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\begin{document} 
\title{Indium Phosphide (InP)}
\author{ Heesuk Rho\\
	 \\
         Department of Physics\\ 
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}
 
\date{Nov. 27, 1995}
\maketitle
\begin{abstract}
In recent years there has been considerable interest in indium phosphide (InP) and related alloys because of their applications in many electronic and photonic devices.\hspace{.1in} This review summarizes bonding structure, band structure, and material properties of Indium Phosphide.
\\[1em]
\end{abstract}

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\section{Bonding Structure}

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\small
\hspace{.4in}$InP$ crystalizes in the cubic zinc blende (sphalerite) structure with density 4.81 $g cm^{-3}$, which consists of two interpenetrating $fcc$ lattices, one shifted by $a/4$ [111] ($a$ $=$ $5.8687 \AA$ $at$ $291.15$ $K$) relative to the other $fcc$ lattice. The two fcc lattices are occupied by two different atoms, $i.e.$, $In$ and $P$. The cubic unit cell of a sphalerite structure is shown in $Figure$ $1$.\cite{refB}

\begin{figure}[htb]

\centerline{\psfig{figure=figure1.ps,height=2.5in}}

\caption[]{\em{(a) The sphalerite structure; (b) stacking of (111) planes in the sphalerite structure. ABC... is the stacking order in one $fcc$ lattice and $\alpha\beta\gamma$ is the stacking order in the other.\cite{refB}}}
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\small
Although the bonding in the sphalerite structure consists of $sp^3$ hybridized orbitals, unlike in diamond cubic semiconductors, there is some charge transfer between the two types of atoms giving rise to a partial ionic character to the bonding. If $f_{i}$ and $f_{c}$ denote the fractions, respectively, of ionic and covalent character in the bond, then the sum $f_{i}$ + $f_{c}$ equals unity. The $f_{i}$ value of $InP$ is $0.421$.\cite{refC}


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\section{Band Structure}
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\small
\hspace{.4in}$InP$ is a direct band gap semiconductor. The $conduction$ $band$ minimum is situated $\Gamma$. Higher conduction band minima at $L$ and $X$ have been detected in optical experiments. The $X$ band minima show no camel's back structure, in contrast to most other $III-V$ compounds with zinc blende structure. The $valence$ $band$ has the structure common to all zinc blende type semiconductors.\cite{refD} The $\Gamma$ point (k = 0,0,0) which is the point of highest symmetry in the Brillouin zone is the most suitable point to use as the basis for a $k \cdot p$ representation to cover the entire zone. Semiconductors with tetrahedral coordination have valence band maxima at the $\Gamma$ point and many have conduction band minima also at the $\Gamma$ point (e.g., $GaAs$, $InP$). The valence band maxima are derived from the $p$-like orbitals $p_{x}$, $p_{y}$, $p_{z}$, which remain degenerate under the 
tetrahedral group of the zinc blende lattice.\cite{refC}
 

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\caption[]{\em{Band structure obtained with a non-local pseudopotential method.\cite{refE}}}
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\section{Material Properties}
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\small
\hspace{.4in}{\bf Elastic properties and sound velocity}
\hspace{.2in}For a cubic crystal, and so for $InP$, there are three independent elastic constants $S_{11}$, $S_{12}$, and $S_{14}$. The elastic compliance relates strain $\epsilon_{i}$ to stress $\sigma_{j}$, $\epsilon_{i}$ $=$ $\Sigma_{j}S_{ij}\sigma_{j}$. In covariant form $\sigma_{i}$ $=$ $\Sigma_{j}C_{ij}\epsilon_{j}$, where $C_{ij}$ are the elastic stiffness constants. For a cubic crystal $C_{ij}$ are related to $S_{ij}$ according to

$$C_{11} = \frac{S_{11}+S_{12}}{S},\ \ \ \ C_{12} = \frac{-S_{12}}{S},\ \ \ \ C_{44} = \frac{1}{S}$$
where $S$ $=$ $(S_{11} - S_{12})(S_{11} + 2S_{12})$. From the point of practical utility, the elastic constants of interests are the Young's modulus $E$ and the Poisson's ratio $\nu$. In the cubic crystals both $E$ and $\nu$ are anisotropic.\cite{refC}
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\hspace{.2in}Sound velocity of $InP$ is strongly dependent on the propagation directions. The bulk sound velocity $v$ (long-wave length, nondispersive) can be calculated by the equation, $$v = \sqrt{\frac{C_{ij}}{g}}$$ where $g$ is the crystal density.\cite{refA}

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\hspace{.2in}{\bf Optical Properties}
\hspace{.2in}$In_{1-x}Ga_{x}As_{y}P_{1-y}/InP$ hetrostructures become increasingly interesting as light sources and detectors for optical fiber communication systems in the $1.4$ $\sim$ $1.7$ $\mu$$m$ wavelength region. Knowledge of optical properties is very important in the design and analysis of such optoelectronic devices.\cite{refA} The optical absorption of solids can be described either by the complex index of refraction $n$ + $ik$ or by the comlpex dielectric function $\epsilon_{1}$ + $i\epsilon_{2}$, where $n$, $k$, $\epsilon_{1}$, and $\epsilon_{2}$ are all real functions of the frequency $\omega$. The functions $\epsilon_{1}(\omega)$ and $\epsilon_{2}(\omega)$ are connected by the Kramers-Kroenig relations, $$\epsilon_{1} = n^{2} - k^{2},\ \ \ \ \epsilon_{2} = 2nk$$The absorption coefficient $\alpha$ is related to $k$ by $\alpha$ = 2$\omega k/c$, where $k$ is the imaginary part of the refractive index and $c$ is the extinction coefficient. For $InP$, refractive index $n$ = 3.41 near the band gap energy $E_{g}$, dielectric constants $\epsilon(0)$ = 12.61 and $\epsilon(\infty)$ = 9.61, and temperature dependence $\frac{1}{n}\frac{dn}{dT}\simeq2.7\times10^{-5} K^{-1}$.\cite{refB}

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\hspace{.2in}{\bf Thermal Properties}
\hspace{.2in} The measured $C_{p}$ for $InP$ is nearly the same as $C_{v}$ at temperatures below 70 $K$, but exceeds $C_{v}$ above 70 $K$ as a result of thermal expansion of the crystal lattice.\cite{refA} Thermal resistivity of semiconductors is an important parameter in the design of power-dissipating devices such as semiconductor lasers. It is also important in calculating the figure of merit of thermoelectric devices. For $InP$ near room temperature, coefficient of thermal expansion is $4.75\times10^{-6}$ $/^{\circ}C$ and thermal conductivity 0.68 $Wcm^{-1}K^{-1}$.\cite{refB}

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\hspace{.2in}{\bf Dispersion Curves}
\hspace{.2in} Research on the crystal dynamics of perfect lattices has aroused considerable interest in compounds having a zinc blende structure. This development is the outcome of many experimental results, particularly inelastic neutron scattering data for phonon dispersion curves.

\begin{figure}[htb]

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\caption[]{\em{Phonon dispersion curve of $InP$.\cite{refA}}}
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In $Figure$ $3$ the phonon frequency ($\nu$) versus reduced wave vector ($q$) results for {\bf q} along the principal symmetry directions [100], [110], and [111] of $InP$ at room temperature, as obtained from these studies. The solid circles are the neutron scattering data, and the solid lines are calculated results. The open circles show the values of zone-center $LO$ and $TO$ phonon frequencies obtained from Raman scattering data. The dashed lines near {\bf q} $\rightarrow$ 0, $\nu$ $\rightarrow$ 0 represent the initial slopes of the acoustic modes as calculated from the sound velocities. It is recognized that the dispersion curves of $InP$ exhibit major features similar to those of other zinc blende crystals, such as $GaAs$, $GaP$, $InSb$, $GaSb$, $CdTe$, $ZnTe$, and $ZnSe$.\cite{refA}

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{\bf Transport Properties}
\hspace{.2in} The transport in n-type samples is determined up to 800 $K$ by the electrons in the $\Gamma_{6c}$-minimum. Only at very high temperatures $L_{6c}$-electrons contribute to the conductivity. In high purity samples polar scattering dominates at room temperature. Mobility values for single crystal and poly crystalline samples are mostly lower. Best values at 300 $K$ are around 4000 $cm^{2}/Vs$. At 77 $K$ values up to 130,000 $cm^{2}/Vs$ have been reported in \cite{refF}.

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{\bf Impurities and Defects}
\hspace{.2in} Discrimination of donor chemical species is made difficult because of the small electron effective mass $(m_{c}^{*}<0.1 m_{0})$ and large dielectric constant $(\epsilon_{s}\simeq12)$. This leads to small values of electron binding energy at donor states. The effective mass donor Rydberg for $InP$ is 7.31 $meV$. Zinc and Carbon are often common residual acceptors in $InP$. The Group $IV$ elements, except for $C$ and $Ge$ do not substitute for $P$, and so show no amphoteric behavior.\cite{refD}

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\begin{thebibliography}{99}

\bibitem{refA} Sadao\ Adachi,\ Physical Properties of III-V Semiconductor Compounds,\ John Wiley \& Sons(1992)
\bibitem{refB}Avishay\ Katz\ (Editor),\ Indium Phosphide and Related Materials,\ Artech House(1992)
\bibitem{refC}V.\ Swaminathan,\ A. T.\ Macrander,\ Material Aspects of GaAs and InP Based Structure,\ Prentice-Hall(1991)
\bibitem{refD}O.\ Madelung\ (Editor),\ Semiconductors Group IV Elements and III-V Compounds,\ Berlin, New York, Springer-Verlag(1991)
\bibitem{refE}J.R.\ Chelikowski,\ M.L.\ Cohen,\  Phys. Rev. B14 (1976) 556
\bibitem{refF}J.A.\ Adamski,\ J. Cryst. Growth 60 (1982) 141 

\end{thebibliography}

\end{document}
