A Note on the Electrochemical Nature of the Thermoelectric Power

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Apertet et al.: A note on the electrochemical nature of the thermoelectric power

The coefficient αAB, obtained for the whole circuit, is related to the Seebeck coefficient of each material through [17]: αAB = αB αA, (2) − where αA and αB are the Seebeck coefficients of the materials 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 quantities. 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 proportionality 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, electrochemical 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 contribution of each potential.

3.1 Deﬁnition of the thermopower

The Seebeck coeﬃcient may be obtained from a microscopic 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. In the following,∇ for the sake of simplicity, we consider a unidimensional system so that the spatial gradient reduces to its x-component: . ∇x µ = µ + µe. (5) Note that the quantitiese we just referred to as potentials 3.2 Distinction between the potentials are actually energies. The electrical contribution µe may be expressed as a function of the electrostatic potential ϕ (a genuine potential contrary to µ and µ) so that the Consider a semiconductor sample at thermal equilibrium electrochemical potential reads: and characterized by a spatially inhomogeneous doping. e As the carrier concentration is nonuniform, a particle cur- µ = µ + qϕ, (6) rent takes place from the region of higher concentration to that of lower concentration: This is the diffusion process where q is the electricale charge of the considered particle. associated with the variation of the carriers’ chemical po- When used in solid state physics, these quantities have tential across the system. This type of electrical current to be related to an energy band diagram. This correspon- is referred to as the diffusion current. The inhomogeneous dence may be found for example in the book of Heikes and electron population in the system thus generates an elec- Ure [8]: Considering the example of an n-doped semicon- tric potential difference and hence a built-in electric field ductor, the electrochemical potential µ corresponds to the which influences the electrons’ motion in such a fashion Fermi level, the electrostatic energy eϕ corresponds to that it tends to curb the diffusion current. The electron the energy level of the bottom of thee− conduction band motion driven by the built-in electric field is the drift cur- while the chemical potential µ corresponds to the dif- rent, which, at thermal equilibrium, exactly cancels the ference between these two quantities and is often called diffusion current, in accordance with the principle of Le Fermi energy These notations are summarized on fig. (2). Chatelier and Braun. In this case, the measured voltage The difference between Fermi level (µ) and Fermi energy across the system always remains zero and there is no (µ) was already highlighted by Wood [9]: “The differ- net electrical current even if the system is short-circuited: ence between the Fermi energy and thee Fermi level should The electric field associated with the electrical potential be noted. The Fermi energy is generally measured from variation is obviously not an electromotive field. However, the adjacent conducting band edge (valence or conduction if the electrons are placed in a non-equilibrium situation band for holes or electrons, respectively), i.e. a reference caused by a thermal bias applied across the system, a non level which may vary in energy, whereas the Fermi level is vanishing electric current may be obtained when the cir- measured from some arbitrary fixed energy level”. This last cuit is closed. This current obviously stems from the un- remark stresses the importance of the choice of an energy compensated contributions of both the diffusion and drift reference, which is a key parameter: To express energies of charge carriers, and it is traditionally related to the in a semiconductor, the bottom of the conduction band gradient of the temperature and to the gradient of the is often used as the reference [10]; however, for studies electrochemical potential. of non-equilibrium phenomena such as thermoelectricity, The electrochemical potential µ of a population of elec- it is mandatory to define an arbitrary fixed energy refer- trically charged particles is the sum of a chemical con- ence independent of the position within the material since tribution µ, the chemical potential,e and of an electrical both µ and µ may vary along the system. It seems the contribution µe [17]: only way to correctly describe the relative displacement e 4 Y. Apertet et al.: A note on the electrochemical nature of the thermoelectric power

µ EC µ n(T )= N exp − = N exp , (7) kBT kBT e with

3/2 2πmeﬀ kBT N =2 , (8) h2

and where EC is the energy level of the bottom of the conduction band, meff is the electron effective mass, kB is the Boltzmann constant and h is the Planck constant. The Seebeck coefficient is associated with non-equilibrium phenomena, and, as such, it is tightly linked to transport properties of electrons inside the material. To take account Fig. 3. Schematic illustration (adapted from Ref. [21]) of the of these properties, we build on the drift-diffusion equation variations of the bottom of the conduction band, EC , the top used to obtain the net electrical current density J : of the valence band, EV , the Fermi level, EF , and the vac- x uum level just outside the material, ǫS , all along the circuit J = enM + eD n, (9) depicted in Fig. 1. The slopes of the lines have been greatly x nEx n∇x exaggerated for clarity, and band bending at the interfaces has where Mn and Dn are the electron mobility and diffusivity, been neglected. and where the electric field is related to the energy level Ex EC through: of these energies. Note that the vacuum level infinitely far xEC xEC x = ∇ = ∇ . (10) from the system, E∞, might be a good and meaningful E − q e energy reference. At first, we assume a situation where the electron diffusiv- Figure 3 illustrates the variations of the different en- ity Dn does not depend on the other parameters, including ergies around the circuit depicted in Fig. 1 in the case of the position. The variation of Dn will be discussed further semiconductor materials. It highlights the difference be- below. tween the slope of the bottom of the conduction band and The Seebeck coefficient is obtained setting Jx = 0. the slope of the Fermi level: The variation of the chemical However this current density should be related first to potential thus differs from the variation of the electro- xT and xµ rather than to x and xn. To do so, we chemical potential. Distinguishing these two energies is, evaluate∇ the∇ gradient of the electronE density∇ given by eq. therefore, crucial to properly evaluate the Seebeck coeffi- (7) consideringe that EC, µ and T may vary along the mate- cient. Figure 3 also displays the vacuum level ǫS just out- rial. This approach is seldom found in the literature as one side the material (different from E∞). This vacuum level often sets EC = 0, thus consideringe the bottom of the con- is related to the bottom of the conduction band through duction as the reference everywhere in the nonequilibrium the affinity χ of the material. The discontinuities in ǫS system. As already stressed, this viewpoint is misleading at the interfaces might be seen as contact potentials. On for thermoelectric phenomena. From eq. (7), the gradient the contrary, the Fermi level EF is continuous along the of electron density reads: system, even at the interfaces. Its variation however un- dergoes a sudden change at the interface, reflecting both changes in temperature gradient (assumed constant in a 3 n xT n xn = ∇ + 2 [T ( xµ e x) µ xT ] . (11) given material) and in Seebeck coefficient from a material ∇ 2 T kBT ∇ − E − ∇ to an other. The thermopower is indeed associated with e bulk material but not to interfaces. A similar figure for a We then use this equality along with Einstein’s relation system made of metals can be found in Ref. [21]. between the electron mobility Mn to the electron diffusivity Dn: M e n = (12) 3.3 From potentials to thermoelectric power: the Dn kBT illustrative case of a non-degenerate semiconductor to modify eq. (9) as follows:

We emphasise the importance of the distinction between xµ kB 3 µ Jx = enMn ∇ + xT . (13) µ and µ on the derivation of the thermoelectric power us- e e 2 − kBT ∇ ing the example of a non-degenerate semiconductor doped e withe electrons. In this case, the expression of the carrier Now, setting Jx = 0 and using the deﬁnition given in concentration n is rather simple: eq. (4), we ﬁnd: Y. Apertet et al.: A note on the electrochemical nature of the thermoelectric power 5

Scattering mechanism Exponent s in the conduction band and may thus be approximated by Acoustic phonon -1/2 its average value, i.e., 3/2kBT . Replacing τ in eq. (17), we Ionized impurity (strongly screened) -1/2 obtain: Neutral impurity 0 Piezoelectric +1/2 D (D )= n (1 + s) (T ) . (18) Ionized impurity (weakly screened) +3/2 ∇x n T ∇x Table 1. Values of the exponent s for diﬀerent scattering Finally, inserting eq. (18) and eq. (11) in eq. (16) yields: mechanisms (adapted from Ref. [22]).

xµ kB 5 µ Jx = enMn ∇ + + s xT , (19) e e 2 − kBT ∇ kB 3 µ e α = , (14) − e 2 − kBT and consequently: with a constant electron diffusivity, which is the expected kB 5 µ α = + s . (20) expression for a non-degenerate semiconductor. Further, − e 2 − kBT this result may also be interpreted by looking at the net thermal energy transported by each carrier transported The contribution of the diffusivity gradient to the thermoelectric power is (1 + s)kB/e and hence depends only on inside the material, i.e., qΠ, where Π is the Peltier coef- − ficient [17]. For electrons, this energy is the scattering parameter s. This term has also been recovered by Cai and Mahan [12] using a Boltzmann equation. 3 Note that this term was also introduced by Ioffe [23] with eΠ = kBT µ, (15) − 2 − the notation αD. However, Ioffe used a different power law: He assumed that the carrier’s mean free path l is r since it corresponds to the energy above the Fermi level proportional to (E EC) . He consequently found that µ and hence to the sum of the average thermal energy for − αD = (1/2+ r)kB/e. This discrepancy is quite easy to free electrons and of the energy between the Fermi level − understand since τ is proportional to l/√E EC. ande the bottom of the conduction band, i.e., µ. We thus − recover the second Kelvin relation relating− the Seebeck and Peltier coefficients: Π = αT . 4 Discussion

3.4 Taking into account diffusivity variation 4.1 An unusual derivation While the result given in Eq. (14) is well-known, its deriva- If we relax the assumption of constant diffusivity Dn, this latter becomes a function of the spatial coordinate x and tion presented here is quite original. Indeed, it was di- we end up with the so-called Stratton equation [22]: rectly obtained from the drift-diffusion equation. Thus, the phenomenological equation associated with thermoelectric transport: Jx = enMn x + e x (Dnn) E ∇ xµ = enM + eD (n)+ en (D ) , (16) Jx = σ ∇ σα xT, (21) nEx n∇x ∇x n e − ∇ e It corresponds to a more general form of the drift-diffusion where σ = enMn is the electrical conductivity, is identi- equation, which contains a third contribution to the car- cal to eq. (19) (or to eq. (13) depending on the hypothesis rier motion, directly linked to the gradient of diffusivity made). This latter appears as a modified form of the drift- along the system. To evaluate its effect on the thermo- diffusion equation, which accounts for the couple of vari- electric power, we may reexpress it as a function of the ables [µ, T], or more precisely their gradients, rather than temperature gradient using the relation between the dif- the traditional couple [n, ϕ]. This modification puts forth fusivity Dn and the relaxation time of the carriers τ. Since the facte that the first term of the right hand side of eq. Mn = eτ/meff , the Einstein relation reads: (21) does not correspond any longer to the genuine local τ form of Ohm’s law since it does not involve the electrical Dn = kBT . (17) field x. In this case, the true electromotive force is given meff by theE gradient of the electrochemical potential as carriers To keep the calculations on an analytical level, we assume experience both diffusion and effects of the electric field. that we deal with low-energy conduction electrons, and The simple derivation of Eq. (19) has been allowed by we express the relaxation time using a power law of the the use as a reference of a fixed energy level, arbitrarily form: τ (E EC)s, where E is the total energy of the chosen but independent of the position along the material, carrier and∝ s −is a characteristic exponent depending on rather than the bottom of the conduction band. This ap- the scattering mechanisms [22]. Some typical values for proach demonstrates that a particular knowledge of both this exponent are given in Table 1. Note that the energy n and is not mandatory to determine the Seebeck ∇x Ex E EC corresponds to the thermal energy of the carriers coefficient of a non-degenerate semiconductor. Indeed, in − 6 Y. Apertet et al.: A note on the electrochemical nature of the thermoelectric power this case, these two contributions to the electrochemical i.e., to eq. (14). This latter term is identical to the so-called gradient seem to always compensate each other in such Kelvin formula of the thermopower [13]. As discussed by a way that the resulting electromotive power is indepen- Shastry [26], this contribution “captures the many body dent of specific assumptions, for example a constraint on density of states enhancements, while missing velocity and the carrier concentration. Equation (19) is thus valid for a relaxation contributions”. It thus justifies the introduction wide range of temperatures: It is correct for the extrinsic of the coefficient S to take into account dynamical effects. regime, i.e., when the carrier concentration is fixed by the We believe however that this coefficient should not be pre- concentration of impurities, but it also remains valid in the sented as effective since it does not reflect the appearance freeze-out regime and in the intrinsic regime where addi- of the electromotive force due to the temperature gradi- tional carriers are thermally generated. However, in this ent. It represents only one of the possible contributions latter regime, the electron hole contribution to thermo- to this electromotive force. From a practical viewpoint, it electric power should also be considered as these minority has recently been demonstrated that the contribution to carriers may no longer be negligible. One may also refer the thermopower from the diffusivity gradient might be to Ref. [24] in which the authors derive the Seebeck co- significant [27]. efficient focusing only on potentials and the electric field rather than using a statistical approach. 4.4 On the contact potentials

4.2 Link with non-equilibrium thermodynamics Finally, we want to point out the inappropriate use of the contact potentials in the derivation of the thermoelectric While Eq. (21) is widely used in solid-state physics, its power sometimes found in the literature. For example, in formulation is slightly different in non-equilibrium ther- Ref. [23], Ioffe obtains Eq. (19) splitting the Seebeck co- modynamics as general forces are traditionally computed efficient into three separate terms, one being αD while 1 from the gradients µ and (1/T ), instead of µ the two others, αn and αϕ, are associated respectively T ∇x ∇x ∇x and xT [17,25]. So, one should then rewrite Eq. (21) to to concentration gradient and to the “temperature depen- get: ∇ e e dence of the contact potential”. However, as demonstrated later by Chambers [21], contact potentials are irrelevant σT 1 2 1 to thermoelectric effects. This latter term is indeed intro- Jx = xµ + σαT x . (22) e · T ∇ · ∇ T duced only to compensate the erroneous expression of αn e stemming from the confusion between ϕ and µ/e. From an With this form, it is possible to identify each term with experimental viewpoint, contact potentials are irrelevant the canonical expression [17], since they cannot be probed by a voltmeter: Ase depicted in Fig. 3, these energy discontinuities concern only the bottom of the conduction band E (or identically the vacuum Jx 1 1 C JN = = L11 xµ + L12 x , (23) level just outside the material ǫ ) but not the Fermi level − e · T ∇ · ∇ T S EF . e to recover the expressions of the kinetic coefficients in the 2 2 thermoelectric case, i.e., L11 = σT/e and L12 = σαT /e. 5 Conclusion

4.3 On the so-called effective Seebeck coefficient In this article, we have discussed the definition of the thermoelectric power with a special emphasis on its re- Let us now turn to the previous analysis of the thermo- lationship to the electrochemical potential. A proper con- electric power in non-degenerate semiconductor. In ref. sideration of all potentials inside the material has led to [11], Mahan introduces an effective Seebeck coefficient S, demonstrate that the phenomenological equation for the distinct from the genuine thermoelectric power eq. (3) ob- electrical current involving thermoelectric coefficients may tained from measurements. In a subsequent article with be derived directly from the drift-diffusion equation. We Cai [12], this effective coefficient is presented as the ra- also shed light on the physical interpretation of the effec- tio between the electric field and the temperature gradi- tive Seebeck coefficient defined by Mahan, showing that it ent. These two different Seebeck coefficients are related is actually related to the gradient of diffusivity along the through the following relation [12]: system.

1 ∂µ α = S + , (24) e ∂T n References A comparison of eq. (16) with the equation (20) of ref. 1. S. R. de Groot Thermodynamics of Irreversible Processes [11] leads to identify the effective Seebeck coefficient to (Interscience Publishers Inc., New York, 1958). the contribution of the diffusivity gradient, i.e., αD. The 2. Y. Apertet, H. Ouerdane, C. Goupil, and Ph. Lecoeur, second term of the right hand side of eq. (24) should then Phys. Rev. E 85, 031116 (2012). be associated with the assumption of constant diffusivity, 3. T. J. Seebeck, Abh. K. Akad. Wiss. Berlin 289 (1821). Y. Apertet et al.: A note on the electrochemical nature of the thermoelectric power 7

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