4.2 Measurement of the phase transition of two-impurity Kondo system and
4.2.2 Phase transition of two-impurity Kondo system
A stability diagram of the double dot – presenting the usual honeycomb-like pattern – is shown in Fig. 4.7b as the function of the plunger gate voltages, VgL and VgR for fixed VgB = −0.22 V. The numbering on the figure indicates the electron number of the quantum dots above an unknown number of closed shells. The dots are weakly coupled to each other, indicated by the straight boundaries between the charge states. And they are well coupled to the leads, which is indicated by smeared edges of the Coulomb diamonds and the zero-bias Kondo resonances measured in the finite bias spectroscopy of the (1,2) and (2,1) states, shown on Fig. 4.7c&d, respectively. The color coded lines on panel b mark the gate voltage settings where the finite bias measurements have been performed.
4.2. Measurement of the phase transition of two-impurity Kondo system and quantum dot-mediated exchange field
N NiFe
gB
a)
gL gR
{
QDL QDR
HfO InAs NW
-0.36 -0.4 -0.44 VgL (V)
-0.11 -0.1 -0.09 0.05 0.1 0.15
VgR (V)
(0,0) (0,2)
(2,2) (2,0)
(1,1) (2,1)
(1,0) (1,2)
(0,1) c) d)
VgL (V) VSD (V)
c) d)
b)
-0.36 -0.44 -0.4
0.05 0.1 0.15 0.2
VgR (V) 0.5
-0.5 0
0.01 0.02 0.03 0.04
-0.1 -0.11
G (G0)
G (G0) G (G0)
Figure 4.7:a) Schematics of the measured device. The InAs nanowire is contacted by a normal (N) and a ferromagnetic lead (NiFe). An array of bottom gates is used to define a double dot in the wire. b) Stability diagram of the double dot as a function of the voltages on the two plunger gates,VgL andVgRmeasured atVB =−0.22 V. The numbering indicates the electron occupation of the dots above a closed shell. c-d) Finite bias measurements along the color coded lines on panel b, in the (1,2) and (2,1) charge states, showing the Kondo-related zero-bias peaks in the odd states. The gray dashed lines indicate the edges of the Coulomb diamonds.
4.2. Measurement of the phase transition of two-impurity Kondo system and quantum dot-mediated exchange field
The summary of the tuning of the interdot tunnel coupling is shown on Fig. 4.8. The left column shows the stability diagram as the function the plunger gate voltages, VgL and VgR. The middle column shows the finite bias spectroscopy along the cut through the (0,2)−(1,1)−(2,0) states, indicated by the gray dashed lines in the left column. Finally, the right column shows the magnetic field dependence of the finite-bias conductance at the center of the (1,1) charge state, measured along the gray dashed lines in the middle column. The interdot tunnel coupling is increased from row to row by increasing the voltage on gB. Note that the single-impurity Kondo effect, similar to the results shown on Fig. 4.7c&d, is present in the even-odd and odd-even charge states for all values of VgB presented here (not shown).
For weak tunnel coupling, with VgB =−0.22 V the stability diagram shows the usual honeycomb pattern. The top left panel of Fig. 4.8 is the same as Fig. 4.7b. The finite bias spectroscopy measured through the two electron states shows a zero-bias Kondo resonance in the (1,1) charge state. For weak interdot coupling the quantum dots individually form a Kondo singlet state with their neighboring electrode. In the transport through the system the interdot barrier serves as a tunnel contact, therefore the Kondo resonances probe each other. At finite bias the voltage mostly drops on the interdot barrier, while the Kondo singlets are pinned to the Fermi energy of the normal leads, and therefore they are not in resonance. Hence a single zero-bias resonance is present in the transport in the (1,1) charge state (see left panel of Fig. 4.6b for the illustration).
IncreasingVgB, the overall conductance significantly increases (see the second and the third row of Fig. 4.8 with VgB =−0.2 and -0.18 V, respectively). A significant difference between the top left panel and the two below is that the conductance is strongly increased in the whole (1,1) charge state. Correspondingly, the Kondo resonance of the (1,1) charge state is more pronounced (see middle column). With increasing VgB the barrier between the dots is lowered, and hence a larger current flows.
Upon increasing the tunnel coupling further (see the fourth row of Fig. 4.8 withVgB =
−0.118 V) the states of QDL and QDR hybridize, and form the bonding and anti-bonding combinations, indicated by the curvy edges of the charge states. In the (1,1) charge sector the singlet combination hybridizes with the (2,0) and (0,2), and gains energy compared to the triplets (see Sec. 2.2.1). Since this antiferromagnetic singlet ground state of the double dot is unique, no Kondo correlation can develop in equilibrium. Correspondingly, in the stability diagram the conductance drops within the (1,1) charge state and in the finite bias measurement instead of a single zero-bias peak, two peaks are present at VSD ≈ ±80 µV (indicated by white arrows). Sometimes this phenomenon is called the splitting of a Kondo resonance. However, the emergent finite-bias resonances are not Kondo peaks anymore, but the inelastic cotunneling lines corresponding to the singlet-triplet excitation, whose amplitudes are enhanced by the strong coupling to the normal leads.
For even stronger interdot tunnel coupling the hybridization of dot states and singlet-triplet spitting increase further (see the last row of Fig. 4.8).
To further illustrate the transition from the Kondo singlet ground state to the exchange coupled singlet state, bias-dependent line cuts at the center of the (1,1) charge state are shown in Fig. 4.9a. For low coupling a single zero-bias peak is present, which splits for voltages VgB > −0.13 V. The Kondo temperatures, TK and the singlet-triplet splittings,
∆EST are determined from the curves as the FWHM of the zero-bias peaks and the peak-to-peak distances of the split peaks, respectively. They are plotted in Fig. 4.9b as
4.2. Measurement of the phase transition of two-impurity Kondo system and quantum dot-mediated exchange field
-0.36 -0.4 -0.44 VgL (V)
-0.4
-0.44 VgL (V)
-0.4 -0.44 -0.48
VgB = -0.22 V VgB = -0.2 V
VgB = -0.18 V
VgB = -0.118 V
VgB = -0.07 V -0.5
-0.55
-0.6
-0.6 -0.65
-0.7 VgL (V)VgL (V)VgL (V)
-0.11 -0.1 -0.09
-0.11 -0.1 -0.09
-0.11 -0.1 -0.12
-0.12 -0.14
-0.13 -0.15
VgR (V)
0 VSD (mV)
-0.5 0.5
0 -0.5 0.5
0 -0.5 0.5
0 -0.5 0.5
0 -0.5 0.5 0.05 0.1 0.15
0.1 0.3 0.5
0.2 0.4 0.6
0.2 0.4 0.6
0.2 0.4
0 VSD (mV)VSD (mV)VSD (mV)VSD (mV)
-0.4 -0.35 -0.45
-0.4 -0.44 -0.48
-0.42 -0.46 -0.5
-0.54 -0.6
-0.6 -0.66
VgL (V)
0 VSD (mV)
-0.5 0.5
0 -0.5 0.5
0 -0.5 0.5
0 -0.5 0.5
0 -0.5 0.5 VSD (mV)VSD (mV)VSD (mV)VSD (mV)
0.5
0 1
0.5
0 1
0.5
0 1
0.5
0 1
0.5
0 1
B (T) 0.05
0.15 0.25
0.1 0.3 0.5
0.2 0.4 0.6
0
0.1 0.3 0.5
0.2 0.4
0 0.05
0.15 0.25 0.1 0.3 0.2 0.2 0.4 0.6
0 0.1 0.3 0.2 0.02 0.06 0.1
Figure 4.8: Transport through the 2IK system at different interdot tunnel couplings. Left column: stability diagrams as the function of the plunger gate voltages. Middle column: finite bias spectroscopy along the dashed line in the left. Right column: Magnetic field dependence of the conductance along the dashed line in the middle column. The interdot coupling is increased from row to row by the more positiveVB voltage. For detailed description see the text.
4.2. Measurement of the phase transition of two-impurity Kondo system and quantum dot-mediated exchange field
black squares and red dots. While the singlet-triplet splitting has a clear dependence on VgB, in accordance with the expectation of the increased splitting at more positive voltages, the Kondo temperature only weakly depends on it. The small increase of the Kondo temperature with increasing VgB is presumably due to broadening induced by the increasing interdot tunnel coupling. The exchange splitting allows for the determination of the interdot tunnel coupling,tLR with the ∆EST = 8t2LR/(UL+UR) relation for the zero-field singlet triplet splitting (see Sec. 4.1.5). The on-site Coulomb energies, UL ≈ 3 meV and UR ≈ 1 meV are determined from finite bias measurement in the even-odd charge sectors (see the gray dashed lines in Figs. 4.6c&d). The obtained tLR values are plotted in Fig. 4.9b as blue triangles.
a)
-0.4 -0.2 0.0 0.2 0.4 0.0
0.5 1.0 1.5 2.0 2.5
G (G 0)
VSD (mV) VgB = -0.07 V
-0.19 V TK 2 ΔEST
-0.20 -0.16 -0.12 -0.08 0
50 100 150 200 250 300
350 kBTK
ΔEST tLR
Energy (μeV)
VgB (V) b)
Figure 4.9: Phase transition of the 2IK system. a) Bias dependence at the center of the (1,1) charge state or different values of VgB. At low interdot couplings a single zero-bias peak is observed, while for large couplings,VgB >−0.13 V, the peak splits. b) The Kondo temperature, TK, the singlet-triplet splitting, ∆EST and the interdot tunnel coupling, tLR determined from the curves on panel a.
The magnetic field is another efficient tool to study the ground state of the double dot.
As I will show, in some sense it is more sensitive than the zero-field spectroscopy. At weak couplings the splitting of the Kondo peak in magnetic field is linear (see top right panel of Fig. 4.8). The slope of splitting corresponds to a g-factor of g ≈2.5. The g-factors of the dots were measured by the splitting of the Kondo resonances in the (1,2) and (2,1) charge states, giving gL ≈ 2.5 and gR ≈ 3, respectively (not shown here). These values are consistent with the splitting in the (1,1) charge state, but one has to keep in mind that the g-factors can depend on the charge state. Upon increasing the interdot tunnel coupling the resonance becomes more pronounced (second panel in the right column), but similarly to the weaker coupling case, the splitting in magnetic field is linear with a similar slope.
In the opposite limit of strong coupling, when the ground state is the antiferromagnetic singlet, the magnetoconductance is strikingly different (see the two bottom panels in the right column). Due to the singlet-triplet splitting two finite bias resonance lines are present, which at first get closer at low fields and finally get apart. Note that on the bottom panel only the positive bias resonance is well-pronounced. This no-crossing behavior is
4.2. Measurement of the phase transition of two-impurity Kondo system and quantum dot-mediated exchange field consistent with the results presented in the previous section for double dots with strong spin-orbit interaction (see Fig. 4.3c).
Now let us turn to the intermediate coupling regime (middle row) withVgB =−0.18 V.
In the zero-field finite-bias spectroscopy no singlet-triplet splitting is observed, only a zero-bias peak is present, but in finite magnetic field the splitting is not linear, instead, resembling the strong coupling limit, it is quadratic-like. The zero-bias peak persists up to 0.5 T. Presence of the zero-bias peak indicates a strong Kondo correlation on the dot, but the non-linear behavior in magnetic field is a signature of the antiferromagnetic singlet ground state.
To sum up, I have presented the observation of the two-impurity Kondo phase transi-tion measured in a double quantum dot in an InAs nanowire.