Better than Slater-determinants: center-of-mass free basis sets for few-electron quantum dots

Error analysis of eigenenergies of the standard configuration interaction (CI) method (right black lines). The left colored lines are obtained by explicitly handling all spurious states.
Error analysis of eigenenergies of the standard configuration interaction (CI) method (right black lines). The left colored lines are obtained by explicitly handling all spurious states. The arrows point out the increasing error of the CI approach with increasing center-of-mass admixing.

Solving the interacting many-body Schrödinger equation is a hard problem. Even restricting the spatial domain to a two-dimensions plane does not lead to analytic solutions, the trouble-makers are the mutual particle-particle interactions. In the following we consider electrons in a quasi two-dimensional electron gas (2DEG), which are further confined either by a magnetic field or a harmonic oscillator external confinement potential. For two electrons, this problem is solvable for specific values of the Coulomb interaction due to a hidden symmetry in the Hamiltonian, see the review by A. Turbiner and our application to the two interacting electrons in a magnetic field.

For three and more electrons (to my knowledge) no analytical solutions are known. One standard computational approach is the configuration interaction (CI) method to diagonalize the Hamiltonian in a variational trial space of Slater-determinantal states. Each Slater determinant consists of products of single-particle orbitals. Due to computer resource constraints,  only a certain number of Slater determinants can be included in the basis set. One possibility is to include only trial states up to certain excitation level of the non-interacting problem.

The usage of Slater-determinants as CI basis-set introduce severe distortions in the eigenenergy spectrum due to the intrusion of spurious states, as we will discuss next. Spurious states have been extensively analyzed in the few-body problems arising in nuclear physics but have rarely been mentioned in solid-state physics, where they do arise in quantum-dot systems. The basic defect of the Slater-determinantal CI method is that it brings along center-of-mass excitations. During the diagonalization, the center-of-mass excitations occur along with the Coulomb-interaction and lead to an inflated basis size and also with a loss of precision for the eigenenergies of the excited states. Increasing the basis set does not uniformly reduce the error across the spectrum, since the enlarged CI basis set brings along states of high center-of-mass excitations. The cut-off energy then restricts the remaining basis size for the relative part.

The cleaner and leaner way is to separate the center-of-mass excitations from the relative-coordinate excitations, since the Coulomb interaction only acts along the relative coordinates. In fact, the center-of-mass part can be split off and solved analytically in many cases. The construction of the relative-coordinate basis states requires group-theoretical methods and is carried out for four electrons here Interacting electrons in a magnetic field in a center-of-mass free basis (arxiv:1410.4768). For three electrons, the importance of a spurious state free basis set was emphasized by R Laughlin and is a design principles behind the Laughlin wave function.

Hot spot: the quantum Hall effect in graphene

Hall potential in a graphene device due to interactions and equipotential boundary conditions at the contacts.

An interesting and unfinished chapter of condensed mater theory concerns the quantum Hall effect. Especially the integer quantum Hall effect (IQHE) is actually not very well understood. The fancy cousin of the IQHE is the fractional quantum Hall effect (FQHE). The FQHE is easier to handle since there is agreement about the Hamiltonian which is to be solved (although the solutions are difficult to obtain): the quantum version of the very Hamiltonian used for the classical Hall effect, namely the one for interacting electrons in a magnetic field. The Hamiltonian is still lacking the specification of the boundary conditions, which can completely alter the results for open and current carrying systems (as in the classical Hall effect) compared to interacting electrons in a box.
Surprisingly no agreement about the Hamiltonian underlying the IQHE exists. It was once hoped that it is possible to completely neglect interactions and still to obtain a theoretical model describing the experiments. But if we throw out the interactions, we throw out the Hall effect itself. Thus we have to come up with the correct self-consistent solution of a mean field potential which incorporates the interactions and the Hall effect.

Is it possible to understand the integer quantum Hall effect without including interactions – and if yes, how does the effectively non-interacting Hamiltonian look like?

Starting from a microscopic theory we have constructed the self-consistent solution of the Hall potential in our previous post for the classical Hall effect. Two indispensable factors caused the emergence of the Hall potential:

  1. repulsive electronic interactions and
  2. equipotential boundary conditions at the contacts.

The Hall potential which emerges from our simulations has been directly imaged in GaAs Hall-devices under conditions of a quantized conductance by electro-optical methods and by scanning probe microscopy using a single electron transistor. Imaging requires relatively high currents in order to resolve the Hall potential clearly.

In graphene the dielectric constant is 12 times smaller than in GaAs and thus the Coulomb repulsion between electrons are stronger (which should help to generate the Hall potential). The observation of the FQHE in two-terminal devices has led the authors of the FQHE measurments to conjecture that hot-spots are also present in graphene devices [Du, Skachko, Duerr, Luican Andrei Nature 462, 192-195 (2009)].

These observations are extremely important, since the widely used theoretical model of edge-state transport of effectively non-interacting electrons is not readily compatible with these findings. In the edge-state model conductance quantization relies on the counter-propagation of two currents along the device borders, whereas the shown potential supports only a unidirectional current from source to drain diagonally across the device.

Moreover the construction of asymptotic scattering states is not possible, since no transverse lead-eigenbasis exists at the contacts. Electrons moving strictly along one side of the device from one contact to the other one would locally increase the electron density within the contact and violate the metallic boundary condition (see our recent paper on the Self-consistent calculation of electric potentials in Hall devices [Phys. Rev. B, 81, 205306 (2010)]).

Are there models which support a unidirectional current and at the same time support a quantized conductance in units of the conductivity quantum?

We put forward the injection model of the quantum Hall effect, where we take the Hall potential as being the self-consistent mean-field solution of the interacting and current carrying device. On this potential we construct the local density of states (LDOS) next to the injection hot spot and calculate the resulting current flow. In our model, the conductivity of the sample is completely determined by the injection processes at the source contact where the high electric field of the hot spots leads to a fast transport of electrons into the device. The LDOS is broadened due to the presence of the electric Hall field during the injection and not due to disorder. Our model is described in detail in our paper Theory of the quantum Hall effect in finite graphene devices [Phys. Rev. B, 81, 081410(R) (2010), free arxiv version] and the LDOS in a conventional semiconductor in electric and magnetic fields is given in a previous paper on electron propagation in crossed magnetic and electric fields. The tricky part is to prove the correct quantization, since the absence of any translational symmetries in the Hall potential obliterates the use of “Gedankenexperimente” relying on periodic boundary conditions or fancy loop topologies.

In order to propel the theoretical models forward, we need more experimental images of the Hall potential in a device, especially in the vicinity of the contacts. Experiments with graphene devices, where the Hall potential sits close to the surface, could help to establish the potential distribution and to settle the question which Hamiltonian is applicable for the quantum Hall effects. Is there anybody out to take up this challenge?

Interactions: from galaxies to the nanoscale

Microscopic model of a Hall bar
(a) Device model
(b) phenomenological potential
(c) GPU result

For a while we have explored the usage of General Purpose Graphics Processing Units (GPGPU) for electronic transport calculations in nanodevices, where we want to include all electron-electron and electron-donor interactions. The GPU allows us to drastically (250 fold !!!) boost the performance of N-body codes and we manage to propagate 10,000 particles over several million time-steps within days. While GPU methods are now rather popular within the astrophysics crowd, we haven’t seen many GPU applications for electronic transport in a nanodevice. Besides the change from astronomical units to atomic ones, gravitational forces are always attractive, whereas electrons are affected by electron-donor charges (attractive) and electron-electron repulsion. Furthermore we have a magnetic field present, leading to deflections. Last, the space where electrons can spread out is limited by the device borders. In total the force on the kth electron is given by \vec{F}_{k}=-\frac{e^2}{4\pi\epsilon_0 \epsilon}\sum_{\substack{l=1}}^{N_{\rm donor}}\frac{\vec{r}_l-\vec{r}_k}{|\vec{r}_l-\vec{r}_k|^3}+\frac{e^2}{4\pi\epsilon_0 \epsilon}\sum_{\substack{l=1\\l\ne k}}^{N_{\rm elec}}\frac{\vec{r}_l-\vec{r}_k}{|\vec{r}_l-\vec{r}_k|^3}+e \dot{\vec{r}}_k\times\vec{B}

Our recent paper in Physical Review B (also freely available on the arxiv) gives the first microscopic description of the classical Hall effect, where interactions are everything: without interactions no Hall field and no drift transport. The role and importance of the interactions is surprisingly sparsely mentioned in the literature, probably due to a lack of computational means to move beyond phenomenological models. A notable exception is the very first paper on the Hall effect by Edwin Hall, where he writes “the phenomena observed indicate that two currents, parallel and in the same direction, tend to repel each other”. Note that this repulsion works throughout the device and therefore electrons do not pile up at the upper edge, but rather a complete redistribution of the electronic density takes place, yielding the potential shown in the figure.

Another important part of our simulation of the classical Hall effect are the electron sources and sinks, the contacts at the left and right ends of the device. We have developed a feed-in and removal model of the contacts, which keeps the contact on the same (externally enforced) potential during the course of the simulation.

Mind-boggling is the fact that the very same “classical Hall potential” has also been observed in conjunction with a plateau of the integer quantum Hall effect (IQHE) [Knott et al 1995 Semicond. Sci. Technol. 10 117 (1995)]. Despite these observations, many theoretical models of the integer quantum Hall effect do not consider the interactions between the electrons. In our classical model, the Hall potential for non-interacting electrons differs dramatically from the solution shown above and transport proceeds then (and only then) along the lower and upper edges. However the edge current solution is not compatible with the contact potential model described above where an external reservoir enforces equipotentials within each contact.