Efficiency as a performance metric for material optimization in thermoelectric generators

The optimization of thermoelectric (TE) materials with respect to carrier concentration, chemical composition, microstructure, etc. is inevitable for maximizing the performance of TE devices. Theoretical performance prediction can speed up this process dramatically as the synthesis and experimental characterization of all relevant combinations is practically impossible. Conventionally, the dimensionless figure of merit ( zT ) is considered as a measure of TE energy conversion capability. However, zT could mislead the search for optimized materials as it is only an intermediate parameter. To resolve this issue, we combined a device performance calculation routine (one-dimensional continuum theory-based, with fully temperature dependent TE properties) with a band structure-based material model. As an example, a study was conducted on p-type Mg2Si1− x Sn x solid solutions for which optimization of carrier concentration ( n ) and composition ( x ) is required. Here, according to previous findings, a single parabolic band (SPB) model was assumed, with an effective mass linearly dependent on composition and carrier concentration, and acoustic phonon and alloy scattering of the charge carriers. It was found that for a cold side temperature of 300 K and a hot side temperature of 500 K (which is well within the validity limits of a SPB model), the optimum n for Mg2Si1− x Sn x based on efficiency was found to be at 4.5 ×1019cm−3 , while based on zT max it was found to be about 20% higher. Additionally, the usage of the temperature average of zT ( zT TAv) for finding the optimum parameters is also analysed. For p-Mg2(Si,Sn), zT TAv predicts the optimum composition and carrier concentration close to the exact efficiency calculation, despite the fact that the efficiency predicted by zT TAv can be quite off from exact efficiency. The usage of zT TAv was further tested for common TE materials such as n-type Mg2Si1− x Sn x x and PbTe and a similar conclusion is obtained. Finally, the reason for this closeness and the importance of using exact efficiency plots is discussed.


Introduction
Devices made of thermoelectric (TE) materials convert a certain fraction of the heat passed through them into useful electrical power or, vice versa, can pump heat from cold to hot, driven by an electric current [1]. A good TE generator material needs to be able to maintain the temperature (T) difference (i.e. should have low thermal conductivity κ) while being able to transport the generated electric power to the external load at low internal loss (high electrical conductivity σ). The Seebeck coefficient (α), which is proportional to the output voltage, is inversely related to electrical conductivity (more specifically to the number of charge carriers) [2]. Due to this inherent, partially reverse coupling, an optimum between these TE properties has to be found for best performance [2]. Materials have to be engineered to meet this optimum, resulting in exploration of several adjustable parameters such as base composition, ratios of the constituting elements, doping level, dopants species, etc, affecting the electronic band structure and thermal transport. For example, tuning the doping species and concentrations in Skutterudites [3], Mg 3 Sb 2 [4,5] or tuning of x in PbTe 1−x Se [6], Bi 2−x Sb x Te 3 [7], Sn 1−x Se [8], Mg 2 Si 1−x Sn x [9] and changing the Zr or Hf concentration in half-Heuslers [10,11] are some strategies to optimize performance.
In order to reduce time-consuming experimental efforts to study the effect of each of the free parameters, modelling material properties and thereby predicting TE device performance is a convenient approach. Conventionally, a dimensionless figure of merit (zT) defined as α 2 σ κ · T is used as a measure of performance, guiding the development of TE materials [1]. However, zT as a measure of efficiency originates from the maximum efficiency expression η max = ∆T ) derived under the constant property model (CPM) assumptions, whereas in reality, the material properties are temperature dependent. Constant averaged material properties entering the CPM model (obtained by averaging the temperature dependent data) must be chosen such that they reflect the actual material behaviour in the device as good as possible. Despite the use of appropriate averages, the efficiency prediction is inaccurate due to asymmetry in distribution of Joule heat to the hot and cold side of the device and assumption of magnitude and distribution of Peltier-Thomson heat in CPM compared to reality [12][13][14][15][16][17]. Additionally, the Carnot efficiency [16,18,19] is not taken into account when considering just the figure of merit z. Therefore, zT alone can be quite misleading for performance estimation [16][17][18][19][20].
To overcome this, here, we analyse the device efficiency (for temperature dependent properties) directly to estimate the optimum parameters. This is done by a custom-made 1D performance calculation tool (based on the solution of the 1D heat balance) that was developed in [17]. The material properties can be obtained from electronic band structure models such as the single parabolic band (SPB) model [21,22], a multiband model [23][24][25] or similar models which give simplified expressions for material properties based on the solution of the Boltzmann transport equation (BTE). The 1D device performance routine is combined with the BTE-based models in a single calculation routine such that the fundamental material parameters are directly coupled to the predicted device efficiency.
As an example for the implementation of the technique, p-type Mg 2 Si 1−x Sn x has been chosen as Mg 2 (Si,Sn) solid solutions have gained popularity due to the high performance of the n-type materials [26,27], availability, low cost and progress in contact development [28,29]. However, the p-type suffers from still limited performance with a maximum reported zT of about 0.5 [9,30] and hence, the performance needs to be improved. Previously, several authors [9,[31][32][33][34] have shown the capability of an SPB model to closely describe the behaviour of p-type Mg 2 Si 1−x Sn x . In a recent study [31], the SPB model was successfully applied to the whole compositional range of p-type Mg 2 Si 1−x Sn x , involving acoustic phonon and alloy scattering as dominant scattering mechanisms. This allowed for an approximate identification of the optimum carrier concentration n opt and optimum composition x opt . However, the optimum was determined based on zT max which is not exact as stated before. Here, we would like to identify refined optimum parameters for p-Mg 2 Si 1−x Sn x using the actual efficiency and compare the results with those obtained kusing averaged zT TAv as an indicator of optimum parameters.

Method
The basic SPB model equations describing the three main TE transport properties (α, σ and κ) are given below [22]: where is the Fermi integral of the order i, k B is the Boltzmann constant, e is the electronic charge, η c = EF kBT is the reduced chemical potential of the charge carriers and E F is the Fermi energy. n is the charge carrier concentration given by equation (4) where m * D is the density of states effective mass, h is Planck's constant, and µ is the mobility. In contrast to the original publication where the experimental data is discussed and a linear fit for m * D is used [31] we have introduced here a bilinear equation for m * D with a dependence both on n and x. The additional weak dependence on n improves the agreement between experimental data and model as discussed in [31]; the best fit is given by m * D (x, n) = (2.14 − 1.39x +0.16 * 10 −20 cm 3 n ) m 0 , where m 0 is the electron rest mass. The fitted 2D plot and the comparison of a purely linear fit of m * D (x) and m * D (x, n) with m * D obtained by comparing the SPB model with experimental data is given in the supplementary information (SI) in figure S1 available online at stacks.iop.org/ JPENERGY/3/044006/mmedia. In p-type Mg 2 Si 1−x Sn x , the mobility is assumed to be governed by acoustic phonon and alloy scattering mechanisms [31]. The exact equations for these scattering mechanisms and the scattering potentials used (from [31] which were obtained by comparing the SPB model with experimental data) are given in the SI.
For predicting the properties using the SPB model, for each n (taken as a free parameter), the chemical potential is calculated from the Fermi integral and with this, α (T) is calculated. The electrical conductivity σ (T) is calculated according to equation (2). The thermal conductivity κ consists of the lattice thermal conductivity κ L and the conductivity associated with the charge transport given by L σ T, where L is the Lorenz number . κ L (T) is an input parameter for an SPB model and, for the considered case, it is obtained from experimental data as in [31]. A 2D polynomial fit function κ L (x, T) covering the whole compositional range as described in [31] was used.
From all of these temperature dependent parameters the TE properties are obtained. Employing x and n as independent variables matches the experimental reality as a change in x corresponds to an isovalent substitution of Si by Sn and n is adjusted by adding small amounts of dopants (Li substituting Mg in this case); hence x and n can be adjusted basically independent of each other [30]. While applying the SPB model, the validity of the assumptions should be discussed. The SPB assumption fails above the temperature at which bipolar conduction becomes relevant i.e. where more than one band contributes significantly to conduction. Conventionally, the validity range of the SPB model can be judged by visual examination of the Pisarenko plot [21] or by comparing experimental results and calculated SPB model output. Here we have used the shape of the zT (T) curve as a criterion to estimate the validity range as it is related to efficiency. An example zT curve calculated using the SPB model (solid lines) for a composition of x = 1, and n = 0 was chosen as the validity limit of the SPB model. dα dT was analysed for comparison and it was found that the results are very similar, with d(zTexp) dT being the more conservative limit on average, see figure S2 in SI. Such data were obtained for a number of samples available from literature [31,[36][37][38] and the validity limit was interpolated over the whole compositional range. In order to avoid unphysical extrapolation, the maximum validity limit outside the known experimental points was set to 700 K.
Considering a single TE leg with a hot side temperature T h and a cold side temperature T c, the device performance, i.e. efficiency and power output, is calculated [17,35] using the material properties obtained employing the SPB. In the steady-state, the exact temperature profile T (x) obtained by solving the TE heat balance equation [39,40] (equation (5)) is used for accurate performance calculation of the TEG. In 1D, the heat balance equation is written as [35], where j is the current density. The term d corresponds to the Fourier heat flux which also compensates for the locally appearing Joule heat ρ (T) j 2 and Peltier-Thomson heat T dα dT ∂T ∂x . The exact solution of equation (5) is obtained using the iterative procedure described in [17]. From the temperature profile, the exact power and efficiency are obtained as follows:  power density to the inflowing heat flux at the hot side of the TEG leg (q in ; equation (7)), where q in is given by equation (8): Here, −κ h · dT dx h is the Fourier heat flux into the leg and j · α h T h is the Peltier heat flux absorbed at the hot side. The suffix h indicates the properties at the hot side, i.e. κ h = κ (T h ) and α h = α (T h ). The maximum efficiency or power is obtained by building the η vs j or p vs j characteristics, respectively, and finding the maximum (1st derivative being zero). This calculation is done for the entire range of composition of p-Mg 2 Si 1−x Sn x and carrier concentrations based on the interpolated microscopic parameters m * D (x, n) and κ L (x, T).

Results
The validity limit was obtained as explained in section 2 (figure 1) using the experimental data available in literature [31,36,38]. The full data used is given in table S1 in the supplementary info. The data was interpolated using the Thin Plate Spline algorithm available in the Origin software [41] and is shown in figure 2. As expected, as we move from Mg 2 Si to Mg 2 Sn, the maximum temperature up to which the SPB model is valid reduces for a fixed n, due to the decreasing band gap, and as we move from lower carrier concentrations to higher, the validity limit increases as the contribution of the minority carriers decreases [31,42]. The partially wavy form of the temperature contour lines is due to the input data, which was obtained from different sources and shows the usual experimental scatter.

Optimum carrier concentration
Contour plots showing the calculated maximum efficiency for a hot side temperature (T h ) of 500 K and a cold side temperature (T c ) of 300 K as a function of x and n are shown in figure 3(a). For comparison to zT max ( figure 3(b)), zT TAv contours are also plotted in figure 3(c), where zT TAv = 1 ∆T T h Tc zT (T) dT (a physically appropriate temperature average for Seebeck coefficient suggested by Ioffe) [1]. Since minority carrier effects are not considered in an SPB model, zT increases monotonously with T and therefore zT max is the result of the SPB model at the considered hot side temperature (here 500 K). The line for the chosen T h of 500 K from the validity plot in figure 2 is superimposed onto these plots (the area with SPB valid only up to less than  500 K indicated by the dimmed region) to clarify what part of the modelling results is physically interpretable. Note that the performance maximum near x = 1 is thus not accessible. It arises due to the low m * D for low n and high x and overcompensates the effect of increasing lattice thermal conductivity for x → 1, but is outside the validity range for the SPB model; practically zT is lower here than predicted by the SPB model due to the influence of minority carriers.
As can be seen from figure 3(d), the optimum carrier density n opt differs between the calculated efficiency and zT max plots, while n opt from the zT TAv plot coincides approximately with that from the efficiency plot. For efficiency, the maximum is found at n = 0.44 × 10 20 cm −3 for zT TAv , while for zT max, the maximum occurs at n = 0.54 × 10 20 cm −3 .

Discussion
The necessity to define a validity range is due to employing an SPB model for the efficiency calculation, which naturally fails at higher temperatures due to bipolar contributions. This could be overcome by a multiband description, but so far no multiband model with a good agreement of experimental and modelling data has been published. To date, most modelling approaches were focused on n-type Mg 2 Si 0.4 Sn 0.6 for which the SPB model is usually valid for a larger temperature range, partially because of the higher mobility of the electrons compared to the holes [4,24,43,44]. Previous approaches to estimate the validity range of an SPB model were based on the analysis of the temperature dependence of the calculated electrochemical potential [31] or approximate two-band modelling [45]; using d(zT) dT as a criterion is a more pragmatic approach. As d(zT) dT decreases towards higher T, the zT of the model will usually be higher than that of the real sample; hence the efficiency from the SPB model will be too high if the chosen temperature interval includes a range close to the experimental zT max . However, we estimate this maximum relative difference in efficiency to about 10%. From figure 4, it can be seen that the possibility to predict n opt , x opt and the efficiency at the optimum fails beyond a hot side temperature of 610 K, since for higher T h the optimum lies outside the valid region of the employed SPB model. Nevertheless, with an expected maximum application T h of ≈ 700 K, the SPB model can be applied to a significant fraction of the relevant temperature range and the modelling results are of practical relevance.
From figure 3, it can be seen that the maximum efficiency plot and the zT max (zT at 500 K) plot do not result in the same optimum region. This is because zT max is one of the less suitable methods of representing the performance of the TE material [17,19,20] and overestimates performance [12]. For T h = 500 K, the optimum n determined by zT max is off from the efficiency prediction by 21% as seen from figure 3. Figures 5(a) and (b) shows the changes in optimum n and x , respectively, for T h = 400 K, 500 K, 600 K with T c = 300 K. It can be seen that the optimum n using zT max is always overestimated and that the discrepancy can be > 20%. Also, as expected, the discrepancy reduces as the temperature interval gets smaller, i.e. tending towards the CPM. A similar trend but much weaker in magnitude is observed in the case of zT TAv . There is no significant difference in optimum x with respect to the exact efficiency and zT TAv and zT max as seen from figure 4(b). Analysing figures 5(a) and 3(d) it can be concluded that the shift in the optimum n with T h is for p-Mg 2 Si 1−x Sn x of minor practical relevance. The maximum of η (n) is wide, and hence a deviation from n opt by even 20% will mean a performance loss of only about 1.5%. Similar for η (x), a mean composition near 0.635 would be suitable for all T h for p-Mg 2 Si 1−x Sn x .
The optimum parameters obtained with zT TAv plots almost coincide with those from the efficiency plots. This is because the basic input SPB parameters (m * D , deformation potential constant (E Def ), κ L ) do not vary erratically with x and n for the considered material. However, to see if the optimum position (x, n) predicted by CPM (using zT TAv or zT max in the efficiency formula) is different from that of exact efficiency, it is important to see how the parameters that determine this discrepancy between CPM and exact calculation vary with n and x, rather than considering the discrepancy in the obtained efficiency value between CPM and the exact calculation. This is because even though the discrepancy between the CPM and exact efficiency might be quite large, the change of the factors that determine this discrepancy, for different n and x, might be constant or negligibly varying, leading to the same optimum position (x, n) in both cases. This means that the parameters to be optimized (n and x in our case) can often nevertheless be obtained with high accuracy using CPM. A large discrepancy might be expected if the difference between CPM and exact efficiency is itself a strong function of n and x.
Additionally, as a comparison, the optimum parameters obtained for n-type Mg 2 Si 1−x Sn x [46] and PbTe [47] using zT TAv and exact efficiency are shown along with p-type Mg 2 Si 1−x Sn x in figure S3 of the SI (normalized efficiency and zT TAv are presented). As can be seen, the width of the η and zT TAv curves vs n varies from case to case, and zT TAv predicts the optimum n very similar to exact efficiency in all cases.
If an SPB model is employed to describe the TE properties, it is presumably rare to see a material case where zT TAv cannot be used to predict optimum parameters since it does not include any intrinsic carrier effects. The accuracy of efficiency prediction by CPM varies with a qualitative change in the slope and curvature of the TE properties on T. In particular, these vary with the relative relevance of bipolar effects [12,16,17]. With constant or moderately changing parameters of a SPB model there will not be qualitative changes in the curve shape of the TE properties: α (T) and ρ (T) will rise approximately linearly and κ (T)  will decrease approximately as κ ∝ c1 T + c 2 [22,48]. The relation between the chosen mode of zT definition and the physically justified Ioffe ZT (from the adequate averages of Seebeck and resistivities) will be quite stable under these conditions and also the distribution of Joule and Thomson heat, as discussed, will not vary a lot, as it is more the change of slope of the properties with T rather than their absolute magnitude which makes the effects here. With that there is good reason to trust in that if zT TAv gives a very good match in two examples it would do so for many more materials; zT TAv can thus be a convenient alternative if the SPB model can be employed. However, in order to achieve higher efficiency, higher temperatures where bipolar conduction occurs are usually employed for applications and hence the SPB assumption and therefore zT TAv can prove to be ineffective due to the stronger change of temperature dependence of the transport properties. Note also, that even though zT TAv apparently can be used to identify the optimum carrier concentration, it can show quite some inaccuracy for the calculation of efficiency itself [12,16,17]. Finally, the approach shown here can relatively easily be upgraded to a p-n couple [22] and electrical and thermal contact resistances can be implemented. Then the approach shown here helps for an accurate estimation of device efficiency and optimized material properties under realistic application conditions. Employing the efficiency derived from the temperature dependent properties (equation (8)) allows for an optimization with respect to n in scenarios when the area adjustment [18] to accommodate for different currents in p and n leg cannot be done for some practical reason or when contact resistances play a large role.

Summary and outlook
A simple and efficient tool to predict the exact optimum composition of the solid solution and optimum carrier density has been shown using p-Mg 2 Si 1−x Sn x as an example. This approach is applicable to any material for which a description of the TE properties is given. This is demonstrated exemplarily by figure S3 in the supporting material, where the efficiency (normalized) vs n curves for PbTe and n-Mg 2 Si 0.4 Sn 0.6 have been provided. The difference in optimum carrier concentration obtained using commonly used indicators such as zT max or zT TAv vs the optimum obtained considering the locally varying TE properties for p-Mg 2 X has been discussed. Even though the discrepancy in calculated efficiency between CPM and exact calculations exists, for some materials like the one considered here, zT TAv can be used to find optimum carrier concentration and composition with good accuracy for practical use.
In practice, to reach maximum efficiency, hot side temperatures entering the range of bipolar conduction are important, and to model such cases, multi band models are necessary. With the shown TEG calculation routine, the effect of having metal contacts and its effect on optimum parameters can be easily implemented and studied. The combination of a semiconductor physics model such as SPB and continuum theoretical efficiency calculation can also be employed to optimize material grading or segmentation for further performance enhancement.

Data availability statement
All data that support the findings of this study are included within the article (and any supplementary files).