1. Can a solid be superfluid?
Symposium on Quantum Phenomena and Devices
at Low Temperatures, 2008
Helsinki Univ. of Technology
Moses Chan - Penn State

2. Outline Introduction Torsional oscillator measurements.
Origin of supersolidity :
zero point vacancies, grain boundaries
glassy regions or dislocations?
Specific heat and dc flow experiments
Conclusions

3. Theoretical ‘consensus’ in 1970s:
Superfluidity in solid is not impossible!
- If solid 4He can be described by a Jastraw-type wavefunction that is commonly used to describe liquid helium then crystalline order (with finite fraction of vacancies) and BEC can coexist.
G.V. Chester, Lectures in Theoretical Physics Vol XI-B(1969);
Phys. Rev. A 2, 256 (1970)
J. Sarfatt, Phys. Lett. 30A, 300 (1969)
L. Reatto, Phys. Rev. 183, 334 (1969)
- Andreev and Liftshitz assume the specific scenario of zero-point vacancies and other defects ( e.g. interstitial atoms) undergoing BEC and exhibit superfluidity.
Andreev & Liftshitz, Zh.Eksp.Teor.Fiz. 56, 205 (1969).

4. R I(T)=Iclassical[1-fs(T)]
The ideal method of detection of superfluidity is to subject solid to dc or ac rotation.
Leggett, “Can a solid be superfluid?” PRL 25, 1543(1970)
Quantum exchange of particles arranged in an annulus under rotation leads to a measured moment of inertia that is smaller than the classical value
Solid Helium R
I(T)=Iclassical[1-fs(T)]
fs(T) is the supersolid or nonclassical rotational inertia (NCRI) fraction.
Its upper limit is estimated by different theorists to range from 10-6 to 0.4; Leggett: 10-4

5. Torsional Oscillator Technique is ideal for the detection of superfluidity
Amp
Drive
Detection
3.5 cm
Torsion rod
Torsion cell
Quality Factor
Q= f0 / f ~106
Stability in the period is ~0.1 ns
Frequency resolution of 1 part in 107
Mass sensitivity of ~10-7 g
f~ 1kHz

6. Torsional oscillator studies of superfluid films
Vycor Δ
I total= I cell+ I helium film,
Above Tc the adsorbed normal liquid film behaves as solid and oscillates with the cell. In the superfluid phase, helium film decouples from oscillation.
Hence Itotal and drops. 
Berthold,Bishop, Reppy, PRL 39,348(1977)

7. Bulk solid helium in annulus
Torsion cell
with helium in annulus
Filling line
Channel OD=10mm
Width=0.9mm
Torsion rod
3.5 cm
Mg disk
Torsion cell
Al shell
Detection
Solid helium in annular channel
Drive

8. E. Kim & M.H.W. Chan, Science 305, 1941 (2004)
Bulk solid helium in annulus
f0=912Hz
51bar
Total mass loading = 3012ns
Measured decoupling, -o=41ns
NCRIF = 1.4%
E. Kim & M.H.W. Chan, Science 305, 1941 (2004)

9. Non-Classical Rotational Inertia Fraction
|v|max
Total mass loading
=3012ns at 51 bars

10. Irrotational Flow Superfluids exhibit potential (irrotational) flow
For our exact dimensions, NCRIF in the blocked cell should be about 1% that of the annulus*
*E. Mueller, private communication.

11. Reversibly blocked annulus Rittner & Reppy, Cornell
Top View: Blocked
Top View: Open
solid inertia
Nonclassical
rotational
inertia
Gap 73 mm, NCRI = 16 %

12. Nonclassical rotational inertia results have been replicated in five other labs.
The temperature dependence of NCRI is reproduced.
However, the magnitude of NCRIF varies from 0.015% up to 16%.
Simulations studies indicate NCRI is not possible in a ‘perfect’ crystal and that quantum zero point vacancies concentration is too low to account for the observations.
(this conclusion not universally accepted).
This leaves other defects, e.g. grain boundaries, glassy regions and dislocations as possible origins or at least ‘enhancer’ for NCRI.

13. Grain boundaries
It has been suggested that the observed NCRI are due to atomically thin superfluid (liquid) films along the grain boundaries, if true, NCRI should scales with total grain boundary area.
Grain size of solid 4He inside Vycor glass is ~ 7nm and in bulk it is ~0.1 mm. This means the grain boundary surface differs by many orders of magnitude, but observed NCRI is about the same, ~1%.
Nevertheless, most bulk solid samples grown by the blocked capillary method is polycrystalline. Does NCRI exists in a single crystal ?

14. E. Kim & M.H.W. Chan, Nature 427, 225 (2004).
Solid helium in Vycor glass
7nm
f0=1024Hz
62bar
Total mass loading = 4260ns
Measured decoupling, -o=17ns
NCRI fraction, or NCRIF = 0.4%
(with tortuosity, 2% )
-*[ns]
E. Kim & M.H.W. Chan, Nature 427, 225 (2004).
*=971,000ns

15. Crystal Growth
High quality single crystals have been grown under constant temperature1 and pressure2
Best crystals grown
in zero temperature
limit
1. O.W. Heybey & D.M. Lee, PRL 19, 106 (1967); S. Balibar, H. Alles & A. Ya
Parshin, Rev. Mod. Phys. 77, 317 (2005).
2. L.P. Mezhov-Deglin, Sov. Phys. JETP 22, 47 (1966); D.S. Greywall, PRA 3, 2106
(1971).

16. Constant T/P growth from superfluid (1ppb 3He) Heat in
Heat out
Q ~ 500,000
Tony Clark and Josh West

17. Solid 4He (1ppb 3He) grown under constant P/T from superfluid

18. Solid 4He (1ppb 3He) grown under constant P/T from superfluid

19. Solid 4He (1ppb 3He) grown under constant P/T from superfluid

20. Solid 4He (1ppb 3He) grown under constant P/T from superfluid

21. Solid 4He (1ppb 3He) grown under constant P/T from superfluid

22. Solid 4He (1ppb 3He) grown under constant P/T from superfluid

23. Samples grown under constant P/T from superfluid collapse onto
one curve for T > 40mK and share common onset temperature, TC ~ 80mK. Superfluid film flow along grain boundary cannot be the ‘origin’ of NCRI.

24. High temperature tail of NCRI
Transition broadened in BC samples (probably “polycrystalline”) and by 3He impurities

25. Superfluid film along the grain boundaries cannot be the mechanism for NCRI.
What is the most likely origin for the large variation in NCRI from cell to cell?
Dislocation lines density in solid 4He ranges from ~10 to 109 cm-2 , sensitive to how the solid is nucleated and grown. But how does dislocations enable or enhance NCRI?

26. Dislocations Two of the common types: edge & screw In solid 4He
Dislocation density, L = ~5  <1010 cm-2

27. 3He impurities at “high” T
Network pinning at the nodes
3He impurities are not ‘attached’ to the dislocation lines.

28. Increasing fraction of 3He atoms condense onto and stiffen dislocation network with decreasing T
Iwasa et.al., J.Phys. Soc. Jpn. 46, 1119( 1979);
Paalanen, Bishop and Dail,
PRL 46, 664(1981)

29. 3He Effect ( bulk samples)
E. Kim, J. S. Xia, J. T. West, X. Lin, A. C Clark, and M. H. W. Chan, , PRL 100, ( 2008)

30. Onset of NCRI tracks the expected temp
Onset of NCRI tracks the expected temp. of 3He condensation onto dislocation

31. Direct measurement of the Shear Modulus
James Day and John Beamish, Nature (London) 450, 853 (2007).

32. Shear Modulus mirrors NCRI
James Day and John Beamish, Nature (London) 450, 853 (2007).

33. The increase in shear modulus and the onset of NCRI at low T must be related, but how?
An increase in shear modulus of solid 4He in TO can result an increase in its resonant freq., ie. the same sign as the observed NCRI. But simulation studies shows the magnitude of the ‘Beamish’ effect is typically orders of magnitudes smaller than that observed in most TOs.
The ‘Beamish’ effect or shear modulus stiffening cannot explain the blocked annulus experiments.
It has been suggested (Anderson) that the onset of supersolidity stiffens the shear modulus.
Simulations show the dislocation lines are ‘superfluid’ and it has been suggested that the onset of NCRI requires a ‘rigid’ dislocation network.

34. What about the superglass model?
Since glass has no crystalline order, it is thought superfluidity is more likely to occur.
If this is the case there should be thermodynamic signatures ( e.g. thermal hysteresis, and linear specific heat)

35. Heat capacity Is there a thermodynamic signature of the transition?
There is no specific heat information on solid 4He below 200 mK. The heat capacity of metal container swamps the contribution of solid 4He

36. Heat Capacity measurements in a Silicon cell
Reasons for Si:
Low heat capacity:
High thermal conductivity:
Helium Volume= 0.926cc Si Al
Glass Capillary
Stycast 2850
Heater
Thermometer
0.1mm ID
1.5 cm
Lattice Materials CorP., MT, USA
Optical Silicon purity : >99.999%
Heat capacity of solid 4He is more than 10 times larger
than the Si cell over the entire temp. range

37. AC Calorimetry KΘ Kb Sample Kh Internal time constant << 1/ω
External time constant >> 1/ω Kb
Sample
Thermal Bath
Thermometer KΘ
Heater Kh
Paul F. Sullivan, G. Seidel, Phys Rev. 173, 679 (1968).
Yaakov Kraftmakher, Physics Reports, 356 (2002)

38. Specific heat of commercially pure 4He (0.3ppm 3He)
log C vs. log T
Constant volume
technique.
20 hours to finish solidification.
No long time constant.
No hysteresis
No annealing effect
No thermal cycle effect

39. Specific heat of solid 4He (0.3ppm & 1ppb)
log C vs. log T

40. C/T vs. T2 Data above 140 mK extrapolate to zero
the data seem to diverge in the low-temperature limit. This behaviour, which is more prominent in the 30 p.p.m. sample, suggests the presence of a temperature-independent term in addition to the Debye contribution.
Data above 140 mK extrapolate to zero

41. ‘Linear T’ term from Pressure measurements ?
PHYSICAL REVIEW B 76,
V. N. Grigor'ev et al
Freshly grown sample
Annealed sample (19.83cc/mole)
— Grüneisen constant ρ — molar volume
CV~Tn P~Tn+1

42. Specific heat with Debye phonon term subtracted
T3 subtracted
~20 mJ mol−1 K−1
~2.5×10-6 kB per 4He atom
Explain the Scatter.

43. Higher resolution in 2008 sample cell
Minimum Temperature 25mK
New preliminary result with a more sensitive thermometer.
Disadvantage of Si?
8 out of 10 cells leaks under high pressure
4 hours to solidify a sample.
Solid sample grown under constant pressure condition has the smallest peak.
T3 subtracted
T3 subtracted

44. Comparison with the Helsinki melting curve measurement.
Constant volume 1ppb 2008
Constant pressure 1ppb 2008
No anomaly in melting curve after correction for instrumental effect.
Discrepancy we don’t understand
Todoschenko et al., PRL 97, (2006); JETP Lett. 85, 555 (2007).

45. dc ( superfluid-like ) flow?
Greywall and Beamish carried out experiments by increasing pressure on one end of a solid ( bulk or in Vycor glass) sample. No evidence of flow.
Balibar and collaborators saw flow in liquid-solid coexistence samples. The observed phenomenon is likely a consequence of liquid film flow through the ‘crack’ in the solid.

46. DC superflow reported by ‘pushing’ on superfluid in contact with solid : ‘UMass Sandwich’ (R.B. Hallock) 3 1 2
R, reservoirs
V, Vycor
S, sample C1 C2

47. Conclusions
NCRI or superfluid like behavior seen in solid 4He in 7 labs.
Shear modulus increases at low temperature, the temperature dependence of the increases mimics that of NCRI
It is possible that either (1) NCRI ‘requires’ a rigid dislocation network, or (2) onset of NCRI stiffens the solid lattice.
No evidence of linear T term in specific heat, but a peak near NCRI onset is seen, size of peak is smaller for a ‘higher’ quality sample.
Preliminary dc superfluid flow through solid helium reported.

48. Xi, Tony, Eunseong, Josh

50. Lindemann Parameter
the ratio of the root mean square of the displacement of atoms to the interatomic distance (da)
A classical solid will melt if the Lindemann’s parameter exceeds the critical value of ~0.1 .
X-ray measurement of the Debye-Waller factor of solid helium at ~0.7K and near melting curve shows this ratio to be
(Burns and Issacs, Phys. Rev. B 55, 5767(1997))
total energy
zero-point energy
Zero-point Energy
Inter-atomic potential

51. Solid 4He at 62 bars in Vycor glass
Period shifted by ns due to mass loading of solid helium
*=966,000ns

52. Strong and ‘universal’ velocity dependence
in all annular samples ω R
vC~ 10µm/s
=3.16µm/s
for n=1
Vortices are important

53. Blocked capillary (BC) method of growing solid samples
heat drain
solid
blocks
fill-line
Be-Cu torsion rod
and fill-line
gravity

54. Supersolid response of helium in Vycor glass
Period drops at 175mK
 appearance of non-classical rotational inertia (NCRI)
size of period drop
- ~17ns 
*=971,000ns

55. 3He impurity effect in Vycor
-*[ns]
*=971,000ns

56. Solid helium in porous gold
490nm
f0=359Hz
27bar
Total mass loading = 1625ns
Measured decoupling, -o=13ns
NCRIF = 0.8%
(with tortuosity, 1.2% )
E. Kim & M.H.W. Chan, JLTP 138, 859 (2005).

58. Specific heat with the temperature independent constant term subtracted

59. Previous solid 4He heat capacity measurements below 1K
Year Low temperature limit
Swenson1 1962, K
Edwards K
Gardner K
Adams K
Hebral K
Clark K
They all observed T3 phonon contribution.
Their sample cells used in these experiments were all constructed with heavy wall metal or epoxy which contribute significantly to the heat capacity at low temperature.
E. C. Heltemes and C. A. Swenson, Phys. Rev. 128, 1512 (1962); H. H. Sample and C. A. Swenson, Phys. Rev. 158, 188 (1967).
D. O. Edwards and R. C. Pandorf, Phys. Rev. 140, A816 (1965).
W. R. Gardner et al., Phys. Rev. A 7, 1029 (1973).
S. H. Castles and E. D. Adams, J. Low Temp. Phys. 19, 397 (1975).
B. Hébral et al., Phonons in Condensed Matter, edited by H. J. Maris (Plenum, New York, 1980), pg. 169.
A. C. Clark and M. H. W. Chan, J. Low Temp. Phys. 138, 853 (2005).

60. Results: 4He at different 3He concentrations in glass capillary cell
Constant volume technique
No long time constant.
No hysteresis
No change due to annealing
No thermal cycle effect

61. C vs T3 10ppm sample 0.7+/-0.2 kB per 3He 30ppm sample
Constant contribution from 3He impurity
10ppm sample
0.7+/-0.2 kB per 3He
30ppm sample
1.7+/-0.3 kB per 3He

62. Is there a linear T term?
For 0.3ppm, T>0.14K T3 relation only

63. NMR measurement of spin (3He) diffusion:
A. R. Allen, M. G. Richards & J. Schratter J. Low Temp. Phys. 47, 289 (1982).
M. G. Richards, J. Pope & A. Widom, Phys. Rev. Lett. 29, 708 (1972).

64. Results: pure 4He (0.3ppm & 1ppb)
Ln C vs. Ln T
What is the deviation?

65. Previous solid 4He heat capacity measurements below 1K
Year Low temperature limit
Swenson1 1962, K
Edwards K
Gardner K
Adams K
Hebral K
Clark K
They all observed T3 phonon contribution.
Their sample cells used in these experiments were all constructed with heavy wall metal or epoxy which contribute significantly to the heat capacity at low temperature.
E. C. Heltemes and C. A. Swenson, Phys. Rev. 128, 1512 (1962); H. H. Sample and C. A. Swenson, Phys. Rev. 158, 188 (1967).
D. O. Edwards and R. C. Pandorf, Phys. Rev. 140, A816 (1965).
W. R. Gardner et al., Phys. Rev. A 7, 1029 (1973).
S. H. Castles and E. D. Adams, J. Low Temp. Phys. 19, 397 (1975).
B. Hébral et al., Phonons in Condensed Matter, edited by H. J. Maris (Plenum, New York, 1980), pg. 169.
A. C. Clark and M. H. W. Chan, J. Low Temp. Phys. 138, 853 (2005).

66. Results from Edwards and Hebral
The background problem
Edwards
Hebral
Effect of changing 1% of the empty cell

67. Temperature scale based on
Results: pure 4He (0.3ppm)
The heat capacity of the empty cell at 0.2 K and 0.05 mK are respectively less than 3 % and 10 % of the heat capacity of solid helium at 27 bar.
0.3ppm
Minimum temperature 40mK
Temperature scale based on
3He melting curve

68. Specific heat peak is found when T3 term subtracted The peak is independent of 3He concentration
Peak height: 20 μJ mol-1 K-1
(2.5 x 10-6 kB per 4He atom)
Excess entropy: 28 μ J mol-1 K-1
(3.5 x 10-6 kB per 4He atom)

69. BC samples can also be grown
Heat out
Q ~ 500,000

70. Granato-Lucke applied to 4He
Dislocations pinned by network, 3He, jogs?
Typical LN in pure 4He crystals ~ 5mm

71. Ln C vs. Ln T

72. Compare with torsional oscillator

74. Solid 4He at various pressures show similar temperature dependence, but the measured supersolid fraction shows scatter with no obvious pressure dependence
NCRIF
NCRIF
NCRIF
NCRIF

75. Pressure dependence of supersolid fraction
Blue data points were obtained by seeding the solid helium samples from the bottom of the annulus.
NCRIF
What are the causes of
the scatter in NCRIF?

76. Pressure dependence of supersolid fraction
Blue data points were obtained by seeding the solid helium samples from the bottom of the annulus.
NCRIF
What are the causes of
the scatter in NCRIF?

77. Thermal history of 1ppb samples
Velocity changes at low temperature lead to interesting behavior…
Protocol followed below:
(1) cooling, (2) velocity increase, (3) warming, (4) cooling 2 1 3 4

78. End

79. Specific heat with T3 term subtracted
Specific heat peak is independent of 3He concentrations.
Assuming 3D-xy universality class (same as the lambda transition in liquid 4He).
Use two-scale-factor universality hypothesis, ρs ~0.06%. 1ppb study of TO found this number lays between 0.03% and 0.3%.
Peak height:
20 μJ mol-1 K-1 (2.5 x 10-6 kB per 4He atom)
Excess entropy:
28 μ J mol-1 K-1 (3.5 x 10-6 kB per 4He atom)

80. Hysteresis in Pressure measurement of phase separation
A.N.Gan'shin, V.N.Grigor'ev, V.A.Maidanov, N.F.Omelaenko, A.A.Penzev, É.Ya.Rudavskii, A.S.Rybalko., Low Temp. Phys. 26, 869 (2000).

82. Dislocations Two of the common types: edge & screw
Dislocation density, L = ~5  <1010 cm-2
3-d network, LN ~ 1 to 10 mm (L~105 to 107)
[LN ~ 0.1 to 1 mm (L~109)]

83. Granato-Lucke applied to 4He
Dislocations intersect on a characteristic length scale of LN ~ 1 5mm
Dislocations can also be pinned by 3He impurities
Distance between 3He atoms (if uniformly distributed):
1ppb  1000a ~ 0.3mm
0.3ppm  150a ~ 45nm
1%  a ~ 15nm

84. 3He-dislocation interaction
Actual 3He concentration on dislocation line is thermally activated
*Typical binding energy, W0 is 0.3K to 0.7K

85. 3He-dislocations interaction
Line considered as crossover from network pinning to 3He impurity pinning of dislocations (LNetwork ~ L3He spacing)
Average length
LNetwork ~ 1 to 10mm
For L ~ 105 to 106cm-2
Smaller lengths
(< 1mm) are expected
for larger dislocation densities

86. Solid helium in Vycor glass
Weak pressure dependence…
from 40 to 65bar
Strong velocity dependence

87. 3He-4He mixtures
-*[ns]
*=971,000ns
Question: If there is a transition between the normal and supersolid phases, where is the transition temperature?

88. 3He-4He mixtures ( inside Vycor)

89. Frequency dependence -TO increases with frequency
-Low temperature NCRIF unchanged
~150mK ~220mK
Aoki, Graves & Kojima, PRL 99, (2007).