Hi,
I am trying to use Van der Pauw geometry to measure mobility of my field effect transistor samples. I know by doing Hall measurement I can get Hall mobility. However, I can only access field effect properties (no magnetic field), so I was wondering whether there is a way to calculate field effect mobility by using Van der Pauw method.
Currently, I know Van der Pauw method can give the sheet resistance (Rs) of sample. And I also found there is a paper that used Van der Pauw method to calculate the mobility. They used the equation as:
mobility=transconductance / Capacitance * ln2 / pi
But I am not quite sure why geometry factors (length, width) are not included in that equation. And I did not found any other paper using the same equation as they do. So I would like to ask the validity of this equation and more information about it.
Thank you!
Well, I had a more detailed answer typed up but it failed to upload. Here's the abridged version.
If you have functioning field-effect devices, why can't you measure the mobility directly from saturation-mode device characteristics? I imagine making a VdP sample is only marginally easier since the contacts should be well-defined and placed at the extremity of the sample.
Regarding the above equation, the ln2/pi factor suggests to me that the authors of the paper assume the two orthogonal resistances in the VdP equation are equal, which causes the VdP equation to simplify to:
Rs = R*pi/ln2
If this is case, it's probably because they chose W = L to allow the simplification, setting the W/L term to 1. However, the equation is still missing the overdrive voltage term (Vds-Vt) that is part of the standard saturation mode device model. If you could upload or link to the paper using this equation, myself or someone else could better assist you here.
The geometrical factors are not included due to the symmetry of the VdP probe. That is, when the resistance measured between two horizontal terminals is equal (to a good approximation) to that measured between two vertical terminals, the geometrical factors disappear from the VdP equation.
Mobility can be derived from the same sheet resistance if you know the thickness of your diffusion and the volume doping level of the same: mu=1/(q Rs Ns) where Ns is obtained multiplying the doping level and the thickness.
I do not know the details of your configuration, but consider that the mobility you measure is measured for carriers moving in the top-down direction (the same of the measured voltage difference)
Dear Wellenius and Curro,
Thank you very much for your kindly reply. The paper I mentioned is attached below. And thanks for your guys answer, I found that, as you said, the paper uses that equation because of the square geometry of sample, which yields L=W and therefore cancel the geometry factor in the equation.
Also, in my case, it seems like that my FET samples were still in linear mode regime, and I could only extract mobility from the slope dG/dVg (transconductance) (if I am correct).
And I got some follow-up questions about your replys:
(1) how about non-square case? if the sample geomerty is not symmetric (e.g. retangular), how do I modify the equation to estimate mobility?
(2)As far as I know, the equation mu=1/(q Rs Ns) depends on the knowledge of Ns, which should be given by Hall measurement. Is there any other way can be used for determining Ns? (because my samples are not intentioally doped, I am not sure the doping concetration)
P.S. I know the capacitance model which can be used in estimation of induced charge carrer by gate electric field: n=capacitance*(Vg-Vt), where Vg is gate voltage and Vt is threshold voltage. But I am not sure whether it is okay to use it as Ns for mobility extraction.
Thank you once again for your help.
Sincerely,
https://arxiv.org/ftp/arxiv/papers/1502/1502.01615.pdf
1) The VdP model which stands behind the method is based on a non-trivial and transcendent equation that can not be analythically resolved but in few high-symmetry cases. Moreover, one of the key requirements to be assured is that the perimetric contacts must be as far one onether as possible (compared to the same contact size). Just check the VdP equation on text books and you will clearly see the difficulties. Of course there is Always the possibility to solve the equation for non-square probe by a numerical iterative procedure... but precision would be partially lost.
2) If you do not know Nd but also the diffusion thickness you can not use the method. That is, even if you find Nd with some other kind of measurement you would still need the diffusion thickness. You say that your diffusion is not intentionally doped... It could be better having some more info about your sample. Perhaps, the needed parameters could be obtained from other process/geometry characteristics
I agree with Giuseppe's answer above. I strongly encourage you build and characterize your FET devices to achieve the best estimate of FE mobility. You can find a good description of FET device models at the link below.
http://ecee.colorado.edu/~bart/book/book/contents.htm
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