1. Singlet-Triplet and Doublet-Doublet Kondo Effect
in an Artificial Atom
S. Sasaki,1 * S. Amaha2, N. Asakawa2 and S. Tarucha1,2,3
1NTT Basic Research Laboratories
2Department of Physics, University of Tokyo
3Mesoscopic Correlation Project, ERATO, JST *

2. Outline
Quantum dot is suitable for the study of the Kondo effect various parameters tunable (gate voltage, magnetic field etc). ··· manipulation of spin state is easy
Advantage of using a vertical quantum dot = artificial atom
··· well defined electron number (down to “0”) and spin state
Experimental results
Dot – lead coupling G~ 400meV
singlet-triplet Kondo for even N
··· similar to our previous report (Nature 405 (2000) 764)
doublet-doublet Kondo for odd N doublet with orbital degeneracy ··· New!
I’d like to talk about experimental study of the Kondo effect in quantum dots. Here are my collaborators at Univ. of Tokyo and ERATO Project. This is the outline of my talk. We have measured two dot samples. I’ll first discuss a dot very strongly coupled to the leads where we saw the Kondo effect for 1 & 2 electrons in the dot. Then I’ll discuss an intermediate coupling dot where we saw the Kondo effect for 8 & 9 electrons.

3. Electronic states in a circular artificial atom
N and S clearly defined!
2D disk shaped dot B 8
degeneracy 6 2p 4 1s 2
Harmonic potential
Our vertical quantum dot is essentially a 2D disk shaped dot, and the lateral confinement is well described by a Harmonic potential. Therefore, equidistant energy levels are formed in the dot with degeneracy 2,4,6 and so on including spin. Such electronic structure is reflected in an addition energy spectrum shown below. Here, large energy gaps are observed at specific electron numbers such as 2, 6, 12 corresponding to the complete filling of each level, or shell filling, just like real atoms. Also Hund’s rule is observed, that the parallel spin is favored when each shell is half-filled for N=4, 9 etc. .
If you apply magnetic field in this direction, degeneracy of each orbital is lifted, and the evolution of the single particle levels is given by this so-called Fock-Darwin states. Here, each level is doubly degenerate due to spin. And if you also take into account Coulomb interaction between electrons, you get the experimental tunnel spectra as shown on the right. Of course, there is no Kondo effect in this case because the tunnel coupling between the dot and the leads is small. This is 1st shell, 2nd shell and so on, and a pair structure is noted due to spin degeneracy. You can also determine the spin state S. For example, the N=2 ground state at low magnetic field is a spin singlet, or S=0, where the electron spins are anti-parallel. However, at high magnetic field, the triplet state S=1 with parallel spins is energetically favored. So spin singlet-triplet transition occurs here and I’ll discuss the Kondo effect at this S-T transition in the first part of my talk. In the second part, I’ll discuss the Kondo effect at similar S-T transition for N=8 observed in another dot.

4. Kondo effect in quantum dots
manipulation of spin state via various parameters (gate voltage, magnetic field etc.) ⇒　 detailed analysis of the Kondo effect
Anti-ferromagnetic spin coupling between dot and leads GL GR E m U
Coherent higher-order tunneling e0
lead
lead
dot
DOS
Gate voltage fixed at S≠0 valley
Increase of S≠0 valley conductance
Vd = 0
N-1(odd)
S=1/2
N (even)
S=0
N+1(odd)
S=1/2
T<TK
In quantum dots, you can manipulate the spin states via various parameters such as gate voltage and magnetic field. Therefore, detailed analysis of the Kondo effect is possible, and several groups including ourselves have reported observation of the Kondo effect in quantum dots.
This is an energy diagram of a dot coupled to the external leads via tunnel barriers. This shows a Coulomb blockade situation where no direct tunneling is allowed because of the charging energy, U. However, if the dot has spin ½, anti-ferromagnetic spin exchange interaction between dot and leads enables coherent higher order tunneling at temperature lower than the Kondo temperature, TK, and a peak develops at Fermi energy in the local density of states of the dot.
So what are the experimental signatures of the Kondo effect in quantum dots? First, in Coulomb oscillation characteristic, conductance of the C.B. valleys with odd electron number and spin 1/2 increases with decreasing temperature because of this higher order tunneling. And when a gate voltage is fixed at such a CB valley, a zero-bias peak develops in differential conductance vs. source-drain bias characteristic reflecting this peak in DOS. And this conductance shows linear log-T dependence around the Kondo temperature Tk, and saturates at low temperature limit.
Conductance
dI/dVd
dI/dVd
T>>TK
logT TK
Gate voltage Vd

5. Sample structure (leveling technique)
T=1.5K
15.0E-9
-1.0E-9
0.0E+0
1.0E-9
2.0E-9
3.0E-9
4.0E-9
5.0E-9
6.0E-9
7.0E-9
8.0E-9
9.0E-9
10.0E-9
11.0E-9
12.0E-9
13.0E-9
14.0E-9
-500.0E-3
-3.0E+0
-2.5E+0
-2.0E+0
-1.5E+0
-1.0E+0 I
Dot
Barriers
Current
Gate
GaAs
AlGaAs
InGaAs B
~0.5 mm
This figure schematically shows the sample structure of our vertical quantum dot. A pillar with a diameter of about 0.5 mm is made from a double barrier structure and a gate is wrapped around the pillar to change the electron number, N. A thin line mesa is attached to the side of the pillar for current injection into the dot formed in the InGaAs well layer. We apply magnetic field in this direction parallel to the pillar. Tunnel coupling to the leads Gamma is 0.95meV for the dot #1, and 350meV for the dot #2. This picture on the left is a SEM image of the actual dot sample. Vg
Good pinch-off characteristics

6. Fabrication process (leveling technique)
EB-litho.
dry & wet etch
AuGe/Ni
PMGI
gate
Ti/Au
AlGaAs barriers
Develop in NMD
AZ5214E
Several seconds in NMD
Ti/Au
to bonding pad

7. B-N diagram with large G (Kondo effect)
New Kondo effect expected for S>1 ?
Chess-board-like S-T and D-D Kondo
n=2
n=1
-0.3
Vac=3mV (Lock-in)
T=60mK
Spin flip
MDD
Vg (V)
10mS
5.38E+0
-10.12E-3
1.49E+0
3.98E+0
Now let us consider two orbitals that make crossing in the presence of B. Each orbital is doubly degenerate due to spin, so you can put up to four electrons to these two orbitals. The right panel schematically shows the magnetic field dependence of the electro-chemical potential, mu, in black lines. For N+1, spin is always 1/2 , and the electro-chemical potential simply goes up and down.
For N+2, spin state is singlet away from the crossing point, but spin triplet state is favored close to the crossing point due to Coulomb interaction, and a downward cusp appears. So S-T degeneracy occurs on these green lines. Mu for N=3 and 4 have the inverted form like this. Now, a strong Kondo effect is expected at such S-T degeneracy because there are four available states. For odd N, spin is ½ and conventional weaker Kondo effect occurs. However, there are also four states on the red lines due to orbital degeneracy, and similar enhancement of the Kondo effect is expected. So we may call this doublet-doublet Kondo effect compared with S-T Kondo.
N=2
N=0
-2.4
B (T) 6

8. Enhanced Kondo at level crossingsB
Single particle energy
Level crossing (n=2)
TKS-T~TKD-D>T >TKD (>>TKT) m
N+4
S=0
N+3
S=1/2
N+2
|1, 1>
|1, 0>
|0, 0>
|1,-1>
Singlet-Triplet
|S, SZ>
S=0
S=1
S=0
S=1/2
N+1
S=0
N (even)
Doublet-Doublet
Now let us consider two orbitals that make crossing in the presence of B. Each orbital is doubly degenerate due to spin, so you can put up to four electrons to these two orbitals. The right panel schematically shows the magnetic field dependence of the electro-chemical potential, mu, in black lines. For N+1, spin is always 1/2 , and the electro-chemical potential simply goes up and down.
For N+2, spin state is singlet away from the crossing point, but spin triplet state is favored close to the crossing point due to Coulomb interaction, and a downward cusp appears. So S-T degeneracy occurs on these green lines. Mu for N=3 and 4 have the inverted form like this. Now, a strong Kondo effect is expected at such S-T degeneracy because there are four available states. For odd N, spin is ½ and conventional weaker Kondo effect occurs. However, there are also four states on the red lines due to orbital degeneracy, and similar enhancement of the Kondo effect is expected. So we may call this doublet-doublet Kondo effect compared with S-T Kondo. B
|1/2, 1/2>
|1/2, 1/2>
Orbital degeneracy
　　　　　⇒ Enhanced Kondo
S-T Kondo (even N)
|1/2,-1/2>
|1/2,-1/2>
D-D Kondo (odd N)

9. S-T and D-D Kondo at orbital crossings
-0.4
2nd Landau level
1st Landau level
Vg (V)
n=2
S=0
S=1
S=1/2
n≦2 B m
N=16
-0.9
0.2
B (T)
1.5
Chess Board ?
10mS
5.40E+0
-10.12E-3
1.00E+0
2.70E+0

10. Coulomb diamonds for S-T Kondo
B=B0
-0.5 D D
Vg (V)
Kondo peak
@Vsd=0 S T S D
N=15
-0.74
0.95
1.4 -2 2
B (T)
Vsd (mV)

11. Temperature dependence of S-T and D-D Kondo peak
T. A. Costi and A. C. Hewson, Phil. Mag. B65 (1992) 1165.

12. Kondo peak splitting with S-T degeneracy lifting
60mK Vg S T S
1.4 mS
S-T 41 12 25
Vsd (mV)
Now, let us see the behavior of the Kondo peak when the S-T degeneracy is lifted. The right panel shows the Kondo peak when magnetic field is swept across S-T transition field. The red curve is for the S-T degeneracy. The ground state is triplet for the lower traces and singlet for the upper traces. When the S-T degeneracy is completely lifted, the Kondo peak is split into two. When Vsd is close to zero, cotunneling involving only the ground state is possible as shown here. However, when the Vsd reaches the S-T energy difference Delta, new cotunneling process involving the excited state becomes possible, and Kondo-like correlation produces peaks at Vsd=Delta. That’s why we observe the Kondo peak splitting when the S-T degeneracy is lifted. In the next viewgraph, I’ll discuss variation of the Kondo temperature near S-T degeneracy where you can see a single Kondo peak. S T S
-1.4
0.95
1.4
B (T)
Coulomb blockade gap

13. Kondo peak splitting with D-D degeneracy lifting
60mK T S Vg D D
1.4
D-D
Vsd (mV)
Now, let us see the behavior of the Kondo peak when the S-T degeneracy is lifted. The right panel shows the Kondo peak when magnetic field is swept across S-T transition field. The red curve is for the S-T degeneracy. The ground state is triplet for the lower traces and singlet for the upper traces. When the S-T degeneracy is completely lifted, the Kondo peak is split into two. When Vsd is close to zero, cotunneling involving only the ground state is possible as shown here. However, when the Vsd reaches the S-T energy difference Delta, new cotunneling process involving the excited state becomes possible, and Kondo-like correlation produces peaks at Vsd=Delta. That’s why we observe the Kondo peak splitting when the S-T degeneracy is lifted. In the next viewgraph, I’ll discuss variation of the Kondo temperature near S-T degeneracy where you can see a single Kondo peak. D
-1.4
1.15
1.35
B (T)
Coulomb gap

14. Mysterious behavior at S=1/2 ⇔3/2 degeneracy
dip at Vsd = 0
No zero-bias peak in dI/dV??
Conductance 1K
B (T)
Vg (V)
3.0
1.5
-0.3
-0.5
N=20
1/2 1
S=3/2
S=1/2
3/2
1st spin flip
70mK
-1.0
Vsd (mV)
1.0
-0.6
S=1/2-3/2 degeneracy
Vsd (mV)
0.6
1.84
2.0
B (T)

15. Summary Magnetic field induced Kondo effect in a vertical quaytum dot:
Kondo effect for even N ・・・ singlet-triplet
Kondo effect for odd N ・・・ doublet-doublet ←New!
Both give similar Kondo temperature due to
four-fold degeneracy
To summarize, in the strong coupling dot, we have observed the conventional spin doublet Kondo for N=1 and S-T Kondo for N=2 in high magnetic field. We have demonstrated that the Kondo effect is greatly suppressed by in-plane magnetic field. In the intermediate coupling dot, we have observed S-T Kondo for N=8 and D-D Kondo enhanced by orbital degeneracy for N=9.
Thank you for your attention!