Letter pubs.acs.org/NanoLett
Electron Pairing in Designer Materials: A Novel Strategy for a Negative Effective Hubbard U Melanie R. Butler,* Bijan Movaghar, Tobin J. Marks, and Mark A. Ratner* Department of Chemistry and the Materials Research Center, Northwestern University, Evanston, Illinois 60208, United States S Supporting Information *
ABSTRACT: We propose a set of design rules with a model Hamiltonian that allows electrons to form attracting pairs through the exploitation of a new combination of resonant band alignment and Coulombic repulsion. The pair bands and single particle bands in various lattices are calculated and compared in energy, and regions of net attraction are identified. This work provides guidelines for the construction of molecular systems, nanocrystals, and nanoparticle arrays with the potential for superconductivity. KEYWORDS: Electron correlation, Hubbard model, nanoparticle arrays, supramolecular lattices, superconductivity
S
nanocrystal arrays are also promising avenues for controlling electronic structure and would be particularly appropriate for the models proposed here due to their energy tunability, high dielectric environment, and therefore small (repulsive) Hubbard U (1−2 eV). The band coupling between nanocrystals has, in the past, tended to be small due to inevitable surface passivation.25,26 However, recent progress made on this problem has produced semiconductor nanocrystal structures with enhanced band coupling. In particular, the recent work of Boneschanscher et al.27 demonstrates the fabrication of selfassembled, covalently bonded nanocrystals. This synthetic methodology shows a great deal of promise for realizing the design parameters suggested by this model. Here we demonstrate a new strategy for the design of materials with a negative effective Hubbard U. We will begin with a traditional, positive Hubbard U representing Coulombic repulsion and demonstrate how the energetic landscape and coupling parameters can be chosen to give rise to an effective U that is negative. To demonstrate this, we begin with the simplest correlation scenario, which is to explore the physics of only two interacting electrons in proposed model systems. We will show that a negative effective U can be derived analytically by comparing the energies of a single electron in the lattice to
ince they have a total spin of zero, bound singlet electron pairs are bosons that behave according to Bose−Einstein statistics. This can give rise to many interesting properties including superfluidity, enhanced diamagnetism, the Josephson effect, and superconductivity.1 The Hubbard and Mott− Hubbard models and variations thereof that include longer range Coulomb coupling2−4 have been used to explain and predict a variety of these states of matter.5−13 Electron pairs that experience an attractive interaction have been previously modeled by taking the on-site electron−electron interaction, called the Hubbard U, to be negative.14,15 Traditionally the origin of this effective attraction is electron−phonon coupling. We introduce here a new pairing mechanism that does not rely on interaction with the lattice, but on careful design of electronic energies and couplings. This model is motivated by the machinery of organic molecular design, synthesis, and self-assembly, as well as great advances in nanoparticle design and fabrication, thin film processing, sheet doping, etc. For an introduction to some of the interesting properties that can arise from such novel structures, the reader is referred to the excellent review by Seo et al.,16 and new work on doping of organic semiconductors such as graphene,17 pentacene,18 or rubrene.19 An important innovation is the discovery of nondestructive surface doping.17,20−22 This is particularly impressive with the technique of liquid electrolyte gates, which allow induced sheet charge densities of up to 1015/cm2.23,24 Nanoparticles and © XXXX American Chemical Society
Received: October 27, 2014 Revised: December 16, 2014
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Figure 1. (a) Illustration of two-dimensional trilayer with a yellow intercalating layer. The inset shows a single isolated triad with labeled sites A, B, and A′. (b) Two-dimensional cubic intercalate structure (Model II).
the paired electron state in the same lattice. We will call the spatially bound electron pair a “bielectron”. The bielectron energy is computed by considering basis states in which the spatial extent of the pair is restricted to a maximum size. This is done to simplify the mathematics and allow analytical solutions. This methodology could also be applied to holes for p-type materials, although we will only discuss the n-type case here. Given the range of synthetic tools at our disposal, and the desire to fabricate new materials that display behavior such as superconductivity at room temperature, we would like to pursue a model that features bound pair formation from purely electronic processes. Thus, we focus on electronic coupling only. Phonon coupling, important and with its own merits, will be incorporated in the future. We begin by studying a few simple model structures. Consider an infinite two-dimensional triad consisting of two different materials A (red monolayers) and B (yellow intercalate layer) as shown in Figure 1a. As an illustration of the model Hamiltonian and proof of principle, we first consider a single three-site triad in isolation (Figure 1a, inset). We will subsequently examine the band structure of the full twodimensional trilayer, which can be thought of as a 2D array of coupled triad sites. The Hamiltonian for a single triad can be written
∑
Htriad =
εiσ ci+σciσ +
i = A,B,A ′ ; σ
+
∑
∑
Figure 2. Illustration of the singlet pair basis states and the distinction between d1 and the novel coupling term d2.
and the subsequent logical development of the pairing argument are different from this work, and we shall not discuss this further here. There are eight possible two-electron basis states for a singlet pair in the triad, neglecting double occupation of the yellow site (Figure 2). We use these basis states to solve for the singlet two-electron ground state eigenfunction analytically, and we call its energy E2S. Similarly, for a single electron with spin σ in the triad there are three possible basis states that mix to give the single electron ground state eigenfunction and its energy, which we call E1σ. An effective U for the triad is then given by the difference Ueff = E2S − (E1σ + E1−σ) = E2S − 2E1σ. Note that, while the Hamiltonian parameters are independent and can take any value, we will be examining an area of the parameter space that is most relevant to our goal of designing a system with a negative effective Hubbard U. In particular, we are proposing to design systems in which states with nearest neighbor coupling d2 (e.g., |ϕ1⟩ and |ϕ2⟩ in Figure 2) are resonant. This requires that 2ε0 + U ≈ ε0 + ε1; in particular, since we take ε0 to be zero, we will discuss the results for ε1 = U here and throughout. Additionally, all plots are in reduced units of d1, so the surfaces demonstrate how the eigenvalues vary as the parameters change relative to d1. E2S and 2E1σ are plotted in Figure 3a as a function of d2 and U = ε1. These results correspond to the intuitive picture of the system as follows. For d2 = d1 = 1, there is no energetic benefit in putting two electrons in a single triad. The large Coulombic repulsion leads to a pair energy that is greater than that of twice the single electron energy. However, as the ratio of d2/d1 increases, the Coulombic penalty is gradually offset and compensated for by the stronger pair coupling. This coupling stabilizes the electron pair relative to single electrons at some critical d2/d1 ratio, depending on the magnitude of the red site U. Thus, for a sufficiently large d2, the triad has a negative Ueff. Justification for the realizability of such a large d2 comes from the fact that it couples resonant states (see below). The reader should note that this argument can be extended to Hamiltonians where electron−electron repulsion with neighboring sites is included. For example with nearest-neighbor
Uni ↑ni ↓
i = A,A ′
(d 2ni , −σ + d1(1 − ni , −σ ))
i = A,A ′ ; σ
× (cB+σciσ + ci+σcBσ )
(1)
where the first term represents the diagonal site energies, for which the red site (A,A′) energies are ε0, chosen to be zero, and the B site energy is ε1 > ε0. The second term is the Hubbard term, which suppresses double occupation on the red sites. Double occupation of the yellow site, while allowed, is not considered here due to its high energy. The third term is the nearest neighbor coupling from the red sites A or A′ to the yellow site B, which is the essential component of the model. For coupling to and from singly occupied sites the coupling is called d1, and coupling involving a doubly occupied site is denoted d2 (Figure 2a). The Hamiltonian thus depends on four independent parameters: ε1, U, d1, d2. Note that, if we take d1 = d2, then the hopping term is independent of occupation and in the single-component system is the usual Hubbard “t” term. The dependence of the hopping term on the site occupation in the above model alloy introduces a new level of complexity that can lead to interesting behavior, and which we now explore. Occupation-enhanced coupling has been studied by several authors, in particular we refer the reader to the work of Marsiglio and Hirsch.10 However, in the work of Marsiglio and Hirsch, the reason for the enhanced correlated hopping term B
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Figure 3. (a) Single triad energy eigenvalues E2S (green) and 2E1σ (blue), and their dependence on d2 and U = ε1 in units of d1. (b) Comparison of bielectron (E2S(trilayer), green) and single electron (2E1σ(trilayer), blue) energies for two-dimensional trilayer in units of d1 for the lowest energy band with |k| = 0 and variable t, d2.
different in energy. As a simple illustration of why d2 can be much larger than d1, we refer to the double square well potential shown in Figure 4. The higher energy state of the red
repulsion U12, the resonant design argument is shifted to require U ≈ ε1 + U12 (for ε0 = 0). Before discussing the consequences of this model for various lattices let us discuss some realistic ranges for these parameters. One-body band couplings such as d1 for noncovalently bonded organics are generally in the range of 0.01−0.5 eV.28,29 The red and yellow site energies, ε0 and ε1, represent the various LUMO (HOMO) levels for n(p)-type organic molecules and generally can be tuned over a large range by altering the molecular size, extent of conjugation, and composition through the use of electron rich or poor units. Actual Hubbard U values, on the other hand, are not simple to estimate mainly due to the Hubbard U’s strong dependence on the molecular environment (gas phase, dielectric, bulk, etc.). For organics, U has been claimed to be as low as 1.0−1.5 eV in potassium-doped fullerenes.30 These values are for bulk materials, however, and would be predicted to be much higher in the gas phase. This is essentially because the doubly charged state will strongly polarize its environment and thereby lower its energy in the bulk. Computational parametrizations31−33 of the Hubbard model for organic materials estimate the magnitude of intermolecular screening on the reduction of U to be on the order of 1−3 eV.34,35 This substantial reduction highlights the importance of estimating the effective Hubbard U rather than the bare U for the isolated molecule. The magnitude of the correlated banding energy d2 is central to the entire logic of the proposed model. We have already shown that a negative Ueff is possible for the single triad if the ratio d2/d1 is large enough. We now examine how plausible this is for real systems. The magnitude of d2 (a charge transfer integral) is strongly molecule−molecule dependent, and its evaluation is a challenge to be addressed by computational science. In most DFT calculations the charge-transfer coupling is determined from the LUMO−LUMO splitting in the neutral system, so the charge polarization effect in the neighbor is neglected.36 In the present model, there can be two extra charges on the red site, which would lead to a large polarization effect in the neighbor. Recently it has been shown that including the charge effects can lead to a considerable improvement in charge transfer integrals at large molecule− molecule distances.37 While polarization effects may be important in some systems, this is not the only factor that determines the final values of d1 and d2. There are simpler considerations, such as the higher energy of the doubly occupied state and the near resonance of states that are coupled by d2 (e.g., |ϕ1⟩ and |ϕ2⟩ in Figure 2). Nearly resonant states generally couple more strongly than states that are very
Figure 4. Illustration of d2 and d1 for a double square well potential.
box has a longer decay length than the lower energy eigenstate of the red box. Thus, it will couple more strongly with the yellow box eigenstate of the same energy. If we use a very simple estimate that only takes the difference in tunneling amplitude into account, d2/d1 ratios in the range of 3−7 can already be expected (see Supporting Information). With the above considerations (polarization and higher energy resonance), it is reasonable to expect that systems can be found, or intentionally designed, in which d2 is as much as an order of magnitude greater than d1, and that potentially d2/d1 could be large enough to overcome the pair Coulombic repulsion. The above discussion summarizes the purely electronic aspects of the model, which can be applied to various dimensions and geometries. Now that the general principle of a negative Ueff via occupation-enhanced coupling has been established, we will return to the trilayer illustrated in Figure 1(a), and examine the band structure of the infinite twodimensional array. Here the red sites couple to each other in the x,y plane with coupling ‘t’, described in the Hamiltonian as ∑,σt(c+jσciσ + hc) where denotes nearest-neighbor redred coupling. The yellow sites only couple to the red sites, as described above for the single triad. Coupling between the yellow sites is not included here for simplicity, but it could easily be added and will not change the qualitative results. We consider the interesting consequences of allowing this coupling later. The model is easily solved for a single particle using Bloch’s theorem (see Supporting Information) and is plotted below (Figure 5). For the two-electron case, we consider what happens when a strongly bound pair is moving in the lattice. Let us define a bielectron to be an electron pair with limited spatial extension (to be described below). The pair is bound (stable) if its energy is less than twice the single electron eigenstate energy for the same lattice. If the bielectron energy is larger than twice the single electron energy, it is unstable, and C
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Figure 5. Comparison of bielectron (E2S(trilayer), green) and single electron (2E1σ(trilayer), blue) band structures in units of d1 for t = 2d1, U = ε1 = 10d1 and (a) d2 = 5d1 and (b) d2 = 17d1.
Figure 6. Bielectron (E2S(cubic), green) and single electron (2E1σ(cubic), blue) band structures for U = ε1 =10d1 and (a) d2 = 3d1 and (b) d2 = 4d1.
the pair will dissociate. We find the lowest eigenvalue for the bielectron by diagonalizing the Hamiltonian using two-electron basis states that meet the following spatial restrictions. The paired electrons are constrained to be either in the same vertical triad or sitting on adjacent red sites in the x,y plane. We again do not allow both electrons to occupy a yellow site simultaneously. An actual bound bielectron pair, if stable, would be expected to extend farther than this in space; however, the strong requirement we impose here serves as an upper bound on the bielectron energy. In analogy to the single triad model we will call the bielectron energy E2S(trilayer), and it will again be compared to twice the single electron Bloch energy in the same lattice (2E1σ(trilayer)). Please see the Supporting Information for further discussion on the methodology and effective U derivation. The lowest bielectron band is plotted in Figure 5 in comparison with the single electron case. Again, the bielectron is found to be more stable than the single electron band for d2/ d1 greater than 17. Notably, the in-plane coupling t discourages bielectron formation (Figure 3b). Thus, a narrow band in the red layers is beneficial for bielectron stabilization. However, red site coupling would be less detrimental to pair stability if we considered less restrictive bielectron basis states, such as allowing the pair to extend over more distant red sites. The above is proof-of-principle demonstrating that pairing is possible. If we treat the two-dimensional trilayer as an array of triad sites (Supporting Information) the Ueff is just E2S(triad) − 2E1σ(triad). The minimum d2/d1 ratio for U = ε1 to give Ueff ≤ 0 is given (for a single triad) by d2 = d1
limits of strong and weak attraction in the framework of a negative U model. We will now consider another type of lattice. Model II (Figure 1b) consists of a two-dimensional regular cubic compound lattice of red and yellow sites, and we call the lowest bielectron and single electron eigenvalues E2S(cubic) and E1σ(cubic), respectively. The definitions of Model I apply here as well, and the correlated banding term d2 now acts from any red site in the doubly occupied state to any neighboring yellow site. Let us start with the strongly coupled bielectron limit. Here the electrons are constrained to be within length a (red−red distance in Figure 1b) of each other. This is the strongest possible restriction; in reality the pair can be more extended, and this lowers the energy of the pair even further by reducing Coulomb repulsion and improving the pair banding. Again, no bends or kinks are allowed in the structure. For the bound pair to be stable, we need E2S(cubic) < 2E1σ(cubic), which is possible in a given range of parameters to be determined by d1, d2, U, ε1. The band structure for the lowest bielectron state is plotted in Figure 6 versus the energy of two single electrons in units of d1. As expected, E2S(cubic) − 2E1σ(cubic) = Ueff ≤ 0 for all momentum values for suitably large d2/d1; in this case, d2/d1 greater than about 4. The cubic intercalate geometry is notably more favorable for bielectron formation than that of the trilayer from Model I. The imposition of an energetic barrier (yellow site) in every direction for the single electron maximizes the utility of the bielectron’s resonant d2 coupling, leading to a negative effective Hubbard U (Supporting Information). It is evident from inspecting the curvature of the band structures for the above models that, although the bielectron energy is less than that of the single electron pair, its effective mass is larger. In terms of potential application this is not a particularly desirable result, since a higher effective mass implies lower particle mobility, making the particle more sensitive to disorder and Anderson localization. The higher mass is to some extent a consequence of the assumptions made as to the length of the bielectron, and the disregard for the doubly occupied yellow sites. In order to demonstrate this we have also solved
24 + 3(ε1/d1)2 − (ε1/d1) 8 + (ε1/d1)2 2 2
(2)
In reality the effective interaction might be just weakly attractive, and the pair will resemble a Cooper pair rather than a strongly bound bielectron bosonic particle (as considered here for illustration). Gyorrfy et al.14 have shown how to treat both D
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Figure 7. Exact one-dimensional intercalate bielectron (E2S, green) and single electron (2E1σ, blue) band structures for U = ε1 = 10d1 and (a) d2 = 3d1 and (b) d2 = 5d1.
In conclusion, we have demonstrated using a simple model that pairing can be attained in systems in which doubly occupied states can couple effectively with higher-energy neighboring bridge sites. Allowing coupling between bridge sites enriches the physics and allows the model to be related to the Anderson and Kondo-like models. As a purely electronic process, the pair attraction may be stronger and less prone to disruption by temperature than pairs formed by lattice polaronic stabilization. Note that the proposed design criteria implemented here in no way contradict or exclude the role of phononic interactions or lattice relaxation. The resonant intercalate design may, in some cases, not suffice to take U into the negative domain, but it can lower U sufficiently for phonon-induced bipolaron relaxation to be significant. If realized, such materials could be applied to the design of high-TC superconductors. Further refinements such as intercalate coupling, next nearest-neighbor host coupling, and diamagnetic response will be investigated in the future. The challenge now is to demonstrate for which materials the necessary parameters such as a small Hubbard U and large d2/ d1 ratio might be attained. We will focus on estimating these quantities in the future through the use of a variety of computational techniques that account for relevant factors such as dielectric environment and bulk structure, and the effect of excess charge on coupling. One-particle energies and couplings have been extensively studied, and reasonable estimates that capture experimental trends are not too challenging. However, these values are often approximated by the neutral system energies, and the effects of charging are disregarded. More rigorous methods that take into account polarization are therefore in order.37,40 Once computational results identify promising species, the synthetic tools available to the materials community are welldeveloped to pursue fabrication of the structures described herein. Nanoparticle synthesis can already provide the desired site energies, as energy tuning can be achieved through altering particle size to select the pairing energy U. Work in selfassembly has already been implemented to fabricate superlattices and crystals of such particles.25,41 Achieving large coupling (tunneling) between nanoparticle sites is a more challenging task, although the amount of activity in this area is promising. Organics, on the other hand, can be tuned to have a range of site energies as well as couplings, though control of the bulk structure is difficult to attain. A more rigorous understanding of how to rationally design and construct binary organic superlattices is necessary before fabrication of materials corresponding to this model can be attempted in a targeted fashion. However, this is also an intensive area of study for the materials and nanotechnology communities and is likely to lead
the band structures for the exact one-dimensional intercalate model, which allows two electrons to occupy yellow sites simultaneously. Here we have still restricted the bielectron to length a, but we have allowed both electrons to simultaneously occupy a yellow site. The Hilbert space for this system is ninedimensional (Supporting Information), and the lowest bielectron band must be solved for numerically. It is plotted in Figure 7 with the single electron band structure. For a large d2/d1 ratio not only is the bielectron more stable than the noninteracting pair, the curvatures of the bands indicate that the bielectron also has a lower effective mass. While the relative energy of the bielectron is captured well by the assumption that both electrons could not be on yellow sites, the shape of the band is greatly affected by including this requirement. This is to be expected since excluding the double yellow occupation requires the pair to move in an inchwormlike progression along the lattice, contracting to the doubly occupied red site before proceeding. This leads to poor effective banding for the bielectron and a high effective mass. There are a great number of lattices where this model can be applied, and one can consider the effective mass of the bound pair to constitute another design parameter in its own right. For Model II (Figure 1b), suppose that there is diagonal coupling between yellow sites that share a red site. We can then identify the band formed by the yellow sites with themselves as analogous to an s-band, and the red sites as analogous to dlevels, so that the coupling d1 is analogous to the Anderson model Vds admixture, usually written as a k−d coupling Vkd. We then have together with the Hubbard U on the d-site the exact analogy to Anderson’s famous magnetic impurity model.38 The periodic repetition of the red sites then gives the periodic Anderson model. Furthermore, one can then use the Schrieffer Wolf transformation39 to generate a periodic Kondo Hamiltonian. Our model would thus seem to be a lower order imitation of these famous and widely used Hamiltonians, were it not for the fact that we are considering the following features: (1) The center of the s-band is roughly at U and (2) importantly and crucially Vd↑↓,s > Vd,s, i.e., the doubly occupied d-state has a stronger coupling to the s-band than that of the single particle. The first feature is obviously possible in the spatial molecular- or nanocrystal-like representation, with U not too large and the neighbor s-orbital level roughly at resonance with U. The second assumption is however not obviously relevant to the original Anderson k−d mixture model, which features a large U and paramagnetic spin intended for metals. Finally we point out that the Kondo singlet condensation phenomenon is a many body effect, and not analogous to the negative U binding of singlet pairs considered here. E
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(24) Ueno, K.; Nakamura, S.; Shimotani, H.; Yuan, H. T.; Kimura, N.; Nojima, T.; Aoki, H.; Iwasa, Y.; Kawasaki, M. Nat. Nanotechnol. 2011, 6, 408. (25) Talapin, D. V.; Lee, J. S.; Kovalenko, M. V.; Shevchenko, E. V. Chem. Rev. 2010, 110, 389. (26) Talapin, D. V. MRS Bull. 2012, 37, 63. (27) Boneschanscher, M. P.; Evers, W. H.; Geuchies, J. J.; Altantzis, T.; Goris, B.; Rabouw, F. T.; van Rossum, S. A. P.; van der Zant, H. S. J.; Siebbeles, L. D. A.; Van Tendeloo, G.; Swart, I.; Hilhorst, J.; Petukhov, A. V.; Bals, S.; Vanmaekelbergh, D. Science 2014, 344, 1377. (28) Bredas, J. L.; Calbert, J. P.; da Silva, D. A.; Cornil, J. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 5804. (29) Coropceanu, V.; Cornil, J.; da Silva, D. A.; Olivier, Y.; Silbey, R.; Bredas, J. L. Chem. Rev. 2007, 107, 926. (30) Gunnarsson, O. Rev. Mod. Phys. 1997, 69, 575. (31) Scriven, E.; Powell, B. J. Phys. Rev. B 2009, 80, 205107. (32) Scriven, E.; Powell, B. J. J. Chem. Phys. 2009, 130, 104508. (33) Nakamura, K.; Yoshimoto, Y.; Kosugi, T.; Arita, R.; Imada, M. J. Phys. Soc. Jpn. 2009, 78, 083710. (34) Cano-Cortes, L.; Dolfen, A.; Merino, J.; Behler, J.; Delley, B.; Reuter, K.; Koch, E. Eur. Phys. J. B 2007, 56, 173. (35) Cano-Cortes, L.; Dolfen, A.; Merino, J.; Koch, E. Physica B 2010, 405, S185. (36) Valeev, E. F.; Coropceanu, V.; da Silva, D. A.; Salman, S.; Bredas, J. L. J. Am. Chem. Soc. 2006, 128, 9882. (37) Reslan, R.; Lopata, K.; Arntsen, C.; Govind, N.; Neuhauser, D. J. Chem. Phys. 2012, 137, 22A502. (38) Anderson, P. W. Rev. Mod. Phys. 1978, 50, 191. (39) Schrieffer, J. R.; Wolff, P. A. Phys. Rev. 1966, 149, 491. (40) Kaduk, B.; Kowalczyk, T.; Van Voorhis, T. Chem. Rev. 2012, 112, 321. (41) Konstantatos, G.; Sargent, E. H. Nat. Nanotechnol. 2010, 5, 885.
to improved control over synthesis and assembly of novel structures in the near future. Overall, this model in conjunction with the computational and synthetic tools at our disposal provides an interesting avenue for the pursuit of materials or material networks that are capable of forming attracting electron pairs.
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ASSOCIATED CONTENT
S Supporting Information *
Eigenvalue derivations for single triad, Model I, and Model II; effective Hubbard U calculations for single triad and Model II. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Corresponding Authors
*E-mail:
[email protected]. *E-mail:
[email protected]. Author Contributions
M.R.B. and B.M. contributed equally in this manuscript. The manuscript was written through contributions of all authors. Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank the Northwestern University Materials Research Science and Engineering Center (MRSEC grant DMR1121262), for funding this research. M.R.B. thanks the NSF for the award of a Graduate Research Fellowship (NSF DGE1324585).
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REFERENCES
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