Non‐Rigid Band Structure in Mg2Ge for Improved Thermoelectric Performance

Abstract Magnesium silicide and its solid solutions are among the most attractive materials for thermoelectric generators in the temperature range of 500–800 K. However, while n‐type Mg2(Si,Ge,Sn) materials show excellent thermoelectric performance, the corresponding p‐type solid solutions are still inferior, mainly due to less favorable properties of the valence bands compared to the conduction bands. Here, Li doped Mg2Ge with a thermoelectric figure of merit zT of 0.5 at 700 K is reported, which is four times higher than that of p‐type Mg2Si and double than that of p‐type Mg2Sn. The reason for the excellent properties is an unusual temperature dependence of Seebeck coefficient and electrical conductivity compared to a standard highly doped semiconductor. The properties cannot be captured assuming a rigid band structure but well reproduced assuming two parabolic valence bands with a strong temperature dependent interband separation. According to the analysis, the difference in energy between the two bands decrease with temperature, leading to a band convergence at around 650 K and finally to an inversion of the band positions. The finding of a combination of a light and a heavy band that are non‐rigid with temperature can pave the way for further optimization of p‐type Mg2(Si,Ge,Sn).


Microstructure
for p-type Mg 2 Ge, assuming a single parabolic band model

Modelling a) Model system with onevalence band (VB) and one conduction band (CB) with different temperature dependent band gaps
We model the minority carrier effects by assuming a 1VB and 1CB system, testing two different temperature dependent band gaps. The first case is according to [1] and describes a weak temperature dependence ( = 5.7 − 1.8 × 10 −4 ) of the band gap, see Figure 2a; the other calculation parameters are listed in the Table 1. Table 1. Input parameters for the modelling of p-type Mg 2 Ge using a 1VB+1CB system.

Parameters VB CB
Density of states effective mass ( * ) [ 0 ] 2.5 0.49 [2] Valley of degeneracy ( ) 2 [3] 3 [4] Deformation potential ( ) 9 [3] 9.8 [4] Elastic constant ( ) [Pa] 1.17 × 10 11 [5] Carrier concentration (p) [cm -3 ] 1.05× 10 20 The input parameters for the 1VB+1CB system are listed in the Table 1. The deformation potential between VB and CB is different due to the different position of point (VBM at  point and CBM at point) [6] . As literature data for Mg 2 Ge is very sparse and no value of E Def,CB has been published previously we have used data from Mg 2 (Si,Sn), the value ( , = 9.8 was obtained from the analysis of the samples published in [2] and similar to the value obtained for Mg 2 Si [4] . The carrier concentration is assumed to be = 1.05 × 10 20 cm −3 . The carrier concentration is assumed so that the model fits with the experimental data. As can be seen from Figure S 4b and c, the contribution of the CB to the total Seebeck coefficient and the total electrical conductivity can be neglected. Figure S 4d shows that temperature dependence of Seebeck coefficient for the 1VB+1CB system increases as temperature increases and approaches a maximum beyond 700 K while experimentally we observe a weak maximum at around 500 K, in agreement with the 2PVB model. Also, the electrical conductivity of the 1CB+1VB model does not resemble our thermoelectric properties,  decreases as temperature increases while in our case,  decreases till a minimum at 425 K followed by a relatively sharp increase at higher T. We furthermore observe that the bipolar thermal conductivity is small for the chosen parameters. For the second case with ( ) = . − × − according to [7] , the decrease in energetical difference between CB and VB is much stronger as temperature increases (Figure S   (empty symbols). As we are considering here three bands there are also three bipolar contributions: that for carrier pairs from SO and the effective HH+LH band, for carrier pairs from SO and CB and from HH+LH and CB. The contribution from SO and HH+LH is small as both carriers have the same sign.
For ( ) = 0.74 − 5.85 × 10 −4 we find agreement between the calculated and the measured bipolar thermal conductivity (see Figure 5b). This is somewhat in between the previously considered cases from literature ( ) = 0.57 − 1.8 × 10 −4 [1] and ( ) = 0.74 − 8 × 10 −4 [7] and justifies that the conclusions we obtained from the analysis of these cases are generally valid for the material system. The temperature dependence of Hall coefficient ( ) is shown in Figure 1(a) and is taken from the cooling data. Figure 1a shows a strong temperature dependent of which is decreasing with increasing temperature. This is a clear indication for the influence of (at least) a second band, either the conduction band or further valence band [8] . The Hall coefficients for the 1CB+1VB and the 2PVB system are calculated according to the equations from the manuscript and the parameters from Table 1.
Comparing the experimental result with the 2PVB system we do not find total agreement ( Figure   1b). The agreement is good at low temperatures, but the temperature dependence of is different between 300 K and 450 K. At higher temperatures both the prediction and the experimental result show a visible, but relatively weak temperature dependence. Quantitatively the values agree within a factor of 10 over the whole temperature range.
Comparison with the 1VB+1CB system (c and d), however, shows a much stronger disagreement. For both assumed descriptions of the band gap, the Hall coefficient shows a change of sign and a very strong change with temperature in the neighboring temperature ranges, both features are not observed experimentally. The change of sign arises from the higher mobility of the electrons compared to those of the holes (see Equation 8), which is a general feature of the Mg 2 X system [4] . The zero crossing for the Hall coefficient is thus a universal feature for a p-Mg 2 X system where the minority carriers start to become relevant and not due to our choice of parameters. Note also that the difference between experimental value and prediction is partially several orders of magnitude for the 1VB+1CB system. We thus deduce that the observed temperature dependence of the Hall coefficient cannot be explained by the influence of the minority carriers.
For a two band system with a convergence of the two relevant bands, is predicted to possibly show a maximum value and a visible, but relatively weak temperature dependence [10] . Note however, that the maximum is not a universal feature and note also that the temperature where it supposed to occur, depends on mobility and carrier concentration ratios [11] . While for the 2PVB system we predict such a maximum (at around 450 K), we do not see this in the experimental data, which decreases monotonously with temperature. However, we note that is close to zero around room temperature, possibly indicating a maximum just below room temperature.
We believe the fact that we do not get a prefect agreement between the experimental result and the prediction from the 2PVB system is the simplified description of the system. P-type Mg 2 X is known to have three valence bands, two of which we treat as one effective band. While this does not affect the modelling of the Seebeck coefficient, it matters for the conductivities and the Hall coefficients, generally the Hall coefficient of a system with two distinct bands is not the same as that of a system with two identical bands. Modelling a 3PVB system would certainly cause a better fit for , however, we do not have sufficient experimental input to avoid a heavily underdefined system, a better agreement because ofmany adjustable parameters would therefore not carry a lot of meaning.

b) A 2PVB model Without Interband separation (IBS)
We have assumed the HH and the LH bands are degenerate and systematically change the effective mass and carrier concentration such that the model fits with the experimental data.
However, the modeled data does not fit well with our data (Figure S8 and S9).

c) A 2PVB model with Interband separation (IBS)
In principle, we have an under defined system with some degree of freedom with the choice of parameters. Thus, we tested different parameters which mostly lead to a poor fit. However, this also implies that the absolute values of the model parameters are subject to significant uncertainty.
The Figures 10 and S11 show that the model reproduces the experimental data only with the right temperature dependent interband separation and lower . ). The model reproduces the experimental Seebeck coefficient data very well however it is not the case for the electrical conductivity. ). The model reproduces both the S and σ experimental data qualitatively well however, there is large discrepancy between the modeled and the experimental S(T) data between 300 and 500 K. Figure S12-S13 show the temperature dependent of transport data to be quite sensitive to ∆ . , * = .
). The temperature dependence of the calculated and are quite sensitive to ΔE thus the behavior of temperature dependent transport data changes with changing in ΔE values. , * = .
). The temperature dependent modeled Seebeck coefficient and electrical conductivity was observed to be quite sensitive to the inter-band separation energy (ΔE). Figure S14 shows that the effect of the second band is negligible for extremely larger inter-band separation. ). The effect of SO band is basically negligible due to large ΔE compared to our value thus the S and σ model is mainly governed by the HH+LH band.
We have also tried to fit the experimental data of the samples with different Li concentration with higher SO effective mass. The model reproduces the electrical conductivity very well however there lies a discrepancy in the modeled and experimental data of the Seebeck coefficient. ). There is an improvement in the agreement of the modeled data to the experimental  data.
We have also tried to fit the lighter SO effective mass band (  ), the ΔE = (-0.26+4×10 -4 T)e/ and = . × − . The temperature dependence of modeled electrical conductivity increases sharply at lower temperatures which is inconsistence with our experimental data.
We have also attempted to model the electrical transport properties by keeping the mass ratio constant with different * and adjust the carrier concentration such that the model fits with the experimental data. Furthermore, we have employed the lower deformation potential ( = 7.5 ) and using the measured carrier concentration. Both of the models fit with the experimental data however the fits are not getting better ( Figure S17-S18).