Parity lifetime of bound states in a proximitized semiconductor nanowire

2.3 MB·English

by  A. P. Higginbotham

Parity lifetime of bound states in a proximitized semiconductor nanowire
A. P. Higginbotham,1,2,∗ S. M. Albrecht,1,∗ G. Kirˇsanskas,1 W. Chang,1,2 F. Kuemmeth,1
P. Krogstrup,1 T. S. Jespersen,1 J. Nyg˚ard,1 K. Flensberg,1 and C. M. Marcus1
1Center for Quantum Devices, Niels Bohr Institute, University of Copenhagen,
Universitetsparken 5, 2100 Copenhagen Ø, Denmark
2Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
(Dated: January 22, 2015)
5 Quasiparticle excitations can compromise the Majoranasystemwithprotectedtotalparity,whereboth
1 performance of superconducting devices, caus- the semiconductor nanowire and the metallic supercon-
0
ing high frequency dissipation, decoherence in ductor are mesoscopic [36, 37].
2
Josephson qubits [1–6], and braiding errors in ThemeasureddeviceconsistsofanInAsnanowirewith

n proposed Majorana-based topological quantum epitaxial superconducting Al on two facets of the hexag-
a
computers [7–9]. Quasiparticle dynamics have onal wire, with Au ohmic contacts (Figs. 1a,b). Four de-
J
been studied in detail in metallic superconduc- vicesshowingsimilarbehaviorhavebeenmeasured. The
1
2 tors [10–14] but remain relatively unexplored in InAs nanowire was grown without stacking faults using
semiconductor-superconductor structures, which molecular beam epitaxy with Al deposited in situ to en-

] are now being intensely pursued in the context sure high-quality proximity effect [38, 39]. Differential
l
l of topological superconductivity. To this end, conductance, g, was measured in a dilution refrigerator
a
h we introduce a new physical system comprised of with base electron temperature T ∼ 50 mK using stan-
- a gate-confined semiconductor nanowire with an dard ac lock-in techniques. Local side gates, patterned
s
e epitaxially grown superconductor layer, yielding with electron beam lithography, and a global back gate
m an isolated, proximitized nanowire segment. We were adjusted to form an Al-InAs HQD in the Coulomb
. identify Andreev-like bound states in the semi- blockade regime, with gate-controlled weak tunneling to
t
a conductor via bias spectroscopy, determine the theleads. Thelowerrightgate,V ,wasusedtotunethe
R
m characteristic temperatures and magnetic fields occupation of the dot, with a linear compensation from
- for quasiparticle excitations, and extract a par- the lower left gate, VL, to keep tunneling to the leads
d
ity lifetime (poisoning time) of the bound state symmetric. We parameterize this with a single effective
n
in the semiconductor exceeding 10 ms.
o
c Semiconductor-superconductor hybrids have been in-
[ a b
vestigated for many years [15–19], but recently have re-

1 ceived renewed interest in the context of topological su- B B VSD
Au
v perconductivity, motivated by the realization that com-
5 biningspin-orbitinteraction, Zeemansplittingandprox- InAs VR
5 imity coupling to a conventional s-wave superconductor Al I VL
1
provides the necessary ingredients to create Majorana 200nm VBG
5
0 modes at the ends of a one-dimensional (1D) wire. Such c -0.2 0 g (e2/h) 0.2 0.4
. modes are expected to show nonabelian statistics, allow-
1
0 ing, in principle, topological encoding of quantum infor- 0.4
5 mation [20–22] among other interesting effects [23, 24].
1 V)
Transport experiments on semiconductor nanowires m
v: proximitizedbyagroundedsuperconductorhaverecently V (SD0 o e o
Xi revealed characteristic features of Majorana modes [25–
28]. Semiconductor quantum dots with superconduct-
r -0.4
a ingleadshavealsobeenexploredexperimentally[29–32],
and have been proposed as a basis for Majorana chains 0.9 VG (V) 1.1
[33–35]. Here, we expand the geometries investigated
FIG. 1: Nanowire-based hybrid quantum dot. a, Scanning
in this context by creating an isolated semiconductor- electron micrograph of the reported device, consisting of an InAs
supercondutor hybrid quantum dot (HQD) connected to nanowire (gray) with segment of epitaxial Al on two facets (blue)
andTi/Aucontactsandsidegates(yellow)onadopedsiliconsub-
normal leads. The device forms the basis of an isolated
strate. b, Device schematic and measurement setup, showing ori-
entation of magnetic field, B. c, Differential conductance, g, as
a function of effective gate voltage, VG, and source-drain voltage,
VSD, at B = 0. Even (e) and odd (o) occupied Coulomb valleys
∗Theseauthorscontributedequallytothiswork. labeled.
2
gate voltage, VG (see Supplement). a g (e2/h) b g (e2/h)
-0.2 0 0.2 0.4 -0.2 0 0.2 0.4
Differential conductance as a function of V and
G
source-drain bias, V , reveals a series of Coulomb dia-
SD 0.2
monds,correspondingtoincrementalsingle-chargestates 1
of the HQD (Fig. 1c). While conductance features at
ohlofiwglhefbtbi-aiasa,nsdVaSrrDiegh<ets-sf0ea.nc2tinimaglVlcy,oniaddedunicstttiacinnaclcteiinvfeeaeetavucerhnes-doiisdadmobopsnaedtrtv,eeradnt. V (mV)SD0 (Δ/e)vSD0
Thisresultsinaneven-oddalternationofCoulombblock-
ade peak spacings at zero bias, similar to even-odd spac-
ings seen in metallic superconductors [40, 41]. However, -0.2 -1
the parity-dependent reversing pattern of subgap fea- EXPERIMENT THEORY
1095 1115 -1 0 1
tures at nonzero bias has not been reported before, to V (mV) v (Δ/e)
G G
our knowledge. The even-odd pattern indicates that a c d e f
pasareilteyc-tsreonnssitiavreebadoudneddtsotattheeiHs QbeDin.g filled and emptied vvS G D == 00 vG = -Ee0 vS D = 2Ee0
Measured charging energy, E = 1.1 meV, and su-
C
perconducting gap, ∆ = 180 µeV, satisfy the condition
(∆ < E ) for single electron charging [42, 43]. Differ-
C
ential conductance at low bias occurs in a series of nar- D S
rowfeaturessymmetricaboutzerobias,suggestingtrans-
portthroughanAndreev-likeboundstate,withnegative FIG.2: Subgap bias spectroscopy, experiment and model.
differential conductance (NDC) observed at the border a, Experimental differential conductance, g, as a function of gate
of odd diamonds. NDC arises from slow quasiparticle voltage VG and source-drain VSD, shows characteristic pattern
including negative differential conductivity (NDC). b, Transport
escape, as discussed below, similar to current-blocking model of a. vG = αVG up to an offset, where α is the gate lever
seen in metallic superconducting islands in the opposite arm. Axis units are ∆/e = 180 µV, where ∆ is the supercon-
regime, ∆>E [44, 45]. ducting gap. See text for model parameters. c, Source and drain
C (gold) chemical potentials align with the middle of the gap in the
To gain quantitative understanding of these features,
HQD density of states. No transport occurs due to the presence
wemodeltransportthroughasingleAndreevboundstate ofsuperconductivity. d,Discretestateinresonancewiththeleads
in the InAs plus a Bardeen-Cooper-Schriffer (BCS) con- atzerobias. Transportoccursthroughsinglequasiparticlestates.
e, Discrete state in resonance with the leads at high bias. Trans-
tinuum in the Al. The model assumes symmetric cou-
portoccursthroughsingleanddouble(particle-hole)quasiparticle
pling of both the bound state and continuum to the
states. f, Discrete state and BCS continuum in the bias window.
leads, motivated by the observed symmetry in V of Transportisblockedwhenaquasiparticleisinthecontinuum,re-
SD
theCoulombdiamonds. Transitionrateswerecalculated sultinginNDC.
from Fermi’s golden rule and a steady-state Pauli mas-
ter equation was solved for state occupancies. Conduc-
tance was then calculated from occupancies and transi- neling, not accounted for by the model.
tion rates (see Supplement). The observation of negative differential conductance
Measured and model conductances are compared in places a bound on the relaxation rate of a single quasi-
Figs. 2a,b. The coupling of the bound state to each particle in the HQD from the continuum (in the Al) to
lead, notingthenear-symmetryofthediamonds, wases- the bound state (in the InAs nanowire). Negative differ-
timated to be Γ =0.5 GHz, based on zero-bias conduc- ential conductance arises when an electron tunnels into
0
tance (Fig. 2d). The energy of the discrete state, E = the weakly coupled BCS continuum, blockading trans-
0
58 µeV at zero magnetic field, was measured using finite port until it exits via the lead. The blocking condition is
bias spectroscopy (Fig. 2e). The normal-state conduc- shownforahole-likeexcitationinFig.2f. Unblockingoc-
tance from each lead to the continuum, g =0.15 e2/h, curswhenthequasiparticlerelaxesintotheboundstate,
Al
was estimated by comparing Coulomb blockaded trans- followed by a fast escape to the leads. NDC thus indi-
port features in the high bias regime (VSD = 0.4 mV). cates a long quasiparticle relaxation time, τqp, from the
The superconducting gap, ∆ = 180 µeV, was found continuum to the bound state. Using independently de-
from the onset of NDC, which is expected to occur at terminedparameters, theobservedNDCisonlycompat-
eVSD = ∆−E0 (Fig. 2f). While the rate model shows iblewiththemodelwhenτqp >0.1µs(seeSupplement).
good agreement with experimental data, some features This bound on τqp is used below to similarly constrain
arenotcaptured,includingbroadeningathighbias,with the characteristic poisoning time for the bound state.
greater broadening correlated with weaker NDC, and Turning our attention to the even-odd structure of
peak-to-peak fluctuations in the slope of the NDC fea- zero-bias Coulomb peaks (Figs. 3a,b), we observed con-
ture. These features may be related to heating or cotun- sistentlarge-smallpeakspacings(Fig.3),associatingthe
3
a c cludes thermal quasiparticle excitations follows earlier
≈
treatments [40, 41, 43], including a discrete subgap state
T = 50 mK
0.1 B = 0 e o e as well as the BCS continuum [41] (Fig. 3d). Even-odd
2e/h) 2/h)0.1 0e( g.05penecaekosfpfarceiengendeirffgeieres,nce, Se −So, depends on the differ-
g (0 o e g (e )h/2 4
0 V (V) 1 0 862 864≈934 936 0 Se−So = αe(Fo−Fe), (1)
G V (mV)
b d G
where α is the (dimensionless) gate lever arm. The free
F
energy difference, written in terms of the ratio of parti-
85
V) tion functions,
m E0
S ( F −F =−k T ln(cid:18)Zo(cid:19), (2)
75 T = 350mK 0 o e o e B Ze
B = 150mT
0 V (V) 1 0 1 N 2 3 depends on D(E), the density of states of the HQD,
G
FIG.3: Even-odd Coulomb peak spacingsa,Measuredzero- Zo =(cid:90) ∞dE D(E)lncoth[E/(2k T)], (3)
bias conductance, g, versusgate voltage, VG, at temperature T ∼ Ze 0 B
50mK,andmagneticfieldB=0. b,Peakspacing,S,versusgate
voltage. Black points show spacings from a calculated using the where D(E) consists of one subgap state and the contin-
peakcentroid(firstmoment), redpointsT =350mKandB =0,
uum. For ∆(cid:29)k T, this can be written
purplepointsB=150mTandT ∼50mK. c,Right-mostpeaksin B
a. Peakmaxima((cid:52))andcentroids((cid:4))aremarked. d,Freeenergy,
F, at T = 0 versus gate-induced charge, N, for different HQD Fo−Fe ≈−kBT ln(Neffe−∆/kBT +2e−E0/kBT), (4)
occupations, where N =CVG/e up to an offset and C is the gate √
capacitance. Parabola intersection points are indicated by circles, where Neff = ρAlV 2πkBT∆ is the effective number of
correspondingtoCoulombpeaks. BCScontinuum(shaded),shown continuum states for Al volume, V, and normal density
foroddoccupancy. OddCoulombdiamondscarryanenergyoffset of states ρ [40, 41] (see Supplement).
E0 forquasiparticleoccupationofthesubgapstate,resultingina Within Athlis model, one can identify a characteristic
differenceinspacingforevenandodddiamonds.
temperature, T∗ ∼ ∆/[k ln(N )], less than the gap,
B eff
above which even-odd peak spacing alternation is ex-
pected to disappear. Note in this expression N it-
larger spacings with even occupation, as expected the- eff
self depends on T, and also that T∗ does not depend
oretically [42, 43] and already evident in Fig. 1. Oc-
on the bound state energy, E . A second (lower) char-
casional even-odd parity reversals on the timescale of 0
acteristic temperature, T∗∗ ∼ (∆−E )/[k ln(N /2)],
hours were observed in some devices, similar to what 0 B eff
which does depend on E , is where the even-odd alter-
is seen in metallic devices [14]. Peak spacing alterna- 0
nation is affected by the bound state, leading to satu-
tion disappears at higher magnetic fields, B, consistent
ration at low temperature [40, 41]. For a spin-resolved
with the superconducting-to-normal transition, and also
zero-energy (E = 0) bound state—the case for unsplit
disappears at elevated temperature, T > 0.4 K, signif- 0
Majoranazeromodes—thesecharacteristictemperatures
icantly below the superconducting critical temperature,
coincide and even-odd structure vanishes, as pointed out
T ∼ 1 K. The temperature dependence is consistent
c in Ref. [36]. In the opposite case, where the bound state
withsimilarbehaviorseeninmetallicstructures[40,41],
reachesthecontinuum(E =∆), thesaturationtemper-
andcanbeunderstoodastheresultofthermalactivation 0
ature vanishes, T∗∗ = 0, and the metallic result with no
ofquasiparticleswithintheHQDwithfixedtotalcharge.
bound state is recovered [40, 41].
AsseeninFig.3c,individualCoulombpeaksareasym- Experimentally,theaverageeven-oddpeakspacingdif-
metric in shape, with their centroids (first moments) on ference,(cid:104)S −S (cid:105),wasdeterminedbyaveragingoveraset
e o
the even sides of the peak maxima. Note that the asym- of24consecutiveCoulombpeakspacings,includingthose
metry leads to higher near-peak conductance in even shown in Fig. 3, at each temperature. Figure 4 shows
valleys, the opposite of the Kondo effect. The asym- even-odd peak spacing difference appearing abruptly at
metric shape is most pronounced at low temperature, T ∼ 0.4 K, and saturating at T ∼ 0.2 K, with a
onset sat
T <0.15K,anddecreaseswithincreasingmagneticfield. saturation amplitude near the value expected from the
The degree of asymmetry is not predicted by the rate measured bound state energy, 4V = 4E /(αe). Fig-
0 0
model, even taking into account the known small asym- ure4showsgoodagreementbetweenexperimentandthe
metryduetospindegeneracy[46]. Intheanalysisbelow, model,Eq.(1),usingadensityofstatesdeterminedinde-
weconsiderpeakpositionsdefinedbothbypeakmaxima pendentlyfromdatainFig.2,withV =7.4×104 nm3 as
and centroids. afitparameter,consistentwiththemicrograph(Fig.1a),
A model of even-odd Coulomb peak spacing that in- and ρ =23 eV−1nm−3 [14].
Al
4
T** T*(B=0)
4V (B=0)
peak max 0
peak centroid
model ( = 0)
model ( = 50 neV)
h)
2e/20
T*(100 mT) T*(40 mT) 3
-0
1
10 g ( 0
4V (40 mT)
0 Se So
0.3 0.4
V) 10 VG (V)
m
‹S (o mV) 4V0 (80 mT)
S-e ‹S (o
‹ S-e 4V0 (100 mT)
‹
0 4V (150 mT)
0 0
0.05 0.1 T (K) 0.4
0.05 0.1 T (K) 0.2 0.5
FIG. 4: Temperature and magnetic field dependence of the even-odd peak spacings. Average even-odd spacing difference,
(cid:104)Se−So(cid:105), versus temperature, T. Spacing between peak maxima (triangle) and centroids (square) are shown. Spacing expected from
lowerZeeman-splitboundstate,4V0(B)=4E0(B)/(αe),indicatedonrightaxis. Quasiparticleactivationtemperature,T∗,andcrossover
temperature, T∗∗, indicated on top axis. Solid curve is Eq. (1) with a HQD density of states measured from Fig. 2 (∆ = 180 µeV,
E0 = 58 µeV, α = 0.013), and the fitted aluminum volume, V = 7.4×104 nm3. Dotted curve includes a discrete state broadening,
γ=50neV,fittothecentroiddata. Leftinset: Sameasmain,butatB=40,80,100,150mT,fromtoptobottom. Curvesarefittotwo
sharedparameters: g-factor,g=6,andsuperconductingcriticalfield,Bc=120mT,withotherparametersfixedfrommainfigure. Right
inset: RepresentativeCoulombpeaksshowingeven(Se)andodd(So)spacings.
Theasymmetricpeakshapecomplicatesmeasurement typical range for InAs nanowires [47, 48], supporting our
of even-odd spacings, as one can either use the centroids interpretation of the bound state residing in the InAs.
or maxima to measure spacings, the two methods giving The fit value of critical field, B =120 mT, is typical for
c
differentresults. Largerpeaktailsontheevenvalleyside this geometry.
causethe centroidstobemore regularly spacedthanthe
Good agreement between the peak spacing data and
maxima. This is evident in Fig. 4, where the centroid
the thermodynamic model (Fig. 4) suggests that the
method shows a decreasing peak spacing difference at
numberofthermallyactivatedquasiparticlesobeysequi-
low temperature, while with the maximum method the
spacing remains flat. The thermal model of S −S can librium statistics, Neq(T) = Ne2ffe−2∆/kBT (see Supple-
e o ment for derivation). Saturation caused by the bound
alsoshow adecreaseat lowtemperatureifbroadeningof
state means that even-odd amplitude loses sensitivity as
the bound state is included (See Methods). We do not
a quasiparticle detector below T . We therefore take
sat
understandatpresentifthelowtemperaturedecreasein
N (T ) ∼ 10−5 (for T ∼ 0.2 K) as an upper bound
eq sat sat
the centroid data is related to the decrease seen in the
for the number of quasiparticles at temperatures below
model when broadening is included. It is worth noting,
T . Thecorrespondingupperboundofthequasiparticle
sat
however,thatthefittothecentroiddatagivesabroaden-
fraction, x = N (T )/(ρ V∆) ∼ 10−8, is compara-
qp eq sat Al
ing γ = 50 neV, reasonably close to the value estimated
ble to values in the recent literature, 10−5 −10−8, for
from the lead couplings, (hΓ )2/∆=20 neV.
0 metallic superconducting junctions and qubits [3–6, 13].
Applied magnetic field (direction shown in Fig. 1b) We now discuss the implications of our measurements
reduces the characteristic temperatures T , T , and for determining the poisoning time, τ , of the bound
onset sat p
saturation amplitudes. Field dependence is modeled by state. For the present geometry, the dominant source
includingZeemansplittingoftheboundstateandorbital of poisoning of the bound state is not tunneling of elec-
reduction of the gap and bound state energy, taking the trons from the leads, which is negligible in the strongly
g-factor and critical magnetic field as two fit parameters blockaded regime, but is rather the continuum in the
appliedtoalldatasets. Thefitvalueg =6lieswithinthe strongly-coupled Al, within the isolated structure itself.
5
Theoretical estimates [8] suggest an inverse relationship where D(E) is the density of states of the HQD,
between τ and the number of available quasiparticles,
p
1 1
with a proportionality that depends on system details. D(E)=ρ (E)+ ρ+(E)+ ρ−(E). (7)
Takingthe bound on single quasiparticle relaxationtime BCS 2 0 2 0
from the continuum into the bound state, τ > 0.1 µs,
qp Wetakeρ (E)tobeastandardBCSdensityofstates,
BCS
fromabove,asthepoisoningtimewhenasinglequasipar-
ticle is present, we estimate τ by scaling for the actual ρ VE
p ρ (E)= Al θ(E−∆) (8)
number of quasiparticles in equilibrium, Neq, giving a BCS (cid:112)E2−∆(B)2
poisoning time τ =τ /N (cid:38)10 ms.
p qp eq
We expect τ to depend weakly on the bound state (θisthestepfunction),andρ tobeapairofLorentzian-
p 0
energy for low-energy bound states [11, 49, 50], includ- broadened spinful levels symmetric about zero,
ing for Majorana zero modes at E = 0. Device geom-
0
γ/2π γ/2π
etry may somewhat alter the number of quasiparticles ρ±(E)= + .
available to relax into the bound state, i.e. by chang- 0 (E−E0±)2+(γ/2)2 (E+E0±)2+(γ/2)2
(9)
ing N , but any increase can be compensated by ex-
eff
Zeemansplittingoftheboundstateandpair-breakingby
ponentially small decreases in the quasiparticle temper-
the external magnetic field are modeled with the equa-
ature. The long poisoning time obtained here suggests
tions
that a large number of braiding operations in Majorana
systems should be readily achievable within the relevant
∆(B) 1
time scale. E0±(B) = ∆ E0± 2gµBB, (10)
Methods (cid:115)
(cid:18)B (cid:19)2
Sample preparation: InAs nanowires were grown in ∆(B) = ∆ 1− , (11)
the[001]directionwithwurzitecrystalstructurewithAl Bc
epitaxiallymatchedto[111]ontwoofthesix{1¯100}side-
where E is the zero-field state energy and ∆ is the zero
facets. Theywerethendepositedrandomlyontoadoped 0
field superconducting gap. In the event that a bound
siliconsubstratewith100nmofthermaloxide. Electron-
state goes above the continuum, E+ > ∆(B), we no
beamlithographicallypatternedwetetchoftheepitaxial s
longer include the state in the free energy. Equation (6)
Alshell(TranseneAlEtchantD,55C,10s)resultedina
was integrated numerically to obtain theory curves in
submicron Al segment (310 nm, Fig. 1a). Ti/Au (5/100
Fig. (4).
nm)ohmiccontactsweredepositedontheendsfollowing
Equations (10) and (11) are reasonable provided the
in situArmilling(1mTorr, 300V,75s), withsidegates
lowerspin-splitstateremainsatpositiveenergy,E− >0.
deposited in the same step. For the present device, the 0
For sufficiently large B , the bound state will reach zero
end of the upper left gate broke off during processing. c
energy, resulting in topological superconductivity and
However, the device could be tuned well without it.
Majorana modes, the subject of future work.
Master equations: The master equations (used for
Fig.1b)considerstateswithfixedtotalparity,composed We thank Leonid Glazman, Bert Halperin, Roman
of the combined parity of quasiparticles in the thermal- Lutchyn and Jukka Pekola for valuable discussions,
ized continuum and the 0, 1, or 2 quasiparticles in the and Giulio Ungaretti, Shivendra Upadhyay and Claus
bound state (see Supplement). Sørensenforcontributionstogrowthandfabrication. Re-
Free energy model: Evenandoddpartitionfunctions search support by Microsoft Project Q, the Danish Na-
in Eq. 2, F −F = −k T ln(Z /Z ), can be written as tional Research Foundation, the Lundbeck Foundation,
o e B o e
sumsofBoltzmannfactorsoverrespectivelyoddandeven the Carlsberg Foundation, and the European Commis-
occupancies of the isolated island. For even-occupancy, sion. APH acknowledges support from the US Depart-
ment of Energy, CMM acknowledges support from the
(cid:88)
Z =1+ e−Ei/kBTe−Ej/kBT +..., (5) Villum Foundation.
e
i(cid:54)=j
where the first term stands for zero quasiparticles, the
second for two (at energies E and E ), and additional
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1
SUPPLEMENTARY INFORMATION:
QUASIPARTICLE PARITY DYNAMICS IN A SUPERCONDUCTOR-SEMICONDUCTOR HYBRID
QUANTUM DOT
1. Effective gate voltage definition
2. Bound on the single quasiparticle relaxation time
3. Detailed interpretation of Coulomb diamonds
4. Derivation of transport and thermal model
5. Comparison of free energy approximations
6. Effect of the bound state on the free energy
1. Effective gate voltage definition
WedefineaneffectivegatevoltageinsoftwaretotunetheHQD.Thephysicalgatevoltages, V andV , arerelated
R L
to the effective gate voltage, V , by
G
V =V +κ V ,
R R,0 G
(cid:112)
V =V + 1−κ2 V ,
L L,0 G
with κ=0.9997 and offset voltages V =−2.41 V, V =−3.96 V.
R,0 L,0
These transformation rules ensure that V2 = (V −V )2 +(V −V )2, so that V can be interpreted as the
G R R,0 L L,0 G
distance from (V ,V ) in the V −V plane.
R,0 L,0 R L
All measurements are performed at backgate voltage V =2.39 V.
BG
2. Bound on the single quasiparticle relaxation time
The effect of quasiparticle relaxation is shown in Figs. S1a-e. Quasiparticle relaxation results in a disappearance
of the negative differential conductance, in combination with the appearance of an extra conductance threshold. We
quantify this observation by introducing the relative conductance ratio
g(cid:48)+g
R= NDC (S1)
g(cid:48)−g
NDC
whereg istheminimumofthenegativedifferentialconductance,andg(cid:48) isthemaximumoftheextraconductance
NDC
threshold that appears when τ →0 (see Fig. S1i). The R-value is a metric for the relative strength of the negative
qp
differential conductance.
Figs. S1f-j show example conductance traces at constant bias and their associated R-values. The traces show that
R≈−1 corresponds to slow quasiparticle relaxation, and R≈1 corresponds to fast quasiparticle relaxation.
Fig. S2 shows the R-value calculated as a function of single quasiparticle relaxation time, τ . Also shown is a
qp
measuredR-valueaveragedoverallnegativedifferentialconductancefeaturesinFig.1ofthemaintext. Themeasured
R-value is consistent with τ >0.1 µs, giving the experimental bound on the single quasiparticle relaxation time.
qp
3. Detailed interpretation of Coulomb diamonds
EachconductancethresholdintheCoulombdiamondplotscanbeinterpretedwiththehelpofthetransportmodel,
as shown in Fig. S3. For example, the highest bias at which NDC is observed occurs at the intersection of black and
green lines, when v =(∆+E )/e.
SD 0
2
a b c d e
0.4
0.2
1
0.2
V) e)
m Δ/
( Vsd 0 (vSD0 0 e( g
2
h/
)

-1
-0.2
EXPERIMENT τqp = ∞ τqp = 0.1 μs τqp = 5 ns τqp = 0 -0.2
f g h i j
g’
0.1 0.1
h) h)
2/e 0 2/e 0
g ( g ( R = -0.95 R = -0.8 g R = 0.1 R = 1
-0.1 -0.1 NDC
1100 1120 -1 0 1 -1 0 1 -1 0 1 -1 0 1
VG (mV) vG (Δ/e) vG (Δ/e) vG (Δ/e) vG (Δ/e)
Fig. S1: Effect of quasiparticle relaxation. a Measured conductance g versus source-drain bias V and gate V . b,
SD G
Transport model of a, with τ =∞. v ≡αV up to an offset, where α is the gate lever arm. Axis units are ∆/e=180 µV.
qp G G
c-e, Model with τ =0.1 µs, τ =5 ns, and τ =0 respectively. f-j, Conductance versus gate at constant bias indicated in
qp qp qp
a. Relative conductance ratio, R=(g(cid:48)+g )/(g(cid:48)−g ), for theory curves is labeled (see text).
NDC NDC
1
R0
-1
0.001 0.01 0.1 1
(μs )
qp
Fig. S2: Single quasiparticle relaxation bound. Relative conductance ratio, R = (g(cid:48)+g )/(g(cid:48)−g ), versus single
NDC NDC
quasiparticle relaxation time τ . Dashed curve is theory derived as shown in Fig. S1. Data is the average over all charge
qp
transitions in Fig. 1, with vertical error the standard deviation of the mean, and horizontal error propagated from vertical.
4. Derivation of transport and thermal model
This section gives a detailed derivation of the transport and thermal model used in the main text.
To describe the electron transport through a metallic superconducting quantum dot we consider the following
model:
H =H +H +H , (S2)
LR D T
3
a b c
2 2 2
0.2
g
e) e) e) (
Δ/ Δ/ Δ/ e2
(0 (0 (0 /h
D D D )
S S S
v v v
0.0
-2 -2 -2
E = 0.1Δ E = 0.3Δ E = 0.5Δ
0 0 0
-1 0 1 -1 0 1 -1 0 1
v (Δ/e) v (Δ/e) v (Δ/e)
G G G
Fig. S3: Interpreting conductance thresholds. a, Calculated conductance g versus v and v . E = 0.1∆, all
SD G 0
other model parameters same as main text. Dotted lines are v /2 = ±(v +∆/e) [black], v /2 = ±(v +E /e) [blue],
SD G SD G 0
v /2 = ±(v −E /e) [green], v /2 = ±(v −∆/e) [red]. b, E = 0.32∆, all model parameters same as main text. c,
SD G 0 SD G 0
E =0.5∆, all other model parameters same as main text. Color scale shared across all plots.
0
where the Hamiltonian
(cid:88)
H = (ε −µ )c† c (S3)
LR ανs α ανs ανs
ανs
describes the normal metallic leads with c† being an electron creation operator in the lead α ∈ {L,R}, with an
ανs
orbital quantum number ν and spin s ∈ {↑,↓}. The leads have chemical potentials given by µ = ±V/2, where V
α
denotes symmetrically applied bias. For the semiconductor-superconductor hybrid quantum dot, we use a simplified
model consisting of a Bardeen-Cooper-Schrieffer (BCS)[S1] continuum and an Andreev bound state for fixed number
of particles [S2]:
(cid:88) (cid:104) (cid:88) (cid:105)
H = E γ† γ + E γ† γ +Ec , (S4)
D 0 0s,p 0s,p n ns,p ns,p N
s,p=e,h n
(cid:104)(cid:88) (cid:88) (cid:105)
Ec =U(N −N )2, N = d† d + d† d , (S5)
N g 0s 0s ns ns
s n
where d† creates an electron in the continuum with quantum number n (e.g. momentum of electrons on the dot),
ns
and d† denotes electron creation in a localized level, which gives rise to a subgap state in the BCS spectrum. The
0s
charging effects on the quantum dot are described by constant interaction model given by the term Ec , where the
N
chargingenergyisgivenbyE =2U andthenumberofelectronsonthedotiscontrolledbyagatevoltageV , which
C g
is parameterized by dimensionless number N = V /E . The operator N gives the total number of electrons on the
g g C
dot. With superconducting pairing, the dot Hamiltonian is diagonal in the basis of the quasiparticle operators γ†,
which are given by [S3–S5]
d =u γ +sv γ† , (S6a)
ns n ns,e n −ns¯,h
γ =u d −sv d† S, (S6b)
ns,e n ns n −ns¯
γ =u S†d −sv d† , (S6c)
ns,h n ns n −ns¯
γ† =S†γ† , γ =S†γ , (S6d)
ns,e ns,h ns,h ns,e