Edge states in graphene quantum dots: Fractional quantum Hall effect analogies and differences at zero magnetic field
We investigate the way that the degenerate manifold of midgap edge states in quasicircular graphene quantum dots with zig-zag boundaries supports, under free-magnetic-field conditions, strongly correlated many-body behavior analogous to the fractional quantum Hall effect (FQHE), familiar from the case of semiconductor heterostructures in high magnetic fields. Systematic exact-diagonalization (EXD) numerical studies are presented for the first time for fully spin-polarized electrons and for total angular momenta in the range of . We present a derivation of a rotating-electron-molecule (REM) type wave function based on the methodology introduced earlier [C. Yannouleas and U. Landman, Phys. Rev. B 66, 115315 (2002)] in the context of the FQHE in two-dimensional semiconductor quantum dots. The EXD wave functions are compared with FQHE trial functions of the Laughlin and the derived REM types. It is found that a variational extension of the REM offers a better description for all fractional fillings compared with that of the Laughlin functions (including total energies and overlaps), a fact that reflects the strong azimuthal localization of the edge electrons. In contrast with the multiring arrangements of electrons in circular semiconductor quantum dots, the graphene REMs exhibit in all instances a single polygonal-ring molecular (crystalline) structure, with all the electrons localized on the edge. Disruptions in the zig-zag boundary condition along the circular edge act effectively as impurities that pin the electron molecule, yielding single-particle densities with broken rotational symmetry that portray directly the azimuthal localization of the edge electrons.
Since the discovery tsui82 () of the fractional quantum Hall effect (FQHE) in two-dimensional (2D) semiconductor heterostructures in the presence of a high perpendicular magnetic field (), phenomena associated with strongly correlated electrons in the lowest Landau level (LLL) have attracted significant and continuous attention. laug83 (); hald83 (); halp84 (); jain89 (); laug99 (); jainbook (); yann02 (); yann03 (); yann04 (); yann07 (); chan05 (); chan06 () Early on, it was realized that the essential many-body physics in the LLL could be most effectively grasped through the use of trial wave functions, with celebrated examples being the Jastrow-type Laughlin laug83 () (JL) and composite fermion jain89 () (CF) trial functions associated with the formation of a special class of quantum-liquid states. laug99 () Later interest in finite 2D electronic systems, like semiconductor quantum dots (QDs) under high , led to the consideration of a different class of analytic trial functions known as rotating electron (or Wigner) molecules yann02 (); yann03 (); yann04 (); yann07 () (REMs or RWMs). An advantage of the REMs is that, while they exhibit good total angular momenta, they directly incorporate the molecular (crystalline) configurations that dominate the anisotropic pair correlation functions revealed through numerical exact-diagonalization studies for a finite number of electrons under high in a disk geometry. The initial derivation yann02 () of the REM trial functions generated a flare of theoretical activity around the question which class of trial functions (or combination of them) is most appropriate for describing the correlated many-body physics in the LLL of a small number of electrons . jainbook (); yann03 (); yann04 (); yann07 (); chan05 (); chan06 () Furthermore, experimental advances in the field of ultracold trapped neutral atoms have been followed by considerable theoretical activity regarding the nature of correlated states in the LLL that are formed during the rapid rotation of the trap; see, e.g., Refs. gunn00, ; gunn01, ; popp04, ; joli05, ; barb06, ; baks07, .
Recent progress in the fabrication of new materials, and in particular in isolating and handling of a single graphene sheet, geim04 (); dehe04 (); geim05 (); stor05 () offers most promising materials for future, post-silicone, miniaturized electronicsgeim07 (); dehe08 () (sometime called nanoelectronics). This expectation is based on the two-dimensional character of graphene, where the electrons are essentially confined in the spatial direction normal to the graphene plane. Fabrication of nanoscale device elements for use in electronics, spintronics and information processing, such as single-electron transistors, quantum point contacts, and quantum dots, would require additional confinement in the other two spatial dimensions. However, to achieve the requested additional confinement, techniques (based on electrostatic gating) developed for the creation of QDs in semiconductors (such as GaAs) cannot be used because of the unique electronic structure of graphene. The difficulty originates from the relativistic, Dirac-like, nature of the low-energy quasiparticles in graphene. In particular, the gapless nature of the electrons in grapheneguin08 () allows them to penetrate unimpeded through a high and wide potential barrier. geim06 () This phenomenon, which is known as the Klein paradox, klei29 (); calo99 () is in fact not a paradox but a consequence of the relativistic character of the electrons, with a sufficiently high potential being repulsive for electrons but attractive for positrons, thus resulting in positron states inside the barrier which can be matched to the electronic continuum states outside, consequently resulting in perfect transmission through the barrier; the underlying property of the Dirac equation is known as the charge-conjugation symmetry.
In light of the above, one wishes to explore alternative, non-electrostatic methods, for fabrication of lower dimensionality graphene nanostructures. One route for achieving the desired added planar confinement consists of etching, or cutting graphene into the desired geometry (e.g., ribbons, avou07 (); han07 (); dai08 () circular disks, or other shapes ihn08 (); ihn09 (); ihn09.2 ()). It is expected that further progress in fabrication, characterization and understanding of the properties of such graphene nanostructures (in particular zero-dimensional QDs) would lead to their use for the study of interesting many-body phenomena, as well as their employment as components in miniaturized electronic devices.
Here we explore theoretically certain properties of circular graphene quantum dots, defined via cutting the desired shape from a two-dimensional extended sheet. In particular, we regard investigations of graphene QDs as providing an opportunity for reexamination (and possibly experimental resolution) of remaining questions concerning the appropriateness of liquid-type vs. molecular-type trial functions for a finite number of 2D electrons. Indeed, it has been known for some time that manifolds of degenerate midgap edge states exist in graphene nanostructures (such as graphene ribbons) when they terminate in a zig-zag boundary. fuji96 (); dres96 () In a recent paper wuns08 () it was noted that the single-particle edge states associated with circular graphene dots with zig-zag boundary conditions, and in the absence of an applied magnetic field, display degeneracies and quantum numbers in close analogy with the manifold of single-particle states that form the familiar LLL in semiconductor heterostructures at high . Furthermore, the numerical calculations of Ref. wuns08, , covering rather limited ranges of electron numbers (i.e., ) and total angular momenta (i.e., ), suggested that the use of quantum-liquid-type trial functions in relation to the graphene LLL (gd-LLL) may be less promising than that of Wigner-crystal-type ansatzes.
In this paper, applying a methodology based on angular-momentum projection techniques that was introduced in Ref. yann02, , we derive analytic REM trial functions appropriate for the gd-LLL. By introducing a single variational parameter, we demonstrate numerically (through systematic comparisons with EXD calculations) that the variational variant of the REM (referred to as vREM) substantially outperforms the Laughlin trial functions (as well as the ansatz of Ref. wuns08, ) for all values of fractional fillings within the expanded angular-momentum range , and for all of the following larger numbers of electrons and 8.
Ii Lowest Landau level for circular graphene dots
ii.1 Single-particle edge states
It is well known ando05 () that the low-energy bandstructure of graphene can be described by a linearized tight-binding Hamiltonian . For a graphene dot with a circular symmetry this linearized Hamiltonian is given wuns08 () in polar coordinates by
where is the Fermi velocity, and specifies the degenerate in energy valleys for the two bands formed at the and points. The index can be considered as a “pseudospin”, which creates a fourfold degeneracy when the spin degree of freedom is also considered. The general solution of the Hamiltonian in Eq. (2) is a two component vector of the form
where and denote the two triangular sublattices of graphene.
The usual volume solutions (which are zero on the graphene boundary but otherwise extend everywhere inside the area enclosed by the graphene dot) have energy , with and components that are expressed via the Bessel functions. Here we are not interested in such volume solutions. Instead we focus on the special edge states with zero energy . These states are eigenfunctions of under the assumption that the graphene boundary exhibits an uninterrupted zigzag edge; fuji96 (); dres96 (); wuns08 () an outline of their derivation from the Hamiltonian is given in Appendix A.
Henceforth we will only need to remember the precise form of the edge states, which is given by
Namely one of the and components is everywhere zero (both on the boundary and inside the dot) and the two valleys and are decoupled even when the two-body Coulomb interaction is considered (which is the main focus of this paper; see below). As a result, in the following, we will drop the sublattice and valley indices. We will also assume that the electrons are fully polarized.
Since the single-particle angular momentum (to guarantee normalizability), the manifold of such model edge states forms a set of degenerate states similar to the lowest Landau level (LLL), familiar from the case of 2D semiconductor devices at very high magnetic fields . We will call the manifold note2 () of degenerate edge states with the graphene-dot lowest Landau level (gd-LLL). The main difference [apart from the normalization constant, see Eq. (3) in Ref. yann02, ] between the two cases is that the single-particle states in the usual LLL exhibit an additional Gaussian multiplicative factor where is the magnetic length. This Gaussian is missing from the expression for the edge states in Eqs. (4) and (5); instead one has for .
It is thus natural to investigate possible similarities related to fractional quantum Hall effect (FQHE) physics.
ii.2 Classes of variational many-body wave functions
FQHE physics in the LLL has been extensively investigated for the case of 2D semiconductor quantum dots.jainbook (); yann07 () A main focus has been the underlying nature of the correlated many-body states, i.e., ’liquid’ (Laughlin, composite fermions) or molecular (’crystalline’, REM). Detailed comparisons of pair-correlations functions between JL/CF and REM states with EXD ones support the view that the molecular (localized electrons) picture in semiconductor QDs provides the most appropriate description. The emergence of a gd-LLL, as described above in graphene dots offers a further area for testing the appropriateness of liquid-type variational wave functions (JL/CF) versus those that describe REMs.
First we will proceed with deriving a modified REM trial wave function that takes into consideration the differences between the single-particle states which span the usual LLL (zero-node 2D-harmonic-oscillator states) and the gd-LLL (edge states).
Iii Derivation of variational REM trial wave functions for graphene dots
iii.1 Intermediary parameter-free REM functions
REM analytical wave functions in the LLL for electrons in two-dimensional semiconductor quantum dots were derived earlier in Ref. yann02, . The physics underlying such a derivation is based on the theory of symmetry breaking at the mean-field level and of subsequent symmetry restoration via projection techniques. yann07 (); note1 () In particular, this approach consists of two steps:
(I) At the first step, one constructs a Slater determinant out of displaced single-particle states , , that represent the electrons localized at the positions , with (omitting the particle indices) and . Note that necessarily all electrons are localized radially on the edge of the graphene dot, so that .
Naturally, for the LLL case of semiconductor QDs, the localized single-particle states (referred to also as orbitals) were taken to be displaced Gaussians with appropriate Peierls phases due to the presence of a perpendicular magnetic field [see Eq. (1) in Ref. yann02, ]. In the case of electrons in graphene dots, however, the gd-LLL is spanned by edge-like orbitals (without a Gaussian factor), i.e.,
and as a result the appropriate localized orbitals are taken to have an exponential form
The localized orbital can be expanded in a series over the basis functions in Eq. (6) in the following way
When constructing the many-body Slater determinant , one considers orbitals representing electrons on a ring of radius (the radius of the graphene dot) forming a regular polygon, i.e.,
The single Slater determinant represents a static electron (or Wigner) molecule (REM or RWM). Using Eq. (8), one finds the following expansion (within a proportionality constant):
where ; the elements of the determinant are the functions , with being the diagonal elements.
(II) Second step: The Slater determinant breaks the rotational symmetry and thus it is not an eigenstate of the total angular momentum . However, one can restore yann02 (); yann07 () the rotational symmetry by applying onto the projection operator
where are the eigenvalues of the total angular momentum.
When applied onto , the projection operator acts as a Kronecker delta: from the unrestricted sum in Eq. (11) it picks up only those terms having a given total angular momentum (henceforth we drop the constant prefactor when referring to angular momenta). As a result the projected wave function is written as (within a proportionality constant)
We further observe that it is advantageous to rewrite Eq. (13) by restricting the summation to the ordered arrangements , in which case we get
The second determinant in Eq. (14) can be shown mathematica () to be equal (within a proportionality constant) to the following product of sine terms times a phase factor (independent of the individual ’s):
Thus, the final result for the REM wave function is (within a proportionality constant):
iii.2 Introducing the variational parameter
As described below, we found that the agreement between the REM in graphene dots and the EXD solutions can be improved in a nontrivial way by introducing variational parameters. In particular, we found that consideration of a single variational parameter serves our purpose remarkably well. Specifically, one replaces the prefactor
in Eq. (16) by the following expression:
We call the -optimized wave functions the variational REM functions (denoted by vREM). When , the vREM coincides with the parameter-free REM expression. We note that a single-parameter variational crystal-type wave function, but with a different dependence on the parameter , has also been employed in Ref. wuns08, . The present choice of variational parameter [see Eq. (18)] produces substantially better results (see below). From a practical point of view, we note that the crystal-type wave function proposed in Ref. wuns08, does not contain a “less-than” ordered-arrangement restriction in the summation indices , and as a consequence it generates an exponentially larger number of expansion terms, thus greatly inhibiting numerical evaluations for larger and .
Iv Exact diagonalization and two-body Coulomb matrix elements
For a circular graphene QD comprising electrons in the gd-LLL, the many-body hamiltonian comprises only the two-particle interelectron Coulomb repulsion, i.e.,
where is the dielectric constant and denotes the relative distance between the and electrons.
The REM wave functions derived in the previous section will be compared to the EXD ones that are solutions of the exact diagonalization of the hamiltonian (19) in the many-body Hilbert space spanned by the Slater determinants
where the single particle functions are given by the edge states of Eq. (6) and the index counts the arrangements with .
Namely, is written as
and the exact diagonalization of the many-body Schrödinger equation
yields the coeffiecients and the EXD eigenenergies .
The matrix elements between the basis determinants [see Eq. (20)] are calculated using the Slater rules szabobook () and taking into account that, in the gd-LLL, the many-body hamiltonian has contributions from the Coulomb interaction only, i.e.,
Naturally, one also needs the two-body matrix elements of the Coulomb interaction in the basis formed out of the single-particle edge states. These matrix elements are given through appropriate analytic expressions. Indeed by defining
where is the generalized hypergeometric function weinbook () at the point , is the Gamma function, and
V Numerical Results
v.1 EXD total energies
In Fig. 2 we display systematic EXD total energies in the range of to edge electrons as a function of the total angular momenta (in the large range ). This large range and the consideration of electrons were not reached in another recent publication;wuns08 () they are, however, essential for unequivocally establishing the proper similarities and differences with the high-magnetic-field physics of semiconductor QDs.
For fully polarized spins considered here, the minimum total angular momentum is , in analogy with the case of semiconductor QDs.yann07 () Furthermore, in analogy again with the case of semiconductor QDs, the total energies decrease on the average as increases. On top of this average trend, one observes prominent oscillations of period . These oscillations reveal that the states with , , are energetically the most stable in their immediate neighborhood. Borrowing the terminology from the literature yann07 (); maks00 () of semiconductor QDs, we refer to these states in graphene QDs as cusp states, and the corresponding total angular momenta (i.e., ) as magic angular momenta. It is well known that cusp states develop to fractional-quantum-Hall-effect (FQHE) states in the thermodynamic limit (), with the corresponding fractional filling factor being .
Following a similar analysisyann07 (); maks00 () with the case of semiconductor QDs, one can conclude that the appearance of the oscillatory period in the total energies (associated with the cusp states) is a reflection of formation of -type Wigner molecules, with all the electrons localized on a single ring (of radius ) at the apices of a regular -polygon. There is a major difference, however, between the present system and the semiconductor quantum dot case. That is, in semiconductor QDs, more than one isomers may form with concentric multiring arrangements occurring for electrons in the dot; such arrangements are denoted as (see Ref. yann07, ), where , are the number of localized electrons on each ring; . In contrast, in the case of graphene dots only the one-ring molecular configuration arises (with no electron residing at the geometrical center of the graphene QD).
For , the EXD energies in Fig. 2 approach the limiting value corresponding to the classical electrostatic energy of point-like electrons in a (0,N) configuration with radius , i.e.,
v.2 EXD densities and pair correlations
The EXD eigenfunctions conserve the total angular momentum and the corresponding electron densities are circularly symmetric. This property “conceals” the presence of the Wigner molecule. The crystalline structure, however, is present in the intrinsic frame of reference of the electron molecule, and it can be revealed through the use of the fully anisotropic pair correlation function , defined as
is often referred to as the conditional probability distribution (CPD), since it is proportional to the probability of finding an electron at under the condition that another one is situated at the point (the socalled fixed, or observation, point).
In Fig. 3 we display the conditional probability distribution for the case of electrons and the magic total angular momentum (, with ), which corresponds to the celebrated fractional filling. One clearly observes six humps (arranged in a single-ring configuration) associated with formation of a (0,7) rotating Wigner molecule. (As is well known from the literature of semiconductor quantum dots, yann07 () the localized electron at the fixed point does not contribute any hump in the CPDs.) Similar CPDs are found for other values of .
v.3 Comparison between vREM and EXD wave functions
We turn now to comparisons between the EXD wave functions and the vREM ones. We first observe that the REM and vREM functions [see Sect. III] correspond to the magic angular momenta , , since all the sine-product coefficients in the expansion (16) are identically zero for . In this section, we will show that the vREM functions represent a high-quality approximation to the EXD eigenfunctions by investigating wave function overlaps and relative errors between the total energies obtained by the two methods; the relative errors are defined as .
We start by displaying in Fig. 4 the relative error of the vREM total energies. The vREM offers an excellent approximation, since the maximum relative error is less than 0.045%. For all sizes examined, the maximum relative error occurs about (see Section V.1), and subsequently it decreases as increases, approaching zero as .
In Fig. 5, we display the overlaps between the vREM functions and the EXD solutions. These overlaps are larger than 0.985 for and larger than 0.95 for and they tend to slowly approach unity as increases.
In Fig. 6, we display the values of the variational parameter that optimize the vREM trial functions for a given number of electrons as a function of . These values are significantly different from unity (which corresponds to the parameter-free REM). In fact the optimal values are smaller than 0.4, and they slowly decrease to about 0.16 for , for all the values . We stress that optimization of is essential for achieving a high quality reproduction of the EXD ground states. Without the additional optimization (i.e., taking only the value ) the behavior of the overlaps is unsatisfactory, since they tend to diminish as increases (see Appendix B). The degradation of the overlaps of the parameter-free REM () as increases in the case of the graphene quantum dot contrasts with the opposite behavior of the overlaps of the parameter-free REM in the case of semiconductor quantum dots. yann02 (); yann03 (); yann07 () This difference is attributed to the absence of translational invariance for the electrons in the graphene quantum dot, which leads to differences in the organization of the EXD excitation spectra.
v.4 EXD versus Laughlin wave functions
It is interesting to compare the accuracy with which the vREM wave functions approximate the EXD ones with that of the Laughlin trial functions. The Laughlin wave functions are restricted to the socalled main (odd) fractions and have played an important role in the FQHE literature of the extended two-dimensional electron gas in semiconductor heterostructures. Their form is
where the Gaussian factors are missing [see Sect. II.1] due to the differences in the single-particle states between semiconductor and graphene quantum dots; .
In Fig. (7), we display the relative error, , of the Laughlin total energies with respect to the ground-state EXD ones as a function of increasing total angular momentum . The Laughlin relative errors are substantially larger (on the average by a factor of 5) than the vREM ones [see Fig. 4]; this is the case even for the celebrated fractional filling.
In addition, the Laughlin overlaps (plotted in Fig. 8) exhibit an unsatisfactory performance compared to that of the vREM overlaps, that is: (i) they become steadily smaller as the angular momentum increases, and (ii) even for , they are smaller than the corresponding vREM overlaps in all instances studied here, i.e., electrons in the graphene dot.
We conclude that the Laughlin functions fail to capture the case of the gd-LLL, while the vREM functions offer an appropriate approximation for graphene QDs.
v.5 EXD versus composite-fermion wave functions
It is also interesting to compare the accuracy with which the vREM wave functions approximate the EXD ones with that of the composite-fermion trial functions, which are more general than the Laughlin functions. Along with the Laughlin functions, they CF trial functions have played a significant role in the FQHE literature of the extended two-dimensional electron gas in semiconductor heterostructures. Their form jain89 (); jainbook () in the disc geometry (case of 2D QDs studied here) are given by the expression,
where and is the Slater determinant of non-interacting electrons of total angular momentum ; it is constructed according to the Independent Particle Model (IPM) from the Darwin-Fockdf () orbitals , where and are the number of nodes and the angular momentum, respectively [for the values of and in the th Landau level in high , see Appendix F].
The single-particle electronic orbitals in the Slater determinant are not restricted to the lowest Landau level. As a result, it is necessary to apply a projection operator to guarantee that the CF wave function lies in the LLL, as appropriate for . We carry the projection according to section 4.3 of Ref. heinbook, . After obtaining the projected CF function in the LLL, the corresponding trial function in the gd-LLL is constructed by simply replacing by the in Eq. (6).
Since the CF wave function is an homogeneous polynomial in the electronic positions ’s, its angular momentum is related to the non-interacting total angular momentum as follows,
Here we will consider the mean-field version of the composite-fermion theory, according to which the Slater determinants are the socalled compact states (see Appendix F for details; the corresponding values of are listed in Table 2). We note that recently several extensions of the CF theory have been formulated jeon07 () that account for residual-interaction effects among the individual composite-fermion states. Consideration of such residual-interaction effects is beyond the scope of the present paper.
In Table 1, we compare total CF and EXD energies (per particle) for electrons in a graphene dot. We also display the corresponding relative errors and overlaps .
In addition to the , magic angular momenta for found from EXD calculations (see Fig. 2), the compact-state CF theory mistakenly predicts the existence of magic angular momenta with , , e.g., for . Furthermore, even for the states with , e.g., (Table 1), the quantitative performance of the compact CF functions (concerning relative errors and overlaps) is rather weak compared to that of the vREM: the CF relative errors are larger roughly by a factor of 10, while the CF overlaps are systematically smaller () than the vREM ones () (see Figs. 2 and 5, and Table 1).
As was the case with the Laughlin functions, we conclude that the compact CF functions are also at a disadvantage compared to the vREM concerning the description of strongly correlated states in the gd-LLL.
v.6 Comparison with the Wigner-crystal ansatz of Ref. wuns08,
Here we compare the vREM total energies with those associated with the Wigner-crystal trial function of Ref. wuns08, , given by
where is a variational parameter.
In Fig. 9, we display the relative errors of the WC-ansatz energies relative to the EXD ones. From a comparison of these results with those displayed in Fig. 4, we conclude that the relative errors of the WC ansatz are on the average at least twice as large as those corresponding to the vREM, reflecting the superior description of the gd-LLL provided by the latter function.
Vi Pinned electron molecules
The zig-zag geometry (on which the boundary conditions are applied; see Section II.1) does not allow formation of a continuous circular edge without some structural or chemical modification of the graphene hexagonal lattice structure. Without such modification, regions along the circular edge satisfying a zig-zag condition must necessarily be disrupted by a number of discrete points associated with arm-chaired conditions.been08 () It has been found that the edge states are robust in this case, dres96 (); been08 () and as a result these discrete set of disruptions act as effective impurities that modify the many-body hamiltonian in Eq. (19). The presence of such impurity terms in the many-body hamiltonian will mix the good-total-angular-momentum REM states, resulting in the formation of pinned electron molecules (PEMs). In contrast to the REMs (whose electron density is uniform along the azimuthal direction, that is, not showing any azimuthal density modulation), the electron density of a pinned electron molecule is expected not to have circular symmetry; it will exhibit angular density oscillations, and the number of humps will equal the number of electrons .
We demonstrate this property of a PEM for two particular cases displayed in Figs. 10 and 11. Fig. 10 displays for the electron density for the linear superposition of two REM states with and , while Fig. 11 displays for the electron density for the linear superposition of two REM states with and . In both cases the expected angular modulation is clearly well formed with seven humps in the former and eight humps in the latter case.
The manifold of degenerate midgap (zero-energy) edge states in circular graphene quantum dots with zig-zag boundaries resembles, under free-field conditions, note3 () the celebrated lowest Landau level, familiar from the case of semiconductor heterostructures in high magnetic fields. The effect of interactions in this graphene-LLL were systematically investigated and were found to generate many-body strongly correlated behavior that exhibits many similarities with the fractional quantum Hall effect.
Numerical exact-diagonalization studies were presented for fully spin-polarized electrons and for total angular momenta in the range of . Moreover, we presented a derivation of a rotating-electron-molecule type wave function based on the methodology introduced earlieryann02 () in the context of the FQHE in two-dimensional semiconductor quantum dots. The EXD wave functions were compared with the derived rotating-electron-molecule and other suggested FQHE trial functions, like the Laughlin function and the Wigner-crystal ansatz of Ref. wuns08, . It was found that a variational extension of the REM offers a better description for all fractional fillings compared with that of the Laughlin and Wigner-crystal ansatz functions (including total energies and overlaps). The success of the REM function reflects the importance of strong azimuthal localization of the edge electrons in graphene quantum dots.
The variational REM functions were derived through the use of a two-step method: (i) first a mean-field-type single Slater determinant constructed out of localized electron orbitals (that break circular symmetry) was considered; this determinant describes the finite analog of a classical static Wigner-crystal, and (ii) a multideterminantal wave function was generated through the subsequent application of projection techniques that introduced azimuthal fluctuations and restored the circular symmetry and good total angular momenta.
In contrast with the multiring arrangements of electrons in circular semiconductor quantum dots, we found that the graphene REMs exhibited in all instances a single polygonal-ring molecular structure. Disruptions in the zig-zag boundary condition along the circular edge behave effectively as crystal-field effects that pin the electron molecule, yielding single-particle densities with broken rotational symmetry that portray directly the azimuthal localization of the edge electrons.
Acknowledgements.This work was supported by the US D.O.E. (Grant No. FG05- 86ER45234).
Appendix A More on edge states
The general solution of the eigenvalue equation corresponding to the linearized tight-binding Hamiltonian (2) is of the form
where ; obviously when occurring in a phase. As a result, the matrix eigenvalue problem is equivalent to the following set of equations involving the vector components:
where we considered only the case for (the case can be treated in a similar way).
We are interested in solutions with (the socalled midgap solutions), in which case the set of equations (34) reduces to
The solutions of these equations are
The boundary condition is that of a zigzag graphene edge that ends always on a site of the same lattice, i.e., the condition
forces the component to vanish everywhere on the sublattice, yielding the final form
The normalization constant is easily calculated and was given in Eq. (4).
Appendix B The parameter-free REM ()
As mentioned in Section V.3, the overlaps between the parameter-free REM waves function in the gd-LLL and the EXD ones behave in an unsatisfactory way, i.e., they decrease as increases. The precise behavior of is displayed in Fig. 12, and it contrasts with that of the variational REM, , displayed in Fig. 6.
The degradation of the reflects the fact that progressively the overlap of the REM wave function with the excited EXD states increases with increasing . On the other hand, the optimized values of correspond to variational REM trial functions that have practically zero overlap with these excited EXD ones. We have found that such optimal values can be found for all studied values of and . This is illustrated in Fig. 13, where the overlaps of the vREM with the first excited EXD state dip toward zero at the optimal values.
Appendix C Single-particle density
We give here the expression for calculating the single-particle density for a single many body state , where the basis wave functions are the Slater determinants defined by Eq. (20).
Specifically, one has
where the edge states ’s are given by Eq. (6) and denotes the single-particle angular momenta associated with the Slater determinant ; naturally . The single-particle density operator connects in principle Slater determinants that differ at most in one orbital. However, in the LLL, the conservation of the total angular momentum implies that there is no pair of Slater determinants in the linear superposition of that differ precisely by one single orbital; thus one sets when deriving Eq. (39).
Appendix D Single particle density for a superposition of two wave functions
In Section VI we discussed how the disruptions in the zigzag boundary conditions create crystal-field effects that pin the rotating electron molecule. The effect of this pinning is described through the linear superposition of two many-body wave functions (EXD and/or REM) with magic good total angular momenta and ; namely, through a wave function such that
where we have dropped the subscript and superscripts ’EXD’ or ’REM’ from the ’s on the r.h.s.
The ’s are known through their expansions over Slater determinants [see Section IV], i.e,
and the Slater determinants and are built out of single particle states having individual angular momenta and such that and .
Using the operator defined by the first line of Eq. (39), the single-particle density is given by
The diagonal terms and are given by Eq. (39). Since is a one-body operator, the cross terms connect Slater determinants that differ precisely by one of orbital;szabobook () we denote by and the corresponding pair of indices. By applying the Slater rules described in Ref. szabobook, (including bringing the two determinants into “maximum coincidence”), one finds:
where depending on the even or odd number of exchanges of two rows (or columns) needed to bring the two determinants into maximum coincidence.
Appendix E Two-particle conditional probability distribution
For the conditional probability density [see Eq. (28)], one has
where the operator is symmetrized;
The matrix elements of between the two Slater determinants and are calculated according to the Slater rules for a two-body operator.szabobook ()
Appendix F More on composite fermions
There is no reason to a priori restrict the Slater determinants to a certain form. jain95 () Following Ref. jain95, , we will restrict the non-interacting to the range , and we will assume that the Slater determinants are the so-called compact ones. Let denote the number of electrons in the th Landau Level (LL) with ; is the index of the highest occupied LL and all the lower LL’s with are assumed to be occupied. The compact determinants are defined as those in which the electrons occupy contiguously the single-particle orbitals  of each th LL with the lowest angular momenta, . The compact Slater determinants are usually denoted as , and the corresponding total angular momenta are given by .
For the CF theory, the magic angular momenta can be determined by Eq. (31), if one knows the non-interacting ’s. For , the CF magic ’s in any interval , , can be found by adding units of angular momentum to each of the ’s. To obtain the non-interacting ’s, one needs first to constructjain95 () the compact Slater determinants. The compact determinants and the corresponding non-interacting ’s are listed in Table 2.
There are nine different values of ’s, and thus the CF theory for predicts that there are always nine magic numbers in any interval between two consecutive JL angular momenta and , , For example, using Table 2 and Eq. (31), the CF magic numbers for in the interval () are found to be the following nine:
In the interval (), the CF magic numbers are:
- (1) D.C. Tsui, H.L. Stormer, and A.C. Gossard, Phys. Rev. Lett. 48, 1559 (1982).
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