Magnetic Field-Induced Giant Enhancement of Electron-Phonon Energy Transfer in Strongly Disordered Conductors

Abstract

Relaxation of soft modes (e.g. charge density in gated semiconductor heterostructures, spin density in the presence of magnetic field) slowed down by disorder may lead to giant enhancement of energy transfer (cooling power) between overheated electrons and phonons at low bath temperature. We show that in strongly disordered systems with time-reversal symmetry broken by external or intrinsic exchange magnetic field the cooling power can be greatly enhanced. The enhancement factor as large as 10 2 at magnetic field B ∼ 10 Tesla in 2D InSb films is predicted. A similar enhancement is found for the ultrasound attenuation.Introduction-A number of recent experiments show that energy transfer (the cooling power) J (T e , T ph ) = J(T el ) − J(T ph ) between overheated electrons with temperature T el > T ph and phonons at low bath temperature T ph may vary by several orders of magnitude when measured per one electron per volume. The out-flux J(T ) = W T p may have different power-law temperature dependence with the exponent p both smaller and larger than the classical result p = 5 valid for pure metals. In disordered metals with complete screening of Coulomb interaction and impurities that are fully involved in the lattice motion one expects [1-3] a power law with p = 6 which corresponds to weaker energy transfer compared to the clean case. This is related with the "Pippard ineffectiveness condition" (denoted as PIC below) [4, 5] formulated for the rate of inelastic electron-phonon scattering. A very accurate experiments in metal films of Hf and Ti [6] confirmed this theoretical expectation, including the value of the pre-factor W in front of T 6 . At the same time, experiments on heavily doped Si [7] which also demonstrated the T 6 behavior, gave at low temperatures the value W/n e (n e is the carrier density) larger by a factor of 10 3 than the theoretical prediction in Ref. [1][2][3]. Surprisingly, the T 6 behavior of the cooling rate with approximately the same values of W/n e as in Ref. [7] were extracted from the recent experiments [8] on amorphous InO films showing weakly insulating behavior in magnetic field of 11T. In this case W/n e was larger by a factor of 5 × 10 4 than the theoretical prediction for a dirty metal approaching the Anderson transition. Clearly, neither of the above cases with anomalously large cooling rate correspond to the piezoelectric type of electron-phonon coupling where the PIC does not hold and the theory predicts T 4 temperature behavior of cooling rate [9,10]. It is also dubious that the model of impurities which are only partially involved in the lattice motion [2] that also leads to enhanced cooling rate with T 4 temperature behavior, is realistic for the cases in question. Thus there was a quest from experiment for a different and more general mechanism of enhancement of cooling rate in strongly disordered conductors.

Evidence for a Finite-Temperature Insulator

“…We followed this theoretical work with a systematic study and obtained a good agreement 30 . We were also able to estimate the T dependence of the e - ph scattering rate, τ e − ph , on the high B side of the insulating peak and found a rather strong dependence of τ e − ph ∼ T −4 , which is in agreement with the modified dirty metal model 29 31 . The success of this theoretical description provides an essential indication that, in our regime of measurements, the electrons are decoupled from the phonons.…”

Electron cooling by phonons in superconducting proximity structures

“…As discussed in Refs. [12,13,[18][19][20], the electronphonon interaction in disordered systems is most conveniently described in a co-moving reference frame, i. e., attached to the oscillating ions of the crystal lattice, since the impurities oscillate together with the lattice. The electron-phonon interaction Hamiltonian is assumed to have the form…”

“…The energy exchange between electrons and phonons has also been studied in bulk BCS superconductors [13,[18][19][20] and superconducting proximity structures [21]. The presence of a gap in the quasiparticle spectrum leads to a significant suppression of the cooling power at low temperatures and makes these systems advantageous for quantum calorimetry applications.…”

Collective modes and ultrasonic attenuation in a pseudogapped superconductor

“…Namely, we consider attenuation rate α(ω, T ) of ultrasound wave propagating in a superconducting material with a well developed pseudogap ∆ P ≫ ∆, at the temperature range T ≪ ∆ k B and at relatively low frequencies ̵ hω ≪ k B T (below the Boltzmann's constant k B is set to unity). In the paper 23 we have shown that ultrasound attenuation rate is intrinsically related to the electron-phonon inelastic energy exchange, which becomes very inefficient at sub-Kelvin temperatures, leading to thermal instabilities and electron overheating 24 in a "neighbouring" insulating state of the same (or similar) materials 5,25,26 . We identified in Ref.…”