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Magnetic properties of (Ge, Pb)1−xMnxTe
Solid State Communications, Vol. 32, PP. 1069—1073. Pergamon Press Ltd. 1979. Printed in Great Britain.
MAGNETIC PROPERTIESOF (Ge, Pb)1~Mn~Te T. Hamasaki* Department of Applied Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152, Japan (Received 22 June 1979 by I. Kanamori) Pseudotemary alloys {(GeTe)1_,(PbTe)~}1...~(MnTe)~ have been prepared and the magnetic and electrical properties of these alloys have been studied. The magnetic interaction in (PbTe)1_~(MnTe)~ is very weak, but it has been found that the {(GeTe)1~(PbTe)~)1_~ (MnTek alloys (y ~ 0.1) are ferromagnetic. In the latter, the increase of the amount of Pb substitution makes both the paramagnetic Curie temperature and the carrier density lower. These experimental results are discussed by using the RKKY interaction. THE ELECTRICAL and magnetic properties of dilute alloys have been studied by many authors [1]. The interaction between the magnetic impurities in these alloys is analyzed by an indirect interaction via conduction electrons, i.e. the Ruderman—Kittel—Kasuya—Yosida (RKKY) interaction [21. Recently, the interaction between magnetic ions diluted in semiconductive materisis has attracted much attention because their carrier density is easily controlled. Typical examples of this type are the pseudobinary alloys (GeTe)1_~(MnTe)~ [3—51and (SnTe)1_~(MnTe)~ [6,7]. It has been reported that these alloys order ferromagnetically, even though MnTe itself is antiferromagnetic. The ferromagnetic ordering of these alloys might be caused by the RKKY interaction. But the magnetic properties of (PbTe)1_~ (MnTe)~have not been reported, though the electrical [8] and optical [9] properties are known. In contrast 3 of GeTe and SnTe, with 1020_1021 carriers (holes) cm 7—lO’8cm3. Therefore, the carrier density of PbTe is 10’ the magnetic properties of (PbTe) 1_~(MnTe)~ might be different from those of (GeTe)1_~(MnTe)~ and (SnTe)1_~(MnTe)~. We investigate the pseudotemary alloys GeTe—PbTe—MnTe which are obtained by substituting Pb for Ge in GeTe—MnTe, in order to examine the carrier density dependence of the RKKY interaction in the dilute semiconductive alloys and also to clarify the difference of the magnetic properties between GeTe—MnTe and PbTe—MnTe. We prepared the samples {(GeTe)1_~(PbTe)~)1_~ (MnTe)~,where x is the mole ratio of MuTe in the samples andy the ratio of PbTe by which a part of GeTe in (GeTe)1_~(MnTe)~ is replaced, from high purity *
Present address: Physics Department, College of ~ Arts, Kyushu Sangyo University, Kashil, Fukuoka 813, Japan.
elements, Ge (five nine purity), Pb (six nine purity), Mn (four nine purity) and Te (six nine purity). The appropriate amounts of the elements were sealed in an evacuated silica tube, the inside of which was coated with carbon. This ampule was then heated above melting point for 24 hr and rapidly cooled to room temperature. The products were crushed and sealed again in an evacuated silica tube and reacted. This procedure was repeated several times in order to obtain a uniform Mn distribution. Finally, the obtained products were powdered and checked by X-ray diffraction and used for the magnetic measurements. We obtained homogeneous samples with x = 0—0.1 andy = 0—0.1 and 1. For the measurements of the Hall coefficient and electrical resistivity the powder samples were pressed in the form of a rectangular prism and sintered for a few hours at about 650°C.The Hall coefficient was measured at room temperature and liquid N 2 temperature. It was that the Hall coefficient depended very little onconfirmed temperature. The magnetic susceptibility was measured by means of Hirakawa-type magnetometer [10] from liquid N2 temperature to 350 K. We obtained the susceptibility of MnTe diluted in host materials by subtracting the diamagnetic susceptibility of the host materials (GeTe, GeTe—PbTe and PbTe) from the measured susceptibility. With the exception of PbTe, the diamagnetic susceptibility of these host materials was independent of temperature. Figure 1 shows the temperature dependence of the inverse susceptibility of (Ge0.95 Pb~)1_~ (MnTe)~.For the samples with low Mn concentration, the magnetic susceptibility follows the Curie—Weiss law well. But for the higher Mn concentrations the susceptibility deviates from the Curie—Weiss law near the Curie temperature 7~.According to Cochrane et aL [4], this deviation
1069
1070
MAGNETIC PROPERTIES OF (Ge, Pb)1..~Mn~Te
Vol. 32, No. 11
6emu/g )_1
l/Z9(l O
(Ge~Pboc
5)i..xMn~Te
1.0
X0.01
-
-
0.02 0
0-5
-
-
0.05 0.1 £
0
100
200 Temperature (K)
300
400
Fig. 1. The temperature dependence of the inverse magnetic susceptibility of(Geo.95Pb~)1_~MnxTe.
6emuIgT1
1/2~(1ci6emu!gT’
1 IZ9(1~Y
Pb
1~Mn~Te
XO1%J?’
2-
-0.2 0.015
0.02
1
-
-
0.1
0.05 0
0
0
I
100
I
200 Temperature (K)
1
300
Fig. 2. The temperature dependence of the inverse susceptibility of (PbTe)1_~(MnTe)~.
0 400
Vol. 32, No. 11
MAGNETIC PROPERTIES OF (Ge, Pb)1~Mn~Te
1071
8 (K) 1
Fig. 3. The dependence of the paramagnetic Curie temperature 0 on the amount of Pb substitution (y) of {(GeTe)1_~(PbTe)~}1_~ (MnTe)~.Squares show they dependence of 0 in x = 0.1, triangles x = 0.05, closed circles x = 0.02 and open circlesx = 0.01. from the Curie—Weiss law seems to be caused partly by a non-uniform Mn distribution because a very small hysteresis ioopT~. in B—H curves untilsamples 20 K higher than For the highwas Mnobserved concentrated with small y, we could observe the spontaneous mag. netization even at liquid N 2 temperature. The paramagnetic Curie temperature becomes higher as the concentcation of Mn increases. Figure 2 shows the temperature dependence of the inverse susceptibility of (PbTe)1_~(MnTe)~. For all samples, the susceptibifity follows the Curie—Weiss law well. The paramagnetic Curie temperatures are nearly equal to OK for all Mn concentrations within experimental error. l’his fact shows that the strength of the exchange interaction is very weak. The dependence of the paramagnetic Curie temperature 0 on the amount of Pb substitution (y) for the samples with constant Mn concentration x = 0.01, 0.02, 0.05 and 0.1 is shown in Fig. 3. As the measured susceptibiity of the samples with x = 0.01 was very small and the correction of the diamagnetic susceptibility of the host materials was large, the error of the paramagnetic Curie temperature was estimated to be ±10K. For x = 0.01, there is no y dependence of the paramagnetic Curie temperature. While, forx = 0.02,0.05 and 0.1,0 decreases asy increases. This means that the magnetic exchange interaction becomes weaker as the concentration of Pb increases. Figure 4 shows the y dependence
of the effective spin value for x = 0.01, 0.05 and 0.1. As y increases, the effective 2~. spin value approaches 5/2 of the spin valueatof Mn As even low Mn concentration, ferromagnetic ordering was observed, the interaction between Mn atoms is clearly long range. Now, we will discuss the experimental results with the RKKY interaction, which is the long range interaction via carriers between localized spins. We have to calculate this interaction by using actual distributions of localized magnetic spins and actual band structures of carriers. According to Lewis [11], the band structure of GeTe is roughly spherical and the free carrier effective mass can be de defmed. On the other hand, the Fermi surface of PbTe consists of four prolate spheroids centered at the L points [121. It is difficult to estimate the RKKY interaction taking into account this complex band structure. But the Pb substitution is low for almost all samples. Therefore, in the present paper, in order to roughly inspect the Pb concentration dependence of the RKKY interaction, we assume that the band structures are spherical. The usual damping type RKKY interaction [13] is written as ~ .111 = ~ J~dF(2kFR~J) e_RiJP~, (1) F 4 and n where F(x) = (sin x x cos x)/x 0 is the number —
1072
MAGNETIC PROPERTIES OF (Ge, Pb)1.~Mn~Te
Vol. 32, No. 11
S~f 2.5
x= 0.01 0.05
_____
~0.1
2.0r~~ (Ge1..~Pb~)1~Mn~Te 0
I
I
0.05
0.1
y
Fig. 4. The dependence of the effective spin value Set~on the amount of Pb substitution (y). Squares show they dependence of Sett mx = 0.1, trianglesx = 0.05 and open circlesx of conduction holes per a primitive cell, EF the Fermi energy, kF the Fermi wave number, X the mean free path of carriers, Jad the s—d exchange constant between the conduction holes and the Mn ions, and R,~the distance between the sites I and / occupied by Mn atoms. In the molecular field approximation, the parainagnetic Curie temperature 0 is expressed as 2S(S+1)J(0) = 3k~ (2) where kB is the Boltzmann constant. We will calculate J(0) in a f.c.c. lattice, because the rhombohedral distortion in the samples from a f.c.c. lattice is very small. We define z1 as the number of neighbouring lattice points at the distance R1 in a f.c.c. lattice, where R, = ..,/(J/2)a0 (j = 1, 2,. ) and a0 is the lattice parameter. If we assume a perfectly random distribution of Mn, it may be written that zrlattice points are occupied by the and 24/2m xz1—Mn atoms. Using h of carriers kF = (3ir2n)”3, where the m isrelations,EF the effective=mass and n the carrier density, we obtain the kF dependence ofJ(0) as follows; . .
J(O)
= ~
mJ~a~
51 2ir3h2 (2kFaO)4 ~I zJF(2kFRJ) ~
(3)
=
0.01.
Table 1. Values ofthe parwnagnetic (Jlsrie temperature 0, the Curie constantper gram Cg and the effective spin value per Mn atom Seff of(Ge1_~Pb~)0.9Mn0.jTe 4 Set 0(K) C5x10 1 _____________________________________________ 0 69 13.0 1.83 0.01 75 16.1 2.08 0.05 60 16.7 2.16 0.1 30 16.5 2.18 1 1 12.5 2.37 In Table 3, the values of the strength of the exchange interaction of (Ge1_~Pb~)~9Mn0.1Te normalized by the Mn concentration x are shown, which are calculated from the experimental data of the paramagnetic Curie temperature and the effective spin value by using equation (2). We calculate J(0)/xm*J~from equation (3), where m*mass. is theThe ratio of the effective mass the free electron calculated values are alsotolisted in Table 3. The value of J(O)/xm*J~becomes larger as the concentration of Pb increases. This is contrary to the experimental results. We can not explain the dependence of the strength of the exchange interaction on the amount of Pb substitution only by the change of the carrier density. As the Fermi surface of PbTe is nonapherical, above
Taking into account the convergence of equation (3), we carried out the summation to R~= 1 5a 0. mentioned discussion is inapplicable to the (PbTe)1_~ In Table 1, the experimental results of (Ge1.~Pb~)o.9(MnTe)~alloys. But, in order to inspect the effect of Mn0,1Te are tabulated. The values of parameters used in the low carrier density of these alloys on the magnetic the calculation are tabulated in Table 2. The mean free properties, we try to calculate the strength of the path A is obtained from the formula A = hkF a/ne, exchange interaction on the basis of the R.KKY interwhere a is the electric conductivity and e the electronic action. The result is also shown in Table 3. The value of charge. As the concentration of Pb increases, the carrier J(O)/xm*J~is smaller one order of magnitude than that (hole) density decreases but the mean free path is almost of the other alloys and this explains that the exchange invariant, interaction of this alloy is very small In order to discuss
Vol. 32, No. 11
MAGNETIC PROPERTIES OF (Ge, Pb)1~Mn~Te
1073
Table 2. Values of the carrier (hole) density n, the Fermi wave number kF, the lattice parameter a0 and the mean free path A at liquid N2 temperature of(Ge1_~Pb~)0,9Mn0,1Te 21cm’1) kF(l08cm~) ao(1O~cm) X(10~’8cm) y n(10 O 2.3 0.41 5.964 17 0.01 1.7 0.37 5.982 19 0.05 1.3 0.34 5.994 16 0.1 0.87 0.30 6.010 19 1 0.00015 0.016 6.418 —
Table 3. Comparison of the experimental results with the numerical calculations of(Ge 1_~Pb~)0.9Mn0.1Te. J(0)/x is obtainedfrom the experimental data of the paramagnetic Curie temperature and the effective spin value. J~i(Ø)/~*J~ is calculated on the basis ofthe RKKY interaction, where m* is the ratio of the effective mass to. the free electron mass 2eV) J~a~(O)/xm*J~ (eV’) y J(O)/x (10 O 1.7 2.1x102 0.01 1.6 2.5 0.05 1.1 2.6 O1 0 56 28 1 0.32
2.
3. 4. 5. 6. 7.
—
8. the magnetic properties of these alloys more precisely, the calculation of the strength of the RKKY interaction based on the actual band structure is desirable. Acknowledgements The author would like to express his sincere thanks to Dr. T. Hashimoto and Professor T. Okada for encouragement through this work. Thanks are due to Miss K. Hayashi for her kind assistance.
9. 10.
—
11. 12. 13.
REFERENCES 1.
For example, J. Kondo, Solid State Phy& 23, 183 (1969); AJ. Heeger, Solid State Phys. 23, 283 (1969).
M.A. Ruderman & C. Kittel,Phys. Rev. 96,99 (1954); T. Kasuya, F~ogr.Theor. Phys. 16,45 (1956); K. Yosida, Phys. Rev. 106, 893 (1957). M. Rodot, J. Lewis, H. Rodot, G. Villers, J. Cohen & P. Mollard,J. Phys. Soc. Japan 21, Suppl. p. 627 (1966). R.W. Cochrane & J.O. StrOm-Olsen, Proc. 13th mt. Conf Press, Low Temp. Phys. p. 427. University of Colorado Boulder (1972). R.W. Cochrane, M. Plischke & J.O. StrOm-Olsen, Phys.Rev.B9,3013 (1974). J. Cohen, A. Glova, P. Mollard, H. Rodot & M.Rodot,J.Phys.C4-142(1968). M.P. Mathur, Deis, C.K. Jones, Patterson, WJ. Carr, Jr. D.W. & R.C. Miller,J. Appl. A. Phys. 41, 1005 (1970). M.N. Vinogradova, N.Y. Kolomoets & L.M. Sysoeva, Soy. Phys. Semicond. 5, 186 (1971). M.N. Vinogradova, l.A. Drabkin. Yu.Ya. Eliseeva & I.V. Nel son Soy. Phys.—Semicond. 6 1283 (1973). K. Hirakawa & K. Hirakawa, Japan J. Appl. Phys. 2, 817 (1963). J.E. Lewis, Phys. Status Solidi 35, 737 (1969). G.M.T. Foley & D.N. Langenberg, Phys. Rev. B! 5, 4850 (1977). P.G. de Gennes,J. Phys. Radium 23, 630 (1962).
MAGNETIC PROPERTIESOF (Ge, Pb)1~Mn~Te T. Hamasaki* Department of Applied Physics, Tokyo Institute of Technology, Oh-okayama, Meguro, Tokyo 152, Japan (Received 22 June 1979 by I. Kanamori) Pseudotemary alloys {(GeTe)1_,(PbTe)~}1...~(MnTe)~ have been prepared and the magnetic and electrical properties of these alloys have been studied. The magnetic interaction in (PbTe)1_~(MnTe)~ is very weak, but it has been found that the {(GeTe)1~(PbTe)~)1_~ (MnTek alloys (y ~ 0.1) are ferromagnetic. In the latter, the increase of the amount of Pb substitution makes both the paramagnetic Curie temperature and the carrier density lower. These experimental results are discussed by using the RKKY interaction. THE ELECTRICAL and magnetic properties of dilute alloys have been studied by many authors [1]. The interaction between the magnetic impurities in these alloys is analyzed by an indirect interaction via conduction electrons, i.e. the Ruderman—Kittel—Kasuya—Yosida (RKKY) interaction [21. Recently, the interaction between magnetic ions diluted in semiconductive materisis has attracted much attention because their carrier density is easily controlled. Typical examples of this type are the pseudobinary alloys (GeTe)1_~(MnTe)~ [3—51and (SnTe)1_~(MnTe)~ [6,7]. It has been reported that these alloys order ferromagnetically, even though MnTe itself is antiferromagnetic. The ferromagnetic ordering of these alloys might be caused by the RKKY interaction. But the magnetic properties of (PbTe)1_~ (MnTe)~have not been reported, though the electrical [8] and optical [9] properties are known. In contrast 3 of GeTe and SnTe, with 1020_1021 carriers (holes) cm 7—lO’8cm3. Therefore, the carrier density of PbTe is 10’ the magnetic properties of (PbTe) 1_~(MnTe)~ might be different from those of (GeTe)1_~(MnTe)~ and (SnTe)1_~(MnTe)~. We investigate the pseudotemary alloys GeTe—PbTe—MnTe which are obtained by substituting Pb for Ge in GeTe—MnTe, in order to examine the carrier density dependence of the RKKY interaction in the dilute semiconductive alloys and also to clarify the difference of the magnetic properties between GeTe—MnTe and PbTe—MnTe. We prepared the samples {(GeTe)1_~(PbTe)~)1_~ (MnTe)~,where x is the mole ratio of MuTe in the samples andy the ratio of PbTe by which a part of GeTe in (GeTe)1_~(MnTe)~ is replaced, from high purity *
Present address: Physics Department, College of ~ Arts, Kyushu Sangyo University, Kashil, Fukuoka 813, Japan.
elements, Ge (five nine purity), Pb (six nine purity), Mn (four nine purity) and Te (six nine purity). The appropriate amounts of the elements were sealed in an evacuated silica tube, the inside of which was coated with carbon. This ampule was then heated above melting point for 24 hr and rapidly cooled to room temperature. The products were crushed and sealed again in an evacuated silica tube and reacted. This procedure was repeated several times in order to obtain a uniform Mn distribution. Finally, the obtained products were powdered and checked by X-ray diffraction and used for the magnetic measurements. We obtained homogeneous samples with x = 0—0.1 andy = 0—0.1 and 1. For the measurements of the Hall coefficient and electrical resistivity the powder samples were pressed in the form of a rectangular prism and sintered for a few hours at about 650°C.The Hall coefficient was measured at room temperature and liquid N 2 temperature. It was that the Hall coefficient depended very little onconfirmed temperature. The magnetic susceptibility was measured by means of Hirakawa-type magnetometer [10] from liquid N2 temperature to 350 K. We obtained the susceptibility of MnTe diluted in host materials by subtracting the diamagnetic susceptibility of the host materials (GeTe, GeTe—PbTe and PbTe) from the measured susceptibility. With the exception of PbTe, the diamagnetic susceptibility of these host materials was independent of temperature. Figure 1 shows the temperature dependence of the inverse susceptibility of (Ge0.95 Pb~)1_~ (MnTe)~.For the samples with low Mn concentration, the magnetic susceptibility follows the Curie—Weiss law well. But for the higher Mn concentrations the susceptibility deviates from the Curie—Weiss law near the Curie temperature 7~.According to Cochrane et aL [4], this deviation
1069
1070
MAGNETIC PROPERTIES OF (Ge, Pb)1..~Mn~Te
Vol. 32, No. 11
6emu/g )_1
l/Z9(l O
(Ge~Pboc
5)i..xMn~Te
1.0
X0.01
-
-
0.02 0
0-5
-
-
0.05 0.1 £
0
100
200 Temperature (K)
300
400
Fig. 1. The temperature dependence of the inverse magnetic susceptibility of(Geo.95Pb~)1_~MnxTe.
6emuIgT1
1/2~(1ci6emu!gT’
1 IZ9(1~Y
Pb
1~Mn~Te
XO1%J?’
2-
-0.2 0.015
0.02
1
-
-
0.1
0.05 0
0
0
I
100
I
200 Temperature (K)
1
300
Fig. 2. The temperature dependence of the inverse susceptibility of (PbTe)1_~(MnTe)~.
0 400
Vol. 32, No. 11
MAGNETIC PROPERTIES OF (Ge, Pb)1~Mn~Te
1071
8 (K) 1
Fig. 3. The dependence of the paramagnetic Curie temperature 0 on the amount of Pb substitution (y) of {(GeTe)1_~(PbTe)~}1_~ (MnTe)~.Squares show they dependence of 0 in x = 0.1, triangles x = 0.05, closed circles x = 0.02 and open circlesx = 0.01. from the Curie—Weiss law seems to be caused partly by a non-uniform Mn distribution because a very small hysteresis ioopT~. in B—H curves untilsamples 20 K higher than For the highwas Mnobserved concentrated with small y, we could observe the spontaneous mag. netization even at liquid N 2 temperature. The paramagnetic Curie temperature becomes higher as the concentcation of Mn increases. Figure 2 shows the temperature dependence of the inverse susceptibility of (PbTe)1_~(MnTe)~. For all samples, the susceptibifity follows the Curie—Weiss law well. The paramagnetic Curie temperatures are nearly equal to OK for all Mn concentrations within experimental error. l’his fact shows that the strength of the exchange interaction is very weak. The dependence of the paramagnetic Curie temperature 0 on the amount of Pb substitution (y) for the samples with constant Mn concentration x = 0.01, 0.02, 0.05 and 0.1 is shown in Fig. 3. As the measured susceptibiity of the samples with x = 0.01 was very small and the correction of the diamagnetic susceptibility of the host materials was large, the error of the paramagnetic Curie temperature was estimated to be ±10K. For x = 0.01, there is no y dependence of the paramagnetic Curie temperature. While, forx = 0.02,0.05 and 0.1,0 decreases asy increases. This means that the magnetic exchange interaction becomes weaker as the concentration of Pb increases. Figure 4 shows the y dependence
of the effective spin value for x = 0.01, 0.05 and 0.1. As y increases, the effective 2~. spin value approaches 5/2 of the spin valueatof Mn As even low Mn concentration, ferromagnetic ordering was observed, the interaction between Mn atoms is clearly long range. Now, we will discuss the experimental results with the RKKY interaction, which is the long range interaction via carriers between localized spins. We have to calculate this interaction by using actual distributions of localized magnetic spins and actual band structures of carriers. According to Lewis [11], the band structure of GeTe is roughly spherical and the free carrier effective mass can be de defmed. On the other hand, the Fermi surface of PbTe consists of four prolate spheroids centered at the L points [121. It is difficult to estimate the RKKY interaction taking into account this complex band structure. But the Pb substitution is low for almost all samples. Therefore, in the present paper, in order to roughly inspect the Pb concentration dependence of the RKKY interaction, we assume that the band structures are spherical. The usual damping type RKKY interaction [13] is written as ~ .111 = ~ J~dF(2kFR~J) e_RiJP~, (1) F 4 and n where F(x) = (sin x x cos x)/x 0 is the number —
1072
MAGNETIC PROPERTIES OF (Ge, Pb)1.~Mn~Te
Vol. 32, No. 11
S~f 2.5
x= 0.01 0.05
_____
~0.1
2.0r~~ (Ge1..~Pb~)1~Mn~Te 0
I
I
0.05
0.1
y
Fig. 4. The dependence of the effective spin value Set~on the amount of Pb substitution (y). Squares show they dependence of Sett mx = 0.1, trianglesx = 0.05 and open circlesx of conduction holes per a primitive cell, EF the Fermi energy, kF the Fermi wave number, X the mean free path of carriers, Jad the s—d exchange constant between the conduction holes and the Mn ions, and R,~the distance between the sites I and / occupied by Mn atoms. In the molecular field approximation, the parainagnetic Curie temperature 0 is expressed as 2S(S+1)J(0) = 3k~ (2) where kB is the Boltzmann constant. We will calculate J(0) in a f.c.c. lattice, because the rhombohedral distortion in the samples from a f.c.c. lattice is very small. We define z1 as the number of neighbouring lattice points at the distance R1 in a f.c.c. lattice, where R, = ..,/(J/2)a0 (j = 1, 2,. ) and a0 is the lattice parameter. If we assume a perfectly random distribution of Mn, it may be written that zrlattice points are occupied by the and 24/2m xz1—Mn atoms. Using h of carriers kF = (3ir2n)”3, where the m isrelations,EF the effective=mass and n the carrier density, we obtain the kF dependence ofJ(0) as follows; . .
J(O)
= ~
mJ~a~
51 2ir3h2 (2kFaO)4 ~I zJF(2kFRJ) ~
(3)
=
0.01.
Table 1. Values ofthe parwnagnetic (Jlsrie temperature 0, the Curie constantper gram Cg and the effective spin value per Mn atom Seff of(Ge1_~Pb~)0.9Mn0.jTe 4 Set 0(K) C5x10 1 _____________________________________________ 0 69 13.0 1.83 0.01 75 16.1 2.08 0.05 60 16.7 2.16 0.1 30 16.5 2.18 1 1 12.5 2.37 In Table 3, the values of the strength of the exchange interaction of (Ge1_~Pb~)~9Mn0.1Te normalized by the Mn concentration x are shown, which are calculated from the experimental data of the paramagnetic Curie temperature and the effective spin value by using equation (2). We calculate J(0)/xm*J~from equation (3), where m*mass. is theThe ratio of the effective mass the free electron calculated values are alsotolisted in Table 3. The value of J(O)/xm*J~becomes larger as the concentration of Pb increases. This is contrary to the experimental results. We can not explain the dependence of the strength of the exchange interaction on the amount of Pb substitution only by the change of the carrier density. As the Fermi surface of PbTe is nonapherical, above
Taking into account the convergence of equation (3), we carried out the summation to R~= 1 5a 0. mentioned discussion is inapplicable to the (PbTe)1_~ In Table 1, the experimental results of (Ge1.~Pb~)o.9(MnTe)~alloys. But, in order to inspect the effect of Mn0,1Te are tabulated. The values of parameters used in the low carrier density of these alloys on the magnetic the calculation are tabulated in Table 2. The mean free properties, we try to calculate the strength of the path A is obtained from the formula A = hkF a/ne, exchange interaction on the basis of the R.KKY interwhere a is the electric conductivity and e the electronic action. The result is also shown in Table 3. The value of charge. As the concentration of Pb increases, the carrier J(O)/xm*J~is smaller one order of magnitude than that (hole) density decreases but the mean free path is almost of the other alloys and this explains that the exchange invariant, interaction of this alloy is very small In order to discuss
Vol. 32, No. 11
MAGNETIC PROPERTIES OF (Ge, Pb)1~Mn~Te
1073
Table 2. Values of the carrier (hole) density n, the Fermi wave number kF, the lattice parameter a0 and the mean free path A at liquid N2 temperature of(Ge1_~Pb~)0,9Mn0,1Te 21cm’1) kF(l08cm~) ao(1O~cm) X(10~’8cm) y n(10 O 2.3 0.41 5.964 17 0.01 1.7 0.37 5.982 19 0.05 1.3 0.34 5.994 16 0.1 0.87 0.30 6.010 19 1 0.00015 0.016 6.418 —
Table 3. Comparison of the experimental results with the numerical calculations of(Ge 1_~Pb~)0.9Mn0.1Te. J(0)/x is obtainedfrom the experimental data of the paramagnetic Curie temperature and the effective spin value. J~i(Ø)/~*J~ is calculated on the basis ofthe RKKY interaction, where m* is the ratio of the effective mass to. the free electron mass 2eV) J~a~(O)/xm*J~ (eV’) y J(O)/x (10 O 1.7 2.1x102 0.01 1.6 2.5 0.05 1.1 2.6 O1 0 56 28 1 0.32
2.
3. 4. 5. 6. 7.
—
8. the magnetic properties of these alloys more precisely, the calculation of the strength of the RKKY interaction based on the actual band structure is desirable. Acknowledgements The author would like to express his sincere thanks to Dr. T. Hashimoto and Professor T. Okada for encouragement through this work. Thanks are due to Miss K. Hayashi for her kind assistance.
9. 10.
—
11. 12. 13.
REFERENCES 1.
For example, J. Kondo, Solid State Phy& 23, 183 (1969); AJ. Heeger, Solid State Phys. 23, 283 (1969).
M.A. Ruderman & C. Kittel,Phys. Rev. 96,99 (1954); T. Kasuya, F~ogr.Theor. Phys. 16,45 (1956); K. Yosida, Phys. Rev. 106, 893 (1957). M. Rodot, J. Lewis, H. Rodot, G. Villers, J. Cohen & P. Mollard,J. Phys. Soc. Japan 21, Suppl. p. 627 (1966). R.W. Cochrane & J.O. StrOm-Olsen, Proc. 13th mt. Conf Press, Low Temp. Phys. p. 427. University of Colorado Boulder (1972). R.W. Cochrane, M. Plischke & J.O. StrOm-Olsen, Phys.Rev.B9,3013 (1974). J. Cohen, A. Glova, P. Mollard, H. Rodot & M.Rodot,J.Phys.C4-142(1968). M.P. Mathur, Deis, C.K. Jones, Patterson, WJ. Carr, Jr. D.W. & R.C. Miller,J. Appl. A. Phys. 41, 1005 (1970). M.N. Vinogradova, N.Y. Kolomoets & L.M. Sysoeva, Soy. Phys. Semicond. 5, 186 (1971). M.N. Vinogradova, l.A. Drabkin. Yu.Ya. Eliseeva & I.V. Nel son Soy. Phys.—Semicond. 6 1283 (1973). K. Hirakawa & K. Hirakawa, Japan J. Appl. Phys. 2, 817 (1963). J.E. Lewis, Phys. Status Solidi 35, 737 (1969). G.M.T. Foley & D.N. Langenberg, Phys. Rev. B! 5, 4850 (1977). P.G. de Gennes,J. Phys. Radium 23, 630 (1962).