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\title{Introduction on the Semiconductor Heterostructures}
\author{ Yong Song\thanks{\tiny Tel:513.556.5450 \hspace{5em} Mail Location:0011}\\
	 \\
         Department of Physics\ \\ 
         \,University of Cincinnati \\
         Cincinnati, Ohio\  45221\\[1em]}
 
\date{March 07,2002 }
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\begin{ }

\noindent
Abstract:The heterostructure physics becomes more and more important. More and more 
people are entering this dynamic field. This article introduces some basic concepts 
on the semiconductor heterostructures. And the application of the hetero\-structure 
is also presented in the end of the article.\\[1em]
\end{ }

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%\markright{{\tiny Solid State Physics,Winter Quarter 2002}}
\section{Introduction}

\noindent
 Today you can't image what the world would be if without the semiconductor 
heterostructures technology. And it is said that near two thirds of all 
research groups in the physics of semiconductors are involve in it 
especially double heterostructures. In 2000 one of the Physics Nobel prize 
owners is Professor Zhores I.Alferov whose major achievment is this field. 
He started to study it in the early 1930s$^{[1]}$ 

\
\

\noindent
Semiconductors are composed of more than one matrial.Semiconductor 
heterostructure, which are simply atomically abrupt interfaces between two 
different semiconductors,can be thought of as building blocks for more 
elaborate nanostructures. Or we can say in another way, semiconductor 
heterostructures are crystalline solids containing one or more 
heterojunctions---interfaces between two different semiconductors. An abrupt 
heterojunction couples electronic states from two different materials with 
different positions of band extrema(i.e. potential energy), or different 
effective masses,etc$^{[2]}$. 

\
\

\noindent
GaAs-Al$_{x}$Ga$_{1-x}$As is always mentioned(/studied) in this field. It 
is usually abbreviated simply to AlGaAs. Sometimes it is called ``sandwich'' 
because the GaAs is between two layers of Al$_{x}$Ga$_{1-x}$As. The whole 
system acts like an elementary quantum well. It has some properties that 
will be introduced later. 

\
\

\noindent 
Variations in composition can be used to control the motion of electrons and 
holes and further determine the properties of the heterostructure. It's 
important for us to know the alignment of bands at a heterojunction, where 
two materials meet.

\section{Classical Heterostructures}
\subsection{General Properties of Heterostructures}

First of all we should use an assumptions. We treat electrons as they are 
free,except for an effective mass. 
\
\
\noindent
We like to study III-V materials and their alloys for their semiconducting 
properties. Minimum band gap and the lattice constant are two important 
properties of those materials. The following figure shows these two 
properties of them$^{[3]}$:

\
\

\noindent
From Fig 1. we can find that the lattice constants of AlAs and GaAs are very 
close. So GaAs and AlAs are always been made into the alloys. 

\pagebreak

\begin{figure}[htbp]

\centerline{\includegraphics*[height=3in]{fig1.ps}}
%\centerline{\includegraphics*[bbllx=0.00in,bblly=0.00in,bburx=3.98in,bbury=3.05in,scale=1.00]{fig-1.ps}}
\caption[]{\em{Plot of the lattice constant of various semiconductors against their minimum band gap E$_{g}$.\ \ (Full Line: Direct Band Gap\ \ \ \  Dash Line: Indirect Band Gap)}}
\end{figure}


\noindent The active regions of heterostructures are typically at or close to 
interfaces. In the real life it is impossible to join the two materials in a 
seamless way. But for III-V heterostructures the more perfect interfaces 
they get, the better performance they will have. If you want to join the two 
materials perfectly to get a nearly `ideal' heterostructure, you should 
first choose the materials that have the same crystal structures(Or at least 
symmetry) and the two lattice constants must be nearly identical. 

\
\

\noindent
The lattice constant(effective lattice constant) of Al$_{x}$Ga$_{1-x}$As is 
given by (ignores alloy scattering)


\begin{center} x a$_{AlAs}$+(1-x) a$_{GaAs }$ \end{center}
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\small \begin{center} (a$_{\tiny AlAs\small }$is the lattice constant of AlAs)\end{center}
\setlength{\parskip 1ex}
\
\normalsize
\noindent
And the averaged crystal potential {\slshape U(r)} is:
\begin{center} X{\slshape U}$_{AlAs}$(r) + (1-x) {\slshape U}$_{GaAs}$(r)\end{center}

\noindent
So generally we can conclude at least three conditions to get an ideal 
heterostructure:

. The two materials' atomic structures must match.

. The interface must have high quality. Namely, it must keep clean and pure and have \-\ \ \ \ \ \ \ perfect lattice structure.

. The layers may be thin so we are able to change the composition of successive layers\\ \- {\hspace{2em}}very rapidly, preferably and easily.

\subsection{The Bond Diagram}


Most heterostructures are formed by heterojunctions. The heterojunctions are 
either flat or have very low curvature on the atomic scale. So we can just 
study the planar heterojunctions. Their properties depend only on one 
longitudinal coordinate(e.g. assume it is the z axis). With this assumption, 
the structures then can be classified in terms of band diagrams.

\
\

\noindent
Consider a semiconducting alloy of the type A$_{x}$B$_{1-x. }$Here the A 
and B are two different semiconductor materials(including alloy 
respectively). Let's focus on a heterojunction between two materials A and 
B. E$_{g}^{A} $ is less than  E$_{g}^{B}$.

\
\

\noindent
Some notations:

E$_{c}$: Energy of Conduction Bands \hspace{5em} E$_{v}$: Energy of Valence Bands


$\Delta$E$_{c}$: Difference between the energies of the conduction band extreme in the two\linebreak
\- \hspace{4em}materials. $\Delta$ E$_{c} \equiv $ E$_{c}^{B}$- E$_{c}^{A} $ = $\chi ^{A}$ - $\chi ^{B}$

$\Delta$E$_{v}$: Difference between the energies of the valence band extreme 
in the two materials.\linebreak
\-{\hspace{4em}}$\Delta$ E$_{v} \equiv $ E$_{v}^{B}$- E$_{v}^{A}$
\
\

\noindent 
The positions of E$_{c}$ and E$_{v}$ are determined by the chemical 
bonding of the atoms and can be measure or calculated by solving 
Schrodinger's equation. These energy levels must be measured relative to 
some reference level. We choose one energy reference level(vacuum levels 
aligned in Fig 2$^{[3]}$ ---- Or label it as E$_{0}$). $\chi $ is the 
electron affinity, the energy needed to remove from the semiconductor of an 
electron located at E$_{c}$ and make it a free electron.
\begin{figure}[htbp]
\centerline{\includegraphics*[height=3in]{fig2.ps}}
%\centerline{\includegraphics*[bbllx=0.00in,bblly=0.00in,bburx=3.98in,bbury=3.05in,scale=1.00]{fig2.ps}}
\caption{Alignment of the bands at a heterojunction between materials:A(left) and B(right)\ \ \ {\small (Note: Anderson's Rule: The Vacuum levels of the two materials of one 
heterostruction should be lined up)}}
\end{figure}
\normalsize
\
\

\noindent 
If the components are two different semiconductors, the band diagram can 
also be constructed from the each known band structure. The relative postion 
of the bands can be determined from the electron affinity. 

\bigskip


\subsection{Four Types of Heterostructure}
\subsubsection{ Type I (Or Straddling Alignment) Heterostructure}



As the Fig 3$^{[2][4]}$ shows: 


\begin{figure}[htbp]
\includegraphics*[height=2in]{fig3.ps}
%\includegraphics*[bbllx=0.00in,bblly=0.00in,bburx=1.45in,bbury=2.57in,scale=1.00]{fig3.eps}
\caption[]{(a) Type I structures have band offsets of opposite sign

\ \ \ {} { } { } { } { } { }(b) Observed band alignments for Al$_{0.3}$Ga$_{0.7}$As
}
\end{figure}




\bigskip

\noindent
The narrower band gap is enclosed within the wider band gap(as the Fig3a 
shows). In this case, the signs of band offsets for the two bands are 
opposite. This is the case for such systems as GaAs/AlGaAs, 
GaSb/AlSb,GaAs/GaP. From Fig3-b, We can find for Al$_{0.3}$Ga$_{0.7}$As-GaAs 
system: $\Delta$E$_{c} $=0.23 ev, $\Delta$E$_{v}$= -0.15ev. The offset in valence bands 
is to be about 40\% of the difference in band gaps. For this material pair, 
the conduction and valence bands of the smaller bandgap semiconductor lie 
completely within the bandgap of the wider bandgap semiconductor. 

\subsubsection{ Type II(Staggered) Heterostructure and Type III(Broken-gap)}


\
\

\noindent 
For Type II heterostructures, the band offsets have the same sign$^{[2]}$. The 
examples are InP/Al$_{0.48}$In$_{0.52}$P , In$_{x}$Ga$_{1-x}$As/Ga$_{x}$ 
Sb$_{1-x}$ As and Al$_{x}$ In$_{1-x}$As/InP systems.

\pagebreak
\begin{figure}[htbp]
\centerline{\includegraphics*[height=2in]{fig4.ps}}
\caption[]{Type II. The band offsets have the same sign}
\end{figure}
\
\

\noindent 
Type III also has the band offsets of the same sign, but the offsets are so 
large that the gaps in the two materials do not overlap(Fig 5$^{[2][4]}$). 
The example is InAs/GaSb.
\begin{figure}[htbp]
\centerline{\includegraphics*[height=3.5in]{fig5.ps}}
%\centerline{\includegraphics*[bbllx=0.00in,bblly=0.00in,bburx=3.98in,bbury=3.05in,scale=1.00]{fig5.ps}}
\caption[]{(a) Band diagram of Type III Heterostructure (b) The valence electrons 
from the right material can penetrate the conduction band of the left 
material and form a double layer of electrons and holes at the interface. 
(c) Band alignments for GaSb and InAs.}
\end{figure}

\subsubsection{ Difference between Types II and III}

Type III heterostructures are sometimes called ``Type II with nonoverlapping 
gaps'' or ``Type II misaligned Heterostructure''. But there is an important 
physical difference between the two types: The Type III components will 
equalize the Fermi levels and leads to large bending (Fig 5b) to get a 
equilibrium point.

\
\

\noindent 
The band diagrams ignore electrostatic potentials. Near the junction the 
mobile rearrange after the semiconductors are placed in contact. when placed 
in contact, electrons move from the semiconductor with the higher 
Fermi-level to the one with lower direction. Then the process will produce 
an electric field to balance this transfer. 

As the Fig. 6$^{[4]}$ or Fig 5b shows.

\begin{figure}[htbp]
\centerline{\includegraphics*[height=4in]{fig6.ps}}
\caption[]{GaSb:InAs/P-n Heterojunction (a) before contact (b) after contact}
\end{figure}

\
\

\noindent 
Where E$_{F}$ is the Fermi Level. Before they are in contract, there are 
separate Fermi levels for the n-type(E$_{FN}$) and p-type 
(E$_{FP}$)semiconductors. But after they are put in contact, the energy 
bands adjust to align the Fermi levels. 

\section{Some Important Applications}

As we have mentioned in the first two paragraphs, heterostructure 
productshave been widely used in the industry and research field$^{[5]}$. You also 
can find its applications in your life.

. Semiconductor lasers (Used in Compact Disk---CD player)

. High-efficiently LED's(heterostructure light-emitting diodes) and 
low-noise, 

\-\ \ high-electron-mobility transistors(HEMTs) used in high-frequency devices

. Solar cells and photodetectors,based on wide-gap window effect.

. Bipolar wide-gap transistors

. High-power diodes and thyristors

. Infrared to visible converters

%\bigskip

\section{Conclusion}

We introduced some basic concepts on the semiconductor heterostructure. 
Although only a few properties are provided in this presentation, Some other 
important properties are almost based on the lattice structure of 
heterostructures included in this article. And some interesting theoretic 
properties and useful industrial technology are not covered because of the 
limit of time and the space. With the basic concepts it is not difficult for 
you to find out why and how the heterostructures are widely used near you. 



\begin{thebibliography}{99}
\bibitem{1} Zh. L. Alferov.\textsl{The history and future of semiconductor heterostructures}.\ Semiconductors\ 32(1), January 1998.
\bibitem{2} Fedor T. Vasko,\ Alex V. Kuznetsov.\textsl{Electronic states and optical transitions in semiconductor heterostructures}.\ Springer Press, 1998.
\bibitem{3} John H. Davies. \textsl{The physics of low-dimensional semiconductors}.\ Semiconductors\ 1$^{st}$ published 1998
\bibitem{1} \textsl{Dr. Mark Lundstrom's lecture notes on heterostructors in Fall 1995}\\
Dr. Mark Lundstrom: Professor of School of Electrical and Computer Engineering and the NSF MRSEC for Technology-Enabling Heterostructures, Purdue University
\bibitem{1} Zh. L. Alferov.\textsl{Review of Modern Physics}.\ Review of Modern Physics\ Volume 73,July 2001.
\end{thebibliography}

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