Far-infrared conductivity measurements of pair breaking in superconducting Nb 0.5 Ti 0.5 N thin

PRL 105, 257006 (2010)
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Far-Infrared Conductivity Measurements of Pair Breaking in Superconducting
Nb0:5 Ti0:5 N Thin Films Induced by an External Magnetic Field
Xiaoxiang Xi,1 J. Hwang,1,2 C. Martin,1 D. B. Tanner,1 and G. L. Carr3
Department of Physics, University of Florida, Gainesville, Florida 32611, USA
Department of Physics, Pusan National University, Busan 609-735, Republic of Korea
National Synchrotron Light Source, Brookhaven National Laboratory, Upton, New York 11973, USA
(Received 16 August 2010; published 16 December 2010)
We report the complex optical conductivity of a superconducting thin film of Nb0:5 Ti0:5 N in an external
magnetic field. The field was applied parallel to the film surface and the conductivity extracted from farinfrared transmission and reflection measurements. The real part shows the superconducting gap, which
we observe to be suppressed by the applied magnetic field. We compare our results with the pair-breaking
theory of Abrikosov and Gor’kov and confirm directly the theory’s validity for the optical conductivity.
DOI: 10.1103/PhysRevLett.105.257006
PACS numbers: 74.78.w, 74.25.Ha, 78.20.e, 78.30.j
Magnetic fields have dramatic effects on superconductors; when they are stronger than the upper critical field,
superconductivity is destroyed. Fields below this critical
value induce supercurrents and also act on the spin and
orbital motion of quasiparticles. An applied field lifts the
spin degeneracy of each electronic state, potentially causing a paramagnetic shift of the quasiparticle density of
states [1], which would give a linear shift of the spectroscopic gap with field [2]. The effect is noticeable only for
materials with small spin-orbit scattering, in which the spin
is a ‘‘good’’ quantum number. The field also alters the
orbitals of single-particle states from which the BCS
ground state is formed, breaking the time-reversal symmetry of the condensate. The result is a finite pair breaking
and an overall weakening of the superconducting state.
This weakening is directly revealed by a reduction in the
single-particle gap and forms the basis of the pair-breaking
theory originally proposed by Abrikosov and Gor’kov [3]
to describe the effect of magnetic impurities on superconductivity. The depairing phenomena can be characterized by a single pair-breaking parameter that depends on
whether the theory describes external magnetic fields,
supercurrents, spin exchange, or other effects. Maki [4]
showed that a thin superconductor in the dirty limit will
exhibit pair breaking, equivalent to that caused by magnetic impurities, when subjected to a homogeneous magnetic field. Because paramagnetism and pair breaking can
both affect the spectroscopic gap, experimental verification
of the pair-breaking effect is simplified in materials with
large spin-orbit scattering [5].
Ordinary metallic superconductors have a gap in their
optical spectrum [6], requiring a minimum of 2 of photon
energy to break Cooper pairs. The gap, which in weak
coupling BCS theory is 2 ’ 3:5kTc , makes the T ¼ 0 real
part of the optical conductivity be zero for photon energies
below the gap. The missing spectral weight in 1 ð!Þ
appears as a delta function at zero frequency [7]. By
the Kramers-Kronig relations, the delta function gives a
dominant 1=! form to 2 ð!Þ and is responsible for
frequency-independent penetration depth L ¼
c= 4!2 . This behavior is observed in most metallic
superconductors (Sn, In, Pb, Hg, etc.) although strongcoupling effects are sometimes necessary for quantitative
agreement [8]. By determining both the real and imaginary
parts of the optical conductivity under an applied magnetic
field, one can test theories for the magnetic field suppression of the gap.
We find it somewhat surprising that magnetic-fieldinduced pair-breaking effects have not been convincingly
verified by optical studies. Such effects have been observed
in tunneling spectra [9] and are hinted at by absorption data
[2]. In addition, the effects of magnetic impurities have
been studied in detail [10]. In this Letter, we report farinfrared transmission and reflection spectra of Nb0:5 Ti0:5 N
under an external magnetic field, applied parallel to the
film surface. The extracted optical conductivity 1 demonstrates a suppression of the gap by the field, in quantitative agreement with the pair-breaking theory. This is the
first time that optical absorption has been employed to test
quantitatively the theory of pair breaking by an external
magnetic field.
We selected for study a 10-nm-thick film on quartz from
a set of NbTiN thin films of varying thicknesses and
substrate materials. A selection criterion was strong spinorbit scattering @=so 1 [5], where the spin-orbit scattering time so ¼ 3:0 1014 s is taken to be that of NbTi
[11] and is the single-particle gap. The films were grown
by reactive magnetron sputtering in argon and nitrogen gas
with a NbTi cathode [12]. Transmittance data (inset in the
upper panel in Fig. 1) give the normal-state sheet resistance: Rh ¼ 146 =h. Magnetic susceptibility measurements with a SQUID magnetometer determine Tc 12 K
and Hc2
15 T. The optical gap for T ¼ 2 K ( Tc ) and
zero field is 28:5 cm1 . The quartz substrate has negligible
absorption in the spectral range of interest (10–110 cm1 ).
Infrared transmission and reflection measurements were
Ó 2010 The American Physical Society
PRL 105, 257006 (2010)
FIG. 1 (color online). Superconducting to normal-state transmission (upper panel) and reflection (lower panel) ratios at 0, 5,
and 10 T. The weak oscillatory features are partially resolved
multiple internal reflections in the substrate. The upper panel inset
is the transmittance for T ¼ 300 K, measured at 3 higher
spectral resolution along with a fit using the optical constants of
quartz [13] (blue curve). The inset in the lower panel is a diagram
of the reflection sample holder. The sample is the thin black slab
near the middle. A polished aluminum roof-type mirror reflects the
input beam onto the sample at a 30 angle and then redirects the
reflected beam back onto the original optical path.
performed at Beam line U4IR of the National Synchrotron
Light Source, Brookhaven National Laboratory. The samples were mounted in a 4 He Oxford cryostat equipped with
a 10 T superconducting magnet; the minimum temperature
is 1.6 K. The spectra were collected by using a Bruker IFS
66-v/S spectrometer and a high-sensitivity, large-area,
B-doped Si composite bolometer operating at 1.8 K;
cooled filters limited the upper frequency to 110 cm1 .
A sketch of the reflection stage is shown in the inset in
the lower panel in Fig. 1. The angle of incidence for the
reflection measurements is about 30 ; for transmission it is
near normal. The magnetic field is applied parallel to the
film surface. The field direction is important when considering the behavior of these type-II superconductors. For a
normal field, vortices appear above Hc1 and form a dense
lattice of lossy core material as it approaches Hc2 . We
avoided this vortex regime by orienting the field parallel
to the film surface. The machining tolerances are such that
the misalignment is less than 0.3 , making the number of
vortices be 0.5% of those in a perpendicular field. In this
case, because the thickness is much smaller than the penetration depth and somewhat smaller than the coherence
length, a significant density of vortices is unlikely.
Nb1x Tix N typically has a penetration depth 100 nm
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and a coherence length 20 nm [14], satisfying = 1. When a magnetic field is applied parallel to the
film surface, the vortex spacing is greater than [15],
which itself is close to the film thickness. Therefore we
do not expect vortex-induced effects to be significant.
Moreover, the field decays according to the much larger
penetration depth, making the average field in the film be
approximately 0.999 of the applied field [16] and, hence,
nearly uniform.
Our goal is to extract the optical conductivity of the thinfilm superconductor from reflection and transmission measurements. Beginning with the pioneering work of Palmer
and Tinkham [8], this approach has been used a number of
times to study both metallic and cuprate superconductors
[17]. In a conventional transmittance or reflectance measurement, one measures separately the sample and a reference having known optical properties—typically, an open
aperture with no sample for transmittance and a known
metal for reflectance. Sample exchange can lead to errors,
especially for the absolute reflection, where sample orientation is critical. To avoid sample exchange errors, we used
the sample in the normal state, and at H ¼ 0 T, for our
reference. Specifically, we measured the sample spectrum
(transmission or reflection) at different fields ranging from
0 to 10 T in the superconducting state (T ¼ 3 K), using the
normal-state (T ¼ 20 K), zero-field spectrum for the reference. If required, the relative measurements can be made
absolute by measuring the normal-state transmittance and
reflectance or by calculating them from the Drude model in
the limit ! 1= (a very good assumption for our films).
The directly acquired data are therefore the ratios of transmittance T S =T N and reflectance RS =RN , where the subscripts S and N denote superconducting state and normal
state, respectively. Figure 1 shows the data at zero, intermediate, and high fields, measured with a resolution of
3:5 cm1 , which does not fully resolve the interference
fringes from multiple reflections inside the substrate. The
peak in T S =T N shifts to lower frequency as the field
increases, suggesting the suppression of the energy gap
due to the field. The reflection data were corrected for the
measured stray light and for the 30 angle of incidence (see
[18]) before calculating the optical conductivity. The fielddependent transmittance is similar to recent data for NbN
measured at a single frequency [19].
The analysis for the thin-film optical conductivity ¼
1 þ i2 begins with the expressions for the normalincidence transmission through, and reflection from, the
front film surface of the sample [8]:
ðZ0 1 d þ n þ 1Þ2 þ ðZ0 2 dÞ2
Rf ¼
ðZ0 1 d þ n 1Þ2 þ ðZ0 2 dÞ2
ðZ0 1 d þ n þ 1Þ2 þ ðZ0 2 dÞ2
where Z0 377 is the vacuum impedance, d is the film
thickness, and 1 and 2 are the real and imaginary parts of
PRL 105, 257006 (2010)
the optical conductivity of the thin film, either in the superconducting state or in the normal state. In practice, we
measure the combination of film and substrate, giving the
external transmittance T ext and external reflectance Rext .
If the substrate surfaces are parallel on the scale of the
wavelength and the measurement resolution is high enough,
these quantities typically show fringes due to partially
coherent multiple internal reflections inside the substrate.
Smoothing high resolution data or taking measurements
with a low resolution produces the incoherent spectrum,
where one may add intensities rather than amplitudes. In
this case, T ext ¼ T f ð1 RQ Þex =ð1 RQ R0f e2x Þ,
where RQ ðn 1Þ2 =ðn þ 1Þ2 is the reflectance of the
quartz surface, is the (small) absorption coefficient of the
quartz, x is the thickness of the quartz, and R0f is the film
reflection from inside the substrate. There is a similar equation for Rext . Quartz has negligible absorption and dispersion over the spectral and temperature range of interest.
Thus we take ¼ 0 and n ¼ 2:12, yielding RQ 0:13.
Our measurements give us the external transmission
and reflection ratios T ext;S =T ext;N and Rext;S =Rext;N , respectively, that include the substrate. For the range of
conductivity values expected for the film, we find that, to
a very good approximation, T ext;S =T ext;N ¼ T S =T N and
Rext;S =Rext;N ¼ RS =RN . The normal-state transmittance
and reflectance can be derived from Eqs. (1) and (2) by
setting 1 ¼ N , 2 ¼ 0, T N ¼ 4n=ðZ0 N d þ n þ 1Þ2 ,
and RN ¼ ðZ0 N d þ n 1Þ2 =ðZ0 N d þ n þ 1Þ2 . Here
N is related to the normal-state sheet resistance of the
thin film Rh ¼ 1=N d, which we have determined from
the normal-state transmittance. Hence we know T N and
RN and may use them to calculate T S and RS from our
measured ratios. Then we invert Eqs. (1) and (2) to find
nR 1 RS T S
¼ h
¼ h
ðZ0 1 d þ n þ 1Þ2
Z0 T S
The normalized optical conductivity at 0, 5, and 10 T are
shown in Figs. 2(a)–2(c). 2 =N has some data points
missing because the term under the square root in Eq. (4)
is not guaranteed to be positive for the measured transmission and reflection when noise is included. A weak
interference fringe in both the transmission and reflection
measurements results in the excess 2 =N over the
40–80 cm1 range. The solid lines are fits to the data using
the pair-breaking theory as extended by Skalski et al. [20]
to calculate 1 =N at 0 K:
1 Z G þ!=2
dq½nðq þ !=2Þnðq !=2Þ
N ! G !=2
þ mðq þ !=2Þmðq !=2Þ
where nðqÞ ¼
for !p
2G and zero otherwise,
Reðu= u 1Þ and mðqÞ ¼ Reð1= u 1Þ. u is the
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FIG. 2 (color online). (a)–(c) The real and imaginary parts of
the T ¼ 3 K superconducting state optical conductivity (normalized to the normal-state conductivity) at three different applied
magnetic fields. The solid lines are fits to 1 =N using the pairbreaking theory. The dashed lines show the corresponding
2 =N as determined by a Kramers-Kronig transform of the
real part. (d) The fitted 1 =N at six fields.
solution to u ¼ q þ iu= u2 1, with the paircorrelation gap. This, in turn, can be determined from the
pair-breaking parameter and the zero-field excitation gap
0 by using lnð=0 Þ ¼ =4 for < . G in the
integration limits is the effective spectroscopic gap: G ¼
½1 ð=Þ2=3 3=2 for < . We fit our H ¼ 0 T results
to determine 0 and then proceeded to fit 1 =N for
H > 0 T by using only as an adjustable parameter. The
imaginary part of the conductivity (dashed lines) was
calculated by a Kramers-Kronig transform of the real
part. The temperature T 0:25Tc is low enough that the
gap has reached its zero-temperature value. The zero-field
case reduces to the standard BCS Mattis-Bardeen [6] description of a dirty-limit superconductor, consistent with
results for the similar compound NbN [21]. Figure 2(d)
shows the fitted 1 =N at six different fields. Clearly, the
absorption edge moves to lower energy as the field increases. The field-induced pair breaking also smears out
the gap-edge singularity in the quasiparticle density of
states [20], so that the initial rise of 1 becomes less abrupt
for increasing fields, as can be seen by comparing the 0 T
and the 10 T results in Fig. 2.
The oscillator-strength sum rule
0 1 ð!Þd! ¼
ne2 =2m, where n is the electron density and e and m
are the charge and mass of the electron, respectively,
requires the area under 1 ð!Þ to be the same for normal
and superconducting states. The ratio 1 =N in Fig. 2 is
always less than unity; the ‘‘missing area’’ condenses to a
PRL 105, 257006 (2010)
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data, the former showing the absorption edge depressed
due to the applied in-plane field. The degree of suppression
is in good agreement with the pair-breaking theory.
This work was supported by U.S. Department of Energy
through DE-ACO2-98CH10886 (NSLS) and DE-FG0202ER45984 (University of Florida) and by Korea NRF
No. 20090074977. We are grateful to P. J. Hirschfeld for
valuable discussions and to G. Nintzel for technical support. We thank P. Bosland and E. Jacques for providing the
NbTiN samples.
FIG. 3 (color online). Field dependence of the pair-breaking
parameter , determined from the experimental optical conductivity (circles) along with a fit to ¼ bH2 (solid line). The inset
shows the pair-correlation gap and the effective spectroscopic
gap G vs field. The solid lines are theoretical predictions using
the pair-breaking theory and the fitted value of b.
function at zero frequency that is a measure of pair
condensate density and is directly related to the paircorrelation gap. Figure 2(d) therefore shows a weakening
of superconductivity as the field increases. There is a limit
in which the absorption edge approaches 0, while the
missing area remains finite. The superconductor enters
a ‘‘gapless’’ region but still maintains superconducting
The quantity describes the strength of pair breaking.
For any perturbing Hamiltonian that breaks time-reversal
symmetry, is proportional to the square of the perturbing
Hamiltonian in the dirty limit [22]. For the range of
fields used here, is quadratic in the field: ¼ bH2 ¼
tr v2f ðeHdÞ2 =18, where tr is the transport collision time
and d is the film thickness [15]. , as extracted from our data
at different fields, is plotted in Fig. 3. The quadratic fit is
good, yielding b ¼ 0:020 cm1 =T2 . We estimate tr from
N ¼ ne2 tr =m and Rh ¼ 1=N d, tr ¼ m=Rh dne2 1:51 1016 s, where Rh ¼ 146 =h, d ¼ 10 nm, and
n 1:61 1023 cm3 is the electron density of NbN [23]
similar to that of NbTiN. If we take the Fermi velocity to
be that of NbN [23], vf 1:95 108 cm=s, then b ¼
0:039 cm1 =T2 , consistent with the fitted value within the
uncertainty of the materials parameters.
The spectroscopic energy gap G and the paircorrelation gap are shown in the inset in Fig. 3. Both
G and drop as the field increases, but the reduction of
G is much greater at any given field. The sample at the
highest attainable field of 10 T is still far away from the
gapless region where G vanishes. The experimental and
theoretical values of G and the pair-correlation gap are
in excellent agreement.
In conclusion, we measured far-infrared transmission
and reflection of a thin-film superconductor in a magnetic
field parallel to the film surface. The real and imaginary
parts of the optical conductivity are derived from these
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