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# Lazy electrons in graphene

Contributed by Eric J. Heller, July 11, 2019 (sent for review May 29, 2019; reviewed by Troy Van-Voorhis and Peter G. Wolynes)

## Significance

The Born–Oppenheimer approximation has had an ambiguous role in condensed matter theory for many years. Without it, solids would seem to be a “pea soup” of electrons and nuclei. On the other hand, using the adiabatic Born–Oppenheimer, one inevitably encounters multiple narrow avoided crossings of electronic energy levels as a function of nuclear positions. “Adiabatic” is key here: there exist “diabatic” Born–Oppenheimer approximations that the time-dependent tight-binding work reported here shows are much more robust. We find that the electrons are “lazy” in that they respond weakly and slowly to nuclear geometry changes. Our results suggest an electron–nuclear diabatic wave function that may have significant impact on analytical and numerical approaches to spectroscopy and dynamics of extended π-electronic systems.

## Abstract

Within a tight-binding approximation, we numerically determine the time evolution of graphene electronic states in the presence of classically vibrating nuclei. There is no reliance on the Born–Oppenheimer approximation within the p-orbital tight-binding basis, although our approximation is “atomically adiabatic”: the basis p-orbitals are taken to follow nuclear positions. Our calculations show that the strict adiabatic Born–Oppenheimer approximation fails badly. We find that a diabatic (lazy electrons responding weakly to nuclear distortions) Born–Oppenheimer model provides a much more accurate picture and suggests a generalized many-body Bloch orbital-nuclear basis set for describing electron–phonon interactions in graphene.

- graphene
- time-dependent quantum mechanics
- nonadiabatic dynamics
- tight-binding
- Born–Oppenheimer approximation

In recent years, graphene has become perhaps the most popular 2D material studied for its optical, electronic, and mechanical properties. Electron–phonon interactions in graphene have become a focus because of their role in optical processes, thermalization of electrons, and superconductivity (1⇓⇓⇓⇓–6). Experimental measurements of electron coherence lengths as well as theoretical studies have suggested that electron–phonon interactions play a key role in relaxation of charge carriers following photoexcitation (7⇓–9). Few theoretical studies investigate the time-dependent dynamics of electrons in the presence of phonons, and graphene electronic structure calculations typically assume static, unbroken perfect symmetry of the lattice. However, even at low temperatures, this symmetry exists only in an average sense if the vibrational modes of the nuclei are unfrozen.

The fixed nuclei Born–Oppenheimer (BO) approximation may of course be implemented for static, broken symmetry geometries. However, degenerate or nearly degenerate adiabatic electronic Bloch waves of high-symmetry nuclear configurations must very strongly mix among themselves, even for configurations within the zero point motion of the nuclei. Such extremely strong, spatially extended electronic mixing for tiny, even local, nuclear displacements cannot be physical. If it were, the relevance of Bloch band structure (10) for high-symmetry configurations would be called into question. Instead, something like the average nuclear positions holds sway, suggesting a diabatic wave function more attuned to the lazy or sluggish response of the electrons to instantaneous nuclear configurations and vindicating (to no one’s surprise) the textbook Bloch wave high symmetry treatments. Clearly, an intuitively diabatic mindset is already built into the canon of solid-state physics.

In this paper, we develop numerical evidence that a diabatic BO (DBO) approximation is much superior to the standard adiabatic one. As with the adiabatic BO (ABO) approximation, the electronic wave function in a diabatic formulation remains a function only of nuclear configuration. Electron–phonon inelastic scattering remains a clear breakdown or correction to the diabatic approximation, as it is in the adiabatic approximation. In molecular physics, electron–phonon inelastic scattering is called vibronic coupling.

Through numerical simulations, we study the explicit time evolution of electronic wave functions in the presence of lattice vibrations. Modeling the nuclei classically, we work within the single-particle picture for the electronic degrees of freedom to construct a time-dependent tight-binding electronic Hamiltonian whose matrix elements change with fluctuations in interatomic bond distances. We then directly solve the resulting time-dependent Schrödinger equation (TDSE) for the electronic states. In doing so, we are exactly solving for the electronic time evolution within the tight-binding approximation, without invoking any form of the BO approximation at all.

We do not include the back-action of the π-electronic states on the nuclear motions. This is justified by the infinitesimal importance of single electrons on the forces experienced by the carbon atoms in a large sample of graphene, and the relative unimportance of the π-electron cloud as against the in-plane orbitals responsible for most of the binding forces in graphene. However, back-action should be taken seriously in the future, especially when considering the full multielectron response to nuclear geometry fluctuations. It will also be clear from our results that the minor adjustments to the nuclear positions due to back-action as they jostle thermally will not qualitatively affect our central conclusion that π electrons cannot and do not follow the details of nuclear motion in graphene.

## Theory and Model

Vibrations break the symmetry of a periodic crystal, but for convenience and computations, symmetry may be partially restored at long range by using a supercell consisting of many primitive unit cells. The thermal motion is then taken to repeat at the supercell level. In this situation, within the ABO approximation, the motion preserves the band index n, where n is an integer running from 1 to N, the number of atoms (with one basis orbital each) in the supercell. Projecting the true solution from the TDSE onto the ABO basis, we are able to understand whether the time evolution of the electron is described by the ABO approximation, which has been frequently used in condensed matter physics and quantum chemistry since its development in 1927 (11, 12). In fact, we will see a DBO approach is much closer to the truth.

We simulate a sheet of graphene with time-dependent nuclear vibrations by modeling the nuclei classically and the electronic degrees of freedom quantum mechanically (13, 14). Using a nearest-neighbor tight-binding model, we construct a time-dependent Hamiltonian for the electronic degrees of freedom. Using this Hamiltonian, we solve the TDSE and compare the true electronic states to ABO basis states.

The classical treatment of nuclear motion may be surprisingly close to reality. The “Schrödinger Correspondence Principle” (15, 16) linking the classical motion of harmonic oscillators (including infinite dimensional ones) to exact quantum evolution supports this statement. That is to say, as is discussed in ref. 16, Schrödinger showed that the mean position and momentum of quantum wave packets in a harmonic oscillator tend to obey the classical equations of motion. Thus, the treatment of nuclei as classical harmonic oscillators is a simplified yet justified approach to the investigating the quantum mechanical many-body problem.

### Classical Nuclei.

In order to consider a wide range of vibrational modes, we construct a supercell for graphene containing

We take the ith carbon nucleus to have position

The classical eigenvalue problem for the normal modes becomes

### Tight-Binding Hamiltonian.

Within the linear combination of atomic orbitals approximation, we write the single-particle electronic wave function

### Electronic Wave Function.

As an initial condition, we use a Bloch state with k such that

Solving the time-dependent tight-binding Hamiltonian numerically as the nuclei vibrate, we calculate the electronic wave function

### TDSE Solution.

The time-evolving electronic wave function is calculated using the TDSE **4**, we have

We also note that the supercell wave vector k, which labels an eigenvalue of the operator that translates a Bloch state by a supercell lattice vector, is conserved in the time evolution due to translational symmetry. Thus, the wave function

### ABO Approximation.

A discussion of the ABO approximation and its application to the time-dependent Hamiltonian *SI Appendix*, *Supporting Information Text*. For reference, we solve the TISE within the BO approximation

Thus, the TISE results in the eigenvalue problem

### Numerical Methods.

In our simulation, we choose amplitudes for the vibrational eigenmodes corresponding approximately to room temperature,

### Γ Point, Lowest Nondegenerate State.

For reference we calculate the ABO band structure at every time

### Γ Point, Nearly Degenerate State.

It is abundantly clear from Fig. 3 that, as a function of time, the band energies fluctuate significantly; nearly degenerate bands suffer avoided crossings in time. In particular, at the Γ point, we find 2 distinct clusters of 9 bands each—1 below and 1 above the Fermi energy—caused by the breaking of degeneracy due to nuclear displacements. The electronic states corresponding to the bands in these clusters are of particular interest; as suggested by time-dependent perturbation theory, we would expect these states to be highly coupled, making electronic time evolution unlikely to be adiabatic. Without loss of generality, we choose to investigate the behavior of the electronic state corresponding to band index

Additionally, we measure the preservation of electronic character by calculating the squared modulus of the autocorrelation function

### Near K Point.

Experimental optical excitation of graphene electrons is commonly performed with photons of energy *SI Appendix*, Fig. S1); bands 18 and 19 exhibit numerous avoided crossings in the time domain.

As with the Γ point case, we solve Eq. **8** for

## Results

In the following sections, we present the results of our simulations, comparing the ABO and true electronic states.

### Γ Point, Lowest Nondegenerate State.

While the results are not plotted here, the overlap probability

### Γ Point, Nearly Degenerate State.

Fig. 4 shows the overlap probability *Left*, given the initial conditions we set, the probability of overlap between the true state and the

Due to the incessant nuclear vibrations, the nearly degenerate manifolds of 9 ABO states are being constantly scrambled. Is there a diabatic (DBO) basis that approximates time evolution of the electronic wave function over much longer intervals, one that could be constructed in this case from the 9 ABO basis states?

From the squared modulus of the autocorrelation

### Near K Point.

We present the overlap probabilities *SI Appendix*, Fig. S2. The scenario is similar to the Γ point case, where we see clustering of bands. Due to the folding of the Brillouin zone,

The autocorrelation function of the time-dependent tight-binding electronic state decays over time, although much less quickly than the ABO states at the Γ point. We can conclude that the electronic state loses its character at a timescale much larger than the vibrational period of the classical nuclear oscillations, in a manner highly suggestive of diabatic time evolution.

### Extension to Larger Graphene Sample.

The eventual decay of the autocorrelation function is reminiscent of the initial sinusoidal behavior of a state that begins in a superposition of eigenstates in a 2-state system. When the eigenbasis is larger, the autocorrelation of any such initial state takes much longer to return to the original state. In a system whose adiabatic states are constantly changing due to the nuclear vibrations, an adiabatic state is no longer an eigenstate of the electronic Hamiltonian once the nuclei have shifted slightly. As a result, we see decay of the autocorrelation, which within any reasonable time, will likely not return to 1.

As described previously, the clustering of ABO bands tends to indicate whether or not an ABO state will mix with states with similar energies in a diabatic time evolution regime. We have confirmed that such clustering—and the avoided crossings that appear and disappear in the time domain—continues to appear with larger choices of supercells (*SI Appendix*, Fig. S3). At

Our representative simulation choosing an initial condition from an ABO state within one of these clusters similarly found that by *SI Appendix*, Fig. S4). Our simulations, otherwise restricted by computational memory limits, demonstrate consistency of our conclusions regarding diabatic time evolution at various size scales of the graphene supercell.

### Many-Body DBO Approximation.

We have said that the electrons in our simulations are lazy in some sense. However, they still evolve in time. Since the nuclear motion is taken to be classical here, with no back-reaction, we are also blind to electron–phonon inelastic scattering. However, we are not under the illusion that any simple guess for many-body wave functions can be essentially exact. What we strive for is something far better than the ABO Ansatz. We believe the numerical results strongly suggest such a guess, which we present now.

The present calculation took the s- and p-orbital backbone to give rise to a set of harmonic interactions, and the

The division into coexisting adiabatic and diabatic worlds makes a great deal of sense when one considers relative timescales: the ability of a closely bound electron to follow a lone nucleus is given by a short timescale roughly on the order of

How do we encode a diabatic nature into an appropriate many-body wave function? Starting with the Bloch wave function,**10** extends Bloch Theorem to situations where it does not strictly apply. We call this the diabatic Bloch wave, or DBW, approach. It is a specification of diabatic, approximate many-body eigenfunctions, introduced here as far superior to traditional ABO states.

The DBW Ansatz is not yet a roadmap for a first principles many-body approach, because we are addressing only a single body electronic wave function, together with a many-body phonon basis. Here, only the Bloch orbitals are treated diabatically.

The DBW wave functions form a complete basis with which to expand more accurate eigenfunctions. Their simple analytical form makes them easy to work with. The real utility of this basis will become clear if it (and its extensions) perform well for spectroscopic matrix elements, bilayer systems, description of excitons, and beyond. These applications of the DBW remain for future work.

## Summary

In this paper, we examined the time evolution of electronic states in graphene in the presence of classically modeled nuclear vibrations. Parameterizing the tight-binding Hamiltonian as a function of classically time-varying interatomic bond distances, we find that the electronic states generally tend to preserve their character over many typical vibrational periods in the presence of vibrations from many modes. This is diabatic, not adiabatic, behavior. The electrons may be described as “lazy,” in that they respond weakly and slowly compared to nuclear geometry changes. The ABO approximation predicts that near these avoided crossings, the electronic state would exhibit rapid changes in character which are not seen in the wave function calculated from the TDSE.

This is in stark contrast to many small molecules, but other molecules are sometimes best described by diabatic rather than adiabatic approximations; there is a substantial literature regarding this (see, for example, ref. 22 and references therein). By solving the TDSE, the true electronic wave function was projected onto the ABO basis. We find rapid decay of the adiabatic states, on the order of one vibrational period, and a few “revivals” over a short period. Not much insight is gained in the adiabatic picture; it is likely to become even less informative as the size of the supercell is taken to be larger. Within the ABO framework, we see that the inclusion of lattice vibrations leads to the appearance of numerous avoided crossings in the supercell band structure.

We have shown that, within one vibrational period, the ABO approximation breaks down because the actual time evolution of the electronic state is highly diabatic. A diabatic basis formed from a superposition of ABO states may provide a much better starting point spectroscopy, electron–phonon interactions and scattering, excitons, and more.

## Acknowledgments

This work was supported by NSF Grant CHE-1800101 and by NSF Center for Integrated Quantum Materials Grant DMR-1231319 (to E.J.H.). V.M. was supported in part by a Harvard University Herchel Smith Summer Research Fellowship, a Harvard University Program for Research in Science and Engineering Student Fellowship, and a Harvard University Program for Research in Science and Engineering Program Assistant Fellowship. We thank Donghwan Kim for helpful discussions.

## Footnotes

- ↵
^{1}To whom correspondence may be addressed. Email: mohanty{at}alumni.harvard.edu or eheller{at}fas.harvard.edu.

Author contributions: V.M. and E.J.H. designed research, performed research, contributed new analytical tools, analyzed data, and wrote the paper.

Reviewers: T.V.-V., Massachusetts Institute of Technology; and P.G.W., Rice University.

The authors declare no conflict of interest.

Data deposition: Code has been deposited in GitHub, https://github.com/mohanty-heller/lazy_electrons_in_graphene.

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1908624116/-/DCSupplemental.

Published under the PNAS license.

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