# Simulation of Field Assisted Sintering of Silicon Germanium Alloys

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## Abstract

**:**

## 1. Introduction

_{2}Te

_{3}, PbTe, etc. [11,12,13]. The main factor responsible for the enhancement of figure of merit (ZT) in these materials is suppression of lattice thermal conductivity due to the increased phonon scattering, as well as electron tunneling and energy filtering effects on grain boundaries [7,14,15,16,17,18,19,20,21].

_{s}and the heating rate are crucial parameters, that have an impact on material grain size and densification rate.

_{2}and MnSi

_{1.4}samples in order to obtain a more realistic picture of the temperature field distribution. In order to trace the sintering temperature difference ΔTs in the samples, two thermocouples were placed onto the sample-plungers interfaces, and a small sector of the die was cut off. For the sintering temperatures of about 1000 °C, a vertical temperature gradient of 55 and 60 °C was found to be formed in MgSi

_{2}and MnSi

_{1.4}samples, respectively. The influence of the temperature gradient on thermoelectric properties of Bi

_{0.5}Sb

_{1.5}Te

_{3}has been reported in Reference [27].

## 2. Materials and Methods

_{80}Ge

_{20}doped with 2% (at.) of boron. Powders were sintered with a respect of parameters optimized in previous work [4]. The samples were compressed at room temperature for two minutes, then the pressure was increased and reached the peak of 60 MPa. The samples were gradually heated to 100 °C and then the temperature was raised up to 1150 °C, with a heating rate of 15 °C/s. The soaking time was 5 min, then the pressure was reduced to 10 MPa, and the samples were slowly cooled to the room temperature. The sintering was performed in vacuum. During the consolidation cycle, the experimental parameters of temperature, applied pressure, current, voltage, and displacement were recorded continuously.

_{el}was measured on bars 1 mm × 3 mm × 12 mm using a homemade transport measuring system (Cryotel Ltd., Moscow, Russia). The accuracy of these measurements were checked against a silver sample of 99.99% purity. The values of the Seebeck coefficient were taken from Reference [4].

_{s}. A line of individual samples has been sintered for each sintering temperature. Values of electrical (σ

_{el}) and thermal (κ) conductivities were obtained as follows. After the sintering, the disc samples were taken from the SPS setup and corresponding measurements (either thermal diffusivity or electrical conductivity) were performed in a wide temperature interval. The magnitude of σ

_{el}or κ at T

_{s}was taken as that one obtained from the measurement data at the corresponding temperature.

_{s}= 500 °C and electrical conductivity up the T

_{s}= 800 °C. At temperatures lower than these values, we used approximation by the nearest function. A drastic increase of σ

_{el}and κ seen for the samples sintered at T

_{s}≥ 800 °C was conditioned by the fact that these samples had rather large relative densities, which approach to the theoretical value for T

_{s}≥ 1100 °C [3,4,5,6,7,8,9].

_{r}are the following [37]:

_{s}(T)ρ

_{r}

^{3.2},

_{r}

^{2},

_{d}is a volume of dense material without pores, and V is the material volume.

## 3. Modeling

#### 3.1. Geometry

#### 3.2. Mathematical Description

#### 3.2.1. Electrical and Thermal Processes

**j**= −σ

_{el}(∇V+S∇T),

**q**= κ∇T + ST

**j**,

_{el}is electrical conductivity, V is the voltage, S is the Seebeck coefficient, T is the absolute temperature, κ is the coefficient of thermal conductivity.

**j**= 0.

_{p}is the heat capacity, ρ is the density, t is the time, Q

_{j}is the Joule heat (Q

_{j}= j∇V), Q

_{h}is the dissipated heat.

#### 3.2.2. Mechanical Processes

_{mech}and an applied force F has the following form:

_{mech}+ F = 0.

**u**:

**u**+ ∇

**u**

^{T}).

_{0}+ C:(ε − ε

_{0}− ε

_{th}),

_{0}is the initial stress, C is the 4th order stiffness tensor or elastic moduli, “:” stands for the double-dot tensor product (or double contraction), ε

_{0}is the initial strain, ε

_{th}is the thermal strain, ε

_{th}= α (T − T

_{0})—thermal strain, where α is the coefficient of thermal expansion, T

_{0}is an initial temperature.

#### 3.2.3. Electrical and Thermal Contacts

_{asp}and slope m

_{asp}. The m

_{asp}was assumed to have the default value 0.4. It was admitted that σ

_{asp}equaled the sum between the half grain size of the materials being in the contact. For the graphite used in the experiment, a half grain size equaled 22.5 μm [41]. Hence, for “graphite–graphite” contacts (interfaces I and III, Figure 2b), the average asperities height was equal to 45 μm. According to a microscopic analysis [3] the average grain size of nanostructured Si-Ge did not exceed 5 μm. A half grain size of Si-Ge was assumed to be 2.5 μm. Hence, the average asperities height for the “Si-Ge–graphite” contacts were equal to 25 μm within the calculations (interfaces II, IV, V, Figure 2b). The value of microhardness was taken for a less harder material (i.e., graphite).

_{c}vs time were calculated for the contacts. These functions were used for the contact resistances evaluation. The model of electrical and thermal contact conductance was described in [42] and presented in this work in Equations (11)–(18).

**j**through the contact interface is determined as:

**n**∙

**j**= −h

_{1}_{e}(V

_{1}− V

_{2}),

**n**∙

**j**= −h

_{2}_{e}(V

_{2}− V

_{1}),

_{1}is voltage on the source boundary, V

_{2}is voltage on the destination boundary.

_{e}of contact interface is:

_{el.cont}is the harmonic mean of contact interface electrical conductivity, H

_{c}is the hardness of the less hard material being in the contact.

**q**through the contact interface is calculated as:

**n**∙

**q**= −h

_{1}_{t}(T

_{1}− T

_{2}),

**n**∙

**q**= −h

_{2}_{t}(T

_{2}− T

_{1}),

_{1}and T

_{2}are the temperatures of the source and destination boundaries, correspondently.

_{t}of contact interface is determined as:

_{t}= h

_{c}+ h

_{g}+ h

_{r},

_{g}and h

_{r}are the thermal conductance of the gap and the radiative conductance, respectively; h

_{g}was assumed to be zero, and:

_{SB}= 5.670 × 10

^{−8}W∙m

^{−2}∙К

^{−4}is the Stefan–Boltzmann constant, ε

_{1}and ε

_{2}are the emissivity factors of the source and destination boundaries being in the contact, κ

_{cont}is the harmonic mean of the contact interface thermal conductivity.

#### 3.2.4. Electric Boundary Conditions

#### 3.2.5. Thermal Boundary Conditions

**n**(κ∇T) = h∙(T

_{ext}− T

_{e}),

^{2}∙K; n is normal vector, T

_{ext}= 20 °C is the water temperature, and T

_{e}is the temperature on electrode surface.

**n**(κ∇T) = ε

_{r}σ

_{SB}(T

_{amb}

^{4}− T

^{4}),

_{amb}= 20 °C is the ambient temperature.

#### 3.2.6. Mechanical Boundary Conditions

#### 3.2.7. Mesh

## 4. Simulation Results and Discussion

#### 4.1. Temperature and Current in the Sample and Setup Elements

_{1.74}; we obtained this value using the approximation of the results reported in Reference [43]. A Seebeck coefficient of Ge-Si was about 210 μV/K at the sintering temperature. Thus, the Peltier coefficient was in the range from 290 to 316 mV for Ge-Si, and from 75 to 80 mV for MnSi

_{1.74}. The sintering temperature difference obtained in the Ge-Si sample with a height of 3 mm and diameter of 20 mm reached 118 degrees in comparison with 65 degrees in the MnSi

_{1.74}sample of the same size. These results seemed to be in rather good correlation. Some additional parameters such as graphite properties or sintering time could have an impact on the final result.

_{TC}in the point corresponding to the thermocouple aperture. The maximum difference between T

_{TC}and T

_{s}in the sample was 54 °C. The difference between T

_{TC}and the average calculated temperature in the sample was 10 °C. The temperature in thermocouple aperture was closer to the colder sample surface temperature. The temperature difference was equal to 12 °C in this case.

#### 4.2. Contact Resistance and Its Impact on the Temperature

_{TC}are higher in the model with contact resistances. This difference reached 35 degrees.

_{s}in the sample, that did not exceed 20 °C. The model with thermoelectric effect (curve 2) results in an additional 50 degrees in the ΔT

_{s}value during the soaking time (from 700 to 900 s). At the same time, no sufficient impact of contact resistances on temperature difference was observed from the calculation.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Dependence of electrical conductivity σ

_{el}, thermal conductivity κ and density ρ on sintering temperature of Si

_{80}Ge

_{20}samples.

**Figure 2.**Front view of the spark plasma sintering (SPS) setup (

**a**), an enlarged view of the specimen and the mold (

**b**); positions I–V are contact interfaces.

**Figure 3.**The dependence of current density on time that was used in the model with thermoelectric effect and contact resistance.

**Figure 5.**Normal current density lines in the sample and surrounding setup elements; the distribution of temperature field is presented by the 2D plot; the results are presented for p–type Si-Ge specimen sintered at 1150 °C.

**Figure 6.**The sintering temperature in the Si-Ge sample and the mold along the radial direction for different z-coordinates for the sintering time t = 900 s. Red circle in the graph indicates the location of thermocouple aperture.

**Figure 7.**Electrical conductance of the contacts. (

**a**) Contacts IV and V; asterisk markers correspond to the contact IV, solid line to contact V. (

**b**) Contacts I–III.

**Figure 8.**Thermal conductance of the contacts. (a) Contacts IV and V; asterisk markers correspond to the contact IV, solid line to contact V. (

**b**) Contacts I–III.

**Figure 9.**The difference between maximum sintering temperatures (1), minimum temperatures (2) and temperatures in the thermocouple (3) obtained from the models with and without contact resistances.

**Figure 10.**Dependence of maximum temperature difference in the sample volume on sintering time: With thermoelectric effect and contact resistances (1); with thermoelectric effect and without contact resistances (2); and without thermoelectric effect and contact resistances (3).

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**MDPI and ACS Style**

Tukmakova, A.; Novotelnova, A.; Samusevich, K.; Usenko, A.; Moskovskikh, D.; Smirnov, A.; Mirofyanchenko, E.; Takagi, T.; Miki, H.; Khovaylo, V.
Simulation of Field Assisted Sintering of Silicon Germanium Alloys. *Materials* **2019**, *12*, 570.
https://doi.org/10.3390/ma12040570

**AMA Style**

Tukmakova A, Novotelnova A, Samusevich K, Usenko A, Moskovskikh D, Smirnov A, Mirofyanchenko E, Takagi T, Miki H, Khovaylo V.
Simulation of Field Assisted Sintering of Silicon Germanium Alloys. *Materials*. 2019; 12(4):570.
https://doi.org/10.3390/ma12040570

**Chicago/Turabian Style**

Tukmakova, Anastasiia, Anna Novotelnova, Kseniia Samusevich, Andrey Usenko, Dmitriy Moskovskikh, Alexandr Smirnov, Ekaterina Mirofyanchenko, Toshiyuki Takagi, Hiroyuki Miki, and Vladimir Khovaylo.
2019. "Simulation of Field Assisted Sintering of Silicon Germanium Alloys" *Materials* 12, no. 4: 570.
https://doi.org/10.3390/ma12040570