a (Å)

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1 Supplementary Figures a Intens ity (a.u.) x=0.09 x=0.07 x=0.06 x=0.05 x=0.04 x=0.03 x=0.02 x=0.01 x=0.00 P 1- x S 2x/3 Se θ (deg.) a (Å) This work Vegard's law x (mol.) Supplementary Figure 1. Phase characterization. The X-ray diffraction patterns (a) and lattice parameters for P 1-x S 2x/3 Se (x = 0~0.09) solid solutions. ΔK (nm - 1 ) x = 0.00 x = 0.01 x = 0.03 x = 0.05 x = K 2 C (nm - 2 ) Supplementary Figure 2. The modified Williamson-Hall plots. The peak roadening analysis y the modified Williamson-Hall plots for P 1-x S 2x/3 Se (x = 0~0.07) solid solutions according to the XRD data.

2 a P 0.96 S Se P 0.96 S Se 1 µm 0.5 µm c P 0.93 S Se d P 0.93 S Se 1 µm 0.5 µm Supplementary Figure 3. Microstructures of P 0.96 S Se and P 0.93 S Se. Dislocations in P 0.96 S Se (a, ) and P 0.93 S Se (c, d), confirming the increased dislocation density with increasing S 2 Se 3 concentration.

3 Accumulative κ L (W m - 1 K - 1 ) a U- and N- s cattering PSe at 300 K Mean- free path (nm) Accumulative reduction in κ L (W m - 1 K - 1 ) Point defects Dis locations Point defects and dis lcations P 0.95 S Se at 300 K Frequency (THz) Supplementary Figure 4. Predictions of lattice thermal conductivity using Born-von Karman approximation. Predicted mean-free path dependent accumulative lattice thermal conductivity for PSe (a), and the predicted frequency dependent accumulative reduction in the lattice thermal conductivity for P 0.95 S 0.33 Se due to point defects and/or dislocations (). The modeling is ased on a Born-von Karman approximation and the predictions are for 300 K.

4 Supplementary Tales Supplementary Tale 1. Equations for phonon relaxation times (τ) associated with different types of scattering processes, where τ U, τ N, τ PD, τ DC and τ DS are the relaxation times due to the scattering of Umklapp processes, Normal processes, point defects, dislocation cores and dislocation strains, respectively. Types of scattering mechanisms Relaxation times (τ (s 1 )) 1 Umklapp processes τ 1 2 k B V0 3γ 2 ω 2 T U = (6π 2 ) 1 3 M5v 3 1 Normal processes τ 1 2 k B V0 3γ 2 ω 2 T N = (6π 2 ) 1 3 M5v 3 Point defects τ PD 1 = V (ω 4 4πv 3. x i i 4 3 V) Dislocation cores τ 1 DC = N D Dislocation strains 12 M i M M ε` 9 a i a a ; 2 < τ 1 DS = A B 2 D γ 2 ω r r v 2 2 L v T v 3 ω3 Supplementary Tale 2. Parameters used for the Deye modeling. Parameters Description Values Ref. β ratio of N- to U- processes 4 1, estimated V" Average atomic volume of P 1-x S 2x/3 Se a 3 i /8 m 3 - M" Average atomic mass for P 1-x S 2x/3 Se M P1-xS2x/3Se /( ) kg - v Average sound speed 1787 m s -1 This work v L Longitudinal sound speed 3150 m s -1 This work v T Transverse sound speed 1600 m s -1 This work γ Gruneisen parameter 1.7 x i Impurities concentration in solid solutions x S 0.07 This work M i Atomic mass of impurities Μ S = g mol -1 - M Atomic mass of matrix Μ P = g mol -1 - ε` Anharmonic parameter 64 a i Lattice parameters for P 1-x S 2x/3 Se x i Å This work A Lattice parameters for PSe Å This work N D Dislocation density of P 1-x S 2x/3 Se [60(x i -0.01)+1] x i 0.01 cm -2 This work B D Burgers vector m This work A Pre-factor for dislocation scattering 0.96 r Poisson s ratio

5 Supplementary Tale 3. Parameters used for the modified Williamson-Hall model. Parameters Description Value Ref. θ B Diffraction angle at the exact Bragg position This work hkl Indices of crystal plane (200) (220) (400) (420) (422) - Δ2θ Full width at half-maximum (FWHM) of the corresponding diffraction peak at θ B This work λ Wavelength of the synchrotron X-ray 6.87 Å - K K=2sinθ B /λ 2sinθ B /λ ΔK ΔK=(Δ2θ)cosθ B /λ (Δ2θ)cosθ B /λ A Parameter determined y the effective outer cut-off radius of dislocations 2.6 B D Burgers vector m This work C Average dislocation contrast factor C h00 (1-q(h 2 k 2 +h 2 l 2 +k 2 l 2 )/(h 2 +k 2 +l 2 ) 2 ) Average dislocation contrast factor C h00 corresponding to the h00 reflection determining This work, y elastic modulus q Parameter determined y the elastic modulus -2.7 This work, c 11 c 12 c GPa Elastic modulus 19.3 GPa 15.9 GPa O Non-interpreted higher-order error terms Not included in this work d Average crystallite size nm This work, fitted N D Dislocation density m -2 This work, fitted 7 8 Supplementary Tale 4. Mechanical strength measured y modified small punch (MSP) technique for several materials at room temperature. Composition of compounds Thickness of samples (mm) Load at failure (N) MSP strength (MPa) PSe Na 0.02 P 0.98 Te P 0.98 S Se P 0.95 S Se P 0.95 S Se P 0.93 S Se P 0.93 S Se

6 Supplementary Discussion To etter understand the mean free path and the frequency dependent lattice thermal conductivity (κ L ) accumulation, it is elieved to e more precise if taking the effect of reduced phonon group velocity at high phonon energies into account 9, 10. This leads the Born-von Karman 11 dispersion relationship to e more reliale than that of Deye model. This improvement has een adopted to understand the lattice thermal conductivity of ulk and low-dimensional thermoelectrics or metals including Si-Ge 12 and PTe 13, silver 14 and Al-Si etc. 15. Using the same method, we modeled the phonon transport for PSe, ased on a Born-von-Karman type phonon dispersion of ω = 2vq/πSin(2q/q c π) rather than the Deye type assuming ω = vq, where ω is the phonon frequency, v is the sound velocity, q is the wave vector and q c is the cut-off wave vector. The predicted accumulative κ L due to Umklapp and Normal scattering in pure PSe, helps us understand the important range of mean free path that contriutes to heat conduction. Furthermore, the predicted phonons frequency dependent accumulative reduction in κ L distinguishes the effect of each scattering mechanism. As shown in the Supplementary Fig. 4a, 50% of the heat in pure PSe is carried y phonon of mean-free path up to 13 nm and the central 80% heat is carried y mean-free path (MFP) etween 4 nm and 400 nm at 300 K. As compared with the availale prediction y first-principles calculations 16, which includes the contriutions of optical phonons (with even shorter MFPs) and results in a higher lattice thermal conductivity, the current model prediction shows a very good agreement on the normalized κ L -accumulation within the overlapped range of MFP. The randomly distriuted dense in-grain dislocations here roughly enale a range of mean-free path to e achieved for reducing the lattice thermal conductivity y 50% at the 300 K (Fig. 3a). It is shown that dense in-grain dislocations, indeed lead to an effective scattering of phonons with mid-frequencies and therefore a significantly reduced lattice thermal conductivity (Supplementary Fig. 4). The mechanical property measurements were carried out y modified small punch (MSP) technique 17, a method has een successfully used to characterize the mechanical strength of thermoelectric materials 18, 19. For the MSP measurements, the disk sample was supported y a die with a center hole of 3.93 mm in diameter, and was punched y a cylindrical pressure head of 2.35 mm in diameter with a speed of 0.05 mm min -1. All of the specimens were fine polished and the load was monitored y a high-accuracy transducer. The MSP strength σ MSP can e calculated via: σ MSP = 3P max /(2πt 2 )[1-(1-ν 2 )/4 2 /a 2 +(1+ ν)ln(a/)], where P max is the measured load at failure, t is the thickness of the sample, ν is the Poisson s ratio which is estimated as for PSe-ased materials and as for PTe-ased materials, a is radius of the center hole and is the radius of pressure head, respectively. The mechanical strength is otained y averaging 3~5 samples for each composition, and the results for P 1-x S 2x/3 Se, PSe and PTe are shown in Supplementary Tale 4. One may claim that the mechanical strength of P 1-x S 2x/3 Se decreases a little with increasing density of dislocations (increasing x), however, the strength for samples with dense dislocations is still comparale to that of PTe without dislocations. Therefore, dense in-grain dislocations here do not degrade the mechanical strength to e unacceptale.

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