Analysis of the magnetic properties of semimagnetic semiconductors : an experimental study

Analysis of the magnetic properties of semimagnetic

semiconductors : an experimental study

Citation for published version (APA):

Denissen, C. J. M. (1986). Analysis of the magnetic properties of semimagnetic semiconductors : an

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10.6100/IR251727

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ANALYSIS OF THE

MAGNETIC PROPERTIES

OF

SEMIMAGNETIC SEMICONDUCTORS

AN EXPERIMENT AL STUDY

C.J.M. DENISSEN

ANALYSIS OF THE MAGNETIC PROPERTIES OF SEMIMAGNETIC

SEMICONDUCTORS

AN EXPERIMENT AL STUDY

PROEFSCHRIFT

TER VERKRIJGING VAN DE GRAAD VAN DOCTOR AAN DE TECHNISCHE UNIVERSITEIT EINDHOVEN, OP GEZAG VAN DE RECTOR MAGNIFICUS, PROF. DR. F.N. HOOGE, VOOR EEN COMMISSIE AANGEWEZEN DOOR HET COLLEGE VAN DEKANEN IN HET OPENBAAR TE VERDEDIGEN OP

VRIJDAG 17 OKTOBER 1986 TE 14.00 UUR

DOOR

CORNELIS JOHANNES MARIA DE NISSEN

Dit proefschrift is goedgekeurd door de promotoren:

Prof.dr.ir. W.J.M. de Jonge en

Prof.dr. J.A. Mydosh

The work described in this thesis was part ofthe research program ofthe Solid State Division, grot,~p "Cooperative Phenomena", Department of Physics, Eindhoven University ofTechnology, Eindhoven, The

TABLE OF <D'ITEliTS

I INI'ROOOCTICfi Referen.ces

II SEMIMAGNEfiC SEMIOONOOCTORS OR DIUITED MAGNEfiC SEMIOONOOCTORS 2.1 2.2 Introduetion Magnetic properties References

I I I TIIEOREfiCAL APPROXIMATiefiS FOR DIUITED MAGNEfiC SYSTEMS WITII LONG-RANGE INTERACTiefiS

3.1 3.2 3.3 3.4 3.5 Introduetion

Simple nearest-neighbour isolated cluster model Extensions of the NN isoLated cluster model Nearest-neighbour pair correlation approximation Mean field approximation

Referen.ces IV EXPERIMENTAL JIEfHODS 4.1 4.2 4.3 4.4 4.5 4.6 4.7 Introduetion

Preparation of (Cd1 _xMnx) 3As2 and _{(Zn1 _xMnx) 3As2} AnaLysis of the sampLes

Calorimetry Magnet izn.t ion

Dynamic susceptibility DC susceptibility References 1 3 4 5 9 11 11 17 24 31 33 34 34 35 39 43 46 47 48

V JIAGNETIC BDIAVIOOR OF 11IE SEMIJIAGNETIC SEMI<llNOOCfOR (Cdl-xJinx}~2 5.1 Introduetion 5.2 Structure 5.3 5.4 5.5 Experimentat resutts Interpretut ion

Con.cluding remarks and discussion Referen.ces

VI MAGNETIC BDIAVIOOR OF 11IE SEMIJIAGNETIC SEMiaxiDUCTOR (Znl-xllnx}3As2 6.1 Introduetion 6.2 6.3 6.4 Experimentat resutts Interpretut ion

Conctuding remarks and discussion References

VII ANALYSIS OF~ MAGNETIC PROPERTIES OF DILUfED JIAGNETIC SEMiaJNIXJCroRS IN<ll.IDING LONG-RANGE IJITERACfl<ftS

7.1 7.2 7.3 Introduetion Results Discussion Referen.ces

VIII SPIN-~ BDIAVIOOR OF 11IE SEJIIJIAGNETIC SEJIIrnNOOCI'ORS Zn 1_xJinxSe and Zn1_xMnxTe

8.1 Introduetion 8.2 8.3 8.4 Experimental resutts Discussion Con.clusions References

APPENDIX A SAJIElWATTING 49 51 54 65 72 77 80 81 90 95 98 100 102 107 108 110 110 114 118 118 120 122

mAPTER 1 INfRO:nt..n'ION

Semimagnetic semiconductors (SMSC's) or diluted magnetic semicon-ductors (DMS's) can be considered as a new class of semiconducting materials. These materials are ternary semiconductor alloys whose lattice is made up in part of substitutional magnetic ions. During the past ten years these materials have been stuclied intensively. New interesting features in both the semiconducting and the magnetic prop-erties have been observed. The presence of magnetic ions in the SMSC's leads to a spin-spin exchange interaction between the localized mag-netic moments and the band electrons. The modified band electron be-haviour can be described by the action of the spin average <Sz> pro-portional to the bulk magnetization.

The presence of the ·magnetic ions distinguishes SMSC' s from ordi-nary semiconductors. On the other hand, the magnetic ions in these systems interact much weaker wi th each other than those in magnetic semiconductors, and therefore the magnetic system in SMSC's responds toa wide range of applied fields. As aresult of the spin-spin inter-action mentioned above and the spin polarization <Sz> interesting physical effects in moderate fields have been observed, i.e., a large Zeeman spli tting of the electronic energy levels and the exci tonic band, a large Faraday rotation, a large negative magnetoresistance, magnetic polarons, etc. [1]. Moreover, the temperature dependenee of <Sz> offers the possibili ty to observe large temperature effects in the electronic properties, e.g., in the quanturn oscillation amplitude [ 1. 2].

Besides these semiconducting prop~rties the SMSC's are of interest because of their magnetic properties as disordered alloys. More spe-cifically one may refer to the nature and the physical mechanism underlying the magnetic exchange interaction and the origin of the spin-glass transition observed at low temperatures. At the start of the investigations reported in this thesis, the range of the magnetic interactions seemed to be limited to nearest-neighoour sites (short-range interactions). The physical origin of these interactions was not understood. The spin-glass formation was attributed to frustation effects and was thought not to exist at low concentrations. Moreover,

dis-The problems in the understanding of the magnetic properties formed the general motivation of the investigations reported in this thesis. Earlier results on (Cd1_xMnx) 3As2 showed that the properties of this compound resembie those of SMSC's in many respects. However, the observation of the spin-glass transition far below the percolation limit indicated that the magnetic interaction was of longer range than in the SMSC' s reported up ti 11 then. For this reason we thought i t worthwhile to study the magnetic properties of (Cd 1_xMnx) 3As2 in more detail. In a later stage, we extended the investigations to the SMSC (Zn1_xMnx) 3As2 and some other SMSC's of the II-VI group in order to obtain additional evidence on the origin of the magnetic interactions. Recently, the general relevanee of SMSC's has been emphasized by new developments in the area of superlattices and heterojunctions. The large Zeeman splittings mentioned above, which can be of the order of 20 meV at magnetic fields of 1 T. suggest the feasibility of magnetie-field-tuning of many electrical and optica! properties of such artifi-cial structures, which are of interest from both fundamental and ap-plied points of view [3].

The organization of this thesis is as follows: In chapter 2 we start with a brief introduetion of the magnetic properties of SMSC's. Chapter 3 deals with theoretica! approximations, which can be used for

the calculation of the thermadynamie properties of a diluted magnetic system. In chapter 4 the preparation and characterization of (Cd 1_xMnx) 3As2 and (Zn1_xMnx)3As2 are reviewed, tagether with the experimental methods used in the investigations. In chapter 5 and 6 we present the magnetic properties of these compounds. We show that the magnetic specific heat Cm, the magnetization M and the susceptibility

x

can be simultaneously described with a model assuming a random

dis-tribution of the magnetic ions and a long-range magnetic interaction. The possible origin of this magnetic interaction is discussed. In chapter 7 we show that also other typical SMSC's of the II-VI group can bedescribed with this model. Finally, in chapter 8 we present the resul t of ac susceptibi 1 i ty measurements on the Zn-compounds of the II-VI SMSC's. These results indicate also a long-range character of the magnetic exchange interaction. The range of the interaction is compared with the results on other SMSC's.

Parts of the chapters 5-8 have been accepted for publication [4,5,6]. We have choosen to embody the corresponding texts in this thesis in essentially the same farm as they will be published. As a consequence some parts of these chapters may seem somewhat redundant for the reader of this thesis. On the other hand, this choice has the advantage that these chapters can be read rather independently.

References

[1] N.B. Brandt, V.V. Moshchalkov, Adv. Phys. 33, 193 {1984).

R.R. Gal~zka, Proceedings of the 14th International Conference on Physics of semiconductors Edinburgh, 1978, Inst. Phys. Conf. Ser. 43 {1979), (Inst. Phys., London-Bristol) p135.

[2] J.J. Neve, Ph. D. thesis, University of Eindhoven (1984), unpublished.

[3] S. Datta, J.K. Furdyna, R.L. Gunshor, Superlatt. and Microstr. 1, 327 ( 1985).

[4] C.J.M. Denissen, H. Nishihara, J.C. van Goal, W.J.M. de Jonge, Phys. Rev. B 33, 7637 {1986).

[5] C.J.M. Denissen, W.J.M. de Jonge, Solid State Commun., {1986). [6] A. Twardowski, C.J.M. Denissen, W.J.M. de Jonge, A.T.A.M. de

Waele, M. Demianiuk, R. Triboulet, Solid State Commun. 59, 199 (1986).

mAPTER 2 SEMIMAGNETIC SEMialNDUCTORS OR DILUfED MAGNETIC SEMI<DNDUCfORS.

2.1 Introduetion

The experiments reported in this thesis will deal wi th the magnet ie properties of semimagnetic semiconductors (SMSC's) or diluted magnetic semiconductors (DMS's). As quoted in chapter 1 the interesting prob-lems in SMSC's are re1ated to both the semiconducting and the magnetic properties. Recently Brandt et al. [1] have extensively reviewed this research field, and hence we like to refer the reader to their paper for more detailed information.

In this chapter we wil! focus our attention to the magnetic prop-erties of SMSC's. We do not pretend to review the research field but merely present a comprehensive introduetion to this subject. Since major developments in the understanding of the behaviour of SMSC's have occured during the period in which the investigations reported in

this thesis were carried out, we have chosen to include in this intro-duetion some rather recent results, which will be discussed in full detail in the chapters 5-8.

ZnS 2 phoses CdS 2 phoses HgS 2 phoses ZnSe 2 phoses Cd Se

HgSe~~~~~~7nr.n~~~ryznr--~

ZnTe Cdle Hgle ~~~~~~~~~~~--~~ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9 1.0 Fig. 2.1 Vl SoLubiL i ty of Mn c haLco-c :E

gentdes with

II-VI

com-pounds and resuLting crystal structures [2]. x: moLe fraction of Mn chalcogenid.e.

Furthermore, our introduetion will be restricted to SMSC's of the II-VI group. The crystal structure and composi tion range of these compounds are schematically given in Fig. 2.1 [2].

2.2 Magnetic properties

The magnetic properties of SMSC's as monitored by susceptibility, magnetization and specific heat measurements all yield evidence for a substantial (antiferromagnetic) interaction between the Mn-ions. In metallic diluted systems such an interaction is known to originate from the polarization of the free electrons through the so-called RKKY (Rudermann-Kittel-Kasuya-Yosida) mechanism. However, since the elec-tron density in these semiconducting systems is at least three orders of magnitude smaller, this particular mechanism cannot be responsible for this interaction and an alternative mechanism must be formulated.

Surprisingly, the experimental evidence indicates a transition to a so-called spin-glass state in SMSC's at low temperatures. The first experiments on the !I-VI SMSC's [3,4,5] s-trongly suggested that such a transition could only be observed above the percolatien limit for nearest-neighbours (x ~ 0.17 in a fee or hcp lattice). The identifi-cation of the spin-glass transition is based on the characteristic cusp exhibited by the low-field magnetic susceptibility at some tem-perature Tf and on the absence of an anomaly in the specific heat at Tf. In addition to this characteristic cusp in the low field ac sus-ceptibility, the SMSC's show other features which are typical of spin-glasses, i.e., a difference in field-cooled and zero-field-cooled de susceptibility measurements [3,4,6]. a frequency dependenee of Tf in ac susceptibility measurements [7], and a remanent magnetization for temperatures T

<

Tf with quite long relaxation times (t ~ 1000 sec)

[6,B].

Originally the origin of the spin-glass transition above the per-colation limit was attributed to the lattice frustration mechanism [9] inherent to the fee or hcp lattice if the magnetic ions are coupled by an antiferromagnetic nearest-neighbour exchange interaction.

p

x'

x/'

I 020 I

I

I I

..

I I 11

,t

x I

'I

IX I I I

s

I 0.40

x,

I

/•

0.60 Fig. 2.2

Tf - x

diagram for

Hg(Mn)Te.

Experimentat resutts:

(&)

Ref. [13], (x) Ref

.

[14],

(•) Ref. [13],

(~)Ref.

[3].

Recently, however, the spin-glass transition was also observed for concentrations far below the percolation limit, i.e., [Cd(Mn)]3As2 [10], Cd(Mn)Te [11], Cd(Mn)Se [12] and Hg(Mn)Te [13,14]. As an example

the magnetic Tf-x diagram of Hg(Mn)Te is shown in Fig. 2.2. From the observation of a spin-glass transition far below the percolation limit it must be concluded that the interaction inducing this transition is of longer range than the nearest-neighbour distance. Several

interac-tion mechanisms inducing this long-range interaction have been pro-posed [11,12,13,14]. We will return to this subject in chapter 5 when we discuss our own results.

Data for the magnetic specific heat are available for several of the known SMSC's [3,4,15,16]. Generally they show an x dependent "bump" in plots of Cm versus T whereas no anomaly at Tf is observed.

The results have been interpreted by assuming that Cm can be composed from the contributions of different nearest-neighbour clusters - pairs

and triples - [3,4]. In Fig. 2.3 a typical resul t is shown [4]. The calculation of the total magnetic heat capacity with cluster probabii-i tprobabii-ies obtaprobabii-ined from a random dprobabii-is trprobabii-ibutprobabii-ion of the magnetprobabii-ic probabii-ions over the lattice essentially differs from the experimental data on Cm {Fig.

160 120 :.::: Q) 0 E ... -.

_g

) (

BO

QJ LJ 0 Hg1_xMnx Te x:Q.Q27 , - , _~

,.--

... ... I \

,

'

I ll.

_'

I I \ I I \ I I \ I ,'2T

'F

I J \ J \ I \ I

.+ ...

\ I \ J I J I B•OT

•".----

---i I J Tl Kl 2 3 Fi.g. 2.3

Magnette specific heat of

Hg 1_xMnxTe for x = 0.027 as a

function of temperature in zero field, and in magnette fields of

1 and 2 T. Dashed curves indicate

the specific heat calculated for a

cluster model using a random dis-tribution of Mn-ions. Solid curves

show the calculated results for a

distribution of Mn-ions deviating

strongly from a random distribu-tion. An antiferromagnetic exchange

constant ]/kB

=

-0.52 K is used in

the calculations.

resul ts could only be achieved when the cluster p-robabi 1 i ties were taken as adjustable parameters (Fig. 2.3). The specific heat results on Cd(Mn)Te and Cd(Mn)Se were interpreted in the same way [3,14]. From

all results it was concluded that considerable chemica! clustering of the magnetic ions occurred.

However, high temperature susceptibi l i ty and recent high field magnetization measurements do not support this model. We will first describe some representative results of the high field magnetization. Magnetization data have been obtained for most SMSC's up till magnetic

fields B ~ 25 T in a wide temperature range 1.5

<

T

<

40 K. The gener-al behaviour of M indicates an antiferromagnetic interaction [17]. Significant progress in the understanding of the magnetization was made by the observation of steps inMat T ~ 1.5 K for magnetic fields B

z

10 T [18,19]. A typical resul t is shown in Fig. 2.4 [18]. The

5 0 5 10 BIT) Cd1_xMnx Se

x=

0.049 T: 1. 53K 15 20 Fig. 2.4

Magnetization Mof _{Cd1 _xMnxSe} as a function of magnetic

field B. Note the broad step near 12.4 T. The theoretical

prediction for the size

oM

of

the step, assuming a random distribution of Mn-ions, is shown for comparison.

apparent saturation at lower magnetic fields (B ~ 10 T} is called technica! saturation by Shapira et al. [18]. The steps could be inter-preeed as the contri bution from the decoupl ing of nearest-neighbour pairs interacting wi th a strong antiferromagnetic interact ion. The magnetic field value at which the step occurs was related directly to the nearest-neighbour interaction. Also in other compounds, Zn(Mn)Te [20], Cd(Mn)Te [21. 22] and Zn(Mn)Se [18], these steps have been ob-served. In all cases the inferred order of magnitude of the nearest-neigbour interaction is J₁~ ~-10K, which is an order of magnitude

larger then the nearest-neighbour exchange obtained from the original cluster model mentioned above.

The magnitude of the steps in M could be theoretically related to the probability of a spin being part of a pair cluster, assuming a random distribution of the magnetic ions. Moreover, the magnitude of the technica! saturation is in ,accordance with the assumption ([18], see also chapter 3) that only singles and 1/3 of the triple clusters contribute to M, whereas the moment of pair clusters is "paired off".

Rel iable high-temperature susceptibi l i ty data on the "canonical" SMSC's like Cd(Mn)Te, Cd(Mn)Se, Hg(Mn)Te and Hg(Mn}Se were in fact not available until very recently [23], Considerable inconsistencies

in these materials the Curie-Weiss temperature changes sign as a func-tion of concentrafunc-tion [24], indicating a change from ferromagnetic to antiferromagnetic interaction with increasing x. These and other anom-alies (e.g., an unexplained concentration dependenee of the effective spin per magnetic ion [24]) in the previously reported magnetic behav-iour of SMSC's were eliminated by Spalek et al. [23] by a systematic analysis and reinterpretation of the susceptibility data in a fashion reminiscent to the analysis we will report in chapters 5 and 6 for [Cd(Mn)]3As2 and [Zn(Mn)]3As2 , respectively.

These recent results on the magnetization as wellas the high temperature susceptibility show that the nearest-neighbour interac-tions are indeed an order of magnitude larger than the interacinterac-tions obtained from the specific heat. These results also yield no need for readjustment of the distribution of the magnetic ions. In chapter 7 we will return to the conflicting results for the SMSC's of the !I-VI group mentioned above and show that i t is possible to explain all observed thermadynamie properties with a consistent set of interaction parameters and a random distribution of the magnetic ions if the long-range character of the interaction is taken into account.

Refereru:es

[1] N.B. Brandt, V.V. Moshchalkov, Adv. Phys. 33, 193 (1984).

[2] A. Pajaczkowska, Prog. Cryst. Growth and Charact. 1, 289 (1978). [3] R.R. Gal~zka, S. Nagata, P.H. Keesom, Phys. Rev B 22, 3344

( 1980).

[4] S. Nagata, R.R. Gal~zka, D.P. Mullin, H. Arbarzadeh,G.D. Khattak, J.K. Furdyna, P.H. Keesom, Phys. Rev. B 22, 3331 (1980).

[5] R.R. Gal~zka, Proceedings of the 14th International Conference on the Physics of Semiconductors, Edinburgh (1978), Inst. Phys. Conf. 43, 133 (1979).

J.K. Furdyna, J. Appl. Phys. 53, 7637 (1982). [6] S.B. Oseroff, Phys. Rev. B 25, 6584 (1982).

[7] M. Ayadi, P. Nordblad, j. Ferre, A. Mauger, R. Triboulet, J. Magn. Magn. Mater. 54-57. 1223 (1986).

[8] E. Kierzek-Pecold, W. Szymanska, R.R. Gal~zka, Solid State Commun. 50, 685 (1984}.

[9] L. de Seze, J. Phys. C 10, L353 (1977).

[10] W.J.M. de Jonge, M. Otto, C.J.M. Denissen, F.A.P. Blom, C. van der Steen, K. Kopinga, J. Magn. Magn. Mater. 31-34, 1373 (1983}. [11] M.A. Novak, O.G. Symko, D.J. Zheng, S. Oseroff, J. Appl. Phys.

57, 3419 (1985}.

(12] M.A. Novak, O.G. Symko, D.J. Zheng, S. Oseroff, Physica 126B, 469 ( 1984}.

[13] N.B. Brandt, V.V. Moshchalkov, A.O. Orlov, L. Skrbek, I.M. Tsidil'kovskii, S.M. Chudinov, Sov. Phys. JETP 57, 614 (1983}. (14] A. Mycielski, C. Rigaux, M. Menant, T. Dietl. M. Otto, Solid

State Commun. 50, 257 (1984} .

. [15] C.D. Amarasekara, R.R. Gal~zka, Y.Q. Yang, P.H. Keesom, Phys.

Rev. B 27, 2868 (1983}.

[16] G.D. Khattak, C.D. Amarasekara, S. Nagata, R.R. Gal~zka. P.H. Keesom, Phys. Rev. B 23, 3553 (1981}.

[17] J .A. Gaj, R. Planel. G. Fishman, Solid State Commun. 29, 435 ( 1979}.

[18] Y. Shapira. S. Foner, D.H. Ridgley, K. Dwight, A. Wold, Phys.

Rev. B 30, 4021 (1984}.

[19] R.L. Aggarwal, S.N. Jasperson, Y. Shapira, S. Foner, T. Sakakibara, T. Goto, N. Miura, K. Dwight, A. Wold, in

Proceedings of the 17th International Conference on the Physics of Semiconductors,

San

Francisco (1984}, edited by J.D. Chadi and W.A. Harrison (Springer-Verlag} Berlin, 1419 (1985).

[20] Y. Shapira, S. Foner, P. Becla, D.N. Dominques, M.J. Naughton, J.S. Brooks, Phys. Rev. B 33, 356 (1986).

[21] R.L. Aggarwal, S.N. Jasperson, P. Becla, R .. Gal~zka, Phys. Rev. B 32, 5132 (1985}.

[22] B.E. Larson, K.C. Hass, R.L. Aggarwal. Phys. Rev. B 33, 1789 (1986).

[23] j. Spalek, A. Lewicki, Z. Tarnawski, J.K. Furdyna, R.R. Gal~zka, Z. Obuszko, Phys. Rev. B 33, 3407 (1986).

[24] U. Sondermann, J. Magn. Magn. Mater. 2, 216 (1976}. U. Sondermann, E. Vogt, Physica 86-SSB, 418 (1977).

mAPTER 3 TIIEORETICAL APPROXIMATIONS FOR DIUITED MAGNEfiC SYSTEMS WITII l.DNG-RANGE INTERACfiONS

3.1 Introduetion

In this thesis (chapters 5, 6 and 7) we will show that the magnet-ie behaviour of diluted magnetic semiconductors {DMS's) indicates an interaction model with the following properties:

i) the magnetic interaction between the magnetic ions is antiferromagnetic and long-range

ii) the magnetic ions are distributed randomly in the host

lattice.

The Hamiltonian descrihing a magnetic model system having the features

mentioned above is very complex and in fact cannot be solved exactly.

In order to perform theoretica! calculations of the thermadynamie properties, we need to simplify this model system. In this chapter we

will introduce some of the corresponding approximation methods.

3.2 Simple nearest-neighbour isolated cluster model

In this section we start with the most simple approximation of a

diluted magnetic system. In this approximation only the interaction

between nearest-neighbour {NN) ions on adjacent lattice sites is

con-sidered and all interactions between magnetic ions separated by non-magnetic ions are neglected [1]. It is further assumed that the mag-netic ions are randomly distributed over the lattice sites. For low

concentrations the majority of magnetic ions will be part of a single,

pair, or triple cluster. A single cluster is defined as a magnetic ion

which has no NN magnetic ions. A pair cluster is defined as a magnetic

ion wi th only one magnet ie NN which i tself has no other NN magnet ie

ions. Open and closed triple configurations may be similarly defined with the aid of Table 3.1.

The probabilities that a magnetic ion belongs toa single, pair

Cluster type single sl

_•

pl J1 pair Tl J1 J1 open triple 0 closed triple Tl

J~1

c

TabLe 3.1 Types of simpLe cLusters, assuming onLy a NN interaction

1_{1 between the magnetic ions [1].}

Table 3.2 [1]. These probabilities for a fee host lattice are shown in Fig. 3.1 as a function of the concentration x.

L:

represents the total probability of the configurations up till three spins. It is obvious that for low concentrations x the vast majority of the spins is ac-counted for in this simple approximation. For concentrations above

x~ 0.05, however, it is - in principle - necessary to consider also more complex configurations of four or more spins.

If we assume a Heisenberg exchange interaction between the nearest neighbours, the Hamiltonians for the single, pair and triple configu-rations can be solved rather straightforwardly. These Hamiltonians and

Cluster type bcc fee hcp s1 (l-x) 6 (l-x) 8 (l-x) 12 (l-xl12 p 1 6x(l-x) 10. 8x(l-x) 14 12X(l-x)18 12x(l-K) 18 T~ 9x2 ( 1-X)lJJ 4+ (1-X)) 12x2 (1-X) 1'7( ) + ) (1-X) 2 + (1-X) 3) 18x 2 ( l-x) 22 [5{1-x)+2} 18x2 ( 1-X) l ) ( 2+5 (1-x)) Tl o 24x2(t-x)22 Jx 2 (1-x)21(1+6(1-x)+(l-x) 2 } c

TabLe 3.2 ProbabiLity that a magnetic atom is in a singLe, pair, or tripLe cLuster (assuming onty NN interaction).

Fig. 3.1

Probabilities of the va-rious cLuster types of the simple

NN

cluster

0.1 model for the fee

structure.

0.01

0.001

their energy eigenvalues are included in Table 3.3. The energy levels of a pair of

Mn

2_{+-ions, (S = 5/2) are schematically presented in Fig.} 3.2. The part i ti on function of a certain cluster type {Zi) can be calculated from the energy-eigenvalues (E . . ) according to

1. J

(3.1)

where the summation runs over all energy values of cluster type {i). The partition function (Z) of the total magnetic system can be written as

z

n

_.

z ..

₁ (3.2)

Type Hamil tonian Eigenvalues -qJ,JBmBZ • 1!111 { s -J 0 {SA (SA +1) -25 {S+l) ) -gJJ8mB2 • 0 { SA i 25, lml ~ SA -J_{0 (S8 (S 8}+l) -SA (S.._ +1) -S (S+l) ]-qJJ8m8l, 0 i SA ~ 25, ISA-SI i s8 ~ SA+S, lml ~ SB -Jn( SB (S 8+1) -JS (5+1) ]-qiJ8mBz, 0 S SA i 25, ISA-51 i s8 ~ SA+S, 1111 i s 8 -J0 (S8(S 8+1) -SA (SA +1) -S (5+1) )-J5(SA (SA +1) -2S(S+l) }-9'J-18rnez. 0 .{ SA ~ 25, ISA-SI { s 8 i SA+S, lml S S 8 T~rn -2Jn (~i. ~j) -2Jn~ CS. i. ~k) -g-IJB (S~•sj+s:) Bz T~rnp -2J0 (,S_i. • S.j) -2Jm CS. i· ~k) -2Jp (S.j · ~);) -g}l8 (S~+sj+S~) Bz

TabLe 3.3 The HamtLtonians and their eigenuaLues of different

cluster types (simpte NN cluster modeL, extended versions

of the cluster model, etc.). The eigenualues of the

Hamiltonians of the cluster type ~ and

ynmp

are

c determined numerically.

The total free energy F is given by

F = -k_T

_·-s

ln Z = _-k8T

L

_.

ln Z. =

L

F.

1 . 1

1 1

(3.3)

where Fi is the free energy of cluster type (i), Fi

30J

s'

5

,,

Fig. 3.2

Energy levels for a pair of

20J

s '

4 9 Mn2•-ions with an antiferromagnetic

coupling and S ₌ 5/2. The ualues of

S characteristic for each level and

12 J

s '

3 7 _{the degeneracy (2S+l) are indicated.}

6J

s '

2 5

2J

_s'

I 3

The other thermodynamic properties easily follow from the usual rela-tions

_[oF

1]

oB

T (3.4)

c .

m,l -T

[C,'J.

(3.5)

[

a~]

[oM1]

BB2 T

=

BB

T (3.6)

M., C . and l(. can be calculated numerically. Some representative

1 m, 1 1

results are shown in Figs. 3.3, 3.4 and 3.5. The total M, C and l( are

m

obtained by summing over all the cluster types (i), weighted with

their probability.

Fig. 3.3

0.6 _{Speci.fic heat per Mn-i.on}

in zero magneti.c fietd for the ctuster types of

the simpte NN ctuster

c .!? modet, usi.ng

s

= 5/2, I 0.4 c L: =2andan antiferro-g co -"' _{magnetic coupti.ng} E LJ ]1/k.B = -1 K. 02 10 15 T!Kl

c . '2 c l: c E

..

c

""

"'

E '-.t::; 0 ~ l : c E

..

c

""

"'

E '-.t::; 0 CCI ... f -c' .'2 c l : >< 3 20 B(T) 30 30 150 Fig. 3.4

Magnetization per Mn-ion

for the cluster types of

the simple

NN

cluster model, using S

=

512,

g = 2, an

nntiferromagnet-ic coupling _J11hB = -1 K

and T = 4.2 K.

Fig. 3.5

Inverse susceptibility for

the cluster types of the

simple

NN

cluster model,

using S

=

512, g

=

2, an

The approximation described above can, in principle, be used to calculate the thermodynamic properties of diluted magnetic systems in which the nearest-neighbour interaction is very large compared wi th the further neighbour interactions. If, however, the further neighbour interactions are not negligible, i t is possible to extend the above approximation to further neighbour interactions using a similar

approach.

The extensions up till the next- and third-nearest-neighbour interactions will briefly be considered in the next section.

3.3 Extensions of the NN isotated cluster model

We will first describe the extension of the cluster approximation to the next-nearest-neighbour (NNN) interaction [2]. In Table 3.4 the

Cluster type single s2

_•

NN pair pl

...h...

NNN pair p2 J2 I NN open triple Tl J1 I Jl I 0 NNN open triple T2 I J;;: I J;;: I 0 NN-NNN open triple T12 I J~ I J2 I 0 NN closed triple Tl

~1

c NNN closed triple

~

~2

c NN-NN-NNN closed triple T112

~1

c NN-NNN-NNN closed triple T122

~2

c

Table 3.4 Types of clusters, assuming a NN interaction 1₁ and a NNN interaction 1₂ between the magnetic ions.

Cluster type ( 1-x) 1B 6x( 1-x) 26 12x(l-x) 28 bee ( 1-x) 14 8X(l-X)lO 6x(l-x) 22 9x2 ( 1-x) H 4Sx2 ( 1-X) 26 1Sx2 ( 1-x) 37 _[ 2+5 ( 1-x)) 9x2 ( 1-x) 29 [ 4+ (1-x)) 72x 2 (1-x) 35 [1+(1-x)] 72x 2 (1-x) 28 24x2 ( 1-x) 35 l6x 2 (1-x) 32 - 0 J6x 2 (1-x) 25 tee ( l-x)18 12X(l-x) 26 6x( 1-x) 30 90x2 ( 1-x} 34 9x2 (1-x) 41[4+(1-x)) 72x2(l-x)37 (1+(1-x)] 24x2(1-x) 31 hr:p ( 1-x) 18 12X( 1-X) 26 6x( 1-x) 30 9x2 ( 1-x) 33 I 1+9( 1-x) I 9x 2 (1-x) 39 [ 2 ( 1-x) 2 •2( 1-x) 1 •11 18x 2 (1-x)J7(l•S(l-x) I 3x2 ( 1-x) 29 ( 1•6(1-x) 2+ ( 1-x) 4 J

Tabte 3.5 ProbabiLity that a magnetic atom is in a singte, pair, or

tripte duster (assuming NN cmd. NNN interact ion [2]).

Fig. 3.6

Probabitities of the

cLuster types of the NNN cLuster modeL for

the fee structure.

2

is the totaL

probabi-Lity of these cLuster

different cluster types (single, pair and triple configurations) are shown. These are defined in the same way as in the farmer section. It is obvious that the number of types (and Hamiltonians) has increased considerably with respect to the NN cluster model. The probabilities that a spin belongs to a certain cluster type are shown in Table 3.5.

In Fig. 3.6 these probabilities are plotted fora fee lattice. One of the most important effects of including exchange interactions between next-nearest-neighbours is the reduction of the probabi 1 i ty of sin-gles. For x

l

0.03 more than 10% of the ions are not included in the clusters of three or less spins and hence for concentrations above

x "' 0.03, in principle, clusters of four or more spins should be taken into account.

The next steps in this approximation are similar as those in sec-tion 3.2. For this reason we continue here with the extension of the

cluster model up till the third-nearest-neighbour (3thNN} interac-tion [3]. The possible cluster types up till three spins are defined according to Table 3.6. Okada [3] calculated the probabilities that a spin belongs to one of these configurations for a fee lattice. These

probabilities are plotted in Fig. 3.7. It is obvious that already for concentrations above x "' 0.01, more than 10 % of the spins are not included in the cluster types up till at least three spins. It is then

- in principle - necessary to include the cluster types of four or more spins. Since, however, the number of different configurations of these higher order clusters increases enormously, the calculations become rather tedious. For this reason Okada [3] proposed to approxi-mate the higher order clusters by the weighted average of the pair and

triple clusters. Within his approach the contribution of the pair and

triple configurations {i} are multiplied with a factor

K

K; {1- P83)/

2

P ..

i(P,T) 1

(3.8)

In Table 3.3 the Hamiltonians for the different cluster types and their eigenvalues are shown. For the mixed open triples T~ and the mixed closed triple T!23 no analytic expression for the eigenfunctions can be obtained. The eigenvalues of these Hamiltonians have been

Cluster type single 83

•

NN pair pl

...h....

NNN pair p2

_•

Jz

•

3thNN pair p3 J3 NN open triple Tl Ji

_•

Jl

_•

0 NNN open triple T2 Jz

_•

Jz

_•

0 th triple T3 J3 J3 3 NN open 0

NN-NNN open triple T12

_•

Jl JOl

_•

0 NN-3thNN open triple T13 Jl J3 0 NNN-3thNN open triple T23 Jz J3 0 NN closed triple Tl

J~l

c NNN closed triple

~

J~2

c 3thNN closed triple T3

_J~3

c NN-NN-NNN closed triple T112

J~l

c NN-NN-3thNN closed triple Tll3

J~l

c NN-NNN-NNN closed triple Tl22

J~2

c th th . T133

_J~3

NN-3 NN-3 NN closed tr1ple c

NNN-NNN-3t~N

closed triple T223

J~2

c th th . T233

_J~3

NNN-3 NN-3 NN closed tr1ple c NN-NNN-3thNN closed triple T123

J~z

c

Table 3.6 Types of clusters, assuming a NN interaction ]1 , a NNN

a.. 0.001 0.01 x 0.1 Fi.g. 3.7 Probabitities of the

cLuster types of the

3thNN cLuster modeL for

the fee structure.

2 is

the totaL probabitity of

these cLuster types.

obtained numerically. The thermadynamie properties can be calculated in the same way as in the case of the simple isolated cluster model (Eqs. 3.3- 3.6). In Figs. 3.8 and 3.9 representative results for the specific heat and the magnetization per rnagnetic ion for the relevant cluster configurations in a fee host lattice are shown.

The total contribution is obtained by summing the contributions of these cluster configurations according to their probabilities. In Fig. 3.10 we plotted the total magnetic specific heat of a diluted magnetic system calculated with the simple NN cluster model (J1 ~ 0,

J)₁ = 0) and with the 3thNN cluster model (J_{1 _2 , 3}

~ _

0), both fora concentratien x = 0.01. The difference between these two

c 0 I c: ::;:: ... co .x 0.6 TIK)

Fig. 3.8 Specific heat per Mn-ion in zero magnetic field of the cluster types of the 3thNN cluster modeL using S

=

512,

g

=

2 and an antiferromagnetic coupLing J_11kB

=

-6.3 K,

J_21kB

=

-1.9 K and _J31kB

=

-0.4 K.

In principle, it is still possible to extend the cluster model by including also 4th. etc. neighbour interactions. From the calculations outlined above, however, it is obvious that by going from the simple NN to the 3th neighbour cluster model the number of different cluster types and thei r Ham i 1 tonians increase cons iderably. Especially the calculation of the probability that a magnetic ion belongs toa cer-tain cluster type becomes rather complex. For this reason we will not extend the cluster model to further neighbour interactions. In the next two sections we will consider two other approximations, in which the long-range character of the interaction can be included.

'

.

:k

':b

~

c

0 I c _1.5 :L c 0 "lïi c Cl

"'

E '-.<= 1.0 0 co :L 0.5

Fig. 3.9 Magnetization per Mn-ion of the cLuster types of the 3thNN cLuster modeL, using S

=

5/2, g

=

2, J_11kB

=

-6.3 K,

0.15.---rl- -- - - . - - 1 - -- - , - 1 - - - ,- ---,

---/ ... / ... /

'

/

'

-I '..._..._ § 0.10r- 1 ..._ -.._--I I c ::!: co E 0 I I 0.5 1.0

---

_---

-I 1.5 2.0 2.5

Fig. 3.10 Comparison of the totaL specific heat per Mn-ion in zero

magnette fieLd for the simpte NN cLuster modeL (soLid

th

curve) and the 3 NN cLuster modeL (dashed curve). We used

a diLuted magnette system with fee structure, x= 0.01,

S

=

5/2, g

=

2 and antiferromagnetic interactions 1,,

12

=

0.3

1,

and

13

=

0.063

1,

.

3.4 Nearest-neighbour pair correLation approximation

The pair correlation approximation has been introduced by

Matho [4] in order to calculate thermadynamie properties of canonical

spin-glasses with the long ranged RKKY-interaction.

The pair correlation approximation resul ts from a single, but

rather drastic assumption: given a fixed, random distribution of

spins, the partition function of the macroscopie system may be

'nearest magnetic neighbour' may cause some confusion since in the preceding sections we used the term nearest-neighbour in a quite dif-ferent sense. Nearest-neighbour is used for the nearest-neighbour lattice si te, whereas the nearest magnetic neighbour is the nearest site where a magnetic ion is located.

We have extended the pair correlation approximation by taking into account one kind of triple correlation: the contribution of the spins which have two nearest magnetic neighbours at the same distance (and thus the same interaction) is calculated separately. The sites of the - -erys-ta-1-l-i-fie--hos-t--struc-tu-re-are ~rranged--i-n-s-lre-l-1-s-at--d-i-stances-

-R {v

=

1,2,3, ... } around the reference site; the vth shell contains V

Nv

sites. Using n V V

2N

1 V for v ) 0 and n 0 0,

the probabi 1 i ty of finding a nearest magnetic neighbour in shell, for the random case is

nv-1 n

Pv(x)

=

(1- x) - (1 -x) v v ~ 1.

(3.9)

the V th

(3. 10)

The probability of finding a spin with two nearest magnetic neighbours in the same shell (spin in a triple) is

(3.11)

The probability that a spin belongs to a pair is taken as

(3.12)

One should note that in this probability Pp also higher order clusters V

are included. In Fig. 3.11 the resulting probability distribution of pairs and triples for the lattice of (Cd 1_xMnx) 3As2 and x

=

2.6% is given as an histogram. In appendix A

Nv

and nv for some other lattice

.D ro .D 0 '-0. 20.---. ---.---.---. 0 ((d1-x Mnxl3 As2 X= 2.6% 15 20

Ftg. 3.11 Probabtttty of ftndtng a nearest magnette netghbour in

shett u [pair, Eq. 3.12] and probabittty of ftnding two

nearest magnette netghbours in shett u [trtpte, Eq. 3.11], represented by the bLank and shaded histogram,

respeetiuety.

structures are given. The probabilities of pairs and triples for other concentrations and for other host lattices can be calculated using Egs. 3.10-3.12.

The Hamiltonians for the pairs and triples defined above are given

by

(3. 13)

(3. 14)

where Ju = f{Ru) for v ~ 1, Rv is in units of the nearest-neighbour distance _R1(R1 = 1).

Table 3.3. The thermadynamie properties like specific heat, magnetiza-tion and susceptibility can be calculated for each pair and triple with Eqs. 3.3- 3.6. The thermodynamica! quantities of the system as a whole we obtained by summing all pair and triple contributions, i.e.,

"'[p

P T T ] C (x) = ~ P (x)·C /2 + P (x)·C /3 ,

m _u=1 v m,v v m,v etc. (3.15)

These calculations can be performed numerically. The summation over the shells (v) in Eq: 3.15 is carried out up till v = Ü fQL Yt.bich _ _

~

(Pp(x) + PT(x)

~

99.5].

V=l V V

For x= 2.6% and (Cd1_xMnx)3As2 the value of v amounts to 17. For

other concentrations and/or other host lattices Ü may have a different

value.

The question may be raised to what extent the extended nearest-neighbour pair correlation approximation (ENNPA) gives correct results for the thermadynamie properties of a diluted magnetic system.

Accord-ing to Matho [4] the pair correlation approximation can be applied in the .. paramagnetic" domain. Tentatively he estimated that this domain is limited in the space (T,B,x) by the relation

(3.16)

This relation is obtained for the RKKY-long-range interaction, of

which the range depends as a power on the distance R (J ~ R-3 ). For interactions of shorter range, depending as a power on the distance R (J ~ R-n. n

>

3) xJ1 in the right term of Eq. 3.16 changes into

xn/3]1 • In case of a shorter ranged interaction and the samemagnitude

of J 1 the "paramagnetic" domain extends towards lower temperatures.

In the following part of this section we wil! illustrate the

ef-fects of the ENNPA for some magnetic systems which can be solved ex

-actly. We start with the comparison of the result of the ENNPA with the exact salution of an open triple configuration (Fig. 3.12). Ac-cording to the definition of the ENNPA the total specific heat of such

TIK)

Fig. 3.12 Comparison of the result of the ENNPA with the exact result for the specific heat in zero magnetic field of a magnetic chain of three spins (S

=

5/2, g

=

2 and ] 1/k.B

=

-1 K).

a magnetic chain is calculated as

cfNNPA m [ p

T p

J

cm.l + cm,2 + cm,3 13 · (3.17)

P T CP

where C _{1 . C 2 . 3 are the specHic heat per}

m, m, m, ion for ion 1,2,3

respectively.

In Fig. 3.12 we plotted the resul t of the ENNPA and the exact

solution. It is obvious that the error induced by the ENNPA is qui te

large in this special example. For other cluster configurations the

Matho [5] found that the differences on the level of individual clus-ters (between the ENNPA and the exact solution) are wasbed out after ensemble averaging. In order to illustrate this point, we campare the ENNPA with the extended cluster model, which was introduced in section 3.3. for a special diluted magnetic system. We assume a diluted mag-netic system, wi th concentration x = 0.01 and interactions J_{1 .} J₂ J 3 # _{0 and Jv>3}

=

0. Under these conditions the extended cluster model

015 c .~ _0.10 I c l: co -"' E u 0.05 0 0.15

..

r·/-,·"""'

/ / "·~ . / " 0.12

/ /

,,

'l'/

.

... ::::::.·.___

0 0.02 004 .

-·..::::::-

.

__

I

/

- · -

---.:::::::._ ~ -~~~~ 0.5 1.0 1.5 2.0 2.5

Fig. 3.13 Comparison of the specific heat in zero magnetic field

ob-tained from the 3thNN cLuster model (dashed curve) and from

the ENNPA (dashed-dotted curve), assuming a diluted

magnetic system with a fee Lattice, x = 0.01, S

g = 2 and antiferromagnetic interactions ]1 , ]2

5/2,

0.3•]1,

1~

=

0.063•]1 and Ju)₃

=

0. The solid curve represents the calcuLation of the specific heat with the ENNPA of the

mag-netic system mentioned above and ]v _{-425·J1 ·exp (-5.2} Ru)

for v ) 3. The inset shows the resuLts of this caLcuLation

yields accurate results. In Fig. 3.13 the results of the specific heat for both roodels are shown. The results of the ENNPA are 5-10% higher than those of the extended cluster model in the represented tempera-ture region. At still higher temperatures the results of the ENNPA and the cluster model are approximately the same. This comparison yields some confidence on the applicability of the ENNPA as in this case it gives satisfactory results even at a microscopie cluster level.

Originally the ENNPA was developed for a diluted magnetic system with long-range interaction. A cluster model cannot be used for such a magnetic system, as quoted already in section 3.3. To demonstrate the effect of a long-range interaction we extended, as an example, the magnet ie system introduced above wi th an interaction of the type Ju~ exp(-aR) for v

>

3. The result of the ENNPA for the specific heat is also shown in Fig. 3.13. It is obvious that the introduetion of the long-range interaction gives an additional contribution to the specific heat.

Finally, as a transition to the mean field approximation, which we will introduce in the next section, we note that from first

princi-ples [6] the zero field specific heat can be expressed as

For ~we take the general Hamiltonian

0 and a

Substitution of Eq. 3.19 in Eq. 3.18 yields

c

=

m

(3. 18)

x,y,z. (3.19)

(3.20)

enter this equation, which "roughly spoken" may explain the observa-tion that the pair approximaobserva-tion yields a meaningful result for the specific heat. In the mean field approximation (MFA) - where in the paramagnetic region [7] in the absence of a field no spin-spin

corre-lations are present - the specific heat in this region is zero. On the other hand, for the magnetization

(M =

~.<S~>)

and the susceptibility (X=

(l/kBT)[<(~.S~)

2

>-<~

.

S~>

2

]).

1 1 11 11

these single spin expectation values form the main contribution. Since these single spin expectation values are- within the framewerk of the

- --·· - meail---f.-i-e-l.d-app.r.ox.i.ma.t.i.on - eva luat.e.d_corr.ec-t.Ly~ _ _the_MEA__may he u sed .aa."s~---­ an alternative for the ENNPA in the analysis of our magnetization and

susceptibility measurements. This approximation will he introduced in the next section.

3.5 Mean fietd approximation [7,8]

In the mean field approximation {MFA) the interaction of a spin with the other spins is approximated by an internal field Bint defined as J{i -2 ~

J

.

.

S. •S. -~ B2 s~ - -~ _Beffz 8z _{i ·} j;o!i 1 J 1 J ext 1 (3.21) 2~

J ..

<S~> z _Bz z _B~ j;o!i 1J J

Beff ext + Bint' 1nt _~

In this equation B2 _{is the applied magnetic field. The introduetion}

ext

of an internal field decouples the individual spins and therefore neglects the effect of short-range spin-spin correlations. For this reason we used a model in which nearest-neighbour interactions are

still included in the exchange term of the Hamiltonian and only the

further neighbour interactions are included in the internal field.

With respect to the nearest-neighbour interaction, which we will treat exactly, we can, as an example, distinguish the same possihle cluster types as in the simple NN cluster model (section 3.2): singles, pairs,

open and closed triples. The probability that a spin belongs to such a cluster type was already given in section 3.2 for several host

lat-tices (Table 3.2).

Assuming the same effective field for all clusters, the corre-sponding Hamiltonians can be written as

2J S S

(s

z Sz. )Bz

- 1 i• j - ~ i+ J eff (3.22)

Xr·

0

-2J1(S.•S. + S.•R)- gu_ (S~ + Sz Sz)Bz

1 J 1 -k - tl 1 j + k eff

The resulting energy levels are tabulated in Table 3.3 for these clus-ter types. For each clusclus-ter type (i} <S~> can be obtained from

1 <S~> l

2:

m . . exp(-E . . ~T) j 1,J l,J

2

exp( -E. _.lk8 T) . j l.J (3.23)

where the summation runs over all Ei,j of cluster type (i}. B~nt can be calculated with

2 - - x

~ <Sz av ) j)l

2:

N. J J .. J (3.24}

<Sz ) is chosen as the weighted average of the expectation value <Sz>

av

N. is the number of neighbours with interaction J .. We like to stress

J J

that· the summatien in Eq. 3.24, in the case that we are dealing with a long-range interaction Jj' extends over the whole system and contains implicitly the distance dependenee of J(R). Eqs. 3.23 and 3.24 forma set of self-consistent equations, from which B7 and <Sz

>

can be

1nt av

found.

The mean field model, examplified above in combination wi th a nearest-neighour cluster model can also be used in combination wi th other models, in which part of the system is treated exactly or

ap-____ proxi mat i ve ly su_çh as_t.be~l'jj'fU ._Such_app.~:.ox.imati.onS-w.i-l.-1-be-emp-leyeà----

-in the chapters 5 and 6. Quali tatively one may conjecture that the results of such an approach will improve the theoretica! predietien of the model, since both the short range order effects and the average long-range interaction effects are taken into account. However, as we quoted above, the quant i tative effect of such a "mixed" approach de-pends on the speci f ie proper ty, the range of the interact i ons, the concentrations of the magnetic ions, temperature and external field, and therefore can hardly be discussed in general terms.

References

[1] R.E. Behringer, J. of Chem. Phys. 29, 537 {1958).

[2] M.M. Kreitman, D.L. Barnett, J. of Chem. Phys. 43, 364 {1965). [3] 0. Okada, J. Phys. Soc. Japan 48, 391 {1980).

[4] K. Matho, J. Low Temp. Phys. 35, 165 {1979).

(5] K. Matho, private communication.

[6] H.E. Stanley, "Introduction to phase trans i ti ons and cri ti cal phenomena" {Oxford: Ciarendon Press). 1971.

[7] J.S. Smart, "Effective field theories of magnetism··, W.B. Saunders Company {Philadelphia & London), 1966.

[8] G. Bastard, C. Lewiner, J. Phys. C: Solid State Phys. 13, 1469

CliAPfER 4 EXPERIMENTAL METIIOilS

4.1. Introduetion

In this chapter we will consider the preparatien of the crystals and the experimental methods used in the various experiments. We will first describe the method that we used to prepare the diluted magnetic semiconductors with a more or less controlled concentratien of substi-tuted magnetic ions and we will review the different techniques that we used to analyse the composi ti on or' these compounds. Next we wi 11 describe the experimental methods by which the magnetic behaviour of the compounds was investigated: specific heat, susceptibility withor without an external magnetic field, and magnetization.

All crystals were prepared by a modified Bridgman method. The appropriate stoichiometrie amounts of the pure elements were sealed. in carbon-coated quartz ampoules at a pressure of about 10-6 _{mbar. In the} case of (Cd 1_xMnx) 3As2 (CMA) the ampoules were heated up till about 750

°e,

_{which is above the melting point of Cd3As2 (721}

°e)

[1]. After a period of several days, long enough to ensure thorough mixing, the ampoules were slowly (20 mm/day) moved out through a temperature pro-file as given in Fig. 4.1. In this way we obtained good single phase crystals of CMA up till a concentration x = 0.12. For x ~ 0.04 the ingots often contained large single-crystals with a typical volume of 10-100 mm3_{. Above x= 0.12 small amounts of MnAs were also formed. As} MnAs is ferromagnetic up till 40

°e

[3], these smal! amounts could be easily detected with susceptibility measurements.

(Zn1_xMnx) 3As2 (ZMA) was also grown by a modified Bridgman method from the pure elements in a carbon-coated ampoule (pressure about 10-6 _{mbar). The ampoules were heated up till 1100}

°e,

_{which is above}

the melting point of _Zn3As2 (1015

°e)

[1]. The moving speed was also 20 mm/day. In this way we obtained single phase crystals up till

~l

ampoule ingot

~

lower 600 T(KJ 800 Fig. 4.1 Temperature profile through which the sche-matically shown ampoule is moued.

several cases large single-crystals, with a typical volume of 0.5-1 cm3 _{were obtained.}

For most experiments, such as, calorimetry, susceptibility and magnetization, polycrystalline samples are adequate. A single-crystal of ZMA, x= 0.15, with a volume of 1 cm3 _{was grown especially for use}

in neutron scattering experiments in the near future.

For the interpretation of the experimental results, which will be discussed later on, a proper chemica! and structural characterization of the samples is generally of crucial importance. X-ray diffraction, electron mieroprobe measurements and chemica! analyses were employed for this purpose. The results will be reported in the next section.

4.3. Analysis of the samples

A major element in the characterization of the samples is an accu-rate determination of the concentrat ion and the homogenei ty of the

sample. The concentration of magnetic ions in the samples has been determined in several ways. Depending on various factors, such as the expected impurities, the homogeneity and the host materials, spectro-scopie or chemica! methods were used.

As an important tool in the analysis, electron mieroprobe measure-ments were used [4]. An electron bombardment (kinetic energy of

10-30 keV) is used to generate X-rays in the sample to be analysed. From the wavelength and intensity of the lines in the X-ray spectrum the elements that are present can be identified and their concentra-tions can be estimated. In a quantitative analysis the intensities of the X-ray lines from the specimen are'compared with those from stan-dard samples of known composition. In this way the concentratien x of

the magnetic ions could be determined with a relative uncertainty Ax/x of 2 %, provided x> 0.01. Smaller concentrations could also be deter-mined, but the relative uncertainty then amounted to Ax/x "" 10 %. Since the electron beam is focussed on a spot of about 1 ~m2 _and pene-trates about 1 ~m in the material, the resul ting analysis is essenti-ally local. By changing the position of the beam along the sample, the homogeneity in the composition can be determined.

Since we preferred to have another independent methad to check the reliability of the electron mieroprobe measurements, the concentratien

0.06 _{Fig. 4.2}

(Cd 1 _x Mnx) 3 As2 _{Concentrat ion dependenee} of the ingot as a Func-tion of d. The sotid ingot

i~

tine represents Eq. 4.1

) (

'

with k.

=

1.2. x

=

0.045 0 ld and L ::: 40 mm. 0 10 40 d(mm)

was also determined by commercial chemica! analysis [5]. The uncer-tainty in the value of the concentration Ax/x amounts to 2 % in this case. The results of both methods were found to agree within their uncertainties.

From prohing the composition of the samples, especially by means of electron mieroprobe measurements, i t appeared that the samples were homogeneaus in the radial direction of the ingot. The results of mea-surements of the composition as a function of the vertical distance d from the bottorn to the top of the ingot for a _{(Cd 1}_xMnx)₃As₂ sample

- ----a-!'e-Show:n-in-E.i.g--~L-2-EI:.om -tbese-I:esul.t.s_.LL.is.-Ob.ll i m's tha.t_~h=..._. _ _ _ _

stantial variation of the concentration as a function of d occurs. This variation can be understood by the model descrihing the normal

freezing in the case of partially mixing in non-volati le 1 iquids, which is expressed by the law [6]

x(d) kx 0 (1 - d/2)k-1 • ( 4. 1)

where x(d) is the concentration of manganese as a function of d, k is the interface distribution coefficient, x the initia! value of the

0

uniform concentration of the solute in the liquid, and 2 is the total lengthof the ingot. The solid curve in Fig. 4.2 represents a fit of Eq. 4.1 to the data and appears to describe the measured variation in x very well.

The samples used for the experiments were always parts of the ingot. Guided by the results for x= 0.05, we determined the concen-tration and checked the homogenei ty of all samples separately. The samples used for the specific heat measurements were the largest, with a typical length of 5-10 mm along the vertical direction of the ingot. The variation in x over the samples in that case was still less than

Ax/x ~ 10 %.

To check whether any structural phase transition as a function of the concentration x occurs in CMA and ZMA, we determined the lattice constants for various values of x with X-ray powde~ diffraction. The results on CMA are shown in Fig. 4.3. The lattice constants a and c of CMA decrease linearly with x, which can be expected from Vegards'

I 25:3 25.2 cXi 12.5-I I 0 005 0.1 0.15 0.2 x

Fig. 4.3 Lattice constunts a and _{c of (Cd1 _xMnx)3As2 as a function of}

the concentration x, obtained from X-ray powder diffraction.

law [7], and the smaller bonding lengthof Mn-As compared with that of Cd-As. Our results for the lattice constants agree very well with the reported results [2].

In Fig. 4.4 the results on ZMA are shown. A smal! linear increase of a and c as a function x is observed. The change in the lattice

constants is smaller than in the case of CMA. Linear extrapolation of a and c of both CMA and ZMA towards x 1 gives the same values for both compounds. which can be expected from Vegards' law as for bath compounds the endpoint x

The results for the lattice constants do not indicate any struc-tural phase transition. As the change in the lattice constantsas a function of x is less than 1 % for x ( 0.2, we did not correct the

118

"'

0

~

--

---~-- ---0.1 x 23.6 23.5 0.2

Fig. 4.4 Lattice constunts a and _{c of (Zn1 _xMn}x_{)3As2 as a function of}

the concentration x, from X-ray powder diffraction.

4.4. Calorimetry

Two calorimeters were available for specific heat measurements on polycrystalline samples or single-crystals. One calorimeter did cover

the temperature range 400

mK

<

T

<

3 K, another the range

1.5 K

<

T

<

50 K. Bath calorimeters are equipped wi th a superconduc-ting solenoid to provide additional external magnetic fields up till

1 T in the low temperature system and up till 3 T in the high

tempera-ture system. The basic construçtion of both calorimeters is largely similar, and hence we will only describe the system for the high tem-perature region.

The calorimeter is schematically drawn in Fig. 4.5. The outer part of the apparatus consistsof a stainless steel vacuum chamber1 , hav1ng a diameter of approximately 60 mm. The calorimeter is immersed in

.

I

Fig. 4.5

Schematic view of the specific heat equipment.

DetaiLs are giuen in the text.

1

2

3

4