s21_crystalsilicon

The Loss Tangent of Crystalline Silicon

Brandon Nguyen

Advisor: Shaul Hanany

Abstract

I report measurements of electromagnetic absorption of crystalline silicon samples as a function of their conductivities. The absorption was determined by measuring the normal incident transmission with and without the sample and by analytically accounting for losses to reflection. For conductivities between 0.01 to 9.8 S/m I found that the loss tangent tan(δ) = (0.021 ± 0.006)σ + (0.012 ± 0.007) which captures the slope of the theoretical contribution to loss tangent from conductivity of 0.0162 ± 0.0006 Ωm within its uncertainty. Assuming the components of the complex permittivity of crystalline silicon are constant over this range of conductivities, then ε''/ε' = 0.012 ± 0.007.

Introduction

The cosmic microwave background (CMB) is a source of microwave radiation emanating uniformly from all parts of the sky. This radiation originates from an epoch roughly 200,000 years after the Big Bang when the universe became transparent to radiation [1]. To detect these wavelengths, optical elements are used to couple the radiation to receivers inside millimeter-submillimeter (MSM) telescopes. Of interest are the refractive lenses used in such telescopes. Crystalline silicon (c-Si) has been identified as an optimal material for these lenses as it has a higher index of refraction than previously used plastics and is easier to machine than materials with a similar index of refraction [2]. However, extremely high purity c-Si, which has the lowest loss, can only be manufactured in diameters up to 45 cm [3]. Thus, to take advantage of c-Si for larger lenses for higher resolution telescopes, the relationship between purity, correlating to conductivity, and absorption must be identified.

The goal of this project was to experimentally verify the theoretical relationship between conductivity and loss tangent by measuring the transmission of millimeter waves through c-Si wafers of varying conductivities. From transmission, the loss tangent of each sample was calculated, and a linear fit of the loss tangent and conductivity was compared to the theoretical contribution to loss tangent from the conductivity alone.

Theory

Transmission is defined as

where P_I is the incident power and P_T is the transmitted power. Loss tangent is defined as

where the complex permittivity έ = ε' - iε'', the complex index of refraction ή = n - iκ, σ is the conductivity, ν is the frequency of incoming radiation, and ε_0 is the vacuum permittivity [4]. I assumed that n = 3.4 for every c-S sample. To relate loss tangent to transmission I used a python function based on the transfer matrix method (TMM). The code makes the following assumption for the complex index of refraction.

which is true when κ is small compared to n. The formula for transmission at normal incidence involving the complex index of refraction used in the TMM code is

where l is the thickness of the wafer, and c is the speed of light in a vacuum [5, 6]. The experimental loss tangent was found by changing the loss tangent input of the TMM code until the transmission output matched the experimental transmission to two significant figures.

Experimental Setup and Method

Figure 2: Acrylic alignment tool shown in position with mirrors and feed horns. Various cutouts allow the acrylic sheet to be lowed vertically into this position.

After the tool was removed from the setup, an iterative alignment process began to maximize power over the remaining four degrees of freedom; the x and y positions of the source and receiver. Starting from the source x position, the maximum power position was found by measuring power over the entire range of the degree of freedom. Every time a new degree freedom was iterated over, the next step would be to return to the source x position and work through every previous degree of freedom before moving on to the next since changes in one degree of freedom may affect the maximum power position in anothe. For example, the order I went in was source x, source y, source x, source y, receiver x, source x ….

Standing Waves

Standing waves were present in the system as seen in figure 2 and could not be reduced as adding a sheet of Eccosorb caused power to be indistinguishable from noise. One sheet of Eccosorb was already in place between M2 and the receiver, it was placed at a slant so that any reflected waves would be sent either into the ceiling or down into the table.

Figure 4: Left a plot of measured incident power for sample A1. Right: A plot of measured transmitted power for sample A1. Every point represents an average of 50 measurements. Each individual measurement took ~0.5s, meaning the points shown are ~50s apart. The y position represents the extension on the translational stage micrometer. The points were taken every 0.04mm.

For every sample, incident and transmitted power were measured one right after the other in the same way that the points in Figure 4 were taken. P_I for sample A1 was taken as the average of all the points on the left plot Figure 4, P_T was taken as the average of all the points on the right plot of Figure 6. For other samples, P_I and P_T were determined in a similar fashion. This was done to reduce the effects of the standing waves and the change in power due to position on the transmission.

Figure 5: Left a four-point probe. Right: The four-probe method used to calculate resistivity. Conductivity is the inverse of resistivity.

Conductivity was measured using a Veeco/Miller FPP5000 four-point probe. 20 measurements were taken per sample then averaged. Several samples had resistivity that were too large and out of range of the probe and were not used in the experiment except for two high-resistivity wafers, which were both labeled as having a resistivity of 100 Ωm.

Results and Analysis

Table 1: Measured incident power, transmitted power, and calculated transmission for each c-Si sample.

Table 2: c-Si sample information and loss tangent. The final two samples had resistivites out of range for the four-point probe, the manufacturer values were taken instead which is why they don’t have uncertainties for the conductivity and σ contribution to tan(δ) columns.