Semiconductor-based thermal wave crystals

Abstract

One-dimensional phononic crystals made of silicon (Si) and germanium (Ge), both of which are materials commonly used in semiconductor devices, are shown to be effective in inducing bandgaps in the dispersion of heat flow at the nanoscale. Numerical approaches are used to understand the dispersion and propagation of thermal waves in Si–Ge phononic crystals. The results show for the first time how nanostructuring could yield band gaps in the dispersion of thermal phonons in the GHz range. We arrive at conditions that can yield bandgaps as high as 40 GHz; this is a bandgap that exceeds the value reported thus far. Variations in the unit cell dimensions are studied to understand the corresponding evolution in the bandgap frequencies. The control of heat using such proposed media holds promise for better heat management solutions for modern electronic devices, nanoscale sensing as well as for novel applications including the development of thermal diodes and thermal cloaks.

Appendix 1

The Catteneo–Vernotte (CV) heat conduction model in 1D can be written as

$$ q + \tau_{{\text{q}}} \frac{\partial q}{{\partial t}} = - \kappa \frac{\partial T}{{\partial x}}, $$

(6)

where $q$ is the heat flux, $T$ is the temperature,$\tau_{{\text{q}}}$ is the Phonon relaxation time, $\kappa$ is the thermal conductivity.

The energy conservation equation with no internal heat generation is given by

$$ \frac{\partial q}{{\partial x}} = - \rho c_{{\text{p}}} \frac{\partial T}{{\partial t}}, $$

(7)

where $\rho$ is the mass density, $c_{p}$ is the specific heat.

Using Eqs. 6 and 7, we get the hyperbolic heat wave conduction equation for 1D, which is

$$ \frac{1}{{\tau_{{\text{q}}} }}\frac{\partial T}{{\partial t}} + \frac{{\partial^{2} T}}{{\partial t^{2} }} = \frac{\kappa }{{\rho c_{{\text{p}}} \tau_{{\text{q}}} }}\frac{{\partial^{2} T}}{{\partial x^{2} }}. $$

(8)

The propagation velocity ($C_{{{\text{CV}}}}$) of this thermal wave is given by

$$ C_{{{\text{CV}}}} = \sqrt {\frac{\kappa }{{\rho c_{{\text{p}}} \tau_{{\text{q}}} }}} . $$

(9)

A periodically layered 1D TWC structure is shown in Fig. 

Fig. 7

Schematic of 1D periodic bilayer TWC

7. Each unit cell of thickness l consists of two layers A and B of thickness l_A and l_B, respectively. All the material properties of each layer are distinguished by subscript A and B while the left end and right end of the unit cell are represented by subscript L and R, respectively. The interface between the A and B layer is designated using the subscript AB. The superscript j represents the jth unit cell. The coordinate (x, y) are shown in Fig. 7.

$$ l = l_{{\text{A}}} + l_{{\text{B}}} ;x_{{\text{L}}}^{{\text{j}}} = jl;x_{{{\text{AB}}}}^{{\text{j}}} = jl + l_{{\text{A}}} ;x_{{\text{R}}}^{j} = \left( {j + 1} \right)l = x_{{\text{L}}}^{j + 1} . $$

(10)

Considering a 1D time-harmonic thermal wave propagating in the 1D periodic structure with angular frequency ω, the temperature and heat flux fields can be written as

$$ \left\{ {T\left( {x,t} \right),q(x,t)} \right\} = \left\{ {\mathop T\limits^{ \wedge } \left( x \right),\mathop q\limits^{ \wedge } (x)} \right\}e^{ - i\omega t} , $$

(11)

with $\mathop T\limits^{ \wedge } \left( x \right)$ satisfying

$$ \mathop {T\prime \prime }\limits^{ \wedge } \left( x \right) + \frac{{\omega^{2} + \frac{i\omega }{{\tau_{q} }}}}{{C_{CV} }}\mathop T\limits^{ \wedge } \left( x \right) = 0, $$

(12)

where $i = \sqrt { - 1}$.

The general solution of Eq. 12 is

$$ \mathop T\limits^{ \wedge } \left( x \right) = A_{1} e^{i\gamma x} + A_{2} e^{ - i\gamma x} , $$

(13)

where A₁ and A₂ are unknown coefficients, and

$$ \gamma = \sqrt {\frac{{\omega^{2} + \frac{i\omega }{{\tau_{{\text{q}}} }}}}{{C_{{{\text{CV}}}} }}} , $$

(14)

of which real part denotes propagating thermal wave while the imaginary part characterizes attenuation.

Using Eq. 7, we get

$$ q(x) = - A_{1} \frac{i\kappa \gamma }{{1 - i\omega \tau_{q} }}e^{i\gamma x} + A_{2} \frac{i\kappa \gamma }{{1 - i\omega \tau_{q} }}e^{ - i\gamma x} . $$

(15)

Introducing state vector,

$$ S(x) = \left\{ {\mathop T\limits^{ \wedge } \left( x \right),\mathop q\limits^{ \wedge } (x)} \right\}^{T} = M(x)\left\{ {A_{1} ,A_{2} } \right\}^{T} , $$

(16)

where

$$ M(x) = \left( {\begin{array}{*{20}c} 1 & 1 \\ { - \frac{i\kappa \gamma }{{1 - i\omega \tau_{q} }}} & {\frac{i\kappa \gamma }{{1 - i\omega \tau_{q} }}} \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {e^{i\gamma x} } & 0 \\ 0 & {e^{ - i\gamma x} } \\ \end{array} } \right). $$

(17)

The above solution is applicable for both layers A and B and is represented by $S_{{\text{A}}}^{j} (x) = M_{{\text{A}}}^{j} (x)\left\{ {A_{1} ,A_{2} } \right\}^{T}$(with $x_{L}^{j} < x < x_{{{\text{AB}}}}^{j}$) in layer A and $S_{{\text{B}}}^{j} (x) = M_{{\text{B}}}^{j} (x)\left\{ {B_{1} ,B_{2} } \right\}^{T}$ (with $x_{{{\text{AB}}}}^{j} < x < x_{{\text{R}}}^{j}$) in layer B.

At the boundaries of layer A and B, state vectors can be written as

$$ \begin{gathered} S_{{{\text{AL}}}}^{j} = M_{{\text{A}}}^{j} (x_{L}^{j} )\left\{ {A_{1} ,A_{2} } \right\}^{T} ;\quad S_{{{\text{AR}}}}^{j} = M_{{\text{A}}}^{j} (x_{{{\text{AB}}}}^{j} )\left\{ {A_{1} ,A_{2} } \right\}^{T} \hfill \\ S_{{{\text{BL}}}}^{j} = M_{{\text{B}}}^{j} (x_{{{\text{AB}}}}^{j} )\left\{ {B_{1} ,B_{2} } \right\}^{T} ;\quad S_{{{\text{BR}}}}^{j} = M_{{\text{B}}}^{j} (x_{{\text{R}}}^{j} )\left\{ {B_{1} ,B_{2} } \right\}^{T} \hfill \\ \end{gathered} $$

(18)

Eliminating $\left\{ {A_{1} ,A_{2} } \right\}^{T}$ and $\left\{ {B_{1} ,B_{2} } \right\}^{T}$, we get

$$ S_{{{\text{AR}}}}^{j} = M_{{{\text{AR}}}}^{j} \left( {M_{{{\text{AL}}}}^{j} } \right)^{ - 1} S_{{{\text{AL}}}}^{j} ;\quad S_{{{\text{BR}}}}^{j} = M_{{{\text{BR}}}}^{j} \left( {M_{{{\text{BL}}}}^{j} } \right)^{ - 1} S_{{{\text{BL}}}}^{j} $$

(19)

As temperature and heat flux are continuous at the interface between two adjacent sub-layers, we get

$$ S_{{{\text{AR}}}}^{j} = S_{{{\text{BL}}}}^{j} ;\quad S_{{{\text{BR}}}}^{j} = S_{{{\text{AL}}}}^{j + 1} . $$

(20)

From Eq. 19 and 20, we get

$$ \begin{gathered} S_{{{\text{AL}}}}^{j + 1} = M_{{{\text{BR}}}}^{j} \left( {M_{{{\text{BL}}}}^{j} } \right)^{ - 1} M_{{{\text{AR}}}}^{j} \left( {M_{{{\text{AL}}}}^{j} } \right)^{ - 1} S_{{{\text{AL}}}}^{j} , \hfill \\ S_{{{\text{AL}}}}^{j + 1} = M_{{{\text{Transfer}}}} S_{{{\text{AL}}}}^{j} , \hfill \\ \end{gathered} $$

(21)

where $M_{{{\text{Transfer}}}} = M_{{{\text{BR}}}}^{j} \left( {M_{{{\text{BL}}}}^{j} } \right)^{ - 1} M_{{{\text{AR}}}}^{j} \left( {M_{{{\text{AL}}}}^{j} } \right)^{ - 1}$.

Using Bloch theorem, for a wave propagating through a periodic structure

$$ S_{{{\text{AL}}}}^{j + 1} = e^{ikl} S_{{{\text{AL}}}}^{j} . $$

(22)

here k is complex Bloch wavenumber and $k = k_{{\text{Re}}} + k_{{{\text{im}}}}$ ($k_{{\text{Re}}}$ being the real part and $k_{{{\text{im}}}}$ being the imaginary part).

Using Eqs. 21 and 22, we get

$$ M_{{{\text{Transfer}}}} S_{{{\text{AL}}}}^{j} = e^{ikl} S_{{{\text{AL}}}}^{j} , $$

(23)

or

$$ \left| {M_{{{\text{Transfer}}}} - e^{ikl} I} \right| = 0. $$

(24)

Using the detailed expression of M_Transfer in Eq. 24, we obtain the eigenvalue equation

$$ \cosh \left( {ikl} \right) = \cosh (i\gamma_{{\text{A}}} I_{{\text{A}}} )\cosh (i\gamma_{{\text{B}}} I_{{\text{B}}} ) + \frac{1}{2}\left( {\frac{{\eta_{{\text{A}}} \gamma_{{\text{A}}} }}{{\eta_{{\text{B}}} \gamma_{{\text{B}}} }} + \frac{{\eta_{{\text{B}}} \gamma_{{\text{B}}} }}{{\eta_{{\text{A}}} \gamma_{{\text{A}}} }}} \right)\sinh (i\gamma_{{\text{A}}} I_{{\text{A}}} )\sinh (i\gamma_{{\text{B}}} I_{{\text{B}}} ), $$

(25)

where $\eta_{{\text{A}}} = \frac{{\kappa_{{\text{A}}} }}{{\left( {1 - i\omega \tau_{{{\text{qA}}}} } \right)}}$ and $\eta_{{\text{B}}} = \frac{{\kappa_{{\text{B}}} }}{{\left( {1 - i\omega \tau_{{{\text{qB}}}} } \right)}}$.

Solving Eq. 25, dispersion curves (ω vs k) for the 1D periodic TWC are obtained.