Ben-Gurion University  
of the Negev
 
 
 

An Israeli Science Foundation Workshop

Strong Interactions    in Quantum Dots

Program

(abstracts available by clicking on the titles).

All talks are 35 minutes long, with a 15 minutes discussion period.


Saturday, October 25th:

8pm: reception.


Sunday, October 26th:

AM (Chairwoman: Ora Entin-Wohlman) :
1000: opening remarks
1015: L. Kouwenhoven: Excitation spectra of circular few-electron quantum dots
1105: coffee break
1125: C. Marcus: Correlations Between Ground and Excited State Spectra in Quantum Dots
1215: B. Shklovskii: Charging spectrum and configurations of a two-dimensional Wigner crystal island

Lunch

PM (Chairman: David Gershoni) :
1630: coffee
1700: E. Braun: Self-Assembly of Nanometer Scale Electronics by Biotechnology
1750: A. Stern: What is the difference between a quantum dot and a big atom ?

Dinner


Monday, October 27th:

AM (Chairman: Thierry Martin) :
0930: M. Heiblum: Phase Measurements and Dephasing in a QD
1020: Y. Meir: Controlled decoherence in tunneling through quantum dots
1110: coffee break
1130: K. Matveev: Coulomb blockade at almost perfect transmission
1220: L. Glazman: Mesoscopic Charge Quantization

Lunch

PM:
1600-1900 poster session

 

Dinner


Tuesday, October 28th:

AM (Chairman: Yosef Imry):
0930: D. Ralph: Effects of electron interactions on the tunneling spectra of aluminium nanoparticles
1020: O. Agam: Nonequilibrium effects in the tunneling conductance of small metallic grains
1110: coffee break
1140: B. Altshuler: Microwave Absorption in Small Metallic Grains

Excursion to Massada


Wednesday, October 29th:

AM (Chairman: Baruch Horovitz) :
0930: H. Grabert: Strong Tunneling and Coulomb Blockade
1020: D. Esteve: Energy redistribution between electrons in a mesoscopic wire
1100: coffee break
1130: G. Schon: Single-electron tunneling in metal junctions and quantum dots
1220: Y. Alhassid: Universal Mesoscopic Fluctuations of the Conductance in Quantum Dots

Lunch

PM (Chairman: Boris Spivak) :
1700: coffee
1730: L. Levitov: to be announced
1820: R. Berkovits: Localization in Fock space: A finite size scaling hypothesis

Dinner


Thursday, October 30th:

AM (Chairman: Scot Renn) :
0930: B. Kramer: Electronic spectra and non-linear transport properties of quantum dots

        1020: S. Tarucha: Coulomb Interactions in Quantum Dot Molecules
1110: coffee break
1140: I. Aleiner: Tunneling Anomaly in Superconducting Quantum Dots above Paramagnetic Limit

Lunch

Excursion to Mamshit and conference dinner


Friday, October 31th:

AM (Chairman: Amnon Aharony) :
0900: R. Haug: Transport Spectroscopy in Quantum Dots and Artificial Molecules
0950: R. Ashoori: Periodic and Aperiodic Bunching in the Addition Spectra of Quantum Dots
1040: coffee break
1100: M. Kastner: Kondo Physics with an Artificial Atom
1150: concluding remarks

 


Abstracts:


Nonequilibrium effects in the tunneling conductance of small metallic grains
 Oded Agam, NEC Research Institute, 

  Recent experiment on the tunneling spectra of small metallic grains revealed an unusual structure of the differential conductance peaks. Namely, resonance peaks appear in clusters, or develop a substructure as the gate voltage is changed. These features are manifestations of nonequilibrium behavior when the applied source-drain voltage is sufficiently large. Electron-electron Coulomb interaction as well as pairing interaction, when the metallic grain is superconducting, play an important role in determining the magnitude and energy scales of the phenomena. For normal grains, each cluster of resonances is identified with one excited single--electron state of the metal particle, shifted as a result of the different nonequilibrium occupancy configurations of the other single--electron states. Assuming the underlying classical dynamics of the electrons to be chaotic, we determine the typical shift to be $\delta/g$ where $\delta$ is the single particle mean level spacing and $g$ is the dimensionless conductance of the grain. For superconducting grains in the odd charging states, the substructure of the first resonance is explained by nonequilibrium ``gapless'' excitations associated with different energy levels occupied by the unpaired electron. These excitations are generated by inelastic cotunneling.


Tunneling Anomaly in Superconducting Quantum Dots above Paramagnetic Limit,
 I.L. Aleiner, NEC Research Institute,

We consider the tunneling density state in the superconductor above the paramagnetic limit, i.e. Zeeman splitting $E_Z$ is larger than $\sqrt{2}\Delta$, with $\Delta$ being the superconducting gap. In such fields superconductor undergoes first order phase transition to the paramagnetic state. One might expect that the Cooper pairing is irrelevant for the properties of the normal paramagnetic phase. We show that, on the contrary, there are clear and observable effects of the pairing in paramagnetic state {\em even far from the transition region}. One of the most fundamental manifestation of the superconductivity is the gap in the tunneling density of states (DoS) around the zero energy. This gap apparently disappears when the system becomes paramagnetic. We find \cite{elsewhere} that at the same time there appears a dip in the DoS. The position $E^\ast$ is remarkably universal:
E^\ast= \frac{1}{2}\left(E_Z +\Omega\right), \quad \Omega =  \sqrt{E_Z^2-\Delta^2}
for $OD$ (grain), $1D$ (strip) and $2D$ (film) cases. The shape of the tunneling DoS $\nu (\omega)_\sigma$ (where $\sigma=\uparrow,\downarrow$ indicates the spin direction along and opposite to the magnetic field) near the singularity depends on the dimensionality, $d$, of the system: \nu(\omega)_{\uparrow,\downarrow}=\nu_0{\cal F}_d\left(\frac{\omega \pm E^\ast}{W_d}\right) , \quad d=0,1,2.
Here  $\nu_0$ is the bare density of states per one spin, and ${\cal F}_d$ are geometry dependent dimensionless functions. For the quantum dots ( $d=0$) we find
W_0=\left(\frac{\bar{\delta}\Delta^2}{\Omega}\right)^{1/2}, where $\bar{\delta}$ is the one-electron level spacing in the grain. In this case DoS shows the hard gap: {\cal F}_0(x)=\frac{\theta(x^2-1)|x|}{\sqrt{x^2-1}}.

I.L. Aleiner and B.L. Altshuler, Phys. Rev. Lett. (in press) and cond-mat/9704194.


Universal Mesoscopic Fluctuations of the Conductance in  Quantum Dots
 Yoram Alhassid, Yale University

     A statistical theory of the conductance peaks in the Coulomb blockade regime of quantum dots is presented for temperatures that are not necessarily small compared with the mean level spacing. The theory assumes the validity of random matrix theory (e.g. ballistic dots with chaotic dynamics). The conductance peak distributions are found to be universal, depending only on the symmetry class and the temperature measured in units of the mean level spacing $\Delta$. When the temperature $T$ is comparable to $\Delta$ several resonances contribute to the same conductance peak and we find significant deviations from the previously known $T \ll \Delta$ distributions. In contrast to the $T \ll \Delta$ case, these distributions show a strong signature of the charging energy and charge quantization on the dot.  In particular, the sensitivity of the conductance distribution to temperature is suppressed when compared with the predictions of a single-particle theory. The calculations require the canonical free energy and occupations at a fixed number of electrons on the dot, which we find by exact particle-number projection. We also calculate the peak-to-peak correlations as a function of $T/\Delta$, and  conjecture that the oscillations in the level density near the Fermi energy, observed in recent self-consistent density functional calculations, might lead to enhanced correlations. The conductance peak distributions
are also studied  in the crossover from conserved to broken time-reversal symmetry. The weak localization peak of the average conductance maximum is derived in closed form as a function of both magnetic field and  $T/\Delta$.

  The statistical theory provides a unified framework in which the mesoscopic fluctuations of both the conductance peaks and conductance minima can be studied.  In particular, the known results (of Aleiner and Glazman) for the conductance minima distributions are reproduced.



Microwave Absorption in Small Metallic Grains,
B.L. Altshuler, NEC Research Institute and Princeton University,

  There are several mechanisms of microwave dissipation  in nanocrystals. The main two are resonance transitions between the electron eigenstates and relaxation dissipation  that involves interaction between electrons and phonons. We demonstrate that the low frequency microwave absorption  of an ensemble of small metallic grains at low temperatures is dominated by a mesoscopic relaxation mechanism. We predict a giant positive magnetoresistance and very strong temperature dependence of the microwave conductivity.


Periodic and Aperiodic Bunching in the Addition Spectra of Quantum Dots,
R. Ashoori, MIT

We study electron addition spectra of quantum dots in a broad range of electron occupancies starting from the first electron. Spectra for dots containing <200 electrons reveal a surprising feature. Electron additions are not evenly spaced in gate voltage. Rather, they group into bunches. With increasing electron number, the bunching evolves from occurring randomly to periodically at about every fifth electron. The periodicity of the bunching and features in electron tunneling rates suggest that the bunching is associated with electron additions into spatially distinct regions within dots.


Localization in Fock space: A finite size scaling hypothesis,
Richard Berkovits, Bar-Ilan University

  The concept of localization in Fock space will be extended to the study of the many particle excitation statistics of interacting electrons in a two dimensional quantum dot. In addition, a finite size scaling hypothesis for Fock space localization, in which the excitation energy replaces the system size, will be developed and tested by analyzing the spectral properties of the quantum dot. This scaling hypothesis, modeled after the usual Anderson transition scaling, fits the numerical data obtained for the interacting states in the dot. It therefore attests to the relevance of the Fock space localization scenario to the description of many particle excitation properties.


Self-Assembly of Nanometer Scale Electronics by Biotechnology,
Erez Braun, Technion

The realization that conventional microelectronics is approaching its miniaturization limits has motivated the search for an alternative approach based on self-assembled nanometer-scale electronics. Considerable efforts have been put in that direction but progress has been impeded by the fact that materials amenable to self-assembly (e.g. biological molecules) display poor electrical characteristics. Here we propose a two- step self-assembly approach that overcomes that difficulty. The inherent molecular recognition capabilities of DNA molecules are first utilized to construct a network that serves as a template for the subsequent assembly of electronic materials into a useful circuit. The concept is demonstrated by the fully self-assembled fabrication of a conductive metal wire. Hybridization of DNA with surface bound oligonucleotides (short DNA sequences) is first employed for stretching it between two gold electrodes. The DNA then serves as a template for vectorial chemical growth of a 12 mm long, 100 nm wide, granular silver wire connecting the two electrodes. The wire constitutes a quasi one dimensional chain of about 200 grains of a 30-50 nm diameter each. The electrical properties of the wire are studied by direct transport measurements. The I-V curves of different wires depend on the silver growth conditions and can be tuned from ohmic to highly non linear ones. For the non linear case, the shapes of the I-V curves depend on the voltage history. Approaching zero voltage from large positive or negative bias, the current vanishes almost linearly with voltage. A zero-current plateau then develops with differential resistance larger than $10^{13}\Omega $. At higher biases, the wire turns conductive again with a differential resistance somewhat lower than in the original bias polarity. The length of the zero bias plateau in different wires can be made as large as 15 volts. The physics behind the history-dependent asymmetry (hysteresis) is not clear yet. A typical grain size inferred from AFM images implies grain charging energies smaller than 0.1eV. Invoking the Coulomb blockade phenomenon, the large zero current plateau would then require simultaneous charging of a large number of grains in series. It is not clear, however, whether such a mechanism can yield a history-dependent I-V curve. Another source of non-linearity might be inter-grain boundary resistance. Since silver tends to oxidize under ambient atmosphere, that resistance could be very large. Under large enough bias, oxygen and silver migration in the oxide layer results in the formation of a nanometer scale p-n junction whose polarity depends on the applied bias.


Energy redistribution between electrons in a mesoscopic wire,
D.Esteve, Quantronics group, SPEC, CEA-Saclay

  We have measured with a tunnel probe the energy distribution function of quasiparticles in metallic diffusive wires connected to two large pads (``reservoirs''), between which a bias voltage was applied. The distribution function in the middle of a $1.5~\mu {\rm m}$-long wire resembles the half sum of the Fermi distributions of the reservoirs. The distribution functions in $5~\mu {\rm m}$-long wires are more rounded, due to interactions between quasiparticles during the longer diffusion time across the wire. From the scaling of the data with bias voltage, we find that the scattering rate between two quasiparticles varies as $epsilon ^{-2} $


Mesoscopic Charge Quantization
L.I. Glazman, University of Minnesota,

The Coulomb blockade in a chaotic quantum dot connected to a lead by a single, partially transmitting channel is studied. We take into account both the non-perturbative charging effects, and a finite level spacing for the electron states within the dot. Mesoscopic fluctuations of the Coulomb blockade exist at any transmission coefficient.  Because of these fluctuations, the Coulomb blockade is not destroyed completely even at perfect transmission. The oscillatory dependence of all the observable characteristics on the gate voltage is preserved, its period is still defined by the charge of a single electron. However, phases of those oscillations are random; because of the randomness, the Coulomb blockade shows up not in the averages but in the correlation functions of the fluctuating observables ({\em e.g.}, capacitance or tunneling conductance). The found amplitude of fluctuations and the magnetic field dependence of their correlation functions differ from the known results obtained by the self-consistent treatment of interactions.


Strong Tunneling and Coulomb Blockade,
Hermann Grabert , Albert-Ludwigs-U. Freiburg

We investigate charging effects on small metallic islands and their suppression by coupling to a lead electrode via a tunnel junction with large conductance. Perturbative results for weak tunneling and nonperturbative path-integral techniques for strong tunneling are bridged by Monte Carlo data. The combination of all three methods gives a rather clear picture of the behavior in the entire range of parameters. Contact with recent experimental results on strong tunneling in the single electron transistor is made.


Transport Spectroscopy in Quantum Dots and Artificial Molecules,
Rolf J. Haug, Institut f\"ur Festk\"orperphysik, Universit\"at Hannover

  Single-electron tunneling through quantum dots allows to do spectroscopy of the different electronic states involved in transport through the system. One can study the many-particle states in single quantum dtos - artificial atoms - in different regimes: the ground and excited states of dots with small numbers of electrons, but also the level statistics of dots with larger numbers of electrons. In addition, by using only the state with one electron in the dot, one can perform spectroscopy of the electronic states in the leads. Due to the localized nature of this state the local density of states in the leads is studied. Here, interactions are of importance. Similar experiments can be performed in using coupled quantum dots instead of single quantum dots. The coupling of the electronic states in the dots leads to the formation of molecular states in these quantum dot molecules.


Phase Measurements and Dephasing in a QD,
M. Heiblum, the Weizmann Institute

Using a 'double path' electron interferometer, with a QD embedded in one of its paths, coherency of the quantum dot (QD) had been demonstrated.  The QD was found to maintain it coherency even for relatively long dwell times.  A similar configuartion was used to measure the phase an electron accumulates as it passes through the QD.  Under resonance conditions the electron's phase evolved according to the Breit Wigner expression, but between consecutive resonances the phase lapsed abruptly, on a scale smaller than the temperature or the resonance width. This unexpected behaviour has no measurable signature in the conductance of the QD. Adding a 'which path' detector near the 'two paths' interferometer (adjacent to the QD) led to partial depahsing of the QD in a well controlled manner.


Kondo Physics with an Artificial Atom
M. A. Kastner, MIT,

 How localized electrons interact with delocalized electrons is a question central to many of the forefront problems in solid state physics. The simplest example is the Kondo problem, in which an impurity with an unpaired electron is placed in a good metal, and the energy of the localized electron is below the Fermi energy.  At low temperatures a spin singlet state is formed between the localized electron and delocalized electrons at the Fermi energy. The confined droplet of electrons in a single electron transistor (SET) interacting with the metallic leads is closely analogous to an impurity atom interacting with the delocalized electrons in a metal. Because the charge and energy of the droplet are quantized, it is often called an artificial atom. In this talk I review the results of my coworkers, D. Goldhaber-Gordon, Hadas Shtrikman, D. Mahalu, D. Abusch- Magder, and U. Meirav.  They have fabricated and measured a new generation of SET's that reveal all the aspects of the Kondo problem: The spin singlet causes an enhancement of the zero-bias conductance when the number of electrons on the artificial atom is odd but not when it is even.  The singlet is altered by the application of electric or magnetic fields or by increasing the temperature, all in ways that are in agreement with theoretical predictions. The observation of Kondo physics is made possible by the smaller size of the new SETs that allow us to increase the coupling energy between the artificial atom and the leads.  The devices have been fabricated at the Braun Center for Submicron Research, Department of Condensed Matter Physics, Weizmann Institute of Science.


Excitation spectra of circular few-electron quantum dots,
L.P. Kouwenhoven, Delft University

  We study ground states and excited states in semiconductor quantum dots containing 1 to 12 electrons. For the first time, it is possible to identify the quantum numbers of the states in the excitation spectra and make a direct comparison to exact calculations. A magnetic field induces transitions between excited states and ground state. These transitions are discussed in terms of crossings between single-particle states, singlet-triplet transitions, spin polarization, and Hund^Rs rule. Our impurity-free quantum dots allow for atomic physics experiments in magnetic field regimes not accessible for atoms. An extreme example that will be discussed is an artificial atom in the quantum Hall regime. Work done together with D.G. Austing, T. Honda, S. Tarucha, T.H. Oosterkamp, M.W.S. Danoesastro, and M. Eto.


Electronic spectra and nonlinear transport properties of quantum dots
B. Kramer, Institut fur Theoretische Physik, Universiat Hamburg,

The electronic spectra and wave functions of few-electron 1D and 2D quantum dots are calculated and used to obtain the non-linear dc-current-voltage characteristics via a rate equation [1]. The spectra show distinct fine structures which can be related to vibrational properties of the electrons and to tunneling in the configuration space, and the spin properties. It is shown how the features of the spectrum influence the electron transport in the region of sequential tunneling when the dot is weakly coupled to ideal quantum leads. The transport at low bias voltage shows Coulomb blockade. At higher bias voltages, spin selection rules can lead to negative differential conductances [2]. Spatial properties of the wave functions can remove or enhance several of the structures in the current voltage characteristics [3]. Possibilities for experimental verification of the predicted effects are discussed.  

[1] D. Weinmann, W. H\"ausler, B. Kramer, Ann. Physik  5 652 (1996)

[2] D. Weinmann, W. H\"ausler, B. Kramer, Phys. Rev. Lett.  74, 984 (1995)

[3] K. Jauregui, W. H\"ausler, D. Weinmann, B. Kramer, Phys. Rev. B 53, R1713 (1996)


Correlations Between Ground and Excited State Spectra in  Quantum Dots,
Charles M. Marcus, Stanford University

  This talk describes recent measurements of ground and excited state spectra of a GaAs quantum dot for successive numbers of electron occupancy.  Using linear and nonlinear magnetoconductance as a `fingerprint' of transport through a particular wave function,  we observe a direct correlation between the $m^{th}$ excited state of the $N$ electron system and the ground state of the $N +m$ electron system for $m$ up to 4.  Results are consistent with a non-spin-degenerate single particle picture of the filling of levels, with some distortions presumably due to electron-electron interaction. Magnetoconductance fingerprints of ground states reveal anticrossings where wave function characteristics  are exchanged between adjacent levels.


Coulomb blockade at almost perfect transmission,
K.A. Matveev, Duke University,

 I will discuss the theory of Coulomb blockade in a quantum dot coupled to the leads by one or two quantum point contacts with transmission coefficient close to unity. The regime of a relatively large dot is considered, in which the level spacing is small compared to the temperature $T$.  The Coulomb blockade oscillations of the charge and conductance of the dot as functions of the gate voltage persist as long as transmission coefficients of all the contacts are less than unity. As the temperature is decreased, the oscillations of charge remain small, but show non-analytic behavior near the points where the charge is half-integer.  This behavior is identical to the non-Fermi liquid properties of the 2-channel spin-$\frac12$ Kondo model. The Coulomb blockade oscillations of conductance $G$ through a dot strongly coupled to two leads are also small. However, they grow at $T \to 0$, and eventually the peak conductance saturates at $G\sim e^2/h$, whereas in the valleys $G \to 0$.


Controlled decoherence in tunneling through a quantum dot,
Yigal Meir, Ben-Gurion University

The ``Which Path?'' interferometer consists of an Aharonov-Bohm ring with a quantum dot (QD) built in one of its arms, and an additional quantum point contact (QPC) located close to the QD. The transmission coefficient of the QPC depends on the charge state of the QD. Hence the point contact %causes controllable  dephasing  of transport through the QD, and acts as a controllable measurement device for which path an electron takes through the ring. We calculate the suppression of the Aharonov-Bohm oscillations which is caused both by measurement dephasing and by the orthogonality catastrophe, {\it i.e.}, respectively, by real and virtual electron-hole pair creation at the QPC.

Thos work was carried out in collaboration with I. Aleiner and N. S. Wingreen


Effects of Electron Interactions on the Tunneling Spectra of Aluminum Nanoparticles,
Dan Ralph, Cornell University

Tunneling measurements of the discrete electronic energy levels in metal nanoparticles provide a new means to study the forces which act on the electrons inside a metal. We will discuss 3 different effects, each of which produces different characteristic changes in the tunneling spectra.

Superconducting pairing: In particles that are sufficiently large, superconducting pairing forces produce characteristic gaps in the level spectra, different for even and odd numbers of electrons. The magnitude of the measured gaps depends little on particle size, in the regime where the mean level spacing is smaller than the bulk superconducting gap. We will comment on the effects of pairing forces in even smaller particles, with larger level spacings. The primary effect of an applied magnetic field is to disrupt superconductivity by means of spin pair-breaking. However the existing of a discrete electronic spectrum can "soften" the superconducting transition, making it continuous instead of the first-order transition observed in thin films.

Non-equilibrium excitations and electron interactions: Non-equilibrium electron-hole-pair excitations within the nanoparticle are a natural consequence of current flow, if tunneling measurements are conducted with a source-drain voltage larger than the level spacing. In the presence of electron-electron interactions, the existence of such excitations can modify the tunneling spectrum, broadening each single-electron level into a cluster of separate resonances. The width of the clusters provides a measure of the variation in the strength of electron-electron interaction between the chaotic electron eigenstates within the particle. This effect interferes with efforts to measure the interaction-induced lifetime broadening of single-electron energy states.

Spin-orbit scattering: The presence of spin-orbit scattering reduces electronic g-factors below the value of 2, and causes avoided crossings when spin-up and spin-down levels approach each other in an applied magnetic field. The changes in the magnitude of current carried by discrete levels in the avoided crossing region provide information about the relative phases of terms in the Hamiltonian.


Single-electron tunneling in metal junctions and quantum dots,
Gerd Schon, Institut fur Theoretische Festkorperphysik, Universitat Karlsruhe

Electron tunneling through low-capacitance tunnel junctions and through quantum dots will be described. In these systems the charging energy is large, which in metal junction systems leads to the well-known single-electron and Coulomb-blockade effects. In the quantum dot, where discrete quantum levels become observable the Coulomb energy suppresses double occupancy of the relevant level.

If the dimensionless conductance $h/(e^2 R_t)$ of the metal junctions is not small, the usual perturbation theory fails. In order to describe this regime we have developed a diagrammatic real-time description, which accounts in a systematic way for higher order processes such as co-tunneling and inelastic resonant tunneling [1]. We find a reduction of the Coulomb oscillations as well as life-time broadening effects [2]. Our results compare well with recent experiments of the Saclay group.

In the quantum dot we concentrate on situations where one or few levels, with or without spin degeneracy, dominate the transport properties. Again a real-time diagrammatic description can be developed to describe the non-equilibrium transport properties. A Kondo-peak in the spectral function leads to zero-bias anomalies in the conductance [3]. We show how these features are modified if a magnetic field lifts the spin-degeneracy. Our results compare well with the experiments of Ralph et al.   Extensions to multi-dot systems [4] or dots with several levels, as well as the effects of the environment, modeled by an oscillator bath are discussed as well.

[1] H. Schoeller and G. Sch\"on, Phys. Rev. B {\bf 50}, 18436 (1994); J. K\"onig, H. Schoeller and G. Sch\"on, Europhys. Lett. {\bf 31}, 31(1995).

[2] J. K\"onig et al., Phys. Rev. Lett. {\bf 78}, 4482 (1997).

[3] J. K\"onig et al., Phys. Rev. Lett. {\bf 76}, 1715 (1996); Phys. Rev. B {\bf 54}, 16820 (1996).

[4] T. Pohjola et al. preprint; N. Andrei, G.T. Zimanyi and G. Sch\"on, preprint.


Charging spectrum and configurations of a two-dimensional Wigner crystal island
Boris Shklovskii, University of Minnesota.

This talk is devoted to the hundredth anniversary of the discovery of electron. The charging spectrum of a clean two-dimensional island is studied in the regime of small concentrations of electrons or in a strong magnetic field when electrons behave as classical particles and tend to form Wigner crystal. In other words, we deal with Thomson two-dimensional atom of classical electrons in a parabolic confinement potential. We found that the energy of the ground state of the atom as a function of the number of electrons experiences strong quasiperiodic oscillations as a function of  number of electrons because of the  filling of shells in real space. The appearance of shells is related to the interesting geometrical problem formulated by Gauss and recently discussed by Dyson et al. In the case of screening by the close metallic gate  shell effect leads to the negative differential capacitance which brings about simultaneous entering the dot by several electrons. Analogy with the shell effect in nuclear physics will be discussed.


What is the difference between a quantum dot and a big atom ?
Ady Stern, Weizmann Institute of Science

Atoms have been the subject of spectroscopy studies for more than 100 years. For typical atoms, hundreds of spectral lines can be resolved, and line broadening is mostly due to the emission of photons. Quantum dots have been the subject of spectroscopy studies in the last decade. Typically, only a few spectral lines are observable, and the main source of broadening is electron-electron interaction. We attempt to understand the source of this difference.

This work was carried out by Kinneret Lindenstrauss, Uri Sivan and Ady Stern.


Coulomb Interactions in Quantum Dot Molecules
Seigo Tarucha, NTT Basic Research Laboratories

  We use vertically coupled disk-shaped dots to study the filling of electrons in quantum dot molecules. When the dots are quantum mechanically strongly coupled, the electronic states in the coupled system are not localized, and the addition energy spectra and the magnetic field dependece of the molecule resemble those of a single quatum dot: Atom-like properties such as a shell structure, Hund's rule and spin single-triplet transition are all observed in the strongly coupled molecule. When the strength of quantum coupling is comparable to or weaker than Coulomb interactions, both of the symmetric and antisymmetric states formed by the quantum coupling come to be filled. We observe significant modification in the way of electron filling for the molecules having various strengths of quantum coupling, and discuss competition of vertical quantum coupling, lateral quantum confinement and Coulomb interactions responsible for the filling of electrons. We also discuss the role of Coulomb interactions in the transport properties under a high magnetic field, which show transitions of ground states or formation of maximum density droplet.