Degenerated Hole Doping and Ultra‐Low Lattice Thermal Conductivity in Polycrystalline SnSe by Nonequilibrium Isovalent Te Substitution

Abstract Tin mono‐selenide (SnSe) exhibits the world record of thermoelectric conversion efficiency ZT in the single crystal form, but the performance of polycrystalline SnSe is restricted by low electronic conductivity (σ) and high thermal conductivity (κ), compared to those of the single crystal. Here an effective strategy to achieve high σ and low κ simultaneously is reported on p‐type polycrystalline SnSe with isovalent Te ion substitution. The nonequilibrium Sn(Se1− x Te x ) solid solution bulks with x up to 0.4 are synthesized by the two‐step process composed of high‐temperature solid‐state reaction and rapid thermal quenching. The Te ion substitution in SnSe realizes high σ due to the 103‐times increase in hole carrier concentration and effectively reduced lattice κ less than one‐third at room temperature. The large‐size Te ion in Sn(Se1− x Te x ) forms weak Sn—Te bonds, leading to the high‐density formation of hole‐donating Sn vacancies and the reduced phonon frequency and enhanced phonon scattering. This result—doping of large‐size ions beyond the equilibrium limit—proposes a new idea for carrier doping and controlling thermal properties to enhance the ZT of polycrystalline SnSe.

eV for x = 0 and 0.38 eV for 0.4, which are almost a half of the bandgaps obtained by diffuse reflectance (Fig. S7), suggesting this range is in the intrinsic T region of semiconductor and supporting that the estimation of bandgap of Sn(Se 1x Te x ) is reasonable. Figure S6. (a) Sn 3d, (b) Se 3d, and (c) Te 3d core level hard X-ray photoemission spectroscopy (HAXPES) spectra for Sn(Se 1x Te x ) bulks with x = 00.4 at RT. There are two peaks in each Sn 3d, Se 3d, and Te 3d core spectrum, corresponding to the spin-orbit-splitting of 3d 5/2 and 3d 3/2 levels. The Sn 3d 5/2 and Te 3d 5/2 core level spectra are fitted by single symmetric peak each. The Se 3d core level spectrum is fitted by two symmetric peaks for Se 3d 3/2 peak (shaded with orange) and Se 3d 5/2 (shaded with green). The black lines and the red lines indicate the background and total fitting curves, respectively.    Chemical composition line profiles between AB in (a) for Sn, Se, and Te elements. The chemical composition mappings of Sn, Se, and Te clearly support the uniformity in the whole region of the bulk.

Section 5. Electronic structure calculation of Sn(Se 1x Te x ) bulks
The relaxed crystal structures of SnSe, SnSe 0.75 Te 0.25 , and SnSe 0.5 Te 0.5 were obtained by DFT with GGA-PBE functional [1] (cut off energy: 600eV, -centered k-mesh: 6×14×14). The stable structures are shown in Fig. S11, where the Se sites are substituted by Te ions. The a, b, c, and V calculated for the Sn(Se 1-x Te x ) primitive cell models are summarized in Fig. S12.
Considering that the ground-state lattice parameters by DFT typically include errors within 2-3%, the calculated results are consistent with the experimental data within the variation of the functionals. This trend was confirmed also by PBEsol functional [2] . The gradient for the    and Te must be less than those of the elementary phases to avoid the formation of the Sn, Se or Te elemental phases, thus ∆ Sn < 0, ∆ Se < 0, and ∆ Te < 0 are required. Furthermore, to suppress the formation of the competing secondary phases SnSe 2 (trigonal, P3m1), [3] and SnTe (cubic, Fm3 m), [4] the following conditions must be satisfied as well. restricted to the red area as shown in Fig. S14, drawn with CHESTA. [5] Defect calculations. For the defect formation enthalpy (ΔH) calculations, a 72-atoms supercell with 1×3×3 primitive cells and a Γ-centered 3×3×3 k-mesh were used, in which atomic internal coordinates were fully relaxed by GGA-PBE functional until all the forces on the atoms became less than 0.03 eV/Å and the total energy difference was smaller than 10 -4 eV with the fixed cell parameters so as to meet the dilute limit condition. The corresponding plane wave cut-off energy for the basis was set to 250 eV. The convergence test of the cut-off energy was shown in Fig. S15. The difference in total energy between the cut-off energy 250 eV and 500 eV is within 1 meV/atom, which is much smaller than the calculation accuracy of VASP (~10 meV). We considered the intrinsic point defects in Sn(Se 1x Te x ) phase, including vacancies (V Sn , V Se, and V Te ), cation substitutions at the Sn site (Se Sn , Te Sn ), anion substitutions at Se site (Sn Se , Te Se ), and interstitials (Sn i , Se i , Te i ).
, where D is the total energy of the supercell with the defect in the charge state q, and h is that of the perfect host supercell. The ( + ) is the additional energy of electrons trapped by the defect at E F . n i indicates the number of the i-th atoms added (n i < 0) or removed (n i > 0), and is the chemical potential of the i-th atom with respect to that of the corresponding elemental phase ( el ) by = el + ∆ . corr. is the correction value for the defect formation enthalpy in order to suppress the considerable supercell finite-size effects. The standard correction methods for finite-size effects were applied, including the potential-alignment correction and the image charge correction [6] to get the accurate value of the total energies for the charged defect-containing models.
The convergence test of the formation energy of V Sn 2 as a function of the supercell size is shown in Fig. S16 for the reason that V Sn is the most important defect in Sn(Se 1x Te x ) and the 2 charge state is affected most seriously by the charge interaction in the finite supercell.
The largest size of supercell is limited to 2×5×5 supercell containing 400 atoms due to the computational constraints. The origins of the relative energy below are taken at the zero corrected energy of the largest supercell. It shows that the difference between the corrected energy of the 1×3×3 supercell and the largest 2×5×5 supercell is within 0.1 eV, which guarantees that the 1×3×3 supercell provides satisfactory converged values. is the formation entropy (we employed the constant entropy approximation with the value of 10k B .), [7][8][9][10] and D is the defect frozen temperature (we take 300 K for pure SnSe and 973.15 K for Te-doped SnSe due to the rapid quenching process which makes the defects frozen in the high temperature state). The E F,e , D, for given and D and the electron concentration (n) in the conduction band and the hole concentration (p) in the valence band were determined so as to satisfy the charge neutrality condition at 300K. The E F,e at RT for given and D were determined so as to satisfy the charge neutrality condition. Figure S17 summarizes the H of defects in SnSe, Sn(Se 0.75 Te 0.25 ), and Sn(Se 0.5 Te 0.5 ) as a function of E F at the chemical potential conditions of A point (Sn-rich/Se-poor chemical condition), B point (Sn-moderate/Se-moderate chemical condition), and C point (Sn-poor/Se-rich chemical condition) in the chemical potential window (Fig. S14). The calculated hole concentrations of SnSe as a function of E F are shown in Fig. S18. The chemical potential condition of pure SnSe was set at Se-moderate condition (B-point in Fig. S14)  and it is slightly compensated by the electrons produced by V Se (8.28×10 6 cm 3 ), thus, yielding the hole concentration of 2.14×10 14 cm 3 . On the othra hand, the chemical potential condition of Sn(Se 0.5 Te 0.5 ) was set at Se-poor condition (A-point in Fig. S14) because the Sn impurities appeared in the XRD patterns ( Fig. 1(a)).    condition) for (d-i) in the chemical potential window (Fig. S14). E F is measured from valence band maximum (E V ) and ranges to conduction band minimum (E C ). The V denotes vacancy, and the subscripts denote defect sites, where i means interstitial sites. The vertical dashed

Section 7. Chemical bonding analysis of Sn(Se 1x Te x )
Chemical bonding analysis. The chemical bonding analysis for the Sn-Se/Te bonds of SnSe and Sn(Se 0.5 Te 0.5 ) was performed by using the crystal orbital Hamiltonian overlap (COHP), [11] calculated by LOBSTER codes. [12] Figure S19 summarizes the bond length and the -iCOHP value (the integrated -COHP up to E F, to estimate the bonding strength) of Sn-Se/Te bonds. In the pure SnSe, the inter-layer Sn-Se bond (Sn1-Se2 and Sn2-Se3 bond in Fig. S19(c)) possesses the shortest bond length than the in-layer Sn-Se bond (Sn1-Se1 and Sn2-Se2 bond in Fig. S19(c)), corresponding to the largest -iCOHP value. By introducing the Te ions, the This variation is consistent with the result that V Sn is more easily formed in Sn(Se 0.5 Te 0.5 ) with the longer bond distance and weaker bond strength for Sn-Te bond. Note that the different bond dissociation energy for Sn-Te (= 338.1 kJ/mol) is smaller than than for Sn-Se (= 401.2 kJ/mol), [13] which also expects that the introduction of Te ions in SnSe lattice leads to the easier V Sn formation. Actually, the SnTe shows much high p > 10 20 cm 3 due to intrinsic V Sn . [14,15]

Section 6. Phonon transport calculations of Sn(Se 1x Te x )
Phonon transport calculations. The phonon transport calculations were performed by solving the Peierls-Boltzmann transport equation within the relaxation time approximation, as implemented in the ALAMODE code. [16] For Sn(Se 1x Te x ), a 2×3×3 supercell (144 atoms) with a Γ-centered 2×3×3 k-mesh was used for the harmonic interatomic force constants (IFCs), and a 1×2×2 supercell (32 atoms) with a Γ-centered 3×4×4 k-mesh was used for the anharmonic 3rd-order IFCs. The harmonic IFCs were fixed to the values determined by the finite-displacement approach. [17,18] We included all allowed interactions for the harmonic IFCs and the 3rd-order IFCs inside the cutoff radii of 12 bohr. The DFT calculations to obtain the force were performed using the PBEsol functional with a plane-wave energy cutoff of 400 eV, a convergence criterion for the electronic self-consistency loop of 10 8 eV, and the Gaussian smearing method with a smearing width of 0.05 eV. The non-analytic correction was included to the dynamical matrix by the mixed-space approach [19] with the Born effective charges of constituent elements and dielectric constant calculated by density functional perturbation theory. [20] The lattice thermal conductivity ( lat ) was calculated by solving the Boltzmann transport equation under the single-mode relaxation time approximation with a 5×13×13 q point mesh. The convergence test in terms of the q-meshes is described in Fig. S20, which confirms the above q-mesh has enough accuracy. Phonon-isotope scatterings are considered for both phases. [21] Figure S20. The convergence test in terms of the q-meshes for the temperature (T) dependence of  lat along a-, b-, and c-axes of (a) SnSe and (b) Sn(Se 0.5 Te 0.5 ).   (Fig. S8), the effect of l  to experimental  lat would be negligible. Figure S23. The calculated scattering phase space (W ± /m 2 ) at T = 300 K for SnSe and Sn(Se 0.5 Te 0.5 ). The phonon scattering emission part largely increases in the 1.5~3 THz and 4~5 THz frequency range for the Sn(Se 0.5 Te 0.5 ). Note that the phonon frequency range, where the phonon scattering emission part increases, overlaps with the decrease in  (Fig. 5(c)), indicating the phonon scattering channels is increased in SnSe by Te ion substitution.