Scanning gate microscopy measurements on a superconducting single-electron transistor
M. Huefner,* C. May, S. Kičin, K. Ensslin, and T. Ihn
Solid State Physics Laboratory, ETH Zürich, 8093 Zürich, Switzerland
M. Hilke
Department of Physics, McGill University, Quebec, Canada H3A 2T8
K. Suter and N. F. de Rooij
Sensors, Actuators and Microsystems Laboratory (SAMLAB), Institute of Microengineering (IMT), Ecole Polytechnique
Fédérale de Lausanne (EPFL), Rue Jaquet-Droz 1, CH-2002 Neuchâtel, Switzerland
U. Staufer
Micro and Nano Engineering PME, 3mE, TU Delft, Mekelweg 2, 2628 CD Delft, The Netherlands

Received 20 November 2008; revised manuscript received 23 February 2009; published 29 April 2009
We present measurements on a superconducting single-electron transistor
SET in which the metallic tip of
a low-temperature scanning force microscope is used as a movable gate. We characterize the SET through
charge stability diagram measurements and compare them to scanning gate measurements taken in the normal
conducting and the superconducting states. The tip-induced potential is found to have a rather complex shape.
It consists of a gate voltage-dependent part and a part which is independent of gate voltage. Further scanning
gate measurements reveal a dependence of the charging energy and the superconducting gap on the tip position
and the voltage applied to it. We observe an unexpected correlation between the magnitude of the supercon-
ducting gap and the charging energy. The change in EC can be understood to be due to screening, however the
origin of the observed variation in
remains to be understood. Simulations of the electrostatic problem are in
reasonable agreement with the measured capacitances.
DOI: 10.1103/PhysRevB.79.134530 PACS number
s: 73.21.La, 73.23.Hk, 74.50.r
I. INTRODUCTION
A broad variety of quantum dots has been investigated
over the last years. Even though fabrication and control of
these zero-dimensional systems are rather complex, it is by
now established to control individual electrons on semicon-
ductor quantum dots.1,2 Most transport experiments measure
macroscopic currents and voltages. These quantities contain
spatial information of wave functions only in an indirect
way. To investigate the local electronic structure of a quan-
tum dot, one needs to take a different approach. An option is
scanning probe microscopy, where a metallic tip is used as a
movable gate. The tip interacts capacitively with the sample.
Since the tip can be moved freely in all three dimensions,
data can be acquired that has a certain spatial resolution.
Measurements employing this technique have been per-
formed on various semiconductor nanostructures3–8 includ-
ing quantum dots.9–12
In semiconductor quantum dots there are two dominant
energy scales, namely, the charging energy, which depends
on the capacitance and therefore the geometry of the dot and
its gate electrodes including the tip, and the single-particle
level spacing arising from the quantum-mechanical confine-
ment of the system. It is a challenge to disentangle these two
energy scales with a scanning gate experiment and to extract
spatial information about individual quantum states.13,14 As
shown in Ref. 13 the tip-induced potential in the plane of the
two-dimensional electron gas in which the quantum dot is
formed can be complex and may consist of two additive
parts, of which only one depends on the voltage applied to
the tip. It was shown that it is possible to observe distinct
features related to a specific quantum state when performing
scanning gate measurements on a semiconductor quantum
dot. It remained, however, an open question if these features
were related to the dissimilar wave functions of the different
quantum states.
In metallic single-electron transistors
SETs the single-
particle level spacing is orders of magnitude too small for
being observable.15 For superconducting single-electron tran-
sistors new energy scales enter the problem, namely, the en-
ergy gap
of the superconductor and the Josephson energy
EJ.
Here we present scanning gate measurements performed
on a superconducting SET. Spatial images of the differential
conductance give insight into the interaction potential be-
tween the tip and the electrons in the SET. We investigate in
detail how the charging energy as well the superconducting
gap of the SET island depend on tip position and the voltage
applied to the tip.
II. SETUP
The aluminum SET is fabricated on a silicon dioxide sub-
strate employing the method of shadow evaporation.16,17 Be-
tween the two evaporation steps the sample is exposed to
oxygen in order to form the tunnel barriers between the SET
and the leads. Figure 1
a shows a scanning electron micro-
scope picture of such a SET. The lateral dimensions of the
SET island are about 6040 nm2. The thickness of the Al is
about 20 nm, the thickness of the oxide barrier about 1 nm.
The SET island is connected to source S and drain D via
PHYSICAL REVIEW B 79, 134530
2009
1098-0121/2009/79
13/134530
8 ©2009 The American Physical Society134530-1

tunnel barriers of an area of approximately 2040 nm2. A
gate G is located about 300 nm away from the SET.
The measurements are carried out with a scanning force
microscope
SFM operating in a 3He cryostat with a base
temperature of around 300 mK. The scanning sensor consists
of an electrochemically etched PtIr tip with an initial radius
of 50 nm, mounted on a piezoelectric tuning fork.18 This tip
induces a local electrostatic potential
called tip-induced po-
tential or tip potential for simplicity in the sample below as
schematically shown in Fig. 1
b. The magnitude of the po-
tential can be changed by applying a voltage VT to the tip or
changing the tip-sample separation in z direction. The scan-
ning gate measurements shown in this paper are carried out
by scanning the tip at a constant height z=50 nm above the
sample surface and recording the differential conductance of
the SET as a function of the tip position. No current flows
from the tip to the sample due to the vacuum gap of 50 nm
between them. The system can be kept at base temperature
for up to 3 days. The sample can be measured up to a couple
of days without visible charge rearrangements. Due to a vi-
bration reduced setup we can expect the distance between the
tip and the sample to be very stable. Standard lock-in tech-
niques are used to measure the conductance of the SET.
III. TRANSPORT MEASUREMENTS
In order to characterize the SET before performing scan-
ning gate experiments, measurements of the differential con-
ductance
dI /dVSD as a function of the source-drain voltage
VSD and the gate voltage VG are carried out. Such charge
stability diagrams
Coulomb blockade diamonds give in-
sight into the transport processes that contribute to the cur-
rent flow.19–22 They also contain information about the charg-
ing energy EC, the superconducting gap
, as well as the
capacitances involved.
While Coulomb blockade diamonds are recorded, the
SFM tip is kept at a constant position 70 nm above the SET
and the sample is kept at a temperature around 700 mK, well
below the critical temperature TC of aluminum which is
around 1.2 K.23 In order to measure a Coulomb blockade
diamond in the normal conducting state a magnetic field of
0.5 T is applied.
Figures 2
a and 2
b show the dI /dVSD data of the SET
in the normal conducting state measured at finite magnetic
field and plotted on a logarithmic scale in
a and a linear
scale in
b. Figures 2
c and 2
d show the corresponding
Coulomb blockade diamonds of the SET in the supercon-
ducting state at B=0 T.
In the dI /dVSD data recorded in the normal conducting
state, diamond-shaped regions are visible, where the differ-
ential conductance is zero. We can estimate the charging en-
ergy EC from the extent of these Coulomb blockade dia-
monds in the direction of VSD and find it to be around 1 meV.
Without the external magnetic field we observe that the
diamonds do not close any more at zero VSD but we rather
observe a gap in VSD direction of about 1.6 mV. This gap as
indicated in Fig. 2
d is a measure of the superconducting
gap
which is one eighth of this total gap which is about 0.2
meV in our case. This behavior has been reported in numer-
ous publications.21,24,25
In order to understand the scanning gate measurements
presented later in this paper, it is important to realize that the
position of the center of a Coulomb blockade diamond, its
extent p along the gate voltage axis see Fig. 2
a, as well as
EC are the same in the superconducting and the normal state.
The most pronounced feature in the superconducting dia-
mond is the transition in VSD direction from the almost insu-
lating regime to the conducting regime, where transport is
dominated by resonant quasiparticle tunneling. This sharp
rise in the current will be referred to as current onset
CO in
the rest of the paper. Due to finite temperature in our setup,
we do not expect to see processes such as Andreev reflec-
tions or Josephson quasiparticle processes inside the Cou-
lomb blockaded regions as they were observed in other mea-
surements at lower temperatures.21,25 The Josephson energy
EJ can be estimated using EJ=
IC /
2e=h
/
8e2R to be
0.1 eV, where IC is the junction critical current and R the
tunnel resistance of a single junction. The thermal energy
kBT is about 0.06 meV which is 1–2 orders of magnitude
smaller then EC and
. Between each of the relevant energy
(a) (b)100nm SET-island
S D
G
I
ti
p
po
te
nt
ia
l
y
x
FIG. 1.
Color online
a Topography of the SET, where S is the
source, D is the drain, G marks the gate, and I the SET island.
b
Schematic of the local potential induced by the SFM tip.
-4
-3
-2
0.2
0.6
1
V
(
V
)
G
-2 0 2
0
0.1
0.2
V (mV)
SD
0
0.03
0.06
-4
-3
-2
-1
-2 0 2
V (mV)
SD
0.2
0.6
1
V
(
V
)
G
(c)
(a)
(d)
(b)
p
gap
FIG. 2.
Color online
a Coulomb blockade diamond of the
SET in the normal conducting state recorded at a magnetic field
B=0.5 T and a temperature T=700 mK. Color bar shows the dif-
ferential conductance log10
dI /dVSD with the differential conduc-
tance dI /dVSD in units of e2 /h plotted on logarithmic scale, whereas

b shows the same Coulomb blockade diamond plotted on a linear
scale. Color bar shows dI /dVSD
e2 /h.
c Coulomb blockade dia-
mond of the SET in the superconducting state recorded at B=0 T
and T=700 mK, plotted on logarithmic scale. Color bar shows the
value of log10
dI /dVSD with the differential conductance dI /dVSD
in units of e2 /h.
d The same Coulomb blockade diamond on linear
scale. Color bar shows dI /dVSD
e2 /h.
HUEFNER et al. PHYSICAL REVIEW B 79, 134530
2009
134530-2

scales EC
kBTEJ there is about 1 order of magnitude
difference. Since ECEJ Josephson processes are strongly
suppressed and transport is dominated by Coulomb block-
ade. The thermal energy available in the system is not neg-
ligible compared to
. Therefore thermally activated quasi-
particle tunneling can take place, as seen in Fig. 2
c.
To visualize the relevant transport processes, schematic
sketches of the energy-level structure are shown in Fig. 3 for
different positions inside the Coulomb blockade diamond.
Each sketch consists of the Fermi level of the source and
drain to the left and right as well as the single levels inside
the SET shown in the middle and labeled with i=n+1,n ,
n−1. Gray-shaded intervals represent the superconducting
gap
. The solid vertical lines represent the tunnel barriers
between the island and the leads. At the center of the Cou-
lomb blockade diamond Fig. 3
d, all levels are detuned.
No energy is available to produce quasiparticle tunneling.
When moving to higher gate voltages, the levels in the SET
move down Fig. 3
c until one of them becomes resonant
with the source and drain levels Fig. 3
b. When moving
from this point to finite source-drain voltages Fig. 3
g, the
source and drain levels shift antisymmetrically compared to
the SET level. This shift becomes large enough to allow
quasiparticle tunneling at the CO. When moving along the
CO to lower VG Fig. 3
h the source level stays at the same
relative position compared to the SET level, while the drain
level is shifted further down. At the outermost peak of the
CO Fig. 3
i the source and drain levels have been shifted
far enough apart that two levels in the SET can contribute to
transport. As shown by Figs. 3
e and 3
f processes inside
the Coulomb blockaded region are also possible due to the
thermal energy available in the system.
From the period p of dI
V /dVSD typical values for the
gate-island and the tip-island capacitances of 0.3 and 0.6 aF,
respectively, are determined. The capacitance of the island
C

is derived from EC using C =e2 /EC. We assume
that each of the two junctions has the same capacitance,
since the fabrication process and the area are the same.
Therefore the junction capacitance can be calculated using
CSD=
C −CG /2. We find it to be around 80 aF. This value
is consistent with the capacitance one would expect when
approximating the junction capacitance using a simple plate
capacitor model with areas of 2050 nm2, which is calcu-
lated to be 77 aF.
Figure 4 shows the dependence of EC and
on the mag-
netic field. The superconducting gap
reduces until the sys-
tem reaches the normal conducting state around 0.2 T Fig.
4
a. However, the charging energy EC remains essentially
constant for both the superconducting and the normal con-
ducting states as seen in Fig. 4
b.
The exact evaluation routine to derive those quantities
from the Coulomb blockade measurements will be described
later on. However, the evaluation routine is slightly changed
when reaching a magnetic field of B=0.2 T, as the em-
ployed model only works for the superconducting state.
Above B=0.2 T the superconducting gap is approximately
zero and only the charging energy was deduced from the
Coulomb blockade diamonds. As can be seen in Fig. 4
b,
when changing the evaluation procedure, EC rises slightly
and then decreases indicating that the superconducting gap
might not have reached completely zero below B=0.4 T.
However for all data points below B=0.2 T this analysis
shows that the charging energy is little if not at all influenced
by the magnetic field, whereas the superconducting gap
clearly is.
IV. SCANNING GATE MEASUREMENTS
In the following we present scanning gate measurements
of the SET in the normal and superconducting states. Be-
cause the SFM tip acts as a movable gate, changing the volt-
age applied to the tip or changing its position should have a
similar effect as changing the voltage applied to the in-plane
gate. The voltage applied to the tip was kept constant for one
single scan, as was the separation z between sample surface
and tip. The influence the tip has on the SET depends on the
xy position in the scan frame Fig. 1
b.
Figure 5 shows scanning gate measurements taken at two
different magnetic fields. Let us start by discussing the SET
in the normal conducting state. Figure 5
a shows a scanning
gate image taken at a magnetic field of B=0.5 T, VT=0 V,
VG=0 V, and VSD=1.5 mV, which corresponds to the
source-drain voltage marked by the dashed line in Fig. 2
a.
We observe concentric ring-shaped features. The island itself
is expected to be located at the center of the concentric rings.
However, an exact position of the island cannot be given due
to the complexity of the tip. The inset shows a cross section
through the scanning gate image taken along the dashed line
in Fig. 5
a. When looking along the line in Fig. 2
a we see
4321
-2 0 2
V (mV)SD
0.2
0.6
1
V
(V
)
G
(a)
b
c
d
e f
h
i
g
(b)
(c)
(d)
(e)
(f) (h)
(i)
(g)
n
n+1
n-1
n-1
n
n+1
n-1
n+1
n
n-1
n
n-1
n
n-1
n+1
n
n-1
n+1
n
n-1
n+1
n
FIG. 3.
Color online
a Coulomb diamond of the SET in the
superconducting state recorded at B=0 T and plotted in logarithmic
scale. Columns
b–
i show schematics of the energy-level struc-
ture to illustrate how electric transport takes place.
0 0.1 0.2
0
0.1
0.2
0.3
B(T)


(m
eV
)
0 0.2 0.4
0.9
1
1.1
1.2
B(T)
E
(m
eV
)
C
(a) (b)
FIG. 4.
Color online
a Decay of the superconducting gap vs
magnetic field.
b Change in the charging energy as a function of
magnetic field. The two different symbols mark two data sets, taken
in different condensation cycles.
SCANNING GATE MICROSCOPY MEASUREMENTS ON A… PHYSICAL REVIEW B 79, 134530
2009
134530-3

regularly spaced Coulomb resonances with a constant period
p in VG direction. For the chosen VSD voltage we do not
expect to cut through the Coulomb blockaded regime. How-
ever, we expect one maximum in the current in each period
p. Furthermore, we find the peak distances to have a mono-
tonic behavior. This can be clearly observed in the inset of
Fig. 5
a.
Figure 5
b shows a scanning gate measurement of the
same scan frame as in Fig. 5
a but in the situation where the
SET is in the superconducting state. Since the two measure-
ments have been carried out one right after another and no
charge rearrangements were observed, we can assume the
SET to be in the same now superconducting state until the
clearly visible charge rearrangement in the last quarter of the
measurement in the superconducting state takes place. The
signature of superconductivity at this source-drain voltage is
the splitting of the resonance rings. Every single ring splits
up into two when the SET is scanned in the superconducting
state. This finding is consistent with the features observed in
the Coulomb diamonds. Looking at the dashed line in Fig.
2
d we can see that for each period p in the superconducting
state we expect to cross the line of current onset twice. When
looking at the insets in Fig. 5
b we see a difference in the
peak height as compared to the normal conducting scanning
gate measurement. This is consistent with the fact that for the
normal conducting state we do not reach the regime of total
Coulomb blockade for the chosen source-drain voltage,
whereas for the superconducting state we cross the line of the
current onset. The observations made in these scanning gate
measurements are consistent with the Coulomb diamonds
discussed before. Since the number of features as well as
their spacing is the same for the superconducting and normal
conducting state, we know that we can controllably load
single electrons onto the SET by scanning the tip.
The noncircular shape of the Coulomb rings is presum-
ably due to a tip shape that is not completely rotationally
symmetric around the z axis.26 This can probably be attrib-
uted to the topography scanning carried out before the scan-
ning gate measurement, which led to slight deformations, or
the attachment of unwanted particles to the tip. Both effects
have been observed before in scanning gate measurements.26
The effect of temperature on the scanning gate measure-
ment is investigated in Fig. 6. We observe that the double
rings found in the superconducting state merge into single
rings when the temperature is sufficiently high to suppress
superconductivity. The position of the rings remains un-
changed by this transition. This is in good accordance with
the measurement at finite magnetic field. However, the reso-
nances in the normal conducting state are broader due to the
higher temperature. Further investigations of the influence of
the bias voltage VSD on the scanning gate measurements are
shown in the Appendix.
V. TIP-INDUCED POTENTIAL
In order to learn more about the tip-induced potential, a
measurement of the differential conductance as a function of
VT is performed while moving the tip along a line across the
SET. The approximate position of this trace is shown as a
dashed line in Fig. 1. The height z of the tip above the sur-
face was constant at 200 nm. At each of the 1950 steps along
this line a trace of the differential conductance as a function
of the tip voltage was recorded. Figure 7
a shows the result
of this measurement.
We see how the positions of the Coulomb peaks change
when altering the tip position relative to the SET. We observe
a concave and a convex part, which means that our tip po-
tential consists of an attractive and a repulsive component.
However, there is no tip voltage value in the investigated
regime, where the tip does not induce any charge on the SET
d
I/
d
V
(e
/
h
)
2
0.05
0.07
0.09
d
I/
d
V
(e
/
h
)
2 0.1
0.05
x(nm)
0 1000
d
I/
d
V
(e
/
h
)
2
0.06
0.05
x(nm)
0 1000
0.07
(b)(a)
400nm
FIG. 5.
Color online Scanning gate measurements for different
magnetic field. The tip-sample distance is around 50 nm, VG
=0 V, and VSD=1.5 mV. The scanned area is 1.511.51 m2
a
at B=0.5 T and
b at B=0 T. Insets show cross sections at the
position of the gray dotted line. White lines in
a mark the approxi-
mate position of source, drain, and SET of the structure.
d
I/
d
V
(e
/
h
)
2
0.05
0.07
0.09
d
I/
d
V
(e
/
h
)
2 0.1
0.05
x(nm)
0 1000
d
I/
d
V
(e
/
h
)
2
0.06
0.55
x(nm)
0 1000
(b)(a)
400nm
FIG. 6.
Color online Scanning gate measurements for different
temperature. The tip-sample distance is around 50 nm, VG=0 V,
and VSD=1.5 mV. The scanned area is 1.511.51 m2
a at T
=2 K and
b at T=700 mK. Insets show cross sections at the
position of the gray dotted line.
b and Fig. 5
b are identical.
FIG. 7.
Color online
a Conductance of the SET vs the volt-
age applied to the tip along one line over the SET. From this we can
see that the tip potential has an attractive and a repulsive part. The
tip-sample distance is around 50 nm, T=700 mK, VSD=0 V, and
B=0 T.
b shows the gate and the tip capacitances obtained from
different Coulomb diamonds in dependence on the position of the
SET.
HUEFNER et al. PHYSICAL REVIEW B 79, 134530
2009
134530-4

at all positions along the line. This is in agreement with the
observations made previously on semiconductor quantum
dots.27 One would expect to observe a least invasive tip volt-
age close to the value one estimates from the work-function
differences. Contrary to this expectation, we do not find a
least invasive voltage here, even though it would be expected
at an offset voltage of 1.4 V because of the work-function
difference between PtIr

Pt5.6 eV and Al

Al
4.3 eV.
The shapes of the single resonance curves are almost
identical. This indicates that the shape of the tip-induced
potential is independent of the voltage applied to the tip. A
change in VT only changes the offset of this potential, not its
shape over the length scale of this measurement. This is the
manifestation of a strong contribution to the tip-induced po-
tential which is independent of VT. It may result from
charged debris attached to the tip.13 The VT-dependent con-
tribution to the tip-induced voltage becomes visible on larger
length scales in the gate and the tip capacitances, as deduced
from Coulomb blockade diamond measurements. Figure 7
b
shows CT and CG as derived from single Coulomb blockade
diamond measurements performed as the tip is positioned on
various points along one line across the SET. We observe a
smoothly shaped single peak dependence for both capaci-
tances, with a width of several micrometers. The change in
CG is due to the fact that the gate gets shielded from the SET
by the presence of the tip. Therefore the change in CG is
much smaller than the change in CT and inverse in sign to it.
However, the change in CT by 0.4 aF is small compared to
the capacitance of the system, which is around 161 aF. It is
interesting to note that even though the tip potential looked
quite complex at the point of time of this measurement, the
capacitances only show a very smooth Gaussian dependence.
This confirms that the tip potential is the sum of two inde-
pendent parts of which only one depends on VT.
VI. ANALYSIS OF THE TIP-INDUCED POTENTIAL
It has been shown in Ref. 13 that for scanning gate ex-
periments on a semiconductor quantum dot fine structure
could be observed that depended on which quantum state the
dot was kept. The exact interpretation, however, remained an
open question. When performing scanning gate measure-
ments on a metallic SET, we do not expect to see features
connected to individual single-particle wave functions since
in metals the single-particle energy scale is negligible.
In order to shine further light on this question, two scan-
ning gate measurements are performed where all settings
were kept exactly the same, except that the gate voltage was
changed by 1 Coulomb diamond period p to reach a different
charge state. Figure 8 shows two such scanning gate mea-
surements in
a and
b. Figure 8
c shows the difference of
those two measurements. We see that some of the rings do
overlap as expected. However, in the center the measurement
in
a shows rings with a smaller radius than the measure-
ment in
b, whereas at the outermost rings the opposite
seems to be the case. Therefore we have to conclude that a
shift in gate voltage by p does not lead to exactly the same
scanning gate image. This is contrary to our expectations that
the Coulomb rings should overlap completely when carrying
out two scanning gate measurements for two different charge
states in a metallic SET. We therefore have to conclude that
the period depends on the position of the tip, i.e., p
=p
x ,y.
As a conclusion, we have shown that the influence the tip
exerts on the SET during scanning gate measurements is
similar to the influence of a planar gate. Although we have a
complex tip potential, we are able to control the occupation
of the SET island on the level of single electrons. The tip-
island capacitance can be changed by about 0.3% of the
value of the total capacitance of the island. Furthermore the
measurement shown in Fig. 8 reveals that the period p is
influenced by the position of the scanning tip.
VII. INVESTIGATION OF THE CHARGING ENERGY
Knowing that the period p and with it the charging energy
depends on the SFM tip position, the next open question is
how other parameters extracted from the Coulomb diamonds
depend on the tip position or voltage. For the first measure-
ment
grid measurement a grid of 36 tip positions is chosen
that covers the scan frame shown in Fig. 8. At each of these
36 positions a Coulomb diamond is recorded and
, EC, CG,
and CT are extracted. In order to reach a good comparability
of these data points all 36 points are measured in the same
condensation cycle of the fridge. Because of time constraints
the Coulomb blockade diamonds are not recorded with the
resolution shown in Fig. 2, but rather reconstructed from VSD
sweeps at as few VG voltages as needed to extract the desired
quantities. In order to cover a larger lateral distance the same
measurement is carried out for a number of positions distrib-
uted along a stretch of 15 m across the SET
line measure-
ment, see Fig. 1.
Since we also want to investigate the influence of the tip
voltage on
, EC, CG, and CT, Coulomb blockade diamond
measurements are carried out for a constant tip position but
changing tip voltages ranging from 0 to 7 V. Two sets of
measurements are carried out in different condensation
cycles, referred to as VT data 1 and 2.
500nm
V =0.287VgateV =0.851Vgate
dI/dV (e /h)2
00.10.2
dI/dV (e /h)2
0.1 0 -0.1
(a) (b) (c)
difference:(b)-(a)
FIG. 8.
Color online Scanning gate measurement for two dif-
ferent Vgate voltages. Color bar is the same for
a and
b shown
here only once for simplicity.
c shows the difference of those two
measurements. One can see that although only one electron was
added onto the SET the scanning gate measurement shows a differ-
ent spacing of the Coulomb rings. The tip-sample distance is around
50 nm, T=700 mK, VSD=1.5 mV, and B=0 T. The scanned area
is 22 m2
SCANNING GATE MICROSCOPY MEASUREMENTS ON A… PHYSICAL REVIEW B 79, 134530
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The charging energy and the superconducting gap can be
read from a Coulomb diamond when the innermost and out-
ermost positions in VSD direction of the CO are known. To
determine those positions the COs are fitted linearly from
four dI /dVSD versus VSD sweeps
using two sweeps to deter-
mine the rising slope and two sweeps to determine the de-
clining slope. As the peak shape of the CO varies, it is not
possible to fit all peaks to determine the position of the CO.
The CO was rather determined to be positioned at the maxi-
mum of each trace. The period p
x ,y of the Coulomb block-
ade diamonds is determined from a dI /dVSD versus VG
sweep. Assuming this period to be constant over a VG range
of several Coulomb blockade diamonds, the two lines that fit
the COs were shifted in VG direction by p in order to deter-
mine the cross sections of the rising and falling slopes of the
CO. Those cross sections correspond to 4
and 4
+EC in
VSD.
Figure 9 shows the most striking result of these measure-
ments. We observe an anticorrelation between EC and
. For
different tip positions and voltages neither EC nor
stay
constant; they rather vary by about 15% and 20%, respec-
tively. For situations where the charging energy is large

small the superconducting gap is small
large. However
because of the complexity of the tip potential it is impossible
to make out a certain spatial trend in this variation. Even
though the variation in the superconducting gap
is unex-
pected, the variation in EC and p itself can be understood, as
we will discuss in detail in the next section. We compare the
cross sections taken at the minimum and the maximum VSD
value of the CO of two Coulomb diamonds, taken at differ-
ent positions. In Fig. 10, the maximum value of VSD of the
CO stays almost the same; the minimum value, however,
shifts.
In order to verify that this correlation does not arise from
a systematic error, such as the fact that the CO does not run
in a completely straight line as a function of gate voltage or
from noise overlaying the peak structure of the diamonds, we
determined the minimum and the maximum possible values
for EC and
that could be extracted when combining the
peak positions of the CO for all cross sections. We found the
values of EC and
to vary by less than 4%. However, the
change in these values as seen in Fig. 9 is more than a factor
of 3 larger.
The correlation between EC and
is highly unexpected.
Since the SET is metallic, screening should occur on the
surface within the first Angstroms of the sample. Further-
more the junctions are buried and should not be subjected to
the influence of the tip. The superconducting properties of
the system have their origin in the volume of the system and
not on the surface and should therefore not be influenced by
the SFM tip.
VIII. NUMERICAL SIMULATIONS
In order to gain further knowledge about the electrostatics
of the complete system
sample and SFM tip and therefore
learn more about the behavior of the capacitances, simula-
tions are carried out with the software tool COMSOL. The
geometry is modeled following the SEM pictures of the
structure
Fig. 1. The tip is approximated as a cone, i.e.,
having a round cross section closed to the sample with a
diameter of 40 nm, which increases when moving further
away from the structure. The tip is positioned at different
positions with respect to the island and the electric field is
calculated. Furthermore the capacitances of the tip, gate
source, and drain are determined with respect to the island.
Figure 11 shows the magnitude of the electric field in
color, whereas the sample outline is marked by the black
lines. The apex of the tip is depicted by the black circle.
Figure 11
a shows the electric field when the tip is posi-
tioned precisely above the island. The sample geometry is
approximately mirror symmetric left to right. This is re-
flected by the electric field distribution, which is also sym-
metric. When moving the tip away from this symmetry axis
toward the left above one lead, the electric field ceases to be
symmetric. Rather a high electric field is now found in the
vicinity of the tip around the left lead. The electric field can
be influenced by the position Figs. 11
a and 11
b and
voltage
not shown applied to the tip. Moving the tip over
source or drain leads to an asymmetric electric field with
respect to the sample symmetry axis.
Although the shape of the tip in our simulations has been
simplified compared to the experimental setup, the capaci-
tances obtained from the simulation show very good agree-
ment with the experimental values. All numbers are within a
factor of 2 of the measured values and their relative magni-
0.18 0.2 0.24
0.8
0.9
1.0
E
(m
eV
)
C

(meV)
grid data
line data
V data part2T
0.22 0.26
1.1
1.2 V data part1T
FIG. 9.
Color online
over EC extracted from Coulomb
blockade diamond measurements taken at different lateral tip posi-
tions and voltages applied to the tip. The distance of the tip from the
surface was kept constant.
V (meV)SD
0 1 2
dI
/d
V
(e
/h
)
2
0
0.4
0.8
position 1
position 2
CO
CO
EC4

FIG. 10.
Color online Cross sections of two different Coulomb
diamonds. Each set of lines of the same color belongs to one Cou-
lomb diamond. Arrows mark the positions of the current onset in
each trace.
HUEFNER et al. PHYSICAL REVIEW B 79, 134530
2009
134530-6

section peak structures of the Coulomb diamonds should also
be observable in the corresponding scanning gate images
Fig. 12 row
a.
When comparing row
b to row
c in Fig. 12 we notice
a good correlation between those two sets of measurements
with respect to peak shape, height, and spacing. Notably the
measurements for VSD=0,0.6,1.0 mV show an almost per-
fect consistency. The correlation seems not quite as good for
the measurements with VSD=0.4,1.6 mV. For VSD
=0.4 mV the scanning gate measurement shows just barely
split double peaks, whereas the cross section taken from the
Coulomb diamond shows double peaks but with minima be-
tween them that have all the same depth. For VSD=1.6 mV
the scanning gate image shows the beginning of a peak split-
ting into double peaks, where we would expect clean single
peaks from the Coulomb diamond measurement.
*huefner@phys.ethz.ch
1 L. P. Kouwenhoven, D. G. Austing, and S. Tarucha, Rep. Prog.
Phys. 64, 701
2001.
2 A. T. Johnson, L. P. Kouwenhoven, W. de Jong, N. C. van der
Vaart, C. J. P. M. Harmans, and C. T. Foxon, Phys. Rev. Lett.
69, 1592
1992.
3 S. H. Tessmer, P. I. Glicofridis, R. C. Ashoori, L. S. Levitov, and
M. R. Melloch, Nature
London 392, 51
1998.
4 M. A. Topinka, B. J. LeRoy, R. M. Westervelt, S. E. J. Shaw, R.
Fleischmann, E. J. Heller, K. D. Maranowski, and A. C. Gos-
sard, Nature
London 410, 183
2001.
5 G. A. Steele, R. C. Ashoori, L. N. Pfeiffer, and K. W. West, Phys.
Rev. Lett. 95, 136804
2005.
6 R. Crook, J. Prance, K. J. Thomas, S. J. Chorley, I. Farrer, D. A.
Ritchie, M. Pepper, and C. G. Smith, Science 312, 1359
2006.
7 A. Baumgartner, T. Ihn, K. Ensslin, G. Papp, F. Peeters, K. Ma-
ranowski, and A. C. Gossard, Phys. Rev. B 74, 165426
2006.
8 C. R. da Cunha, N. Aoki, T. Morimoto, Y. Ochiai, R. Akis, and
D. K. Ferry, Appl. Phys. Lett. 89, 242109
2006.
9 A. Pioda, S. Kicin, T. Ihn, M. Sigrist, A. Fuhrer, K. Ensslin, A.
Weichselbaum, S. E. Ulloa, M. Reinwald, and W. Wegscheider,
AIP Conf. Proc. 772, 781
2005.
10 S. Kicin, A. Pioda, T. Ihn, M. Sigrist, A. Fuhrer, K. Ensslin, M.
Reinwald, and W. Wegscheider, New J. Phys. 7, 185
2005.
11 A. Pioda, S. Kicin, T. Ihn, M. Sigrist, A. Fuhrer, K. Ensslin, A.
Weichselbaum, S. E. Ulloa, M. Reinwald, and W. Wegscheider,
Phys. Rev. Lett. 93, 216801
2004.
12 P. Fallahi, A. C. Bleszynski, R. M. Westervelt, J. Huang, J. D.
Walls, E. J. Heller, M. Hanson, and A. C. Gossard, Nano Lett. 5,
223
2005.
13 A. E. Gildemeister, T. Ihn, M. Sigrist, K. Ensslin, D. C. Driscoll,
and A. C. Gossard, Phys. Rev. B 75, 195338
2007.
14 A. E. Gildemeister, T. Ihn, R. Schleser, K. Ensslin, D. C.
Driscoll, and A. C. Gossard, J. Appl. Phys. 102, 083703
2007.
15 M. Furlan, T. Heinzel, B. Jeanneret, S. V. Lotkhov, and K. Ens-
slin, Europhys. Lett. 49, 369
2000.
16 G. J. Dolan, Appl. Phys. Lett. 31, 337
1977.
17 E. H. Visscher, S. M. Verbrugh, J. Lindeman, P. Hadley, and J. E.
Mooij, Appl. Phys. Lett. 66, 305
1995.
18 T. Ihn, Electronic Quantum Transport in Mesoscopic Semicon-
ductor Structures, Springer Tracts in Modern Physics 192

Springer, New York, 2004.
19 A. Maassen van den Brink, G. Schön, and L. J. Geerligs, Phys.
Rev. Lett. 67, 3030
1991.
20 T. A. Fulton and G. J. Dolan, Phys. Rev. Lett. 59, 109
1987.
21 R. J. Fitzgerald, S. L. Pohlen, and M. Tinkham, Phys. Rev. B 57,
R11073
1998.
22 D. V. Averin, A. N. Korotkov, A. J. Manninen, and J. P. Pekola,
Phys. Rev. Lett. 78, 4821
1997.
23 J. F. Cochran and D. E. Mapother, Phys. Rev. 111, 132
1958.
24 T. A. Fulton, P. L. Gammel, D. J. Bishop, L. N. Dunkleberger,
and G. J. Dolan, Phys. Rev. Lett. 63, 1307
1989.
25 P. Hadley, E. Delvigne, E. H. Visscher, S. Lahteenmaki, and J. E.
Mooij, Phys. Rev. B 58, 15317
1998.
26 A. E. Gildemeister, T. Ihn, M. Sigrist, K. Ensslin, D. C. Driscoll,
and A. C. Gossard, Appl. Phys. Lett. 90, 213113
2007.
27 A. Pioda, S. Kicin, D. Brunner, T. Ihn, M. Sigrist, K. Ensslin, M.
Reinwald, and W. Wegscheider, Phys. Rev. B 75, 045433

2007.
HUEFNER et al. PHYSICAL REVIEW B 79, 134530
2009
134530-8