![Phonon dispersions tell us how the
normal modes of vibration along
different symmetry directions within
the 1st BZ (pictured to the right).
Above is a plot of the dispersion with
different amounts of biaxial strain.
From it we can take note of shifts in
frequencies and velocities, since
velocities are simply the slopes of
each branch. (see Sec. II.b)
Introduction
Applying mechanical strain to silicon has been widely
shown to be a promising avenue of improving device
performances [1,2]. Below is an image from [1] where
they’ve grown a strained silicon thin-layer over a
relaxed Si1-xGex layer.
We can also see applications of strained silicon from
[2], where compressive strain (a) is used to increase
hole mobility and tensile strain (b) is used to increase
electron mobility. Although the strain dependence on
the electronic properties of silicon has been widely
studied, less is known about the response of the
thermal properties of silicon films that involve strain.
[3,4]
Lattice thermal conductivity in strained silicon thin-films
from first-principles
Cameron Foss, University of Massachusetts Amherst
Nanoelectronic Theory and Simulation Lab, Faculty: Prof. Zlatan Aksamija
Abstract
Electronic device designs have hit a developmental wall due to fundamental problems of heat dissipation, which is due to the downscaling of
semiconductor devices towards the nanometer range. The research being done in the Nanotechnology Simulation and Theory Lab observe how modifications
to crystal structure and composition, such as strain, alloying, and nanostructuring affects phonon behavior in a range of semiconductor materials. In this
poster we examine the strain dependence on the lattice thermal conductivity in silicon thin films. Phonon normal mode vibrations are calculated from first-
principle Density Function Perturbation Theory and are used in calculating the thermal conductivity by solving the phonon Boltzmann Transport Equation
(pBTE). We report that the in-plane [100] thermal conductivity has a weaker dependence on strain than thermal conductivity in the cross-plane direction
[001]. Furthermore, thermal conductivity increases with compressive strain and decreases with tensile strain in the [001] direction.
Remarks: All DFT/DFPT calculations are done with the open-source software suite Quantum-ESPRESSO. Phonon Boltzmann
Transport equation is solved with a MATLAB code. Images of crystal structures above were generated using the crystalline and
molecular structure visualization program XCrySDen.
References
[1] Isaacson David, et al. Strained-Silicon on Silicon and strained-silicon on silicon-germanium on silicon by relaxed buffer bonding, Journal of Electrochemical Society, MIT 2006
[2] Thompson Scott E., et al. A 90-nm Logic Technology Featuring Strained-Silicon, IEEE Transactions on Electron Devices vol. 51 No. 11 2004
[3] Li X, et al., Strain effects on the thermal conductivity of nanostructures, Phys. Rev. B 81 245318 (2010)
[4] Paul A. and Klimeck G., Strain effects on the phonon thermal properties of ultra-scaled Si nanowires, Applied Phys. Lett. 99 083115 (2011)
[5] http://www2.warwick.ac.uk/fac/sci/physics/current/postgraduate/regs/mpags/ex5/phonons/
[6] Baroni S., de Gironcoli S., Dal Corso A., and Giannozzi P., Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. 73, 515-562 (2001)
[7] Z. Aksamija and I. Knezevic, Anisotropy and boundary scattering in the lattice thermal conductivity of silicon nanomembranes, Phys. Rev. B 82 045319 2010
Results
a) b)
Crystal Structure and Strain
• Lattice constant:
Si: a = 5.4309 Å
Ge: a = 5.658 Å
• Face-center cubic
• N=2 atom basis, 3N
phonon branches
• Strain Tensor:
𝝐 =
𝜖 𝑥 0 0
0 𝜖 𝑦 0
0 0 𝜖 𝑧
Methodology
Above is a plot of the thermal conductivity in the direction
of transport for different amounts of strain. We can see
that strain has a small overall effect on the value of thermal
conductivity. This tells us that the thermal conductivity is
being dominated by internal scattering mechanisms rather
than boundary scattering like we see in the prior plot.
The cross plane thermal conductivity is highly dependent
on boundary scattering, which in turn is dependent on
phonon velocities. We can also see that velocities increase
with compressive strain and decrease with tensile strain,
and it is that trend that we see here in the thermal
conductivity. However this does not tell us how strain
affects thermal conductivity in the direction of transport.
b_dir
t_dir
Quantum Mechanics + Semi-Classical
(DFT/DFPT) (BTE)
a
Eqs. From [6]
Eqs. From [7]
I) Density Functional Perturbation Theory:
𝑖ℏ
𝜕Φ 𝒓, 𝑹; 𝑡
𝜕𝑡
= −
𝐼
ℏ2
2𝑀𝐼
𝜕2
𝜕𝑹𝐼
2 −
𝑖
ℏ2
2𝑚
𝜕2
𝜕𝒓𝑖
2 + 𝑉 𝒓, 𝑹 Φ 𝑟, 𝑅; 𝑡
a) Born Oppenheimer Approximation:
- mass of nucleus >> mass of electron
−
𝑖
ℏ2
2𝑚
𝜕2
𝜕𝒓𝑖
2 + 𝑉 𝑟, 𝑅 Ψ 𝒓 𝑹 = 𝐸 𝑹 Ψ 𝐫 𝐑
−
𝐼
ℏ2
2𝑀𝐼
𝜕2
𝜕𝑹𝐼
2 + 𝐸 𝑹 Φ 𝐑 = 𝐸Φ(𝑹)
b) E(R) is called the Born Oppenheimer Potential
Energy surface and by taking its 2nd derivative we
get the Interatomic force constants:
𝐶𝐼𝐽
𝛼𝛽
≡
𝜕2 𝐸 𝑹
𝜕𝑅𝐼
𝛼
𝜕𝑅𝐽
𝛽
c) From which we can find the normal mode
frequencies 𝜔 through solving:
II) Phonon Boltzmann Transport Equation:
a) Thermal conductivity
𝐾 𝛼𝛽
𝑇 = 𝑘
𝑗 𝒒
ℏ𝝎𝒋 𝑞
𝒌𝑇
𝟐
𝝏𝑓 𝑀𝐵
𝜕𝐸
× 𝜏𝑗 𝒒 𝑣𝑗
𝛼
𝒒 𝑣𝑗
𝛽
(𝒒)
b) Group velocities
𝑣𝑗
𝛼,𝛽
=
𝜕𝜔𝑗 𝒒
𝜕𝒒
;
𝑤ℎ𝑒𝑟𝑒 𝜕𝒒 = 𝜕𝒒 𝑥, 𝜕𝒒 𝑦, 𝜕𝒒 𝑧
c) Total Relaxation time
1
𝜏𝑗 𝒒
=
1
𝜏𝑗,𝑁(𝒒)
+
1
𝜏𝑗,𝑈(𝒒)
+
1
𝜏𝑗,𝐼 𝒒
+
1
𝜏𝑗,𝐵 𝒒
𝑡,𝛽
𝐶𝑠𝑡
𝛼𝛽
𝒒 − 𝑀𝑠 𝜔2 𝒒 𝛿𝑠𝑡 𝛿 𝛼𝛽 𝑈𝑡
𝛽
= 0
A phonon is the
fundamental particle of
vibration within a solid.
Phonons are
represented as plane
waves with wave-vector
k (or q)
Phonons have 4 modes
of vibration:
1) Longitudinal Acoustic
2) Transverse Acoustic
-and 2 more for optical-
(6 modes in 3D systems
since there are 2 transverse
directions)
[5]
Optical Mode
Acoustic Mode
• Here we consider biaxial strain where:
𝜖 𝑥 = 𝜖 𝑦 ≠ 𝜖 𝑧
• Naturally Biaxial Strain breaks cubic symmetry.
Conclusions:
In-plane thermal conductivity for a 20nm
thin-film is not dominated by boundary scattering
and has a small dependence on strain. Thermal
transport is anisotropic between the [100] and
[001] directions, resulting in a decrease by a
factor of 2 in the [001] direction.](https://image.slidesharecdn.com/ugradconffoss2015-160324095424/85/Ugrad-conf-foss_2015-1-320.jpg)
Ugrad conf foss_2015
- 1. Phonon dispersions tell us how the normal modes of vibration along different symmetry directions within the 1st BZ (pictured to the right). Above is a plot of the dispersion with different amounts of biaxial strain. From it we can take note of shifts in frequencies and velocities, since velocities are simply the slopes of each branch. (see Sec. II.b) Introduction Applying mechanical strain to silicon has been widely shown to be a promising avenue of improving device performances [1,2]. Below is an image from [1] where they’ve grown a strained silicon thin-layer over a relaxed Si1-xGex layer. We can also see applications of strained silicon from [2], where compressive strain (a) is used to increase hole mobility and tensile strain (b) is used to increase electron mobility. Although the strain dependence on the electronic properties of silicon has been widely studied, less is known about the response of the thermal properties of silicon films that involve strain. [3,4] Lattice thermal conductivity in strained silicon thin-films from first-principles Cameron Foss, University of Massachusetts Amherst Nanoelectronic Theory and Simulation Lab, Faculty: Prof. Zlatan Aksamija Abstract Electronic device designs have hit a developmental wall due to fundamental problems of heat dissipation, which is due to the downscaling of semiconductor devices towards the nanometer range. The research being done in the Nanotechnology Simulation and Theory Lab observe how modifications to crystal structure and composition, such as strain, alloying, and nanostructuring affects phonon behavior in a range of semiconductor materials. In this poster we examine the strain dependence on the lattice thermal conductivity in silicon thin films. Phonon normal mode vibrations are calculated from first- principle Density Function Perturbation Theory and are used in calculating the thermal conductivity by solving the phonon Boltzmann Transport Equation (pBTE). We report that the in-plane [100] thermal conductivity has a weaker dependence on strain than thermal conductivity in the cross-plane direction [001]. Furthermore, thermal conductivity increases with compressive strain and decreases with tensile strain in the [001] direction. Remarks: All DFT/DFPT calculations are done with the open-source software suite Quantum-ESPRESSO. Phonon Boltzmann Transport equation is solved with a MATLAB code. Images of crystal structures above were generated using the crystalline and molecular structure visualization program XCrySDen. References [1] Isaacson David, et al. Strained-Silicon on Silicon and strained-silicon on silicon-germanium on silicon by relaxed buffer bonding, Journal of Electrochemical Society, MIT 2006 [2] Thompson Scott E., et al. A 90-nm Logic Technology Featuring Strained-Silicon, IEEE Transactions on Electron Devices vol. 51 No. 11 2004 [3] Li X, et al., Strain effects on the thermal conductivity of nanostructures, Phys. Rev. B 81 245318 (2010) [4] Paul A. and Klimeck G., Strain effects on the phonon thermal properties of ultra-scaled Si nanowires, Applied Phys. Lett. 99 083115 (2011) [5] http://www2.warwick.ac.uk/fac/sci/physics/current/postgraduate/regs/mpags/ex5/phonons/ [6] Baroni S., de Gironcoli S., Dal Corso A., and Giannozzi P., Phonons and related crystal properties from density-functional perturbation theory, Rev. Mod. Phys. 73, 515-562 (2001) [7] Z. Aksamija and I. Knezevic, Anisotropy and boundary scattering in the lattice thermal conductivity of silicon nanomembranes, Phys. Rev. B 82 045319 2010 Results a) b) Crystal Structure and Strain • Lattice constant: Si: a = 5.4309 Å Ge: a = 5.658 Å • Face-center cubic • N=2 atom basis, 3N phonon branches • Strain Tensor: 𝝐 = 𝜖 𝑥 0 0 0 𝜖 𝑦 0 0 0 𝜖 𝑧 Methodology Above is a plot of the thermal conductivity in the direction of transport for different amounts of strain. We can see that strain has a small overall effect on the value of thermal conductivity. This tells us that the thermal conductivity is being dominated by internal scattering mechanisms rather than boundary scattering like we see in the prior plot. The cross plane thermal conductivity is highly dependent on boundary scattering, which in turn is dependent on phonon velocities. We can also see that velocities increase with compressive strain and decrease with tensile strain, and it is that trend that we see here in the thermal conductivity. However this does not tell us how strain affects thermal conductivity in the direction of transport. b_dir t_dir Quantum Mechanics + Semi-Classical (DFT/DFPT) (BTE) a Eqs. From [6] Eqs. From [7] I) Density Functional Perturbation Theory: 𝑖ℏ 𝜕Φ 𝒓, 𝑹; 𝑡 𝜕𝑡 = − 𝐼 ℏ2 2𝑀𝐼 𝜕2 𝜕𝑹𝐼 2 − 𝑖 ℏ2 2𝑚 𝜕2 𝜕𝒓𝑖 2 + 𝑉 𝒓, 𝑹 Φ 𝑟, 𝑅; 𝑡 a) Born Oppenheimer Approximation: - mass of nucleus >> mass of electron − 𝑖 ℏ2 2𝑚 𝜕2 𝜕𝒓𝑖 2 + 𝑉 𝑟, 𝑅 Ψ 𝒓 𝑹 = 𝐸 𝑹 Ψ 𝐫 𝐑 − 𝐼 ℏ2 2𝑀𝐼 𝜕2 𝜕𝑹𝐼 2 + 𝐸 𝑹 Φ 𝐑 = 𝐸Φ(𝑹) b) E(R) is called the Born Oppenheimer Potential Energy surface and by taking its 2nd derivative we get the Interatomic force constants: 𝐶𝐼𝐽 𝛼𝛽 ≡ 𝜕2 𝐸 𝑹 𝜕𝑅𝐼 𝛼 𝜕𝑅𝐽 𝛽 c) From which we can find the normal mode frequencies 𝜔 through solving: II) Phonon Boltzmann Transport Equation: a) Thermal conductivity 𝐾 𝛼𝛽 𝑇 = 𝑘 𝑗 𝒒 ℏ𝝎𝒋 𝑞 𝒌𝑇 𝟐 𝝏𝑓 𝑀𝐵 𝜕𝐸 × 𝜏𝑗 𝒒 𝑣𝑗 𝛼 𝒒 𝑣𝑗 𝛽 (𝒒) b) Group velocities 𝑣𝑗 𝛼,𝛽 = 𝜕𝜔𝑗 𝒒 𝜕𝒒 ; 𝑤ℎ𝑒𝑟𝑒 𝜕𝒒 = 𝜕𝒒 𝑥, 𝜕𝒒 𝑦, 𝜕𝒒 𝑧 c) Total Relaxation time 1 𝜏𝑗 𝒒 = 1 𝜏𝑗,𝑁(𝒒) + 1 𝜏𝑗,𝑈(𝒒) + 1 𝜏𝑗,𝐼 𝒒 + 1 𝜏𝑗,𝐵 𝒒 𝑡,𝛽 𝐶𝑠𝑡 𝛼𝛽 𝒒 − 𝑀𝑠 𝜔2 𝒒 𝛿𝑠𝑡 𝛿 𝛼𝛽 𝑈𝑡 𝛽 = 0 A phonon is the fundamental particle of vibration within a solid. Phonons are represented as plane waves with wave-vector k (or q) Phonons have 4 modes of vibration: 1) Longitudinal Acoustic 2) Transverse Acoustic -and 2 more for optical- (6 modes in 3D systems since there are 2 transverse directions) [5] Optical Mode Acoustic Mode • Here we consider biaxial strain where: 𝜖 𝑥 = 𝜖 𝑦 ≠ 𝜖 𝑧 • Naturally Biaxial Strain breaks cubic symmetry. Conclusions: In-plane thermal conductivity for a 20nm thin-film is not dominated by boundary scattering and has a small dependence on strain. Thermal transport is anisotropic between the [100] and [001] directions, resulting in a decrease by a factor of 2 in the [001] direction.