Stability of superfluid 3He-B in compressed aerogel

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by  J. I. A. Li

Stability of superfluid 3He-B in compressed aerogel
J.I.A. Li,∗ A.M. Zimmerman, J. Pollanen,† C.A. Collett, W.J. Gannon,‡ and W.P. Halperin§
Northwestern University, Evanston, IL 60208, USA
(Dated: January 22, 2014)
Inrecentworkitwasshownthatnewanisotropicp-wavestatesofsuperfluid3Hecanbestabilized
within high porosity silica aerogel under uniform positive strain [1]. In contrast, the equilibrium
phaseinanunstrainedaerogel,istheisotropicsuperfluidB-phase[2]. Herewereportthatthisphase
stability depends on the sign of the strain. For negative strain of ∼20% achieved by compression,
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the B-phase can be made more stable than the anisotropic A-phase resulting in a tricritical point
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forA,B,andnormalphaseswithacritical fieldof∼100mT.FrompulsedNMRmeasurementswe
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2 identify these phases and the orientation of theangular momentum.

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a The natural state of superfluid 3He is completely free sample was then compressed to a strain ε=−19.4%, di-
J
of impurities. In zero magnetic field it has two phases, rectedalongthefield,εkH,Fig.1a. Werefertothissam-
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one of which has an isotropic order parameter ampli- ple as sample 2. Sample 3, was grown and characterized
2
tude, the B-phase, and the other, the A-phase, is axi- inthesamewayaswassample1,andwascompressedto
] ally anisotropic. Near the superfluid transitiontempera- a strain ε=−22.5%oriented with ε⊥H, Fig. 1b.
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o tureTc,energeticsfavortheA-phaseatpressuresgreater PulsedNMR measurements were performed at a pres-
c than21bar. However,inthepresenceofquencheddisor- sure P=26.3 bar in magnetic fields ranging from H=
- derfromhigh-porosity,silicaaerogel,theoryindicates[3] 49.1 to 196 mT. After an RF pulse that tips the nuclear
pr thatisotropicscatteringofthe3Hequasiparticleswilltip magnetizationofthe3Heatomsbyanangleβ awayfrom
u the stabilitybalance infavorofthe isotropicphase. Fur- the external field, Fourier transformation of the free in-
s thermore, it is predicted that anisotropic scattering will duction decay signal was phase corrected to obtain ab-
.
at have the opposite effect, stabilizing anisotropic phases sorption spectra, as shown in the inset to Fig. 1c. The
m like the A-phase. These predictions havebeen confirmed magnetic susceptibility, χ, was determined from the nu-
experimentallyforisotropic[2,4,5],andanisotropicscat- merical integral of the spectrum. The thermodynamic
-
d tering [1], where scattering anisotropy was achieved by transition between superfluid states was precisely deter-
n stretching the aerogel during growth producing a posi- mined from the susceptibility discontinuity on warming
o
tive uniaxial strain. In this Letter we report our discov- (see Fig. 1c) displayed in phase diagrams in Fig. 2. On
c
[ ery that negative strain from compression, contrary to cooling the supercooled transitions appear ≈300µK be-

1 the theory, stabilizes the isotropic phase and suppresses low Tc for all samples and magnetic fields, as indicated
the A-phase. The competition between the strain and inFig.1c. Thefrequency shiftofthe spectrum,∆ω,was
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3 the magneticfieldrevealsa new criticalpoint(Hc,Tc)in calculated from the first moment of each spectrum rela-
5 the magnetic field-temperature plane. Furthermore, we tivetothe Larmorfrequency,ω ,givenbytheresonance
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1 report that the orientation of the angular momentum in frequency in the normal state. From the spectra in the
5 this destabilized A-phase is perpendicular to the direc- inset of Fig. 1c, it is clear that εkH has a positive shift
.
1 tion of strain contrary to theoretical predictions [6, 7]. and ε⊥H has a negative shift in the A-phase. We infer
0 Consequently, the angular momentum has a continuous thattheorientationofthestrainaxisrelativetothefield
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rotational symmetry with respect to the strain axis and controls the sign of the frequency shift; this implies spe-
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becomes a two-dimensional superfluid glass. cific directions for the angular momentum which we will
:
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In the present work we compare three different sam- discuss later in the context of theory [7, 11].
i
X plesofnominally98%porositysilicaaerogelintheshape In sufficiently large magnetic fields, an equal-spin-
r of cylinders 4.0 mm in diameter with unstrained length pairing (ESP) state, i.e. having the same susceptibility
a
of 5.1 mm. After their growth all samples were found as the normal state, is stabilized near T , Fig. 2. For
c
to be isotropic and homogeneous using optical birefrin- sample 1 we have established earlier [2] that this ESP
gence following established procedures [8] [9]. Sample phase is the A-phase. The equilibrium phase boundaries
1 was the subject of several previous reports [2, 5, 10] T (H) at P = 26.3 bar from B to A-phases on warm-
BA
where we identified the p-wave superfluid states to be A ing, are shown in Fig. 2 for all three samples as a func-
and B-phases,with a suppressedorderparameter ampli- tion of magnetic field. For the isotropic aerogel, ε=0
tude,inarangeofmagneticfieldsandpressuresconclud- (sample 1, green squares), the extrapolation of T for
BA
ing that for an isotropic unstrained aerogel in zero field H2 → 0 occurs precisely at T /T = 1. This is ex-
BA c
there is but one superfluid state, irrespective of pressure pectedforaquadraticfieldcontributiontotheGinzburg-
and temperature, and this is the isotropic B-phase. The Landau free energy following exactly the same behavior
superfluid transitions were sharp ∆T /T . 0.2% and as is well-established for pure 3He below the polycritical
c c
precisely reproduced in several cool downs. This exact point, P < 21 bar [4, 12]. However, for the anisotropic
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a b 1.05
z z
ε H ε H Hc P = 26.3 bar
1.00
y y A
x 0.95
x Tc
ε = -19.4% ε = - 22.5% T/ e
sample 2, || sample 3, ⊥ 0.90 e T H
|| H B
e
c 0.85 = 0
1.0 0.80
0 0.01 0.02 0.03 0.04
al
m 2 2
or 0.8 Δω (kHz) H (T )
n
χ -0.4 -0.2 0 0.2
χ/
FIG. 2. (Color online). Superfluid phase diagram TBA/Tc
0.6 ⊥ versus H2 for three samples at a pressure of P = 26.3 bar
from warming experiments. Two of the three have negative
0.4 strain, ε<0, and one is isotropic, ε=0. All three have the
common feature ofa quadraticdependenceon magneticfield
0.7 0.8 0.9 1.0 1.1 for the B to A-phase transitions. Remarkably, for negative
strain thereappears to bea critical field, Hc.
T/T
c
FIG. 1. (Color online). a) and b) Schematics of the ex- be 250 and 220 nm respectively, all of which are very
perimental arrangements for samples 2 and 3. Sample 2,
reasonable values for a 98% porous aerogel [12].
ε=−19.4%,hasstrainaxisparalleltothemagneticfield. For
The existence of a critical field, H , is unexpected.
sample 3, ε=−22.5%, thestrain axis is perpendicularto the c
It indicates that anisotropic impurity scattering with
magneticfield. c)Liquidsusceptibilitynormalizedtothesus-
ceptibility of the normal state on warming (open red circles) ε < 0 introduces a new additive term in the free en-
and cooling (solid blue circles) versus reduced temperature ergy which depends on strain. Furthermore, contrary
for sample 3 in H =196 mT. Similar behavior was observed to theory [3, 15, 16], it appears that aerogel anisotropy
in sample 2, but was omitted for clarity. Inset: NMR spec- induced by uniaxial compression enhances the B-phase
tra for 3He in aerogel: sample 3 in the normal state (black
stability relative to the A-phase, countering the effect of
solid curve) and A-phase (blue dashed-dot curve); sample 2
an applied magnetic field.
in theA-phase(reddashed curve). Thenormalstate spectra
Basedonsymmetryargumentstherearetwopossibili-
havethesamelineshape. Thespectraforsamples2(3)were
obtained in H = 95.6(79.2) mT respectively and have been ties for the directionof the angular momentum, ˆl, in the
scaled toa common field and temperature for comparison. presence of a large uniaxial strain which determine the
sign of the frequency shift in the anisotropic A-phase.
One model requires that the angularmomentum be par-
aerogels, ε < 0 (sample 2, blue circles and sample 3, allelto the strainˆlkε, calledthe easy-axismodel, which
red triangles) the B-phase stability is enhanced with re- was predicted for ε <0 by two different theories [7, 11].
specttotheA-phaseasaresultofapositiveoffsetofthis The other,the easy-plane model, allowsthe angularmo-
quadratic field dependence described by, mentum to be in a plane perpendicular to the strain,
ˆl⊥ε, not favored by either theory if ε<0.
T H2−H2 H 4
1− BA =g c +O , (1) Using NMR frequency shift measurements as a func-
Tc BA(cid:18) H02 (cid:19) (cid:18)H0(cid:19) tion of temperature and NMR tip angle we have investi-
gatedtheorientationoftheangularmomentumforthese
thereby defining a critical field H at T /T =1. Here
c BA c two models. The dipole energy in the A-phase is [17],
Htivceilsy.8H8.06amnTdgaBnAd a9r7e.8sumpTerflfouridsacmonpslteasn2tsafnrdom3trhesepGecL- FD = −21Ω2AχγA2 ˆl·dˆ 2 where ΩA is the longitudinal
(cid:16) (cid:17)
theory [3, 13]. resonance frequency, γ is the gyromagnetic ratio, χ is
A
We have calculated the quasiparticle mean free path, the nuclear magnetic susceptibility and dˆis a spin-space
λ, by fitting our results to Eq. 1 using the homogeneous vector constrained to be perpendicular to the spin an-
isotropic scattering model [3, 14]. For sample 1, the gular momentum, sˆ, while minimizing F . The relative
D
isotropic aerogel, H =0 and we find λ=210 nm. Simi- orientation of the A-phase order parameters, ˆl and dˆ,
c
larly,forthecompressedaerogelsamples2and3wehave can be parametrized by two angles θ and φ, where θ is
calculated λ from the slope of the phase line in Fig. 2 to the angle between ˆl and the magnetic field H; and φ is
3
200
a e || H b e || H
100
)
z
H
( 0
Dw -100
-200
200
c e T H d e T H
100
)
z
H
( 0
Dw -100
-200
0.80 0.85 0.90 0.95 1.00 1.05 0 50 100 150 200 250 300 350
T/T b (degree)
c
FIG.3. (Coloronline). NMRfrequencyshiftsasafunctionoftemperatureandtipangle. Themagnitudeoftheshiftspresented
herehasbeenscaledtoacommonfieldofH =196mT,andthetipangledependencehasbeenscaledtoacommontemperature
ofT/Tc=0.8basedonourresultsfromsample1. Inallthepanels,thedipole-lockedconfiguration,ˆl⊥H isshownbytheblack
solid curves, and the black dashed curvesare the dipole-unlocked configuration, ˆlkH, with ΩA taken from the measurements
for sample 1 [2]. Comparison of frequency shifts in theA-phase: a, b) for εkH and c,d) ε⊥H. a, c) Dependenceon reduced
temperature with small tip angle, β < 20◦. a) warming in H = 174 mT, yellow crosses; cooling in H = 174 mT, green x;
cooling in H =95.6 mT, bluecircles. c) cooling inH =79.2 mT,red circles. b,d)Dependenceontipangle aftercooling from
the normal state. b) T/Tc = 0.87, H = 49.1 mT, green squares; T/Tc = 0.82, H = 196 mT, blue circles. d) T/Tc = 0.78,
H =95.6 mT, red circles. The blueand red solid curves are calculated for theeasy-plane, 2D glass model, see text.
the angle between dˆand the projection of ˆl in the plane Our frequency shift data in the small tip angle limit,
perpendicular to the field. We note that Ω should be Fig. 3a and 3c have signs opposite to Eqs. 3 and 4.
A
approximately the same for all three samples since the Furthermore, the magnitude of the shifts observed are
quasiparticle mean free paths are very similar [12]. The smaller than the predictions for the easy-axis model,
dependenceofthefrequencyshiftontipangle,β,isgiven which are the maximum (minimum) possible shifts for
by [11, 18]: theA-phaseoftenreferredtoasdipole-locked(unlocked)
configurations[17],shownasblacksolid(dashed)curves.
2 Consequently the easy-axis model is incorrect.
Ω 7 1
∆ω = A −cosβ+ cosβ+ sin2θ(r)
2ω (cid:18) (cid:18)4 4(cid:19) (cid:19)
L (cid:10) (cid:11)
2
Ω 1
− A (1+cosβ) sin2φ(r) sin2θ(r) , (2)
2ω (cid:18)2 (cid:19)
L (cid:10) (cid:11)(cid:10) (cid:11)
where sin2θ(r) and sin2φ(r) are spatial averages For the easy-plane model, ˆl⊥ε, there is a continuous
oeavseyr-aaxd(cid:10)isipmoloedleeln,(cid:11)εgtkhˆlξ,Dw(cid:10)eathalovecaθti=o(cid:11)n0rfo.rAsacmcoprldein2g(εtokHthe) symmetry for the angular momentum in the plane per-
pendiculartothestrain. Thiswillleadtoa2-dimensional
and θ = π, φ = 0 for sample 3 (ε⊥H). For these two
2 (2D) superfluid glass phase similar to the 3D glass state
orientations of the strain relative to the magnetic field
observed for sample 1 [5]. In this case, for sample 2,
Eq. 2 becomes: θ = π and sin2θ =1; whereasfor sample 3, 0<θ < π
2 2
2 1
and sin θ(cid:10)= .(cid:11)To calculate ∆ω we must constrain φ
2 2
Ω
∆ωεkH(β)=−2ωAL cosβ, (3) annadrio(cid:10)w,ethceond(cid:11)sˆi-dveercttowroispoaslssoibdleissocrednearreidoso.nInatlheengfitrhstscscaele-
Ω 2(3cosβ+1) smaller than a dipole length then, 0 < φ < 2π and
A
∆ωε⊥H(β)= 2ωL 4 . (4) sin2φ = 21, and Eq. 2 gives,
(cid:10) (cid:11)
4
B-phaseovertheA-phaseinamagneticfield,incontrast
with existing theories [3, 15, 16]. Furthermore, the an-
2
ΩA 1 gular momentum in the A-phase is not aligned with the
∆ωεkH(β)= 2ω (cid:18)2cosβ(cid:19), (5) strain axis as has been predicted [6, 7], rather it forms a
L
2 2D glass phase in the plane perpendicular to the strain.
Ω 1
A
∆ωε⊥H(β)= − cosβ . (6) We are grateful to V.V. Dmitriev, J.M. Parpia, J.A
2ω (cid:18) 4 (cid:19)
L
Sauls,G.E.Volovikforhelpfuldiscussionandforsupport
from the National Science Foundation, DMR-1103625.
Our calculations for this case, shownas solidblue and
red curves in Fig. 3b and 3d, are in excellent agreement
with our measurements as a function of tip angle and
coincidewiththemeasuredtemperaturedependencesfor
both orientations of the strain relative to the magnetic ∗ [email protected]
field, Fig. 3a and 3c. From this calculation, we obtained † Present Address: Condensed Matter Physics, California
the value of Ω at T/T = 0.8, Ω = 51.0, 54.0 and InstituteofTechnology,Pasadena,California91125,USA
A c A
51.2 kHz for sample 1, 2 and 3 respectively, within the ‡ PresentAddress: DepartmentofPhysicsandAstronomy,
Stony Brook University, Stony Brook, New York 11794,
combined measurement errors of 10%. The consistency
USA
in the values of Ω strongly supports this scenario.
For the secondAscenario, the dˆ-vector has a uniform § [email protected]
[1] J. Pollanen, J. I. A. Li, C. A. Collett, W. J. Gannon,
orientationonlengthscaleslongerthanthedipolelength. W. P. Halperin, and J. A. Sauls, Nature Phys. 8, 317
Then Eq. 5 holds for sample 2; however, for sample 3, (2012).
φ=0 and sin2φ =0, and from Eq. 2, [2] J.Pollanen,J.I.A.Li,C.A.Collett,W.J.Gannon, and
W. P. Halperin, Phys. Rev.Lett. 107, 195301 (2011).
(cid:10) (cid:11)
[3] E. V. Thuneberg, S. K. Yip, M. Fogelstro¨m, and J. A.
2 Sauls, Phys.Rev.Lett. 80, 2861 (1998).
Ω
∆ωε⊥H(β)= A (1−cosβ). (7) [4] G.Gervais,K.Yawata,N.Mulders, andW.P.Halperin,
8ωL Phys. Rev.B. 66, 054528 (2002).
[5] J.I.A.Li,J.Pollanen,A.M.Zimmerman,C.A.Collett,
Atsmalltip anglesthe frequency shift fromEq.7 is zero W. J. Gannon, and W. P. Halperin, Nature Physics 9,
foralltemperatures,inconsistentwiththedatainFig.3c. 775 (2013).
[6] G. E. Volovik,J. Low Temp. Phys.150, 453 (2008).
Additionally,thetipangledependencefromEq.7,shown
[7] J. A.Sauls, Phys.Rev.B. 88, 214503 (2013).
as a blue dash-dotted curve in Fig. 3d, is inconsistent
[8] J. Pollanen, K. R. Shirer, S. Blinstein, J. P. Davis,
with the data indicating that the first scenario better
describes the dˆ-vector orientation. We infer that the HH.alpCehroini,, JT..NMon.-CLriypsptmalalinn,e SLo.liBds. 3L5u4r,io4,668an(2d00W8).. P.
easy-plane model correctly describes the orientation of [9] A positively strained aerogel exhibits optical birefrin-
theangularmomentumtobe constrainedtoaplaneper- gence but becomes non-birefringent under compression
pendicular to the strain-axis for a compressed aerogel, at exactly the value expected for compensation of the
giving rise to a 2D glass phase in that plane. originalgrowth-inducedpositivestrain.Thecompression
processisreversible,meaningthatafterthecompression
Recent measurements of the superfluid density, ρ /ρ,
s of an isotropic aerogel is released it revertstobeinguni-
in zero applied field with a torsional oscillator using a
formly isotropic with its original dimensions.
similarly preparedaerogelsample with 10%compression [10] J.I.A.Li,J.Pollanen,C.A.Collett,W.J.Gannon, and
[19] found evidence of a stable phase just below Tc. In W.P.Halperin,J. Phys.: Conf. Ser. 400, 012039 (2012).
themagneticfield-temperatureplane,suchastablephase [11] G. E. Volovik,JETP Lett. 84, 533 (2006).
might occupy the shadedarea,Fig. 2,resulting in a pos- [12] W. P. Halperin, H. Choi, J. P. Davis, and J. Pollanen,
J. Phys.Soc. Jpn.77, 111002 (2008).
itive slope of the phase boundary to the B-phase. We
[13] H. Choi, J. P. Davis, J. Pollanen, T. M. Haard, and
cannotmakeadirectcomparisonwiththeseresultssince
W. P. Halperin, Phys. Rev.B 75, 174503 (2007).
our NMR experiments have not been performed in suf- [14] J. A. Sauls and P. Sharma, Phys. Rev. B 68, 224502
ficiently low magnetic fields. However, according to the (2003).
Clausius-Clapeyron relation such a stable phase could [15] K. Aoyama and R. Ikeda, Phys. Rev. B 73, 060504(R)
not be an ESP state. (2006).
[16] I. A. Fomin, JETP Lett. 77, 240 (2003).
Insummary,wehaveinvestigatedthe natureofsuper-
fluid3He-Aintwouniformly anisotropicaerogelsamples [17] D. Vollhardt and P. W¨olfle, The Superfluid Phases of
Helium 3 (Taylor and Francis, 1990).
with negative strain achieved with uniaxial compression [18] Y. M. Bunkov and G. E. Volovik, Europhys. Lett. 21,
of∼20%andwehavecomparedthemwithisotropicaero-
837 (1993).
gel. Wediscoveredacriticalfieldcorrespondingtoanew [19] R.G.Bennett,N.Zhelev,E.N.Smith,J.Pollanen,W.P.
critical point (T ,H ) at a pressure of P =26.3 bar and Halperin, andJ.M.Parpia,Phys.Rev.Lett.107,235504
c c
thatnegativestrainenhancesthestabilityoftheisotropic (2011).