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79ISSN 2072-5981 doi: 10.26907/mrsej
aänetic Resonance in Solids
Electronic Journal
Volume 21 Special Issue 4 Paper No 19412 1-26 pages
2019
doi: 10.26907/mrsej-19412
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Yoshio Kitaoka (Osaka University,
Osaka)
Boris Malkin (KFU, Kazan) Alexander Shengelaya (Tbilisi State University, Tbilisi) Jörg Sichelschmidt (Max Planck Institute for Chemical Physics of Solids, Dresden) Haruhiko Suzuki (Kanazawa University, Kanazava) Murat Tagirov (KFU, Kazan) Dmitrii Tayurskii (KFU, Kazan) Valentine Zhikharev (KNRTU,
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* In Kazan University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.
Short cite this: Magn. Reson. Solids 21, 19412 (2019)
doi: 10.26907/mrsej-19412
Magnetic and magnetoelastic properties of non-conducting rare-earth single crystals LiLnF4 (Ln = Tm, Tb, Ho, Dy)
(Review)
I.V. Romanova1'*, M.S. Tagirov1,2
1Kazan Federal University, Kremlevskaya 18, Kazan 420008, Russia
2Institute of Applied Research, Tatarstan Academy of Sciences, Levobulachnaya 36a, Kazan 420111, Russia
* E-mail: [email protected]
(Received May 23, 2019; accepted May 28, 2019; published June 6, 2019)
We review the magnetic and magnetoelastic properties of rare earth tetrafluorides LiLnF4 recently studied at Kazan Federal University in collaboration with the Laboratory of Low Temperatures, Kanazawa University, Japan.
PACS: 31.15.Ne, 75.50.-y, 75.80.+q, 76.30.Kg.
Keywords: crystal electric field parameters, magnetization, magnetostriction, magnetoelastic interaction.
With great gratitude to Malkin Boris Zalmanovich in honor of his 80th birthday, to the very best, sophisticated, sympathetic and attentive Master, with wishes of a long life, the most intelligent students and the great achievements in science
1. Introduction
The study of magnetic properties of rare-earth double fluorides LiLnF4 (Ln = Tm, Tb, Ho, Dy) performed over the past two decades significantly contribute to further development of the theory of magnetoelastic effects in rare-earth magnetic materials [1-3]. Their unique optical, electronic, magnetic and catalytic properties are also important for practical applications [3]. The unit cell of LiLnF4 contains two magnetically equivalent sites of lanthanide Ln3+ ions with the S4 point symmetry. The crystal structure of LiLnF4 compounds is shown in Fig. 1 [4].
Magnetic properties of LiLnF4 are presented in Table 1. LiTbF4 and LiHoF4 are dipolar Ising-like ferromagnets with magnetic moments of Tb3+ and Ho3+ ions directed along the crystallographic c-axis and Curie temperatures Tc = 2.885 K and 1.53 K, respectively [4]. LiDyF4 is an easy-plane anti-ferromagnet with magnetic moments of Dy3+ ions arranged perpendicular to the crystal symmetry axis and an ordering temperature Tn = 0.62 K. LiTmF4 is a Van Vleck paramagnet [4]. Magnetic dipolar interactions play the dominant role in the spontaneous low-temperature ordering of magnetic moments of Ln3+ ions in these compounds. Quantum phase transitions driven by transverse magnetic fields were observed in LiHoF4 at temperatures below Tc [5]. LiTmF4 exhibits a giant forced magnetostriction at liquid helium temperatures [6].
Figure 1. The crystal structure of LiLnF4.
Table 1. Magnetic properties of LiLnF4 compounds.
LiTmF4 LiTbF4 LiHoF4 LiDyF4
4f I2 4f 8 4f10 4f 9
Van Vleck Ising-like dipole Ising-like dipole dipole
paramagnet ferromagnet ferromagnet antiferromagnet
- Tc = 2.89 K Tc = 1.53 K Tn = 0.62 K
2. Experimental technique
Single crystals of LiDyF4, LiTbF4, LiTmF4 and LiHoF4 were grown by Dr S.L. Korableva using Bridgeman-Stockbarger method and oriented by means of X-ray diffractometer. Magnetization measurements were performed on spherical samples with known demagnetization factor. The rotation of the sample in strong magnetic fields was prevented by fixing in Stycast 1266 epoxy with an accuracy in orientation ±3°. The temperature dependences of the magnetization of single crystals in the temperature range 2—300 K and the magnetization curves in magnetic fields 0—5T applied along and perpendicular to the c-axis were measured using dc-SQUID magnetometer MPSM-2 (Quantum Design). Angular dependences of the magnetization in the basal plane of LiDyF4 and LiHoF4 were measured by the inductance method with the magnetic field B up to 2 T applied perpendicular to the c-axis at the temperature 4.2 K. The inductance of the coil with the sample was measured on forward and backward field sweeps by LCR-meter E7-14 using the inductance bridge circuit balanced at zero field at the frequency of 1kHz. The background from the empty coil was measured independently and subtracted from the total signal. The sample was glued inside the capsule and could be rotated inside the coil with an accuracy of ±5°. Magnetization curves M(B) were extracted from the experimental data by integrating the field scans of the derivatives dM/dB (in arbitrary units) at various sample orientations. The obtained curves were calibrated using magnetization data for LiDyF4 and LiHoF4 single crystals measured with a dc-SQUID [7].
3. Theory
The Hamiltonian of a rare-earth magnetic system can be expressed by taking into account an interaction with crystal lattice with free surface, a crystal field and Zeeman energy, and interaction of a rare-earth ion with homogeneous macro- and microdeformations, magnetic dipolar and exchange interactions between ions and electron-phonon energy [8]. The Hamiltonian of a rare-earth magnetic system taking into account an interaction with a crystallographic lattice with a free surface can be written as:
H = £ HLyS + Hat + £££ T^Bp (s, q j) exp(iqRLs)Okp (Ls)Q(qj), (1)
Ls Ls q j pk
where Hl,s is the Hamiltonian of a rare earth ion with the radius-vector Rl,s from the sublattice sth (s = 1,2) in the Lth unit cell in static crystal field, Hlat is an energy of the crystal lattice in harmonic approximation with normal coordinates Q(qj) (q is a wave phonon vector with the frequency wqj-, j is the number of oscillation spectrum branch), the last part in (1) represents electron-phonon energy in linear approximation on ion displacements from equilibrium positions, N is a cell number, Op(L, S) are linear combinations of spherical tensor operators [8] acting in the space of electron states of a rare-earth ion, Bp(s, qj) are interaction parameters determined as the derivatives of the corresponding crystal field parameters with respect to displacements of
ions [9]. Considering corrections to the energy of rare-earth ions in the second order to electron-phonon interaction at low temperature (kBT ^ fiw, kB is a Boltzmann constant, w0 is a boundary phonon frequency) and taking into account only the mixing of electronic states with the energy difference A ^ hw0, we can write the effective Hamiltonian of interaction between Ln3+ ions [9]:
AH = -1 E - ÔLL'^) E (ss', Rll(Ls)°k;(L's')
LL'ss'
where
and
pp'kk'
Rss' D D
LL' — RLs — RL's' ,
kk' pp'
(ss', RLl' ) — W-B (s, qj)Bk (s', -qj) exp(iqRLL ).
(2)
(3)
(4)
q?
Two-particle interactions can be considered in a self-consistent field approximation, neglecting second-order terms with respect to deviations of the operators Op:(Ls) from their average values
(Ok(Ls)) = (O^) (the last equation is valid if all magnetic ions are equivalent as in case of Ln3+ ion in LiLnF4). The free energy of an elastically distorted single crystal in the presence of an applied magnetic field B can be written in the following form:
F — V 2
eC'e + 2^ eb(r)w(r) + ^^ w(r)a(r, r')w(r')
n
+ n
E (°pk^ (o?) + AF,
pkp' k'
(5)
AF — - E kßT lnTrexp(-Heff ,sAb T),
where v is the elementary cell volume, C is the tensor of "probe" elastic constants, e is the deformation tensor, w(r) is the vector of the sublattice displacement r, b(r) is the relation's constants tensor of macro- and microdeformations, aa,p(r, r' ) is the dynamic matrix of the lattice at the center of the Brillouin zone, A^' is the matrix of the relation's constants through phonon field, n is the number of equivalent magnetic sublattices (n = 2 for LiLnF4), Heff,s is the effective single-ion Hamiltonian defined below:
Heff — H(0) + H(P ), H(0) — Ho + Hcf + Hz ,
H(P) — E Vaß+ E Va (s)wa (s) - E ^
aß a,s pkp'k'
kk' pp'
Okp) O?
(6)
(7)
(8)
The first term in (7) is the Hamiltonian of a free ion H0, the second term is the crystal field energy, the third term is the Zeeman energy hz = ^bB(L + 2S); here ^b is the Bohr magneton. First and second terms in (8) define linear interaction of rare-earth ion with homogeneous macro-and microdeformations, correspondingly. Electronic operators Va@ and Va(s) can be presented as linear combinations of spherical tensor operators:
Vaß
E Bp,aß Op,
Va(r) = £ Dp,a (r)Opk. (9)
pk pk
Taking into account the translational symmetry of the lattice we can obtain from equations (2)
and (3) the following equation for elements of the A matrix in last term in (8):
1
\ kk' _ 1
App' — n ^
E B?(s, °jo)B? (s', 0jo)
jo
w,
0 jo
Sss' N
E Bp;(_s, qj)Bpk''(s', -qj)
q?
W
q?
(10)
In the first term, the summation is performed only over the optical branches of the vibrational spectrum. The single ion contribution to the free energy (5) can be written with accuracy up to second-order terms with respect to deformations characteristic of the lattice and deviations of average values of operators Opk from corresponding equilibrium values in a zero magnetic field (simplified equations for O^ are presented below). Considering operator (8) as perturbation one can rewrite H(P) = A^O^ = A : O, and obtain
pk
AF
n
Fo(B) + (H(P))Q + 2A : q : A
(11)
where F0(B) is the free ion energy with Hamiltonian H(0), symbol (...)0 means averaging with equilibrium density matrix p = exp(-H(0)/£;bT)/Tr(exp(-H(0)/kBT)). The matrix elements q equal to
qkk -
Vpp'
kBT
OP
ok
i
<№ | pokp <^ij ot m
+
(12)
+ E(£i - )-1 M pok > <№ i opk; j^i> + <^ij popk; ^ > | ok j^i> i,j=i
Here ei and ^ are eigenvalues and eigenfunctions of operator Heff. Taking into account thermodynamic equilibrium conditions dF/d (O^) = 0 and dF/dwa(r) = 0, and the designation of the convolution over the indexes p and k, introduced above, one can write the free crystal energy as
v n
F - 2eCe + nFo - <o>o :
A : (1 + q : A)-1 + n ^ D(r)a-1(r, r')D(r')
: <o>o +
+ n <o>o : (1 + A :
1
B - b(r)â-1(r, r')D(r')
(13)
where Da(r) = (1 + A : q) 1 : Da(r), C = C' — ^ b(r)a 1(r, r')b(r') is the elastic constants ten-
rr'
sor. The following notation for the quantities that are renormalized by the electron-deformation and electron-phonon interactions is introduced:
n
àaf(r, r') - a«f(r, r') + -Da(r) : q : (1 + A : q)-1 : Df(r')
n
baftl(r) - bapri(r) + -: q : (1 + A : q) : D7(r),
v n
caf7<5 = caf7<5 + - B«f : q: (1 + A : q) 1 : B7<5
Da(r) - (1 + A : q)-1 : D«(r), Bap - (1 + A : q)-1 : B,
1
Jaf.
(14)
(15)
(16) (17)
It should be noted that, with an accuracy of up to terms of the second order with respect to the parameters of the electron-deformation interaction, the dependence of the elastic constants on temperature and magnetic field can be represented by the expression:
n
Cafil&(T, B) - Cafil&(T - 0, B - 0) + -Baf : q : (1 + A : q)-1 : B7<5,
(18)
where the contributions of the magnetic subsystem are determined by the second term and the renormalized constants of coupling with macrodeformations are given by
Bkp,af - Bpaf - £ baffr((r)a-1(r, r')^(r'). (19)
rr;jS
1
o
o
e
A magnetic field-induced relative change in the dimension of the crystal in the direction that is specified by the unit vector with the direction cosines na equals Al/l = ^a/3 nan@ea@, where components of deformation tensor are defined from minimum free energy condition (13):
n
e(B) = - - [SBeff : <0>oIb - SBeff : (O)0|
/0|B=0J
(20) Ion's
here Beffp,aß = Y.p'k' [(1 + A : q) ^kk : Bk',aß, S = C 1 is the compliance tensor. energy levels and their magnetic moments in external magnetic field are determined by effective Hamiltonian, introduced above, where perturbation operator is:
H(P) = Beff e - <O>
0
^ D(r)a-1(r, r')D(r') + A : (1 + q : A)
1
: O. (21)
Equations defined above for elastic constants induced by magnetic field and for effective Hamiltonian of paramagnetic ion are used for calculation of LiLnF4 single crystal parameters.
4. Elastic constants, magnetization and magnetostriction of LiTmF4
The results that are presented below were obtained using the numerical diagonalization of the Hamiltonian of Tm3+ ions in the full space of states of the electronic configuration 4f12 (the number of states is 91). The Hamiltonian of the free ion was written in the standard form taking the parameters of the electrostatic, spin-orbit, and interconfigurational interactions from [10]. In the crystallographic coordinate system, the crystal field that acts on the Tm3+ ion in the LiTmF4 crystal is determined by a set of seven parameters Bpk:
Hcf = b°o° + B4°O4° + B44O4 + b4-4o4-4 + B°O° + B64O4 + B-4 O-4. (22)
The values of the crystal field parameters that were used in this work are presented in Table 2.
The calculated energy levels of the Tm3+ ion (see Table 3, where the sublevels of the lowest multiplet 3H6 are given) agree well with experimental data. The ground state of the Tm3+ ions in the crystal field is the singlet r2(1), the nearest levels are the doublet r34(1) and the singlet r1(1) (the wave functions of different states of the electron shell 4f12 of Tm3+ ions are transformed according to the irreducible representations rk of the S4 point group). Comparison of the calculated temperature dependence of the longitudinal magnetic susceptibility with experimental data (Fig. 2) indicates that the investigated specimens of LiTmF4 contain paramagnetic impurities. Their contribution to the magnetization increases considerably at temperatures lower than 10 K.
Table 2. Interaction parameters of Tm3+ ions with a crystal field and deformations of the Bg symmetry (in cm-1).
p k k p k 4d)k (Bg) BeffP (Bg)* bP (Bg) Beffp(Bg2) *
2 0 184 2 2 1914 1571 4194 3646
4 0 -90 2 -2 1408 1337 -1030 -1169
4 4 -669 4 2 -600 -468 -1615 -1393
4 -4 -638 4 -2 1292 1141 1237 986
6 0 -4.06 6 2 -192 -164 -492 -443
6 4 -328 6 -2 -569 -480 -590 -424
6 -4 -294 6 6 -963 -858 -1375 -795
- - - 6 -6 -1067 -974 -1500 -910
:In a zero magnetic field at a temperature of 4.2 K
Table 3. Energy levels (in cm-1) of the 3H6 multiplet of the Tm3+ ion in a crystal LiTmF4.
Symmetry Experiment Calculation
[11] [12] LiYF4:Tm3+ [13] This work
T2(1) 0 0 0 0
r34(1) 32 - 30 31
T1(1) 57 71 55 62
T2(2) - 281 275 282
T2(3) - 319 305 313
T1(2) - 363 - 362
T34(2) - - - 381
T34(3) - - - 402
T1(3) - - - 415
r2(4) - - - 430
Figure 2. Measured in [14] (symbols) and calculated (solid lines) temperature dependences of the magnetization of a LiTmF4 single crystal: (a) in a magnetic field B = 0.1 T parallel to the c axis; (b) in magnetic fields B = 5.5 T (1) and 3 T (2, 3), perpendicular to the c axis and constituting an angle of 12.5 degrees with the a axis. Line 3 represents the magnetization in the 3T field, calculated without taking into account the multipole interaction between thulium ions.
Calculations are significantly simplified if the symmetry properties of the system are taken into account. Let us introduce linear combinations of components of the deformation tensor e^) = ezz, e(A;;) = (exx + eyy)/2, e(Bg) = exx - eyy, e(Bg) = exy, ei(Eg) = exz, e2(Eg) = eyZ, which transform according to the irreducible representations r = Ag ,Bg , Eg of the factor group symmetry of the lattice. The magnetic field that is directed along the c axis induces only totally symmetric deformations Ag, whereas field that is directed in the ab plane induces totally symmetric and rhombic deformations Bg. Corresponding microscopic deformations of the Ag and Bg symmetry are determined by three and five independent linear combinations of displacements of sublattices wn(r), respectively. In the basis of symmetrized deformations, the operator of the electron-deformation interaction has the following form:
Bp% (ru )e, (ru) + D
n(ru )w„ (ru)
Ok
(23)
Hel-def = ¿^1^
run pk
where en(ru) and wn(ru) are independent variables, u and n are the number and the row of the representation r, respectively. Results of calculations of coupling constants in operator (23) in terms of the model of exchange charges [9] and optical piezospectroscopy of the LiTmF4 crystal show that the interaction of Tm3+ ions with deformations of the Bg symmetry plays a dominant role. As a consequence, we may expect that dynamic deformations of the Bg symmetry make the main contribution to the multipole interaction between Tm3+ ions. The interaction can be represented by linear combinations of eight electronic operators: O2 = O1, O-2 = O2, O| = O3,
O-2 = O4, O2 = O5, O-2 = O6, O6 = O7, and O-6 = Og. In accordance with Eq. 9, after neglecting the dispersion of optical branches of the vibrational spectrum, we obtain the following expression for the elements of the matrix Ai;j (i, j = 1:8), which is determined in the space of the operators indicated:
2
6
Aj =
£
u=1:5
Bi (Bgu )Bj (Bgu) w(BU )2
£
u=1:4
Bi(AU)Bj (AU)
W(AU)2
v Bi(1, qj)Bj (1, -qja)
N ¿r1
q3a
w
2
q?'a
(24)
Multipole interaction parameters (24) were estimated using frequencies of gerade w(AUTO) and transverse ungerade w(A^TO) optical vibrations of the Bg and Au symmetry, respectively, of the LiTmF4 crystal at the center of the Brillouin zone, which were measured in [9], and coupling constants of thulium ions with corresponding vibrations which were found from the analysis of temperature and magnetic field effects in Raman spectra [9,15], and calculations in terms of the exchange charge model. The summation over acoustic vibrations in the last term in Eq. 24 was performed in terms of the long wavelength approximation, which allow to relate the constants of the electron-phonon interaction with the electron-deformation interaction parameters (Eq. 19). The corresponding contributions to elements of the matrix Ai;j are given by (in units of 10-6 cm-1)
A A3 = - { 4.48B(d)(Bg)Bf (Bg) + 2.29 |B(d)(Bg)Bf (Bfl2) + B(d)(Bfl2)Bf (Bg)
+
+1.89Bf(Bg2)Bf (Bg2)} .
(25)
The coupling constants with homogeneous deformations of the Bg symmetry are given in Table 2 [8]. Table 4 presents parameters of the multipole interaction, which were reduced twofold compared to calculated ones in order to match calculation results of temperature dependences
Table 4. Parameters of the multipole interaction for Tm3+ ions Aj j (cm
1
1 2 3 4 5 6 7 8
1 -32.4 -24.6 21.82 -24.7 4.58 11.92 10.13 13.48
2 28.34 -7.99 4.90 -3.21 1.582 7.23 -2.96
3 -0.90 2.06 -1.42 -0.22 -6.14 -2.20
4 -12.8 -0.80 2.15 6.34 0.512
5 -0.87 -0.52 -1.87 -0.76
6 -0.87 -2.49 -3.06
7 -2.95 -4.07
8 -2.83
j_i_i_i_i_u 1 9 Lj_i_i_i_i_u LJ_i_i_i_i_u
0 20 40 60 80 100 ' 0 20 40 60 80 100 0 20 40 60 80 100
T (K) T (K) T (K)
Figure 3. Measured (symbols) [16] and calculated (curves) temperature dependences of the elastic constants (a) Ci6, (b) C66, and (c) Cii — C12 of a single crystal LiTmF4.
of the elastic constants Cn — C12, C16, C66. Elastic constants that determine a change in the lattice energy caused by deformations of the Bg symmetry and the experimental data of [16] are shown in Fig. 3.
We note that contributions to multipole interaction parameters that are caused by the interaction of Tm3+ ions with acoustic vibrations (the last term in Eq. 24) and that induce long-range correlations between states of paramagnetic ions to play a dominant role. Due to renormalization (Eq. 17) the absolute values of the parameters of the effective electron-deformation interaction decrease compared to the parameters (Table 2). In calculations of magnetic field effects, magnetic dipole-dipole interactions were taken into account in terms of the self-consistent field approximation. The local magnetic field that acts on Tm3+ ions in the sth sublattice was given by
B
loc
(s) = B + £
4n
Q(s, s0 - ^Nm
M(s' ),
(26)
where M(s') is the magnetic moment of the ion in the sth lattice, Q(s, s') are the lattice sums, which were calculated by the Ewald method; and Nm is the demagnetization factor. The main attention was focused on effects that manifest themselves in magnetic fields oriented in the base plane of the lattice and that induce deformations of the Bg symmetry. The relative change in the dimension of the crystal along the magnetic field direction at 0 angle with the crystallographic
a-axis is:
1
Al/l = e(A2) + - cos 20 e(Bl) + sin 20 e(B2),
2
(27)
Values of Al/l that were obtained upon substitution into Eq. 27 of the components of the deformation tensor calculated in accordance with Eq. 20 satisfactorily agree with the longitudinal magnetostriction data of cylindrical LiTmF4 specimens with their generatrices along with directions [100] and [110] that were measured at the temperature of 4.2 K in fields with a strength of up to 3 T (see Fig. 4a).
The inverse magnetostrictive effect (a change in the crystal field as a consequence of deformations of the crystal lattice) is most clearly pronounced in splittings of non-Kramers doublets in fields with the same directions as a zero g-factor. In this case, the direct (Zeeman) interaction of a paramagnetic ion with the magnetic field splits the Kramers doublet only in the second order
- 9
" (a) i . w i i _____' . i
0 1 2 3
£ o
I
Figure 4. (a) Field dependences of the longitudinal magnetostriction of a single crystal LiTmF4 at a temperature of 4.2 K in external magnetic field B: (1) B || [100] and (2) B || [110]; (b) field dependence of the splitting of doublet r34(1) in a magnetic field that oriented in the base plane of a single crystal LiTmF4 at an angle of ^ = 10° to the a axis. Results of calculations are shown by solid curves, and symbols correspond to experimental data from [4].
0
180
Figure 5. Measured (symbols) and calculated (curve 1) orientational dependence of the splitting of doublet r34(1) with an energy of 31 cm-1 in external magnetic field B = 4.05 T in the base plane of a single crystal LiTmF4; T = 4.2 K. Curve 2 was obtained without taking into account the multipole interaction [8].
of the perturbation theory. Considerably larger splittings can occur as a consequence of field-incident changes in the lattice structure. In particular, anomalous splittings dv of doublet r34(1) of thulium ions, which significantly change with the orientation of the magnetic field in the base plane of the lattice, were revealed in optical spectra of the crystal LiTmF4 [17]. As can be seen from Fig. 5, the neglection of multipole interactions leads to a strong discrepancy between the results of calculations performed with the use of parameters of the electron-deformation interaction that was determined from piezospectroscopic measurements [8] and the experimental data (the calculated maximal splitting is more than two times greater than the measured splitting).
90
Figure 6. Measured (symbols) [18] and calculated dependences of the magnetization of a single crystal LiTmF4 on the orientation of external magnetic field B ((1) — 0.3, (2) — 1.0, (3) — 2.0, (4) — 3.0, and (5) — 5.5 T) in the base plane of the lattice; T = 4.2 K. The dashed curve shows the results of the calculation (B = 3 T) that was performed without taking into account the multipole interaction.
However, if the operator (21) which contains renormalized effective constants of the electron-deformation interaction is used, the calculated field Fig. 4b and orientational Fig. 6 dependences of splitting dv of doublet r34(1) agree well with the measured dependences.
Changes in the electronic structure of paramagnetic ions that are caused by lattice deformations in a magnetic field give rise to corresponding additional contributions to the field-induced magnetization. As in the case of the splitting of doublet r34(1) considered above, we obtained overestimated values of magnetic moments of thulium ions in magnetic fields that were perpendicular to the c axis and the strength of which was above 1T (Fig. 4b) without taking into account multipole interactions. The measured temperature, field, and orientational dependences of the magnetization is reproduced well only in calculations performed using parameters of the electron-deformation and multipole interactions that are the same to used ones in the above-considered calculations of elastic constants, magnetostriction, and spectrum of thulium ions.
5. Magnetic properties of the ising dipole ferromagnet LiTbF4
In the crystal field the fundamental term 7F6 of the Tb3+ ion is split into seven sublevels corresponding to the irreducible representations r (three singlets), r2 (four singlets), and r34
(three doublets) of point-group symmetry S4. The ground state of the Tb3+ ion is characterized by a quasi-doublet consisting of two singlets (k = 1,2) with the initial splitting 5 & 1cm-1; in this case, the A energy of the nearest neighbour excited state (the r34 doublet) is of the order of 100 cm-1 [3]. At low temperatures (kBT ^ A) only the fundamental quasi-doublet is virtually filled. Since sublevels of the r2 symmetry are mixed only with the component of the external magnetic field Bo that is oriented along the [001] axis of the crystal lattice (we used the crystal system of coordinates in which the z-axis is aligned parallel to the c axis), the magnetic properties of the LiTbF4 single crystal can be considered within the model of an Ising three-dimensional magnet in a transverse magnetic field whose role is played by the initial splitting 5 of the quasi-doublet. The Hamiltonian of the system can be represented in terms of the operators of the effective spin moment components Sla of the i-th Tb3+ ion (S = 1/2) in the following form:
= -1E
2
(gll№)2 K..+J..'
1 ij + Jij
szSj - E № + gmbBzSj). (28)
Here the first sum corresponds to the pair magnetic dipole-dipole and exchange interactions, Kjj = v(3zj — r2j)/r5j, v = a2c/2 is the volume of the unit cell, and rj is the vector connecting two Tb3+ ions. The intrinsic magnetic moment of the Tb3+ ions is characterized by only the z component oriented along the symmetry axis of the crystal lattice. By virtue of the large effective g factor (gy = 2gL = 17.85 [10], where gL = 3/2 is the Lande splitting factor and J is the total angular momentum of the ion), the magnetic dipole-dipole interaction plays a dominant role in magnetic ordering. Since the magnetic dipole-dipole interaction is a long-range interaction, the calculation of the low-temperature magnetic characteristics of the LiTbF4 single crystal within the self-consistent field approximation is justified.
6. Exchange interaction in LiTbF4 single crystal
In the self-consistent field approximation the local magnetic field acting on Tb3+ paramagnetic ions in a single-crystal sample along the [001] direction can be represented as the sum of the external magnetic field B0z, the molecular field Bm = AM, and the demagnetizing field Bd = —DM, where M is the magnetization of the sample, D is the demagnetizing factor (for a spherical sample, D = 4n/3), A = Ad + Aj is the molecular field constant, and AD = Dj/2 = 4n/3 ■ 1.3965 is the Lorentz field constant obtained by summation over the lattice (in our case, the unit cell contains two paramagnetic ions). The exchange field constant AJ = vJ2j Jj/2(g||№)2 is determined by the exchange interaction between the nearest neighbour ions [19]. In particular, based on analyzing the data on quasi-elastic neutron scattering by a LiTbF4 crystal, estimated the exchange integrals for interactions of terbium ions with the nearest neighbour ions (for four ions located at a distance of 0.375 nm, J1 = —0.3624 ± 0.125 cm-1) and with the next-to-nearest neighbour ions (for ions located at a distance of 0.518nm, J2 = 0.07 ± 0.14 cm-1). In this case the molecular field constant is estimated to be A = 5.23 ± 0.56. In the self-consistent field approximation the molecular field constant A can be related to the observed temperature of magnetic ordering through the relationship
1 = A ^ th^kBTC • (29)
By substituting 5 = 1.1 cm 1 and Tc = 2.88 K into relationship Eq. 29, we can estimate the molecular field constant from below: A > 4.32. For the paramagnetic phase at sufficiently low temperatures (Tc < T < A/kB) and in weak magnetic fields B0z ^ 5/(gy^B) ~ 0.1 T, the magnetization along axis c of the lattice can be represented by the expression Eq. 30
Mz = b
0z
H th(<V2fcBT) - 1|+ D
-i
(30)
In the case when the dipole-dipole interaction between paramagnetic ions dominates in the crystal, a decrease in temperature below the Curie point leads to the formation of domains with a zero demagnetizing factor (i.e., long thin cylinders with a generatrix along the magnetization axis are formed) [19]. Note that the total magnetic moment of the crystal MV (V is the crystal volume) remains zero. In a weak external magnetic field, the domains are partially reoriented; as a result, the demagnetizing field BDz completely compensates for the external magnetic field B0z and the magnetization of the crystal is equal to the limiting magnetization defined by expression Eq. 30; i.e., Mz = B0z/D at T ^ Tc and under the condition B0z < M0/D = Bc. Here, M0 is the spontaneous magnetization at a specified temperature of the sample, which satisfies the
B(T)
Figure 7. Magnetization of the LiTbF4 single crystal as a function of the external magnetic field strength (M || B0 || c) at temperatures of (1) - 2, (2) - 5, (3) - 10, (4) - 20, and (5) - 77 K. Symbols indicate the experimental data, and solid lines represent the results of calculations.
T"1 (K-1)
Figure 8. Temperature dependences of the magnetization of the LiTbF4 single crystal in the magnetic field B0 || c. Magnetic field strength B0: (1) - 5.0, (2) - 1.0, (3) - 0.6, (4) - 0.3, (5) - 0.1, and (6) - 9 • 10-4 T. Symbols indicate the experimental data, and solid lines represent the results of calculations using (32). Spontaneous magnetization is taken from [4].
following relationship
S th[S2 + (ff||№AMo)]1/2/2fcBT
[S2 + (g|№ AMo)]1/2 th(S/2kBTc) • ( j
Thus, the dependence of the induced magnetization on the external magnetic field is linear at T < Tc and the saturation of magnetization occurs upon reorientation of all the domains involved. As can be seen from Fig. 7, the magnetization measured at a temperature of 2 K is a linear function of the external magnetic field (straight line 1); in this case, the demagnetizing factor is estimated to be D = 3.91 ± 0.2.
The experimental temperature dependence of the magnetization measured in a very weak magnetic field is shown in Fig. 8 (curve 6). The observed increase in the magnetization with a decrease in the temperature is adequately described by expression Eq. 30 with the molecular field constant Aexp = 4.525 and demagnetizing factor D = 4.11, which correspond to the hori-
zontal portion of the curve Mz. The solid lines in Fig. 7 and Fig. 8 (except for curve 7 in Fig. 8) represent the results of our calculations with the same molecular field constant Aexp = 4.525. Therefore, the ratio of the energy of exchange (antiferromagnetic) interaction between the nearest neighbour ions to the energy of their magnetic dipole-dipole interaction can reach a value of -0.42. The derivative dMz(T, B0z)/dT has a discontinuity at the point of intersection of the curves Mz (T) and M0(T), which represent the temperature dependences of the induced magnetization and the spontaneous magnetization, respectively. Curve 7 in Fig. 8 depicts the temperature dependence of the spontaneous magnetization of the LiTbF4 crystal according to the magnetic neutron scattering data obtained in [4]. It can be seen from the temperature dependences of the magnetization measured in magnetic fields of 9 • 10-4 and 0.1 T (Fig. 8) that, in actual practice, similar information on the spontaneous magnetization of a dipole ferromagnet can be obtained from significantly simpler measurements of the induced magnetization.
7. Parameters of the crystal field in LiTbF4 single crystal
For high temperatures (at which the populations of the excited Stark sublevels of the fundamental term 7F6 for the Tb3+ ions should be taken into account) and strong magnetic fields with considerable mixing of the wave functions of the fundamental quasi-doublet and excited sublevels, the magnetization can be calculated as a function of the temperature and the local magnetic field Bioc = B0 + BM + BD in the framework of the single-ion approximation; that is,
Here, H is the single-ion Hamiltonian operator involving the energy of the rare-earth ion in the crystal field and the energy of the Zeeman interaction:
The dependence of the magnetization on the external magnetic field B0 can be obtained from relationship Eq. 32 by comparing M(Bjoc) and B0 = Bjoc - (A - D)M(Bjoc). It should be noted that there is a wide scatter of the data available in the literature on the crystal field parameters for terbium ions in isostructural crystals LiTbF4 and LiYF4 : Tb3+ (Tb3+ ions substitute for Y3+ ions) [19]. Since the form of Hamiltonian Eq. 22 remains unchanged upon an arbitrary rotation of the coordinate system about the [001] quantization axis and, in particular, the x and y axes can be chosen in the basal plane of the crystal lattice in such a way that one of the four parameters B±4, B±4 becomes zero. Information regarding the g factors and relative energies of the sublevels of the fundamental term of the paramagnetic ion, which is derived from investigations into the linear magnetic susceptibility and optical spectra, makes it possible to determine only six of the seven parameters Bpk involved in the Hamiltonian operator (28). With the aim of determining the parameters of the crystal field in LiLnF4 crystals (Ln = Tb, Dy, Ho, Er, and Yb), Nevald and Hansen [20] studied the nonlinear magnetic susceptibility anisotropy in the basal plane of the crystal system and analyzed the results of magnetic measurements in terms of the single-ion Hamiltonian Eq. 33. However, the magnetic anisotropy of concentrated rare-earth paramagnets at low temperatures is characterized by a substantial contribution of the magnetoelastic interaction. Therefore, in the general case, the Hamiltonian operator Eq. 33 should contain terms accounting for the interaction between the rare-earth ion and the lattice strain induced by the magnetic field. In LiTbF4 crystals the magnetoelastic interaction is suppressed by virtue of the large difference between the energy of the ground state (quasidoublet) and the energy of the nearest neighbour excited states [6]. For the same reason, the Van Vleck paramagnetic susceptibility in the ground state is small and its anisotropy in the basal
M
Sp[J exp(-H/fcßT)] v Sp[exp(-H/fcßT)] '
(32)
H = Hcf + gL№JBloc.
(33)
T(K)
Figure 9. Temperature dependences of the magnetization of the LiTbF4 single crystal in the magnetic field B0 ^ c. Magnetic field strength B0: (1) — 5, (2) — 3, and (3) — 1 T. Symbols indicate the experimental data. Theoretical temperature dependences of the magnetization is calculated using the crystal field parameters obtained in this work (solid lines) and (at B0 = 5 T) the parameters taken from (a) [20] (dotted line), (b) [3] (dashed line), and (c) [19] (dot-dashed line).
plane is virtually absent. Nonetheless, the transverse magnetization can change significantly with an increase in the temperature and upon filling of excited non-Kramers doublet states, which are characterized by a considerable mixing with closely spaced singlet states in the magnetic field Bo ± c. Figure 9 presents the results of our measurements of the transverse magnetization of the LiTbF4 single crystal over a wide temperature range in magnetic fields with inductions of 1, 3, and 5 T. Owing to the very large difference (up to two orders of magnitude in a magnetic field with an induction of 1 T) between the longitudinal and transverse magnetic susceptibilities of the terbium ions, even an insignificant deviation of the external magnetic field direction in the basal plane may introduce large errors into the results of the magnetic measurements. We compared the results of magnetic measurements at different orientations of the external magnetic field with respect to the symmetry axis of the crystal lattice and estimated the error ó< < 0.8° in the orientation of the magnetic field with respect to the (x, y) plane in the experiments, the results of which are presented in Fig. 9. An insignificant increase in the magnetization with an increase in temperature from 3 to 20 K is caused by a decrease in the contribution from the longitudinal component of the magnetic moment of the terbium ions (Fig. 8). With a further increase in temperature, the transverse magnetization increases substantially and reaches a maximum at a temperatures of 80-90 K. The magnetization M y B0 ± c was calculated according to relationships Eq. 32 and Eq. 33.
The results of these calculations are presented in (Fig. 9). The non-monotonic behaviour of the temperature dependence of the transverse magnetization can be explained by the anomalously large field-induced magnetic moment at the lower sublevel of the non-Kramers doublet r34 (nearest to the ground state) with an energy of 108 cm-1. As can be seen from (Fig. 9), the results of magnetization calculations with the use of the crystal field parameters available in the literature (see Table 5) differ significantly from the results of magnetization measurements.
The small correction to the crystal field parameters calculated in [19] within the model of exchange charges made it possible to describe adequately both the energy of Stark sublevels of the TF6 term (in particular, we calculated the splitting of the fundamental quasi-doublet ó = 1.09 cm-1, the splitting factor gy = 17.90, and the energy of the nearest neighbour doublet
Table 5. Crystal field parameters Bp (cm-1), acting on Tb3+ in LiTbF4 and LiYF4:Tb3+ single crystals.
[21] [16] [22]
B0 237 237 227.5
b4 -54.1 -54.1 -121.2
B0 B6 4 4 -30.3
B44 -1130 -854.2 -946.5
B4-4 0 -739 0
b64 -522 -477 -84.5
b-4 -201 -291 ±93.8
[21] LiYF4:Tb3+ [23] This work
222.5 200 200
-95 -100 -100.5
0.25 -3.56 -2
-1171 -1103 -694
0 0 -829
-534 -529 -435
-427 0 -283
5 = 1.09 cm-1), the temperature and field dependences of the magnetization. The results of our calculations are shown by solid lines in Figs. 7-9. In conclusion, we note that the results obtained in the present work can be useful for the systematization of information on the exchange interaction and crystal field parameters for crystals of double rare-earth fluorides.
To determine ratios of the parameters B±4, B±4, we analyzed presented in [24] dependences of the longitudinal magnetization in LiTbF4 on the external pulsed magnetic fields applied to the oriented samples (rectangular rods with dimensions of 1.5 x 1.5 x 2.5 mm3) in the ab-plane along the [100] and [110] directions (coinciding with the long edge of a rod) at the temperature 1.4 K. The magnetic field varied from zero up to 50 T during 20 ^s, and we calculated the magnetization assuming the adiabatic process similarly to the authors of [24]. Changes of the temperature AT of the magnetic subsystem induced by the increase of the field with steps AB were calculated using the following expression
,(Heff (B)m)-(Heff (B))(m) (Heff(B)2) - (Heff(B))2
There are two terbium ions, Tb1 and Tb2, with coordinates r(Tb1) = (0, 0, c/2), r(Tb2) = (0, a/2, —c/4) in the unit cell. According to neutron scattering data [25] at 100 K, the lattice constants of LiTbF4 equal a = 0.5181 nm, c = 1.0873 nm. In the magnetically ordered state, the local field at the terbium ion Tbs (s = 1, 2) is written as follows
at = —t ( eff; Cm r ^;(2) Ab. (34)
4n
^ Qap(Tbs, Tbs') + 2—(Ais — N«)5«p
s'=1,2
m (Tb). (35)
(Tbs) = Ba + ^
Here the demagnetizing factors of the rods are Nx = Ny = 0.68 [22], Nz = 0 (for the domains in the system with the dominant dipole-dipole interactions), the lattice sums QTbs,Tbs(xx) = QTbs,Tbs(yy) = 1.18053, QTbs,Tbs(zz) = 0.63894, QTbi,Tb2(xx) = QTbi,Tb2(yy) = 0.41738, QTb1,Tb2(zz) = 2.16525 (in units of 4n/3v) have been computed by the Ewald method, the value of the isotropic exchange (antiferromagnetic) field constant A = —0.316 have been determined previously in [26]. The small initial external field of 0.01 T has been introduced to account for the ordering of magnetic moments along the c-axis. With the increasing magnetic field normal to the magnetic easy axis [001], the gap between the lower sublevel of the first excited r34 state and the ground quasi-doublet quickly diminishes (see Inset (a) in Figure 10), and the peaks in the differential magnetic susceptibility are observed at the anticrossing point. As an example, evolutions of the spontaneous magnetic moment with the increasing transversal external field and of the induced moment along the field are shown in Inset (b) in Fig. 10. As it is seen in Fig. 10, the differential susceptibilities computed with our set of the crystal field parameters match
e(T)
Figure 10. Differential magnetic susceptibilities of LiTbF4 single crystals in magnetic fields B || [100] and B || [110] (in red and blue, respectively), T =1.4K. Symbols - experimental data digitized from Ref. [26], solid and dashed curves represent the results of calculations carried out with or without taking into account the magnetoelastic interactions, respectively. Inset (a) - calculated energies of the three lower sublevels of the 7F6 multiplet relative to the ground state vs. the magnetic field B || [110]. Inset (b) - calculated magnetization components along (induced moment Mx) and normal (spontaneous moment Mz) to the external field B | [100] vs. the field intensity.
satisfactorily experimental data. However, the widths of the calculated peaks are much broader than the measured ones. It is evident that more elaborated study of the quantum phase transition in LiTbF4 in the transversal magnetic fields beyond the mean field approximation is necessary. A comparison of the solid and dashed curves in Fig. 10 shows that magnetoelastic interactions contribute substantially into the magnetization if the magnetic field is strong enough.
8. Magnetization and magnetostriction in a single crystal of LiHoF4
The LiHoF4 crystal is considered in many works as a model Ising dipole ferromagnet with an easy magnetization axis that coincides in direction with the crystallographic axis c. The literature presents several different sets of parameters of the crystal field for holmium ions in LiHoF4 crystals found from the analysis of optical spectra (see Table 6).
Table 6. Crystal field parameters Bp (cm-1), acting on Ho3+ in LiHoF4 single crystals.
[27] [28] [29] This work
B0 219.7 190.35 190 200.35
B0 -87.3 -78.25 -78 -84.25
B0 -3.55 -3.25 -3.2 -3.5
B4 -710 -657.2 869 -667.24
B-4 -612 -568.6 0 -578.63
B4 -387 -364 427 -364.04
b-4 -253.7 -222.3 105 -222.3
I.V. Romanova, M.S. Tagirov Table 7. Experimental and calculated energy levels of holmium ions in LiHoF4.
2s+1lj r Experimental Calculation
34 0 0
2 7.4 7.35
2 23.1 18.88
1 49 49.68
1 61 58.72
34 72.1-78 68.88
5l8 1 202.14
34 259.12
1 259.23
2 267.2
34 280.86
1 282.47
2 295.85
Table 8. The parameters of the interaction of Ho3+ ions with deformations of Ag, Bg symmetry used in the work (cm-1).
P k B^ (A]) Bj (Ag) P k bP (Bg) Bj(Bg)
2 0 -398.2 -1692.9 2 2 2045 4114
4 0 308 1100 2 -2 2305 -886
4 4 5061.3 5280.4 4 2 -854 -1554
4 -4 3761.8 5602.5 4 -2 1449 1423
6 0 235.4 -166.1 6 2 135 -528
6 4 1930.8 3194.9 6 -2 -427 -523
6 -4 950.8 3009.7 6 6 6 -6 -606 -691 -927 -1046
The Stark structure of energy levels is shown in Table 7 in comparison with the experimentally measured values [18] and calculated by diagonalization the above Hamiltonian Eq. 22 on a complete basis of 1001 states. The basic term 5/8 of the electron shell 4f10 is split into 13 levels: 9 singlets and 4 doublets. The ground state is the non-Kramers doublet r34 and the two overlying excited states are singlets r2. The parameters of the Hamiltonian of free Ho3+ ions used in this work are presented in [30]. From Table 7 it can be seen that good agreement is obtained between the experimental data and the calculation.
The magnetostriction in a LiHoF4 single crystal was measured in Ref. [2]. The change in the dimensions of the single crystal was measured along the direction of the crystallographic axis c, and the external magnetic field was applied in the basal plane ab. When calculating the magnetostriction, the refined parameters of the electron-deformation interaction were used. The values of the parameters of the electron-deformation interaction used in this work are listed in Table 8.
The changes in the crystal size Al/l = ^ a/3 nan/ea/ calculated using the above crystal field parameters and the electron-deformation interaction parameters are shown in Fig. 11. It can be seen that the qualitative agreement is very good, but at 1.6 K the calculation gives underestimated values close to the values at 1.8 K.
The field dependences of the magnetostriction were calculated with the refined crystal field parameters and the parameters of the electron-deformation interaction along the direction of the external magnetic field given by the direction cosines na. The results are in qualitative agreement with the measurement data (see Fig. 12).
The results of calculations of the magnetization of a LiHoF4 single crystal as a function of the magnetic field directed along the a and c axes at different temperatures and the angular dependence of the magnetization upon rotation of the field in the basal plane are compared with experimental data in Figs. 13-15. The field and temperature dependences of the magnetization
B2 (T2)
Figure 11. Magnetostriction in a single crystal of LiHoF4 along the crystallographic c axis in a magnetic field applied in the ab plane at different temperatures, depending on the magnitude of the square of the external magnetic field. Points - experimental data [2], solid lines - calculation.
o
0 1 2 B, T
Figure 12. Field dependences at a temperature of 4.2 K of longitudinal magnetostriction in an external magnetic field B (1 - B || [110], 2 - B || [210]) of a single crystal of LiHoF4. The results of the calculations are represented by lines, the symbols correspond to the experimental data. (Experimental data were obtained by V.I.Krotov, personal communication.)
B(T) 1/T (K1)
Figure 13. Field (a) and temperature (b) dependences of the magnetization along the a axis of a single crystal of a LiHoF4. Solid lines are theoretical results, symbols are experimental data [6].
B(T) T1 (1/K)
Figure 14. Field (a) and temperature (b) dependences of the magnetization along the c axis of a single crystal of a LiHoF4. Solid lines are theoretical results, symbols are experimental data [6].
of a LiHoF4 single crystal were measured using a SQUID magnetometer, and the angular dependences of the magnetization were obtained by the inductive method. As follows from the calculations, the significant anisotropy of the magnetization is mainly due to the magnetoelastic interaction, which makes a relatively large contribution to the magnitude of the magnetization at temperatures less than 10 K and in fields B > 0.5 T.
When calculating the magnetization along the crystallographic axis a, an error was taken into account in the orientation of the single crystal (due to the small ball diameter of 2 mm), which was 6° in the ac plane of the crystal lattice. Figure 15 shows the angular dependence of the magnetization of a single crystal of LiHoF4 in the basal plane. It is seen that the magnetization anisotropy is almost absent. The contribution of the electron-phonon interaction is not significant and was not taken into account during the calculation. The anisotropy of
90
LiHoF
4
T=4.2K B ±[001]
1
1
o
270
Figure 15. Angular dependences of the magnetization in the basis plane of LiHoF4 single crystal at T = 4.2K. Solid lines correspond to the results of calculations, symbols correspond to experimental data.
magnetization in the basal plane is very weakly expressed and the magnetoelastic interaction is much weaker than in LiTmF4 and LiDyF4, which is confirmed by the large magnitude of magnetostriction in these single crystals.
The parameters of the crystal field in a single crystal of LiHoF4 and the parameters of the electron-deformation interaction are determined from an analysis of the obtained data and calculations of the magnetization taken into account with the electron-deformation interaction. A self-consistent description of magnetization, the field dependences of magnetostriction in different directions of the applied magnetic field is presented.
9. Magnetic and magnetoelastic properties of LiDyF4 single crystals
A LiDyF4 single crystal is a dipole XY antiferromagnet with magnetic moments of Dy3+ ions directed perpendicular to the axis of the crystal lattice [7]. An analysis of the magnetic dipole and exchange interactions between the Dy3+ ions and the magnetic structure was carried out on the basis of measurements of the low-temperature heat capacity in [31]. The effects caused by the intersection of the energy levels of the Dy3+ ion in a LiDyF4 single crystal in strong pulsed magnetic fields were also investigated previously [32]. In this work, the magnetization of a LiDyF4 single crystal is calculated. The magnetic dipole interaction between the Dy3+ ions was taken into account by the introduction of a local magnetic field Eq. 26. The crystal field on the Dy3+ ion in a LiDyF4 single crystal in the crystallographic coordinate system is determined by a set of 7 parameters (Table 9).
The values of the parameters of the electron-deformation interaction used in this work are listed in Table 10. The parameters of multipole interactions are given in Table 11.
The results presented below were obtained using the numerical diagonalization of the Hamil-tonian of the Dy3+ ions in the complete state basis of the 4f9 electron configuration (the number of states is 2002). The free ion Hamiltonian is written in the standard form with the parameters of electrostatic, spin-orbit and interconfiguration interactions taken from [8]. The calculated magnetization behaviour in an external magnetic field rotating around an arbitrary axis is shown
Table 9. Crystal field parameters Bp (cm-1), acting on Dy3+ in LiDyF4 single crystals.
Dy3+ (4f9) &H15>/2
p k [32] [4] [12] This work
2 0 182 170 165 160±3
4 0 -53 -85 -88 -83 ±3
6 0 1.5 -4.2 -4.4 -4.6±0.2
4 4 -806 -721 -980 -751±5
4 -4 -380 -661 0 -641±5
6 4 -654 -390 -427 -360±5
6 -4 -272 -248 -65 -243±5
Table 10. Interaction parameters between Dy3+ ions and lattice deformations of Bg symmetry in a LiDyF4 single crystal (cm-1). The effective parameters are given for a zero magnetic field and a temperature T = 4.2 K.
pk 22 2 -2 42 4 -2 62 6 -2 66 6 -6
Bk(Bg) 1997 2355 -887 1200 -135 -427 -731 -691
Beffk (Bfl1) 1617 3817 -1497 1766 -322 -549 -614 -1022
Bpfc(Bg) 4114 -1144 -1120 1652 -528 -375 -841 -878
Beffk (B2) 4614 -4406 337 168 -126 -4 -997 -98
Table 11. Parameters of the multipole interaction A j (cm 1) in a LiDyF4 crystal.
1 2 3 4 5 6 7 8
1 -72 -54.8 48.49 -54.8 10.18 26.5 22.5 29.96
2 62.98 -17.8 10.88 -7.13 3.516 16.07 -6.56
3 -2.0 4.572 -3.15 -0.48 -13.6 -4.88
4 -28.6 -1.77 4.771 14.09 1.137
5 -1.94 -1.16 -4.15 -1.7
6 -1.93 -5.53 -6.8
7 -6.56 -9.05
8 -6.29
in Figure 16. The direction of the axis of rotation was determined by the angle 9 between the axis of rotation and the crystallographic axis c and the angle a between the projection of the axis of rotation to the ab plane of the crystal and axis 9. It can be seen that a small deviation of the external magnetic field from the crystallographic axes of the crystal leads to significant distortions of the angular dependence of the magnetization in the basal ab plane of the lattice.
The temperature and magnetic field dependences of the magnetization of a LiDyF4 single crystal were measured using SQUID magnetometer. The experimental data are well described by a theoretical model, within which interionic multipole interactions are taken into account.
90
270
Figure 16. The calculated angular dependence LiDyF4 single crystal magnetization in an external magnetic field in different planes (B = 2T) : 1 - в = 0, a = 0; 2 - в = 5°, a = 30°; 3 - в = 5°, a = 45°; 4 - в = 5°, a = 60° [7].
The results of magnetization calculations using the set of proposed parameters of the crystal field and with the renormalized coupling constants between macro and micro deformations are shown in Figures 17, 18.
The observed non-monotonic behaviour of magnetization at low temperatures with a maximum close to 25 K and a minimum of about 5 K (see Figure 17a) is explained by the specific relationship between the corresponding components of the g-tensor of the main (gy = 1.15) and first excited (gy = 5.3) the sublevel of the 6H15/2 multiplet of the Dy3+ ion in the crystal field, and also taking into account the redistribution of the occupations of the Kramers doublets with a change in temperature. From the comparison of the calculated and measured temperature dependences of the magnetization in the field B || c (see Figure 17), we clarified the values of the parameters of the crystal field (see Table 9). We note that the calculated energy of 18.6 cm-1 of the first excited sublevel of the main 6H15/2 multiplet differs noticeably from the corresponding energy of 14.2 cm-1 of Dy3+ ions in a LiYF4 crystal [22]. We determined the parameters of the crystal field and compared with the literature data in Table 9. The significant role of electron-deformation interaction in the formation of magnetization in concentrated rare-earth paramagnets at low temperatures becomes evident when we analyze the magnetization of LiDyF4 induced by a magnetic field perpendicular to the c axis (Figure 18). In this case, as shown in Fig. 18b, taking into account only the Zeeman interaction, we observe highly underestimated magnetic moments of the Dy3+ ion in fields exceeding 1 T.
It should be noted that the multipole interaction introduces a noticeable renormalization to the electron-deformation interaction, and one can expect a strong softening of the LiDyF4 crystal lattice, similar to that observed in DyVO4 [32]. The calculated temperature dependences of the elastic constants LiDyF4, which are determined by the induced Bg symmetry deformations, are shown in Fig. 19. In fact, the elastic constants change significantly with decreasing temperature to 10 K and subsequently remain unchanged at low temperatures.
At low temperatures a giant magnetostriction is induced in a magnetic field ~ 1T in a LiDyF4 single crystal at T = 4.2 K. The calculated changes in the crystal size along the direction of
50
^ 40 O)
£ 30
^ 20
10
0
■ (a)
7 5T \
- ^ 3.5T \ \
2T \ \\
1
50
40
30
20
10
0
" (6) 25 K
/11 K
_ CTl Ts
/ or ■ / p / / 0
- / y</n/ p /// 77 K
- p/y f/X 100 K
jfife-i . i . 300 K I.I.I.
10 100 0 1 2 3 4 5
T (K) B (T)
Figure 17. Measured (symbols) and calculated (solid lines) temperature (a) and field (b) dependences of the magnetization in LiDyF4 for different values of the magnetic field (B || c) and temperature [7].
O)
E
CD
0.0 0.1 0.2 0.3 0.4 0.5 0 1 2 3 4
r1 (K-1) B (T)
Figure 18. Magnetic fields dependences (solid lines - calculations, symbols - experiment) of the magnetization in LiDyF4 in a magnetic field. The dashed curve shows the calculated magnetization at T = 5 K without taking into account the electron-deformation interaction [7].
the external magnetic field given by the guide cosines Al/l = ^a^ nan@eap, using the above parameters of the crystal field, are in good agreement with the measurement data (see Figure 20) when taking into account temperature and magnetic field-dependent contributions to elastic constants due to the electron-deformation interaction.
The temperature and field dependences of the magnetization and field dependences of the strongly anisotropic magnetostriction in LiDyF4 single crystals were measured. The giant magnetostriction of «10-3 was observed in a rather small magnetic field of 1 T directed along the [110] crystallographic axis at liquid helium temperature. The sets of parameters which define interactions of the Dy3+ ions with the crystal field and the crystal lattice uniform deformations
T(K)
Figure 19. Calculated temperature dependences of elastic constants in a zero magnetic field. The values of the corresponding elastic constants of the LiYF4 single crystal are shown by the symbols [7,33].
B(T)
Figure 20. Field dependence of the magnetostriction in the two directions of the applied magnetic field in LiDyF4 single crystal at T = 4.2 K. The solid lines - the theory, symbols - experimental data [7].
as well as the parameters of the multipole interactions between the Dy3+ ions were determined. The results of the measurements are well reproduced by simulations. The magnetoelastic interaction in LiDyF4 contributes essentially to the magnetization in external magnetic fields oriented in the basis plane of the crystal lattice at low temperatures.
10. Conclusion
We have measured and calculated elastic, spectral, and magnetic characteristics of a single crystals LiLnF4 taking into account the electron-deformation and multipole interactions. It has been shown that the entire set of experimental data available at present time on the field and temperature dependences of elastic constants, magnetization, magnetostriction, and energy levels can be reproduced not only qualitatively, but also quantitatively, with a satisfactory accuracy in terms of a self consistent theory that uses a unified set of parameters of the crystal field and electron-deformation and multipole interactions.
Acknowledgments
The research was supported in part by RFBR (project No. 18-42-160012r_a). The authors are
thankful to Dr. SafiullinK.R. for the help during article preparation.
References
1. Boldyrev K.N., Popova M.N., Malkin B.Z., Abishev M.N. Phys. Rev. B 99, 041105 (2019)
2. Dunn J.L., Stahl C., Macdonald A.J., Liu K., Reshitnyk Y., Sim W., Hill R.W. Phys. Rev. B 86, 094428 (2012)
3. Vasyliev V., Villora E. G., Nakamura M., Sugahara Y., Shimamura K. Opt. Express 20, 14460 (2012)
4. Aminov L.K., Malkin B.Z., Teplov M.A. Handbook on the Physics and Chemistry of Rare Earths 22, 295 (1996)
5. R0nnow H., Jensen J., Parthasarathy R., Aeppli G., Rosenbaum T.F., McMorrow D.F., Kraemer C.W. Phys. Rev. B 75, 054426 (2007)
6. Al'tshuler S.A., Krotov V.I., Malkin B.Z. JETP Lett. 32, 214 (1980)
7. Romanova I.V., Korableva S.L., Krotov V.I., Malkin B.Z., Mukhamedshin I.R., Suzuki H., Tagirov M.S. J. Phys.: Conf. Ser. 478, 012026 (2013)
8. Romanova I.V., Malkin B.Z., Tagirov M.S. Opt. Spectrosc. 116, 897 (2014)
9. Aminov L.K., Malkin B.Z. Dynamics and Kinetics of Electronic and Spin Excitations in Paramagnetic Crystals, Kazan State University, Kazan (2008) (in Russian)
10. Eyring L., Gschneidner K.A., Lander G.H. Handbook on the Physics and Chemistry of Rare Earths 32 (2002)
11. Christensen H.P. Phys. Rev. B 19, 6573 (1979)
12. Kupchikov A.K., Malkin B.Z., Rzaev D.A., Ryskin A.I. Fizika Tverdogo Tela 24, 2373 (1982) (in Russian)
13. Klimin S.A., Pytalev D.S., Popova M.N., Malkin B.Z., Vanyunin M.V., Korableva S.L. Phys. Rev. B 81, 045113 (2010)
14. Abubakirov D.I., Matsumoto K., Suzuki H., Tagirov M.S. J. Phys.: Cond. Mat. 20, 395223 (2008)
15. Abdulsabirov R.Y., Kazantsev A.A., Korableva S.L., Malkin B.Z., Nikitin S.I., Stolov A.L., Tagirov M.S., Tayurskii D.A., van Tol J. XI Feofilov Symposium on Spectroscopy of Crystals Activated by Rare-Earth and Transition Metal Ions 4766, 59 (2002)
16. Al'tshuler S.A., Malkin B.Z., Teplov M.A., Terpilovskii D.N. Sov. Sci. Rev.: Sect. A 6, 61 (1985)
17. Abdulsabirov R.Y., Kazantsev A.A., Korableva S.L., Malkin B.Z., Nikitin S.I., Stolov A.L. J. Lumin. 117, 225 (2006)
18. Abubakirov D.I., Kuzmin V.V., Suzuki H., Tagirov M.S., Tayurskii D.A. J. Phys.: Conf. Ser. 51, 135 (2006)
19. Romanova I.V., Malkin B.Z., Mukhamedshin I.R., Suzuki H., Tagirov M.S. Phys. Solid State 44, 1544 (2002)
20. Hansen P.E., Nevald R. Phys. Rev. B 16, 146 (1977)
21. Christensen H.P. Phys. Rev. B 17, 4060 (1978)
22. Nevald R., Hansen P.E. Physica B+C 86, 1443 (1977)
23. Liu G., Jacquier B. Spectroscopic Properties of Rare Earths in Optical Materials 83, 1443 (Springer Science and Business Media, 2006)
24. Bumagina L.A., Krotov V.I., Malkin B.Z., Khasanov A.K. Zh. Eksp. Teor. Fiz 80, 1543 (1981) (in Russian)
25. Kjaer K., Als-Nielsen J., Laursen I., Larsen F.K. J. Phys.: Cond. Mat. 1, 5743 (1989)
26. Kazei Z.A., Snegirev V.V., Broto J.M., Abdulsabirov R.Y., Korableva S.L. J. Exp. Theor. Phys. 115, 1029 (2012)
27. Romanova I.V., Abdulsabirov R.Y., Korableva S.L., Malkin B.Z., Mukhamedshin I.R., Suzuki H., Tagirov M.S. Magn. Reson. Solids 8, 1 (2006)
28. Shakurov G.S., Vanyunin M.V., Malkin B.Z., Barbara B., Abdulsabirov R.Y., Korableva S.L. Appl. Magn. Res. 28, 251 (2005)
29. Christensen H.P. Phys. Rev. B 19, 6564 (1979)
30. Romanova I.V., Klochkov A.V., Korableva S.L., Kuzmin V.V., Malkin B.Z., Mukhamedshin I.R., Suzuki H., Tagirov M.S. Magn. Reson. Solids 14, 12203 (2012)
31. Beauvillain P., Chappert C., Renard J.P., Griffin J.A., Laursen I. J. Magn. Magn. Mat. 15, 421 (1980)
32. Kazei Z.A., Snegirev V.V., Chanieva R.I., Abdulsabirov R.Y., Korableva S.L. Phys. Solid State 48, 726 (2006)
33. Blanchfield P., Saunders G.A. J. Phys. C: Solid State Phys. 12, 4673 (1979)
