Research Projects
Conductance Peak Spacing Distributions in the Coulomb Blockade Regime
Research conducted by: Steven Tomsovic, Tatsuro Nagano, Denis Ullmo (LPTMS, Orsay and Duke University)
Introduction
There exist many kinds of quantum dots.
Each dot differs in shape, size, material, and fabrication process.
We are interested in the semiconductor quantum dot generated
in a two-dimensional electron gas (2DEG) which forms at the interface of
a GaAs/Al_x Ga_{1-x}As heterostructure due to confining potential barries from applied
negative gate voltages.
When an isolated region is weakly coupled to two leads and
controlled with the gate electrode to adjust the energy of the region,
the current through this region is suppressed (at low temperatures, this is
known as the Coulomb blockade).
The theory of the Coulomb blockade was developed for a metal
by Kulik and Shekhter in the classical regime (Delta <
In semiconductor quantum dots, the situation is different from a metal.
Due to the small system size, the energy of the
isolated region ``dot'' becomes discrete, and at low enough
temperature, one has to take the effects of the discreteness
of the energy levels into account.
In this 'quantum' regime, the charging energy, the single-particle energy spacing,
and the thermal energy are related by e^2/C approx Delta >> k_B T.
Whereas in the classical regime,
the current flow occurs at the degeneracy point,
there is an additional condition for quantum systems
in order to have current flow.
In the limit of zero-temperature and
small, applied source-drain bias V_{sd},
the gate voltage must be tuned such that
the first unoccupied energy level of the dot
matches the Fermi energy of the electron reservoirs.
As a consequence, an electron can move into the dot and out to the drain
by resonant tunneling,
and current will flow in response to the bias voltage.
Based on the Bohigas-Giannoni-Schmit conjecture,
the single-particle quantum system, whose classical counterpart
represents chaotic dynamics, can be investigated in terms of Random Matrix Theory (RMT)
Because many of the dots come with disordered or irregular shape,
the energy spectrum of the dot can be approximated by the eigenvalues
of a random matrix.
In addition, quantum dot experiments normally proceed by collecting
an ensemble of measured data. This indicates that the important physical
quantities are not system specific information, such as shape for example,
but more generic properties such as symmetry classes.
Time-reversal symmetry within a dot is controlled by applying a magnetic field.
The presence (absence) of magnetic field breaks (preserves) the time-reversal
symmetry.
Our work
We sought to answer the question of how the nature of the dynamics influences the
single particle ground state occupancies when spin is taken into account.
We consider a system of quasi-particles confined by an effective potential and
consider a system of quasi-particles confined by an effective potential interacting through
a screened Coulomb interaction V(r - r') ~ delta(r - r') / N(0).
We introduce a simple, two-dimensional, coupled quartic potential to represent the effective
confinement.
This potential can be tuned so that the dynamics range from integrability to pure chaos.
Our primary concerns were
(1) to investigate how well the previous random matrix theory predictions capture the behavior of pu
rely chaotic dynamical systems and
(2) to investigate how the properties depend on the nature of the dynamics when the coupling paramet
er varies from the chaotic to integrable limiting cases. The implications for the
statistical behavior of the ground state spin and Coulomb blockade peak spacing in nanostructure qua
ntum dots are considered.
Our results are:
(1)Based on the Gaussian ensemble, the distribution of the CB peak spacing (as given by the
combination of Delta-function and the Wigner surmise) shows the strong bimodal structure of the
electron spin effect.
(2)The statistics obtained from the CI +RMT predictions are significantly different than those
obtained from experimental measurements of peak conductances in irregular shaped quantum
dots in the Coulomb blockade.
-The distributions are symmetric and well described by a Gaussian fit
except for the long tails on both sides.
-The width (fluctuation) of distributions are larger than predicted, and
the fluctuations are of order 0.1 - 0.15 measured in units of the
classical charging energy.
-no bimodal structure has been observed as if the statistics are independent of the electron spin.
Figures
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| Schematic illustration of Coulomb blockade in quantum dots. Fermi energy of source and drain is E_F with small bias V_sd, and the discrete energy levels are shown. Left figure: the charging energy creates effective gap e^2/C between the highest filled level and the lowest unoccupied level. Since the chemical potential of the dot does not match the the Fermi levels of the source and drain, the electron tunneling process is prohibited. The number of electrons inside the dot is fixed. Right figure: the gate voltage is tuned to lower the chemical potential of the dot matching the Fermi energy of reservoirs. This situation lifts the Coulomb blockade and allows an electron to tunnel through the barrier one at a time. As a consequence, the conductance shows a peak. |
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| Probability density (top) and cumulative distribution (bottom) of the fluctuating part of the peak spacings (scaled by mean level spacing). Distributions are obtained using both even and odd spacings combined. The dynamical regimes are set to near integrable (dotted line), mixed (dashed), and chaotic (solid) for the TRNI quartic oscillator system. |
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| Electron orbital occupation scenario for the TRNI quartic oscillator system in the near integrable regime (\lambda=0.20, reflection symmetry (+,+), and zeta = 0.8). The main (left) figure shows orbital occupations of the ground states for the N electron systems with N from 124 to 135. The horizontal line represents energy levels, and the vertical axis is adjusted such that the location of each line corresponds to its energy value. Each triangular point represents an electron, and head of triangle indicates spin orientation, up or down. The right figure shows scaled M_{ii} diagonal elements. The magnitude of M_{ii} is given by the horizontal axis, and the orbital number i is adjusted to match the vertical axis of the left figure. (The figure is modified from the original generated by D. Ullmo with permission.) |