Thermoelectricity

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Encyclopedia of Physical Science and Technology

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Thermoelectricity Timothy P. Hogan Michigan State University

I. II. III. IV. V. VI.

Introduction and Basic Thermoelectric Effects Thermodynamic Relationships Thermodynamics of an Irreversible Process Statistical Relationships Applications Summary

GLOSSARY Boltzmann equation An equation based on the Fermi distribution equation under nonequilibrium conditions. The Boltzmann equation describes the rate of change of the distribution function due to forces, concentration gradients, and carrier scattering. Fermi distribution A function describing the probability of occupancy of a given energy state for a system of particles based on the Pauli exclusion principle. Fermi level The energy level which exhibits a 50% probability of being occupied. Joule heating Heating due to I 2 R losses. Onsager relations A set of simultaneous equations that describe the macroscopic interactions between “forces” and “flows” within a thermoelectric system. Peltier effect Absorption or evolution of thermal energy at a junction between dissimilar materials through which current flows. Seebeck effect Open-circuit voltage generated by a circuit consisting of at least two dissimilar conductors when a temperature gradient exists within the

circuit between the measuring and the reference junctions. Thermocouple A pair of dissimilar conductors joined at one set of ends to form a measuring junction. Thermoelectric cooler A heat pump designed from thermoelectric materials typically configured in an array as a series of thermocouples with the junction exposed. Thermopower This is defined here as the absolute Seebeck coefficient and corresponds to the rate of change of the thermoelectric voltage with respect to the temperature of a single conductor with a temperature gradient between the ends. Thompson effect The absorption or evolution of thermal energy from a single homogeneous conductor through which electric current flows in the presence of a temperature gradient along the conductor.

THE FIELD of thermoelectricity involves the study of characteristics resulting from electrical phenomena occurring in conjunction with a flow of heat. It includes flows of electrical current and thermal current and the interactions between them.

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I. INTRODUCTION AND BASIC THERMOELECTRIC EFFECTS In 1822, Seebeck reported on “the magnetic polarization of metals and ores produced by a temperature difference” (Joffe, 1957). By placing two conductors in the configuration shown in Fig. 1, Seebeck observed a deflection of the magnetic needle in his measurement apparatus (Gray, 1960). The deflection was dependent on the temperature difference between junctions and the materials used for the conductors. Shortly after this, Oersted discovered the interaction between an electric current and a magnetic needle. Many scientists subsequently researched the relationship between electric currents and magnetic fields including Amp`ere, Biot, Savart, Laplace, and others. It was then suggested that the observation by Seebeck was not caused by a magnetic polarization, but due to a thermoelectric current flowing in the closed-loop circuit. Seebeck did not accept this explanation, and in an attempt to refute it, he reported measurements on a number of solid and liquid metals, alloys, minerals, and semiconductors. The magnetic polarization hypothesis was incorrect as can be seen in the open-circuit configuration of his experiment. Experimentally, a voltage (V ) at the open-circuit terminals is measured when a temperature gradient exists between junctions such that  T2 V = SAB dT , (1) T1

where SAB is the Seebeck coefficient for the two conductors, which is defined as being positive when a positive voltage is measured for T1 < T2 . The voltage is measured across terminals maintained at a constant temperature T0 . For this voltage to appear in the open-circuit configuration (Fig. 2), there must exist a current which flows in the closed-circuit configuration. Furthermore, in the opencircuit configuration, Seebeck would no longer observe a deflection of the magnetic needle, which is not expected if a magnetic polarization is taking effect. The diligence of his measurements was vertified by the confirmation of his values years later by Justi and Meisner

FIGURE 2 The open-circuit Seebeck effect.

as well as by Telkes, who showed, 125 years after Seebeck’s measurements, that the best couple for energy conversion was formed using ZnSb and PbS, which were two materials examined by Seebeck. Twelve years after Seebeck’s discovery, a scientist and watchmaker named Jean Peltier reported a temperature anomaly at the junction of two dissimilar materials as a current was passed through the junction. It was unclear what caused this anomaly, and while Peltier attempted to explain it on the basis of the conductivities and/or hardness of the two materials, Lenz removed all doubt in 1838 with one simple experiment. By placing a droplet of water in a dimple at the junction between rods of bismuth and antimony, Lenz was able to freeze the water and subsequently melt the ice by changing the direction of current through the junction. In a way, Lenz had made the first thermoelectric cooler. The rate of heat ( ) absorbed or liberated from the junction was later found to be proportional to the current, or =  · I,

where the proportionality constant () was named the Peltier coefficient. Near this time, the field of electromagnetics was being formed and captured much attention in the scientific community. Therefore, another 16 years passed before Thomson (later called Lord Kelvin) reasoned that if the current through the two junctions in Fig. 1 produced only Peltier heating, then the Peltier voltage must equal the Seebeck voltage and both must be linearly proportional to the temperature. Since this was not observed experimentally, he reasoned that there must be a third reversible process occurring. This third process is the evolution or absorption of heat whenever current is passed through a single homogeneous conductor along which a temperature gradient exists, or in equation form, = I

FIGURE 1 Closed-circuit Seebeck effect.

(2)

dT , dx

(3)

is the rate of heat absorbed or liberated along the where conductor, is the Thomson coefficient, I is the current through the conductor and dT /d x is the temperature gradient maintained along the length of the conductor.

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Thomson then applied the first and second laws of thermodynamics to the Seebeck, Peltier, and Thomson effects to find the Kelvin relationships  = SAB T,

(4)

d SAB A− B = , (5) dT T where the subscripts A and B correspond to the two materials in Fig. 2. The second Kelvin relation suggests that the Seebeck coefficient for two materials forming a junction can be represented as the difference between quantities based on the properties of the individual materials making up the junction. Integration of the second Kelvin relation gives     A− B A B d SAB = dT = dT − dT T T T (6) or

 SAB =

A

T

 dT −

B

T

dT.

microscopic analyses can be used in deriving many useful formulas for calculating thermoelectric properties of various materials. The following sections are dedicated to developing the macroscopic and microscopic analyses.

II. THERMODYNAMIC RELATIONSHIPS As shown by the Seebeck effect, when a temperature gradient is placed over the length of a sample, carrier flow will be predominantly from the hot side to the cold side. This indicates that a temperature gradient, T , is a force that can cause a flow of carriers. It is well known that applying the force of an electric potential gradient, V , can also induce carrier flow. In 1931, Onsager developed a method of relating the flows of matter or energy within a system to the forces present. In this method the forces are assumed to be sufficiently small so that a linear relationship between the forces, Xi , and the corresponding flows, Ji , can be written. J1 = L 11 X1 + L 12 X2 + · · · L 1n Xn ,

(7)

Defining the first term on the right-hand side as the “absolute” Seebeck coefficient of material A and the second term as the “absolute” Seebeck coefficient of material B, we find that the Seebeck coefficient for a junction is equal to the difference in “absolute” Seebeck coefficients of the individual materials making the junction. This is a very significant result, as measurements of the individual materials can be used to predict how junctions formed from various combinations of materials will behave, thus removing the need to measure every possible combination of materials. The “absolute” Seebeck coefficient or thermoelectric power of a material, hereafter referred to simply as the thermopower of the material, can be found for material A if the thermopower of material B is known or if the thermopower of material B is zero. A material in the superconducting state has a thermopower of zero, and once a material is calibrated against a superconductor, it can then be used as a reference material to measure more materials. This has been done for several pure materials such as lead, gold, and silver (Roberts, 1977; Wendling et al., 1993). Further understanding of the basic thermoelectric properties and the relationships between them can be found through comparisons of macroscopic and microscopic derivations. The Onsager relations formulate various flows (consisting of matter or energy) as functions of the forces that drive them, thus describing macroscopic observations of materials. Another useful technique for understanding these basic thermoelectric properties utilizes semiclassical statistical mechanics to describe the microscopic processes. Comparisons between the macroscopic and the

J2 = L 21 X1 + L 22 X2 + · · · L 2n Xn ,

(8)

J3 = L 31 X1 + L 32 X2 + · · · L 3n Xn , or Ji =

n 

L im Xm

(i = 1, 2, 3, . . . , n).

(9)

m=1

For carrier and heat flow as described above, the Onsager relationships can be written J = L 11 ∇V + L 12 ∇T, J Q = L 21 ∇V + L 22 ∇T,

(10)

where J is the current density (electric charge flow), and J Q is the heat flux density (heat flow). Without a temperature gradient (T = 0), a heat flux of zero would be expected, contrary to what Eq. (10) would indicate. It is, therefore, important to understand further the primary coefficients L ii and interaction coefficients L i j (i = j) linking these equations. To do so requires a consideration of the thermodynamics of an irreversible process (one in which the change in entropy is greater than zero  S¯ > 0).

III. THERMODYNAMICS OF AN IRREVERSIBLE PROCESS The general application of the Onsager relationship was derived by Harman and Honig (1967) and is summarized here. For a constant electric potential, V , throughout the sample,

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d Q = T d S¯ = dU + P dV −



µi dn i ,

(11)

i

where Q is the heat energy density, S¯ is the entropy density, U is the internal energy density, P is the pressure, µi is the chemical potential of the particle species, and n i is the particle density. The magnitude of the differential volume, dV, is zero since each quantity has been specified per unit volume. This can be combined with the total energy density, E, given by  E =U +V Z i qn i , (12) i

where q is the magnitude of the electronic charge [1.602 × 10−19 (C)], V is an externally applied bias, and Z i is the number and sign of the charges on the ith particle species. For example, an electron would have charge Z e q, where Z e = −1. The time derivative of Eq. (12) gives  ∂E ∂n i ∂U ∂V  Z i qn i + V Zi q = + . (13) ∂t ∂t ∂t i ∂t i From Eq. (11) with dV = 0, dU = T d S¯ +



µi dn i .

(14)

i

Taking the time derivative of (14) gives ∂U ∂ S¯ ∂ T  ∂n i  ∂µi µi ni =T + S¯ + + . ∂t ∂t ∂t ∂t ∂t i i

(15)

Using (15) in (13) yields ∂E ∂ S¯ ∂ T  ∂n i  ∂µi µi ni =T + S¯ + + ∂t ∂t ∂t ∂t ∂t i i  ∂V  ∂n i + . Z i qn i + V Zi q ∂t i ∂t i

(16)

This can be simplified by considering the Gibbs–Duhem relation (Guggenheim, 1957), ∂ T  ∂µi S¯ ni + =0 (17) ∂t ∂t i and µ ¯ i = µi + Z i q V,

(18)

where µ ¯ i is the electrochemical potential, µi is the chemical potential, and Z i q V is the electrostatic potential energy. The relationship among the chemical potential, µ, the electrochemical potential, µ, ¯ and the temperature for electrons is shown in Fig. 3, where the right side of the sample is at a potential of −V1 relative to the left. Equation (16) then reduces to ∂E ∂ S¯  ∂n i ∂V  µ ¯i Z i qn i . (19) =T + + ∂t ∂t ∂t ∂t i i

FIGURE 3 The density of states for a metal at a temperature T1 > 0 K on the left and at a lower temperature, T2 < T1 , on the right.

The rate of change in the particle density n i , is governed by the equation of continuity,   ∂n i ∂n i − ∇ · Ji , (20) = ∂t ∂t s which states that the total rate of change in n i is equal to the local particle generation rate, or source rate, minus the transport of the ith species across the boundary of the differential volume (or local system) of interest. The first term on the right-hand side of the equation is the source term and represents the particle generation (or capture) rate through chemical reactions, for example. The last term is found using Gauss’s theorem,   Ji · nˆ d A = ∇ · Ji dV, (21) where Ji is the flux vector equal to the number of particles of type i moving past a unit cross section per unit time in the direction of Ji , and nˆ represents a unit vector outward normal from an element of area d A on the boundary surface. This represents the total outward flux of the ith particle species from the differential volume of interest. The particle species over which the summation in Eq. (19) is evaluated includes core species, L, which form the host lattice; neutral donors, D; ionized donors, D + ; neutral acceptors, A; ionized acceptors, A− ; electrons in the conduction band, n; and holes in the valence band, p. Therefore,  1  ∂n i ∂n L ∂n D 1 = µ ¯L +µ ¯D µ ¯i T i ∂t T ∂t ∂t ∂n D+ ∂n A ∂n A− +µ ¯A +µ ¯ A− ∂t ∂t ∂t  ∂n p ∂n n +µ ¯n +µ ¯p . (22) ∂t ∂t

+µ ¯ D+

Equation (20) can now be used for each term on the righthand side of (22). Some simplification can be readily

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made, however, when the species L, D, D + , A, and A− are assumed to be immobile such that J L = J D = J D+ = J A = J A− = 0.

(23)

Furthermore, the lattice will not be affected by local transformations, and   ∂n L ∂n L = = 0. (24) ∂t s ∂t Additional relationships can be found to simplify (22) further by identifying the different mechanisms for generation of electrons, n, or holes, p, as follows: (25)  D ⇒ D + + n, (26) A ⇒ A− + p,  ⇒ n + p. (27)

∂ S¯ 1 ∂E ¯ n ∇ · Jn =T + {AI νI + AII νII + AIII νIII − µ ∂t ∂t T ∂V  −µ ¯ p ∇ · Jn } + Z i qn i , (31) ∂t i where AI , AII , and AIII affinities are defined as ¯D +µ ¯ D+ + µ ¯ n, AI ≡ −µ AIII ≡ µ ¯n +µ ¯ p.

AII ≡ −µ ¯ A + µ A− + µ ¯ p, (32)

These general derivations can now be applied to more specific cases by solving for the energy flux term on the left-hand side of the equation using the appropriate approximations for the material under consideration. A. Metals

These reactions are reversible such that the time rate of change of ionized and unionized donors and acceptors must be considered in (22). Identifying the reactions in (25), (26), and (27) as I, II, and III, respectively, the following reaction velocities can be written        ∂n D ∂n D+ ∂n n  − = = = νI ,     ∂t s ∂t ∂t I  s          ∂n p ∂n A ∂n A− − = = = νII , (28)  ∂t s ∂t s ∂t II          ∂n p ∂n n   = = νIII .  ∂t s ∂t III

In metals, the energy density term ∂ E/∂t can be viewed as composed of four contributions.

Therefore, (22) becomes

    1  ∂n i ∂n D ∂n D+ 1 µ ¯i +µ ¯ D+ = µ ¯D T i ∂t T ∂t s ∂t s     ∂n A ∂n A− +µ ¯A +µ ¯ A− ∂t s ∂t s     ∂n n ∂n n +µ ¯n + − ∇ · Jn ∂t I ∂t III     ∂n p ∂n p +µ ¯p + − ∇ · Jp . ∂t II ∂t III

As an externally applied electric field accelerates charged carriers, they do not continue to increase in velocity as they would in free space, but attain some average drift velocity. Therefore, an internal force must exist to counterbalance the external force. This internal force is caused mainly by collisions of the carriers with the lattice, thus providing a mechanism of energy transfer from the applied electric field to the lattice. The first contribution is given by   ∂E = E · (−n n qvn + n p qv p ) = J · E = −J · ∇V ∂t I

(29)

where Jn and J p represent the particle flux densities, while J represents the current density such that

From the relations in (28), Eq. (29) can be written in terms of the reaction velocities, νI , νII , and νIII as follows: 1  ∂n i 1 = {(−µ ¯D +µ µ ¯i ¯ D+ + µ ¯ n )νI + (−µ ¯A T i ∂t T +µ ¯ A− + µ ¯ p )νII + (µ ¯n +µ ¯ p )νIII −µ ¯ n ∇ · Jn − µ ¯ p ∇ · Jn } This can be substituted into (19) to give

(30)

r The rate at which an externally applied field delivers

energy to the local system.

r Two terms arise from the rate of change in the

electrostatic energy either due to a change in the charge concentration or due to a change in the potential, V . r Electrons in the higher-energy states [the energies above µ(T1 ) in Fig. 3] can transition to the available lower-energy states by giving up this excess energy to the lattice, resulting in a heat flux, J Q .

= q(Jn − J p ) · ∇V,

J = q(J p − Jn ).

(33)



(34)

The electrostatic energy density is given by V i Z i qn i . The time rate of change of the electrostatic energy density is     ∂ ∂V  ∂n i Z i qn i = Z i qn i + V Zi q V . ∂t ∂t ∂t i i i (35)

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or, after cancellation and using (20), assuming no generative sources, −∇ · J E = T

FIGURE 4 The density of states at a finite temperature. Only the excitation energy can be transferred to the lattice.

This gives the second and third contributions to the total rate of change in energy density,   ∂E ∂V  = Z i qn i , (36) ∂t II ∂t i    ∂E ∂n i = −V (∇ · J) =V Zi q ∂t III ∂t i = q V [∇ · (Jn − J p )],

(37)

where Eq. (20) was used, with the assumption of no generative sources. The fourth contribution comes from the excitation energy depicted in Fig. 4, which gives rise to a heat flux, J Q . Relative to the bottom of the conduction band, the total heat flux density, Ju , is µ Ju = J Q − J, (38) q thus giving the fourth contribution to energy flow through Fourier’s law of heat conduction,     ∂E µ = −∇ · Ju = −∇ · J Q − J . (39) ∂t IV q Summing contributions I through IV gives the total energy density rate of change as ∂E ∂V  Z i qn i − ∇ · Ju = −J · ∇V − V (∇ · J) + ∂t ∂t i ∂V  = −∇ · V J + Z i qn i − ∇ · Ju ∂t i ∂V  = Z i qn i − ∇ · J E , (40) ∂t i where

J E = Ju + V J (41) is the total energy flux density. Substituting (40) into (19) gives ∂V  ∂ S¯  ∂n i + Z i qn i − ∇ · J E = T µ ¯i ∂t i ∂t ∂t i +

∂V  Z i qn i , ∂t i

(42)

µ ¯ ∂ S¯ ∂ S¯ + ∇ ·J= T + µ∇ ¯ · Jq . ∂t q ∂t

(43)

The rate of change of entropy can, therefore, be written     ∂ S¯ −∇ · J E µ ¯ JE Jµ ¯ = − ∇ · J = −∇ · −∇ · ∂t T qT T qT     1 µ ¯ + JE · ∇ +J·∇ , (44) T qT or using an entropy flux, Js¯ , defined as T Js¯ = J E +

µ ¯ J, q

(45)

gives ∂ S¯ ∂ S¯ 0 ∂ S¯ s = + ∂t ∂t ∂t = −∇ · Js¯ + J E · ∇

    1 µ ¯ +J·∇ , T qT

(46)

where the total entropy is given by the sum of the equilibrium entropy plus additional entropy sources, or S¯ = S¯ 0 + S¯ s . The irreversible process for which  S¯ = ( S¯ − S¯ 0 ) = S¯ s > 0 then consists of the last two terms in the above equation such that     ∂ S¯ s 1 µ ¯ = JE · ∇ +J·∇ . (47) ∂t T qT Using Eq. (45) to substitute for J E , (47) becomes. ∂ S¯ s −Js J = · ∇T + · ∇ µ, ¯ ∂t T qT

(48)

or using µ ¯ = µ + q V along with (41) and (45) to give T Js¯ = J Q ,

(49)

∂ S¯ s J −J Q · ∇T + = · ∇ µ. ¯ 2 ∂t T qT

(50)

then Eq. (48) becomes

These three equations, (47), (48), and (50), could each be written in the general form of  ∂ S¯ s Ji · Xi . = ∂t i

(51)

This is a necessary condition for using the Onsager reciprocity relation that L 12 = L 21 in Eq. (10). Three sets of

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Onsager relations can then be written, by extracting the forces, Xi , from Eqs. (47), (48), and (50).     Z11 1  µ ¯ J= ∇ + Z12 ∇ ,  q T T  (52)     Z21 1  µ ¯   JE = ∇ + Z22 ∇ , q T T  B11 B12 J= ∇µ ¯− ∇T,   qT T (53)  B21 B22   Js¯ = ∇µ ¯− ∇T, qT T  L11 L12 J= ∇µ ¯ − 2 ∇T,   qT T (54)  L21 L22   JQ = ∇µ ¯ − 2 ∇T, qT T thus relating the electrical current density, J, to the energy flux density, J E , the entropy flux density, Js¯ , and the heat flux density, J Q . Equations (52), (53), and (54) can now be used to identify various thermoelectric properties.

Within crystalline materials, electron behavior can be described by the wave nature of electrons and Schr¨odinger’s equation, 2m −h ∂E , (E − V ) = 2 h j ∂t

(55)

where is the electron wave function, E is the total energy, and V is the potential energy of the electrons. The solution to this equation is (r, t) = ψ(r)e− jωt ,

(56)

where is the time-independent solution to Schr¨odinger’s equation. This solution forms a wave packet with a group velocity, v, equal to the average velocity of the particle it describes, such that 1 ∂ω 1 ∂E v = ∇k ω = = ∇k E = , ∂k h h ∂k

(57)

where the use of Planck’s relationship, E = hν = hω, was made. Force times distance is equal to energy, or with a time derivative, v·F=

∂E ∂k 1 ∂E = ·h , ∂t h ∂k ∂t

(58)

giving F=h

∂k . ∂t

px x ≥ h,

p y y ≥ h,

pz z ≥ h,

(60)

where px , p y , and pz are the momentum uncertainties in the x, y, and z directions, respectively. The positional uncertainties in the three directions are given by x, y, and z. It is possible to utilize these uncertainties to define the smallest volume (in real space, or momentum space) that represents a discrete electronic state. Within a cube of material with dimensions L × L × L, the maximum positional uncertainty for a given electron would be x = y = z = L, since the electron must be located somewhere within the cube. This would correspond to the minimum px , p y , and pz given by pxmin =

h h = , x L

p ymin =

h h = , y L

(61)

h h pzmin = = . z L Thus the product

IV. STATISTICAL RELATIONSHIPS

∇ 2 +

The electron wave function, (r, t), itself does not have physical meaning, however, the product of ∗ (r, t) (r, t) represents the probability of finding an electron at position r and time t. As a probability implies, there is a factor of uncertainty, which was quantified in 1927 by Heisenberg.

(59)

h3 (62) L3 gives the minimum elemental volume in momentum space to represent two discrete electronic states (one for spin-up and one for spin-down). The number of states, dg, per unit volume in an element d px d p y d pz of momentum space can be written 1 dg = 3 d px d p y d pz . (63) h Schr¨odinger’s time-independent equation for a free electron (V = 0) is pxmin p ymin pzmin =

∇ 2ψ +

2m Eψ = 0, h2

(64)

which has the solution ψ = Ae jk·r .

(65)

Substituting back into (64) gives E=

h2 2 p2 k = , 2m 2m

(66)

where p 2 = hk was used. Within a crystal, a similar formula can be found when the concept of effective mass, m ∗ , is utilized to account for internal forces on the electrons due to the ion cores at each lattice point. Electrons with energies below some value E are then √defined by a sphere in momentum space with radius p = 2m ∗ E. The number

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of electronic states within the material cube (L × L × L) is found by dividing the total momentum space volume by the volume per state, or   √ 4/3πr 3 4/3π ( 2m ∗ E)3 N =2 =2 h 3 /L 3 h 3 /L 3 8π(2m ∗ E)3/2 3 L . (67) 3h 3 The factor of 2 is included to account for electrons of both spin-up and spin-down. The density of states is defined as the number of states per unit energy per unit volume, or =

(d N /d E) 4π (2m ∗ )3/2 1/2 = E L3 h3   1 2m ∗ 3/2 1/2 = E , 2π 2 h2

g(E) =

(68)

where h = h/2π is Plank’s reduced constant. This equation describes the number of available states for electrons to go into, but it does not describe the way the electrons fill those available states. A. The Fermi Distribution To determine the number of electrons in a given band, it is necessary to find the probability of a given state being occupied by an electron and then integrate over all available states. A more realistic result for metals, which does not assume spherical constant energy surfaces in k-space, thus allowing for the electron energy to deviate from E = h2 k 2 /2m ∗ would be found using the density of states from (63). Within a crystalline material, charge carriers are known to follow the Pauli exclusion principle, which states that only one electrons can occupy a given energy state. The probability that an electron occupies an energy state can be found by considering a simple statistical exercise. If a system is defined to have three allowed energy levels (E1 , E2 , and E3 ), two electrons, and a total energy of 4 eV as shown in Fig. 5, with the three energy levels defined as E1 = 1 eV, E2 = 2 eV, and E3 = 3 eV, it would be expected that 80% of the time, a distribution of one electron in energy level E1 , zero electrons in E2 , and one electron in E3 , or a distribution of (1, 0, 1), would occur. The entropy of a system is related to the most probable arrangement, Wm , of the particles through Boltzmann’s definition, S¯ = k ln Wm .

(69)

When only electrons are considered, the entropy is related to the internal energy of the system, U , the total number of electrons, N , and the volume, V, of the system through Euler’s equation

FIGURE 5 The number of ways, W, two electrons can be distributed in three energy levels to obtain a total energy of 4 eV.

U = T S¯ − PV + µN − q VN ,

(70)

where V represents the internal electrostatic potential. For a simple system with just two available energy states (energy = 0 or energy = E), the probability of finding the system with energy E to that of finding it with energy 0 is W (U0 − E) e S(U0 −E)/k (E) . = = S(U (0) W (U0 ) e ¯ 0 )/k ¯

(71)

¯ 0 − E) ≈ S(U ¯ 0 ) − E( ∂ S¯ ) Using the approximation S(U ∂U0 and   ∂ S¯ 1 ∂ U + PV − µN + qVN = = (72) ∂U T ∂U T simplifies (71) to ¯

(E) e( S(U0 )/k)−(E/kT ) = e−E/kT . = ¯ 0 )/k (0) e S(U

(73)

To determine the probability of a system in energy state E, and that the state is occupied by an electron, then the influence of the total number of electrons, N , must also be taken into consideration. Then the ratio of the probability that the system is occupied by one electron at energy E to the probability that the system is unoccupied with energy 0 is W [(U0 − E), (N0 − 1)] (1, E) = (0, 0) W [U0 , N0 ] e S[(U0 −E),(N0 −1)]/k . (74) ¯ 0 ,N0 ]/k e S[U ¯ 0 − E), (N0 − 1)] ≈ S[U ¯ 0 , N0 ] − E(∂ S/∂U ¯ Using S[(U 0) − ¯ (∂ S/∂ N0 ) yields ¯

=

(1, E) e( S[U0 ,N0 ]/k) − (E/kT ) + ((µ−qζ )/kT ) = e(E F −E)/kT , = ¯ 0 ,N0 ]/k (0, 0) e S[U (75) where the Fermi level is defined as E F = µ − q V . Since (1, E) + (0, 0) = 1, ¯

(1, E) = f (E) =

1 1+

e((E F −E)/kT )

.

(76)

This is the Fermi–Dirac distribution and represents the probability of occupancy of an energy state in equilibrium.

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B. Carrier Concentrations In a free-electron approximation, the total number of electrons in a given energy band can then be found by integrating the product of the density of states and the probability of occupancy of the state for spherical energy surfaces:  Etop (2mkT )3/2 n= g(E) f (E) dE = F1/2 (η), (77) 2π 2 h3 E bottom where η = E F /kT , and Fν (η) is the Fermi–Dirac function,  ∞   ην Fν η = d x. (78) 1 + e x−η 0 Here, the bottom of the energy band was taken to be zero energy corresponding to k = 0, and the integration was allowed to extend to ∞ since the Fermi–Dirac distribution falls to zero at high energy levels. In the degenerate limit, when E F  kT , a series expansion of (78) leads to the following approximations for metals:  2  π 2 kT E F ≈ E F0 1 − + ··· , 12 E F0 (79)      (2m)2/3 E F 3/2 π 2 kT 2 n≈ 1 + + ··· . 3π 2 h3 kT 8 EF At T = 0 K, E F = E F0

  π 2 h2 3n 2/3 = . 2m π

(80)

For nonspherical energy surfaces, the number of electrons per unit volume within an element of momentum space, dpx dp y d pz , is found using (63) and the relation h p = hk = 2π k, 1 2 f (p, r) d px d p y d pz = f (k, r) dk x dk y dk z , 3 h 4π 3 (81) where the factor of 2 accounts for two electrons of opposite spin. The total electron density is then found by integration.

dn =

C. The Boltzmann Function If the material is disturbed from equilibrium, then the distribution will vary, in general, as a function of wavevector, k, position, r, and time, t, or f (k, r, t). At a time t + dt, the probability that a state with wavevector k + dk is occupied by an electron at position r + dr can be found, using Eq. (59), to be f (k + dk, r + dr, t + dt)   1 = f k + Ft · ∇k dt, r + v dt, t + dt . (82) h

The total rate of change of the distribution function near r is then ∂f 1 df = Ft · ∇k f + v · ∇r f + , (83) dt h ∂t which is Boltzmann’s transport equation. The first term on the right-hand side of this equation accounts for contributions from forces, Ft , including externally applied forces, F, and collision forces, Fc . The middle term adds the contributions from concentration gradients, and the last term is the local changes in the distribution function about the point r. Equation (83) is equal to zero since the total number of states in the crystal is constant, thus ∂f −1 Ft · ∇k f − ν · ∇r f = ∂t h −1 1 = Fc · ∇k f − F · ∇k f − ν · ∇r f h h   ∂f 1 = − F · ∇k f − ν · ∇r f. (84) ∂t c h With external forces applied, the distribution function, f , will be disturbed from the equilibrium value, f 0 . Upon the removal of those external forces, equilibrium will be reestablished through collisions, (∂ f /∂t)c . Calculation of this collision term is a formidable task dependent largely on the scattering mechanisms for the material investigated. For small disturbances, however, a relaxation-time approximation is often used which assumes that   ∂f −( f − f 0 ) − f1 = = , (85) ∂t c τk τk where τk is the momentum relaxation time. In steady state, ∂ f /∂t = 0 and Eq. (84) becomes  − f1 1  0 = − F · ∇k f − v · ∇r f   τk h (86) or  τk   f 1 = − F · ∇k f − τk v · ∇r f. h The electric and heat current densities are given by   q J = −qvn = −q v dn = − 3 f 1 (k) dk, 4π (87)  1 JQ = v(E − E F ) f 1 (k) dk. 4π 3 Substituting Eq. (86) into Eqs. (87) starting with the electric current density, J, gives    q −τk J=− 3 vF · ∇k f − τk vv · ∇r f dk. (88) 4π h Assuming parabolic bands, the gradient of the distribution function in k space can be written ∂f ∂f ∇k f = ∇k E = hv. (89) ∂E ∂E

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690 Also, the following can be shown by direct substitution of the Fermi–Dirac distribution (76):      ∂f ∂f ∂ 1 ∂ EF = T E − , (90) ∂x ∂E ∂x T ∂x T with similar results in the y and z directions. Substituting these results into Eq. (88) gives   q ∂f −τk ∂ f J=− 3 T vF hv − vvτk 4π h ∂E ∂E      1 EF × E∇r − ∇r dk (91) T T or  q ∂f J= τk vvF dk 4π 3 ∂E    1 ∂f + q τk vv T (E − E F )∇r dk. (92) ∂E T For example, the applied force might include a contribution from an external electric field (−qE), plus a contribution caused by a temperature gradient (see Fig. 3), or in general as ∇ µ ¯ = ∇(µ + q V ) = ∇µ − qE. Then (92) would be  q ∂f J= τk vv (∇µ − qE) dk 3 4π ∂E  ∂f 1 −q τk vv(E − E F ) (93) ∇r T dk, ∂E T where ∇r (1/T ) = −(1/T 2 )∇r T was used. The electrical current density can be simplified and put into a format similar to the Onsager relations as shown in (54) by using transport integrals defined as  ∂ f0 1 Kn = − 3 τk vv(E − E F )n dk, (94) 4π ∂E where it is assumed that the deviations from equilibrium are small, such that ∂ f /∂ E in Eq. (93) may be replaced with ∂ f 0 /∂ E. This leads to an electrical current density of q J = −qK0 ∇ µ ¯ + K1 ∇T. (95) T Similarly, the heat current density, J Q , follows the same derivation to arrive at  1 ∂f JQ = (∇µ − qE)(E − E F ) dk τk vv 3 4π ∂E  1 ∂f 1 − 3 τk vv(E − E F )2 ∇r T dk (96) 4π ∂E T or 1 J Q = −K1 ∇ µ ¯ + K2 ∇T. (97) T These derivatives form the link between the macroscopic Onsager equations and the atomistic derivations from

Thermoelectricity

Boltzmann’s equation. Comparison of Eqs. (95) and (97) with Eq. (54) shows the following relations:   L11   = −qK0 ,   qT      L11 = −q 2 T K0 ,   −L12 q     K = , 1   L = −qT K , T2 T 12 1 (98) or   L = −qT K L21 21 1,    = −K1 ,     L22 = −T K2 .  qT     −L22 1   = K2 ,   T2 T This shows the Onsager reciprocity relation, in that L12 = L21 . The thermoelectric properties can now be determined through an evaluation of the transport integrals and the appropriate boundary conditions of isothermal (∇T = 0), isoelectric (∇V = −E = 0), static (J = 0), or adiabatic (J Q = 0). For example, under isothermal conditions, where ∇T = 0 and thus ∇µ = 0 (for a homogeneous metal), J = q 2 K0 E = σE,

(99)

and the electrical conductivity is σ = q 2 K0 .

(100)

The electronic contribution to the thermal conductivity is defined for static conditions as J E | J =0 = − κe ∇T , or when J E = J Q as in a one-band material, J Q = κe ∇T , where (95) becomes 0 = −qK0 ∇ µ ¯+

q K1 ∇T. T

Solving for ∇ µ ¯ and substituting into (97) gives   1 K1 1 J Q = −K1 ∇T + K2 ∇T T K0 T   1 K1 K1 = K2 − ∇T T K0 or   1 K1 K1 κ= . K2 − T K0

(101)

(102)

(103)

The absolute Seebeck coefficient, or thermopower, S, can also be found from the static condition, where the use of Eq. (101) gives 1 ∇µ 1 K1 ¯ S= . (104) = q ∇T qT K0 The Peltier coefficient, , can be found by evaluating the heat current density, Eq. (97), for isothermal conditions: J Q = qK1 E.

(105)

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Thermoelectricity TABLE I Combined Results from Macroscopic and Atomistic Analysis Thermoelectric property

Transport integral

σ = q 2 K0

K0 =

1 K1 qT K0   1 K1 K1 κe = K2 − T K0

S=

σ q2 σ K1 = T S q

JQ =

K1 J = ΠJ, qK0

(106)

where the proportionality constant is simply the Peltier coefficient, . Comparing the Peltier coefficient (106) with the thermopower (104) leads to Kelvin’s second relation: Π = T S.

(107)

Results of the transport integrals are summarized in Table I. Thus Eqs. (54) can be rewritten in terms of the thermoelectric properties as  −σ  J = ∇µ ¯ + σS∇T,   q (108)    −T  JQ = σS∇ µ ¯ + κe + T σS2 ∇T.  q Substitution of the transport integrals can be used to evaluate further the thermoelectric properties. Estimations can be made through a series expansion of the transport integrals using a Sommerfeld expansion,  ∞ ∂ f0 Kn = − φn (E) dE ∂E 0 = φn (E F ) +

π2 d2 (kT )2 φn (E F ) + · · · . 6 d E F2

For the electrical conductivity,  q2 ∂ f0 σ = q 2 K0 = − 3 τk vv dk. 4π ∂E

(109)

(110)

In its simplest form for cubic symmetry, this reduces to nq 2 τk , (111) m∗ where n is the electron density with energies near E F , and m ∗ is the effective mass of the electrons. Both the electron density near the Fermi level and the relaxation time are functions of energy, such that the electrical conductivity can be approximated as σ = const · E ξ , where ξ is some number. σ =

L11 = −σT L12 = L21 = −T 2 σS

K2 = κe T + T 2 σS2

Substituting for the electric field, E, from Eq. (99) gives the direct relationship between heat current density and electric current density,

Onsager coefficient

L22 = −T 2 κe − T 3 σS2

A relationship between the electrical conductivity and the thermopower can be found by series expansion K1 , which gives   π2 2 ∂σ  K1 = − 2 (kT ) , (112) 3q ∂ E  E=E F along with σ = q 2 K0 and substituting into (104). This leads to the Mott–Jones equation (Barnard, 1972):   −π 2 k 2 T ∂ ln σ Sd = . (113) 3 q ∂ E EF A distinction of the diffusion thermopower, Sd , has been made here to separate it from a low-temperature effect that has not been considered above. The low-temperature effect typically appears as a peak in the measured thermopower (near 60 K for monovalent noble metals) and is the result of an increased electron–phonon interaction. When a temperature gradient exists across a crystal, heat will flow from the hot side to the cold side through lattice vibrations (phonons) and through electron flow. Various interactions among phonons, lattice defects, and electrons can be described by scattering times for each type of interaction. At high temperatures, phonon–phonon interactions are more frequent than electron–phonon interactions (τ p, p < τ p,e ). At these high temperatures (above the Debye temperature, T > θD ), τ p,e is approximately temperature independent, while τ p, p ∝ 1/T . Under these conditions, the total thermopower is dominated by the diffusion thermopower as given in Eq. (113). At low temperatures (T < θD ), τ p,e ∝ 1/T and τ p, p ∝ eθ D /2T , therefore, as the temperature drops, τ p, p increases more rapidly than τ p,e . When this occurs, τ p, p > τ p,e and electron– phonon interactions will occur more frequently, causing electrons to “dragged” along with the phonons. This gives rise to a larger gradient of carrier concentration across the sample and is additive to the diffusion thermopower such that S = Sd + Sg , where Sg is the phonondrag component of the thermopower described above. At still lower temperatures, phonon-impurity interactions can dominate, causing the magnitude of the thermopower to decrease toward zero. For the remainder of this chapter, the

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temperature is assumed to be much higher than the Debye temperature, such that S ≈ Sd and the diffusion subscript is dropped. When the electrical conductivity can be written σ = const · E ξ , this can be used in the Mott–Jones equation to give   −π 2 k 2 T T µV S= . (114) ξ = −0.0245 ξ 3 q EF EF K 1. Normal Metals In monovalent noble metals (Cu, Ag, and Au), ξ ≈ − 32 has been measured, giving the positive quantity   T µV S = 0.03675 . (115) EF K It is instructive to compare the thermopower of noble metals to the electronic heat capacity, Cel , which is dependent on the density of states, g(E F ), evaluated at the Fermi level. Substituting ξ ≈ − 32 into Eq. (114) gives S=

π 2 k2T 2 qE F

(116)

and Cel =

π2 π 2 N k2T g(E F )k 2 T = , 3 2 EF

(117)

where N is the total number of carriers. Then it can be seen that the electronic heat capacity per carrier is simply the electronic charge times the thermopower, Cel = q S. N

(118)

2. Transition Elements The electronic properties of transition metals are usually considered to have contributions from two bands that overlap at the Fermi level: the s-band, from the s levels of the individual atoms, and the d-band, consisting of five individual overlapping bands. The s-band is broad and typically approximated as free electron-like, while the d-band is narrow, with a high density of states and high effective mass, thus the s electrons carry most of the current. The relaxation time is, however, greatly affected by the high density of states of the d-band. This comes about through the inverse proportionality of the relaxation time to the probability of scattering from one wavevector, k, to another, k . The occupancy and availability of each of these wavevectors are, in turn, proportional to the density of states at the Fermi level. This leads to the relationship of the inverse proportionality of the relaxation time to the density of states:

1 (119) ∝ g(E)| E=E F . τ Due to the relatively high density of states in the d-band, the relaxation time of the highly responsive s-band electrons is dominated by s–d transitions, or 1 1 ≈ ∝ gd (E)| E=E F . τs τs−d

(120)

Neglecting the d-band contribution to the electrical conductivity and rewriting Eq. (110) in terms of the density of states gives 2 2 2 q νs τs gs (E)| E=E F 3  gs (E)  = const · νs2 . gd (E)  E=E F

σ =

(121)

Defining the bottom of the s-band as zero energy, and the partially filled d-band in terms of the holes in the band so it can be referenced to the top of the d-band, such that E 0 is the energy at the top of the d-band, and gd (E) = const · (E 0 − E F )1/2 , then approximating the s-band electrons as free electrons gives  ∂ ln σ  3 1 . (122) = +  ∂ E E=E F 2E F 2(E 0 − E F ) Typically E F  (E 0 − E F ), such that approximating the above equation as the second term on the right-hand side and using this in the Mott–Jones equation (113) gives S=

−π 2 k2T . 6 q(E 0 − E F )

(123)

Again, the electronic heat capacity can be compared to find Cel =

π 2 N k2T , 6 (E 0 − E F )

(124)

and the relationship between the magnitude of the electronic heat capacity and the thermopower remains Cel = q S. N

(125)

3. Semimetals The petavalent elements of As, Sb, and Bi are semimetals with rhombohedral crystal structures. This leads to nonspherical Fermi surfaces and anisotropic scattering such that τ ∝ ksx for a given crystallographic direction, where s accounts for the anisotropy. Likewise, the density of states g(E) ∝ k3x , and kx ∝ (E 0 − E F )1/2 . Using the density of states and the relaxation time for the electrical conductivity in an equation similar to (121) gives

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σ = const · (E 0 − E F )(3+s)/2

(126)

or, in (113), −π k T S∼ (3 + s), = 6 q(E 0 − E F ) 2

using (109) and substituting the transport integrals into (103). Series expansion of K2 gives K2 = −

2

(127)

where (3 + s) < 0. There is an exception to (127) in bismuth, which shows the expected anisotropic thermopower, but an unexpected negative thermopower (S⊥ ≈ −50 µV/K, and S|| ≈ −100 µV/K at 273 K). For bismuth, a value of ξ = (3 + s)/2 should be used in (114) describing electron conduction, instead of using Eq. (127), which is for conduction by holes. 4. Alloys Matthiessen’s rule states that the total resistivity of an alloy formed by two metals can be found by 1 = ρi + ρ j , (128) σ where ρi is the resistivity of the pure solvent metal due to scattering of carriers by thermal vibrations, and ρ j represents scattering of carriers from impurities. This rule is often used for approximations but is not widely applicable since many cases exhibit anisotropic scattering of carriers, causing a large deviation from (128). Assuming the validity of Matthiessen’s rule, (113) can be written   π 2 k 2 T ∂ ln (ρi + ρ j ) S= , (129) 3 q ∂E EF ρ=

which can be written in terms of the difference between S for the alloy and the thermopower of the pure solvent metal, Si , or S = S − Si leads to 1 − (x j /xi ) S =− , S 1 + (ρi /ρ j )

π 2 k2T 2 σ. 3 q2

(133)

The thermal conductivity is given by (103), repeated here for convenience:   1 K1 K1 κe = K2 − . (103) T K0 In metals, (∂/∂ E)σ (E)| E=E F ≈ (σ/E F ), thus K1 ≈ −(π 2 /3q 2 )(kT )2 (σ/E F ), or  2 2 (π /3q 2 )(kT )2 (σ/E F ) K1 K1 ≈ K0 σ/q 2  2  2 π π (kT )2 2 , (134) = (kT ) σ 3q 2 3 E F2 giving

  π 2k2 T π 2k2 T π 2 (kT )2 κe ≈ ≈ σ 1 + σ, 3q 2 3 E F2 3q 2

(135)

where the approximation of (π 2 /3)((kT )2 /E F2 )  1 was used, thus arriving at the Wiedemann–Franz law, or κe /σ T = 2.443 × 10−8 ((W · )/K2 ). The total thermal conductivity, κ, must also include a lattice contribution, κ L , such that κ = κ L + κe .

(136)

The lattice thermal conductivity for metals is generally much lower than the electronic contribution. D. Semiconductors

Using the Gorter–Nordheim relation forthe impurity com ponent of Mattheissen’s rule, ρi = C X 1 − X , where C is the Nordheim coefficient and X is the atomic fraction of the solute atoms in a solid solution, yields a more useful relationship: ρi S = S j + (Si − S j ), (132) ρ

The above analysis is applicable to normal metals, where it is assumed that the carriers are electrons and ∇µ is a function of temperature only. Furthermore, the Onsager relations were developed using four contributions, (33), (36), (37), and (39), to the energy density rate of change, however, two additional contributions exist for semiconductors. These contributions account for transitions of electrons across the bandgap, or the rate of change in carrier concentrations in each band, and for positional gradients of the band edges (valence band and conduction band). The last contribution could arise from temperature gradients and/or compositional variations, for example. These additional contributions have the form   ∂E = −qC (−∇ · Jn ) + qV (−∇ · J p ) (137) ∂t V

where S j is the thermopower for the impurity. The third thermoelectric parameter listed in Table I is thermal conductivity. This can likewise be determined

and   ∂E = Jn · ∇qC − J p · ∇qV , ∂t VI

where



xi = −

∂ ln ρi ∂E

(130) 

 and EF

xj = −

∂ ln ρ j ∂E

 . EF

(131)

(138)

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where −qC and qV represent the internal potential energies of the electrons and holes at the bottom of the conduction band and the top of the valence band, respectively. This leads to the Onsager relations for a two-band model, where, in a steady-state condition (Harman and Honig, 1967), J Q = L 11 X Q + L 12 Xn + L 13 X p , J− = L 21 X Q + L 22 Xn + L 23 X p ,

(139)

J+ = L 31 X Q + L 32 Xn + L 33 X p , where 1 ∇T, T2 1 µ ¯C Xn = − ∇ϕC + ∇ , T qT

XQ = −

V. APPLICATIONS A. Thermocouples Thermocouples are the most common application of thermoelectric materials. Application of the Seebeck coefficient (1), along with the Thompson relation (7), allows one to determine the open-circuit potential for a circuit containing temperature gradients by integrating over temperature as one traverses through the circuit from one terminal of the open circuit to the other. For example, in the circuit shown in Fig. 6 the open-circuit voltage can be written  T1  T2  T3 V = S A dT + S B dT + SC dT T0

(140)

=



σ = σn + σ p ,

+

T5

SC dT +

T3 T1

T2



 S A dT +

T5

Also, µ ¯ C and µ ¯ V represent the difference between the chemical potential energy and the internal potential energy of the carriers in the two bands. The total potential energy of the carriers in an applied field for a semiconductor must include the potential energy from the field as well as the internal potential energies −qC and qV , from the band edges. Contributions to the electrical current density come from electrons, J− = −qJn , and from holes, J+ = qJ p , for the total current density given by J = J− + J+ . Applying the same procedure for this case as followed for metals above, with the additional consideration of the relative potential energies of the band edges using µ ¯ V = −(E F + E V ) and µ ¯ C = E F − E C , gives the following formula for a two-band semiconductor:

Sn σn + S p σ p , σn + σ p

T4

+ 

1 µ ¯V X p = − ∇ϕV − ∇ , T qT

S=

T1





T4 T2

 S B dT +

T1 T5

T0

S D dT +

S A dT T5 T4

SC dT T2

S D dT .

When measuring this potential difference, care must be taken to include the contribution from the leads of the meter. This can be minimized by assuring that the thermocouple-circuit open terminals (in Fig. 6) are at a constant temperature T0 and the terminals on the voltage meter are also at a constant temperature (not necessarily T0 ). B. Generators and Coolers Lenz first demonstrated a thermoelectric cooler by freezing water at the junction between two conductors formed by rods of bismuth and antimony; however, a more common configuration for a thermoelectric cooler is shown in

(141)

 2 p σn σ p K 1n K 1 κ = κ L + κn + κ p + + + (E C − E V ) . σ T q 2 K 0n K 0p Of course, as a semiconductor is doped n-type or p-type, the corresponding contributions, subscripted n or p, respectively, above will dominate. The last term in the thermal conductivity formula, when multiplied by −∇T, would relate to the transport of bandgap energy along the negative temperature gradient and is defined as an ambipolar transport mechanism.

(142)

T4

FIGURE 6 Thermocouple circuit.

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FIGURE 7 Thermoelectric cooler.

Fig. 7. Here the cooling (or warming) junction is made more accessible for device cooling (or heating). Since current is defined as positive in the direction of positive carrier flow (hole flow), and likewise for the direction opposite to negative carrier flow (electron flow), by using one p-type leg and one n-type leg to the cooler, the highest efficiency can be achieved (Fig. 8). In this situation, all carriers flow in the same physical direction (either top to bottom or bottom to top) in both legs. Since charge carriers also carry heat as shown through the Onsager relations, heat will flow through the device in the direction of the carriers. Although the configuration shows a pn junction, these devices do not behave as diodes and electrical current is reversible. This is due to the fact that each of the legs is doped to degeneracy, or near-degeneracy, such that ohmic contacts with the metals are exhibited. The goal in making a thermoelectric cooler is to maximize the coefficient of performance, ϕ, of the device, defined as Q˙ 0 ϕ= , (143) W where Q˙ 0 is the rate of heat absorbed from the object being cooled over the amount of power, W , it takes to drive the cooler. Assuming that the thermopower of materials A and B in Fig. 7 do not vary significantly over the temperature range T0 to T1 , then the Thompson heat may be neglected, and Q˙ 0 = Q˙  − Q˙ T , (144)

where Q˙  is the Peltier heat absorbed at the cold junction and Q˙ T is the thermal losses down the arms of the cooler. The Peltier heat absorbed is Q˙  =  · I , and the thermal losses down the arms consist of thermal conduction losses, K (T0 − T1 ), where K is the thermal conductance of the arms, and Joule heating losses, 12 I 2 R. A factor of 12 on the Joule heating losses is due to half of this heat flowing to the cold end and half flowing to the warm end of the cooler. Substituting gives Q˙ 0 =  · I − 12 I 2 R − K (T0 − T1 ).

(145)

Maximizing Q˙ 0 with respect to current yields  = I · R, or Imax = /R. Using the Kelvin relations, Imax =

(S A − S B )T1 . R

(146)

In steady state, Q˙ 0 = 0, and the maximum temperature gradient Tmax = (T0 − T1 ) is 1 (S A − S B )2 2 1 (147) T1 = Z T12 , 2 RK 2 where Z is defined as the figure of merit for the cooler. Equation (147) clearly shows that the maximum temperature gradient is increased by choosing materials with the largest difference in thermopower values. Therefore, the logical choice is to use one n-type and one p-type material as mentioned previously. Continuing with the evaluation of the coefficient of performance for the cooler, the power absorbed by the device is simply the product of the current and voltage supplied to the cooler, or Tmax =

W = I V = I {I R + (S A − S B )(T0 − T1 )},

(148)

where the voltage across the device includes the resistive and thermoelectric voltage drops. Dividing this into Q˙ 0 yields the coefficient of performance, ϕ=

I − 12 I 2 R − K (T0 − T1 ) . I 2 R + (S A − S B )(T0 − T1 )I

FIGURE 8 Thermoelectric cooler current flow.

(149)

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Taking the derivative with respect to current and setting it equal to zero gives

refrigeration systems is 1.2 to 1.4, for a refrigerator operating at a cold temperature of 263 K while the outside (hot

  (−IR)[ST · I + I 2 R] − [ST + 2IR] I − 12 I 2 R − K (T0 − T1 ) dϕ , =0= dI [ST · I + I 2 R]2 where the substitutions S = (S A − S B ) and T = (T0 −T1 ) were used. After expansion and cancellation in the numerator, dϕ =0 dI   I 2 −R − 12 R · ST + I [2KR(T0 − T1)] + K · ST 2 = . [ST · I + I 2 R]2 (151) Substituting  = (S A − S B )T1 for the Peltier heat removed at the cold junction gives    0 = I 2 −R(S A − S B ) T1 + 12 (T0 − T1 ) + I [2K R(T0 − T1 )] + K (S A − S B )(T0 − T1 )2 . (152) Solving this quadratic equation yields the maximum coefficient of performance at the optimum current, Iopt =

(S A − S B )(T0 − T1 ) , √ R( 1 + Z T¯ − 1)

(153)

where T¯ is the average temperature 12 (T0 + T1 ). Using this in Eq. (143) yields √ 1 + Z T¯ − (T0 /T1 ) T1 ϕopt = , (154) √ (T0 − T1 ) 1 + Z T¯ + 1

(150)

temperature) is at 323 K. Freon-based cooling systems have coefficients of performance that would correspond to a thermoelectric device with Z T between 3 and 4. Also shown is the COP for the present value of Z T ∼ 1. The advantages of thermoelectric devices includes size scalability without loss of efficiency, robustness, low maintenance, a relatively small electromagnetic signature, and the ability both to heat and to cool from a single device, and they are environmentally cleaner than conventional CFCbased coolers. Many thermoelectric companies presently exist, indicating an existing market such that any increase in Z T through a new material and/or configuration could have a direct impact; however, a significant increase in the market is anticipated for an increase in Z T to 2. This, therefore, represents the current goal in Fig. 9. These devices are heat pumps, in that it is also possible to remove the electrical power source, and force a temperature gradient across the thermoelectric device, by contacting one end of it to an external heat source. With a load connected to the device instead of the electrical power source, it then functions as a thermoelectric generator. Thus, the application of an electrical potential gradient causes the generation of a temperature gradient (thermoelectric cooler) and the application of a temperature gradient causes the generation of electrical power (thermoelectric generator).

where the first term represents the coefficient of performance for an ideal heat pump. This shows that both ϕ and T are directly dependent on the figure of merit, Z . Thus maximizing the figure of merit for the individual materials, Z=

S2 S2σ = , ρκ κ

(155)

maximizes the efficiency of the cooler. Desirable materials have large-magnitude thermopowers, S (one n-type and one p-type), and low electrical resistivities, ρ, or, equivalently, high electrical conductivities, σ , and low thermal conductivities, κ. Since the figure of merit has units of K−1 , the unitless quantity of Z T is often reported. It should also be noted that the Peltier heat, Q˙  =  · I , is either absorbed or liberated based on the current direction. Therefore, the same configuration can be used as either a thermoelectric cooler or a heater. For comparison, and to illuminate the present challenge, the coefficient of performance for standard Freon-based

FIGURE 9 The figure of merit versus the coefficient of performance.

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In the case of a generator, the efficiency, η, of the device is defined as the ratio of the power supplied to the load to the heat absorbed at the hot junction: √ TH − TC 1 + Z T¯ − 1 η= . (156) √ TH 1 + Z T¯ + (TC /TH ) This is again dependent on the figure of merit of the device. Through Thompson’s relations we can split the figure of merit for the device into a figure of merit for each of the two legs. When each of these has been maximized individually, then the total device figure of merit will also be maximized assuming that one leg is n-type and one p-type. C. New Directions Traditional materials used in thermoelectric devices are listed in Table II. Near-room-temperature devices have been designed largely for cooling applications, while higher-temperature materials have been generally used in electrical power generation. Research on thermoelectrics was highly active during the decade following 1954, with the United States showing a great interest in hightemperature power generation applications, such as the Si–Ge-based generators used on the satellites Voyager I and II. Recently there has been a resurgence of interest in thermoelectrics, spurred on partly by predictions of the high ZTs possible in quantum confined structures (Hicks and Dresselhaus, 1993). It was predicted that in such structures, both the electrical conductivity and the thermopower could be simultaneously increased due to the sharpening of the density of states as confinement increases from 3D → 2D → 1D → 0D (Broido and Reinecke, 1995). The influence of such sharpening can be seen clearly within the Mott–Jones equation for thermopower (113). An indication of the effect from a rapidly varying density of states comes from mixed-valent compounds such as CePd3 and YbAl3 , which have shown the largest power factor, σ S 2 , among all known materials. Unfortunately, the high thermal conductivity in these materials prevents them from having a correspondingly high figure of merit. An additional increase in Z T for quantum confined materials comes from a decrease in the thermal conductivity due to confinement barrier scattering. Another avenue for investigating thermoelectric materials has been coined the “phonon glass electron crystal” TABLE II The Most Widely Used TE Materials Z max (K−1 )

Useful range (K)

T max (K)

Bi2 Te3 PbTe

3 × 10−3 1.7 × 10−3

<500 <900

650

Si–Ge

1 × 10−3

<1300

1100

300

TABLE III Desirable Material Properties for Thermoelectric Applications (Kanatzidis, 2001) 1. 2. 3. 4. 5. 6.

Many valley bands near the Fermi level, but located away from the Brillouin zone boundaries. Large atomic number elements with large spin–orbit coupling. Compositions with two or more elements such as ternaries and quaternaries. Low average electronegativity differences between elements. Large unit cells. Energy gaps near 10 kT.

(PGEC) method (Slack, 1995), in which short phonon mean free paths and long electron mean free paths are simultaneously sought in a material. A suggested way for a material to exhibit PGEC behavior is making a material that incorporates cages and/or tunnels in its crystal structure large enough to accommodate an atom. The caged atom provides strong phonon scattering by rattling within the cage. Electrons would not be significantly scattered by such “rattlers” since the main crystal structure would remain intact. Additional guidance has been provided by identifying a B parameter defined as   1 2kT 3/2 √ k2 B=γ 2 mx m ymz µx , (157) 2 3π h qκ L where γ is the degeneracy parameter (Hicks and Dresselhaus, 1993). This function should be maximized for optimal Z T . With a large number of valleys within a band, fewer carriers can exist in each valley, thus increasing the contribution to the thermopower from that valley. At the same time, the total number of carriers can be maintained for a high electrical conductivity. High degeneracy parameters are generally found in highly symmetric crystal systems. Large effective masses, or large effective mass components in the axes perpendicular to the current flow, allow for a high electrical conductivity in the direction of interest while maintaining a high B parameter. Equation (157) also indicates that high mobilities, µx , in the transport direction, and a low lattice thermal conductivity are also desirable. It has recently been shown that semiconductors with bandgaps of approximately 10 kT best satisfy these criteria (Mahan, 1998). Six properties of thermoelectric materials that give the best results are listed in Table III.

VI. SUMMARY Macroscopic and atomistic derivations of the thermoelectric properties of electrical conductivity, thermoelectric power, and thermal conductivity have been presented, with approximations for various material systems. The derivations outlined have considered external forces of electric

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698 fields and temperature gradients. Additional effects are realized when more forces such as magnetic fields are included. These include Hall, magnetoresistance, Nernst, Ettingshausen, and Righi–Leduc effects. These magnetic fields also affect the operation of thermoelectric coolers, with significant enhancements of the efficiency possible under strong fields.

ACKNOWLEDGMENT I wish to thank Sangeeta Lal, of Bihar University, for her helpful review of the manuscript.

SEE ALSO THE FOLLOWING ARTICLES ELECTROMAGNETICS • ELECTRONS IN SOLIDS • SEMICONDUCTOR ALLOYS • SUPERCONDUCTIVITY • THERMODYNAMICS • THERMOMETRY

BIBLIOGRAPHY Barnard, R. D. (1972). “Thermoelectricity in Metals and Alloys,” Halsted Press (Division of John Wiley & Sons), New York.

Thermoelectricity Broido, D. A., and Reinecke, T. L. (1995). “Thermoelectric figure of merit of quantum wire superlattices,” Appl. Phys. Lett. 67(1), 100–102. Gray, P. E. (1960). “The Dynamic Behavior of Thermoelectric Devices,” Technology Press of the Massachusetts Institute of Technology/John Wiley & Sons, New York. Guggenheim, E. A. (1957). “Thermodynamics,” 3rd ed., North-Holland, Amsterdam. Harman, T. C., and Honig, J. M. (1967). “Thermoelectric and Thermomagnetic Effects and Applications,” McGraw–Hill, New York. Hicks, L. D., and Dresselhaus, M. S. (1993). “Effect of quantum-well structures on the thermoelectric figure of merit,” Phys. Rev. B 47(19), 12 727–12 731. Ioffe, A. F. (1957). “Semiconductor Thermoelements and Thermoelectric Cooling,” Infosearch, London. Kanatzidis, M. G. (2001). The role of solid-state chemistry in the discovery of new thermoelectric materials. In “Solid State Physics” (H. Ehrenreich and F. Spaepen, eds.), Vol. 69, pp. 51–100, Academic Press, New York. Mahan, G. D. (1998). Good thermoelectrics. In “Solid State Physics” (H. Ehrenreich and F. Spaepen, eds.), Vol. 51, pp. 82–157, Academic Press, New York. Roberts, R. B. (1977). “The absolute scale of thermoelectricity,” Philos. Mag. 36(1), 91–107. Slack, G. A. (1995). New materials and performance limits for thermoelectric cooling. In “CRC Handbook of Thermoelectrics” (D. M. Rowe, ed.), CRC Press, New York. Wendling, N., Chaussy, J., and Mazuer, J. (1993). “Thin gold wires as reference for thermoelectric power measurements of small samples from 1.3 K to 350 K,” J. Appl. Phys. 73(6), 2878–2881.