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J. Korean Ceram. Soc. > Volume 56(3); 2019 > Article
Kim: Effect of Density-of-States Effective Mass on Transport Properties of Two Converging Valence Bands

Abstract

Band convergence is known to be effective in improving thermoelectric performance by increasing the Seebeck coefficient without significantly reducing electrical conductivity. Decoupling of the Seebeck coefficient and electrical conductivity in converged bands is the key requirement. Yet, the degree of decoupling depends on the band parameters of the converging bands. Herein, we report theoretical transport properties of two valence bands as their energy difference changes from 0.25 eV to 0 eV. In order to demonstrate the effect of band parameters in transport, we first conducted calculations for the case where the two bands have the same parameters. Then, we conducted the same calculation by doubling the density-of-states effective mass of one valence band. Given that there are two bands, each band’s effective mass was doubled one at a time while the other band’s effective mass remained constant. We found that the decoupling was strongest when the bands participating in convergence had the same band parameters.

1. Introduction

The farthest man-made object from Earth (the spacecraft Voyager 1) runs on radioisotope thermoelectric generators.1) The radioactive radiation (heat) is converted to electricity by the thermoelectric materials surrounding the radioisotope heat source. This reliable material can also convert an applied electrical potential to a temperature gradient (for cooling application). The efficiency of the conversion (from ΔT to ΔV, and from ΔV to ΔT) is characterized by a dimensionless figure-of-merit zT = (S2 × σ)/κ × T, where S, σ, κ, and T are the Seebeck coefficient, electrical conductivity, thermal conductivity, and the temperature in Kelvin, respectively. At a glance, creating a high-performance thermoelectric material does not seem too demanding: the material should have a high S and σ, and a low κ. However, raising zT becomes non-trivial because of the interconnections among S, σ, and κ.
One of the most popular routes to a high zT value is to suppress the thermal conductivity contributed by lattice vibrations (κ).2) Because κl is not correlated to any other parameters in zT, reducing κl is the most straightforward way to improve performance. zT can also be improved by increasing S and σ simultaneously. Typically, S is inversely proportional to σ. For example, when the carrier concentration of a thermoelectric material is decreased, S increases while σ decreases. The density-of-states effective mass (m*) of a material is closely related to S. The heavier the m*, the higher the S. Since the mobility of charged particles (electrons or holes) is inversely proportional to m*, materials with high m* (high S) have low σ. Here, m* = Nv2/3 × mb*, where Nv and mb* are the valley degeneracy and the band mass of a single valley, respectively.3) To be more specific, mb* is inversely proportional to the mobility. Therefore, if it is possible to increase m* without increasing mb* (i.e., by increasing Nv), then S can be improved without deteriorating σ. A simultaneous increase in S and σ can be acquired by converging nearby bands (in terms of energy). When two bands converge, the effective m* is increased because of the higher Nv. Recently, the highest room temperature zT in the (Bi0.25Sb075)2Te3 alloy was explained by two valence bands converging at the (Bi0.25Sb075)2Te3 composition.4)
Although we now know that converged bands are much better than single band participation in transport, the manner in which the band parameters of the converging bands affect the resulting transport properties is unknown. In this study, we perform two-band model calculations where two valence bands with the same band parameters converge from an energy difference of 0.25 eV to 0 eV. Then, we intentionally double m* of one of the bands and repeat the same two-band model calculations. Since two bands exist, we increase m* one at a time while keeping the other band’s m* constant. When converged, the case where both bands have the same parameters shows a much better power factor (S2 × σ) than that where one of the bands has twice as heavy m* as the other.

2. Experimental Procedure

A two-band model, which is an extension of the single parabolic model, was employed to calculate the resulting transport properties of two converging valence bands.5) Band structures that were postulated in the two-band calculations are described in Fig. 1. Three cases were considered to examine the effect of m* on the transport of the two converging bands. Firstly, we considered the band parameters m* and deformation potential (Edef) of the two participating valence bands (VB1 and VB2 in Fig. 1) to be equal, as shown in Fig. 1(a) (Case I). The energy difference (ΔE) between the upper valence band (VB1) and the lower valence band (VB2) was varied from 0.25 eV to 0 eV (when ΔE = 0 eV, VB1 and VB2 were converged). Secondly, only m* of VB2 (in Fig. 1(a)) was doubled while keeping the other parameters the same as those in the first case (Fig. 1(b), Case II). Lastly, m* of VB1 (in Fig. 1(a)) was doubled while keeping the other parameters the same as those in the first case (Fig. 1(c), Case III). The valley degeneracies (Nv) of the two valence bands (VB1 and VB2) were assumed to be 6 (the same as those of Bi2Te3). When the two bands were converged (ΔE = 0 eV), the total Nv of the converged bands increased to approximately 12.
The energy gap (Eg) between the upper valence band (VB1) and the lowest conduction band (CB) was not changed in all three cases. CB illustrated in Fig. 1 was not included in the two-band model calculations. Only Eg was taken into consideration to account for the change in m* with the modification in Eg (as Eg is reduced, m* is known to decrease as well because of the interaction between the highest valence band and the lowest conduction band according to the Kane dispersion relation).6) We utilized the same band gap (0.16 eV, as in Table 1) for all three cases. Using the band structures described in Fig. 1, Boltzmann transport equations were computationally solved as a function of chemical potential for each valence band (VB1 and VB2). Conductivity-weighted averages of the calculated thermoelectric parameters (Seebeck coefficient, electrical conductivity, power factor, mobility, and carrier concentration) of individual bands were computed to acquire the combined transport properties of the two converging valence bands.
The band parameters employed in the two-band model calculations are listed in the Table 1. Eg and the band parameters of VB1 in Case I (Fig. 1(a)) are similar to those of Bi2Te3. However, ΔE and the parameters for VB2 in Case II (VB1 in Case III) are artificial. These values were carefully selected to efficiently illustrate the effect of m* in the band convergence model calculation.

3. Results and Discussion

Band convergence is one of the most powerful strategies to enhance the electrical properties of thermoelectric materials. While the majority of methods that are known to improve electrical transport increase the Seebeck coefficient (or electrical conductivity) significantly with accompanying electrical conductivity (or Seebeck coefficient) reductions to a lesser magnitude, band convergence enhances the Seebeck coefficient and electrical conductivity simultaneously. The simultaneous increase in both the Seebeck coefficient and electrical conductivity is only possible because the number of valley degeneracies (Nv), and not the single band mass (mb*), are increased in the band convergence. Consequently, even when the total density-of-states effective mas (m* = Nv2/3 × mb*) is increased as a result of the convergence (increase in Nv), its mobility (inversely proportional to mb*), and of course, the electrical conductivity, remain undamaged.
Furthermore, at the convergence, the electrical conductivities of both bands are summed to produce a higher total electrical conductivity. This can be expressed as Eq. (1) below.
(1)
σtotal=σVB1+σVB2
Therefore, we expect the electrical conductivity of the converged bands (σtotal) to be larger than the electrical conductivity contributions of the individual bands (σVB1 or σVB2). Here, the electrical conductivity of an individual band can be calculated using Eq. (2), under the single parabolic band (SPB) model, with acoustic phonon scattering being dominant.
(2)
σSPB=μHRH
where
(3)
μH=3π4vl2d2Edef2m*5/2(kT)3/2F-1/2(η)2F0(η)
and
(4)
RH=3h33π(2m*kT)3/2F-1/2(η)2F02(η)
μH and RH in Eq. (2) are the Hall mobility and Hall coefficient, respectively. e, vl, d, and k in Eqs. (3) and (4) are the electric charge, average phonon velocity, density, and the Boltzmann constant, respectively. Lastly, Fj(η) is the Fermi integral, as detailed below (η is the reduced chemical potential).
(5)
Fj(η)=0ɛjdɛ1+exp(ɛ-η)
The total Seebeck coefficient of the converged bands (Stotal) can be expressed as a conductivity-weighted average of the Seebeck coefficient contributions of the individual bands (SVB1 or SVB2), as in Eq. (6). Again, the Seebeck coefficient of an individual band can be calculated using Eq. (7), under the SPB model, with acoustic phonon scattering being dominant.
(6)
Stotal=(σVB1SVB1+σVB2SVB2)/(σVB1+σVB2)
(7)
SSPB=ke(2F1(η)F0(η)+η)
We first carried out the two-band model calculations for the two valence bands with equal band parameters (density-of-states effective mass, deformation potential, and number of valley degeneracies). For the upper valence band (VB1 in Fig. 1(a)), which is fixed in terms of energy, the position of the lower valence band (VB2 in Fig. 1(b)) with respect to VB1 was varied from 0.25 eV to 0 eV. When the energy difference between VB1 and VB2 (ΔE) is 0.25 eV, the location of VB2 is 0.25 eV below VB1 in terms of energy. When ΔE is 0 eV, the two bands (VB1 and VB2) converge. More specifically, VB2 now has the same energy as that of the fixed VB1.
According to Fig. 2(d), the Seebeck coefficient of the converged bands (0.0 eV, blue solid line) is the highest, as we expected. It is interesting to note that when ΔE = 0.1 eV (yellow-green solid line), the lower valence band (VB2) begins influencing the transport properties. In other words, if the lower valence band is more than 0.1 eV apart from the highest valence band, that band will not participate seriously in transport (this can be changed if the parameters of the bands are altered). Except for the 0.0 eV line (converged blue solid line), the gradients of the other lines for ΔE > 0 eV were changed at higher Hall carrier concentrations (nH). When ΔE is the largest (0.25 eV, red solid line), the slope of the line flattens at nH values exceeding 2E20 (cm−3). On the contrary, if ΔE is only 0.05 eV (green solid line), the change in the slope starts early at 2E19 (cm−3). A larger ΔE means that the lower valence band (VB2) is far from the upper valence band (VB1). In that case, the chemical potential corresponding to the tip of the lower valence band (VB2) is much larger than that corresponding to the tip of the upper valence band (VB1). Therefore, when we sweep through the band structure with varying chemical potentials, if ΔE is large, increasing the chemical potential to reach the tip of VB2 will take a long time. Since the Hall carrier concentration (nH) is directly proportional to the chemical potential, the change of the slope in the Seebeck Pisarenko relation (Fig. 2(d)) indicates the instant when the chemical potential encounters the tip of VB2. When ΔE = 0.05 eV (green solid line in Fig. 2(d)), VB2 starts contributing to the transport from nH ~ 2E19 (cm−3), and when ΔE = 0.25 eV (red solid line in Fig. 2(d)), VB2 starts contributing to the transport only at nH ~ 2E20 (cm−3). In both cases, the lines flatten (the rate of the Seebeck coefficient decrease with increasing nH slows down) with the advent of the lower valence band (VB2).
The Hall mobilities of the converging valence bands were also calculated (Fig. 2(b)). Analogically, when bands are converged, we can think of it as increasing the number of lanes on a highway along which charged particles (holes or electrons) are travelling. Therefore, the mobility of the holes (or electrons) in the converged bands will improve. Indeed, this was observed in the calculated Hall mobilities (μH) of the two converging bands with the same parameters. According to Eq. (1) above, the total electrical conductivity of the converged band (0.0 eV, blue solid line in Fig. 2(c)) is much greater than that of the large ΔE (0.25 eV, red solid line in Fig. 2(c)). At nH > 4 × 1019 (cm−3), the electrical conductivity of ΔE = 0.05 eV (green solid line in Fig. 2(c)) is higher than that of the converged one (ΔE = 0.0 eV, blue solid line in Fig. 2(c)). A similar phenomenon is observed in the nH-dependent Hall mobilities graph (Fig. 2(b)). Starting at the same nH (4 × 1019 (cm−3)), the position of the Hall mobility for ΔE = 0.05 eV (green solid line) and that for the converged one (0.0 eV, blue solid line) is reversed. For the converged case (0.0 eV, blue solid line), individual Hall mobilities from each band and the combined Hall mobility from the converged bands are the same. This is because the tips of the two bands have the same chemical potentials. However, for ΔE = 0.05 eV (green solid line), the chemical potential of VB1 maximum is smaller than that of VB2 maximum (Fig. 1(a)). If we are to compare the converged case and that of ΔE = 0.05 eV, the chemical potential of VB2 maximum for the case of ΔE = 0.05 eV is equal to those of the VB1 and VB2 maxima of the converged case. Simultaneously, the chemical potential of VB1 maximum for the case of ΔE = 0.05 eV is the smallest among all the bands in both cases. This shows that the highest individual Hall mobility (as a function of chemical potential) occurs for VB1 with ΔE = 0.05 eV according to Eq. (3) (the individual Hall mobility from VB2 with ΔE = 0.05 eV is equal to those of VB1 and VB2 from the converged case). Therefore, the combined Hall mobility (function of chemical potential) for ΔE = 0.05 eV is greater than that of the converged case at all chemical potentials. When the combined electrical conductivity is plotted as a function of chemical potential, the combined electrical conductivity of the converged case is higher than that of ΔE = 0.05 eV. The calculated combined Hall carrier concentration of the converged case is higher than that of ΔE = 0.05 eV at all chemical potentials. Since electrical conductivity, Hall mobility, and Hall carrier concentration are interdependent (σ = μH × nH × e), observed trends in the converged case and for ΔE = 0.05 eV can be understood as a combined effect among their different chemical potential dependences. Simultaneous improvement in both the Seebeck coefficient and the electrical conductivity of the converged bands results in a substantial increase in its power factor (S2 × σ), as shown in Fig. 2(a).
Apparent decoupling of the Seebeck coefficient and electrical conductivity becomes less obvious once the density-of-states effective mass (m*) of the lower valence band (VB2) is doubled (Case II in Fig. 1(b)). The power factor of the converged bands (Case II) is still the highest when compared to those of the other bands with ΔE > 0.0 eV. However, the absolute value of the power factor (for Case II: m* of VB2 = 2 × m* of VB1) is approximately 22% lower than that computed for Case I (m* of VB2 = m* of VB1). We now know that the convergence of bands with the same m* is more beneficial to electrical transport than that of two bands whose values of m* differ by a factor of 2 (assuming that the other band parameters are the same).
The behavior of the Seebeck Pisernko relation for Case II (Fig. 3(d)) is similar to that of Case I (Fig. 2(d)). The converged band (0.0 eV, blue solid line) has the highest Seebeck coefficient. However, its Hall mobility (Fig. 3(b)) is lower than those of other bands with ΔE > 0.0 eV at nH < 9E19 (cm−3). This is reasonable as the Hall mobilities of the other bands with ΔE > 0.0 eV mostly originate from the light VB1 (hence the high Hall mobility) at nH < 9E19 (cm−3). Similarly, the electrical conductivity of the converged band (0.0 eV, blue solid line) is lowest among all the bands with different ΔE values. Fortunately, however, the difference in the electrical conductivity is small, which leads to the highest power factor for the converged bands (0.0 eV, blue solid lines in Fig. 3(a)).
Lastly, other two-band model calculations were performed for the case where m* of VB1 is twice as large as that of VB2 (Case III in Fig. 1(c)). The situation of the converged bands in Case III is exactly the same as that in Case II. When ΔE > 0 eV, Cases II and III differ. Since VB1 is heavier than VB2, the total Seebeck coefficient of the converged bands (Stotal) (0.0 eV, blue solid line in Fig. 4(d)) calculated from a conductivity-weighted average of individual Seebeck coefficients (SVB1 and SVB2 from Eq. (6)) can be smaller than the individual Seebeck coefficient of the heavier VB1 (which contributes mainly to transport) (Fig. 4(d)). In a similar vein, the exact opposite reasoning can be applied to explain the highest Hall mobility and electrical conductivity of the converged bands (blue solid line in Fig. 4(b) and 4(c)).

4. Conclusions

Although band convergence facilitates higher electrical transport properties, its effect may vary depending on the details of the band parameters of the participating bands. If the band parameters of the converging bands are the same, the decoupling of the Seebeck coefficient and electrical conductivity is the strongest at the highest power factor.

Acknowledgments

This work was supported by the 2019 Hongik University Research Fund.

Fig. 1
Schematic diagrams of band structures used in the two-band calculations: (a) Case I (equal m*), (b) Case II (lower valence band m* doubled), (c) Case III (upper valence band m* doubled).
kcers-2019-56-3-12f1.jpg
Fig. 2
Calculated transport properties of Case I: (a) power factor, (b) Hall mobility, (c) electrical conductivity, and (d) Seebeck coefficient as a function of Hall carrier concentration.
kcers-2019-56-3-12f2.jpg
Fig. 3
Calculated transport properties of Case II: (a) power factor, (b) Hall mobility, (c) electrical conductivity, and (d) Seebeck coefficient as a function of Hall carrier concentration.
kcers-2019-56-3-12f3.jpg
Fig. 4
Calculated transport properties of Case III: (a) power factor, (b) Hall mobility, (c) electrical conductivity, and (d) Seebeck coefficient as a function of Hall carrier concentration.
kcers-2019-56-3-12f4.jpg
Table 1
Band Parameters Adopted in the Two-Band Model Calculations
Band parameters Case I Case II Case III
Upper Valence Band (VB1) m* (in m0) 0.7 0.7 1.4
Upper Valence Band (VB1) Edef (eV) 18.7
Upper Valence Band (VB1)
Valley degeneracy, Nv
6
Lower Valence Band (VB2) m* (in m0) 0.7 1.4 0.7
Lower Valence Band (VB2) Edef (eV) 18.7
Lower Valence Band (VB1)
Valley degeneracy, Nv
6
Band gap Eg (eV) 0.16
Energy difference between VB1 and VB2 ΔE (eV) 0-0.25

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