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Computational design of in vivo biomarkers

This content has been downloaded from IOPscience. Please scroll down to see the full text. 2014 J. Phys.: Condens. Matter 26 143202 (http://iopscience.iop.org/0953-8984/26/14/143202) View the table of contents for this issue, or go to the journal homepage for more

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 143202 (17pp)

doi:10.1088/0953-8984/26/14/143202

Topical Review

Computational design of in vivo biomarkers ´ Balint Somogyi1 and Adam Gali1,2 1

Department of Atomic Physics, Budapest University of Technology and Economics, Budafoki u´ t 8., H-1111, Budapest, Hungary 2 Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, PO Box 49, H-1525 Budapest, Hungary Received 15 January 2014 Accepted for publication 3 February 2014 Published 20 March 2014 Abstract

Fluorescent semiconductor nanocrystals (or quantum dots) are very promising agents for bioimaging applications because their optical properties are superior compared to those of conventional organic dyes. However, not all the properties of these quantum dots suit the stringent criteria of in vivo applications, i.e. their employment in living organisms that might be of importance in therapy and medicine. In our review, we first summarize the properties of an ‘ideal’ biomarker needed for in vivo applications. Despite recent efforts, no such hand-made fluorescent quantum dot exists that may be considered as ‘ideal’ in this respect. We propose that ab initio atomistic simulations with predictive power can be used to design ‘ideal’ in vivo fluorescent semiconductor nanoparticles. We briefly review such ab initio methods that can be applied to calculate the electronic and optical properties of very small nanocrystals, with extra emphasis on density functional theory (DFT) and time-dependent DFT which are the most suitable approaches for the description of these systems. Finally, we present our recent results on this topic where we investigated the applicability of nanodiamonds and silicon carbide nanocrystals for in vivo bioimaging. Keywords: ab initio, density functional theory, time-dependent density functional theory, nanocrystal, point defects (Some figures may appear in colour only in the online journal)

their relatively long fluorescent lifetimes allow the distinction between the signal and the autofluorescence of the background [1]. Up to date, fluorescent QDs comprising a large variety of semiconductor compounds were successfully synthesized. The most notable compound materials are CdS, CdSe, CdTe, CdHgTe, InP, InAs, ZnS, ZnSe or PbSe [4, 5]. Non-compound group-IV semiconductor nanoparticles such as diamond, Si, SiC and Ge QDs were also successfully manufactured, but their applicability as fluorescent biomarkers were not investigated as thoroughly until recent years [6]. The biosensing applications of QDs can be divided into two separate families: (i) in vitro applications where the experiments are carried out under laboratory conditions, and (ii) in vivo applications where the QDs are introduced into a living organism. Labeling cells, intracellular molecules, nuclear antigens, microtubes and actin filaments with QDs can be considered as routine under in vitro conditions.

1 Introduction 1.1 Fluorescent quantum dots and in vivo applications

Fluorescent imaging is an extremely powerful tool for a wide range of biosensing applications. Small, molecule-sized fluorescent probes can be used for real-time imaging of the targeted molecules, proteins and cells. Compared to the traditional fluorescent contrast agents (organic dyes and fluorescent proteins), semiconductor nanocrystals or quantum dots (QDs) possess far superior optical properties [1–3], thus they became the standard in fluorescent imaging. QDs have a broad absorption and a narrow emission band, and their photostability is outstanding compared to the bleachingprone organic fluorophores [1]. The emission wavelengths of the QDs depend on their diameter allowing size-tunable emission from ultraviolet to the near-infrared region, and 0953-8984/14/143202+17$33.00

1

© 2014 IOP Publishing Ltd Printed in the UK

J. Phys.: Condens. Matter 26 (2014) 143202

Topical Review

silicon-based CCDs are not sensitive in this range. Nowadays CCD sensors made of lower band-gap materials such as InGaAs and HgCdTe are becoming more and more affordable at resolutions suitable for bioimaging. Another QD related phenomenon is often called ‘blinking’ [23–25] due to their intermittent fluorescent intensity. The probability distributions of the bright, fluorescent ‘on’ and the dark, non-fluorescent ‘off’ times follow a power law t −γ where γ ∼ 1.5 [26]. The physical mechanism behind the blinking process has not yet been fully understood. Most of the studies explain the suppression of the radiative process with the non-radiative Auger recombination of the exciton, which is either caused by the light-induced charging of the QDs, or by trap states on the surface of the QDs [27–29]. Blinking is usually considered as a major disadvantage for bioimaging applications. We note briefly that there are special imaging approaches which benefit from blinking [30, 31], where resolution beyond the diffraction limit can be achieved. Nevertheless, the intermittent fluorescence reduces the effective quantum efficiency (and thus the fluorescent intensity) of the probe, and it can prevent the successful optical tracking of single QDs. Great efforts are made to suppress the blinking of QDs [32], usually by the elimination of the surface traps either by flushing the QDs in special antiblinking reagents [33] or by the growth of core/shell QDs with very thick shell structures (over 20 monolayers of overcoating shell material) which results in the undesired radial increase of over 5 nm [34, 35]. Another proposed approach is the band-gap engineering of QDs where a smooth confining potential is utilized to suppress the Auger recombination rate (in contrast to the conventional core/shell structures), which could be obtained from a graded alloy structure [32, 36]. Despite the recent advances in the reduction of the fluorescent intermittency, the blinking of QDs is still one of the big challenges in efficient in vivo applications. For commercial usage, the cost of production is also an important factor. The solution-phase synthesis method is the usual fabrication procedure for compound semiconductor QDs, which offers a relatively cheap way to produce QDs in large quantities. It is worth mentioning that a novel and promising alternative to NIR QDs exists for biological imaging. Noble metal nanoclusters (most notably Ag and Au) typically have a diameter of less than 2 nm, and have bright, stable visible–NIR fluorescence. Noble metal nanoparticles are generally considered as non-toxic, for example colloidal gold has been successfully used to treat rheumatoid arthritis [37]. Gold nanoparticles possess a size-independent plasmon absorption band if their diameter is larger than 2 nm. Below that threshold they behave as molecules, and exhibit sizedependent fluorescent wavelength [38]. Fluorescent noble metal nanoclusters have been employed for various biological applications, including bioimaging [39, 40]. In [41], a successful application of gold nanoclusters was demonstrated for tumor fluorescence imaging. Noble metal nanoclusters posses various favorable properties but they also suffer from fluorescence intermittency [42, 43] and have lower quantum efficiency than that of semiconductor QDs [42].

However, in vivo fluorescent imaging offers a cost-effective alternative to more conventional imaging methods such as magnetic resonance imaging, photo emission tomography, and x-ray computer tomography. QDs were successfully applied under in vivo conditions for cell tracking, vasculature imaging, and most notably tumor imaging [7]. Generally QDs for use in in vivo applications have to meet strict requirements, thus their research poses a great challenge. The most important issue is toxicity. Biologically inert QDs without any undesirable short- or long-term side effects are needed for in vivo applications. The toxicity assessment is probably the most studied problem in the research area related to the biological applications of QDs but due to the ambiguous results of these experiments, no consensus has yet been reached on the toxicity of different semiconductor QDs [7]. According to the studies, the material, size, surface treatment and environment are all important factors in the toxicity of these QDs [7–9]. Although this issue is not fully understood yet, the chemical degradation of the QDs leading to the release of heavy metal ions is considered as the most important source of toxic effects [7]. The full elimination of toxicity has not yet been achieved despite many attempts to reduce the chemical degradation of heavy metal-containing QDs such as the ZnS shell coating method [10]. Another very important requirement for in vivo applications is the colloidal stability of the QDs in an aqueous environment. QDs made of most of the aforementioned semiconductor compounds are hydrophobic and tend to aggregate in water, which has to be prevented by appropriate treatment of their surface, i.e. by adding long carbon chains [11]. The size of the QDs is also a very important parameter: recent studies showed that the hydrodynamical diameter of the nanoparticles should be smaller than 5.5 nm for efficient clearance from the body through urine and bile [12–14]. It is worth mentioning here that the coating of QDs to eliminate chemical degradation or to make water soluble QDs enlarges their hydrodynamical size. The fluorescent wavelength of the QDs determines the efficiency of signal detection. For in vivo applications, emission in the near-infrared (NIR) window is highly desirable to allow the imaging of deeper regions of the body, since in this optical window (700–1350 nm) the absorption, autofluorescence and Rayleigh scattering of the body is minimal [15, 16]. The diameter and the emission wavelengths of the QDs are coupled to each other through the wellknown quantum-confinement effect: the smaller the size of a semiconductor nanoparticle the higher its emission energy is [17]. Most of the appropriately small-sized compound semiconductors are not able to emit in the near-infrared region [4, 5], with the exception of QDs made of InP [18], PbSeb [19, 20], PbS [20, 21] and InAs [22]. Although their optical properties are very promising for in vivo applications, their toxic side-effects still pose a major issue. It is worth noting that the application of QDs with fluorescent wavelengths longer than 1000 nm has been stemmed by the lack of sensitive and high-resolution chargecoupled device (CCD) cameras, because the conventional 2


Topical Review

and the system has to be calculated at quantum mechanical level. In most of the cases the relativistic effects are negligible or small meaning that the Schrödinger equation should be solved for the many-electron system. The spin–orbit coupling may have to be taken into account in the case of large atoms. In most of the cases the Born–Oppenheimer approximation can be applied, where it is assumed that the wavefunctions of the molecule can be decoupled as the product of an electron and nuclei wavefunctions, where the nuclear coordinates are only parameters for the electron wavefunction. Considering the high dimensionality of the electron wavefunction, the Schrödinger equation is numerically not solvable for more than a couple of electrons, thus some approximation is needed.

1.2 Properties of semiconductor QDs ‘ideal’ for in vivo fluorescent biomarkers

We summarize the most important necessary properties of QDs to be employed in in vivo bioimaging applications: (i) they should be non-toxic and biologically inert, (ii) their hydrodynamical diameter should be less than 5.5 nm, (iii) they should have stable fluorescent emission in the nearinfrared range, (iv) they should be produced cost-efficiently in large quantities. Intense on-going experimental research aims to find appropriate semiconductor QDs for in vivo bioimaging; however, no such hand-made solution has yet been found that meets all the criteria as listed above. Here we propose that modern ab initio methods can significantly contribute to the design of suitable semiconductor QDs.

2.1 Ground state properties

Historically, the first successfully applied method for electron structure calculations was the Hartree–Fock approximation. In the Hartree–Fock approximation, the many-electron wavefunction is written as an antisymmetric product of single electron wavefunctions which is known as the Slater determinant. While this approach provided useful results in the early days of electronic structure calculations, the Hartree–Fock method is not considered as an accurate and reliable method for electronic structure calculations because the Hartree–Fock Hamiltonian does not contain the so-called correlation energy. Several quantum chemical methods were developed based on the Hartree–Fock method, and many of them offer a very accurate and reliable way to perform electronic structure calculations for relatively small systems: many-body perturbation theory (MBPT), configuration interaction (CI) and coupled-cluster (CC) methods offer a systematic way to improve the approximation of the correlation energy, at the expense of rapidly increasing computational resources and time. In the case of the CI method, the wavefunction is written as the linear combination of the ground state and excited state Slater determinants. The accuracy and the computation costs depend on the highest order excitations considered in the wavefunction, but generally CI methods are considered computationally very demanding, and suffer from the size-consistency error [45]. The MBPT is a diagrammatic method for tackling the problem of the correlation energy based on the Rayleigh–Schrödinger perturbation theory. It has to be taken into consideration that higher and higher order diagrams improves the accuracy of the calculation, but also leads to increasing computation time. For example the fourth-order approximation MBPT(4) already scales as n7 where n is the number of the electrons in the system. Probably, the most successful electron-correlation method in quantum chemistry is the CC method, where the ansatz wavefunction can be written in the following way: ˆ ψCC = eT φ0 where φ0 is the ground state Slater determinant ˆ and T is the infinite sum of the first, second, third. . . order ˆ excitation operators: Tˆ = The CC method can i Ti . be considered as a resummation of the CI method, where the size-consistency problem does not occur. The accuracy

1.3 Our strategy to design ideal fluorescent agents

Our approach is to identify nanocrystalline materials with great potential for biological applications, and then to alter their unfavorable physical properties while leaving their suitable attributes unchanged. We have chosen such nanocrystals which meet all of the important criteria of fluorescent imaging except the emission in the NIR window. Then we have investigated the possibility of modifying their optical properties either by some special treatment of their surface or by the introduction of color centers into them. We mention here that our approach resembles the idea of the so-called ‘inverse band-gap problem’ proposed by Franceschetti and Zunger [44] where such atomic configurations are sought-after, which produce the targeted electronic or optical properties. In finding in vivo QDs we are interested in the optical properties of objects with a size of some nanometers. Only ab inito methods are able to provide reliable results because quantum mechanical effects are essential in the molecule-sized objects. In finding the appropriate atomic configurations we rather relied on our intuition or knowledge about point defects in bulk counterparts. In the next sections we review the ab initio methods to calculate the electronic and optical properties of semiconductor nanocrystals to design in vivo biomarkers, then we briefly present recent and new results on this topic, and finally we conclude. 2 Computational methods for searching in vivo semiconductor nanoparticles

The most important purpose of the numeric simulations of nanostructures is to understand the physical phenomena behind the results of our experiments, and to make predictions about the properties of experimentally not-yet-realizable systems. To this end, simulation methods with predictive power are needed which do not rely on any empirical parameters. Since the typical size range of the investigated systems is smaller than a couple of nanometers the quantum mechanical effects can have drastic effects on the essential properties of the nanostructures, 3


Topical Review

mechanical effects, hence it is the most fundamental concept in DFT. The xc-potential can be derived from the xc-energy, which is a functional of the electron density. If the xcfunctional was known then the Kohn–Sham DFT (KS-DFT) would be, in principle, exact. Unfortunately, there are only approximations of the xc-functional, and the accuracy of a KS-DFT calculation is very closely related to the applied xcfunctional. The simplest approximation of the xc-functional is the local density approximation (LDA), where the local xcenergy is approximated with the xc-energy of the homogeneous electron gas. A more advanced xc-functional is the family of generalized gradient approximation (GGA), where not only the electron density but also its gradient is taken into consideration. The LDA is a local functional, the GGAs are semi-local functionals, but the exact xc-functional is, in principle, non-local. To overcome this unphysical restriction, orbital dependent xc functionals were introduced into the framework of the generalized Kohn–Sham approach. In the generalized Kohn–Sham scheme, the Levy–Lieb constrainedsearch formulation of the Hohenberg–Kohn functional, is used which allows the xc-functional to not only depend on the total electron density but also on the Kohn–Sham orbitals. This generalization opened the way to the construction of non-local xc-functionals. Although many different classes of orbital-dependent functionals exist [51], here we only mention the hybrid functional approach, which is probably the most successful non-local approach to approximate the xc-functional in practice. In the case of hybrid functionals, a local or semi-local xc-functional is combined with the socalled exact exchange which is the Hartree–Fock exchange calculated from the Kohn–Sham orbitals. There are multiple ways to parametrize the mixing of the semi-local functional and the exact exchange, but the most fundamental one (based on the adiabatic connection theorem) with only one parameter is the following [52]:

of the calculation depends on where the infinite sum for Tˆ is truncated. For example, in the case of the CCSD(T) method only the single and double excitations are kept in Tˆ , and treat the triple excitations perturbatively. This approximation is already very accurate, and often considered as the gold standard in quantum chemistry but at the expense of n7 scaling. Another approach to the many-electron problem is the family of quantum Monte Carlo (QMC) methods [46] which solves the electronic Schrödinger equation with the help of Monte Carlo techniques. The most established QMC method is the diffusion QMC method (DMC), which utilizes Monte Carlo methods for both solving the Schrödinger equation and evaluating the expectation values. The DMC method determines the ground state wave function by propagating the time-dependent Schrödinger equation in imaginary time. QMC methods are considered to be one of the most accurate tools for studying molecular quantum mechanics. They scale exceptionally well with parallelization, and can be accelerated with graphical processing units (GPUs). The computation time for each Monte Carlo step scales as O(N 3 ), and the number of required QMC steps to reach a required statistical error scales as O(N 3/2 ). There are several techniques to further improve the computational performance of QMC methods, resulting in a much more favorable scaling compared to other high accuracy ab initio methods such as CCSD(T). Despite the aforementioned favorable properties of QMC methods, they are still considered as one of the most expensive tools in electronic structure calculations because the stochastic nature of MC methods adds a massive prefactor to the calculations. Grossman et al [47, 48] were the first researchers to utilize QMC methods in order to study small silicon and carbon clusters. Nowadays, ground state QMC calculations of hundreds of electrons can be considered as routine. QMC calculations of excited states are generally not as straightforward as for the ground state, although vertical excitation energies of small molecules were successfully calculated recently using DMC methods [49, 50]. Density functional theory (DFT) is the most successful approach to study the ground state of many-electron systems. It proved to be a very efficient, reliable and relatively accurate method for studying the electronic and optical properties of a wide range of materials including atoms, molecules, nanostructures, crystalline and disordered solids. With DFT the physical properties of a many-electron system are a functional of the electron density. This is very advantageous in terms of computational complexity because the threedimensional density is a much simpler function than the 3 × n dimensional wave function of an n-electron system. The standard and most practical formalism of DFT is the Kohn–Sham theory, where the problem of many interacting electrons in an external potential is reduced to a much simpler problem of non-interacting electrons in an effective potential. The effective potential is the sum of the external potential, the Hartree potential representing the classical Coulomb repulsion between electrons, and the so-called exchangecorrelation (xc) potential which contains all the quantum

hybrid Exc = EcGGA + bExexact + (1 − b)ExGGA .

(1)

Perdew, Ernzerhof and Burke argued [53] that b = 1/4 is a theoretically well-founded choice which provides good results for the set of molecules when PBE (Perdew–Burke– Ernzerhof) [54] is chosen as the semi-local GGA functional. This functional is referred to as PBE0 [55]. There exists other hybrid functionals too, most notably the B3LYP and B3PW91 [56–59] but these are three parameter hybrids where the parameters are fitted empirically, thus they are often considered as semi-empirical functionals. Although DFT yields excellent results for the energy and electron density of the ground state, it does not provide reliable quasi-particle spectra. In fact, the Kohn–Sham oneelectron orbitals are just mathematical constructs, meaning that the Kohn–Sham eigenvalues do not represent measurable quantities, with the exception of the energy of the highest occupied Kohn–Sham orbital, which is the negative of the ionization energy when the KS-DFT is functional is exact. Although the one-particle energies obtained with KS-DFT do not have a strict physical meaning, they are often interpreted 4


Topical Review

as quasi-particle energies. In particular, in the case of hybrid functionals, the Kohn–Sham band structure for solids and the one-particle energies for finite systems were proved to closely reproduce experimental data [60–63]. Since the xc-functional contains all the quantum mechanic effects the quality of the applied approximation governs the accuracy of the results. Probably, the largest shortcoming of KS-DFT is that no method is known to improve the xcfunctional in a systematic manner. While determinant methods in quantum chemistry (CI, MBPT, CC) offer a relatively simple way to improve the accuracy of the results at the expense of increasing computation cost, finding a better xc functional for DFT is somewhat a trial-and-error process. A possible approach to improve the quality of DFT calculations is the MBPT where the results of the DFT calculations are used as zeroth-order approximation. One can write a diagrammatic expansion of the self-energy in terms of the screened Coulomb interaction W , which leads to a faster convergence compared to that of the conventional MBPT which applies the much stronger bare Coulomb interaction. This scheme was developed by Hedin [64], who derived a set of coupled equations which must be solved self-consistently. This closed set of integral equations incorporates the self-energy, screened Coulomb interaction, the polarization and the vertex function, of which the quantities offer some intuition about the physical ingredients of the manyelectron problem [65]. In practice, the Hedin equations are impossible to solve for systems containing many electrons, so approximations are introduced. In practice, the GW approximation (G and W stands for the Green’s function and for the screened Coulomb interaction) is the most current, where the vertex function is approximated in the simplest possible manner. GW calculations are usually performed using the converged DFT Kohn–Sham orbitals to build up the zerothorder approximation of the Green’s function, and then the GW equations are solved in a fully-, partially- or non-self-consistent manner. The GW method became very popular in the ab initio world because it provides reliable results and can be improved by the application of more sophisticated approximations of the vertex function. On the other hand, GW calculations of systems the size of hundreds of atoms are still very challenging computationally, thus GW methods are not very practical for large systems, especially, when computationally less demanding DFT methods are able to describe the given system properly.

the following: ∂ 1 2 − ∇ + veff [ρ](r , t) ψi (r , t) = i ψi (r , t) 2 ∂t ρ(r , t) =

N

|ψi (r , t)|2 ,

(2)

(3)

i=1

where veff [ρ](r , t) = vH [ρ](r , t) + vxc [ρ](r , t) + vext (r , t) is the effective time-dependent potential, which consists of the time dependentHartree, xc- and external potential. The Hartree potential is d3 r ρ(r , t)|r − r |−1 , the vxc [ρ](r , t) is the functional derivative of the time-dependent xc-functional with respect to the time-dependent density and the external potential is often decomposed as the sum of a static and a time-dependent potential: vext (r , t) = vstat (r ) + vTD (r )f (t). vxc [ρ](r , t), in principle, is non-local in time and can have memory effects, however, this is usually neglected due to the difficulty of creating such xc-functionals. Instead, the socalled adiabatic approximation is applied where the explicit time-dependence of vxc is neglected and the density at time t is plugged into a ground-state functional: vxc [ρ](r , t) = vxc (ρ(r , t)). Although this approximation is very drastic, it gives satisfying results in most of the cases. By propagating the wavefunction in real time according to the time-dependent Kohn–Sham equations, the full spectrum of the investigated system can be calculated. It is important to note that the timedependent Kohn–Sham equation is an initial value problem, thus it depends greatly on the initial ground state orbitals. One can only expect reliable results from TDDFT if the ground state is well described by DFT. If the time-dependent component of the external potential is weak, linear response theory can be applied. In linearresponse TDDFT (LR-TDDFT), one assumes that the change in the time-dependent density is approximately proportional to the time-dependent perturbing potential and the higher order terms can be neglected: ∞ (1) t dr χ (t, t , r , r )vTD (r )f (t ), (4) ρ (r , t) = 0

where χ (t, t , r , r ) is the linear density–density response function which describes the excitation spectra of the system. It can be shown that the density–density response function in the Kohn–Sham formalism can be determined by solving a Dyson-like equation: χ [ρGS ](r , r , ω) = χKS [ρGS ](r , r , ω) + dr1 dr2 χKS [ρGS ](r , r1 , ω) 1 × + fxc [ρGS ](r1 , r2 , ω) × χ [ρGS ](r , r1 , ω), |r1 − r2 | (5) where ρGS is the ground state density and the Kohn–Sham density response function χ (r , r , ω) can be composed of the Kohn–Sham orbitals φk (r ), energies k and occupation numbers fk :

2.2 Excited state

The standard DFT and GW methods give reliable results for the ground state, but are, in principle, not applicable for excited states, thus are not able to predict the optical properties of a given many-electron system. The time-dependent density functional theory (TDDFT) is the time-dependent extension of the DFT. The time-dependent Kohn–Sham equations for non-interacting quasi-particles are

2.2.1 Calculation of the UV–VIS absorption spectra.

0+

→ η

k,j

χ (r , r , ω) = lim

(fk − fj )δσk σj

φk∗ (r )φj (r )φj∗ (r )φk (r ) ω − (j − k ) + iη

,

(6) 5


Topical Review

fxc ((r1 , r2 , ω)) is the TDDFT kernel in real time defined as the functional derivative of the xc-functional: δvxc [ρGS ]((t, r , )) fxc [ρGS ](t, t , r , r ) = . (7) δρ(t , r ) ρ=ρGS

excitation energies in the case of Rydberg-like excitations or the so-called charge-transfer excitations, where the electron– hole distance is relatively large. In these cases, the GGA and hybrid kernels tend to severely underestimate the excitation energies which is well documented in the literature [71–73]. To solve this problem, long-range corrected density functionals were developed, where the exchange part of the xc-potential is separated into a long-range part and a short-range part. This is done by partitioning the Coulomb operator 1/r, usually in the following manner [74]:

The TDDFT kernel in frequency space can be obtained as the Fourier transform of equation (7), and it is frequency independent in the adiabatic approximation. This equation can be solved self-consistently starting from χ (0) = χKS but the Casida method is usually applied in practice, where equation (5) is transformed into a matrixeigenvalue problem, M(ω)u = ω2 u,

1 − erf(ωr) erf(ωr) 1 + , = r r r

short -range long-range

(8)

(10)

where ω is the range-separation parameter and erf(r) is the error function. By introducing this scheme, one can obtain an xc-functional which preserves the GGA or hybrid exchange at short range, while leads asymptotically to the Hartree–Fock exchange, which has the correct −1/r decay. In the case of range-corrected hybrid functionals, the formula for the xcenergy can be written in the following manner:

where the matrix elements of M are defined as √ Mkk (ω) = ωk2 δkk + 4 ωk ωk 1 × d 3 r d 3 r Φ k (r ) ( r , r , ω) Φk (r ) + f xc |r − r | (9) and k, k are double indexes for a transition between an occupied i and an unoccupied a KS orbital, ωk = a − i and Φk (r ) = φi∗ (r )φa (r ). The eigenvalues are the square of the excitation energies, and the oscillator strengths can be calculated from the eigenvectors. The TDDFT method is extremely successful, and it became the most favorable approach for calculating excitedstate properties of molecules. However, even TDDFT methods fail in some cases: since the TDDFT is, in principle, exact, all of the shortcomings of the method originate from the applied approximations. In the weak field regime, the linearresponse approximation is physically sound, meaning that the source of errors is the approximate xc-functional Exc , and the fxc TDDFT kernel. The sources of errors in TDDFT calculations can be separated into two qualitatively different categories [66]: (i) errors originated from the adiabatic approximation, (ii) errors due to the incorrect (time- and frequency-independent)fxc (r , r ). The TDDFT kernel is frequency independent in the adiabatic approximation, and works well for single excitations. It was shown [67] that fxc must have strong frequency dependence when single and double excitations lie close in energy to each other. Thus, TDDFT is restricted to one-electron excitations, although postadiabatic TDDFT methods have been developed very recently for model systems [68, 69]. Another crucial issue with the common density functionals and the derived fxc kernels is that they have incorrect long range behavior. It was shown [70], that the xc-potential should decay as ∼ −1/r far away from finite systems or surfaces. GGA functionals describe shortrange exchange and correlation effects correctly but yield an incorrect ∼−e−r asymptotic behavior. In the case of hybrid functionals, the incorporated Hartree–Fock exchange partially remedies this problem but the ∼ b/r (where b is the Hartree–Fock mixing parameter) asymptotic behavior of hybrid functionals is still not satisfying. The faulty longrange behavior of the common functionals results in poor

LRC SR SR LR = Ec + (1 − b)Ex,GGA + bEx,HF + Ex,HF , Exc

(11)

where Ec is the correlation energy, b is the Hartree–Fock SR SR and Ex,HF are the mixing parameter for short range, Ex,GGA GGA and Hartree–Fock elements of the short-ranged part of LR is the long-range Hartree–Fock the functional, while Ex,HF exchange. The b and ω parameters can be chosen in an ab initio manner, where they are optimized to obey Koopmans theorem [74]. An alternative route is to choose such parameters which yield the best results for a benchmark based on a large set of molecules [75]. Other methods may be applied to tackle the problem of charge-transfer excitations. One of the most successful approaches is the constrained DFT (CDFT) method [76], where constraints are forced on the electron- or spindensity with the help of a Lagrange multiplier in the energy minimization. Among various other fields of applications, CDFT is proved to be a reliable method for the description of charge-transfer excitations [76]. A somewhat similar, but not equivalent, method is called SCF, and also known as DFT or excited-state DFT and it is even referred to as constrained DFT in some cases. In the SCF method, the constraint acts on the occupation numbers of the Kohn–Sham states. By forcing non-Aufbau occupations of the Kohn–Sham states, the charge density and the calculated total energy will belong to the excited state [77]. Levy and Perdew proved [78] that all the stationary densities of the exact functional are physically relevant and correspond to eigenvalues of the true Hamiltonian of the system, whether they belong to the ground state or not. In a recent paper [77], Van Voorhis et al showed that the stationary Levy–Perdew densities and the self-consistent solutions of the Kohn–Sham SCF equations are identical if the SCF solutions give the minimum Kohn–Sham kinetic energy. These theories support the idea of using the SCF method for calculating the excited states. In many cases, the SCF method tends to give a better description of lowlying charge-transfer states and Rydberg excitations than the 6


Topical Review

LR-TDDFT method does, even with semi-local functionals [77, 79, 80]. In the case of short-range excitations, the error originated from the incorrect long-ranged decay of the xcenergy is not significant. This is the reason why GGA and, especially, hybrid functional kernels, provide reliable results in the LR-TDDFT scheme. A different route to describe the excited states of manyelectron systems is the Bethe–Salpeter equation on top of the GW method (GW+BSE) [65]. The Bethe–Salpeter equation is the extension of the independent quasi-particle approximation, in order to include the electron–hole interaction. Formally, the Bethe–Salpeter equation is similar to the Dyson-like equation of TDDFT, but with some important differences between the LR-TDDFT and GW+BSE approaches. The independent quasi-particle approximation of the density–density response function (χ0 ) is constructed from the (interacting) quasiparticle Green’s functions in the GW + BSE approach, while χ0 is built up from the non-interacting Kohn–Sham orbitals in LR-TDDFT. The LR-TDDFT kernel is a two point kernel derived from the exchange-correlation energy which only depends on the electron density. On the contrary, the Bethe– Salpeter kernel is originated from the Green’s functions, thus it cannot be contracted and remains a four-point function. The GW+BSE approach proved to be very successful in calculating excited-state properties, especially for bulk materials, where TDDFT methods often fail [65]. However, the GW + BSE method is still very expensive computationally3 , thus it cannot be applied to systems with hundreds of atoms.

Figure 1. The energy as a function of the configuration coordinate is shown for the excitation process of a defect in the Franck–Condon approximation: Eg and Ee are the minima in the quasi-parabolic energy surfaces of the defect in the ground and excited states respectively, while qg and qe are the corresponding configuration coordinates. The energy ladder represents the phonon energies with the phonon ground states at n = 0 and m = 0. At low temperatures, the zero-phonon line (ZPL, blue arrow) dominates both the absorption and emission. At higher temperatures, the high energy phonon states can be occupied, and induce the vertical absorption (A → B, green arrow) and vertical emission (C → D, red arrow). The Stokes shift can be approximated as the energy difference between the vertical absorption and emission (|EA→B | − |EC→D |).

The absorption spectra of molecules can be obtained from TDDFT calculations, but one has to calculate the geometry of the system in its excited states in order to obtain the emission spectra. When the electron density of the molecule changes due to an electronic excitation, the forces between the nuclei also change, which leads to deformation of the atomic configuration of the ground state. In the new excited-state geometry the energy difference between the electronic ground and excited states is different compared to the case of the ground state geometry, so the energy of the emitted photon is not the same as the energy of the absorbed photon. The energy difference between the absorbed and emitted photon is called the Stokes shift. In this case the emitted photon has less energy than the absorbed one, and the rest is usually ‘lost’ as heat. In biological applications, the emission should be studied at room temperature. At room temperature the maximum of the emission spectra appears in the vertical transition between the energy minimum in the potential energy surfaces of the excited state and the ground state at this geometry of the studied object (see C → D transition in figure 1). The geometry of a molecule in its excited-state can be determined with the TDDFT method [81] or GW + BSE [82], but these approaches can be very demanding computationally for large systems. A much less expensive method to obtain

excited-state geometries is the SCF method [83, 84], where a non-Aufbau occupation is chosen in the Kohn–Sham iteration, and the geometry is relaxed in this constrained groundstate of the system, which represents the excited state of the non-constrained system. One has to take care when choosing the artificial occupation because the occupation numbers cannot be determined that correspond to the desired excited state of the system. A possible method to choose the non-Aufbau occupation numbers relies on the results of a TDDFT calculation, e.g. if the TDDFT shows HOMO → LUMO excitation then one can occupy the LUMO and empty the HOMO by a single electron, where HOMO (LUMO) stands for the highest occupied (lowest unoccupied) molecular orbital. Generally, this constrained geometry optimization proved to provide reliable results if the excited state can be well described with a single Slater determinant and the symmetry of the ground and excited states are different, avoiding the hybridization between them [85].

2.2.2 Excited-state geometry and Stokes shift.

3 Results

As we briefly mentioned in section 1, it is pre-requisite to use bioinert nanoparticles for safe in vivo bioimaging. Thus, we considered diamond and silicon carbide nanoparticles. Diamond contains only carbon atoms that are biocompatible with the living organism. Silicon carbide is stable in an aqueous environment and also contains silicon atoms. In the case of an unexpected event when a silicon atom is emitted from silicon carbide, then this silicon atom will form small

3 The GW + BSE approach is usually utilized in simulation of bulk material. Thus, most of the available GW + BSE capable codes use plane-wave basis sets, which is usually more expensive computationally, while there are plenty of codes with localized basis-set capable of TDDFT calculations.

7


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SiO4− 4 ions in cells that can leave the organism [86], thus silicon carbide can be considered as bioinert material. For clearance, the small size is critical. Small nanodiamonds can be made down to 3 nm [87, 88] but it is very difficult to engineer photo-stable fluorescent nanodiamonds smaller than 5 nm in diameter [89, 90]. Very recently, stable color centers in meteorite nanodiamonds of ∼2 nm diameter have been found [91]; however, how to artificially fabricate these fluorescent nanodiamonds has not yet been explored. Thus, we rather concentrated on available molecule-sized diamond cages, called diamondoids, and attempted to tune their optical properties by altering their surface termination. However, ultrasmall fluorescent silicon carbide nanoparticles can be routinely made nowadays [92–95], where relatively simple point defects acting like color centers can presumably be introduced in high concentrations, either in growth or after growth, by different techniques. In the following sections we describe our recent and new results on diamondoids and silicon carbide nanoparticles where we slightly modified their atomic compositions in order to tune their optical properties to the desired emission wavelengths.

[106]. We utilized the TDDFT method in the linear-response approximation. We used the PBE0 [53] functional as the adiabatic xc-kernel for the excited-state calculations. In our study, we investigated the effect of the sulfurization on the smallest diamondoid (adamantane—C10 H16 ) and on a larger one (pentamantane [1(2,3)4]—C26 H32 ). There are carbon atoms on the surface of diamondoids which bond to two hydrogen atoms. The two hydrogen atoms can be substituted by a sulfur atom, resulting in a C=S bond on the surface. We calculated the lowest energy excitation of the sulfurized diamondoids as a function of the number of sulfur atoms on the surface, where we considered the lowest energy configuration in each case. The lowest energy excitations obtained from TDDFT calculations are shown in figure 2. The sulfurization has a drastic effect on the lowest excitation energy: as the number of sulfur atoms on the surface was increased, the absorption edge shifted toward the lowest energy region. To understand the mechanism behind this effect, first we analyzed the effect of a single C=S double bond in adamantane. In pristine adamantane, the HOMO is a bonding combination of sp3 bonds between C and H atoms (σ state), while the LUMO is a 3s-type Rydberg state [98, 107]. The system has Td symmetry, and the first excitation energy comes from the dipole-allowed HOMO–LUMO transition, at 6.7 eV excitation energy calculated with TDDFT using the PBE0 functional. If two hydrogen atoms are substituted for a single S atom then the symmetry is reduced to C2v , and the optical gap is decreased to 5.0 eV. The HOMO is a lone electron pair on the S atom (n state), while the LUMO is the antibonding combination of the double bond between the neighboring C and S atom (π ∗ state). However, the HOMO–LUMO transition is dipole-forbidden, and the first transition is between the HOMO and LUMO + 1, where the LUMO + 1 is the second lowest unoccupied orbital, which is a 4s Rydberg state in this case. This system, known as adamantanethione, has been synthesized [108] and its absorption spectra has been measured [109] allowing direct comparison with the experiment. In [109], it was claimed that the first dipole-allowed transition is between the bonding and antibonding combination of the double bond between the neighboring C and S atom with energy 5.3 eV (π ← π ∗ transition), and the second lowest transition is between the lone pair on the S atom and the 4s-like Rydberg state (n ← 4s). Our calculations correctly described the bonding–antibonding excitation as having the energy of 5.3 eV, but underestimated the energy of the n ← 4s by 0.4 eV. The reason for this inconsistency is that a Rydberg state was involved in the second excitation, leading to a large electron–hole separation. The PBE0 functional (and any other xc-functional without proper long-range correction) has an incorrect long-range behavior [75], and may lead to an underestimation of the excitation energies of charge-transfer transitions [75]. Next we added more S atoms to the system, ensuring that the distance between the S atoms is maximal because we found the C=S double bonds to be slightly polarized, thus the sulfur atoms repel each other. In the case of two sulfur atoms, the relevant single particles’ energies are close to those of adamantanethione, but the HOMO–LUMO transition is dipole-allowed because of the D2d symmetry, resulting in the lowering of the optical gap to

3.1 Diamondoids with sulfurized surface

Ultrasmall diamond nanoparticles, i.e. diamondoids, are hydrocarbon molecules whose dangling bonds at the surface are terminated by hydrogen atoms. Diamondoids containing 10–26 carbon atoms have been isolated from petroleum and purified to 99% purity by a combination of reversedphase high-performance liquid chromatography [96]. Their optical properties have been measured by ultraviolet absorption spectroscopy [97] and were studied by various ab initio methods [48, 98–100]. Nanodiamonds are extremely stable chemically, and they are about the same size as typical dye molecules, thus diamondoids have great potential in biological imaging. However, their fluorescent emission falls into the ultraviolet spectral range rendering them unusable for in vivo bioimaging applications. Our goal was to eliminate their non-favorable optical properties and to propose a superior alternative to the currently existing QDs used for biological imaging. In [101] we showed that the substitution of the appropriate hydrogen atoms by sulfur atoms at the surface of the diamondoids results in a remarkable redshift of their first excitation energy. We determined the groundstate geometries of the nanostructures utilizing the PWSCF code (part of the QUANTUM ESPRESSO package [102]), by employing the PBE [54] exchange-correlation functional within DFT, using ultrasoft pseudopotentials [103]. The plane-wave cutoff was set to 35 Ry, while we used a 10 times larger cutoff for the charge density, which is usually needed because of the augmentation charges that arise in the ultrasoft scheme. To avoid the artificial nanocrystal– nanocrystal interaction introduced by the periodic boundary conditions, the size of the supercell was chosen to allow at least 1 nm separation between the images of the diamondoids. The absorption spectra of nanoparticles were calculated by the cluster code TURBOMOLE [104, 105], with augmented correlation-consistent polarized valence triple zeta basis set 8


Topical Review

Figure 2. The effect of S=O bonds at the surface on the optical gap of diamondoids in case of (a) adamantane and (b) pentamantane[1(2,3)4]. The cyan balls depict carbon atoms. The =S substitutions are indicated by numbers on the upper x-axis of the plots for adamantane and by letters for pentamantane[1(2,3)4], where the dictionary for the letter–number correspondence is the following: a = 1; b = 1, 12; c = 1, 9, 10; d = 1, 3, 10, 12; e = 1, 4, 5, 7, 10, 11; f = 1, 4, 5, 711; g = 13, 612; h = 112. The numbers on the structures next to the graphs represent C atoms to which S atoms bond. This figure was reproduced from [101], copyright (2012) by the American Physical Society.

diamondoids with more than one C=S bond can be fabricated: adamantane-dithione, which is an adamantane molecule with two C=S double bonds is a well-known structure [111], and adamantane with three C=O bonds at the surface has been produced [112]; this can be a starting point for creating multiple C=S double bonds on it. In summary, we showed that sulfurization of ultrasmall diamond nanocrystals can lead to a significant decrease in the optical gap. In the case of pentamantane[1(2,3)4], the optical gap can be shifted to the NIR region, resulting in a very potent candidate for in vivo bioimaging.

3.6 eV. In the case of four S atoms, the energy of the dipoleallowed HOMO–LUMO transition is further lowered to 2.4 eV. Finally, in the case of full sulfurization, the six C=S bonds reduced the optical gap to 2.6 eV, which is slightly larger than that with only four C=S double bonds. This effect is due to symmetry reasons (see [101] for details) and does not interfere with the explanation of the feature of sulfurization as explained below. We found that the shape of the HOMO and LUMO does not change as we increase the number of C=S bonds, but the charge transfer from the S atoms toward the C atoms increases. We investigated the effect of this polarization on the HOMO and LUMO single-particle energies, applying firstorder perturbation theory. In this approximation, the correction 2 1 1 to the states is φ| 4πe 0 ( |r−r − |r−r )|φ , where × e is the S| C| polarization charge, rC and rS are the coordinates of the C and S atoms, and φ is the HOMO or LUMO wavefunction obtained from our DFT calculations on adamantanethione. We found that the change in the HOMO–LUMO gap obtained by this simple model was in the same order of magnitude as obtained from our DFT calculations, so the charge transfer is responsible for the lowering of the optical gap. The charge transfer is caused by steric interaction between close sulfur atoms, and it increases with the number of sulfur atoms. We also investigated the effect of sulfurization on pentamantane[1(2,3)4], which has Td symmetry and 6.0 eV optical gap. After introducing one C=S double bond, the dipole-allowed transition between the n HOMO and π ∗ LUMO states has an excitation energy of 2.6 eV, which is already a very large decrease compared to the pristine one. As can seen in figure 2, the introduction of two, three, four, and six C=S bonds does not lead to any significant further redshift in the optical gap. If we further increase the number of C=S bonds, the close sulfur atoms start to repel each other, leading to an increase in the charge transfer between the sulfur and carbon atoms. The increased charge transfer reduces the HOMO– LUMO gap, which is correlated with the first excitation energy. According to previous measurements, the =S functionalized adamantanes are stable in solution [110]. In addition, in the presence of S−H functional groups it can form an S–S bond between two diamondoids, indicating that the modified diamondoids can covalently bond to biomolecules without any special surface treatment. Experimental studies indicate that

3.2 Point defects in SiC nanocrystals

Silicon carbide (SiC) is widely considered to be bioinert. Bulk and porous SiC has already been applied in several biological applications. Recently molecule-sized fluorescent 3C-SiC4 nanocrystals were fabricated [92–95], offering great potential for fluorescent bioimaging. They have strong, stable fluorescence, and a strong two-photon cross-section [113]. Although the toxicity of SiC nanocrystals is not studied to the same extent as the conventional compound semiconductors, the results are very promising [113, 114] so far. Considering the lack of toxic elements and the chemical robustness of SiC, SiC nanocrystals are almost ideal for in vivo imaging. However, SiC QDs emit light in the visible–ultraviolet interval where the surface chemistry of the SiC nanocrystals has a great impact on the optical properties [115]. Our goal was to tune the emission of the SiC nanocrystals toward the NIR region which is independent from surface chemistry. To this end, we investigated several point defects that might act as appropriate color centers in SiC nanocrystals. Several studies have considered donor–acceptor pair (DAP) defects in bulk SiC [116, 117]. Most of them focus on their white light-emitting diodes applications. The most common dopants are nitrogen, boron and aluminum, thus we included boron–nitrogen and aluminum–nitrogen DAPs in our calculations. In 3C-SiC, two types of DAPs exist: type I (when the donor and the acceptor both substitute the same host atom) and type II (when the

3.2.1 Donor–acceptor pairs.

4

9

3C is the cubic polytype of SiC.


Topical Review

accurate calculations in SiC nanocrystals, yielding Kohn– Sham eigenvalues within 0.1 eV of the reference energies obtained from plane-wave calculations. We utilized the TDDFT method in the linear-response approximation. We used the PBE0 [53] functional as the adiabatic xc-kernel for the excited-state calculations. The calculated lowest excitation energies are shown in figure 3. Figures 3(a) and (b) show the first excitation energies as the function of the diameter: in figure 3(a) the donor–acceptor distance is minimal, while in figure 3(b) the donor and acceptor atoms are well separated. Figure 3(c) shows the first excitation energy as the function of the NC –BC distance in a 1.93 nm SiC nanocrystal, compared to that of the pristine nanocrystal. As is apparent, the DAPs lower the optical gap of SiC nanocrystals, even for the case when the distance between the donor and acceptor dopants is minimal. The optical gap of the defect-containing QDs approximately follows the trend of the pristine ones, but the DAP induced redshift in the first excitation energy slightly decreases as the size of the nanocrystals increases. This can be attributed to the more effective screening of the Coulomb interaction. The type I B–N DAP lowers the absorption edge to a larger extent compared to the type II Al–N DAP, which is expected because BC is a deeper acceptor by ∼0.4 eV than AlSi . The nature of the lowest energy transition is in good agreement with our expectations: the first dipole-allowed transition occurs between the HOMO and LUMO, which can be identified as the acceptor and donor levels, respectively. In the ground state, the donor donates its fifth valence electron to the acceptor. This electron hybridizes with the other three valence electrons of the acceptor, and the four dangling bonds of the neighboring carbon/silicon atoms, resulting in four sp3 bonds. These four sp3 bonds split into an a1 and t2 level in the crystal field because the substitutional point defect has Td symmetry in cubic SiC. The Td symmetry of the system is broken in nanocrystals but the t2 level remains quasidegenerate because the local symmetry of the point defect is still very close to Td . The HOMO is a nearly-degenerate t2 level localized on three sp3 bonds between the acceptor atom and the neighboring carbon/silicon atoms, while the LUMO is an s-like state that is quasi delocalized and centered around the donor atom. This LUMO splits from the ‘conduction band’ of the pristine SiC QD due to the presence of an N-donor. In the excited state, this s-like state is occupied by a single electron while a hole is left on the acceptor level. All in all, we can conclude that the donor–acceptor excitation takes place between two states that are centered at distant places. These excitations (usually referred to as chargetransfer excitations) can hardly be described properly using LR-TDDFT. Although the long-range corrected functionals proved to be a major improvement in the description of chargetransfer excitations in organic molecules, they are not able to provide satisfying results for the SiC nanocrystals5 . An

donor and acceptor substitute different host atoms). In the case of N and Al dopants, only type II DAP exists because the nitrogen atom always substitutes the carbon atom (NC ) while the aluminum atom always substitutes the silicon atom (AlSi ). In contrast, the boron atom can substitute both the carbon and silicon atoms, with the BC defect being the deeper acceptor [118]. In a bulk semiconductor, the fluorescence energies of a given DAP defect can be described with the formula [119] h ¯ ω = Eg − (ED + EA ) − EC − EvdW ,

(12)

where Eg is the gap, ED and EA are the ionization energies of the donor and acceptor, EC is the Coulomb interaction between the donor and acceptor ions after the electron–hole recombination, and EvdW is the interaction between neutral donor and acceptor atoms before the recombination. If the multipole terms were neglected then the Coulomb-interaction energy can be written as EC = −e2 /r, where is the static dielectric constant and r is the donor–acceptor distance. The van der Waals term is usually written as EvdW = (−e2 /)(α 5 /r 6 ) and can be neglected at large r, where α is the effective coefficient. This formula gives an accurate description for the donor–acceptor emission energies for relatively distant pairs, which was confirmed in experiments, e.g., [116, 117, 119]. The situation is much more complicated in nanocrystals. The gap depends on the size of the system, and the dielectric constant is not known, thus the Coulombinteraction energy cannot be easily estimated. We carried out TDDFT and SCF calculations for DAPs in spherical, hydrogen-terminated SiC nanocrystals with a diameter of 1.4–2.3 nm. We identified the three most important factors which determine the excitation (and emission) energies of the system: (i) the defect-related HOMO–LUMO gap depends significantly on the diameter of the nanocrystal, thus the size of SiC QD matters; (ii) the donor–acceptor separation is also important, just like in the case of bulk SiC, and the smaller screening of the nanocrystals makes the Coulomb attraction larger between the hole and the electron; (iii) the position of the defects in the nanocrystal can also be important due to the proximate surface. In our study, we particularly investigated (i) and (ii), and due to (iii) we were forced to restrict our calculations to geometries where both the donor and acceptor is relatively far from the surface, in order to extrapolate our results for larger SiC QDs. We investigated the type I B–N and type II Al–N DAP defects in spherical SiC nanocrystals. The dangling bonds on the surface of our model nanocrystals were terminated with hydrogen atoms. The ground-state geometries of the defect-containing nanocrystals were calculated utilizing the plane-wave VASP [120, 121] code, with the DFT-PBE [54] xc-functional. VASP employs the plane-wave basis set with projector augmented wave (PAW) projectors to describe the effect of the core electrons. The excitation energies were calculated by the cluster code TURBOMOLE [104, 105], with double-zeta plus polarization (DZP) basis set and the Stuttgart effective core potentials (ECPs) for the Si and C atoms [106]. The use of larger basis sets is prohibited for larger nanocrystals. We found that the DZP basis set is already sufficient for fairly

5 We performed some calculations using the long-range corrected functionals LC-ωPBE [122, 145] and CAM-B3LYP [146], utilizing the Gaussian [147] cluster code. We found, that these functionals are not able to give accurate excitation energies for the Rydberg states of small diamondoids, or even for short-ranged excitations of SiC nanocrystals. This is probably due to the fact that their parameters were empirically tuned to give good results for certain organic-molecule benchmark sets.

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First excitation energy [eV]



5 4 3 2 pristine Al-N (type I) r = 0.20 nm B-N (type II) r = 0.31 nm

1 0 1

1.2

1.4

1.6

1.8

5 4 3 2 pristine Al-N (type I) r = 0.62 nm B-N (type II) r = 0.65 nm

1 0

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Nanocrystal diameter [nm]

Fluorescent emission [eV]


3.5 3 pristine (TDDFT) B-N DAP - TDDFT B-N DAP - ∆SCF 0.2

0.4 0.6 0.8 1 1.2 Donor-acceptor distance [nm]

1.8

2

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(b)

4

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Nanocrystal diameter [nm]

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4 3 2 1 Al-N (type I) r = 0.62 nm B-N (type II) r = 0.65 nm

0 1.4

1.2

(c)

1.4 1.6 1.8 2 2.2 Nanocrystal diameter [nm]

2.4

(d)

Figure 3. The calculated first excitation energies of DAP containing SiC QDs. The introduction of DAP effects lowers the optical gap in all cases. The optical gap as the function of the QD diameter in the case of (a) minimal, (b) moderate donor–acceptor distance. (c) The calculated TDDFT and SCF values for the optical gap as the function of the NC –BC distance. The red line shows the first-excitation energy of the pristine SiC nanocrystal of the same size. (d) The predicted fluorescent emission energy as the function of the QD diameter for the two DAP defects. We obtained these values by subtracting the Stokes shift from the calculated optical gaps. We only calculated the Stokes shift for the nanocrystal with 1.93 nm diameter, and assumed that the size dependency of the Stokes shift is negligible.

alternative is the SCF method (see section 2.2.1 ). We relied on the results of the TDDFT calculations to determine the non-Aufbau occupation numbers in our SCF calculations: in the excited state, one electron from the nearly-degenerate HOMO level is promoted to the LUMO level as explained above. In figure 3(c) the TDDFT and SCF results are compared. The SCF results are in very good agreement with the excitation energies obtained with the TDDFT method in the whole donor–acceptor distance range. This result validates our decision to perform SCF calculations and also shows that the partially incorrect asymptotic decay of the applied PBE0 xc functional does not have a significant effect on the excitation energies calculated with TDDFT even for larger donor–acceptor distances. As discussed in section 2.2.1, the ∼1/r decay of the xc-function is only valid for molecules and surfaces, while the decay is much faster (exponential) in bulk materials. This is the reason for the great success of the short-ranged functionals, such as the HSE06 functional [122]. The few-nanometer nanocrystals included in our calculations can be considered as an intermediate case, thus PBE0 may perform well even for charge-transfer excitations. It is also important to note that the Coulomb screening in the nanocrystals weakens the interaction between the donor and the acceptor dopants, so the excitation energy depends less on the donor–acceptor distance than that in molecules, where the screening is negligible.

As expected according to the behavior of DAPs in bulk SiC (equation (12)), the optical gap decreases as the donor– acceptor distance r increases. However, the r-dependence of the optical gap clearly does not follow a Er→∞ + C/r 2 curve (C is a constant), as one would expect according to equation (12). This can be attributed to the fact that the fewnanometer nanocrystals are relatively small, and donor and acceptor atoms reside closer to the surface as we increase the donor–acceptor distance. To gain information about the fluorescent emission, we calculated the Stokes shift in some characteristic cases. We obtained the excited-state geometries by forcing non-Aufbau occupation in the Kohn–Sham iteration, where we relied on the TDDFT results again for the excited-state occupation numbers. The Stokes shifts were determined as the difference of the first excitation energy calculated with TDDFT in the ground and excited state geometries. In 1.93 nm diameter SiC nanocrystals with a donor–acceptor distance of ∼0.6 nm, the calculated Stokes shifts were 0.42 eV and 0.40 eV for B–N and Al–N DAPs, respectively. In ∼2.0 nm SiC nanocrystals this results in stable yellowish and violet emission at room temperature for NC –BC and N–Al DAPs, respectively. This emission is expected to be photo-stable and not surface related but still lacks the requirement of NIR emission for ‘ideal’ in vivo biomarkers. Thus, we further investigated other point defects that may act as NIR color centers in few-nanometer SiC nanocrystals. 11


Topical Review

M ); (b) adjacent Si- and C-vacancies, i.e. divacancy Figure 4. (a) Si-vacancy (◦) and a single metal atom (M) substituting Si-vacancy ( M◦ ◦) with M = vanadium (V), molybdenum (Mo) and tungsten (W). The shaded area represents the ( ◦) and M substituting divacancy ( desired region of fluorescence for in vivo bioimaging. This figure is reproduced from [129].

a few calculations where we moved the point defects closer to the surface in order to evaluate the effect of the position of the defect. We applied the same methods and utilized the same computer codes to calculate the ground and excited states properties as we did in the case of the donor–acceptor defects. Figure 4 shows the calculated first excitation energies for various point defects and nanocrystal diameters. We found that the introduction of the chosen point defects leads to a drastic redshift of the absorption edge. Silicon vacancy, divacancy and all of the investigated transition-metal-related point defects lowered the absorption edge to the NIR region, offering a possible way to alter the fluorescent emission wavelength of the SiC nanoparticles. We found that the investigated point defects introduce occupied and empty levels in the gap of the pristine SiC QDs. The transition occurs between these localized states. To gain a deep insight into the electronic structure of these defects, symmetry analysis was carried out by applying the group theory. Silicon vacancy and divacancy were analyzed M and M◦ defects in detail in [130], and the metal-containing were studied in Ref. [129]. Here we only discuss the case of M defects to provide an example but the same methodology the can be straightforwardly applied to the other defects as well. M defect. Figure 5 shows the electronic structure of the The sp3 hybridized C-dangling bond states of Si-vacancy and the valence s and d electrons of the metal atom split in the M defect, crystal field according to the local Td symmetry of the and the states that belong to the same irreducible representation of the point group are able to interact and create defect states. The sp3 dangling bonds split to levels with t2 and a1 symmetries while the s and d electrons of the metal atom split to three

3.2.2 Vacancy, divacancy and transition-metal-related point defects. Several deep level point defects are known in bulk

SiC that act as fluorescent color centers. Silicon vacancy and divacancy (which comprises a neighboring carbon and silicon vacancy) are two intrinsic defects in bulk SiC, and emit light in the NIR window [123–125]. Several transition metal-related point defects are also known to act as NIR fluorescent centers in bulk SiC [126, 127]. In addition to the silicon vacancy and divacancy, we included tungsten (W), niobium (Nb) and molybdenum (Mo) related point defects in our calculations because these metals are known to be biologically inert, and promise to achieve the NIR emission in molecule-size SiC nanocrystals. In the case of metal related point defects, we considered two different configurations: the metal atom is substituted for the silicon atom (we will refer to these defects M , where M stands for the chemical symbol of the metal as impurity). We also investigated the complex of carbon vacancy adjacent to the substitutional point defect (which we refer to M◦ ). These defects can be especially significant if the metal as atoms are introduced to the SiC by means of ion implantation, which leads to the rise of vacancies at a high concentration. We V◦ defect in our calculations because it has did not include the a significantly high formation energy according to previous calculations [128]. Other forms of these impurities also have high formation energies and are not considered in our study [128]. In our calculations, we considered spherical SiC nanocrystals with varying diameter in the 1.1–2 nm range, and the dangling bonds on the surface were terminated by hydrogen atoms again. In most of our calculations, we placed the point defects in the center of the nanocrystal but we also performed 12


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Figure 5. The electronic structure of metal dopants (M) in cubic SiC nanocrystals. ‘VBM’ and ‘CBM’ represents schematically the position M defect and (b) the corresponding localized of the HOMO and LUMO states of the host cubic SiC QD without the color centers. (a) HOMO and LUMO single particle states in the smallest SiC QD. The most dominant contribution in the electronic excitation is the e → t2∗ transition between the HOMO and LUMO. Color codes for (b): small cyan, larger yellow and tiny white balls represent carbon, silicon and hydrogen atoms, respectively. The isosurface of the square of the corresponding single particle wave functions are depicted. The occupation of states in (a) are relevant for Mo and W atoms, while the HOMO state has one electron less for the V atom. This figure is reproduced from [129].

levels with t2 , e and a1 symmetries. The HOMO is a double degenerate e-state with atomic-like properties due to the fact that only the valence electrons of the metal atom split to a level with e symmetry, which is not able to interact with the electrons of the dangling bonds because they do not show e character in the crystal field. In the case of Mo and W atoms two electrons will occupy this e-state resulting in a high spin (S = 1) ground state. The LUMO is a three-fold degenerate t2∗ state, which is an antibonding combination of levels with t2 symmetry from the dangling-bond electrons of the vacancy and the valence electrons of the metal dopant. According to the group-theoretical analysis, the LUMO is not purely an atomiclike state, but it is still very localized on the defect state as is shown in figure 5(b). The e → t2∗ transition is dipoleallowed, and found to be the lowest energy excitation according to TDDFT calculations. Only a single electron occupies the HOMO e-state in the V defect, which becomes Jahn–Teller unstable, with a slight distortion in the symmetry to D2d , but the nature of the HOMO and LUMO states remains the same. In D2d symmetry, the HOMO e-state splits as a1 +b1 , and the b1 state will be occupied by a single electron. The LUMO t2∗ -state splits as b2 + e. The b1 → b2 transition is dipole-forbidden while b1 → e is allowed. We found that the lowest energy excitation is indeed related to the transition between b1 and e. The most important features are the same for Si-vacancy, M ◦ defects too: the introduction of the defect divacancy, and creates deep levels in the gap which are well localized on the point defects. The lowest energy excitation occurs between these well localized defect levels in all of the investigated cases.

As apparent in figure 4, the quantum-confinement effect is visible as the optical gap lowers with the increase of the diameter of the nanocrystal. However, the size dependency of the optical gap is much weaker compared to the case of the pristine SiC QDs, which is a consequence of the localized nature of the relevant defect levels. We found that the energies of these excitations are 1–2 eV smaller than the oneparticle HOMO–LUMO energy difference, indicating a strong excitonic binding between the electron and hole. We note that the spin–orbit coupling is generally not negligible for metals but we found that the inclusion of the spin–orbit interaction in the calculations have a very small effect on the Kohn–Sham eigenvalues: the change in the HOMO–LUMO energy difference due to spin–orbit splitting was smaller than 0.02 eV in all of the investigated cases. It is important to understand the role of the position of the defects in the SiC QDs because the point defects can be assumed to be uniformly distributed in the volume of the nanocrystals [129]. We calculated the lowest excitation energy at different defect positions, and we found that the value of the lowest excitation energy is almost independent of the position of the defect resulting in a typically 0.04 eV broadening of the optical gap. We also calculated the Stokes shift for Si-vacancy, divacancy and the metal-related point defects in a medium sized (d = 1.4 nm) nanocrystal. We applied the same methodology as for the DAP defects. The calculated Stokes shifts were 0.10 eV, 0.12 eV, 0.14 eV, 0.3 eV, and 0.6 eV for SiW , Mo , W◦ and Mo ◦ defects, respectively. vacancy, divacancy, 13


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Our results indicate that all these color centers emit light in the desired wave lengths for in vivo bioimaging. In summary, we found that small SiC nanocrystals are able to emit light in the NIR region if appropriate color centers are introduced into them. It is important to note that the majority of these defects have a high spin ground state due to the partial occupation of degenerate states. It has been shown recently that the spin state of Si-vacancy [124, 131] and divacancy [132, 133] can be manipulated by optical excitation in bulk SiC in a similar fashion to the nitrogen-vacancy center in diamond [134]. It has already been demonstrated that a combination of the fluorescence and magnetic properties of the nitrogen-vacancy center in diamond can be utilized as a nanoscale magnetic probe [135–140], electric probe [141] or thermometer [142–144] that may be applied to track the biological processes at molecular level. Si-vacancy or divacancy in SiC QD has a similar potential in this field which makes these color centers extremely valuable.

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4 Conclusions

In our review, we attempted to list the most important requirements for making an ‘ideal’ in vivo bioimaging agent from semiconductor nanocrystals. We demonstrated that wellchosen advanced ab initio methods can be used to find such fluorescent nanoparticles to fulfill these stringent criteria. In particular, the introduction of appropriate color centers into molecule-sized silicon carbide nanocrystals would be highly beneficial for biologists, according to our findings. Acknowledgments

The authors acknowledge support from the MTA Lendület program of the Hungarian Academy of Sciences, the FP7 Diamant project of the EU Commission, and the OTKA K-101819 project of the Hungarian Scientific Research Fund. References [1] Resch-Genger U, Grabolle M, Cavaliere-Jaricot S, Nitschke R and Nann T 2008 Quantum dots versus organic dyes as fluorescent labels. Nature Methods 5 763–75 [2] Frangioni J V 2003 In vivo near-infrared fluorescence imaging Curr. Opin. Chem. Biol. 7 626–34 [3] Bruchez M, Moronne M, Gin P, Weiss S and Alivisatos A P 1998 Semiconductor nanocrystals as fluorescent biological labels Sci. 281 2013–16 [4] Michalet X, Pinaud F F, Bentolila L A, Tsay J M, Doose S, Li J J, Sundaresan G, Wu A M, Gambhir S S and Weiss S 2005 Quantum dots for live cells, in vivo imaging, and diagnostics Science 307 538–44 [5] Medintz I L, Uyeda H T, Goldman E R and Mattoussi H 2005 Quantum dot bioconjugates for imaging, labelling and sensing Nature Mater. 4 435–46 [6] Fan J and Chu P K 2010 Group IV nanoparticles: synthesis, properties, and biological applications Small 6 2080–98 [7] Wang Y, Hu R, Lin G, Roy I and Yong K-T 2013 Functionalized quantum dots for biosensing and bioimaging and concerns on toxicity ACS Appl. Mater. Interfaces 5 2786–99 14


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Personalized Computational Models as Biomarkers.

de novo computational enzyme design.

Computational Design of Biomimetic Phosphate Scavengers.

RiboMaker: computational design of conformation-based riboregulation.

Design, Assessment, and in vivo Evaluation of a Computational Model Illustrating the Role of CAV1 in CD4(+) T-lymphocytes.

Computational Methods Applied to Rational Drug Design.

Torsional ultrasonic transducer computational design optimization.

Computational design of molecular motors as nanocircuits in Leishmaniasis.

Computational design of molecular motors as nanocircuits in Leishmaniasis.

Enantioselective enzymes by computational design and in silico screening.

Advances and computational tools towards predictable design in biological engineering.

Convergence in parameters and predictions using computational experimental design.

Computational solvent mapping in structure-based drug design.

Personalized radiotherapy: concepts, biomarkers and trial design.

Computational design of novel peptidomimetic inhibitors of cadherin homophilic interactions.

Design and computational analysis of single-cell RNA-sequencing experiments.

Computational design of nucleic acid feedback control circuits.

Computational design of allosteric ribozymes as molecular biosensors.

R2oDNA designer: computational design of biologically neutral synthetic DNA sequences.

Computational approaches for rational design of proteins with novel functionalities.

Computational design of water-soluble α-helical barrels.

Computational materials design of defect-induced ferrimagnetic MnO.

Towards the computational design of protein post-translational regulation.

Direct Calculation of Protein Fitness Landscapes through Computational Protein Design.