Temperature Dependent Young's Modulus of Si and Ge
Authors: Vanhellemont Jan , Swarnakar Akhilesh Kumar , Van der Biest Omer 
Published: February 2020

1. Introduction. Mechanical and thermal stresses play an important role in many of the processes used to produce electronic devices. For example, bulk Si and Ge single crystals are grown from a melt. Near the melt/solid interface high thermal stresses develop that can lead to dislocation formation and the loss of the crystal. As recently demonstrated, also thermal stress levels below the yield stress have an important impact on intrinsic point defect cluster formation in the crystal [1,2]. It is therefore surprising that few reports can be found on Young's modulus Eof Si and Ge at high temperatures [3].
In the present paper, the impulse excitation technique (IET) [4] is used to determine E

<100>
, E

<110>
and E

<111>
between room and melt temperature of Si and Ge. IET measures the resonant frequency of a specimen by exciting it mechanically with an impulse tool. The resonance frequencies are characteristic for the sample material and are related to its stiffness, mass and geometry. The amplitude decay of a free vibration can be related to the internal friction, i.e. dissipation of the vibration energy or damping. Measurement of the resonance frequency allows the calculation of E.

2. Experimental. Czochralski-grown Si and Ge specimens are prepared with typical dimensions of about 45×7×2 mm3. To remove the mechanical damage, the samples are mirror etched. Before and after the IET measurement, the geometry and weight of each sample is measured accurately.
The measurements are performed using a IMCE HTVP 1750 IET set-up equipped with automated impulse excitation and vibration detection devices. The samples are suspended in flexural mode inside the furnace using carbon wires at the bending nodes, which have minimum motions at these points. Mechanical excitation is performed by periodically hitting the sample with a ceramic rod. The acoustic wave generated in the sample is guided to a microphone outside the furnace using a alumina tube The samples are heated in argon atmosphere from room temperature up to melt temperature using heating rates between 3 and 7 ºC/min.

3. Results and discussion. The Young's moduli obtained in the present study are slightly above those of Burenkov et al as illustrated in the Figure showing the normalised E

<111>
versus the normalized temperature T /T

m
for Si and Ge obtained in this study [5.6], compared with the results of Burenkov et al [3]. The figure illustrates also that the temperature dependence of the elastic constants of both materials is very similar when expressed as function of temperature normalized with respect to T

m
as is also the case for some of the other material properties [7]. The experimental data can be reproduced well using a second order polynome

E

<ijk>

= A

0
+ A

1

T + A

2

T
2, (1)
and the results of the best fits are listed in Table 1. The listed coefficients allow calculating the Si and Ge temperature dependent mechanical properties to be used in device and process simulations.
Table 1: Coefficients of Eq. (1), with T in ºC and E

<ijk>
in GPa, obtained from best fits to experimental data [5,6].

A

0

A

1
(x10-3)

A

2
(x10-6)

Si

<100>
127.5
- 6.3
- 4.4

<110>
161.26
- 7.00
- 5.60

<111>
181.8
- 7.9
- 5.9

Ge

<100>
101.86
- 8.78
- 5.57

<110>
136.33
- 10.85
- 8.28

<111>
152.91
- 11.76
- 9.54

References

[1] J. Vanhellemont, E. Kamiyama and K. Sueoka, ECS J. Solid State Sci. Technol.
2, P166 (2013).
[2] K. Sueoka, E. Kamiyama, J. Vanhellemont and K. Nakamura, submitted for publication in ECS Solid State Letters.
[3] Yu. A. Burenkov, S. P. Nikanorov and A. V. Stepanov, Soviet Phys. - Solid State
12, 1940 (1971).
[4] G. Roebben, B. Bollen, A. Brebels, J. Van Humbeeck and O. Van der Biest, Rev. Sci. Instrum.
68, 4511 (1997).
[5] J. Vanhellemont and E. Simoen, J. Electrochem. Soc.
154, H572 (2007).
[6] A. K. Swarnakar, O. Van der Biest and J. Vanhellemont, Phys. Status Solidi C
11 , 150 (2014).
[7] A. K. Swarnakar, O. Van der Biest and J. Vanhellemont, to be presented at Symposium X “Materials research for group IV semiconductors: growth, characterization and technological developments” of the 2014 EMRS Spring Meeting, to be published in Phys. Status Solidi C .