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\title{    Fourier transform spectroscopy and the study of the optical quantities  by far-infrared Reflectivity measurements }
\author{Pradeep Bajracharya\\
     \\
         Department of Physics\\
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}

\date{30 Nov 2001}
\maketitle
\begin{abstract}
The infrared reflectivity measurement was studied with fourier transform spectroscopy for different frequency range at various temperature and calulated the conductivity with kramer's kroning analysis.The results are analysed as changed of metallic phase and insulator phase of Si:B.


\end{abstract}

\pagebreak

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\bf   Introduction \rm
 Fourier transform spectroscopy is well recognized method for analytic spectroscopic measurement in uv,visible and infrared region .It can scan the entire spectral region between$ 10-1000 cm^{-1}$.The FTIR spectrometer Digilab FTS-14 has been the first commercially available fast scanning FTIR instrument operating in the frequency range $15-10000cm^{-1}$[1].Its application in science and industry are extensive .In an attempt to study the nature of the metal-insulator transition ,we carried out far-infrared reflectivity measurementWe will here study the heavily doped silicon considered as a random system model described by a Fermi liquid Model of non-interacting system .The temperature dependent behaviors of this system (Si:B) were studied mainly through the analysis of optical conductivity $\sigma_{1}(\omega) $ in the far-infrared range.
\vspace{0.1in}

\bf BASIC PRINCIPLE AND INSTRUMENTATION \rm

\vspace{0.1in}

FTS obtains spectral information in the entire frequency region by measuring the interferogram collected through the interference of two equally divided beams .The beam of light emitted from the source (so) is directed to the beam splitter (BS) which is designed to allow half of the beam pass through and reflect the other half.The reflected half travels to the fixed mirror (FM) and travels back to the beam splitter (BS) with total path length 2L ,while the trasmitted half travels to the movable mirror (MM) and travels back with total path length$ 2L+ x $.Thus when two beams are recombined within the beam splitter, the recombined beam exhibits an interference pattern depending on the path difference x.After the recombined beam is directed to the sample (Sa) ,the reflected or transmitted beam is calculated in the detector (De) as shown in fig(1)[2].

\begin{figure}[htb]
\centerline{\psfig{figure=Fig1.ps,height=3in}}
\caption{\em{ schematic diagram of fourier transform spectroscopy .}}
\end{figure}

\vspace{0.1in}

The detailed experimental setup is as shown in schematic diagram(fig.2).For the reflectivity measurement,a special kind of set up was made .A vacuum shroud is installed within the sample chamber of Bruker instrument  in order to create the vacuum environment of $ 10^{-6} to ~10^{-7} $Torr . A helium flow refrigerator is inserted inside of vacuum shroud.A sample holder made with $0_{2} $ free copper is attached at the end of the refrigerator.The whole assembly is installed on an $x-y-z $stage .Six flat mirrors and two circular mirrors are arranged to achieve both focussing and near normal incidence of the impinging light .The angle of incidence was set at $8^{0}$.In order to get absolute value of reflectivity,the radation reflectd from the sample is compared to the radiation reflected from aluminium reference mirror.The polythene window used in the far-infrared range is wedged to 
avoid multiple interference effects between two parallel surfaces.[3]
\begin{figure}[htb]
\centerline{\psfig{figure=pf1.ps,height=3in}}
\caption{\em{Diagram of fourier transform spectrometer }.}}
\end{figure}

\vspace{0.1in}

\bf Theoretical background[4]: \rm
\vspace{0.1in}
In FTS ,the detector measured an intensity $I(x)$ which is called interferogram.This interferogram is related with spectral decomposition $I{(\omega)}$ .
\begin{eqnarray}
 I(x)& =& \int_{0}^{\infty}(1+\cos{\omega x})\frac{I{(\omega)}}{2}d\omega\\
& =&\int_{0}^{\infty}\frac{I{(\omega)}}{2}d\omega + \int_{0}^{\infty}\frac{(e^{i\omega x} +e^{-i\omega x})}{2}\frac{I(\omega)}{2}d\omega
\end{eqnarray}
In fourier Transformation,By definition of negative frequencies $I^{\ast}{(\omega)} = I{(-\omega)}$

For the real spectrum,now 
\begin{eqnarray}
I(x) &= &\frac{1}{2}\int_{0}^{\infty}I{(\omega)}d\omega + \frac{1}{4}\int_{-\infty}^{\infty}I{(\omega)}e^{i\omega x}d\omega \\
&=&\frac{1}{2}I(0) + \frac{1}{4}\int_{-\infty}^{\infty}I(\omega)e^{(i\omega x)}d\omega
\end{eqnarray}
Here the $4I(x)-2I(0)$ and $I{(\omega)} $ constitute Fourier transform pair ,the spectral information I{(\omega)} becomes

\begin{equation}
I{(\omega)}=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i\omega x} [4I(x)-2I(0)]dx
\end{equation} 
The spectrum,$I{(\omega)} $is calulated by I(x) found experimentally
    
\vspace{0.1in}

Kramer's-Kroning relations:
This relation enable to find the real part of the response of a linear passive system if the imaginary part of the response is known for all frequencies and vice versa.Then we can determine all the optical properties from the reflectivity data.
The linear passive system in Fourier Transform be 
$\Re{(\omega)} = \alpha{(\omega)}\mathcal{F}(\omega)$
where the fourier component of a "response"$ \Re{(\omega)} $and "cause" $\mathcal{F}(\omega)$ are linked by 'generalized susceptibility' $\alpha{(\omega)}$.
To get phase relation between $\Re{(\omega)}$ and $\mathcal{F}(\omega)$ $\alpha{(\omega)} $be taken as  complex 
\begin{equation}
 \alpha{(\omega)} = \alpha_{1}(\omega) +i \alpha_{2}(\omega)
\end{equation}
Taking Fourier transform of above equation, we get
\begin{equation}
R(t) = \int_{-\infty}^{\infty}\alpha(t-t')F(t)dt
\end{equation}
where R(t) at time t is resultant response of all cause F(t).
The causality can be written as
\begin{equation}
\alpha(t-t') = 0 &for &t-t'<0
\end{equation}
 
Equivalently $\alpha(\omega) = \int_{\infty}^{\infty}e^{i\omega t}dt$
                            $=\int_{0}^{\infty}\alpha(t)e^{i\omega t}dt$
where $\alpha(\omega)$has no singularity in upper half plane as $|\omega|\rightarrow\infty$.
Applying cauchy's theorem and taking an integral we get
\begin {equation} 
\alpha{(\omega)} = \frac{1}{i\pi}P\int_{-\infty}^{\infty}\frac{\alpha(\omega^')}{\omega^{'}-\omega}d\omega
\end{equation}
using $\alpha{(-\omega)} = \alpha^\ast{(\omega)},$
KK Relations become
\begin{eqnarray}
 \alpha_{1}(\omega) &= &\frac {1}{\pi}P\int_{-\infty}^{\infty}\frac{\alpha_{2}(\omega^{'})}{\omega^{'}-\omega}d\omega^{'}\\
 &=&\frac {2}{\pi}P\int_{0}^{\infty}\frac{\omega^{'}\alpha_{2}(\omega^{'})}{(\omega^{'})^2-(\omega)^2}d\omega^{'}
\end{eqnarray}
\begin{eqnarray}
\alpha_{2}(\omega)}&= &-
\frac {1}{\pi}P\int_{-\infty}^{\infty}\frac{\alpha_{1}(\omega^{'})}{\omega^{'}-\omega}d\omega^{'}\\
& = &-\frac{2\omega}{\pi}P\int_{0}^{\infty}\frac{\alpha_{1}(\omega^{'})}{\omega^{'}^2-\omega^2}d\omega^{'}
\end{eqnarray}
\vspace{0.1in}
Calculations:
The reflectivity $R(\omega)$ for normal incidence is 
$R(\omega) = r(\omega)r^\ast(\omega) =|\frac{n(\omega)-1+ik(\omega)}{n(\omega)+1+ik(\omega)}|$
where $r(\omega)= \sqrt{R(\omega)}e^{i\theta(\omega)}$ is complex reflectivity amplitude.
\begin{equation} 
\ln r(\omega)=\ln\sqrt{R(\omega)} + i\theta(\omega)
\end{equation}
Then KK relations become
\begin{equation}
\theta(\omega) = -\frac{\omega}{\pi}\int_{0}^{\infty}\frac{\ln{R(\omega^{'})}}{(\omega^{'})^2-(\omega)^2}d\omega
\end{equation}
It has singularity at$\omega^{'} = \omega$.
Then
$0=-\frac{\omega}{\pi}\int_{0}^{\infty}\frac{\ln{R(\omega)}}{(\omega^{'})^2-(\omega)^2}}d\omega$
Therefore
\begin{equation}
\theta(\omega) = -\frac{\omega}{\pi}P\int_{0}^{\infty}\frac{\ln\frac{R(\omega^{'})}{R(\omega)}}{(\omega^{'})^{2}-(\omega)^{2}}d\omega 
\end{equation}
Then we can calculate the optical properties from these equations
\begin{eqnarray}
n(\omega)&=&\frac{1-R(\omega)}{1-2R^{0.5}\cos\theta+R(\omega)}\\
k(\omega)&=&\frac{2R^{0.5}(\omega)\sin\theta(\omega)}{1-2R^{0.5}(\omega)\cos\theta +R(\omega)}\\
\epsilon(\omega)&=&\epsilon_{1}(\omega)+i\epsilon_2(\omega)\\
                &=&n^2(\omega)-k^2(\omega) +i2n(\omega)k(\omega)[17]
\end{eqnarry} 
\vspace{0.1in}
\bf  Results and discussion \rm
\vspace{0.1}
In this particular study,commercial si:B single crystal wafers(500\microm thick) grown along the $<111>$ direction were used.The infrared reflectivity was measured using aFourier transform spectrometer (Bruke IFS113v) for frequencies ranging from $20 cm^{-1} to $5000 cm^{-1}$.The complex dielectric function $& \epsilon(\omega)&=&\epsilon_{1} +\frac{4\pi i\sigma_{1}}{\omega} $was calculated from the reflectivity data using a Kramers'-Kroning anlysis[7].
\vspace{0.1in}
\begin{figure}[htb]
\centerline{\psfig{figure=pf5.ps,height=3in}}
\caption{\em{far-infra red and mid-infrared reflectivity data of Si:B.}}
\end{figure}
The infrared reflectivity of Si:B at various temperature is displayed in fig{3}[5].It shows a remarkable change with temperature .At 300K,a typical Drude -like metallic response was found ;The reflectivity approaches unity in the zero frequecy limit and shows the reflectivity minimum at$ \thickapprox $400 cm^{-1} $resulting from the plasma behavior of carriers.As the temperature is lowered below 300k,the overall reflectivity systematically changes its behavior ;The reflectivity above $100cm^{-1}$ increase while the reflectivity below  $ 100cm^{-1}$decreases as the temperature decrease.This shows systematic oscillator strength transfer from the free carriers to the localized carriers as the teperature is lowered .
The real part of frequency-dependent conductivity $\sigma_{1}(\omega)$,calculated from the reflectivity data via KKrelations as in fig {4} for frequencies between $2 cm^{-1} $and $1000  cm^{-1}$.Transition from a metallic state to a localized state is clearly seen in this figure .As the temperature is lowered ,the spectral weight of $\sigma_{1}(\omega)$ below $ 130cm^{-1}$ is gradually transferred to the conductivity above $130 cm^{-1}$.The critical frequency $\omega_{c} $\thickapprox $130 cm^{-1} $can be identified as a crossover frequency, above which the photon field cannot distinguish between the carriers in metallic states and the carriers in insulating states[6,7].The cross over frequency is almost temperature independent ,which implies the carrier concentration of the insulating phase remains constant.
\vspace{0.1in}
The$\epsilon_{1}(\omega) $also gradually changes from high temperature metallic behavior to low temperature localized behavior .At high temperature ,the $\epsilon_{1}(\omega)$ approaches to a negative value at zero frequency limit, crosses zero and gradually to a positive value $\epsilon_{\infty} $ \thickapprox $12 $ at the high frequency limit .As the temperature is lowered ,the $\epsilon_{1}(\omega)$ below $320 cm^{-1}$ is increased sustantially so that it approaches to a positive value without zero crossing which is a typical behavior of localized systems .
\begin{figure}[htb]
\centerline{\psfig{figure=pf3.ps,height=3in}}
\caption{\em{ optical conductivity and dielectric function at various temperature.}}
\end{figure}
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\bibitem{ref2} R.J Bell, \it Introductory Transform spectroscopy \em , Academic press,1972.
\bibitem{ref3} K.Kim, \it Far-infrared studies of highly correlated system \em , thesis  1996.
\bibitem{ref4} J.m Ziman, \it Principle of theory of solids \em CU Press,1972.
\bibitem{ref5} C.kittel, \it Introduction to solid states physcis \em , J.willey 1986.
\bibitem{ref6} F.wotten, \it Optical Properties to modern optics \em , Holt 1975.
\bibitem{ref7} K.H.kim et.al., \it J.Physics:condensed matter \em , 10(1998) 2963-2971.
\bibitem{ref8} Boris shapiro, \it Physical review B \em , 25,4266(1982).
\end{thebibliography}
\end{document}
