## Abstract

We observe spin blockade due to Pauli exclusion in the tunneling characteristics of a coupled quantum dot system when two same-spin electrons occupy the lowest energy state in each dot. Spin blockade only occurs in one bias direction when there is asymmetry in the electron population of the two dots, leading to current rectification. We induce the collapse of the spin blockade by applying a magnetic field to open up a new spin-triplet current-carrying channel.

Current flow through an electronic system between contact leads (source and drain leads) is determined by the transition rates between the source lead and the drain lead, via states in the electronic system. When inversion symmetry is absent in the system, the electrical current can provide bias-dependent rectification. This was first suggested for a single molecule with appropriately configured molecular orbitals by Aviram and Ratner (1) many years ago, and very recently for multiple dots suffering from the Coulomb blockade effect (2). In both of these cases, the rectifying effect depends on bias-dependent transition rates and is characterized by irreversible trapping in a state from which current can no longer proceed.

Transition rates in electronic systems depend in general on the Pauli exclusion principle, which prevents two electrons of parallel spin from occupying a single spatial orbital. Observation of the Pauli effect in condensed matter systems depends, however, on low temperatures and reduced dimensions such that the energy spacing of states at the Fermi level is comparable to, or larger than, the thermal energy. One recently observed manifestation of this exchange (i.e., Pauli exclusion) effect in one-dimensional (1D) systems is the suppression of the current noise due to anti-bunching of electrons (3, 4). Here, we show that in transport through a series of 0D sites, with symmetry under inversion appropriately broken, the Pauli effect, in combination with the Coulomb blockade, can be used to block current altogether in one direction while permitting it to flow in the opposite direction, thereby realizing a fully controllable spin-Coulomb rectifier.

Quantum dots are often referred to as artificial atoms because the electrons inside are well confined in 0D states and the electronic properties are analogous to those of real atoms (5,6). The electronic configuration and the number of electrons in the quantum dot are parameters that can be easily manipulated (6–8). A quantum dot containing one electron in a single orbital state is just like a hydrogen atom. We use two weakly coupled hydrogen-like quantum dots to construct a diatomic molecule that forms the two-electron two-site system. We can then observe the consequences of Pauli exclusion directly on single-electron tunneling via the two-electron states.

Suppose we have two sites, site 1 and site 2, weakly coupled together, and one electron is permanently localized on site 2 (Fig. 1A). Now consider transport of a second electron through the system between two contact leads. The number of electrons on site 1 (2), *N*
_{1}(*N*
_{2}), varies between 0 and 1 (1 and 2). An electron can only be transported through the system when the transition between the two-electron states (*N*
_{1},*N*
_{2}) = (0,2) and (1,1) is allowed. This condition is met when the necessary energy cost to add one more electron to the system is compensated for by the action of a nearby plunger gate voltage, or the voltage between the leads, which is a familiar single-electron charging phenomenon (5).

Crucially, spin effects also markedly influence electron transport. Because the tunnel coupling between the two sites is sufficiently weak, the (*N*
_{1},*N*
_{2}) = (1,1) spin-singlet and spin-triplet states are practically degenerate (9). Additionally, for (*N*
_{1},*N*
_{2}) = (0,2), only a spin singlet is permitted because of Pauli exclusion. Therefore, electron transport is only allowed for a channel made from the (1,1) and (0,2) singlet states. This always holds true for reverse bias when the chemical potential of the left lead, μ_{ℓ}, nearest site 1, is lower than that of the right lead, μ_{r}, nearest site 2, because only an antiparallel spin electron can be injected onto site 2 from the right lead (Fig. 1A). On the other hand, for forward bias, μ_{ℓ} > μ_{r}, either the (1,1) singlet or triplet can be populated with more or less the same probability by injection of an electron onto site 1 from the left lead. If the (1,1) singlet is populated, a single-electron tunneling current can flow through the singlet state. Once the triplet is populated, however, subsequent electron transfer from site 1 to 2 is blocked by Pauli exclusion. Note that an electron arriving on site 1 usually cannot go back to the left lead because of the fast relaxation of the hole state left behind in the lead (10). Thus the (1,1) triplet will sooner or later be occupied on a time scale sufficiently longer than the electron tunneling time between the leads, and this should lead to clear current suppression, for example, in dc measurement. Because this blockade is due to spin and not charge, we hereafter refer to this process as “spin blockade,” and this provides current rectification. Blockade of single-electron transport associated with spin has been reported for single-dot (11–13) and double-dot (14,15) systems. In most of these cases, blockade appears only when the difference in the total spin between the *N* and*N* + 1 electron ground states (GSs) is greater than 1/2. Such a large difference cannot be made up by single-electron tunneling events, but this itself is not related to Pauli exclusion.

Our double-dot device is made by vertically coupling two circular quantum dots that are located between two contact leads called the source and drain (Fig. 1B) (16, 17). The lateral confinement in each dot imposed by the surrounding Schottky gate is well approximated by a 2D harmonic potential (6,18). A quantum dot with such a potential has atomlike electronic properties: shells composed of 1*s*, 2*p*, 3*s, *3*d*, … orbitals and the filling of near degenerate states in accordance with Hund's first rule (6,7). The typical characteristic energy, ℏ︀ω_{0}, of the lateral confinement in each dot is about 4 meV for the lowest 1*s* orbital state (18). Located ℏ︀ω_{0} above is the excited 2*p* orbital state. In our vertical device configuration, the total number of electrons in the whole double-dot system, *N* (=*N*
_{1} + *N*
_{2}), can be varied one-by-one as a function of gate voltage,*V*
_{G}, starting from *N* = 0 (6, 17). Here, we label the two dots “dot 1” and “dot 2,” and they correspond to site 1 and site 2 in Fig. 1A, respectively. Similarly, *N*
_{1}(*N*
_{2}) is now the number of electrons in dot 1 (dot 2). The transmission coefficients for all the tunnel barriers (dot-contact lead and dot-dot) are sufficiently weak that electron transport can be discussed just in terms of sequential tunneling between the source and drain.

The general situation of Fig. 1A can be reproduced in our structure (see Fig. 2A, potential diagrams) if there is an appropriate potential offset, 2δ, between the two quantum dots at zero source-drain voltage, *V* = 0 V (19, 20). Then, just one electron is trapped in the 1*s* orbital state of dot 2, and the two-electron GS is either (*N*
_{1},*N*
_{2}) = (1,1) or (0,2). For (1,1), the singlet and triplet states are nearly degenerate because the coupling between the two dots is very weak (9). For (0,2), only the singlet GS is initially relevant. A (0,2) triplet excited state (ES) can be formed by putting two parallel spin electrons in the 1*s* and 2*p* states in dot 2. However, its energy is normally much higher than that of the (1,1) and (0,2) GS, so it does not influence the transport in our discussion for the moment. When viewing Fig. 1A, we can see that electrons can be transported by the (0,2) and (1,1) singlet states for reverse bias, whereas for forward bias, the (1,1) triplet can be populated, leading to the blockade of electron transport. We now define the electrochemical potential of the (*N*
_{1}, *N*
_{2}) GS to be μ(*N*
_{1}, *N*
_{2}) and the chemical potential of the source (drain) lead to be μ_{r} (μ_{ℓ}). Spin blockade in Fig. 1A can appear when electrons are injected from the drain to the (1,1) triplet in the nonlinear transport (21). We assume that spin is conserved in the electron tunneling throughout our double-dot system and that there are no spin flips on a time scale sufficiently longer than the electron tunneling time. This condition actually holds for our experiments, which we will explain later.

We measure the dc current, *I*, flowing vertically through the two dots as a function of *V*
_{G} and *V* to study the linear and nonlinear electron transport.*V*
_{G} and *V*, respectively, are used to change the electrostatic potential of the two dots together, and the potential offset between the two dots. By adjusting these voltages and using a double-dot sample with an appropriate potential offset 2δ (19), we can realize the situation in Fig. 1A. d*I*/d*V* – *V*
_{G} measured for small *V* (∼0 V) shows clear Coulomb oscillation peaks (Fig. 2A, lower right inset). *N* increases one-by-one, starting from *N* = 0 every time a current peak is crossed as*V*
_{G} is made more positive. The first peak (X) is very small but definitely present at *V*
_{G} ∼ –2 V, and this indicates transport through the double-dot system for*N* fluctuating between 0 and 1. The second peak (P) and the third peak (Q) are much larger. This implies that tunneling is elastic between the source and drain leads for *N* = 1↔2 and 2↔3. However, because of the potential offset between the two dots, this is not the case for *N* = 0↔1. Electrons must tunnel through the lowest state in dot 2, but since there is no available state for tunneling in dot 1, second-order tunneling through dot 1 is necessary (22). We then study nonlinear transport through the two-electron states by measuring the*I*-*V* for *V*
_{G} fixed at the second Coulomb oscillation peak (Fig. 2A, main curve). The data reveal a large, nearly constant d*I*/d*V* at*V* ∼ 0 V for both bias polarities, and this is due to elastic tunneling in the linear response regime. Tunneling here occurs via the (1,1) and (0,2) singlet states, which are aligned at*V* ∼ 0 V when we take into account the effect of the potential offset 2δ. For ∣*V*∣ ≥ 1 mV, transport is nonlinear (see potential diagrams in Fig. 2A). The current is clearly suppressed in forward bias because of spin blockade, whereas a large current still flows in reverse bias because of inelastic tunneling via the singlet states. The spin blockade is lifted for *V*> 7 mV when an electron can be ejected from the lowest state of dot 2 to the source. The spin blockade is further confirmed from measurements of nonlinear transport for a wide range of *V*
_{G}values (Fig. 2B).

The spin blockade should appear everywhere the forward bias situation of Fig. 1A is satisfied in the (*V*,*V*
_{G}) plane. Such a region is indeed observed in the experimental data of d*I*/d*V* versus*V* and *V*
_{G}, which is plotted on a logarithmic scale in Fig. 2B. The central three diamond-shaped regions of d*I*/d*V* ∼ 0 siemens identify the*N* = 1 to 3 regions of Coulomb blockade. Note that the diamonds touch each other at *V* ∼ 0 V for*V*
_{G} values corresponding to the Coulomb oscillation peaks in Fig. 2A. Application of a sufficiently large*V* will usually lead to the lifting of Coulomb blockade by the addition of an electron to dot 1 (dot 2) from the drain (source), or the removal of an electron from dot 1 (dot 2) to the drain (source). At the threshold for lifting the Coulomb blockade, a large inelastic tunneling current (or large d*I*/d*V*) appears when two states, one for each dot, are present between μ_{ℓ} and μ_{r}. This holds true for all of the*N* > 1 regions of Coulomb blockade, but only partly for the *N* = 1 region. This is because the presence of the potential offset will mean only one state can lie between μ_{ℓ} and μ_{r}. In addition, we can also see that the right boundary of the *N* = 2 Coulomb blockade diamond is marked by a weak d*I*/d*V* threshold. Located to the right is an extended region of small d*I*/d*V*. This is the region where nonlinear transport involving two-electron states occurs. The (1,1) (singlet and triplet) and (0,2) (singlet) states are all located between μ_{ℓ} and μ_{r} of the contact leads, and so the conductance suppression is assigned to the spin blockade (see Fig. 2A, forward bias potential diagram). The spin blockade is lifted when the forward bias is sufficiently large that the electron initially trapped in dot 2 is ejected to the source, or one more electron is injected into dot 1 from the drain. The former (later) is given by the condition μ(0,1) ≥ μ_{r} (μ_{ℓ} ≥ μ (2,1)), which determines the lower right (upper right) boundary of the spin blockade region. Note that to the left of the*N* = 2 Coulomb blockade diamond, there is no such current suppression, because the transition (0,2)→(1,1) via the singlet states is always allowed (see Fig. 2A, reverse bias potential diagram).

In Fig. 2C, with the aid of a simple model, we reproduce important features in the experimental data of Fig. 2B (23). The bold lines enclosing the diamond-shaped regions indicate the thresholds for lifting Coulomb blockade in both dots. The faint dotted lines inside and below the *N* = 1 Coulomb diamond indicate thresholds due to the second-order tunneling (24). The spin blockade region is located to the right of the *N* = 2 Coulomb diamond. In addition, the potential offset 2δ can be “tuned out” to allow elastic tunneling of electrons through the (1,0) and (0,1) states by application of an appropriate reverse bias. This occurs along the vertical dashed line. Figure 2C reproduces quite well the shape and size of all the important features in Fig. 2B, and this allows us to be confident that we correctly identify the measured spin blockade region.

The main *I*-*V* curve in Fig. 2A is actually used to make the data section at *V*
_{G} = –1.854 V in Fig. 2B (horizontal white dashed line). From the level of the current in the spin blockade region (black curve in the upper left inset to Fig. 2A), we estimate the lifetime of the spin triplet to be*e*/*I* ∼ 100 ns. This is markedly longer than the time taken for just single-electron inelastic tunneling between the contact leads (*e*/*I* ∼ 1.6 ns) obtained from the current level for *V* < –1 mV. From Fig. 2B we also find that the magnitude of the current in the spin blockade region is more or less the same as that observed in the vicinity of the lower part of the *N* = 1 Coulomb blockade region. This suggests that the current in the spin blockade region due to the decay of the (1,1) triplet state is also caused by second-order tunneling processes (25) and that the intrinsic spin-flip relaxation time is longer than 100 ns. An even longer relaxation time for spin has recently been reported for a single quantum dot (26). The upper left inset to Fig. 2A also shows the*I* versus *V* curve that cuts across the center of the *N* = 2 Coulomb blockade region (red curve). Here, the current level in the spin blockade region is the same as that for the black curve, but the current level is much lower in the region of Coulomb blockade (<100 fA). This clearly demonstrates that current leakage is negligible in our system.

To explore further the spin blockade region in Fig. 2, A and B, we study the influence of a magnetic field (*B* field) applied parallel to the current flowing through the two dots. In our model, spin blockade in forward bias can be lifted if a (*N*
_{1},*N*
_{2}) = (0,2) triplet is available, and it can be made lower in energy than the (1,1) triplet (Fig. 3A). Then, a spin-triplet channel is opened and current can flow. The (0,2) triplet at low *B* field is the ES with two parallel spin electrons in dot 2, one in the 1*s* state and the other in the 2*p* state. At *B* = 0 T, the 2*p*single-particle state has a twofold orbital degeneracy. The 2*p*
^{+} and 2*p*
^{–} states have orbital angular momentum quantum numbers of +1 and –1, respectively, due to the rotational symmetry in the lateral confinement. When the*B* field is initially increased, the 2*p*
^{+} and 2*p*
^{–} states split apart, and so the energy of the 2*p*
^{+} state approaches that of the 1*s* state. In addition, the Coulomb energy of the two-electron states is increased due to magnetic confinement of the wave functions, but it occurs at a slower rate for the (0,2) triplet than for both the (1,1) triplet and (0,2) singlet, because the 2*p* state is more spatially extended than the 1*s* state. Therefore, the (0,2) triplet becomes energetically favored relative to the other two-electron states (27). To track the *B*field–induced change in energy of the (0,2) triplet relative to the (1,1) triplet, we investigate the related resonance, which gives rise to a current peak in forward bias. The 1*s*-2*p*
^{+}(1*s*-2*p*
^{–}) resonance peak is identified by the line labeled ▴(▵) at *B* = 0 T inFig. 2B (28, 29). The fact that these two resonance lines are slightly separated is probably due to small asymmetry in the lateral confinement. Fig. 3B shows the *B*field–induced downshift of the 1*s*-2*p*
^{+} resonance, which is identified by the current peak labeled ▾ in *I*-*V* curves that cut across the spin blockade region. In Fig. 3C, the plotted shift of both the 1*s*-2*p*
^{+} and 1*s*-2*p*
^{–} resonances reflects directly the expected trend for 1*s* and 2*p* Fock-Darwin states (6, 29). As the *B* field is increased toward 5 T, the 1*s*-2*p*
^{+}current peak approaches the *V* = 0 V axis as it progressively enters the spin blockade region. Spin blockade is then lifted, and we can see enhanced current on and to the right of the current peak. We note that for plots like Fig. 2B, but for*B* ≥ 5 T, part of the 1*s*-2*p*
^{+} resonance line actually enters the *N* = 2 Coulomb blockade region and immediately disappears. This *B* field signature also clearly demonstrates that the spin blockade observed in Fig. 2, A and B, is spin-related rather than charge-related.

Our experiments on a weakly coupled double quantum dot system reveal directly the importance of Pauli exclusion on spin transport. The observed current rectification has possible applications to spin memories, filters, and transistors.

↵* To whom correspondence should be addressed. E-mail: tarucha{at}phys.s.u-tokyo.ac.jp