We consider the possibility of adding a stage to a dilution refrigerator to
provide additional cooling by ``filtering out'' hot atoms. Three methods are
considered: 1) Effusion, where holes having diameters larger than a mean-free
path allow atoms to pass through easily; 2) Particle waveguide-like motion
using very narrow channels that greatly restrict the quantum states of the
atoms in a channel. 3) Wall-limited diffusion through channels, in which the
wall scattering is disordered so that local density equilibrium is established
in a channel. We assume that channel dimension are smaller than the mean-free
path for atom-atom interactions. The particle waveguide and the wall-limited
diffusion methods using channels on order of the de Broglie wavelength give
cooling. Recent advances in nano-filters give this method some hope of being
practical.Comment: 10 pages, 3 figures. Corrected typos and made some minor wording
  change

J. M. Ziman

Neal Kalechofsky

R. K. Pathria

William J. Mullin

English

arXiv

This is the pre-published version which is collected from arXiv. The published version is at 
http://prl.aps.org/abstract/PRL/v97/i1/e010801We consider the possibility of adding a stage to a dilution refrigerator to provide additional cooling by “filtering out” hot atoms. Three methods are considered: (1) effusion, where holes having diameters larger than a mean-free path allow atoms to pass through easily; (2) particle waveguidelike motion using very narrow channels that greatly restrict the quantum states of the atoms in a channel; (3) wall-limited diffusion through channels, in which the wall scattering is disordered so that local density equilibrium is established in a channel. We assume that channel dimensions are smaller than the mean-free path for atom-atom interactions. The particle waveguide and the wall-limited diffusion methods using channels on order of the de Broglie wavelength give cooling. Recent advances in nanofilters give this method some hope of being practical.

Mullin, WJ

Kalechofsky, N

ScholarWorks@UMass Amherst

Theory of cooling by flow through narrow pores

We consider the possibility of adding a stage to a dilution refrigerator to provide additional cooling by “filtering out” hot atoms. Three methods are considered: (1) effusion, where holes having diameters larger than a mean-free path allow atoms to pass through easily; (2) particle waveguidelike motion using very narrow channels that greatly restrict the quantum states of the atoms in a channel; (3) wall-limited diffusion through channels, in which the wall scattering is disordered so that local density equilibrium is established in a channel. We assume that channel dimensions are smaller than the mean-free path for atom-atom interactions. The particle waveguide and the wall-limited diffusion methods using channels on order of the de Broglie wavelength give cooling. Recent advances in nanofilters give this method some hope of being practical

University of Massachusetts AmherstScholarWorks@UMass AmherstPhysics Department Faculty Publication Series Physics2006Theory of cooling by flow through narrow poresWJ Mullinmullin@physics.umass.eduN KalechofskyFollow this and additional works at: https://scholarworks.umass.edu/physics_faculty_pubsPart of the Physics CommonsThis Article is brought to you for free and open access by the Physics at ScholarWorks@UMass Amherst. It has been accepted for inclusion in PhysicsDepartment Faculty Publication Series by an authorized administrator of ScholarWorks@UMass Amherst. For more information, please contactscholarworks@library.umass.edu.Recommended CitationMullin, WJ and Kalechofsky, N, "Theory of cooling by flow through narrow pores" (2006). PHYSICAL REVIEW LETTERS. 34.Retrieved from https://scholarworks.umass.edu/physics_faculty_pubs/34arXiv:cond-mat/0605072v2  [cond-mat.other]  16 May 2006Theory of cooling by flow through narrow poresWilliam J. Mullin∗Physics Department, Hasbrouck Laboratory, University of Massachusetts, Amherst, MA 01003Neal Kalechofsky†Oxford Instruments America Inc., 130A Baker Avenue, Concord, MA, 01742, USA(Dated: February 6, 2008)We consider the possibility of adding a stage to a dilution refrigerator to provide additionalcooling by “filtering out” hot atoms. Three methods are considered: 1) Effusion, where holes havingdiameters larger than a mean-free path allow atoms to pass through easily; 2) Particle waveguide-like motion using very narrow channels that greatly restrict the quantum states of the atoms ina channel. 3) Wall-limited diffusion through channels, in which the wall scattering is disorderedso that local density equilibrium is established in a channel. We assume that channel dimensionare smaller than the mean-free path for atom-atom interactions. The particle waveguide and thewall-limited diffusion methods using channels on order of the de Broglie wavelength give cooling.Recent advances in nano-filters give this method some hope of being practical.We investigate here the possibility cooling a gas by “fil-tering out” hot atoms, perhaps as an adjunct to a dilutionrefrigerator with solutions of 3He in liquid 4He.[1] Ourmethod involves passing a degenerate Fermi gas throughnarrow constrictions formed by pores in a partition. Un-der certain conditions we find that this particle “leakage”allows a lowering of the temperature of the remaining gas.Our study adds to many recent experimental and theo-retical analyses of quantum size effects in the behavior ofparticles in nanopores (as in Refs. [2, 3, 4]).An initial idea of how to remove the hot atoms issuggested by a technique used in electron heterojunc-tion physics,[5, 6, 7, 8] where the electron gas is passedthrough a narrow constriction formed by a gate potential.Because the constriction is narrow the bands of states al-lowed in this “particle waveguide” are widely separated,which means that not all energies are allowed through.Adjusting the states in the channel can allow selectivepassage particles in states at the Fermi energy, so thatone removes only hot atoms. In such an approach theconstriction must be of order of the de Broglie wave-length of atoms at the top of the Fermi surface, which isroughly the separation between fermions. Because of thelimited states (that is, bands) we characterize channelsof this size as “narrow.” We will see that this approachcan be made to work under appropriate conditions on thenano-pores.An alternative possibility involves channels or poreswith diameters much larger than the de Broglie wave-length but still less than the mean-free path of thefermions in the gas, which can be large due to its 1/T 2behavior. We call these “wide” pores.We will consider different kinds of flow through theholes of the two sizes mentioned: effusion, waveguideflow, and wall-limited diffusion. In effusion, the holesare by definition wide and the states in the channel re-main three-dimensional with no banding. Moreover thewalls are sufficienly smooth that the particles undergo noback scattering nor do they come into equilibrium withthe channel walls. Effectively all that the pores do is toallow particles already directed in the positive z-directionto pass through the membrane. We might hope that thiswould allow cooling because the intensity of fast atomspassing through a hole is larger than that of the slowatoms and these, on average, carry more energy. Indeedit is a standard textbook exercise[10] to show that thisworks for a Boltzmann gas. But does it work for a de-generate Fermi gas?In waveguide flow the channels are narrow enough tohave well-defined bands. Again the particles are assumedto undergo no back scattering nor do they come into ther-mal equilibrium with the channel walls. Nevertheless be-cause of the bands only particles with certain energiesare allowed in the channel. If, for example, the Fermienergy of the gas in the container is coincident with thebottom of the lowest band one might expect that onlyhigh energy particles would get through the holes andthe remaining gas would be cooled.A third situation is wall-limited diffusion or Knudsenflow, which could occur in wide or narrow pores. Thereis scattering at the walls including back scattering. Therate of diffusion in the channel depends on the diam-eter of the pore, the particle velocity, and the densitygradient that maintains the flow. In this and the abovemethods we will assume the density difference is main-tained by pumping away the particles that pass throughthe membrane. Both effusion and wall-limited diffusionwere considered by one of the authors some time ago forenhancing polarization.[9]If the Fermi gas in the cooling cell is, say, a 1% solutionof 3He in liquid 4He at millikelvin temperature, then theFermi temperature is 124 mK, the de Broglie wavelengthabout 3 nm, and the mean-free path is about 1 µm atT = 15 mK.Suppose the number of atoms in a box (B1) is N withdensity n. The atoms pass through a membrane and en-2ter a second box (B2) maintained (by pumping) at amuch smaller particle number N2 and density n2. Themembrane contains a great number M of holes each ofdiameter d and of total hole area A. (See Fig. 1.) Wewill compute particle current JN and energy current JEpassing through the holes. The rates of particle and en-ergy change, dN/dt = AJN and dE/dt = AJE in thecontainer B1 combine indEdt= CVdTdt+(∂E∂N)T,VdNdt(1)to give the cooling power CV dT/dt where CV is the heatcapacity at constant volume in B1.FIG. 1: Schematic diagram of the apparatus. 3He in solu-tion with liquid 4He enters the cooling chamber B1, passesselectively through the membrane into B2. A gas at loweredtemperature remains in B1. The 3He density in B2 is keptlow by pumping. The gas is recycled, after being recooled bya dilution fridge, back into B1.In the case of effusion the particle current isJN (1 →2) = 2h3∫ +∞−∞dpx∫ +∞−∞dpy∫ +∞0dpzpzmnp (2)where np is the momentum distribution function andthe factor of 2 accounts for spin degeneracy. There isa similar expression for JE . For a classical gas passingthrough wide holes we find a cooling power given by−nkBAT (kBT/2πm)1/2 . The minus sign implies cool-ing in agreement with the usual textbook treatment.[10]In the degenerate limit effusive cooling fails.For particle waveguide motion (in narrow holes) weneed to take into account the banding of the states due tothe limited transverse motion; the energies are given byǫnz = ǫn+ǫz with ǫz = ~2k2z/2m.We will assume a squarecross section with width d so that ǫn = ǫ0ηn where ǫ0 =π2~2/(md2) and ηn = 1, 2.5, ...(n2x + n2y)/2, .... (Using acircular cross section gives no significant differences inour final results.) In this case Eq. (2) is replaced byJN (1→2) = 2d2h∑n∫ +∞0dpzpzm1(eβ(ǫz+ǫn−µ) + 1)(3)with µ the chemical potential (≈ ǫF at low T ). Interms of the one-dimensional Fermi integrals Fl(γ) =∫∞0 dzzl1/(ez−γ + 1), the cooling power for the waveg-uide case isCVdTdt= − 2Ad2hβ2pW (4)wherepW ≡∞∑n=1[F1(αn)− (ξ − βǫn) ln(1 + eαn)] . (5)The filling of the bands in the channels is determined byαn = β(µ−ǫn) with β = (kBT )−1and ξ = β (∂E/∂N)T,Vis a property of the gas in the cooling chamber (B1). Forcooling pW must be positve. If γ = βµ and t = T/TFthen the relation between density and chemical potentialleads to the usual transcendental equation for µ :32t3/2F1/2(γ) = 1 (6)Further, taking the derivative of E versus N givesξ =G3/2(γ)G1/2(γ)(7)where Gl(γ) =∫∞0 dzzlez−γ/(ez−γ + 1)2 = lFl−1. Givena value of t from the temperature and the concentration,we must solve Eq. (6) for γ; we put that into Eq. (7) toget ξ. We find it convenient to introduce the the ratioof lowest band edge to Fermi energy y ≡ ǫ0/ǫF so thatβǫn = yηn/t.For numerical calculations we fix T at 15 mK as a rea-sonable incoming temperature in the cooling cycle. Then,if the number of holes in the membrane is M =1011, theprefactor in Eq. (4) is 13 µW. In the case of wide holes(where waveguide flow becomes effusion), the sum overbands in Eq. (4) becomes a double integral over trans-verse momenta and in this continuum limit we findpW ≈ π2ty[F2(γ)− ξF1(γ)] continuum limit (8)We have made no assumptions here about whether thesystem is degenerate or not—just that the channel statesare now continuous. If the factor in square brackets ispositive, then it says that the smaller y, the larger is thecooling. This result stems from just having bigger andbigger holes, allowing more hot atoms out. But of coursethere is a limit to how big the holes can be to maintaina pressure differential across the membrane.Next consider the highly degenerate limit of the lastform, for which Fn = γn/n and ξ = γ, so thatpW ≈ − π12tyγ3 degenerate continuum limit (9)which agrees with a direct calculation—no cooling occursin that case, because there are many channel states belowthe Fermi energy allowing low energy atoms to escape.3Indeed the gas left behind can then end with a highertemperature.For the fully classical limit we take eγ to be very largeso thatpW ≈√π31t1/2yclassical continuum limit. (10)This result again agrees with the textbook classical effu-sion result.In Fig. 2 we show numerical results based on Eq. (5). Itis useful to plot ypW rather than pW because then all theplots collapse onto a single curve for t > 1, as implied byEq. (8). The t dependence for large t is t−1/2 and we areindeed in the asymptotic region described by Eq. (10).We see that y = 1 is a special point with the lowest bandedge lined up with the Fermi energy. (Recall however,that the cooling power prefactor, which is not includedin pW contains a T2 so all cooling powers go to zero withdecreasing T .) When the lowest band edge is lower thanthe Fermi energy (y < 1) we lose cooling at very low T(as in Eq. (9)), but surprisingly we regain it for largerT due to the contributions of the higher bands . Finallyfor all band edges above the Fermi energy (y > 1) we getcooling for all T , but, because the curves in the figureare multiplied by the factor y, the actual cooling poweris diminished when we divide out this factor, and thesecases are not as useful.-1.0-0.50.00.5yp2.42.22.01.81.61.41.21.00.80.60.40.2t        y 21.51 0.5 0.3 FIG. 2: Reduced waveguide cooling power as a function oftemperature parameter t for various band-edge y values fora square cross section. Because of the multiplicative factor yall curves collapse onto the same classical limit curve at larget.The parameters we have used are not independent. Be-cause TF depends on x2/3, where x is the concentration,and ǫ0 depends on d−2, then for T = 15 mK we find therelationt = 1.8× 10−2y (d(nm))2 (11)From the plot, we see that if we have y < 1 we get a 1/yenhancement in cooling power, but to gain that we mustalso have t & 0.4 to avoid heating. For y > 1 we cango to lower t values but we lose cooling power because ofthe 1/y factor and because of the dip in the curves.Consider d = 10 nm, M = 1011 which is perhapswithin practical reach. The value t = 0.4 (TF = 35 mKand x = 0.2%) implies y = 0.2 and the cooling would beon the order of a 50 µW in this ballistic-flow waveguidecase. Such a result would be quite a remarkable cool-ing rate, but the assumption of perfectly smooth walls isoptimistic. However, the inevitable coating of the porewalls by a couple of layers of solid 4He will enhance thesmoothness.It perhaps seems more likely that the walls of the chan-nel would cause scattering, including back scattering, in-terband transitions, etc. Thus we consider next a sim-ple wall-limited diffusion or Knudsen flow model. Ourstarting point is a kinetic equation for the distributionfunction in collision-time approximation[11],[12]vp · ∇np = −1τδnp (12)where τ is the time between collisions with the wall. Theleft side can be written as −vpz∂n(0)p /∂ǫp∂µ/∂z, wheren(0)p is the local equilibrium distribution function andδnp is the correction to local equilibrium. The gradi-ent in chemical potential ∂µ/∂z is proportional to thegradient in density. A much more rigorous approach tosuch a kinetic equation is described in Ref. [13]. We con-sider again the case of a very narrow channel containingbanding. We solve for δnp and use it to compute, say,the particle flux asJN =2d2h∑n∫ +∞−∞dpzpzmδnp (13)From Eq. (1) the cooling power is found to beCVdTdt= −(2Ad2hβ2)pK (14)where we have introduced the same prefactor as in thewaveguide case so thatpK = ddγdz[ty 〈ηn〉]1/2∑n[G3/2(αn)− ςnG1/2(αn)].(15)with ζn = ξ− yηn/t. In this equation the collision time τis writen in terms of the transverse velocities in a channel.That is, we write τ ∼ d/(2v¯) with v¯ =√2 〈ǫn〉 /m where〈ǫn〉 ≡∑nǫn∫ +∞−∞dpzn(0)p (ǫn)[∑n∫ +∞−∞dpzn(0)p]−1(16)If we divide this quantity by the lowest band edge, thenwe have 〈ǫn〉 /ǫ0 = 〈ηn〉 used in Eq. (15).We have to evaluate the derivative dγ/dz. What wemean by this quantity is dγ/dz = (dγ/dn)(dn/dz) since4it is the gradient in density n that drives the flow. Wecan find this by taking the derivative of the self-consistentexpression, Eq. (6). In that equation t depends on nbecause ǫF does. We finddγdz=2t3nF1/2tG1/2dndz≈ 49G3/2G1/21L=49ξ1L(17)where in the middle form we have taken the density gra-dient as n/L with L the length of a channel. What wecompute then ispK =49dLξ[ty 〈ηn〉]1/2∑n[G3/2(αn)− ςnG1/2(αn)].(18)We again plot the product, y pK . The results areshown in Fig. 3; note that the ratio d/L is not includedin the plots. The curves differ from those of the waveg-uide in a several ways. The curves are always positive,i.e., represent cooling, and the y = 1 degenerate casediverges (the T 2 in the prefactor cures that). We havechecked the numerical calculations by doing various lim-iting relations analytically. For example, the continuum(small y) case in the limit t→ 0 is pK → 2.41d/Ly.43210yp L/d1.00.90.80.70.60.50.40.30.20.1t          y 1.5 1 0.5 0.3 0.1 FIG. 3: Reduced Knudsen cooling power as a function oftemperature parameter t for various y values for a squarecross section. The factor of d/L in the overall cooling poweris not included here.As in the waveguide case, small y gives amplifiedcooling. Indeed, for M = 1011 holes, y = 0.1 givespKL/d ≈ 1.6/y = 16. Again consider d = 10 nm. Thecooling power is 13 µW×16d/L = 208d/L µW. A mem-brane width of L = 1 µm gives a cooling of just 2.1 µW.By Eq. (11) we have t = 0.18 with the y value chosen orTF = 83 mK and x = 0.5%. This cooling power mightbe useful if the width L used is not too optimistic. Thecooling would be enhanced if a largerM value were avail-able. For small y values the continuous degenerate limitgives an upper limit on the cooling power within theseconditions. We havepK . 132.4ydLµW =31.2ydLµW (19)In summary, we have examined here the physics of par-ticle flow through narrow pores and estimated the pos-sibility of cooling by this method and have found somepotential for success. As nano-technology improves, thepossibilities may increase. A very smooth-walled chan-nel that would provide waveguide type flow would givethe greatest cooling. The more probable situation ofKnudsen flow, while providing cooling over all parameterranges, has the factor d/L reducing the cooling power.However, even that circumstance does not make it im-possible. The numerical results give hope that this ap-proach can lead to a add-on device to extend the rangeof a dilution refrigerator. Experiments are being plannedto test the potential of the method.[14] While we haveconsidered the possibility of a practical application ofnanopores here, experiments on this kind also provideinteresting physics, namely detecting quantum size ef-fects in the narrow channels and the resultant restrictionof states as already seen experimentally.[2, 3]We thank R. Hallock, J. Owers-Bradley, and G. Pickettfor useful conversations.∗ Electronic address: mullin@physics.umass.edu† Electronic address: kalechofsky@ma.oxinst.com[1] A preliminary version of this report is given in W. J.Mullin and N. Kalechofsky, Proceedings of LT24, AIPConference Proceedings (accepted for publication).[2] Junko Taniguchi, Akira Yamaguchi, Hidehiko Ishimoto,Hiroki Ikegami, Taku Matsushita, Nobuo Wada, SilvinaM. Gatica, Milton W. Cole, Francesco Ancilotto, ShinjiInagaki, and Yoshiaki Fukushima, Phys. Rev. Lett. 94,065301 (2005).[3] S. L. Phillipson, A. M. Guenault, S. N. Fisher, G. R.Pickett, and P. J. Y. Thibault, Nature 395, 578 (1998);and P. A. Reeves, A. M. Guenault, S. N. Fisher, G.R. Pickett, and G. Tvalashvili, Physica B 284-288, 319(2000). .[4] M. W. Cole, Vincent H. Crespi, G. Stan, C. Ebner, JacobM. Hartman, S. Moroni, and M. Boninsegni , Phys. Rev.Lett. 84, 3883 (2000); F. Ancilotto, M. M. Calbi, S. M.Gatica and M. W. Cole, Phys. Rev. B 70, 165422 (2004).[5] N. G. Einspruch and W. R. Frensley, eds. Heterostruc-tures and Quantum Devices (Academic Press, 1994).[6] A. Szafer and A. D. Stone, Phys. Rev. Lett. 62, 300(1989)[7] M. Bu¨ttiker, Phys. Rev. B 41, R7906 (1990).[8] P. F. Bagwell, Phys. Rev. B 41, 10354 (1990).[9] W. J. Mullin, Phys. Rev. Lett. 57, 2710 (1987).[10] R. K. Pathria, Statistical Mechanics (Butterworth-Heinemann, 1996).[11] J. M. Ziman, Electrons and Phonons (Oxford at theClarendon Press, 1960).[12] M. J. Rice, Phys. Rev. 165, 288 (1968).[13] A. E. Meyerovich and I. V. Ponomarev, Phys. Rev. B65, 155413 (2002); Yiying Cheng and A. E. Meyerovichcond-mat/0512614.[14] John Owers-Bradley, private communication.

Theory of cooling by flow through narrow pores

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