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Monolayer Ge2X4S2 (X = P, As) are novel two-dimensional (2D) layered materials with suitable optical absorption properties in the visible range and high carrier mobility, so they possess broad application prospects in the photoelectric and thermoelectric fields. In this work, their thermoelectric properties are systematicly evaluated by using the first-principles and Boltzmann transport theory. For monolayer Ge2As4S2 and Ge2P4S2, their smaller phonon group velocities, low relaxation times and the large Grüneisen parameters result in ultra-low lattice thermal conductivities, which are 3.93 W·m–1·K–1 and 3.19 W·m–1·K–1 in the armchair direction, 4.38 W·m–1·K–1 and 3.79 W·m–1·K–1 in the zigzag directions at 300 K. Their electronic band structures reveal that the monolayer Ge2As4S2 is a semiconductor with a direct band gap of 1.21 eV, while the single-layer Ge2P4S2 owns an indirect band gap of 1.13 eV. Meanwhile, the twofold degeneracy of valence band provides a large p-type Seebeck coefficient that is 1800 μV·K–1 for Ge2P4S2 and 2070 μV·K–1 for Ge2As4S2 in the armchair direction. Obviously, monolayer Ge2X4S2 has smaller lattice thermal conductivity and higher power factor, thus it is worth exploring their thermoelectric properties. The results prove that monolayer Ge2As4S2 and Ge2P4S2 have outstanding thermoelectric performances at 500 K when they are treated by optimal n-type doping. The maximum ZT values of monolayer Ge2As4S2 and Ge2P4S2 are 3.06 (armchair direction) and 3.51 (zigzag direction), as well as 3.21 (armchair direction) and 2.54 (zigzag direction), indicating that monolayer Ge2X4S2 can be a potential candidate in the medium-temperature thermoelectric applications.Disclaimer: The English version of this article is automatically generated by iFLYTEK Translation and only for reference. We therefore are not responsible for its reasonableness, correctness and completeness, and will not bear any commercial and legal responsibilities for the relevant consequences arising from the English translation.
1. Introduction
To achieve the ambitious goal of "double carbon", the key lies in vigorously promoting clean energy and improving energy efficiency.In the industrial field, the low-quality waste heat in the medium temperature zone (60-225 ℃) accounts for about 15% of the total energy consumption of the whole society. A large number of industrial waste heat resources are not effectively utilized and directly discharged into the environment with the medium, resulting in serious environmental pollution and energy waste.Thermoelectric materials are environmentally friendly new energy materials that can directly convert heat energy into electric energy. Their thermoelectric power generation effect has great advantages and application prospects in the recovery and utilization of low-quality industrial waste heat. Their thermoelectric conversion efficiency can be expressed by the dimensionless thermoelectric figure of merit ZT to describe [1,2]:
ZT=S2σTκl+κe, (1) In the formula, molecule S 2 σ is the power factor (including the Seebeck coefficient S and conductivity σ); T is the absolute temperature; denominator is the lattice thermal conductivity κ l and electronic thermal conductivity κ Sum of e.In general, standards for the commercial application of thermoelectric materials require their ZT has a value higher than 1 and has a high Thermoelectric materials with a ZT value usually have either a high power factor or a low lattice thermal conductivity, or both [3-5].The conversion efficiency of traditional thermoelectric materials in the intermediate temperature region is not high enough, which seriously restricts their large-scale application. Therefore, it is of great scientific significance to find materials with excellent thermoelectric properties in the intermediate temperature region.In recent years, due to the development of nanotechnology, new thermoelectric materials have been discovered, among which layered materials have attracted much attention because of their special crystal structure and transport properties.
In recent years, layered materials have attracted much attention in the field of thermoelectricity due to their unique crystal structure and excellent electronic properties. [6-11].It is found that the reduced dimension not only enhances the interface scattering, but also reduces the lattice thermal conductivity. [12,13]. At the same time, it will narrow the valence band (conduction band), increase the carrier effective mass and Seebeck coefficient, and then improve the thermoelectric performance. [14-16].For example, the bulk phase Bi-based ternary compound Bi 2O 2S [17] and Bi 2O 2Se [18,19], they have very small lattice thermal conductivity, but the two-dimensional monolayer structure has smaller lattice thermal conductivity and better thermoelectric properties.Huang et al [20] predicted monolayer Mg 3Sb 2 exhibits a small lattice thermal conductivity and a large ZT value (> 2 at 600 K), its thermoelectric properties are also much higher than those of the corresponding bulk structure.Therefore, the introduction of low-dimensional layered materials provides a new way to control the thermoelectric properties, and greatly stimulates the exploration of other two-dimensional layered thermoelectric materials.
Recently, Zhu et al. [21] Theoretically Predicted Phosphorene Derivatives — Monolayer Ge 2 X 4S 2 ( X = p, as), with similar Gep The layered hexagonal crystal structure of 3 [22,23].Currently known similar structures, such as single-layer SnP 3 [24], InP 3 [25], SbP 3 and GaP 3 [26] all show low lattice thermal conductivity and excellent thermoelectric properties, which greatly improves the study of monolayer Ge 2 X 4S 2 Interest in thermoelectric properties.Based on the first-principles and Boltzmann transport theory, the phonon and electronic transport properties of these two materials are systematically studied.The results show that single, layer Ge 2P 4S 2 and Ge 2As 4S 2 has a lower lattice thermal conductivity along the armchair direction at 300 K, which is 3.93 W·m –1·K –1 and 3.19 W · m –1·K –1. In addition, the double degeneracy of the valence band makes both materials have large Seebeck coefficients.Monolayer Ge 2 X 4S 2 exhibits excellent thermoelectric properties based on its weak phonon transport ability and good electronic transport properties.Among them, the monolayer n-type doped Ge 2P 4S 2 Max at 500 K The value of ZT reaches 3.51 (zigzag direction), and the excellent thermoelectric performance in the medium temperature region indicates that it has a good application prospect in industrial waste heat power generation.The theoretical and computational studies in this paper will also stimulate the interest of experimental scientists in the preparation, characterization and control of thermoelectric properties of these materials.
2. Calculation method
Based on first-principles and density functional theory, combined with Vienna Ab-initio Simulation Package (VASP) software [27] calculated the monolayer Ge 2 X 4S 2 ( X = P, As).In order to avoid the coupling between the layers, along the A vacuum layer with a thickness of 30 Å was set in the Z direction, and by the vdW-DF2 method [28] Corrected van der Waals interaction.In the calculation, the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional in Generalized Gradient Approximation (GGA) is used [29,30].Plane-wave cutoff energy of 500 eV and 11 × 11 × 1 at the time of structural relaxation were used K-point mesh with convergence criteria for total energy and each interatomic force set to 10, respectively –7 eV·Å –1 and 0.01 eV · Å –1.
Monolayer Ge is calculated by solving the Boltzmann transport equation 2 X 4S 2 ( Phonon transport properties of X = P, As).Where in a 3 × 3 × 1 supercell a 3 × 3 × 1 K point grid, using VASP combined with Phonopy software [31] to obtain the second order interatomic force constant matrix (2 nd IFCs), and the phonon dispersion curve is calculated.Finite-difference based method, using the thirdorder. Py program [32] to obtain the third-order force constant (3 rd IFCs). In this paper, the interaction of the seventh nearest neighbor atoms is considered.Will 2 nd and 3 rd IFCs as input, the lattice thermal conductivity of the material was solved and obtained by the ShengBTE software package [33]. In order to obtain more accurate lattice thermal conductivity, a 50 × 50 × 1 compact K-point grid.
In this paper, the electronic transport properties are also studied by Boltzmann transport theory and rigid band approximation.The electronic transport properties represented by Seebeck coefficient and conductivity are in the BoltzTrap software package Implemented in [34], the calculation here selects a 45 × 45 × 1 K point grid.The electronic band structure is based on the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional [35] Calculated to obtain.In addition, based on the deformation potential theory The carrier mobility and relaxation time of these two materials are also calculated by using the relaxation time approximation method. This method has little change under the influence of energy scale, and has accurately verified the thermoelectric properties of many materials. [38,39].
3. Results and Discussion
3.1 Crystal structure and phonon dispersion
Fig. 1(a) and Fig. 1(b) presents the monolayer Ge 2 X 4S The geometry of 2 (space group is
Pˉ3m1 , No.164) and the corresponding Brillouin zone.Monolayer Ge can be seen from Fig. 2P 4S 2 and Ge 2As 4S 2 all exhibit InP similar to monolayer The wrinkled structure of 3, which can increase phonon scattering and reduce lattice thermal conductivity.Each primitive cell of these two materials consists of two Ge atoms, four X atoms and two S atoms, where each Ge atom is simultaneously associated with two X atom linked to an S atom.Structural parameters after full relaxation such as 表1, which is similar to Zhu et al. The theoretical values of [21] are in good agreement with each other. In order to understand the bonding characteristics of the two materials, the electron localization functions (ELFs) were calculated.Such as As shown in Fig. 1(c), when ELF = 0, there is no electron, ELF = 0.5 corresponds to a free electron gas, and ELF = 1 indicates complete localization. It can be found that the ELF value of all atomic bond regions is 0.5, especially the ELF value of P — P (As — As) bond is above 0.75, indicating that they are composed of strong covalent bonds, which greatly improves the stability of the crystal [40-42].In order to calculate Ge more accurately 2 X 4S Thermal and electrical conductivities of 2 and compared with other monolayer materials, applied literature [ The method of 24] calculated the monolayer Ge 2 X 4S Effective thickness of 2.Such as Shown in 图(1), Ge 2 X 4S 2 The atoms in the uppermost and lowermost layers are Ge atoms, and the vertical distance between the two layers of Ge atoms plus 2 times the average van der Waals radius of Ge atoms is defined as a single layer of Ge. 2 X 4S Effective thickness of 2.The average van der Waals radius of Ge atoms is 2.32 Å [43], the monolayer Ge can be calculated 2P 4S 2 and Ge 2As 4S The effective thickness of 2 is 6.57 Å and 6.49 Å.
Figure 1. (a) The structure of monolayer Ge2X4S2; (b) schematic diagram of the Brillouin zone; (c) electron localization function, where the purple, red and blue balls are Ge atoms, P (As) atoms and S atoms, respectively.Table 1. The parameters of optimal structure.晶格常数/Å 键长/Å Ge—S Ge—As (P) As—As (P—P) Ge2As4S2 7.10 2.48 2.60 2.48 Ge2P4S2 6.77 2.50 2.49 2.21 | Show Table
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Fig. 2(a) and Fig. 2(b) is monolayer Ge 2 X 4S Phonon dispersion curve and phonon density of States (PhDOS) of 2. It can be clearly seen that the phonon spectra of these two materials have no imaginary frequency, which indicates that their structures are dynamically stable.Monolayer Ge is shown by using ab initio molecular dynamic (AIMD) simulations 2 X 4S 2 can remain thermodynamically stable at 800 K [21].Fig. 2(a) and It can also be seen in Fig. 2(b) that the phonon dispersion curve consists of 3 acoustic branches (including longitudinal mode LA, transverse mode TA and out-of-plane mode ZA) and 21 optical branches.As the mass of As atoms is larger than that of P atoms, the monolayer Ge 2As 4S The frequencies of the phonon dispersion curves of 2 are all lower than those of monolayer Ge 2P 4S 2, which means that the monolayer Ge 2As 4S 2 has a smaller phonon group velocity.Further analysis shows that the monolayer Ge 2As 4S 2 Acoustic Branch and Lowest Optical Branch The M points are degenerate together, which means strong optical-acoustic coupling and small lattice thermal conductivity [44-47].Fig. 2(c) is monolayer Ge 2 X 4S Vibration modes of ZA, TA, LA modes and the lowest optical branch of 2.From It can be seen from Fig. 2(c) that all atoms of ZA mode move in the out-of-plane direction at low frequency, and all atoms of TA mode and LA mode move in the same direction in the plane.While in the lowest optical mode OP T In 1, Ge and S atoms move in opposite directions in the middle layer, and P and As atoms move disorderly in the plane. This mixed motion of multiple atoms will greatly suppress the transmission of phonons and reduce the lattice thermal conductivity.
Figure 2. Phonon Spectrum and density of state for monolayer Ge2P4S2 (a) and Ge2As4S2 (b), the corresponding vibration modes of acoustic phonon branches (ZA, TA, and LA) and the lowest optical breach (Opt1) near and at Г point.3.2 Phonon transport properties.
The lattice thermal conductivity of a material is the sum of the contributions of all phonon modes [48,49], is one of the important factors affecting its thermoelectric performance, which can be obtained by solving the Boltzmann transport equation [50]
κl=1V∑λCλvλτλ, (2) Among C is heat capacity; ν is the phonon group velocity; τ is the relaxation time; λ is the phonon mode.Based on the Boltzmann transport equation and the ShengBTE software package, the monolayer Ge is calculated. 2 X 4S Plot of lattice thermal conductivity as a function of temperature for 2, as Shown in Fig. 3(a).In the temperature range of 300 to 800 K, there is an obvious inverse relationship between the lattice thermal conductivity and temperature, and its value decreases with the increase of temperature.This phenomenon is mainly due to the enhancement of phonon scattering with increasing temperature, which has been demonstrated in many materials [51,52].Monolayer Ge at 300 K 2P 4S The lattice thermal conductivities of 2 along the armchair and zigzag directions are 3.93 W · m –1·K –1 and 4.38 W·m –1·K –1, while monolayer Ge 2As 4S The corresponding values for 2 are 3.19 W · m –1·K –1 and 3.79 W·m –1·K –1, which is much lower than some classical layered materials, such as MoS 2(26.2 W·m –1·K –1) [53] and phosphorene (83.5 W·m –1·K –1) [54]. Fig. 3(b) is a plot of the cumulative thermal conductivity versus phonon frequency at room temperature, which reflects the contribution of phonons of different frequencies to the total lattice thermal conductivity.The results show that the monolayer Ge 2P 4S 2 and Ge 2As 4S The phonon branch of 2 (low frequency region < 2 THz) contributes more than 90% to the total lattice thermal conductivity in both directions.In order to further study the contribution of each phonon branch to the lattice thermal conductivity, we calculate the lattice thermal conductivity as The relative contribution of each phonon mode to the total lattice thermal conductivity shown in Fig. 3(c).Pair monolayer Ge 2P 4S 2, when the heat flow is along the armchair direction, the contribution of each branch to the total lattice thermal conductivity is in the order of TA > LA > ZA > Opt, while the contribution along the zigzag direction is in the order of LA > TA > Opt > ZA; monolayer Ge 2As 4S The order of 2 in armchair direction is LA > TA > Opt > ZA, and the order of 2 in zigzag direction is TA > LA > Opt > ZA.From It can be seen from Fig. 3(c) that the thermal transport of these two materials is mainly contributed by TA and LA modes. In addition, the size effect on monolayer Ge is also studied in this paper. 2 X 4S 2 Effect of Thermoelectric Properties.Fig. 3(d) is a plot of the cumulative lattice thermal conductivity versus the mean free path (MFP) at 300 K. It can be seen that as the MFP increases, the cumulative lattice thermal conductivity increases significantly until it reaches a maximum.It is worth noting that phonons with a free path of more than 100 nm contribute more than 50% to the thermal conductivity, which indicates that the lattice thermal conductivity can be further reduced by size effect and nanocrystallization.
Figure 3. (a) Lattice thermal conductively with respect to temperature, (b) cumulative lattice thermal conductivity as function of frequency, (c) the contribution of phonon acoustic mode (ZA, TA, LA) and optical modes (Opt) to the total lattice thermal conductivity, (d) cumulative lattice thermal conductivity as a function of mean free path (MFP) of monolayer Ge2X4S2.To further analyze the lattice thermal conductivity, the monolayer Ge was calculated using 2 X 4S Phonon group velocity of 2 at room temperature:
vλ,q=∂ωλ,q∂q, (3) Among ω λ, q is the phonon frequency; q is the magnitude of the phonon wave loss.Observation Fig. 4(a) and Fig. 4(b) found that both materials have small phonon group velocities in the low frequency region, where the group velocities of the phonon modes are in the order of LA > TA > ZA, which is consistent with The phonon modes in Fig. 2 are related to the different slopes of the phonon dispersion curves.We can also see that the monolayer Ge 2As 4S The phonon group velocity of 2 is smaller, which is an important reason for its smaller lattice thermal conductivity.
Figure 4. (a), (b) The phonon group velocity, (c), (d) Grüneisen parameters, and (e), (f) phonon relaxation time of monolayer Ge2X4S2. The inset of Fgiure (e) and Figure (f) is three-phonon scattering phase space.Another key parameter that affects the lattice thermal conductivity is the Grüneisen parameter ( γ), which is often used to describe the anharmonic interaction, and the anharmonic interaction determines the strength of the phonon-phonon interaction. The larger the absolute value of the Gruneisen parameter, the weaker the phonon transport ability.The Gruneisen parameter can be calculated by the following formula:
γλ,q=−Vωλ,q∂λ,q∂V. (4) Such as Fig. 4(c) and Shown in Fig. 4(d), monolayer Ge 2 X 4S 2 The absolute value of the Gruneisen parameter is very large in the low frequency region, especially for the ZA mode, which greatly suppresses the phonon transport capability and is very beneficial to obtaining a low lattice thermal conductivity.
The phonon relaxation time is also calculated by solving the scattering processes, as Fig. 4(e) and Fig. 4(f), it is found that the monolayer Ge 2 X 4S The low-frequency phonon branch of 2 has a small relaxation time, which can be compared with the single-layer SnP 3 comparison [24], this is related to its three-phonon scattering phase space (P 3) has a lot to do with..P 3 refers to the available space for three-phonon scattering processes [55], is an important parameter determining the phonon relaxation time. Higher P The 3 value means that there is more room for three-phonon scattering processes, which is usually a sign of small phonon relaxation time and low thermal conductivity [56].From Fig. 4(e) and From the illustration of Fig. 4(f), it is found that the three acoustic branches all have higher P 3 value.In-depth analysis of phonon group velocity, Grüneisen parameter and phonon relaxation time reveals that the monolayer Ge 2 X 4S 2 The reason for the smaller lattice thermal conductivity.
3.3 Electron transport properties.
The electronic band structures and the corresponding electronic densities of States (DOS) of the two materials are calculated. Fig. 5, a single layer of Ge can be clearly seen 2P 4S The conduction band minimum and valence band maximum of 2 are located at Г points and K, the band gap value is 1.13 eV, which indicates that it is an indirect band gap semiconductor. 2As 4S 2 is a direct band gap semiconductor with a band gap value of 1.21 eV.Band degeneracy is an important characteristic of materials with excellent thermoelectric properties, which is often achieved by band engineering. [57].The valence bands of these two materials show obvious double degeneracy near the Fermi level, which means that they have large Seebeck coefficients, which is beneficial to obtain excellent thermoelectric properties.Observing their electronic density of States, we can find that the region corresponding to the valence band is mainly controlled by Ge atoms, while the part corresponding to the conduction band is mainly contributed by the other two kinds of atoms. Fig. 5(c) also proves this.From the charge density corresponding to the lowest conduction band, it can be observed that the charge is concentrated near the P (As) atom and S atom, while the charge of the highest valence band can be observed near all atoms, but more near the Ge atom.In addition, the density of States shows a steep peak near the Fermi level, which is an important sign that the Seebeck coefficient increases sharply here. [58], which is conducive to achieving a high power factor.
Figure 5. The electronic band structure and partial density of states for monolayer Ge2As4S2 (a) and Ge2P4S2 (b), the charge density of the valence and conduction bands near the Fermi level (c).Accurate carrier mobility and relaxation time are important bases for evaluating electron transport properties. Based on deformation potential theory [36], the carrier mobility of these two materials was calculated by the following equation [59,60]:
μ2D=2eℏC2D3kBT|m∗|2E2l, (5) Where, C 2D, k B, m * and E l represent the two-dimensional elastic modulus, Boltzmann constant, carrier effective mass and deformation potential constant, respectively.These parameters and the electron relaxation time ( Concrete values of
τ=μm∗/e ) such as 表2.From It can be seen in 表1 that the mobility and relaxation time of electrons in these two materials are much higher than those of holes in both armchair and zigzag directions, where the monolayer Ge 2P 4S The value of 2 along the zigzag direction is 3277.22 cm –2·V –1·s –1, which is much higher than that of monolayer δ-InP 2684 cm of 3 –2·V –1·s –1[61], which is very beneficial to obtain good thermoelectric properties.Table 2. Calculated elastic modulus C 2D, DP constant El, effective mass m*, carrier mobility μ and scattering time τ for electron and hole in monolayer Ge2X4S2 at 300 K.方向 类型 C 2D/(J·m–2) El /eV m* /me μ/(cm–2·V–1·s–1) τ/ps Ge2P4S2 armchair electron 55.46 2.04 0.22 3277.22 0.41 hole 55.46 3.32 0.65 131.23 0.05 zigzag electron 48.44 1.97 0.22 3676.88 0.46 hole 48.44 3.23 0.65 162.32 0.06 Ge2As4S2 armchair electron 48.94 2.30 0.19 3146.80 0.34 hole 48.94 6.56 0.24 220.71 0.03 zigzag electron 43.25 2.13 0.19 1943.61 0.21 hole 43.25 6.23 0.24 275.50 0.04 | Show TableDownLoad:
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Monolayer Ge is calculated by BoltzTrap software based on Boltzmann transport theory and relaxation time approximation 2 X 4S The effect of doping on the electronic transport properties is also studied by the rigid band approximation theory.Fig. 6 presents the electron transport parameters and chemical formulas from 300 to 500 K Diagram of μ. Where, when When μ is negative, it represents p-type doping, otherwise it is n-type doping. The Seebeck coefficient and conductivity are generally described by the following equations [34]:
Figure 6. Electronic transport parameters of monolayer Ge2X4S2 include the Seebeck coefficient ((a), (b)), electrical conductivity ((c), (d)), electronic thermal conductivity ((e), (f)), and power factor ((g), (h)) as function of the chemical potential.Sαβ(T,μ)=1eTVσαβ(T,μ) ×∫∑αβ(ε)(ε−μ)[−∂fμ(T,μ)∂ε]dε, (6) σαβ(T,μ)=1V∫∑αβ(ε)[−∂fμ(T,μ)∂ε]dε, (7) ∑αβ(ε)=e2N0∑i,kτvα(i,k)vβ(i,k)δ(ε−εi,k)dε, (8) Among V is the volume of the unit cell;
∑αβ(ε) is the transport distribution function;fμ(T,μ) is the Fermi-Dirac distribution function.From Fig. 6(a) and It can be seen in Fig. 6(b) that the Seebeck coefficient has a significant temperature-dependent behavior and decreases with increasing temperature. At room temperature, the monolayer Ge 2P 4S 2 along armchair direction and zigzag direction at – 0.The 06 eV site has the largest p-type Seebeck coefficient of 1800 μV · K –1 and 1680 μV · K –1, less than monolayer Ge 2As 4S 2070 μV · K of 2 –1 and 2040 μV · K –1; simultaneous monolayer Ge 2P 4S 2 and Ge 2As 4S 2 at 0.Large n-type Seebeck coefficient is also obtained for 06 eV, where the maximum is – 1720 μV · K, respectively –1 (zigzag direction) and – 1940 μV · K –1 (armchair direction).Such a large Seebeck coefficient of these two materials can be compared with that of single-layer InP. 2000 μV · K of 3 Compared with –1, it is beneficial to obtain a large power factor.Obviously, the Seebeck coefficient of p-type is significantly higher than that of n-type, which is mainly related to the steep peak of the density of States near the Fermi level.In addition, the maximum of Seebeck coefficient appears in the low chemical potential region, which is related to the energy dependence of the density of States, and they follow the Mott relation. [62], appearing as
S=π2k2BT3e[1ndn(ε)dε+1μdn(ε)dε]ε=μ. (9) Therefore, according to the formula, the Seebeck coefficient can be further improved by adjusting the carrier concentration.
Fig. 6(c) and Fig. 6(d) is monolayer Ge 2 X 4S 2 plot of conductivity versus chemical potential. From Fig. 6(c) and It can be seen in Fig. 6(d) that the conductivity maximum occurs in the higher chemical potential region due to the Fermi-Dirac distribution.Different from the Seebeck coefficient, the n-type value of the electrical conductivity is much larger than its p-type value, which is mainly attributed to its high electron mobility and relaxation time. The effect of the electronic thermal conductivity on the thermoelectric properties at different temperatures is also studied.The electronic thermal conductivity follows the Wiedemann-Franz law [63], can be obtained by the following calculation
κe=LσT, (10) Among L is the Lorentz number (
L=π2k2B/3e2 ). Because the electronic thermal conductivity is proportional to the electrical conductivity, it can be observed that Fig. 6(e) and The electronic thermal conductivity shown in Fig. 6(f) has a very similar function curve to the electrical conductivity, which also makes them of the same doping type.The difference between the two transport parameters is that the electronic thermal conductivity is more sensitive to temperature and increases with temperature.From Fig. 6(e) and It can also be seen in Fig. 6(f) that the monolayer Ge 2P 4S 2 shows a large n-type electronic thermal conductivity in the zigzag direction, which is mainly related to its large n-type conductivity.Monolayer Ge is calculated from the obtained Seebeck coefficient and conductivity 2 X 4S Power factor of 2 ( S 2 σ).Such as Fig. 6(g) and Fig. 6(h), the power factor also shows a temperature dependent behavior similar to the Seebeck coefficient, but its value increases with increasing temperature.Under the influence of large n-type conductivity, the power factor obtains a large n-type value at 500 K, which means that they may be potential n-type thermoelectric materials.
3.4 Thermoelectric figure of merit
With the phonon transport parameters and electron transport parameters obtained before, Ge is calculated 2 X 4S The thermoelectric figure of merit of 2 ZT, evaluate its thermoelectric performance.At Fig. 7(a)— (d), a single layer of Ge can be seen 2 X 4S 2 in both armchair and zigzag directions The values of ZT have similar curves, both of which increase with the increase of temperature, and their maximum values show that the n-type is greater than the p-type.At optimal n-type doping, monolayer Ge 2P 4S 2 and Ge 2As 4S 2 along armchair direction at 500 K The ZT values are 3.06 and 3.21, with values of 1.75 and 1.86 at room temperature, respectively, which exceeds that of phosphorene at the same temperature ZT value; while the maximum values of these two materials in the zigzag direction are 3.51 and 2.54.By comparison, it is found that due to the monolayer Ge 2P 4S 2 It has a larger power factor in the zigzag direction, so it shows a larger ZT value.The effect of carrier concentration on Seebeck coefficient and conductivity is completely opposite, that is, the increase of carrier concentration will lead to the decrease of Seebeck coefficient and the increase of conductivity, and vice versa. Therefore, the appropriate carrier concentration can maximize the power factor.Fig. 7(e)— (h) at a certain temperature ZT value versus carrier concentration.The results show that the optimal hole (electron) concentration is 1 × 10 10 to 1 × 10 12 cm –2, where the maximum n-type (p-type) can be obtained ZT value.
Figure 7. The ZT values with respect to chemical potential (a)–(d) and carrier concentration (e)–(h) for monolayer Ge2X4S2.4. Summary
In this paper, monolayer Ge is systematically studied by first-principles and Boltzmann transport theory 2 X 4S Thermoelectric properties of 2.The results show that these two materials have small phonon group velocity, large Gruneisen parameter and low phonon relaxation time, which lead to very low lattice thermal conductivity.The electronic band structure indicates that the monolayer Ge 2P 4S 2 and Ge 2As 4S 2 have band gap values of 1.13 eV and 1.21 eV semiconductor, and the double degeneracy of the valence band provides a large p-type Seebeck coefficient for these two materials. Based on the weak phonon transport ability and good electronic transport properties, the best thermoelectric performance is obtained in the optimal n-type doping.Monolayer Ge at 500 K 2P 4S 2 and Ge 2As 4S Maximum of 2 ZT values are 3.51 (zigzag direction) and 3.21 (armchair direction), which means that they have excellent thermoelectric performance in the medium temperature region and have application prospects in the field of industrial waste heat power generation.
[1] Yang J, Xi L L, Qiu W J, Wu L H, Shi X, Chen L D, Yang J H, Zhang W Q, Uher C, Singh D J 2016 npj Comput. Mater. 2 1
Google Scholar
[2] Ding Z D, An M, Mo S Q, Yu X X, Jin Z L, Liao Y X, Esfarjani K, Lü J T, Shiomi J, Yang N 2019 J. Mater. Chem. A 7 2114
Google Scholar
[3] Hicks L D, Dresselhaus M S 1993 Phys. Rev. B 47 16631
Google Scholar
[4] Hicks L D, Harman T C, Dresselhaus M S 1993 Appl. Phys. Lett. 63 3230
Google Scholar
[5] Zhao L-D, Chang C, Tan G J, Kanatzidis M G 2016 Energy Environ. Sci. 9 044
[6] Liu Y C, Wang W X, Yang J, Li S 2018 Adv. Sustainable Syst. 2 1800046
Google Scholar
[7] Tan Q, Zhao L D, Li J F, Wu C F, Wei T R, Xing Z B, Kanatzidis MG 2014 J. Mater. Chem. A 2 17302
Google Scholar
[8] Yan X, Poudel B, Ma Y, Liu W S, Joshi G, Wang H, Lan Y C, Wang D Z, Chen G, Ren Z F 2010 Nano Lett. 10 3373
Google Scholar
[9] Zhao L D, Dravid V P, Kanatzidis M G 2014 Energy Environ. Sci. 7 251
Google Scholar
[10] Chhowalla M, Shin H S, Eda G, Li L J, Loh K P, Zhang H 2013 Nat. Chem. 5 263
Google Scholar
[11] Xie W J, He J, Kang H J, Tang X F, Zhu S, Laver M, Wang S Y, Copley J R D, Brown C M, Zhang Q J, Tritt T M 2010 Nano Lett. 10 3283
Google Scholar
[12] Dresselhaus M S, Dresselhaus G, Sun X, Zhang Z 1999 Phys. Solid State 41 679
Google Scholar
[13] Rashid Z, Nissimagoudar A S, Li W 2019 Phys. Chem. Chem. Phys. 21 5679
Google Scholar
[14] Snyder G J, Toberer E S 2008 Nat. Mater. 7 105
Google Scholar
[15] Guo S D, Li H C 2017 Comput. Mater. Sci. 139 361
Google Scholar
[16] Zare M, Rameshti B Z, Ghamsari F G, Asgari R 2017 Phys. Rev. B 95 045422
Google Scholar
[17] Zhang R Q, Zhou Z Z, QI N, Zhao B, Zhang Q K, Zhang Z Y, Chen Z Q 2019 J. Mater. Chem. C 7 14986
Google Scholar
[18] Ruleova P, Drasar C, Lostak P, Li C-P, Ballikaya S, Uher C 2010 Mater. Chem. Phys. 119 299
Google Scholar
[19] Yu J B, Sun Q 2018 Appl. Phys. Lett. 112 053901
Google Scholar
[20] Huang S, Wang Z Y, Xiong R, Yu H Y, Shi J 2019 Nano Energy 62 212
Google Scholar
[21] Zhu Y L, Yuan J H, Song Y Q, Xue K H, Wang S, Lian C, Li Z N, Xu M, Cheng X M, Miao X S 2019 Int. J. Hydrogen Energy 44 21536
Google Scholar
[22] Jing Y, MA Y D, LI Y F, Heine T 2017 Nano Lett. 17 1833
Google Scholar
[23] Zeng B W, Long M Q, Zhang X J, Dong Y L, Li M J, Yi Y G, Duan H M 2018 J. Phys. D:Appl. Phys. 51 235302
Google Scholar
[24] Zhu X L, Liu P F, Zhang J R, Zhang P, Zhou W X, Xie G F 2019 Nanoscale 11 19923
Google Scholar
[25] Ouyang T, Jiang E L, Tang C, Li J, He C Y, Zhong J X 2018 J. Mater. Chem. A 6 21532
Google Scholar
[26] Sun Z H, Yuan K P, Chang Z, Bi S P, Zhang X L, Tang D W 2020 Nanoscale 12 3330
Google Scholar
[27] Kresse G, Furthmüller J 1996 Phys. Rev. B 54 11169
Google Scholar
[28] Lee K, Murray É D, Kong L Z, Lundqvist B I, Langreth D C 2010 Phys. Rev. B 82 081101
Google Scholar
[29] Blöchl P E 1994 Phys. Rev. B 50 17953
Google Scholar
[30] Kresse G, Joubert D 1999 Phys. Rev. B 59 1758
[31] Togo A, Oba F, Tanaka I 2008 Phys. Rev. B 78 134106
Google Scholar
[32] Li W, Lindsay L, Broido D A, Stewart D A, Mingo N 2012 Phys. Rev. B 86 174307
Google Scholar
[33] Li W, Carrete J, Katcho N A, Mingo N 2014 Comput. Phys. Commun. 185 1747
Google Scholar
[34] Madsen G K H, Singh D J 2006 Comput. Phys. Commun. 175 67
Google Scholar
[35] Heyd J, Scuseria G E, Ernzerhof M 2003 J. Chem. Phys. 118 8207
Google Scholar
[36] Bardeen J, Shockley W 1950 Phys. Rev. 80 72
Google Scholar
[37] Chaput L, Pécheur P, Scherrer H 2007 Phys. Rev. B 75 045116
Google Scholar
[38] Price P J 1981 Ann. Phys. 133 217
Google Scholar
[39] Xi J Y, Long M Q, Tang L, Wanf D, Shuai Z G 2012 Nanoscale 4 4348
Google Scholar
[40] Becke A D, Edgecombe K E 1990 J. Chem. Phys. 92 5397
Google Scholar
[41] Nalewajski R F, Köster A M, Escalante S 2005 J. Phys. Chem. A 109 10038
Google Scholar
[42] Savin A, Nesper R, Wengert S, Wengert S 1997 Angew. Chem. Int. Ed. 36 1808
Google Scholar
[43] Batsanov S S 2001 Inorg. Mater. 37 871
Google Scholar
[44] Gao Z B, Tao F, Ren J 2018 Nanoscale 10 12997
Google Scholar
[45] Gao Z B, Zhang Z F, Liu G, Wang J S 2019 Phys. Chem. Chem. Phys. 21 26033
Google Scholar
[46] Wee D, Kozinsky B, Marzari N, Fornari M 2010 Phys. Rev. B 81 045204
Google Scholar
[47] Zhou Z Z, Liu H J, Fan D D, Cao G H, Sheng Y 2019 Phys. Rev. B 99 085410
Google Scholar
[48] Li W, Carrete J, Mingo N 2013 Appl. Phys. Lett. 103 253103
Google Scholar
[49] Peng B, Zhang H, Shao H Z, Lu H L, Zhang D W, Zhu H Y 2016 Nano Energy 30 225
Google Scholar
[50] McGaughey A J H, Landry E S, Sellan D P, Amon C H 2011 Appl. Phys. Lett. 99 131904
Google Scholar
[51] Xie G F, Guo Y, Wei X L, Zhang K W, Sun L Z, Zhong J X, Zhang G, Zhang Y W 2014 Appl. Phys. Lett. 104 233901
Google Scholar
[52] Xie G F, Ju Z F, Zhou K K, Wei X L, Guo Z X, Cai Y Q, Zhang G 2018 npj Comput. Mater. 4 1
Google Scholar
[53] Wei X L, Wang Y C, Shen Y L, Xie G F, Xiao H P, Zhong J X, Zhang G 2014 Appl. Phys. Lett. 105 103902
Google Scholar
[54] Zhu L Y, Zhang G, Li B W 2014 Phys. Rev. B 90 214302
Google Scholar
[55] Li W, Mingo N 2015 Phys. Rev. B 91 144304
Google Scholar
[56] Pandey T, Polanco C A, Lindsay L, Parker D S 2017 Phys. Rev. B 95 224306
Google Scholar
[57] Pei Y Z, Wang H, Snyder G J 2012 Adv. Mater. 24 6125
Google Scholar
[58] Zhu X L, Yang H Y, Zhou W X, Wang T B, Xu N, Xie G F 2020 ACS Appl. Mater. Interfaces 12 36102
Google Scholar
[59] Cai Y Q, Zhang G, Zhang Y W 2014 J. Am. Chem. Soc. 136 6269
Google Scholar
[60] Zhang L C, Qin G Z, Fang W Z, Cui H J, Zheng Q R, Yan Q B, Su G 2016 Sci. Rep. 6 19830
[61] Yi W C, Chen X, Wang Z X, Ding Y C, Yang B C, Liu X B 2019 J. Mater. Chem. C 7 7352
Google Scholar
[62] Heremans J P 2005 Acta Phys. Pol. A 108 609
Google Scholar
[63] Jonson M, Mahan G D 1980 Phys. Rev. B 21 4223
Google Scholar
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图 1 (a) 单层Ge2X4S2的几何结构; (b) 对应的布里渊区; (c) 局域电荷密度. 其中紫色、红色以及蓝色球分别是Ge原子、P(As)原子和S原子
Figure 1. (a) The structure of monolayer Ge2X4S2; (b) schematic diagram of the Brillouin zone; (c) electron localization function, where the purple, red and blue balls are Ge atoms, P (As) atoms and S atoms, respectively.
图 2 单层Ge2P4S2 (a)和 Ge2As4S2 (b)的声子色散和对应的声子态密度, 以及它们对应的振动模式(c)
Figure 2. Phonon Spectrum and density of state for monolayer Ge2P4S2 (a) and Ge2As4S2 (b), the corresponding vibration modes of acoustic phonon branches (ZA, TA, and LA) and the lowest optical breach (Opt1) near and at Г point.
图 3 单层Ge2X4S2的晶格热导率与温度的关系(a), 累积晶格热导率与频率的关系(b), 各分支对累积晶格热导率的贡献(c), 累积晶格热导率与声子平均自由程的关系(d)
Figure 3. (a) Lattice thermal conductively with respect to temperature, (b) cumulative lattice thermal conductivity as function of frequency, (c) the contribution of phonon acoustic mode (ZA, TA, LA) and optical modes (Opt) to the total lattice thermal conductivity, (d) cumulative lattice thermal conductivity as a function of mean free path (MFP) of monolayer Ge2X4S2.
图 4 单层Ge2X4S2的声子群速度(a), (b), 格林艾森参数(c), (d), 声子弛豫时间和三声子散射相空间(e), (f), 图 (e)和图(f)中的插图为三声子散射相空间
Figure 4. (a), (b) The phonon group velocity, (c), (d) Grüneisen parameters, and (e), (f) phonon relaxation time of monolayer Ge2X4S2. The inset of Fgiure (e) and Figure (f) is three-phonon scattering phase space.
图 5 单层Ge2P4S2(a) 和Ge2As4S2 (b) 的电子能带结构和对应的电子态密度, 以及价带和导带在费米能级附近的电荷密度 (c)
Figure 5. The electronic band structure and partial density of states for monolayer Ge2As4S2 (a) and Ge2P4S2 (b), the charge density of the valence and conduction bands near the Fermi level (c).
图 6 单层Ge2X4S2的电子输运参数(a), (b)塞贝克系数; (c), (d)电导率; (e), (f) 电子热导率; (g), (h) 功率因子
Figure 6. Electronic transport parameters of monolayer Ge2X4S2 include the Seebeck coefficient ((a), (b)), electrical conductivity ((c), (d)), electronic thermal conductivity ((e), (f)), and power factor ((g), (h)) as function of the chemical potential.
图 7 单层Ge2X4S2的ZT值分别与化学势((a)—(d))和载流子浓度((e)—(h))的关系
Figure 7. The ZT values with respect to chemical potential (a)–(d) and carrier concentration (e)–(h) for monolayer Ge2X4S2.
表 1 弛豫后的结构参数
Table 1. The parameters of optimal structure.
晶格常数/Å 键长/Å Ge—S Ge—As (P) As—As (P—P) Ge2As4S2 7.10 2.48 2.60 2.48 Ge2P4S2 6.77 2.50 2.49 2.21 
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表 2 300 K下单层Ge2X4S2的弹性模量C 2D, 形变势常数El, 有效质量m*, 载流子迁移率μ和弛豫时间τ
Table 2. Calculated elastic modulus C 2D, DP constant El, effective mass m*, carrier mobility μ and scattering time τ for electron and hole in monolayer Ge2X4S2 at 300 K.
方向 类型 C 2D/(J·m–2) El /eV m* /me μ/(cm–2·V–1·s–1) τ/ps Ge2P4S2 armchair electron 55.46 2.04 0.22 3277.22 0.41 hole 55.46 3.32 0.65 131.23 0.05 zigzag electron 48.44 1.97 0.22 3676.88 0.46 hole 48.44 3.23 0.65 162.32 0.06 Ge2As4S2 armchair electron 48.94 2.30 0.19 3146.80 0.34 hole 48.94 6.56 0.24 220.71 0.03 zigzag electron 43.25 2.13 0.19 1943.61 0.21 hole 43.25 6.23 0.24 275.50 0.04
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[1] Yang J, Xi L L, Qiu W J, Wu L H, Shi X, Chen L D, Yang J H, Zhang W Q, Uher C, Singh D J 2016 npj Comput. Mater. 2 1
Google Scholar
[2] Ding Z D, An M, Mo S Q, Yu X X, Jin Z L, Liao Y X, Esfarjani K, Lü J T, Shiomi J, Yang N 2019 J. Mater. Chem. A 7 2114
Google Scholar
[3] Hicks L D, Dresselhaus M S 1993 Phys. Rev. B 47 16631
Google Scholar
[4] Hicks L D, Harman T C, Dresselhaus M S 1993 Appl. Phys. Lett. 63 3230
Google Scholar
[5] Zhao L-D, Chang C, Tan G J, Kanatzidis M G 2016 Energy Environ. Sci. 9 044
[6] Liu Y C, Wang W X, Yang J, Li S 2018 Adv. Sustainable Syst. 2 1800046
Google Scholar
[7] Tan Q, Zhao L D, Li J F, Wu C F, Wei T R, Xing Z B, Kanatzidis MG 2014 J. Mater. Chem. A 2 17302
Google Scholar
[8] Yan X, Poudel B, Ma Y, Liu W S, Joshi G, Wang H, Lan Y C, Wang D Z, Chen G, Ren Z F 2010 Nano Lett. 10 3373
Google Scholar
[9] Zhao L D, Dravid V P, Kanatzidis M G 2014 Energy Environ. Sci. 7 251
Google Scholar
[10] Chhowalla M, Shin H S, Eda G, Li L J, Loh K P, Zhang H 2013 Nat. Chem. 5 263
Google Scholar
[11] Xie W J, He J, Kang H J, Tang X F, Zhu S, Laver M, Wang S Y, Copley J R D, Brown C M, Zhang Q J, Tritt T M 2010 Nano Lett. 10 3283
Google Scholar
[12] Dresselhaus M S, Dresselhaus G, Sun X, Zhang Z 1999 Phys. Solid State 41 679
Google Scholar
[13] Rashid Z, Nissimagoudar A S, Li W 2019 Phys. Chem. Chem. Phys. 21 5679
Google Scholar
[14] Snyder G J, Toberer E S 2008 Nat. Mater. 7 105
Google Scholar
[15] Guo S D, Li H C 2017 Comput. Mater. Sci. 139 361
Google Scholar
[16] Zare M, Rameshti B Z, Ghamsari F G, Asgari R 2017 Phys. Rev. B 95 045422
Google Scholar
[17] Zhang R Q, Zhou Z Z, QI N, Zhao B, Zhang Q K, Zhang Z Y, Chen Z Q 2019 J. Mater. Chem. C 7 14986
Google Scholar
[18] Ruleova P, Drasar C, Lostak P, Li C-P, Ballikaya S, Uher C 2010 Mater. Chem. Phys. 119 299
Google Scholar
[19] Yu J B, Sun Q 2018 Appl. Phys. Lett. 112 053901
Google Scholar
[20] Huang S, Wang Z Y, Xiong R, Yu H Y, Shi J 2019 Nano Energy 62 212
Google Scholar
[21] Zhu Y L, Yuan J H, Song Y Q, Xue K H, Wang S, Lian C, Li Z N, Xu M, Cheng X M, Miao X S 2019 Int. J. Hydrogen Energy 44 21536
Google Scholar
[22] Jing Y, MA Y D, LI Y F, Heine T 2017 Nano Lett. 17 1833
Google Scholar
[23] Zeng B W, Long M Q, Zhang X J, Dong Y L, Li M J, Yi Y G, Duan H M 2018 J. Phys. D:Appl. Phys. 51 235302
Google Scholar
[24] Zhu X L, Liu P F, Zhang J R, Zhang P, Zhou W X, Xie G F 2019 Nanoscale 11 19923
Google Scholar
[25] Ouyang T, Jiang E L, Tang C, Li J, He C Y, Zhong J X 2018 J. Mater. Chem. A 6 21532
Google Scholar
[26] Sun Z H, Yuan K P, Chang Z, Bi S P, Zhang X L, Tang D W 2020 Nanoscale 12 3330
Google Scholar
[27] Kresse G, Furthmüller J 1996 Phys. Rev. B 54 11169
Google Scholar
[28] Lee K, Murray É D, Kong L Z, Lundqvist B I, Langreth D C 2010 Phys. Rev. B 82 081101
Google Scholar
[29] Blöchl P E 1994 Phys. Rev. B 50 17953
Google Scholar
[30] Kresse G, Joubert D 1999 Phys. Rev. B 59 1758
[31] Togo A, Oba F, Tanaka I 2008 Phys. Rev. B 78 134106
Google Scholar
[32] Li W, Lindsay L, Broido D A, Stewart D A, Mingo N 2012 Phys. Rev. B 86 174307
Google Scholar
[33] Li W, Carrete J, Katcho N A, Mingo N 2014 Comput. Phys. Commun. 185 1747
Google Scholar
[34] Madsen G K H, Singh D J 2006 Comput. Phys. Commun. 175 67
Google Scholar
[35] Heyd J, Scuseria G E, Ernzerhof M 2003 J. Chem. Phys. 118 8207
Google Scholar
[36] Bardeen J, Shockley W 1950 Phys. Rev. 80 72
Google Scholar
[37] Chaput L, Pécheur P, Scherrer H 2007 Phys. Rev. B 75 045116
Google Scholar
[38] Price P J 1981 Ann. Phys. 133 217
Google Scholar
[39] Xi J Y, Long M Q, Tang L, Wanf D, Shuai Z G 2012 Nanoscale 4 4348
Google Scholar
[40] Becke A D, Edgecombe K E 1990 J. Chem. Phys. 92 5397
Google Scholar
[41] Nalewajski R F, Köster A M, Escalante S 2005 J. Phys. Chem. A 109 10038
Google Scholar
[42] Savin A, Nesper R, Wengert S, Wengert S 1997 Angew. Chem. Int. Ed. 36 1808
Google Scholar
[43] Batsanov S S 2001 Inorg. Mater. 37 871
Google Scholar
[44] Gao Z B, Tao F, Ren J 2018 Nanoscale 10 12997
Google Scholar
[45] Gao Z B, Zhang Z F, Liu G, Wang J S 2019 Phys. Chem. Chem. Phys. 21 26033
Google Scholar
[46] Wee D, Kozinsky B, Marzari N, Fornari M 2010 Phys. Rev. B 81 045204
Google Scholar
[47] Zhou Z Z, Liu H J, Fan D D, Cao G H, Sheng Y 2019 Phys. Rev. B 99 085410
Google Scholar
[48] Li W, Carrete J, Mingo N 2013 Appl. Phys. Lett. 103 253103
Google Scholar
[49] Peng B, Zhang H, Shao H Z, Lu H L, Zhang D W, Zhu H Y 2016 Nano Energy 30 225
Google Scholar
[50] McGaughey A J H, Landry E S, Sellan D P, Amon C H 2011 Appl. Phys. Lett. 99 131904
Google Scholar
[51] Xie G F, Guo Y, Wei X L, Zhang K W, Sun L Z, Zhong J X, Zhang G, Zhang Y W 2014 Appl. Phys. Lett. 104 233901
Google Scholar
[52] Xie G F, Ju Z F, Zhou K K, Wei X L, Guo Z X, Cai Y Q, Zhang G 2018 npj Comput. Mater. 4 1
Google Scholar
[53] Wei X L, Wang Y C, Shen Y L, Xie G F, Xiao H P, Zhong J X, Zhang G 2014 Appl. Phys. Lett. 105 103902
Google Scholar
[54] Zhu L Y, Zhang G, Li B W 2014 Phys. Rev. B 90 214302
Google Scholar
[55] Li W, Mingo N 2015 Phys. Rev. B 91 144304
Google Scholar
[56] Pandey T, Polanco C A, Lindsay L, Parker D S 2017 Phys. Rev. B 95 224306
Google Scholar
[57] Pei Y Z, Wang H, Snyder G J 2012 Adv. Mater. 24 6125
Google Scholar
[58] Zhu X L, Yang H Y, Zhou W X, Wang T B, Xu N, Xie G F 2020 ACS Appl. Mater. Interfaces 12 36102
Google Scholar
[59] Cai Y Q, Zhang G, Zhang Y W 2014 J. Am. Chem. Soc. 136 6269
Google Scholar
[60] Zhang L C, Qin G Z, Fang W Z, Cui H J, Zheng Q R, Yan Q B, Su G 2016 Sci. Rep. 6 19830
[61] Yi W C, Chen X, Wang Z X, Ding Y C, Yang B C, Liu X B 2019 J. Mater. Chem. C 7 7352
Google Scholar
[62] Heremans J P 2005 Acta Phys. Pol. A 108 609
Google Scholar
[63] Jonson M, Mahan G D 1980 Phys. Rev. B 21 4223
Google Scholar
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