Abstract
Double quantum dots are convenient solidstate platforms to encode quantum information. Twoelectron spin states can be detected and manipulated using quantum selection rules based on the Pauli exclusion principle, leading to Pauli spin blockade of electron transport for triplet states. Coherent spin states would be optimally preserved in an environment free of nuclear spins, which is achievable in silicon by isotopic purification. Here we report on a deliberately engineered, gatedefined silicon metaloxidesemiconductor double quantum dot system. The electron occupancy of each dot and the interdot tunnel coupling are independently tunable by electrostatic gates. At weak interdot coupling we clearly observe Pauli spin blockade and measure a large intradot singlettriplet splitting > 1 meV. The leakage current in spin blockade has a peculiar magnetic field dependence, unrelated to electronnuclear effects and consistent with the effect of spinflip cotunneling processes. The results obtained here provide excellent prospects for realising singlettriplet qubits.
Introduction
Gatedefined semiconductor quantum dots enable the confinement and manipulation of individual electrons and their spin^{1}. Most of the relevant parameters – electron filling, energy splittings, spin states, exchange interaction – can be tuned in situ by electric and magnetic fields. Because of this exquisite level of control, quantum dots are being investigated as candidate systems for spinbased quantum information processing^{2}. In group IIIV semiconductors such as GaAs, the development of highly tunable double quantum dots has allowed the study of both singleelectron and twoelectron spin dynamics^{3,4,5,6,7}. However, the nuclear spins always present in these materials produce strong decoherence of the electron spin degree of freedom and result in phase coherence times T_{2} of below 1 ms^{8,9}. Conversely, groupIV semiconductors such as silicon, silicongermanium and carbon can be isotopically purified, leaving only spinless isotopes. The weak spinorbit coupling^{10} and the absence of piezoelectric electronphonon coupling^{11} allow for extremely long spin relaxation times T_{1} of order seconds, as already demonstrated in several experiments^{12,13,14}. The phase coherence times have not been measured yet, but they are expected to reach ∼ 1 s as well, in highly purified ^{28}Si substrates with low background doping concentration^{15}.
A widely successful method to observe and control spin phenomena in quantum dots^{1} consists of defining a double quantum dot in a series configuration and tuning the potentials such that sequential electron transport requires a stage where two electrons must occupy the same dot. The eigenstates of a twoelectron system are singlet and triplet spin states, separated by an energy splitting Δ_{ST} which can be large in tightly confined dots. The electron transport then becomes spindependent and can be blocked altogether when the twoelectron system forms a triplet state^{5,16}. This phenomenon, known as Pauli spin blockade, has been extensively exploited to investigate the coherence of singlespin^{4} and twospin states^{3} in GaAs and InAs^{17} quantum dots. Therefore, observing and controlling spin blockade in silicon is a key milestone to unravel the full potential of highly coherent spin qubits. Preliminary success has been obtained in Si^{18} and SiGe^{19} devices, but in each case the double dot system under study resulted from local variations in the potential of a lithographicallydefined single dot, making it difficult to control individual dot occupancies or interdot coupling. Spinbased quantum dot qubits require exquisite control of these parameters, so a highly tunable doubledot system in silicon is essential. For singlettriplet qubits in multivalley semiconductors it is also crucial to ensure that a large valleyorbit splitting is present, to avoid the lifting of Pauli blockade due to valley degeneracy^{20,21}.
Here we present an engineered silicon double quantum dot which shows excellent tunability and robust charge stability over a wide range of electron occupancy (m, n). The silicon metaloxidesemiconductor (MOS) structure utilizes an AlAl_{2}O_{3}Al multigate stack that enables very small dots to be defined, each with independent gate control, together with gatetunable interdot coupling. Such multigate stacks have previously been used to construct single Si quantum dots with the ability to achieve single electron occupancy^{22}. The double dot presented here exhibits spin blockade in the fewelectron regime, from which we are able to extract a large singlet–triplet energy splitting and also investigate a new mechanism giving rise to singlet–triplet transitions in the weakcoupling regime.
Results
Device architecture
Figure 1 shows a scanning electron micrograph (SEM) and crosssectional schematic of the device, which incorporates 7 independently controlled aluminium gates. When a positive bias is applied to the lead gates (L1 and L2) an accumulation layer of electrons is induced under the thin SiO_{2}, to form the source and drain reservoirs for the double dot system. A positive voltage on the plunger gate P1 (P2) causes electrons to accumulate in Dot 1 (Dot 2). Independent biasing of P1 and P2 provides direct control of the doubledot electron occupancy (m, n). The tunnel barriers between the two dots and the reservoirs are controlled using the barrier gates: B1, B2 and B3. The middle barrier gate B2 determines the interdot tunnel coupling. The electrochemical potentials of the coupled dots can also be easily tuned to be in resonance with those of the source and drain reservoirs. As shown in Fig. 1(b), gates L1 and L2 extend over the source and drain n^{+} contacts and also overlap gates B1 and B3. The upperlayer gates (P1 and P2) are patterned on top of the lead and barrier gates. The lithographic size of the dots is defined by the distance between adjacent barrier gates (∼30 nm) and the width of the plunger gates (∼50 nm), as shown in Fig. 1(a).
Interdot tunnel coupling tunability
Figure 2 shows the measured differential conductance of the device as a function of the plunger gate voltages, V_{P1} and V_{P2}, with all other gate voltages held constant, together with sketches of the energy landscape of the double dot. The chargestability maps moving from Fig. 2(a) to 2(c) clearly show the effects of an increasing interdot coupling as the middle barriergate voltage V_{B2} is increased, lowering the tunnel barrier between the dots. Fig. 2(b) shows the characteristic honeycombshaped stability map representing intermediate interdot coupling^{23}, obtained at V_{B2} = 1.32 V. At lower middle barriergate voltage, V_{B2} = 1.20 V, we observe a checkerbox shaped map [Fig. 2(a)], since the middle barrier is opaque enough to almost completely decouple the two dots. In contrast, the stability map in Fig. 2(c) shows the formation of diagonal parallel lines at V_{B2} = 1.40 V. Here the two dots effectively merge into a single dot due to the lowering of the middle barrier [Fig. 2(f)]. Increasing V_{B2} further results in stronger differentialconductance lines, suggesting a simultaneous increase in dotlead couplings. The transport measurements shown here do not allow a precise determination of the electron occupancy (m, n) in the dots, since it is possible that electrons remain in the dots even when I_{SD} is immeasurably small. For the regime plotted in Fig. 2 there were at least 10 electrons in each dot, based on our measurement of Coulomb peaks as we further depleted the system. An absolute measurement of dot occupancy would require integration of a charge sensor into the system^{7}. These results nevertheless demonstrate that the multigated structure provides excellent tunability of coupling while maintaining charge stability over a wide range of electron occupancy.
Capacitances and charging energies
Application of a DC sourcedrain bias V_{SD} causes the triplepoints in the weaklycoupled regime [Fig. 2(a)] to extend to form triangular shaped conducting regions [Fig. 3(a)] from which the energy scales of the double dot system can be determined^{23}. From a triangle pair, we extract the conversion factors between the gate voltages and energy to be α _{1} = eV_{SD}/δV_{P1} = 0.089e and α _{2} = eV_{SD}/δV_{P2} = 0.132e, where δV_{P1} and δV_{P2} are the lengths of the triangle edges, as shown in Fig. 3(a). The total capacitances of Dot1 and Dot2 can then be calculated^{23}, giving C_{1} = 16.3 aF and C_{2} = 14.5 aF. The accuracy of these values is around 5%, limited by the accuracy with which our data can be fitted by the superimposed (dotted) triangles in Fig. 3(a).
From the above analysis, we find the charging energies of the two dots to be E_{C,1} = e^{2}/C_{1} = 9.8 meV and E_{C,2} = e^{2}/C_{2} = 11 meV, indicating that the left dot is slightly larger than the right dot. We note that a previous study for a single quantum dot^{22} reported a charging energy of 6 meV at an electron occupancy of ∼ 40 electrons. In Fig. 3 we estimate an electron occupancy of 10 or less in each dot and the larger charging energies measured here are consistent with the lower occupancy and hence smaller size, of these dots.
Pauli spin blockade
Figure 3 shows the current I_{SD} through the double dot as a function of the two plunger gate voltages when measured with both positive [Fig. 3(a)] and negative [Fig. 3(b)] sourcedrain biases. Here we observe a suppression of current at one bias polarity, the characteristic signature of Pauli spin blockade^{6,7}. At V_{SD} = +2.5 mV we observe a pair of overlapping full bias triangles, as shown in Fig. 3(a). Resonant transport through the ground state and the excited states in the double dot occurs when the states within the dots are exactly aligned, leading to peaks in the current which appear as straight lines parallel to the triangle base in Fig. 3(a). The nonresonant background current level at the centre of the triangle is attributed to inelastic tunneling. The nonzero current throughout the triangular region indicates that electrons from the reservoir can tunnel freely from the S(0,2) singlet state to the S(1,1) singlet state, as depicted in the cartoon (red box in Fig. 3). Note that here we define (m, n) as the effective electron occupancy^{18}, while the true electron occupancy is (m+m_{0}, n+n_{0}). The Pauli blockade expected for twoelectron singlet and triplet states occurs when the total electron spin of each dot is zero in the (m_{0}, n_{0}) state.
At the complementary negative bias of V_{SD} = −2.5 mV we observe strong current suppression in the region bounded by the dashed lines in Fig. 3(b). The suppression arises because the transition from T(1,1) to S(0,2) is forbidden by spin conservation during electron tunneling. Once the T(1,1) triplet state is occupied, further current flow is blocked until the electron spin on one dot reverses its orientation via a relaxation process (green star box in Fig. 3)^{6,7}.
Note that for both positive V_{SD} [Fig. 3(a)] and negative V_{SD} [Fig. 3(b)] the current I_{SD} increases as V_{P1} and V_{P2} increase, leading to apparent asymmetry in the bias triangles, with the highest currents in the topright of both figures. For V_{SD} = −2.5 mV this leads to a weak conducting region (light red) at the base of the bias triangle. The asymmetry indicates that the double dot system is more strongly coupled to the drain contact than to the source.
Singlettriplet splitting
In a magnetic field B there are four accessible spin states: the singlet S; and three triplets T_{−}, T_{0} and T_{+}, corresponding to S_{Z} = −1, 0, +1. The singlet–triplet splitting Δ_{ST} is the energy difference between the blockaded ground state S(0,2) and the excited state T_{−}(0,2)^{7,18}. Here we study Δ_{ST} as a function of B, applied parallel to the substrate, by measuring spin blockade at a negative bias. Figures 4(a–c) show the bias triangles in the spin blockade regime at increasing magnetic fields B = 2, 4 and 6 T, with the splitting Δ_{ST} marked in Fig. 4(a). For B > 700 mT the current in the spinblockaded region is fully suppressed and so, in order to identify the base of the effective bias triangles [dotted lines in Figs. 4(a–c)], we use the dimensions of the (nonblockaded) bias triangles for V_{SD} = +2.5 mV and align these to the visible peaks of the triangles for V_{SD} = −2.5 mV. The measured splitting Δ_{ST} decreases linearly with increasing B [Fig. 4(d)], as expected, since the triplet states split linearly by the Zeeman energy, E_{Z} = ±S_{Z}gµ_{B}B, where µ_{B} is the Bohr magneton and S_{Z} is −1, 0, +1. A linear fit through Δ_{ST}(B) yields a Landé gfactor of 2.1 ± 0.2, consistent with electrons in silicon.
We observe a very large value of the (0,2) singlettriplet splitting at B = 0, Δ_{ST} ≈ 1.4 meV. If this were a true twoelectron double quantum dot, the result would imply that the nearest valleyorbit state was at least 1.4 meV above the ground state. The first excited valleyorbit state should be a combination of the ±z valleys and would lift the spin blockade^{20,21}, showing no remarkable energy shift in a magnetic field, however, since the electron occupancy in our dots is unknown and larger than two, no conclusive statements can be made on the absence of lowlying valleyorbit states.
Leakage current in blockade regime
If some mechanism exists to mix the singlet and triplet states or to induce transitions between them, then the spin blockade can be lifted, leading to a measurable leakage current^{6}. Here, we observe leakage currents in the spin blockade regions for low values of magnetic field, B < 700 mT. Fig. 5(a) shows the surface plot of the leakage current I_{SD} as a function of both detuning and magnetic field B, while Figs. 5(b) and 5(c) show line traces of I_{SD} as a function of B at zero detuning and I_{SD} as a function of at zero magnetic field, respectively. We find that the leakage current has a maximum at B ≈ 0 and falls to zero at B ∼ 700 mT. As discussed below, we find that the transition from triplet to singlet is well explained by spinflip cotunneling^{24}, resulting in a nonzero timeaveraged leakage current via the mechanism illustrated in Figure 5(d).
Discussion
The suppression of leakage current by an applied magnetic field has been observed in GaAs double quantum dots^{6} and attributed to the effect of hyperfine coupling between the electron spins and the surrounding bath of nuclear spins. In that case the width δB of I_{SD}(B) yields the average strength of the hyperfine field. For an unpolarized nuclear spin bath , where δB_{max} is the hyperfine field assuming fully polarized nuclei and N is the number of nuclei overlapping with the electron wave function. For a typical GaAs dot overlapping with ∼ 10^{6}–10^{7} nuclei, δB_{max} ∼ 6 T ⇒ δB ∼ 2–6 mT^{6,7,25}. In natural silicon, however, the hyperfine interaction is much smaller than in GaAs, with δB_{max} ≈ 1.9 mT^{26}. Here instead we found that B > 300 mT is necessary to suppress the leakage current. Therefore, hyperfine coupling can be ruled out as a mechanism for the lifting of spin blockade.
An alternative mechanism for a transition from triplet to singlet has been recently proposed, where the spin flip is caused by inelastic cotunneling involving one of the leads^{24}. The spinflip rates due to cotunneling from the spinpolarized triplet states, T_{±}(1,1), are exponentially suppressed when the Zeeman energy is large compared to the thermal broadening of the electron states in the leads (i.e., for gµ_{B}B > k_{B}T, where T is the electron temperature and B is the applied magnetic field). This is because the excitation processes on the dot require the removal of an electron from above the lead Fermi level and the creation of an electron below the Fermi level [see diagram in Fig. 5(d)].
A rateequation analysis accounting for the energy dependence of the spinflip cotunneling rates^{24} then gives a simple form in the limit of weak interdot tunneling t and weak cotunneling compared to the tunnel rates Γ_{S,D} between a dot and its nearby source or drain lead (,):
Here, the B = 0 spinflip cotunneling rate (for and , eV_{SD}) is:
with mutual (interdot) charging energy U′ and Δ = α _{1}δV_{P1} + α _{2}δV_{P2} for plunger gate voltages δV_{P1,P2} measured from the effective (0, 1) – (1, 1) – (0, 2) triple point (lowerleft corner of the bias triangle in Fig. 3(b)). Eq. (2) accounts for virtual transitions between effective (1, 1) and (0, 1) (first term) as well as effective (1, 1) and (1, 2) charge states (second term).
In the present case, Δ ≃ eV_{SD} ≫ U′. The higher current level in the upper right corner of Fig. 3(b) further suggests Γ_{D} ≫ Γ_{S}, giving (for this particular experiment):
Using the above expression for , we then use Eq. (1) to fit to the I_{SD}(B) data in Fig. 5(b), giving us Γ_{D} = 30 µeV for the tunneling rate and T = 155 mK for the electron temperature.
The B = 0 spinflip cotunneling rate is energyindependent in the limit . However, the leakage current does acquire a dependence on the energy detuning, , when the escape rate from the doubledot due to resonant tunneling is suppressed below the spinflip cotunneling rate. This leads to a Lorentzian dependence of the current on detuning with a tdependent width :
Eq. (4) is valid in the same limit (,) as Eq. (1). In the strongtunneling limit, , the theory predicts that I() should show a strong resonanttunneling peak of width ∼ t, followed by a slowlyvarying Lorentzian background described by Eq. (4) at large . The absence of a strong resonanttunneling peak in the data of Fig. 5(c) confirms that the device is operating in the regime , justifying our use of Eqs. (1) and (4) to analyse the data.
A nonlinear fit to the I_{SD}() data [for V_{B2} = 1.1176 V in Fig. 5(c)] using Eq. (4) yields t = 0.5 µeV for the interdot tunneling rate, using our previously determined values Γ_{D} = 30 µeV and T = 155 mK. These parameter values are well within the experimentally expected range. The small value of t indicates weak interdot tunnel coupling, consistent with the results shown in Fig. 3(b). By independently tuning the interdot tunnel coupling via control of the middle barrier gate voltage, we have verified that by increasing V_{B2} to 1.1186 V, the leakage current acquires a higher absolute value and a broader Bdependence [blue trace in Fig. 5(c)], as predicted. We note that as V_{B2} is made more positive, the interdot coupling t increases as expected and the dotlead couplings Γ_{D} also increase somewhat. We conclude that the spinflip cotunneling mechanism provides a consistent explanation of the observed leakage current in the spin blockade regime. The mechanism could be applied to reanalyse previous experiments in group IV semiconductors^{27} where the nature of the leakage current was not fully understood.
In conclusion, we have presented a lithographicallydefined double quantum dot in intrinsic silicon showing excellent charge stability and low disorder. The multigate architecture provides independent control of electron number in each dot as well as a tunable tunnel coupling. We observed Pauli spin blockade in an effective twoelectron system from which we extracted the singlet–triplet splitting. The leakage current in the spin blockade regime is well explained by a spinflip cotunneling mechanism, which could be of widespread importance in groupIV materials with weak hyperfine coupling. The results obtained here provide a pathway towards investigation of spin blockade in silicon double quantum dots with true (1,1) and (2,0) electron states. Towards this end, we are planning future experiments incorporating a charge sensor to monitor the last few electrons^{28}. We anticipate that such an architecture will provide excellent prospects for realising singlet–triplet qubits in silicon^{29}.
Methods
Fabrication steps
The devices investigated in this work were fabricated on a 10 kΩcm n–type high resistivity 〈100〉 silicon wafer using standard microfabrication techniques. The n^{+} source and drain ohmic contacts regions in Fig. 1(b) were produced via high concentration phosphorus diffusion at ∼ 1000°C, resulting in peak dopant densities of ∼ 10^{20} cm^{−3}. Next, the highquality SiO_{2} of 10 nm thickness was grown via dry thermal oxidation in the central region at 800°C in O_{2} and dichloroethylene. The barrier gates were first patterned on the thin SiO_{2} region using electron beam lithography (EBL) followed by thermal evaporation of 40 nm thick aluminium and liftoff process. Before the next EBL step, the barrier gates are exposed to air for 10 mins at 150°C to form ∼ 4 nm of Al_{2}O_{3} acting as a dielectric layer. This process was repeated for lead gates and plunger gates layers with aluminium thicknesses of 40 nm and 120 nm respectively. A final forming gas anneal (95% N_{2} and 5% H_{2}) was performed for 15 mins to achieve a low density of SiSiO_{2} interface traps, of order 10^{10} cm^{−2} eV^{−1}, as measured on a similarly processed chip^{30}. The low trap density is clearly reflected in the device stability and the low level of disorder observed in the transport data shown in the results section.
Experimental setup
Electrical transport measurements were carried out in a dilution refrigerator with a base temperature T ∼ 100 mK. We simultaneously measured both the DC current and the differential conductance dI/dV_{sd}, the latter using a sourcedrain AC excitation voltage of 100 µV at 87 Hz.
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Acknowledgements
The authors thank D. Barber and R. P. Starrett for their technical support and acknowledge the infrastructure support provided by the Australian National Fabrication Facility. This work was funded by the Australian Research Council, the Australian Government and by the U. S. National Security Agency and U.S. Army Research Office (under Contract No. W911NF0810527). W.A.C. acknowledges the funding from the CIFAR JFA. F.Q. acknowledges funding from NSERC, WIN and QuantumWorks.
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Contributions
N.S.L. fabricated the devices. N.S.L., W.H.L. and C.H.Y. designed and performed the experiments. W.A.C. and F.Q. modelled the spinflip cotunneling rate. N.S.L., F.A.Z., W.A.C., F.Q., A.M. and A.S.D. wrote the manuscript. A.S.D. planned the project. All authors discussed the results and commented on the manuscript at all stages.
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Lai, N., Lim, W., Yang, C. et al. Pauli Spin Blockade in a Highly Tunable Silicon Double Quantum Dot. Sci Rep 1, 110 (2011). https://doi.org/10.1038/srep00110
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DOI: https://doi.org/10.1038/srep00110
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