Optimizing the thermoelectric performance of low-temperature SnSe compounds by electronic structure design
A. J. Hong,aL. Li,aH. X. Zhu,aZ. B. Yan,aJ.-M. Liu*aand Z. F. Renb
Recently, the SnSe compound was reported to have a peak thermoelectricfigure-of-merit (ZT) of2.62 at 923 K, but the ZTvalues at temperatures below 750 K are relatively low. In this work, the electronic structures of SnSe are calculated using the density functional theory, and the electro- and thermo- transport properties upon carrier density are evaluated by the semi-classic Boltzmann transport theory, revealing that the calculated ZT values along the a- andc-axes below 675 K are in agreement with reported values, but that along theb-axis can be as high as 2.57 by optimizing the carrier concentration to n 3.6 1019 cm3. It is suggested that a mixed ionic–covalent bonding and heavy-light band overlapping near the valence band are the reasons for the higher thermoelectric performance.
I. Introduction
Thermoelectric (TE) materials and devices have been receiving much attention over the past few decades due to their capabil- ities for direct and reversible conversion between electrical energy and heat energy.1–10The search for proper TE materials with high conversion efficiency is important for developing advanced TE technologies.11For a TE device, the thermal power PF¼S2sand the dimensionlessgure of meritZT¼S2sT/ktot
are two core parameters,1whereS,T,s,ktot,kl, andkerepresent, respectively, the Seebeck coefficient, absolute temperature, electrical conductivity, total thermal conductivity, and lattice and electronic components ofktot(ktot¼kl+ke). Nevertheless, for realistic TE materials, simultaneously achieving high PF and lowktot remains a contradictory issue due to the competing dependences of the parameters (S,s,kl, andke) on chemical and electronic structures,5,9,10 making maximization of the TE performance extremely challenging. Because of such complex- ities, theoretical predictions from the full-scalerst-principles calculations have been of interest in guiding TE materials synthesis.
It has been proposed that a combination of electronic crystal and phononic glass in one material is a promising approach.1,3–7 This requires the simultaneous and delicate design of the crystalline and electronic structures. The electrical conductivity sdepends on the carrier density and mobility, which are both determined by electronic structure, while the Seebeck coeffi- cientSis essentially determined by the gradient of density of
states (DOS) of the conduction band near the Fermi level (chemical potential).11Surely, the lattice thermal conductivitykl
can be substantially suppressed by modulating the material microstructure,3 but the electronic thermal conductivity keis again highly dependent on the electronic structure. In a general sense, one is in a good position to optimize the TE performance of a material by tentatively designing the electro- and thermo- transport behaviors based on the electronic structure. In other words, electronic structure engineering has become a major branch of TE materials science.
To engineer the transport behaviors, one of the effective and oen-employed approaches is to modulate the carrier density (n) by chemical substitution and charge doping (thus varying the chemical potential m) without seriously distorting the topology of the electronic structure. Such carrier density modulation may lior lower the Fermi level (i.e.m) so that the ZTcan be optimized. A common strategy is to choose some TE parent compounds with one or more outstanding TE parame- ters and then improve the other parameters in order to reach optimized PF andZTvalues. Along this line, a series of parent compounds have been explored, followed by extensive investi- gations on the consequence of various doping trials for each compound.
Recent studies9,12–14 revealed that selenium-based compounds are promising TE candidates. While polycrystalline b-CuAgSe exhibited very lowktot(<0.5 W m1K1) and theZT reaches up to0.95 at 623 K (ref. 14), polycrystalline Cu2Se was found to have a ZT value as high as 1.5 at T ¼ 1000 K.12 Surprisingly, the single-crystal SnSe compound was reported to have aZTvalue as high as 2.62 along theb-axis and 2.30 along thec-axis atT923 K.9These values represent the highest ones reported so far, and thus allow SnSe to be a good platform for exploring the possibilities for even better TE performance.
aLaboratory of Solid State Microstructures and Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China. E-mail: liujm@nju.
edu.cn; Tel: +86 25 83596595
bDepartment of Physics and TcSUH, University of Houston, Houston, TX 77204, USA Cite this:J. Mater. Chem. A, 2015,3,
13365
Received 6th March 2015 Accepted 20th May 2015 DOI: 10.1039/c5ta01703c www.rsc.org/MaterialsA
Journal of
Materials Chemistry A
PAPER
Similar to Cu2Se, the SnSe compound has two phases.15,16The high-Tphase possesses ultralow kl (<0.25 W m1 K1 at T >
800 K) and moderate PF values, thus leading to surprisingly high ZT values.9 Very differently, the low-T phase, however, shows much lower TE properties, and itsZT value along the b-axis is only1.0 atT750 K, the transition point between the two phases.
Considering the requirement for intermediate-T applica- tions and the fact that the low-T SnSe compound has high Seebeck coefficient (S> 500mV K1atT< 675 K andS>
400mV K1atT¼300–750 K), the promotion of its overall TE performance is highly desired. Experimentally reported carrier density of SnSe alloys in the intermediateTrange is relatively low (n 1018cm3),9which leaves sufficient space for carrier density modulation by means of low-level carrier doping.
II. Approach and theory
In this work, we intend to perform a full-scalerst-principles electronic structure calculation on the low-TSnSe phase and subsequently evaluate the electro- and thermo-transport properties atniteTby means of the semi-classic Boltzmann transport theory. In this scheme, all the thermodynamic properties relevant to electron transport can be calculated quantitatively so long as chemical potential m (or doping carrier densityn) is given. Along this line, one is allowed to investigate the consequence of charge/carrier doping (low level) without seriously distorting the topology of the elec- tronic structure. Similar schemes have been extensively employed for TE materials design and performance optimi- zation.17–20 Our calculations indeed predict the signicant impact of carrier density variation on the TE properties of low- TSnSe phase.
We start from the experimentally determined crystal struc- ture of the low-TSnSe phase.15It has the space group Pnma (#62), as shown in Fig. 1, and the lattice constants area¼11.58
˚A,b¼4.22˚A, andc¼4.40˚A. The Sn and Se atoms are located on two different planes with the dihedral angle of20. We employ the density functional theory (DFT) scheme with the full- potential linearized augmented plane-wave (LAPW) method implemented.21The WIEN2k package can offer high-precision and accurate calculations on electronic structure with relatively low efficiency. The exchange and correlation interactions are described using the generalized gradient approximation (GGA) and the Perdew–Burke–Ernzerhof (PBE) functional modied by the Becke–Johnson potential (mBJ).22 The mBJ is a local approximation of an atomic“exact-exchange”potential and a screening term. The mBJ method allows the calculation of band gaps with accuracy similar to the very expensive GW calcula- tions22 and can improve the band gap in sp-bonded semi- conductors.23The muffin-tin radii are set as 2.5 a.u. for both Sn and Se atoms with a well-converged basis set determined byRMT Kn¼7.0, corresponding to 5907 plane waves. In our calcu- lations, wex the lattice symmetry and lattice constants, and then all atoms in the conventional cell are allowed to relax sufficiently until the minimization of forces is less than 0.5 mRy per a.u.
Given the whole set of electronic structure data and varying m, we employ the semi-classic Boltzmann transport theory to calculate the electro- and thermo-transport behaviors. The calculated transport coefficients are found to be converged using a shied 2055 53kmesh. The originalk mesh is interpolated onto a mesh ve times as k dense. All of the calculations are implemented by solving the Boltzmann trans- port equation with the constant relaxation time approxima- tion.24 In details, the electrical conductivity tensors and electronic thermal conductivity tensors at non-zero electric current are obtained by the following equations:24
sabðT;mÞ ¼ 1 U ð
sabð3Þ
vf v3
d3 (1)
k0abðT;mÞ ¼ 1 e2TU
ð
sabð3Þð3mÞ2
vf v3
d3 (2)
wheree,U,3, andfare the electron charge (e), reciprocal space volume (U), carrier energy (3), and Fermi distribution function (f), respectively. The conductivity tensorssabcan be expressed as:
sabð3Þ ¼e2s N
X
i;k
nanbdð33i;kÞ
d3 (3)
wheresandna(nb) are the relaxation time and electron group velocity;kis the wave-vector. In the standard procedure, theS andkeat zero electric current can be obtained:
Sij(T,m)¼(s1)aicaj (4) kij¼k0ijTcib(s1)abcaj (5) where
cabðT;mÞ ¼ 1 eTU
ð
sabð3Þð3mÞ
vf v3
d3: (6)
It should be mentioned that relaxation time s is weakly dependent on the band index and k-direction, and is thus spatially anisotropic for most cases. However, this dependence is quite trivial and can be neglected safely without much damage to the results, even in the quantitative sense. Earlier studies25,26did indicate that thesis orientation-independent for most materials, i.e. approximately isotropic. Even for super- conducting cuprates whose electrical conduction is known to be anisotropic, this relaxation time remains almost isotropic.
Fig. 1 A schematic drawing of the lattice structure of low-TSnSe compound.
Therefore, realistic calculations oen take this approximation, although these properties may be spatially anisotropic.
As is well known, doping carrier densitynis given by:
n¼n0Ð
f(3,m,T)D(3)d3 (7) wheren0is the valence electron number and D(3) is the total DOS as a function of3 as evaluated from the electronic struc- ture. There is a one-to-one correspondence between doping carrier densitynand chemical potentialmat a givenT. To this stage, a self-consistent calculation based on the Boltzmann transport theory is immediate, and no details of the practical calculation procedure are repeated here.
III. Results and discussion
A. Electronic structures
The calculated band structures and DOS along the high symmetry lines are shown in Fig. 2. The reciprocal spacea*,b*, andc*axes are parallel to the real spacea-,b-, andc-axes. All energies are referenced to the middle of the band gap. It can be seen that the conduction band minimum (CBM) is located at theGpoint, and there is a local CBM at (0.000, 0.329, 0.000) along theG–Yline. The rst and second valence band maxi- mums (VBMs) are located at (0.000, 0.000, 0.354) and (0.000, 0.000, 0.444) along the G–Z line. There is the third VBM at (0.000, 0.316, 0.000) along theG–Yline, implying two indirect band gaps,Eg¼0.849 eV and 0.862 eV, along theG–Yline. Along theG–Zline, there are also two indirect band gaps,Eg¼0.842 eV and 0.828 eV. The similar gaps allow similar transport proper- ties along thea- andb-axes. As is well known for an indirect bandgap semiconductor, electrons cannot shifrom the VBM to the CMB without momentum change. Fig. 2 shows that the VBM and CBM have nearly equal momentum at gap➁. Namely, electrons are more easily excited into the conduction band at band gap➁than those at other band gaps.
It is noted that gap ➁(Eg ¼ 0.862 eV) almost equals the previously measured value of 0.86 eV.9We then focus on the upper part of the highest valence band. The dispersion along theG–Zline is greater than that along the planarA–Z. Theat part of the highest-valence band near the VBM is benecial for high Seebeck, and the conductivity is determined mostly by the steep band. In fact, earlier work27–30indicated that a mixture of the heavy and light bands near the valence edge is favorable for high TE performance, because the light band allows good
electrical conduction and the heavy band benets the high Seebeck coefficient. In addition, Fig. 2 shows that the total DOS increases more rapidly near the VBM than that near the CBM.
The major DOS contribution to the CBM comes from the Sn atoms, while the Se atoms contribute more to the DOS near the VBM.
The orbital-decomposed band structures are presented in Fig. 3(a) and (b), where the coarseness of curves scales the DOS intensity. It is seen that the VBM mainly comes from the Se 4p states, and the Sn 5p states only have weak contribution.
This suggests that the p-type doping at the Sn site will increase the carrier density so as to improve the electrical conductivity, while the VBM shape can be roughly maintained so as to keep the high Seebeck coefficient. The projected DOS in the [10.0 eV, 10.0 eV] interval is shown in Fig. 3(c). A comparison of Fig. 3(c) with 2 allows us to conclude that the bands from the Fermi level to 5.0 eV are mainly from the Sn 5p and Se 4p states, while the bands from5.0 eV to the Fermi level primarily come from the Se 4p states. The highest bonding peaks near the VBM show the characteristics of the Se 4p electrons but also contain contributions from the Sn 5s electrons, implying the weak s–p hybridization between the Sn and Se atoms. Such s–p hybridization can lead to dramatic DOS variation near the VBM, which is favorable for a high Seebeck coefficient.
Fig. 2 Calculated band structure and DOS spectra for Sn atoms and Se atoms as well as the total DOS for pure SnSe compound.
Fig. 3 Calculated orbital-decomposed band structures for Sn 5p orbital (a) and Se 4p orbital (b). The projected DOS spectra for Sn 5s, Sn 5p, Se 4s, and Se 4p orbitals are plotted in (c).
We also calculate the electron density and charge density difference on the Se–Se–Se plane (schematically shown in Fig. 1 by the shadow plane), as displayed in Fig. 4(a) and (b), respec- tively. The Sn–Se bond has a weak covalent component, which again conrms the s–p hybridization between the Se 4p and Sn 5s. The weak tendency for Se atoms to accumulate charge from the surrounding Sn atoms can also be seen in Fig. 4(b). It is thus suggested that the bond between the Sn and Se atoms is more or less of mixed ionic–covalent nature.
B. Thermoelectric properties
Subsequently, we investigate the electro- and thermo-transport behaviors. In Fig. 5(a)–(e) are plotted several calculated parameters as a function of n, respectively, along the three major axes atT¼675 K. TheS(n) curves show the single-peaked pattern and the peak location and height shialong different major axes. The peak values ofS(n) along thea-,b- andc-axes reach up to 544.07, 690.37, and 655.13mV K1, respectively, atn 9.81019, 2.31019, and 2.31019cm3.
To calculate other parameters, relaxation timesis needed, but extracting its value from the ab initio calculation is still challenging. Usually, the constant relaxation time approxima- tion is used,31,32and we takes¼0.51014s in the present calculations. Thes(n) dependences along the three axes are all monotonous at high doping level with small differences along theb- andc-axes, but thes(n) along thea-axis decreases with increasingnat low doping level. It can be seen from Fig. 5(a) that thes(n) along the b-axis is much larger than along the a-axis at high doping level. S(n) and s(n) exhibit opposite dependences, resulting in the PF peaks along these axes roughly atn1020to 1021cm3. The PF along theb-axis is about twice as large as that along thec-axis and almost ten times larger than that along the a-axis. The ke(n) along all three axes is very sensitive ton. Fig. 5(d) presents that thekealong the b- and c-axes rst decrease and then increase with increasing n.
However, the ke along the a-axis rst increases, and then decreases, and again rapidly increases with increasing n. In
essence, the ke(n) dependence is determined by the band structure of SnSe crystals.
As an example, the evaluated ZT(n) curves along the three major axes atT¼675 K for the p-type doped systems are pre- sented in Fig. 5(e), where measuredklvalues were taken from ref. 9. One sees thatZTis sensitive to the p-type carrier density, and a variation ofnover two orders of magnitude is sufficient to modulate theZTbetween the minimal and maximal. TheZT(n) dependence is also anisotropic, yielding the relationZTb-axis>
ZTc-axis>ZTa-axis, in agreement with experimental results.9The predicted highestZTvalue of2.57 occurs at the p-type carrier density n 3.6 1019 cm3 along the b-axis, which needs experimental checking. We also predict the ZT(n) of poly- crystalline SnSe at the same temperature, as shown in Fig. 5(e).
The highestZTvalue can reach up to 1.86 at the p-type carrier densityn4.21019cm3.
C. Comparison with experiments
Finally, we compare our calculated data with measured data. So far, measuredS,s, and,ktotdata along the three major axes of SnSe single crystals as a function ofTare available.9For such a comparison, one needs measured n(T) or m(T) data for our calculations. Given the data in ref. 9 and the assumption of constant n (¼5 1017 cm3) over the whole T range, the as-evaluated S(T) and s(T) data are plotted in Fig. 6(a)–(f), focusing on theT-range from 300 K to 700 K. The calculated data coincide reasonably well with measured data along all the three axes, although the discrepancy becomes slightly remark- able at both extremes of theTrange, particularly fors(T) along the b- and c-axes. The discrepancy is believed to most likely originate from the assumption of a constant relaxation time. In Fig. 4 Calculated valence electron charge density (a) and electron
density difference on the Se–Se–Se plane (b).
Fig. 5 Calculated TE parametersS,s,ke, PF, andZTas a function of hope-carrier densitynatT¼675 K, and theZT(n) dependence for a polycrystalline sample is also presented.
addition, we extract the measuredktot(T) data in ref. 9 to eval- uate theZT(T) data, and the results are presented in Fig. 6(g)–(i) in comparison with measured ZT(T) data.9 Again, we see consistency between the calculated and measured ZT data, particularly in the low-Trange.
The capability of the present computational scheme may be highly appreciated considering the current status of quantita- tive predictions for TE performances. It allows comprehensive design and evaluation of the TE performance for a realistic material. One can always start from the stoichiometric compound for electronic structure calculation, and then opti- mize the TE parameters by carefully tuning the chemical potential on condition of relatively low level doping so that the electronic structure topology remains qualitatively unchanged.
This strategy is no doubt helpful for guiding the practical synthesis and substitution/doping processes for better TE materials and performance.
IV. Summary
In summary, we have calculated the electronic structure and TE properties of the low-TSnSe compound using therst-principles calculations plus the semi-classic Boltzmann transport theory. It is revealed that the high Seebeck coefficient and good electrical conductivity are attributed to the s–p hybridization and the mixed heavy-light band structure near the VBM. It is predicted that proper modulation of the chemical potential or p-type carrier density can remarkably enhance the power factor PF and
gure-of-merit factorZT. The calculated results are well consis- tent with experimental data reported recently. When the p-type carrier density is enhanced to 3.6 1019 cm3, optimal ZT values up to2.57 along theb-axis atT¼675 K are predicted.
Acknowledgements
This work is supported by the National 973 Projects of China (Grant no. 2015CB654602), the Natural Science Foundation of
China (Grant no. 51431006), and “Solid State Solar Thermal Energy Conversion Center (S3TEC)”, an Energy Frontier Research Center funded by the U.S. Department of Energy, Office of Science, Office of Basic Energy Science under award number DE-SC0001299.
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