Millisecond Electron-Phonon Relaxation in Ultrathin Disordered Metal Films at Millikelvin Temperatures

Abstract

We have measured directly the thermal conductance between electrons and phonons in ultra-thin Hf and Ti films at millikelvin temperatures. The experimental data indicate that electron-phonon coupling in these films is significantly suppressed by disorder. The electron cooling time $\tau_\epsilon$ follows the $T^{-4}$-dependence with a record-long value $\tau_\epsilon=25ms$ at $T=0.04K$. The hot-electron detectors of far-infrared radiation, fabricated from such films, are expected to have a very high sensitivity. The noise equivalent power of a detector with the area $1\mum^2$ would be $(2-3)10^{-20}W/Hz^{1/2}$, which is two orders of magnitude smaller than that of the state-of-the-art bolometers. Comment: 13 pages, including 3 figures

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Millisecond Electron-Phonon Relaxation in Ultrathin Disordered Metal Films
at Millikelvin Temperatures

M. E. Gershensona), D. Gong, and T. Sato
Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08854

B.S. Karasik
Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109

A.V. Sergeev
Dept. of Electrical and Computer Engineering, Wayne State University, Detroit, MI 48202

We have measured directly the thermal conductance between electrons and phonons in ultra-thin Hf
and Ti films at millikelvin temperatures. The experimental data indicate that electron-phonon coupling in
these films is significantly suppressed by disorder. The electron cooling time τε follows the T -4-dependence
with a record-long value τε = 25 ms at T = 0.04K. The hot-electron detectors of far-infrared radiation,
fabricated from such films, are expected to have a very high sensitivity. The noise equivalent power of a
detector with the area 1 µm2 would be (2-3)×10-20 W√Hz, which is two orders of magnitude smaller than
that of the state-of-the-art bolometers.


a) Author to whom correspondence should be addressed
E-mail: gersh@physics.rutgers.edu

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Future space far-infrared (FIR) radioastronomy missions will require significant improvement in
the sensitivity of radiation detectors in the 40-500 µm wavelength range and integration of the detectors
in large arrays for faster sky mapping 1. The photon-noise-limited noise equivalent power (NEP) of a
detector combined with a cooled space telescope is expected to be ~ 1x10-19 W/√Hz 2, or even lower
for narrow band applications. The state-of-the-art conventional bolometers currently demonstrate
NEP ~ 10-17 W/√Hz at 0.1 K, along with the time constant τ ~ 10-3 s 3,4.
Recently, we proposed a concept for a submm/FIR direct detector based on disorder-
controlled electron heating in superconducting microbridges 5. The hot-electron direct detectors
(HEDDs) operating at millikelvin temperatures can offer unparalleled sensitivity, along with the simplicity
of fabrication on bulk substrates, integration with planar antennas, and large-array scalability.
For HEDDs, both decrease of the device volume and increase of the time constant improve the
sensitivity 5. The time constant of HEDDs is determined by the electron cooling time, τε, due to electron-
phonon relaxation. To achieve high sensitivity, the materials with long time constant τε are needed. The
slowest energy relaxation is expected in properly designed disordered superconducting films 5. Since the
first measurements of the electron-phonon relaxation time in metal films at T = 25-320 mK 6, not much
has been added to our knowledge of the electron-phonon processes at millikelvin temperatures. In this
Letter, we present measurements of the electron-phonon cooling time at T < 1 K in disordered Hf and
Ti thin films, which are promising candidates for the HEDD sensor elements.
We have measured the electron-phonon cooling time in thin films of Hf and Ti on sapphire
substrates (the parameters of three samples are listed in Table 1). The films with the critical temperature
Tc = 0.3-0.5K and the superconducting transition width ∆Tc ≈ 2-7 mK were deposited by dc
magnetron sputtering. They were lithographically patterned into 5 µm wide and 10 cm long strips

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shaped as meanders. The large length ensures that one can neglect the outdiffusion of hot electrons into
cold leads 7 and that the electron-phonon coupling is the only mechanism for electron cooling.
For measuring τε, we used the well-developed hot-electron technique (see, e.g. 8). At T < 1K,
the electron-electron scattering rate exceeds the electron-phonon one by many orders of magnitude 9,
and the non-equilibrium distribution function for electrons is well-thermalized. The thermal conductivity
between electrons and phonons, Ge-ph = Ce/τε , can be found from the energy balance equation 8
( )
T
()
()()
K
TT
a
dT
TC
t
V
P
e
T
∫
T
e
e
12
)(
22
e
aa
e
t
g
+
−
===
++
, (1)
where Ce = γT is the electron heat capacity, τε = τε(1K)⋅T
-α, P = I2R is the Joule power dissipated in a
thin film of volume V, Te is the non-equilibrium electron temperature, and T is the equilibrium phonon
(bath) temperature. By applying different amounts of the Joule power to the film at a fixed bath
temperature, and measuring the corresponding increase of Te one can find tε from Eq. 1 [for Te –
T << Te, Eq. 1 can be linearized as
PTT
(
V
)
TCT
ee
/ )()(
−=
e
t
].
In our experiment, the resistance of a sample was measured at a very small ac current, Iac, by a
resistance bridge as a function of the bath temperature and the heating dc current Idc. As the electron
“thermometer” above the critical temperature (T > TC), we used the temperature dependence of the
quantum corrections to the normal-state resistance 9. Below TC, the sample was driven into the resistive
state by the magnetic field B. The resistive state is very sensitive to the electron overheating; this allows
to measure τε with an unparalleled accuracy.
We have verified that, when the measurements are performed at T < Tc in the resistive state,
the extracted cooling time does not depend on B. Only in this case the non-linear effects in

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superconductivity, e.g., depairing, can be safely ignored. Comparison between the data obtained in the
normal and resistive states is shown in Fig. 1. Firstly, the dependence τε(T) was extracted from the hot-
electron measurements in the resistive state at different values of B, which corresponded to the sample
resistance R ~ (0.2 –0.9)RN, RN is the resistance in the normal state. Secondly, the resistance was
measured as a function of Idc at a fixed T < Tc in the normal state, when superconductivity was
completely suppressed b y the magnetic field. To obtain the dependence R(Te), shown in Fig. 1, the
electron temperature Te for each value of Idc was calculated from Eq. 1 using the τε data obtained from
the measurements in the resistive state. Figure 1 shows that the dependence R(Te) coincides with the
temperature dependence R(T) measured in equilibrium (Idc = 0). The latter dependence is due to the
logarithmic quantum corrections to the resistance in a two-dimensional film 9 (dashed line is a guide to
the eye). This coincidence rules out non-thermal effects in the resistive state, and allows us to facilitate
measurements by taking advantage of a very high sensitivity of the resistive state to electron overheating.
The temperature dependences of the thermal conductivity between electrons and phonons, Ge-
ph, measured for two Hf samples with different Tc are shown in Fig. 2. With lowering the temperature,
Ge-ph decreases rapidly as T 5. For comparison, we also plot the theoretical temperature dependence of
the thermal conductivity between the film and sapphire substrate 10, Gb = (Rbd)-1 ~ T 3 (Rb is the thermal
boundary resistance between the film and the substrate, d is the film thickness). Figure 2 shows that the
lower the temperature, the easier to realize the hot-electron regime, Ge-ph << Gb, when the electron-
phonon coupling is the bottleneck of the energy transfer from hot electrons to the environment.
In order to calculate the electron cooling time from Ge-ph, we used the “bulk” values of the electron
heat capacity for Hf and Ti (see Table 1). The temperature dependences of τε for these films are shown
in Fig. 3. The cooling time exceeds 1 ms at T ~ 0.1 K, and becomes record-long, up to 25-30 ms, at

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the lowest temperatures T = 30-40 mK. Saturation of the temperature dependence of τε below 0.1 K,
observed for sample 3, can be tentatively attributed to electron overheating by the electronic noise in the
experimental set-up. Indeed, the noise power, which is sufficient to overheat the electrons by ~ 10 mK
at T = 50 mK, does not exceed ~ 3⋅10-14 W (the corresponding noise current ~ 0.3 nA) even for our
“macroscopic” samples.
The experimental dependence τε(T) ∝ T –4 is consistent with predictions of the theory of electron-
phonon interaction i n disordered metals for the “dirty” limit, when the electron scatterers (impurities,
defects, and film/substrate interface potential) vibrate the same way as the host atoms 11. The condition
of the “dirty” limit, qTl << 1 (qT is the wave number of thermal phonons, l is the electron mean free
path), is satisfied in the studied films over a broad range T < 50 K. In disordered films, the transverse
phonons govern the electron-phonon coupling 12. Acoustic impedances of Hf and Ti are close to the
impedance of sapphire substrate (rut ? 2.6×107 g/m2s), so vibrations of the film-substrate interface are
expected to be identical to the phonon modes in the film. Under these conditions, the electron cooling
rate is given by 13
5
t
4
22
4
11
)(
20
),(
B
r
F
F phe
u
Tk
e
v
aTa
s
p
ette
h
==
−
−
−
, (2)
where τe-ph-1(T,εF) is the electron-phonon scattering rate for an electron at the Fermi level (ε = εF), σ is
the film conductivity, ut is the transverse sound velocity, vF is the Fermi velocity, ρ is the density of the
film. The numerical factor, a, describes energy averaging ( a = 0.107 for the T 4-dependence of the
relaxation rate
14). The calculated dependences τε(T) are shown in Fig. 3. For both materials, an
excellent agreement between the experimental data and the theory was obtained with no fitting
parameters.