MEMS/NEMS in (super)fluids Eddy Collin, ULT Institut Néel – CNRS Cryocourse 25/09/2016

MEMS/NEMS in (super)fluids Content Brief historical introduction: mechanical probes in quantum fluids Motivations: what are MEMS/NEMS, and why in helium(s)? Basics of mechanics and fluidics 2 µm 60 µm Eddy Collin, ULT Institut Néel – CNRS Cryocourse 25/09/2016

MEMS/NEMS in (super)fluids Content Brief historical introduction: mechanical probes in quantum fluids Motivations: what are MEMS/NEMS, and why in helium(s)? Basics of mechanics and fluidics 2 µm 2 µm 60 µm Eddy Collin, ULT Institut Néel – CNRS Cryocourse 25/09/2016

Brief historical introduction: mechanical probes in quantum fluids What’s a quantum fluid? A liquid with large zero point (kinetic) energy compared to binding (potential) energy 𝜆= 𝐸 𝑘𝑖𝑛 𝐸 𝑝𝑜𝑡 bigger than 1 for H2 (1.7), 4He (2.6) and 3He (3) Binding in Helium is due to Van der Waals forces: very small! 4He liquefies at 4.2 K and 3He at 3.2 K under 1 atmosphere Both do not solidify under saturated vapor pressure By comparison: H2 (larger polarizability) liquefies at 20 K but solidifies at 14 K

Brief historical introduction: mechanical probes in quantum fluids What’s a quantum fluid? 3He, Fermi surface k-space Extremely simple systems: model for condensed matter Copper, Fermi surface k-space (valence 1) 4He and 3He only liquids down to absolute 0 K No spin, Boson Nuclear spin 1/2, Fermion

Brief historical introduction: mechanical probes in quantum fluids What’s a quantum fluid? Extremely clean systems: model for condensed matter 4He and 3He only liquids down to absolute 0 K No spin, Boson Nuclear spin 1/2, Fermion 6N Silver nanowire 5N 0.3 ppm Mn ! 1 ppm Mn F. Pierre et al., Phys. Rev. B 68, 085413 (2003) Unique experimental realization of the many-body strongly interacting quantum system (see lecture V. Tsepelin)

Brief historical introduction: mechanical probes in quantum fluids What’s a quantum fluid? 4He and 3He only liquids down to absolute 0 K No spin, Boson Nuclear spin 1/2, Fermion Both have a superfluid state: 4He: Bose condensation below ≈ 1 K 3He: BCS p-wave pairing below ≈ 1 mK (macroscopic coherent state with no viscosity) Unique experimental realization of the many-body strongly interacting quantum system (see lecture V. Tsepelin)

Brief historical introduction: mechanical probes in quantum fluids How does one measure a quantum fluid? Bulk (thermodynamic) properties like specific heat, magnetization (for 3He), Elementary excitation (spectrum) from neutron scattering, Transport properties like heat conduction, spin diffusion (for 3He), Friction with the fluid at low velocities (laminar flow) leads to definition of viscosity Friction at larger velocities (turbulent flow) occurs when eddies form within the fluid; defines quantum turbulence

Brief historical introduction: mechanical probes in quantum fluids How does one measure a quantum fluid? Bulk (thermodynamic) properties like specific heat, magnetization (for 3He), Elementary excitation (spectrum) from neutron scattering, Transport properties like heat conduction, spin diffusion (for 3He), Friction with the fluid at low velocities (laminar flow) leads to definition of viscosity When the flow occurs in fluid with particles travelling very long distances (long mean-free-path), the flow is molecular; defines ballistic regime of elementary excitations Friction at larger velocities (turbulent flow) occurs when eddies form within the fluid; defines quantum turbulence

Brief historical introduction: mechanical probes in quantum fluids How does one measure a quantum fluid? These properties (and the ones related to them) can be measured by mechanical probes: Immersed in the fluid “vibrating wire” Or containing the fluid “Andronikashvili” torsional oscillator When the flow occurs in fluid with particles travelling very long distances (long mean-free-path), the flow is molecular; defines ballistic regime of elementary excitations Friction with the fluid at low velocities (laminar flow) leads to definition of viscosity Friction at larger velocities (turbulent flow) occurs when eddies form within the fluid; defines quantum turbulence

Brief historical introduction: mechanical probes in quantum fluids How does one measure a quantum fluid? These properties (and the ones related to them) can be measured by mechanical probes: Immersed in the fluid “vibrating wire” Or containing the fluid “Andronikashvili” torsional oscillator Probes sensing intrinsic property of the fluid…. But need a theory of the interaction solid/liquid! J. Tough, W. McCormick, J. Dash, Phys. Rev. 132, 2373 (1963) M. Black, H. Hall, K. Thompson, J. Phys. C: Solid St. Phys. 4, 129 (1971) A.M. Guénault, V. Keith, C.J. Kennedy, S.G. Mussett, G.R. Pickett, J. of Low Temp. Phys. 62, 511 (1986) Sup. 4He Norm. 3He Sup. 3He

Brief historical introduction: mechanical probes in quantum fluids How does one measure a quantum fluid? These properties (and the ones related to them) can be measured by mechanical probes: Immersed in the fluid “vibrating wire” Or containing the fluid “Andronikashvili” torsional oscillator Probes sensing intrinsic property of the fluid…. But need a theory of the interaction solid/liquid! Basic properties of both 4He and 3He very well known http://pages.uoregon.edu/rjd/vapor1.htm (R.J. Donnelly) J. Wilks, Introduction to liquid Helium, 1987 http://www.physik.uni-augsburg.de/theo3/helium3/ (D. Vollhardt) Thanks to many experiments! http://spindry.phys.northwestern.edu/he3.htm (W. Halperin) W.P. Halperin & L.P. Pitaevskii (Eds), Helium Three (1990) D. Vollhardt & P. Völfle, The Superfluid Phases of Helium Three, (1990)

MEMS/NEMS in (super)fluids Content Brief historical introduction: mechanical probes in quantum fluids Motivations: what are MEMS/NEMS, and why in helium(s)? Basics of mechanics and fluidics

Motivations: what are MEMS/NEMS, and why in helium(s)? Context: friction at the smallest scales MEMS:micro-electro-mechanical systems Y. Lee, Florida E.C., Grenoble 60 µm “Andronikashvili” torsional oscillator M. Chan, Penn State NEMS: nano- E.C., Grenoble S. Kafanov, Lancaster 2 µm “Conventional”: BIG! (bulk) 1 nm 100 nm 10 µm 1 mm 10 cm

Motivations: what are MEMS/NEMS, and why in helium(s)? Context: friction at the smallest scales Y. Lee, Florida MEMS:micro-electro-mechanical systems E.C., Grenoble “Andronikashvili” torsional oscillator M. Chan, Penn State NEMS: nano- E.C., Grenoble S. Kafanov, Lancaster New unique possibilities: Probing very small lengthscales Versatility of fabrication 60 µm 2 µm 1 nm 100 nm 10 µm 1 mm 10 cm

Motivations: what are MEMS/NEMS, and why in helium(s)? What are these new possibilities? From bulk… Same as in electronic solid-sate devices! Phase diagram Lamp Transistor Integrated processor Mesoscopic (e-) physics, Quantum electronics!

Motivations: what are MEMS/NEMS, and why in helium(s)? What are these new possibilities? From bulk… Same as in electronic solid-sate devices! Phase diagram Go meso in 4He and 3He! Unravel old questions, Tackle radically new problems Integrated processor Mesoscopic (e-) physics, Quantum electronics!

Motivations: what are MEMS/NEMS, and why in helium(s)? What are these new possibilities? Relevant lengthscales: In both normal/superfluid 4He At lower T Phase diagram ~ µm @ K & MHz (decay length of non-prop. sound wave) (h viscosity and r mass density) Can couple MEMS devices to a wall; size effects Creates slippage at boundaries; cross-over viscous/molecular lMean-free-path ~ 100 µm @ K Lde Broglie ~ 10 nm @ K (“size” of quasi-particle) Resolve size of excitations; Young’s experiment? 10 nm 100 nm 1 cm (calculated at 100 mK) 2 cm 6 cm Electrons in metal (Aharonov-Bohm) C60F48 Stefan Gerlich et al., Nature Comm. 2, 263 (2011) Why not? R.A. Webb et al., Phys. Rev. Lett. 54, 2696 (1985)

Motivations: what are MEMS/NEMS, and why in helium(s)? What are these new possibilities? Relevant lengthscales: In both normal/superfluid 3He At lower T l F ~ few A atomic ° Not relevant, too small Phase diagram (Fermi wavelength) Can be used to couple MEMS devices through a sound mode; Landau’s transverse sound? A. Duh et al., J. of Low Temp. Phys. 168, 31 (2012) ~ few µm @ mK & MHz (decay length of non-prop. sound wave) (h viscosity and r mass density) Creates slippage at boundaries; cross-over viscous/molecular lMean-free-path ~ few 10 µm @ mK Resolve size of excitations; Young’s experiment? 10 nm 100 nm 1 cm (calculated at 100 µK) 2 cm 6 cm Lde Broglie ~ 15 nm @ mK (“size” of quasi-particle)

Motivations: what are MEMS/NEMS, and why in helium(s)? What are these new possibilities? Relevant lengthscales: Now, in the superfluids For 4He, coherence length x0 ≈ 1 A ° Not relevant, too small Phase diagram For 4He, other lengthscale: Typical size of defects’ texture of order parameter: tangles of vortices (mass vortex itself ≈ x0 too small) Study quantum turbulence down to about 100 nm; versatility of fabrication! Example: “Bolometric camera” with nano-mechanics? Typical: 100 nm x 100 nm x 100 µm Enabling to image quasiparticles, turbulence,… S. L. Ahlstrom et al., JLTP 175, 725 (2014) e.g. for 3He superfluid, ULANC

Motivations: what are MEMS/NEMS, and why in helium(s)? What are these new possibilities? Relevant lengthscales: Now, in the superfluids Resolve size of Cooper pair; friction at the scale of the pair? (in “dirty superfluid” superfluidity suppressed when disorder ≈ x0) J.V. Porto, J.M. Parpia, Phys. Rev. Lett. 74, 4667 (1995) D. T. Sprague et al., Phys. Rev. Lett. 75, 661 (1995) For 3He, coherence length x0 ≈ 100 nm (size of paired quasiparticles in BCS; p-wave pairing) Phase diagram r In r-space: (Not to scale) Oscillation: kF Size: Study quantum turbulence down to the size of a vortex: resolve Kolmogorov/ Richardson cascade! D. I. Bradley et al., PRL 96, 035301 (2006) How is energy dissipated at the scale of a vortex? For 3He, other lengthscale: Typical size of defects of order parameter: e.g. vortices ≈ x0 & rich zoology of defects! (because of 3He order parameter)

Motivations: what are MEMS/NEMS, and why in helium(s)? What are these new possibilities? Relevant lengthscales: Now, in the superfluids Resolve size of Cooper pair; friction at the scale of the pair? (in “dirty superfluid” superfluidity suppressed when disorder ≈ x0) J.V. Porto, J.M. Parpia, Phys. Rev. Lett. 74, 4667 (1995) D. T. Sprague et al., Phys. Rev. Lett. 75, 661 (1995) For 3He, coherence length x0 ≈ 100 nm (size of paired quasiparticles in BCS) Phase diagram For 3He, other lengthscale: Typical size of boundary layer: Andreev bound states in ≈ x0 (BCS gap suppressed at boundaries) e.g. Viewpoint in physics, Shou-cheng Zhang, Physics 1, 6 (2008) Xiao-Liang Qi, Rev. of Mod. Phys. 83, 1057 (2011) Andreev bound states in 3He-B are Majoranas Fermions! In 3He-A are Weyl Fermions! (symmetries of order parameter) x0 Confinement: 3He “slab” experiments (see lecture J. Saunders) Y. Tsutsumi et al., PRB 83, 094510 (2011) G. Volovik, Pis'ma v ZhETF 90, 440 (2009)

Motivations: what are MEMS/NEMS, and why in helium(s)? What are these new possibilities? Main aspect: Complementary to other systems! V. Mourik et al., Science 336, 1003 (2012) Stevan Nadj-Perge et al., Science 346, 602 (2014) e.g. with solid-state: topological superconductor, quest of Majorana particles efficient local probes, but imperfect, disorder, impurities, interfaces… MEMS/NEMS can do the job! Phase diagram In superfluid 3He: ultra-clean, well-known order-parameter, But not yet efficient local probes! G. Volovik, Pis'ma v ZhETF 90, 440 (2009)

MEMS/NEMS in (super)fluids Content Brief historical introduction: mechanical probes in quantum fluids Motivations: what are MEMS/NEMS, and why in helium(s)? Basics of mechanics and fluidics

Basics of mechanics Reduced from continuum mechanics Basic mechanical objects: Beams/string, i.e. essentially 1D systems Simplest (linear) theory of flexure: Euler-Bernoulli E: Young’s modulus (Pa) T: tension (linear, N) r: mass density (kg) Ix,y: second moment of area Ix = w h3/12 Iy = h w3/12 (m4) A: section A= w h (m2) 2 families Phase diagram Simplest (linear) theory of flexure: Kirchhoff-Love D: flexural rigidity (Pa m3) D= E h3/[12 (1-v2)] v: Poisson’s ratio T: tension (surface, N/m) r: mass density (kg) h: thickness (m) 1 family Plates/membranes, i.e. essentially 2D systems

Basics of mechanics Basic mechanical objects: Beams/string, i.e. essentially 1D systems E: Young’s modulus (Pa) T: tension (linear, N) r: mass density (kg) Ix,y: second moment of area Ix = w h3/12 Iy = h w3/12 (m4) A: section A= w h (m2) G:torsion rigidity (Pa) G= E/[2(1+v)] Plus: longitudinal mode, and torsion Phase diagram (angle) ; Plates/membranes, i.e. essentially 2D systems Plus: two in-plane l,m: Lamé coefs. (Pa) l=E v /[(1+v)(1-2v)] m=G v: Poisson’s ratio r: mass density (kg) Primary-wave (longitudinal) Secondary-wave (transverse) 2 families solved with

Basics of mechanics Basic mechanical objects: Same procedure for 2D: define mode shapes Beams/string, i.e. essentially 1D systems E: Young’s modulus (Pa) T: tension (linear, N) r: mass density (kg) Ix,y: second moment of area Ix = w h3/12 Iy = h w3/12 (m4) A: section A= w h (m2) G:torsion rigidity (Pa) G= E/[2(1+v)] Two flexure, one longitudinal mode, and one torsion Phase diagram ; Discrete solutions: n=0,1,2,… eigenfrequency eigenshape Solving: harmonic solution + boundary condition Condition on at ends, e.g. doubly clamped

Basics of mechanics Basic mechanical objects: For more: A.N. Cleland, Foundations of Nanomechanics, Springer (2003) Basic mechanical objects: Beams/string, i.e. essentially 1D systems Example of out-of-plane flexure Phase diagram n=0 fundamental, first flexural mode Remark: n=0 is not the DC deflection, Which is strictly speaking the superposition of the DC response of all modes (see following discussion) Ansys numerics; very close analytics n=4 fifth mode

Basics of mechanics Some objects are indeed a bit more complex: Plate attached with microfabricated springs Y. Lee, Florida M. Gonzalez et al., PHYSICAL REVIEW B 94, 014505 (2016) Then, mainly relies on numerical simulations for the mechanics (Comsol, Ansys,…)

Basics of mechanics Consider a small excitation: Doubly-clamped type Along dir. , Integr. over structure, Here, normal to chip surface Mechanical susceptibility: Linear response theory: c(w) = Re[c(w) ] + i Im[c(w) ] = c’ + i c’’ x(w) = c(w) F(w) dispersion absorption w c’’ c’ Out-of-plane flexure: Same ideas apply to other mode families Mode 1 Mode 2 Mode 3 Static Etc… Cantilever type

Basics of mechanics Consider a small excitation: Doubly-clamped type Along dir. , Integr. over structure, Here, normal to chip surface Mechanical susceptibility: Linear response theory: c(w) = Re[c(w) ] + i Im[c(w) ] = c’ + i c’’ x(w) = c(w) F(w) dispersion absorption w c’’ c’ Out-of-plane flexure: Same ideas apply to other mode families Mode 1 Mode 2 Mode 3 Static Etc… Cantilever type

Basics of mechanics Consider a small excitation: Doubly-clamped type Along dir. , Integr. over structure, Here, normal to chip surface Mechanical susceptibility: Linear response theory: c(w) = Re[c(w) ] + i Im[c(w) ] = c’ + i c’’ x(w) = c(w) F(w) dispersion absorption w c’’ c’ Out-of-plane flexure: Same ideas apply to other mode families Mode 1 Mode 2 Mode 3 Static Etc… Cantilever type Static deflection is the superposition of the adiabatic response of all modes. It is small, need a strong force/good detection scheme. Usually prefer to work in resonant mode, win the factor Q (quality factor f0/Df).

Basics of mechanics Consider a small excitation: Doubly-clamped type Here, normal to chip surface For each mode n, Mechanical susceptibility: Linear response theory: cn (w) = Re[cn (w) ] + i Im[cn (w) ] = cn’ + i cn’’ xn(w) = cn(w) Fn(w) dispersion absorption w c’’ c’ Out-of-plane flexure: Mode 1 Mode 2 Mode 3 Static Etc… Cantilever type xn(w) e.g. max. deflection Same ideas apply to other mode families

Basics of mechanics Consider a small excitation: 4.2 K vacuum Doubly-clamped type Here, normal to chip surface For each mode n, Mechanical susceptibility: Linear response theory: cn (w) = Re[cn (w) ] + i Im[cn (w) ] = cn’ + i cn’’ xn(w) = cn(w) Fn(w) dispersion absorption w c’’ c’ Out-of-plane flexure: Mode 1 Mode 2 Mode 3 Static Etc… Cantilever type xn(w) e.g. max. deflection simple 1D harmonic oscillator In the linear regime, each mode n has: mode parameters Can be calculated from theory/numerics Same ideas apply to other mode families

Basics of mechanics Consider a small excitation: Doubly-clamped type Here, normal to chip surface For each mode n, Mechanical susceptibility: Linear response theory: cn (w) = Re[cn (w) ] + i Im[cn (w) ] = cn’ + i cn’’ xn(w) = cn(w) Fn(w) dispersion absorption w c’’ c’ Out-of-plane flexure: Mode 1 Mode 2 Mode 3 Static Etc… Cantilever type xn(w) e.g. max. deflection In the linear regime, any complex dynamics can be obtained by simple superposition of all modal responses BUT: all of this not true anymore in nonlinear regime, i.e. when excitation is too large… Same ideas apply to other mode families

Basics of mechanics At larger excitations… Doubly-clamped type The resonances are not Lorentzian anymore: “Duffing” type resonances n=0 4.2 K vacuum Cantilever type Cantilever type 36

Basics of mechanics At larger excitations… Doubly-clamped type The resonances are not Lorentzian anymore: “Duffing” type resonances Plus hysteresis (in frequency)! Cantilever type Cantilever type n=0 4.2 K vacuum Bistable range 37

Basics of mechanics At larger excitations… Doubly-clamped type The resonances are not Lorentzian anymore: “Duffing” type resonances Plus hysteresis (in frequency)! Cantilever type Cantilever type And modes are (dispersively) coupled together!! n=0 4.2 K vacuum Coupling term Position of mode depends on motion of the others 38

Basics of mechanics At larger excitations… Doubly-clamped type The resonances are not Lorentzian anymore: “Duffing” type resonances Most cases: nonlinearities seen as a limitation…. I. Kozinsky et al., APL 88, 253101 (2006) Cantilever type Cantilever type But it can be used in new experimental Schemes! e.g. amplification: R. Almog et al., APL 88, 213509 (2006) 39

Basics of mechanics Actuation/detection schemes The two simplest ones: just require metallic contacts Capacitive: -) rely on for the actuation -) rely on for the detection (with flat gates, or combs) Fixed Mobile g Example: mobile planar capacitor

Basics of mechanics Actuation/detection schemes The two simplest ones: just require metallic contacts Capacitive: -) rely on for the actuation -) rely on for the detection (with flat gates, or combs) g Mobile Example: mobile planar capacitor Fixed x

Basics of mechanics Actuation/detection schemes The two simplest ones: just require metallic contacts Capacitive: -) rely on for the actuation -) rely on for the detection (with flat gates, or combs) g Mobile Example: mobile planar capacitor Surface S Fixed Strongly nonlinear! x

Basics of mechanics Actuation/detection schemes The two simplest ones: just require metallic contacts Capacitive: -) rely on for the actuation -) rely on for the detection (with flat gates, or combs) g Mobile Example: mobile planar capacitor Surface S Fixed Lowest order…. Again, higher (nonlinear) term give rise to new physics, e.g. parametric pumping x

Basics of mechanics Actuation/detection schemes The two simplest ones: just require metallic contacts Capacitive: -) rely on for the actuation -) rely on for the detection (with flat gates, or combs) g Mobile Example: mobile planar capacitor Surface S Fixed Lowest order x

Basics of mechanics Actuation/detection schemes The two simplest ones: just require metallic contacts Capacitive: -) rely on for the actuation -) rely on for the detection (with flat gates, or combs) g Mobile Example: mobile planar capacitor Surface S Fixed Lowest order x

Basics of mechanics Actuation/detection schemes The two simplest ones: just require metallic contacts Capacitive: -) rely on for the actuation -) rely on for the detection (with flat gates, or combs) In practice, geometry is more complex than 2 planes: integrate over structure and Example: mobile planar capacitor With Lowest order

Basics of mechanics Actuation/detection schemes The two simplest ones: just require metallic contacts Magnetomotive: -) rely on for the actuation -) rely on for the detection (needs a coil for the B field) Example: mobile rigid bar Mobile

Basics of mechanics Actuation/detection schemes The two simplest ones: just require metallic contacts Magnetomotive: -) rely on for the actuation -) rely on for the detection (needs a coil for the B field) Mobile Example: mobile rigid bar x

Basics of mechanics Actuation/detection schemes The two simplest ones: just require metallic contacts Magnetomotive: -) rely on for the actuation -) rely on for the detection (needs a coil for the B field) Mobile Example: mobile rigid bar x

Basics of mechanics Actuation/detection schemes The two simplest ones: just require metallic contacts Magnetomotive: -) rely on for the actuation -) rely on for the detection (needs a coil for the B field) In practice, geometry is more complex than rigid bar: integrate over structure Example: mobile rigid bar and With Linear scheme!

Basics of mechanics Actuation/detection schemes For more: J.A. Pelesko and D.H. Bernstein, Modeling MEMS and NEMS, Chapman & Hall CRC (2003). Actuation/detection schemes The two simplest ones: just require metallic contacts Capacitive: -) rely on for the actuation -) rely on for the detection (with flat gates, or combs) Magnetomotive: -) rely on for the actuation -) rely on for the detection (needs a coil for the B field) And also: -) Dielectric gradient forces (looks like capacitive) -) Piezoelectric/piezoresitive (needs right material) -) Optics (see Marquardt & Sillanpää)

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) Device has a Lorentzian response Linearity: laminar flow (not turbulent) Continuum mechanics: Navier-Stokes equation P: pressure in fluid (Pa) h: viscosity of fluid (dynamic viscosity, Pa s) r: mass density fluid (kg) Ingredients: -) pressure P driving the flow, -) inertia of fluid r -) friction h (links shear force to gradient of velocity in perp. dir.) Incompressible fluid: Simplify the problem: G.G. Stokes, Mathematical and physical papers (1901) D.C. Carless, H.E. Hall, J.R. Hook, JLTP 50, 583 (1983) Computing the force exerted by the fluid on a solid Simplest case: cylindrical solution, harmonic motion

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) Depends on a single parameter: viscous penetration depth Fluid velocity field, in-phase with cylinder velocity Magnitude velocity field y/a x/a (normalized to cylinder radius a) (cylinder moves in dir.) d/a =2

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) Depends on a single parameter: viscous penetration depth Fluid velocity field, in-quadrature with cylinder velocity Magnitude velocity field (normalized to cylinder radius a) (cylinder moves in dir.) y/a y/a x/a d/a =2 x/a

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) Depends on a single parameter: viscous penetration depth “size” of the flow field Compute the force on cylinder from integrating the fluid stress tensor on the surface in-phase Dissipative effect: energy lost in friction i.e. broadening of mechanical resonance Inertial effect: drag the fluid mass around i.e. frequency shift of mechanical resonance (which is negative) y/a x/a in-quadrature Measuring these, one computes d, and then computes h x/a y/a

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) Depends on a single parameter: viscous penetration depth Possible errors/corrections to this result: Length L >> d to validate “infinite length” limit (in practice usually easy to match) Device not cylindrical! But rectangular… Numerical fits for corrections, not a big effect (15 % most) J.E. Sader JAP 84, 64 (1998) Finite size corrections if d is not much smaller than container Flow couples to other walls in cell… Fluid is not incompressible! Can radiate sound waves…. k: compressiblility (assuming no sound absorption, i.e. h=0) In practice, small effect for small size/not too high frequency (but be cautious) Philip M. Morse and K. Uno Ingard, Theoretical acoustics, Princeton University Press (1986)

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) M. Black, H. Hall, K. Thompson, J. Phys. C: Solid St. Phys. 4, 129 (1971) An example (vib. wire) with a quantum fluid: Normal liquid 3He, viscosity Measurement with a (planar) MEMS: Y. Lee’s group M. Gonzalez et al., PHYSICAL REVIEW B 94, 014505 (2016)

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) What happens in a superfluid? Two-fluid model: Superfluid fraction rs, viscosity hs=0 (superfluid condensate: “renormalised vacuum”) Normal fraction rn, viscosity hn≠0 (excitations, bath of quasi-particles) Of course rs + rn = r density of fluid Need to know: how many excitations (rn) how much they dissipate (hn) Phenomenological, but why does it work? Why can we treat a quantum fluid with classical Navier-Stokes equations in the first place? Because N-body interaction in fluid is quantum, but is effectively described by non-interacting particles which are thus non-correlated! And the device itself is classical!

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) What happens in a superfluid? P = 0 bar An example (vib. wire) with a quantum fluid: Superfluid 4He Tl M. Morishita et al., JLTP 76, 387 (1989) An example (vib. wire) with a quantum fluid: Superfluid 3He Tcs A.M. Guénault et al., J. of Low Temp. Phys. 62, 511 (1986)

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) What happens in a superfluid? P = 0 bar An example (vib. wire) with a quantum fluid: Superfluid 4He Tl M. Morishita et al., JLTP 76, 387 (1989) An example using a NEMS: Superfluid 4He S. Kafanov, Lancaster, work in progress

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) What happens in a superfluid? P = 0 bar An example (vib. wire) with a quantum fluid: Superfluid 4He Tl M. Morishita et al., JLTP 76, 387 (1989) But why this data is not displayed? Damping drops, and tends to zero… An example (vib. wire) with a quantum fluid: Superfluid 3He Tcs A.M. Guénault et al., J. of Low Temp. Phys. 62, 511 (1986)

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) At some point, continuum mechanics breaks down: Navier-Stokes not valid anymore! Navier-Stokes requires: Mean free path of excitations/(quasi-)particles lmfp << d (i.e. small volume of fluid within d3 has well defined properties) Mean free path of excitations/(quasi-)particles lmfp << a (i.e. the moving object can be treated as a boundary condition) Usually, when it breaks down: a < lmfp << d at first, Which can be described by a slippage correction (device “sees less” the fluid) Introduce slip length as a first order correction (modify the flow on the surface, allowing velocity object ≠ velocity fluid) H. Hojgaard et al., JLTP 41, 473 (1980)

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) When lmfp >> d the regime is definitely different: one enters molecular (or ballistic) flow The device “sees” (quasi-)particles bouncing off its surface They are two limits: The reflection can be specular (perfect transfer of normal momentum; energy preserved) The reflection can be diffusive -) either due to asperities (energy is preserved) -) either due to interaction with surface excitations (energy is not preserved microscopically, but is preserved on average)

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) When lmfp << d the regime is definitely different: one enters molecular (or ballistic) flow The device “sees” (quasi-)particles bouncing off its surface Need to know: how many excitations (rn in superfluid) how they interact with surface (specularity 0 < s < 1) For a classical gas, Force/damping prop. to density (pressure) independent of frequency (not very sensitive to parameter s) V. Kara et al., Nano Lett. 15, 8070 (2015)

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) When lmfp << d : molecular (or ballistic) flow An example (vib. wire) with a quantum fluid: Superfluid 4He Ballistic phonons, density prop. T4 M. Morishita et al., JLTP 76, 387 (1989) Ballistic Bogolioubov quasi-particles, Andreev-reflected! Density prop. Exp[D/T] An example (vib. wire) with a quantum fluid: Superfluid 3He A.M. Guénault et al., J. of Low Temp. Phys. 62, 511 (1986)

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) For the “magic” there, discuss with Lancaster people! QP QH When lmfp << d : molecular (or ballistic) flow An example (vib. wire) with a quantum fluid: Superfluid 4He Ballistic phonons, density prop. T4 M. Morishita et al., JLTP 76, 387 (1989) QH Ballistic Bogolioubov quasi-particles, Andreev-reflected! Density prop. Exp[D/T] An example (vib. wire) with a quantum fluid: Superfluid 3He QP A.M. Guénault et al., J. of Low Temp. Phys. 62, 511 (1986)

Basics of fluidics Immersing e.g. a beam in a fluid We’ll consider only linear processes: small excitations (displacement, velocity) When lmfp << d : molecular (or ballistic) flow Superfluid 3He reproduced with a MEMS: Goalpost-shaped Si structure M. Defoort et al., JLTP 183, 284 (2016) Collaboration Grenoble - ULANC An example (vib. wire) with a quantum fluid: Superfluid 3He A.M. Guénault et al., J. of Low Temp. Phys. 62, 511 (1986)

Eddy Collin, ULT Institut Néel – CNRS Any questions? Everything perfectly clear? Really? Then, thank you 2 µm 60 µm Eddy Collin, ULT Institut Néel – CNRS Cryocourse 25/09/2016