我正在写实验报告，在最后一页，我的浮点数格式不正确。我留下了巨大的空白，我无法摆脱它们，而且它占用了超过页数。我还收到错误：“浮点数卡住了（无法放置）；在输入行 158 上尝试类选项 [floatfix]。”但我不知道哪个浮点数卡住了，因为它们都出现在文档中。任何帮助都非常感谢，我会附上代码。谢谢
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\begin{document}
\title{Hall Effect in Semiconductors:}
\author{10398927
Third Year Physics Student
Lab Report}
\date{\today}% It is always \today, today,
% but any date may be explicitly specified
\begin{abstract}
The Hall effect was investigated in an Indium Antimonide (InSb) sample, where the mobility, Hall coefficient and charge concentration were found as a function of temperature. The sample was found to be n-type doped, and the energy band gap was found to be (0.24\textpm0.01)eV, which was greater than the true value.
\end{abstract}
\maketitle
%\tableofcontents
\section{\label{sec:level1}Introduction:\protect}
This report discusses the effect of temperature on various properties of a doped semiconductor, including conductivity, charge carrier concentration and resistance. Semiconductors have a unique temperature dependence, compared to conductors and insulators, whereby depending on its temperature regime, it will exhibit considerably different behaviour.
The Hall effect occurs in a conductive sample when a current is applied perpendicular to an external magnetic field. Charge carriers (positive for holes, negative for electrons) accumulate on one side of the sample, and produce an emf, also known as the Hall voltage. Figure 1 highlights how the different forces and charge carrier movements produce a Hall Voltage across a sample.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=2.0]{hall effect diagram.png}
\caption{An annotated diagram showing the forces, fields and charges with their corresponding directions.}
\label{fig:diagram}
\end{figure}
The Hall voltage is inversely proportional to the charge carrier density in the sample, and therefore is very useful in investigating how charge carrier density varies with temperature, and how this relates to the valence-conduction band gap and the dopant band gap. Additionally, the sign of the Hall coefficient (found from the Hall voltage), in both the low and high temperature regimes determines whether the sample exhibits p or n type doping.
\section{\label{sec:level2}Theory:\protect}
Electrons in a current move from one side to another with a velocity. When an external magnetic field, is applied perpendicular to this current, the charge carriers experience a Lorentz force acting on them \begin{equation}F \propto I\textsubscript{s}B_z \end{equation}
where I\textsubscript{s} is the sample current, and B\textsubscript{z} is the external magnetic field.\\
The system reaches equilibrium when the magnetic force is equal to the electric force, with the equation for this being
\begin{equation}
V_{H}d = \frac{I_{s}B_{z}}{ne} = \frac{-I_{s}B_{z}}{pe}
\end{equation}
where n is the concentration of electrons (m\textsuperscript{-3}), d is the thickness of the sample (m), e is the absolute value of electric charge (C) and p is the concentration of holes (m\textsuperscript{-3}).
The Hall coefficient is defined as
\begin{equation}
R_{H} = \frac{1}{ne}
\end{equation}
and can be found experimentally. From this, the electron concentration can be calculated, and from its sign, the type of dopant atom can be determined.
Semiconductors have a vast amount of uses, due to their unique transport properties which depend on the temperature of the sample. At 0K, the conductivity of all semiconductors is zero, as it is impossible for any electrons to have been excited. Beyond this temperature, the charge carriers follow the Fermi - Dirac occupation function, which determines the probability of finding a fermion within a given state. By integrating this function with the density of states, we find the electron concentration at a given temperature to be
\begin{equation}
n \propto T^{3/2}exp(\frac{-E_{G}}{2K_{B}T})
\end{equation}
where T is the temperature (K) and E\textsubscript{G} is the band gap.\\
The T\textsuperscript{3/2} term comes from the density of states in the conduction band, while the exponential term is due to the occupation function for a semiconductor.\\
The charge carrier concentration at low temperatures for a semiconductor is very low, but dopant atoms can be introduced to increase the conductivity in the low temperature regime. Doping involves introducing another type of atom into a material, which has a much smaller energy band gap, and so at low temperatures the dopant can donate an electron or electron hole to contribute to the charge carrier flow. A p-type dopant contains an empty energy level just above that of the valence band, meaning electrons may be excited, leaving positive electron holes in the valence band. Conversely, an n-type dopant contains a fully populated energy level just below that of the conduction band, so electrons may be excited to the conduction band.\\
Thermocouples are an effective way of measuring temperature in an experimental setup. The temperature difference between the two probes produces an emf, which is proportional to the difference in temperature between the two probes. If one probe is kept at a known, constant temperature, and the thermocouples are properly calibrated, the temperature at the second probe can be accurately measured.
\section{\label{sec:level3}Experimental Methods:\protect}
Before any measurements are taken, it is necessary to set the zero value for the Hall voltage. This is accomplished by setting the magnetic field to zero, and varying the Hall voltage slider until a minimum value is read on the multimeter. Subsequently, the reference temperature containers (as shown in figure 2), should be filled with ice and a small amount of water, so that the reference temperature of 273.15 K is kept constant, to ensure the thermocouples are measuring the correct temperature.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=0.25]{Experimental setup.png}
\caption{A schematic diagram showing the main apparatus involved in the experimental setup.}
\label{fig:experimental setup}
\end{figure}
Starting at room temperature, the current should be set to a large value on the dial, around 80mA, and the magnetic field set to 0 T. The multiplexer should be switched to the number corresponding to current, and the current should be recorded from the multimeter. This procedure should be repeated to measure the voltage. To accurately measure the Hall voltage however, an average value is required, as the values can fluctuate massively on the multimeter. Values for the min, max, average and count number are taken, and are used to calculate the uncertainty by substituting into
\begin{equation}
uncertainty = \frac{Min.\: value - \:Max.value}{Count\:number}
\end{equation}
Once the sample current, sample voltage and Hall voltage have been recorded, the current should be varied, and this process is repeated until the current reaches a large negative value, around -80mA.The multimeter used had a high precision, recording to 4 siginificant figures, producing standard uncertainties of \textpm0.2mA for sample current, and \textpm0.03mV for sample voltage at high temperatures. Sample voltage uncertainty increased to 3mV at low temps, as the readings became much greater.
Following this, the same measurements should be recorded for increasing values of the magnetic field. This is important, as V\textsubscript{H} will be plotted against I\textsubscript{s} x B, so with more data points for different magnetic fields, a more accurate average can be taken for the Hall coefficient and electron concentration.
Additionally, the heater and cryostat should be utilised to change the sample temperature, and the experiment should be repeated at many different temperatures. Low temperature readings of roughly 125K[reference] and lower are necessary to investigate the dopant contribution to the current, while the high temperature readings (around 300K-400K) will be plotted and compared to equation (4), to investigate how the data matches the theoretical behaviour. Furthermore, equation (4) can be re-arranged to give
\begin{equation}
ln(\frac{AT^3/2}{n}) = \frac{E_{G}}{2K_{B}T}
\end{equation}
where A is just a constant, and this can be plotted to determine the valence band-conduction band energy gap.
\\
By comparing the sign of the Hall coefficient in the low and high temperature regimes, the type of doping in the sample can be found. As mentioned in the theory section, an n-type dopant contributes electrons, so if the sign of the Hall co-efficient is consistent in both regimes, the sample is n-type. Alternatively with a p-type dopant, the sign of the Hall coefficient would be opposite in the low and high temperature regime.
\section{\label{sec:level4}Results and Data Analysis:\protect}
Firstly, values for V\textsubscript{s} were plotted against I\textsubscript{s} to investigate the resistance, and as expected, this produced a linear relationship with the gradient being equal to the resistance, R. The plots for 350.5K and 81K are shown in figure 3 - note that error bars have been plotted but are too small to be visible on the plot.
\begin{figure}[!htbp]
\centering
\includegraphics[scale=0.25]{voltage against current plot.png}
\caption{In a plot of sample voltage against sample current, resistance was calculated as (0.156\textpm0.001)\textohm\:for 350.5K, and (11.80\textpm0.02)\textohm\: for 81K.}
\label{fig:resistance}
\end{figure}
For 350.5K this produces an almost perfect linear fit, but for 81K it is clear from the plot that the resistance is increasing as the magnitude of the current increases. This is due to the high current producing a heating effect in the semiconductor, as the rate of collisions increases due to the charge carriers moving at a faster speed.
This effect is much more noticeable at low temperatures, as the thermal energy is lower, so the heat transfer due to vibrations has a greater effect. Additionally, the resistance for the 81K case was calculated by plotting just the middle four points, however, when fitted, the reduced chi squared value was still very large (around 1000), implying the relationship is not linear even at small current values. To rectify this, more data points should have been taken in the low current range in an attempt to find a linear fit. In general though, these two plots exhibit the behaviour we expect from the theory, with the resistance decreasing at higher temperature as the number of charge carriers increases.\\
The Hall coefficient for a given temperature and magnetic field was calculated by finding the gradient of $V_{H}d$ against $I_{s}B_{z}$. The mean was then taken of the Hall coefficient values for a given temperature. The sign of the Hall coefficient at both high and low temperatures was unvaried, so the sample is therefore n-type doped.
From the values for the Hall coefficient and equation (3), the electron concentration can also be calculated as a function of temperature and these are plotted as a log scale in figure 4.
\begin{figure}[!hbp]
\centering
\includegraphics[scale=0.5]{electron conc temp.png}
\caption{A log plot showing how electron concentration varies with temperature.}
\label{fig:logplot}
\end{figure}
The electron concentration for low temperatures is close to constant and doesn't exhibit exponential behaviour, so it is likely this is the saturation region, where all the dopant electrons have been excited to the conduction band. At higher temperatures, the electron concentration increases exponentially, following the proportionality shown in equation (4). Equation (6) can be utilised to obtain a fit for the valence band-conduction band energy gap, where the value was calculated as (0.24\textpm0.01)eV
\begin{figure}[!hbp]
\centering
\includegraphics[scale=0.5]{electronic band gap.png}
\caption{A plot using equation (6) to calculate the electronic band gap.}
\label{fig:band gap}
\end{figure}
In literature, the band gap for InSb is quoted as 0.17eV [reference], which means the experimentally calculated value is roughly 40\% greater than the quoted value.\\
The calculated value for the band gap depends on the electron concentration values calculated prior, so the discrepancy between the experimental value for the band gap compared to the real value may be attributed to this term. As seen in figure 5, the data is a good fit, so the error is likely systematic. One possible cause may be a non-linearity in the best fit plots when finding the Hall coefficient, so the gradient calculated may actually be inaccurate.
\nocite{*}
\bibliography{apssamp.bib}
\end{document}
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