Supplemental Materials. Advanced Thermoelectrics Governed by Single Parabolic Band Model:

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1 Electonic Supplementay Mateial (ESI) fo Phyical Chemity Chemical Phyic. Thi jounal i The Royal Society of Chemity 04 Supplemental Mateial Advanced Themoelectic Govened by Single Paabolic and Model: Mg Si 0. Sn 0.7, a Canonical Example Wei Liu,, Hang Chi, Hui Sun, Qiang Zhang, Kang Yin, Xinfeng Tang *, Qingjie Zhang and Ctiad Uhe * Depatment of Phyic, Univeity of Michigan, Ann Abo, Michigan 4809, USA State Key Laboatoy of Advanced Technology fo Mateial Synthei and Poceing, Wuhan Univeity of Technology, Wuhan 40070, China *Coepondence (): C.U., cuhe@umich.edu, X.T., tangxf@whut.edu.cn. Thee autho contibuted equally to thi wok.

2 In the Single Paabolic and (SP) model, with an enegy dependent elaxation time τ τ ε, (S) 0 all tanpot coefficient can be obtained by olving the oltzmann tanpot equation -. Specifically, the Seebeck coefficient i given by ( ) ( ) ( ) ( ), k S (S) q whee k i the oltzmann contant, q -e i the electon chage, (E /k T) i the educed emi level meaued fom the conduction band minimum, -/ fo any catteing mechanim whoe mean-fee path i independent of enegy (MPIE), uch a acoutic phonon (AP) catteing and alloy catteing (AS) found elevant in thi eeach, and ( n ) i the emi integal given by: 0 n n ( ) ξ dξ. ξ (S) e The expeimentally acceible n H ( /qr H, whee R H i the Hall coefficient) i elated to the fee caie denity n via the Hall facto H : n H n H, (S4) whee ( ) ( ) ( ) ( ) ( ) H, (S) and

3 ( m k ) / T ( ). 4π n (S6) h Hee m* i the denity-of-tate effective ma given a a function of the ingle valley effective ma m and the aociated valley degeneacy N v a / m N v m. (S7) Conideing only the dominant catteing mechanim of AP and AS, the oveall µ H i given by the Matthieen ule µ. AS AP (S8) µ H H µ H Noting that µ Ψ ( ), (S9) µ H d, 0 whee Ψ ( π ) i given by Ψ ( ) π 4 Γ ( ) ( ) ( ), (S0) with ( AS, AP) indicating diffeent catteing mechanim. µ epeent the dift d, 0 mobility in the non-degeneate limit which can be calculated, fo the MPIE cae, with the knowledge of the coeponding mean-fee path fom MPIE d, 0 m MPIE d,0 4 q MPIE d,0 ( πm k T ) / (S) The liteatue eult 4 fo AS i m AS d,0 9(π ) / x 4 64e N 0 ( x) E ( m ) / ( k T ) a /, (S)

4 whee x ( 0.7 fo Mg Si 0. Sn 0.7 ) i the Sn faction in the olid olution, N 0 i the numbe of atom pe unit volume, and E a i the paamete chaacteizing the alloy potential fluctuation in Mg Si 0. Sn 0.7. The eult fo AP i ( ) ( ) ( ) / / 4 /,0 8 T k m E v e d l AP d ρ π m (S) whee v l i the longitudinal velocity of ound and ρ i the denity. E d i the defomation potential. The electonic themal conductivity κ e can be detemined by uing the Wiedemann-anz law, κ e LσT, whee L i the Loenz numbe, σ i the electical conductivity and T the abolute tempeatue. L and it tempeatue dependence ae given by 6 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 7 q k L (S4) 4

5 igue S: (a) XRD patten and (b) DSC cuve of Mg (Si 0. Sn 0.7 ) -y i y (0 y 0.04) olid olution. Reult indicate that, imila to the othe Mg Si -x Sn x -baed olid olution, 7-0 all XRD patten of Mg (Si 0. Sn 0.7 ) -y i y can be indexed to the Mg Si 0. Sn 0.7 olid olution with the cubic tuctue and pace goup of mm, except fo one light peak belonging to MgO. In addition, DSC cuve evolve gadually with tempeatue in the ange of 0-80 K and no anomalou themal effect ae obeved, atteting to Mg (Si 0. Sn 0.7 ) -y i y being table duing meauement.

6 igue S: ack Scatteing Image (a), elemental mapping ditibution of Mg (b), Si (c), Sn (d) and i (e) fo Mg (Si 0. Sn 0.7 ) 0.97 i 0.0. All the contituent element ditibute athe evenly in the micon ize cale. We hould note that, even i doping amount i a high a y 0.0, we do not obeve obviou egegation in the matix of Mg (Si 0. Sn 0.7 ) 0.97 i

7 igue S: (a) coelation between the Seebeck coefficient S and the Hall caie denity n H at oom tempeatue fo Mg Si 0.4 Sn 0.6 (b), Mg Si 0. Sn 0. (b), and Mg Si (c), epectively. 7-9, -4. The ingle paabolic band (SP) model i utilized to geneate the denity-of-tate effective ma m * fom S and n H. The cloe the compoition of Mg Si - xsn x to the band convegence at Mg Si 0. Sn 0.7, the bette i the coincidence between the expeimental S and the Piaenko cuve baed on the SP model, epecially when n H > cm -. 7

8 . E. H. Putley, The Hall Effect and Related Phenomena, uttewoth, London, H. J. Goldmid, Electonic Regigeation, Pion, London, 98.. A.. May, E. S. Tobee, A. Saamat and G. J. Snyde, Phy. Rev., 009, J. W. Haion and J. R. Haue, Phy. Rev., 976,, J. adeen and W. Shockley, Phy. Rev., 90, 80, H. J. Goldmid, Electonic Refigeation, Pion, London, W. Liu, Q. Zhang, K. Yin, H. Chi, X. Zhou, X. Tang and C. Uhe, J Solid State Chem, 0, 0, W. Liu, X. Tang, H. Li, K. Yin, J. Shap, X. Zhou and C. Uhe, J. Mate. Chem., 0,, W. Liu, X. Tang, H. Li, J. Shap, X. Zhou and C. Uhe, Chem. Mate., 0,, W. Liu, X. Tang and J. Shap, Jounal of Phyic D: Applied Phyic, 00, 4, Q. Zhang, J. He, T. J. Zhu, S. N. Zhang, X.. Zhao and T. M. Titt, Appl. Phy. Lett., 008, 9, H. Gao, T. Zhu, X. Liu, L. Chen and X. Zhao, J. Mate. Chem., 0,, S. K. ux, M. T. Yeung, E. S. Tobee, G. J. Snyde, R.. Kane and J.-P. leuial, J. Mate. Chem., 0,, G. S. Nola, D. Wang and M. eekman, Phy. Rev., 007, 76, 04. 8