### Exciton-polariton Bose-Einstein condensate at room temperature and heff hierarchy

Ulla gave in my blog a link to a very interesting work about Bose-Einstein condensation of quasi-particles known as exciton-polaritons. The popular article tells about a research article published in Nature by IBM scientists.

Bose-Einstein condensation happens for exciton-polaritons at room temperature, this temperature is four orders of magnitude higher than the corresponding temperature for crystals. This puts bells ringing. Could heff/h=n be involved?

One learns from Wikipedia that exciton-polaritons are electron hole pairs- photons kick electron to higher energy state and exciton is created.These quasiparticles would form a Bose-Einstein condensate with large number of particles in ground state. The critical temperature corresponds to the divergence of Boltzmann factor given by Bose-Einstein statistics.

1. The energy of excitons must be of order thermal energy at room temperature: IR photons are in question. Membrane potential happens to corresponds to this energy. That the material is organic, might be of relevance. Living matter involves various Bose-Einstein condensate and one can consider also excitons.

As noticed the critical temperature is surprisingly high. For crystal BECs it is of order .01 K. Now by a factor 30,000 times higher!

2. Does the large value of heff =n×h visible make the critical temperature so high?

Here I must look at Wikipedia for BEC of quasiparticles. Unfortunately the formula for n1/3 is copied from source and contains several errors. Dimensions are completely wrong.

It should read n1/3= (ℏ)-1 (meffkTcr)x, x= 1/2.

[not x=-1/2 and 1/ℏ rather than ℏ as in Wikipedia formula. This is usual: it would important to have Wikipedia contributors who understand at least something about what they are copying from various sources].

3. The correct formula for critical temperature Tcr reads as

Tcr= (dn/dV)y2/meff , y=2/3.

[Tcr replaces Tc and y=2/3 replaces y=2 in Wikipedia formula. Note that in Wikipedia formula dn/dV is denoted by n reserved now for heff=n×h].

4. In TGD one can generalize by replacing ℏ with ℏeff=n ×ℏ so that one has

Tcr→ n2Tcr .

Critical temperature would behave like n2 and the high critical temperature (room temperature) could be understood. In crystals the critical temperature is very low but in organic matter a large value of n≈ 100 could change the situation. n≈ 100 would scale up the atomic scale of 1 Angstrom as a coherence length of valence electron orbitals to cell membrane thickness about 10 nm. There would be one dark electron-hole pair per volume taken by dark valence electron: this would look reasonable.

One must consider also the conservative option n=1. Tcr is also proportional to (dn/dV)2, where dn/dV is the density of excitons and to the inverse of the effective mass meff. meff must be of order electron mass so that the density dn/dV or n is the critical parameter. In standard physics so high a critical temperature would require either large density dn/dV about factor 106 higher than in crystals.

Is this possible?

1. Fermi energy E is given by almost identical formula but with factor 1/2 appearing on the right hand side. Using the density dne/dV for electrons instead of dn/dV gives an upper bound for Tcr ≤ 2EF. EF varies in the range 2-10 eV. The actual values of Tcr in crystals is of order 10-6 eV so that the density of quasi particles must be very small for crystals: dncryst/dV≈ 10-9dne/dV .
2. For crystal the size scale Lcryst of the volume taken by quasiparticle would be 10-3 times larger than that taken by electron, which varies in the range 101/3-102/3 Angstroms giving the range (220-460) nm for Lcryst.
3. On the other hand, the thickness of the plastic layer is Llayer= 35 nm, roughly 10 times smaller than Lcryst. One can argue that Lplast ≈ Llayer is a natural order of magnitude for Lcryst for quasiparticle in plastic layer. If so, the density of quasiparticles is roughly 103 times higher than for crystals. The (dn/dV)2-proportionality of Tcr would give the factor Tcr,plast≈ 106 Tcr,cryst so that there would be no need for non-standard value of heff!

But is the assumption Lplast ≈ Llayer really justified in standard physics framework? Why this would be the case? What would make the dirty plastic different from super pure crystal?

The question which option is correct remains open: conservative would of course argue that the now-new-physics option is correct and might be right.

For background see the chapter Criticality and dark matter.