Electric Field Effects on Curved Graphene Quantum Dots

Citation: de-la-Huerta-Sainz, S.; Ballesteros, A.; Cordero, N.A. Electric Field Effects on Curved Graphene Quantum Dots. Micromachines 2023, 14, 2035. https://doi.org/10.3390/ mi14112035 Academic Editor: Aiqun Liu Received: 1 September 2023 Revised: 28 October 2023 Accepted: 29 October 2023 Published: 31 October 2023 Copyright: © 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/). micromachines Article Electric Field Effects on Curved Graphene Quantum Dots Sergio de-la-Huerta-Sainz 1, Angel Ballesteros 1and Nicolás A. Cordero 1,2,3,* 1Physics Department, Universidad de Burgos, 09001 Burgos, Spain; [email protected] (S.d.-l.-H.-S.); [email protected] (A.B.) 2International Research Center in Critical Raw Materials for Advanced Industrial Technologies (ICCRAM), Unversidad de Burgos, 09001 Burgos, Spain 3Institute Carlos I for Theoretical and Computational Physics (IC1), 18016 Granada, Spain *Correspondence: ncorder[email protected] Abstract: The recent and continuous research on graphene-based systems has opened their usage to a wide range of applications due to their exotic properties. In this paper, we have studied the effects of an electric field on curved graphene nanoflakes, employing the Density Functional Theory. Both mechanical and electronic analyses of the system have been made through its curvature energy, dipolar moment, and quantum regeneration times, with the intensity and direction of a perpendicular electric field and flake curvature as parameters. A stabilisation of non-planar geometries has been observed, as well as opposite behaviours for both classical and revival times with respect to the direction of the external field. Our results show that it is possible to modify regeneration times using curvature and electric fields at the same time. This fine control in regeneration times could allow for the study of new phenomena on graphene. Keywords: graphene; nanoflake; electric field; quantum revival; DFT 1. Introduction There is no doubt that there has been a long-lasting wave of advancements and innovation revolving around graphene, with strange phenomena and potential applications appearing on a yearly basis. Almost every field seems to have an application where the extraordinary properties of graphene (either pristine or oxidised) can shine, from engineering [1–4] to catalysis [ 5 – 8 ], medicine [ 9 – 13 ], sensing [ 14 – 16 ], optics/electromagnetics [ 17 – 20 ], hydraulics [21,22], and energy management [23–25], to cite a few examples. In spite of the mechanical properties of graphene being quite remarkable, the electronic ones are by far the main factor responsible for its attractiveness as a research topic and the basis for several groundbreaking applications. The peculiar band structure of graphene, calculated by Wallace many decades ago [ 26 ] by an approximated yet still useful method, presents the electrons as massless Dirac quasiparticles [ 27 ], allowing them to reach high speeds [28] that result in its incredible electrical conductivity [ 29 , 30 ]. This, alongside the ballistic transport [ 31 , 32 ] observed on non-perturbed systems, makes graphene a great replacement for common conductive materials. Also, due to its near-relativistic behaviour, graphene is often used as an experimental analogue in order to replicate otherwise nearly impossible to measure phenomena, like the Klein paradox [ 33 – 35 ], Zitterbewegung [ 36 – 39 ], or the properties of spacetime with a negative curvature [40–43]. Perhaps one of its most interesting applications derives from the interaction between graphene and an external electric field. As the band structure is so important for its electronic properties, it is quickly deduced than they can be modified and tuned easily by adjusting the field, allowing feats such as moving the Fermi energy between bands and changing the type and density of charge carriers—the ambipolar electric field effect [44,45]—or widening the gap between valence and conduction bands to obtain an insulating behaviour, if needed. Taking also into account the high mobility of electrons in this material and its Micromachines 2023,14, 2035. https://doi.org/10.3390/mi14112035 https://www.mdpi.com/journal/micromachines Micromachines 2023,14, 2035 2 of 17 nanometric scale, graphene is a very promising alternative material for fast and incredibly small electronic components, such as field-effect transistors [ 46 ]; the ballistic transport would ensure a quick response and reduced heating, and the robustness and transparency of the material could extend its use to flexible devices and screens [47–49]. A known effect of an electric field on graphene is the appearance of the curvature. When a transversal field is applied, the material bends alongside the direction of the field, as has been both predicted and observed recently [ 50 – 53 ]. While curved graphene is not a strange view, as it naturally appears in the material itself even when suspended and unperturbed [29,54–58] , the effects of the deformation on the electronic behaviour are notable. Intrinsic ripples are known to disturb the otherwise free electron paths [ 27 ], and local deformations, known as nanobubbles, can be easily produced in the material, allowing for a localised tuning of electric conductivity and more exotic phenomena, like pseudo-magnetic field generation [ 59 , 60 ]. It has been recently shown that double quantum dots can be created by these pseudo-magnetic fields in nanobubbles and that their quantum states can be manipulated to create a controllable qubit [61]. Thus, our aim in this paper about graphene nanoflakes is to study the effects of an electric field on a spherically deformed system, analysing the electronic spectrum via temporal evolution through the quantum regeneration phenomenon—the partial or total recovery of the initial state of a wavepacket after a certain amount of time, known as the regeneration or revival time. An atomistic method, like the Density Functional Theory (DFT), will serve as the main tool, as it has previously for nanoflakes with spherical geometries [62] , as well as those with cylindrical and hyperbolical geometries [ 63 ], in the absence of an external field. Some of those results will be used here for comparative purposes. 2. Materials and Methods As this work is a continuation of our previous research about quantum revivals in graphene, the methodology, as well as the system, will remain the same: a hexagonal graphene nanoflake of 10 benzenic rings per edge, with zig-zag edges and hydrogen passivated; the shape will be achieved through the application of the spherical surface equation, Equation (1), for several values of Rbetween 30 Å and 1000 Å: z=qR2−x2−y2. (1) This flake will be subjected to a structural optimisation calculated within the Local Density Approximation (LDA) using the Gaussian 16 package [ 64 ] until forces on all atoms are below 0.0045 Hartree/Bohr and displacements below 0.0018 Bohr. While other common functionals, such as B3LYP or Generalised Gradient Approximations (GGAs), can be used for smaller systems, LDA results for graphene systems are of a similar or higher quality [65,66] , they require lower computational work, and this functional has been successfully used to study carbon nanostructures [ 67 – 73 ]. Meta-GGAs provide better accuracy than LDA but the computing time is much higher. Taking into account that some of our calculations needed 1 year of CPU time at the LDA level, using those functionals was out of the question. Therefore, we followed the advice in a recent benchmark paper on ab initio descriptions of carbon nanomaterials using 15 different functionals [ 74 ] and used LDA. The basis set will remain 6-31G** (with p and d functions as polarisation aids) [ 75 ], as it has provided good quality results on all calculations made so far. Highly spherical nanobubbles can be achieved with external electric fields [ 51 ], so a perfect spherical deformation could be enough for this study; however, for the sake of experimental realism and fair comparisons, two different sets of geometric restrictions will be applied to the nanoflake (Figure 1): (i) maximal restriction, forcing all carbon atoms to lie on the spherical surface and (ii) minimal restriction, forcing only the 12 carbon atoms on the vertices to remain on the spherical surface. The first case may seem purely academic, but it is a first approximation to a possible experimental setup. One method to stabilise graphene for electronic applications is using hexagonal boron nitride (h-BN) as a Micromachines 2023,14, 2035 3 of 17 substrate [ 76 – 78 ]. A spherical boron nitride fullerene [ 79 – 81 ] could be used to support a spherical graphene nanoflake. (a) Fixed surface (b) Fixed vertices Figure 1. Top (upper row) and side (lower row) views of the two boundary conditions for a spherical nanoflake with initial R= 40 Å . Clear differences between both optimised structures can be seen, especially on the edges. The atoms fixed in the second case are highlighted in blue. Images generated with Gaussview 6 [82]. Regarding the electric field, considering the original flat nanoflake was constructed in the xy plane, the fields applied will be in the direction of the zaxis—and thus perpendicular to the unperturbed flake plane—with three intensities (0.0050 a.u., 0.0100 a.u., and 0.0250 a.u.) in two opposite directions (one in the same direction as the flake deformation and the other in the opposite direction) each, labelled as ± 50, ± 100, and ± 250, with the positive sign corresponding to the field in the same direction as the flake deformation. These fields could seem to have an astonishing intensity, considering that a field of 1 atomic unit equals 5 ×1011 Vm−1 , but at the nanometric scale such fields can be achieved by setting the electrodes close enough. 3. Results The study of the effect of an electric field has been carried out from both mechanical and electronic points of view, the first through the curvature energy and the latter by calculating the temporal evolution of a wavepacket and checking if quantum regeneration occurs. To this end, spherical surfaces of various curvature radii have been used to deform the flake, performing an optimisation on each combination of radius, restriction set, and electric field. 3.1. Curvature Energy As was the case in our previous works, the mechanical study will use the total energy of each nanoflake, taking the flat configuration as the origin of energies, i.e., E=Ecurved −Eflat , (2) where Ecurved is the energy of the curved flake and Eflat is the energy of the planar configuration (both energies calculated with the same electric field). Figure 2shows the curvature energy plotted against 1 /R2 —which is the Gaussian curvature of the flake for the maximal restriction case—for different field strengths. In order to make figures easier to understand, we have used point markers for results corre- Micromachines 2023,14, 2035 4 of 17 sponding to maximal restriction and dashed lines for results corresponding to minimal restriction. In this latter case, the lines are merely guides for the eye connecting the results corresponding to consecutive values of 1 /R2 (we have used the same set of values as for the maximal restriction case). While all curves show that, as expected, flakes with higher curvatures have higher energies, the trends vary depending of the relative direction of the field; for zero fields, the plot has a nearly parabolic shape, with the flat case being the lowest energy geometry; for positive fields, the plots are also parabolic, but the minimal energy is reached for a non-planar structure, which showcases the curvature induced on the material as a transversal field is applied; lastly, negative fields result in an initial rise in energy at lower curvatures, as the field is trying to induce a deformation opposite to the pre-existing one, an effect countered by bending as curvature grows. Figure 2. Curvature energy vs. 1 /R2 plots for different values of field intensity and direction—with positive fields in red tones and negative ones in blue tones—with markers and dotted lines for total and minimal restriction, respectively. Stronger fields lead to analogous behaviours, but the effects are bigger as the field increases. The optimal structure is more curved and less energetic for positive fields, while the initial energy increase appears earlier and is greater for negative fields. However, a strong enough negative field can override the curvature effect and produce a maximum in energy. 3.2. Electric Dipole Moment Due to its close interaction with external fields, the behaviour of the electric dipole moment has also been studied. From the flake shape and symmetry, a null moment would be expected for the flat case and a non-zero moment on the negative zaxis for the spherical ones; this moment would be intensified with positive fields, as the charge distribution that originated it becomes more pronounced and decreases or even flips with negative fields. In Figure 3, we have plotted the zcomponent of the dipole moment against 1 /R2 for each of the seven field values. As expected, in general, the dipole along the zdirection grows in absolute magnitude with an increasing curvature, since a bigger curvature allows for a greater charge displacement. In the absence of an external electric field (green points), the planar structure (1 /R2= 0) has no global dipole moment due to the symmetry of the electron cloud with respect to the graphene plane. Micromachines 2023,14, 2035 5 of 17 Figure 4shows the dependence of one particular case with the field, showing an almost perfectly linear trend, as expected. Without an electric field, the dipole moment is not exactly zero because curvature causes the centre of charge of the electron cloud (negative) to not be at the exact same position as the centre of charge of the atomic nuclei (positive). The first is slightly below the second and this fact creates a very small downwards dipole. With a positive electric field, i.e., the field pointing upwards, atomic nuclei tend to move up (and with them the centre of positive charge) while the electronic cloud tends to move down (and with it the centre of negative charge). This translates into a larger negative dipole moment that increases with the strength of the field. The opposite happens when the applied electric field is negative, i.e., when the field points downwards, atomic nuclei tend to move down, while the electronic cloud tends to move up. This results in a larger positive dipole moment that increases with the field strength. Figure 3. zcomponent value of the electric dipole moment vector against 1 /R2 for all seven field cases. Figure 4. zcomponent of the electric dipole against external field, for a nanoflake with R = 150 Å. Because of the relatively small variation within the same field, only a single case is displayed here, for clarity. Micromachines 2023,14, 2035 6 of 17 It is interesting to see the behaviour for negative fields in Figure 3, where in this case the dipole component initially decreases before growing. For positive fields, the charge shift induced by the field and the curvature work together to achieve a greater downwards dipole; for negative fields, they oppose, and as charges can move longer distances in these conditions—the centre and edges are farther away—the curvature achieves a totally opposite effect. The direction of the dipolar moment suggests a higher electron density on the sphere inner region, something already studied on larger curved carbon nanostructures [ 30 , 83 ], in which the curvature reduces the effective distance between atoms and provokes a rehybridisation on them, with a higher overlap and a consequent energy decrease. While there are works studying the connection between the local charge distribution and graphene rippling [ 84 ], these are oriented to a more continuous and extended system; alas, it is not justified to assume a direct connection between our small, static system and a much bigger and more dynamic phenomenon like graphene corrugation. 3.3. Regeneration Times Quantum regeneration phenomena will be studied using the electronic spectrum of each nanoflake as a starting point. For that purpose, we define a wavepacket as a linear combination of eigenfunctions unwith varying coefficients cn: |Ψ(0)i= ∞ ∑ n=0 cn|uni. (3) We can calculate the state of the packet after an arbitrary time t in order to study its time evolution: |Ψ(t)i= ∞ ∑ n=0 cn|unie−i ¯hEnt, (4) with En being the eigenvalue corresponding to the |uni eigenstate. The weights of each eigenfunction will follow a Gaussian distribution defined by level numbers—positive integers for unoccupied states and negative for occupied ones—centred around the LUMO+4 ( n0 = 5) and a width σ of 0.7, ensuring a wavepacket narrow enough to observe quantum regeneration : cn=1 σ√πe−(n−n0)2 2σ2. (5) Using level numbers allows us to use an analytical approximation for the regeneration times, as described in [ 85 ]. This approximation begins with a Taylor expansion of the spectrum E(n)around the central level n0: En=En0+E0 n0(n−n0) + 1 2!E00 n0(n−n0)2+1 3!E000 n0(n−n0)3+. . . . (6) By combining Equations (4) and (6), we reach an expanded evolved state: |Ψ(t)i= ∞ ∑ n=0 cn|unie−i ¯h(En0+E0 n0(n−n0)+ 1 2! E00 n0(n−n0)2+1 3! E000 n0(n−n0)3+...)t, (7) in which each term on the exponential function constitutes a temporal scale. This expression allows for the definition of several regeneration times, directly connected to the spectrum derivatives: TZb =π¯h |En0|, (8) TCl =2π¯h |E0 n0|, (9) Micromachines 2023,14, 2035 7 of 17 TRe =2π¯h |E00 n0|/2! , and (10) TSup =2π¯h |E000 n0|/3! . (11) The main times of interest here will be TCl , the classical time, and TRe , the revival time, as they will be easier to observe and rationalise. Being dependent on the first and second derivatives, respectively, the classical time gives information about the absolute value of the energy differences and thus the compression of the spectrum, while the revival time informs about the similarity between energy differences and thus about the regularity of the spectrum. TZb is related to Zitterbewegung, a relativistic phenomenon only observed on particular wavepackets (cat states), different from ours. Here, it is just a mere phase factor. In contrast, TSup , the super-revival time, would correspond to a higher scale oscillation on the temporal evolution, which is difficult to detect due to the decreasing value of higher-order derivatives and as such will not be considered either. The interpolating function selected to calculate the derivatives appearing in these times is a second-degree polynomial, obtained by a least squares fit to the three central levels of the wavepacket. By using three points—the level numbers and their energies—an exact expression for the polynomial can be obtained: E(n) = b−a 2(n2−n2 0) + 1 2(a+b+2an0−2bn0)n+En0, (12) with n0 being the central orbital number, En0 its energy, and a and b the absolute energy differences with the lower ( n0− 1) and upper ( n0+ 1) levels, respectively. Both first and second derivatives can be analytically computed easily: E0(n) = (b−a)n+1 2(a+b+2an0−2bn0)(13) and E00(n) = b−a. (14) These expressions will be useful in later sections. As for the actual temporal evolution, it can be monitored through the autocorrelation function, defined as the overlap of initial (t=0) and later (t>0) states: A(t) = hΨ(0)|Ψ(t)i= ∞ ∑ m=0 ∞ ∑ n=0 c∗ mcnhum|unie−i ¯hEnt(15) Working with orthonormal states in the wavepacket makes it possible to substitute Kronecker’s delta functions for the overlap integrals and cancel all cases where n6=m: A(t) = ∞ ∑ m=0 ∞ ∑ n=0 c∗ mcne−i ¯hEntδmn = ∞ ∑ n=0|cn|2e−i ¯hEnt. (16) The physically sound quantity is the squared modulus of the autocorrelation function: |A(t)|2= ∞ ∑ m=0 ∞ ∑ n=0|cm|2|cn|2e−i ¯h(En−Em)t. (17) Taking into account the definition of the cosine function, we can rewrite this expression as follows: |A(t)|2= ∞ ∑ n=0|cn|4+ ∞ ∑ n=0 ∞ ∑ m<n 2|cm|2|cn|2cos(ωmnt), (18) Micromachines 2023,14, 2035 8 of 17 where ωmn = (Em−En)/¯h . This formula relates the temporal evolution to the energy differences between different levels on the packet, rather than the energies themselves. As a sum of cosines, a complete revival is guaranteed (if the spectrum is commensurable) after a long enough time, when the least common multiple of all individual periods is reached. We can focus on our particular wavepacket to further simplify this expression. While is it true that the wavepacket comprises five levels, the coefficient distribution and their squares on the general formula makes the three central ones (4, 5, and 6) the main contributors, so the general expression can be simplified. We can ignore any terms involving other levels. We can go even further and remove also the ω46 term, as the distribution is narrow enough to make its contribution barely noticeable, resulting in the following expression: |A(t)|2≈2|c4|2|c5|2cosa ¯ht+2|c5|2|c6|2cosb ¯ht, (19) where we have removed the first sumatory and proportionality constants, as the periodicity of the function remains the same, and a and b are, like before, the energy differences between level 5 and levels 4 and 6, respectively. Being a sum of cosines, and considering c4 and c6 are identical due to the distribution’s symmetry, it can be easily converted to a product of cosines: |A(t)|2≈4|c4|2|c5|2cosa+b 2¯htcos a−b 2¯ht. (20) The two new arguments found here are interesting: the first one is the average of both main energy differences, and the second gives information about how similar they are, which makes them perfect candidates for the origin of both classical and revival times, respectively . Finally, Figure 5shows a typical view of |A(t)|2 plotted against time, with a clear pattern of a high-frequency oscillation modulated by a low-frequency one, whose periods correspond to the values of TCl and TRe , respectively; the former is located at the first maximum of the time evolution and the latter at the first maximum of the enveloping curve. This corresponds to the shape or a general sum (or product) of cosines, corroborating our previous mathematical explanation. Figure 5. Temporal evolution plot for a fixed surface nanoflake, with R= 150 Å in a −50 field, as an exemplary case. Dotted lines represent the analytical values calculated for both TCl and TRe. Micromachines 2023,14, 2035 9 of 17 In general, the observation of both times is an easy task, but it is not guaranteed: as both times approximate in value, the interference between them increases, resulting in a larger shift of the first maximum, quickly deviating from the TCl analytical value, making the visualisation of TRe harder, if not impossible. Summarising, there are two different ways to calculate regeneration times. The first one is looking for the maxima in the modulus of the autocorrelation function by numerically searching for the first maximum of the function (for TCl ) or the first maximum of its enveloping curve (for TRe ). The second one is using the analytical expressions given in Equations (9) and (10). 3.3.1. Classical Time Numerical and analytical values of classical time are plotted against 1 /R2 in Figure 6 for the fixed surface case and in Figure 7for the fixed vertices case. Taking the fixed surface as an ideal situation, we can see that, without any field applied, both numerical and analytical values of TCl grow with 1 /R2 for low curvatures; this trend does not last, as a heavy deviation between them arises as we progress to more curved flakes, resulting in an overall monotonic growth for analytical TCl and a seemingly parabolic plot for numerical TCl . The discrepancy between the two values is attributed to the interference from TRe . Analytical TCl , being inversely related to the first derivative, shows a compression of the spectrum in the levels studied. The small distortion on the highest positive field is due, as will become clear later, to an important shift on the levels of the packet. This base trend is intensified with positive fields, with the discrepancy happening at lower curvatures and lower values of TCl with stronger fields, reaching a limit value at a certain 1 /R2 . For negative fields, both values start with a decrease—with a good agreement between them—and seem to stabilise with an increasing curvature. These limit values barely change with the field intensity but do with its direction. The fixed vertices case is similar, though there are sudden changes on the plot due to the shift to a more optimal structure as the curvature grows. However, the main effects are still observable, so previous analysis and conclusions still apply. Figure 6. Classical time against 1 /R2 for all fixed surface calculations, with markers for numerical values and dotted lines for analytical ones. Micromachines 2023,14, 2035 16 of 17 27. Katsnelson, M.I.; Geim, A.K. Electron scattering on microscopic corrugations in graphene. Philos. Trans. R. Soc. A Math. Phys. Eng. Sci. 2008,366, 195–204. [CrossRef] 28. Bolotin, K.; Sikes, K.; Jiang, Z.; Klima, M.; Fudenberg, G.; Hone, J.; Kim, P.; Stormer, H. Ultrahigh electron mobility in suspended graphene. 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