Bergqvist, L.

Dederichs, P. H.

Juelich Shared Electronic Resources

John von Neumann Institute for ComputingDilute Magnetic SemiconductorsL. Bergqvist, P. H. Dederichspublished inNIC Symposium 2008,G. Mu¨nster, D. Wolf, M. Kremer (Editors),John von Neumann Institute for Computing, Ju¨lich,NIC Series, Vol. 39, ISBN 978-3-9810843-5-1, pp. 153-160, 2008.c© 2008 by John von Neumann Institute for ComputingPermission to make digital or hard copies of portions of this work forpersonal or classroom use is granted provided that the copies are notmade or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwiserequires prior specific permission by the publisher mentioned above.http://www.fz-juelich.de/nic-series/volume39Dilute Magnetic SemiconductorsLars Bergqvist and Peter H. DederichsInstitute of Solid State PhysicsResearch Centre Ju¨lich, 52425 Ju¨lich, GermanyE-mail: {l.bergqvist, p.h.dederichs}@fz-juelich.deWe present first principles calculations for prospective magnetic materials for future applica-tions in spintronics. Spintronics combine the charge and spin degree of freedom of the electronsmaking it possible to create novel devices with increased functionality compared to semicon-ductor devices used today. The material studied is focused on dilute magnetic semiconductors(DMS) like Ga1−xMnxAs that play a key role in semiconductor spintronics. Due to their fer-romagnetic properties they can be used in magnetic sensors and as spin injectors. The basicproblem for applications are, however, the relatively low Curie temperatures of these systems.We therefore focus on understanding the magnetic properties and on a reliable calculation ofCurie temperatures from first principles. We have developed a theoretical framework for cal-culating critical temperatures by combining first principles calculations and statistical methodslike Monte Carlo simulations.1 IntroductionMagnetic materials have been known to mankind for a very long time. Perhaps the mostfamous example is the compass needle but today magnets are used everywhere, as perma-nent magnets in electrical motors in our cars, in harddrives on our computers and mp3-players, in loadspeakers, credit cards etc. Meanwhile, semiconductor based devices thatuse the charge of the individual electrons as the information carrier, have extensively beenused in the last century in integrated devices like microprocessors, memory modules anddiodes used in computers, amplifiers, cell phones etc. The performance of these deviceshas increased by several orders of magnitude over the last decades (according to Moore’slaw, the performance on transistors double every 18 mounts). This performance gain isdue to, among other things, development in manufacturing and fabrication which makesit possible to shrink the dimension of the transistors and pack them more densely in orderto make a faster device which simultaneously runs cooler and more efficiently. However,nowadays this route to improve performance seems to face severe difficulties, because thedimensions are becoming so tiny that classical physics no longer is applicable and quan-tum effects start to become important, like tunneling through the gate oxide layer causingmalfunction of the transistor. Of course, this has been known for a long time and so farthe industry has all the time found new clever ways to overcome the problem and makethe devices go faster. For instance, Intel replaced the gate oxide from silicon dioxide to arare earth based material (hafnium silicide) in their latest 45nm process, which improvesnot only the switching speed of the transistor but also reduces the power needed. How-ever, all this development will eventually come to an end where it is simply not possible toshrink the dimensions further. It is here spintronics enter. The basic idea is to make deviceswhere quantum mechanics is actually employed and not fought against. The electrons notonly carry electrical charge but also spin, a purely (relativistic) quantum effect that can notbe explained by classical physics. The spin is then manipulated in the spintronic device,153instead of the charge, schematically shown in Figure 1.Figure 1. Schematic overview of spintronics which combines both charge (electronics) and spin (magnetism)into a novel field of research and applications.The most successful application of spintronics to date is without any doubt the gi-ant magneto resistance (GMR) read head in harddrives, an effect that was discovered byP. Gru¨nberg and A. Fert who received the 2007 Nobel Prize in Physics. In very simpleterms, the effect arises in a multilayer that consists of alternative magnetic and nonmag-netic materials in which the electrical resistivity depends on the magnetic configuration ofthe magnetic layers.Dilute magnetic semiconductors like Ga1−xMnxAs are considered as ideal materialsfor spintronics. Only a few percent of Mn impurities are needed to achieve ferromag-netism. Thus the hope is that they can substitute the metallic ferromagnets to achieveall-semiconductor spintronics. Moreover they are half-metallic exhibiting a 100% spin po-larization at the Fermi level EF , being ideal for spin dependent devices. Their biggestdisadvantage is that the reported Curie temperatures are well below room temperature.Here we will present some results of ab initio calculations, showing the complexity ofthese systems.2 Computational MethodsThe main goal of this study is to calculate critical temperatures Tc of several DMS sys-tems. For applications the Tc needs to exceed room temperature by some margin sinceit determines in a way the upper working temperature of the device. To calculate Tc forrealistic systems using only quantum mechanics methods is a formidable task which hasnot yet been solved. We have therefore used an alternative approach where we have splitthe problem into two parts, here called a two-step approach. In the first step, we applyquantum mechanical calculations using density functional theory to calculate the total en-ergy of the system. These calculations do not have any adjustable parameters nor need154input from experiments. The total energy is then mapped to a simplified solvable model,in this case a classical Heisenberg model, which we solve in the second step by applyingstatistical methods to estimate Tc. Although simplified, this is by no means simple and inorder to obtain reliable results, large scale calculations are called for.0 64128 256 512 1024 204801234567x 10−3No of processorsPerformance t CPU−1 (s−1 )Parallel performance of VASP on the IBM Blue Gene/L and IBM p690Ideal BG/LIBM BG/LIdeal p690IBM p690Eff. 52 %Eff. 88 %Eff. 77 %Eff. 55 %Eff. 35 %Figure 2. Parallel performance of VASP on the IBM p690 cluster (JUMP) and the IBM Blue Gene/L (JUBL) atthe Research Centre Ju¨lich.The quantum mechanical calculations are based on density functional theory (DFT)in combination with the local spin density approximation (LSDA). These calculations areable to predict many physical quantities of real materials using only the lattice geometryand atomic numbers as input parameters. In practice, many different implementations existbased on various degrees of sophistication. We use the program VASP (Vienna Ab initioSimulation Package) of J. Hafner and G. Kresse1–3 for our studies. VASP is a completepackage for performing ab initio molecular dynamics (MD) simulations using either pseu-dopotentials or the projector-augmented wave method (PAW) and a plane wave basis set.The VASP code is spin polarized and fully relativistic and can therefore treat most systemsin the periodic table. The program is written in Fortran90 and is massively parallelizedusing the MPI library. The parallelization is done over the bands and/or over plane wavecoefficients to improve scaling. Moreover, the program relies a lot on Fast Fourier Trans-forms (FFT) which is also done in parallel. From a code optimizing point of view, thecode steps through time, and performs a Fast Fourier Transform (FFT) and a matrix diago-nalization using iterative routines (RMM-DIIS or blocked Davidson) every time step. Thediagonalization is optionally performed using SCALAPACK routines. The VASP programhas been ported to both the IBM p690 cluster (JUMP) and to the IBM Blue Gene/L (JUBL)at the Research Centre Ju¨lich.In Figure 2 we show a scaling plot of the parallel performance of VASP on the JUMPand the JUBL computers. The total number of atoms in the unit cell was 250, which is atypical number for a large-sized problem. The program on JUMP does scale acceptable upto 256 processors with the efficiency above 0.5. Due to the faster interconnect on the BlueGene/L computer, the program scales much better than on JUMP and show acceptable155scaling of more than 50% up to 1024 processors. Given that each individual processoron JUBL is slower than on JUMP, one needs approximately twice as many processorson JUBL compared to JUMP for the same performance but this is a minor issue due tothe larger resources and the relatively good scaling on JUBL. A profiling of the parallelperformance on the JUBL reveals that the matrix diagonalizing routines which use theparallel SCALAPACK library do not scale well above 512 processors which hinders a goodoverall scaling (above 512 procs.). However, it should be said that the most time consumingroutine in VASP, which is the iterative matrix diagonalizing routines using the RMM-DIISalgorithm, scales perfectly linearly up to 2048 processors. The memory requirement ofVASP is by all means quite large, which has the implication that one usually needs to runthe program in coprocessor mode on JUBL. However, the memory requirement per nodedecreases with increasing number of processors due to data distribution over the nodes.Moreover, with the new Blue Gene/P computer this should be a minor issue due to thelarger available memory on that computer.Figure 3. Speedup of parallel MC on the IBM p690 cluster (JUMP) at the Research Centre Ju¨lich.Once the classical Heisenberg model has been constructed from our first principlescalculations, we apply statistical methods to obtain finite temperature properties from thecalculated exchange coupling constants. We are using Monte Carlo (MC) simulations forthis. MC gives a numerical solution in which both the positional disorder and the spinfluctuations are treated exactly. The simulations are performed on a fixed lattice using theMetropolis or the heat bath algorithm. The fixed lattice allows for fine grained paralleliza-tion using domain decomposition of the lattice using MPI which in turn allows for studieson very large simulation cells. Technical details are found in Ref.4. Magnetic exchangeinteractions as far out as 25 shells of neighbours from a central atom can be treated, whichis crucial for real metallic systems where the interactions typically are long ranged as wellas for diluted systems.In Figure 3, scaling of the MC program as a function of number of processors on theJUMP computer are displayed. As seen from the figure, excellent scaling, in fact super-156linear scaling for some sizes, are obtained up to 125 processors. The program has a verysmall memory requirement (a few megabytes) and also run very well on the Blue Gene/L.3 Diluted Magnetic Semiconductors (DMS)A diluted magnetic semiconductor (DMS) can be realized by alloying of magnetic ele-ments in a regular semiconductor like GaAs. Some of the Ga cations will then be replacedby magnetic atoms like Mn which carry a local magnetic moment. Moreover, if the va-lence of the substituted atom is different from the cation, holes are introduced which canmediate (ferro) magnetic interactions between the Mn atoms. An important consequenceof this substitution is that the magnetic atoms are diluted and have a random distribution.DMS like Ga1−xMnxAs are one of the most hopeful materials for an all semiconductorspintronics. An enormous amount of papers and strong progress has been produced in thisfield, but the bottleneck for applications is still the availability of DMS with Curie tem-perature Tc above room temperatures. Thus understanding, predicting and realizing DMSwith Tc larger than room temperature is one of the most important problems in spintronics.The highest conclusively reported Tc of DMS is around 170 K for 8% Mn doped GaAs,which is too low for applications.0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09050100150200250300350xT c (K)MFA−VCAMCExpFigure 4. Calculated Curie temperatures of Mn-doped GaAs as a function of Mn concentration x using the meanfield approximation (MFA-VCA) and Monte Carlo simulations (MC). The exchange coupling parameters arecalculated from LDA.Over the years, more and more knowledge has been gathered on DMS systems. For ex-ample, it has been demonstrated that electron doping strongly reduces the ferromagnetismleading to a disordered local moment state (DLM)5. The dominating exchange mechanismleading to ferromagnetic order, i.e. Zeners’s p−d-exchange and Zener’s double exchange,have been clarified by ab initio calculations and are now well understood6, 7. Moreover,it has been demonstrated that percolation and disorder effects play a crucial role in thesesystem by strongly reducing the ferromagnetism8–10. As an example, in Figure 4 calcu-lated Curie temperature of Mn-doped GaAs is displayed as a function of concentration ofsubstituted Mn-atoms employing the LDA. The most accurate method, namely MC sim-ulations, yields Tc values that due to percolation problems are considerably lower than157previous mean field estimates (MFA-VCA), e.g. by Dietl et. al., so that percolation seemsto present a general obstacle for high Tc in DMS. This is particularly true for DMS like(Ga,Mn)N or (Zn,Cr)Te, in which due to the wide band gap the interaction is very shortranged. However, even for (Ga,Mn)As, where the interaction is fairly long ranged, this isa significant effect (see Figure 6).0 50 100 150050010001500Na)x=0.03x=0.05x=0.080 50 100 150 200 2500200040006000Nb)x=0.03x=0.05x=0.080 20 40 60 80 100 120 140 160 180 200010002000heff (meV)Nc)x=0.05x=0.10x=0.20Figure 5. Distributions of local magnetic fields for a) Mn-doped GaAs, b) Mn-doped GaN and c) Cr-doped ZnTe.x denotes the concentration of magnetic impurities.In order to get a more complete picture why the mean field approximation togetherwith the virtual approximation (MFA-VCA) sometimes completely fails, but at other timesgives reasonable results we have performed an analysis of the local effective magneticfields heff in DMS materials. To be more specific, the heff is defined asheffi =∑jJ0j〈Szj 〉, (1)for each magnetic site i and at T = 0K , 〈Szj 〉 = 1. In a non-random system, each sitei has the same value of heff . However, in contrast, in random system like DMS each sitehas a different local environment and we will instead obtain a distribution of local effectivemagnetic fields as displayed in Figure 5 for Mn-doped GaAs, Mn-doped GaN and Cr-dopedZnTe. The distribution is obtained for huge systems with around 105 magnetic impuritiesto make sure that basically all local environments are included. If a deeper analysis isperformed, one can conclude that the MFA-VCA method is reasonable if the distributionis close to a Gaussian. If not, then the MFA-VCA method will fail spectacularly.158The distribution for Mn-doped GaAs (Figure 5a) does not differ dramatically from aGaussian, especially for the higher concentrations. Consequently, of the three differentDMS systems considered here it is the system where the MFA-VCA estimate is closest tothe exact MC value (approximately a factor of 2 too large). The situation in GaN is verydifferent (Figure 5b). The distribution is very far from a Gaussian but instead it consistsof several peaks (arising from the dominant interactions in the 110-direction). It is clearfrom this distribution that the MFA-VCA estimate will be very wrong and this is indeedthe case (the exact MC values are about one order of magnitude lower). An intermediatecase is Cr-doped ZnTe, in which the distribution shows several smaller peaks. On the otherhand, the concentration of magnetic atoms is larger than in the two cases above making theMFA-VCA estimate slightly improved.Figure 6. Exchange interactions in 5% Mn-doped GaAs using cluster embedding in CPA medium as a functionof distance. The inset shows the distribution of nearest neighbour interaction.Another important question is: How well can the exchange coupling constants Jijbetween two impurities at sites i and j be described by a unique value Jij independent ofthe local environments, i.e. the location of other magnetic impurities in the vicinity of the(i, j) pair? In the standard approach by Lichtenstein this effect is thought to be included byusing averaged Green’s functions based on the coherent potential approximation (CPA).We have investigated his problem in detail by calculating Jij for a series of disorderedlocal environments (clusters) embedded in the CPA medium. The results presented inFigure 6 show large fluctuations of the Jij data, depending strongly of the positions ofthird and fourth Mn impurities in the neighbourhood of i and j. The inset shows thefrequency distribution of nearest neighbour values J01. The configurational average isin disagreement with the Lichtenstein’s results based on the CPA, at least for the nearest159neighbour interactions. However, it agrees well with the exact results for two impuritiesi and j embedded in the CPA medium, showing that the disorder is well described by theCPA, provided that the 2-impurity problem is solved correctly.4 Concluding RemarksWe have presented a theoretical investigation of prospective materials for future appli-cations in spintronics, namely diluted magnetic semiconductors, using a combination ofquantum mechanical calculations and statistical methods. Unfortunately, the critical tem-peratures are still too low for practical use.AcknowledgmentsL.B. acknowledges support from the EU within the framework of Marie Curie Actions forHuman Resources and Mobility.References1. G. Kresse and J. Furthmu¨ller, Phys. Rev. B 54, 11169 (1996).2. G. Kresse and J. Furthmu¨ller, Comput. Mat. Sci. 6, 15-50 (1996).3. http://cms.mpi.univie.ac.at/vasp/.4. L. Bergqvist, Ph.D. Thesis, Department of Physics, Uppsala University (2005).5. 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).6. K. Sato, P. H. Dederichs and H. Katayama-Yoshida, Europhys. Lett. 65, 403 (2003).7. K. Sato, P. H. Dederichs, H. Katayama-Yoshida and J. Kudrnovsky´,J. Phys.:Cond. Matter 16, S5491 (2004).8. L. Bergqvist, O. Eriksson, J. Kudrnovsky´, V. Drchal, P. A. Korzhavyi and I. Turek,Phys. Rev. Lett. 93, 137202 (2004).9. L. Bergqvist, O. Eriksson, J. Kudrnovsky´, V. Drchal, A. Bergman and I. Turek,Phys. Rev. B 72, 195210 (2005).10. K. Sato, W. Schweika, P. H. Dederichs and H. Katayama-Yoshida, Phys. Rev. B 70,201202(R) (2004).160

Dilute Magnetic Semiconductors

http://juser.fz-juelich.de/record/152962/files/FZJ-2014-02627.pdf

Dilute Magnetic Semiconductors

Abstract

Similar works

Full text

Available Versions

Juelich Shared Electronic Resources