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begin{abstract}
In this module we derive the quantum of thermal conductance, and along the way, concepts related to phonon density of states and spatial dimensionality are discussed and applied. The end result is a simple expression for how much heat a given acoustic phonon branch, or polarization, can carry between two isothermal reservoirs divided by the temperature difference between the reservoirs.

This module will be posted as a wiki page on thermalHUB.org as soon as the wiki interface is complete (hopefully within one month).
end{abstract}
maketitle
% ----------------------------------------------------------------
section{Energy of a Collection of Phonons}
The total energy $U$ of a set of phonons in a solid at equilibrium can be expressed as the sum over all allowable wavevectors ($textbf{K}$) and polarizations ($p$) as (see cite[p. 452]{ashcroft76} and cite[p. 117]{kittel96}).


U = sum_textbf{K} sum_p
left( langle n_{textbf{K},p} rangle + frac{1}{2}right) hbar omega_{textbf{K},p}
label{eq:energy1}

where $langle n_{textbf{K},p} rangle$ represents number of occupied normal modes (or number of phonons) given by the Planck (or Bose-Einstein) distribution function, and each pair ($textbf{K},p$) represents a so-called normal mode (emph{i.e.}, a phonon) corresponding to a given wavevector (wavelength) and polarization. The Planck distribution is given by

begin{equation}
langle n_{textbf{K},p} rangle = left[expleft(frac{hbar omega_{textbf{K},p}}{k_B T}right) - 1 right]^{-1}
label{eq:planck}
end{equation}

Often, we are more concerned with the energy emph{density} ($u$), rather than the emph{total} energy ($U$), and this new quantity can be calculated simply by dividing Eq. ref{eq:energy1} by the system's `volume'. However, we emphasize that the present concept of `volume' depends on the dimensionality of the system, and this point often raises some confusion. Here, let us think of `volume' as length for a 1-D (quantum wire) system, as area for a 2-D (quantum well) system, and as the ordinary volume for a bulk solid.

Our interest is to study transport processes, and as such, the double summation in Eq. ref{eq:energy1} is cumbersome. In particular, we'd like to turn the summation over textbf{K}-space (emph{i.e.}, over all active phonon wavelengths and directions) into an integral. The summation over polarization branches (emph{e.g.}, longitudinal-acoustic, transverse-optical, etc.) normally must remain (or else fall victim to approximation). Therefore, we introduce the concept of density of states. This concept is used primarily to convert summations or integrals over textbf{K}-space to integrals over frequency. Kittel's textbook (chapter 5) does a particularly nice job of explaining this concept. Here, we will focus on 1-D systems.

In brief, each allowable normal mode in $textbf{K}$-space is separated by a distance $2pi/L$, where $L$ is a characteristic macroscopic length of the medium (in this case a quantum wire) containing the phonons. One subtlety of the foregoing development and its description in some textbooks is that $textbf{K}$ can take both positive and negative values, and therefore, if we define $K$ as the absolute magnitude of $textbf{K}$ (emph{i.e.}, $K = |textbf{K}|$), then there is one allowable wavevector for each increment of $pi/L$ in $K$-space (which, by definition, is strictly positive). In other words, the number of allowed phonon states in a 1-D region from 0 to $K$ (which, recall, is the absolute magnitude of $textbf{K}$) is

begin{equation}
N = frac{K}{pi/L}
label{eq:1D_number}
end{equation}
Once this number of allowed modes or phonons is known, the density of such `states' can be expressed as
begin{equation}
D_{1D}(K)= frac{dN}{dK} = frac{L}{pi}
label{eq:1D_density_k}
end{equation}

More often, the density of states is described with respect to the phonon frequency instead of wavevector. This transformation is made quite readily using the chain rule and the definition of phonon group velocity

begin{equation}
D_{1D}(omega) = frac{dN}{domega} = frac{dN}{dK} frac{dK}{domega} = frac{L}{v_gpi}
label{eq:1D_densityofstates}
end{equation}
where $v_g = domega / dK$ is the phonon group velocity and does, in general, depend on frequency (or wavevector).

Once we have established the density of states, we can convert the textbf{K}-space summation of Eq. ref{eq:energy1} to an integral in energy emph{density} form as
begin{eqnarray}
u =  frac{U}{V} & = & frac{1}{V} sum_{textbf{K}} sum_p left( langle n_{textbf{K},p} rangle + frac{1}{2}right) hbar omega_{textbf{K},p} nonumber \
 & = & frac{1}{V} sum_p int dtextbf{K} D(K) left( langle n_{textbf{K},p} rangle + frac{1}{2}right) hbar omega_{textbf{K},p} nonumber \
 & = & frac{1}{V} sum_p int_{0}^{infty}domega D(omega) left[ n(omega,T) + frac{1}{2} right] hbar omega
label{eq:energy_density_general}
end{eqnarray}
where we have used $n(omega, T) = langle n_{textbf{K},p} rangle$ to emphasize this function's dependence on frequency and temperature (see Eq. ref{eq:planck}). We have also dropped the textbf{K} and $p$ subscripts on $omega$, recognizing that the frequency of a phonon depends implicitly on its wavevector and polarization. We re-state that the `volume' $V$ can be a length, area, or regular volume, depending on the dimensionality of the system. The foregoing equations actually apply to any dimensionality. For the the 1D systems of interest here, the energy density (emph{i.e.}, energy per unit length) is
begin{eqnarray}
u =  frac{U}{L} & = & frac{1}{L} sum_p int_{0}^{infty}domega D_{1D}(omega) left[ n(omega,T) + frac{1}{2}right] hbar omega nonumber \
 & = & sum_p int_{0}^{infty}domega frac{1}{v_{g} pi} left[ n(omega,T) + frac{1}{2}right] hbar omega
label{eq:1D_energy_density}
end{eqnarray}
The foregoing equations give the phonon energy per unit length in a quantum wire.

section{Energy Flux}
The energy density derived above can be carried by phonons that move through the quantum wire with group velocity $v_g$. Noting that, at a given location on the wire, half of the phonons move leftward while the other half move rightward, we can write the leftward energy flux as
begin{eqnarray}
J_{Q,Rrightarrow L} & = & frac{1}{2} sum_p int_{0}^{infty}domega frac{1}{v_{g} pi} v_g left[ n(omega,T) + frac{1}{2}right] hbar omega nonumber \
 & = & frac{1}{2 pi} sum_p int_{0}^{infty}domega left[ n(omega,T) + frac{1}{2}right] hbar omega
end{eqnarray}
A similar expression would be used for the rightward heat flux. We note here that while heat flux normally has units of energy flow rate per unit area (emph{e.g.}, W/m$^2$), the units of $J_{Q,Rrightarrow L}$ above are Watts because the textbf{K}-space is one-dimensional for the quantum wire. For a general 3D system, the units of $u$ would be energy per volume (emph{e.g.}, W/m$^3$) and the energy flux would have the usual units of energy flow rate per area.

begin{figure}
%requires usepackage{graphicx}
  includegraphics[width=200pt]{quantum_cond_wire.eps}\
  caption{Schematic of a quantum wire with a device section between two thermal reservoirs}label{fig:quantum_wire}
end{figure}

Our primary purpose is to study the emph{net} flow of heat through a 1D wire whose ends are held at constant temperatures, as shown in Fig. ref{fig:quantum_wire}. citet{rego98} formulated the net flow of heat from right to left as
begin{equation}
J_{Q,net} = frac{1}{2 pi} sum_p int_{0}^{infty}domega left[ n(omega, T_R) - n(omega, T_L) right] hbar omega Xi_p (omega)
label{eq:1D_qnet}
end{equation}
where $Xi_p(omega)$ is the probability of transmission of a phonon with frequency $omega$ and polarization $p$.
section{Thermal Conductance}
Thermal conductance $h$ is defined as the ratio of heat flux to temperature difference. In the present context, it becomes the ratio of $J_{Q,net}$ in Eq. ref{eq:1D_qnet} to the temperature difference $Delta T = T_R - T_L$. For the simple case of only acoustic phonons (emph{i.e.}, those with zero frequency at zero wavevector), citet{rego98} showed that for perfect transmission (emph{i.e.}, $Xi_p(omega) = 1$) the integrals can be evaluated analytically, and the thermal conductance can be expressed as
begin{eqnarray}
h_{1D} & = & frac{1}{2 pi} sum_{p=1}^{n_p} int_{0}^{infty}domega left[ frac{n(omega, T_R) - n(omega, T_L)}{Delta T} right] hbar omega nonumber \
 & = & frac{k_{B}^{2} pi}{6 hbar} left( frac{T_R + T_L}{2} right) N_p
label{eq:1D_conductance}
end{eqnarray}
where $N_p$ is simply the number of active phonon polarizations. Therefore, if we assume a small temperature difference between the reservoirs and express their arithmetic average with the symbol $T$, the following equation defines the quantum of thermal conductance
begin{equation}
h_q = frac{k_{B}^{2} T pi}{6 hbar} left( = T times 9.462 times 10^{-13} textrm{W/K$^2$} right)
label{eq:quantum_cond}
end{equation}
This important result provides an upper limit on the amount of heat that can flow between two reservoirs connected by a quantum wire by means of a given acoustic phonon polarization. Notably, this upper limit exists even though we have assumed perfect transmission (emph{i.e.}, $Xi_p(omega) = 1$). In order to increase the thermal conductance, one would need to add more polarizations (citet{rego98} derive results for higher-energy, non-acoustic phonon modes as well) or increase the wire's cross-sectional area such that the effective textbf{K}-space becomes two-dimensional and more modes are available to traverse from one side to the other. For our immediate purposes, the purely 1D representation suffices and provides a convenient model system for the study of lattice dynamics and phonon transport.