confusions about calculating seebeck coefficient

[wot19920302]

Dear QuantumWise staffs :
          As we know, at low temperature, seebeck coefficent can be discribed by the equation in figure1: it's proportial to the slope of InT(E) at fermi level. However, in VNL, when I use thermoelectrical plugin, I can get seebeck as the function of Fermi level which can't be explained by the equation in figure1???.
          Figure2 is electron transmission spectrum and fig3 is seebeck coefficent as the function of fermi level at low
temperature(100K). As  seen from fig2, the transmission is  0 at the range from -0.5eV to 0.5eV, while seebeck coefficient  exhibts negative infinite values between 0eV-0.3eV in fig3....shouldn't these seebeck coefficent be zero because of 0 slope ?  When fermilevel shifts to 1eV, according to the equation, the slope of the transmission at this point is positive infinite value, so the seebeck coefficent should be  negative, but indeed seebeck coefficient in the plugin shows 0 at 1eV.......I wonder wether I misunderstand the relationship between transmission spectrum and seebeck coefficent ?
         I am looking forward your replies.
         sincerely

[zh]

To understand the equation in your figure 1, please read the following review paper (pages 17-19):
Heat flow and thermoelectricity in atomic and molecular junctions
Yonatan Dubi and Massimiliano Di Ventra
Rev. Mod. Phys. 83, 131 – Published 28 March 2011

As for how the Seebeck coefficient S is calculated from the transmission spectrum, you can refer to the script in the following discussion:
http://quantumwise.com/forum/index.php?topic=3411.0

And the thesis (page 61) by  Po-Hao Chang: Thermoelectric and thermospintronic transport in Dirac material-based nanostructures
http://udspace.udel.edu/handle/19716/17764

[wot19920302]

dear zh
       thanks for your guide. It helps me to know how seebeck coefficent is calculated in VNL thermoelectrical plugin. However, I still have  two questions about seebeck calculating:
     (1) the script of calculating seebeck coefficient you remind me reading http://quantumwise.com/forum/index.php?topic=3411.0 seems "missing" a minus:It should be S=-1/(eT)*(K₁/K₀) instead of S=1/(eT)*(K₁/K₀). When fermi level shifts to LUMO (E-Ef>0), seebeck shows a positive value instead of a negetive value because of "missing" minus. I don't know if I misunderstand that formula
     (2) I think the equation I offered in fig1 can be applied to analyse relationship between transmission and seebeck coefficient in linear response regime after reading this review： Heat flow and thermoelectricity in atomic and molecular junctions. ....In fact, I have read some papers which use this formula to analyse seebeck even at room temperature. But  the slope of transmission still can't explain the behaviour of seebeck as a function of fermi level I offerd in fig2 and fig3.What"s wrong with my thought?
      I really need your help .
      Best

[zh]

The negative sign in your equation 1 should be removed.  Please refer to the Rev. Mod. Phys. paper (eq.24) in my last reply.

S= 1/(eT) * K1/K0.   Also no negative sign.

[wot19920302]

But sir, in this link: http://science.sciencemag.org/content/sci/315/5818/1568.full.pdf, there is  a negative sign in eq3     What"s more,  in this paper:Seebeck coefficient of thermoelectric molecular junctions: First-principles calculations, S=-1/(e*T)*(K1/K0) 

[zh]

Thanks for reminding this. 

The definition for the sign of Seebeck coefficient  in literature may be messed.   In convention, S is defined as  - (\Delta V)/ (\Delta T). That is, if S is positive, the end with the higher temperature has the lower voltage, and vice versa. The voltage gradient in the material will point against the temperature gradient. From this definition, we may  know that in p-type semiconductors (which have only positive mobile charges, electron holes), S is positive. While in n-type semiconductors (which have only negative mobile charges, electrons), S is negative.

For the calculations of S by using the equation of  1/(e*T) * (K1/K0), the sign of  e (electron charge) is also messed in literature.  More exactly, it may be more properly to write the e  as charge 'q'.   

[Anders Blom]

Very good example of the importance of not only looking at how the formula is written, but also what the symbols mean. Even more so, to understand what positive/negative values mean physically.

Thanks, ZH and wot19920302.

[wot19920302]

thanks for your help, Zh and Anders Blom

[wot19920302]

dear Quantumstaffs:           I inspect the script from this link : http://quantumwise.com/forum/index.php?topic=3411.0   and find this block I can"t understand:    

Code

 t1 = transmission_spectrum.evaluate(spin=spin)
                 t1 = SplineInterpolation1D(E, t1)
                 t2 = transmission_spectrum.evaluate(spin=Spin.Up)+transmission_spectrum.evaluate(spin=Spin.Down)
                 t2 = SplineInterpolation1D(E, t2)

       It seems that  spin-resolved seebeck coefficient should be calculated by 1/eT*(K1/K0), where K1 contains single spin transmission spectrum and K0 contains the sum of transmission of spin.up and spin.down.     But in this paper: Spin-dependent Seebeck effects in graphene-based molecular junctions (PHYSICAL REVIEW B 93, 195426 (2016) ),K1 and K0 seems only contains one type spin(up or down), do I misunderstand the script  offered?                 Best