Nonequilibrium Cotunneling through a Three-Level Quantum Dot

0.79 MB·English

by  S. Schmaus

Nonequilibrium Cotunneling through a Three-Level Quantum Dot
S. Schmaus,1 V. Koerting,2,∗ J. Paaske,3 T. S. Jespersen,3 J. Nyg˚ard,3 and P. W¨olfle4
1Institut fu¨r Theorie der Kondensierten Materie,
Universit¨at Karlsruhe (TH), D-76128 Karlsruhe, Germany
2Department of Physics, University of Basel, Klingelbergstrasse 82, CH-4056 Basel, Switzerland
3The Niels Bohr Institute & Nano-Science Center,
University of Copenhagen, DK-2100, Copenhagen, Denmark
4Institut fu¨r Theorie der Kondensierten Materie,
Universit¨at Karlsruhe, D-76128 Karlsruhe (TH), Germany
(Dated: January 9, 2009)
9
We calculate the nonlinear cotunneling conductance through a quantum dot with 3 electrons
0
occupyingthethreehighestlyingenergylevels. Startingfroma3-orbitalAndersonmodel,weapply
0
a generalized Schrieffer-Wolff transformation to derive an effective Kondo model for the system.
2
Within this model we calculate the nonequilibrium occupation numbers and the corresponding

n cotunnelingcurrenttoleadingorderintheexchangecouplings. Weidentifytheinelasticcotunneling
a thresholds and their splittings with applied magnetic field, and make a qualitative comparison to
J
recent experimental data on carbon nanotube and InAs quantum-wire quantum dots. Further

9 predictionsof themodelsuch as cascade resonances and amagnetic-field dependenceof theorbital
levelsplittingarenotyetobservedbutwithinreachofrecentexperimentalworkoncarbonnanotube
] and InAsnanowire quantum dots.
l
e
- PACSnumbers: 72.10.Fk,73.63.Nm,72.15.Qm,73.23.Hk,71.70.Ej
r
t Keywords: electronictransport,non-equilibrium,spin-orbitcoupling,carbonnanotube, InAsnanowire
s
.
t
a
The Kondo effect has been observed in a number of detailsofinterorbitaltransitionsbetter inthese systems.
m
different quantum dot (QD), and single molecule de- This is what we set out to do in the present paper.
d- vices1,2,3,4,5. The effect is manifested as a sharp con- In the following, we study the case of a quantum dot
n ductance peak at zero bias-voltage,developing when the occupiedbyanoddnumberofelectrons,featuringaspin-
o temperature is lowered beyond the characteristic Kondo doublet groundstate, giving rise to a zero-bias Kondo
c temperature. It relies on a spin-degenerate ground-state peak,butwithadditionalorbitals/levelsleadingtoflank-
[ on the quantum dot, which gives rise to logarithmically ing inelastic cotunneling steps or peaks. The basic 3-
2 singular spin-flip scattering of the conduction electrons orbital Anderson model is illustrated in Fig. 1 and will
v traversing the dot. Meanwhile, for quantum dots with beshowntohostavarietyofdifferentI-V-characteristics
9 sufficiently small level spacings or slightly broken de- depending on the relative magnitudes of the 6 different
6 generacies, the Kondo-peak at zero bias voltage can be tunneling-amplitudes. This 3-orbital model was briefly
9
flanked by two or more satellite steps or even peaks discussedbysomeofthepresentauthorsinRef.5(cf.in-
1
in the nonlinear conductance. Such inelastic cotunnel- set in Fig. 4b), where it was invoked to explain the rel-
.
9 ing features are often seen in both GaAs6, Carbon nan- atively sharp peaks at finite bias flanking a zero-bias
0 otube7,8,9 (CNT)andInAs-wire5,10 quantumdots,butin
8
most cases they are masked by charge-excitations which
0
can be nearby in energy and for this reason they have
:
v not received much attention.
i
X Single-moleculetransistors,onthe otherhand, exhibit
a much larger charging energy, E (∼ 100 meV instead
r C
a of 5 meV, say, for a typical quantum dot). At the same
time, these molecular systems often display a number of
degeneracies which are weakly broken once the molecule
is contacted by source, and drain electrodes, thus induc-
ing splittings of the order of a few meV. Most recently,
this was seen in Refs. 11,12, where junctions holding an
OPV5, or a C molecule, respectively, showed a very
60
clear singlet-triplet splitting on the scale of 1 meV to-
gether with a charging energy of the order of 100 meV. FIG. 1: Illustration of the setup. We consider the three
This is a very convenient separation of energy scales highest-lyinglevelsofaquantumdot,separatedbyenergiesδ,
which raises the experimental resolution of inelastic co- and δ′. Both orbitals are connected to source (left lead) and
tunneling phenomena to new standards. Nevertheless, drain (right lead) via 6 different tunnel couplings tαn. The
given the simpler level-structure of most conventional dot is in the Coulomb blockade regime with the three levels
QD-devices, it is desirable to revisit and understand the adjusted bya gate toaccommodate exactly 3 electrons.
3
V]
m
[d0
s
V
-3 0T 0.5T
2
-2.5 -2.2 -2.5 -2.2
Vg [V] Vg [V]

0.15
0.2
0.10
0 T
)h
2G [e/h]0.1 2 d (/Ve0.05
0T d/I0.00
0.0 0.9T
10 T
-0.05
-1 0 1
-2 -1 0 1 2
Vsd [mV]
V (mV)
FIG.2: (color online) Differential conductance,G, asa func- FIG. 3: (color online) Differential conductance, dI/dV, as a
tionofbiasvoltage,Vsd,foranInAs-wirebasedquantumdot function of bias voltage, V, for a carbon nanotube quantum
at T = 0.3 K. The data were taken at magnetic fields (per- dot at T = 0.08 K. The data were taken at magnetic fields
pendicular to the wire) B = 0 (thick), 0.1 (dotted), ..., 0.9 (perpendicular to the tube) B = 0 (thick), 0.1 (dotted), 1
T (red) and the curves were offset by 0.008 e2/h for clarity. (thin), 2, 3, ..., 9, 10 T (red) and the curves were offset by
The data were taken for an odd occupied Coulomb diamond 0.008 e2/h for clarity. The data were taken for an odd oc-
at gate voltage Vg =−2.35 V5. cupied Coulomb diamond at gate voltage Vg = −4.96 V8.
(Note that at finite magnetic fields features are broadened
dueto noise induced by themagnet power supply).
Kondo-effect observed in an InAs quantum-wire dot5.
The measured nonlinear conductance curves for varying
The leads are described by the non-interacting Hamilto-
applied magnetic fields are shown in Fig. 2. This exper-
nian
iment constitutes one of the rare cases where such side-
ppleea,kFsigco.u3ldshaocwtsuasilmlyilbaerdreastoalvreedco.rdAesdoonneafusrinthgeler-wexaallmed- Hlead = (εk−µα)c†αkσcαkσ (2)
α=XL,R
carbon nanotube (CNT) quantum dot (QD). Both sets kσ
ofmeasurementsshowwell-definedpeakswhichsplitinto
weakthresholdsonapplyingamagneticfield. Noticethe where α = L,R labels the leads, assumed to be in equi-
different field-strengths needed in the two experiments, librium at chemical potentials µα. The operator c†αkσ
reflecting the roughly 4 times largerspin-orbit enhanced createsanelectroninleadαofmomentumkandspinσ.
g-factor in InAs as compared to the CNT. A simple constant interaction model is used to describe
the quantum dot itself in terms of three non-degenerate
The bulk of this paper deals with leading order
orbitals with one common Coulomb repulsion U(∼E ):
nonequilibrium cotunneling for the 3-orbital Anderson C
model in the Kondo-regime. First we derive an effective
cotunneling, or Kondo model for a system with three Hdot = εnσfn†σfnσ
electrons distributed on the three orbitals. From this ef- Xnσ
fective low-energy model we then proceed to calculate +1U f† f f† f −1 , (3)
the I-V characteristics to leading (second) order in the 2 Xnσ mXσ′ nσ nσ(cid:16) mσ′ mσ′ (cid:17)
cotunneling amplitude. In section III we discuss some of
the salient transport features of this system and in sec- where n,m = 1,2,3 label the orbitals on the dot and
tion IV we discuss the CNT and the InAs data shownin f† creates anelectroninorbitaln with spinσ andwith
nσ
Figs. 2 and 3. In this context we point out additional energy ǫ =ǫ − 1σgµ B where ǫ ≡ǫ +δ, and ǫ =
nσ n 2 B 2 1 3
properties of such samples which can be read off the dI- ǫ +δ′,withlevel-splittingsdenotedbyδ andδ′. Finally,
2
dV characteristic. the tunneling Hamiltonian,
H = t c† f +h.c. ,
tun αn ασ nσ
α=XL,R(cid:0) (cid:1)
nσ
I. EFFECTIVE LOW-ENERGY KONDO MODEL
describes the coupling of the leads to the dot via six in-
dependent tunneling amplitudes t , see illustration in
αn
The 3-orbital Anderson model corresponding to the
Fig. 1. For later convenience, we have introduced the
setup in Fig. 1 we write as local conduction electron operators c† = c† .
ασ k αkσ
We now restrict our attention to the KoPndo-regime in
H =H +H +H . (1) which all three orbitals are sufficiently narrowcompared
lead dot tun
3
and the 4-particle states, that is:
E2e− =2ε1+U >E3e−, (6)
E4e− =2ε1+2ε2+6U >E3e−.
In the following, we shall choose ε = −5/2 U, which
2
places the system at the particle-hole symmetric point
where E4e− −E3e− =E2e− −E3e− =U/2, with E3e− =
E as the ground state.
sσ
In order to eliminate charge-fluctuations from the 3-
orbital Anderson model (1), we employ a generalized
FIG. 4: Illustration of the three lowest lying energy states Schrieffer-Wolff transformation13 which serves to elimi-
thataretakenintoaccountintheeffectivecotunnelingmodel nate the tunneling-term, H , andretains only terms of
tun
of the 3-orbital Anderson model. From left to right: spin, second order in the tunneling-amplitudes in the form of
hole and particle states. an effective cotunneling, or Kondo Hamiltonian:
H =H + E |aσihaσ|+H . (7)
eff lead aσ int
to the charging energy to effectively suppress all charge- Xaσ
fluctuations, leaving the dot with a well defined occupa-
The effective interaction now takes the form of a
tionof3electronsinthreeorbitals. Sincetheexperimen-
spin/orbital exchange-term:
taldatainFigs.2and3donotresolvehigherlyingpeaks
corresponding to excited states with two electrons in or- 1
bital3, weshallsimplify ourmodel further by alsoomit- Hint = Jαaβb (cid:20)2Sab·τσ′σ−Pˆabδσ′σ(cid:21)c†ασ′cβσ, (8)
ting these higher lying 3-particle states. In the experi- αXβ,ab
ments, the cotunneling thresholds for these higher lying σ′σ
states must be comparable to the charging-energy and in terms of the vector of Pauli matrices, τ, the spin-
are therefore masked by charge-fluctuations. As men- operator for the quantum-dot
tioned earlier, this is a typical problem of the relatively
1
large quantum dots (compared to single-molecule junc- Sab = |aσiτσσ′hbσ′|, (9)
tions) which provide a rather poor separationof energy- 2
Xσσ′
scales. Altogether, we are now left with an effective
low-energy Hilbert-space spanned by the six lowest ly- andthepotentialscatteringtermPˆ =Pˆinel+Pˆel which
ab ab ab
ing three-electron states: consistsofaninelasticscatteringwhichinvolvesachange
in the orbital state
|sσi=f2†σf1†↓f1†↑|vaci, Pˆinel =1 δ δ +δ δ −δ δ −δ δ
|hσi=f† f† f† |vaci, (4) ab 4 a,s b,p a,p b,s a,s b,h a,h b,s
2↓ 2↑ 1σ (cid:0) (cid:1)
|pσi=f† f† f† |vaci. × |aσ′ihbσ′|. (10)
3σ 1↓ 1↑ Xσ′
We label these three-body states by indices a,b = and the elastic scattering which occurs via empty levels,
{s,h,p} (i.e. {spin, hole, particle}), together with the i.e. level 3 for |si and |hi and level 2 for |pi,
spin-index σ = {↑,↓}. The notation is obvious from the
1
illustration of the states in Fig. 4. The energies of these Pˆel = δ δ δ +δ δ +δ δ
states are ab 4 a,bXa˜ (cid:0) a˜,s a,p a˜,h a,p a˜,p a,s(cid:1)
× |a˜σ′iha˜σ′|. (11)
1
Esσ =2ε1+ε2+3U − σgµBB, Xσ′
2
1 To lowest order a transition between the excited states
E =ε +2ε +3U − σgµ B, (5)
hσ 1 2 B
2 |pi and |hi is not possible.
1 A constantenergy offsetarisingin the Schrieffer-Wolff
E =2ε +ε +3U − σgµ B.
pσ 1 3 2 B transformation is neglected and so are further potential
scattering terms. These lead only to a constant offset in
In the case of equidistant energy levels E = E and the differential conductance of the order of δ/U2, which
hσ pσ
the two excited states, corresponding to respectively an can be neglected in our qualitative study. Details of
electron moved to orbital 3 or a hole moved to orbital the calculation,including the expressionforthe coupling
1, are degenerate. The energy ε is set by tuning the functions Jab, are given in appendix A.
2 αβ
voltage on a gate and here we choose it such that the Note that when calculating the effective cotunneling
3-particle state will be lower in energy than both the 2, amplitudes in the appendix, we retain the differences
4
in energy-denominators of the different amplitudes, i.e. a simple delta-function:
at this stage we do not make the approximation that
U ≫ δ(′). The according differences in amplitudes ex- Aaa,σ(ω)=2πδ(ω−ωaσ). (15)
presses the broken orbital symmetry of the model, even
with allsix tunneling-amplitudes being equal. Neverthe- The lesser function is found from the quantum Boltz-
less, in order for the Schrieffer-Wolff transformation to mann equation (QBE), or generalized Kadanoff-Baym
be meaningful, we must demand these differences to be equation:
small, and in all our numerical calculations we therefore
Γ (ω)G< (ω)=A (ω)Σ< (ω). (16)
choose a large charging energy (U ∼ 100δ) which effec- aa,σ aa,σ aa,σ aa,σ
tively makes all energy denominators equal. In other
words,retainingδ(′) inthedenominatorsofthecotunnel- Within the delta-function (quasiparticle) approximation
for the spectralfunction, this equation can be solved us-
ingamplitudeswouldrequireamorecarefultreatmentof
ing the following ansatz:
charge-fluctuations.
G< (ω)=in A (ω), (17)
aa,σ aσ aσ
II. NONEQUILIBRIUM PERTURBATION
through which the QBE takes the form of a simple rate-
THEORY
equation.
The conduction electrons are assumed to remain in
When the bias voltageis largeenoughto populate the thermalequilibriumandarethereforecharacterizedsim-
excited states on the dot, these will no longer be ther- ply by their respective chemicalpotentials together with
mally occupied14,15. This effect is incorporated in the a simple flat-band approximation for the momentum-
Keldysh component Dyson equation expressed in terms summed spectral functions, or local conduction electron
of nonequilibrium Green functions16,17 with self-energies density of states (DOS) at the contact,
calculated to leading (second) order in the effective co-
tunneling amplitudes or exchange couplings νFJαaβb ≪1. A(ω)=2πνFθ(D−|ω|), (18)
Following the approach taken in Ref. 15, we employ a
pseudo-fermion representation for the dot-states, with in terms of the DOS at the Fermi surface, ν , and half
F
operators defined by d† |0i ≡ |aσi and h0|d ≡ haσ|, bandwidth D.
aσ aσ
and subject to the constraint Q = d† d = 1.
a,σ aσ aσ
The constraint is enforced with the Paid of a Lagrange
multiplier λ, included as an additional term, H = A. Nonequilibrium occupation numbers
λ
λ d† d , in the Hamiltonian18. The exact projec-
a,σ aσ aσ
tioPntothephysicalHilbertspaceiseffectedbytakingthe In writing the QBE in Eq. (16), we have tacitly as-
limitλ→∞. Weshallneedthenon-equilibriumGreen’s
sumedthe pseudo-fermionself-energiesto be diagonalin
functions:
both spin, and orbital indices. Neglecting spin-orbit in-
teractions spin is a conserved quantum number and the
Gab,σσ′(τ,τ′)=−i TCK daσ(τ)d†bσ′(τ′) (12) self-energy will therefore be diagonal in this index. This
D (cid:0) (cid:1)E does not hold for the orbitalquantum number a and off-
Gαβ,σσ′(τ,τ′)=−iDTCK(cid:0)cασ(τ)c†βσ′(τ′)(cid:1)E (13) dreiaggimoneasltutderiemds,heΣreabf,oσr, wcahnichartihsee.levNeelvseprltithteinlegsss,δ,inδ′tahree
with TCK being the time ordering operator along the assumed to be much larger than level broadenings, Γa,
Keldysh contour. Calligraphic letters denote pseudo- energy conservation suppresses such off-diagonal terms
fermion, italic letters conduction electron Green’s func- andtheselfenergycansafelybeassumedtobediagonal.
tions. For an example in which off-diagonal contributions re-
The retarded and advanced Green’s functions can be mainimportantwerefertheinterestedreadertoRef.19.
calculated like in the equilibrium case directly from the Thus neglecting off-diagonal self-energies and using
Dysonequation. Thepseudo-fermionspectralfunctionis Eq. (17), the QBE in Eq. (16) takes the following form:
obtainedfromtheimaginarypartoftheretardedGreen’s
function, Σ< (ω )
n =i aσ aσ . (19)
aσ
Γ (ω )
aσ aσ
1
A =−2 Im , (14)
aa,σ (cid:20)ω−ω −ΣR (cid:21) The self energy Σ< itself depends on the occupation
aσ aa,σ aσ
numbersandthisthereforeconstitutesasetofsixcoupled
and takes the approximate form of a Lorentzian at the equations for six unknown occupation numbers. These
resonancefrequency ω =ε +λ and ofwidth Γ = equations are underdetermined and should therefore be
aσ aσ aa,σ
−2Im[ΣR ]. Sincethespectralfunctionappearsinlater solved together with the constraint Q = n = 1.
aa,σ aσ aσ
evaluations only in convolution with functions that vary More details of the actual calculation of pPseudo-fermion
onalargerenergyscalethanΓ ,weapproximateitby self-energies are given in appendix B.
aa,σ
5
B. Cotunneling current
Weobtainthecurrentoperatordirectlyfromthetime-
derivative of the density operator at the contact in the
left lead, say (see Ref. 15 and references therein). With
n = c† c , one finds
L σ Lσ Lσ
P
dn ie
L
j =e = [n ,H ]
L dt ~ L int
1
=α,aXb,σ′σ(cid:26)2Sab·τσσ′(cid:16)JLabRc†Lσ′cRσ −JRabLc†Rσ′cLσ(cid:17)
+Pˆabδσ′,σ(cid:16)JLabRc†LσcRσ −JRabLc†RσcLσ(cid:17)(cid:27). (20)
In terms of the correlationfunctions,
Dαspβin(τ,τ′)=−ihTCK JαaβbSab(τ)21τσ′σc†ασ′(τ′)cβσ(τ′)i,hFoIGle.a5n:d p(caorltoicrleonstliantee)s iOncctuhpea3ti-olenvenluAmnbdeerrssoonf mthoedeslpfionr,
aXb,σ′σ symmetric coupling to the leads, tαn = 0.1pU/νF. For
Dpot (τ,τ′)=−ihT JabPˆ (τ)c† (τ′)c (τ′)i, a clear separation of energy scales the temperature is cho-
αβ,σ CK αβ ab ασ βσ sen small T/δ = 0.001 and the levels are clearly separated
Xab δ′/δ = 2. The dashed lines indicate the transition energies
DLtoRt(τ,τ′)=DLspRin(τ,τ′)+DLpoRt(τ,τ′), (21) {±gµBB,δ,δ±gµBB,δ′,δ′±gµBB}in{blue,black,red}for
a finitemagnetic field of B/δ=0.25.
the expectation value of the current is given in lowest
order in the couplings Jab by
LR
A. Occupation numbers
4πe
hj i=− Re Dtot,>(τ,τ)
L h LR
(cid:2) (cid:3)
π 2e ResultsfortheoccupationnumbersareshowninFig.5.
= ν2 Jab Jba
2 h F LR RL Forconvenienceweperformthecalculationswithasmall
(cid:16) (cid:17) aXb,σσ′n thermal smearing. Nevertheless, the temperature is cho-
× 2τσ′στσσ′ + δa,sδb,c+δa,cδb,s senfarsmallerthananyotherenergyscaleintheproblem
(cid:16) c=X{p,h}(cid:0) (cid:1)(cid:17) (T =0.001δ throughout) and all plots can be thought of
as corresponding to T =0. For finite magnetic field and
× naσY(εbσ′ −εaσ−eV)−nbσ′Y(εaσ−εbσ′ +eV) zerobias-voltagethegroundstate,|s↑i,isoccupiedwith
(cid:16) (cid:17) probability one. As illustratedin Fig. 5, the other states
+eV JLaaRJRaaL na˜,σ δa,pδa˜,s+δa,pδa˜,h+δa,sδa˜,p become populated for larger voltages and in the limit
Xa˜ (cid:0) (cid:1)o V →∞ all states will be equally occupied. Inter-orbital
(22) transitions play a role as soon as the voltage becomes
larger than δ or δ′, which is the energy needed to excite
where Y(x) = x nB(x) and nB(x) = 1/(exp[x/2T]−1) respectivelyanelectronintoahigher-lyinglevel,orahole
denotestheBosedistribution. Thesumgoesoverallpos- into a lower-lying level. For equidistant levels (δ = δ′)
sible states a,b={s,h,p},whichareweightedaccording these excitations are identical and we find n =n .
hσ pσ
to the nonequilibrium occupation numbers nbσ′. In Fig. 6 we show the same occupation numbers as in
Fig. 5, but now calculated for different tunnel-couplings
to the left lead. Again, the ground-state is partially
III. DISCUSSION OF RESULTS depleted with increasing voltage, but due to the asym-
metric couplings a clear asymmetry in bias is observed.
Tuning the six different tunneling amplitudes, the two For negative bias, only the p-states are being populated,
different level-spacings and external magnetic field al- due to a strong |si → |pi (cotunneling) amplitude pro-
lowsforalargevarietyofcotunnelingI-Vcharacteristics. portional to t t = 3. The |si → |hi amplitude is
L2 R3
With an eye towards the experiments mentioned in the proportional to t t = 1/3 and therefore only hardly
L1 R2
introductionweshalldiscussafewofthesalientfeatures. pumped at all. For positive bias, the situation is re-
Sincewerestrictourcalculationstoleadingorderpertur- versedsince hp|H |si∝t t =1is nowmuchsmaller
int L3 R2
bationtheory,wedonotaddressthe interestingquestion than hh|H |si ∝ t t = 3, whereby it is n which
int R1 L2 h
of Kondo-correlations in neither the zero-bias peak, nor rises with voltage. Notice also that since hh|H |pi = 0
int
any of the finite-bias conductance peaks. there are no direct transitions between the two excited
6
states, whereby an incipient occupation of |hi, say, will
be accompanied by a deletion of |si and thereby further
prohibit the pumping of |pi.
Fortheasymmetrictunnel-couplingschosenforFig.6,
one can also observe a population inversion at positive
bias. That is, for eV & δ+gµ B, n increases rapidly
B h↑
and stays larger than the ground-state occupation n
s↑
for an extended bias-range. The positive bias demands
electrons to predominantly jump from the left lead onto
the dot and from the dot into the right lead. Therefore,
t ≫t willmakethe|si→|himuchmorelikelythan
L2 L1
the |hi→|si transition at positive bias and the dot gets
stuck in the excited state |hi. As we shall argue later,
thisexcitedstatedoesnotsustainashighacurrentasthe
groundstate and therefore this population inversionwill
infactleadtoregionsofnegativedifferentialconductance
(NDC).
FIG. 6: (color online) Occupation numbers of the spin, hole
The’direct’thresholdforpopulating|h,↑i,say,isgiven and particle states in the 3-level Anderson model for non-
by eV = δ. Nevertheless, from Fig. 6 it is seen that symmetriccouplingtotheleads,i.e.tL2/tRn =3,tL3/tRn =1
nh↑ starts growing already at eV = δ − gµBB. This and tL1/tRn = 1/3 where tRn = 0.1pU/νF. With the same
slightlylower’indirect’thresholdisduetoacascadeeffect finite magnetic field B/δ =0.25, level splitting δ′/δ =2 and
in which |s,↓i is populated for V > gµ B and from temperature T/δ = 0.01 as in Fig. 5 we observe an occupa-
B
therethethresholdto|h,↑iisnowonlyδ−gµBB instead tion inversion around eV ≈ δ+B since nh↑ becomes higher
of δ. Further cascade effects are observed at eV = δ occupied than the ground state ns↑. This blocking of transi-
′
(instead of the direct threshold δ+gµBB) which is the tionsandthecascade effectappearingatδ()−gµBB aredis-
threshold for the transition |s,↓i → |h,↓i. A similar cussedinthemaintext. Gridlinesrefertovariousthresholds:
effect for n is observed at negative bias, with indirect ±gµBB(blue),{δ,δ±gµBB}(black),and{−δ′,−δ′±gµBB}
p,σ
thresholds already at eV = −(δ′ − gµ B) and eV = (red).
B
−δ′. Notice that these effects are just barely visible for
symmetric couplings as in Fig. 5. As we shall see in the
next subsection, these cascade effects show up as small showsextrastepsbothateV =δ−gµBB(|s,↓i→|h,↑i)
and at eV =δ′−gµ B (|s,↓i→|p,↑i). Notice that the
steps in the nonlinear conductance. B
cascade effect is only visible for B 6= 0, and disappears
when 2gµ B > δ(δ′) since n (n ) is then occupied
B h,↑ p,↑
before n .
B. Cotunneling conductance s,↓
Moreinvestigationsarenecessaryinclarifyingtowhat
extent these cascade-features in the conductance will be
In Fig. 7 we show the differential conductance for the
enhanced by the Kondo-effect, but even a small addi-
same parameter set as in Fig. 5. As expected, we find
tional step as seen in Fig. 7 can have important bear-
a step in dI/dV whenever a new level enters the volt-
ings for interpreting experiments in which the internal
age window. Furthermore, as pointed out in Refs. 15,20,
dot, or molecule states are not known in advance. Judg-
thevoltage-dependenceofthenonequilibriumoccupation
ing from the magnetic field dependence, the three tran-
numbers promotes these conductance steps to pointed
sitions at eV = δ−gµ B,δ,δ+gµ B seen in Fig. 7
cusps. Since we are limiting our calculations to second B B
(imagineforamomentthatδ′ ≫δ)couldinfactbemis-
order perturbation theory, these plots do not take into
interpreted as direct transitions between a spin-doublet
accountthe possible Kondo-enhancementofthese cusps.
ground state and an excited S=3/2 state, as illustrated
Nevertheless, the tendency is known from previous cal-
by the energy-diagrams in Fig. 8.23 This example shows
culations: each cusp in the differential conductance to
that a spin-spectrum read off from B-dependent inelas-
secondorderislogarithmicallyenhanced15 andtheselog-
tic cotunneling-lines in a diamond-plot (dI/dV vs. V
arithmicdivergencesarecontainedroughlybytheinverse g
andV )shouldbeinterpretedwithsomecare,especially
life-time of the excited state involved in the relevant in- sd
when enhancing the lines by plotting higher derivatives
elastic cotunneling process21,22.
of the current.
1. Cascade induced side-peaks
2. Negative differential conductance
EvenforthesymmetriccouplingsinFig.7,weobserve
extra steps reflecting the aforementioned cascade effect. In Fig. 9 we plot the conductance for the parameters
For example, the red(dashed) curve with gµ B =0.25δ used in Fig. 6 and with varying values for the tunneling
B
7
FIG. 9: (color online) Differential conductance dI/dV of the
FIG. 7: (color online) Dependence of the differential con- non-symmetric3-levelAndersonmodelwith3electronforthe
ductance dI/dV on the applied bias voltage in units of parameters given in Fig. 6. The four different set of param-
eV/δ for various strengths of the magnetic field gµBB/δ = eters correspond to the ratio of tL2/tRn = {3,2,1.5,1} with
0,0.25,0.5,0.75. The quantum dot is coupled symmetrically tL1/tRn =1/(tL2/tRn)={1/3,1/2,2/3,1},respectively. The
totheleadwith theparametersgiveninFig.5. Thekinksat parameter set with tL2/tRn = 3 and tL1/tRn = 1/3 corre-
±(δ−gµBB)and±(δ′−gµBB)originatefromtheoccupation sponding to Fig. 6 shows NDC at the voltage values corre-
of excited states dueto the cascade effect. sponding toan inversion of occupation probabilities.
amplitudes fromthe left leadto orbitals 1 and 2,respec- strong coupling t = 3. In contrast, starting in |si one
L2
tively. As inFig.6,weobserveapronouncedasymmetry can also fill in an electron in orbital 2 from the left lead
in bias-voltage, and with the largest difference in cou- first and then tunnel out into the right lead from either
plings toorbitals1and2 (black/solidcurve),weobserve orbital 2 or from orbital 1, thus allowing more current-
a sudden drop to negative differential conductance for carrying tunneling processes involving t =3.
L2
eV &δ. As mentioned before, the NDC is driven by the Thusa sufficiently strongdifference incouplingto two
population inversion between |si and |hi. As argued in of the orbitals will cause a population inversion with a
the previoussection,the positive biasdrivesthe dotinto concomitant decrease in the current as bias is increased.
the excited state |hi, and since (for these parameters) For intermediate coupling asymmetry, we see from the
this state leads to a lower current than |si, the current other curves in Fig. 9 that the NDC disappears while an
decreasesandweobservetheNDC.Thepoorcotunneling asymmetry in bias remains.
acrossthe dot in the state |hi is a simple consequence of
thedoubleoccupancyofthewell-coupledorbital2,which
demands that orbital 2 is emptied to the right lead be-
IV. DISCUSSION OF EXPERIMENTS
fore it can be filled from the left, thus making use of the
A. Experiments on Carbon Nanotube
S=1/2 S=3/2 The data shown in Fig. 3 were recorded on the same
3/2 sample as was discussed in Ref. 8. In fact, this is the
neighboring charge-state to the even number occupied
1/2 1/2
statestudiedinthatwork. InRef.8,someofthepresent
-1/2 -1/2 authors obtained a very gratifying fit to the inelastic co-
-3/2 tunneling line reflecting a strong transition from singlet
δ δ
ground- to an excited triplet state. In particular, it was
1/2 1/2 argued that the very sharp cotunneling line could only
-1/2 -1/2 arise from the joint effect of nonequilibrium pumping
(finite-biasoccupations)ofthetripletstateandsubstan-
tial logarithmic enhancements from the nonequilibrium
FIG. 8: Comparison of direct and indirect transitions with
a doublet groundstate and an excited doublet or quartet, re- Kondo-effect.
spectively,atfinitemagneticfield. Cascadeeffectprovidesan As already stated, we shall not embark on higher
indirect transition in the former, at energy δ−gµBB, which order perturbation theory in this paper, which means
matches a direct transition in thelatter. that we should not expect to obtain any quantita-
8
(b) 3 (c)
0.25T 4
V]
m
[sd0 2
V
V]
-3 m
(a) 3 [d0
0T s
V] V
m
[d0 -2
s
V
-4
-3
-2.5 -2.2
Vg [V] Vg
FIG.11: (a)and(b)PlotofdI/dV vs. gate,andbias-voltage,
for the InAs-wire quantum dot with B = 0 T and 0.25 T,
respectively. Strong tunnel-coupling perturb the Coulomb-
blockade diamond and particularly strong sequential tunnel-
ing lines are clearly present (thick dark lines) in the region
with a mixture of 2 and 3 electrons on the dot. c) Calcula-
FIG. 10: Modeling of the CNT QD analyzed in Ref. 8
tion of the sequential tunneling lines from solving semiclas-
as measured in the neighboring diamond, i.e. Fig. 3,
sical rate-equations (conductance in arbitrary units) bring-
where orbital scattering is observed at δ = 1.5 meV.
ing out some of the main lines and indicating the states
Further parameters of the plot are {tL1,tL2,tR1,tR2} =
which are involved in the dominant transitions and which
{0.032,0.028,0.108,0.063}pU/νF, B = 0,0.14,...1.4 meV,
are involved in the cotunneling for N = 3. Parameters are
and T = 81mK ∼ 0.008meV. The conductance is given in
δ = 1.4 meV, δ′ = 1.1 meV, T = 300mK, U = 3 meV,
arbitrary unitsand thecurvesare offset for clarity.
tL1 =tR1=tL3 =tR3 =1, tL2=0.4, and tR2 =2.5.
tive agreement with the data in Fig. 3. Nevertheless,
strong peak close to eV = −1.4 meV together with
since basically all parameters were fixed by the fit in
a somewhat weaker peak close to eV = 1.1 meV
Ref. 8, we shall assume that these remain largely un-
(cf. Fig. 2). This asymmetry in bias-voltage was not
changedwhenpassingontotheneighboringcharge-state
addressed in Ref. 5, but as we shall argue it can read-
and use them as input for a numerical evaluation of
ilybe understoodintermsofanasymmetryintunneling
the nonlinear conductance there. The result is shown
amplitudes of orbital 2 to source and drain. The excited
in Fig. 10 and the gross features such as bias-voltage
state giving the strong sequential tunneling line in the
asymmetry and B-field splittings match those of Fig. 3
upper left corner of Fig. 11 (a) is ascribed to the excited
quite well, albeit with a complete lack of sharp finite-
3-particle state with one electron in orbital 3 instead of
bias peaks, as expected. The parameters from Ref. 8
orbital 2. This line connects to the inelastic cotunneling
were tunnel couplings of the two lowest lying levels of
line at positive bias inside the N=3 diamond (|si → |pi
{t ,t ,t ,t } = {0.032,0.028,0.108,0.063} U/ν
L1 L2 R1 R2 F transition). From this we infer that δ′ ≈ 1.1 meV. No-
(see Fig. 4 in Ref. 8), an orbital splitting of δp≈ 1.5
tice that a sequential tunneling line corresponding to a
meV, a charging energy of U ≈ 3.0 meV, and a higher
transition from the N=2 ground-state to the N=3 |hi-
lying level at δ′ =∆−δ ≈3.1 meV, ∆ being the single-
state is only possible to higher order in the tunneling
particle level spacing in the nanotube. Since δ′ & U,
amplitudes and is therefore strongly suppressed in the
charge-excitations set in before orbital 3 is reached and
experimentaldata. Theinelasticcotunnelinglineatneg-
this latter orbital is therefore not included in the calcu-
ative bias can now be ascribed to an (N=3) |si → |hi
lation.
transition at energy δ ≈ 1.4 meV. This is again consis-
tentwiththelowerleftcornerofFig.11(a)showingthat
this line connects to a very strong sequential tunneling
B. Experiments on InAs-nanowires
linewhichmustcorrespondtoatransitionfromtheN=4
ground-state to the N=3 |hi-state. At this point it is
The data shown in Fig. 2 constitute the first observa- the (N=3) |pi-state which is not directly coupled to the
tionofKondo-effectinInAs-nanowirequantumdotsand N=4 ground-state and hence suppressed in the data. In
were discussed by some of the present authors in Ref. 5. Fig.11(c),weshowtheresultofsolvingasetofsemiclas-
Alreadyinthatpaperthe3-orbitalmodelwasinvokedto sical rate-equations. The dark lines correspond to high
explainthefinite-biaspeaksflankingthezero-biasKondo conductance (arbitrary units) due to sequential tunnel-
peak and, as mentioned in the introduction, this exper- ing only. We do not attempt a 6 parameterfit of Fig. 11
iment was the main motivation for this more detailed (a), and tunneling amplitudes are therefore chosen with
investigation of that model. a very simple asymmetry, assuming that only orbital 2
Apart from the zero-bias Kondo peak, it is the sharp is coupled in a special way. This choice of parameters
finite-bias peaks which dominate these data: A very captures some, but not all, of the gross features of the
9
FIG. 12: (color online) Plot of dI/dV vs. bias voltage, FIG. 13: (color online) Plot of dI/dV vs. bias voltage, V,
V, calculated within second order perturbation theory. Pa- calculated within second order perturbation theory for sym-
rameters are chosen so as to approximately match the volt- metriccoupling totheleads andneglecting theparticle state
age asymmetry brought out by the experiment (compare (δ′ > EC). The g-factors are different for the two involved
Fig. 11). Parameters are B = 0, δ = 1.4 meV, δ′ = 1.1 levels: g = 2g . The orbital peak moves towards higher
2 1
meV and T = 0.001 meV (blue/solid curve). The black voltages suchas ∼(1−g1/g2)g2µBB/2 with increasing mag-
(dashed-dotted)curvecorrespondstoallequaltunnelingam- netic field g2µBB/δ = {0,0.25,0.75}. The red (dashed)
plitudes and the red (dashed) curve is the same as the blue curve at g2µBB/δ = 0.75 illustrates that the shift is also
(solid)butwithtemperaturecorrespondingtotheexperiment observableatatemperaturecorrespondingtotheexperiment
(T =300mK∼0.025meV). (T =300mK∼0.025 meV).
exp exp
that two neighboring levels could have g-factors of 1.9
data in Fig. 11 (a).
and 10, respectively. It is interesting to note that such a
The question remains why there is only one and not
difference in g-factors would give rise to an apparent B-
twoinelasticcotunnelingpeaksatbothpositiveandneg-
dependenceofthelevel-spacings. Inourmodel,wewould
ative bias. As demonstratedby the calculationshown in
expect the following B-dependent inelastic cotunneling
Fig.12,however,theobservedasymmetryinbias-voltage
thresholds:
can be understood quite simply as orbital 2 being asym-
metrically coupled to the source and drain electrodes. δ˜(B)=E −E =δ+(g −g )µ B/2,
h↑ s↑ 2 1 B
NoticethatitisthenearlyequidistantlevelsintheInAs- δ˜′(B)=E −E =δ′+(g −g )µ B/2.
wire quantum dot which prompts us to incorporate the p↑ s↑ 2 3 B
effects of both h and p excited states. For asymmetric InRef.5,a detailedinvestigationofthe splitting ofboth
couplings the δ′ = 1.1meV and δ = 1.4meV peaks in thezero-biasKondopeakandthefinite(positive)-biasco-
Fig. 12 dominate for positive and negative voltages, re- tunneling peak with magnetic field indeed revealed two
spectively,incontrasttosymmetriccouplingwhereboth different g-factors. Interpreting the finite-bias cotunnel-
can be observed. The small steps observed for zero tem- ingasabove,we canthus infer fromRef.5 thatorbital2
peratue (blue/solid curve) are washed out for large tem- has g = 7.7 and orbital 3 has g = 8.5. This difference
2 3
perature smearing (red/dashed curve) and the nonlinear is rather small and would cause a largely negligible shift
conductance is now asymmetric with respect to voltage. of roughly 0.02 meV of δ′ at largest applied fields (0.9
Foralargerdifferenceintheeffectiveδ andδ′,asistypi- Tesla). In Fig. 13 we show the result of a calculation
calforcarbonnanotube quantumdots whereδ wouldbe with slightly different g-factors by a factor of 2. This is
the subband-splittingandδ′ wouldbe ofthe orderofthe the cause of the slight shift of the main kink in the blue
single-particle level-spacing (∆ = δ′ +δ), the |pi state (solid)curvetoavalueaboveeV =δ. Theobservationof
would most often be masked by charge-fluctuations, i.e. such B-dependent cotunneling thresholds thus provides
δ′ will often be comparable to the charging-energy,EC. an interesting consistency check on the difference in g-
AsdemonstratedrecentlyintheexperimentbyCsonka factors of two neighboring orbitals.
et al.10, the strong spin-orbit coupling in InAs can give Inclosing,wenotethatthediamond-plotintheupper
rise to very different g-factors for neighboring quantum left panel of Fig. 11 has a few strong-coupling irregu-
dot orbitals or energy-levels. In that experiment this larities which we do not address in this paper. First of
was brought out particularly clear by a modulation of all,the inelastic cotunneling lines display a slightdepen-
the orbitals by an additional top-gate. Fixing the top- dence on gate-voltage. This is a feature which was in-
gate voltage and adjusting the back-gate, it was shown vestigated in detail for a carbon nanotube quantum dot
10
in Ref. 9, where it was explained in terms of second or- orbitals. In the present experiment this difference was
der tunneling-renormalizationwith energydenominators too smallto give a noticeable effect, but with differences
depending on gate-voltage. Second, the inelastic cotun- as quoted in Ref. 10 it should be a pronounced effect24.
neling line at positive bias, which we assigned to the ex- Possiblefurtherinfluenceofstrongspin-orbitcouplingon
cited p-state, appears to continue straight through the thecotunnelingspectroscopyofinelasticspinandorbital
sequentialtunnelingregionmixingN=3andN=4states. transitionsconstitutes an interestingproblem onits own
At this point we merely speculate that this is related to right and will be addressed in a separate publication.
a very strongly coupled orbital 3, which might allow the
partially populated N=4 states to conduct via virtual
transitions to the N=5 state. How, and if, this would Acknowledgments
work out in detail will be assessed in future work.
We thank M. Mason, L. DiCarlo, C. M. Marcus
(Harvard University), M. Aagesen, and C. B. Sørensen
V. CONCLUSIONS (Niels Bohr Institute) for contributions to the experi-
ments. Furthermorewe liketo thank S. Csonka,M.Trif,
In this paper we have analyzed the inelastic cotunnel- D. Stepanenko, V. Golovach and B. Trauzettel for valu-
ing spectroscopy of a system with three electrons dis- able discussions.
tributed among three orbitals. Starting from the 3- This work has been supported by the DFG-Center for
orbitalAndersonmodelintheCoulombblockaderegime, Functional Nanostructures (CFN) at the University of
wehaveprojectedoutcharge-fluctuationsandderivedan Karlsruhe(S.Sch.andP.W.),theInstituteforNanotech-
effectivecotunneling(Kondo)modeldescribingelasticas nology, Research Center Karlsruhe (P. W.), the Danish
well as inelastic tunneling induced transitions between Agency for Science, Technology and Innovation and the
different orbital and spin states of the impurity or quan- DanishNaturalScienceResearchCouncil(J.P.,T.S.J.,
tum dot in the N=3 charge-state. By treating all tunnel J. N.), the Swiss NSF and the NCCR Nanoscience
couplingsindependentlywecanstudyasymmetricbehav- (V. K.).
ior inherent to most of the experimental samples.
All calculations were performed within second order
APPENDIX A: SCHRIEFFER-WOLFF
nonequilibriumperturbationtheory,andthe bias depen-
TRANSFORMATION AND COUPLING
dent nonequilibrium occupations of the impurity-states
CONSTANTS
were shown to give rise to marked cusps at voltages
matching the different transitions. We demonstrate how
a bias induced cascade in the occupations can lead to In this work we concentrate on the physics inside a
Coulomb diamond where the 3-level quantum dot is oc-
indirect transitions between two excited states. Further-
cupied by exactly three electrons. All virtual processes
more, we find that certain asymmetries in the tunneling
amplitudes of the three orbitals can lead to a population to two or four electron states can be treated perturba-
inversion, which in turn can be accompanied by lines of tively and leadto a Kondo-likeinteractionandpotential
negative differential (cotunneling) conductance (NDC). scattering terms.
Using a Schrieffer-Wolff transformation13 we find to
We have revisited the carbon nanotube data from
second order in the tunneling Hamiltonian H the in-
Ref. 8 with focus on an odd occupied charge-state. Im- tun
teraction Hamiltonian
porting the tunnel-couplings inferred from Ref. 8, we
confirm the asymmetry in the conductance strength at 1
pstoespitsivaet fiannidte-nbeigaastaivtethbeiaos.rbiTtahletsreanpsaitriaomnse,tearnsdlewaedbteo- Hint =α,Xβ;n,mntαmt∗βnc†ασfmσEaσ−E4e−fn†σ′cβσ′
lieve that the experimentally observed peaks are mainly σ,σ′
1
due to logarithmic enhancements from a nonequilibrium +t∗ t f† c c† f .
Kondo-effect. αm βn mσ ασEaσ −E2e− βσ′ nσ′o
We have also analyzed the InAs-wire data from Ref. 5
which can be rewritten as
ingreaterdetailandprovidedanexplanationofthebias-
asymmetry in peak-positions. This was understood as a t t∗ t∗ t
slightdifferenceinthetwolevel-spacingstogetherwithan Hint = (cid:26)εαm+3βUn − εβn+α2mU(cid:27)c†ασcβσ′fn†σ′fmσ
asymmetry in the tunnel-coupling of orbital 2 to source α,Xβ;n,m n m
and drain electrodes. This asymmetry in couplings was σ,σ′
rthouegNh=ly3codniasmistoenndt,waisthcotnhfiermoveedrabllysatasbeimlitiycladsisaigcraalmraftoer- − δn,mδσσ′εtαm+t3∗βUn c†ασcβσ′ (A1)
α,Xβ;n,m n
equation calculation. Finally, we have pointed to an ad- σ,σ′
ditional consequence of strong spin-orbit coupling in the
InAs-wires, namely the apparent B-dependence of the The Kondo exchange interaction for the levels n,m =
level-spacing caused by different g-factors for different {1,2,3}andthusfortheorbitalstatesa,b={s,h,p}can