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Submitted on 1 Jan 1988
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THEORY OF THE NUCLEAR MAGNETIC
ORDERING IN Cu IN A FIELD
Per-Anker Lindgård
To cite this version:
JOURNAL DE PHYSIQUE
Colloque C8, Supplement au no 12, Tome 49, dkembre 1988
THEORY OF THE NUCLEAR MAGNETIC ORDERING IN Cu IN A FIELD Per-Anker Lindghd
Rise National Laboratory, DK-4000 Roskilde, Denmark
Abstract. - The phase diagram for the nuclear magnetic ordering of Cu in a magnetic field is theoretically found to exhibit three distinct phases in accordance with recent measurements. It is a consequence of the frustrated ground state properties of an antiferromagnetic fcc structure for quantum spins with nearest neighbour Heisenberg and dipolar interactions, a model of relevance also for the resonating valence bond problem.
Magnetic ordering of the nuclear spins for Cu has been observed below 60 nK, using susceptibility mea- surements [I]. Recent neutron scattering measure- ments [2] has confirmed this and revealed that the structure is a simple type I antiferromagnet. There is, however, observed three distinct phases as a func- tion of an external field. The problem is that of a frustrated antiferromagnetic ground state [3]. Cup- per has the fcc lattice structure and the nuclei of Cu have the spin I = 312. These interact via the dipolar interaction and the Ruderman Kittel [4] interaction, which is dominant, as calculated from first principles 151. Furthermore, the nearest neighbour (nn) interac- tion is dominant by an order of magnitude [5] and gives in the mean field theory rise to a simple type I anti- ferromagnetic (-4FM) structure. Since all (nn) bonds form triangles the system is very frustrated and the AFM structure is in simple theories infinitely degen- erate with respect to linear combinations of ordering vectors k, = ~ / a l where L is a unit vector along the cubic directions x, y or z, a is the lattice constant. The dipolar interaction requires only that the spins are perpendicular to the ka vector. In a field the general structure is then described by any of the degenerate 1-k, 2-k or 3-k linear combinations (denoted I, I1 or 111). We shall now demonstrate that non-linear ef- fects lift the degeneracies and stabilize different phases as observed experimentally. The phases are distinct because of different quantum mechanical ground state correlations.
Consider a nearest neighbour fcc antiferromagnet with (nn) Heisenberg interactions J and dipolar inter- actions D per bond in an external field H = H (0, 0 , l )
where a and b represent sums over the four sublat- tices in the fcc structure, and P is the interconnect- ing unit vector. The moments M for the four spins at a = 1, 2, 3 and 4 are canted towards the field with different relative canting angles for the different multi- kstructures. The Hamiltonian is now rotated so that the local quantization axis are along the lo- cal canted moments. In the mean field (MF) theory the energy levels are Zeemann split. The population
factor is p = exp (-@JIM) or for H
>
4 ~ l M p =exp
(-@
(H-
~ J ' M ) ),
where J' = 4 J+
D. Then the average magnetic moment M = (3+p-$-3p3&/ 2$, and the reduced partition function Z = 1+
p+
p+
p,
where
p
= l/kT and k is the Boltzmann constant. The (MF) free energy per spin is in the paramagnetic phaseF~~~~ =
-
(3/2) J ' M (M+
1)-
kT In Z-
(312) H' and in the AFM phasewhere H' = (H
-
~ J ' M ).
The entropy assumes the form S / k = /3J1M (3/2-
M)+
In 2. At the critical field H, = 4J1M there is a second order phase bound- ary. The structures I, I1 and I11 are degenerate for all fields H<
4J1M, within the mean field theory. This was the previously found result [I]. The energy levelsin the paramagnetic phase are equidistant with a sep- aration H
-
3J1M. IfkT
decreases as H -3J1M, it fol- lows that the population factor p is constant as well as the entropy S / k . The isentrop in a (T, H) plot is thus simply a straight line through H = 3J1M. The slope is determined by the initial polarization M, which is also constant. At the phase boundary to the AFM struc- ture, the field is replaced by the constant internal field JIM. The isentrops are therefore independent of field in the ordered phase, within mean field theory. Exper- imentally one finds [I], however, two distinct crossings of phase lines in the ordered phase, when following a constant entropy curve. This is incompatible with the above results of the mean field theory.Let us now go beyond mean field theory and consider the effect of off-diagonal terms in the Hamiltonian (1). For example, the isotropic interaction Hamiltonian has the form for relatively canted spins
This Hamiltonian can both move a spin flip from site to site, and simultanously flip two adjacent spins against the local molecular fields. The latter process is similar t o local crystal field transitions and should be included before considering the former, low energy spin wave type excitations which do not contribute at T = 0. Two important features can now be noted a) the in- teractions bring in a field dependence'of the thermo- dynamic quantities in the ordered region, where there
C8 - 2052 JOURNAL DE PHYSIQUE
was none in the mean field theory, and b) the struc- tures I, I1 and I11 differ qualitatively because of the different relative canting angles. The pair interactions can therefore determine the phase diagram and distin- guish between the various structures I, I1 and 111.
Let us evaluate the effect of the two spin flip terms (I:
I?
+
1; I c ) using second order perturbation the- ory for a cluster of the four sublattice spins. The ground state with energy 0, say, has a wave func- JGion 10) 3- 111, 1 2 , 1 3 , 14) with maximum momentI, = 312 along the various directions of the local fields of strength J'M. This is coupled to states with energy 2J'M having two spins flipped and wave functions of the 12) = 111
-
1, 1 2-
1,4,
14),
and various per-mutations. There is no coupling to the one flipped cluster states 11) at energy J'M. Second order per- turbation theory shifts the cluster levels of energies (0, J'M, ~ J ' M ) by (-An, 0, +An), where n = I, I1 or 111. The state n with the largest An is the ground state, An is found to be of the order of 10 % of J'M. The cluster calculation is similar in spirit to Ander- son's randomly distributed singlet cluster theory [3]. All possible 1-k and 2-k structures and the 3-k structures with two spins along the field were inves- tigated. It was found that only the structures with the spin projections in the high symmetry directions in the x- y plane are relevant. The relative stability of the four sublattice, simple type one, antiferromagnetic multi-k structures were studied for different ratios of D / J (in the form of D' = 3 DIJ'). At H = 0 the result agrees with a recent spin wave calculation [6], how- ever there the 3-k structure I11 was not investigated. The strength of the Ruderman Kittel interaction [4] in Cu is measured by a quantity R defined in refer- ence [I]. From the first principle calculations (51 we find R = -0.34. Using this value for Cu (D' = 0.9) we predict a transition between the I, I1 and I11 structures as a function of the magnetic field. The experimental value [I] for Cu is R = -0.42 (D' = 0.8)
.
For this value the stable phase for intermediate fields fluctu- ates between the I1 and a different I11 structure. The conclusion is that for a relevant range of R values the magnetic structure of Cu is predicted t o show at least three multi-k phases as a function of magnetic field along the (001) direction [7]. This is in agreement with the expe~imental results [I]. For a field along the (110) direction the phase diagram is simpler and we obtain a smooth transition between a 1-
k, and a 1 - Ic, structure via a 2-k structure, see figure 1.The neutron scattering measurements [2] for this field direction find two type I AFM phases separated by a transition region with low intensity at intermediate fields.
For finite T the effect of spin wave excitations terms
12
IT in all phases I, I1 and I11 is to lower thetransition tem$erature TN relative to the mean field
TN
(MF).
However, a recent Monte Carlo computerFig. 1. - Calculated regions of stability for different type one AFM phases a t T = 0 as a function of field H (101) and D' = 3015' (D' is between 0.8 and 0.9 for Cu). An incommensurate I'K-s.tructure, with the ordering vector in the rK-direction, might penetrate along the phase bound- ary at H
-
0.3 Hc (in mean field theory [5] the r K phase is stable for I)'>
1.1).simulation study on a two-dimensional model system for Cu [81 suggests that the isentrops are not reduced significantly in temperature. A temperature renormal- ized phase diagram with the mean field isentrops is in good agreement with that found experimentally by Huiku et al. [I].
The magnetic phase diagram for Cu has been dis- cussed using a simple model. It is concluded that the quantum spin fcc lattice with nearest neighbour anti- ferromagnetic Heisenberg and dipolar interactions ex- hibits several distinct multi-k ground states as a func- tion of a magnetic field.
Acknowledgement
It is a pleasure to thank Matti Huiku for inspiring discussions.
[l] Huiku, M. T., Jyrkkio, T. .4., Kyynkainen, J . M., Loponen, M. T., Lounasma, 0. V. and Oja, A. S., J. Low Temp. Phys. 62 (1986) 433;
A recent summary of the experimental and theo- retical status is given by Oja, A. S., Phys. Scr. 36 (1987) 462.
[2] Jyrkkio, T. A., Huiku, M. T., Lounasmaa, 0. V., Siemensmeyer, K., M. Kakurai, Steiner, M., Clausen, K. N. and Kjems, J. K., Phys. Rev. Lett. 60 (1988) 2418.
[3] Anderson, P. W., Mater. Res. Bull. 8 (1973) 153; Anderson, P. W., Science 235 (1987) 1196. [4] Rudermann, M. A. and Kittel, C., Phys. Rev. 96
(1954) 99.
[53 Lindghd, P.-A., Wang, X.-W. and Harmon, B.
N., J. Magn. Magn. Mater. 54-57 (1986) 1052. [6] Viertio, H. E. and Oja, A. S., Phys. Rev. B 36
(1987) 3805 using the complete interactions; See also the earlier spin wave analyses by ter Haar, D. and Lines, M. E., Philos. Trans. A 255 (1962) 1.
[7] Lindghrd, P.-A., Phys. Rev. Lett. 6 1 (1988) 629. [8] Lindgbd, P.-A., Viertio, H. and Mouritsen, 0 .
