Coulomb Blockade Peak Spacings Research Articles

Coulomb-blockade peak spacing statistics of graphene quantum dots on SiO2

Extrinsic disorder strongly affects the performance of graphene-based quantum dots. The standard SiO2 substrate is generally considered to be one major factor besides edge-induced disorder. In this report we present the fabrication of lithographically defined quantum dots on SiO2 with short and narrow constrictions and different central island sizes. Low temperature transport measurements display distinct Coulomb-blockade peaks with amplitudes exceeding what is commonly observed experimentally. The analysis of the normalized Coulomb-blockade peak spacing shows a size dependence, which has not previously been observed for devices on SiO2. Furthermore, a quantitative comparison of the peak spacing distribution to the literature shows that one of the two devices compares favorably to a similar sized dot placed on hexagonal boron nitride, which is known to reduce the substrate disorder. Our findings suggest that the other sources of extrinsic disorder, such as lithography residues, may play an important role for the performance of large graphene quantum dots.

Coulomb blockade doppelgängers in quantum Hall states

In this paper, we ask the question: how well can Coulomb blockade experiments correctly identify and distinguish between different topological orders in quantum Hall states? We definitively find the answer to be: quite poorly. In particular, we write the general expression for the spacing of resonance peaks in a simple form that explicitly displays its dependence on the conformal scaling dimensions of the systems' edge modes. This form makes transparent the general argument that the Coulomb blockade peak spacings do not provide a strongly indicative signature of the topological order of the system since it is only weakly related to the braiding statistics. We bolster this general argument with examples for all the most physically relevant non-Abelian candidate states, demonstrating that they have Coulomb blockade doppelg\"angers--candidate states at the same filling fraction with identical Coulomb blockade signatures but dramatically different topological orders and braiding statistics.

Transport properties of double-walled carbon nanotube quantum dots

The transport properties of quantum dot (QD) systems based on double-walled carbon nanotubes (DWCNTs) are investigated. The interplay between microscopic structure and strong Coulomb interaction is treated within a bosonization framework. The linear and nonlinear current-voltage characteristics of the QD system are calculated by starting from the Liouville equation for the reduced density matrix. Depending on the intershell couplings, an eight-electron periodicity of the Coulomb blockade peak spacing in the case of commensurate DWCNT QDs and a four-electron periodicity in the incommensurate case are predicted. The contribution of excited states of DWCNTs to the nonlinear transport is investigated as well.

Coulomb blockade peak spacings: Interplay of spin and dot-lead coupling

For Coulomb blockade peaks in the linear conductance of a quantum dot, we study the correction to the spacing between the peaks due to dot-lead coupling. This coupling can affect measurements in which Coulomb blockade phenomena are used as a tool to probe the energy level structure of quantum dots. The electron-electron interactions in the quantum dot are described by the constant exchange and interaction (CEI) model while the single-particle properties are described by random matrix theory. We find analytic expressions for both the average and rms mesoscopic fluctuation of the correction. For a realistic value of the exchange interaction constant J_s, the ensemble average correction to the peak spacing is two to three times smaller than that at J_s = 0. As a function of J_s, the average correction to the peak spacing for an even valley decreases monotonically, nonetheless staying positive. The rms fluctuation is of the same order as the average and weakly depends on J_s. For a small fraction of quantum dots in the ensemble, therefore, the correction to the peak spacing for the even valley is negative. The correction to the spacing in the odd valleys is opposite in sign to that in the even valleys and equal in magnitude. These results are robust with respect to the choice of the random matrix ensemble or change in parameters such as charging energy, mean level spacing, or temperature.

Statistics of wave functions in disordered systems with applications to Coulomb blockade peak spacing

Despite considerable work on the energy-level and wavefunction statistics of disordered quantum systems, numerical studies of those statistics relevant for electron-electron interactions in mesoscopic systems have been lacking. We plug this gap by using a tight-binding model to study a wide variety of statistics for the two-dimensional, disordered quantum system in the diffusive regime. Our results are in good agreement with random matrix theory (or its extensions) for simple statistics such as the probability distribution of energy levels or spatial correlation of a wavefunction. However, we see substantial disagreement in several statistics which involve both integrating over space and different energy levels, indicating that disordered systems are more complex than previously thought. These are exactly the quantities relevant to electron-electron interaction effects in quantum dots; in fact, we apply these results to the Coulomb blockade, where we find altered spacings between conductance peaks and wider spin distributions than traditionally expected.

Ground-state energy and spin in disordered quantum dots

We investigate the ground-state energy and spin of disordered quantum dots using spin-density-functional theory. Fluctuations of addition energies (Coulomb-blockade peak spacings) do not scale with average addition energy but remain proportional to level spacing. With increasing interaction strength, the even-odd alternation of addition energies disappears, and the probability of nonminimal spin increases, but never exceeds $50%.$ Within a two-orbital model, we show that the off-diagonal Coulomb matrix elements help stabilize a ground state of minimal spin.

Interactions in chaotic nanoparticles: Fluctuations in Coulomb blockade peak spacings

We use random matrix models to investigate the ground state energy of electrons confined to a nanoparticle. Our expression for the energy includes the charging effect, the single-particle energies, and the residual screened interactions treated in Hartree-Fock. This model is applicable to chaotic quantum dots or nanoparticles--in these systems the single-particle statistics follows random matrix theory at energy scales less than the Thouless energy. We find the distribution of Coulomb blockade peak spacings first for a large dot in which the residual interactions can be taken constant: the spacing fluctuations are of order the mean level separation Delta. Corrections to this limit are studied using the small parameter 1/(kf L): both the residual interactions and the effect of the changing confinement on the single-particle levels produce fluctuations of order Delta/sqrt(kf L). The distributions we find are significantly more like the experimental results than the simple constant interaction model.

Coulomb-blockade peak-spacing distribution: Interplay of temperature and spin

We calculate the Coulomb Blockade peak spacing distribution at finite temperature using the recently introduced ``universal Hamiltonian'' to describe the e-e interactions. We show that the temperature effect is important even at kT~0.1\Delta (\Delta is the single-particle mean level spacing). This sensitivity arises because: (1) exchange reduces the minimum energy of excitation from the ground state and (2) the entropic contribution depends on the change of the spin of the quantum dot. Including the leading corrections to the universal Hamiltonian yields results in quantitative agreement with the experiments. Surprisingly, temperature appears to be the most important effect.

Fluctuations of interactions and the peak spacings distribution in Coulomb blockade quantum dots

Experimental and theoretical investigations of the distribution of Coulomb blockade peak spacings in chaotic quantum dots have shown a significant deviation from the Wigner–Dyson distribution predicted by the constant interaction model. Recently we introduced a random interaction matrix model for chaotic or diffusive dots that describes the crossover of the peak spacing statistics from a Wigner–Dyson distribution to a Gaussian-like distribution. Here we use this model to study the dependence of the peak-spacing distributions on the underlying space-time symmetries of the one-body Hamiltonian, and in particular we compare the cases of conserved and broken time-reversal symmetry. We also study the dependence of the peak spacing fluctuations on disorder strength in diffusive dots using a standard tight-binding Hamiltonian with Coulomb interactions. The fluctuations are found to increase with disorder and can be related to the disorder-enhanced fluctuations of the interaction matrix elements.

Statistics of Coulomb-blockade peak spacings for a partially open quantum dot

We show that randomness of the electron wave functions in a quantum dot contributes to the fluctuations of the positions of the conductance peaks. This contribution grows with the conductance of the junctions connecting the dot to the leads. It becomes comparable with the fluctuations coming from the randomness of the single-particle spectrum in the dot while the Coulomb-blockade peaks are still well-defined. In addition, the fluctuations of the peak spacings are correlated with the fluctuations of the conductance peak heights.

Addition Spectra of Chaotic Quantum Dots: Interplay between Interactions and Geometry

We investigate the influence of interactions and geometry on ground states of clean chaotic quantum dots using the self-consistent Hartree-Fock method. We find two distinct regimes of interaction strength: While capacitive energy fluctuations $\delta \chi$ follow approximately a random matrix prediction for weak interactions, there is a crossover to a regime where $\delta \chi$ is strongly enhanced and scales roughly with interaction strength. This enhancement is related to the rearrangement of charges into ordered states near the dot edge. This effect is non-universal depending on dot shape and size. It may provide additional insight into recent experiments on statistics of Coulomb blockade peak spacings.

Statistics of Coulomb-blockade peak spacings within the Hartree-Fock approximation

We study the effect of electronic interactions on the addition spectra and on the energy level distributions of two-dimensional quantum dots with weak disorder using the self-consistent Hartree-Fock approximation for spinless electrons. We show that the distribution of the conductance peak spacings is Gaussian with large fluctuations that exceed, in agreement with experiments, the mean level spacing of the non-interacting system. We analyze this distribution on the basis of Koopmans' theorem. We show furthermore that the occupied and unoccupied Hartree-Fock levels exhibit Wigner-Dyson statistics.

Fluctuation of conductance-peak spacings in large semiconductor quantum dots

Fluctuation of Coulomb blockade peak spacings in large two-dimensional semiconductor quantum dots are studied within a model based on the electrostatics of several electron islands among which there are random inductive and capacitive couplings. Each island can accommodate electrons on quantum orbitals whose energies depend also on an external magnetic field. In contrast with a single island quantum dot, where the spacing distribution is close to Gaussian, here the distribution has a peak at small spacing value. The fluctuations are mainly due to charging effects. The model can explain the occasional occurrence of couples or even triples of closely spaced Coulomb blockade peaks, as well as the qualitative behavior of peak positions with the applied magnetic field.

Fluctuation of inverse compressibility for electronicsystems with random capacitive matrices

This article is concerned with the statistics of the addition spectra of certain many-body systems of identical particles. In the first part, the pertinent system consists of N identical particles distributed among K<N independent subsystems, such that the energy of each subsystem is a quadratic function of the number of particles residing on it with random coefficients. On a large scale, the ground-state energy E(N) of the whole system grows quadratically with N, but in general there is no simple relation such as EN = aN+bN 2. The deviation of E(N) from exact quadratic behaviour implies that its second difference (the inverse compressibility) XN ≡E(N+1)−2E(N)+E(N−1) is a fluctuating quantity. Regarding the numbers XN as values assumed by a certain random variable X, we obtain a closed-form expression for its distribution F (X). Its main feature is that the corresponding density P (X)=dF (X)/d X has a maximum at the point X=0. As K→∞ the density is Poissonian, namely, P(X)→e−X This result serves as a starting point for the second part, in which coupling between subsystems is included. More generally, a classical model is suggested in order to study fluctuations of Coulomb blockade peak spacings in large two-dimensional semiconductor quantum dots. It is based on the electrostatics of several electron islands among which there are random inductive and capacitive couplings. Each island can accommodate electrons on quantum orbitals whose energy depends also on an external magnetic field. In contrast to a single-island quantum dot, where the spacing distribution between conductance peaks is close to Gaussian, here the distribution has a peak at small spacing value. The fluctuations are mainly due to charging effects. The model can explain the occasional occurrence of couples or even triples of closely spaced Coulomb blockade peaks, as well as the qualitative behaviour of peak positions with the applied magnetic field.

Statistics of Coulomb Blockade Peak Spacings

Distributions of Coulomb blockade peak spacings are reported for large ensembles of both unbroken (magnetic field $B\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$) and broken ( $B\ensuremath{\ne}0$) time-reversal symmetry in GaAs quantum dots. Both distributions are symmetric and roughly Gaussian with a width of $\ensuremath{\sim}2%--6%$ of the average spacing, with broad, non-Gaussian tails. The distribution is systematically wider at $B\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}0$ by a factor of $\ensuremath{\sim}1.2\ifmmode\pm\else\textpm\fi{}0.1$. No even-odd spacing correlations or bimodal structure in the spacing distribution is found, suggesting an absence of spin degeneracy. There is no observed correlation between peak spacing and peak height.

Parity Effect in Ground State Energies of Ultrasmall Superconducting Grains

We study the superconductivity in small grains in the regime when the quantum level spacing $\delta\varepsilon$ is comparable to the gap $\Delta$. As $\delta\varepsilon$ is increased, the system crosses over from superconducting to normal state. This crossover is studied by calculating the dependence of the ground state energy of a grain on the parity of the number of electrons. The states with odd numbers of particles carry an additional energy $\Delta_P$, which shows non-monotonic dependence on $\delta\varepsilon$. Our predictions can be tested experimentally by studying the parity-induced alternation of Coulomb blockade peak spacings in grains of different sizes.

Fluctuations of Conductance Peak Spacings in the Coulomb Blockade Regime: Role of Electron-Electron Interaction

We study influence of electron-electron interaction on statistics of Coulomb blockade peak spacings in disordered quantum dots. It is shown that the interaction combined with fluctuations of eigenfunctions of the Fermi sea enhances the peak spacing fluctuations, in accordance with recent experiments. In addition, account of the spin degrees of freedom leads to a pronounced odd-even structure for weak interaction ( ${e}^{2}/\ensuremath{\epsilon}\ensuremath{\ll}{v}_{F}$); in the opposite case ( ${e}^{2}/\ensuremath{\epsilon}\ensuremath{\gtrsim}{v}_{F}$) this structure is washed out.