# Phonon Bridge Effect in Superlattices of Thermoelectric TiNiSn/HfNiSn With Controlled Interface Intermixing

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Schematic representation of the investigated TiNiSn/HfNiSn superlattice designs. Black dots correspond to the common elements Ni and Sn, while Ti and Hf are symbolized by red and green dots, respectively. The annotation gives the superlattice period in units cells (UC) and the amount of artificial intermixing.

**Figure 2.**Heat flow caused by the heater structure for a substrate with a thin film (

**a**) and for a bare substrate (

**b**). As the heater is much broader than the overall film thickness the heat flow is quasi 1-dimensional and the edge effects are negligible.

**Figure 3.**Typical XRD diffraction pattern in the vicinity of the (002)-peak. The investigated sample is a 6 by 6 UC superlattice with 1 UC artificial intermixing. The red lines correspond to a simulation using a homemade algorithm [27].

**Figure 4.**HR-STEM images of superlattice samples with differing degrees of artificial intermixing: (

**a**) Pristine superlattice with only intrinsic intermixing. (

**b**) Superlattice with an added intermixing of one unit cell, i.e., 0.6 nm.

**Figure 5.**Thermal resistivities ${\rho}^{\alpha}=1/\kappa $ of superlattices of different period length and Ti${}_{0.5}$Hf${}_{0.5}$NiSn interlayer thickness of $\alpha =$ 0, 0.5 and 1 unit cells (UC), respectively. (

**a**) Experimental ${\rho}_{ex}^{\alpha}$ thermal resistivities together with the fit ${\rho}_{fit}^{\alpha}$. The blue bars indicate the thin film thermal resistivies of the constituent materials TiNiSn, HfNiSn, and Ti${}_{0.5}$Hf${}_{0.5}$NiSn. (

**b**) Breakdown of interfacial ${\rho}_{intf}^{\alpha}$ and bulk-like ${\rho}_{bulk}^{\alpha}$ contribution to the thermal resistivity. The point of equal contribution is marked with an arrow. The bulk-like contributions correspond to a weighted average of the bulk resistivities of the constituent materials. (

**c**) The rocking curve widths $\omega $ as a proxy for crystal quality.

**Figure 6.**Overview over the transmission function and the finite size terms for different values of the period length d and specularity parameter. (

**a**) The transmission function in dependence on the specularity parameter. $p=$ 0 corresponds to a fully diffusive interface interaction, which is described by the diffusive mismatch model (DMM). The fully specular case is described by the acoustic mismatch model (AMM). (

**b**) Finite size term in the DMM case, which quantifies the modification of the bulk thermal resistivity by phonon confinement effects. (

**c**) Finite size term in the AMM case.

**Figure 7.**Contributions to the effective interface thermal resistance. (

**a**) The thermal boundary resistance describes the effect of incomplete transmission of heat-carrying phonons across an interface. (

**b**) The influence of the finite size term per superlattice period. As the finite size term depends on the period length, two values for 1 and 10 nm are given together with the mean value. (

**c**) The total effective interface thermal resistance per superlattice period. The experimental values are marked for superlattices with intermixing layers of 0, 0.5, and 1 unit cells (UC), i.e., 0, 0.3 and 0.6 nm.

**Table 1.**Literature values used in the estimation of the effective interface thermal resistance. The Debye velocity has been calculated from the band structure simulations given in [12].

Debye Velocity ${\mathit{v}}_{\mathit{d}}$ (m/s) | Heat Capacity ${\mathit{C}}_{\mathit{i}}$ ($\times {10}^{6}$ J/m${}^{3}$K) | Debye Temperature ${\mathit{\theta}}_{\mathit{D}}$(K) | |
---|---|---|---|

Source | [12] | [22] | [22] |

TiNiSn | 3560 | 2.31 | 380.1 |

HfNiSn | 3090 | 2.65 | 316.2 |

**Table 2.**Material equivalents for different interface designs. The material equivalent expresses how much additional material corresponds to an individual interface in terms of thermal resistance.

Artificial Intermixing Layer Thickness (Unit Cells) | Interface Material Equivalent ${\mathit{d}}_{\mathit{equi}}$ (nm) | Effective Thermal Interface Resistance ($\times {10}^{-9}$ Km${}^{2}$/W) |
---|---|---|

0 | 3.0 | 0.81 |

0.5 | 3.8 | 1.08 |

1 | 2.3 | 0.75 |

© 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

Heinz, S.; Angel, E.C.; Trapp, M.; Kleebe, H.-J.; Jakob, G. Phonon Bridge Effect in Superlattices of Thermoelectric TiNiSn/HfNiSn With Controlled Interface Intermixing. *Nanomaterials* **2020**, *10*, 1239.
https://doi.org/10.3390/nano10061239

**AMA Style**

Heinz S, Angel EC, Trapp M, Kleebe H-J, Jakob G. Phonon Bridge Effect in Superlattices of Thermoelectric TiNiSn/HfNiSn With Controlled Interface Intermixing. *Nanomaterials*. 2020; 10(6):1239.
https://doi.org/10.3390/nano10061239

**Chicago/Turabian Style**

Heinz, Sven, Emigdio Chavez Angel, Maximilian Trapp, Hans-Joachim Kleebe, and Gerhard Jakob. 2020. "Phonon Bridge Effect in Superlattices of Thermoelectric TiNiSn/HfNiSn With Controlled Interface Intermixing" *Nanomaterials* 10, no. 6: 1239.
https://doi.org/10.3390/nano10061239