### 1. Introduction

Replacing current vapor-compression refrigerators with thermoelectric refrigerators can reduce the rate of climate change. Unlike vapor-compression refrigerators, which utilize harmful greenhouse gases as refrigerants, thermoelectric refrigerators do not require any greenhouse gases to operate. Thermoelectric refrigerators can produce cooling using a two-dimensional array of alternating

*p*- and*n*-type thermoelectric materials connected in series via the Peltier effect [1]. The efficiency of thermoelectric refrigerators depends on the performance of the thermoelectric materials, which is characterized by a dimensionless figure-of-merit,*zT*=*S*^{2}*σT*/*κ*, where*S*,*σ*,*κ*, and*T*are the Seebeck coefficient, electrical conductivity, thermal conductivity and the absolute temperature, respectively. However, currently the*zT*of known thermoelectric materials is not high enough to compete with vapor-compression refrigerators.Considerable efforts have been devoted to improving

*zT*. Band engineering is frequently adopted to maximize*S*^{2}*σ*(power factor). Enhancing the power factor is challenging because of the trade-off between*S*and*σ*. If*S*is increased by making the band mass of a single Fermi surface pocket (*m*_{b}^{*}) heavier, the relevant*σ*will be decreased (band flattening). However, band convergence is an effective approach to enhance the power factor, as it enhances*S*(via increasing valley degeneracy (*N*) with a constant_{v}*m*_{b}^{*}) without deteriorating*σ*[2-4]. Identifying a band parameter that is not coupled to both*S*and*σ*is key to successful power factor improvement.Although it has often been neglected to avoid complication, the types of band masses that determine

*S*and*σ*are different. While*S*is directly proportional to the product of*N*_{v}^{2/3}and*m*_{b}^{*}, which is equivalent to density-of-states effective mass (*m*_{d}^{*}=*N*_{v}^{2/3}*m*_{b}^{*}),*σ*is inversely proportional to conductivity effective mass (*m*_{c}^{*}) [5-7]. For a spherical Fermi surface pocket, its*m*_{c}^{*}is identical to*m*_{b}^{*}. However, for an ellipsoidal pocket, the*m*_{c}^{*}becomes smaller than the*m*_{b}^{*}[8]. Therefore, controlling Fermi surface nonparabolicity (*K*) can decouple*σ*from the*S*.Here, we theoretically demonstrate that control of

*K*is another effective band engineering approach to improve*zT*. The effects of engineering*K*on each of the electronic transport properties are calculated using a two-band model, where fitted band parameters of Bi-Sb-Te (the best candidate for thermoelectric refrigerators) are used. We find that*K*engineering enhances the power factor by increasing*σ*while keeping*S*constant.### 2. Experimental Procedure

The two-band model (solutions of the Boltzmann transport equations) was fitted to the experimental

*S*, and*σ*of a state-of-the-art*p*-type Bi-Sb-Te ingot measured as a function of*T*to obtain the band parameters (*m*_{b}^{*}and deformation potential, Ξ) of the valence and conduction bands in the Bi-Sb-Te (spherical Fermi surface pockets assumed) [9]. For 300 K, 0.3, and 0.6*m*(electron rest mass) were adopted for_{e}*m*_{b}^{*}, corresponding to the valence and conduction bands, respectively. The*N*of both bands were set to be six, and an energy gap (_{v}*E*) of 0.19 eV was adopted in the two-band model [10-12]. It was assumed that acoustic phonon scattering limited the carrier mobility of both bands in the Bi-Sb-Te. Details on the two-band model fitting can be found in our previous works [4,13,14,15]. While keeping the aforementioned parameters (_{g}*m*_{b}^{*}, Ξ,*N*, and_{v}*E*) constant, electronic transport properties were calculated for different valence band_{g}*K*(= 0.2, 0.5, 1, 2.5, and 4) via the two-band model (*K*= 1 was assumed for the conduction band). Here, each electronic transport property was calculated as a function of the chemical carrier concentration (*n*) for*T*= 300 K. The effect of*K*engineering was evaluated by comparing the electronic transport properties calculated for*K*≠ 1 to those obtained for*K*= 1 (where Fermi surface pockets are spherical).### 3. Results and Discussion

### 3.1. Decoupling *m*_{c}^{*} from *m*_{b}^{*} with *K* ≠ 1

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To theoretically demonstrate how changing the shape of Fermi surface pockets affects the thermoelectric performance of

*p*-type Bi-Sb-Te, its electronic transport properties were calculated using varying valence band*K*with constant band parameters (*m*_{b}^{*}, Ξ,*N*, and_{v}*E*) under the acoustic phonon scattering assumption. The band parameters of Bi-Sb-Te were first obtained via the two-band model while assuming its Fermi surface pockets were spherical (_{g}*K*= 1 for both bands). The*K*is defined as a ratio of*m*_{b}^{*}along the longitudinal ellipsoid direction (*m*_{b,∥}^{*}) to that along the transverse ellipsoid direction (*m*_{b,⊥}^{*};).In terms of

*K*, the degree of decoupling*m*_{c}^{*}from*m*_{b}^{*}is determined as below.When

*K*in Eq. (2) equals to unity, the*m*_{c}^{*}and*m*_{b}^{*}are identical, but for*K*≠ 1, the*m*_{c}^{*}becomes lighter than*m*_{b}^{*}as presented in Fig 1(a) [8]. The solid black line in Fig 1(a) represents*m*_{c}^{*}/*m*_{b}^{*}as a function of*K*. The symbols indicate*m*_{c}^{*}/*m*_{b}^{*}for each*K*in the range from 0.2 to 4. The specific*K*s were chosen so that their corresponding*m*_{c}^{*}/*m*_{b}^{*}ratios in the valence band were all different. Engineering*K*to be less than 1 (*m*_{b,∥}^{*}being heavier than*m*_{b,⊥}^{*};) is more effective for decoupling*m*_{c}^{*}from*m*_{b}^{*}than changing*K*to be larger than 1, according to Fig 1(a). In other words, decoupling*σ*from*S*becomes possible for a*K*other than 1.The

*μ*of Bi-Sb-Te was calculated as a function of_{d}*n*for varying valence band*K*(0.2, 0.5, 1, 2.5, and 4) to evaluate the effect of engineering*K*in drift mobility (*μ*) as shown in Fig 1(b). Commonly, the_{d}*μ*was described to be inversely proportional to_{d}*m*_{b}^{*5/2}assuming*K*= 1. However, the*μ*is more accurately described by a product of_{d}*m*_{c}^{*-1}and*m*_{b}^{*-3/2}for*K*≠ 1, as given in Eq. (3).If the

*m*_{c}^{*}in Eq. (3) is expressed in terms of*m*_{b}^{*}and*K*(as in Eq. (2)), Eq. (3) becomesAccording to Eq. (4), the

*μ*is inversely proportional to the_{d}*m*_{c}^{*}/*m*_{b}^{*}ratios, which depend on*K*(Fig 1(a)). In Fig 1(b), the*μ*calculated with_{d}*K*= 1 is the lowest for all ranges of*n*(green triangle and line). In contrast, the*μ*calculated with_{d}*K*= 0.2 is the highest (orange square and line). Because the*m*_{c}^{*}/*m*_{b}^{*}ratio for*K*= 0.2 (*m*_{c}^{*}/*m*_{b}^{*}= 0.73) is smaller than that for*K*= 1 (*m*_{c}^{*}/*m*_{b}^{*}= 1), the*μ*for_{d}*K*= 0.2 is calculated to be higher than that for*K*= 1. For other*K*, the corresponding*μ*also depends on the reciprocal of the_{d}*m*_{c}^{*}/*m*_{b}^{*}ratio (Eq. (4)). Therefore, a higher*μ*is expected with a_{d}*K*that gives a smaller*m*_{c}^{*}/*m*_{b}^{*}ratio.### 3.2. Effect of varying *m*_{c}^{*}/*m*_{b}^{*} on *σ* via *K* engineering

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To examine the effect of engineering

*K*on*σ*, the*σ*of Bi-Sb-Te was calculated for varying valence band*K*(0.2 – 4), as shown in Fig 2(a). Because the*σ*is directly proportional to*μ*as given below (e is the electric charge),_{d}a similar trend was observed in

*μ*for different_{d}*K*(Fig 1(b)), and this is also expected for*σ*. According to Fig 2(a),*σ*for*K*other than unity were calculated to be higher than that for*K*= 1. Again, it is the value of*m*_{c}^{*}/*m*_{b}^{*}determined from a given*K*, and not the*K*itself that is correlated to*σ*. The*m*_{c}^{*}/*m*_{b}^{*}ratios which correspond to*K*(0.2 – 4) are provided in Table 1 (Eq. (2)).From Table 1, the

*K*arranged in order of decreasing*m*_{c}^{*}/*m*_{b}^{*}are: 1, 0.5, 2.5, 4, and 0.2. The calculated*σ*is also increased with decreasing*m*_{c}^{*}/*m*_{b}^{*}(*K*: 1 → 0.5 → 2.5 → 4 → 0.2) as shown in Fig 2(a). Engineering*K*to reduce*m*_{c}^{*}/*m*_{b}^{*}was found to be beneficial to increase*σ*.A minor difference between

*σ*and*μ*is that_{d}*σ*only depends on*m*_{c}^{*}. Because*n*is directly proportional to*m*_{b}^{*3/2},*σ*becomes inversely proportional to*m*_{c}^{*}only as below.In other words,

*σ*is the electronic transport parameter, which can be separated from*S*most effectively with relevant*K*engineering.The effect of varying

*m*_{c}^{*}/*m*_{b}^{*}(via*K*engineering) on electronic thermal conductivity (*κ*) was also studied. The heat-carrying carrier conductions (_{e}*κ*) contribute to the total_{e}*κ*along with heat conductions via lattice vibrations (lattice thermal conductivity,*κ*) and bipolar conductions (bipolar thermal conductivity,_{l}*κ*) as in Eq. (7)._{bp}Among

*κ*,_{l}*κ*, and_{e}*κ*, the_{bp}*κ*is the most directly related to_{e}*σ*via the Wiedemann-Franz law, as below, where*L*is the Lorenz number [16].This is because the

*L*is independent of both*m*_{b}^{*}and*m*_{c}^{*}, the*κ*in Eq. (8) is directly proportional to_{e}*σ*at constant*T*. Therefore, a higher*κ*is expected with a_{e}*K*that corresponds to a lower*m*_{c}^{*}/*m*_{b}^{*}like*σ*, but the higher*κ*decreases_{e}*zT*. Fig 2(b) shows the*κ*for varying_{e}*K*as a function of*n*. The*κ*with_{e}*K*= 0.2 (the lowest*m*_{c}^{*}/*m*_{b}^{*}, orange square and line) was calculated to be the largest, and that with*K*= 1 (the maximum*m*_{c}^{*}/*m*_{b}^{*}, green triangle, and line) was calculated to be the smallest. The inset in Fig 2(b) shows the*L*calculated for varying*K*as a function of*n*. Because the*L*is not a function of*m*_{c}^{*}, the calculated*L*are all identical for different*K*. The discrepancy among*κ*for different_{e}*K*is calculated to increase at higher*n*. For the range of*n*where the thermoelectric performance of most materials is optimized (~10^{19}cm^{-3}), the increase in*κ*due to_{e}*K*engineering is negligible. Although decreasing*m*_{c}^{*}/*m*_{b}^{*}increases both*κ*and_{e}*σ*, the improvement in*σ*outweighs the increase in*κ*._{e}### 3.3. Effect of varying *m*_{c}^{*}/*m*_{b}^{*} on *S* via *K* engineering

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To examine how varying

*m*_{c}^{*}/*m*_{b}^{*}(via*K*engineering) affects*S*, the*S*of Bi-Sb-Te was calculated for varying valence band*K*(0.2 – 4) at 300*K*, as shown in Fig 3(a). The*S*for different*K*are almost identical for*n*≥ ~5 × 10^{18}cm^{-3}, but they are different for*n*< ~5 × 10^{18}cm^{-3}. Although*S*calculated from a single band is independent of*m*_{c}^{*}(but dependent on*m*_{d}^{*}instead), the overall*S*calculated from two bands does depend on*m*_{c}^{*}via*σ*contributions from each band. The overall*S*is calculated by Eq. (9) below, where*S*and_{i}*σ*are the partial contributions from each band (_{i}*i*= VB (valence band), CB (conduction band)).When

*σ*_{VB}and*σ*_{CB}in Eq. (9) are comparable at low*n*, the difference in*σ*_{VB}due to varying valence band*K*is well captured in the overall*S*. For example, the*σ*_{VB}for*K*= 0.2 is the highest among*K*in the range from 0.2 to 4 (Fig 2(a)). Hence, the*S*for*K*= 0.2 is calculated to be the highest for*n*< ~5 × 10^{18}cm^{-3}, as in Fig 3(a) (*S*_{VB},*S*_{CB}, and*σ*_{CB}being constant with respect to*K*). In Fig 2(a), we simply stated that the overall*σ*increased because of the decrease in the*m*_{c}^{*}/*m*_{b}^{*}ratio. To be more specific, the increase in the overall*σ*was due to the increase in*σ*_{VB}originating from a decrease in the valence band*m*_{c}^{*}/*m*_{b}^{*}ratio (*K*of the conduction band was fixed). However, once the*σ*_{VB}becomes much greater than*σ*_{CB}at high n, the*S*in Eq. (9) converges to*S*_{VB}, as shown below.Because

*S*_{VB}is independent of*m*_{c}^{*}, the*S*is calculated to be approximately invariant with respect to*K*at*n*≥ ~5 × 10^{18}cm^{3}, as in Fig 3(a). Given that the optimum properties of thermoelectric materials are frequently realized at*n*~ 1 × 10^{19}cm^{-3}, it can be concluded that the*S*is not affected by varying*K*.The effect of varying

*m*_{c}^{*}/*m*_{b}^{*}(via*K*engineering) on*κ*was also examined. Unlike_{bp}*κ*, both_{e}*S*and_{i}*σ*(i = VB, CB) are required to calculate_{i}*κ*, as shown below._{bp}In Eq. (11), the

*S*_{VB}and*S*_{CB}are independent of varying*K*, as they do not depend on*m*_{c}^{*}. And because we assume that the*K*of the conduction band is unity for simplicity, only the*σ*_{VB}is changed with varying*K*. Therefore, it is expected that the*κ*with a low_{bp}*m*_{c}^{*}/*m*_{b}^{*}will be high because of high*σ*_{VB}, as in the case of*S*. However as calculated in Fig 3(b), the discrepancy in*κ*due to different_{bp}*K*is negligible for all*n*. Hence, varying*K*does not affect*κ*._{bp}### 3.4. Effect of varying *m*_{c}^{*}/*m*_{b}^{*} on power factor via *K* engineering

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To demonstrate the effect of varying

*m*_{c}^{*}/*m*_{b}^{*}(via*K*engineering) on the power factor, the power factor of Bi-Sb-Te was calculated for varying valence band*K*(0.2 – 4) at 300*K*as shown in Fig 4(a). Since suppressing*m*_{c}^{*}/*m*_{b}^{*}is beneficial to*σ*(Fig 2(a)), while causing a negligible change in*S*(Fig 3(a)), an improvement in the power factor is expected. According to Fig 4(a), the highest power factor was calculated for the*K*= 0.2 (orange square and line), which corresponds to the lowest*m*_{c}^{*}/*m*_{b}^{*}(Table 1), and the lowest power factor was calculated for*K*= 1 (green triangle and line) which corresponds to the maximum*m*_{c}^{*}/*m*_{b}^{*}. The calculated power factor increased with decreasing*m*_{c}^{*}/*m*_{b}^{*}(*K*: 1 → 0.5 → 2.5 → 4 → 0.2), as observed in*σ*in Fig 2(a). Therefore, engineering*K*to be other than unity will always improve the power factor.Lastly, to investigate the effect of varying

*m*_{c}^{*}/*m*_{b}^{*}(via*K*engineering) on*zT*, the*zT*of Bi-Sb-Te was calculated for varying valence band*K*(0.2 – 4) at 300*K*as shown in Fig 4(b). For the*zT*calculation,*κ*is required in addition to the_{l}*κ*and_{e}*κ*calculated in Fig 2(b) and 3(b), respectively. Because the_{bp}*κ*cannot be calculated using the two-band model, the experimental_{l}*κ*of a state-of-the-art Bi-Sb-Te ingot was adopted from the literature (_{l}*κ*= 0.8 W/m-K at 300_{l}*K*) [9]. The highest*zT*was calculated for*K*= 0.2 for all*n*, as shown in Fig 4(b). Its optimum*zT*was achieved near*n*~ 10^{19}cm^{-3}. The sharp decrease in*zT*for*n*> 10^{19}cm^{-3}is attributed to the increasing*κ*and_{e}*κ*at high_{bp}*n*. This confirms that engineering*K*to decrease*m*_{c}^{*}/*m*_{b}^{*}is another effective approach to improve*zT*.Engineering

*K*is another effective band engineering approach to improve the power factor of thermoelectric materials. Making*K*other than unity decouples*m*_{c}^{*}from*m*_{b}^{*}, which enables us to break the trade-off between*S*and*σ*. As*m*_{c}^{*}/*m*_{b}^{*}decreases,*σ*is improved without deteriorating*S*. Although*κ*is also increased with decreasing_{e}*m*_{c}^{*}/*m*_{b}^{*}, the*κ*increase is negligible compared to the_{e}*σ*increase. Therefore,*zT*improvement is guaranteed if the*m*_{c}^{*}/*m*_{b}^{*}is reduced by engineering*K*.We assumed that the

*K*of both the valence and conduction bands were unity when obtaining the band parameters of the state-of-the-art Bi-Sb-Te ingot from the two-band model fitting. However, it has been reported that the valence band*K*of Bi-Sb-Te was optically determined to be approximately 2 [17]. Consequently, the determined band parameters of Bi-Sb-Te (*m*_{b}^{*}, Ξ) with*K*= 1 can be marginally different from what they should be (with*K*= 2). The discrepancy between the fitted Ξ and the actual Ξ will be higher than that expected between the fitted and actual*m*_{b}^{*}because*K*only affects*m*_{c}^{*}-related parameters. Nonetheless, the fact that engineering*K*improves power factor remains intact because the beneficial effect of decreasing*m*_{c}^{*}/*m*_{b}^{*}on power factor applies to any set of band parameters.Both decreasing

*m*_{c}^{*}/*m*_{b}^{*}by engineering*K*and decreasing Ξ have similar impacts on*μ*as the_{d}*μ*is inversely proportional to the product of Ξ_{d}^{2},*m*_{c}^{*}and*m*_{b}^{*3/2}. In addition, because*S*is independent of Ξ, the effect of suppressing Ξ on electronic transport properties is almost identical to the effect of engineering*K*on electronic transport properties. Therefore, an additional optical study that measures*m*_{b}^{*}at different principal axes is required to confirm that an improvement in power factor originates from*K*engineering.No experimental approach to engineer only

*K*has been suggested yet. Although alloying is known to alter*K*, it also changes other band parameters at the same time, making the evaluation of the effect of*K*change difficult [17]. As an alternative, materials with innately low*m*_{c}^{*}/*m*_{b}^{*}have been theoretically sought as potential candidates for new high-performance thermoelectric materials [18].Although the effect of engineering

*K*was investigated using narrow band gap Bi-Sb-Te, its effect will be even greater for large band gap materials since the potential increase in*κ*due to_{bp}*K*engineering will be minimized because of their substantially low*κ*[19]. Moreover, since we know that_{bp}*K*other than unity can significantly alter the electronic transport properties of thermoelectric materials, accurate*K*input (obtained via measurements or simulations) is important to accurately determine materials’ band parameters via band modeling.### 4. Conclusions

A new band engineering approach that by-passes the trade-off between

*S*and*σ*has been suggested. Engineering the shape of Fermi surface pockets (*K*, non-parabolicity factor) to be ellipsoidal was found to decouple conductivity effective mass (*m*_{c}^{*}) from the band mass of a single Fermi pocket (*m*_{b}^{*}). When*K*is engineered to lower the*m*_{c}^{*}/*m*_{b}^{*}ratio, the*σ*is calculated to increase while*S*remains constant. It was calculated that the thermoelectric figure-of-merit (*zT*) of a Bi-Sb-Te ingot can be improved to higher than 1.3 if the nonparabolicity factor is engineered to be 0.2. Although alloying is known to change*K*, it also simultaneously alters other band parameters. However, it is suspected that alloying, combined with carefully designed doping, might only change*K*so that it improves the power factor of materials effectively.