Phonon diffraction and dimensionality crossover in phonon-interface scattering

Abstract

Thermal resistance at interfaces and grain boundaries is important in a range of fields from microelectronics to energy materials. Standard models treat interfaces as structureless even though at the nanoscale they are often better described as arrays of linear defects. Here, we examine several characteristics of heat transport that arise when considering such a structure at the interface. When heat carrying phonons scatter off linear defect arrays, diffraction of phonons occurs. Furthermore, a dimensionality crossover is observed in diffusive transport. Phonons transition from seeing a structureless planar defect when their wavelength is longer than the defect spacing, $\lambda \gtrsim D$, to seeing the interface as a collection of independently scattering linear defects when $\lambda \lesssim D$. By applying this theory to grain boundary strain-field scattering, we show that this dimensionality crossover can explain the frequency dependence of grain boundary scattering and transmissivity, which results in the T² temperature dependence observed in the low-T thermal conductivity of poly/nanocrystalline materials.

Introduction

Heat conduction across interfaces is an inherently complicated topic of great scientific and technological importance. Advancement in microelectronic technologies (e.g. microprocessors, high-power electronics¹) demands ever more efficient removal of the heat generated^2,3, making the reduction of phonon scattering at interfaces an important engineering challenge. In the opposite direction, maximizing phonon scattering by interfacial defects is essential for engineering thermal transport in state-of-the-art thermal barrier coatings^4,5, and thermoelectrics^6,7.

The current standard models for describing phonon-interface interactions are the acoustic and diffuse mismatch models (AMM and DMM)⁸, which define the interface as a structureless planar defect where the transmissivity or scattering rate is determined by the bulk properties on either side of the interface. Specifically, within the AMM phonons interact with interfaces through an acoustic analog of the refraction of light, where a change in elastic properties (i.e. acoustic impedance) determines the probability that an incident phonon will reflect or transmit across the interface^9,10. In the DMM every phonon scatters randomly or diffusely at the interface and the probability of transmission is determined by the density of phonon states on either side of the boundary¹⁰. The commonly used gray model also assumes diffuse scattering and simply limits the phonon mean free path (MFP) to the average grain size of a polycrystal. This is essentially an extension of the Casmir limit which was initially proposed for single crystals where the MFP was limited to the size of the sample¹¹. Each of these models predicts frequency independent scattering of phonons at low temperatures, and thus predicts a T³ temperature dependence in the lattice thermal conductivity, κ_L, stemming from the temperature dependence of the heat capacity.

In contrast, studies of the low-T κ_L of polycrystals provided by Wang et al.¹², Watari et al.¹³, and Berman¹⁴ show a κ_L ∝ T² temperature dependence (Fig. 1) which is characteristic of a grain boundary (GB) scattering rate that is linear in frequency, τ⁻¹ ∝ ω. An empirical expression for the spectral transmissivity function of phonons at GBs with the form t(ω) = (1 + αω/ω_max)⁻¹ (where α is a constant on the order of unity, and ω_max is the maximum phonon frequency), was presented to account for the observed κ_L ∝ T² (τ⁻¹ ∝ ω) behavior in polycrystals¹² and has been used in a number of computational and experimental studies^15,16,17. Additionally, Hua et al.¹⁸ have recently measured the spectral phonon transmissivity of an Al–Si interface and experimentally showed that the interface of dissimilar materials can have a transmissivity function which is ω-independent at the lowest frequencies and decreases above some critical crossover frequency¹⁸. Providing mechanistic explanations for these phenomena is a key step in understanding phonon-interface interactions at a fundamental level.

Fig. 1

The low temperature lattice thermal conductivity (κ_L) of polycrystalline samples compared to that of single crystals. In polycrystals where the phonon scattering is dominated by grain boundary interactions, κ_L scales as T². In single crystals where scattering is dominated by phonon sample surface interactions, T³ behavior is observed. The κ_L and T values are normalized for comparison. Un-normalized data are shown in Fig. 8

The frequency independence of the phonon scattering in the AMM, DMM, and grey models stems (in part) from their definition of the interface as a structureless planar defect. However, real interfaces between similar (GBs) and dissimilar (phase boundaries) materials are not structureless at the nanoscale but are better described as arrays of linear defects of various types (refs.^{19,20,21,22,23,24} and Section 2.3 of ref. ²⁵). In fact, prevailing models of GB defect energies and structure have essentially extended the classic Read–Shockley model²⁴, which defines the GB structure as an array of lattice dislocations, to many GB types (including GBs at high angle) despite the complexity and variety of GB structures^26,27,28. Further, interfaces between dissimilar materials are known to accommodate the mismatch in lattice parameter by the formation of arrays of misfit dislocations spaced periodically (e.g. GaAs heterostructures^29,30). X-Ray and electron diffraction peaks arising from the periodic dislocation structures in semi-coherent phase boundaries and GBs have been experimentally observed and can be used to study interfacial structure^31,32. We note that the wavelength of X-rays and electrons used in these studies are comparable to the wavelengths of heat carrying phonons. The interface between two crystalline materials with structural periodicity will tend to have structural periodicity itself. We propose that the description of phonon-interface scattering should be grounded in a definition which contains this structural information.

Phonon diffraction conditions arising from periodically spaced dislocations have been discussed previously by authors, such as Klemens³³, Carruthers³⁴, and are formally considered by Omini and Sparavigna³⁵. Previous works, however, have not examined the dimensionality crossover effects that diffraction engenders. We have previously suggested that dislocation strain may dominate phonon-GB scattering processes³⁶. Here we develop a rigorous theory for this and identify several interesting phenomena that arise from treating the GB dislocation arrays collectively. (This study focuses only on phonon-interface interactions, and does not consider electron–phonon interactions which can be important at metal–semiconductor interfaces).

To that end, we first discuss the general effect of defect dimensionality on phonon scattering in order to establish concepts that will be used to interpret the theory that follows. We then derive a general formula to calculate the phonon lifetime due to an array of linear defects, where by analyzing the conservation laws and kinematics of this system, diffraction conditions, and a crossover in defect dimensionality can be observed. Next, this general formula is applied to the specific case of phonon-GB strain field scattering by defining the linear defects as the strain field from edge dislocations which collectively define a symmetric tilt GB. A simple, semi-empirical expression is provided as an excellent approximation of the full analytical theory which embodies the crossover in defect dimensionality observed in this calculation. Finally, this scattering theory is applied to standard phonon transport models, where the phonon-GB strain field scattering and inherent dimensionality crossover provide a mechanistic explanation for the experimentally observed κ_L ∝ T² of polycrystalline and nanocrystalline materials^12,13,14. This power law analysis provides evidence that the dominate phonon-GB scattering mechanism is through GB strain fields. It may also provide an explanation for crossover effects in the frequency dependence of phonon transmissivity at phase boundary interfaces¹⁸.

Results

Dimensionality and phonon scattering rate

We start by considering the effect of just the dimensionality of the scattering defect on the ω-dependence of the phonon lifetime. We follow the basic precepts of the theory of phonon scattering and its contribution to lattice thermal conductivity as laid out, for example, by Ziman³⁷. By applying these precepts to elastic scattering from a defect with scattering potential V (described in more detail in Supplementary Note 1), the scattering rate or inverse lifetime (Γ(k) = τ(k)⁻¹) of a phonon with wavevector k and frequency ω may be written as a product of three factors,

$$\Gamma ({\mathbf{k}}) = n_{\bar n\mathrm{d}}{\kern 1pt} g_{\bar n\mathrm{d}}(\omega ){\kern 1pt} \overline{\left| {M_{\bar n\mathrm{d}}} \right|^2} .$$

(1)

The three factors n, g, and $\overline {\left| M \right|^2}$ are the spatial density of the defect in the crystal, the phonon density of states (pDOS), and a term containing the square of a scattering matrix element $M = \left\langle {{\mathbf{k}}^\prime \left| {{\cal H}^\prime } \right|{\mathbf{k}}} \right\rangle = (L_xL_yL_z)^{ - 1}{\int} {\mathrm {d}}^3{\mathbf{r}}{\kern 1pt} V({\mathbf{r}}){\mathrm {e}}^{i({\mathbf{k}} - {\mathbf{k}}^\prime ) \cdot {\mathbf{r}}}$, where V(r) is the scattering potential and L_xL_yL_z is the volume of the crystal containing the defect. The bar in $\overline {\left| M \right|^2}$ indicates that |M|² has been weighted by the forward scattering parameter and summed over possible final states k′. The essential point, indicated by the subscript ‘$\bar n\mathrm{d}$’, is that all three factors depend on $\bar n$, the codimension of the defect. That is,

$$\bar n = 3 - d_{\mathrm {d}}$$

(2)

with d_d being the dimensionality of the defect. Thus, for a one-dimensional defect such as a dislocation line, d_d = 1 and $\bar n = 2$, for a point defect d_d = 0 and $\bar n = 3$, and so on.

We can understand Eq. (1) as follows. For example consider a line defect, so $\bar n = 2$. Suppose the defect is parallel to the z-axis. Then, in addition to the energy, the z component of the phonon wavevector must also be conserved in the scattering event. This results in the phonon scattering onto a circle in k space, and when summing over all possible final state wavevectors the density of accessible phonon states is effectively that of a system with two real spatial dimensions, written as g_2d. At the same time, the matrix element of the perturbation reduces to an integral over the coordinates perpendicular to the defect, x and y, and since this is now a two-dimensional integral, we write it as M_2d. Finally, the density of line defects n_2d is an areal density, or a number per unit area with dimensions (length)⁻².

It is easy to see that the same reasoning applies to planar ($\bar n = 1$) and point ($\bar n = 3$) defects. All three cases are illustrated in Fig. 2 where the scattering diagram is shown on top (analogous to an Ewald sphere) and the corresponding density of final states (Debye), $g_{\bar n\textrm{d}}$, is given on the bottom, which for acoustic phonons obeys the following relation:

$$g_{\bar n\textrm{d}} \propto \omega ^{\bar n - 1}.$$

(3)

Fig. 2

Phonon scattering diagrams for defects with different dimensionality. a A phonon scattered elastically by a point defect scatters into the 3d phonon density of states (pDOS). b A phonon scattered by a linear defect (scattering potential, V(x, y)) conserves phonon momentum in the z-direction and thus scatters into the 2d pDOS, contributing ω to the phonon scattering rate. c A phonon scattered by planar defect (V(x)) conserves phonon momentum within the defect plane (yz-plane) and scatters into the 1d pDOS which is ω-independent. v_g and v_p are the phonon group and phase velocities

To reiterate, a phonon interacting with a point defect scatters onto a shell in k space, a phonon interacting with a line defect scatters onto a circle, and one interacting with a planar defect scatters onto two states (forward and backward scattering) on a line in k space.

Our objective in writing the rate as a product of $n_{\bar n\textrm{d}}$, $g_{\bar n\textrm{d}}$, and $\overline {\left| {M_{\bar n\textrm{d}}} \right|^2}$, is to highlight those aspects of the answer that are most important for practical purposes. It is useful to refer to the quantities $g_{\bar n\textrm{d}}$ and $\overline {\left| {M_{\bar n\textrm{d}}} \right|^2}$ as the ‘phase-space’ and ‘matrix-element’ contributions, respectively. For scattering of acoustic phonons from dislocation strain fields (at low frequencies), the matrix-element contribution is essentially independent of frequency. Thus, its τ⁻¹ ∝ ω frequency dependence^33,34,38 comes solely from the phase-space contribution. It may also be useful to examine Eq. (1) by simple dimensional analysis. Denoting the dimensions of any quantity X by [X], we have

$$\left[ {n_{\bar n\textrm{d}}} \right] = \frac{1}{{({\mathrm {length}})^{\bar n}}},\quad \left[ {g_{\bar n\textrm{d}}} \right] = \frac{{{\mathrm {time}}}}{{({\mathrm {length}})^{\bar n}}},\quad \left[ {\overline {\left| {M_{\bar n\textrm{d}}} \right|^2} } \right] = \frac{{({\mathrm {length}})^{2\bar n}}}{{({\mathrm {time}})^2}},$$

(4)

and Γ(k) correctly has dimensions of (time)⁻¹. It is important to note that, while the isotropic approximation is implied in Fig. 2, these dimensionality and phase space arguments hold even when anisotropy in the phonon dispersion is considered.

Dimensionality and thermal conductivity

The effects of these dimensionality arguments on the temperature dependence of the lattice thermal conductivity can be seen by applying them to standard thermal transport models, such as the Callaway model for phonon thermal conductivity (Section 4.9, Eq. (107) of ref.³⁹)

$$\kappa _{\mathrm{L}} = \frac{1}{3}\intop_0^{\omega _{{\mathrm {max}}}}C_{\mathrm{s}}(\omega )\tau (\omega )v_{\mathrm{g}}(\omega )^2{\mathrm {d}}\omega ,$$

(5)

in the isotropic, single mode approximation, where C_s(ω) is the spectral heat capacity (Supplementary Eq. (S61)) and v_g is the phonon group velocity. At low temperatures, when phonon scattering is dominated by microstructural defect scattering with a frequency independent matrix-element contribution, it can be found that κ_L varies with T as³⁹

$$\kappa _{\mathrm{L}} \propto T^{4 - \bar n}\,{\mathrm{or}}\,T^{1 + d_{\mathrm {d}}}.$$

(6)

Scattering from an infinite array of linear defects

Kinematics and diffraction conditions

Due to the wavelike nature of phonons, their interaction with periodic perturbations leads to diffraction conditions which depend on the wavelength of the phonon and the spacing of the perturbations. We now consider the case of linear defects parallel to the z-axis periodically spaced by a distance D. The x and y coordinates of the defects are

$$x = 0,\;y = jD\quad (j = 0, \pm 1, \pm 2, \ldots ){\kern 1pt} .$$

(7)

Defect arrays fitting this definition describe a variety of interfaces, such as grain boundaries and semi-coherent phase boundaries which can be defined as arrays of linear dislocations^26,27,40.

Scattering from such an array will conserve phonon energy and momentum in the z-direction, k_z. In addition, diffraction conditions will be observed from the periodic structure in the y direction, such that k_y will change by integer multiples of 2π/D. This in turn means that for any incident phonon wavevector k, the scattered phonon wave vector k′ will take only a discrete set of values. Then, with the definition

$$q_m = \frac{{2\pi }}{D}m,$$

(8)

we can solve for the final phonon states by casting the constraints on k′ as

$$k^\prime = k,\quad k_z^\prime = k_z,\quad k_y^\prime = k_y - q_m\quad \quad ({\mathrm{m}} = {\mathrm{integer}}).$$

(9)

The condition k′ = k is equivalent to k′²=k², or, ${k_x^\prime}^{ 2} = k_x^2 + k_y^2 - {k_y^\prime}^{2}$, which leads to

$$k_x^\prime = k_{x,m \pm }^\prime \equiv \pm (k_x^2 + 2k_yq_m - q_m^2)^{1/2}.$$

(10)

However, $k_x^\prime$ must be real, so only a limited number of values of m are allowed for any given k, i.e. only a finite number of final phonon states (k′) exist.

Figure 3 provides a visualization of the conservation laws given in Eq. (9). Figure 3b illustrates the conservation of k_z, meaning the incident phonon will scatter onto a circle in k space. Figure 3c shows this scattering circle. Due to the conservation of k_y modulo q_m (as well as the conservation of energy, k′ = k), only intersections of this scattering circle with the dashed lines give valid final wavevectors. As |k| increases (i.e. phonon frequency increases), the size of the circle increases, while the spacing between the dashed lines is fixed. When |k| becomes large enough such that another term in the sum over m becomes available, a fresh diffraction event is encountered. These diffraction peaks will be observed in our calculation of GB strain field scattering.

Fig. 3

Kinematics and conservation laws leading to diffraction conditions and dimensionality crossover. a An illustration of the array of linear defects which scatters a phonon k into a state k′. b Scattering from linear defects requires the conservation of momentum in the k_z direction, $k_z = k_z^\prime$. This results in a scattering circle that lies parallel to the xy-plane. c Because of further conservation laws arising from the periodic nature of the scattering potential (given in Eq. (9) in the text), only intersections of this circle with the dashed lines are valid. For small |k|, only the m = 0 line will give valid solutions (and a phonon density of states (pDOS), g_1d). When |k| is large, essentially the entire circle is accessible (pDOS, g_2d)

Dimensionality crossover

When |k| is smaller than π/D, the only allowed solution is m = 0, for which $k_y^\prime = k_y$, $k_z^\prime = k_z$, and $k_x^\prime = \pm k_x$. In other words, only forward scattering and specular reflection are allowed, and only the latter contributes to τ⁻¹. This is the solution found by Klemens when solving for phonon scattering at GBs (Eq. (73) of ref. ³³). In this case, the interface behaves like a structureless planar defect, and k_y is conserved just as k_z, making the final pDOS one-dimensional (g_1d). When |k| is much larger than 2π/D, on the other hand, much of the scattering circle is accessible and there is effectively no constraint on $k_y^\prime$. Now the interface behaves like a collection of independently scattering line defects, and the density of final phonon states is two-dimensional (g_2d). The transition between these two limits gives rise to a ‘dimensionality crossover’ and has important implications for the lifetime and thermal conductivity.

We show in Supplementary Note 2 that the expression for the phonon scattering rate from an array of linear defects takes the form

$$\Gamma \left({\mathbf{k}}\right) = \frac{{n_{1d}}}{{\hbar ^2v_{\mathrm{g}}D^2}}\mathop {\sum}\limits_{m,\sigma = \pm } \left| {\tilde V_1\left(q_{x,m\sigma }, - q_m\right)} \right|^2(1 - \widehat {\mathbf{k}} \cdot \widehat {\mathbf{k}{\prime}} )J\left({\mathbf{k}},m\right),$$

(11)

where $q_{x,m\sigma } = k_{x,m\sigma }^\prime - k_x$; $\tilde V_1$ is the Fourier transform of the scattering potential associated with one line defect at (x, y) = (0, 0), i.e.,

$$\tilde V_1(q_x,q_y) = \int{}\int {\mathrm {d}}x\,{\mathrm {d}}y V_1(x,y){\mathrm {e}}^{ - i(q_xx + q_yy)};$$

(12)

and $J({\mathbf{k}},m) = k\left( {k_x^2 + 2k_yq_m - q_m^2} \right)^{ - 1/2}$ is a Jacobian or volume rescaling factor associated with re-expressing the conservation laws (Eq. (9)) in terms of q_x, q_y, and q_z. The sum in Eq. (11) is over accessible final wavevectors as shown in Fig. 3c, and each term in the sum corresponds to a separate diffraction peak.

GB strain field scattering

In this section, we consider the specific problem of scattering from the strain field of grain boundaries, using the concepts developed in the previous two sections. An edge dislocation array (each with Burgers vector $b\widehat {\mathbf{x}}$) with the geometry shown in Fig. 4a is considered. This array describes a symmetric tilt grain boundary (STGB) with a misorientation angle θ_GB such that,

$$2{\mathrm{sin}}\left( {\frac{{\theta _{{\mathrm{GB}}}}}{2}} \right) = \frac{b}{D}.$$

(13)

This is the same microscopic picture behind the extended Read–Shockley model which is used to describe the GB energy of many types of GBs, including GBs at high angle^26,28. Figure 4a schematically illustrates this interfacial defect next to its strain fields which are shown in Fig. 4b.

Fig. 4

A periodic array of edge dislocations describing the structure of a symmetric tilt grain boundary. a Schematic illustration of an array of edge dislocations (⊥) periodically spaced by D with Burgers vector $b\widehat {\mathbf{x}}$ forming a grain boundary of angle θ_GB. Careful examination of the atoms highlighted in red reveals that they are displaced. b Grain boundary strain maps of hydrostatic strain ε_Δ, pure shear strain ε_S, and rotation ε_R, calculated via Eq. (18) with parameters given in Table 1. Scale bar shows the percent strain (10⁻²)

The displacement field u(r) and the distortion tensor ∂u_i/∂x_j are then independent of z, and the strain state can be formulated in terms of three quantities²⁷,

$$\varepsilon _{\mathrm{\Delta }} = \frac{{\partial u_x}}{{\partial x}} + \frac{{\partial u_y}}{{\partial y}},\quad \varepsilon _{\mathrm{S}} = \frac{1}{2}\left( {\frac{{\partial u_x}}{{\partial y}} + \frac{{\partial u_y}}{{\partial x}}} \right),\quad \varepsilon _{\mathrm{R}} = \frac{{\partial u_x}}{{\partial y}} - \frac{{\partial u_y}}{{\partial x}},$$

(14)

which describe the dilation strain, shear strain, and rotation, respectively. Each type of deformation constitutes an independent scattering channel, which contributes additively to the scattering rate.

For the perturbation V due to this strain, we employ the same form as that used by Ziman (Section 6.4 of ref. ³⁷). We will take the perturbation from the array of dislocations as the sum of contributions from each individual dislocation for each deformation type. More specifically, denoting the contribution from a single dislocation along the line x = y = 0 by a subscript ‘1’ as in Eq. (11),

$$V_{1,a} = \hbar \omega \gamma _a\varepsilon _a(x,y),\quad (a = {\mathrm{\Delta }},{\mathrm{S}},{\mathrm{R}}),$$

(15)

so that the total perturbation from each type of deformation is

$$V_a = \hbar \omega \gamma _a\mathop {\sum}\limits_{n = - \infty }^\infty \varepsilon _a(x,y - nD).$$

(16)

Here, ω is the phonon angular frequency, and γ_a is a coefficient relating the type of strain or rotation a to a change in phonon energy. The change in phonon energy with strain, γ_Δ and γ_S, is defined through the generalized Grüneisen parameter (Eq. (11.75) of ref. ⁴¹). The change in phonon energy with a rotation of the lattice, γ_R, is determined by the anisotropy of the phonon dispersion and elastic properties.

In this work, we use a single averaged Grüneisen parameter, γ = γ_Δ = γ_S = γ_R instead a full phonon-mode and deformation-specific representation, which is in line with historical treatments of Ziman³⁷, Klemens³³, and Carruthers³⁴, and more recent ones by Meng et al.⁴², for example. Further discussion about the Grüneisen parameter, and extensions beyond this approximation are provided in Supplementary Note 3.

With this perturbation defined, we can now use the formalism developed in Supplementary Note 2 and Eq. (11) to calculate the phonon lifetime. It can be seen from Eq. (11) that this will require the Fourier transform of the single dislocation perturbation which can be found for each deformation type to equal

$$\tilde V_{1,a} = \hbar \omega \gamma \tilde \varepsilon _a({\mathbf{q}}){\kern 1pt} .$$

(17)

The explicit forms of these deformations are (Section 3-4 of ref. ²⁷)

$$\varepsilon _{\mathrm{\Delta }} = \frac{{ - b}}{{2\pi }}\frac{{(1 - 2\nu )}}{{(1 - \nu )}}\frac{y}{{(x^2 + y^2)}},\quad \varepsilon _{\mathrm {S}} = \frac{b}{{4\pi (1 - \nu )}}\frac{{x(x^2 - y^2)}}{{(x^2 + y^2)^2}},\\ \quad \varepsilon _{\mathrm {R}} = \frac{b}{\pi }\frac{x}{{(x^2 + y^2)}},$$

(18)

and thus their Fourier transforms are given by

$$\tilde \varepsilon _{\mathrm{\Delta }}({\mathbf{q}}) = ib\frac{{(1 - 2\nu )}}{{(1 - \nu )}}\frac{{q_y}}{{q^2}},\quad \tilde \varepsilon _{\mathrm{S}}({\mathbf{q}}) = - ib\frac{1}{{(1 - \nu )}}\frac{{q_xq_y^2}}{{q^4}},\\ \quad \tilde \varepsilon _{\mathrm{R}}({\mathbf{q}}) = - 2ib\frac{{q_x}}{{q^2}},$$

(19)

where b is the Burgers vector for one dislocation, and ${\nu}$ is the Poisson ratio. Note that because all forms of $\tilde \varepsilon _a$ scale as 1/q, and since q (the momentum transfer) scales as k = ω/v_p, the ω dependence cancels in $\tilde V_{1,a}$ from Eq. (17) (at low ω). This results in a matrix-element contribution that is essentially ω independent for phonon-strain field scattering, in concordance with our assertion in ‘Dimensionality and phonon scattering rate’. When the three channels are added together, we get

$${\mathrm{\Gamma }}_{{\mathrm {gbs}}}({\mathbf{k}}) = \frac{{n_{1{\mathrm {d}}}\gamma ^2}}{{v_{\mathrm {g}}D^2}}\mathop {\sum}\limits_{m,\sigma } \omega _k^2\left[ {\left| {\tilde \varepsilon _{\mathrm{\Delta }}} \right|^2 + \left| {\tilde \varepsilon _{\mathrm {S}}} \right|^2 + \left| {\tilde \varepsilon _{\mathrm {R}}} \right|^2} \right](1 - \widehat {\mathbf{k}} \cdot \widehat {\mathbf{k}}\prime )J({\mathbf{k}},m).$$

(20)

This expression will be averaged over incident phonon direction (Eq. (S59)) and be applied to Eq. (5) to examine its implications for κ_L.

Rotation and specular reflection

It leads to greater physical insight to examine the m = 0 (specular) term in the sum in Eq. (20) separately from the m ≠ 0 (nonspecular) terms. For the m = 0 term, q_y = 0, and we see from Eq. (19) that $\tilde \varepsilon _{\mathrm{\Delta }} = \tilde \varepsilon _{\mathrm{S}} = 0$, so only rotation is nonzero (see the inset of Fig. 5). Thus, for k < π/D, only specular reflection survives and only rotational deformation contributes to scattering. This deformation arises because the crystals on the two sides of the GB are rotated with respect to each other, and the deformation is long-ranged for the same reason (see ε_R in Fig. 4b). The coupling coefficient for rotation is determined by the anisotropy of the crystal and is only a function of the second-order force constants⁴³. Thus, the m = 0 term contains the same physics as the AMM, and γ_R can be regarded as a descriptor of anisotropy.

Fig. 5

Phonon diffraction peaks arising from the periodic nature of a grain boundary strain field. The scattering rate is calculated from Eq. (20), for a phonon at normal incidence (${\mathbf{k}}||\widehat {\mathbf{x}}$) at different magnitudes of the phonon wavevector, k = |k|. Scattering at low k is non-zero as shown in the inset, and diffraction peaks are observed as singularities at 2πm/D, where m is an integer

Phonon diffraction from GB strain fields

Scattering of wave-like phonons from the GB strain fields shown in Fig. 4 with material values for Si (given in Tables 1 and 2) is calculated using Eq. (20) and the results for phonons at normal incidence (${\mathbf{k}}||\widehat {\mathbf{x}}$) are shown in Fig. 5.

Table 1 Intrinsic scattering parameters entering $\tau _{{\mathrm{pp}}}^{ - 1}$, $\tau _{{\mathrm{pd}}}^{ - 1}$, and $\tau _{{\mathrm{Casimir}}}^{ - 1}$ obtained by fitting to single crystal data^13,14,58

Table 2 Material parameters used for grain boundary strain scattering, $\tau _{{\mathrm{gbs}}}^{ - 1}$

Diffraction events for normal incidence occur when k = 2πm/D, where m is an integer, and these events appear as singularities in Γ_gbs. The exact position of the diffraction peaks depends on the phonon angle of incidence. This diffraction may be observable for ballistic phonons if the periodic structure of the interface is maintained for a length scale longer than the wavelengths of the phonons considered. In a real system where the GB structure is not infinitely periodic, one would expect these diffraction peaks to broaden in a manner analogous to Scherrer-broadening of X-ray diffraction peaks due to particle size effects⁴⁴.

Dimensionality crossover in diffuse heat conduction

For diffuse (bulk) thermal conduction, a dimensionality crossover is observed as phonons transition from seeing the GB as a 2d plane, to a collection of individually scattering linear defects. In bulk thermal conduction, phonons transport diffusively, arriving at the interface in random directions rather than one specific angle of incidence. Thus, the diffraction observed for one specific k (Fig. 5) is washed out when all the possible directions of incidence are included. We define τ(ω) in Supplementary Eq. S59 as a lifetime averaged over incident direction which can be directly applied to Eq. (5). By setting Γ(k) in Eq. S59 to Γ_gbs(k), we obtain a directionally averaged GB strain scattering rate τ_gbs(ω)⁻¹ which is shown in Fig. 6. This can be used to describe the ω-dependence of phonon-GB scattering in diffuse heat conduction, and thus be used to interpret the temperature dependence of the lattice thermal conductivity of bulk polycrystalline and nanocrystalline materials.

Fig. 6

Dimensionality crossover effects observed in phonon-GB strain field scattering in diffuse heat conduction. a The vertical dashed line denotes the crossover frequency, below which the phonon scatters off the GB as a 2d defect (ω independent, τ ∝ ω⁰), and above which scatters off the grain boundary as an array of 1d defects (τ ∝ ω⁻¹). D is the linear defect spacing (Fig. 4a), and θ_GB can be interpreted as the magnitude of angular deviation from a special boundary²⁶. b The spectral scattering rate is shown where the dashed line compares the approximate formula given in the main text (Eq. (21)) to the exact formula (solid line). c Phonon-GB strain field scattering cast in terms of a spectral transmissivity function (Eq. (24)) shows a frequency independent transmissivity at low-ω and a decrease above the crossover frequency

The change in slope, or crossover in ω-dependence, observed in Fig. 6 is an important result of this study. When the phonon wavelength is larger than the microscopic structure of a GB (D in this work), the phonon will see it as planar defect and the phase space contribution to the phonon lifetime will be ω-independent (g_1d). When the wavelength is smaller than the microscopic structure of the GB, the phonon lifetime will obtain ω-dependence through the phase space contribution (g_2d). In the case shown above for GB strain field scattering, the matrix-element contribution is ω-independent so the ω-dependence of the phonon lifetime comes exclusively from phase-space contributions. We note that, while the diffraction conditions shown in Fig. 5 require structural periodicity of the GB structure, the dimensionality crossover discussed here is purely a phase-space effect and is thus expected to be preserved even if perfect periodicity is not maintained in a real GB structure.

By recognizing the dimensionality crossover behavior, the following semi-empirical expression can be inferred through careful analysis:

$$\frac{1}{{\tau _{{\mathrm{gbs}}}(\omega )}} = A{\kern 1pt} n_{1{\mathrm{d}}}v_{\mathrm{s}}\gamma _{\mathrm{R}}^2\left( {\frac{b}{D}} \right)^2 + B\left( {\frac{{n_{1{\mathrm{d}}}}}{D}} \right)\gamma ^2b^2(\omega - \omega ^ \ast )\Theta \left( {\omega - \omega ^ \ast } \right),$$

(21)

where

$$\omega ^ \ast = \frac{{4\pi v_{\mathrm{s}}}}{{3D}}{\kern 1pt} ,$$

(22)

is the dimensionality crossover frequency (averaged over the incident phonon direction) and Θ(x) is the Heaviside step function. The best fit of Eq. (21) to Eq. (20) is obtained with A = 8/3 and B = 0.93 K_ν, where

$$K_\nu = 1 + \frac{{(1 - 2\nu )^2}}{{4(1 - \nu )^2}} + \frac{1}{{32(1 - \nu )^2}},$$

(23)

and ν is Poisson’s ratio. Analytical justification of the dependencies, functional form, and magnitudes of coefficients A and B in Eq. (21) is given in Supplementary Note 2. Eq. (21) provides an excellent approximation (dashed lines in Figs. 6 and 7) for the numerical solution of the full analytical expression in Eq. (20) (solid lines). We note that the first term in Eq. (21) is the m = 0 (specular) term in the sum in Eq. (20). It is ω-independent and proportional to the linear density of interfaces n_1d, a misorientation factor b/D (Eq. (13)), and a coefficient characterizing the crystal anisotropy γ_R. The second term in Eq. (21) contains m≠0 (nonspecular) terms and is proportional to the areal density of GB dislocations n_1d/D and the Burgers vector squared b², and a coefficient characterizing the anharmonicity of the crystal γ. These dependencies stem directly from the dimensionality arguments presented while discussing Figs. 2 and 3. As discussed in ‘Rotation and specular reflection’, the first term in Eq. (21) depends only on rotation, while the second term depends on rotation, hydrostatic, and shear strain. Thus, if the scattering coefficients of each deformation type (a = Δ, S, R) are treated separately the relative magnitudes of the first and second term will change and the dependence of $\tau _{{\mathrm{gbs}}}^{ - 1}$ with GB angle will also change.

Fig. 7

General features of the influence of phonon-grain boundary (GB) strain-field scattering on low temperature lattice thermal conductivity. The figure is based on parameters appropriate for Si with a Debye dispersion as described in the text. Curves are shown for several values of GB dislocation spacing (D), corresponding to GB angles (θ_GB) shown in the legend. When D is small, phonons see the grain boundary as a planar defect ($\bar n = 1$) resulting in ω-independent scattering and κ_L ∝ T³. When D is large the phonons see the grain boundary as an array of independently scattering line defects ($\bar n = 2$) which gives $\tau _{{\mathrm{gbs}}}^{ - 1} \propto \omega$, leading to κ_L ∝ T² behavior at low-T. The full and dashed lines are calculated using the full summation (Eqs. (20) and (S59)) and the semi-empirical formula (Eq. (21)), respectively

The scattering theory presented here characterizes the phonon GB interaction as a phonon lifetime τ(ω), which is a convenient way to understand dimensionality and phase space effects. In Fig. 6c we cast our results in the form of a spectral phonon transmissivity using the expression provided by Dames and Chen⁴⁵

$$t(\omega ) = \frac{{v_{\mathrm{g}}n_{1{\mathrm{d}}}\tau (\omega )}}{{\frac{3}{4} + v_{\mathrm{g}}n_{1{\mathrm{d}}}\tau (\omega )}}{\kern 1pt} .$$

(24)

This demonstrates how the dimensionality crossover of the phonon-interface interaction results in a transmissivity that is ω-independent at low ω, and decreases above the critical frequency ω^*, which is a function of the linear defect spacing.

Low-T thermal conductivity of polycrystals

Now, we discuss how GB strain scattering influences thermal transport in polycrystalline materials. Specific analysis is given for polycrystalline Si, AlN, and Al₂O₃. We use the Callaway model for phonon thermal conductivity (Eq. (5)) and modeling procedure given by Wang et al.^12,46 wherein the net relaxation rate is given by

$$\tau ^{ - 1} = \tau _{{\mathrm{pp}}}^{ - 1} + \tau _{{\mathrm{pd}}}^{ - 1} + \tau _{{\mathrm{gbs}}}^{ - 1}{\kern 1pt} .$$

(25)

The three contributions are due to phonon–phonon (τ_pp), intrinsic point defect (τ_pd) and GB strain (τ_gbs) scattering. We take $\tau _{{\mathrm{pp}}}^{ - 1} = C_1\omega ^2Te^{ - C_2/T}$ (same form as that given by Slack⁴⁷), $\tau _{{\mathrm{pd}}}^{ - 1} = C_3\omega ^4$, but account for GB scattering in polycrystalline materials using Eq. (21). Since $\tau _{{\mathrm{pp}}}^{ - 1}$ and $\tau _{{\mathrm{pd}}}^{ - 1}$ are considered intrinsic to a given crystal system, the coefficients C₁, C₂, and C₃ are fit to single crystal data (Table 1) and fixed for subsequent modeling. The parameters for Si are the same as those of Wang et al.¹².

In Fig. 7, we show the temperature dependence of κ_L that results from the modeling procedure described above for the specific case of Si. This dependence is shown for various values of D, the GB dislocation spacing (Fig. 4a), with the remaining material parameters for Si given in Table 2. The central point of the figure is that depending on D, the T dependence of κ_L at low temperature (below the temperature of maximum κ_L) can vary from T² to T³. At such temperatures the dominant scattering in this model is from GB strain and therefore, the variation from T² to T³ stems directly from the dimensionality crossover argument embedded in our semi-empirical formula for τ_gbs (Eq.(21), Fig. 6). As GB dislocation spacing decreases, the crossover frequency ω^* (Eq. 22) increases. For reference, the peak in the phonon occupation number at ~50 K occurs at $\omega /\omega _{\mathrm{D}} \simeq 0.2$. For the case of D = 1 nm, ω^*/ω_D = 0.37, so below 50 K most phonons see the GB as a planar defect with $\bar n = 1$ (τ independent of ω) resulting in κ_L ∝ T³ (see lower most curve in Fig. 7). For larger dislocation spacings the crossover shifts downward in frequency and a significant number of phonons see the GB as a collection of linear defects with $\bar n = 2$ and τ⁻¹ ∝ ω, leading to κ_L ∝ T². This crossover in τ⁻¹ also means that for any fixed D there is a transition from κ_L ∝ T² to κ_L ∝ T³ at a crossover temperature which is related to ω^*.

We now wish to compare our model for GB strain scattering with experimental thermal conductivity data of real materials. We choose polycrystalline Si¹², AlN¹³, and Al₂O₃¹⁴ as model systems. The data for AlN given by Watari et al. is specifically included because the sample was synthesized with clean GBs with almost no GB oxide phase. The intrinsic phonon scattering parameters are given in Table 1 as before. In addition, material parameters related to GB strain scattering are required to calculate τ_gbs. These parameters are given in Table 2 and are obtained as follows. Literature values are used for the average speed of sound, Grüneisen parameter, and Poisson’s ratio. When applied to a bulk polycrystal, τ_gbs in Eq. (21) is meant to embody phonon scattering off an ensemble of GB structures within the polycrystal. Thus, one may interpret D as the characteristic length scale of GB structure in the polycrystal and b as an average GB dislocation Burgers vector. Recognizing the energetic considerations which require dislocations to have Burgers vectors which are integer multiples of a primitive unit cell vector (see Figure 9–2 in ref. ²⁷), a good approximation (and lower bound) for the average GB Burgers vector is b_GB = (VN)^1/3, where V is the volume per atom and N is the number of atoms in the primitive unit cell. When chosen in this way, b_GB is fixed and the only adjustable parameters are D, and the average grain size $\bar d$. The latter is related to the linear density of interfaces by calculating the GB area over volume assuming cubic grains, i.e., $n_{1{\mathrm {d}}} = 3/\bar d$. The agreement between the experimental grain sizes and the values used in this modeling study is reasonable, as can be seen in Table 2.

The results of this comparison are shown in Fig. 8. The figure includes data for both single and polycrystalline materials. The single crystal data of AlN and Al₂O₃ show classic T³ power laws at low-T, indicative of Casimir scattering from crystal surfaces. Thus, in the theoretical modeling, ω-independent Casimir scattering is incorporated by adding a term $\tau _{{\mathrm{Casimir}}}^{ - 1} = v_{\mathrm{g}}/L_{{\mathrm{sc}}}$ in the total scattering rate. Here, L_sc is the size of the single crystal (Table 1). By contrast, all polycrystalline data show T² dependence at low-T. This dependence is captured by our GB strain scattering model and demonstrates the importance of defect dimensionality considerations. The figures also show a comparison of the data with the commonly used gray model, where instead of using $\tau _{{\mathrm{gbs}}}^{ - 1}$ in Eq. (25) we use $\tau _{{\mathrm{gray}}}^{ - 1} = v_{\mathrm{g}}/\bar d$. As discussed in the Introduction τ_gray is ω-independent (like the AMM and DMM at low temperatures) since it is a direct extension of Casmir scattering to polycrystals and thus predicts a T³ power law which is not seen. In addition, the low-T magnitude of κ_L given by the gray model is too low by several orders of magnitude. We note that the value of $\bar d$ used to produce the dashed lines in Fig. 8 is adjusted to match the magnitude of the roll-over κ_L. This value differs from that shown in Table 2 by at most a factor of two. Were this adjustment not made the comparison with experimental data would be worse. We note here that for the theoretical modeling we have used both Debye and Born-von Karman (BvK) dispersions (Supplementary Note 4) and as can be seen the differences are not significant except for the softest material considered here, Si. In particular the low-T power law behavior is not affected when dispersive phonons are considered.

Fig. 8

Thermal transport modeling of polycrystalline (pc) (a) Si, (b) AlN, and (c) Al₂O₃. Data for these materials is taken from refs.^12,13,14. The intrinsic scattering parameters are fit to single crystals (sc) and have the values shown in Table 1. The solid lines follow from the theoretical modeling as described in the text. The data for polycrystals shows a clear T² power law in accord with our GB strain field scattering theory and defect dimensionality arguments. The dashed lines show a comparison of the data with the gray model. The error in the literature data is smaller than size of the data points on this logarithmic scale

The characteristic length scale parameter D deserves further discussion. We can understand the values used in our theoretical model by considering the case of Si. The modeling results show that GBs with characteristic lengths of $D \gtrsim 3\,{\kern 1pt} \,{\mathrm{nm}}$ will display κ_L ∝ T² like behavior above ~15 K, as is observed experimentally. By relating D to a GB angle using Eq. (13), we can estimate that GBs within 5 to 10° of a special boundary should have spacings larger than ~3 nm. This includes low-angle GBs, as well as GBs in the vicinity of special GBs at higher angle (e.g. Σ5 at 36.9° for symmetric tilt and twist boundaries in cubic materials)^19,20,21. Given that this window spans a significant range of possible GB angles, and that such GBs are low in energy²⁶, we argue that a sufficient proportion of GBs that occur naturally in a polycrystalline material would scatter with τ⁻¹ ∝ ω, and result in κ_L ∝ T². While the results of this modeling study agree with the experimental data available, controlled computation and experimental studies containing spectral information of the phonon–GB interaction would help further validate the effects discussed here.

Structural information of interfaces can indeed be included when calculating the thermal boundary resistance by computational methods, such as molecular dynamics⁴⁸ or Green’s function methods^49,50, and much progress has been made regarding these approaches in recent years³. These detailed methods are invaluable for progress in the field of heat transfer as the interfacial structure can be defined and systemically controlled, and spectral information can be obtained which is difficult to obtain experimentally⁵¹. Due to the inevitable complexity of interfaces in real materials (e.g. the GB character distribution in a bulk polycrystal), detailed theoretical and computational treatments should be used to establish engineering design principles which are generalizable and can thus be applied to real systems. This work shows, through standard scattering theory, that general kinematic arguments can explain measurable phenomena which emerge from this complexity. We hope this work can help guide more detailed computational simulations and experiments.

Phonon-dislocation interactions have also been studied using molecular dynamics⁵², and even ab initio methods⁵³. The ab initio technique of Wang et al.⁵³ extends the T-matrix formalism to phonon scattering on an array of dislocation quadrupoles, with the periodicity imposed by periodic boundary conditions⁵³. The structure calculated in ref. ⁵³ however eliminates the long range (1/r) strain field⁵⁴ which is characteristic of dislocations and is known to be the dominant phonon-dislocation interaction^{37,38,43,55,56,57}. This long range strain field is required to give the τ⁻¹ ∝ ω and thus κ_L ∝ T², so it is not surprising that this study does not find κ_L ∝ T². We emphasize that the dislocation configuration considered in ref. ⁵³ does not describe a GB, and since it does not contain the 1/r strain field it does not describe isolated lattice dislocations.

Discussion

Phonon scattering at interfaces is an inherently complex phenomenon where many physical processes are at play simultaneously. This work focuses specifically on effects that arise when considering interfacial structure at the nanoscale by defining the interface as an array of linear defects, rather than treating the interface as a structureless planar defect. This definition is sufficiently general such that it includes many types of grain boundaries (particularly those of low energy) and semi-coherent phase boundaries. Indeed the interface between two materials with structural periodicity will tend to have structural periodicity itself, and we suggest that the standard model for phonon-interface scattering in crystalline materials should not assume perfect disorder at the interface. Several emergent phenomena arise from this analysis stemming directly from this structural definition. These include phonon diffraction conditions arising from the periodic structure of the interface and the wavelike nature of phonons, as well as a crossover in the ω dependence of the phonon lifetime stemming from dimensionality and phase space considerations. The general analytical expression derived was applied to the specific case of a symmetric tilt GB where the linear defects were defined as the strain field from edge dislocations. The result is a phonon-GB strain lifetime (τ_gbs) that is independent of ω below a critical frequency $\omega ^ \ast \simeq 4\pi v_{\mathrm{s}}/3D$ which depends on the GB dislocation spacing D, and $\tau _{{\mathrm{gbs}}}^{ - 1} \propto \omega$ above ω^*. A simple semi-empircal expression is provided as an excellent approximation of the full analytical expression, which embodies this dimensionality crossover effect. This scattering theory is applied to standard phonon transport models and is shown to explain the κ_L ∝ T² temperature dependence of polycrystalline materials at low temperatures. This power law analysis provides evidence that the dominant phonon GB scattering mechanism is through GB strain fields and that interfacial structure and strain energy are important.

Code availability

The Mathematica scripts used for numerical computation in this study are provided at https://github.com/rileyhanus/Phonon-scattering-and-transport-in-polycyrstals.