%This is a LaTex template for the materials review part of the course
%It allows for the insertion of postscript figures into the text
%complete with captions. In addition, it will generate a title page
%with an abstract,title, and authors name.

\documentstyle[12pt]{article}
\input epsf
\addtolength{\evensidemargin}{-6ex}
\addtolength{\oddsidemargin}{-6ex}
\addtolength{\textwidth}{+14ex}
\addtolength{\textheight}{+27ex}
\setlength{\parindent}{+5ex}
\setlength{\topmargin}{-1em}

\def\baselinestretch{1.0}

\begin{document}
\title{Method of measurement of energy gaps in semiconductor}
\author{Kapila P Hewaparakrama\\
        \\
        Department of Physics\\
        University of Cincinnati\\
        Cincinnati, Ohio 45221\\[1em]}
\date{June 4,2001 }
\maketitle

%\newpage
\begin{abstract}
This paper presents a particular  experimental  method to determine the
band gaps of Semiconductors through the optical  absorption spectra. The 
procedure of analysis of the spectra is discussed for two cases, i.e for 
direct and indirect band gaps. Two examples of experimental results have 
been presented. It is shown that $SrTiO_3$ has an indirect band energy of
3.16 eV at 10 K and the direct band gap of CdTe thin film decreases with
the increasing thickness.\\
\end{abstract}

\newpage

\section{\bf{Introduction}}

Characteristic of semiconducting and insulating material is a band gap
between valence and conduction bands.The Energy band gap of a
semiconductor material plays an important role in determining the
photo-conversion efficiency of a solar cell.  There are many methods to
determine the energy band gaps namely optical absorption method,
photoelectrochemical methods, intrinsic conduction measurement at high
temperature, photoemission and band structure calculation. Here, optical 
absorption method is used to determine the energy band gaps of
semiconductor.\\  

\subsection{\bf{Absorption of Light in Semiconductor}}

When incident photon energy exceeds the band gap of a semiconductor,
electrons can be excited from the valence to the conduction band. Removal
of an electron from the valence band creates a positively charged vacancy
called a hole. The light passing through the material is absorbed then,
and the number of generated electron-hole pairs depends on the number of
incident photons $\Gamma_0(\nu)$ (per unit area, unit time and unit
energy).
Inside the semiconductor, the photon flux at position x ,$\Gamma(x,\nu)$,
can be
expressed as a function of absorption coefficient $\alpha(\nu)$,

\begin{eqnarray}
\Gamma(x,\nu) & = & \Gamma_0(\nu)exp(-\alpha x) 
\end{eqnarray}

\noindent
The abosption coefficient $\alpha(\nu)$ depends on the band structure of
the semiconductor.There are two major processes of absorption involved in
determining $\alpha$, they are called direct and indirect absorptions.   


\subsection{\bf{Direct and Indirect band gaps in semiconductor}}

In the semiconductors, the band gap can be classified as direct or
indirect -a band gap is said
to be ``direct'' when the energy minimum (the bottom) of the conduction
band lies directly above the energy maximum (the top) of the valence band
in reciprocal k-space (Fig. 1a)  Otherwise, we call it an ``indirect''
band gap (Fig. 1b).

\noindent
At the top of the valence band and at the bottom of the
conduction band
have the same wave vector in a direct band gap solid, but different wave
vectors in an indirect band gap material.  Whether the gap is direct or
indirect is important in the application of inorganic semiconductors (such
as Si, Ge and GaAs) to photoconduction and 
electroluminescence\cite{refA}\\.



\begin{figure}[h]
%\epsfxsize=3.72667in
\epsfysize=3.0in
\centerline{\epsffile{kfig2.ps}}
\caption[]{\em{Schematic representations of electronic band structures
near the Fermi level for  (a) a semiconductor with direct band gap and
(b) a semiconductor with n indirect band gap. $\epsilon_{F}$ is the Fermi
level having the Fermi wave vector $K_{F}$.\cite{refA}}}
\end{figure}  

 

\subsubsection{\bf{Direct Optical Absorption in semiconductor}} 

The direct optical absorption process is illustrated in Fig. 1a. It is a
first order process involving only the absorbed photon, which can be
represented approximately as a vertical line on energy vs. wave vector
plot(K). If $ h\nu $ is the photon energy and $\Delta E$ is the energy
gap of the semiconductor.Then\\   
\begin{eqnarray}
\Delta E & = & h\nu 
\end{eqnarray}

\noindent
The expected variation of $\alpha $ with photon energy near the minimum
energy for absorption $E^d_g$ is given by 
\begin{eqnarray}
\alpha(h\nu) & = & c(h\nu-E^d_g)^{1/2} 
\end{eqnarray}
where c is a constant.

\noindent
Therfore a plot of $\alpha^2 $ as a function of photon Energy $(h\nu)$
yields a straight line for a direct absorption(fig. 2a). Direct Energy band
gap can be obtained from that plot.

\subsubsection{\bf{Indirect Optical Absorption in semiconductor}} 

An indirect optical absorption process is illustrated in Fig 1b. It is a
second order process involving both the absorbed photon and a
simultaneously absorbed or emitted phonon, which occurs when the minimum
of the conduction band and the maximum of the valence band occur at
different values of wave vector K. The absorption transition conserves
energy\\
\begin{eqnarray}
\Delta E & = & h\nu \pm E_p
\end{eqnarray}
where $h\nu$ is the photon energy and $E_p$ is the phonon energy.
The absorption-edge of the indirect gap is well described by\cite{refF}\\
\begin{eqnarray}
\alpha(h\nu) & = & A\Bigg[\frac {(h\nu - h\nu_{p}
-E^i_g)^2}{1-\exp^{-E_p/{KT}}} +
 \frac{(h\nu + h\nu_{p} -E^i_g)^2}{\exp^{E_p/{KT}}-1}\Bigg]   
\end{eqnarray}
where A is a constant, K is the Boltzmann constant,$h\nu_{p}(=E_p)$ is the
phonon energy participating
in the transition, and $E^i_g$ is an indirect band gap energy.
A plot of $\alpha^{1/2}$ as a function of photon Energy $h\nu$ gives the
two values at $\alpha^{1/2}$=0 from the two dotted straight lines as shown in the fig.2b. Then, indirect band gap of the semiconductor is \\
\begin{eqnarray}
E_g^i & = & \frac{E_1 + E_2}{2} 
\end{eqnarray}

\begin{figure}[h]
%\epsfxsize=3.72667in
\epsfysize=3.5in
\centerline{\epsffile{Fig4.ps}}
\caption[]{\em{The graph of the (a)$\alpha^2$ versus photon energy for
direct band gap semiconductor  and (b) $\alpha^{1/2}$ versus photon energy
verses for indirect band gap semiconductor.\cite{refB}}}
\end{figure} 

\section{\bf{Experimental set-up}}

The essential components of a spectrophotometer include: (1) a stable source
of radiant energy, (2) a monochomator to resolve the radiation into
component
wavelengths or bands of wavelengths, (3)sample compartment and (4) a radiation detector.
\begin{figure}[h]
%\epsfxsize=3.72667in
\epsfysize=3.5in
\centerline{\epsffile{Fig3.ps}}
\caption[]{\em{Block diagram of absorption apparatus.\cite{refC}}}
\end{figure}

\subsection{\bf{Sources of  Radiation}}

A tungsten filament lamp is the most satisfactory and inexpensive source
of visible and infrared radiation. The filament is heated by a stabilized
d-c power supply, or by a storage battery. The tungsten filament emits
continuous radiation in the region between 350 nm and 2500 nm. 
For source of ultraviolet radiation, the hydrogen lamp and deuterium lamp
are used. They consist of a pair of electrodes which are
enclosed in a
glass tube provided with a quartz window and filled with hydrogen or
deuterium gas at low pressure. When a stabilized high voltage is applied
to the electrodes, an electron discharge occurs which excites other
electrons in the gas molecules to high energy states. As the electrons
return to their ground state they emit radiation which is continuous in
the region roughly between 180 nmand 350 nm. Similarly, a xenon discharge
lamp is used as a source of ultraviolet radiation. The xenon lamp produces
higher intensity radiation, but it is not as stable as the hydrogen lamp.
It also emits visible radiation which may appear as stray radiation in
ultraviolet applications.\cite{refC}

\subsection{\bf{Monochromators}}

The souces of radiation  commonly used emit continuous radiation over 
wide ranges of wavelengths. However, narrow band widths have many 
advantages: (1) narrow band radiation will allow the resolution of 
absorption bands which are quite close to each other,(2) with narrow 
band radiation a peak may be measured at its absorption maximum, thus 
increasing the sensitivity, and (3) the absorption of narrow band 
radiation will tend to show greater adherence to Beer's law     
because only that radiation which can be absorbed is measured. 
Monocromators resolve wide band polychromatic radiation from the source
into narrow bands or monochromatic radiation with effective band widths
varying from 35 $m\mu$ to 0.1 $m\mu$. 
The components of monochromator include: (1) an entrance slit which admits
polychromatic radiation from source; (2) a collimating device, either a lens
or a mirror; (3) a dispersion device, either a prism or grating, which
resolves the radiation into component wavelengths; (4) a focusing lens or 
mirror; and (5) an exit slit.  The effective band width of 
radiation emerging from the monochromator depends on several factors, 
including the dispersing element and the slit widths of both the entrance
and exits slits. Narrow slit widths isolate narrow bands; however, the
slit width also limits the radiant power which reaches the detector. 
Therefore, the minimum band width may be determined by the sensitivity
of the detector.\cite{refC} 
  
\subsection{\bf{Detection Devices}}

Any detector absorbs the energy of the photons which strike it and converts
this energy to a measurable quantity such as the darkening of a photographic
plate ,an electric current,or thermal changes. Most modern detectors generates
an electrical signal.  Any detector must generate a signal which is 
quantitatively related to the radiant power striking it. The noise of a detector
refers to the background signal generated when no radiant power from the sample
reaches the detector. This noise may be caused either by random changes 
within the detector itself or by electrical pick up of other signals in the 
vicinity of the detector unit. Important requirements  for detectors include:
(1) high sensitivity with a low noise level in order to allow the detection
of low levels of radiant power, (2) short response time,(3) long term 
stability to insure quantitative response, and (4) an electronic signal which
is easily amplified for typical readout apparatus.
and exits slits. Narrow slit widths isolate narrow bands; however, the
slit width also limits the radiant power which reaches the detector. 
Therefore, the minimum band width may be determined by the sensitivity.\cite{refC}  

\subsubsection{\bf{Ultraviolet and Visible Radiation Detectors}}

Ultraviolet and Visible photons possess enough energy to cause photoejection
of electrons when they strike surfaces which have been treated with specific
types of compounds. Their absorption may also cause bound, non-conducting 
electrons to move into conduction bands in certain semiconductors. Both 
processes generate an electric current which is directly propotional to the 
radient power of the absorbed photons. Device which employ these systems
are called photoelectric detectors and are sub-classified as phototubes and
photovoltaic cells.\cite{refC}

\subsection{\bf{General Procedure of Measurement and Analysis}}

In principle we want to detect the spectrum that comes purely from the
sample. In practice, however we cannot avoid of having the spectrum of
our sample get mixed with that from other sources such as from the
substrate, or from some noice existing in the  sample compartment. We
therefore need to do two measurements, one without the sample and the
other one with the sample. The spectrum, namely the quantity
$\Gamma(\nu,x)$, that we will analyze is then the difference between
$\Gamma(\nu,x)$ without sample and $\Gamma(\nu,x)$ with the sample.
In the analysis we extract $\alpha(\nu)$ from $\Gamma(\nu,x)$ by means of
equation (1). $\alpha(h\nu)$ is the plotted against $h\nu$. Finally by
looking at the graphs and refering to equation (3) (for direct gap) or
equation(5) (for indirect gap) we can extract the value of $E_g$.



\section{\bf{Examples of experimental results.}}

From the absorption mesurements above 77 K, it is reported that $SrTiO_3$
has an indirect band gap\cite{refD} in accordance with the band
calculation\cite{refE}.
Fig.4 shows absorption-edge spectra in the square-root scale. A simple
linear shape at 10-60 K indicates that $\alpha(h\nu)$ has a quadratic
dependance on the phonon energy, which means that $SrTiO_3$ has an
indirect band gap.The absorption edge of the indirect gap is well
discribed by eq.(5).Experimental results are well reproduced by eq.(5)
with $E^i_g=3.16 eV$. 


\begin{figure}[h]
%\epsfxsize=3.72667in
\epsfysize=1.8in

\centerline{\epsffile{FIG6.ps}}
\caption[]{\em{$\alpha(h\nu)$ is plotted in the square-root
scale below 120 K. Solid line shows the calculated curves at 10,60,110,120
K using eq.(2).\cite{refF}}} 
\end{figure}  
  
\noindent
The $({\alpha}h\nu)^2$ versus $h\nu$ plots for CdTe films of different
thicknesses but deposited at identical conditions are shown in Fig.5. It
has been observed that the direct band gap decreases with increasing
thickness.\cite{refG}

\begin{figure}[h]
%\epsfxsize=3.72667in
\epsfysize=2.0in

\centerline{\epsffile{FIG7.ps}} 
\caption[]{\em{Plot of $\alpha(h\nu)^{1/2}$ vs. $h\nu$ for CdTe films of different thicknesses.\cite{refG}}} 
\end{figure}




\begin{thebibliography}{99}
\bibitem{refA} Dong-Kyun Seo, R. Hoffmann , \"Theoritical Chemistry
Accounts 102(1999) 23.
\bibitem{refB} N.B.Hannay,\"Semiconductor 443(1959) 
\bibitem{refC}H.H.Willard,L.L.Merritt,  \"INSTRUMENTAL METHODS OF
ANALYSIS. 
\bibitem{refD}M.Capizzi, A.Frova,\"Phys.Rev.Lett. 25(1970) 1298.
\bibitem{refE}A.H.Kahn, A.J.Leyendecker,\"Phys. Rev.135(1994)A1321.
\bibitem{refF}T.Hasegawa, M.Shirai, K.Tanaka,\"Journal of Luminescence
87(2000)1217.
\bibitem{refG}X.Mathew, P.J.Sebbastian, \"Solar Energy Materials and Solar
Cells 59(1999)85.
43(2000)66.
\end{thebibliography}
\end{document}































