Open Access Article

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M. Yazdani-Kachoei and
S. Jalali-Asadabadi*

Department of Physics, Faculty of Sciences, University of Isfahan (UI), Hezar Gerib Avenue, Isfahan 81746-73441, Iran. E-mail: saeid.jalali.asadabadi@gmail.com; sjalali@sci.ui.ac.ir; Fax: +98 31 37934800; Tel: +98 31 37932435

Received
27th September 2019
, Accepted 18th October 2019

First published on 6th November 2019

Experimental evidences show that Ce-based compounds can be good candidates for thermoelectric applications due to their high thermoelectric efficiencies at low temperatures. However, thermoelectric properties have been studied less than the other properties for CeRhIn_{5}, a technologically and fundamentally important compound. Thus, we comprehensively investigate the thermoelectric properties, including the Seebeck coefficient, electrical conductivity, electronic part of thermal conductivity, power factor and electronic figure of merit, by a combination of quantum mechanical density functional and semiclassical Boltzmann theories, including relativistic spin–orbit interactions using different exchange–correlation functionals at temperatures T ≤ 300 K for CeRhIn_{5} along its a and c crystalline axes. The temperature dependences of the thermoelectric quantities are investigated. Our results reveal a better Seebeck coefficient, electrical conductivity, power factor and thermoelectric efficiency at T ≪ 300, in agreement with various other Ce-based compounds, when a high degree of localization is considered for the 4f-Ce electrons. The Seebeck coefficient, power factor and thermoelectric efficiency are made more efficient near room temperature by decreasing the degree of localization for 4f-Ce electrons. Our results also show that the thermoelectric efficiency along the a crystalline axis is slightly better than that of the c axis. We also investigate the effects of hydrostatic pressure on the thermoelectric properties of the compound at low and high temperatures. The results show that the effects of imposing pressure strongly depend on the degree of localization considered for 4f-Ce electrons.

Our results show that the maximum value for the Seebeck coefficient occurs at low temperatures for CeRhIn_{5}, consistent with previous experimental reports for other Ce-based compounds,^{13,20,21,24,48} when the 4f-Ce electrons lie in the high localized regime. Furthermore, based on our results the Z_{e} value of our studied case in lower temperatures is more than that of higher temperatures.

Experimental results^{24,25,49} have shown that applying pressure can affect the thermoelectric efficiency of the other rare earth based compounds. In addition, experimental measurements^{41,43} have shown that imposing pressure can affect the electrical conductivity and thereby the thermoelectric efficiency of CeRhIn_{5}. Moreover, Shishido et al.^{50,51} have experimentally verified that the degree of localization of 4f-electrons in CeRhIn_{5} is decreased by imposing pressure. Thus, these experimental evidences motivated us to investigate the effects of pressure as well as the degree of localization of 4f-Ce electrons on the thermoelectric properties and thermoelectric efficiency of CeRhIn_{5}. For this purpose, we have performed our calculations in different volumes using various exchange–correlation functionals (XCFs) with different degrees of localization for 4f-Ce electrons, including LDA+U^{52} and hybrid^{53,54} approaches as well as PBE-GGA. Our results show that the Seebeck coefficient of CeRhIn_{5}, its power factor and Z_{e} value increase as the degree of localization decreases at high temperatures. This shows that the Seebeck coefficient, power factor and Z_{e} value are made more efficient near room temperature by decreasing the degree of localization of 4f-Ce electrons. Our results also reveal that the effect of hydrostatic pressure on the thermoelectric parameters strongly depends on the considered degree of localization for 4f-Ce electrons, in accordance with our previous work.^{55} The electronic structures of CeRhIn_{5}, including its density of states (DOS) and band structure are also investigated in this study.

The transport properties are calculated by BoltzTraP code^{61} which is based on the Boltzmann theory.^{46,47} Since this code needs a very high k-mesh, a denser mesh of 150000 k-points is considered for the thermoelectric calculations. The Seebeck coefficient, electrical conductivity and the electronic part of the thermal conductivity tensors are calculated by the following formulas:^{61,62}

(1) |

(2) |

(3) |

(4) |

σ_{αβ}(i,k) = e^{2}τ_{i,k}υ_{α}(i,k)υ_{β}(i,k),
| (5) |

In many previous works, the thermoelectric parameters have been studied at fixed temperature versus chemical potential (μ) or at fixed μ versus temperature (T). However, as eqn (1)–(3) in the manuscript show, the elements of electrical conductivity (σ_{αβ}(μ,T)), Seebeck coefficient (S_{αβ}(μ,T)) and thermal conductivity (κ_{αβ}(μ,T)) tensors are the functions of two parameters, i.e. μ and T. The BoltzTraP code calculates these parameters versus T and μ. In fact, BoltzTraP changes the value for μ step by step and calculates σ_{αβ}(μ,T), S_{αβ}(μ,T) and κ_{αβ}(μ,T) at fixed μ versus temperature at each step. Thus, a change of the μ or T values can change σ_{αβ}(μ,T), S_{αβ}(μ,T), κ_{αβ}(μ,T). Therefore, to optimize the thermoelectric parameters, μ and T should be considered simultaneously. For this, we follow a new strategy to analyze the outputs of BoltzTraP code. In our strategy, we change the temperature step by step (1 K in our calculations) and find the maximum of thermoelectric parameters for fixed temperature at each step from the outputs of BoltzTraP code, i.e., case.condtens and the value of μ or doping level which leads to this maximum value. We encounter too many numbers in the case.condtens. Therefore, to find the maximum values of σ_{αβ}(μ,T), S_{αβ}(μ,T) and κ_{αβ}(μ,T) at fixed μ, we use a simple program, max-conduct. To calculate the maximum values of power factor (PF = σS^{2}) and electronic figure of merit , we use another program, i.e. max-PF. This program calculates the diagonal elements of PF = σS^{2} and versus μ and T, then finds the maximum values of these parameters the same as S, σ and κ.

(6) |

(7) |

σ = 1/ρ = neμ, | (8) |

All the above evidences explicitly confirm that the thermoelectric parameters depend on the electronic structures of the materials. Thus, the electronic structure plays a key role and a discussion in this respect can give physical insight into the thermoelectric properties of the system in question. Before discussing the electronic structure results, however, it is important to consider and keep in mind the following three points. The first point is that to perform accurate electronic structure calculations, it is essential to be aware of the degree of localization of the system for selecting an appropriate functional to satisfactorily deal with the exchange–correlation term. The following evidences may assist with the first point. Previous dHvA measurements show that 4f-Ce electrons in CeRhIn_{5} are localized at zero pressure.^{50,51} Experimental measurements^{73} performed at zero pressure by Fujimori et al. also confirm that the 4f-Ce electrons in CeRhIn_{5} have localized character. Furthermore, the dynamical mean field theory calculations^{30} demonstrate that 4f-Ce electrons of CeRhIn_{5} are more localized than those of CeCoIn_{5} and CeIrIn_{5} at zero pressure. The latter cases (CeCoIn_{5} and CeIrIn_{5}) themselves are also demonstrated by ARPES measurements to be localized systems.^{74,75} All these theoretical and experimental evidences clearly show that the 4f-Ce electrons are localized in CeRhIn_{5} at zero pressure. The second point which should be considered for producing accurate electronic structures concerns the validity of the functional used for the exchange–correlation term. It is well-known that the standard GGA often fails to reproduce the correct electronic structures for strongly correlated f-electron systems. For highly correlated systems, the thermoelectric properties, especially the thermopower S, can be strongly influenced by the location of the narrow DOSs with respect to the Fermi level. Therefore, in this work, we use the band-correlated GGA+U and hybrid B3PW91 approaches to investigate the electronic structures of CeRhIn_{5}. The third point, which is also crucial for an accurate prediction of the electronic structure, concerns the pressure dependence of the localization degree. Although the 4f-Ce electrons are localized in CeRhIn_{5} at zero pressure, the dHvA measurements^{50,51} show that the localization degree of 4f-Ce electrons in CeRhIn_{5} is pressure dependent. This implies that the localization degree of these electrons can be reduced by imposing pressure. Furthermore, we discussed in our recent work,^{55} in agreement with previous works,^{76} that the exchange–correlation energy of Ce-based compounds could not be satisfactorily described only by a single functional for every pressure. Therefore, for a specific pressure range an appropriate functional must be selected; band-correlated (band-like) functionals are more appropriate for low (high) pressures.^{55} Thus, following this strategy, in addition to the band-correlated GGA+U and hybrid B3PW91 approaches, the band-like PBE-GGA functional is also used for the electronic structure investigation.

Now, we can discuss the electronic structure of the system by considering the above points. To this end, we have calculated the spin up and spin down band structures of CeRhIn_{5} compound using PBE-GGA, PBE-GGA+U with U_{eff} = 5.5 eV and hybrid B3PW91 with α = 0.3 for the three volumes introduced in Section 2. Here, the band structures are shown only for the first volume (which corresponds to the experimental volume) together with the available experimental results^{60} for comparison to validate our calculations in Fig. 1. The complete results including band characters are presented in detail in Fig. 2 and 3 of the ESI.† The comparison confirms that our calculated band structures are consistent with the experimental results.

Fig. 1 Spin up and spin down band structures calculated by PBE-GGA, GGA+U with U_{eff} = 5.5 eV and hybrid B3PW91 with α = 0.3 for CeRhIn_{5} using the experimental volume. In the right figure, the ARPES experimental result^{60} is also shown for comparison. |

We have found that the pressure cannot considerably change the band structures of the system for spins up and down using PBE-GGA. We have also observed that the number of bands crossing the Fermi level is 5 and not changed by pressure using PBE-GGA. To validate these observations, the maximum and minimum energies and bandwidths, as well as occupation numbers, are extracted from the calculated band structures and tabulated in Table 1. The results show that the PBE-GGA bands data can be only slightly changed by pressure. Although the widths of the 5 PBE-GGA bands are increased by pressure (due to the small decrease of the minimum energies and the small increase of their maximum energies), these increments are not very considerable. We noticed that in contrast to the pressure, however, the degree of localization could more considerably affect the band structures. If instead of the band-like PBE-GGA, the band-correlated PBE-GGA+U and B3PW91 functionals are used, more bands (6 bands) cross the Fermi level. The latter number of band crossing (6 bands) is different from that obtained using the band-like PBE-GGA (5 bands). This difference confirms a more significant effect of the localization degree than pressure in complete agreement with the result reported for CeIn_{3} in our recent work.^{55} This result can be reconfirmed if we consider the fact that the degree of localization predicted by PBE-GGA is lower than those predicted by GGA+U and B3PW91. Keeping the latter fact in mind, we see in Table 1 that, in contrast to pressure, the bands data are substantially affected by the degree of 4f localization. The results show that the minimum energy (E_{min}) decreases as the degree of localization increases and inversely the results show that the maximum energy (E_{max}) increases as the degree of localization increases. Therefore, the bandwidths increase for the five bands, but this time, the increments of the bandwidths are remarkable and should be considered. The difference between the effect of pressure and localization degree can be also seen in the band structures presented in the ESI,† if we follow the effects of localization degree and pressure on the distributions of the 4f-Ce states and their locations with respect to the Fermi level. These results are in good agreement with previous works.^{32}

XCF | Band | First volume | Second volume | Third volume | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

E_{min} (eV) |
E_{max} (eV) |
Width (eV) | Occup | E_{min} (eV) |
E_{max} (eV) |
Width (eV) | Occup | E_{min} (eV) |
E_{max} (eV) |
Width (eV) | Occup | ||

PBE-GGA | γ1 | −1.3783 | 0.0729 | 1.4512 | 0.9728 | −1.4055 | 0.0753 | 1.4808 | 0.9727 | −1.4462 | 0.0791 | 1.5253 | 0.9730 |

γ2 | −1.2030 | 0.0940 | 1.2970 | 0.9269 | −1.2328 | 0.0968 | 1.3296 | 0.9247 | −1.2783 | 0.1021 | 1.3804 | 0.9211 | |

γ3 | −1.1729 | 0.3588 | 1.5317 | 0.602 | −1.2010 | 0.3579 | 1.5589 | 0.6015 | −1.2445 | 0.3537 | 1.5983 | 0.6001 | |

γ4 | −1.0897 | 0.3931 | 1.4828 | 0.2954 | −1.1154 | 0.3916 | 1.5070 | 0.2948 | −1.1539 | 0.3891 | 1.5430 | 0.2944 | |

γ5 | −1.0506 | 0.4999 | 1.5505 | 0.2033 | −1.0747 | 0.4939 | 1.5686 | 0.2068 | −1.1114 | 0.4832 | 1.5947 | 0.2119 | |

GGA+U (U_{eff} = 5.5 eV) |
γ1 | −1.5310 | 0.5243 | 2.0553 | 0.9186 | −1.5491 | 0.5341 | 2.0832 | 0.9154 | −1.5754 | 0.5479 | 2.1233 | 0.9116 |

γ2 | −1.5050 | 0.5513 | 2.0563 | 0.9060 | −1.5229 | 0.5584 | 2.0812 | 0.9037 | −1.5450 | 0.5733 | 2.1183 | 0.8982 | |

γ3 | −1.3006 | 1.1148 | 2.4155 | 0.3807 | −1.3278 | 1.1328 | 2.4606 | 0.3821 | −1.3719 | 1.1543 | 2.5263 | 0.3848 | |

γ4 | −1.2836 | 1.1211 | 2.4047 | 0.3747 | −1.3113 | 1.1350 | 2.4463 | 0.3765 | −1.3526 | 1.1556 | 2.5082 | 0.3789 | |

γ5 | −1.1880 | 1.5800 | 2.7680 | 0.2143 | −1.2106 | 1.6159 | 2.8265 | 0.2151 | −1.2501 | 1.6610 | 2.9112 | 0.2175 | |

γ6 | −1.1670 | 1.5914 | 2.7584 | 0.2057 | −1.1902 | 1.6314 | 2.8216 | 0.2071 | −1.2255 | 1.6943 | 2.9198 | 0.2089 | |

B3PW91-α = 0.30 | γ1 | −1.5011 | 0.4869 | 1.9880 | 0.9267 | −1.4417 | 0.2728 | 1.7145 | 0.9509 | −1.4703 | 0.3187 | 1.7890 | 0.9463 |

γ2 | −1.4829 | 0.5660 | 2.0489 | 0.9014 | −1.4185 | 0.5566 | 1.9750 | 0.9006 | −1.4463 | 0.5589 | 2.0052 | 0.9005 | |

γ3 | −1.2863 | 0.9585 | 2.2448 | 0.3821 | −1.2466 | 0.6066 | 1.8533 | 0.3952 | −1.2786 | 0.6362 | 1.9148 | 0.3913 | |

γ4 | −1.2731 | 1.0699 | 2.3430 | 0.3725 | −1.2262 | 0.8002 | 2.0264 | 0.3598 | −1.2573 | 0.8453 | 2.1025 | 0.3633 | |

γ5 | −1.1719 | 1.3956 | 2.5675 | 0.2097 | −1.1208 | 0.9026 | 2.0234 | 0.2074 | −1.1495 | 0.9265 | 2.0760 | 0.2096 | |

γ6 | −1.1580 | 1.4800 | 2.6380 | 0.2076 | −1.0938 | 0.9960 | 2.0899 | 0.1859 | −1.1218 | 1.0333 | 2.1551 | 0.1890 |

The above results on the difference between the effects of localization degree and pressure are also supported and reconfirmed by the total and partial up and down DOSs calculated by PBE-GGA, GGA+U and B3PW91 for the three considered volumes of the compound. The DOSs are presented in Fig. 4 and 5 of the ESI,† and here we only quantitatively present the values of the total DOS^{tot}(E_{F}) in Table 2, since the negligible effects of pressure can be straightforwardly seen in the total DOS^{tot}(E_{F}). As clearly seen in Table 2, the DOS^{tot}(E_{F}) using PBE-GGA only very slightly decreases as pressure increases. A similar trend can be seen in this table using GGA+U and hybrid B3PW91 approaches. However, the DOS^{tot}(E_{F}) is much more considerably affected by GGA+U and B3PW91 than PBE-GGA. This theoretical investigation shows that the effect of the degree of localization is more considerable than the weak effect of pressure.

XCF | Spin | First volume | Second volume | Third volume |
---|---|---|---|---|

PBE-GGA | Up | 5.63 | 5.52 | 5.34 |

Down | 1.68 | 1.65 | 1.63 | |

GGA+U (U_{eff} = 5.5 eV) |
Up | 0.99 | 0.97 | 0.96 |

Down | 1.04 | 1.03 | 1.01 | |

B3PW91−α = 0.30 | Up | 1.63 | 1.54 | 1.36 |

Down | 1.07 | 1.04 | 1.02 |

Despite all the above discussed theoretical differences between the effects of pressure and localization degree, we should note that in experiment, however, the effect of the localization degree is tightly related to the effect of pressure. This is so because the degree of localization is itself changed by pressure for strongly correlated systems in nature.^{55,76} So why is pressure not as effective as localization degree in our above theoretical discussion? Do the theoretical results contradict the experimental results? The answer is that when we apply pressure experimentally, the degree of localization is also naturally changed, but the degree of localization theoretically depends on the functional used. Therefore, when we apply pressure theoretically the degree of localization is not remarkably changed automatically, because in the current available DFT approaches the degree of localization should be applied manually by selecting an appropriate functional and tuning by hand its parameter such as U parameter in GGA+U or α parameter in B3PW91 for the exchange–correlation term.^{55} Thus, the theoretical results can be consistent with the experimental results, if a proper degree of localization is considered. This reconfirms our recent report on the pressure dependency of localization degree in CeIn_{3}.^{55} This point will be considered in the subsequent sections.

The doping levels corresponding to the maximum values of the spin up positive (negative) xx and zz components of the hole-like (electron-like) Seebeck coefficients, N^{max-pos} (N^{max-neg}), are shown as functions of temperature in Fig. 2(ai2) and (bi2) (Fig. 3(ai2) and (bi2)) for i = 1–3. In all these hole- and electron-like figures, regardless of the carrier type, the unit of the doping level is evidently electron per unit cell, e per uc, even though the negative carriers are electrons and the positive carriers are holes. We limited the range of the doping level to between −10^{21} and 10^{21} carriers per cm^{3} so that it can be realized in experiments. As can be clearly seen from Fig. 2(a12) and (a22), in the first volume the carriers related to spin up S^{max-pos}_{xx} are electrons, i.e., N^{max-pos} < 0, at most of the temperature range within GGA+U and B3PW91 approaches. The same results are observed for the doping level related to spin up S^{max-neg}_{xx} within GGA+U and B3PW91 approaches, see Fig. 3(a12) and (a22). However, the value of N^{max-pos} is typically higher than the value of N^{max-neg} for the xx component in all of the considered temperature range using both GGA+U and B3PW91. For S^{max-pos}_{zz} the related carriers are holes, i.e., N^{max-pos} > 0, up to T ≃ 45 K within both GGA+U and B3PW91 functionals. However, in the first volume and at T ≃ 45 K, a considerable gap suddenly occurs and the zz-component of the doping level N^{max-pos} drops down from around 1.5 e per uc to about −0.017 e per uc (−1.5 e per uc) within GGA+U (B3PW91), and hence after this temperature the carriers change their type and become electrons, see Fig. 2(b12) (Fig. 2(b22)). After T ≃ 45 K, the zz-component of the doping level N^{max-pos} changes slightly up to room temperature within the B3PW91 functional, see Fig. 2(b22). But, within GGA+U, the zz-component of the doping level N^{max-pos} again experiences a considerable gap at T ≃ 270 K and the doping level drops from around −0.03 e per uc to about −1.6 e per uc. The GGA+U approach predicts that for the electron-like spin up S^{max-neg}_{zz} in the first volume the related carriers are holes up to T ≃ 20 K, while at T ≃ 20 K a considerable gap occurs and the carriers change their type to electrons up to room temperature, see Fig. 3(a12). In contrast to GGA+U, B3PW91 does not predict a regular behavior for the zz-component of spin up N^{max-neg}, as can be seen in Fig. 3(b22); there are several considerable gaps in the zz-component of N^{max-neg} within the B3PW91. In summary, in most of the temperature range, the predominant carriers related to the xx-components of S^{max-pos} and S^{max-neg}, i.e., N^{max-pos} and N^{max-neg}, are electrons using GGA+U and B3PW91, see Fig. 2(ai2) and 3(ai2) for i = 1 and 2. This result holds for the carriers related to the zz-components of S^{max-pos} within the GGA+U and B3PW91, as shown in the (bi2) panels of Fig. 2 for i = 1 and 2, as well as for the zz-components of S^{max-neg} within GGA+U, see Fig. 3(b12). Available experimental reports^{24,25,49} show that applying hydrostatic pressure can improve the Seebeck coefficient of the other Ce-based compounds near room temperature. Furthermore, for strongly correlated f-electron systems the thermoelectric properties, especially the thermopower S, are strongly influenced by the position of a very narrow maximum in the density of states relative to the Fermi level. In this case, it is necessary to analyze the calculated S versus the change of the Fermi level, due to the high gradient of the density of states at the Fermi level. This analysis can be performed through the imposing pressure, because imposing pressure can change the Fermi level and the DOSs relative to the Fermi level, see Table 2. This motivated us to investigate the effect of pressure on the thermoelectric parameters of the compound under study. Therefore, we have performed our calculations for the two other considered volumes in addition to the first one. For convenience, these volumes are called the second and third volumes from now on. We recall that the second and third volumes are 2% and 5% smaller than the first volume, while the first volume corresponds to the experimental volume. This means that the second and third volumes are under pressure. The results displayed in Fig. 2(a11) and (b11) clearly show that applying pressure within GGA+U does not change S^{max-pos}_{xx} and S^{max-pos}_{zz} drastically for low temperatures, but it makes worse (better) the S^{max-pos}_{xx} (S^{max-pos}_{zz}) at high temperatures. Similar results can be obtained for S^{max-neg}_{xx} and S^{max-neg}_{zz}. The (a11) panels of Fig. 2 and 3 indicate that within GGA+U the S^{max-pos}_{xx} and S^{max-neg}_{xx} values of the first volume are higher than those of the second and third volumes for T > 35 K and T > 100 K, respectively. On the contrary, within GGA+U, S^{max-pos}_{zz} and S^{max-neg}_{zz} in the first volume are less than those of the second and third volumes for T > 65 K.

As per the result obtained by GGA+U, the volume reduction changes the S^{max-pos}_{xx}, S^{max-neg}_{xx} and S^{max-neg}_{zz} values very slightly at low temperatures using B3PW91, however, in contrast to the result obtained by GGA+U, the effect of volume reduction on S^{max-pos}_{zz} at low temperatures is more than high temperatures using B3PW91, see Fig. 2(b21). The maximum values of S^{max-pos}_{zz} in the second and third volumes are about 5.8 and 4.2 μV K^{−1} lower than the maximum value of S^{max-pos}_{zz} in the first volume at T ≃ 7 K using B3PW91. For high temperatures, the B3PW91 predicts very different results compared to the GGA+U by applying pressure. In contrast to the GGA+U, the B3PW91 predicts that the S^{max-pos}_{xx} (S^{max-neg}_{xx}) in the first volume is less than that of second and third volumes for T > 20 K (T > 65 K). This effect of pressure is consistent with the experimental report on the Seebeck coefficient of CeIrIn_{5}.^{24} Moreover, based on Fig. 3(b21), S^{max-neg}_{zz} is larger in the first volume than that in the second volume but lower than that in the third volume for T > 65 K. The volume reduction effect on S^{max-pos}_{zz} is negligible at high temperatures. The effect of volume reduction on both components of N^{max-pos} (N^{max-neg}) doping levels is significant only at low temperatures using GGA+U, see the (a12) and (b12) panels of Fig. 2 (Fig. 3). The xx and zz components of N^{max-pos} (N^{max-neg}) remain approximately unchanged by the volume reduction for T > 35 K (T > 20 K) and T > 45 K (T > 36 K), respectively. On the other hand, B3PW91 predicts that both components of N^{max-pos} and N^{max-neg} doping levels in the first volume are significantly different from the second and third volumes in most of the considered temperature range, see the (a22) and (b22) panels of Fig. 2 and 3. But, the doping levels in the second and third volumes are very similar for both components and both electron- and hole-like Seebeck coefficient using B3PW91.

Shishido and coworkers,^{50,51} using dHvA experiments, have shown that the degree of 4f-electron localization has been decreased by imposing pressure on CeRhIn_{5}. Hence, it is interesting to investigate the localization effects on the thermoelectric parameters of CeRhIn_{5}. Moreover, variation of the Hubbard U parameter can strongly affect the calculated thermoelectric properties. Thus, we have calculated these parameters using three different functionals with three different degrees of localization, i.e., PBE-GGA, GGA+U and B3PW91, to study the effects of the degree of localization on the thermoelectric parameters. The degree of 4f-Ce electron localization is predicted to be much lower by PBE-GGA than GGA+U and B3PW91 schemes. The xx and zz components of S^{max-pos} (S^{max-neg}), as calculated by PBE-GGA, are shown in the (a31) and (b31) panels of Fig. 2 (Fig. 3) for the three considered volumes. PBE-GGA predicts a peak with a value of about 6.7 μV K^{−1} (4.9 μV K^{−1}) for S^{max-pos}_{xx} (S^{max-pos}_{zz}) at T ≃ 7 K (T ≃ 10 K) in the first volume, as shown in Fig. 2(a31) (Fig. 2(b31)). This prediction of the low localized PBE-GGA is very close to the predictions of the high localized GGA+U and B3PW91. After this temperature (T ≃ 7 K), where the maximums occur in the Seebeck curves, the S^{max-pos}_{xx} (S^{max-pos}_{zz}) decreases to a value of about 5.4 μV K^{−1} (4.3 μV K^{−1}) at around T ≃ 20 K, and then it increases again as temperature increases up to the room temperature. These results indicate that the behavior of S^{max-pos}_{xx} and S^{max-pos}_{zz} predicted by the low localized PBE-GGA is the same as those predicted by the high localized GGA+U and B3PW91 at low temperatures. Consequently, it can be concluded that at low temperatures the degree of 4f-Ce localization has no considerable effect on the S^{max-pos}_{xx} and S^{max-pos}_{zz} values. But, the low localized PBE-GGA predicts much higher values for the S^{max-pos}_{xx} and S^{max-pos}_{zz} values at high temperatures compared to the high localized GGA+U and B3PW91. This can be clearly seen for the xx [zz] component by comparing the (a31) [(b31)] panel with the (a11) [(b11)] and (a21) [(b21)] panels of Fig. 2. This comparison also indicates that within the low localized PBE-GGA, the values of S^{max-pos}_{xx} at high temperatures are about four times larger than those at low temperatures, while its values are in the same range at low and at high temperatures using the high localized GGA+U and B3PW91. This result also holds for S^{max-pos}_{zz}. Therefore, the maximum values of S^{max-pos} components as predicted by PBE-GGA at low temperatures can be approximately neglected compared to the 4 times higher values of S^{max-pos} components at high temperatures. In this case, we can assume that the PBE-GGA Seebeck curves monotonically increase by temperature.

The monotonic increase of the Seebeck curves by temperature is a well-known character of normal metals. Therefore, CeRhIn_{5} is estimated by PBE-GGA to behave almost like a normal metal, if the small peaks at low temperatures are omitted in the (a31) and (b31) panels of Fig. 2 compared to the large values of the Seebeck components at high temperatures. The effect of volume reduction also depends on the functional used so that it is differently predicted by PBE-GGA compared to GGA+U and B3PW91. Fig. 2(a31) and (b31) reveal that volume reduction makes worse both the xx and zz components of S^{max-pos} at high temperatures. The same result can be also seen for the xx component at low temperatures, because the PBE-GGA xx component of S^{max-pos} in the first volume is more than those in the second and third volumes, see Fig. 2(a31). Thus, volume reduction decreases the xx component of S^{max-pos} and as a result makes it worse. But, the zz component is slightly improved by volume reduction at low temperatures using PBE-GGA. In contrast to S^{max-pos}, the S^{max-neg} components are completely zero at high temperatures using PBE-GGA, see Fig. 3(a31) and (b31). This zero value implies that it may be impossible to find the electron-like Seebeck coefficient using PBE-GGA at the considered doping level range. Fig. 3(a31) and (b31) show that PBE-GGA, the same as GGA+U and B3PW91, predicts a maximum (negative minimum) for the xx and zz components of S^{max-neg} at low temperatures, i.e., T ≃ 7 K in the first volume. The latter figures also show that the volume reduction slightly improves the xx and zz components of S^{max-neg} at low temperatures within PBE-GGA, but does not change their zero values at high temperatures. In summary, a comparison of the (ai1) [(bi1)] panels of Fig. 3 for i = 1–3 demonstrates that reduction of the localization degree considerably influences S^{max-neg}_{xx} (S^{max-neg}_{zz}) at high temperatures, but it is not very remarkable at low temperatures. The doping levels related to the xx and zz components of S^{max-pos} (S^{max-neg}) predicted using PBE-GGA, are displayed in the (a32) and (b32) panels of Fig. 2 (Fig. 3). Both components of N^{max-pos} are approximately fixed at most of the temperature range, except for a considerable gap at low temperatures, as shown in Fig. 2(a32). The latter figures also show that the volume reduction does not considerably change the doping levels. For most of the temperature range, PBE-GGA predicts about 1.5 e per uc electron doping for the S^{max-pos} components in the three considered volumes. The xx [zz] component of N^{max-pos} is not considerably affected by decreasing the localization degree, see the (a32) [(b32)] panel and the (a12) [(b12)] and (a22) [(b22)] panels of Fig. 2.

We also see considerable gaps in the N^{max-neg} components at low temperatures, within PBE-GGA in three volumes, see Fig. 3(a32) and (b23). We do not show any doping level, N^{max-neg}, in the latter figures for temperatures in which the S^{max-neg} components are zero, because the doping level for this situation is meaningless. At the end of this section and based on the presented results, we can conclude that for the first volume corresponding to the experimental volume, both xx and zz Seebeck coefficient components of CeRhIn_{5} show maximum values at low temperatures around the T ≃ 7 K using the three considered XCFs. Based on our results, the volume reduction does not affect the Seebeck coefficient components considerably at low temperatures, except for S^{max-pos}_{zz}. But, this is not the case for high temperatures. The effect of volume reduction depends on the functionals used, in agreement with our recent report.^{55} The volume reduction makes worse the Seebeck coefficient component along the a crystalline axis within the GGA+U and PBE-GGA approaches at high temperatures. Along the c crystalline axis, volume reduction improves the Seebeck coefficient component at high temperatures within GGA+U, but makes it worse within PBE-GGA. On the contrary, the xx components of the hole- and electron-like Seebeck coefficients, as calculated by B3PW91, along the a (c) crystalline axis in the first volume are less (more) than those in the second and third volumes at high (low) temperatures. Our results also show that the degree of localization for the 4f-Ce electrons has a significant effect on the Seebeck coefficient components at high temperatures, but at low temperatures these effects are negligible. Decreasing the Seebeck coefficient causes the hole like Seebeck coefficient components of heavy fermion CeRhIn_{5} to almost behave like the normal metals. Moreover, the doping levels related to the maximum values of the hole-like and electron-like components are electrons at most of the considered temperature range in all the considered volumes and XCFs.

The doping levels related to the components of the maximum electrical conductivity are presented in Fig. 4(ai2) and (bi2) for i = 1–3. As shown in Fig. 4(a12) and 5(a12), the doping carriers related to σ^{max}_{xx}/τ are electrons for all the temperature range in the three considered volumes using GGA+U. B3PW91 predicts that the doping levels corresponding to the σ^{max}_{xx}/τ are electron (hole) for all the temperature range in the first (second and third) volume(s), see Fig. 4(a22) and 5(a22). PBE-GGA predicts the hole doping levels corresponding to the σ^{max}_{xx}/τ for all the temperature range for the three considered volumes, as shown in Fig. 4(a32) and 5(a32). These results confirm that the volume reduction effects on the doping levels related to σ^{max}_{xx}/τ depend on the degree of localization for 4f-Ce electrons. For σ^{max}_{zz}/τ, the doping levels are holes using all the considered XCFs at all the temperature range, see (bi2) panels of Fig. 4 and 5 for i = 1–3.

The values of doping levels corresponding to the maximum values of κ^{max}_{e}/τ components are presented in the (ai2) and (bi2) panels of Fig. 6 for i = 1–3. As shown in Fig. 6(a12), the electron doping levels can be attributed to the xx component of κ^{max}_{e}/τ for all the temperature range in the three considered volumes using the GGA+U approach. The same result can be observed using B3PW91 for the xx component of κ^{max}_{e}/τ in all the temperature range in the first volume, see Fig. 6(a22). However, this is completely different for the xx component of κ^{max}_{e}/τ in the second and third volumes using B3PW91 XCF, as shown in Fig. 6(a22). As the latter figure shows, in all the considered temperature range, the hole doping is related to the xx component of κ^{max}_{e}/τ in the second and third volumes using B3PW91. The hole doping levels are related to the xx component of κ^{max}_{e}/τ in the three considered volumes using PBE-GGA in all the considered temperature range, as shown in Fig. 6(a32). Similar results can be observed for the electrical conductivity. All the considered XCFs predict the hole doping levels for the zz component of κ^{max}_{e}/τ in the three considered volumes, as shown in the (bi2) panels of Fig. 6 for i = 1–3.

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## Footnote |

† Electronic supplementary information (ESI) available: The electronic structure results together with two Fortran programs and a flowchart provided by the authors for extracting the maximum values of the thermoelectric parameters calculated by the WIEN2k and BoltzTrap codes. See DOI: 10.1039/c9ra07859b |

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