We cleave the initial structures of the 2D In–VA materials from the (110) face of the ZB semiconductor In–VA compounds. The atomic structures and lattice constants of the cleaved 2D structures then are both fully optimized. Figure 1 shows 2D planar and buckled honeycomb structures of In–VA. The array of atoms of PL InN resembles the honeycomb hexagonal structure of graphene, forming a planar layer with an In atom and an N atom bonded in a single plane and their π–π bonding is quite strong. In their stable states, other InX (X = P, As, Sb, Bi) form buckled hexagonal structures, since they tend to be more energetic for sp³ hybridization. The group-III elements In can form trigonal sp² bonded molecular structure of the plane with D_3h symmetry, while the most common configuration for the group-V elements is of trigonal pyramid.^[23] In general, the nature of InN is sp² hybridized, however, the heavier group-V elements P, As, Sb, and Bi elements exhibit the sp²–sp³ hybridization due to their preference for sp³ bonding.

Fig. 1.

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	Fig. 1. (color online) Top and side views for 2D (a) planar InN and (b) buckled honeycomb InX (X = P, As, Sb, Bi) structures. In the planar structure, atoms are located on the same plane. In the buckled structure, the alternating atoms are located in two different parallel planes. The buckling Δ is the distance between these two planes. θ refers to the value of angle between neighboring bonds.

Figure 1(a) illustrates this PL hexagonal structure of InN with space group PM2, and figure 1(b) shows the low-buckled (LB) hexagonal nature of InX with space group P3M1. Each group-V atom is covalently bonded with one In atom, forming a simple hexagonal unit cell. For LB structure, atoms of In and VA lie in different planes having a buckling distance, Δ, as shown in Fig. 1(b). The obtained equilibrium lattice constant a₀ and buckling height Δ are listed in Table 1. The calculated values of equilibrium lattice constant a₀ in LB phase of InP, InAs, InSb, and InBi are 4.25 Å, 4.38 Å, 4.68 Å, and 4.80 Å for the pristine buckled honeycombs, which are all larger than 3.57 Å of InN. In buckled geometries, as the position of In atom rises, the value of angle between neighboring bonds VA–In–VA decreases. These structural properties are in good agreement with those of previous theoretical work.^[23–35] To date, there have been no theoretical or experimental results about the structural or electrical transitions of 2D In–VA in our investigated temperature range. Therefore, the obtained structures here are used for calculating the TE properties below.

Table 1.

Table 1.

Table 1. Calculated structural and electrical properties of 2D hexagonal In–VA. Stable structures are identified as planar and buckled geometries, respectively, showing the values of lattice constant (a), buckling parameter (Δ), angle between neighboring bonds (θ), energy gap (Eg), calculated within PBE. The electron effective mass, , θ, and Δ are described in Fig. 1. . a/Å Δ/Å θ/(°) Eg/eV InN 3.57 0 120 0.54 0.46 3.63a 0.43a 3.57b InP 4.25 0.54 115.51 0.44 1.16 4.25a 0.52a 0.37a 4.17b 0.51b 115.8b InAs 4.38 0.67 113.60 0.41 0.85 4.38a 0.67a 0.32a 4.28b 0.62b 114.1b InSb 4.68 0.77 112.82 0.33 0.79 4.68a 0.76a 0.28a 4.57b 0.73b 113.2b InBi 4.80 0.85 111.76 0.25 0.17 4.81c 0.85c a Ref. [23] b Ref. [24] c Ref. [35].

Table 1. Calculated structural and electrical properties of 2D hexagonal In–VA. Stable structures are identified as planar and buckled geometries, respectively, showing the values of lattice constant (a), buckling parameter (Δ), angle between neighboring bonds (θ), energy gap (Eg), calculated within PBE. The electron effective mass, , θ, and Δ are described in Fig. 1. .

The calculated band structures of single-layer In–VA and their corresponding total densities of states (DOSs) are plotted in Figs. 2 and 3, respectively. At the PBE level, the bottom of the conduction band is located at Γ point. Single-layered InN and InP are predicted to be indirect-band-gap semiconductors with band gaps of 0.46 eV and 1.16 eV, respectively. The InAs, InSb, and InBi are direct-band-gap semiconductors with band gaps of 0.85 eV, 0.79 eV, and 0.17 eV, respectively, following a decreasing trend. For InX with buckling height, the energy band gap shrinks as atomic number increases.

Fig. 2.

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	Fig. 2. (color online) Energy bands of monolayer honeycomb structures of In–VA.

Fig. 3.

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	Fig. 3. (color online) Total DOSs of In–VA around the Fermi level.

Figure 3 shows the total DOSs for all five In–VA compounds near the Fermi level, which are mainly associated with TE properties.^[36] Due to the total DOS increasing more rapidly near the VBM than near the CBM, thermal power factors for p-type cases will be much larger than for n-type cases. Dramatic DOS variation near the VBM is favorable for high Seebeck coefficients.^[37] Sharp peaks in the electronic DOS near the Fermi energy can enhance the TE performances in monolayer systems, which could partially be attributed to the improving electrical conductivity and imposing asymmetry between holes and electrons transport.^[38,39]

The dynamical stability of the hexagonal 2D In–VA is determined by the phonon spectra shown in Fig. 4. None of the five In–VA materials shows imaginary frequencies, demonstrating that all the structures are dynamically stable. The lattice thermal conductivity is mainly governed by anharmonic phonon–phonon scattering.^[40] It is one optical branch of InN that has low frequencies and overlaps with the acoustic branches, which is different from LB structural InX, and can be seen as an indication of soft mode and strong anharmonicity.^[41] Consequently, low thermal conductivities are obtained. The optical branches of the LB InX structures move gradually to lower frequencies, as the atomic number of X increases. The main reason for the production of different flexural phonon modes is anticipated to be due to the buckling structures in InP, InAs, InSb, and InBi, which leads to strong coupling between out-of-plane modes and in-plane modes. Besides, the strengths of In–X (X = P, As, Sb, Bi) bonds are much weaker than those of the In–N bonds, leading to a significantly lower group velocity of the ZA mode in InX.

Fig. 4.

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	Fig. 4. (color online) Calculated vibration frequencies of phonon modes versus k of monolayer In–VA.

According to the calculated electronic structures, we are able to evaluate the electronic transport coefficients by using the Boltzmann transport equation within constant relaxation time approximation. The Seebeck coefficient S can be calculated to be independent of the relaxation time τ, while the electrical conductivity σ is divided by τ, so what we obtain is σ/τ. The relaxation time is usually obtained by fitting to the experimental data.^[42] Since there are no clear experimental values against which these compounds can be measured, we assume τ = 2 × 10⁻¹⁴ s in the present calculations, which is reasonable.^[43] The Seebeck coefficient S and electrical conductivity σ are given by^{[44, 45]}

The Seebeck coefficients of the single-layer In–VA as a function of chemical potential (μ) at different temperatures are plotted in Fig. 5(a). The positive and negative μ correspond to n-type and p-type doping of the system, respectively. As expected, under n-type and p-type doping, S, σ/τ, and PF/τ are all enhanced. In Fig. 5(a), a sharp change with two opposite peaks of Seebeck coefficients around the Fermi level occurs, and there is a downward trend in the value of S with the increase of the temperature T, indicating that an optimal carrier concentration is favorable for improving TE performance. The peak values of Seebeck coefficient at room temperature are 1015 μV/K for the InN single-layer, 1786 μV/K, 1396 μV/K, 1272 μV/K, and 289 μV/K, for the LB InX single-layer, respectively, which are all for the p-type doping. It is seen that S plots of In–VA are asymmetric for p-type and n-type doping, and the former is more preferable, which may be due to the asymmetry of valence and conduction bands of In–VA.

Fig. 5.

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	Fig. 5. (color online) Calculated TE properties: (a) Seebeck coefficients, (b) electrical conductivities, (c) power factors, and (d) electrical thermal conductivities relative to relaxation time, of 2D InX as a function of chemical potential at 300 K (black line), 600 K (red line), and 900 K (blue line), respectively.

In Fig. 5(b), σ/τ increases gradually with the changing of chemical potential near Fermi level. When the doping level is relatively small, with band gap increasing, σ/τ generally decreases in similar manners and close values, which is in contrast to the tendency of the Seebeck coefficient. The highest value of σ/τ is attained for InBi in the vicinity of fermi level. This behavior can be attributed to a small effective mass associated with InBi. Unlike the Seebeck coefficients, electrical conductivities are less sensitive to temperature. For the p-type doped In–VA monolayer, the electrical conductivities are significantly increased with the doping level, which are larger than those of the n-type ones.

In Fig. 5(c), we plot the power factors of the In–VA monolayer under the conditions of different chemical potentials and temperatures. In the case of chemical potential, the maximum PF generally increases with band gap decreasing, which could be attributed to the increased electrical conductivity exceeding the negative effect of the Seebeck coefficients. The PF is larger for p-type doping level than for n-type doping level for both PL and LB monolayer systems and will increase with doping level increasing. At the same time, the high DOS near Fermi energy provides large enough electrical conductivity. Consequently, the PF (S²σ) of the single-layered PL InN shown in Fig. 5(c) is larger than that of LB InX materials.

The electronic thermal conductivity is addressed in Fig. 5(d). As expected, these values increase with the increase of doping level and the temperature, showing a similar trend to that of electrical conductivity σ. The effect of temperature on electronic thermal conductivity is more apparent than on electrical conductivity.

Next, we will investigate the doping effect on the thermal transport property of the In–VA monolayer. The intrinsic lattice thermal conductivities of the single-layered In–VA at different temperatures are estimated by using the ShengBTE^[35] code based on the phonon Boltzmann transport equation, and the results are shown in Fig. 6. For both cases, κ_l decreases with the reciprocal of temperature (T⁻¹) increasing. This implies that the dominant scattering mechanism is the Umklapp process.^[38] The lattice thermal conductivity of PL InN is evidently larger than that of the LB structure. From InP to InBi, with the increasing of buckling height and decreasing of phonon scattering, the lattice thermal conductivities diminish.

Fig. 6.

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	Fig. 6. (color online) Lattice thermal conductivities of 2D In–VA as a function of temperature, from 300 K to 900 K.

With all the transport coefficients available, ZT values of the single-layered In–VA as a function of chemical potential at different temperatures are obtained as shown in Fig. 7. At 900 K, the ZT values reach the maximum values for In–VA whether it is PL or LB structure, at p-type doping levels. The maximal ZT values of 0.779, 0.583, 0.696, 0.727, and 0.373 for InN, InP, InAs, InSb, and InBi are observed at μ = −0.306 eV, μ = −0.449 eV, μ = −0.281 eV, μ = −0.231 eV, and μ = −0.232 eV, respectively. Attributed to the p-type, the doped power factors are greater than the n-type doped ones, and p-type doped ZT values of In–VA are higher than those of n-type doping levels. Thus the properly p-type doped In–VA should be potential TE materials working at medium-high temperature.

Fig. 7.

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	Fig. 7. (color online) ZT values as a function of the carrier concentration of the In–VA monolayer at 300 K (black line), 600 K (red line), and 900 K (blue line), respectively.

In order to investigate TE properties and their temperature relationships of In–VA more intuitively, we plot Seebeck coefficients, electrical conductivities, PF trend, ZT values versus temperature at a carrier density of 4.8 × 10²⁰ cm⁻³ as shown in Fig. 8.

Fig. 8.

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	Fig. 8. (color online) Calculated TE properties: (a) Seebeck coefficients, (b) electrical conductivities, (c) power factors relative to relaxation time, and (d) ZT, of 2D InN (black lines), InP (red lines), InAs (blue lines), InSb (pink lines), and InBi (green lines) as a function of temperature, respectively.

Figure 8(a) illustrates the tendencies of the Seebeck coefficients, which increase with temperature rising. Our measurements of Seebeck coefficients consistently indicate that appropriate doping to In–VA induces electron carriers, beneficial to the improvement of the TE performance. There is a slightly downward trend in the value of σ/τ with the increase of temperature which is opposite to the trend of the Seebeck coefficient. Increased electron carrier concentration due to p-type doping is responsible for high electrical conductivity in In–VA. At the same doping level and same temperature, InN which possesses the maximal electron effective mass exhibits the biggest Seebeck coefficient and the smallest electrical conductivities in these five materials. Figure 8(c) shows the PF with the constant relaxation time, which is used to calculate the ZT value. In relatively highly p-type doped samples, we observe that PF increases with temperature rising up to 900 K.

The ZT values as a function of temperature are shown in Fig. 8(d). The maximum ZT values of In–VA are 0.779, 0.504, 0.554, 0.494, and 0.308 at 900 K with carrier densities being all 4.8 × 10²⁰ cm⁻³, respectively. For the p-type doping, the maximal ZT value of InN is larger than that of LB structural InX, which is mainly attributed to the prominently maximal S (about 283 μV/K at 900 K) in this case. It is observed that the maximal ZT value of In–VA at 900 K is much larger than that at 300 K with the corresponding doping concentration, which is caused due to the increase of the electrical conductivity and the decrease of the lattice thermal conductivity with the increase of temperature. It manifests that 2D In–VA is indeed a medium-high temperature TE material.