Joachim Sonntag
email: sonntag.joachim45@gmail.com

Bertrand Lenoir
email: bertrand.lenoir@univ-lorraine.fr

Composites to Produce a Material with Zero Absolute Thermopower S = 0 or a Thermopower Switch between S = 0 and S = 0

Keywords: seebeck coefficient, seebeck coefficient standard reference, composites, Effective Medium Theory, amorphous alloys, phase separation, thermopower switch PACS: 71.23.-k, 71.55.Jv, 72.10.Bg, 72.15.-v

may come

Composites to Produce a Material with Zero Absolute Thermopower S = 0 or a Thermopower Switch between S = 0 and S ̸ = 0

Introduction

Experimental data of thin a-Cr 1-x Si x ("a" in a-Cr 1-x Si x stands for "amorphous") films [1,[START_REF] Sonntag | Connectivity Sensors Germany GmbH[END_REF] produced by ion beam sputtering have shown that the concentration dependence of the thermoelectric power S shows a discontinuity at x = 0.49. Coming from x < 0.49, S is negative and increases monotonically approaching S = 0 µV/K at x = 0.49, where S jumps suddenly to S = 2.0 µV/K (see Figure 1). With further growing x, S continues to increase monotonically. Originally, the reason for this discontinuity was not understood. Only the assumption, that a-Cr 1-x Si x films could be composed of two different amorphous phases (differing by short range order) gave a possible answer. Indeed, theoretical calculations based on Effective Medium Theory (EMT) have shown that these two properties, S = 0 µV/K and the step can be explained, if the coexistence of two different phases is assumed [START_REF] Sonntag | Disordered electronic systems. III. Thermoelectric power in alloys with phase separation[END_REF], and that such a step occurs independently of the temperature. The experimental confirmation of the presence of two amorphous phases in a series of amorphous transition-metal-metalloid alloys took place in the 1990s. Applying Raman spectroscopy, infrared absorption, extended X-ray absorption fine structure (EXAFS) and anomalous small-angle X-ray scattering (ASAXS) it has been confirmed for a number of amorphous transition metal-metalloid alloys that these indeedly consist of two different amorphous phases [START_REF] Edwards | The metal-insulator transition: A perspective[END_REF][START_REF] Edwards | An Investigation of the Structure of Amorphous Si1-xNix through the Metal-Insulator Transition[END_REF][START_REF] Lorentz | Structural studies of the phase separation of amorphous FexGe100-x alloys[END_REF][START_REF] Van Raap | Evidence of phase separation in amorphous FexSi1-x films[END_REF][START_REF] Regan | Anisotropic phase separation through the metal-insulator transition in amorphous alloys[END_REF]. Regan et al. [START_REF] Regan | Anisotropic phase separation through the metal-insulator transition in amorphous alloys[END_REF] found for co-sputtered a-W 1-x Ge x , a-Fe 1-x Ge x , a-Fe 1-x Si x and a-Mo 1-x Ge x films phase separation regions of the order of 1 nm in the growth plane and 1.5-2.0 nm in the growth direction. They could show that their measure-ments are in agreement with the assumption of two coexisting amorphous phases, a-Ge or a-Si, on the one side and a metallic phase with a-FeGe 2 , a-FeSi 2 , or a-MoGe 3 compositions for the last three systems, respectively, on the other side. Raap et al. [START_REF] Van Raap | Evidence of phase separation in amorphous FexSi1-x films[END_REF] found amorphous phases separation in co-sputtered a-Fe 1-x Si x films into regions of a-Si and an intermetallic close in composition to a-FeSi 2 with 0.6 nm in the film plane and 1 nm in size in the growth direction. For the case a-Cr 1-x Si x , the two identified amorphous phases are a-Cr 3 Si and a-Si [START_REF] Sonntag | Disordered electronic systems: Concentration dependence of the dc conductivity in amorphous transition-metalmetalloid alloys (metallic regime)[END_REF]. The bonding state in these two phases differs in terms of orbital structure, sp for the metal-rich phase and sp 3 for a-Si.

The two properties, S = 0 µV/K and the discontinuity observed in a-Cr 1-x Si x , Figure 1, deserve special interest because they can be used to produce a reference standard for the absolute thermopower (Seebeck coefficient Standard Reference material-SRM) and a thermopower switch between S = 0 µV/K and S = 0 µV/K. SRM's used in practice to measure the absolute thermopower of materials are generally not very accurate (see Appendix A). A decisive advantage of the proposed new SRM based on a-Cr-Si is that it can be used to set S = 0 µV/K very precisely. x for a-Cr 1-x Si x at T = 300 K calculated by Equation (1), respective Equation [START_REF] Sonntag | Disordered electronic systems: Concentration dependence of the dc conductivity in amorphous transition-metalmetalloid alloys (metallic regime)[END_REF] and Equations ( 2)- [START_REF] Bjorneklett | Effective Medium Theory and the thermal conductivity of plasmasprayed ceramic coatings[END_REF] for the phases A (= a-Cr 3 Si) and B (= a-Si), where ∂E C,A /∂T = ∂E V,B /∂T = 0 was set. This curve agrees with those of Figure A.2 in [START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF] for c = 2.6 eV. The experimental data are taken from Gladun et al. [START_REF] Gladun | Electrical transport properties of high resistance Cr-Si-O thin films[END_REF] (diamonds), Weser [1] (open triangles) and Sonntag [START_REF] Sonntag | Connectivity Sensors Germany GmbH[END_REF] (full triangles).

The discontinuity in S(-) at x = x s = 0.49 corresponds to the discontinuities in Figures 234. 2)-( 13) for a hypothetical composite with S 0 A = -13.0 µV/K, S 0 B = +1.5 µV/K, κ e,A = 8.5 mW/cmK, and κ e,B = 12.7 mW/cmK (corresponding to n = 10 22 cm -3 and p = 2 × 10 22 cm -3 and T = 300 K, where dµ/dT, Equation [START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF], is calculated for ∂E C,A /∂T = ∂E V,B /∂T = ∂ϕ i /∂n i = 0). It is important to underline that the fabrication conditions of a-Cr 1-x Si x films play a pivotal role on the appearence of the discontinuity in S at S = 0 µV/K. Actually, a co-sputtering approach (from a Cr target and a Si target) leads to completely different results from those found originally; both the resistivities were essentially higher and the discontinuity was not found. The difference between these two data groups was not understood. Apparently, this difference is caused by the fact that the experimental original data was produced by single deposition, i.e., for each x separately, whereas the experimental data produced by co-sputtering [START_REF] Sonntag | Journal of Physics: Condensed Matter High-throughput characterization of the Seebeck coefficient of a-(Cr1-xSix)1-yOy thin film materials libraries as verification of the extended thermopower formula[END_REF], i.e., during the sputtering process all the different samples with different x were electrically connected with each other. It is also the aim to find the reason of this difference. As it will be argued in this paper, the reason is apparently a general phenomenon, which, until now, has not yet been considered in the experimental practice, which has consequences with respect to both the dynamics of formation of the structure during production/deposition the films and, more over to the electronic transport properties as thermopower, the electrical and thermal conductivity and the Hall coefficient of the samples. It will be shown that these properties are essentially determined by the demand that (spatial) continuity of the entropy and energy flux densities and their gradients is saved during the solidification process. This has far-reaching practical consequences in the search for new materials with new properties, for which the method of "high-throughput characterization" by co-sputtering is often used today. The aim of this paper is thus twofold, first, to make known the highly exciting property of a step in S with S = 0 µV/K and its consequences (SRM and switch), and second, to recognize that co-sputtering and single-coating can give completely different results, due to the different dynamics during the layer deposition process.

As a theoretical support, the EMT formalism, which has been widely used to explain the transport properties of composites, including ceramics [START_REF] Bjorneklett | Effective Medium Theory and the thermal conductivity of plasmasprayed ceramic coatings[END_REF][START_REF] Smith | Thermal Resistance of Grain Boundaries in Alumina Ceramics and Refractories[END_REF][START_REF] Li | Effective Thermal Conductivity of Ultra-High Temperature Ceramics with Thermal Contact Resistance[END_REF] and thermoelectrics [START_REF] Ghosh | Thermoelectric properties of Co4Sb12 with Bi2Te3 nanoinclusions[END_REF][START_REF] Sharma | Influence of nanostructuring and heterogeneous nucleation on the thermoelectric figure of merit in AgSbTe2[END_REF][START_REF] Vaney | Effective Medium Theory based modeling of the thermoelectric properties of composites: comparison between predictions and experiments in the glass-crystal composite system Si10As15Te75-Bi0.4Sb1.6Te3[END_REF], will be implemented.

The paper is structured as follows. The thermopower formula and its application are described in Section 2. Section 3 treats the effect of the carrier densities and band edges on the concentration dependence of S. The results of Sections 2 and 3 and what is the cause of the difference of the two data groups described previously are discussed in Section 4 and summarized in Section 5. In the Appendix A the state of the art on SRM's is described, and in Appendix B our EMT formula applyed in the present paper is compared with EMT formulas derived earlier by other authors.

Formulas for the Calculation of the Seebeck Coefficient

The thermopower formula applyed for the calculation of S vs. x shown in Figure 1 reads [START_REF] Sonntag | Thermoelectric power in alloys with phase separation (composites)[END_REF][START_REF] Sonntag | Effective Medium Theory based modeling of the thermoelectric properties of composites: comparison between predictions and experiments in the glass-crystal composite system Si10As15Te75-Bi0[END_REF]

∑ i υ i κ e,i /S i -κ e /S κ e,i /S i + 2κ e /S = 0, (1)
where S and κ e are the Seebeck coefficient and the electronic contribution to the thermal conductivity of the composite, κ. S i , κ e,i and υ i are the corresponding parameters and the volume fraction of the phase i (A, B, . . .). The thermopower formula Equation ( 1) is derived on basis of the Effective Medium Theory (EMT).

For the derivation of Equation ( 1) the following assumptions were made: The alloy is a composite consisting of two different phases which form spherical phase grains, randomly arranged and in a symmetrical fashion. This assumption is only an approximation, but we believe that this is a good description of the principle behavior of the current flow densities through the composite consisting only of amorphous phase grains. There are also attempts to consider deviations from a spherical shape in the formulas, for instance the Generalized Effective Medium Theory (GEMT), based on a phenomenological model with the addition of elements of the percolation theory to the EMT. This idea was applied by Vaney et al. [START_REF] Vaney | Effective Medium Theory based modeling of the thermoelectric properties of composites: comparison between predictions and experiments in the glass-crystal composite system Si10As15Te75-Bi0.4Sb1.6Te3[END_REF] to S for crystalline composites.

Each phase is characterized by its own transport coefficients. At the boundary face between a single phase grain and the surrounding 'effective medium' continuity of the current densities and potentials and their gradients are saved, and the additional condition J = J i = 0 is to be fulfilled. J i and J are the electrical current density in a single phase grain and the surrounding 'effective medium', respectively.

S vs. υ i can be calculated by Equation (1) if S i , κ e,i and κ e are known. S i can be calculated by [START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF]

S i = S 0 i + 1 |e| dµ dT , ( 2
)
where µ is the electrochemical potential, and T the absolute temperature. S 0 i and 1 |e| dµ dT are the "scattering term" in the phase i and the "thermodynamic term". The "thermodynamic term" is identical for all the phases in the composite.

For metallic phases, S 0 i and κ e,i are given by

S 0 i = π 2 k 2 B T(1 + r i ) 3e i E F,i , (3)
κ e,i = 16π 3 9

m i L i E F,i h 3 k 2 B T, (4)
following from the Boltzmann transport equation (BTE) in the approximation of nearly free electrons (NFE). |e| is the elementary charge; e i = -|e| and +|e| for electrons and holes, respectively. m i and L i are the effective mass and the mean free path, respectively, in the phase i. r i characterizes the energy dependence of L i according to L i ∝ E r i . h is the Planck's constant, and k B the Boltzmann constant. E F,i is the Fermi energy given by

E F,i = h 2 8m i 3 π 2/3 n 2/3 i ( 5
)
for a phase i with electron conductivity having the electron density n i . The approximation in Equation ( 4) is surely a good one for metallic phases, if the phases form macroscopic clusters. The formula for κ (total thermal conductivity) derived by Odelevskii [START_REF] Odelevskii | Raschet obobshchennoy provodimosti geterogennykh system[END_REF] reads

∑ i υ i κ i -κ κ i + 2κ = 0. ( 6
)
Therefore we assume that

∑ i υ i κ e,i -κ e κ e,i + 2κ e = 0 (7)
is valid as well.

The Effect of the Carrier Densities and Band Edges on S vs. υ B

Let us consider a two-phase composite where the phase A has electron conductivity with n ≡ n A , whereas the phase B has hole conductivity with the hole density p, for which the Fermi energy reads

E F,B = h 2 8m B 3 π 2/3 p 2/3 (8)
(characterized by Figure 1c in [START_REF] Sonntag | Disordered electronic systems. III. Thermoelectric power in alloys with phase separation[END_REF]). Equation (1) has two solutions, S(+) and S(-), S = S(±) = 4κ e ± 2 + 8(κ e,A /S A )(κ e,B /S B )

with = (3υ A -1)κ e,A /S A + (3υ B -1)κ e,B /S B and υ A + υ B = 1. In Figure 2 an example of calculation is shown for a hypothetical composite with S 0 A = -13.0 µV/K, S 0 B = +1.7 µV/K, κ e,A = 8.5 mW/cmK, and κ e,B = 12.7 mW/cmK. These transport parameters correspond to n = 10 22 cm -3 and p = 2 × 10 22 cm -3 if calculated by Equations (3)- [START_REF] Edwards | An Investigation of the Structure of Amorphous Si1-xNix through the Metal-Insulator Transition[END_REF] and Equation ( 8) [START_REF] Kirejew | Physik der Halbleiter[END_REF], p. 348) at T = 300 K. m 0 is the bare electron mass. d i is the interatomic distance in the phase i (d A = 0.25 nm and d B = 0.234 nm). For calculating dµ/dT in Equation ( 2),

with m A = m 0 , m B = 0.2 × m 0 , r i = 2 and L i = 4/π × d i ([
dµ dT = ∂E C,A ∂T + ∂µ 0 A ∂T - ∂µ 0 A ∂T + ∂µ 0 B ∂T + ∂E C,A ∂T - ∂E V,B ∂T 1 + υ A ∂µ 0 B ∂p -|e| ∂ϕ B ∂n B υ B ∂µ 0 A ∂n -|e| ∂ϕ A ∂n ( 10
)
has been applied with

∂µ 0 i ∂T = - π 2 k 2 B T 6E F,i , ( 11
)
∂µ 0 A ∂n = 2E F,A 3n , ( 12
)
∂µ 0 B ∂p = 2E F,B 3p (13)
(following from the Fermi-Dirac-statistics), where the contributions by the band edges (E C,A , E V,B ) and those of the electrostatic potential (ϕ i ) to dµ/dT were still neglected, i.e., ∂E C,A /∂T = ∂E V,B /∂T = ∂ϕ i /∂n i = 0 was set. ϕ i and µ 0 i are the electrostatic potential and the chemical potential, respectively, in the phase i. E C,A and E V,B are the band edges of the conduction band (CB) in the phase A and the valence band (VB) in the phase B, respectively. Note that dp = -dn B . With this dµ/dT, S i and S(±) are calculated by Equation ( 2) and Equation ( 9), respectively. The result is shown in Figure 2: At υ B = 0.63 there is a discontinuity in the S(-) and S(+) curves, where the lower kink occurs exactly at S(-) = S(+) = 0. An analogous result is also obtained for other values of n and p. Figure 3 shows S(±) vs. υ B for different values of p, while n is hold constant, n = 10 22 cm -3 , and the other parameters m i , r i , L i are identical to those applied in Figure 2. With decreasing the value of p, the size of the discontinuity (the distance between the upper and lower kinks) decreases continuously, until it disappears completely at a critical value of p and a gap opens in the concentration range where there are no real solutions S(±), because the square root in Equation ( 9) becomes imaginary. For the example composite shown in Figure 3, this critical value is p crit = 1.2149 × 10 22 cm -3 : For p ≤ p crit , Equation ( 9) has no longer real solutions for the entire concentration range.

If n is varied, while p, m i , r i and L i , are identical to those applied in Figure 2, we get a similar result: Here the critical value is n crit = 1.24 × 10 22 cm -3 . For n ≥ n crit (and p = 2 × 10 22 cm -3 ), Equation ( 9) has no longer real solutions for the entire concentration range.

Until now fixed band edges have been assumed. However, the thermopower of the phases S i , Equation ( 2), depend on dµ/dT, and dµ/dT depends on ∂E C,A /∂T and ∂E V,B /∂T, according to Equation [START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF]. In Figure 4 the effect of ∂E C,A /∂T on the solutions of Equation ( 9) is shown. For this variation, in Equation ( 2) ∂E V,B /∂T = ∂ϕ i /∂n i = 0 is set, and the other parameters are identical to those applied in Figure 2. The discontinuities in the S(-) curve (bold line) and the S(+) curve (thin line) are shifted to lower υ B as ∂E C,A /∂T increases. Here the critical value is (∂E C,A /∂T) crit = 0.5849 × 10 -6 eV/K. For ∂E C,A /∂T > (∂E C,A /∂T) crit there are no real solutions for the entire concentration range 0 < υ B < 1.

A variation of ∂E V,B /∂T has a similar effect as a variation of ∂E C,A /∂T. There the critical value is (∂E V,B /∂T) crit = 1.8967 × 10 -6 eV/K. For ∂E V,B /∂T > (∂E V,B /∂T) crit there are no real solutions for the entire concentration range 0 < υ B < 1.

When we vary the parameters n, p, ∂E C,A /∂T and ∂E V,B /∂T, the lower kink of S(±) at the discontinuity occurs at S(-) = S(+) = 0, as long as we are not beyond the critical values mentioned.

The two S(-) curves in Figure 4 for (∂E C,A /∂T) crit = 0.5849 × 10 -6 eV/K and (∂E C,A /∂T) = +0.3 × 10 -6 eV/K are very similar to the experimental data for amorphous a-Cr 1-x Si x thin films, deposited under different deposition conditions: For the a-Cr 1-x Si x films sputtered from different Cr 1-x Si x targets in single manufacturing processes, there is a significant step in the S versus concentration dependence as shown in Figure 1, whereas a-Cr 1-x Si x films sputtered from a Cr target and a Si target arranged separately from each other (co-sputtering) does not show such a step [START_REF] Sonntag | Journal of Physics: Condensed Matter High-throughput characterization of the Seebeck coefficient of a-(Cr1-xSix)1-yOy thin film materials libraries as verification of the extended thermopower formula[END_REF]. The essential difference between these two methods of deposition is the fact, that in the co-sputtered films there is a common dµ/dT for all the x realized during the deposition run, whereas for the films sputtered from different Cr 1-x Si x targets, dµ/dT is different for the different x. Between these two series of a-Cr 1-x Si x films there are also considerable differences regarding their specific resistivity ρ versus x dependences ( [START_REF] Sonntag | Electronic Transport in Alloys with Phase Separation (Composites)[END_REF], Figure 7 therein).

Discussion

As shown in Section 3 for the hypothetical composite, S(-) and S(+) approach always the value "0" at the lower kink of the discontinuity provided that the carriers in the two phases have different signs, electrons and holes (the concentration where this kink occurs will be noted υ B = υ B,s and x = x s for the corresponding atomic concentration). This property can be used to construct "Seebeck coefficient Standard Reference with 0 µV/K absolute". In order to use this property, the following questions are still to be answered:

(1) Does the experimental Seebeck curve S exp vs. x (respectively S exp vs. υ B ) of a real two-phase alloy follow one of the solutions, S(-) or S(+), or does it follow a smooth curve changing between S(-) to S(+) at the discontinuity? (see Figure 2) (2) If S exp vs. x follows one of the solutions, S(-) or S(+), does then the lower kink of the discontinuity in the S exp vs. x curve occur at S exp = 0? (3) Which effect do the electrostatic potentials have on the discontinuity in S(±)? (4) Which meaning or consequence has the fact that Equation (1) (respectively Equation ( 9))

does not have real solutions for the entire concentration range if the carrier densities or the band edge shifts are beyond the critical values specified earlier?

Let us start with the point (1). If the thermopower changed between S(-) and S(+) at the discontinuity, no discontinuity in S exp vs. x is expected experimentally. However, the experimental data in Figure 1 apparently follow the calculated S(-) curve on both sides of the discontinuity at x s = 0.49, i.e., they does not cross over from S(-) to S(+) or vice versa. Contrarily to that, for the two limiting cases "υ B = 0" and "υ B = 1" it follows from Equation [START_REF] Sonntag | Disordered electronic systems: Concentration dependence of the dc conductivity in amorphous transition-metalmetalloid alloys (metallic regime)[END_REF] with Equation ( 7) that S(+) is the physical solution.

Let us move now to the second question, (2): As can be seen in Figure 1, at the lower kink of the S(-) curve at x s , S(-) = 0. Actually, coming from the left-hand side of this discontinuity, the experimental thermopower data approach zero precisely at the lower kink at x s . This experimental finding corresponds to the calculated S(-) curves for the hypothetical composite specified in Section 2 drawn in Figures 2 and3. (Note that υ B increases monotonically with x provided that the phase compositions, x A and x B , are constant.)

We emphasize that the jump in the thermopower at S = 0 is not the result of approximations. It also does not depend on the choice of the model applied for calculation of the transport coefficients of the phases A and B ("parabolic band model with an effective mass and power law scattering rates"). The jump at S = 0 always follows purely mathematically from Equation (1), independent of the chosen numbers for S i and κ e,i , if Equation (1) has a mathematical solution at all for S(-) and S(+).

It is clear that if one of the two phases is present as separate islands, the band model can no longer be applied to this phase. This is true for υ B < 1/3 for phase B and, on the other hand, also for υ A < 1/3 for phase A. In amorphous composites this limit, which separates "island structure" and "continuous structure" from each other, is very precisely at υ B = 1/3 with respect to phase B. The same is true for phase A. Details to this problem are discussed in detail in [START_REF] Sonntag | Thermoelectric power in alloys with phase separation (composites)[END_REF].

In Figure 5, the same curve S(-) of Figure 1 is shown once more, however with the difference that S B = 0 and κ e,B = 0 are set for υ B < 1/3, and S A = 0 and κ e,A = 0 are set for υ A < 1/3. For υ i < 1/3 the phase i exists as islands separated from each other, where the electronic bands in the phase i are not formed. The curve S(-) in Figure 5 agrees relatively well with the experimental data. It is noteworthy that at υ A = 1/3 and at υ B = 1/3 two more steps appear, which are also reflected in the experimental data.

If Equation (1) does not provide a mathematical solution, we conclude that the underlying physical model (the "EMT"-assumption that "the phase grains are spherical with no preferred orientations and arranged in a symmetrical manner") is not a good approximation to describe the structure of the alloy. More on this in the next points, (3) and (4) that are treated in the following paragraphs.

The electrostatic potential affects the thermopower of a composite, but does not affect it in a homogeneous material [START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF]. For the example alloy a-Cr 1-x Si x it is shown that the discontinuity in S(-) shifts to larger υ B as the electrostatic parameter c increases and one may conjecture that such a situation where Equation (1) does not have solutions for the entire concentration range is not realized in the nature, because the electrostatic potentials act contrarily to the effect of ∂E C,A /∂T (respectively ∂E V,B /∂T) [START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF]. For instance, assuming that ∂E C,A /∂T > 0, then a temperature depending electron transfer occurs from the phase A to the phase B leading to an increase of the electrostatic potential difference between the phases counterbalancing the effect of ∂E C,A /∂T on the solutions of Equation ( 1). 1, where however S B = 0 and κ e,B = 0 are set for υ B < 1/3 and S A = 0 and κ e,A = 0 are set for υ A < 1/3. For υ i < 1/3 the phase i exists as islands separated from each other. (The curve for S(-) agrees with that obtained by Sonntag [START_REF] Sonntag | Thermoelectric power in alloys with phase separation (composites)[END_REF]).

However, such a counterbalancing is surely incompletely, and one can conjecture that situations are possible in nature (i.e., experimental conditions) where Equation (1) does not have solutions for the entire concentration range. For concentration ranges, where Equation (1) does not have a mathematic solution, we speculate that the structure and arrangement of the phase grains realized is different from that assumed for the derivation for Equation (1), for instance that the phase grains are not spheric or/and arranged in an asymmetrical fashion or/and there are preferred orientations of them. The reason is the following: Equation (1) has been derived under the condition that at the boundary face between a single spherical phase grain and its surroundings, (spatial) continuity of the entropy-flux density, J S , and its gradient is fulfilled [START_REF] Sonntag | Thermoelectric power in alloys with phase separation (composites)[END_REF]. This condition must also be fulfilled during the solidification process of the alloy. If Equation (1) does not have a solution (for a certain x), then the condition of continuity of J S at the boundary faces between the different phases cannot be fulfilled leading to the fact that such a structure as specified cannot be realized. Instead, another atomic structure is favored, where the continuity of J S and its gradient can be fulfilled. This other atomic structure can be, for instance, characterized by non-spheric phase grains or other values for x A (and x B ) corresponding to a situation where the phase compositions become a function of x. Saving continuity of J S and its gradient, the structure of the alloy is matched at the especial conditions prevailing during the solidification of the alloy. This discussion can also be executed considering the energy flux density J E and its gradient.

In other words, the structure realized in any alloy does not only depend on the diffusion mobility of the atoms and whether there exists a (relative) minimum in the Gibbs' free energy. The structure realized depends also on the fact whether it is compatible with the demand that continuity of the entropy and energy flux densities and their gradients is saved during the solidification process.

This conclusion is a consequence of the Gibbs equation and its conditions of validity. Gibbs equation reads

dU = TdS + ∑ i µ 0 i dn i , (14)
where n i is the particle density of species i. U is the internal energy density, S is the entropy density, and µ 0 i is the chemical potential of the ith species present in the system. Equation ( 14) holds both for electronic carriers and atoms and ions [START_REF] Harman | Thermoelectric and Thermomagnetic Effects and Applications[END_REF][START_REF] Schulze | [END_REF]. Introducing

E = U + ϕ s q (15)
and considering the Gibbs-Duhem relation, which is applicable to the function U, it follows

S ∂T ∂t + ∑ i n i ∂µ 0 i ∂t = 0, ( 16
)
where t is the time. With the electrochemical potential µ i we get

∂E ∂t = T ∂S ∂t + ∑ i µ i ∂n i ∂t + q ∂ϕ s ∂t , ( 17
)
where ϕ s is the electrostatic potential. q is the net charge density with q = ∑ i q i . q i is the net charge density of the species i. Equation ( 17) is the basic formula for deriving the transport equations to describe the electronic transport processes, but also for diffusion processes in solids, expressed by the transport coefficients, to bring the system into equilibrium. Equation ( 17) is based on the continuity of the energy and entropy flow densities and their derivations according to place and time so that they do not break off. For our above conclusion it is now important under which conditions Equation ( 17) holds. "From an empirical viewpoint, use of the Gibbs equation is justified on the basis that, except for turbulence and shock-wave phenomena, it leads to excellent agreement with experiment. Therefore, we take the view that Equation ( 17) remains applicable, as long as local deviations from equilibrium are sufficiently small." (quoted from Harman and Honig [START_REF] Harman | Thermoelectric and Thermomagnetic Effects and Applications[END_REF], p. 10 therein). One can now object that this condition, that the local deviations from the equilibrium state during the layer deposition, are not small, because the process of layer growth is certainly far away from an equilibrium state. However, this means nothing other than that the resulting phase distribution of the growing layer on the substrate can more or less deviate from an uniform distribution and that also the shape of the phase grains can more or less deviate from a spherical shape. This means that the assumption for the derivation of the EMT approximation made for Equation (1), "the phase grains are spherical with no preferred orientations and arranged in a symmetrical fashion" is not or only partially fulfilled. This is all the more serious if there are not only microscopic concentration gradients on the substrate (single target deposition), but a macroscopic, pronounced concentration gradient over the entire substrate in the horizontal direction (co-sputter deposition). Because Equation ( 17) is also the basis for the derivation of Equation ( 1), it cannot apply if the deposition conditions deviate too much from the equilibrium. A situation, where Equation (1) has no solution, is equivalent to a situation, where the demand of this continuity of the entropy and energy flux densities and their gradients cannot be fulfilled by such a structure as assumed leading to a modified grain structure and structural arrangement.

Such a modified structure could be for example such that the phase grains are not spherical or such that there are additional holes or spaces at the phase boundaries. In this last case, the resistivity of the alloy is expected to be higher than without these defects.

For the assumption of holes or spaces speeks the fact, that the resistivities of the films produced by co-sputtering are essentially higher than the films produced by single sputtering. The question, which structure is actually realized, can be answered by specific structural investigations.

Independently of this open question, we state that for all the cases considered in Figures 23456, the lower kink at the discontinuity in the calculated S(-) curves is exactly at S(-) = 0, provided that the critical values for n, p, ∂E C,A /∂T, ∂E V,B /∂T are not yet exceeded. x calculated for different temperatures: T = 100, 200, 300, and 400 K. The discontinuity at x = 0.27 disappears if S 0 B = 0 and κ e,B = 0 is set for υ B < 1/3, which is a better approximation than given by Equation ( 3) and Equation ( 4), because the phase B does no longer form a macroscopic cluster through the alloy; at υ B = 1/3 the energy spectrum changes from a quasi-continuous spectrum (υ B > 1/3) to a discrete energy spectrum (υ B < 1/3) typical for separate phase grains [START_REF] Sonntag | Thermoelectric power in alloys with phase separation (composites)[END_REF].

Thermopower Switch

As shown in Figure 1, coming from x < 0.49 and increasing x, at room temperature the thermopower jumps at x = x s = 0.49 from S = 0 µV/Kto S = 2.0 µV/K. Now let us ask whether or not the value of x s is temperature dependent? By applying Equation (1) one can show, that the calculated value of x s is actually a function of T. This is shown for two choosen examples in Figures 7 and8, where several values are assumed for the physical parameters which are contained in the formulas summarized in Sections 2 and 3. Indeedly, for both cases there is a jump in S vs T. That is why, we assume that such a temperature dependence can also be expected for real composites or nanocomposites.

By choosing of a suitable composition x very close to x s , we have the possibility to produce a reference standard for the Seebeck coefficient S, where S = 0 µV/K absolute. Until now, such (theoretically established) reference standards for S = 0 µV/K were restricted to superconductors below the critical temperature.

For a-Cr 1-x Si x , S(-) = 0 at x s , independently of the temperature, as shown in Figure 6. This finding is especially noteworthy, because (contrarily to that) the thermopower in a homogeneous metal depends on T. For the case of temperature independent band egde, in a homogeneous "NFE"-metal, S depends even linearly on T (in correspondence to [START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF], Equations ( 38) and (39) therein). x s itself (characterized by S(-) = 0), depends, however, on temperature if ∂E C,A /∂T = 0 or ∂E V,B /∂T = 0, i.e., for two different temperatures the values for x s are different. Only for the hypothetical special case that ∂E C,A /∂T = ∂E V,B /∂T = 0, x s turns out to be independent of T. This fact seems to be a disadvantage for a realization of a highly precise reference standard with S = 0. On the other hand, it can also be an advantage: By choice of an appropriate composition x close to x s one could produce a "thermopower switch", which swiches between S = 0 µV/K and a finite S = 0 at a certain temperature, the switching temperature T s . If this "switching property" is actually realized for a given x close to x s , the value S = 0 µV/K could be adjusted arbitrarily precisely by variation of T approaching T s , so that this material could serve, after all, as a very precise reference standard for S = 0 µV/K absolutely. While this formal discussion is, to a certain extent, speculative, let us calculate such a "thermopower switch" for the hypothetical composite specified in Section 2. In Figures 7 and8 such a "swiching" property is shown for ∂E C,A /∂T = +0.2 × 10 -6 eV/K and -0.2 × 10 -6 eV/K, respectively, where simultaneously ∂E V,B /∂T = ∂ϕ i /∂n i = 0 was set. In the Figures 7a and8a, the υ B,s and x s , related by

x s = x A N A (1 -υ B,s ) + x B N B υ B,s N A (1 -υ B,s ) + N B υ B,s , (18)
are drawn vs. T. N i and x i are the atomic density and the atomic concentration in percent in the phase i, respectively, which were set to be N A =7.9 × 10 22 cm -3 , N B = 5.0 × 10 22 cm -3 , x A = 0.25, and x B = 1.00. According to Figure 7a, a composite with ∂E C,A /∂T = +0.2 × 10 -6 eV/K and x = 0.48 would show a switching temperature of T s = 312.41 K with a switching property as indicated in Figure 7b. According to Figure 8a, a composite with ∂E C,A /∂T = -0.2 × 10 -6 eV/K and x = 0.615 would show a switching temperature of T s = 311.62 K with a switching property as indicated in Figure 8b. In both Figures 7b and8b, at the lower kink of the discontinuity, S(-) = 0 precisely. As can be seen in the insets of Figures 7b and8b, on the side where S(-) is close to 0, S(-) changes very slowly with T with rates of -0.0037 (µV/K) per 1 K and 0.001 (µV/K) per 1 K, respectively.

However, we emphasize that such a "switching property" of S is a fictitious result following from an application of the thermopower formula Equation (1) to the two-phase composite specified in Section 2 with the additional property that E C,A depends on T. Note that for the calculations shown in Figures 7 and8 the effect of the electrostatic potentials has been neglected. Taking into account these contributions, the "switching properties" are expected to be modified, and there remains the open question whether such a hypothetical discontinuity occurs really in the S vs. T dependence or whether the effect of the electrostatic potential counterbalances this effect of band edge shift. Therefore, it is not at all clear whether or not there are real composites with such a "switching property".

Suitability of a-Cr 1-x Si x films as reference standard: Amorphous alloys are generally in a relatively unstable state. In contrast to this general experience, for not too small x, a-Cr 1-x Si x thin films, adequately annealed below the crystallization temperature T k , can be very stable as long as the temperature of application is essentially lower than T k . This high stability is assumed to be caused especially by an extremely thin a-Cr-Si-O passivation film at the surface of the a-Cr 1-x Si x film as well as by the p-d bonds at the phase boundaries (as discussed in [START_REF] Sonntag | Disordered electronic systems. II. Phase separation and the metal-insulator transition in metal-metalloid alloys[END_REF], Section 2 A therein). Close to x s , the crystallization temperature is T k ≈ 550 K or a little higher (depending on the annealing time), and thus a practical application of an a-Cr 1-x Si x film as a SRM up to temperatures T ≈ 500 K seems to be reasonable.

The method for designing SRM's proposed in the present paper has three advantages: (i) A SRM is now available with 0 µV/K absolute above superconducting temperatures and thus can also be applied at room temperature or even above, (ii) the assumption of reversible processes [which is the basis for Equation (A1)] is no longer necessary, and (iii) now we have the possibility to determine the thermopower very precisely by a method which is completely independent of the classical method (applied by Borelius et al. [START_REF] Borelius | Thermoelectric Power and Electrical Resistivity of Dilute Alloys of Mn, Pd, and Pt in Cu, Ag, and Au[END_REF][START_REF] Borelius | Measurements on thermoelectric forces down to temperatures obtainable with liquid or solid hydrogen[END_REF]).

Regarding the advantage (iii) we state, when once a SRM with exactly 0 µV/K absolute is designed for one material, then the Thomson coefficient can be measured for all other interesting materials using Equation (A1). These measuring results can be compared directly with the experimental data of the Thomson coefficient of any other material. In other words, with this new SRM we have the possibility to check the question mentioned earlier, to what extent the processes acting on a real measurement of the Thomson coefficient τ (described in the Appendix A) are actually reversible, because Equation (A1) holds only for the case that only reversible processes act.

We close by a short statement on the formulas Equations ( 2) and (3) in connection with Equation [START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF] applied for the present calculations: The BTE formula S 0 i , Equation ( 3) is assumed to describe exclusively the scattering contribution to S i . The correctness of this assumption is, however, not yet confirmed. The question of whether or not in S 0 i the effect of "∂µ 0 i /∂T" (respectively "- [START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF]). However, this question does not influence the conclusions of the present paper, especially that the lower kink of S(±) at the discontinuity would occur at S(-) = S(+) = 0. If in S 0 i , Equation (3), the effect of "∂µ 0 i /∂T" (respectively "-

π 2 k 2 B T 6E F,i ") is contained indirectly is still open (see the discussion in
π 2 k 2 B T 6E F,i
") would be contained indirectly, the S(±) vs. υ B curves would be shifted in Figures 234; however, in this case the discontinuity would also occur at S(-) = S(+) = 0.

Conclusions

Applying the EMT to composites with different kinds of carriers in the different phases (electrons and holes), it is concluded that a discontinuity can be expected in the concentration dependence of the Seebeck coefficient which coincides exactly with the transition from negative values to positive ones. Such a dicontinuity (kink) is actually found experimentally in a-Cr 1-x Si x thin films sputtered from different Cr 1-x Si x targets representing a composite with the phases a-Cr 3 Si and a-Si. This feature can be applied to make reference standards for an absolutely zero Seebeck coefficient at room temperature and even at higher temperature. The experimental confirmation of this feature for a-Cr 1-x Si x alloys supports the theory of composites developed in the last years [START_REF] Sonntag | Disordered electronic systems. III. Thermoelectric power in alloys with phase separation[END_REF][START_REF] Sonntag | Disordered electronic systems: Concentration dependence of the dc conductivity in amorphous transition-metalmetalloid alloys (metallic regime)[END_REF][START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF][START_REF] Sonntag | Thermoelectric power in alloys with phase separation (composites)[END_REF][START_REF] Sonntag | Disordered electronic systems. II. Phase separation and the metal-insulator transition in metal-metalloid alloys[END_REF]. Under certain conditions such a kink at S = 0 µV/K can also be expected in the temperature dependence of S.

The experimentally finding that the electronic properties of amorphous Cr-Si films can differ fundamentally depending on the prevailing fabrication conditions suggests that the structure realized in any alloy is not only determined by the diffusion mobility of the atoms and that for this structure a (relative) minimum of the Gibbs' free energy exists, but it depends also on the fact whether this structure is compatible with the demand that (spatial) continuity of the entropy and energy flux densities and their gradients is saved during the solidification process. concentration, where the discontinuity occurs J i electric current density in the phase i J Q,i thermal current density in the phase i J S,i entropie flux density in the phase i

Symbols

Appendix A. Seebeck Coefficient Standard Reference Materials

Absolute values of the thermoelectric power (Seebeck coefficient) of any material cannot be measured directly. It is always differences between two materials which are measured. That is why, in practice, Seebeck coefficient Standard Reference materials (SRM's) are designed which allow to measure the absolute Seebeck coefficient of any material by a direct comparison with it.

SRM's used in practice to measure the absolute thermopower of materials are generally not very accurate. An exception are superconductors, whose thermopower is exactly S = 0 µV/K. However, these can only be used for temperatures T smaller than the critical one T c . For higher temperatures special SRM's are used, which are, however, only approximations for an absolute value of S, which are the worse the higher the desired comparison temperature or reference temperature. Examples for such SRM's are Pb, Sn or a special silver-normal-alloy (Ag + 0.37at%Au) (Borelius et al. [START_REF] Borelius | Thermoelectric Power and Electrical Resistivity of Dilute Alloys of Mn, Pd, and Pt in Cu, Ag, and Au[END_REF][START_REF] Borelius | Measurements on thermoelectric forces down to temperatures obtainable with liquid or solid hydrogen[END_REF][START_REF] Borelius | Establishment of an absolute scale for the thermo-electric force[END_REF], Christian et al. [START_REF] Christian | Thermo-electricity at low temperatures. VI. A redetermination of the absolute scale of thermo-electric power of lead[END_REF]). For this last strategy the fact was used that for a certain material the absolute Seebeck coefficient S is related with τ, the Thomson coefficient, by (first Kelvin relation)

S = T 0 τ T dT, ( A1
)
where T is the absolute temperature. The precision of such a SRM is however limited: First, Equation (A1) holds only if all the processes involved are reversible, but it is not yet completely clear to what extent this condition is fulfilled under the conditions of real measurement of τ. Second, the precision of the measured Thomson coefficients is restricted, because there are a series of sources for systematic uncertainties in measuring the Thomson coefficient τ (see, for instance Borelius et al. [START_REF] Borelius | Thermoelectric Power and Electrical Resistivity of Dilute Alloys of Mn, Pd, and Pt in Cu, Ag, and Au[END_REF]). Another practical method for supplying a SRM has been realized by Lowhorn et al. [START_REF] Lowhorn | Development of a Seebeck coefficient Standard Reference Material[END_REF][START_REF] Lowhorn | Development of a Seebeck coefficient Standard Reference Material TM[END_REF]. By two different, independent experimental techniques Lowhorn et al. have developed a special SRM (Te-doped Bi 2 Te 3 ) which is designed as a comparison material for thermoelectric research in different laboratories. For this SRM uncertainties in the Seebeck coefficient are given by Lowhorn et al. [START_REF] Lowhorn | Development of a Seebeck coefficient Standard Reference Material[END_REF] in the range of ≈1 . . . 6 µV/K in the temperature range T = 10 . . . 391 K. It remains a problem that absolute values for the Seebeck coefficient can only be measured relatively imprecisely, especially at room temperature and at higher temperatures.

Appendix B. Comparison with Earlier EMT Formulas

There are a series of thermopower formulas published in the scientific literature, [START_REF] Webman | Thermoelectric power in inhomogeneous materials[END_REF][START_REF] Airapetiants | Thermal Electromotive Force and Additional Thermal Conductivity of Statistical Mixtures[END_REF][START_REF] Halpern | The thermopower of binary mixtures[END_REF][START_REF] Balagurov | Theory of Thermoelectric Properties of Two-Component Media[END_REF][START_REF] Bergman | Thermoelectric properties of a composite medium[END_REF], which all are different from our formula Equation (1), and they do not provide such a discontinuity in the α versus x dependence. On the first look all these formulas look different. However, comparing their formulas in detail, all provide an identical concentration dependence ([23], Section 2.5.1. therein), α versus x, and what is especially remarkable, all these formulas contain α i , σ i and κ i . Reason for this fundamental difference to the formula Equation ( 1) is the fact, that the authors start with an approximation of the thermal current density, J Q , for instance Webman et al. [START_REF] Webman | Thermoelectric power in inhomogeneous materials[END_REF],

J Q = -κ • gradT + α • σ • gradϕ. (A2)
κ and κ i are the total thermal conductivities in the composite and the phase i, respectively. α and α i are the Seebeck coefficients in the composite and the phase i, respectively. (Why we distinguish between S and α becomes clear further below.) In formula Equation (A2) the quadratic term in α and the contribution of the chemical potentials of the phases, µ 0 i , are not contained. An additional difference is that J Q , Equation (A2), represents the total thermal current density (containing both the electronic part and the lattice part). However, if we leave aside this difference with respect to κ, we are still left with the fundamental difference that neglecting the quadratic term of α results in all three transport coefficients, α i , σ i and κ i , appear in the EMT formula. This is not the case if the quadratic term in α is considered. If this point and the contribution of the chemical potentials are considered in J Q,i (formula [START_REF] Ghosh | Thermoelectric properties of Co4Sb12 with Bi2Te3 nanoinclusions[END_REF] in [START_REF] Sonntag | Disordered electronic systems. III. Thermoelectric power in alloys with phase separation[END_REF]), then in the EMT formula only κ e,i and α i occur. Starting with J i and J Q,i and correcting these inaccuracies contained in Equation (A2), then it follows for the EMT formula (formula [START_REF] Christian | Thermo-electricity at low temperatures. VI. A redetermination of the absolute scale of thermo-electric power of lead[END_REF] in [START_REF] Sonntag | Disordered electronic systems. III. Thermoelectric power in alloys with phase separation[END_REF]), where J i and J Q,i are the local electric current density and the local thermal current density (without the lattice contributions), respectively, of the phase i. The angular brackets in Equation (A3), . . . , characterize an average. Equation (A3) looks very complicated, especially because of the term . . . . A simplier formula we get, if J i and J S,i is used as the starting equations ( [START_REF] Sonntag | Thermoelectric power in alloys with phase separation (composites)[END_REF]), where J S,i is the entropie flux density in the phase i: In Equation (A4) dµ 0 i /dT (which are different for the different phases) do not occur, insteadly the common dµ/dT.

Replacing α i + (dµ/dT)/|e| and α + (dµ/dT)/|e| by S i and S, respectively, we get Equation (1), i.e., that S 0 i in Equation ( 2) is identical with α i . This replacing was the decisive step for geting the final formula Equation (1). In the earlier practice of thermoelectricity, α was generally used as the thermopower, respective Seebeck coefficient, of a material, where the "thermodynamic term" (dµ/dT)/|e| did not occur in all the earlier thermopower formulas. This decisive step α -→ S, respective α i -→ S i , i.e., the introduction of the additional "thermodynamic term" (dµ/dT)/|e| to the thermopower formulas founded in [START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF] has important consequences. In [START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF] it is also shown, that this contribution gives the answer on the question unsolved until now, why there are simple metals with positive thermopower as Cu, Ag, Au, and Li.

The difference between Equations (A4) and (A3) reflects a general problem for multiband systems or multiphase systems: Applying J i and J Q,i , problems can arise, which can be avoided when J i and J S,i is used. ( [START_REF] Harman | Thermoelectric and Thermomagnetic Effects and Applications[END_REF], pages 28 and 40). So, Equation (A4), respective Equation (1), can be considered as the exact EMT formula under the special conditions ("the phase grains are spherical without preferred orientations and arranged in a symmetrical fashion".)

In the formulas Equation (A4), Equations (A3) and ( 1) only the electronic part of the thermal conductivity occurs, but not the lattice part. This is in agreement with the seminal work of Harman and Honig [START_REF] Harman | Thermoelectric and Thermomagnetic Effects and Applications[END_REF], especially formula (1.17.1a) on page 40 and the explanatory text. In the electronic transport equations the lattice thermal conductivity does not appear.

Figure 1 .

 1 Figure 1. S(-) vs. x for a-Cr 1-x Si x at T = 300 K calculated by Equation (1), respective Equation (9) and Equations (2)-(13) for the phases A (= a-Cr 3 Si) and B (= a-Si), where ∂E C,A /∂T = ∂E V,B /∂T = 0 was set. This curve agrees with those of Figure A.2 in[START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF] for c = 2.6 eV. The experimental data are taken from Gladun et al.[START_REF] Gladun | Electrical transport properties of high resistance Cr-Si-O thin films[END_REF] (diamonds), Weser[1] (open triangles) and Sonntag[START_REF] Sonntag | Connectivity Sensors Germany GmbH[END_REF] (full triangles). The discontinuity in S(-) at x = x s = 0.49 corresponds to the discontinuities in Figures2-4.

Figure 2 .

 2 Figure 2. S(-) (bold line) and S(+) (thin line) vs. υ B calculated by Equations (2)-(13) for a hypothetical composite with S 0 A = -13.0 µV/K, S 0 B = +1.5 µV/K, κ e,A = 8.5 mW/cmK, and κ e,B = 12.7 mW/cmK (corresponding to n = 10 22 cm -3 and p = 2 × 10 22 cm -3 and T = 300 K, where dµ/dT, Equation[START_REF] Sonntag | The effect of the band edges on the Seebeck coefficient[END_REF], is calculated for ∂E C,A /∂T = ∂E V,B /∂T = ∂ϕ i /∂n i = 0).

Figure 3 .Figure 4 .

 34 Figure 3. Same as Figure2, for varying values of p: p = 1.5, 2, 3, and 5 (in units of 10 22 cm -3 ), while ∂E C,A /∂T = ∂E V,B /∂T = ∂ϕ i /∂n i = 0 is set. The discontinuity in the curves shifts to lower υ B as p decreases. For p < 1.2149 × 10 22 cm -3 (and n = 10 22 cm -3 ), Equation (9) does not have real solutions for the entire concentration range. For p > 1.2149 × 10 22 cm -3 the lower kink of S(±) at the discontinuity occurs always at S(-) = S(+) = 0.

Figure 5 .

 5 Figure 5. Same as Figure1, where however S B = 0 and κ e,B = 0 are set for υ B < 1/3 and S A = 0 and κ e,A = 0 are set for υ A < 1/3. For υ i < 1/3 the phase i exists as islands separated from each other. (The curve for S(-) agrees with that obtained by Sonntag[START_REF] Sonntag | Thermoelectric power in alloys with phase separation (composites)[END_REF]).

3 Figure 6 .

 36 Figure 6. Same as Figure1, however S(-) vs. x calculated for different temperatures: T = 100, 200, 300, and 400 K. The discontinuity at x = 0.27 disappears if S 0 B = 0 and κ e,B = 0 is set for υ B < 1/3, which is a better approximation than given by Equation (3) and Equation (4), because the phase B does no longer form a macroscopic cluster through the alloy; at υ B = 1/3 the energy spectrum changes from a quasi-continuous spectrum (υ B > 1/3) to a discrete energy spectrum (υ B < 1/3) typical for separate phase grains[START_REF] Sonntag | Thermoelectric power in alloys with phase separation (composites)[END_REF].

Figure 7 .Figure 8 .

 78 Figure 7. (a) x s and υ B,s vs. T calculated for the example of Figure 4 with ∂E C,A /∂T = +0.2 × 10 -6 eV/K. For x = 0.48, T s = 312.41K. (b) S(-) vs. T for x = 0.48. The inset shows the same data with a better resolution. At the lower kink of the discontinuity, S(-) = 0 precisely; S(-) decreases with increasing T very slowly at a rate of -0.0037 (µV/K) per 1 K.

  α i -(dµ 0 i /dT)/|e| -κ e α-(dµ 0 i /dT) /|e| κ e,i α i -(dµ 0 i /dT)/|e| + 2

  i α i +(dµ/dT)/|e| -κ e α+(dµ/dT)/|e| κ e,i α i +(dµ/dT)/|e| + 2 κ e α+(dµ/dT)/|e| = 0 (A4)

Data Availability Statement:

The data that support the findings of this study are available on reasonable request to J.S.

Author Contributions: J.S. developed the theoretical formalism and conceived of the presented idea. J.S. and B.L. discussed the results and contributed to the final version of the manuscript. All authors have read and agreed to the published version of the manuscript.

Funding: This research received no external funding.

Institutional Review Board Statement: Not applicable.

Informed Consent Statement: Not applicable.

Conflicts of Interest:

The authors declare no conflict of interest.


Bartlomiej Wiendlocha
email: wiendlocha@fis.agh.edu.pl

Shantanu Misra

Anne Dauscher

Bertrand Lenoir

Christophe Candolfi
email: christophe.candolfi@mines-nancy.univ-lorraine.fr

Residual resistivity as an independent indicator of resonant levels in semiconductors

Distortion of the density of states induced by specific impurities, a mechanism known as resonant level (RL), is an efficient strategy to enhance the thermoelectric performances of metals and semiconductors. So far, experimental signatures identifying the resonant nature of an impurity have relied on the so-called Ioffe-Pisarenko plot that enables visualizing the induced thermopower enhancement at specific carrier concentrations. However, this method cannot solely discern RL from other possible band-structure-related sources of thermopower enhancement such as band-shape modifications or band convergence. An independent method of resolving this problem is proposed here. A detailed theoretical and experimental analysis of the low-temperature electrical resistivity r0 and carrier mobility µ0 of the resonant-level system SnTe doped with In is presented as a function of the impurity concentration x. By comparing to non-resonant cases of SnTe doped with I, Mn, and Ga, we demonstrate that the construction of r0(x) and µ0(x) plots allows to distinguish between resonant and non-resonant impurities, even when some of them induce similar thermopower enhancements. This methodology is further confirmed by analyses performed for Na-and Tl-doped PbTe, illustrating how the combination of transport measurements at low temperatures can be used to determine the resonant nature of an impurity.

Introduction

Doping semiconductors with impurities is the cornerstone of modern electronics and conversion energy applications, providing a straightforward mean to tailor their electronic properties. Depending on the nature of the impurities, "rigid-like" or "non-rigid-like" with respect to the host semiconductor, their presence can induce either a mere shift in the chemical potential or more profoundly reshape the electronic bands. The electronic band structure of semiconductors can be thus advantageously manipulated by both types of impurities.

Among functional materials used in energy conversion technologies, thermoelectric semiconductors are a class of materials combining unique transport properties used in electrical power generators and Peltier cooling modules. The efficiency with which they directly convert heat into electrical power, and vice versa, at an absolute temperature 𝑇, is proportional to the figure of merit 1,2 𝑍𝑇 = 𝛼 ! 𝑇/𝜌𝜅, a dimensionless parameter that combines material-specific transport coefficients: thermopower a, electrical resistivity r and total thermal conductivity 𝜅.

Over the last decades, significant efforts were devoted to optimize the performance of thermoelectric materials via dedicated strategies focusing on either optimizing the power factor 𝛼 ! /𝜌 through doping or lowering the total thermal conductivity 𝜅. [START_REF] Zhu | [END_REF]4 In the former approach, band structure engineering has been a particularly successful strategy to push the ZT values beyond unity at high temperatures. 4,[START_REF]Advanced Thermoelectrics: Materials, Contacts, Devices, and Systems[END_REF] Among the mechanisms that allow to manipulate the electronic bands beyond the rigidlike scheme, resonant levels (RLs) provide a textbook example. [START_REF] Zhu | [END_REF]4,[START_REF]Advanced Thermoelectrics: Materials, Contacts, Devices, and Systems[END_REF][START_REF] Heremans | [END_REF] Used as an efficient route to enhancing the thermopower of metals and semiconductors, a RL, or "virtual bound state", 7 may be formed upon doping a material at energies where a "real" bound state would be formed in the absence of neighboring atoms. [START_REF] Györffy | Electrons in disordered metals and metallic surfaces[END_REF] In the specific case of semiconductors, this resonant energy has to overlap with either the valence or the conduction bands. One of the manifestations of a resonant state is the formation of a sharp peak in the impurity's density of electronic states (DOS) at the resonance energy, leading to a distortion of the total DOS, the energy dependence of which varies rapidly over a narrow energy window. The hybridization of the resonant states with the host electronic states results in a band-smearing effect, [START_REF] Wiendlocha | [END_REF]10,11 enabling an increase in the thermopower when the Fermi level resides near this distortion. Thus, the position of the RL with respect to the band edge and the width of the smeared bands are the key parameters to achieve optimized ZT values.

So far, the main experimental strategy to unravel the resonant nature of an impurity in semiconductors was tied to the variation of the thermopower a as a function of the charge carrier concentration p. Known as the Ioffe-Pisarenko plot, the a(p) relation is characteristic of the electronic structure of the host semiconductor, and is frequently well captured by models based on solutions of the Boltzmann transport equation. In most cases, doping induces a rigidlike shift of the Fermi energy in the conduction or valence bands, giving rise to smooth variations in a(p) that follow the Ioffe-Pisarenko plot of the host material. In contrast, the formation of a RL manifests as a significant departure from the predicted trend, signaling nonrigid-band behavior. Experimentally, the conclusion as for the resonant nature of the impurity can be drawn when the thermopower is significantly enhanced compared to the material specific a(p) curve, as observed for Pb1-xTlxTe, 12,13 Sn1-xInxTe, 11,[START_REF] Zhang | Proc. Natl. Acad. Sci[END_REF] Bi2-xSnxTe3, 15 As2-xSnxTe3 16 or Ge1-xInxTe. 17 However, upon doping, deviations from the rigid-band trend in a(p) may originate from other effects, such as variations in the dominant scattering mechanism or in the relative position of band extrema possibly leading to band convergence. Through substitutions or the formation of solid solutions, this last mechanism also efficiently enhances the ZT values due to the convergence of two, or more, band extrema that increases the number of valleys 𝑁 " and hence, the density-of-states effective mass 𝑚 #$% * (𝛼 ∝ 𝑚 #$% * = 𝑁 " !/( 𝑚 ) * where 𝑚 ) * is the band effective mass). Thus, an analysis of the character of an impurity solely based on the Ioffe-Pisarenko plot may be inconclusive. Here, we demonstrate a straightforward way of getting a clear and independent experimental proof of the formation of a RL, allowing for the distinction between resonant and non-resonant impurities, regardless of their effect on the thermopower.

Our analysis is based on a combination of low-temperature electrical and galvanomagnetic measurements to probe the emergence of resonant scattering, a mechanism specific to the presence of a RL 18,[START_REF] Ravich | CRC Handbook of Thermoelectrics[END_REF] but not generally used to establish the resonant nature of an impurity. By performing systematic experimental and theoretical studies on the binary SnTe, we propose a novel methodology based on the construction of the compositional dependence of the residual resistivity r0(x) and charge carrier mobility µ0(x), analogue to the Ioffe-Pisarenko plot. We hereafter consider the p-type chalcogenide semiconductor SnTe doped with the resonant impurity In 11,[START_REF] Zhang | Proc. Natl. Acad. Sci[END_REF] and the non-resonant impurities I, Ga, and Mn chosen as counter-examples. As a further proof-of-principle, the analysis for SnTe is supplemented by a similar one performed for the best-known resonant thermoelectric system Pb1-xTlxTe.

Results and discussion

Figure 1 shows the Ioffe-Pisarenko plot of SnTe calculated using a two-valence band model [START_REF] Brebrick | [END_REF]21 (see Supplementary Information file for the equations and physical parameters used).

The thermopower measured for self-doped and iodine-doped SnTe samples generally follows the model prediction. "Self-doped" samples correspond to pristine binary SnTe samples that contain naturally-formed defects, which are Sn vacancies acting as double-hole acceptors. 22,23,24,25 This behavior suggests that both the Sn vacancies and iodine substitutional impurity modify the valence band structure of SnTe in a rigid-like manner. In contrast, a substantial enhancement of the a values above the Ioffe-Pisarenko line is observed experimentally for In-, Mn-, and Ga-doped samples. Although all these three elements lead to a basically similar enhancement of a, different mechanisms are at play: In forms a resonant level, while Mn induces band convergence of the two main valence bands governing the transport in SnTe, 26,[START_REF] Li | Novel Thermoelectric Materials and Device Design Concepts[END_REF] similarly to Ga, reported to increase the number of degenerate valleys in the Brillouin zone. [START_REF] Nikitin | [END_REF] Thus, SnTe is an excellent and sensitive test-case to our proposed methodology, not only capable of confirming the resonant behavior of In but also to provide a clear distinction between In and non-resonant Ga/Mn, as the analysis solely based on thermopower enhancement is inconclusive.

The key difference between resonant and non-resonant impurities, serving as a basis for our methodology, is the distinct way with which they affect the electronic bands. To visualize this effect, a magnification of the electronic band structure of SnTe containing different doping atoms, is shown in Figure 2 in the region of interest in the Brillouin zone (BZ; see Figure S1 in Supplementary Information). Electronic structure calculations were performed using the KKR-CPA method 29,[START_REF] Ebert | The Munich SPR-KKR package, version 7.7.3[END_REF] , with the band structure being visualized using the Bloch spectral density functions (BSF), further details of which are given in the Supplementary Information. Figure 2a shows the valence bands of SnTe with 0.32% of Sn vacancies. This concentration of vacancies, inherent to SnTe, is equivalent to a hole concentration p of 10 20 cm -3 as typically observed experimentally. SnTe samples are thus self-doped by Sn vacancies, which act as intrinsic rigid-band-like dopants, that is, they change the position of the Fermi energy but do not significantly modify either the density of states (DOS) or the electronic dispersion relations, inducing only a small band smearing (see below). In In contrast, iodine (Figure 2d), being a monovalent donor in SnTe, 21 only rigidly shifts EF towards the valence band edge, compensating the acceptor behavior of Sn vacancies.

Furthermore, while Ga (Figure 2e,f) is isoelectronic to In, it does not form RL but slightly increases the DOS near EF and reduces the energy offset between the two band extrema at the L and S points, due to its non-rigid-like behavior.

Band smearing is an effect of prime importance to understand the difference in the transport properties between resonantly and non-resonantly doped samples. A deeper analysis of the dopant-induced band smearing is shown in Figure 3, in which we compare the shape of the last valence band, lying between the L and S points of the BZ (panels a-c and g-i). Thanks to the BSF technique, the strength of the band smearing, which is proportional to the strength of electron scattering on impurities, may be quantified. In a typical alloy systems, the BSF for a single wave vector k adopts the shape of the Lorentz function (see, Supplementary Information), and the electronic relaxation time t is in such a case related to its full width at half maximum 31 D as 𝜏 = Lorentzian character of BSF is retained, even for 2% of Ga. In sharp contrast, the addition of a minute amount of In as low as 0.5% is sufficient to generate a non-Lorentzian BSF with shoulders. Such a modification is characteristic of resonant levels [START_REF] Györffy | Electrons in disordered metals and metallic surfaces[END_REF][START_REF] Wiendlocha | [END_REF] and strongly enlarge the width of the BSF. This effect is dramatically enhanced upon increasing the In concentration to 2% for which the BSF exhibits a double-bell shape. Rough estimates of t at the same k1-point drops to 2´10 -14 s for 0.5% In, and to 3´10 -15 s for 2% In, i.e. an order of magnitude lower than that of Ga and two orders of magnitude lower than those obtained for Sn vacancies and I. This huge difference in t opens the possibility of an experimental detection of RL through electrical transport measurements since the electric conductivity s and carrier mobility µ are inversely proportional to t. Moreover, our results clearly show that the relaxation time is no longer welldefined for such non-Lorentzian BSF shapes, [START_REF] Butler | [END_REF] resulting in differences in actual mobilities that will be even larger than differences in t suggest. To precisely quantify the effect of impurityinduced scattering and take into account the contributions from the whole Brillouin zone, electric conductivity needs to be calculated. Due to the strong smearing of the electronic bands, the Boltzmann formalism within the relaxation time approximation would fail to describe appropriately the transport properties of this system, due to the impossibility to derive the velocity of the charge carriers defined as the gradient of the band energy with respect to the wave vector. These difficulties can be overcome by using the Kubo-Greenwood formalism 29,[START_REF] Butler | [END_REF]32,33 that enables to calculate the residual electrical resistivity and charge carrier mobility at the temperature of 0 K, taking into account the impurity-induced scattering. The comparison of calculations with measurements performed at liquid helium temperatures, that is, in the limit where impurity scattering largely outweighs phonon scattering, may thus clearly distinguish between resonant and non-resonant impurities. are gathered in Supplementary Information). In calculations, the experimental and theoretical hole concentrations were matched by adding an appropriate number of Sn vacancies, which act as double-hole acceptors (all the data are gathered in Table 1). The comparison of the theoretical and experimental r0 values indicates an overall good agreement between both sets of data. In particular, the calculations capture qualitatively the two main characteristics of r0, that is, its linear increase with increasing x for substitution levels lower than 1% and its tendency to level off at higher In concentrations. Furthermore, below 0.5%, the computed values agree quantitatively with the experimental data while above this concentration, the calculations tend to overestimate the experimental values, most likely due to overestimation in the density-ofstates effective mass, as discussed in Ref. [11].

A straightforward visualization of the strong scattering of holes on the RL is provided by the rapid decrease in the hole mobility µ0 measured by Hall effect (Figure 4b). Despite the very low substitution levels covered in the series Sn1.03-xInxTe, µ0 drops by two orders of magnitude from 3405 cm 2 V -1 s -1 for x = 0.0 to 34 cm 2 V -1 s -1 for x = 0.02. This trend is well captured by our calculations over the entire concentration range, which confirms that it is an intrinsic property of the material, related to the RL. Note that the polycrystalline nature of our samples results in grain boundary scattering that contributes to limit the hole mobility in the x = 0 sample, which is likely the reason for the larger disagreement between calculations and measurements in this case. Nevertheless, the overall agreement achieved herein suggests that this mechanism only weakly influences µ0 at very low temperatures compared to resonant scattering. It is important to recall here that the strong reduction in mobility at such a low temperature does not rule out the beneficial influence of the RL on enhancing the thermoelectric properties at high temperatures. Above room temperature, the strengths of the electron-phonon and temperature-independent RL scattering start to be of similar magnitudes, significantly reducing the difference in mobilities between resonantly-and non-resonantly-doped SnTe. In particular, as shown in Ref. [11], the ratio of electrical resistivities of 2% In doped and 0% In doped SnTe samples dropped to only 1.3 at 800 K, leading to comparable mobilities in both samples. This allows for an enhancement in the peak ZT values in SnTe with resonant In dopant, 11,[START_REF] Zhang | Proc. Natl. Acad. Sci[END_REF] in spite of the low-temperature reduction in mobility. Moreover, due to the lower sensitivity of resonant-doped system on temperature-induced electron-phonon scattering when compared to non-resonant doping, the ZT enhancement was observed to persist in a broader temperature range, leading to enhanced average ZT values. 11 Similar situation was also evidenced in Tl-doped PbTe. 13 The benefit of constructing residual resistivity r0(x) and residual mobility µ0(x) plots becomes evident when comparing the above-mentioned trends with those of I and Ga-doped SnTe (only calculated values because no low-temperature resistivity measurements are available in literature), as well as Mn-doped SnTe (calculated and experimental 34,35,[START_REF] Inoue | Memoirs of the Faculty of Engineering[END_REF]37 values, see also Figure S3 in the Supplementary Information file for an additional mobility plot for Mndoped SnTe with a constant vacancy concentration). Despite the similar substitution levels in all the four series of samples, In induces a striking enhancement in r0 and reduction in µ0 compared to the rigid-band-like I and non-rigid-band-like, non-resonant Mn and Ga dopants, clearly highlighting the resonant character of In in SnTe. Thus, the r0(x) and µ0(x) plots provide a direct and convenient visualization of the resonant nature of In while an analysis solely based on the Ioffe-Pisarenko plot would remain inconclusive in this case.

To successfully apply this methodology to new semiconductors and impurities of an unknown character, more than two different series of samples should be considered, paying a particular attention to the low impurity concentration range. As seen in Fig. 4, the drop of µ0(x)

and increase in r0(x) for SnTe are the steepest up to 1% of impurity concentration. In this compositional range, the difference between the rigid-band-like I and non-rigid, non-resonant Ga and Mn (which scatter more strongly than I) is not very pronounced, below an order of magnitude, in contrast to the two-orders-of-magnitude difference between I and In.

It is important to emphasize that µ0(x) is a more sensitive indicator of the presence of a RL compared to the enhancement in a typically observed in the Ioffe-Pisarenko plot, as the difference between RL and non-RL doping cases reaches one-to-two orders of magnitude.

Moreover, the sole electrical resistivity analysis could be also misleading for impurities that compensate the main charge carriers, as is the case of I that behaves as a donor impurity in ptype SnTe. The relatively rapid increase in r0 with x seen for I-doped SnTe in Figure 4a is mainly caused by the decrease in the hole concentration (see Table 1), rather than by the appearance of strong scattering, as evidenced by the weak µ0(x) dependence (see Figure 4b).

Thus, both physical quantities should be analyzed in parallel to draw firm conclusions.

Furthermore, the methodology proposed herein should be used to analyze the mobility in samples doped with similar impurity and charge carrier concentrations, and for which, scattering on point-defects (i.e. on impurities) is the main source of electrical resistivity in addition to phonon scattering. In particular, the ρ0 and µ0 values may be affected by both microstructural and disorder-related effects that notably include grain boundary scattering and the disorder induced by the dopants and possible intrinsic defects. For these reasons, this approach may not be applicable to nanostructured samples for which, the small grain size may be the dominant source of electrical resistance. Although concerns related to charge localization at the grain boundaries can be avoided by performing systematic studies on single crystals, our present results show that our proposed methodology is equally applicable to polycrystalline samples. As demonstrated in our prior study on the series Sn1.03-xInxTe, 11 tends to strongly lessen the development of energy barriers at the grain boundaries. 44,45 In addition to intrinsic defects, dopants inevitably further enhance the level of disorder, which also decreases the charge carrier mobility in semiconductors. However, the comparison of the influence of In on the electrical resistivity and Hall mobility with those observed for other dopants shows that resonant scattering largely outweighs the effect of disorder.

The effectiveness of our methodology is further confirmed by literature results gathered for Tl-and Na-doped PbTe in Figure 5. Using available experimental data [START_REF] Giraldo-Gallo | [END_REF]39,40,41,[START_REF] Matsushita | Superconductivity and mixed valency in thallium doped lead telluride[END_REF][START_REF] Keiber | [END_REF] we have constructed the residual resistivity and the residual mobility plots, which evidence the same behavior as in the case of SnTe. The resonant character of Tl against non-resonant Na is highlighted by the increase in r0(x) and decrease in µ0(x) for similar impurity concentrations, and supplements the conclusion of the Tl resonance drawn from the Ioffe-Pisarenko plot and band structure calculations. [START_REF] Heremans | [END_REF]10,12

Conclusions

In conclusion, we have shown that the compositional dependences of the residual electrical resistivity r0 and charge carrier mobility µ0 can provide a clear experimental signature of the resonant character of the introduced dopant in semiconductors. At low temperatures, both physical quantities successfully capture the fundamental differences between resonant and conventional doping impurities, despite both of them may lead to seemingly similar thermopower enhancements. This conclusion is firmly confirmed by calculations of the electrical conductivity of the resonant system Sn1-xInxTe using the Kubo-Greenwood formalism that enabled us to make a direct connection between our experimental observations and the RLinduced resonant scattering that governs r0 and µ0 at low temperatures. The simple method of constructing r0(x) and µ0(x) plots, analogous to the Ioffe-Pisarenko plot, provides an independent tool to characterize the nature of a given impurity in semiconductors, in which scattering on impurities is the main source of the residual electrical resistivity. This reinforces the interest of probing the transport properties of doped semiconductors down to liquid helium temperatures. for Mn were taken from Ref. [34] (all concentrations except for 1%, 2.2%, 3%), Ref. [35] for 1 and 3%, and Ref. [36] for 2.2%. From Ref. [34], data for annealed samples, available for x ≤ 0.88%, were taken. Experimental Hall carrier concentrations for I were taken from Ref. [21].

In the case of Mn-doped SnTe, due to transition to the ferromagnetic state, an anomaly in the electrical resistivity is observed at low temperatures [34]. However, this anomaly only changes the electrical resistivity by few percent, so that it does not affect our conclusions. (f) 2% of GaSn. TDOS is the total DOS per formula unit, whereas dotted and dashed lines show partial DOS of impurity atoms with their dominating angular momentum contribution, calculated per atom (that is, not multiplied by its concentration). (ranging from 0.4 up to 1%), each of them acting as a double-hole acceptor (see Table 1). For Ga, a concentration of 0.5% of Sn vacancies was kept constant across the entire Ga concentration range. The data measured on single-crystalline and polycrystalline Mn-doped SnTe were taken from Refs. [34,35,36], where they were measured at temperatures below 5 K.

In panel (a) lines and symbols are the same as in panel (b). Lines are guides to the eye in both panels. : [38] (for Na concentrations of up to 0.62%), [39] (1% Na), [40,41,42] (Tl concentrations up to 1.4%), [43] (1.6% Tl). Relatively large error bars for the electrical resistivity in Pb1-xTlxTe are taken after [42], for 1.6% Tl-doped PbTe the error bar has been estimated from the point size in Ref. [43]. Enhancement of 𝜌 . (𝑥) and drop in 𝜇 . (𝑥) are similar to those observed for SnTe:In versus SnTe:Mn, SnTe:Ga, and SnTe:I, confirming the ability of the residual resistivity and residual mobility plots to distinguish between resonant and non-resonant impurities, in addition to the Ioffe-Pisarenko plot.

  Figure 2b,d the valence band structure is shown with an additional doping of 0.5% of In and I. The difference between rigid-band-like (Sn vacancies and I) and resonant (In) doping is evident. Resonant state is formed on 5s orbitals of In, which is evidenced by i) the formation of a narrow peak in the partial In DOS; ii) an increased total DOS near the valence band edge, and iii) a strong valence band smearing. All these effects are intensified when the concentration of In increases, as presented in Figure 2c.

ℏ+

  . In Figure3(d-f) and (j-l) the BSF is plotted for k1 ≈ (0.469, 0.469, 0.00) !, -(a is the lattice parameter of the cubic crystal structure of SnTe), equivalent to 1/4 of the L -S distance, that is, near the point where the valence band is crossing the Fermi level EF in panel a). Points show the calculated BSF while the line is a fit to a Lorentzian function. As shown in Figure3d,j, the Lorentzian describes very well the shape of the BSF when nonresonant impurities (Sn vacancies and I) are present, with a magnitude of the relaxation time of 𝜏 ~ 3-4´10 -13 s at k1. Slightly larger smearing is induced by Ga (Figure3h-i,k-l), but still a

Figure 4

 4 shows such comparison with the electrical resistivities r0 and hole mobilities µ0 measured at 5 K in the series of polycrystalline samples Sn1.03-xInxTe (0 ≤ x ≤ 0.02) reported as a function of the In concentration (the excess Sn concentration introduced in the nominal composition was used to partly compensate the formation of Sn vacancies, further experimental details and numerical values

  the careful control of the synthesis process, and notably of the mass-to-volume ratio in the silica ampules, ensured very similar microstructure and Sn vacancy concentrations, resulting in the observed step-like variations in the transport properties over the In concentration range covered. The possible presence of energy barriers at the grain boundaries, observed in various chalcogenide semiconductors and giving rise to activated-like behavior of the Hall mobility, can be ruled out in the series Sn1.03-xInxTe due to the conventional temperature dependences of the Hall mobility and the metallic character of the electrical resistivity (see Figure S4 in Supplementary Information). In the specific case of SnTe, the very high dielectric constant of around 1700 (Ref. 38; G. Nimtz and B. Schlicht in Narrow-Gap Semiconductors (Springer, Berlin, 1983))

Figure 1 .

 1 Figure 1. Ioffe-Pisarenko plot of SnTe showing variations in the thermopower a as a function of the hole concentration p. The black solid curve stands for the theoretical dependence calculated using a two-valence-band model.11,21,23 The experimental data points are taken from the literature for In-doped,11,[START_REF] Zhang | Proc. Natl. Acad. Sci[END_REF] , I-doped,21 Mn-doped,[START_REF] Li | Novel Thermoelectric Materials and Device Design Concepts[END_REF] Ga-doped,[START_REF] Nikitin | [END_REF] and self-doped (that is, by Sn vacancies)[START_REF] Zhang | Proc. Natl. Acad. Sci[END_REF]22,23 SnTe samples.

Figure 2 .

 2 Figure 2. Density of states (DOS) and electronic band structure around the L point of the Brillouin zone (see Figure S1 in Supplementary Information) based on Bloch spectral functions (BSF) of SnTe doped with: (a) 0.32% of Sn vacancies (VacSn); (b) 0.32% of VacSn and 0.5% of InSn; (c) 2% of InSn; (d) 0.32% of VacSn and 0.5% of ITe; (e) 0.32% of VacSn and 0.5% of GaSn;

Figure 3 .

 3 Figure 3. (a-c, g-i) Two-dimensional projections of Bloch spectral functions (BSF), plotted in logarithmic scale, for the last valence band between the L and S points of the Brillouin zone for SnTe containing (a) 0.32% of VacSn; (b) 0.32% of VacSn and 0.5% of InSn; (c) 2% of InSn; (g) 0.32% of VacSn and 0.5% of ITe; (h) 0.32% of VacSn and 0.5% of GaSn; (i) 2% of GaSn. In these plots, the black color corresponds to a BSF value greater than 300 a.u., that is, 22 eV -1 . In panels(d-f) and (j-l) the BSF is plotted for k1 ≈ (0.469, 0.469, 0.00) !, -(a is the lattice parameter of the cubic crystal structure of SnTe), equivalent to 1/4 of the L -S distance, that is, near the point where the valence band is crossing the Fermi level in panel a). Points show the calculated BSF while the line is a fit to a Lorentzian function. While the fit is very good in the non-resonant case, the Lorentzian function is no longer a good approximation of the BSF shape in e) and especially f). The electronic lifetime, estimated from the width of the fitted Lorentzian, is shown in the figures.

Figure 4 .

 4 Figure 4. (a) Residual electrical resistivity r0 and (b) residual hole mobility µ0 as a function of the dopant concentration x in SnTe. In calculations for In, I and Mn cases, hole concentrations were fixed to the experimental values by adding an appropriate concentration of Sn vacancies

Figure 5 .

 5 Figure 5. Residual resistivity 𝜌 . (𝑥) and residual mobility 𝜇 . (𝑥) plots for PbTe doped with resonant impurity Tl and non-resonant impurity Na. Experimental data were taken at temperatures below 4 K in Refs.:[38] (for Na concentrations of up to 0.62%),[39] (1% Na),

Table 1 .

 1 Experimental and calculated values of the residual resistivity 𝜌 . and residual mobility (qe is the elementary charge, p is the measured Hall carrier concentration in the case of experimental data and a nominal carrier concentration in the case of calculated values) for SnTe doped with In, Mn, Ga (on Sn site), and I (on Te site), as a function of the dopant concentration x. The experimental values of p for the In series were measured at 5 K. Data are presented in Figure4. Sn vacancy concentrations VacSn were used in the calculations to match the experimental and theoretical hole concentrations. For SnTe doped with In and Mn, for which the experimental data were available, the hole concentration p in the calculations was fixed to the experimental value (shown in the second column) by adding an appropriate number of Sn vacancies, as shown in the last column. Each VacSn acts as a double-hole acceptor, whereas In and Ga provide a single hole to SnTe, and Mn is a neutral impurity if substituted for Sn in SnTe (all the holes come from VacSn in this case). For SnTe doped with Ga, for which we are not aware of any low temperature resistivity and carrier concentration measurements in the literature, a constant concentration of 0.5% of Sn vacancies was assumed. Experimental data

	𝜇 . =	/ 0 ! 12 "

2 V -1 s -1 ) % VacSn

				In
	0.00	1.05	0.19	0.09	3400	6558	0.33
	0.05	1.41	0.33	0.26	1440	1702	0.42
	0.15	1.63	0.78	0.61	547	627	0.44
	0.25	1.76	1.12	1.06	316	335	0.43
	0.35	2.14	1.52	1.24	195	235	0.50
	0.45	2.14	1.85	1.94	160	150	0.45
	0.60	2.35		3.16		84	0.44
	0.75	2.58	3.26	4.15	83	58	0.44
	0.85	2.61		5.21		46	0.40
	1.00	2.76	3.19	5.94	73	38	0.37
	1.50	3.55		5.95		30	0.37
	2.00	4.50	4.08	5.41	32	26	0.42
	2.50	5.23		5.13		23	0.40
	3.00	6.02		4.81		22	0.40
				Mn
	0.04	1.70	0.20	0.14	1800	2585	0.41
	0.13	1.40	0.18	0.16	2510	2723	0.44
	0.22	2.10	0.25	0.21	1175	1430	0.66
	0.35	1.50	0.23	0.25	1829	1690	0.47
	0.44	1.20	0.26	0.29	1996	527	0.38
	0.88	2.80	0.36	0.45	617	527	0.88
	1.00	2.66	0.27	0.45	870	527	0.84
	1.50	2.54		0.58		422	0.80
	2.20	2.87	0.36	0.75	600	289	0.91
	3.00	3.14	0.86	0.93	231	214	0.99
				Ga
	0.00	1.58		0.12		3329	0.50
	0.50	2.38		0.15		1774	0.50
	1.00	3.17		0.15		1287	0.50
	1.50	3.96		0.15		1057	0.50
	2.00	4.75		0.16		847	0.50
	2.50	5.55		0.16		721	0.50
	3.00	6.34		0.16		608	0.50
				I
	0.50	0.82		0.25		3054	0.51
	1.00	0.60		0.45		2303	0.69
	1.50	0.35		0.82		2183	0.86
	2.00	0.32		1.00		1969	1.10

Acknowledgements B.W. and S.M. have contributed equally to this work. B.W. was partly supported by the National Science Centre (Poland), project No. 2017/26/E/ST3/00119.

Conflicts of interest

There are no conflicts to declare.