This is the author’s peer reviewed, accepted manuscript.

However, the online version of record will be different from this version once it has been copyedited and typeset. Thermoelectric properties of penta-InP5: A first-principles and
machine learning study
Nguyen Thanh Tien,a‡ , Pham Thi Bich Thao a , Duy Khanh Nguyen,b,c , Le Nhat Thanh a , and Vo Khuong Dien d,e†
a College of Natural Sciences, Can Tho University, 3-2 Road, Can Tho City 94000, Vietnam
b Laboratory for Computational Physics, Institute for Computational Science and Artificial Intelligence, Van Lang University, Ho
Chi Minh City, Vietnam
c Faculty of Mechanical - Electrical and Computer Engineering, School of Technology, Van Lang University, Ho Chi Minh City,

Vietnam
d Engineering Research Group, Dong Nai Technology University, Bien Hoa City, Vietnam.
e Faculty of Engineering, Dong Nai Technology University, Bien Hoa City, Vietnam.

Smart wearable devices that harvest energy from ambient sources, such as body heat, are gaining significant attention due
to their potential in diverse applications. Thermoelectric (TE) materials, which convert thermal energy to electrical power,
are critical for these devices, yet achieving both high TE performance and mechanical flexibility remains a significant
challenge. Here, we investigate the TE properties of the penta-InP5 monolayer, a novel two-dimensional material, using
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0251741

first-principles calculations integrated with machine learning potentials. We show that penta-InP5 achieves a remarkable
figure of merit, with values of 0.51 and 0.42 for hole and electron doping, respectively, at room temperature. Additionally,
the material demonstrates remarkable mechanical properties, with an in-plane stiffness of 52 N/m and a fracture strain of
23% for the uniaxial strain. These findings suggest that penta-InP5 is a promising candidate for flexible, high-performance
TE applications, advancing the potential of wearable energy-harvesting devices.

1 INTRODUCTION ues at room temperature 9 . Bulk SnSe achieves exceptional

ZT values but requires high operating temperatures around
Smart wearable devices have attracted considerable inter-
900 K 10 . Consequently, there is a growing demand for new
est globally for applications in communication, medicine,
materials that combine high room-temperature TE perfor-
healthcare, and other fields 1–3 . These devices rely on en-
mance with mechanical flexibility for wearable electronics.
vironmental energy harvesting to eliminate the need for ca-
bles and batteries. Thermoelectric (TE) conversion, driven In 2014, the theoretical prediction of a novel carbon
by the Seebeck effect, has emerged as a highly efficient allotrope, penta-graphene, sparked significant interest in
method for harvesting thermal energy without additional the scientific community 11 . This discovery prompted
mechanical or electromagnetic inputs. To optimize TE per- further investigations into pentagonal structures, includ-
formance, the dimensionless figure of merit (ZT ) is a crit- ing penta-CN2 12,13 , penta-SiC2 14,15 , penta-PdSe2 16 , penta-
ical parameter, and efforts often focus on balancing elec- TMBs/Cs 17 , and penta-silicene 18–20 . Notably, several of
trical and thermal properties to enhance efficiency. How- pentagonal 2D materials have demonstrated strong poten-
ever, the inherent trade-offs between the Seebeck coefficient tial as high-performance TE materials. For instance, Y.
(S) and electrical conductivity (σ ) present a persistent chal- Zhao and colleagues reported a power factor as high as 1.5
lenge in achieving high ZT values 4 . mWm−1 K−2 for a PdSe2 flake with a thickness of 5 nm 21 .
Similarly, penta-silicene exhibited impressive ZT values of
Wearable TE devices demand materials with high
approximately 3.04 for electron doping and 3.43 for hole
room-temperature ZT values and strong mechanical
doping at 300 K 20 . Another member of the 2D pentago-
stability 5 . Conducting polymers, such as poly(3,4-
nal materials family, penta-InP5 , has attracted considerable
ethylenedioxythiophene):poly(styrenesulfonate) 6 , provide
research interest due to its tunable electronic bandgap 22 ,
flexibility and low thermal conductivity but suffer from low
potential applications in photocatalysis 23 , and suitability
power factors due to limited electrical conductivity. While
for linear and nonlinear optical applications 24 . However,
Bi-Te alloys exhibit promising ZT = 1.7 at 300 K 7 , their
the thermoelectric performance of penta-InP5 remains rela-
mechanical limitations hinder their use in wearable applica-
tively unexplored.
tions 8 . Nanostructured Te-based compounds improve me-
chanical properties but often experience diminished ZT val- In this study, we examine the TE properties of the penta-
InP5 monolayer using on-the-fly machine learning poten-
†Email: nttien@ctu.edu.vn tials (FMLP) combined with first-principles calculations.
Corresponding author: ‡ vokhuongdien@dntu.edu.vn We integrate FMLP within VASP and phono3py to reduce

1
 the computational cost of simulating lattice thermal trans- respectively.

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port. For electronic transport, we calculate electron re-
laxation times, τnk , for each electronic band index n and The second- and third-order interatomic force constants
wave vector k. These relaxation times are then used to (IFCs) are calculated using the FMLP method in combina-
solve the Boltzmann transport equation, allowing us to eval- tion with the Phonopy and Phono3py codes 25,26 , employing
uate the electronic contributions to the figure of merit. By a supercell size of 5×5×1. The lattice thermal conductivity
combining these contributions with the thermal conductiv- is computed using the following expression 25? :
ity of the lattice, we evaluate the overall TE performance of
penta-InP5 . Our results indicate that at room temperature,
it achieves maximum ZT values of 0.51 for hole doping 1
∑ f0 (ωλ ) ( f0 (ωλ ) + 1) (h̄ωλ )2 vαλ vλ τλ ,
αβ β
and 0.42 for electron doping. Additionally, the penta-InP5 κl =
kB T 2 ΩN λ
has a high in-plane stiffness of approximately 52 N/m and (1)
a fracture strain of up to 23% for the uniaxial strain. These in this context, kB , T , Ω, h̄, and N refer to the Boltzmann
findings highlight penta-InP5 as a promising candidate for constant, temperature, unit cell volume, Planck’s constant,
wearable TE devices. and the number of wave vector points in the first Brillouin
zone, respectively. The terms ωλ , vλ , and τλ denote the
2 COMPUTATIONAL DETAILS
angular frequency, group velocity, and relaxation time of
The estimation of the figure of merit in TE materials phonon mode λ , while f0 (ωλ ) represents the Bose-Einstein
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0251741

presents several challenges. First, lattice thermal conduc- distribution function. A 100 × 100 × 1 k-point mesh is uti-
tivity is typically calculated using ab initio methods that lized in Phono3py to ensure the convergence of thermal
account for three-phonon scattering 25,26 . However, these conductivity calculations.
methods are time-consuming and computationally expen-
sive. Second, electrical transport is often modeled using
the Boltzmann transport equation with the constant relax-
2.2 Electronic Transport Properties
ation time approximation (CRTA). In this approach, the
relaxation time τ is derived from single-mode deforma-
tion potential theory (SDPT) 27–29 , which assumes isotropic The electron-phonon Wannier (EPW) module 35 , imple-
electron-phonon (el-ph) coupling and considers only acous- mented in the QUANTUM ESPRESSO package 36 , is used
tic phonon scattering. These assumptions can lead to in- to calculate electrical transport properties. The calculations
accuracies in predicting electronic transport properties. In utilize norm-conserving (NC) pseudopotentials 37 and the
this study, we present a framework that combines FMLP PBE exchange-correlation functional 32 . The kinetic energy
with comprehensive el-ph coupling in first-principles cal- cutoff for wave functions is set to 80 Ry, and a Monkhorst-
culations. This approach enables us to efficiently capture Pack k-point grid of 8×8×1 is employed. The plane-wave
both lattice and electronic transport properties, allowing forbasis is then transformed into the Wannier basis using an
accurate estimation of TE performance. interpolation technique. In this stage, the electron-phonon
2.1 Lattice thermal conductivity matrix elements are calculated by applying a very fine k-
mesh of 160×160×1 in momentum space for both electron
The training of the FMLP model is carried out using and phonon states.
the machine learning module within VASP 30,31 . The
Perdew-Burke-Ernzerhof (PBE) functional is employed for The charge carrier mobility is determined from the fol-
32
the exchange-correlation term , while the projector aug- lowing expression 38 :
mented wave (PAW) method is utilized to model the inter-
∂f
actions between valence electrons and ionic cores 33 . Cal-
R
∑i BZ τik vik,α vik,β ∂ Eikik dk
culations are performed with a plane-wave energy cutoff of µαβ = e R . (2)
∑i BZ fik dk
500 eV and a Monkhorst-Pack k-point grid of 2×2×1 for a
6×6×1 supercell. To generate a reliable dataset for train-
ing the FMLP model for penta-InP5 , ab-initio molecular dy- where α and β refer to Cartesian directions, e is the ele-
namics (AIMD) simulations 34 at 300 K are performed. A mentary charge, τik is the relaxation time for the electronic
canonical ensemble is used with 10,000 steps, each with a state indexed by i and wave vector k, v is the electronic
time step of 1 fs, to obtain configurations close to equilib- band velocity, f is the Fermi-Dirac distribution at equilib-
rium. During the training process, the weights for energy, rium and E is the electronic energy. The scattering rates
forces, and stresses are assigned values 1, 0.1, and 0.001, 1/τik of all phonons are computed as 38 :

2
 addition to the thermodynamic stability discussed hereafter,

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thermal stability of penta-InP5 is verified in terms of the
1 2 Im Σnk 2π dq
Z
2
= =∑ g(ik, jk′ , λ q) calculations of molecular dynamics (AIMD). After 10 ps
τik h̄ λ , j BZ
h̄ ΩBZ
n of AIMD simulations, the penta-InP5 monolayer retained
×


f jk′ + nλ q δ (Eik − E jk′ + h̄ωλ q )δk+q,k′ +G good structural integrity at both 300 K and 700 K, as shown

o in Fig. S1. This suggests that penta-InP5 monolayers may
+ 1 + nλ q − f jk′ δ (Eik − E jk′ − h̄ωλ q )δk−q,k′ +G . be stable in ambient to high-temperature environments.
(3) We begin our analysis by evaluating the reliability of the
′
Here, g(ik, jk , λ q) is the electron-phonon coupling FMLP for phonon property calculations. Figures 1(b) and
strength, n is the Bose-Einstein distribution, ωλ q is the 1(c) show the per-atom energy and atomic forces for various
phonon frequency, ΩBZ is the volume of the Brillouin zone, configurations of penta-InP5 , calculated using both FMLP
and G is the reciprocal lattice vector. The two delta func- and DFT. A strong linear correlation is observed between
tions ensure the conservation of energy and momentum. the results from FMLP and DFT, with a slope close to 0.5.
The root mean square errors (RMSE) for energy and forces
2.3 Thermoelectric Performance are 0.102 meV/atom and 0.022 eV/Å, respectively, indicat-
Based on the lattice and electronic transport properties, the ing that FMLP achieves a high level of precision compara-
TE performance of penta-InP5 is calculated using our in- ble to DFT in reproducing potential energy surfaces.
house code. The TE transport coefficients for electrical con- In combination with Phonopy and Phono3py 25,26 , FMLP
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ductivity (σ ), Seebeck coefficient (S), and electronic ther- was applied to compute the phonon dispersion of penta-
mal conductivity (κe ) are computed as follows 38,39 . InP5 , as shown in Fig. 1(d). The excellent agreement be-
tween the FMLP (black line) and DFT (red dashed line) re-
1 ∂ fik
Z
σαβ (T, E f ) = −e2 vik,α vik,β τik dk. (4) sults further validates the accuracy of the energy and force
Aarea ∑
i BZ ∂ Eik predictions. Using the FMLP generated in this study, we are
able to quickly and accurately determine the thermal con-
R ∂ fik
1 ∑i BZ (Eik − E f )vik,α vik,β τik ∂ Eik dk ductivity of penta-InP5 monolayer. There are no imaginary
Sαβ (T, E f ) = − ∂f
.
eT frequencies in the dispersion diagram, confirming the ther-
R
∑i BZ vik,α vik,β τik ∂ Eikik dk
(5) modynamic stability of the materials 47 . Eighteen phonon
branches are present, including the three acoustic and fif-
teen optical branches, which correspond to the six atoms
2
κe,αβ (T, E f ) = −T Sαβ (T, E f )σαβ (T, E f )− in the unit cell. We highlight three acoustic branches that
1
Z
∂ fik (6) significantly influence lattice thermal conductivity: two in-
(Eik − E f )2 vik,α vik,β τik dk.
TAarea ∑
i BZ ∂ Eik plane acoustic modes [transverse acoustic (TA) and longi-
tudinal acoustic (LA) modes], and one out-of-plane acous-
where Aarea is the surface area of the primitive cell and E f tic mode [flexural acoustic (ZA) mode]. The penta-InP5
is the chemical potential. monolayer has a small phononic band gap between 40 meV
The figure of merit (ZT) is calculated as: and 50 meV, which originates from the slight difference in
atomic masses between the In and P atoms. It is impor-
2 (T, E )σ (T, E )
Sαβ f αβ f tant to note that a large frequency gap between acoustic and
ZTαβ (T, E f ) = T. (7) optical phonons (a-o gap) generally results in high κl . In
κe,αβ (T, E f ) + κl
contrast, a small a-o gap is highly desirable for designing
3 RESULTS AND DISCUSSIONS materials with ultralow κl 48 . The small a–o gap observed in
penta-InP5 suggests strong acoustic-optical phonon scatter-
3.1 Phonon transport properties ing, indicating expected ultralow κl values in this material.
The optimized atomic structure of the penta-InP5 monolay- The calculated κl of penta-InP5 monolayer over the tem-
ers is presented in Fig. 1(a). Both the top and side views perature range from 100 to 700 K is shown in Fig. 2(a).
reveal that penta-InP5 exhibits similar structural character- The data display a typical κl ∼ 1/T dependence, which is
istics to penta-graphene 11 , featuring a five-membered ring mainly due to the increasing phonon scattering at higher
composed of four phosphorus (P) atoms and one indium temperatures. Fig. 2(a) further highlights the contribu-
(In) atom. The optimized geometric parameters, including tions of different phonon branches to the total thermal con-
the lattice constants a (b) and the thickness of the mono- ductivity, with the majority of heat transport being carried
layer h, are found to be 5.06 Å and 2.93 Å, respectively, by acoustic phonons. In contrast to graphene 49 , the ZA
which aligns well with previously reported values 22–24 . In mode is not the dominant contributor to thermal conduc-

3
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Figure 1 (a) The optimized structures of penta-InP5 monolayers are shown in both top and side views. The purple balls represent
phosphorus (P) atoms, while the cyan balls indicate indium (In) atoms within the penta-InP5 monolayer. The dashed rectangular box
outlines the size of the unit cell. (b) A comparison of energy and (c) atomic force calculations for multiple configurations of the
penta-InP5 monolayer using FMLP and DFT methods. (d) The phonon dispersion of the penta-InP5 monolayer is presented based
on DFT and FMLP calculations. The abbreviations correspond to the transverse acoustic (TA), longitudinal acoustic (LA), and one
out-of-plane (ZA) acoustic modes.

4
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Figure 2 Thermal conductivity of penta-InP5 : (a) Branch-dependent thermal conductivity, with the inset bar chart showing the
contribution of phonon modes to thermal conductivity at T = 300 K; (b) energy-dependent phonon group velocity; (c)Grüneisen
parameter; and (d) phonon scattering rate of the penta-InP5 monolayer at T = 300 K.

5
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Figure 3 Electronic transport properties of penta-InP5 monolayers. (a) Electronic band structure of the penta-InP5 monolayer cal-
culated using the PBE functional. The Fermi level is shifted to zero. (b) Scattering rates of holes with energies within 0.3 eV of
the VBM. (c) Scattering rates of holes with energies within 0.3 eV of the CBM. The black dotted line represents the electron/hole
scattering rates calculated using the CRTA method. (d) El-ph scattering strength for hole states. (e) El-ph scattering strength for elec-
tron states. (f) Corresponding atomic vibrations for the phonon mode depicted in (d) and (e), with arrow directions and magnitudes
indicating atomic movements.

6
 derstanding the behavior of lattice thermal conductivity. A

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large |γ| indicates strong anharmonicity, which typically
leads to reduced κl . In Fig. 2(c), we plot the distribution
of the Grüneisen parameters for penta-InP5 . Notably, the
high Grüneisen values for the acoustic modes, particularly
for the ZA vibration, suggest that the acoustic branches in
the penta-InP5 monolayer exhibit significant anharmonicity.
As a result, the phonon-phonon scattering rates (1/τ ph−ph )
for the ZA and LA/TA phonons are one to two orders of
magnitude larger than those for optical phonons, as shown
in Fig. 2(d). The combination of low group velocities (vi,q )
and relatively high 1/τ ph−ph further supports the conclusion
that penta-InP5 monolayer has intrinsically low κl .

3.2 Electronic transport properties

Figure 4 Comparison of electron and hole mobilities in the The band structure of the penta-InP5 monolayer (Fig. 3(a)
penta-InP5 monolayer, calculated using the CRTA method and and Fig. S2) reveals key electronic features. The valence
incorporating full electron-phonon coupling effects, at tempera- band maximum (VBM) is located at the M point, while
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tures ranging from 100 to 700 K and a carrier doping concentra-

the conduction band minimum (CBM) lies between the Γ
tion of approximately 1013 cm−2 .
and M points, resulting in an indirect bandgap of 2.70 eV,
consistent with previous findings 22–24 . The valence band
tivity in penta-InP5 , due to the lack of reflectional sym- shows four prominent peaks at M and X, with the energy
metry in its buckled crystal structure 49 . At room temper- difference between them being less than 0.1 eV, while the
ature, the intrinsic κl of penta-InP5 monolayer is around conduction band exhibits four valleys at the CBM, indi-
1.82 W·m−1 ·K−1 , which is significantly lower than that of cating multivalley characteristics. These multivalley fea-
most conventional 2D materials. For example, the room- tures suggest strong inter-valley scattering, leading to en-
temperature κl of monolayer MX2 (M = Mo, W; X = hanced electron-phonon interactions, which likely impact
S, Se) exceeds 50 W·m−1 ·K−1 50–52 , while that of penta- carrier mobility. Additionally, the relatively flat band dis-
graphene surpasses 150 W·m−1 ·K−1 15,53 . Therefore, penta- persion near the band edges reflects a larger effective mass,
InP5 holds promise as a potential candidate for TE applica- influencing the density of states (DOS) around the Fermi
tions. level. This behavior is crucial for the enhanced Seebeck co-
To gain deeper insight into the microscopic mechanisms efficient, contributing to the excellent TE performance ex-
behind the ultralow κl in penta-InP5 , we conducted an in- pected in the 2D InP5 monolayer.
depth analysis of key parameters that influence κl , includ- We now examine the ab initio el-ph coupling rate [Eq.
ing phonon velocities, the Grüneisen parameter, and phonon (3)] with both acoustic and optical phonon modes in penta-
scattering rates, with particular emphasis on the contribu- InP5 . For comparison, we calculate the effective masses,
tions of the three acoustic modes. elastic modulus, and deformation potential constants, and
The phonon group velocity of penta-InP5 , obtained from further derive the carrier mobilities and scattering rates us-
the first derivative of energy with respect to the wave vector ing the CRTA method, as shown in Table S1. In typical non-
q, vi,q = ∂∂ωq , is illustrated in Fig. 2(b). Based on Eq. (1), the polar materials, where there is a symmetric charge distribu-
reduced velocities of the acoustic phonons contribute to the tion within the unit cell lattice, electron-acoustic phonon
suppression of the κl . The group velocities, vi,q , of the 15 interactions primarily dominate the scattering rates 27,54 .
highest optical phonon modes are considerably lower than However, in polar materials such as penta-InP5 , this sce-
those of three acoustic phonons. Notably, vi,q for penta- nario changes significantly. The optical phonon scattering
InP5 is exceptionally low, with a maximum value of only 5 channel, which is usually not significant, becomes strongly
km.s−1 . For all directions, the highest value of vi,q is sig- enhanced due to the polar nature of the material 55,56 . Our
nificantly lower than that of the InP3 - a promising TE can- calculations for penta-InP5 show that electron scattering by
didate, and is much smaller compared to common 2D ma- optical phonons is much stronger than scattering by acoustic
terials such as graphene (21.71 km/s) and monolayer MoS2 phonons, indicating an enhanced electron-optical phonon
(6.4 km/s). coupling in this system. Moreover, due to the consider-
The Grüneisen parameter (γ) quantifies the anharmonic able variation of electron occupations in the transport en-
interactions within a crystal and is a critical factor in un- ergy regime, the transport coefficients may be sensitive to

7
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Figure 5 Thermoelectric performance calculated considering full el-ph couplings. (a) Electrical conductivities, (b) electronic thermal
conductivity, (c) Seebeck coefficients, (d) power factors, and (e) ZT values at 300 K, 500 K, and 700 K. (f) Maximum ZT values
at 300 K for various TE materials including WSTe 40 , InP3 38 , PbTe 41 , Bi2 Te2.7 Se0.3 :Cu 42 , Ag2 Se 43 , Antimonene 44 , SiGe 45 , PbSe-
based 46 , and SnSe 10 . The data are compiled from experimental results or theoretical calculations that incorporate full electron-
phonon coupling.

8
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Figure 6 Mechanical properties of penta-InP√5 . (a)√Orientation-dependent Young’s modulus and (b) Poisson’s ratio for penta-InP5 .
(c) Equilibrium energy-strain curves the 2 × 2 supercell under tensile strain ε⊥ . The arrows indicate the equilibrium strain
√ for √
magnitude εk . The inset shows the 2 × 2 supercell model with strain ε⊥ applied along the diagonal direction of the primitive cell.
(d) Stress–strain relationship under biaxial and uniaxial tensile strain, with the red arrow indicating the maximum strain.

9
 changes in scattering strength. The scattering rates com- mobility values µ determined from the CRTA approach dis-

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puted using the CRTA method are represented by dashed cussed previously (see Table S1 and Fig. S3). In Fig. 5, we
lines. When compared with the total scattering rates that present the resulting electronic transport characteristics us-
take into account full electron-phonon interactions, the scat- ing full el–ph coupling, including the Seebeck coefficient
tering rates based on the CRTA method are notably overes- S, electrical conductivity σ , electronic thermal conductivity
timated. κel , power factor PF and ZT values.
To thoroughly explain the underlying mechanisms of Electrical conductivity σ , which depends on carrier con-
this scattering phenomenon, we examine the el-ph cou- centration and mobility, shows distinct behavior at low
pling strength, calculated at a temperature of 300 K. Our and high doping levels. For doping concentrations below
focus is on the acoustic and optical modes that signifi- |EF | = 0.96 eV, σ increases with temperature as carrier con-
cantly contribute to the scattering processes. The vibra- centrations increase. However, above this doping level, the
tional characteristics of these modes are shown in Fig. 3(f). decrease in mobility offsets any increase in concentration,
For penta-InP5 monolayer, the optical phonon branch OP causing σ to decline. According to the Wiedemann–Franz
(14) demonstrates the strongest electron coupling, with a law, κel = Lσ T (with L being the Lorenz number), κel
coupling strength |g| that diverges as 1/q when the wave should be proportional to σ . However, Fig. 5(b) shows
vector q approaches the Γ point. This divergence results that κel deviates slightly from this proportionality, reflecting
from the Fröhlich interaction 57 , yielding a typical coupling the semiconductor nature of penta-InP5 , where the Wiede-
strength of 160 meV, as illustrated in Figs. 3(a-b). In con- mann–Franz law does not strictly apply.
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trast, the acoustic modes—specifically the ZA, TA, and LA The Seebeck coefficient S is determined by the logarith-
modes—show relatively low el-ph coupling strengths, indi- mic energy derivatives of both the density of states D(ω)
cating their minor contributions to the scattering rates ob- and the relaxation time τ(ω) at the Fermi level E f . This
served in the penta-InP5 monolayer. Additionally, it is im- relationship is described by the Mott equation 58 :
portant to note that the el-ph coupling strengths at VBM
π 2 kB2 T ∂ ln D(ω) ∂ ln τ(ω)
 
and CBM are significantly lower than those observed at the S=− +
Γ point. This reduction suggests a weaker scattering rate in 3e ∂ω ∂ω Ef
these energy regions compared to deeper energy levels. The lattice thermal conductivity is a crucial factor for TE
Based on calculated relaxation times, the electron and efficiency. It decreases inversely with temperature, yield-
hole mobilities in penta-InP5 monolayer are precisely deter- ing values of 1.820, 1.082, and 0.770 W m−1 K−1 at tem-
mined across the temperature range of 100–700 K, as illus- peratures of 300 K, 500 K, and 700 K, respectively. Con-
trated in Fig. 4. Carrier mobility exhibits a decreasing trend sequently, the maximum figure of merit ZT increases with
with increasing temperature, attributable to enhanced el-ph temperature, reaching values of 0.51, 1.12, and 1.77 for p-
coupling at high temperatures. Notably, the comprehensive type doping and 0.42, 0.84, and 1.26 for n-type doping at
inclusion of el-ph interactions yields total scattering rates the same temperatures. Below 700 K, ZT values show that
for electrons and holes that surpass those calculated under p-type doping has slightly higher values with full electron-
the CRTA, as shown in Fig. 3(b). Consequently, mobilities phonon coupling. Our calculations indicate that penta-InP5
derived from full el-ph couplings are approximately an or- achieves one of the highest ZT values among TE materi-
der of magnitude lower than those obtained via the CRTA, als at room temperature [see Fig. 5(f)]. It is important to
as depicted in Fig. 4. Moreover, the CRTA often fails to note that the CRTA method tends to overestimate ZT for
capture the temperature dependence of mobility. Specifi- both doping types due to its simplified scattering assump-
cally, within the CRTA, µ(T ) strictly follows a 1/T depen- tions [Fig. S3].
dence. In contrast, the el-ph method reveals a modified 1/T
trend, which differs for electrons and holes. Thus, the trans- 3.4 Mechanical properties
port properties of penta-InP5 monolayers are significantly With the superior room-temperature TE performance, we
modulated by the el-ph coupling rate, particularly due to now examine the mechanical flexibility of monolayer penta-
optical phonon contributions at higher temperatures. InP5 , a critical parameter for flexible electronics applica-
tions. Flexibility in 2D materials can be assessed by their
3.3 Thermoelectric performance
in-plane stiffness and fracture toughness, where the former
To examine the TE performance of penta-InP5 , we com- indicates the material’s ability to resist the load deforma-
pute the chemical potential-dependent properties using both tion, and the latter provides a measure of its fracture resis-
energy- and momentum-dependent relaxation times that tance.
consider full el-ph couplings, and CRTA method. The con- To quantify in-plane stiffness, we calculate the
stant relaxation times are calculated by τ = µ.m∗ /e, with orientation-dependent Young’s modulus E and Poisson’s ra-

10
 tio ν via 59–61 : Specifically, penta-InP5 has an in-plane stiffness of approx-

This is the author’s peer reviewed, accepted manuscript. However, the online version of record will be different from this version once it has been copyedited and typeset.
imately 52 N/m, which indicates its strong resistance to de-
∆C
E(θ ) =   , formation. Additionally, it exhibits a large fracture strain of
C11 m4 +C22 n4 + C∆C
66
− 2C12 m2 n2
about 23%, ensuring structural stability under mechanical
  stress. These mechanical characteristics, along with its high
C11 +C22 − C∆C m2 n2 −C12 (m4 + n4 ) ZT, make penta-InP5 suitable for smart wearable devices.
66
ν(θ ) = −   ,
C11 m4 +C22 n4 + C∆C − 2C12 m2 n2 SUPPLEMENTARY MATERIAL
66

2 , m = cos(θ ), and n = sin(θ ), with

Refer to the supplementary material for details on thermal
where ∆C = C11C22 −C12 stability, electronic properties, and a comparison of TE per-
θ defining the angle between the strain direction and lattice formance calculated using full electron-phonon coupling
orientation. Through polynomial fitting of the strain-energy and the constant relaxation time approximation (CRTA).
curves, we find the elastic constants C11 = C22 = 48.45 N/m,
C12 = 11.22 N/m, and C66 = 22.91 N/m. Satisfaction with ACKNOWLEDGMENTS
Born-Huang criteria 62,63 , where C11 > 0, C66 > 0, and This research is funded by Vietnam National Foundation for
C11C22 > C12 2 , indicating mechanical stability.
Science and Technology Development (NAFOSTED) under
Figures 6(a) and 6(b) illustrate the anisotropy of E and grant number 103.01-2023.06
ν, showing clear deviations from the circular symme-
try. Specifically, E ranges from 45.81 N/m at θ = 0◦ to References
PLEASE CITE THIS ARTICLE AS DOI: 10.1063/5.0251741

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