The magnetic behavior of diluted magnetic semiconductors
Citation for published version (APA): Swagten, H. J. M. (1990). The magnetic behavior of diluted magnetic semiconductors. Technische Universiteit Eindhoven. https://doi.org/10.6100/IR340473
DOI: 10.6100/IR340473
Document status and date: Published: 01/01/1990
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Download date: 09. Oct. 2021 THE MAGNETIC BEHAVlOR OF DILUTED MAGNETIC SEMICONDUCTORS
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H.J.M. Swagten THE MAGNETIC BEHAVlOR OF DILUTED MAGNETIC SEMICONDUCTORS
PROEFSCHRIFT
ter verkrijging van de graad van doctor aan de Technische Universiteit Eindhoven, op gezag van de Rector Magnlflcus, prof. Ir. M. Tels, voor een commissie aangewezen door het College van Oekanen in het openbaar te verdedigen op dinsdag 20 november 1990 te 16.00 uur
door
HENRICUS JOHANNES MARIA SWAGTEN geboren te Roermond Dit proefschrift is goedgekeurd door de promotor: prof. dr. ir. W.J.M. de Jonge.
The work described In this thesis was part of the research program of the Soliel State Divlslon, group Cooperative Phenomena, of the Department of Physics at the Eindhoven Univarsity of Technology, and was supported financlally by the Foundation for Fundamental Research on Matter (FOM). TABLE OF CONTENTS
CHAPTERI. INTRODUCTION 1
CHAPTER ll. MAGNETIC BEHAVlOR OF 11-VI GROUP 23 DILUTED MAGNETIC SEMICONDUCTORS, BASED ON Mn
1. Magnetic beha.vior of the diluted magnetic semiconductor Znt-x:MnxSe, p. 24 2. Ma.gnetic properties of Znt-xMnx:S, p. 39
CHAPTER m. MAGNETIC BEHA VlOR OF 11-VI GROUP DILUTED MAGNETIC SEMICONDUCTORS, BASED ON Fe
1. Magnetic properties of the diluted magnetic semiconductor Znt-:xFexSe, p. 46 2. Low-temperature specific heat of the diluted magnetic semiconductor H&t-x:-yCdyFexSe, p. 60 3. Ma.gnetic susceptibility of iron-based semima.gnetic semiconductors: high-temperature regime, p. 74 4. Ma.gnetiza.tion steps in iron-based diluted ma.gnetic semiconductors, p. 83
CHAPTER IV. CARRIER-CONCENTRATION-DEPENDENT 89 MAGNETIC BEHA VlOR IN IV-VI GROUP DILUTED MAGNETIC SEMICONDUCTORS
1. Preparatien a.nd carrier concentra.tions of Pbt-x-ySnyMn:x Te and Snt-:xMnxTe, p. 90 2. Hole density and composition dependenee of ferroma.gnetic ordering in Pbt-x-ySnyMnxTe, p. 96 3. Carrier-concentration dependenee of the ma.gnetic interactions in Snt-xMn:xTe, p. 101
SAMENVATTING 105 LIST OF PUBLICATIONS 106 CHAPTER ONE
Introduetion
In this chapter the diluted ma.gnetic semiconductors (DMS) are introduced. First, the general physical properties, such as the composi tion range and band structure are briefly reviewed. Subsequently, some attention is paid to the 6p-d exchange, which is the background of all the phenomena specific for DMS. The major part of this chapter, however, deals with the magnetic properties of DMS, and serves as an introduetion to the other parts of this thesis. With respect to the Mn-containing DMS we will focus on the d-d exchange (J) between Mn ions. In particular, the strength of J, as well as the relevanee of long-range interactions, are emphasized. The possible origin of these intera.ctions is analyzed on the basis of existing exchange models. Next, a new family of DMS, containing Fe instead of Mn, is introduced. In Fe-based materials new physical effects are ob served, a.rising from Fe levels located above the valenee bands, which are not populated in the Mn case. More importantly, we wil! discuss the possible influence of this on the magnetic properties. Finally, we will introduce the IV-VI group DMS. In comparison with II-VI and II-V materials, the carrier concentration in these systems is very high, lead ing, in some cases, to ferroma.gnetic RKKY interactions. Moreover, it has been shown that the ferromagnetic interactions in Pbt-x-ySnyMnxTe and Snt-xMnxTe strongly depend on the carrier density. This phenom enon, the so-called ca.rrier-induced ferromagnetism, is introduced, and also some recent developments are reported. Chapter I INTRODUCTION 2
I. GENERAL state (paramagnetic, spin-glass, antiferro magnetic, ferromagnetic). Diluted magnetic semiconductors (DMS) The flexibility with which the lattice - or semimagnetic semiconductors (SMSC) constants and band gaps can be varied within - are ternary or quaternary semiconductors the wide range of magnetie-ion concentration whose cations are partially replaced by a con in, mostly, Mn2+-containing DMS, as well as trolled amount of divalent transition-metal the unique tunability of the electronk levels ions and recently, also rare--earth ions. A with magnetic field and temperature, make canonical example of such system is Cd 1_x DMS materials good candidates for applica Mnx Te, belonging to the most extensively tions in, for example, field-tunable far-infra studied II-VI group alloys. Most of the effort red sensors and lasers, and optica! devices has been devoted to the substitution of Mn2+, based on the large Faraday rotation. Finally, partially because physical properties are not one of the most spectacular developments complicated by nonvanishing orbital momen concerns the MBE growth of heterostruc ta (S = 5/2, L = 0) but, more importantly, tures, superlattices and thin films based on because of the technological ease of Mn2+ DMS. Besides the promising preparation of implantation in the semiconducting host (in new, hitherto nonexisting structures such as 2 striking contrast withother ions such as Fe + Ga1_xMnxAs and MnTe it is with MBE poss and Co2+). ibie to bring the physics of DMS, and es The scientific interest for DMS materials pecially the characteristics known as band has been triggered by the observation of a gap engineering, into the rich and exciting variety of exciting effects, in the field of semi phenomena of the Quanturn W ell. conductor physics, as well as on physics of This first part of the thesis presents a magnetism of solids. As for the semiconduc physically oriented introduetion to the pres ting effects, they all stem from the interac ent status in this new class of materials. As tion between d electrans (the spins) and the for the structural and electronk properties of s- and p-like orbitals of the band electrons, DMS, we will, however, confine ourselves to the so-called sp-d exchange (Jsp-d)· This a rather brief overview, since a review by interaction amplifies so to say the Zeeman J.K. Furdyna1 and a recent issue of Metals splitting of electronk levels, and, consequent and Semiconductors 2, as well as more dated ly, interesting effects are observed, such as a reviews 3, are providing excellent insights in giant Faraday rotation, an extremely large the physical properties of DMS. In these re negative magneto-resistance, and field-de cent reviews 1-2, the magnetism of DMS, pendent metal-insulator transitions. which is the content of this thesis, is focused The magnetic properties of DMS are merely on II-VI group materials containing intimately related to the indirect spin-spin the conventionally Mn2+. A braader introdue interaction between the random distributed tion to this subject will therefore be pre magnetic ions (Jd-d)· The interaction Jd-d sented here, emphasizing current topics in originates from indirect electron processes this field: the relevanee of long-range interac between two localized moments and their tions, the substitution of Fe2+ instead of intermediating band electrans and as a result, Mn2+, and the observation of carrier-concen Jd-d is strongly coupled to Jsp-d· The experi tration-dependent magnetic interactions in mental contributions to unravel these under IV-VI group materials. Furthermore, this lying physical mechanisms are mainly dealing part will offer the reader an introductory with the strength, character and radial de guideline to the forthcoming chapters of the pendence of Jd-d> and the magnetic phase thesis, in which these subjects will be treated diagram in which temperature and concentra in detail. tion of magnetic ions trigger the magnetic Chapter I INTRODUCTION 3
IT. CRYSTAL STRUCTURE, COMPOSITION AND BAND STRUCTURE Telluride DM S 4.2 I<
In Ta.ble I, see page 4, the crysta.l struc > ture and the composition range of all known .. Mn- and Fe-containing DMS are ta.bulated. a. In most cases DMS crystals of high quality < and large dimensions can he synthesized 2 by (!) > using the Bridgman (-Stockbarger) method (!) a: or, toa lesser extent, chemica! transport. The liJ z incorporation of Fe, and very recently also lil Co in the semiconductors is not so straight forward as in the Mn case, which, tentative 1 ly, has been ascribed to the resemblance -~~~~~~~~~~~~ 0.0 0.2 0.4 0.6 0.8 1.0 Mn CONCENTRATION x
FIG. 2. Energy gap Eg vs. Mn contentration z for telluride DMS at low temperatures. A Iinear extrapola.tion of Eg to z 1 gives an energy ga.p of o t..J uz between the electronic configuration of Mn ~ (f) and the cations (both half or completely 0 filled in contrast to Fe or Co). z The availa.ble structural investigations 2 1- for the systems in Table I reveal a. Vegard <( u type linea.r dependenee of the la.ttice constant z' with the concentration of magnetic ions. This 0 linear behavior has been used for the deter u~ 4.0 mination of the Mn concentration, besides z other more direct methods like electron probe w<( micro ana.lysis (EPMA). For II-VI com ::::1! pounds on the basis of Mn2+ the la.ttice para meters have been established by Furdyna and co-workers14 , and are shown in Fig. 1. The Vegard-type departure of the lattice para 3 7 · 0.0 0.2 0.4 0.8 1.0 meter from that of the semiconducting host MOLE FRACTION X is, for all substances, clearly illustrated. One should, however, realize that standard x-ray diffraction only probes the average bond FIG. 1. Mean cation-cation dista.nces d as a lengths and not the microscopie features of function of the Mn mole fraction z for At-xMnxB the structure. Very recently1•15, extended a.lloys originating from the II-VI group. The lattice x-ray absorption fine structures (EXAFS) parameters can be obta.ined from d a.s follows, zinc studies on Zn 1_xMnxSe convincingly demon blende: a= t1/2, wurtzite: a = d and c = (8/3)112d; strated that on microscopie scale strong dis ana.lytic expressions corresponding to the figure are tortions from the zinc blende phase are vis given in Ref. 1 and Ref. 14. ible. Moreover, both the Zn-Se and Mn-Se C':i b'" I» "CC... TABLE I. Crystal structure and composition range for Mn- and Fe-based ~ diluted magnetic semiconductors. .... Material Type I Ref. Crystal structure Composition range Zn 1_xMnxS II-VI zinc blende 0 Zn1_xFexS II-VI zinc blende 0 bond length remain constant over the entire the anomalous field-dependent physical be concentration regime, which is of great im havior of DMS, involves the band structure porta.nce for the Mn2•-Mn2+ exchange con as well as the local magnetic ion 3d5 levels, stauts in DMS. We will return to this in which are superimposed on the band struc Sec. IV on d-d exchange. ture. Fig. 3 shows a schematic mustration of As we quoted before, one of the attract these levels in an idealistic zinc blende band ive features in DMS is the tunability of band structure, which is a representative example parameters upon substitution. The most well for II-VI DMS having the direct gap at the r known example for this is the proportionality point. The majority (spin up) and minority of the energy gap with composition, which for (spin down) Mn levels are in reality narrow almost all DMS has been established. For bands due to mixing or hybridization with s Te-containing II-VI compounds the tunabil and p orbitals of the semiconductor. Accord ity of the gap by composition is plotted in ing to Larson et al. 16 , the location and degree Fig. 2, illustrating the "opening" of the gap of hybridization of these levels are extremely in Hg 1.xMnx Te. important for the strength of the exchange The sp-d exchange, which as we pointed integrals in DMS, which we will discuss in out before, is the driving mechanism bebind some detail further on. Ueff in Fig. 3 is the energy necessary to add one electron to a Mn2+ (3d5 -+ 3d6) and amounts to:::: 3.4 eV in Cd1.xMnx Te, and it should be distinguished E from intra-ion transitions involving spin-flip processes. For example, the flipping of one spin (ifTrT to nn !) requires approximately 2.2 eV and is the dominating mechanism in opt ical processes t. m. sp-d EXCHANGE All phenomena in DMS arise from the sp-d exchange between the spins ( d elec trons) and the band electrans (s or p elec trous ). For the description of the effect of the interaction on the electronic system one ex ploits the extension of the electron wave functions - the mobile electron interacts simultaneously with a large number of spins - in order to transform the Heisenberg-ex change Hamiltonian as follows ------~------K (1) FIG. 3. Schematic band structure for a. DMS with the direct gap at the r point. The Mn levels ed j (occupied) a.nd ed! (unoccupied) are split by the energy Ueff (:::: 7 eV). The effect of p~d hybri where u and S are the spin operators for band diza.tion a.nd crysta.l field on the Mn levels is ne electron and 3d5 electron, respectively, Jsp-d glected in this schematic representation, since the is the electron-ion exchange constant, and r effect is smal! on the scale of the figure. and Ri the coordinates of band electron and Chapter I INTRODUCTION 6 Mn2+, respectively. Two approximations are g-factors in DMS can become extremely ma.de in this transforma.tion, (i) the spin op large, in some cases exceeding 100, which is erator S has been replaced by its therma.l unique for semiconductors. Some examples of average in the z direction ported in this thesis. First of all, we will con In a DMS, the magnetic behavior sider the Mn-eontaining DMS originating should, obviously, be interpreted as arising from the II-VI group. In these compounds from the interaction between Mn2+. Original antiferromagnetic (AF) long-range interac ly, so-called cluster models have been de tions between Mn~H ions have been observed veloped to describe the behavior. In these in all existing compounds, irrespective of the models the interaction is essentially re wide variety of intrinsic properties, such as stricted to Mn2+ ions at neighboring lattice band gap, lattice constant, covalency and sites. However, on the basis of the cluster carrier concentration. Apart from the charac models it was not possible to describe all the ter and strength of the interaction we will available data without the questionable need pay much attention to the spherical exten to adjust the random distribution of the Mn sion of the interaction, and the relation with ions. These inconsistencies could only be clar the underlying physical mechanism. Subse ified by allowing a long-range tail for the d-d quently, we will introduce a relatively new interactions. The observation of spin-glass class of DMS materials on the basis of Fe. freezing for Mn concentrations well below the Through the 3d6 configuration of Fe2+ the percolatien limit for nearest-neighbor (NN) magnetism of these alloys is different from interaction (~ 0.195 for an fee lattice) ulti Mn-type DMS, and, moreover, in the context mately proved the existence of an interaction of exchange mechanisms, the Fe2+ can alter extending over many lattice sites. the phenomena we observed in the Mn case. Besides the complication of the spherical Finally, we will present an introduetion to extension of Jd-d• it is in the random diluted the IV-VI compounds containing Mn2+. The system rather difficult to extract the interac- carrier concentration in these systems is so high that the so-called RKKY interaction (i.e., a spin polarization of the carriers) can become effective and induces ferromagnetic 30 _...;.11:.:..:..11_H interactions. Moreover, it will be shown that in some of these compounds the ferromagnet ic behavior of the system can be switched on or off, just by variation of the carrier concen (9) tration, the so-called carrier-induced ferro 20 magnetism. ::::! >. 2 2 -0"1 [Mn +- Mn +[ A. Mn-containing DMS r... "'c: (71 j{.:: -2J "' 12 I s..s:I J The research on the magnetism of DMS is also almast exclusively restricted to the Mn case. Apart from the relative ease to ob 6 !SI tain high-quality crystals with a wide range of Mn concentrations, the 3d5 electrans of 2 !31 Mn2+ add up to a simple, theoretically at 0 6 (11 tractive spin-only S512 ground state. Though H crystal-field effects might affect this state by mixing with higher levels, it is generally af firmed [e.g. by electron paramagnetic reson ance (EPR) of Mn2+ in semiconductors2] that FIG. 4. Energy levels for a pair of Mn2+ ions these effects are very small leaving the with an antiferromagnetic coupling J and S = 5/2. ground state six-fold degenerate with S = For ea.ch level the degenera.cy is indicated between 5/2 and L = 0. brackets. Chapter I INTRODUCTION 8 tion from experimental data, since a variety of cluster types (pairs, triples, quartets, ... ) are present on the basis of random statistics, each exhibiting its own energetica! structure. By allowing a long-range Jd-d even every spin is coupled with any other spin! Never 20 theless, for sufficiently small Mn concentra tions, typically below 10%, the number of 12 pairs become large enough to dominate the magnetic properties and to trace the charac 6 ter and strength of the Mn-Mn exchange. 2 lml The energy-level scheme for such a Mn-Mn o~o+~~~~~------o pair, coupled antiferromagnetically, is shown in Fig. 4. Fortunately, and due to the spin only state of Mn2+, the exchange-induced splitting directly correspond to the coupling constant IJl. Accurate determination of the interaction strength between adjacent Mn2+ ions, the so-called nearest-neighbor interac tion Jn, can be obtained from prohing the M gap between the singlet ground state and the first excited state (21 Jl, see Fig. 4), e.g., by inelastic neutron diffraction 19, Raman scat tering2 and, somewhat less directly, by the high-field magnetization20, see Fig. 5. Other techniques, such as the high-temperature susceptibility, specific heat and EPR, extract Jn indirectly in the sense that other clusters FIG. 5. (a.) Energy level diagram, in units ( triples, quartets and long-range coupled I JNN I, for a pa.ir of NN Mn2+ ions at B = 0. (b} ions) interfere with the phenomena character Zeeman splitting of the energy levels in a. ma.gnetic istic for JNN, and serious errors are to be ex field B; note the level crossings at Bn. (c) Illustra pected. In Table II all investigations on Jn tion of the ma.gnetiza.tion curve at low tempera.tures, are gathered. showing the a.ppa.rent sa.turation (M = M5), foliowed We already argued that it is by now by the ma.gnetiza.tion steps due to pairs. The ma.g commonly accepted that, in general, the ex netiza.tion steps due to triples or other clusters, as change integral is not restricted to Mn ions wel! as long-ra.nged coupled ions a.re not shown at neighboring cation sites only, but extends (after Ref. 20). also to more distant neighbors. From a sealing analysis of the concentration depend enee of the spin-glass freezing temperatures, To explain the magnitudes of J and the it was possible to derive how Jd-d decays observed tendendes - such as the increasing with distance between the Mn ions. It was IJn I on going from Te via Se to S and from pointed out that the interaction beyond the Cd to Zn - we should consider theoretica! nearest neighbors (NN) is rapidly decreasing exchange models. Basically, forthespin-spin in strength, and, from the experimental point interaction in semiconductors three mechan of view, hard to extract quantitatively. So isms are available, i.e., superexchange59, far, reliable data, though very scarce, could Bloembergen-Rowland (BR)&O and Ruder only be obtained on JNNN• the next-nearest mann-Kasuya-Kittel-Yosida (RKKY)&t. All neighbor exchange constant, see Table II. mechanisms can be expressed in terms of vir- Chapter I INTRODUCTION 9 TABLE II. Nearest--neighbor a.nd next-nearest-neighbor exchange constants (JNN• JNNN) for Mn- and Fe-based diluted magnetic semiconductors. Material -JN!I (K) Techn. Ref. -JNNM (K) Techn. Ref. Zn 1_xMnxS ~10 16.9±0.6 MAGN 21,22 0.6 MAGN 22 16.1±0.2 NEUT 23 4.8±0.1 susc 24 13.0±1.5 EPR 25 ~0.7 Tr(x:) 26 Zn 1_xMnxSe :::13 9.9±0.9 MAGN 21,27 2.4 3±2.4 susc 29,30 d2.6 12.2±0.3 28,20 ~0.7 Tr(x:) 31 12.3±0.2 NEUT 23 18 13.5±0.95 susc 21,29 Zn 1_xMnxTe 10.0±0.8 10.1±0.4 MAGN 32,33 0.6 MAGN 35 9.25±0.3 8.8±0.1 34,27 4.6±0.9 3.6±2 susc 30 9.0±0.2 20 9.52±0.05 7.9±0.2 NEUT 36,31 12 11.85±0.25 susc 32,37 Cd1_xMnxS >4 8.6±0.9 MAGN 21,38 5.2±0.3 MAGN 39 10.5±0.3 10.6±0.2 28,39 9.65±0.2 11.0±0.2 22 Cd1.xMnxSe 8.3±0. 7 7.9±0.5 MAGN 21,40 1.6±1.5 MAGN 40 8.1±0.2 39 4.6±2 5.4±1.5 susc 30 9 10.6:t:0.2 susc 21,37 7.7±0.3 RAM 41 Cdt-xMnxTe :::10 6.3±0.3 MAGN 21,40 1.9:t:l.l 1.1±0.2 MAGN 40,42 6.1±0.3 6.2±0.2 28,20 0.67 NEUT 43 ~7.5 6.7 NEUT 44,43 1.2±1 susc 30 6.9±0.15 susc 37 0.9 LARS 16 7.7±0.3 EXCI 46 0.55±0.05 SPEC 46 6.1±0.2 RAM 41 8 LARS 16 Hgt-xMnxSe 6±0.5 5.3±0.5 MAGN 47,35 10.9±0.7 susc 37 0.1 EPR 48 Hg1_xMn,.Te 5.1±0.5 4.3±0.5 MAGN 47,35 1 susc 49 15 15.7 7.15±0.25 susc 50,49,37 0.7±0.3 SPEC 51 (Znt-xMnxhAs2 l:llOO Tr(x) 5,51 ;::2 Tr(x) 5,51 (Cd1.xMnxhAs2 :::30 Tr(x) 5,52 ~5 Tr(x) 5,52 Pb1_xMnxS 0.537 MAGN* 53 0.0537 MAGN 53 1.28 susc 54 Pb1.xMnxSe ~1 MAGN* 55 1.67 susc 54 Pb1_xMnxTe d MAGN* 55 0.84 susc 54 Znt-xFexSe 22±2 susc 56 Cd1_xFexSe 11.25±1.5 18.8±2 susc 57,56 Hgt-xFexSe 15±1 18±2 susc 57,58 EPR electron paramagnetic resonance; EXCI = high-field exciton splitting; LARS Larson's superexchange model; MAGN = steps in the high-field magnetization; MAGN* = high-field magnetization, fitted with pair approximations; NEUT = inelastic neutron scattering; RAM = Raman scattering; SPEC = low-temperature magnetic specific heat; SUSC high-temperature susceptibility; Tr(x) spin-glass freezing temperatures, combined with thermadynamie properties. Chapter I INTRODUCTION 10 tua.l electron transitions ( described by ex (Ed), see Fig. 3, together with the hybridiz change andfor hybridization) between band ation parameter Vdp' completely determine electron or hole, and the d5 electrons. In Fig Jn: ure 6 it is schematica.lly illustrated that in superexchange 2-electron ( or hole) processas Jn = -8.8 Vdp 4[(Ed+ UerrEv)"2Uef(1 (3) take place, the so-called interband transi 3 tions. In BR on the other hand, only intra + (Ed+UerrEv)" ] band transitions are allowed, and finally, RKKY accounts for an interband process at Since in this 3-level model Jn is determined the Fermi level. As a consequence, we may by hybridization of d levels with the valenee state, roughly speaking, that the only prin band states only, one would expect that {J, ciple difference between them is that super the exchange parameter for the valenee exchange applias for open-gap materials (iso bands, exhibits the same chemica! trends as lators, semiconductors ), BR for smali-gap Jn. This is indeed corroborated by experi materials (semiconductors) and RKKY for mental data, although one should realize that partially fitled bands (metals, semimetals). fJ is composed not only of the p-d hybridiz By a quantitative analysis1& Larson ation term but also of a much smaller term of et al. were able to demonstrate that in open opposite sign, representing the 1f r exchange gap DMS [(Zn,Cd)t.xMnx(S,Se,Te)] the ma which tend to align the d-electron spin with jor contribution comes from the antiferro the band-electron spin. The absence of hybri magnetic superexchange (> 95%), in which dization with conduction band states there only 2-hole processas in the valenee band are fore explains why a, the integral for the con allowed, see Fig. 6. Furthermore, they estab duction bands, is small compared to fJ and of lished the inferior role of the BR mechanism opposite sign. and the absence of RKKY. In addition, with The extension of the exchange beyond a simplified so-ca.lled three-level superex the NN in Larson's model is expressed16 by a change model they could predict the observed material-independent Gaussian form f(R), chemica! trends encountered in Table II. In yielding this model the valenee band energy Ev, the difierence between the minority and majority J(R)"' exp(-aR2) (4) Mn level ( Uerr) and the Mn majority level Material-independent means independent of electronic details, in our case applicable for the wide-gap Mn compounds. In Chapter II we will illustrate on the basis of an experi u mental study that for those systems, with ernphasis on Zn 1.xMnxSe and Zn 1.xMnxS, a long-range tail of J should be present. From an aualysis of the paramagnetic to spin-glass transition temperaturas a power-law decay of SE BR RKKY J(R) will be suggested, J(R)"' R·n, with n:::: 7. (5) FIG. 6. Schematic representation of the ex change processes superexchange (SE), Bloembergen It will be shown experimentally that, quite Rowland (BR) and Rudermann-Kittel-Kasuya remarkably \see the Discussion in Chapter Yosida (RKKY). The filled valenee bands, empty II.l), this R· decay is also material-insensi conduction bands, Mn spin states and tra.nsitions are tive for all wide-gap DMS. However, the shown (based on Ref. 62). power-law decay in Eq. 5 is, particularly at Chapter I INTRODUCTION 11 larger distances, essentially different from Larson's approach. Nevertheless, for the first NN f Larsen et aL few neighbors (NN, next NN, next-next NN) -[Jee exp(-4.89R2ta2J the difference is rather small, and, moreover, + de Jonge et al. by camparing Eq. 4 and 5 with existing data J cx1/ R7 on JNN, JNNN and JNNNN• see Fig. 7, strong evidence is found for the usefulness of both 10 the approximations in this regime. NNN More support for the extension of J be yond the NN will be given in Chapter II, + ~ where we will closely examine the therma c dynamie properties ( susceptibility, magnetiz ....0 u NNNN ation, specific heat) of DMS, exemplified for "'.... Zn _xMnx(Se,S). It will he convincingly dem .... 1 "'c onstrated that the overall behavior of the f properties cannot be understood on the basis of short-range interactions and therefore the long-range character of the interaction should be taken into account. The so-called o Cd(MnJTe extended nearest-neighbor pair approxima • Cd!MnJSe tion (ENNP A), developed by Denissen al. 5 et a Zn(MniS for long-range coupled random diluted spins, is capable to describe all the magnetic prop 0.1 • Zn(MnJTe erties sim uit aneously. Finally, there exists some very recent 1/{2 results on the d-d exchange which, in prin Ria ciple, should be included in our current understanding and rnadeling methods for DMS. First, it was pointed out on page 3 FIG. 7. Radial dependenee of the exchange in that strong distortions of the Mn2+ -anion - tera.ction IJl ba.sed on (i) La.rson 's three-level Mn2+ bond angles are observed by EXAFS, superexchange modeJ16 (Eg. 3), a.nd (ii) ba.sed on a. capable to influence to strength of the inter sealing a.nalysis of the spin-glass freezing tempera action by varying the Mn concentration. A ture TF(z) (Eq. 5) by de Jonge and co-workers thorough examination of e(x)37, this is the (Ref. 5,31,63), compared with experimenta.l data concentratien dependenee of the Curie-Weiss (Ref. 20,22,40 and Ta.ble II). constant prohing interaction strengths, and also neutron diffraction studies prohing JNN 19, seem to substantiate these tendencies. In the for a Dzyaloshinsky-Moriya anisotropic ex latter case, even a linear variation of JNN change with concentratien has been suggested. Des pite all this, the calculations for DMS sys J{DM - L D(Rij)• SixSj , (6) tems are commonly confined to low concen i:j:j trations, typically below 10%, where no dras tic effects can be expected. IDNN !Ju I being roughly between w-a and Recently, some investigations were deal w-1 depending on the specific system. ing with the the anisotropic part of the ex Though relatively small compared with the change. Larson et al. t6 were able to relate the isotropie part of the interaction, this aniso observed tendendes of the EPR linewidth to tropy can become of vital significanee for the their superexchange model, yielding evidence magnetic properties at low temperatures Chapter I INTRODUCTION 12 where effects due to J11 can be safely ig nored. Moreover, the characteristic upturn of the magnetic specific heat at temperatures below 1 K (see Chapter II and Ref. 26,31) strongly suggest the existence of DM aniso tropy, which therefore should be included in future calculations. In this section we did not pay much at tention to small gap DMS materials such as ( Cdt-xMnx)3As2 and Hg1-xMnxSe. Existing • =0.0 ... 10-4 data on these compounds5,5t,s2,&3,64 pointed to a much slower decay of the interaction {between R-3 and R-5 instead of R-1), which, tenta.tively, indicates an increasing role of the FIG. 8. A schematic picture of the band struc BR mechanism and can bedescribed in terros ture of Hst-xFexSe near the r point for severa.l of a small or vanishing fundamental energy values of ~. The Fe2+ level is shown as a resonant gap. There is, however, actually no conclus donor. There are approximately 5·1018 cm-3 con ive theoretica! support to explain the ob duction band statea availa.ble below the Fe2+_level. served tendencies, neither for the strength of For ~ > 3·10-4, all these levels are filled by ioniz Jn (being very large for II-V DMS and not ation of Fe2+ ions to Fe3+ ions. This provides a. accurately known for small gap 11-Vl's, see na.turallimit for the Fermi level (after Ref. 1). Table 11) nor for the more extended tail of the interaction. In the Discussion of Chapter II.1 some additional comments will be put of 3d5), forming an occupied Fe level abóve forward to bring this subject into a broader the valenee band. For zero-gap Hg 1_xFexSe context. this implies the level is lying within the con duction band (±230 meV from the bottom10) B. Fe-based DMS providing conduction band states by ioniz ation of Fe2+ (Fe2+-+Fe3++e-). The concen As we quoted earlier, the majority of tration of these so-called resonant donors experimental studies so far have been de thus determines the Fermi level, and, by in voted on the semiconductors containing creasing the number of Fe2+ ions, the Fermi Mn2+. Recently, however, new systems based level will shift to higher energies until, at a on Fe2+ have been investigated11. Apart from dopant level of 1018 cm-3 (= x~ 0.0005) the the relative short history of this class of DMS level is pinned to the Fe d level. This phe - the first papers65 on magnetic properties nomenon is depicted in Fig. 8 and has at appeared in 1985 -, this is also due to the tracted much attention, also because the ion crystal growing process. Up to now II-VI ized Fe3+ donors are believed, according to compounds with Fe concentrations in bulk Mycielskï11, to form a so--<:alled perfectly crystals never exceed 26%; see Table I. In the "charge-superlattice". case of Zn 1_xFexSe, however, it was by means Also from the magnetic point of view the of MBE growth of thin layers possible to interest for the Fe compounds is created by cover the entire range of Fe concentrations, the specific 3d levels of Fe 2+. From the previ from x= 0 (ZnSe) to x= 1 (FeSe), con ous subsection we learned that these levels taining promising perspectives for future re play, for instanee in Larson's superexchange searchli6,67. model, a decisive role in the magnetic d--d One of the new and challenging oppor interaction. Any alteration within these tunities in Fe DMS is created by the surplus levels, such as provided by the Fe2+ com of one electron compared to Mn (3d6 instead pared with Mn 2+, can therefore be of great Chapter I INTRODUCTION 13 influence on the exchange mechanism in lying Fe levels, since it is known10,S 7,70 that DMS. In particular, the Fe level located for y 0.35 0.40 the Fe level is located just above the valenee band might affect the mag at the bottorn of the conduction band; see netic properties, since this implicitly means Fig. 9. for zero-gap DMS (e.g. Hg 1_xFexSe) a level in On the whole, the magnetic properties of the conduction band whereas for open-gap Fe-based DMS, probed by standard tech systems (e.g. Cd1_xFexSe and Zn 1_xFexSe) it niques such as magnetization, susceptibility, is located between conduction and valenee and specific heat, display other characteris band. tics than we observed for Mn2+, stemming In that context, some experimental from the nonzero orbital momenturn of Fe2+, studies indicating striking differences be since, according to Hund's rules, the 3d6 con tween zero- and open-gap systems appeared figuration of Fe2• implies S 2 and L 2. in recent years 11 ·5 7·68 . In the study of the The crystal field and spin-orbit coupling in specific heat69 presented in Chapter III.2 we duce a magnetically inactive ground state, will, however, report an analogous low-tem separated from higher-lying levels with en perature magnetic behavier for all Fe-based ergies typically 10 - 30 K, see Fig. 10, ex II-VI group DMS. Therefore, the magnetic hibiting Van Vleck-type paramagnetism (at properties seem to be predominantly deter low temperatures, the susceptibility is con mined by mainly Fe levels below the valenee stant). Only in the presence of an external band, which, similar to Mn, hybridize much magnetic field, mixing with other states in strenger with the band electrans than Fe duces a nonzero magnetic moment (in other levels above the valenee band. Moreover, to words, a magnetic-field-induced paramag strengthen this claim we will not only con netism). For fields strong enough to aver sider the boundary materials Hg 1_xFexSe and eome the cubic symmetry of the wave func Cd1_xFexSe, but also its quaternary alloy tions, magnetic anisotropy is observed in the Hg 1_x-yCdyFexSe. These alloys represent an magnetization. These phenomena can be ideal testcase to unravel the role of the high- understood on the basis of an existing crys tal-field model first developed by Low and Weger 71 , and later adjusted72 by Slack et al., and several others. The validity of this model in the presence of relatively high Fe-concen trations we encounter in DMS materials was unambiguously verified by recent far-infrared (FIR) absorption data on some representative l 7 5 ---Fe'' I systems H . As an illustration, the FIR I transmission of Zn1_xFexSe is presented in Fig. 11. 1.o~ o In Chapter III.l and Ref. 76 a thorough -0.4 analysis of the thermadynamie properties of Cd Se HgSe Zn .xFexSe will be presented, and, apart from y----- 1 the phenomena characteristic for isolated Fe ions listed above, conclusive evidence for an FIG. 9. The schematic band structure of antiferromagnetic interaction between Fe2• Hgl-x-yCdyFexSe solid solutions ranging from HgSe ions, similar to the Mn DMS, will be found. to CdSe. The position of the 3é level of Fe2• ions In distinction to Mn, the interaction is in crystals of different compositions is taken from masked by the Van-VIeek behavior of non the paper of Mycielski (Ref. 70). The picture is valid interacting or weakly coupled ions, and un for all accessible concentrations :e, i.e., for fortunately, the strength as well as the poss 0 < z < 0.15. ibie tail of J is therefore extremely hard to Chapter I INTRODUCTION 14 c: 0 0.8 'ti ï;; 1§"' 111 T.2SOOOK c: 0.6 ....~ 0.4 ~L-~ 0 wavenumber (cm-•J FIG. 11. The transmission of Znt-xFexSe, :t = 0.0148, at T = 1.5 K; the At -+ Tt transition is indicated by the arrow, whereas the broad dip at FIG. 10. Energy level diagram for isolated Mn2+ higher wave numbers can he associated with the and Fe2+ ions. Effects of crysta.I field, spin-orbit At - Ta transition (see Fig. 10). interaction and magnetic field are shown (not to sca.Ie ). The energy distances marked in the figure correspond to a ZnSe host lattice (after Ref. 12). Chapter III.4 prediets the existence of steps in the low-temperature magnetization of Fe compounds, analogously to the Mn case extract from these data. However, a theoreti (Fig. 5) and almost insensitive of crystal cal study of the high-temperature suscepti field and spin-orbit parameters (6>.2/fl, see bility56, see Chapter III.3, offers the possibil Fig. 10). Though experimental evidence for ity to extract Jn by oomparing data with this is still lacking, these calculations might theoretica! expressions ior the Curie-Weiss offer the opportunity to obtain reliable quan temperature. We reeall that for the simple titative data on Jn. spin-only case J is extracted from the well known Curie-Weiss law: C. IV-VI componnds X= Cf(T-8) , The number of papers devoted to IV-VI group DMS is completely outnumbered by a= 2xS(S+l) E Zt1t/3kB , (7) those on the Mn-containing II-VI group ma i terials and only since very recently a rapidly with Ji = J11, for i 1, growing scientific interest can be observed9. J1NN ior i 2, etc. Among the IV-VI DMS most of the attention z1 = number of ions coupled by J1 has been focused on the Pb-containing com pounds9·53-55·79-82, e.g. Pbt-xMnxTe. The In the Fe case similar expressions can be magnetic properties of these alloys closely found, although supplemented with addi resembie the behavior of the II-VI Mn-eon tional terms originating form the crystal taining DMS described in one of the previous field effects, yielding exchange constants ap sections, i.e., the interactions are antiferro proximately two times larger than ior Mn; magnetic, spin-glass phases have been de see Table IL Unfortunately, these expressions tected, and, in a first order approximation, include some model-dependent parameters the thermadynamie properties can be fairly which can possibly obscure this approxi well described by pair approximations. mation. However, the strengthof the AF interac 77 Finally, a numerical study presented in tion in Pb1_xMnxTe is much weaker than in Chapter I INTROD U CT ION 15 we already argued that for the II-VI com pounds considered earlier, the small concen tration of charge carriers rules out the indi rect RKKY mechanism between the Mn2+ ions, leaving the antiferromagnetic super exchange dominant in the exchange pro cesses. In contrast to this, the interactions observed in the IV-VI compounds Sn1_x 89 MnxTe84-88, Pb1_x·yGeyMnxTe and Ge1.x Mnx Te90 are ferromagnetic, and should be explained by assuming a RKKY-type of in teraction. The oscillating RKKY interaction, H (tesla) brought about by the high carrier concentra tions in these systems, of the order 1021 cm·3, FIG. 12. Magnetization of Pbt-xMnxTe at T is indeed ferromagnetic, at least for the first 4.2 K. The circles repreaent the data and the solid few neighbors (see Fig. 13). Moreover, the lines were obtained from three-parameter fits (after R-3 decay of J.RKKY implies, on the basis of Ref. 54). sealing arguments, that Tc should be propor tional with x, since the II-VI's, leading, as an example, to very (7) small negative Curie-Weiss temperatures and a stepless, quickly saturating magnetization which is indeed experimentally observed87. (see Fig. 12). This can be seen also in Table The carrier concentrations for all DMS II, where IJn I is for all cases below 2 K. are schematically shown in Fig. 14, where we Gorska et al. 55 ascribed this strong reduction see that for the 11-VI compounds the concen compared to the II-VI's primarily to the sep trations are so small that the ferromagnetic aration (dMn-a) between Mn and an adjacent RKKY interactions can be neglected and anion (in this case Te). Within existing only AF coupling is observed. On the right models on superexchanget6,59 the interaction is very sensitive for this separation through the dMn -a ·4 or dMn -a ·71 2 dependenee of the hybridization parameter Vdp' yielding 0 0 0 0 0 0 0 ojlooooo 0 0 0 0 t 0 0 (8) 0 0 Q 0 0 0 0 0000000 We should, however, not exclude that the 00,0000 0 0 0 0 0 0 0 observed differences are above all induced by ooooootf R the Mn - anion - Mn bond angle. According 0 0 0 0 0 0 0 oooftoo to a quanturn mechanica! treatment of an 0000000 idealized three-site molecule37 ·59, the NN interaction varies with cos 2cp being minimal for the 1rj2 angle in the rocksalt structure of FIG. 13. Schematic sketch of magnetic moments Pb1_xMnxTe. On the other hand, Ginter randomly distributed in a matrix, and the resulting et al. sa indicated that exchange constants at RKKY exchange integral plotted as a function of the L point, where the direct gap is located in the distance. For the carrier concentrations we IV-VI DMS, are substantially diminished usually encounter in Pbt-x-ySnyMnx Te and Snt-x compared to the r point in the II-VI's. Mnx Te several positions for magnetic ions are avail In the context of exchange mechanisms able in the first strong ferromagnetic regime. Cha.pter I INTRODUCTION 16 Fig. 15. It is clear that this effect provides the link between the antiferromagnetic a.nd ferromagnetic regimes, depicted in Fig. 14. The modified RKKY modeJ95,9 4 - modified in the sense that the conventional RKKY is supplemented with a realistic band structure Pb Te of Pb1_ySnyTe - enabled us in 1988 to understand the existence of Pcrit• as well as the strong ferromagnetic coupling above Pcrit· In fact, the strong ferromagnetic coup tt ling is provided by the shift of the Fermi level with charge density, which, at Pcrit• enters a second set of valenee bands, located along the E direction in the Brillouin zone. 1020 1022 This model will be treated in Chapter IV.2. carrier density (cm- 31 Some recent improvements on this modei96, taking into account the degeneracy of the E bands, alter the calculations presented in Chapter IV .2 to some extent, and, therefore, FIG. 14. Schematie representa.tion of carrier eon the most up to date experimental data and eentrations in DMS ba.aed on Mn2+. The antiparallel calculations in the modified RKKY model a.nd parallel a.rrows indieate the observed a.ntiferro are shown in Fig. 15. ma.gnetie a.nd ferroma.gnetic interactions in those Referring once more to the plot of the regimes, respeetively. In Pbt-x-ySnyMnxTe, carrier carrier concentrations in DMS (Fig. 14), and coneentrations in both regimes a.re eovered. given the fact that Pb1_x_1SnyMnx Te sharply separates the ferro- and antiferromagnetic hand side of Fig. 14, at the highest possible regime by Pcrit• one would expect for Pb1_x carrier densities, the ferromagnetic IV-VI Mnx Te and Sn1_xMnx Te, both situated in the compounds are gathered. Consequently, one vicinity of Pcrit• the same effect as in Pbt-x-y can see that in between Pb1_xMnxTe and SnyMnx Te if the carrier density can be Sn 1_xMnx Te a natura! boundary exists at changed in that amount that Pcrit is in carrier concentrations of roughly 1020 cm -a, cluded. For Pb1_xMnx Te it is extremely com dividing the Mn-containing DMS into an plicated by standard annealing procedures to antiferromagnetic and ferromagnetic interac raise the density to Pcrit or even higher. For tion regime. Sn 1_xMnx Te, however, it is possible to de In 1986 Story et al. 91,92 shed new light crease the carrier density in such amount on the role of the carrier density by the dis that Pcrit eau be reached 97• As a consequence, covery of the so-called carrier-concentration it cou1d be shown by us9 8 that the interac induced ferromagnetism in Pb0•25Sn0•72- tions rapidly vanish in the vicinity of the Mn0.03Te, a quaternary blend between Pb1_x critical carrier concentration. This result does MnxTe and Sn1_xMnxTe. In these compounds not only generalize the occurrence of a carrier it is possible to cover a wide range of carrier phase line in DMS, but it also provides sup concentrations, from 1019 cm-3 up to at least port on the validity of the modified two-band 1021 cm-3. At a critical carrier concentration RKKY model. These results on the ternary 20 3 (Pcrit il$ 3·10 cm- ) a steplike increase to Sn1_xMnx Te will be presented in Chapter ferromagnetic interactions was observed, IV.3. In this part we will not only consider whereas below Pcrit the interactions beoome the carrier-concentration-dependent magnet very small and weakly antiferromagnetic, as ie interactions. Also a close examination of reported in more recent investigations94; see the low-temperature magnetic phases will be Chapter I INTRODUCTION 17 Though several studies succeeded the initial report of Story et al. 91 extensions have 100 been made to Pb0 .25Sn0 •72Mn0.08Te and 101 Pbo.s2Snu5Mn0 •03Te , and spin-glass-like phases have been detected101 for p < Pcrit the three-dimensional ( T,x,p) phase diagram for these compounds is far from complete and hardly any quantitative information on ex 2 change constant (e.g., a, {3, Jn, JNNN) is yet available8 . Magneto-optical investigations employed in II-VI and II-V group DMS to 0 determine a and {3, cannot be applied for Pb1_x-ySnyMnxTe and Sn 1_xMnxTe due to the Pcrit strong carrier absorption, and therefore other 0 15 techniques, such as EPR102 or nuclear mag netic resonance, will be indispensable in fu ture investigations. As for the d-d exchange parameters, up to now no adequate tooi has FIG. 15. Curie-Weiss temperature [8 x (0.03/z), been found to correlate experimental phe used for small variations in Mn concentration] as a nomena (such as the magnetization steps in fundion of the carrier concentration p of Pbo.25- the II-Vl's) directly to the coupling con Sno.72Mno.oaTe; data are taken from Ref. 91,94 and stants (Ju, Jnlf, and so on). 95. The calculations are performed within the two Besides the unique phenomenon of Pcrit. band RKKY model (Ref. 95), supplemented with new effects of fundamental interest may be the multi valley character of the relevant band 95 expected from the delicate interplay of structure (parameters m1* = 0.05, m2* = 1.00, RKKY interaction, magnetic ion concentra lsp-d = 0.32 eV). tion and carrier concentration. For example, very preliminary data on PbusSn0. 72Mn0.02- Te and Snu8Mn0 .02Te by de Jonge et al.1os made, since, somewhat mysteriously, reports indicate a carrier-induced ferromagnetic to by Mauger et al. 85 claimed for p > Pcrit and spin-glass transition at carrier densities well x between 0.03 and 0.06 the existence offer above Pcrit• at a fixed concentration of Mn2+! romagnetic phases mixed with spin-glass On the basis of the RKKY interaction, this features, a so--called reentrant spin-glass. intriguing phenomenon can, tentatively, be From our data on Sn 1_xMnxTe in this regime brought about by an increasing competition we will show that it is extremely hard to dis between ferro-- and antiferromagnetic coupled criminate between the magnetic phase transi ions by increasing the charge density. tion, ferromagnetic or reentrant spin-glass. By means of neutron-scattering experiments performed very recently by Vennix et al. 99 , it was for the first time possible to rule out 1 J.K. 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Chapter I INTRODUCTION 22 SCOPE OF THE THESIS In this thesis the results of experimental studies on diluted magnetic semiconductors are presented. In Chapter II the low-tem perature 1Ilagnetic properties of both Zn 1_x MnxSe and Zn 1_xMnxS will be analyzed. The relevanee of long-range interactions and how to incorporate them into model calculations, and the underlying physical mechanisms for the Mn-Mn interaction, will be the central items in this chapter. Subsequently, an anal ogous treatment will be made in Chapter Hl of the low-temperature behavior of DMS on the basis of Fe2+ instead of Mn2+. The elec tronic configuration of Fe2+ necessitates a more complicated treatment, and the Fe in teraction, though observable, will be hard to extract from the available data. The high temperature susceptibility and high-field magnetization will be analyzed in order to obtain quantitative information on the Fe2+ Fe2+ interaction. Finally, Chapter IV will deal with carrier-concentration-dependent interactions observed in the low-temperature magnetic behavior of Pbt-x-ySnyMnxTe and Sn 1_xMnx Te. A model will be presented in which these phenomena are related to the exchange mechanism in combination with the band structure of the host semiconductor. The contents of Chapters II, lil and IV have already been published elsewhere. We have chosen to embody the corresponding texts in this thesis in essentially the same form as they have been published. As a con sequence, some parts of these chapters may seem somewhat superfluons for the reader of this thesis. On the other hand, this choice has the advantage that these parts can be read rather independently. CHAPTER TWO Magnetic behavior of 11-VI group diluted magnetic semiconductors, basedon Mn The magnetic susceptibility and specific heat of the diluted mag netic semiconductor Znt-xMnxSe (paragraph 1) and Znt-xMnxS (para graph 2) have been measured in the temperature range 10 mK < T < 40 K for 0.01 < z < 0.53. A paramagnetic-11pinglass transition was ob served in the whole contentration range. The contentration dependenee of the freezing temperature Tr was found to be compatible with a radial dependenee of the exchange interaction between manga.nese ions of the type J(R)"' R-6·8 for Znt-xMnxSe and J(R) "' R-1·6 for Znt-xMnxS. Ba.sed on this observation we calcula.ted thermadynamie properties with the extended nearest-neighbor pair a.pproxlmation. lt appears tha.t this a.pproximation provides a good simultaneous destription of the specific heat, high-field magnetization as well as the high-temperature suscepti bility with parameters J0fkB = -13 K {nearest-neighbors interaction) and Ji(R)/fes -7/R 6·8 K (distant-neighbors interaction) for Znt-x MnxSe, and parameters JofkB =-16K and Ji(R)/kB =-10/R 1·6 K for Znt-xMnxS· A compa.rison is made with other diluted magnetic semicon ductors and the possible origin of the exchange mechanism is then dis cussed. Chapter 11, § 1 MAGNETIC BEHAVIOR ..... , BASEDON Mn 24 Chapter II, para.graph 1 Magnetic behavior of the diluted magnetic semiconductor zn1.xMnxSe [ publisbed in Phys. Rev. B 36, 7013 {1987) ] I. INTRODUCTION observation of the spin-glass {SG) transition only a.bove the percolation limit (xc) of the During the past years extensive investi host lattice4.5. It was suggested then that the gations have been performed on the magnetic SG transition was brought about by short behavior of diluted magnetic semiconductors range [nearest-neighbor (NN)] AF interaction (DMS) (i.e., II-VI or II-V compounds con causing topological frustration effects due to taining controlled quantities of randomly the high symmetry of the host lattice. Recent substituted magnetic ions)1. So far data have results, however, for low Mn concentrations been obtained almost exclusively on systems in Hgt-xMnxTe1, (Cdt-~MnxhAs2 2 , (Znt-x 7 of the type At-xMnxB, such as Hg 1_xMnx Te, MnxhAs23, Cdt-xMnxTe, Cdt-xMnxSe and 8 Cdt-xMnx Te, Cd1_xMnxSe, Zn1_xMnx Te and very recently Zn 1.xMnxTe, Zn 1.xMnxSe re (Cdt-xMnx)aAs2, (Znt-xMnx)3As22.3. From veal the existence of SG phase also below Xe. these data, as far as available, a rather typi The situation for {Cd1.xMnxhAs2 is even cal behavior is observed. This behavior eau more pronounced since in this case for x > Xe be characterized as follows. the NN frustration mechanism is also ex cluded due to the simple cubic symmetry of ( 1) Curie-Weiss behavior of the magnet the host lattice2• Moreover, subsequent calcu ie susceptibility x at high temperatures indi lations on the basis of NN interactions only, cating antiferromagnetic (AF) Mn-Mn inter gave rise to a wide spread of exchange para a.ctions. meters deduced from various sets of data and (2) A cusp or kink in the low-tempera the need to adjust the random distribution of ture x indicating a spin-glass-like transition the magnetic ions. To our knowledge no con at a temperature depending on the Mn con sistent set of parameters explaining all the centration x. data simultaneously has been obtained on (3) A magnetic contribution to the speci this basis. fic heat Cm with a broad maximum shifting It is our claim that these discrepancies to higher T with x. are mainly due to the fact that the long-range ( 4) A field dependenee of the magnetiz character of the interactions is not taken into ation M indicating AF interactions, usually account as we have shown recently for 2 accompanied with steps in high fields. (Cd1.xMnx) 3As2 and, though less extensive, forsome other systems as well 9• Moreover, as Oiiginally this magnetic behavior was inter we argued beforeto, detailed knowledge about preted as arising from interactions between the range or radial dependenee of this long Mn ions situated at the nearest-neighbor range interaction might yield valuable infor (NN) sites in the host lattice4.5. This conjec mation about the driving physical mechan ture was strongly supported by the original isms bebind the Mn-Mn interaction in DMS. Chapter 11, § 1 Magnetic behavior of ..... Znt-xMnxSe 25 In view of thls we thought it worthwhile to study the magnetic properties of Zn1_x MnxSe in some detail. Preliminary results 50 have been reported recently8• We will report susceptibility and specific-heat results in a wide composition range (0.01 < x < 0.53) and we will try to interpret these data ( to gether with magnetization11 data and high temperature susceptibility12) simultaneously on the basis of one model incorporating short range as well as long-range interaction in a random array. II. EXPERIMENTAL RESULTS 20 The samples of Zn 1_xMnxSe were grown by the modified Bridgman metbod under the pressure of a neutral gas. The crystalline structure of this material was reported13 to temperature IK) be cubic for x ::::; 0.06, polytypic for 0.06 ::::; x 5 0.12 and hexagonal for x > 0.12. In a pre vions magneto-optical investigation14 of this FIG. 2. de susceptibility of Zno.usMno.t54Se material no essential influence of polytypism measured after zero-field cooling (ZFC, H < 1 G) was found although some scattering of energy and field cooling (FC, H = 20 G) as a function of gap for x;::: 0.2 was reported15• temperature. The arrow indicates freezing tempera ture Tr as obtained from ac susceptibility. Inset: similar data for Zno. 72Mno.2sSe (Ref. 12). The Mn concentrations x of the investi gated samples as measured by mieroprobe analysis were: x= 0.014±0.001, 0.023±0.002, 0.056±0.002, 0.064±0.006, 0.103±0.005, 0.154± 0.004, 0.254±0.003 and 0.53±0.02. Generally these actual concentrations were considerably larger ( typically 50%) than the nomina! con centrations of the starting materials. A. Low-temperature susceptibility The ac susceptibility was measured with a conventional mutual inductance bridge op FIG. 1. ac sUllCeptibility of Znt-xMnxSe [x = erating in the region 100 < f < 2000 Hz and 0.023 (e), 0.064 (o), 0.154 (t;), 0.245 t:J and 0.53 (X)] fields less than 1 G. Same representative sus as a function of temperature for different Mn com ceptibility data for various x are shown in positions. The arrows indicate the freezing tem Fig. 1. The results below 1.5 K were obtained peratures Tr. Note the different vertical scales; the in a dilution refrigerator for which no ad data are multiplied by the factor indicated in the equate absolute calibration of x was avail figure. able, which may result in some devia.tion of . Chapter 11, § 1 MAGNETIC BEHAVlOR ..... , BASED ON Mn 26 tions and kinklike for low concentrations. A TA.BLE 1. The frening temperatul't!ll Tr for simnar situa.tion was reported for Cd1_xMnx Znt-x:MnxSe. Te where a well-pronounced cusp for x > 0.3 (Ref. 4) and a kink for x< 0.15 (Ref. 6) was Tr(ll') ll' observed. In contrast to the susceptibility, as 0.022 0.023 we will see later on, no anomalous behavior 0.225 0.064 can be detected in the specific heat. 0.55 0.103 de susceptibility data, field-cooled as 1.7 0.154 5.7 0.245 well as zero-field-cooled, are shown in Fig. 2 24 0.53 for concentrations above and below the perco lation limit. This characteristic behavior sup ports the interpretation of the anomaly in the susceptibility as a transition to a spin-glass the data a.t low temperatures. Figure 1 clear state. The resulting phase diagram, Tr - x, in ly shows an a.nomalous behavior of the sus the range 0.023 ~x~ 0.53 is shown in ceptibility at a. certain temperature Tr Fig. 3. Inspeetion of this phase diagram (freezing temperature) depending on the con shows that Tr-+ 0 when x-+ 0. From this centration of Mn ions (see Table I). The experimental observation one may already anoma.lies are cusplike for high concentra- conjecture that the interactions inducing this spin-glass transition are relatively long ranged, since otherwise no freezing should have been observed for x below the percola 1 I tion limit, which amounts to xc = 0.18 in this I case. I I I B. Specific heat J I Specific-heat data were obtained with a I I conventional adiabatic heat-pulse calorime I ter in the temperature range 0.4 < T < 20 K. p I I The magnetic contribution Cm to the specific .{ heat was obtained by subtraction of the ·lat I tice contribution of pure ZnSe and the nu I l clear hyperfine contribution of the Mn ions. I The results for Cm in zero external mag I netic field are shown in Fig. 4. As quoted / above, no anomaly was observed at the tem .I SG perature Tr, indicated by arrows in the fig S I ure. The overall behavior of Cm is similar to I I that observed for the other DMS: a broad I maximum shifting to higher temperatures , I with increasing x. In our case this maximum is not observed for x~ 0.06 since it is located 0 contentration x at temperatures lower than 0.4 K. In the presence of a magnetic field the specific-heat data show a shift of the maximum to higher FIG. 3. Phase diagram for Znt-xMnxSe ( e our temperatures with increasing field, as shown data; 0 Ref. 12). The dashed line is a guide to the in Fig. 5 for Zn0.986Mn0.014Se. These data eye only. look very similar to earlier resultsl& in the Chapter II, § 1 Magnetic beha.vior of .••.. Znt-xMnxSe 27 temperature range 0.3 ~ T ~ 3.5 K on Zn 1.x MnxSe with a nominal Mn contentration of 0.01. Quantitative comparison is, however, difficult since the actual Mn contentration is not known. C. High-tempera.ture susceptibility E u a.nd ma.gnetiza.tion The magnetization has been measured up to 15 kOe and was reported earlier by onè of the present authors11• For comparison some selected results will be shown later on. The high-temperature susceptibility results12 show a Curie-Weiss behavior with E> = -45 K for x 0.05 at high temperatures, in temperature !K) dicating AF interactions between the Mn2• ions. Moreover, e was found to be a linear function of the concentration12 suggesting a FIG. 4. Magnetic specific heat of Znt-xMnxSe random distribution of Mn 2• ions. [:a: = 0.014 {n), 0.023 ( +), 0.056 (x), 0.103 (0) and 0.154 (O)]. Inset: magnetic specific heat of Znt-x ID. INTERPRETATION MnxSe (:t = 0.154 and 0.245 as wel! as specific heat of pure ZnSe). The arrows indicate the freezing tem A. The spin-glass transition peratures Tf. Among the data reported above, the existente of a spin-glass transition for vanish ingly small concentrations of x is the most obvious indication of the existence of long 2 0.08 range interactions between the Mn • ions. As a basis for the interpretation we would like to focus our attention on this freezing transition since the contentration dependenee of this transition can be con ,J o.o sidered as a probe of the radial dependenee of the interaction strength between the impur .::: 0.02 ities10 . ,:;.. 5} In this respect it is relevant to note that the experimental data on the transition 0 strongly support the spin-glass nature of the temperature !K) transition. These data include the cusp or kink in the ac susceptibility, the continuons behavior of the specific heat, and the hyster FIG. 5. Magnetie-field dependenee of the mag esis observed in the de susceptibility. These netic specific heat of Zno.gssMno.ot4Se for B = observations match perfectly the phenom 0.00, 0.494, 0.857, 1.03, 2.04 and 2.80 T. The solid enological characteristics which are oommon lines are obtained from the ENNPA using Jo/leB= ly applied to define a transition to a canoni -13K and Ji/lcB -1/R6.8 K. = calspin-glass17. If one accepts the nature of the transi- Chapter II, § 1 MAGNETIC BEHAVIOR ..... , BASEDON Mn 28 tion as a spin-glass freezing (which is, how ever, disputed as we will argue in the dis cussion), then a sealing analysis should be applicable. Such a sealing analysis generally exploits the fact that for a continuons ran dom distribution it is assumed that Rijax = const, where R1j denotes a typical distance between the ions. Implementation of this expression in a model for spin-glass freezing, given a known functional form for the radial dependenee of the exchange interaction, then yields a theoretica! prediction for Tr(x) which FIG. 6. Natura! logarithm of the freezing tem can he compared with experimental data. perature Tras a function of (a) ln:ll and (b) :ll-113 for This procedure is elaborated in the Ap Znt-xMnxSe (e, our data, 0 Ref. 12). The straight pendix for a continuons as well as a discrete solid lines have slopes of (a) 2.3 and (b) -4.1. distribution of ions. In the spirit of earlier analysest8,19 the spin-glass-freezing condition is based on the existence of a critica! fraction should not exclude the possibility that the x of blocked or frozen ions at the freezing tem dependenee of the freezing temperature Tr perature. The fraction of these ions is deter can he different above and below the percola mined by the probability of finding at least tion limit. Therefore one should he carelul in one ion within a sphere of radius R1(T), im drawing definite conclusions from Fig. 6. plicitly given by J(R1)S2 = kBT. The results demonstrata the applicability of this ap B. Magnetic properties proach also outside the limit of the very di Jute regime to which it is usually restricted. It follows from the inspeetion of the ex Given a powerlike or exponential radial de perimental data and the analysis of the spin pendence of the interaction, the concentra glass transition that the Mn-Mn interaction tion dependenee of Tr can he expressed as in Zn 1.xMnxSe is AF and that it is rather long ranged, decaying as R-6.8. The relevant lnTr"' !n lnx for J(R)= J 0R-n thermadynamie properties can he described with the so-ealled pair-approximation model, or which is an approximative calculation method, particularly useful for random arrays lnTr N a x-113 for J(R)= J 'exp(-aR) 0 with long-range interaction. It has been in troduced by Matho2° for canonical metallic In Fig. 6 the experimental data T1(x) for SG's and was recently successfully used for Zn1_xMnxSe are plotted in the coordinates DMS as well2,3,9. This approximation is suitable for power and exponential depend based on the assumption that the partition ence, respectively. A comparison between function of the system can he factorized into these shows that the simple power depend contributions of pairs of spins. We will con enee J(R) "' R-n seems to describe the data in sider two roodels in some detail: extended the whole concentration range better than nearest-neighbor pair approximation2 the exponential decay [J(R)"' exp(-aR)]. (ENNP A) and hierarchy pair approximation The exponent n deduced from Fig. 6 is !:! 6.8. (HPA) of Rosso2 1• In the ENNPA each spin We would like to stress here that although it is considered to he coupled by an exchange is clear that a power law yields a better fit to interaction J1 only to its nearest ma.gnetic the experimental data than an exponential neighbor, which may he located anywhere at decay in the concentration range from far a distance R from a reference site. The stat- below to far above the percolation limit, one 1 Chapter II, § 1 Magnetic behavior of ..... Znt-xMnxSe 29 istical weight of these pair configurations (al with various R1 (which can only take on dis crete values depending on the symmetry of the host lattice) is assumed to he determined by the random distribution of the ions. The Hamiltonian for a pair is given by J{i = -2J1S1·Sj -gJLB(Siz+S/)Bz, (1) where Ji = J(Ri) and R1 denotes the distance between the sites i and j. If N1 is the number (b) of lattice sites in a shell with radius R1 and using ni = j =tLi Nj for j > 0 and n0 = 0, the probability for a pair formation can betaken as the probability of finding at least one nearest spin in the ith shell ( assuming all j < i shells empty) and reads for a random distribution as (2) FIG. 7. Pair formation in the (a) ENNPA and (b) HP A. The arrows repreaent spin-spin interac tions. On the other hand in HP A the spins are arranged in a collection of separate pairs or dered by decreasing interactions21 • The calcu in both roodels are shown in Fig. 8. As could lation of the probability distribution for pairs be expected, the number of pairs with small in this case is simHar to that used in the R1 (i $; 7) is enhanced in ENNP A, with re ENNPA: P 1(x) is a product of (a) the prob spect to HPA. This situation is reversed for ability for a spin to have a magnetic neighbor more distant pairs. Since the probability dis at a distance R1 and (b) the probability that tributions are known, the thermodynamic both spins do not belong to a pair with a properties (Am) can be obtained by summing shorter R1. Wethen obtain: the respective pair contributions: n i V 1 Am = L !AmiPi(x) · (4) P 1(x) = [1-(I-x)N l [1-l Pj(x)r i=l j=l (3) (jeRt) Each pair contribution (Ami) contains the exchange parameter J 1. Following the results where the summation runs over sites in the of previous section we take Ji = J 0/R16.8, sphere R1; some sites are skipped in the sum where Ri is in units of the NN distance in the matien because the probabilities that ion i host lattice. The summation over the shells i and ion j do not belong to a pair with shorter is carried up to shell i for which 1.J:v P1 ~ 21 R1 are not independent . The principal dif 0.995. For low concentrations (x < 0.03) ference between ENNP A and HP A is shown usually v < 20, being even less for higher in Fig. 7 for a system of four ions. The calcu concentrations. lated probability distributions for x= 0.023 The results of the specific-heat calcula- Chapter II, § 1 MAGNETIC BEHAVlOR ..... , BASEDON Mn 30 fee lattice ••• x •0.023 "' '*e 0.10 -.... temperafure IK! FIG. 9. Zero-field magnetic specific heat calcu lated in the ENNPA (solid linea) and HPA (dashed ilshelll linea) together with experimental data.. FIG. 8. Probability of finding at least one mag netic neighbor in shell i represented by the blank nearest-neighbor interaction J0 is inferred histogram [Eq.(2)] and probability for a spin to have from independent experiments a.nd the radial at least one magnetic neighbor in shell i under the dependenee from Tf(x), we inserted J 0 as a condition that both spins do not belong to a pair constant value ( = -13 K) a.nd assumed for 6 8 with a shorter distance [Eq. (3)] represented by the further neighbors Ji = J 1/Rt • (i> 0) where shaded histogram. J 1 is the only adjustable parameter. J 1 was chosen to obtain the best overall agreement tion for both the ENNP A a.nd HP A are shown in Fig. 9 tagether with the experimen . tal data. For J0 (i.e., NN interaction) we • have taken -13K as indicated by high-field i 0.10 magnetization data22. Recent inelastic neu tron scattering data23 yielded a comparable value, although slightly lower. One can no tice that at low temperatures ( T ~ 0.2 K) HP A gives larger valnes than ENNP A, whereas for higher temperatures the situation is reversed. This results from the discussed difference in probability distributions (Fig. 8). The difference is much more signifi cant foi higher concentrations (x l:i 0.06) than temperafure (KJ for lower (x l:i 0.01 ), for which ENNP A and HP A nearly coincide. It should he stressed that the specific-heat curves shown in Fig. 9 FIG. 10. Magnetic specific heat of Znt-xMnxSe. were simply calculated with no fitting to the The dasbed lines represent calculations with the experimental data. Presumably a better ENNPA as described in the text using Jo/ka = 6 8 agreement will he obtained if J1 valnes are -13 .K and Ji/ka = -7/R • K. The solid linea re treated as adjustable parameters. Since the present the ENNP A with triples included. Cha.pter II, § 1 Magnetic beha.vior of ..... Znt-xMn:xSe 31 for all three experimental quantities: Cm, M and X· For the final calculations we have extended our ENNP A similarly as it was done for (Cdt-xMn:x)aAs 2 (Ref. 2) and (Zn,_:x MnxhAs2 (Ref. 3); we considered not only 3.0 pairs but also "triples" (i.e., configurations in .e~ Ql which two spins are located at the same dis c: i? tanee from the reference site). e The results of ENNP A calculations with ~co J0 = -13 K and J 1 = -7 K are shown in Figs. 5 and 10 (Cm), 11 and 12 (M). The high-temperature susceptibility calculations Zn 0.95 Mn 0.05 Se for Zno.g 5Mn 0.05Se yield a Curie-Weiss tem r~ taK perature e = -45 K which is in perfect agree ment with the experiment12. The high-field magnetization, shown in Fig. 12, has been =------_!______1_ ___ _ recently fitted with a model including a near 0 10 20 30 magnel ie field m est-neighbor interaction J 0 = -9.9 K and a mean field24. Our results also show that a satisfactory description can be obtained with the same set of parameters as used for the FIG. 12. Steplike ma.gnetization of Zno.gsMno.or other thermadynamie quantities. Se [Ref. 24] together with the ENNP A prediction The ENNP A may be further extended (solid line, lofles ::::-13K and J1/les :::: -7/R6·8 K; by combining it with a mean-field approxi dasbed line, lofleB :::: -12 K and lilles :::: mation to account for the average interaction -7/R6.8 K). Since no absolute va.lue of ma.gnetiz of a spin with the other spins not belauging ation is given in Ref. 24 the ma.gnetiza.tion was to a pair (i.e., spins with i > v). The results sca.led a.t 25 T. generally confirm this conjecture, although the obtained correction is rather small. We conclude that, although these extensions do not significantly imprave the results and some systematic deviations remain, the gen eral agreement shows that it is, in principle, possible to explain the behavior of specific heat, magnetization and susceptibility simul taneously, without the need to adjust the random distribution of the Mn ions. As we quoted before, recently some Cm data on Zn 1_xMnxSe with nominal Mn concentration of x= 0.01 were publisbed by Keesom16. As suming only contributions from singles, pairs magnelic field (T) and triples he was able to describe the data rather well. However, since essentially both FIG. ll. High-field magnetiza.tion of Zno.95- the total Mn concentration as well as the Mn0.05Se. The solid lines represent calculations with statistica! distribution were used as a fitting the ENNPA using lofleB ::::-13K and lilles:::: parameter, a oomparisou with these results -7/R6.8 K. does not seem very significant. Chapter II, § 1 MAGNETIC BEHA VIOR ..... , BASED ON Mn 32 IV. DISCUSSION • lZn Mn) Se •IZn Mn) Te The analysis of the concentration de • !Cd Mn) Te pendence of the freezing temperature, as per o !Cd Mnl Se formed for Zn _xMnxSe in the preceding sec •IZn Mnl3As2 1 g •ICd Mnl As ~· tion, can also he applied to other DMS. The .= 3 2 ,.4'+-''"' • available data on Tf(x) fora number of them are gathered in Fig. 13. In all cases it appears that a description of Tf(x) which is basedon 01 ~/+·~ •o a power-law dependenee of J(R) = J 0R-n fits the data in the whole concentration range // /,t . rather well, in contrast with the description O.o1 0.01 0.1 based on an exponential decay of J(R). Whether this is indicative of a specific mech anism remains to he seen, however, since (as FIG. 13. Freezing tempera.ture Tf as a function was argued beforeS) it is not a priori clear of the Mn concentration z for various DMS on log whether the same mechanism is responsible arithmic scale. The straight lines are fitted to the for the spin-glass freezing below and above data yielding the power dependenee J(R)"' R-n as the percolation limit. tabulated in Table II. References on the origin of The exponents n deduced from Fig. 13 the data are given in the text. are tabulated in Table II. The various sys tems in Table II are arranged in the order of decreasing gap. Reviewing the table gives rise Taniguchi et al. 27 obtained information about to the following comments. The considerable the Mn 3d density of states and p-d hybridiz increase of the range ("' 1/n) of the interac ation in the series Cd1_xMnx Y (Y = S, Se and tion going from wide-band-gap materials to Te). They concluded that for this series the small-band-gap materials is obvious. This degree of p-d hybridization increases on go fact is not inconsistent with the expectations ing from Te to S. However, the data tabu based on the Bloembergen-Rowland ex lated in Table II do not reflect the expected change interaction25 as the driving mechan systematic change in range of the interaction. ism behind the spin-glass freezing. In this In fact, a most remarkable feature of the case, however, an exponential decrease data on Tf versus x as shown in Fig. 13 is the [J{R)"' exp(-oR)] should have been expected surprising universa! behavior of the II-VI and the apparent universa! range of the ex wide-band-gap materials. Not only is the change for the II-VI compounds, irrespective con centration dependenee analogous ( and of the appreciable variation of the gap magni thus the range 1/n), but the absolute magni tude, is somewhat surprising. Moreover, if tude of the freezing temperature is also the superexchange would he the drivinf mechan same, taking into account the scattering of ism as suggested by Larson et al. 2 , at least the data. In view of the fact that considerable for the nearest-neighbor interactions, an in differences exist between the lattice para creasing range of the interaction would he meters of the II-VI compounds in this series related to an increasing covalent character of (up to 20%), a variation of the freezing tem the bonding26. Roughly speaking, such an perature by a factor of 2 or 3 might have increasing covalency may indeed be expected been anticipated within the concept of our when the II-VI systems are compared with model, given the pronounced radial depend II-V and IV-VI systems. More detailed infor enee of the interaction strength J{R). The mation can he obtained from the location of data apparently do not support this conjec the energy of the Mn d levels with respect to ture. This might he considered as an indica the top of the valenee band. Recently tion that, besides the interaction strength, Chapter Il, § 1 Magnetic behavior of ..... Znt-xMnxSe 33 TABLE II. Type, concentration range, band gap, nearest-ileighbor dÎiltance and exponent n of various DMS. Material Type x range Eg (eV) NN distance (Á) ll Ref. Zn1_xMnxS II- VI 0.3 -0.4 ~3.8 3.83 ~6.8 31 Znt-xMnxSe II- VI 0.02-0.5 2.8-3 4.00 6.8 This paper,12 Zn 1_xMnxTe II- VI 0.07 0.6 2.4-2.8 4.31 6.8 8,32 Cdt-xMnxSe II-VI 0.05-0.5 1.8-2.6 4.28 ~6.8 7,30 Cd1_xMnxTe II- VI 0.01-0.6 1.6-2.5 4.58 ~6.8 4,19,6,33 Hgt-xMnxTe II-VI 0.02-0.5 ~o- 1.1 4.55 =5 5,34,35 Hgt-xMnxSe II VI 0.02-0.3 ~o 4.30 1!15.0 37 (Zn 1_xMnx) 3As 2 II-V 0.005-0.1 d 2.94 4.5 3 (Cdt-xMnx),As2 II-V 0.005-0.2 0-0.2 3.17 3.5 2 Pb 1.xMnxTe IV -VI 0.03-0.1 0.2-0.4 4.56 3 36 the freezing process is also determined by needed to activate a clearcut transition. More topological criteria. specifically, it was suggested for impurity As we quoted in the Introduction, the spins in III-V semiconductors that the ran overall characteristics of DMS include, dom anisotropy of the indirect exchange is among others, AF long-ranged interactions the driving force towards a spin-glass state, and spin-glass formation for a wide range of irrespective of the sign of the exchange inte concentrations. It has been questioned gral39. whether real spin-glass formation is possible Experimental evidence of such an addi in a random diluted array coupled by long tional anisotropy in DMS is scarce, though ranged isotropie AF interactions only, since not completely absent. For the present com in that case the driving mechanisms of frus pound Zn1_xMnxSe, electron-spin-resananee tration or competition would not be effec results were reported for dilute samples, indi tive17. To start with, we would like to em cating a uniaxial single-ion anisotropy D = phasize that the present experiments on 0.1 K39• Moreover, inspeetion of the specific Zn1_xMnxSe did yield the observation of typi heat data as shown in Fig. 4 indicates an in cal spin-glass characteristics. These include crease in Cm at the lowest temperatures, the cusp in the susceptibility, a continuons which is not reflected in the calculations. specific heat, and a difference in zero-field This is by no means unique for Zn1_xMnxSe caoled and field-cooled magnetizations, both and has been observed in a number of DMS. for concentrations above as well as below the Earlier attempts have been made in Cd 1_x percolation limit. We feel that with these MnxSe to explain this behavior in terms of a data the canonkal spin-glass nature of the single-ion anisotropy, although in that case transition is strongly supported17. no single-ion splitting was observed in ESR 31 With respect to the fundamental ques experiments . For Zn 1_xMnxSe, the reported tion about the spin-glass freezing in DMS as value of D ~::~ 0.1 K can, as calculations of the such, we would like to point out that aniso resulting Schottky anomaly have shown, ex tropy might play an important role. From plain, in principle, the increase of the low Monte Carlo calculations and renormaliz temperature Cm. Further direct evidence on ation group treatments28,29 , it has been sug this anisotropy is, however, difficult to ob gested that, in general, an additional aniso tain. Preliminary experiments on a single tropy in an isotropie system will lower the crystal with 1 at.% Mn2• in an external field critica! dimension and a small anisotropy is applied along the principal axis showed no Chapter Il, § 1 MAGNETIC BEHAVlOR ..... , BASED ON Mn 34 macroscopie preferred direction. This, how ever, can be understood by assuming random -7 • JIRl- R local axes, in agreement with the ESR re o JIRl~e-SJR snlts39. In order to establish the presence of ' J(R)~ R-3 these anisotropic terros and their influence on the freezing process, further research is necessary. Vi ~ 0.1J0 APPENDIX In this Appendix we present a simple c: !? model which corroborates the usefulness of ~ 0.01J0 the sealing analysis relating the freezing tem 8 perature to the concentration. This model is 17 based generally on the idea of Smith and 0 • Escorne et a.l. 18 defining the SG freezing as the process of cluster blocking. 0.001J 0 We consider a partienlar magnetic ion ' . which may be blocked (frozen) by coupling ' ' with its magnetic neighbors if the exchange 1N 2N 3N 4N energy J(R )S2 is larger than the thermal en 0.0001J0 • t • • ergy k8 T. On the other. hand, an ion is C?n 1 2 sidered as being "free" (1.e., freely responding RINN distance i to the external magnetic field) if it bas no magnetic neighbor inside the sphere of radiU,S J(R) Rt defined by FIG. 14. Exchange constant and radial de pendenee for J(R) "' R-7 (e) and J(R) "' exp(-ó.lR) (o). The prefactors are ehosen so tha.t J{Ru) =Jo. (Al) Arrows indiea.te nearest neighbors (lN), next-nearest neighbors (2N), and so on. For The probability that a partienlar ion is free is eomparison also J(R)"' R-3, corresponding to given by dipole-dipole interaction, is also shown. The prefactor for this case was ehosen to be 0.01 Jo, an n· Prree = (1-x) 1 (A2) overestimation of the dipole-dipole interaction in DMS. where n1 is the total number of lattice sites 3 inside a sphere with volume 41fRi /3. fora fee structure). Equation (Al) relates R to the tempera 1 For DMS materials the exchange con ture and Prree may he calculated directly for stant is supposed to he relatively long ranged a partienlar lattice, as we will show below. Before that we consider the approximation (i.e., extending also for further, ~ot only t~e nearest, neighbors) and dependmg on dis fora very dilute system which results in ana tanee lytica! solution. For a very dilute system as (i.e., semicontinous distribution of ions) we J(R) = JoR-n have (A4a) or (A3) J(R) = J 0'exp{-aR). (A4b) In Fig. 14 both relations (A4a) and {A4b) are where A is volume per one lattice site (A= shown for n = 7 and a= 5.1 as proposed for a.3 for a simpie-cubic structure and A= a3/4 Cha.pter II, § 1 Ma.gnetic beha.vior of ..... Znt.xMnxSe 35 ftr; Lathce JIRI•J"R7 fee: lattice 10 temperature {Jo) FIG. 15. Probability that a. magnetic ion in a fee FIG. 17. Proba.bility tha.t a. ma.gnetic ion in a fee la.ttice is not blocked for z = 0.01, 0.05 a.nd 0.10 a.s la.ttice is not blocked as a function of temperature a funetion of tempera.ture for continuous distribu for J(R) "' R" 7 a.nd z = 0.01, 0.05, 0.10, 0.20 a.nd tions. Solid line, J(R)"' exp(-5.1R) ((A5b)J; dasbed 0.30. At the nea.rest-neighbor dista.nee J{R) = J0. line, J(R) "' R-7 [(A5a)]. Temperature is in Jo units, where Jo is the interaction va.lue for the nearest neighbors. fcc latlire x-0.0'5 fc( lattice t( .. o.os - exp {-S.1Rl R-7 ·······R-4 0.5 0.001 0.001 10 temperature !Jol temper~ture !Jol FIG. 18. Proba.bility tha.t a. ma.gnetic ion in a. fee FIG. 16. Probability that a magnetic ion in a. fee lattice is not blocked for z = 0.05 a.nd J(R) "' R-7. lattice is not blocked a.s a. function of temperature Solid steplike line, distribution the same a.s in for z = 0.05. Solid line, J(R)"' exp(-5.1R); da.shed Fig. 17; solid line, the same distribution but con line, J(R) "' R"1; dotted line, J(R) "' R-4. All inter voluted with exponentia.l function with 'Y = 0.4 J0; a.ctions are chosen so that all of them have the same da.shed line, continuous approximation [Eqs. (A5) va.lue (Jo) at nea.rest-neighbor dista.nce. and (A6)]. Chapter Il, § 1 MAGNETIC BEHAVlOR ..... , BASED.ON Mn 36 TABLE 111. Freezing temperatures Tf (in units of JoS2) for various Pr values a.nd concentrations. Concentration J(R) Prree 0.01 0.02 0.05 0.10 0.15 0.20 0.30 0.10 3.77xlO-I 1.39xl0'4 4.62al0'3 2.45al0'2 0.0773 0.139 0.208 e·l.lr 0.05 8.llxl0·7 3.49110'1 1.58al0'1 1.19xl0'2 0.0284 0.0619 0.105 0.01 3.94a10·8 2.69.to·• 1.95al0·4 2.18al0'3 0.00615 0.119 0.0228 0.10 1.3hl0... 6.55.w-• 5.4hl0"3 0.0218 0.0657 0.127 0.205 l/R7 0.05 5.82xl0·5 2.90.10'4 2.as.to·a 0.0106 0.0249 0.0549 0.103 0.01 1.25al0'5 5.90al0·5 4.88aiO·t 0.00225 0.00542 0.0105 0.0221 wide-band-gap DMS8.9. Finally we get from The obtained distributions are shown in (A1)-(A4), Fig. 15 for a fee lattice, showing a gradual decrease of the probability that an ion is not 4a/3[J S2/T]3/n !rozen with decreasing temperature. We as 0 (A5a) Prree = (1-x) 1 sume that fora sufficiently small Prree = Pr, where Pr is an arbitrary chosen constant de Prree = (1-x)4a/3[1/aln(Jo'S2JT)J3. (A5b) pending on the specific mechanism, the ion system may be considered !rozen, the tem perature at which Pr is reached defined as the x 0.02 0.05 0.1 02 03 freezing temperature Tr [Prree(Tr) = Pr]. Then, from (A5), one may obtain the sealing -n J!Rl•J0 R laws: n=7.0 lnTr"" ln lnx for (A4a) 1 (A6a) In Tr"" a x·113 for (A4b) . (A6b) These relations have also been reported by other authorslB and are applied in this ar -8 ticle. We now abandon the limitation to the very dil u te case ( continuons distribution as sumption) and extend our model to higher x. In Fig. 16 the probability distribution versus temperature is shown for ·x = 0.05 for both ·3 ·1 ln x interactions (A4a) and (A4b). As can be no ticed1 there is only slight difference between power (A4a) and exponential decay (A4b), FIG. 19. Logarithm of the freezing tempera.ture since for T ~ 0.003 J 0 (and a= 5.1, n = 7) reauiting from probability distribution (u discussed both dependencies are practically the same in the text) u a. function ofloga.rithm of contentra (cf. Fig. 14). It is also evident that for a tion for J(R) = JoR·n, n = 7, for different freezing longer-range interaction (such as R-4) the ion proba.bilities Pr = 0.01, 0.05 a.nd 0.10. The slope of system freezes faster at higher temperatures the lines yields a.n exponent n = 6.8. than for a shorter-range interaction ( such as Chapter II, § 1 Magnetic beha.vior of ..... Zn1-xMnxSe 37 R-7). The steplike structure of the Pfree dis pinga, J. Ma.gn. Magn. Mater. 31-34, 1373 tribution is a consequence of the "sharp11 (1983). freezing condition (Al). The steps correspond 3 C.J.M. Denissen, Sun Da.kun, K. Kopinga to the passing through consecutive discrete W.J.M. de Jonge, H. Nishihara, T. Sakakiba.ra. coordination spheres. In Fig. 17 the Prree a.nd T. Goto, Phys. Rev. B 36, 5316 (1987). distributions are shown for 0.01 ~ x ~ 0.30. 4 R.R. Ga.la.zka., S. Nagata. a.nd P.H. Keesom, It follows from this figure that for low con Phys. Rev. B 22, 3344 (1980). centrations (x~ 0.05) the ion system freezes S. Na.gata, R.R. Ga.la.zka., D.P. Mullin, H. Ar gradually whereas for higher x (x~ 0.10) barzadeh, G.D. Khattak, J.K. Furdyna. and rather abrupt freezing may be observed. For P.H. Keesom, Phys. Rev. B 22, 3331 (1980). x> 0.3 (Le., exceeding the validity of our M.A. Novak, O.G. Symko, D.J. Zheng and model) the Prree distribution practically does S. Oseroff, J. Appl. Phys. 57, 3418 (1985). not depend on x, as could be expected. To 7 M.A. Novak, O.G. Symko, D.G. Zheng and S. obtain a more realistic model we may Oseroff, Physica. 126B, 469 (1984). "smooth" the freezing condition (Al) by con A. Twardowski, C.J.M. Denissen, W.J.M. de voluting the obtained distributions with a Jonge, A.T.A.M. de Waele, M. Demia.niuk and Gaussian, Lorentz, or an exponential func R. Triboulet, Solid State Commun. 59, 199 tion. The result for exponential function (0.5x (1986). exp(-pog Tl h) where 1 is the full width at 9 C.J.M. Denissen a.nd W.J.M. de Jonge, Solid half maximum parameter} is shown in Fig. 18 State Commun. 59, 503 (1986). for x= 0.05 and arbitrary chosen as 1 = 0.4>< 10 W.J.M. de Jonge, A. Twardowski and C.J.M. J 0. Freezing temperatures Tr found from this Denissen, in Diluted Ma.gnetic Semiconductor.f, convoluted distributions are tabulated in edited by R.L. Aggarwal, J.K. Furdyna a.nd S. Table III for various Pf values. It may be von Monar (Materials Research Society, Pitts noticed that the Tr valnes obtained for Pr = burgh, 1987), Vol. 89, p. 153. 0.05 are well comparable with experimental 11 A. Twa.rdowski, M. von Ortenberg, M. Demi results for Zn 1_xMnxSe (J0 = -13 K and S = a.niuk a.nd R. Pauthenet, Solid State Commun. 5/2). They also obey formulas (A6) quite 51, 849 (1984). well. An example is shown in Fig. 19 for 12 J.K. Furdyna., private communications. 7 13 J(R) = J0R- for Pf = 0.10, 0.05, and 0.01. E. Micha.lski, M. Demia.niuk, S. Ka.czmarek Good linearity may be observed for x~ 0.20. a.nd J. Zmija., Electron Technology 13, 4, 9 The n value deduced from the slope is l:i 6.8, (1982). which compares favorably with the inserted 14 A. Twa.rdowski, T. Dietl a.nd M. Demia.niuk, value n = 7. We therefore feel confident in Solid State Commun. 48, 345 (1983). using sealing laws in the form (A6) to de 15 R.B. Bylsma, W.M. Becker, J. Kossut and U. scribe our data, even for a discrete lattice and Debska., Phys. Rev. B 33, 8207 (1986). at somewhat higher concentrations. 16 P.H. Keesom, Phys Rev. B 33, 6512 (1986). 17 J.A. Mydosh, in Hyperfine Interactions, Vol. 31 of Lecture Note11 in Phy!ÎC8 (Springer, New York, 1986}, p. 347. See the following review papers: N.B. Brandt, 18 D.A. Smith, J. Phys. F. 5, 2148 (1975). V.V. Moshcha.lkov, Adv. Phys. 33, 193 (1984); 19 M. Escorne, A. Mauger, R. Trihoulet a.nd J.L. J.K. Furdyna., J. App!. Phys. 53, 8637 (1982); Tholence, Physica. 107B, 309 (1981). J.A. Gaj, J. Phys. Soc. Jpn. Suppl. 49, A797 20 K. Matho, J. Low Temp. Phys. 35, 165 (1979). (1980). 21 M. Rosso, Phys. Rev. Lett. 44, 1541 (1980) 2 C.J .M. Denissen, H. Nishihara., J.C. van Gooi P.A. Thomas and M. Rosso, Phys. Rev. B 34, a.nd W.J.M. de Jonge, Phys. Rev. B 33, 7637 7936 (1986}. (1986); W.J.M de Jonge, M. Otto, C.J.M. 22 Y. Shapira., S. Foner, D.H. Ridgley, K. Dwight Denissen, F.A.P. Blom, C. v.d. Steen, K. Ko- a.nd A. Wold, Phys. Rev. B 30, 4021 (1984). Chapter IJ, § 1 MAGNETIC BEHAVlOR ..... , BASED ON Mn 38 23 T.M. Giebultowicz, J.J. Rhyne and J.K. Fur 30 C.D. Amarasekara, R.R. Galazka, Y.Q. Yang d.yna, in Proceediags of U.e tb.irtp-fin' •••Wil and. P.H. Keesom, Phys. Rev. B 21, 2868 cos/erenee os magndîsm and magnetû: ma (1983). teriaJI, Baltimore, 1986, edited by N.C. Coon, 31 Y.Q. Yang, P.H. Keesom, J.K. Furdyna and J.D. Adam, J.A. Beardsley andD.L. Huber (J. W. Giriat, J. Solid State Chem. 49, 20 (1983). Appl. Phys. 61, 3537 (1987); 61, 3540 (1987)]. 32 S.P. McAlister, J.K. Furdyna and W. Giriat, 24 J.P. Lascaray, M. Nawrocki, J.M. Broto, M. Phys. Rev. B 29, 1310 (1984). Raltoto and M. Demianiuk, Solid State 33 M. Escorne and A. Mauger, Phys. Rev. B 25, Commun. 61, 401 (1987). 4674 (1982). 25 N. Bloembergen and T.J. Rowland, Phys. Rev. 34 N.B. Brandt, V.V. Moshchalkov, A.O. Orlov, 97, 1679 (1955); G. Bastard and C. Lewiner, L. Skrbek, I.M. Tsidil'kovskii and S.M. Chu ibid. 20, 4256 (1979). dinov, Sov. Phys. JETP 57, 614 (1983). 26 W. Geertsma, C. Haas, G.A. Sawatzky and G. 35 A. Mycielski, C. Rigaux, M. Menaut, T. Dietl Vertogen, Physica 86-88A, 1039 (1977); G.A. and M. Otto, Solid State Commun. 50, 257 Sawatzky, W. Geertema and C. Haas, J. Magn. {1984). Magn. Mat. 3, 37 (1976); B.E. Larson, K.C. 36 M. Escome, A. Mauger, J .L. Tholence and R. Haas, H. Ehrenreich and A.E. Carlsson, Solid Triboulet, Phys. Rev. B 29, 6306 (1984). State Commun. 56, 347 (1985). 37 R.R. Galazka, W.J.M. de Jonge and A.T.A.M. 27 M. Taniguhi, M. Fujimori, M. Fujisawa, T. de W aele and J. Zeegers, Solid State Commun. Mori, I. Soumaand Y.Oka, Solid State Com 6, 1047 (1988). mun. 62, 431 (1987).' 38 N.P. ll'in and I.Ya. Korenblit, Zh. Eksp. Teor. 28 A.J. Bray, M.A. Moore and A.P. Young, Phys. Fis 81, 2070 (1981) [Sov. Phys. - JETP 54, Rev. Lett. 56, 2641 (1986). 1100 (1982)]. 29 A. Chakrabarti and Ch. Dasgupta, Phys. Rev. 30 J. Kreissl and W. Gehlhoff, Phys. Status Solidi Lett. 56, 1404 (1986). (a) 81, 701 {1984). Chapter II, § 2 Magnetic properties of Znt-xMnxS 39 Chapter II, paragraph 2 Magnetic properties of zn1_xMnxS [ published in Solid State Commun. 66, 791 (1988) ] I. INTRODUCTION pared with the Cd or Zn selenides or tel lurides, but also it has been reported that the The magnetic properties of diluted mag tendency for hybridization of the Mn d levels netic semiconductors (DMS) or semimagnetic increases going from the Te via Se to S; see semiconductors (SMSC) have attracted con Taniguchi et al. 7. Both aspects are believed siderable attention during the past years. to he of possible importance in determining Most of the investigations have been devoted the strength and range of the Mn-Mn ex to the class of the II-VI compounds such as change8. Cd 1_xMnxTe, Znt-xMnxTe, Hg 1_xMnxTe, and Since no data in the low-eoncentration the conesponding selenides. Among others, regime are yet available for Zn 1_xMnxS we the phase diagram, including the transition thought it worthwhile to study the magnetic to the spin-glass (SG) phase, the interaction properties of this compound in some detail. between the Mn2• ions and the carrier sys To that purpose we have performed ac sus tem, as well as the nature and strength of the ceptibility and specific-heat capacity 2 interaction between the Mn • ions, have been measurements on Zn 1_xMnxS for concentra extensively studiedl. tions x below 10%. From the concentration In some recent publications2-6 we em dependenee of Tf the specific decay of the phasized the relevanee of the long-range in Mn2•-Mn2+ interaction with the interatomie teractions between the magnetic ions beyond distance wîll be derived. Subsequently we the nearest neighbor. It has been demon will try to interpret the specHic-heat data, strated that only when the long-range char tagether with earlier magnetization and sus acter of the d-d interaction is taken into ac ceptibility data on the basis of one model count, a simultaneous and consistent descrip incorporating short-range as well as long tion of all the thermadynamie properties can range interactions in a random array. be obtained. Moreover, the range and strength of this long-range interaction can he IT.RESULTS used to relate the interaction to the under lying physical mechanism. In addition, it has The ac susceptibility and specific-heat been shown6 that the freezing temperature Tr experiments were performed on Zn 1_xMnxS which marks the transition to the spin-glass with x= 0.028, 0.047, 0.092 and 0.097. All state, can be used as a probe to determine concentrations were checked by mieroprobe the spatial range of the interaction J(R) in analysis. The susceptibility was measured this class of compounds. with a mutual inductance bridge operating at In various aspects, Zn 1_xMnxS represents 90 - 9000 Hz and an excitation field of less an extreme case. Not only does it possess the than 1 G. The low-temperature results are shortest interatomie lattice distance, as com- shown in Fig. 1. The well-pronounced cusp Chapter II, § 2 MAGNETIC BEHAVlOR ..... , BASED ON Mn 40 TABLE I. The freesing temperatures Tf for Zn1-xMnxS. .. - Tr(x) x "" 10 \ G~ ... .5 0.023 0.028 0.050 0.047 "' -2 0.28Ó 0.092 +,+< -4 /.: 7.97 0.30 8 12 /', 11.46 0.35 .-213 //. Zn Mn S . *- I-X X 13.97 0.40 g' ·;;; ti 0.1 -7.6 (or kink) in the ac susceptibility is attributed J: /f/K J-R // to the paramagnetic to spin-glass transition. />i> In Table I the freezing temperatures as de 0.01 duced from Fig. 1 are tabulated, including /.// some earlier data for higher concentrations OJ)1 0.1 coocentration x obtained by Yang et a1.1o. It seems obvious that also in this case a freezing transition extends to very dilute sys FIG. 2. Freezing tempera.ture Tr of Znt-xMnxS as a function of the Mn concentration 11:. The full tems, far below the nearest-neighbor (NN) circles are data. obtained from Yang et a.l. (Ref. 10). The solid line and the dashed-dotted line repreaent n = 7.6 and n = 6.8, respectively, with logTr "' I fnlogz. In the inset In Tf is plotted as a function of ..... ~:· 2 /3 (see the text} . percolation limit. As quoted earlier, the con centration dependenee of the freezing tem • perature Tr(x) can be related to the radial . dependenee of the exchange interaction trig • x= 0.028 .. gering the transition. By assuming a power .. law behavior J(R)"' R·n, Tr(x) can be ex pressed6 as x •• - t x •• x ••• logTr(x)"' in logx . (1) x .. OJ)47 x .... ·,. In Fig. 2 we plotted Tr versus x, both on a "x - logarithmic scale. The figure shows indeed a OJ)92 d>o "'\.. linear dependenee between logTr and logx omooo oom o o o (t)O oo o~ characterized by n 7.6. This result differs '~>q,~ = t slightly from results in related materials like OJ)1 0.1 A1-xMnxB in which A = Zn,Cd and B = Se, ternperature [ K 1 Te, where for all the systems n = 6,8 has been found ( indicated by a dashed-dotted curve in Fig. 2). We will return to this point in the discussion. 1. FIG. Low--temperature ac susceptibility of The specific heat of Zn1.xMnxS has been Zn1-xMnxS; f = 1000 Hz, Hac llj 0.5 G. The arrows measured with an adiabatic heat-pulse calori indica.te the freesing temperature Tf. meter between 0.4 and 15K. The magnetic Chapter II, § 2 Magnetic properties of Znt-xMnxS 41 0.25r----r---.---...,-----, dynamic properties reileet the typical behav ior for the II-VI wide gap DMS. The magnet ization, as exemplified for Zno.g 58Mn0.042S in 0.20 Fig. 4, reveals at low temperatures so-ealled technica! saturation in :fields up to 15 T, well below M8 (5/2)gp,B (l:j 12 emu/g in this particular case) for uncoupled paramagnetic - 0.15 ions. This behavior supports strong antiferro magnetic (AF) coupling between the Mn2+ V ions, which is common for related DMS. In -·~ 0.10 Fig. 5 the susceptibility of Zn 1_xMnxS is VI shown. For high temperatures the susceptibil :.::V ity obeys typical Curie-Weiss behavior. The '"g Curie--Weiss temperature is negative, again E 0.05 indicating antiferromagnetic coupling, and scales with x, while x deviates from Curie Weiss law at lower temperatures. It follows from our data that the d~d 4 12 16 exchange interaction is antiferromagnetic and temperature [KJ rather long ranged, decaying like R-7.6. Con sequently, the system consists of a random FIG. 3. Magnetic specific heat (Cm) of Znt-x array of Mn2+ ions all coupled with an anti MnxS· The solid linea repreaent the ENNP A calcula ferromagnetic interaction whose strength tions with JNNI~ = -16 K and J(R)/kJI depends on the speci:fic Mn-Mn dista.nce. The -lOR-7.6 K. behavior of such a.n extremely complicated contribution Cm, obtained by subtracting the 10.-----.-----~------~----~ lattice speci:fic heat of ZnS from the total speci:fic heat, is shown in Fig. 3. The scat tering of the data points at higher T is brought about by the fact that in that tem perature region the magnetic contribution Cm is only a small fraction of the total specific heat, especially for low concentrations x. In contrast to the specific heat measured in re lated compounds, the structure observed in Cm is rather pronounced. As we will show later on, this is due to the extremely high value of the NN interaction and the fast de cay of the interaction as indicated by n = 7.6, resulting in well-separated values for the 0 10 20 30 nearest-neighbor (NN), next-nearest-neigh field [ T J bor (NNN) and further magnetic-neighbor interactions. Magnetization ( M) and high-tempera FIG. 4. High-field magnetization {M) of ture susceptibility (x) measurements have Zno.9ssMno.042S for several temperatures (Ref. 11). been reported by Twardowski et a1. 11 and The solid lines represent the ENNPA calculations Brumage et a1. 12 , respectively. Both thermo- with JNNfkn =-16K and J(R)/~ = -10R"7.6 K. Cha.pter II, § 2 MAGNETIC BEHAVlOR ..... , BASED ON Mn 42 network can only he solved in an a.ppro:xi A = E tAi(Ji)Pi(x) , (3) mative way. In the dilute limit the so-ca.lled i extended nearest-neighbor pair a.pproxima. tion (ENNPA), introduced by Mathots, has where P1(x) is the probability of finding at been successfully used recently for other least one magnetic neighborat a distance R1. DMS2-6. It is ba.sed on the a.ssumption that The summation is performed until 99.5% of the partition function of such a system can be the pairs is taken into account. This ENNP A factorized into contributions of pairs. In the model may be extended by including 3-spin ENNP A each ion is considered to be coupled clusters (open triples} in order to account for by an exchange interaction J 1 only to its the increasing probability to find clusters of nearest magnetic neighbor, loca.ted anywhere more than 2 spins, with increasing x. at the distance R1. This interaction is treated The only input parameters for our model in the conventional isotropie Heisenberg are the exchange constanis J1 for the differ Hamiltonian: ent possible pairs. For NN we take JnfkB = -16K resulting from recent neutron-diffrac (2) tion studies14, whereas the long-range inter action we assumed in the form deduced from The statistica! weight of the pair configur SG freezing, J(R} = 1t1R-7.6. hR is not ation is assumed to be determined by the necessarily equal to J11 since one cannot a random distribution of the ions. Thus any priori exclude different interaction meeban thermodynamic function of the crystal can be isros between nearest and further neighbors. evaluated as In fact h1. was our only adjustable para meter chosen to obtain the best overall agree ment with the data and amounts to -10 K, which means Jrrn/kB = -0.72 K, J1nnudkB = -0.15 K and so on. Let us now consider the numerical re sults for Cm, X and Mof Zn1_xMnxS· As we quoted above, the NN interaction strength is 20 times as large as the NNN interaction strength. This is convincingly demonstrated by the theoretica! as well as the experimental magnetic contribution to the specific heat (Fig. 3}, where the contribution of Jn, at higher temperatures, is well separated from the low-temperature contribution of more distant pairs. From the figure it is clear that the specific heat of magnetic neighbors be yond the NN cannot be neglected. At higher coiicentrations quantitative deviations be 300 400 tween experiment al data and the ENNP A temperature ! KI calculations occur, which is inherent to the pair appro:ximation. The magnetization of Zn 1.xMnxS is also well predicted within the concepts of our FIG. 5. High-tempera.ture susceptibility (X) of model, as shown in Fig. 4. In this case we Znt-xMnxS (Ref. 12). The solid lines repreaent the considered each pair to be influenced by the ENNPA ca.lcula.tions with Jn/lts =-16K and effective field Berr arising from all ions more J(R)/lts = -lOR-1.6 K. distant than Ril Chapter JI, § 2 Magnetic properties of Znt-xMnxS 43 tematic decrease of the nearest-neighbor (JNN) and further-neighbor interaction strength (hRR-n) in the series going from sulfides to telinrides is obvious. This trend is in accordance with the systematic decrease of It is also illustrated in the figure that the the p-d hybridization in such a series as ob inclusion of triples is highly necessary to ob served by Taniguchi et al. 7 by synchroton tain a correct magnitude of M. In fact, the radlation photoemission. According to Larsou ground state of an open triple is degenerate et al. 8 in their treatment of superexchange, and responds to relatively low fields, whereas such a decrease of hybridization ( denoted by the ground state of a pair is nondegenerate the parameter Vdp) yields a decrease of J and contributes to M only at sufficiently high since their results indicate that J N vdp 4. fields (B > 2IJI/gf.LB), when it becomes mag With respect to the R dependenee of the in netically active, leading to the well-known teraction Ehrenreich et al. 15 state that, on steps in the magnetization. The first step in the basis of the same superexchange mechan the magnetization of the NN pairs in Zn 1_x ism within a class of materials, the R depend MnxS is predicted at a field of roughly 25 T, enee is expected to remain largely independ far outside the present experimental data ent of the actual atomie constituents. Despite range. the slight increase of n for Zn 1_xMnxS it Finally, also the susceptibility agrees seems that the results tabulated in Table II fairly with the ENNP A prediction. In Fig. 5 qualitatively confirm this expectation. It the theoretica! curves for x-1 bend from should be noted, ho wever, that these authors Curie-Weiss law at lower Tand the extrapo predict a Gaussian decay of the interaction lation from the high-temperature regime to out to about the fourth NN [J"' 0.2exp(--4.89 the T axis reflects the AF net interaction. •R2fa2)(eV) where a is the lattice constant], However, quantitative comparison does not instead of the power dependenee for all dis seem very significant, since the magnitude of tauces ("' R-n) employed in the present analy the susceptibility has been fitted to the Mn sis. Such a Gaussian decay for all R would concentration 12. yield a proportionality between lnTf and x-213 which is not supported by the data (see III. DISCUSSION AND CONCLUSIONS the inset in Fig. 2). One might argue, how ever, that Tf(x) probes specifically the long Summarizing the results shown before range interactions, which is also implicitly we feel that the magnetic properties of Zn1.x assumed when continuons sealing laws are MnxS, in partienlar in the dilute limit, can be applied. simultaneously described with a single set of A final comment concerns the freezing parameters, provided the long-range charac transition. It has been noted before6 that the ter as deduced from the concentration de actual observed freezing temperature Tf as a pendence of the freezing temperature, is function of the concentration x seems surpris- taken into account. In that respect, we would like to emphasize that no fitting procedure has been employed. Although such a pro TABLE IL Exchange interaction parameters for cedure certainly would yield a better descrip Znt-xMnxS, Znt-xMnxSe and Znt-xMnxTe1,6. tion of some of the data, it remains to be seen, however, whether such a procedure n would give physically meaningful results. In Table II we have tabulated the ex Zn1_xMnxS -16 -10 7.6 -13 change parameters of Zn _xMnxS together Znt-xMnxSe -7 6.8 1 Zn 1_xMn.Te -9 6.8 with the closely related Se and Te. The sys- Chapter II, § 2 MAGNETIC BEHAVlOR ..... , BASEDON Mn 44 ingly universa.! for a large class of wide-gap edited by R.L. Agga.rwal and J.K. Furdyna Mn-type DMS A1-xMnxB where A= Cd,Zn and S. von Molnar (Materials Research So and B = Se,Te. The present results on Zn 1_x ciety, Pittsburgh, 1987}, Vol. 89, p. 153. MnxS seem to strengthen this impression. As 6 A. Twa.rdowski, H.J.M. Swagten, W.J.M. de we quoted in the Introduction, Zn1.xMnxS Jonge and M. Demianiuk, Phys. Rev. B 36, represents an extreme case with respect to 7013 (1987). hybridization, band gap and lattice constant. 7 M. Taniguchi, M. Fujimori, M. Fujisawa, T. The resulting exchange constants, as we saw Mori, I. Souma and Y. Oka, Solid State Com in Table 11, are almast a factor two larger for mun. 62, 431 (1987). 8 Zn1_xMnxS than for the corresponding Zn1_x W. Geertsma, C. Ha.ss, G.A. Sawatzky and G. Mnx Te. Despite all this, the freezing transi Vertogen, Physica 86-88A, 1039 (1977); G.A. tion Tf(x) is very close to the other wide-gap Sawatzky, W. Geertsma and C. Ba.ss, J. Ma.gn. materials and in the very dilute limit even Magn. Mater. 3, 37 (1976); B.E. Larson, K.C. lower (see Fig. 2) which seems to be at vari Hass, H. Ehrenreich and A.E. Carlsson, Solid anee with the common wisdom that the State Commun. 56, 347 (1985). freezing transition should be proportional to 9 A. Twa.rdowski, C.J.M. Denissen, W.J.M. de the driving interaction. Therefore, it seems Jonge, A.T.A.M. de Waele, M. Demia.niuk and that these observations point to alternative R. Triboulet, Solid State Commun. 59, 199 triggering mechanisms of the spin-glass tran (1986). sition such as anisotropy, topological effects 111 Y.Q. Qang, P.H. Keesom, J.K. Furdyna. and or system-independent long-range interac W. Giriat, J. Solid State Chem. 49, 20 (1983). tion mechanisms, which have not been con 11 A. Twa.rdowski, M. von Ortenberg, M. Demia sidered so far. niuk and R. Pauthenet, Acta Phys. Pol. A 67, 339 (1985). 12 W.H. Brumage, C.R. Ya.rger and C.C. Lin, Phys. Rev. 133, A765 (1964). See the following review papers: N.B. Brandt ta K. Ma.tho, J. Low Temp. Phys. 35, 165 (1979). and V.V. Moshchalkov, Adv. Phys. 33, 193 14 T.M. Giebultowicz, J.J. Rhyne a.nd J.K. Fur (1984); J.K. Furdyna, N. Samarth, J. Appl. dyna, in Proceeding/1 of the Thirty-First An Phys. 61, 3526 (1987). nual Conference on Magnetism o.nd Ma.gnetic 2 C.J.M. Denissen and W.J.M. de Jonge, Solid Ma.terials, Baltimore, 1986, edited by. N.C. State Commun. 59, 503 (1986). Coon, J.D. Adam, J.A. Bea.rdsley a.nd D.L. 3 C.J.M. Denissen, H. Nishihara, J.C. van Gool Huber [J. Appl. Phys. 61, 3537 (1987); 61, and W.J.M. de Jonge, Phys. Rev. B 33, 7637 3540 (1987)]. (1986). H. Ehrenreich, K.C. Hass, B.E. La.rson and 4 C.J.M. Denissen, Sun Dakun, K. Kopinga., N.F. Johnson, in Diluted Magnetic SemicOfl. W.J.M. de Jonge, H. Nishihara, T. Sakakibara ductors, edited by R.L. Agga.rwal a.nd J .K. and T. Goto, Phys. Rev. B 36, 5316 (1987). Furdyna. and S. von Molnar (Materials Re W.J.M. de Jonge, A. Twardowski and C.J.M. search Society, Pittsburgh, 1987), Vol.89, p. Denissen, in Diluted Magnetic Semiconductors, 187. CHAPTER THREE Magnetic behavior of 11-VI group diluted magnetic semiconductors, basedon Fe The magnetic specific heat and !ow-field ac susceptibility of the iron-based diluted magnetic semiconductor (DMS) Znt-xFexSe in the temperature range 0.4 - 20 K are reported in paragraph 1. Antiferromagnetic d-d interactions and indications of spin-glass freezing are observed. These data, together with high-temperature susceptibility and high-field magnetization, are interpreted in the extended nearest-neighbor pair approximation, with an antiferromagnetic exchange between the Fe2+ ions. Fair agreement between theory and experiment is obtained for a.ll the thermadynamie quantities with one set of parameters provided by independent experiments. The strength and range of the interactions, as well as the freezing mechanisms, are com pared with DMS containing Mn2+, and the results are discussed in rela tion with the existing models. In paragraph 2 the specific heat of the diluted magnetic semicon ductors Hgt-x-yCdyFexSe will be presented for temperatures below 25 K. The excess specific heat (Cm) has been extracted by sealing of nonmag netic lattice contributions. The resulting Cm of these systems reveal a broad maximum at T :ll 10 K, whereas for low temperatures a Schottky type exponential decay of Cm can be observed. These phenomena can be fairly well understood on the basis of a simple crystal-field model and a random distribution of Fe2+. On the other hand, no drastic effect of the energy gap is clearly reflected by our data and therefore, we suggest that only Fe2+ states invalving the valenee bands are dominating the anti ferromagnetic d-d interactions, analogously to the Mn2+ case. The high-temperature magnetic susceptibility of random, diluted magnetic systems is discussed in paragraph 3, for magnetic ions having not only spin momenturn but also nonvanishing orbital momentum. In particular, the magnetic susceptibility of Znt-xFexSe (Fe2+, S = 2, L = 2) is reported in the temperature range 2 - 300 K. The nearest-neighbor exchange Fe-Fe interaction is estimated as: Jni.fkn = -22 K for Znt-x FexSe and JNN/kB = -17.8 K for Cdt-xFexSe. In paragraph 4 the high-field magnetization of diluted magnetic semiconductors has been calculated for magnetic ions with spin- and orbital momenta. It is shown that, in addition to the spin-only case (Mn2+), also for Fe-based DMS cha.racteristic steps in the lew-tempera ture rna.gnetiza.tion should he detectable. The position of first step in the magnetization, being almest insensitive for crystal-field parameters and sample orientation, can be used to establish the nearest--neighbor inter action (JNN ), provided that I Jn I exceeds a critica! value. Chapter III, § 1 MAGNETIC BEHAVIOR ..... , BASEDON Fe 46 Chapter m, paragraph 1 Magnetic properties of the diluted magnetic semiconductor zn1_xFexSe [ publisbed in Phys. Rev. B 39, 2568 (1989) ] I. INTRODUCTION cally attractive case, since all the phenomena which involve orbital momenturn are absent. Diluted magnetic semiconductors (DMS) In contrast, substitutional iron (Fe2+) or semimagnetic semiconductors (SMSC) - can serve as a much more general case, since i.e., II-VI, II-V, or IV-VI semiconductors it possesses both spin and orbital momenta with a controlled amount of magnetic ions (S = 2 and L = 2). Fe2+ has a d6 electronic substituted for nonmagnetic cations - have configura.tion. The ground state of an Fe2+ been extensively studied during the recent free ion (6D) is split by a tetrabedral crystal years because of their interesting magneto field into a li E orbital doublet and a higher optical and magnetic properties1• Since DMS lying 5T orbital triplet (separated from 5E by can generally be synthesized with a wide 6. = lODq, where Dq is the crystal-field range of concentrations of magnetic ions, parameter). The spin-orbit interaction splîts they offer an exceptional possibility to study the liE term into a singlet A 11 a triplet T 11 a both the very diluted and the concentrated doublet E, a triplet T2 and a singlet A 2 (the regimes in one material. In fact, the existence energy separation between these states is of paramagnetic, spin-glass and antiferro approximately equal to 6>.2/lODq, where >. is magnetic phases was reported in these ma the spin-orbit parameter)U. The ground terials depending on the concentration and state is a magnetically inactive singlet A 1 temperature range2. resulting in Van Vleck-type paramagnet So far the research has been devoted ismH. The properties of Fe-based DMS are, mainly to DMS containing Mn ions. The however, relatively unexplored yet7, although magnetic behavior of these systems show recently some attempts to understand the common characteristics which can be under magnetic behavior of these compounds were stood on the basis of a random array of local reported8·11. In view of that, we thought it ized Mn ions coupled by long ranged, iso worthwhile to study the magnetic properties tropie antiferromagnetic ·exchange interac of Zn1.xFexSe in some detaiL tions2.3 which are mediated by the carriers. In this paper we report the results of The underlying microscopie mechanisms in magnetic specific heat and low-field suscepti relation to the band structure, as well as the bility measurements in the temperature range nature of and the driving force bebind the 0.4 - 20 K. Together with the high-tempera observed phase transition and the role of the ture susceptibility obtained by us 11 very re anisotropy, are, however, still somewhat ob cently, and high-field magnetization dataB, scure and object of further investiga.tionsl-3. we were able - for the first time for Fe In that respect, substitutional Mn2+ with based DMS- to describe all these magnetic 6 its degenerate A1 spin-only ground state properties within one model (extended near represents a rather simple, though theoreti- est-neighbor pair approximation) and obtain Chapter III, § 1 Magnetic properties of ..... Znt-xFexSe 47 ---··------ a fairly good agreement between experiment I I I and theory. . 1- ~ Zn 1_xFexSe + x~ 0.014 8 • 0 - •• 0 x x~ 0.042 11. EXPERIMENTAL RESULTS .!:' ' 0 . x~ 0.062 ::> " 0 0 E . 0 X= 0.144 <11 0 6r- .. . x= 0.207 - The samples of Zn _xFexSe were grown b 1 "' .. 0 by the modified Bridgman methad under the ,., x 0 pressure of a neutral gas, in the nominal con .... ~x~ ...... 0 *x . 4 •• x centration range 0.001 < x < 0.55. All the :e +- .. . • ingots were checked by mieroprobe analysis a. •"' :.'.: ...~· and it was found that the iron concentration ::I"' + + x_ never exceeded x= 0.22. Growing crystals Vl 2 -· with higher iron content only yielded an in + crease of precipitations of iron selenide. Crys I I tals with precipitations were not used for 5 10 15 20 further study. temperature IK l The crystalline structure of this material was studied by x-ray diffraction and was FIG. 2. ac susceptibility of Znt-xFexSe a.s a found to be cubic in the whole concentration function of temperature. range. Variation of the resulting lattice con stant with x is shown in Fig. 1. Results re 11 ported for Zn 1_xFexSe thin layers are also The concentration of these impurities was inserted in this figure. typically less than 0.01% of the actual Fe2+ The electron paramagnetic resonance content. The Fe concentration of the crystals (EPR) analysis performed on some of our reported in this paper was x= 0.014, 0.04, samples revealed the presence of paramagnet 0.062, 0.14 and 0.21. ie impurities (mainly Mn 2•) originating from the souree materials used for crystal growing. A. Low-temperature susceptibility The ac susceptibility x was measured with a conventional mutual inductance bridge operating in the region 100 < f < 2000Hz and fields less than 0.0001 T. Some representative susceptibility data are shown in Fig. 2. The data for low concentrations (x < 0.07) differ from susceptibility results previously reported for Fe2•-doped crystals&: instead of temperature-independent suscepti bility at low temperatures (i.e., typical Van Vleck-type paramagnetism), we abserve a monotonons increase of the susceptibility. We believe that such a behavior is due to para magnetic impurities present in our crystals, as mentioned earlier. We estimated this addi FIG. 1. Lattice constant of Znt-xFexSe as a tional contribution to x as a difference be function of concentration x. The solid line denoting tween the x value at 4 K and x measured 2 a(z) (5.666±0.001) + (0.051±0.01)z Á, is obtained below 4 K. For Zn 0 •958Fe0.042Se the Mn + from a least-5quares fit. The dashed line represents impurity concentration was estimated this results from Ref. 12. way as ximp 0.0002, in fair agreement with Chapter III, § 1 MAGNETIC BEHAVlOR ..... , BASED ON Fe 48 EPR data for this crystal. For the crystals with higher x one can observe a maximum in x, which is well pro nounced for x = 0.21 at Tr = 9 K, whereas for x= 0.144, one can only notice that it ap 10 pears at roughly 2 K. Since no anomalies oc cur in the specific heat for these samples at the corresponding temperatures, we tenta tively ascribe these maxima to a paramag netic to spin-glass phase transition, in anal ogy with the Mn systems. We should stress, c:n however, that this conjecture requires further c study. In particular the zero-field--cooled and ·~ field--cooled de susceptibility should be .t • Zn 1_x Fe x Se o Cd _x Fe x Se measured, and a wider composition range 1 A Hg 1_xFexSe should be investigated. The latter one de 0.1 pends, however, strongly on improverneut in the crystal growing technology. Nevertheless, if one accepts the nature of the susceptibility anomaly as a spin-glass 0.04 0.06 0.1 0.2 0.4 freezing, then a sealing analysis should be applicable. Such a sealing a.nalysis generally exploits the fact that for a continuons ran FIG. 3. Freezing temperature as a function of 3 dom distribution it is assumed that R1j x is concentra.tion z for Znl-xFexSe, Cdt-xFexSe constant, where R1j denotes a typical dis (Ref. 10) and Hgl-xFexSe (Ref. 10). The solid line tanee between the ions. lmplementation of indicates a radial decay of the exchange integral this expression in a model for spin-glass J tv R-n with n ::1::: 12; for comparison the dashed freezing, given a known functional form for dotted line illustrates n = 6.8; see Ref. 3. the radial dependenee of the exchange inter action, then yields a theoretica! prediction for Tr(x) which can be compared with experi action between Fe ions than observed in the mental data. Mn case. We should stress, however, that This procedure was successfully used for this result can be considered only as an iudi Mn-based DMS 2•3•13, where Tr was found to cation of the possible difference in interaction obey the relation range for Fe and Mn ions, since the data available up till now are too limited. We will lnTr"' !n lnx . return to this point in the Discussion. This is compatible with a radial dependenee B. High-tem.perature susceptibility of the exchange interaction between Mn ions of the type: J(R)"' R-n, where n = 6.8 for The high-temperature susceptibility of 11 wide-gap materials as Cd 1_xMnxTe or Zn 1_x Zn 1.xFexSe was recently studied by us in MnxSe and n = 5 for lower-gap materials order to compare the experimental data with (Hg 1_xMnxTe and Hg1_xMnxSe)3. We per a hightemperature series expansion (HTE) formed a similar procedure for our data as derived for Fe2+. The susceptibility was found shown in Fig. 3, where we also inserted all to obey a Curie-Weiss law in the tempera the available data for other Fe-based DMS10. ture range 70 - 300 K, with a negative Curie The exponent n deduced from Fig. 3 is n :lj Weiss temperature indicating antiferromag 12, indicating a much shorter range of inter- netic coupling between Fe2+ ions. Based on Chapter III, § 1 Magnetic properties of ..... Znt-xFexSe 49 these results we evaluated the nearest 12 neighbor exchange integral Jn -22 K, which is substantially larger than in Zn _x Zn 1_xFexSe B:::: 2.75 T 1 10 x= 0.015 MnxSe [-12.6 K (Ref. 14)]. Such a strong 0 exchange interaction seems compatible with oo "' 'u:rP(lJ the freezing temperature Tr = 9 K for Zn0.79- u.."' 8 n"? o ~0 0 Fe0.21Se which is twice as large as that ob 0 <0 "' E oo0 0 Eöo ::; served3 in Zn1_xMnxSe with similar concen 6 ~q" tratien x. We will return to the detailed 0.062 .?:' analysis of the susceptibility for both low and E high temperatures later on. LJ 4 C. Specific heat 2 Specific-heat data were obtained with a conventional adiabatic heat-pulse calorime 0 ter in the temperature range 0.4 - 20 K. The magnetic contribution Cm to the specific heat was obtained by subtraction of the lattice FIG. 5. Magnetic specific heat of Znt-xFexSe for contribution of pure ZnSe. The results for Cm B = 2.75 T. (per mol Fe) in zero external magnetic field are shown in Fig. 4. As quoted earlier, no anomaly was observed at the temperature Tr as indicated by arrows in the figure. The overall behavior of Cm resembles a Schottky type anomaly for the lowest concentrations for which a well pronounced, braad maxi 12 mum is observed at about 10 K. For x > 0.14 no maximum is observed and Cm is practical ly a linear function of the temperature. In 10 distinction to the Mn-based DMS15,16,3,17 no - structure is observed below 2 K: Cm tends to "'ru u.. 8 zero monotonously with decreasing tempera 0 ture. No additional contribution at lower E ::; temperatures is expected since in the 6 measured range already ~ 87% of the entropy ~ E of the ten-level system is reeavered for x w 4 0.015 (assuming CPN T-2 forT> 20 K). The decrease of Cm/x with x indicates the relevanee of the Fe2•-Fe2• interaction. If the only contribution to Cm would be from single (isolated) ions, as proposed previous ly8, Cm would scale with x. We found that temperature the specific heat is not appreciably influenced by a magnetic field as shown in Fig. 5: only a very slight shift of Cm to higher tempera tures, hardly exceeding the experimental ac FIG. 4. Magnetic specific heat of Znt-xFexSe in curacy, can be observed. Such a behavior is the absence of magnetic field. The arrows indicate again in contrast with results reported for the freezing temperatures Tf. manganese DMS 3, 15. Chapter III, § 1 MAGNETIC BEHAVlOR .•... , BASED ON Fe 50 D. Magnetization space is split into four subspaces: 5E,.sE, 5Ex5T, 5Tx5E, 5Tx5T. IB all wide-g&p ma. The magnetization has been measured terials used as host lattices for DMS, the up to 15 T and was reported earlier by one of crystal-field splitting is of the order of the present authors8• For comparison, some 3000 cm -t and is much larger than the other relevant results will be shown later on. terms in Hamiltonian {1) (typically 2 orders of magnitude). It is therefore reasanabie to m.THEORY limit our considerations to the 5Ex 5E sub space. In fact, such a limitation means that Very recently we proposed a model to we neglect 5E-5T mixing caused by Hexch and describe the thermadynamie properties of H8 . We have checked that even for very Fe-based DMS9. We will reeall here the gen strong exchange interactions (as J = -200 K) eral idea behind this model and discuss it in the corrections to the energies resulting from some detail for Zn1_xFexSe. We start by con this limitation do not exceed a few percent. sirlering the Fe2+-Fe2+ pair problem. Mixing due to spin-<>rbit interaction and The Hamiltonian of an Fe-Fe pair, crystal field is taken into account exactly by taking into account crystal field, spin-<>rbit suitable choice of the single-ion wave func interaction, magnetic field and exchange in tions19. Within the 5Ex5E subspace the teraction between the Fe2+ ions, which is as Hamiltonian (1) is solved numerically by sumed to be isotropie and Heisenberg-type, diagonalization of the corresponding lOOx 100 can be expressed in the following form: H = Her + Hso + Hexch + Ha Oq = 293 cm- 1 where: {1) A =-SS cm-1 À =-103 cm1 Hexch = -2JS1• S2 , Hso= ÀSt•Lt+ÀS:~·L2 , 60 f- 1- 6- 3- H8 = (Lt+L2+2St+2S2)· J.taB , 12- 3- z- ;~ 3- 3- J j ...... 33- and Her is the crystal-field term. is the :.::: 3- exchange constant28, S and L are spin and - 40 1- 4_ 3- orbital momentum operators, respectively. 9 3- The indices 1 and 2 refer to the ions of the 1- z- pair. 6- 2 1- The full Fe2•-Fe • pair wave function is 20 6- 2- 3-- taken as a linear combination of products of 2- 3- single ion wave functions 3- 2- 7/J = L ll!nmfn{l)gm(2) , (2) 0 1- 1- 1- 1- 1 1-- n,m J=O J=-5K J=-10K J=-15K J=O J=-SK where /0 (1), gm(2) are single-ion wave func tions. Since 5E, 5T terms are 10-fold and 15- fold degenerate, respectively, we are dealing FIG. 6. Low-lying energy levels for a single finally with 625 pair basis functions. To sim Fe2• ion (J = 0) and pairs of Fe 2+ ions with ex plify the problem we have chosen f0 (1), gm(2) change interactions J = -5 K, -10K and -15K for according to Slack et al. (wave functions AS (a) Dq = 293 cm-1 and À = -85 cm-1 (b) Dq = 293 in Ref. 5). These wave functions diagonalize cm-1 and À = -103 cm-1 (i.e., free ion value). The Her and K80, which means that our Bilhert numbers denote the degeneracy of the levels. Chapter III, § 1 Magnetic properties of ..... Znt-xFexSe 51 erties of an Fe2•-Fe2+ pair. In Fig. 8 we show the specific heat of a pair for different ex change interactions. For all presented J values, at low temperatures a well pro nounced maximum in the specific heat is ob served, similarly to a single Fe2+ ion (J = 0). This maximum shifts to lower temperatures as the exchange interaction increases and slightly changes its shape. Application of moderate magnetic fields influences the speci ,.. fic heat only slightly (less than 5% for B < ~ 6 T), which is a consequence of the small ..c: "' 0 changes of the energy level separations with magnetic field ( d. Fig. 7). In Fig. 9 the calculated magnetization of an Fe2•-Fe2+ pair is shown for different tem 2 Fe-Fe pair peratures. In distinction to the free Fe + ion magnetization (Fig. 9, inset), the pair mag netization shows no saturation with magnetic field (we checked it up to 60 T). Moreover, no characteristic steps are observed, which occur for Mn 2•-Mn2+ pairs 20-22 and are due to crossings of excited pair levels with the magnetic field ground state. In the present case the ground FIG. 7. Splitting of the pair energy levels in r------.------.------.------.12 magnetic field (J -5 K, Dq:::: 293 cm-1 and À == = Fe-Fe pair --85 cm-1). c: Hamiltonian matrix. The resulting energy 0 pair diagram is shown in Fig. 6 for some "" ..."' 0 values of J. For the crystal-field splitting ';;" 0.6 E and spin-orbit parameter, we adopted the x ::; 6 1 values reported for ZnSe:Fe (Dq = 293 cm- , E E u u À -85 cm-1). To demonstrate the influence of the spin-orbit coupling, the energy levels for À -103 cm- 1 (i.e., the free ion value) are also shown in Fig. 6(b ). It can be noticed that the exchange interaction does not essen tially change the iron energy level structure, in the sense that the ground state is still a temperature { K l singlet one. However, the first excited state approaches the ground state when J in creases. In the presence of a magnetic field FIG. 8. Specific heat of an Fe2•-Fe2+ pair as a the degeneracy of the Fe2•-Fe2+ pair is lifted function of ternperature for J:::::: -5, -10 K, and as shown in Fig. 7 for fields up to 20 T. -15K (Dq 293 crn-1, >. = --85 cm-1) in the ab The energy level scheme can be used for sence of a magnetic field. For comparison the behav the calculation of all thermadynamie prop- ior for noninteracting ions (J:::: 0) is also shown. Chapter III, § 1 MAGNETIC BEHAVlOR ..... , BASEDON Fe 52 Fe-Fe pair - -;6 4 u.. ~ N ';!. 1- u..QI ö 3 j:: 0 E c: ....-~ .:"' 2 Ql c: 50 100 150 temporoture ( K) E"' Fe-"Fe pair J=-5 K 0 200 temperature I K l 0 5 10 15 20 magnet ie field ( T ) FIG. 10. Susceptibility of an Fe2•-Fe2+ pair FIG. 9. Magnetization of an Fe2+_Fe2+ pair coupled by an exchange interaction J = -5 K as a 1 coupled by an exchange interaction J = -5 K as a function of temperature (Dg = 293 cm- , À = function of magnetic field for düferent temperatures 85 cm1· For comparison the behavior of noninter (Dg= 293 cm-1, À = -85 cm-1). Inset: ma.gnetiz acting ions (J = 0) is also shown. Inset: inverse a.tion of a single, isolated Fe2+ ion (J = 0} as a susceptibility as a. function of temperature. function of temperature for the same parameters Dq and À. modynamic properties of a "real" Zn 1_xFexSe crystal in the so-called extended nearest state is coupled to the excited states by the neighbor pair approximation {ENNPA)13, magnetic field and is then pusbed down in which was recently successfully used for de stead of being crossed by excited states scription of the magnetic properties of Mn (Fig. 7). Therefore no steps are predicted based DMS13,3. even at very low temperatures. The ENNP A is an approximative calcu.:. Finally, in Fig. 10 the susceptibility as a lation metbod particularly useful for random, function of temperature is shown. A typical diluted systems with long-ranged interac Curie-Weiss behavior [x"' (T-E>)-1] is pre tions. It is based on the assumption that the dicted for the high-temperature limit when partition function of the whole crystal can be an interaction is present in distinction to the factorized into contributions of pairs of ions. Curie-type behavior (x "' T-1) for a single Fe In this metbod each ion is considered to be ion in a crystal field. coupled by an exchange interaction J 1 only to its nearest magnetic neighbor, which may be IV. INTERPRETATION located anywhere at a distance R1• The stat istica! weight of pair configurations with dif Using the results of the preceding sec ferent R1 is given by random distribution of tion, one can now describe the relevant ther- the magnetic ions, and therefore any thermo- Chapter III, § 1 Magnetic properties of ..... Znt-xFexSe 53 dynamic quantity eau be expressed in the suming the same interaction range as re following form ported3 for Znt-xMnxSe (Ji == JnR-6•8). The ENNP A calculations were per A = E !Ai(Ji)Pi(x) , (3) formed numerically. The summation over i i shells in (3) was carried out up till i = v for which where A1 represents the specific heat, suscep tibility or magnetization of a pair and de V pends on h P1(x) is the probability of find E Pi(x) > 0.99 (5) ing at least one nearest Fe ion in the ith shell i=l and corresponds for a random distribution to: In practice v = 12 was quite sufficient for all u·1 1 u·1 x. Pi(x) = (1-x) - - (1-x) , (4) The calculated specific heat, tagether with the experimental data, is shown in where ui is the number of lattice sites inside Fig. 11 for x == 0.015, and Fig. 12 for x = a shell of radius Ri· 0.062. It is obvious that if the exchange inter For the actual calculations we have to action between the Fe ions is taken into ac insert valnes for the parameters: Dq, >., J 1, count, a much better description of the ex- J 2, J 3, ...••• For the crystal-field splitting and spin-orbit parameters we have taken the valnes /:::.. == 2930 cm -1 ( or Dq == 293 cm -1) and >. = -85 cm -t reported 6 for ZnSe doped with Fe. These valnes resnlt from combining 0.20 Z n0.985 Fe 0.015 Se spectroscopie data for the 5E--+ 5T optica! 1 1 transition24 with susceptibility results 5, as -- NN LR= O 25 suming Harn's rednetion factor to equal 1. 1NN We notice that the crystal-field parameter 0.15 obtained in that way is practically deter mined by the energy of the 5E--+ 5T transi tion, since it depends only very weakly on the value of >. (20% variatien of >. results in 1% 1 [ NN variation of /:::.. ), whereas the spin-orbit para 6 8 meter results from the fitting procedure. The 1LR 1NN R- • exchange integral for nearest neighbors is JNN=-22K,JLR 0 [ provided by recent high-temperature suscep À = ·97 cm·1 tibility data11 : JNN = -22 K. The interaction range is suggested to be J = JnR-12 (see 1 5 10 15 20 Sec. II). This means, in practice, that only the nearest-neighbor interaction is relevant, temperature ( K l since more distant neighbors are so weakly coupled that their magnetic properties differ FIG. 11. Specific heat of Zn0. 11 ssFe0.015Se in the very little from those of single ions. However, ENNPA as a function of temperature without a as we already discussed in Sec. II, the data magnetic field for Dq = 293 cm -1, ). = -85 cm -1, concerning the spin-glass freezing in Fe- and various exchange interaction parameters. The based DMS are not sufficient to provide reli crosses show the influence of a magnetic field B = abie information about this interaction range. 2.64 T for JNN = -22 K and JLR. = 0. Also the We will therefore discuss two cases: only influence of a different spin-orbit parameter, ). = nearest neighbor, short-ranged interaction -97 cm-1, is shown. The points are experimental (NN) and a long-range interaction (LR), as- data: solid circles B = 0, open circles B = 2. 75 T. Chapter III, § 1 MAGNETIC BEHAVIOR ..... , BASED ON Fe 54 first~xcited state. The systematic deviation of the calculations from the experimental data in the low-temperature region indicates that the actual lowest energy gap is some 0.6 what larger than resulting from the parame ters taken in the present calculation - (l:l 18K). An increase of this gap to:::: 22 K is ""' necessary to explain the observed specific ö heat. Preliminary far-infrared transmission E 0.4 (FIR) experiments2&, where we observed the -..... optica! transitions At-+ Tt and At-+ T2, E w have confirmed this suggestion. To explain the experimental E(T1)- E(At) and E(T2) - 1NN=·2ZK • 1tR=O E(A 1) energy differences within the frame JNN = ·22 K workof Slack's model5, one has to adjust the ----- [ LR = NNR·6.8 1 1 spin-orbit parameter to -97 cm·1 keeping Dq _,_,_,_ [ JNN =·22 K 1JLR" 0 unchanged. The resulting specific heat is >.=·97cm· shown in Figs. 11 and 12. One can notice that this la.rger value of À indeed improves the 15 20 description, although, especially for the temperature ( Kl higher concentration, significant deviations FIG. 12. Specific heat of Zno.938Feo.0&2Se in the ENNP A as a function of temperature without a magnetic field for Dq = 293 cm·1, À = -85 cm·1, 5 and various exchange interaction parameters. The crosses show the influence of a magnetic field B = 2.64 T for J:rnf = -22 K and Jta = 0. Also the !!' 4 T = 2 K influence of a different spin-orbit parameter, À = ..ê -97 cm·1, is shown. The points are experimental • •• data: solid circles B = 0, open circles B = 2.75 T. • .9 3 1;; :;:N ..c Cl\ro perimental data can be obtained than for the E 2 noninteracting ion model ( J = 0 in Figs. 11 and 12). The latter description is reasonable for x= 0.015 whereas a substantial discrep ancy is found for x= 0.062. We should note, however, that the dilute limit which is essen tial in the ENNP A may not be satisfied for x= 0.06. In fact, for Mn-based DMS only 0 5 10 15 20 crystals containing less than 5% magnetic magnetic field I T I ions could be properly described by the 13 3 ENNPA • . FIG. 13. Magnetization of Zno.g5Fe0.05Se in the In the dilute limit, where the single-ion ENNP A as a. function of magnetic field a.t T = 2 K contribution is dominant, the exponential and T = 30 K for Dq = 293 cm·t, À= -85 cm-1, onset of Cm clearly marks the influence of the and various exchange interaction parameters. The energy gap between the ground state and the points represent experimenta.l data. (Ref. 8). Chapter III, § 1 Magnetic properties of ..... Znt-xFexSe 55 interaction J(R), is not very significant. This results from the fact, as we quoted before 9, that the further-neighbors exchange interac tions are too weak to alter the Fe2+ level scheme significantly. Consequently, no perti nent conclusions about the interaction range can be drawn from the present data. Summarizing the present results we feel :.0 that the model presented above provides a 15. "' reasonable, simultaneons description of all ~ 20 0 0 the magnetic properties with a single set of "' 0 0 "'~ parameters. We believe that the observed "'> .5 deviations at higher Fe concentrations are mainly due to the fact that these concentra tions are somewhat outside the diluted limit, to which the ENNP A is specifically appli 100 200 cable. temperature [ Kl We would like to emphasize that no fit ting parameters are employed in our calcula FIG. 14. Inverse susceptibility of Znl-xFexSe for tions and comparison with the data. Such a z :::; 0.022 and z =0.072 in the ENNP A as a procedure would certainly yield a better de function of temperature for Dq = 293 cm -1, À = scription of some of the experimental data, -85 cm-1, and different exchange interaction para but, in view of the large number of parame meters. The points are experimenta.l data (Ref. 11). ters, it would not be necessarily physically meaningful. remain. V. DISCUSSION AND CONCLUSIONS High-field magnetization results are shown in Fig. 13 for Zno.g 5Feu5Se. Also in As quoted earlier, specific-heat data and this case it is evident that ion-ion interac preliminary FIR experiments indicated that tions have to be taken into account. the energy gap between the ground state and The magnetic susceptibility calculations the first-excited states is somewhat larger are presented in Fig. 14 for Zn0 .978Fe0.022Se than the calculated gap resulting from the and Zn 0 .928Fe0.072Se (Ref. 11). The typkal reported crystal-field and spin-orbit parame curvature of the calculated susceptibility, ters. To describe this splitting within Slack's which is also observed in the experimental model, the spin-orbit coupling parameter had data, is obviously due to exchange interac to increase from --85 cm -1 to -97 cm -1. One tions and is neither observed for semi-iso has to note, however, that this is just a for lated Fe2+ ionss, nor reproduced by calcula mal description. Physically, it seems more tions for isolated ions as shown in Fig. 14 likely that substitution of magnetic ions re (J = 0). Again we cbserve an increasing dis sults in a local lowering of the cubic symme crepancy between calculations and experi try which may lift the orbital degeneracy to mental data when the concentratien in start with. However, data to support this creases. conjecture are not available and we therefore For all magnetic properties considered refrained from implementing these effects. here, we cbserve (cf. Figs. 11-14) that the Furthermore we would like to point out difference between calculations in which the that our use of an isotropie exchange interac exchange interaction is implemented by the tion term of the Heisenberg type must be nearest neighbors alone, or by a long-range considered as a first approximation. Genera!- Chapter 111, § 1 MAGNETIC BEHAVlOR ..... , BASED ON Fe 56 TABLE I. Nearest-neighbor interaction strength (Ju), radial dependenee of the interaction (n with J"' R-n) and p-d exchange parameter (No) for several Mn- and Fe-type DMS. Mn Fe Jn(K) n N0(eV) Jn(K) n N0(eV) CdSe -1oa 6.8C -1.27e -1sg.h ZnSe -12.6& 6.sc -1.30f -22h 12 -1.6! ZnS -16b 7.6d a Referente 28 R.eference 29 b R.eference 14 g R.eference 10 c R.eference 3 h R.eference 11 d R.eference 17 R.eference 30 e R.eference 27 ly speaking, anisotropic terms will be present first few neighbors 31·32. Although calculations for Fe2 +-Fe2+ interactions, specifically when of Wei et al. 33 seem to indicate an increase of the actual calculations are restricted (as in the hybridization for Fe2+ (or Mn+), which the present case) to the lower subset of would be consistent with the observed in levels. crease of Jn as well as the p-d exchange Finally, we would like to comment on parameter N 0 (cf. Table I), we feel that perti the exchange interaction between Fe2+ ionsin nent conclusions would be rather speculative relation to Mn2+_Mn2+ exchange. In TableI at the present stage. we collected the available data concerning Fe With respect to the range of the long and Mn exchange in similar DMS. Inspeetion range interaction ("' 1/n), as probed by the of this table shows that no drastic differences freezing temperature as a function of x, a are observed between Mn and Fe exchange. oomment must be made. The relation be An antiferromagnetic, long-range exchange is tween Tr and concentration x is based on a found in all cases, although the nearest sealing analysis and the conjecture that Tr is neighbor interaction is stronger for Fe and related to the interaction energy at the aver seems to be of shorter range than for Mn. age distance (Rav) between the magnetic Assuming that superexchange is the domi wns• 3, e.g., nant mechanism, as was rather well estab lished for Mn-type DMS, the simultaneons (6) increase of strength and decrease of range is not completely in accordance with theory. In In the case that we are dealing with a singlet general an increasing strength is expected to ground state with a quenched magnetic mo be accompanied with an increasing range31 or ment, the interaction energy is less effective the range is considered as largely independent and should be modified, imptementing the of the atomie constituents32. These ten fact that the interacting ions have only an dencies, however, mainly refer to compari induced moment. This situation is somewhat sons between Mn-type DMS. The substitu analogous to the case studied by Moriya34 tion of Fe instead of Mn may well result in who showed, in a mean-field approximation, changes in the band structure and d energy that an ensemble of Ni2+ ions with singlet levels, which complicate a meaningful com ground state cannot sustain spontaneons or parison since both the location of the unoc der unless the exchange interaction is, rough cupied d levels with respect to the valenee ly speaking, strong enough to overcome the band, as well as the degree of hybridization, ground-state splitting. determine the strength of Jn, at least for the We will illustrate this behavior by con- Chapter III, § 1 Magnetic properties of ..... Znt-xFexSe 57 x 0.1 -4 A 0 !V>"' ,.;;--- V T ~ 1 K ~ -2 _s -6 0 -2 interaction strength ( K) FIG. 15. · Ca.lculated low-temperature pair eerre -12 lation function used: J N R-6.8. For the pairs with the degen Commun. 65, 235 (1988). erate ground state, the resulting Tr(x) indeed 10 A. Lewicki, J. Spalek and A. Mycielski, J. shows the implemented range of interaction Phys. C 20, 2005 (1987). (1/n). However, for pairs with a singlet 11 A. Twardowski, A. Lewicki, M. Arciszewska, ground state, the range, as probed by Tr(x), W.J.M. de Jonge, H.J.M. Swagten and M. is virtually suppressed due to the rednetion of Demianiuk, Phys. Rev. B 38, 10749 (1988}. 12 Denissen, in Diluted Magnetic Semiconductor&1 considered in Ref. 9, since no information is edited by R.L. Aggarwal, J .K. Furdyna and S. provided about it. von Molnar (Materials Research Society, Pitts 19 In our previous paper (Ref. 9) we used a burgh, 1987), Vol. 89, p. 153. simpler basis, given by Slack et al. (Ref. 5) in 3 A. Twardowski, H.J.M. Swagten, W.J.M. de their Tables VIII and IX. In that way 5E-5 T Jonge and M. Demianiuk, Phys. Rev. B 36, mixing due to Kso was treated in second-order 7013 {1987). perturbation method. It should be stressed, 4 W. Low and M. Weger, Phys. Rev. 118, 1119 however, that no substantial difference was {1960). found between present and the later results. 5 G.A. Slack, S. Roberts and J.T. Wallin, Phys. 20 R.L. Aggarwal, S.N. Jasperson, Y. Shapira, Rev. 187, 511 (1969). S. Foner, T. Sakibara, T. Goto, N. Miura, 6 J.P. Mahoney, C.C. Lin, W.H. Brumage and K. Dwight and A. Wold, in Proceeding11 of the F. Dorman, J. Chem. Phys. 53, 4286 (1970). 17th International Conference on the Physics 7 See the review paper A. Mycielski in Diluted of Semiconductor11, San Franci&co, 1984, edited Magnetic Semiconductor&, edited by R.L. Ag by J.D. Chadi and W.A. Harrison (Springer, garwal, J .K. Furdyna and S. von Molnar (Ma New York, 1985), p. 1419. terials Research Society, Pittsburgh, 1987), 21 Y. Shapira, S. Foner, D.H. Ridgley, K. Dwight Vol. 89, p. 159. and A. Wold, Phys. Rev. B 30, 4021 (1984). 8 A. Twardowski, M. von Ortenberg and M. 22 J.P. Lascaray, M. Nawrocki, J.M. Broto, M. Demianiuk, J. Cryst. Growth 72, 401 (1985). Rakoto and M. Demianiuk, Solid State Com A. Twardowski, H.J.M. Swagten, T.F.H. v.d. mun. 61, 401 (1987). Wetering and W.J.M. de Jonge, Solid State 23 K. Matho, J. Low Temp. Physics 35, 165 Chapter III, § 1 Magnetic properties of ..... Znl-xFexSe 59 ------ (1979). (1987). 24 J.M. Bara.nowski, J.W. Allen a.nd G.L. Pea.r 31 G.A. Sa.watzky, W. Geertsma a.nd C. Hass, J. son, Phys. Rev. 160, 627 (1967). Magn. Magn. Mater. 3, 37 (1976); C.E.T. Gon 25 F.S. Ha.rn, Phys. Rev. 138, A1727 (1965). ca.lves da. Silva. and L.M. Falicov, J. Phys. C 5, 26 A. Twardowski, H.J.M. Swa.gten a.nd W.J.M. 63 (1972); A.N. Kocharyan a.nd D.I. Khomskii, de Jonge, in Proceeding11 of tke tutk Internar Fiz. Tverd. Tela (Leningrad) 17, 462 (1975) tional Conference on Pky11icll of Semiconduc [Sov. Phys. - Solid State 17, 290 (1975)]. tors, edited by W. Za.wa.dski (Warsaw, 1988), 32 H. Ehrenreich, K.C. Hass, B.E. Larson a.nd p. 1543. N.F. Johnson, in Diluted Magnetic Semicon 27 M. Arciszewska and M. Nawrocki, J. Phys. ductor&, edited by R..L. Aggarwal, J .K. Fur Chem. Solids 47, 309 (1986). dyna and S. von Molnar (Materia.ls Research 28 J. Spa.lek, A. Lewicki, Z. Tarnawski, J .K. Fur Society, Pittsburgh, 1987), Vol. 89, p. 187; dyna, R.R. Galazka. a.nd Z. Obuszko, Phys. B.E. Larson, K.C. Hass, H. Ehrenreich and Rev. B 33, 3407 (1986). A.E. Carlsson, Solid State Commun. 56, 347 29 A. Twardowski, M. von Ortenberg, M. Demia. (1985). niuk a.nd R. Pauthenet, Solid State Commun. 33 Su-Huai Wei a.nd A. Zunger, Phys. Rev. B 35, 51, 849 (1984). 2340 (1987). 30 A. Twardowski, P. Glod, W.J.M. de Jonge a.nd T. Moriya, Phys. Rev. 117, 635 (1960). M. Demia.niuk, Solid State Commun. 64, 63 Chapter III, § 2 MAGNETIC BEHAVlOR ..... , BASEDON Fe 60 Chapter III, paragraph 2 Low-temperature specific heat of the dlluted magnetic semiconductor Hg1_x-yCdyFexSe [ published in Phys. Rev. B 42, 2455 (1990) ] I. INTRODUCTION strength of these interactions, since the Fe2+ d6 level may be located far above the valenee Diluted magnetic semiconductors (DMS) band in contrast to the Mn2+ levels. The or semimagnetic semiconductors are current properties of Fe-based DMS are, however, ly attracting considerable attention because still relatively unexplored 4, although recently of their interesting transport, magneto-op some optica! and magnetic data were re tical and magnetic propertiesl. So far, re ported for some DMS4-to. The optica! prop search has been focused mainly on DMS con erties, and in partienlar the band splitting of taining Mn2+ ions. The magnetic behavior of Fe-based DMS (which follow macroscopie these systems can be understood on the basis magnetization9) show a similar behavior as of a random array of localized Mn ions the analogous quantities for the Mn-type coupled by long-ranged isotropie antiferro materials, and, in that sense seem not very magnetic exchange interactionsl-3 which are sensitive to the orbital momentum. mediated by the carriers. It has been shown On the other hand, the magnetic prop that the dominant microscopie mechanism erties are expected to be substantially differ causing d-d exchange interaction between the ent from Mn-type DMS. This difference can Mn ions is probably superexchange (at least he ascribed to the different energy scheme of for the nearest neighbors). However, the rela Mn2+ and Fe2+ ions. Substitutional Fe2+ has a tion to the band structure, as well as the na d6 electronk configuration. The ground state ture of and the driving force behind, the ob of the Fe2+ free ion (5D) is split by a tetra served phase transitions and the role of aniso bedral crystal field into a SE orbital doublet tropy are still somewhat obscure, and subject and a higher lying 5T orbital triplet (separ for further investigations2,3. ated from 5E by lODq, where Dq is the crys In a way the substitutional Mn2+ with tal-field parameter). Spin-orbit interaction 6 5 its degenera te A 1 spin-only ground state, splits the E term into a singlet At. a triplet represents a rather simple, although theoreti T11 a doublet E, a triplet T2 and a singlet A 2 cally attractive case, since all the phenomena (the energy separation between these states is which involve the orbital momenturn are ab approximately equal to 6).2/lODq, where À is sent. In contrast, substitutional iron Fe2+ the spin-orbit parameter)11,12. Thus the (d6) can serve as a much more general case, ground state is magnetically inactive singlet since it possesses both spin and orbital mo A1 resulting in Van Vleck-type paramagnet menta (S = 2 and L = 2). Moreover, the d-d ism12·13. Moreover, isolated Fe2+ ions contrib exchange is intimately related to the location ute to the specific heat even in the absence of of the magnetic ion levels with respect to the a magnetic field, in contrast with isolated band structure1. Therefore, substitution of Fe Mn2+ ions, where only Mn-Mn pairs and instead of Mn might affect the character and larger clusters can contribute to the specific Cha.pter III, § 2 Low-temperature ..... Hgt-x-yCdyFexSe 61 heat for B 0. ll.EXPERIMENTAL So far some experimental data reported 4 10 15 on Zn 1_xFexSe6.1 , as well as Cd1_xFexSe • , The samples of Hgt-xFexSe, Hgt-x-yCdy seem to corroborate this model of Fe-based FexSe ( cubic) and Cd1_xFexSe (hexagonal) DMS. Reported data on the zero-gap DMS investigated in the present paper are listed in 8 Hg 1_xFexSe are, however, at varianee with Table I. They were grown by a modified such a model, since the specific heat was re Bridgman technique, and their concentration ported to he much smaller than for Zn 1_xFex and hornogeneity was checked by mieroprobe Se and the low-temperature susceptibility analysis. The specific heat ( Cp) was reveals a rather typical Brillauin-type para measured by a standard heat-pulse method rnagnetic behavior instead of Van Vleck be in the temperature range 1.5 - 25 K and in havior. It was suggestedB that this behavior the absence, as well as in the presence, of a of Hg 1_xFexSe results frorn the specific loca magnetic field (up toB= 2.75 T). The speci 6 tion of the Fe d level in relation to the HgSe fic-heat results for Hg 1_xFexSe, Hgt-x-yCdy bands (in this material the Fe level is situ FexSe and Cdt-xFexSe are shown in Figs. 1, 2 ated 0.2 eV above the bottorn of the conduc and 3, respectively. The magnet ie con tribu tion band, whereas in wide-gap materials like tion to the total specific heat can he well ZnSe or CdSe it is located in the gap between observed in low temperatures. We notice that conduction and valenee band5). CP for all the compounds is practically field Very recently it was suggested that some independent (at least for B < 3 T), which is of these discrepancies between wide and zero exemplified in Fig. 4 for Hgt-xFexSe. This gap rnaterials rnentioned earlier, might he suggests a singlet ground state for Fe ions for rernoved by a more careful analysis of the 1 specific-heat data 6 and by taking into ac 2 count the contribution of the Fe3• ions, which 17 10 1 are always present in Hg 1_xFexSe crystals , to the susceptibility. In view of this we 5 thought it worthwhile to investigate the mag netic properties of open as well as zero-gap 2 Fe~ontaining DMS in a more detailed way. 0 As we will see later these results will allow us 1C ' to estirnate the nearest-neighbor interactions "" 5 2 ' Hg 1_.re.se between Fe * ions. 0 E 2 2 a w-\ u X"Ü TABLE I. Ma.teria.ls used in experiment. Q x=0.0!5 A X"0.025 Material x (Fe) y (Cd) 1-x-y (Hg) 0 x=0.045 ~-···--····-··· ; Hg, .•Fe,Se 0.015 0 0.985 0.024 0 0.976 0.045 0 0.955 5 i Hg 1••. yCdyFe.Se om 0.43 0.56 0 5 10 15 20 0.044 0.446 0.51 T (K) Cd 1•• Fe.se 0.008 0.992 0 0.034 0.966 0 0.080 0.920 0 FIG. 1. Specific heat ( Cp) of Hgt-xFexSe, for several concentra.tions z. Chapter III, § 2 MAGNETIC BEHAVlOR ..... , BASEDON Fe 62 2 to 1 101 5 5 2 2 100 100 5 I ' ;f 5 ;f I Hg 1_x-yCdvfexSe ...... ' 0 0 2 -e e 2 2 2 10-1 c. 10-1 Q, u x=O.O , y=O. 490 u 5 x=O 5 ' x=O.O!O, y=0.430 x=o. ooa x=0.044, y=0.446 , x=0.034 . 2 2 x=O.OBO 10-2 10-2 i ,I 5 ! 5 0 5 10 15 20 0 5 10 15 20 T (K) T (K) FIG. 2. Specific heat (Op) of Bgt-x-yCdyFexSe, FIG. 3. Specitic heat (Op) of Cdt-xFexSe, for for several concentrations. several concentrations z. all the compounds. At higher temperatures, that substitution of heavy Hg or Cd atoms by CP is dominated by the lattice contribution. relatively light Fe atoms changes the lattice In the case of Hg 1_xFexSe, Cp of pure HgSe vibrational modes considerably, and, conse exceeds CP for the diluted system with iron quently, the lattice specific heat of the mixed for sufficiently high T. Such a situation was or diluted system differs significantly from observed earlier by us16 and was also reported that of the nondiluted system. This may be 18 for Hgt-xMnxTe . well observed in Fig. 5, where we show the H the magnetic contribution Cm is esti specific heat of pure ZnSe (as a close approxi mated by subtraction of the undiluted lattice, mation of FeSe ), CdSe, HgSe and a mixed i.e., compound Hg0_51Cdo.4 9Se. The increasing difference in atomie mass of Zn, Cd and Hg Cm(Hgt-xFexSe) = Cp(Hgt-xFexSe) as compared to Fe, which in the Debye ap (1) proach yields an increasing difference in 9D, - Cp(HgSe) , is clearly reflected in Cp· It is obvious that for Hg _xFexSe, and to a lesser extent for this would lead to the physically unaccept 1 Cd1_xFexSe, a more realistic procedure of able situation of negative magnetic specific extracting the magnetic contribution from at higher temperatures. This situation is heat the total specific heat should be used. extreme for Hg 1_xFexSe but not unique. There are several possible procedures of Moreover, for Cd1_xFexSe, with 8% of Fe, the simulating the lattice heat capacity of an magnetic specific heat obtained by Eq. {1) actual crystal. We have tested some of them can become negative at high T. (based mainly on the Debye approach or lin- Such a situation results from the fact Cha.pter III, § 2 Low-tempera.ture ..... Hgt-x-yCdyFexSe 63 (2) ACp(HgSe,T) + BCp(CdSe,T) + xCp(ZnSe, T) , ;;::;" 2 where the prefactors A, Band x are chosen in ' ?f accordance to the actual x and y concentra ' 100 '""""'a tion (Table I). ..E, In ordertotest this procedures we calcu lated the specific heat of Hg0_51Cdo.4 9Se using u 0 x=O. 024 , B=O the experimental data on the HgSe and CdSe c x=0.024, B=2.BT 2 lattices. The result of these attempts are 10-1 16 (a) :: . ZnSe 0 Cd Se HgCdSe .. 12 0 HgSe 0 5 10 15 20 - PLS l T (K) *" ~' 8 ~ 2 FIG. 4. Specific heat ( Gp) of Hgt-xFexSe (:r. c. = u ..... 0.024), for B = 0 a.nd B 2.8 T. ~ ·•·•······•· ear interpolation of nonmagnetic lattices), 0 300 and we have found that the validity of all of (b) them is rather limited to the temperature range where the magnetic contribution is 250 somehow comparable to the lattice Cp· At higher temperatures where Cm«: Cp, simula ...... 200 tion of lattice CP requires very accurate infor g mation about the crystal composition. Such accuracy can hardly be obtained for the sys (J)" tems like Hg 1.xFexSe or Hgt-x-yCdyFexSe with the help of standard mieroprobe or 100 chemical analysis. Finally, whatever pro cedure was used, we were not able to obtain 50 so fine high-temperature results as were 0 5 10 15 20 25 found for Zn 1.xFexSe. We should stress that T (K) the low-temperature data, which provide the most important physical information, are reliable for all the compounds. FIG. 5. (a.) Specific heat of ZnSe, CdSe, Hgo.u With respect to the remarks given above Cdo.4gSe and HgSe; the solid line represents calcula we decided to use a simple phenomenological tions within PLS (see the text). {b) The correspond interpolation procedure for the lattice simula ing Debye-temperatures [9D(T)] of the nonmagnet tion (PLS): ie lattices. Chapter III, § 2 MAGNETIC BEHAVlOR ..... , BASED ON Fe 64 shown in Fig. 5(a) together with the actual 25 experimental data. Our procedure, although Cd _.Fe.se it has no physical background, seems to pro 1 vide a very reasonable approximation for the 20 lattice specific heat of the mixed crystal. x•O .008 I x•O .034 The results obtained for Cm are given in 2f . x•O .080 ,__,' 15 Figs. 6 (Hgt-xFexSe) and 7 (Cdt-xFexSe), 0 where Cm/x is displayed as a function of tem E ~ perature. For completeness, in Fig. 8 the re 10 x sults for Zn 1_xFexSe (Ref. 14) are given, ob E tained by a simple lattice subtraction [Eq. 'u (1)] which is equivalent to PLS for this com 5 pound. Due to the delicate dependenee of CP on the actual composition of Hg and Cd for the quaternary compound Hg 1_x-yCdyFexSe, we could not obtain the reliable Cm data in 5 10 15 20 this case. Therefore, this compound will not T (K) he analyzed quantitatively. We only note that the overall behavior of the Hg 1_x-yCdy FexSe heat capacity is the same as that of the FIG. 7. Magnetic specHic heat per mole Fe other systems (cf. Figs. 1-3). (Cm/z) of Cdt-xFexSe, using PLS [Eq. (2)). The It can he observed that the overall shape solid line shows Cmfz for z = 0.034 at B = 2.8 T. of Cm is similar for all Fe-based DMS and The dashed line displays the calculated Cm/z (Refs. the magnetic specific heat calculated per one 21-26) assuming no interaction between Fe ions. Fe ion (Cm/x) generally decreases with in creasing iron concentration. However, there is 0 ' Zn 1 _.Fe.se ~/, '' 'r/!h~':'i'~ '~'~"', 0 0 8 8 :0 0 ooo"''\,: .,'/"' 00 x=0.015 " r:P' a dJ """ "a "' / 2f' 2f' : ::~:~:~~ "o ' ....: •••.•••• •••. ,__,' 6 ,__,' 6 x=O. 207 .,."._-.-._ • 0 0 E E ~ ~ I 4 4 x .. x ' E ' E i u .· u 2 2 L 0 0 0 5 10 15 20 0 5 10 15 20 T (K) T (K) FIG. 6. Magnetic specHic heat per mole Fe FIG. 8. Magnetic specHic heat per mole Fe (Cmfz) of Hgt-xFexSe, using PLS [Eq. (2)). The (Cmfz) of Znt-xFexSe; see Ref. 14. solid line shows Cm/z for z = 0.025 at B = 2.8 T. Chapter III, § 2 Low-temperature ..... Hgt-x-yCdyFexSe 65 practically no magnetie-field dependenee of state and a threefold degenerate first-excited Cm for B < 3 T (see Figs. 4, 6 and 7). From state at .ó.E, which schematically represents this behavior one may conclude that the ma the lowest levels of the Fe2+ ion in a cubic jor contribution to Cm originates from Fe2+ crystal field with spin-orbit interaction 12: ions in a singlet ground state as we showed before14. In that case one cou1d also ex:pect rough ly the same absolute values of Cm at the maximum, which is appa.rently not observed. (3) These large differences ma.y only pa.rtially resu1t from the fact that our procedure of with gtfg0 = 3. Within the framework of simulating the lattice is not perfect. On the obtained below the maximum of Cm with a whole, however, it seems rather well estah slope determined by .ó.E. Extrapolation to lisbed that the absolute value at the maxi T = oo will yield the value Jor g1/g0• De mum of the magnetic specific heat of Cd1.x tailed calculations for the real energetica! FexSe is much higher than that of Zn1.xFex structure of an isolated Fe ion (i.e., five Se. We return to this point later. levels12) reveal a simila.r linear behavior of In order to visualize the ex:ponential ln(CmT2) versus T-1 (Fig. 12). onset of the magnetic specific heat we plotted The data for the various compounds in the low-temperature data of Zn 1.xFexSe deed show a linear part below Cmax yielding, (Fig. 9), Cdt-xFexSe (Fig. 10) and Hgt-xFexSe for low Fe concentrations where Eq. (3) (Fig. 11) as In( Cm T2) versus T·1• The specific should apply, .ó.E in the range 20-25 K, in ordinates are chosen in accordance to the accordance with data from spectroscopie ex: low-temperature behavior of a Schottky periments, and g tf g0 in the ~ánge 2-4. The anomaly originating from a singlet ground devia.tions at ex:tremely low temperatures 10,-r-.-~.-.-.--r-.-.~~--~~ 10 9 9 Cd 1_.Fe,Se 8 8 •• 0.006 7 7 6 6 5 4 3 B 3 c c rl 2 ~ 2 c 0 -1 ... 1 A -2 -2 -3 -3 0.0 0.1 0 0 3 0.~ 0. 0.0 0. 1 0.2 0.3 0 4 0,5 0.6 0.7 1/T (K-1) 1/T (K- 1) FIG. 9. Magnetic specific heat per mole Fe FIG. 10. Magnetic specific heat per mole Fe (Cmfz) of Znl-xFexSe, plottedas ln(Cm~/z) versus (Cm/'JJ) of Cdt-xFexSe, plottedas ln{CmT2/z) versus 1/ T; the lines are described in the text. Line A 1/ T; the lines a.re described in the text. Line A yields LlEsingle = 21K and line B LlEpair < 13 K. yields LlEsingle = 19K and line B LlEpair < 9 K. Chapter III, § 2 MAGNETIC BEHAVlOR ..••. , BASED ON Fe 66 (see Figs. 9-11) result from the fact that in 10 our erystals we are dealing not only with iso lated Fe ions [to which Eq. {3) applies], but [)q = 293 cm- 1 also with Fe-Fe pairs as well as larger clus À = -95 cm- 1 ters. We reeall tha.t for x = 0.015 only 83% of the Fe ions ca.n be regarded as isolated in the 5 sense tha.t they have no magnetic nearest neighbor (NN). It was shown7,t4 that the ~ energetica! structure of a Fe-Fe pair, coupled ~ by exchange interaction, yields a situation 0 essentially similar to that of a.n isolated Fe " x = 0.015 ...... " ' ion: the pair ground state is a singlet, how = ' ever, the energy gap between the ground and ' ' the first-acited sta.tes is strongly reduced ' ' [typically by a factor of two (Refs. 7 a.nd -5 ' 14)]. Therefore, if Fe-Fe pairs are present in x = 0.00 the crystal, one may expect an additional eontribution to the Cm which will be dominant at low temperatures since only the -1QL-~~~--~~~--~~~~ first excited pair states will be thermally 0.0 0.2 0.4 0.6 O.B 1. 0 popula.ted. This pair eontribution ca.n be observed in Fig. 12, where we piotted the calcula.ted Cm assuming only NN interactions a.nd a statistièal distribution of Fe ions in the FIG. 12. Calculated magnetic specific heat per crystal o.(for details of the calculations we mole Fe (Cm/z), plotted as ln(CmT2/z) versus 1/T refer to Refs. 7 and 14). We note that al- (Dq = 293 cm-1, À = -95 cm-1, Jn = -22K). The line for z = 0.00 corresponds to the case of noninter 10 acting Fe ions. 9 8 7 though our model calculations result from a rather simple Ha.miltonia.n descrihing Fe-Fe 6 pairs and single ions7,14, the behavior of the x 5 ru real experimental data ca.n be well reeovered. ';- 4 2 E It is also apparent that plotting ln(CmT /x) !::! 3 1 c versus T- enables one to estimate energy .-. 2 gaps for a.n isolated Fe ion as well as for the Fe-Fe pair from the appropriate linear parts 0 of the plot. In the present case the tempera -1 A ture is still somewhat to high to restore the -2 linear part in the pair contribution eomplete -3 ly. Therefore the "pair" straight lines in Figs. 0 0 0. 1 0.2 0.3 0.4 0.5 0.5 0 7 9-11 give only an estimate of the upper limit 1/T (K-1) for the pair energy gap. Actually, we find a rednetion of the pair energy gap by a factor FIG. 11. Magnetic speeifie heat per mole Fe 1.6- 2 (in respect to the isolated Fe ion en (Cm/z) of Hgt-xFexSe, plottedas ln(Cmx4/z) versus ergy gap) instead of 2.44 resulting from the 1/T; the lines are described in the text. Line A calculations ( assuming exchange interaction yields 6Esingle = 16K and line B 6Epair < 10 K. between nearest neighbors of the order of Chapter III, § 2 Low-temperature ..... Hgt-x-yCdyFexSe 67 -20 K). briefly that in this model we factorized the For completeness we would like to add whole magnetic system into pairs coupled by that the contributions resulting from nuclear (principally) long-range exchange interaction hyperfine interaction, the presence of para [so-called extended nearest-neighbors pair magnetic impurities in our crystals ( mainly approximation (ENNPA)19,20,3]. The basic Mn2+ with concentrations n = 1011- 1019 element in this model is based on the numeri cm "3) and the linear term in Cm brought cal salution of an Fe-Fe pair, where the crys about by the free carriers (n < 1019 cm-3), are tal field ( cubic or hexagonal), spin-orbit in too small to be observed in this specific ex teraction, magnetic field and the Heisenberg periment. type exchange interaction are taken into ac count simultaneously. It was shown 7 that the m. DISCUSSION ground state of both an isolated Fe ion and an Fe-Fe pair is a singlet with a first (ex From the experimental data presented cited) threefold degenerate state at about we coneinde the following. 10-20 K, leading to a Schottky-type speci (i) The magnetic contribution to the fic heat, which is very weakly magnetic field specific heat of our crystals is fairly large at dependent, in accordance with the experimen low temperatures, whereas at higher tempera tal evidence. tures it is only a small fraction of the lattice However, this model also prediets a simi contribution. Therefore, the high-tempera lar maximal absolute value of the specific ture Cm of some materials is very sensitive to heat (practically independent of crystal-field the extraction method and in some cases and spin-orbit interaction parameters), (Hgt-x-yCdyFexSe) reliable data cannot he whereas our experimE]ntal maximal valnes obtained (at least not without a separate differ largely in magnitude. In Fig. 7 we show study of the Hg 1.x-yCdyZnxSe lattice specific the calculated specific heat Cm/x for Cd 1.x heat). However, the most important physical FexSe assuming no interaction between Fe information results from the low-temperature ions, which is the upper limit for Cmfx21-26. data, where there are no problems with ex Inspeetion of Fig. 7 shows that the maximal tracting Cm from the total heat capacity. specific heat of Cd 1_xFexSe is apparently con (ii) The magnetic contribution to the sistently too large irresp~ctive of the specific specific heat shows similar overall behavior procedure used to estimate the lattice27. for all the Fe-type DMS investigated so far In order to substantiate this fact we cal (in spite of some uncertainty in the absolute culated the total entropy contained in the value). experimental specific heat. The data in the (iii) The exponential low-temperature temperature range 0 < T < 1.5 K were ap increase of Cm clearly indicates an energy gap proximated by a linear extrapolation, where between the ground state and the excited as for T > 20 K a high temperature T-2 ap states, of the order 20 K. proximation was used. The reeavered experi · (iv) The very weak field dependenee of mental entropy is always 50-60% larger than Cm strongly indicates that the ground state of expected for an Fe2+ ten-level system28 S = the Fe ion is a singlet one. RlnlO. This suggests that in fact we are (v) The decrease of the magnetic specific dealing with more than the ten levels con heat per one Fe ion (Cm/x) with increasing tained in the 5 E multiplet. Since the term concentration, suggests an antiferromagnetic structure of Fe is rather well established for d-d exchange interaction between Fe ions in both Cd1-xFexSe and Hg 1.x-yCdyFexSe from all cases. the 5E-+ 5T optical transitwn5•25, the only These observations are in agreement mechanism which can supply additional en with our "crystal-field" model developed ergy levels seems inclusion of electron recently for Fe-based DMS7,14. We reeall pbonon interaction, i.e., Jahn-Teller effect29. Cha.pter UI, § 2 MAGNETIC BEHAVlOR ..•.• , BASED ON Fe 68 The Jahn-Teller effect on the 5E term of Fe2+ has been studied by several authorsao~aa. DQ : 293 cm-1 A = -95 cm-• Generally speaking, modifications of the 5E energy structure due to electron-pbonon in 15 teraction depend on the coupling phonon en 1 ergy. H the phonon interaction is larger than t 50 the spin-orbital splitting of the 5E term, the 0 10 main effect of the Jahn-Teller effect is a re 5 0 duction of the energy intervals between the >< ) 5E term levels32. This reduction, however, 5 can be accounted for by a small change in the crystal-field and spin-orbit parameters and is thus not very apparent. Only if the phonon energy is small enough (in practice smaller 5 10 15 20 25 than spin-orbital splitting of the 5E term), T (K) this electron-pbonon coupling modifles the energy levels seriouslyao. In partienlar, new FIG. 13. lnfluence of a.dditiona.l 11 phonon-replica.11 energy levels appear in the energy spectrum energy levels (simula.ting the Jahn-Teller effect) on as a result of coupling of 0,1,2, ...... phonons the specHic hea.t in the crystal field model, using to the electronic levelsao. For crystals like Dq = 293 cm·l, À= -95 cm·l. ZnS or ZnSe, phonons which can couple to the electronic states [TA(L)] have sufficiently high energies34 and produce no noticeable phonons of energy hw can be coupled. For effects. The far-infrared spectroscopie data simplicity we assumed no coupling between for ZnS:Fe and Zn 1 ~xFexSe can be satisfactor these vibronic levels, i.e., no true Jahn-Teller ily described by considering only the crystal effect. The resulting specific heat is shown in field and spin-orbit interactionas. On the Fig. 13 for different phonon energies. As con other hand, for instanee in CdTe, the TA(L) jectured, the maximal value of the specific phonon has an energy small enough (35 cm-1) heat increases with decreasing phonon en to modify the energetica! structure and op ergy, i.e., with increasing number of vibronic tica! selection rules 'considerablyao. The re levels which can be thermally populated. ported TA phonon energy for CdSe36 is Although the phonon energies, which produce 43 cm-1, which is a considerably smaller a. substantial increase of Cm, are apparently value than the total splitting of the 5E term, too low as compared to the reported Bril suggesting that the Jahn-Teller effect should louin-zone--edge TA phonon [43 cm·1 (Ref. be taken into account for this materiaL To 36)], one cannot exclude the possibility that our knowledge nothing is known about elec phonons over a considera.ble range inside thè tron-pbonon coupling for the 5E term in entire Brillottin zone contribute to electron Hg 1.yCdySe or HgSe. Only very recently it pbonon coupling, which may result in smaller was suggested from Mössbauer experiments37 "effective phonon 11 energy. We should stress, that a Jahn-Teller distortion might be ex however, that these calculations should be pected for Hg 1_xFexSe. Since no Jahn-Teller regarded only as an illustration suggesting a calculations have been reported for CdSe:Fe possible souree of the anomalous high speci (in particular, taking into account its hexag fic-heat values. A detailed discussion of this onal structure), we performed a simple calcu problem requires precise calculations of the lation to demonstrate the influence of Fe2+ energy levels with crystal-field, spin 11 phonon replica11 energy levels on the specific orbit, and Ja.hn-Teller interactions taken heat. We considered five equally spaeed Fe into account. One can, however, expect that electronic levels11, to which up to ten for the Fe-Fe pair, such calculations would Chapter lil, § 2 Low-temperature ..... Hgt-x-yCdyF'exSe 69 lead to rather large Hamiltonian matrices. Summarizing the remarks listed, we no tice that no essential differences are observed for the specific heat for Hg 1_xFexSe- Hgt-x-y CdyFexSe - Cd1_xFexSe series. The general behavior of Cm(Hg 1_x-yCdyFexSe) is also simi 1 lar to that found for Zn1_xFexSe 4. Recent analysis of the magnetic susceptibility of Hg 1_xFexSe (Ref. 17) shows that the reported Brillauin-type behavior in Ref. 8 is mainly due to the presence of Fe3• ions in all the investigated Hg 1_xFexSe crystals: a result of - valenee self-ionization of Fe2• ions4. Fe3+ ions have a band d5 electronk configuration (the same as Mn2+ ions) resulting in a Curie-type susceptibility. FIG. 14. Schematic representation of the elec Although the concentration of Fe3+ ions is tronic Fe levels within the band structure of a II-VI semiconductor, based on recent calculations on the small [n = 5·1018 cm-3, constant for all x > 0.01 (Refs. 4 and 5)], their contribution Mn containing CdTe (Ref. 38). to the total susceptibility may be dominant at low temperatures (T < 3 K) and may type DMS (d5), the five electrons of Mn oc completely mask the Van Vleck-type tem cupy e. and t. orbitals (e+2t+ 3 configuration) perature-independent susceptibility of Fe2+ located deeply in the valenee band. For Fe ions. In Ref. 17 it was shown that below type materials ( d6) the additional sixth elec: 3-4 K, the temperature-dependent suscepti tron must occupy an e_ orbital, yieldîng a bility is independent of the Fe concentratien e. 2t +3 e_1 configuration. The reported vari and can be well described as the Curie-type ation of the Fe-level position with Cd con susceptibility of n = 5·1018 cm-3 Fe3+ ions. A tent in Hg 1_:x:-yCdy:FexSe (Ref 5) concerns similar effect was also reported in Zn 1_x precisely electrans from the e_ orbital. How FexSe9, although in tha.t case the paramag ever, e orbitals in tetrabedral crystals hybrid netic ions were impurities in the material ize very weakly with p-type bands: they do used for crystal growing (mainly Mn2+ as not form strong a-type honds; only weak detected by electron paramagnetic reson 1r-type bonding is formed. Therefore,; no first ance9). order effects are expected with vafiation of' Summarizing our discussion on the re the Fe-level position relatively to the band sults of specific-heat and susceptibility data structure39 . Consequently, we feel confident so far, we are tempted to conclude that these to assume that the description of the magnet data do not reveal any anomalous behavior ie properties of all the Fe-based DMS investi which can be related to the location of the gated so {ar, can be based in first approxima Fe2+ energy level relative to the band struc tion on a simple "crystal field model'1 7 ture. This does not exclude, in principle, (eventually augmented by electron-phonon hybridization of the resonant Fe2• level with interaction). the conduction band 8, but the effect of such a In that respect one can now estimate the hybridization does, in our opinion, not mani d-d exchange interaction parameters for the fest itself in the present data. This weak Fe series Hgt-x-yCdyFexSe. This can be done level dependenee may be understood on the usîng high-temperature series expansion for basis of recent calculations of the Cd 1_xMnx the magnetic susceptibility developed recent Te energy structure38• In Fig. 14 we repro ly40 and based on the "crystal-field model" duce a schematic diagram for the d levels in of Fe-based DMS. In this model the Curie II-VI semiconductors38. In the case of Mn- Weiss temperature determined from high- Chapter 111, § 2 MAGNETIC BEHAVlOR ..... , BASEDON Fe 70 temperature susceptibility data, can be ex TABLE II. Parameters for the crystal pressed as 40 field model. e(x) = TABLE III. Curie-Weiss temperatures and d-tl interaction strength for II-VI Fe--eontaining DMS; a comparison is made with Mn compounds. Material e(K) Jn(K) Jn (Mn) (K) Hgu9Fe0•01Se -9±5 (8) -18:t2 -11 (45), -{i ( 42) Hgu5Fe0.nSe -38±5 (41) -18:t2 -11 (45), -{i (42) Hgo.s4Cdo.4tFeo.osSe -35±5 (8) -18:t2 Cdu5Fe0•05Se -36 (8) -18.8±2 -9 (43) Zn 1.xFexSe -22:t2 (40) -13 (44) Chapter III, § 2 Low-temperature ..... Hgt-x-yCdyFexSe 71 DMS (Ref. 51) can be also adopted for Fe change interaction in these materials awaits type materials giving some insight in chemi further study. Magnet(H)ptical experiments ca! trends observed in Table Hl The "three providing information about N0{J and ultra level model" is based on kinetic superex violet photoemission studies determining Ed change49,51 and simplifies the actual situation would also be very usefuL by consirlering only three levels: the valenee band (at energy Ev), the six-electron oc IV. CONCLUSIONS cupied d level (Ed), and an unoccupied level to which electron from the valenee band can In the present study of the specific heat he transferred. Such an electron transfer re of Hg 1.x-yCd FexSe DMS we found that all sults in an increa.se of the correlation energy, the Fe-type bMS investigated so far demon so a seven-electron configuration ha.s energy strate generally similar magnetic properties, 1 Ed + U. Thus the analytic expression for J which suggest that the ground state of both can he written as51 the isolated Fe ion and the Fe-Fe pair is a singlet (separated from the excited states by J = -2Vpd4 [(Ed+U-Ev)-2 U·t at a bout 20 cm -t for an isolated ion and (5) 10 cm·1 for Fe-Fe pair). Antiferromagnetic + (Ed+U-Ev)-3)]f(R) , coupling between Fe ions is observed. No experimental evidence was found that the where Ypd is the hybridization parameter magnetic properties of narrow-gap DMS dif descrihing the probability of a valenee band fer from those of wide-gap materials and, electron hopping to the Fe d level; f(R) de consequently, no Fe-level position depend scribes the dependenee on the Fe-Fe dis enee could he deduced. A simple crystal-field tance. The parameter Ypd may be related to model ( eventually complemented by electron the s-d exchange constant for the valenee pbonon interaction) seems to he sufficient to band N {J51 0 describe the magnetic properties of all the Fe-type DMS. The d-d exchange parameters for Hgt-x-yCdyFexSe systems are substantial (6) ly larger than those of Hgt-x-yCdyMnxSe, in agreement with the situation encountered for Zn .xFexSe and Zn .xMnxSe. At the moment knowledge about the 1 1 parameters occurring in Eqs. (5) and (6) is rather poor. In particular, N0{J is known only for Znt-xFexSe (Ref. and Cdt-xFexSe 9) See, Diluted Magnetic Semiconductors, Vol. 25 (Ref. 52). For these materials one finds that of Semiconductors and Semimetals, edited by N {J is larger in Zn _xFexSe by a factor of 1.3 0 1 J.K. Furdyna and J. Kossut (Academie, New in respect to Zn _xMnxSe 9•53 {1.5 in the case 1 York, 1988); Diluted Magnetic Semiconductors, of CdSe). Assuming that this increase of N {J 0 edited by R.L. Aggarwal, J .K. Furdyna and S. results mainly from a stronger hybridization, von Molnar (Materails Research Society, Pitts one can expect a larger d-d exchange para burg, 1987), Vol. 89; J.K. Furdyna, J. AppL meter by a factor 1.7 (2.1 for CdSe), which Phys. 64, R29 (1988). compares favorably with the experimental 2 W.J.M. de Jonge, A. Twardowski a.nd C.J.M. value for J (Jn(Zn .xFexSe) = -22 K ~ 1.7" 1 Denissen in Diluted Magnetic Semiconductors, = -19K JNN(Znt-xMnxSe), JNN(Cdt-xFexSe) edited by R.L. Aggarwal, J.K. Furdyna. and S. ~ 2.1 x JNN(Cd _xMnxSe)]. The higher Jn 1 von Molnar (Ma.tera.ils Research Society, Pitts values obtained for HgSe and Hg .yCdySe 1 burg, 1987), Vol. 89, p.156. with iron suggest a simHar situation also for A. Twardowski, H.J.M. Swa.gten, W.J.M. de the Hg _x-yCdyFexSe series. Detailed analysis 1 Jonge and M. Demianiuk, Phys. Rev. B 36, of the chemical trends in d-d and s-d ex- Chapter III, § 2 MAGNETIC BEHAVIOR ..... , BASED ON Fe 72 7013 (1987). hexagonal distartion in a wa.y more common in See the review papers, A. Mycielski, J. Appl. the literature (see Ref. 23}: Ht = -2/3B4x 3 Phys. 63, 3279 (1988); A. Twardowski, J. (04°+20/204 ) + B2°02° + B4°04°, and the Appl. Phys. 67, 5108 (1990). z axis was chosen along [111J direction (i.e., 5 A. Mycielski, P. Dzwonkowski, B. Kowalski, parallel to crystal c axis). The presence of B.A. Orlowski, M. Dohrowolska, M. Arcis trigonal distortion in the Hamiltonian causes zewska, W. Dobrowolski and J .M. Baranowski, splitting of triplets Tb T2 into singlets and J. Phys. C 19, 3605 (1986}. doublet (Tt- A2,B and T2- At>E) (Ref. 13). 6 A. Twardowski, M. von Ortenberg and M. Since the Harniltonia.n matrix is pa.rametrized Demianiuk, J. Cryst. Growth 72, 401 (1985). hy material constants Dq (describing cubic 7 A. Twardowski, H.J.M. Swagten, T.F.H. v.d. crystal field}, 11, 11' (both descrihing hexagonal Wetering and W.J.M. de Jonge, Solid State distortion), and À (spin-orbit interaction), the Commun. 65, 235 (1988). obtained energies can he used for the estima. 8 A. Lewicki, J. Spalek and A. Mycielski, J. tion of these parameters for Cdt-xFexSe. We Phys. C 20, 2005 (1987). used for that very recent measured energies of 9 A. Twardowski, P. Glod, W.J.M. de Jonge and optica! transition At-+ A2 [13 cm-1 (Ref. 24)J, M. Demianiuk, Solid State Commun. 64, 63 At-+ E [17.6 cm-1 (Ref. 24)J as well as zero (1987). pbonon line energy for SE-+ ST transition 10 D. Heima.ri:, A. Petrou, S.H. Bioom, Y. Sha [2375 cm-1 (Ref. 25)). The best fit was found pira, E.D. Isaacs and W. Giriat, Phys. Rev. for Dq = 257 cm-1, À = -93.8 cm-1, 11 = Lett. 60 (18}, 1876 (1988). 31 cm-1 and 111 = 38 cm-1. 11 W. Low and M. Weger, Phys. Rev. 118, 1119 22 A. Twa.rdowski, Solid State Commun. 68, 1069 (1960). (1988). 12 G.A. Slack, S. Roberts and J.T. Wallin, Phys. 23 A. Abragam and B. Bleaney, Electrcrn. Para Rev. 187, 511 (1969). magnetic Re1onance of Tramition Metal Ion1 J. Mahoney, C. Lin, W. Brumage and F. Dor (Clarendon, Oxford, 1970}. man, J. Chem. Phys. 53, 4386 (1970). 2( D. Scalbert, J. Cernogora, A. Mauger and C. H.J.M. Swagten, A. Twardowski, W.J.M. de Benoit a la. Giullaume, A. Mycielski, Solid Jonge and M. Demianiuk, Phys. Rev. B 39, State Commun. 69, 453 (1989). 2568 (1989}. 25 M. Arciszewska and A. Mycielski (private com 15 A. Lewicki and A. Kozlowski, Acta. Phys. Pol. munication). A 73, 435 (1988}. 26 Parameters used for this calculation are ta.bu 16 H.J.M. Swa.gten, F.A. Arnouts, A. Twardow la.ted in Table n and were choosen for Cd1_x ski, W .J .M. de Jonge and A. Mycielski, Solid FexSe, to fit spectroscopie data, and for Hgt-x State Commun. 69, 1047 (1989). FexSe and Hgt-x-yCdyFexSe, to provide rea 17 M. Arciszewska A. Lenard T. Dietl, W. Ple sonable description of the low-temperature Cm· siewicz, T. Skoskiewicz and W. Dohrowolski, 27 One should not he confused hy data for higher Acta Phys. Pol. A 17, 155 (1990). Fe concentrations, which compares reasonably Ui R.R. Galazka, S. Nagata a.nd P.H. Keesom, with the calculations. The calculated specific Phys. Rev. B 22, 3344 (1980). heat shown in Figs. 6-8 corresponds to the 19 K. Matho, J. Low Temp. Phys. 35, 165 (1979). very dilute limit, in which interactions between 20 C.J.M. Denissen, H. Nishihara, J.C. van Gooi Fe ions can be neglected. For higher a:, AF and W.J.M. de Jonge, Phys. Rev. B 33, 7637 interaction between Fe ions results in lowering (1986). of Cmfz. 21 The 11 crystal-field 11 model for hexagonal ma 28 We notice that an Fe-Fe pair contributes in terials has been recently reported (see Ref. 22) the same way: Spair = tRln(10xlO) = RlnlO. with rela.tively simple form of the hexagonal 29 M.D. Sturge in Solid State Pb.y1ic1, edited by term in the Hamiltonian. In this paper we used F. Seitz, D. Turnbull and H. Ehrenreich (Aca basically the same model, however, expressing demie, New York, 1967), Vol. 20. Chapter III, § 2 Low-temperature ..... Hgt-x-yCdyFexSe 73 30 J.T. Vallin, Phys. Rev. B 2, 2390 (1970). J.D. Chadi a.nd W.A. Harrison (Springer, New 31 G.A. Slack, S. Roberts a.nd F.S. Ham, Phys. York, 1985), p. 1419. Rev. 155, 170 (1967). T.M. Giebultowicz, J.J. Rhyne, J.K. Furdyna, 32 G.A. Slack, F.S. Ham a.nd R.M. Chrenko, J. Appl. Phys. 61, 3537 (1987); 61, 3540 Phys. Rev. 152, 376 (1966). (1987). 33 D. Buhma.nn, H.J. Schultz and M. Thiede, 44 J. Spalek, A. Lewicki, Z. Tarnawski, J.K. Fur Phys. Rev. B 24, 6221 (1981). dyna., R.R. Gala.zka, Z. Obuszko, Phys. Rev. B 34 J.C. Irvin a.nd J. LaCombe, Ca.n. J. Phys. 50, 33, 3407 (1986). 2596 (1972} 46 M.A. Ruderman a.nd C. Kittel, Phys. Rev. 96, 35 A.M. Witowski, A. Twardowski, M. Pohlmann, 99 (1954); K. Yoshida, Phys. Rev. 106, 893 W.J.M. de Jonge, A. Wieck, A. Mydelski and (1957); T. Kasuya, Prog. Theor. Phys. 16, 45 M. Dernia.niuk, Solid State Commun. 70, 27 (1956). (1989). 47 N. Bloembergen and T.J. Rowland, Phys. Rev. 36 R. Beserma.n, Solid State Commun. 23, 323 97, 1679 (1955) . (1977}. 48 C. Lewiner a.nd G. Bastard, J. Phys. C 13, 37 I. Nowik, E.R. Bauminger, A. Mycielski a.nd H. 2347 (1980); C. Lewiner, J.A. Gaj a.nd G. Bas Szymcza.k, Physica B 153, 215 (1988). ta.rd, J. Phys. (Paris) Colloq. 41, C5-289 38 Su-Hua.i Wei a.nd A. Zunger, Phys. Rev. B 35, (1980). 2340 (1987). 49 P.W. Anderson in Soli.d State Physic11 14, 99 39 It is worthwhile to notice then, that in both (1963), edited by F. Seitz and D. Turnbull Mn-type a.nd Fe-type DMS, the same elec (Academie, New York, 1963), Vol. 14, p. 99. trons (occupying t + orbital) are responsible for 50 G.A. Sawatzky, W. Geertsrna a.nd C. Hass, J. a-d interaction. Magn. Magn. Mater. 3, 37 (1976); C.E.T. Gon 40 A. Twa.rdowski, A. Lewicki, M. Arciszewska, calves da Silva a.nd L.M. Falicov, J. Phys. C 5, W.J.M. de Jonge, H.J.M. Swagten a.nd M. 63 (1972}. Demianiuk, Phys. Rev. B 38, 10749 (1988). 51 B.E. Larson, K.C. Hass, H. Ehrenreich and 41 J. Spalek (private communication). A.E. Carlsson, Solid State Commun. 56, 347 42 R.R. Gala.zka, W. Dobrowolski, J.P. Lascara.y, (1985); Phys. Rev. B 37, 4137 (1988). M. Nawrocki, A. Bruno, J.M. Broto, J.C. Ous 52 A. Twardowski, K. Pakula, M. Arciszewska set, J. Mag. Mag. Mater. 72, 174 (1988). a.nd A. Mycielski, Solid State Commun. 73, 43 R.L. Aggarwal, S.N. Jasperson, Y. Shapira, S. 601 (1990). Foner, T. Sa.kibara, T. Goto, N. Miura, K. 53 A. Twardowski, in Diluted Magnetic Semicon Dwight and A. Wold, in Proceedings of the ductorl, edited by M. Jain (World Scientific, 17th International Conference of the Physics of Singapore, 1990). Semiconductors, Sa.n Francisco, 1984, edited by Chapter III, § 3 MAGNETIC BEHAVIOR ..... , BASED ON Fe 74 Chapter m, para.graph 3 Magnetic susceptibility of iron • based semimagnatie semiconductors: High - tempersture regime [ publisbed in Phys. Rev. B 38, 10749 (1988) ] I. INTRODUCTION magnetic situation, because Fe2• possesses both spin and orbital momenta ( S = 2 and Semimagnetic semiconductors (SMSC) L = 2)3-5. or diluted magnetic semiconductors1 are ma Although some attempts to understand terials ba.sed on the well known 11-VI or 11-V the magnetic behavior of these compounds compounds in which some ca.tions are substi were already reported6-8, crudal information tuted by magnetic ions. The concentration of about the Fe-Fe interaction (strength and magnetic ions can be controlled in a very range) is still lacking. The only estimate for wide range (from doped to mixed crystals) the nearest-neighbors interaction (Jnd is which allows one to study in one material derived from the high-temperature suscepti both the very dilute and highly concentrated bility8 assuming L = 0. We therefore thought regimes. Because of that, these alloys have it worthwhile to study the L # 0 case more attracted considerable attention during the carefully in order to be able to evaluate the last decade and interesting magneto-optical d-d exchange parameters more reliable. We and magnetic properties were reported 1• The also report in this paper new results of the strong s--d exchange interaction between magnetic susceptibility of Zn1_xFexSe in the band electrous and localized d electrous of temperature range 2-300 K. Finally, we 8 the magnetic ions results in effects like giant reexamine previous results for Cd1_xFexSe. free-exciton Zeeman splitting or formation of magnetic polaron. On the other hand, due to ll. DERIVATIONOFTHE the antiferromagnetic d-d interaction be HIGH-TEMPERATURE tween magnetic ions, various magnetic phases SUSCEPTIDILITY FOR L # 0 (paramagnetic, spin-glass and antiferromag netic) were found, depending on the concen In this section we calculate the high tration and temperature range. temperature expansion (HTE) of the static So far, most of the studies were devoted magnetic susceptibility following the metbod to manganese-based SMSC, such as Cd1_x applied in Ref. 9 for the spin-only case Mnx Te, Hg1_xMnx Te or Zn1_xMnxSe. From a (L = 0). We obtain a Curie-Weiss law for magnetic point of view these materials are the general case of magnetic ions with both particular since Mn2+ bas a ds electronic con S # 0 and L # 0, resulting in a relationship figuration resulting in a spin- a random, dilute system can be expressed in the following way: J{ = -1 E 2JiJsi.sjeiej + P.BBz E (Lzi+gSzï)Çï + E Hcnei + E .\L1-Si6 , (1) i,j i i i where sums over i and j run over alllattice sites (i# J), and Ç1 is 0 or 1, depending on whether the cation site is occupied by a nonmagnetic (Zn, Cd or Hg) or a magnetic ion, respectively. S1; L1 are the atomie-spin and orbital-momenta operators, respectively. The first term in Hamil tonian (1) describes the d-d exchange interaction (J1j is the exchange integra.l), the secoud the influence of the magnetic field, the third the crystal-field splitting and the last the spin-orbit interaction. A given sequence { Ç!} determines a specific dis tribution of magnetic ions on the lat tice sites. The static susceptibility per unit volume is defined by: (2) where V is the volume of the system and F is its free energy, given by (3) In this expression P( {{i; .... , {N}) is the probability distribution of a given sequence ({11 Ç2, ...... , ÇN ), Tr denotes trace, Z is the pa.rtition function for a pa.rticula.r configuration { Ç!i ..... , {x}, ( -) means averaging over configurations and {3 = 1/kBT. In the high-temperature regime the partition function Z can be expanded as follows: (4) Performing double derivation of free energy (3), one finds _kBT[18 2 Z 2 and (5) X- V Z 8B z [i f8~] 1 , :Bz = -{3 TrM + [32 Tr(MH)- {33 Tr(MH2)/2 , (6) z (7) where we introduced M =8H/8Bz and used the relation Tr(ABC) = Tr(BCA) = Tr( CAB). Sub stituting (6) and (7) into (5) yields, to first order in H, _ jj_[Tr(M2) + _ri!'~(M2)Tr(H) _ {3Tr(M2H) '!'!M}2 X-v Trl TrlTrl Tr1 [TrfJ _ {3TrM [Tr(M)Tr(H) hlA!1llJ] 2 Tr1 Tr1Trl -rrrfl (8) Chapter lil, § 3 MAGNETIC BEHAVlOR .•..• , BASED ON Fe 76 2 - 2(3 (9) One can now eva.luate the traces in (9) and get where Et= >.LtSi + Hcfi is the single-ion crystal-field and spin-orbit operator, and Mzi = Lzi + gSzi is the single ion magnetic moment operator. It follows from (10) and (9) that 2 2 2 2 X= (J.ts /kBTV) E {Dij [ Performing an averaging over configurations yields 2 2 2 2 X= (J.ts /ksTV)xN [ V= = x for i = j , e1ej = _ (13) [ çiej = x2 for i # j . Equation (12) can be expressedas follows: X= /B'~~xN and for high temperatures can be expan~ed as a Curie-Weiss law C(x)/[T-E>(x)] wtth Curie constant and a Curie-Weiss temperature w ~ 20 = A + 0 0x , (16) ~ 1:·~-100 ~'-Ll'---~,-----~---:;3;;;;'00 0 100 200 which reduces toa sta.ndard HTE (Ref. 9) for temper ature ( Kl L = 0. If the distribution of magnetic ions is truly random [Eq. (3)], then both the Curie constant and the Curie-Weiss temperature FIG. 1. Inverse magnetic susceptibility of Znt-x are a linear function of temperature. However Fe xSe as a function of temperature. Straight. lines - in distinction from the spin-only case9 - show Curie-Weiss law with parameters accordmg to the Curie-Weiss temperature may be nonzero Table I. even for x~ 0 (noninteracting ions). The values of averages in (12) and (13) can be evaluated for a partienlar ion in a partienlar the inverse susceptibility is plotted versus crystal field if the ion eigenfunctions are temperature. For clarity of the figure the known. We will return to this point for Fe2+ low-temperature part (below 77 K) is shown in the Discussion. only for x = 0.022 and 0.072. At high tem peratures all the samples reveal .a typi~al ID. EXPERIMENTAL RESULTS Curie-Weiss behavior with a negattve Cune Weiss temperature. These temperatures We have measured the magnetic suscep e(x) as determined from extrapolation of tibility per unit mass Xm [related to Eq. (2) Xm -1 'versus T, are tabulated in Table I. Be by Xm = Xv/ p, p being density of the crystal] low ~ 70 K a well-pronounced downward Of Zn Fe Se with x 0.022±0.002, 0.049± 1-x x ' = bending of Xm -t can be observed. This 0.003, 0.057±0.003, and 0.072±0.005 as bending - as distinct from Mn-based ma checked by mieroprobe analysis. Xm was terials11 - changes its slope as T ~ 0, al studied in the temperature range 2 - 300 K, though even at the lowest measured temper~ in two stages. Below 77 K we used a standard tures no temperature-independent Xm-1 IS ac mutual inductance method, whereas above observed (i.e., typkal Van Vleck-type behav 77 K the Faraday metbod was applied. In the ior is absent). This is in agreement with our latter case the magnetization Mm of the previous results12 and is probably due to samples was measured, which .was. found to paramagnetic impurities present in ~he souree depend linearly on the magnetlC fteld up to iron material used for crystal growmg 12 • The 1 T. The susceptibility was obtained from the deviation from Van Vleck-type behavior in relation x = Mm/ B. All ~he data v:e.r~ cor the low-temperature range seems to be pro rected for the diamagnette susceptlbthty of 6 portional to the iron content, which corrobor ZnSe (Xd = -0.32·10- emu/9)1° .. ates such an assumption. The results are shown m Ftg. 1, where Chapter lil, § 3 MAGNETIC BEHAVIOR ..... , BASED O.N Fe 78 IV. DISCUSSION 5T ns1 A. High-tempera.ture susceptibility The behavior of the experimenta.l data ~::. • 29Jlcm- 1 presented above as Xm( T) in the range of high (4240KI temperatures may be compared with the re sults of HTE as derived in Sec. II. In Fig. 2, S(x) and C(x) are shown as functions of x. Bath quantities are proportiona.l with concen tration x, in agreement with Eqs. (15) and (16). Moreover, the Curie-Weiss temperature tends to zero ( within experimental accuracy) as x -+ 0 suggesting that parameter A from Eq. (16) vanishes in our case. The detailed fit of the data presented in Fig. 2 by Eqs. (15) and {16) yields the following values: C0 = vr1i ion + 15 FIG. 3. Energy-level diagram for an isolated Fe2+ ion. Effects of crystal field, spin-orbit interac tion and magnetic field are shown (not to scale). (212±40)·10-4 emuK/g, ( TABLE I. Relevant parameters for Znt-xFexSe a.nd Cdt-xFexSe. Material x C (10' emuK/g) e (K) Zn1.zFexSe 0.022 5.25±0.1 -18.5±2.1 9.2 0.049 11.60±0.1 -46.5±3.5 9.1 0.057 12.53±0.1 -52.2±1.3 8.4 0.072 13.93±0.2 -64.5±2.5 7.4 !8'2Serr( Serr+l) Cd1.xFexSe 0.05 -36±6 10.6 (Ref. 8) 0.15 -102±8 10.1 (Ref. 8) 8 available (see Table I). Cd1.xFexSe crystal lizes . in hcp structure, which re1mlts in lowering of the cubic symmetry, and conse where Serf is the so-called effective spin and quently, the threefold degeneracy of T 1 and is derived from the experimental value C(x). T2 levels is removed. In order to account for Thus the factor ig2Seff(Serr+1) plays the 2 this effect, we completed single-ion crystal same role as 60 observed for x = 0.022, whereas a substantial discrepancy exist for x= 0.072. We should Znl-x Fex Se 1 stress, however, that for x= 0.072, the dilute ... limit as considered in the ENNP A, may not §"' be approached. In fact, in all the cases when ~40 the ENNP A was applied for SMSC, a reason :.a +=a. able description of the experimental data was e: obtained only for low concentrations (x< ~ X=0.072 o 3 0 0.05)1 -14. We therefore find the situation .. presented in Fig. 4 quite satisfactory. V. CONCLUSIONS We generalized the high-temperature temperature !KI expansion of the magnetic susceptibility of SMSC for the case of magnetic ions with non vanishing orbital momenta. We obtained a FIG. 4. Inverse ma.gnetic susceptibility of Curie-Weiss law with Curie constant and Znt-xFexSe a.s a function of temperature. Solid lines Curie-Weiss temperature linearly dependent repreaent the ENNP A calculations with parameters on the concentra.tion of the magnetic ions. JNN = -22 K, Jp = Jn/Rp6.8, lODq = 2930 cm-1 This result is in agreement with high-tem and .À= -85 cm-1. perature experimental data for Zn1_xFexSe and Cd 1.xFexSe. From the Curie-Weiss tem perature we estimated the exchange integrals 14,17. This metbod is basedon the assumption JnfkB = -22 K for Zn1.xFexSe and Ju/ka that the partition function of the system can = -17.8 K for Cd1_xFexSe, which are both be factorized into contributions of pairs of systematically higher than those observed in ions. In the ENNP A each ion is considered to the conesponding Mn compounds. Using be coupled by an exchange interaction h these exchange constants, we were able to fit only to its nearest magnetic neighbor, which the susceptibility in the whole temperature may be located anywhere at the distance RL range by applying the ENNP A. from a reference site. The statistica} weight [PL(x)] of pair configurations with various RL is assumed to be determined by the random distribution of the ions. Knowing the N.B. Brandt and V.V. Moshalkov, Adv. Phys. susceptibility of an Fe pair, coupled by an ex 33, 193 (1984), J.K. Furdyna, J. Appl. Phys. change integral h, one can evaluate the sus 53, 8637 (1982), J .A. Gaj, J. Phys. Soc. Jpn. ceptibility of the whole crystal as 49, 797 (1980). 2 A. Mycielski, in Di.lv.ted Magneti.c Semi.conduc tor&, edited by S. von Molna.r, R.L. Agga.rwal and J.K. Furdyna, (Materials Research Society, Pittsburgh, 1987), Vol. 89, p. 159. where the summation is performed until 98% 3 W. Low and M. Weger, Phys. Rev. 118, 1119 of the pairs are included. For more details of (1960). the calculations we refer to Ref. 7. G.A. Sla.ck, S. Roberts and J.T. Wallin, Phys. Results of our calculations for Zn 1_xFex Rev. 187, 511 {1969). Se (x= 0.022 and x= 0.072) are shown in 5 J. Mahoney, C. Lin, W. Brumage and F. Dor Fig. 4. Fairly good agreement between the man, J. Chem. Phys., 53, 4386 (1970). experimental and calculated susceptibility is 6 A. Twardowski, M. von Ortenberg and M. Chapter III, § 3 MAGNETIC BEHAVlOR ..... , BASEDON Fe 82 Demianiuk, J. Cryst. Growth 72, 401 (1985). (1986). 7 A. Twardowski, B.J.M. Swagten, T.F.B. van 14 A. Twardowski, B.J.M. Swagten, W.J.M. de de Wetering and W.J.M. de Jonge, Solid State Jonge and M. Demianiuk, Phys. Rev. B 36, Commun. 65, 235 (1988). 7013 (1987), W.J.M. de Jonge, A. Twardowski 8 A. Lewicki, J. Spalek and A. Mycielski, J. and C.J .M. Denissen, in Diluted Ma.gnetic Phys. C 20, 2005 (1987). Semiconductor1, edited by S. von Molnar, R.L. 9 J. Spalek, A. Lewicki, Z. Tarnawski, J.K. Fur Aggarwal and J .K. Furdyna, (Materials Re dyna, R.R. Gala.zka and z. Obuszko, Phys. search Society, Pittsburgh, 1987), Vol. 89, p. Rev. B 33, 3407 {1986). 156. 10 D.J. Chadi, R.M. White and W.A. Barrlson, 15 T.M. Giebultowicz, J.J. Rhyne and J.K. Fur Phys. Rev. Lett. 35', 1372 (1975). dyna, J. Appl. Phys. 61, 3537 (1987)i 61, 3540 11 R.R. Gala.zka, S. Nagata and P .B. Keesom, (1987). Phys. Rev. B 22, 3344 (1980), S. Nagata, R.R. 16 Such a value provides a sirnilar Tt-level split Gala.zka, D.P. Mullin, B. Arbarzadeh, G.D. ting as reported (Ref. 5) for CdS:Fe. We Khattak, J.K. Furdyna and P.B. Keesom, ibid. should stress, however, that the exchange 22, 3331 (1980). integral Jn is not very sensitive for Op: 12 A. Twardowski, P. Glod, W.J.M. de Jonge and increasing Op 5 times to 150 cm -1 results only M. Demianiuk, Solid State Commun. 64, 63 in a 3% increase of Jn· (1987). 17 K. Matho, J. Low. Temp. Physics 35, 165 13 C.J .M. Denissen, B. Nishlhara, J.C. van Gooi (1979). and W.J.M. de Jonge, Phys. Rev. B 33, 7637 Chapter III, § 4 Magnetization steps in iron-based ..... 83 Chapter III, para.graph 4 Magnetization steps in iron- based diluted magnetic semiconductors [ publisbed in Phys. Rev. B 41, 7330 (1990) ] I. INTRODUCTION as a direct probe of the determination of Jrnr 5. In this report we will show that, despite Diluted magnetic semiconductors (DMS) the complexity of the Fe2• energy-level generally consist of a nonmagnetic semicon scheme, in this case magnetization steps ducting host in which a controlled fraction of should also be detectable and could be used cations are replaced by isovalent magnetic to determine Ju, provided that I Jn I ex ions. Up till recently the investigations were ceeds a certain critical interaction. Moreover, almost exclusively restricted to DMS con we will show that the position of the first taining Mn 2 + as the magnet ie ion 1• An in magnetization step is almost independent of creasing number of papers at the moment, sample orientation and crystal-field para however, deal with DMS containing other meters. These results therefore offer experi transition elements, such as Co2+ or Fe2•. mentalists the opportunity to establish the Magnetically the latter ions represent a more Fe2+_pe2+ exchange interaction from the genera!, though more complex, case since, in high-field magnetization. addition to the spin, also orbital momenta are involved2. ll. THE·Fe-FePAIR Notwithstanding the increasing number of investigations devoted to these compounds, The quantum-mechanical treatment of the experimental status is still somewhat the isolated Fe2• ion, substituted in a semi poor. In contrast to Mn DMS, for instance, conducting host, has been elaborately evalu no experimental data have been reported yet ated by Slack, Roberts, and Vallin and which enable one to deterroine the nearest others6• In addition to their treatment for neighbor (NN) interaction strength Jn in a Fe-doped crystals, a Beisenberg-type Fe2•_ straightforward way. The strength and sign Fe2• interaction has been included for DMS of JNN are of great fundamental interest and (see our previous paper7): are intimately related to the band structure and the location of the electronic levels with :K =KeF + J >.. (al are separated roughly by 6>..2/ a, where represents the spin-orbit parameter and a (or lODq) the crystal-field splitting. For the IJ =-10K I calcula.tions we used the parameters for Znt-xFexSe: >.. = -95 cm·1, a = 2940 cm·1; see Sec. III. In the presence of the external field, mixing with higher sta.tes repulses the magnetically inactive ground state, thereby -150 10 inducing a. net ma.gnetic moment. For small fields the ground state varles with -B2, yield ing a.t low tempera.tures the so-called Van Vleck-type of parama.gnetism. For antiferromagnetically coupled pairs ·175 (J < 0) the energy schemes become very com plicated and it follows, somewhat remark able, that a critica! interaction strength IJcritl (Jcrit ~~: -13/-17 K depending on the lal @11 [00111 B (Jl 0 {b) FIG. 1. Low-lying energy 'levels and magnetiz Q; tL zo 0 ation, at T = 0 (see insets), of the Fe2+ pair for B E alo"ng the crystallographic [111] axis; (a) J -10 K, t::, = .., (b) J = -22 K, the arrow indica.tes the position of the magneti.Zation step. z (cl 6 7 (HeF + 1{80) • • Frà1!t, the field dependenee of the eigenvalnes the -Jl1agnetiza.tion for one 20 mole Fe2+ ionscan be càlcula.ted by 10 I B 11 [ 1111 I 0 100 150 B !Tl with i and j running from 1 to 100. FIG. 2. Magneti.Zation, at T = 0, of an Fe2+ For uncoupled pairs (J = 0) the situ single ion (J = 0, -·-), and an Fe2+ pair (J = ation is rela.tively simple. For B = 0 we re -22 K, --), for B along three crystallographic call that the low-lying Ah T11 E, T2 a.nd A2 axes: (a) [001], (b) [101], (c) [111]. Chapter III, § 4 Magnetization steps in iron-based ..... 85 field-direction) is present, above which steps trated in Fig. 2 for B along the [001], [lOl], in the magnetization are visible. The occur and [111] axes. We want to emphasize that, rence of such a critical interaction results as far as the first step is concerned ( see Figs. from the interplay of the terros in the full 2 and 3) the field direction does not influence Hamiltonian ( 1): crystal field in cl u ding spin its position significantly ( always within orbit coupling, exchange, and magnetic field. 10%), but only the size of the step. Quite For IJl < I Jcrit I the relatively large degree surprisingly, it also appears that, despite the of mixing repulses the ground state in that complex energy scheme of the Fe2+ pair, a amount that excited states cannot intersect. close resemblance with the spin only situ The downward bending of the ground state is ation (gJ.tBBstepl = 2kB IJl) is obtained, even quantified by the zero-temperature suscepti for J closely approaching Jcrit· Furthermore, bility x which depends on the crystal-field we checked the sensitivity of the first-step parameters >. 2/!::. and the interaction J. For position on the crystal-field parameter >.. An increasing J the susceptibility of the pair de increase from -95 to -105 cm-1, which actual creases and in the limit of large J ( with an ly increases the single--ion splitting with 22%, antiferromagnetic ground state) x becomes induces a step shift of roughly 5%. zero. This rednetion of x with increasing J creates the crossing phenomenon. As an illus ill. THE ACTUAL CRYSTAL tration, energy levels and magnetization for J -10K and J = -22 K are plotted in Using the results of Sec. II one is now Fig. 1. We will return to this subject in the able to describe the magnetization of an a.c Discussion. t.ua.l DMS crystal. We will confine ourselves In general, the position of the steps de to Zn1_xFex:Se, which, from the magnetic pend strongly on the field direction as illus- point of view, is representative for several Fe--containing DMS originating from the U VI group [e.g., Cd1_xFex:Te (Ref. 8)]. More ao.-----~-----.-----.----~.-----. over, the crysta.l-field parameters are quite well establishedll·1•9 and in addition, from the [~11 [001Jj high-T susceptibility Jn [-22 K (Ref. 4)] has 60 IB 11 [111J] been estimated. To ca.lculate the magnetization of such a ~ ( random array of Fe ions, one usually5 decom poses the array in singles (no nearest neigh bors), nearest-neighbor pairs, triples and higher-order arrays as determined by statis tics (cluster approximation). This approach, ho wever, is valid in case the exchange inter action is restricted to nearest neighbors only. In principle the long-range character of J 0 -10 ·20 -30 -40 ·50 should be taken into account, which couples J IK) every spin with others. In such a case the system can be described within the so-called extended nearest-neighbor pair approxima FIG. 3. Position of the first rnagnetization step tion (ENNPA)10 • For Zn1_xFexSe it has been as a fundion of interaction strength, for B along the suggested7 that J"' R-6.8. The evidence, how crystallographic (001) and [111] axis; the spin-only ever, is rather weak and a sharper decrease case (YJ.tBBstep1 = 2~ IJl) and the results within with distant, which reduces J to Jn only, the introduced model [see the inset and Eq. (4)] are cannot be excluded7. also shown. In Fig. 4 the results for both cases, i.e., Chapter III, § 4 MAGNETIC BEHAVlOR ..... , BASED ON Fe 86 the ENNP A and cluster approximation (re tion. In contrast to the Mn case the step size stricied to singles and pairs), are drawn for is not completely determined by statistles x= 0.05, showing the characteristic step in only, but also depends on parameters such as the [111] magnetization. Only minor differ J, >.2/ tl, and field orientation. As an illustra. ences between the two approximations can be tion, for Ju = -22 K and B 11 [111] it follows distinguished due to the energetically close that 8M 111 30xPJ>air(x) emu/g, Ppair(x) based resemblance between isolated and weakly on ra.ndomly distributed Fe ions [= 12x coupled ions. Since in fact these differences x(l-x)18]. For x= 0.05 this results in a step are restricted to small fields, where 8JLBB ~:~ size of ~ 0.35 emufg. As a. consequence, êM ks IJl for J beyond the NN, we may conclude ca.n serve as a probe for the randomness of that the steps are insensitive to the tail of J. the magnetic system, which, by this means, Neglect of internal fields so far, however, can has been verified for Mn-eontaining DMS5. possibly affect the position of the steps seri To our knowledge no experimental data ouslyu. The influence of a nonzero tempera exist on the magnetization in the regime ture is a.lso depicted in Fig. 4. lt is clear that where, on the basis of the present calcula due to thermal population of excited states, tions, steps should occur. For B < 15 T data which is relatively important at fields in the are reported for Cd1_xFexTe (Ref. 8) and vicinity of Bstep• a broadening of the steps Zn1_xFexSe (Ref. 7), and a reasonable descrip will be induced. tion within the pair approximation can be The actual change in the magnetization obtained; see the supplementary data in at Bstep• brought about by the nearest-neigh Fig. 4. However, since these data do not ex bor paus, is independent of any approxima- tend to the magnetization step, which is the subject of this report, we will not attempt any further interpretation. IV. DISCUSSION The ca.lculations revealed the existence of a critical value IJcritl below which no steps in the magnetization can be observed. In addition to the comments made before, we will introduce a simple model which explains the observed phenomena. in a more transpar ent way. The part of the complicated energy scheme of the Fe pair, which intuitively is responsible for the occurrence of magnetiz ation steps, can be approximated by the fól• lowing assumptions: (i) the singlet ground state varies quadratically with field: E0 = 40 60 -Xpair(T=O)B2, obeying Van Vleck-type of B !Tl paramagnetism; (ii) the first intersection with E0 will be brought about by sta.tes close ly following the spin-only eigenfunctions of a FIG. 4. Magnetization of Zn0.95Fe0,05Se for B pair: Et = ms,pair8J.tBB, ms.pair = -1; and a.long the crysta.llographic [111] a.xis, uaing the (iii) for B = 0, E0 and Et are separated by ENNPA with J = -22R-6.8 K (T = 0, --, T = 21 Jl , which is in principle only valid for spin 1.5 K, --- -) and the cluster approxirnation with only eigenfunctions of a pair ( J > >. 2/ Ä). The J= -22 K (T= 0, -·-·-); the squares repre difference between the ground and the first sent data from Ref. 7. excited state can be written as: Chapter III, § 4 Magnetization steps in iron-based ..... 87 Eo-Et = -Xpair( T=O)B2 + gJ.LBB- 2kB IJl , netization should be subject to can be sum marized as follows: (a) temperature should be (4) kept well below 4.2 K for an optima! popula tion of the ground state; (b) observation of at where the pair susceptibility can be obtained least the first magnetization step would re by treating J{ for vanishing small fields ..F.ig quire fields of the order of 50 T; ( c) the use of ure 3 is supplemented with the first posttlve oriented single crystals is preferable, though root of Eq. (4); the second root is unphysical not essential, for the position of the first step; ly large. The existence of a critica! interac fields parallel to the [111] axis should pro tion strength {Jcrit = -10 K), though som~ duce maximal step sizes; and ( d) an Fe-con what smaller in magnitude than observed m centration x ::: 0.052 should be chosen to the exact treatment {Jcrit = -13/-17 K), is maximize the NN pairs. indeed predicted. It is clear from Fig. 3 that this simple model apparently contains the essential elements from both the crystal-field part and the spin-only part of J{, thereby J.K. Furdyna, J. Appl. Phys. 64, R29 (1988) inducing the phenomena observed by treat ment of the full Hamiltonian. 2 A. Mycielski, J. Appl. Phys. 63, 3279 (1988) 3 B.E. Larson, K.C. Hass, H. Ehrenreich and Experimentally, the existence of the step magnetization in Fe-based DMS is not as A.E. Carlsson, Phys. Rev. B 37, 4137 (1988) 4 A. Twardowski, A. Lewicki, M. Arciszewska, evident as suggested by the present calcula W.J.M. de Jonge, H.J.M. Swagten and M. Jn tions since obviously, if I I < I Jcrit I, Demianiuk, Phys. Rev. B 38, 10749 (1988) which is not excluded12, no characteristic See for instance, S. Foner, Y. Shapira, D. steps will be observed. For IJNN I > JJcrit I , Heiman, P. Becla, R. Kershaw, K. Dwight and we already discussed the close relatwn be A. Wold, Phys. Rev. B 39, 11793 (1989) tween randomness and the height of the mag 6 W. Low and M. Weger, Phys. Rev. 118, 1119 netization step. Apart from that we should (1960); G.A. Slack, S. Roberts and J.T. Val also consider the width of the step as a limit lin, ibid. 187, 511 (1969); J.P. Mahoney, C.C. ing factor. The influence of the temperature Lin, W.H. Brumage and F. Dorman, J. Chem. can be reduced by performing the experi Phys. 53, 4286 (1970) ments well below 4.2 K. Additional broad 7 H.J.M. Swagten, A. Twardowski, W.J.M. de ening of the step could be indu~ed any ?Y Jonge and M. Demianiuk, Phys. Rev. B 39, extra matrix element in the Hamlltoman. As 2568 (1989) an example, it has been shown very recently 8 C. Testelin, A. Mauger, C. Rigaux, M. Guillot for Mn13, that a DM anisotropy, though very small would lift the degeneracy of the and A. Mycielski, Solid State Commun. 71, 923 (1989) groun'd state at Bstep• thereby wirlening the M. Hausenblas, L.M. Claessen, A. Wittlin, A. step. Further, also internal fields 11 and even Twardowski, M. von Ortenberg, W.J.M. de the experiment al conditions ( static versus Jonge and P. Wyder, Solid State Commun. 72, pulsed field)13 are believed to affect the struc 27 (1989). ture of the magnetization step. 10 C.J.M. Denissen, H. Nishihara, J.C. van Gool Despite the complicating factors listed and W.J.M. de Jonge, Phys. Rev. B 33, 7637 above we feel encouraged by our present (1986); A. Twardowski, H.J.M. Swagten, treatment of the Fe2•-Fe2+ pair, nume~ical W.J.M. de Jonge and M. Demianiuk, ibid. 36, that steps in the magnetization of Fe-<:on 7013 (1987) taining DMS should be detectable, t~ereby 11 B.E. Larson, K.C. Hass and R.L. Aggarwal, offering the unique possibility to estabhsh the Phys. Rev. B 33, 1789 (1986) exchange interaction between adjacent Fe2+ 12 We already argued that the expression which ions in a rather straightforward way. The determines JNN from the high-temperature experimental conditions the high-field mag- Chapter III, § 4 MAGNETIC BEHAVlOR ..... , BASEDON Fe 88 expansion of X includes numerical model-de pendent expressións which could oliscure this approximation; further, in compounds other than Znt-xFexSe, having slightly different crystal-field parameters, the value of lcrit itself will be somewhat different apart from the fact that also Jn is a priori different from that in Znt-xFexSe (Ref. 4). 13 Y. Shapira, S. Foner, D. Heiman, P.A. Wolff and C.R. Mdntyre, Solid State Commun. 70, 355 (1989) CHAPTER FOUR Carrier- concentratien - dependent magnetic behavier in IV-VI group diluted magnetic semiconductors In paragraph 1 of this chapter we will describe the preparatien of the IV-VI group compounds Pbt-x-ySnyMnxTe and Snt-xMnxTe, used for the investigations presented in paragraphs 2 and 3. The systems are synthesized using the Bridgman technique, yielding good-quality crytr tals, consisting of monocrystals with dimensions up to 1 cm3• The effects of nonzero segregation on the composition is analyzed. Secondly, we will pay attention to the manipulation of the carrier density in these ma terials, by applying isothermal annealing processes in a proper environ ment. The influence of the concentratien of charge carriers on the ferro magnetic phase transition of the diluted magnetic semiconductor Pbt-x-ySnyMnxTe for various compositions is reported in paragraph 2. A critica! density of carriers above which a ferromagnetic transition can take place, is observed. A simple modified RKKY mechanism for semi conductors is proposed in which carriers from two valenee bands located in different regions of the Brillouin zone contribute and a finite mean free path is implemented. An excellent quantitative agreement with the data is obtained. In paragraph 3 it is shown that for the diluted magnetic semi conductor Snt-xMnx Te also a critica! carrier density (Pc :::: 3 · 1020 cm -3) can be found above which ferromagnetic behavior is displayed. This behavior can be explained within the two-band RKKY model. Prelimi nary experiments on the nature of the low-temperature phase are pre sented. Chapter IV, § 1 CARRIER-CONCENTRATION-DEPENDENT ..... 90 Chapter IV, paragraph 1 Preparation and carrier concentrations of Pb1.x-ySnyMnxTe and Sn1_xMnxTe For the research on IV-VI group alloys1 and vapor growth, can be employed. Never the preparatien of high-quality crystalline theless, the vast majority of IV-VI com structures, if possible large monocrystals, is pounds can be conveniently synthesized by necessary. From the point of view of semi applying Bridgman single-crystal growth3. conducting effects, it is, for instance, often Very recently also thin films and layered desirabie to synthesize crystals with extreme structures containing IV-VI group semicon purity, in order to obtain mobilities as high ductors and/or DMS have been prepared by as possible. Also from the magnetic point of means of MBE4. view, it is desirabie to investigate large single As for the bulk systems used in the pres crystals, where additional effects from grain ent investigation ( see Chapter I and the fol bounda.ries, crystal imperfections or impur lowing para.gra.phs) they were all prepared by ities are, tentatively, reduced to a minimum. the conventional Bridgman technique. In this Moreover, for some magnetic techniques, process the melt is contained in a growth such as neutron scattering, it is often necess ampoule and is solidified by lowering the ary to have mono-crystals of considerable ampoule through the negative temperature size. For this purpose several crystal-growing gradient of the furnace. In our vertical fur techniques2, such as Czochralski, Kyropoulos, nace this gradient is supplied by the natural TABLE I. Materials grown with the Bridgman single-crystal growth apparatus; the purity of the constituents is given in between brackets (e.g., 6N fr: 99.9999%); m is the maas of the eonstituents; fis the metalfnonmetal fraction, whieh for Snt-xMnxTe has been ehosen according to existing phase diagrams on SnTe1; 1halcing denotes the time interval the melt has been shaken before Brigdman growth; V is growth rate. Material Constituents m (g) f shaking (h) V(mm/h) (Pb0•25Sn0.72Mn0•03)rTe Pb{6N),Sn(5N) 20 1 6 1.4 T~5N),MnTe 2(3N) (Pb0•12Sn0.45Mn0•03)rTe Pb 6N),Sn(5N) 20 1 6 1.4 Te(5N),MnTe2(3N) {Sn0•87Mn0•03)rTe Sn(5N),Te(5N) 20 0.984 5 1.4 MnTe2{3N) (Sno.a4Mno.oe)rTe Sn{5N),Te(5N) 40 0.984 5 1.4 MnTe2(3N) Chapter IV, § 1 91 drop in temperature from the middle section to the bottorn of the furnace; see Fig. 1. The shape of the crucible is such that single-crys u tal growth can start at a conical tip without the need of a crystal seed, and, if appropriate conditions are satisfied, see below, the mono crystal can extend steadily throughout the ~~ crucible. There exist two alternative Bridg !""" man techniques, one where the furnace is pulled over the ampoule and one where both • the ampoule and furnace are fiXed whereas 20 the temperature is slowly being decreased. For completeness, we would also like to men tion the Bridgman-Stockbarger variant5 where instead of a slowly varying tempera 40 ture a steplike gradient is used, provided by a • two-zone furnace. The Bridgman technique has several . disadvantages which are primarily induced • 60 by the contact between the melt and the cru • cible. First, impurities can be picked up from the wall, and, in the case of the quartz am • poules& we used, it is known that chemical - -· reactions with the melt can take place. To prevent this, the ampoules are, after careful cleaning and etching, carbon-coated by ...... ,.. pyrolysis with acetone. Secondly, the wall = can act as a undesirable nucleation site of the 750 800 850 solid state, and, finally, during the cooling T ("Cl process after the growth, the contraction of the ampoule wall can seriously damage the crystal or create dislocations and other crys tal imperfections. In our equipment, a cooling FIG. 1. Bridgman single-crystal growth equip rate less than 0.5 °C/min seems to be suffi ment. The constituents are sealed under 10-6 Torr ciently low to prevent the ampoule or crystal in a carbon-coated quartz ampoule. After heating from cracking. the crucible up to approximately 50 Oe above the The quality of the crystals is governed melting point, the melt is shaken for severa.l hours in by a delicate balance between the speed of order to stir it a.nd to prevent vapor inclusions. Sub the solidification interface and the steepness sequently, the a.ctual Bridgman growth starts by of the temperature gradient normal the inter moving the ampoule slowly downwa.rd through the face. It is well-known that a shallow gradient negative temperature gradient of the furnace (see can cause so-called constitutional super the right-hand side of the !~gure). Solidification cooling, this is a breakdown of the sharp takes place at an interface where Tfurnace = T melt• solid-liquid interface yielding uncorrelated exemplified for Sn 1-xMnx Te. The temperature is solidification tbraughout the melt. According controlled by using a computer based on a 68000 toa macroscopie transport model 2 where con microproceBBOr. This system also activatea two step vection and radial concentration gradients ping motors, one for sha.king the melt, and one, are neglected, the temperature gradient together with a retardation device, for lowering the d T / dx and the growth ra te V should, in or- ampoule at a.ny rate below 15 mm/h. Chapter IV, § 1 CARRIER-CONCENTRATION-DEPENDENT ..... 92 (a} der to avoid superrooiing in a compound A - _ , B11 be chosen such that the following condi tion is satisfied: dT/dx > mC0 1-k (1) V D k ' with m : absolute value of the liquidus slope C0 : starting melt concentration (y) k : segregation coefficient ( =C 5/ C1) between A and B at C1 = y D : diffusion coefficient of liquid A in liquid AyB 1.y . The left-hand side of Eg. 1 can be controlled by a proper choice of experimental condi tions1 wherea.s the right-hand part camprises intrinsic properties of the alloy. As an ex Sn ample for Pbu8Sn0•12Te, the parameters k and m can be inferred from the phase dia 1 gram of Pb1_ySnyTe depicted in Fig. 2, yield {b) ing k ~ 0.88 and m ~ 13ooc per mole fraction solute. By inserting D = 7·10-5 cm2/s (re 7 ported for SnTe in Pb0•8Sn0•2Te ) and dT/dx = 5 OCfcm for the furnace in Fig. 1, we find for that V should not exceed 2.5 mm/h. Depending on the specific system and available equipment, Bridgman growth usual ly takes place at rates between 0.1 and 10 mmfh. In our growing apparatus these veloe ities are realized by a stepping motor to gether with a retardation device, resulting in discrete steps always smaller than 1 J.tm. Any arbitrary velocity below roughly 15 mm/h 1 Mole troelion of Sn Te in Pb1.,sn.Te x .x• can be chosen. Before Bridgman growth starts, a second stepping motor shakes the melt (f = 50 Hz, ampl. = 1.6 mm) for homo genization and for the removal of vapor in cluded in the melt ( mostly near the ampoule wall). Unfortunately, for the Mn-eontaining FIG. 2. (a) Temperature - composition phase Pb .x-ySnyMnxTe systems, no information is diagram for the Pb-Sn-Te ternary system. (b) 1 yet available on the solubility properties of Ternperature - composition phase diagram for Mn. We are therefore not able to estimate Pbl-xSnxTe; after Ref. 1. the growth rate necessa.ry to exclude super cooling and we were forced to ada.pt growth 8 rates reported for Pb1.x-ySnyMnxTe and Sn1_xMnxTe, which usually resulted in good- Cha.pter IV, § 1 93 quality large monocrystals. The growth para tion no fluctuations in composition have been meters used to synthesize our systems, as found. Along the longitudinal direction well as other preparatory details, are though, we found a smooth variation of the gathered in Table I. The crystal structure of composition as depicted in Fig. 3 for Pb0.52- these systems have been investigated by Sn0.45Mn0.03Te, which we will consider in x-ray techniques. No noticeable deviations some more detail. from the reported rocksalt structure ( see The variation in Sn ( and Pb) content is Chapter I) could be detected, and, in some a common feature for these compounds1 and cases, it was checked that the ingot consists can be ascribed to the nonzero segregation of monocrystals with dimensions up to 1 cm3• between solidus and liquidus line, which we The composition of the crystals (Pb,Sn,Mn saw in Fig. 2. By assuming a constant segre and Te) is determined by electron probe gation factor k, a large diffusion compared to micro analysis (EPMA). In the radial direc- the growth rate, no diffusion in the solid part of the crystal, and no convection in the liquid, the concentration profile for a conical ampoule (see Fig. 3) follows2 C8{x) = kC0(1-g)k-t , (2) .,o...., r•,L~+L2 ~ with g the fraction of material solidified: L1 _14_ 0 Sn-concent rat ion I y) The Sn profile in Pb0.52Snu5Mn0•03Te shown in Fig. 3 has been fitted to this expression, • yielding for the best overall agreement with the data k = 0.83, which is surprisingly close to the value deduced from the PbTe- SnTe phase diagram in Fig. 2 (approximately 0.78 for Pb0 •55Snu5Te). This implies that appar ently the growth conditions were satisfactor 3.5 Mn-concentration I x) ily and supercooling did not occur, but also that the role of the substitutional Mn ions on ~ e... 3.0 the phase relations in the Pb1_1SnyTe sys ,. tems can be neglected. From the composi 19 tional profile of the Mn [Fig. 3 ( c)] we have to 2.5 0 conclude that segregation of MnTe in Pb1_y SnyTe, similar to SnTe in PbTe, cannot be 0 0.2 0.4 0.6 0.8 1.0 neglected too. The segregation coefficient for x I !L1+l2l the MnTe solubility, obtained from fitting the data in Fig. 3( c) with Eq. 2, is practically equal to that for SnTe in PbTe, being 0.86. FIG. 3. Compositiona.l profiles for Pbo.52Sno.45- From the concentration profiles in Fig. 3 we Mn0.03Te obtained by EPMA analysis. The data may deduce that for a sample taken from the have been fitted to Eq. 2 (the solid lines), yielding part of the ingot where x/(Ll+L2) < 0.8, the for the best overall agreement with the data, Ie = Sn and Mn variation over a typical direc 0.83 for the Sn and k 0.86 for the Mn concentra tional sample size (~ 2 mm) will never ex tion. ceed 4%. As a condusion from this, we may Cha.pter IV, § 1 CARRlER-cONCENTRATION-DEPENDENT ..... 94 state that though the most profitable situ ation occurs when k is very close to unit imd 0.6 ~::.....c.::=-- ---==------...... -=----i concentrations are constant, the segregation K-1 of the substituents in our systems is small 10 enough to prevent undesirable compositional profiles over the lengthof the crystals. More over, from the magnetic point of view, it is not only advantageous to know exactly what Mn concentration is in the system but the concentration should also be as constant as possible throughout the sample dimensions. We have checked that in both Pb1_ySn1- Te and SnTe, Mn concentrations of at least FIG. 4. Apparent carrier concentrations at 77 K 10% can be substituted in the semiconducting (300 K for SnTe) versus isothermal annealing tem host, without clear signs of a rednetion of the pera.ture for Pbt-xSnxTe from various authors; after crystal quality. Nonsupercooling conditions Ref. 1. are apparently satisfied and no drastic changes in the segregation coefficient (be tween MnTe and Pb1_1Sn1Te) may therefore ionic nature of the chemical bond in these be expected. Nevertheless, due to the lack of compounds. This also explains why carrier any detailed phase diagrams of the SnTe - concentrations in the more covalent semi MnTe or PbTe - Sn Te - MnTe system, the conductors (e.g., Cd Te, ZnSe; see Fig. 14 in crystal growing processes in IV-VI group Chapter I) are considerably smaller. DMS are still far from controllable and pre One of the consequences of the high vent the crystal grower from manipulation of number of vacancies is the fact that PbTe the growth parameters by pure physical ar and SnTe will never exist in pure roetal and guments. nonroetal stoichiometry. In PbTe1 both n The carrier concentration in IV-VI and p-type conductivity is observed and this group semiconductors1,9 is, in generàl, rela implies that the solidus area extends to both tively high (1018- 1022 cm-3). By appropriate the nonroetal and metal-rich side, from annealing procedures it is possible to control Pbo.g 988Te to Pbt.o0008Te, respectively. In the density of carriers, which, as we will see, SnTe on the other handt, the solidus area is is rather crucial in our studies of magnetic completely shifted to the Te-rich side, due to properties~ For PbTe, SnTe and Pb1_ySn Te the relatively small formation enthalpy for it is well established 1 that these high den Sn vacancies in comparison with Te va sities originate from a natural abundance of ncancies9, and, consequently, only p-type metal- or nonroetal vacancies, which create conductivity is observed with carrier concen 20 holes and electrons, respectively. trations roughly between 10 (Sn0.9996Te) · The density of defects is governed by the and 1021 cm-3 (SnussTe). These phase formation enthalpy (Hv) of the vacancies, boundaries are drawn in Fig. 4, together with which, in a very simplified model, is deter supplementary data for the ternary Pb1_ySny mined byl: Te, showing that for increasing Sn concentra tion the carrier densities gradually transform (3) from PbTe to SnTe. By isothermal annealing of Pb1_ySnyTe where Nis the density of defects and N0 is one is able to control the number of va the concentration of lattice cells. In IV-VI cancies. At a constant temperature the crys compounds these enthalpies are small with tal is kept under roetal or nonroetal vapor for respect to other semiconductors due to the several days ( depending on temperature and Chapter IV, § 1 95 sample size) and afterwards, when an equilib Phyllic11 of IV- VI compounàll and alloy&, edited rium has been reached, a rapid quench to by S. Ra.bii (Gordon and Breach, London, room-temperature will freeze in the achieved 1974); D.R. Lovett, SemimetaLs and narrow defect structure. For instance, annealing of bandgap 11emiconductorll1 (Pion, London, 1977); as-grown Sn Te (apparent carrier concentra G. Nimtz, B. Schlicht and R. Domhaus, Nar 3 tion PaooK :::! 8·1020 cm- ) at T = 650 oe un row-gap llemiconductorll, (Springer, Berlin, der a Sn-rich atmosphere will yield p 300K :::! 1983); Narrow-gap &emiconductorll and related 2·1020 cm-3, as can he deduced from Fig. 4. material.s, edited by D.G. Seiler and C.L. Litt The annealing procedures we applied for sev ler (Hilger, Bristal, 1990). eral magnetic samples listed in Table I are 2 See for instance, J.C. Brice, Selected topiel in basically performed according to the recipes 11olid state phyllic11: The growth of cry&tal.s from given by BrebrickiO and Hewes et al. u for liquids, (North Holland, Amsterdam, 1973), nonmagnetic systems. The presence of Mn2+ Vol. XII; A.W. Vere, O'I"!J&tal growth, priciple11 had no visible effect on the phase boundary and progreu, (Plenum PrellS, New York, 1987). depicted in Fig. 4, though in some cases a 3 P.W. Bridgman, Proc. Amer. Acad. Arts Sci. small rednetion of the number of Mn2+ ions 60, 305 (1925). was observed after annealing. 4 See for instance, G. Bauer and H. Clemens in In order to obtain very small carrier Narrow-gap semiconductor/I and related ma concentrations in SnTe, even smaller than terials, edited by D.G. Seiler and C.L. Littler PsooK :::! 2·1020 cm-3 (Fig. 4), we had to apply (Hilger, Bristal, 1990), p. Sl22. an alternative procedure developed by Inoue 5 D.C. Stockbarger, Rev. Sci. lnstr. 7, 133 et al. 12. They were able to rednee the carrier (1936). density to below 1·1020 cm -a by isothermal 6 The melting point of the IV-VI compounds is annealing under Zn vapor, at T = 650 oe. sufficiently low to permit the use of quartz Tentatively, this is due to the more covalent a.mpoules; at approximately 1500 Oe quartz Zn-Te bond with respect to Sn-Te, and becomes soft and can no longer serve as cru consequently, a great number of additional eibie for Bridgman growth Sn vacancies can be occupied by Zn atoms. In 7 A.L. Fripp, R.K. Crouch, W.J. Debnam Jr, contrast to the narrow solidus area in SnTe, l.O. Clark and J.B. Wagner, J. Cryst. Growth the Pb1_ySnyTe solid phase boundaries 73, 304 (1985). (Fig. 4) are such that a very wide range of 8 See references in Chapter I of this thesis, and densities can be achieved without the use of references in G. Bauer, in Diluted Magnetic doping with other elements. Nevertheless, in Semiconductors, edited by R.L. Aggarwal, J.K. future research on IV-VI group DMS it Furdyna and S. von Molnar (Materials Re might be necessary to extend the current search Society, Pittsburgh, 1987), Vol. 89, p. carrier-density regime, for instance, to 107. concentrations exceeding 10 21 cm- 3. In that 9 F.F. Sizov and S.V. Plyatsko, J. Cryst. Growth case, more effort should be employed on the 92, 571 (1988); O.A. Pankratov and P.P. effects of doping, since besides the Povarov, Solid State Commun. 66, 847 (1988); manipulation of the vacancy density by H.M. Polatoglou, Physica Scripta 39, 251 (isovalent) doping with Zn, and also ed1•9, it (1989}. is possible to add nondivalent foreign ions, 10 R.F. Brebrick, J. Phys. Chem. Solids 24, 27 e.g. Na.13, to the system. The donor or (1963). acceptor levels these dopants might create 11 C.R. Hewes, M.S. Adler and D. Senturia, J. within the band structure can supply the Appl. Phys. 44, 1327 (1973). additional charge carriers. 12 M. Inoue, M. Tana.be, H. Yagi and T. Tat sakuwa, J. Phys. Soc. Jpn. 47, 1879 (1979). 13 M. Ocio, Phys. Rev. B 10, 4274 (1974). Chapter IV 1 § 2 CARRIER-CONCENTRATION-DEPENDENT ..... 96 Chapter IV 1 paragraph 2 Hole density and composition dependenee of ferromagnetic ordering in Pb1_x-ySnyMnx Te [ published in Phys. Rev. B 371 9907 (1988) ] I. INTRODUCTION cannot be understood by the RKKY model, originally modeled for free (parabolic) elec Diluted magnetic or semimagnetic semi trons. conductors - i.e., semiconducting alloys In this paper, we will present new ex with substituted magnetic ions - exhibit a perimental data on the phase diagram of variety of interesting transport, magneto Pbt-x-ySnyMnx Te (PSMT). A model will be optical, and magnetic properties1. Recently presented which explains quantitatively the Story, Gal~t-zka, Franke! and W olff2 reported typical carrier concentration dependenee of the observation of a carrier-concentration the Curie-Weiss temperature 0 and Tc· In induced ferromagnetic transition in Pb0.2r particular, we will show that the steplike Sn0.72Mn0.03Te above a critica! carrier con increase of e at a critica! hole density can be centration Pcrit· This can be considered as a understood on the basis of the modified demonstration of the strong effect of carrier RKKY interaction mechanism extended with concentration on the magnetic properties of a more realistic two-valence-band approxi diluted magnetic semiconductors, resulting in mation and a finite mean free path for the a magnetic phase diagram which includes the carriers. With this modification, a new exten carrier concentration as a parameter. sion for the applicability of the RKKY inter Application of pressure on the system in action in the diluted magnetic semiconduc the ferromagnetic regime yielded a shift in tors is suggested. the transition temperature (Tc) which, ac Pbt-x- Sn Mnx Te is a quaternary alloy cording to Suski, Igalson and Story3, could be with a rocksJt fee lattice in which the Mn qualitatively understood on the basis of the ions are randomly distributed. The samples Ruderman-Kittel-Kasuya-Yosida (RKKY) were grown by a Bridgman method. The interaction mechanism4. It was pointed out, composition and homogeneity were character however, that a redistribution of carriers due ized by x-ray diffraction and mieroprobe to the pressure dependenee of the specific measurements. All investigated samples were band structure should betaken into account. single phased (0.005 terial, obeys a Curie--Weiss law at tempera a tmes above roughly 10K, as shown in Fig. 1(b). The Curie--Weiss temperature 0, >2 ...x which is proportional to the average exchange 0 n § interaction between the Mn ions, matches Tc ..... "'3 within a. few tenths of a degree and is propor e c u ;;:; tional to x as expected for a random system with long-range interactions [see inset in Fig. 1(b )]. Figure 2 shows the carrier concen tration dependenee of e for several composi tions, supplemented with the earlier results of Story et aJ.2. Above Pcrit an abrupt, al most steplike increase of 0 from approxi mately zero to large positive valnes is ob served, indicating a net ferromagnetic inter action. The data in Fig. 2 suggest a rather universa! dependenee on the carrier concen tration, only slightly dependent on the com position. However, we like to note that in Pb0.50Sn0.47Mn0.03 Te a rednetion of Pcrit (~ 1-1020 cm-3), with respect to the systems with y = 0.72 is suggested, although a larger 0 10 20 30 set of samples is necessary to obtain more T [ K l pertinent conclusions. To explain the abrupt transition to a FIG 1. (a) Low-temperature magnetic specific ferromagnetic phase as the hole density ex heat ( 0 111) and ac susceptibility (Xac) of Pbo.28-x ceeds a critica! density, we will introduce a Sno.72MnxTe, z = 0.027 and p 7·1020 cm-3, simple modeL We consider a system of local showing the transition to a ferromagnetic state. {b) Inverse ac susceptibility (Xac -l) for z ~ 0.03 and severa.l carrier concentrations (p); the Curie-Weiss tempera.ture (0) as a fundion of the Mn concentra. tion (z), in the ferroma.gnetic regime, is shown in the in set. • Pb 1_x-ySnyMnxTe x y and the values for the carrier concentration quoted here were measured at T = 79 K. ac susceptibility and specific-heat measurements have been performed in the temperature range 0.4 - 50 K. Above the 0 4 16 critical carrier concentration Pcrit ~ 3x 1020 cm·3 a magnetic phase transition was observed. An mustration of the low-tempera ture behavior is shown in Fig. l(a). Magnet FIG. 2. Curie-Weiss temperature, scaled on ization and de susceptibility confirmed the z = 0.03 [0(0.03/:z:)] of Pbt-x-ySnyMnxTe as a ferromagnetic nature of the transition. The fundion of the carrier concentration (p). The dark high-temperature part of Xac• corrected for circles are data obta.ined from Story et al. (Ref. 2). the diamagnetic contribution of the host ma- The solid line is a guide to the eye. Chapter IV, § 2 CARRIER....CONCENTRATION-DEPENDENT ..... 98 ized magnetic moments, the Mn ions in the structure (see Ref. 1). We take into account Pb 1 _ySn~Te matrix, interacting with the free only the highest two valenee b8.Ilds aecording holes. The following assumptions will be to different band structure calculations&-9 for made: (i) intraband RKKY interaction be Sn Te and Pb1_ySnyTe with y > 0.4. tween the magnetic moments; (ii) mean-field Although the assumptions listed above approximation for the spin system; {iii) a are rather crude, we believe that this simple finite mean free path for the free holes; and model contains the relevant characteristies of (iv) representation of the complicated band the magnetie subsystem as well as the hole structure of PSMT by a set of two parabalie subsystem. Within this model, the Curie valenee bands with different effective masses, Weiss temperature is given by as shown in Fig. 3. The first assumption in eindes the use of second-order perturbation 2 theory, although for PSMT the exchange e = 2S(S+l)x L L 11err(Rj) , (la) constant J might well be of the same order of 3kB i=l j~l magnitude as the Fermi energy (Er) of the free holes ( see, however, the discussion of where J1eff(Rj) is the exchange integral Mauger and Godart in Ref. 5). The mean field approximation completely neglects the (lb) dynamics of the spin system. The RKKY interaction was originally modeled for metals, i.e., for free electrans with parabalie bands. The applieation to semiconductors requires at least two essential modifieations, with which are expressed in the third and fourth assumption. First, the mean free pathof free F(zjl) = [sin(zj')-zjlcos(zjl)J , (le) carriers eannot be taken to be infinite. Sec ondly, the detailed band strueture of a semi and with z/ = 2kr1Rj, where kri is the Fermi conductor might play a decisive role. We wave number for the ith band. Rj is the dis used the band structure of Pb1_ySn1Te as a tanee between two lattiee points in the fee reference model for the band structure of lattice (R0 = 0 and R1 is the nearest-neigh PSMT. It is assumed that the presence of Mn bor distance ). S is the magnitude of the spin ions does not significantly alter the band (S = 5/2 for Mn2+), J1 the exchange constant for VBi, and a.0 is the lattice parameter. The effect of a finite mean free path ( >.) is incor porated along the lines proposed by de Gennesto, by inclusion of an exponential term in Jieff(Rj). ~I Next, some results of our model calcula Eg' tion for Pb0•25Sn0•72Mn0•03Te will be pre l!.VB sented. We would like to stress that the para meters are not obtained from a fit to the ex ~ perimental data. The lattiee parameter a0 = L 6.347 A was estimated from the Pb1_x:MnxTe 11 and Sn1.xMnxTe data •12 by linear Vegard type interpolation. For the exchange con stants, it was assumed that J 1 = J 2 = 0.3 eV, FIG. 3. Schematic illustration of the band struc- from a linear interpolation between the ex 13 ture of Pbt-x:Snx:Te used in the model calculations perimental values for Pb1.x:MnxTe as in text. described the (0.07 eV) and for Sn1_x:Mnx:Te {0.40 eV). Chapter IV, § 2 Hole density ..... in Pbt-x-ySnyMnx Te 99 However, it is a priori not excluded that the contributions from both valenee bands. Also, exchange integtals differ for both valenee the theoretica! curves corresponding to the bands14. For the effective masses we take the case that only VB1 or VB2 is filled are in 15 values from Ocio for Pb0.53Sn0.nTe: m1* = serted, for an infinite mean free path. The 0.05m6 and m 2* = l.OOme· Figure 4(a) shows difference between these contributions is the Curie-Weiss temperature as a function of mainly caused by the considerable difference hole density, as calculated from Eq. (1) with between the effective masses m1* and m2*, see Eq. 1(b). In Fig. 4(b) the model calcula tions are compared with the experimental 8 data. The different curves correspond to dif (a) ferent values of the mean free path -\ of the holes. It is clear from this plot that the effect 6 of>. can he large. We used realistic valnes for l<:: -\, i.e., 7 and 13 Á, corresponding to the ex perimental observed mobilities in the range )( 4 15-50 cm2/Vs. The critica! hole concentra -", tion in Figs. 4(a) and 4(b), Pcrit ~ 2.4x ~ 1020 cm-a, has been determined from a two (!) 5 2 band Kane model1 , with as additional para meters16 E = 0.15 eV and ~EVB = 0.6 eV (see Fig. 3f. Experimental values for the critical carrier concentration Pcrit15 ·11, de duced from the temperature dependenee of the Hall factor, vary roughly between 2 and 3·1020 cm-3, which is consistent with the Kane-approximation. The excellent agreement between the calculations and the data shown in Fig. 4(b) is obvious. The observed steplike increase of 9 to positive (ferromagnetic) valnes at a critical carrier density as well as the magni tude of 9 above and below that critical con 0 centration are correctly predicted. At this point, however, it should be noted that the 0 4 8 12 16 qua.ntita.tive agreement may be somewhat p [ 10 20 cm-3 1 fortuitons since it depends also on the value for J 1 and J 2, the p-d exchange constants, which in a first approximation we have as FIG. 4. (a) Curie-Weiss temperature scaled on sumed to be equal to 0.3 eV by linear inter :c = 0.03 [9(0.03/z)] as a function of the carrier polation. A further verification of the validity concentration (p), as calculated from Eq. (1). The of the model can possibly be found in an ex full lines cortespond to contribution from both va tension of the composition range (Pb-Sn) lence bands, for À = m and À = 7 Ä, whereas the which would yield a slight shift in Pcrit dashed-dotted a.nd dashed line repreaent the contri brought about by the variation in~~. The bution of VBl and VB2, respectively, for À = m. (b) observed quasi-universal behavior of El(p) in 9(0.03/z) as a. function of p, for >. = 7 Ä and À = the restricted composition range studied so 13 Ä, in combination with the experimental data for far, is in accordance with the existing data on Pbt-x-ySnyMnxTe; see Fig. 2 for the description of ~EvB in that range15,17. the symbols. It is worthwhile to note that on the basis Chapter IV, § 2 CARRIER-CONCENTRATION-DEPENDENT ..... 100 of the present model the contrasting magnet 3 T. Suski, J. Igalson and T. Story, in Proceed ie behavior of Snt-xMnx Te and Pbt-xMnx Te ingl of tAe 18tA International Conference on can be explained by the difference in carrier the Ph1111icl of Semiconductor1, Stockholm, concentration. Ferromagnetism was found in 1fJ86, (World Scientific, Singapore, 1987), p. 18 2 3 Sn1.xMnxTe (with p ~ 5·10 0 cm- ) while 1747; J. Magn. Magn. Mat. 66, 325 (1987). 2 3 4 Pb1_xMnxTe (p < 10 0 cm- ) is paramagnatie M. Rudermann and C. Kittel, Phys. Rev. 96, down to very low temperatures where recent 99 (1954); T. Kasuya, Progr. Theor. Phys. 16, ly a spin-glass transition was observed19. The 45 (1956); K. Yoshida, Phys. Rev. 106, 893 reported antiferromagnetic (AF} sign of the (1957). interactions in Pb1_xMnx Te is probably due A. Mauger and C. Godart, Phys. Reports 141, to the increasing relevanee of other than 51 (1986). intraband mechanisms which might be AF 6 Y. Ota and S. Rabii, in The Ph11lic• of Semi for low carrier concentrations. In fact, one metall and NarrotJJ Gap Semiconductor1, edited might conjecture that also in PSMT for low by D.L. Carterand R.T. Bate (Pergamon, New concentrations a spin-glass transition at very York, 1971) p. 343. low temperatures would occur. 7 J.O. Dimmock, in The Phy11ic1 of Semimetall Finally, we would like ·to add that one of and NarrotJJ Gap Semiconducf.or1, edited by the essential results of the present study, i.e., D.L. Carter and R. T. Bate (Pergamon, New the first indication of separate contributions York, 1971) p. 319. from carriers in different regions of the Bril 8 G. Martinez, M. Schluter and M.L. Cohen, louin zone might also be relevant for calcula Phys. Rev. B 11, 651 (1976). tions of interband interaction mechanisms 9 Tung and M.L. Cohen, Phys. Rev. 180, 823 which have been proposed for diluted mag (1969). netic semiconductors. 10 P.J. de Gennes, J. Phys. Radium 23, 630 (1962). Note: ihe calcula.tions we presented include 11 V.G. Vanyarkho, V.P. Zlomanov and A.V. the assumption of an isotropie RKKY inter Novoselova, Izv. Akad. Nauk. SSSR Neorg. action between the Mn ions, yielding a cor Mater. 6, 1534 (1970). rect prediction ofS(p); however, somerecent U. Sondermann, J. Magn. Ma.gn. Mat. 2, 216 results point to the relevanee of the E- band (1976). symmetry, which, possibly, ca.n alter the cal 13 Landolt-Bornstein: Phy11ic1 of II-VI and cula.tions performed in this pa.ra.graph of the I-VII comp011.nd1, 1emimagnetic •emiconduc• thesis. torl, edited by 0. Madelung (Springer, Berlin, 1982), Vol. 17, Pt. b. 14 J. Ginter, J.A. Gaj and Le Si Dang, Solid State Commun. 48, 849 (1983). N.B. Brandt and V.V. Moshchalkov, Adv. 15 M. Ocio, Phys. Rev. B 10, 4274 (1974). Phys. 33, 193 (1984); J.K. Furdyna and N. 16 G. Nimtz, B. Schlicht and R. Dornhaus, Nar Samarth, J. of Appl. Phys. 61, 3526 (1987); row Gap Semicondv.ctor1, (Springer, Berlin, R.R. Gala.zka, in Proceeding1 of the 3rd Con 1983). ference on the Phy•ic1 of Magnetic Material1, 17 A. Aladgadgyan ànd A. Toneva, Phys. Statua Szczyrk-Bila, Poland, 1986, edited by W. Solidi (b) 85, K131 (1978). Gorzkowski, H.K. Lachowicz and H. Szymczak 18 A. Mauger and M. Escorne, Phys. Rev. B 35, (World Scientific, Singapore, 1987), p. 26. 1902 (1987). 2 T. Story, R.R. Gala.zka, R.B. Frankeland P.A. UI M. Escorne, A. Mauger, J.L. Tholence and R. Wolff, Phys. Rev. Lett. 56, 777 (1986). Triboulet, Phys. Rev. B 29, 6306 (1984). Chapter IV, § 3 Carrier-<:oncentra.tion dependenee ..... in Snt-xMnxTe 101 Chapter IV, paragraph 3 Carrier- concentratien dependenee of the magnetic interactions in Sn 1_xMnx Te ( published in Semicond. Sci. Technol. 5, S131 (1990)) I. INTRODUCTION carrier density. Sn1_xMnxTe is, in that re spect, an extreme case since it commonly The IV-VI group compounds Pbt-x-y possesses such high densities of carriers that SnyMnx Te reptesent up till now a rather the second set of valenee bands is populated, umque class of materials since the density of yielding strong ferromagnetic interactions. carriers (p) is included in the magnet ie phase Rednetion of the carrier density below Pc diagram, demonstrating a strong coupling would transfarm the system into a paramag between the magnetic and electronk struc net, if our model applies. ture. As was shown by Story, GalQ.Zka, So far, no st rong effect of the carrier 1 Frankel and W olff , strong ferromagnetic density in Sn 1_xMn:x:Te has been reported and interactions between the Mn2+ ions are de we therefore thought it worthwhile to study veloped when the density of carriers exceeds the influence of p on the low-temperature a critica! value Pc(~ 3·1020 cm-3). magnet ie properties of Sn1_xMnx Te in order Recently we have shown2 that this in to corroborate our conjecture and observe the triguing behavior can be explained by taking expected breakdown of the ferromagnetic into account the contributions to the RKKY interactions. Furthermore, we will focus our interaction mediated by carriers located in attention on the low-temperature phase for different regions of the Brillouin zone. The p > Pc, in order to characterize this phase, dramatic increase of the interactions at Pc is which in earlier reports has been denoted as then brought about by the entrance of the ferromagnetic, spin-glass, as well as reen Fermi level into a set of heavz-hole valenee trant spin-glass3-6. bands, located along the 2.. axis, which strongly enhances the long-range RKKY in ll.RESULTS teraction. On the basis of this model one could A. Influence of carrier concentration speenlate about the contrasting magnetic behavior of Pb1_xMnx Te and Snt-xMnx Te. The experiments were performed on Although both materials have a similar band Sn0•97Mn0•03Te and Sn0•94Mn0.06Te samples structure and have been studied rather thor grown by a Bridgman method. The as-grown oughly3, their entirely different low-tempera carrier concentration varies between 5 and 10 ture magnetic properties ( antiferromagnetic times 1020 cm-3, due to slight deviations from for Pb1_xMnxTe and ferromagnetic for Sn1_x stoichiometry. Lower carrier concentrations Mnx Te) seem somewhat puzzling. From the can be obtained by isothermal annealing in basic result of the modilied two-band RKKY Zn atmosphereB. model one might conjecture that this behav Figure 1 shows the ac susceptibility, the ior is caused by the considerable difference in magnetization, and specific heat of a sample Chapter IV, § 3 CARRlER-CONCENTRATION-DEPENDENT •.••. 102 0.10 0.15 .. p 110 20 cm- 3 1 (al . 0.08 ,.,· . . 0 9.4 .. .. . 7.0 "'; .. 0 4.5 "' 0.06 .. 2. 7 . 0.10 ffi" .. : 0.04 I- ... 0 "'i5 0.02 . . :t ... >< 0.05 Q ~ 0.6 0.5 (b) ...... - 0.4 . ';" 0 2 3 4 5 6 7 "' T !KI "ffi 0.3 -"t: 0.2 FIG. 2. ac ausceptibility of Sn . Mno.osTe for 0.1 0 97 , aeveral carrier concentrations; driving ac field: 0 ... . 0.016 G, 920 Hz. 0.4 0.3 fel "; "' .0 E 0. 2 ::::: ...,. o. 1 6 0 2 3 4 5 6 7 T (KI ~4 I "'0 ó i FIG. 1. Sn . Mn • Te, p 7-10·1020 cm·3 0 91 0 03 = x 2 j (a) ac sllliCeptibility, (b) magnetization for 00 i Hnc = 5 G, and, ( c) magnetic specific heat. o'--? with carrier concentrà.tions in the range 0 2 6 10 7-10·1020 cm·3, the range to which earlier investigations were restricted. In this range, a transition to a "ferromagnetic"-like ordered state is observed. When the carrier concen FIG. 3. Curie-WeiBS temperature (scaled on tration is reduced (notably below 5·1020 :c = 0.03) as a function of the carrier concentration; cm·3), the transition shifts considerably to the open squares are the data for Sno.97Mno.osTe, lower temperatures (see Fig. 2) and changes whereas the dashed-t10lid line only represents a its shape till, at the lowest density p ~ 3x guide to the eye. A prelimina.ry data point for z = 1020 cm·3, a second transition in the suscepti 0.06 has also been inaerted (full square). Calcula bility seems to appear at T = 100 mK. These tiona within the modified RKKY model are given by data clearly demonstrate the carrier-eoncen- the solid line. Chapter IV, § 3 Carrier-(;oncentration dependenee ..... in Sn1_xMnxTe 103 tration dependenee of the phase transition. In order to obtain more quantitative 1.6 20 G results on the interaction strength, the high temperature susceptibility was measured. A clear Curie-Weiss behavior was observed 1. 2 above 10K. The Curie-Weiss constante, ~ which monitors the total interaction "' 10 G strength, is plotted in Fig. 3 and a dramatic, ~ 0.8 almast steplike decrease of e from positive l: (ferromagnetic) values to zero is observed 5 G 0.4 near Pc ~ 3 · 1020 cm -a. The same figure con tains the results of a calculation of e (or EJij) based on the tw to a (reentrant) spin-glass. However, if do ma.ins are taken into account, a ferromagnet 0.08 ic transition cannot be ruled out on the basis of the available evidence. Moreover, the Mn - concentrations used in the present study \,. 0.06 (x = 0.03 and 0.06) imply average Mn-Mn distances rather close to the first ferromag ;: 0.04 netic-antiferromagnetic oscillation in the >< RKKY-interaction (for carrier concentrations around 1021 cm-3), which might obscure the O.Q2 general tendencies. Further, exploration of the effect of both Mn concentration and car rier concentration on the ( T,x,p) magnetic 0 2 3 4 5 TI Kl phase diagram of Sn1_xMnx Te seems therefore necessary. T. Story, R.R. Galazka, R.B. Frankeland P.A. !! Wolff, Phys. Rev. Lett. 56, 777 (1986). "ë 2 :::1 H.J.M. Swagten, W.J.M. de Jonge, R.R. 1:' Galazka, P. Warmenbol and J.T. Devreese, .t Phys. Rev. B 37, 9907 (1988). ~ " 3 For some recent reports see, A. Mauger and M. ;: >< Escorne, Phys. Rev. B 35, 1902 (1987); M. Escorne, A. Mauger, J.L. Tholence and R. Triboulet, ibid. 29, 6306 (1984). 4 U. Sondermann, J. Magn. Magn. Mater. 2, 216 0 0.5 1.0 1.5 TI Tc (1976). 5 M.P. Mathur, D.W. Deis, C.K. Jones, A. Pat terson, W.J. Carr and R.C. Miller, J. Appl. FIG. 5. ac susceptibility in the presence of ex Phys. 41, 1005 (1970). 2 6 temal de fields; (a) Sn0•97Mn0•03Te, p =9.4·10 0 M. Inoue, M. Tanabe, H. Yagi and T. Tat cm -3; (b) model calculations with the SK model, sukawa J. Phys. Soc. Jpn. 47, 1879 (1979). using 1J = 0.95; de fields are in units 10-3Hnc/Tc· 7 G. Nimtz, B. Schlicht and R. Dornhaus, Nar row gap Semicondv.ctor& (Springer, Berlin, 1983). is hard to decide about the specific mechan 8 I. Maartense and G. Williarns J. Phys. F 6, ism. The behavior of the ZFC and FC mag 1121 (1976). netization in low fields and the field depend 9 D. Sherrington and S. Kirkpatrick, Phys. Rev. enee of the dynamic susceptibility could be Lett. 35, 1792 (1975). considered as characteristic features related SAMENVATTING 105 SAMENVATTING Dit proefschrift beschrijft een experimen sen 0.01 en 0.25, zijn experimenteel bepaald. teel onderzoek naar het magnetische gedrag Ook in deze sytemen bestaat er een AF inter van verdunde magnetische halfgeleiders. Dit actie, tezamen met indicaties voor spinglas zijn nieuwe verbindingen van conventionele overgangen bij lage temperaturen, en kunnen halfgeleiders zoals CdTe of ZnSe, waarin een de thermodynamische grootheden simultaan bekend gedeelte van de cation-roosterplaat beschreven worden met de ENNP A. Met be sen (x) is vervangen door random gesubsti trekking tot het mechanisme van de interac tueerde magnetische ionen, zoals het divalen tie zijn er geen aanwijzingen gevonden voor te Mn2+. In Hoofdstuk I wordt een overzicht de invloed van Fe toestanden die gelegen zijn gegeven van de fysische eigenschappen van boven de valentieband en die voortkomen uit verdunde magnetische halfgeleiders, waarbij de specifieke 3d6 configuratie van Fe2+. Door de nadruk ligt op het magnetische gedrag in de invloed van kristalveld en spin-baan kop relatie met de overige delen van het proef peling is de sterkte en dracht van de Fe2+ schrift. Fe2• interactie moeizaam te achterhalen. Via In Hoofdstuk II wordt de lage-tempera uitdrukkingen voor de Curie-Weiss tempera tuur soortelijke warmte en susceptibiliteit tuur voor deze systemen kan een schatting van de U-VI groep verdunde magnetische gemaakt worden voor de interactie tussen halfgeleiders Zn 1.xMnxSe en Zn 1.xMnxS ge naburige Fe ionen: Jn = -22 K voor Zn 1.x rapporteerd, voor Mn concentraties tussen FexSe, -18.8 K voor Cd 1.xFexSe, en -18 K 0.01 en 0.53. Voor alle systemen wordt een voor Hg 1.xFCxSe. Berekeningen van de hoge spinglas overgang waargenomen (bij de zoge velden magnetisatie van Fe-houdende ver naamde freezing temperatuur Tr) die op dunde magnetische halfgeleiders tonen aan grond van schalingsanalyse compatibel is met dat JNN wellicht direct geëxtraheerd kan wor een antiferromagnetische (AF) lange-dracht den uit de positie van stapvormige veran interactie tussen Mn2• ionen van het type deringen bij lage temperaturen. J(R) "'R·n, met n = 6.8 voor Znt-xMnxSe en De invloed van de concentratie van la 7.6 voor Zn 1.xMnxS. Hierop gebaseerd zijn de dingsdragers op de ferromagnetische fase van thermodynamische grootheden beschreven de verdunde magnetische halfgeleiders met de extended nearest-neighbor pair ap Pbt-x-ySnyMnxTe en Sn 1_xMnxTe (x< 0.10) proximation (ENNPA), die in een bevredi wordt gerapporteerd in Hoofdstuk IV. In deze gende simultane beschrijving voorziet van systemen is er een kritische gatendichtheid alle beschikbare experimentele gegevens. In waarneembaar, waarboven de aanvankelijk de vergelijking met overige verdunde magne geringe interacties getransformeerd worden tische halfgeleiders wordt er onder meer aan naar een sterke ferromagnetische koppeling. dacht geschonken aan het universele gedrag Met behulp van een simpel Ruderman-Kit van Tf(x) in systemen met een brede band tel-Kasuya-Yosida model, dat gemodificeerd afstand, en wordt de relatie gelegd met theo is voor de bijdrage van ladingsdragers vanuit retische exchange modellen die aan de Mn2•• verschillende gebieden uit de Brillouin zone, Mn2• interactie ten grondslag liggen. is het gedrag van de Curie-Weiss tempera Hoofdstuk III beschrijft de magnetische tuur te beschrijven, zowel wat betreft het eigenschappen van een nieuwe klasse verdun bestaan van Pcrit als de sterkte van de ferro de magnetische halfgeleiders, op basis van magnetische koppeling boven Pc rit. Er is bo Fe2• in plaats van Mn2•. De soortelijke vendien een aanvang gemaakt met de karak warmte en susceptibiliteit van Zn 1.xFexSe en terisering van de geobserveerde magnetische Hg 1.x-yCdyFexSe, met Fe concentraties tus- fasen. LIST OF PUBLICATIONS 106 LIST OF PUBLICATIONS Ma.gnetic properties of Zn 1_xMnxS. Ma.gnetic properties of the diluted ma.gnetic H.J.M. Swagten, A. Twa.rdowski, W.J.M. de semiconductor Zn 1_xFexSe. Jonge, M. Demia.niuk a.nd J.K. Furdyna., H.J.M. Swagten, A. Twardowski, W.J.M. de Solid State Comm. 66, 791 {1988). Jonge and M. Demianiuk, Phys. Rev. B 39, 2568 (1989). Hole density and composition dependenee of ferroma.gnetic ordering in Pb-Sn-Mn-Te. Ma.gnetiza.tion steps in Ftrba.sed diluted H.J.M. Swagten, W.J.M. de Jonge, R.R. ma.gnetic semiconductors. Gal~tzka., P. Warmenbol a.nd J.T. Devreese, H.J.M. Swagten, C.E.P. Gerrits, A. Twar Phys. Rev. (Rapid Communication) B 37, dowski and W.J.M. de Jonge, Phys. Rev. 9907 (1988). (Rapid Communication) B 41, 7330 (1990). The ma.gnetic properties of II-VI group Low-tempera.ture specific hea.t of Ftrba.sed Ftrtype diluted ma.gnetic semiconductors. DMS. H.J.M. Swagten, A. Twardowski, F.A. Ar H.J.M. Swagten, A. Twardowski, E.H.H. nouts, W.J.M. de Jonge a.nd M. Demia.niuk, Brentjens, W.J.M. de Jonge, M. Demianiuk in 11 Proceedings of the International Confer and J.A. Gaj, in "Proceedings of the 20th ence on Magetism", edited by D. Givord International Conference on the Physics of (France, 1988); J. de Physique C 8, 873 Semiconductors", Thessaloniki, 1990, to be (1988). published. Hole density a.nd composition dependenee of Ma.gnetic beha.vior of the diluted ma.gnetic the diluted ma.gnetic semiconductor PbSn semiconductor Zri t-xMnxSe. MnTe. A. Twardowski, H.J.M. Swagten, W.J.M. de H.J.M. Swagten, H~ v.d. Hoek, W.J.M. de Jonge, and M. Demianiuk, Phys. Rev. B 36, Jonge, R.R. Gal~tZka, P. Warmenbol a.nd J.T. 7013 (1987). Devreese, in "Proceedings of the 19th Inter national Conference on Physics of Semicon Thermadynamie properties of iron-ba.sed ductors", edited by W. Zawadski, (Warsaw, 11-VI semima.gnetic semiconductors. 1988), p. 1559. A. Twardowski, H.J.M. Swagten, T.F.H. v.d. Wetering, W.J.M. de Jonge, Solid State Carrier-concentra.tion-dependent ma.gnetic Comm. 65, 235 (1988). properties of the diluted ma.gnetic semi conductor Sn(Mn)Te. Ma.gnetic susceptibility of iron-based semi H.J.M. Swagten, S.J.E.A. Eltink and W.J.M. ma.gnetic semiconductors: high-tempera.ture de Jonge, in ''Growth, characterization a.nd regime. properties of ultra.thin magnetic films and A. Twardowski, A. Lewicki, M. Arciszewska, multilayers'', edited by B.T. Jonker, J.P. W.J.M. de Jonge, H.J.M. Swagten and M. Heremans and E.L. Marinaro (Materials Re Demianiuk, Phys. Rev. B 38, 10749 {1988). search Society, Pittsburgh, 1989), Vol. 151, p. 171. Carrier induced ma.gnetic interaction in the diluted ma.gnetic semiconductor PbSnMnTe. Ma.gnetic specHic hèa.t of the diluted mag W.J.M. de Jonge, H.J.M. Swagten, R.R. netic semiconductor Hg 1_xFexSe. Gal~tZka, P. Warmenbol and J.T. Devreese, H.J.M. Swagten, F.A. Arnouts, A. Twar IEEE transactions on Magnetics 24, 2542 dowski, W.J.M. de Jonge and A. Mycielski, (1988). Solid State Comm. 69, 1047 (1989). LIST OF PUBLICATIONS 107 The d-d exchange interaction in the semi Magnetization study of Fe-based II-VI group magnetic semiconductor ZnFeSe. diluted magnetic semiconductors. A. Twardowski, H.J.M. Swagten and W.J.M. W.J.M. de Jonge, H.J.M. Swagten and A. de Jonge, in "Proceedings of the 19th Inter Twardowski, in "Properties of II-VI semi national Conference on Physics of Semicon conductors: bulk crystals, epitaxial films, ductors", edited by W. Zawadski (Warsaw, quanturn well structures, and dilute magnetic 1988), p. 1543. systems", edited by F.J. Bartoli Jr., H.F. Schaake and J.F. Schetzina (Materials Re Correlation between electronic and magnetic search Society, Pittsburgh, 1990), Vol. 161, properties in the IV-VI group diluted mag p. 485. netic semiconductors. S.J.E.A. Eltink, H.J.M. Swagten, N.M.J. Low-temperature specHic heat of the diluted Stoffels and W.J.M. de Jonge, J. Magn. magnetic semiconductor Hg1_x-yCdyFexSe. Magn. Mater. 83, 483 (1990). A. Twardowski, H.J.M. Swagten and W.J.M. de Jonge, Phys. Rev. B 42, 2455 (1990). Specific heat study of Fe-based narrow and wide gap diluted magnetic semiconductors. Neutron diffraction on the diluted magnetic W.J.M. de Jonge, H.J.M. Swagten, C.E.P. semiconductor Sn 1_xMnxTe. Gerrits and A. Twardowski, in "Narrow-gap C.W.H.M. Vennix, E. Frikkee, H.J.M. Swag semiconductors and related materials", ten, K. Kopinga and W.J.M. de Jonge, in edited by D.G. Seiler and C.L. Littler (Hil "35th Annual Conference on Magnetism and ger, Bristol, 1990); Semicond. Sci. Technol. Magnetic Materials", San Diego, 1990, to be 5, S270 (1990). published. Carrier concentration dependenee of the mag EPR study of s-d exchange in PbSnMnTe. netic interactions in Sn(Mn)Te. T. Story, C.H.W. Swuste, H.J.M. Swagten, W.J.M. de Jonge, H.J.M. Swagten, S.J.E.A. R.J.T. van Kempen and W.J.M. de Jonge, in Eltink and N.M.J. Stoffels, in "Narrow-gap "35th Annual Conference on Magnetism and semiconductors and related materials", Magnetic Materials", San Diego, 1990, to be edited by D.G. Seiler and C.L. Littler (Hil published. ger, Bristol, 1990); Semicond. Sci. Technol. 5, S131 (1990). Diluted magnetic semiconductors. W.J.M. de Jonge and H.J.M. Swagten, SpecHic heat of Zn 1_xCoxSe semimagnetic in "Proceedings of the NATO Institute on semiconductors. the Science and Technology of N anostruc A. Twardowski, H.J.M. Swagten, W.J.M. de tured Materials", Greece, 1990, to be pub Jonge and M. Demianiuk, Acta Phys. Polon. lished. A 77, 167 (1990). DANK voor de waardevolle bijdrage aan dit proefschrift, Jos v. Amelsvoort Hans Heijligers Frank Amouts Hans v.d. Hoek Gerrie Baselmans Erik Janse Pascal Bloemen Wim de Jonge Paul Boomkamp Rob v. Kempen Rob Braun Klaas Kopinga Eric Brentjens Edward Maesen Joop v.d. Broek Jef Noijen Rinus Broekmans Peter Nouwens Hans Dalderop Marc v. Opstal Wil Delissen A. v.d. Pasch Cees DeniBBen Marco Poorts J. Devreese Helga Pouwels Jerome Dubois Cees v.d. Steen Noucha Duncker Nico Stoffels Paul Eggenkamp Tomasz Story Menno Eisenga Coen Swuste Stepban Eltink Andrzej Twardowski J. Flokstra Suzanne te Velthuis R. Ga!azka Kees Vennix Gerard v. Gastel Rianne van Vinken Karel Gerrits Peter Warmenbol Bob Groeneveld Eric v.d. Wetering Ruth Gruyters Marc Willekene Frank Haarman Jos Zeegers faculteitswerkplaats Natuurkunde faculteit Natuurkunde, Universiteit van Antwerpen faculteit Scheikundige Technologie, vakgroep Fysische Chemie faculteit Technische Natuurkunde, Universiteit Twente glastechnische groep CTD groep Cooperatieve Verschijnselen, vakgroep Vaste Stof groep Halfgeleiderfysica, vakgroep Vaste Stof groep Lage Temperaturen, vakgroep Vaste Stof heliumbedrijf faculteit Natuurkunde Institute of Experimental Physics, University of Warsaw Institute of Physics, Polish Academy of Sciences, Warsaw Institute of Technica! Physics, Warsaw secretariaat vakgroep Vaste Stof Shell Nederland B.V. stichting FOM DE AUTEUR H.J.M. Swa.gten, gehuwd met J .M.C. Swa.gten-Mestrom. 1962 Geboren op 26 juli te ·Roermond, Boon va.n W.J.M. Swagten en A.M. Swagten-Pulles. 1974-1980 Middelbaar onderwijs aan het Bisschoppelijk College "Broekhin 11 te Roermond. 30 mei 1980: examen Atheneum - B. 1980-1987 Universitair onderwijs aan de Technische Universiteit Eindhoven, Faculteit der Technische Natuurkunde. Afstudeeronderzoek in de Vakgroep der Vaste Stof, groep Cooperatieve Verschijnselen. 11 februari 1987: ingenieursexamen (met lof). Titel: "Het magnetische gedrag van Pbt-x-ySnyMnxTe". 1987- 1990 Promovendus (als Onderzoeker In Opleiding, stichting Fundamenteel Onderzoek der Materie) aan de Technische Universiteit Eindhoven, Faculteit der Technische Natuurkunde, in de Vakgroep der Vaste Stof, groep Cooperatieve Verschijnselen. 20 november 1990: promotie. Titel: "The magnetic behavior of diluted magnetic semiconductors". STELLINGEN behorende bij het proefschrift THE MAGNETIC BEHAVlOR OF DILUTED MAGNETIC SEMICONDUCTORS door H.J.M. Swagten 1. Bij de interpretatie van de lage-temperatuur soortelijke warmte van verdunde magnetische halfgeleiders moet men terdege rekening houden met de modificatie van de rooster soortelijke warmte in vergelijking met die van het halfgeleidende gastmateriaalt. In enkele gevallen, en met name Hg 1_xMnxSe en Hg1_xFexSe, kan de veronachtzaming van dit effectl leiden tot apert onjuiste conclusies. t dit proefschift, Hoofdstuk III, paragraaf 2 + R.R. Galazka, S. Nagata and P.H. Keesom, Phys. Rev. B 22, 3344 (1980); A. Lewicki, J. Spalek and A. Mycielski, J. Phys. C 20, 2005 {1987). 2. Het "three-level-superexchange" model van Larson, Hass, Ehrenreich en Carlssont, en met name de voorspelling omtrent het lange-dracht karakter van de magnetische interactie in II-VI groep verdunde magnetische halfgeleiders met een grote bandafstand, wordt bevestigd door het geobserveerde universele gedrag van de spinglas overgangstemperatuurt in deze materialen. t B.E. Larson, K.C. Hass, H. Ehrenreich en A.E. Carlsson, Phys. Rev. B 37, 4137 (1988) i dit proefschrift, Hoofdstuk I en Hoofdstuk II, paragraaf 1. 3. Escorne, Godinho, Tholence en Maugert relateren het gedrag van de dynamische susceptibiliteit van Sn1_xMnxTe, met Mn concentraties variërend tussen 0.03 en 0.06, aan een zogenaamdere-entrant spinglasfase. Deze interpretatie is onjuist, omdat in hun experimenten niet voldaan is aan de laag-veld limiet voor de magnetisatie. t M. Eseorne, M. Godinho, J.L. Tholence en A. Mauger, J. Appl. Phys. 57, 3424 (1985). 4. Bij de berekeningen van de Curie-Weiss temperatuur in IV-VI groep verdunde magnetische halfgeleiders, op basis van het twee valentiebanden RKKY-modelt, dient rekening gehouden te worden met de symmetrie-eigenschappen van de E-band. t dit proefschrift, Hoofdstuk I en IV 5. Het door Hagen en Griessent opgestelde inversieschema om de verdelingsfunctie voor thermisch geactiveerde fluxbeweging in supergeleiders te bepalen uit de relaxatie van de magnetisatie is in principe alleen juist in de limiet van een kleine spreiding in de activeringsenergie. t C.W. Hagen en R.. Griessen, Phys. Rev. Lett. 62, 2857 (19811). 6. Aan de basis van elk fysisch onderzoek aan de vaste stof staat de synthese van geschikte kristalstucturen. Dit strookt niet met de ondergeschikte aandacht voor preparatieve details in veel wetenschappelijke publicaties. 7. Het verdient aanbeveling de benaming "verdunde magnetische halfgeleiders" te 11 hanteren in plaats van "semimagnetische halfgeleiders , omdat laatstgenoemde de suggestie wekt dat de magnetische verschijnselen die geobserveerd worden in deze materialen, slechts ten dele of zelfs quasi-magnetisch zouden zijn, hetgeen niet juist is. 8. Bij de vaststelling van snelheidslimieten voor motorboten op stromende vaarwegent, is de toevoeging van het coördinatenstelsel, stilstaand of meebewegend ten opzichte van het water, op zijn plaats. t zie bijvoorbeeld, Bijzonder reglement snelle motorboten Maas, Stb. 664, 1345 (1971). 9. De beweegredenen van de onderzoeker om wetenschappelijke resultaten te publiceren en de mate waarin de onderzoeker publiceertt, hangen niet noodzakelijkerwijze samen met de kwaliteit en de relevantie van de publicatie. Bij de beoordeling daarvan doen deze beweegredenen zelfs niet ter zake. t P.A.J. Ackermans, proefschrift TUE 1990 (stellingen). 10. Het inzicht in de arbeids-en inkomensverhoudingen op universiteiten kan verrijkt worden door het besef dat, weliswaar op basis van een structureel draagvlak van wetenschappelijke stafleden, de daadwerkelijke uitvoering van het wetenschappelijke onderzoek in handen ligt van promovendi. 11. Middelen tegen kaalhoofdigheid gaan voorbij aan de esthetische verrukking die er van het kalende mannenhoofd uit kan gaan. Roermond, 3 oktober 1990.