Thermoelectric Elements with Negative Temperature Factor of Resistance

*Yuri Bokhan*

## **Abstract**

The method of manufacturing of ceramic materials on the basis of ferrites of nickel and cobalt by synthesis and sintering in controllable regenerative atmosphere is presented. As the generator of regenerative atmosphere the method of conversion of carbonic gas is offered. Calculation of regenerative atmosphere for simultaneous sintering of ceramic ferrites of nickel and cobalt is carried out. It is offered, methods of the dilated nonequilibrium thermodynamics to view process of distribution of a charge and heat along a thermoelement branch. The model of a thermoelement taking into account various relaxation times of a charge and warmth is constructed.

**Keywords:** thermoelement, ferrites, thermistor, nanoceramics, model

## **1. Introduction**

Perspective direction of building of thermal cells with a high Z-factor is use nanomaterial's and nanocomposite [1].

The quantity of works, theoretical and experimental, devoted to research thermoelectric nanomaterial's, steadily grows. The received results are optimistically enough, at least, from the point of view of fundamental science.

The augmentation of thermoelectric quality factor in nanomaterial's is bound to two physical phenomena [2]:


For building of ceramic semi-conductor stuffs, instead of doping by certain impurities it is possible to use a furnacing method in a reducing atmosphere [3, 4]. Such method, together with use of stuffs with NTRC, will allow to raise quality

factor of thermoelectric stuffs and to raise efficacy of thermal cells, especially in the field of high temperatures where ceramic stuffs, are the steadiest. Such method probably building n - and p - phylum of conductivity a choice of composition of gas atmosphere and furnacing temperatures in one shot.

### **2. Material's**

In thermoelectric nanocomposites the size of grain does not exceed several tens nanometers. It is obvious that for increasing of thermoelectric performance efficiency, the following condition is necessary: the size of grain should be less, than the average length of free run of phonons, but more than the average length of free run of charge carriers (electrons or holes). In this case phonons on intercrystallite borders, disperse more effectively and it leads to stronger reduction of thermal conductivity (at the expense of reduction of grid contribution), in comparison with electric conductivity reduction, providing total increase of thermoelectric quality factor.

It is obvious that in nanocomposites the lobe of intercrystallite borders will increase with the reduction of aggregate size, it will lead to consecutive depressing of thermal conductivity of the material. It is natural that dispersion of electrons on intercrystallite borders will take place leading to the reduction of their motility. However, the thermal conductivity reduction in volume nanocomposites can be more essential, than electric conductivity reduction. Thus, volume nanocomposites, consisting from nano the grains of the thermoelectric material that are divided by intercrystallite borders, can potentially possess high thermoelectric efficiency. They will have electric conductivity high and low thermal conductivity simultaneously.

Oxidic ceramic materials may be transferred into semiconductor state by means of process of operated valence. For this purpose, different methods are used, such as, a restoration method, i.e. ceramics furnacing in the regenerative medium [3, 4], unisovalent substitution are routinely used. In this case it is possible to receive comprehensible conductivity, at conservation of low thermal conductivity. In ceramic materials probably substantial growth of the dispersion mechanism, at low thermal conductivity, but it leads to quality factor augmentation.

*Thermoelectric Elements with Negative Temperature Factor of Resistance DOI: http://dx.doi.org/10.5772/intechopen.98860*

So, for example, some series of spinel's, at isomorphous substitution are transferred in to a state of the semi-conductor conductivity, possessing conductivity with NTRC (**Figure 1**) [5–8].

Oxide doping of nickel by lithium leads to sharp augmentation of conductivity at the expense of changing lithium ions into nickel ions in octahedral positions. The formation of the solid solution with uncompensated charge allows to create different types of conductivity by the variation of lithium concentration.

$$\left(\mathbf{x}/2\operatorname{Li}\_2\mathbf{O} + (1-\mathbf{x})\operatorname{NiO} + \mathbf{x}/4\operatorname{O}\_2 \rightarrow \left(\operatorname{Ni}^{2-}\operatorname{I}\_{1-2\mathbf{x}}\operatorname{Li}^{+}\operatorname{Xi}^{3+}\mathbf{x}\right)\mathbf{O}\right)$$

Similar reaction takes place when CO is replaced. If it is possible to omit superfluous oxygen, the solution of oxides receives the additional uncompensated charge in octahedrons spinel, and that process leads to conductivity augmentation. It follows that having combined doping with roasting in the recovery medium, it is possible to receive ceramic materials with adjustable conductivity. You should mind that thermal ceramic conductivity is defined by the phonon mechanism with characteristic wave length � 5–10 microns, and creating necessary grain frame of ceramics it is possible to achieve substantial increase of quality factor of the material.

Joint roasting in the recovery medium of ferrite on the basis of spinel and dielectrics, such as solid solutions on the basis of perovskite families of titanates of barium, strontium and lead, allows to create multilayered frames [3].

Let us consider as model of spinels recovery NiMn2O4 and CoMn2O4 in a gaseous medium received by conversion СO2 and H2O over Carboneum. The calculation shows that such reactions happen in some stages. It is difficult to analysis all reactions, we will illustrate calculation on the reactions defining composition of a gaseous environment (**Figure 2**).

Reaction СO2 + C ! 2CO takes place with ΔZ0=43462+6,121TlgT-62,746 T – is potential change of Gibbs at temperature T and lgKp <sup>=</sup> � 9500 T�<sup>1</sup> - 1,338lgT + 13.715 - an equilibrium constant. Reaction C + 2H2O!CO + H2 has with ΔZ0=35730 +6,121TlgT-55,403 T and lgKp <sup>=</sup> � 7811 T�<sup>1</sup> - 1,338lgT + 12.170 accordingly [6]. The joint analysis of the received expressions shows that thermodynamically reactions are resolved with Т ≈ 500К, and kineticly proceed with sufficient speed with Т ≈ 700К.

The recovery of nickel oxides and cobalt thermodynamically is possible with Т ≈ 700К. Thus, kineticly, recovery reactions take place in demanded atmosphere and there is no necessity of a pre-treatment of medium. Using diagrammes of Ellingem-Richardson-Dzheffez (**Figure 3**), we see that at the yielded temperatures cobalt and nickel recovery descends simultaneously. It allows to create simple adjustment of temperature necessary degree of restoration in spinels.

It is necessary to notice that at such temperatures there is a restoration to manganese metal. Calculation shows that manganese is reduced to the bivalent state and does not variate the position in a grid. At the same time nickel and cobalt, are restored to a monovalent state and provide n and р conductivity accordingly.

**Figure 2.** *The schema of installation for roasting in a reducing atmosphere.*

#### **Figure 3.**

*Diagramme of Ellingem-Richardson-Dzheffez for equilibriums metal - metal oxide: Points of M and In - temperatures of elements phase changes.*

Thus, it is possible to create semi-conductor branches of thermal cells with various types of conductivity in one technological process is possible.

It is obvious that when the size of grain will decrease, the area of borders of grains will increase. It will lead to higher degree of restoration of a ceramic material on border of grains. Accordingly, conductivity of borders will differ from conductivity actually grains. Naturally, dispersion electron's in grain borders will take place, and their mobility will be reduced. However, on the average, change of conductivity of a material will not be so essential because of certain uniformity of a material. Thus, there is a possibility at the expense of essential reduction phonon's heat conductivity to raise good quality of a material. Therefore, use of ceramic materials with NTCR allows to create high-temperature thermoelements with good quality [9].

#### **3. Microscopic model**

Known microscopic models of thermoelectric effect in semiconductors, to some extent, are grounded on the solution of the equation of Boltzmann's.

Let us spot thermoelectromotive force (TEF) in the presence of a temperature lapse rate (∇*Т* 6¼ 0) from the stationary kinetic equation of Boltzmann's in relaxation time approach:

*Thermoelectric Elements with Negative Temperature Factor of Resistance DOI: http://dx.doi.org/10.5772/intechopen.98860*

$$(\bar{V}, \nabla\_{\bar{\mathcal{U}}} f) + \frac{1}{\hbar} (\bar{F}, \nabla\_{\bar{\mathcal{U}}} f) = -\frac{f - f\_0}{\tau(\bar{k})} = -\frac{f^{(1)}(\bar{k})}{\tau(\bar{k})}.\tag{1}$$

were *f* � *f* <sup>0</sup> = *f* ð Þ1 *k* , а *F* – exterior force, *τ k* – a relaxation time. In case of the undegenerated semiconductor the equilibrium distribution

function for electrons is accepted in a view:

*f on* ¼ *e* �*Е*�*<sup>μ</sup> <sup>k</sup>*0*<sup>T</sup>* for electrons, *f op* ¼ *e μ*0�*E <sup>k</sup>*0*<sup>T</sup>* ¼ *e* �*E*þ*μ*þ*ΔЕ <sup>k</sup>*0*<sup>T</sup>* for electron defects. Here *<sup>μ</sup>*<sup>0</sup> ¼ �*Δ<sup>Е</sup>* � *<sup>μ</sup>*, *<sup>E</sup>* <sup>¼</sup> <sup>2</sup>*k*0<sup>2</sup> 2*m*<sup>∗</sup> *p* ,.

Approving of the allowance to equilibrium function *f* ð Þ<sup>1</sup> *k* by a small, on the left of Eq. (1) exchange *f* на *f*0. Then we will have:

$$\nabla\_{\vec{\eta}} f \approx \nabla\_{\vec{r}} f\_0 = \frac{\partial f\_0}{\partial T} \nabla T + \frac{\partial f\_0}{\partial \mu} \nabla \mu = \frac{\partial f\_0}{\partial E} \left( \frac{\mu - \mathcal{E}}{T} \nabla T - \nabla \mu \right), \tag{2}$$

$$
\nabla\_{\bar{k}} f \approx \nabla\_{\bar{k}} f\_{\; \; 0} = \frac{\partial f\_{\; \; 0}}{\partial E} \nabla\_{\bar{k}} \mathbf{E} = \hbar \frac{\partial f\_{\; \; 0}}{\partial E} \bar{\mathbf{V}}.\tag{3}
$$

Substituting expressions (2) and (3) in the Eq. (1), and being restricted to a case when the electric field *E* ¼ �∇*φ*, where φ - electrostatic potential, we will have for electrons operates only:

$$\bar{V}\_n \frac{\partial f\_0}{\partial \mathcal{E}} \left\{ \frac{\mu - \mathcal{E}}{T} \nabla T - \nabla \mu \right\} + \epsilon \frac{\partial f\_0}{\partial \mathcal{E}} \bar{V}\_n \nabla \phi = -\frac{f^{(1)}(\bar{k})}{\pi\_\epsilon(\bar{k})}.\tag{4}$$

From the Eq. (4) we will spot *f <sup>n</sup>* ð Þ<sup>1</sup> *k* :

$$f\_{\
u}^{\ \ (1)}\left(\bar{k}\right) = \tau\_{\epsilon}\left(\bar{k}\right)\frac{\partial f\_{\ 0}}{\partial \mathcal{E}}\left\{\frac{\mu - \mathcal{E}}{T}\nabla T - \nabla(\mu - \epsilon\phi)\right\}\bar{V}\_{n}.\tag{5}$$

for electron defects:

$$\left\{f\_p\right\}^{(1)}\left(\bar{k}'\right) = \tau\_p \frac{\partial f\_0}{\partial E} \left\{\frac{\mathcal{E} + \mu + \Delta\mathcal{E}}{T} \nabla T - \nabla(\mu - e\phi)\right\} \bar{V}\_p. \tag{6}$$

From relations (2)–(6) it is visible that the equilibrium distribution function is supposed nonuniform, i.e. in an equilibrium state there are processes of transport of heat and particles, and transport of particles is carried out not only at the expense of an external field. Usually it is considered that the dispersion mechanism can be considered through a relaxation time.

For example, dispersion of charge carriers is carried out at interaction with ultrasonic oscillations of a crystalline lattice. In this case the free length *l* ¼ *Vτ* does not depend on energy of carriers, and it is possible to express a relaxation time through *l*:

$$
\tau = \frac{l}{V} = \frac{m\_n^\* l}{\hbar} k^{-1}.\tag{7}
$$

Having entered a label *<sup>Е</sup>* <sup>¼</sup> *<sup>k</sup>*0*Tα*, here *<sup>α</sup>* <sup>¼</sup> *<sup>d</sup><sup>ε</sup> dT* – specific TEF, equal to the relation TEF to an individual difference of temperature.

Then

$$\bar{j}\_n = n u\_n \left\{ \nabla (\mu - \epsilon \phi) + \left( 2k\_0 - \frac{\mu}{T} \right) \nabla T \right\},\tag{8}$$

$$\bar{j}\_p = p u\_p \left\{ \nabla (\mu - e\phi) - \left( 2k\_0 - \frac{\mu + \Delta E}{T} \right) \nabla T \right\}. \tag{9}$$

In this expression <sup>4</sup>*el* 3 2*πm*<sup>∗</sup> *<sup>n</sup>* ð Þ *<sup>k</sup>*0*<sup>T</sup>* <sup>1</sup> *=*<sup>2</sup> ¼ *un*ð Þ *T* – Mobility of electrons.

The full density of a current we will spot expression:

$$\bar{j} = nu\_n \left\{ \nabla(\mu - \mathbf{e}\phi) + \left(2k\_0 - \frac{\mu}{T}\right) \nabla T \right\} + $$
 
$$ pu\_p \left\{ \nabla(\mu - \mathbf{e}\phi) - \left(2k\_0 - \frac{\mu + \Delta \mathcal{E}}{T}\right) \nabla T \right\} \tag{10} $$

For a finding TEF it is necessary to spot a potential difference at a broken circuit. Having equated *j* ¼ 0, from (10) equality follows:

$$\nabla \left(\frac{\mu}{e} - \phi\right) = -\frac{k\_0}{e} \left(2 - \frac{\mu}{k\_0 T}\right) \nabla T.$$

Specific TEF α it is spotted as

$$a = \frac{\left|\nabla\left(\phi - \frac{\mu}{\varepsilon}\right)\right|}{\left|\nabla T\right|}. \tag{11}$$

For the natural semiconductor *<sup>n</sup>* <sup>=</sup> *<sup>p</sup>* <sup>=</sup> *ni*, *<sup>μ</sup>* ¼ � *<sup>Δ</sup><sup>Е</sup>* <sup>2</sup> and the relation (11) will look like:

$$a = \frac{k\_0(b-1)}{e(b+1)} \left( 2 + \frac{\Delta \mathcal{E}}{2k\_0 T} \right). \tag{12}$$

here *<sup>b</sup>* <sup>¼</sup> *un up* , *<sup>Δ</sup>*<sup>Е</sup> <sup>2</sup> ¼ *μ*.

From the gained relation (12) it is visible that quantity TEF for the natural semiconductor is spotted only by forbidden band breadth ΔE and a relation of mobility of charge carriers.

Expression (4) is gained, actually, in approach, when derivative of function much less than the function. It means that times much major, then warmth and charge relaxation times are considered. During too time presence of dependence of temperature and potential from co-ordinates specify in presence of interior lapse rates. Therefore, it is not absolutely correct use of an equilibrium distribution function as it corresponds to concept of an equilibrium state of local approach.

Consecutive viewing of transport of a charge and heat for the systems which are in lapse rates of temperature and potential can be spent a method of the nonequilibrium statistical operator [10].

The method of the nonequilibrium statistical operator allows to write down a uniform fashion the equations featuring a kinetics, taking into account a principle of impairment of correlations, and to gain expressions for relaxation times.

As method bottom the combined equations - the dilated equations Neumann's background serves:

$$\frac{\partial \rho}{\partial t} + \frac{i}{\hbar}[H, \rho(t)] = -\varepsilon \left(\rho(t) - \rho\_{eq}(t)\right) \tag{13}$$

*Thermoelectric Elements with Negative Temperature Factor of Resistance DOI: http://dx.doi.org/10.5772/intechopen.98860*

where H - a Hamiltonian of system's, *ρeq*ð Þ*t* - the quasi-equilibrium statistical f operator, and *ϵ* – - the infinitesimal radiant providing irreversibility, which *ϵ* ! 0 after thermodynamic transition.

Hamiltonian we will choose in a view [11]:

$$\begin{aligned} \mathbf{H} &= \mathbf{H}\_0 + \mathbf{H}\_1 + \mathbf{H}\_2, \end{aligned} \tag{14}$$

$$\begin{aligned} \mathbf{H}\_0 &= \sum\_k E\_k a\_k^+ a\_k + \sum\_m E\_m a\_m^+ a\_m + \sum\_{m'} I\_{mm'} a\_{m'}^+ a\_m + \sum\_q \hbar a\_{m'}^+ c\_q, \\\\ \mathbf{H}\_1 &= \sum\_{k m q} \left( V\_q + M\_q a\_m^+ a\_m \right) e^{iqR\_m} a\_{k+q}^+ a\_k \\\\ &+ \frac{1}{2} \sum\_{k k' q} \frac{4\pi c^2}{q^2} a\_{k+q}^+ a\_{k'} a\_{k'-q}^+ a\_k \\\\ &+ \sum\_{kk'm q} \left( R\_{m k q} a\_m^+ a\_{k'} a\_{k'-q}^+ a\_k + R\_{m k q}^\* a\_k^+ a\_{k'} a\_{k'-q}^+ a\_m \right), \end{aligned}$$

$$\mathbf{H}\_2 = \sum\_{k \neq m} \mathbf{V}\_{k \neq m} \left( c\_q^+ + c\_{-q} \right) \left( a\_{k+q}^+ a\_k + a\_m^+ a\_{m'} + a\_{k+q}^+ a\_m \right),$$

Here *a*<sup>þ</sup> *m k*ð Þ *am k*ð Þ � �- operators of a creation (annihilation) of electrons corresponding localized m (nonlocalized - k) to states, *c*<sup>þ</sup> *<sup>q</sup> c*�*<sup>q</sup>* � � - operators of a creation (annihilation) of phonons with a wave vector q, *Jmm*<sup>0</sup> - a matrix element of a jump between the localized states, *Vq* - potential of scatterers, *Mq* ≈ <sup>4</sup>*πe*<sup>2</sup> *q*2 P *q* Ψ∗ <sup>0</sup> ð Þ *<sup>p</sup>* <sup>Ψ</sup>0ð Þ *<sup>p</sup>* <sup>þ</sup> *<sup>q</sup>* , *Rmkq* <sup>≈</sup> <sup>4</sup>*πe*<sup>2</sup> *<sup>q</sup>*<sup>2</sup> *<sup>e</sup>i k*ð Þ <sup>þ</sup>*<sup>q</sup>* <sup>∙</sup>*Rm* <sup>Ψ</sup>0ð Þ *<sup>k</sup>* <sup>þ</sup> *<sup>q</sup>* , V*kqm*- - a matrix

elements the interaction electron–phonon, obvious expression for which is spotted by the concrete mechanism of dispersion.

The formal solution (13) we will write down in a view:

$$\rho(t) = \rho\_{eq}(t) - \lim\_{\varepsilon \to +0} \int\_{-\infty}^{t} dt' e^{-\varepsilon(t-t')} e^{-\frac{i(t-t')\mathbb{H}}{\hbar}} \left( \frac{\partial \rho\_{eq}(t')}{\partial t'} + + \frac{1}{i\hbar} \left[ \rho\_{eq}(t'), \mathbb{H} \right] \right) e^{\frac{i(t-t')\mathbb{H}}{\hbar}}.\tag{15}$$

Boundary conditions we will accept, in view of independence of a Hamiltonian of time, in a standard view [10]:

$$\operatorname{Tr}\left(\rho\_{\text{eq}}\mathbf{n}\_{\text{m}}\right) = \operatorname{Tr}(\rho\mathbf{n}\_{\text{m}}) = <\mathbf{a}\_{\text{m}}^{+}\mathbf{a}\_{\text{m}} >,$$

$$\operatorname{Tr}\left(\rho\_{\text{eq}}\mathbf{n}\_{k}\right) = \operatorname{Tr}(\rho\mathbf{n}\_{k}) =  \tag{16}$$

At the initial moment of time <*a*<sup>þ</sup> *<sup>k</sup>*,*<sup>m</sup>ak*,*<sup>m</sup>* <sup>&</sup>gt;*<sup>t</sup>*¼<sup>0</sup> <sup>=</sup> *<sup>e</sup>*ð Þ *<sup>E</sup>*ð Þ� **<sup>k</sup>** *<sup>μ</sup> <sup>=</sup>kT* <sup>þ</sup> <sup>1</sup> � ��<sup>1</sup> - an equilibrium distribution function of electrons with temperature T.

Expressions for streams of a charge and heat we will choose in a view:

$$j(r,t) = -\frac{2e}{\left(2\pi\right)^3} \int v\_{\varnothing} f(\mathbf{k}, r, t) d\mathbf{k},\tag{17}$$

$$W(r,t) = \frac{2}{\left(2\pi\right)^{3}} \left[ \left(E(\mathbf{k}) - \mu\right) \upsilon\_{\mathbf{k}} f(\mathbf{k}, r, t) d\mathbf{k},\right] \tag{18}$$

#### *Thermoelectricity - Recent Advances, New Perspectives and Applications*

where μ - chemical potential. It is necessary to score that in an equilibrium state when *f*ð Þ¼ **k**, *r*, *t f* <sup>0</sup>ð Þ **k**, *r*, – an equilibrium distribution function, streams are equal to zero. Thus, the problem solution consists in a finding and, after substitution, expressions for streams of a charge and warmth (17) and (18).

We choose a distribution function in a view:

$$f(\mathbf{k}, r, t) =  \, \, \,,\tag{19}$$

The select of a distribution function in the form of (19) is caused by majority carriers of a charge taking into account possibility of transition of electrons from non-local in the localized states. Actually, it means possibility of an interference of states in basic one-particle states [11]:

$$\mathbf{1} = \sum\_{\mathbf{m}} |\mathbf{m} > < \mathbf{m}| + \sum\_{\mathbf{k}} |\mathbf{k} > < \mathbf{k}|$$

Such select is caused by definition of dynamic variables in method nonequilibrium statistical operator as these variables spot charge and warmth transport. At the expense of a phonon subsystem it is difficult to express transport of heat in the form of a stream. However, the shape of a Hamiltonian (14) allows to express < *c*<sup>þ</sup> *<sup>q</sup> cq* > through < *a*<sup>þ</sup> *<sup>k</sup>*,*<sup>m</sup>ak*,*<sup>m</sup>* > in the second order on a constant an interaction electron–phonon.

As it is known [11] to write down the transport equations relation performance is necessary:

$$[\mathbf{P}\_{\mathbf{k},}\mathbf{H}\_{\mathbf{0}}] = \sum\_{\mathbf{l}} \mathbf{c}\_{\mathbf{kl}} \mathbf{P}\_{\mathbf{l}}.$$

Using commutation relations of pairs of fermi-operators we will gain for a *P*<sup>11</sup> ¼ *a*<sup>þ</sup> *<sup>k</sup> ak*, *P*<sup>22</sup> ¼ *a*<sup>þ</sup> *<sup>m</sup>*0*am*, *P*<sup>12</sup> ¼ *a*<sup>þ</sup> *<sup>k</sup> am*, *P*<sup>21</sup> ¼ *a*<sup>þ</sup> *mak* relation:

$$\begin{aligned} \left[\mathbf{P}\_{\mathbf{1}1}, \mathbf{H}\_{0}\right] &= \mathbf{0}; \left[\mathbf{P}\_{22}, \mathbf{H}\_{0}\right] = \mathbf{0};\\ \left[\mathbf{P}\_{12}, \mathbf{H}\_{0}\right] &= -\sum\_{\mathbf{m'}} \left[\left(\mathbf{E}\_{\mathbf{k}} - \mathbf{E}\_{\mathbf{m}}\right) - \mathbf{J}\_{\text{mm'}} \delta\_{\text{mm'}}\right] \mathbf{a}\_{\mathbf{k}}^{+} \mathbf{a}\_{\mathbf{m}};\\ \left[\mathbf{P}\_{21}, \mathbf{H}\_{0}\right] &= \sum\_{m'} \left[\left(E\_{\mathbf{k}} - E\_{m}\right) - \mathbf{J}\_{mm'}\right] a\_{m}^{+} a\_{k}; \end{aligned} \tag{20}$$

Thus from (20) it is visible that in processes of transport the dominant role is played by <*a*<sup>þ</sup> *<sup>k</sup> am* >, <*a*<sup>þ</sup> *mak* >, < *a*<sup>þ</sup> *<sup>k</sup> ak* > addends and serves as the tank having the temperature and chemical potential, depending on co-ordinates and time. Therefore, it is possible to spot from the kinetic equations and coefficients from (17) to (18).

Let us write down system of the kinetic equations for < *P*<sup>12</sup> > and < *P*<sup>21</sup> > in a view:

$$\begin{split} &\frac{\partial < \mathbf{P}\_{12} > -\frac{1}{\mathbf{i}\hbar} \sum\_{\mathbf{m'}} \left[ (\mathbf{E}\_{\mathbf{k}} - \mathbf{E}\_{\mathbf{m}}) - \mathbf{J}\_{\text{mm'}} \delta\_{\text{mm'}} \right] < \mathbf{P}\_{12} \geq \ &= \\ &= \mathbf{S}\_{12}^{(1)} + \mathbf{S}\_{12}^{(2)}; \\ &\mathbf{S}\_{12}^{(1)} = \frac{1}{\mathbf{i}\hbar} < \left[ \mathbf{P}\_{12}, \mathbf{H}\_{\text{int}} \right] >; \\ &\mathbf{S}\_{12}^{(2)} = -\frac{1}{\hbar^2} \int\_{-\infty}^t dt' e^{r(t'-t)} < \left[ H\_{\text{int}}, [H\_{\text{int}}, \mathbf{P}\_{12}] + i\hbar \, P\_{12} \frac{\partial \mathbf{S}\_{12}^{(1)}}{\partial < P\_{12}} \right] >, \end{split} \tag{21}$$

*Thermoelectric Elements with Negative Temperature Factor of Resistance DOI: http://dx.doi.org/10.5772/intechopen.98860*

Where *Hint* ¼ *H*<sup>1</sup> þ *H*2*:* Similar equation registers and for < *P*<sup>21</sup> > to within transposition of coefficients.

Carrying out commutated in collisional members, we will gain the kinetic equations for < *P*<sup>12</sup> > and < *P*<sup>21</sup> > in the second order on force constants and quadratic on <*P*<sup>12</sup> > , <*P*<sup>21</sup> >*:*

Thus writing down the kinetic equation for and carrying out integration on p we will gain the Eq. (21) in which the right part is spotted through.

Without giving bulky expressions for streams, taking into account dispersion mechanisms, we will give dependences of relaxation times for the elementary case of transfer in the first order.

As the free length usually, I use various relaxation times enters into relations for mobilities. At high temperatures i.e. when energy of a phonon ℏ*ω*<sup>0</sup> is much less than energy of an electron *k*0*T* dispersion it is possible to consider elastic. Considering that at dispersion on ultrasonic phonons ℏ*ω*<sup>0</sup> < < *k*0*T* also we will gain:

$$
\pi = \frac{\sqrt{2}}{4\pi} \frac{M a^3 (\hbar a v\_0)^2 \mathcal{E}^\natural}{Z^2 e^4 m^{\*\natural} k\_0 T}. \tag{22}
$$

Free length

$$l = \tau V = \frac{V}{2\pi} \frac{M}{m^\*} \left(\frac{\hbar a\_0}{\mathcal{Z}^2 \slash\_a}\right)^2 \frac{\mathcal{E}}{k\_0 T},\tag{23}$$

Were *V* – velocity of an electron.

In case of low temperatures, ℏ*ω*<sup>0</sup> > >*k*0*T*, The electron–phonon interaction becomes inelastic. In this case processes of uptake of phonons are possible only, and to enter a relaxation time it is impossible. But in case of low temperatures if to consider a requirement ℏ*ω*<sup>0</sup> > >*k*0*T*, the majority of the electrons immersing energy of a phonon, transfer in an energy interval from ℏ*ω*<sup>0</sup> to 2ℏ*ω*0. Such electrons will almost instantaneous let out phonons, since the relation of probability of emission to probability of uptake *Nq*þ<sup>1</sup> *Nq* <sup>≈</sup> exp <sup>ℏ</sup>*ω*<sup>0</sup> *<sup>k</sup>*0*<sup>T</sup>* > >1. As a result of such uptake energy of an electron does not change almost. It allows to view interaction of an electron with optical oscillations of a lattice at very low temperatures as elastic and to enter a relaxation time.

Calculation for a relaxation time gives view expression [12]:

$$\tau\_{\rm on} = \frac{3\sqrt{2}}{4\pi} \frac{M a^3 (\hbar o\_0)^{3/2}}{Z^2 e^4 m^{\* \dagger \frac{1}{2}}} e^{\frac{\hbar o\_0}{k\_0 T}}. \tag{24}$$

From (15) it is visible that at low temperatures the relaxation time on optical

phonons does not depend on energy, and on temperature depends exponentially.

Free length

$$l = \tau V = \frac{3a}{2\pi} \frac{M}{m^\*} \left(\frac{h\nu\_0}{Z\epsilon^2/\_a}\right)^2 \epsilon^{\frac{h\nu\_0}{K\_0 T}} \sqrt{\frac{\mathcal{E}}{k\_0 T}}.\tag{25}$$

It is necessary to score that because of presence exp <sup>ℏ</sup>*ω*<sup>0</sup> *k*0*T* � � free length always more than interatomic distance, i.e. *l* >>*a*.

From the presented analysis follows that charge relaxation times have essentially non-linear temperature dependence and the dispersion mechanism. All it demands the approach which is distinct from the traditional.

### **4. A method of the extended irreversible thermodynamics**

In the standard approach of modeling of distribution of warmth in a thermoelement [4] as a rule, the classical equations of balance of transport of warmth are used:

$$q = a\dot{q}T - \frac{j^2 rL}{2} \tag{26}$$

where q - a specific heating capacity, T - temperature теплоотдающей mediums, α, r - TEF and a thermomaterial specific resistance, j - current density, L - thermobranch length.

Prominent feature of model (26) is lack of a time dependence q and temperature ρ. In case of use as a branch of a thermoelement of a material with NTCR [6], the temperature dependence is essential and demands the account at modeling of parameters of a thermoelement. The consecutive account of relaxation processes can lead to occurrence of waves of warmth and essentially changes character of distribution of heat at an initial stage of process.

Other feature of modeling is comparison of effects of model with experiment. Enough compound circuit of excitation and temperature measuring's is observationally used. The most attractive the plan with use as a radiant of warmth and a charge of an impulse of a current (**Figure 4**) looks.

In this case, measuring temperatures in points Ti, at excitation by a current impulse (point T1) it is possible to realize the direct plan of distribution of warmth along a thermoelement branch.

As it was already specified earlier, use of classical thermodynamics invokes certain fundamental problems because of presence in system of a thermal cell of streams of heat and a charge. Therefore, the thermodynamic approach possesses intrinsic discrepancy. Such problems can be avoided, using the approach of extended irreversible thermodynamics [13]. Extended irreversible thermodynamics far from local balance uses as new explanatory variables dissipation streams, i.e. heat stream q, mass flux J and stress tensor P. Thus, in nonequilibrium system entropy S is function not only classical variable, but also dissipation streams:

$$\mathbf{S} = \mathbf{S} \left\{ \mathbf{U} \left( \mathbf{x}, \mathbf{t} \right), \mathbf{v} \left( \mathbf{x}, \mathbf{t} \right), \mathbf{C} \left( \mathbf{x}, \mathbf{t} \right), \mathbf{q} \left( \mathbf{x}, \mathbf{t} \right), \mathbf{I} \left( \mathbf{x}, \mathbf{t} \right) \right\}.$$

Introduction of streams as explanatory variables quite defensible from the physical point of view. Really, if in system there is any stream it means the directed

**Figure 4.** *Plan of modeling of a thermobranch.*

*Thermoelectric Elements with Negative Temperature Factor of Resistance DOI: http://dx.doi.org/10.5772/intechopen.98860*

locomotion of carriers of heat or mass. Hence, entropy which, as it is known, is a measure of affinity to an equilibrium state, specifies directions of locomotion of all system. Extended irreversible thermodynamics, in difference from classical is irreciprocal, introduces into processes time, as a variable that leads to the differential equations for dissipation streams of evolutionary (relaxation) phylum. In the elementary case relaxation times enter into such equations (Maksvella-Kataneo), in our case of heat and a charge.

Hence using methods of extended irreversible thermodynamics and the relaxation times, received from modeling representations about interactions in stuffs, it is possible to construct the consistent theory of the thermoelectric phenomena.

The initial combined equations can be written down in a view [13].

$$q + \text{tr}\_T \frac{\partial q}{\partial t} = -\lambda \nabla T + q\_0(t, \mathbf{x}) \tag{27}$$

$$
\tau\_t \frac{\partial \dot{i}}{\partial t} = -(\dot{i} - \sigma E') \tag{28}
$$

where: *q*0ð Þ *t*, *x* – source of heat, *λ* is the thermal conductivity coefficient, *<sup>E</sup>*<sup>0</sup> <sup>¼</sup> *<sup>E</sup>* � *<sup>T</sup>*<sup>∇</sup> *<sup>T</sup>*�<sup>1</sup> *μe* , *Е* is the electric field strength, *μ<sup>e</sup>* is the chemical potential, *τT*, *τ<sup>e</sup>* is the relaxation time of heat and charge, *σ<sup>e</sup>* is conductivity,*T* is temperature.

Thus, the inclusion of dissipative flows in the series of independent variables leads to the fact that these flows are no longer determined by the gradient of the corresponding transfer potential, as in the classical local-equilibrium case, but they are solutions of the evolution Eqs. (27) and (28). These equations describe the process of relaxation of dissipative flows to their local-equilibrium values.

While analyzing the system of Eqs. (27) and (28), we use the following approximations. We assume that the coefficient of thermal conductivity and the relaxation time of the heat are constant and temperature is independent. Such assumption is correct in connection with the fact that the calculation of the heat relaxation time must be carried out taking into account the propagation of heat in the system. In other words, in the case of the heat propagation, the problem is self-consistent. Taking into account that the distribution of the heat and charge front may be considered in a single grain, we can assume that the spread of non-locality is rather weak, and the process describes the approach of the permanent *τ<sup>T</sup>* and *λ*.

The *τ<sup>e</sup>* - is an expression for the relaxation time of conduction electrons of a nondegenerate atomic semiconductor τe�ε �1/2 Т�<sup>1</sup> [12], where ε is an energy of the width order of the forbidden band of the semiconductor. Such an expression for the charge relaxation time is an approximation that has a temperature dependence. It is necessary to solve the kinetic equation for the charge propagation taking into account the dispersion law in the conduction mechanism [8]. However, in our case, such task is complicated by the fact that it is necessary to consider the flow of the charge along the grain surface. It complicates the solution of the kinetic equation, which must be solved taking into consideration the percolation flow model.

Let us converse (28) considering communication of reciprocity coefficients of Onsager's with phenomenological relations [13]:

$$
\nabla T^{-1} = \frac{1}{\lambda T^2} q - \frac{\mu\_e - aT}{\lambda T^2} i; \tag{29}
$$

$$E - \nabla \mu\_{\epsilon} = \frac{a}{\lambda}q - \left(a\frac{\mu\_{\epsilon} - aT}{\lambda} - r\right)i;\tag{30}$$

Here Thomson's relation is used *αT* ¼ �*P*, were P – сcoefficient of Pelte. As a result of (28) transfers in

$$\frac{\pi\_T}{\pi\_\varepsilon} \frac{\partial \dot{t}}{\partial \pi} = -\frac{\mu\_\varepsilon - aT}{r\lambda T} q + \frac{(\mu\_\varepsilon - aT)^2}{r\lambda T} \dot{t} \tag{31}$$

Time scale in (31) we will choose concerning relaxation times [10]. Here in expression (31) replacement is yielded *t =<sup>τ</sup><sup>e</sup>* ¼ *<sup>τ</sup><sup>T</sup>=τe <sup>τ</sup>*; *<sup>τ</sup>* <sup>¼</sup> *<sup>t</sup> τT* . Thus, the time dependence scale is spotted by the relation of relaxation times of warmth and a charge [12]. Conformity introduction between a return relaxation time and a thermal

conductivity *<sup>τ</sup>T*�1' � *<sup>λ</sup> λ*2 *q* , were λ<sup>q</sup> =2l – the doubled path length *λ<sup>q</sup>* ¼ *τV* ¼ � �<sup>2</sup>

3*a π M m*<sup>∗</sup> *ω*0 *Ze*<sup>2</sup> *=a e ω*0 *k*0*T* ffiffiffiffiffiffi *Е k*0*T* q , <sup>λ</sup> – thermal conductivity. *<sup>E</sup>*<sup>0</sup> <sup>¼</sup> *<sup>E</sup>* � *<sup>T</sup>*<sup>∇</sup> *<sup>T</sup>*�<sup>1</sup> *μe* � �, *τ<sup>e</sup>* – charge relaxation time, *σ* – conductivity, *μ<sup>e</sup>* – chemical potential, *Е* – electric intensity,*T* –

temperature, *ε*� energy of the order of breadth of a forbidden band of a material.

Similarly, we will converse the Eqs. (27) and (31). We use a relation (29) and having presented *<sup>λ</sup>*∇*<sup>T</sup>* ¼ �*λT*<sup>2</sup> <sup>∇</sup>*T*�<sup>1</sup> � �, let us gain:

$$
\pi\_T \frac{\partial q}{\partial t} = -(\mu\_\varepsilon - \alpha T)i + q\_0(t, \mathbf{x}) \tag{32}
$$

$$\frac{\pi\_T}{\pi\_\epsilon} \frac{\partial \dot{t}}{\partial \pi} = \frac{\mu\_\epsilon - aT}{r} \frac{\nabla T}{T} \tag{33}$$

Further we use the law of conservation of energy and (32). It is as a result had:

$$\frac{1}{4\pi} \frac{\partial^2 T}{\partial \mathbf{r}^2} = -\frac{a}{c\_v \rho} i \nabla T + \frac{1}{c\_v \rho} \frac{\partial}{\partial \mathbf{r}} q\_0(\mathbf{r}, \mathbf{x}) \tag{34}$$

were ρ – material density, c*<sup>v</sup>* - specific heat capacity.

Thus, the Eqs. (33) and (34) feature distribution of a charge and temperature allocation to a thermoelement under the influence of a current impulse.

Warmth radiant we will choose in a view:

$$q\_0(t, \mathbf{x}) = r i^2 \tag{35}$$

I.e. a radiant is Joule heat. Thus, we consider that the current does not depend on co-ordinate and is spotted only by dependence from τ. Carrying out differentiation on τ and using (33), we will gain:

$$\frac{1}{\tau\_T} \frac{\partial^2 T}{\partial \tau^2} = \frac{1}{c\_v \rho} [\gamma \mu\_\epsilon - (1 + \gamma) a T] i \frac{\nabla T}{T} \tag{36}$$

were γ =*τe=τT*.

As a result, we gain the combined equations featuring model of allocation of temperature along the sample at excitation by an impulse of a current.

$$\frac{\partial i}{\partial \tau} = \gamma \frac{\mu\_{\epsilon} - aT}{r} \frac{\nabla T}{T} \tag{37}$$

$$\frac{1}{\tau\_T} \frac{\partial^2 T}{\partial \tau^2} = -\frac{i}{c\_v} \left[ a \nabla T - \gamma (\mu\_{\epsilon} - aT) \frac{\nabla T}{T} \right]$$

*Thermoelectric Elements with Negative Temperature Factor of Resistance DOI: http://dx.doi.org/10.5772/intechopen.98860*

Prominent feature of model (37) is presence of the addends proportional to the relation of relaxation times of a charge and heat γ which acts as natural parameter little. The system (37) looks like a series development on, to the first order though at its deduction of any guesses about little it was not supposed. Thus, in the equation for change of temperature the addend of the zero order on γ is spotted by a thermoelectric stream of warmth. It shows that in the course of warmth distribution, to the initial moment of time, there is a heating at the expense of a current, as the most prompt.

Initial and boundary conditions for system we will choose in a view

$$i(\mathbf{0}) = I;\tag{38}$$

$$T(\mathbf{0}, \mathbf{0}) = T\_1; T(\mathbf{0}, L) = T\_2; \frac{\partial T}{\partial \tau}(\mathbf{0}, \mathbf{0}) = \mathbf{0}. \tag{39}$$

The predominant model of the conductivity of thermistors with NTCR is a model of hopping conductivity in the approximation of the "nonadiabatic" polaron of a small radius leading to the temperature dependence of conductivity [6]:

$$
\sigma = \pi^{\frac{3}{2}} \frac{e^2 l^2 f^2 E^{-1/2}}{h(kT)^2} \exp\left(-E/kT\right) \tag{40}
$$

where: *l* is an effective hopping length, *J* is a parameter of jamping, *E* is an energy of hop activation,*T* is a temperature. Such nonlinear temperature dependence of electrical conductivity leads to a substantial nonequilibrium process of the heat and charge transfer in the branches of the thermoelement. It should be noted that in the case of a thermoelectric effect the process has a nonlocal character both in the coordinate and time. It is usually assumed that the chemical potential does not depend on temperature and is approximately equal to the Fermi energy. However, for a nondegenerate semiconductor with a temperature conductivity dependence (40), essentially nonlinear, it is necessary to consider the temperature dependence of the chemical potential [12], which has a logarithmic temperature dependence:

$$
\mu\_{\epsilon} = kT \ln \left[ \frac{4}{3\sqrt{\pi}} \left( \frac{\varepsilon}{kT} \right)^{\frac{3}{2}} \right] < 0 \tag{41}
$$

where k is a Boltzmann constant.

Thus, the problem of calculating the heat transfer in this system is nonstationary. To solve it, we assume the model to be one-dimensional, and dismiss the second-order terms in the temperature gradients. The initial and boundary conditions are assumed to be standard [14]. Such assumptions allow us to make a qualitative analysis of the nature of the propagation of heat and charge in the system. We examine the model at the distances of the grain size order. The generalization of the sample dimensions requires the establishment of an averaging procedure, which differs from the standard method i.e. the introduction of certain average or effective parameters requires additional considerations and cannot be carried out by simple averaging.

#### **5. A construction and manufacturing of thermoelements**

For manufacturing of structural thermoelements it is convenient to use the known production technology of chips-inductances.

Formation of structure of chips-thermoelements is carried out by level-by-level drawing ferrite and conductor layers. On a surface of a flat substrate superimpose thin (to several tens micron) a stratum of paste on the basis of the dielectric powder immixed with a binding material and solvents. On a surface of the dried stratum shape a printing expedient current-carrying (ferrite NiMn2O4 and CoMn2O4) drawing in the form of a semicoil, then superimpose the stratum coating 1/2 areas of preparation and leaving unclosed extremity of a semicoil. The following part of current-carrying drawing (semicoil) is superimposed so that the extremities of conductors were imposed against each other. Similar operations are iterated the necessary number of times that spending drawings in the form of semicoils were joined consistently, forming a spiral (**Figure 5**). Deficiencies of a "stage" expedient of manufacturing concern low reliability of switching of coils in places of transition of spending paste from the inferior plane on the upper.

Other, simpler plan of switching of the extremities of semicoils through a hole in the dielectric stratum (**Figure 6**) is developed also.

As linking of semicoils is yielded each time on half of interturn distance that twice reduces probability of formation of flaws on this critical site, thanks to the free diffluence of paste linking of coils trustier, structural structure trustier.

Structural thermoelements devices are made by a group expedient, i.e. simultaneously agglomerated a considerable quantity of devices on a substrate by the area 100�100 mm. After the termination of the making up the group package is slited on separate devices which are exposed to sintering.

Making of regenerative atmosphere allows to spend formation of a firm solution to furnaces with not compensated charge that leads to various type of conductivity depending on an initial composition.

The intermixture of gases СО - СО<sup>2</sup> can be gained two expedients. The first is grounded on interaction of gases СО<sup>2</sup> and the prosir-butanovoj of an intermixture, the second expedient - on restoration СО<sup>2</sup> at its gear transmission through a stratum of the heated coal. According to the first variant regenerative medium gained by conversion the prosir-butanovoj of an intermixture and СО2. It occurs at temperature 800–1000°С. Formation of gases is accompanied by following responses:

$$\begin{aligned} \text{C}\_{\text{m}}\text{H}\_{2\text{m}+2} &\rightarrow \text{mC} + (\text{m}+\text{1})\text{ H}\_{2}, \text{C} + \text{H}\_{2}\text{O} \rightarrow \\ \rightarrow \text{CO} + \text{H}\_{2}, \text{C} + 2\text{H}\_{2}\text{O} &\rightarrow \text{CO}\_{2} + 2\text{H}\_{2}, \\ \text{C} + \text{CO}\_{2} &\rightarrow 2\text{CO} \end{aligned}$$

The conducted examinations have shown that most full responses proceed at temperature 1000°С.

Advantage of the given method is that the rate of flux of gases is simply enough set with the help concentration of gas metter. However, enough high temperature is

**Figure 5.** *The plan of the making up switching through a step.*

*Thermoelectric Elements with Negative Temperature Factor of Resistance DOI: http://dx.doi.org/10.5772/intechopen.98860*

**Figure 6.**

*The plan of the making up switching through a hole.*

necessary for making of demanded atmosphere with an optimum exit of gases �1000°С and the materials applied at manufacturing of the converter, should possess ability to work in regenerative medium (the quartz, special hightemperature grades of a steel). Besides, in this case takes place many-stage chemical responses that demands application of the special data units checking a composition of atmosphere on an exit of the converter.

On the second expedient a gas intermixture gained by gear transmission СО2 through the ceramic pipe filled with heated coal. There is an oxidizing of coal and restoration СО<sup>2</sup> on response:

СО<sup>2</sup> þ С ! 2СО

In an equilibrium state at atmospheric pressure in a gas intermixture on a converter exit the following concentration СО in % contains: 2; 15; 58; 94; 99,3 accordingly at temperature 300, 500, 600, 700, 800°С. Advantage of this method is the low temperature at which there is a formation of regenerative medium. In our case it makes 650–700°С. Regulation of a composition of medium is carried out by change of temperature of the converter. It is necessary to carry necessity of a periodic fill of the converter to deficiencies coal.

Advantage of this method is the low temperature at which there is a formation of regenerative medium. In our case it makes 650–700°С. Regulation of a composition of medium is carried out by change of temperature of the converter. It is necessary to carry necessity of a periodic fill of the converter to deficiencies coal.

As builders it is possible to use restoration шпинелей NiMn2O4 and CoMn2O4 in the gas medium gained by conversion СO2 and H2O over carbon.

The offered procedure allows with sufficient repeatability to gain ceramic semiconductor materials with n and p conductivity types at simultaneous roasting.

## **6. Results and discussion**

The results of numerical simulation are shown in **Figure 7**. The system analysis (37) was carried out for various relations τT/τ<sup>e</sup> and the dimensionless time t/τ<sup>T</sup> was used.

The result of the simulation is presented on **Figure 7a**, provided that the times of relaxation of heat and charge are close. In this case, the propagation of heat occurs almost simultaneously with the charge density. It has a character close to a solitary wave. Such result is quite obvious, since in this case Joule heat is released simultaneously with heat transfer and the increase of the charge current occurs with the velocity that is close to the velocity of propagation of heat V1. A characteristic

feature of such propagation is a formation of the wave on the length of the conductivity hop.

In the case when there is τ<sup>T</sup> >>τ<sup>e</sup> (**Figure 7b**), the break of propagation front happens and the Joule heat wave V2 outpaces the actual heat transfer wave due to the temperature gradient. Thus, in this case two waves are formed, which are spatially separated. At the same time, the relaxation of the heat does not occur during the hopping of the charge, and the system is in a locally nonequilibrium state. In other words, charge transfer generates a locally nonequilibrium state in which the charge flow is a fast variable.

In the case when there is τT< <τ<sup>e</sup> (**Figure 7c**), the heat relaxation occurs faster than the charge transfer, and a heat propagation front coinciding with the charge transfer is formed. It should be noted here that the steepness of the front is determined by the mechanism of hopping conductivity and the approximations in the calculation. When there is more correct calculation there will be no gaps on the front.

The mode of heat transfer will be especially manifested in functional gradient materials [3], especially along the grain boundaries. By creating a regular structure with the required relaxation time ratios, it is possible to achieve the wave character of the heat transfer and charge transfer. It will allow to create devices that simultaneously measure and regulate the temperature.

Investigating movement of a charging and temperature wave it is possible to estimate the relation of times of a relaxation. It will allow, at qualitative level to draw certain conclusions about the carrying over mechanism. Such possibility is very actual for ceramic materials.

The first and obvious expedient of pinch of thermoelectric quality factor *ZT* of materials is optimization of their properties spotting thermoelectric efficiency. Such expedient allows to raise, though and is in most cases inappreciable, thermoelectric properties of traditional thermoelectric materials which are well studied and for which the physical analogues allowing purposefully to spot a direction of optimization of properties of materials are developed. As the basic directions of such optimization it is possible to ooze:


Optimum concentration of charge carriers (i.e., a select of an optimum level of a doping in the course of synthesis of a thermoelectric material) allows to provide the peak value of a thermoelectric quality factor. Physically, existence of optimum concentration of charge carriers is related by that at magnification of concentration the direct-current conductivity *σ* grows, and value termo-EDS, on the contrary, decreases. Dependence *σ* from concentration of charge carriers *n* is obvious and directly follows from direct-current conductivity definition.

Dependence termo-EDS from concentration of charge carriers is caused by a gas degeneracy of carriers at magnification of concentration of the item. For degenerated electronic gas Fermi level *EF* gets to a conduction band (for the electronic semiconductor), and requirement *Ef-Ec> kT*, where *EC* - energy of a bottom of conduction band, *k* - a Boltzmann constant is satisfied.

In this case energy and velocity of electrons are spotted by value of a Fermi level and practically do not depend on temperature. For such semiconductor, in the presence of a lapse rate of temperatures on its opposite extremities, streams of

*Thermoelectric Elements with Negative Temperature Factor of Resistance DOI: http://dx.doi.org/10.5772/intechopen.98860*

**Figure 7.** *Distribution of heat in a thermoelements for various ratios τT, τe, L.*

## *Thermoelectricity - Recent Advances, New Perspectives and Applications*

electrons from the cold and hot extremities feebly differ, hence, volume termo-EDS will be inappreciable. Termo-EDS, and, hence, and a thermoelectric quality factor, it is possible to achieve pinch in semiconductors and semimetals in requirements when there is the strong degeneration, but concentration of charge carriers is great enough.

Optimization of breadth of forbidden band *Eg* is fundamental parameter of an electronic spectrum of the semiconductors which change allows to optimize their thermoelectric properties. As a result of a series of examinations [15] it has been shown that from the point of view of reception of a material with the best thermoelectric performances (for the unregenerate semiconductor) performance of following requirements is necessary:


#### **7. Conclusion**

For the first time features of carrying over of warmth and a charge in semiconductor branches of a thermoelement are considered. The model of ceramic semiconductor branches is constructed of materials with negative temperature factor of resistance for thermoelements. Methods of the expanded irreversible thermodynamics spend modeling of distribution of temperature along the sample. On the basis of the spent modeling the technique of registration of impulses of temperature and definition of the relation of times of a relaxation in materials with negative temperature factor of resistance is offered. It is received that depending on a parity of times of a relaxation of processes of carrying over of a charge and warmth, various operating modes of a thermoelement can be realized.

It is offered to use by manufacture of thermobatteries the known production technology the chips-inductances. Feature of the specified technology is packing of separate elements in assemblage which then is divided into separate elements. Thus, instead of the semi-conductor materials containing rare and often ecologically dangerous materials, it is offered to use oxides ceramics. Roasting in regenerative atmosphere of the furnace allows to make layered thermoelements for one cycle of roasting.

Modeling of thermoelements from the ceramic materials possessing negative temperature in factor of resistance is spent.

*Thermoelectric Elements with Negative Temperature Factor of Resistance DOI: http://dx.doi.org/10.5772/intechopen.98860*

## **Author details**

Yuri Bokhan Vitebsk Branch Educational Establishment, The Belarus State Academy of Communication, Vitebsk, Belarus

\*Address all correspondence to: yuibokhan@gmail.com

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

## **References**

[1] Rowe D.M. (ed.) CRC Handbook of Thermoelectrics. CRC. Boca Ration.- 1995. - 701 p.

[2] Rowe D.M. (ed.) Thermoelectrics handbook: macro to nano. CRC Press. Taylor & Francis group. 2006. - 954 p.

[3] Anatychuk, L.I. Current Status and some prospects of Thermoelectricity// Journal of Thermoelectricity. 2007. №2. P.7-20.

[4] Dmitriev, A.V., Zvyagin, I.P. Current trends in the physics of thermoelectric materials // Uspekhi Fizicheskikh Nauk. 2010. Vol. 180. №8. P. 821–838. DOI: 10.3367/UFNr.0180.201008b.0821

[5] Clarke, D.R. Oxide Thermoelectric Devices: A Major Opportunity for the Global Ceramics Community. // 5th International Congress on Ceramics, Beijing, August 2014.

[6] Feteira, A. Negative Temperature Coefficient Resistance (NTSR) Ceramic Thermistors: An Industrial Perspective. // J. Am. Ceram. Soc. 2009. vol.92. №5. p.967-983. DOI: 10.1111/ j.1551-2916.2009. 02990.x

[7] Koshi Takenaka Negative thermal expansion materials: technological key for control of thermal expansion // Sci. Technol. Adv. Mater. 2012. Vol. 13. P.1-10.DOI:10.1088/1468-6996/13/1/ 013001/

[8] Terasaki, I. High-temperature oxide thermoelectrics. //J.Appl. Phys. 2011.- 110.- 053705.

[9] Bokhan, Y. I., Varnava, A. A. Roasting of Ceramic Materials with the Negative Temperature Resistance Coefficient of Recovery Atmosphere. // Journal of Materials Sciences and Applications. 2018.Vol 4. No 3. p. 47-50. ISSN: 2381-1005 (Online)

[10] Röpke, G. Electrical Conductivity of Charged Particle Systems and Zubarev's Nonequilibrium Statistical Operator Method. Theor. Math. Phys.2018.194. p. 74–104. Doi:10.1134/ S0040577918010063

[11] Röpke, G. Electrical conductivity of a system of localized and delocalized electrons. Theor. Math. Phys.1981. 46. p. 184–190 Doi:10.1007/BF01030854.

[12] Veljko Zlatic, Rene Monnier Modern Theory of Thermoelectricity. Oxford. University Press. 2014. 289 p.

[13] Jou D., Casas-Vázquez J. and Lebon G. Extended Irreversible Thermodynamics, 3rd ed. Springer, Berlin. 2001.– 571 p.

[14] Bokhan, Y. I., Varnava, A. A. Thermoelectric ceramic element with a negative temperature factor of resistance //Journal of Thermoelectricity. 2018. No 1.p. 40-47.

[15] Melnikov, A.A., Kostishin, V.G., Alenkov, V.V. Dimensionless Model of a Thermoelectric Cooling Device Operating at Real Heat Transfer Conditions: Maximum Cooling Capacity Mode.// Journal of Electronic Materials, 2017.Vol. 46. No. 5. p.2737-2742. DOI: 10.1007/s11664-016-4952-0.

## **Chapter 8**

## Quantum Physical Interpretation of Thermoelectric Properties of Ruthenate Pyrochlores

*Sepideh Akhbarifar*

## **Abstract**

Lead- and lead-yttrium ruthenate pyrochlores were synthesized and investigated for Seebeck coefficients, electrical- and thermal conductivity. Compounds A2B2O6.5+z with 0 ≤ z < 0.5 were defect pyrochlores and *p*-type conductors. The thermoelectric data were analyzed using quantum physical models to identify scattering mechanisms underlying electrical (*σ*) and thermal conductivity (*κ*) and to understand the temperature dependence of the Seebeck effect (*S*). In the metal-like lead ruthenates with different Pb:Ru ratios, *σ* (*T*) and the electronic thermal conductivity *κ<sup>e</sup> (T)* were governed by 'electron impurity scattering', the lattice thermal conductivity *κ<sup>L</sup> (T)* by the 3-phonon resistive process (Umklapp scattering). In the lead-yttrium ruthenate solid solutions (Pb(2-x)YxRu2O(6.5*<sup>z</sup>*)), a metal–insulator transition occurred at 0.2 moles of yttrium. On the metallic side (<0.2 moles Y) 'electron impurity scattering' prevailed. On the semiconductor/insulator side between x = 0.2 and x = 1.0 several mechanisms were equally likely. At x > 1.5 the Mott Variable Range Hopping mechanism was active. *S* (*T*) was discussed for Pb-Y-Ru pyrochlores in terms of the effect of minority carrier excitation at lowerand a broadening of the Fermi distribution at higher temperatures. The figures of merit of all of these pyrochlores were still small (≤7.3 <sup>10</sup><sup>3</sup> ).

**Keywords:** Ruthenate pyrochlores, thermoelectricity, scattering mechanisms, metal–insulator transition, glass-like thermal conductivities

#### **1. Introduction**

The discovery of high thermoelectric performance of NaxCoO4 [1] has triggered renewed interest in oxide thermoelectric materials, though the figure of merit (*zT*) of this and other oxides is still too small for widespread application.

Pyrochlore is an oxide mineral [(Na,Ca)2Nb2O6 (OH,F)] that forms brown to black, glassy octahedral crystals. In natural occurrences, the A and B atom sites can be occupied by many elements. Apart from naturally occurring compounds, over 500 compositions have been synthesized [2]. This wide-spread interest in pyrochlores is due to their large spectrum of properties, which include electronic, magnetic, electro-optic, piezoelectric, catalytic and more. The structure of pyrochlores is usually described by the topology and the geometric shape of coordination polyhedra. The structural formula is VIIIA2 VIB2 IVX6 IVY. The roman numerals show the coordination numbers. The crystal structure of pyrochlores is

face-centered cubic (fcc). The space group is *Fd*3*m*, the lattice parameter is a = 0.9-1.3 nm. There are 8 formula units in the unit cell. X is oxygen (O2�) and Y can be oxygen, hydroxyl, fluoride (O2�, OH�, F�). The unit cell contains larger A (*rA* = 0.087 – 0.151 nm) and smaller B cations (*rB* = 0.040 – 0.078 nm) [3] surrounded by oxygens; in minerals by some OH� and/or F�. The B atoms are accommodated in distorted, corner-sharing BO6 octahedra. A network of BO6 octahedra forms the backbone of the structure [4]. The larger A atoms are located inside of slightly distorted hexagonal rings formed by six BO6 octahedra. The structure can be regarded as two interpenetrating networks of B2O6 and A2O<sup>0</sup> units. The pyrochlore structure tolerates vacancies on the A and O<sup>0</sup> sites ('defect pyrochlores'), which can be represented by the general formula A1-2B2X6Y0-1. Work in this chapter deals with synthesizing lead ruthenate derivatives and lead-yttrium ruthenate solid solutions, measuring thermoelectric properties, and understanding the data based on the pyrochlore structure, A- and B-site occupancy, ligand and crystal fields as well as quantum-physical explanation of scattering mechanisms.

## **2. Electronic properties of ruthenate pyrochlores**

Pyrochlores exhibit various electronic properties. Ruthenate pyrochlores with 4*d* transition elements on the B-site are of interest because their electronic properties can change when changing the A-site atom. The electrical conductivity is affected by the network of the corner-sharing RuO6 octahedra, i.e., the B2O6 sublattice [4]. Bi2Ru2O7 and Pb2Ru2O6.5 show metal-like electrical conductivity to the lowest measured temperatures of a few degrees Kelvin [5, 6]. Bi2Pt2O7 is an insulator [7]. RE2Ru2O7 (rare earth RE = Pr to Lu), and Y2Ru2O7 are Mott insulators with a spinglass ground state [5, 8, 9]. Some pyrochlores show metal–insulator transitions (MIT), depending on temperature or pressure, for example Hg2Ru2O7 and Tl2Ru2O7 [10, 11]. Tl2Ru2O7 shows a MIT at 120 K [9], likely due to the formation of a spin gap in the one dimensional Haldane chain [12]. Therefore, the electronic structure of a pyrochlore, especially near the Fermi level (*EF*), must be affected by the element on the A-site via the A–O–Ru bond and the unoccupied O<sup>0</sup> sites. Defect pyrochlores are cationic conductors, i.e., they are solid electrolytes. Some of them with 4*d* or 5*d* atoms on the B-site have been used as oxygen electrodes, because of their ionic (O2�) and electronic conductivity [4].

There are at least two factors that may contribute to the electronic properties of Ru(IV) pyrochlores: 1) The Ru–O–Ru bond angle, which affects the Ru4+ *t2g* band width and varies with the size of the A-site cation [13]. 2) Hybridization of unoccupied states of A-site cations (e.g., Tl, Bi, and Pb) with the Ru 4*d* states via the oxygen framework [14]. Changes of the Ru–O–Ru bond angle depend also on the Ru–O bond length, which is affected by the size of A and the resulting changes in orbital overlap and bandwidth [13, 15]. Respective studies have shown that pyrochlores with metallic behavior have greater Ru–O–Ru bond angles than that of insulating ruthenate pyrochlores [16]. There seem to be only small structural changes in A2Ru2O7 pyrochlores when comparing metallic and semiconducting members. Electrical transport properties may just be positioned near the edge of localized and itinerant electrons. This positioning at the edge of the metal–insulator divide may help finding promising thermoelectric materials. High Seebeck coefficients are on the insulator side and potentially acceptable electrical conductivity on the metallic side, providing a balance in properties [4, 5].

Lee et al. [13] computed the band width of Ru *t2g* block bands for various pyrochlore ruthenates using the extended Hückel tight binding method and showed that metallic phases have a wider bandwidth than semiconducting phases. The

*Quantum Physical Interpretation of Thermoelectric Properties of Ruthenate Pyrochlores DOI: http://dx.doi.org/10.5772/intechopen.99260*

authors concluded that the metal-versus-insulator behavior of ruthenium pyrochlores can be explained in terms of the Mott-Hubbard mechanism of electron localization. Lee et al. [13] showed that there is a linear relationship between the ionic radius of the A cation and the Ru–O–Ru bond angle. I used the software *ATOMS V6.4.0* to calculate the Ru–O–Ru bond angle, which is 136.24°. The ionic radius for 8-coordinated Pb2+ is 0.129 nm. This is the highest bond angle and the largest ionic radius compared with the data shown in Figure 3a in [13]. The data confirm that Pb2+ follows as well the trendline established for cations on the A-site in ruthenate pyrochlores. Based on the high bond angle of 136.24°, Pb2Ru2O6.5 can be expected to fall in the group of metallic ruthenate pyrochlore phases with a high Ru *t2g* band width as shown by Lee et al. in Figure 3b in [13].

Lead ruthenate Pb2Ru2O6.5 shows metal-like conductivity at room temperature. The ceramic has been used as catalyst for fuel cells, organic syntheses, and charge storage capacitors [4, 17]**.** As with all pyrochlores, the B atom, here Ru, is six-fold coordinated: each Ru atom is located at the center of a slightly distorted octahedron of equidistant oxygen atoms. Lead is VIII-coordinated with oxygen. Pb2Ru2O6.5, or more precisely, Pb2Ru2O6O<sup>0</sup> 0.5 forms an ordered defect-pyrochlore structure in which every other O<sup>0</sup> site is empty [18]. The vacancies in the O<sup>0</sup> sites are ordered. Following Hsu et al. [14] discussion, the metallic behavior of lead ruthenate is most likely due to the formation of an extended Pb 6*p* band overlapping with the Ru 4*d* band and is attributed to electron transport mainly within the octahedral network of BO6 units, the backbone of the pyrochlore structure [4, 14, 19]. The fivefold Ru 4*d* levels are divided into an unoccupied *eg* band (� 2-5 eV above the Fermi energy *EF*) and a partially occupied *t2g* band (� 1 eV below to 1 eV above *EF*). The *t2g* band is broad and mainly metallic due to covalent admixture with O 2*p* states. The Pb 6 *s* state is too deep (� 8.5-10 eV below *EF*) to be mixed with the Ru 4*d* states (the Fermi levels are in the same partially filled Ru 4*d* band). However, the unoccupied Pb 6*p* bands (only 5-9 eV above *EF*) are close enough to mix with the Ru 4*d* states.

## **3. Quantum physical background**

Following semiconductor physics, transport theory of metals, and degenerate semiconductors (parabolic band, energy-independent scattering approximation [20]) the Seebeck coefficient *S*, the electrical resistivity *ρ* [21], and the total thermal conductivity *κ* can be expressed as

$$
\sigma = \frac{1}{\rho} = \pm e n \mu \tag{1}
$$

$$S = \frac{8\pi^2 k\_B^2}{3eh^2} m^\* \, T \left(\frac{\pi}{3n}\right)^{2/3} \tag{2}$$

$$
\kappa = \kappa\_{\epsilon} + \kappa\_{L} \tag{3}
$$

where *e* is the charge of a carrier, *n* is the carrier concentration (1/m3 ), *μ* the carrier mobility (m<sup>2</sup> /V. s), and the plus or minus sign in Eq. (2) depends on the type of charge carrier, holes (+) or electrons (�); *kB* is the Boltzmann constant, *h* is Planck's constant,*T* is the absolute temperature. The *m*\* in Eq. (2) refers to the effective mass of the charge carriers. Heavier carriers (higher *m*\*) will move more slowly. Hence, less mobility leads to lower electrical conductivity but a higher Seebeck coefficient (Eq. (2)). In other words, both, a smaller carrier concentration and a higher carrier's effective mass decrease electrical conductivity. The exact relationship between effective mass and carrier mobility depends on the electronic

structure of a given material, on whether the material is isotropic or anisotropic and on scattering mechanisms [22].

The Wiedemann-Franz law (Eq. (4)) provides an expression for *κ<sup>e</sup>* in Eq. (3):

$$
\kappa\_{\epsilon} = \sigma LT = \epsilon n \mu LT \tag{4}
$$

where *σ* is the electrical conductivity, *L* the Lorenz number,*T* is the absolute temperature. For metals and degenerate semiconductors *L* assumes the Sommerfeld value of 2.44 � <sup>10</sup>�<sup>8</sup> W. Ω/K<sup>2</sup> . For non-degenerate, single parabolic band materials, and acoustic phonon scattering conditions, *<sup>L</sup>* drops to 1.49 � <sup>10</sup>�<sup>8</sup> <sup>W</sup>. Ω/K<sup>2</sup> [23]. The Wiedemann-Franz law is based on the assumption that free electrons (an electron gas) transport heat and electricity in metals, for which the total thermal conductivity is approximately equal to *κe*. The value of the Lorenz number varies among materials and depends, e.g., on band structure, on the position of the Fermi level, and on temperature; for semiconductors *L* relates to the carrier concentration [24]. The Lorenz number can vary, particularly with carrier concentration, e.g., in lowcarrier concentration materials it can be about 20% lower than for metals [25]. Important deviations from the Wiedemann-Franz law are seen, e.g., in multi-band materials [26, 27], in nanowires [28], in superconductors [29], and in the presence of disorder [30]. The latter will be important for the materials investigated here. *κ* and *σ* are always determined experimentally. An accurate determination of *κ<sup>e</sup>* is critical, since *κ<sup>L</sup>* is often calculated from the difference between *κ* and *κ<sup>e</sup>* (Eq. (3)).

Electronic transport processes can be analyzed using the density functional theory (DFT), electronic band-structure calculations, and the Boltzmann transport theory [31–33]. The Fermi-Dirac distribution function can be included in the computations to deal with temperature effects. To get a more complete description of electrontransport or electron scattering in a solid material, the energy (ε) dependence of the relaxation time of electrons near the Fermi energy is needed. To calculate the *σ* (*T*), *S* (*T*), and *κe*(*T*) the Boltzmann transport equation framework is applied.

A scattering mechanism includes scattering from charge carriers and phonons. Scattering from charge carriers includes ionized impurities, acoustic phonons, the phonon deformation potential and scattering at crystal boundaries. Phonon scattering consists of phonon–phonon-, point-defect- and phonon-carrier scattering. To calculate the total scattering rate as the sum of the individual contributions Eq. (5), Mathiessen's rule is used:

$$\frac{1}{\tau\_{\text{total}}} = \sum \frac{1}{\tau i} \tag{5}$$

where *τ<sup>i</sup>* is the relaxation time for each scattering mechanism and τtotal is total relaxation time. Note that because (*ε*) is integrated in the Boltzmann framework, the temperature dependence of conductivity may not be the sum of the temperature dependence of simple scattering processes. For example, if scattering process *A* has a relaxation time of *τA*(ε) and scattering process *B* has a relaxation time of *τB*(*ε*), then the total *ε* dependence will be

$$\frac{1}{\tau\_{\mathsf{T}}(\varepsilon)} = \frac{1}{\tau\_{\mathsf{A}}(\varepsilon)} + \frac{1}{\tau\_{\mathsf{B}}(\varepsilon)}\tag{6}$$

$$
\pi\_\mathbf{t}(\varepsilon) = \frac{\mathsf{\tau}\_\mathbf{A}(\varepsilon) + \mathsf{\tau}\_\mathbf{B}(\varepsilon)}{\mathsf{\tau}\_\mathbf{A}(\varepsilon)\mathsf{\tau}\_\mathbf{B}(\varepsilon)} \neq \mathsf{\tau}\_\mathbf{A}(\varepsilon) + \mathsf{\tau}\_\mathbf{B}(\varepsilon) \tag{7}
$$

As shown in Eq. (7), the combination of two scattering mechanism is not simply a summation, unless other approximations apply.

#### *Quantum Physical Interpretation of Thermoelectric Properties of Ruthenate Pyrochlores DOI: http://dx.doi.org/10.5772/intechopen.99260*

**Table 1** shows that many different scattering mechanisms have been published. Here, I provide a list of examples of scattering mechanisms of electrical-, and thermal conductivities, as well as the Seebeck coefficient for one case. *TF* is the Fermi temperature, and *TBG* is the Bloch-Grüneisen temperature.

Umklapp or inharmonic scattering is present in crystalline materials at high temperatures and is the scattering of phonons by other phonons in threephonon resistive processes. The significance of the process increases as the temperature increases, i.e., when more phonons with larger wave vectors are excited [38].


#### **Table 1.**

*Scattering mechanism of electrical-, thermal conductivities, and Seebeck coefficient.*

In many semiconductors that show 'glass-like' behavior, there is a change in the conduction mechanism with temperature from thermally activated to 'Variable Range Hopping' (VRH) conduction [39, 40]. According to Mott, electron hopping between nearest neighbor sites is not always favored at low temperatures as the sites may be significantly different in energy. It is possible that electrons prefer to move to an energetically similar but more remote site. In this regime, the following conduction law is expected for the variation of the conductivity of glass-like disordered systems [41, 42]:

$$
\sigma = \sigma\_0 e^{-(\mathbf{T}\_0/\mathbf{T})^\mathbf{I}} \tag{8}
$$

where σ<sup>0</sup> is the pre-exponential factor,*T*<sup>0</sup> is a characteristic temperature. The condition 0 < *Υ* < 1 is fulfilled, if the VRH mechanism dominates the conduction. *Υ* = ¼ corresponds to a Mott VRH, if the density of states around the Fermi level can be assumed to be constant; *Υ* = 1/2 correlates with an Efros-Shklovskii VRH, if there is a gap at the Fermi level [41, 42]; *Υ* = 1 agrees with a Nearest Neighbor Hopping (NNH) mechanism.

The most proper way to determine the accuracy of *Υ* in Eq. (8) is to analyze the electrical conductivity versus temperature by using the approach of Zabrodskii and Zinoveva [43] as follows. Let *W* be defined as

$$W(T) = \ln\left[\frac{\mathrm{d}\ln\sigma(\mathrm{T})}{\mathrm{d}\ln T}\right] \tag{9}$$

By inserting Eq. (8) into Eq. (9), *W* (*T*) can be written as Eq. (10)

$$W(T) = \ln|\Upsilon + \Upsilon| \ln|T\_0 - \Upsilon| \ln|T|\tag{10}$$

*W*(*T*) can also be used to determine metallic and insulating behavior. If the slope of ln [*W*(*T*)] vs. ln *T* is negative, the material is an insulator. However, if the slope is positive, the material behaves like a metal [37]. Generally, in semiconductors (doped or un-doped) *σ*(*T*) and *S*(*T*) have opposite temperature dependencies. In Mott's Variable Range Hopping the electrical conductivity depends on temperature as *σ* ∝ *e*�ð Þ <sup>1</sup>*=<sup>T</sup>* <sup>1</sup>*=*ð Þ *<sup>d</sup>*þ<sup>1</sup> and the Seebeck coefficient as *S* ∝ *T* (d�1)/(d+1), where d is the dimensionality of the system [44]. In our work, the system is 3D; *σ*(*T*) and *S*(*T*) vary with temperature as *σ* ∝ *e*�ð Þ <sup>1</sup>*=<sup>T</sup>* <sup>1</sup>*=*<sup>4</sup> and *S* ∝ *T* ½.

#### **4. Selecting and making lead- and lead-yttrium ruthenates**

Substitutions of atoms on the A- and/or B-site in pyrochlore, here Pb2Ru2O6.5, change the thermoelectric properties. I have selected and synthesized two sets of ruthenate pyrochlore compounds, one with variable Pb:Ru ratio, the other constitutes solid solutions of Pb- and of Y-ruthenate (**Table 2**). Changing the Pb:Ru ratio changes A and B site occupancy and the defect concentrations. For example, reducing the number of Pb2+ ions creates more vacancies in the A2O<sup>0</sup> sublattice and changes the properties of the already existing oxygen vacancies, which are occupied by Pb 6s<sup>2</sup> electron lone-pairs in Pb2Ru2O6.5. Less than two Pb2+ ions mean less electrons for the vacancies. Reducing Ru should affect electrical conductivity, which is mainly due to the RuO6 backbone structure of ruthenate pyrochlores.


#### **Table 2.**

*Lead ruthenate derivatives and lead yttrium ruthenate solid solutions.*

Partial Pb-Y substitutions have been studied and showed a transition from metal-like electrical conductivity to semiconducting, prior to becoming an insulating material [45]. Since Pb in Pb2Ru2O6.5 causes the width of the Ru *t2g* block band to be fairly wide and that of yttrium to be narrower [13], then yttrium is an interesting element when expecting property changes for partial substitutions. Obviously, VIIIY3+ with its much smaller ionic radius than VIIIPb2+ reduces the Ru–O–Ru bond angle [13], thereby making the increasingly Y-rich compound less metallic.

The pyrochlores were made as follows. Equally sized powders, about 10 μm in size were cold pressed into pellets, reacted at high temperature (1173 K to >1273 K), crushed, pressed again into pellets, sintered, cooled, and then measured. Only Pb2Ru2O6.5 and Y2Ru2O7 have been studied by others. The electrical conductivity of Pb2Ru2O6.5 has been measured frequently, as shown in **Table 3**.

The results depended on the method of synthesis and fluctuated between 120 � 30 S/cm (298 K) [46] and 4651.2 S/cm (300 K) [6]. Temperatures higher than 598 K have also been studied [17, 47].


*Quantum Physical Interpretation of Thermoelectric Properties of Ruthenate Pyrochlores DOI: http://dx.doi.org/10.5772/intechopen.99260*

#### **Table 3.**

*Known temperature dependences of electrical conductivity* σ*(*T*) of lead ruthenate.*

## **5. Lead ruthenate derivatives**

X-ray powder diffraction measurements and phase identification were performed. Search-match routines in *JADE9* software (Materials Data. Inc.) were used. The powder patterns were calibrated using a corundum (α-Al2O3) standard. X-ray powder diffraction patterns of lead ruthenate (Pb2Ru2O6.5) and six derivatives are shown in **Figure 1**. All *d*-spacing and intensities match the isometric pyrochlore crystal structure. Lattice constants a0 were calculated for all samples. The lattice constant of pure lead ruthenate (a0 = 1.0257 nm) was in good agreement

#### **Figure 1.**

*X-ray diffraction patterns of lead ruthenate pyrochlore [51] and six derivatives (below: Reference pattern 00-034-0471 of Pb2Ru2O6.5; 'c' = unidentified contaminant).*

**Figure 2.** *SEM image showing octahedral crystals of our Pb2Ru2O6.5.*

with Jade reference # 00-034-0471 by Horowitz et al. [52] with a value of 1.0252 nm.

Substitutions affect *a*0, but the isometric unit cell is maintained in all cases. A closer look at the X-ray spectra at 2θ = 28° showed that traces of unreacted RuO2 may be present, particularly in Pb:Ru = 1.7:2.3. The yield of all synthesized pyrochlores is close to 100%.

Solid-state synthesis of pyrochlores involved a series of manipulations, which introduced ZrO2 from milling containers, SiO2 and Al2O3 from glass mortars and pestles. The impurities were analyzed by semi-quantitative X-ray fluorescence and amounted to ≈ 1 wt.%. When present, unreacted RuO2 amounted to ≈ 1 wt.% as well. Suggesting an overall purity of our pyrochlores of at least 98 wt.%. **Figure 2** shows a secondary electron image of octahedral Pb2Ru2O6.5 crystals.

## **5.1 Electrical- and thermal conductivity and Seebeck coefficients of lead ruthenate derivatives**

In this section I present and discuss thermoelectric properties of derivatives of lead ruthenate (Pb(2+x)Ru(2-x)O(6.5*<sup>z</sup>*) and Pb(2-x)Ru2O(6.5*<sup>z</sup>*)), **Table 2**). Comparisons are made with the published results of lead ruthenate [48].

**Figure 3a** shows electrical conductivity as a function of temperature for lead ruthenate derivatives. All conductivities decrease with increasing temperature, suggesting metal-like performance of all ceramics. The lead ruthenate derivative with a deficiency of lead, i.e., Pb1.8Ru2O6.5<sup>z</sup> exhibits the highest electrical conductivity. The lowest conductivity was seen for the derivative with an excess of lead and a deficiency of ruthenium, i.e., Pb2.9Ru1.1O6.5z. The oxygen content of the derivatives was not measured.

**Figure 3.** *(a) Electrical conductivity and (b) Seebeck coefficients of lead ruthenate derivatives.*

*Quantum Physical Interpretation of Thermoelectric Properties of Ruthenate Pyrochlores DOI: http://dx.doi.org/10.5772/intechopen.99260*

Electrical conductivity is attributed essentially to transport of electrons in and through BO6 octahedra, the backbone of the pyrochlore structure [53, 54]. The Pb 6*s* <sup>2</sup> electron lone-pair is highly localized in defect pyrochlores like lead ruthenate with oxygen deficiency (O<sup>0</sup> vacancies) in the sublattice. Hence, the electron lonepair is stereo-chemically active. This results in the formation of ordered oxygen vacancies and the displacement of Pb closer to the vacancy [14]. In this way, the pyrochlore structure is stabilized vis-à-vis the perovskite structure [18]. Pb may be present in lead ruthenates in two valence states as Pb2+ 1.5Ru4+2Pb4+0.5O6.5 [9]. Alternatively, Ru could be present in two valence states as Ru4+ and Ru5+, i.e., Pb2Ru4+1.0Ru5+ 1.0O6.5. Even with some Pb4+ present, the pyrochlore structure would still be sufficiently stabilized by the Pb2+ 6*s* <sup>2</sup> electron lone-pair.

Removal of Ru, i.e., decreasing the overlap of the Pb 6*p* with the Ru 4*d* bond, should result in a decreasing electrical conductivity, relative to pure lead ruthenate, which is confirmed by the experimental results (B-site stoichiometry <2). For charge balance reasons, it is assumed that Pb enters the B-site as Pb4+, which also means absence of some Pb 6*p* electrons. Decreasing Pb2+ on the A-sites (x = 1.7) without substitutions increased the electrical conductivity significantly, which may be the result of creating more oxygen vacancies and vacancies in the A2O<sup>0</sup> sublattice. The relatively strong increase of electrical conductivity due to increasing Ru to 2.3 at the expense of Pb to 1.7 is less easy to explain. The replacement of 2 Pb2+ by one Ru4+ on the A site would cause an A-site vacancy, which could increase electrical conductivity. Ru4+ is smaller than Pb2+. This should cause rattling of Ru, which would increase phonon scattering and thus decrease thermal conductivity. This will be discussed further in context with thermal conductivity and scattering mechanisms.

**Figure 3b** shows that the Seebeck coefficients go through a maximum near 423 K for all lead ruthenate derivatives. The dependence of the Seebeck effect on the composition of these pyrochlores is not very pronounced. The highest and lowest coefficients deviate from each other by about 15%, independent of temperature. A surprising finding was that *S*(*T*) increased initially with increasing temperature and then decreased with further increasing temperature. Since *S*(*T*) is positive the majority of charge carriers are holes, i.e., lead ruthenate pyrochlore and its derivatives exhibit *p*-type conductivity. The kind of temperature dependence of *S* found for these pyrochlores has been observed in other systems as well. For example, Y2-xBixRu2O7 show a broad maximum at 170 K for x = 1.6 [9], but the authors gave no explanation. Paschen [51] observed a maximum of *S*(*T*) in a clathrate-like material, i.e., in Eu3Pd20Ge6, at about 120 K. The author attributed the maximum to valence fluctuations of Eu (3<sup>+</sup> and 4<sup>+</sup> ). Sidharth et al. [55] measured a broad maximum of *S*(*T*) for tin chalcogenides, SnSe1-xTex between 400 K and 600 K. These authors discuss their findings in terms of the influence of carrierphonon-, carrier-carrier scattering and conductivity-limiting bipolar conduction at high temperature. Tse et al. [56] reported a maximum of *S* for NaxSi46 clathrates at x = 16 for 200 K. Certain clathrates are metallic, but their thermal conductivity resembles that of glass-like solids, as with lead ruthenate [48] and the derivatives studied here. Tse et al. [56] ascribed the trend of *S*(*T*) as a function of sodium content to the band profile of the silicon framework. The rise of *S*(*T*) with increasing temperature to a maximum and the decrease with further increasing temperature was qualitatively reproduced by raising the Fermi level. A maximum in the temperature dependence of *S*(*T*) was also seen in bismuth metal thin films. For a 33 nm thick film, the maximum occurred near 425 K [57]. The authors of [57] discussed the phenomenon by taking into account the temperature dependence of the Fermi energy, which cannot be neglected because bismuth has a low Fermi energy. Hence, increasing and decreasing Seebeck coefficients have been observed

in a variety of materials and in different temperature ranges. Explanations of respective data for different materials vary.

The Seebeck coefficient of these *p*-type pyrochlores is inversely proportional to the carrier concentration (Eq. (2)). Before the onset of intra-band minority carrier (electron) excitation at lower temperatures, *S*(*T*) increases with *T* (**Figure 3b**), confirming metallic behavior of all pyrochlores. At higher temperatures the Fermi distribution broadens, which leads to an exponential increase in minority electrons, due to thermal excitation and resulting in a reduction of the Seebeck coefficient [58]. The Seebeck coefficient reaches a maximum, in this case at 423 K (**Figure 3b**) for all derivatives.

**Figure 4** shows the results of thermal conductivity measurements of all lead ruthenate pyrochlores. Thermal conductivity increases with temperature and was found to have increased in all derivatives above that of lead ruthenate. Note that the compounds 1.7:2.3 and 1.8:2.0 showed the highest electrical conductivities (**Figure 3a**), while their thermal conductivities are least affected (**Figure 4**). This will be addressed in the next section. I noticed an increase of the total thermal conductivity with temperature in the crystalline lead ruthenate and its derivatives, which is typical of glass-like materials. A glass-like behavior of all these pyrochlores is likely due to an electronic contribution from an increasing concentration of minority electrons with temperature, which also contributed to the downturn of the Seebeck coefficients.

The presence of structural defects and a large unit cell create pronounced anharmonicity, which lowers the thermal conductivity [59]. Pb2Ru2O6.5 possesses intrinsic disorder characteristics, i.e., vacancies in the A2O<sup>0</sup> sublattice and stereochemically active 6*s* <sup>2</sup> electron lone-pairs on Pb. Similar defects can be expected to prevail in our derivatives. These defects cause a glass-like temperature dependence of thermal conductivity of our lead ruthenate derivatives. This was also observed, e.g., in chalcogenides, clathrates, and yttrium-stabilized zirconia (YSZ) [38], and materials with intrinsic disorder [60–63]. The thermal conductivity of lead ruthenate is close to that of vitreous silica [48], which is 1.93 W/m. K at 373 K. With increasing temperature, the 3-phonon resistive process (Umklapp scattering) and excitation of more phonons with larger wave vectors become increasingly important [64], thereby lowering the lattice thermal conductivity of a crystal [59].

**Figure 4.** *Thermal conductivity of lead ruthenate derivatives.*

#### **5.2 Scattering mechanisms in lead ruthenate derivatives**

The measured thermal conductivity *κ* is the sum of electronic *κ<sup>e</sup>* and lattice thermal conductivity *κ<sup>L</sup>* (Eq. (3)). The electronic and lattice contributions are

#### *Quantum Physical Interpretation of Thermoelectric Properties of Ruthenate Pyrochlores DOI: http://dx.doi.org/10.5772/intechopen.99260*

plotted in **Figure 5a** and **b**, respectively. The electronic thermal conductivity increased with temperature and decreased with increasing Pb content. The Pb: Ru = 1.8:2.0 and the Pb:Ru = 1.7:2.3 pyrochlores, respectively, have the highest electronic thermal conductivities. With one exception, the lattice thermal conductivities are almost independent of temperature but vary with composition. Partial removal of Pb from Pb2Ru2O6.5 lowers the lattice thermal conductivity. Using the Lorenz number of 2.44 � <sup>10</sup>�<sup>8</sup> <sup>W</sup>Ω/K<sup>2</sup> for metals to calculate *<sup>κ</sup><sup>e</sup>* yielded negative values for *<sup>κ</sup>L*, which is meaningless. To avoid this, I used 1.49 � <sup>10</sup>�<sup>8</sup> <sup>W</sup>Ω/K<sup>2</sup> , which has been applied for non-degenerate, single parabolic band materials, and acoustic phonon scattering conditions [62].

As expected from electrical conductivities (**Figure 3a**), the compounds with a lead deficiency show *κ<sup>e</sup>* values higher than in pure lead ruthenate (**Figure 5b**). However, the *κ<sup>L</sup>* values are the lowest of all. These compounds are presumably the ones with the highest defect concentrations of all lead ruthenate derivatives, which are assumed to increase phonon scattering and thus lower *κL*.

#### **Figure 5.**

*(a) Lattice- and (b) electronic thermal conductivity of lead ruthenate derivatives.*

The electrical and thermal conductivity data of lead ruthenate derivatives have been analyzed for underlying scattering mechanisms. The temperature dependence of the scattering mechanisms listed in **Table 1** were least-squares-fitted to the experimental data. The goodness of the fits was analyzed by using the reduced chisquared statistic (χ<sup>2</sup> /*DOF*) (Eq. (11)) and the best fit was assumed to have identified the most likely scattering process.

$$\chi^2/DOF = \frac{\text{Sum of squared errors}}{DOF} \tag{11}$$

where *DOF* is the number of degrees of freedom, which is the number of data points minus the number of fit parameters. For all seven synthesized pyrochlores, the best fits for *σ*(*T*), *κe*(*T*), and *κL*(*T*) were *T* �1/2,*T* 1/2, and *T* �<sup>1</sup> , respectively. All fits for all pyrochlores pointed at the same underlying scattering mechanisms. A graphical representation of these dependencies can be found in [48] for pure lead ruthenate. According to **Table 1**, the underlying scattering mechanism is 'electron impurity scattering'. Since this mechanism prevails only at *T* >> *TF* (**Table 1**), the excellent match between data and the fits suggests that the Fermi temperature *TF* of lead ruthenate and its derivatives is below room temperature. Other joint scaling shapes rule out other combinations of dominant scattering mechanisms. This is evidence that traditional electron-acoustic phonon scattering is suppressed and thus most scattering is due to intrinsic disorder and impurities.

The lattice thermal conductivity *κL*(*T*) of lead ruthenate and all derivatives varies with *T* �<sup>1</sup> . This dependence suggests that the 3-phonon resistive process (Umklapp scattering, **Table 1**) is responsible for the decrease of *κL*(*T*) in all compositions. This finding supports the electron-impurity scattering mechanism for *σ*(*T*) and *κe*(*T*) (**Table 1**) [64], because transport properties are sensitive to structural disorder such as site substitutions, vacancies and localized impurities, all of which are present in these pyrochlores at different concentrations. These defects can lower the thermal conductivity to glass-like behavior [59]. In this regard, compounds with stereochemically active electron lone-pairs associated with constituent atoms (here Pb2+ on the pyrochlores' A-site) have attracted significant attention. Other examples are Cu3SbSe3, AgSbSe2, and other chalcogenides. The 3-phonon resistive process will be addressed again in Section 6.2. The same causes as here hold there for Umklapp scattering for lead-yttrium ruthenate solid solutions. The electronic contribution *κe*(*T*) to the total thermal conductivity *κ*(*T*) increased with increasing temperature and was estimated to be about 72% at 25°C and 85% at 300°C, respectively. The total thermal conductivity *κ*(*T*) results mainly from electronic thermal conductivity.

## **6. Lead yttrium ruthenate solid solutions**

We have reported recently [45] on electrical conductivity and Seebeck coefficients of lead yttrium ruthenate solid solutions. The focus of that investigation was on finding and explaining a metal insulator transition in a suite of Pb-Y ruthenate solid solutions between the metal-like paramagnetic Pb2Ru2O6.5 and the antiferromagnetic Mott insulator Y2Ru2O7. Only 10 mol% of Pb needed to be replaced by yttrium to reach the point of transition from a metal-like to a semiconducting ceramic. Following the Mott-Hubbard model, yttrium opens the Mott-Hubbard gap and fills the lower Mott-Hubbard band with localized *t2g* electrons thereby changing the mechanism of electron transport [45].

In this section I report on the hitherto unknown thermal properties of lead yttrium ruthenates and analyze these together with our previously published measurements on electrical conductivity and the Seebeck effect [45] in terms of quantum-physical scattering.

Structure and purity of all samples (**Table 2**) have been determined by X-ray diffraction and X-ray fluorescence analysis, respectively. The X-ray diffraction patterns of pure lead and yttrium ruthenate are known (Jade reference # 00-034- 0471, *a*<sup>0</sup> = 1.0252 nm and # 00-028-1456, *a*<sup>0</sup> = 1.0139 nm, respectively). The solid solutions are all of isomeric structure and follow Vegard's law, i.e., the unit cell parameter decreases linearly with increasing concentration of Y3+ (*r* = 0.1019 nm), which is smaller than Pb2+ (*r* = 0.129 nm) [45].

#### **6.1 Thermal conductivity of lead yttrium ruthenate solid solutions**

The results of thermal conductivity measurements are shown in **Figure 6**. Thermal conductivity *κ* increases with temperature. Pure lead ruthenate has the highest thermal conductivity (2.35 W/m. K) of the pyrochlores at all temperatures [48]. As the Y concentration increases, *κ* and its temperature dependence decrease. Pure yttrium ruthenate has the lowest thermal conductivity.

*κ* increases with temperature up to an yttrium concentration of x = 1.0. At and above x = 1.5, *κ* is very low and nearly temperature independent. These pyrochlores are insulators as indicated by low electrical conductivity as well [45]. The increase of *κ* with increasing *T*, and the low values of *κ* are typical of glass-like behavior, although all these pyrochlores are crystalline. As with lead ruthenate derivatives, glass-like behavior is probably due to a contribution to *κ<sup>e</sup>* of an increasing

*Quantum Physical Interpretation of Thermoelectric Properties of Ruthenate Pyrochlores DOI: http://dx.doi.org/10.5772/intechopen.99260*

concentration of minority electrons with increasing temperature, which also contributed to the decrease of the Seebeck coefficient for x = 0 to 1.5 [45].

**Figure 6.** *Thermal conductivity of Pb- and Y-ruthenate and solid solutions.*

#### **6.2 Scattering mechanisms in lead yttrium ruthenate solid solutions**

The electronic thermal conductivity *κ<sup>e</sup>* was calculated with Eq. (4), the lattice thermal conductivity *κ<sup>L</sup>* with Eq. (3) and plots are shown in **Figure 7a** and **b**.

The electronic thermal conductivity *κ<sup>e</sup>* (**Figure 7a**) decreases with increasing Y content but increases with temperature. The slope of the temperature dependence of *κ<sup>e</sup>* decreases with increasing concentration of yttrium. **Figure 7a** (lower diagram) contains the data for the two insulators Pb(2-x)YxRu2O(6.5+z) with x = 1.5 and 2.0 on a different scale, to show that there is still a temperature dependence of *κe*. Y2Ru2O7 shows the lowest electronic thermal conductivity.

The lattice thermal conductivity *κ<sup>L</sup>* decreases with increasing temperature for x ≤ 0.2 (**Figure 7b**). Between x = 0.4 and x = 2.0 *κ<sup>L</sup>* becomes temperature independent. *κ<sup>L</sup>* increases initially with temperature in pure lead ruthenate (x = 0).

In this section electrical- and thermal conductivity and Seebeck coefficients of the pyrochlores Pb(2-x)YxRu2O(6.5+z) (0 ≤ x ≤ 2, 0 ≤ *z* ≤ 0.5) will be analyzed for the underlying scattering mechanisms. To do so, the temperature dependence of the scattering mechanisms listed in **Table 1** were fitted to the experimental data. The goodness of all fits was analyzed by using Eq. (11), which yielded the most likely scattering mechanisms.

**Figure 7.** *(a) Electronic- and (b) lattice thermal conductivity of Pb- and Y-ruthenate and solid solutions.*

As shown in **Table 1**, for most scattering mechanisms at least two properties, for example, *σ*(*T*) and *κe*(*T*) must be fitted together to show the respective temperature dependencies. For Pb(2-x)YxRu2O(6.5+z) with x = 0 and 0.1 the best fits of electrical conductivity data *σ*(*T*) and of the electronic component of thermal conductivity *κe*(*T*) vary with *T* 1/2 and *T* 1/2, respectively (see **Figure 8a** and **b**). According to **Table 1** the underlying scattering mechanism is 'electron-impurity scattering' at *T* >> *TF* [37]. The lattice thermal conductivity *κL*(*T*) varies with *T* <sup>1</sup> suggesting that the 3-phonon resistive process (Umklapp scattering, **Table 1**) is responsible for the decrease of *κL*(*T*) in the pyrochlores with x = 0 and x = 0.1 moles of yttrium. This finding supports the electron-impurity scattering mechanism for *σ*(*T*) and *κe*(*T*) (**Table 1**) [41]. The experimental data and the least squares fits are shown in **Figure 8a**–**c**.

For Pb(2-x)YxRu2O(6.5+z) with x = 0.2 and x = 0.4 the best fits of the electrical conductivity data *σ*(*T*) and of the electronic component of thermal conductivity *κe*(*T*) did not allow for an unambiguous determination of the scattering mechanism. The goodness of two fits was practically identical. Two variations of *σ*(*T*) and *κe*(*T*) with temperature are equally likely: σ(*T*) varies with *T* 1/2 or with *T* <sup>1</sup> and *κe*(*T*) varies with *T* 1/2 or with *T* <sup>0</sup> . Based on **Table 1** the underlying scattering mechanisms are 'electron-impurity scattering' at *T* << *TF* provided that *σ*(*T*) varies with *T* 1/2 and *κe*(*T*) with *T* 1/2. If *σ*(*T*) varies with *T* <sup>1</sup> and *κe*(*T*) with *T* <sup>0</sup> then 'electron– phonon scattering' at *TBG* << *T* << *TF*, is the mechanism (**Table 1**) [37]. The lattice thermal conductivity *κL*(*T*) varies with *T* <sup>1</sup> implying Umklapp scattering and supports the electron-impurity scattering mechanism. The actual scattering process could also be a combination of the two afore-mentioned mechanisms for which the temperature dependence would be unknown. In principle, more information could be obtained from an evaluation of Eq. (5), which has not been done yet. The fits for

x = 0.4 are the electron-impurity scattering as shown in **Figure 8**. **Figure 8a–c** show

#### **Figure 8.**

*Measured (a) electrical conductivity, (b) electronic-, and (c) lattice thermal conductivity fits as a function of* T *for Pb(2-x)YxRu2O(6.5+z) with x = 0, 0.1, 0.2, and 0.4 moles of Y.*

*Quantum Physical Interpretation of Thermoelectric Properties of Ruthenate Pyrochlores DOI: http://dx.doi.org/10.5772/intechopen.99260*

Pb(2-x)YxRu2O(6.5+z) with x = 0, 0.1, 0.2, and 0.4 fitted with *T* 1/2,*T* 1/2, and *T* <sup>1</sup> for *σ*, *κe*, and *κL*, respectively.

For Pb(2-x)YxRu2O(6.5+z) with x = 1.0 a clear selection of a mechanism cannot be made. The selection criteria, i.e., best least squares fit and goodness of the fit, allow for three scattering mechanisms. *σ*(*T*) could vary with *T* 1/2 and *κe*(*T*) with *T* 1/2 or *σ*(*T*) varies with *T* <sup>1</sup> and *κe*(*T*) with *T* <sup>0</sup> . Based on **Table 1** the underlying scattering mechanism would be electron-impurity scattering at *T* << *TF* or electron–phonon scattering at *TBG* <<*T* <<*TF*, or both (**Table 1**) [34]. The lattice thermal conductivity *κL*(*T*) varies with *T* <sup>1</sup> implying Umklapp scattering and supports the electron-impurity scattering mechanism. The third possibility is that the Mott Variable Range Hopping (MVRH) mechanism [20] is active, which requires a *T* 1/2 dependence of *S*(*T*) and for *σ*(*T*) an exp(1/*T*) 1/4 dependence. Fitting *T* 1/2 to *S*(*T*) yielded a satisfactory goodness of the fit. The same was the case when fitting the exponential function to the *σ*(*T*) data. As I mentioned for x = 0.2 and 0.4, here with an yttrium content of 1.0 mole there could be a combination of the scattering mechanisms for which the temperature dependence could be determined using Eq. (5). This pyrochlore, like the ones in the previous section are in the MIT zone. This explains why there may be more than one scattering process active. The MVRH fitting is shown in **Figure 9**.

For Pb(2-x)YxRu2O(6.5+z) with x = 1.5 and 2.0 the best fits of electrical conductivity *σ*(*T*) varies with exp(1/*T*) 1/4 and the Seebeck coefficients with *T* 1/2. As for x = 1.0, MVRH can be suggested. Mott's variable range hopping model describes hopping conduction between localized states with electron energies close to the Fermi level [41, 65, 66]. Hopping takes into account both thermally activated hopping over an energy threshold and phonon-assisted tunneling between localized states. For thermally activated hopping the Boltzmann factor exp(-*W*/*kBT*) applies, where *W* is the energy barrier between the localized states. The electrical conductivity of the pyrochlores on the insulator side of the MIT shows thermally activated conduction characteristics (Figure 3 in [45]). This behavior indicates transport by the MVRH [67]. It could be due to a random potential caused by yttrium substitution [9]. **Figure 9** shows fitting of Pb(2-x)YxRu2O(6.5+z) with x = 1.0, 1.5, 2.0 by MVRH for *σ* and *S*.

The figure of merit *zT* of a thermoelectric material determines its value for practical applications. Respective values have been calculated for lead ruthenate derivatives and for Pb-Y ruthenate solid solutions. All *zT* values are still more than two orders of magnitude below those achievable today with other ceramics. The highest *zT* for all pyrochlores studied here was 7.3 <sup>10</sup><sup>3</sup> at 523 K for Pb1.9Y0.1RuO6.5z. Since the focus of this work was on quantum physical

#### **Figure 9.**

*Measured electrical conductivity (a) and Seebeck coefficients (b) with respective fits as a function of temperature for Pb(2-x)YxRu2O(6.5+z) with x = 1.0, 1.5, and 2.0.*

interpretation of the pyrochlores' thermoelectric properties, no effort was made yet to search for pyrochlores with higher figures of merit.

## **7. Summary and conclusions**

Thermoelectric properties of ceramics are determined by their crystal structure and chemical composition. Important structural details affecting the transport mechanisms of heat and electricity are, e.g., vacancies, impurities, lattice site occupancy, lone pair electrons, and substitutions. The nature of these structural details can determine or change scattering mechanisms that determine the thermoelectric performance of the material. A selection of published mechanisms has been compiled. The way of testing these mechanisms by fitting respective mathematical functions to experimental data and the quantitative evaluation of the fits have been shown in this chapter, based on a large number of experimental data.

All thermoelectric properties of isomorphic lead ruthenate pyrochlores (defect pyrochlores) with different Pb:Ru atom ratios were measured and analyzed in quantum-physical terms to interpret transport mechanisms of thermal and electrical conductivity and to understand the temperature dependence of the Seebeck effect. All seven pyrochlores were *p*-type, exhibiting common features such as site substitutions, vacancies and some impurities, to which transport mechanisms are sensitive. Therefore, the same scattering mechanisms were seen for all pyrochlores, specifically, electrical conductivity *σ*(*T*), which varied with *T*1/2, the electronic part of thermal conductivity *κe*(*T*) varied with *T*1/2 and the lattice thermal conductivity *κL*(*T*) with *T*<sup>1</sup> . Hence, *σ*(*T*) and *κe*(*T*) were governed by electron-impurity scattering and *κL*(*T*) by the 3-phonon resistive process (Umklapp scattering), which supports the electron-impurity scattering mechanism. The Seebeck effect was inversely proportional to the carrier concentration.

Measurements and quantum-physical analyses were also done with lead-yttrium ruthenate solid solutions, which are defect pyrochlores as well. The temperature dependence of the Seebeck coefficient was qualitatively the same as for the lead ruthenate and derivatives. The same interpretation applies. A metal–insulator transition (MIT) occurred, if 0.2 moles of Pb were replaced by Y. The endmember Pb2Ru2O6.5 shows metal-like behavior. Y2Ru2O7 is an insulator. Impurity scattering prevailed until the MIT was reached. The temperature functions of electrical conductivity and electronic thermal conductivity were the same as for the lead ruthenates. On the semiconductor/insulator side of the MIT, up to about one mole of yttrium, several scattering processes were equally likely, e.g., electron-impurity- and electron–phonon scattering. From one to two moles of yttrium, the Mott-Variable-Range Hopping mechanism was active, as suggested by *σ*(*T*) varying with exp(1/*T*) 1/4. All figures of merit *zT* were small, about 7.3<sup>10</sup><sup>3</sup> maximum at 523 K.

Based on these findings the following conclusions are provided:


*Quantum Physical Interpretation of Thermoelectric Properties of Ruthenate Pyrochlores DOI: http://dx.doi.org/10.5772/intechopen.99260*


## **Acknowledgements**

I would like to thank Dr. Werner Lutze, my former PhD adviser, for his constant support and consult and many discussions while I was writing this chapter. Many thanks go to Drs. Nicholas A. Mecholsky and David McKeown for discussions and Dr. Marek Brandys for building the instrument to measure electrical conductivity and Seebeck coefficients. I thank Dr. Ian Pegg for his encouragement and financial support, which made it possible for me to do this work.

## **Conflict of interest**

The author does not have any conflict of interest.

*Thermoelectricity - Recent Advances, New Perspectives and Applications*

## **Author details**

Sepideh Akhbarifar The Catholic University of America, Vitreous State Labratory, Washington D.C., USA

\*Address all correspondence to: sepideha@vsl.cua.edu

© 2021 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

*Quantum Physical Interpretation of Thermoelectric Properties of Ruthenate Pyrochlores DOI: http://dx.doi.org/10.5772/intechopen.99260*

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## *Edited by Guangzhao Qin*

Next-generation energy sources are crucial for combating the world's energy crisis. One such alternative energy source is thermoelectricity, which is cost-efficient and environmentally friendly. This book presents a comprehensive overview of the progress made in thermoelectrics over the past few years with a focus on charge and heat carrier transport from both theoretical and experimental viewpoints. It also presents new strategies to improve thermoelectricity and discusses device physics and applications to guide the research community.

Published in London, UK © 2022 IntechOpen © pricelessphoto / iStock

Thermoelectricity - Recent Advances, New Perspectives and Applications

Thermoelectricity

Recent Advances, New Perspectives

and Applications

*Edited by Guangzhao Qin*