# Electronic measurements of entropy in meso- and nanoscale systems

Eugenia Pyurbeeva<EMAIL_ADDRESS>School of Physics and Astronomy,
Queen Mary University of London, Mile End Road, London E1 4NS, UK Jan A. Mol
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

Entropy is one of the most fundamental quantities in physics. For systems with
few degrees of freedom, the value of entropy provides a powerful insight into
its microscopic dynamics, such as the number, degeneracy and relative energies
of electronic states, the value of spin, degree of localisation and
entanglement, and the emergence of exotic states such as non-Abelian anyons.
As the size of a system decreases, the conventional methods for measuring
entropy, based on heat capacity, quickly become infeasible due to the
requirement of increasingly accurate measurements of heat. Several methods to
directly measure entropy of mesoscopic quantum systems have recently been
developed. These methods use electronic measurements of charge, conductance
and thermocurrent, rather than heat, and have been successfully applied to a
wide range of systems, from quantum dots and molecules, to quantum Hall states
and twisted bilayer graphene. In this Review, we provide an overview of
electronic direct entropy measurement methods, discuss their theoretical
background, compare their ranges of applicability and look into the directions
of their future extensions and applications.

###### Contents

  1. I Introduction
  2. II Alternative entropy measurements in continuous-charge nanodevices
  3. III Quantised-charge nanodevices
    1. III.1 The effects of charge quantisation
    2. III.2 Thermodynamic parameters of a quantised-charge system
    3. III.3 Maxwell relations in quantised-charge systems
    4. III.4 Quantised-charge entropy measurement methods
    5. III.5 Limitations of entropy measurement methods
  4. IV Current applications
    1. IV.1 Detection of non-Abelian anyons
    2. IV.2 Twisted bilayer graphene
    3. IV.3 Single electron transistors and single molecule devices
  5. V Outlook and Conclusions

##  I Introduction

The connection between macroscopic observables and microscopic dynamics was
first made by Maxwell when he attributed part of the heat capacity $C$ of a
gas to the rotational degrees of freedom of its moleculesMaxwell (1860). The
name _entropy_ was coined by Clausius in 1865 to describe the non-usable
energy increase in a steam engine’s exhaust as a function of its temperature
$T$ through the relation $dS=CdT/T$. Boltzmann and Gibbs later gave entropy
its statistical basis that connects the observable averaged state functions to
the microscopic dynamics of a system described by the number of microstates
$\Omega$ and their probabilities, epitomised by Boltzmann’s constant
$k_{\rm{B}}$ in his definition of entropy $S=k_{\rm{B}}\ln\Omega$. Since then,
entropy measurements based on heat capacity and temperature have been used to
probe the microscopic structure of bulk materials, such as the disorder in
alloys Bragg and Williams (1934, 1935); Nix and Shockley (1938), phase
transitions in molecular crystals Guthrie and McCullough (1961) and the
magnetic arrangements of spin-ice Ramirez  _et al._ (1999).

The development of nanofabrication, in particular molecular beam epitaxy and
electron beam lithography, has made it possible to confine electrons in one or
more dimensions. This has enabled the measurement of heat capacity and entropy
in mesoscopic systems such as two-dimensional (2D) electron gasses in GaAs
quantum well structures Gornik  _et al._ (1985); Wang  _et al._ (1988); Bayot
_et al._ (1996) and of fractional quantum Hall systems Schulze-Wischeler  _et
al._ (2007); Schmidt  _et al._ (2017). These experiments present significant
difficulties as the contribution to the heat capacity of the confined
electronic states is generally far smaller than that of the vibrations, or
phonons, in the substrate material. To overcome this challenge, the electronic
contribution to the heat capacity was increased by increasing the number of
quantum wells Gornik  _et al._ (1985); Bayot  _et al._ (1996); Wang  _et al._
(1988) and by exploiting the difference in thermalisation time of the
electronic and phononic systems Schulze-Wischeler  _et al._ (2007); Schmidt
_et al._ (2017). Despite the ingenuity of these solutions, further down-
scaling of entropy measurements based on heat capacity to few-electron
nanodevices is practically impossible as the small numbers of electrons and
electronic states involved require the measurement of increasingly small
amounts of heat and heat capacities. For comparison, the maximum heat capacity
of a single electronic spin is of the order of $10^{-24}$J/K, while the
minimal values of heat capacity per particle experimentally measured in
Schmidt  _et al._ (2017) are three orders of magnitude greater.

Yet, it is precisely in systems containing only a few particles that entropy
measurements can reveal the most about microscopic dynamics. For example, the
von Neumann entropy of the entangled state
$\left(|00\rangle+|11\rangle\right)/\sqrt{2}$ is $k_{\text{B}}\ln 2$, while
the entropy of the product state
$\left(|00\rangle+|01\rangle+|10\rangle+|11\rangle\right)/2$ is zero Potts ;
Kosloff (2013). There is therefore a strong motivation to measure the entropy
of few-electron quantum systems that has led to the development of alternative
ways that do not rely on detecting the prohibitively small flow of heat.
Electronic entropy measurements have recently emerged to address this
challenge. Unlike heat, charge is a conserved quantity, which makes it much
easier to measure. Charge sensitivities down to level of
$10^{-6}\times$e/$\sqrt{\text{Hz}}$ can be achieved in few-electron devices,
and charge currents as small as $10^{4}$ e/s can be detected. Both these
measures, of charge state and current, can be used to determine the entropy of
a system. Electronic entropy measurements are thus ideally suited to probe
microscopic dynamics of few-electron quantum systems that are weakly coupled
to their environment.

This Review will summarize the thermodynamic theory underpinning both charge-
and current-based entropy measurements of mesoscopic quantum systems, and
discuss their application to several material systems and devices, including
twisted bilayer graphene Rozen  _et al._ (2021); Saito  _et al._ (2021),
Kondo-systemsHan  _et al._ (2021), quantum Hall states, and quantum dots and
quantum dot systems Hartman  _et al._ (2018); Kleeorin  _et al._ (2019) as
well as single-molecule devices Gehring  _et al._ (2021); Pyurbeeva and Mol
(2021). In all these systems electronic entropy measurements may serve a dual
purpose. On the one hand, electronic entropy measurements provide insight into
the internal quantum structure of matter, and might reveal exotic quasi-
particles such as Majorana fermions. On the other hand, knowledge can be
gained about the operation of an entire new class of nanoscale thermodynamic
systems that include single-electron heat engines that operate close to the
Curzon-Alhborn limit Josefsson  _et al._ (2018), stochastic Maxwell demons
Koski  _et al._ (2015); Sánchez, Samuelsson, and Potts (2019); Sánchez,
Splettstoesser, and Whitney (2019) and Szilard engines Kim  _et al._ (2011),
and clocks Erker  _et al._ (2017); Pearson  _et al._ (2021).

In Section II of this review, we will discuss different approaches to
determine entropy of bulk materials that do not rely on heat measurements. In
Section III we will extend this discussion to systems with a quantised charge,
and in Section IV we will discuss how these entropy measurements can be used
to reveal microscopic dynamics of quantum systems under investigation.
Finally, we will provide an outlook on the future of nanoscale thermodynamics
and entropy measurements of quantum systems.

##  II Alternative entropy measurements in continuous-charge nanodevices

Figure 1:  Experimental setups for measuring entropy in nanoscale systems. a)
A nanotube-based single electron transistor (SET) measuring the charge density
and compressibility in a continuous-charge system (twisted bi-layer graphene,
tBLG) under an external magnetic field $B_{||}$Rozen  _et al._ (2021). The
graphene is encapsulated from both sides by insulating hexagonal boron nitride
(hBN) layers (blue). The graphite back gate (purple) is used for setting the
chemical potential of the graphene. b) A charge-state based entropy
measurement setup in a GaAs quantum dotHartman  _et al._ (2018). The
electrodes coloured in yellow electrostatically define the circuit in the 2D
electron gas. The quantum dot (QD) exchanges electrons with a thermal electron
reservoir (red). The quantum point contact on the left is used to measure the
charge ($n$ (d.c.) and $\Delta n$ (a.c.), see Figure 2) of the quantum dot.
This charge can be tuned by the voltage $V_{\mathrm{p}}$. c) A transport
entropy measurement setup in a single-molecule deviceGehring  _et al._ (2021).
A molecule bridges the gap formed by electromigration of gold contacts. Each
side of the device has a heater (blue), while the back-gate (purple) is used
to control the energy levels of the molecule. Simultaneous measurement of
current, conductance and thermocurrent is possible due to the double
demodulation technique – the components of current through the molecule are
taken at zero frequency, the oscillation frequency of the bias voltage and
double the frequency of the heater bias.

Since conventional approaches to entropy measurements are not easily
applicable to mesoscopic systems, there has been great effort in developing
approaches that rely on observables that are experimentally accessible in
these systems, with electric measurements being the prime candidate. The
thermoelectric effect, which connects heat (and therefore entropy) and charge
transport has been proposed as an entropic probe for non-Abelian statesYang
and Halperin (2009). Indeed, for a free electron gas, the low-temperature
Seebeck coefficient is directly proportional to the entropy per unit
chargeChaikin and Beni (1976); Goupil  _et al._ (2016). While this relation
holds true in a wide range of bulk materialsBehnia, Jaccard, and Flouquet
(2004); Zlatić  _et al._ (2007), it breaks down, for example, in highly
anisotropic transportMravlje and Georges (2016) and should thus be used with
caution in mesoscopic systems.

Going beyond the thermoelectric relation between heat and charge transport,
Maxwell relations can be used to connect the experimentally controllable
magnetic field, $B$, and temperature, $T$, and the observable magnetisation,
$m$, electron density, $n$, and chemical potential, $\mu$, of a quantum Hall
system to its entropy per unit area, $s$Cooper and Stern (2009):

$\left(\frac{\partial m}{\partial T}\right)_{B}=\left(\frac{\partial
s}{\partial B}\right)_{T}$ (1) $\left(\frac{\partial\mu}{\partial
T}\right)_{n}=-\left(\frac{\partial s}{\partial n}\right)_{T}$ (2)

This approach was initially proposed as an alternative to interferometry
methods to detect signatures of non-Abelian quantum Hall statesSarma,
Freedman, and Nayak (2005); Stern and Halperin (2006); Bonderson, Kitaev, and
Shtengel (2006); Feldman  _et al._ (2007), however it can equally be applied
to any electronic system with temperature- and field-dependent magnetisation,
such as spin-ordering transitions in low density electron gases in Wigner
crystal or Luttinger liquid states. Maxwell relations have also been used to
study magic-angle twisted bi-layer graphene to reveal a Pomeranchuk-type
effectRozen  _et al._ (2021); Saito  _et al._ (2021) (see Section IV.2) in a
setup as shown in Figure 1a.

While the thermoelectric and Maxwell relation approaches that we have
described thus far are generally system-independent they rely on the implicit
assumption of large particle numbers such that any change in state variables
such as entropy and magnetisation is linear with the change in particle
numberChaikin and Beni (1976); Zlatić  _et al._ (2007); Yang and Halperin
(2009); Mravlje and Georges (2016). This assumption does not necessarily hold
for meso- and nanoscale systems with small numbers of particles. In the
remainder of this Review we will discuss the conceptual and experimental
differences between entropy measurements in what we term continuous-charge
devices and quantised-charge devices.

##  III Quantised-charge nanodevices

Figure 2:  Entropic effects in charge state and charge transport measurements
in quantised-state nanodevices. a) A schematic of a charge state measurement
setup. An energy level coupled to a thermal bath, with the charge state of the
level independently determined. b) The mean excess population $n$ of the
energy level in the case of the non-degenerate and two-fold degenerate energy
level ($\Delta S=0$ or $\Delta S=\pm\ln 2$). c) A model of experimental time-
resolved measurement of the charge state of the device. Information of entropy
(or degeneracy) can be extracted from the proportion of time the device spends
in each of the charge statesHofmann  _et al._ (2016). d) A model of the
experiment performed in Hartman  _et al._ (2018) – mean population is measured
as a function of gate voltage (energy level), where an overall shift of the
distribution is observed on top of its spreading. The exact value of the
entropy difference can be found from fitting the difference between the
populations at the two temperatures. e) A schematic of a charge transport
measurement setup. The energy level is coupled to two thermal baths, with a
potential and/or a temperature difference across. The current through the
system is measured. f) The current through the nanodevice for a non-degenerate
and two-fold degenerate energy level ($\Delta S=0$ or $\Delta S=\pm\ln 2$). g)
A model of the experimental thermal shift of the conductance peak of a device,
as observed in Harzheim  _et al._ (2020); Gehring  _et al._ (2021). h) A model
of the thermocurrent and power factor measurement of a quantised-charge
nanodevice. The value of the entropy difference can be extracted from the
asymmetry.

###  III.1 The effects of charge quantisation

We define quantised-charge devices as those meso- and nanoscale systems for
which partial derivatives with respect to particle number $N$ are undefined.
It is important to note that these devices do not necessarily contain a
smaller total number of particles than a continuous-charge devices, but rather
that they have a non-linear energy-particle relation and typically contain a
smaller number of available particle number states. The most common example of
quantised-charge devices are meso- or nanoscale systems that exhibit Coulomb
blockadeNazarov and Blanter (2009), meaning the electrostatic Coulomb
repulsion is the dominant energy scale and only a small number of charge
states (values of particle number in the system) are energetically accessible.
Coulomb blockade is regularly observed in devices based on lithographically-
defined quantum dotsHanson  _et al._ (2007); Thijssen and Van der Zant (2008)
(see Figure 1b), and in single-atomFuechsle  _et al._ (2012) and single-
molecule devicesOsorio  _et al._ (2008); Brooke  _et al._ (2015) (see Figure
1c).

The change of the electrostatic energy depends on the particle number itself
as

$E_{el}(N+1)-E_{el}(N)=2e^{2}/C\left(N+\frac{1}{2}-N_{0}\right)$ (3)

where $E_{el}$ is the electrostatic energy of the system, $C$ is its
capacitance, and $N_{0}$ is the background charge. Therefore, a partial
derivative of energy with respect to particle number cannot be defined for
systems in which the electrostatic energy dominates, even though they may
contain a large number of particles, $N_{0}$. Moreover, in Coulomb-blocked
devices where only two charge states are energetically accessible and $\Delta
N$=1, no linearisation of any state variable $X$ can be justified, since

$\left(\frac{\partial X}{\partial N}\right)_{Y}\neq\frac{X(N+1)-X(N)}{\Delta
N}=X(N+1)-X(N).$ (4)

Additionally, the partial derivative at constant $Y$ cannot be defined, as
each charge state is associated with its own value of $Y$. This means that the
approach developed for continuous-charge devices cannot be applied directly to
quantised-charge devices.

###  III.2 Thermodynamic parameters of a quantised-charge system

One of the main strengths of thermodynamics is system-independence. A
thermodynamic treatment of quantised-charge devices, analogous to the Maxwell
relations for continuous-charge devices, allows for the development of entropy
measurement methods that describe the general quantum dynamics of such
systems. To do this, we will first specify the parameters (state variables)
used in the thermodynamic description of small quantised-charge systems,
namely the temperature $T$, chemical potential $\mu$, particle number $N$, and
entropy $S$.

Temperature. While a quantised-charge system, or its electronic state dynamics
may or may not be sufficiently large to have its own definable temperature,
the Maxwell equation approach taken in section II requires equilibrium
conditions. Following this, we may take the temperature of the nanoscale
system to be equal to the well-defined temperature of the bath with which  it
exchanges electrons Blundell and Blundell (2009); Ford (2013); Potts . In the
case of more than one bath being present, equilibrium is only possible when
the temperature differences between the baths are infinitesimally small.

Chemical potential. In quantum dot literature, the energy difference between
the charge states $\epsilon=U(N+1)-U(N)$ is often referred to as the chemical
potential $\mu$Hanson  _et al._ (2007). However, this does not agree with the
canonical definition of the chemical potential $\mu=(\partial U/\partial
N)_{S}$ due to reasons outlined after equation 4. Instead, for quantised-
charge systems, one should take the approach similar to that of temperature
Pyurbeeva and Mol (2021); Child  _et al._ (2021a) – if a system is in
equilibrium with a bath regarding particle exchange, its chemical potential is
equal to that of the bath Blundell and Blundell (2009); Ford (2013). And if
several baths are present, for quasi-equilibrium conditions the difference in
chemical potentials must be infinitesimally small. For all realistic
temperatures at which nanoelectronic measurements are performed, the electron
gas that forms the bath is highly degenerate and the chemical potential of the
bath and the system are to a high degree of accuracy equal to the Fermi
energy, $\mu=E_{F}$.

Particle number. When we consider a quantised-charge device that exhibits
Coulomb blockade, the particle number constantly fluctuates between $N$ and
$N+1$ with occupation probabilities $p_{N}$ and $p_{N+1}$ that depend on
external parameters (temperature, chemical potential, magnetic field, etc.)
similarly to $N$ in continuous-charge devices. The problem of temporal charge
fluctuations and the discontinuity of state variables can be overcome by using
the mean particle number

$\bar{N}=p_{N}N+p_{N+1}(N+1)=N+n,$ (5)

where $n=p_{N+1}$ is defined as the mean excess number of electrons , which
varies between 0 and 1.

Entropy. In line with the time-independent mean particle number we can define
the mean entropy for a state with a given mean excess number of electrons $n$.
The entropy of a system in each of the charge states, $S_{N}$ or $S_{N+1}$, is
easily defined, however in the case of fluctuating charge, an additional term
is needed to account for the uncertainty of the charge state itself  and the
mean entropy of the system is given byPyurbeeva and Mol (2021)

$\bar{S}=S_{0}+nS_{N+1}+(1-n)S_{N},$ (6)

where $S_{0}$ is the “coarse-grained” Gibbs entropy of the charge state
uncertainty $S_{0}=-n\ln n-(1-n)\ln(1-n)$.

###  III.3 Maxwell relations in quantised-charge systems

Taking the considerations for the treatment of independent thermodynamic
parameters outlined in Section III.2, the standard Maxwell relation should be
interpreted as

$\left(\frac{\partial\mu}{\partial
T}\right)_{\bar{N}}=-\left(\frac{\partial\bar{S}}{\partial\bar{N}}\right)_{T}$
(7)

where $\mu$ is the chemical potential of the bath, $\bar{N}$ is the mean time-
averaged population of the system and $\bar{S}$ is the Gibbs entropy
incorporating both charge state uncertainty and the entropy of each charge
state.  However, since it is the difference between the single-particle energy
of the system and the chemical potential of the bath that is proportional to
the externally controlled gate voltage applied to a quantised-charge device,
$\epsilon-\mu\propto V_{G}$, it is more useful to derive a Maxwell relation
for this difference rather than for the chemical potential.

Such a relation, which we name the _Microscopic Maxwell Relation_ can be
derived by considering the mean free energy of the system,
$\bar{F}=U(N)+\varepsilon n-T\bar{S}$, and the general relation relation for
the chemical potential $\mu=\left(\partial\bar{F}/\partial\bar{N}\right)_{T}$,
which yields

$\varepsilon-\mu=T\left(\frac{\partial\bar{S}}{\partial\bar{N}}\right)_{T}.$
(8)

Since $N$ is constant we can replace all derivatives with respect to $\bar{N}$
with derivatives with respect to $n$ and arrive at what we call the
_Microscopic Maxwell Relation_ Pyurbeeva and Mol (2021)

$\frac{\varepsilon-\mu}{T}=k_{\rm{B}}\ln\frac{1-n}{n}+\Delta S.$ (9)

In a different approach, which we name the _Integral Maxwell Relation_ the
Maxwell relation (equation 7) can be reformulated in terms of the time-
averaged particle number $\bar{N}$ and the chemical potential of the bath
asSela  _et al._ (2019); Child  _et al._ (2021b, a)

$\Delta
S_{\mu_{1}\rightarrow\mu_{2}}=\int\limits_{\mu_{1}}^{\mu_{2}}\frac{d\bar{N}}{dT}d\mu.$
(10)

The benefit of Equation 10 is that it is completely equivalent to the Maxwell
relation and therefore holds true for all systems. It has to be noted,
however, that in it $\Delta S_{\mu_{1}\rightarrow\mu_{2}}$ is the total
entropy difference, with no distinction between the entropy due to the charge
state uncertainty and the entropy of the pure charge states. Thus, in order to
find $\Delta S$ between the charge states, one has to shift $\mu$ (or,
equivalently $\varepsilon$) from the point of a pure $N$ charge state to that
a pure $N+1$ charge state: in the approximation of only two states being
available, from $-\infty$ to $\infty$, but in reality between the two
consecutive Coulomb-blocked regions.

###  III.4 Quantised-charge entropy measurement methods

Figure 3:  a) Experimental results for the entropy difference between the $N$
and $N+1$ charge states as a function of magnetic field Hartman  _et al._
(2018). b) Experimental results for the entropy difference between the $N+1$
and $N+2$ charge states as a function of magnetic field Hartman  _et al._
(2018). c) The entropy difference between two redox states of a single-radical
molecule found through thermoelectric spectroscopy Pyurbeeva  _et al._ (2021).
The points show the experimental data, while the solid line gives the fit to a
doublet $\leftrightarrow$ singlet + triplet transition model (panel d). The
model parameters, such as the singlet-triplet splitting can be extracted from
the fit. d) The fitting model for the entropy measurements in Fig. 3 c.

Entropy measurement protocols can be divided into two groups – those based on
charge state measurement and those based on charge transport.

Charge-state based measurements The first demonstration of an electronic
entropy measurement methodHartman  _et al._ (2018) was based on a charge-state
technique that employed the thermal shift of the charge degeneracy ($n=1/2$)
point (see Fig 2b) as a measure for entropy difference between the charge
states. A quantum point contact (QPC) was located close to a quantum dot in a
GaAs lithographically-defined device and used as a mean charge sensor (Fig.
1b). The dependence of mean charge of the dot on the transport energy level
was measured at two temperatures, and the entropy difference was found by
fitting the asymmetry of the difference curve (see Fig. 2d).

The energy $\epsilon$ at which the probabilities of both charge states are
equal, denoted $\varepsilon_{1/2}$, is given by Equation 9 with $n=1/2$

$\varepsilon_{1/2}-\mu=T\Delta S$ (11)

which gives the thermal shift

$\frac{\partial\varepsilon_{1/2}}{\partial T}=\Delta S.$ (12)

Building on the first charge-state based entropy measurement method Hartman
_et al._ (2018), the applicability of the entropy measurement method for the
detection of Majorana zero modes has been theoretically shownSela  _et al._
(2019) (see Section IV.1) using the integral Maxwell relation formulation.
Further experimental developments concerned the design of a bespoke device and
an optimised measurement protocol improving thermal equilibration in the
systemChild  _et al._ (2021b), and the measurement of the entropy as a
function of coupling strength (lifetime of a charge state), that could be
controlled by gates forming the quantum dotChild  _et al._ (2021a).

Charge transport based measurements The first charge transport entropy
measurement protocol that was experimentally demonstrated used simultaneous
measurements of conductance and thermocurrent and inferred the entropy from
fitting the thermocurrent to the following equationKleeorin  _et al._ (2019)

$L(\varepsilon,T)=C(T)L_{NI}(\mu+\Delta(T),T)+G(\mu,T)\frac{\Delta(T)}{T},$
(13)

where $L$ is the thermoelectric conductance $L=\partial I/\partial\Delta T$ of
the device, $L_{NI}$ is the thermoelectric conductance of the non-interacting
(non-degenerate) level, which is symmetric in $\varepsilon$, $G$ is the
electrical conductance and $\Delta(T)$ is a function of entropy such that
$\partial\Delta(T)/\partial T=\Delta S/2$.

The first transport-based entropy measurement approach was subsequently
generalised using the observation that Equation 9 can be written to give an
expression of the excess number of electrons as a shifted Fermi distribution,
$n=f(\epsilon-T\Delta S)$Pyurbeeva and Mol (2021), which yields the following
equations for the conductance

$G\propto\frac{1}{T}\left[1-f(\varepsilon)\right]f(\varepsilon-T\Delta S),$
(14)

and thermoelectric conductancePyurbeeva  _et al._ (2021)

$L\propto\frac{\varepsilon}{T^{2}}\left[1-f(\varepsilon)\right]f(\varepsilon-T\Delta
S).$ (15)

From this it follows that the energy for which the conductance is maximum,
$\epsilon_{G}$, is given by

$\varepsilon_{G}-\mu=\frac{T\Delta S}{2},$ (16)

and that therefore the thermal shift of the $G(\epsilon)$ peak position (see
Fig. 2g) isGehring  _et al._ (2021)

$\frac{\partial\varepsilon_{G}}{\partial T}=\frac{\Delta S}{2}.$ (17)

Moreover, the entropy difference between two charge states can be directly
measured from the asymmetry of the thermoelectric conductance $L(\epsilon)$
(see Fig. 2h).

###  III.5 Limitations of entropy measurement methods

Both realisations of the Maxwell relation in microscopic systems (equations 9,
10) follow from the standard thermodynamic Maxwell relation and therefore
apply to a wide range of systems. However, some physical properties
manifesting in nanodevices are usually not considered in standard
“macroscopic” thermodynamics. The two most obvious ones are vibrational
effects and lifetime broadening.

It has been experimentally shown that mechanical degrees of freedom
significantly affect charge transport in weakly coupled molecular devices
Seldenthuis  _et al._ (2008); Thomas  _et al._ (2019). This was described by
introducing a new energy parameter – the reconfiguration energy $\lambda$ –
which quantifies the energy that is lost to the phonon bath due to the
relaxation preformed by the molecule after an additional electron is added
Sowa, Mol, and Gauger (2019). This energy can be associated with an entropy
change $\Delta S_{ph}=\lambda/T_{ph}$ of the phonon bath, where $T_{ph}$ is
its temperature. However, this loss of energy to the bath means that each of
the charge states can no longer be attributed to a particular entropy value.
Therefore, systems with strong coupling between electronic and mechanical
degrees of freedom are not described by the existing theory. At the same time,
they offer a potential avenue of further work into electronic characterisation
of mechanical states in systems with strong electronic-mechanical coupling.

In the rate equation approach lifetime broadening is represented by a
Lorentzian-shaped peak replacing the $\delta$-function for the energy of the
transport level Harzheim  _et al._ (2020). From the quantum-mechanical point
of view, a finite electron lifetime in the charge states means that the number
and energy operators, $\hat{N}$ and $\hat{\varepsilon}$, no longer commute.
This invalidates any equation containing $\varepsilon$ and $n$ simultaneously,
such as equation 9. However the integral form of the Maxwell relation
(equation 10) appears to hold, as it only uses the chemical potential of the
bath, and the dependence of entropy on coupling strength (lifetime) has been
experimentally measured Child  _et al._ (2021a). A fundamental explanation of
the measured entropy remains an interesting theoretical problem.

Charge-transport based entropy measurement methods possess more limitations
than the charge-state ones. Unlike mean charge, any transport property –
conductance, thermocurrent, power factor, etc. – is fundamentally a non-
equilibrium quantity and therefore is not necessarily linked to the state
function entropy. Furthermore, the derivation of all existing charge transport
entropy measurement methods assume small level splitting – a requirement that
all energy states of the system observe the same electron population. This
requirement was well-formulated in the SI to Kleeorin  _et al._ (2019):_“one
of the levels dominates the transport, or the levels are degenerate, or at
high temperatures ( $T\gg\delta\varepsilon$) or at low temperatures
($T\ll\delta\varepsilon$)”_. At the same time, a strength of transport-based
entropy measurement methods is that they can be performed on devices that are
otherwise widely studied, usually for thermoelectric research. In some cases
Kim  _et al._ (2014); Dorsch, Fahlvik, and Burke (2021), they can even be used
as a post-factum analysis of existing experimental data.

##  IV Current applications

###  IV.1 Detection of non-Abelian anyons

One of the motivating factors behind the development of electronic entropy
measurement methods Hartman  _et al._ (2018); Kleeorin  _et al._ (2019) was to
detect non-Abelian anyons, or Majorana modes. Theory predicts that  Majorana
modes have the potential to speed up and increase precision in quantum
computingKitaev (2003); Lian  _et al._ (2018): Majorana qubits support non-
Abelian statistics, braiding and fault tolerant operation.

Several physical systems have been predicted to host Majorana modes, but their
detection and control is very challenging and has not yet been achieved in a
way that could be taken as proof beyond reasonable doubt. Most attempts focus
on electrical fingerprintingSarma, Freedman, and Nayak (2005); Stern and
Halperin (2006); Bonderson, Kitaev, and Shtengel (2006); Feldman  _et al._
(2007), but a key drawback of this approach is that the minute conductance
signatures of Majorana modes could be easily misinterpreted or confused with
those originating from other mechanisms, especially in disordered samplesThorp
(2021); Kayyalha  _et al._ (2020); Zhang  _et al._ (2021); Sills and Thorp
(2021).

However, it has been predicted that nonlocal Majorana zero mode (MZM) states
can be detected through their entropic signatures Cooper and Stern (2009);
Ben-Shach  _et al._ (2013): while a Majorana qubit – a nonlocal two-level
system formed by two MZMs – has a trivial entropy of $k_{\rm{B}}\ln 2$, a
single MZM should possess an universial fractional entropy of
$\frac{1}{2}k_{\rm{B}}\ln 2$ Smirnov (2015) – which is difficult to explain
through different means.

The main challenge is creating a system in which the entropy of a single
quasiparticle can be measured in a charge measurement setupHartman  _et al._
(2018); Sela  _et al._ (2019). The finite coupling between two MZMs forming
the Majorana qubit can be reduced either by increasing their spatial
separation or by controlling the tunneling phases of the MZMsSmirnov (2021a)
via, for example, local gates controlling the tunneling barriers between
topological superconductors and quantum dotsGau  _et al._ (2020a, b).
Alternatively, experiments can be simplified by increasing the temperature,
which diminishes the dependence of entropy on the MZM tunneling phases, and by
involving multiple MZMs. In the latter case the sensitive adjustment of the
tunneling phases becomes obsolete and an entropy of $\frac{n}{2}k_{\rm{B}}\ln
2$, where $n$ is the number of MZMs, is expected.Smirnov (2021b)

A single spin-1/2 impurity that is anti-ferromagnetically coupled to a single
conduction electron channel gives rise to single channel Kondo effect. Such
Kondo effect is universal and its universality has recently been verified by
thermocurrent spectroscopyHsu  _et al._ (2022). At high temperatures the
impurity is free and acts as an unpaired spin with entropy $k_{\rm{B}}\ln 2$,
while at low temperatures it is screened by conduction electrons (a "Kondo
cloud"), forming a many-body-singlet with zero residual entropy. However, in
the case of multiple conduction electron channels the impurity can be
"overscreened" and a finite (non-zero) residual entropy is expected. It has
been predicted that such systems can host Majorana fermions with a residual
impurity entropy of $k_{\rm{B}}\ln\sqrt{2}$ when two channels are present, or,
in the case of more – Fibonacci anyons with $k_{\rm{B}}\ln\phi$, where
$\phi=\frac{1}{2}(1+\sqrt{5})$ is the golden ratio. A promising experimental
realisation for the multi-channel Kondo model are charge Kondo quantum dot
devicesIftikhar  _et al._ (2015, 2018) which hold the potential for detection
of non-Abelian anyons through entropy measurements.Han  _et al._ (2021)

###  IV.2 Twisted bilayer graphene

Recently electronic entropy measurements have been conducted in twisted
bilayer graphene Rozen  _et al._ (2021); Saito  _et al._ (2021). This is a
continuous-charge system, so the extension of theory allowing for charge
quantisation (Section III) was not required, and instead the Maxwell relation
approach outlined in Section II was used.

Typically, thermodynamic states with high entropy are more stable at higher
temperatures than states with low entropy. The two recent entropy measurements
in twisted bilayer grapheneRozen  _et al._ (2021); Saito  _et al._ (2021)
revealed an exotic phenomenon which seemingly violates this intuitive picture.

Both studies measured the temperature-dependent change of the electrochemical
potential $\mu$ of one-quarter-filled twisted bilayer graphene, either
directly Saito  _et al._ (2021), or by integrating the inverse local
electronic compressibility, which was measured by a nanotube-based scanning
single electron transistorRozen  _et al._ (2021) (see Figure 1a). The entropy
of the system could then be determined by using a Maxwell relation (see
Equation 2).

It was found that the entropy per electron of the high-temperature,
electrically insulating phase is greater than that of the low-temperature,
metallic phase by an amount corresponding to a single free spin. The suggested
explanation for this is that the high-temperature insulating phase adopts a
ferromagnetic order, in which the iso-spins (a combination of valley and spin
degree of freedom) are globally aligned in one preferred direction, but the
constraint on their alignment is weak. In the metallic phase, on the other
hand, the constraint on iso-spin alignment is high ensuring a non-magnetic
state with an equal number of iso-spins with opposite orientations.
Consequently the stability of the ferromagnetic insulating state is favoured
at elevated temperatures due to its slightly higher entropy. A similar effect
is known as the Pomeranchuk effect in He3, which solidifies when the
temperature is increased.

###  IV.3 Single electron transistors and single molecule devices

The recent resurgence in interest towards electronic entropy measurement
methods is focused on single electron transistors (SETs) – devices in which
electrical transport is dictated by Coulomb repulsion. SETs have been realised
in a wide range of systems: from electron gasesMeirav, Kastner, and Wind
(1990), semiconductor nanostructuresReed  _et al._ (1988), hetero-
nanowiresThelander  _et al._ (2003) to single molecule junctionsLiang  _et
al._ (2002). The theoretical description of quantised-charge entropy
measurements in Section III is system-independent and only relies on few
charge-states being energetically available.

Entropy measurements based on charge sensing were demonstrated as proof of
concept in Hartman  _et al._ (2018) (see Figure 1b). To this end, the entropy
differences were measured for the $N=0\leftrightarrow N=1$ and
$N=1\leftrightarrow N=2$ charge state transitions. A precise value of
$k_{\rm{B}}\ln 2$ of a single spin 1/2 was found in the first transition
(Figure 3a), while a singlet to triplet transition was observed in the $N=2$
charge states at high magnetic fields (Figure 3b). Such precise entropy
measurements can be used for probing more exotic systems like those with non-
Abelian statistics (see Section IV.1 "Detection of non-Abelian anyons").

Molecular electronics, the concept of using single molecules as building
blocks of nanodevices, gained enormous theoretical and experimental interest
over the last decades. Molecules are fundamentally perfectly reproducible and
their length scales of 1-3 nm offer a platform for downscaling and surpassing
the limitations of conventional silicon-based technologiesGehring, Thijssen,
and van der Zant (2019) by harnessing the achievements of chemical
synthesisPark, Kang, and Yoon (2019); Widawsky  _et al._ (2012); Lambert,
Sadeghi, and Al-Galiby (2016). Thermoelectric research of molecular devices
has received continuously growing attention in recent years for potential
applications in energy conversion and solid-state cooling Reddy  _et al._
(2007); Rincón-García  _et al._ (2016); Cui  _et al._ (2017); Gehring  _et
al._ (2017); Gehring, Thijssen, and van der Zant (2019); Wang, Meyhofer, and
Reddy (2020).

Degeneracy of energy levels in single-molecule devices is of interest, as it
has recently been suggested that molecular systems with highly degenerate
ground states can lead to exceptionally high thermoelectric power
factors.Sowa, Mol, and Gauger (2019) A conductance-based entropy measurement
in a Gd-terpyridine complex (Figure 1c), which followed the method depicted in
Figure 2g, indeed shows an enhacement of the power factor of the
thermoelectric junction by spin entropy,  originating from energy level
degeneracy.Gehring  _et al._ (2021)

Beyond finding strategies to enhance the thermoelectric efficiency,
thermoelectric experiments on single-molecule junctions can furthermore serve
as a novel kind of spectroscopy, which allows to deduce the energy level
structure of a molecule from its thermoelectric properties. As proof of
concept, this was demonstrated on a radical molecule Pyurbeeva  _et al._
(2021) (see Fig. 3c,d) revealing the presence of a triplet state in the
reduced state of the molecule, as well as extracting the energy of the
singlet-triplet transition.

##  V Outlook and Conclusions

This review has examined the latest chapter in the long history of entropy
measurements as a macroscopic probe for microscopic dynamics. Unlike
conventional entropy measurements based on heat, these latest measurements
probe entropy directly from electronic measurements in nanodevices. This idea
was first proposed in the early 2000s and has gained attention in the past few
years.

Currently, the field is in transition from proof-of-concept experiments,
measuring the entropy of a simple degenerate level Hartman  _et al._ (2018);
Kleeorin  _et al._ (2019) and conceptual theoretical work Sela  _et al._
(2019) to practical applications, which involved the demonstration of a
Pomeranchuk-type effect in twisted bilayer graphene Rozen  _et al._ (2021);
Saito  _et al._ (2021) and the measurement of the singlet-triplet transition
energy in a single-molecule device. In the area of detection of non-Abelian
anyons the theoretical background is in place and current work is focused on
improving experimental devices for greater experimental accuracy Child  _et
al._ (2021b).

As electronic entropy measurements complete the transition to yet another tool
in the experimentalist’s toolbox, they will have two distinct applications.
First, entropic singatures can be used for detection, as it is proposed for
non-Abelian anyons. Secondly, the value of entropy can allow to determine
microscopic dynamics of a system with no a-priori knowledge, as it was
demonstrated in twisted bilayer graphene Rozen  _et al._ (2021); Saito  _et
al._ (2021) and in a single-molecule device Pyurbeeva  _et al._ (2021).

In this review we have attempted to develop a taxonomy for existing electronic
measurement techniques, and to reveal their underlying thermodynamic
principles, as well as their limitations.

The consideration of limitations itself opens up new questions and areas of
research. Examples of these include studying mechanical degrees of freedom in
systems with strong electron-vibrational coupling and devices in the rarely-
considered intermediate case between the Landauer regime and weakly-coupled
sequential tunnelling, in which lifetime in the device is small and has a
significant contribution to entropy Child  _et al._ (2021a).

Finally, the ability to measure thermodynamic parameters, primarily entropy,
in nanodevices opens them up as an experimental platform for studying
stochastic, and especially quantum thermodynamics. This is in contrast with
the usual bottom-up experimental approach to the field Micadei  _et al._
(2019), but has the benefit of years of technological advancement of
nanodevices for practical applications, as well as ease of experimentation.
This approach would require additional theoretical work, as the term “open
quantum systems” usually implies systems open to energy, but not particle
exchange. However the experimental accessibility of electrical nanodevices
which include both justifies this effort.

In summary, to date, the field has focused on developing novel measurement
techniquesand conceptual demonstration of entropy measurements in a range of
systems. Moving forwards, the next stage will be to apply these techniques in
a systematic way to uncover microscopic dynamics of quantum correlated
systems, potentially discover new and exotic quasi-particles in condensed
matter systems, as well as expand the theoretical background for more
fundamental applications.

## Acknowledgements

P.G. acknowledges financial support from the F.R.S.-FNRS of Belgium (FNRS-
CQ-1.C044.21-SMARD, FNRS-CDR-J.0068.21-SMARD, FNRS-MIS-F.4523.22-TopoBrain),
from the Federation Wallonie-Bruxelles through the ARC Grant No. 21/26-116 and
from the EU (ERC-StG-10104144-MOUNTAIN). This project (40007563-CONNECT) has
received funding from the FWO and F.R.S.-FNRS under the Excellence of Science
(EOS) 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.

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