We present an improved nuclear refrigerator reaching 0.3 mK, aimed at microkelvin nanoelectronic experiments, and use it to investigate metallic Coulomb blockade thermometers (CBTs) with various resistances R. The high-R devices cool to slightly lower T, consistent with better isolation from the noise environment, and exhibit electron-phonon cooling ~ T^5 and a residual heat-leak of 40 aW. In contrast, the low-R CBTs display cooling with a clearly weaker T-dependence, deviating from the electronphonon mechanism. The CBTs agree excellently with the refrigerator temperature above 20 mK and reach a minimum-T of 7.5 +/- 0.2 mK.
We demonstrate experimentally a precise realization of Coulomb Blockade Thermometry (CBT) working at temperatures up to 60 K. Advances in nano fabrication methods using electron beam lithography allow us to fabricate a uniform arrays of sufficiently small tunnel junctions to guarantee an overall temperature reading uncertainty of about 1\%
We investigate temperature errors of Coulomb blockade thermometer (CBT) arising from inevitable non-uniformities in tunnel junction arrays. The errors are proportional to the junction resistance variance in the universal operation regime and this result holds approximately also beyond this originally studied high temperature range. We present both analytical and numerical results, and discuss briefly their implications on achievable uniformity based on state-of-the-art fabrication of sensors.
We have investigated one and two dimensional (1D and 2D) arrays of tunnel junctions in partial Coulomb blockade regime. The absolute accuracy of the Coulomb blockade thermometer is influenced by the external impedance of the array, which is not the same in the different topologies of 1D and 2D arrays. We demonstrate, both by experiment and by theoretical calculations in simple geometries, that the 1D structures are better in this respect. Yet in both 1D and 2D, the influence of the environment can be made arbitrarily small by making the array sufficiently large.
We investigate Coulomb blockade thermometers (CBT) in an intermediate temperature regime, where measurements with enhanced accuracy are possible due to the increased magnitude of the differential conductance dip. Previous theoretical results show that corrections to the half width and to the depth of the measured conductance dip of a sensor are needed, when leaving the regime of weak Coulomb blockade towards lower temperatures. In the present work, we demonstrate experimentally that the temperature range of a CBT sensor can be extended by employing these corrections without compromising the pr…
The charge localization of single electrons on mesoscopic metallic islands leads to a suppression of the electrical current, known as the Coulomb blockade. When this correction is small, it enables primary electron thermometry, as it was first demonstrated by Pekola et al. (Phys. Rev. Letters, 73, 2903 [1994]). However, in the low temperature limit, random charge offsets influence the conductance and limit the universal behavior of a single metallic island. In this work, we numerically investigate the conductance of a junction array, and demonstrate the extension of the primary regime for larg…
We have measured current-voltage characteristics of two-dimensional arrays of small tunnel junctions at temperatures from 1.5 K to 4.2 K. This corresponds to thermal energies larger than the charging energy. We show that 2D-arrays can be used as primary thermometers in the same way as 1D-arrays, and even have some advantages over 1D-arrays. We have carried out Monte Carlo simulations, which agree with our experimental results.
We present a scheme and demonstrate measurements of a Coulomb blockade thermometer (CBT) in a microwave transmission setup. The sensor is embedded in an $LCR$ resonator, where $R$ is determined by the conductance of the junction array of the CBT. A transmission measurement yields a signal that is directly proportional to the conductance of the CBT, thus enabling the calibration-free operation of the thermometer. This is verified by measuring an identical sensor simultaneously in the usual dc setup. The important advantage of the rf measurement is its speed: the whole bias dependence of the CBT…
We present a measurement and analysis scheme for determining traceable thermodynamic temperature at cryogenic temperatures using Coulomb blockade thermometry. The uncertainty of the electrical measurement is improved by utilizing two sampling digital voltmeters instead of the traditional lock-in technique. The remaining uncertainty is dominated by that of the numerical analysis of the measurement data. Two analysis methods are demonstrated: numerical fitting of the full conductance curve and measuring the height of the conductance dip. The complete uncertainty analysis shows that using either …
We investigate a wafer scale tunnel junction fabrication method, where a plasma etched via through a dielectric layer covering bottom Al electrode defines the tunnel junction area. The ex-situ tunnel barrier is formed by oxidation of the bottom electrode in the junction area. Room temperature resistance mapping over a 150 mm wafer give local deviation values of the tunnel junction resistance that fall below 7.5 % with an average of 1.3 %. The deviation is further investigated by sub-1 K measurements of a device, which has one tunnel junction connected to four arrays consisting of N junctions (…
We have measured the third derivative of the current-voltage characteristics, d^3I/dV^3, in a two-dimensional array of small tunnel junctions using a lock-in amplifier. We show that this derivative is zero at a voltage which scales linearly with the temperature and depends only on the temperature and natural constants, thus providing a primary thermometer. We demonstrate a measurement method which extracts the zero crossing voltage directly using a feedback circuit. This method requires only one voltage measurement, which makes it substantially faster than the original Coulomb blockade thermom…
We study Coulomb blockade effects in the thermodynamic quantities of a weakly disordered metallic grain coupled to a metallic lead by a tunneling contact with a large conductance $g_T$. We consider the case of broken time-reversal symmetry and obtain expressions for both the {\em ensemble averaged} amplitude of the Coulomb blockade oscillations of the thermodynamic potential and the correlator of its {\em mesoscopic fluctuations} for a finite mean level spacing $δ$ in the grain. We develop a novel method which allows for an exact evaluation of the functional integral arising from disorder aver…
We consider mesoscopic fluctuations of the Coulomb blockade oscillations in a disordered metallic grain in an external magnetic field coupled to a metallic lead by a tunneling contact with a large conductance $g_T$ and capacitively coupled to a gate. For a finite mean level spacing $δ$ in the grain and at temperatures $T \gg δ$ we obtain the oscillatory part of the correlator of thermodynamic potentials at different values of the gate voltage. The correlations decay algebraically with the gate voltage difference in contrast to the exponential decay that would result from the mean-field treatme…
We study electronic transport through metallic multi-island Coulomb-blockade systems. Based on a diagrammatic real-time approach, we develop a computer algorithm that generates and calculates all transport contributions up to second order in the tunnel-coupling strengths for arbitrary multi-island systems. This comprises sequential and cotunneling, as well as terms corresponding to a renormalization of charging energies and tunneling conductances. Multi-island cotunneling processes with energy transfer between different island are taken into account. To illustrate our approach we analyze the c…
We use a novel technique to experimentally explore transport properties through a single metallic nanoparticle with variable coupling to electric leads. For strong dot-lead coupling the conductance is an oscillatory function of the gate voltage with periodicity determined by the charging energy, as expected. For weaker coupling we observe the appearance of additional multi-periodic oscillations of the conductance with the gate voltage. These harmonics correspond to a change of the charge on the dot by a fraction of an electron. This notion is supported by theoretical calculations based on diss…
The current-voltage characteristics through a metallic quantum dot which is well coupled to a metallic lead are measured. It is shown that the I-V curves are composed of two contributions. One is a suppression of the tunneling conductivity at the Fermi level and the second is an oscillating feature which shifts with gate voltage. The results indicate that Zero-Bias-Anomaly and Coulomb Blockade phenomena coexist in an asymmetric strongly coupled quantum dot.
Properties of one-dimensional (1D) arrays of low Ohmic tunnel junctions (i.e. junctions with resistances comparable to, or less than, the quantum resistance $R_{\rm q}\equiv h/e^2\approx 25.8$ k$Ω$) have been studied experimentally and theoretically. Our experimental data demonstrate that -- in agreement with previous results on single- and double-junction systems -- Coulomb blockade effects survive even in the strong tunneling regime and are still clearly visible for junction resistances as low as 1 k$Ω$. We have developed a quasiclassical theory of electron transport in junction arrays in th…
Access to lower temperatures has consistently enabled scientific breakthroughs. Pushing the limits of \emph{on-chip} temperatures deep into the microkelvin regime would open the door to unprecedented quantum coherence, novel quantum states of matter, and also the discovery of unexpected phenomena. Adiabatic demagnetization is the workhorse of microkelvin cooling, requiring a dilution refrigerator precooling stage. Pulse-tube dilution refrigerators have grown enormously in popularity due to their vast experimental space and independence of helium, but their unavoidable vibrations are making mic…
We have observed anomalous transport properties for a 50 nm Bi dot in the Coulomb-blockade regime. Over a range of gate voltages, Coulomb blockade peaks are suppressed at low bias, and dramatic structure appears in the current at higher bias. We propose that the state of the dot is determined self-consistently with the state of a nearby two-level system (TLS) to which it is electrostatically coupled. As a gate voltage is swept, the ground state alternates between states of the TLS, leading to skipped Coulomb-blockade peaks at low bias. At a fixed gate voltage and high bias, transport may occur…
We study a simple model of a self-assembled, room temperature Coulomb-blockade nanostructure containing a metallic nanocrystal or grain connected by soft molecular links to two metallic electrodes. Self-excitation of periodic grain vibrations at 10 - 100 GHz is shown to be possible for a sufficiently large bias voltage leading to a novel `shuttle mechanism' of discrete charge transfer and a current through the nanostructure proportional to the vibration frequency. For the case of weak electromechanical coupling an analytical approach is developed which together with Monte Carlo simulations sho…
Weakly coupled quantum dots in the Pauli spin blockade regime are considered with respect to spin-dependent transport. By attaching one half-metallic and one non-magnetic lead, the Pauli spin blockade if formed by a pure triplet state with spin moment $S_z=1$ or -1. Furthermore, additional spin blockade regimes emerge because of full occupation in states with opposite spin to that of the half-metallic lead.
We study the effect of the exchange interactions in small superconducting grains near the Coulomb blockade regime. We extend the standard description of the grain in terms of a single collective variable, the charge and its conjugated phase, to include the spin degree of freedom. The suppression of spin fluctuations enhances the tendency towards Coulomb blockade. The effective charging energy and conductance are calculated numerically in the regime of large grain-lead coupling.
The shot noise of the current through a single electron transistor (SET), coupled capacitively with an electronic box, is calculated, using the master equation approach. We show that the noise may be sub-Poissonian or strongly super-Poissonian, depending mainly on the box parameters and the gate. The study also supports the idea that not negative differential conductance, but charge accumulation in the quantum dot, responds for the super-Poissonian noise observed.
We focus on a metallic quantum dot coupled to a reservoir of electrons through a single-mode quantum point contact and capacitively connected to a back-gate, by taking into account that the gate voltage can exhibit noise; This will occur when connecting the gate lead to an external transmission line with a finite Ohmic resistance. The voltage (charge) fluctuations at the back gate can be described through a Caldeira-Leggett model of harmonic oscillators. For weak tunneling between the lead qnd the dot, exploiting the anisotropic Bose-Fermi Kondo model, we show that zero-point fluctuations of t…
We investigate theoretically the electronic transport properties in narrow graphene ribbons with an adatom-induced defect. It is found that the lowest conductance step of a metallic graphene nanoribbon may develop a dip even down to zero at certain values of the Fermi energy due to the defect. Accompanying the occurrence of the conductance dip, a loop current develops around the defect. We show how the properties of the conductance dip depend on the parameters of the defect, such as the relative position and severity of the defect as well as the width and edges of the graphene ribbons. In part…
We analyze an interplay between Coulomb blockade and quantum fluctuations in a coherent conductor (with dimensionless conductance $g \gtrsim 1$) attached to an Ohmic shunt. We demonstrate that at T=0 the system can be either an insulator or a metal depending on whether its total resistance is larger or smaller than $h/e^2\approx 25.8$ k$Ω$. In a metallic phase the Coulomb gap is fully suppressed by quantum fluctuations. We briefly discuss possible relation of this effect to recent experiments indicating the presence of a metal-insulator phase transition in 2d disordered systems.
We non-perturbatively analyze the effect of electron-electron interactions on weak localization (WL) in relatively short metallic conductors with a tunnel barrier. We demonstrate that the main effect of interactions is electron dephasing which persists down to T=0 and yields suppression of WL correction to conductance below its non-interacting value. Our results may account for recent observations of low temperature saturation of the electron decoherence time in quantum dots.
We study resonant tunneling through an interacting quantum dot coupled to normal metallic and superconducting leads. We show that large Coulomb interaction gives rise to novel effects in Andreev transport. Adopting an exact relation for the Green's function, we find that at zero temperature, the linear response conductance is enhanced due to Kondo-Andreev resonance in the Kondo limit, while it is suppressed in the empty site limit. In the Coulomb blockaded region, on the other hand, the conductance is reduced more than the corresponding conductance with normal leads because large charging ener…
Theoretical description of the field emission of electrons from nanoscale objects weakly coupled to the cathode is presented. It is shown that the field- emission current increases in a step-like fashion due to single-electron charging which leads to abrupt changes of the effective electric field responsible for the field emission. A detailed consideration of the current-voltage characteristics is carried out for a nanocluster modelled by a metallic spherical particle in the close vicinity of the cathode and for a cylindrical silicon nanowire grown on the cathode surface.
We investigate the transport properties of a new class of ratchets. The device is constructed by applying an ac voltage to the metallic single electron tunneling transistor, and a net transport current is induced by the time-dependent bias-voltage, although the voltage value is on the average zero. The mechanism underlying this phenomenon is the Coulomb blockade of the single electron tunneling. The directions and the values of the induced net currents can be well controlled by the gate-voltage. A net transport current has also been observed even in the absence of the external bias-voltages, w…
We report on a systematic study of the Coulomb blockade effects in nanofabricated narrow constrictions in thin (Ga,Mn)As films. Different low-temperature transport regimes have been observed for decreasing constriction sizes: the ohmic, the single electron tunnelling (SET) and a completely insulating regime. In the SET, complex stability diagrams with nested Coulomb diamonds and anomalous conductance suppression in the vicinity of charge degeneracy points have been observed. We rationalize these observations in the SET with a double ferromagnetic island model coupled to ferromagnetic leads. It…
We report measurements on single-electron pumps, consisting of two metallic islands formed by three tunnel junctions in series. We focus on the linear-response conductance as a function of gate voltage and temperature of three samples with varying system parameters. In all cases, strong quantum fluctuation phenomena are observed by a log(k_B T/(2 E_co)) reduction of the maximal conductance, where E_co measures the coupling strength between the islands. The samples display a rich phenomenology, culminating in a non-monotonic behavior of the maximal conductance as a function of temperature.
We study full counting statistics of electron tunneling in Coulomb-blockade systems in the limit of short measuring-time intervals. This limit is particularly suited to identify correlations among tunneling events, but only when analyzing the charge-transfer statistics in terms of factorial cumulants $C_{{\rm F}, m}(t)$ rather than ordinary ones commonly used in literature. In the absence of correlations, the short-time behavior of the factorial cumulants is given by $C_{{\rm F}, m}(t) \propto (-1)^{m-1}t^m$. A different sign and/or a different power law of the time dependence indicates correl…
The purpose of this article is to present an accurate way, based on a physical description, to simulate Coulomb blockade devices. The method underlying the simulations depends only on fundamental parameters of the system and does not require the use of high level fitting parameters as tunneling conductances contrary to number of current Coulomb blockade simulators. It lies mainly on the transfer Hamiltonian formalism and Bardeen's formula within the framework of effective mass tensor. It can be applied to metallic Coulomb blockade devices as well as semiconductor ones. The details of this meth…
We study Coulomb blockade (CB) oscillations in the thermodynamics of a metallic grain which is connected to a lead by a tunneling contact with a large conductance $g_0$ in a wide temperature range, $E_Cg_0^4 e^{-g_0/2}<T<E_C$, where $E_C$ is the charging energy. Using the instanton analysis and the renormalization group we obtain the temperature dependence of the amplitude of CB oscillations which differs from the previously obtained results. Assuming that at $T < E_Cg_0^4 e^{-g_0/2}$ the oscillation amplitude weakly depends on temperature we estimate the magnitude of CB oscillations in the gr…
A magnet with precessing magnetization pumps a spin current into adjacent leads. As a special case of this spin pumping, a precessing macrospin (magnetization) can assist electrons in tunneling. In small systems, however, the Coulomb blockade effect can block the transport of electrons. Here, we investigate the competition between macrospin-assisted tunneling and Coulomb blockade for the simplest system where both effects meet; namely, for a single tunnel junction between a normal metal and a metallic ferromagnet with precessing magnetization. By combining Fermi's golden rule with magnetizatio…
A general expression is given for the change in free energy when a charge tunnels through a junction in a one-dimensional array of N metallic islands with arbitrary capacitances and arbitrary background charges. This is used to obtain expressions for the (average) threshold voltage of the Coulomb blockade for a few characteristic geometries. We find that including random background charges has a large effect on the N-dependence of the threshold voltage: In an array with identical junction capacitances C and gate capacitances C_g, the threshold voltage, averaged over the background charge, is p…
We study admittance and energy dissipation in an out-of-equlibrium single electron box. The system consists of a small metallic island coupled to a massive reservoir via single tunneling junction. The potential of electrons in the island is controlled by an additional gate electrode. The energy dissipation is caused by an AC gate voltage. The case of a strong Coulomb blockade is considered. We focus on the regime when electron coherence can be neglected but quantum fluctuations of charge are strong due to Coulomb interaction. We obtain the admittance under the specified conditions. It turns ou…
We study an interacting single-level quantum dot weakly coupled to three electrodes. When two electrodes are biased by voltages with opposite polarities, while keeping the third lead (the stem) grounded, the current through the stem is a measure of electron-hole asymmetry of the dot. In this setup we calculate the stem current for both metallic and ferromagnetic (collinearly polarized) leads and discuss how the three-terminal device gives additional information compared to the usual two-terminal setup. We calculate both the sequential and cotunneling contribution for the currents. For the latt…
High quality single wall carbon nanotube quantum dots have been made showing both metallic and semiconducting behavior. Some of the devices are identified as small band gap semiconducting nanotubes with relatively high broad conductance oscillations for hole transport through the valence band and low conductance Coulomb blockade oscillations for electron transport through the conduction band. The transition between these regimes illustrates that transport evolves from being wave-like transmission known as Fabry-Perot interference to single particle-like tunneling of electrons or holes. In the …
Cooling nanoelectronic devices below 10 mK is a great challenge since thermal conductivities become very small, thus creating a pronounced sensitivity to heat leaks. Here, we overcome these difficulties by using adiabatic demagnetization of \emph{both} the electronic leads \emph{and} the large metallic islands of a Coulomb blockade thermometer. This reduces the external heat leak through the leads and also provides on-chip refrigeration, together cooling the thermometer down to 2.8$\pm$0.1 mK. We present a thermal model which gives a good qualitative account and suggests that the main limitati…
We demonstrate radiofrequency thermometry on a micrometer-sized metallic island below 100 mK. Our device is based on a normal metal-insulator-superconductor tunnel junction coupled to a resonator with transmission readout. In the first generation of the device, we achieve 90 μK/Hz^1/2 noise-equivalent temperature with 10 MHz bandwidth. We measure the thermal relaxation time of the electron gas in the island, which we find to be of the order of 100 μs. Such a calorimetric detector, upon optimization, can be seamlessly integrated into superconducting circuits, with immediate applications in quan…
We report on a new type of single-electron transistor (SET) comprising two highly resistive Cr thin-film strips (~ 1um long) connecting a 1 um-long Al island to two Al outer electrodes. These resistors replace small-area oxide tunnel junctions of traditional SETs. Our transistor with a total asymptotic resistance of 110 kOhm showed a very sharp Coulomb blockade and reproducible, deep and strictly e-periodic gate modulation in wide ranges of bias currents I and gate voltages V_g. In the Coulomb blockade region (|V| < 0.5 mV), we observed a strong suppression of the cotunneling current allowing …
We study the low temperature electrical transport behaviour of a silicon single electron transistor. The island and leads are defined by patterned phosphorus doped regions achieved by ion implantation through a polymer resist mask. In the device a 50 nm diameter island, containing ~600 donors and having a metallic density of states, is separated from source and drain leads by undoped silicon tunnel barriers. The central island and tunnel barriers are covered by a surface gate in a field effect transistor geometry allowing the coupling between the leads and island to be controlled. Coulomb bloc…
We have theoretically studied the thermoelectric properties of serially coupled quantum dots (SCQD) embedded in an insulator matrix connected to metallic electrodes. In the framework of Keldysh Green's function technique, the Landauer formula of transmission factor is obtained by using the equation of motion method. Based on such analytical expressions of charge and heat currents, we calculate the electrical conductance, Seebeck coefficient, electron thermal conductance and figure of merit (ZT) of SCQD in the linear response regime. The effects of electron Coulomb interactions on the reduction…
We discuss electron transport in individual nanometer-scale metallic grains at dilution refrigerator temperatures. In the weak coupling regime, the grains exhibit Coulomb blockade and discrete energy levels. Electron-electron interactions lead to clustering and broadening of quasiparticle states. Magnetic field dependences of tunneling resonances directly reveal Kramers degeneracy and Lande g-factors. In grains of Au, which have strong spin-orbit interaction, g-factors are strongly suppressed from the free electron value. We have recently studied grains in the strong coupling regime. Coulomb b…
We describe a technique for connecting a nanometer-scale gold grain to leads by atomic-scale gold point contacts. These devices differ from previous metallic quantum dots in that the conducting channels are relatively well-transmitting. We investigate the dependence of the Coulomb blockade on contact resistance. The high-resistance devices display Coulomb blockade and the low-resistance devices display a zero-bias conductance dip, both in quantitative agreement with theory. We find that in the intermediate regime, where the sample resistance is close to $h/e^2$, the I-V curve displays a Coulom…
The distribution of nearest-neighbour spacings of Coulomb blockade oscillation peaks in normal conducting aluminum single electron transistors is found to be non-Gaussian. A pronounced tail to reduced spacings is observed, which we attribute to impurity-specific parametric charge rearrangements close to the transistor. Our observation may explain the absence of a Wigner-Dyson distribution in the experimental nearest-neighbour spacing distributions in semiconductor quantum dots.
Correlated states emerge in low-dimensional systems owing to enhanced Coulomb interactions. Elucidating these states requires atomic scale characterization and delicate control capabilities. In this study, spectroscopic imaging-scanning tunneling microscopy was employed to investigate the correlated states residing in the one-dimensional electrons of the monolayer and bilayer MoSe2 mirror twin boundary (MTB). The Coulomb energies, determined by the wire length, drive the MTB into two types of ground states with distinct respective out-of-phase and in-phase charge orders. The two ground states …
We find that tunnel junctions are not needed to provide single electron effects in a metallic island. Eventually the tunnel junction may be replaced by an arbitrary scatterer. It is important that even a {\it diffusive} scatterer provides a sufficient isolation for single electron effects to persist. To formulate this in exact terms, we derive and analyze the effective action that describes an arbitrary scatterer. We also consider the fluctuations of the effective charging energy.
Disordered granular systems, at temperatures where charging effects are important, are studied, by means of an effective medium approximation. The intragrain charging energy leads to insulating behavior at low temperatures, with a well defined Coulomb gap. Non equilibrium effects can give rise to a zero temperature transition between a metallic, gapless phase, and an insulating phase.
We study the tunneling transport through a nanojunction in the far-from-equilibrium regime at relatively low temperatures. We show that the current-voltage characteristics is significantly modified as compared to the usual quasi-equilibrium result by lifting the suppression due to the Coulomb blockade. These effects are important in realistic nanojunctions. We study the high-impedance case in detail to explain the underlying physics and construct a more realistic theoretical model for the case of a metallic junction taking into account dynamic Coulomb interaction. This dynamic screening furthe…
We develop a theory of the exchange interaction between ferromagnetic (FM) metallic grains embedded into insulating matrix by taking into account the Coulomb blockade effects. For bulk ferromagnets separated by the insulating layer the exchange interaction strongly depends on the height and thickness of the tunneling barrier created by the insulator. We show that for FM grains embedded into insulating matrix the exchange coupling additionally depends on the dielectric properties of this matrix due to the Coulomb blockade effects. In particular, the FM coupling decreases with decreasing the die…
We study the low frequency admittance of a small metallic island coupled to a gate electrode and to a massive reservoir via a \emph{multi channel} tunnel junction. The ac current is caused by a slowly oscillating gate voltage. We focus on the regime of inelastic cotunneling in which the dissipation of energy (the real part of the admittance) is determined by two-electron tunneling with creation of electron-hole pairs on the island. We demonstrate that at finite temperatures but low frequencies the energy dissipation is ohmic whereas at zero temperature it is super-ohmic. We find that (i) the c…
We study charge transport of a topological superconductor connected to different electromagnetic environments using a low-energy description where only the Majorana bound states in the superconductor are included. Extending earlier findings who found a crossover between perfect Andreev reflection with conductance $2e^2/h$ to a regime with blocked transport when the resistance of the environment is larger than $2e^2/h$, we consider Majorana bound states coupled to metallic dots. in particular, we study two topological superconducting leads connected by a metallic quantum dot in both the weak tu…
