Crossover temperature in electron–phonon heat exchange in layered nanostructures

Abstract

We study theoretically the effect of the effective dimensionality of the phonon gas distribution on the heat exchange between electrons and phonons in layered nanostructures. If we denote the electrons temperature by Te and the phonons temperature by T ph , then the total heat power P is proportional-in general-to T x e −T x ph , the exponent x being dependent on the effective dimensionality of the phonon gas distribution. If we vary the temperature in a wide enough range, the effective dimensionality of the phonon gas distribution changes going through a crossover around some temperature, TC . These changes are reflected by a change in x. On one hand, in a temperature range well below a crossover temperature TC only the lowest branches of the phonon modes are excited.They form a (quasi) two-dimensional gas, with x = 3.5. On the other hand, well above TC , the phonon gas distribution is quasi three-dimensional and one would expect to recover the three dimensional results, with x = 5. But this is not the case in our layered structure. The exponent x has a complicated, non-monotonous dependence on temperature forming a "plateau region" just after the crossover temperature range, with x between 4.5 and 5. After the plateau region, x decreases, reaching values between 3.5 and 4 at the highest temperature used in our numerical calculations, which is more than 40 times higher than TC .

Modification of electron-phonon coupling by micromachining and suspension

“…In the numerical calculation of Ref. 12, the local power-law exponent ∂ lnQ ∂ ln T el is found to lie between 4.5 and 4.7 for T /T C = 2 . .…”

“…A negative P 0 indicates that the background heating was higher in the phonon temperature measurement configuration compared to the electron temperature measurement. The theoretical work that most closely models our experimental system is that by Anghel and coworkers 12,28,29 , wherein they consider a sheet consisting of finite-thickness metal and dielectric layers. In Ref.…”

“…In Ref. 12 in particular, they extend their analysis to cover the range of temperatures where the phonon dimensionality changes from two to three. A key parameter of their model is the crossover temperature, for which they give the homogeneous media approximation T C ≈ c t /(2k B d), where c t is the transversal (shear wave) speed of sound, and d is the total thickness of the membrane.…”

“…Our main finding is a modified electron-phonon heat flowQ ∝ T n el −T n ph with n ≈ 4.6, where T el and T ph denote the electron and phonon temperatures, respectively. Very recent theoretical analysis 12 of the metal-dielectric bilayer system has predicted a "plateau region" where 4.5 < n < 5 over a wide range of temperatures where the phonon system undergoes a 2D to 3D transition. Our work appears to confirm this prediction.…”

Electron-phonon coupling in copper intercalated Bi$_{2}$Se$_{3}$

“…The power law k is sensitive to the dimensionality of the sample [19], disorder [12,15,20], type of phonons [12], screening [21] and chirality [22] of charge carriers. Here we distinguish the heat flux mechanism, which is determined by the transverse phonons at T dis T T BG [14],…”