Bielectron vortices in gated graphene C. A. Downing1 and M. E. Portnoi1, 2, ∗ 2
1 School of Physics, University of Exeter, Stocker Road, Exeter EX4 4QL, United Kingdom International Institute of Physics, Universidade Federal do Rio Grande do Norte, 59012-970 Natal, RN, Brazil (Dated: June 16, 2015)
We study the formation of bound two-particle states in gapless monolayer graphene in gated structures. We find that, even in the regime of massless Dirac fermions, coupling can occur at zero-energy for different or same charge quasiparticles. These bipartite states must have a non-zero internal angular momentum, meaning that they only exist as stationary vortices. We propose a new picture of the experimentally seen Fermi velocity renormalization as a manifestation of these pairs, and suggest the possibility of a condensate of these novel quasiparticles.
arXiv:1506.04425v1 [cond-mat.mes-hall] 14 Jun 2015
PACS numbers:
The introduction of novel quasiparticles in condensed matter physics is one of the most powerful tools to simplify the quantum mechanical many-body problem. The historic examples of phonons, excitons and Cooper pairs paved the way to the modern day collective excitations such as wrinklons [1] and dropletons [2]. Furthermore, the recently introduced plasmarons [4] and levitons [3] are amongst the quasiparticles that have been discussed in relation to the two-dimensional (2D) Dirac material, graphene [5, 6], which itself has the rather exotic singleparticle Hamiltonian H1 = vF σ · p, where vF is the Fermi velocity. In this work, we predict theoretically gated monolayer graphene as a host material for another type of quasiparticle, a stationary bielectron vortex. Our treatment can also be easily reworked to describe chargecarrier pairing in any 2D Dirac-Weyl material [7] with a short-range interaction between massless Dirac fermions. Pair formation in graphene was considered in connection to an excitonic insulator, discussed well before the isolation of graphene [8] and revisited soon thereafter [9, 10]. Upon assuming the existence of a gap, it is possible to estimate the exciton binding energy, which in turn helps one to self-consistently find the value of the gap. With the help of a gap, several groups have studied BoseEinstein condensation [11–13] and superfluidity [14, 15] in spatially-separated two-layer graphene. However, the excitonic size obtained from the above-mentioned selfconsistent calculations exceeds hundreds of graphene lattice constants, whereas a single-particle gap requires a potential difference between the neighboring graphene atoms. Unsurprisingly, no gap has as yet been observed experimentally in gated graphene structures [16] and a question remains: can two charge carriers bind together in an ideal 2D system with a linear dispersion? It is been claimed excitons do not exist in gapless graphene [17], and so considerations of trigonal warping, which effectively introduces an angular-dependent single-particle mass, have been suggested as a route towards pair formation [18–20]. It is a common knowledge that electrostatic confinement of 2D massless Dirac fermions is impossible as a
result of the Klein paradox [21, 22], whereby there is a perfect transmission for normally incident particles. An argument is usually made that conservation of pseudospin, the dot product of the Pauli spin matrix and the momentum unit vector, forbids bound states. However, at the Dirac point pseudospin vanishes, a fact we exploit in this work. Indeed, unlike the case of a finite energy, zero-energy bound states may form at the apex of the Dirac cone [23–25]. Mathematically, this is because at finite energy the effective Schrodinger equation at long range maps on to the problem of scattering states in a non-relativistic system (with solutions decaying like the square root of distance) [26]; whereas at zero energy solutions exist which decay algebraically depending on the angular momentum quantum number m. When m is nonzero the solutions are fully square-integrable, such that they are rotating ring-like states, which always avoid the Klein tunneling due to their vorticity resulting in a nonzero momentum component along the potential barrier. It should be emphasized that the existence of these fully-confined states does not require introducing any effective mass for the quasiparticles via either imposing the widely discussed sublattice asymmetry resulting in the finite band gap or considering trigonal warping terms which can be treated as angularly-dependent parabolic additions to the linear dispersion. The only requirement for the existence of zero-energy vortices in a strong enough (beyond a critical strength) radially-symmetric external potential is a long distance asymptotic decay of this potential to be faster than Coulomb. In practice the latter condition always takes place in realistic quasi-twodimensional Weyl systems since a metallic gate in a close proximity to the 2D electron gas is required to maintain the Fermi level at the Dirac point. At the Dirac point (zero energy) the sign of the external potential becomes unimportant for the confinement. Fully confined zero-energy vortices should be clearly distinguished from the widely discussed ‘atomic collapse’ peculiarities in the graphene density of states in a supercritical attractive Coulomb potential, since the potential decaying as 1/r cannot support square-integrable
2 states. Notably, the experimentally observed maximum in graphene’s density of states in the presence of supercritical impurities [27], which is attributed to the wavefunction collapse, can be better explained using the zero-energy vortices picture in conjunction with optimal screening. Indeed, for the ‘collapsed’ states the observed peaks in the density of states are too close to the Dirac points, and the spatial extent of the measured induced charge density around the impurities is of the order of tens of graphene lattice constant, which is easier to explain in terms of the large-size vortices rather than the short-scale wavefunction collapse at the impurity center. Furthermore, the recently discovered ‘whisperinggallery modes’ in scanning tunneling microscopy of gated graphene [28] can be also explained in terms of zeroenergy vortices. In this Letter, we generalize the principles behind the aforementioned single particle picture of confinement to the two-body problem. We show that electrostatic binding of particles into vortices is possible, energetically favorable and can arguably provide an alternative explanation of several experimental findings for gated graphene. A consideration of two particles with interaction potential, in the framework of a four-by-four Dirac-Weyl Hamiltonian, shows that at zero-energy the sign of the potential is irrelevant in analyses of confinement, as the interaction potential only appears as a logarithmic derivative or squared. Thus, forming bielectron vortices is as much a possibility as binding into excitons. The binding of the repelling particles is a consequence of the symmetric gapless band structure of graphene, such that the negative kinetic energy can fully compensate electrostatic repulsion. The considered bound pairs have to be static as two particles only bind if they have a zero total wavevector K; thus we deal with ‘pinned’ vortex pairs. This is because a nonzero K ensures that angular momentum m is no longer a good quantum number and necessitates one to seek a solution as a linear combination of relative motion wavefunctions with all possible values of m. However, this expansion would include the nonsquare-integrable component corresponding to m = 0, which acts to deconfine the whole state. It is important to consider gated structures, which modifies the situation from a pure Coulomb problem [29–31] for which no square-integrable solutions exist. The necessity for metallic gates inevitably leads to image charges resulting in short-range interactions [32, 33] and an Ohno-type cutoff as is known in studies of polymer systems [34]. Of course, in this setup the dielectric environment is still of great importance [35], as is the geometry of the device which both contribute to the effective strength of the interaction. As we demonstrate below, the seemingly rigid conditions on the strength and extent of the inter-particle potential, required to maintain the total energy at zero, are in fact easily satisfied for large-size vortices by linear screening provided by a
small number of residual free carriers. Previous theoretical works on excitonic effects in Dirac materials have approached the problem in either a BetheSalpeter formalism [36] or in the language of a twobody matrix Hamiltonian [37–39], which we will utilize here. The two-body Hamiltonian can be written as the Kronecker sum of the single-particle Hamiltonians H = H1 ⊕ H2 , or explicitly (as there are two sublattices and two particles) as the 4 × 4 matrix px2 H = vF px1
0 p x2 + ipy2 + ipy1 0 px1
− ipy2 px1 0 0 + ipy1 px2
0 − ipy1 , − ipy2 0
− ipy1 0 px1 0 px2 + ipy2
(1) where the subscripts 1 and 2 refer to the two particles. The matrix Hamiltonian given by Eq. (1) is written for two electrons belonging to the same Dirac valley. It can be easily modified for the particles of different charge (electron and hole) and for two particles belonging to different valleys. Here and in what follows we also neglect spin, which is in principle important as it governs the parity of the relative motion function for the same-valley electrons. However, our immediate aim is to demonstrate the existence of bound states leaving classification of all possible pairs for the future work. The Hamiltonian acts upon a two-particle wavefunction constructed via the Kronecker product Ψ(r1 , r2 ) = ψi (r1 ) ⊗ ψj (r2 ), where i, j = (A, B). In the absence of an interaction potential IU (r1 − r2 ), diagonalization of Eq. (1) yields four eigenenergies: E = ±vF p2x1 + p2y1
1/2
± vF p2x2 + p2y2
1/2
. (2)
As is usual with two-body problems, utilizing center of mass and relative motion coordinates: X = (x1 + x2 )/2, Y = (y1 + y2 )/2, x = x1 − x2 , y = y1 − y2 ; and assuming a translationally-invariant system such that the centerof-mass momentum ¯hK is a constant of motion, one can employ the ansatz Ψi (R, r) = exp(iK · R)ψi (r), where the index i = (1, 2, 3, 4) numerates the four components of the wavefunction, spanning the two sublattices and two particles. Now the eigenenergies can be rewritten as E/¯hvF = ± (KX /2 + kx )2 + (KY /2 + ky )2
1/2
± (KX /2 − kx )2 + (KY /2 − ky )2
1/2
, (3)
where kx,y and KX,Y are the wavevectors along the relative coordinates x, y and center-of mass coordinates X, Y respectively. As shown in Ref. [37], when K = 0 one can rewrite the relative motion coordinates (x, y) into polar coordinates (r, θ) reducing Eq. (1) into a system of three
3 where we have used to estimate the value of U0 the standard Ohno cut-off on-site energy of 11.3 eV [34]. Weak ∂r + m 0 screening by a small number of mobile uncoupled carriφ (r) h ¯ vF r 1 U (r)−E m+1 = 0, ers allows the system to adjust the inter-particle interacφ (r) 2 ∂ + 2 −∂r + m−1 2 r r h ¯ vF r U (r)−E tion potential so that it satisfies the strength condition φ3 (r) 0 −∂r + m r h ¯ vF given by Eqs. (9) to support bound states, resulting in (4) an energetically-favorable drastic reduction in the chemwith m = 0, ±1, ±2, ... and where one can take φ4 = 0. ical potential of the many-electron system. Indeed, pair Let us now consider a model potential given by U (r) = formation due to doping is a well-known mechanism [42]. 2 U0 /(1 + (r/d) ), with Ohno-like on-site energy U0 , and the long-range cut-off d is given by distance from the graphene layer to the metallic gate. This model poten1.5 1.5 10 tial, which as we will see can be solved exactly, provides 8 a reasonable approximation [25] to the more realistic po10 3 tential decaying at large distances as 1/r . Notably this functional form is well known in optics as the spatially inhomogeneous Maxwell’s fish-eye lens [41], and remark0 0 ably is the simplest exactly solvable model, as the square -1.5 -1.5 well does not admit a nontrivial solution [39]. The sys1.5 -1.5 1.5 -1.5 tem of Eqs. (4) can be reduced to a second order differential equation for φ2 , which admits an analytical soluFIG. 1: (Color online) Radial probability densities for the tion for the chosen interaction potential. This solution pair states with quantum numbers (n, m) = (0, 429), (1, 428) is square-integrable only if E = 0. The same is true for left-to-right, where the coordinates are measured in units of any potential decaying faster than the Coulomb potend. tial, which does not support square-integrable solutions altogether, so from now on we consider zero-energy states The other wavefunction components φ1,3 (r) are readily only. When r ∼ 0, one finds the usual short-range behavobtainable from Eqs. (4), and their long range behavior ior φ2 ∼ r|m| , whilst the asymptotic behavior as r → ∞ r → ∞ tells us that the m = 0 √ state is non-squareT is given by the decay φ2 ∼ r|m|−2η , where T − 1+|m|2 1, r−1 , 1 . integrable, (φ1 , φ2 , φ3 ) → r √ Thus, the pair states are rotating-ring like modes. |m| + 1 + m2 + 1 . (5) η= The requirements for bielectron vortex formation fol2 low from Eq. (9). The long-range cut-off d can be esThus, we seek a solution in the form timated from the separation between the graphene and |m| the gate where the image charges reside. The typical A (r/d) φ2 (r) = × f (r), (6) graphene-gate separation for experimentally attainable 2 η d (1 + (r/d) ) graphene samples is around 100 nm; however for high quality graphene on a boron nitride substrate [43], where where A is a normalization constant and f (r) is a polythe role of the gate is played by another highly doped nomial in r that does not affect the short- and long-range graphene layer, it could as low as just 10 nm and can behavior. Upon substituting Eq. (6) into Eqs. (4), elimibe even smaller for a graphene layer on top of metallic nating φ1,4 (r), and changing the variable with ξ = (r/d)2 graphite [48]. For the most common case of d = 100 nm, we arrive at the following equation for f (ξ) we obtain U0 d/¯hvF ≈ 1718; thus, with a tiny adjustment of the potential strength by free-particle screenξ(1 + ξ)2 f 00 (ξ) + (1 + ξ) [m + 1 + (m + 2 − 2η)ξ] f 0 (ξ) h i ing, many different pair states can exist corresponding to + ( 14 h¯Uv0Fd )2 − η 2 f (ξ) = 0, (7) (n, m) = (0, 429), (1, 428), ..., (427, 2), (428, 1). Probability density plots are displayed in Fig. 1 for the two lowest which is a form of the Gauss hypergeometric equation node states with n = 0, 1. [40]. Its solution regular at ξ = 0 is The average particle separation as a function of m is shown in Fig. 2, illustrating that the minimum size of ξ (8) f (ξ) = 2 F1 −n, −n + 12 h¯Uv0Fd ; |m| + 1; 1+ξ , the state is d, which is found for high m states, and the maximum size is 2d. For low-m states the size of the pair where we have terminated the power series to ensure deis 1.6d. Thus, for d = 100 nm, zero-energy bielectron caying solutions. This leads to the following condition states can exist for carrier densities up to several units of for bound bielectron pairs 1010 cm−2 , whereas at higher densities they are destroyed by the Mott transition caused by overlap of the pairs. U0 d = 17.2 × d[nm] = 4(n + η), n = 0, 1, 2... (9) To avoid pair overlap we require nd2 < 1, leading to the hvF ¯ equations only, U (r)−E
8
10
8
6
6
4
4
2
2
0
0
640246
640246
4
2
d
2
1.5
1
1
0
0
200
200
400
m
400
FIG. 2: A plot of the average size of the pair state as a function of quantum number m.
above Mott density estimate. Much higher densities of vortices can be sustained for lower values of d. As we already mentioned, the presence of a metallic gate suggests an interaction potential with a dipole-like (1/r3 ) decay due to image charges, this potential does not support an analytic solution. The problem can instead by solved numerically by expanding the eigenfunctions in a Fourier-Bessel series, defined by √ ∞ an 2X Jm (αn Lr ), (10) φ2 (r) = L n=1 Jm+1 (αn ) where αn are roots of the Bessel function and the radial functions with different m satisfy orthonormality over a length L, which is taken to be large enough such that the confined state wavefunctions are insensitive to the boundary condition φ2 (L) = 0. Evaluating the consequent matrix elements and solving the secular equation numerically, one finds that the eigenvalues for the more realistically decaying potential U (r) = U0 /(1 + (r/d)3 ) are h¯Uv0Fd = 7.47, 10.86, 12.70 and so on. Another approach would be to do develop the variable phase method [44] for a two-particle Dirac problem. One can speculate that the introduced bielectron vortices have already been observed in the range of experiments on gated graphene. Indeed, a reservoir of silent, stationary, zero-energy bielectron vortices offers an alternative explanation of the experimentally-seen Fermi velocity renormalization in gated graphene structures [45– 48] which is observed instead of the widely theorized gap. According to our picture, the observed Fermi velocity renormalization is an artifact of overestimating the number of charge carriers defining the position of the Fermi level; whereas, a large number of them disappear into a mostly silent many-body ground state of bosonic vortices. The best-known experiment [45] on Fermi velocity renormalization is based on measuring the cyclotron mass, given by mc = h ¯ (πn)1/2 /vF∗ . However, if a large amount of the carriers condense into a reservoir of zeroenergy bosonic vortices, the corrected lower density n∗ of remaining free fermions should be substituted into the formula for the cyclotron mass, then the smaller observed
cyclotron mass can be explained without renormalizing vF to a higher value. The same is true for the quantum capacitance measurements [47], since the presence of the charged boson reservoir changes drastically the Fermi energy dependence on the total carrier density from the expected relation, which is used to estimate renormalized vF∗ . Notably, the setup used in Ref. [47] is highly suitable for observation of zero-energy bielectron vortices since the graphene-gate separation in this experiment is only 20 − 30 nm, allowing for a wider range of carrier densities when the vortices exist. Notably, both the historic [49] (before the isolation of graphene) work on Fermi velocity renormalization in free-standing graphene and its later corrections [50] is essentially based on the particular long range behavior of the unscreened Coulomb potential resulting in logarithmically divergent corrections at small n. Therefore we believe that the applicability of these theoretical results should be taken with caution for gated structures. Another manifestation of the proposed bipartite states could be a significant increase in the apparent carrier concentration in low-carrier-density graphene in a quantizing magnetic field compared to the low-field measurements for the same structures. This effect has been observed experimentally by several groups [51–55] and could be another indicator of our proposed pair states, which form a dark-matter-like reservoir in the absence of a magnetic field. This carrier density increase in the quantum Hall regime was observed both for strongly disordered structures [51–55] grown on SiC and high quality graphene on graphite [48]. Whereas the existing explanation [51] based on high density of defects is perfectly suited for some structures, the vortex-based explanation works for all gated structures. In the bipartite-vortex-based model, when the magnetic length becomes smaller than the size of a vortex, the pair breaks up and the behavior of charge carriers is totally defined by a strong magnetic field. The best system to observe our novel quasiparticles is a structure containing a narrow layer of boron nitride between a low carrier density graphene layer and a high density layer, which can be considered as a metallic gate, the system for which the metal-dielectric transition was observed [43]. It would be interesting to probe the condensate directly by microwave/terahertz excitations of the gated, low-density graphene. One should expect to see the silent reservoir of zero-energy vortices (a peak in the density of states at E = 0) in low carrier density gated graphene by increasing interband absorption otherwise suppressed by the Pauli blocking at an energy of excitation equal to the Fermi energy EF , instead of 2EF anticipated for the case of high carrier density graphene where all zero-energy vortices are screened out [56]. A double-layer graphene system, with the high carrier density layer acting as a gate governing the range of the inter-particle interaction in the low-density layer [43] is, in our view, again the most promising candidate for such
5 experiments. A practical application of the introduced zero-energy quasiparticles could be found in nonlinear optics, where a reservoir of charged vortices at the Dirac point will contribute noticeably to optical nonlinearities at low frequencies, which have recently been declared [57] of little significance in graphene due to the commonly assumed vanishing density of carriers when the Fermi level is near the Dirac point where the nonlinearities are strongest. Less practical but perhaps more interestingly, the reservoir of bielectron vortices can play a similar role to the superconductor in proximity to a Weyl semimetal by enforcing electron-hole symmetry. Indeed, adding an electron to the condensate of vortices is equivalent to adding another zero-energy vortex and a hole, which makes the system a promising candidate in the on-going search of Majorana modes in solids. The analogue of a quantum wire needed for manipulating Majorana fermions is offered by zero-mode graphene waveguides [44, 58]. In conclusion, we have demonstrated that a gapless Dirac-Weyl 2D system with a short-range inter-particle interaction favors the existence of zero-energy charged bound pairs. In our view, the best candidate for observing these hitherto overlooked quasiparticles is gated monolayer graphene. These bielectron states are immobile vortices, defined by a nonzero angular momentum, and are bosonic in nature, opening up the possibility for the gapless condensate which is arguably a new state of matter. The seemingly rigid requirements on the strength and spatial extent of the interaction potential to support such two-particle vortices should be naturally favored to reduce the total energy of the system, with the potential strength easily adjusted via weak linear screening by a small number of residual free carriers. We speculate that the reservoir of zero-energy charged vortices provide an alternative explanation to a number of experiments in gated graphene. This work was supported by the UK EPSRC (CAD), the EU FP7 ITN NOTEDEV (Grant No. FP7-607521), and FP7 IRSES projects CANTOR (Grant No. FP7612285), QOCaN (Grant No. FP7-316432), and InterNoM (Grant No. FP7-612624). We thank R. J. Nicholas, L. A. Ponomarenko and B. I. Shklovskii for fruitful discussions.
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