Towards electrical domain-wall control in polyacetylene-based electronic nanodevices
Leandro M. Arancibia Andrés I. Bertoni Cristián G. Sánchez Alejandro M. Lobos

Quantum optical classifier with superexponential speedup
    Seth Lloyd 0000-0003-0353-4529
    April 29, 2024
========================================================


We theoretically propose a polymer-based nano-device consisting of a single trans-polyacetylene (tPA) molecule capacitively coupled to external voltage gates. We model the integrated device using a Su-Schrieffer-Heeger (SSH)-like Hamiltonian, and we demonstrate the emergence of localized domain walls (DWs) with quantized charges (i.e., soliton excitations) localized at the gates. Interestingly, by increasing the applied voltage, multiple discrete charges can be accumulated, which may be useful for potential technological applications. Exploiting the topological character of the solitonic excitations of tPA, this device can be considered as an organic-based quantum dot with a very large and robust quantized charge.




§ INTRODUCTION



The emergent area of synthetic organic metals based on polymers is an important field at the intersection of chemistry and physics, with interest
both from the perspective of fundamental research as well as for organic electronics and other technological applications  <cit.>. Historically, trans-polyacetylene (tPA), a linear chain of carbon atoms with alternating single and double bonds, was identified as the first organic conductor<cit.>. 

A key aspect of tPA is its double-degenerate ground state corresponding to the two possible ways to accommodate the (Peierls) dimerization pattern of single and double bonds. In its ground state, the system is an insulator characterized by a (Peierls) gap in the single-particle spectrum of excitations. However, tPA can be excited to a soliton-like state

which features an excess charge bound to lattice deformations known as domain walls (DWs), which bridge two different dimerization patterns, and intra-gap single-electron states linked to the solitonic nature of the excitation. The first experimental evidence of  soliton excitations in tPA was indirectly established via optical spectroscopy<cit.> and magnetic electron paramagnetic resonance (EPR) experiments<cit.>.

Recent progress in nano-fabrication methods has shown the possibility to synthesize linear chains of tPA on clean metallic surfaces<cit.>. In particular, in Ref.<cit.> it was possible for the first time to address and study a single tPA molecule deposited on a Cu(110) surface, via scanning tunneling microscopy (STM) techniques. Experimental signatures in the STM differential conductance dI/dV indicate the presence of a trapped electronic state at the DW, constituting the first experimental evidence of a charged soliton. Furthermore, the ability to control and manipulate DWs in bi- and tri-layer graphene has been experimentally demonstrated in recent works<cit.>. This type of synthesis stimulated recent proposals of single-molecule nano-devices that use soliton excitations as the main mechanism for electronic transport<cit.>.

Motivated by these recent advances, in this work we theoretically demonstrate the possibility to capacitively induce charged DWs (CDWs) in a single tPA molecule, and therefore to control its electronic intra-gap spectrum, allowing interesting functionalities for this class of devices. The proposed nano-device, depicted schematically in Fig. <ref>, consists of a linear tPA molecule deposited on an insulating film (e.g., SiO_2) under which gate voltages are applied. These gate voltages are assumed to locally modify the chemical potential of the exposed molecular segments via capacitive coupling and induce the nucleation of CDWs, which can be regarded as a local doping of the molecule at the gate region. In other words, we propose that the strong electron-phonon coupling present in organic π-conjugated molecules can be exploited to generate charge carriers or spin excitations at any desired region within the polymer. Although this device effectively behaves as a molecular quantum dot (QD) that hosts quantized charges, we show that it is markedly more localized and superior in robustness to disorder and other local perturbations over a wide range of parameters due to the topological origin of the DWs.



§ MODEL


Illustrated in Fig. <ref>, our model considers a finite, linear tPA chain deposited on an insulating thin film, e.g. silica, with two voltage gates placed underneath so that they are only capacitively coupled to separate regions of the molecule. These gates are connected to external independent voltages that are able to locally control the electron density of the molecular segments by modifying their electro-chemical potential. We model this device by the means of the Su-Schrieffer-Heeger (SSH) Hamiltonian<cit.>, with slight modifications to treat open-ended molecular chains<cit.> and to describe the effect of the voltage gates on the electron density. The SSH Hamiltonian is a celebrated model introduced to describe the interaction of π-conjugated electrons with the classical ionic degrees of freedom in linear polyalkenes. This rather simple model successfully accounts for many of the experimentally observed properties of tPA, such as the existence of a Peierls dimerization gap, and theoretically explains the emergence of fractional solitons <cit.>. The SSH model is one of the most paradigmatic models in condensed-matter physics, since it realizes the simplest example of a topological insulator in one dimension<cit.>. 

Explicitly, we have


    Ĥ = Ĥ_el + Ĥ_latt  ,


where the electronic Hamiltonian, Ĥ_el, describes both the electron-lattice and the electron-gate interaction, and the classical Hamiltonian, Ĥ_latt, describes the elastic interaction between the (CH) groups:


    Ĥ_el =    ∑_n=1,s^N_s-1[ -t_0 + α( u_n+1 - u_n) ] c_n,s^†  c_n+1,s + H.c. + ∑_n=1,s^N_sμ_n   c_n,s^†  c_n,s  , 
    Ĥ_latt =    ∑_n=1^N_sp_n^2/2M + ∑_n=1^N_s-1K/2( u_n+1 - u_n)^2 - ∑_n=1^N_s-1Γ  (u_n+1 - u_n)   .


The first term of Eq. (<ref>) describes the electrons in the π bands, arising from the 2p_z orbitals of a tPA molecule with N_s (CH) moieties, each one considered a discrete “site’’ in the chain. The operator c_n,s (c^†_n,s) annihilates (creates) an electron with spin s at site n. The parameter t_0 = 2.5  eV is the hopping integral, α = 4.1  eV/Å is the electron-lattice coupling parameter, and u_n are the classical variables describing the deviation from equilibrium of the n-th site along the chain axis. Essentially, the term in square brackets in Eq. (<ref>) is what results from expanding the full hopping integral up to first order, with a small parameter (u_n+1 - u_n ).

The last term of Ĥ_el introduces the effect of the external gate voltages. In order to simplify the theoretical calculations, we choose to work in the microcanonical ensemble, where the number of electrons N_el has been set assuming an overall half-filling condition, i.e. N_el = N_s with N_s being the total number of sites in the chain; this condition is ensured by constraining the electronic solutions in all the steps of the variational minimization process. As a result of imposing global half-filling (i.e., charge neutrality under our assumptions), the effect of the gate voltages is limited to the redistribution of electronic charge within the system, favoring an excess of electron density at some sites at the expense of its depletion at other points in the chain. The microcanonical constraint is equivalent to disallowing electron transfer events, or any other direct interaction, through or with the insulating film, which therefore acts as a decoupling layer. While a single-gate version of the device may be  simpler to realize from the  experimental point of view, it becomes theoretically more complicated as it necessarily would involve interaction and equilibration with an external charge reservoir (i.e., grand-canonical ensemble). However, placing the gates sufficiently separated from each other allows to avoid direct interaction between them and enables us to infer the properties of a single-gate device. In addition, in what follows we assume exact anti-symmetric gates (i.e, identical to each other but with opposite voltages). Accordingly, as shown in Fig. <ref>(b), we define two distinct gate regions g^+ and g^-, each spanning w_g sites, which are associated with constant voltages +V_g and -V_g, respectively, with V_g > 0. In Eq. (<ref>), these external gate voltages are encoded in a phenomenological site-dependent chemical potential μ_n, such that μ_n = +V_g,  n ∈ g^+ and μ_n = -V_g,  n ∈ g^-, being zero for non-gate sites (μ_n = 0 ,  n ∉{g^+, g^-}). The anti-symmetry of the external perturbation is crucial for extrapolating our results to the case of a single gate in open conditions, since it would ensure the occurrence of simultaneous and (anti-)symmetric events involving electrons and lattice.

The first term in Eq. <ref> provides a classical description for the lattice ions, where M = 2.152 × 10^-26 Kg is the mass of each site (i.e., of the (CH)_n groups) and K = 21  eV/Å^2 is the effective spring force constant of the C-C bonds. As was already noted by Su et al.<cit.>, and by Vanderbilt et al.<cit.>, in the open-boundary SSH model there is a tendency for the chain to shrink under the effect of the unconstrained electronic linear coupling to the π-electrons. This is of course an artifact of the SSH model, rooted at the harmonic approximation, that should be solved by more sophisticated treatments (e.g., proper ab initio potentials). In order to compensate for this undesired effect while keeping the simplicity of the theoretical modeling, we followed Refs. <cit.> and introduced an additional term, ∑_n=1^N_s-1Γ  (u_n+1 - u_n) = Γ  (u_N_s - u_1), which applies an explicit “stretching’’ force to the ends of the chain to compensate for the spurious contraction. We chose Γ = 5.15  eV/Å in order to reproduce the same bulk properties seen in experiments (i.e. values of the Peierls distortion, gap size, etc.). Since the stretching term only introduces distortions that are localized at the ends of the chain, it does not affect our conclusions based on sites that are away from the edges of the molecule. 

For this work, we considered a tPA molecule containing two hundred carbon atoms (i.e., N_s = 200) that is capacitively coupled to a pair of anti-symmetric voltage gates of width w_g = 40   a ≈ 5  nm at distance of L_s = 80   a ≈ 10  nm; these values are within experimental reach using, e.g., electron-beam lithography techniques. Throughout this study, we have adjusted the gate voltages up to V_g = 1.5  eV.

Previous experimental works using chemical doping techniques reported evidence of charged intra-gap states associated with solitonic or polaronic excitations appearing at random positions along the tPA chain<cit.>. Although the underlying physical mechanism is similar, the crucial difference with our case is that we propose a way to induce the electronic doping in a controlled manner, via the nucleation of DWs facilitated by tunable external voltage gates at specific locations of the chain (see Section <ref>). This effect could be exploited in the design of novel organic quantum electronic devices. Due to the topological origin of the induced intra-gap states, we argue that our phenomenological description, although simplistic, is sufficient to capture the main features of the voltage-controlled doping. We also speculate that more sophisticated (even microscopic) descriptions of this system will not qualitatively change our main conclusions.



§ METHODOLOGY


In order to solve the static problem we focus on the zero temperature case, and employ the Hellmann-Feynman theorem<cit.>:


    ∂ E_GS/∂ R_n = ⟨Ψ_GS| ∂Ĥ/∂ R_n| Ψ_GS⟩ = 0   ,


where |Ψ_GS⟩ is the electronic ground state wavefunction, E_GS is the total ground state energy (i.e., Ĥ  |Ψ_GS⟩ = E_GS  |Ψ_GS⟩), and R_n = na + u_n is the position of the n-th site along the chain axis, with lattice parameter a = 1.22  Å. Eq. (<ref>) leads to the following system of coupled equations:


    +K  ( R_2 - R_1 - a ) - Γ + α∑_s∑_ν=1^occ[ ψ_ν,s^*(2)  ψ_ν,s(1) + H.c.] =      0   , 
    
    K  ( R_n+1 - 2 R_n + R_n-1) - α∑_s∑_ν=1^occ[ ( ψ_ν,s^*(n+1) - ψ_ν,s^*(n-1) )  ψ_ν,s(n) + H.c.] =      0  
    ∀  n such that  1<n<N_s  , 
    
    -K  ( R_N_s - R_N_s-1 - a ) + Γ - α∑_s∑_ν=1^occ[ ψ_ν,s^*(N_s-1)  ψ_ν,s(N_s) + H.c.] =      0   .


In the above equations, ψ_ν,s(n) is the probability amplitude of the ν-th eigenstate at the n-th site, i.e. |ψ_ν,s⟩≡∑_n=1^Nψ_ν,s(n) c^†_n,s |0⟩, which satisfies the single-particle eigenvalue equation:


    Ĥ_el  |ψ_ν,s⟩ = ϵ_ν  |ψ_ν,s⟩  ,


with ϵ_ν as the corresponding single-particle energy. Moreover, note that summations run over the doubly occupied (“occ’’) one-electron states (since N_el^↑ = N_el^↓ due to the SU(2) symmetry of the model). Note also that Eq. (<ref>) is solved for a given specific configuration of the lattice deformation field R_n; therefore, the set of Eqs. (<ref>)-(<ref>) form a complete set of coupled non-linear equations that must be solved self-consistently. Finally, note that for our model of non-interacting electrons, the electronic ground state is simply written as | Ψ_GS⟩ = ∏_s ∏_ν=1^N_el c_ν,s^† |0⟩, with the fermionic operators c_ν,s defined as c_ν,s = ∑_n=1^Nψ_ν,s(n)   c^†_n,s.

We have implemented a numerical routine for the energy minimization that starts with a random distribution of ion positions around their equilibrium values R_n^(0). To obtain the electron-lattice solution of minimum energy, we start by diagonalizing the Hamiltonian Ĥ_el, built from the initial positions, to obtain its electronic eigenstates, which are then used to calculate the new ion positions, R^(1)_n, by solving the linear system on R_n of Eqs. (<ref>)-(<ref>). We iterate this process until a convergence criterion is met, namely when the root-mean-square deviation (RMSD) of the ion positions with respect to the preceding frame falls below 10^-8  a.

It is important to highlight that each step of the optimization routine ensured that the total number of electrons remained constant (hence keeping the system neutral), as this corresponds to the microcanonical ensemble under the half-filling condition.



§ RESULTS AND DISCUSSION




Our main finding is the occurrence of discrete ground-state transitions in tPA, as we continuously varied the external gate voltages (V_g∈[0,1.5]   eV), characterized by the voltage-induced accumulation of quantized charge in the gate regions. These quantized jumps are related to the appearance of DWs within the gate regions.

As is well known<cit.>, the intra-gap electronic properties of tPA are intimately related to the Peierls distortion profile of the unperturbed ground-state. It is customary to define the staggered distortion as follows:


    y̅_n≡     (-1)^n[ u_n+1 - u_n] 
    
    =      (-1)^n[ R_n+1 - R_n - a ]  ,


where its fast oscillations have been removed via the factor (-1)^n. As a result, y̅_n is a smooth field on the scale of the lattice parameter a, which allows to identify the regions with a specific alternation pattern, and therefore the presence of DWs in the system (i.e., where y̅_n crosses zero). Due to the symmetry of the SSH model,under the change u_n → -u_n with a simultaneous gauge transformation c_n,s→ (-1)^n c_n,s, the values ± u_0 are degenerate. Consequently, a region with a constant staggered field of y̅_n = ± 2 u_0 corresponds to one of the two possible Peierls-dimerization patterns of tPA. Either of these two uniform solutions leads to a gap of Δ_0 = 4 α| u_0| between the filled valence band and the empty conduction band, at T = 0. For the present case, we have determined the Peierls gap at V_g = 0 to be Δ_0 = 0.628  eV; this value will be used as a reference throughout this work.

In the absence of external voltages, and using the aforementioned parameters, we recovered a uniform profile with |y̅_n| = 0.0755  Å. This single-pattern configuration is robust to small changes in the chemical potential of the molecule, as this remained to be the only solution in the range of very low voltages, up to ≈ 0.4  eV (see Fig. <ref>(a)). Note that, even for the unperturbed system, we observe small depressions near the edges; these distortions are introduced by the stretching term (proportional to Γ) and should be ignored as they are of no physical significance far away from the edges.

As the external voltage V_g is increased beyond a critical value of V_g^[1]≈ 0.411  eV, the system undergoes an abrupt transition to a different distortion profile containing a single DW per gate (see Fig. <ref>(b)). The emergence of a DW corresponds to a voltage-induced quantum phase transition<cit.>, which arises from the crossing of energy levels as the external perturbation is continuously varied. The energy crossing occurs at critical values of V_g where two solutions, differing in the number of DWs per gate, become  degenerate. The abrupt transitions are a consequence of the minimization process, which always selects the solution with the lowest energy for a given value of V_g. In line with this, the solutions with a number of DWs different from that of the ground state can be regarded as molecular excited states. It is important to remark that these  excited states are local minima (i.e., stable solutions with energies greater than the lowest possible) that can be accessed by choosing appropriate initial seeds for the ion positions in the minimization process.

It is worth mentioning that, in tPA, a single DW can be modeled in the continuum limit a → 0,   N_s→∞, by the means of the massive Dirac equation<cit.>, in which the distortion profile y̅_n→Δ(x), with x = na, can be interpreted as an effective sign-changing “mass’’ term. For an infinite system with a DW centered at x=0, this mass term adopts the simple functional form Δ(x) = 2 u_0tanh (x / ξ ), where ξ = ħ v_F / Δ_0 is the width of the DW (with v_F being the Fermi velocity of the massless Dirac fermions<cit.>. For x →±∞, Δ(x) is a solution that interpolates between the constant values ± 2 u_0 corresponding to single Peierls dimerization patterns. This model predicts topological intra-gap electronic states that are localized at the DW within a distance ξ and emerge at the Fermi energy (i.e., at ϵ = 0) in the electronic spectrum. These bound states are able to accommodate an extra charge and/or spin<cit.>, giving rise to the celebrated soliton excitations in tPA. The distortion profile of Fig. <ref>(b), which contains one DW per gate separated by a distance L_s≫ξ, can be successfully reproduced with the ansatz:


    Δ (x) ≈ -2 u_0tanh( x - x_g^+/ξ) . tanh( x - x_g^-/ξ),


where x_g^+ and x_g^- refer to the center of the respective gates. In other words, perturbing the tPA molecule via an external voltage favors the accumulation of excess charge bound to the gate regions, via the generation of CDWs. For the case at hand (see Fig. <ref>(b)), a positive charge Q_g^+ = +e becomes localized at the DW on g^+ and, because of the anti-symmetry of the external potential profile, a compensating negative CDW also emerged at g^- with charge Q_g^- = -e, maintaining the overall charge neutrality. Interestingly, although the charge estimated from the solution electron density was found to be localized at the lattice sites of the DWs, as in what we recognize as CWDs, it was not possible to identify individual one-electron (bound) intra-gap state responsible for the excess charge in the molecular solution provided by our tight-binding SSH model, demonstrating the collective nature of the charge localization phenomenon. 




Moreover, we find that this mechanism is not limited to the generation of a single excess charge per gate. We note that, by increasing the voltage even further, new critical values appear (V_g^[2],   V_g^[3],  …), all corresponding to quantum transitions to solutions with an increasing integer number of CDWs per gate, i.e. with the respective number of localized, quantized excess charges per gate, as shown in Fig. <ref>(c-h).

To better visualize the abrupt nature of the quantum phase transitions, in Fig. <ref> we show the evolution of the electronic spectrum around the energy gap, as a function of V_g. Here, the quantum phase transitions can be seen as abrupt changes in the electronic single-particle eigenenergies, with changes being more pronounced for the intra-gap states. Note that although the electronic spectrum varies even in regions of phase stability, the (lattice) distortion profile remains robust and visibly unaffected, meaning that its ability to accumulate charge via CDWs remains intact. Only a quantum transition, as the external potential is increased, can lead to a new configuration of ions capable of localizing more charge.



To emphasize the quantized nature of the accumulated charge, as an argument in favor of the topological origin of these transitions being the formation of voltage-induced CWDs, in Figs. <ref>(a) and (b) we show, respectively, as functions of the applied voltage V_g, the total energy of the system (i.e., electronic plus lattice) and the absolute net charge contained at each gate region. In panel (a) we can visualize the energy of the lowest configuration (thick lines) for each voltage stability range. In addition, note the occurrence of level crossings giving rise to quantum phase transitions, at (critical) voltage values, as we discussed above. In panel (b) we can see that the charge contained in each gate region is a strictly constant integer value between transition voltages. These quantized charges are associated with different solitonic phases of tPA and are as robust to changes in V_g as the lattice profile of the  stable configuration, which is consistent with the gate net charge coming from the CWDs. There exists a simple analytical link between the two panels of Fig. <ref>, as the absolute excess charge per gate Q_g can be obtained by deriving the total energy E_GS with respect to the external potential V_g, making use of the Hellman Feynman theorem:


    ∂ E_GS/∂ V_g =    ⟨Ψ_GS| ∂Ĥ_el/∂ V_g| Ψ_GS⟩  , 
    
    =     -2   Q_g,


where the factor 2 is obtained due to the symmetry of the perturbation, as shown in Section A of the supplementary material (SM). The remarkable simplicity of this expression is related to the fact that the capacitive coupling term contributes to the total energy as the electrostatic potential energy of the charges accumulated per gate due to the applied external voltage. 

In Section B of the SM, we provide evidence that the perturbed molecular solutions are not only robust over a range of voltages (see e.g. Fig. <ref>(b)), but also to strong noise in the applied voltage profile μ_n, which may originate from normal room temperature fluctuations or aspects related to the non-ideality of an experimental setup.



§ SUMMARY AND CONCLUSIONS


From a theoretical perspective, we have discussed the design and feasibility of a proposed tPA-based nano-device capable of accumulating localized discrete charges in a controllable manner via external voltage gates. We attributed the quantized nature of the charges on each gate to the voltage-induced generation of CDWs, occurring thanks to the large intrinsic electron-phonon coupling of π-conjugated molecules. The appearance of CDWs is associated with quantum phase transitions due to energy level crossings between stable solutions with different numbers of DWs in their lattice staggered field profiles. We found that although these solutions are stable for a range of voltages and even to strong noise in the external perturbation, their electronic spectra are highly sensitive to changes in the external potential, providing a way to induce and control the energy of the topological intra-gap states. Overall, the proposed device is comparable to a semiconductor quantum dot (QD), but more robust and with highly localized charges. A significant difference with the so-called “Coulomb diamonds” in QD circuits is that the stability of the quantized charges does not come from Coulomb interactions, but from the robustness of the DWs.

Although building single-molecule electronic devices is certainly a real experimental challenge, we believe that the proposed device may be within reach in the near future, as there have been several recent advances in the on-surface synthesis of π-conjugated molecules<cit.>, which can be observed and controlled with modern EBL-, AFM-, and STM-based techniques<cit.>. Recent theoretical modeling also supports the possibility of designing single-molecule electronics<cit.>, which could even be based on conductive polymers other than tPA<cit.>.

While the anti-symmetry of the external voltage profile was just a trick to simplify our theoretical calculations, we emphasize that this is not a necessary condition for any experimental setup. In other words, our results can be easily extrapolated to the case of a single gate in open conditions, in which case the tPA molecule would have to be weakly coupled to an electron reservoir that allows N_el to fluctuate (e.g., a metallic contact). Grand-canonical ensemble calculations would require the inclusion an explicit coupling to a reservoir; this is the subject of our subsequent line of research which is currently underway. Although our simple model neglects an explicit treatment of electron-electron interactions, given the remarkable energy difference between the ground- and low-lying excited-states, we do not expect our results (or conclusions) to change qualitatively, even in cases with strong correlations; this is also an argument for the stability of the configurations in a room-temperature scenario. In fact, rather than being a detrimental feature, the ability to induce strong correlations could open up new interesting possibilities for the device, involving exotic ground states, such as the generation of localized magnetic moments or the Kondo effect. For example, recent experiments have shown the presence of superconductivity in α-terphenyl polymers upon doping <cit.>
, and strong correlations have been invoked to explain these interesting results.

In conclusion, and in light of our results, it becomes tempting to imagine the many exciting ways in which the presented mechanism for generating an arbitrary number of quantized excess charges at desired locations could be exploited in the design and development of next-generation tPA-based nano-devices.



§ ACKNOWLEDGEMENTS

This work was supported by CONICET and Agencia
I + D + i under PICT 2017-2081, Argentina.


10

Asboth16_Short_course_on_TIs
J.K. Asbóth, L. Oroszlány, and A.P. Pályi.
A Short Course on Topological Insulators: Band Structure and
  Edge States in One and Two Dimensions.
Lecture Notes in Physics. Springer International Publishing, 2016.

Bernevig_book_TI_TSC
B. Andrei Bernevig and Taylor L. Hughes.
Topological Insulators and Topological Superconductors.
Princeton University Press, 2013.

blanchet1983photoexcitations
Graciela B Blanchet, C. R. Fincher, T. C. Chung, and A. J. Heeger.
Photoexcitations in trans-(ch) x: a fourier-transform infrared study.
Physical review letters, 50(24):1938, 1983.

Chiang77
C. K. Chiang, C. R. Fincher, Y. W. Park, A. J. Heeger, H. Shirakawa, E. J.
  Louis, S. C. Gau, and Alan G. MacDiarmid.
Electrical conductivity in doped polyacetylene.
Phys. Rev. Lett., 39:1098–1101, Oct 1977.

cirera2020tailoring
Borja Cirera, Ana Sánchez-Grande, Bruno de la Torre, José Santos,
  Shayan Edalatmanesh, Eider Rodríguez-Sánchez, Koen Lauwaet, Benjamin
  Mallada, Radek Zbořil, Rodolfo Miranda, et al.
Tailoring topological order and π-conjugation to engineer
  quasi-metallic polymers.
Nature Nanotechnology, 15(6):437–443, 2020.

Farchioni_Grosso_Organic_electronic_metals_book
R. Farchioni, G. Grosso, and (Eds).
Organic Electronic Materials. Conjugated Polymers and Low
  Molecular Weight Organic Solids.
Springer-Verlag, Berlin Heidelberg GmbH, Germany, 2001.

feynman39
R. P. Feynman.
Phys. Rev., 56:340, 1939.

goldberg1979electron
IB Goldberg, HR Crowe, PR Newman, AJ Heeger, and AG MacDiarmid.
Electron spin resonance of polyacetylene and asf5-doped
  polyacetylene.
The Journal of Chemical Physics, 70(3):1132–1136, 1979.

Grill2007
Leonhard Grill, Matthew Dyer, Leif Lafferentz, Mats Persson, Maike V. Peters,
  and Stefan Hecht.
Nano-architectures by covalent assembly of molecular building blocks.
Nature Nanotechnology, 2(11):687–691, Nov 2007.

Han21_Surface_assisted_fabrication_low_dimensional_carbon_nanostructures
Dong Han and Junfa Zhu.
Surface-assisted fabrication of low-dimensional carbon-based
  nanoarchitectures.
Journal of Physics: Condensed Matter, 33, 07 2021.

Heeger88_Solitons_in_conducting_polymers
A. J. Heeger, S. Kivelson, J. R. Schrieffer, and W. P. Su.
Solitons in conducting polymers.
Rev. Mod. Phys., 60:781–850, Jul 1988.

HernangomezPerez20_Solitonics_with_PA
Daniel Hernangómez-Pérez, Suman Gunasekaran, Latha Venkataraman, and
  Ferdinand Evers.
Solitonics with polyacetylenes.
Nano Letters, 20(4):2615–2619, 2020.
PMID: 32125870.

Huang19_Superconductivity_in_p_terphenyl
Ge Huang, Guo-Hua Zhong, Ren-Shu Wang, Jia-Xing Han, Hai-Qing Lin, and Xiao-Jia
  Chen.
Superconductivity and phase stability of potassium-doped
  p-quinquephenyl.
Carbon, 143:837–843, 2019.

Jackiw76_Jackiw_Rebbi_soliton
R. Jackiw and C. Rebbi.
Solitons with fermion number ½.
Phys. Rev. D, 13:3398–3409, Jun 1976.

Jiang18_Manipulation_of_DWs_in_bi_and_trilayer_graphene
Lili Jiang, Sheng Wang, Zhiwen Shi, Chenhao Jin, M. Iqbal Bakti Utama, Sihan
  Zhao, Yuen-Ron Shen, Hong-Jun Gao, Guangyu Zhang, and Feng Wang.
Manipulation of domain-wall solitons in bi- and trilayer graphene.
Nature Nanotechnology, 13(3):204–208, Mar 2018.

park2022creation
Jae Whan Park, Euihwan Do, Jin Sung Shin, Sun Kyu Song, Oleksandr Stetsovych,
  Pavel Jelinek, and Han Woong Yeom.
Creation and annihilation of mobile fractional solitons in atomic
  chains.
Nature nanotechnology, 17(3):244–249, 2022.

sachdev1999quantum
Subir Sachdev.
Quantum phase transitions.
Physics world, 12(4):33, 1999.

sethna1982photoinduced
James P Sethna and Stephen Kivelson.
Photoinduced soliton pair production in polyacetylene: An instanton
  approach.
Physical Review B, 26(6):3513, 1982.

Shen17_Frontiers_on_surface_review
Qian Shen, Hong-Ying Gao, and Harald Fuchs.
Frontiers of on-surface synthesis: From principles to applications.
Nano Today, 13:77–96, 2017.

Su80_Dynamics_of_solitons_in_PA
W. P. Su and J. R. Schrieffer.
Soliton dynamics in polyacetylene.
Proceedings of the National Academy of Sciences,
  77(10):5626–5629, 1980.

Su79_Solitons_in_polyacetylene
W. P. Su, J. R. Schrieffer, and A. J. Heeger.
Phys. Rev. Lett., 42:1698–1701, Jun 1979.

Takayama80_Continuum_model_for_PA
Hajime Takayama, Y. R. Lin-Liu, and Kazumi Maki.
Continuum model for solitons in polyacetylene.
Phys. Rev. B, 21:2388–2393, Mar 1980.

Vanderbilt80_Disorder_in_tPA
David Vanderbilt and Eugene J. Mele.
Effects of disorder on the electronic structure of undoped
  polyacetylene.
Phys. Rev. B, 22:3939–3948, Oct 1980.

Vos96_SSH_model_finite_length
Fernando L. J. Vos, Daniel P. Aalberts, and Wim van Saarloos.
Su-schrieffer-heeger model applied to chains of finite length.
Phys. Rev. B, 53:14922–14928, Jun 1996.

Wang19_Solitons_in_individual_PA_molecules
Shiyong Wang, Qiang Sun, Oliver Gröning, Roland Widmer, Carlo A. Pignedoli,
  Liangliang Cai, Xin Yu, Bingkai Yuan, Can Li, Huanxin Ju, Junfa Zhu, Pascal
  Ruffieux, Roman Fasel, and Wei Xu.
On-surface synthesis and characterization of individual polyacetylene
  chains.
Nature Chemistry, 11(10):924–930, Oct 2019.

weinberger1980electron
BR Weinberger, E Ehrenfreund, A Pron, AJ Heeger, and AG MacDiarmid.
Electron spin resonance studies of magnetic soliton defects in
  polyacetylene.
The Journal of Chemical Physics, 72(9):4749–4755, 1980.

Yan19_Superconductivity_in_p_quaterphenyl
Jia-Feng Yan, Guo-Hua Zhong, Ren-Shu Wang, Kai Zhang, Hai-Qing Lin, and
  Xiao-Jia Chen.
Superconductivity and phase stability of potassium-intercalated
  p-quaterphenyl.
The Journal of Physical Chemistry Letters, 10(1):40–47, 2019.

Yao19_Unconventional_nanofabrication_supramolecular_electronics
Yifan Yao, Lei Zhang, Emanuele Orgiu, and Paolo Samorì.
Unconventional nanofabrication for supramolecular electronics.
Advanced Materials, 31(23):1900599, 2019.

Zhong18_Superconductivity_in_p_terphenyl
Guo-Hua Zhong, Dong-Yu Yang, Kai Zhang, Ren-Shu Wang, Chao Zhang, Hai-Qing Lin,
  and Xiao-Jia Chen.
Superconductivity and phase stability of potassium-doped biphenyl.
Phys. Chem. Chem. Phys., 20:25217–25223, 2018.



