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Removing Heat from Si with a “Thermal Circuit”: An Ab-Initio Study

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Electronic devices generate unwanted heat. The removal of this heat often involves a layer of a high‐thermal‐conductivity material X deposited at a strategic location on the Si chip. The layer is then cooled using a mechanical fan or more sophisticated means. The authors present here the results of ab‐initio molecular‐dynamics simulations which examine the thermal interactions between heat generated in Si and three materials X: Ge, C, and Si itself. The geometry is a very thin wire of atoms X located within a Si slab. Heat is periodically removed from the wire as if it were connected to a heat sink. The wire then absorbs heat from the warmer Si slab, thus cooling it. The rate at which the Si temperature drops depends on how efficiently heat crosses the Si|X boundary and then is absorbed by the wire. The interactions involve the coupling of Si‐Si vibrational modes with Si‐X (interface) and then X‐X (layer) modes. The worst performance occurs for XC as lower‐frequency Si‐Si modes must decay into higher‐frequency Si‐C then CC modes. This involves slow two‐ (or more) phonon processes. The fastest cooling rate occurs when the wire is made of Si itself: there is no Si|Si interface and the vibrational interactions involve only Si bulk modes which couple to each other resonantly. The Ge wire performs quite well, as the Si‐Si modes easily decay into lower‐frequency Si‐Ge and Ge‐Ge modes.
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Removing Heat from Si with a Thermal Circuit:An
Ab-Initio Study
Stefan K. Estreicher,* Trey M. Vincent, and Christopher M. Stanley
Electronic devices generate unwanted heat. The removal of this heat often
involves a layer of a high-thermal-conductivity material X deposited at a
strategic location on the Si chip. The layer is then cooled using a mechanical
fan or more sophisticated means. The authors present here the results of ab-
initio molecular-dynamics simulations which examine the thermal interac-
tions between heat generated in Si and three materials X: Ge, C, and Si itself.
The geometry is a very thin wire of atoms X located within a Si slab. Heat is
periodically removed from the wire as if it were connected to a heat sink. The
wire then absorbs heat from the warmer Si slab, thus cooling it. The rate at
which the Si temperature drops depends on how efficiently heat crosses the
Si|X boundary and then is absorbed by the wire. The interactions involve the
coupling of Si-Si vibrational modes with Si-X (interface) and then X-X (layer)
modes. The worst performance occurs for X55C as lower-frequency Si-Si
modes must decay into higher-frequency Si-C then CC modes. This involves
slow two- (or more) phonon processes. The fastest cooling rate occurs when
the wire is made of Si itself: there is no Si|Si interface and the vibrational
interactions involve only Si bulk modes which couple to each other
resonantly. The Ge wire performs quite well, as the Si-Si modes easily decay
into lower-frequency Si-Ge and Ge-Ge modes.
1. Introduction
Removing the excess heat generated by electronic devices is a
tricky issue facing modern electronics. A common approach is to
deposit a layer of some material X on the semiconductor
substrate, in the present case Si. This layer is then connected to a
heat sink or positioned near a mechanical fan. This material X is
often selected for its high thermal conductivity. However, before
being removed, heat must rst efciently penetrate this layer,
which is the focus of the present contribution.
We restrict ourselves to the phonon contribution to heat ow,
that is, to non-metallic materials X such as C, Ge, and Si itself.
The source of the excess heat is the active region of the device,
often a very small and highly-doped volume in which charge
carriers may dominate heat transport. However, the much larger
slabinside of which this active area is
builtis normally high-resistivity Si in
which heat is mostly carried by phonons
around room T. Thus, the problem dis-
cussed here involves Si-Si oscillators inter-
acting with the vibrational modes
associated with the Si|X interface, which
in turn decay into X-X modes.
At the atomic level, these interactions are
mostly independent of the details of the
geometrical conguration, unless it is heavily
strained (and therefore intrinsically unsta-
ble). For example, the frequency ranges of Si-
Cvibrationalmodesare quite similar for a
wide range of relaxed congurations.
Our thermal circuitconsists of a very
thin (for computational reasons) straight
wire of material X fully embedded within a
2D-periodic slab of Si, with X55C, Ge, or Si
itself. This choice covers materials with
higher-, lower-, and identical-frequency
vibrational modes as the Si host slab.
The wire geometry is theoretically conve-
nient but we do not suggest that it is the
optimal conguration or is easily imple-
mented in actual devices. The geometry of
the Si|X interface must be optimized: at
layers, indentations, etc. This issue is not addressed here. We
focus here on how efciently heat transfers from a Si substrate to
the layer X around room T.
Note that our wire is small, with only a few X-X vibrational
modes. Most of the interactions involve the Si|X interface. Such
defect-related interactions are best done at the rst-principles
level. Considerably larger systems can be handled with empirical
potentials, but it is not clear that such potentials correctly
describe the interface dynamics. The present work focuses on
these dynamics at the rst-principles level.
Section 2 contains an overview of the issues involved when a
heat front encounters a defect, as predicted by earlier ab-initio
molecular-dynamics (MD) simulations. Section 3 describes the
methodology and the supercell. Section 4 contains our results for
X¼Ge, C, or Si. Section 5 summarizes the key points.
2. Heat Flow and Defects
Real materials always contain defects, which are dened here as
any disruption to the perfect crystalline order: point defects and
impurities, precipitates, dislocations, grain boundaries, surfaces,
etc. In the vicinity of a defect (as well as in nanostructures), the
Prof. S. K. Estreicher, T. M. Vincent, Dr. C. M. Stanley
Physics Department
Texas Tech University
Lubbock, TX 79409-1051, USA
E-mail: stefan.estreicher@ttu.edu
The ORCID identification number(s) for the author(s) of this article
can be found under https://doi.org/10.1002/pssa.201800427.
DOI: 10.1002/pssa.201800427
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periodicity of the underlying crystal is lost. The concept of a
Brillouin zone becomes less useful and Umklapp processes are
no longer dened. The interactions are better described in direct
rather than reciprocal space.
The phonon contribution to heat ow involves the inter-
actions between the normal vibrational modes (modes) of the
system. There are two types of modes in a real material.
[1,2]
The
host crystal (or bulk) modes are delocalized. A bulk-mode
excitation involves the motion of many (sometimes all) the host
atoms. If such a mode is excited above the background
temperature, it decays fast via a one-phonon process. Indeed,
bulk modes exist in semi-continuous bands: all the modes have
numerous neighborswith near-identical frequencies with
whom they can couple efciently. In contrast, defect-related
modes are localized in space at or near the defect. They involve
the motion of just a small number of atoms. We call them
Spatially-Localized Modes or SLMs. Once excited, they decay
slowly: SLMs do not couple efciently with delocalized modes,
even if they have the same frequency.
[2]
When discussing oscillators, the words fastand sloware
dened in terms of the number of periods of oscillation. A fast
process occurs within a single period of oscillation or so, while a
slowprocess involves many periods. Calculations
[2]
of the
vibrational lifetimes of many SLMs and bulk modes shows that
bulk modes often decay within a single period of oscillation (a
few tenths of a ps) while the decay of SLMs require dozens or
even hundreds of periods (510 ps for two-phonon processes and
much longer when more phonons are involved).
[3]
The capture of phonons in defect-related SLMs for such
meaningful lengths of time is called phonon trapping. It is the
reason why defects hinder the ow of heat: they behave like tiny
sponges that capture small amounts of energy in SLMs. More
importantly, since the excitation survives for a long time, the
decay of the trapped phonons depends on the availability of
receiving modes rather than on the origin of the excitation. The
optical analogy is uorescence.
These processes are temperature dependent. We dene the
temperature window [T
0
,T
hf
] as the interval between the
background temperature T
0
and the temperature of the heat
front T
hf
as it arrives at the defect. Using k
B
T¼ħω, this becomes
the frequency window [ω
0
,ω
hf
]. If the defect has SLMs with
frequencies within that window, then phonon trapping occurs
fast by resonant coupling, a one-phonon process. If there are no
SLMs within that window, then phonon trapping by higher-
frequency SLMs requires two-(or more) phonon processes which
are at least one order of magnitude slower.
[3]
This is of particular importance when the defect is the interface
between two materials, in the present case Si and a material X. When
heat is generated in the Si region, it propagates via the coupling of
many Si-Si modes. Once the heat front reaches the Si|X interface, it
traps in Si-X SLMs. And then, the decay of the trapped phonons
depends on the number of receiving modes on both sides of the
interface. For example, in the case of a Si|Ge interface and around
room T, there are many more receiving modes on the Ge side than on
the Si side since Ge is heavier than Si and has more low-frequency
modes than Si. As a consequence, around room T, most of the heat
trappedattheSi|GeinterfaceeasilypenetratestheGelayerandthe
temperaturedifferencebetweentheSiandtheGesideremains
small. The opposite holds for the Si|C interface: most of the heat
trapped at the Si-C interface decays back into Siwhich contains many
more low-frequency receiving modes.Now,heataccumulatesonthe
Si side of the Si|C interface and a much larger temperature
discontinuity results. This results in a large Kapitza (or interface)
resistance. This is discussed in greater detail elsewhere.
[4]
Thus, the optimal material X in the present problem is not
necessarily the one with the highest thermal conductivity.
Indeed, heat must rst cross the Si|X interface and then be
absorbed by that material X within the temperature windows of
interest. Only then can it be usefully removed by some heat sink.
Finally, rst-principles calculations have also shown that the
reduction in the thermal conductivity associated with the surface
of the material is also associated with the surface SLMs. In the
case of a H-saturated Si nanowire,
[5]
the surface SLMs are the Si-
H stretch and wag modes. The former are very high in frequency
(2000 cm
1
) and are not thermally excited around room T, but
some of the latter (900 cm
1
) can become excited in the hot
region of the nanowire when a T gradient is set-up. Since these
modes are much higher in frequency than any Si-Si mode
(530 cm
1
), they require two-phonon processes to interact in
any way with the Si-Si modes in the bulk: the surface Si-H wag
modes couple resonantly to each other much faster than they can
decay into (or be excited by) Si-Si modes. The result is that heat
propagates independently in the bulk and on the surface. Thus,
the surface reduces the thermal conductivity because one must
wait for its contribution to arrive before reaching thermal
equilibrium. There is precious little interaction between bulk
and surface oscillators: the frequencies are too far apart.
3. Theoretical Approach
Our electronic-structure calculations are based on local density-
functional theory
[6,7]
as implemented in the SIESTA package.
[8,9]
The electronic core regions are replaced by ab-initio norm-
conserving pseudopotentials
[10]
optimized for SIESTA.
[11]
The
valence states are described with numerical pseudo-atomic basis
sets. We use double-zeta orbitals for H and C and add a set of 3ds
for Si and Ge. The surface orbital radii are optimized using the
approach proposed by Garcia-Gil et al.
[5,12]
The host material is a H-saturated Si slab in a box which is
periodic in the {x,y} plane. The box is larger than the nanowire in
the third (z) direction where the surface Si dangling bonds are
saturated with H. This construction allows the system (the slab
with or without an embedded wireof atoms X) to be fully
relaxed. The Si
230
X
20
H
128
supercell is shown in Figures 1 and 2,
for X¼Ge and X¼C, respectively.
The localization L
2
α
of the interface and wire vibrational
modes are shown in Figures 3 and 4, where αis the sum over all
the Ge and C atoms, respectively, as well as the interface Si atoms
(those directly bound to a Ge or C atom). This localization
number is obtained from the orthonormal eigenvectors e
α
is
of
the dynamical matrix, where s numbers the modes, αthe nuclei,
and i is the Cartesian coordinate. Since these eigenvectors are
normalized, the sum over all the atoms Σ
α
L
α
2
¼Σ
α
(e
α
xs
)
2
þ(e
α
ys
)
2
þ(e
α
zs
)
2
¼1. The localization
[13,14]
of mode s on the
group of atoms αis Σ
α
L
α
2
<1. Here, L
2
is plotted versus the mode
frequency ω. The modes with L
2
<0.05 are not shown (they are
not localized). Note that in the case of the Si|Ge interface, there
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are many low-frequency localized modes while the Si|C interface
is characterized by much higher-frequencies.
The ab-initio MD simulations are initiated with the supercell
preparation technique.
[13]
The eigenvalues and eigenvectors of
the dynamical matrix are used to prepare the slab in a linear
combination of normal modes with random mode phases and a
random distribution of mode energies that corresponds to the
desired initial temperature prole. Each specic set of phases
and energies is an initial microstate. No thermostat is used. This
process results in small T uctuations starting with MD time
step 1. However, there is an innite number of initial microstates
corresponding to the same macrostate and it is necessary to
average over ndifferent initial microstates. We average here over
n¼30 MD runs.
4. Results
The system is prepared in thermal equilibrium at 300 K. We
assume that the wire is connected to a thermal bath which
removes heat from the wire every 1.5 ps (without touching any Si
atoms) in such a way that the total temperature of the entire slab
drops by 8 K. This is achieved by stopping the MD run, re-scaling
the kinetic energy of all the atoms within the wire, and then
restarting the MD run. The total temperature of the supercell is
plotted versus time in Figure 5.
Every time heat is removed from the wire, the system goes out
of equilibrium as the Si slab becomes warmer than the wire. As
the MD run continues, heat moves from Si, through the Si|X
interface, and into the wire atoms X, and the temperature of the
Si atoms in the slab drops at some rate which depends on how
efciently heat moves from Si into the wire. This rate depends on
on the relative frequencies of the Si-Si, Si-X, and X-X modes.
Figure 6 illustrates the process in the case where X¼Ge. The
temperatures of all the Ge atoms (i.e., the Tof the wire) and of all
the Si atoms are shown. Every 1.5 ps, T
Ge
drops as we cool the
wire, and then rebounds toward the new equilibrium Tas the
Figure 1. The Si
230
Ge
20
H
128
slab is placed inside a 2D-periodic box (not
shown) which extends above and below the slab. The surface H atoms are
white, the Si atoms are blue, and the Ge atoms are magenta.
Figure 2. The Si
230
C
20
H
128
slab is placed inside a 2D-periodic box (not
shown) which extends above and below the slab. The surface H atoms are
white, the Si atoms are blue, and the C atoms are black.
Figure 3. Localization of the vibrational modes associated with the Ge
atoms in the wire (solid red lines) in the wire and the Si atoms directly
bound to a Ge atom (dashed blue lines) versus the mode frequency.
Figure 4. Localization of the vibrational modes associated with the C
atoms in the wire (solid black lines) in the wire and the Si atoms directly
bound to a C atom (dashed blue lines) versus mode frequency. Only the
modes up to 1000 cm
1
are shown. Note that most interface modes are at
much higher frequencies than in the case of Ge.
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wire sucks heat from the surrounding Si atoms. The net result is
that the Si slabs cools down.
We have tested three materials X for the thermal wire: Ge, C,
and Si itself. Ge is heavier than Si. Therefore, the Si-Si modes
have higher frequency than Si-Ge modes, which themselves have
higher frequencies than the corresponding Ge-Ge ones. The
exact opposite holds for C, which is lighter than Si.
4.1. Si Wire
The Si-Si modes in the bulk, at the interface,and in the wire
are identical: there is no interface and no Kapitza resistance, and
all of the modes are of the bulk type which couple to each other
faster than any impurity-related mode (SLM) can couple to bulk
modes. Therefore, we expect Si (in Si) to be the most efcient
material to remove heat from the slab.
4.2. Ge Wire
There are no deep gap levels associated with Ge impurities in Si.
Ge is not a recombination center for charge carriers. Then, the
Ge-Si bond length does not differ much from the Si-Si one, and
the Ge wire generates only weak strain in the Si slab (Figure 1). A
comparison of bond lengths shows that Ge-Si/Si-Si is off by only
1.6% while C-Si/Si-Si is off by some 18%. Thus, a Ge wire
would not become a strong gettering center for other impurities
(such as fast-diffusing transition-metals) which could have an
unwanted impact of the electronic properties of the material.
Finally, Ge is commonly mixed with Si and is an easy material to
play with.
4.3. C Wire
Interstitial C is an unwanted impurity in Si. Substitutional C is
electrically inactive but traps light impurities such as H and O
and then forms electrically centers. The main reason for this is
the C-Si bond is not only quite strong relative to the Si-Si bond
but also much shorter: the presence of C generates a
considerable amount of lattice distortion in the crystal (Figure 2).
However, diamond (and re-crystallized a-C) has a very high
thermal conductivity and it is easy of remove heat from a C layer
(or in our case, a C wire). However, as discussed above, heat
penetrates a C layer only reluctantly at low to moderate
temperatures. Figure 4 shows that the process would become
efcient only above 700 K or so.
Figure 7 compares the temperature of the Si, Ge, and C wires,
respectively. In each case, heat is removed from the wire every
1.5 ps in such a way that the total temperature of the entire cell
drops by 8 K. The T of the wire returns toward thermal
equilibrium the fastest in the case of Si while the C wire needs
much longer times.
The temperature of the Si slab T
Si
(t) is shown in Figure 8.We
applied a 75-point running average because the raw tempera-
ture data exhibit larger oscillations and the curves tend to
overlap. The rate at which the slab cools down depends of
course on the geometry and the amount of heat removed every
1.5 ps. Therefore, one should not extract quantitative informa-
tion from these curves. But the qualitative features show that
the most efcient material is Si itself while C is the least
Figure 6. Every 1.5 ps, the temperature of the Ge atoms (magenta curve)
drops as heat is removed from the wire. Then T
Ge
rebounds as it absorbs
heat from the surrounding Si atoms. The temperature T
Si
of the Si atoms
in the slab (blue curve) drops at a rate that depends on the type of atoms
in the wire.
Figure 7. Comparison of the recovery times required by the Si (blue,
upper curve), Ge (magenta, middle curve), and C (black, bottom curve)
wires. Heat is removed every 1.5 ps, and then the wires absorb heat from
the Si slab.
Figure 8. Temperature of the Si atoms in the slab as we remove heat from
the wire every 1.5 ps. The black curve correspond to a C wire (top), the
magenta curve to a Ge wire (middle), and the blue curve to a Si wire
(bottom). A 75-time-steps running average was used.
Figure 5. Total temperature of the slab versus time. Every 1.5 ps, heat is
removed from the wire is such a way that the temperature of the entire
system drops by 8 K. Note that the T fluctuations are very small.
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efcient material for extracting heat from Si. These results are
consistent with the heat ow study in ref.
[15]
. In particular, a
strong correlation exists between the rate at which Si cools
down and the relative frequencies of the X-related vibrational
modes.
5. Key Points
The supercell preparation technique was used to perform these
ab-initio MD simulations without thermostat in a Si
230
X
20
H
128
slab containing a thermal wire made of a material X. The MD
runs are averaged over 30 initial microstates. No thermostat or
empirical parameters are used.
We have calculated how fast the temperature of Si slab drops
(starting around room T) if heat is periodically removed from the
wire as if it were connected to a heat sink. Every time heat is
removed from the wire, the temperature of the Si slab
surrounding it begins to drop as heat crosses the Si|X interface
and enters the colder wire. The same calculations are repeated
with wires made of Ge, C, or Si itself. The purpose is to evaluate
which material is most efcient at removing heat from Si when
connected to it. Once the material has been identied, the details
of the geometry must of course be optimized.
We nd that the most efcient material is not the one with the
highest thermal conductivity, but the one where the vibrational
modes couple the most efciently to the bulk modes in the host
material (Si). This result is consistent with the heat-ow study on
ref.
[14]
. From an experimental perspective, it would be
interesting to compare the cooling efciency of a Si device via
a C versus a Ge layer, with the same layer geometry and at the
same operating temperatures.
The issue is to transfer energy from Si (Si-Si modes) through
Si|X interface (Si-X modes) into the material X (X-X modes). If X
is different from Si, there is a Kapitza resistance (R
K
) at the Si|X
interface. The calculation of this resistance directly from the ab-
initio MD data for the Si|SiO
x
interface is discussed elsewhere.
[4]
Since it is much easier for Si-Si modes to decay into lower-than
into higher-frequency modes, R
K
is much smaller when heat
propagates from Si toward a heavier material than a lighter one.
The recommendations or suggestions are to 1) minimize the
number of interfaces and 2) move heat from higher- toward
lower-frequencies materials. The thermal conductivity of
material X matters only after it has absorbed the heat.
One should keep in mind that, for computational reasons, our
wires are extremely thin. Therefore, there are more Si-X interface
modes than XX ones. In more realistic situations, such as a
thicker X layer on the Si substrate, the importance of XX modes
will substantially increase and we expect that heat transfer from
Si into C will be worse than from Si into Ge than we nd here.
These calculations are in the planning stage and may allow us to
draw more general conclusions.
Acknowledgments
The authors are most thankful for the generous amounts of computer
time provided by Texas Techs High Performance Computer Center and by
the Texas Advanced Computer Center.
Conflict of Interest
The authors declare no conflict of interest.
Keywords
ab-initio molecular dynamics, heat flow, interfaces, lattice thermal
transport
Received: June 6, 2018
Revised: July 20, 2018
Published online:
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Pseudopotential-based Density-Functional Theory (DFT) permits the calculation of material properties with a modest computational effort, besides an acknowledged tradeoff of generating and testing pseudopotentials that reproduce established benchmark structural and electronic properties. To facilitate the needed benchmarking process, here we present a pragmatic method to optimize pseudopotentials for arbitrary materials directly from eigenvalue sets consistent with all-electron results. This method thus represents a much needed pragmatic route for the creation and assessment of sensitive pseudopotentials for DFT calculations that has been exemplified within the context of the SIESTA code. Comprehensive optimized pseudopotentials, basis sets, and lattice parameters are provided for twenty chemical elements in the bulk, and for both LDA and GGA exchange–correlation potentials. This method helps addressing the following issues: (i) the electronic dispersion and structural properties for Ge, Pd, Pt, Au, Ag, and Ta better agree with respect to all-electron results now, (ii) we provide the expected metallic behavior of Sn in the bulk – which comes out semiconducting when using available pseudopotentials, (iii) we create a validated pseudopotential for LDA-tungsten, and (iv) we create the first Bi pseudopotential for SIESTA that reproduces well-known electron and hole pockets at the L and T points. We investigated the transferability of these pseudopotentials and basis sets, and predict a new phase for two-dimensional tin as well.
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