Abstract
Graphene quantum dots are attractive candidates for solid-state quantum bits. In fact, the predicted weak spin-orbit and hyperfine interaction promise spin qubits with long coherence times. Graphene quantum dots have been extensively investigated with respect to their excitation spectrum, spin-filling sequence and electron-hole crossover. However, their relaxation dynamics remain largely unexplored. This is mainly due to challenges in device fabrication, in particular concerning the control of carrier confinement and the tunability of the tunnelling barriers, both crucial to experimentally investigate decoherence times. Here we report pulsed-gate transient current spectroscopy and relaxation time measurements of excited states in graphene quantum dots. This is achieved by an advanced device design that allows to individually tune the tunnelling barriers down to the low megahertz regime, while monitoring their asymmetry. Measuring transient currents through electronic excited states, we estimate a lower bound for charge relaxation times on the order of 60–100 ns.
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Introduction
Since the seminal work by Loss and DiVincenzo1, quantum dots (QDs) have been extensively tested as building blocks for solid-state quantum computation. Today, the most advanced implementations of QD qubits are realized in III/V heterostructures2,3,4. Nevertheless, the strong hyperfine interaction in these compounds poses fundamental limits to the spin coherence time, stimulating the search for alternative host materials, especially within the elements of group IV (ref. 5). In this context, graphene is a particularly promising candidate thanks to its low nuclear spin densities and weak spin–orbit interaction6,7,8. However, the gapless electronic band structure and the Klein tunnelling phenomenon9 make it challenging to confine electrons electrostatically. This limitation can be circumvented by etching nanostructures in graphene, thus introducing a disorder-induced energy gap that allows the confining of individual carriers10,11,12,13,14,15,16,17. So far, excitation spectrum18,19,20,21, spin-filling sequence22 and electron-hole crossover23 have been studied in graphene QDs. However, these devices still lack the tunability of GaAs QDs, in particular with respect to the transparency of the tunnelling barriers, which hinders our ability to address the relaxation dynamics of individual quantum states experimentally.
In this article, we present QDs with highly tunable barriers formed by long and narrow constrictions. This design is motivated by our recent study on the aspect ratio of graphene nanoribbons17, and it proved to be optimal to realize tunnelling barriers that can be tuned down to the low MHz regime. The enhanced control over the tunnelling barriers, combined with the information on their asymmetry provided by integrated charge detectors24, enable us to reach regimes where it is possible to investigate the excited states (ESs) relaxation dynamics via pulsed-gate transient current spectroscopy25,26. Our measurements indicate a lower bound for the charge relaxation time of about 60–100 ns.
Results
Device design and low-bias transport characterization
Our devices consist of an etched graphene island (diameter, d=110 nm) connected to source and drain leads by long and narrow constrictions serving as tunnelling barriers (Fig. 1a). Two graphene nanoribbons, located symmetrically on each side of the island, can be used simultaneously as gates to tune the transparency of the barriers and as detectors sensitive to individual charging events in the dot26,27. The electrostatic potential of the QD is controlled by a central gate, on which a bias-tee mixes alternate current (AC) and direct current (DC) signals (Fig. 1b). This allows performing pulsed-gate experiments with the same gate used for DC control. The device is tuned by a back gate in the low charge-carrier density regime, where transport is dominated by Coulomb blockade effects18,19,20 (Supplementary Fig. S1a).
The QD is first characterized by low-bias transport measurements (Fig. 1c) as a function of the voltage applied to the two side gates controlling the transparency of the barriers18. The optimized design of our device allows us to individually tune the tunnelling rates of both barriers (ΓL, ΓR) over a wide range. By suppressing these rates we can access the sequential tunnelling regime, where the dot current exhibits approximately equally spaced Coulomb resonances as a function of the voltage VCG applied to the central gate, with peak currents up to a few hundred fA (Fig. 1d, lower curve). A current of 200 fA corresponds to an overall dot transmission rate of Γ<1.25 MHz. In correspondence of each Coulomb peak, the current through the graphene nanoribbon on the right side of the QD shows characteristic sharp kinks (Fig. 1d, upper curve), which indicate that the nanoribbon acts as a sensitive charge detector for the QD27,28.
Excited states spectroscopy and charge sensing
The addition energy of the dot and its ES spectrum are then probed by finite-bias spectroscopy. Figure 2a show the current and the differential conductance through the QD. These Coulomb diamond measurements give an estimate of the addition energy of the QD, Eadd≈10.5 meV. In a simple disc-capacitor model, this corresponds to a QD diameter of 120 nm, in good agreement with the geometric size of our device. Clear signatures of transport through well-defined ESs can be already observed in the current (for example, see Fig. 2b). They become even more evident in the differential conductance (Fig. 2c), from which we extract a level spacing of about Δ=1.5–2.5 meV. This is in agreement with the electronic single-particle level spacing given by meV, where N, the number of carriers on the dot, is assumed to be on the order of 10–20 (ref. 20).
Important information on the asymmetry of the barriers can be inferred by plotting dICD/dVCG (ref. 24), where ICD is the current flowing through the charge detector (see Fig. 2d). Regimes where the coupling to the leads is comparable in magnitude are characterized by the appearance of regions of high dICD/dVCG in correspondence of both edges of a Coulomb diamond (cf. the diamond centred around VCG≈4.2 V). Vice versa, regimes where one of the tunnel barriers is dominating are signalled by a strongly one-sided dICD/dVCG (cf. diamond centred around VCG≈4.45 V). This knowledge permits to identify regimes where the relaxation dynamics of the ESs can be investigated by pulsed-gate spectroscopy25,26. This technique requires in fact the pulse rise time (τrise) to be the fastest timescale in the system, that is, both tunnel rates (ΓL, ΓR) need to be much slower than τrise−1. To ensure that this condition is satisfied, we tune the dot into a low current regime where the tunnelling barriers are roughly equal. Conductance measurements in such a regime are shown in Fig. 2e. Note that the charge detector is such a sensitive electrometer that QD ESs can be clearly resolved in the differential transconductance dICD/dVSD (Fig. 2f).
Pulsed-gate spectroscopy and transient currents
To investigate the relaxation dynamics of ESs in the graphene QD, we employ a pulsed-excitation scheme similar to the one introduced by Fujisawa et al.25,26,29. The basic idea is to probe transient phenomena in a QD by measuring the averaged DC-current flowing through the system in the presence of a small (fixed) bias voltage VSD and of a square voltage-pulse applied to one of the gates (see Fig. 3a). The crucial requirement is that the rise time of the pulse has to be much shorter than the inverse of the tunnel rates, so that the occupation of the QD cannot follow adiabatically the change of the potential and, when the pulse-level switches from high to low (or vice versa), the system is brought in a state of non-equilibrium. If the frequency of the pulse is low (100 kHz in Fig. 3b), the square-wave modulation of the gate voltage results simply in the splitting of the Coulomb resonance into two peaks30. These peaks stem from the QD ground state (GS) entering the bias-window at two different values of VCG, one for the lower pulse-level (A), and one for the upper one (B). At larger frequencies (800 kHz in Fig. 3d), together with the splitting there is an additional broadening of the peaks. The situation changes dramatically at higher frequencies (from a few to tens of MHz), where a number of additional peaks appear, due to transient transport through QD ESs (see Fig. 3e). Each of these additional resonances corresponds to a situation in which the QD levels are pushed well outside the bias-window in the first half of the pulse, and then brought into a position where transport can occur only through the ESs in the second one, see Fig. 3f. No current can flow in any of these two configurations in the stationary state. However, in the second half of the pulse electrons can tunnel from one lead to the other via the ES, as long as the GS remains unoccupied (see Supplementary Fig. S2). Once the GS gets filled, either because of tunnelling from the leads or relaxation from the ES, transport is blocked. The blockade is released by returning the pulse-level to its initial value. In these conditions, a square-wave modulation of the gate voltage results in a train of short current pulses, which can be resolved in our DC-current measurements only if the frequency of the pulse is higher than the characteristic rate γ of the blocking processes. The lowest frequency at which signatures of transport through ESs emerge provides, therefore, an upper bound for γ.
Probing relaxation times
A more accurate estimate of the relaxation time of the ESs can be obtained with a different pulse scheme (see Fig. 4a), where TA is varied while keeping TB and VPP fixed25,26,31. Such a measurement is shown in Fig. 4b for the same Coulomb resonance investigated in Fig. 3b–e. The two outermost peaks correspond to transport via the dot GS when this is in resonance with the bias-window during TA (right peak) and TB (left peak). The inner peak results from transport via an ES, as indicated in the scheme sketched in Fig. 4c. For each of these peaks, we estimate the average number of electrons tunnelling through the device per cycle ‹n›=I (TA+TB)/e. Figure 4d shows the number of electrons transmitted via the GS (blue) and via the ES (red) as a function of the pulse length TA. While the first one increases linearly with TA, the second tends to saturate, indicating a transient effect. Fitting this data set with n(TA)=nsat [1-exp(−γ TA)], where nsat is the saturation value for long TA (ref. 25), we extract the characteristic rate of the blocking processes γ=12.8 MHz. As both tunnelling and relaxation lead to the occupation of the GS, γ is approximately given by γ~Γ+1/τ, where τ is the intrinsic relaxation time of the ES and Γ the characteristic tunnelling rate. This in turn gives a lower bound τ>78 ns for the lifetime of the QD ES. By studying further electronic ESs with energies in the range of 1.7–2.5 meV (see Supplementary Fig. S3–S5), we estimate a lower bound for the relaxation time in the range 60–100 ns.
Discussion
The experimentally observed relaxation timescale is likely related to the lifetime of charge excitations, which is limited by electron–phonon interaction. Interestingly, the extracted charge relaxation times are a factor 5–10 larger than what has been reported in III/V QDs25,26,31,32. This might be a signature of reduced electron–phonon interaction in sp2-bound carbon, where piezoelectric phonons are absent. To test this hypothesis, we perform a preliminary estimate of relaxation times in graphene QDs based on simplified assumptions for the QD and the phonon density of states (see Methods). Assuming that the coupling to longitudinal acoustic (LA) phonons via the deformation potential is the most dominant relaxation mechanism33, we obtain relaxation times in the range 40–400 ns for an ideal circular graphene QD with 110 nm diameter (see Supplementary Fig. S6). This is in reasonable agreement with the lower bound for the relaxation time τ extracted from the pulsed-gate measurements. It should, however, be kept in mind that these estimates are based on a very simplified model. More realistic calculations should take into account the effects of the destroyed graphene lattice, which breaks the A-B lattice symmetry and leads to a phonon density of states that is strongly dependent on the exact shape of the QD itself.
In summary, in this article we experimentally address the dynamics of ESs in graphene QDs. This is achieved by employing an innovative device design that allows realizing highly tunable graphene QDs, suitable for investigations with pulsed-gate spectroscopy. Measuring transient currents through ESs, we obtain a lower bound of 60–100 ns for the relaxation time of electronic excitations. Our work indicates a clear strategy of optimizing the sample design for pulse-gate experiments, and represents a fundamental step towards the investigation of spin life-times and coherence times in graphene QDs.
Methods
Device fabrication
The sample is fabricated by mechanical exfoliation of natural bulk graphite. Graphene flakes are transferred to highly doped silicon substrates with a 295-nm silicon oxide top-layer. Single-layer flakes are selected via Raman spectroscopy. The Raman spectrum of the measured graphene flake shows a 2D-peak with full width at half maximum (FWHM) of 34 cm−1 (cf. Supplementary Fig. S1b). The peak amplitude of the 2D-line is almost twice as high as the G-line, proving that the flake is one monolayer thick34. The flake is then patterned by electron beam lithography, followed by Ar/O2 reactive ion etching. A second electron beam lithography and evaporation (5/50 nm Cr/Au) step is performed to create source and drain electrodes and gate contacts on the graphene device.
Transport measurements
All measurements are performed in a dilution refrigerator, at a base temperature of ~20 mK. We use home-built low-noise DC amplifiers (amplification factor of the order 108, bandwidth <1 kHz, integration time 200 ms) to measure currents with a precision better than 50 fA. For the pulsed-gate experiments, we use the arbitrary waveform generators Tektronix AWG520 and Tektronix AWG7082C, and the bias-tee Anritsu K251 to mix AC and DC signals on the central gate. Two attenuation stages (10 dB attenuator at room temperature and 20 dB attenuator at 4 K) and impedance mismatch reduce the amplitude of the pulse at the sample by a factor 15.8 (24 dB) with respect to the one produced by the generator. The pulse shape remains, however, almost unperturbed (cf. Supplementary Fig. S7). The rise time detected close to the sample is about 250–300 ps. Measurements were performed in two different devices with the same lithographic design. Both devices show similar behaviour. Data from the second sample are shown in the Supplementary Fig. S8.
Estimating relaxation times in graphene QDs
A preliminary estimate of charge relaxation time τ in graphene QDs can be obtained in terms of Fermi’s golden rule , by making some simplified assumptions on the electron-phonon coupling Hamiltonian Hel−ph and the shape of the electronic wave functions Ψi and Ψf. To this end, we model the graphene QD as a circular disc of radius R=55 nm confined by an infinite mass term35. The dot states are then eigenfunctions of the following Hamiltonian: , where the first term on the right hand side is the Dirac-like graphene Hamiltonian and the second one includes a mass-related potential energy term V(r) (κ is here the valley index), which is zero inside the QD but tends to infinity towards the edges, that is, V(r)=0 for r<R and V(r)→∞ for r≥R (refs 35, 36). The wave functions of such a Hamiltonian are given by:
Where Jm(x) are Bessel functions of the first kind, are normalization constants and kn are the solutions of the equation Jm(kR)=κJm+1(kR) (36).
Assuming the coupling to the LA phonon mode via deformation potential to be the dominant cause of relaxation (see Supplementary Note), and that at the low temperature of the experiment the phonons are in their ground state, the electron-phonon coupling Hamiltonian take the form37
where D is the deformation potential, A the area of the graphene QD, ρ=7.5 × 10−7 kg m−2 the mass density and vs=2 × 104 m s−1 the sound velocity of the LA mode. The creation operator describes the emission of one phonon.
With these ingredients, the inverse relaxation time can be written as
where and q0=Δ/vs, is the wave vector of a phonon whose energy matches the QD level spacing Δ. This expression can be computed numerically and in Supplementary Fig. S6, we show the calculated relaxation times τ as function of the level spacing Δ. We find that for a level spacing in the range of 1.5–2.5 meV, coupling to LA phonons results in relaxation times on the order of 40–400 ns, in reasonable agreement with our experimental findings.
Additional information
How to cite this article: Volk, C. et al. Probing relaxation times in graphene quantum dots. Nat. Commun. 4:1753 doi: 10.1038/ncomms2738 (2013).
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Acknowledgements
We acknowledge Stefan Trellenkamp for support with electron beam lithography and Uwe Wichmann for help with the low-noise measurement electronics. We thank Hendrik Bluhm, Guido Burkard, Fabian Hassler, Thomas Ihn and Markus Morgenstern for helpful discussions. Support by the DFG (SPP-1459 and FOR-912) and the ERC are gratefully acknowledged
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C.V., C.N. and C.S. planned and designed the experiment. C.V, S.F. and S.E. fabricated the samples. C.V. and C. N. carried out the measurements and C.V., C. N., S.K., F.H. and A.M. analysed the results. All authors discussed the results. The manuscript was written by F.H., A.M., C.V. and C.S. with inputs from C.N., S.K. and S.E.
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Supplementary Figures S1-S8, Supplementary Note 1 and Supplementary References (PDF 2750 kb)
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Volk, C., Neumann, C., Kazarski, S. et al. Probing relaxation times in graphene quantum dots. Nat Commun 4, 1753 (2013). https://doi.org/10.1038/ncomms2738
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DOI: https://doi.org/10.1038/ncomms2738
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