Final Draft Lab 3 (LL and NZ) Determination of Carrier Density through Hall Measurements and Determination of Transition Temperature (\(Tc\)) in a High-Tc Superconductor

An electromagnet was used to provide a magnetic field to 3 different conducting samples: n type Geranium (n-Ge), p type Geranium (p-Ge), and silver (Ag). A calibrated Hall probe was used to obtain the current (\(\vec{I}_{mag}\)) to magnetic field (\(\vec{B}\)) calibration of the iron-core electromagnet. The Hall voltages (\(V_H\)) produced by each of the three samples were plotted against \(B\), and a linear line was produced, as expected. The slope (\(\frac{\Delta V_H}{\Delta B}\)) of each of the graphs were used to calculate the Hall coefficient for each sample, which we found to be \(-4.99\cdot 10^{-3}\pm -0.0998 \cdot 10^{-3}\frac{\textrm{Vm}}{\textrm{AT}}\), \(5.64 \cdot 10^{-3}\pm 0.11 \cdot 10^{-3} \frac{\textrm{Vm}}{\textrm{AT}}\), \(-2.24 \cdot 10^{-10}\pm -0.04 \cdot 10^{-10} \frac{\textrm{Vm}}{\textrm{AT}}\) respectively. These do not really agree with given values of \(-5.6\cdot 10^{-3}\frac{\textrm{Vm}}{\textrm{AT}}\) for n-Ge, \(6.6\cdot 10^{-3}\frac{\textrm{Vm}}{\textrm{AT}}\) for p-Ge, and \(-8.9\cdot 10^{-11}\frac{\textrm{Vm}}{\textrm{AT}}\) for silver by the manufacturer. Using the Hall coefficients, we found their carrier densities to be \(-1.25\cdot 10^{21}\pm 0.025 \cdot 10^{21} \textrm{m}^{-3}\) for n-Ge, \(1.11\cdot 10^{21}\pm 0.02 \cdot 10^{21} \textrm{m}^{-3}\) for p-Ge, \(-2.79\cdot 10^{28}\pm 0.06\cdot 10^{28} \textrm{m}^{-3}\) for silver, which are all in the same order of magnitude as the given absolute values of \(1.2 \cdot 10^{21} \textrm{m}^{-3}\), \(1.1 \cdot 10^{21} \textrm{m}^{-3}\), \(6.6 \cdot 10^{28} \textrm{m}^{-3}\).

A current was applied to the superconductor \(Bi_2 Sr_2 Ca_2 Cu_3 O_{10}\) which was cooled in liquid nitrogen until it became superconducting, and was allowed to warm slowly. Its voltage and temperature were monitored in the warming process which we used to produce a graph of voltage against temperature. The graph showed a transition temperature of about \(118\textrm{K}\pm 2\textrm{K}\), similar to the provided critical temperature of \(108\textrm{K}\).

The Hall Effect is an important part of finding out about charge transport in a material.In this experiment, we aim to determine the charge density in Ge n-type, Ge p-type semiconductor samples, and in a Ag (silver) sample. We will also be determining the transition temperature (\(Tc\)) of the high-Tc superconductor \(Bi_2 Sr_2 Ca_2 Cu_3 O_{10}\) (aka \(Bi2223\)).

Hall Effect was discovered by Edwin Herbert Hall, an American physicists, in 1879. This phenomenon can be used to determine the sign of the charge carrier in electrical conductors such as semiconductors and superconductors. The fundamental idea of Hall Effect can be illustrated in Figure \ref{fig:HallIllustration}: