Enhancement of the thermoelectric efficiency of PbTe by selective site doping: Effect of group VA impurities

Computational Materials Science 97 (2015) 159–164

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Enhancement of the thermoelectric efficiency of PbTe by selective site doping: Effect of group VA impurities K. Xiong, R.C. Longo ⇑, W. Wang, R.P. Gupta, B.E. Gnade, K. Cho Materials Science & Engineering Dept., University of Texas at Dallas, Richardson, TX 75080, USA

a r t i c l e

i n f o

Article history: Received 15 August 2014 Received in revised form 9 October 2014 Accepted 12 October 2014 Available online 9 November 2014 Keywords: Thermoelectricity Density-functional theory Spin–orbit coupling

a b s t r a c t The electronic structure of group VA impurities in PbTe is investigated by first principles calculations. Our findings show that these impurities can act as either donors or acceptors, depending on whether they occupy Pb or Te lattice sites. The impurities introduce a defect state at 0.3 eV above the conduction band minimum if they stay at Pb sites, whereas they introduce a state at the edge of the PbTe valence band if they stay at Te sites. The implications of our calculations on the thermoelectric efficiency of PbTe are also discussed. Ó 2014 Elsevier B.V. All rights reserved.

1. Introduction Thermoelectric materials (TE) are currently the interest of many research efforts, in order to develop a new generation of energy technologies to solve the growing environmental and sustainability problems associated to fossil fuels [1,2]. TE materials can convert heat into electricity, but their development needs to overcome the low energy conversion efficiency of the current TE materials. The performance of a TE material is well characterized by a parameter named the ‘‘figure of merit’’, which can be expressed as:

ZT ¼

S2 rT

j

ð1Þ

where S is the Seebeck coefficient, r is the electrical conductivity, and j is the thermal conductivity. The main commercial TE material, Bi2Te3, has a ZT value of 1, but practical applications will need ZT  3, in order to get high enough energy conversion efficiency [3]. As can be seen in Eq. (1), in order to have a high ZT value, the TE material should have a large S, a high r, and a low j. The lead chalcogenide PbTe TE material is a semiconductor of the IV–VI group with a small band-gap. This compound has been found to show the best thermoelectric performance at low-medium temperatures (400–800 K) [4,5]. PbTe shows a larger figure of merit than Bi2Te3 in the temperature interval required for most of the relevant industrial applications (450–600 K) [6,7]. Some of them include waste heat recovery and solar and geothermal power heat transformation into electricity. This finding has motivated ⇑ Corresponding author. http://dx.doi.org/10.1016/j.commatsci.2014.10.027 0927-0256/Ó 2014 Elsevier B.V. All rights reserved.

numerous research investigations to further improve the TE performance of PbTe, including forming superlattices, fabricating nanostructures, engineering band structure, etc. [5,8,9]. Group VA elements, in particular Sb and Bi, are well-known dopants for PbTe-based compounds. There have been a number of recent experimental studies that attempt to maximize the ZT of PbTe by incorporating Sb and Bi [10–12]. The behavior of these impurities in PbTe is known to be complex and has been investigated for a long time. Strauss reported for the first time that As and Sb are amphoteric in PbTe, but the case of Sb is not yet conclusive, while Bi acts as a donor in PbTe [13]. Recent experiments confirmed the amphoteric behavior of Sb in PbTe [14]. Moreover, the energy level of impurity-induced states is of crucial importance, because it has been recently shown that Tl-doped PbTe has a ZT value twice higher than that of undoped PbTe. This increase of figure of merit is due to the resonant states induced by the Tl impurity in a position close to the Fermi level of the PbTe, which enlarge the Seebeck coefficient. The increase of the Seebeck coefficient (S) is directly proportional to any change in the density of states (DOS), as long as the low doping concentration do not vary the effective masses of the electrons and/or holes (charge carriers). By using the Mott equation, it is possible to correlate the Seebeck coefficient and any local change in the DOS (given by dn (E)/dE) [15],

  d½lnðrðEÞÞ dE 3 q E¼EF   2 p kB 1 dn ðEÞ 1 dl ðEÞ ¼ þ kB T n dE 3 q l dE E¼EF

S¼

p2 kB

kB T

ð2Þ

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where kB is the Boltzmann constant, n is the electron/hole (charge carrier) density and l represents their respective mobility. This finding motivates us to explore other dopants that could have a similar effect. This requires a detailed understanding of electronic structures of these doped systems. During the last few years many theoretical studies have been dedicated to investigate the effects of dopants on the electronic structure of PbTe [16–20]. The impact of Pb-substituted group VA impurities on the PbTe electronic structure and transport coefficients has been investigated in detailed through first principles calculations [18,20]. But, since in PbTe group VA impurities are amphoteric, it is also necessary to see the effects of these impurities in PbTe when they substitute the Te sites. In this work, we present a systematic study of the effects of both Pb- and Te-substituted group VA impurities on the electronic structure of PbTe by ab initio calculations. 2. Computational methods We performed ab initio calculations using density-functional theory (DFT) with plane-wave basis sets and projector augmented wave (PAW) pseudopotentials, as implemented in VASP code [21]. The exchange and correlation energies were obtained with the local density approximation (LDA) functional. An energy cutoff of 400 eV and a 4  4  4 k-point mesh are chosen in our calculations. The structural relaxations were performed until the difference in the energies and forces were less than 104 eV and 0.01 eV/Å, respectively. Since it is well known the effect of the spin–orbit coupling (SOC) on the band gap of PbTe, SOC corrections were also included in our calculations. For the calculations of the doped-PbTe system, we built PbTe 3  3  3 supercells containing 216 atoms (the formula of the supercell is Pb108Te108), large enough to avoid the interactions between the dopants and their replica images of the periodic supercells [18,20]. Thus, the impurity-induced band structure will be less dispersive and it will be possible to determine the defect state in the electronic band gap of PbTe with more precision. 3. Results and discussion The bulk PbTe has a fcc structure with a lattice parameter of 6.379 Å, as obtained in our LDA calculations (the experimental value is 6.462 Å [22]). Fig. 1 shows the band structure and projected density of states (PDOS) of the PbTe unit cell without (Fig. 1(a) and (e)) and with SOC (Fig. 1(b) and (f)). The obtained electronic band gap without SOC corrections is 0.6 eV (see Fig. 1(a)), and it is of direct type (at the high-symmetry point L). Including SOC in the calculations narrows the band gap to 0.18 eV (see Fig. 1(b)), in excellent agreement with the experimental result (0.22 eV [23]). At the L point, both the valence band maximum (VBM) and the conduction band minimum (CBM) are two-fold degenerate if the calculation is spin-polarized. The band structures of the undoped 3  3  3 supercell are shown in Fig. 1(c) and (d). Both VBM and CBM are now located at the high-symmetry point R (instead of L), due to the Brillouin zone folding. At the R point, the CBM and the VBM are both eightfold degenerate. As we used the 3  3 supercell to obtain the doping behavior of the PbTe TE material, the band structures shown in Fig. 1(c) and (d) will be our reference states. Our obtained PDOS of the PbTe unit cell (see Fig. 1(e) and (f)) shows that both conduction band (CB) and valence band (VB) consist of a mixture of Pb 6p and Te 5p states. We now examine the electronic structure of Pb-substituted group VA impurities (the trivalent impurity in PbTe). The calculated PDOS of X0Pb (X = P, As, Sb and Bi) in PbTe without SOC are shown in Fig. 2. From the plot, we can see that, for each impurity X, there is a sharp peak located at 10 eV, approximately. The

PDOS of the neighboring Te atom shows a similar feature. These peaks are contributed by X p states and Te s states. On the other hand, there is also a peak close to the CB edge, which comes primarily from the p states of the impurity [18]. Since X has one more valence electron than Pb, the Fermi level now locates at the CB edge. To see more clearly the impact of X0Pb on the electronic structure of PbTe, in Fig. S1 [24] we compare the calculated total DOS of X0Pb doped PbTe to that of the undoped PbTe. To make the DOS of the two systems comparable, the energy of the DOS of the undoped PbTe is shifted slightly by a small amount (less than 0.1 eV) [17], to match the core states of the two systems. The picture clearly shows that the Pb-substituted impurity modifies the DOS of CB edge. Fig. 3 shows the band structures of X0Pb doped PbTe, including SOC. Comparing with the band structure of the undoped PbTe (Fig. 1(d)), it is clearly shown that for all the VA dopants, the impurity-induced band locates in the CB, in agreement with previous theoretical work [20]. In order to identify the position of an specific impurity-induced band, we obtained the average value of the energy of that band. In our calculations, the impurity level is found to be located above the CBM by 0.32 eV, 0.33 eV, 0.27 eV, and 0.31 eV, for P, As, Sb, and Bi, respectively. This impurity-induced levels stem from a strong interaction between the impurity p level (which lies 0.3 eV above the CBM. Although not directly comparable, an slightly different value of 0.6 eV has been obtained using GGA [20]) and the conduction band edge states of the PbTe host. As previously stated, the CBM is eight-fold degenerate (for a spin polarized calculation). The impurity splits the CBM into a six-fold degenerate level and a two-fold degenerate level, which belongs to the impurity-induced band. As shown in Fig. 3, all the impurity-induced bands shift downwards, closing the band gap (with respect to the undoped PbTe). Since the impurities have one more electron than Pb, the Fermi level now lies above the CBM. Thus, X0Pb acts as a donor in PbTe and releases its electron to the CB bottom. All the obtained band gaps (after doping) lie within the 0.1–0.15 eV range (see Fig. 3), slightly larger than those obtained for Sb- and Bidoped PbTe using GGA (60 and 20 meV [20]). The band splitting obtained using 3  3  3 supercells is approximately two times lower than the one obtained using smaller supercells [20], because the effect of the impurity becomes weaker in larger systems, but at the same time (although this rigid band model may not be the most appropriate to describe the impurity-induced levels accurately) it allows us to obtain a more clear physical picture of the energy bands near the gap region (without obtaining unrealistic metallic behaviors). Now we turn to the electronic structures of Te-substituted X (X = P, As, Sb and Bi) in PbTe. Since X has five valence electrons, it will create a hole if it replaces Te (group VIA) in PbTe. Figs. 4 and S2 [24] show the calculated PDOS of X0Te in PbTe without including SOC. Similar to the XPb cases, there is an impurityinduced core peak in the PDOS at 10 eV, approximately (see Fig. 4), which mainly arises from the X s states. Meanwhile, at this deep level of energy there are also some contribution from s states of neighboring Pb and Te atoms. The states below 10 eV are identified to be Te s states. Fig. 4 shows that there is another peak at the top of the PbTe partial VB, corresponding to the hybridization between the impurity p states and nearby Pb s states (see Fig. S2 [24]). It should be noted that the intensity of this impurity-induced state increases as the atomic number of the impurity increases. As shown in the total DOS plot (see Fig. S3 [24]), for X0Te (X = P and As) in PbTe this state does not cause a large distortion on the VB DOS of PbTe, whereas Sb and Bi introduce a sharp peak at the VBM of PbTe. Fig. 5 shows the calculated band structures of X0Te in PbTe with SOC. Comparing with Fig. 1d, we can see clearly the trend of the change of the PbTe band structure by these impurities. It is shown that PTe and AsTe bring less impact on the electronic structure of

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161

Fig. 1. Calculated band structure and partial density of states (DOS) of PbTe. (a) Band structure of PbTe primitive cell without including SOC. (b) Band structure of PbTe primitive cell with SOC. (c) Band structure of PbTe 3  3  3 supercell without including SOC. (d) Band structure of PbTe 3  3  3 supercell with SOC. (e) Local DOS of PbTe primitive cell without including SOC. (f) Local DOS of PbTe primitive cell with SOC. The Fermi level is set to 0 eV.

PbTe than SbTe and BiTe. Indeed, SbTe and BiTe split the bands at the top of the VB (see Fig. 5c and d), also changing the position of the VBM from R to C high-symmetry point (c.f. Fig. 5c and d). To further understand this finding, we analyzed the band structures of Sb- and Bi-doped PbTe without considering SOC, as shown in Fig. S4 [24]. It reveals that Te-substituted Sb and Bi indeed cause the splitting of the two valence bands, even without including the SOC (although the VBM is still at the R point). The SOC makes the splitting more significant because of the drastic reduction of the PbTe band gap. Therefore, this is not an artifact of DFT–LDA, but indicates that the doping concentration of our system is still too high. Unfortunately we could not further increase the supercell size in our calculations due to the computational cost. By obtaining the average of the impurity-induced energy band, the positions of the impurity levels were found to be located below the VBM at 96 meV, 86 meV, 60 meV and 7 meV, for P, As, Sb, and Bi, respectively. Thus, these impurities act as acceptors in PbTe, introducing additional charge carriers (holes) at the top of the VB. Finally, we will discuss the doping effect on the TE efficiency of PbTe. The amphoteric behavior of these impurities in PbTe increases the carrier concentration and, subsequently, the electri-

cal conductivity. On the other hand, the incorporated impurities such as Sb or Bi, could also reduce the thermal conductivity of PbTe due to the different atomic size and atomic mass. However, the positions of the energy levels that are introduced by these impurities are not enough to increase the Seebeck coefficient of PbTe [18]. As shown in Eq. (2), in order to enhance the Seebeck coefficient of PbTe two basic requirements must be satisfied. The first one is that the impurity-induced level should be located in the band extrema of PbTe. The second characteristic is that the energy level of the impurity should be easily reached with a reasonable doping concentration, i.e. it must located in the band extrema but relatively close to the Fermi level. In our previous work, we showed that the impurity-induced level (doping range) moves from 0.09 eV below the VBM (p-type) to 0.15 eV above the CBM (n-type) [25]. Thus, for all group VA impurities, the XPb-induced energy level seems to be relatively deep into the PbTe CB, far above the Fermi level. In comparison, the XTe-induced energy level, which lies at the top of the PbTe VB, meets both requirements. However, for P and As doping in PbTe, as shown in Fig. S2 [24], the XTe states are not sharp enough to distort the DOS of PbTe sufficiently and hence effectively increase the derivative dn/dE, as shown in

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K. Xiong et al. / Computational Materials Science 97 (2015) 159–164 4.0

(c) 4.0

0

PPb:PbTe w/o SOC

3.5 3.0

Te bulk

2.5

Pb bulk

Density of States (a.u.)

Density of States (a.u.)

(a)

2.0 1.5 1.0 0.5

Te near P Pb near P P

EF

0.0 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

1

2

3 4

5

0

SbPb:PbTe w/o SOC

3.5 3.0

Te bulk

2.5

Pb bulk

2.0

Te near Sb

1.5 1.0 0.5

Pb near Sb Sb

6

3.0

Te bulk

2.5

Pb bulk

Density of States (a.u.)

Density of States (a.u.)

(d)

0

AsPb:PbTe w/o SOC

3.5

2.0 1.5 1.0 0.5

Te near As Pb near As As

1

2

3 4

5

6

Energy (eV)

Energy (eV)

(b) 4.0

EF

0.0 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

EF

0.0 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

1

2

3

4

5

6

Energy (eV)

4.0

0

BiPb:PbTe w/o SOC

3.5 3.0

Te bulk

2.5

Pb bulk

2.0

Te near Bi

1.5 1.0

Pb near Bi

0.5

EF Bi 0.0 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

1

2

3

4

5

6

Energy (eV)

Fig. 2. Local DOS of (a) P0Pb, (b) As0Pb, (c) Sb0Pb, and (d) Bi0Pb doped PbTe, obtained without considering SOC. The Fermi level is set to 0 eV.

Fig. 3. Band structure of (a) P0Pb, (b) As0Pb, (c) Sb0Pb, and (d) Bi0Pb doped PbTe. The band structures were obtained considering SOC. The Fermi level is set to 0 eV.

Eq. (2). On the contrary, both Sb and Bi impurities cause the required distortion of the DOS at the top of PbTe VB. Thus, they meet both requirements for enhancing the PbTe Seebeck coefficient. However, the two basic requirements are a necessary but not sufficient condition, as discussed by Jaworski et al. [14]. In their work, it is found that, in PbTe, neither Pb-substituted Sb nor Te-

substituted Sb would lead to the enhancement of PbTe Seebeck coefficient. For their PbSbxTe1x samples, they observed localized hole carriers, which do not contribute to transport. Therefore, even though Te-substituted Sb in PbTe brings a large peak in the DOS, this impurity-induced state will not cause enhancement of the Seebeck coefficient if it is a localized state.

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K. Xiong et al. / Computational Materials Science 97 (2015) 159–164

(a) 4.0

2.5

Te bulk Pb bulk

2.0 1.5 1.0 0.5

Te near P Pb near P P

EF

1

2

3

4

5

3.0

Te bulk

2.5

Pb bulk

2.0 1.5 1.0 0.5

Te near Sb Pb near Sb Sb

6

3.0

Te bulk

2.5

Pb bulk

Density of States (a.u.)

Density of States (a.u.)

(d)

0

AsTe:PbTe w/o SOC

3.5

2.0 1.5 1.0 0.5

Te near As Pb near As As

1

2

3

4

5

6

Energy (eV)

Energy (eV)

(b) 4.0

EF

0.0 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

0.0 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

0

SbTe:PbTe w/o SOC

3.5

Density of States (a.u.)

Density of States (a.u.)

3.0

(c) 4.0

0

PTe:PbTe w/o SOC

3.5

EF

0.0 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

1

2

3

4

5

6

Energy (eV)

4.0

0

BiTe:PbTe w/o SOC

3.5 3.0

Te bulk

2.5

Pb bulk

2.0 1.5 1.0

Te near Bi Pb near Bi

0.5

Bi EF 0.0 -11 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0

1

2

3

4

5

6

Energy (eV)

Fig. 4. Local DOS of (a) P0Te, (b) As0Te, (c) Sb0Te, and (d) Bi0Te doped PbTe, obtained without considering SOC. The Fermi level is set to 0 eV.

Fig. 5. Band structure of (a) P0Te, (b) As0Te, (c) Sb0Te, and (d) Bi0Te doped PbTe. The band structures were obtained considering SOC. The Fermi level is set to 0 eV.

4. Conclusions In summary, in this work we have obtained the electronic structure and energy levels of group VA impurities in PbTe TE material by ab initio calculations. These impurities act as donors if they substitute Pb atoms, whereas they act as acceptors if they substitute

Te atoms. We find that the Pb-substituted impurities introduce a defect state at 0.3 eV above the CBM, while the Te-substituted impurities introduce a state at the edge of the PbTe VB. Our results indicate that these impurities can increase the charge carrier concentration of PbTe and, hence, enhance the electrical conductivity of PbTe. Moreover, the Pb-substituted impurities may not be suit-

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able to enhance the PbTe Seebeck coefficient and, consequently, the TE efficiency. In comparison, for Te-substituted impurities in PbTe, P and As do not cause significant change in the PbTe DOS and, thus, they do not increase figure of merit of PbTe by enhancing the Seebeck coefficient or the electrical conductivity, whereas Sb and Bi could enhance the PbTe Seebeck coefficient (and, consequently, ZT) if the impurity-induced states are delocalized. Acknowledgements The authors thank the II-VI Foundation, a private foundation, for financial support of this work. Calculations were done at the Texas Advanced Computing Center (TACC). Appendix A. Supplementary material Supplementary data associated with this article can be found, in the online version, at http://dx.doi.org/10.1016/j.commatsci.2014. 10.027. References [1] G.J. Snyder, E.S. Toberer, Nat. Mater. 7 (2008) 105. [2] M.S. Dresselhaus, G. Chen, M.Y. Tang, R. Yang, Z. Ren, Adv. Mater. 19 (2007) 1043. [3] J. Yang, T. Caillat, MRS Bull. 31 (2006) 224. [4] J.H. Dughaish, Physica B 322 (2002) 205.

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