Anisotropic phases of superfluid 3He in compressed aerogel

4.7 MB·English

by  J. I. A. Li

Anisotropic phases of superfluid 3He in compressed aerogel
J.I.A. Li,∗ A.M. Zimmerman, J. Pollanen,† C.A. Collett, and W.P. Halperin‡
Northwestern University, Evanston, IL 60208, USA
(Dated: January 15, 2015)
Ithasbeenshownthattherelativestabilitiesofvarioussuperfluidstatesof3Hecanbeinfluenced
by anisotropy in a silica aerogel framework. We prepared a suite of aerogel samples compressed up
to30%forwhichweperformedpulsedNMRon3Heimbibedwithintheaerogel. WeidentifiedAand
B-phases and determined their magnetic field-temperature phase diagrams as a function of strain.
From these results we infer that the B-phase is distorted by negative strain forming an anisotropic
superfluid state more stable than the A-phase.
5
1
0
2 The A and B-phases of superfluid 3He are unconven-
a b c d
tionalCooperpairedstateswithorbitalangularmomen-
n
tumL=1. Theyarejusttwoofmanyp-wavestateseach
a
J distinguished by unique broken symmetries and these
0 are the only stable states in pure 3He in low magnetic
1 field[1]. However,newphasesinthismanifoldhavebeen b0 b12 b20 b30
reportedforsuperfluidconfinedwithinananisotropicex- ε = 0% ε = -11.7% ε = -19.8% ε = - 30.0%
]
n ternal framework; in one case chiral states within posi-
o tively strained 98% porous, silica aerogel [2] and in an- 200
c other, equal spin pairing states within nematically ori- e
-
r ented alumina hydrate aerogels [3]. Additionally, for a
p 150 a0
negativelystrainedaerogel, obtainedbycompression, we
su foundthattherelativethermodynamicstabilitybetween opy ba1192
t. the A and B-phases can be reversed resulting in an un- otr100 b20
a usual tricritical point between A, B, and normal phases s
m ni b30
with a critical magnetic field of H ≈100 mT [4]. A
c
- Our observation of preferential B-phase stability rel- 50
d
ative to the A-phase [4] appears at first glance to be
n
o inconsistent with theoretical predictions where it was es-
0
c tablishedthatanisotropicscatteringof3Hequasiparticles
[ would favor the anisotropic A-phase, over the isotropic - 0.3 - 0.2 - 0.1 0.0

1 B-phase [5–7]. To address this problem we have investi- Strain
v gated the dependence of the A-to-B phase diagram as a
4 function of negative strain.
FIG. 1. (Color online). Anisotropy introduced by compres-
8
Herewereportthatthesquareofthiscriticalfieldisdi- sive strain, ε, in two batches of 98% porosity aerogel, a and
3
rectlyproportionaltotheaerogelanisotropywhichwede- b. a) to d) Optical birefringence images of the transmitted
3
0 termined from optical birefringence. Based on our mea- light intensity through the aerogel samples. e) Anisotropy in
. surements of the strain dependence of the A-to-B phase theaerogeldefinedastheintensityoftheopticalbirefringent
1
signal versus strain. The black solid line is a linear fit for
0 diagramwehavedeterminedthattheB-phaseentropyis
samples from batch b.
5 increasedrelativetotheunstrainedB-phasewhichtrans-
1 forms under negative strain to a new type of anisotropic
v: superfluid, which we call a distorted B-phase, that can
i effectivelycompetewiththeA-phaseinamagneticfield. negative strain −19.4% (a19) and −22.5%(a23) intro-
X
Wepreparedtwodifferentbatchesofaerogelsamplesof duced into isotropic samples from the same batch (a0);
r
98%porosity,batchaandb,intheshapeofcylinders4.0 a19 (originally a0) was measured with the strain axis
a
mm in diameter with unstrained length of 5.1 mm. For paralleltothemagneticfield,ε(cid:107)H,anda23withε⊥H.
clarity we label the sample by the batch number as well In the present work three new samples are taken from
as the nominal amount of negative strain introduced by batch b prepared using the same methods as batch a.
compression, e.g., a sample from batch a with −19.4% Optical birefringence was performed before introducing
negative strain is labeled as sample a19. Some results anisotropy by compression and the samples were found
with samples from batch a have also been reported pre- to be homogeneous and isotropic, Fig. 1(a) [11, 12]. We
viously[4,8–10],wherethesuperfluidphaseswereidenti- strained these samples −11.7% (b12), −19.8% (b20),
fied and the amplitude of the order parameter was mea- and−30.0%(b30)bymechanicalcompressionmeasured
sured. A critical field was discovered for samples with as a displacement at room temperature. The strain in-
2
condensation energy and the symmetry of the super-
1.05
fluid states, and α(T) ∝ 1 − T/T . The B-phase has
Normal phase c
a quadratic field dependent free energy compared to the
1.00
normalstate,whereastheA-phasefreeenergyisfieldin-
A-phase
dependent. From Eq. 1 and 2, the field suppression of
T / T c BA 00..9950 a0 TBA in theTGL limit ca(cid:18)nHb2e−wrHitt2e(cid:19)n as, [1(cid:18)5,H16(cid:19)]4
b12 B-phase 1− TBA =gBA H2 c +O H , (3)
0.85 b20 c 0 0
b30 where H = 0 for pure 3He and for 3He in an isotropic
c
0.80 aerogel [8]. H0 and gBA are constants from the GL the-
0 0.01 0.02 0.03 0.04 ory[5,17]. However,thepresenceofstrainintheaerogel
H 2 ( T 2 ) breaks the orbital rotation symmetry of the superfluid
to which we attribute the appearance of a critical field,
H , that we find depends systematically on the magni-
c
FIG. 2. (Color online). Superfluid phase diagram T /T
BA c tudeofthestrain. Infactallanisotropicaerogelsamples
versus H2, with ε(cid:107) H for anisotropic b samples compared
with negative strain exhibit this behavior as exemplified
to the isotropic sample a0 [10] at a pressure of P =26.5 bar
in Fig. 2. The existence of a critical field is indicative
fromwarmingexperiments. Thesuperfluidtransitiontemper-
atures,T ,fora0,b12,b20,andb30are2.213,2.152,2.118, of a new term in the relative free energy between the
c
and 2.118 mK. The critical fields are indicated by arrows: A and B-phase which is not captured by Eq. 1 and 2.
Hc =45.5 mT, 91.8 mT, and 120.2 mT for these samples. To determine the origin of Hc we must isolate how each
of the phases is affected by strain. For this purpose we
measure the longitudinal resonance frequency of the su-
ducedanisotropywascharacterizedbytheintensityfrom
perfluid A and B-phases, Ω and Ω , which are directly
transmittedopticalbirefringence,Fig.1(b-d),takenfrom A B
related to their respective order parameter amplitudes,
a 2 × 2 mm2 central portion of the images. The inte-
∆ and ∆ , and correspondingly to their free energy.
grated intensity as a function of strain for the two differ- A B
For the isotropic sample a0, Ω and Ω satisfy the
entbatchesareconsistentwitheachother,Fig.1(e),both A B
Leggett relation, Eq. 4, which expresses the unique or-
ofwhichshowadelayedonsetofbirefringencewithstrain
der parameter symmetries of the A and B-phases in the
of ε ≈ −8%, due to the intrinsic mechanical behavior
Ginzburg-Landau regime given by,
of the aerogel (see supplementary information). Beyond
this onset, the transmitted intensity increases linearly. 5 Ω2 χ ∆2
= B B A (4)
All b samples were oriented with the strain axis parallel 2 Ω2 χ ∆2
tothefield,cooledsimultaneously,andfilledwith3Hesu- A A B
percritically. We performed pulsed NMR measurements where we take ∆2 ≈∆2 [8, 18] [19] and the susceptibil-
A B
atapressureP =26.5barinmagneticfieldsrangingfrom ities χ and χ are measured.
A B
H= 49.1 to 214 mT with small tip angles, β ∼20◦. De- The maximum frequency shift of the A-phase from
tailedproceduresforthesemeasurementsandtheiranal- sample a0 is shown in Fig. 3 as open black diamonds,
yses have been described earlier [2, 8, 10]. where the entire sample is in the dipole-locked config-
Thefielddependenceofthethermodynamictransitions uration (dˆ(cid:107)ˆl⊥H) ensured by warming from the B-
onwarming,T (H2)areshowninFig.2forsamplea0, phase [10], where dˆ is the direction along which the
BA
b12, b20, and b30. Quadratic field dependences are spin projection is zero, and ˆl is the direction of the or-
observed for the field suppression of T in all samples, bital angular momentum. The longitudinal resonance
BA
which can be qualitatively understood by analogy with frequency can be calculated from the high field approx-
the free energy difference between A and B-phases for imation: Ω 2 = 2ω ∆ω with ∆ω the measured shift
A L A A
pure 3He in the Ginzburg-Landau limit given by, in frequency from the Larmor frequency ω . When our
L
isotropic aerogel is subjected to sufficient uniaxial nega-
1 (cid:0)α2(T)+2α(T)g H2(cid:1) tive strain the A-phase becomes a two-dimensional (2D)
F −F =− z +O(H)4, (1) orbitalglassphase[4]withfrequencyshiftatsmallNMR
B N 36 β + 1β
12 3 345 tip angles β given by [20, 21],
FA−FN =−316αβ22(4T5). (2) ∆ω2D = Ω4ωAL2 β ≈0. (5)
The β and g are material parameters in the Ginzburg- Using Ω determined from the isotropic sample a0 and
i z A
Landau functional [13, 14] that determine the superfluid Eq. 5, the frequency shift for a 2D glass phase with the
3
150 3.0
A-phase a0
100 b12 2.5 B-phase
a19
2π(Hz) 500 bb3200 π (kHz) 21..05 aaa02193
∆ω / A-50 dipole-locked ∆ω /2 B 1.0
2D-glass

-100 3D-glass 0.5
dipole-unlocked
0
-150
0.90 0.95 1.00 1.05 0.70 0.80 0.90 1.00
T/T T / Tc
c
FIG. 3. (Color online). NMR frequency shifts of the A- FIG. 4. (Color online). NMR frequency shifts ∆ω of the
B
phase, ∆ω , as a function of reduced temperature measured B-phaseasafunctionofreducedtemperaturemeasuredwith
A
with small tip angle, at P ≈ 26 bar, adjusted to a common small tip angle, at P = 26.3 bar. The frequency shifts are
field of H = 196 mT. On warming from the B-phase, sam- from a peak in the NMR spectrum [8] corresponding to the
plea0isinthedipole-locked(dˆ(cid:107)ˆl⊥H)configurationgiving texture, (cid:96)ˆ⊥H, that determines Ω , Eq. 6. This texture ap-
B
a maximum frequency shift (black diamonds) [8]. The black pears above (below) a crossover temperature T ∼0.85T for
x c
solid curve is for the frequency shift of pure 3He-A with a sample a19 (a23); see supplementary information. The fre-
scaling factor of 0.42, corresponding to an order parameter quency shifts for the anisotropic samples a19 and a23, mea-
suppressionof0.35. Thefrequencyshiftofatwo-dimensional sured at H = 95.6 mT, are significantly larger than for the
glassphaseatsmalltipangleiscalculatedusingEq.5,black isotropic sample a0 indicating a change in the order param-
dashedcurve,ingoodagreementwiththedatafora19at174 eter for strain ε ∼ −20%. The frequency shift for sample
mT(orangesquares)andforb20(redcrosses)andb30(blue a0 was measured at H = 196 mT and scaled to a common
triangles) at 196 mT. The negligible shift for b12 (green cir- fieldof95.6mTusingEq.6. Acomparisonwithsamplesb20
cles)withnolinewidthincrease(notshown),ischaracteristic andb30wasnotpossibleowingtotextureinhomogeneityfor
of a three-dimensional glass phase [10]. T >T .
x
bital quantization axis (cid:96)ˆperpendicular to the magnetic
same order parameter amplitude is plotted as the black
field,forexampleconstrainedbysamplewalls,hasamax-
dashed curve in Fig. 3. The frequency shift for samples
imum frequency shift for small β directly related to the
a19, b20 and b30, Fig. 3, agree well with the black
longitudinalresonancefrequencyandcanbeidentifiedas
dashed curve, showing that the longitudinal resonance
aprominentpeakinthespectrumatlargefrequency [8],
frequencyoftheA-phase,Ω ,isnotaffectedbynegative
A
strain up to 30%. We conclude that the order parameter
2Ω2
amplitude of the A-phase, ∆ , is insensitive to aerogel ∆ω = B β ≈0. (6)
A B 5 ω
anisotropy induced by negative strain and consequently L
the free energy is also unaffected. Additionally, a zero We refer to the spin dynamics of this texture as the
frequency shift and zero linewidth increase are observed (cid:96)ˆ⊥H-mode. In previous work we measured ∆ω , for
B
in sample b12, Fig. 3, indicative of the existence of a this mode in sample a0 [8], shown in Fig. 4 together
three-dimensional (3D) glass phase [10]. As shown in with its extension to T as a black dashed curve. When
c
Fig.1(b)and(e), sampleb12isstrainedsignificantlyby we compare this to the frequency shift in the A-phase,
ε = −11.7% but only displays a weak optical birefrin- Fig. 3, we obtain the factor of 5/2 given in Eq. 4; this
gence due to the delayed onset of anisotropy at ∼ −8% factor expresses the characteristic symmetries of the ax-
strain. Therefore, the observation of a 3D glass phase ial and isotropic p-wave states (A and B-phases respec-
suggests a threshold aerogel anisotropy below which the tively) as is the case for pure superfluid 3He. On the
continuous3Drotationalsymmetryoftheorderparame- other hand, for the negatively strained aerogels a19 and
ter,andthusthe3Dglassphase,ispreserved(seesupple- a23, the (cid:96)ˆ⊥H-mode frequency shifts are significantly
mentaryinformation). Consequently,thetransitionfrom increased. The black dash-dotted curve is a fit to our
a3D-glassphasetoa2D-glassphasemustoccurwithin- data for H = 95.6 mT found by multiplying the black
creasing(negative)strainwithinasmall7%window. The dashed curve by 1.7. We conclude that the B-phase in
NMR tip angle behavior of the 3D glass phase in sample strained aerogel is not an isotropic state and might have
b12 is shown in the supplementary information. adifferentorderparametersymmetry. Thisisanewtype
In the B-phase the order parameter texture with or- ofanisotropicsuperfluidwhichcontinuouslyevolvesfrom
4
theisotropicB-phasewithincreasinganisotropyinduced
1.6
by the compressive strain in the aerogel.
a
In Eq. 3, the two parameters affected by strain de- 1.2
duced from the AB-phase diagram are the square of the 2T )
criticalfield,Hc2,andtheslopeoftheBtoA-phasetran- -2 0 0.8 a0
saitfiuonnc,tgioBnA/oHf a02n.iIsnotFroigp.y5f,owrethpelotthtrheeeseaetwroogeqlubanstaitmiepslaess 2H (1c 0.4 bba112920
comparedwiththeisotropicsamplea0andfindthatthey b30
areproportionaltotheanisotropyinducedbystrain. We 0.0
understand that the mechanism by which strain modi-
fies the superfluid is to create anisotropic quasiparticle 52
scattering in the superfluid. Consequently we can infer b
thatanisotropicscatteringsmoothlyincreaseswithaero-
48
gel anisotropy and is responsible for creating a distorted 2H 0
Bfic-apthiaosne.ofAthltehonuegwhswtaetehaitveisnnootnmetahdeleesasscpleeacrififcroimdenotuir- g / BA 44
susceptibility measurements that it is a non-equal spin
40
pairing state (NESP) with the same susceptibility as the
B-phase. The latter comparison was made between a19
and in the unstrained aerogel a0 at the same temper- 36
0 50 100 150 200
ature, pressure, and field on their respective AB-phase
Anisotropy (arb. units.)
boundaries [22]followingtheproceduredescribedinRef.
[4, 8, 10].
FIG. 5. (Color online). (a) The critical field squared H2
Withthisinputwecancalculatethechangeofentropy c
and (b) g /H2 for sample a0 (open black diamond), a19
BA 0
of the new phase, compared to the isotropic B-phase,
(open orange square), b12 (open green circles), b20 (red
using the Clausius-Clapeyron equation, crosses)andb30(openbluetriangles)asafunctionofaerogel
anisotropy taken from measurements of the optical birefrin-
gence intensity. H 2, and g /H2 show a strict proportion-
c BA 0
dT 1χ −χ
=− A new, (7) alitytoaerogelanisotropy,withtheexceptionofsamplea19.
dH2 2S −S Sample a19 does not lie on the same line as the batch b
A new
samples for the second parameter, g /H2.
BA 0
where χ and S are the susceptibility and the entropy of
the two phases along the phase equilibrium line. Since
the slopes of these lines, Fig. 2, are increasingly more
negative with increasing strain we deduce that the new
∗ [email protected]
phase has a higher entropy compared with the isotropic
† Present Address: Institute for Quantum Information
aerogel B-phase.
and Matter (IQIM), California Institute of Technology,
In summary, we report NMR measurements as a func- Pasadena, California 91125, USA
‡ [email protected]
tion of compressive strain in aerogel that determine the
[1] D. Vollhardt and P. Wo¨lfle, The Superfluid Phases of
phase diagrams of the superfluid phases within the aero-
Helium 3 (Taylor and Francis, 1990).
gel framework. The anisotropy in the aerogel stabilizes
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anewanisotropicnon-equalspinpairingsuperfluidstate W. P. Halperin, and J. A. Sauls, Nature Phys. 8, 317
which evolves continuously from the isotropic B-phase (2012).
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[4] J.I.A.Li,A.M.Zimmerman,J.Pollanen,C.A.Collett,
is proportional to the anisotropy. Additionally, the en-
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tropy of this distorted B-phase is larger than that of the
112, 115303 (2014).
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Sauls, Phys. Rev. Lett. 80, 2861 (1998).
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[6] K. Aoyama and R. Ikeda, Phys. Rev. B 73, 060504(R)
Dmitriev, Yoonseok Lee, J.M. Parpia, and G.E. Volovik
(2006).
for helpful discussion and to W.J. Gannon for help in
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theearlyphasesofthisworkandtotheNationalScience N.Mulders, andY.Lee,Phys.Rev.B72,094519(2005).
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5
W. P. Halperin, Phys. Rev. Lett. 107, 195301 (2011).
40
[9] J.I.A.Li,J.Pollanen,C.A.Collett,W.J.Gannon, and
W.P.Halperin,J.Phys.: Conf.Ser.400,012039(2012). a
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W. J. Gannon, and W. P. Halperin, Nature Physics 9, py
o
775 (2013). otr 20
[11] J. Pollanen, K. R. Shirer, S. Blinstein, J. P. Davis, s
ni
H. Choi, T. M. Lippman, L. B. Lurio, and W. P. A
Halperin, J. Non-Crystalline Solids 354, 4668 (2008). 10
[12] A.M.Zimmerman,M.G.Specht,D.Ginzburg,J.Polla-
nen,J.I.A.Li,C.A.Collett,W.J.Gannon, andW.P.
Halperin, J. Low Temp. Phys. 171, 745 (2013). 0
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(1976). 140 b
[14] J. W. Serene and D. Rainer, Phys. Rev. B 17, 2901 120
(1978). m )2 100
[15] G.Gervais,K.Yawata,N.Mulders, andW.P.Halperin, c
Phys. Rev. B. 66, 054528 (2002). s (g/ 80
[16] W. P. Halperin, H. Choi, J. P. Davis, and J. Pollanen, s 60
e
J. Phys. Soc. Jpn. 77, 111002 (2008). Str 40
[17] H. Choi, J. P. Davis, J. Pollanen, T. M. Haard, and
W. P. Halperin, Phys. Rev. B 75, 174503 (2007). 20
[18] A. J. Leggett, Rev. Mod. Phys. 47, 331 (1975). 0
[19] The free energy difference between the two phases is -0.15 -0.10 -0.05 0
much smaller than the superfluid condensation energy
Strain
[23].
[20] V. V. Dmitriev, D. A. Krasnikhin, N. Mulders, A. A.
FIG. 6. (Color online). Optical birefringence characteriza-
Senin, G. E. Volovik, and A. N. Yudin, JETP Lett. 91,
tionandsimultaneousstress-strainmeasurementofanaerogel
599 (2010).
samplefrombatchbduringcompressionupto≈−20%. The
[21] J.Elbs,Y.M.Bunkov,E.Collin, andH.Godfrin,Phys.
aerogel sample has the same dimension as the samples used
Rev. Lett. 100, 215304 (2008).
[22] J. I. A. Li, Transverse Pulsed NMR of Superfluid 3He inour3HeNMRexperiments. (a)Aerogelanisotropydefined
as the intensity of the optical birefringence is shown versus
in Aerogel: Engineering Superfluid States with Disorder,
negative strain. There is a delayed onset of anisotropy with
Ph.D. thesis, Northwestern University (2014).
increasingstrainatε∼−7%. (b)Stressversusstrainduring
[23] J.E.BaumgardnerandD.D.Osheroff,Phys.Rev.Lett.
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93, 155301 (2004).
[24] J. Pollanen, Transverse Pulsed NMR of Superfluid 3He attribute to a surface effect, after which the stress increases
linearly with strain.
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[25] G. E. Volovik, JETP Lett. 63, 301 (1996). develops and the sample lights up as shown in Fig. 1(b)-
[26] A. I. Larkin, JETP Lett. 31, 784 (1970). (d). Therefore aerogel anisotropy can be defined as the
[27] Y. Imry and S. Ma, Phys. Rev. Lett. 35, 1399 (1975). intensity of the optical birefringent signal.
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Since the global anisotropy was introduced with me-
[29] W. F. Brinkman and H. Smith, Phys. Lett. 51, 449
chanical strain, simultaneous stress-strain measurements
(1975).
[30] J. A. Sauls and P. Sharma, Phys. Rev. B 68, 224502 were performed along with the optical characterization,
(2003). onisotropicaerogelsampleswithdifferentgrowthcondi-
tions,todeterminetherelationshipbetweenstress,strain
andthetransmittedlightintensity,Ref.[12]. Theaerogel
SUPPLEMENTARY INFORMATION samples used in Ref. [12] have cylindrical shape, ∼ 7.5
mm in diameter and ∼ 7.5 mm long, and the growth
SI.1 Stress-strain characterization condition is controlled by varying ammounts of ammo-
nia catalyst. It was discoverd that independent of the
In previous work at Northwestern, Ref. [11, 24], it has growth condition, the stress and strain response obey a
been shown that optical cross-polarization studies pro- linear relationship for all the samples, while the trans-
vide good measures of aerogel anisotropy. The absence mitted intensity versus strain displays a delayed onset
of any optical axis was taken as evidence of a homoge- at small negative strain, ∼ −3%. The sound velocity
neous, globally isotropic aerogel. Such a sample remains in the aerogel sample, calculated from the slope of the
dark in the optical birefringence measurement, as shown stress-strain curve, was shown to be rather sensitive to
in Fig. 1(a). As mechanical strain is introduced into the thecatalystconcentrationduringthesamplegrowthpro-
nominally isotropic sample, a well defined optical axis cess, Ref. [12], suggesting that the microscopic structure
6
Sample Batch Porosity Strain ε Orientation T (mK)
40 ca
a0 a 98.2% 0 - 2.213
a19 a 97.8% −19.4% ε (cid:107) H 2.194
30
a23 a 97.7% −22.5% ε ⊥ H 2.186
y
p
o b12 b 98.0% −11.7% ε (cid:107) H 2.152
sotr 20 b20 b 97.8% −19.8% ε (cid:107) H 2.118
ni
A b30 b 97.4% −30.0% ε (cid:107) H 2.118
10
TABLE I. Samples used in the experiments.
0
0 40 80 120 160
Stress (g/cm 2 ) 200
FIG.7. (Coloronline). Transmittedintensityversusapplied
100
stress for a batch b aerogel sample. The threshold stress of
∼40 g/cm2 is similar to our observations from other sample Hz)
batches, Ref. [12]. π ( 0
2
ω/
∆
-100
of the isotropic aerogel samples from different batches b12 T/Tc = 0.83
can be significantly different, due tovaryinggrowth con- P = 26.5 bar H = 196 mT
dition. Therefore sample characterization is important -200
0 50 100 150 200 250 300 350
before we make comparison between samples from batch
β (degree)
a and b.
Weperformedopticalbirefringencecharacterizationon
FIG. 8. (Color online). Tip angle behavior of the A-phase
an isotropic aerogel sample from batch b with the same
on cooling from the normal state in sample b12 at P =26.5
dimensions as the samples used in the experiment, 4.0 bar, H = 196 mT and T/T = 0.83. The blue dash-dotted
ca
mm in diameter and 5.1 mm long. Simultaneous stress- lineisthetheoryforthe3Dglassphase,wherethesuperfluid
strainmeasurement,Fig.SI6(b),onsetsasnegativestrain order is completely hidden, Ref. [10, 25], as indicated by the
isappliedtothesample,providingaccuratemeasurement data. The expected tip angle behavior for the dipole-locked
configuration, ˆl⊥H, is shown by the black solid curve, and
for the zero of strain. With zero of strain accurately de-
the black dashed curve is the dipole-unlocked configuration,
termined, aerogel anisotropy shows a strain dependence
ˆl(cid:107) H,withΩ takenfromthemeasurementsforsamplea0,
A
similar to Fig. 1(e), which is linear with increased strain
Ref. [8].
fornegativestrainofε<−10%. Afittothelinearpartof
the birefringence intensity curve, Fig. SI6(a), intercepts
with zero intensity at a delayed onset of ∼−7% strain.
The stress and strain response shown in Fig. SI6(b) SI.2 Superfluid glass in the A-phase of b12
exhibits a delayed onset ε (cid:46) −3%. A possible reason
for this nonlinearity can be related to the fact that the Itwaspredictedthattheorientationalorderofthe3He
surface of the aerogel sample has a different stress-strain A-phaseinaerogelwouldbedestroyedbyrandomfluctu-
response compared to the bulk of aerogel. This surface ations of local anisotropy on a spatial scale less than the
effect is less significant for the larger samples used in dipole length, ξ ∼ 10µm, thereby forming a superfluid
D
Ref. [12]. We note that the surface effect can be elimi- glass, Ref. [25]. The glass phase is fundamentally differ-
nated by plotting aerogel anisotropy versus the applied entfromadisorderedA-phaseinducedbyinhomogeneity
stress. For the batch b sample, the transmitted light of the aerogel on long length scales. In order to reveal
intensity as a function of stress, Fig. SI7, displays a non- thecharacteristicsofathreedimensional(3D)superfluid
linear behavior at small stress ∼40 g/cm2 similar to the glass it is necessary to eliminate structural inhomogene-
longer samples used in Ref. [12]. ity requiring careful sample preparation and character-
In the anisotropy versus strain relation shown in ization as was demonstrated in Ref. [10]. In that work
Fig. 1(e), sample a19 agrees well with the fit to the theglassphaseof3He-Awasinvestigatedintheisotropic
batch b samples, providing evidence that samples from samplea0cooledfromthenormalstateusingNMRmea-
batchaandbweregrownundersimilarconditions. This surements with small tip angle β. The superfluid glass
justifies the comparison between samples from batch a was identified by zero frequency shift and zero linewidth
and b. In Table. I, we list detailed information of the broadening as compared to the normal state, consistent
samples from batch a and b. with theoretical prediction Ref. [25].
7
The key conceptual basis of an orientational glass
3.0
phase, according to the original theoretical works of
Larkin, Ref. [26], and Imry and Ma, Ref. [27], is the 2.5 B-phase
eraxmisteetnercetohfatalceoandtsintuooutshesydmemstreutrcytioonfaovfetchteorsourpdeerrflupiad- kHz) 2.0 a23 ε || H
order on a local scale, a situation that was first noted 2π ( 1.5 a19 ε ⊥ H
by Volovik, Ref. [28], to be applicable to 3He in aero- ω / B
gel. We reported earlier, Ref. [4], that negative strain ∆ 1.0
introduced by compression ∼ 20% restores the compo-
0.5
nent of the superfluid order in the direction along the
strain axis, stabilizing a 2D glass phase. However, in 0
the present work on sample b12 with ε = −11.7% we 0.70 0.80 0.90 1.00
foundalmostnofrequencyshift,Fig.8,andnolinewidth T / Tca
broadening compared to the normal state. This result
stronglyindicatestheformationofa3Dglassphaseeven FIG. 9. (Color online). B-phase frequency shift at small
β, ∆ω , versus reduced temperature at P = 26.3 bar, and
inthepresenceofwell-characterizedglobalanisotropy,al- B
H = 95.6 mT. The cross-over transition is indicated by the
beit quite small as indicated in Fig. 1(e). We performed
sharp variation in the frequency shift in both samples.
tip angle measurements of the frequency shift in sample
b12 at T/T = 0.83, deep in the superfluid A-phase.
ca
The absence of a frequency shift evident in Fig. SI3 for 2.3
all β up to ∼ 360◦, matches well with the theory for a Tca
3Dglassphase(dash-dottedblueline). Theexpectedtip
anglebehaviorforthedipole-lockedconfiguration,ˆl⊥H, 2.1 A
is shown by the black solid curve, and the black dashed K)
curveisthedipole-unlockedconfiguration,ˆl(cid:107)H,withΩ m TBAa
A T (
takenfromtheresultsforsamplea0,Ref.[8]. Thisisthe
1.9
first measurement of the NMR tip angle response of the
frequency shift in a 3D superfluid glass. B
Despite some global anisotropy in this sample, the
1.7
observation of the 3D glass phase indicates that the
0 0.01 0.02 0.03 0.04
local random field anisotropy is dominant and that
H 2 (T 2 )
there exists a threshold in global anisotropy required to
stabilize a 2D glass. Since a 2D glass phase is observed
FIG. 10. (Color online). Superfluid phase diagram
in sample b20 and b30, Fig. 3, the threshold global T/T (H2) at P =26.3 bar, showing the field dependence of
ca
anisotropy is between ε =−12% and 20%, including the thecross-overtransitiontemperatureintheanisotropicsam-
delayed onset in anisotropy of ∼−8%. ples. The black solid line is the superfluid transition in the
aerogel,T . TheopenredcirclescorrespondtoBtoA-phase
ca
SI.3 Cross-over transition in the distorted transitionsonwarming,TBAa,measuredinsamplea23. The
texturaltransitionsintheB-phase,T ,forsamplea19(sam-
B-phase x
ple a23) are shown as the blue open squares (triangles). No
fielddependenceforT isobservedineitheranisotropicsam-
x
A textural crossover is ubiquitous in the B-phase for ple,andnodependenceontheorientationofthefieldrelative
all aerogels we have studied. We plot the frequency shift to strain.
of sample a19 and a23 versus temperature in Fig. SI4.
The textural cross-over is marked by the abrupt change
in the frequency shift data as a function of temperature. quency shift of the (cid:96)ˆ⊥ H-mode texture of the distorted
For all anisotropic samples the sharply defined crossover B-phasecanonlybemeasuredforT <Txforsamplea23,
temperature is Tx/Tc ∼0.85, independent of the relative and above the transition temperature, Tx <T <Tca for
orientation between the strain and the magnetic field. sample a19 as shown in Fig. 4.
On warming through the transition, the orientation of The orbital quantization axis of the B-phase can be
theorbitalangularmomentumquantizationaxisswitches influenced by the magnetic field, the wall of the sample
from (cid:96)ˆ (cid:107) ε to (cid:96)ˆ ⊥ ε. In sample a19, the NMR spec- cell, as well as aerogel anisotropy. Consequently a
tra are dominated by the Brinkman-Smith-mode below possible explanation for the textural cross-over transi-
the cross-over temperature, T ∼1.9 mK, and above the tion is the competition between these three orienting
x
transition temperature the texture shifts to (cid:96)ˆ⊥ H, and effects. In Fig. 10 we show the field dependence of the
the Brinkman-Smith-mode is the main texture at high cross-over transition temperature, T (H2), in samples
x
temperature. Due to the cross-over transition, the fre- a19 and a23. In the magnetic field range of 31.1 to
8
2.5 2.0
ε || H T/Tc a23 ε T H
2.0 a19 H = 49.1 mT 0.76 1.5
a19 H = 49.1 mT 0.91
Hz) 1.5 b30 H = 95.6 mT 0.76 kHz) 1.0 4995..16 mmTT
ω/2π (k 1.0 ω/2π ( 0.5 119560 mmTT
∆ 0.5 ∆
0
0
-0.5
-0.5
0 50 100 150
0 50 100 150
β (degree)
β ( o )
FIG. 11. (Color online). Frequency shift versus tip angle FIG. 12. (Color online). Magnetic field dependence of the
for sample a19 and b30 in the distorted B-phase. The mag- tip angle measurements for a23 in the distorted B-phase at
nitude of the shifts have been adjusted to a common field of T/T =0.46. Themagnitudeoftheshiftshavebeenadjusted
c
H =95.6 mT, a common temperature of T/T =0.76. The to a common temperature of T/T = 0.78 and a common
ca ca
tip angle behavior of the isotropic sample a0 is represented field of H = 95.6 mT. The solid black curve is the tip angle
by the solid black curve for the Brinkman-Smith-mode and behaviorfortheBrinkman-Smith-modeandthedashedcurve
thedashedcurveforthetexturewith(cid:96)ˆ⊥H,Ref.[8]. Thetip forthe(cid:96)ˆ⊥H-modeoftheisotropicsamplea0,Ref.[8]. Inthe
angle behavior of the anisotropic samples matches well with magnetic field range of our measurements, the distorted B-
the black solid curve below the textural cross-over, whereas phase behavior deviates significantly from the shift observed
above the cross-over it displays larger frequency shifts than insamplea0. ThebluesolidcurveisafittothedataatH =
themaximumfrequencyshiftsobservedintheisotropicsam- 196 mT (solid blue circles), using a simple cosine function.
ple a0.
squares) is significantly increased. This result for small
196 mT, the cross-over transition temperatures show no β was noted in Fig. 4.
obvious field dependence, suggesting that the effect of
On the other hand, when the strain axis is perpendic-
varyingthemagneticfieldisnotsignificantinorienting(cid:96)ˆ.
ulartothemagneticfield, ε⊥H, thetipanglebehavior
of the scaled frequency shift deviates significantly from
SI.4 Tip angle dependence in the distorted either mode as shown in Fig. SI12. At small tip angle β,
B-phase the scaled shift measured in sample a23, is independent
of the magnetic field, and significantly larger than the
To explore the details of the dipole interaction of maximum frequency shift measured in the B-phase of
the distorted B-phase, we compare the tip angle depen- the isotropic sample a0 (black dashed curve), Ref. [8].
dence of the frequency shift, scaled to a common field of This is consistent with the small β measurement shown
H =95.6mTusingEq.6,totheB-phaseintheisotropic in Fig. 4. At large tipping angle, a magnetic field
aerogel, Ref. [8]. In the latter case, as in the present dependence is observed. At H = 49.1 mT, the large
work, there is a crossover between two order parameter tip angle frequency shift displays a sharp minimum at
textures as a function of temperature each with charac- β = 90◦ and small onset at β > 90◦, similar to the (cid:96)ˆ⊥
teristic spin dynamics that can be identified from NMR H texture of the isotropic aerogel (black dashed curve).
tipanglemeasurements. Fortheisotropicaerogelthetwo With increasing magnetic field, these features in tip
modesofthespindynamics,stabilizedbythesetextures, angle behavior evolve at H = 95.6 mT and completely
are the Brinkman-Smith-mode, Ref. [29], at high tem- disappear at H = 196 mT. The blue solid curve in
peratures, and below the crossover temperature, the (cid:96)ˆ⊥ Fig. SI12 is a numerical fit to the H = 196 mT data
using a cosine function, demonstrating that at H = 196
H-mode induced by influence of the walls on the orbital
quantization axis, (cid:96)ˆ. For the negatively strained aerogels mT, the tip angle behavior, and consequently the dipole
interaction, is markedly different from the isotropic
with strain axis parallel to the magnetic field, ε (cid:107) H,
state (black solid and dashed curves) where neither of
the tip angle behavior in sample a19 (orange squares)
and b30 (blue triangles) for the Brinkman-Smith-mode the B-phase modes (Brinkman Smith or (cid:96)ˆ⊥ H-mode)
matches well with that of the B-phase measured in the have magnetic field dependence of their scaled frequency
isotropic aerogel (black solid curve), Ref. [8], showing shifts, Ref. [8].
that this mode is unaffected by negative strain parallel
tothemagneticfield. However,thescaledfrequencyshift SI.5 Mean-free path analysis from scattering
dependenceonβ forthe(cid:96)ˆ⊥H-modeinsamplea19(red theory
9
the symmetry of the isotropic state, gives quasiparti-
It is noteworthy that fitting the AB-phase diagram cle mean free paths, λ, from g /H2 to be λ =
BA 0 a0
slopesusingthehomogeneousisotropicscatteringmodel, 210nm,λ = 250nm,λ = 215nm,λ = 240nm
a19 b12 b20
Ref. [5, 30], assuming that the distorted B-phase has and λ =270nm.
b30