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Superhigh thermoelectric figure of merit in silver halides AgCl and AgBr from first principles

Searching for the high-performance thermoelectric (TE) materials has always been a long-held dream in the thermoelectricity field. Recently, it is found in experiments that the largest figure of merit ZT of $2.6$ can be reached in SnSe crystals at 923 K and Cu2Se sample at 850 K, which arouses the enormous interest of seeking high-ZT materials. Based on first-principle calculations and Boltzman transport equation (BTE), we report in this letter that silver halides (AgCl and AgBr) in rocksalt structure have excellent TE performances. A superhigh ZT of about 7.0 at mid-temperature (~600K) is obtained in the p-type doped AgCl and AgBr crystals, which far exceeds the ZT values of all current bulk TE materials. This record-breaking ZT value is attributed to the ultralow intrinsic lattice thermal conductivity \kappa_L (e.g. \kappa_L ~0.10 and 0.09 Wm^{-1}K^{-1} for AgCl and AgBr at 600 K, respectively). Our results may be a feat that could revolutionize the field of the heat energy conversion.


I. INTRODUCTION
The thermoelectric (TE) technology provides a simple and environmentally friendly solution for direct conversion from heat to electricity, which has drawn a good deal of attentions [1][2][3][4][5][6][7]. However, due to lacking significant progress in lead-free, efficient TE devices, the influence of TE technology has mainly remained within a small sphere of niche applications [8,9]. The efficiency of a TE material is determined by the figure of merit ZT defined as where S is the Seebeck coefficient, σ is the electrical conductivity, T is the temperature in kelvin, κ e represents thermal conductivity of the charge carrier and κ L is that of the lattice. Currently, the ZT values of most typical bulk TE materials, such as PbTe, n-Type skutterudites CoSb 3 , PbSe, Bi 2 Te 3 , nanocomposites and p-type half-Heusler are in the range of 1.5 ∼ 2.5 [10][11][12][13][14][15][16]. Due to the interdependent relationship in σ, S and κ, it is usually difficult to improve an average ZT well above 2.5. In the past decade, there are several approaches proposed to improve ZT, including the enhancement of electronic properties (S and σ) (by doping electron or hole [17,18], introducing the resonant states in the vicinity of Fermi level [19][20][21], and band convergence [22,23]) and the reduction of the κ L (by enhancing phonon scattering through disorder within the unit cell [24,25] or forming solid solutions [22,26]).
So far, the largest ZT value of 2.6 in experiment can be reached in SnSe crystals at 923 K and Cu 2 Se sample at 850 K [8,9]. In SnSe crystals the excellent TE performance stems from the ultralow thermal conductivity, such as κ L ∼ 0.23 ± 0.03 Wm −1 K −1 at 973 K, while the remarkable TE performance in Cu 2 Se sample is attributed to the localization of Cu + induced by the incorporation of indium (In) into the Cu 2 Se lattice, which enhances the σ and reduces the κ L of the nanocomposites simultaneously [8,9]. A common feature in these materials is the presence of low κ L , which is a crucial ingredient of high-ZT materials. Therefore, searching for TE materials with low intrinsic κ L is our striving directions. In 1986, M. V.
Smirnov et al. reported that molten alkali halides and their mixtures have ultralow κ L , e.g., the κ L ∼ 0.2 − 1.4 W/mK in the temperature range of 900 ∼ 1300 K [27], which inspires us to study the heat transport and TE properties of halide materials. In this work, we present that the silver halides AgCl and AgBr crystals materials may be the best candidates for 2 high TE performance.
For AgCl and AgBr, the theoretical and experimental studies are only focused on the electronic structure properties, ionic transport properties, optical absorption and response or other chemical properties in the past few decades [28][29][30][31][32][33]. In the practical applications, AgCl is a common reference electrode in electrochemistry, while AgBr is widely used in photographic films. Nevertheless, to date the study of the heat transport properties and TE properties in these materials is lacking, which is may be due to the low melting point (728 K for AgCl, 701 K for AgBr) [28,29]. In this paper, we systematically investigate the heat and electronic transport properties of AgCl and AgBr, and conclude that they have ultralow intrinsic κ L , high S, and consequently remarkable TE performances.

II. METHODOLOGY
Using first-principle calculations and Boltzman transport equation (BTE), we study the electronic structure, lattice thermal transport and electron transport properties of AgCl and To obtain the electron transport properties, we use the rigid-band approach and the semiclassical Boltzmann theory, which is performed in the BOLTZTRAP code [39]. In this approach, the constant scattering time approximation τ is used, which is the only parameter that can be tuned. The value of τ from 1 to 8 fs is used to obtain relatively reasonable results.
The electronic structure is recalculated by VASP on the dense k-points of 80 × 80 × 80 to  acquire precise derivatives of the Kohn-Sham eigenvalues. 4

III. RESULTS AND DISCUSSION
AgCl and AgBr crystals are stable in the rocksalt structures (as shown in Supplemental S. 1) below their melting point (728 K for AgCl, 701 K for AgBr) [28,29]. Their optimized lattice constants a are 5.60Å and 5.85Å, respectively, as shown in TABLE I, which are slightly larger than the experimental values of 5.55Å for AgCl and 5.774Å for AgBr [30], this is due to the fact that GGA often overestimates the lattice constant. Usually, the phonon spectrum can inspect the structure stability, which lies in the fact that for each phonon mode the frequency should be a real quantity and not imaginary [41].  K, respectively.) [10]. Furthermore, to gain insight into the heat transport mechanism and decide which phonon modes provide the primary heat conductivity, the accumulative lattice thermal conductivity κ a scaled by the total κ L as a function of frequency, which exhibits the summed contributions from the phonon modes below the specified frequency, is also calculated, as shown in Supplemental S.2. For AgCl (AgBr), more than 80% of the heat transport is induced by the phonons with the frequency below 3 (2) THz. Based on the combination of S.2 and Fig. 1, we can find that three acoustic phonon modes and two TO modes dominate the heat transport. In addition, the ultralow κ L usually hints excellent TE performance if their power factor S 2 σ (PF) and electronic transport properties are good enough. To evaluate the electronic transport and TE properties, the S, σ/τ and κ e were calculated by the BOLTZTRAP code [39] as a function of temperature and carrier concentration n.
The calculations with the scissors shift of 2.33 and 2.0 eV (E GW 0 gap -E GGA gap ) for AgCl and AgBr crystals, are also performed. The results show that band gap underestimate have no effect on electronic transport properties. Additionally, the best TE performances are obtained in p-type doped AgCl and AgBr crystals, while n-type doping cases show bad TE performance, as shown in Supplemental S. 5, thus we no longer care about the n-type doping cases for these materials in the following discussion.
The calculated thermal power S is shown in Supplemental S. 6. The S increases with temperature at the same carrier concentration and decreases with carrier concentration at the same temperature, similar to the tendency in most of semiconducting TE materials [3].
The S values are much high. For instance, the values of S for AgCl (AgBr) are in the range of 350 ∼ 430 (330 ∼ 410) µV /K at n∼ 10 20 cm −1 as the temperature increases from 300 to 600 K. These values of S are much larger than PbTe, such as the values of S are in the range of 100 ∼ 300 µV/K at 300 ∼ 600 K [50]. In addition, the values of S of AgCl are higher than that of AgBr at the same temperature, indicating that a possible higher ZT in AgCl. Although the σ and κ e values are not confirmed at present, the combination of large S and ultralow κ L suggest a possible high ZT in AgCl and AgBr. To calculate σ and κ e , we should estimate the amplitude of electronic scattering times τ and consider the effect of lattice vibration on practical τ , thus we use τ from 1 to 8 fs to obtain a reasonable result, as shown in Supplemental S. 7 and S. 8. We find that the σ values does not rely on temperature, which is in accordance with the electronic Boltzmann theory [39]. The values of σ of AgCl are slightly smaller than that of AgBr. Moreover, to obtain the accuracy ZT values, it is necessary to calculate the κ e , although the values of κ e are much lower than the κ L . These results are presented in Supplemental S. 8 at 300, 500 and 600 K for AgCl and AgBr, respectively. One can find that the values of κ e of AgBr are slightly higher than that of AgCl in the temperature range 300 ∼ 600 K.
Next, the power factor (PF) and ZT for p-type doped AgCl and AgBr as a function of carrier concentration at 600 K are shown in Fig. 3(a) and (b). In these panels, the lower can lead to a high ZT, meanwhile being subject to κ e and κ L , which indicates a compromise between the PF and κ and a complex competition mechanism within the TE materials. Our results suggest that AgCl and AgBr crystals in rocksalt structure have unprecedented large ZT values, which are highest than that of all current bulk TE materials. Finally, we must point out that due to the property of unusual sensitivity to light for these materials, a black shell is needed to design TE devices, meanwhile ensuring that the operating temperature of devices is lower than their melting point.

IV. CONCLUSION
To summarize, we have calculated electronic structure, lattice thermal transport and electronic transport properties of rocksalt structure AgCl and AgBr crystals, which is employed first principles and phonon (electron) Boltzmann transport theory. The ultralow κ L of 0.202 and 0.185 Wm −1 K −1 of AgCl and AgBr are obtained at the room temperature.
Usually, the ultralow κ L indicates the excellent TE performances, hence a combination of the first principle calculations and the semiclassical analysis was used to investigated the TE properties for these materials. The electronic transport properties are determined by the electronic energy band structure. We find that the flat band leads to high S and highly dispersive band results in good σ. Therefore, the unprecedentedly large values of ZT of 7.2 and 7.1 are obtained at 600 K in the p-type doped AgCl and AgBr, which is defeated the ZT values of all current bulk TE materials. These results indicate that AgCl and AgBr are excellent mid-temperature (500 − 900 K) power generation materials, although we need a black shell and the operating temperature of devices is below their melting point.