Hot electrons in low-dimensional phonon systems

A simple bulk model of electron-phonon coupling in metals has been partially successful in explaining experiments on metal films that actually involve surface- or other low-dimensional phonons. However, by an exact application of this standard model to a semi-infinite substrate with a free surface, making use of the actual vibrational modes of the substrate, we show that such agreement is fortuitous, and that the model actually predicts a low-temperature crossover from the familiar $T^5$ temperature dependence to a stronger $T^6\log T$ scaling. Comparison with existing experiments suggests a widespread breakdown of the standard model of electron-phonon coupling in metals.

Figure 1

(Inset) Conducting film of thickness $d$ attached to insulator. The top surface of the metal is stress free. (Main) Temperature dependence of the thermal power exponent $x$ for a $10\mathrm{nm}$ (solid curve) and $100\mathrm{nm}$ (dashed curve) Cu film.

Figure 2

$\Phi \left(y\right)$ function.

Figure 3

One of the eight circular patches centered on the ⟨111⟩ and equivalent directions of the Cu Fermi surface. The shaded region represents a section of a cylindrical neck of radius $r_a$ that crosses a hexagonal Brillouin zone boundary. In our Fermi surface model, the Fermi energy changes linearly from $v_a=0.40$ to $v_b=0.75$ times the free-electron value in the annular region $r_a<r<r_b$, and remains 0.75 beyond $r_b$.