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High superconducting critical temperatures depend systematically on the electron-
phonon interaction strength
C. Gadermaier1*, V. V. Kabanov1, A. S. Alexandrov1,2,3, L. Stojchevska1, T. Mertelj1, C.
Manzoni4, G. Cerullo4, N. D. Zhigadlo5, J. Karpinski5, Y.Q.Cai6, X. Yao6, Y. Toda7, M. Oda8,
S. Sugai9,10, and D. Mihailovic1
1Department of Complex Matter, Jozef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia.
2Department of Physics, Loughborough University, Loughborough LE11 3TU, United
Kingdom.
3Instituto de Física ‘Gleb Wataghin’/DFA, Universidade Estadual de Campinas-UNICAMP
13083-859, Brazil.
4IFN-CNR, Dipartimento di Fisica, Politecnico di Milano, Piazza L. da Vinci 32, 20133
Milano, Italy.
5Laboratory for Solid State Physics ETH Zurich, 8093 Zurich, Switzerland.
6Department of Physics, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai
200240, China.
7Department of Applied Physics, Hokkaido University, Sapporo 060-8628, Japan.
8Department of Physics, Hokkaido University, Sapporo 060-0810, Japan.
9Department of Physics, Art and Science, Petroleum Institute, P.O. Box 2533, Abu Dhabi,
UAE.
10Department of Physics, Faculty of Science, Nagoya University, Furo-cho, Chikusa-ku,
Nagoya 464-8602, Japan.
The origin of high critical temperature (Tc) superconductivity is still remarkably elusive.
To gain insight into the high- Tc mechanism, we need experiments which identify the
parameters that determine Tc and link them to the interaction(s) that establish the
superconducting state. Here we show that for pnictides, cuprates, and bismuthates Tc
depends systematically on the primary electron energy relaxation rate 1/τ1. We find that
1/τ1 is a direct experimental measure of the strength of the electron-phonon interaction
(EPI) and correlates with structural parameters, in particular the length of the
crystallographic a-axis. Tc(1/τ1) is a non-monotonic function with the maximum at
intermediate relaxation rates (~16 ps-1), suggesting that EPI provides the attractive
interaction for the high-Tc pairing mechanism, where the highest Tc occurs in the
crossover region between weak and strong EPI.
By far the most studied parameter for high critical temperatures is the doping level x and
related quantities such as the superfluid density1,2. Tc(x) shows a ubiquitous arc shape, where
the height of the arc, i.e. the maximum obtainable Tc is different for each compound and
hence determined by at least one other parameter. As the second parameter structural factors
have been proposed, in particular the length of the crystallographic a-axis3-5. Tc depends on
these lengths non-monotonically, with one characteristic maximum at an intermediate value.
Unfortunately, so far it was not clear how this parameter is linked to an interaction involved in
the superconducting mechanism.
Here we show that Tc depends systematically on the primary femtosecond electron energy
relaxation rate. We have studied the electron relaxation dynamics of 14 samples from the
pnictide, cuprate, and bismuthate compound families, using a purpose-built femtosecond
pump-probe spectroscopy system that enables pump and probe wavelengths from 500 to 700
nm with a pulse repetition rate of 250 kHz and sub-30 fs time resolution. This set-up uniquely
combines the time resolution necessary to resolve all the relevant relaxation processes and the
sensitivity to work at low excitation fluences (<20 µJcm-2), avoiding non-linear, intensity
dependent relaxation processes and sample heating or degradation. The transient reflectivity
response at different wavelengths and temperatures is used to deduce the primary relaxation
rate 1/τ1.
We measured the transient reflectivity of the selected compounds at room temperature (Fig.
1b). They show a primary relaxation rate 1/τ1 ranging from 3 to 25 ps-1 (i. e. relaxation time τ1
between 40 and 330 fs, Table 1). For the cuprates and bismuthates we also identify a slower
relaxation component 1/τ2. Some of the transient reflectivity signals show strong coherent
phonon oscillations6, whose strength depends on the shape of the transient spectra and does
not give direct information on the EPI strength. The τ1 values for YBa2Cu3O6+x,
Bi2Sr2CaCu2O8.14, and the pnictides agree well with room temperature data from the
literature7-10, while for HgBaCa2CuO4.1 and Ba1-xKxBiO3 they are measured for the first time,
thanks to our improved time resolution. For La2-xSrxCuO4 our recent results obtained with
much higher pump intensity11 are confirmed.
The superconducting critical temperature Tc of all these compounds depends systematically
on 1/τ1 (Fig. 1c). The data follow an arc Tc(1/τ1) with a distinct maximum around 1/τ1 ~ 16 ps-
1. For the secondary relaxation rate 1/τ2 we find no such systematic (see supplementary Fig 6)
and do not discuss it any further. The femtosecond pump-probe experiment thus directly
yields a parameter - 1/τ1 - that is uniquely related to Tc.
The relaxation of photoexcited electrons proceeds by transferring the electrons’ excess energy
to phonons and possibly other bosonic excitations. The electron-phonon relaxation time τe-ph
is directly related to the EPI strength expressed as the second moment λ<ω2> of the
Eliashberg spectral function12-14:
232
ωλπτ LB
pheTk
h=− (1)
with TL being the sample temperature. Early studies on conventional metallic superconductors
have shown good agreement between the measured 1/τe-ph and λ<ω2> obtained by other
methods15. Electron energy relaxation via interaction with higher energy bosons, i.e.
collective electronic excitations, has been inferred to model the transient spectra16 in
Bi2Sr2CaCu2O8+x, but has never been observed directly. The linear relation between τ and TL
in Eq. 1 is limited to the regime where ħω kBTL. At room temperature, for our compounds
the characteristic phonon energy is of the order of kBTL and the linearity is expected for
phonons but not for higher energy bosons. We confirmed the linearity for La1.85Sr0.15CuO4 in
the range TL = 140 ÷ 310 K (See Supplementary Figure 5). Hence, following the unanimous
assignment in the literature8-11,15, 17, 18, we ascribe τ1 to electron-phonon interaction (τ1 = τe-ph).
We therefore conclude that, for a given sample temperature, 1/τ1 is a direct experimental
measure of the EPI strength.
As the other parameter governing Tc besides doping the length a of the a-axis has been
proposed3-5, or equivalently some related structural parameter, such as the Cu-O in-plane
bond length or the anion height in the pnictides. In the cuprates, superconductivity occurs
only if doping- or pressure-induced strain shrinks a below its equilibrium value of a = 3.94 Ǻ,
with a maximum Tc at a = 3.84 Ǻ (Fig. 2a). We find that 1/τ1 and hence λ<ω2> increases
linearly with decreasing a (Fig 2b), meaning that a and the EPI strength are closely correlated
parameters. This correlation suggests that the strain in the CuO2 plane enhances the EPI.
We may extrapolate that also in the pnictides the EPI is related to a. In the cuprates a varies
significantly with doping, by about 0.1 Ǻ when going over the whole superconducting arc of
the phase diagram3, corresponding to a variation of the relaxation rate by 8 ps-1. On the other
hand19, a in Ba(Fe1-xCox)2Se2 changes by less than 0.002 Ǻ, which translates to less than 1 ps-
1. This explains why for Ba(Fe1-xCox)2Se2 the two parameters doping and 1/τ1 are largely
independent (See Fig 1c and Table 1), while for cuprates 1/τ1 changes appreciably with
doping9.
The EPI strength is often expressed via the dimensionless parameter λ (the zero-order
moment of the Eliashberg spectral function) rather than λ<ω2>. To calculate λ from λ<ω2>,
one would need to know the complete Eliashberg spectral function. As a crude approximation,
λ<ω2> is often simply divided by the square of an effective phonon frequency ω*. A
commonly used value for cuprates11,18 is 40 meV. Using this value, we estimate that the
maximum Tc in Fig. 1c is obtained at λ ~ 0.4. This estimate agrees well with λ obtained from
tunnelling spectroscopy, neutron scattering, photoemission, and calculations (See Refs 9-11,
17, 18 for references to pertinent works). Our experimental results differ from the
conventional Bardeen-Cooper-Schrieffer (BCS) prediction in two important aspects: Tc(λ) is
non-monotonic and the measured Tc values are significantly higher.
The finding that the EPI strength is a crucial parameter in determining Tc suggests that EPI is
a fundamental interaction in the high-Tc mechanism. The highest Tc for each compound
family is found in the crossover region between weak and strong EPI. Such non-monotonic
Tc(λ) with much higher critical temperatures than BCS is described in polaronic
superconductivity20 and percolative intersite bipolaron models21. Alternatively, there may be
two parallel correlations: the EPI strength with a, and a different interaction that actually
causes superconductivity with a. For such a scenario to work, at least one of the model
parameters22 must correlate with a, or an experiment must yield a systematic Tc(α), with α
being a direct experimental signature of the proposed interaction.
Acknowledgments This work was supported by the Slovenian Research Agency (ARRS)
(grants 430-66/2007-17, BI-CN/07-09-003, and BI-IT/11-13-001), ROBOCON 2011-2012,
the Royal Society (grant JP090316), and by the European Commission (grant ERG-230975
and the European Community Access to Research Infrastructure Action, Contract RII3-CT-
2003-506350 (Centre for Ultrafast Science and Biomedical Optics, LASERLAB-EUROPE)).
We thank A. J. S. Chowdhury, I. R. Fisher, and J. W. Hodby for samples and D. Brida, P.
Kusar, and D. Polli for discussions.
Author contributions V. V. K., A. S. A., and D. M. conceived the project and contributed
the original scientific ideas. C. M. and G. C. purpose-built the high-repetition rate NOPA for
this work. L. S. determined the Tc values and helped setting up the temperature dependent fs
experiment. Y. Q. C., J. K., T. M., M. O., S. S., Y. T., X. Y., and N. D. Z. provided samples.
C. G. performed the fs experiments, analysed the data, and wrote the manuscript with critical
input from all authors.
Author information The authors declare no competing financial interest. Please contact C.
G. at [email protected] for correspondence and reprint requests.
Methods
We measure the characteristic time scales τi of the various electron energy relaxation
processes using femtosecond optical pump-probe spectroscopy, where a first laser pulse (the
pump) excites the electrons and a second (weaker) pulse probes the relative change ∆R/R in
the reflectivity of the sample (See Figure 1a), which directly traces the electron energy
relaxation. In order to disentangle the various processes and determine their τi we use
different probe wavelengths, where the individual contributions appear with different spectral
weight. Since the τi depend on the sample temperature (See Equation 1), and additional
processes appear in the presence of low-temperature order23, we measure all samples at room
temperature. To avoid non-linear, intensity dependent relaxation processes and sample
heating or degradation, we keep the pulse fluence below 20 µJcm-2, with a pulse repetition
rate of 250 kHz. To provide the necessary time resolution and sensitivity, we purpose-built a
system that enables pump and probe wavelengths from 500 to 700 nm and sub-30 fs time
resolution (see supplementary information). Sub-25 fs pump pulses centred around 535 nm
with a repetition rate of 250 kHz are provided by a non-collinear optical parametric amplifier
following the lay-out of Ref. 24 (see supplementary information for a detailed description).
Probe pulses are provided by a white light continuum generated in 2.5 mm thick sapphire.
Adequate time resolution is ensured via spectral filtering of the chirped probe pulses25. In
several of our samples, this allowed the identification of previously unresolved relaxation
processes.
The sample growth and doping is described elsewhere26-32. Samples are glued to a copper
support using G Varnish. When needed, samples are cleaved or polished to obtain a reflecting
surface with low scattering. The reflectivity and the relative spectral weight of the signal
components may vary across the surface, but the relaxation times do not. To eliminate the
influence of the sample thickness we only used samples much thicker than 100 µm, while our
experiment probes less than the first 100 nm. The superconducting critical temperature Tc (see
Table 1) was determined by measuring the temperature dependence of the AC magnetic
susceptibility using a superconducting quantum interference device (Quantum Design MPMS-
XL-5).
t
0.0 0.2 0.4 0.6 0.8
0
1
2
3
4
5
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
0 5 10 15 20 25
0 180 360 540 720 900
0
20
40
60
80
100
120
140
Ba(Fe1-xCox)2As2
SmFeAsO1-xFx
YBa2Cu3O6+x
HgBa2CuO4+x
BiSr2CaCu
2O
8+x
La2-xSrxCuO4
Ba1-xKxBiO3
λ<ω2> (meV2)
T c (K
)
1/τ1 (ps-1)
Figure 1. a) Schematics of the pump-probe experiment. b) Transient differential reflection at
295 K for Ba(Fe0.93Co0.07)2As2, SmFeAsO0.8F0.2, YBa2Cu3O6.9, HgBa2CuO4.1,
Bi2Sr2CaCu2O8.14, La1.85Sr0.15CuO, and Ba0.64K0.36BiO3 (top to bottom). SmFeAsO0.8F0.2 and
La1.85Sr0.15CuO are probed at 620 nm, all others at 580 nm. Thin solid lines are fits as
described in the supplementary information. c) The superconducting transition temperature Tc
as a function of the primary relaxation rate 1/τ1. Large symbols indicate (almost) optimally
doped samples.
3.76 3.80 3.84 3.88 3.92 3.960
20
40
60
80
100
120
140
1
23
4
5
6
7
8
910
1112
13 14
1516
17
18
T c (K
)
crystallographic a-axis (A)3.75 3.80 3.85 3.90 3.95 4.000
5
10
15
20
25
1/τ 1 (p
s-1)
crystallographic a-axis (A)
Figure 2. a) Critical temperature for cuprates as a function of the a-axis (two times the in-
plane Cu-O distance as published in Ref. 3). Large symbols are our (almost) optimally doped
cuprate samples (same symbol/colour coding as Fig. 1), small squares (values and numbering)
are from Ref. 3, line is a fit to a Lorentzian b) Electron-phonon relaxation rates of the
optimally doped compounds as a function of the a-axis length. Solid lines are linear fits for
cuprates (blue) and pnictides (red).
Table 1. Stoichiometries, superconducting critical temperatures, and electron-phonon
relaxation times and rates of the investigated samples.
Sample Tc (K) τe-ph (fs) 1/τe-ph (ps-1)
BaFe2As2 0 300 3.3
Ba(Fe0.975Co0.025)2As2 0 330 3.0
Ba(Fe0.949Co0.051)2As2 20 320 3.1
Ba(Fe0.93Co0.07)2As2 23 300 3.3
Ba(Fe0.89Co0.11)2As2 10 300 3.3
SmFeAsO0.8F0.2 49 190 5.3
YBa2Cu3O6.5 63 100 10
YBa2Cu3O6.9 90 77 13
HgBa2CuO4.1 98 62 16
Bi2Sr2CaCu2O8.14 80 49 20
La1.9Sr0.1CuO4 30 42 24
La1.85Sr0.15CuO4 38 45 22
Ba0.64K0.36BiO3 35 47 21
Ba0.55K0.45BiO3 25 47 21
Supplementary Material
A. Methods
A1 femtosecond pump-probe spectroscopy:
In femtosecond optical pump-probe spectroscopy a first laser pulse (the pump) creates a non-
equilibrium electron energy distribution and a second (weaker) pulse probes the change in the
reflectivity R or transmittivity T of the sample as a function of pump-probe delay tPP. The
relative change ∆R/R(tPP) or ∆T/T(tPP) directly tracks the electronic relaxation processes33,34.
In order to yield useful data, a pump-probe experiment must meet the following requirements:
(i) provide sufficient time resolution to track all relevant processes, (ii) allow reliable
disentanglement of the different processes, (iii) avoid sample degradation and heating, and
(iv) provide a good compromise between the signal-to-noise ratio and the time needed for
alignment and data acquisition. The time-resolution is given by the cross-correlation between
the pump and probe pulse intensity profiles if both are transform-limited, i.e. as short as their
spectral width ∆ω allows (τ ~ 1/∆ω). Non-transform limited pulses are chirped, i.e. different
wavelengths are delayed with respect to each other. The time-resolution of a pump-probe
experiment using chirped pulses can be improved compared to the cross-correlation by
spectral filtering35 of the probe, which can be realised by placing a wavelength selective filter
after the sample; selecting a spectral slice of the probe pulse for detection also selects a
temporal slice. This way the time resolution can be almost as high as for transform-limited
pulses of the same spectral width of the unfiltered probe pulse35. We use nearly transform-
limited pump pulses of about 20 fs duration, modulated at ~1.5 kHz with a mechanical
chopper, and broadband probe pulses with partially compensated chirp and with a
monochromator placed between the sample and detector (photodiode using phase-sensitive
detection with a lock-in amplifier), giving sub-30 fs time resolution. The occurrence of
relaxation times as short as 40 fs shows that the high time resolution was crucial in enabling
our study.
Since the relaxation of excited electrons proceeds via different processes, the time traces
∆R/R(tPP) are never simple single-exponential decay curves, even if the individual relaxation
processes are. Their contribution to the signal varies with wavelength, therefore a probe pulse
covering a broad spectral range (i.e. having a broad spectrum and/or tunable centre
wavelength) is needed to capture and disentangle all processes involved.
In order to avoid sample degradation, heating, or intensity-dependent non-linear relaxation
phenomena, we kept excitation density below 20 µJcm-2. This requires averaging over a large
number of pulses, hence we used a 250 kHz set-up and a custom built non-collinear optical
parametric amplifier (NOPA), based on the design in Ref 36, which is able to run at such low
pump fluences (see section A3 for a detailed description).
A2 the high-repetition rate non-collinear optical parametric amplifier
We use 2.5 µJ, 50 fs pulses centred at 800 nm at a repetition rate of 250 kHz generated by a
regenerative amplifier (Coherent RegA 9000). About 10% of the pulse energy is used to
generate the white light continuum of the probe beam and another 10% to generate the white
light continuum to seed the NOPA. The remaining 2 µJ are frequency doubled to pump the
NOPA with approximately 750 nJ pulse energy at 400 nm. Unlike for the typical 1 kHz
NOPAs that run on much higher pump energies, here the β-barium borate (BBO) crystal is
placed in the Rayleigh range of the focussing lens to achieve the necessary pump intensities.
Pump and seed beams overlap in the 2 mm BBO crystal cut for type I phase matching at 3.7°.
At this angle phase matching is ensured over a large bandwidth and the bandwidth of the
output is given by the temporal overlap between the pump and the chirped seed pulse. Our
NOPA yields pulses of approximately 60 nm bandwidth, tuneable between centre
wavelengths of 500 to 650 nm by changing the pump-seed delay. At 535 nm centre
wavelength, which we use for our experiments, the amplified pulse energy is 20 to 30 nJ. A
prism compressor compensates most of the chirp and leads to pulse durations around 20 fs.
Figure 1. The high-repetition rate NOPA set-up.
A3 The fitting procedure
To extract the relaxation times, we fit the data to the following model curve: (i) As the
generation term we use a Gaussian, whose duration is a fit parameter. The duration is assumed
to be that of the convolution of the pump intensity profile with the transform limit of the
probe intensity profile or slightly longer (see section A2), which we find confirmed in all our
data. (ii) Relaxation processes with time constants 30 fs – 3 ps are modelled as single- or
double-exponential decays. (iii) slower relaxation processes are modelled as a plateau. (iv)
oscillatory components are modelled with the same generation term as the main signal, and
leaving as fit parameters amplitude, phase, frequency, and damping time. We measure all
samples at different wavelengths (however, due to wavelength dependent reflectivity and
signal strengths we do not always have a useful signal at all wavelengths we try) and two
perpendicular orientations. Although we let all fit parameters vary freely between different
wavelengths, for the same sample the best fits for the relaxation times vary only within the
experimental error. From this observation we conclude that at different wavelengths we look
at the same relaxation processes, only with varying spectral weight. Below we show three
examples of data and fits.
0.0 0.2 0.4 0.6 0.8-0.2
0.0
0.2
0.4
0.6
0.8
1.0
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)0.0 0.2 0.4 0.6 0.8
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 2. Normalised transient differential reflection of HgBa2Ca2Cu3O8.2 at 580 (left) and
620 nm (right) fitted with two exponentially decaying contributions (left: τ1 = 67 fs, τ2 = 1.5
ps, right τ1 = 57 fs, τ2 = 1.4 ps) and a plateau.
0.0 0.2 0.4 0.6 0.8-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)0.0 0.2 0.4 0.6 0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 3. Normalised transient differential reflection of Bi2Sr2CaCu2O8.14 at 580 (left) and 620
nm (right) fitted with two exponentially decaying contributions (left: τ1 = 47 fs, τ2 = 78 fs,
right τ1 = 52 fs, τ2 = 83 fs) of opposite sign and a plateau.
0.0 0.2 0.4 0.6 0.8-0.2
0.0
0.2
0.4
0.6
0.8
1.0
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)0.0 0.2 0.4 0.6 0.8
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 4. Normalised transient differential reflection of Ba(Fe0.949Co0.051)2As2 at 580 (left) and
620 nm (right) fitted with one exponential decay (left: τ1 = 350 fs, right τ1 = 320 fs), a plateau
and one oscillatory component of 170 cm-1.
The complete data and fits are shown in section C.
B. Interpretation of the observed relaxation behaviour and implications for the HTS
pairing mechanism
B1. Determining the EPI strength from the relaxation time
Our primary observation relates the dominant EER to Tc. The EER can be related to
fundamental EPI using Allen’s reformulation37 of the two-temperature model (TTM)38:
23 ωλπτ eB
ETk
h= (1)
where λ<ω2> is the second moment of the Eliashberg spectral function, Te is the initial
electronic temperature and τE is the energy relaxation time. The factor λ<ω2> is of
fundamental interest, as it quantitatively describes the weighted coupling of electrons to the
lattice39,40. (In traditional electron-phonon models of superconductivity, it is directly related to
Tc). The model was used in numerous studies on conventional and unconventional
superconductors to determine λ<ω2>, with reasonable agreement, particularly in conventional
metallic superconductors33 giving values between λ = 0.08 (for Cu) and λ = 1.45 (for Pb).
The main assumption of Allen’s model was that the photoexcited electrons (and holes) relax
on a timescale τe-e which is much faster than the electron-phonon relaxation rate τe-ph, which
was justified with the statement that “experimental monitors are so far consistent with the
notion that even if the electron and phonon distribution functions are athermal, they are still
adequately characterised by a thermal distribution at equivalent energy”. However, recent
time-resolved ARPES experiments41 showed that the hot electron distribution departs from
the thermal one, particularly at high energies.
Allen's TTM made one additional prediction: τe-ph should be strongly dependent on Te, and
thus also on pump laser intensity, but this important dependence was never confirmed by
experiments, suggesting that the model's applicability to the high-temperature
superconductors may need to be reexamined.
Recently Kabanov and Alexandrov (KA) (2008) suggested that the original assumption that
τe-e << τe-ph may not be valid in bad metals such as oxides, and obtained an exact analytical
solution to the Boltzmann equations without making this a-priori assumption42. They
calculated the e-e scattering rate time, which for a typical HTS material gives τe-e ~ 1 ps (here
we have used µc ~ 1 and EF = 0.5 eV), which is much longer than the measured relaxation
times. Their expression for the electron phonon energy relaxation is very similar to Allen's
formula, with the exception of a numerical factor of 2, and the important replacement of Te by
the lattice temperature TL:
232
ωλπτ LB
ETk
h= (2)
The theory eliminates the problem of determining the initial electron temperature, opening the
way to systematic direct and accurate measurements of λ<ω2>, since TL can be accurately
determined. Importantly, the model also predicts that τe-ph should be independent on pump
laser intensity, which is eminently verifiable. Making a rough estimate of the expected
relaxation time, using Eq. 2, the predicted τe-ph = 40-400 fs, depending on λ<ω2> indicates
that experimental resolution beyond the 70 fs used so far is required to systematically
investigate the EPI. Importantly, the model applies to the high-temperature state, which is
assumed to be described by Boltzmann’s equation, with electron correlations taken into
account with a pseudopotential. Thus, experiments measuring λ<ω2> need to be performed at
high temperatures in order to avoid correlation effects. Note that the KA formula differs from
the one by Allen only by a factor of 2 and the fact that the EER is given in terms of TL rather
than Te, which is accurately measurable.
To decide whether the TTM is an acceptable approximation, we recently determined τe-ph for
YBCO and LaSCO with pump intensities in the range 50 – 500 µJcm-2, which significantly
heats the electron system but not the lattice43. We estimated effective electron temperatures
after excitation between 400 and 800 K, depending on the excitation density. However, the
measured τe-ph showed no dependence on the pump intensity. Hence while Equation 2 reliably
yields the same λ<ω2> for data taken at different excitation densities, the TTM estimates
depend on the inferred initial electron temperature after excitation. This shows that the
consequences of the inappropriate TTM assumption in Allen’s model τe-e << τe-ph are not
limited to a simple numerical correction factor, but lead to qualitative disagreement with
experiment.
All theoretical description of EER, to date, has been done in the framework of Fermi liquid
theory (FLT). Since the applicability of FLT to cuprates is controversial, we briefly assess the
implications on our data analysis. FLT predicts that at low temperatures τe-ph~ TL -2, while at
higher temperatures τe-ph ~ TL, as stated in Equation 2. We observed the relaxation behaviour
of optimally doped LaSCO in the temperature range 140 – 310 K and found a linear increase
of τe-ph with TL (see Fig 5). For lower temperatures the spectral weight of the fast component
becomes too low to be fitted reliably. Hence, Equation 2 adequately describes EER in HTS
compounds at least on a qualitative level. Any possible deviation from FL behaviour in this
temperature range is thus quantitative at most and would appear as a prefactor. Such a
prefactor may depend on the strength of electronic correlations and hence on the doping level.
However, within one compound family 1/τe-ph varies only moderately with different doping
levels (see Fig. 2 of the main paper). While we cannot presently ascertain any systematic
doping dependent prefactor or doping dependent EPI strength, the small variations suggest
that for the range of doping levels we investigated 1/τe-ph is an appropriate measure for
comparing the EPI strength of our compounds. Finally, we emphasise that irrespective of
which model is used to ultimately connect τE with the EPI, our presented systematics relate
two experimentally measured quantities, τE and Tc without prejudice as to the exact model
relating τE to the microscopic EPI strength. Since the dominance of the EPI in EER has so far
not been challenged, we conclude - in agreement with previous works - that the main mode of
EER is via the EPI.
0 40 80 120 160 200 240 280 3200
10
20
30
40
50
τ 1 (fs
)
sample temperature (K)
0 40 80 120 160 200 240 280 3200
50100150200250300350400450500550
τ 2 (fs)
temperature (K)
Figure 5. The energy relaxation times of La1.85Sr0.15CuO as a function of the sample
temperature, determined from ∆T/T at 580 nm.
B2 the second relaxation time τ2
The data for the pnictide samples can be fitted very well with a single-exponential decay and
a plateau, plus one coherent oscillation. Hence, over the observed temporal window up to 800
fs after excitation, no appreciable second decay process could be identified. For the cuprates
and bismuthates, a second decay component is found. Its dependence on the sample
temperature, as measured for La1.85Sr0.15CuO (see Fig 5, right panel), suggests that it also
obeys a relation analogous to Equation 2. However, as shown in Figure 6, there is no apparent
correlation between the time constant of this relaxation and Tc. Hence this relaxation pathway,
whose assignment is beyond the scope of this paper, does not seem to be important in
determining Tc.
0 2 4 6 8 10 12 140
20
40
60
80
100
120
140
YBa2Cu3O6+x
HgBa2Ca2Cu3O8+x
Bi2Sr2CaCu2O8+x
La2-xSrxCuO
Ba1-xKxBiO3
criti
cal t
empe
ratu
re (K
)
relaxation rate (ps-1)
Figure 6 Superconducting critical temperatures as a function of the second fastest relaxation
time.
C. Raw data and fits
In the following we display the data for all samples, normalised and fitted as described in
section A4. Probe wavelengths are indicated in the figures; where the signal quality was
sufficient, two orthogonal sample orientations were measured to average out possible
anisotropies.
0.0 0.2 0.4 0.6 0.8
0.0
0.4
0.8
1.2
1.6
2.0 580 nm 620 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (fs)
Figure 7. Normalised transient differential reflection of BaFe2As2.
0.0 0.2 0.4 0.6 0.8
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8
3.2
3.6 580 nm 620 nm 660 nm 700 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 8. Normalised transient differential reflection of Ba(Fe0.975Co0.025)2As2.
0.0 0.2 0.4 0.6 0.8
0.0
0.4
0.8
1.2
1.6
2.0
2.4 580 nm 620 nm 660 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 9. Normalised transient differential reflection of Ba(Fe0.949Co0.051)2As2.
0.0 0.2 0.4 0.6 0.8-0.20.00.20.40.60.81.01.21.41.61.82.0 580 nm
620 nm 660 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 10. Normalised transient differential reflection of Ba(Fe0.93Co0.07)2As2.
0.0 0.2 0.4 0.6 0.8-0.20.00.20.40.60.81.01.21.41.61.82.0
580 nm 620 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 11. Normalised transient differential reflection of Ba(Fe0.89Co0.11)2As2.
0.0 0.2 0.4 0.6 0.8-0.20.00.20.40.60.81.01.21.41.61.82.0
620 nm 660 nm 700 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 12. Normalised transient differential reflection of SmFeAsO0.8F0.2.
0.0 0.2 0.4 0.6 0.8-0.20.00.20.40.60.81.01.21.41.61.82.02.22.42.6
580 nm 620 nm 660 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 13. Normalised transient differential reflection of YBa2Cu3O6.5.
0.0 0.2 0.4 0.6 0.8
0.0
0.4
0.8
1.2
1.6
2.0
2.4
2.8 580 nm 620 nm 660 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 14. Normalised transient differential reflection of YBa2Cu3O6.9.
0.0 0.2 0.4 0.6 0.8-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6 580 nm 620 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 15. Normalised transient differential reflection of HgBa2Ca2Cu3O8.2.
0.0 0.2 0.4 0.6 0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.6
580 nm 620 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 16. Normalised transient differential reflection of Bi2Sr2CaCu2O8.14.
0.0 0.2 0.4 0.6 0.8-0.20.00.20.40.60.81.01.21.41.61.82.02.22.42.6
580 nm 620 nm 660 nm 700 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 17. Normalised transient differential reflection of La1.85Sr0.15CuO4.
0.0 0.2 0.4 0.6 0.8-0.20.00.20.40.60.81.01.21.41.61.82.0 580 nm
620 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 18. Normalised transient differential reflection of La1.9Sr0.1CuO4.
0.0 0.2 0.4 0.6 0.8-0.8-0.6-0.4-0.20.00.20.40.60.81.01.21.41.6
580 nm 620 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 19. Normalised transient differential reflection of Ba0.64K0.36BiO3.
0.0 0.2 0.4 0.6 0.8
-1
0
1
2
3
4
580 nm 620 nm 660 nm
∆R/R
(arb
. uni
ts)
pump-probe delay (ps)
Figure 20. Normalised transient differential reflection of Ba0.55K0.45BiO3.
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