Spin Unrestricted Excited State Relaxation Study of Vanadium(IV

Article pubs.acs.org/JPCC

Spin Unrestricted Excited State Relaxation Study of Vanadium(IV)-Doped Anatase Stephanie J. Jensen,† Talgat M. Inerbaev,‡,§ and Dmitri S. Kilin*,†,∥ †

Department of Chemistry, The University of South Dakota, Vermillion, South Dakota, United States Faculty of Physics and Technical Sciences, L. N. Gumilyov Eurasian National University, Astana, Kazakhstan § National University of Science and Technology “MISIS”, Moscow, 119049 Russian Federation ∥ Department of Chemistry and Biochemistry, North Dakota State University, Fargo, North Dakota, United States ‡

S Supporting Information *

ABSTRACT: Atomistic modeling of light driven electron dynamics are important in studies of photoactive materials. Spin-resolved electronic structure calculations become necessary when dealing with transition metal, magnetic, and even some carbon materials, intermediates, and radicals. An approximate treatment can be pursued in the basis of spincollinear density functional theory. Most transition-metal compounds exhibit open shell nonsinglet configurations, necessitating special treatment of electrons with α/β spin projections. By separate treatment of electronic states with the α/β spin components one is able to describe a broader range of materials, identify new channels of relaxation and charge transfer, and provide knowledge for rational design of new materials in solar energy harvesting and information storage. For this methodology, named spin-resolved electron dynamics, spin-polarized DFT is used as the basis to implement nonadiabatic molecular dynamics. At ambient temperatures, the thermal lattice vibrations results in orbital and energy fluctuations with time. Nonadiabatic couplings are then calculated, which control the dissipative dynamics of the spin resolved density matrix. Different initial excitations are then analyzed and used to calculate relaxation dynamics. Spin-resolved electronic dynamics approach (SREDA) is applied to study vanadium(IV) substitutionally doped bulk anatase in a doublet ground state. The results show that a difference in the electronic structure for α and β spin components determines consequences in optical excitations and electronic dynamics pathways experienced by electrons with α and β spin projections. Specifically, the lone occupied V 3d α-orbital increases the range of absorption and defines the rates and pathways of relaxation for both holes and electrons with α-spin projection. Optical excitations involving occupied V 3d α-orbital are responsible for IR-range absorption, followed by nonradiative relaxation. Certain transitions involving orbitals of α-spin component occur in the visible range and induce localization of a negative charge on the V ion for an extended time period. The slower nonradiative relaxation rate of α-excitations is rationally explained as a consequence of difference of electronic structure for α and β spin projections and specific pattern of energy levels contributed by doping. Specifically, excitations involving orbitals with α-projection of spin experience transitions through larger subgaps in the conduction band compared to the ones experienced by similar excitations involving orbitals with β-projection of spin. It is anticipated that this methodology can be broadly implemented on multiple applications of transition metal based materials, including optoelectronics, information storage, laser crystals, dyes, photovoltaic materials, and metal oxides for photoelectrochemical water splitting.

I. INTRODUCTION Titanium dioxide (TiO2) is a promising semiconducting material for photocatalysis, but it suffers several drawbacks on large scale practical applications. At first, its wide band gap (3.2 eV) restricts the photocatalytic property to UV radiation, making it ineffective for visible light. Second, the relatively high rate of electron and hole recombination in TiO2 tends to decrease its photocatalytic efficiency.1 Therefore, to address these two main issues, a great deal of attention has been paid to the functionalization of titania by (i) varying size, shape, and surface or introducing a mesoporous form of titania,2 (ii) by sensitization titania surface with dyes, including metalorganic dyes,3 and by (iii) introduction of mid band gap cationic or anionic dopants into the TiO2 crystal structure. In this respect, © 2016 American Chemical Society

transition metals, rare earth metals, and nonmetals were examined as dopants that improve the visible light photoresponse.4−9 The intriguing and promising properties of transition metal doped oxide semiconductors inspire an atomistic modeling effort, for identifying details and mechanisms of basic processes contributed and controlled by transition metal doping. The rest of the section overviews (a) unique properties contributed by any transition metal doping and, particularly, by vanadium ions, (b) available approaches and challenges in atomistic modeling of photoexcited properties in semiconductors doped by ions with Received: December 12, 2015 Revised: January 28, 2016 Published: January 28, 2016 5890

DOI: 10.1021/acs.jpcc.5b12167 J. Phys. Chem. C 2016, 120, 5890−5905

Article

The Journal of Physical Chemistry C

as spintronics,28 are increasingly commanding the attention of researchers. Promising properties of materials containing transition-metal doping can be evaluated at three types of methodologies: (i) spin unrestricted electronic structure, (ii) excited state dynamics limited to closed-shell configurations, and (iii) excited state dynamics with open-shell configurations enabled by combining the developments of (i) and (ii). (i). Spin Unrestricted Electronic Structure. Spin-polarized density functional theory provides the ground state electronic structure, separating α (spin up) and β (spin down) electrons. In application to titania, spin-polarized DFT and TDDFT has allowed comparison of excitations of different multiplicity in titania nanoparticles.29 Spin-unrestricted electronic structure2 was found useful in describing ferromagnetic materials, including TiO2 doped by transition metals,30 as well as vanadium,31 and TiO2 with oxygen vacancy converting Ti4+ to Ti3+ resulting in a local magnetic moment at Ti3+ ions.32 The success of using the ground state spin polarized electronic properties has been shown in adequate predicting electronic configurations matching Russell−Saunders terms of lanthanide-doped laser crystals33,34 and properties of metal−organic frameworks.35 This article uses the spin-polarized DFT ground state electron computations as a basis for studying spin-unrestricted nonadiabatic dynamics. (ii). Excited State Dynamics Limited to Closed-Shell Configurations. Nonadiabatic dynamics calculations reflect the interaction of electronic and nuclear degrees of freedom, beyond the Born−Oppenheimer approximation. In a broader sense, nonadiabatic dynamics treats coupled and entangled dynamics of electronic and nuclear degrees of freedom, for example, in photochemical reactions.36,37 In a narrower sense, nonadiabatic dynamics describes electronic transitions induced by nuclear motion. Nonadiabatic dynamics calculations typically combine the calculation of nonadiabatic couplings38 and the propagation of the electronic state of the model in time.39 The nonadiabatic dynamics have been implemented via a range of methods, including Tully’s surface-hopping-based methods40,41 and density-matrix-based methods.42 Most of the available implementations of nonadiabatic dynamics are adjusted to describe closed-shell models. Nonadiabatic dynamic calculations allow analysis of the relaxation rates and pathways for excitations, specifically those that correlate to the peaks in the optical absorption spectrum. Nonadiabatic dynamics at different levels of implementation have been very helpful in describing photoexcited processes in titania9 and, specifically, for dynamics in titania sensitized by broad variety of dyes.43−47 Earlier, spin-restricted nonadiabatic dynamics were successfully used to model relaxation rates and pathways in transition metal doped anatase for low spin or antiferromagnetic configurations, as an approximation.48−52 However, as mentioned previously, spinspecific electronic characteristics can affect the overall behavior of a material, and as computational capacity and demand for more accuracy increases, every attempt should be made to provide closer-to-reality simulations. In this case, it requires evolving from spin restricted to spin unrestricted nonadiabatic dynamics calculations, especially for transition metal and other materials where altering the number of unpaired electrons can change the properties drastically. (iii). Excited State Dynamics for Open-Shell Configurations. To date, only a few studies were performed for the exploration of nonadiabatic electron dynamics in open-shell systems. Recently, time-dependent density functional theory was used to study intense, short, laser-pulse-induced demagnetization in bulk Fe, Co, and Ni. The time evolution of the spin

open-shell configuration, and (c) an overview of the practical tasks of this paper. I.a. Unique Properties Contributed by Transition Metal Doping Such As a Vanadium Ion. Transition metal doping is one of the most effective approaches to extend the absorption edge of TiO2 to the visible light region because of the unique d electronic configuration and spectral characteristics of transition metals. Dopant atoms either insert a new band into the original band gap or modify the conduction band or valence band, improving the photocatalytic activity of TiO2 to some degree and changing its magnetic susceptibility.10 Theoretical analysis reveals that the localized energy level due to Co doping was sufficiently low to lie at the top of the valence band, while dopants such as V, Mn, Fe, Cr, and Ni produce the midgap states.11 Experimentally, it was demonstrated that V-doping into the TiO2 lattice leads to a change of the bandgap of TiO2, resulting in an extension of the absorption edge to the visible light region and thereby improving the visible lightdriven photocatalytic activity of TiO2.12 Vanadium and nitrogen codoped TiO2 was, for example, synthesized by the sol−gel method, and the catalyst showed high visible light photocatalytic activity for the degradation of RhB.13 The visible light absorption efficiency of V−N-codoped TiO2 was better than that of V or N singly doped TiO2 due to the narrowing of the band gap, induced by the simultaneous incorporation of V and N into the TiO2 lattice. Vanadium and yttrium codoped titania also provides lower gap and visible range absorption.14 Vanadium atoms replace lattice titanium forming substitutional impurity rather than having separate phases of vanadium oxides and titania.15−17 Doping with vanadium narrows the band gap of TiO2, henceforth making it possible to absorb visible light.18,19 As a result, vanadium-doped TiO2 shows excellent photocatalytic activity under daylight due to excitation of visible light.19,20 TiO2/V−TiO2 composite photocatalysts were generally shown to have a much higher photocatalytic destruction rate than that of undoped TiO2, which is ascribed mainly to the electrostatic-field-driven electron−hole separation in TiO2/V−TiO2 composite photocatalysts.21 Previous experimental studies have mainly focused on vanadium oxide surface doped TiO2 with others showing the vanadium oxide to be a better photocatalyst than V(IV) substitutional doped TiO2.22−26 This is possibly due to V (IV) acting as a recombination center, which is not conducive for photocatalytic applications according to Martin, Morrison, and Hoffmann.22 However, more recently, there has been success codoping V(IV) with carbon, which absorbs more visible light causing the single 3d electron of V(IV) to excite into the conduction band.27 One of the purposes of computational modeling of V(IV)doped titania is to prove right or wrong if the vanadium(IV) is able to act as efficient recombination center. Another purpose of the modeling is exploration of ability of vanadium-doped titania to participate in formation of charge transfer states upon photoexcitation. If visible light absorption is increased (e.g., with dye-sensitizing), the charge could be collected via electrodes in a photoelectrochemical cell. I.b. Atomistic Modeling of Properties of Photoexcited Semiconductors Doped by Ions with Open-Shell Configuration. In order to accurately describe electron dynamics of transition metals, spin polarized modeling is essential due to the majority of transition metal elements in typical oxidation states often have unpaired electrons. The necessity to separate α- and β-spins is becoming more essential as areas, such 5891


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occupation numbers for each orbital f i,σ, density for each spin ⇀ ⇀ component ρα( r ), ρβ( r ), and total energy ETOT[ρα, ρβ] are all ⎯→ ⎯ parametrically dependent on the positions of the ions { RI }. The contribution of each KS orbital to total density is defined ⇀ ⇀ 2 either via occupation numbers ρσ( r ) = Σi f i,σ|φKS σ,j ( r )| or via ⇀ KS ⇀ KS ⇀ ( r )φm,σ *( r ). Desired density matrix ρσ( r ) = Σlmρσ,lmφl,σ electronic properties are then computed as indicated below and analyzed for photocatalytic characteristics. Optical properties are analyzed based on the transition dipole moment matrix elements

states was taken into consideration through the functional dependence of the exchange-correlation potential on the density and the magnetization density of the system at the current and all previous times that effectively take into account the history of the spin state’s propagation.53 The two-component, timedependent, noncollinear density functional theory was used for the description of nonadiabatic propagation of spin-polarized electrons within a mean-field framework.54 This method was applied to study the spin evolution during the H2 and O2 dissociation.54 Trajectory surface hopping calculations in the time-dependent Kohn−Sham representation were performed for modeling the spin-dependent charge carrier nonadiabatic relaxation dynamics in dilute magnetic semiconductor Mn2+:ZnO quantum dots.55 A preliminary and approximate treatment was reported for excitations of trivalent cerium cation in a dielectric host matrix.56 I.c. Practical Aspects of Modeling Photoinduced Excited State Dynamics in Anatase Doped by OpenShell Elements. This paper presents new details in processing of spin polarization in nonadiabatic dynamics of open shell models. As a natural progression from spin-restricted electronic calculations, to spin-restricted dynamics, to spin unrestricted electronic calculations, spin unrestricted electron dynamics isolates α and β electron states in both (i) basic energy structure calculations, (ii) electron coupling calculations, and (iii) nonadiabatic dynamics. As a result, α and β relaxation dynamics of electrons and holes are analyzed separately, allowing distinct relaxation pathways and differing charge transfers for α/β as opposed to only one pathway and identical charge transfers done in previous calculations.48−51 The paper is organized as follows: methodology is introduced in section II, with subsections II.a, theoretical approaches, including equations and highlights of new methodology, and II.b, computational details. Results in section III are organized in several subsections: hypothetical pathways of electronic relaxation are introduced in subsection III.a, numerical characteristics of electron-to-nuclei interaction of the model are presented in subsection III.b, while numerical arguments in support of hypothesized scenarios of excited state dynamics and their analyses are reported in subsection III.c. Open questions are discussed in section IV, and conclusions are summarized in section V.

∫ φσKS,i *r φ⃗ σKS,i d r ⃗

Dσ⃗ , ij = e

(1)

for transitions between the initial state σ,i and final state σ,j. The transition dipole moment is then used for calculating oscillator strength. fσ , ij = |Dσ⃗ , ij |2

4πmevσ , ij 3ℏe 2

(2)

Here, me is the mass of an electron, ℏ is Planck’s reduced constant, vσ,ij is resonant frequency for ij transition with spin component σ, and e is the charge of the electron. The oscillator strength calculates the probability of absorbing light, which is then used as a weight to calculate absorption aσ (ε) =

∑ fσ ,ij δ(ε − Δεσ ,ij)

(3)

σ , ij

Electronic structure of atomic models was explored at equilibrium-optimized geometry and then along nuclear trajec⎯→ ⎯ tory { RI (t)} modeling system interfacing a thermostat. Positions of ions enter DFT equations as parameters as described above. dρσ , jk

( )

Electronic dissipative transitions

dt

between KS

diss

orbitals with indices σ,i and σ,k are computed along mole⎯→ ⎯ cular dynamics trajectory for positions of ions { RI (t)} with ⎯→ ⎯

initial conditions for positions { RI (t = 0)} and velocities

{

⎯→ ⎯ d R (t dt I

= 0)

} representing ambient temperature. Molecular →̈

⎯→ ⎯

dynamics due to Newton equation of motion MIR I = F I ⎯→ ⎯ determines trajectory, { RI (t)}, which results in time-dependent changes of electronic structure. Specifically, orbitals, φα,i and

II. METHODOLOGY II.a. Theoretical Approaches. As V(IV) only has one 3d electron, it is imperative to use spin-unrestricted calculations in order to isolate α and β states for accurate electronic calculations and analysis of relaxation dynamics. For the following methodology, the approximation of excluding spin flip transitions is implemented due to negligible spin−orbit coupling. As an approximation, Kohn−Sham orbitals are used to describe excited states when studying relaxation dynamics.48,50−52,57 In TDDFT approach, excited states are composed as superposition of multiple electron−hole pairs and typically have smaller excitation energy due to exciton binding.58 Our approach of independent orbitals is a first step toward correlated excitation. The electron− hole pairs are elementary contributions, building blocks for correlated excited states, coming out of TDDFT calculations. Computation of electronic structure uses the position of ions {R⃗ I} as parameters, as explained in SI, eqs S1 and S2. For the following equations, let σ = α or β to indicate ⎯→ ⎯ spin. Kohn−Sham (KS) orbitals φKS σ,j , their energies εσ,i ({ R I }),

⎯→ ⎯

⎯→ ⎯

φβ,i, and their corresponding energies, εα,i({ RI }) and εβ,i({ RI }), experience change and fluctuation with time. On-the-fly nonadiabatic couplings are computed along nuclear trajectory as Vσ , ij(t ) =−

iℏ 2Δt

∫ dr φ⃗ σKS,i *({R⃗I(t )}, r ⃗)φσKS,j ({R⃗I(t + Δt )}, r ⃗) + hc (4)

We process the autocorrelation function Mσ,ijkl(τ) of nonadiabatic couplings as they represent second order perturbation in respect to electron−nuclear interaction and provide the first nonvanishing contribution in time-dependent perturbation theory for electronic degrees of freedom.59 In what follows, averaging is performed along the duration of the trajectory, T. Mσ , ijkl(τ ) = 5892

1 T

∫0

T

Vσ , ij(t + τ )Vσ , kl(t )dt

(5a)


Article

The Journal of Physical Chemistry C Γ +σ , ijkl =

1 T

∫0

Γ−σ , ijkl =

1 T

∫0

T

T

e−iωijt Mσ , ijkl(t )dτ

(5b)

e−iωklt Mσ , ijkl(t )dτ

(5c)

operators are spin-specific, and by casting operators in the form of matrix elements in a specific basis this definition i reads (Lσ ρσ )ij = − ℏ ∑k (Fσ , ikρσ , kj − ρσ , ik Fσ , kj). A linear combination of density operator eigenvectors with (ξ) relevant expansion coefficients ⟨ρ(a,b) σ,ij (0)|ρσ ⟩ allows to solve eq 8a for any initial condition (eq 7) as follows:

A Fourier transform of coupling autocorrelation function Γ+σ,ijkl and Γ−σ,ijkl, where δlj and δik are Kronecker δ symbols, provide components of Redfield tensor, Rσ,ijkl, which control dissipative dynamics of density matrix.60,61 R σ , ijkl =

Γ +σ , ijkl

+

Γ−σ , ijkl

+ δlj ∑ m

⎛ dρσ , jk ⎞ ⎜⎜ ⎟⎟ = ⎝ dt ⎠diss

Γ +σ , ijkl

− δik ∑

Γ−σ , ijkl

m

ρσ , ij (t ) =

Total density of electrons for spin projection σ can be composed of partial contributions from orbitals ϕKS σ,l with density matrix elements playing the role of weight coefficients KS KS ρσ(r) = Σlmρσ,lmφσ,l φσ,m*, this equation serves as formal definition of density matrix. Specific initial excitations by a photon, ℏΩab,σ = εb,σ − εa,σ occur between orbitals a and b of spin projection σ at time t < 0. This excitation is described by density matrix at time t = 0, where δ is the Kronecker delta symbol and fσ,i is equilibrium occupation of (σ,i) orbital prior to photoexcitation, ⎧1, i ≤ HOσ , defined by Fermi− fσ,i = ρσ,ii (t < 0) = ⎨ ⎩ 0, i ≥ LUσ Dirac distribution.

nσ″(ε , t ) =

∑ (Fσ ,ikρσ ,kj σ ,k

dρσ , ij

( ) dt

(7)

⟨Δεσe⟩(t) =

⟨Ee , σ ⟩(t ) = (8a)

are facilitated by thermal fluctuations of

(Lσ + R σ )ρσ(ξ) = Ω(σξ)ρσ(ξ)

(8c)

ρσ , ii (t )εσe(t)

i ≥ LUσ

(10b)

⟨Δεσe⟩(t ) − ⟨Δεσe⟩(∞) ⟨Δεσe⟩(0) − ⟨Δεσe⟩(∞)

(10c)

(σ , ab) ΔnCB (z , t ) =

ions. Solution of eq 8a is based on the Markov approximation resulting in Redfield tensor elements being time independent. In this limit, eq 8a can be cast with use of superoperators (8b)

∑

The expectation values of a photoexcited hole are defined analogously. The partial charge density distributions in the conduction (CB) and valence (VB) bands as functions of z and time t can be obtained from the average of the density operator after integration over x and y and are given by

diss

ρσ̇ = (Lσ + R σ )ρσ

(10a)

This equation describes the dynamics of a population gain when Δn > 0 and a population loss when Δn < 0 at energy ε, which corresponds to the electron and the hole parts of an excitation. The expectation energy of a photoexcited electron can be expressed for a chosen spin projection as follows:

Here, elements of Fock matrix represent adiabatic contribution to energy of electronic subsystem Fσ,ik = δikεσ,i. Electronic transitions

(9)

Δnσ (ε , t ) = nσ″(ε , t ) − nσ′ (ε)

Time evolution of electronic degrees of freedom is calculated by solving the equation of motion as follows: i ℏ

(8d)

The difference of nonequilibrium distribution presented in eq 8a and the equilibrium distribution provides the comprehensive explanation of electron and hole dynamics as a function of energy and time. The change of population with respect to the equilibrium distribution is then expressed as

⎪

ρσ̇ , ij = −

∑ ρσ ,ii (t )δσ(ε − εσ ,i) i

⎪

⎛ dρσ , ij ⎞ ⎟⎟ − ρσ , jk Fσ , ki) + ⎜⎜ ⎝ dt ⎠diss

t

Solution of eq 8a provides time dependent elements of density matrix ρσ,ij(t). Diagonal elements, ρσ,ii(t), determine time-dependent occupations of Kohn−Sham orbitals. Dynamics of charge density distribution, rate of energy dissipation, and rate of charge transfer are all calculated with the above information. Specifically, the distribution of charge as a function of energy reads

(5d)

(6)

ρσ(a, ij, b)(0) = δij(fσ , i − δia + δib)

(ξ)

ρσ(a, ij, b)(0)|ρσ(ξ) ρσ(ξ)e Ωσ

ξ

∑ R σ ,jklmρσ ,lm σ , lm

∑

∑

(ρσ(σ, ij, ab)(t ) − ρσref, ij )

∫ dxdy(φσKS,i (r)·φσKS,j *(r)),

ij ϵ CB

ΔnCBσ , ab > 0 (σ , ab) ΔnVB (z , t ) =

(11)

∑

(ρσref, ij − ρσ(σ, ij, ab)(t ))

∫ dxdy(φσKS,i (r)·φσKS,j *(r)),

ij ϵ VB (σ , ab) ΔnVB >0

(12)

→ φKS σ,i ( r )

where is the ith,jth KS orbital, and the indices i and j belong to the conduction or valence bands. The reference density operator ρref describes the system in a steady state before light is turned off at time t = 0. These densities describe the dynamics of charge rearrangement over time after light is removed and the system eventually decays back to equilibrium. In what follows, we focus on the long-time limit t > τcoh, τcoh ≈ 10 fs, which leads to decoherence and the limit ρσ,ij → 0, for i ≠ j. These distributions reflect the time evolution of the charge transfer along the z direction in the system. Combining

with Lσ + Rσ being Liouville and Redfield superoperators for σ spin projection, while Ω(ξ) and ρ(ξ) stand for eigenσ σ values and eigenvectors of the combined superoperator. Solutions of eqs 8b and 8c are counted by index of excited states ξ. Liouville superoperator acting on arbitrary operator Â is defined as follows and is equivalent to commutator of argument operator Â with Fock operator (Hamiltonian) i i ̂ ̂ − AF ̂ )̂ . In application of F̂ , LÂ = − ℏ [F ̂ , Â ] = − ℏ (FA Liouville operator to density operator (matrix), in case the 5893


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Figure 1. Schematic representation of excitation (A) and relaxation (B) pathways for α and β electron components in vanadium-doped titania. Energies of Kohn−Sham orbitals are represented by bars, electrons occupying such orbitals are symbolized by short up and down arrows for α and β electrons. Character of orbitals is labeled. (A) Possible optical excitations are represented by red straight, solid lines and labeled by roman numerals i−iv, as introduced in the text. (B) Possible nonradiative relaxation pathways are symbolized by dashed wavy arrows. Hole relaxation has been excluded due to triviality. A β-hole will relax quickly to the HO as all VB states are O 2p. Different excitations are marked by lower case roman numerals correlating to eqs 15.i−15.iii, while relaxation pathway steps are indicated with upper case roman numerals correlating to eq 16. An α-hole will likely relax quickly through the O 2p states of the VB and stall at the energy jump to the single V state (HOα).

one state for a longer time in comparison to the other states involved in the relaxation. The trap can be due to an increased energy subgap to overcome or the transfer of population between two states with low coupling.62 The mechanism of trapping is hypothetically related to concepts of energy gap law and phonon bottleneck: Electronic transitions are most efficient in case energies of orbitals are offset by a value Δεσ,i = εσ,i+1 − εσ,i matching a normal-mode frequency Δεσ,i ≈ ℏωvib. However, in case such energy offset exceeds any of available vibrational frequencies, Δεσ,i > ℏωvib the transition becomes less probable. II.b. Computational Details. The model used is anatase with a substitutional vanadium dopant at the center of the simulation cell. The crystal model was created using crystallographic unit cell information to get the Cartesian coordinates and then duplicated to create a bulk section of Ti36O72 size. The simulation cell is V1Ti35O72, with periodic boundary conditions imposed in x-, y-, and z- directions. The doping percentage is then 1.77 wt % vanadium. Electronic structure, accounting for the Coulomb, correlation, and exchange

eqs 11 and 12, one obtains the variation of the total density as a function of time and position, (σ , ab) (σ , ab) (σ , ab) Δntotal (z , t ) = ΔnCB − ΔnVB

(13)

Calculated rates of relaxation for electrons and holes are computed as follows: −1

∞

ke = {τ e}−1 =

{∫

k h = {τ h}−1 =

{∫0

0

⟨Ee⟩(t )dt

}

(14a) −1

∞

⟨E h⟩(t )dt

}

(14b)

where ⟨Ee⟩(t) and ⟨Eh⟩(t) are the energy expectation values of a photoexcited charge carrier defined in eqs 10b and 10c and are assuming an exponential fit, which is most applicable in two cases: relaxation between two states and relaxation through states equally spaced in energy. The considered model is expected to exhibit what is called a trapping state, when population occupies 5894


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with value of U − J = 4 eV.69 Calculations were done using PAW formalism, which was developed using the concept of pseudopotentials in the plane-wave basis supplied by VASP.70−72

III. RESULTS III.a. Electronic Structure and List of Hypothetical Processes. Since vanadium is modeled as a substitutional dopant in the anatase model, it is expected to take on an oxidation state of four. The formation energy of the doped model is computed as energy needed to replace one titanium atom to one vanadium atom Eformation = E(doped anatase) − E(bulk anatase) − μ V + μTi

where chemical potentials of vanadium μV and titanium μTi were estimated via bulk metal calculations μV = E(bulk vanadium metal)/Nions and μTi = E(bulk titanium metal)/Nions of dopant material per ion energy. This energy of replacement of titanium by vanadium has been found to be Eformation = 2.09 eV. Figure 1 shows the schematic representation of spin unrestricted band structure composed of α and β states for the studied model. As expected, there is only 1 vanadium 3d orbital occupied by an electron in the ground state configuration, which is in the α set of energy bands. Additionally, four sets of bands are identified, listed from lowest to highest energy: O 2p, V 3d, both hybridized Ti/V and solely Ti 3d intermixed. Energies of orbitals of V/Ti and Ti character are arranged in subsequent order, with V/Ti being “sandwiched” between Ti. These are visualized in Figure 2, which shows the density of states along with partial charge density representations for each aforementioned group of energy bands. We hypothesize that level of hybridization between V and Ti orbitals may depend on choice of exchange-correlation functional and on use of DFT+U correction.73 When considering the vanadium electronic states, α has one occupied (HOα), and both α and β have two unoccupied, energy degenerate LUσ and LUσ + 1 states, where

Figure 2. Electronic structure of V(IV) doped anatase in ground state of doublet configuration. Spin-polarized density of states for α (red, top) and β (blue, bottom) states and corresponding partial charge density isosurface images for certain interval of energy. Filled states and blue partial charge density isosurfaces represent occupied states whereas no fill and yellow partial charge density isosurfaces represents unoccupied states. There is a general trend of orbital's character change as orbital energy increases for both α and β orbitals: O 2p to V to V/Ti to Ti. Density of states for states with α-projection of spin, however, has one occupied vanadium state in the vicinity of CB that is expected to alter the energy of electronic states compare to that of β, where all vanadium states are unoccupied.

electron−electron interactions and the interaction of electrons with ions, is computed with density functional theory by selfconsistent solving of the Kohn−Sham equations,63 as implemented in the Vienna ab initio simulation package (VASP),64−67 with details provided in Supporting Information and in ref 33.33 In this work, Perdew−Burke−Ernzerhof (PBE) exchange-correlation functional under the generalized gradient approximation (GGA)68 was used. The on-site Coulomb correlation of Ti d-electrons was taken into account by employing Hubbard corrections in Dudarev parametrization

Figure 3. Absorption spectrum for the vanadium (IV) doped anatase model, computed according to eqs 1−3, is shown along with transition types for each set of peaks, described by eqs 15.i−15.iii. The peaks attributed to TiO2 are in the range of 200−500 nm and involve both α- and β-transitions. It is predicted the higher energy peaks are the result of transitions of types i and ii due to larger transition energy and both α- and β-peaks occurring at equal energy and similar amplitude. Additionally, low energy absorption peaks attributed to α electrons are in the range of 500−1500 nm shown in inset A. The largest peak of the system, at about 1900 nm is in inset B and also due to α electrons. All the peaks greater than 500 nm are due to the lone d electron of V (IV) being excited from HOα and, henceforth, must be transition type iii or iv. 5895


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Figure 4. Numerical characteristics of electron−phonon coupling. (A) Fluctuations of KS orbital energies εα,i(t) and εβ,i(t) induced by molecular ⎯→ ⎯

dynamics of ions { RI (t)}. The highest amplitudes of oscillations/fluctuations are shown by the V states, followed by the Ti/V states, while both O and Ti states, of the valence and conduction bands respectively, have less variation and are densely packed. The band gap Δεσgap = ΔεσLU − ΔεσHO, σ = α or β) is much larger for β, than that of α (and, additionally, than the jump needed for HO − 1 oxygen state to get to HO vanadium state in α). (B) Autocorrelation of nonadiabatic couplings for pairs of orbitals near gap with time, computed by eq 5a. (C) Rates of population transfer from states i to j, Rσ,iijj for states around the band gap, calculated according to eqs 5d. A large peak indicates a very fast relaxation, while no peak indicates a trapping state or the possibility of the transfer from state i to state j being skipped altogether for a more efficient pathway. An interesting difference in transitions at the bottom of the conduction band exists for α and β. While α has low rates, β has much higher rates of transitions, likely due to the occupations of a vanadium state in the α band structure.

Electron relaxation for α and β are visualized in Figure 1B and is characterized by the following equation with the starting point (I, II, or III) depending on the aforementioned optical transition type:

σ = α or β. Valence bands for both α and β are made up of O 2p states with the exception of the lone α V 3d single electron state. From this, four optical transitions types are hypothesized to occur and visualized in Figure 1A, with types iii and iv only allowed for α; they are listed below along with redox half reactions: Type i: O 2p to Ti 3d transitions, the only type available in bulk titania, e− + Ti4 + → Ti 3 +, h+ + O2 − → O−

Ti 3 + + V 4 + → Ti4 + + V3 + → V 4 + + e−/h+ I → II → III

Optical absorption spectra, Figure 3, show near identical α and β peaks at lower wavelengths, and solely α peaks past 450 nm, at lower energy. As the only difference between α and β states being the V 3d electron, these lower energy peaks must be attributed to optical transition types iii and iv. While optical transitions types i and ii have similar transition energies, the relaxation pathway for electrons and holes will vary slightly. Both α and β have degenerate LUσ and LUσ+1 states, but α-electrons have a slightly larger energy difference between these orbitals to get to the final relaxation states compared to that of β. The larger sub gap will likely slow down relaxation of an α-electron. For holes, relaxation varies drastically. The β-hole is expected to relax very quickly due to the entire valence band being made up of only densely packed oxygen 2p states. The α-hole is theorized to relax much slower as the HOα−1 O 2p

(15.i)

Type ii: O 2p to V 3d transition e− + V 4 + → V3 +, h+ + O2 − → O−

(15.ii)

Type iii: V 3d to Ti 3d transition e− + Ti4 + → Ti 3 +, h+ + V 4 + → V 5 +

(16)

(15.iii)

Type iv: V to V 3d intra-atomic transition, which does not change oxidation state. These d−d transitions, are expected to be less intense/sensitive to optical perturbations due to the Laporte selection rule, forbidding transitions, which do not match atomic orbital momentum of Δl = ±1.74 The relaxation pathway for both α and β electrons should be similar as the conduction band structures are very similar. 5896


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Figure 5. Comparison of α- and β-transition at similar wavelengths and oscillator strengths. (A) Occupation of states by electron during relaxation with time, ρii,σ(t) directly obtained as a numerical solution of eq 8d. (B) Population of states during hole relaxation with time, also obtained from eq 8a. (C) Change in population of states with energy vs time, computed according to eqs 9 and 10. (D) Change in population along the Z-axis with time, computed by eq 13. For (C) and (D), the color legend in reference to occupation by an electron/hole (n) is as follows: green, Δn = 0, no change in occupation; red, Δn < 0, negative, gaining an electron; blue, Δn > 0, positive, gaining a hole.

relaxation through these states will likely be very quick. The states closest to the conduction band edge are the aforementioned V 3d, hybridized V/Ti 3d and Ti 3d intermixed states. These states have higher amplitudes of fluctuation of energy with time, with the solely V 3d states having the highest amplitudes. Both α/β degenerate LUσ/LUσ+1 states are extremely close in energy and their energies evolve in time in a very similar way, coherent to each other. Additionally, the HOα state is coherent to the LUα/LUα+1 α states. This observation of energies of several orbitals evolving in time with the same frequency suggests that these electronic orbitals are likely coupled to the same normal mode. Figure 4B shows the autocorrelation of nonadiabatic couplings for pairs of orbitals near the gap with time, computed by eq 5a. Rates of population transfer from states i to j around the band gap, represented by elements

state is expected to trap the hole, slowing down relaxation to the HOα V 3d state. Optical transition types iii and iv are predicted to have electron relaxation rates similar to i and ii, while the holes have no relaxation due to already being located on the HOα. III.b. Numerical Characteristics of Nonadiabatic Dynamics. In general, nonadiabatic dynamics implies coupled evolution of electronic and nuclear degrees of freedom, where dynamics of electronic degrees of freedom is expected to modify nuclear trajectory.75 Here, we adopt an approximation, that nuclear motion provides perturbative correction to electronic state evolution. The influence of electronic state dynamics onto nuclear dynamics is neglected.76,77 Molecular dynamics affect orbital energies, as shown in Figure 4A. The lower energy valence band O 2p states are densely packed, as are the higher energy conduction band Ti 3d states. The 5897


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Table 1. Table of Computed Electronic Characteristics for the Highest Oscillator Strength α-Transitions Correlating with Peaks in the Optical Absorption Spectrum (Figure 3)a α-transitions and relaxation rates donor

acceptor

A B C D E F

HO−8 HO−7 HO−37 HO−33 HO−26 HO−8

LU+3 LU+3 LU+6 LU+27 LU+5 LU+2

G

HO−5

LU+1

H I J K

HO HO HO HO

LU+2 LU+3 LU+38 LU+6

OS

eV

Type i α-Optical Transitions 14.9855 3.101 5.69115 3.0873 3.41140 4.2126 3.24056 4.4179 2.91422 4.0244 2.19815 2.8396 Type ii α-Optical Transitions 0.31262 2.3437 Type iii α-Optical Transitions 62.5930 0.654 1.38525 0.9154 0.61411 1.7606 0.58850 1.3047

λ (nm)

Ke

Kh

399.871 401.645 294.355 280.676 308.120 436.681

0.1981 0.1981 0.2431 0.2764 0.2343 0.3472

0.1129 0.1104 0.1978 0.1940 0.1857 0.1129

529.078

−5.5511

0.1088

1896.02 1354.59 704.305 950.410

0.3472 0.1981 0.3048 0.2431

a

Oscillator strength (OS) is calculated by eq 2, and relaxation rates are calculated by eqns 14 and 15 for electrons and holes, respectively, in units of inverse picoseconds. Hole relaxation is relatively constant from transition to transition due to the HOα−1 trapping state; holes always relax slower than their electron counterparts for each transition (aside from type iii, where no hole relaxation occurs). Electron relaxation is slightly more complicated in that the energy gap law is not always abided. Electron excited to the LUα+3 state have slower relaxation than those excited either near the same energy as LU+3 or much higher (e.g. LU+27). It is theorized that because LUα+3 is a solely Ti state, it is harder to relax to the V LUα/HOα+1 degenerate states compared to, for example, LUα+5, which is a Ti/V mixed state that more readily relaxes the electron to the final relaxation states. Additionally, for the deep conduction band transitions, solely Ti states, like LUα+3, can be skipped all together, allowing faster relaxation than if the state was involved in the relaxation pathway.

Table 2. β-Transitions Correlating with Peaks in the Optical Absorption Spectrum (Figure 3) are Computationally Analyzed for Electronic Characteristicsa β-transitions and relaxation rates donor

acceptor

A B C D E

HO−8 HO−8 HO−32 HO−37 HO−11

LU+3 LU+4 LU+28 LU+7 LU+20

F G

HO−13 HO−4

LU LU+1

OS

eV

Type i β-Optical Transitions 14.17975 3.0918 15.13703 3.1204 3.835472 4.4333 3.401683 4.2518 0.745598 3.7532 Type ii β-Optical Transitions 0.249131 2.8036 0.248852 2.7432

λ (nm)

Ke

Kh

401.0609 397.385 279.7014 291.6412 330.3847

1.3379 0.9910 0.4822 0.4892 0.5034

2.5318 2.5318 0.5998 0.5761 1.9223

442.2885 452.0268

−4.8356

1.5184 10.1419

a Oscillator strength (OS) is calculated by eq 2, and relaxation rates are calculated by eqs 14 and 15 for electrons and holes, respectively, in units of inverse picoseconds. Both β electron and hole relaxation rates are faster than all α electron and hole relaxation rates. The results for β clearly indicate vanadium dopant would act as a “recombination center” with such fast relaxations for both electrons and holes.

of Redfield tensor Rσ,iijj are computed according to eq 5d and visualized in Figure 4C. While the rates deep in the valence and conduction bands are consistently high, α and β rates around the band gap vary considerably. For α-optical transitions, relaxation from HOα−1 to HOα is very slow, almost zero peak, while relaxation is also slow for almost all states close to the band gap. For β, certain states have fast relaxation between themselves, some moderate, and some slow. Low or zero peaks usually indicate that either a trapping state or transfer state defined in what follows will be involved in the relaxation. For this paper, a trapping state is defined as a state where the electron or hole occupies a state during relaxation for an extended period of time without significant occupation of adjacent states. For instance, since HOα−1 to HOα transition probability is almost zero and involves a transfer of the hole from O 2p to V 3d, it is likely that HOα−1 would serve as a trap state. A transfer state is a state where occupation

of the state occurs for an extended time period, during relaxation, but there is significant occupation of adjacent states. Transfer states are likely mixed or hybridized states with near degenerate energies. For β, relaxation of electrons to LUβ/LU+1β likely involves a transfer state, a hybridized V/Ti state, which would allow easier transition to the V LUβ/LU+1β degenerate states. The extended time period while a transfer state is occupied allows the relaxation pathway to skip purely Ti states and allows a pathway of Ti → Ti/V → V rather than Ti → Ti/V → Ti → V, where the latter would be energetically unfavorable. A comparison between α/β-optical transitions with similar transition energies highlights the differences in electronic dynamics between the two, which are both classified as type i optical transitions (O 2p to Ti 3d optical transitions). Figure 5 shows the relaxation dynamics following for near exact optical transitions in regards to transition energy, number of states involved, and probability of the transition to occur. Tables 1 5898


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Figure 6. Type i optical transitions in the α-band structure, characterized as O 2p → Ti 3d transition. Each transition in rows B−F is characterized by dynamics of certain observables, organized in four columns: First column, populations of orbitals with i > H0α computed by eq 8d; Second column, populations of orbitals with i > LUα; Third column, distribution of charge density as function of energy, by eq 10a; Fourth column, distribution of charge density as function of position in space, by eq 13. Each transition in rows B−F correlates to a peak in the optical absorption spectra, with computed data listed in Table 1. For electron's initial excitation close to the conduction band edge (B, E, F), relaxation occurs readily to degenerate LUα/LUα+1 states, while those excited to deeper in the conduction band use LUα+5 V/Ti state as a transition state to relax to the V LUα/LUα+1 degenerate states. Hole relaxation is consistent throughout. The hole is trapped on the O 2p HOα−1 state for an extended time period until relaxing to the HOα vanadium state. During the hole trapping, the vanadium dopant experiences an extended time period of negative charge (Δn(z,t)) as the electron relaxes to its final states more quickly than the hole (Δn(ε,t)).

(α) and 2 (β) show that both α and β A transition is HOσ−8 → LUσ+3, transition energy is about 3.1 eV (400 nm), and oscillator strength differs by about 5%. While the transition properties are near identical, the relaxation dynamics vary considerably. As theorized previously, the electron relaxation dynamics act similarly for α and β due to the conduction bands having the same general makeup, but the rate of relaxation differs, with α at 0.1981 and β at 1.3379 ps−1 (Tables 1A and 2A, respectively). This means the α electron relaxes to degenerate LUσ/LUσ+1 after about 10 ps, while β only takes about 1 ps. It is possible the LUα+3 state’s larger sub gap compared to that of LUβ+3 causes the difference in the electron relaxation rate.

Figure 5A shows a similar electron relaxation pathway for both, where final relaxation occurs to the LUσ/LUσ+1 degenerate states, albeit at different rates. Hole relaxation occurs with a larger difference in rates, with α at 0.1129 and β at 2.519 ps−1. Figure 5B shows the difference in the relaxation pathway is the HOσ−1 state. For α, HOα−1 acts as a trap state due to the necessity of the hole transferring from O 2p to the V 3d HOα state, whereas the HOβ−1 is nearly skipped as the hole moves quickly through the O 2p states of the β valence band. The electron/hole distribution with respect to energy and time, computed according to eq 10, are shown in Figure 5C, which emphasizes the differences in the α/β band gap and the 5899


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Figure 7. Type ii α-optical transitions is a O 2p → V 3d transition. Columns follow the pattern of Figure 6. Electron relaxation is immediate, as it is excited to the degenerate LUα/LUα+1 states. Electronic occupation at later stages of evolution is shared between LUα+1 and LUα+1 due to their near degeneracy. The relevant occupation values are not literally equal, ρα,LU,LU(t) ≠ ρα,LU+1,LU+1 since relevant orbital energies are not literally equal εα,LU(t) ≠ εα,LU+1 and experience rare crossings events. The hole relaxes much like that of transition type i: Ti is trapped on the HOα−1 O 2p state until finally relaxing to the HOα V 3d. This causes a significant time period of negative charge on the vanadium dopant, while the hole is relaxing through the VB and trapped on the HOα−1 state.

Figure 8. Type iii α-optical transitions are V 3d to Ti 3d transitions. Rows and columns are organized in the same way as in Figure 6. As the hole is already located on HOα, charge separation occurs due to the electron relaxing through the conduction band. Since no trapping states were noted in analysis, deeper transitions into the conduction band (J, K) will allow for longer charge separation while transitions near the CB edge (H, I) will be short-lived. A noticeable difference between types i and ii is the charge on vanadium during relaxation is positive.

differing relaxation rates, with α relaxation taking much longer for both electron and hole compared to that of β. The dynamics of the distribution of charge along the z-axis with time, computed according to eq 13, is shown in Figure 5D for both α and β. The α-optical transition has holes initially on O 2p states throughout the crystal and the electron on Ti states. Once the electron relaxes to the LUα/LUα+1 vanadium 3d final

relaxation states, the hole is still trapped on HOα−1 O 2p state; this trapping causes a negative charge on the V atom for about 70 ps until the hole is able to finally relax to the HOα vanadium 3d state where recombination is expected to follow. The β-optical transition is very different, as the electron is on the V throughout the entirety of relaxation and the hole relaxes very quickly to HOβ O 2p state. 5900


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Figure 9. Type i β-transitions of O 2p → Ti 3d character. Rows and columns are organized in the same way as in Figure 6. For holes, the relaxation is directly dependent on the energy gap; the further into the valence band the hole is, the longer it takes it to relax. All hole relaxations are still significantly faster compared to those of α-relaxation rates. B acts much like the type i β-transition A, presented earlier with very fast electron relaxation. (C−E) Electron relaxation is slightly slower as a trap state is noted as a yellow peak in the figures of energy distribution dynamics. This delay causes a short time period of charge separation as the electron relaxes to LUβ/LUβ+1 degenerate states, but it is not of the same magnitude as those of type i α-transitions.

Type i α-optical transitions are illustrated in Figure 6B−F, all of which use the same electron relaxation pathway as Aα, a nearly direct transfer of the electron from the initial excited state to LUα/LUα+1 degenerate states during relaxation. α-Optical transitions C and D (Figure 6) use a transfer state, identified as LUα+5, a Ti/V 3d hybrid state. The mixed character of the transfer state allows for easier relaxation to the final resting place of the vanadium ion. It is important to note the LUα+5 state is considered a transfer state as no considerable increase in the electron relaxation rate occurs when comparing to the other type i transitions that do not have LUα+5 in their relaxation pathway (Table 1). Relaxation rates for the electron for α type i optical transitions range from 0.1981 ps−1 (A/B) to 0.3472 ps−1 (F). The slower electron relaxation of A/B is due to LUα+3 being a pure Ti 3d state, which needs more time to transfer charge from Ti ions to the V dopant ion. Hole relaxation for all type i optical transitions (A−F) are nearly identical. The hole travels quickly through all O 2p states, then is trapped on the final 2p state of HOα−1 until finally relaxing to the HOα vanadium 3d state. This trapping causes a charged state for all type i α-transitions, as shown in the final column of Figure 6, illustrating dynamics in

In conclusion to the comparison of excitations of α and β character, the initial characteristics of the transition are the same when considering energy of excitation, likelihood of the transition occurring, and the categorized type of transition; however, the hot carrier relaxation dynamics show considerable differences. The α electron relaxes more slowly compared to that of β due to a larger sub gap to overcome. The α hole encounters a trapping state at HOα−1, causing a significant localized negative charge on the vanadium dopant for an extended time period, while the β hole readily relaxes through the O 2p valence band states to HOβ. The result is the β Aβ transition having a very quick charge separation of the electron and hole prior to recombination while the α Aα transition has extended charge separation with localized negative charge on the V dopant. III.c. Comparison of Alternative Pathways of Photoexcitation. The rest of the optical transitions corresponding to peaks in the absorption spectra are grouped together by transition type, as introduced in Figure 1. Type i α-optical transitions all behave similarly to the Aα discussed above, except for those that occur deeper into the conduction band, which utilize a transfer state for faster relaxation to LUα/LUα+1. 5901


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Figure 10. Type ii β-transitions that are O 2p → V 3d in character. Rows and columns are organized in the same way as in Figure 6. Type ii βtransitions of relevancy (have correlating peaks in the absorption spectrum), only those where the electron is initially located on the degenerate LUβ/ LUβ+1 are analyzed. In both cases, electron relaxation is instant and hole relaxation happens very quickly, with F slower than G due to the electron coming from deeper in the valence band.

rate to be 0.5998 ps−1. Hole relaxation is directly dependent on the number of states involved in the relaxation pathway. For type ii transitions illustrated in Figure 10F,G, electron/hole relaxation is practically instantaneous due to the electron being excited to the degenerate LUβ/LUβ+1 states and the hole quickly relaxing through the O 2p states of the valence band. Therefore, it is predicted that β-transitions would all relax very quickly and, as they do not show an extended localized charge in the crystal lattice like that of type i α transitions, would be less effective as a photocatalyst compared to that of the α transitions.

space Δn(z,t). Hole relaxation for type i α-transitions ranges from 0.1104 to 0.1978 ps−1, so very little disparity exists. The lone α type ii transition (O 2p → V 3d) is shown in Figure 7, but it is one of the most interesting cases. The transition occurs at 530 nm, slightly past the TiO2 attributed peaks. The electron is excited to LUα+1, which is a degenerate state of LUα, so immediate “relaxation” of the electron occurs. However, the hole takes the longest time to relax to HOα despite the initial hole being located on HOα−5. Nearly all of the hole density populates the metastable state HOα−1 before final relaxation to HOα, the vanadium dopant. As a result, the dopant has a very intense negative charge for well over 100 ps. This transition is extremely important as no β-transition competes at the same wavelength, and a very long lasting negative charge on V could be exploited for catalysis by targeting the specific wavelength of this transition. Finally, α also shows type iii optical transitions, those of V → Ti 3d, shown in Figure 8H−K. These transitions all occur at energies below the TiO2 peaks of the absorption spectra. Since an electron in the ground state occupies only one vanadium d-orbital, all of these transitions occur with the hole already located on HOα and the V ion. Henceforth, this localized positive charge is predicted to be less effective as a photocatalyst when the electron shows quick relaxation, Table 1, which is the case for H, J, and K, since both the electron and hole will be on the V dopant ion rather fleetingly. However, I shows a slower electron relaxation rate, similar to that of A/B, due to LUα+3 being Ti 3d in character, slowing down the relaxation to LUα/LUα+1 V 3d states. While the type i transition may show some potential for photocatalysis, most transitions in this energy range are not considered promising. β-Optical transitions, whether type i or ii, show results, which are very similar to each other. For type i, shown in Figure 9B− E, the electron relaxes to LUβ/LUβ+1 degenerate states through a transfer state (Ti/V 3d hybridized state which aids in relaxation from Ti to V states) similar to those of α-relaxations. Additionally, since the entire valence band is O 2p states, hole relaxation is very rapid. Table 2 shows the slowest electron relaxation rate to be 0.4822 ps−1 and the slowest hole relaxation

IV. DISCUSSION For an α-transition, in general, the localized charges are longer lived compared to those of β. A longer relaxation, and henceforth charge separation, increases the likelihood of the charge catalyzing a reaction or being harvested for light energy purposes. The longer relaxations for α-holes and electrons are directly related to two things: the hole trapping HOα−1 state and the larger sub-gap to overcome during electron relaxation. These indicated conflicting properties for the system as a whole. While β-transitions show very fast relaxation, which could correlate to vanadium being coined as a “recombination center” in an experimental-based article,26 α-transitions show promising properties as a catalytic material. One problem with the conflicting results is the α-transitions that show the longestlifetime localized negative charge, types i and ii, directly compete for absorption with β types i and ii transitions. Therefore, it would be very difficult to separate the α- and β-transitions without using polarized light or magnetic moments in experiment. Another property to note is the fact that V can act as either an nor p-type dopant, where it takes on a negative charge for types i and ii transitions and a positive for type iii transitions. As βtransitions are all n-type for the dopant V, however, short-lived charge, there is no competing processes for this interesting property. The transition where V is a p-type dopant is solely due to an α-transition and the energy difference compared to that of types i and ii transitions is rather significant. It would allow an experiment to exist, where negative charge on vanadium is created by exciting in the 200−450 nm wavelength range and a positive 5902


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transition metals as well as altering the application to include spin orbit coupling calculations, in the framework of noncollinear density functional theory. Spin-resolved electronic structure calculations become necessary when dealing with transition metal, magnetic, and even some carbon materials, intermediates, and radicals. It is anticipated that this methodology can be broadly implemented on numerous transition metal applications, including optoelectronics, information storage, laser crystals, dyes, photovoltaic materials, and metal oxides for photoelectrochemical water splitting.

charge for the 700+ nm wavelengths, thus enabling control of charge transfer direction by tuning the wavelength of incident light. While no surface is available for charge transfer to occur in this specific model, it is predicted that a vanadium(IV) dopant in anatase at or near a surface would embody similar properties mentioned above. Specifically, for α-transitions, a photoelectrochemical cell ideally could harvest the charge (negative for types i and ii transitions, positive for type iii) for use in reducing water into hydrogen and water or other catalytic processes. Additionally, any impurities in the crystal, which cause vanadium lose an electron or gain an electron would alter the results. Losing an electron would cause the loss of the lone V 3d electron and theoretically the HOα−1 would no longer be a trapping state.

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ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcc.5b12167. Additional details on methods and spin polarized density (PDF).

V. CONCLUSIONS A methodology for nonadiabatic dynamics of spin-polarized models is introduced and applied to vanadium doped TiO2, with doublet ground state configuration. An original hypothesis of the existence of several classes of pathways selected by excitation and spin polarization has been confirmed by numerical data. Types i (oxygen-to-titanium) and ii (oxygento-vanadium) transitions were observed for both α and β spin components, while the type iii (titanium-to-vanadium) optical transition was observed for an α-optical absorption peak; no definitive type iv intra-atomic vanadium transitions were observed. It was found that α- and β-transitions, even when similar in transition energy, have completely different relaxation dynamics. A photon of a given wavelength can therefore induce two independent and noncoinciding pathways of photoinduced dynamics. The difference in the pathways is dictated by the difference in electronic structure: sub gaps and band energy alignment. Specifically, α type i transitions would be ideal for photocatalysis in that they have an extended relaxation time, which causes spatial charge separation where a localized negative charge on the vanadium could be used for catalysis. Dynamics for these type i transitions show extended trapping states. Additionally, the type iii α excitations cause a local positive charge on the vanadium in respect to ground state density distribution, and has no competing β-transition in the optical absorption spectra. For β-transitions, the very quick relaxations to the lowest excitation are dictated by the values of sub-gaps in the electronic structure. It would be difficult to exploit the type i transitions as they are all paired to β peaks in the optical absorption spectra. At this stage of research, there is no simple way to excite only α- or only β-transitions, as in nature, they would simply coexist. However, recognizing the aspects of the α- and β-pathways is useful in that it not only confirms previous experimental results,26 but also suggests the material may be more useful for photocatlysis as the α types i and iii pathways may be detected due to recent advances of lab techniques and increase in sensitivity of instruments. Explored scenarios exhibited features in agreement with the Energy Gap Law, that is, long relaxation occurs over larger sub gaps and an equal distribution of population over degenerate energy levels. The new method detailing processes of spin polarization in nonadiabatic dynamics was successfully implemented on the simple model of vanadium(IV) doped anatase. In its present form, the method can adequately describe compounds composed of first row transition metals and models in nonsinglet configurations (doublet, triplet, quadruplet, etc.). Future applications include replacing the vanadium with other first row

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]; [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS This research was supported by NSF awards EPS-0903804 and CHE-1413614 for methods development and by DOE, BES Chemical Sciences, NERSC Contract No. DE-AC02− 05CH11231, allocation Award 89959 “Computational Modeling of Photocatalysis and Photoinduced Charge Transfer Dynamics on Surfaces.” Authors acknowledge the use of computational resources of USD HPC cluster managed by Douglas Jennewein. S.J.J. thanks DGE-0903685 South Dakota IGERT program for support. D.S.K. acknowledges support form NDSU Department of Chemistry and Biochemistry and College of Science and Mathematics. Authors thank Yoshiyuki Kawazoe (Japan) for inspiring discussion on aspects of photocatalytic applications of doped titania. D.S.K. thanks Andrei Kryjevski for discussions on code implementation and Ge (Hugo) Yao for feedback on transition metal spectroscopy. Authors thank Wendi Sapp and Peng Cui for editorial suggestions and discussions. T.M.I. thanks the Center for Computational Materials Science, Institute for Materials Research, Tohoku University (Sendai, Japan) for their continuous support of the SR16000 M1 supercomputing system. Calculations were partially performed on the High Performance Cluster of National Open Research Laboratory of Information and Space Technologies of KazNRTU after K.I. Satpayev.

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REFERENCES

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