VOLUME 93, N UMBER 1

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PHYSICA L R EVIEW LET T ERS

Enhanced Kondo Effect via Tuned Orbital Degeneracy in a Spin 1=2 Artificial Atom
S. Sasaki,1,* S. Amaha,2 N. Asakawa,2 M. Eto,3 and S. Tarucha1,2,4
1

NTT Basic Research Laboratories, NTT Corporation, Atsugi-shi, Kanagawa 243-0198, Japan2
Department of Physics, University of Tokyo, Bunkyo-ku, Tokyo 113-0033, Japan
3
Faculty of Science and Technology, Keio University, Kohoku-ku, Yokohama 223-8522, Japan
4
Mesoscopic Correlation Project, ERATO, JST, Atsugi-shi, Kanagawa 243-0198, Japan(Received 12 November 2003; published 2 July 2004)

A strong Kondo effect is observed for a vertical quantum dot holding an odd number of electrons andspin 1=2 when an orbital degeneracy is induced by magnetic field. The estimated Kondo temperature forthis ‘‘doublet-doublet’’ degeneracy is similar to that for the singlet-triplet degeneracy with an evenelectron number, indicating that a total of fourfold spin and orbital degeneracy accounts for theenhancement of the Kondo temperature. The experimental observation is qualitatively reproduced byscaling calculations using an SU(4) model at the orbital degeneracy.
DOI: 10.1103/PhysRevLett.93.017205

PACS numbers: 75.20.Hr, 73.23.Hk, 73.63.Kv

A quantum dot offers unprecedented opportunities ofrealizing and manipulating a single magnetic impurity
interacting with a Fermi sea of conduction electrons. Rich
aspects of the Kondo physics [1] have been exploredowing to the tunability of relevant parameters in quantum
dots. Pioneering works on the Kondo effect in quantum
dots [2 –6] proceed within the framework of the conventional spin 1=2 Anderson impurity model for a spindegenerate single level. The Kondo effect is observed
only when the number of electrons, N, is odd with aspin S  1=2. However, exceptions to this odd-even parity behavior have been noted in a lateral quantum dot [7]where the Kondo effect is observed for both odd and even
N. This has been followed by a more detailed reportof the Kondo effect for even N in a ‘‘vertical’’ quantum
dot having well-defined N and S [8]. This novel Kondo
effect occurs when a magnetic field induces a spin
singlet (S  0)-triplet (S  1) degeneracy in the multiorbital ground state [9–15].
In this Letter, we present a new multilevel mechanism
which enhances the Kondo temperature, TK , using a vertical quantum dot. We observe a strong Kondo effect whena magnetic field induces an orbital degeneracy for odd N
and S  1=2. The estimated TK is similar to that for the
singlet-triplet (S-T) Kondo effect for even N, indicatingthat a total of fourfold spin and orbital degeneracy forboth cases contributes to enhance TK .
It is known that the orbital degeneracy plays an important role in the Kondo effect of magnetic impurities
with f electrons. When the total degeneracy factor is
Nd for spin and orbital degrees of freedom, the Kondo
effect is described by the Coqblin-Schrieffer model ofSUNd  symmetry [16]. The factor Nd increases TK as
D0 e1=Nd J with exchange coupling J, where D0 andare the bandwidth and density of states of conduction
electrons. We examine a model which has the SU(4)symmetry at the orbital degeneracy, yielding a larger
TK than in a single-orbital case of SU(2) symmetry,explaining the experimental results [17].

The inset of Fig. 1 schematically shows the verticalquantum dot structure used in the present study. It is in theform of a submicron-sized circular mesa fabricated froman AlGaAs/InGaAs/AlGaAs double barrier structure.
Current flows vertically through the mesa, and N ischanged with a gate electrode wrapped around the mesa.
S and N can be unambiguously determined in this type ofvertical quantum dots owing to the highly symmetriclateral confinement potential and a built-in dot-lead
coupling  via AlGaAs barriers [18].  for the present
dot device is 400 eV for the first electron entering the
dot and gradually increases as N increases. All the transport measurements have been performed in a dilution

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 2004 The American Physical Society

0031-9007=04=93(1)=017205(4)$22.50

-0.5
A

Vg(V)

-1.0
ν =2
-1.5

N=6
B

-2.0

N=2
Gate
Current

N=0
0

1

GaAs
Dot AlGaAs
InGaAsAlGaAs
GaAs

2

3

4

B(T)
FIG. 1 (color). Color-scale plot of the linear conductance as afunction of the gate voltage and magnetic field. Blue corresponds to zero conductance, white to G  25 S, and red toG  50 S. All the electrons occupy the first Landau level on
the right-hand side of the dashed line (filling factor  2).
The inset shows a schematic diagram of the dot structure madefrom AlGaAs/InGaAs/AlGaAs double barrier tunnel diode.

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refrigerator with a base temperature of ’ 60 mK, using astandard lock-in technique with an ac excitation voltage
between the source and drain of 3 V. We apply a magnetic field parallel to the current.
Figure 1 shows a color-scale plot of the linear conductance, G, as a function of gate voltage, Vg , and magnetic
field, B, at base temperature ’ 60 mK. Conductanceshows maxima at specific Vg where N is allowed tochange by one for a fixed B (Coulomb oscillation peaks).
Therefore, each stripe in Fig. 1 represents a B dependenceof the N-electron ground state. Large Coulomb blockade
gaps at N  2 and 6 at B  0 reflect the shell filling of 1sand 2p orbital states, respectively, formed in a diskshaped circular artificial atom [18]. Overall evolution of
stripes with B agrees with successive filling of the single
particle levels, or Fock-Darwin states [19] En;l , by spinup and spin-down electrons. En;l is shown in Fig. 2(a); q
l1 1
 c  n   jlj h 4!20  !2c ; (1)En;l   h!
2
2 2
where n  0; 1; 2; . . . is the radial quantum number andl  0; 1; 2; . . . is the angular quantum number. h!
 0 isthe lateral confinement energy and h!
 c  eB=m . The
pairwise motion of the stripes in Fig. 1 is due to the spin
degeneracy of each En;l state.
When the Kondo effect occurs at T TK , the conductance in the Coulomb blockade gap increases. In ourquantum dot device,  gradually increases with increasing Vg . Therefore, TK generally increases as Vg increases,
and the enhancement of the conductance due to the
Kondo effect is noticeable in the Coulomb blockade
gaps for Vg > 1:0 V.
The solid lines in Fig. 2(b) schematically show a magnetic field dependence of the electrochemical potential
for electron numbers from N  1 (odd) to N  4 (even)when electrons are added to the two crossing orbitals.
Coulomb interaction favors a spin triplet state (S  1) forN  2 in the vicinity of the magnetic field where a single
particle level crossing occurs, and the ground stateshows a downward cusp [20]. The S-T Kondo effect is
expected on the dotted lines with a considerably higher
Kondo temperature, TKS-T , than the conventional S  1=2
Kondo temperature, TKD , because of the larger degeneracy [8].
Figure 2(c) shows detailed measurement conducted inregion A marked in Fig. 1. The last orbital crossings occur
between En;l states with n; l  0; 1 and 0; l l > 1on the dotted line, and all the electrons occupy the ground
Landau level at higher B (  2). A spin triplet state isobserved at N  16 and B ’ 1:2 T, where states n; l 
0; 1 and (0, 7) are occupied by electrons having parallel spins [see circle in Fig. 2(a)]. In the N  16 Coulomb
blockade gap, the conductance is enhanced by the S-T
Kondo effect at B ’ 1:1 and B ’ 1:3 T, corresponding to
the dotted lines in Fig. 2(b). As for N  15 and 17, theconventional S  1=2 Kondo effect is expected. However,017205-2

FIG. 2 (color). (a) Fock-Darwin states calculated withh!
 0  1 meV. The circle denotes crossing between orbital
state n; l  0; 1 (dotted line) and (0, 7) (thick solid line)
where the Kondo effect for N  15 and 16 is studied in detail.
(b) Schematic magnetic field dependence of the electrochemical potential for electron numbers from N  1 (odd) to N  4(even) occupying two crossing orbitals. A skew toward theupper-left direction takes into account the effect of decreasingh!
 0 with increasing N. (c) Detailed measurement conducted inregion A in Fig. 1. The color scale is same as in Fig. 1. S, T, andD denote states with S  0 (singlet), S  1 (triplet), and S 
1=2 (doublet).

the conductance enhancement in the Coulomb blockade
gaps is not clearly observed except in the regions corresponding to the dash-dotted lines in Fig. 2(b), where two
S  1=2 states with different total angular momentum,
M, are degenerate. When such an orbital degeneracy ispresent for odd N, a total of four states, i.e., M  M1 ; M2
M1  M2 , SZ  1=2, are involved in forming the
Kondo singlet state (if one can neglect the Zeeman splitting), and enhancement of TK similar to the S-T Kondo
effect is expected. We define this new multilevel mechanism for odd N as ‘‘doublet-doublet’’ (D-D) Kondo effect.
Because TKD is much lower than the D-D Kondo temperature, TKD-D , only a slight conductance enhancement isobserved in the odd N Coulomb blockade regions when
there is no orbital degeneracy. Then, a honeycomb pattern
is formed in a B-N diagram, as is clearly captured inFig. 2(c), due to the S-Tand D-D Kondo effects that occur
consecutively for different orbital crossings, provided
TKD < T < TKS-T , TKD-D . Our honeycomb pattern is different from the ‘‘chessboard pattern’’ discussed in a lateral quantum dot [21]. In a vertical quantum dot, one
can assume that the orbital quantum numbers are conserved in tunnel processes between the dot and leads dueto their same rotational symmetry. Hence we expect ‘‘two
017205-2

0.5

0.5
0.0

D

D

-0.5

0.8

(c)
0.5

1.3

1.2

1.4

B (T)

N=16, S-T
N=15, D-D

(e)
00

-0.5
0.0

(d)
D-D

-0.5

S-T
0.0

D-D
D-D
-20
-20
S-T
S-T
S-T
(triplet side)
-40
-40
0
0

10
20
10
20
ln ∆ / T (0)K

ln ∆ /T K (0)

30
30

0.1

0.8

2

dI/dVsd (e /h)ST∆B (T)

0.7
N=15
D-D Kondo

G/G0

0.8

1.0

0.0

-0.1

1.0

S

B (T)

2

(b)

N=16
S-T Kondo

0.0
-0.5

1.2

∆G (e /h)0.9

Vsd (mV)p
fitted to the data, where
21=s  1 [4,24].
The TK for the S-T and D-D Kondo effect estimatedfrom the curve fitting are 700 mK and 490 mK, respectively. The higher TK for the S-T Kondo effect may be dueto the larger , and it is difficult to experimentally
determine which is larger, TKS-T or TKD-D , for the same .
The fitted values of the parameter s are larger than s ’
0:2 for a conventional spin 1=2 system. However, theestimation of s is less reliable because it changes substantially with the chosen fitting range. The expected Zeeman
splitting of ’ 30 eV at B ’ 1:2 T is smaller than TK
estimated above. Therefore, no Zeeman splitting of the
TK0  TK =

(b)

(a)

ln T
KK((∆)∆/)/Tln T
TK (0)K (0)(2)

Vsd (mV)

G=G0  fTK02 =T 2  TK02 gs ;dI=dVsd Kondo peak is observed, and we treat all four S-T
and D-D states as quasidegenerate.
Next, let us investigate the effect of lifting the S-T and
D-D degeneracy by changing B. Figures 4(a) and 4(b)show a color-scale plot of dI=dVsd in B-Vsd plane for the
D-D Kondo effect (N  15) and for the S-T Kondo effect(N  16), respectively, with Vg fixed in the center of therespective Coulomb blockade gap. Conductance peaks at
Vsd  0 are observed near the degeneracy field, B0 .
Because the S-T or D-D degeneracy is lifted as jBj 
jB  B0 j increases, the Kondo effect is broken and thezero-bias peak is suppressed. At large jBj, a peak or step
is observed at eVsd  , where the color suddenly
changes. Here,  is the B-dependent energy difference
between the singlet and triplet states, or between the two
doublet states. This peak/step is due to cotunneling associated with the two states separated by  [8], and therefore observed within the Coulomb blockade gap(’0:6 meV).
Figure 4(c) shows Vsd values of the conductance peak/
step as a function of B. Peak/step positions for the S-T(B0  1:12 T) and the D-D (B0  1:25 T) Kondo effectVsd (mV)

channels’’ of conduction electrons in the leads when two
orbitals are relevant in the quantum dot; each channel
couples to only one of the two orbitals. In a lateral quantum dot, on the other hand, a ‘‘single channel’’ in the leads
preferentially couples to the outer orbital in the dot.
Figure 3(a) shows temperature dependence of the differential conductance dI=dVsd versus Vsd for the S-T (N 16) and D-D (N  15) Kondo effect [solid and open
triangle in Fig. 2(c), respectively] with the gate voltage
fixed in the center of the respective Coulomb blockade
gap. A clear Kondo peak at Vsd  0 is observed at low
temperatures, whose height decreases with increasing
temperature as expected for the Kondo effect. Temperature dependence of the Kondo peak height normalized by
the low temperature limit value, G0 , is shown in Fig. 3(b).
A background from non-Kondo cotunneling is subtracted
from each trace in Fig. 3(a) [22]. The Kondo peak conductance shows almost linear logT dependence around’400 mK and saturates at lower a temperature. The unitary limit conductance of 2e2 =h [23] is not reached in our
device probably because of the asymmetry in the twotunnel barriers. Dotted lines in Fig. 3(b) are the theoretical curves(a)week ending
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PHYSICA L R EVIEW LET T ERSVOLUME 93, N UMBER 1

0.7

0.6
0.4

0.6

0.2

N=16, S-T
TK=700mK,S=0.8
N=15, D-D
TK=490mK,S=1.1
2

-0.8

-0.4

0.0

0.4

Vsd (mV)

0.8

3

4 5 6

100

1000

T (mK)

FIG. 3. (a) Temperature dependence of the differential conductance dI=dVsd versus Vsd for the S-T and D-D Kondo effectfrom T  60 mK (thick solid line) to 1.5 K (dashed line).
(b) Temperature dependence of the Kondo peak height normalized by the low temperature limit value. Dashed lines are thetheoretical curves fitted to the data.

017205-3

FIG. 4 (color). Color-scale plot of dI=dVsd in B-Vsd plane for(a) the N  15 D-D Kondo effect and (b) the N  16 S-T
Kondo effect with Vg fixed in the center of the respective
Coulomb blockade gap. Blue corresponds to G  10 S andred to G  35 S. (c) Vsd values of the conductance peak or
step obtained from (a) and (b) as a function of the magnetic
field difference B  B  B0 , where B0 is the degeneracy
magnetic field. (d) Relative conductance measured from thedegeneracy (B  0) at Vsd  0, for the D-D (solid line) andthe S-T (dotted line) Kondo effect. (e) Calculated results of
TK =TK 0 as a function of =TK 0, on a log-log scale. TK 0 is
the Kondo temperature at   0. For the S-T Kondo effect(triplet side), TK  deviates from a power law for large  (seeRef. [10]; we choose J1  J2 and C2  0).

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almost coincide, indicating that they involve the same
orbital states, namely, n; l  0; 1 and (0, 7).
For the D-D Kondo effect, we examine an SU(4)model. The dot state is characterized by spin Sz 
1=2 and pseudospin Tz  1=2 corresponding to two
orbitals,  Sz ; Tz . Considering the second-order tunneling processes to the leads with two channels [25], we
obtainX~  S;H  "k cyk; ck;  JS  s  B
(3)
k;

P
1

^ d , s  1 P 0 P cy0 ^ c ,with S  2
dy 
k;
2 kk
k;
y
y
where d and d (ck; and ck; ) are the creation andannihilation operators of dot state (conduction electron^ are generators of the Liek; ). Fifteen components of 
algebra Su(4), f&x ; &y ; &z ; I  )x ; )y ; )z ; Ig  fI  Ig,
where &i ()i ) are the Pauli matrices in the spin (pseudospin) space and I is the unit matrix. The energy difference~ , the I  is expressed by the fictitious magnetic field B)z component of whichis  and the other components areP~  S  =2 S dy dS ;1=2  dy
0 [BSz ;1=2 z
Sz ;1=2 dSz ;1=2 ].
z
We evaluate TK as a function of  using the ‘‘poor
man’s’’ scaling method [26]. We find that (i) TK  is
maximal around   0, and (ii) TK  decreases withincreasing  by a power law, TK   TK 0 
TK 0=+ with +  1 [27]. In the S-T
pKondo effect,
TK  obeys a power law with +  2  5 on the triplet
side, whereas TK  drops to zero suddenly on the singlet
side [10,12]. Figure 4(d) shows the experimentally observed relative conductance, G, measured from thedegeneracy (B  0) at Vsd  0, as a function of B.
The conductance drops abruptly on the singlet side (B <0) in the S-T Kondo effect, whereas it drops more slowly
and symmetrically in the D-D Kondo effect, in qualitative agreement with the theory [Fig. 4(e)]. These behaviors are also present for other D-D and S-T Kondo effects
at different N, shown in Fig. 2(c).
In summary, we have presented a new multilevelmechanism for the Kondo effect in an artificial atom.
We have shown that an orbital degeneracy for odd N andS  1=2 dramatically enhances the Kondo effect as in the
S-T Kondo effect reported before.
The authors acknowledge valuable discussions with
T. Sato, W. Izumida, L. I. Glazman, H. Hyuga,
M. Stopa, and J. Kondo. The work is supported by thefinancial support from the Grant-in-Aid for Scientific
Research A (Grant No. 40302799), the Nano-Photonic
and Electron Devices Technology Project, the Focused
Research and Development Project for the Realization of
the World’s Most Advanced IT Nation, and SORST-JST.
The power law of TK  in D-D Kondo effect was
originally derived by K. Yamada et al. [28].

*Electronic address: satoshi@nttbrl.jp

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[16] A. C. Hewson, The Kondo Problem to Heavy Fermions(Cambridge University Press, Cambridge, England,1993).
[17] For previous work of the multi-orbital Kondo effect inquantum dots, see T. Inoshita et al., Phys. Rev. B 48,
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[21] M. Stopa et al., Phys. Rev. Lett. 91, 046601 (2003), andreferences therein.
[22] L. I. Glazman and M. Pustilnik, in New Directions in
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[23] W. G. van der Wiel et al., Science 289, 2105 (2000).
[24] T. A. Costi and A. C. Hewson, Philos. Mag. B 65, 1165
(1992).
[25] We assume that the strength of the tunnel coupling V isequivalent for two orbitals, which seems to be the experimental situation. The exchange coupling J in Eq. (3)is given by J  V 2 =Ec , where 1=Ec  1=E  1=Ewith E being addition and extraction energies.
[26] We obtain scaling equations (i) dJ=d lnD  4 J2 forthe energy scale D   [SU(4) Kondo effect], and(ii) dJ=d lnD  2 J2 for D   [conventional SU(2)Kondo effect]. is the density of states in the leads. When  TK 0, we match the solutions of (i) and (ii) at D  [9,10] and obtain a power law of TK .
[27] An equivalent situation to our SU(4) model has been
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Lett. 90, 026602 (2003).
[28] K. Yamada et al., Prog. Theor. Phys. 71, 450 (1984).

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