% prelimary report on ARPES results on CdHgTe, two column
% by Matthias Kreier, see more at 
% http://people.physik.hu-berlin.de/~kreier/

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\title{Determination of effective hole mass in 
Hg\textsubscript{1-X}Cd\textsubscript{X}Te  by ARPES}
\author{Matthias Kreier\\
{\small Institut für Physik, Humboldt Universität zu Berlin}\\
{\small \texttt{matthias.kreier@physik.hu-berlin.de}}}
\date{} % December 1st, 2007

\begin{document}
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\twocolumn[
  \begin{@twocolumnfalse}
    \maketitle
    \begin{abstract}
	  Several samples of Hg\textsubscript{1-X}Cd\textsubscript{X}Te with X values 
	  between 0.07 and 0.4 were investigated by ARPES.
	  Because this method is very surface sensitive, we will also consider surface
	  problems and requirements.
	  Using the methods of Laue, LEED, SEM and AFM we checked the surfaces that
	  were preparated by cleaving. Based on a proven good surface quality we 
	  represent our photo emission spectra in normal emission, taken at BESSY at 
	  photon energies between 44 and 125 eV. Thereafter the fitted spectra were assigned to 
	  the representing points inside the Brillouin zone. Concludingly we can directly 
	  fit the dispersion and derive the effective hole mass of different bands as well 
	  as the energy separation by the spin-orbit interaction. These results are compared
	  to already well known effective mass values
    \end{abstract}
	\bigskip
  \end{@twocolumnfalse}
]

\renewcommand{\figurename}{Fig.}
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\section{Introduction}

The method of \underline{A}ngle \underline{R}esolved \underline{P}hoto \underline{E}lectron \underline{S}pectroscopy (APRES) is a unique method to observe the bandstructure of a sample. With this technique it is possible to compare directly the theoretically calculated bandstructure with the real states of electrons inside a material. Other methods having an indirect approach depend on the used model and the right interpretation of the results.

The following measurements were taken with the High Resolution Photo Electron Spectrometer (HIRES-PES) of the EES group of Prof. R. Manzke at the Humboldt University to Berlin. It uses a Omicron AR65 electron analyser with a achievable resolution of 10 meV. The different photon energies were supplied by the BUS-beamline at the BESSY synchrotron.

\begin{figure}[htbp]
  \centering
  \begin{minipage}{8cm}
    \includegraphics[width=\textwidth]{pic/fig3}
    \caption{Comparison of the bandgap vs. lattice constant variation with the lattice constant for a varity of III-V compounds.}
    \label{norton3}
  \end{minipage}
\end{figure}

The samples of Hg\textsubscript{1-X}Cd\textsubscript{X}Te were gradually provided by the Faculty of Low Temperature Physics and Supraconductors at the Moscow State University. Special thanks belongs to Dr. \ru{Никифоров Владимир Николаевич} %V. Nikiforov.
and Prof. \ru{В. Г. Средин}. %V. G. Sredin

\section{Crystal Quality}

In our investigation we analysed the effect of different X values in the composition of Hg\textsubscript{1-X}Cd\textsubscript{X}Te to the bandstructure. CdTe is a semiconductor with 1.6 eV bandgap. HgTe is reffered to be a semimetal. More precise it is described as a zero-gap semiconductor with an inverted bandstructure. HgTe has a negative band gap of -0.34 eV. Both materials have a direct bandgap and very similar lattice constants (HgTe 6.445 \AA , CdTe 6.488 \AA ). They can be mixed into alloys of any ratio between Cd\textsubscript{X} and Hg\textsubscript{1-X}. By varying the X value of the composition the bandgap can be adjusted to a deserved value (see figure \ref{norton3}, after \cite{norton02}). Regarding the lattice match the alloy Cd\textsubscript{0.96}Zn\textsubscript{0.04}Te is a good substrate for MWIR and LWIR applications.

Supplied by the Moscow institute we got a variety of alloys with different X values from 0.07 to 0.4. An overview of these six investigated samples is given in table \ref{samples1}. The material HgTe has already been investigated by N. Orlowski in our workgroup \cite{orlowski00}. The respective X value would be zero.

\begin{table}[htbp]
 \centering
 \begin{tabular}{|c|c|c|c|c|c|c|}
 \hline
 Sample & value of X & weight & orientation \\\hline
 \bf I  & 0.07   & 186.3 mg & [111] \\\hline
 \bf II & 0.4    & 130.3 mg &       \\\hline
 \bf III& 0.2    & 158.5 mg & [110] \\\hline
 \bf IV & 0.183  & 129.2 mg & [110] \\\hline
 \bf V  & 0.1955 & 248.5 mg & [110] \\\hline
 \bf VI & 0.105  & 108.9 mg & [110] \\\hline
 \end{tabular}
 \caption{Data of investigated samples}
 \label{samples1}
\end{table}

For angle resolved measurements the orientation of the samples plays a vital role. We first checked the orientation of all 6 samples by the Laue reflection method. The measurements has been done in Moscow by \ru{Телегина Инна Васильевна} at the \ru{кафедре физики твёрдого тела или кафедре физики конденсированных сред}. All pictures were taken at 32 kV and 30 mA with an exposure time of 10 minutes. The film was located 40 mm from the sample. One example can be seen in figure \ref{m6}.

The Laue method is also sensitive to polychrystallic parts. But on the photograph no disturbing spots of different crystal orientation can be seen. We observe clear and bright spots, which proves a good single crystal quality.

\begin{figure}[htbp]
  \centering
  \begin{minipage}{8cm}
    \includegraphics[width=\textwidth]{pic/laue/m6}
    \caption{Laue reflection picture of [110] surface of sample II.}
    \label{m6}
  \end{minipage}
\end{figure}

The distribution of the Laue reflexes belongs to a [110] surface that is tilted by some 2 degrees. The [110] is the natural cleavage direction of ZnS crystals and the surface we investigated. It has no dangeling bonds but is electrically neutral.

Next we checked the surface using the AFM (\underline{A}tomic \underline{F}orce \underline{M}icroscope) of the Moscow Biomedical Institute at Kaschirskoje. In the following picture the remaining fragments of the electro erosion saw can be seen. On the left we see a channel of 1$\mu$m width and 50nm depth. If we zoom in further we see many more parallel structures that remained from the fragmentation of the incidend big bulk crystal.

Spicer \textit{et. al.} \cite{spicer83} reportet a movement of dislocation states from the bulk to the surface at roomtemperature due to the weak Hg bond. In the AFM picture they can be seen inside the channel. The parallel periodic structure are disturbed by some bumps of 10nm height and 30nm diameter.

\begin{figure}[htbp]
  \centering
  \begin{minipage}{8cm}
    \includegraphics[width=\textwidth]{pic/afm/5-2}
    \caption{This surface has been preparated mechanically by a electropyrolytic 
	saw for dividing bigger monocrystal blocks. Thereafter the surface was cleaned
	chemically. One can see disslocations that moved to the surface.}
    \label{afm5-2}
  \end{minipage}
\end{figure}

The small valley in diagonal direction seems to be a remaining scratch from the sample preparation with tweezers. This is another sign for the fragile surface of HgCdTe and the attention it requires to work with this alloy.

\section{Surface preparation}

The ARPES method is only sensitive to the first few \AA  of the sample, but the results should represent the bulk properties. The main reason is the short mean free path of excitated electrons inside the bulk material (see figure \ref{3-mfw}) at UPS energies. Investigating a material using this method several surface effects have to be minimized or eliminated. Therefore a clean surface plays a vital role. All measurement are taken in UHV (\underline{U}ltra \underline{H}igh \underline{V}acuum, $P < 10^{-9} mbar$) conditions. Using a simple thermodynamic approximation one can show that in this case there are several hours until adsorption or covering processes change the surface.

\begin{figure}[htbp]
  \centering
  \begin{minipage}{8cm}
    \includegraphics[width=\textwidth]{pic/3-mfw}
    \caption{Mean free path $\lambda_m$ of electrons in matter, depending on kinetic energy.}
    \label{3-mfw}
  \end{minipage}
\end{figure}

A crystal transfered from normal air pressure conditions into vacuum has to be prepared before measurement. One common method for metals is the combination of sputtering and annealing. Yet this procedure does'nt apply for HgCdTe as the surface crystal structure is disturbed in a extrem way \cite{spicer83}. The roughtness reaches several Nanometer. The crystal structure changes and differes some hundered Nanometer from the bulk crystal structure. And even the composition changes in a dimension that can't be neglected. One reason is the high partial pressure of Hg. 

Therefore we used a cleavage mechanism for surface preparation, as did other groups (\cite{silberman82}, \cite{shih87} and \cite{shih87-2}) prior to us. For the investigation of HgTe N. Orlowski constructed a special cleavage chamber. The Moscow samples are that small and thin, that the cleavage can be done directly inside the sample holder, using a shop glued to the top of the sample. Our group uses this method to cleave van-der-Waals crystals. For these small crystals with covalent bondings this method also applies.

Our first cleavage we performed at room temperature. The results we checked thereafter with a scanning electron microscope. Some steplike structure in the $\mu$m range can be observed. Another surface sensitive method is LEED, but we were unsuccessfull in getting a picture of the uncooled cleaved sample.

\begin{figure}[htbp]
  \centering
  \begin{minipage}{8cm}
    \includegraphics[width=\textwidth]{pic/leed156}
    \caption{Successfull LEED picture at 156 eV after cooled cleavage with liquid nitrogen.}
    \label{leed156}
  \end{minipage}
\end{figure}

In June 2007 finally we did a cooled cleavage inside the vacuum. The sample was mounted into a sample holder on a manipulator, that can be cooled by liquid nitrogen and helium. We used $\rm N_2^{fl}$ and reached a temperature of 100K at the sample. After the cleavage we were able to see the first LEED picture (see figure \ref{leed156}) of our sample.

From this picture one can derive the lattice constants, using the incident electron energy and geometry of the LEED analyser. With picture \ref{leed156} we derived the lattice constants a=3.5 \AA {}  and b=2.4 \AA  {} , which is consistant with the [110] surface of HgCdTe.

\begin{figure*}[hbtp]
  \centering
  \begin{minipage}{17cm}
    \includegraphics[width=\textwidth]{pic/overview1}
    \caption{Overview Photo Electron Spectra of HgCdTe, taken at the BUS beamline at BESSY}
    \label{overview1}
  \end{minipage}
\end{figure*}

\section{ARPES spectra of HgCdTe}

At photon energies usually avaliable on a Synchrotron as BESSY one gets a photo electron spectra as shown in picture \ref{overview1}. The core levels of Te (4d) and Hg (5d) show now dispersion and are not considered in the following. In the energy band from -6.5 to -4 eV some Hg6s states can be observed, as can be seen in figure \ref{disp4}. These level only reveal some intensity modulation due to matrix element effects rather than any dispersion. Therefore we will only consider the energy range of -4 to 0 eV, where zero is defined by the fermi energy.

All the following spectra were taken in normal emission. The recorded data where very noisy, so they were smoothed several times \cite{shirley72} using the formula 

\begin{eqnarray}\label{smooth}
  I_S(N) = \frac{1}{4}[I(N-1) + 2I(N) + I(N+1)] ~~.
\end{eqnarray}

Here we named intensity after the smoothing operation $I_S(N)$. The intensities were normalised.

Our first measurement at the synchrotron was with a sample of x=0.4 value. The result of an uncooled cleavage is shown in figure \ref{disp4}. The features at 43 eV photon energy is a reflection of Te core levels excitated by the second order of the monochromator at the BESSY beamline. In this spectra we also recorded the data of the Hg6s levels (4 to 6 eV). They don't show dispersion.

\begin{figure}[htbp]
  \centering
  \begin{minipage}[b]{80mm}
    \includegraphics[width=\textwidth]{pic/disp4}
    \caption{Dispersion of sample with x=0,4}
    \label{disp4}
  \end{minipage} 
\end{figure}

Although we measured up to 120 eV, no valence band maximum appeared at this energy. The only usable dispersion is seen between 80 and 50 eV and appears to be at 65 eV. Regarding energy and momentum conservation this point belongs to the $X$ point.

In august 2007 at a different beamtime we got a successfull cooled cleavage and observed a dispersion as shown in figure \ref{disp16} with a sample of x=0.16 value. Based on the experience gathered at the first BESSY beamtime we only measured until 4 eV binding energy. This waterfall spectra shows that the valence band dispers only in this energy range and reaches its maxima (e.g. $\Gamma$ and VBM) at energies of 120 eV and below 44 eV. Between these energies there is another maxima at approximately 76 eV, it can be seen from 80  to 68 eV. It probably belongs to the $X$ point.

\begin{figure}[htbp]
  \centering
  \begin{minipage}[b]{80mm}
    \includegraphics[width=\textwidth]{pic/disp16}
	\caption{Dispersion of sample with x=0,163}
	\label{disp16}
  \end{minipage}
\end{figure}

Our final measurement we did with a sample of x=0.07. In this case we exceeded the photon energy beyond 120 eV to 125 eV, although this was already limited by the beamline and the used grating. But from symmetry conditions it is clear to be seen that the $\Gamma$ point is located at 120 eV.

\begin{figure}[htbp]
  \centering
  \begin{minipage}[b]{80mm}
    \includegraphics[width=\textwidth]{pic/disp06}
	\caption{Dispersion of sample with x=0,07}
	\label{disp06}
  \end{minipage} 
\end{figure}

In this spectra, though, the $X$ ooint at 65 eV is clear to be seen and shows dispersion from 55 to 75 eV. We took this part also into consideration.


\section{Peak fit of the spectra}

The next step is to fit peaks into the spectra, according to electronic bands inside the material. We used Gaussian peaks that are folded with the temperature dependant Fermi function. One might expect a Voigt profile instead of a Gaussian fit. The reason is simply a practical one. Using Voigt it was not possible to recieve sufficiant results. 

The dispersion near the $\Gamma$ point at 120 eV has been fitted by several gaussian peaks for each spectra simultanously. The results for x=0.16 are shown in the table \ref{peakfit1}. Here $h\nu$ is the excitation photon energy at the BESSY Synchrotron Storage Ring, provided by the BUS-Beamline Monochromator.

\begin{table}[htbp]
 \centering
  \begin{tabular}{|c|c|c|c|c|c|}
  \hline
  $h\nu$ & $V_1$ & $V_2$ & $V_3$ & $V_4$ & $|\vec{K}|$\\
   eV    &  eV   &  eV   &   eV  &  eV   & \AA$^{-1}$\\
   \hline\hline
   120   & 0.413 & 0.947 & 1.810 & 3.226 & 0      \\\hline
   115   & 0.476 & 1.046 & 1.554 & 2.603 & 0.0194 \\\hline
   110   & 0.568 & 1.141 & 1.302 & 3.167 & 0.078  \\\hline
   105   & 0.717 & 1.115 & 1.148 & 3.097 & 0.175  \\\hline
   100   & 1.531 & 1.897 & 2.424 & 3.078 & 0.311  \\\hline
    95   & 1.740 & 1.984 & 2.366 & 2.757 & 0.486  \\\hline
  \end{tabular}
  \caption{Fitting results of spectra in figure \ref{disp16}, x=0.16}
  \label{peakfit1}
\end{table}

We assign the $\Gamma_8$ band to the volume band V$_1$ and the $\Gamma_7$ band to the volume band V$_2$. The next fitting results are taken for a x=0.07 sample. Results are shown in table \ref{peakfit2}

\begin{table}[htbp]
 \centering
  \begin{tabular}{|c|c|c|c|c|c|c|c|}
  \hline
  $h\nu$ & $V_1$ & $V_2$ & $V_3$ & $V_4$ & $V_5$ & $|\vec{K}|$ \\
   eV    &  eV   &  eV   &   eV  &  eV   &   eV  & \AA$^{-1}$  \\
   \hline\hline
   120   & 1.053 & 1.605 & 2.148 & 2.447 & 2.910 &  0.085 \\\hline
   115   & 0.998 & 1.329 & 1.864 & 2.151 & 3.162 & -0.009  \\\hline
   110   & 1.121 & 1.649 &       & 2.248 & 3.058 & -0.094 \\\hline
   105   & 1.513 &       & 1.712 & 2.291 & 3.029 & -0.187 \\
   \hline\hline
   75    & 1,224 & 1,674 & 1,949 & 2,169 & 2,375 & 0.312 \\\hline
   70    & 0,883 & 1,056 & 1,708 & 2,374 & 2,323 & 0.161 \\\hline
   65    & 1,111 & --    & 1,005 & 1,534 & 1,967 & 0.005 \\\hline
   60    & 1,09  & 1,788 & 2,223 & 2,524 & 2,995 & -0.157 \\\hline
   55    & 1,389 & 1,591 & 1,889 & 2,356 & 3,121 & -0.326 \\\hline
  \end{tabular}
  \caption{Fitting results of spectra in figure \ref{disp06}, x=0.07}
  \label{peakfit2}
\end{table}

Finally we fited the $X$-point features of the x=0.4 spectra in figure \ref{disp4}. Results are shown in table \ref{peakfit3}.

\begin{table}[htbp]
 \centering
  \begin{tabular}{|c|c|c|c|c|c|}
  \hline
  $h\nu$ & $V_1$ & $V_2$ & $V_3$ & $V_4$ & $|\vec{K}|$\\
   eV    &  eV   &  eV   &   eV  &  eV   & \AA$^{-1}$\\\hline
   80    & 1.386 & 2.017 & 2.218 & 2.149 & 0.457 \\\hline
   70    & 1.282 & 1.703 & 2.014 & 2.710 & 0.161 \\\hline
   60    & 1.413 & 1.863 & 2.753 & 2.908 & -0.157 \\\hline
   55    & 1.335 & 1.839 & 2.209 & 2.726 & -0.326 \\\hline
   50    & 0.908 & 1.529 & 2.205 & 2.703 & -0.502 \\\hline
  \end{tabular}
  \caption{Fitting results of spectra in figure \ref{disp4}, x=0.4}
  \label{peakfit3}
\end{table}

\section{Transfer to K-space}

In the previous tables we already assigned a fitting K-value to each photon energy $h\nu$. In first approach we tried to fit the resulting curve with a quadratic expression.

To fullfill momentum conservation as well as energy conservation usually the fittet points are taken into a structure plot to assign each point to a fitting free electron value. In this fitting we add certain primitive lattice vectors to the final state. We already know from former measurements, that most values around 115 eV belong to $\vec{G}_1 = \frac{2\pi}{a}(4,4,0)$ and therefore respecting the $\Gamma$ point. The dispersion and local valence band maximum around 65 eV belongs to the $X$ point and a lattice vector $\vec{G}_2 = \frac{2\pi}{a}(3,3,1)$. For this prelimary results this is sufficiant. Further details about the structure plot can be found in \cite{orlowski00}, page 64 to 68.

\subsection{Parabolic fit}

Commonly in books the law of dispersion for electrons in bulk materials is given by the expression 

\begin{eqnarray}\label{dispersion1}
  E = E_0 + \frac{\vec{p}^2}{2m_{eff}} = E_0 + \frac{\hbar^2}{2m_e m^*}K^2~~.
\end{eqnarray}

If following we assign the y axsis to the energy in $V$, measure $E_0$ also in $V$ and the x values in \AA$^{-1}$. Formula \ref{dispersion1} transforms thereby into

\begin{eqnarray}\label{dispersion2}
  y = E_0 + \frac{\hbar^2}{2m_e e}\frac{1}{m^*}x^2 
       = E_0 + \frac{3.806}{m^*}x^2~~.
\end{eqnarray}

We can now use $m^*$ as fit parameter to obtain the results of the next section.

\subsection{Nonparabolic fit}

According to the Kane model for narrow gap semiconductors the behavior of electrons on top of the valence band is more hyperbolic that quadratic. We fittet our K-values therefore again with the hyperbolic formulae

\begin{eqnarray}\label{hyperbolic}
  y = E_0 + b\sqrt{\frac{x^2}{m^*} + 1}~~.
\end{eqnarray}

The resulting masses differ slightly from the quadradic approximation, as the following section shows. We have to remind, that the Kane approximation is only valid for energy gaps below 0.35 eV.

\section{Results of effective mass}

\subsection{Dissertation Orlowski: X=0 (HgTe)}

First we analyse the reciprocal lattice values N. Orlowski measured in his dissertation \cite{orlowski00} on HgTe. In our approach this would represent exactly the x=0 point.

\begin{table}[htbp]
  \centering
  \begin{tabular}{|c|c|c|}
  \hline
             &$M_{eff}$& $\Delta$E  \\\hline
  $\Gamma_8$ &  -1,15  &    0   eV  \\\hline
  $\Gamma_6$ &  -0,49  &  -0,33 eV  \\\hline
  $\Gamma_7$ &  -0,24  &  -0,92 eV  \\\hline
  \end{tabular}
  \caption{Effective Mass of HgTe.}
  \label{Meff0}
\end{table}

This data have been directly been digitalisised from figure \ref{hgte1} and been fitted quadratic.

\begin{figure}[htbp]
  \centering
  \begin{minipage}{8cm}
    \includegraphics[width=\textwidth]{pic/hgte1}
    \caption{Bandstructure of HgTe along $\Gamma\Sigma$K direction \cite{orlowski00}.}
    \label{hgte1}
  \end{minipage}
\end{figure}

According to the size of the Brillouin zone the point K here represents a momentum of 1.031 \AA$^{-1}$ and defines hereby the x axis.


\subsection[HgCdTe: X=0,07 (BESSY, 26.8.07)]{Hg\textsubscript{1-X}Cd\textsubscript{X}Te: X=0,07 (BESSY, 26.8.07)}

In the next approcimation we investigated the data of table \ref{peakfit2}. Here we have two local maxima of the valence band: one at 115 eV and a second at 65 eV. The first one belongs to the $\Gamma$ point (Umklappvektor [4,4,0]), the latter one to the $X$ point(Umklappvektor [3,3,1]). The results are shown in table \ref{Meff07}

\begin{table}[htbp]
  \centering
  \begin{tabular}{|c|c|c|}
  \hline
             &$M_{eff}$& $\Delta$E  \\
			 \hline\hline
  heavy holes ($\Gamma_8$) & 0.251 & 0.977 eV  \\\hline
  light holes ($\Gamma_7$) & 0.097 & 1.309 eV  \\
  \hline\hline
  heavy holes ($X_8$)& 1.403 & 1.118 eV  \\\hline
  light holes ($X_7$)& 1.267 & 1.671 eV  \\\hline
  \end{tabular}
  \caption{Effective Mass of Hg\textsubscript{0.93}Cd\textsubscript{0.07}Te.}
  \label{Meff07}
\end{table}

The fitting has been done with xmgrace. The measure points and the fitting is shown in figure \ref{dis07}.

\begin{figure}[htbp]
  \centering
  \begin{minipage}{8cm}
    \includegraphics[width=\textwidth]{pic/dis07}
    \caption{Dispersion fit in the K space according to the peak positions fitted in table \ref{peakfit2}.}
    \label{dis07}
  \end{minipage}
\end{figure}

\subsection[HgCdTe: X=0,16 (BESSY, 25.8.07)]{Hg\textsubscript{1-X}Cd\textsubscript{X}Te: X=0,16 (BESSY, 25.8.07)}

In this zero-gap case we were unable to distinguish an additional $\Gamma_6$ band, so we could only determine two dispersive bands and two respective effective masses. The fit was done to the data shown in table \ref{peakfit1}.


\begin{table}[htbp]
  \centering
  \begin{tabular}{|c|c|c|}
  \hline
             &$M_{eff}$& $\Delta$E  \\\hline
  $\Gamma_8$ &   0.55  &   0,48 eV  \\\hline
  $\Gamma_7$ &  0.857  &   1.08 eV  \\\hline
  \end{tabular}
  \caption{Effective Mass for X=0.163}
  \label{Meff2}
\end{table}

Surprisingly the mass of the split-off band $\Gamma_7$ is heavier than the heavy hole band on the surface of the valence band. The difference in $\Delta$E results in a different investigation of the measured data. Orlowskli \textit{et. al.} used the E$_{VBM}$ as reference and zero where we measured the fermi energy and used this result as zero.


\begin{table*}[htbp]
  \centering
  \begin{tabular}{|c|c|c|c|c|c|c|}
  \hline
   X   & Te4d$_{5/2}$ & Te4d$_{3/2}$ & Cd4d$_{5/2}$ & Cd4d$_{3/2}$ & Hg5d$_{5/2}$ & Hg5d$_{3/2}$ \\\hline
  0.07 & 39.873 & 41.357 &        &        & 7.977 & 9.807 \\\hline
  0.17 & 39.938 & 41.416 & 10.529 & 11.239 & 8.025 & 9.827 \\\hline
  0.40 & 39.968 & 41.429 & 10.689 & 11.398 & 8.191 & 9.934 \\\hline
  \end{tabular}
  \caption{Core level positions for Tellurium, Cadmium and Mercury, depending on the composition.}
  \label{corelevel1}
\end{table*}






\subsection[HgCdTe: X=0,4 (BESSY, 3.2.07)]{Hg\textsubscript{1-X}Cd\textsubscript{X}Te: X=0,4 (BESSY, 3.2.07)}

As already shown in picture \ref{disp4}, the uncooled cleavage of the HgCdTe single crystals lead to undeserved surfaces. In this case we measured not only normal emission but also a superposition of many angles of the surface. Therefore we got to a certain extend energy dependant DoS (Density of States) picture. No valence band maximum can be seen at the expected $\Gamma$ point. But the $X$ point at 65 eV shows dispersion and we did a fit for comparision with the x=0.07 results.

\begin{table}[htbp]
  \centering
  \begin{tabular}{|c|c|c|}
  \hline
       &$M_{eff}$& $\Delta$E  \\\hline
  $X_8$& 3.994   & 1.242 eV  \\\hline
  $X_7$& 2.259   & 1.717 eV  \\\hline
  \end{tabular}
  \caption{Effective Mass of Hg\textsubscript{0.6}Cd\textsubscript{0.4}Te.}
  \label{Meff4}
\end{table}

\subsection{Dependance of $m^*$ to X value?}

Now I can compare the different effective masses $m^*_{hh}$ and $m^*_{lh}$ as well as the energy difference to the split-off band (due to spin-orbit interaction) in dependance of the composition value X. The result for the $\Gamma$ point is shown in table \ref{final1}.

\begin{table}[htbp]
  \centering
  \begin{tabular}{|c|c|c|c|}
  \hline
   X   & $\Delta E_{SO}$ & $m^*_{hh}$ & $m^*_{lh}$ \\\hline
  0.00 & 0.92 eV & 1.15  & 0.24  \\\hline
  0.07 & 0.33 eV & 0.251 & 0.097 \\\hline
  0.16 & 0.60 eV & 0.55  & 0.857 \\\hline
  \end{tabular}
  \caption{Effective masses and SO-energy in dependance of X.}
  \label{final1}
\end{table}

Additionally I calculated the respecting values for the $X$ point. The values for X=0 are the same as at the $\Gamma$ point (see picture \ref{hgte1}).

\begin{table}[htbp]
  \centering
  \begin{tabular}{|c|c|c|c|}
  \hline
   X   & $\Delta E_{SO}$ & $m^*_{hh}$ & $m^*_{lh}$ \\\hline
  0.00 & 1.02 eV & 1.15  & 0.24  \\\hline
  0.07 & 0.55 eV & 1.40  & 1.27  \\\hline
  0.40 & 0.48 eV & 3.99  & 2.26 \\\hline
  \end{tabular}
  \caption{Effective masses and SO-energy at the $X$ point.}
  \label{final2}
\end{table}

REMARK: In the advanced laboratory I did the Hall experiment with a sample of Hg\textsubscript{1-X}Cd\textsubscript{X}Te (X=0.203). Our results were a Hall resistivity of $R_H = (772 \pm 8) cm^3/As$ at 77K. For charge carrier concentration (p-type) we got $p = (8.09 \pm 0.08) 10^{15} cm^3$. Deriving the arising of the values with temperature and performing a linear approximation we got to values for [Anstieg]. Comparison of coefficients resultet in 

\begin{eqnarray}\label{hall}
  E_g = 2ka
\end{eqnarray}

and therefore in the following energy gap results:\\
$E_g = (0.0960 \pm 0.0009) eV$ from conductivity\\
$E_g = (0.1143 \pm 0.0001) eV$ from Hall coefficiant.

The mobility of holes was $\mu = (390 \pm 9) \frac{cm^2}{Vs}$. The impulse relaxation time $\tau = m^*\mu/e$ resulted in $\tau = m^*(2.22 \pm 0.05)10^{-13} s$. Finally the mobility ratio was $b = \frac{\mu_n}{\mu_p} = (45 \pm 1)$.

\section{Core Level Shift}

In former UPS studies of HgCdTe Shih \textit{et. al.} reported a core level shift for the Hg5d and Cd4d bands \cite{shih87}. With the photon energy at BESSY avaliable we checked the core levels of Te4d as well. Our results for different compositions are shown in table \ref{corelevel1}.

Interestingly the spin-orbit split energy of Te and Hg changes with the composition, while the Cd value remains almost constant. As can be seen in table \ref{corelevel2} for Te the difference reaches $\Delta_{Te} = 23 meV$ and for Hg $\Delta_{Hg} = 87 meV$.

\begin{table}[htbp]
  \centering
  \begin{tabular}{|c|c|c|c|}
  \hline
   X   & Te4d  & Cd4d  & Hg5d  \\\hline
  0.07 & 1.484 &       & 1.830 \\\hline
  0.17 & 1.478 & 0.710 & 1.802 \\\hline
  0.40 & 1.461 & 0.709 & 1.743 \\\hline
  \end{tabular}
  \caption{Spin orbit split energy, depending on x.}
  \label{corelevel2}
\end{table}

The relative positions shift too. From table \ref{corelevel3} we see a shift of up to $\Delta_{Te-Cd} = 146 meV$ for Cadmium and Tellurium, where the shift between the core levels of Tellurium and Mercury values up to $\Delta_{Te-Cd} = 136 meV$.

\begin{table}[htbp]
  \centering
  \begin{tabular}{|c|c|c|c|c|}
  \hline
   X   & Te-Cd  & Te-Cd  & Te-Hg & Te-Hg  \\\hline
  0.07 &        &        & 31.896 & 31.550 \\\hline
  0.17 & 29.409 & 30.177 & 31.913 & 31.589 \\\hline
  0.40 & 29.279 & 30.031 & 31.777 & 31.495 \\\hline
  \end{tabular}
  \caption{Relatice energy positions of core levels.}
  \label{corelevel3}
\end{table}

In the work of Silberman \textit{et. al.} \cite{silberman82} they measured values for the Hg5d core levels at 7.8 eV and 9.5 eV. The values for Cd4d were 10 eV and 10.7 eV. Depending on the composition Shih \cite{shih87} reported a shift of 0.1 eV for Hg core levels and 0.25 eV for Cd core levels, respectively.

\section{Conclusion}

As we've seen the ARPES method is a powerful tool to determine the electronical bandstructure of a given material. Because of its momentum resolving possibilities we have direct access to the dispesion of the valence bands inside the crystal.

Our results also show, how filigrant this method is to a good prepared surface. Therefore it is nessesary to check the surface quality as well as the single crystal quality using other methods. As for HgCdTe bulk crystals, that are not grown inside the vacuum, a clean UHV surface can only be achieved by cooled cleavage.

The effective mass for the heavy holes should be in the regime of $0.45 < m^* < 0.55$ in a nonparabolic approximation. In some cases we got similar results by ARPES, while in other cases these values differ about more than 50 \% . This may be to the different ways these values are derived. For composition values $0 < X < 0.6$ the mass of light holes should symetric and in between 0.02 and 0.05 in a temperature range from 77 to 300 Kelvin. For x=0.07 we got a value even lower than that with our result $m^* = 0.097$ (see table \ref{final1}). 


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\end{document}