SUPPORTING ONLINE MATERIAL
Reverse bias measurements
A common signature of Pauli blockade is the strong suppression of current in one bias
direction (1, 2). In a double dot structure tuned to the Pauli blockade regime, the states (0,1),
(1,1), and (0,2) are accessible, where (m, n) specifies the number of electrons on the left and
right dots. The relevant (0,2) and (1,1) states are illustrated in the schematic of Fig. S1A. Note
that the single dot singlet-triplet splitting, between S(0,2) and T(0,2), is much larger than the
double dot singlet-triplet splitting, between S(1,1) and T(1,1).
The main panels in Figs. S1C and D each show two blurred triangles that are partially superimposed. As a guide to the eye these are outlined in white. Together, the triangles define
the area in which transport is energetically allowed (3). The two triangles correspond to two
different current cycles, commonly known as the electron cycle and hole cycle. All of the data
in this paper correspond to transport by the electron cycle, (0, 1) → (1, 1) → (0, 2) → (0, 1).
The hole cycle, (1, 2) → (1, 1) → (0, 2) → (1, 2), exhibits features similar to those visible in
the electron cycle, although slight differences exist.
Under forward bias, blockade due to the T(1,1) state (see text) is clearly visible in Fig. S1C
(as in Fig. 1C), both for the electron and hole cycles. Pauli blockade is lifted in the upper left
of the triangles, where the transition from T(1,1) to T(0,2) is energetically allowed (see also the
level diagram outlined in red).
Reversing the voltage bias across the device gives the opposite cycles: (0, 1) → (0, 2) →
(1, 1) → (0, 1) for the electron cycle and (1, 2) → (0, 2) → (1, 1) → (1, 2) for the hole cycle.
In these cases the system is in (0,2) before it is in (1,1), so it can never get stuck in T(1,1) and
Pauli blockade does not occur.
1
Definition of detuning
Throughout this work, detuning is defined so that the energy of T(1,1) remains constant as
the energy of the (0,2) states vary. The energy of T(1,1) includes both the left and right dot
electrostatic potentials. Therefore, in order to keep the T(1,1) energy constant while changing
the energy of the (0,2) states, both VL and VR have to be changed, but in opposite directions.
This axis in parameter space is shown by the gray line in Figs. 1B and S1C. The T(1,1) electrochemical potential (the energy needed to transition from (0,1) to T(1,1)) moves when changing
detuning (Fig. S1B), even though the energy of the T(1,1) state remains the same. This is because detuning changes the energy of the (0,1) state (3).
Other processes that lead to leakage currents
Spin exchange with the leads allows a blocked T(1,1) state to change into S(1,1), which
can then move to S(0,2). Alternatively, a second order virtual tunnelling process can bring
T(1,1) through the energetically forbidden T(0,2) state. Both effects were observed, when the
dot levels were close to the lead chemical potentials or to the T(0,2) chemical potential respectively (for example outside the dotted lines in Fig. 1A). However, the measurements presented
in Figs. 3 and 4 were taken with dot parameters at which these processes were insignificant. As
further evidence that these processes are not relevant in our measurements, neither would have
a strong dependence on in-plane magnetic field. A spin-orbit interaction may also couple triplet
to singlet states, but the rate for this process is very slow for states close together in energy (4).
Instead, the field dependence observed in our measurements is consistent with an inhomogeneous field between the two dots of order 2 mT; the magnitude of this field, combined with
field hysteresis and the observation of long relaxation times demonstrate that the origin of the
leakage current is the hyperfine interaction with the nuclear spins of the semiconductor lattice.
2
Model for the magnetic field dependence of the inelastic leakage current
A model for the transport cycle in the Pauli blockade regime in the presence of an inhomogeneous nuclear field has been developed by O. Jouravlev and Y. Nazarov. This model is based
on a density matrix approach. When an electron tunnels into the left dot from the left lead (with
rate ΓL ), the probability to form a singlet or any of the three triplet states is defined to be the
same (P=0.25 for each state). The transition from S(1,1) to S(0,2) is described as an incoherent
process with rate Γi , and the tunnel rate of the electron to the right lead ((0,2) to (0,1)) is defined
by ΓR . The degree of mixing of the singlet and triplet states is calculated from the set of eigenstates of the system hamiltonian. This hamiltonian includes the coherent coupling t between the
S(1,1) and S(0,2) state, the inhomogeneous and homogeneous components of the nuclear field,
and the external (homogeneous) magnetic field.
The nuclear field is thought to have a correlation time of order 100µs (5). Because this
time is much longer than the time it takes for an electron to move through the two dots, it is
appropriate to consider a fixed nuclear field for each electron transport cycle. In the experiment,
we measure the current through a 10 Hz filter, and collect a data point once every 100 ms. Each
point in the measured current traces therefore represents an average over 1000 different nuclear
fields.
The time-averaged fluctuating nuclear field is incorporated in the model by assuming independent Gaussian distributions of the field in each of the two dots, centered around 0 mT and
with the same standard deviation σ for the x̂, ŷ and ẑ components such that 3σ 2 = hBN 2 i.
Here, BN is the rms strength of the nuclear field in a single dot. With these definitions, the
relation between σ and the rms strength of the inhomogeneous component of the nuclear field,
p
p
√
as defined in the text, is h∆BN 2 i = 2hBN 2 i = 6σ.
The model predicts that in the limit of (∆ST , Γi ) EN , the peak shape depends only on
3
σ. This is consistent with the qualitative picture presented in the text and the inset of Fig.2C, as
well as with the experimental results shown in Fig. 3B.
The width of the calculated peak scales linearly with the parameter σ. The best fit with the
p
data gives σ = 0.705 ± 0.008 mT corresponding to h∆BN 2 i = 1.73 ± 0.02 mT. This value is
used for the calculated peak shape shown in Fig. S2, overlaid with the experimentally measured
leakage current. We note that the model fits well not only to the sides and tails of the measured
peak, but also reproduces the small dip near 0 mT.
Bistability of the nuclear polarization
Multiple stable values for the leakage current (see e.g. Fig. 4B) suggest multiple stable
configurations of the electron-nuclear spin system. The existence of multiple stable configurations can arise from the competition between several polarization and depolarization processes.
Dynamic nuclear polarization close to the |T+ i − |Si transition tends to enhance the external
field, while the |T− i − |Si transition opposes the field. At 150 mK, nuclear spin diffusion and
spin-lattice relaxation processes always lead to depolarization.
In contrast to previous reports on bistability of the electron-nuclear system when sweeping
field (as in (6, 7)), we observe that the system can fluctuate in time between stable points. We
believe that the electron-nuclear system can be kicked out of a stable point for instance by
background charge fluctuations or statistical fluctuations in the nuclear field, and reach another
stable point through dynamic nuclear polarization. When such processes occur repeatedly, they
lead to a randomly fluctuating leakage current.
We stress that charge fluctuations alone cannot explain our observations. Such an explanation would be inconsistent with the observations of Figs. 4D and S3, as explained in the main
text.
4
Fig. S1
A
B
T(0,2)
S(0,2)
T(1,1) S(0,2)
T(1,1)
T(1,1) S(0,2)
S(1,1)
T(1,1)
Detuning<0
Detuning=0
C
Detuning>0
Blockade, Bias=800mV
Ho
1000
le C
ycl
e
T
800
S
I (fA )
600
on
ctr
Ele
400
cle
Cy
VR (mV )
S(0,2)
T
200
Bext=100mT
S
T
S
0
V L (mV)
D
No Blockade, Bias=-800mV
Ho
le C
ycl
e
-1000
T
on
ctr
cle
Cy
-400
-200
axes reversed
Bext=0mT
S
I (fA )
-600
Ele
VR (mV )
-800
T
S
T
S
0
V L (mV)
Figure 1: The Pauli blockade observed with forward bias is absent for reverse bias. (A) The
level diagram shows the electrochemical potentials for transitions from the (0,1) state to the
(0,2) and (1,1) singlet and triplet states. (B) The electrochemical potentials move with detuning
as shown (see online text). (C) Color-scale plot of the current through two coupled dots as
a function of the left and right dot potentials (voltage bias 800 µeV, Vt = −181 mV). White
triangles illustrate the electron and hole cycles. Pauli blockade is visible in the lower part of
each triangle. The schematics to the right of the data depict the electrochemical potentials for
three different settings of VL and VR (colored dots). (D) Similar data for opposite bias (-800
µeV). No blockade is visible and current flows throughout the area defined by the two triangles.
These data are taken at zero field, but no field dependence was observed up to 100 mT for this
bias direction.
Fig. S2
2
= 1.73mT
I (fA)
∆BN
Bext (mT)
Figure 2: The calculated peak shape (solid red line) shows excellent agreement with the measured inelastic leakage current (crosses, Vt = −206 mV, ∆LR = 50 µeV).
Fig. S3
DLR +12 meV
1500
DLR +8 meV
1200
DLR +4 meV
900
I (fA)
DLR +0 meV
600
D LR -4 meV
300
DLR -8 meV
0
0
100
50
Time (sec)
Figure 3: Time dependence of the resonant leakage current at several values of ∆LR , with
the nuclear system initialized to zero polarization for each trace (the position of ∆LR = 0 is
approximate, but the step size is accurate). The fluctuations change character sharply when
∆LR is changed by only 4 µeV, which corresponds to a change in VL and VR of only 50µV
(Vt = −149 mV, Bext = 200 mT).
References and Notes Supporting Online Material
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(2004).
3. W.G. van der Wiel et al., Rev. Mod. Phys. 75, 1 (2003).
4. A.V. Khaetskii, Y.V. Nazarov, Phys. Rev. B 61, 12639 (2000).
5. R. de Sousa, S. Das Sarma, Phys. Rev. B 67, 033301 (2003).
6. M. Dobers, K. v. Klitzing, J. Schneider, G. Weimann, K. Ploog, Phys. Rev. Lett. 61, 1650
(1988).
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1984).