Superconductivity, including high-temperature superconductivity
CALCULATION OF THE MAGNETIZATION OF THE
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CALCULATION OF THE MAGNETIZATION OF THE CsDy „
4 …
CRYSTAL AT T Ä 0 K. PHASE TRANSITIONS CAUSED BY AN EXTERNAL MAGNETIC FIELD The results of the previous Section imply that in the absence of an external magnetic field at T ϭ0 K the magnetic structure of the RE ions in CsDy ͑MoO
4 ) 2 should correspond to the 3AF-6 configuration. For it the angles of rotation of the magnetic moments of the RE ions for their ‘‘own’’ and
the ‘‘other’’ subsystems are
1,2
ϭ0.458° and 1,2 ϭϮ3.974°, respectively ͑Fig. 3a͒. Thus the structure un- der consideration is a four-sublattice structure, made up of sets of ferromagnetic chains of RE ions lying along the c direction of the crystal, and the orientation of the magnetic moments in the chains of ions of types 1 and 2 is also close to this direction. To study the evolution of the magnetic structure in an external magnetic field directed along the orthorhombic axes a, b, and c, we must determine the configuration that will have the minimum energy at the given value of the magnetic field. This might be the initial configuration 3AF-6 or a canted configuration with a resultant magnetic moment in the direction of the field H. Here one expects that a symmetric canted phase will be realized, since the field will be applied in directions of symmetry. The magnetic structure of this angular configuration can be determined by finding the mini- mum angular dependence of its energy E 0 ␣ ( , ) in the
external magnetic field: E 0 ␣ ͑ , ͒ϭE dd ␣ ͑ , ͒ϩE dm ␣ ͑ , ͒ϩE L ␣ ͑ , ͒ϩE H ␣ ͑ , ͒, ␣ ϭa,b,c, ͑10͒ where E dd ␣ ( , ) ϭ͚
Ͻ j ͓( i
)r
2 Ϫ3( i r i j )( j r i j ) ͔/r i j 5 is the energy of the dipole–dipole interaction of the RE ions of FIG. 3. Magnetic structure of CsDy ͑MoO 4
2 in the ordered state for T ϭ0 K in the absence of an external magnetic field H ͑the 3AF-6 configuration͒ ͑a͒ and for H ϾH
: H ʈc ͑b͒; Hʈb ͑c͒; Hʈa ͑d͒. 565 Low Temp. Phys. 26 (8), August 2000 Anders et al.
the canted configuration; E dm ␣ ( , ) ϭN ␣ ␣ 2 ( , )n 0 /2 is the energy of the demagnetizing fields for a sample of macro- scopic dimensions, N ␣ is the demagnetizing factor in the direction ␣ ; E L ␣ ( , ) ϭϪ2
␣ 2 ( , )n 0 /3 is the Lorentz en- ergy; and E
␣ ( , ) ϭϪ ␣ ( , )H ␣ is the energy of the magnetic moment in the external field. Here the functions ␣
, ) have the form:
͑
, ͒ϭ B S †g
sin
͑ ͒ϩg
cos ͑
͒sin͑
͒ ϩg
cos ͑
͒cos͑
͒ ‡;
͑
, ͒ϭ B S †g
sin
͑ ͒ϩg
cos ͑
͒sin͑
͒ ϩg
cos ͑
͒cos͑
͒ ‡; ͑11͒
z ͑ , ͒ϭ B S †g
sin
͑ ͒ϩg
cos ͑
͒sin͑
͒ ϩg
cos ͑
͒cos͑
͒ ‡, where
ϭ1,2.
The minimum of the energy E 0 ␣ as a function of the angles
and
that specify the direction of the magnetic moments of the RE ions in the sites in an applied external field was found by a numerical calculation. Here it was as- sumed that if the condition E 0 ␣ ϾE 6 is holds, then the mini- mum energy of the magnetic system will correspond to the 3AF-6 configuration, and if E 0 ␣
6 then it would corre- spond to a canted configuration. By determining the direction of the magnetic moments of the RE sites in this way and, hence, their projection ␣ on the field direction, one can construct the magnetization as a function of the external magnetic field applied in the corresponding direction. Figure 4 shows the results of a calculation of the func- tions M
(H), M b (H), and M a (H) for a spherical sample and a two-dimensional ͑in the bc plane͒ slab with a ratio of sides equal to 1:100 ͑in the experiments whose results will be used for comparison, the samples were generally in the form of slabs ͒. When the external magnetic field is directed along the c axis ͑Fig. 4͒ and H c ϽH c1 c , where H c1 c ϭ1.2 kOe, the mag- netic structure of the RE ions in CsDy ͑MoO
4 ) 2 will corre- spond to the 3AF-6 configuration. At the field value H c ϭH c1 c there occurs a phase transition from the antiferromag- netic configuration 3AF-6 to a canted configuration with a resultant magnetic moment in the direction of the magnetic field ͑see Fig. 3b͒. Here the magnetic moments of the RE ions of different sublattices will be rotated in a symmetric way for a spherical sample and a slab, by 1,2
ϭϪ0.3° and 1,2 ϭϮ5.8°, and the projection of the magnetic moment of the ions on the coordinate axes will have the following val- ues:
c ϭϪ0.02
B ; y c ϭϮ0.6
B ; z c ϭ6.67
B . Since for a spherical sample the demagnetizing factor in this direction is nonzero, the magnetic structure of the sample after the phase transition will be of the domain type. The interval of field values in which the domain structure will exist is determined by the demagnetizing field H dm
ϭ1.3 kOe. At a value of the external magnetic field H c ϭH c2 c ϭH c1 c ϩH dm
ϭ2.5 kOe, the domain structure vanishes. Further increase in the magnetic field will lead to additional rotation of the magnetic moments of the RE ions and, hence, to a small increase in the longitudinal component of the re- sultant moment. In the limit H c →ϱ, the magnetic moments of the RE ions will be directed in such a way that the value of z c will be maximum. In this case the magnetic moments of the RE ions of the two sublattices will be rotated by 1,2 ϭ0.1° and 1,2 ϭϮ4.6°, respectively, and the components of the magnetic moments of the ions are
ϭ0.01
B ; y c ϭϮ0.6
B ; z c ϭ6.68
B . The critical field parameters calculated above for samples of different shapes and the values of the magnetic moment projections and the angles and
, characterizing the behavior of the CsDy ͑MoO
4 ) 2 system for a field orien- tation H ʈc, are presented in Table II. The system behaves in an analogous way when the ex- ternal magnetic field is along the a and b axes, since the plane in which the magnetic moments lie for the four- sublattice structure under study does not coincide with the basal planes of the orthorhombic unit cell. The behavior of the system is also characterized by the presence of critical fields H c1 and H c2 , the values of which are given in Table II FIG. 4. Calculated curves of the magnetization of CsDy ͑MoO
4 ) 2 versus the magnetic field for T ϭ0 K ͑———͒, and experimental curves 12 for T ϭ0.6 K and H ʈc ͑– – – –͒ for a sphere ͑a͒ and for a two-dimensional ͑in the bc plane͒ slab with a 1:100 ratio of sides ͑b͒. 566
Low Temp. Phys. 26 (8), August 2000 Anders
et al. along with the other magnetic parameters. The magnetic structure of the RE ions for H ϾH
is shown in Fig. 3c (H ʈb) and 3d (Hʈa). It is of interest to compare the results obtained in this calculation and in experimental studies of the magnetic prop- erties of CsDy ͑MoO
4 ) 2 , which have been done by a number of authors. Here it should be kept in mind that the resultant magnetization of the crystal, examined above for different orientations of the external magnetic field, can also be char- acterized by the value of the equivalent G factor in accor- dance with the expression M i ϭG i
S for an effective spin of the Dy 3 ϩ
ϭ1/2. In this case the values of the equiva- lent G factor depend on the value of the external field, and in high fields H ϾH c2 one has the values G c ϭ13.36, G b ϭ1.66, and G a ϭ3.66. It is this parameter that should be compared with the effective g factor obtained experimen- tally.
In studying the magnetic susceptibility of CsDy ͑MoO
4 ) 2 in the paramagnetic state, for a field orientation H ʈc a value of g
ϭ12.4 was obtained 4 for the effective g factor of the Dy 3 ϩ ions. The magnetic component of the heat capacity of this crystal in a magnetic field with this same orientation is described by an effective g factor g
ϭ12.8.
3 These values are close to that calculated by us: G
ϭ13.36.
As can be seen in Fig. 4, the magnetization of CsDy
͑MoO 4 ) 2 in the magnetically ordered state as a func- tion of the magnetic field in the orientation H ʈc ͑Ref. 12͒ also has much in common with our predictions. In addition, in Ref. 3, in a study of thermograms of the heat capacity of this crystal in the magnetically ordered state for H ʈc, anoma- lies were noted at H Ϸ1.7 kOe and 5 kOe. These fields can be compared with the calculated values H c1 a and H c2 a . Be-
sides the natural causes due to growth inhomogeneities of the crystals and scatter in the parameters of the internal interac- tions, the numerical differences between the results of the calculations and experiments done at finite temperatures may be due both to computational errors related to the accuracy with which the angles of rotation of the principal axes of the effective g-factor tensors of the RE ions are determined and also to errors in the orientation of the samples in the heat-capacity and magnetic-susceptibility experiments in CsDy
͑MoO 4 ) 2 . A magnetic field in the H ʈb orientation gives rise to a magnetic moment in the magnetically ordered state of this crystal, the value of which is approximately an order of mag- nitude smaller than for the orientation H ʈc. 12 This also cor- responds to the relationship that we found between the pa- rameters G c and G b . For the orientation H ʈa one observes an appreciable scatter of the experimental values of the effective g factor: the values obtained in Refs. 3 and 4 were g
ϭ14.2 and g a ϭ8.9, respectively. In any case, however, they are quite a bit higher than the value that we obtained: G
ϭ3.65. The reason for this difference is still not clear to us.
1. According to the calculations done in the dipole ap- proximation for the CsDy ͑MoO
4 ) 2 unit cell containing two magnetically inequivalent RE ions, the configuration that should be realized in the ordered state is 3AF-6. The energy per ion of the dipole–dipole interaction is E 6 ϭϪ1.28794 K, and the magnetic moments of the two sublattices are rotated by 1,2 ϭ0.458° and 1,2
ϭϮ3.974°, respectively. 2. A calculation of the field dependence of the magneti- zations M
, M b , and M c shows that phase transitions should be observed in an external magnetic field as a result of a change of the magnetic structure of the RE ions from the initial 3AF-6 configuration to a canted configuration with a resultant magnetic moment in the direction of the external field. 3. A comparison of the calculated values of G i with the
effective g factors obtained from the experimental data on the heat capacity and susceptibility of CsDy ͑MoO 4
2 in an
external magnetic field shows that they are in satisfactory agreement. TABLE II. Parameters characterizing the behavior of CsDy ͑MoO
4 ) 2 in an external magnetic field. ͑The values of the critical fields are given in kOe, the magnetic moment components in Bohr magnetons, and the angles in degrees. ͒ 567 Low Temp. Phys. 26 (8), August 2000 Anders
et al. a ͒ E-mail: aanders@ilt.kharkov.ua 1 A. G. Anders, V. S. Bondarenko, S. B. Bordovsky, A. Feger, and A. Orendacheva, Fiz. Nizk. Temp. 23, 1195 ͑1997͒ ͓Low Temp. Phys. 23, 895 ͑1997͔͒.
2 A. G. Anders, V. S. Bondarenko, A. Feher, and A. Orendaceva, Fiz. Nizk. Temp. 22, 1042 ͑1996͒ ͓Low Temp. Phys. 22, 794 ͑1996͔͒. 3 P. Shtefaniji, A. Orendaceva, A. Feger, E ´ . E. Anders, and A. I. Zvyagin, Fiz. Nizk. Temp. 15, 1105 ͑1989͒ ͓Sov. J. Low Temp. Phys. 15, 615 ͑1989͔͒.
4 E. N. Khats’ko and A. S. Cherny , Fiz. Nizk. Temp. 11, 540 ͑1985͒ ͓Sov. J. Low Temp. Phys. 11, 296 ͑1985͔͒. 5 V. A. Vinokurov and V. P. Klevtsov, Kristallografiya 17, 127 ͑1972͒. 6 N. M. Nesterenko, V. I. Fomin, V. I. Kut’ko, and A. I. Zvyagin, Preprint, 26–82 ͓in Russian͔, Institute for Low Temperature Physics and Engineer- ing, Academy of Sciences of the USSR, Kharkov ͑1982͒.
7 J. M. Luttinger and L. Tisza, Phys. Rev. 70, 954 ͑1946͒. 8
͑Amsterdam͒ 57, 281 ͑1972͒. 9 T. Niemeyer and H. W. J. Blote, Physica ͑Amsterdam͒ 67, 125 ͑1973͒. 10 S. K. Misra, Phys. Rev. B 8, 2026 ͑1973͒. 11 S. B. Feodos’ev, E. S. Syrkin, I. A. Gospodarev, V. P. Popov, A. A. Gurskas, and N. M. Nesterenko, Fiz. Tverd. Tela ͑Leningrad͒ 31, 186 ͑1989͒ ͓Sov. Phys. Solid State 31, 102 ͑1989͔͒. 12 A. S. Cherny , Candidate’s Dissertaion ͓in Russian͔, B. Verkin Institute for Low Temperature Physics and Engineering, National Academy of Sci- ences of Ukraine, Kharkov ͑1994͒.
Translated by Steve Torstveit 568
Low Temp. Phys. 26 (8), August 2000 Anders
et al. LOW-DIMENSIONAL AND DISORDERED SYSTEMS Surface electromagnetic modes in layered conductors in a magnetic field V. M. Gvozdikov Department of Physics, Kharkov State University, Svobody sq. 4, 61077, Kharkov, Ukraine ͑Received October 18, 1999, revised March 1, 2000͒ Fiz. Nizk. Temp. 26, 776–786 ͑August 2000͒ A transfer-matrix approach is developed for studies of the collective electromagnetic modes in a semi-infinite layered conductor subjected to a quantizing external magnetic field perpendicular to the layers. The dispersion relations for the surface and bulk modes are derived. It is shown that the surface mode has a gap in the long-wavelength limit and exists only if the absolute value of the in-plane wave vector q exceeds the threshold value q * ϭϪ1/(a ln͉⌬͉). Depending on the sign of the parameter ⌬ϭ(Ϫ
0 )/(
0 ϩ), the frequency of the surface mode
(q, ⌬) goes either above ͑for ⌬Ͼ0͒ or below ͑for ⌬Ͻ0͒ the bulk-mode frequency (q,k) ϭ (q,k ϩ2 /a) for any value of k. At nonzero magnetic field H the bulk mode has a singular point q 0 (H) at which the bulk band twists in such a way that its top and bottom bounds swap. Small variations of q near this point change dramatically the shape of the dispersion function (q,k) in the variable k. The surface mode has no dispersion across the layers, since its amplitude decays exponentially into the bulk of the sample. Both bulk and surface modes have in the region qa ӷ1 a similar asymptotic behavior ϰq 1/2 , but
s (q, ⌬) lies above or below (q,k), respectively, for ⌬Ͼ0 and ⌬Ͻ0 ͑a is the interlayer separation; 0
stand for the dielectric constants of the media outside the sample and between the layers; q and k are the components of the wave vector in the plane and perpendicular to the layers, respectively ͒. © 2000 American Institute of Physics. ͓S1063-777X͑00͒00508-9͔ Download 2.75 Mb. Do'stlaringiz bilan baham: 
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