Abstract
Electron transport in nanoscale structures is strongly influenced by the Coulomb interaction that gives rise to correlations in the stream of charges and leaves clear fingerprints in the fluctuations of the electrical current. A complete understanding of the underlying physical processes requires measurements of the electrical fluctuations on all time and frequency scales, but experiments have so far been restricted to fixed frequency ranges, as broadband detection of current fluctuations is an inherently difficult experimental procedure. Here we demonstrate that the electrical fluctuations in a singleelectron transistor can be accurately measured on all relevant frequencies using a nearby quantum point contact for onchip realtime detection of the current pulses in the singleelectron device. We have directly measured the frequencydependent current statistics and, hereby, fully characterized the fundamental tunnelling processes in the singleelectron transistor. Our experiment paves the way for future investigations of interaction and coherenceinduced correlation effects in quantum transport.
Introduction
The electrical fluctuations in a nanoscale conductor reveal a wealth of information about the physical processes inside the device compared with what is available from a conductance measurement alone^{1,2,3}. Zerofrequency noise measurements are now routinely performed using standard techniques, but, more recently, the detection of higher order current correlation functions, or cumulants, has attracted considerable attention in experimental^{4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19} and theoretical^{20,21,22} studies of charge transport in manmade submicron structures. Measurements have, for example, been encouraged by the intriguing connections between current fluctuations and entanglement entropy in solidstate systems^{23,24} as well as by the possibility to test fluctuation theorems^{25,26,27} at the nanoscale that are now a topic at the forefront of nonequilibrium statistical physics.
Experiments on current fluctuations have mainly focused on zerofrequency current correlations^{4,5,6,7,8,9,10,11,12,13,14,15,16} whereas measurements of finitefrequency cumulants of the current have remained an outstanding experimental challenge. However, noise detection in a restricted frequency band gives only partial information about correlations and measurements that cover the full frequency range are necessary to access all relevant timescales that characterize the transport process. In a zerofrequency measurement, correlation effects are integrated over a long period of time and important information about characteristic timescales are lost. To observe the dynamical features of the correlations, finitefrequency measurements are required^{2,3,28,29,30,31,32,33,34}. It has even been shown theoretically that the frequencydependent third cumulant (the skewness) of the current may contain further information about correlations and internal timescales of a system compared with the finitefrequency noise alone, for instance in chaotic cavities^{29} and diffusive conductors^{30}.
Figure 1a–c shows a typical time trace of the currents in our nanoscale singleelectron transistor (SET) consisting of a quantum dot (QD) coupled through tunnelling barriers to source and drain electrodes. The QD is operated close to a charge degeneracy point in the Coulomb blockade regime, where only a single extra electron at a time can enter from the source electrode and leave through the drain electrode. The applied voltage bias eV is larger than the electronic temperature k_{B}T which prevents electrons from tunnelling in the opposite direction of the mean current. While a stream of electrons is driven through the QD, a separate current is passed through the nearby quantum point contact (QPC) whose conductance is highly sensitive to the presence of extra electrons (Δq=0,1) on the QD^{35,36}. By monitoring the switches of the current through the QPC, we can thus detect, in realtime, single electrons tunnelling through the left (L) and right (R) barriers and thereby determine the corresponding pulse currents I_{L}(t) and I_{R}(t). The time trace of the currents illustrates the electrical fluctuations of interest here.
We now analyse the current fluctuations obtained from data measured over 24 h during which 300 million electrons passed through the SET. Our highquality measurements allow us to develop a complete picture of the noise properties of the SET, which not only focuses on the zerofrequency components of the fluctuations, but also contains the full frequencyresolved information about the noise and higher order correlation functions. From the measured finitefrequency noise spectrum, we extract the correlation time of the current fluctuations. The noise spectrum, however, is a onefrequency quantity only which does not reflect correlations between different spectral components of the current. To observe such correlations, we employ bispectral analysis and consider the finitefrequency skewness (or bispectrum) of the current. The skewness shows that the current fluctuations are nongaussian on all relevant time and frequency scales owing to the nonequilibrium conditions imposed by the applied voltage bias. Our measurements are supported by model calculations that are in excellent agreement with the experimental data. The results presented here provide a fundamental understanding of the electrical fluctuations in SETs that are expected to constitute the basic building blocks of future nanoscale electronics.
Results
Noise spectrum
The measured noise spectrum for the right tunnelling barrier is presented in Figure 1d. The finitefrequency currentcorrelation function is defined in terms of the noise power as^{1}
where , α=L,R, is the Fouriertransformed current and double brackets denote cumulant averaging over many experimental realizations. Figure 1d shows the Fano factor for the right tunnelling barrier with the mean current being constant in the stationary state. The noise is symmetric in frequency, , and results are shown for positive frequencies only. For uncorrelated transport, the noise spectrum would be white, that is, F^{(2)}(ω)=1 on all frequencies, corresponding to a Poisson process. Our measurements, in contrast, show a clear suppression of fluctuations below the Poisson value at low frequencies. This is due to the strong Coulomb interactions on the QD that introduce correlations in the stream of electrons propagating through the SET: each electron spends a finite time on the QD during which it blocks the next electron entering the QD. The measured noise spectrum has a Lorentzian shape whose width is determined by the inverse dynamical timescale of the correlations. Our measurements of the finitefrequency noise thereby enable a direct observation of the correlation time τ_{c}≃55 μs of the transport (Fig. 1d).
To corroborate our experimental results, we calculate the noise spectrum of the schematic model in the inset of Figure 1d. Single electrons tunnel from the left electrode onto the QD at rate Γ_{L} and leave it through the right electrode at rate Γ_{R}. The Fano factor is then^{1}
where theoretically τ_{c}=(Γ_{L}+Γ_{R})^{−1} is identified as the correlation time. This expression qualitatively explains the measured noise spectrum. Quantitative agreement is obtained by also taking into account the finite detection rate Γ_{D} of the QPC chargesensing scheme^{13,37}. The three parameters Γ_{L}, Γ_{R}, and Γ_{D} can be independently extracted from the distribution of waiting times between detected tunnelling events^{14,38,39}. The model calculations (see Methods) are in excellent agreement with measurements over the full range of frequencies, demonstrating the high quality of our experimental data.
Crosscorrelations
Crosscorrelations between the left I_{L}(t) and the right I_{R}(t) currents can also be measured (Fig. 1d). For the schematic model, the (crosscorrelation) Fano factor reads
which in the zerofrequency limit coincides with the noise spectrum, , as a consequence of charge conservation on the QD. The two currents are clearly correlated at frequencies that are lower than the inverse correlation time , but the crosscorrelator eventually reaches zero at higher frequencies. Interestingly, the crosscorrelations of the detected pulse currents are slightly negative at high frequencies. This is due to the finite resolution of the QPC chargesensing protocol that is not able to distinguish current pulses that are separated in time by an interval that is shorter than the inverse detector rate .
Higher order cumulants
We now turn to measurements of higher order finitefrequency cumulants. The mth finitefrequency current correlator corresponding to a timedependent current I(t) is defined as
where translational invariance in time implies frequency conservation as indicated by the Dirac delta function δ(ω) and S^{(m)}(ω_{1},...,ω_{m−1}) is the polyspectrum^{40,41}. In the case m=2, the polyspectrum yields the noise power spectrum S^{(2)}(ω), whereas the skewness (or bispectrum) is given by m=3 with the corresponding Fano factor F^{(3)}(ω_{1},ω_{2})=S^{(3)}(ω_{1},ω_{2})/e^{2}I. We focus here on the frequencydependent skewness S^{(3)}(ω_{1},ω_{2}), although our experimental data in principle allows us also to extract cumulants of even higher orders.
The measured skewness (Fig. 2a), shows a much richer structure and frequency dependence compared with the noise spectrum. The skewness obeys several symmetries following from the definition (Fig. 2b). We exploit the mirror symmetry with respect to interchange of frequencies, S^{(3)}(ω_{1},ω_{2})=S^{(3)}(ω_{2},ω_{1}), to compare measurement and model calculations: experimental results are presented above the diagonal (ω_{1}=ω_{2}), while model calculations are shown below (Fig. 2a). The nonzero bispectrum indicates nonGaussian statistics on all frequencies and shows strong correlations between different spectral components of the current. Importantly, these correlations are not a consequence of nonlinearities in the detection scheme, but are solely due to the physical nonequilibrium conditions imposed by the applied voltage bias. The pulse currents are directly derived from the tunnelling events, and the influence of external noise sources, including the amplification of the QPC current, is thereby explicitly avoided. Intuitively, one would expect the correlations to vanish, if the observation frequency is larger than the average frequency of the transport. Surprisingly, however, a certain degree of correlation persists even if one frequency is large, while the other is kept finite. This is in stark contrast to the second Fano factor F^{(2)}(ω) that approaches unity in the highfrequency limit (Fig. 1d), corresponding to uncorrelated tunnelling events.
For the schematic model in Figure 1d, the finitefrequency skewness reads^{32}
having defined , and ω_{3}=ω_{1}+ω_{2}. This expression qualitatively explains the measured finitefrequency skewness and shows that the skewness has a complex structure that is not just a simple Lorentzianshaped function of frequencies. We note that the third Fano factor F^{(3)}(ω_{1},ω_{2}) reduces to the zerofrequency limit of the noise F^{(2)}(0) in the limit ω_{2}=0 and . This is immediately visible in Figure 2c. Again, quantitative agreement is achieved by including the finite detection rate Γ_{D} in the model calculations, as illustrated explicitly in Figure 2c–e. The analytic expression shows that the frequency dependence of the skewness, unlike the auto and crosscorrelation noise spectra, is not only governed by the correlation time τ_{c}. The skewness has an involved frequency dependence, given by several frequency scales, which cannot be deduced from the noise spectrum alone.
Higher order crosscorrelations
In Figure 3a, we finally consider the crossbispectrum , measuring the thirdorder correlations between tunnelling electrons entering and leaving the SET. The three nonredundant permutations of the left and right pulse currents lead to a reduced symmetry of the crossbispectrum, as visualized in comparison with the autocorrelation bispectrum in Figure 3a. The crossbispectrum coincides with the autobispectrum on the line ω_{2}=0, where they approach the zerofrequency correlation of the shot noise for (Fig. 3b). In contrast, for ω_{2}≠0, the crossbispectrum shows a frequency dependence similar to the crosscorrelation shot noise and the anticorrelations due to the detection process become visible (Fig. 3c).
Discussion
We have measured the current statistics of charge transport in an SET and directly determined the dynamical features and timescales of the current fluctuations. From the measured frequencydependent noise power, we extracted the correlation time of the fluctuations. The noise power is a singlefrequency quantity only, and to investigate the correlations between different spectral components of the current using bispectral analysis, we measured the frequencydependent third order correlation function (the skewness). The skewness shows that the current statistics are nongaussian on all frequencies owing to the applied voltage bias. Our experimental results are supported by model calculations that are in excellent agreement with measurements. The results presented here are important for future applications of SETs in nanoscale electrical circuits operating with single electrons. Our accurate and stable experiment also facilitates several promising directions for basic research on nanoscale quantum devices. These include experimental investigations of fluctuation relations at finite frequencies and higher order noise detection of interaction^{42} and coherenceinduced correlation effects^{43} in quantum transport under nonequilibrium conditions.
Methods
Device fabrication
The device was fabricated by local anodic oxidation techniques using an atomic force microscope on the surface of a GaAs/AlGaAs heterostructure with electron density 4. 6×10^{15} m^{−2} and a mobility of 64 m^{2}/Vs. The twodimensional electron gas residing 34 nm below the heterostructure surface is depleted underneath the oxidized lines, allowing us to define the QD and the QPC.
Measurements
The experiment was carried out in a ^{3}He cryostat at 500 mK with an applied bias of 900 μV across the QD to ensure unidirectional transport and to avoid the influence of thermal fluctuations. The QPC detector was tuned to the edge of the first conduction step. The current through the QPC was measured with a sampling frequency of 500 kHz. The tunnelling events were extracted from the QPC current using a step detection algorithm and converted into timedependent pulse currents: Time was discretized in steps of Δt=40 μs and in each step the number of tunnelling events Δn in (out) of the QD was recorded. The current into (out of) the QD at a given time step is then I(t)=(−e)Δn/Δt.
Error estimates
We estimated the errors of the measured spectra by dividing the experimental data into 30 separate batches. The spectra were determined for each batch individually and their standard deviation was used as a measure of the experimental accuracy.
Finitefrequency cumulants
To define the finitefrequency cumulants of the current, we consider the mtime probability distribution^{32,33} P^{(m)}(n_{1}, t_{1};...;n_{m}, t_{m}) that n_{k} electrons have been transferred during the time span [0, t_{k}] for k=1,...,m. The corresponding cumulant generating function is
with χ=(χ_{1},...,χ_{m}), t=(t_{1},...,t_{m}), and n=(n_{1},...,n_{m}). (An equivalent definition uses only a single but timedependent counting field^{44}.) The mtime cumulants of P^{(m)}(n; t) are then
with the corresponding mtime cumulants of the current
These equations define the cumulant averages denoted by double brackets . In the Fourier domain, the current cumulants can be expressed as
as the sum of frequencies is zero in the stationary state. The Fouriertransformed current is denoted as and S^{(m)}(ω_{1}, ω_{2},..., ω_{m−1}) is the polyspectrum^{40}, which for m=2 and m=3 yields the noise spectrum S^{(2)}(ω) (second cumulant) and the bispectrum (third cumulant) S^{(3)}(ω_{1}, ω_{2}), respectively. We note that the noise spectrum S^{(2)}(ω) in the stationary state is a singlefrequency quantity that can be related directly to the onetime probability distribution P^{(1)}(n, t) according to MacDonald's formula^{45}
The bispectrum F^{(3)}(ω_{1},ω_{2}) in contrast is a twofrequency quantity that reflects correlations of the current beyond what is captured by P^{(1)}(n, t) alone.
Theoretical model
We consider the probability vector p(t)〉=[p_{00}(t), p_{10}(t), p_{01}(t), p_{11}(t)]^{T}, where the first index denotes the number of (extra) electrons on the QD, i=0, 1, and the second index denotes the detected number of (extra) electrons on the QD as inferred from the current through the QPC, j=0, 1. The probability vector evolves according to the rate equation
having introduced separate counting fields χ_{L} and χ_{R} that couple to the number of detected electrons that have passed the left and the right barriers, respectively^{44}. The matrix M(χ_{L}, χ_{R}) reads^{13,14,37}
where factors of in the offdiagonal elements correspond to processes that increase by one the number of detected electrons that have tunnelled across the left (right) barrier (Fig. 1c,d). The detector rate Γ_{D} of the QPC detector scheme tends to infinite for an ideal detector only, but is finite otherwise.
Calculations
For the calculations of the finitefrequency cumulants it is useful to write the matrix as
where M_{0}=M(0, 0) and I_{L(R)} is the super operator for the detected current through the left (right) barrier^{46}. Additionally, we need the stationary state , found by solving and normalized such that , where . We define the projectors^{46} and Q=1−P and the frequencydependent pseudoinverse^{47} R(ω)=Q[iω+M_{0}]^{−1}Q, which is welldefined even in the zerofrequency limit ω→0, since the inversion is performed only in the subspace spanned by Q, where M_{0} is regular. These objects constitute the essential building blocks for our calculations. The frequencydependent second and third cumulants, S^{(2)}(ω) and S^{(3)}(ω_{1},ω_{2}), can then be evaluated following refs 47,48 and 33, respectively.
Additional information
How to cite this article: Ubbelohde, N. et al. Measurement of finitefrequency current statistics in a singleelectron transistor. Nat. Commun. 3:612 doi: 10.1038/ncomms1620 (2012).
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Acknowledgements
We thank M. Büttiker, C. Emary, and Yu. V. Nazarov for instructive discussions. W. Wegscheider (Regensburg, Germany) provided the wafer and B. Harke (Hannover, Germany) fabricated the device. The work was supported by BMBF through nanoQUIT (C. Fr., N. U., F. H., and R. J. H.), DFG through QUEST (C. Fr., N. U., F. H., and R. J. H.), the Villum Kann Rasmussen Foundation (C. Fl.), and the Swiss NSF (C. Fl.).
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All authors conceived the research. N.U., C.Fr. and F.H. carried out the experiment and analysed data. All authors discussed the results. C.Fl. developed theory and performed calculations. N. U., C.Fr., and C.Fl. wrote the manuscript. F.H. and R.J.H. supervised the research. All authors contributed to the editing of the manuscript.
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Correspondence to Rolf J. Haug.
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Ubbelohde, N., Fricke, C., Flindt, C. et al. Measurement of finitefrequency current statistics in a singleelectron transistor. Nat Commun 3, 612 (2012). https://doi.org/10.1038/ncomms1620
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