Optimization of the Electronic Band Structure and the Lattice Thermal

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Optimization of the electronic band structure and the lattice thermal conductivity of solid solutions according to simple calculations: a canonical example of the Mg2Si1-x-yGexSny ternary solid solution Kang Yin, Xianli Su, Yonggao Yan, Yonghui You, Qiang Zhang, Ctirad Uher, Mercouri G. Kanatzidis, and Xinfeng Tang Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.6b02308 • Publication Date (Web): 20 Jul 2016 Downloaded from http://pubs.acs.org on July 21, 2016

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Chemistry of Materials

Kang Yin,† Xianli Su,†* Yonggao Yan†, Yonghui You†, Qiang Zhang†, Ctirad Uher‡, Mercouri G. Kanatzidis and Xinfeng Tang†* †

State Key Laboratory of Advanced Technology for Materials Synthesis and Processing, Wuhan University of Technology, Wuhan 430070, China ‡ 

Department of Physics, University of Michigan, Ann Arbor, Michigan48109, USA

Department of Chemistry, Northwestern University, Evanston, Illinois 60208, USA

Supporting Information ABSTRACT: The dependence of the electronic band structure of Mg2Si0.3-xGexSn0.7 and Mg2Si0.3GeySn0.7-y (0 ≤ x, y ≤ 0.05) ternary solid solutions on the composition and temperature is explained by a simple linear model, and the lattice thermal conductivity of solid solutions with different Si/Ge/Sn ratio is predicted by the Adachi model. The experimental results show excellent consistency with the calculations which suggests that the approach might be suitable to describe the electronic band structure and the lattice thermal conductivity of other solid solutions using these simple calculations. Beyond this, it is observed that the immiscible gap in the Mg2Si1-xSnx binary system is narrowed with the introduction of Mg2Ge. Moreover, for the Sb doped solid solutions Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤ y ≤ 0.05), the energy offset between the light conduction band and the heavy conduction band at higher temperatures (500-800 K) will decrease with the increasing Ge content, thus making a contribution to the conduction bands degeneracy, and enhancing the power factor in turn. Meanwhile, mass fluctuation and strain field scattering processes are enhanced when Ge substitutes for Sn in Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤ y ≤ 0.05) due to the large discrepancy between the mass and size of Ge and Sn, and the lattice thermal conductivity is decreased as a consequence. Thus, the thermoelectric performance is improved, with the figure of merit ZT larger than 1.45 at about 750 K and the average ZT value between 0.9 to 1.0 in the range of 300800 K, which is one of the best results in the Sb doped Mg2Si1-x-yGexSny systems with single phase.

Given the world’s unrelenting thirst for coal, oil, natural gas and other fossil fuels which leads to the depletion of energy reserves as well as the deterioration of the eco-environment due to an enormous amount of CO2 emission, thermoelectric (TE) power generation has drawn considerable interest as a green and clean candidate technology for waste heat recovery in vehicles and industrial processes in general.1, 2 This fully solid-state technology is capable of directly converting waste heat into usable electricity and, in its inverted function, can also facilitate cooling and air conditioning without working fluids. The efficiency of TE materials is governed by the dimensionless figure of merit, ZT = S2T/, where S is the Seebeck coefficient,  is the electrical conductivity, T is the absolute temperature, and  is the thermal conductivity. Typically, there are two approaches about how to improve the dimensionless figure of merit: one is to focus on enhancing the power factor (S2) by doping3-6 and energy band engineering,7-11

while the other relies on lowering the thermal conductivity by alloying,12-14 filling,15-17 forming lower dimensional structures,1821 and nanostructuring.22-25 Mg2IV (IV = Si, Ge, Sn) based compounds are materials intended for TE power generation in the temperature range of 300800 K. The attribute of these compounds is not only a good TE performance but also the fact that the chemical elements are plentiful, inexpensive and environmentally friendly.26 All the compounds Mg2Si, Mg2Ge, and Mg2Sn share a similar electronic band structure consisting of two closely lying conduction bands (Figure 1 and Table 1).27-29 However, while the bottom of the conduction band in Mg2Si and Mg2Ge is dominated by the light electron band (CL band) which has its origin in the 3p orbital of Mg hybridized with the s orbital and d-eg orbital of Si or Ge, the bottom of the conduction band of Mg2Sn is dominated by the heavy electron band (CH band) arising from hybridization between the 3s orbital of Mg and the d-t2g orbital of Sn. 30-35 This implies that by forming solid solutions of Mg2Si1-xSnx and Mg2Ge1-xSnx, band

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convergence must take place at some particular value of x where the band edges of the light and heavy conduction bands attain the comparable energy. When this happens, the band degeneracy Nv of the system is increased which leads to a significant enhancement of the density-of-states (DOS) effective mass m* = Nv2/3mb* without increasing the band effective mass mb*. When this condition is achieved, a large increase in the Seebeck coefficient is expected without any deterioration in the electrical conductivity. Thus, by rationally designing solid solutions with the appropriate composition, the power factor and the overall TE performance can be enhanced.26, 36-43 It is important to note that the composition where the energy band convergence takes place is not the same as the one which yield the lowest lattice thermal conductivity in the Mg2Si1-xSnx system.44, 45 Thus, the lattice thermal conductivity can be further reduced by substituting Si (or Sn) with Ge taking the composition corresponding to the band convergence as the starting point.36-38, 46, 47 However, reports on the electronic band structure of Mg2Si1-xyGexSny (0 ≤ x, y, x+y ≤ 1) ternary system are sparse and the detailed mechanism how substitution of Ge reduces the lattice thermal conductivity has not been provided yet. To shed further light on this issue, we have selected the Mg2Si0.3Sn0.7 (the composition resulting in the convergence of the two conduction bands) as an example,48-51 and carefully explored the effect of Ge substitution on sites of either Si (Mg2Si0.3-xGexSn0.7) or Sn (Mg2Si0.3GeySn0.7-y) (0 ≤ x, y ≤ 0.05). The changes in the electronic band structure and the transport properties of such solid solutions are characterized in detail. At the same time, it is found that the immiscible gap in the Mg2Si1-xSnx (0 ≤ x ≤ 1) soild solutions is narrowed as Ge substitutes for Si or Sn. The relationship between the electronic band structure and the composition as well as the temperature is described by a simple linear model. The lattice thermal conductivity is described by the Adachi model.52 In addition, the dependence of different phonon scattering parameters on the content of Ge is explained quantitatively based on the Callaway model.53-55 To optimize the carrier concentration, Sb doping is applied and a high power factor of 4.7 mWm-1K-2 is achieved for Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤ y ≤ 0.05) owing to the band convergence in the temperature range of 500-800 K. Together with the reduced lattice thermal conductivity due to enhanced alloy scattering (arising from the large mass and size differences between Ge and Sn), a peak ZT value of 1.45 is achieved at around 750 K for Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02, demonstrating an increase of about 20% compared with that of Mg2.16(Si0.3Sn0.7)0.98Sb0.02. The experimental data agree well with the predictions by this linear model and the Adachi model, which bodes well for exploring other solid solution systems with potentially high TE performance by the aforementioned simple computation approach.

High purity powders of Mg (99%), Mg2Si (99.99%), Ge (99.99%), Sn (99.9%), and Sb (99.999%) were used in the twostep solid state synthesis of Mg2Si0.3-xGexSn0.7, Mg2Si0.3GeySn0.7-y, Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 and Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤ x, y ≤ 0.05), where the samples were placed in the pyrolytic boron nitride (PBN) crucibles and sealed in the silica ampoules, with the first and the second reaction steps carried out at 837 K for 24 h and 973 K for 24 h, respectively.56 The final consolidation process was done by the spark-plasma-sintering process (SPS) with the uniaxial pressure of 35 MPa at 933 K for about 10 min.

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An extra 8 at.% of Mg was added in order to compensate for the evaporation loss of Mg during the synthesis.57 The phase composition of different solid solutions was checked on a PANalytical X’Pert Empyrean X-ray diffractometer using Cu K radiation (= 1.5406 Å), with the generator voltage and the tube current to be 40 KV and 40 mA, respectively, and the scan range was 10-80 degrees with a scan step size of 0.02 degree. The lattice parameter was then deduced from the Rietveld refinement. The actual chemical composition and the back scattering images of the ingots were characterized by the electron probe microanalysis (EPMA, JEOL JXA-8230 system) with a wavelength dispersive spectrometer (WDS). The morphology of the fractured surface was determined by the field-emission scanning electron microscopy (FESEM, Hitachi SU8020). The electrical conductivity and the Seebeck coefficient S in the range of 300– 800 K were acquired on a commercial ZEM-3 system (Ulvac Sinku-Riko) in helium by a standard DC four-probe configuration combined with a differential voltage/temperature technique. The Hall coefficient RH at 300 K was measured by the five probe method using a physical property measurement system (PPMS, Quantum Design), with a magnetic field of ±2 T. The carrier concentration nH and the carrier mobility H were obtained with the aid of relations nH = 1/(eRH) and H = RH, respectively. The total thermal conductivity was determined from Total = Cp, where is the thermal diffusivity acquired by the laser flash method (Netzsch LFA 457), is the density measured by the Archimedes method in alcohol, and Cp is the specific heat obtained by a differential scanning calorimeter (TA DSC Q20) in argon. The uncertainty in the experimental values was due, in large part, to the uncertainties in the determination of the sample sizes. The overall errors in measurements of the electrical conductivity, the Seebeck coefficient and the thermal conductivity are estimated to be ± 5%, ±3%, and ±5%, respectively. It is worth mentioning that Mg2.16(Si0.3Ge0.05Sn0.65)0.98Sb0.02, which showed the best TE performance, was also used to check the thermal stability at elevated temperatures. The process consisted of coating the ingot with a BN spray and annealing the structure in air at 773 K for 15 days.58 Transport properties were then re-evaluated in detail.

Figure 2(a) illustrates the temperature dependence of Seebeck coefficient for Mg2Si0.3-xGexSn0.7 and Mg2Si0.3GeySn0.7-y (0 ≤ x, y ≤ 0.05) with different Ge content. And Figure 2(b) shows the electrical conductivity as the Arrhenius’ equation of = 0exp[Eg/(2kBT)]. Figure 2(c) and Figure 2(d) depicts the energy band gap Eg and the conduction band offset E0 of Mg2Si0.3-xGexSn0.7 and Mg2Si0.3GeySn0.7-y as a function of the Ge content using a simple linear model. Experimental values of Eg acquired from the slope in Figure 2(b) are also shown in Figure 2(c). Deducing from the experimental data of published papers about Mg2Si-Mg2Sn and Mg2Ge-Mg2Sn binary systems,59, 60 the energy band gap Eg of Mg2A1-xBx (A = Si, Ge, B = Sn, 0 ≤ x ≤ 1) roughly follows a linear dependence on the A/B ratio in the miscible range: Eg ( x )  (1  x ) Eg (Mg2A)  x Eg (Mg2B)

(1)

However, few reports have investigated the relationship of the conduction band offset E0 with the composition of the solid solutions. As shown in Figure 1, the conduction band offset is the gap between the two conduction bands ( C1 band VS. C2 band),


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and the electronic band gap is the gap between the edge of the conduction band and the valence band (C1 band (in Mg2Si and Mg2Ge) or C2 band (Mg2Sn) VS. V1 band). Logistically, if the band gap of the solid solution follows the linear relationship, so will the conduction band offset. If the value of E0 is defined positive when the C1 band dominates the edge, while it is negtive in the case of C2 band, there follows the relation:

E0 ( x )  (1  x ) E0 (Mg2Si)  x E0 (Mg2Sn)

(2)

By taking the data given in Table 1 into the above two equations and setting the conduction band offset E0(x) = 0, we obtain the result x = 0.71. This means that the two conduction bands in Mg2Si1-xSnx have converged when the composition is Mg2Si0.29Sn0.71, in excellent agreement with the experimental results of Liu et al.50 A similar result, x = 0.78, is obtained for the convergence of conduction bands in Mg2Ge1-xSnx (0 ≤ x ≤ 1), again, nicely matching the experimental value of x = 0.75.40, 61 Such excellent agreements imply that the linear description of the conduction band offset is really applicable, and the electronic band structure of the ternary Mg2Si1-x-yGexSny (0 ≤ x, y, x+y ≤ 1) system might also be described in this way (if taken it in the form of Mg2Si1-xSnx-Mg2Ge1-ySny), i.e., Eg ( x, y )  (1  x y ) Eg (Mg2Si)  x Eg (Mg2Ge)  y Eg (Mg2Sn)

(3)

E0 ( x, y )  (1  x y ) E0 (Mg2Si)  x E0 (Mg2Ge)  y E0 (Mg2Sn)

(4)

Calculations in Figure 2(c) predict that the energy band gap of Mg2Si0.3GeySn0.7-y will increase with the growing Ge content, while that of Mg2Si0.3-xGexSn0.7 will remain more-or-less constant. Experimental values of the energy band gap are highly consistent with the calculations, demonstrating the validity of this simple linear model. Although the conduction band offset shows a positive correlation with the increasing Ge content (notably for Mg2Si0.3GeySn0.7-y), it remains smaller than the value of 1kBT at 750 K (the temperature where these solid solutions doped with Sb or Bi show the peak ZT value).62-64 Thus, the single parabolic band (SPB) model is still useful when discussing the charge transport process.65 Furthermore, the Adachi model is employed to characterize the relationship between the lattice thermal conductivity and the composition of Mg2Si1-x-yGexSny ternary system,52, 66



1 Lattice

( x, y )  (1  x  y )

1 Mg 2 Si

 x

1 Mg 2 Ge

 y

1 Mg 2 Sn



x (1  x  y ) CMg2Si  Mg2 Ge  y (1  x  y ) CMg2Si  Mg2Sn  xyCMg2 Ge  Mg2Sn

(5)

Here, (Mg2IV) is the lattice thermal conductivity of Mg2IV (IV = Si, Ge, Sn) derived from literature26 and CMg2X-Mg2Y is the term arising from the lattice disorder due to a random distribution of X and Y atoms (X, Y = Si, Ge, Sn) on the sublattice sites. The values of CMg2Si-Mg2Ge, CMg2Si-Mg2Sn and CMg2Ge-Mg2Sn are 0.78 W1 mK, 1.25 W-1mK and 1.00 W-1mK, respectively, obtained from calculations.66 Plots of the calculated lattice thermal conductivity at 300 K for Mg2Si0.3-xGexSn0.7 and Mg2Si0.3GeySn0.7-y based on the Adachi model are shown in Figure 3(a). It is clear that Ge substituting Sn in Mg2Si0.3GeySn0.7-y is expected to significantly reduce the lattice thermal conductivity and, indeed, the experimental data (the two lines are fitted values, while the dots are the measured values) in Figures 3(b) confirm this prediction. Based on the research of Liu et al,64 Sb doping is applied to optimize the carrier concentration. Figure 4(a) shows the powder XRD patterns of Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 and Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤ x, y ≤ 0.05), together with the

recalculated reference pattern of Mg2Si0.3Sn0.7 according to JCPDS#01-089-4254. Figure 4(b) provides an expanded view of Figure 4(a) within the angular range of 68-75 degrees. Judging from the data, all samples are single phase structures, and the XRD peaks of Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 shift to higher angles as the content of Ge increases while those of Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 are just the opposite. This change is further verified by the refined lattice parameters shown in Figure SI1, which match well with the Vegard’s law, documenting that Ge is successfully substituted on the sites of Si and Sn, respectively. Beyond the regular XRD analysis, further characterization is adopted to check whether the structures are, indeed, of the single phase nature. EPMA results (shown in Table 2 later) indicate that the actual chemical composition of the samples strictly follows the nominal stoichiometry. Figure 5(a) depicts a number of darker contrast spots of the Mg2Si-rich phase (highlighted as the circled regions) which appear on a back scattering image of the Mg2.16(Si0.3Sn0.7)0.98Sb0.02 solid solution, consisting with the data reported by Q. Zhang et al.67 They arise because the composition of this solid solution is near the edge of the miscibility gap of the Mg2Si-Mg2Sn binary system.48, 49 Back scattering images in Figand taken on ures 5(b) Figure 5(c) Mg2.16(Si0.25Ge0.05Sn0.7)0.98Sb0.02, as well as Mg2.16(Si0.3Ge0.05 Sn0.65)0.98Sb0.02 (two typical examples of Mg2.16(Si0.3-xGex Sn0.7)0.98Sb0.02 and Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤ x, y ≤ 0.05)), are clean and devoid of such spots, indicating that the Ge-doped solid solutions are single phase structures, as already noted when discussing the XRD data. The transition from multiphase to single phase structures is closely related to the Mg2Si-Mg2Ge-Mg2Sn ternary phase diagram, the sketch map of which at about 973 K is shown in Figure 5(d).68 Similar to the Mg2Si-Mg2Sn and Mg2Ge-Mg2Sn binary systems,48, 69 a miscibility gap still exists in the ternary system, however, it becomes much narrower in the presence of Ge. At a given temperature, and when the Si (or Sn) ratio remains unchanged, the solid solution will shift from the immiscible region of the phase diagram to the miscible region as the content of Ge increases, shown by the black dots in Figure 5(d). This is a welcome news regarding the fabrication of high performance single phase Mg2IV (IV = Si, Ge, Sn) solid solutions. Figures 6(a)-6(d) show the electrical conductivity , the Seebeck coefficient S, the power factor PF, and the Lorenz number L of different solid solutions as a function of temperature. The electrical conductivity of all samples decreases with the increasing temperature, as evidenced in Figure 6(a), which is characteristic of the conduction process in degenerate semiconductors. The electrical conductivity of Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 with different Ge contents is almost the same, though a bit lower than the value of Mg2.16(Si0.3Sn0.7)0.98Sb0.02, while that of Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 is identical with the Ge-free solid solutions of the same stoichiometry. Together with the higher Seebeck coefficient in the temperature range of 500-800 K, the power factor of Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 is greatly improved. The improvement is a consequence of the modified band structure. As described by Equations. 3 and 4, the band structure of Mg2Si1x-yGexSny shows a strong dependence on the Si/Ge/Sn ratio. However, this reflects only the situation at 0 K. For much higher temperatures, the equations should be modified as: EgT ( x , y )  (1  x y ) EgT (Mg 2Si)  x EgT (Mg 2Ge)  y EgT (Mg 2Sn)

EgT (Mg 2 IV)  Eg0K (Mg 2 IV) 


dEg dT

T

(6)

(7)


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E0T ( x, y )  (1  x y ) E0 (Mg 2Si)  x E0 (Mg 2Ge)+ y E0 (Mg 2Sn)+

dE0 T (8) dT

where EgT(Mg2IV) is the energy band gap of Mg2IV at a certain temperature T, dEg/dT is the temperature dependent differential of the energy band gap, E0(Mg2IV) is the conduction band offset at 0 K, and dE0/dT is the temperature dependent differential of the conduction band offset. All band structure parameters are given in Table 1, except the value of dE0/dT which is deduced from the fitting data of the electrical conductivity and the Seebeck coefficient as suggested by M. Fedorov et al. 44 The values turn up to be dE0/dT ≈ - 7×10-5 eVK-1 for Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 and ≈ - 5×10-5 eVK-1 for Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤x, y ≤ 0.05). 3D plots of the energy band gap as well as the band offset are displayed in Figure 7. The results clearly indicate that the absolute value of the conduction band offset |E0| for Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02) in the temperature range of 500800 K will be reduced with the increasing content of Ge when 0≤ y ≤0.05. In other words, the presence of Ge in the structure will make it easier to achieve band convergence at elevated temperatures and thus improve the power factor. The key transport parameters, such as the room temperature dependence of the carrier concentration nH and the carrier mobility H on the content of Ge are plotted in Figure 8(a) and Figure 8(b). Room temperature values of the Seebeck coefficient S and the carrier mobility H as a function of the carrier concentration nH are shown in Figures 8(c) and Figure 8(d), respectively. As the data reveal, the carrier concentration of all solid solutions is basically the same. The carrier mobility of Mg2.16(Si0.3GeySn0.7y)0.98Sb0.02 decreases with the increasing Ge content while that of Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 remains unchanged. The SPB model is employed to investigate the effect of the composition change on the charge transport process. Assuming only acoustic phonon scattering and alloy scattering as important, the carrier mobility of the material is given as70 1

H

=

1

Hph



1

(9)

Hal

The mobility limited by acoustic phonons can be written as 71-73

Hph 

2 e 4  L2  F0 ( ) 3/ 2 2 3( kB T )  ( m*b )5 / 2 F1/ 2 ( )

(10)

where Fi() stands for the Fermi integral given by 

xi d x ,  EF / kB T 0 1  exp  x   

Fi    

(11)

with EF being the Fermi energy and  is the reduced value. The velocity of longitudinal sound waves, vL, is taken as 5300 ms1 62 ,  is the density,  is the deformation potential, and mb* is the averaged single-valley band effective mass (related to the density of states effective mass as m* = Nv2/3mb* where Nv is the orbital degeneracy, i.e. the number of symmetry related carrier pockets, which equals to Nv = 6 here). 62 Regarding the alloy scattering, the carrier mobility can be written as 74, 75 Hal 

N0 F0 ( ) 16 e 4 (12) 2 * 5/ 2 9 2 [ x (1- x - y )+ y (1- x - y )+ xy ]( kB T )1/ 2 U ( mb ) F1/ 2 ( )

where N0 is the number of atoms per unit volume, and U is the alloy scattering potential. The undetermined key parameters in equation (10) and (12) are the deformation potential and the alloy

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scattering potential U. The deformation potential of Mg2Si1-xyGexSny is estimated by the linear relationship from the values of Mg2Si, Mg2Ge and Mg2Sn which are 17 eV, 18.7 eV, and 10.1 eV, respectively.73, 76, 77 The plots in Figure 8(b) demonstrate that the model including acoustic and alloy phonon scattering processes quantitatively describes the experimental results with the values of  = 12.17-12.26 eV and U = 0.25-0.30 eV, both parameters in good agreement with other reports.62 Because the deformation potential of Mg2Ge is much larger than that of Mg2Sn, its value in Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 increases rapidly as the content of Ge increases, thus leading to the reduced carrier mobility in this solid solution. Apart from the band convergence at higher temperatures, weak electron-phonon coupling and modest atomic disorder also play an important role in improving the power factor. At the same time, as the Pisarenko plot (m*/m0 = 2.25) in Figure 8(c) indicates, the effective mass of the solid solutions shows no obvious dependence on the increasing Ge content within the small range of 0 ≤ x, y ≤ 0.05. The carrier mobility roughly follows the H∝nH-1/3 dependence on the carrier concentration, which is common in degenerate semiconductors.78 Following the detailed study of electronic transport properties discussed above, the thermal transport is also carefully scrutinized, as shown in Figure 9. The total thermal conductivity Total as well as the sum of the lattice thermal conductivity and the bipolar part Lattice+Bipolar of different solid solutions are presented in Figures 9(a) and 9(b), respectively. And the latter is acquired from the relation Lattice + Bipolar = Total – LT, where L, , and T are the Lorenz number (calculated by the SPB model under the relaxation time approximation, shown in Figure 6(d) ),62 the electrical conductivity and temperature, respectively. Comparing these two figures, it follows that the difference in the total thermal conductivities of these solid solutions is caused mainly by changes in the lattice thermal conductivity which approximately follows the Lattice∝T-1/2 power law, signaling the dominance of alloy scattering,79, 80 see the inset in Figure 9(b). The Callaway model is applied to gain a clear picture of the relative importance of different scattering processes in the above solid solutions. The relevant parameters are listed in Table 2 and represent the scaling parameter u, the disorder scattering parameter expt obtained from the experimental lattice thermal conductivity data, the mass fluctuation scattering parameter M and the strain field scattering parameter S, and the strain field related adjustable parameter S for the Si/Ge/Sn sublattice (it is a function of the Grüneisen parameter  characterizing the anharmonicity of the lattice).53, 54, 81, 82 The values of S and S were obtained by fitting the data expt = calc = M + S,81-83 detailed information for the calculation procedures can be found in the supporting information. Due to a large difference in atomic masses and atomic radii of Ge and Sn, M and S increase rapidly with the Ge content in Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 and thus enhance the alloy scattering process and notably reduce the lattice thermal conductivity. In contrast, the lattice thermal conductivity of Mg2.16(Si0.3xGexSn0.7)0.98Sb0.02 show almost no difference with respect to the content of Ge owing to quite similar crystal properties of Mg2Ge and Mg2Si. Meanwhile, the microstructure of Mg2.16(Si0.3xGexSn0.7)0.98Sb0.02 and Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 solid solutions is quite similar (shown in Figure SI2) with no nanostructures being observed, implying rationality in describing the phonon transport process by the Callaway model without considering any influence of nanostructuring. By the way, thanks to a wider energy band gap in Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (shown in Figure 2(c)), the bipolar effect will make a contribution at a much


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higher temperature, which also helps in maintaining low thermal conductivity. 47, 84 Figure 10(a) plots the ZT value of different solid solutions as a function of temperature, and Figure 10(b) shows the average value in the temperature range of 300-800 K. On account of the improved power factor at high temperatures (500-800 K) as well as the significantly reduced thermal conductivity due to enhanced alloy scattering, the TE performance of Mg2.16(Si0.3GeySn0.7y)0.98Sb0.02 (0 ≤ y ≤ 0.05) solid solutions is greatly improved, reaching the peak ZT value of 1.45 at about 750 K. The average ZT value over the 300-800 K range is boosted to an impressive range of 0.9-1.0. In addition, these solid solutions also displayed good thermal stability when annealed at 773 K in air (shown in Figure SI3), which is of vital importance for long-term service applications in thermoelectric modules.

of Ge content (Figure S1), The microstructure of Mg2.16(Si0.3xGexSn0.7)0.98Sb0.02 (x=0.05) and Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (y=0.05) (Figure S2), The thermoelectric properties of Mg2.16(Si0.3Ge0.05Sn0.65)0.98Sb0.02 before and after annealing at 773 K with BN coatings in air. (Figure S3. This material is available free of charge via the Internet at http://pubs.acs.org.

Email: [email protected] * Email: [email protected]

The authors declare no competing financial interests.

Mg2Si0.3-xGexSn0.7, Mg2Si0.3GeySn0.7-y, Mg2.16(Si0.3and Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤ x, y ≤ 0.05) solid solutions with different Ge contents were prepared by a two-step solid state reaction method followed by SPS consolidation. Structural characteristics of these solid solutions were thoroughly examined, the electronic and the thermal transport properties were measured over a broad range of temperatures. Dependence of the band structure parameters of Mg2Si0.3-xGexSn0.7 and Mg2Si0.3GeySn0.7-y (0 ≤ x, y ≤ 0.05) ternary systems on the composition and temperature was quantitatively explained by a simple linear model. Lattice thermal conductivity of the solid solutions with the different Si/Ge/Sn ratio was predicted by the Adachi model. The experimental data agree well with the calculated results which shows that it is reasonable to describe the band structure and the lattice thermal conductivity of Mg2Si1-x-yGexSny (0 ≤ x, y, x+y ≤ 1) solid solutions by such simple models. Moreover, it is observed that the immiscible range in the Mg2Si1-xSnx (0 ≤ x ≤ 1) binary system becomes narrower with the introduction of Mg2Ge for the fixed Si or Sn ratio. At the same time, the energy offset between the CL and CH bands in the Mg2.16(Si0.3GeySn0.7y)0.98Sb0.02 (0 ≤ y ≤ 0.05) solid solutions with optimized carrier concentration decreases with the increasing content of Ge at elevated temperatures (500-800 K), making the band convergence easier to realize and thus improve the power factor. Meanwhile, mass fluctuation and strain filed scattering processes are enhanced when Ge substitutes for Sn in Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 because of a large difference between atomic masses and radii of Ge and Sn atoms. This leads to a reduction in the lattice thermal conductivity. As a consequence, the dimensionless figure of merit is enhanced to values as high as 1.45 at 750 K for solid solutions with the composition y = 0.04-0.05. Perhaps even more important, the average value of ZT in the range of 300-800 K reaches very close to 1.0, which is one of the best results for the Sb-doped Mg2Si1-x-yGexSny system. It is also noteworthy that these high performance compositions show very good thermal stability when coated. The general approach of using a simple linear extrapolation of band parameters of binary solid solutions and applying it to ternary solid solutions may be helpful in explorations of other solid solution systems with good prospects for thermoelectricity. xGexSn0.7)0.98Sb0.02

Supporting Information. The calculated lattice parameter for Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 and Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤ x, y ≤ 0.05) as a function

This work is financially supported by the National Basic Research Program of China under project 2013CB632502, the Natural Science Foundation of China (Grant No. 51402222, 51172174, 51521001), the 111 Project of China (Grant No. B07040), the Fundamental Research Funds for the Central Universities (WUT: 2013-YB-013), and the CERC-CVC joint U.S.-China Program supported by the U.S. Department of Energy under the Award Number DE-PI 0000012 in verification of high temperature transport property measurements.. We would like to thank Tingting Luo and Rong Jiang for their help in the Materials Research and Test Center of WUT.

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A Table of Contents graphic

Figure 1. The sketch map of the electronic band structure for Mg2IV (IV = Si, Ge, Sn): (a) Mg2Si and Mg2Ge; (b) Mg2Sn.

Table 1. Parameters of the electronic band structure for Mg2IV (IV = Si, Ge, Sn).26, 28 Compound

Eg (0 K) (eV)

dEg/dT (10-4eVK-1)

E0 (0 K) (eV)

Mg2Si

0.77

-6

0.4

Mg2Ge

0.74

-8

0.58

Mg2Sn

0.35

-3.2

0.16


9


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Figure 2. The electrical propereties of Mg2.Si0.3-xGexSn0.7 and Mg2Si0.3GeySn0.7-y (0 ≤ x,y ≤ 0.05): (a) the temperature dependent Seebeck coefficient S; (b) the natural logarithm value of the electrical conductivity lnas a function of 1/(2kBT) after intrinsic excitation; (c) the energy band gap Eg of different solid solutions (dots: the experimental data derived from Figure 2(b), lines: the data predicted by the linear model.); (d) the predicted value of the conduction band offset E0.


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Figure 3. (a) The lattice thermal conductivity of Mg2Si0.3-xGexSn0.7 and Mg2Si0.3GeySn0.7-y (0 ≤ x, y ≤ 0.05) at 300 K predicted by the Adachi model, plotted as a function of the Ge content. (b) the experimental valuie of the sum of the lattice thermal conductivity and the bipolar part Lattice+Bipolar of the solid solutions as a function of temperature.

Figure 4. (a) The powder XRD patterns of Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 and Mg2.16(Si0.3GeySn0.7y)0.98Sb0.02 (0 ≤ x, y ≤ 0.05); (b) the expanded view in the range of 68-75 degrees.


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Figure 5. (a)-(c): The back scattering images of Mg2.16(Si0.3Sn0.7)0.98Sb0.02, Mg2.16(Si0.25Ge0.05Sn0.7)0.98Sb0.02 and Mg2.16(Si0.3Ge0.05Sn0.65)0.98Sb0.02, respectively; (d) the schematic phase diagram of Mg2Si-Mg2Ge-Mg2Sn tenary system at 973 K.


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Figure 6. (a)The electrical conductivity ; (b) the Seebeck coefficient S; (c) the power factor PF; (d) and the Lorenz number L, of different solid solutions as a function of temperature.


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Figure 7. (a) and (b): The composition and temperature dependence of the electronic energy band gap Eg as well as the conduction band offset E0 for the solid solution Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 (0 ≤ x ≤ 0.05); (c) and (d): The composition and temperature dependence of the electronic energy band gap Eg as well as the conduction band offset E0 for the solid solution Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤ y ≤ 0.05).


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Figure 8. The electrical properties of different solid solutions: (a) the carrier concentration nH; (b) the carrier mobility H; (c) the Pisarenko plot (S-nH) at room temerature (300 K); (d) the nH-H plot at room temperature (300 K).


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Figure 9. (a) The total thermal conductivity kTotal; (b) the sum of the lattice thermal conductivity and the bipolar part Lattice+Bipolar, of Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 and Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤ x, y ≤ 0.05) with different composition. Inset: the Lattice-T -1/2 plot before the intrinsic excitation process.

Table 2. Actual composition (determined by EPMA), lattice thermal conductivity, kLattice, disorder scaling parameter, u, disorder scattering parameters expt, M, S, and the strain field related adjustable parameter for the Ge sublattice, S. kLattice Ge content

Actual composition

u

expt

M

S

S

(Wm-1K-1) x, y=0

Mg2.14(Si0.29Sn0.71)0.98Sb0.018

2.2693

-

-

-

-

-

x=0.02

Mg2.14(Si0.27Ge0.02Sn0.71)0.98Sb0.022

2.1662

0.3850

0.0075

0.0017

0.0058

184.0423

x=0.03

Mg2.15(Si0.27Ge0.03Sn0.71)0.98Sb0.021

2.1677

0.3820

0.0074

0.0024

0.0050

100.1284

x=0.04

Mg2.14(Si0.26Ge0.04Sn0.70)0.98Sb0.019

2.1691

0.3790

0.0073

0.0031

0.0042

60.8257

x=0.05

Mg2.15(Si0.26Ge0.05Sn0.69)0.98Sb0.020

2.1706

0.3760

0.0071

0.0037

0.0035

38.8783

y=0.02

Mg2.15(Si0.31Ge0.02Sn0.77)0.98Sb0.021

2.1860

0.3432

0.0060

0.0045

0.0015

0.7048

y=0.03

Mg2.14(Si0.30Ge0.03Sn0.67)0.98Sb0.019

2.0806

0.5402

0.0148

0.0067

0.0082

2.6081

y=0.04

Mg2.14(Si0.29Ge0.04Sn0.67)0.98Sb0.018

2.0381

0.6094

0.0189

0.0088

0.0101

2.4536

y=0.05

Mg2.15(Si0.29Ge0.05Sn0.66)0.98Sb0.020

1.9600

0.7304

0.0272

0.0109

0.0163

3.2181

Figure 10. (a) The temperature dependent dimensionless figure of merit of solid solutions with different composition; (b) the average value in the temperature range of 300-800 K.


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TOC graphic


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Figure 1. The sketch map of the electronic band structure for Mg2IV (IV = Si, Ge, Sn): (a) Mg2Si and Mg2Ge; (b) Mg2Sn. 85x57mm (300 x 300 DPI)


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Figure 2. The electrical propereties of Mg2.Si0.3-xGexSn0.7 and Mg2Si0.3GeySn0.7-y (0 ≤ x,y ≤ 0.05): (a) the temperature dependent Seebeck coefficient S; (b) the natural logarithm value of the electrical conductivity lnσ as a function of 1/(2kBT) after intrinsic excitation; (c) the energy band gap Eg of different solid solutions (dots: the experimental data derived from Figure 2(b), lines: the data predicted by the linear model.); (d) the predicted value of the conduction band offset E0. 177x132mm (300 x 300 DPI)



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Figure 3. (a) The lattice thermal conductivity of Mg2Si0.3-xGexSn0.7 and Mg2Si0.3GeySn0.7-y (0 ≤ x, y ≤ 0.05) at 300 K predicted by the Adachi model, plotted as a function of the Ge content. (b) the experimental valuie of the sum of the lattice thermal conductivity and the bipolar part κLattice+κBipolar of the solid solutions as a function of temperature. 177x72mm (300 x 300 DPI)


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Figure 4. (a) The powder XRD patterns of Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 and Mg2.16(Si0.3GeySn0.7y)0.98Sb0.02 (0 ≤ x, y ≤ 0.05); (b) the expanded view in the range of 68-75 degrees. 177x66mm (300 x 300 DPI)



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Figure 5. (a)-(c): The back scattering images of Mg2.16(Si0.3Sn0.7)0.98Sb0.02, Mg2.16(Si0.25Ge0.05Sn0.7)0.98Sb0.02 and Mg2.16(Si0.3Ge0.05Sn0.65)0.98Sb0.02, respectively; (d) the schematic phase diagram of Mg2Si-Mg2Ge-Mg2Sn tenary system at 973 K. 177x133mm (300 x 300 DPI)


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Figure 6. (a)The electrical conductivity σ; (b) the Seebeck coefficient S; (c) the power factor PF; (d) and the Lorenz number L, of different solid solutions as a function of temperature. 177x147mm (300 x 300 DPI)



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Figure 7. (a) and (b): The composition and temperature dependence of the electronic energy band gap Eg as well as the conduction band offset E0 for the solid solution Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 (0 ≤ x ≤ 0.05); (c) and (d): The composition and temperature dependence of the electronic energy band gap Eg as well as the conduction band offset E0 for the solid solution Mg2.16(Si0.3GeySn0.7-y)0.98Sb0.02 (0 ≤ y ≤ 0.05). 177x154mm (300 x 300 DPI)


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Figure 8. The electrical properties of different solid solutions: (a) the carrier concentration nH; (b) the carrier mobility µH; (c) the Pisarenko plot (S-nH) at room temerature (300 K); (d) the nH-µH plot at room temperature (300 K). 177x144mm (300 x 300 DPI)



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Figure 9. (a) The total thermal conductivity κTotal; (b) the sum of the lattice thermal conductivity and the bipolar part κLattice+κBipolar, of Mg2.16(Si0.3-xGexSn0.7)0.98Sb0.02 and Mg2.16(Si0.3GeySn0.7y)0.98Sb0.02 (0 ≤ x, y ≤ 0.05) with different composition. Inset: the κLattice-T -1/2 plot before the intrinsic excitation process. 177x72mm (300 x 300 DPI)


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Figure 10. (a) The temperature dependent dimensionless figure of merit of solid solutions with different composition; (b) the average value in the temperature range of 300-800 K. 177x73mm (300 x 300 DPI)



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TOC graphic 83x35mm (300 x 300 DPI)


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Table of contents entry for the abstract 87x76mm (300 x 300 DPI)


Optimization of the Electronic Band Structure and the Lattice Thermal

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