Pis'ma v ZhETF, vol.91, iss. 11, pp.669-675 © 2010 June 10
Orbital glass and spin glass states of 3He-A in aerogel
V. V. Dmitriev, D. A. Krasnikhin, N. Mulders+, A. A. Senin, G. E. Volovik*A, A. N. Yudin P.L. Kapitza Institute for Physical Problems RAS, 119334 Moscow, Russia + Department of Physics and Astronomy, University of Delaware, Newark, 19716 Delaware, USA *Low Temperature Laboratory, Aalto University, FI-00076 AALTO, Finland AL.D. Landau Institute for Theoretical Physics, 119334 Moscow, Russia Submitted 29 April 2010
Glass states of superfluid A-like phase of 3He in aerogel induced by random orientations of aerogel strands are investigated theoretically and experimentally. In anisotropic aerogel with stretching deformation two glass phases are observed. Both phases represent the anisotropic glass of the orbital ferromagnetic vector 1 - the orbital glass (OG). The phases differ by the spin structure: the spin nematic vector d can be either in the ordered spin nematic (SN) state or in the disordered spin-glass (SG) state. The first phase (OG-SN) is formed under conventional cooling from normal 3He. The second phase (OG-SG) is metastable, being obtained by cooling through the superfluid transition temperature, when large enough resonant continuous radio-frequency excitation is applied. NMR signature of different phases allows us to measure the parameter of the global anisotropy of the orbital glass induced by deformation.
1. Introduction. Superfluidity of 3He in high porosity aerogel [1, 2] allows to investigate influence of impurities (i.e. aerogel strands) on superfluidity with nontrivial Cooper pairing. It was found that like in bulk 3He two superfluid phases (called by analogy A-like and B-like) can exist in 3He in aerogel in a weak magnetic field [3]. The B-like phase is analogous to bulk 3He-B and has the same order parameter [3, 4]. As for the A-like phase then for many years the situation was not clear. It was proposed that the A-like phase in aerogel is described by the Larkin-Imry-Ma [5, 6] (LIM) model [7]. The random orientations of silicon strands induces spatially random distribution of 3He-A order parameter. However, properties of nuclear magnetic resonance (NMR) in various aerogel samples were different and did not correspond to the properties of the fully randomized bulk A phase. Situation became much more clear when it was realized that even weak anisotropy of aerogel can influence the NMR properties and was found experimentally that in squeezed by aerogel the A-like phase behaves as the bulk 3He-A but with the orbital vector I fixed along the deformation axis [8]. This observation was in agreement with LIM model developed for the case of nonzero global anisotropy [9]. It was also found that intrinsic anisotropy in some samples can be large enough to orient I and all NMR properties of the A-like phase in such samples correspond to the bulk A phase order parameter oriented along some fixed axis [10].
Here we consider consequences of the theory developed in [7, 9] for different values and types of global
anisotropy and report results of detailed NMR investigations of A-like phase in three aerogel samples with different anisotropy.
2. Theory. In superfluid 3He, the spin-orbit interaction is small compared to other characteristic energy scales. That is why the superfluid phases of 3He consist of two nearly independent subsystems of orbital and spin degrees of freedom. The bulk 3He-A is characterized by nematic ordering in the spin subsystem [11] and by the ferromagnetic ordering in the orbital subsystem. Its order parameter is the matrix
Aaj = Ada(e) + ie2j). (1)
Here d is unit vector describing the nematic spin order. Orthogonal unit vectors e1 and e2 describe orbital fer-romagnetism with ferromagnetic moment along the unit vector I = e1 x e2.
Depending on value and type of anisotropy several possible structures of the order parameter (1) can be realized in the A-like phase of 3He in aerogel. For isotropic aerogel, the orbital vector I is randomized due to the quenched local anisotropy provided by random orientations of aerogel strands [7]:
(i) = 0, (ll) = </2) = (It) = 1/3. (2)
This glass state is the realization of the LIM phenomenon in 3He-A. In this state the space average of the order parameter (1) is zero, (Aaj) = 0, and
(AaiA*ßj)
(AaiAßj) = 0, 2 / C.C. = -A2Sij (Saß
xhß^j ■
(3)
Here h is unit vector along magnetic field H which keeps d in the plane normal to h, where d is randomized due to spin-orbit interaction with chaotic orbital momentum 1. We call this configuration the OG-SG state, since it is the combination of the orbital glass (OG) and spin-nematic glass (SG). The OG-SG states produced by random anisotropy of aerogel strands provide the experimental realization of the random anisotropy glasses discussed in different systems [12], such as random anisotropy Heisenberg spin glasses in magnets [13, 14] and nematic glasses in liquid crystals [15-17].
Uniaxial deformation adds more states of superfluid 3He-A in aerogel. Chaotic spatial distribution of the orbital vector 1 is modified under deformation and becomes anisotropic. For squeezing or stretching of aerogel we obtain the orbital glass states with global anisotropy:
(i)
= 0, (i2z) =
1 + 2 q
<0 = <0 =
I-«
(4)
Here the axis of deformation is z and we introduce parameter q of global anisotropy, which is positive in squeezed aerogel, negative in the stretched case and q = 0 in isotropic aerogel.
Stretching of aerogel gives the global easy plane anisotropy with ^0.5 < q < 0 and 0 < (l2z) < 1/3. In the limit of large stretching, q approaches the value q = ^0.5, where (l\) = 0, i.e. 1 is kept in x — y plane and the planar LIM state is formed. This orbital glass is described by the random anisotropy XY model.
Squeezing of aerogel gives the global easy axis anisotropy with 1 > q > 0 and 1/3 < (l\) < 1. However, in this state q may not reach 1, because for deformations greater than some critical value [9] the orbital ferromagnetic (OF) state should be restored. In the OF state, 0 and is parallel to the deformation axis z.
At large squeezing, (lz) approaches +1 or — 1. Properties of the OF state correspond to the bulk A phase but with I fixed along the axis of deformation. Observations of such ferromagnetic state were reported in [8, 10, 18]. For the intermediate squeezing deformations, the transverse components (/2) and (/2) may be substantial and the ferromagnetic order, (lz) ^ 0, is supplemented by the glass state for lx and ly components.
3. NMR frequency shift. The frequency shift of transverse NMR from the Larmor value is given by [19, 20]:
Aw/Awo = 7cos/3 + l/? w £ 2 1-
duD
■(lx h)2
dcosß cos/3
(I • dxh) — cos/3, (5)
UD = sin2 /3 + -(1 + cos/3)2(I • d x h)2 ^(Icos2/3+icos/3^i)(ixh)2,
(6)
where Awo = n^/(2w) is maximal possible value of the shift, is Leggett frequency, /3 is the tipping angle of magnetization, Ud is the normalized spin-orbit (dipoledipole) energy averaged over fast precession of magnetization. The unit vector d here is the result of averaging over fast precession, it coincides with the original spin-nematic vector d only for /3 = 0.
In the LIM state, the dipole energy should be averaged over space. The LIM characteristic length is expected to be much smaller than the dipole length, which characterizes the spin-orbit interaction, so the NMR line should not be broadened due to inhomogeneities of 1.
We consider the general case when H is tilted by an angle fi to anisotropy axis z:
h = cos ß z + sin ß x.
(7)
If the global orientation of I is capable to orient d (to minimize Ud) then in squeezed aerogel we get:
d sin ß z ^ cosß x, q > 0, and in stretched aerogel:
d = y, q< 0.
(8)
(9)
These states in deformed aerogel combine anisotropic orbital glass (OG) and the ordered spin nematic (SN) and we denote them as anisotropic OG-SN states.
If I is not able to orient d then the chaotic distribution of spin vector d is realized:
d = cos (sin /tz - cos ß x) + sin y,
(10)
with random <Iv This is an anisotropic OG-SG state -the anisotropic orbital glass accompanied by spin glass. Note that the OG and SG subsystems have different characteristic length scales.
3.1. Anisotropic OG-SN in squeezed aerogel. In this state the spin nematic is regular, and we should average only over the orbital glass state. In squeezed aerogel, the averaging over space of (8) gives:
((i.dxh)2) = <Z2) = i(l^</2)), (11)
((ixh)2) = l^) + isinV(3</^l). (12)
Then the NMR frequency shift is
f • 2 7cos/? + l\ ,,0,
Awo = ^ V cos@ + sm M | J ■ d3)
Equation (13) with q m 1 is applicable also for the orbital ferromagnetic (OF). Therefore NMR can not distinguish between OF and OG-SN states in squeezed aerogel with q = 1. The latter state may be considered as multidomain OF state or the Ising orbital glass.
3.2. Anisotropic OG-SN in stretched aerogel. For stretched aerogel, from (9) we obtain:
1 cm
J, 2
((i.dxh)2) = i(l^{/2)) + isin2/t(3{/2)^l)
and the corresponding NMR frequency shift is
Aw ( n . 2 5 cos 0 — V --= q I — cos 0 + sm u---
Aw0 V 4
(14)
(15)
3.3. Anisotropic orbital glass + spin glass. In the OG-SG state the spin-nematic vector d in (10) is random, and we obtain:
{(i.dxh)2) = i((ixh)2), with ^(1 x h)2^ from (12). Then Aw is given by:
Aw
Au>0
= q cos 0 [ - sin2 /t — 1
(16)
(17)
which is valid for both squeezed and stretched aerogel.
4. Conditions of experiments. Three different aerogel samples with porosity of 98.2% and with different types and values of anisotropy were used. All the samples had cylindrical form with axes oriented along z. Diameter and height of the samples (in mm) were the following: 4 and 4 (sample 1), 3.8 and 5 (sample 2), 6 and 3 (sample 3). The experimental chamber (Fig.l) was made from epoxy resin "Stycast-1266". The upper cell (1 and 2 on Fig.l) was used first for sample 1 and then for sample 2. Samples 1 and 2 had gaps ~0.1 mm from side walls of the ce
Для дальнейшего прочтения статьи необходимо приобрести полный текст. Статьи высылаются в формате PDF на указанную при оплате почту. Время доставки составляет менее 10 минут. Стоимость одной статьи — 150 рублей.