3.5.9 Onsager Relations

The relationship between Seebeck and Peltier coefficient has been identified in the pioneering work by Kelvin and formulated in the first Kelvin relation. However, Onsager formulated an extended theory valid for general systems consisting of several mutually dependent irreversible processes [12]. His theory is based on thermodynamics and the general principle of time symmetry and holds for systems around their equilibrium. Every flux of a certain quantity $x_i$ within a system is given by the linear combination of according driving forces

$$\ensuremath{\ensuremath{\mathitbf{j}}}_{x_i} = \sum_j L_{ij} \ensuremath{\xi}_j \,,$$ (3.123)

while each flux has an assigned driving force defined by the derivative of the system's entropy with respect to the according quantity [94]

$$\ensuremath{\xi}_j = \ensuremath{\partial_{x_j} S} \,.$$ (3.124)

Following (3.123), the affinities $\ensuremath{\xi}_j$ describe the deviation from equilibrium, which is characterized by zero currents. In equilibrium, the entropy $S$ reaches a maximum. Onsager's reciprocal relations state the identities of the cross coefficients

$$L_{ij} = L_{ji}$$ (3.125)

for vanishing magnetic fields. The cross coefficients $L_{ij}$ are a measure for the coupling of the single transport phenomena within the system. A system with $L_{ij}=0$ consists of independent irreversible processes, where every driving force only affects its connected flux. A convenient form to identify the according affinities to given fluxes is obtained by considering the time derivative of the entropy

$$\ensuremath{\ensuremath{\partial_{t} S}} = \sum_j \ensuremath{\xi... ...bla_{\!r}}}}\ensuremath{\cdot}}\ensuremath{\ensuremath{\mathitbf{j}}}_{x_i} \,.$$ (3.126)

For the thermoelectric case, the basic relations are given from irreversible thermodynamics [9]. In local thermodynamic equilibrium, the differential total energy is obtained as the sum of products of corresponding internal variable and the differential external variable

$$\d U = \ensuremath{T}\d\ensuremath{S}- p \d V + \mathrm{q}\ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}^\mathrm{c}}\d N$$ (3.127)

with the number of particles within the system $N$ and the chemical potential $\ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}^\mathrm{c}}$ , which is the difference of the electrochemical and electrostatic potentials. Since the electrostatic potential is connected to the carrier densities as well as doping densities by Poisson's equation, it is not an independent variable in the thermodynamic sense. Neglecting thermal expansion, the $\d V$ -term vanishes. Thus, expression of the differential entropy results in

$$\d S = \frac{1}{\ensuremath{T}} \d U - \frac{\mathrm{q}\ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}^\mathrm{c}}}{\ensuremath{T}} \d N \,.$$ (3.128)

Introducing the according current densities, the entropy flux becomes

$$\ensuremath{\ensuremath{\mathitbf{j}}}_s = \frac{1}{\ensuremath{T... ...nu}}^\mathrm{c}}}{\ensuremath{T}} \ensuremath{\ensuremath{\mathitbf{j}}_\nu}\,.$$ (3.129)

Formulating balance equations for the entropy, the total energy, and the particle density, as well as assuming steady state conditions, the change of entropy for the thermoelectric case is obtained as

$$\ensuremath{\ensuremath{\partial_{t} S}} = \ensuremath{\ensuremat... ...nu}}^\mathrm{c}}}{\ensuremath{T}} \ensuremath{\ensuremath{\mathitbf{j}}_\nu}\,.$$ (3.130)

According to the definition of heat $\d Q = \ensuremath{T}\d S$ , the heat flux equation can be obtained from the energy flux as

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{q}}= \ensuremat... ...hi}_{\ensuremath{\nu}}^\mathrm{c}}\ensuremath{\ensuremath{\mathitbf{j}}_\nu}\,.$$ (3.131)

Thus, the change of entropy with respect to the heat flux instead of the total energy flux reads

$$\ensuremath{\ensuremath{\partial_{t} S}} = \ensuremath{\ensuremat... ...nu}}^\mathrm{c}}\ensuremath{\cdot}\ensuremath{\ensuremath{\mathitbf{j}}_\nu}\,.$$ (3.132)

From (3.130), it follows that if particle current and total energy current are considered as fluxes, the according affinities are $\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}(1/\ensuremath{T})$ and $\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}(\ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}^\mathrm{c}}/\ensuremath{T})$ . A more convenient choice can be extracted from (3.132), where the affinities $\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}(1/\ensuremath{T})$ and $(1/\ensuremath{T}) \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}} \ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}^\mathrm{c}}$ follow from the particle current and the heat current as chosen fluxes. In the sequel, the combination of particle flux and heat flux has been chosen for the analysis of several transport models derived. The general equations for particle and heat flux with the Onsager coefficients $L_{ij}$ read

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu}= L_{11} \frac{1}{\ensu... ...h{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\left( \frac{1}{\ensuremath{T}} \right)$$ (3.133)

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{q}}= L_{21} \fr... ...ensuremath{\mathitbf{\nabla_{\!r}}}}}\left( \frac{1}{\ensuremath{T}} \right)\,.$$ (3.134)

The expansion of the temperature derivatives results in a more convenient form which can be used for a direct coefficient comparison with the transport equations obtained previously

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu}= L_{11} \frac{1}{\ensu... ...2} \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}$$ (3.135)

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{q}}= L_{21} \fr... ...\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}\,.$$ (3.136)

Particle current and heat current obtained by Bløtekjær's approach read

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu}= - \ensuremath{\mathrm... ...t) \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}$$ (3.137)

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{q}}= - \ensurem... ...ath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}}$$ (3.138)

$$\quad - \ensuremath{\ensuremath{\mu}_\nu}\ensuremath{\nu}k_\ensur... ...\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}\,.$$

A coefficient comparison between (3.135) and (3.137) as well as (3.136) and (3.139) yields the Onsager coefficients

$$L_{12} = \ensuremath{\ensuremath{\mu}_\nu}\ensuremath{\nu}\frac{k... ... \frac{5}{2} - \ln \frac{\ensuremath{\nu}}{\ensuremath{N_\mathrm{c,v}}} \right)$$ (3.139)

$$L_{21} = \ensuremath{\ensuremath{\mu}_\nu}\ensuremath{\nu}\frac{k... ...rac{5}{2} - \ln \frac{\ensuremath{\nu}}{\ensuremath{N_\mathrm{c,v}}} \right)\,.$$ (3.140)

As a consequence of the macroscopic relaxation time approximation, where one relaxation time for every current is introduced, different mobility definitions enter the particle current and heat current. Thus, the model obtained by Bløtekjær's approach does not inherently fulfill Onsager's reciprocity theorem.

Next, the model obtained by Stratton's approach is analyzed. The equations for particle and heat flux are obtained for homogeneous materials as

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu}= - \ensuremath{\mathrm... ...t) \ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}$$ (3.141)

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{q}}= - \ensurem... ...ath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}}$$ (3.142)

$$\quad - \ensuremath{\ensuremath{\mu}_\nu}\ensuremath{\nu}\frac{k_... ...ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T} \,.$$

Analogously, the Onsager coefficients are identified by a coefficient comparison between (3.135) and (3.141) as well as (3.136) and (3.142) as

$$L_{12} = L_{21} = \ensuremath{\ensuremath{\mu}_\nu}\ensuremath{\n... ...th{r_\nu}}- \ln \frac{\ensuremath{\nu}}{\ensuremath{N_\mathrm{c,v}}} \right)\,.$$ (3.143)

For the Stratton model, Onsager's reciprocity theorem holds due to the application of the microscopic relaxation time approximation, where a single relaxation time enters all fluxes.

Finally, the model derived in Section 3.5.7 is analyzed. In contrast to the other models, the stochastic part is described by a linear combination of all fluxes taken into account. The final particle current and heat current equations are

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu}= - \frac{\ensuremath{Z... ...ath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}}$$ (3.144)

$$\quad -\frac{\left(\ensuremath{Z_{11}}-\frac{5}{2}k_\ensuremath{\... ...nu}\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}$$

$$\ensuremath{\ensuremath{\mathitbf{j}}_\nu^\mathrm{q}}= -\frac{\fr... ...ath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{\ensuremath{\Phi}_{\ensuremath{\nu}}}$$ (3.145)

$$\quad -\left( \frac{\left( \frac{5}{2} k_\ensuremath{\mathrm{B}}\... ...ath{Z_{00}}\ensuremath{Z_{11}}- \ensuremath{Z_{10}}\ensuremath{Z_{01}}} \right.$$

$$\quad \left. - k_\ensuremath{\mathrm{B}}\ensuremath{T}\ln \frac{\... ...\ensuremath{\ensuremath{\ensuremath{\mathitbf{\nabla_{\!r}}}}}\ensuremath{T}\,.$$

The Onsager coefficients are identified in the usual way by a coefficient comparison between (3.135) and (3.145) as well as (3.136) and (3.146), respectively. Thus, the according Onsager coefficients are

$$L_{12} = \frac{\left(\ensuremath{Z_{11}}-\frac{5}{2} k_\ensuremat... ...}- \ensuremath{Z_{10}}\ensuremath{Z_{01}}} \ensuremath{\nu}\ensuremath{T}^2 \,,$$ (3.146)

$$L_{21} = \frac{\frac{5}{2} k_\ensuremath{\mathrm{B}}\ensuremath{T... ...11}}- \ensuremath{Z_{10}}\ensuremath{Z_{01}}} \ensuremath{\nu}\ensuremath{T}\,.$$ (3.147)

A comparison of the relevant parameters results in the relation

$$\ensuremath{Z_{10}}+ \frac{5}{2} k_\ensuremath{\mathrm{B}}\ensure... ...eft( k_\ensuremath{\mathrm{B}}\ensuremath{T}\right)^2 \ensuremath{Z_{01}}= 0\,.$$ (3.148)

Briefly summarized, Stratton's equations inherently fulfill the Onsager relations, while the Onsager conformity of Bløtekjær's equations depends on the choice of the model parameters. This is a consequence of the extended degrees of freedom due to the additional model parameters introduced by the macroscopic relaxation time approximation. For the non-diagonal ansatz, the reciprocity principle is fulfilled, when the model is parametrized obeying the relation (3.148).

M. Wagner: Simulation of Thermoelectric Devices