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overall thermal resistance in advanced properties of the materials at different the Boltzmann transport equation (BTE)
packages. The introduction of backside dimensional scales and levels of detail. (Figure 1b). The model additionally
power delivery schemes (moving the Each model can be used by itself, captures thermal effects at the nanoscale.
entire power distribution network depending on the use case of interest. The material properties of metals
to the chip’s backside) and novel The output of each of the models can and dielectrics, as characterized on
transistor architectures (such as gate- also be used as input for the next, dedicated test structures, are used
all-around nanosheets) in the logic enabling a complete analysis of a full as an input for the model. For the
roadmap may impact the temperature BEOL stack. The modeling work is dielectrics, for example, the so-called
of the metal lines as well—in a good combined with experimental data 3ω method was used to experimentally
or bad sense. obtained on test vehicles with industry- ext ract the ther mal conductivit y
A more comprehensive modeling relevant materials and dimensions. The of r eleva nt d iele ct r ic m at e r ia l s
framework is needed to capture the measured data are fed into the models, very precisely, revealing a value of
impact of these innovations on heat allowing for accurate calibration and 1.15W/mK for SiO 2 and 0.3W/mK for
propagat ion a nd to en ha nce ou r prediction of future scenarios. OSG3.0 (an organosilicate glass with 3.0
fundamental understanding of how dielectric constant).
heat propagates in nar row BEOL An overview of the different (sub-) In the third step, the researchers
structures. This framework should models zoom out to a larger par t of the
allow us to ident if y t he biggest In the first step, the researchers BEOL layout. A full BEOL stack’s
contributors to the warming up, predict investigate the materials at the atomic thermal properties are modeled using
how the BEOL’s thermal resistance level using the density-functional theory a 3D finite element modeling (FEM)
evolves with newer technology nodes, (DFT) (Figure 1a). With this model, approach, calibrated with self-heating
a nd m a ke r e c om me nd at ion s for they derive fundamental properties of measurements (Figure 1c). This final
thermal-aware interconnect design. electrons and phonons, i.e., the heat step gives a thermal conductivity/
carriers that move the energy inside resistance mapping of the BEOL stack
A multi-step approach using the material. and its individual layers, allowing for a
calibrated models In the next step, the heat conduction fast evaluation of the temperature rise
We have taken a modular approach within the materials is modeled for in the interconnect structure.
to assess the thermal properties of the varying material dimensions, from µm To f ully estimate how the heat
BEOL and developed different types to nm scale. An in-house developed propagates in the BEOL, the model
of models that capture the thermal modeling tool is used that is based on must also accurately account for the
heat exchange between the FEOL
and BEOL. We have developed a
methodology for estimating this heat
exchange. This ther mal coupling
is added as an extra “layer” to the
FEM model.

Applying the models to different
use cases: trends and useful
insights
The next sections discuss various use
cases when applying the models shown in
Figure 1.
T he ther ma l conduc t i v it y of
elemental metals drops at <10nm line
width. A typical outcome of the BTE
modeling work is an evolution of the
thermal conductivity of the materials with
decreasing line widths. For all evaluated
elemental metals, the thermal conductivity
significantly drops when line widths scale
into the 10nm region, which is relevant
for the local level of interconnects at
Figure 1: Schematic representation of the proposed hybrid thermal modeling approach: a) (top) Density- advanced technology nodes. This is
functional theory (DFT), b) (middle) Boltzmann transport equation (BTE), and c) (bottom) finite element illustrated in the graph in Figure 1b.
modeling (FEM) based models capture thermal properties at different dimensional scales.

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