NANO LETTERS
Shot Noise Measurements on a Single Molecule
2006 Vol. 6, No. 4 789-793
D. Djukic and J. M. van Ruitenbeek* Kamerlingh Onnes Laboratorium, UniVersiteit Leiden, Postbus 9504, NL - 2300 RA Leiden, The Netherlands Received January 19, 2006; Revised Manuscript Received March 2, 2006
ABSTRACT We report measurements of shot noise in the current through a single D2 molecule. The molecular junctions were formed by means of the mechanically controllable break junction technique. The configuration of the D2 molecule bridging the gap between two Pt tips is verified by use of point contact spectroscopy. Maintaining the same junction shot noise measurements were performed and the observed quantum suppression shows that conductance is carried dominantly by a single, almost fully transparent conductance channel. This observation allows us to decide between conflicting model calculations for this system, and this may serve as a benchmark for further computations on molecular junctions.
Although the idea of exploiting molecules as functional units in electronic circuits has existed since 19741 and many transport experiments on single molecules have been reported,2-10 the agreement between the theory and experiment is not always satisfactory. Both theory and experiment are exploring new territory and there is a clear need for welldefined test systems. In the experiment reported here we concentrate on a simple molecular junction device and bring together two very sensitive and easily interpretable measurements: point contact spectroscopy and shot noise measurements. By means of point contact spectroscopy we verify that the junction is formed by a D2 molecule. The observed vibration modes are characteristic of the Pt-D2-Pt junction, and isotope substitution allows us to verify the origin of the signal. Shot noise is used to analyze the decomposition of the total transmission in terms of individual conductance channels and shows that the conductance is carried dominantly by a single channel. This allows us to definitely distinguish between the calculations of refs 11 and 12 and those of refs 13-15 and to confirm that the junction is formed by just a single molecule. The hydrogen (deuterium) molecular junction may thus serve as a benchmark for molecular electronics computations. Because of the discreteness of the electron charge, the current through any system fluctuates around the average value. This effect was originally discovered by Schottky16 in vacuum tube diodes but can be observed in many other systems including semiconductor p-n junctions,17 metalinsulator-metal tunnel junctions,18 and a wide variety of mesoscopic structures.19 If individual electrons are transmitted randomly from one side to the other, as in vacuum diodes, * Corresponding author. E-mail:
[email protected]. 10.1021/nl060116e CCC: $33.50 Published on Web 03/16/2006
© 2006 American Chemical Society
then the noise is Poissonian, having a frequency-independent spectral density SI ) 2e|I|, where I is the average current through the device and -e is the electron charge. Correlations between the electrons will influence that randomness and therefore affect the shot noise for the structure under study. Because the size of a mesoscopic system is comparable to some typical lengths that determine the level of correlation between the electrons, such as the electron coherence length, the electron-phonon, or the electron-electron scattering length, the size of the system will affect the level of shot noise significantly. Thus, by measuring the level of shot noise, one can gain insight into mesoscopic electronic properties that are not accessible by standard conductance measurements. For the single-molecule junctions considered here, the length of the system is much smaller than all other lengths, and only comparable to the Fermi wavelength. In this regime one may have a single perfectly transmitting channel for which the electrons are in a coherent state across the entire structure and for which the shot noise vanishes. For a transmission probability between 0 (high-barrier tunnel junction) and 1 (perfect contact) the noise level is determined by the partition noise: the degree of freedom for the electron to choose between being transmitted or reflected. This allows us to determine the channel transmission experimentally,20,21 which is the primary interest of this work. Using the mechanically controllable break junction (MCBJ) technique for Au atomic contacts, ref 22 showed that shot noise is strongly suppressed when the conductance is close to the conductance quantum, G0 ) 2e2/h. The value 1G0 for Au corresponds to a single-atom contact, or a chain of single atoms, which then carries one completely transparent conductance channel. In contrast, for Al shot noise is not
suppressed near G ) 1G0 because partially transmitting channels are involved in the conductance.23,24 By exploiting the superconducting subgap features, it is possible for one to determine the full set of transmission probabilities {τi} for a given contact from which the shot noise level can be quantitatively predicted.25 A full expression for the noise level of a quantum point contact has been derived for finite temperature T and applied bias voltage V19,26
SI ) 2eV coth
( )
eV 2e2
2kT
h
∑i
τi(1 - τi) + 4kT
2e2 h
∑i τ2i
(1)
For zero voltage this expression simplifies to the one for Johnson-Nyquist noise, SI ) 4kTG. In the limit of low temperatures (kT , eV) eq 1 reduces to
( ) N
SI ) 2eI 1 -
τ2i ∑ i)1 N
) 2eI F(τ1, ..., τN)
(2)
τi ∑ i)1
Here, F is the so-called Fano factor that measures the degree to which the classical Poissonian shot noise level is modified and which depends on the number of conducting channels and their transmission. We will exploit this property for the analysis of the conductance channels in a Pt-D2-Pt bridge. The molecular junctions were formed at 4.2 K using the MCBJ technique, as described previously.14,27 The cryostat together with the amplifiers was placed in an acoustically shielded Faraday cage to prevent coupling to external vibrations or electromagnetic fields. A schematic drawing of the experimental setup is shown in Figure 1. The experiment consists of a few steps and combines dc twopoint voltage-biased conductance measurements, ac differential conductance measurements, and current-biased shot noise measurements. Before the measurements, the sample chamber containing a clean Pt wire MCBJ device is evacuated to about 10-6 mBar at room temperature and lowered into a liquid helium dewar. The chamber is fitted with active charcoal for cryogenic pumping so that the pressure in the sample chamber during the measurement is expected to be well below 10-12 mBar. Once cold, the Pt wire is broken at the precut notch by mechanical bending of the substrate to which it is fixed (more details can be found, e.g., in the review ref 28). First, we collect a clean Pt conductance histogram in order to verify that the surface is not contaminated by residual gas condensed on the contact. This is done by repeatedly breaking and reforming contacts, a few times per second, and collecting the digitized curves of the evolution of conductance into a histogram. Histograms are typically built from many thousands of breaking traces, and all recorded curves are included, without exception. A histogram for a clean junction has a peak near the typical conductance of a single Pt atom, ∼1.5G0, and a low count below 1G0, and it 790
Figure 1. Schematic outline of the setup used for simultaneous measurements of point contact spectroscopy and shot noise on a single-molecule junction at 4.2 K. The dipstick with the sample chamber is under vacuum and inserted in the cryostat filled with liquid He. The cryostat together with all amplifiers are put inside a Faraday cage. A set of switches, S1 and S2, is used to change between point contact spectroscopy and shot noise measurements. For current-biased shot noise measurements (S1 open, S2 closed), the current through the contact is controlled by a divider on a battery. The signal is picked up by two separate Cu wire twisted pairs and then is amplified 105 times by two parallel stages of lownoise preamplifiers. The two outputs are sent to a network signal analyzer, which takes a Fourier transform of both signals and calculates the cross spectrum of the two signals. The setup is converted for point contact spectroscopy measurements by closing S1 and opening S2.
is a standard procedure to verify this at the start of each new measurement run. Next, we inject a small amount of molecular D2 (or H2) gas and observe the conductance histogram changing from the clean Pt characteristics, with a pronounced conductance peak near 1.5G0, into one typical for a Pt-D2-Pt bridge, having a peak near 1G0 and a high count at conductance values below this value. The origin of much of the structure at lower conductance is still under investigation and it is likely due to other configurations involving several hydrogen molecules around the junction. Here we focus on the structure responsible for the peak near 1G0 for which we can obtain the most detailed information (see below). This procedure of characterizing the junction by recording histograms only involves dc voltage-biased conductance measurements, with bias voltages in the range from 10 to 150 mV. After this transformation of the conductance histogram is observed, we change from dc to ac voltage bias conductance Nano Lett., Vol. 6, No. 4, 2006
Figure 2. Differential conductance (dI/dV) curves for various stages of stretching of a Pt-D2-Pt junction at 4.2 K (top part). The curves have not been displaced so that one can observe a small decrease of the total conductance as a function of stretching. One can recognize two or three vibration mode features, but the one at ∼80 mV is most pronounced. This mode corresponds to the longitudinal vibration of the D2 molecule between the Pt banks in a configuration as illustrated schematically in the inset. The lower panel shows the numerical derivative of the dI/dV curves that have been shifted vertically with respect to each other for clarity. Because the shift of the curves and the increment in stretching of the contact is fixed between each successive curve, the dashed lines illustrate that the shift of the energy of the mode is approximately linear with stress on the junction, decreasing with stretching.
measurement using a lock-in technique. As was reported previously,14,27 the peak near 1G0 is associated with the formation of a stable single-molecule junction, which can be verified by point contact spectroscopy29 of the molecular vibration modes. When the bias voltage crosses the energy required for exciting vibration modes of the molecule, eV > pωi, we observe a decrease in the conductance. The identification of the vibration modes can be further scrutinized by repeating the experiments for isotopes H2, D2, and HD and by following the vibration mode energy as a function of stretching of a contact.14,27 Differential conductance curves, dI/dV, are recorded for fixed contact configuration, using an ac modulation of 1 mVrms amplitude and a frequency of 7.777 kHz, while slowly ramping the dc bias between -100 and +100 mV. We carefully verified the transfer characteristic of the ac circuit by measuring known resistances at different frequencies. The low-frequency part coincides with the dc measured value within 0.1%, whereas the ac conductance at 7.777 kHz requires a correction of +1.6% because of roll-off. A series of dI/dV measurements for D2 is shown in Figure 2. Each curve is taken for the same configuration as the previous but stretched over a small fixed incremental distance. The total stretching distance is approximately 0.5 Å, but note that the change in the Pt-D bond distance is much smaller than this because a large part of the displacement results in elastic deformation of the Pt metal leads. The decrease of the vibration mode energy with stretching demonstrates that the mode is a longitudinal vibration of the D2 molecule between the Pt leads, in agreement with the Nano Lett., Vol. 6, No. 4, 2006
previous observations and the calculations by Thygesen and Jacobsen.14 Note that there is some variation of the observed vibration mode energies between various realizations of the molecular junction and that the one in Figure 2 is on the low side of the distribution shown in ref 14. This particular molecular junction needs to be stable for about 20 min for recording the full series of dI/dV spectra and shot noise measurements. The series of shot noise measurements reported here are done on a configuration equivalent to the one identified by the bold curve in Figure 2. For measurement of shot noise the noise signal from the molecular contact is first amplified 105 times by two parallel sets of low-noise preamplifiers. Because we are interested in noise levels on the order of 10-9 V/xHz, which is smaller than the noise level of the low-noise preamplifiers,30 we use a parallel arrangement of two wide-band amplification circuits and feed the output signals to a two-channel spectrum analyzer (Stanford SR780), which allows us to form an averaged cross spectrum of the two signals. By taking the cross spectrum and averaging, we reject the noise that does not have a common source and therefore does not originate from the molecular contact. Once we have adjusted a stable contact close to a desired value by dc measurement (∼1G0 for the Pt electrodes bridged by a D2 molecule) we take a dI/dV curve using the lock-in technique. The dI/dV measurements are used here as a signature of the particular contact. Any configurational change will result in a different shape of the dI/dV curve, and because our goal is to take a noise power spectrum on the same contact for different bias currents, a reproducible shape of the dI/dV curves ensures us that the contact remains unchanged during the experiment. Once we have identified a stable contact with a clear vibration mode signal, we first record a zero-bias noise spectrum between 1 and 100 kHz, which shows the thermal (Johnson-Nyquist) noise that is also present when the sample is biased and that we want to remove. The thermal noise for a contact with a resistance of -9 G-1 V/ 0 = 12.9 kΩ at 4.2 K is expected to be 1.73 × 10 xHz. The measured noise signal of ∼1.87 × 10-9 V/xHz corresponds to a contact temperature of 4.9 K. This is slightly above the bath temperature as a result of imperfect thermalization of the junction. Better thermalization can be achieved by admitting helium contact gas, but we want to avoid this in view of possible contaminations. The measured thermal noise is removed from our signal by subtracting it from the current biased curves. The cross spectra are averaged 10 000 times, which takes about 1 min. After each recording of a noise curve, we verify the contact by taking a dI/dV curve. The total noise at finite bias consists of a white shot noise contribution, including the thermal and the nonequilibrium components (eq 1), plus 1/f-noise, which is dominant at low frequencies and is believed to originate from the motion of defects in the leads. The low frequency part is very sensitive to any configurational changes of the contact, and the size of this signal varies strongly between different realizations of the molecular junction. In the high frequency part of the noise spectrum we find several sharp spikes due to electromagnetic pickup, despite the shielding by the Faraday cage, 791
Figure 3. Excess noise, that is, the difference between the measured white noise level with and without current, as a function of the applied current for a Pt-D2-Pt junction having a vibration mode spectrum as shown in Figure 2. The Poissonian shot noise level is indicated by the curve marked “full shot noise”, and we observe a strong quantum suppression of the noise in our junction. The zero bias conductance for this junction is G ) (1.010 ( 0.005)G0, and we obtain a good description of our data with theory assuming essentially a single fully open channel, {τ1,τ2} ) {0.995, 0.015}. Within the error margins set by the conductance, the channel transmissions range between {τ1,τ2} ) {0.992, 0.013} and {0.997, 0.018}. The dashed curve illustrates the sensitivity of the noise to the distribution of transmission probabilities for a slightly different choice, τ1 ) 0.985 and τ2 ) 0.025.
but they are removed by subtracting the thermal noise, because they are common to all spectra. The noise spectra show a roll-off at higher frequencies due to the low-pass transfer characteristics of the electronic circuit. Because the thermal noise is white and depends only on the previously determined resistance of the contact, we use the zero bias noise spectrum to calibrate the transfer characteristics. By fitting the thermal noise curve with a low-pass filter transfer function in the higher frequency domain, we obtain the parameters for correction of the roll-off. After dropping the low-frequency part dominated by 1/f-noise, compensating for the roll-off and subtracting the thermal noise for a series of bias currents on a given contact, we obtain the dependence of the excess noise as a function of applied bias. We have analyzed such data for two independent measurements done on two contacts having a conductance slightly higher and slightly lower than 1G0. Figure 3 shows the results for a junction with a zero bias conductance of G ) (1.010 ( 0.005)G0, obtained from the dI/dV measurements. From the conductance we obtain ∑nτn ) 1.010 ( 0.005, and the sum must have at least two contributions. The fit in Figure 3 shows good agreement for τ1 ) 0.995, τ2 ) 0.015, corresponding to a Fano factor F ) 0.020. We do not have much freedom to redistribute the total conductance over these, or additional, channels. The noise increases as soon as we reduce the transmission of the almost fully open channel and increases further when we transfer this transmission to the other channel(s). This sensitivity is illustrated in Figure 3 for an example of τ1 ) 0.985, τ2 ) 0.025. By moving only 1% of the transmission from the almost fully open channel to the other channel the Fano factor increases to F ) 0.039, which is clearly too high to explain the data. We may, of course, assume a third channel but this will again increase the noise unless we redistribute the small transmis792
sion of the second channel over the two small ones. For a set of transmissions {τ1 ) 0.995, τ2 ) 0.008, τ3 ) 0.007} the fit to the data is equally good and the Fano factor is nearly the same. The accuracy to which the transparency of the first channel is fixed is approximately 0.3% and is limited by the accuracy of the ac conductance measurement. In a second experiment we focused on a Pt-D2-Pt junction having a clear vibration mode at 45 meV, which belongs to the transverse hindered rotation mode of the molecule. In this case the zero bias conductance is slightly lower than 1. We have G ) 0.98G0 so that ∑nτn ) 0.98, and we obtain a good description with τ1 ) 0.975, τ2 ) 0.005, corresponding to a Fano factor of F ) 0.030. Again, we obtain a dominant contribution to the conductance of a single, nearly open channel, and only very small additional channels. Our results allow us to distinguish with high confidence between the various computational methods applied to this molecular device. Garcı´a et al.11 (see also Cuevas et al.12) used density-functional-theory-based calculations and obtained a conductance for the linear configuration (sketched in the inset of Figure 2) of only G ) 0.2G0. They propose an alternative structure, having two H (or D) atoms arranged perpendicular to a Pt-Pt atomic contact, for which the conductance in their computation is close to the quantum unit of conductance. However, this conductance results from contributions of three channels, with transmissions given approximately as {τ1 ) 0.7, τ2 ) 0.2, τ2 ) 0.1}. The Fano factor for this set of transmissions is 0.46, more than an order of magnitude larger than the measured values. In contrast, the computations by Thygesen and Jacobsen13,14 and by Garcı´a-Sua´rez et al.15 find that the linear arrangement forms a stable structure with a well-defined total transmission very close to 1G0 over a wide energy range around the Fermi energy, implying that it is not very sensitive to the details of the atomic arrangement. The conductance is carried dominantly by a single conductance mode, in agreement with the present experiment. This observation reinforces the agreement obtained between the observed vibration modes and their stretching dependence for the hydrogen molecule with those in the calculations by Thygesen and Jacobsen.14 The arrangement proposed by Garcı´a et al.11 is also found in the other calculations,13-15 at a lower degree of stretching of the contact, but the conductance in the later calculations for this structure comes out to be much higher, 1.5-2G0, which excludes it as the configuration of the experiment. Although the experiment does not allow us to exclude possible further alternatives, the number of observed parameters imposes severe limitations on other interpretations. It is interesting to investigate the question of what determines the differences between the various computational approaches. The differences may, among other factors, be due to the arrangement chosen for the Pt atoms in the leads, the size of the Pt atom cluster taken into account, the way the connection to the leads is introduced, and in the choice of the exchange-correlation functional. When the answer to this question becomes clear it may be of relevance to computations of metal-molecule-metal junctions in general. Nano Lett., Vol. 6, No. 4, 2006
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