Spectral invariants

Bose-Einstein condensation of excitons. In contrast to electrons and electron holes which are fermions, their combination to excitons renders these bosons as it was first proposed for helium-4 to explain its observed superfluidity at the $\lambda$-point (2,17K) of liquid helium. This condensation occurs when all atomic particles fall into the same, lowest quantum state. Excitons should therefore also condensate, as proposed by Böer and colleagues in 1961, assuming a separate thermodynamic phase when the main distance between the excitons reaches the de Broglie wavelength ${\lambda }_{dB}$:

or estimating the critical temperature Tcr for exciton condensation as Template:NumBlk with Template:NumBlk

Here $n$ is the density of excitons, $m_{\mathit{e}\mathit{f}\mathit{f}}$ their effective mass, and $g$ the optical generation rate of excitons, $\hslash =h/2\pi$, and $k$ the conventional Planck and Boltzmann constant. When entering the values of the tabulated constants $\hslash =1.0547\times 10^{-34}(W{s}^2)$; Template:Pad $k=1.381\times 10^{-23}(Ws/K)$ and assuming the exciton mass of the order of the electron mass $m_{\mathit{e}\mathit{f}\mathit{f}}=9\times 10^{-35}(Wc{m}^3{s}^{-2})$ one estimates from (Template:EquationNote) and (Template:EquationNote) $T_{cr}=(g\tau {)}^{3/2}{\hslash }^2/(k{m}_{\mathit{e}\mathit{f}\mathit{f}})=(g\tau {)}^{3/2}\times 10^{-11}$ that means that for a critical temperature of $0.01K$ one needs $g\tau =10^6$ that can be achieved with a reasonable generation rate of $10^5$ by pumping with a tunable laser at the exciton wavelength, and a life time of 10 seconds in an ultrapure single crystal of $C{u}_{2}0$.

Condensate excitons are superfluid and will not show any interaction with phonons while the normal exciton absorption is broadened by phonons, in a superfluid, this interaction disappears and the optical absorption degenerates to a line. An experimental observation must await better equipment to reach lower temperatures and tunable lasers that can be used for pumping.

More detailed calculations were proposed by J. Keldysh and later by D. Snoke et al starting a large number of experimental verifications.

See also

• Bose–Einstein condensate

References

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