arXiv:cond-mat/0507172v2 [cond-mat.supr-con] 9 Oct 2005
Quantum size effects on the perpendicular upper critical field in ultra-thin lead films
Xin-Yu Bao,1Yan-Feng Zhang,1Yupeng Wang,1Jin-Feng Jia,1Qi-Kun Xue,1X. C. Xie,1,2and Zhong-Xian Zhao1, ∗
1Beijing National Laboratory for Condensed Matter Physics,
Institute of Physics, Chinese Academy of Sciences, Beijing 100080, China
2Department of Physics, Oklahoma State University, Stillwater, OK 74078, USA
(Dated: Submitted 14 July 2005, Revised 7 October 2005)
We report the thickness-dependent (in terms of atomic layers) oscillation behavior of the perpen-
dicular upper critical field Hc2⊥ in the ultra-thin lead films at the reduced temperature (t = T/Tc).
Distinct oscillations of the normal-state resistivity as a function of film thickness have also been ob-
served. Compared with the Tc oscillation, the Hc2⊥shows a considerable large oscillation amplitude
and a π phase shift. The oscillatory mean free path caused by quantum size effect plays a role in
PACS numbers: 74.78.-w, 73.20.At, 74.20.-z, 74.62.Yb
There is a long history of scientific research on super-
conducting thin films. In particular, theoretical and ex-
perimental studies have been carried out to understand
how the film thickness affects the superconducting prop-
erties. It seems that the reported experimental results
of thin films can be explained by the existing theories
of superconductivity [1, 2, 3, 4, 5, 6, 7]. However, most
previously studied superconducting films were still rel-
atively thick, normally over several tens of nanometers,
and the film morphology was usually poor. If the film
surface is atomically uniform and the thickness is further
reduced to several nanometers so that the quantum size
effects become apparent, a natural question arises: will
some unexpected new phenomena emerge? In particular,
does the conventional theory still work?
Previous theoretical works have predicted many pos-
sible prominent physical properties modulated by quan-
tum size effects: electronic structure, critical tempera-
ture, electron-phonon interaction, resistivity, Hall con-
ductivity and so on [8, 9, 10, 11, 12, 13].
are also some related important experimental results
[14, 15, 16, 17, 18, 19, 20, 21], such as the Tcoscillations
in ultra-thin Pb films, which are caused by the density
of states oscillations in confined quantum well structures
[22, 23] and by the electron-electron interaction medi-
ated by quantized confined phonons [11, 24]. However,
the properties of the upper critical field affected by the
quantum size effect have not been reported in previous
work. In this Letter, we report our experimental observa-
tion of the oscillatory Hc2⊥through magneto-transport
measurement of ultra-thin Pb films. The oscillations are
similar to those of Tcbut the motivations are more com-
plex. Besides the factors for Tcoscillation, we interpret
this unexpected phenomena by the oscillatory mean free
path in ultra-thin superconducting films caused by the
quantum size effect.
The 3mm×10mm sized Si(111) wafers were used as
substrates and prepared by the standard cleaning pro-
cedure to obtain the clean Si(111)-7 × 7 surface. The
base pressure of the UHV-MBE-STM-ARPES (Angle
Resolved Photoemission Spectroscopy) combined system
we used was about 5 × 10−11Torr. The Si substrate
was cooled down to 145K during the MBE layer-by-layer
growth of the Pb films. The growth rate was controlled
at 0.2 ML/Min (Monolayer/Minute) and a RHEED (Re-
flection High Energy Electron Diffraction) was used for
real time monitoring of the growth. After deposition, the
sample was warmed up slowly to room temperature and
transferred to the analysis chamber where the STM and
ARPES were used to investigate the surface topography
and the electronic structures, respectively . For ex-
situ magneto-transport measurements, all the Pb films
were covered with a Au protection layer of 4ML before
being taken out of the UHV system.
The R − H measurements were carried out in a very
short time after the samples were taken out of the vac-
uum. The applied field was perpendicular to the sample
surface and the temperatures were set near and below Tc.
To avoid trapping flux in, the magnet was discharged to
zero in oscillate mode and the sample was warmed up
to 8K before the R − H measurement for each temper-
ature. Then the perpendicular upper critical field Hc2⊥
at different temperatures was obtained from the R − H
measurements at the field where the resistance reached
half of the normal state resistance RN. The resistance
approaches RNvery gradually because of the magnetore-
sistance effect. So we took RN as the resistance where
the resistance variation ratio is within 0.1%.
Figure 1 shows the R−H curves of a 21 ML sample at
different temperatures. The arrow points out the defined
perpendicular upper critical field Hc2⊥at 4.7K. The inset
of Fig.1 shows Hc2⊥vs. temperature for the 21 ML film.
It shows a perfect linear dependence on T near Tc, which
is a typical property of a superconductor with a high
value of the Ginzburg-Landau parameter κ. The inset
of Fig.1 can be used to determine the zero field critical
temperature Tcby extrapolating the plot to Hc2⊥= 0. Tc
determined in this way is shown as a function of thickness
in Fig.4(a). Normally, a direct way of determining critical
temperature is through the R − T measurement at zero
FIG. 1: R-H curve of the 21 ML sample. The magnetic field
is perpendicular to the sample surface. The black arrow in-
dicates the determined upper critical field at 4.70K. The in-
set shows the Hc2⊥ as a function of T for this sample. The
plot is linearly extrapolated with dashed lines to both high
and low temperature sides. The measurements were carried
out with a Quantum Design Magnetic Property Measurement
FIG. 2: The reduced resistances of Pb films as a function of
temperature are shown in (a). The resistances are normalized
by the normal state resistance at T = 8K.
oscillation of normal state resistivity at 8K as a function of
(b) shows an
field. We find that the critical temperatures determined
by both methods show a consistent oscillation behavior
and the values are quite close for every thickness.
The reduced R − T curves of Pb films from 21 ML to
28 ML are shown in Fig.2(a). The normal state resistiv-
ity ρnoscillation with film thickness at T = 8K is shown
in Fig.2(b). The similar resistivity oscillations caused by
quantum size effect have been reported in single crys-
talline Pb and Pb-In thin films at T = 110K . But
in polycrystalline films, oscillations of the normal state
resistivity have not been observed although Tchas been
found to oscillate with film thickness . In our ex-
periment, both Tc and ρn oscillations are observed. It
indicates that the quantum size effects show up in both
superconducting state and normal state but the intensi-
ties and mechanisms may vary in different way depending
on sample conditions.
Figure 3(a) shows Hc2⊥ as a function of the reduced
temperature t = T/Tc. For every thickness Hc2⊥shows
a good linear dependence on t near t = 1. Hc2⊥ ver-
sus film thickness for t=0.90 and 0.95 are shown respec-
tively in Fig.3(b). It is shown that with the film thickness
variation, Hc2⊥exhibits an oscillation behavior, which is
similar to the reported Tcoscillation . However, the
oscillation of Hc2⊥are π out of phase to that of Tc, i.e.,
peaks appear in the odd layer samples where dips appear
in the even layer samples, which is opposite to the Tc
oscillation shown in Fig.4(a).
In the early theories proposed to understand the mag-
netic properties of thin film superconductors, the TGS
(Tinkham, de Gennes and Saint James) theory [3, 5] was
validated as showing a good agreement with the former
experimental results [6, 7]. According to TGS theory, the
upper critical fields Hc2⊥near Tcshould monotonically
increase when the film thickness decreases, which can be
described in the following form :
√2κ(T,∞)Hc(T)(1 + b/d),
∞(T). Here Hc(T) is the thermodynamic
critical field, λL is the London penetration depth, λ∞
is the bulk weak field penetration depth, Φ0 is the flux
quantum (Φ0= hc/2e = 2.07 × 10−15Wb) and d is the
film thickness. In Fig.3(b), the dashed lines, calculated
using Eq.(1) and the related parameters in previous work
 with film thicknesses appropriate to our samples, show
the same tendency as the experimental curves if the os-
cillations are ignored. The measured Hc2⊥values of our
samples are about three times larger than the calcu-
lated values (note the different scales on the two sides
of Fig.3(b)), which may be caused by stronger interface
or impurity scattering in our films that gives rise to a
large resistivity, thus large Hc2⊥(see discussion below).
The linear dependence on t shown in Fig.3(a) also gives
an information that for a given film thickness, the tem-
perature dependence follows reasonably well with Eq.(1)
whether that particular film is at the peak or valley of
The TGS theory above includes surface scattering ef-
fects but does not consider the quantum size effects that
occurs in ultra thin films. The absent Hc2⊥ oscillation
from TGS theory means that the thickness depended
quantum size effect is the original source of the Hc2⊥os-
cillation. According to the G-L (Ginzburg-Landau) the-
ory, Hc2⊥is determined by the in-plane coherence length
ξ?. In a three dimensional anisotropic superconductor,
the perpendicular upper critical field near Tcis given by
[4, 7]: Hc2⊥ =
than 10nm which is much smaller than the Pipard coher-
ence length of a bulk Pb superconductor (ξbulk
∞(T)/Φ0 and b=
?. Our ultra-thin films are thinner
FIG. 3: (a) shows the perpendicular upper critical field versus
the reduced temperature t. The oscillation behavior at t=0.90
and 0.95 are plotted in (b). The dashed lines correspond to
the calculated results using Eq.(1).
thickness, which is defined by the way shown in the insert of
Fig.1. The rescaled ρnTc variation is shown in (b), which is
defined in the following way: ∆ρnTc = (ρnTc− ρ′
(a) shows the oscillation behavior of Tc with film
cis the value of ρnTc for the 21ML film.
we can use the quasi two dimensional formula :
For the linear dependence on t near t = 1 shown in
Fig.3(a), Hc2⊥ has the same oscillation behavior with
thickness as that of −?dHc2⊥
tem should be considered as a dirty-limit superconductor
because of the strong scattering. For dirty superconduc-
tors near Tc, ξ2
?≈ ξ0l where ξ0 is the Pipard coherence
length and l is the mean free path for a film [2, 4]. Ac-
cording to BCS theory, ξ0∝ 1/Tc, therefore we can get
Hc2⊥ ∝ Tc/l at a certain t. In Fig.3(b) and Fig.4(a),
it is shown that the oscillation amplitude of Hc2⊥ and
Tc are about 40% and 10% respectively. On the other
hand, the mean free path l and the normal state resistiv-
ity ρnhave the following relation: l ∝ 1/ρn, from which
at a certain t. The sys-
we can derive Hc2⊥ ∝ ρnTc. In Fig.2(b), the ρn oscil-
lation shows a big amplitude about 60% and the same
phase as Hc2⊥. The rescaled variation of ρnTcis shown
as ∆ρnTcin Fig.4(b), which fits well with the oscillation
behavior of Hc2⊥. It implies ρn oscillation dominates
over the Tcoscillation in Hc2⊥and gives rise to a π phase
shift between Tcand Hc2⊥oscillations. In earlier works,
the oscillatory conductivity in quantized thin films has
been observed  and the effects of impurity and sur-
face and interlayer roughness on quantum size effects in
thin films have been discussed [12, 26]. Though at the
moment we do not have a complete answer to the oscil-
lations of ρnwith thickness for our films with atomically
uniform surfaces, the previous experiments on the layer-
spacing oscillation provides a strong indication that
the modulation of the interface roughness with thickness
may play an important role. In that experiment they
found that the interlayer spacings oscillate with a period
of quasi double layer and even-monolayer samples have
shorter interlayer spacings. This is also supported by the
binding energy modulation observed [22, 24]. It indicates
the lattice feels at home with the conduction electrons
for the even-monolayer samples while it is not so for the
odd-monolayer samples. The unaccommodating lattice
and conduction elections in the odd-layer samples could
induce some lattice distortion and therefore enhance the
interface roughness. This enhanced interface roughness
must induce a higher resistivity.
We believe the experimental findings in our ultra-thin
films are due to a variety of combined quantum size ef-
fects from ultra-thin film thickness. The quantum size
effect can show up either as a modulation of the in-
terface roughness induced by the interlayer spacings as
well as the modulation of the phonon modes and the
electron-phonon couplings which both affect the normal
state transport properties of the samples, of course, also
causing the wave vector quantization along the thickness
direction. Under the circumstance, only the components
of electronic wave vector in the surface plane, i.e., the
x-y plane, have a continuous distribution.
the electron density distribution is rather inhomogeneous
along the z direction. The modulation of the electron-
densities may further feedback to the electron-interface
and electron-phonon scattering processes and therefore
to the mean free path. Another relevant issue is that
the G-L theory is only a mean-field theory in which all
the short-distance fluctuations are integrated out. For
our ultra-thin films, to give an adequate description of
all the electronic states and the scattering processes, we
must go back to the microscopic theory of BCS super-
conductivity within the subband framework and derive
the multi-band G-L theory. The G-L order parameter
Ψ perpendicular to the film is limited to quantized val-
ues and may also show modulation with the interlayer-
spacing modulation. Each subband may have a different
value of coherence length ξ in the x-y plane, namely ξn,d,
4 Download full-text
where n is the subband index and d is the number of the
monolayers. In general, Hc2⊥is determined by a matrix
equation with m being the size of the matrix in which m
is the number of subbands below the Fermi energy. In
the limit that one of the ξn,dis much smaller than all the
others, Hc2⊥ is predominantly determined by this min-
imum value, which could be much higher than that of
the bulk. The story here is similar to that of the newly
discovered superconductor MgB2, where only two bands
are involved . If the film becomes thicker, the number
of subbands will increase. The interaction of subbands
will weaken the quantum size effects and the coherence
length will be close to the average one. The oscillation
behavior of Hc2⊥will eventually disappear beyond a large
In conclusion, a large oscillation of Hc2⊥in the ultra
thin lead films are observed as a function of film thick-
ness. The Hc2⊥ oscillation is opposite to that of Tc in
phase and cannot be simply attributed to the modula-
tion of the density of states and Tc. A large value of
Hc2⊥is also observed. Considering the interface and sur-
face scattering and the modulation of coherence length
and mean free path induced by the quantum size ef-
fect, a possible mechanism is proposed to explain both
the anomalous oscillation of Hc2⊥ and its large value.
We believe that a quantitative description for the find-
ings in our experiments must be based on the combined
quantum size induced modulation effects on the inter-
layer structures, electronic structures, phonons, electron-
phonon and electron-interface scattering processes. Fur-
ther consideration about the flux dynamics is also neces-
sary by including the interface and surface scattering ef-
fects and two-dimensional fluctuations in the multi-band
We are grateful to Professors Lu Yu, Hai-Hu Wen,
Dong-Ning Zheng, Jue-Lian Shen, Michal Ma and Li Lu
for useful discussions. We thank Shun-Lian Jia, Wei-
Wen Huang and Hong Gao for their help in measure-
ment. We also thank Tie-Zhu Han and Zhe Tang for their
help in sample preparation. This work was supported by
the National Science Foundation, the Ministry of Science
and Technology of China and the Knowledge Innovation
Project of the Chinese Academy of Sciences. X.C. Xie
is supported by US-DOE under Grant No. DE-FG02-
04ER46124 and NSF-MRSEC under DMR-0080054.
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