Quantum capacitance of a superconducting subgap state in an electrostatically oating dot-island Filip K. Malinowski1R. K. Rupesh2Luka Pave si c3Zolt an Guba4Damaz de Jong1Lin Han1

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Quantum capacitance of a superconducting subgap state in an electrostatically

ﬂoating dot-island

Filip K. Malinowski,1, ∗R. K. Rupesh,2Luka Paveˇsi´c,3Zolt´an Guba,4Damaz de Jong,1Lin Han,1

Christian G. Prosko,1Michael Chan,1Yu Liu,5Peter Krogstrup,5Andr´as P´alyi,4, 6 Rok ˇ

Zitko,3and Jonne V. Koski7

1QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands

2School of Physics, University of Hyderabad, Hyderabad, India

3Joˇzef Stefan Institute & Faculty of Mathematics and Physics, University of Ljubljana, Ljubljana, Slovenia

4Department of Theoretical Physics, Institute of Physics, Budapest University

of Technology and Economics, M˝uegyetem rkp. 3., H-1111 Budapest, Hungary

5Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark

6MTA-BME Quantum Dynamics and Correlations Research Group, M˝uegyetem rkp. 3., H-1111 Budapest, Hungary

7Microsoft Quantum Lab Delft, Delft University of Technology, 2600 GA Delft, The Netherlands

(Dated: October 6, 2022)

We study a hybrid device deﬁned in an InAs nanowire with an epitaxial Al shell that consists of a

quantum dot in contact with a superconducting island. The device is electrically ﬂoating, prohibiting

transport measurements, but providing access to states that would otherwise be highly excited and

unstable. Radio-frequency reﬂectometry with lumped-element resonators couples capacitatively

to the quantum dot, and detects the presence of discrete subgap states. We perform a detailed

study of the case with no island states, but with quantum-dot-induced subgap states controlled

by the tunnel coupling. When the gap to the quasi-continuum of the excited states is small, the

capacitance loading the resonator is strongly suppressed by thermal excitations, an eﬀect we dub

“thermal screening”. The resonance frequency shift and changes in the quality factor at charge

transitions can be accounted for using a single-level Anderson impurity model. The established

measurement method, as well as the analysis and simulation framework, are applicable to more

complex hybrid devices such as Andreev molecules or Kitaev chains.

Andreev bound states [1] and Yu-Shiba-Rusinov

states [2] are the most familiar types of subgap states

(SGSs) observed in Josephson junctions [3, 4], at atoms

on a superconducting surface [5] or in semiconduct-

ing quantum dots (QDs) coupled to superconductors

(SCs) [6, 7]. SGSs, just like electronic states in QDs,

are well localized in space and for odd electron occupancy

have a spin which can be manipulated [8–10]. The ground

state (spin singlet or doublet) depends on the microscopic

details [11, 12]. Electrostatic ﬂoating of such a device

consisting of a dot coupled to a superconductor ﬁxes the

total charge, so that the SGS cannot undergo the singlet-

doublet phase transition, thereby enabling access to the

regimes beyond reach of conventional transport measure-

ments. Furthermore, forcing a ﬁxed charge of the system

largely eliminates quasiparticle poisoning that challenges

the realization of qubits based on SGSs [9, 13–15].

In this work we study a SGS formed in a QD coupled

to a SC island deﬁned in an InAs nanowire. The sys-

tem is galvanically isolated and the total charge is ﬁxed.

We couple the QD capacitively to a radio-frequency res-

onator [16] and study the device through its eﬀects on the

resonator response. We propose a simple device model

that is solved using the density matrix renormalization

group (DMRG). We establish that the reactive part of

the device response predominantly originates from the

charge dispersion, i.e. the charge susceptibility of the

instantaneous eigenstates (quantum capacitance). The

∗f.k.malinowski@tudelft.nl

tunneling capacitance, related to the redistribution of oc-

cupancies between the eigenstates during a driving cycle,

is signiﬁcantly smaller, but the associated relaxation pro-

cess leaves a footprint on the dissipation in the resonator

(Sisyphus resistance) [17].

The device under study consists of a nanowire with

a two-facet epitaxial Al shell [18, 19] selectively etched

away (Fig. 1(a)). Wrapped gates are used to electrostat-

ically divide the wire into segments. The left segment,

1.8 µm long, is operated as a SC island that is tuned by

gate voltage VS. The right segment, 500 nm long, forms

a QD that is tuned by gate voltage VD. The dot and

the island are tunnel coupled with coupling strength t

that is controlled by gate voltage VB(Fig. 1(a,c)). The

side barrier gate voltages VL/R are set to large negative

values (<−2 V) to galvanically disconnect the device

and ﬁx its total charge on a timescale of several minutes

to days. Additionally, the QD plunger gate is attached

to an oﬀ-chip spiral inductor resonator [20] (inductance

L= 570 nH; resonance frequency f0≈368 MHz; internal

and external quality factors Qint ≈4000 and Qext ≈285,

respectively). Near an interdot charge transition the elec-

tron tunneling between the QD and SC island is enabled,

loading the resonator with an additional capacitance C

and conductance G, see Fig. 1(b) for the eﬀective RLC

network model of the setup[21]. The resonator loading

manifests as a shift of the resonant frequency f0and a re-

duction of the internal quality factor Qint (Appendix A).

In order to study the formation of a SGS in the QD we

ﬁrst investigate features indicating whether there are ad-

ditional discrete SGSs formed in the SC island itself. We

arXiv:2210.01519v2 [cond-mat.mes-hall] 5 Oct 2022

resonator device

FIG. 1. (a) False-colored SEM of the device nominally iden-

tical to the one measured, schematically illustrating the res-

onator circuit with a bias-tee. (b) An equivalent resonator

circuit. The ﬁxed inductance, capacitance and resistance rep-

resent the spiral inductor resonator. The variable capacitance

and resistance represent the loading of the resonator due to

quantum and tunneling capacitance, and losses in the cou-

pled dot-island system. (c) Cartoon illustrating the density

of states in a superconducting island, and their coupling to

the quantum dot. (d) Schematic energy diagram illustrat-

ing the evolution of the discrete ground state with increasing

tunnel coupling. The gray shaded area represents a quasi-

continuum of states, with an unpaired quasiparticle on the

superconducting island. The red shaded area represents a

range within which the system will likely be thermally ex-

cited to the quasi-continuum.

start at a moderately positive value of VS≈0.25 V and

measure a charge stability diagram by sweeping VSand

VD(Fig. 2(a)). The measurement reveals a pattern of

alternating wider and narrower regions of stable charge

(labeled “E” and “O” , respectively), separated by charge

transitions that are weakly asymmetric with respect to

their maximum. Following Ref. 22 we interpret that nar-

row stability regions “O” correspond to an odd-occupied

island, since SC pairing favors even occupancy of the is-

land. Next, we apply a large negative island gate voltage

VS≈ −2 V. We expect this to deplete the semiconductor

wire under the Al shell, eliminating any potential subgap

states [19, 23, 24]. The resulting charge stability diagram

shown in Fig. 2(b) exhibits a similar pattern of narrower

and wider regions of charge stability, but the capacitance

at charge transitions has a much smaller magnitude and

exhibits very strong asymmetry, with a sharp edge on

the side of the charge stability regions “O” . Finally,

we tune the barrier gate voltage VBmore positive while

keeping VS≈ −2 V. Fig. 2(c) shows that the resonance

periodicity remains unchanged, however the number of

observed interdot charge transitions is halved, the tran-

sitions are symmetric, and the added capacitance at the

charge transition is increased.

We interpret the three tunings of the device as follows.

For VS≈0.25 V, there are one or several discrete subgap

states in the island (Fig. 2(a)). Since the lowest of these

states as well as the QD-induced subgap state are well

separated from the SC continuum, the charge transitions

exhibit many of the same features as those in double QD

devices [22]. If the semiconductor is depleted, however,

the QD state hybridizes only with the quasicontinuum

above the gap.

Fig. 2(b) represents the case of weak hybridization

(small t) that allows the QD-induced subgap state to

approach the quasicontinuum within <

∼kT . This en-

ables thermal excitation to one of the many states with a

single quasiparticle that is decoupled from the QD, sup-

pressing the capacitance (“thermal screening”). Since for

an odd-occupied island the ground state approaches the

quasicontinuum much more closely (c.f. Fig. 1(d)) the

thermal screening leads to strong asymmetry of charge

transitions.

Fig. 2(c) corresponds to a strong hybridization (large

t) in which case the dot-induced discrete ground state

becomes well separated from the quasicontinuum. This

results in the vanishing of stability regions “O” , leaving

wide charge transitions separating states diﬀering by 2 in

the QD occupancy. In the following, we ﬁx VS=−2 V,

thereby eliminating unintended subgap states, and study

the transition between the weak and strong hybridization

regime in more detail.

Fig. 3 presents a transition between the hybridization

regimes in a single charge stability diagram [25], mea-

sured with respect to dot and barrier gate voltages, VD

and VB. A range of VBis chosen so that the barrier

gate tunes the tunnel coupling with only relatively small

change in VB. The shrinking and vanishing of the stabil-

ity region “O” with increasing VBis highlighted by taking

line cuts through the charge stability diagram (Fig. 3(c)).

As the region “O” shrinks, the magnitude of capacitance

Cat the charge transition increases and becomes maxi-

mal when the pair of charge transitions merges.

We propose an intuitive understanding of the region

“O” through an analogy to the singlet-doublet quan-

tum phase transition in the case of the QD coupled to

a grounded SC [6]. In that case, the QD charging energy

Ucompetes with the tunnel coupling Γtto the SC lead.

As the QD level εis tuned, the limit of ΓtUfavors

increments of the QD occupancy in steps of one electron,

switching between the singlet states at even ﬁlling and

doublet states at odd ﬁlling. In contrast – large tun-

nel coupling (ΓtU) favors increases of QD occupancy

in steps of two electrons and the system remains in the

singlet state at all times. In ﬂoating devices with ﬁxed

total charge the total system parity cannot change and

therefore the quantum phase transition does not occur.

Nonetheless, the parity of the QD may change provided a

suﬃcient amount of thermal energy is available to excite

a quasiparticle to the quasicontinuum of the SC states

with high multiplicity N. As illustrated in Fig. 1(d), for

FIG. 2. Charge stability diagrams revealed by capacitance C. Panels (a-c) represent three regimes: (a) undepleted semicon-

ductor under the Al shell; (b) depleted semiconductor under the Al shell and small tunnel coupling; (c) depleted semiconductor

and large tunnel coupling. Shared power-law normalization of color maps enables direct comparison. Top panels show the cuts

along the dotted lines, while insets illustrate schematically the energy diagram in each regime (cf. Fig. 1(d)). (d) Close-ups on

peak patterns in the three regimes; color scheme as in the top panels of (a-c).

suﬃciently small tunnel coupling the quasicontinuum is

separated from the discrete SGS by δ<

∼ln(N)kBT. Qua-

sicontinuum states do not couple to the resonator, hence

we interpret the sharp edges of the capacitance peaks,

illustrated in Fig. 3(a,c,f), to be due to such thermal

excitations. The shrinking of region “O” is due to in-

creasing δfor increasing Γt. For suﬃciently strong Γt

relatively to charging energies, so that δ > ln(N)kT for

any value of VD, the charge transitions merge. In the

following, we model the capacitance and its suppression

to lend support to this interpretation.

We employ a model of a single-level Anderson impurity

coupled to a ﬁnite-sized SC, with the Hamiltonian of the

form [12]

H=εˆn+Un↓n↑+X

εkˆnk+ES

2 X

ˆnk−ng!2

−ξX

k,q c†

k↑c†

k↓cq↓cq↑+ H.c.+VX

k,σ c†

kσ dσ+ H.c.,

(1)

where dσand ˆn(σ)are annihilation and electron num-

ber operators for the impurity, σ=↑,↓.ckσ is the an-

nihilation operator for island orbital kwith energy εk.

ε=αVDis the energy of the impurity level; α– the

lever arm; U– the impurity charging energy; ES

C– the

SC island charging energy; ξ– the SC pairing strength;

V– the impurity-bath hopping. The model is solved

using the DMRG method (Supplementary Sec. B). We

compute the charge susceptibilities χg,e for the two low-

est energy states (ground state gand excited state e),

separated in energy by δ. We do not explicitly compute

higher excited states, but instead assume high multiplic-

ity Nfor the excited state e[26]. We expect N<

∼108,

estimated based on the number of Al atoms composing

the SC shell.

The quantum capacitance is given by

Cq=Pgχg+ (1 −Pg)χe,(2)

where Pg= (1 + Ne−δ/kBT)−1is the equilibrium oc-

cupancy of the ground state at temperature T. Since

χeχgin the relevant gate voltage range (Supplemen-

tary Fig. C.1), for δ<

∼ln(N)kBTthe quantum capac-

itance is strongly suppressed. Assuming the tunneling

capacitance to be small, we ﬁt the experimental data

with a model that takes into account solely the quantum

capacitance contribution, see Fig. 3(c). In the simultane-

ous ﬁt to all curves we use ﬁxed values of ∆ = 250 µeV,

U= 502 µeV and ES

C= 196 µeV(estimated from data

presented in Suppl. Fig. F.4), common free parameters α

and T, and separate free parameter Γtfor each cut. Fur-

thermore, the ﬁt includes a constraint of Γteﬀectively

enforcing its monotonous increase with VB. The ﬁt re-

sults are presented with black dashed lines in Fig. 3(c),

and yield α= 0.87, T= 168 mK and N= 1.6×103.

Fig. 3(d-f) illustrates for the case of Γt= 30 µeV how a

variable δtranslates into Pgthat in turn determines the

contribution of the ground-state quantum capacitance to

the total value of C.

In contrast, assuming no thermal excitations we are

neither able to reproduce the magnitude of Cqfor all data

sets nor the degree of asymmetry for the most negative

values of VB(see Supplementary Fig. C.1). The obtained

value of T= 168 mK is larger than the base temperature

of our setup (30 mK), which may be related to eﬀectively

increased temperature due to rf excitation, imprecision of

describing the quantum dot as a single-level impurity, or

strong covariance between Tand Nin the nonlinear ﬁt.

As a consistency check of our interpretation we cal-

culate RCqdVD=Qtot across charge transition pairs,

to yield a quantity we dub the “charge signature” of the

transition (Fig. 3(a,b); Supplementary Section D). In the

absence of thermal screening we expect a charge signa-

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Quantumcapacitanceofasuperconductingsubgapstateinanelectrostaticallyoatingdot-islandFilipK.Malinowski,1,R.K.Rupesh,2LukaPavesic,3ZoltanGuba,4DamazdeJong,1LinHan,1ChristianG.Prosko,1MichaelChan,1YuLiu,5PeterKrogstrup,5AndrasPalyi,4,6RokZitko,3andJonneV.Koski71QuTechandKavliInstituteofNanoscien...

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Quantum capacitance of a superconducting subgap state in an electrostatically oating dot-island Filip K. Malinowski1R. K. Rupesh2Luka Pave si c3Zolt an Guba4Damaz de Jong1Lin Han1.pdf

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Quantum capacitance of a superconducting subgap state in an electrostatically oating dot-island Filip K. Malinowski1R. K. Rupesh2Luka Pave si c3Zolt an Guba4Damaz de Jong1Lin Han1

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