Influence of Phonon dimensionality on Electron Energy Relaxation
Influence of Phonon dimensionality on Electron Energy Relaxation
J. T. Karvonen and I. J. Maasilta
Nanoscience Center, Department of Physics, P.O. Box 35, FIN-40014 University of Jyväskylä, Finland.
We studied experimentally the role of phonon dimensionality on electron-phonon (e-p) interaction
in thin copper wires evaporated either on suspended silicon nitride membranes or on bulk substrates,
at sub-Kelvin temperatures. The power emitted from electrons to phonons was measured using
sensitive normal metal-insulator-superconductor (NIS) tunnel junction thermometers. Membrane
thicknesses ranging from 30 nm to 750 nm were used to clearly see the onset of the effects of two-
dimensional (2D) phonon system. We observed for the first time that a 2D phonon spectrum clearly
changes the temperature dependence and strength of the e-p scattering rate, with the interaction
becoming stronger at the lowest temperatures below ∼ 0.5 K for the 30 nm membranes.
PACS numbers: 63.22.+m, 63.20.Kr, 85.85.+j
It is an established fact that at sub-Kelvin tempera-
tures the thermal coupling between conduction electrons
and the lattice becomes very weak [1]. This has signifi-
cant implications for the operation of low-temperature
detectors and coolers [2], or for any solid-state sys-
tems where dissipation and cooling are relevant. Low-
temperature electron-phonon (e-p) interaction has been
studied widely during the past decades, but mostly only
for the case in which the phonons are fully three di-
mensional (3D) [3, 4, 5, 6]. However, due to signifi-
cant advances in fabrication of thin suspended structures,
many practical devices and detectors exist in which the
phonons are expected to move freely only within the
plane of a membrane, forming a quasi-2D system [7].
The question how the two-dimensionality of the phonon
modes influences e-p interaction has been addressed the-
oretically for certain cases [8, 9, 10], but no clear exper-
imental observation of the effect has been reported to
date, although several attempts have been made [11, 12].
In this paper, we show for the first time experimen-
tally that the electron-phonon interaction clearly changes
depending on the dimensionality of the phonons, as ex-
pected from theory. E-p coupling was measured with the
help of sensitive NIS tunnel junction thermometry [13],
for thin Cu wires on suspended silicon nitride (SiNx)
membranes with thickness varying from 30 nm to 750
nm, which spans the transition from 2D to 3D phonons.
In addition, samples with identical Cu wires on bulk
substrates were also measured for comparison. For the
thinnest membranes, the e-p interaction was strengthened
in comparison with the bulk samples, and its tempera-
ture dependence changed significantly, as is predicted by
the theory [8, 9, 10]. The change was large enough to
give indirect evidence that the dispersive (ω ∼ k2), flex-
ural modes of the membrane likely play a major role in
the e-p interaction.
In the presence of stress-free boundaries, the bulk
transversal and longitudinal phonon modes (with sound
velocities ct and cl, respectively) couple to each other
and form a new set of eigenmodes, which in the case
of a suspended membrane are known as the horizontal
shear modes (h), and symmetric (s) and antisymmet-
ric (a) Lamb modes [14]. The frequencies ω for the h
modes are simply ω = ct
+ (mπ/d)2, where k‖ is the
wave vector component parallel to the membrane sur-
faces, d is the membrane thickness and the integer m is
the branch number. However, the dispersion relations of
the s and a Lamb modes cannot be given in a closed an-
alytical form, but have to be calculated numerically. The
lowest three branches, dominant for thin membranes at
low temperatures, have low frequency analytical expres-
sions: ωh = ctk‖, ωs = csk‖, and ωa =
k2‖, where
cs = 2ct
− c2t )/c
is the effective sound velocity of
the s mode, and m⋆ = ~
− c2t )/3c
is an
effective mass for the a-mode ”particle”. This lowest a-
mode with its quadratic dispersion is mostly responsible
for the non-trivial behavior of the e-p interaction [9, 10].
Note that already a single free surface affects the modes
[15] and the e-p interaction [16], as the bulk modes cou-
ple and form another new set of eigenstates, including
the surface localized Rayleigh-mode. Thus, the widely
observed result for e-p power flow P = ΣV (T 5e − T
from a metal volume V with Te the electron and Tp the
phonon temperature, is not expected to hold even for
thin enough films on bulk substrates.
A schematic of the Cu wire samples on suspended sil-
icon nitride membranes and the used measuring circuit
is shown in Fig. 1. 17 samples were made on either sus-
pended membranes or bulk substrates, where nitridized
(100) Si wafers with 30, 200 and 750 nm thick low-stress
SiNx top layers were used as the substrate for both cases.
The suspension of the SiNx membranes (size 600×300
µm2) was achieved by anisotropic backside wet etching of
the silicon substate in KOH, and the metallic structures
were fabricated using standard e-beam lithography and
multi-angle shadow mask evaporation techniques. As the
e-p interaction strength is sensitive to the thickness and
disorder level of the metal [17], we minimized its effect
by evaporating the Cu wires of a specific thickness on
all the different substrates simultaneously. Ultrathin Cu
layers (t=14-30 nm) were used to strengthen the effect of
the thin membranes. The oxide layer forming the tun-
nel junction barriers was produced by thermal oxidation
of Al. Table I presents the essential dimensions of the
http://arxiv.org/abs/0704.0336v2
samples discussed in this paper, measured by scanning
electron (SEM) and atomic force (AFM) microscopies.
The electron mean free path l was determined from the
resistance of the wire at base temperature 60 mK, using
the accurately measured dimensions of the wire.
TABLE I: Parameters for samples. M= suspended SiNx mem-
brane and B= bulk substrate. B6 had an oxidized Si sub-
strate.
Sample SiNx d Cu t V l τ (0.2K) τ (0.8K)
(nm) (nm) [(µm)3] (nm) (µs) (µs)
M1 30 14 2.71 5.7 2.6 0.16
B1 30 14 2.46 4.9 7.1 0.030
M2 200 14 2.44 4.6 15.0 0.11
B2 200 18 3.67 4.1 6.4 0.045
M3 30 19 5.50 11.2 2.2 0.30
B3 30 19 4.62 9.8 4.3 0.034
M4 750 22 6.09 10.3 3.1 0.030
B4 750 22 5.87 8.7 3.9 0.013
M5 30 32 6.09 22 1.8 0.31
B5 30 32 5.09 19 2.7 0.038
B6 - 32 7.10 22 1.6 0.031
CuAl Nb/Al
FIG. 1: (Color online) A Schematic of the suspended samples
and the measuring circuit. Red lines are the normal metal
Cu, light gray Al for SINIS-junctions and dark gray Al or Nb
for SN-junctions.
We used the hot-electron technique [3] to measure the
e-p interaction by overheating the electrons by Joule heat
power P and measuring the resulting electron tempera-
ture Te. All the samples had two electrically isolated
Cu normal metal wires next to each other (Fig. 1). The
longer wire (L = 500µm) was heated by applying a slowly
ramping voltage across the pair of superconducting Nb
(or Al) leads in direct metallic contact to Cu, forming
SN junctions. These junctions provide excellent electri-
cal, but very poor thermal conductance due to Andreev
reflection, as the junctions are biased within the super-
conducting gap ∆. Thus, due to the lack of outdiffu-
sion of electrons and the long length of the wire, input
heat is distributed uniformly in the interior of the wire
and the electron gas cools dominantly by phonons, in-
stead of diffusively [18] or by thermal photons [19]. Since
L >> Le−e, the electron-electron scattering length, elec-
tron temperature is also well defined without complica-
tions from non-equilibrium [20]. In our sample geometry
the electron temperature is measured with two additional
Al leads forming a NIS tunnel junctions pair (SINIS) in
the middle of heated wire, as a function of input Joule
power P = IV measured in a four probe configuration.
The purpose of the short Cu wire, with additional SI-
NIS thermometer on it, is to give an estimate of the local
phonon temperature Tp, as the e-p power flow depends
on both Te and Tp.
The current-biased Al SINIS thermometer is ideally
suited to measure temperature below a few Kelvins, [2]
due to its high sensitivity (in our DC measurement ∼ 0.1
mK at 0.1 K) and low power dissipation. In addition,
for all the data here, the SINIS voltage vs. temperature
response follows the BCS theory without fitting param-
eters very accurately at least down to ∼ 0.2 K, where
typically saturation sets in. This saturation depends on
the strength of the e-p interaction (size of thermometer
and type of substrate) and the amount of filtering, and
thus we conclude that it is most likely caused by external
noise heating. For this reason we take the most conser-
vative approach and assume that all saturation is caused
by it, in which case we can use BCS theory to convert
the measured voltage data for all temperatures.
Even if the electrons lose their energy overwhelmingly
to the phonons in our sample geometry, it is still pos-
sible that the measured temperature is not only deter-
mined by the e-p interaction. This is because the emitted
phonons could be removed so ineffectively from the mem-
brane that the phonon transmission becomes a bottleneck
for the energy flow. Bulk scattering of phonons at low
temperatures is very weak [7], even for thin disordered
membranes [21], as is boundary resistance for thin films
on bulk substrates [22, 23]. In contrast, almost noth-
ing quantitative is known about the boundary resistance
between a thin metal film and a thin 2D membrane, or
between a thin 2D membrane and a bulk substrate. How-
ever, it seems clear that if the combined metal film and
membrane thickness is below the thermal wavelength of
the phonons, the phonon modes in the two materials are
strongly coupled, leading to an effectively non-existent
boundary resistance. Hence, if we check that the mem-
brane temperature Tp is not too high compared to Te
(effective enough hot phonon removal), we can be confi-
dent that the measured Te reflects the e-p interaction.
Figure 2 shows the main result of the measurements,
with Te and Tp plotted vs. the heating power density
p = P/V for all membrane thicknesses (30 nm, 200 nm
and 750 nm). In addition, data from a few represen-
tative bulk samples are shown. Compared to the cor-
responding bulk substrate sample (B4), Te of the 750
nm membrane (M4) shows no difference at all, and it
effectively behaves as bulk. This is reasonable, because
for the 750 nm membrane the estimated dimensional-
ity cross-over temperature [24, 25] Tcr = ~ct/(2kBd) is
∼ 30 mK, with ct = 6200 m/s for SiN. The phonon
temperatures Tp, however, show a big difference: The
0.1 1 10 100 1000
 of M3      T
 of M2
 of M4       T
 of B1-B6
 of M1
 of M2
 of M4
 of B1 and B2
 of B4
Heating power density [pW / ( m)3] 
FIG. 2: (Color online) Measured electron and phonon tem-
peratures Te and Tp versus the applied heating power density
in log-log-scale.
bulk samples show almost no response from the satura-
tion value of the thermometer ∼ 190 mK, whereas the
membrane phonons heat up measurably, most likely due
to the boundary resistance between the membrane and
the bulk. Nevertheless, this increase in Tp for all sam-
ples is small enough not to influence the e-p interaction.
For the 200 nm thick membrane (M2) (Tcr ∼ 110 mK),
at low heating power densities [p < 40 pW/(µm)3] the
temperature dependence follows the behavior of the bulk
sample (B2), although with a difference in the absolute
value. This shows that the strength of the e-p coupling
weakens compared to the bulk. At higher powers and
temperatures (p > 40 pW/(µm)3, where Te > 0.6 K),
Te starts to increase more rapidly in the membrane sam-
ple, most likely due to the boundary resistance effects.
The phonons in the 30 nm thick membrane sample (M1)
are expected to be in the 2D limit at low temperatures
(Tcr ∼ 0.5K), and a clear sign of this can be seen in
Fig. 2 as a strongly different behavior of the measured
Te vs. p curve with respect to all other samples. Below
∼ 6 pW/(µm)3 the e-p coupling is notably stronger (Te
lower) than in the corresponding bulk (B1) or any other
sample, but again at highest temperatures the influence
of other effects starts to dominate over the e-p coupling.
To study the temperature dependence of the data in
Fig. 2 more accurately, we plot the logarithmic deriva-
tives d(log p)/d(logTe) in Fig. 3 (a)-(c). For low heat-
ing powers (T ne >> T
p ) Pe−p ≈ T
e , where n is the
power law of the e-p interaction, thus in that regime
d(log p)/d(logTe) = n. Typically this exponent is n ≈ 5
for thicker (t > 30 nm) metal films on bulk substrates
[3, 4, 17], if the disorder in the film is not too strong
[26, 27, 28]. From Fig. 3 (a) we first of all see that
for the 30 nm membrane sample M1, the difference to
the bulk sample B1 is very clear. The M1 data has a
0.1 1 10 100 1000
 M1   B1
 M2   B2
 M4   B4
Heating power density [pW / (  m)3]
FIG. 3: (Color online) Numerical logarithmic derivatives of
the measured data in Fig. 2. (a) Te data for M1 and B1, (b)
Te data for M2 and B2, (c) Te data for M4 and B4.
plateau of n ∼ 4.5 between p = 0.1 - 6 pW/(µm)3, while
for B1, n continuously decreases from much higher val-
ues. Note that the strong increase of d(log p)/d(logTe)
below p ∼ 0.1 pW/(µm)3 is caused by the saturation
of the Te measurement, and not by the e-p interaction.
The point where n starts deviating from n = 4.5 cor-
responds to Te ≈ 0.4 K, which is surprisingly consistent
with the estimated Tcr ∼ 0.5 K. In contrast, the tempera-
ture dependence of the 200 nm membrane (M2) and bulk
(B2) samples [Fig. 3 (b)] are identical with each other
and with the 30 nm bulk sample (B1), as long as the
e-p interaction is dominant (up to 40 pW/(µm)3). The
750 nm membrane (M4) and bulk (B4) samples also give
identical values of n [Fig. 3 (c)]. The difference between
sample pairs M4,B4 and M2,B2 is caused by the Cu wire
thickness, which is expected to influence the temperature
dependence strongly [16, 27].
Finally, we discuss the effect of the Cu wire thickness
on the measured e-p interaction. The results for the
thinnest 30 nm membrane samples, with Cu thickness
t = 14,19 and 32 nm are shown in Figs 4 (a) and (c).
It is apparent that the metal film thickness has only a
minor effect on the e-p interaction on thin membranes,
and only influences the boundary resistance in the 3D
limit, by increasing its effect for thicker t, as expected.
However, for wires on bulk substrates, Figs 4 (b) and (d),
the effect of the Cu wire thickness on e-p interaction is
more profound. The thinner the Cu film, the more its
temperature dependence deviates from n = 5, which, for
comparison, is observed for a more typical t = 32 nm
Cu wire on oxidized Si (B6). This behavior is qualita-
0.1 1 10 100 1000
0.1 1 10 100 1000
(a) (b)
Heating power density [pW /( m)3]
FIG. 4: (Color online) (a) Te versus p = P/V for 30 nm mem-
brane samples M1,M3,M5. (b) Te versus p for bulk samples,
from top to bottom B1 (top), B3, B5 and B6 (bottom). (c)
d(log p)/d(log T ) of the data in (a). (d) d(log p)/d(log T ) of
the data in (b). From top to bottom: green line B1 (top),
magenta B3, blue B5, Red B6 (bottom). In (d) noise has
been filtered to help the eye.
tively consistent with the predicted effect of the surface
phonon modes [16], but could also depend on the disor-
der, as the thickening of the film increases the mean free
path l (Table I) and pushes the sample closer to the clean
limit. An apparent exponent as high as ∼ 7 could pos-
sibly be explained by the combination of strong disorder
and surface modes, but again, detailed theory is lacking.
In conclusion, we have obtained the first clear evidence
that the electron-phonon interaction at low tempera-
tures changes quite significantly when the phonon modes
become two-dimensional. To quantify the effects, the
electron thermal relaxation times τ = γV Te/(dP/dTe),
where γ = 100 J/K2m3 for Cu, are presented in Table I
for all the samples at two temperatures Te = 0.2 and 0.8
K. At Te < 0.5 K, the thinnest membranes can have a
a factor 2-3 strengthening effect, whereas at higher tem-
peratures the thermal relaxation from membranes can be
an order of magnitude weaker compared to bulk samples.
The membrane close to transition region (d=200 nm) was
shown to have a weaker (∼ factor of two) e-p interaction
strength than the bulk samples. Thinning the metal film
on bulk substrates also leads to a sizeable weakening of
the e-p interaction. The observed power law exponent
for the 2D limit is consistent with n ≈ 4.5, and is much
smaller than the corresponding bulk exponent n = 6..7.
A reduction by more than a factor one gives indirect evi-
dence of the importance of the flexural, dispersive Lamb-
modes for the membrane electron-phonon interaction, in
agreement with theory [9, 10].
Discussions with T. Kühn and A. Sergeev and tech-
nical assistance by H. Niiranen are acknowledged. This
work was supported by the Academy of Finland project
Nos. 118665 and 118231, and by the Finnish Academy
of Sciences and Letters (J.T.K.).
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http://arxiv.org/abs/0705.1936
http://arxiv.org/abs/cond-mat/0702542
