Magnetic structure of vortex lattice state in the adsorbed second layer of 3He films

Physica B 165&166 (1990) 711-712 North-Holland

Magnetic Mitsutaka

Structure FUJITA

of Vortex Lattice and Kazushige

State in the Adsorbed

Second Layer of 3He Films

MACHIDAA

Institute of Materials Science, University of Tsukuba, Tsukuba ADepartment of Physics, Kyoto University, Kyoto 606, Japan.

305, Japan.

Stimulated by a recent discovery of a sharp specific heat anomaly at 2.5mK in a second layer of adsorbed 3He on grafoil by Greywall et al., we study a lattice gas model or Hubbard model on a triangular lattice within the mean field approximation. In the case of nearly half fill’m g s, we find a novel type of the magnetic structure forming vortex lattice states, which is non-collinear and characterized by triple Q vectors keeping the three-fold symmetry. A mysterious phase transition signaled by a sharp specific heat peak around T=2.5mK in 3He films adsorbed on grafoil is found by Greywall and Busch’ and arouses much attention2. The experiment was done at surface coverages p = 0.160,0.175, 0.184atoms/A2 above the first layer is completed (p=pr ~0.114), but before the third layer promotion (p N 0.24). Since Franc0 et afs already found a discernible negative deviation of the magnetization expected from the free spin value in this coverage regime, this phenomenon is certainly related to some kind of nuclear magnetic ordering, possibly antiferromagnetic, of the second layer 3He atoms sat on the first layer which itself forms a triangular lattice incommensurate with the underlying graphite. As for the solidification of second layer, Van Sciver and Vilches4 showed a melting peak to the p = 0.18 coverage in their specific heat experiment. Abraham et al5 found a registry solid phase in the low density by a numerical Monte Carlo simulation. Although the neutron experiment6 on higher coverage p = 0.203 failed to detect solidification of second layer, it sounds that depending upon the density the solid phases, which have different type of registry with the triangular lattice of the first layer, exist for very narrow ranges of coverage. These lower commensurate solid phases should play a role as a reference system for nearby densities. For example, say p2=p-pr =0.064 the system finds the fi x fi registry phase because p2/p1 almost equals to 417 7. Since the density pa is rather randomly chosen in an actual experimental situation, all 3He atoms would never be precisely accommodated in an appropriately selected registry solid phase. There generally exist excess or deficit atoms in this registry phase. From the view point that in the 3He bulk the higherorder cyclic exchanges make significant contributions’ to the magnetism, it is worth examining this two dimensional magnetic phenomenon from the itinerant picture rather than the localized one7sg. We study here a lattice gas model or Hubbard model at T=O on a triangular lattice; H = -t &;J>s &c~,, + U Ci niTnil where

0921-4526/90/$03.50

t(=l) is the hopping integral between the nearest neighbor sites and U is the on-site hard core repulsion. We confine our discussion to coverages well before the second layer is completed and ignore the first layer, thus consider only the second layer. If the number of 3He are equal to the lattice sites of a registered triangular lattice, it corresponds to the “half-filling”. A deviation from that atom number inevitably introduces a few excess or deficit atoms, namely defects. This corresponds to the “nearly half-filling case” of the Hubbard model. We take here the simplest mean field approach under the assumption that there is a long-range order in the ground state. In fact, as stressed by Friedman et al.” the dipole interaction, although its coupling energy is extremely weak (- O.l@‘), guarantees its existence and makes the nuclear spins confine in the plane of the substrate, therefore we take account of only two spin components S’ and S’. Then the mean field Hamiltonian is given by

H= -t

c 4&a+ ; C{(n; - .y)?l,T +

0

-S:(CffCil + h.C.)} -

(n, + ,y+,_

I

r C(7lf - (sS:)~- (AS~)~}

(1)

I

where the order parameters; the moments Sf,.SF and atom density 12; depend on the ith site, are determined self-consistently by S[ =< nir > - < 71,~>, SF =< Cfyc,~> and n, =< “if > + (pg = 1). To seek a stable spin and density configuration by increasing the unit cell size of a periodic pattern under a given filling, we numerically solve (1) and self-consistent equations iteratively, starting from various kinds of initial configurations {S:,S:,ni}(a = l,...N) under the periodic boundary condition. As for the initial condition both cases of the collinear and non-collinear spin patterns are considered. For larger unit cell, we restrict ourselves to the case that the system holds three-fold symmetry, namely the pattern is characterized by the triple Q wave vectors. We note that the & x fi sub lattice is characterized by the single Q vector. + < CflC,~>

@ 1990 - Elsevier Science Publishers B.V. (North-Holland)

712

M. fijita,

Let us first examine the half-filling case where the atom number per site is n = 1. There are several possible magnetic phases: (1) antiferromagnetic phase (AF) with the 2 x 1 sublattice, (2) ferromagnetic phase (FM), (3) ferrimagnetic phase (FR) with the 6 x fi sublattice , (4) the 120” structure (I$) in which the three spins on a unit triangle take three distinctive directions differed by 120” each other and other collinear and non-collinear cases with a larger unit cell. We examine the relative stability among these states and find that V, is the lowest among them for a wide range of U (2 U, = 5.5)‘l. For example, when U = 10.0 the relative energies per site are -2.9231(G), -2.8925(FR), -2.8817 (AF) and -2.4997 (FM). In the V, phase, the six bands, which come from up and down spins and three sublattice structure, group together into two separate bands and open a gap only in the middle of them, making V, energetically favorable for n = 1 but not for n # 1. Below U, the indirect gap between them vanishes and V, is destabilized. Having seen that the non-collinear V, is stable over the other collinear states for n = 1, we seek a generic magnetic structure with larger unit cell sizes for nearly half-filling cases where defects should be neatly accommodated. We display a typical example of the stable magnetic ordering pattern in the self-consistent solution in Fig.1 for the case thatN = 9 x 9 lattice with 93 atoms, i.e. n = 31/27 when U = 8.0. It forms &? x &? sub lattice structure. We see that (1) there are three vortices contained in a unit cell and they form a vortex lattice structure”. There are two kinds of vorticity or winding number; one is $1 situated at the center and three corners of the hexagon while the other is -2. The total vorticity vanishes. (2) The gross feature of this vortex lattice structure (VLS) can be constructed by the triple Q wave vectors whose directions are l”Z apart and whose magnitudes 1Q I= 8a/9 in this case. (3) All the moments on each site almost equal except for the vortex sites where the moment vanishes and excess (or deficit) density exclusively accumulated, forming a localized object or a triple Q soliton lattice. This implies that higher harmonics play an important role. (4) The sum of all the moments vanishes, thus there is no net ferromagnetic component in the system. (5) The spin configuration far from the vortex sites tends to refer the V, configuration which is the most stable state for the half-filling case. In other fillings near n = 1 we can always find a selfconsistent solution of VLS with the differernt sizes of the unit cell, provided that the combination between the triple Q wave vectors and the atomic filling is properly chosen. As the deviation from n = 1 becomes small, the unit cell of the VLS becomes large by changing the fundamental wave vectors 1Q 1 to accommodate defects. In conclusion, the second 3He layer is found to be an interesting test ground for the study of two-dimensional

K. Machida

,,___+’

9

,,L f (\

1 ,

\

1

,

’ A<* \

’ \’ / ’

\y:

‘/

/

‘\\,

/-,

I, 1

1

1

”

,‘,‘I

’

YJ; ’

y,L ’ t l,l

‘A

1

’ :>

L.~__J___, ’ 1 ‘/ ’ ’ \’

‘/

,y-,

’ ,

, ,‘.‘I , \

’ I’

\

,

,’

/’ \

Fi .l M netic structure of Vortex lattice state with the 27 x 27 sublattice when U =8.0 and n=31/27. The ?49 broken line indicates the unit cell. spin 5 system on a frustrated triangular lattice with defects whose number and interaction strength U/t can be controlled by varying coverages. We have found a novel type of the periodic ordering pattern of the vortex lattice structure for accommodating defects which normally exist in an actual experimental situation of the absorbed 3He films on grafoil. The present study might help to understand the magnetism in itinerant electron systems such as fee Sd-transition metal al10ys~~ where because of non-bipartite crystalline structure there appear various non-collinear types of magnetic patterns with triple Q vectors. The physical origin for this largely remains unexplored. References 1) D.S. Greywall and P.A. Busch, Phys.Rev.Lett. 62, 1868 (1989). 2) P.V.E. McClintock, Nature 340, 98 (1989). 3) H. France, R.E. Rapp and H. Godfrin, Phys.Rev.Lett. 57, 1161 (1986). 4) S.W. Sciver and O.E. Vilches, Phys.Rev. B18, 285 (1978). 5) F.F. Abraham, J.Q. Broughton, P.W. Leung and V. Elser, preprint. 6) C. Tiby, H. Wiechert, H.J. Lauter and H. Godfrin, Physica 107B, 209 (1981). 7) V. Elser, Phys.Rev.Lett. 62, 2405 (1989). 8) D.M. Ceperley and G. Jacucci, Phys.Rev.Lett. 58, 1648 (1987). 9) M. Roger, Phys.Rev.Lett. 64, 297 (1990). 10) K. Machida and M. Fujita, preprint. 11) L.J. Friedman et al., Phys.Rev.Lett. 62, 1635 (1989). 12) See for example, T. Jo and K. Hirai, J.Phys.Soc.Jpn. 53, 3183(1984). S. Kawarazaki et al., Phys.Rev.Lett. 61, 471 (1988). C.L. Henley, ibid 62, 2056 (1989).