Thermal conductivity in higher-order generalized hydrodynamics: Characterization of nanowires of silicon and gallium nitride

doi:10.1016/j.physe.2014.01.031

Physica E: Low-dimensional Systems and Nanostructures

Volume 60, June 2014, Pages 50-58

https://doi.org/10.1016/j.physe.2014.01.031 Get rights and content

Highlights

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It is specified a cylindrical geometry in samples of Si and GaN in the macro to nano-scales.
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The study of the thermal conductivity and its influence on the figure-of-merit in thermoelectric devices.
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The thermal conductivity is strongly dependent on the cylinder radius.

Abstract

An analysis of the influence of geometry and size on the thermal conductivity in semiconductors, particularized to the study in Si and GaN, is presented. This is done in the framework of a higher-order generalized hydrodynamics (HOGH) of phonons in semiconductors, driven away from equilibrium by external sources. This HOGH is derived by the method of moments from a generalized Peierls–Boltzmann kinetic equation built in the framework of a Non-Equilibrium Statistical Ensemble Formalism. We consider the case of wires (cylindrical geometry) exploring the effect of size (radius), particularly in the nanometric scale when comparison with experiment is done. Maxwell times, which are quite relevant to define the hydrodynamic movement, are evidenced and characterized.

Graphical abstract

The thermal conductivity is strongly dependent on the ratio of the cylinder radius R to a characteristic length ℓ, the latter given approximately by the velocity of sound times a kind of relaxation time (energy Maxwell time) which has values in the order of hundreds of nanometers.

Introduction

It has been noticed [1] that the ceaseless innovation in semiconductor design creates a demand for a better understanding of the physical processes involved in materials with constrained geometries and functioning in far-from-equilibrium conditions. A particular question is the one of thermal transport in small semiconductor devices [2] and data centers [3], used in refrigeration processes in microprocessors [4], [5]. The heat generated by silicon chips in integrated circuits must be efficiently removed once the performance of modern electronic devices degrades as the temperature increases. One approach for providing active cooling in chips consists in the use of thermo-electric materials, which effectively transport heat via charge-current flow [6].

These questions belong to the area of nonequilibrium phonon dynamics [7] or, more precisely, to the subject of phonon hydro-thermodynamics (that is, hydrodynamics associated to nonequilibrium “irreversible” thermodynamics) [8], [9], [10]. The hydro-thermodynamics of phonons, driven away from equilibrium by external sources, is built resorting to a theory, the one used in the present work, which, for the sake of completeness is summarized in the next section. It is based a nonlinear quantum kinetic theory [11], [12], [13] built on the basis of a Non-Equilibrium Statistical Ensemble Formalism (NESEF for short) [14], [15], [16], [17]. It follows from the solution via the moments method of a generalized NESEF-based Peierls–Boltzmann kinetic equation for the single-phonon distribution function, to obtain such higher-order phonon hydro-thermodynamics. In a contracted description, meaning using the one of order 1, the solution of the evolution equations of the hydrodynamic motion is obtained; this is described in Section 3. In Section 4 we consider a particular constrained geometry and the thermal conductivity in nanowires is analyzed and compared with experiments. Maxwell times, which are quite relevant to the definition of the hydrodynamic motion, are evidenced and characterized.

Section snippets

Summary of a phonon mesoscopic hydro-thermodynamics

We consider a system of longitudinal acoustic la phonons in a semiconductor in anharmonic interaction with the accompanying transverse acoustic ta phonons. The sample is in contact with a thermostat at temperature T₀. An external pumping source drives the la phonon system out of equilibrium. The system is characterized at the microscopic level by the Hamiltonian $\hat{H} = {\hat{H}}_{OS} + {\hat{H}}_{OB} + {\hat{H}}_{SB} + {\hat{H}}_{SP},$ which consists of the Hamiltonian of the free la phonons ${\hat{H}}_{OS} = \sum_{q} ℏ ω_{q} (a_{q}^{†} a_{q} + 1 / 2),$ where $ω_{q}$ is the frequency

Phonon mesoscopic hydro-thermodynamics of order 1

First we notice that in the evolution equations the two families in the sets of Eqs. (8), (13) are coupled by cross-terms that account for thermo-striction effects. In cases where these effects are not particularly relevant they can be disregarded and we obtain two independent sets of evolution equations, one for the n-family and the other for the h-family.

For dealing with the kind of experiments on heat transport in semiconductors in the conditions of constrained geometries we are considering,

The case of constrained geometries

We consider now the effect of geometry and boundary conditions on the thermal conductivity of the sample. Let us consider the steady state when Eq. (41) becomes $[1 + ℓ^{2 [2]} \otimes [\nabla : \nabla]] I_{h} (r) = M^{[2]} \nabla J_{hS} (r),$ where $M^{[2]} = θ_{h} θ_{I} (A_{h}^{[2]} - b_{L}^{[2]}),$

First we notice that Eq. (40) tells us that $I_{h} (r, t) = - θ_{I} (A_{h}^{[2]} (t) - b_{L}^{[2]} (t)) \nabla h (r, t) + θ_{I} J_{IS} (r, t),$ where the right side has the role of a thermodynamic force that drives the flux. On the other hand, using Eqs. (37), (14) we arrive at $\nabla h (r, t) = \sum_{q} ℏ ω_{q} ν_{q} (r, t) [ν_{q} (r, t) + 1] \nabla F_{q} (r, t),$ and taking

Heat conductivity in nanowires

The results derived in the previous section are applied to the analysis of heat conductivity in nanowires, that is cylinders with a small radius (nanometer scale) meaning $R / ℓ ≲ 1$ . In this case the mode frequency $ω_{q}$ is dependent on an integer index (for the motion in the constrained perpendicular circular plane) and a continuous index q_z (for the motion in the longitudinal direction). Introducing the approximate expression, written in a Debye model, $ω_{nqz} \approx s \sqrt{q_{z}^{2} + {(π n / R)}^{2}}$ , and the sum over q consists

Final remarks

Present day advances in technology and the associated industrial processes require improvements in the theory of several areas of condensed matter physics [46]. We may mention hydrodynamic processes that do not fall within the domain of the standard (Onsagerian) linear hydrodynamics which applies to movements characterized by long wavelengths in space and low frequencies in time, linear relationship between fluxes and thermodynamic forces, and weak fluctuations. This is, particularly the case

Acknowledgments

The authors would like to acknowledge partial financial support received from the São Paulo State Research Agency (FAPESP), Goiás State Research Agency (FAPEG) and the Brazilian National Research Council (CNPq): The authors are CNPq Research Fellows.

In Memoriam: We very much regret to report the deceased of our dear colleague Professor Dr. Áurea Rosas Vasconcellos, a genuine, devoted and extremely competent Teacher and Researcher with fervent dedication to Theoretical Physics in the Condensed

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