I am very interested in predicting the RF reflection and transmission
(at a frequency of 2.6 GHz) through a thin layer of Molybdenum (Mo)
that may be sputtered onto a 2 mil plastic substrate. The Mo
sputtering process will be very slow: over a cumulative time span of
several thousand hours the thickness will build up to as much as 500
Angstrom. I'm interested in looking at the RF properties as a
function of the thickness of the layer from 0 to 500 Angstrom, and as
a function of temperature from 90 to 400 K.

I know how to do the calculation given the value of the conductivity of Mo: for a thin layer of good conductor the metal can be modeled as a sheet resistance R = 1/(sigma*t), where sigma is the conductivity and t is the thickness. For a thicker layer one can use a more rigorous transmission line analogy. However, from browsing the literature, I've seen that the effective conductivity depends on several factors, including temperature, film thickness, and the manner in which the Mo is arranged: single crystal, polycrystalline, or amorphous.

This deposition is going to occur in a vacuum, due to ion bombardment of a molybdenum surface, at temperatures that vary over the range of 90 to 400 K.

I have found a paper (R. C. Hansen and W. T. Pawlewicz, "Effective conductivity and microwave reflectivity of thin metallic films," IEEE Trans. Antennas Propagat., vol 30, no 11, Nov 1982) that shows how to calculate the effective conductivity of a thin metallic layer given the bulk conductivity sigma_0 and the electron mean free path length L (in the bulk metal). The calculation is based on earlier work by Fuchs, Sondheimer, and Campbell. Hansen and Pawlewicz do not provide any comparison with measurements, but state that "this model fits polycrystalline films reasonably well" along with the claim that "most thin films will be polycrystalline." They provide an example calculation for a gold (Au) film, using the values of sigma_0 = 4.1e7 S/m and L = 570 Angstrom, which I assume are both valid at room temperature, approx. 300 K.

My questions:

1. Should I expect the deposited Mo layer to be polycrystalline, so that the Hansen/Pawlewicz formulas are valid? If not, how to proceed?

2. What is the electron mean free path length for Mo? Does this depend on temperature?

3. Is it true that the bulk conductivity of metals is inversely proportional to temperature over my working range (90K to 400K)?

4. Any pointers to other useful books or papers? I have only a minimal undergraduate EE background in solid state theory from 25 years ago!

Thanks very much,

Peter Simon peter underscore simon at ieee dot org (return email address is a spam trap)

I know how to do the calculation given the value of the conductivity of Mo: for a thin layer of good conductor the metal can be modeled as a sheet resistance R = 1/(sigma*t), where sigma is the conductivity and t is the thickness. For a thicker layer one can use a more rigorous transmission line analogy. However, from browsing the literature, I've seen that the effective conductivity depends on several factors, including temperature, film thickness, and the manner in which the Mo is arranged: single crystal, polycrystalline, or amorphous.

This deposition is going to occur in a vacuum, due to ion bombardment of a molybdenum surface, at temperatures that vary over the range of 90 to 400 K.

I have found a paper (R. C. Hansen and W. T. Pawlewicz, "Effective conductivity and microwave reflectivity of thin metallic films," IEEE Trans. Antennas Propagat., vol 30, no 11, Nov 1982) that shows how to calculate the effective conductivity of a thin metallic layer given the bulk conductivity sigma_0 and the electron mean free path length L (in the bulk metal). The calculation is based on earlier work by Fuchs, Sondheimer, and Campbell. Hansen and Pawlewicz do not provide any comparison with measurements, but state that "this model fits polycrystalline films reasonably well" along with the claim that "most thin films will be polycrystalline." They provide an example calculation for a gold (Au) film, using the values of sigma_0 = 4.1e7 S/m and L = 570 Angstrom, which I assume are both valid at room temperature, approx. 300 K.

My questions:

1. Should I expect the deposited Mo layer to be polycrystalline, so that the Hansen/Pawlewicz formulas are valid? If not, how to proceed?

2. What is the electron mean free path length for Mo? Does this depend on temperature?

3. Is it true that the bulk conductivity of metals is inversely proportional to temperature over my working range (90K to 400K)?

4. Any pointers to other useful books or papers? I have only a minimal undergraduate EE background in solid state theory from 25 years ago!

Thanks very much,

Peter Simon peter underscore simon at ieee dot org (return email address is a spam trap)