Carrier induced magnetic interaction in the diluted magnetic semiconductor PbSnMnTe

Carrier induced magnetic interaction in the diluted magnetic

semiconductor PbSnMnTe

Citation for published version (APA):

Jonge, de, W. J. M., Swagten, H. J. M., Galazka, R. R., Warmenbol, P., & Devreese, J. T. (1988). Carrier

induced magnetic interaction in the diluted magnetic semiconductor PbSnMnTe. IEEE Transactions on

Magnetics, 24(6), 2542-2547. https://doi.org/10.1109/20.92168

DOI:

10.1109/20.92168

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Published: 01/01/1988

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2542 IEEE TRANSACTIONS ON MAGNETICS, VOL. 24, NO. 6, NOVEMBER 1988 Carrier Induced Magnetic Interaction in the Diluted Magnetic Semiconductor PbSnMnTe

W.J.M. de Jonge, H.J.M. h g t e n

Department of Physics, Eindhouen University of Technology, 5600 MB Eindhouen, The Netherlands

R.R.

GalazJza

Institute of Physics, Polish Academy of Sciences,Al. Lotnikow 32/46, 02-668 Warsaw, Poland P . Warmenbol. J.T. Deureese

Department of Physics, Uniuersity of Antwerp, Uniuersiteitsplein 1 . B-2610 Wilrijk-Antwerpen, Belgium Low temperature ac susceptibility and specific heat measurements have been performed to study the influence of the concentration of charge carriers on the ferromagnetic phase transition of Pb,-,-ySnyMn,Te for various compositions ( 0 . 0 0 5 < x < O . l . 0.4< y i 1 . 0 ) . A critical density of carriers (pcrit) above whlch a fexomagnetic transition can take place is observed. This behaviour is also reflected in the Curie-Weiss temperature 8 (probing

Bj

. ) , obtained from the high temperature susceptibility data.

A simple modified RKK$ mechanism for semiconductors is proposed in which carriers from two valence bands located in different regions of the Brillouin zone contribute. Also the effect of a finite mean free path of the carriers is taken into account. On the basis of this new model an excellent agreement with the data is obtained without the use of any fitting parameter.

I . Introduction

Diluted Magnetic Semiconductors (DMS) or

Semimagnetic Semiconductors (SMSC) have attracted considerable attention during the past years. The fundamentally as well as technologically interesting and characteristic features of DMS are directly related to the coupling between the two interacting subsystems: the electronic system of the carriers and the (diluted) magnetic system of paramagnetic ions [ l ] . Most of the investigations have been devoted to the class of the 11-VI compounds diluted with Mn. such as Cd,-,Mn,Te, Zn,_,Mn,Te, Hg,_,Mn,Te and corresponding selenides and sulfides although also compounds like (Cd,_,Mn,),As, and (Zn,-,Mn,),As, have been studied [2]. Some recent results are devoted to Fe-based

DMS

[3].

As for the semiconducting properties, striking effects were observed such as: a giant faraday rotation, anomalous and large magneto-resistance

effects as well as significant bandstructure

modification in moderate magnetic fields [ 4 ] . Generally spoken, these effects can be traced back to the field enhancement and polaron formation brought about by the interaction between carriers and magnetic ions.

Magnetically the behavior of these systems originates from the interactions between the magnetic

ions. Typically these interactions are

antiferromagnetic (AF) and long ranged (with

significant and systematic variations in range depending on covalency or bandgap) [5]. A spin-glass phase is found at low magnetic ion concentration (Mn2+ as well as Fez+) extending to the very dilute limit. At higher concentrations sometimes AF ordered regimes are reported. Since the carrier concentrations is in general relatively low ( n . p <

lo"

cm-") a number of physical mechanisms involving different bands might be responsible for these interactions. A substantial number of papers have been devoted to this subject [SI. Recent theoretical results however, seem to indicate that, at least for the interactions between nearest neighbors, superexchange might be the dominant mechanism [7]. These nearest neighbor interactions cannot, however, be responsible for the observed spin-glass transition in the diluted limit which is claimed to be triggered primarily by the long range part of the exchange interaction [5,8].

As we quoted above the carrier concentration in the 11-VI DMSs is generally rather low. Several IV-VI DMS however exhibit a much higher carrier concentration in the range of 10"-102' caused by deviations from stoichiometry. In contrast to the 11-VI DMSs some of these systems, such as Sn,-,Mn,Te, Ge,-,Mn,Te and (PbCe),-,Mn,Te, show ferroimgnetic ordering fenomena, implicating a rather different exchange mechanism [9.10]. Furthermore a slight dependence of the ferromagnetic transition temperature T , with carrier concentration was observed.

Recently Story et al. [11] reported the

observation of a carrier concentration induced ferromagnetic transition in Pbo.2,Sno,,,Mno.o,Te (PSMT)

above a critical concentration pcrit. This striking demonstration of the influence of the carrier concentration on the magnetic properties is facilitated by the significant range of carrier concentration which can be realised in this system. A shift in the transition temperature

(T,)

was observed upon application of pressure on the system in the ferromagnetic regime and could be qualitatively understood on the basis of the Ruderman Kittel Kasuya Yoshida (RKKY) interaction mechanism [ l 2 ] . It was pointed out. however, that a redistribution of carriers due to the pressure dependence of the specific band structure should be taken into account. A RKKY interaction between localized moments, resulting from intra-band scattering of the charge carriers by the magnetic moments, seems to be the only interaction that is strong enough and sufficiently long ranged to explain the magnitude of the observed critical temperatures in these

IV-VI

compounds. The abrupt change in T , at the critical hole density pcrit in PbSnMnTe. however, cannot be understood by the original RKKY model, modelled for free (parabolic) electrons.

In this paper we will focuss on the behavior of the latter system. New experimental data on the phase diagram of PbSnMnTe will be presented and a model will be introduced which explains quantitatively the typical carrier concentration dependence of the Curie-Weiss temperature 8 and

T,.

In particular, we will show that the steplike increase of 8 at a critical hole density

(pcrit) can be understood on the basis of a modified RKKY interaction mechanism extended with a more realistic two valence band approximation and a finite mean free path for the carriers. With this modification a new extension for the applicability of the RKKY interaction in the diluted magnetic semiconductors is suggested.

The organization of the present paper is as follows. In section 2 and 3 the sample preparation and

results of specific heat and susceptibility

measurements on PSMT are presented. Samples with different compositions are considered and the hole density is varied over more than an order of magnitude.

A simple model, which is based on the RKKY-model and which contains the essential elements of the bandstructure of PSMT, is introduced in section 4 to describe the hole density dependence of the critical temperature in PSMT. Our conclusions are presented in section 5 .

11. Sample preparation

The PSMT samples were grown by a modified Bridgman method. Stoichiometric amounts of the pure elements were sealed under vacuum in carbon coated quartz ampoules, at temperatures between 810 and 920'C. depending on the composition. The composition and homogeneity of the samples were characterized by x-ray diffraction and micro-probe measurements. Single-phase compounds were obtained in all the investigated samples, i.e. with composition parameter x between

0018-9464/88/1100-2542$01.00@1988 IEEE

1

2543

observed. In particular, in Pbo.~oSno.,7t4no.o.Te a reduction of pcrit (= 1*1OZo cm-= ) , with respect to the systems with y=0.72. could be inferred from inspection of Fig. 2 , although a wider range of samples is necessary to obtain more pertinent conclusions.

0.10

-

a08

i 0.005 and 0.10 and y between 0.40 and 1.0. The ingots generally contained large single crystals with dimensions up to 10 mn. The Mn- as well as the Sn- (or Pb-) concentrations varied slightly over the ingot. The relative accuracy of the composition is therefore estimated as 10%.

The carrier concentration of PSMT was varied between 8.1Oi9 and 10" by isothermal annealing in a Te- or Sn-rich atmosphere: for the proper procedure we refer to Hewes et al. [13]. The carrier density in PSMT is temperature independent below 100

K

while for higher temperatures a gradual decrease has been reported [14]. This is corroborated by the Van der Pauw experiments we performed on

PSMT

at

T = 7 9

and 300

K.

Obviously, the figures for the carrier concentration quoted in this paper refer to the values at liquid nitrogen temperature.

111. Experimental Results a.c. susceptibility

The a.c. susceptibility (XaC) has been measured with a conventional mutual inductance bridge in the region 9 O < u < 9 O O O Hz. between 1.2 and 40

K.

The driving a.c. field typically amounts to approximately 0.5 C . 3 W ol E

-

.

3

I &m 2 v u 0. 4.0 OD28 1

0

-10 0 10

20

30 T [ K l

Figure 1 . The high temperature inuerse a.c.

susceptibility of Pbo.ze~xSno.7.JnxTe, for x N 0.03 and

uarious carrier concentrations. The insert shows the

Curie-Weiss temperature 8 as function of the Nn

concentration in the ferromagnetic regime

(p

>

3.1OZ0 m - " ) .

The high temperature part of

xac.

at low Mn

concentrations corrected for the diamagnetic

contribution of the host material, obeys a Curie-Weiss law for all the investigated samples. at temperatures T > l O

K;

some examples are shown in Figure 1. From this plot of the inverse susceptibility we may estimate the temperature 8 which is proportional to the average exchange interaction between the Mn ions. Figure 2 displays the carrier concentration dependence of 0 supplemented with the earlier results of Story et al. [ll]. For comparison the data have been scaled on x = 0 . 0 3 anticipating on the linearity of 8 with x , see insert of Fig. 1. Above p c r i t ? 3 * 1 O z 0 cm-= an abrupt, almost steplike increase to positive 8 is observed. indicative for a net ferromagnetic interaction. Below pcrit 0 is considerably smaller and within our experimental accuracy we cannot distinguish between a zero or slightly negative 8 ( ~ - 0 . 1

K).

The data seem to reflect a rather remarkable universal dependence on the carrier concentration, irrespective of the appreciable variation in the composition parameters. Nevertheless. it is not excluded that a small shift of pcrit with composition parameters x or y may be

5l- 0

-I

3 pb,-x-ysny MnxTe

-

X Y

-

0

o

0.03 0.72

m 0.005

0.72

1

v

0.09

0.72

-

+ 0.03 0.64 x 0.03

0.47

I I

0.05 0.72

Figure 2. The Curie-Weiss temperature 0 scaled on

x = 0.03 of Pb,-.ShyMn,Te as function of the carrier

concentration p, for seueral compositions. The black

circles are data tciken from Story et al. [11]. The

solid line is a guide to the eye.

-

-? 0.06

z

-

x- 0.04 a02 0 O.WO3 0.0002 4.0.1020 0 P 0 P

sp

0 X I I I

(bq

X' 0.1 2 Pb0.25 '"0.72 Mn0.03Te a, , i 2.3 G

HR0@?

Ea,= 0.2 G 0 .* . O o o / o o o o B 1 2 3 4 5 T I K I

Figure 3. Low temperature a.c. susceptibility of

Pbo.zs~xSno.7zMnxTe for x

=

0.03. The upper p r t (a) shows the dependence of

x '

on the hole concentration taken with a driuing field

B,,

= 0 . 5 G ; the ,Lower part shows both components (X' and X") of the susceptibility

and illustrates the dependence o n the driving field

Bac. The snuzll temperature shift of

T,

between the two

figures is caused by a slightly different hole- and

Mn-concentration.

- ._

2544

The a.c. susceptibility diverges in the systems with high carrier density (p)pCrit. the ferromagnetic regime) at a temperature T, a few tenth of a Kelvin below 0. After reaching a maximum. comparable with the inverse demagnetization factor (l/N),

Xac

decreases more or less gradually and tends to zero at the lowest temperatures. Some representative results are shown in Fig. 3 . The shape of

Xac,

especially below the anomalous divergence at the transition temperature, is similar to the observations of Escorne et al. [9] on Sn,-.Mn,Te with x > 0 . 0 3 and looks like a reentrant

ferro-spinglass phase rather than a typical

ferromagnet, in which saturation to 1/N is expected. Preliminary experiments yield also strong irreversible

effects, reflected by a nonzero imaginary

susceptibility and the dependence of

Xac

on the driving field. However. also anisotropic magnetic exchange might be the driving force behind the typical shape of the susceptibility in PSMT.

It is obvious that a further analysis of the low temperature magnetic phase of PSMT might be relevant. However, in this paper we confine ourselves further to the interpretation of the intriguing carrier dependence of

Tc

and 0 . So. a detailed analysis of the observations as examplified in Fig. 3 is not in the scope of this article and will be published elsewhere. Specific heat

performed with an adiabatic heat pulse calorimeter in the range 0.4<T< 10

K.

The magnetic contribution C? to the specific heat has been obtained by subtraction of the lattice contribution of PbSnTe and the nuclear hyperfine contribution of the Mn ions.

The specific heat experiments have been

I I I I I

o M3.0.72.7*iOrn cm.]

0 2 4 6

T I K I

Figure 4 . The zero fieLd magnetic specific heat Cm of Pbl-,-ySny~n,Te f o r uarious compositions and hole concentrattons.

Some representative results for the magnetic specific heat of PSMT are shown in Fig. 4. In the ferromagnetic regime there is a magnetic transition at helium temperatures, in excellent agreement with the position of the divergence of the susceptibility. Again, below the transition temperature the data do not

resemble a good ferromagnet. Instead of an

exponentional, concave low temperature part. Cm of PSMT varies somewhat convex, almost linear. Analogous observations have been reported on the dilute magnetic alloy PdMn [15], which exhibit mixed ferro-spinglass properties for Mn concentrations roughly between 0.02 and 0.05. Within the experimental temperature range no anomalous behavior has been observed so far in PSMT with low carrier concentrations (p<pqrit). The dominant magnetic contribution, to be associated with the average interaction energy. is shifted to very low

temperatures with respect to the systems with high hole density. This is consistent with the tendency in the susceptibility as quoted above.

IV.

Model calculation

In order to clarify the abrupt transition to a ferromagnetic phase as the hole density exceeds a critical density, a simple model is introduced below. We consider a system of localized magnetic moments, the Mn ions in the matrix PbSnTe. interacting with the free holes. The following approximations will be made: 1. intra-band

RKKY

interaction between the magnetic 2. mean-field approximation for the spin system, 3. for the free holes an finite mean free path is

assumed,

4 . representation of the complicated bandstructure of PSMT by a set of two parabolic valence bands: one located at the L-point of the Brillouin zone (VB1) and one located along the >direction (VB2). The top of VB1 is higher in energy than the top of VB2 and

ml*, the effective hole mass of VB1 is much smaller than m2* of VB2; see Fig. 5.

moments,

Pb,

,Sn,Te

\ r

V B I

----

-

-

AvB/\

rn;=0.05me

Figure 5. Schematic illustration of the band structure

of Pb,-,Sn,Te as used in the model calculation described in the text.

Point ~ 1 includes the assumption of isotropic

scattering by a deltafunction type potential (Fermi contact interaction). The attempts to improve this approach [16] however make an analytical treatment of the models difficult and necessitate a numerical solution. In point 1 also the validity of second order perturbation theory is assumed, although for PSMT the exchange constant J might well be of the same order of magnitude as the Fermi energy ( e ~ ) of the free holes. The non-Heisenberg nature of such a system is manifested by a non-analytic dependence of the critical temperature on J . as was recently pointed out by Nagaev [17] for the case of EuO and EuS with However, Mauger and Godart [18] showed that the convergence of the corresponding Wigner-Brillouin expansion is fast enough to justify the truncation to terms in second order, while the Raleigh-Schrodinger expansion is not necessarily convergent. The mean field approximation of point 2 amounts to neglecting the dynamics of the spin-system entirely. The RKKY interaction was originally modellled for metals. The application to semiconductors requires at least two essential modifications. Firstly, the mean free path of free carriers cannot be taken to be infinite for a semiconductor (point 3 ) . Secondly, the detailed bandstructure of a semiconductor (point 4) might play a decisive role. The bandstructure of Pb,_,Sn,Te is taken as a reference model for the bandstructure of PSMT. It is assumed that the presence of Mn ions does not significantly alter the bandstructure (see ref. 1). We take into account only the highest two valence bands according to different bandstructure calculations [19]

2545 for SnTe and Pb,...Sn,Te with x>0.4. Moreover

nondegeneracy may lead to finite temperature effects [18]. Here we take the degenerate limit ( T = O

K).

Although the assumptions we made are rather crude, we believe that this simple model contains the relevant characteristics of the magnetic subsystem as well as the hole subsystem. Within our' simple model the Curie-Weiss temperature, for a finite mean free path, is given by

J:ff(Rj)

is the exchange integral, which can be written as:

with

and z . ~ = ~ J z F ~ R . where kFi is the Fermi wavevector for the ?-th band. R . is the distance between two lattice points in the FCC fattice (R,=O and

R l

is the nearest neighbor distance). Ji is the exchange constant for VBi. me the electron mass and a,, is the lattice parameter. The finite mean free path (A) is introduces by inserting the exponentional term exp(-RjA) in (lb) as proposed by de Gennes [20].

Applying the model to Pb~.z&no.72Mno.oaTe requires a choise of the parameters to be used in the calculations. For the lattice constant we have taken a, ~ 6 . 3 4 7 as estimated from linear interpolation between the PbMnTe and SnMnTe data. For the effective masses we have adopted the values given by Ocio [14] for Pbo.5aSno.4,Te: m,*=0.05nce and m2*=l.iXme. The exchange constants Ji and J 2 for the two valence bands are not known. The experimental exchange integrals for PbMnTe and SnMnTe. interpreted as arising from a single band, are reported to be 0.07 eV and 0.40 eV [21], respectively. In a first approximation we have taken the two interactions identical and equal to the

V B I

( A =

W )

0

4

8

12

p 1020cm-31

Figure 6. The contribution of VBl and VB2 to the scaled Curie-Weiss temperature e(O.O3/x) as function of the hole Concentration p assuming t h a t only one band is populated. The IWM free path is tahen as infinite.

interpolated value J l = J 2 = 0 . 3 0 eV. It is, however, not excluded a priori that the exchange integrals should be different for both valence bands [22].

The calculated Curie-Weiss temperature 8 as a function of hole density, according to Eq. (1). in the assumption that only one of the two valence bands is occupied is shown in Fig. 6. The dashed line corresponds to the case when only

VB1

is filled (the highest band in the L-point). The full line is for

VB2

(the highest valence band in the >direction). For a fixed hole density, the Curie-Weiss temperature for VB1 (0,) is much smaller than the corresponding O2 for VB2. The ratio between 8% and OZ is roughly equal to the ratio between the masses mi* and m,*. which is obvious from Eq. (lb) where Jieff(Rj) depends linearly on the effective hole mass. It is precisely this large difference in 8 between the two valence bands which, in this simple model, results in a steep increase of the effective ferromangetic interactions when the hole density is increased above pcrit and the Fermi level enters VB2 (see Fig. 5). From a two-band Kane model [14] with additional parameters -0.15 eV and AEvB=0.6 eV (see Fig. 5) a critical%p- o e concentration

is obtained This value is

consistent with our observations as well as the values reported [23] from the the temperature dependence of the Hall factor. which vary between 2 and 3*1OZ0 cut-=.

f"

h = 1 2 A

/ /

1 -

I I I I

1

0

4

8 12 16

p

[ 102ocm-3

1

Figure 7. The scaled Curie-Weiss temperature e(O.O3/x)

as calculated from eq. 1 , c o m p r e d with the data. The lines correspond to the contribution from both valence bands simultaneously for a realistic mean free path as indicated by the mobility.

Fig. 7 displays the Curie-Weiss temperature as a function of hole density, as calculated from Eq. (1) with contributions from the two valence bands simultaneously. The calculations were performed for h = m as well as for some realistic values of h in the range indicated by the experimentally observed mobilities (p"50 c m 2 N s ) .

Comparison of the calculations with the data in Fig. 2 shows that an excellent agreement both qualitatively as well as quantitatively can be obtained. The observed step-like increase of 8 to ferromagnetic values at a critical hole density is correctly reproduced, as well as the magnitude. For

p < p C f i t the present model gives a small but non-zero positive 0 (typically 0.2

K).

while the experimental 8

2546

is smaller if non-zero at all. The reason for this discrepancy is not clear at present. There are two effects which candidate to explain this discrepancy: 1) the additional contribution of direct AF exchange as in the case of 11-VI semimagnetic semiconductors [7] and 2) the contribution of inter-band [24] RKKY interaction, which may be antiferromagnetic o r

ferromagnetic, depending on details of the

bandstructure [25] and on the mean free path [20,26]. Since PSMT is a small gap semiconductor and since the inter-band RKKY exchange energy roughly goes like candidate is likefy to be the best one.

[24,25] ~X~(-R.[~€G(~CB*+~"B*)]-"') this last

V. Discussion and conclusions.

I t is gratifying to conclude that a rather crude and simple two band RKKY model is capable of explaining one of the most intriguing features of this quartenary alloy, i . e . the carrier induced ferromagnetism. We have to stress however at this point that, although no fitting procedure has been applied, the quantitative agreement. specifically in the ferromagnetic regime, may be somewhat fortuitous. The absolute magnitude of 8 in that region depends also on the value for the p-d exchange constants J l and J 2 , which in a first approximation we assumed to be equal to 0.3 eV. A further verification of the validity of the model can possibly be found in an extension of the composition range (Pb-Sn) which would yield a slight shift in pcrit brought about by the variation in AEvB.

PbMnTe and SnMnTe can be considered as the limiting cases in the composition range of PbSnMnTe. On the basis of our model the contrasting magnetic behavior of these compounds might be understood from the difference in hole concentration. Sn,-,Mn,Te, with x>0.03 and Q-5.1OZo ~ m - ~ , was reported to display a ferromagnetic transition [27]. while Pb,-.Mn,Te. with p <

lozo

cm-3, is paramagnetic down to very low temperatures where recently a spinglass transition was observed [ 2 8 ] . Moreover the reported 0 values for Sno.966Mno.03,Te (p=5.1OZ0 ~ m - ~ ) as well our own results for Sn,. 9 ~ ~ M n o . ozgTe (p = 7.1OZ0 ~ m - ~ ) closely match the quasi-universal curve shown in Fig. 2. We expect therefore that a reduction of the hole density in SnMnTe (if possible) would also transfer the system to a paramagnet. The same mechanism might also be responsible for the substantial decrease of the (ferromagnetic) 0 in (Gel~x~yPby)o.~Mno.lTe with increasing Pb content. partlcularly since it was reported to be accompanied with a simultaneous decrease of the carrier concentration

[lo].

The RKKY d-d exchange interaction which is shown above to be effective in PSMT for p > p C r i t is in general a radially oscillating function decaying with R-3 and changing sign with a typical periodicity of ~ / 2 k ~ . see Eq. 1. This alternating behavior gives rise to the existence of a critical concentration of Mn, above which the short-range ferromagnetic interactions outweights the oscillatory behavior. In the more dilute limit competition might be expected giving rise to a spin-glass behavior. For carrier concentrations in the range of 10" this critical Mn concentration for the structure of PSMT can be estimated as 5%. So far no direct evidence of a spin-glass transition has been obtained down to ~ ~ 0 . 0 3 . However, the hysteresis effects in the susceptibility for low concentrations strongly supports the existence of such a transition, which is currently under investigation.

Concluding we would like to mention that the essential result of the present study, i . e . the explicit

demonstration of the relevance of separate

contributions from carriers in different regions of the Brillouin zone, might also be relevant for alternative interaction mechanisms which have been proposed f o r other classes of diluted magnetic semiconductors.

Acknowledgements.

Acknowledgements are due to C. v.d. Steen. H. v.d. Hoek, H.J.M. Heyligers and A. Szczerbakow for their experimental assistence and sample preparation. Part of this work was supported by the Stichting voor Fundamenteel Onderzoek der Materie (FOM) that forms part of the Netherlands Organization for the Advancement of Research ("0). c11 c21 c31 c41 c5i C6l c71

C81

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