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\begin{document}
\title{ Gallium Phosphide}
\author{Ramesh Paudyal\\
     \\
         Department of Physics\\
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}

\date{March 6, 2002}
\maketitle
\begin{abstract}

Gallium phosphide is commercially one of the most promising III-V semiconductor because of
its application to opt-electronics due to wide band gap and thermal stability. A wide variety
of theroetical and experimental works have given detailed information about the physical 
properties of the material. This paper will discuss some of the physical properties of the
gallum phosphide semiconductor.
\end{abstract}

\pagebreak

\def\baselinestretch{1.1}
\vspace{0.1in}
\bf Introduction \rm

Some of the atoms from group III of the periodic table combine  with the atoms of group V to
form "III-V" semiconducting compound, in a particular gallium phosphide (GaP) semiconductor. 
The gallium phosphide is commercially one  of the most important "III-V" semiconductor because
of its application to electroluminesent devices.

\vspace{0.1in}

\bf Structure of Gallium Phosphide \rm
\vspace{0.1in}

Most of the compound semiconductor of groups "III-V" crystallize into the  cubic zinc blende
form. The gallium phosphide (GaP) compound possesses the cubic zinc blende structure. Where each
atom is at the centre of the regular tetrahedron, at the four corners of which lie atoms of the 
other kind.

\vspace{0.1in}

\begin{figure}[htb]
\centerline{\psfig{figure=zinc.ps,height=2in,width=3in}}
\caption{\em{Cubic Unit cell of the zinc blende Structure containing eight atoms \cite{ref2}.}}
\end{figure}

\vspace{0.1in}
The unit cell of the cubic zinc blende structure is same as the diamond form except that the two
different kinds of the atom occupy alternate positon in the lattice. The space group is
F$\bar4$3m (schoenflies, $T_{d^2})$ and the point grroup is $ \bar43m (T_{d})$. The two  sub 
lattices being displaced relative to each other by one of quarter of the body diagonal of the 
cube. The cubic zinc blend structure which is simplest crystals lacking a center of symmetry and
hence capable of exhibiting pizeoelectric and related effects depending on a polar symmetry. The
lattice constant a is the distance between the lattice points of primitive cubic refrence lattice.
For Gallium phosphide (GaP) lattice constant 'a' is $5.45\AA$. The lattice constant of a 
semiconductor can expand or contract when impurity atoms are incorporated. The crystal density is
one of the simplest and most important parameter. There are four molecules in a unit cell of the  
zinc blende lattice.

\vspace{0.1in}

\begin{tabular}{|r|l|}
\hline

Crystal structure  &  zinc blende\\
Density  &  $4.14 gm cm^{-3}$\\ 
Number of atoms in $cm^{-3}$  &   $4.94\times 10^{22}$\\              
Melting point  &  $1480^{0}C$\\
Refractive index  &  $3.37$\\
Lattice constant  &  $5.45 \AA$ \\
Thermal conductivity  & $1.1 W/cm C$\\
Specific heat capacity  $(C_{p})$ &  0.313 J/g K \\
\end{tabular}


\vspace{0.1in}
\bf Properties of Gallium Phosphide\rm

\vspace{0.1in}
\bf 1. Electronic Enegry-Band Structure \rm

\vspace{0.1in}
The atoms gallium and phosphorus have 3 and 5 electrons respectively outside  a core of closed
shell with an $s^{2}p^{1}$ and $s^{2}p^{3} $ electronic configuration. Between them, therefore 
atoms have an average of four valence electrons per atom avilable for binding. These are three 
possible bonding in combination of gallium and phosphorus.
1. covalent 2. ionic and 3. neutral

\vspace{0.1in}

For covalent bonding each phosphorus atom donates an electron to gallium atom each with four
valence electrons. These combine to form $sp^{3}$ hybrid and tetrahedral bond. For pure ionic
bonding we may suppose that gallium donates 3 electrons to phosphorus, forming $Ga^{+3}$ and
$P^{3-}$ ions, each with spherically symmetrical closed shells configuration. These ions would be
hold together in the crystal by purely electrostatic forces. The propertiesof a material may
provide information on its ionic character. If there is a transfer of charge which is related to 
elctronegativity of two kinds of atoms. In the neutral bond, gallium and phosphorus retain their
electrons so that there is no charge difference between these atoms.

\vspace{0.1in}

\begin{figure}[htb]
\centerline{\psfig{figure=band.ps,height=3in,width=3in}}
\caption{\em{Band structure of GaP\cite{ref3}.}}
\end{figure}

The transitions between the states which are not vertical in an energy band diagram are called
indirect tansitions. Gallium phosphide is known to be a more suitable material to study some of
the indirect band gap optical process. Since it has three an indirect band gap with energy gaps
$E_{g} =2.885$ ev(direct) and $E_{g} =2.338$ ev (indirect) at 0 K, $E_{g}=2.338$ ev (direct)
and $E_{g}=2.61$ ev(indirect) at 300 K respectively. These energy are just large enough to 
sustain resonably efficient luminescent for red, yellow and grenish light, even at 300 K. 
The temprature dependence of the direct energy gap is given by the relation,

$E_{g}(T)=E{g}(0)-(\alpha T^{2})/(\beta+T)$      Where, $E_{g}(0)$ is band gap at 0 K. 
The temprature depndence $E_{g}$ is mainly due to the electron phonon interactions and $\beta$ is 
proportional to debye temprature. The effective density of conduction and valence bands are, 
$3.4\times 10^{15}T^{3/2}cm^{-3}$ and $3.6\times 10^{15}T^{3/2}cm^{-3}$ respectively.

The band gap and effective masses depend on a pressure. The dependencece of band gap with pressure
is given by the relation,
 
$E_{g}(p)= E_{0}+ap+bp^{2}$, where pressure  is in K bar. The spin orbit
splitting energy in this material is very small.

\vspace{0.1in}

\bf 2. Thermal Properties\rm

\vspace{0.1in}

Study of the thermal properties of solid gives connection with the fundamental physical properties
like a thermal energy  of a solid. The heat capacity at constant pressure $c_{P}$ versus temprature
for GaP is shown below. The specific heat of the GaP increases monotonically with temprature. 
At low temprature $ C_{p}$ and $ C_{v} $ are nearly the same, but $c_{p}$  exceeds $c_{v}$ at 
higher temprature as result of thermal expansion of the crystal lattice. The specific heat
capacity of GaP is 0.313J/g K at 300 K.
\vspace{0.1in}

\begin{figure}[htb]
\centerline{\psfig{figure=specific.ps,height=2.5in,width=3in}}
\caption{\em{Temprature dependence of specific heat capacity \cite{ref3}.}}
\end{figure} 

\vspace{0.1in}
The debye temprature $\theta_{D}$ can be used in charaterzing the  excitation of phonons and to
describe various lattice theramal phenomena. The  debye tempartures of GaP are $\theta_{D}$= 495 
at $300^0$ K and $\theta_{D}$ =446 at 0 K. strain tensor$[e]$ by the relation,

Strain tensor is proportional to the Temprature(T). The proportionally constant is $\alpha$ is  the linear
thermal expansion coefficient.
 
This is given by the relation ,
$\alpha_{ th} =(1/a)( \frac{\partial a}{\partial T})$ , Where a is crystal lattice parameter
In the negative expansion  coefficient is due to the entropy contribution of the Gibb's free 
energy.

Lattice thermal conductivity results essentially from interaction  between phonons and from the 
scattering of phonons by crystalline imperfection. Thermal conductivity of semiconductor plays an
important role in the design of power dissipation.
\vspace{0.1in}

The temprature dependence of thermal condcutivity of Gap is shown below. This graph shows that
the the thermal conductivity of pure single crysystal is zero at 0 K and rises approximately
exponetially to a maximum near 30 K, falls some how faster than 1/T and then varies approximately
as 1/T to the melting point. Thermal conductivity of GaP is 1.1 W /cm C at 300 K.

\vspace{0.1in}

\begin{figure}[htb]
\centerline{\psfig{figure=thermal.ps,height=2.5in,width=3in}}
\caption{\em{Temprature dependence of thermal conductivity\cite{ref3}.}}
\end{figure}

\bf3. Optical properties\rm

The band structure  of a semiconductor is closely related to the dielectric function
$\epsilon(\omega)$ of a material. Studies of the optical properties of solid have proved to be a
powerful tool in understaning of electronic structure of the solids. The complex dielectric
function of $\epsilon(\omega)$ of real and imaginary parts are related to the Kramers Kroning 
relation. Bascially, it describes the optical properties of the medium at all photon energies.
However, the Knowledge of optical properties is very important in the design and analysis of such 
optoelectronics devices. since GaP is an indirect band gap semiconductor. The dielectric constant 
of Gap depends on temprature since it increases from 10.75-11.1 in the static dielectric constant of GaP
between 1.6 K and room temprature. The dielectic constant, $(\epsilon)$ consists of two parts, 
one of the contribution is from the  electronic and the other is lattice. The contribution of
the electronic part of the static dielectric constant is obtained by measuring the refractive 
index ($\epsilon$ is $n^2$) as a function of energy

\vspace{0.1in}

\begin{figure}[htb]
\centerline{\psfig{figure=refractive.ps,height=2.5in,width=3in}}
\caption{\em{Refractive-index dispersion for GaP at 300K \cite{ref3}.}}
\end{figure}

\vspace{0.1in}
Figure 5 shows that the experimentally determined refractive index at 300 K versus photon energy. The peak is due to the transition of electrons.

\vspace{0.1in}

\bf Piezoelectricity\rm

\vspace{0.1in}
This is generated due to electronic polarization by application of stress to a crystal lacking a 
centre of symmetry. The gallium phosphide lacks centre of symmetry or inversion. Piezoelectricity 
tensor has only one component $e_{14}$. The experimental calculated value for Gap is$e_{14}=-0.1 C/m^{2}$.

\vspace{0.1in}

\bf Application of Gallium Phosphide\rm

The semiconductor gallium phosphide(GaP) due to its large energy gap $E_{g}$ 2.78 ev at 300k and
good thermal stability is potential matieral for optical and high temprtute device, light emiting
red and green diodes and transistors, optoelectronics devices, bipolar transistors(BJTs) and
rectifiers as semiconductor and lasers and electronic refrigeration.

\begin{thebibliography}{99}
\bibitem{ref1} Sadacho Adachi, Physical properties of III-V Semiconductor Compounds \em, John Wliley Sons,1992 
\bibitem{ref2} C. Hilsum and A.C. Rose-Innes, Semiconducting III-V compounds \em, Pregamon Press, 1961 
\bibitem{ref3} Ioffe Institute,  Properties of Gallium Phosphide \em, http://www.ioffe.rssi.ru, 2002                                        
\end{thebibliography}
\end{document}
