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A Note on the Electrochemical Nature of the Thermoelectric Power

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A Note on the Electrochemical Nature of the Thermoelectric Power EPJ manuscript No. (will be inserted by the editor) A note on the electrochemical nature of the thermoelectric power Y. Apertet1a, H. Ouerdane23, C. Goupil4, and Ph. Lecoeur5 1 Lyc´ee Jacques Pr´evert, F-27500 Pont-Audemer, France 2 Russian Quantum Center, 100 Novaya Street, Skolkovo, Moscow region 143025, Russian Federation 3 UFR Langues Vivantes Etrang`eres, Universit´ede Caen Normandie, Esplanade de la Paix 14032 Caen, France 4 Laboratoire Interdisciplinaire des Energies de Demain (LIED), UMR 8236 Universit´eParis Diderot, CNRS, 5 Rue Thomas Mann, 75013 Paris, France 5 Institut d’Electronique Fondamentale, Universit´eParis-Sud, CNRS, UMR 8622, F-91405 Orsay, France Received: date / Revised version: date Abstract. While thermoelectric transport theory is well established and widely applied, it is not always clear in the literature whether the Seebeck coefficient, which is a measure of the strength of the mutual interaction between electric charge transport and heat transport, is to be related to the gradient of the system’s chemical potential or to the gradient of its electrochemical potential. The present article aims to clarify the thermodynamic definition of the thermoelectric coupling. First, we recall how the Seebeck coefficient is experimentally determined. We then turn to the analysis of the relationship between the thermoelectric power and the relevant potentials in the thermoelectric system: As the definitions of the chemical and electrochemical potentials are clarified, we show that, with a proper consideration of each potential, one may derive the Seebeck coefficient of a non-degenerate semiconductor without the need to introduce a contact potential as seen sometimes in the literature. Furthermore, we demonstrate that the phenomenological expression of the electrical current resulting from thermoelectric effects may be directly obtained from the drift-diffusion equation. PACS. 84.60.Rb Thermoelectric, electrogasdynamic and other direct energy conversion – 72.20.Pa Ther- moelectric and thermomagnetic effects 1 Introduction similar materials as the junctions between these materials were maintained at different temperatures. While Seebeck Thermoelectricity is a mature yet still very active area of interpreted the observed phenomenon as a thermomag- research covering various fields of physics, physical chem- netic effect, Oersted soon reexamined Seebeck’s work and istry, and engineering. The large interest in thermoelec- showed that in this case the magnetic field was an indi- tric systems is mostly due to the promising applications rect effect as it originated in the presence of an electro- in the field of electrical power production from waste heat motive force induced by the temperature difference [4]. as thermoelectric devices may be designed for specific pur- The proportionality coefficient between this electromotive poses involving powers over a range spanning ten orders force and the temperature difference across the system is of magnitude: typically from microwatts to several kilo- the thermoelectric power, which has also been coined as watts. Further, thermoelectricity also provides model sys- “Seebeck coefficient”. arXiv:1502.05697v2 [cond-mat.mtrl-sci] 2 Apr 2016 tems that are extremely useful in the development of the- The definition of the thermoelectric coupling has later ories in irreversible thermodynamics [1,2]. been extended from that derived from the first experi- The discovery of the thermoelectric effect is usually ments to both thermodynamic [5] and microscopic [6,7,8, attributed to Seebeck. In 1821, he published the results 9] properties of materials. However, as of yet, there still and analysis of his experiments aiming at establishing a is no clear consensus on its relationship with the vari- magnetic polarization in a metallic circuit simply by per- ous thermodynamic potentials and their variations (see, turbing the thermal equilibrium across this latter [3]. More e.g., refs. [10,11,12,13,14,15,16]). Indeed as the terminol- precisely, Seebeck described the appearance of a magnetic ogy and conventions may vary from a discipline to an- field within a closed electrical circuit made of two dis- other, say, e.g., solid-state physics and electrochemistry, it is not always straightforward to establish a clear distinc- Send offprint requests to: Y. Apertet tion or relevant associations between Fermi energy at zero a Email address: [email protected] or finite temperature, electrochemical potential, voltage, 2 Y. Apertet et al.: A note on the electrochemical nature of the thermoelectric power The coefficient αAB, obtained for the whole circuit, is re- lated to the Seebeck coefficient of each material through [17]: αAB = αB αA, (2) − where αA and αB are the Seebeck coefficients of the ma- terials A and B respectively. From an experimental viewpoint, the presence of the material B (= A) is mandatory as it is associated with the probe’s wires6 (see, e.g., ref. [18]). However, if its See- beck coefficient αB is sufficiently small to be neglected, the measurement may be used to determine directly the Seebeck coefficient of material A. In this case, one gets: V2 V1 αA = − . (3) −T2 T1 − Note the presence of a minus sign in the expression above: It is often overlooked in the literature but, fortunately, Fig. 1. Determination of the Seebeck coefficient for a circuit that omission is most of the time compensated by the ab- composed of two dissimilar materials. sence of a clear sign convention for the measured voltage. Let us now turn to the analysis of the measured quanti- ties. While the temperature is not subject to questioning, Fermi level relative either to the conduction band mini- the voltage obtained from a voltmeter must be defined mum or to the vacuum, and chemical potential. unambiguously. Indeed, it appears that its connection to In this article, we discuss the definition of the See- the microscopic and thermodynamic properties of mate- beck coefficient focusing particularly on the distinction rials has remained unclear for quite some time, leading between chemical and electrochemical potentials. First, in Riess to publish in 1997, hence fairly recently, an article sect. 2, we address the experimental determination of the untitled “What does a voltmeter measure ?” [19]. In that Seebeck coefficient in order to identify the quantities of in- paper, Riess demonstrated that the voltage measured by terest. Next, the purpose of sect. 3 is to demonstrate that a a voltmeter between two points in a circuit is the differ- clear physical picture of thermoelectric phenomena at the ence of electrochemical potentials µ at the two considered microscopic scale may be obtained on the condition that points divided by the elementary electric charge e, but not the potentials are carefully introduced. For this purpose, the difference between the electrostatice potentials ϕ alone. we review the standard definitions given in the literature The potential V might thus be defined as V = µ/e. This to remove any confusion between the chemical and elec- result is recovered when one measures the voltage− at the trochemical potentials before we present and discuss our ends of a pn junction at equilibrium: While there ise a built- derivation of the Seebeck coefficient for a non-degenerate in electric field associated with the depletion layer, the semiconductor. measured voltage remains zero. The Seebeck coefficient thus appears as a link between the applied temperature difference and the resulting difference of electrochemical 2 Experimental determination of the potential between the two junctions. thermoelectric power The simple technique presented here is not the only one used to determine the thermoelectric power of a given material. Indeed, since the measurement always involves The determination of the Seebeck coefficient traditionally a couple of materials, the absolute Seebeck coefficient of involves components made of dissimilar materials, which the second material has to be known accurately. To obtain we label A and B respectively. The two materials are com- this value, it is possible to use low temperature measure- bined to obtain two junctions as depicted in fig. 1. These ment to reach superconducting state where α = 0 and then junctions are then brought to different temperatures T1 derive higher temperatures values using the Thomson co- and T2. An isothermal voltage measurement at a tem- efficient that can be measured for a single material. For a perature T3, is performed between the free ends of the detailed presentation of the Seebeck coefficient metrology, component B. The voltage thus measured is V2 V1 (this the reader may refer to the instructive review by Martin − notation allows to clearly define a direction for the volt- et al. [20]. age) and the Seebeck coefficient αAB associated with the global system, i.e. the couple AB, is defined as the pro- portionality coefficient between the resulting voltage and 3 Relationship between the thermoelectric the applied temperature difference: power and the electrochemical potential V2 V1 αAB = − . (1) In order to better understand the influence of each po- T2 T1 tential, we identify the respective effects of temperature − Y. Apertet et al.: A note on the electrochemical nature of the thermoelectric power 3 bias, concentration difference, and electric charge, and we discuss the relationship between chemical potential, elec- trochemical potential and the band diagram of materials. We then derive the Seebeck coefficient in the simple case of a non-degenerate semiconductor to illustrate the con- tribution of each potential. 3.1 Definition of the thermopower The Seebeck coefficient may be obtained from a micro- scopic analysis of the considered materials, with the local version of Eq. (3), in open-circuit condition, i.e., with a vanishing electrical current: µ α = ∇ , (4) e T ∇e where µ and T are respectively the local electrochemical Fig. 2. Energy levels in an n-type semiconductor highlighting potential and temperature, defined at each point of the the notations used in this article (adapted from ref. [8]). The energy EG refers to the bandgap energy. system.e The notation is associated with the gradient of each quantity.
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