Supporting Information

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1 Supporting Information Thermal Transport Driven by Extraneous Nanoparticles and Phase Segregation in Nanostructured Mg (Si,Sn) and Estimation of Optimum Thermoelectric Performance Abdullah S. Tazebay 1, Su-In Yi 1, Jae Ki Lee, Hyunghoon Kim 3, Je-Hyeong Bahk, Suk Lae Kim 1, Su- Dong Park, Ho Seong Lee 3, Ali Shakouri, Choongho Yu 1, * 1 Department of Mechanical Engineering, Texas A&M University, College Station, TX 7783, USA Korea Electrotechnology Research Institute, Changwon, Korea 3 School of Materials Science and Engineering, Kyungpook National University, 8 Daehak-ro, Buk-gu, Daegu 7-71, Korea Birck Nanotechnology Center, Purdue University, West Lafayette, IN 797, USA *Corresponding author: chyu@tamu.edu - 1 -

2 Table S1. Atomic ratios of raw elements and volume fractions of TiO nanoparticles in 1 different kinds of solid solutions made of magnesium silicide and magnesium stannide. Note that 1% excessive magnesium was used to compensate loss during synthesis processes. Sample no. Volume fraction Atomic ratio TiO Sn Sb Si As Mg 1 % 1% 3 % % 5 % 6.1% 7.% % 9 1% % 11 5% 1 1% - -

3 Figure S1. Scanning electron microscope (SEM) image with energy dispersive spectroscopy (EDS) results on dark and grey regions

4 Figure S. High resolution transmission electron microscope (TEM) images used to determine distribution of nanograin sizes. The scale bar indicates nm (continued to the next page). - -

5 Figure S. (continued from the previous page) - 5 -

6 Figure S3. SEM images (upper row) used to estimate the ratio of phase segregation percentage and binary masks (lower row) created from those figures in the first row by using ImageJ. a b c 1 nm Figure S. (a) TEM-EDS elemental mapping of Sample 6 (% TiO ). (b) Colored mapping results of Si, Sn, As, Mg, Sb, Ti, and O

7 Figure S5. Electrical conductivity (a), thermopower (b), and thermal conductivity (c) of Sample 5~1 containing TiO nanoparticles (~1 vol%) and 1.5% Sb dopants. Figure S6. Thermoelectric power factor of Sample 5~1 (a) and Sample 1~ (b). Thermoelectric figure of merit (ZT) of Sample 5~1 (c)

8 Electrical Conductivity and Thermopower Calculation Electron (or maority) carrier concentration (N e ) and hole (or minority) carrier concentration (N h ) can be calculated with the Fermi levels as input parameters: E mx3 / E mx1 / exp E EF, X 3 / kbt 1 exp E EF, X1 / kbt 1 (S1) Ne NV, X 3 de NV, X1 de E mhh / E mlh / exp E EF, HH / kbt 1 exp E EF, LH / kbt 1 (S) Nh NV, HH de NV, LH de Here we considered two X-valleys in the conduction band (labeled as X1 and X3) and two -valleys in the valence band (labeled as HH, and LH), as shown in Fig. S6. 1 Each integral was calculated with respect to the band edge of each valley (i.e., E = at the band edge). For convenience, we used the band edge of X3 valley as a reference, the four valleys can be expressed with the Fermi level (E F ) as: E E F X E (S3), 3 F F, X 1 EF E (S) F, HH F, LH G F E E E E (S5) where E is the energy offset (E Edge,X1 E Edge,X3 ) and E G is the band gap as shown in Fig. S6. X3 X1 E G E LH HH Figure S7. Multi valley schematic of Mg Si. Sn.6. With the effective mass (m * ) in Table S1, N e and N h were found, and then ionized impurity concentration (N II ) was calculated by: NII Ne Nh (S6) - 8 -

9 Thermopower (S) can be calculated with electrical conductivity () by: S S (S7) S d, ( E) E EF, qt de (S8) df ( E) d, ( E) q D ( E) ( E) v ( E) de (S9) 1.5 ( m ) D ( E) E (S1) 3 1 f ( E) exp E EF, / kbt ( E) ( E) ( E) ( E 1 AC, II, POP, NI, ) (S11) ( E) (S1) E v ( E) (S13) 3m N V, d, * ( E) de (S1) (S15) where q, T, E, m *, N V are electron charge, absolute temperature, carrier energy with respect to band edge, carrier effective mass, and valley degeneracy, respectively. The index indicates valley (X1, X3, HH, and LH). The subscripts AC, POP, II, and NI in scattering relaxation time represent acoustic phonon, ionized impurity, polar optical phonon, non-ionized impurity, respectively. The material parameters employed are summarized in Table S. The mobility of electron (μ e ) and hole (μ h ) carriers can be obtained by: X1 X 3 e (S16) qn e HH LH h (S17) qn h The mean free path of electronic carrier (l ) as a function of energy in valley can be obtained by: l E * m (S18) - 9 -

10 Differential conductivity / 1 5 (Sm -1 ev -1 ) X3, 8 K Energy (ev) Mean free path (nm) Differential conductivity / 1 5 (Sm -1 ev -1 ) X1, 8 K Energy (ev) Mean free path (nm) Differential conductivity / 1 5 (Sm -1 ev -1 ) X3, K Mean free path (nm) Differential conductivity / 1 5 (Sm -1 ev -1 ) Energy (ev) Energy (ev) Figure S8. Calculated mean free path of electron carriers in X3 and X1 valley as a function of energy for no TiO nanoparticle sample at 8 K and K using the velocity and relaxation time X1, K Mean free path (nm) - 1 -

11 Table S. Parameters used for calculation. 1- The atomic concentration of Sn is y. Parameters Values m * X3 (m ).38 m * X1 (m ) T m * HH (m ) 1.5 m * LH (m ) 1. E G (ev) ( T ) (1- y) + ( T ) y E (ev) (.) (1- y) + (-.165) y ε (F/m) ( (1- y) y) ε (F/m) (13.3 (1- y) + 17 y) ħω (mev) (1- y)+8.8 y ρ NP (g/cm 3 ) 3.78 d (nm) 11 ρ M (g/cm 3 ) 1.88 (1- y)+3.59 y D e (ev) 7 D h (ev) 1 C l (N/m ) (.15 (1- y)+3. y) 1 1 ω C,L (THz) 5.3 (1- y)+. y ω C,T (THz) 9.7 (1- y) y v L (m/s) 77 (1- y)+ 9 y v T (m/s) 9 (1- y)+ 3 y γ.5 (1- y)+ 1.7 y α for TiO %, 1%, %, 5% Respectively.65,.6,.55,.5 β for TiO %, 1%, %, 5% Respectively 1,.8,.6,.5 N V, X3, N V, X1 3 N V, HH, N V, LH

12 Thermal Conductivity Calculation Electronic thermal conductivity (k e ) and bipolar thermal conductivity (k bi ) can be calculated by: The Lorenz number is expressed as: k L T (S19) e L, ( ) d E E EF, d, ( E) qt de E E qt F, de (S) e h kbi ( Se Sh) T e h (S1) where e X1 X 3; h HH LH (S) S S S X 1 X 1 X 3 X 3 e ; e S S S HH HH LH LH h (S3) h Lattice thermal conductivity was calculated by using a modified Callaway model that separately considered longitudinal and transverse modes of phonons. 1 kl ( kl kt ) (S) 3 kt ki 3 B 3 1 I i Ii1 vi Ii3 The subscript i stands for either longitudinal mode (L) or transverse mode (T). I I x i / T xe i1 p, i x I (S5) dx (S6) ( e 1) dx (S7) x i / T pi, xe i x pn, i ( e 1) dx (S8) x i / T pi, xe i3 x pn, i pr, i ( e 1) where x=ħω/(k B T) is the reduced energy of phonon and ϴ i is the Debye temperature (S9) p, i pn, i pr, i The relaxation time corresponding to the normal scattering is described as: 3 kb V ( ) xt (S3) pn, L 5 MvL - 1 -

13 kb V ( ) xt (S31) pn, T 5 MvT where k B, γ, V, M, v L, v T, ħ and T are Boltzmann constant, Grüneisen parameter, average volume of an atom, average mass of an atom, longitudinal sound velocity, transverse sound velocity, reduced Planck constant, and absolute temperature, respectively. Umklapp scattering is expressed as: 1 kb 3 i /3T ( pu, i ) x T e Mvii (S3) where θ i is the Debye temperature which is equal to ħω C /k B. The scattering due to random alloy between Si and Sn is 1 V kt B pa, i x 3 vi ( ) M Si, Sn MSi M Si, Sn MSn M Si, Sn 3 (1 y) y M M M M Mg Si, Sn Si, Sn Si, Sn (S33) (S3) MSi, Sn (1 y) MSi ym (S35) Sn where y is the atomic ratio of Sn, which is.6 for Mg Si. Sn.6, and M Si,Sn is the average atomic mass of Si and Sn accordingly with the ratio y. Phonon scattering by electronic carriers was considered by using: ( ) ( ) ( ) (S36) pe, i pe, i, X 3 pe, i, X 1 EF, (, /(16, ) /) * 3 i x i x k ( ) BT D 1 e m vi 1 e pe, i, NV, x EF, i, ( i, x /(16 i, ) x/) kbt ( ) ln 1 e (S37) * mv i i, (S38) kt where D e and ρ are electron deformation potential and density of Mg Si. Sn.6, respectively. We ignored phonon scattering due to holes (minority carriers) since its influence on the total relaxation time is very small. Phonon scattering by nanoparticles was considered by using: 6 1 i NP NP B pnp, i kbdnpx T 3 vi ( ) B v N D k x T (S39) NP M 1 exp M (S)

14 where N NP is the concentration of nanoparticles and D NP is the diameter of nanoparticles with the assumption of a sphere shape. λ accounts for the density difference between the matrix material and TiO nanoparticles. λ approaches 1 when the density difference is infinity. N NP and D NP are related each other since the volume percent of TiO nanoparticles is given. N NP X /1 DNP 3 3 (S1) where X is the volume percent of TiO. N NP was found to be , , and cm -3 respectively for samples with 1%, %, and 5% TiO nanoparticles. Figure S9. Undoped single crystalline Mg Si and Mg Sn data -5 parameters (.5 and 1.7) by considering normal scattering and Umklapp scattering. was fitted to find the Grüneisen Phase segregation was calculated by using the area percentage of Mg Sn (A MgSn ) obtained from ImageJ software. 6 The segregated Mg Si (A MgSi ) phase was calculated by multiplying A MgSn and both atomic ratio (.6:. for Sn:Si) and areal ratio (square of lattice constant). A a (S) MgSi.6 Mg ( ) Si AMg Sn amg. Sn where a MgSi and a MgSn are lattice constants of Mg Si and Mg Sn, respectively. The total segregated portion was obtained by adding the two segregated areas. A A A (S3) Total MgSi MgSn It should be noted that the areal ratio for the two phases is the same as the volumetric ratio, assuming that - 1 -

15 the thickness is the same for both. The relaxation time of holes (minority carriers) was adusted with a multiplying factor β, ranging from to 1 in order to consider the reduction of bipolar thermal conductivity due to the minority carrier scattering by TiO nanoparticles. For the sample without TiO nanoparticles, β= ,,,, ( E) ( E) ( E) ( E) ( E) (S) HH AC HH II HH POP HH NI HH ,,,, ( E) ( E) ( E) ( E) ( E) (S5) LH AC LH II LH POP LH NI LH Error Estimation Experimental errors related to thermal conductivity, electrical conductivity, and thermopower measurements were obtained mainly from uncertainty in instrumentation and dimensions. For electrical conductivity, uncertainty related to distance measurement between two probes, width, and thickness was estimated as ±%. For thermopower measurements, uncertainty was estimated by using the largest and lowest slope as an upper (+8%) and lower bound (-%) from the temperature-voltage relation. Thermal conductivity (k) was obtained by using the formula k = c, where c,, and are specific heat, mass density, and thermal diffusivity, respectively. For the density measurement, estimated uncertainty was ±3% and ±% respectively corresponding to volume and mass measurements. For thermal diffusivity, ±3% was obtained from three consecutive measurements with the flash apparatus. Therefore, the overall uncertainty corresponding to thermal conductivity measurements was estimated to be as ±8%. Overall uncertainty of the thermoelectric figure-of-merit was then calculated using the error propagation formula shown below. ZT S k (S6) ZT S k References 1. Bahk, J.-H.; Bian, Z.; Shakouri, A., Electron Transport Modeling and Energy Filtering for Efficient Thermoelectric Mg Si 1-x Sn x Solid Solutions. Phys. Rev. B: Condens. Matter Mater. Phys. 1, 89, Wang, S.; Mingo, N., Improved Thermoelectric Properties of Mg Si x Ge y Sn 1-x-y Nanoparticle-in- Alloy Materials. Appl. Phys. Lett. 9, 9, Yi, S.-i.; Yu, C., Modeling of Thermoelectric Properties of SiGe Alloy Nanowires and Estimation of the Best Design Parameters for High Figure-of-Merits. J. Appl. Phys. 15, 117, Akasaka, M.; Iida, T.; Matsumoto, A.; Yamanaka, K.; Takanashi, Y.; Imai, T.; Hamada, N., The Thermoelectric Properties of Bulk Crystalline n- and p-type Mg Si Prepared by the Vertical Bridgman Method. J. Appl. Phys. 8, 1,

16 5. Martin, J., Thermal Conductivity of Mg Si, Mg Ge and Mg Sn. J. Phys. Chem. Solids 197, 33, Schneider, C. A.; Rasband, W. S.; Eliceiri, K. W., NIH Image to ImageJ: 5 Years of Image Analysis. Nat. Methods 1, 9,