Introduction

Thermal conduction in solids is one of the most fundamental physical processes. It reveals the nature of lattice dynamics as well as phonon scattering mechanisms. Thermal conductivity of solids also influences many technologically important topics including thermal insulation and management of energy storage and conversion systems, microelectronics, data storage devices; efficiency of thermoelectric materials; and stability of sensors and actuators. For semiconductors the low temperature thermal conductivity is not substantially distinct from those of insulators; at elevated temperatures, however, it becomes interesting and yet intriguing due to the vital roles of charge carriers and their interactions. A signature of electron-hole coupling in semiconductors is the bipolar thermal conduction at elevated temperatures, when the calculated lattice thermal conductivity (κ-LσΤ, where κ is the total thermal conductivity, L the Lorenz number, σ the electrical conductivity and T the absolute temperature) is significantly higher than the T−1 temperature dependence expected for phonon-phonon interaction dominated thermal conductivity1,2,3,4,5,6,7. Similar effect has also been found in semimetals8,9,10. For intrinsic semiconductors, it is well recognized that the mobility ratio between electrons and holes () determines the bipolar thermal conductivity (κb), which maximizes when b = 111,12. Consequently, κb is insignificant for InSb, primarily due to its very large mobility ratio (b > 100)13. In the case of heavily doped semiconductors, the mobility ratio however is no longer a valid guide for understanding or predicting κb, due to the substantially different majority and minority carrier concentrations. For example, recent experiments showed significant κb in p-type heavily-doped skutterudites despite of the mobility ratio between two carriers being greater than 10 (hole mobility ~1–5 cm2/V-s with a concentration of ~1021 cm−3 and electron mobility ~30–50 cm2/V-s with a concentration of ~1018–1019 cm−3 at 800 K, according to our numerical analyses which are presented below)14,15,16,17,18, while the n-type skutterudites do not show appreciable κb, consistent with the rather small b value (~1/50)19,20,21,22,23,24,25,26,27,28,29. Similar observations have been reported for many other semiconductors30,31,32,33,34,35,36,37,38,39. These intriguing results necessitate comprehensive understanding of κb in semiconductors. A recent report attempted to model κb in doped Bi2(Te0.85Se0.15)3 crystals but was unable to capture the specific roles of electronic band structure and carrier scattering mechanisms on κb35.

In this study we report a combined experimental and computational effort that focused on unraveling the general behavior of κb in semiconductors. A numerical method for modeling the temperature dependence of κb for intrinsic as well as extrinsic (heavily doped) semiconductors encompassing a wide range of band gap and electronic band structure has been developed. We find that κb in semiconductors is in general “conductivity-limiting”. In analogous to the bipolar ionic conduction and multiple-step diffusion processes, in which the overall kinetics are determined (limited) by the lower rate species or processes, the bipolar thermal conduction is limited by the charge carrier with lower partial electrical conductivity40,41. Therefore, it is determined by the minority carrier partial electrical conductivity and by the mobility ratio (“mobility-limiting”) in extrinsic and intrinsic semiconductors, respectively. In order to validate these findings, we experimentally demonstrated κb reduction based on electronic band structure modulation and preferential minority carrier scattering. These results largely broaden our understanding of thermal conduction in semiconductors as well as offer insights for optimizing thermoelectric properties of narrow gap semiconductors.

Results and Discussion

Bipolar thermal conductivity in semiconductors can be expressed as1,2,3,4

where σi and αi (subscript i = n, p) are the partial electrical conductivity and Seebeck coefficient for electrons and holes, respectively. For a single parabolic band, the Seebeck coefficient of each carrier can be written as42

where kB is the Boltzmann constant, e the free electron charge, ξ the reduce Fermi energy, λ the carrier scattering parameter, Fx the Fermi integral of the order of x. Therefore , where Eg is the band gap4. For acoustic phonon scattering (λ  = −1/2), the term can be written as which is associated with the total energy carried by electron-hole pairs (band gap energy and kinetic energies). The electrical conductivity of each carrier is

where i = n, p designates the carrier concentrations of electron and hole, respectively.

“Conductivity-Limiting” Bipolar Thermal Conductivity

To elucidate the bipolar thermal conduction behavior in semiconductors, we may rearrange Eq. (1) into (assuming acoustic phonon scattering λ = −1/2, which is valid for most thermoelectric materials)

For a given material at a fixed T, the variation of as a function of ξp or ξn is rather negligible, while the carrier concentrations and the partial electrical conductivity σi (right side of Eq. (4)) could change by several orders of magnitude because of the activation behavior of the charge carriers. Here and are the reduced kinetic energies of holes () and electrons (), respectively, which only slightly change their numerical values when varying the Fermi level43. To verify these analyses, numerical data for p-type skutterudites (RxFe3NiSb12) with Eg = 0.2 eV, mp* = 5 m0, mn* = 2 m0, at 800 K are plotted in fig. 1, where mp*, mn* and m0 are the effective mass of holes, effective mass of electrons and free electron mass, respectively. The details of the calculations will be discussed below. As shown in fig. 1(a), with increasing ξp from –1 (weakly-degenerate) to 2 (degenerate), only increases from 4.2 to 5.3, ~25% increases; whereas p increases by a factor of ~10 and the minority carrier partial conductivity σn decreases by a factor of ~20. These suggest that for semiconductors in general, Eq. (1) or (4) can be approximated as , therefore κb in semiconductors is actually “conductivity-limiting”, analogous to the rate-limiting phenomena in kinetic diffusion processes41. For intrinsic semiconductors, since n = p, Eq. (4) can be further approximated to be , consistent with the large body of literature already developed. In the case of extrinsic semiconductors (n ≫p or p ≫n) κb is primarily determined by the partial electrical conductivity of the minority carriers, not by the mobility ratio. A linear dependence of κb vs. σn at 800 K for p-type doped skutterudites, as shown in fig. 1(b), further substantiates our proposed “conductivity-limiting” concept for bipolar thermal conduction in semiconductors.

Figure 1
figure 1

(a) Numerically calculated total reduced kinetic energy for holes and electrons, hole (majority carrier) concentration and electron (minority carrier) partial electrical conductivity as a function of the reduced Fermi level (ξp); (b) calculated κb as a function of minority carrier partial electrical conductivity σn. The dashed line is a guide for eye. The calculations were carried out for p-type skutterudites (RxFe3NiSb12) with Eg = 0.2 eV, mp* = 5m0, mn* = 2m0, at 800 K.

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Numerical Modeling

Data presented in fig. 1 were calculated by our numerical method for modeling the temperature dependence of κb in semiconductors. Our numerical method aimed at discerning the underlying physics that controls κb, including the electronic band structure features and carrier scattering mechanisms. We use the experimental carrier concentration values as those of the majority carriers. Based on the majority carrier concentration and Seebeck coefficient at room temperature and the maximum Seebeck coefficient value at elevated temperatures, we can determine the Fermi level, the majority carrier effective mass and Eg44,45. The minority carrier effective mass is used as an adjustable parameter. The majority and minority carrier concentrations and their temperature dependences are calculated based on semiconductor statistics46. In order to obtain the T dependence of mobility, we first modeled its carrier concentration dependence at room temperature. We then assumed that the carriers are predominantly scattered by the acoustic phonons, therefore and . For example, the room temperature carrier mobility of n-type and p-type 3d transition metal-based skutterudite antimonides Rx(Fe,Co,Ni)4Sb12 as a function of carrier concentration is shown in fig. 2, where R represents fillers and x the filling fraction. The data were taken from the literatures14,15,16,17,18,19,20,21,22,23,24,25,26,27,47,48,49,50,51,52,53,54,55,56,57,58,59 and were well represented by an empirical expression (the solid lines in fig. 2)60

Figure 2
figure 2

Room temperature carrier mobility as a function of carrier concentration for (a) p-type and (b) n-type Rx(Fe,Co,Ni)4Sb12 skutterudites. The solid lines are least squares fits to the data using Eq. (5). Data used here are taken from Refs. 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27 and 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59.

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where iRef is the reference carrier concentration, approximately where degeneracy sets in, a is a fitting parameter and and are the minimum and maximum possible mobility, respectively. In general, the carrier concentration dependence of mobility for all semiconductors studied in this work can be well accounted by this phenomenological formula and the fitting parameters are summarized in the Table S1 (Supporting Information, SI).

Based on Eqs. (1)–(3) and (5) and the aforementioned method, we were able to numerically fit the temperature dependence of κb in intrinsic and extrinsic semiconductors with a large variation of band gap, the Fermi level and effective mass values. Figure 3(a) shows the excellent agreement between the experimental data (symbols) for intrinsic Si single crystal7 and degenerate polycrystalline skutterudite Yb0.7Fe3NiSb12 in a wide temperature range. Figure 3(b) shows the calculated () and experimental () values of κb for a variety of materials at various temperatures, including (Bi,Sb)2(Te,Se)337,61, skutterudites, Si and Ge6,7. The dashed line in fig. 3(b) represents . These results suggest that our method well accounts for the temperature dependence of κb in semiconductors (all relevant parameters used in our calculations are summarized in Table S2, SI). Since κb is determined by the minority carrier partial electrical conductivity in doped semiconductors, the minority carrier effective mass and its mobility, as well as Eg will have strong influence. The extent to which these parameters affect κb is illustrated in Fig. S1 (SI). This also suggests that κb modification can be achieved by manipulating these parameters.

Figure 3
figure 3

(a) Experimental (symbols) and fitted (solid lines) bipolar thermal conductivity of intrinsic Si single crystal and degenerate Yb0.7Fe3NiSb12 vs. T. (b) Experimental (κbExp) and calculated (κbCal) bipolar thermal conductivity for intrinsic Si and Ge single crystals and degenerate Bi2Te3-based zone melted (ZM) compounds and p-type skutterudites at various temperatures. The dashed line represents κbCal = κbExp.

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Bipolar Thermal Conductivity Reduction

In order to examine the validity of the minority carrier dominated bipolar thermal conduction in heavily doped semiconductors and to utilize the concept of modifying κb presented, we investigated ways of κb reduction motivated by the recent quest for high efficiency thermoelectric materials that necessitate low thermal conductivity62,63,64,65. It is well known that in filled skutterudites66, the triple degenerate conduction band minimum (CBM) is primarily composed of d-orbitals from the transition metals (TMs), with some contribution from Sb p-states (p-d hybridization). Thus the density of states (DOS) at the CBM can be effectively adjusted by varying the TMs. Our first principles calculations reveal that in the p-type Ba-filled skutterudites, DOS at the CBM decreases significantly with decreasing Fe/Co ratio on the TM sites from 2:2 to 1:3, as shown in fig. 4(a), mainly due to the higher energy and thus more contribution of 3d orbitals of Fe as compared with those of Co. The distinct DOS of minority carrier band further suggests that κb for p-type Ba0.5Co3FeSb12 should be smaller than BaCo2Fe2Sb12 due to the minority carrier partial conductivity reduction. Data for 800 K κb vs. the majority carrier (hole) concentration for a series of BaxCo3FeSb12 and BayCo2Fe2Sb12 samples are plotted in fig. 4(b) and the lines represent fitting to the data using the minority carrier effective masses and , respectively. This electronic band modulation induced κb reduction substantiates the dominant role of the minority carriers. Because of the commonly triple-degenerate and 3d-orbital-dominated nature of the CBM, the minority carrier effective masses of the p-type skutterudites are usually much higher than those of the n-type, in which the minority carrier band is mainly composed of single-degenerate Sb p-orbital-featured light bands67. Therefore, the predominant underlying reason for large differences in κb between the n- and p-type skutterudites is actually due to the effective mass differences between the corresponding conduction and valence (minority) bands.

Figure 4
figure 4

(a) The density of states around the CBM for BaCo2Fe2Sb12 and BaCo3FeSb12. (b) Bipolar thermal conductivity at 800 K as a function of hole (majority carrier) concentration for BaxCo2Fe2Sb12 and BayCo3FeSb12. The lines in (b) are fits to the data using different minority carrier effective mass values.

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Our second example of κb reduction takes the advantage of preferential scattering of the minority carriers. Normally in heavily doped semiconductors, the minority carriers are non-degenerate. Given the electronic band structure of a material and the Fermi level (determined by the majority carrier concentration), one can calculate the range of minority carrier wavelength46. For example, the electron wavelength in a heavily-doped p-type Bi2Te3 (p = 3.5 × 1019 cm−3) is approximately between 10 nm and 50 nm, as shown in fig. 5(a). We compare κb of p-type zone melted (ZM) and nanostructured (Nano) Bi0.5Sb1.5Te3 prepared by the melt spinning combined with subsequent spark plasma sintering (MS-SPS) technique61. Figure 5(b) shows, at comparable majority carrier concentrations between the ZM and Nano samples, a significant κb reduction is achieved when nanoprecipitates are introduced into the sample. The minority carrier partial electrical conductivity is determined by Eg, minority effective mass and mobility. The estimated small Eg variation between ZM and Nano is only responsible for 20% of the κb reduction. For a large system like nanostructured Bi0.5Sb1.5Te3, a full electronic band structure calculation is computationally unfeasible. It is difficult to directly determine mn* (minority carrier) at the CBM. The estimated mn* values of n-type doped ZM and Nano Bi2Te2.7Se0.3 are 1.0 m0 and 1.1 m0, respectively38,68. If we assume comparable mn* at CBM between the ZM and Nano samples, the major part of κb reduction between the p-type ZM and Nano Bi0.5Sb1.5Te3 with comparable majority hole concentrations could be attributed to the reduction of minority carrier mobility (μn) corroborated by our κb fittings, where μn = 4095 cm2/V-s for the ZM and 1115 cm2/V-s for the Nano. The TEM image (inset of fig. 5(b)) shows that the sizes of nanoprecipitates closely match those of the minority electron wavelengths. Given the majority hole wavelength is estimated to be ~2 nm, we postulate a strong preferential minority carrier scattering by the nanoprecipitates in the Nano Bi0.5Sb1.5Te3. Similar κb reduction can also be observed in nanostructured n-type Bi2(Te,Se)3 compounds38,69,70. Extensive recent studies have established the role of nanostructure on lattice thermal conductivity reduction63,65, we propose an “preferential minority carrier scatterings” for κb reduction, which is partially responsible for the thermoelectric performance gains reported, especially at elevated temperatures61,71. Recent theoretical work has also demonstrated that similar κb reduction via heterostructure barriers scattering is possible72. Finally we caution that nanostructure induced band structure modulation reported in AgPbmSbTe2-m might be possible for Bi0.5Sb1.5Te373, which could be responsible for part of the κb reduction.

Figure 5
figure 5

(a) The calculated electron wavelength and the product of the Fermi-Dirac distribution function and electronic density of states f(E) g(E) vs. energy with the zero point corresponding to the conduction band minimum (Ec). (b) The experimental and modeled bipolar thermal conductivity vs. temperature, for p-type zone melted (ZM) and nanostructured (MS-SPS) Bi0.5Sb1.5Te3. The inset is a TEM picture of the MS-SPS bulk sample which shows 10–50 nm nanoprecipitates. (The room temperature minority carrier mobilities of ZM and Nano samples are μn = 4095 and 1115 cm2/V-s, respectively).

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Summary

To conclude, our combined theoretical analysis and experimental measurements have established that in semiconductors bipolar thermal transport is in general a “conductivity-limiting” phenomenon, which is controlled by the carrier mobility ratio and the minority carrier partial electrical conductivity for the intrinsic and extrinsic cases, respectively. The numerical method we developed quantifies the role of electronic band structure and carrier scattering mechanisms. We have also demonstrated feasible strategies for manipulating the bipolar thermal conductivity in doped semiconductors via electronic band structure modulation and/or preferential minority carrier scatterings. We expect our study to be beneficial to the current interests in optimizing thermoelectric properties of narrow gap semiconductors.

Methods

Samples in this study were synthesized by a combination of induction melting and long-term high-temperature annealing, by zone melting, or by MS-SPS and the details of which were documented elsewhere19,61. High-resolution transmission electron microscopy (TEM) images were collected using a JEM-2100F TEM. Electrical conductivity (σ) and Seebeck coefficient (α) were simultaneously measured by an Ulvac ZEM-3 under a low-pressure helium atmosphere. Thermal conductivity was calculated from the measured thermal diffusivity (D), specific heat (Cp) and density (d) using the relationship κ = DCpd. Thermal diffusivity D was tested by laser flash diffusivity method using a Netzsch LFA-457 system and Cp was measured by a Netzsch DSC 404F1 using sapphire as the reference. The accuracy of the κ measurements is estimated to be ~10% and the precision <5%. κb were extrapolated from κb + κL = κ-LσT by assuming lattice thermal conductivity κL is inversely proportional to T. Hall measurements were performed on a Janis cryostat equipped with a 9 Tesla superconducting magnet. The carrier concentration of electron (n) or hole (p) and the corresponding Hall mobility μn or μp (subscript n represents the electron and p the hole) were estimated from the measured Hall coefficient (RH) and electrical conductivity by the relation and , respectively.

The first-principles electronic band structure calculations were performed with the generalized gradient approximation functional of Perdew, Burke and Ernzerhof74, with projected augmented wave method75,76, as implemented in Vienna ab initio simulation package (VASP)77. The computational techniques are similar to those published previously66,78. The de Broglie wavelengths (λ) is defined as, λ = h/m*v, where h, m* and v are the Planck constant, carrier effective mass and drift velocity, respectively. m*, v and λ of degenerate majority carriers are almost energy independent (kBT within the Fermi level), while for non-degenerate minority carriers these values are energy dependent, which are derived from band structure. The detailed calculation method is shown in Supporting Information and the calculated density of state, mn* and vn of electrons for p-type Bi2Te3 (ξp = 0.25, mp* = 1.3 m0) are shown in figure S2 (SI).

Additional Information

How to cite this article: Wang, S. et al. Conductivity-limiting bipolar thermal conductivity in semiconductors. Sci. Rep. 5, 10136; doi: 10.1038/srep10136 (2015).