Effects of alkali metal element substitution on the thermoelectric properties of β-Zn4Sb3

The effects of alkali metal elements Na, Li, and K on the electronic structure and thermoelectric properties of β-Zn4Sb3 were extensively studied based on first-principles electronic structure calculations and Boltzmann transport theories. We found that these alkali metal elements with s orbitals could bring resonant distortion in the electronic density of states (DOS). The effects of these resonant peaks on the thermoelectric properties of β-Zn4Sb3 were analyzed. And the results showed that a ∼13/13/20-fold increase on the room-temperature Seebeck coefficient of β-Zn4Sb3 is achieved due to Na/Li/K doping respectively. Accordingly, the optimizing power factor/thermoelectric figure of merit of β-Zn4Sb3 gets an about 1.69/1.43-fold increase at room temperature upon Na doping. And their corresponding optimal Fermi levels of three doped compounds all lie near and below the valence band maximum of the host.


Introduction
Because of its potential applications of power generation or cooling environmental-friendly, thermoelectric materials have been attracted widespread attention in recent years. [1][2][3][4][5][6] The energy conversion efficiency of thermoelectric materials is characterized by the dimensionless figure of merit / s l = ZT S T , 2 where T is the absolute temperature, S is the Seebeck coefficient, s is the electrical conductivity, ( ) l l l = + L C is thermal conductivity with both the lattice (l L ) and carrier (l C ) contributions.
The standard procedure for boosting ZT of β-Zn 4 Sb 3 has been simultaneous optimization of carrier concentration and thermal conductivity through doping [18][19][20][21][22][23][24][25][26][27][28][29][30]. However, the results showed that this method has reached its limits and that further significant progress will be achieved by focusing on the power factor s S . 2 Recently, Mahan and Sofo [31] theorized that the introduction of resonant distortion in the electronic density of states (DOS) could significantly enhance the Seebeck coefficient S without sacrificing the electrical conductivity s. Heremans et al [32] showed that after Tl doping, ZT of PbTe is doubled due to the increase in S. This is ascribed to the DOS distortion of the host. Similarly, the phenomenon was also observed in Al-doped PbSe, Indoped SnTe, or Sn-doped Bi 2 Te 3 [33]. In this paper, we showed that alkali-metal-element doping can also cause resonant distortion in DOS of β-Zn 4 Sb 3 . These alkali metal elements all have one valence electron in s orbital of the outermost layer. Thus resonant peaks close to the Femi level in DOS are formed after that the s levels of these elements overlap or hybridize with the host band. Accordingly, how these resonant levels affect thermoelectric properties after doping alkali metal elements (here Na, Li and K) in β-Zn 4 Sb 3 was discussed.

Computational details
The electronic structure of Zn-substituted compounds MZn 35 Sb 30 (M=Na/Li/K) has been investigated using the Vienna ab initio simulation package (VASP) with the projector augmented wave (PAW) scheme and Perdew, Burke and Ernzerhof generalized gradient approximation (GGA-PBE) for the exchange-correlation functional [34][35][36]. The calculations of the undoped system has been performed within a framework of Zn 36 Sb 30 (see figure 1 of supplemental material is available online at stacks.iop.org/MRX/6/1265j6/mmedia) for simplicity. Zn-substituted systems have been treated in the framework of MZn 35 Sb 30 (M=Na/Li/K). Based on test calculations, the energy cutoff for the plane wave expansion is 350 eV, and the k-point density is 2×2×2 for structural optimization and 4×4×4 for DOS calculations with the Monkhorst-Pack method. The calculated equilibrium lattice parameters values of the undoped system (a=21.299 Å, c=17.444 Å) were set as the initial values for the Zn-substituted systems. The ionic coordinates and the unit cell's size and shape were optimized simultaneously to eliminate structures with internal stress until the forces converge to less than 0.001 eV/Å. Based on the computed DOS, we used Boltzmann transport theory to calculate s, S and l C of Znsubstituted compounds with MATLAB software program as follows [37]: where e, f 0 , E f , υ(E), k, g(E), and τ is the electron charge, Fermi distribution function, Fermi energy, electron velocity, electron wave vector, density of states, and total relaxation time, respectively. Deformation potential scattering of acoustic and optical phonons, and polar scattering by optical phonons [38,39] have been considered, as listed below: Values for material constants used in these equations are given in table I of supplemental material. More calculation details can be found in our earlier works [40].

Results and discussion
In consistent with the earlier model calculations, [14] the   The positive Seebeck coefficient indicates p-type conduction in all four compounds. Moreover, S of the three Na/Li/K-doped cases is boosted greatly when compared with that of pristine β-Zn 4 Sb 3 in the interesting carrier concentration range. And their peak value of S relative to that of β-Zn 4 Sb 3 is nearly 13/13/20 times at n≈5.6×10 27 m −3 , n≈4.4×10 27 m −3 , and n≈5.2×10 27 m −3 , respectively. According to Mott formula [39] the resonant peaks brought by Na/Li/K elements would be more and more δ-function-like in the DOS as the amount of Na/Li/K elements decreases. And thus, a sharper increase in S would be expected.
The electrical conductivity σ as a function of the carrier concentration n is shown in figure 2(b). As expected, with the increase of carrier concentration, σ for three doping cases all increase monotonously and show much less than that of undoped β-Zn 4 Sb 3 in the interesting carrier concentration range due to the enhancement of carrier scattering by point defects. Particularly σ of the KZn 35 Sb 30 compound is much lower, almost less than half that of β-Zn 4 Sb 3 at n≈8×10 27 m −3 . Exceptionally, σ of Na-substituted system starts to outnumber that of pristine β-Zn 4 Sb 3 as n>8.5×10 27 m −3 , and on that account, as shown in figure 2(c), Na-substituted case gains a 1.69-fold increase in optimizing power factor of β-Zn 4 Sb 3 with n≈5.7×10 27 m −3 when T=300 K. However, optimizing power factors of Li/K-substituted systems, both decrease about 9%/25% compared with that of unsubstituted case, when n≈4.6×10 27 m −3 , and n≈5.5×10 27 m −3 , respectively.
The carrier thermal conductivity l C of MZn 35 Sb 30 (M=Na/Li/K) as a function of carrier concentration n is shown in figure 3(a). One can see that the carrier concentration dependence of l C of MZn 35 Sb 30 (M=Na/ Li/K) similar to that of the electrical conductivity: (a) l C of all three substituted cases increase monotonously with the increase of carrier concentration; (b) l C for three substituted cases degrade largely compared with that of pristine β-Zn 4 Sb 3 due to the enhancement of carrier scattering by point defects. (c) l C of Na-doped system changes to outnumber that of pristine β-Zn 4 Sb 3 as n>7.7×10 27 m −3 . However, despite all this, Na doping system achieves a maximum ZT of 1.43 times as large as that undoped case (ZT≈0.15 at n≈4.5×10 25 m −3 ) as shown in figure 3(b). Figure 3(b) shows that carrier concentration dependence of room temperature ZT assuming l = --0.6 Wm K L 1 1 (an average experimental value for undoped β-Zn 4 Sb 3 ) for β-Zn 4 Sb 3 doped with Na/Li/K. Na-substituted system shows the best performance and its maximum ZT reaches 0.22 with n≈5.6×10 27 m −3 , attributed by the elevation of power factor coincident with the decrease of carrier thermal conductivity. As for the Li/K-substituted cases, because of the degraded power factor, their maximum ZT values are also degraded, which is 0.13 and 0.1 at n≈4.6×10 27 m −3 or 5.5×10 26 m −3 with T=300 K, respectively. For three-substituted cases, their peaks of ZT all locate just in the range of carrier concentration for the optimizing power factors. If in conjunction with other methods (such as nanocomposite [41]) to lower lattice thermal conductivity, we believe that there still has scope for ZT to improve further.
To further figure out the effects of Na/Li/K doping on thermoelectric performance, room temperature carrier concentration n as a function of the Fermi energy E F for MZn 35 Sb 30 (M=Na/Li/K) is shown in the figure 4. At a fixed T, carrier concentration n increases as E F shifts upward into the deeper lying valence bands. According to the results of figure 3(b), we know that when n≈5.6×10 27 m −3 , n≈4.6×10 27 m −3 and 5.5×10 26 m −3 , ZT of three Na/Li/K-substituted compounds will be maximized, respectively. And we found that their optimal E F at this point are: E F (Na)=−0.11 eV, E F (Li)=−0.09 eV and E F (K)=−0.09 eV, being about 0.192/0.132/0.36 eV away from their resonant peak, respectively, and all lying near the maximum of the host valence band. This is in agreement with the results of recent research on resonant states [42,43].

Conclusions
We have investigated the effects of doping Na, Li and K on the electronic structure and thermoelectric properties of β-Zn 4 Sb 3 and demonstrated the enhancement of the thermoelectric properties as a result of resonant doping. All three alkali metal elements could cause resonant DOS distortion, a boost in the Seebeck coefficient, and  drastic reduction of conductivity and carrier thermal conductivity. A 69%/43% increase of optimal power factor and ZT could be achieved by doping Na. The enhancement of thermoelectric performance for Nasubstituted case results from the elevation of power factor and the simultaneous decrease of carrier thermal conductivity. Moreover, the optimal E F corresponding to the maximal ZT is about 0.192 eV away from its resonant peak, and near the maximum of the host valence band. These findings provide a better understanding of resonant doping and useful reference to design high-performance thermoelectric materials.