Bose-Einstein Condensation in Multilayers

Abstract

The critical BEC temperature Tc of a non interacting boson gas in a layered structure like those of cuprate superconductors is shown to have a minimum Tc,m, at a characteristic separation between planes am. It is shown that for a < am, Tc increases monotonically back up to the ideal Bose gas T0 suggesting that a reduction in the separation between planes, as happens when one increases the pressure in a cuprate, leads to an increase in the critical temperature. For finite plane separation and penetrability the specific heat as a function of temperature shows two novel crests connected by a ridge in addition to the well-known BEC peak at Tc associated with the 3D behavior of the gas. For completely impenetrable planes the model reduces to many disconnected infinite slabs for which just one hump survives becoming a peak only when the slab widths are infinite.Since London first suggested 1 that superfluidity in liquid 4 He might well be a manifestation of Bose-Einstein condensation (BEC) of the helium atoms before interatomic interactions are "switched on," BEC in layered systems began to be studied to understand helium films 2,3,4 . The discovery of high-T c superconductivity stimulated renewed interest in compounds with layered structures 5 in which a BEC mechanism seems to be an essential feature to explain high critical temperatures 6 . The so-called "Uemura plot" 7 of data from muon-spin relaxation (µSR), neutron and Raman scattering, and angle-resolved photoemission (ARPES) measurements exhibits T c vs Fermi temperatures T F ≡ E F /k B where E F the Fermi energy and k B the Boltzmann constant. Empirical T c s of many cuprates straddle a line parallel to the Uemura-plot diagonal line associated with the simple BEC formula T 0 ≃ 3.31 2 n 2/3 B /mk B ≃ 0.218T F corresponding to an ideal gas of bosons of mass m = 2m * and number density n B = n s /2 where m * is the individualcharge-carrier effective mass and n s their number density. The parallel line of data is shifted down from T 0 by a factor of 4-5. This has been judged 8 a "fundamental importance of the BEC concept in cuprates." In addition, the possibility of creating BECs or superfluidity of ultracold fermions 9 in optical lattices 10 , along with the expected observation of BEC of excitons (electron-hole pairs) in semiconductors 11 , have further revived theoretical and experimental 12,13 efforts to better understand the behavior of quantum gases in layered geometries.Most models based on layered structures 6,15 simulating quasi-2D high-T c superconductors, or other to study BEC 16,17,18 , rely on a single-boson hopping interaction term producing nearest-interlayer couplings in one spatial dimension while moving freely in the other two directions. The energy spectrum is typically of the form ǫ k = 2 (k 2 x + k 2 y )/2m + ǫ kz with ǫ kz = ( 2 /M a 2 )(1 − cos k z a) where a is the plane separation and the constant 2 /M a 2 is a measure of the bosonic Cooper pair hopping probability between planes. In the case of CuO 2 planes in cuprate superconductors a bos...

Boson Gas in a Periodic Array of Tubes

“…Although at very low (or zero) temperatures and/or high densities the interaction between particles cannot be neglected, we focus on an interactionless boson gas to study the effects of a periodic confining potential on the properties of the system. We show that this simple model captures qualitatively the properties of real systems, including the emergence of thermal phase transitions and/or dimensional crossovers 38,39 . In the following section we describe our system model.…”

“…As the separation between planes is lowered, the critical temperature reaches a P 0 -dependent minimum value and then it is expected 38,39 to increase again towards T 0 . For systems with a x,y > λ 0 the numerical calculations for the critical temperature and specific heat are very sensitive to the number of energy bands considered.…”

Dimensional crossover of a boson gas in multilayers

“…In Ref. [15] it was shown that a layered structure with plane separation a is revealed in the specific heat behavior as a function of temperature. For closely separated planes (a λ 0 ≡ h/ √ 2πmk B T 0 ), a minimum at T ≡ T min whose corresponding thermal wavelength scales with a as λ 2a is found in the C V vs T curves, where T 0 is the ideal gas BEC critical temperature.…”

Trapping Effect of Periodic Structures on the Thermodynamic Properties of a Fermi Gas

“…(1) for has been extensively analized in Refs. 13,14 and 15 , where the allowed and forbidden energy-band structure is shown, and the importance of taking the full band spectrum has been demonstrated.…”