Thermoelectric Performance of Two-Dimensional AlX (X = S, Se, Te): A First-Principles-Based Transport Study.

By using the first-principles calculations in combination with the Boltzmann transport theory, we systematically study the thermoelectric properties of AlX (X = S, Se, Te) monolayers as indirect gap semiconductors. The unique electronic density of states, which consists of a rather sharp peak at the valence band maxima and an almost constant band at the conduction band minima, makes AlX (X = S, Se, Te) monolayers excellent thermoelectric materials. The optimized power factors at room temperature are 22.59, 62.59, and 6.79 mW m-1 K-2 under reasonable electronic concentration for AlS, AlSe, and AlTe monolayers, respectively. The figure of merit (zT) increases with temperature and the optimized zT values of 0.52, 0.59, and 0.26 at room temperature are achieved under moderate electronic concentration for AlS, AlSe, and AlTe monolayers, respectively, indicating that two-dimensional layered AlX (X = S, Se, Te) semiconductors, especially AlSe, can be potential candidate matrices for high-performance thermoelectric nanocomposites.

Introduction

With the global energy crisis and environmental impact of fossil fuels, a compelling need exists for thermoelectric (TE) materials which can directly convert waste heat into electric power. Generally, the thermally driven electrical performance of TE materials is measured by the power factor PF = S2σ, in which S and σ are, respectively, Seebeck coefficient and electrical conductivity. The conversion efficiency of TE materials is gauged by the dimensionless figure of merit zT = S2σT/κ, where T is the absolute temperature, κ is the sum of electronic (κe) and lattice (κl) thermal conductivity. A high-performance TE material not only requires a high power factor but also has to possess a low thermal conductivity. Unfortunately, it is complex to efficiently achieve waste heat recovery for the interdependency of transport parameters (S, σ, and κe). S and σ unusually interweave and behave in an opposite trend.[1] The improvement of TE devices thus strongly depends on the optimization of electronic and thermal transport properties. A strategy for achieving high TE performance is to identify materials with intrinsically high TE performance, which can be applied as potential matrices for high-performance TE nanocomposites. Since the arrival of graphene,[2] the large number of the two-dimensional (2D) layered materials, such as black phosphorus (BP),[3,4] transition metal dichalcogenides (TMDs),[5−8] and group III metal chalcogenides (GIIIAMCs),[9−13] have been synthesized and characterized. Unlike graphene, these 2D layered materials are semiconductors and can potentially play key roles in the fabrication of next-generation nano-electronic devices. In particular, these 2D semiconductors could alleviate the coupling between S and σ because of the quantum confinement effects, consequently enhancing TE performance.[14−18]

Indeed, TMDs such as MoS2, MoSe2, WS2, and WSe2 exhibit thickness-dependent TE properties[19] because of the thickness-dependent electronic band structure[20,21] and maximum power factor of about 340 and 150 mW m–1 K–2 for n-type monolayer MoSe2 and p-type MoS2 monolayers,[22] respectively, which are much higher than those of 20 and 30 mW m–1 K–2 for n-type bulk MoSe2 and p-type bulk MoS2, respectively. Experimentally, the Seebeck coefficient up to 30 mV K–1 for single-layer MoS2 at room temperature is observed,[23] which is significantly larger than the one observed in bulk MoS2 (∼7 mV K–1),[24] and moreover can be tuned between 0.4 and 100 mV K–1 by an external electric field using a field effect transistor.[25] As for BP, theoretical calculations in different approaches indicate that the BP monolayer shows a strong spatial anisotropy in electrical and thermal conductivities, which makes zT in the armchair direction larger than that in the zigzag direction,[26,27] and using the electric-double-layer transistor configuration experimentally, the Seebeck coefficient of ion-gated BP reached 510 μV K–1 at 210 K in the hole-depleted state,[28] which is much higher than the reported 335 μV K–1 of single crystal BP at 300 K.[29]

Among promising TE 2D layered semiconductors, monolayer layered GIIIAMCs such as InSe, GaS, GaSe, or GaTe have entered the spotlight. These monolayer layered semiconductors have unique electronic structures, having a flat band at the valence band maxima (VBM) and a parabolic band at the conduction band minima (CBM), thus resulting in a rather sharp peak at VBM and an almost constant band at CBM in the density of states (DOS).[30,31] Experimentally, the monolayer or few layers of InSe,[11,32,33] GaS,[12,34,35] GaSe,[13,36−39] and GaTe[10] have been synthesized, and much effort has been dedicated to reveal the electronic and optical properties and potential applications in many fields.[9] It was recently reported that the carrier mobility in few-layer InSe was more than 103 cm2 V–1 s–1 at room temperature,[40] which is much larger than 321 cm2 V–1 s–1 of n-type InSe polycrystalline samples with the PF of about 250 μW m–1 K–2 and zT of 0.42 at 700 K.[41] The thickness-dependent TE properties of GaS, GaSe, InS, InSe monolayers have been theoretically investigated based on a constant relaxation time,[42] and by more precise relaxation time calculation, the InSe monolayer has both a large S and a large σ, giving the maximum PF of 49 (43) mW m–1 K–2 for the p-type (n-type) InSe monolayer at room temperature in the armchair direction.[43] Otherwise, it was theoretically reported that monolayer InSe exhibited unusual phonon behavior and ultralow thermal conductivities.[44] The TE performance of two-dimensional layered semiconductor InSe was further effectively enhanced by reducing film thickness and modulating electron density.[45] The study on the temperature-dependent TE properties of GaS, GaSe, and GaTe monolayers by the Boltzmann transport theory based on constant relaxation time showed the high zT of GaS, GaSe, and GaTe monolayers at low temperature.[46] Thus, it seems that monolayer GIIIAMCs such as InSe, GaS, GaSe, or GaTe are promising matrices for high-efficient TE nanocomposites.

Recently, other group III–VI monolayers such as AlX (X = S, Se, Te) have been proposed to have similar band structures with InSe, GaS, GaSe, or GaTe monolayers and high mechanical stability.[47] We thus expect that AlX (X = S, Se, Te) monolayers could be suitable candidates for applications in electronic and TE devices. However, relevant works about κl and TE properties of AlX (X = S, Se, Te) are lacking till date. In this work, we present a detailed study on TE properties of 2D layered semiconducting AlX (X = S, Se, Te) by using first-principles in combination with the Boltzmann transport theory, which enables systematic analysis and comparison on TE performance for possible III–VI 2D semiconducting configurations.

Results and Discussion

Crystal and Electronic Structures

The structure of AlX (X = S, Se, Te) monolayers is modeled based on the geometry of InSe, GaSe, and GaS monolayers, which have already been synthesized experimentally,[12,37,40] for they all belong to group III binary monolayers. All the AlX (X = S, Se, Te) compounds crystallize in a honeycomb structure belonging to space group D3. Figure presents the top view and side view of the monolayer AlX unit cell which consists of four atoms (two Al and two X atoms). In a monolayer AlX, there exist four sublayers which stack in the order of X–Al–Al–X with X = S, Se, and Te. After fully relaxing the internal coordinates of the atoms, the optimized configurations which have the lowest energy are considered. Table summarizes the calculated structural parameters, together with available theoretical results. The optimized lattice parameters (a) for AlS, AlSe, and AlTe monolayers are respectively 3.46, 3.70, and 4.10 Å, which compare favorably with those reported previously.[47] The calculations show that our approach is reliable for these monolayers, which is the precondition to accurately predict the transport coefficients. In AlX (X = S, Se, Te) monolayers, the distance between Al and X atoms (dAl–X) increases as the atomic radius of the X element increases. The optimized lattice parameters and the distance between X and X atoms (dX–X) exhibit a similar trend as dAl–X. Otherwise, the distance between Al and Al atoms (dAl–Al) shows a downward trend and is independent of the type of X atoms. By fitting the total energy (E) of the system to the lattice dilation of Δl/l0, the two-dimensional elastic modulus (C2D) is listed in Table .

Figure 1

Ball-and-stick model of the AlX (X = S, Se, Te) monolayers from the top view (a) and side view (b).

Table 1

Optimized Lattice Constant (a), Bond Length (dAl–Al, dX–X, dAl–X), Bond Angle (θ), and Band Gap (Eg) of AlX (X = S, Se, Te) Monolayersa

AlX	a (Å)	dAl–Al (Å)	dX–X (Å)	dAl–X (Å)	θ (deg)	Eg (eV)
AlS	3.46 (3.57)	2.59 (2.59)	4.82 (4.73)	2.29 (2.32)	98.22 (100.4)	2.05 (2.10)
AlSe	3.70 (3.78)	2.57 (2.57)	4.95 (4.90)	2.45 (2.47)	98.31 (99.75)	2.01 (1.99)
AlTe	4.10 (4.11)	2.55 (2.58)	5.11 (5.14)	2.69 (2.70)	99.30 (99.30)	1.86 (1.84)

The values in brackets are theoretical results in ref (47).

Table 2

Elastic Constant (C2D), Effective Mass (m*), DP Constant (E1), Carrier Mobility (μ), and Relaxation Time (τ) of AlX (X = S, Se, Te) Monolayers at Room Temperature

AlX	carrier type	C2D (N/m)	m* (me)	E1 (eV)	μ (×103 cm2 V–1 s–1)	τ (fs)
AlS	electron	125.23	1.45	0.85	1.65	1357.37
	hole	125.23	2.68	4.47	0.03	44.04
AlSe	electron	106.03	2.30	1.09	0.46	600.73
	hole	106.03	3.74	2.65	0.02	46.52
AlTe	electron	86.59	2.72	1.39	0.15	239.22
	hole	86.59	4.87	0.80	0.03	96.86

Figure a–c show the band structures and total DOS for AlX monolayers with X = S, Se, Te, indicating that all the monolayers exhibit as indirect band gap semiconductors. The obtained band gap for each system is given in Table , which is in good agreement with the previous theoretical calculation.[47] The minima of the conduction band (CBM) sits at the highly symmetric M point and the maxima of the valence band (VBM) is seated along the Γ–K direction, and as we go down the period, the VBM moves toward higher energy, resulting in the decrease of the band gap from 2.05 for AlS to 1.86 eV for AlTe monolayers. It is clear from band structures that the top of the valence band is quite flat and the bottom of the conduction band shows a parabolic band, and consequently resulting in an unusual combination of a rather sharp peak at VBM and an almost constant band at CBM, which is the character of a good TE property similar to renowned PbTe1–Se bulk TEs.[48]Figure d–f show the atom-resolved local DOS of three monolayers. It is observed that the states below the Fermi energy mainly originate from the X atoms and the states above the Fermi energy are composed by admixing of X and Al atoms. With the purpose of further analyzing the features around the CBM and VBM, the s- and p-orbitals’ projected DOS (PDOS) profiles of all the atoms are shown in Figure g–l. The analysis of the PDOS demonstrates that VBM and CBM for AlS/AlSe/AlTe monolayers are both formed by the S/Se/Te p-orbitals with some admixture from Al p-orbitals. Based on the VBM and CBM, we can evaluate the effective mass (m*) of electron and hole carriers by

Figure 2

Band structures and total DOS (a–c) and projected DOS (d–l) of AlX (X = S, Se, Te) monolayers.

The obtained values of m* for the two carriers in AlX (X = S, Se, Te) monolayers are collected in Table . Expectedly, the effective mass of electrons is smaller than that of holes in all the three monolayers, which is mainly attributed to a steeper CBM (for electrons) compared with a VBM (for holes). Table also summarizes the calculated deformation potential (DP) constant (E1), obtained carrier mobility (μ), and relaxation time (τ) at room temperature. It is obvious to find that the carrier mobility of electrons is larger than that of holes resulting from the unusual DOS feature seen in Figure . The temperature dependence of the relaxation time (τ) is presented in Figure .

Figure 3

Calculated carrier relaxation time for electrons (a) and holes (b) as a function of temperature in AlX (X = S, Se, Te) monolayers.

TE Properties

The carrier concentration (n) dependence of transport coefficients including the Seebeck coefficient (S), electrical conductivity (σ), power factor (PF = S2σ), and electronic thermal conductivity (κe) at 300, 500, and 700 K for AlX (X = S, Se, Te) monolayers are presented in Figure (for n-type systems) and Figure (for p-type systems). As can be seen from Figures and 5a–c, the absolute Seebeck coefficients decrease for n-type and p-type AlS, AlSe, and AlTe monolayers with increasing carrier concentration, but they show an increasing trend when increasing the temperature at constant carrier concentrations, reflecting that S is inversely proportional to n, but is proportional to T. Moreover, the absolute S of n-type AlS and AlSe monolayers is smaller than that of the p-type ones. For example, the absolute values of S for n-type AlS are 322.89 (at 300 K), 367.61 (at 500 K), and 397.79 μV K–1 (at 700 K) at the carrier concentration of 1011 cm–2, while for the p-type systems, they are 652.39 (at 300 K), 704.82 (at 500 K), and 717.92 μV K–1 (at 700 K), which are about twice of those for the n-type ones at the same condition. Also, the absolute S are 489.73 (at 300 K), 554.62 (at 500 K), and 578.94 μV K–1 (at 700 K) for n-type AlSe, which are also smaller than 641.74 (at 300 K), 680.89 (at 500 K), and 700.56 μV K–1 (at 700 K) for p-type AlSe. Otherwise, the absolute S of n-type AlTe monolayer are 350.18, 402.69, and 445.74 μV K–1 at the temperature of 300, 500, 700 K, which are larger than 316.47, 298.24, and 321.46 μV K–1 for the p-type AlTe monolayer at the same carrier concentration and temperature. The consequent electrical conductivity (σ) is shown in Figure d–f (for n-type systems) and Figure d–f (for p-type systems). Contrary to the Seebeck coefficient, the electrical conductivity increases with n in the whole concentration range. The behavior of σ with respect to temperature, however, is completely inverse. All the three n-type systems hold bigger σ rather than p-type systems at the same carrier concentration, originating from the larger carrier mobility of electrons because of σ = neμ. An ideal TE material not only requires to minimize the thermal conductivity but also simultaneously needs a maximized power factor (PF). However, transport coefficients are usually coupled with each other. As can be seen from above discussion, the presence of large effective mass is required for obtaining a high Seebeck coefficient, but electrical conductivity is inversely proportional to the value of m*. One should make a balance between the Seebeck coefficient and electrical conductivity for a high power factor (PF). The calculated PF for n-type and p-type AlS, AlSe, and AlTe monolayers is presented in Figures and 5g–i. The curves first increase with carrier concentration to a maximum and then go down with further increasing in n. For the n-type AlS monolayer, the power factor can be optimized to be as high as 22.59 (300 K), 22.45 (500 K), and 23.39 mW m–1 K–2 (700 K), which are all larger than that for the p-type system. This indicates the superior TE behavior of the n-type system with respect to the p-type system. The largest power factor (62.59 mW m–1 K–2) among the three systems is obtained in the n-type AlSe monolayer at the carrier concentration of 5.5 × 1012 cm–2 at T = 300 K because of the largest electrical conductivity. As discussed above, the largest electrical conductivity is mainly attributed to the highest carrier mobility of electrons in the AlSe monolayer (in Table ).

Figure 4

Seebeck coefficient (S) (a–c), electrical conductivity (σ) (d–f), power factor (PF) (g–i), and electronic thermal conductivity (κe) (j–l) as a function of carrier concentration for n-type AlS, AlSe, and AlTe monolayers at 300, 500, and 700 K.

Figure 5

Seebeck coefficient (S) (a–c), electrical conductivity (σ) (d–f), power factor (PF) (g–i), and electronic thermal conductivity (κe) (j–l) as a function of carrier concentration for p-type AlS, AlSe, and AlTe monolayers at 300, 500, and 700 K.

We then turn to explore the phonon transport properties of AlX (X = S, Se, Te) monolayers, as the thermal transport in semiconductors is mainly contributed by the phonon. Figure a–c shows the dispersion of phonon modes of AlX (X = S, Se, Te) monolayers, no imaginary frequencies in all phonon branches confirm the dynamical stability of the AlX (X = S, Se, Te) monolayers, which are consistent with the previous theoretical work.[47] The calculated phonon dispersions for the three systems are all composed of three acoustic modes and nine optical ones. The three acoustic modes contains two linear modes [longitudinal acoustic (LA) and transverse acoustic (TA)] for in-plane vibrations and a flexural one (ZA) for out-of-plane vibrations. The dispersion of all the three acoustic modes is quite stronger away from the zone center, thus giving higher velocity modes than typical optical phonons and possesses most of the heat. Besides, it is found that the acoustic modes are more dispersive from AlS to AlSe to AlTe monolayers, leading to a higher group velocity and thus an enhanced lattice thermal conductivity.[49−51] The group velocity of ZA, TA, and LA phonons near the Γ point for the AlX (X = S, Se, Te) monolayers is calculated and listed in Table . For comparison, the corresponding results of MoS2, GaSe, and InSe monolayers are also presented.[49,52] As a fact, the ZA branch plays a crucial role in κl of 2D materials.[53,54] For instance, 75% κl is derived from the ZA mode in graphene.[55] In this study, the larger phonon group velocities for the ZA branch in the AlX (X = S, Se, Te) monolayers compared with those of MoS2 monolayer will contribute to the lower κl in the MoS2 monolayer. However, the calculated κl of the MoS2 monolayer is much larger than that of AlX (X = S, Se, Te) monolayers, and thus suggesting lower Debye temperature in the AlX (X = S, Se, Te) monolayer. In Slack’s model,[56] κl has cubic dependence on Debye temperature, and accordingly, a higher Debye temperature corresponds a higher κl. A comparison of κl for three selenide-based monolayers, AlSe, GaSe, and InSe, is illuminating. The atomic mass of Al (27) is much smaller than that of Ga (70) and In (115), and thus, one might expect a higher κl for the AlSe monolayer. Nevertheless, the calculated κl of the AlSe monolayer is the smallest among the three monolayers. To explain the unusually low κl of the AlSe monolayer, we consider the mechanical properties of the three selenide-based monolayers. Young’s modulus (E) of AlSe, GaSe, and InSe monolayers are, respectively, 26.71, 28.62, and 27.96 GPa. Poisson’s ratio (σ) of AlSe, GaSe, and InSe monolayers is listed as 0.24, 0.24, and 0.29.[47] The AlSe monolayer has very small E and σ, indicating lower vibrational strength.[57] From the phonon dispersion of the AlSe, GaSe, and InSe monolayers shown in other literature,[47] the ZA modes are more and more flat from AlSe to GaSe to InSe monolayers. Therefore, we conclude that the AlSe monolayer has the smallest Debye temperature among the three selenide-based monolayers as discussed above. Therefore, a small Debye temperature together with low Poisson’s ratio always means a weak interatomic bonding, which will decrease the κl of the AlSe monolayer.[58] As a result, the AlSe monolayer has a relative κl compared to GaSe and InSe monolayers. The calculated lattice thermal conductivity (κl) of the AlX (X = S, Se, Te) monolayers as a function of temperature in the range from 300 to 700 K is shown in Figure d–f. One can clearly see that the lattice thermal conductivity of all the three AlS, AlSe, and AlTe monolayers decreases as the temperature increases. The values of κl for AlS, AlSe, and AlTe monolayers are respectively 3.00, 4.21, and 4.45 W m–1 K–1 at room temperature. When taking the thickness into consideration, the thermal sheet conductances of two-dimensional AlS, AlSe, and AlTe are, respectively, 2.39, 3.61, and 3.86 nW K–1 (300 K), which are comparable to those of high zT bulk materials such as PbTe[48] but much smaller than those of Mo- and W-based dichalcogenides[59−63] and GIIIAMCs such as InSe[64] at the same temperature. The low thermal conductivity of AlS, AlSe, and AlTe monolayers suggests that these three monolayers could have favorable TE performance.

Figure 6

Phonon dispersion relations (a–c) and lattice thermal conductivity (κl) as a function of temperature (d–f) for AlX (X = S, Se, Te) monolayers.

Table 3

Calculated Group Velocities (vg) (km/s) of ZA, TA, and LA Phonons Near the Γ Point for the AlX (X = S, Se, Te) Monolayers with MoS2, GaSe, and InSe Monolayers for Comparison

monolayers	ZA	TA	LA
AlS	1.726	1.745	2.967
AlSe	2.092	2.518	4.525
AlTe	3.036	3.811	4.882
GaSe		2.487	4.162
InSe		1.853	3.272
MoS2	1.40	3.96	6.47

Based on the obtained TE transport parameters (S, σ, and κ) of AlX (X = S, Se, Te) monolayers, the TE performance can be evaluated. Figure plots the zT values as a function of carrier concentration and temperature for n-type and p-type monolayers. For p-type AlS, the optimal value of zT at room temperature is 0.33 at the carrier concentration of 1.9 × 1013 cm–2. At a high temperature of 700 K, a maximal zT of 0.62 appears at the carrier concentration of 2.4 × 1013 cm–2. The largest zT can be further improved to 0.52 (at T = 300 K and n = 1.5 × 1011 cm–2) and 0.67 (at T = 700 K and n = 1.1 × 1011 cm–2) for n-type AlS. As such, the n-type AlS monolayer exhibits a more favorable TE performance rather than the n-type system. Similar to the AlS case, the n-type AlSe monolayer has the largest zT of 0.59, 0.69, and 0.74 at the temperature of 300, 500, and 700 K because of the extremely large power factor. Notably, even at a lower temperature of 300 K, zT can also compete with that at higher temperatures. Thus, the n-type AlSe monolayer can exhibit promising TE performance even at relatively low temperatures. Similar to AlS and AlSe monolayers, n-type AlTe can be used as a promising TE material. The optimal zT at 300 and 700 K for the n-type AlTe monolayer are, respectively, 0.26 and 0.73 at the carrier concentration of 5.4 × 1011 and 2.7 × 1012 cm–2. Including the results of AlS, it can be concluded that the n-type AlSe monolayer exhibits the largest TE figure of merit among the studied AlX (X = S, Se, Te) monolayers because of its intrinsic band structure. Although the DP theory has been widely used to calculate the relaxation time for two-dimensional systems, it still may be unstable for the DP model which fully ignores electron–optical–phonon interaction, polar scattering, and other scattering mechanisms. In order to make a comparison to the previous work easier, the zT values of AlX (X = S, Se, Te) monolayers at 300 K with a relaxation time range centered around the calculated ones from the DP model are plotted in Figure . The solid black lines stand for zT using the relaxation time obtained from the DP model. The zT values obtained on the basis of estimated relaxation time from the DP model listed in Table are nearly equal to the largest values for both n- and p-type AlX (X = S, Se, Te) monolayers. Assuming the same relaxation time, our obtained values of zT are even larger than those of the previously promising TE material such as the Pd2Se3 monolayer.[65] Therefore, our study indicates the high performance of AlX (X = S, Se, Te) monolayers, especially the AlSe monolayer when used as potential candidate matrices for TE nanocomposites.

Figure 7

Figure of merit (zT) as a function of carrier concentration for n-type and p-type AlS (a–b), AlSe (c–d), and AlTe (e–f) monolayers at 300, 500, and 700 K.

Figure 8

Figure of merit (zT) values for n-type and p-type AlS (a–b), AlSe (c–d), and AlTe (e–f) monolayers at 300 K with a relaxation time range centered around the calculated ones from the DP model.

Computational Method and Process

In the present work, the first-principles calculations on AlX (X = S, Se, Te) monolayers are conducted within the VASP code[66−68] using the projector-augmented wave method.[69] The combination of Perdew–Burke–Ernzerhof functional scheme and generalized gradient approximation[70] describes the electronic exchange–correlation potential. After a strict convergence test, cutoff energy of the plane wave is chosen as 600 eV and the Monkhorst–Pack uniform k-point sampling[71] is selected as 24 × 24 × 1 in the whole Brillouin zone. The energy and force convergence criteria are, respectively, taken to be 10–6 eV and 0.01 eV/Å. By conducting the energy minimization calculations on the basis of the conjugate gradient method, we determine the optimized configuration of the systems. A vacuum slab of 20 Å is enough to eliminate the interactions between the adjacent AlX (X = S, Se, Te) monolayers.

Based on the calculated accurate electronic structures and carrier relaxation time, the electronic transport properties are then calculated from the semiclassical Boltzmann theory.[72] The obtained DFT results are used as an input of BoltzTraP to estimate TE performance of AlX (X = S, Se, Te) monolayers. In this approach, the temperature (T) and chemical potential (μ) dependence of the Seebeck coefficient (Sαβ), electrical conductivity (σαβ), and electronic thermal conductivity (καβe) are expressed as[73,74]in which subscripts α and β represent the two axis directions in the momentum space (or according to real space). Here, Ω is the volume of the unit cell and fμ stands for the Fermi–Dirac distribution function. The σαβ(ε) as a function of ε is in the form of

In this formula, i and donate the band index and wave vector, respectively. Parameters N, e, and τ express the number of k-points samples, electron charge, and carrier relaxation time, respectively. The carrier group velocity vα(vβ) along the α(β) direction can be calculated from

The electrical conductivity (σ) and electronic thermal conductivity (κe) are proportional to the relaxation time (τ). Therefore, the accurate treatment of τ is of great importance. Many earlier theoretical calculations on TE properties were performed based on a constant relaxation time[42,75,76] and the value is generally overestimated.[27,43,77] In this study, the relaxation time is evaluated by adopting the DP theory,[78] which is widely used to calculate the relaxation time for two-dimensional systems.[26,79−83] Accordingly, the relaxation time is calculated byin which the effective mass (m*) is obtained from the accurate band structure. For a 2D crystal, the carrier mobility (μ) can be expressed as

In the formula, the two-dimensional elastic modulus (C2D) is calculated fromwhere l0 is the optimized lattice parameter of the unit cell, Δl = l – l0 is the lattice variation when the compressed and expanded systems are compared with the optimized system, and E is the corresponding energy of the compressed and expanded systems. md is the average density-of-states effective mass dominated by . E1 is the DP constant defined as

The anisotropic in-plane lattice thermal conductivity under the relaxation time approximation can be calculated as the sum of the contributions of all phonon model λ with different wave vectors and branch indexeswhere V is the crystal volume, Cλ is the specific heat per mode, vλα and τλ are the velocity components along the α direction and the relaxation time of the phonon mode λ. The lattice thermal conductivity (κl) can be obtained by solving the phonon Boltzmann transformation related to the harmonic and anharmonic interatomic force constants (IFCs) as performed by the ShengBTE code.[84−86] The inputs for the ShengBTE are the second- and third-order IFCs. For this, a 4 × 4 × 4 supercell with 5 × 5 × 1 k-point sampling is used to calculate the second-order IFCs and phonon frequencies by the Phonopy package.[87] The third-order IFCs are obtained by using the thirdorder.py module using a 4 × 4 × 4 supercell and Γ-point only calculations. Note that we use eq to calculate the electronic thermal conductivity (κe) because the Wiedemann–Franz law is only suitable for the system where the scattering of electrons in the material is dominated by elastic collision.[88] Here, the effective van der Waals thickness of AlX (X = S, Se, Te) monolayers is 7.97, 8.57, and 8.68 Å, which is defined as the summation of the buckling distance and two van der Waals radii of the outermost surface atoms of structures.[89]

Conclusions

To summarize, we have presented the electronic, phonon, and TE properties of AlX (X = S, Se, Te) monolayers by the density functional theory in combination with the Boltzmann transport theory. All the AlX (X = S, Se, Te) monolayers are indirect band gap semiconductors with unique electronic structures, having a flat band at VBM and a parabolic band at CBM, and consequently, a rather sharp peak at VBM and an almost constant band at CBM in DOS. No imaginary frequencies in all phonon branches verify the dynamical stability of the AlX (X = S, Se, Te) monolayers. A detailed study of TE properties as a function of carrier concentration and temperature is carried out. The largest power factors are around 22.59, 62.59, and 6.79 mW m–1 K–2 at room temperature and 23.40, 54.36, 10.42 mW m–1 K–2 at 700 K under reasonable electronic concentration for AlS, AlSe, and AlTe monolayers, respectively. The figure of merit (zT) increases with temperature, and the optimized zT values reach 0.52, 0.59, and 0.26 at room temperature and 0.67, 0.74, and 0.73 at 700 K under reasonable electronic concentration for AlS, AlSe, and AlTe monolayers, respectively, thus suggesting that the n-type AlX (X = S, Se, Te) monolayers, especially the AlSe monolayer can be used as potential candidate matrices for high-performance TE nanocomposites.