Distribution Of 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.

Etched graphene quantum dots on hexagonal boron nitride

We report on the fabrication and characterization of etched graphene quantum dots (QDs) on hexagonal boron nitride (hBN) and SiO2 with different island diameters. We perform a statistical analysis of Coulomb peak spacings over a wide energy range. For graphene QDs on hBN, the standard deviation of the normalized peak spacing distribution decreases with increasing QD diameter, whereas for QDs on SiO2 no diameter dependency is observed. In addition, QDs on hBN are more stable under the influence of perpendicular magnetic fields up to 9 T. Both results indicate a substantially reduced substrate induced disorder potential in graphene QDs on hBN.

Peak spacing statistics in silicon single-electron transistors: Size and gate oxide thickness dependence

We measured the fluctuations of the addition spectra versus the size of the quantum dot in low-mobility gated nanowires etched in silicon-on-insulator thin films. The standard deviation of the peak spacing distribution in the high electron concentration regime is found to scale as the inverse surface area of the dot and its absolute value is comparable to the energy spacing Δ of the one-particle spectrum. We have also varied the gate oxide thickness for dots with the same size. Away from the first oscillations we observe that the gate capacitance is remarkably constant over a large gate voltage range, whatever the dimensions of the dot and the gate oxide thickness. In these nanowires the confinement of the electronic wavefunctions is not provided by intentional tunnel junctions but by the quasi-1D nanowire itself, which behaves at low temperature like a disordered insulator. Coulomb blockade diamonds give a direct measurement of the small capacitance of those access regions.

Cotunnelling versus sequential tunnelling in Coulomb blockade metallic double quantumdot structures

The cotunnelling is systematically studied in comparison to sequential tunnelling inCoulomb blockade metallic double quantum dot structures, using the standardmaster equation approach. In the case of zero gate voltage, we are able to deriveanalytical expressions of threshold voltages for both tunnelling processes:Vth(s) for the sequentialtunnelling and Vth(2) for the lowest-order inelastic macroscopic quantum tunnelling (cotunnelling). Taking intoaccount the gate and temperature effects, numerical solutions of the master equationshow that an increase of the inter-dot capacitance leads to a decrease of the ratioVth(2)/Vth(s) and at the same time to a decrease of cotunnelling conductance compared to the sequentialone. An oscillation of cotunnelling conductance is observed in classical Coulomb blockedregions. In comparison with the sequential tunnelling conductance spectroscopy, the peakheight of cotunnelling conductance is about three orders of magnitude smaller and the peakspacing distribution is far from regular. Increase of temperature raises the current anddestroys the Coulomb gap. Its relative influence is more important at lower bias.

Electron-electron interactions in isolated and realistic quantum dots: A density functional theory study

We use Kohn-Sham spin-density-functional theory to study the statistics of ground-state spin and the spacing between conductance peaks in the Coulomb blockade regime for both 2D isolated and realistic quantum dots. We make a systematic investigation of the effects of electron-electron interaction strength and electron number on both the peak spacing and spin distributions. A direct comparison between the distributions from isolated and realistic dots shows that, despite the difference in the boundary conditions and confining potential, the statistical properties are qualitatively the same. Strong even/odd pairing in the peak spacing distribution is observed only in the weak e-e interaction regime and vanishes for moderate interactions. The probability of high spin ground states increases for stronger e-e interaction and seems to saturate around $r_s \sim 4$. The saturated value is larger than previous theoretical predictions. Both spin and conductance peak spacing distributions show substantial variation as the electron number increases, not saturating until $N \sim 150$. To interpret our numerical results, we analyze the spin distribution in the even $N$ case using a simple two-level model.

Spin ande−einteractions in quantum dots: Leading order corrections to universality and temperature effects

We study the statistics of the spacing between Coulomb blockade conductance peaks in quantum dots with large dimensionless conductance g. Our starting point is the ``universal Hamiltonian''--valid in the g->oo limit--which includes the charging energy, the single-electron energies (described by random matrix theory), and the average exchange interaction. We then calculate the magnitude of the most relevant finite g corrections, namely, the effect of surface charge, the ``gate'' effect, and the fluctuation of the residual e-e interaction. The resulting zero-temperature peak spacing distribution has corrections of order Delta/sqrt(g). For typical values of the e-e interaction (r_s ~ 1) and simple geometries, theory does indeed predict an asymmetric distribution with a significant even/odd effect. The width of the distribution is of order 0.3 Delta, and its dominant feature is a large peak for the odd case, reminiscent of the delta-function in the g->oo limit. We consider finite temperature effects next. Only after their inclusion is good agreement with the experimental results obtained. Even relatively low temperature causes large modifications in the peak spacing distribution: (a) its peak is dominated by the even distribution at kT ~ 0.3 Delta (at lower T a double peak appears); (b) it becomes more symmetric; (c) the even/odd effect is considerably weaker; (d) the delta-function is completely washed-out; and (e) fluctuation of the coupling to the leads becomes relevant. Experiments aimed at observing the T=0 peak spacing distribution should therefore be done at kT<0.1 Delta for typical values of the e-e interaction.

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.

Signatures of spin pairing in chaotic quantum dots.

Coulomb blockade resonances are measured in a GaAs quantum dot in which both shape deformations and interactions are small. The parametric evolution of the Coulomb blockade peaks shows a pronounced pair correlation in both position and amplitude, which is interpreted as spin pairing. As a consequence, the nearest-neighbor distribution of peak spacings can be well approximated by a modified bimodal Wigner surmise, in which interactions are taken into account beyond the constant interaction model.

Scanning tunneling spectroscopy of InAs nanocrystal quantum dots

Scanning tunneling spectroscopy is used to investigate single InAs nanocrystals, 20--70 \AA{} in diameter, in a highly asymmetric double barrier tunnel junction configuration. The $I\ensuremath{-}V$ characteristics reflect contributions of both single-electron charging and the atomiclike level structure of the quantum dots. The spectra are simulated and well described within the framework of the ``orthodox model'' for single-electron tunneling. The peaks in the tunneling spectra display a systematic broadening with the reduction of dot diameter, from 40 to 150 meV over the studied quantum dot size range. This is assigned to a decreased electron dwell time on the dot, due to reduction of the barrier height, induced by the blueshift of the quantum-confined levels. The distribution of peak spacings within charging multiplets in the tunneling spectra is found to be Gaussian, resembling observations on metallic quantum dots.

Random matrix model for quantum dots with interactions and the conductance peak spacing distribution

We introduce a random interaction matrix model (RIMM) for finite-size strongly interacting fermionic systems whose single-particle dynamics is chaotic. The model is applied to Coulomb blockade quantum dots with irregular shape to describe the crossover of the peak spacing distribution from a Wigner-Dyson to a Gaussian-like distribution. The crossover is universal within the random matrix model and is shown to depend on a single parameter: a scaled fluctuation width of the interaction matrix elements. The crossover observed in the RIMM is compared with the results of an Anderson model with Coulomb interactions.

Correlation Effect on Ground State Energy Statistics in Disordered Quantum Dots

Distribution of peak spacings of the Coulomb oscillation is theoretically investigated in disordered quantum dots. Taking account of the electron–electron interaction by the perturbation, we show that a pair-type (BCS-type) correlation plays an important role in determining the distribution in the diffusive region. The correlation effect can result in a sharp peak structure of the distribution, deviating drastically from the prediction of the random matrix theory, in the absence of a magnetic field. Numerical results support the proposed mechanisms. The pair-type correlation does not work in the presence of a magnetic field.

Absence of Bimodal Peak Spacing Distribution in the Coulomb Blockade Regime

Using exact diagonalization numerical methods, as well as analytical arguments, we show that for the typical electron densities in chaotic and disordered dots the peak spacing distribution is not bimodal, but rather Gaussian. This is in agreement with the experimental observations. We attribute this behavior to the tendency of an even number of electrons to gain on-site interaction energy by removing the spin degeneracy. Thus, the dot is predicted to show a non trivial electron number dependent spin polarization. Experimental test of this hypothesis based on the spin polarization measurements are proposed.

Coulomb Blockade Peak Spacing Fluctuations in Deformable Quantum Dots: A Further Test of Random Matrix Theory

We propose a mechanism to explain the fluctuations of the ground state energy in quantum dots in the Coulomb blockade regime. Employing random matrix theory we show that shape deformations may change the adjacent peak spacing distribution from Wigner-Dyson to nearly Gaussian even in the absence of charging energy fluctuations. The distribution is determined by the average number of anticrossings between successive conductance peaks and the presence or absence of a magnetic field. Our mechanism is tested in a dynamical model whose classical dynamics is chaotic. The results are in good agreement with experiments and apply to spin resolved or spin degenerate states. [S0031-9007(98)06645-9] PACS numbers: 73.23.Hk, 05.45. + b Most phenomena in mesoscopic electronic transport are well understood invoking the picture of a single quantum particle moving coherently in a disordered or complex potential. In the case of semiconductor quantum dots, the complexity of the background or confining potential is manifest in the chaotic nature of the underlying electronic classical dynamics. A large amount of literature [1] indicates that the statistical properties of the spectra of chaotic systems are successfully modeled by random matrix theory (RMT) [2]. When applied to transport in open ballistic quantum dots of irregular shape, this theory predicts a strong universal behavior for the statistical fluctuations of the conductance, similar to that found in disordered systems [3]. Over the last few years, several experiments confirmed this behavior, making open electronic cavities one of the paradigms of quantum chaos [4,5].