Valley blockade and multielectron spin-valley Kondo effect in silicon

5.5 MB·English

by  A. Crippa

Valley blockade and multielectron spin-valley Kondo effect in silicon
A. Crippa,1,2,∗ M. L. V. Tagliaferri,1,2 D. Rotta,1,2 M. De Michielis,2 G.
Mazzeo,1,2 M. Fanciulli,1,2 R. Wacquez,3 M. Vinet,3 and E. Prati4,2,†
1Dipartimento di Scienza dei Materiali, Universit`a di Milano Bicocca, Via Cozzi 53, 20125 Milano, Italy
2Laboratorio MDM, CNR-IMM, Via C. Olivetti 2, 20864 Agrate Brianza (MB), Italy
3CEA-LETI-MINATEC CEA-Grenoble, 17 Rue des Martyrs 38054 Grenoble, France
4Istituto di Fotonica e Nanotecnologia, CNR, Piazza Leonardo da Vinci 32, 20133 Milano, Italy
We report on the valley blockade and the multielectron Kondo effect generated by an impurity
atominasiliconnanofieldeffectdevice. Accordingtothespin-valleynatureoftunnellingprocesses,
5
andconsistentlywiththoseallowedbythevalleyblockaderegime,themanifestationofKondoeffect
1
obeys to the periodicity 4 of the electron filling sequence typical of silicon emerging at occupation
0
N =1,2,3. Thespin-valleyKondoeffectemergesunderdifferentkindsofscreeningdependingonthe
2
electron filling. By exploiting the valley blockade regime, valley index conservation in the Kondo

r SU(4) is deduced without the employment of an external magnetic field. Microwave irradiation
p
suppresses the Kondo effect at occupancies up to three electrons.
A

9
I. INTRODUCTION amount of valley mixing in contact-impurity tunnelings
2
gives rise to different kinds of Kondo screening from

] N = 1 to N = 3 at the relatively high temperature
l
l Thecontrolofindividualelectronsandimpurityatoms of 4.2 K. In agreement with theory, standard Coulomb
a
in silicon nano Field Effect Transistors (FETs) has deep blockade is observed at N = 4 where Kondo effect is
h
impact on the field of valleytronics1, which consists of forbidden. The fourfold degeneracy is usually revealed
-
s exploiting the orbital part of the electron wave function by splitting the orbital screening from the spin screen-
e
as additional degree of freedom with respect to more ing with a magnetic field6,20,21. Differently, in our
m
conventional charge and spin states. Valley-related experiment the valley blockade of first-order transport
.
t effects have been lately reported, from the valley filling (N =0→1→0) influences the Kondo processes within
a sequence in silicon quantum dots (Si-QDs)2–4 and the N =1 Coulomb diamond: it selects donor-lead tran-
m
lifetime-enhanced transport as indirect observation of sitions between identical valley states, which beautifully
- spin and valley blockade5, to a tunable valley Kondo confirms the selection rule at the base of the SU(4)
d
n effect in a As atom at single electron filling (N = 1)6. symmetry. The Kondo perturbed regime is obtained at
o Valley-based qubits7 and pure valley blockade8 have the base temperature at filling up to 3 electrons and it
c been proposed in analogy to spin qubits and Pauli spin is suppressed by means of microwave irradiation.
[ blockade respectively. The direct experimental obser- The paper is organized as follows. In Sec. II the

2 vation of the valley blockade, namely the suppression device and the experimental methods are described.
v of transport determined by the orthogonality of orbital In Sec. III the experimental results are presented and
5 degrees in a reservoir and at an impurity site, was till discussed. The Appendix connects the spin-valley states
6 now unaddressed. Valley quantum numbers may also mentioned in Sec. III with the established formalism of
6
influence higher-order tunneling processes involving two Ref. 22. In Supplemental Material details about device
2
ormoreparticles,asinthecaseoftheKondoeffect. The electrical characterization and data analysis procedure
0
simplestKondotheoryforIII-Vsemiconductorquantum are reported.
.
1 dots accounts for the emergence of Kondo effect only
0
with uncoupled spins in the dot9–11. Generalized to the
5
case of Si-QDs and group-V donors, the Kondo physics
1
: predicts that both spin and valley indices of a confined
v II. DEVICE AND METHODS
electron can be screened by a Fermi sea through virtual
i
X spin-valley flip processes, allowing for highly symmetric
SU(4) Kondo states12. The tri-gate FET is fabricated from fully depleted
r
a Wealreadydemonstratedsingle/fewdonorsbasedsilicon silicon-on-insulator (FDSOI) technology. The silicon de-
transistors operability13–15 and microwave irradiation vicelayerisphosphorousimplanted(1018cm−3)andthen
effects in single atom based devices for spectroscopy of etchedtodefinethenanowirechannel(Fig. 1a). Thegate
excited states at zero bias16 as well as in single interface is then formed on the three sides of the narrow channel
defects17,18. with a 5 nm thermal silicon dioxide of isolation. Silicon
Here we investigate a system of a P atom strongly nitride (Si N ) spacers around the gate protect the ac-
3 4
coupled to the leads of a silicon ultra-scaled tri-gate tive region from Arsenic implantation of highly doped
FET (Fig. 1a)19. Peculiarities of valley dependent (metal-like) source-drain electrodes. Nominal channel
quantum transport arise both in sequential tunneling width of the tested device is 50 nm, gate length is 20
processes and in Kondo-like cotunneling events. The nm and channel is 8 nm thick. A dc voltage V sets the
g
2
III. RESULTS AND DISCUSSION
A. Valley blockade
The valley blockade arising at single electron filling is
observed by the asymmetry of the stability diagram of
D+ ↔ D0 transitions in Figure 2a. The lack of con-
qd qd
duction in the section of the map involving the ground
state only at negative bias voltages is the hallmark of
valley blockade regime predicted in Ref. 8 as reported in
Figure 2b. It is determined by perpendicularity of the
FIG. 1. (Color online) (a) Scanning Electron Micrograph orbital states at the Fermi energies of the contacts and
image of a tri-gate FET identical to measured device. Be- the ground state of the donor, which is labeled with the
low the sketch of the active region with the two conductive odd parity index o. The first excited state of the donor,
paths. BrowndenotesthegatecontactwhereasSi3N4spacers labeled with the even e parity index, defines in addition
around the gate are yellow. Heavily-doped regions of reser- the inelastic cotunneling region delimited by the hori-
voirs are highlighted in blue. (b) From the stability diagram zontal dashed lines. The valley splitting ∆ is 6.2 meV24.
two Coulomb blockade patterns appear, each with its own
The blockade is lifted at V =−6.2 mV when the donor
lever-arm factor α. Red-orange (high current) diamonds are sd
excited state is resonant with the e states at the Fermi
attributedtothedonorD stronglycoupledwithleads. Blue
qd
level in the left lead.
(lowcurrent)diamondsarerelativetoaweaklycoupledquan-
tum dot. The origin of the valley blockade regime is ascribed to
the random graining of the As atoms of the contacts in
proximityoftheactiveregion,whichoriginatessingulari-
tiesinthereservoirdensityofstates26. Asthevalleypar-
ity of the wave functions depends on low dimensionality
of randomly diffused dopant distribution27, the electron
states at the Fermi levels of source and drain result un-
intentionally of opposite valley.
Such very special condition is confirmed by the stability
gatepotentialtotunethenumberofconfinedelectrons,a
diagram of the weakly coupled quantum dot in Figure
bias V controls the difference between the Fermi levels
sd 2d. The same asymmetry as the D+ ↔ D0 resonance
of the electron reservoirs at the contacts. The differen- qd qd
provesfromanindependentmeasurementthatthevalley
tial conductance dI/dV is measured by using a stan-
sd blockade occurs because of the opposite valley polariza-
dard lock-in excitation of 40 µV at 116 Hz applied to
rms tion of the contacts.
the source electrode. The microwave line consists of a
Theeffectivenessofthevalleyblockadeoverthecompet-
3.5-mm-diameter beryllium in stainless-steel coaxial line
ingCoulombblockadeisquantifiedthroughasimplerate
(UT-141) and it is ended by an unmatched dipole an-
equation model: tunneling events between identical val-
tenna.
leys turn out to be more than 10 times faster than those
As in similar doped15 and undoped23 devices with tri-
between states of different valley. The current across a
gate geometry, the onset of quantum transport takes
potentialbarrier,forexampletheleftone,inthesequen-
place via two parallel paths at the corners of the
tial regime is I = e(Γ Γ )/(Γ +Γ ), where Γ
nanowire. Twosetsofpeaksdifferingbyordersofmagni- left in out in out in
andΓ denotetheratesinandoutofthedonor28. Any
tudeinconductivityprovidedistinctcapacitivecouplings out
temperature dependence is neglected since we are inter-
with the gate (Figs. 1a and 1b).
ested in a ratio between tunneling rates and not in their
The highly conductive path D is attributed to a donor
qd absolute values. When n levels of the donor enter in the
close to the edge of the silicon nanowire with electronic biaswindow,Γ hastobereplacedwith(cid:80)n Γi 28. Re-
states hybridized with the interface states of the cor- in i=1 in
ferringtothebanddiagrams(b)and(c)ofFigure2b,we
ner. For the exhaustive discussion see Supplemental
define ΓX as the rate of tunneling events between levels
Material24,inagreementwithvaluesofdonorsembedded in
with opposite parity (i.e. from the left lead to the donor
in etched-silicon channels6,20,25 Its first two (subthresh-
groundstate). Analogously,Γ0 correspondstotunneling
old) peaks D0 and D− are associated to transport via in
qd qd processes preserving the valley index (i.e. from the left
neutral and negative charge states of the phosphorous
lead to the donor excited state). In particular, along the
atom respectively. At higher gate fields the potential
lines marked by (cid:52) and ♦ in the inset in Figure 2c, the
barriers at the ends of the tiny channel originate quan-
tum dot-like states visible as above-threshold resonances currents are I(cid:52) = eΓΓXinXin+ΓΓoouutt and I♦ = e(ΓΓ0in0in++ΓΓXinXin+)ΓΓoouutt re-
and Coulomb diamonds with N ≥ 3 electrons confined. spectively. Since the valley blockade selects only tunnel-
Such donor based quantum dot system is referred as the ing events entering the donor, relaxations are admitted
donor hereafter. inthetraversingtime,givingtoauniqueoutcomingrate
3

0.11 0.60 0.0105 1.0
2 e/h) VsNd= =-7 1mV 0.55 VsNd= = 0 2mV VsNd= = 0 3mV 0.0100VsNd= = 0 4mV 0.8 2 e/h)
-2 0dV(10sd.10 0.50 0.0095 0.6 -2 dV(10sd
dI/ 0.45 0.4 dI/
0.09
0.40 0.0090
0.2
0.35
0.0085
0.08 0.0
4 6 81012 4 6 81012 4 6 81012 4 6 81012
T (K)
FIG.3. (Coloronline)Conductanceatdifferenttemperatures
spanning the first spin-valley shell, for reference see Fig. 1b.
Theincreaseofconductanceatlowtemperaturesisthefinger-
printofemergingKondophysics. ThreeKondotemperatures
T =(8.2±1.3)K,T =(3.0±0.4)KandT =(3±1)K
K1 K2 K3
areobtainedatN =1,2,3electronfillingrespectively. Inthe
Kondo perturbed regime the fitting function adopted is suit-
ableforbothSU(2)andSU(4)frameworks11,20,21. ForN =4
the Kondo trend is not observed as the shell is complete.
B. Kondo effect
We now focus on the experimental investigation of
FIG. 2. (Color online) (a) Stability map of the D+ ↔ D0 the strong coupling regime to the leads in the first sil-
qd qd
icon spin-valley shell (i.e. from N = 1 to N = 4) of
resonanceandfirstCoulombdiamond(N =1). Dashedlines
indicate the onset of cotunneling via the excited state. For a the donor-dot system. In III-V semiconductor quantum
better visualization the derivative of the differential conduc- dots the presence of the Kondo effect stems from the
tanceiscomputed. (b)Expectedstabilitymapinpresenceof paired-unpaired spin filling sequence at odd occupation
inelastic cotunneling and valley blockade and, on the right, numbers9–11;differently,heretheKondoperiodicityisal-
energydiagramsforsequentialtransitions. Thindashedlines teredbythepresenceofvalleydegreesoffreedom. Asili-
representemptystatesavailableinthereservoirs. (c)Relative consystemwithanorbitalspacinglargerthanthevalley
comparison of tunneling rates between states with identical
splitting, such as an isolated donor, is predicted to have
and opposite valley composition. Inset: current stability dia-
a Kondo periodicity of four29, as in carbon nanotubes30.
gramfromwhichthevalueI(cid:52)andI♦usedinEquation(1)are Figure 3 provides the first experimental signature of a
obtained. (d) Transport data before the onset of conduction
spin-valley Kondo effect reflecting the fourfold degener-
through the donor. The sensitivity is tuned to magnify the
current contribution of QD. The first peak shows the asym- acy of the first orbital shell. The hallmark of the Kondo
metry due to the opposite valley polarization of the source perturbed transport lies in the increase of conductance
and drain electrodes and the consequent valley blockade. GS by lowering the temperature down to 4.2 K at N = 1,
andESdenotethegroundandtheexcitedstaterespectively. 2 and 3. Such fingerprint is not observed at N = 4 as
no unpaired spin-valley degrees can be screened by the
Fermiseaoftheleads. InFigure4atheKondoresonance
at V = −7 mV is marked with a green rhombus. The
sd
non-zero conductance background from |V | (cid:38) 5 mV is
sd
attributed to inelastic cotunneling. To discern the con-
Γ . tribution to conductance due to cotunneling from that
out
At the stationary state I =I ≡I. Because of the Kondo-related, the opposite temperature dependence of
left right
valley blockade, we consider ΓX (cid:28) Γ0 . By further as- the two second-order transport mechanisms is exploited.
in in
suming similar couplings between the donor and the two ByheatingthesampletheKondoresonanceprogressively
reservoirs (Γ0 ∼Γ ), we obtain: dampstillvanishingat24K,whilethecotunnelingback-
in out ground is enhanced24.
I 1 The neutral charge configuration of the donor D0 is
(cid:52) (cid:39)1− (1) qd
I♦ 1+2ΓXin/Γ0in shown in detail in the right part of Figure 2a. The selec-
tion rule on valley parity underlying the valley blockade
I(cid:52) and I♦ are evaluated from the data plotted in the regimeisreflectedinsecond-orderKondotransport: only
inset of Figure 2c, so that the ratio ΓX/Γ0 is extracted those processes for which the parity index is conserved
in in
for different values of gate voltage. clearly stand out from the cotunneling background. Fig-
4
by V for tunneling amplitude of processes preserving
0
the valley index and V for exchanges between different
X
valleys (m and m¯ indicate opposite parity indices).33
Thelackofazero-biasKondopeakrevealsanintervalley
couplingV smallerthantheintravalleytermV . Kondo
X 0
processes at V = −7 mV preserve the electron valley
sd
parity: the Hamiltonian in Equation (2) reduces to the
highsymmetricformwithjusttheV intravallycoupling
0
term. ItcorrespondstoaSU(4)-symmetricKondomodel
arising from a twofold spin degeneracy combined with
a twofold valley degeneracy; as the cotunneling onset
marks a non-zero valley splitting weakly dependent on
V (Fig. 2a), the condition of nearly-degenerate valleys
g
is here achieved by setting a bias equal or larger than ∆.
Spin-valley Kondo usually associates with spin-valley
blockade, as reported for carbon nanotubes21,30 where
reservoirs and dot are formed within the same tube.
Here, as the wave functions in the leads are built from
electron states of As with same orbital symmetry as
the electron state at the P in the channel, the exper-
iment reveals that in the device under investigation
such maintenance of symmetry is achieved thanks to
the specific distribution of As atoms in the reservoir
FIG. 4. (Color online) Conductance as a function of bias in areas. In III-V semiconductors SU(4) symmetric Kondo
the N = 1 and N = 2 configurations. On the right relative emerges either from spin-charge entanglement in two
Kondo processes. (a) If the valley parity is conserved the electrostatically coupled quantum dots34 or by inducing
Kondo resonance is observed (Vsd = −7) mV, whereas at multiorbitalgroundstatesinverticaldots35. Inourcase,
V =0,+7mVtheKondoeventsarepreventedbythevalley
sd two important clues point towards the SU(4) symmetry:
blockade. (b) Schematics describing spin-SU(2) Kondo effect
firstly, at operating temperatures of the order of the
in presence of valley mixing.
Kondo temperature T the SU(4) physics dominates
K1
over the SU(2); secondly, the energy difference between
the Kondo peak and the excited state (∼ 0.8 meV) is
ure4areportsasectionoftheCoulombdiamond. Kondo comparabletoT 32. Thecombinedemergenceofvalley
K1
eventswithandwithoutvalleyflipsoftheelectronbound blockade and SU(4) Kondo effect leads to a unified
to the donor (expected at Vsd = +7 mV and 0 mV picture of quantum transport based on valley parity
respectively20,29,31, marked by black arrows) do not ap- conservation during tunneling.
pear. Differently from N = 1, the Kondo effect at N = 2
The general Anderson Hamiltonian including valley de- emerges as a zero bias resonance. The participation of
grees of freedom reads as32 the excited state e would imply a Kondo temperature
T ∼ ∆/k , where k is the Boltzmann constant,
(cid:88) (cid:88) K2 B B
H = εkc†kmσckmσ+ εmσd†mσdmσ yielding TK2 ∼70 K, a value that can be easly excluded
kmσ mσ thanks to the experimental data reported in Figure
+ (cid:88)U n n + (cid:88)V (c† d +d† c ) 3. Therefore the single-particle level picture has to
mm(cid:48) m↑ m(cid:48)↓ 0 kmσ mσ mσ kmσ
be replaced with two-electron states obtained from
mm(cid:48) kmσ
(cid:88) Slater determinants. Following the formalism previously
+ V (c† d +d† c )
X km¯σ mσ mσ km¯σ adopted, the orbital part of such determinants is a
kmσ combination of e and o states, and the spin component
(2)
is either a singlet |S(cid:105) or a triplet |T(cid:105). Their total
antisymmetric tensor products provides the two-particle
wherec†,ccreateandannihilatenon-interactingfermions √
states: |S(cid:105)|e,e(cid:105), |S(cid:105)|o,o(cid:105), |T(cid:105)(|o,e(cid:105) − |e,o(cid:105))/ 2 and
in the leads, ε is the single-particle energy level of √
mσ |S(cid:105)(|o,e(cid:105)+|e,o(cid:105))/ 2. The electron-electron interaction
the donor localized state with valley m and spin σ, d†
further mixes the orbital components associated to
and d are the creation and annihilation operators of
the three spin singlets. As a result, both singlets and
this state, U the intra (m = m(cid:48)) or inter (m (cid:54)= m(cid:48))
mm(cid:48) triplets have as orbital counterpart a linear combination
valley Coulomb repulsion and n = d†d is the number
of different valley parities (see Appendix). The Kondo
operator; c†d and d†c describe tunneling events between
transport takes origin when an electron in the reservoir
a spin-valley state in the reservoir and a spin-valley level
with a well-defined valley parity tunnels through a
of the donor and viceversa. The channels involving the
barrier by changing its valley index to form the com-
coupling of the donor with the reservoirs are weighted
5
binations of mixed valley eigenstates above mentioned. fromthermal-activatedorphoton-assistedbackground24.
For this reason these Kondo exchanges do not suffer any Next, the Kondo resonance suppression is investigated
effect of the valley blockade previously discussed: the byvaryingthefrequency. Thezerobiasresonanceatthe
V mixingchannelgovernstheKondotransitionswithin lowest filling at which it is observed, namely N = 2 at
X
the N = 2 diamond. The exchange integral between V = 34 mV, is studied under microwave irradiation at
g
wave functions with different combinations of valleys different frequencies. Above 30 GHz the photon energy
is negligible22: in the limit of non-interacting spins |S(cid:105) approaches k T and the microwave-induced suppres-
B K2
and |T(cid:105) are degenerate in energy. Hence, each level sion curves collapse over a single trend24, revealing that
admits spin flip Kondo processes at zero bias (sketch in this regime the microwave field perturbs the coherent
in Figure 4b). The ground and the first two-electron spinflipeventsunderlyingtheKondotransport. Because
excited level are coupled with the conduction electrons ofthehighworkingtemperatureanappreciablesuppres-
of the leads through two orthogonal scattering channels, sion is achieved when many photons cooperate to bring
leading to a two-level SU(2) Kondo effect. That one theKondostateoutofequilibrium,eV >hν. Weunder-
ω
with higher Kondo temperature is activated here. The linethatsuchsuppressivetrendstandsoutindependently
vanishingoftheexchangeinteractionhampersaunivocal from the choice of the background fitting function24.
labeling of our Kondo at N = 2 as under, double-fully The systematic comparison among the Kondo reso-
or overscreened type. nances at N = 1,2,3 by varying the power of a 30
Such an anomalous Kondo effect at even occupancy GHz excitation is carried out. In Figure 5 the ratio
is inherently related to the non-zero momentum of (dI/dV ) /(dI/dV ) is plotted as a function of
sd peak sd dark
silicon conduction band edges. In III-V direct-gap the parameter eV /k T . Here ”dark” corresponds to
ω B K
semiconductor devices integer total spin states may thetotalabsenceofanyexternalmicrowavesignal. Such
lead to the appearance of Kondo peaks at even electron renormalization allows a direct comparison of the three
fillings. In Ref. 36 a magnetic field brings singlet and cases N = 1,2,3 taking into account for the different
triplet into degeneracy, giving rise to singlet-triplet Kondo temperature. Microwave irradiation affects the
Kondo transitions at zero-bias. Alternatively, by means Kondo resonance for all the three occupancies. Such
of an electric field the orbital spacing of two adjacent suppression can be ascribed to the quenching effect of
levels can be tuned37 to promote the spin triplet state the microwave photons rather than to heating, that is
as the ground state. In the system under investigation excludedbytheestimationoftheelectronictemperature
suchconditionisexperimentallyruledout, asbyvarying by means of Coulomb blockade thermometry. Under the
V the Kondo resonance remains pinned at zero bias. assumption of temperature-broadened peak, i.e a cosh−2
g
Within the N = 3 diamond the contribution to lineshape,weregisteranincreaseof∼0.8Katthemaxi-
zero-bias conductance due to the Kondo effect (third mumpower(-5dBmnominally)withrespecttothebase
panel of Fig. 3 and Fig. 4c) is attributed to the temperature in dark conditions. The weak thermal sup-
unpaired spin of the third electron fully screened by pression expected in this range of temperatures cannot
themany-bodyspinoftheelectronsinthereservoirs9–11. account for the complete destruction of the Kondo reso-
nance,whichhastobeattributedtothemicrowavefield.
At amplitudes of energy oscillations smaller or compa-
rable to the Kondo binding energies, such behavior is
ascribable to adiabatic variations of the bias combined
C. Microwave suppression of Kondo effect
withweakmodulationsofthegatepotential41,42,though
their separate contribution is difficult to evaluate. By
Finally the attention is turned to the perturbation of
increasing the power of irradiation the rate of photon-
the system by means of a tunable microwave irradia-
induced events increases as well, progressively inhibiting
tion. In this section the experimental microwave sup-
the Kondo processes.
pression of the Kondo effect, previously observed in di-
Theresponsetomicrowavesignaldiffersamongthecases
rect bandgap semiconductors38–40, is experimentally ad-
N = 1,2,3, as shown in Figure 5. The slope of the
dressed in a multi-valley semiconductor like Si.
suppression curve for N = 1 is smaller than that for
As the setup is equipped with an unmatched coaxial
N = 2,3, which by contrast are comparable. An exten-
cable employed as an antenna, the random microwave
sive theory41,42 has been developed for a single spin 1/2
field distribution induces oscillations both to bias and to
intheKondoregime(i.e. T (cid:28)T )perturbedbyphotons
donor chemical potential. An empirical conversion be- K
with hν > k T . Though a well-established theoretical
tween nominal power P and effective amplitude of the B K
analysis is still lacking for the mentioned working condi-
perturbation V on the donor site38 is established for
ω tions (multi-valley semiconductor, hν ∼ k T ) we can
eachelectronoccupancy(N =1,2,3)(detailsreportedin B K
infer valuable information from our experimental data.
Supplemental Material24). However, the microwave cou-
ThedifferentsuppressiveratesofFigure5suggestacon-
plingwiththequantumsystem, aswellasthelinetrans-
nection between the Kondo symmetries of the three oc-
mission efficiency, varies also at different frequencies.
cupancies. Itisnotablethattheoccupancieswithsimilar
For this reason, we first developed a robust background
dampingrateshavecomparableKondotemperaturesand
subtraction procedure to isolate the Kondo resonances
6

1.0
k 0.8
ar
d
d
s
V 0.6
d
dI/
/
d 0.4
s
V
d
dI/ 0.2
N = 1
N = 2
N = 3 = 30 GHz
0.0
1 10
eV /kBTK
FIG.6. (Coloronline)(a)DifferentialconductanceoftheD
qd
FIG.5. (Coloronline)NormalizedKondopeakamplitudesfor resonance without microwave irradiation. (b) Same stability
N = 1,2,3 as functions of eVω/kBTK for the three different diagrammeasuredat-17.5dBm,-10dBm,-7.5dBm,-5dBm,
occupancies. -2.5 dBm and 0 dBm nominal powers with a frequency of
30 GHz. The asymmetry due to the valley blockade is not
influenced by the microwave irradiation.
CONCLUSIONS
the same underlying symmetry. Such suppressive trends
ValleyblockadeandmultielectronKondorelatedtrans-
arequalitativelyinagreementwithpreviousobservations
in SU(2) spin Kondo effect38,40. port are observed in a single P donor silicon transistor.
The Kondo perturbed regime arises at partial filling of
The reasons of the different rate for N = 1 are not un-
thefirstspin-valleyshellofsilicon,N =1,2,3. AtN =1
ambiguouslyaddressed. Ontheonehand,themicrowave
occupancy the pure valley blockade occuring between
suppressioncouldbelessefficientonSU(4)thanonSU(2)
donor and leads selects Kondo spin-valley processes at
symmetry due to different couplings of the microwave
non-zero bias consistent with a SU(4) symmetry. Mixed
field to spin and to valley degrees. On the other hand,
even-odd valley states originate spin SU(2) Kondo reso-
as T > T ,T , stating the different symmetry at
K1 K2 K3
nances at zero bias when 2 and 3 electrons are confined.
N =1,theenergycarriedby30GHzphotonsmayresult
The Kondo effect is not observed when the shell is com-
tobelesseffectiveinperturbingtheKondoeffectbecause
plete at N = 4, as predicted. Finally, microwave driven
of the higher binding energy k T .
B K1
oscillationssuppressallthethreeKondoresonanceswhile
Overall, our results show the effectiveness of microwave
leaving the valley blockade phenomenology unaffected.
suppression over the first spin-valley shell, suggesting a
possible extension of the theory of Refs. 41, 42 beyond
the context of standard spin 1/2 Kondo effect.
ACKNOWLEDGEMENTS
Finally,weinvestigatetheeffectofmicrowaveirradiation
on the valley blockade regime. The valley selection rules
are not perturbed by the 30 GHz microwave irradiation. The device has been designed and fabricated by the
In Figure 6 no significant variations with respect to the AFSiD Project partners under the coordination of M.
dark condition appear in the asymmetric D+ ↔ D0 Sanquer. The authors acknowledge S. Cocco for techni-
qd qd
peak by increasing the microwave power. At the Fermi cal assistance and M. Belli, A. Chang and D. Culcer for
levels of the reservoirs the valley parity electronic states fruitful discussions.
are not altered and the valley blockade phenomenology
is fully preserved8. Such experimental result suggests
thatthemicrowavesuppressionoftheKondoresonanceis APPENDIX
mainly due to spin coherence breaking, while valley fluc-
tuationsremainunaffected. TheKondopeakatN =1is In this Appendix the spin-valley states are explicitly
effectively suppressed at V = 2 mV24. Non-zero peaks expressed. In Table I we connect our formalism to the
ω
of Figure 5 can be therefore ascribed to competing deco- effective-mass theory of Hada and Eto22 by writing the
herent mechanisms like thermal fluctuations or inelastic states in both bases. For the two-particle states the su-
cotunneling. perscript 1 labels the first electron wave function and 2
the second electron wave function.
7
By taking into account the Coulomb interaction the triplet apart, remain. An example of the orbital
1 between the two electrons as a perturbation, part of a spin singlet is √1 e2iθ(cid:2)−isin2θ(|e,e(cid:105)+|o,o(cid:105))+
t4hπe(cid:15)(cid:15)0e|irg1e−nrf2u|nctions can be written as linear combination cos2θ(|o,e(cid:105)+|e,o(cid:105))(cid:3), whi2ch shows how both even and
of Slater determinants. The Coulomb interaction is spin odd valley parities contribute.
independent, so mixed singlet-triplet states annihilate,
and just combinations between the three singlets, and
Spin Orbital
Even-odd formalism Ref. 22 formalism
|e(cid:105) √1 (ψz+e−iθψ−z)
2
Single
|↓(cid:105), |↑(cid:105)
particle
|o(cid:105) √1 (ψz−e−iθψ−z)
2
|ee(cid:105) e−iθ(ψ(1)ψ(2)+ψ(1)ψ(2))+ 1(ψ(1)ψ(2)+ψ(1)ψ(2)e−2iθ)
2 z −z −z z 2 z z −z −z
|S(cid:105) |oo(cid:105) −e−iθ(ψ(1)ψ(2)+ψ(1)ψ(2))+ 1(ψ(1)ψ(2)+ψ(1)ψ(2)e−2iθ)
2 z −z −z z 2 z z −z −z
√12(|eo(cid:105)+|oe(cid:105)) √12(ψz(1)ψz(2)−ψ−(1z)ψ−(2z)e−2iθ)
Two particles
|T (cid:105)
+
|T−(cid:105) √12(|eo(cid:105)−|oe(cid:105)) e√−i2θ(ψ−(1z)ψz(2)−ψz(1)ψ−(2z))
|T (cid:105)
0
TABLE I. Conversion table between the even-odd formalism adopted in the main text and the formalism of Ref. 22
.
SUPPLEMENTAL MATERIAL
Estimated energies of the P atom
InFigure1bofthemaintexttwoseriesofconductancepeaksdifferingbyanorderofmagnitudeappear. Red-orange
(i.e. high-current) diamonds give a unique lever-arm factor α, thus a single coupling with the gate, observed up to
N = 5. The blue diamonds of Figure 1b have a different lever-arm factor. The pair of interlaced diamond patterns
with different lever-arm factors suggests a circuital scheme with two conductive paths in parallel (D and QD) with
qd
a negligible inter-conductance43. The first is associated to a phosphorus atom, labeled as D , while the second to a
qd
disorder-assisted quantum dot (QD), like previously observed in similar samples. D localized states arise from the
qd
hybridization of Coulomb potential of a donor, with the gate-induced potential well at the Si/SiO interface44.
2
Accordingtothedopingconcentration,weexpect6-12donorsinthevolumeofthechannel. Byloweringtemperature
thermal transport is progressively quenched so that the threshold voltage V can be easly extracted from the
TH
linear part of the G(V ) characteristics, see Figure 7a. At 300 K the subthreshold current is ascribed to thermally
g
broadened tunnelings through the dopants of the active region45 (Fig. 7b). At cryogenic temperatures only those
donorssufficientlycoupledwithbothsourceanddrainmaybeobservedfromquantumtransport. Similarlytoprevious
reportsonsingleatomtransistorswithetchedchannels16,46,47,thefirstadditionenergyofD liesintherange15−40
qd
meV (here U = 26.4 meV). The neutral charge state D0 is 44.7 meV below the conduction band edge marked by
qd
the threshold voltage. This ionization energy is very close to the 45 meV of P atoms in bulk silicon48, indicating
the proximity of the atom to the gate oxide45. Such a vicinity enhances the confinement in the gate-field direction,
delocalizing the electron wavefunction along the tunneling direction. The consequent strong tunnel coupling with
8
FIG. 7. (a) The threshold voltage of the D conductive path is extracted from the linear part of the G(V ) characteristics at
qd g
relatively high temperatures. The contribution to G of the QD island is here neglected being one order of magnitude weaker.
(b) The presence of donors in the channel is confirmed by the subthreshold current at room temperature.
FIG. 8. (a) Raw data of the Kondo resonance for N = 2 at different microwave power at 30 GHz frequency. Dashed arrows
identify the background enhancement due to photon assisted tunneling, whereas the solid arrow highlights the suppressive
trend of the Kondo resonance. (b) Same data after background subtraction.
reservoirs allows for cotunneling and Kondo processes.
The conclusive argument to attribute the high conductance peaks to a donor is provided by the valley splitting ∆.
It is estimated from the dI/dV curves as the center of the cotunneling step, see data analysis in Subtraction of the
sd
Background section. The value of about 6.2 meV is indeed inconsistent with the typical valley splitting of (cid:46) 1 meV
of a Si/SiO quantum dot49.
2
Subtraction of the background
In this section we show that the height of the Kondo peak is irrespective from the choice of the background model.
ThequantitativeanalysisoftheKondocontributiontoconductanceisperformedbytakingintoaccountbothinelastic
cotunnelingandthermalactivatedtransport. ByincreasingmicrowavepowertheKondopeakamplitudedecreases,as
a result of oscillating bias and decoherent microwave-induced processes like spin flip cotunneling; on the other hand,
thebackgroundconductanceinCoulombblockedregionsincreasesduetophotonassistedtunnelling. Thecotunneling
step enhances as well, as predicted by Flensberg50 and experimentally confirmed in Refs. 51 and 52. To discriminate
the effects, the data analysis reported in Ref. 53 allows to isolate the Kondo physics from other competitive effects.
In Figure 8 the zero bias Kondo resonance for N =2 is shown at different microwave power values, namely the dark
signalandatthenominalappliedpower-17dBm,-15dBm,-11dBmand-8dBm,beforeandafterdataanalysis,with
9
Data
N = 3 Parabola
2.50 2-Exponential
Dark signal P = -25 dBm 2-Boltzmann
2.25

)h
2/
e
3
-0 2.00
1
( d 2.50
s
V
d P = -20 dBm P = -15 dBm
dI/
2.25

2.00
-2 -1 0 1 -2 -1 0 1
Vsd (mV)
FIG. 9. Examples of the background fitting using three different functions: a parabola (red), double exponential (blue) and
double Boltzmann (green). The data refer to the N =3 Coulomb diamond with a 30 GHz radiation except the top-left panel.
At -15 dBm the Kondo resonance is completely suppressed.
the microwave frequency fixed at 30 GHz. Without any background subtraction at the higher microwave power, i.e.
from -11 to -8 dBm, it is difficult to distinguish the Kondo resonance from the Coulomb valley bottom. In Figure 8b
theKondopeakisdecoupledfrombackgroundaccordingtothefollowingprocedure. Wefirstsystematicallytestthree
different background functions for N = 1,2,3. A parabolic, a double exponential and a double Boltzmann fittings
are imposed in the proximity of the resonances53. As an example we report in Fig. 9 the data analysis for N = 3.
Generallyparabolaoradoubleexponentialfunctionsreturnalowerboundforexperimentalbackgroundsignal, while
the double Boltzmann gives a fit closer to data. Table II reports the adjusted coefficients R2 obtained by fitting the
data of Figure 9. It confirms the qualitative conclusions previously derived: the parabolic fit is characterized by the
lowest R2 among the tested functional forms, thus signaling the biggest deviation from the experimental data. The
double exponential returns a R2 value close, but generally lower, to the double Boltzmann. Analogous results, not
shown here, are obtained for N =1,2. According to these results the data in Figure 5 of the main text and Figs. 8,
Parabola Double exponential Double Boltzmann
0.982 0.985 0.985
0.986 0.987 0.992
0.980 0.982 0.982
0.965 0.981 0.986
TABLE II. : Adjusted R2 for the fits shown in Fig. 9.
10, 11, 12, 14 are extracted after a 2-Boltzmann background subtraction. Once determined the best functional form
of the background signal, the next step is the data analysis of the resulting Kondo resonance. To evaluate the height
of such resonances for N = 2,3 we use a simple Lorentzian (the theoretical shape of the Kondo peak at non-zero
temperatures is still debated):
2A w
g =g + (3)
0 π 4(V −V )2+w2
sd c
10
Background: 2-Boltzmann
0.3 g0 = (-1.90 – 0.05)•10-4 Data
Vsd = -0.120 – 0.006 Lorentzian
32 e/h ) 0.2 wA r==ed (6=4. 2.244. 1 ––3 060. .1•01)9•01-05 -3
- 0
1
(
d 0.1
s
V
d
dI/
0.0
N = 2
-6 -4 -2 0 2 4 6
Vsd (mV)
FIG. 10. Example of a Lorentzian fit performed after background subtraction. The data shown are taken at a nominal
microwave power of -19 dBm at 15 GHz, in the N =2 valley.
Background: 2-Boltzmann
Kondo Data
1.25 Cotunneling step fit
Kondo peak fit
h) Global fit
2 /
e 1.00 Cotunneling step
-3 0
(1d
s
V 0.75
d
dI/
0.50
N = 1
-10 -8 -6 -4 -2 0
Vsd (mV)
FIG. 11. Example of data analysis for N =1. The non-zero bias resonance is characterized by a non trivial shape, showing a
shoulder at -6.2 mV due to the cotunneling step.
where the fitting parameters are g , A, w and V . The Lorentzian fits adequately the data, as reported in Fig. 10
0 c
where a -19 dBm irradiation is applied at 15 GHz.
Inordertoevaluateanddiscriminatespin-valleyKondoeffectfrominelasticcotunnelingatN =1weadoptamodified
approach. Thecotunnelingsignalispredictedtobeastepsmoothedbytemperature,superimposedonabackground;
in some cases it can be characterized by a cusp-like resonance54. In absence of an overall functional form describing
the simultaneous presence of Kondo effect, inelastic cotunneling and background signal we choose to fit our data
with the sum of two Lorentzians (one fits the Kondo resonance and one fits the cotunneling resonance) superimposed
to each of the three fitting functions mentioned above. At 4.2 K and in dark conditions, the two Lorentzian result
centered at V =−6.2 mV and V =−7 mV, which are identified as the cotunneling and Kondo peak respectively.
sd sd
The best R2 (= 0.92) are obtained with the double Boltzmann choice, returning the satisfactory fits of Figure 11.
As shown in Fig. 13, the Kondo resonance height behaviour is insensitive to the chosen background function. It is
worthy to note that the fitting procedure with double Lorentzians returns good results when the two resonances are
comparable. As soon as one prevails on the other this procedure will never return a complete suppression, producing
an artificial saturation of the height of the decreasing peak.
We finally remark with Figure 12a that the Kondo peak decreases by increasing of microwave power whereas the
cotunneling behaves oppositely. The net result consists in a shift of the position of the conductance maximum
towards less negative biases, as visible in Figure 12b.