Abstract
From femtosecond spectroscopy (fs-spectroscopy) of metals, electrons and phonons reequilibrate nearly independently, which contrasts with models of heat transfer at ordinary temperatures ([Formula: see text] K). These electronic transfer models only agree with thermal conductivity [Formula: see text] data at a single temperature, but do not agree with thermal diffusivity [Formula: see text] data. To address the discrepancies, which are important to problems in solid state physics, we separately measured electronic (ele) and phononic (lat) components of [Formula: see text] in many metals and alloys over [Formula: see text]290–1100 K by varying measurement duration and sample length in laser-flash experiments. These mechanisms produce distinct diffusive responses in temperature versus time acquisitions because carrier speeds [Formula: see text] and heat capacities [Formula: see text] differ greatly. Electronic transport of heat only operates for a brief time after heat is applied because [Formula: see text] is high. High [Formula: see text] is associated with moderate [Formula: see text], long lengths, low electrical resistivity, and loss of ferromagnetism. Relationships of [Formula: see text] and [Formula: see text] with physical properties support our assignments. Although [Formula: see text] reaches [Formula: see text] near 470 K, it is transient. Combining previous data on [Formula: see text] with each [Formula: see text] provides mean free paths and lifetimes that are consistent with [Formula: see text] K fs-spectroscopy, and new values at high [Formula: see text]. Our findings are consistent with nearly-free electrons absorbing and transmitting a small fraction of the incoming heat, whereas phonons absorb and transmit the majority. We model time-dependent, parallel heat transfer under adiabatic conditions which is one-dimensional in solids, as required by thermodynamic law. For noninteracting mechanisms, [Formula: see text]. For metals, this reduces to [Formula: see text] above [Formula: see text]20 K, consistent with our measurements, and shows that Meissner’s equation [Formula: see text] is invalid above [Formula: see text]20 K. For one mechanism with multiple, interacting carriers, [Formula: see text]. Thus, certain dynamic behaviors of electrons and phonons in metals have been misunderstood. Implications for theoretical models and technological advancements are briefly discussed.
Highlights
In metals and alloys, heat can be conducted by both electrons and lattice vibrations.[1,2,3] Because electron–phonon interactions occur when heat is conducted through a metal, as evidenced in thermoelectric power[4] as well as during resistance heating, studying this phenomenon has played an important role in historic and modern investigations of the physics of metals.[3,4,5] Electrons have a very low heat capacity but high speeds which are thought to compensate during heat transfer
For many different metallic elements and alloys, over some range of the temperatures explored, we observed a small rise overlapping with or immediately following the laser pulse (Fig. 6). Because such rise characteristics are expected for fast electron transport, and because distinct signals for electrons and phonons should be resolvable (Sec. 2.2), we proceeded to measure many different metals and alloys at varying lengths (Table 1; Sec. 3.1) to understand this phenomenon. Both thermal rises are often observed in short samples [e.g., Figs. 6 and 8(a)], quantifying thermal diffusivity associated with the rapid rise generally requires longer samples and shorter collection times than are conventionally applied to metals (Figs. 8–10)
We quantified electron behavior in these metals within uncertainties set by the limitations of our equipment, which largely result from the low heat capacity and high speed inherent to the electron
Summary
Heat can be conducted by both electrons and lattice vibrations.[1,2,3] Because electron–phonon interactions occur when heat is conducted through a metal, as evidenced in thermoelectric power[4] as well as during resistance heating, studying this phenomenon has played an important role in historic and modern investigations of the physics of metals.[3,4,5] Electrons have a very low heat capacity but high speeds (near the Fermi velocity) which are thought to compensate during heat transfer Both properties stem from the Pauli exclusion principle, which only allows electrons near the Fermi energy to be promoted upon receipt of small amounts of heat energy.[4,6,7] Phonons, being bosons, are not subject to this restriction, and contain the majority of the heat content of the metal, but move comparatively slowly, similar to sound speeds.[8] Electronic and phononic processes should be largely independent.[2,3,4,5,6,7,8] Ultrafast optical spectroscopic experiments on thin metal films[9] and surfaces (e.g., Ref. 10) confirm this deduction. Of particular importance is that thermal fields associated with each particle type can differ in accordance to their statistics: for example, in response to high frequency, high intensity laser pulses, electrons are immediately promoted to high energy levels, they thermalize to a Fermi distribution which may be ∼500 K different from ambient.[15,16] The electrons may experience magnetic interactions, before reequilibrating with the phonon gas.[11,13,14,26] After presenting our results (Sec. 4), conditions under which Eq (10) is valid will be stipulated (Sec. 5)
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