A. J. Keller 1 , S. Amasha 1,† , I. Weymann 2 , C. P. Moca 3,4 , I. G. Rau 1,‡ , J. A. Katine 5 , Hadas Shtrikman 6 , G. Zar´ and 3 , and D. Goldhaber-Gordon 1,* 1 Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA 2 Faculty of Physics, Adam Mickiewicz University, Pozna´ n, Poland 3 BME-MTA Exotic Quantum Phases “Lend¨ ulet” Group, Institute of Physics, Budapest University of Technology and Economics, H-1521 Budapest, Hungary 4 Department of Physics, University of Oradea, 410087, Romania 5 HGST, San Jose, CA 95135, USA 6 Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 96100, Israel † Present address: MIT Lincoln Laboratory, Lexington, MA 02420, USA ‡ Present address: IBM Research – Almaden, San Jose, CA 95120, USA * Corresponding author; [email protected] Contents S1 Full LBTP survey 2 S2 Summary of NRG calculations 8 S2.1 NRG calculations ............................. 8 S2.2 Choosing NRG parameters ........................ 9 S3 Extracting LBTP cuts from 2D data sets 11 S4 G( 1 , 2 ) for all measured T 12 S5 Temperature dependence details 17 S6 Empirical Kondo forms 17 1 Emergent SU(4) Kondo physics in a spin–charge-entangled double quantum dot SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS2844 NATURE PHYSICS | www.nature.com/naturephysics 1 © 2013 Macmillan Publishers Limited. All rights reserved.
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Supplemental information for
“Emergent SU(4) Kondo physics in a spin-charge-entangled
double quantum dot”
A. J. Keller1, S. Amasha1,†, I. Weymann2, C. P. Moca3,4, I. G. Rau1,‡, J. A. Katine5,
Hadas Shtrikman6, G. Zarand3, and D. Goldhaber-Gordon1,*
1Geballe Laboratory for Advanced Materials, Stanford University, Stanford, CA 94305, USA2Faculty of Physics, Adam Mickiewicz University, Poznan, Poland
3BME-MTA Exotic Quantum Phases “Lendulet” Group, Institute of Physics, Budapest University
of Technology and Economics, H-1521 Budapest, Hungary4Department of Physics, University of Oradea, 410087, Romania
5HGST, San Jose, CA 95135, USA6Department of Condensed Matter Physics, Weizmann Institute of Science, Rehovot 96100, Israel
†Present address: MIT Lincoln Laboratory, Lexington, MA 02420, USA‡Present address: IBM Research – Almaden, San Jose, CA 95120, USA
*Corresponding author; [email protected]
Contents
S1 Full LBTP survey 2
S2 Summary of NRG calculations 8
S2.1 NRG calculations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
S2.2 Choosing NRG parameters . . . . . . . . . . . . . . . . . . . . . . . . 9
S3 Extracting LBTP cuts from 2D data sets 11
S4 G(ε1, ε2) for all measured T 12
S5 Temperature dependence details 17
S6 Empirical Kondo forms 17
1
Emergent SU(4) Kondo physics in a spin–charge-entangled double quantum dot
SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS2844
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© 2013 Macmillan Publishers Limited. All rights reserved.
S7 Bias spectroscopy 22
S7.1 Ne = 3 LBTP, zero magnetic field . . . . . . . . . . . . . . . . . . . . 22
S7.2 Ne = 1 LBTP, 1.0 T field . . . . . . . . . . . . . . . . . . . . . . . . . 22
S8 Technical details 26
S8.1 Electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
S8.2 Electron temperature calibration . . . . . . . . . . . . . . . . . . . . . 26
S8.3 Magnetic field calibration . . . . . . . . . . . . . . . . . . . . . . . . . 28
S8.4 g-factor calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
S8.5 Bias spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2
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S1 Full LBTP survey
The data presented in Fig. 1b are only a subset of the full survey of conductance
around lines between triple points (LBTPs). The full survey, shown in Fig. S1, demon-
strates that 11/12 of Ne = 1 or Ne = 3 LBTPs exhibit higher conductance towards
the adjacent (1,1) hexagon. In addition, twelve (1,1)/(2,0) or (0,2)/(1,1) LBTPs were
surveyed: these should possess a five-fold degeneracy assuming the (2,0) ground state
is a singlet rather than triplet. The Ne = 1 and Ne = 3 LBTPs differ qualitatively
from the (1,1)/(2,0) and (0,2)/(1,1) LBTPs in that the latter class of LBTPs do not
exhibit a simple pattern of which end of the LBTP has higher conductance. Experi-
mental parameters Γ1, Γ2 and peak conductances are extracted from each data set and
summarized in Table S1.
Because we claim that the Ne = 1 and Ne = 3 LBTP data reflect the particle-hole
symmetry of a four-fold degenerate state, it is natural to expect that the pattern is
destroyed when the four-fold degeneracy is broken. Fig. S2 shows the Ne = 1 and
Ne = 3 LBTPs surveyed again in an in-plane magnetic field of 2.0 T, corresponding
to EZ = gµBB = 0.051 meV for g = 0.44. Here, EZ > Γ1, Γ2 for all of the surveyed
LBTPs. With the Zeeman splitting having broken the spin degeneracy at the LBTPs, a
periodic pattern is no longer discernible. Table S2 summarizes the extracted parameters
for each data set, as in Table S1.
Fig. S3 shows how a small but finite VSD affects the observed asymmetry at an
Ne = 1 LBTP. The LBTP measured here corresponds to the same absolute electron
occupation numbers as data set 553 shown in Fig. S1. Only for negative VSD approach-
ing −10 µV does the conductance near (0,0) exceed that near (1,1). For positive VSD,
the pattern of higher conductance nearer to (1,1) than (0,0) is actually exaggerated.
The effect of finite VSD is similar regardless of whether it is applied to dot 1 or 2. Input
offset voltages from current amplifiers could obscure our observed pattern, were it not
for our ability to stabilize these voltages to within 1 µV (see section S8.1).
3
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-177
-176
-175
P2 (mV)
-331
-329
P1 (
mV)
(o,e
) (e,o
)-1
80
-179
-178
P2 (mV)
-265
-263
P1 (
mV)
(e,o
)
(o,e
)
-184
-183
-182
P2 (mV)
-182
-180
P1 (
mV)
(e,o
)
(o,e
)
-215
-214
-213
P2 (mV)
-214
-212
P1 (
mV)
(e,o
)
(o,e
)-2
10
-209
-208
P2 (mV)
-285
-283
P1 (
mV)(o,e
)
(e,o
)
-206
-205
-204
P2 (mV)
-372
-370
P1 (
mV)(o,e
)
(e,o
)
-264
-263
-262
P2 (mV)
-265
-264
-263
P1 (
mV)
(o,e
) (e,o
)
-270
-269
-268
P2 (mV)
-176
-175
-174
P1 (
mV)
(o,e
) (e,o
)-2
58
-257
-256
P2 (mV)
-324
-323
-322
P1 (
mV)
(o,e
) (e,o
)
-308
-307
-306
P2 (mV)
-369
-368
-367
P1 (
mV)
(e,o
)
(o,e
)-3
10
-309
-308
P2 (mV)
-283
-282
-281
P1 (
mV)
(e,o
)
(o,e
)-3
17
-316
-315
P2 (mV)
-217
-216
-215
P1 (
mV)
(e,o
)
(o,e
)
-314
-313
-312
P2 (mV)
-262
-261
-260
P1 (
mV)
(o,o
)
(e,e
)
-309
-308
-307
-306
P2 (mV)
-322
-321
-320
P1 (
mV)
(o,o
)
(e,e
)
-174
-173
-172
P2 (mV)
-375
-374
-373
P1 (
mV)
(o,o
)
(e,e
)
-180
-179
-178
P2 (mV)
-290
.5-2
88.5
P1 (
mV)
(o,o
)
(e,e
)
-185
-184
-183
P2 (mV)
-218
-216
P1 (
mV)
(o,o
)
(e,e
)
-219
-218
-217
P2 (mV)
-180
-178
P1 (
mV)(e,e
)
(o,o
)
-213
-212
-211
P2 (mV)
-260
-258
P1 (
mV)(e,e
)
(o,o
)
-208
-207
-206
-205
P2 (mV)
-326
-324
P1 (
mV)
(o,o
)
(e,e
)
-257
-256
-255
P2 (mV)
-372
-371
-370
P1 (
mV)
(e,e
) (o,o
)-2
61
-260
-259
P2 (mV)
-286
-284
P1 (
mV)
(e,e
) (o,o
)-2
67
-266
-265
P2 (mV)
-219
-218
-217
P1 (
mV)
(e,e
) (o,o
)
-320
-318
P2 (mV)
-174
-173
-172
P1 (
mV)
(o,o
)
(e,e
)
_658
_672
_642
_688
_553
_704
_737
_729
_716
_664
_649
_678
_501
_695
_709
_743
_732
_722
_754
_773
_787
_758
_766
_780
1.00
0.90
0.80
Frac
tion
of m
ax G
(ad
just
ed p
er p
lot)
Figure
S1:
Experim
entalzero
biasconductan
ceG
=G
1+G
2forasurvey
of24
LBTPs,at
zero
mag
netic
fieldan
dat
T=
20mK.
Electronoccupation
numbersare
labeled
herebytheirparity(e
=even
,o=
odd).
Eachsquareof
measureddatacorrespon
dsto
a
regionspan
ning3mV
inVP1andVP2.Thecolorscales
foreach
squarehavebeenindividually
setso
that
only
databetween75
%
and10
0%ofthemax
imum
conductan
cearevisible.Eachdatasetis
identified
byanumber
inthebottom-leftof
each
plot.
Set
766
(markedbytriangle)
correspondsto
thebottom-leftplotin
Fig.1d
.Allsets
appearingin
Fig.1d
haveagray
backgrou
nd.Of
thefourother
oddparitydatasets,only
one(672
)does
not
show
theexpectedasymmetry;it
has
noclearasymmetry
atall.
4
4 NATURE PHYSICS | www.nature.com/naturephysics
SUPPLEMENTARY INFORMATION DOI: 10.1038/NPHYS2844
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Data set Γ1 Γ2 γ1 γ2 Data set Γ1 Γ2 γ1 γ2
664 24 32 0.82 0.89 743 27 31 0.43 0.80
672 29 36 0.69 0.66 737 27 29 0.66 0.77
678 35 39 0.63 0.68 732 27 31 0.56 0.78
688 28 36 0.54 0.67 729 28 31 0.59 0.79
695 32 42 0.67 0.70 722 30 33 0.69 0.80
704 33 32 0.73 0.78 716 32 34 0.74 0.80
658 26 27 0.51 0.89 758 28 33 0.48 0.68
649 30 27 0.66 0.88 754 30 31 0.64 0.67
642 28 27 0.75 0.90 766 27 30 0.51 0.64
501 26 31 0.51 0.84 773 27 34 0.59 0.68
553 28 29 0.82 0.89 780 29 34 0.73 0.67
709 30 33 0.74 0.92 787 29 34 0.77 0.66
Table S1: For each data set shown in Fig. S1, experimentally controllable parameters
Γ1, Γ2, γ1, and γ2 are extracted by fitting a Lorentzian lineshape to a Coulomb blockade
(CB) peak neighboring the LBTP. Γ1(2) corresponds to the FWHM of the CB peak
in dot 1 (2), in units of µeV. The width in gate voltage is converted to an energy
using conversion factors derived from bias spectroscopy, taken near each LBTP. γ1(2)
are defined to equal the conductance at the CB peak of dot 1 (2) in e2/h. For these
data it is not known whether the source or drain lead is more coupled for either dot.
In all cases, the electron temperature Te = 20 mK.
5
NATURE PHYSICS | www.nature.com/naturephysics 5
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-182
-181
-180
P2 (mV)
-187
-186
-185
P1 (
mV)
(o,e
) (e,o
)
_126
0
-172
-171
-170
P2 (mV)
-335
-334
-333
P1 (
mV)
(o,e
) (e,o
)
_124
0
-175
-174
-173
P2 (mV)
-257
-256
-255
P1 (
mV)
(o,e
) (e,o
)
_124
8
-223
-222
-221
P2 (mV)
-217
-216
-215
P1 (
mV)
(e,o
)
(o,e
)
_118
4
-216
-215
-214
P2 (mV)
-294
-293
-292
P1 (
mV)
(e,o
)
(o,e
)
_121
3
-210
-209
-208
P2 (mV)
-370
-369
-368
P1 (
mV)
(e,o
)
(o,e
)
_122
6
-278
-277
-276
-275
P2 (mV)
-336
-335
-334
P1 (
mV)
(o,e
) (e,o
)
_115
7
-284
-283
-282
P2 (mV)
-253
-252
-251
P1 (
mV)
(o,e
) (e,o
)
_116
3
-285
-284
-283
P2 (mV)
-188
-187
-186
P1 (
mV)
(o,e
) (e,o
)
_117
6
-327
-326
-325
P2 (mV)
-213
-212
-211
P1 (
mV)
(e,o
)
(o,e
)
_113
0
-321
-320
-319
P2 (mV)
-288
-287
-286
P1 (
mV)
(e,o
)
(o,e
)
_113
5
-314
-313
-312
P2 (mV)
-365
-364
-363
P1 (
mV)
(e,o
)
(o,e
)
_114
4
1.00
0.90
0.80
Frac
tion
of m
ax G
(ad
just
ed p
er p
lot)
Figure
S2:
Experim
entalzero
source-drain
biasconductan
ceG
=G
1+
G2forasurvey
oftw
elve
Ne=
1an
dN
e=
3
LBTPs,
inan
in-planemagnetic
fieldof
2.0T
atT
=20
mK.TheLBTPssurveyed
correspon
dto
thesameab
solute
electron
occupationsas
theLBTPsof
Fig.S1.
Thedataarepresentedas
described
inthecaption
ofFig.S1.
Set
1135
(markedbytriangle)
correspon
dsto
thebottom-leftplotin
Fig.1b
,an
dset766in
Fig.S1.
6
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Data set Γ1 Γ2 γ1 γ2
1240 29 33 0.98 0.70
1248 29 34 0.88 0.65
1260 31 36 0.90 0.80
1226 32 32 0.62 0.78
1213 30 35 0.70 0.83
1184 31 32 0.76 0.87
1157 34 32 0.94 0.95
1163 31 35 0.94 0.99
1176 35 31 0.88 0.98
1144 32 29 0.58 1.02
1135 32 31 0.75 0.99
1130 30 35 0.79 0.97
Table S2: For each data set shown in Fig. S2, experimentally controllable parameters
Γ1, Γ2, γ1, and γ2 are extracted and reported as in Table S1.
7
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-216
-215
-214
-213
VP2 (mV)
-213
-212
-211
-210
V P1
(mV)
-213
-212
-211
-210
V P1
(mV)
-213
-212
-211
-210
V P1
(mV)
-213
-212
-211
-210
V P1
(mV)
-216
-215
-214
-213
VP2 (mV)
-213
-212
-211
-210
V P1
(mV)
-213
-212
-211
-210
V P1
(mV)
-216
-215
-214
-213
VP2 (mV)
-213
-212
-211
-210
V P1
(mV)
-216
-215
-214
-213
VP2 (mV)
-213
-212
-211
-210
V P1
(mV)
(1,1
)
(0,0
)
1.1
1.0
0.9
0.8
0.7
G (e
2 /h)
-216
-215
-214
-213
VP2 (mV)
-213
-212
-211
-210
V P1
(mV)
Zero
bia
s
V SD
(1) =
+5
µV
V SD
(1) =
+10
µV
V SD
(2) =
+10
µV
V SD
(2) =
+5
µV
V SD
(1) =
-5 µ
V V S
D (1
) = -1
0 µV
V SD
(2) =
-10
µVV S
D (2
) = -5
µV
Figure
S3:
Experim
entalconductan
ceG
=G
1+
G2measuredat
anN
e=
1LBTP,withsm
allbutfiniteVSD.Allcolor
scales
show
from
0.70
to1.12
e2/h
,em
phasizingtheconductan
cealon
gtheLBTPnear(0,0)an
d(1,1).
Center:
VSD=
0
foreach
dot.Thisdatasetwas
takenim
mediately
aftercheckingtheinputoff
setvoltageof
bothcurrentam
plifiers.
Top
row:FiniteVSDisap
plied
across
dot
1on
ly.Bottom
row:FiniteVSDisap
plied
across
dot
2on
ly.
8
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S2 Summary of NRG calculations
S2.1 NRG calculations
In our numerical calculations the double quantum dot (DQD) system is modeled by
the following Hamiltonian
H = HDQD +HTun +HLeads, (1)
where
HDQD =∑jσ
εjnjσ +∑j
Ujnj↑nj↓
+ U ′∑σσ′
n1σn2σ′ + gµBBzSz, (2)
describes the two dots, with njσ = d†jσdjσ the occupation number operator of dot
j = 1, 2 for spin σ, εjσ the energy of a spin-σ electron residing on dot j. Uj (U ′)
denotes the intradot (interdot) Coulomb correlations, while Bz is the magnetic field
applied along the z-direction and Sz is the z-component of the double dot’s spin. The
tunneling Hamiltonian HTun reads
HTun =∑αk
∑jσ
tαj(c†αjkσdjσ + d†jσcαjkσ), (3)
where c†αjkσ is the creation operator of an electron in lead α = L,R coupled to dot j,
with momentum k and spin σ of energy εαjk. Tunneling processes between the dots
and leads are described by hopping matrix elements tαj. Tunneling between the two
dots is suppressed by tuning gates in our experiment, and hence is omitted from the
model. The leads are described by noninteracting quasiparticles
HLeads =∑αjkσ
εαjkc†αjkσcαjkσ. (4)
Due to the coupling to external leads, the dots’ levels acquire a width described by
∆αj = πραj|tαj|2, with ραj the density of states of lead α coupled to dot j.
We performed the full density-matrix numerical renormalization group calculations
(fDM-NRG) [1, 2, 3, 4], employing the Budapest Flexible DM-NRG code [5]. For ef-
ficient calculations, we used the charge U(1) and the spin SU(2) symmetries in each
9
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channel, resulting in four symmetries altogether. When considering the effect of ex-
ternal magnetic field Bz, the spin invariance is reduced to the U(1) symmetry for the
spin z-component in each channel. In our computations we retained 2500−5000 states
at each iteration depending on the exploited symmetries and used the discretization
parameter Λ = 2.
We calculated the linear conductance through dot j using the following formula
Gj =e2
hαj∆j
∑σ
∫dω πAjσ(ω)
(−∂f(ω)
∂ω
), (5)
where f(ω) is the Fermi-Dirac distribution function and αj = 4∆Lj∆Rj/(∆Lj +∆Rj)2
is the left-right asymmetry factor for dot j, with ∆j = ∆Lj +∆Rj. Ajσ(ω) denotes the
spectral function of the j-th dot level for spin σ, Ajσ(ω) = − 1πImGR
jσ(ω), with GRjσ(ω)
the Fourier transform of the retarded Green’s function, GRjσ(t) = −iΘ(t)〈{djσ(t), d†jσ(0)}〉.
To improve the quality of the spectral functions and reduce the effects related with
broadening of Dirac delta functions, we also used the z-averaging trick [6].
S2.2 Choosing NRG parameters
Most of the parameters used in NRG calculations may be extracted from routine mea-
surements of the two dots. To a good approximation, a small decrement in the dot level
is proportional to a small increment in gate voltage. The proportionality constant, as
well as the charging energies U ′, U1, and U2, are measured directly by routine bias
spectroscopy. U ′ may be extracted from the change in ε1 of dot 1’s Coulomb blockade
peak position as an electron is added to dot 2, or vice versa. U1 and U2 are determined
from Coulomb blockade diamonds taken over a wider range of energy. Results of the
conductance calculations around the LBTP are, however, largely insensitive to values
of U1 and U2 as they are much greater than U ′. Therefore, in addition to U ′ extracted
from the stability diagram, we need to determine four parameters as precisely as pos-
sible to characterize conductance around the LBTP: the coupling strengths ∆1 and ∆2
(or linewidths) for dot 1 and 2 in the underlying Anderson model and the asymmetry
parameters α1 and α2.
∆1 may be extracted by taking cuts away from the LBTP on a mixed valence peak
of dot 1 (side of charge stability hexagon). There, for large intradot interactions U1
and U2, the FWHM of the conductance curve Γ1, divided by T, must be a universal
10
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function of ∆1/T , and likewise for dot 2. We produced a first estimate for ∆1(2) by
computing these universal curves with NRG, but managed to determine ∆1(2) only with
an accuracy of about 5 − 10%, as noise and other effects (perhaps Fano interference
at zero magnetic field, or neglected internal states of the dots, etc.) also affect the
widths of the measured peaks. Although this accuracy seems to be very good, it
is not sufficient: combined with the ∼ 5% accuracy of U ′ it amounts in a ∼ 15%
error in the ratios ∆1(2)/U′, which can give rise to a factor of ∼ 1.5 difference in the
Kondo temperature along the LBTP, the latter being exponentially sensitive to this
ratio. We therefore fine-tuned these values of ∆1(2) further by applying a complex
fitting procedure (for details, see Section S3), where we computed full two-dimensional
conductance plots at a fixed temperature (T = 40 mK) and fixed ∆1(2) by analyzing
various cuts of these. In this way, the value of the ratios ∆1(2)/U′ could be fixed with
a 1− 3% accuracy. The asymmetry parameters α1 and α2 were selected such that the
calculations reproduce the experimentally observed height of the mixed valence peaks
of dot 1 and 2, as well as the temperature dependent conductance in other regions of
parameter space. We estimate the accuracy of these parameters to be around ∼ 5%.
Since we do not use any broadening procedure in the conductance calculations but
calculate G(T ) directly from the spectral peaks, the only possible source of error in the
NRG calculations is due to the finite number of kept states. We checked, however, that
the number of kept states in our calculations was sufficient and changing it did not
influence the accuracy of the computed conductance curves. The NRG conductance
curves can thus be considered as “numerically exact.” For the 2D conductance plots,
we kept N = 2500 states, and for the G(T ) traces we kept N = 8000 states. The
spectral function calculations, however, contain an additional broadening parameter,
which typically reduces the accuracy of the computations at high and intermediate
frequencies. Therefore, curves presented in Figs. S11c and S11e are less accurate and
should not be considered as quantitative regarding the precise width and shape of the
predicted (and observed) high-energy features.
In Fig. 4, most of the parameters used for the spectral function calculation were
unchanged from those used in NRG calculations earlier in the paper. However, in
the calculation we set α1 = α2 = 1 for simplicity, as it would only contribute a scale
factor otherwise. For each value of experimental EPZ , the values of ε1, ε2 used in the
calculation are shown in Table S3.
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EPZ (meV) ε1 (meV) ε2 (meV)
0 -1.33667 -1.63833
0.012 -1.33001 -1.64499
0.018 -1.32721 -1.64779
0.026 -1.32300 -1.65200
0.036 -1.31784 -1.65716
Table S3: Parameters ε1 and ε2 used for each value of experimental EPZ in Fig. 4.
For the data of Fig. 4, the precise values of ∆1 and ∆2 were not determined, as
the tuning of the device was different from when the data of Figs. 2 and 3 were taken.
Nonetheless, the ∆ values should be similar and the spectral functions describe the
data remarkably well.
S3 Extracting LBTP cuts from 2D data sets
The zero-detuning cuts presented in Fig. 2c and 2d were extracted numerically from
2D data sets. The cuts are highly sensitive to cut direction such that adjusting the
endpoints by even a few µeV can result in significantly different conductances along the
cut. With experimental data alone, this poses a significant problem, since the line of
zero detuning cannot be exactly identified. Moreover, it is difficult to control for shifts
of the LBTP unrelated to renormalization as the temperature is varied. Physically
meaningful shifts of the mixed-valence peaks with temperature are to be expected, but
undesirable shifts, predominantly from random charge transitions in the donor layer of
the heterostructure, may also contribute.
To address these concerns, for fixed NRG parameters we compare the 2D exper-
imental data sets to the 2D NRG calculations, at each measured temperature. The
pseudospin-resolved conductances from the experimental data and from NRG were fit
to Lorentzians to find the peak positions. The experimental data were then offset
such that the peak positions matched those in the NRG data. Some manual shifts of
0.005 meV or less were used following the fitting procedure to provide best agreement
along the LBTP cuts. Note that the scale factor between gate voltage and energy is
experimentally determined, and only the offsets of the axes are adjusted.
12
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S4 G(ε1, ε2) for all measured T
Fig. 2a and 2b show respectively the measured and calculated conductanceG = G1+G2
as a function of ε1 and ε2 at T = 40 mK. We reiterate that the scale factor converting
gate voltage to energy is experimentally determined, with an offset determined for each
axis by comparing with NRG calculations. The validity of this assumption—using NRG
to determine offsets for ε1 and ε2—is borne out by the impressive agreement throughout
the 2D conductance maps for T ≥ 40 mK. Fig. S4 shows experimental measurements
and NRG calculations of G(ε1, ε2, T ). Cuts through each plot for both fixed ε1 and
fixed ε2 are shown in Figs. S5 (ε1, ε2 = 0 meV), S6 (ε1, ε2 = -0.05 meV), and S7 (ε1, ε2
= -0.1 meV). Apart from the 22 mK and 30 mK data sets, we find agreement between
theory and experiment over a wide range of gate voltages and temperatures.
13
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0.15
0.10
0.05
0.00
-0.0
5
-ε2 (meV)0.
150.
00 -ε1
(meV
)
22 m
K
0.15
0.00 -ε
1 (m
eV)
0.15
0.10
0.05
0.00
-0.0
5
-ε2 (meV)
0.15
0.00 -ε
1 (m
eV)
55 m
K
0.15
0.00 -ε
1 (m
eV)
0.15
0.10
0.05
0.00
-0.0
5
-ε2 (meV)
0.15
0.00 -ε
1 (m
eV)
99 m
K
0.15
0.00 -ε
1 (m
eV)
0.15
0.10
0.05
0.00
-0.0
5
-ε2 (meV)
0.15
0.00 -ε
1 (m
eV)
30 m
K
0.15
0.00 -ε
1 (m
eV)
0.15
0.10
0.05
0.00
-0.0
5
-ε2 (meV)
0.15
0.00 -ε
1 (m
eV)
65 m
K
0.15
0.00 -ε
1 (m
eV)
0.15
0.10
0.05
0.00
-0.0
5
-ε2 (meV)
0.15
0.00 -ε
1 (m
eV)
112
mK
0.15
0.00 -ε
1 (m
eV)
0.15
0.00 -ε
1 (m
eV)
0.15
0.10
0.05
0.00
-0.0
5
-ε2 (meV)
0.15
0.00 -ε
1 (m
eV)
147
mK
0.15
0.00 -ε
1 (m
eV)
0.15
0.10
0.05
0.00
-0.0
5-ε2 (meV)
0.15
0.00 -ε
1 (m
eV)
77 m
K
0.15
0.00 -ε
1 (m
eV)
0.15
0.10
0.05
0.00
-0.0
5
-ε2 (meV)
0.15
0.00 -ε
1 (m
eV)
87 m
K
0.15
0.00 -ε
1 (m
eV)
0.15
0.10
0.05
0.00
-0.0
5
-ε2 (meV)
0.15
0.00 -ε
1 (m
eV)
49 m
K
0.15
0.00 -ε
1 (m
eV)
0.15
0.10
0.05
0.00
-0.0
5
-ε2 (meV)
0.15
0.00 -ε
1 (m
eV)
40 m
K
100
8060
4020
0
221.
855
1.4
991.
130
1.7
651.
3511
21.
0540
1.5
771.
2514
70.
9549
1.45
871.
2
%%
T (m
K)G
(e2 /h
)
Figure
S4:
Con
ductan
ceG
=G
1+G
2through
thedotsas
afunctionof
temperature,ε 1,an
dε 2.For
each
temperature,experim
ental
conductan
ce(left)
ispairedwithNRG-calcu
latedconductan
ce(right).Eachpairshares
anindividually-set
colorscale.
Thecolor
scalesallrange
from
0e2/h
tothevalueindicated
inthetable.
14
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1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
22 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
55 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
99 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
30 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
65 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
112
mK
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
40 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
77 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
147
mK
0.10
0.00 -ε
2 (m
eV)
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
87 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
49 m
K
Figure
S5:
Cuts
through
theexperim
entalan
dcalculatedG
ofFig.S4.
Experim
entaldataareden
oted
byredcrosses,
andNRG
calculation
sbysolidblack
lines.Eachpaircorrespon
dsto
atemperature
stated
within
theplot.
Intheleft
(right)
plotof
each
pair,
−ε 2
(1)=
0meV
.
15
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1.6
1.2
0.8
0.4
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
22 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
0.8
0.4
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
55 m
K
0.10
0.00 -ε
2 (m
eV)
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
99 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
0.8
0.4
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
30 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
65 m
K
0.10
0.00 -ε
2 (m
eV)
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
112
mK
0.10
0.00 -ε
2 (m
eV)
1.2
0.8
0.4
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
40 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
77 m
K
0.10
0.00 -ε
2 (m
eV)
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
147
mK
0.10
0.00 -ε
2 (m
eV)
0.10
0.00 -ε
2 (m
eV)
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
87 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
0.8
0.4
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
49 m
K
Figure
S6:
Cuts
through
theexperim
entalan
dcalculatedG
ofFig.S4.
Experim
entaldataareden
oted
byredcrosses,
andNRG
calculation
sbysolidblack
lines.Eachpaircorrespon
dsto
atemperature
stated
within
theplot.
Intheleft
(right)
plotof
each
pair,
−ε 2
(1)=
0.05meV
.
16
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1.5
1.0
0.5
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
22 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
0.8
0.4
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
55 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
99 m
K
0.10
0.00 -ε
2 (m
eV)
1.5
1.0
0.5
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
30 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
0.8
0.4
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
65 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
112
mK
0.10
0.00 -ε
2 (m
eV)
1.6
1.2
0.8
0.4
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
40 m
K
0.10
0.00 -ε
2 (m
eV)
1.2
0.8
0.4
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
77 m
K
0.10
0.00 -ε
2 (m
eV)
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
147
mK
0.10
0.00 -ε
2 (m
eV)
0.10
0.00 -ε
2 (m
eV)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
87 m
K
0.10
0.00 -ε
2 (m
eV)
1.6
1.2
0.8
0.4
0.0
G (e2/h)
0.10
0.00 -ε
1 (m
eV)
49 m
K
Figure
S7:
Cuts
through
theexperim
entalan
dcalculatedG
ofFig.S4.
Experim
entaldataareden
oted
byredcrosses,
andNRG
calculation
sbysolidblack
lines.Eachpaircorrespon
dsto
atemperature
stated
within
theplot.
Intheleft
(right)
plotof
each
pair,
−ε 2
(1)=
0.1
meV
.
17
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S5 Temperature dependence details
As stated in the main text, the point ε = -0.03 meV was chosen for the temperature
dependence because it is a point where TK is large compared to experimentally ac-
cessible temperatures. However, apart from the saturation observed at T = 40 mK
that prevents observation of the low-T rollover, the experimental data are consistent
with both SU(4) universal scaling and NRG calculations for our device configuration at
other points along the LBTP. In Figs. S8 and S9 we show the temperature dependence
at ε = -0.04 meV and ε = -0.05 meV, respectively.
Uncertainties in the experimental conductances of Fig. 3 are likely dominated by
the uncertainty in maintaining constant ε1 and ε2 between data taken at different
temperatures, rather than conductance noise. We extract the conductances from the
2D maps of Figs. 2a and 2b and similar maps at other temperatures. The offsets (but
not the scale) of the ε1 and ε2 experimental axes of Figs. 2a and 2b are set using
the theoretical calculations, and this considerably reduces this uncertainty. After this
alignment procedure, the remaining uncertainty in ε1 and ε2 may be conservatively
taken as the pixel spacing of ε1 and ε2 in our 2D conductance maps, approximately
0.003 meV.
In determining error bars, experimental points in the 2D conductance map neigh-
boring ε1 = ε2 = −0.03 meV are considered to be independent measurements of the
conductance at ε1 = ε2 = −0.03 meV, with a Gaussian weight: wi = exp[−((ε1 −(−0.03))2+(ε2− (−0.03))2)/σ2], where σ = 0.003 meV. The error bars then reflect the
standard deviation of the weighted mean, and are largest at low temperatures where
the conductance varies the most rapidly in any direction in ε1 and ε2. The (unbiased)
standard deviation of the weighted mean, s, is given by:
s2 =V1
V 21 − V2
ΣNi=1wi(xi − µ∗)2 (6)
where µ∗ is the weighted mean, V1 = ΣNi=1wi, and V2 = ΣN
i=1w2i .
S6 Empirical Kondo forms
The empirical Kondo form was introduced by D. Goldhaber-Gordon, et al. [7] and
provides a convenient approximation of conductance through a quantum dot in the
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1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
G (e
2 /h)
12 3 4 5 6
102 3 4 5 6
1002 3 4 5 6
1000Temperature (mK)
}
G(T=TK)
Experimental data NRG calculations of device SU(4) Anderson model TK = 152 mK SU(2) Anderson model
SU(2) Anderson model TK = 182 mK
Figure S8: Experimental data for the temperature dependence of the conductance
(circles) at ε1 = ε2 = −0.04 meV in Fig. 2d. Experimental data are compared with
NRG calculations of the device (solid black line), as well as with the SU(4) and SU(2)
Anderson models in the Kondo regime (Ne = 1). The blue dash-dotted SU(4) scaling
curve and the green dotted SU(2) scaling curve have TK = 152 mK fixed to that
identified in the device calculations, where G(T = TK) = G(T = 0)/2. The red
dashed SU(2) scaling curve is for a best-fit TK = 182 mK. Parameters for the NRG
computations were: B = 0, U1 = 1.2 meV, U2 = 1.5 meV, U = 0.1 meV, ∆1 = 0.017
meV, ∆2 = 0.0148 meV, α1 = α2 = 0.875. These are the same used in Fig. 3.
19
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1.6
1.4
1.2
1.0
0.8
0.6
0.4
0.2
G (e
2 /h)
12 3 4 5 6
102 3 4 5 6
1002 3 4 5 6
1000Temperature (mK)
}
G(T=TK)
Experimental data NRG calculations of device SU(4) Anderson model TK = 113 mK SU(2) Anderson model
SU(2) Anderson model TK = 121 mK
Figure S9: Experimental data for the temperature dependence of the conductance
(circles) at ε1 = ε2 = −0.05 meV in Fig. 2d. Experimental data are compared with
NRG calculations of the device (solid black line), as well as with the SU(4) and SU(2)
Anderson models in the Kondo regime (Ne = 1). The blue dash-dotted SU(4) scaling
curve and the green dotted SU(2) scaling curve have TK = 113 mK fixed to that
identified in the device calculations, where G(T = TK) = G(T = 0)/2. The red
dashed SU(2) scaling curve is for a best-fit TK = 121 mK. Parameters for the NRG
computations were the same as in Fig. S8.
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SU(2) crossover regime as a function of temperature:
G(T ) = G0
(1 + (21/s − 1)
(T
TK
)n)−s
,(7)
where s = 0.22, n = 2, G0 is the conductance attained at zero temperature, and TK
is the Kondo temperature. This form is purely phenomenological and was invented
to describe succinctly the numerically-calculated spin-1/2 SU(2) universal scaling [8].
With such a formula it is convenient to estimate TK from experimental results using
nonlinear regression, however care must be taken in its application. Importantly, for
s = 0.22 and n = 2 this formula does not describe the universal SU(4) scaling. Various
papers have nonetheless used the empirical SU(2) form (7) to fit data for which the
applicability is not clear. In the absence of an alternative, this is a reasonable heuristic
since the differences between the SU(4) and SU(2) scaling are subtle, but this procedure
is not strictly justified.
In particular, the leading-order temperature dependence of (7) is quadratic by de-
sign at T � TK in order to describe SU(2) Kondo scaling, but conformal field theory
predicts the SU(4) Kondo state to have a leading-order cubic temperature depen-
dence at T � TK , despite retaining a Fermi liquid character (normally associated with
quadratic dependence) [9]. Therefore, both parameters s and n must be changed to
expect a nice agreement for T � TK , where the empirical form is designed to apply.
Fig. S10 shows how s = 0.22, n = 2 describes SU(2) universal scaling in the crossover
regime. Changing s alone is seen to be insufficient to describe the SU(4) universal
scaling especially for temperatures T < TK , where the fitting is most sensitive. How-
ever, a good fit to the SU(4) universal scaling may be obtained with s = 0.20, n = 3.
We must emphasize that although (7) provides an accurate fitting in the full crossover
region, it fails at temperatures T � TK , where it does not reproduce the well-known
logarithmic behavior characteristic of the Kondo problem.
From our experiences with analyzing the experimental data in this paper, empir-
ical forms must be used with great care and supported by other methods. A blind
application to our data would yield spurious conclusions, owing to the saturation at
T = 40 mK. Also, as can be seen from the NRG results for our device, there are
some expected deviations from the universal scaling, particularly at T > TK , where
the empirical forms become less accurate.
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1.0
0.8
0.6
0.4
0.2
0.0
G/G 0
0.012 3 4 5 6 7
0.12 3 4 5 6 7
12 3 4 5 6 7
10T/TK
SU(2) universal scaling SU(4) universal scaling s = 0.22, n = 2 s = 0.33, n = 2 s = 0.20, n = 3
Figure S10: Universal SU(2) (red) and 1/4-filling SU(4) (blue) scaling curves for the
conductance as a function of temperature. TKSU(2) and TKSU(4) are both defined such
that G/G0 = 0.5. Also shown are empirical fits in the form of (7): s = 0.22, n = 2
describes SU(2) (black dotted); s = 0.33, n = 2 best approximates the SU(4) form
without changing n (solid black); s = 0.20, n = 3 provides a good approximation of
the SU(4) form.
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S7 Bias spectroscopy
S7.1 Ne = 3 LBTP, zero magnetic field
Fig. S11 shows the orbital state-resolved bias spectroscopy and calculated spectral
functions for the Ne = 3 LBTP of Fig. 4, but with zero magnetic field (Fig. S11a).
As in Fig. 4, dot 2 only exhibits hole-like processes, but since EZ = 0, there is only
a single peak expected at ω = −EPZ . The NRG calculations (Fig. S11e) corroborate
the naive expectation, and the experimental data (Fig. S11d) show a peak evolving
into a shoulder that tracks with EPZ .
In dot 1, an electron-like process may be expected at ω = EPZ , but owing to the
unpaired electron, a peak should also appear at ω = 0. This is clearly seen in the
NRG calculations (Fig. S11c), but not evident in the experimental data (Fig. S11b).
Rather, a peak near ω = 0 for EPZ = 0 appears to move towards positive ω as EPZ
increases. Because the shift is small, the increasing spectral weight at ω > 0 together
with limitations in measurement resolution may explain this observation. The peak
near ω = −30 µeV is unexpected but may be due to a low-lying excited state.
When EPZ = 0, the width of the peak near ω = 0 should change noticeably as a
magnetic field is applied, reflecting an SU(4) to SU(2) crossover with lower TK when
the four-fold degeneracy is broken. Experimentally, making such a comparison with
confidence is challenging. Maintaining a particular tuning long enough to perform
bias spectroscopy with compensation for capacitances (see section S8.5) at both zero
magnetic field and finite magnetic field places extreme demands on the stability of
the device being measured. In our experiment, we needed to adjust the gate voltages
controlling the dot-lead tunnel barriers in between measuring the zero and finite mag-
netic field data, and thus a direct comparison between Fig. 4 and Fig. S11 is invalid.
Additionally, supposing the device were stable indefinitely, there could in principle be
an uncontrolled orbital effect from an in-plane magnetic field, owing to the finite thick-
ness of the 2DEG. This would change ∆ and therefore TK , regardless of the degeneracy
being broken.
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0.05
0.04
0.03
0.02
0.01
0.00
G 1 (
e2 /h)
-40 -20 0 20 40-eVSD (1,2) (µeV)
0.05
0.04
0.03
0.02
0.01
0.00
G 2 (
e2 /h)
-40 -20 0 20 40-eVSD (1,2) (µeV)
EPZ
EPZ = 0 µeV 14 µeV 20 µeV 30 µeV
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
1A1
-40 -20 0 20 40 (µeV)
1.4
1.2
1.0
0.8
0.6
0.4
0.2
0.0
2A2
-40 -20 0 20 40 (µeV)
EPZ
b
c
d
eEPZ
2
1a
Figure S11: (a) Inelastic transitions between pseudo-Zeeman-split states of the double
dot at an Ne = 3 LBTP, in zero magnetic field. An electron of either spin can occupy
dot 1. (b) Experimental conductance G1 for dot 1. The four traces correspond to
different values of EPZ , with EPZ > 0 meaning dot 1 is favored to hold the unpaired
electron. (c) Calculated spectral function A1 for dot 1. (d) Experimental conductance
G2 for dot 2. (e) Calculated spectral function A2 for dot 2. For all panels, Γ1,Γ2 ≈0.04 meV. Γ1S and Γ2S were both tuned to be∼ 2–3% of Γ1D and Γ2D, respectively, such
that the biased leads probe the equilibrium local density of states on their respective
dot. The bias is applied to both dots simultaneously. The parameters used for the
calculations were T = 40 mK, B = 1 T, U1 = 1.2 meV, U2 = 1.5 meV, U = 0.1 meV,
∆1 = 0.017 meV, ∆2 = 0.0148 meV. Note that α1 = α2 = 1 serve only as normalization
factors in the calculation. The ε1, ε2 used are in Table S4.
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EPZ (meV) ε1 (meV) ε2 (meV)
0 -1.3185 -1.62015
0.014 -1.3115 -1.62715
0.020 -1.3085 -1.63015
0.030 -1.3035 -1.63515
Table S4: Parameters ε1 and ε2 used for each value of experimental EPZ in Fig. S11.
S7.2 Ne = 1 LBTP, 1.0 T field
Fig. S12 shows the orbital state-resolved bias spectroscopy and calculated spectral
functions at an Ne = 1 LBTP, in a 1.0 T Zeeman field. By considering the cartoon
of Fig. S12a, and identifying each electron-like process with a corresponding hole-like
process in Fig. 4a, the relationship between the Ne = 1 LBTP and Ne = 3 LBTP
becomes clearer. We again consider ω and −eVSD as equivalent.
In dot 2, all of the expected features are observed (Fig. S12d): a weak peak at
ω = EZ , a peak (threshold) that tracks with EPZ for EPZ < EZ , and a purely orbital
Kondo peak at ω = 0 for EPZ = 0. The overall shapes of the curves are in rough
qualitative agreement with the spectral functions in Fig. S12e, although the relative
peak heights may differ.
However, in dot 1 (Fig. S12b), the purely orbital Kondo peak at ω = 0 for EPZ = 0
is obscured by poorly understood background conductance at positive ω. Additionally,
an unexpected feature is observed at ω = −30 µV that does not track with EPZ . It is
tempting to suggest that the LBTP being measured is actually a (1,1)/(2,0) LBTP. In
this interpretation, both dots could hold an unpaired electron, and both dots should
exhibit a peak at ω = ±EZ . In other words, the spectral functions for both dots should
look similar to Fig. S12e, with ω → −ω for dot 1. However, the increasing conductance
at positive ω in Fig. S12b is in qualitative agreement with Fig. S12c, and would not be
expected in this alternate explanation. Additionally, our ability to maintain electron
occupation number assignments is supported by Fig. S1. Therefore, the unexpected
feature is instead likely associated with a low-lying excited state.
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0.12
0.10
0.08
0.06
0.04
0.02
0.00
G 2 (
e2 /h)
-40 -20 0 20 40-eVSD (1,2) (µeV)
0.12
0.10
0.08
0.06
0.04
0.02
0.00
G 1 (
e2 /h)
-40 -20 0 20 40-eVSD (1,2) (µeV)
0.8
0.6
0.4
0.2
0.0
1A1
-40 -20 0 20 40 (µeV)
EPZ = 0 µeV 12 µeV 18 µeV 26 µeV 36 µeV
0.8
0.6
0.4
0.2
0.0
2A2
-40 -20 0 20 40 (µeV)
b
c
d
e EZ
– EZ –
EPZ-EPZ
EZ-EZ
-EZ
-EPZ
EPZ
EZ
EZ
EPZ1
2
1
2EPZ
a
Figure S12: (a) Inelastic transitions between Zeeman-split states of dot 1 and dot 2 at
an Ne = 1 LBTP. (b) Experimental conductance G1 for dot 1 in a 1.0 T Zeeman field.
The five traces correspond to different values of EPZ , with EPZ > 0 meaning dot 2 is
favored to hold the unpaired electron. (c) Calculated spectral function A1 for dot 1.
(d) Experimental conductanceG2 for dot 2. (e) Calculated spectral function A2 for dot
2. For all panels, Γ1,Γ2 ≈ 0.04 meV. Γ1S and Γ2S were both tuned to be ∼ 2–3% of Γ1D
and Γ2D, respectively, such that the biased leads probe the equilibrium local density
of states on their respective dot. The bias is applied to both dots simultaneously. The
parameters used for the calculations were T = 40 mK, B = 1 T, U1 = 1.2 meV, U2 =
1.5 meV, U = 0.1 meV, ∆1 = 0.017 meV, ∆2 = 0.0148 meV. Note that α1 = α2 = 1
serve only as normalization factors in the calculation. The ε1, ε2 used are in Table S5.
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EPZ (meV) ε1 (meV) ε2 (meV)
0 -0.06333 -0.06167
0.012 -0.05667 -0.06833
0.018 -0.05387 -0.07113
0.026 -0.04966 -0.07534
0.036 -0.0445 -0.0805
Table S5: Parameters ε1 and ε2 used for each value of experimental EPZ in Fig. S12.
S8 Technical details
S8.1 Electronics
For the data taken in Fig. 1b and 4 of the paper, custom current amplifiers designed
by Y. Chung of Pusan National University (early version of that which is presented in
[10]) were used in place of commercial Ithaco / DL Instruments 1211 current amplifiers,
which have been previously employed in our measurement setup [11]. The custom
amplifiers are crucial to this experiment in that the input offset voltage of the current
amplifiers must remain stable over a period of days to avoid applying an uncontrolled
source-drain bias across the dot. Over a continuous interval of 2.8 days, the standard
deviation of the input offset voltage was measured to be 1.0 µV for the amplifier
attached to dot 1, and 0.6 µV for the amplifier attached to dot 2. The amplifiers
were characterized in the same locations where they were used for measurement, as
no active temperature control of the amplifiers was performed during measurement or
characterization.
S8.2 Electron temperature calibration
Our electron temperature was calibrated by Coulomb blockade peak thermometry,
using the same device and during the cool-down when the data of Figs. 2 and 3 were
measured. Only a single dot was formed and measured during the electron temperature
calibration. In Fig. S13, we show a temperature-limited Coulomb blockade peak.
Experimental data (black circles) are compared with a theoretical lineshape (solid blue
line) describing the limit ∆E � kBT � �Γ, where ∆E is the level spacing [12]. The
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0.12
0.10
0.08
0.06
0.04
0.02
0.00
G (e
2 /h)
-255.0 -254.8 -254.6 -254.4
VP1 (mV)
Measured Temperature limited Including finite Γ
-150
-100
-50
0
50
100
150
V SD
(µV)
-256.0 -255.0 -254.0VP1 (mV)
G (e2/h)
0.120.080.040.00
Figure S13: Left: Temperature-limited Coulomb blockade peak (black circles). The
solid blue line is a fit to (8), and the dashed red line is a fit to a convolution of that
and a narrow Lorentzian lineshape. Right: Bias spectroscopy on this peak gives a
clean Coulomb blockade diamond, where the slopes may be confidently extracted to
determine αg.
lineshape is given by:
G(Vg) = y0 +G0
4kBT
1
cosh2(
αgeVg
2kBT
) (8)
where αg determines the conversion between gate voltage and energy, e is the
electron charge, Vg is the gate voltage away from resonance, G0 is a temperature-
independent prefactor, and y0 allows for a small offset due to instrumentation. We
extract and fix αg = 0.0572 from the slopes of lines seen in bias spectroscopy. The
temperature extracted from the fit is insensitive to y0.
Additionally, we fit to a convolution of the inverse-cosh-squared lineshape and a
Lorentzian, to describe both finite T and Γ (dashed red line). Agreement is seen with
both lineshapes. Neglecting Γ we find Te = 23 mK, or Te = 20 mK with Γ = 1.1 µeV.
The uncertainty in temperature is small, and becomes smaller at higher temperatures.
A temperature dependence was performed on this peak by heating the mixing cham-
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ber of our dilution refrigerator, and the electron temperatures extracted from the fits as
a function of temperature were recorded. By comparing with a ruthenium-oxide resis-
tance thermometer on our measurement probe, we establish a correspondence between
the resistance reported on the resistance thermometer and the electron temperature.
S8.3 Magnetic field calibration
0.33
0.32
0.31
0.30
0.29
0.28
B z (
T)
-1.0 0.0 1.0By (T)
R (Ω)
1200
1100
1000
900
800
Figure S14: Four-wire resistance as a function of the y-axis (in-plane) and z-axis (per-
pendicular) magnetic fields. The slopes of the solid white and dashed white lines are
m = −0.0206 and m = −0.0203, respectively. This corresponds to a 1.2◦ misalignment
between the y-axis field and the plane of the sample.
Because of small but uncontrolled sample tilt with respect to axes defined by the
two-axis magnet in our experimental dewar, energizing only the in-plane coil will give
rise to a perpendicular component as seen by the sample, and vice versa. To apply a
magnetic field precisely in the plane of the sample, as is done in Fig. 4, we calibrate in
situ using a four-wire current-biased measurement of Shubnikov-de Haas oscillations in
resistance, as a function of both the nominally perpendicular field Bz and nominally
in-plane magnetic field By.
Fig. S14 shows the Shubnikov-de Haas oscillations observed near a perpendicular
magnetic field of 0.3 T, and how they track with an added in-plane field. The geometry
of the 2DEG mesa is not well defined, so both even and odd components of magnetore-
sistance contribute to the measured resistance. The observed stripes correspond to a
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Magnetic field (T) Splitting (µeV) |g|1.0 — —
2.0 51 0.44
3.0 80 0.46
4.0 104 0.45
Table S6: Approximate spin state splittings and corresponding g-factors as a function
of magnetic field.
constant perpendicular field. The slope of the stripes gives a compensation factor such
that any perpendicular component introduced by the in-plane magnetic field may be
cancelled out by application of an added perpendicular field to within a few percent.
Even an applied field in the plane of the sample will subtly modify orbital states
because of the finite extent of the electronic wavefunctions normal to the plane, an
effect we neglect in our analysis.
S8.4 g-factor calibration
The Zeeman energy EZ is related to the magnetic field B by EZ ≡ |g|µBB, where µB
is the Bohr magneton and g is the g-factor. Among GaAs/AlGaAs heterostructures,
the g-factor can vary considerably, and so we calibrate in situ for our device by looking
for a Zeeman splitting in the bias spectroscopy as we vary an in-plane magnetic field.
Fig. S15 displays conductance through dot 2, demonstrating the Zeeman splitting. A
splitting is seen to emerge by B = 1.0 T, though the exact splitting is not resolved
owing to the width of the level. As the field is increased, we can extract the splitting
by reading off the value of VSD(2) above which the source-drain voltage drop is large
enough to allow for inelastic spin flip scattering processes. From this value, any offset
for true zero bias is then subtracted (usually a few µV or less). Table S6 summarizes
the extracted splittings and corresponding g-factors. We find |g| consistent with that
of bare GaAs, |g| = 0.44, and take this value in calculating EZ for given B.
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-300
-200
-100
0
100
200
300
V SD
(2)
(µV)
-254 -252 -250 -248VP2 (mV)
B = 2.0 T
-300
-200
-100
0
100
200
300
V SD
(2)
(µV)
-252 -250 -248VP2 (mV)
B = 3.0 T
-300
-200
-100
0
100
200
300
V SD
(2)
(µV)
-250 -248 -246 -244VP2 (mV)
B = 4.0 T
-300
-200
-100
0
100
200
300
V SD
(2)
(µV)
-254 -252 -250 -248VP2 (mV)
B = 1.0 T
G2 (e2/h)
0.3
0.2
0.1
0.0
Figure S15: Conductance G2 as a function of source-drain bias VSD(2) across dot 2
and gate voltage VP2, at in-plane magnetic fields of B = 1.0 T (top-left), B = 2.0 T
(top-right), B = 3.0 T (bottom-left), and B = 4.0 T (bottom-right). The color scale
is fixed for all four values of magnetic field, which are labeled in the upper-left of each
plot. Blue solid lines correspond to the alignment of the source lead Fermi energy with
the ground state, and blue dotted lines correspond to alignment of the drain lead Fermi
energy with the ground state. White arrows denote where VSD(2) is read off to extract
the splitting.
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S8.5 Bias spectroscopy
To apply and maintain a particular EPZ while changing the applied source-drain biases
VSD1(2) across dot 1 (2) requires some care. Gates P1 and P2 as well as leads S1
and S2 all have capacitances to both dot 1 and dot 2. These capacitances must all
be characterized every time the W gates or magnetic field are changed. Once the
capacitances are known, electrostatic gating of the dots by the biased source leads
may be compensated by changes in VP1 and VP2. Further details have been published
previously [13].
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