# Seebeck Coefficient from the BTE

I am trying to model the Seebeck coefficient and the electrical conductivity and lattice thermal conductivity for a thermoelectric material from the BTE using the RTA. I have two questions.

1. I am going to calculate each band separately and add. The electrical conductivity per band can be added together to find the total conductivity. However, the I saw in one paper that the total Seebeck coefficient is weighted by the electrical conductivity weighted average of the of the Seebeck coefficient per unit band. Why is this?

2. I am working with a material that has many grain boundaries. After I calculate the bulk crystalline properties, is there any way that I can include the effect of the grain boundaries on the lattice thermal conductivity? I know the average grain size and I would like a way to model this.

I am trying to model the Seebeck coefficient and the electrical conductivity and lattice thermal conductivity for a thermoelectric material from the BTE using the RTA. I have two questions.

1. I am going to calculate each band separately and add. The electrical conductivity per band can be added together to find the total conductivity. However, the I saw in one paper that the total Seebeck coefficient is weighted by the electrical conductivity weighted average of the of the Seebeck coefficient per unit band. Why is this?
I did not understand the second line of that question. Could you please revise it? Also, could you cite the paper you were referring?

2. I am working with a material that has many grain boundaries. After I calculate the bulk crystalline properties, is there any way that I can include the effect of the grain boundaries on the lattice thermal conductivity? I know the average grain size and I would like a way to model this.
This is a very crude approach: you can try and determine the mean free path due to the grain boundary scattering i.e $\lambda_{grain}$. Then you can find the effective means free path ($\lambda_{eff}$) using the resistors in parallel formula, where the mean free path due to each scattering mechanism is analogous to one of the resistors in the network.

$\frac{1}{\lambda_{eff}} = \frac{1}{\lambda_{grain}} + \frac{1}{\lambda_{impurity}} + ... etc.$

Now, if you can determine the reflection probability of your carriers, on average, at a grain boundary then you can estimate the mean free path due to the grain boundary scattering; see this probability is “p.” Then the carrier has to pass, on average, “1/p” grain boundaries before it gets completely reflected. If you multiply this by the average size of the grain then you can determine the mean free path due to grain boundary scattering.

I don't know if anyone uses this method, nor do I know of it's accuracy. This is what made sense to me. If, however, you want to be more certain and/or want to be more rigorous than I suggest you take a look at “effective medium theory.” This is a very fancy method commonly used to model grain boundaries in carrier transport.

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Satyala, N. and D. Vashaee (2012), Journal of Electronic Materials 41(6): 1785-1791.

The total Seebeck coefficient they used was the following -

$S_{tot}$ = $\frac{\sum_{i} σ_i S_i }{\sum_i σ_i}$

Why not just use $S_{tot} = \sum_{i} S_i$

Is the seebeck coefficient dependent on the electrical conductivity for each band.. I am quite confused

Satyala, N. and D. Vashaee (2012), Journal of Electronic Materials 41(6): 1785-1791.

The total Seebeck coefficient they used was the following -

$S_{tot}$ = $\frac{\sum_{i} σ_i S_i }{\sum_i σ_i}$

Why not just use $S_{tot} = \sum_{i} S_i$

Is the seebeck coefficient dependent on the electrical conductivity for each band.. I am quite confused
Seebeck coefficient is a function of energy; just like conductivity. Therefore when you want the total Seebeck coefficient, it has to be weighted by their energy dependent conductivities.