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\title{(Ga,Mn)As\\and other Ferromagnetic Semiconductors}
\author{ Gregory A. Braun\\
	 \\
         Department of Physics\\ 
         University of Cincinnati \\
         Cincinnati, Ohio 45221\\[1em]}
 
\date{ }
\maketitle
\begin{abstract}

	Ferromagnetic semiconductor are a relatively new class of materials. These materials are of interest because they combine the electric current controling aspects of semiconductors with the memory capabilities of ferromagents.  (Ga,Mn)As is very similar to the well known semiconductor GaAs, and so they posses similar properties, but only the former is ferromagnetic.  These similarities includes the Bravis lattice and the lattice constant, which will be shown to vary linearly with the concentration of Mn.  The fabrication of this material is explained, as are other ferromagnetic semiconductors.  The origin of ferromagnaism in (Ga,Mn)As is discussed, the answer to which remains open, and finally some possible applications are mentioned.\\[1em]
\end{abstract}

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% INTRODUCTION
Semiconductors are the materials utilized in virtually all electronics today.  They are used to control the flow of electric charge in a manner specific to each of their respective uses.  Specifically, they are used in the processing aspect of computing.  Ferromagnetic materials, on the other hand, are also used in electronics, but for the purpose of information storage.  In general, these two classes of materials do not overlap.  Recently, using new techniques, ferromagnetic semiconductors (FMS) have been successfully synthesized, one of the most important of which is (Ga,Mn)As.  The source of ferromagnetism, and sometimes antiferromagnetism, in these materials is of intrest to physicists, and is not yet entirely understood.  In addition, these FMS have the possibility of carrying electrons while keeping track of their spin, thus opening the door to the field of magnetoelectronics.  


\section{Structure of (Ga,Mn)As}

(Ga,Mn)As is similar in many respects to the well known semiconductor GaAs because of their obviously similar makeup.  The Bravis lattice structure of GaAs is that of the zinc-blende lattice, or ZnS lattice, which can be thought of as two interpenetrating face centered cubic lattices offset by (1/4,1/4,1/4)\cite{refD}.  This lattice has a tetrahedral structure, with each of the Ga atoms surrounded by four As atoms, as can be seen in Figure 1.  Note that if the two types of atoms are the same, this becomes the diamond lattice.  The GaAs crystal has a lattice constant of 0.566 nm.

\begin{figure}[htb]
\centerline{\includegraphics*[height=2.5in]{f1.eps}}
\caption[]{\em{The tetrahedral structure of this Bravis lattice is shown by the four lines surrounding the dark colored atom.  Note that only the four necessary atoms of the lattice cell are shown to aid visibility of this tetrahedron.}}
\end{figure}

The structure of (Ga,Mn)As is basically that of GaAs, with a small fraction of the Ga atoms replaced with Mn atoms at random.  The chemical formula for this material is often written Ga$_{1-x}$Mn$ _{x}$As, where $x$ is the relative concentration of Mn atoms.  As the Ga atoms are replaced by Mn atoms, the lattice structure remains the same, until the concentration of Mn becomes too large, at which a second phase of MnAs will form with a NiAs lattice structure\cite{refC}.  This disrupts the zinc-blende lattice and limits the amount of Mn which may be added, and therefore the amount of ferromagnetism possible in the semiconductor.  This can be overcome to some extent by the use of MBE, discussed later.  While the lattice structure remains the same, the lattice constant does not, but increases linearly with x, the concentration of Mn, as seen in figure 2, from \cite{refA}.  This is measured using x-ray diffraction.  Also seen in this figure, the lattice Constants for (Ga,Mn)As and (In,Mn)As approach each other as $x$ goes to unity, where the formula would me MnAs for both materials.  This shows that the two measurements are consistent, and that the lattice constant for a fictitious MnAs crystal with a zinc-blende Bravis lattice would be about 0.60 nm.  

\begin{figure}[htb]
\centerline{\includegraphics*[height=2.5in]{f2.eps}}
\caption[]{\em{This shows the linear dependence of the lattice constant $a$ on the Mn concentration $x$, as determined by x-ray diffraction.  The insert is a close up of the (Ga,Mn)As data.  Note that the two extrapolated linear fits are in rough agreement as $x$ approaches one.\cite{refF}}}
\end{figure}

   
\section{Fabrication of (Ga,Mn)As}

	Synthesis of (Ga,Mn)As with significant ferromagnetic properties has not traditionally been possible due to the limited solubility of Mn, or other magnetic elements in GaAs.  Recently (Ga,Mn)As with concentrations of Mn as high as $x=0.07$ has been created using Molecular Beam Epitaxy, or MBE, at low temperatures.  MBE requires growing samples under ultra high vacuum conditions, or UHV, where the pressure is in the range of 10$^{-11}$ torr.  The molecular beams are formed by heating compounds until they vaporize, and then allowing them to escape through a small hole.  Under this vacuum, the molecules can travel without collisions, and thus remain in a tightly collimated beam.  The beams are then sent onto a substrate, where they deposit, thus growng the desired sample \cite{sbook}.

	Fabrication of (Ga,Mn)As uses specifically Low Temperature MBE, where the substrate is held at a relatively lower temperature.  At this lower temperature, the forming crystal is not able to reach equilibrium, and so a lattice structure that would otherwise be impossible is created.  Instead, the lattice reaches a sort of local minimum, but does not have enough energy to reach the true minimum.  This local minimum is the desired zinc-blende structure, while the true minimum has MnAs as a second phase, in the form of the NiAs Bravis lattice.  So, only with MBE it is possible to create (Ga,Mn)As with a zinc-blende lattice and a significant portion of Mn.

	The substrate used is GaAs, and is held at the "low" temperature of about 250$^{o}$ C.  The sources for the molecular beams are elemental solids, with As in excess.  First a layer of GaAs or (Ga,Al)As is grown on the substrate to provide a buffer 100 nm thick, with a substrate temperature of 250$^{o}$ C for GaAs, or 650$^{o}$ C for (Ga,Al)As.  Then the substrate temperature is lowered to 250$^{o}$ C (if not there already), and the Mn beam is switched on to begin the growth of the (Ga,Mn)As layer, which is between 150 and 200 nm thick.  No other changes are required to begin growth of (Ga,Mn)As layer.  Once begun, the (Ga,Mn)As growth rate is typically 0.6 $\mu$m/hour \cite{refC}.

	The magnetic properties of Ga$_{1-x}$Mn$ _{x}$As will depend on the concentration $x$, which is determined by the rate of Mn sent onto the substrate.  If this rate is too high, then the Mn and As atoms will form a second phase of MnAs in the NiAs structure, as discussed above.  This is what limits the concentration to $x=0.07$, which is still high enough to observe ferromagnetic properties.	 During the growth process, Reflection High-Energy Electron Diffraction, or RHEED, is used to monitor the crystalline structure at the growing surface.  This allows the experimenter to notice if conditions, such as the Mn concentration, are causing the MnAs second phase to form \cite{refA}.


\section{Other ferromagnetic semiconductors}

	(Ga,Mn)As is certainly not the only ferromagnetic semiconductor that had been successfully made.  In the 1960's and 70's much work was done on semiconducting spinels and europium chalcogenides, both which posses both ferromagnetic and semiconducting properties.  Instead of adding magnetic atoms at random sites in a semiconductor, as is done with (Ga,Mn)As, these materials contain magnetic atoms in a regular periodic array.  The difficulty with these materials is that they are very different in crystalline structure from other semiconductors, and are very difficult to grow \cite{refA}.  Thus, they are not good candidates for technological uses of FMS.  The studies of these materials did, however, lead to the current interest in the heavy fermion system \cite{refF}.

	Another approach to FMS is to replace atoms in a non-magnetic semiconductor with magnetic ones, such as Mn. These materials are referred to as diluted magnetic semiconductors, or DMS, and include (Ga,Mn)As \cite{refF}.  The first of these inspected used semiconductors of the type II-VI, where the roman numerals indicate the periodic table groups of the constituent atoms.  The reason for this choice was due to the fact that Mn$^{++}$ has the same valance as the group II cations \cite{refF}.  The problem with type II-VI DMS is that they are difficult to dope them into either p- or n-type semiconductors, as is done to make semiconductors useful in electronics.  Additionally, without p or n carriers, the antiferromagnetic interchange between Mn atoms dominates the ferromagnetic interactions, and the material is not ferromagnetic, but instead is paramagnetic, antiferromagnetic, or spin-glass like in behavior \cite{refA}.  As doping technology advances, however, these materials are becoming more promising \cite{refF}. 

	In the late 1980s MBE was used to create the first type III-V DMS, (In,Mn)As.  The high concentration of Mn allowed ferromagnetism, but with a low Curie temperature below 50 K \cite{refB}.  Above the Curie temperature, the material is no longer ferromagnetically ordered, due to thermal excitations.  Inspection of (In,Mn)As led to the discovery of p-type carrier based ferromagnetism, discussed later.  Along with explaining the ferromagnetism observed, this created interest in other III-V DMS, and thus to the development of (Ga,Mn)As.

	The specific interest in (Ga,Mn)As among all the other FMS is due to two factors.  First (Ga,Mn)As has a much higher Curie temperature of up to 110 K; this makes experimental investigation much easier.  Also, however, (Ga,Mn)As is studied because of its similarity to GaAs, which is already used in a wide variety of semiconductors today, including the semiconductor lasers in CD players among other things, microwave transistors used in cell phones, and other applications \cite{refA}.  This lends hope to possibly one day utilizing FMS in everyday electronics.

	While the Curie temperature of (Ga,Mn)As is high enough for scientific examination, it is still well below the temperature that would be needed for FMS to be used without cooling.  Other FMS may prove to have Curie temperatures above room temperature.  Curie temperatures were calculated for several p-type semiconductors with diluted amounts a Mn, as can be seen in Figure 3 \cite{refB}.  Note that most of these FMS have note been synthesized, and some assumptions made have not been verified experimentally.  In particular, (Ga,Mn)N may prove to be a useful room temperature FMS in the future.

\begin{figure}[htb]
\centerline{\includegraphics*[height=2.5in]{f3.eps}}
\caption[]{\em{Here the curie temperatures are listed for several p-type DMS, with 0.5\% concentrations of Mn, and 3.5x10 $^{\mathit{20}}$ holes per cm $^{\mathit{3}}$.  The vertical line indicates room temperature. \cite{refB}}}
\end{figure}


\section{Origin of Ferromagnetism in DMS}

	In order to get an idea of how ferromagnetism is carried in dilute magnetic semiconductors, it is useful to look at a number of such materials.  First of all, we must look into how semiconductors carry electric current.  Usually, semiconductors such as silicone or GaAs are doped with some foreign atoms to make them either n-type or p-type.  This refers to the charge, negative or positive , by which current is carried in the material.  In n-type materials, it is the electrons which are free to move, and they carry the current.  In p-type semiconductors, the current is carried by positively charged "holes", which are unfilled states where electrons could be.  By forming junctions between n- and p-type semiconductors, the current can be controlled in a specific way.  For example, a diode consists of a p-type and n-type junction, which allows (positive) current to flow easily from the p-type to the n-type, but not in the opposite direction \cite{refG}.

	When Mn atoms are added to GaAs, they act as electron acceptors, and therefore leave freely moving holes, which act as the charge carriers.  This material is observed to be ferromagnetic.  If the (Ga,Mn)As is doped with Sn atoms, which act as an electron donor, the hole concentration can be compensated for, leaving no free holes.  This fully compensated (Ga,Mn)As is observed to actually be antiferromagnetic, i.e., the unpaired electron spins tend to align opposite from one another \cite{refH}.  In addition, n-type (In,Mn)As is also observed to be antiferromagnetic \cite{refF}.  Only p-type (In,Mn)As is ferromagnetic, and in fact it was in (In,Mn)As that the source of ferromagnetism in type III-V DMS was first found.

	So, these type III-V DMS appear to be antiferromagnetic unless there are holes present, in which case they are ferromagnetic.  This suggests two things.  First, the direct interaction between the Mn atoms themselves makes the material antiferromagnetic, because their spins anti-align.  Second, holes, or p-type carriers, somehow induce ferromagnetism in the material, and this ferromagnetism overpowers the antiferromagnetic effect.  This interaction must be long-ranged enough to allow ferromagnitism to exist in a dilute material, where the Mn atoms are far apart.  In this way, while there is a direct antiferromagnetic interaction between neighboring Mn atoms, the ferromagnetic interaction extends over many atoms, and is the dominant effect, when present \cite{refB}.

	Exactly how the holes mediate the ferromagnetism in type III-V DMS is not yet decide on.  One major candidate is the Ruderman-Kittel-Kasura-Yosida (RKKY) interaction, which is fairly well known \cite{refF}.  This theory uses a cut-off length as a parameter, and gives both ferromagnetism and antiferromagnetism at different length scales.  Due to the low hole concentration of (Ga,Mn)As, the antiferromagnetic behavior shows up only at distances greater than this cut-off length, and so does not manifest itself \cite{refA}.  The RKKY model can be used to determine the value of the Curie temperature, but it depends somewhat on the chosen cutoff of the model.  Since the model gives at least a qualitative measurement of the Curie temperature, it is considered by many to be the most likely interaction \cite{refA}.  Another similar possible model is the double-exchange picture.  Experimental studies of the nature of the p-type carriers may help differentiate between these models.

	Other physicists believe that the holes mediate the ferromagnetism through spin waves.  This model allows for arbitrary iterant-carrier spin polarization as well as dynamic correlations, both of which are absent in the RKKY model.  The creators of this theory claim that these are essential to explaining the properties of type III-V DMS, and thus that the RKKY model must be flawed \cite{refE}.


\section{Possible Applications: Magnetoelectronics}

	Virtually all electronics today use semiconductors, and depend only on the charge of the electron, not its spin.  A given semiconductor device will behave differently under different charge conditions such as voltage and current, but will treat electrons of different spin identically.  The reason for this is that semiconductors are not, as a rule, ferromagnetic, and have no spin ordering and do not differentiate between spins.  Thus traditionally, the spins are averaged over, and spin plays effectively no part in semiconductor electronics \cite{spin}.

	The development of Giant MagnetoResisatance (GMR) in the last decade has started to change the role of spin in electronics.  In GMR, spin polarized electrons are sent from one ferromagnetic material to another, through a thin layer of nonmagnetic material sandwiched between the ferromagnetic materials, usually metal.  The resistance of the electron transport through this device then depends on the spin of the incoming electrons.  Thus, the electronics can differentiate between spins, and a new degree of freedom is introduced into the electronics.  This technology is already used in sensors and read heads, and magnetic RAM is on its way to the market in the years to come \cite{spin}.

	The problem with GMR devices is that they use ferromagnetic metals, and some sort of magnetic-semiconductor interface in required to connect this device with the rest of the semiconductor circuit.   Thus they do not work well in the integrated circuit technology of semiconductors.  This is where FMS come into play.  By instead sandwiching a nonmagnetic semiconductor between ferromagnetic semiconductors, an entirely semiconductor device could be fabricated.  This then would work well in integrated circuits with other semiconductors \cite{spin}

  As mentioned before, one particular application of magnetoelectronics is Magnetic Random Access Memory, or MRAM.  Magnetic RAM requires far less power than traditional DRAM, and does not require power to retain its information, i.e. it is non-volatile.  Because it is non-volatile, computers would retain the information stored in RAM even after the power is shut down.  On turning the computer on, it would instantly return to the state it was already in, without having to wait for the computer to boot up.  Since MRAM requires far less power than DRAM, it will be particularly useful in laptop computers, where power consumption is at a premium \cite{MRAM}.  


\section{Conclusion}

	In this essay, a the structure and fabrication of (Ga,Mn)As has been discussed, in order to understand the basic nature and feasibility of the material itself.  Also, (Ga,Mn)As was compared with other ferromagnetic semiconductors to establish its usefulness and importance in this field.  The physical source of the ferromagnetism was discussed, which is theoretically interesting, and still being evaluated.  Finally, the potential applications in magnetoelectronics were mentioned, to demonstrate its importance within a technological framework.  (Ga,Mn)As is certainly an intriguing material, both scientifically and technologically.

\pagebreak

\begin{thebibliography}{99}
\bibitem{refA} H.\ Ohno, Science, {\bf 281}, 951(1998).
\bibitem{refB} Dietl, T., Jrnl. App. Science, {\bf 11} (11), 2001.
\bibitem{refC} H.\ Ohno, A.\ Shen, F.\ Mastukura, A. Oiwa, A. Endo, S. Katsumoto, and Y. Iye, Appl. Phys. Lett., {\bf 69} (3), 15 July 1996.
\bibitem{refD} Solid-State Physics, Harald Ibach and Hans L\"uth, Springer-Verlag, New York, 1993.
\bibitem{refE} K\"onig, J, Lin, H, MacDonald, A, Physica E 10 (2001) 139-42.
\bibitem{refF} H. Ohno, J. Magn. Magn. Mater., 200 (199) 110-129.
\bibitem{refG} Analog Electronics for Scientific Application, Barnaal, Denis, Waveland Press, Prospect Heights, IL, 1989.
\bibitem{refH} Introduction to Ferromagnetism, Aharoni, A.,Clarendon, Oxford, 1996.
\bibitem{sbook} Fundamentals of Semiconductors, Yu, P. and Cardona, M., Springer, New York, NY, 2001.
\bibitem{spin} Schmidt, G. and Molenkamp, L., Jrn. Appl. Phys., 89 (2001), 19, pp. 7743-7. 
\bibitem{MRAM} Butler, W., Zhang, X., Schulthess, T., Nicholson, D., Oparin, A., and MacLaren, J., Jrnl. App. Phys., 85 (8), 1999.

\end{thebibliography}
\end{document}

