Conductance via Multi orbital Kondo Effect in Single Quantum Dot

0.17 MB·English

by  Rui Sakano

Conductance via Multi orbital Kondo Effect in Single Quantum Dot
Rui Sakano and Norio Kawakami
Department of Applied Physics, Osaka University,
Suita, Osaka 565-0871, Japan
6 (Dated: February 6, 2008)
0
0 We study the Kondo effect in a single quantum dot system with two or three orbitals by using
2 the Bethe-ansatz exact solution at zero temperature and the non-crossing approximation at finite
temperatures. Forthetwo-orbitalKondoeffect,theconductanceisshowntobeconstantatabsolute
n
zeroinanymagneticfields,butdecreasemonotonicallywithincreasingfieldsatfinitetemperatures.
a
In the case with more orbitals, the conductance increases at absolute zero, while it features a
J
maximumstructureasafunctionofthemagneticfieldatfinitetemperatures. Wediscusshowthese
5
characteristic transport properties come from themulti orbital Kondo effect in magnetic fields.
2

PACSnumbers: 73.63.Kv,73.23.-b,71.27.+a,75.30.Mb
]
l
l
a
h I. INTRODUCTION from each other: the decrease of the conductance in the
- two-orbital case is due to the decrease of the effective
s
Kondo temperature, in contrast to the ordinary SU(2)
e TheKondoeffectisatypicalelectroncorrelationprob-
m lem, which was originally discovered in dilute magnetic Kondoeffect. Moreover,we showthat inthree- ormore-
orbital cases, the conductance even features a maximum
. alloys as a resistance minimum phenomenon [1, 2]. Al-
t structureinmagneticfields,whichisnotexpectedinthe
a though the essence of the Kondo effect was already clar-
m ified, it is quite interesting to systematically study the ordinary spin Kondo effect.
This paper is organized as follows. In Sec. II, we in-
- Kondo effect in artificial systems. Recently, the Kondo
d effect observedin quantum dot systems, which have rich troduce a generalized impurity Anderson model for our
n multi-orbital quantum dot system, and outline how to
tunable parameters [3, 4], has attracted considerable at-
o treat transport properties by means of the Bethe-ansatz
tention [5, 6, 7]. This has stimulated intensive investiga-
c
exactsolutionatabsolutezeroandtheNCAmethodatfi-
[ tions on electron correlations. Indeed, a lot of the work
nitetemperatures. Weinvestigatethetwo-orbitalKondo
have been done both theoretically and experimentally.
1 effectinSec. III,andthenmovetothecasewithmoreor-
Among intensive studies, the Kondo effect due to the
v bitalsinSec. IVbytakingthethree-orbitalKondoeffect
8 orbitaldegreesoffreedomis ofparticularinterest,where
as an example. We discuss characteristic magnetic-field
9 effectiveorbitalsaregivenbyahighlysymmetricshapeof
dependence of the conductance due to orbital effects at
5 thedot,multiplycoupledquantumdots,etc[8,9,10,11,
1 12, 13, 14]. In such systems, not only the ordinary spin zero and finite temperatures. A brief summary is given
0 Kondo effect [15, 16] but also the orbital Kondo effect inSec. V,wherethemulti-orbitalKondoeffectsarecom-
6 paredwith the ordinarysingle-orbitalspin Kondo effect.
due to the interplay between orbital and spin degrees of
0
freedomoccurs;theKondoeffectinmultiple-dotsystems
/
t [17, 18, 19, 20, 21, 22, 23, 24, 25, 26], the singlet-triplet
a
m Kondoeffect[27,28,29,30],andthe SU(4)Kondoeffect II. MODEL AND METHOD
in single-dot systems [31, 32, 33, 34, 35].
-
d Inthispaper,wefocusontheKondoeffectinasingle- A. Multi orbital quantum dot
n dotsystemwithmultipleorbitals,andexplorecharacter-
o
istictransportpropertiesatzeroandfinitetemperatures. Let us consider a single quantum dot system with N-
c
: Tothisend,weusetheBethe-ansatzexactsolutionofthe degenerateorbitalsinequilibriumstate. AsshowninFig.
v impurity Anderson model and the non-crossing approxi- 1,weassumethateachenergy-levelsplittingbetweenthe
i
X mation (NCA). To be specific, we deal with a quantum orbitals is ∆orb in the presence of magnetic field and the
dotsystemwithtwoorthreeorbitalsasatypicalexample Zeemansplitting is much smaller than it, so that we can
r
a of the multi-orbital Kondo effect. ignore the Zeeman effect: the degenerate levels are split
Experimentally, the two-orbital Kondo effect (SU(4) into N spin doublets. The energy level of each orbital
Kondo effect) in single-dotsystems has already been ob- state is specified as
served in transport properties, where the conductance
shows similar field-dependence to the usual SU(2) spin ε =ε +l∆ (1)
σl c orb
Kondoeffect: inbothcasestheconductanceismonotoni- N −1 N −3 N −1
(l =− ,− ,··· , )
callysuppressedinthepresenceofmagneticfields[31,34].
2 2 2
One of the main purposes of the preset paper is to
demonstrate that although the field-dependent conduc- where ε denotes the centre of the energy levels and l(σ)
c
tance shows analogous behavior, its origin is different represents orbital(spin) index.
2
This type of orbital splitting has been experimen- rule, we can evaluate the zero-bias conductance in the
tally realized as Fock-Darwin states in vertical quantum Kondo regime at absolute zero [39, 40],
dot systems [8, 36, 37] or clockwise & counter-clockwise
2e2
states in carbon nanotube quantum dot systems [34], G= sin2(πn ), (5)
σl
where the orbitalsplittings are proportionalto the mag- h Xl
nitude of magnetic fields.
wherethe occupationnumberoftheelectronn isgiven
σl
by the Bethe-ansatz solution.
dot
C. NCA method
lead lead
∆ The above technique to calculate the conductance
ε orb by the exact solution cannot be applied to the finite-
c
∆ temperature case. Therefore, we complimentarily make
orb
use of the NCA for the analysis at finite temperatures
[2, 41]. The NCA is a self-consistent perturbation the-
ory, which summarizes a specific series of expansions in
FIG. 1: Energy-level scheme of a single quantum dot with
threeorbitals coupled to two leads. thehybridizationV. Thismethodisknowntogivephys-
ically sensible results at temperatures around or higher
thanthe Kondotemperature. Infact, itwas successfully
applied to the Ce and Yb impurity problem mentioned
above, for which orbital degrees of freedom play an im-
B. Bethe-ansatz exact solution
portantrole[42,43,44,45]. Therefore,bycombiningthe
exact results at zero temperature with the NCA results
We model the above quantum dot system by assum- atfinitetemperatures,wecandeducecorrectbehaviorof
ing that the Coulomb repulsion U between electrons is the conductance for the model (4).
sufficiently strong at the quantum dot, so that we ef- The NCA basic equations can be obtained in terms of
fectively put U → ∞. We focus on the N-degenerate coupledequationsfortwotypesofself-energiesΣ (z)and
0
orbital states in the quantum dot, which are assumed to Σ (z) of the resolvents R (z)=1/(z−ε −Σ (z)),
σl α α α
hybridize with the correspondingconductionchannels in
theleadsforsimplicity. Intheseassumptions,oursystem 2Γ D f(ε)
Σ (z) = dε ,(6)
0
canbe describedby anSU(N) multi-orbitalextensionof π Z z−ε +ε−Σ (z+ε)
Xl D σl σl
the impurity Anderson model in the strong correlation
Γ D 1−f(ε)
limit (U →∞). The Hamiltonian reads Σ (z) = dε , (7)
σl
π Z z−ε +ε−Σ (z+ε)
D 0 0
H = H +H , (2)
0 I for the flat conduction band of width 2D. At finite tem-
H = ε |σlihσl|+ ε c† c , (3) peratures, the conductance is given as [46],
0 σl k kσl kσl
Xσ,l Xkσl 2e2 df(ε)
G= Γ dε − A (ε), (8)
HI = (Vk|σlih0|ckσl +h.c.), (4) h Xl Z (cid:18) dε (cid:19) σl
Xkσl
where f(ε) is the Fermi distribution function and A (ε)
σl
where c† (c ) creates (annihilates) a conduction elec- is the one-particle spectral function including the effect
kσl kσl
tron with wavenumber k, spin σ(= ±1/2) and orbital l. of the above self-energies.
An electron state at the dot is expressed as |σli, which
is supplemented by the unoccupied state |0i.
III. TWO-ORBITAL KONDO EFFECT
In spite of its simple appearance, the Hamiltonian (4)
is difficult to treat generally because of strong correla-
tions coming from the U → ∞ condition. Fortunately, Webeginwithaquantumdotsystemwithtwoorbitals
thismodelwasextensivelystudiedtwodecadesagointhe (N =2),whereitisassumedthattheorbitaldegeneracy
contextofelectroncorrelationsfor Ce andYb rare-earth is lifted by a splitting ∆orb while the spindegeneracyre-
impuritysystems. Inparticular,theexactsolutionofthe mains in the presence of magnetic field. As mentioned
model was obtained by the Bethe ansatz method under above, the two-orbital Kondo effect, often referred to as
the condition that the density of states for electrons in the doublet-doublet or SU(4) Kondo effect, has been ex-
the leads is constant (wide-band limit) [38]. This solu- perimentallyobservedinverticaldotsystemsandcarbon
tion enables us to exactly compute the static quantities, nanotube dot systems [31, 34]. We will be mainly con-
but not transport quantities in a direct way. However, cerned with the Kondo regime, where the characteristic
by combining the Landauer formulaand the Friedelsum energy scale is given by the Kondo temperature TSU(4).
K
3
0.6
A. Exact results at absolute zero
0.5
0.4
(a) h)
1 2/ 0.3
e
2
0.8 total G/( 0.2
l=-1/2
h) 0.6 l= 1/2 0.1
2/
e
2 0
G/( 0.4 0 0.2 0.4 0.6 0.8 1
0.2 ∆ /(Γ/π)
orb
0
0 1 2 3 4 5 FIG. 3: Finite-temperature conductance computed by the
∆ /TSU(4) NCAasafunctionoftheenergysplittingfortheSU(4)model
orb K in theKondoregime: kBT =0.1Γ/π whereΓ is thebareres-
(b)
1 onance width. The energy level of the dot is ǫc = −20Γ/π,
which gives theSU(4) Kondotemperature TSU(4) ∼0.1Γ.
K
0.8
4)
U(
SK 0.6 of the orbital splitting. As seen in Fig. 2(b), it mono-
T
eff/ K 0.4 tonically decreases, and is inversely proportional to the
T orbitalsplittingintheasymptoticregion: Teff ∼1/∆
K orb
0.2 [22, 33, 47, 48].
0
0 1 2 3 4 5
B. Analysis at finite temperatures
∆ /TSU(4)
orb K
Itisremarkablethattheconductancedoesnotdepend
FIG. 2: (a) Zero-temperature conductance calculated by the on magnetic fields at absolute zero, however, we should
exact solution as a function of the energy splitting for the recallthattheconductanceisexperimentallyobservedat
SU(4) Anderson model in the Kondo regime. The total con- finite temperatures. Since the Kondo temperature pro-
ductance is constant, although the contributions from two
vides the characteristic energy scale of the system, the
orbitals are changed monotonically. (b) The effective Kondo
decreaseof the effective Kondo temperature shouldhave
temperature Teff.
K a tendency to suppress the Kondo effect at a given finite
temperature. Therefore, we naturally expect from the
It is known that the conductance for the SU(4) case zero-temperature analysis that the conductance mono-
(∆ =0) andthe SU(2) case(∆ =∞) is both given tonically decreases with increasing magnetic fields at a
orb orb
by 2e2/h at absolute zero. However, the conductance finite temperature. This can be confirmed by the NCA
in the intermediate coupling region (∆ = finite) has numerical calculations at finite temperatures. The re-
orb
not been discussed so far. The calculated conductance sults are shown in Fig. 3, from which we indeed see the
for each orbital is shown in Fig. 2(a), which changes monotonic decrease of the conductance. The obtained
smoothly around the Kondo temperature TSU(4). An results qualitatively agree with the experimental obser-
K vation of the SU(4) Kondo effect in the quantum dot
important point is that the total conductance is al-
ways 2e2/h, which is independent of magnetic fields. systems [31, 34]. We wish to emphasize again that the
magnetic-field dependence found here is different from
In fact, by putting the occupation number n =
σ,±1/2
that discussed so far for the ordinary spin Kondo ef-
1/4∓δn(∆ ) into Eq. (5), we can readily derive the
orb
fect without orbitals: the decrease of the conductance
constant conductance in the presence of magnetic fields.
is caused by the decrease of the Kondo temperature in
This result is characteristic of the SU(4) case, which is
our scenario for the SU(4) case.
consistent with those deduced for the double-dot system
with a special setup [18].
It should be noted here that there is another impor-
IV. THREE-ORBITAL KONDO EFFECT
tant effect due to magnetic fields: the effective Kondo
temperature changes with magnetic fields. The effective
Kondo temperature, which is defined by the inverse of In the previous section, we have considered the con-
the spin susceptibility for the lowest doublet, gives the ductance for the two-orbital Kondo effect. We now gen-
width of the effective Kondo resonance in the presence eralizeourtreatmenttothemultiorbitalcaseswithmore
4
than two orbitals. In contrast to the two-orbital case, energy splitting (or magnetic field), and approaches the
where the conductance at finite temperatures shows be- valueof2e2/hforlargesplittings(SU(2)case). Itisnote-
haviorsimilar to the usual spin Kondoeffect, we will see worthy that the conductance is enhanced by the mag-
that the conductance for three- or more- orbital systems netic field in contrast to the ordinary spin Kondo effect
shows qualitatively different behavior in the magnetic- withoutorbitaldegreesoffreedom,forwhichtheconduc-
field dependence. This provides us with an efficient way tanceissimplysuppressedbythemagneticfield. Inmore
tocharacterizethemulti-orbitalKondoeffectexperimen- generalcases with N orbitals,the conductance increases
tally. Here, we discuss the characteristictransportprop- from (2Ne2/h)sin2(π/(2N)) to 2e2/h with the increase
erties by taking the three-orbital Kondo effect (N = 3) of the magnetic field since the electron number of the
as an example. lowest-orbital state changes from 1/(2N) (zero field) to
1/2(largefields). Thisanalysisatzerotemperature nat-
urally leads us to predict the following behaviorat finite
temperatures. Since the magnetic field decreases the ef-
A. Exact results at absolute zero
fective Kondo temperature, there appears the competi-
tionbetweenenhancementandsuppressionoftheKondo
Exploiting the Bethe-ansatz exact solution and the
effect at finite temperatures. As a consequence, a max-
Landauerformula,wecalculatetheconductanceatT =0
imum structure should appear in the conductance as a
forthesingle-dotsystemwiththreeorbitalsintheKondo
function of the magnetic field in generalcases with more
regime. We note that Schlottmann studied a similar
than two orbitals. This unique feature distinguishes the
model for Ce-impurity systems in a crystalline field to
three- or more-orbital Kondo effect from the two-orbital
discuss its magnetic properties [49]. The conductance
Kondo effect.
(a)
1
total
0.8 l =-1 B. Analysis at finite temperatures
l = 0
h) l = 1
2e/ 0.6 Here,wewishtoconfirmtheconductancemaximumat
2
G/( 0.4 finite temperatures predicted from the zero-temperature
analysis. The conductance for the three-orbital Kondo
0.2 effect computed by the NCA at finite temperatures is
shown in Fig. 5. At low temperatures, the predicted
0
maximum structure indeed emerges in the conductance
0 0.5 1 1.5 2 2.5 3 3.5
as a function of the energy splitting. It is seen that the
∆orb/TSKU(6) height of the maximum grows and its position gradually
(b)
1 shifts towardthe regionwith larger splitting ∆ as the
orb
temperature decreases. The shift reflects the fact that
0.8 the suppression of the Kondo effect due to the decrease
SU(6)K 0.6 olofwtehreteKmopnedroattuermeps,e.rature becomes somewat weaker at
T
eff/ K 0.4 To see how the conductance maximum structure
T emerges for the three-orbital Kondo effect, we show the
0.2 one-particle spectral density A(ω) computed at k T =
B
0.083Γ/π in Fig. 6. For the case of ∆ = 0 (three de-
orb
0
generate orbitals), there is a single SU(6) Kondo peak,
0 0.5 1 1.5 2 2.5 3 3.5
while for ∆ = 0.5Γ/π, it splits into three peaks. For
∆orb/TSKU(6) ∆orb = 2.0Γo/rbπ, we can see that the SU(2) Kondo peak
is developed around the Fermi level ω = 0, for which
FIG. 4: (a) Conductance calculated by the exact solution at only the lowest energy level (l = −1) is relevant. Note
absolutezeroasafunctionoftheenergysplitting∆orb. Each that further increase of the energy splitting shoud have
linerepresentsthetotalorthecontributionsfromeachorbital a tendency to suppress the above SU(2) Kondo effect
state. The system is in Kondo regime: the total number of at a given temperature because of the decrease of the
theelectrons in thequantumdot isn =1. (b)Theeffective
d effective Kondo temperature. The enlarged picture of
Kondotemperature TKSU(6). the spectral density around the Fermi level is shown in
Fig. 6(b), which determines the linear conductance at
computed by the exact solution is shown as a function low temperatures. It is seen that the spectral weight
of the energy splitting in Fig. 4. It is seen that the aroundtheFermilevelonceincreasesandthendecreases
totalconductanceatzerotemperature monotonicallyin- with increasing energy splitting, which gives rise to the
creases from 3e2/2h (SU(6) case) with the increase of maximum structure in the conductance.
5
TABLE I: Comparison of the conductance among different Kondo effects in the presence of magnetic field at zero and finite
temperatures.
temperature spin Kondoeffect two-orbital Kondo effect three- or more-orbital Kondoeffect
T =0 decrease constant 2e2/h increase upto 2e2/h
T 6=0 decrease decrease maximum
0.5
(a)
kT=0.083Γ/π
2e/h) 00..34 kkkkkBBBBBBTTTTT=====00000.....111250265005700ΓΓΓΓΓ/////πππππ ΓωA()on 00 ..168 ∆∆oorrbb==20∆..o05rbΓΓ=//ππ0
(2 0.2 cti
/ n
G u 0.4
0.1 al f
ctr 0.2
e
0 p
0 0.5 1 1.5 2 2.5 s 0
∆ /(Γ/π) -2 0 2 4 6
orb ω/(Γ/π)
(b)
FIG. 5: Conductance for thethree-orbital Kondoeffect com- 0.4
puted by the NCA calculation at different temperatures as ω) ∆orb=0
a function of the orbital splitting ∆orb. The parameter is A( ∆orb=0.5Γ/π
εc = −25Γ/π, which gives TKSU(6) ∼ 0.2Γ. At low tempera- Γn ∆orb=2.0Γ/π
tures, the predicted maximum structure appears. Note that o
for ∆orb = 0 and kBT = 0.083Γ/π, the total number of the ncti 0.2
electrons in the quantumdot is nd ∼0.82. al fu
r
ct
e
p
V. SUMMARY s 0
-0.2 -0.1 0 0.1 0.2
ω/(Γ/π)
We have discussed transport properties via the multi-
orbital Kondo effect in a single quantum dot system by
using the Bethe-ansatz exact solution at zero temper- FIG.6: (a)NCAresultsforthetotalspectraldensityA(ω)for
ature and the NCA at finite temperatures. It has been different orbital splittings at kBT = 0.083Γ/π. (b)Zoom of
A(ω) around the Fermi level of electrons in the leads ω =0.
shownthattheorbitalKondoeffectgivesrisetosomere-
markable transport properties, which are different from The bare energy level is εc =−25Γ/π.
the ordinary spin Kondo effect. The results are summa-
rized in TABLE I in comparison with the usual SU(2)
field-dependent conductance due to the SU(4) Kondoef-
spin Kondo effect.
fect.
For the two-orbital SU(4) Kondo effect, it has been
found that the conductance at absolute zero is constant In the three-orbital or more-orbitalcases, the conduc-
(2e2/h) irrespective of the strength of magnetic fields. tance at absolute zero increases in magnetic fields, and
However, the effective Kondo temperature, which gives is then saturatedatthe value of 2e2/h. Inthis casealso,
theenergyscaleatlow-temperatures,decreasesmonoton- the effective Kondo temperature decreases as the field
ically. Therefore, if the conductance is observedat finite increases, so that the conductance features a maximum
temperatures, it decreases with the increase of the mag- structure at finite temperatures. Such unusual behav-
netic field, in accordance with the recent experimental ior in the conductance may be observed experimentally
results for the conductance in the vertical quantum and in vertical quantum dot systems by tuning multi-orbital
the carbon nanotube quantum dot system [31, 34]. We degenerate states systematically [50].
stressagainthatalthoughthemagnetic-fielddependence In this paper, we have considered a simplified model
at finite temperatures seems similar to the conventional forthequantumdotwithmultiple orbitals: e.g. thetun-
SU(2) spin Kondo effect at first glance, the mechanism nelingmatrixelementsbetweentheleadsandthedotare
is different: the decrease of the conductance in the two- assumedto be constantand independent of orbitals. We
orbital SU(4) Kondo effect is due to the decrease of the believe that in spite of such simplifications the present
effective Kondo temperature. This fact should be prop- modelcapturesessentialpropertiesinherentinthemulti-
erly taken into account for the detailed analysis of the orbital Kondo effect at least qualitatively. It remains an
6
interestingproblemtoimproveourmodelbyincorporat- H. Yonehara for advices on computer programming,and
ingthedetailedstructureofthequantumdotparameters, T. Ohashi and T. Kita for discussions.
which is now under consideration.
Acknowledgments
We wouldlike to express our sincere thanks to M. Eto
and S. Amaha for valuable discussions. R.S. also thanks
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