Review of 2D superconductivity: the ultimate case of epitaxial monolayers
Christophe Brun
1
,
∗
Tristan Cren
1
, and Dimitri Roditchev
1,2
1
Institut des Nanosciences de Paris, Sorbonne universit´e,
UPMC univ Paris 6 and CNRS-UMR 7588, 4 place Jussieu, 75252 Paris, France and
2
Laboratoire de physique et d’´etudes des mat´eriaux,
LPEM-UMR8213/CNRS-ESPCI ParisTech-UPMC, 10 rue Vauquelin-75005 Paris, France
(Dated: September 30, 2016)
The purpose of this review is to focus from an experimental point-of-view on the new physical
properties of some of the thinnest superconducting films that can be fabricated and studied in situ
nowadays with state-of-the-art methods. An important characteristic of the films we address is
that the underlying electronic system forms a two-dimensional electron gas (2DEG). Up to now
there are only few of these systems. Such true 2D superconductors can be divided into two classes:
surface-confined or interface-confined films. Because the second types of films are burried below
the surface, they are not accessible to purely surface-sensitive techniques like angular-resolved pho-
toemission spectroscopy (ARPES) or scanning tunneling spectroscopy (STS). As a consequence the
bandstructure characteristics of the 2DEG cannot be probed nor the local superconducting proper-
ties. On the other hand, in situ prepared surface-confined films are nowadays accessible not only to
ARPES and STS but also to electrical transport measurements. As a consequence surface-confined
systems represent at present the best archetypes on which can be summarized the new properties
emerging in ultimately thin superconducting films hosting a two-dimensional electron gas, probed
by both macroscopic and microscopic measurement techniques. The model system we will widely
refer to consists of a single atomic plane of a conventional superconductor, like for example lead
(Pb), grown on top of a semiconducting substrate, like Si(111). In the introductory part (I) we first
introduce the topic and give historical insights into this field. Then in the section II, we introduce
useful concepts worked out in studies of so-called ”granular” and ”homogeneous” superconducting
thin films that will be necessary to understand the role of non-magnetic disorder on 2DEG su-
perconductors. In this section, we also briefly review the superconducting properties of crystalline
Pb/Si(111) ultrathin films grown under ultrahigh vacuum (UHV) conditions in order to illustrate
their specific properties related to quantum-size effects. In the next section (III) we review the
growth methods and structural properties of the presented 2DEG surface-confined superconductors.
In section (IV), we review the electronic structure and Fermi surface properties as measured by
macroscopic ARPES and confront them to ab initio DFT calculations based on the characterized
atomic structures of the monolayers. The following section (V) reviews the macroscopic properties
inferred from in situ electrical transport measurements methods, including attempts to study the
Berezinsky-Kosterlitz-Thouless 2D regime. In the last section (VI), we summarize the emerging
local spectroscopic properties measured by STS. These latter demonstrate variations of the local
superconducting properties at a scale much shorter than the superconducting coherence length
due to a combined effect of non-magnetic disorder and two-dimensionality. Further peculiar local
spectroscopic effects are presented giving evidence for the presence of a mixed singlet-triplet super-
conducting order parameter induced by the presence of a strong Rashba spin-orbit coupling term
at the surface. These local signatures will be discussed along with ARPES and transport measure-
ments in parallel high magnetic field on closely related systems. Finally, we present in anisotropic
Pb and In monolayers the peculiar role played by atomic steps on vortex properties, leading to the
observation by STS of mixed Abrikosov-Josephson vortices in agreement with in situ macroscopic
transport measurements. From the overview of all recent experimental and theoretical results it
appears that these surface 2D superconductors, such as one monolayer of Pb on Si(111), are ideal
templates to engineer and realize topological superconductivity.
PACS numbers: 74.45.+c, 74.55.+v, 74.78.-w, 74.78.Na
Contents
I. Introduction 2
II. Basic properties of ultrathin disordered
s-wave superconductors 4
A. Granular superconducting films 4
B. Disordered homogeneous superconducting
films 5
C. Crystalline ultrathin superconducting films 7
III. Monolayer films growth and structural
properties 9
IV. Monolayers electronic properties from
ARPES measurements confronted to ab
initio DFT calculations 11
V. Superconducting properties from in situ
magneto-transport measurements 13
A. General transport properties in
2
perpendicular magnetic field 13
B. Critical current and role of atomic steps 14
C. magneto-transport measurements in parallel
magnetic field : hint for a spin-triplet
component of the order parameter 15
D. Berezinski-Kosterlitz-Thouless transition in
monolayer superconducting films 17
VI. Local superconducting properties from
scanning tunneling spectroscopy
measurements 19
A. New inhomogeneous superconducting
properties in monolayer superconductors
induced by 2D electronic dimensionality :
short-range spatial variations of the height
of coherence peaks 19
B. In-gap states induced by non-magnetic
disorder in monolayer superconductors :
hint for a mixed singlet-triplet
superconducting order parameter induced by
strong Rashba spin-orbit coupling 20
C. Vortex properties and role of step edges in
monolayer superconducting films 23
1. zero magnetic field properties at step
edges 23
2. magnetic field properties at step edges 24
VII. Conclusion 25
VIII. Acronyms 26
IX. Acknowledgments 26
References 26
I. INTRODUCTION
Early in 1964 Ginzburg and Kirzhnits have con-
jectured the possible existence of superconductivity
being confined at the very surface of a solid
1
. They
had in mind a truly two-dimensional superconductor
in the sense that the electron gas itself would be
localized on the surface of the material and would form
a two-dimensional electronic system. As experimental
realizations Ginzburg and Kirzhnits imagined two
systems. In the first one the bulk material would be
a dielectric hosting Tamm surface states which filling
could be induced and controlled by charge transfer. An
attractive electron-phonon coupling should then develop
in the surface layer to induce superconductivity, possibly
involving surface phonons. As a second system they
imagined a bulk metal also hosting Tamm surface states.
In order to enable revealing intrinsic surface supercon-
ductivity, the attractive coupling between electrons and
phonons should be larger for surface electrons than for
bulk electrons
2
. Very interestingly, due to a softening of
phonon modes at the surface, Ginzburg envisioned the
possibility to obtain high-Tc surface superconductors in
both of these systems. Surface engineering with suitable
light-element-molecules was also mentioned as a way to
enhance the critical temperature, following the ideas of
Little
3
.
Historically, the study of clean surface or interface
systems was long hindered by technical limitations
in the cleanliness of the preparation processes. First
of all, the search for the thinnest superconductors
began with the study of ultrathin superconducting films
deposited on an insulating or semiconducting substrate,
using the physical vapor deposition method through
resistive Joule heating
4
. These films usually consisted
of nanocrystallites or crystalline grains, structurally
and electrically coupled togoether. Depending on the
electrical coupling between neighboring grains, the
obtained superconducting films can be divided into two
classes of systems, so-called granular and homogeneous
systems.
As we explain below in section II, the development of
this field brought forward important new concepts and
clarifications about the role of non-magnetic disorder in
conventional ultrathin superconductors. Nevertheless,
the preparation conditions did not enable fabricating
true 2D-electron-gas based superconducting systems.
This does not mean that 2D-superconductivity could not
be reached : on the contrary, as soon as the film thickness
d is smaller than the superconducting coherence length
in the direction perpendicular to the film’s surface,
2D-superconductivity is realized. Typically the thinnest
granular or homogeneous films studied there are about
a few nanometers thick and do not superconduct above
a critical disorder. This threshold for the destruction of
superconductivity occurs when k
F
l
e
decreases toward
unity (k
F
being the Fermi wave vector and l
e
the elastic
3
electronic mean-free-path). In terms of the macroscopic
sample resistivity, superconductivity disappears when
the square resistance of the film becomes compara-
ble to the quantum of resistance for electron pairs
(R
square
' h/4e
2
= 6.45kΩ). R
square
corresponds to the
resistance of a square film, thus R
square
= ρL/dw where
ρ is the electrical resistivity d, w, L being respectively
the sample thickness, width, and length. For a square
film l = w and thus R
square
= ρ/d.
The advent of molecular beam epitaxy (MBE) and
surface physics methods, with the use of ultrahigh
vacuum (UHV) techniques and highly purified source
materials, enabled bridging the gap. Favorable growth
conditions were obtained in order to create clean surface
and interface systems. Up to now, the widely studied
semiconducting III-V of II-VI heterostructures, like for
instance the very well-known GaAs/AlGaAs 2D electron
gas system (2DEG), which allowed the discovery of
the integer and fractionnal quantum Hall effect, did
not reveal any superconducting behavior. The first
system that appeared to be close to a true 2DEG system
and showed a superconducting transition below 1K is
LaAlO
3
/SrTiO
3
5,6
. This oxyde heterostructure is built
out of two insulating parts. At the interface, a charge
transfer occurs to release strong electric polarization
effects in the upper LaAlO
3
layer. As a result an
interface electron gas is formed about 10 nm below the
surface, which electro-chemical potential can be tuned
by electrostatic gating. This gating allows the system to
be tuned from a superconducting state with a maximum
critical temperature to an insulator when the 2DEG is
fully depleted. The microscopic pairing mechanism in
these systems is still debated and under study. A recent
overview of superconductivity in oxydes interfaces can
be found by Gariglio et al.
7
As these types of interface films are burried below
the surface, they are not accessible to purely surface-
sensitive techniques like angular-resolved photoemission
spectroscopy (ARPES) or scanning tunneling spec-
troscopy (STS). As a consequence the bandstructure
characteristics of the 2DEG cannot be probed nor the
local superconducting properties. On the other hand,
in situ prepared surface-confined films grown by MBE
are nowadays accessible not only to in situ ARPES and
STS but also to in situ magneto-electrical transport
measurements and structural characterization tools like
electronic or optical surface diffraction. As a conse-
quence surface-confined systems represent at present the
best archetypes on which can be summarized the new
properties emerging in ultimately thin superconducting
films hosting a two-dimensional electron gas, because
they can be probed by both in situ macroscopic and
microscopic measurement techniques. The model system
we will widely refer to in this review consists of a single
atomic plane of a conventional superconductor, like for
example lead (Pb), grown on top of a semiconducting
substrate, like Si(111)
8
. By conventional supercon-
ductivity we mean that the pairing interaction is of
electron-phonon type.
We decided also in this paper not to focus on re-
cent advances in studying few layers, bilayers or
monolayers of dichalcogenides materials. In these
systems conventional superconductivity is most prob-
ably at play. This includes for example the study of
ultrathin MoS
2
or 2H-NbSe
2
films, using various surface
encapsulation techniques to protect the layers and/or
gating techniques to reach the 2D electronic regime
of superconductivity
9–14
. We believe that although
very interesting results were obtained in this field re-
cently, these measurements consist in almost exclusively
magneto-electrical transport experiments. This is way
too lacunar yet to furnish an overview of their precise
superconducting properties from a multi-techniques
approach. For similar reasons we will not address
recent results obtained by inducing superconductivity
at the surface of SrTiO
3
15
or layered ZrNCl
16
using an
electric-double-layer gating in an organic electrolyte, or
at the surface of graphene using proximity effects
17,18
.
We also mention that it is beyond the scope of
this review paper to cover the recent advances obtained
in the field of unconventional superconductivity, where
the pairing mechanism is not mediated by an electron-
phonon interaction. In particular very promising results
were obtained by fabricating in UHV ultrathin cuprates
films
19
. In the field of iron-based superconductivity,
trumendous effects of the control of the interface and
substrate termination were demonstrated for FeSe films
using an SrTiO
3
substrate
20,21
. A critical tempera-
ture above 100 K was reported for one unit cell of
FeSe/SrTiO
3
, which is one order of magnitude larger
than the bulk FeSe critical temperature. There is thus
hope in these fields both to better study and understand
the unconventional superconducting properties and
to engineer a higher critical temperature by playing
suitably with the substrate.
We will adopt in our review the following plan. In
the next section II, we will introduce useful concepts
worked out in studies of so-called ”granular” and
”homogeneous” superconducting thin films that will
be necessary to understand the role of non-magnetic
disorder on 2DEG superconductors. In this section, we
will also briefly review the superconducting properties
of crystalline Pb/Si(111) ultrathin films grown under
ultrahigh vacuum (UHV) conditions in order to illustrate
their specific properties related to quantum-size effects
and thickness reduction. In the section (III) we will
review the growth methods and structural properties
of the presented 2DEG surface-confined superconduc-
tors. In section (IV), we will present the electronic
structure and Fermi surface properties as measured by
macroscopic ARPES and confront them to ab initio
DFT calculations based on the characterized atomic
4
structures of the monolayers. The following section
(V) will summarize the macroscopic properties inferred
from ex situ and in situ magneto-electrical transport
measurements, including recent attempts to study the
Berezinsky-Kosterlitz-Thouless 2D regime. In the last
section (VI), we will summarize the emerging local
spectroscopic properties measured by STS. These latter
demonstrate new short-range spatial variations of the
local superconducting properties due to a combined
effect of non-magnetic disorder and two-dimensionality,
and peculiar in-gap states giving evidence for the pres-
ence of a mixed singlet-triplet superconducting order
parameter induced by the presence of a strong Rashba
spin-orbit coupling term at the surface. These local
signatures will be discussed along with ARPES and in
situ transport measurements in parallel high-magnetic
field on closely related systems. Finally, the peculiar
role played by atomic steps in anisotropic monolayers on
vortex properties, leading to the observation of mixed
Abrikosov-Josephson vortices in agreement with in situ
macroscopic transport measurements will be presented.
In a last section, we will conclude and summarize the
key-results obtained using both local and macroscopic
measurement techniques. We will also figure out the
remaining important issues to be clarified in monolayer
superconductors. From the overview of all recent
experimental and theoretical results it will appear that
these surface 2D superconductors, such as one monolayer
of Pb on Si(111), are ideal templates to engineer and
realize topological superconductivity.
II. BASIC PROPERTIES OF ULTRATHIN
DISORDERED S-WAVE SUPERCONDUCTORS
Real materials are not pure and contain impurities or
structural defects which may deeply alter their electronic
properties. It was shown by Abrikosov, Gorkov
22
and
Anderson
23
that the presence of non-magnetic disorder
in s-wave superconductors does not alter their thermody-
namic properties, upon the assumption of time-reversal
symmetry. Later on, detailed experimental and theo-
retical investigations have addressed over decades the
effect of disorder on the properties of superconductors,
in particular through thickness reduction. As a result,
several complicated situations may occur, depending on
the interplay between structural and electronic prop-
erties of the material. Some of these aspects are re-
viewed briefly by Goldman and Markovic
24
and thor-
oughly by Feigel’man et al.
25
and by Gantmakher and
Dolgopolov
26
. In general a transition to a metallic or in-
sulating state occurs beyond a critical disorder, which is
achieved when the film’s square resistance becomes com-
parable to R
square
' h/4e
2
.
Ga
Bi
a) b)
FIG. 1: Illustration of the different behaviors observed
in so-called granular and homogeneous disordered su-
perconducting ultrathin films. Evolution of the tempera-
ture dependence of the square resistance of Gallium and Bis-
muth ultrathin films. Each curve corresponds to a deposited
thickness given in
˚
A. (a) Granular Ga films deposited by the
quench-condensed method on a quartz substrate. Data repro-
duced from Jaeger et al.
31
. (b) Bi films grown by the same
method but deposited on a Ge-coated layer. Data reproduced
from Haviland et al.
34
. The thickness increments are respec-
tively of 0.05-0.1
˚
A for Ga and 5.4
˚
A for Bi.
A. Granular superconducting films
Historically, so-called granular materials were syn-
thetized first using the quench condensation method
4
.
Usually these thin films were synthetized by evaporating
a metallic source onto a cold dielectric substrate consist-
ing of glass or quartz. These films consist of grains or
nanocrystallites oxydized at their surface because they
were prepared in a rather high vacuum pressure. Thus
these grains are coupled together by low-transparency
tunnel junctions: the electrons can not directly hop from
one grain to the next but should tunnel to be transferred
from one nanocrystal to the other. As a consequence the
electrical coupling between neighboring grains is rather
poor and Coulombic effects start to play an important
role through the charging energy associated to each grain.
The grain’s charging energy is E
c
= e
2
/2C, where e is the
electronic charge and C the grain’s effective capacitance.
It represents the Coulombic energy barrier necessary to
inject (extract) one electron into a particular grain
27
.
5
In granular materials, the charging energy E
c
com-
peats with another energy scale, the Josephson energy
E
J
, related to the establishment of superconductivity
in the whole film
28–31
by coupling the superconducting
phases of neighboring grains. The Josephson energy is
given by E
J
= Φ
0
I
c
, Φ
0
= h/2e being the flux quantum
and I
c
the critical current, i.e. the maximum supercur-
rent, of a Josephson junction. As the grains are elec-
tronically poorly coupled to each other, superconductiv-
ity develops in each grain labelled by the index j with
its own order parameter ∆
j
exp(iΦ
j
) (amplitude ∆
j
and
phase Φ
j
) and couples from one grain to the next by the
Josephson effect.
The ratio between the Josephson and the charging en-
ergy characterizes how Cooper pairs can tunnel from one
grain to the neighboring one and establish a long range
phase coherence of the superconducting order (E
J
E
c
)
or in the contrary be blocked on grains by charging ef-
fects (E
c
E
J
). The typical evolution of the critical
temperature as a function of film thickness d is seen on
Fig. 1(a). The temperature onset, where the resistance
starts to decrease from above, does not change with thick-
ness. This indicates that the superconducting transition
temperature remains roughly constant and over a thick-
ness change of less than one monolayer the films go to
an insulating state. To summarize the granular super-
conductors’s properties: there is a competition between
Coulomb blockade and Josephson coupling and the T
c
of
the whole film is given by the elementary grain’s T
c
which
does not change much upon thickness reduction.
B. Disordered homogeneous superconducting films
With the progress in deposition techniques, so-called
disordered homogeneous superconducting films could be
grown
24,25,32–34
. As shown in Fig. 1(b), they are charac-
terized by a gradual reduction of their critical tempera-
ture T
c
with decreasing thickness d. This behavior is at
variance with the one seen in Fig. 1(a). In contrast to the
synthesis process used forgranular films described above,
a germanium few monolayer thick film is deposited first
on top of the dielectric substrate. This provides dan-
gling bonds at the surface making the growth of metal
films much more smooth and homogeneous on the Ge-
coated substrate. These disordered homogeneous films
consist of nanocrystals well-coupled electrically to each
other: in contrast to granular films there are no weak
links between the neighboring nanocrystal. This situa-
tion is in practice achieved in many ultrathin films: Pb,
In, Bi, Al, MoGe, nitrides like TiN or NbN
24,25,32,33
. In
all these systems (about 2 nm thick for the thinnest) the
superconducting properties are two-dimensional (2D), as
the film thickness d is smaller than the low-temperature
superconducting coherence length ξ
0
, but the underlying
electronic wavefunctions are still three-dimensional (3D)
as λ
F
remains significantly smaller than d.
The mechanisms driving the film from a super-
FIG. 2: Spatial gap variations observed in so-called
homogeneous disordered superconducting ultrathin
films. The color map represents the superconducting gap
measured locally by STS in a 3.6 nm thick TiN film grown
ex situ on Si/SiO
2
substrate. The superconducting inhomo-
geneities show up on a scale of few tens of nanometers. The
film’s temperature dependent electrical resistivity and typical
STS spectrum are shown in Fig. 4 for film TiN1. Adapted
from Sac´ep´e et al.
39
.
conducting to an insulating or metallic state are a
complex combination of Coulombic effects and Anderson
localization effects, both being induced by increasing
disorder
24,25
.
Coulombic effects alone lead to the so-called Fermionic
scenario
35
. It is characterized by enhanced Coulomb
repulsion between delocalized electrons because in-
creased scattering due to high disorder provokes a
reduced electronic screening. This reduced electronic
screening was originally predicted by Altshuler and
Aronov in disordered metallic systems. They showed
that this effect results in a spectroscopic signature :
there is a reduced electronic density of states (DOS)
at the Fermi level. This effect is now referred to as
the Altshuler-Aronov anomaly
36
. In the Finkelstein
scenario, the Altshuler-Aronov effect leads to a reduced
electron-phonon coupling constant through the increased
electronic repulsion term, and thus to a reduced T
c
for
increasing disorder.
On the other hand, Anderson localization effects
alone induced by strong disorder (i.e. neglecting
Coulombic effects) lead to the localization of single-
electron wavefunctions on a localization length L
loc
(see
25
and references therein). This localization process
introduces a new energy scale δ
loc
= 1/(ν
0
L
3
loc
), ν
0
being
the DOS at E
F
per spin. δ
loc
represents the average
electronic level spacing in a volume of size L
3
loc
where
electrons are localized. In principle, Cooper pairing of
electrons in a localized volume is still possible if δ
loc
is
smaller than T
c
, as it is the case for single isolated small
