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\documentclass[a4paper, 12pt,oneside]{article}
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%Début du document
\begin{document}
\title{\normalsize{Lab Work Report - Group N$^\circ$\\ XX - Experiment}}
\date{\normalsize{\today}}
\author{\normalsize{Name} 1\and \normalsize{Name 2}}
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\begin{center}
\large\textbf{\sffamily Experiment N$^\circ$ H6 : Hall Effect}\\%
\large\sffamily Group N$^\circ$16: Ancarola Raffaele, Cincotti Armando\\%
\large\sffamily \today\qquad Yikai Yang\\%
\end{center}
% Introduction
\section{Introduction}
The {\bf Hall Effect} is a {\it galvanomagnetic} effect that can be observed when an induction field is applied
to a conductor or semi-conductor. The following report will study this effect on different samples, in order to determine differents
electromagnetic behaviours of the studied materials.
The Hall effect is used in many applications such as the manufacture of magnetic sensors and concatless position sensors.
The main purpose of this experiment is to measure the Hall constant $R_H$ of many materials, in order to get to interesting conclusions.
\section{A brief Theoretical basis}\label{th}
\subsection{Hall Effect}\label{hallth}
\begin{minipage}{\textwidth}
\hspace{-0.02\textwidth}
\begin{minipage}{0.52\textwidth}
The {\bf Hall effect} can be observed in a conducting or semi-conducting material,
when a current $\vec{I}$ [\si{\ampere}] is applied on it while a magnetic
field $\vec{B}$ [\si{\tesla}] is applied perpendicularly to the current.
The effect is detected as a tension $V_H$ appears on the edges of the conductor,
in the direction perpendicular to $\vec{B}$ and $\vec{I}$.
In order to simplify the notation, a reference system as the one in Figure (\ref{hallfig}) is chosen, so that $B$ is considered as
the magnitude of $\vec{B}$ in z direction, and $j$ the magnitude of the current density in x direction,
defined as $j = I/S$. $S = a\cdot b$ is the section of the conductor.
\end{minipage}
\hspace{0.02\textwidth}
\begin{minipage}{0.52\textwidth}
\includegraphics[width = \textwidth]{effetto_hall.PNG}
\captionof{figure}{Hall effect applied to a sample having a section of $a \cdot b$}
\label{hallfig}
\end{minipage}
\vspace{0.5cm}
\end{minipage}
The tension $V_H$ appears because, once the forces in the conductor are balanced, an electric field
$\vec{E_H}$, the {\it Hall field}, appears to compensate the {\it Lorentz Force} $\vec{F_l}$ \cite{ref1}:
\begin{equation}\label{lorentz}
\vec{F_l} = I \cdot \vec{l}\times \vec{B} = j S \cdot \vec{l}\times \vec{B}
\end{equation}
where $\vec{l}$ is a leght vector in the direction of the current.
By relation (\ref{lorentz}), knowing that $\vec{E_H}$ is opposite to $\vec{F_l}$,
$V_H$ is such that $V_H \propto j \cdot B \cdot b $, supposing that $E_H$ is homogeneous . This means that exists a constant
$R_H$, the Hall constant, such that \cite{ref2}:
\begin{equation}\label{prop}
V_H = R_H j B b \Rightarrow V_H = R_H \frac{I}{S} B b = \frac{R_H}{a} I \cdot B
\end{equation}
$R_H$ can be computed as :
\begin{equation}\label{rh}
R_H = \frac{aV_H}{IB}
\end{equation}
Furthermore, appling the Ohm law $\vec{j} = \sigma \vec{E}$, where $\sigma$ is the electric conductivity of the material traversed by $\vec{j}$,
an expression for $E_H$ can be found as \cite{ref2}:
\begin{equation}\label{eee}
E_H = \frac{1}{qN}jB
\end{equation}
where $N$ and $q$ are respsectively the number of charge carrier per unit of volume and their charge.
In the following report, to simplify the computations, it is hypothised that the only charge carrier are electrons and so $q = e =1.6\cdot 10^{-19}$ C, where $e$ is the elementary charge.
To finish with, relations (\ref{rh}) and (\ref{eee}), and knowing that $V_H = E_H b$ for an homogeneous $E_H$, imply :
\begin{equation}\label{qn}
R_H = \frac{1}{qN}
\end{equation}
\subsection{Hysteresis Cycle for ferromagnetic materials}\label{hysth}
For a perfect solenoid with a non-ferromagnetic core, the induction field $\vec{B}$ follows a linear relation on $\vec{H}$, the magnetic field generated
by the coil. In order to obtain a bigger induction field, a ferromagnetic core is used (see our last report \cite{ref3}). In this case the function
$B(H)$ describes an Hysteresis, so the relation between $B$ and $H$ isn't linear.
Nevertheless, if the Hysteresis is negligible, function $B(H)$ stays linear in a certain domain, before $B$ gets to a saturation value.
As for a perfect solenoid $H\propto I_B$, where $I_B$ is the current passing through the coil, in the linear domain is possible to find a bijection between $I_B$ and $B$.
\section{Experimental procedure}
\subsection{Experimental setup}
\begin{minipage}{\textwidth}
\hspace{-0.02\textwidth}
\begin{minipage}{0.52\textwidth}
Firstly, two coils with a ferromagnetic core are positionned in order to obtain a certain induction field $B$ when a current $I_B$ passes through them.
$I_B$ is provided by a current generator, and an amperemeter is attached in serie to the circuit in order to measure $I_B$. Then, samples of different materials are placed between the two coils so that the induction field can be applied to them. $B$ is measured by a Teslameter placed between the coils. This first setup
is shown on Figure (\ref{coilsfig}).
\end{minipage}
\hspace{0.02\textwidth}
\begin{minipage}{0.52\textwidth}
\includegraphics[width = \textwidth]{coils.PNG}
\captionof{figure}{Coils montage}
\label{coilsfig}
\end{minipage}
\vspace{0.5cm}
\end{minipage}
Secondly, the studied samples are connected to a current generator, attached in serie to an Amperemeter in order to measure the current
$I$ passing through the sample. Then a Voltemeter is also attached to the sample in order to measure $V_H$.
The configuration of these circuits depends on the studied sample: the cross samples in Figure (\ref{FourHonor}), where the circuit
can be attached on 4 different points of the material, and the samples with 5 connections in Figure (\ref{TakeFive}), where a potentiometer can be
attached to the circuit in serie to the Voltmeter.
\begin{minipage}{\textwidth}
\hspace{-0.08\textwidth}
\begin{minipage}{0.48\textwidth}
\includegraphics[width = \textwidth]{montaggio4.png}
\captionof{figure}{Four contacts circuit}
\label{FourHonor}
\end{minipage}
\hspace{0.02\textwidth}
\begin{minipage}{0.48\textwidth}
\includegraphics[width = \textwidth]{montaggio5.png}
\captionof{figure}{Five contacts circuit}
\label{TakeFive}
\end{minipage}
\end{minipage}
\subsection{Potetiometer} \label{potentiometer}
\begin{minipage}{\textwidth}
\hspace{-0.02\textwidth}
\begin{minipage}{0.55\textwidth}
Even though no induction field $B$ crosses the sample surface, a remanent tension $V_r$ can be measured. The potentiometer
is an electronical device composed by two variable and one fixed resistances, as in Figure (\ref{potentiofig}), that allows to modify the resistance applied at the connection to the sample. This device is used to adjust this resistance, in order to remove the remanent tension $V_r$ so that it doesn't interfere in the measurements.
\end{minipage}
\hspace{0.02\textwidth}
\begin{minipage}{0.45\textwidth}
\includegraphics[width = \textwidth]{potentiometer.png}
\captionof{figure}{Illustration of a potentiometer}
\label{potentiofig}
\end{minipage}
\vspace{0.5cm}
\end{minipage}
\subsection{Determination of $R_H$ and $N$} \label{detRH}
Once the sample is installed between the coils, varying $I$ under a fixed induction field $B$, or varying $I_B$ in order to obtain different fields $B$ on a fixed
$I$, allows to study $V_H$ in function of $I$ or $B$. It also allows to obtain two computations for $R_H$ using relation (\ref{rh}).
The width $a$ of the sample is given with this one. Once obtained $R_H$, the number $N$ of charge carrier is computed by relation (\ref{qn}) knowing their charge $q$.
\section{Results and discussion}
\subsection{Determination of the linear domain $B \propto I_B$}
\begin{minipage}{\textwidth}
\hspace{-0.02\textwidth}
\begin{minipage}{0.52\textwidth}
\resizebox{\textwidth}{!}{
\input{graphs/hysteresis.tex}
}
\captionof{figure}{Current $I_B$ applied on the coils in function of the induction field $B$}
\label{hysteresis}
\end{minipage}
\hspace{0.02\textwidth}
\begin{minipage}{0.52\textwidth}
The graph (\ref{hysteresis}) shows that the linear domain for the relation
$B(I_B)$ is given by $I \in [-3, 3]$. Then, making a linear fit on that
interval the following relation is obtained:
\begin{align}\label{rel_BI}
B &= m_{IB} \cdot I_B + b_{IB} \\
m_{IB} &= (0.123 \pm 0.002) \text{ \si{\tesla\per\ampere}} \\
b_{IB} &= (-0.011 \pm 0.019) \text{ \si{\tesla}}
\end{align}
By looking at graph in Figure (\ref{hysteresis}), it can be noticed that hysteresis phenomenon is in fact negligible
and the considerations of section \ref{hysth} can be applied in this case.
\end{minipage}
\end{minipage}
\subsection{Four contacts samples: \ce{InP}:\ce{Si}}
Provided that the circuit configuration (Figure \ref{FourHonor}) does not allow to
eliminate the parassite {\bf residual tensions} $V_r$, it's a good practice to check
whether these terms could be influential or not.
The table (\ref{residual}), shows how $V_r$ behaves for 4 cyclic permutations of the configuration in Figure (\ref{FourHonor}).
In fact, the table shows clearly that $V_r$ values are in the order of
the \si{\milli\volt}, when the mesured data in figures (\ref{FourContVB}) and (\ref{FourContVI}) are much
higher (order of \si{\volt}). This means that these tensions are negligible.
To finish with, this parasite tensions are caused by imperfections in the setup, as the current passing through the sample isn't perfectly aligned on the
chosen reference system as in Figure (\ref{hallfig}).
\begin{table}[H]
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
Configuration & {\bf 1234} & {\bf 2341} & {\bf 3412} & {\bf 4123} \\
\hline
{\bf Residual tension} $V_r$ & $-35$ mV & $69$ mV & $-35$ mV & $69$ mV \\
\hline
\end{tabular}
\caption{Experimentally computed $R_H$ for four contacts samples}
\label{residual}
\end{table}
The graphs below (Figures (\ref{FourContVB}) and (\ref{FourContVI})) show that
both the current $I$ and the induction field $B$, when varied, produce the expected behaviour
on the tension $V_H$ as described in the relation (\ref{prop}). Then, applying a linear fit,
the slope of the provided line corresponds to the ratio $\frac{R_H}{a}$.
\begin{minipage}{\textwidth}
\hspace{-0.08\textwidth}
\begin{minipage}{0.52\textwidth}
\resizebox{\textwidth}{!}{
\input{graphs/Si_VB.tex}
}
\captionof{figure}{$V_H$ in function of $B$, $I$ fixed to $1$ \si{\milli\ampere}}
\label{FourContVB}
\end{minipage}
\hspace{0.02\textwidth}
\begin{minipage}{0.52\textwidth}
\resizebox{\textwidth}{!}{
\input{graphs/Si_VI.tex}
}
\captionof{figure}{$V_H$ in function of $I$, $B$ fixed to $400$ \si{\milli\tesla}}
\label{FourContVI}
\end{minipage}
\end{minipage}
As it can be seen, the tension $V_H$ raises when $I$ or $B$ are increased, which
implies a positive value for $R_H$. In fact, the table (\ref{rh_semicondtab})
shows that both the values obtained by different procedures are positive and
practically equivalent.
On the other hand, it can be noticed that $R_H$ depends on the section following an inverse
proportionality contrary to what was expected (relation (\ref{rh})). This means
that the ratio $\frac{V_H}{IB}$ depends on the section $a$. Probably this dependance is due to the fact that in practice samples aren't ideal
, and also by any change in temperature of the sample.
\begin{table}[H]
\centering
\begin{tabular}{cc|c|c|c|c|}
\cline{3-6}
& & \multicolumn{2}{c}{Relation $V_H(B)$} & \multicolumn{2}{|c}{Relation $V_H(I)$} \\
\hline
\multicolumn{1}{|c|}{\bf Medium} & Section $a$ [\si{\micro\metre}] & $R_H$ [\si{\deci\metre^3\per\coulomb}] & $N$ [\si{10^{21}.1\per\metre^3}] & $R_H$ [\si{\deci\metre^3\per\coulomb}] & $N$ [\si{10^{21}.1\per\metre^3}] \\
\hline
\multicolumn{1}{|c|}{InP:Si} & 1 & $2.0 \pm 0.1$ & $3.1 \pm 0.1$ & $1.9 \pm 0.1$ & $3.2 \pm 0.1$ \\
\hline
\multicolumn{1}{|c|}{InP:Si} & 2 & $1.3 \pm 0.1$ & $4.7 \pm 0.1$ & $1.1 \pm 0.1$ & $5.6 \pm 0.1$ \\
\hline
\end{tabular}
\caption{Experimentally computed $R_H$ for four contacts samples}
\label{rh_semicondtab}
\end{table}
\subsection{Five contacts samples: Silver, Copper and \ce{In_2O_3}:\ce{SnO_2}}
In this case, the circuit montage (Figure \ref{TakeFive}) gives a supplementary vantage on the
measurement precision: provided that the residual tensions (Table \ref{residual})
are no more negligibile, a {\it potentiometer} is used in order to eliminate
them (description in Section \ref{potentiometer}).
\begin{minipage}{\textwidth}
\hspace{-0.08\textwidth}
\begin{minipage}{0.52\textwidth}
\resizebox{\textwidth}{!}{
\input{graphs/Ag_Cu_VB.tex}
}
\captionof{figure}{$V_H$ in function of $B$, $I$ fixed to $2$ \si{\ampere}}
\label{FiveContVB}
\end{minipage}
\hspace{0.02\textwidth}
\begin{minipage}{0.52\textwidth}
\resizebox{\textwidth}{!}{
\input{graphs/Ag_Cu_VI.tex}
}
\captionof{figure}{$V_H$ in function of $I$, $B$ fixed to $400$ \si{\milli\tesla}}
\label{FiveContVI}
\end{minipage}
\end{minipage}
Note that in graphs (\ref{FiveContVI}) and (\ref{FiveContVB}) there is still a small deviation.
This is caused by a lack in sensitivity of the potentiometer, which makes it
impossible to perfectly set an offset up to zero. This inconvenient can be
easily bypassed by taking only the slope term of a linear fit instead of the
entire result, because the affine term is supposed to give that deviation.
\begin{table}[H]
\centering
\begin{tabular}{cc|c|c|c|c|}
\cline{3-6}
& & \multicolumn{2}{c}{Relation $V_H(B)$} & \multicolumn{2}{|c}{Relation $V_H(I)$} \\
\hline
\multicolumn{1}{|c|}{\bf Medium} & Section $a$ [\si{\micro\metre}] & $R_H$ [\si{\milli\metre^3\per\coulomb}] & $N$ [\si{10^{28}.1\per\metre^3}] & $R_H$ [\si{\milli\metre^3\per\coulomb}] & $N$ [\si{10^{28}.1\per\metre^3}] \\
\hline
\multicolumn{1}{|c|}{Silver} & 1.9 & $-0.078 \pm 0.008$ & $-8.0 \pm 0.2$ & $-0.058 \pm 0.001$ & $-11 \pm 1$ \\
\hline
\multicolumn{1}{|c|}{Copper} & 1.6 & $-0.037 \pm 0.005$ & $-17.0 \pm 0.1$ & $-0.052 \pm 0.001$ & $-12 \pm 1$ \\
\hline
\multicolumn{1}{|c|}{\ce{In_2O_3}:\ce{SnO_2}} & 0.15 & $-3.0 \pm 0.4$ & $-0.21 \pm 0.01$ & $-2.8 \pm 0.1$ & $-0.22 \pm 0.01$ \\
\hline
\end{tabular}
\caption{Experimentally computed $R_H$ for five contacts samples}
\label{rh_metaltab}
\end{table}
Contrary to what had been seen for the silicium case, the values of $R_H$ shown
on Table (\ref{rh_metaltab}) are negative.
It can be also noticed, that $R_H$ value for \ce{In_2O_3}:\ce{SnO_2} is bigger than for metals, meaning that it is more
affected by the induction field $B$.
\subsection{$N$ and $R_H$ sign, and considerations on their values}
As it is shown on tables (\ref{rh_semicondtab}) and (\ref{rh_metaltab}), semiconductors have a positive $R_H$, and conductors a negative $R_H$.
Nevertheless, \ce{In_2O_3}:\ce{SnO_2} is a semi-conductor and its $R_H$ is negative, because in semi-conductors $R_H$ depends on their doping.
This means that the moving charges in a InP semiconductor are positive, meaning that a {\it hole} is moving \cite{ref2}, as a non-negative charge (a non-electron) can be considered as a positive one. On the other hand, $R_H$ value is negative, meaning that electrons are moving in conductors.
Comparing both the table, it can be noticed that $N$ is bigger for conductors than for semi-conductors, meaning that the latter are worst conductors.
On the other hand, $R_H$ is bigger in semi-conductors, showing that they are affected the most by variations in the induction field $B$.
To finish with, $N$ has been computed under the hypothesis that in the studied samples there's only one type of charge carrier. Any error on the computed values is in fact due to this hypothesis, and also to the fact that Ohm Law is completely consistent only if there's no temperature gradient in the sample, which is practically unlikely to be true.
\section{Conclusion}
In conclusion, the report shows how the Hall effect can be measured and studied in laboratory for different materials. The expetiments allowed to conclude
that conductors are in fact better in {\it conducting} than semi-conductors, which is quite intuitive. But also, on the other hand, it allowed to conclude that the latter are more sensitive to the induction field $B$, meaning that they can be used as better sensors of magnetic fields. This experimet is also really important in the field of particle physics, as it permits for exemple, to identify the sign of charge carrier in electric conductors.
To finish with, it can be suggested to consider different charge carrier in the study of semi-conductors,
as it can lead to better results and better understanding of this kind of materials.
\section{Annexes}
\subsection{\ce{In_2O_3}:\ce{SnO_2} plots}
\begin{minipage}{\textwidth}
\hspace{-0.08\textwidth}
\begin{minipage}{0.52\textwidth}
\resizebox{\textwidth}{!}{
\input{graphs/InSn_VB.tex}
}
\captionof{figure}{$V_H$ in function of $B$, $I$ fixed to $100$ \si{\milli\ampere}}
\end{minipage}
\hspace{0.02\textwidth}
\begin{minipage}{0.52\textwidth}
\resizebox{\textwidth}{!}{
\input{graphs/InSn_VI.tex}
}
\captionof{figure}{$V_H$ in function of $I$, $B$ fixed to $400$ \si{\milli\tesla}}
\end{minipage}
\label{InSnPlots}
\end{minipage}
\subsection{Linear regression}
\paragraph{Variance and covariance}
Let $X$, $Y$ be sets of data sized $N$, then the covariance $cov_{XY}$ is given by:
\begin{align}
cov_{XY} &= \frac{1}{N} \sum\limits_{k = 1}^N (X_k - \bar{X})(Y_k - \bar{Y}) \\
\bar{X} &= \frac{1}{N} \sum\limits_{k = 1}^N X_k \\
\bar{Y} &= \frac{1}{N} \sum\limits_{k = 1}^N Y_k \\
\end{align}
Then the variances $\sigma_X^2$ and $\sigma_Y^2$ are given by:
\begin{align}
\sigma_X^2 &= cov_{XX} \\
\sigma_Y^2 &= cov_{YY}
\end{align}
\paragraph{Fit}
Let $X$ and $Y$ be as above, then the linear regression $f(x)$ is given
by:
\begin{align} \label{linear_reg}
f(x) &= m_{XY} \cdot x + b_{XY} \\
m_{XY} &= \frac{cov_{XY}}{\sigma_X^2} \\
b_{XY} &= \bar{Y} - m_{XY} \cdot \bar{X}
\end{align}
\paragraph{Application to $V_H(B)$ and $V_H(I)$}
Following the relation (\ref{linear_reg}), just replace the set
$Y$ with the taken data of $V_H$, analogly replace $X$ with
$B$ or $I$ depending the context:
\begin{align}
V_H(B) &= m_{BV_H} \cdot B + b_{BV_H} \\
V_H(I) &= m_{IV_H} \cdot I + b_{IV_H}
\end{align}
\paragraph{Application to $B(I)$}
\begin{equation}
B = m_{IB} \cdot I + b_{IB}
\end{equation}
\subsection{About the error}
\subsubsection{Error on fit slope \cite{ref4}}
\paragraph{Quadratic deviation from $f$}
Let $X$, $Y$ and $f(x)$ be as above, then the quadratic deviation $\epsilon_{XY}^2$
from $f(x)$ is given by :
\begin{equation}
\epsilon_{XY}^2 = \frac{1}{N - 2} \sum\limits_{k = 1}^N (Y_k - f(X_k))
\end{equation}
\paragraph{Error on $m_{XY}$ and $b_{XY}$}
\begin{align}
\Delta m_{XY} &= 2 \frac{\epsilon_{XY}}{\sigma_X} \\
\Delta b_{XY} &= 2 \epsilon_{XY} \sqrt{\frac{1}{N} + \frac{s_X^2}{\sigma_X^2}} \\
s_X^2 &= \frac{1}{N} \sum\limits_{k = 1}^N X_k^2
\end{align}
\paragraph{Application to $V_H(B)$ and $V_H(I)$}
Same as before: replace $Y$ by $V_H$ and $X$ with $B$ or $I$ and then
compute the error on the slope.
\subsubsection{Error on $R_H$}
\paragraph{$V_H$ depends on $B$}
\begin{equation}
\Delta R_H = a \abs*{\frac{\Delta m_{BV_H}}{I}} + a \abs*{\frac{m_{BV_H}}{I^2} \Delta I}
\end{equation}
\paragraph{$V_H$ depends on $I$}
\begin{equation}
\Delta R_H = a \abs*{\frac{\Delta m_{IV_H}}{I}} + a \abs*{\frac{m_{BV_H}}{B^2} \Delta B}
\end{equation}
\subsubsection{Error on $B$}
\begin{equation}
\Delta B = \abs*{m_{IB}} \cdot \Delta I + \abs*{I} \cdot \Delta m_{IB} + \abs*{b_{IB}}
\end{equation}
\section{Literature References}
% Bibliographie
\begin{thebibliography}{99}
\bibitem{ref1}
HARALD, Brune, Cours de Physique III, Faculté des Sciences de Base, Section de Physique, Ecole Polytechnique Fédérale de Lausanne.
\bibitem{ref2}
Notices des TP de physique H6 : Effet Hall.
\bibitem{ref3}
ANCAROLA, Raffaele, CINCOTTI, Armando, Experiment N G1 : Hysteresis Cycle. Lausanne 3rd of December 2018, Ecole Polytechnique Fédérale de Lausanne.
\bibitem{ref4}
Uncertainty Slope Intercept of Least Squares - Faith A. Morrison - \url{http://pages.mtu.edu/~fmorriso/cm3215/UncertaintySlopeInterceptOfLeastSquaresFit.pdf}
\end{thebibliography}
\end{document}
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