
$\Delta S=0$, we observe an anti-symmetric thermoelectric response
($I_{\mathrm{th}}(\varepsilon)=-I_{\mathrm{th}}(-\varepsilon)$) as there is no
difference in free energy between the two redox states. We expect $\Delta S$
to reach its positive maximum at $B\sim 12$ T, outside of our experimental
range, when the $|T_{+}\rangle$ and $|S\rangle$ states cross and the $N+1$
entropy is maximum $S_{N+1}\approx k_{B}\ln 2$ while $S_{N}$ tends to zero.

This analysis demonstrates that thermocurrent spectroscopy is a sensitive tool
to access the fundamental thermodynamic properties of nanoscale devices. It
has several advantages over previously employed methods, because, unlike e.g.
the approach taken in 11, 12 our method does not rely on charge state
measurements and therefore does not require the fabrication of complicated
charge sensors and is applicable to any devices on which thermoelectric
measurements can be performed. Furthermore, it does not require measurements
of the device at multiple temperatures to find the entropy, as was the case
one of the previous approaches 14. We applied the method to a single-molecule
device and found that the direct measurement of the entropy difference with
the change of the redox state reveals the presence of a low-lying triplet
state in the molecule and allows us to quantify the exchange coupling between
the free radical and the extra added electron. This serves as further evidence
for the potential of practical applications of entropy measurement methods.

We thank M. van der Star for his support in sample fabrication. This work is
part of the Organization for Scientific Research (NWO) and the Ministry of
Education, Culture, and Science (OCW). P.G. (research associate) acknowledges
financial support from the F.R.S.-FNRS of Belgium and a Marie Skłodowska-Curie
Individual Fellowship under Grant TherSpinMol (ID: 748642) from the European
Union’s Horizon 2020 research and innovation programme. J.A.M. was supported
through the UKRI Future Leaders Fellowship, Grant No. MR/S032541/1, with in-
kind support from the Royal Academy of Engineering. H.v.d.Z, C.H., M.M. and
D.V. acknowledge funding by the EU (FET-767187-QuIET). M.M. acknowledges
support from the Swiss National Science Foundation (SNF grant numbers
200020-178808) and the 111 project (90002-18011002). C.W. thanks the
Independent Research Fund Denmark for an international postdoctoral grant
(9059-00003B).

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