ARTICLE
Thermal conductivity and air-mediated losses in
periodic porous silicon membranes at high
temperatures
B. Graczykowski1,2,3, A. El Sachat1,4, J.S. Reparaz1,8, M. Sledzinska1, M.R. Wagner 1,9, E. Chavez-Angel1,Y.Wu
5,
S. Volz5,6,Y.Wu
5, F. Alzina1& C.M. Sotomayor Torres 1,7
Heat conduction in silicon can be effectively engineered by means of sub-micrometre porous
thin free-standing membranes. Tunable thermal properties make these structures good
candidates for integrated heat management units such as waste heat recovery, rectification
or efficient heat dissipation. However, possible applications require detailed thermal char-
acterisation at high temperatures which, up to now, has been an experimental challenge. In
this work we use the contactless two-laser Raman thermometry to study heat dissipation in
periodic porous membranes at high temperatures via lattice conduction and air-mediated
losses. We find the reduction of the thermal conductivity and its temperature dependence
closely correlated with the structure feature size. On the basis of two-phonon Raman spectra,
we attribute this behaviour to diffuse (incoherent) phonon-boundary scattering. Furthermore,
we investigate and quantify the heat dissipation via natural air-mediated cooling, which can
be tuned by engineering the porosity.
DOI: 10.1038/s41467-017-00115-4 OPEN
1Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain. 2NanoBioMedical Centre,
Adam Mickiewicz University, ul. Umultowska 85, PL-61614 Poznan, Poland. 3Max Planck Institute for Polymer Research, Ackermannweg 10, 55218 Mainz,
Germany. 4Department of Physics, Universitat Autonoma de Barcelona, Campus UAB, Bellaterra, 08193 Barcelona, Spain. 5Laboratoire dEnergetique
Moleculaire et Macroscopique, Combustion, CNRS, CentraleSupelec, Grande Voie des Vignes, 92295 Chatenay-Malabry, France. 6Laboratory for Integrated
Micro-Mechatronics Systems, CNRS UMI2820, Institute of Industrial Science, University of Tokyo, 4-6-1 Komaba, Meguro-ku, Tokyo 153-8505, Japan.
7ICREA Pg. Lluís Companys 23, 08010 Barcelona, Spain.
8
Present address: Institut de Ciència de Materials de Barcelona, ICMAB-CSIC, Campus Universitari
de Bellaterra, E-08193 Bellaterra, Spain.
9
Present address: Institute of Solid State Physics, Technische Universität Berlin, Hardenbergstr. 36 10623 Berlin,
Germany. Correspondence and requests for materials should be addressed to B.G. (email: [email protected])
NATURE COMMUNICATIONS |8: 415 |DOI: 10.1038/s41467-017-00115-4 |www.nature.com/naturecommunications 1
Progress in the last few decades in nano-scale thermal
transport has enabled a significant degree of control over
heat and sound propagation by lattice vibrations—phonons.
The latest investigations on the thermal properties of silicon,
the most common material in electronics, micro-electro-
mechanical system and nano-electro-mechanical systems
(NEMS) and photonics, have pointed to nanostructuring as a
highly efficient approach to acoustic phonon engineering1–7.In
this context, the reduction of the characteristic sizes in nanowires
and ultra-thin membranes, surface nano-engineering and
patterning of holes in thin membranes have been used to modify
the acoustic phonon dispersion and lifetimes of relevance for
hypersound and heat transport2,5,7. On the one hand, structures
with periodically arranged holes have shown coherent features,
such as a modified phonon dispersion relation with band folding,
and therefore they can be termed as truly phononic crystals
(PnCs). On the other hand, single crystalline porous silicon
membranes, with sub-micrometre feature sizes, suppressed lattice
thermal conductivity κand potentially encouraging power factor,
are considered as a practical realisation of the phonon-glass
electron-crystal concept and therefore are expected to enhance
the thermoelectric performance or the figure of merit (ZT) of
silicon8. Moreover, they offer the possibility of modifying the
temperature dependence of κby nanostructuring and thereby
provide a tool to control the directional heat flow to enable,
e.g., heat rectification schemes9–11. Nevertheless, the phenom-
enon of the reduced thermal conductivity in periodic porous
membranes (or PnCs) and its physical origin, i.e., the contribu-
tion of coherent (wave-like) and incoherent (particle-like)
processes, the role of structure porosity, lattice parameter, dis-
order and surface roughness are still a matter of a lively debate
based on a variety of theoretical models and experimental tech-
niques. Up to now all the experimental efforts, made mostly by
using suspended heater-thermometer platforms8,12–15 or, more
recently, time domain thermoreflectance16,17, have focused on a
temperature range close or below room temperature (RT). For
example, the results obtained so far clearly show that the
reduction of κis correlated with the material porosity, which
cannot be explained by considering simply volume removal.
However, both techniques are burdened with assumptions
regarding interface thermal resistance and boundary conditions
that require additional modelling and free-fitting parameters.
Furthermore, they are not suitable for the high temperature
investigation of the thermal properties, crucial for most applica-
tions with a heat sink set at RT, and the possible contribution of
air to heat dissipation. In the regime where the surface-to-volume
ratio is high, the latter may dominate heat flow resulting in
thermal losses. Therefore, the balance between the influence of air
on the efficiency of the thermoelectric modules and the cost of
using evacuated modules have to be considered. On the other
hand, passive and efficient air-mediated cooling could be
considered as an advantage for heat dissipation in nanodevices.
The miniaturisation and resulting size-induced reduction of the
thermal conductivity may have adverse consequences to remove
fast and efficiently the redundant heat caused by Joule or radiative
heating. This may lead to overheating of the functional units in
silicon-based electronics, NEMS, photonic and optomechanical
devices, and thereby result in, e.g., a decrease of computational
performance, signal processing efficiency and the quality factor or
in a drift of the operational frequency.
In this work we report a new experimental insight into the
in-plane thermal properties of periodic porous silicon mem-
branes, with a special focus on the reduction of the thermal
conductivity and its evolution at high temperatures. Furthermore,
we investigate natural air-mediated cooling and demonstrate its
impact on heat dissipation in the considered structures.
To overcome disadvantages coming from techniques requiring
electrical contacts, transducers or data analysis with free-fitting
parameters, we employ the contactless technique of two-laser
Raman thermometry (2LRT)18,19.
Results
Experimental techniques. 2LRT has recently been shown to be a
suitable method for thermal mapping with sub-micrometre
spatial resolution and for determining κof ultra-thin silicon
membranes18, including the effect of the native oxide19.In
principle, it can be applied to any material in membrane format,
where the absorbed power can be considered uniform along the
thickness, exhibiting a detectable temperature-dependent phonon
Raman scattering signal.
The fabrication of periodic porous membranes made of a
square lattice of cylindrical holes (Fig. 1) in the membrane was
based on electron beam lithography and reactive ion etching. As
the basis and reference sample we used commercially available,
single crystalline silicon (001), t=250 nm thick membranes with
a window size of 3.2 × 3.2 mm2placed on a thick silicon square
frame. Three samples labelled S1, S2 and S3 of an intentionally
similar hole diameter of about d=135 nm differ in the lattice
constant ameasured as 300, 250 and 200 nm, respectively.
Figure 1b displays a schematic top view of the sample designed to
keep radial symmetry about the x
3
axis running through the
heating spot. The heating island of a diameter about 5 μm has no
holes to avoid the diffraction of the heating beam and subsequent
uncertainties in the absorbed power measurements. This island is
surrounded by the periodic porous membrane with a total
diameter of 100 μm. In general, the radial symmetry of the sample
simplifies measurements to a single line scan in the x
1
direction,
which determines a temperature profile. Figure 1c and d shows
typical scanning electron microscope (SEM) images of the
samples with a visible heating island. For details see Methods
Section.
Si membrane
Heating island
Heating spot
Probe spot
Scan
[110]
Periodic porous
membrane
x2
x1
aa
nd
t
[110]
ab
cd
Fig. 1 Samples for two-laser Raman thermometry experiment. aSchematic
picture of the periodic porous membrane—square lattice of cylindrical holes
in the free-standing membrane, where t=250 nm is the membrane
thickness, dis the hole diameter, ais the lattice parameter and nstands for
the neck. bTop view of the sample design depicting the temperature scan
direction and crystallographic orientation. The porous area is enclosed by
two circles with diameters of about 5 and 100 μm. c,dScanning electron
microscope images of sample S2 (a=250 nm and d=140 nm). Scale bars
in c,dare 20 and 2 μm, respectively
ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/s41467-017-00115-4
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Thermal conductivity of pristine membrane. Although the
reduction of the thermal conductivity in silicon membranes or
thin films is well studied, there is a lack of experimental data at
temperatures above 400 K20. Moreover, the reported values of κat
RT vary over a wide range, e.g., for 250 nm thick membranes the
values reported go from 60 to 100 W m−1K−120–25. Thus, we first
characterise the bare membrane pointing out at the same time the
principles of the experimental approach. Figure 2a displays two
steady-state heat flow temperature profiles obtained at the same
heating power P
0
=8.125 mW, but under different ambient
conditions. The plots labelled vacuum and air correspond to the
ambient pressure of 10−3and 103mbar, respectively. These
measurements were also performed in the porous samples to
ascertain the effect of the air-mediated cooling discussed later. It
follows from Fig. 2a that by applying a relatively small power to
the membrane in vacuum a temperature difference from 920 to
420 K is established over a distance of 500 μm which, in practice,
allows us to determine κas a function of temperature over a wide
range by a single measurement. To do so we assume the
temperature gradient to be zero in the x
3
direction and diffusive
in-plane heat flow, for which the radial heat is governed by
Fourier’s law: P
0
/(2πrt)=−κ(T)dT/dr, where ris the distance
from the centre and tis the membrane thickness. Then, taking
rdT=dr¼dT=dðln rÞ¼ξðrÞwe obtain the following expression
1100
900
Temperature (K)
700
500
300
Position r (μm) In (r)
–400 –200 200 400 –12 –11 –10 –9 –8
Position r (μm)
S1
S2
S3
FEM
Ref. 29
Ref. 30
0.0 0.2 0.4 0.6 0.8
–50
–50
–25
–25
0
0
25 25
50
50
X2 (μm)
In (r)
1000
800
600
400
200
–12–13 –11 –10
0
1100
900
700
500
300
–40 –20 20 400
1.0
0.8
0.6
0.4
0.2
0.0
Temperature (K)
Volume reduction factor
Temperature (K)
PnC edge
Filling fraction
X
1
(μm)
400 K
900 K
ab
cd
ef
Fig. 2 Two-laser Raman thermometry results. Linear and corresponding logarithmic temperature profiles of a,bpristine 250 nm thick silicon membrane
and c,dsample S3 with lattice parameter of a=200 nm and hole diameter of d=130 nm. Red circle-line and blue square-line plots indicate experimental data
obtained in vacuum and air, respectively. Solid lines in b,ddenote theoretical fits using Eq. (2). Error bars in a,b,c,drepresent experimental uncertainties of
the measured temperature (see Methods Section). eVolume reduction factor εas a function of porosity calculated by FEM and using analytical
expressions. fMeasured temperature map of the sample S1 with lattice parameter a=300 nm and hole diameter d=135 nm
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for the thermal conductivity as a function of temperature:
κTðÞ¼ P0
2πtξrðÞ
:ð1Þ
Consequently, knowing P
0
and t, the value of κ(T) can be
extracted from the logarithmic temperature profile T(ln r) shown
in Fig. 2b. For the simplest case where κ≠f(T), κwould be
directly obtained from Eq. (1) and the slope of the linear fit of the
logarithmic temperature profile. Nevertheless, T(ln r) plotted in
Fig. 2b is clearly nonlinear and, for convenience, we use a specific
function to fit the experimental data. Assuming that κ(T) of the
membrane and the porous samples resembles the behaviour of
bulk silicon at high temperatures, we put κ(T)=αT−βin Fourier’s
law26, which after integrating, gives the function:
T¼AP01βðÞ
2πtαln r
1=1βðÞ
;ð2Þ
where Ais a constant of integration while the fitting parameters
βand αdetermine κ(T). The curve in Fig. 2b is obtained by fitting
Eq. (2) to the experimental data points. We note that this
procedure is valid for a data range starting at a distance of a few
micrometres from the central point r=0. This is due to the
Gaussian shape of the heating source and possible effects of the
photo-generated free charge carriers at high concentration, which
may diffuse away from the illuminated area and contribute to the
total thermal conductivity with two counteracting effects27. The
data points that do not match a preliminary fit using Eq. (2), and
therefore from the sample area where the assumed dependence of
κ(T) does not apply are thus omitted. In this study we use the data
obtained for r>3μm, for which we assume the role of the lattice
thermal conductivity is predominant and we impose its power
function dependence. The effect of thermally and photo-generated
electron–hole pairs is discussed in Methods Section. The
determined κ(T) is plotted in Fig. 3a and assuming as before κ
(T)=αT−β, extrapolated to close the ends of the 300−1000 K
range. An unambiguous reduction of the thermal conductivity and
of its dependence on temperature with respect to bulk silicon26 is
clearly observed. This behaviour is commonly attributed to the
phonon mean free path (MFP) suppression due to the phonon
diffuse boundary scattering in addition to phonon–phonon
scattering processes7,19,21,22,24,25. From Fig. 3a we obtain
κ
mem
=78 ±6Wm
−1K−1for the pristine membrane at 300 K,
which corresponds to a two-fold reduction of κwith respect to
bulk Si. This value is in good agreement with recent transient
thermal grating measurements, on silicon membranes25. Further-
more, it is consistent with other existing data obtained for silicon
thin films of similar thicknesses by means of the harmonic Joule
heating technique23. A further decrease of κcan be achieved by
reducing the membrane thickness and by surface nano-
engineering down to a value of 8 ±2Wm
−1K−1for 9 nm thick
membrane, as recently reported19,24.
Thermal conductivity of periodic porous membranes. Now we
shall move on to the results obtained in porous membranes in
vacuum. As previously formulated for the case of the bare
membrane, we use the temperature line scan and the corre-
sponding temperature profile in logarithmic scale to determine
κ(T). These are shown in Fig. 2c and d, respectively, for the case
of the sample S3 (Table 1). For the in-vacuum measurements we
notice that by heating with less power than for the pristine
membrane, namely P
0
=0.41 mW, we create a similar tempera-
ture rise at r=0. Simultaneously, the created temperature
gradient covers a range of about 300–1000 K over a distance of
only 50 μm. This suggests a reduction of κcaused by the lattice of
holes in the membrane. In principle, this effect could result from
volume removal, while the intrinsic material properties such as
the thermal conductivity might be preserved. Therefore, κ
exp
(T)
of the porous membranes determined from the experimental data
and Eq. (2) has to be scaled by a factor, which takes into account
the specific porosity of the membrane. This can be done by using
a volume correction factor εthat can be obtained analytically or,
more accurately, numerically by solving the diffusive heat trans-
port model using the finite element method (FEM)28. Figure 2e
shows a plot of εas a function of the sample filling fraction ϕ
obtained from FEM, which we use here, compared with two
analytical expressions by Eucken29 and later by Hashin and
100
Bulk Si26
Measured
Extrapolated
250 nm Si membrane
S1
S2
S3
10
Thermal conductivity (Wm–1 K–1)
1
300
1.0 1.0
0.8
0.6
0.4
0.2
0.0
0
0.1
0.01
100 200 100 300200
Neck n (nm)
T
Neck n (nm)
400 500 600 700 800 900
Temperature (K)
~T –0.31
~T –0.47
~T –0.55
~T –0.85
~T –1.31
norm = /mem
a
bc
Fig. 3 Temperature dependence of the thermal conductivity. aThermal
conductivity of porous membranes (S1, S2 and S3) and 250 nm thick
membrane as a function of temperature. Solid and dashed lines denote
measured and extrapolated κ, respectively. The shaded areas indicate
experimental uncertainty derived in Methods Section. bNormalised
thermal conductivity of porous membranes as a function of the neck
size nfor three example temperatures; circles—300 K, triangles—600 K,
squares—900 K. The error bars are experimental uncertainties of κderived in
Methods Section. The dashed lines are guides to the eye connecting data
points of the same temperature. The solid line is a reference plot indicating
n2dependence. cThe exponent βgoverning the temperature dependence
of κas a function of the neck size n. The dashed line is a guide to the eye and
the arrow indicates βof the 250 nm thick membrane. Horizontal and vertical
error bars indicate the experimental uncertainties of βand ndetermined
from the data fit using Eq. (2) and from scanning electron microscope
images, respectively
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Shtrikman30. Once κ
exp
(T) and ϕof the porous membrane are
obtained, we calculate the intrinsic thermal conductivity κ(T)
from the formula: κ(T)=κ
exp
(T)/ε(ϕ). It is worth mentioning that
Eq. (2) used to determine κ
exp
(T) is valid when the radial sym-
metry of the temperature field is preserved. That symmetry is
preserved for the pristine silicon membranes as reported in ref. 18.
In such a case the in-plane thermal conductivity of the membrane
follows κof bulk silicon in diffusive regime that is isotropic. This
is not obvious for the periodic porous membranes with the
in-plane structural anisotropy driven by the square lattice of
holes. Nevertheless, in our case, the 2D thermal map obtained for
the sample S1 and shown in Fig. 2f clearly demonstrates radial
symmetry with no signature of the structure driven in-plane
thermal anisotropy, and thus validates the use of Eq. (2). We note
that the probe laser spot measures an average temperature over its
size, which comprises several unit cells. In this way we obtain an
effective κof porous membranes, which is the thermal con-
ductivity that would correspond to a homogeneous membrane in
the Fourier approximation. Figure 3a shows a comprehensive plot
of κ(T) obtained for three porous samples and the 250 nm thick
bare membrane. These results show the reduction of the thermal
conductivity in the porous samples with respect to the pristine
membrane, which depends on the sample feature size and tem-
perature. It is commonly accepted that the observed reduction of
the thermal conductivity in the periodic porous membranes is
dominated by the diffuse phonon-boundary scattering, but the
contribution of coherent effects seems to remain a matter of
ongoing debate2,5,13,15,28,31. It is clear that phonons with MFP
shorter than the lattice parameter can be treated purely as par-
ticles governed by the bulk dispersion relation. Otherwise, the
wave-like nature can be manifested as, e.g., dispersion relation
zone folding, flattening of branches and band gaps, all of which
have implications on the phonon group velocity and density of
states (DOS) and consequently on the thermal conductivity. In
such a case periodic porous membranes are synonymous to 2D
PnCs. In bulk silicon, MFPs span a broad range with MFPs longer
than 1 μm contributing to ~ 50% of the total thermal con-
ductivity25,32–34. Furthermore, as predicted theoretically, in the
case of 250 nm thick silicon films, phonon MFPs exceeding 300
nm contribute to about 20% of the total thermal conductivity35.
In this work we investigate structures with lattice parameters
smaller than 300 nm. Therefore, it might be reasonable to con-
sider the suppression of the thermal conductivity due to coherent
Bragg scattering. Indeed, the results of Brillouin light scattering
(BLS) from similar structures showed unambiguously the mod-
ification of the phonon dispersion, exhibiting typical features of
PnCs band diagram, such as zone folding, modification of group
velocity and band gap opening36. However, BLS probes the
coherence in a relatively low frequency range (up to tens of GHz),
which becomes relevant at very low temperatures, where most of
the heat is carried by long wavelength phonons5,17,37. At high
temperatures their contribution is negligible and, following recent
theoretical work, most of the heat in bulk silicon is carried by
phonons of frequencies around 5 THz31,32. Although this range
is beyond BLS, the change in the phonon dispersion relation in
periodic porous membranes (or PnCs), if any, can be captured
indirectly by two-phonon (second order) Raman scattering38–42.
The second-order Raman spectrum, which in bulk silicon is
dominated by overtones and, thus, has a striking similarity with
the two-phonon DOS, is sensitive to the phonon dispersion
relation along the Brillouin zone. Although it cannot be used to
determine the dispersion relation, any change (additional dips or
peaks) may be a clear indication of the modified phonon dis-
persion. In Fig. 4we compare second-order Raman spectra of the
pristine membrane and S1 measured in x3x1x1
ðÞx3scattering
geometry. As we can notice the measured range of 100–1100 cm
−1that corresponds to 1.5–16.5 THz shows no detectable singu-
larities in the Raman spectra of the S1 compared to the pristine
membrane. Furthermore, all spectral features are typical of bulk
Si. Thus, it is unlikely that coherent effects play a role in the
observed reduction of κ. We note that the latest results obtained
at RT provide contradictory conclusions and the comprehensive
understanding of this problem is still elusive15,43. As follows from
Fig. 3a, at first glance κ(T) decreases together with the sample
neck size, namely, the shortest distance separating holes, given by
n=a−d. Figure 3b depicts explicitly this behaviour in a log–log
plot of the normalised thermal conductivity κ
norm
at three
example temperatures as a function of the neck size. From these
data we can conclude that over a broad range of temperatures the
dependence of κ
norm
on the neck size can be approximated by a
function k
norm
∝n2. Interestingly, the same correlation function
has been found recently for similar structures, albeit smaller
feature size8. This trend holds for three example temperatures
and for a particular neck size the rate of decrease of κbecomes
smaller with increasing temperature. Since in our case the thermal
conductivity depends on temperature as κ(T)∝T−βwe can con-
clude that this behaviour is governed by the exponent β. The
latter parameter is plotted as a function of neck size in Fig. 3c.
Table 1 Characteristic sizes, coefficients and example experimental data for the porous membranes and the 250 nm silicon
membrane: a—lattice spacing, d—hole diameter, ϕ—filing fraction, ε—volume reduction factor, κ—thermal conductivity
(W m−1K−1).
Sample a (nm) d (nm) ϕε κat 300 K κat 600 K κat 900 K βκ
bulk
/κat 300 K
Membrane –0 0 1 77.9 ±8.1 42.5 ±4.3 30.1 ±3.2 0.85 ~2
S1 300 135 0.159 0.725 21.9 ±1.9 14.6 ±1.1 11.9 ±0.9 0.55 ~7
S2 250 140 0.246 0.604 8.5 ±0.9 6.2 ±0.6 5.1 ±0.5 0.47 ~18
S3 200 130 0.332 0.499 3.9 ±0.4 3.2 ±0.3 2.8 ±0.3 0.31 ~40
Experimental uncertainties of κare calculated using the error propagation described in Methods Section.
Intensity (a.u.)
TA(X)
2TA(L)
2TA(X)
2TA(W)
TO(X)+TA(X)
TO(Σ)+TA(Σ)
LO(Γ)
TO+LA
2TO
S1
Membrane
200 400 600 800 1000
Raman shfit (cm–1)
Fig. 4 One- and two-phonon Raman spectra. Data obtained for pristine
250 nm membrane and S1 at room temperature in x3x1x1
ðÞx3scattering
geometry. The arrows indicate critical points of the first Brillouin zone of
bulk silicon, where TA and TO are transverse acoustic and optical modes,
respectively, and LO are longitudinal optical modes38
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As can be seen, the temperature dependence of κbecomes weaker
with decreasing distance between holes. This points to the
diminishing role of phonon–phonon processes, which are over-
whelmed by the temperature-independent diffuse phonon-
boundary scattering. From Fig. 3b and c we deduce that both,
the reduction and temperature dependence of the thermal con-
ductivity, are related to the geometrical features of the samples. In
other words, by changing the neck size, the thermal properties
can be designed and tuned in a simple and efficient manner. All
the results discussed above are summarised in Table 1. We point
out that the thermal conductivity of the porous membranes can
be reduced at most by a factor of about 40 with respect to bulk
silicon at 300 K, thus approaching the amorphous limit of silicon
κ
a-Si
=1.7 W m−1K−134.
Air-mediated cooling. Up to this point, we have considered only
measurements performed in vacuum, i.e., where air-mediated
thermal transport via convection and conduction is negligible.
Air-mediated heat transport becomes rather important for the
porous membranes as operational building blocks in real-life
applications, besides those for outer space. In principle, air has a
poor thermal conductivity of about 2.623−6.763 × 10−2Wm
−1K−1
between 300 and 900 K44. However, in structures with high
surface-to-volume ratio and reduced thermal conductivity heat
dissipation via air has to be taken into account. This effect was
previously observed in graphene45, photonic crystals46 and nano
beams47. To analyse this issue in porous membranes we go back
to Fig. 2a–d, where we compare 2LRT results obtained in vacuum
and air. At first glance, the temperature profiles of the pristine
membrane and the example porous membrane shown in Fig. 2a
and c, respectively, clearly indicate heat losses caused by the
presence of air. Foremost, the temperature rise at r=0 for both
samples in air are significantly smaller than those measured in
vacuum. Consequently, if one were to use the conventional single
laser Raman thermometry the thermal conductivity measured in
air would be overestimated by factors of about 1.4 and 1.9 for the
membrane and S3, respectively. Likewise, κ(T) obtained from the
corresponding logarithmic temperature profiles recorded in air
(Fig. 2b and d) would be inaccurate.
In what follows, we use these data and propose an approach to
quantify the heat dissipation resulting from natural air-mediated
cooling in 2D systems. Let us return to Eq. (1), which determines
κin vacuum conditions at a temperature Tusing the measured
P
0
and ξ(r). The latter, considered simply as the slope of the
logarithmic temperature profile at r, is different for the vacuum
and air data, showing larger nonlinearity for data taken in air, as
seen in Fig. 2b and d. The only reason for this difference comes
from the air-mediated losses, denoted P
loss
, which occur in a
distance from 0 to r. The thermal conductivity κ(T) and thickness
tremain unchanged, thus in the case of the data obtained in air
Eq. (1) can be rewritten as:
κT0
ðÞ¼
P0Ploss
2πtξ0rðÞ;ð3Þ
where we denote the temperature and the slope of the
logarithmic temperature profile at any arbitrary point r
determined in air by T′and ξ′(r), respectively. By combining
Eqs (1) and (3) we obtain a simple expression that determines the
relative air-mediated losses:
Ploss
P0
¼1ξ0rðÞ
ξrðÞ
κT0
ðÞ
κTðÞ
;ð4Þ
Figure 5a depicts relative losses as a function of the surface-to-
volume ratio calculated for the membrane and the three porous
samples. Here, we compare heat dissipation from the lateral area
defined by the porous membrane, thus Eq. (4) is derived for
r=50 μm. The losses are seen to vary from about 8 to 85%,
while the general trend is clearly nonlinear. The latter may result
from the porosity and specific orientation of the samples,
in particular the holes, with respect to the gravity vector, which
allows free air flow through the holes. Consequently, the losses
grow faster than it would be expected from the increasing surface-
to-volume ratio. In general, one can conclude from these data that
air results in significant heat dissipation which, for the porous
and thin membranes, cannot be neglected. Unfortunately, this
limits the applicability of these structures in low-cost thermo-
electric devices, which have to operate at atmospheric pressure
with a reasonable performance. Nevertheless, this clear dis-
advantage might be favourable in cases where fast and efficient
cooling is required like in silicon-based electronics, mechanical
resonators, photonics and optomechanics. To analyse this issue
we plot in Fig. 5a the absolute value of losses P
loss
as a function of
surface-to-volume ratio. In this case, the use of relative losses may
result in a misleading conclusion, where the optimum surface-to-
volume ratio would need to be as large as possible. As shown
before the measured samples differ in the thermal conductivity,
therefore, to get approximately the same temperature rise at the
hot spot we applied a different heating power for each case. From
Fig. 5awefind that the porosity of S1 increases the absolute losses
by about 60% with respect to the pristine membrane, but this
100
1.0
0.8
0.6
0.4
0.2
0.0
1.0
0.8
0.6
0.4
0.2
0.0
0.010 0.015 0.020
Surface-to-volume ratio (nm–1)
40
30
20
10
0
80
Relative losses Ploss/P0 (%)
/ mem
Absolute losses Ploss (mW)
Thermasl conductivity (Wm–1 K–1)
60
40
20
0
a
b
Fig. 5 Air-mediated heat losses. aRelative (red diamonds,left axis) and
absolute (blue triangles,right axis) losses caused by air-mediated cooling as
a function of surface-to-volume ratio. bMeasured κ
exp
=κε (red circles) and
intrinsic κ(blue triangles) thermal conductivity at 600 K as a function of
surface-to-volume ratio. The dashed lines in a,bare guides to the eye.
The errors bars are derived using the error propagation of κderived in
Methods section
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trend is reversed when the surface-to-volume ratio is further
increased (samples S2 and S3). To understand this behaviour we
examine the measured κ
exp
and the intrinsic κthermal
conductivity, which are plotted at example temperature of 600
K in Fig. 5b. The extensive structure property such as the thermal
conductance is governed not only by the intrinsic κbut also by
the volume removal factor ε, thus the effective thermal
conductivity values of the porous membranes are even lower
and given by κ·ε. Now, if we compare Fig. 5a and b we see that
increasing surface-to-volume ratio results in two opposite effects:
the increase of the relative heat losses due to the presence of air
and the reduction of the thermal conductivity. In other words, if
efficient heat removal is the main priority then the effect of the
reduced κhas to be minimised. In practice this boils down to
making the porous membrane neck sufficiently larger than the
thermal phonon MFP.
Discussion
In summary, we investigated heat transport carried via lattice
conduction and air convection and conduction in porous
membranes made of square array of holes in 250 nm thick
monocrystalline silicon membranes by 2LRT. We show that the
in-plane thermal conductivity of silicon and its temperature
evolution from RT to about 900 K can be effectively reduced and
tuned by means of the sample geometrical feature, i.e., the neck
size. We also demonstrate that the thermal conductivity at RT can
be decreased down to about 4 W m−1K−1. This value reaches the
amorphous limit of silicon and to be achieved in a pristine
monocrystalline silicon membrane it would require a thickness
below 9 nm19,24. The measured two-phonon Raman spectra of a
pristine membrane and of periodic porous membranes show no
significant difference, and therefore we attribute the observed
reduction to the shortening of the phonon MFP due to diffuse
(incoherent) phonon-boundary scattering. Furthermore, we
determine the heat dissipation in porous membranes resulting
from the presence of air and find it to be significant and tunable
depending on the sample surface-to-volume ratio. These results
provide a new insight into the heat transport at the nanoscale,
with potential implications for future design of silicon-based
devices for, e.g., energy harvesting, effective heat dissipation in
sensors, optomechanics, photonics, requiring operation at high
temperatures.
Methods
Two-laser Raman thermometry. Figure 6shows schematically the concept of the
2LRT experiment performed on porous membranes. Light emitted by the fibre-
coupled continuous wave laser operating at λ
h
=405 nm is focused onto the sample
from the bottom by a long working distance microscope objective (×50 and
NA =0.55) acting as a Gaussian heat source with a waist size of about 2 μm. The
absorbed power P
0
is measured on site for each sample as the difference between
incident and transmitted plus reflected light intensities probed by a calibrated
system based on a cube non-polarising beam splitter (BS) with an error of ΔP
0
=
2%. The second laser beam with λ
p
=488 nm is focused on the sample from the top
by the microscope objective (×50 and NA =0.55) and works as a temperature
probe. The sample is mounted in a vacuum chamber, which is fixed to the heating
1000
ab
800
600
Temperature T (K)
Intensity (a.u.)
400
200
0
508
Raman shift ΔR (cm–1)Raman shift ΔR (cm–1)
512 516 520 524 500 510 520 530
Fig. 7 The two-laser Raman thermometry calibration. aTemperature as a function of silicon longitudinal optical phonon frequency. The circles indicate
experimental data points. The solid line stands for the linear fit with slope of -43.43 ±0.05 K(cm−1)−1calculated for data ranging between 300 and 870 K.
bRepresentative Raman spectra obtained in vacuum for S1 at different distances rfrom the heating spot; black,blue and red circles correspond to r=10, 20
and 40 μm, respectively. Solid lines indicate Lorentzian function fits of the corresponding experimental data
To raman spectrometer
Probe laser
p
= 488 nm
Tramsmitted beam
Vacuum
Reflected beam
Heating laser
h
= 405 nm
Scanning direction
Reference beam
BS
x
3
x
150×
NA=0.55
50×
NA=0.55
Sample
Fig. 6 Schematics of the two-laser Raman thermometry experiment. The
setup is based on the triple-grating Raman spectrometer (T64000, Horiba)
and the vacuum temperature controlled microscope stage (THMS350V,
Linkam). The non-polarising cube BS and three powermeters are used to
determine the absorbed power P
0
from intensities of the incident,
transmitted and reflected laser beams
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beam unit and motorised scanning stage. The latter is fixed in x
3
(vertical) direction
and allows scanning in the x
1
x
2
(Fig. 1b) plane with a spatial resolution of 0.2 μm.
The latter value, together with the probe beam spot diameter defined as 1.22λ
p
/
NA ≈1.08 μm, results in the total spatial uncertainty of the experiment Δr≈0.55
μm. The spectral position of the longitudinal optical phonon (LO) of silicon
(Δω
R
(300 K) =520.7 cm−1) is used as a temperature reference. Figure 7a displays a
2LRT temperature calibration curve from which the temperature coefficient
between 300 and 870 K was determined as dT/dΔω
R
=-43.43 ±0.05 K (cm−1)−1.
Figure 7b shows Raman spectra and their Lorentzian fits of the S1 sample at
different relative positions rbetween the heating and probe lasers. The total
temperature uncertainty was calculated using the error of the temperature coeffi-
cient and standard error of the Lorentzian function fit performed for each point.
The uncertainties in the measurements were assessed through error propagation of
the equation determining the thermal conductivity of the membranes κ=αT−β,
where α,βand their standard errors Δαand Δβ, respectively, are determined by
fitting experimental data by means of Eq. (2). The total error of the thermal
conductivity was determined from a formula Δκ=((∂κ/∂α)2Δα2+(∂κ/∂β)2Δβ2)1/2.
To minimise the influence of the probe laser beam on the measurement, its power
is set below the value (typically <2% of P
0
) that results in a measurable temperature
rise. In principle, for membranes placed on a thick non-circular frame the
assumption of the 2D map radial symmetry is valid when the line scan is sub-
stantially shorter than the membrane lateral size.
Thermally generated charge carriers. 2LRT enables measurements of κat high
temperatures where thermally excited free charge carriers may contribute to the
total measured thermal conductivity in two counteracting effects27.Thefirst
decreases the lattice thermal conductivity due to electron–/hole–phonon scattering,
while the second increases the total thermal conductivity due to the presence of
additional heat carriers. Both effects strongly depend on the carrier concentration.
In our case, according to bulk silicon data48 the expected carrier concentration at
about 900 K would be below 1018 cm−3. This value, according to ref. 27 renders the
electron–/hole–phonon scattering contribution negligible. At the same time the
additive effect of the electronic contribution is expected to be small26. If the elec-
tronic contribution has no size/pores-induced reduction it would account for about
0.3 W m−1K−1of the total thermal conductivity at 900 K, which is within the
experimental error for the data of all samples. This is the upper expected value
which, due to the high surface-to-volume ratio, can be further decreased. In prin-
ciple, for bulk silicon the electronic contribution to the thermal conductivity is
considered significant at temperatures above 1000 K. In our case this effect would be
manifested in the 2LRT logarithmic temperature profile as a mismatch between data
points and the dependence given by Eq. (2) gradually increasing with temperature.
Photo-generated charge carriers. The electron–hole lifetime τ
b
in bulk
silicon, depending on doping, purity and temperature can take values from
micro to milliseconds42,49,50. Nevertheless, the effective charge carrier lifetime
1/τ
eff
=1/τ
b
+1/τ
s
can be much shorter due to the surface recombination
characterised by the surface lifetime τ
s
. For the 250 nm thick membrane, carrier
diffusivity D=25 cm2s−1and surface recombination velocity S=8700 cm s−1
we estimate τ
eff
≈τ
s
=1.4 ns49,50. Thus, the diffusion length takes value of
L=ffiffiffiffiffiffiffiffiffi
DTeff
p=1.9 μm. Therefore, and due to the size of the heating spot, we
determine κfrom the experimental data obtained for r>3μm. In the case of the
probe spot, the absorbed laser power is typically lower than P
p
=50 mW then,
considering a 100% efficiency, we estimate a generation rate of G=P
p
/E
ph
≈
1.2×1014 s−1, where E
ph
≈4.1×10−19 J is the photon energy at λ
p
=488 nm. Taking
the probe spot diameter as 1 μm, membrane thickness 250 nm and carrier lifetime
as 1.4 ns we estimate the excess carrier concentration to be about 8.8×1017 cm−3.
This value goes below the critical value of 1019 cm−3, above which electron–/
hole–phonon scattering contribution has to be considered27. We note that both
τ
eff
and Lcan be further decreased by the presence of holes in porous membranes.
Sample preparation. The positive electron beam resist AR-P 6200 (Allresist) was
spun at 4000 r.p.m. for 1 min, followed by 1 min bake at 150 °C on a hot plate.
Electron beam lithography (Raith 150-TWO) was performed at 30 kV acceleration
voltage. After development in AR 600-546 (Allresist) for 1 min, the samples were
post-baked for 1 min at 130 °C on a hot plate. This step additionally hardened the
mask before the reactive ion etching. The pattern was transferred to silicon using
the Bosch process (Alcatel AMS-110DE). The source power was set to 500 W and
the flow of SF
6
and C
4
F
8
gases was of 150 and 100 sccm, respectively, and the
etching time was of 60 s. Finally, after pattern transfer the samples were placed in a
plasma system (PVA Tepla), and cleaned in 50 sccm O
2
at 400 W for 1 min. The
rms surface roughness of the holes was measured by high-resolution SEM
(Magellan 400L) to be about 2 nm. The diameter variation of the holes was
determined from SEM images to be Δd=±5 nm. Further details regarding the
fabrication process are reported in ref. 51.
Data availability. The data that support the findings of this study are available
from the corresponding authors upon request.
Received: 29 July 2016 Accepted: 2 June 2017
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Acknowledgements
We acknowledge the financial support from the FP7 project MERGING (Grant
No. 309150) and QUANTIHEAT (Grant No. 604668); the Spanish MINECO projects
nanoTHERM (Grant No. CSD2010-0044) and PHENTOM (FIS2015-70862-P), the
programme Severo Ochoa (Grant SEV-2013-0295) and funding from the CERCA Pro-
gramme / Generalitat de Catalunya. B.G. acknowledges the support from the Foundation
for Polish Science (Homing/2016-1/2) and Alexander von Humboldt foundation.
Author contributions
B.G., F.A. and C.M.S.T.: Conceived the work and wrote the manuscript. J.S.R., B.G.,
M.R.W. and E.C.A.: Designed and built the two-laser Raman thermometry. B.G. and
F.A.: Designed the samples. M.S.: Fabricated the samples and acquired the SEM images.
B.G., A.E.S. and J.S.R.: Performed the thermal conductivity measurements. S.V. and Y.W:
Contributed to the experimental data interpretation. C.M.S.T., B.G. and F.A.: Identified
the research topic and formulated the research questions. All authors contributed to the
discussion and interpretation of the results.
Additional information
Competing interests: The authors declare no competing financial interests.
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