Ph. D Tomas Mancal
email: mancal@karlov.mff.cuni.cz

Jan Olsina
email: olsina@karlov.mff.cuni.cz

Jan Olšina

Tomáš Mančal

Coherence Dephasing in Excitonic Molecular Complexes: Role of Markov and Secular Approximations

Keywords: Coherence dephasing, molecular excitons, reduced density matrix, Markov approximation, secular approximation Coherence dephasing, molecular excitons, reduced density matrix, Markov approximation, secular approximation PACS 05.60.Gd, 05.30.-d, 78.47.jh, 82.53.Kp

We compare four different types of equations of motion for reduced density matrix of a system of molecular excitons interacting with thermodynamic bath. All four equations are of second order in the linear system-bath interaction Hamiltonian, with different approximations applied in their derivation. In particular we compare time-nonlocal equations obtained from so-called Nakajima-Zwanzig identity and the time-local equations resulting from the partial ordering prescription of the cummulant expansion. In each of these equations we alternatively apply secular approximation to decouple population and coherence dynamics from each other. We focus on the dynamics of intraband electronic coherences of the excitonic system which can be traced by coherent two-dimensional spectroscopy. We discuss the applicability of the four relaxation theories to simulations of population and coherence dynamics, and identify features of the two-dimensional coherent spectrum that allow us to distinguish time-nonlocal effects.

Response to Reviewers: We thank both referees for their effort

Also, for excitonic systems such as the one we study, diagonal peaks oscillate only in the non-rephasing part. The total spectrum has thus also the advantage of larger number of oscillating spectral features. 3. Following the referee's suggestion to rewrite the paper in a more purposeful manner, we have removed several intermediate formulas and presented only their results, as these can be found in standard textbooks. Overall we have tried to concentrate on the purpose of the paper which is to demonstrate difference between approximations applied to second order relaxation theories. We kept section introducing some level of details of the 2D spectroscopy. We think that details of the theory of non-linear spectroscopic methods might be useful for a reader of the Journal of Molecular Modeling, which has its main readership in different theoretical fields 4. We have tried our best improve our English throughout the manuscript.

Reviewer #2:

We have numbered the seven points raised by the referee successively. Points 1 -3: All sentences were corrected according to the referee's suggestion. Point 4: We agree with the referee that the paragraph he/she mentions is not very clearly written, so we rewrite and shorten it. Point 5: Sentence corrected according to the referee's suggestion. Point 6: We add a citation to Wilkie and Wong, J. Phys. A: Math. Theor. 42 (2009) 015006, where the positivity of time non-local equations are discussed, and an example of positivity breakdown is given. 7. We define the term "coherence transfer" in the Section 3.3. We added a note in the text (second paragraph of section 5.1) which explains our conclusion about the coherence transfer. The presence of non-secular terms causes small oscillations of the populations in Fig. 6.

Introduction

Modeling molecular properties related to their non-equilibrium dynamics requires various theoretical approaches depending on the particular microscopic processes related to the observed molecular features. Since the dawn of quantum mechanics, properties of molecules and solids have been studied theoretically in ever greater detail. This has led in recent years to a state in which dynamics of complex systems with multitude of degrees of freedom (DOF) is accessible to quantitative theoretical study [START_REF]Quantum Dynamics of Complex Molecular Systems[END_REF]. Many properties of molecular systems are directly related to the equilibrium or time dependent conformations of nuclear DOF for which electronic states play the role of a background contributing to the nuclear potential energy surfaces. Problems like these are the realm of molecular dynamics (MD) in its classical, quantum or mixed versions and quantum chemistry (QC), where impressive qualitative and quantitative results have been achieved in recent years. For certain types of dynamical problems, however, less expensive model approaches are the preferred choice due to the scale of studied system or due to the physical nature of studied processes. A good example of such a problem is ultrafast photo-induced excited state dynamics of small molecular systems and their aggregates [START_REF] Van Amerongen | Photosynthetic Excitons[END_REF]. Here, most of the relevant experimental information is only available through ultrafast non-linear spectroscopy, and thus the theory has to span the whole distance between the microscopic dynamics of the molecular system, and the macroscopic description of experimental signals [START_REF] Mukamel | Principles of nonlinear spectroscopy[END_REF]. Typical field in which such an approach has yielded deep understanding of the relevant physico-chemical processes is the study of primary processes in photosynthesis. The related quantum mechanical problem is usually formulated in terms of a model describing the relevant DOF of the system (electronic states of photosynthetic molecules), and a thermodynamics bath (the protein environment). Parameters for such models can be supplied by experiment [START_REF] Renger | [END_REF], QC studies [5,6], MD modeling [7], or are a result of suitable simplified models [START_REF] Mukamel | Principles of nonlinear spectroscopy[END_REF].

Recent advances in non-linear spectroscopy have opened a wide new experimental window into the details of ultrafast photo-induced dynamics of molecular systems. Experimental realization of two-dimensional (2D) coherent spectroscopy in the visible and near IR regions [8][9][10][11] has enabled to overcome some of the frequency-vs. timeresolution competition problem otherwise faced by ultrafast spectroscopy, and yielded thus unprecedented experimental details of the time evolution of molecular excitations. Most importantly, it was predicted that the presence of certain oscillatory features in 2D spectra is a manifestation of coherences between molecular excited states [12,13]. It was also concluded that these oscillations should be present in the 2D spectrum of photosynthetic Fenna-Matthews-Olson (FMO) chromophore-protein complex [12]. Experimental results not only confirmed this prediction [14], but yielded also surprising results such as unexpectedly long life time of these coherences, as compared to the predictions of standard dephasing rate theory. Furthermore, while possible coherence transfer between the pairs of electronic levels was ignored by the relaxation theory used in Ref. [12], the experiment provided some evidence for its role in excitation energy transfer. It was speculated that photosynthetic systems might use the coherent mode of energy transfer to more efficiently channel excitation energy by scanning their energetic landscape in a process similar to quantum computing [14]. More experiments have recently reported coherent dynamics in photosynthetic systems [15] and conjugated polymers [16], and the field of energy transfer in photosynthesis has seen an increased interest from theoretical researchers from previously unrelated fields [17][18][19][20].

Theoretical basis for the description of the decoherence phenomena in excitation energy transfer has been developed long ago in the framework of the reduced density matrix (RDM) [21,22]. Equations of motion (EM) resulting from this scheme are characterized by the presence of time retarded terms responsible for energy relaxation and decoherence processes. Equations of this type will be denoted as time non-local in this paper. Later, an alternative approach to the derivation of the EM for the RDM has emerged which yields time local equation of motion [23,24]. Both theories express the relaxation term in form of an infinite series in terms of the system-bath interaction Hamiltonian, but differ in time ordering prescriptions for the cumulant expansion of the evolution operator. The time local equations correspond to so-called partial time ordering prescription of the cummulant expansion, while the time non-local equations result from so-called chronological time ordering [25,26]. Although the two schemes yield formally different EM for the RDM, they are in fact equivalent as long as the complete summation of the corresponding infinite series is performed. When the infinite series are truncated at a finite order, the two theories yield equations that predict different RDM dynamics. This is a result of different statistical assumptions about the bath that are implicitly made in the two cases [25,26]. In all orders of expansion, so-called Markov approximation can be used to transform the time non-local equation of motion into a certain time-local form. This has to be regarded as an additional approximation which simplifies the numerical treatment of the time non-local equations. Interestingly, in the second order the time-local equations and the time non-local equations with Markov approximation have exactly the same form.

Until recently, most experiments were not sensitive to coherence between electronic levels. This allowed a host of further approximations to simplify EM. Most notably, the secular approximation, which amounts to decoupling RDM elements oscillating on different frequencies from each other, has limited the energy transfer phenomena to separate dynamics of population transfer and coherence dephasing [START_REF] May | Charge and Energy Transfer Dynamics in Molecular Systems[END_REF]. Even on very short times scale, experiments aimed at studying population dynamics (pump probe) are not sensitive enough to coherence between electronic levels to require non-secular theory for their description. It was however suggested that measured relaxation time can be distorted by non-secular effects [START_REF] Čápek | [END_REF]. Consequently, most of the theory developed for evaluation of experiments has been aimed at improving calculation of the population relaxation rates [29][30][31]. With experiments now uncovering new details about the role of electronic coherence, theoretical methods beyond rate equations for probabilities which are both accurate and numerically tractable are required. Although schemes for constructing EM for the RDM beyond second order, based on co-propagation of the RDM with auxiliary operators, seem feasible and promising [32,33], second order theories might still be the only option for treatment of extended molecular systems. It was suggested previously that second order perturbation theory with respect to systembath coupling provides a suitable framework for development of such methods [34]. This notion is also supported by the fact that in the special case of so-called spin-boson model, second order time-local equation of motion already represents an exact equation of motion for the RDM [35]. Recently, non-secular Lindblad rate equations which are valid for arbitrarily strong system-bath coupling were parametrized by secular time local second order (Redfield) rate theory [36]. Such approach seems to reproduce effects observed in experiment of Ref. [14]. However, due to the time-scale of the experiment, constant rate equations might not be appropriate for the description of the short time dynamics. Consequently, we concentrate exclusively on theories with time dependent relaxation rates, here.

In this paper we study the following four different second order theories: (a) full time non-local (full TNL) equation of motion resulting from the Nakajima-Zwanzig identity or equivalently from the chronological ordering prescription in the cummulant expansion, (b) the full time local (full TL) equation of motion resulting from the partial ordering prescription in the cummulant expansion, or equivalently from Markov approximation applied to TNL equation, (c) time non-local equation with secular approximation (secular TNL), and (d) time local equation with secular approximation (secular TL). We discuss the applicability of these equations to the description of the energy relaxation and decoherence dynamics in small systems of molecular excitons with the emphasis of on recent 2D spectroscopic experiments and the dynamics of coherence between electronic excited states. Note that, in this paper, full refers to the equations where no secular approximation has been applied. These equations are still of second order perturbation theory in the system-bath coupling.

The paper is organized as follows. The next section introduces Hamiltonian description of an aggregate of small molecules embedded in a protein or solid state environment. In Section 3 we describe the details of four different EM for the RDM describing electronic states of a molecular aggregate. Two-dimensional coherent spectroscopy, and non-linear spectroscopy in general are introduced in Section 4. In Section 5 we present and discuss numerical results comparing different theories of relaxation on calculations of coherence life time and 2D spectra.

Model Hamiltonian

The investigated molecular system is an aggregate composed of N monomers embedded in protein environment. Let us first consider a monomeric molecule (a chromophore) embedded in the environment, but insulated from interaction with its neighboring monomers. The monomer Hamiltonian has a form

H m = ε (m) g + T (Pm) + V (m) g (Qm) |gm gm| + ε (m) e + T (Pm) + V (m) e (Qm) |em em| , (1)
where |gm , |em denote electronic ground and excited states, and ε

(m) g , ε (m) e
represent electronic energies of these states. The kinetic term T (Pm) and the potential terms

V (m) g (Qm), V (m) e
(Qm) represent the intra-molecular DOF and the protein environment (bath or reservoir) interacting with these states. By Qm (Pm) we denote the (possibly macroscopic) set of coordinates (impulses) describing both the intramolecular nuclear DOF of the m-th monomer as well as the DOF of its surroundings. The total Hamiltonian of the monomer can be split into the system, reservoir and the system-reservoir coupling terms

H m S ≡ ε (m) g |gm gm| +(ε (m) e + V (m) e (Qm) -V (m) g (Qm) eq )|em em| , (2)
H m R ≡ [T (Pm) + V (m) g (Qm)] × (|gm gm| + |em em|) = [T (Pm) + V (m) g (Qm)] ⊗ 1, (3)
H m S-R ≡ V (m) e (Qm) -V (m) g (Qm) -V (m) e (Qm) -V (m) g (Qm) eq |em em| ≡ ∆Φm(Qm)|em em| . (4)
Here, A(Q) eq represents averaging of an arbitrary Q-dependent operator over equilibrium state of the bath. By this choice of the splitting we have assured that ∆Φ (m) (Qm) = 0 for the system in equilibrium. To simplify the notation, we redefine electronic energy of the excited state to include the equilibrium average of the potential energy difference between the electronic excited and ground states, ε(m

) e = ε (m) e + V (m) e (Qm) -V (m) g
(Qm) eq and we drop the tilde over ε (m) e further on in this paper. An aggregate built out of these monomers can be represented on a Hilbert space composed of collective aggregate states. We define the aggregate ground state

|g = N m=1 ⊗|gm , (5)
states with a single excitation

|un = n-1 m=1 ⊗|gm ⊗ |en N m ′ =n+1 ⊗|g m ′ , (6)
and multi-excited states in an analogical manner. We drop the sign ⊗ in further consideration for the sake of brevity. The Hamiltonian of the aggregate is constructed using the energies of collective states

H non-int S = εg|g g| + n (∆εn + Ω) |un un| + h. e. t., (7)
where εg = n ε

(n) g , Ω = N -1 n ε (n)
e , and ∆εn = ε

(n) e -Ω + m =n ε (m)
g . The abreviation h. e. t. denotes higher excitonic terms. Due to the fact that the monomers are positioned in a tight aggregate, we have to account for the interaction energy between their excited states. The interaction energy between states |um and |un will be denoted Jmn, and the corresponding contribution to the total Hamiltonian reads

H int S = n =m (Jnm|un um| + c.c.) + h. e. t.. (8)
Due to the off-diagonal terms Jmn the collective states defined in Eq. ( 6) are not eigenstates of the total Hamiltonian

H S = H non-int S + H int S .
Although the basis of the states |g , |un and multiple-excitation states of the aggregate provides efficient means for defining the Hamiltonian, it is more practical to switch into the basis of eigenstates of the Hamiltonian H S . One of the reasons is that while matter interacts with light, the differences between eigenenergies of H S define the resonant transition frequencies. The system-bath coupling part of the aggregate Hamiltonian reads

H S-B = n ∆Φn|un un| + h. e. t., (9)
and thus no terms in the total Hamiltonian couple the ground state with the first excited state or higher excited state bands. In fact, the total Hamiltonian splits into blocks separated approximately by the energy Ω (see Fig. 1). This reflects the neglecting of all adiabatic couplings, which are supposed to be so weak that they do not lead to transitions on the time scale of interest (femto and picoseconds). This property is well justified e.g. for chlorophyll systems.

For the subsequent use in this paper, we denote the eigenstates of the total electronic Hamiltonian H S by |ūa , a = 1, . . . , N , for single exciton states formed as linear combinations of single excitation states |un and | Ūa , a = N + 1, . . . , N + N (N -1)/2, for two-exciton states formed from the linear combination of pairs of single excitation states.

Second order relaxation theories

In this section we consider interaction of the electronic system described by the Hamiltonian H S with a macroscopic bath composed of the DOF of the molecular surroundings. Standard approach to such a problem is to derive EM for reduced density operator

ρ(t) = tr B W (t), ( 10
)
where tr B is a trace over the bath DOF and W (t) is the total density operator. The derivation is conveniently achieved via projection operator technique [START_REF] May | Charge and Energy Transfer Dynamics in Molecular Systems[END_REF][START_REF] Fain | Irreversibilities in Quantum Mechanics[END_REF]. Two general schemes exist. First, so called Nakajima-Zwanzig identity leads to integrodifferential, i.e. time-non-local master equation for the RDM with a convolution memory term. Alternatively, a different projector operator identity can be used to derive EM which is time-local, leading to so-called convolutionless master equation [START_REF] Fain | Irreversibilities in Quantum Mechanics[END_REF]. Interested reader can refer to Refs. [25,26,[START_REF] Fain | Irreversibilities in Quantum Mechanics[END_REF] for details of the derivations and comparison of the two schemes. Below, we will use specific second order approximation to the general equations.

System-Bath Coupling

We will now assume the interaction Hamiltonian in a form of Eq. ( 9) where index n now runs through all relevant single-exciton and multi-exciton states

H I = n ∆ΦnKn. (11)
Correspondingly, Kn = |un un| for single excitonic states. the system-bath interaction contribution can be conveniently expressed via so-called bath (or energy gap) correlation functions defined as

Cmn(τ ) = tr Q {U B (-τ )∆ΦmU B (τ )∆Φnweq}. ( 12
)
Here, we have chosen a specific form of the bath density matrix w ≡ weq, where weq is the equilibrium density matrix of the bath DOF. Defining also an operator

Λm(τ ) = n Cmn(τ )U S (τ )KnU † S (τ ) (13)
Fig. 1 Illustration of the level structure of an excitonic system. The excited states |en of N monomers with transition frequency Ω (left part of the figure) split due to the resonance interaction into N one exciton states |ūn (right). Absorption spectra of ensembles of noninteracting (left) and interaction (right) monomers. The system also exhibits higher excitons states (two-exciton states | Ūn are depicted here), with Ω being the mean transition frequency from the one-to two-exciton bands. A pictorial 2D spectrum with peaks resulting from transitions between the ground-and one-exciton states (red arrows) and one-and two-exciton states (blue arrows) is presented in the upper left corner of the figure. The transitions between the ground-and one-excitons states lead to positive contributions to the 2D spectrum (absorption and ground state bleach), while the transitions between the one-and two-exciton states result in a negative contribution (excited state absorption).

and a superoperator M (2) (τ ) such that

M (2) (τ )A = m [Km, Λm(τ )U S (τ )AU S (-τ ) -U S (τ )AU S (-τ )Λ † m (τ )] -, (14)
the EM for the RDM can be rewritten either is a time local form as

∂ ∂t ρ(t) = -iL S ρ(t) - m t-t0 0 dτ [Km, Λm(τ )ρ(t) -ρ(t)Λ † m (τ )] - (15)
or in a time non-local form as

∂ ∂t ρ(t) = -iL S ρ(t) - m t-t0 0 dτ [Km, Λm(τ )U S (τ )ρ(t -τ )U S (-τ ) -U S (τ )ρ(t -τ )U S (-τ )Λ † m (τ )] -. (16)
It is important to note that the two equations and their solutions differ. They are differnt second order approximations to the exact equation of motion for the RDM Provided we can supply a model for the correlation function Cmn(τ ) we are in position to write down the EM for RDM in terms of known quantities. The last step necessary to implement these equations is to represent them in the basis of the eigenstates of the aggregate Hamiltonian. We define

ρ ab (t) = ūa|ρ(t)|ūb , a, b = 1, . . . , N, (17)
ρ ab (t) = Ūa|ρ(t)| Ūb , a, b = N + 1, . . . , N + N (N -1)/2, (18)
and in a similar manner for matrix elements of other operators and superoperators. This leads to

∂ ∂t ρ ab (t) = -iω ab ρ ab (t) - cd R abcd (t)ρ cd (t), (19)
with R abcd (t) being the matrix elements of the superoperator defined by the r. h. s. of Eq. ( 15), and

∂ ∂t ρ ab (t) = -iω ab ρ ab (t) - cd t-t0 0 M abcd (τ )ρ cd (t -τ ), (20)
with M abcd (τ ) the matrix elements of the superoperator defined by the r. h. s. of Eq. ( 16). All the quantities needed to calculate the matrix elements R abcd (t) and M abcd (t) are known provided the energy gap correlation function is know.

Energy gap correlation function

As a suitable model of the energy gap correlation function we choose so-called multimode Brownian oscillator (BO) [START_REF] Mukamel | Principles of nonlinear spectroscopy[END_REF]. In general, Brownian oscillator model can interpolate between underdamped intra-molecular DOF and (usually) overdamped bath DOF representing the immediate surroundings of the molecule. In this paper, we assume the correlation function of the energy gap of each molecule in the aggregate to be the same, and independent of neighboring molecules, i.e.

C ab (t) = C(t)δ ab . (21)
The correlation function C(t) is taken in a form of the overdamped BO model

C(t) = -i λΛe -Λ|t| sgn t +λΛ coth β Λ 2 e -Λ|t| + 4Λλ β ∞ n=1 νne -νn|t| νn 2 -Λ 2 , ( 22
)
with νn ≡ 2πn β , β ≡ 1 k B T , Λ ≡ 1 τc . ( 23
)
Here, λ is the reorganization energy, νn are so-called Matsubara frequencies, k B is the Boltzmann constant, T is the thermodynamic temperature and τc is the so-called bath correlation time. The BO form of the correlation function satisfies all general constraints put of a correlation function by thermodynamics [START_REF] Mukamel | Principles of nonlinear spectroscopy[END_REF]. Apart of the temperature which we assume to be T = 300 K in all calculations in this paper, the BO model is determined by two parameters only; by the reorganization energy λ which is experimentally related to the Stokes shift S = 2λ and by the bath correlation time τc. BO is a widely used, well physically motivated, but not the only possible model for the bath correlation function. Implications of other forms of the correlation function for the RDM dynamics will be studied elsewhere.

Secular and constant relaxation rate approximations in the energy eigenstate basis

Eqs. ( 19) and ( 20) are systems of coupled (integro-) differential equations for the elements of the RDM. From the first terms on the r. h. s. we deduce that the element ρ ab (t) oscillates with a frequency close to ω ab . It is often justified to assume that two terms oscillating on different frequencies are independent of each other. For their envelopes ρab (t) = e iω ab t ρ ab (t) we have

∂ ∂t ρab (t) = - cd R abcd (t)e i(ω ab -ω cd )t ρcd (t), (24)
and integration over time has therefore a relatively smaller contribution when ω abω cd = 0. Neglecting these contributions, usually termed secular approximation [START_REF] May | Charge and Energy Transfer Dynamics in Molecular Systems[END_REF], leads to setting corresponds to the total transition rate from the level a to all other levels. The terms R abab (t) (a = b) are rates of the damping of a coherence element ρ ab (t). In secular approximation, the dynamics of populations of electronic levels is thus decoupled from the dephasing of coherences. The terms other then population tranfer and dephasing rates will be refered to as non-secular terms, here. The processes related to these terms, such as coherence to coherence transfers, or coherence to population transfers, will be refered to as non-secular processes or coherence transfer processes. The above arguments for the secular approximation apply also to the integrodifferential equation (20), and we can thus define four different second order EM for the RDM, with different levels of approximation. From the perspective of our derivation, the most general second order equation is Eq. ( 20), which we have denoted full TNL. The convolutionless Eq. ( 19) denoted full TL can be regarded its approximation, but it can also be alternatively viewed as derived by different cummulant approximation, see Refs. [25,26]. The set of four methods investigated here is completed by applying secular approximation to the full TNL and full TL equations.

R abcd (t) = 0, (25)
All four sets of EM we consider here are extensions to the two well-known constant relaxation rate theories. To arrive at the well-known Redfield equations [START_REF] May | Charge and Energy Transfer Dynamics in Molecular Systems[END_REF], one can assume certain coarse graining of the RDM dynamics so that all significant changes to the ρ(t) occur on a time scale much longer than the correlation time τc. Then time t 0 in Eq. ( 15) can be put to -∞ and the integration limits are then from zero to infinity. The relaxation tensor R thus becomes time independent. If we, on the other hand, consider the decay of C(t) to be much faster than even the transition frequencies between electronic levels, we can assume C(t) ≈ C 0 δ(t) and Eq. ( 15) has the well-known Lindblad form [START_REF] Lindblad | [END_REF][START_REF] May | Charge and Energy Transfer Dynamics in Molecular Systems[END_REF]. Only for the Lindblad form and for the Redfield equations in secular approximation, it can be shown that the diagonal elements of ρ(t) are always positive. For all other equations we have derived here, this assertion cannot be proven in general. This is a consequence of the fact that they are derived in a low order of perturbation theory.

Non-linear Spectroscopic Signals

Non-linear spectroscopic signals are very well described by time-dependent perturbation theory [START_REF] Mukamel | Principles of nonlinear spectroscopy[END_REF]. The EM, Eqs. ( 15) to (16), can be extended by semiclassical light-matter interaction term. This yields

∂ ∂t ρ(t) = -iL S ρ(t) -D[ρ(t)](t) + iVρ(t)E(t), (26)
where

E(t) = n • E(t)
is the projection of the external electric field vector E(t) on the normal vector n in direction of the molecular transition dipole moment. The symbol D[ρ(t)](t) represents the relaxation term chosen from the full TNL, full TL, secular TNL or secular TL equations of motion. The superoperator V is a commutator with the dipole moment operator µ = nµ, so that for an arbitrary operator A we have

VA = 1 [µ, A] -. (27)
4.1 Third-order non-linear response theory Non-linear optical signals are related to the RDM via polarization

P (t) = tr{µρ(t)}. (28)
In particular, for the third order non-linear signal E

s (t) one can write

E (3) s (t) ≈ iωP (3) (t) = iω tr{µρ (3) (t)}, (29)
where the upper index (3) denotes that the quantity is of the third order of the perturbation theory with respect to the external electric field E(t). By defining the evolution superoperator U(t) which fulfills Eq. ( 26) with E(t) = 0 we can write the third order perturbation term as

ρ (3) (t) = -i ∞ 0 ∞ 0 ∞ 0 dτ 3 dτ 2 dτ 1 U(τ 3 )VU(τ 2 )VU(τ 1 )Vρ 0 ×E(t -τ 3 )E(t -τ 3 -τ 2 )E(t -τ 3 -τ 2 -τ 1 ). ( 30
)
In experiment, the laser field is often prepared in a form of three incident pulses

E(t) = A 1 (t -t 1 )e -iΩ1(t-t1)+ik1r +A 2 (t -t 2 )e -iΩ2(t-t2)+ik2r +A 3 (t -t 3 )e -iΩ3(t-t3)+ik3r + c.c., (31)
with different k-vectors k 1 , k 2 and k 3 . In the rest of the paper we assume

Ω 1 = Ω 2 = Ω 3 ≡ Ω, A 1 (t) = A 2 (t) = A 3 (t) ≡ A(t).
The expression obtained by inserting Eq. (30) into Eq. ( 29) can be significantly simplified in cases where the system consists of a ground-state and a band of excited states, with the transition frequency close to resonance with the laser pulse frequency Ω, and by assuming the laser pulses are ultra short, i.e. A(t) ≈ E 0 δ(t). For an experiment which detects non-linear signal emitted in the direction -k 1 + k 2 + k 3 , the third order signal has a frequency ≈ Ω and it is obtained from just a handful of response functions that represent certain contributions to the triple commutator in Eq. (30). The details of the derivation can be obtained e.g. in Ref. [11].

If the delays between the pulses are selected such that τ denotes the delay between the first (k 1 ) and the second (k 2 ) pulses, and T denotes the delay between the second and third (k 3 ) pulse (e.g. t 3 = 0, t 2 = -T and t 1 = -Tτ ) we can write for the time and the delay dependent signal field

Es(t, T, τ ) ≈ R 2g (t, T, τ ) +R 3g (t, T, τ ) -R * 1f (t, T, τ ), τ ≧ 0, ( 32
)
Es(t, T, τ ) ≈ R 1g (t, T, |τ |) +R 4g (t, T, |τ |) -R * 2f (t, T, |τ |), τ < 0. ( 33
)
The absolute value in Eq. ( 33) originates from the fact that response functions R are defined for positive time arguments only, and negative τ is achieved by switching the order of the k 1 and k 2 pulses. The individual response functions R are listed in the

Numerical Results and Discussion

In this section we study dynamics of model aggregate viewed via population and coherence dynamics and via 2D coherent spectrum. We define a simple model aggregate for which we calculate excited state dynamics including evolution of coherences between electronic states, linear absorption and 2D spectra at chosen population times. Calculations of linear absorption, which require only knowledge of the time evolution of optical coherences, are performed using the secular time local equation, since it is known to yield exact result at least for some models [35]. Population dynamics is calculated using all four methods we discussed in Section 3, and the results are compared.

The simplest model of an aggregate that can exhibit all effects observed in Ref. [14] is a trimer. The geometry of the studied models, together with the meaning of the parameters is presented in Fig. 2. In Tab. 1 we summarize the main parameters of the model. Because we are not interested in the absolute amplitude of the absorption or 2D spectra we assume the transition dipole moments dn to be taken relative to some value d 0 . All three resonance couplings J between the molecules are set to J = 200 cm -1 for the calculations presented here.

The values of the transition dipole moments, together with the exciting light intensity, determine the initial condition for the population dynamics. We assume that the excitation light intensity and the value of the transition dipole moment are such that the system is only weakly excited. The total population of the excited state band is normalized to 0.01. The relative values of the transition dipole moments are chosen so that the linear absorption spectrum (see Fig. 3) shows peaks of roughly the same height. Two peaks originating from the energetically lowest and the energetically highest states dominate the spectrum, the third level contributes as a shoulder to lowest energy peak.

Two parameters that influence the coupling for the model system to the bath are reorganization energy λ and correlation time τc. We vary these parameters in the range that can conceivably represent chlorophylls in photosynthetic complexes (see e. g. Refs. [39,40]).

Population relaxation and evolution of coherences

First, we compare relaxation dynamics of populations of excited state of our aggregate after excitation by an ultrashort laser pulse. TL equations of motion where solved by standard numerical methods for ordinary differential equations provided by the Mathematica R software. For the TNL equations we used fast Fourier transform method. Figure 4 presents the first 1 ps of the population dynamics after a δ-pulse excitation of the trimer from Tab. 1 at the temperature T = 300 K. Reorganization en-Fig. 2 Geometry and parameters of a trimer aggregate. One monomer is chosen to be positioned at the origin of the coordinate system, with the transition dipole moment pointing along the x axis. The positions of the transition dipole moments of the other two molecules in space are characterized by their distance h 2 and h 3 from the origin of coordinates and by the angles α 2 and α 3 . Orientations and lengths of the dipoles are given in Tab. 1. In our example we assume that the aggregate is planar.

ergy λ = 120 cm -1 and correlation time τc = 50 fs are the same at all three monomers. The dynamics with the same parameters for a selected coherences element ρ 13 (t) is presented in Fig. 5. The overall conclusion is that all four methods yield a similar general behavior for the populations, with some difference at the short time evolution and also slightly different long time equilibrium. Examination of the Figure 5 leads us to the conclusion that the methods yield two different results -a short coherence life time for the time local methods, and a relatively longer life time in case of the time non-local methods. The behavior of the coherence ρ 13 (t) represents a general tendency that we have observed for all electronic coherences over a wide range of parameters.

Let us now concentrate on short time behavior of the populations and coherences in more retail. In the short time evolution of the coherences the four methods group into two distinct groups with short (TL methods) and long (TNL methods) coherence life time. Whether the underlying equation is secular or not seems to have only a little influence on the coherence dynamics. Fig. 6 shows the short time (0 -400 fs) compar- ison of the population calculated by the four EM. In addition, short time population dynamics calculated by constant rate secular and non-secular Redfield equations are presented. We can clearly see that the results can be naturally grouped according to the presence of fast oscillatory modulation of the population relaxation dynamics. In the one group we have the full TL and full TNL methods, where such oscillations clearly occur, the second group comprises the two secular methods with no oscillations present. Thus, it can be concluded that the non-secular terms in the EM, which represent the coherence transfer and transfer between populations and coherences, are the cause of these oscillations. This is also supported by comparison of the population dynamics of the full TL and full TNL equations from Fig. 4 (e.g. the population of the state 1). The oscillation on the full TNL curve last longer than those of the full TL one, which reflects the longer coherence life time we have found for the TNL equations. In comparison to the four time dependent theories, the calculation with constant rates yields a completely different dynamics at at short times. This is consistent with the idea of the slipage of initial conditions [41] during the interval roughly corresponding to the length of bath correlation time.

Let us now discuss the long time limit of the time evolution. As expected, the two secular theories yield the same equilibrium at long population times. This equilibrium corresponds to the canonical distribution of population among the excitonic levels at T = 300 K. In both secular TNL and secular TL cases, coherences have relaxed to zero at long times as the inset of the Fig. 5 demonstrates. The non-secular TNL and TL equations yield non-zero, stationary coherences at long times, and correspondingly, the long time equilibrium populations do not correspond to the canonical thermal equilibrium. Although both non-secular theories converge to results different from the canonical equilibrium, the full TNL equation yields populations that are physical at all times for the studied system parameters, i.e. they are always positive. The full TL equation on the other hand fails to keep probabilities positive at long times, and the occupation probability of the highest electronic level becomes negative after 200 fs for the parameters used on Fig. 4.

In light of recent experiments [14], the conclusion that time non-local theories lead to a longer coherence life time than the time-local ones (i.e. also longer than the standard constant rate theories) is the most interesting. We have performed calculations of the RDM dynamics while varying the reorganization energy and the correlation time. The absolute values of the coherence ρ 13 (t) elements were fitted by a single exponential to estimate coherence life-time. The results are summarized in Fig. 7. The Fig. 7A shows the results for secular TL and secular TNL equations. Clearly, with growing correlation time τc, the full TNL equations lead to a increasing coherence life time. The full TL equation shows only a very weak dependence of the coherence life time on correlation time. Another interesting observation is that for correlation time longer then 50 fs, the dependence of the coherence life time on the reorganization energy λ is different for full TNL and TL methods. Time local theory, in accordance with the standard rate theories, predicts decrease of the coherence life time with λ. The full TNL theory predicts (within the parameter range studied here) an opposite tendency. The Fig. 7B shows similar conclusion for the non-secular versions of the theories, with the same difference between TL and TNL theory. The dependence of the coherence life time on λ in case of TNL equations is not monotonous.

Two-dimensional spectrum

As discussed in the Introduction, the secular TL equation of motion yields an exact result for the dephasing of an isolated optical coherence [35]. One can show, by comparison of the absorption spectra calculated by secular TL and TNL methods (see Fig. 3), that the TNL theory leads to certain artifacts (second peak) and is therefore not suitable for the description of the optical coherence evolution. Consequently, by the TL theories one can hope to obtain valid results only for the evolution superoperators at the first and the third time interval of the third order response functions by the TL theories. In Ref. [42] it was shown that non-secular terms in the TL equations for optical coherences lead to temperature dependence of the positions of excitonic bands in absorption spectra. This dependence was shown to be strong when the electronic states involved are characterized by significantly different reorganization energy [42,34]. Indeed it can be shown for homodimer that the non-secular terms are exactly zero in second order TL theory if the monomers exhibit the same reorganization energies [34]. We can therefore expect the non-secular effects in the optical coherences to be weak in our case, and we choose secular TL to calculate the evolution superoperators in the first and the third time interval of the response function, Eq. (34).

Concerning the population interval, the situation is somewhat different. As we have shown above, the non-secular TL theory leads to dynamics that breaks the positivity condition for the population probabilities at long times. At the same time, short time dynamics is very similar to the full TNL. Both theories predict population oscillations during the life time of the electronic coherences. The full TNL equation, however, preserves positivity, at least for the parameters studied here, and can be therefore used to calculate meaningful 2D spectra. For the same reason, both secular theories can also be successfully used to calculate 2D spectrum. As the oscillation of the populations predicted by non-secular theories are too small to be reliably observed in 2D spectrum (only a small change of the crosspeak amplitude due to the population transfer is observed after 140 fs of relaxation in 2D spectrum of Fig. 8 ) we expect only a small difference of the 2D spectrum to appear between the secular and full TNL theories. For the calculation of the representative 2D spectrum we therefore choose the secular TL and the full TNL theories. These two differ from each other mainly in the prediction of the life time of the electronic coherences. The observable difference in the calculated 2D spectra should therefore predominantly result from the different life time of the electronic coherence.

Fig. 8 presents 2D spectra for λ = 30 cm -1 and τc = 100 fs. These parameters lead to a rather slow relaxation and consequently to narrow spectral peaks in both absorption (see Fig. 3) and 2D spectra. This allows us to clearly see characteristic T -dependent oscillations of the peaks in 2D spectrum. At T = 0 fs, both methods provide the same 2D spectrum, with four peaks. Two diagonal peaks arise when all three perturbations of the system by electric field occur on the same level, while two crosspeaks appear from interactions occurring on different levels. Negative peaks correspond to excited state absorption (see Fig. 1). For two molecules that are not excitonically coupled, all contributions to the crosspeaks cancel out exactly, while if two molecules are excitonically coupled non-zero crosspeaks appear. The shapes of the peaks are influenced by the phase evolution of the coherence elements of RDM during the population time T . On the upper left figure of Fig. 8 we have marked the elongation of the diagonal and off-diagonal peaks by arrows. The elongation can be best judged by looking at the zero contour (in black). This particular elongation is characteristic for the phase of the ρ 13 (t) element (see upper right figure of Fig. 8) at T = 0. At T = 20 fs the phases of the ρ 13 (t) calculated by both methods are opposite to the phase at T = 0. The 2D spectra calculated by the two different methods at T = 20 fs differ only in the precise positions of the contours. This phase of the coherence element is characterized in 2D spectrum by a different orientation of the peaks. Interestingly, at T = 140 fs the two methods predict ρ 13 (t) that have mutually opposite phases and as a consequence the 2D spectra at T = 140 fs calculated by different methods differ in the orientation of their crosspeaks. Since the secular TL theory predicts a simple dephasing of the coherence and a regular oscillation with a single frequency proportional to the energy difference between corresponding energy levels, it is in principle possible to distinguish, even experimentally, deviations from this prediction. Our conclusion is that such a deviation should be a consequence of the memory effects in the reduced system time evolution. The coherence element ρ 13 (t), which is mainly responsible for the oscillatory behavior of the crosspeaks, is presented in the upper right corner of the figure. The 2D spectrum at T = 0 fs is the same for both methods and is therefore presented only once. The population times are selected so that they represent different phases of the ρ 13 (t) element (denoted by arrows on the coherence element figure). Arrows in the 2D spectra denote the orientation of the peaks. All spectra are normalized to 1 with contour step of 10 %. Positive features are in full red line, negative features are represented by dashed blue line, and the zero contour is depicted by the full black line.

Validity of secular and Markov approximations

Several conclusions about the applicability of the secular and Markov approximations can be drawn from the above results. As pointed out in Ref. [35], Markov approximation, which in the second order in system-bath coupling converts the TNL equations to the TL ones, leads accidentally to an exact result for an optical coherence element interacting with the harmonic bath. It has been also pointed out previously [33,43] that in the same case, the TNL equations lead to artifacts. When studying relaxation dynamics of the populations and electronic coherences in excitonic systems, full TL theory leads to a breakdown of the positivity of the RDM, while none of the secular theories suffer from this problem. In principle, the full TNL theory suffers from this problem, too [44]. However, it has been found less susceptible to it here. The secular theories lead to canonical density matrix at long times, while the full TNL results in a stationary state characterized by non-zero (but constant) coherences. Such result corresponds to an additional renormalization of the electronic states by the interaction with bath, and has to be expected even at a weak coupling limit [45]. It is important to note in this context that the canonical equilibrium is to be expected for the system consisting of the molecule and the bath as a whole, not for its parts [45].

For the population dynamics we are therefore forced to conclude that the full TNL theory represents the best candidate for a correct description of relaxation phenomena in the second order of the system bath interaction. It predicts similar population transfer times as other methods, it is much less sensitive to the breakdown of the positivity than its TL counterpart, and it leads to a bath renormalization of the canonical equilibrium. Most interestingly however, it predicts longer coherence life time than the TL theory. It was recently established by Ishizaki and Fleming [33] that this is to be expected from a higher order theory.

In the light of the above conclusions about the dynamics of optical coherences and the populations and coherences of the one exciton band, we suggest a hybrid approach to calculating 2D spectra, which consists of the application of the TL method on optical coherences (first and third time interval) and the full TNL method on the calculation of the RDM dynamics in the one exciton band during the population time T .

Conclusions

In this paper we have compared four different theories of excitation energy transfer and relaxation in molecular aggregate systems, with a special attention paid to lifetime of electronic coherences. Second order time non-local and time local theories with and without secular approximation were studied. For our specific model of an aggregate we have concluded that time non-local theories can account for experimentally observed electronic coherence life time that is significantly longer than the one predicted by the standard time-local secular relaxation rate theory. Markov approximation to time local EM was found to be responsible for the reduction of the coherence lifetime, while the influence of the secular approximation on the life time was found rather weak. The time local theory secular is found to break positivity of the occupation probabilities in the range of parameters studied here. We conclude that time-local second order theory is not suitable for simulating the coherence transfer effects. Simulations of two-dimensional spectra show that the time non-local effects can be experimentally identified based on the analysis of the oscillations of the cross peaks.

  for all term except when a = c and b = d , or a = b and c = d. The interpretation of the remaining non-zero terms is simple. The terms R aabb (t) for a = b represent rates of transition from level denoted by index b to a level denoted by a. The term Raaaa(t)

Fig. 3

 3 Fig.3Linear absorption spectrum of the model trimer for various parameters of the system bath interaction: (a) λ = 120 cm -1 , τc = 50 fs, (b) λ = 30 cm -1 , τc = 100 fs, calculated by the secular TL theory (full lines) and the secular TNL theory (dashed lines).

Fig. 4

 4 Fig.4First 1000 fs of the excited state population dynamics of a trimer with parameters λ = 120 cm -1 , τc = 50 fs, calculated by all four methods. For these particular parameters, the full TL equation breaks positivity of the RDM diagonal elements after 200 fs. Its prediction for the populations of the lowest and highest levels is significantly different from the other three methods.

Fig. 5

 5 Fig.5First 500 fs of the dynamics of the RDM coherence element ρ 13 (t), with parameters from Fig.4, calculated by all four methods. Detail of the long time part of the time evolution is presented in the inset.

2 Fig. 6 Fig. 7 Fig. 8

 2678 Fig.6First 400 fs of the population dynamics of the trimer with parameters λ = 30 cm -1 and τc = 50 fs. Results of full TL and TNL theories are presented in upper subfigure (A), the secular results are found the the lower subfigure (B). In both figures, magenta lines represent population dynamics calculated by constant rate Redfield theory.

Table 1

 1 Parameters of the model trimer. The parameter ǫn represents the transition energy of n-th monomer, transition dipole moments dn are taken relative to some value d 0 . Parameters hn and αn are explained in Fig. 2.

	n	ǫn cm -1	dn,x |dn|	dn,y |dn|	dn,z |dn|	|dn| d 0	hn	αn grad
	1	9850	1	0	0	0.65	0	0
	2	10000	-0.94	0.34	0	2.15	10	60
	3	10150	-0.94	0.34	0	0.9	10	120

Acknowledgements This work was supported by the grant KONTAKT ME899 from the Ministry of Education, Youth and Sports of the Czech Republic. Two-dimensional spectra were produced using the NOSE package available under GNU Public License at http://www.sourceforge.net.

Appendix A. Most importantly, they consist of series of propagation of the density matrix blocks by evolution operators obtained from the solution of EM. We have e.g.

where the evolution superoperators

ab act on an arbitrary operator A as a dipole operator µ ab from the right, i.e. V (R)

ab is defined analogically with the action of µ ab from the left. The indices e and g denote electronic bands as denoted in Fig. 1. Thus, the above operators and the action of superoperators on an arbitrary operator A are expressed in the basis of Hamiltonian eigenstates as

Eqs. ( 35) to [START_REF] Lindblad | [END_REF] together with the Appendix A enable us to calculate expected nonlinear signal from the knowledge of the matrix elements of the evolution superoperator. This type of knowledge can be obtained from solutions of the four different EM that we presented in Section 3.

Two-dimensional Coherent Spectroscopy

Two-dimensional coherent spectrum, Ξ(ω t , T, ωτ ), is obtained from the non-linear signal by Fourier transforming the time and pulse delay dependent signal electric field E S (t, T, τ ) along the t and τ variables [8,11] as

The Fourier transform in τ yields an ωτ dependence that is formally similar to linear absorption spectrum, while the transform in t yields generalized absorption and stimulated emission from a non-equilibrium state created by the first two laser pulses. 2D spectrum thus represents a 2D absorption/emission and absorption/absorption correlation plot. During the pulse delay time T the system evolves both in the electronically excited state and in the ground state, but no optical signal is generated. Relaxation of populations in the electronically excited band leads to evolution of non-diagonal 2D spectral features, so-called cross-peaks. Cross-peaks appearing at T = 0 are a signature of excitonic origin of the observed excited states. The 2D cross-peaks oscillate in T as long as the corresponding electronic coherence elements of the reduced density matrix are oscillating. The life time of the electronic coherences can thus be estimated directly from the T dependent sequence of 2D spectra [12,14].

A Third Order Response Functions

In this appendix we list the third order response function used in calculating the impulsive 2D spectra. The first index of the response function follows the standard notation of Ref. [START_REF] Mukamel | Principles of nonlinear spectroscopy[END_REF]. The second index is g for pathways not involving the two-exciton band, while all pathways denoted by f include a two-exciton contribution (see e.g. Ref. [11]).

Operators and superoperators used in this appendix are defined in Section 4.1.


K Zehani

F Mazaleyrat

V Loyau

E Labouré

Effect of temperature and time on properties of Spark Plasma Sintered NiCuZn: Co ferrite

Keywords: numbers: 75.50.Gg, 75.75.Cd, 81.20.Ev, 81.40.Rs Spark Plasma Sintering, Spinel ferrite, Grain size, Complex real permittivity, Complex initial permeability I

Spark Plasma Sintering is a powerfal method to produce fine grain dense ferrite at low temperature. However, the process, usually conducted in neutral atmosphere in a carbon die, yields carbon surface deposition and reduction of F e 3+ into F e 2+ . It's shown that subsequent annealing in air can remove carbon and, under some conditions, produce complete oxidation of F e 2+ ions. Regular values of the resistivity and permittivity (resp. 1 MΩm, 13 ε 0 ) are recovered for most samples annealed not higher than 750 • C. Relatively high value of the permeability (up to 240) and high merit factor (µ s × f r > 5 GHz) have been achieved.

The latest research in power electronics have focused on the miniaturization of electronic boards in particular by reducing the size of active components. Today, there is a tremendous demand of passive components reduction, particularly magnetic ones usually based on spinel ferrite. These ferrites attracted a lot of attention because of their magnetic behavior and their applications in power electronics devices for from the radio frequency (300 kHz to 3 MHz) to microwave (3 MHz to 20 GHz). The NiZn ferrite possesses a relative high initial permeability up to several hundred MHz, and a high resistivity, but the conventional sintering temperature is too high for co-sintering with silver or copper conductors (Sintering temperature at 1250 • C). However, if copper is substituted in a proportion of 0.1-0.2, the conventional sintering temperature is decreased down to 950 • C 1 . This study will focus on studying the effect of temperature and sintering time using the Spark Plasma Sintering process 2 on structural, dielectric and magnetic properties of NiCuZn ferrite doped with cobalt.

II. EXPERIMENTAL:

Powders of basic oxides F e 2 O 3 , NiO, CuO and ZnO and Co 2 O 3 were used for the preparation of NiCuZn ferrite using the standard technique of ceramics: jar mixture of precursor oxides, adding Co 2 O 3 -in order to decreased the magneto-crystalline anisotropy constant -, calcination at 760 • C for two hours and grinding by attrition for 30 min. The molar ratio was adjusted to achieve the following composition: N i 0.195 Cu 0.2 Zn 0.599 Co 0.006 F e 2 O 4 .

a) Electronic mail: karim.zehani@satie.ens-cachan.fr This composition was checked after sintering by SEM-EDS which confirms this composition within the accuracy of the method (Co was not detected). The sintering was carried out at different temperatures to determine the optimum properties at low sintering temperature. To investigate the structural and dielectric properties different pellets ∅8 have been produced by SPS between 650 and 800 • C in steps of 50 • C under a uni-axial pressure of 50 MPa, in neutral atmosphere (Argon) with a heating rate of 160 • Cmin -1 followed by a plateau of 0 to 15 min and cooling 160 • Cmin -1 . After sintering, the two faces of each sample were polished to remove graphite pollution from the pistons. For each sample, a second was made under the same conditions but with additional treatment at 500 • C under air in order to burn residual carbon an reoxyde the surface. The crystal structure of the different sintered samples was studied by Xray diffraction using a PANalytical X'Pert PRO MPD diffractometer, with cobalt source. The area swept in this study is between 20 • and 80 • in steps of 0.001 • . The relative density was calculated as the ratio of the experimental density measured by Archimedes method and the theoretical density deduced from lattice parameter. To determine the crystallite size, the spectra were refined by the Rietveld method using the software Maud 4 . The electrical resistivity was measured using a multimeter by the four points method. The dielectric permittivity and complex initial permeability as a function of frequency (between 100 kHz and 500 MHz) have been determined by impedance spectroscopy (HP 4195 B). To make the measurements of dielectric properties before and after annealing, we have deposited a gold layer about 100 nm by Sputtring. The permeability measurements were performed in coaxial cell, with ring samples (∅ ext = 8 mm, ∅ int = 4 mm and h = 3 mm). X-ray diffraction patterns taken from the powder after calcination shows all peaks characteristic of the spinel phase and no trace of parasitic phase or basic oxides. Refinement of the spectra using an inverse spinel type ferrite structure confirms this conclusion and yields a grain size of 100 nm. For sintered samples, no difference is visible -the lattice parameter is always 841.7 pm-except the crystallite size (Fig. 1) which increases with sintering temperature. It is noticeable that the sintering time has little influence on this parameter. In contrast the relative density strongly depends on annealing step duration (see in set Fig. 1). The strongest density ( 95%) can be achieved even at 700 • C if the temperature is maintained at least 5 min.

B. Dielectric properties:

The electrical resistivity and complex permittivity were measured before and after burning in air to clarify the effect of carbon on ferrite. Figure 2 shows the resistivity versus sintering time for different sintering temperatures before and after burn. The large increase of re- sistivity after burn may be explained by 2 phenomenons. First, SPS in a graphite die produces carbon evaporation and deposition at the surface of the sample, that short circuits the ferrite. This carbon layer is visible on XRD spectrum when recorded on as-sintered samples. Burn under air easily burns the surface deposited carbon. Secondly, the evaporated carbon can react with ferrite and after this treatment, the resistivity recovers the high value expected for NiZn ferrites, except for sample sintered at 800 • C which are probably reduced in the bulk and cannot be reoxidised because the porosity is closed. To confirm the reduction of F e 3+ in to F e 2+ we investigated the dielectric properties of these ferrite in a frequency range between 1 MHz and 500 MHz at room temperature. Figure 3 shows the different spectra of complex permittivity for different plateau length and temperatures before and after burning (500 • C). For samples sintered at (700 • C) and burnt, the dielectric permittivity is constant -in agreement with literature -and the decrease as a function of frequency is no longer observed. This confirms the burn of carbon and re-oxidization of the surface. By opposition samples sintered at 750 • C for 10 and 15 min still show after burning the decrease in ε (f) probably because the porosity is closed (the relative density is larger than 96 %) so F e 2+ can't be oxidized back into F e 3+ in the bulk. The high permittivity at low frequency has been observed in CCTO and BTO ferroelectrics [5][6][7][8][9][10][11][12][13] . The Internal Barrier Layer Capacitor (IBLC) effect is related to the micro-structure, composed of (semi)conducting grains separated by insulating grain boundaries. The effective relative permittivity given by ε ef f = ε gb D/t depending on the grains size (D) and the average thickness of grain boundary (t).

C. Complex initial permeability:

Table 1 shows the values of initial permeability, relaxation frequency and merit factor for different samples sintered at 650, 700, 750 and 800 • C. There is either an increase of complex permeability according to the sintering temperature and sintering duration. This increase is due to grain growth and porosity reduction 14,15 . Compared to the permeability values obtained by A. Lucas [START_REF] Lucas | Study and development of broadband RF transformers[END_REF] for the same types of ferrites sintered at 935 • C for 2 h (conventional sintering), we note that the initial permeability is much lower, between 20 and 250, compared to 560 for conventional method. This is due to a much smaller grain size and probably to the single domain structure of the particles. The complex permeabilty spectra depicted in fig. 3 are showing the typical Snoek features as the real parts converges at high frequency. It is remarked that all of them exhibits a marked resonance in the real part characteristic of materials having a good homogeneity in terms of grain size and composition.

Tables 1 shows that the real part of the initial permeability is roughly proportional to the relaxation frequency f r , which confirms Snoek's law [START_REF] Nakamura | [END_REF] . Although the permeability is lower than that of conventionally sintered materials, the merit factor µ i ×f r is finally very close (up to 5.32 compared to 5.8 GHz).

In conclusion, this study has shown that SPS is very efficient for NiCuZn ferrite processing. These dense and well crystallized samples exhibits very fine grain size. The dielectric and magnetic properties can be controlled by sintering temperature and time, keeping a high value of the merit factor. Samples annealed at 750 • C for 5 min shows a good compromise in term of permeability, relaxation frequency (f r )and resistivity when submitted to re-oxidation treatment. Higher permeability can be obtained but only at the price of larger dielectric loss. Hence, SPS should be used for the fabrication of NiCuZn ferrite based integrated magnetic component using LTCC process.

FIG. 1 .

 1 FIG. 1. Grain size and Relative density as a function of temperature and time of sintering: Square (0 min), Circle (1 min), Up Triangle (5 min), Diamond (10 min) and Star (15 min).

FIG. 2 .FIG. 3 .

 23 FIG. 2. Electric resistivity as a function of sintering time for different temperatures before and after burning

FIG. 4 .

 4 FIG.4. Real and imaginary parts of the permeability for selected samples

TABLE I .

 I Initial permeability, relaxation frequency and merit factor of SPS ferrite toroids

			Sintering at 650 • C	Sintering at 700 • C	Sintering at 750 • C	Sintering at 800 • C
	Time	5 min 10 min 15 min 5 min 10 min 15 min 5 min 10 min 15 min 5 min 10 min 15 min
	µ i f r (MHz)	17,3 169	17.4 169	17.4 170	34 131	39 115	52 89	126 37	156 32	178 27	152 33	213 24	242 22
	µ i × f r (GHz)	2.92	2.94	2.96	4.45	4.48	4.62	4.66	4.99	4.80	5.02	5.11	5.32
		600
		500
	Grain size ( nm )	300 400
		200
		600 100	650	700	750	800	850
				Sintering time ( min )


S N Kane
email: kane_sn@yahoo.com

M Coisson
email: m.coisson@inrim.it

P Tiberto

F Vinai

F Mazaleyrat

On the influence of Joule heating induced nanocrystallization on structural and magnetic properties of Co 64 Fe 21 B 15 alloy

Keywords: Nanocrystalline alloys, Mössbauer spectroscopy, XRD, magnetic properties PACS : 75.60.E, 61.18.K, 61.10.N, 75.60.E

Structural and magnetic properties of Joule heated Co 64 Fe 21 B 15 alloy has been studied. Crystallization of the specimens starts after Joule heating with 1.6 ampere current, leading to the formation of nanocrystalline Co 80 Fe 20 phase with grain diameter ranging between 19 to 25 nm. Crystallization of the specimens is found to detoriate the soft magnetic properties. Observed decrease of the hyperfine field of the amorphous component upon Joule heating suggests the increase of the boron near-neighbours to iron and the changes in the hyperfine field of the crystalline component are attributed to the changes in the environment around Fe-atom in the specimen. The coercivity is found to obey a domain wall pinning process instead of random anisotropy, as expected in soft magnetic nanocrystalline magnetic materials.

Introduction

The soft magnetic properties of amorphous ferromagnets are generally improved by low-temperature anneals, which produces relaxation of the atomic structure and a reduction of the quenched-in internal stresses [START_REF] Luborsky | Amorphous Metallic Alloys[END_REF].

Conversely, the development of a dominant crystalline fraction by performing high-temperature thermal treatments is generally seen to destroy the soft magnetic properties of amorphous alloys [START_REF] Baricco | [END_REF]. However, the soft magnetic properties of a certain Fe-based alloys have been improved by early Crystallization stages [3]. In the eighties, in addition to conventional furnace treatment having heating rate of the order of 0.1 to 1 K/s, a new method exploiting the heat released to the sample by an electrical current flow has been employed [4,5]. The current annealing techniques are characterised by rather high heating rates (10 4 to 10 6 K/s) allowing the sample to crystallize at higher temperatures with respect to furnace annealing. As a consequence, fast heating techniques [6] are currently employed in order to obtain metallic materials with attractive physical properties starting from amorphous ribbons. Among these, dc Joule heating is a convenient technique to induce structural changes [7] in a shorter time with respect to conventional furnace annealing treatments [8]. Typically, electrical current density values ranging in the interval 1  10 7 to 1  10 8 A/m 2 for time varying between 1 and 1000s are exploited. This technique allows to perform annealing under a self-generated magnetic field and, consequently, to induce a magnetic anisotropy simply by using low-current densities in order to reach temperatures below the Curie temperature of the studied alloy [9]. In this technique, heating rate reaches  100 K/s, which is lower than the maximum values provided by pulse heating technique; however it remains two orders of magnitude higher with respect to conventional annealing treatment. The amorphous-to-crystalline transformation occurs several seconds after reaching a steady-state temperature, and allowing to carryout proper measurements of selected physical properties (e. g. -variation of the resistance of the specimen by passing dc current through it, providing information on crystallization of the specimen) during the structural transformation [10]. The time scale of the process is made suitable for on-line physical measurements by using an electrical current of the order of few amperes. In this case, sample heating has been monitored by measuring a structure sensitive property such as the electrical resistivity.

Both X-ray diffraction (XRD) and Mössbauer spectroscopy provide structural information on the specimen under investigation. It is of value to note that, X-rays can not differentiate between Fe and Co atoms, as their scattering amplitudes are very similar. Thus, XRD essentially provides information about the average TM-TM pair correlation function (TM -Transition Metal). On the other hand, transmission Mössbauer spectroscopy measurements are sensitive to the Fe atoms only and hence they provide information about the local environment around Fe atoms. Furthermore, the Fe hyperfine field is more sensitive to the boron near neighbour (both number and the distances) as compared to the Co near-neighbours and thus hyperfine field would provide information about the variations in the metalloid near-neighbour environment.

In the present work we investigate the influence of dc Joule-heating induced nanocrystallization on structural and magnetic properties of Co 64 Fe 21 B 15 amorphous ribbons, using X-ray diffraction (XRD), Mössbauer spectroscopy and hysteresis measurements.

Experimental details

Amorphous Co 64 Fe 21 B 15 ribbons (width 6 mm and thickness19 m) were obtained by rapid solidification technique on a rotating drum. In order to carefully evaluate the electrical current value, which induces the amorphous-to-crystalline transformation, one sample has been submitted to a current ramp and the electrical resistance was continuously monitored. The electrical resistance evolution as a function of electrical current 'I' shows an increase of resistance up to I 1.9 A and decreases for higher currents, indicating the onset of microstructural changes within the amorphous matrix (i. e. sample crystallization). Selected strips with a length of 10 cm have been submitted to dc Joule-heating in vacuum [7] with 'I' ranging from 0 to 2 A for 30 minutes in order to induce structural changes in the specimens. Hysteresis loops were measured using a conventional induction technique at 10 Hz, which were used to obtain hysteresis parameters and the relative permeability. Xray diffraction (XRD) measurements were carried out at room temperature using Cu-K  radiation. XRD patterns were analyzed by fitting a crystalline and one amorphous component using pseudo-voigt profiles. The average grain size D is obtained by Scherrer's formula using integral width of the (110) line and the lattice parameter a is calculated using Nelson-Taylor-Sinclair correction in order to take into account the peak shift due to sample offset. For the amorphous phase, the first nearest neighbour distance 'X m ' is calculated using the expression : X m = 1.227 /2 sin [START_REF] Guinier | X-ray Diffraction in Crystals, Imperfect crystals and Amorphous Bodies[END_REF]. Apparent area is obtained using expression -

a c c A A A A   * (1)
where A c -area of the crystalline phase (110) peak, A a -area of amorphous phase main halo.

Volume of the crystallites is given by -

1 1 * * 1    a c f f A A x V (2)
where f c -scattering factor of the crystalline component and f a -the scattering factor of the amorphous component. These factors having dependence on the concentration of electrons, we can consider in first

approximation f c /f a  1 / (1-C B )
, where C B is the concentration of Boron in residual amorphous matrix. Because calculated and a new value of V x is decreased. After 3 to 4 iterations it converges to better than 1%.

Volume fraction of the crystalline component is thus calculated by -

B B XRD C A C A V * * 1 ) 1 (    (3)
where C B is the concentration of Boron in residual amorphous matrix.

Transmission Mössbauer spectra were recorded at room temperature in constant acceleration mode using 57 Co:Rh source and were fitted assuming a distribution of hyperfine fields using NORMOS program [START_REF] Brand | [END_REF]. For Mössbauer measurements, X -the Co % in Fe-Co phase (Co x Fe 1-x ) is deduced as follows :

1 1 *   x Fe V C A X (4)
where V x is the volume fraction of crystallites obtained from XRD measurements, A * is the area of the crystalline phase obtained from Mössbauer measurements, C Fe is the concentration of Fe atoms in the parent alloy.

The apparent area is obtained using expression -

aq Fe cr Fe C C A  * (5)
where C Fe Cr is the concentration of Fe in crystalline phase and C Fe aq is the concentration of Fe in as-cast specimen.

Volume fraction of the crystalline component from Mössbauer measurements

V x Möss is thus obtained by - x - 1 A * Möss  x V (6)
where x -Co % deduced from average hyperfine field

Results and discussions

Figure 1 depicts the XRD patterns of the selected Joule heated specimens. Perusal of figure 1 shows that after annealing with a Joule heating current of 1.6 Amp. for 30 minutes, the crystallization starts and for higher Joule heating currents, crystalline fraction increases as evidenced by the presence of sharp crystalline peaks at 45.2 and 65.8  (on two theta scale). Parameters obtained by analysing XRD data are depicted in table 1. For the studied specimens, within experimental errors the X m ( 0.246  0.001 nm) remains unchanged, so no definite conclusion can be drawn about the change in the mass density of the specimen from the XRD data. Table 1 also shows that the grain size of the nanocrystalline phase ranges between 19 and 25 nm and the corresponding volume fraction varies between 3.4 and 31.4 %. Volume fraction shows linear dependence with increase in the Joule heating current. Grain size of the studied samples is of the order of nanometer, although the studied samples do not contain Cu, which is a well-known nucleating element [13]. It is worth noting that in a recent paper Ping et al. [14] have shown that in Co-rich alloy systems, Fe-Co nanocrystalline grains do not need Cu as a nucleating agent to precipitate nanocrystalline grains in the amorphous precursor; this result well matches with the present studies. Obtained lattice parameter 'a' values suggest that the formed nanocrystalline phase is Co 80 Fe 20 , and its composition almost remains almost constant with increase in Joule heating current, although volume fraction of the formed nanocrystalline phase increases, for higher Joule heating current. It should be noted that in the crystalline Fe and Co, B is soluble in a very little extent only, and probably it is expelled from the developing nanocrystalline phase into the residual amorphous phase. Table 1 depicts the Joule heating current dependence of V x obtained by XRD measurements. For comparison, A is also given (obtained from transmission Mössbauer measurements). V x is higher as compared with A, which can be attributed to the fact that: X-rays can not differentiate between Fe and Co atoms, as their scattering amplitudes are very similar. Thus, XRD essentially provides information about the average TM-TM pair correlation function, whereas Mössbauer measurements are sensitive to the Fe atoms only and hence they provide information about the local environment around Fe atoms.

Additionally this difference in the area of the crystalline phase obtained from XRD and Mössbauer measurements can be attributed to the presence of an interface zone along with crystalline component and residual amorphous matrix as mentioned by Miglierini et al. [15], which has major contribution especially in the early stages of crystallization. It is worth noting that, for higher Joule heating current treatment, A * agrees more with V XRD , which can be attributed to smaller contribution from the interface.

Figure 2 shows the Mössbauer spectra and the corresponding hyperfine field distribution as obtained from the least square fit to the experimental data using NORMOS program [START_REF] Brand | [END_REF]. Mössbauer spectra of the crystallized specimens consist of two components, one each corresponding the crystalline component and a residual amorphous matrix component. Mössbauer results also show that crystallization of the specimen starts after Joule heating with 1.6 Ampere current. Hyperfine field distribution of amorphous component is broad, while that of crystalline component is sharp. It is well known that the broad hyperfine field distribution in transition metalmetalloid metallic glasses essentially reflects the distributions of boron near-neighbours of Fe atoms, and the hyperfine field of a given iron atom decreases almost linearly with increasing number of boron near-neighbours [16].

Figure 3 shows the dependence of average hyperfine field for crystalline and amorphous phase with Joule heating current. With increase in Joule heating current, the hyperfine field for crystalline phase increases, indicating that the crystalline phase is becoming richer in Fe and the observed values of hyperfine fields for crystalline component (varying between 32 T to 33.5 T) suggest that the hyperfine field of 32 T corresponds to 95 % of Co in the crystalline phase and hyperfine field of 33.5 T corresponds to 80 % of Co in the crystalline phase [17]. It should be noted that lattice parameter values show that Co content in the nanocrystals remains constant ( 80 % ) with increase in Joule heating current. Obtained values of hyperfine field for the crystalline phase with highest Joule heating current can be explained assuming out-diffusion of boron from the crystalline phase, increasing the hyperfine field as observed from Mössbauer measurements. Decrease of the hyperfine field of the residual amorphous phase with increase of Joule heating current indicates the changes in the environment around Fe atom and supports the assumption of out-diffusion of Boron atoms from the developing nanocrystalline phase. It should be noted that Fe hyperfine field is more sensitive to the boron near neighbour as compared to the Co near-neighbours. Also, B is soluble in a very little extent in the crystalline Fe and Co, major portion of boron remains in the residual amorphous matrix, as also indicated by XRD measurements. So, as crystallization proceeds, the average number of boron near-neighbours to iron (in the amorphous phase) increases, thus reducing the iron magnetic moment, resulting in the reduction of hyperfine field corresponding to the residual amorphous phase.

Figure 4 shows the hysteresis loops of as-cast and Joule heated samples (shown with x-offset in order to be clearly visible). The measurements have been performed at a constant induction (J  1.2 T) to better evaluate the coercive field. The observed tilt in these loops for the as-cast specimen and for Joule heated specimen can be partially ascribed to the demagnetization effect occurring in them (measured specimens were ~ 100 mm long, the pick up coils were ~ 30 mm long and were placed on the sample region treated at a constant temperature) [7].

The tilt is seen to increase with increasing the electrical current value at which the thermal treatment has been performed clearly indicates an increase of the transverse magnetic anisotropy induced by the heating current itself.

Variation of coercivity H c calculated from the hysteresis loops for the as-cast and Joule heated specimens are shown in figure 5(a), whereas the variation of relative permeability is depicted in figure 5(b). Perusal of figure 5(a) shows that for Joule heating current above 1.6 Ampere, where crystallization of the specimen starts, H c shows an appreciable increase. As crystallization proceeds, growing crystallites act as pinning centres hindering the motion of domain walls, thus increasing the coercive field. Figure 5 (b) shows that the relative permeability is almost constant as long as the annealing current does not exceed 1.5 A. Then, coherently with the coercivity data and the other results reported in the manuscript, the permeability goes down quite rapidly (this is accompanied with an increase of the crystalline fraction and increase of coercivity).

For Joule heating current above 1.6 A, where crystallization of the specimen starts, H c shows an appreciable increase. As crystallization proceeds, growing crystallites enhances the effective anisotropy, thus increasing the coercive field. Using the random anisotropy model proposed by Herzer [18], the average anisotropy < K >, <K>/J s were calculated for crystalline samples (reported in table 2) using :

< K > =V x 2 ( K 1 4 D 6 ) / A 3 ( 7 )
and

H c ~ p < K > / J s ( 8 )
where V x -volume fraction of crystallites, D -grain size , K 1 -first anisotropy constant ( 43300 J m -3 ) [START_REF] Bozorth | Ferromagnetism[END_REF], Aexchange stiffness ( 2.2 × 10 -11 Jm -1 ) calculated from virtual Curie temperature ~ 1450 K [START_REF] Weiss | [END_REF] and p -a dimensionless parameter.

As H c does not follow the Randon Anisotropy Model, a different explanation can be given for the same, as given below. Let us consider that the H c is due to domain wall pinning on Fe-Co nanoparticles due to the local demagnitising field. Using two dimensional wall bowing [START_REF] Bertotti | Hysteresis in Magnetism[END_REF], H c is given by:

 

     2 2 0 2 2 0 2 2 6 6 D K D J d D J d D D E d s s p              ( 10 )
where -J s = J s crystal -J s amorphous .

Substituting value of E p from eq. ( 10) in eq. ( 9) and writing

3         d D D V x yields after reduction to D V J K C H x s d c  2  ( 11 )
Taking J s  1.5 T and J s cryst.  2 T, J s amorph.  1.5 T ( V x << 1) and  ~ 10 -3 -10 -4 Jm -2 ; the order of magnitude is fine with C ~ 10 -1 -10 -2 . Equation [START_REF] Guinier | X-ray Diffraction in Crystals, Imperfect crystals and Amorphous Bodies[END_REF] clearly shows that H c varies linearly with V x D (H c  V x D). Perusal of figure 6 clearly shows that H c varies linearly with V x D , as is shown by equation [START_REF] Guinier | X-ray Diffraction in Crystals, Imperfect crystals and Amorphous Bodies[END_REF].

CONCLUSIONS

X-ray diffraction, transmission Mössbauer spectroscopy and hysteresis measurements have been used to investigate the influence of Joule heating on structural and magnetic properties of Co 64 Fe 21 B 15 alloys. Joule heating leads to the formation of nanocrystalline Co 80 Fe 20 phase with variation in grain size ranging between 19 -25 nm, which can be used effectively for tailoring the soft magnetic properties. Decrease of hyperfine field for the residual amorphous phase after Joule heating suggests increase of boron near neighbours to the iron, whereas the variation of hyperfine field for the crystalline phase is attributed to the formation of Co 80 Fe 20 phase and changes in the environment around Fe-atom. The coercivity was found to result from domain wall pinning by developing nanoparticles in relation with the large magnetization contrast between the two phases.

Table 1

Parameters obtained from Mössbauer and XRD measurements. Lattice parameter (a), mean grain diameter (D) and volume fraction of nano-grains (V x ) and area of the crystalline phase (A) for as cast and Joule heated specimens obtained using XRD and Mössbauer measurements.

From Mössbauer measurements

From XRD measurements Joule heating current (Amp.)

B hf (T)  0.40 X (%) (Co x Fe 1-x) A* (%)  1 D (nm)  2 a (nm) 0.0001 X (%) A * (%) (Co x Fe 1-x) V XRD (%)  2

,

  D -average grain diameter, d -average distance between neighbouring nanograins.,  -wall energy ( AK 4  ),  p -interaction range between wall and defect for an elementary volume wall and defect is the product of surface fraction of particles multiplied by the demagnitizing energy of particles.

Figure 1 :

 1 Figure 1:X-ray diffractograms of selected Joule heated specimens.

Figure 2 :

 2 Figure 2:Mössbauer spectra and the corresponding hyperfine field distributions for as-cast and Joule heated specimens.

Figure 3 :

 3 Figure 3:Annealing current dependence of hyperfine field for amorphous and crystalline phase. Line connecting points are guide to the eye.

Figure 4 :

 4 Figure 4:Hystresis loops of as-cast (A. C.) and Joule heated samples, with treatments shown with each hysteresis loop.

Figure 5 :

 5 Figure 5: Dependence of (a) coercive field and (b) relative permeability with annealing current.Line connecting points are guide to the eye.

Figure 6 :

 6 Figure 6 :Dependence of coercive field with V x D and linear fit to the points.

  Figure 1

  Figure 4

  Figure 5

Table 2

 2 Parameters obtained for crystallized specimens using the random anisotropy model. For comparison, values of J s and H c are also given.

	1.6	32.22 98	7	19	0.2836	79	4	3.5
	1.7	31.75 98	13	25	0.2838	78	18	15
	2.0	33.2	85	25	25	0.2837	78.5	37	31

A* -Relative area, x -deduced Co %, V XRD -crystalline volume fraction obtained from XRD measurements

ACKNOWLEDGEMENTS

S. N. Kane acknowledges gratefully one month hospitality as invited professor from the Ecole Normale Supérieure de Cachan, Cachan (France), during November-December 2010.