Thermoelectric Properties of Pristine Graphyne and the BN-Doped Graphyne Family.

In this paper, we have investigated the thermoelectric properties of BN-doped graphynes and compared them with respect to their pristine counterpart using first-principles calculations. The effect of temperature on the thermoelectric properties has also been explored. Pristine γ-graphyne is an intrinsic band gap semiconductor and the band gap significantly increases due to the incorporation of boron and nitrogen atoms into the system, which simultaneously results in high electrical conductivity, a large Seebeck coefficient, and low thermal conductivity. The Seebeck coefficient for all these systems is significantly higher than that of conventional thermoelectric materials, suggesting their potential in thermoelectric applications. Among all the considered systems, the "graphyne-like BN sheet" has the highest electrical conductance and lowest thermal conductance, ensuring its superiority in thermoelectric properties over the other studied systems. We find that a maximum full ZT of ∼6 at room temperature is accessible in the "graphyne-like BN sheet".

Introduction

The rapid advancement of human civilization during recent days demands a huge energy requirement, which is not able to be fulfilled from the natural resources available and thus leads to the global energy crisis. To overcome this crisis, various ways have been adopted, among which thermoelectric energy conversion plays a very vital role and is an essential requirement for fulfilling the demand for next-generation nanoelectronic devices.[1] Thermoelectric technology is one of the most effective methods for energy harvesting since it provides a pavement that can convert waste heat into electricity and vice versa. That is why thermoelectric materials have gained significant attention among the research community during the present days.[1,2] The performance of a thermoelectric material is characterized by a dimensionless quantity, namely, the figure of merit[3] ZT and is defined as , where G is the electrical conductance, S is the Seebeck coefficient, T is the temperature, ke is the electronic part of thermal conductance, and kph is the phonon contribution of thermal conductance. To achieve a high ZT, a thermoelectric material should possess a large Seebeck coefficient and electrical conductance (a high value of power factor S2G) and simultaneously a low thermal conductance. In general, it is difficult to fulfill the above criteria as all these transport coefficients S, G, and k are coupled with each other in traditional thermoelectric materials. However, recent studies show that low-dimensional systems can possess a high ZT due to the quantum confinement effect.[4,5] Thus, searching for low-dimensional nanomaterials with high thermoelectric efficiency is becoming a challenging task for the research community. Literature survey reveals that a carbon-based material is a great choice for designing and fabricating efficient thermoelectric materials.[6−13] Among these carbon-based materials, graphene[14] finds vast applications[15,16] but the presence of zero band gap results in a very small Seebeck coefficient of graphene along with high thermal conductivity, which significantly reduces its thermoelectric performance.[17] To reduce its thermal conductance, enormous efforts have been put in such as defect engineering,[18,19] isotope engineering,[20] and a superlattice structure.[21] Another two-dimensional (2D) carbon allotrope, namely, graphyne,[22] is formed by inserting acetylenic linkages (sp hybridization) in between two carbon atoms of graphene. By varying the number and position of acetylenic linkages, several substructures of graphyne can be obtained such as α-, β-, γ-, 6,6,12-graphyne, and graphdiyne.[23,24] Among these, α-, β-, and 6,6,12-graphyne show characteristics of a Dirac material while γ-graphyne and graphdiyne are intrinsic semiconductors in nature.[24] The presence of versatile characteristics in γ-graphyne makes it a suitable candidate for electronic,[25,26] optoelectronic,[27−29] gas sensing,[30,31] energy storage,[32] and spintronic devices.[33] The presence of a band gap also leads to a significant enhancement of its Seebeck coefficient[13] and the presence of acetylenic linkages reduces the thermal conductivity significantly.[13] All these surveys prompt that γ-graphyne is a promising candidate for a future thermoelectric material, as already reported by several researchers theoretically.[10,11,13] Recent advances in experimental synthesis of γ-graphyne[34] open a new door for fabrication of thermoelectric devices based on graphyne. As the band gap is responsible for the large Seebeck coefficient, the thermoelectric performance of graphyne can be tuned by engineering its band gap. One of the effective methods for tuning the band gap is doping with suitable atoms. Interestingly, it has been observed that the band gap of γ-graphyne can be increased by co-doping with B and N atoms at different positions.[28,29,35,36]

Motivated by the above findings, here for the first time, we investigate the thermoelectric properties of BN-doped γ-graphynes and compare its performance with its pristine counterpart. Interestingly, it has been observed that doping with the B or N atom increases the band gap, thus leading to a large Seebeck coefficient and high thermoelectric performance of BN-doped γ-graphynes compared to pristine γ-graphyne. Incorporating B and N atoms in other systems such as holey graphene, graphene oxide, and so forth significantly enhances the thermoelectric performance,[37,38] which is another reason for choosing B and N atoms as dopants in our present study. For any thermoelectric device, p-type and n-type materials need to be arranged in a tandem device to increase its thermal voltage output. This is achieved in our study by co-doping with B and N atoms.[39]

Computational Methodology

To obtain the optimized geometry and ground-state Hamiltonian of all these materials, we have used SIESTA 3.2 computational package[40] using density functional theory (DFT). The Perdew–Burke–Ernzerhof method[41] is used to account for the exchange and correlation functional of generalized gradient approximation. Troullier–Martins-type norm-conserving pseudopotentials[42] are used to account for the core electrons and linear combinations of atomic orbitals to construct the valence states. A double-ζ-plus polarized numerical atomic orbitals type basis set along with a real-space grid having a mesh cut-off energy of 600 Ry is considered for the calculation. The Brillouin zone is sampled using a 1 × 12 × 12 Monkhorst–Pack grid k-point, and a maximum force tolerance of 0.01 eV/Å is used in the calculation. A vacuum of 15 Å is kept along X-direction to avoid the interaction between two periodic images.

Transport Calculation

The mean-field Hamiltonians obtained from the converged DFT calculations are combined with the quantum transport code GOLLUM.[43,44] The transmission coefficient, T(E), for electrons passing from the source to the drain with energy E is calculated through the relationwhere ΓL,R(E) = i[ΣL,R(E) – ΣL,R†(E)] describes the broadening due to the coupling between the central scattering region (SR) and left (L) and right (R) electrodes. ΣL,R(E) are the retarded self-energies associated with this coupling. GR = (ES – H – ΣL – ΣR)−1 is the retarded Green’s function where H is the Hamiltonian and S is the overlap matrix.

Thermoelectric properties such as the electrical conductance (G), Seebeck coefficient (S), and electronic part of thermal conductance (ke) of the device as a function of temperature are calculated using the following relations[43,45]where f(E) is the Fermi–Dirac probability distribution function, T is the temperature, EF is the Fermi energy, G0 = 2e2/h is the quantum conductance, e is the charge of the electron, and h is Planck’s constant.

After obtaining the transmission as a function of energy [T(E)], we have calculated the integral L(EF, T) using eq . This equation depends on temperature and Fermi energy. To calculate the electrical conductance G(EF, T), we use the Landauer formula as given in eq where the term L0 is obtained from eq . We then calculate G versus the Fermi energy for fixed temperatures as shown in Figure . The other thermoelectric parameters such as the Seebeck coefficient (S) and electronic part of thermal conductance (ke) also depend on the integral L(EF, T) and can be calculated using the relations 3 and 4.

Figure 4

Variation of electrical conductance with Fermi energy for (a) pristine γ-graphyne; (b) γ-graphyne with BN at the linear chain; (c) γ-graphyne with BN at the hexagonal ring; and (d) γ-graphyne-like BN sheet at different temperatures.

It is worth mentioning that we calculate the thermoelectric properties of structures under the assumption that the charge state of the structures and their electronic structure are not changed by changes of Fermi energy (EF) or temperature (T), which is valid for small changes in EF and T.

Results and Discussion

In this work, we have studied the thermoelectric properties of 2D BN-doped γ-graphynes and compared them with pristine γ-graphyne. For BN-doped systems, we have considered three configurations as “γ-graphyne with BN at the hexagonal ring”, “γ-graphyne with BN at the chain” and “γ-graphyne-like BN sheet”. The device comprising the SR sandwiched between two electrodes is presented in Figure and the corresponding coordinates of the structures are provided in Table S1 of the Supporting Information. State-of-the-art theoretical calculations by several research groups already confirmed the stability of the above-mentioned systems.[28,35,36,46,47] Thus, we considered pristine γ-graphyne and its BN-derivatives for making the device.

Figure 1

Representative device model showing two electrodes and the SR (a) for pristine γ-graphyne nanojunction; (b) for γ-graphyne nanojunction with BN at the linear chain; (c) for the γ-graphyne nanojunction with BN at the hexagonal ring (d) for γ-graphyne-like BN sheet nanojunction. LE and RE represent the left electrode and right electrode, respectively.

We then calculate the thermoelectric properties such as Seebeck coefficients, the electronic contribution to the thermal conductance, and the electronic part of figure of merit for pristine and BN-doped γ-graphyne and discuss their variation with temperatures. All of these parameters are obtained from the energy dependence of the electron transmission coefficient.

Figure shows the electron transmission spectra [T(E)] at zero bias voltage as a function of E within the energy range [−3.0, +3.0] eV. There is an energy band gap in all transmissions that follows “pristine γ-graphyne” < “γ-graphyne with BN at the chain” < “γ-graphyne with BN at the hexagonal ring” < “γ-graphyne-like BN sheet”.[35] The transmission spectra equal to the number of open channels and show stepwise behavior.

Figure 2

Variation of zero-bias transmission spectra with energy for (a) pristine γ-graphyne; (b) γ-graphyne with BN at linear chain; (c) γ-graphyne with BN at hexagonal ring; and (d) γ-graphyne-like BN sheet. T(E) describes the transmission probability of electrons with energy E traversing from one side of the device to the other side. This is combined with eq to calculate temperature-dependent quantities such as the conductance and the Seebeck coefficient (see the Computational Methodology section).

Figure depicts the variation of the Seebeck coefficient (S) for the pristine and BN-doped γ-graphyne systems with Fermi energy at different temperatures (200, 300, and 500 K). The temperature gradient in a material initiates the flow of current between hot and cold electrodes and as a result, an electric field is developed across the two ends and hence a voltage, known as the Seebeck voltage. As observed in Figure a, the highest value of S is 1.01 × 10–3 V/K at 200 K (Table ), corresponding to EF = ±0.03 eV, which is much higher than that of a conventional thermoelectric material.[48] The S value decreases with an increase in temperature. The magnitude of S for pristine γ-graphyne at room temperature is 0.64 × 10–3 V/K (Table ) and is in good agreement with the previously reported result[11] and the magnitude of S at room temperature is also much higher than that of the conventional thermoelectric material such as Bi2Te3.[48] Now for γ-graphyne with BN at the chain position, the highest value of S is 3.07 × 10–3 V/K (Table ) at 200 K and the maximum values of S are found at the Fermi energy of −0.32 and +0.05 eV, respectively (Figure c). When BN is at the ring position of γ-graphyne, the highest value of S is 3.09 × 10–3 V/K at T = 300 K (Table ) and the maximum value of S is recorded corresponding to the Fermi energy of −0.64 and +0.02 eV, respectively (Figure b). Finally, for the “γ-graphyne-like BN sheet”, the maximum value of S is found at the Fermi energy +0.18 and +1.17 eV (Figure d) and the highest value of S is 3.13 × 10–3 V/K (Table ). Interestingly, for the “γ-graphyne-like BN sheet” as the temperature increases, the asymmetric energy distribution of electrons around the Fermi level also increases, which leads to an increase in the magnitude of S with the temperature rise. The Seebeck coefficient of pristine γ-graphyne possess two peak values around the Fermi level irrespective of temperature and both of them have nearly the same value, indicating the isotropic nature of S. Although the magnitude of the two peaks is the same for BN-doped systems, for “γ-graphyne with BN at the chain” (at 300 and 500 K) and “γ-graphyne with BN at the ring” (at 500 K), both the peaks move toward the negative energy side, whereas the peaks shift toward the positive energy side for the “γ-graphyne-like BN sheet” (at 500 K) with respect to pristine γ-graphyne. This asymmetric energy distribution of electrons around the Fermi level for BN-doped systems leads to a greater value of the Seebeck coefficient. A system with a wide band gap generally has a large S value as S is related to Egvia the relation .[49] The wide band gap of pristine γ-graphyne, which is already reported by various groups,[25,35] contributes to the significant S value in this system. For pristine γ-graphyne, the calculated S value is higher than graphene[50] due to its semiconducting nature.[13] When BN has doped, the magnitude of S increases significantly compared to that of pristine γ-graphyne. The magnitudes of S at 300 and 500 K for all these systems increase in the following manner “pristine γ-graphyne” < “γ-graphyne with BN at the chain” < “γ-graphyne with BN at the ring” < “γ-graphyne-like BN sheet”. This result is supported by the band gap characteristic of these systems. The Seebeck coefficient for all systems studied is significantly higher than for some other reported materials such as the boron arsenide sheet,[51] graphdiyne,[52] phosphorene,[53] MoSe2,[54] WSe2,[54] monolayer bismuth,[55] 1L-ZnPSe3,[56] selenene, and tellurene.[57] The experimental and theoretical studies already confirmed that a material to be used in thermoelectricity should have thermoelectric conversion performances of around 230 μV/K[58] and as all our studied material exhibits, a much higher Seebeck coefficient compared to the usual value, clearly suggesting the possibility of using these materials in thermoelectric applications. We found that the Seebeck coefficient is generally high in the structures studied. This is due to sharp features in the transmission functions. It is because the Seebeck coefficient is proportional to the slope of the transmission coefficient[59]T(E) as S ∝ −∂LT(E)/∂E at EF. This means that a flat transmission gives a zero Seebeck coefficient, while zones with large slopes give a high S. Furthermore, the sign of S changes for zones with positive and negative slopes. For example, the sign of S for pristine γ-graphyne is positive (negative) for energies around −0.2 eV (0.2 eV).

Figure 3

Variation of the Seebeck coefficient with Fermi energy for (a) pristine γ-graphyne; (b) γ-graphyne with BN at the linear chain; (c) γ-graphyne with BN at the hexagonal ring; and (d) γ-graphyne-like BN sheet at different temperatures.

Table 1

Calculated Seebeck Coefficient for Pristine and BN-Doped γ-Graphynes at Different Temperatures

system	T (K)	EF (eV)	S (V/K)
pristine γ-graphyne	200	–0.03	1.01 × 10–3
		0.03	–1.01 × 10–3
	300	–0.05	0.64 × 10–3
		0.05	–0.64 × 10–3
	500	–0.07	0.36 × 10–3
		0.07	–0.36 × 10–3
γ-graphyne with BN at the chain	200	–0.32	3.07 × 10–3
		0.05	–3.07 × 10–3
	300	–0.20	2.50 × 10–3
		–0.08	–2.50 × 10–3
	500	–0.23	1.46 × 10–3
		–0.05	–1.46 × 10–3
γ-graphyne with BN at the ring	200	–0.95	3.07 × 10–3
		0.33	–3.07 × 10–3
	300	–0.64	3.09 × 10–3
		0.02	–3.09 × 10–3
	500	–0.41	2.33 × 10–3
		–0.20	–2.34 × 10–3
γ-graphyne-like BN sheet	200	–0.77	3.07 × 10–3
		2.12	–3.07 × 10–3
	300	–0.45	3.10 × 10–3
		1.80	–3.10 × 10–3
	500	0.18	3.13 × 10–3
		1.17	–3.13 × 10–3

For a material to be applicable in thermoelectric applications, in addition to a high value of the Seebeck coefficient, a large magnitude of electrical conductance (G) is also needed. The electrical conductance (Figure ) is highest for the “γ-graphyne-like BN sheet” and the magnitude is lowest for the “γ-graphyne with BN at the ring”. The magnitude of G increases with temperature in the gap (e.g., E = [−0.2 to +0.2]); however, it decreases on resonances (e.g., E = [−0.8 to +0.4]) for all these systems. However, for the “γ-graphyne-like BN sheet”, the decrement in G is more prominent compared to the rest of the systems.

The electronic part of thermal conductance is small for all these materials (Figure ). The highest ke is observed for pristine γ-graphyne while ke is lowest for the “γ-graphyne-like BN sheet”. The magnitude of ke increases with an increase in T. The thermal conductance shows a similar trend with the electrical conductance in agreement with the Wiedemann–Franz law.

Figure 5

Variation of the electronic part of thermal conductance with Fermi energy for (a) pristine γ-graphyne; (b) γ-graphyne with BN at the linear chain; (c) γ-graphyne with BN at the hexagonal ring; and (d) γ-graphyne-like BN sheet at different temperatures.

After obtaining all the transport coefficients, we have finally calculated the electronic thermoelectric figure of merit of pristine and BN-doped γ-graphyne systems using the relation . Here, we have considered only the electronic part of the figure of merit. Figure S1 shows the variation of ZTe as a function of the Fermi energy for different temperatures. For all these systems, two obvious peaks around the Fermi level have been observed. ZTe for “pristine γ-graphyne” and “γ-graphyne with BN at the chain” decreases with an increase in temperature, while for “γ-graphyne with BN at the ring” it increases first (200 to 300 K) and then decreases (300 to 500 K) (Table S2). In the case of the “γ-graphyne-like BN sheet”, ZTe increases with an increase in temperature and it also has the highest ZTe among the considered systems. The phonon thermal conductivity of pristine γ-graphyne is predicted to be about 75 W/mK at room temperature.[11] From this, we estimate the phonon contribution to thermal conductance in our pristine γ-graphyne junction as kph = 30 pW/K. Using this value, the full ZT = ZTe/(1 + kph/ke) is obtained for the pristine γ-graphyne junction as shown in Figure a. Phonon band structure of these systems calculated using first-principles methods including all phonon modes are shown in Figure S2 of the Supporting Information. This indicates that they a have similar dispersion relation. The Debye frequency of “pristine γ-graphyne” is amongst the highest. It is expected that it should have higher thermal conductance (kph) than the “γ-graphyne-like BN sheet” and “γ-graphyne with BN at the chain”. Its thermal conductance (kph) also should be close to the kph of “γ-graphyne with BN at the hexagonal ring”, especially the small difference in Debye frequency happens to be in high frequencies with a small contribution to room temperature kph. Figure shows the estimated full ZT. Clearly, ZT is high in these materials. This is an indication that these systems hold great potential for thermoelectricity. Among all the considered systems, the “γ-graphyne-like BN sheet” possesses the highest maximum ZT and a maximum ZT of ∼6 is observed at room temperature. This is the minimum ZT expected as the Debye frequency of the “γ-graphyne-like BN sheet” is lower than that of “pristine γ-graphyne”, and kph for the “γ-graphyne-like BN sheet” is expected to be lower than that used to estimate the full ZT and therefore full ZT is expected to be higher than these values. The other factors which attribute to the highest ZT in the “γ-graphyne-like BN sheet” are the largest Seebeck coefficient (S), largest electrical conductance (G), and lowest electrical thermal conductance (ke) of the “γ-graphyne-like BN sheet” compared to other systems.

Figure 6

Variation of the figure of merit with Fermi energy for (a) pristine γ-graphyne; (b) γ-graphyne with BN at the linear chain; (c) γ-graphyne with BN at the hexagonal ring; and (d) γ-graphyne-like BN sheet at different temperatures.

A previous study[60] by our group shows that a double vacancy in γ-graphyne results in a wide band gap semiconductor with a band gap of 1.82 eV for majority spin while the band gap for minority spin is 1.34 eV. This wide band gap indicates that double vacancy in γ-graphyne will lead to a large Seebeck coefficient. Literature survey also reveals that the introduction of the defective environment in the γ-graphyne nanoribbon system significantly enhances the thermoelectric performance of the material.[61−63] Therefore, we expect further enhancement with defects in our structures too.

Conclusions

In conclusion, within the framework of DFT in association with the quantum transport technique, the thermoelectric properties of BN-doped γ-graphynes with varying temperatures have been investigated and compared with respect to their pristine analogue. The Seebeck coefficient of all these doped systems is significantly higher than those of conventional thermoelectric materials, ensuring their potentiality for thermoelectric applications. The large Seebeck coefficient originates from the modulation of the band gap achieved by the incorporation of BN atoms. This leads to high full ZT of ∼6 for the γ-graphyne-like BN sheet at room temperature. Our present study shows that these materials have huge potential for thermoelectricity. We hope that our report of a large value of ZT will motivate the experimentalists to characterize BN-doped graphyne systems.