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\begin{document} 
\title{Gallium antimonide (GaSb)}
\author{Volodymyr Nahirnyak\\
	 \\
         Department of Physics\\ 
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}
 
\date{\today }
\maketitle
\begin{abstract}

Although GaSb has not been widely utilized in technological applications, its  energy band structure  does allow various interesting physical experiments to be performed on intervalley transfer mechanisms and thus the nature of L and X energetic minima in the Ga series of III-V semiconductor compounds can be clarified.    \\[1em]
\end{abstract}

\pagebreak

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\section{Crystalline structure}

 The crystal lattice of  CaSb belongs to the zinc blende lattice and is constituted by two $fcc$  sublattices  formed by atoms Ga and Sb. The density of the crystal is   5.61 $g\cdot cm^{-3}$ and the lattice constant $a_{o}$ = 6.09593 $\dot{A}$ [1]  and the melting point $T$ = 985 K [2] and  Debye temperature $\theta_{D}$ - 266K.\\ 
The zinc blende structure  of GaSb is based on the diamond lattice  but the two different atoms in primitive cell.The two $fcc$ sublattices can be defined by basic translational vectors: $a_{o}$/2(1,1,0),
$a_{o}$/2(0,1,1), $a_{o}$/2(1,0,1) and are displaced by vector $d$=$a_{o}$/4(1,1,1) with respect to one another. The bonding forces in such semiconductor compounds are partially covalent  and partially ionic. Although the zinc blende structure is closely related to the diamond structure , there is an important difference between the two: the diamond lattice has a center of inversion midway between two atoms in primitive cell, whereas the zinc blende lattice does not. This  has implications for the nonlinear optical properties of GaSb: only crystals lacking a center of inversion can be strongly nonlinear[4].
Besides the translations there are  24 symmetry operations of the group T$_{d}$  which transform the zinc blende lattice into itself.
\begin{figure}[htb]


\centerline{\includegraphics*[height=1.5in]{pic1.ps}}

\caption[]{\em{Crystalline structure of GaSb.\cite{refB}}}

\end{figure}
Since the translational symmetry of the lattice is that of the $fcc$ sublattices, the values of the wave vector $k$   on which energies and wave functions depend can be defined within a volume of reciprocal space which is obtained by bisecting with 	perpendicular planes the reciprocal lattice of basic vectors 2$n$/a$_{o}$(1,1,1), 2$n$/$a_{o}$(1,1,1), 2$n$/$a_{o}$(1,1,1) ,where$n$ - integer number.
The points of high symmetry $\Gamma$ [$k$=0],L[$k$=2n/$a_{o}$(1/2,1/2,1/2)] and X[$k$=2n/$a_{o}$(1,0,0) are represented in the Fig.2. along with the lines $\Lambda$ and  $\Delta$ which connect $\Gamma$ with L and X .There are three X - points and four L – points. The path from the center of zone $\Gamma$ to the points of high symmetry is very significant for description of the band structure of the crystal.The reduced zone (first Brillouin zone ) is displayed in Fig.2. and the symmetry point and symmetry lines are indicated [1]. 

\begin{figure}[htb]

\centerline{\includegraphics*[height=1.5in]{pic2.ps}}

\caption[]{\em{First Brillouin zone \cite{refA}.}}
\end{figure}



\section{Energy band structure}

The conduction band in CaSb is characterized by three sets of minima the lowest minima at $\Gamma$, slightly higher minima at the L - points at the surface of the Brillouin zone and even higher minima at the X - points. The valence band has the structure common to all zincblende  semiconductors
 (Fig. 3.).      

\begin{figure}[htb]
\centerline{\includegraphics*[height=1.5in]{pic3.ps}}

\caption[]{\em{Band structure of GaSb obtained with  a non-local pseudopotential calculation \cite{refB}.}}
\end{figure}





In the direct band material such as GaSb the position of minima at X and L can be determined by Hall effect and by study of the pressure dependence of resistivity. The measurements of ultraviolet reflectivity helps to determine the band gaps.

\section{Elastic constants and lattice dynamics in GaSb}

Below the experimentally measured elastic constants are presented (Table 1.). The values of elastic constants were determined on the base of measurements of transit time pulses of 30 $Mhz$ longitudinal and transverse ultrasonic waves in sulfur –doped $n$ –type single crystals of GaSb down to 4.2 K [7]. A pulse –echo-overlap method was used to measure the transit times of 2 $\sim\mu$sec – wide pulses of longitudinal and transverse ultrasonic waves in samples to which the appropriate quartz transducers were bounded.
The velocity of each type  of ultrasonic wave was deduced from the measured transit time and the sample length. The elastic constants were, in turn, deduced from the velocities $v$ and the sample density $\rho$. The relations used to obtained the elastic constants from the velocities are:
\begin{equation} \rho v_{[110],[110]}^{2} = \frac{1}{2} \cdot\ (c_{11}+c_{12}+2c_{44}) ,
\end{equation}
\begin{equation} \rho v_{[110],[\bar{1}10]}^{2} = \frac{1}{2}\cdot (c_{11}-c_{12}) ,
\end{equation}
\begin{equation} \rho v_{[100],\bot[100]}^{2} =  c_{44} .
\end{equation} 

\begin{center}
\begin{tabular}{|c|c|c|c|}
\hline Elastic moduli (10$^{11} dyn/cm^{2}$)&T=296 K, &T=77 K,  &T=4 K, \\  &$\rho$ = 5.619 $g/cm^{3}$&$\rho$ = 5.638 $g/cm^{3}$ & $\rho$ = 5.638 $g/cm^{3}$ \\
\hline $c_{11}$&8.834&9.052&9.082 \\
\hline $c_{12}$&4.023&4.113&4.131 \\
\hline $c_{44}$&4.322&4.435&4.447 \\
\hline
\end{tabular}
\end{center}
\begin{center}
Table 1. Elastic  constants for GaSb \cite{refG}.
\end{center}
\section{Dispersion of acoustic waves}

The acoustical (A) and optical (O) modes of transverse (T) and longitudinal (L) branches of the phonon dispersion curves which were measured on the base of inelastic neutron scareting experiments are presented on the figure below.Three different direction of the wave propagation have been investigated.$\xi$=a$_{0}k/2\pi$  ,where a$_{o}$ – crystal constant= 6.0959 $\dot{A}$[7].The measurements were conducted at room temperature on  a single crystal of undoped GaSb.

\begin{figure}[htb]

\centerline{\includegraphics*[height=1.5in]{pic4.ps}}

\caption[]{\em{Phonon dispersion curves in GaSb crystal\cite{refH}.}}
\end{figure}



The above frequency dispersion data were used to calculate the lattice contribution to the heat capacity $C_{v}$ (Fig.5.) and the Debey tempetature $\theta_{D}$. The temperature dependance of both the heat capacitance and Debey temperature are presented on the Fig.5. The results obtained are in good accordance with those theoretically calculated.
 
\begin{figure}[htb]
\centerline{\includegraphics*[height=1.5in]{pic5.ps}}

\caption[]{\em{Temperature dependence of heat capacitance C$_{v}$ and Debey temperature $\theta_{D}$ in GaSb, measured and calculated \cite{refH}.}}
\end{figure}


\section{Transport properties}

By transport properties we mean the response of the electrons to applied forces.The proper approach to the problem of  electronic motion in a periodic lattice would be to solve the time dependent Schrodinger equations.It's known that the electrons moving in a perfect periodic potential with no applied forces has a constant velocity and is not scattered by the atoms of the crystal. Only cystal imperfections serve  to scater electrons. In order to get correct equations of motions we have to take into account the "effective masses "of electrons.
Effective mass is one of the important concepts in the analysis of semiconductors , in the theory of scattering. The important point is that an electron in a periodic potential is accelarated relative to the lattice in an applied electric or magnetic field as if the mass of electron were equal to an effective mass which differs from the mass of free electron  [4]. \\
In GaSb the conduction band lowest valley occurs at the same  $k$ value as  the valence band maximum (for instance $k$=0). It means that GaSb belongs to a direct gap semiconductors. 
The curvatures (or effective masses) of valence and conduction band edges are another important   features of transport properties of GaSb is illustrated in Fig. 6.

\begin{figure}[htb]
\centerline{\includegraphics*[height=1.5in]{pic6.ps}}

\caption[]{\em{Effective masses in the energetic valleys of GaSb \cite{refE}.}}
\end{figure}




The large curvature of the band structure in $\Gamma$ valley results in a low effective  mass and corresponding high mobility of the conduction electrons. High electron mobility is used to good purpose in galvanomagnetic devices, based on either the Hall effect (proportional to $\mu$ ,electron mobility) or magnetoresistance (proportional to $\mu^{2}$). The high mobility also implies that conduction electrons acquire a high drift momentum in  an electric field and hence there is strong energy  transfer to the electron assembly leading to pronounced ``hot electron" effects [4].
Although the $\Gamma$ valley is spherically symmetrical in the energy bands of GaSb there is an essential deviation from the simple relation:\\ 
\begin{equation} E = \frac{\hbar^{2}k^{2}}{2m_{c}^{*}} ,
\end{equation}
between the energy $E$ and the electronic crystal momentum $\hbar k$ ($m_{c}^{*}$ being the effective mass of carriers  in  the $\Gamma$ valley).The   deviation  arises from the interaction of the conduction band with the nearest valence band and given via  $(kp)$  perturbation
method by formula:\\
\begin{equation}\frac{m}{m_{c}^{*}}= 1+\frac{P^{2}}{3}\cdot
\bigg( \frac{2}{E_{g}}+\frac{1}{E_{g}+\Delta}\bigg)  ,
\end{equation}\\
where $m$ is free electron mass , $E_{g}$ is the fundumental band gap,and $\Delta$ is the valence band spin – orbit splitting. For GaSb ,the experimental parameters ,$m^{*}_{c}$= $m^{*}_{e}$ = 0.041$m$ ($\Gamma$ valley),$E_{g}$= 0.726 eV, $\Delta$ =0.80 eV. The effective masses of electrons (in units of $m$ ) are presented in Table. 2 [2]. 


\begin{center}
\begin{tabular}{|c|c|c|}
\hline $\Gamma$ valley&L valley& X valley \\
\hline 0.041&0.11&0.22 \\
\hline
\end{tabular}
\end{center}      
\begin{center}
Table 2. Effective masses of electrons in energetic valleys of GaSb.
\end{center}

\section{Carrier mobility}

The carrier mobility $\mu$  defined as the magnitude of carrier drift velocity $v$ per unit electric field $\mathcal{E}$,may be expressed in terms of the carrier relaxation time  $\tau$  for various scattering processes in the form [5]:\\ 
\begin{equation}\mu =\frac{v}{\mathcal{E}}=\frac{q \tau}{m^{*}}  ,
\end{equation}\\ 
where $m^{*}$ - conductivity effective mass. The effective masses of electrons $m_{n}^{*}$   and  holes $m^{*}_{p}$ in semiconductors can be determined by cyclotron resonance [9]. The electrical conductivity $\sigma$ in the presence of both electrons and holes is given by the sum of separate contributions:
\begin{equation}\sigma =q(n\mu_{e}+p\mu_{h}) , 
\end{equation}
where $n$,$p$ - concentration of electrons and holes [4].The typical value of $ \mu_{e}$ and  $\mu_{h}$  for GaSb is given in Table 3.For intrinsic semiconductor GaSb the concentration of electrons and  holes are equal $n$=$p$.This permits us to write the \textsl{dc} conductivity in the following form [5]: \\ 
\begin{equation}\sigma=2q\bigg( \frac{kT}{2\pi\hbar^{2}}\bigg)^{3/2} (m_{e}m_{h})^{3/4} exp \bigg\{-\frac{E_{g}}{kT}\bigg\} \cdot (\mu_{e}+\mu_{h}) .
\end{equation}
\\ 
By using the temperature dependence of carrier mobility we can determine the energy gap $E_{g}$ of GaSb. The temperature dependence of $\mu_{e}$ and $\mu_{h}$ for GaSb is well studied \cite{refB}.
GaSb possesses the cubic zincblende structure and a direct energy gap. However , the four equivalent L$_{1c}$ minima lie energetically very near the lowest conduction band minimum ( $\sim$ 0.07 eV above $\Gamma_{1c}$),so that most electrons populate L$_{1c}$  at room temperature. Because of this complicated transport situation , and because of the relatively small energy gap (0.0726 eV) at $\Gamma$ , GaSb has not been widely utilized in technological applications. Nevertheless ,the band structure of GaSb does allow various interesting physical experiments to be performed on intervalley transfer mechanisms and the nature of L$_{1c}$ and X$_{1c}$ minima in the Ga series of III-V crystals[5].

GaSb has direct energy gap and low electron effective mass and due to this a high electron mobility. For the $n$-type GaSb all three conduction bands with minima situated at L and X contribute to entire conductivity . X – band contributes to conductivity mostly above 180 C$^{0}$. The mobilities of electrons and holes in the sample of GaSb with concentration of $n$ = 1.49$\times 10^{18}$ $cm^{- 3}$ at room temperature  are presented in the following table.
\begin{center} 
\begin{tabular}{|c|c|}
\hline Electron mobility $\mu_{e}$($cm^{2}/V\cdot s$)&Hole mobility $\mu_{h}$($cm^{2}/V\cdot s$) \\
\hline 3750($\Gamma$ minima)&680\\
\hline 482(L minima)&2500 (77 K)\\
\hline 107(X minima)& \\
\hline
\end{tabular}
\end{center}
\begin{center}
Table 3. Electron and hole mobilities in GaSb\cite{refB}.
\end{center}

\section{Conclusions}

The band structure , effective masses,valence crystal binding and transport properties are the most important features which  describe the general nature of the GaSb
semiconductor compound.The increased interest to this material may be explained  by the unique feature of  GaSb – very narrow energy gaps, high mobility , and small
effective masses of carriers and due to this the possibilities of practical applications in high frequency semiconductor devices (Gunn diode).
There  is also considerable scientific interest in all mixed III-V compounds of Ga series. The continuous variability of parameters in such kinds of materials makes possible the design of a material with virtually any gap width within the given range. The designability of such materials is especially useful for infrared or optical devices in which operation over particular wavelength ranges is required ,such as in infrared sensors and injection lasers[3].




\begin{thebibliography}{99}

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\\ Semiconductors and semimetals. ,edited by R.K.Willardson,A.C.Beer.,(500),v.1.,1966.

\bibitem{refB} Otfried Madelung.\ Semiconductors.{\bf 116}, (317),1996.


\bibitem{refC} D.Long.Energy band structure of mixed crystals of III-V compounds. Semiconductors and semimetals. , edited by R.K.Willardson,A.C.Beer.,{\bf 145} ,(500),v.1.,1966.

\bibitem{refD}Charls\ Kittel.\ Introduction \ to Solid State Physics.\ {\bf 311,371},(766), 1971. 
\bibitem{refE} Alexander O.E.Animalu. Intermediate quantum theory of crystalline solids.{\bf 293}, (516) 1997.
\bibitem{refF} D.L.Rode.Low-field electron transport. p.75. Semiconductors and semimetals. ,edited by R.K.Willardson,A.C.Beer.,v.10 (312),1975.
\bibitem{refG} W.F.Boyle, R.J.Sladek. Physical Rev.,B,v.11,{\bf2933},1975.
\bibitem{refH} M.K.Farr,\ J.G.Traylor ,\ S.K.Sinha. Lattice dynamics of GaSb. \ Physical Rev.,B,\ v.11,{\bf1587},1975.
\bibitem{refI}\ H.Ibach,\ H.L$\ddot{u}$th.\ Solid State Physics. {\bf 295}, (341), 1990.
 

\end{thebibliography}



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