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Interacting electrons in polygonal quantum dots

The low-lying eigenstates of a system of two electrons confined within a two-dimensional quantum dot with a hard polygonal boundary are obtained by means of exact diagonalization. The transition from a weakly correlated charge distribution for small dots to a strongly correlated"Wigner molecule"for large dots is studied, and the behaviour at the crossover is determined. In sufficiently large dots, a recently proposed mapping to an effective charge-spin model is investigated, and is found to produce the correct ordering of the energy levels and to give a good first approximation to the size of the level spacings. We conclude that this approach is a valuable method to obtain the low energy spectrum of few-electron quantum dots.

This feature of quantum dots containing low density electrons has been exploited recently 11 as a starting point complementary to perturbative or mean field approaches that are reliable at weak interaction strengths. As the electron density in these systems localizes around the Wigner lattice points 12 , it is sensible to construct a many-particle basis from antisymmetrized combinations of non-orthogonal one-particle states localized at these sites. Orthogonalization leads to a lattice model of strongly correlated electrons of Hubbard type, and the subsequent elimination of the high-energy states of this model was then shown to yield an effective "tJV " Hamiltonian. This effective Hamiltonian has a very much smaller Hilbert space than that of the original quantum dot enabling, in principle, the treatment of systems of many more electrons than can be handled by direct diagonalization methods. The ordering of the low energy levels as well as their relative separations agree quite satisfactorily up to a scaling factor with data obtained by exact diagonalization for a one-dimensional dot containing up to four electrons 11 . Here we examine the validity of the tJV -approach in two-dimensional hard wall boxes. In the case of two (or higher) dimensions, it should be noted that the number of energy minima in configuration space, i.e. the number of peaks N 0 in the charge-density distribution in the low density limit, can be larger than the number of electrons. For example, for two electrons in a square box the ground-state will consist of a superposition of states in which the electrons are located on diagonally opposite vertices of the square, and so will exhibit a four-peak structure. In this paper we consider three different box shapes: triangular, square and hexagonal. For the case of two electrons the eigenenergies and eigenfunctions are obtained by exact diagonalization, and then compared with the predictions of the tJV model.
We consider the Hamiltonian: where we assume that the electrons can be described within the parabolic band approximation by an effective mass m * . The shape of the dot is set by the confining potential V (r), which is zero within the dot and infinite outside.
The full wavefunction of the two-electron system may be written as a product of a spinor and a spatial function: where for a singlet (triplet) state ψ is symmetric (antisymmetric) under particle exchange. As the Hamiltonian (1) does not contain any spin dependent terms it is only necessary to consider its action on the spatial component of the wavefunction to obtain its eigenvalues. We choose to use a basis of position eigenstates (a finite-difference method) as opposed to the momentum eigenbasis used for example in 2,13 . An advantage of using this basis is that it is a simple matter to impose the required homogeneous Dirichlet boundary conditions for dots regardless of their shape, which can be very difficult to achieve in a momentum-space basis. An exception to this is the case of the square shape, which we used to check the reliability of the finite-difference approach by comparing with results obtained using a momentum-space basis. The agreement was found to be excellent.
Using the basis of position eigenstates amounts to replacing the spatial continuum (r 1 ; r 2 ) with a four-dimensional mesh. The wavefunction ψ is then only evaluated at a discrete set of points: and the spatial derivatives in Eq. (1) are replaced with symmetric difference approximations: The Hamiltonian can now be represented as a N 4 × N 4 matrix, where N is the number of mesh points per dimension.
Since the Coulomb interaction is diagonal in this representation, and the kinetic terms (4) only operate between neighbouring mesh points, this matrix will clearly be extremely sparse. This allows it to be stored in a very economical fashion, and also permits an highly efficient use of Lanczos diagonalization routines to obtain the lowest few eigenstates. As N is increased, and the separation between the mesh points decreases, the spectrum of the discrete system will approach that of the continuum model and the form of the mesh will be irrelevant. Finite-size effects can be minimized, however, by choosing a mesh which matches the symmetries of the potential as far as possible. For the case of a square dot we therefore chose a square mesh, and a triangular mesh for the triangular and hexagonal dots.
On an Alpha workstation as many as 24 4 lattice points could be used, and in all cases we checked the convergence of the eigenvalues as the number of lattice points was increased to this level.
In Fig.1 we present the ground state charge-distributions for the three types of polygonal boundary for a selection of dot sizes L, where L is the side-length of the polygon. In each case the dot material was taken to be GaAs, with an effective mass m * = 0.067m e , and a relative permittivity ǫ r = 10.9, resulting in a Bohr radius of a B = 8.8nm. For small dots the Coulomb interaction is weak on the scale of the kinetic energy, and the two-particle ground state resembles the non-interacting ground state, with the charge distribution being peaked at the centre of the dot. Conversely, in very large dots the charge distribution is strongly localized near the vertices of the dot where the interaction is minimized, and the charge distribution practically vanishes away from these maxima. In analogy to the formation of a Wigner crystal in an infinite electron system, this configuration is referred to as a Wigner molecule 13 . It is an important question to clarify the characteristic scale r c of the mean electron separation for which the crossover between these two extreme cases takes place, since the actual value of r c cannot be obtained reliably using analytical arguments. In 1D boxes r c ≈ 1.5a B has been found from tracing the charge density distribution of the exact ground state 13 . Using this technique we detect the change from the non-interacting situation by observing the point at which the charge distribution first shows a local minimum at the centre of the dot instead of a maximum. In this way we estimate from Fig.1 that r c ≈ 10a B for all three dot geometries, which is almost an order of magnitude larger than the values found in one dimension. This rough estimate is in reasonable agreement with the critical value of r c ≈ 35a B found by Ceperley for crystallization of the 2D electron gas 9 . A somewhat larger value for r c in two dimensions has been conjectured 14 due to the considerably enhanced tendency for the electrons to surround one another. Although r c marks the transition from the non-interacting regime, an additional qualitative change occurs at a somewhat longer length scale, at which the charge distribution develops N 0 sharply defined, well separated maxima. It is to be expected that as the number of vertices is increased, and the polygon becomes closer to a circle, that this transition will occur at increasingly low densities. Accordingly we can observe this change at ∼ 30a B for the triangle and square, but only at the larger value of ∼ 100a B for the hexagon. Before this transition occurs in the hexagonal dot the charge density shows a ring-like structure (as would occur in a circular dot), and in this regime the spectrum indeed resembles that of a diatomic rotor composed of spin 1/2 fermions, as can be seen in Fig.6b.
was proposed to describe the low energy physics. Here P is a projector eliminating doubly-occupied lattice sites, and t and J are the standard hopping and Heisenberg terms between nearest neighbour sites. The V term accounts for the nearest neighbor Coulomb repulsion and may be written V ≈ e 2 /4πǫ 0 ǫ r L for large L. Although first principles calculations of the energies t and J are difficult, one can easily estimate the ordering of their magnitudes: For the three geometries considered here H tJV can be diagonalized analytically to obtain the energy levels in terms of t, J and V , together with their corresponding spins. Let us first consider the case of the square dot, which is described by two electrons moving on a four-site lattice. Here the V term in (5) is important to discriminate energetically between two electrons sitting on diagonally opposite vertices as opposed to being on adjacent vertices. The lowest energy manifold of states was derived in Ref. 11 and, setting the ground-state energy to zero, the manifold consists of a singlet ground state, two degenerate triplet states at (2∆ + J), and a singlet at an energy of 4∆ where ∆ = 2t 2 /V , and ∆ ≫ |J|. The next set of levels are separated from this lowest energy multiplet by an energy gap of the order of V . The transition from the almost constant density of states found in small dots to a spectrum consisting of multiplets separated by large energy gaps of this sort can be regarded as a signature for the crossover to the Wigner regime 15 .
In Fig.2 we plot the lowest energy levels of the square dot as a function of the dot size, normalized to the energy of the highest singlet state. The overall structure of the spectrum agrees with that of the tJV Hamiltonian (5), and quantitatively reproduces the numerically exact spectra obtained by Bryant for large dots 2 . The doubly-degenerate triplet state lies between the two singlet states and, asymptotically, the spectrum becomes equidistant as L → ∞.
Remarkably, the triplet levels are below 1/2 for finite L corresponding to a ferromagnetic J < 0. This is somewhat unexpected, since pair exchanges of electrons would be expected to have antiferromagnetic couplings 16  The dominant energy scale ∆ can be roughly estimated using the pocket state picture 14 together with the WKB approximation: where the classical Euclidian action: is taken along the path q : (r 2 ) → (r 1 , r 2 ) that carries n electrons between two adjacent energy minimum configurations '(1)' and '(2)' (n = 2 for the example of the square and n = 1 for the triangle). Here, v( q) = e 2 /4πǫ 0 ǫ r |r 1 − r 2 |. For the square the path corresponds to a π/2 rotation of the two electrons around the centre of the square, yielding: The parameter r s is the typical scale of electron separation (which for two electrons is equal to the dot size L). In Fig.3b we show ∆ versus √ L on a semilogarithmic plot and verify that ∆ ∼ e − √ L/ξ . From the slope we extract a value for ξ = 1.64a B that is in excellent agreement with the pocket state prediction of ξ = 1.60a B from Eq.9.
The triangular dot is described by two electrons moving on a three-site lattice, and hence the V term is irrelevant as it will just give rise to an overall shift in energy levels. The resulting tJ model has a more complicated ground-state manifold than for the square dot, and is shown in Fig.4a. This is also exactly the ground-state manifold for the hexagonal dot, where the V term is again of importance. We can again employ the pocket state WKB theory to estimate the magnitude of the dominant energy scale t. For the triangle we have: for one electron hopping along the edge to the empty site. In Fig.5 we present a semilogarithmic plot of t versus √ L for the triangle which again confirms that the scaling of the dominant energy is of the form t ∼ e − √ L/ξ . The value of ξ = 4.45a B measured from this plot compares reasonably with ξ = 5.67a B as predicted by WKB theory from Eq.10.
In Fig.6 we present the energy levels obtained by the diagonalizations (again scaled by the energy of the highest singlet state) for the triangle and the hexagon. The sequence and the asymptotic ratios of the energy separations again agree with that predicted by the tJV model. In contrast to the earlier result, however, it is not possible to fit these results with a single J. For example, for both geometries the higher triplet approaches its asymptotic value 4/3 from above, implying J > 0, whereas the lower triplet approaches 1/3 from below, requiring a ferromagnetic J. This behaviour indicates that although the tJV Hamiltonian is able to adequately predict the gross features of the low energy spectrum, it does not, in fact, provide a complete description of the dynamics occurring in the triangular and hexagonal dots. As this Hamiltonian was obtained as a reduced version of a more general model of Hubbard type 11 this raises the possibility that during the reduction procedure some terms were dropped which may be of importance in these situations. To investigate this further we again consider the lattice model with three sites forming an equilateral triangle and occupied by two electrons. With one orthogonal state per site the Hamiltonian is a generalised Hubbard model: where t ij are the one electron matrix elements coming from the kinetic energy and the confining potential and U ijkl are Coulomb matrix elements. We now consider all contributions to the low-lying manifold for which there is no double occupation of sites. (These are much higher in energy and their effect may be accounted for by second-order perturbation theory where they give rise to an antiferromagnetic superexchange term, as described in Ref. 11 ). The largest intersite Coulomb term has matrix elements U ijij ≡ V and the corresponding (ferromagnetic) exchange term has matrix elements U ijji ≡ J F /2. These terms are already included in the tJV Hamiltonian. All other terms in (11) involve hopping of electrons and fall into the following three classes: This is a spin-dependent hopping term which always takes us to a high energy state since it involves double occupation (e.g. electron hops from site 1 to site 2 provided site 2 is occupied with an electron of opposite spin).
It thus contributes to superexchange by lowering the energy of singlets. This term, which renormalizes J, is also discussed Ref. 11 .
This is potentially an important term since it operates in the ground manifold. However, for two electrons in a triangle, we can set n i = 1 (ie site i is always occupied for states in which c † jσ c lσ does not give zero) and hence this term merely renormalizes the kinetic energy (t)-term.
3. (i = l : j = k = i) where the last step follows since all U 's are the same (K = U/2).
This last term is potentially important since it operates in the ground manifold and it cannot be accounted for simply as a renormalization of the other parameters (i.e. J and t). It is also spin dependent, behaving differently when operating on singlet states from triplet states. Consider a base state in which there is a spin up electron on site i, a spin down electron on site k and no electron on site j. This operator will then move an electron from site k to site j and then flip the spins of sites i and j. It is thus a combined hop and spin-flip, as shown in Fig.7a. On the other hand, when it operates on a state where both electrons have the same spin it only performs the hop. Furthermore, an estimate of the magnitude of K shows that it is comparable with J.
Retaining all terms gives the effective model: where K is a positive coupling and the V -dependence has been dropped since it gives the same energy contribution for all states. Introducing this additional term alters the eigenenergies as shown in Fig.4b, and this alteration is of exactly the correct form to account for the energy level separations obtained from the numerical diagonalizations. For both dot geometries, fitting the results with these parameters yields a positive value for K as expected, and a positive (antiferromagnetic) J of similar magnitude.
We can also show that a K-term is also important for the low-lying manifold of other polygonal quantum dots.
Consider, for the example, the regular hexagonal geometry for which the exact numerical low-lying spectrum was In fourth-order we get the usual superexchange (J) term but, in addition, we get a K-term which also corresponds to a simultaneous rotation of ±π/3 but now also involves a spin-flip, as shown in Fig.7b. The final effective Hamiltonian has the form: where the summation is over all six pairs of opposite sites. The effective Hamiltonian (14) describing the low-lying states of the hexagon is isomorphic to the effective Hamiltonian for the triangular dot (Eq.12) with ∆ playing the role of t. This may be seen explicitly by writing down expressions for the effective Hamiltonian matrices in their respective localized bases. Thus the low-lying eigenstates of the triangular and hexagonal dots are in one-to-one correspondence, in agreement with the numerical results.
Note that by a similar reasoning we can generate effective K-terms for any polygonal dot. With the exception of the square dot, such terms are necessary in order to give quantitative agreement with the exact low-lying spectrum. This is not the case for the square since, as shown in Fig.7c, a rotation of π/2 followed by a spin-flip is equivalent to a rotation of −π/2 with no spin-flip. The effect is, therefore, to merely renormalize the second-order ring processes.
In conclusion we have examined the behaviour of the lowest energy levels of two electrons confined to two-dimensional polygonal quantum dots. For sufficiently large dots the ground-state charge distribution shows a quasi-crystalline structure, which can be used as the basis for mapping the system to an effective tJV lattice model. This model is found to give the correct ordering of energy levels and to give a good first approximation to the energy level spacings. The dominant energy scale of the systems can be estimated semi-classically to good accuracy, and although it is hard to obtain estimates for the remaining parameters, they decrease exponentially with the size of the dot and so this description becomes increasingly reliable in large dots. For the case of triangular and hexagonal dots, however, it was evident that it was necessary to retain the three-site terms normally neglected in the derivation of the effective model to account for the detailed behaviour of the lowest energy levels. The lattice description of the dot considerably simplifies the calculation of the energy spectrum, and provides an appealing interpretation of the low-energy excitations occurring in these structures. It is still desirable to investigate different electron numbers and dot geometries to check the universality of our results concerning the critical density characterizing the cross-over into the Wigner regime.