Stability of Bose Einstein condensates of hot magnons in YIG
We investigate the stability of the recently discovered room temperature Bose-Einstein condensate (BEC) of magnons in Ytrrium Iron Garnet (YIG) films. We show that magnon-magnon interactions depend strongly on the external field orientation, and that the BEC in current experiments is actually metastable - it only survives because of finite size effects, and because the BEC density is very low. On the other hand a strong field applied perpendicular to the sample plane leads to a repulsive magnon-magnon interaction; we predict that a high-density magnon BEC can then be formed in this perpendicular field geometry.
In a remarkable very recent discovery demo06 (), a Room Temperature Bose-Einstein condensate (BEC) of magnon excitations was stabilized for a period of roughly in a thin slab of the well-known insulating magnet Yttrium Iron Garnet (YIG). The density of magnons was quite low ( per lattice site), and the density of the BEC was apparently unknown, but . This result can be understood naively in terms of a weakly-interacting dilute gas of bosons, provided that (i) one assumes that the number of magnons is conserved, so their chemical potential may be non-zero, and (ii) the interactions between them are repulsive (attractive interactions favour depletion of the BEC, causing a negative compressibility and instability of the BEC BEC ()). In the experiment, the magnon dispersion was controlled both by the sample geometry and external magnetic field, in such a way that magnon-magnon collisions conserved magnon number - the decay of the BEC was then attributed to spin-phonon couplings demo06 (). The magnons in the experiment had rather long wavelengths, of order a m - hitherto such excitations have been treated entirely classically walker ().
This experiment raises a number of important questions, not least of which concern the kind of superfluid properties possessed by a BEC of such unusual objects. However a much more basic question about the stability of the system must first be answered. In fact we find the rather startling result that under the conditions of the experiments reported so far, these interactions were actually attractive, ie., the BEC ought to be unstable! We shall see that because of the particular geometry used, the BEC is actually metastable to thermally activated or tunneling decay, and that it only survives because its density is low - above a critical density, given below, it is absolutely unstable. However we also find that by changing the field configuration in the system one can make the interactions repulsive, and the BEC should then stabilize at a much higher density - opening the way to much more interesting experiments.
YIG is one of the best characterized of all insulating magnets YIG (). It is cubic, with lattice constant , ordering ferrimagnetically below . At room temperature the long-wavelength properties can be understood using a Hamiltonian with ferromagnetic exchange interactions between effective ’block spins’ , one per unit cell, whose magnitude , with , is defined by the experimental saturated magnetisation density ( at room temperature; with , one has ), along with dipolar couplings between these; the resulting lattice Hamiltonian takes the form:
where the sums , are taken over lattice sites at positions , etc., denotes nearest-neighbor spins, , and . The nearest-neighbour dipolar interaction . The isotropic exchange is determined experimentally from at room temperature. One then has , and .
In what follows we set up a theoretical description of the BEC, taking into account the external field, dipolar and exchange interactions, and boundary conditions in the finite geometry. We evaluate the interactions and the BEC stability for 2 different field configurations; the general picture then becomes clear.
(i) In-plane field: All experiments so far have had in the slab plane. The combination of exchange, dipolar, and Zeeman couplings then leads to a magnon spectrum shown in Fig.1, in which the competition between dipolar and exchange interactions leads to a finite- minimum in at a wave-vector , where is the slab thickness. To completely specify and the inter-magnon interactions one needs boundary conditions, which can involve partial pinning of the surface spins rado59 (); patton (); KALSLAV (). Demokritov et al demo06 () assume free surface spins, implying that (i) when is at the surface; here is the normal to the surface, and (ii) that the allowed momenta along (see Fig.1, inset) are , leading to different magnon branches labelled by . For now we assume a continuous in-plane momentum, and later discuss the effect of in-plane quantization; and we assume , taking the lowest-energy magnon branch.
here is the internal field, the demagnetization factor (for a slab in the -plane, , ), (for YIG, at room temperature), and . The form of the dimensionless function depends on the direction of . In the important case when is parallel to , ie., along , one has
In the experiment demo06 () magnons are argued to condense at the minima ; when , and , one has .
We now set up a theoretical description of the BEC, including all 4-magnon scattering processes (3-magnon scattering is excluded by the kinematics), using a generalized Bogoliubov quasi-average technique Belyaev58 () to incorporate the BEC. Defining magnon operators , a magnon BEC at , with condensed magnons, has quasi-averages
corresponding to a condensate wave-function . More generally is multiplied by a phase factor , which is crucial to the BEC dynamics, but not necessary for a stability analysis of the BEC.
We make a Holstein-Primakoff magnon expansion HP40 () up to 4th-order in magnon operators, including contributions from both (2 in - 2 out) and (3 in - 1 out) magnon scattering processes sparks61 () (we ignore multiple-scattering contributions here, which are relative to the leading terms). Then, taking quasi-averages, we can write the Hamiltonian in the form
where the interaction term takes the form
Here and are the four-magnon scattering amplitudes between states and respectively. For the sample geometry in Fig.1, with , these scattering amplitudes are found to be
with , and , where and and are given by and respectively. Higher-order multiple-scattering contributions to are relative to the leading terms given here.
We plot the combinations in Figures 2 and 3. Both these amplitudes are sensitive to the external field and the film thickness. The first amplitude, , becomes negative in the entire region of fields at . The second amplitude is positive at .
Near the energy minimum (when ), one has , and the Bogoliuibov transformation is straightforward because is symmetric when and . The spectrum thus has 4 branches, with excited quasiparticle energies
where , and .
The inset in Fig.3 shows the ”phase diagram” of the quasi-two dimensional YIG for different values of and . One immediately sees a paradox: the amplitudes are never both positive, so bulk BEC should not exist - yet BEC has been observed demo06 () in samples with an in-plane field , in which .
The paradox is resolved by noting that in a finite geometry, the energy gap from the condensate to excited modes can be larger than the scattering amplitude, leading to a potential barrier to decay. For weakly-interacting Bose gases this yields BECmeta () an upper critical number of condensate particles, above which the barrier disappears; one has , where is the s-wave scattering length, and the characteristic size of the BEC wave function. The constant , and depends on the sample geometry.
In the present case we can write the critical density as , where is the minimum quasiparticle energy in the presence of the BEC; below this critical density the BEC is metastable to tunneling or thermal activation. If the BEC were to spread through the entire slab, then , where the length depends on the direction of the field relative to the slab axes. Taking this result literally for the experiment demo06 (), with a slab measuring in the plane, one finds (for fields along the long and short sides of the slab respectively). However this result is certainly too low, since it assumes a perfectly uniform BEC - in reality disorder and edge effects will smear the magnon spectrum and restrict the size of the BEC, and a more realistic estimate for is then . This yields for the experiment. Thus we conclude that for in-plane fields, it will be impossible to raise above this value; this could be checked experimentally (eg., by increasing the pumping rate).
(ii) Perpendicular field: Now the results are very different. Consider first an infinite thin slab, which is simple to analyze. The competition between the external field and the demagnetization field (which favours in-plane magnetisation) gradually pulls the spins out of the plane; below a critical field , the Free Energy is degenerate with respect to rotation around and so the magnons are gapless, but at they align with and a gap opens up. The minimum in the magnon spectrum is always at (Fig.4).
The inter-magnon scattering amplitude is now always positive; neglecting a very small exchange contribution one finds
where is the momentum in the plane, and takes the form (3) but with . This radically changes the situation - now a BEC is stable with no restriction on the condensate density. There are however restrictions on ; when the system has a “kinetic instability” Melkov (), in which the pumping of the magnons at one frequency destabilizes the magnon distribution, along with strong microwave emission.
In a real finite sample things are much more complicated. Even without surface anisotropy the spins near the slab edge are put out of alignment with the bulk spins by edge demagnetisation fields; and surface anisotropy does the same to spins on the slab faces. However in the central region of the sample, at distances further from the surface than the exchange length, the spectrum returns to the infinite plane form. For fields well above , eg., for , all of the spins will be aligned along , and (10) will then be valid everywhere.
In this case we have the striking result that a BEC of pumped magnons should be possible with densities much higher than present. To give an upper bound is complicated since the problem then becomes essentially non-perturbative (similar to, eg., liquid ), beyond the range of higher-order magnon expansions. However there appears to be no obstacle in principle to raising . At present the highest achievable density is probably limited by experimental pumping strengths rather than any fundamental restrictions. Such a high-density BEC existing at room temperature would be extremely interesting, and certainly possess unusual magnetic properties.
Remarks: The 2 cases studied above are actually limiting cases of a more general situation in which one can manipulate the inter-magnon interactions by varying the field direction and strength, and vary the upper critical density for BEC formation by changing the sample geometry. Thus the analysis can be easily generalized to long ’magnetic wires’ or whiskers, and we also expect that BEC will be stabilized there when the external field is perpendicular to the sample axis, but unstable or metastable when the field is parallel to the sample axis. Further details of the various possible cases will be published elsewhere.
Acknowledgements: We acknowledge support by the Pacific Institute of Theoretical Physics, the National Science and Engineering Council of Canada, the Canadian Institute for Advanced Research, the Louisiana Board of Regents (Contract No. LEQSF (2005-08)-RD-A-29), the Tulane University Research and Enhancement Fund, and the US Air Force Office of Scientific Research (Award no. FA 9550-06-1-0110). We would also like to thank B. Heinrich and D. Uskov for very useful conversations.
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