# Gate-tunable direct and inverse electrocaloric effect in trilayer graphene

Natalia Cortés<EMAIL_ADDRESS>Departamento de Física, Universidad
Técnica Federico Santa María, Casilla 110V, Valparaíso, Chile  Oscar Negrete
Departamento de Física, Universidad Técnica Federico Santa María, Casilla
110V, Valparaíso, Chile  Centro para el Desarrollo de la Nanociencia y la
Nanotecnología, 8320000 Santiago, Chile  Francisco J. Peña  Departamento de
Física, Universidad Técnica Federico Santa María, Casilla 110V, Valparaíso,
Chile  Patricio Vargas  Departamento de Física, Universidad Técnica Federico
Santa María, Casilla 110V, Valparaíso, Chile  Centro para el Desarrollo de la
Nanociencia y la Nanotecnología, 8320000 Santiago, Chile
<EMAIL_ADDRESS>
###### Abstract

The electrocaloric (EC) effect is the reversible change in temperature and/or
entropy of a material when it is subjected to an adiabatic electric field
change. Our tight-binding calculations linked to Fermi statistics, show that
the EC effect is sensitive to the stacking arrangement in trilayer graphene
(TLG) structures connected to a heat source, and is produced by changes of the
electronic density of states (DOS) near the Fermi level when external gate
fields are applied on the outer graphene layers. We demonstrate the AAA-
stacked TLG presents an inverse EC response (cooling), whereas the EC effect
in ABC-stacked TLG remains direct (heating) regardless of the applied gate
field potential strength. We reveal otherwise the TLG with Bernal-ABA stacking
geometry generates both the inverse and direct EC response in the same sample,
associated with a gate-dependent electronic entropy transition at finite
temperature. By varying the chemical potential to different Fermi levels, we
find maxima and minima of the DOS are located near the extremes of the
electronic entropy, which are correlated with sign changes in the differential
entropy per particle, giving a particular experimentally measurable electronic
entropy spectrum for each TLG geometry. The EC effect in quantum two-
dimensional layered systems may bring a wide variety of prototype van der
Waals materials that could be used as versatile platforms to controlling the
temperature in nanoscale electronic devices required in modern portable on-
chip technologies.

Today

##  I Introduction

The rapid increase in the need for efficient, environmentally friendly and
low-cost materials with the capability of cooling, has opened interesting
possibilities to explore material properties beyond the traditional vapor-
compression method applied in household and industrial refrigeration Scott
(2011); Shi  _et al._ (2019). As a clean and efficient alternative to tuning
the systems temperature, can be the implementation of caloric effects on
solid-state materials, driven by applied pressure, electrical, magnetic and/or
mechanical fields, conducting to the barocaloric, electrocaloric (EC),
magnetocaloric and elastocaloric effect, respectively Moya  _et al._ (2014);
Shi  _et al._ (2019); Li  _et al._ (2019). Thermal response on solid systems
is produced by either of those external field acting on and/or removed from
the sample under adiabatic conditions Valant (2012), leading to large
temperature changes $\Delta T$ associated with isothermal entropy changes
$\Delta S_{T}$, generally near phase transitions Moya  _et al._ (2014);
Otoničar and Dkhil (2020).

In the last decades, the EC effect—the change in temperature and entropy as an
electric field is applied—has been widely studied in three-dimensional
dielectric multilayer capacitors (MLC), by considering a different number of
layers with diverse thicknesses in the samples, typically of the order of
micrometers Kar-Narayan and Mathur (2010); Moya  _et al._ (2018), and finding
thickness-dependent thermal responses Moya  _et al._ (2018). In single-crystal
MLC, the temperature change reaches $\sim 0.9$ K near ferroelectric phase
transitions when moderate electric fields are applied Moya  _et al._ (2013).
In high-quality samples of MLC, the temperature change increases up to 5.5 K
Nair  _et al._ (2019); Otoničar and Dkhil (2020). As films of MLC scales down
to hundreds of nanometers, a giant EC response of $\sim 12$ K is manifested
close phase transitions Mischenko  _et al._ (2006). Theory predicts larger
cooling power in thin-film MLC of ceramics and polymers by varying the number
of layers of the systems Kar-Narayan and Mathur (2009). The feasible
scalability of MLC systems to reduced dimensionality may allow expanding the
EC effect to novel materials for electrical refrigeration at the atomistic
level Lisenkov and Ponomareva (2009); Ponomareva and Lisenkov (2012).

Nanoscale systems including van der Waals multilayers and heterostructures,
comprise a wide variety of quantum solids Novoselov  _et al._ (2016) that
could be used as prototype miniaturized materials to get caloric effects. A
giant EC response of $\sim 23$ K has experimentally been achieved in a
ferroelectric heterobilayer because of interface-induced interactions Shirsath
_et al._ (2019). Theoretical studies on graphene nanoribbons under an electric
and magnetic field show controllable entropy changes magnitude and entropy
sign changes due to the applied magnetic field Reis and Soriano (2013). First-
principle and thermodynamic calculations in a two-dimensional (2D) monolayer
of MoTe2, have reported a temperature change of $10-15$ K near the structural
phase transition that occurs when the monolayer is subject to electrostatic
gating Rehn  _et al._ (2018).

We explore here the thermal response of atomically thin layered
nanostructures, focusing on gated trilayer graphene (TLG). Both experimental
and theoretical studies have revealed the electronic band structure and
density of states (DOS) of TLG samples strongly depend on the stacking pattern
they possess, showing significant changes under applied electrical potentials
Aoki and Amawashi (2007); Avetisyan  _et al._ (2009); Koshino and McCann
(2009a); Avetisyan  _et al._ (2010); Craciun  _et al._ (2009); Lui  _et al._
(2011); Yankowitz  _et al._ (2013); Zou  _et al._ (2013); Wang  _et al._
(2016). This makes TLG materials suitable to inspect the EC effect and novel
thermodynamic quantities such as the differential entropy per particle (DEP)
Kuntsevich  _et al._ (2015); Tsaran  _et al._ (2017); Grassano  _et al._
(2018); Shubnyi  _et al._ (2018); Galperin  _et al._ (2018); Sukhenko  _et
al._ (2020). Through Bloch electrons in parametrized tight-binding models
linked to Fermi statistics, we show how the thermal response varies within
each TLG stacking as a result of the quantum-thermodynamic processes involved
at low-energies. As each system is gated, we find entropy changes for AAA-
stacked TLG linearly increase with temperature, showing an inverse EC effect
(cooling) up to room temperature. In contrast, the entropy changes for ABC TLG
non-linearly increase as temperature increases, giving a direct EC effect
(heating). Remarkably, the ABA-TLG arrangement presents a dual EC response as
cools and heats within the same sample, displaying gate-dependent maxima for
entropy changes. We also analyze the number of electrons thermally excited of
ABA-TLG at low and high temperatures, unveiling its direct connection to the
gate-tunable electronic entropy and dual EC effect.

##  II Quantum-Thermodynamic Model

We simulate TLG structures with simple hexagonal AAA, Bernal ABA, and
rhombohedral ABC stacking order by using a $\pi$-orbital tight-binding (TB)
model, that captures the main low-energy symmetry properties around the Fermi
level for each system Partoens and Peeters (2006, 2007); Koshino and McCann
(2009b); Avetisyan  _et al._ (2009); Koshino and McCann (2009a); Avetisyan
_et al._ (2010); Yuan  _et al._ (2010); Mohammadi  _et al._ (2014). To
construct the TB Hamiltonians, we consider the unit cells of the TLG with
three pairs of atoms {$A1$-$B1$}, {$A2$-$B2$} and {$A3$-$B3$}, which are
located respectively in the bottom, central and top graphene layers, as shown
in Fig. 1. We use different parametrization (hoppings) depending on the
stacking type of the TLG. For each system we have first taken a minimal set of
parameters from fitting photoemission spectroscopy spectra to TB models Bao
_et al._ (2017). These parameters include a strong intralayer hopping
$\gamma_{0}$ between $Ai$ and $Bi$ nearest neighbors within each graphene
layer ($i=1,2,3$ is the layer index), and one interlayer hopping connecting
direct vertical nearest-layer carbon sites with strong coupling $\gamma_{1}$,
as used for AAA-stacked TLG in Fig. 1(a). For ABC and ABA-stacked TLG, this
parametrization also incorporates a weaker nearest-layer coupling
$\gamma_{3}$, Fig. 1(b) and 1(c) respectively. In particular, for ABA-TLG, we
also have taken a full set of parameters adopting the Slonczewski-Weiss-
McClure parametrization Dresselhaus and Dresselhaus (2002), which contains one
extra nearest-interlayer hopping $\gamma_{4}$ as well as two next-nearest
interlayer hoppings $\gamma_{2}$ and $\gamma_{5}$. An asymmetric onsite-
potential energy difference with magnitude $2V_{g}$, induced by a bias voltage
between the external graphene layers, can act as an external gate potential on
each carbon atom of top $+V_{g}$ and bottom $-V_{g}$ graphene layers of the
TLG structures, similar to bilayer graphene under an external voltage Hao and
Lee (2010). This asymmetric potential is a crucial quantity to tune the
electronic DOS and to produce a caloric response near the Fermi level of TLG
systems.

The Hamiltonians for the TLG samples
$\mathcal{\hat{H}}^{\alpha}(\boldsymbol{k})=\hat{H}^{\alpha}_{0}(\boldsymbol{k})+\hat{V}$
($\alpha=\mathrm{AAA,ABA,ABC}$) include a diagonal matrix $\hat{V}$ of the
gate potential proportional to $V_{g}$. The matrices of hopping
$\hat{H}^{\alpha}_{0}(\boldsymbol{k})$ can be obtained with the matrix
elements
$H_{jj^{\prime}}^{\alpha}(\boldsymbol{k})=\sum_{\boldsymbol{R}}e^{i\boldsymbol{k}\cdot\boldsymbol{R}}E_{jj^{\prime}}^{\alpha}(\boldsymbol{R})$,
where
$E_{jj^{\prime}}^{\alpha}(\boldsymbol{R})=\left\langle\phi_{j}(\boldsymbol{r})\right|\hat{H}_{0}^{\alpha}\left|\phi_{j^{\prime}}(\boldsymbol{r-R})\right\rangle$
is the hopping integral between the atomic orbitals
$\left|\phi_{j}\right\rangle$ at 0 and $\left|\phi_{j^{\prime}}\right\rangle$
at lattice vector $\boldsymbol{R}$ with $j(j^{\prime})=Ai,Bi$.
$\boldsymbol{R}_{1}=a_{\text{C-C}}(1,0)$,
$\boldsymbol{R}_{2(3)}=a_{\text{C-C}}\big{(}-1/2,+(-)\sqrt{3}/2\big{)}$ are
in-plane nearest-neighbor vectors with $a_{\text{C-C}}=1.42$ Å the carbon-
carbon distance within a graphene layer, and
$\boldsymbol{k}=(k_{x},k_{y},k_{z}$) is the momentum. The TLG Hamiltonians are
represented in the basis with components
$\\{\psi_{A1},\psi_{B1},\psi_{A2},\psi_{B2},\psi_{A3},\psi_{B3}\\}$.

Figure 1: Schematic representation of trilayers graphene with diverse stacking
patterns, hexagonal AAA (a), rhombohedral ABC (b) and Bernal ABA (c). Blue
(red) sphere show $A$ ($B$) carbon sublattice connected through intralayer
$\gamma_{0}$ and interlayer hoppings $\gamma_{1}$, $\gamma_{2}$, $\gamma_{3}$,
$\gamma_{4}$ and $\gamma_{5}$ as indicated; $\pm V_{gate}$ is the gate
potential applied to the outer graphene layers on each system, the orange
square enclosing each structure represent the thermal source at temperature
$T$. (d) Shows the Brillouin zone in momentum space for all trilayers
graphene, where green lines highlight the triangular $\bm{k}$-path with
corners located at high symmetry points $\Gamma=(0,0)$,
$K=\frac{2\pi}{a}\big{(}0,\frac{2}{3}\big{)}$ and
$K^{\prime}=\frac{2\pi}{a}\big{(}\frac{\sqrt{3}}{3},\frac{1}{3}\big{)}$, whit
$a=\sqrt{3}a_{\text{C-C}}=2.46$ Å the 2D graphene lattice constant.

The density of states $D(E,V_{g})$ is a fundamental gate-tunable electronic
quantity, which allows us to calculate the electronic thermodynamics
properties of the TLG systems. The DOS depends of both, the electronic level
with energy $E$ and the gate potential $V_{g}$ applied on the outer carbon
atoms of each TLG. As the TLG structures can be considered as quasi 2D
systems, we numerically calculate the DOS using a 2D Brillouin zone (BZ) in
reciprocal $\boldsymbol{k}$-space (i. e., $k_{z}=0$). We use a fine mesh of
about ten million of $\boldsymbol{k}$ points in the area enclosed by the green
triangle of Fig. 1(d), and for every $\boldsymbol{k}$-state we evaluate the
energy levels coming from each band of
$\mathcal{\hat{H}}^{\alpha}(\boldsymbol{k})$. By calculating the DOS, we can
obtain the number of electrons $N$ in each unit cell of the trilayers

$N(V_{g},T,\mu)=\int_{E_{l}}^{E_{h}}D(E,V_{g})n_{\text{F}}(E,T,\mu)dE,$ (1)

where $E_{l(h)}$ is the lowest (highest) electronic energy eigenvalue of the
considered Hamiltonian, $n{{}_{\text{F}}}(E,T,\mu)=1/[e^{\beta(E-\mu)}+1]$ is
the Fermi-Dirac function distribution with $\beta=1/k_{\text{B}}T$,
$k_{\text{B}}$ is the Boltzmann constant, $\mu$ is the chemical potential and
$T$ temperature. At $T=0$ K, the electronic levels are occupied up to the
Fermi energy $E_{\text{F}}$, and $n{{}_{\text{F}}}$ it converts in the
Heaviside function, so that we obtain
$N=\int_{E_{l}}^{E_{\text{F}}}D(E,V_{g})dE=6$ for the unit cell of each TLG.
For finite temperatures $T>0$, we can calculate the chemical potential
$\mu(V_{g},T)$ by inversion of Eq. 1.

The total entropy of the system $S=S_{\text{latt}}(T)+S_{e}(V_{g},T)$ includes
two terms, the entropy of the lattice $S_{\text{latt}}(T)$, giving account of
the phonon contribution, in which we assume it is only dependent on
temperature and not on $V_{g}$ Kutnjak  _et al._ (1999). In our approximation
we neglect $S_{\text{latt}}(T)$ and use $S\sim S_{e}(V_{g},T)$, where the
electronic entropy

$S_{e}(V_{g},T)=-k_{\text{B}}\int_{E_{l}}^{E_{h}}D(E,V_{g})\mathcal{F}(n_{\text{F}})dE,$
(2)

depends on the gate potential and temperature, and is calculated in the
triangular area of the BZ of Fig. 1(d), where the number of electrons is fixed
to $N=6$. In Eq. 2,

$\mathcal{F}(n_{\text{F}})=n_{\text{F}}\ln
n_{\text{F}}+(1-n_{\text{F}})\ln(1-n_{\text{F}}),$ (3)

its approximated by a Lorentzian-like function
$L(E,T,\mu)=C/[e^{(|E-\mu|/2k_{\text{B}}T)^{3/2}}+1]$. By considering $C=1.4$,
low and high $T$ values, we obtain excellent agreement between Eq. 3 and
$L(E,T,\mu)$ with $-\mathcal{F}(n_{\text{F}})\approx L(E,T,\mu)$, so that Eq.
2 transforms as

$S_{e}(V_{g},T)\simeq k_{\text{B}}\int_{E_{l}}^{E_{h}}D(E,V_{g})L(E,T,\mu)dE.$
(4)

The main contribution of $L(E,T,\mu)$ to the electronic entropy is given by
their temperature-dependent width, which increases as temperature increases.
Within the range of temperature we work, $k_{\text{B}}T\ll\gamma_{1}$, the
chemical potential remains constant at $\mu=0$ eV in order to fulfill $N=6$
for the unit cell of each TLG.

The EC effect is in general quantified by either the temperature change that
occurs when the electric field changes adiabatically, or the entropy change
induced by isothermal application or removal of the electric field. Typically
there are two ways to measure the caloric effects in solid materials: (i)
through the variation of temperature in an adiabatic thermodynamic path
$\Delta T$ (direct measurement), and (ii) by means of the entropy change in an
isothermal path $\Delta S_{T}$ (indirect measurement), where the subindex $T$
indicates constant temperature. It is important to note that is experimentally
challenging to obtain $\Delta T$ as compared to $\Delta S_{T}$, as direct
measurements generally require precision calorimetry Scott (2011).

The EC effect in our TLG structures is then obtained through the indirect way,
by means of electronic entropy changes calculations as a function of $T$
through Eq. 4 at $\mu=E_{\text{F}}=0$,

$-\Delta S_{e,T}(T)=S_{e}(V_{g}=0,T)-S_{e}(V_{g},T).$ (5)

In case to obtain $-\Delta S_{e,T}>0$, we are in presence of the direct EC
effect, that is the TLG system is capable to heat as $V_{g}$ increases. In the
opposite case when $-\Delta S_{e,T}<0$, the system present an inverse EC
effect and so the sample cools down.

###  II.1 Differential electronic entropy per particle

To get essential insights into the microscopic origin of the EC effect in TLG
structures, we have calculated a fundamental thermodynamic quantity, the
differential electronic entropy per particle $s$ (DEP). The DEP is an
experimentally measurable quantity in gated 2D electron systems, allowing
superior sensitivity in comparison to a.c. calorimetry Kuntsevich  _et al._
(2015), and can be directly linked to the thermoelectric power or
alternatively to the Seebeck coefficient Goupil  _et al._ (2011). The DEP,
$s=\partial S_{e}/\partial N$ is found from the Maxwell relation as Kuntsevich
_et al._ (2015); Tsaran  _et al._ (2017); Grassano  _et al._ (2018); Shubnyi
_et al._ (2018); Galperin  _et al._ (2018); Sukhenko  _et al._ (2020)

$s(\mu,V_{g},T)=\left(\frac{\partial S_{e}}{\partial
N}\right)_{T}=-\left(\frac{\partial\mu}{\partial T}\right)_{N},$ (6)

and its related to the DOS through $N$ in Eq. 1 by means
$s(\mu,V_{g},T)=(\partial N/\partial T)_{\mu}(\partial
N/\partial\mu)^{-1}_{T}$.

To interpret our results, we make a useful transformation of the form
$s(\mu,V_{g},T)=(\partial S_{e}/\partial N)_{T}=(\partial
S_{e}/\partial\mu)_{T}(\partial\mu/\partial N)_{T}$. The function that
predominantly governs the DEP corresponds to the entropy’s derivative with
respect to $\mu$. Consequently, we expect that at maxima and minima of the
electronic entropy as a function of $\mu$, the DEP vanishes, giving account of
the thermodynamic fingerprints for each TLG arrangement.

As the number of particles $N$ can vary in a doped/gated system, we obtain a
caloric effect directly connected to the DEP, which could be used as a
feasible procedure to determine the entropy changes given by fluctuating
charge carriers, see Appendix for more details.

##  III The electrocaloric response in trilayers graphene

###  III.1 AAA-stacked TLG

AAA-stacked TLG has been recently exfoliated Bao  _et al._ (2017), preserves
metallic character in the presence of external electric fields Wu (2011) as in
the monolayer graphene case, and possesses mirror reflection symmetry with
respect to the central layer. In this structure the {$Ai$,$Bi$} sublattices
match to the nearest-neighbor layer carbon sites {$Ai+1$,$Bi+1$} with vertical
hopping $\gamma_{1}$. The whole system effectively can be seen as a direct
superposition of three graphene monolayers as shown in Fig. 1(a). Because of
the crystal symmetry, the AAA-TLG Hamiltonian

$\mathcal{\hat{H}}^{\mathrm{AAA}}(\boldsymbol{k})=\begin{pmatrix}-V_{g}&\gamma_{0}f_{0}&\gamma{{}_{1}}f_{1}&0&0&0\\\
\gamma_{0}f_{0}^{*}&-V_{g}&0&\gamma{{}_{1}}f_{1}&0&0\\\
\gamma{{}_{1}}f_{1}^{*}&0&0&\gamma_{0}f_{0}&\gamma{{}_{1}}f_{1}&0\\\
0&\gamma{{}_{1}}f_{1}^{*}&\gamma_{0}f_{0}^{*}&0&0&\gamma{{}_{1}}f_{1}\\\
0&0&\gamma{{}_{1}}f_{1}^{*}&0&V_{g}&\gamma_{0}f_{0}\\\
0&0&0&\gamma{{}_{1}}f_{1}^{*}&\gamma_{0}f_{0}^{*}&V_{g}\end{pmatrix},$ (7)

allows analytical eigenvalues connecting monolayer-graphene modes
$\varepsilon_{1,2}$ and AA-bilayer graphenelike modes $\varepsilon_{3,4,5,6}$

$\displaystyle\varepsilon_{1,2}$
$\displaystyle=\pm\gamma_{0}\left|f_{0}\right|\text{,}$ (8a)
$\displaystyle\varepsilon_{3,4}$
$\displaystyle=\pm\gamma_{0}\left|f_{0}\right|-\sqrt{V_{g}{{}^{2}}+2\gamma_{1}^{2}\left|f_{1}\right|^{2}}\text{,}$
(8b) $\displaystyle\varepsilon_{5,6}$
$\displaystyle=\pm\gamma_{0}\left|f_{0}\right|+\sqrt{V_{g}{{}^{2}}+2\gamma_{1}^{2}\left|f_{1}\right|^{2}}\text{.}$
(8c)

$f_{0}(k_{x},k_{y})=e^{-ik_{x}a_{\text{C-C}}}+2\cos({\frac{\sqrt{3}}{2}k_{y}a_{\text{C-C}}})e^{ik_{x}a_{\text{C-C}}}$
and $f_{1}(k_{z})=e^{ik_{z}c}$ are respectively an in-plane and out-of-plane
momentum-dependent functions with $c=3.3$ Å the interlayer distance Aoki and
Amawashi (2007).

Figure 2: Low-energy electronic spectra for AAA-stacked trilayer graphene. (a)
Band structure in the vicinity of the $K$ point. (b) Density of states
obtained for the triangular area enclosed by green lines in Fig. 1(d).
Vertical dashed line in (a) highlight the $K$ point, horizontal dashed lines
in (a)-(b) are fixed at $E_{\text{F}}=0$ eV. The hopping parameters are
$\gamma_{0}=3.2$ eV and $\gamma_{1}=0.18$ eV Bao  _et al._ (2017).

Figure 2(a) shows the low-energy electronic structure given by Eqs. 8 near the
$K$ point for different values of the gate potential $0\leq V_{g}\sim
2\gamma_{1}$. Due to the structural symmetry, the Dirac points
$K$,$K^{\prime}$ are degenerate, where around each one of them there are three
pairs of linear branches creating a diamond structure with Dirac cones
throughout energy-momentum space Wu (2011). At the Fermi level
$E_{\text{F}}=0$ and momentum $K$, or charge neutrality point (CNP), one of
the Dirac cones remains constant regardless of the $V_{g}$ strength,
resembling the monolayer graphene modes [Eq. 8(a)]. When $V_{g}=0$ (thin dark
blue solid lines) and because of interlayer interaction $\gamma_{1}$, a pair
of Dirac cones shift away from the CNP in a symmetric way by
$E=\pm\sqrt{2}\gamma_{1}\simeq 0.25$ eV. As $V_{g}$ is turned on, a pair of
cones disperse to $E=\pm\sqrt{V_{g}{{}^{2}}+2\gamma_{1}^{2}}$, Eq. 8(b) and
8(c). The effect of the gate potential therefore can be represented as a
renormalization of the interlayer hopping energy Redouani  _et al._ (2018),
preserving metallic character for the bands as in the unperturbed case
($V_{g}=0$). At low-energies, the DOS preserves electron-hole symmetry
[$D(E,V_{g})=D(-E,V_{g})$] in Fig. 2(b), and shows a global minimum at the
CNP, which causes the major contribution to the EC response at low and high
temperatures. At the energies given by the shifted Dirac cones from the Fermi
level, the DOS shows two symmetric discontinuities, not providing states to
the EC effect as the $L$ function in Eq. 4 does not capture those states up to
room temperature.

The EC effect (or entropy changes) for AAA-stacked TLG is presented in Fig.
3(a). The $-\Delta S_{e,T}$ response is linear as a function of $T$, following
the linear behavior of the electronic spectra near $E_{\text{F}}$ of Fig. 2.
The EC effect is inverse for all $V_{g}$ and the entire temperature range from
0 to 300 K, and reaches a high value $\simeq-0.23$ $\mu$eV/K at room
temperature for large gate voltage ($V_{g}=0.4$ eV). This result can be
explained directly from the plots of the electronic entropy as a function of
the chemical potential presented in Fig. 4(a) at $T=30$ K and Fig. 4(b)
$T=300$ K. The electronic entropy at $T=30$ K almost preserves the DOS shape
of Fig. 2(b) as the $L$ function slowly smears the DOS at low temperature,
while at $T=300$ K the entropies are largely smoothed with one magnitude order
higher than $T=30$ K because of higher $T$. The electronic entropy at $\mu=0$
and $V_{g}=0$ for both temperatures $T=30$ K and $T=300$ K is the smallest
entropy with respect to the other entropies with larger $V_{g}$. This means
$S_{e}(V_{g}=0,T)<S_{e}(V_{g}\neq 0,T)$, hence we always obtain $-\Delta
S_{e,T}<0$ from Eq. 5, then the AAA-stacked TLG cools down for all $V_{g}$ we
have considered here.

The DEP as a function of $\mu$ is shown in Fig. 4(c) and 4(d) for the same
temperatures as for the electronic entropy calculations. The DEP for AAA TLG
its an odd function of $\mu$, that is $s(-\mu)=-s(\mu)$, and at $\mu=0$
vanishes for all gate fields. At low temperature in Fig. 4(c), the DEP shows a
stairslike shape, mainly determined by the slope $\left(\partial
S_{e}/\partial\mu\right)_{T}$, with maxima (minima) localized near the
discontinuities of the electron (hole) regime of the electronic entropy. The
states of the maxima and minima of the DEP are around the shifted Dirac cones
of the electronic dispersion of Fig. 2(a), which do not contribute to the EC
effect as we mention above. At room temperature in Fig. 4(d), the DEP is
smoothed and one magnitude order higher than $T=30$ K, as also seen for the
electronic entropy, demonstrating a general linear behavior for the electronic
and thermodynamics quantities in AAA-stacked TLG.

Figure 3: The isothermal entropy change $-\Delta S_{e,T}$ as a function of $T$
for different gate potentials as indicated. (a) AAA-stacked and (b) ABC-
stacked trilayer graphene. The chemical potential is set to $\mu=0$. Dashed
vertical lines indicate $T=30$ K and $T=300$ K, where we calculate the
electronic entropy and DEP in Fig. 4 and Fig. 6. Figure 4: Electronic
entropies (a)-(b) and differential entropy per particle (c)-(d) as a function
of the chemical potential for AAA-stacked trilayer graphene. Left panels stand
for $T=30$ K, right panels $T=300$ K. Vertical dashed lines indicate $\mu=0$.

###  III.2 ABC-stacked TLG

Rhombohedral ABC-stacked trilayer graphene is one of the commonly stable
crystal obtained in experimental procedures Bao  _et al._ (2011); Lui  _et
al._ (2011); Bao  _et al._ (2017), which can be considered as a zero-gap
semiconductor material in the unperturbed case, and as a semiconductor when
external electric-field potentials are applied Aoki and Amawashi (2007). This
stacking geometry possesses inversion symmetry, but lacks mirror symmetry Lui
_et al._ (2011) as is schematized in Fig. 1(b). The TB Hamiltonian considering
the hopping interactions in Fig. 1(b) is given by

$\mathcal{\hat{H}}^{\mathrm{ABC}}(\boldsymbol{k})=\begin{pmatrix}-V_{g}&\gamma_{0}f_{0}&0&\gamma_{3}f_{3}&0&0\\\
\gamma_{0}f_{0}^{*}&-V_{g}&\gamma{{}_{1}}f_{1}&0&0&0\\\
0&\gamma{{}_{1}}f_{1}^{*}&0&\gamma_{0}f_{0}&0&\gamma{{}_{3}}f_{3}\\\
\gamma{{}_{3}}f_{3}^{*}&0&\gamma_{0}f_{0}^{*}&0&\gamma{{}_{1}}f_{1}&0\\\
0&0&0&\gamma_{1}f_{1}^{*}&V_{g}&\gamma_{0}f_{0}\\\
0&0&\gamma{{}_{3}}f_{3}^{*}&0&\gamma_{0}f_{0}^{*}&V_{g}\end{pmatrix},$ (9)

where $f_{3}(\boldsymbol{k})=f_{0}(k_{x},k_{y})e^{ick_{z}}$. The band
structure and DOS from Eq. 9 considering $0\leq V_{g}\leq\gamma_{1}$ are
plotted in Fig. 5(a) and Fig. 5(b) respectively. The electronic spectra
preserve electron-hole symmetry around $E_{\text{F}}=0$ for all $V_{g}$. In
the electronic structure there are valence and conduction bands dispersing
according to the hoppings between the different atoms. For $V_{g}=0$ (thinnest
gray lines), the lower conduction band and higher valence band touch at
$E_{\text{F}}$ near the $K$ point, where the DOS shows a local maximum.

Figure 5: (a) Band structure near the $K$ point, (b) DOS from the green
triangle area in Fig. 1(d) for ABC-stacked trilayer graphene with diverse gate
potential $V_{g}$ as indicated. Horizontal (vertical) dashed lines indicate
$E_{\text{F}}=0$ eV ($K$ point). The hopping parameters are $\gamma_{0}=3.10$
eV, $\gamma_{1}=0.4$ eV and $\gamma_{3}=0.2$ eV Bao  _et al._ (2017).

Furthermore when $V_{g}=0$, two other pairs of bands shift away from
$E_{\text{F}}$ and are degenerate, crossing near $E\simeq\pm 0.4$ eV because
of the strongly interlayer coupling $\gamma_{1}$ between the direct bonded
$B1$-$A2$ and $B2$-$A3$ dimers. The DOS increases at these crossing-band
energies and presents a pair of local sharp peaks, not providing states to the
EC effect for $T>900$ K. When $V_{g}$ is turned on, an energy gap between the
lower electron (conduction band) and higher hole branch (valence band) opens
at the $K$ point as well as in their vicinity because of the potential energy
difference between the bottom and top graphene layersGuinea  _et al._ (2006);
Latil and Henrard (2006); Aoki and Amawashi (2007); Koshino and McCann
(2009b); Avetisyan  _et al._ (2010); Wu (2011). The gaps non-linearly increase
as $V_{g}$ increases Wu (2011); Wang  _et al._ (2016), the DOS vanishes for
the gap energies as expected, and shows two high symmetric peaks (van Hove
singularities) near the valence and conduction band edges, whose states are
mainly contributed by the carbon atoms of the top and bottom graphene layers
Wang  _et al._ (2016). The bandgap edges for low-gate potential $V_{g}=0.05$
eV ($V_{g}=0.1$ eV) provide states to the EC effect for temperatures starting
from $\sim 80$ K ($140$ K), while the states contribute for $T>240$ K when
high-gate fields $V_{g}=0.2,0.4$ eV are applied.

Figure 3(b) shows the EC effect for ABC-TLG structure. Here we observe a non-
linear direct response (heating) for the whole range of temperatures and gate
potentials. This occurs as the reference electronic entropy [$S_{e}(V_{g}=0)$]
is higher than the other gate-dependent entropies, because the states of the
DOS evaluated at $D(E_{\text{F}}=0,V_{g}=0)$, are fully capturated by the $L$
function at low and high temperatures. We also have checked that there is not
a direct-inverse transition [$-\Delta S_{e}(V_{g},T)=0$] up to $T=500$ K.
Moreover, an interesting effect occurs for temperatures below 80 K for all
$V_{g}$, where the entropy changes are constant and do not present differences
in the caloric response because of the presence of the band gap. As the
temperature increases in Fig. 3(b), the maxima of the entropy changes
correspond to $-\Delta S_{e}(V_{g}=0.05\ \text{eV},T=137\ \text{K})\simeq
0.14$ $\mu$eV/K and $-\Delta S_{e}(V_{g}=0.1\ \text{eV},T=236\ \text{K})\simeq
0.20$ $\mu$eV/K. For high $V_{g}$ and $T>0$, $-\Delta S_{e}$ equally increases
for $V_{g}=\gamma_{1}/2=0.2$ eV and $V_{g}=\gamma_{1}=0.4$ eV, reaching a
caloric response of $\simeq 0.28$ $\mu$eV/K at room temperature.

These behaviors are consistent with the electronic entropy as a function of
$\mu$ in Fig. 6(a) at $T=30$ K and Fig. 6(b) $T=300$ K. For both temperatures
at $\mu=0$, the electronic entropy fulfills the condition
$S_{e}(V_{g}=0,T)>S_{e}(V_{g}\neq 0,T)$, giving the direct EC response, then
the ABC-TLG sample heats, opposite to the AAA-TLG case. At low temperature in
the vecinity of $\mu=0$ in Fig. 6(a), the only entropy contributing to the EC
effect is that for $V_{g}=0$, while the entropies when $V_{g}\neq 0$ vanish
and do not provide states to the EC effect as expected. For temperatures lower
than $80$ K, we have verified that $-\Delta S_{e}(V_{g}\neq 0)\simeq
S_{e}(V_{g}=0)$ because of the band gap. At room temperature in Fig. 6(b), the
electronic entropies are nearly five times larger than entropies at $T=30$ K,
showing non null values for all $V_{g}$ in the gap region. The entropy is
almost flat for $V_{g}=0$, while the other entropies have a parabolic shape as
$V_{g}$ increases, with almost the same values for high gate-potential fields
$V_{g}=0.2,0.4$ eV near $\mu=0$, as expected from the EC response in Fig.
3(b).

Figure 6: Electronic entropies (a)-(b) and differential entropy per particle
(c)-(d) as a function of $\mu$ for ABC-stacked trilayer graphene. Left panels
stand for $T=30$ K, right panels $T=300$ K. Vertical dashed lines indicate
$\mu=0$, and vertical dash-dotted lines in (a)-(b) at $\mu=\pm 0.127$
($\mu=\pm 0.18$) eV highlight the maxima at $T=30$ ($T=300$) K for
$S_{e,T}(V_{g}=0.4)$, which correlate with sign changes of the DEP in (c)-(d).

The DEP of ABC-stacked TLG also is an odd function of $\mu$ at low and high
temperatures, as shown in Fig. 6(c) and Fig. 6(d) respectively. At $T=30$ K
the peak value of $|s(\mu\simeq 0)|$ is three magnitude orders lower than that
of AAA TLG, and increases as the gate field increases. As the EC effect in
Fig. 3(b) at $T=30$ K has not dependence on $V_{g}$, the DEP presents sizable
dip-peak structures in the band gap zone, with large slopes in the vicinity of
$\mu=0$ for $V_{g}\neq 0$, while $s(\mu)=0$ for $V_{g}=0$ except near
$E_{\text{F}}$. The chemical potential dependence for $s(\mu)$ gives zeros
when $S_{e,T}(\mu)$ reach maxima, as indicated with vertical dash-dotted lines
particularly for $V_{g}=0.4$ eV. At low temperatures, zeros of $s(\mu)$ and
maxima of $S_{e,T}(\mu)$ are located near the band edge Van Hove singularities
of the electronic spectra in Fig. 5. This behavior is similar to the
thermopower of biased bilayer graphene Hao and Lee (2010) and other gapped 2D
materials such as germanene under an external electric field Grassano  _et
al._ (2018) and semiconducting dichalcogenides Shubnyi  _et al._ (2018). At
room temperature, the DEP in Fig. 6(d) decreases one magnitude order along
with smoother resonances, showing a sinelike shape about $E_{\text{F}}$ for
all $V_{g}\neq 0$. Near $\mu=0$ the slopes decrease as the gate potential
decreases. The slope reverses for $V_{g}=0$ because of the maximum of
$S_{e,T}(\mu=0)$ instead a minimum. These results are in agreement with the EC
response in Fig. 3(b).

###  III.3 ABA-stacked TLG

Trilayer graphene with Bernal-ABA stacking is the most stable geometry as
resembles a graphitelike atomic structure Aoki and Amawashi (2007). The ABA
TLG possesses mirror symmetry with respect to the middle graphene layer. This
structural symmetry is broken in the presence of electrostatic potentials
Guinea  _et al._ (2006); Koshino and McCann (2009a), driving to semimetallic
character with tunable overlap of the electronic bands Aoki and Amawashi
(2007); Craciun  _et al._ (2009); Wang  _et al._ (2016), as well as
controllable gaps at $K$ point and near it Avetisyan  _et al._ (2009); Wu
(2011); Wang  _et al._ (2016). The TB Hamiltonian considering all hoppings in
Fig. 1(c) is given by

$\mathcal{\hat{H}}^{\mathrm{ABA}}(\boldsymbol{k})=\begin{pmatrix}-V_{g}&\gamma_{0}f_{0}&\gamma_{4}f_{3}&\gamma_{3}f_{3}&\gamma_{2}f_{1}&0\\\
\gamma_{0}f_{0}^{*}&-V_{g}&\gamma{{}_{1}}f_{1}&\gamma_{4}f_{3}&0&\gamma_{5}f_{1}\\\
\gamma_{4}f_{3}^{*}&\gamma{{}_{1}}f_{1}^{*}&0&\gamma_{0}f_{0}&\gamma_{4}f_{3}^{*}&\gamma{{}_{1}}f_{1}\\\
\gamma{{}_{3}}f_{3}^{*}&\gamma_{4}f_{3}^{*}&\gamma_{0}f_{0}^{*}&0&\gamma{{}_{3}}f_{3}^{*}&\gamma_{4}f_{3}^{*}\\\
\gamma_{2}f_{1}^{*}&0&\gamma_{4}f_{3}&\gamma_{3}f_{3}&V_{g}&\gamma_{0}f_{0}\\\
0&\gamma{{}_{5}}f_{1}^{*}&\gamma_{1}f_{1}^{*}&\gamma_{4}f_{3}&\gamma_{0}f_{0}^{*}&V_{g}\end{pmatrix}.$
(10)

We analyze first a simple case for ABA TLG, labeled nearest-layer (NL) TB
model, including hopping parameters $\gamma_{0}$, $\gamma_{1}$ and
$\gamma_{3}$, with $\gamma_{2}=\gamma_{4}=\gamma_{5}=0$ from spectroscopy
experiments Bao  _et al._ (2017). We take different values of the external
gate potential in the limit $0\leq V_{g}\gtrsim\gamma_{1}$ for the electronic
spectra and EC calculations.

Figure 7(a) show the electronic band structure around $K$ point, Fig.7(b)
correspond to the DOS. Both quantities are shown in the vicinity of
$E_{\text{F}}=0$. The energy spectra are symmetric about zero energy with and
without the presence of the gate field, preserving electron-hole symmetry
Koshino and McCann (2009a). For $V_{g}=0$ (solid black lines), two bands mimic
the $k$-linear electronic dispersion of monolayer graphene at the $K$ point,
and one pair of parabolic bands overlap along $KM$ $k$-axis near
$E_{\text{F}}$. The whole unperturbed bands ($V_{g}=0$) give a superposition
of states of monolayerlike and Bernal-bilayerlike graphene states Guinea  _et
al._ (2006); Partoens and Peeters (2006, 2007). The DOS for $V_{g}=0$ shows a
sharp minimum at $E_{\text{F}}$ with two symmetric peaks surrounded it [see
inset of Fig.7(b)]. The major contribution of the edge peaks its from the AB-
bilayerlike electron and hole band edges, whose states are fully captured by
the $L$ function for $T>10$ K.

The gate field breaks mirror reflection symmetry with respect to the central
layer and mixes the linear and parabolic bands in the region of $E_{\text{F}}$
Guinea  _et al._ (2006); Koshino and McCann (2009a), so that when $V_{g}=0.05$
eV (red dotted lines) and $V_{g}=0.1$ eV (gray solid lines), the monolayerlike
linear bands are shifted from $E_{\text{F}}$, and the bilayerlike bands remain
almost unperturbed. The DOS for these cases presents a minimum at
$E_{\text{F}}$, and shows large electron and hole edge peaks as more states
are overlapped near $E_{\text{F}}$, where the electronic states contribute to
the EC effect for $T>30$ K. When the gate potential increases comparable to
$\gamma_{3}\sim V_{g}=0.3$ eV (dashed green lines) and $\gamma_{1}\simeq
V_{g}=0.4$ eV (solid purple lines), linear dispersion is seen at the $K$ point
as well as along the $KM$ direction near $E_{\text{F}}$, while a gap opens
along $K\Gamma$ as $\gamma_{3}$ is non zero Avetisyan  _et al._ (2009); Bao
_et al._ (2017). For these large-gate field potentials, the DOS almost
vanishes at $E_{\text{F}}$ and shows one pair of symmetric peaks centered near
the gap edges, contributing to the EC response when $T\geq 70$ K. Other pair
of peaks are centered in the vicinity of the minima of the highest valence
band and maxima of the lowest conduction band, not providing states to the EC
effect for $T<170$ K.

Figure 7: Electronic spectra for ABA-stacked trilayer graphene near the Fermi
level using nearest layer NL (a)-(b) and next-nearest layer NNL (c)-(d) TB
models. Left panels show the electronic band structure about $K$ point in the
Brilloin zone. Right panels ilustrate the density of states for the triangular
area enclosed by green lines in Fig. 1(d). (a)-(b) $\gamma_{0}=3.15$ eV,
$\gamma_{1}=0.39$ eV and $\gamma_{3}=0.25$ eV Bao  _et al._ (2017). (c)-(d)
$\gamma_{0}=3.16$ eV, $\gamma_{1}=0.39$ eV, $\gamma_{2}=-0.020$ eV,
$\gamma_{3}=0.315$ eV, $\gamma_{4}=0.044$ eV and $\gamma_{5}=-0.04$ eV Koshino
and McCann (2009b). Horizontal (vertical) dashed line indicates $E_{\text{F}}$
($K$ point). Inset in (b) shows a zoomed area for the spectra of the NL TB
model around $E_{\text{F}}=0$.

When all the interactions between the atoms are taken into account in the ABA
TLG system, labeled as next-nearest layer (NNL) TB model, the energy spectra
are no longer electron-hole symmetric about $E_{\text{F}}$ as in the NL case
because of the additional electron hoppings. However, the full band structure
for $V_{g}=0$ can still be considered as a combination of the band diagram of
a single graphene layer and that of AB-bilayer graphene as shown in Fig. 7(c).
At $K$ point, the valence and conduction parabolic bands open a gap near
$E_{\text{F}}$ for $V_{g}=0$, where the DOS shows a flat minimum in the hole
zone ($E_{\text{F}}\lesssim 0$), Fig. 7(d). The electron and hole peak edges
from the AB-bilayerlike bands in this case, provide states to the EC effect
for higher temperatures than in the NL case, $T>60$ K. When $V_{g}=0.05$ eV
(dotted red lines), the gate potential causes anticrossing of the parabolic
bands, similar as seen when negative and positive low-charged gates act on a
suspended ABA TLG Avetisyan  _et al._ (2009). At $E_{\text{F}}=0$ the DOS
increases, showing nearly a maximum instead of a minimum as in the NL case.
The electronic states belonging to this maximum are fully captured by the $L$
function for $T>30$ K. For $V_{g}=0.1$ eV, the bands are similar to the NL
model, but opening a gap at the $K$ point, and showing a highly antisymmetric
DOS, whose states near $E_{\text{F}}$ contribute to the EC effect for $T>60$
K. Large gate potentials $V_{g}=0.3$ eV and $V_{g}=0.4$ eV open several gaps
at the $K$ point as well as along both $K\Gamma$ and $KM$ $k$-axis, similar to
the effect of gate-induced high charge density on ABA TLG Avetisyan  _et al._
(2009). The DOS for these high $V_{g}$ values show minima at zero energy, as
in the band structure there are allowed states only near the $K$ point,
contributing to the EC response for $T\geq 30$ K.

In Fig. 8(a) and 8(b), we present respectively the EC response for both the
NL-ABA and NNL-ABA TB model. Both cases are qualitatively similar as they show
a dual behavior seen as a combination of the direct and inverse EC effect,
whose gate-dependent inversion $-\Delta S_{e,T}(V_{g})=0$ occurs at relatively
low temperatures. Moreover, noticeable differences are observed for the
entropy changes depending on the magnitude of the applied gate field in each
ABA-TB case. We observe a small direct response zone (heating) for low gate-
fields $V_{g}=0.05$ (dotted red lines) and $V_{g}=0.1$ eV (solid gray lines)
in both ABA-TB models, with maxima $-\Delta S_{e}\leq 0.005$ $\mu$eV/K at
$T\leq 12$ –see insets of Fig. 8. As temperature increases for these low-gate
field values, the entropy changes become inverse for $T\leq 21$ and remain
constant as temperature increases, cooling down up to $T=300$ K.

The EC effect for high-gate fields $V_{g}=0.25,0.3,0.4$ eV is very different,
as the direct caloric response zone (the maxima and inversion entropy changes)
occurs at higher temperatures as compared to low-gate field values. For the
ABA-NL case in Fig. 8(a), the maxima of the entropy changes are almost
constant, reaching $-\Delta S_{e}\simeq 0.011$ $\mu$eV/K at $T\simeq 23$ K.
The entropy inversion temperature also is nearly the same for all high
$V_{g}$, occurring at $T\sim 45$. The entropy changes maxima for high-gate
fields increase almost twice in the ABA-NNL model, with higher temperatures in
the range $26\ \text{K}\leq T\leq 30$ K. This is because the $L$ function
captures a maximum instead of a minimum of the reference DOS [$D(V_{g}=0)$] in
the NNL-ABA model. Remarkably, the inversion temperature in the ABA-NNL model
varies with high-gate fields, $56\ \text{K}\leq T\leq 69$ K. As the
temperature increases, the entropy changes increase up to room temperature,
cooling more for high gate field values. The inversion temperatures for the
ABA-NNL system also can be obtained in zone II of Fig. 10(a), where the
reference entropy (black line) crosses with all other entropies.

The direct-inverse response in ABA-TLG is captured by the electronic entropy
as a function of $\mu$ in Fig. 9(a) and 9(b). We present here only the ABA-NNL
plots as both ABA system EC results are equivalent. Figure 9(a) shows
$S_{e,T}$ at $T=20$ K resembles the DOS shape of Fig. 7(d) with broken
electron-hole symmetry. At $\mu=0$, the reference entropy for $V_{g}=0$ is
lower than the entropy for $V_{g}=0.05$ eV, and larger than the entropies for
all other gate field values, see Fig. 10(a). Consequently, the sample cools
for $V_{g}=0.05$ eV, and heats for higher gate fields at $T=20$ K, as expected
from the NNL-EC response in Fig. 8(b). When the temperature increases up to
$T=150$ K in Fig. 9(b), the electronic entropies are nearly four times the
entropies at $T=20$ K, and are smoothed because of higher temperature. The
reference entropy shows a flat line shape near $\mu=0$ and is smaller than all
other entropies, indicating the system cools for all gate fields at $T=150$,
as also seen from the EC response.

Figure 8: The isothermal entropy changes $-\Delta S_{e,T}$ as a function of
temperature $T$ for different gate potentials as indicated. (a) ABA-stacked
trilayer graphene considering NL hoppings, (b) ABA-stacked trilayer graphene
with NNL hoppings. The chemical potential is set to $\mu=0$ eV. Horizontal
dashed line indicates $-\Delta S_{e,T}=0$, while vertical dashed lines stand
for $T=20$ K and $T=150$ K, where the entropies are calculated as a function
of $\mu$ in Fig. 9. Insets show a zoom for each ABA-EC response at relatively
low temperatures. Figure 9: Electronic entropies (a)-(b) and differential
entropy per particle DEP (c)-(d) as a function of $\mu$ for ABA-stacked
trilayer graphene considering the NNL-TB model. Left panels stand for $T=20$
K, right panels $T=150$ K. Vertical dashed lines indicate $\mu=0$ for all
plots. Vertical dash-dotted lines at $\mu=-0.083$ eV and $\mu=0.1$ eV in (a);
and at $\mu=-0.065$ eV, $\mu=0.083$ eV in (b) highlight the maxima for
$S_{e,T}(V_{g}=0.4)$, which correlate with sign changes of the DEP in (c) and
(d) respectively. Figure 10: (a) Electronic entropy $S_{e}(T)$, (b) density of
thermally excited electrons $n_{th}(T)$ versus temperature calculated at
$E_{\text{F}}=0$ in ABA-stacked TLG within the NNL-TB model for different gate
potentials $V_{g}$. In (a), I: $S_{e}(V_{g}=0)\geq S_{e}(V_{g}\neq 0)$ , II:
$S_{e}(V_{g}=0.05,0.1\ \text{eV})\geq S_{e}(V_{g}=0)\geq
S_{e}(V_{g}=0.25,0.3,0.4\ \text{eV})$, III: $S_{e}(V_{g}=0)\leq
S_{e}(V_{g}\neq 0)$. Zone I, II and III in (b) corresponds to the same
temperature range as in (a).

Figure 9(c) and 9(d) show the DEP for the ABA-NNL model at $T=20$ K and
$T=150$ K respectively. $s(\mu)$ is not a symmetric function of $\mu$ because
of broken electron-hole symmetry for all gate-potential fields. Near zero
chemical potential and $V_{g}=0,0.3,0.4$ eV at 20 K, where the sample heats
[see inset of Fig. 8(b)], the DEP shows large spikes, which can be attributed
to the semimetal character and the imbalance between electron and hole
carriers of the electron spectra in Fig. 7(c) and 7(d). There are two other
smaller resonances when $\mu$ lies in the vicinity of the discontinuities of
the electronic entropy, shifted to larger $\mu$ as $V_{g}$ increases. These
entropy discontinuities at 20 K are located near the minima of the valence
band and maxima of the conduction band of the electron dispersion. $s(\mu)$
decreases for low gate fields $V_{g}=0.05,0.1$ eV in the vicinity of
$E_{\text{F}}$, near where the EC response displays a sign change at $\sim$21
K [see inset of Fig. 8(b)]. As the ABA TLG gets cool for all $V_{g}$ at 150 K
[Fig. 8(b)], the DEP in Fig. 9(d) reduces by almost half near $\mu=0$, and the
resonances are now spread out, mixed with the shifted ones close to $S(\mu)$
discontinuities.

Because the ABA-stacked TLG is capable to heats and cools within the same
sample as a direct and inverse EC response are obtained in Fig. 8, it is
interesting to know how the electronic entropy $S_{e}$ is related to the
charge density at the microscopic level. We calculate the number of carriers
thermally excited of ABA-stacked TLG through the NNL-TB model. We determine
the density of excited electrons $n_{th}$ from the valence band to the
conduction band of Fig. 7(c) as a function of temperature for selected values
of the gate potential. In a similar way to Eq. 1, $n_{th}$ above the Fermi
level at a given temperature is given by
$n_{th}(V_{g},T)=A^{-1}\int_{\mu}^{E_{h}}D(E,V_{g})n_{\text{F}}(E,T,\mu)dE$,
where $A$ is the unit cell area of the TLG, and we take $\mu=0$ as the Fermi
energy at $T=0$.

The temperature dependence of $S_{e}(T)$ and $n_{th}(T)$ is shown in Fig.
10(a) and Fig. 10(b) respectively. The fraction of excited electrons follows a
very similar pattern as the electronic entropy because charge carriers
contribute to the entropy by filling electronic states as temperature
increases. In Fig. 10(b), electrons are excited at low temperature for low
gate potentials $V_{g}=0.05,0.1$ eV, where the electronic dispersion and DOS
of the parabolic bands present a negligible bandgap at $E_{\text{F}}=0$, see
Fig. 7(c) and 7(d). As the gate potential increases, the energy gaps of ABA-
TLG increase, and a higher temperature is needed to excite electrons from the
valence band to the conduction band. This produces a suppression (non-linear
behavior) of $n_{th}$ at low temperatures for high electric potentials. From
$\sim 70$ K, $n_{th}$ linearly increases with temperature at high
temperatures, which may correspond to a monotonic DOS as a function of energy,
that is gapless parabolic electronic bands near $E_{\text{F}}$. Similar
behavior for the density of excited electrons has been experimentally observed
in graphene multilayers including AB-bilayer and ABA-TLG, correlated with a
theoretical temperature-dependent gap approximation, where both procedures
give account of a finite-temperature electronic phase transition Nam  _et al._
(2018). In our case, large gate-field values $V_{g}=0.25,0.3,0.4$ eV induced
overlap and gaps in the electronic structure of ABA-TLG, which compete with
temperature in the proximity of $E_{\text{F}}$. This causes the direct EC
effect $-\Delta S_{e}>0$ with maxima and inversion values located in zone II
of Fig. 10(a), which correspond to the charge suppression zone II of Fig.
10(b), that is gaped electronic bands. On the contrary, the inverse EC effect
$-\Delta S_{e}<0$ is located in zone III of $S_{e}(T)$, where electrons are
thermally excited in a linear way as a function of temperature, indicating
gapless bands near $E_{\text{F}}$.

##  IV Conclusions

Gated trilayer graphene structures show an electrocaloric response that
depends on the stacking pattern and the electronic character they possess near
the Fermi level. Trilayer graphene with AAA stacking remains metallic under
the applied gate field, presenting large entropy changes as the gate potential
increases, linearly cooling up to room temperature. In striking contrast, ABC-
stacked trilayer graphene converts into a semiconductor when the gate field is
turned on, non-linearly heating as temperature increases. The energy spectra
of ABA-stacked TLG geometry present semimetallic as well as gapped character
with gate voltage, cooling and heating in the caloric response in a linear and
non-linear mixed way, giving a combination of AAA and ABC electrocaloric
effect. We verified that gate-dependent thermally excited charge at
$E_{\text{F}}=0$ for ABA-TLG can be directly linked to the electronic entropy
as well as to the EC response, where charge suppression is related to the
direct EC effect, and linearly excited electrons correspond to the inverse EC
effect.

The differential entropy per particle shows particular signatures for each TLG
structure at low and high temperatures. Measuring this thermodynamic quantity
could be used as a cheaper and novel experimental technique to identify the
stacking pattern in trilayers graphene and-or similar layered 2D materials. It
is worth mentioning that a new caloric effect based on the entropy per
particle and EC effect connection can be obtained, see Appendix. We refer to
this mechanism as charge-carrier caloric effect, as the variable quantity are
charge carriers instead of an external field. We propose this effect may be
measured by varying the charge density and keeping constant the electric field
in the sample, similar to a 2D electron gas experimental setup Kuntsevich  _et
al._ (2015).

##  V Acknowledgments

N.C. acknowledges support from ANID Fondecyt Postdoctoral Grant No. 3200658,
P.V. and O.N. acknowledge support from ANID PIA/Basal AFB18000. F.J.P.
acknowledges support from ANID Fondecyt, Iniciación en Investigación 2020
grant No. 11200032, and the financial support of USM-DGIIE.

## Appendix: Charge-Carrier Caloric Effect

The connection between the electrocaloric effect and the differential entropy
per particle $s$ defined in Sec. II.1, can be obtained by analyzing the
Helmholtz free energy $F$ and internal energy $U$ of a general system that
includes a variable number of particles $N$ in its formulation.

First, we define the differential internal energy of the system as

$dU=TdS+\textbf{E}\cdot d\textbf{P}+\mu dN,$ (11)

where the first term corresponds to heat $\delta Q=TdS$, with $S$ the total
entropy; the second term is the definition of electric work
$W=\int\textbf{E}\cdot d\textbf{P}$, where P is the electric polarization and
E is the electric field. The last term $\mu dN$, where $\mu$ is the chemical
potential, is the energy needed to add or remove one particle of the system.
On the other hand, $F$ is defined as

$F=U-TS-\textbf{E}\cdot\textbf{P},$ (12)

and its differential is

$dF=-SdT-\textbf{P}\cdot d\textbf{E}+\mu dN,$ (13)

where we have used Eq. 11. From Eq. 13, we can obtain the following Maxwell
relations:

$\left(\frac{\partial
S}{\partial\textbf{E}}\right)_{T,N}=\left(\frac{\partial\textbf{P}}{\partial
T}\right)_{\textbf{E},N},$ (14)

and

$\left(\frac{\partial S}{\partial
N}\right)_{T,\textbf{E}}=-\left(\frac{\partial\mu}{\partial
T}\right)_{\textbf{E},N},$ (15)

where the last equation is the same as presented in Eq. 6, but including the
thermodynamic variable associated with the electric field, and the subscripts
refer to variables held constant during partial differentiation.

To take into account the electronic entropy variations in our system, it is
convenient to specify the total differential entropy, whose expression is
given by

$dS=\left(\frac{\partial S}{\partial
T}\right)_{\textbf{E},N}dT+\left(\frac{\partial
S}{\partial\textbf{E}}\right)_{T,N}d\textbf{E}+\left(\frac{\partial
S}{\partial N}\right)_{T,\textbf{E}}dN.$ (16)

If we calculate the entropy change in an isothermal process ($dT=0$) along a
path of the form $S(\textbf{E}_{1},N_{1})\rightarrow
S(\textbf{E}_{2},N_{1})\rightarrow S(\textbf{E}_{2},N_{2})$, we obtain

$\Delta S_{T}=\int_{\textbf{E}_{1}}^{\textbf{E}_{2}}\left(\frac{\partial
S}{\partial\textbf{E}}\right)_{T,N=N_{1}}d\textbf{E}+\int_{N_{1}}^{N_{2}}\left(\frac{\partial
S}{\partial N}\right)_{T,\textbf{E}=\textbf{E}_{2}}dN.$ (17)

Employing the Maxwell relations of Eq. 14 and Eq. 15, we finally obtain

$\Delta
S_{T}=\int_{\textbf{E}_{1}}^{\textbf{E}_{2}}\left(\frac{\partial\textbf{P}}{\partial
T}\right)_{\textbf{E},N}d\textbf{E}-\int_{N_{1}}^{N_{2}}\left(\frac{\partial\mu}{\partial
T}\right)_{N,\textbf{E}=\textbf{E}_{2}}dN.$ (18)

This last equation occurs for an isothermal process in which the electric
field and the number of particles vary in the process. The terms associated
with the DEP corresponds to the integrand in the second term of Eq. 17 and Eq.
18. This means that the DEP is directly connected to the electrocaloric
phenomena. The charge-carrier caloric effect will correspond to the
measurement of a process in which the electric field remains fixed, but a
variation in the number of particles is present in the initial and final
stages of the process. Hence, the pure caloric effect obtained from the DEP is
given by

$\Delta
S_{T}^{\text{DEP}}=-\int_{N_{1}}^{N_{2}}\left(\frac{\partial\mu}{\partial
T}\right)_{N,\textbf{E}}dN.$ (19)

It is essential to highlight that when the number of particles remains fixed,
the usual expression of the electrocaloric effect is recovered from Eq. 18

$\Delta
S_{T}=\int_{\textbf{E}_{1}}^{\textbf{E}_{2}}\left(\frac{\partial\textbf{P}}{\partial
T}\right)_{\textbf{E},N}d\textbf{E}.$ (20)

As the differential entropy per particle is an experimentally measurable
quantity, we believe that the associated charge-carrier caloric effect can be
performed through indirect measurements, not requiring precision calorimetry.
A similar procedure has been recently reported by a direct measurement of
$\partial\mu/\partial T$, obtaining very small electric currents, caused by a
periodic temperature modulation in a 2D diluted electronic system Kuntsevich
_et al._ (2015).

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