In nanoscale thermal transport, how to accurately describe the heat flow of electrons or phonons between two reservoirs (thermal baths) from a microscopic perspective has become a research hotspot. Landauer theory was initially proposed to analyze electron transport in quantum wires, but later this concept was extended to bosonic systems like phonons, revealing the essence of transport in nano- or micro/nano-structures.
The core idea of Landauer theory in heat transport is: the transport process can be understood as quantum states being scattered or transmitted through a relatively short device region, and the upper limit of transport is determined by the number of available modes (transport channels) and the probability that these modes can transmit through the device (i.e., transmission function $\tau(\mathbf{k})$).
A simple physical picture is:
The heat flux density from the left to the right reservoir (unit: W/m$^{d-1}$) is generally written as:
\[J_{Q,L\to R}(T_1) = \frac{1}{L^d}\sum_p\sum_{\mathbf{k};k_x>0} v_{gx,p}(\mathbf{k}) \tau_p(\mathbf{k}) [E_{i,p}(\mathbf{k}) - \mu][f_i^0(E_{i,p}(\mathbf{k}), T_1) + c_0]\]where:
Similarly, the heat flow from right to left can be written, and the net heat flow is:
\[J_{Q,\text{net}} = \frac{1}{L^d}\sum_p\sum_{\mathbf{k};k_x>0} v_{gx,p} \tau_p [E_{i,p} - \mu][f_i^0(T_1) - f_i^0(T_2)]\]Note that the zero-point energy $c_0$ cancels out in the net heat flow and thus has no net contribution.
For better analysis, the k-space summation is often converted into an integral (Landauer integral). The forms differ with dimensionality:
Angle $\theta$ denotes the angle between the wave vector and the transport direction $x$. It’s often assumed that $\tau$ is direction-independent.
By introducing the density of states (DOS), the k-space integral can be converted into an integral over frequency $\omega$ or energy $E$.
Using frequency DOS $D(\omega)$ and $E = \hbar\omega$ with $\mu = 0$:
With nonzero chemical potential $\mu$ and energy DOS $D(E)$:
To combine quantitative analysis with intuitive understanding, the Landauer theory introduces the concept of number of modes.
This idea can also be used to analyze heat flow $Q$ (in watts), which is the product of heat flux density $J_Q$ and “cross-sectional area”:
For phonons, the general expression for heat flow (or thermal power) is:
\[Q_{\text{ph}} = \frac{1}{2\pi} \int_0^\infty M(\omega)\tau(\omega)\hbar\omega [f_\text{BE}^0(T_1) - f_\text{BE}^0(T_2)]\,d\omega\]where:
The number of modes varies with dimensionality:
By comparing the relations $J_Q(1D)=Q$, $J_Q(2D)=Q/W$, $J_Q(3D)=Q/A$ with the expressions for heat flux density, we can derive the relationship between mode density $M(\omega)$ and density of states $D(\omega)$:
\[M_{\text{1D}}(\omega) = 1 = \pi v_g(\omega) D_{\text{1D}}(\omega)\] \[M_{\text{2D}}(\omega) = \frac{K(\omega)}{\pi} = \pi \cdot \frac{2v_g(\omega)}{\pi} D_{\text{2D}}(\omega)\] \[M_{\text{3D}}(\omega) = \frac{K^2(\omega)}{4\pi} = \pi \cdot \frac{v_g(\omega)}{2} D_{\text{3D}}(\omega)\]Here, $v_g(\omega)$ is the group velocity, and $K(\omega)$ is the wave vector magnitude.
We know the phonon DOS is:
\[D_{\text{1D}}(\omega) = \frac{1}{\pi v_g(\omega)}\] \[D_{\text{2D}}(\omega) = \frac{K(\omega)}{2\pi v_g(\omega)}\] \[D_{\text{3D}}(\omega) = \frac{K^2(\omega)}{2\pi^2 v_g(\omega)}\]Substituting these into the expressions for number of modes:
\[M_{\text{1D}}(\omega) = \pi v_g(\omega) \cdot \frac{1}{\pi v_g(\omega)} = 1\] \[M_{\text{2D}}(\omega) = W \cdot \pi \cdot \left[ \frac{2v_g(\omega)}{\pi} \right] \cdot \frac{K(\omega)}{2\pi v_g(\omega)}\] \[M_{\text{3D}}(\omega) = A \cdot \pi \cdot \left[ \frac{v_g(\omega)}{2} \right] \cdot \frac{K^2(\omega)}{2\pi^2 v_g(\omega)}\]The terms in square brackets above represent the spatially averaged x-direction group velocity at a given frequency:
The number of modes $M(\omega)$ can also be interpreted geometrically as the number of half-wavelengths fitting into the cross-section:
For electrons, a similar analysis gives the energy-integrated thermal flow as:
\[Q_{\text{el}} = \frac{1}{\pi\hbar} \int_0^\infty M(E) \tau(E)(E - \mu)[f_\text{FD}^0(T_1) - f_\text{FD}^0(T_2)]\,dE\]According to Lundstrom and Jeong (2013), for electrons with a parabolic band, the number of modes is:
\[(\text{1D}) :\quad M_\text{1D}(E) = H(E - E_c)\] \[(\text{2D}) :\quad M(E) = W \cdot g_v \cdot \frac{\sqrt{2m^*(E - E_c)}}{\pi\hbar} \cdot H(E - E_c)\] \[(\text{3D}) :\quad M(E) = A \cdot g_v \cdot \frac{m^*(E - E_c)}{2\pi\hbar^2} \cdot H(E - E_c)\]Where:
Thermal conductance $G_Q$ is defined as heat flow per unit temperature difference:
\[G_Q = \frac{Q}{T_1 - T_2}\]For small temperature differences, the differential form is:
\[G_Q(T) = \frac{Q(T + \delta T/2, T - \delta T/2)}{\delta T}\]Here, $M$ is the number of modes, $\tau$ is the transmission function, and $f^0$ is the equilibrium distribution.
The integrand of $G_Q$ represents the contribution to thermal conductance at each frequency or energy, known as the spectral conductance $G_Q’$:
These functions show that the energy transport capacity depends on both the number of available modes $M$ and the thermal sensitivity $\partial f / \partial T$.
To simplify analysis, one can define a normalized spectral conductance by dividing out constants like $k_B$, $M$, and assuming ideal transmission $\tau = 1$:
\[\tilde G_Q' = \frac{G_Q'}{C_0k_BM\tau} = (f_i^0)^2e^xx^2\]with $x = \hbar\omega / k_B T$ as the dimensionless energy.
In the phonon case, for low-frequency modes, the normalized conductance approaches 1 — indicating that each mode contributes equally at low frequencies, until the energy exceeds thermal energy $k_BT$.
In the electron case, low-energy modes contribute little due to the Pauli exclusion principle: electrons must be thermally excited across the Fermi level, so most contribution comes from states near $E \approx \mu$.