Metals at High Temperatures:

Thermoelectric Power

Thermopower or, more broadly, thermoelectric phe-nomena play an important role in basic science, as wellas in a wide variety of applications of metals. A uniquefeature of thermoelectric phenomena is that theyoriginate from an energy dependence of conductionelectron properties, such as their mobility and con-centration (this is not true for phonon drag thermo-power, which is important only at low temperaturesand will not be discussed here). The electrical con-ductivity and the electronic thermal conductivity,however, only depend on the average magnitude of theconduction electron mobility and concentration.Thermopower, therefore, provides complementaryinformation on the electronic properties and thescattering mechanisms in conductors. This article is areview of the state of experimental knowledge ofthermopower in metallic materials at high tempera-tures. It is assumed that the boundary between thehigh- and the low-temperature region is determined bythe characteristic energy of the most importanttemperature-induced scattering mechanism. In crys-talline conductors the most common temperature-induced scattering mechanism is the scattering ofconduction electrons with lattice vibrations. Thecharacteristic temperature of lattice vibrations (calledthe Debye temperature) of most metals falls in therange 100400K. This is why in practice the boundarybetween low and high temperatures is about 300K.

1. Theoretical Introduction

Thermopower, or historically the Seebeck effect, con-sists of the generation of an electric field, E

!, in the

presence of a temperature gradient, ~!T, such that E

!fl

S~!T (S is the thermopower coefficient), in a conductor

under the condition of zero net current: jfl 0. Ingeneral, the coefficient S is a tensor of second rank; inthe case of isotropic media the tensor degenerates intoa scalar (Barnard 1972). A microscopic considerationof electronic transport phenomena is usually based onthe semiclassical Boltzmann equation for the electrondistribution function f (a detailed discussion is given inBoltzmann Equation and Scattering Mechanisms). Asolution of this equation in a linear approximation fora stationary and homogeneous temperature gradientreveals the following expression for the diffusionthermopower:

Sflfi1

rerT

1

r(T )&

!

x(e, T ) (efil)E

F

fif !

e

G

H

de (1)

where electrical conductivity, r(T ), is given by

r(T )fl&

!

x(e, T )

E

F

fif !

e

G

H

de (2)

and

x(e, T )fle#

12p$q

l(k,k,T )r~e(k)r

dA (3)

with f !, l(k, k, T ), , and l being the FermiDiracdistribution function, the free path (which depends onthe wave vectors of the Bloch electrons and on thetemperature), the drift velocity, and the electrochemi-cal potential, respectively. The integration in Eqn. (3)is taken over a constant energy surface e(k)fl const. inthe wave vector k-space. Equation (1) for the thermo-power is general in the scope of the Boltzmannformalism. It can be applied irrespective of a de-generacy of the conduction electron system. Usuallyone employs a further approximation associated withthe fact that the conduction electron gas in a metal isstrongly degenerated. The function fif !}e has asharp peak at efll, thus the integrand in Eqns. (1)and (2) is nonzero in a narrow (as compared to l)energy region near efll. Expanding the functionx(e, T ) in a Taylor series in the vicinity of efll andtaking only the first nonvanishing terms of thisexpansion, the following expressions for the electricalconductivity and the thermopower can immediately bewritten:

r(T )flx(l, T ) (4)

and

Sflfip#

3

kB

rerk

BT

E

F

1

x

x

e

G

H e=l

(5)

For phonon-induced scattering processes Eqn. (4)leads to the well-known BlochGru$ neisen law, whichpredicts for the phonon-induced electrical resistivity alinear dependence of q(T ) above about the Debyetemperature (Dugdale 1977).

Equation (5) is known as Motts expression for thethermopower. According to Eqn. (5) the thermopoweris also a linear function of the temperature. This is themain and the only general theoretical result for thediffusion thermopower of metals at high temperatures.Equation (5) is valid at sufficiently low temperatures:k

BTe

c, with e

cdepending on the electronic structure

of the conductor. Within the free-electron model, ec

coincides with the Fermi energy: ecfll. Since l values

are typically about 5eV (corresponding to a tem-perature of about 610%K), the criterion for theapplicability of Motts formula is satisfied at all inpractice achievable temperatures. However, for metalswith complex electronic band structures (such astransition metals), e

ccan be considerably smaller than

the Fermi energy, and the Mott expression fails todescribe the thermopower at high temperatures. As is

1

Metals at High Temperatures: Thermoelectric Power

T (K)

S (

V K

1)

Figure 1Temperature dependencies of the thermopower of metalsof group I:V, Li; E, Na; D, K; , Rb; ^, Cs. Thedotted line denotes the thermopower according to the free-electron model. The arrows (at the end of the S(T )curves) indicate the change of the thermopower at thecorresponding melting points. The broken line in the caseof lithium is a linear extrapolation of the experimentalresults from 150K to 250K towards the melting point. Forcaesium, S has been measured at one temperature belowthe melting point (300K) and at one just above.

shown below, the experimental data frequently do notconfirm the linear relationship given by Eqn. (5). Thethermopower at high temperatures exhibits in mostinvestigated metals and alloys a complex temperaturedependence which cannot be described by the simplelinear relationship that follows from Motts formulaeven to a first approximation.

Experimentally it is found that transition metalsbelonging to the same group in the periodic table showa very similar temperature variation of the thermo-power at high temperatures. However, for metals fromdifferent groups of the periodic table qualitativelyquite different S vs. T dependencies have been found(Vedernikov 1969). This and other observations(Burkov and Vedernikov 1995) indicate that thetemperature variation of thermopower is intimatelyconnected with details of the density of states aroundthe Fermi energy. A microscopic theory able todescribe this connection and providing a realisticdescription of high-temperature thermopower has notbeen formulated. The situation is naturally morecomplicated in disordered alloys because there areserious difficulties in understanding of the electronicstructure of disordered metal systems. Therefore, there

exist no sufficiently general theoretical models able todescribe, even qualitatively, electrical resistivity andthe thermopower of disordered alloys.

For the analysis of experimental resistivity data, theso-called Matthiessen rule is frequently used. It statesthat the total electrical resistivity of a dilute alloy canbe represented in the form

q(T )fl qimp

qph

(T )

where qph

(T ) is the temperature-dependent part of theelectrical resistivity due to electronphonon scatteringand q

impis the temperature-independent impurity

contribution. Matthiessens rule is well validated forvery dilute alloys and for a weak scattering potential ofan impurity (see Boltzmann Equation and ScatteringMechanisms). A combination of the Matthiessen rulefor the resistivity and Motts expression together withthe WiedemannFranz law finally gives theNordheimGorter rule, which relates the thermo-power and resistivity of an alloy for a given tem-perature:

Sfl (Sph

fiSimp

)qph

qph

qimp

Simp

(6)

where S, Sph

, and Simp

are the total thermopower, thethermopower due to electronphonon scattering, andthe thermopower due to impurity scattering. FromEqn. (6) it follows that the dependence of the totalthermopower, S, of an alloy on q

ph}(q

phq

imp) is

linear. From the slope of the function S vs.qph

}(qph

qimp

) and the intercept on the vertical axis,one can determine the values S

phand S

imp. The

derivation of the NordheimGorter rule ( just as withthe theoretical derivation of the Matthiessen rule),however, is based on fairly rigid assumptions (see alsoBoltzmann Equation and Scattering Mechanisms),namely:

independence of the electronic structure on alloycomposition, and

independence of the electronphonon and elec-tronimpurity scattering.Both assumptions can apparently be realized in verydilute alloys.

Metallic compounds in general have a more com-plex electronic structure and consequently the scatter-ing processes of the conduction electrons cause astronger temperature dependence of the transportcoefficients. This is especially true for the thermo-power.

2. Experimental Data

The periodic table provides a natural classificationscheme of the properties of metals and their alloys.The selected thermopower data of metals and theiralloys are arranged according to their position in theperiodic table.

2

Metals at High Temperatures: Thermoelectric Power

T (K)

S (

V K

1)

Figure 2Temperature dependencies of the thermopower of metalsof group II:V, Be along the c-axis; D, Be normal to thec-axis; , Ca; , Sr; E, Ba. The arrows indicatetemperatures of polymorphic transformations for calciumand strontium. The lines are guides for the eye.

2.1 Pure Metals

The metals from the first group of the periodic table,lithium, sodium, potassium, rubidium, and caesium(francium has no stable isotopes, the transport proper-ties are not known), have comparatively simple elec-tronic band structures, and it is known that theirFermi surfaces have the shape of a slightly distortedsphere. Therefore, it is expected that their propertieswill be close to those of a free-electron gas system. Tosome extent, this expectation is confirmed by thetemperature variation of the thermopower, since athigh temperatures the thermopower is an almost linearfunction of the temperature. However, the magnitudeof the thermopower is considerably larger than thevalue expected for a free-electron system (see Fig. 1).Moreover, lithium and caesium (the latter at lowtemperatures and in the liquid state) show positive Svalues. A detailed discussion and a theoretical in-terpretation of the thermopower behavior of group 1metals can be found in Barnard (1972).

Divalent metals belonging to group 2, beryllium,magnesium, calcium, strontium, and barium (radiumhas no stable isotopes), have Fermi surfaces of morecomplex shapes than the metals of group 1. This hasan immediate impact on the temperature variation ofthe thermopower of these metals. The S vs. T curves ofall metals of group 2 take much larger values and areessentially nonlinear up to the highest temperatures

T (K)S

(V

K1

)

T-(Zr)

T-(Ti)

Figure 3Temperature dependencies of the thermopower of metalsof the titanium group: D, Ti; E, Zr; , Hf. The lines areguides for the eye. Titanium and zirconium showpolymorphic transformations indicated by the arrows.

(see Fig. 2). Beryllium and magnesium crystallize inthe hexagonal structure. S(T ) of beryllium, measuredalong the principal crystallographic directions a and c,shows a pronounced anisotropy. The S vs. T curve ofmagnesium (not included in Fig. 2) is known up toroom temperature only. From 100K to 300K S vs. Tis similar to that of strontium, the anisotropy is,however, smaller in this temperature range.

The physical properties of the transition metals aredominated by the d-electron density of states at theFermi energy. These electrons in the solid state formbands; however, these bands are much narrower thanthe bands derived from s or p atomic orbitals.Therefore, the d electrons cannot be treated as freeelectrons, even to a first approximation. Reflecting thecomplicated electronic structures, the thermopower oftransition metals reveals a variety of complex tem-perature dependencies. There is, however, one gen-erality: transition metals belonging to the same groupof the periodic table have qualitatively similar tem-perature dependencies of thermopower. Exceptions tothis rule are themetals of group III: scandium, yttrium,and lanthanum. They show rather different S(T )dependencies, probably due to the increasing im-portance the f electrons play in lanthanum metal. Thesimilarity of the S vs. T curves among the elements inthe titanium group is shown in Fig. 3. RepresentativeS(T ) data of other transition metals are given in Fig.4. The thermopower of the three ferromagnetic tran-

3

Metals at High Temperatures: Thermoelectric Power

T (K)

S (

V K

1)

Figure 4Temperature dependencies of the thermopower oftransition metals and of copper: D, Sc; E, Nb; , Mo;^, Ru; , Ir;V, Pt; , Cu. The lines are guides for theeye.

sition metals, iron, cobalt and nickel, is depicted inFig. 5. Below the Curie temperatures (indicated by thearrows) the S vs. T variation is mainly due to theevolution of the spin-polarized electronic structure asthe splitting of the spin-up and spin-down sub-bandsdecreases when approaching the Curie temperature.Above the Curie temperature the thermopower, how-ever, shows a temperature dependence similar to thatof the other metals from the corresponding group.Note thatS vs.Tof nickel above theCurie temperature(631K) is very similar to that of palladium.

Near room temperature the noble metals, copper,silver, and gold, all exhibit very similar S values to theother transition metals; however, S vs. T of silverincreases at high temperatures faster than linear,whereas S vs. T of gold shows a saturation tendency.

The polyvalent metals to the right of the noblemetals in the periodic table all have comparativelysmall thermopower, within the range 3lVK" tofi2lVK" at room temperature. Of these metals, leadis of a particular importance for thermoelectricmeasurements. Lead is used as the primary referenceto form the Absolute Thermoelectric Scale. Thethermopower of lead is determined from Thomsonheat measurements. The most precise data for leadhave been published by Roberts (1977, 1981). Theother metals that are used as the primary referencematerials (especially at high temperatures) are copperand platinum (see Roberts 1981, Roberts et al. 1985).

The lanthanide group of metals includes the 14elements from cerium to lutetium. Most of thesemetals exhibit a thermopower similar with respect tothe magnitude and the temperature dependence. Astypical examples, the thermopower of dysprosium andholmium are shown in Fig. 6 (Burkov et al. 1996). Themeasurements presented in Fig. 6 span a rather widetemperature range, also in the liquid state of bothmetals. For a review on the transport properties ofliquid rare-earth metals see Van Zytveld (1989). Thetwo divalent elements among the lanthanides, eu-ropium and ytterbium, show basically the same S(T )dependence as the trivalent lanthanides. However, themagnitude of the thermopower of europium andytterbium at maximum is about 30lVK", which is anorder of magnitude larger than that of the otherlanthanides. The thermopower of cerium is a mono-tonic function of temperature, decreasing with in-creasing temperature from 80K to 1000K, and beingabout 7lVK" at room temperature. The exper-imental data of the thermopower of rare-earth metalsat high temperatures are reviewed in Vedernikov et al.(1977).

The actinides, elements from actinium to lawren-cium, include mostly unstable elements. Thermopowerdata exist only for thorium, uranium, neptunium, andplutonium (Foiles 1985).

2.2 Binary Alloys

Because of the practically unlimited variety of metallic

T (K)

S (

V K

1)

Figure 5Temperature dependencies of the thermopower of theferromagnetic transition metals: , Fe; E, Co; D, Ni.The lines are guides for the eye. The arrows indicate theCurie temperature.

4

Metals at High Temperatures: Thermoelectric Power

T (K)

S (

V K

1)

Figure 6Temperature dependencies of the thermopower of tworare-earth metals: , Dy; E, Ho. The lines are guides forthe eye. The arrows indicate the melting temperature. Bothmetals (as well as other rare-earth metals) have apolymorphic transformation from hexagonal to the b.c.c.structure just below the melting temperature. However,this transformation is not resolved in these measurementsowing to a large temperature gradient across the samples.

alloys, generally valid tendencies can be given only forthe simplest cases, such as continuous binary solidsolutions (CBSSs). There are about 30 binary metallicsystems known with complete solid solubility. TheCBSS systems can be divided into two types, namelyalloys formed of metals from the same group of theperiodic table (isoelectronic alloys) and alloys con-sisting of metals from different groups (noniso-electronic alloys). The thermopower of isoelectronicalloys shows a comparatively weak variation with theconcentration and the S(T ) behavior is in most casesnot much different from that of the elements of whichthe alloys consist. A notable exception to this rule isthe AuAg system, where the S(T ) curves differmarkedly from those of the pure metals. An exampleof the thermopower for an isoelectronic alloy system(VNb) is shown in Figs. 7 and 8. In alloys ofnonisoelectronic metals, the thermopower variesstrongly with the alloy composition. The concen-tration dependence of the thermopower of the NbMosystem at 293K and 1700K is given in Fig. 7. Apronounced concentration dependence of the thermo-power with an extremum in the middle concentrationrange is frequently observed in these alloy systems. Incontrast to isoelectronic alloy systems, in noniso-

at.% of NbS

(V

K1

)

Figure 7Thermopower of VNb and MoNb alloys dependent onthe alloy composition (on niobium content): , VNb at293K; , VNb at 1000K; E, MoNb at 293K; D,MoNb at 1700K. The lines are guides for the eye.

electronic systems remarkable changes in the S(T )curves occur for different concentrations. Usuallythese changes in S(T ) are not continuous. In mostnonisoelectronic alloys, the S vs. T curves preservefeatures characteristic of the corresponding puremetals within a broad region of composition on theside of each alloy component. Qualitative changes inthe character of the temperature dependence occur ina comparatively narrow region of alloy composition.A classical example of these types of alloys is theAgPd alloy system. Transport properties of thesealloys have been the subject of comprehensive studies.A summary of the results and interpretations is givenin Dugdale (1977). As an example, the temperaturedependence of the thermopower above room tem-perature for selected concentrations of the NbMosystem is given in Fig. 9. A detailed discussion ofexperimental data of the resistivity and the thermo-power of CBSSs is given in Burkov and Vedernikov(1995). For the thermopower of further alloys seeFoiles (1985).

2.3 Compounds

Metallic compounds, or ordered alloys, possess analmost unlimited variety of properties. There are heavyfermion compounds, magnetic and nearly magneticcompounds, intermediate valence compounds, high-temperature superconductors, and others. Among

5

Metals at High Temperatures: Thermoelectric Power

T (K)

S (

V K

1)

Figure 8Temperature dependencies of the thermopower of VNballoys: D, V; *, 28% Nb; , 58% Nb; E, 100% Nb.The composition is given in at.%. The lines are guides forthe eye.

T (K)

S (

V K

1)

Figure 9Temperature dependencies of the thermopower of NbMoalloys: E, Nb; , 36% Mo; *, 64% Mo; , 84% Mo;D, 100% Mo. The composition is given in at.%. The linesare guides for the eye.

them, the thermopower of heavy fermion compoundshas probably been studied most extensively (Brandtand Moshchalkov 1984, Bauer 1991).

These studies, however, as a rule have been perf-ormed only at low temperatures (see also Kondo Sys-tems and Heay Fermions: Transport Phenomena andRare Earth Intermetallics: Thermopower of Cerium,Samarium, and Europium Compounds). Thermo-power of a family of nearly magnetic and magneticcompounds, RCo

#(where R designates a rare-earth

element), has been investigated in detail up to 1000K(Gratz et al. 1995).

Bibliography

Barnard R D 1972 Thermoelectricity in Metals and Alloys.Taylor and Francis, London

Bauer E 1991 Anomalous properties of CeCu and YbCu-based compounds. Ad. Phys. 40, 417534

Blatt F J, Shroeder P A,Foiles C L,GreigD1976 ThermoelectricPower of Metals. Plenum, New York

Brandt N V, Moshchalkov V V 1984 Concentrated Kondosystems. Ad. Phys. 33, 373467

Burkov A T, Kolgunov D A, Hoag K, Van Zytveld J 1996Thermopower and electrical resistivity of liquid and crystallineDy and Ho at temperatures 3002000K. J. Non-Cryst. Solids205/207, 3327

Burkov A T,VedernikovM V 1995 Electrical and thermoelectricproperties of metallic binary continuous solid solutions. In:Srivastava S K, March N H (eds.) Condensed Matter andDisordered Solids. World Scientific, Singapore, pp. 361424

Dugdale J S 1977 The Electrical Properties of Metals and Alloys.Edward Arnold, London

Foiles C L 1985 Thermopower of pure metals; and Thermo-power of dilute alloys. In: Landolt-BoX rnstein. Numerical Dataand Functional Relationships in Science and Technology. NewSeries. Group III, Vol. 15, Metals. Springer-Verlag, NewYork, pp. 48104; 123209

Gratz E, Resel R, Burkov A T, Bauer E, Markosian A S,Galatanu A 1995 The transport properties of RCo

#com-

pounds. J. Phys.: Condens. Matter 7, 6687705Roberts R B 1977 The absolute scale of thermoelectricity. Phil.

Mag. 36, 91107Roberts R B 1981 The absolute scale of thermoelectricity II.

Phil. Mag. 43, 112535Roberts R B, Righini F, Compton R C 1985 The absolute scale

of thermoelectricity III. Phil. Mag. 52, 114763Van Zytveld J B 1989 Liquid metals and alloys. In: Gschneidner

K A, Eyring L (eds.) Handbook on Physics and Chemistry ofRare Earths. North-Holland, Amsterdam, Vol. 12, pp.357407

Vedernikov M V 1969 The thermoelectric powers of transitionmetals at high temperatures. Ad. Phys. 18, 33770

Vedernikov M V, Burkov A T, Dvunitkin V G, Moreva N I1977 The thermoelectric power, electrical resistivity, and Hallconstant of rare earth metals in the temperature range801000K. J. Less-Common Met. 52, 22145

A. T. Burkov

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Metals at High Temperatures: Thermoelectric Power

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Encyclopedia of Materials : Science and TechnologyISBN: 0-08-0431526

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