Computational method

Abstract

The dilute magnetic semiconductor (Ga,Mn)As , which is the most interesting and promising material for spintronics applications, has been theoretically studied by using Density Functional Theory. First of all, calculations on GaAs were done and it was found that GaAs is a semiconductor with a direct band gap. The calculated value of the band gap is ~ 0.5eV. Secondly, the material iron was

considered and it was confirmed that iron is a ferromagnetic metal with 2.2µB net magnetic moment.

Then a magnetic impurity of manganese, Mn was introduced in the nonmagnetic GaAs and it became ferromagnetic with a net magnetic moment of 4µB. The origin of the ferromagnetic behaviour is discussed and also the Curie temperature TC of the material. It appeared that (Ga,Mn)As is a suitable material for DMS but TC has to be increased before (Ga,Mn)As could be used for spintronics

applications and on that account some methods of increasing TC are considered at the end.

Sammandrag

Den magnetiska halvledaren (Ga,Mn)As som är det mest intressanta och lovande materialet för spinelektroniska tillämpningar har teoretiskt undersökts med hjälp av Täthetsfunktionalteorin. Först gjordes beräkningar på GaAs och det visade sig att GaAs är en halvledare med direkt bandgap. Det beräknade värdet på bandgapet är ca 0.5eV. Sedan var det järn som undersöktes och det blev bekräftat att järn är en ferromagnetisk metall med netto magnetisk moment lika med 2.2μB. Då magnetiska störningar i form av mangan atomer, Mn, infördes i det omagnetiska GaAs blev halvledaren ferromagnetisk med netto magnetisk moment lika med 4μB. Orsakerna till den

ferromagnetiska ordningen diskuteras och även Curie temperaturen TC för materialet. Det visade sig att (Ga,Mn)As är ett lämpligt material för tillverkning av magnetiska halvledare men TC måste ökas innan (Ga,Mn)As skulle kunna användas i spinntroniska tillämpningar och av det skälet anges i slutet vissa metoder för att öka TC.

Contents

Introduction ··· 5

Computational method ··· 5

What are the typical properties of the electronic band structure of a semiconductor? ··· 6

What makes a metal ferromagnetic? ··· 9

How does a magnetic defect alter the density of states of a semiconductor? ··· 12

Applications of DMS ··· 14

Can (Ga,Mn)As be used in electronics today? If not, what can be changed? ··· 14

Discussion and outlook ··· 16

Acknowledgements ··· 17

References ··· 19

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Introduction

All electronic technology we use today is based on the charge and spin of electrons. Although

electronics is developing all the time the charge and spin of an electron are used only separately. But the novel field of spintronics makes use of both and opens the possibility of developing an entirely new technology with new capabilities and opportunities for applications. One such application is magnetic semiconductors. To make nonmagnetic semiconductors ferromagnetic has been a challenge for researchers for a long time, but many studies have been done and ferromagnetic semiconductors have been realized. As it is implied that spintronics is an expanding field in physics with many interesting applications and that is why I chose to do this project.

The purpose of this project is to acquire knowledge of the electronic band structure of

semiconductors and ferromagnetic materials and understand what is happening when magnetic elements are introduced into a nonmagnetic semiconductor. Furthermore, I will try to find out what is the limitation and weakness of today’s ferromagnetic semiconductors and what can be done to make them suitable for spintronics applications. My part of the project consists of answering some basic questions about the properties of materials by reading literature and simultaneously

performing density functional calculations. The calculations are done on one of the most used semiconductor, which is gallium arsenide, GaAs and the metals iron, Fe and manganese, Mn.

In the following report I first discuss the used computational method. Secondly, I explain the concept of the band structure of a solid and finally I discuss, based on my results and previous research, the problems in obtaining successful spintronic materials.

Computational method

The method used in the calculations is called Density Functional Theory, DFT, [1] which is based on the time independent Schrödinger equation. The general solution of the Schrödinger equation contains all stationary information about a system, but it is impossible to solve for most systems and thus approximations have to be used. DFT does not take into account the motion of the nuclei and the system of interacting electrons is approximated by a system of non-interacting electrons moving in an average potential from all the other electrons. Now the time independent Schrödinger equation may be written as the Kohn-Sham equation:

1

2 V

( ) r 

 

     

 

 

where Veff is the average potential, Ψ is the wavefunction, ε is the energy eigenvalues and the index i indicates one of the infinite number of solution. For determining the average potential one more approximation is used, namely the general gradient approximation, GGA [2].

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What are the typical properties of the electronic band structure of a semiconductor?

The electronic band structure of a solid determines its electronic and optical properties. The band structure describes ranges of energy that an electron is allowed or forbidden to have. The allowed regions in energy are called energy bands. By solving the Schrödinger equation for a particular potential we get the energy eigenvalues as a function of the reciprocal space vector k, i.e. ε(k) and the plot of ε(k) is actually the electronic band structure.

In a single isolated atom the electrons are only allowed to occupy certain discrete set of energy levels. If two atoms are moved close together to form a molecule, the outer electrons start to interact and each atomic orbital split in two molecular orbitals. When a large number of atoms, typically of the order of 10²⁰ or more, are brought close to each other to form a solid, the atomic orbitals split into an extremely large number of molecular orbitals. The energy difference between the orbitals is so small that the orbitals can be treated as continuous bands of energy, see Fig.1.

However, in some intervals of energy there will not occur any bands, leaving a forbidden region in energy.

Fig.1: As the number of atoms increases and the spacing between them decreases the discrete energy levels split and form energy bands. In the figure you can see that as the spacing decreases the band gap becomes bigger.

In a semiconductor, the band up to which all of the states are full at absolute zero temperature is called valence band and the unoccupied band above the valence band is called conduction band.

Electrons which are present in the valence band are at their ground state and are bound to an atom while those in the conduction band are excited and move freely within the solid. The energy

difference between the top of the valence band and the bottom of the conduction band is the definition of the band gap. If the top of the valence band and the bottom of the conduction band occur at the same k-point in the Brillouin zone then the gap is called direct band gap. Otherwise the gap is called indirect. One more important concept in semiconductor physics is the Fermi level. It is the chemical potential of the electrons and at T=0 the Fermi level is in the middle of the band gap for an intrinsic, pure, semiconductor. But in my calculations the Fermi level is determined as the highest occupied energy state and therefore from now on by Fermi level I mean the maximum of the valence band.

- 7 - The wavefunction of an electron satisfies the Schrödinger equation and each wavefunction

corresponds to a different electronic state. The crystal structure and the boundary conditions of a material restrict the wavelength of the electron and thus the electron states may have only certain energies. But some states have the same wavelength and thus at certain energies there may be many states available. However, some wavelengths are not allowed and therefore no states are available at these energies. The number of states at each energy level is described by the density of states of a system. A high density of states for a particular energy level means that there are many available states for occupation whereas a density of states equal to zero signifies that there is a band gap and no states are present at that energy level.

The electronic structure specifies the type of a material. There are three major types of materials, which are metals, semiconductors and insulators. Metals are distinguished from semiconductors and insulators because metals do not have any band gap at the Fermi level. Metals have partly filled valence band, while in a semiconductor and an insulator the valence band is completely full. The only difference between insulators and semiconductors is the size of the band gap, in insulators the band gap is larger.

Fig.2: Calculated total DOS of GaAs.

The total density of states and partial DOS for Ga and As were calculated, and the partial DOS is selected in s, p and d orbitals. The graphs of those are presented in Fig.2 and 3. Also, the band structure of GaAs was plotted along the high symmetry k-points L, gamma and X and is shown in Fig.4. It should be pointed out that the energy in all graphs in this report is relative to the Fermi level.

Then it can be seen in Fig.2 that the total DOS at the Fermi level is zero. Also, in the electronic band structure there are no energy bands at the Fermi level. Hence there is a band gap which confirms that GaAs is a semiconductor. The band gap occurs at the gamma point and so it is a direct band gap.

The value of the band gap I calculated with DFT is ~ 0.5eV and the experimental value is 1.52eV at

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T=0 K [3]. Moreover, in the graph of partial DOS for Ga and for As we see that near the Fermi level s and p orbitals are dominating. This may be explained by the following. Ga has three valence electrons and As has five valence electrons. Therefore GaAs is a III-V compound with eight valence electrons. In GaAs the eigenfunctions are built of linear combinations of sp³–hybrid. A sp³–hybrid means that there are four orbitals that are linear combinations of s, px, py and pz orbitals. In each orbital there is place for two electrons and so in the four sp³ orbitals there is place for eight electrons. Hence all orbitals in GaAs are completely full and that explains why GaAs is a semiconductor. As mentioned GaAs has eight valence electrons and each band contains two states hence GaAs has four valence bands. There are only three valence bands in Fig.4 which ought to mean that two of the bands are degenerate, i.e. they have the same energies. A narrow band corresponds to narrow peak in the graph of DOS and that is equivalent to localized states. On the other hand a broader band

corresponds to DOS that is spread over large energy interval and thus is equivalent to delocalized states. So if DOS and the band structure are analysed carefully it can be seen that they contain the same information.

Fig.3: Calculated partial DOS for Ga and As.

Semiconductors are widely used in electronics. Devices are made from semiconductors and not metals because the properties of a semiconductor are easily manipulated. The number of charge carriers and the conductivity of a semiconductor can be easily controlled and changed by adding impurities. The high conductivity and concentration of carriers in a metal are not really appropriated for devices and therefore are not preferred in the production of devices.

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Fig.4: Calculated band structure of GaAs.

What makes a metal ferromagnetic?

The electronic structure of metals differs from that of semiconductors. A characteristic of metals is that they contain partially filled bands. That means that the Fermi level occurs in allowed region in energy and thus there is no band gap. Semiconductors have full bands and because of that the net magnetic moment of a semiconductor is always zero, i.e. the electrons are in spin up/down pairs. But when the band of metals is partly filled the number of spin up and spin down electrons may not be the same. That means that charge density of spin up minus charge density of spin down,





There are three types of magnetism, paramagnetism, diamagnetism and ferromagnetism. The total magnetic field in an external magnetic field is

 

0

1

B  B     M

Where B⁰ is the external field, M is the magnetization of the material, or the magnetic moment per unit volume, ₀ is the relative permeability and _m is the magnetic susceptibility.

For paramagnetic materials _m is a small positive number, for diamagnetic materials _m is negative and for ferromagnetic materials _m is a large positive number. A negative _m means that the internal magnetic field in the material is opposite in direction to that of the external field while for a

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positive m the additional and the external field are in the same direction. So if an external magnetic field is applied to a ferromagnetic material the total field will increase. Actually, a ferromagnetic material has a magnetic moment even in no external magnetic field, i.e. it has a spontaneous magnetic moment. That indicates that the electron spins are aligned parallel. There are also

antiferromagnetic materials which have ordered magnetism, but in this case the magnetic moments are aligned antiparallel. Fig.5 illustrates different arrangement of the magnetic moments.

Fig.5: Arrangement of dipoles in different types of materials.

In most of the materials the magnetic moments are randomly orientated but in few materials the magnetic moments tend to align. That is due to the exchange interaction. The outermost electrons in a solid are delocalized, they move freely in the crystal and may interact with electrons from

neighbouring atoms which have the same wave function. If the atoms have local magnetic moment the delocalized electrons may mediate the magnetic moment among neighbouring atoms and in that way causing alignment of the atomic magnetic moments. It would result in a net magnetic moment.

Materials in which the magnetic moments tend to align parallel are ferromagnetic.

As the temperature increases, the thermal fluctuations increase so much that the ferromagnetic order of the magnetic moments destroys. The temperature above which the spontaneous magnetization in ferromagnetic materials vanishes is called Curie temperature, TC. The Curie temperature depends on the properties of the material, but the stronger the interaction between the magnetic moments the higher is the Curie temperature. For example, the transition metals iron, nickel, and cobalt are ferromagnetic with TC of 1043, 627 and 1388K, respectively [3]. In

ferromagnetism the Curie temperature is a fundamental property and is of great importance for applications.

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Fig.6: Calculated DOS of iron. Positive DOS represents spin up states and negative DOS represents spin down states.

According to the calculations iron is magnetic with a net magnetic moment of 2.2µB per Fe atom and this value agrees with experimental data. This time the calculations were done with regard to spin direction. In Fig. 6 is represented the calculated density of states, DOS, for iron and the spin up and spin down states are plotted separately. It is seen that DOS for spin up and spin down electrons is not the same. The DOS for spin down electrons is shifted to the right which means that below the Fermi level there are more electrons of spin up than spin down. As a result, the net magnetic moment is not zero. Some materials like iron have high density of states at the Fermi level, but that is not energetically favourable and in order to decrease the total energy a splitting between the spin up and spin down states occur. As a result the energy of electrons with spin up is lower than for those with spin down and consequently some of the electrons with spin down flip their spin and move to lower energy state. The shift of the DOS for spin up and spin down states, the principle, is illustrated in Fig.7.

Fig.7: The differences in DOS between nonmagnetic and ferromagnetic metal.

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The calculated band structure of iron, along the k-points gamma, H, P, N, gamma and P, is shown in Fig.8. There is no band gap at the Fermi level, so iron is a metal. The figure shows that the bands for spin up and spin down electrons are divided. The bands for spin up are below corresponding bands for spin down which implies that the spin up states are energetically favoured.

Fig.8: Calculated band structure of iron.

How does a magnetic defect alter the density of states of a semiconductor?

The properties of a semiconductor can be changed by adding impurities into the material.

Introducing magnetic defects into a nonmagnetic semiconductor can make it magnetic and even ferromagnetic [4]. This type of semiconductors is called diluted magnetic semiconductors, DMS, Fig.9b, which consist of alloys of nonmagnetic semiconductor, Fig.9a, and few fragments with magnetic defects.

There are three types of defects, which are substitutional, interstitial and clusters of magnetic impurity. The first one means that impurities replace some of the original atoms in the

semiconductor. The interstitial defects sit among the original atoms and the last one tends to form clusters or inclusions in the lattice [5]. It has been shown that only the substitutional defects are responsible for the ferromagnetism in a III-V semiconductor [6]. For that reason I chose to insert only substitutional defects in the calculations.

Two types of DMS have been studied by researchers throughout the world, namely II-VI and III-V semicunductors. II-VI DMSs are easier to prepare since the number of valence electrons of the positive ions matches that of the impurities’. Also, the concentration of impurity can be quite high

- 13 - compared to that of III-V DMS. But the magnetic interaction in II-VI DMSs is dominated by

antiferromagnetic exchange resulting in paramagnetism or antiferromagnetism. Additionally, the Curie temperature TC for II-VI semiconductors is very low. On the other hand, III-V DMSs show ferromagnetic behaviour at higher temperatures and regardless of the fact that these are difficult to prepare, because of the low solubility of the magnetic defects, they are more suitable of being good DMSs [5].

Fig.9 The white circles represent one type of atoms and the grey circles represent another type of atoms. In figure 9a there is a semiconductor with nonmagnetic atoms while in figure 9b some of the grey atoms are replaced by red atoms with magnetic moment.

Magnetic atoms were introduced in the lattice of the material with the aim to make the nonmagnetic III-V semiconductor GaAs magnetic. GaAs is by far the most studied semiconductor and manganese, Mn, appears to be a promising magnetic material for making GaAs ferromagnetic [7]. Because of that I used the 3d transition metal Mn in the calculations.

A bulk calculation with an 8 atom cell was performed. In the original cell, Ga4As4, one Mn atom was substituted into a Ga site in the GaAs lattice, so the new cell was Ga3MnAs4. That corresponds to 25%

substitutional Mn defects. Mn has seven valence electrons while Ga has only three. Three of the Mn electrons go to bonding with As and the remaining four electrons are expected to provide the semiconductor with local magnetic moment of 4µB per Mn atom. After the calculations were done it appeared that the computed net magnetic moment agrees with the expected value and because this is the net and not the local magnetic moment that means that there is a long range magnetic order in the material. Hence by doping the nonmagnetic GaAs with Mn the semiconductor became magnetic, and even ferromagnetic.

In Fig.10 is presented the density of states for (Ga,Mn)As, namely the total DOS, the DOS of d-orbitals of Mn and p-orbitals of As. At the Fermi level the DOS for spin down electrons is zero, but is finite for spin up electrons. Thus the spin up channel is metallic, there is no band gap, but the spin down channel has a band gap at the Fermi level. Hence, all the conduction electrons will have the same spin and therefore 100% spin polarized current is expected. Materials which have that property are called half-metals. These materials are suitable for electronics applications. Furthermore, in the figure one can see that the d-orbitals of Mn and p-orbitals of As are hybridized, that is believed to be responsible for the ferromagnetic behaviour in (Ga,Mn)As [4,5].

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Fig.10: Calculated DOS of (Ga,Mn)As. Positive DOS represents spin up states and negative DOS represents spin down states.

Applications of DMS

The information technology is based on the charge and spin of electrons, but today these are used only separately. Semiconductor devices manipulate the charge of electrons, while the electron spin is used for mass storage of information. For long time people have tried to make use of both the charge and spin of electrons and that new field in electronics is called spintronics. There are many

advantages of spintronics and one is the creation of persistent devices which have the capability of mass storage of information. Other applications of spintronics are spin injection and sensors. If spin polarized current is injected into semiconductors it will be possible to carry out quantum bit

operations required for quantum computing. The ultimate materials for spin injection are half-metals with 100% spin polarization. Also, GMR (Giant Magneto Resistance) read heads used in hard disks are one successful application of sensors [8]. Peter Grünberg and Albert Fert got the Nobel Prize in Physics in 2007 for the discovery of GMR.

Can (Ga,Mn)As be used in electronics today? If not, what can be changed?

In order to use DMS devices in electronics, the Curie temperature TC has to be above room

temperature. Theoretical predictions based on the Zener model suggest that TC above 300K could be attained in (Ga,Mn)As if the concentration of substitutional Mn is higher than 10% [9]. But so high concentration of Mn is difficult to achieve in a laboratory and today the highest reported TC in

- 15 - (Ga,Mn)As is 185K for 11-13% Mn concentration[10]. Such a low Curie temperature is not sufficient for spintronics applications and for that reason a lot of research is going on with the aim of increasing the value of TC.

Experimental and theoretical work agree on the conclusion that TC of (Ga,Mn)As depends on the concentration of Mn atoms that has substituted Ga, i.e. the substitutional Mn [7]. But during the crystal growth of (Ga,Mn)As, substitutional Mn is not the only formed defect, other defects like interstitial Mn and As antisites will also be formed [7]. As antisites are atoms substituted on the Ga lattice and are found to be the most common defects in (Ga,Mn)As [11]. Both interstitial Mn and As antisites counteract the substitutional Mn, which provide (Ga,Mn)As with ferromagnetic moment.

Thus the ferromagnetism and the Curie temperature TC will increase if the amount of interstitial Mn and As antisites is reduced. That could be done by preparing the crystal at low temperature and annealing [7].

The ferromagnetism in (Ga,Mn)As is mediated by carriers, holes, provided by the substitutional Mn atoms [4]. Since the magnetic exchange interaction and hence TC depends on the concentration of the holes, additional impurities have been used to see if the carrier concentration would increase.

Berryllium, Be, has been used as additional impurities in (Ga,Mn)As, but it actually lowers TC because adding holes prevents the incorporation of substitutional Mn [5].

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Discussion and outlook

This work has contributed to an understanding of the electronic structure of different types of materials and specially semiconductors. GaAs, iron and (Ga,Mn)As were simulated by using Density Functional Theory, DFT and the General Gradient Approximation, GGA. The calculated band gap of GaAs does not agree with the experimental value, the difference is ~ 1ev. But underestimation of the band gap is a well known error within DFT [12]. The concentration of the introduced magnetic defect was 25%. That concentration is not realistic, (Ga,Mn)As with so high amount of Mn is impossible to grow. Nevertheless, the magnetic defect provides GaAs with magnetic moment and change in the density of states could be observed. As to the rest all results agreed with previous calculations and all results were realistic.

In conclusion, my calculations show that nonmagnetic semiconductors become ferromagnetic when magnetic defect are added. But before III-V DMSs could be used for spintronics applications the Curie temperature TC has to be increased drastically. Hence future research should aim at finding DMS with high TC, higher than 300K. It is only the substitutional defects in III-V semiconductors that provide ferromagnetism, so the aim is to get rid of other defects. Therefore potential studying would be to codope GaAs and examine whether it increases the substitutional defects or not. Another suggestion would be to study II-VI DMS. They are not intrinsic ferromagnetic, but great concentration of magnetic elements can be attained and it is known that the TC is proportional to the

concentration. That is a useful property and it is interesting to see if it is possible to make II-VI DMS ferromagnetic with high TC.

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Acknowledgements

I would like to show my gratitude to all of those who supported me in any respect during the completion of the project.

First of all I would like to thank my two supervisors Susanne Mirbt and Biplab Sanyal for providing me with creative ideas and always supporting me. Their constructive feedback on my paper made the hard part of the project easier for me.

I am grateful for the help I got from Taizo Shibuya, who introduced me to the computer programs and helped me to get started. I would also like to thank all the kind people I have met at the Division of Materials Theory. It was very nice to work with you!

Most of all I want to thank my wonderful friends for the unforgettable years in Uppsala and forgiving me every time I was late because of working with my thesis.

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References

[1] W. Kohn, Rev. Mod. Phys. 71, 1253 (1999).

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[3] С Kittel, Introduction to Solid State Physics , J. Wiley & Sons 7th Ed, 1996.

[4] H.Ohno, Science 281, 951 (1998).

[5] T. Dietl, D. D. Awschalom, M. Kaminska, H. Ohno, Spintronics, Volume 82 (Semiconductors and Semimetals), academic press, 2008.

[6] F. Máca and J. Mašek, Phys. Rev. B 65, 235209 (2002).

[7] P. A. Korzhavyi, I. A. Abrikosov, E. A. Smirnova, L. Bergqvist, P. Mohn, R. Mathieu, P. Svedlindh, J.

Sadowski, E. I. Isaev, Yu. Kh. Vekilov, and O. Eriksson, Phys. Rev. Lett. 88, 187202 (2002).

[8] L. Bergqvist, “Finite Temperature Magnetism and Its Application to Spintronics”, licentiate thesis, Uppsala University, 2003.

[9] T. Dietl, H. Ohno, F. Matsukura, J. Cibert, D. Ferrand, Science 287, 1019 (2000).

[10] K. Olejník, M. H. S. Owen, V. Novák, J. Mašek, A. C. Irvine, J. Wunderlich, and T. Jungwirth, Phys.

Rev. B 78, 054403 (2008).

[11] B.Grandidier et al., Appl. Phys. Lett. 77, 4001 (2000).

[12] B. Arnaud and M. Alouani, Phys. Rev. B 63, 085208 (2001).

Figures

Fig.1: http://image.tutorvista.com/content/semiconductor-devices/energy-levels-splitting- process.jpeg

Fig.5: Encyclopædia Britannica, article Ferromagnetism

Fig.7: Atomic Scale Design Network(ASDN): Educational Webportal, article Spintronics