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aänetic Resonance in Solids
Electronic Journal
Volume 19, Issue 1 Paper No 17101, 1-17 pages 2017
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Hugo Keller (University of Zürich, Zürich) 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) Valentin Zhikharev (KNRTU, Kazan)
In Kazan University the Electron Paramagnetic Resonance (EPR) was discovered by Zavoisky E.K. in 1944.
Angular and temperature dependencies of EPR linewidth and Gorter relaxation rates in concentrated paramagnets: Application to La0.9Sr0.1MnO3 and La0.875Sr0.125MnO3
E.Kh. Khalvashi 1, *, N.P. Fokina 2, M.O. Elizbarashvili 3 1 Batumi Shota Rustaveli State University, Ninoshvili str. 35, Batumi 6010, Georgia 2 Department of Science, Georgian Technical University, Kostava str. 77, Tbilisi 0175, Georgia
3 Vladimir Chavchanidze Institute of Cybernetics, Georgian Technical University, Sandro Euli str. 5, Tbilisi 0186, Georgia
*E-mail: [email protected]
(Received December 23, 2016; revised February 2, 2017; accepted February 3, 2017)
The angular and temperature dependencies of the electron paramagnetic resonance (EPR) linewidth in the weak constant field and of the relaxation rates measurable by the Gorter type experiments in zero constant magnetic field (Gorter RRs) are analytically investigated in magnetically concentrated paramagnets with the dominating exchange interaction. The consideration is restricted to the experiments, where the EPR linewidth is both anisotropic and linear over the temperature. It is suggested that under such conditions the EPR broadening is caused by the spin-lattice relaxation of the anisotropic interaction via the one-phonon mechanism. The analytical results are brought to the form suitable for the extraction of the Dzyaloshinsky-Moriya and the crystal field interaction constants from the experiment. It is shown that the EPR linewidth at the constant field direction along any crystal axis is equal to the half sum of the zero-field RRs for the two other crystal axes. The obtained results are successfully used for the interpretation of the EPR experimental results in the La0.9Sr0.1MnO3 and La0.875Sr0.125MnO3 single crystals in the Jahn-Teller strongly distorted phase in the definite temperature interval. The angular dependencies of the Gorter RRs and the EPR linewidth are presented graphically at the constant field continuous rotations in the three crystallographic planes in La0.9Sr0.1MnO3.
PACS: 76.30.-v, 75.30.Gw, 76.60.Es.
Keywords: EPR linewidth angular dependence, anisotropic relaxation rates, one-phonon spin-lattice relaxation.
1. Introduction
The lightly doped manganites La1_xSr.MnO3 show in the paramagnetic phase intense exchange-narrowed electron paramagnetic resonance (EPR) signals with a large variation of the EPR linewidth as a function of the temperature. Moreover, the single crystals exhibit a pronounced anisotropy of the linewidth in the Jahn-Teller strongly distorted O /1, PM phase and a less anisotropy in weakly distorted O /1, PM phase (see the x - T phase diagram in Refs. 1, 2).
The systematic weak field EPR investigations on La1-xSrxMnO3 eight single crystals with the Sr dopings 0 < x < 0.2 were carried out in [2]. The EPR linewidth angular dependence for x = 0.05 and temperature dependencies for all samples were presented there. The angular dependence was interpreted as caused by the Dzyaloshinsky-Moriya (DM) antisymmetric interaction along the antiferromagnetically coupled b axis (we use here the axis notations according to the papers [3]). Further, in the detailed paper [3], which were the refinement of the paper [2] for La0 95Sr0 05MnO3 sample, it was shown that one has to take into account also the DM interaction HDM within the ferromagnetically coupled ac plane. Besides, another source of EPR broadening, namely, the interaction with the crystal field (CF) HCF was taken in the Ref. 3 into account. It should be mentioned that the linewidth temperature dependence in [3] was sufficiently weak (of the form (TL -dCW)/ TL , where TL is the lattice temperature, and 6CW is the compound Curie-Weiss
temperature). Therefore, the pure spin origin of the EPR broadening was suggested there, without taking the spin-lattice relaxation (SLR) into account. Since the strong exchange narrowing takes place
in these concentrated paramagnets, the EPR linewidth was determined by the second moment conditioned by the anisotropic interactions Han = HDM + HCF divided by the exchange frequency a>x [4]. Differing from the weak temperature dependence in [3], the linear temperature dependence AHepr (Tl ~) = AHepr (dCW) + b (TL -0cw ) of the anisotropic linewidth was observed in some samples
in [2] within the definite temperature range. For instance, there is a temperature interval for samples with x = 0.1 and x = 0.125 , where the EPR linewidth is both anisotropic and linear. The linear increase of the linewidth was ascribed in [2] to the one-phonon SLR, as it was suggested earlier in Refs. 5 and 6 for the same range of the doping. However, the one-phonon SLR of a Zeeman subsystem of magnetic ions should have Tl directly proportional dependence and should not be anisotropic. Besides, in [5] the EPR linewidth values for the three crystals ( x = 0.1, x = 0.2, x = 0.3) were essentially the same at the frequencies v = 9 GHz and v = 36 GHz . If the one-phonon SLR of a Zeeman subsystem would be the reason of the EPR broadening, there would be v2 proportional dependence of its rate. It should be mentioned here that recently we suggested the weak field EPR broadening model conditioned by the one-phonon mechanism of the SLR most of all of the Han freedom degrees rather than these of the Zeeman one for the concentrated paramagnets with the dominating exchange interaction [7]. It was demonstrated there that the corresponding SLR rate and, consequently, the EPR linewidth are proportional to the anisotropic second moments conditioned by H an rather than proportional to v2. On the other hand, just the anisotropic second moments are able to cause the angular dependence of the EPR linewidth observed in [2, 3]. So, to our mind, the one-phonon SLR mechanism of the EPR broadening proposed in [7], in which the linewidth is anisotropic and does not depend on v, should be appropriate both for the interpretation of the experiments of Refs. 5, 6 and experiments on the single crystals La09Sr01MnO3 and La0 875Sr0125MnO3 in the definite temperature intervals in Ref. 2.
The same mechanism should be valid also for the relaxation rates (RR) measurable in the Gorter type experiments on La09Sr01MnO3 and La0875Sr0125MnO3 in the zero constant field ("zero-field" RR in terms of Refs. 8, 9). It was shown in [7] that the weak field EPR linewidth and the Gorter RRs are connected with one another by the simple linear relation. The particular case of this relation was revealed in Ref. 10 (see also the references therein) in the experiments on another magnetically concentrated crystal CuCl2 • 2H2O: the EPR half width on the half height at the constant field direction along any crystal axis is equal to the half sum of the zero-field RR measured by means of the Gorter method for the two other crystal axes (further, "Gorter RRs"). There was also an approach of Ref. 11 (see also [12] and [13]), where the different RRs were ascribed to the magnetization components along the different axes transverse to a constant field in the laboratory frame of reference in the case of the arbitrary anisotropic interaction.
Possessing the analytical temperature and angular dependencies of the EPR linewidth and the zero-field RRs, both including DM and CF parameters, it is possible to duplicate by the Gorter type experiments the obtaining of the interaction constants from EPR. We would like especially attract the attention of the experimentalists to such a possibility. In the given paper, we suppose to refine the results of our paper [7] and to apply them to the temperature and angular dependence interpretation of the EPR linewidth in [2] and to the description of these dependencies for the "zero-field" Gorter RRs in the La09Sr01MnO3 and La0875Sr0125MnO3 compounds within the definite temperature range.
2. The statement of the problem
We consider here the spin-system of the single crystals of the magnetically concentrated compounds La09Sr01MnO3 and La0875Sr0125MnO3 interacting with the lattice in the weak constant field (the condition of the EPR constant field weakness will be presented in Appendix 1).
The Hamiltonian of the problem is as follows:
h = h0 + hsph, h0 = hi + hm + hz + hph, (1)
where the isotropic Heisenberg superexchange between Mn ions via the oxygen ions H"x = -jI Si ■ Sj
•< j
is the dominant interaction (J is the superexchange constant), and the EPR anisotropy is conditioned by the antisymmetric Dzyaloshinsky-Moriya (DM) interaction hdm and the interaction with the crystal field (CF) hcf (their sum is denoted by han). So, we'd like to bring the explicit expressions of the DM interaction hdm between i and j spins and the one-spin interaction hcf:
Hdm (i, j) = d j [Si X S j ], Hcf (j) = i ^sjs;,
m,n
where d^ is the Dzyaloshinsky-Moriya vector for the pair of i, j spins S,, Sj; D^ are the tensor components of the CF interaction constant; m, n = X, Y, Z. The axes X, Y, Z of the laboratory frame of reference (LFR) are determined in the EPR experiments by the external magnetic field along the Z axis and the varying field along the X axis. hph is the phonon Hamiltonian; we assume
that the phonons are at the lattice temperature. As in the case of EPR in the sited experimental papers, here the Zeeman interaction of Mn ions hz is supposed to be the smallest one. A model Hamiltonian is taken for the spin-lattice interaction
Hph = I ea,gnapsn = I ^ {g^sX + Gl^ + gzapsz}, (2)
n,a ,p a ,p
where e^^ is the component of the crystal deformation tensor, g"^^ is the component of the spin-phonon bond tensor.
We suppose that in the weak constant field it is the spin subsystem hnn + hz ~ hnn, which exchange energy with the lattice. The interaction h'sx >> han provides the fast modulation of the local magnetic fields, therefore h eisx cannot participate in the spin-lattice relaxation, as a separate reservoir (see Ref. 14). Since the stated problem is essentially many-particle, we shall proceed using the thermodynamical model. At that, we suppose that due to the strong spin-spin interactions the subsystem hnn + hz ~ hnn reaches the internal equilibrium faster than SLR can occur. Therefore, the Zubarev method [15] of the nonequilibrium statistical operator was applied by us leading to the following formula for the SLR rate (see Appendix 1 for details)
17 Tsph={$ ipw^Trntl te)2 (Mn I (-)(TL-ecw(3)
where n = X, Y, Z; a>m is the Debye frequency; pcr is the crystal density; tc = h / J is the exchange correlation time; v is the average sound velocity; haan is the mean-square quantum of han; a is the spin-orbit coupling parameter, and A is the distance of the excited orbit level from the ground one; (mn2) («) is the second moment calculated with the n-th component of the magnetization of the
exchange non-coupled spins - in our case, effectively, at the infinitely large temperature. The expressions for the second moments (m2x) (®), (m\) («) conditioned by the interactions hdm
and hcf, which enter the Eq. (3), with their angular dependencies in the crystallographic frame of reference (CFR) are brought in the Section 4 (some raw results were presented in Ref. 16).
3. Gorter RRs, EPR linewidth, and their connection
Let us consider the Gorter type experiment with the low-frequency field directed along the definite axis. Suppose that the Gorter low-frequency field is directed along the X or Y axis of our LFR. Then SX or SY will be the only non-zero variable spin component in Eq. (2), correspondingly. Just SX or SY will determine then the form of Hsph: in the first case Hph = Y ed pQxdpSx, while in the
a,P
second case HSph = Y ex,pGXpSY.
x,p
Therefore, the following expressions for the relaxation rates measurable in the corresponding experiments of the Gorter type (Gorter RRs)
1/Txy =l^2 ^2 ^
y (qXp) (m2xy ) hkr t -eCW), AJ 2x2h1p 3a1 £p^xp> K 'any ' bKl cw'
y r cr an x ,p
(4)
are following from Eq. (3). It was shown in Ref. 7 with the help of the macroscopic consideration that the T-1, T— relaxation rates are at the same time the relaxation rates of the X and Y components of the macroscopic magnetization.
On the other hand, at the EPR experiment the both S X and S Y are the non-zero variable spin components. Then Hsph = Y ex: p'{§Xcp SX + GXXp SX ] and the EPR relaxational broadening
x ,p x'p x x
A^Z™ = (1/ 2)Ts-1 [17], corresponding to the SLR rate T^, is equal to
A<r =|-
G2
(O.T,. 1
2 Y ( )2 (M2X ) (-) + (xp )) (MY ) (»)] kB (Tl - ecw). (5)
A J 2„~~k-pcrV5 3a;, -xp
Comparing (5) with (4), we get the known from the literature [11, 12] relation (further,"Gorter-EPR" relation) between the EPR linewidth and the Gorter RRs in the LFR
AaEPR =(1/2)( + T- ),
(6)
which is confirmed also by the macroscopic consideration (see Ref. 7). It should be mentioned that the Gorter-EPR relation is valid at the arbitrary anisotropic interactions.
We would like to note that the EPR linewidth and Gorter RR angular dependencies as well as their inter-relation expressed by the Eq. (6) were obtained by us earlier for the KCuF3 compound [13]; however, only the spin-spin interactions determine there the EPR broadening.
At the constant field arbitrary direction relatively to the CFR, there is the one-to-one correspondence between the angles, which H0 makes with the axes of CFR, and the angles, which the Gorter field H1
measuring T-1, and the Gorter field H1 measuring TY_1, make with the CFR axes (see Fig. 1 and Table 1).
Figure 1. Connection between the rotation of the LFR relatively to the CFR and the obtaining Ta l, Tb l, Tc 1 from TX1, T- (see Appendix 2 for comments).
Table 1. Coordinates of the Gorter field vector H1 (0, p) which is able to measure relaxation rates of X (T- (d,p() and Y (T- (0,p)) components of magnetization drawn in Fig. 1 (see Appendix 2 for comments).
Position of LFR respectively to CFR Coordinates of the Gorter field H1 (<9,p) recording Tx (d,p( Coordinates of the Gorter field H1 (0,p) recording T- (0,p)
Fig. 1(a) H1 (0,0) = (7/1,0,0) ^ T- (0,0) = T- H1 (0,0) = (0, H1,0) ^ T-1 (0,0 ) = Tb 1
Fig. 1(b) H1 (0, 0) = (H1 cos 0, 0, -H1 sin 0) H^n/2,0) = (0,0,-H1) ^ Tx >/2,0) = T-1 H1 (0,0) = ( 0,H1,0) ^ T-1 (0,0) = Tb 1
Fig. 1(c) H1(n/2,p) = (0,0, -H1) ^ T-1 (n/2,p) = T-1 H1(n/2,p) = (-H1sinp, H1cosp,0) H1(n/2,n/2) = (-H1,0,0) ^ T->/2, n/2) = T-1
Fig. 1(d) H1(0, n / 2) = (0, H1 cos 0, -H1 sin 0) H1 (0, n/2) = (0, Hx,0) ^ Tx 1 (0, n/2 ) = Tb 1 H1(0,n/2) = (-H1, 0,0) ^ T-1 (0,n / 2 ) = T;1
The relation (6) coincides with the corresponding relations of Refs. 8-10 in the particular cases when the LFR axes X, Y, Z coincide with the CFR axes a, b, c in the arbitrary order. It should be mentioned that a simple method of T"1, Tbl, T- obtaining from EPR follows from (6): if the EPR linewidths are measured at the constant field directed in turn along the three crystallographic axes, then the solution of the three "Gorter-EPR" relations with respect to T-1, Tbl, T- will give their values. The relaxation rates T~l, Tbl, Tc_1, which enter such relations, are nothing but the "zero-field" relaxation rates measured in the zero constant fields in the Gorter type experiments known from the literature [8-10]. We would like to present the method of their value calculation using the obtained by us Tx\ T- expressions for the cases of the constant magnetic field rotations within the three crystallographic planes (the LFR rotations relatively to the CFR). These rotations can be depicted in the following form:
It is seen from Fig. 1 that in the case of the arbitrary anisotropic interactions, the analytical expressions for T"1, Tbl, T- can be obtained from these for TXl, T- (Eqs. (4)) after the transition to CFR at the following rotation angles:
T- = T- (0,p = 0) ,at H0 rotation in the plane ac;
T- = Txl(d = n/2), at H0 rotation in the plane ab; (7)
T- = T- (0, p = n/ 2) ,at H0 rotation in the plane bc.
Here, the following angular coordinates are used: the polar angle dof the LFR Z axis (H0 direction) with respect to the crystallographic axis c, and the corresponding azimuthal angle p which is counted from the a axis.
Having in mind the comparison with the experimental data, we take into account in (5) the Debye frequency value = 6n2cv3, where c is the magnetic ion concentration, the ratio c / pr = NA /M,
where NA is Avogadro number, M is the molar mass of the compound under consideration. Then the EPR linewidth in kOe can be written in the following form:
AHfR
(k°e) = Cph ^ Y ( )2 )mX ) (C + )(p )2 M ) (co)] (Tl -Ocw ), (8)
&Mb xp v )
whe'e C„-(f)2 ^^
v y an
4. Calculation of the second moments and the mean-square frequencies
Firstly, we would like to bring some comments to the calculation of the second moments in the compounds under consideration. The investigations of Deisenhofer et al [3] show that the contributions of the DM and CF interactions enter the full moments additively: (Mn) = (Mn ) + (Mn ). Similar to Ref. 3, we shall present the second moments (M2n) and the
value co2m, as the expressions averaged over the r magnetically nonequivalent ions of the magnetic
r r
structure (Mn) =(1/ r)Y(ML)., co2an =(1/ r)Y(al). . Here the values with the j subscript are
an j=1 j j=1 j
calculated for the one j-s position ion. At that the DM interaction with the q nearest i-neighbors of the j-s ion contribute to (m,dm ) ^ and (®2>M ) , in the form of the sums over i = 1... q. For the crystals of
the distorted perovskite structure, - the materials under our consideration, r = 4; q = 6 . It should be noted that the summation over i is not required for the one-spin CF interaction. The partial second moments of DM interaction in the LFR are equal to
M ™ =3ijY:)2 +d)2], m ym =-§jdXi +«)2!. (9)
where a = S(S +1) and the summation way is as described above. The following should be
mentioned in connection with the a value. As it is demonstrated in [2], the EPR signal in the doped lanthanum manganites is in accordance with the behavior expected within an intermediate octahedral ligand field. Such field splits the five d orbitals of Mn ions into a lower triplet t2 and an
upper doublet eg . The dominant Hund's coupling causes the parallel spin alignment of the four d
electrons of the Mn3+ spins and leads to the so-called high-spin state S = 2. Then, if the surrounding is ideally octahedral, the Zeeman effect is rather complicated and the observation of an EPR signal is rather unlikely. On the other hand, the Jahn-Teller distortion splits the orbital doublet into two singlets, where the ground state is a spin quintuplet with the usual Zeeman effect. The three d electrons of Mn4+ occupy the triplet t2g with the parallel spin alignment S = 3/2. The ground state is
an orbital singlet with a spin quadruplet. So, within the Jahn-Teller strongly distorted O' phase the EPR signal is due to the both Mn3+ and Mn4+ spins. However, experiment shows the non-zero contribution of the Mn3+ ions even after the transition O'^O. This fact witnesses that the weak orthorhombic distortion in O phase is still enough strong to retain the EPR signal of Mn3+ ions. Therefore, we assume that for the these phases of the lanthanum manganites with the x doping the value S(S + 1) is the linear superposition of the both Mn3+ and Mn4+ spin contributions: S (S +1)| = 6 (1 - x) + 3.75 x. At that, we do not take into account that according to [1] the effective
moments are considerably higher than the spin-only values. The authors of [1] believe that this increase is due to the formation of the superparamagnetic clusters (the so-called "giant" spins).
For the obtaining the angular dependencies of the different second moments, the transition from the LFR to the CFR is needed.
Accomplishing the transition to the CFR, we take into account the known from Ref. 3 fact that
all sums over the cross terms dldj with a^ j , where a,j = a,b,c, vanish. Then the second
moments (9) take the form:
M2XDM = -hr x{(d; )2 (sin2 p + sin2 0cos2 p) + (db )2 (cos2 p + sin2 0sin2 p) + ()2 cos2 0}, (10)
3 h ~Tij
M 2dm = ^ I {(d; )2cos2 p + (db )2sin2 p + (dc )2}, (11)
where da,d^,dc are the DM vector dj components along the CFR axes. At the calculation of the sums I(dyb,c) for the compounds under our consideration, we use the results of the Ref. 3. It is
shown there that these sums are equal for all of j and can be presented as the functions of only the two parameters d1, d2.
The values of M 2nCF in the LFR have the form
MX _ 4a - 3x^1/ ~ () ~ () (2 () ~ (2
20rr J=1
I {(¿J + DD^)- 2 j + ((- j
-2 (¿J + ¿Yj-2j((- ¿>>) + 4 (( + DJ + 4Dj )},
(12)
M* =
4a-31 {(( + ¿j)- 2DJ )2 +((- D(Y))2
j =1 v
2CP 20h2r j=1
+2(dj + DYj-2d^)(dj -£№)+4(( + DJ + 4DJ)}.
(13)
Here DJ are the components of the CF interaction tensor in the LFR, which are linearly connected [3] with those in the CFR DJ. The values of M!^CP in the CFR will be written only for the H0 rotation in the ab plane of the crystal because of their inconvenience. They have the following form
MXcp (ab plane) = £ [D) -DbJ)) + 4(¿b ) + (dJ )2 + (dJ )
MYf (ab plane) = I {o.5 (2DJ -¿a -DbJ) )2 + 0.5 [( -DbJ) )2 + 4() + 5()2 + (dJ)2) + [(( -Dj -D<>>)(( -Dj) + 3[(^(J(¿0)
cos 2^
(14)
(15)
Note that it is possible to obtain the expressions for M^F in the ac and bc planes from (14, 15), if the replacements X — Y are made in the l.h.s. and replacements p^ 0 ; a, b, c ^ c, a, b and a, b, c ^ c, b, a, - in the r.h.s. of the above expressions, respectively.
The values of (M 2+-) in the CFR, which are necessary for the EPR linewidth angular dependence
V 'an
obtaining, can be easily found from the known (M2) .
2 an
Also, the straightforward calculation gives the following expression for the DM mean quantum squared Magnetic Resonance in Solids. Electronic Journal. 2017, Vol. 19, No 1, 17101 (17pp.) 7
1=1
4a -
)2=4a s )2+k )2+w )2}.
(17)
The expression of (h«DM )2 contains the square of the DM vector length; hence, it is angle independent in the CFR and has the following form there:
(»«dm )2=3a £ {(a )2+(< )2)21 •
As to the CF interaction mean quantum squared, it is equal in the LFR to
(«)2 = -L£{(4a-3)-D^f +4Dj + 2(( + D(f)
j=i
(j )2
(18)
+ (8a - 1)(DDXXX + + 4 (2a + 1)(( + D^Ôg + 4 (3a - 1)DZZ After the transition to the CFR for the H0 rotation in the ab plane (18) takes the form ( hvCF )2 ( ab plane) = -0-£ j(4a - 3) (3 / 4) [ (D(bj - Djj )2 + 4D0b)2
20r
) - D(j)
■( 1/4) ( Dbb>- Daa)) - 4Dab
(j )2
cos^
(19)
+ 3(ô2")2 + D^)2) + (D£)2 -Ôj)2)cos2^ + (8a - 1))ôbb) + Daf)2 + 4 (2a +1))^ + D^) + 4 (3a -1)Dj
We have made sure with the help of the computer calculations that the terms of (%aCF )
containing angular dependence practically does not influence the value of (%aCF ) (their contribution
is less than 3%), so, in (19) they can be neglected, and for the numerical calculations we have used Eq. (19) without them.
We would like to make a definite note regarding the spin-phonon bond tensor components Ç"a,p, . They can be considered as the crystal electric field potential arising from the crystal lattice distortions [18, p. 241; 19, p. 619]. Further the rough evaluation « 3e2r2 / 4xs0R3 = |g| from [18, p. 241]
'afj
will be used for Qhp, where r2 is the mean distance of the d-electrons from the nucleus squared; R = {(8/ 3)(Rf + Rm3 + Rf )j is the nearest ion O2- average distance from the Mn ion; Rj, Rm, Rs
are the long, the middle and the short octahedron axes; s0 is the electric constant. So, we suppose that the approximate equality of Q^. , Qrg,p. , Q^, parameters takes place, their mutual value being denoted further, as G •
Knowing the above-calculated second moments, the Gorter RRs T~l, Tbl, T- (4) can be written in the following explicit form:
T- = CsphQ2 (Tc )ef (TL -®cw )' 4a-3 5%2 (cc - Dbb )2- ^4Dbc- ^Dl- ^ ; 4a In2 [(db)2- >(d; )2 ;
T- = CsphQ2 (rc )ef (TL -°cw y 4a-3 5%2 (Dcc - D )2- aa f t- 4 D2 ac + Dl- +D2b ; 4a 3%2 [(d; )2 ;
T- = CsphQ2 (Tc )ef (TL -@cw ) 4a-3 5%2 ~(Daa - Dbb )) + 4 Dlb + Di +Di ; 4a [(d; )' ]
, (20)
where we write (rc )f instead of rc for the case when the narrowing of the EPR linewidth is caused not only by the isotropic exchange (see below).
5. Comparison with the experiment
Comparison of the analytical angular and temperature dependencies of the EPR linewidth obtained in the refined model of the EPR spin-phonon broadening with the experimental data for the compounds La09Sr01MnO3 and La0875Sr0125MnO3 will be carried out in the temperature interval 0CW < TL < Td
Here 0CW is the Curie-Weiss temperature of the O' phase transition to the ferromagnetic phase; ToO is
the temperature of the structural transition from the O' phase to the O phase. The temperature interval reflects the requirement of the strong Jahn-Teller distortions leading to the DM interaction (TL < Tdo),
and the condition 0CW < TL is necessary for the validity of the Eq. (A11) (see Appendix 1). As is shown below, the EPR linewidth angular and temperature dependencies following from this model agree with the experimental data. So, to our mind, the angular and temperature dependencies of the Gorter RRs presented are also adequate for the pointed compounds.
Firstly, we would like to make some general notes regarding the result (8). The Eq. (8) is anisotropic and linear over the temperature, what is in agreement with the experimental results of Ref. 2 for the samples La09Sr01MnO3 and La0 875Sr0125MnO3 in the mentioned temperature interval. At that, Eq. (8) does not depend on the generator frequency. This fact is in accordance with the observation in Ref. 4: the measured EPR linewidths in the similar compounds were essentially the same at the generator frequencies v = 9 GHz and v = 36 GHz . Note, that if it would be the Zeeman subsystem which relaxed to the lattice, then its relaxation rate and, correspondingly, the EPR linewidth determined by it would be proportional to v2. On the other hand, the EPR linewidth (8) is proportional to (M2 -) . Hence, our suggestion that the EPR linewidth is determined in the given case rather by the
2 an
relaxation to the lattice of the tfan interaction, and not of the Zeeman one, leads to the result agreeing with the experiment.
It is shown in Ref. 3, bearing on the experiment, that in a material, where the both DM and CF interactions present, their contributions to EPR linewidth are summed up additively. Our formulae confirm this fact. Really, since the joint heat capacity of these interactions is proportional to co2an =a>2DM + CF and the equality (M2n) =(Mn) +(M2,) takes place, the relation
AcEapR =A®DMM +A®CFr is also seen from our Eq. (8). Obviously, the same is valid for the Gorter RRs, too. Besides, it is demonstrated by the authors of Ref. 3 that the angular dependence of AaP^ is proportional to that of (M + ) . Such regularity also follows from our Eq. (8) in the case, when
2 an
approximate equality of glj , glj , glj parameters takes place, since c2an is practically angle independent, and the identity (MX ) (<*>) +(MY ) (<*>) = 2 (Mr) (<x>) is valid.
V / an ^ ' V / an ^ ' V / an ^ '
Let now begin with the comparison of the EPR linewidth temperature dependence (8) with the experimental dependencies in the samples with x = 0.1, 0.125 from the paper [2], where it is of the linear
form: AHEPR = a + bTL. Following the Ref. 5, we rewrite this dependence in a somewhat different form:
AHEPR (Tl ) = AHEpr (0CW )+ b (Tl -0CW ) , (21)
where AHEPR (0CW) = a + b0CW. Note that the contribution to the EPR linewidth directly proportional
to the temperature was observed in Ref. 17 in the A-site ordered compound YBaMn2O6 with the mixed valence of the Mn ions (Mn3+ and Mn4+). This contribution was successfully interpreted there, as the
result of the Korringa relaxation of the collective eg electrons. For the quantitative agreement with the
experiment it was assumed that this relaxation channel was the bottleneck of the relaxation to phonons of the united system of eg electrons and Mn4+ spins. The presence of the highly mobile collective eg
electrons on the Mn4+ background was the necessary condition for such a relaxation. This condition was fulfilled in the A-site ordered compound YBaMn2O6. Differing from that situation, our consideration is restricted to the region O /1,PM of the x - T phase diagram of the Sr doped lanthanum manganites [1, 2]. Here the conductivity of the isolator type (see, for instance Ref. 3) takes place by means of the thermally activated hopping of the eg electrons between Mn3+ and Mn4 sites [2].
Therefore, the Korringa relaxation cannot be realized here.
The experimental linewidth temperature dependence in the samples La09Sr01MnO3 and La0 875Sr0125MnO3 of Fig. 3 from the Ref. 2 can be presented in kOe in the form:
AH (T Lo,=02+0 0041(-156)
AH(T)| = 0.09 + 0.0043( -190).
In (22) the values of the Curie-Weiss temperature of the O' phase dCW (x = 0.1) = 156 K and
dCW (x = 0.125) = 190 K are substituted, which are known from Ref. 1. From the comparison of (22)
with the Eq. (8) obtained by us it is seen at once that the Eq. (8) corresponds to the second term of the experimental temperature dependencies (21), (22), while the first term of (21), (22) is not present in the Eq. (8). The origin of AHER (dCW 0 in Ref. 5 was connected with the spin-spin broadening.
However, such contribution calculated by the Kubo-Tomita formula [4] under the conditions of the strong exchange narrowing, contains the second moment, which in the concentrated paramagnets should depend on the temperature according to (A11) (see Appendix 1). Following the papers [20], and [21], we suggest that the non-zero temperature independent contribution AH^ (0cw ) to the EPR linewidth in La09Sr01MnO3 and La0875Sr0125MnO3 single crystals is conditioned by the inhomogeneous broadening, which is known to be temperature independent. Naturally, the same inhomogeneous contribution should be added also to the Gorter RRs (4), so, the correlation (6) remains valid. The large inhomogeneous broadening A* = AHEPR (dCW) (~200 Oe) can follow from the noted in Ref. 2 twin structure of the investigated crystals.
Finally, the suggested by us EPR linewidth temperature dependence is of the form
AHEPR (Tl) = A* + b(TL-0CW), (23)
where the value b is equal to
b = C hQ2 (r )e# (M2+-) (oo). gMs n
Here the value (rc replaces the value rc = h / J, its appearance will be explained below.
We would like to compare the temperature dependence (23) with the experimental ones for the samples with x = 0.1; 0.125 from Ref. 2 also quantitatively. This comparison faces difficulties because
the DM and CF interactions parameters, which are necessary for the calculation of (M^-) and o1an
V 'an
(see Section 4), are not known to us for the compounds with x = 0.1; 0.125. However, the DM and CF interactions parameters were extracted from the experiment in Ref. 3 for x = 0.05. We see the way out in the fact that b contains the ratio of the values (M2 ) and ®«n, each of which
an
is some linear combination of these parameters squared. Therefore we suppose that
(m r)
\ 'a
a
(m r)
a
a
(m r)
a
a
. Not allowing for the latter assumption, b can be written
in the more general form:
b = 20^m2k Q2XffN,(M2+-)an(o)
gMs VA
h2v2 M
a
(Mr) (o) V / an N /
x =0.05 a2 an
Though it seems to us that the ratio in the parenthesis is of the unity order, further, to be on the right side, one should take into account that the plots of Fig. 2 are really built in the units
(Mr) (oo)
\ ' an N /
a
(Mr) (oo)
an
a
x=0.1, x=0.125
Note that in (24) we have replaced the correlation time tc = h / J conditioned by the
usual isotropic exchange for (zc )eff, which is
determined by us by the fit of the Eq. (23) to the experimental temperature dependence plots from Ref. 2. The point is that the correlation time calculated with the help of the expression tc = 2S (S +1) h / kB0CW leads to the value of the
constant b, which is an order of magnitude larger than the experimental one. Since as a result of the fit it turned out that (rc)ef «rc /30 for the
both dopings x = 0.1 and x = 0.125, the angular dependence plots of the Fig. 2 are built with this value of (rc )eff. The values lTxl, lTY-l, where
l = 10h / g^B , are plotted in kOe for the relaxation rates of the X and Y components of Mn ion magnetization, correspondingly, there.
We explain the effective decrease of the correlation time in the samples under consideration analogously to the Ref. 5. Namely, we suppose that this decrease reflects the additional motional narrowing of the EPR linewidth due to eg electron hopping between
Mn3+ and Mn4+ sites. The motional narrowing was absent in La0.95Sr0.05MnO3, but, to our mind, plays role for the increased eg electron hopping
intensity in the samples under consideration with the increased doping.
We have obtained the following curves of the angular dependencies of the transverse magnet-
(M2+tn (O)
a
x=0.1, x =0.125
. (24)
0.48
45 90 135
Angle (degree) Figure 2. Dashed green and dotted blue curves describethe angular dependencies of the relaxation rates lTx 1 lTY-1 in kOe of the X and Y components of
Mn ion magnetization, correspondingly, at the constant magnetic field rotation relatively to the crystallographic axes a, b, c of the sample within its three crystallographic planes, l = 10h / g/uB . Solid red curve describes the angular dependence of the EPR linewidth ahepr in kOe. The plot is built for La09Sr01MnO3 at TL = 200 K.
3 -
o
□ □□ x - 0.1 x = 0.125
1 -
0
0
600
200 400
TL(K)
Figure 3. Temperature dependencies of the EPR line-width in La09Sr01MnO3 and La0 875Sr0125MnO3. The squares and triangles depict the experimental data from Ref. 2. The fit according to Eq. (23) of the given paper is indicated by a solid line.
ization relaxation rates and that of the EPR linewidth in kOe in La09Sr01MnO3 compound (see Fig. 2; here, as in Ref. 3, the b axis is perpendicular to the ferromagnetically ordered ac planes).
The comparison of the obtained temperature dependencies with the experimental ones are depicted on the Fig. 3. It is seen that there is a good agreement with the experiment in the definite temperature interval.
Though the roughness of the used model spin-phonon Hamiltonian and the corresponding value evaluations, together with the presence of the fitting parameter, do not enable us to make unambiguous quantitative conclusions, the suggested model agreement with the experimental temperature dependence seems to be reasonable.
6. Conclusion
Summarizing, the following results are obtained for the magnetically concentrated paramagnets under conditions of the dominating exchange interaction and for the weak constant magnetic field EPR broadening caused by the one-phonon spin-lattice relaxation. The main result is that the analytical angular and temperature dependencies of the EPR linewidth obtained agree with the experimental data of the Ref. 2 for the La09Sr01MnO3 and La0875Sr0125MnO3 single crystals. So, one may hope that the angular and temperature dependencies of the zero-field Gorter relaxation rates obtained are also adequate. Further, as it was shown in our recent paper [7], in such materials the EPR half width on the half height at a weak constant field H0 along Z axis can be presented as the half sum of the relaxation
rates Tx\ T- of the magnetization components along X and Y axes measurable in the Gorter type experiments. These relaxation rates can be measured in the experiments of the Gorter type in the zero constant field with the definite (pointed out in the Table 1) directions of the low-frequency field relatively to the CFR. The angular dependencies of the half sum of T-1, TY- coincide with the experimentally observed EPR linewidth angular dependencies, thus demonstrating the continuous "Gorter-EPR relation". This continuous "Gorter-EPR relation" coincides with the discrete ones, which were confirmed experimentally, in the particular cases of the axes X, Y, Z (LFR) coincidence with the axes a, b, c (CFR). Our investigation combines the continuous and the discrete approaches, differing both from the discrete consideration of the Refs. 8-10, restricted to the CFR, and the continuous consideration of Refs. 11, 12 restricted to the LFR.
The refined model of Ref. 7 is used for the interpretation of the experimental results of Ref. 2 for the single crystals La09Sr01MnO3 and La0 875Sr0125MnO3 in the temperature interval, where the EPR linewidth is simultaneously anisotropic and linearly dependent on the temperature. At that, the anisotropic interaction consisting of the DM and CF interactions effectively relaxes to the lattice. It is supposed that all the relaxation rates contain the temperature independent contributions from the inhomogeneous broadening. At such a supposition, the temperature dependence of the EPR linewidth qualitatively agrees with the experiments of Ref. 2. The angular dependence of the EPR linewidth qualitatively agreeing with the experiment of Ref. 2 is presented analytically and for the single crystal La09Sr01MnO3 graphically. For the quantitative agreement of the obtained results with the experiments of Ref. 2, it was necessary to suppose that the correlation time of the interaction narrowing the EPR line is approximately an order of magnitude smaller than the one given only by the isotropic exchange. We suppose that the correlation time decreasing is caused by the additional motional effect due to the eg -electron hopping between the Mn3+ and Mn4+ sites.
The obtained microscopic analytical expressions for the zero-field relaxation rates measurable in Gorter type experiments and for the EPR linewidth in a weak constant field connect the experimentally obtainable macroscopic values of the relaxation rates with the spin structure. Since all these quantities are expressed via the constants of the interactions causing magnetic anisotropy, duplication of the getting of information on these constants in the new technological materials becomes possible.
Acknowledgements
The authors wish to express their deep gratitude to Prof. V.A. Atsarkin for the valuable discussions. One of the authors (E.Kh.) is grateful to M.E. Eremin and D.V. Zakharov for the courteous rendering of the "Report on the scientific work, Kazan, 2004". The financial support of Shota Rustaveli National Science Foundation, Georgia (grant № FR/299/6-110/14) is gratefully acknowledged.
Appendix 1. Calculation of the one-phonon spin-lattice relaxation rate
At the calculation of the spin-lattice relaxation rate, we follow Eq. (15, 29) in Ref. 22:
-1 o
1/Tsph =(1/2){Ha2n)~ { dt(KanKan (t)) , (A1)
—o
where
Kan = (ih)—1 [Han , Hph ] ; (Han) = (®an ) (( ) (A2)
the triangular brackets mean the equilibrium thermodynamic averaging; haan is the mean-square quantum of H an . Further,
Kan (t) = exp((i / h)(Han + Hph)t)Kan exp(—(i / h)(Han + H ph)t) . (A3)
It should be mentioned that the presence of the strong exchange interaction H'esx will be taken into account later so, as it was done in Ref. 7.
The following inequalities are fulfilled in the experiments under consideration
H0 << J / g^0,x/HI, (A4)
KTL >rmax ={2S(S + 1)n0Jn2n}1/2, (A5)
where H0 is the external constant magnetic field in EPR; -^H^ =(g^B^0) ^Sp (H2an) / Sp (Sz2) is
the mean-square local field of H an; TL is the lattice temperature; n0 is the number of the nearest spin neighbors and Jnn = J is the exchange integral for them; for the doping x one has S(S + 1) = 6(1 — x) + 3.75x; kB and u0 are the Boltzmann and magnetic constants, ¡uB is the Bohr magneton.
The weak field condition (A4) provides the strong exchange narrowing of the EPR line and guarantees that it is the anisotropic interaction H an , which relaxes to the lattice during EPR rather than the Zeeman interaction. The inequality (A5), according to Huber [23], provides the efficiency of the one-phonon spin-lattice mechanism of the EPR broadening and the EPR linewidth linear temperature dependence even at sufficiently high temperatures.
Basing on the arguments brought in the Introduction, we suggest further that the relaxational EPR broadening takes place in the single crystal lanthanum manganites with the Sr doping x = 0.1 and x = 0.125 in the definite temperature interval. Spin transitions under the influence of the lattice vibrations, shortening the lifetime of the magnetic ions on the spin levels, serve as the broadening cause of these levels, and consequently, of the EPR line in the compounds under consideration, the corresponding linewidth being [14] Aa>EER =(1/2)7^. We suggest also that the SLR is of the one-
phonon type in La09Sr01MnO3 and La0 875Sr0125MnO3 and is realized by the crystal electric field
modulation at the vibrations of the O2- ions nearest to the magnetic ions (O2- ions are situated in the vertices of the octahedrons surrounding Mn ions). Since our quantitative calculations will have the evaluation character, we proceed from the model spin-lattice Hamiltonian (2). There n = X, Y, Z are the LFR axes; aare the O2- ion coordinates in the local Cartesian frame of reference with the origin in the magnetic ion and with the axes along the O2- octahedron axes; G*'p',Z are the components of the spin-phonon bond tensors;
\l/2
2 yMj ^^ ( exp ( p R-% exp ( pR ))(■*■) (A6)
ea\p[
is the crystal deformation tensor component; Mcr is the crystal mass; ak and vk are correspondingly the frequency and the velocity of a p-s branch phonon possessing wave vector k ; a* and ak are the birth and annihilation operators of such a phonon; Xa are the directing cosines of the phonon polarization vector; f is the wave vector ort.
Further, the fact is used that in the case of the one-phonon SLR Hamiltonian (2) Kan can be presented in the form
Kan X Gna,K"nn , where Kln = (1/ih )[ ^ an ] . (A7)
a ,p n=x,y,z
It follows from (A3) and (A7) that
(Kan Kan (t)) = X ((, )2( a a (t))(K"n Kl (t), (A8)
a ,P
where Kln (t) = exp^hHantJK"an exp ^-hHmtj.
It should be mentioned that the SLR rate in Ref. 7 was obtained in the high temperature approximation over all the energies, including the phonon energy. If one does not do the latter approximation and passes to the continual limit in the phonon spectrum, then the spin-phonon relaxation rate in the long wave approximation in frames of the Debye model has the following form
1 1
T , 16n p ha ,
sph r cr an J
3P a Xl((K:)2W(sn) Ife)2J
a\p'
(A9)
i™ , . , ... ha a
daK" (a) J " da a 3coth—--^-.
any 'Jo 2k T i ' V , 2
¿kdit (a -a) * a
a
Here am is the Debye frequency; pcr is the crystal density. Since Knm (a) is the Fourier-transform of
the normalized correlation function(Knnn Knan (t) / ^(Knnn, one has J™ da Knnn (a) = 1. Note that the
influence of the strong exchange interaction with the exchange frequency aex = J / h, which is suggested to be larger than all other frequencies in the problem under consideration is already taken into account in the equation (A9) in the same way, as it was done in Ref. 7. Further, in the considered model the following equation is valid:
(Kn )2)/((Sn f) =-[Sn , Han ]2) h-2((Sn )2); =(M2n )an (TL ) (A 1 0)
E.Kh. Khalvashi, N.P. Fokina, M.O. Elizbarashvili Following the result of Ref. 24 and Ref. 25 at the calculation of ((Sn) = (1/2))S+S- }T in the
second moment denominator we use the relation ^S+S ^ /^S+S~) = %(TL)/z0 (-L), where z0 (TL)
is the single ion static susceptibility, and z(Tl ) corresponds to the behavior of the exchange coupled Mn system. Similar to Ref. 24, we assume that the single ion susceptibility z0 (-L) obeys the Curie law z0 (TL ) = C / TL, while z(Tl ) can be approximated by the Curie-Weiss behavior Z(TL) = C /(TL -dCW), where C is the Curie constant and dCW is the Curie-Weiss temperature. Then the following relation is valid at TL > dCW:
(tL)=(»). (AID
L
Note that the Eq. (A11) was used in Ref. 7 at TL
Further, evaluating the term
L ' •
2
in (A9), we make use of the
ufe)2 íd q 2
_a\P' 2 _
corresponding value , which is brought in [18], p. 231. However, we shall use it in the
simplified form - assuming that the longitudinal and the transverse velocities of the acoustic waves are equal to one another v, = vt = v. We also suppose that the approximate equality of , gra.p., g^,
parameters takes place, their mutual value being denoted further, as g. At
that,
K)2 íd" 2 (r+v- ■)
= 4ng /3. Note also that if one passes to the variable x = co' / a>ex, which was introduced in Ref. 23, in the integral over da> , then this integral takes the form
Íú)m / 0)ex 3 XJ 1
dx • x3 coth--2-. (A 12)
0 2kBTL (x-»/)2 +1
The expression (A12) is analogous to the function
( x2 >
f (kb-l / rmax ) = f" dx • x3 coth^k--- exp ^^^ J, (A13)
from Ref. 23, in spite of the fact that the different model of the exchange interaction influence on the spin-lattice relaxation rate is used there (note that from (A5) ymax x J). Because of the difficulties to take the integral (A 12) exactly analytically, we approximate (A 12) as
2k„TIa2 ram/me 2kKTj coi -1 .
—B ex I dx • x =—B-L——. At that, we use the fact that cex = zr is the largest frequency in
h hcx 3
the problem under consideration and that according to Ref. 23, at kBTL > ymax the expression (A13) is
directly proportional to the temperature. Further, the coefficient (X/ A)2 should be added to -h,
where X is the spin-orbit coupling parameter, and A is the distance of the excited orbit level from the main one [14]. This is the consequence of the well-known fact that for the case of iron group ions (the case under our consideration) the crystal field reduces an orbital magnetic moment X/ A times [14].
Taking into account all the above mentioned and using the definition of the exchange correlation time rc = h / J, we obtain the expression (3) for the one-phonon spin-lattice relaxation rate in the main text.
Appendix 2. Comments to Fig. 1
Fig. 1(a). On this figure one of the three versions of the classical "axial" implementation of the Eq. (6) (see also [10]) is presented: EPR is observed at H01| Z || c , while the Gorter RR measurements are
observed at the low-frequency field along the a (X || a, Tx = T-) and b (Y || b, T- = Tbl) axes. The other two "axial" implementations are as follows: H 0 || Z || a , X || -c (T- = T-), Y || b (T- = Tbl) and H 0 || Z || b , X || - c (T- = T- ), Y || - a (T- = T- ). The values of T- , T- and T- can be calculated with the help of the expressions (20), and the EPR linewidth - by the formula AafPR = (1/2) (1 + T- 1) ,where i, j, k = a, b, c.
Fig. 1(b). The H0 rotation from the c axis to the a axis (ac plane) is shown by the arrow. At that, the X axis of the LFR turns also in the ac plane (from a to -c), and it is possible to measure Tx 1 (0,p = 0) for any value of 0 (at p = 0) from the value T~1 up to the value T~1 (rotation axis Y
stays immovably, and T- (0,p = 0) = Tb 1).
Fig. 1(c). H0 turns from the a axis to the b axis (ab plane). Here the Y axis turns in the ab plane (from b to -a), and it is possible to measure Ty1 (0 = j2,p) between the values Tb 1 and T~1 (rotation axis X stays immovably along the -c axis, and Tx (0 = j2,p) = T~1).
Fig. 1(d). Finally, the H0 returns to the c axis, X axis turns from -c to b, and in the bc plane the value of Tx :(0,p=n/2) can be measured within the limits T~ \ Tb 1), while Ty1 (0,p = n/2) = T-(Y axis does not turn).
It should be noted that the relaxation rates for any 0 and p values can be calculated by the formulae (4) with the taking into account the expressions (10, 11, 14, 15) for (MX,Y ) and the
V 'an
additional expressions obtained from (14, 15) with the help of the replacements specified nearby. References
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