F J Rodríguez

L Quiroga

C Tejedor

M D Martín

L Viña

R André

Control of non-Markovian effects in the dynamics of polaritons in semiconductor microcavities

We report on time-resolved photoluminescence from semiconductor microcavities showing that an optically controllable mechanism exists to turn on and off memory effects in a polariton system. By increasing the laser pumping pulse intensity we observe revivals of the decaying time-resolved photoluminescence signal, a manifestly non-Markovian behavior of the optically active polaritons.

Based on an open quantum system approach we perform a comprehensive analytical and numerical study of the coupling of optically active polaritons to a structured reservoir to confirm the origin of the observed features. Our findings show that negative detunings and strong excitation should occur simultaneously for memory effects to take place.

I. INTRODUCTION

Semiconductor microcavities (SM) have attracted a great deal of attention in recent years due to the opportunity they bring to create and manipulate, in a controlled way, many-bosons systems in a solid-stateenvironment 1,[START_REF] Deveaud | Special issue on Microcavities[END_REF] . Bose-Einstein condensation signatures of radiation-matter quasiparticles (polaritons) have been recently identified in such systems 3,4,5,6 . On the other hand, due to the rich possibilities of tailoring the matterradiation interaction in SM, optically controlled dynamics of elementary electronic excitations is within reach. In particular, the understanding of the ultrafast dynamics of polaritons becomes crucial for interpreting important quantum control experiments in SM as it has been demonstrated recently 7 . A convenient and versatile way of monitoring the ultrafast dynamics of polaritons is provided by time-resolved photoluminescence (tr-PL) experiments.

Here we focus our attention on II-VI SM where a strong exciton-LO-phonon coupling produces a rapid relaxation for non-resonantly created polaritons, with large excess energies 8 . It has been reported that, under certain conditions, tr-PL following a non-resonant pulsed excitation of a CdTe-based microcavity shows an oscillatory emission dynamics strongly depending on the detuning and the initial excitation density. Furthermore, spin-related effects have been observed when the tr-PL is analyzed into its co-and cross-polarized components after excitation with circularly polarized pulses 9,10,11 . The nonlinear coupling of optically active and dark states has been invoked as a possible mechanism to explain exist-ing experimental results 12 . However, these unusual experimental features seem still to challenge conventional theoretical approaches.

Most theoretical models rely on the Born-Markov approximation to describe the polariton dynamics 13,14,15 . The main feature of these approaches is to neglect memory effects, that is, the behavior of the polariton system at some time t is only determined by its configuration precisely at the same time. The validity of this assumption requires the environment characteristic correlation time to be small as compared with the relaxation time of the system. The fingerprint of a Markov process is an exponential decay, while deviations from a Markovian behavior cause a non-exponential time evolution, with eventually superimposed oscillations implying some degree of correlation between the quantum system and its environment.

The main purpose of the present paper is to describe an optically controllable mechanism of memory effects in SM and to report on tr-PL results which provide its quantitative verification. Several parameters may be used to control the presence or absence of memory effects in the dynamics of a polariton system. Some of them are of a static nature, such as the detuning between the cavity optical mode and the bare exciton resonance, whereas others are of a dynamic nature, such as the intensity of the optical pumping. Here we emphasize on the importance of the latter ones.

The analysis is carried out within the framework of the theory of open quantum systems. In the usual scenario for studying open quantum systems, a central system of interest is directly coupled to a large system usually labeled as the bath. In contrast, here we consider instead a three systems framework: the quantum system of interest (the emitting polaritons), a high-energy (exciton-like) polariton bath and an intermediate system (at the "bottleneck" region). The last system provides control over the indirect coupling between optically active polaritons and the bath of high energy polaritons. We show that, initial excitation intensity behaves as a controllable memory mechanism for producing a rich variety of features in tr-PL signals. Spin effects are not included in the present study.

The paper is organized as follows: in Section II we present our three coupled systems model. We start by briefly discussing some aspects of a master-like equation approach with memory effects that may provide insights on the proper dynamics of the emitting polariton system. However, polariton-polariton interactions together with polariton-loss effects are hard to include and thus make this simple approach unpractical to provide an appropriate description of the problem. In order to obtain a good quantitative agreement with experimental data a microscopic Heisenberg-Langevin approach, including memory effects, has been adopted [START_REF] Rodríguez | AIP Conference Proceedings[END_REF] . In Section III we present and discuss the experimental data, which corroborate our theoretical predictions. A summary is presented in Section IV.

II. THEORETICAL BACKGROUND

A. The model

After their creation in the upper polariton (UP) branch by a pulsed laser, polaritons rapidly relax to the lower polariton (LP) branch states. Once in the LP branch, one of the main mechanisms for polariton scattering in SM is the extremely efficient parametric-down-conversion. By this process two polaritons are simultaneously scattered: one of them goes to a -→ k ∼ 0 or "signal" particle state while the other one (conserving energy and linear momentum) goes to a high | -→ k | exciton-like state, the socalled "idler" particle state. Thus, we start by identifying three regions of interest in the LP branch, as sketched in Figure 1: (i) The bottom of the trap (signal states), -→ k ∼ 0, with energy dispersion E( k), described by operators a k ; (ii) the intermediate or "bottleneck" polariton region, described by operators c k and energy dispersion ǫ( k), where polaritons accumulate after a rapid relaxation from the UP branch, and finally (iii) the excitonlike bath (idler states) described by operators b k and energy dispersion ω( k). The Hamiltonian is then given by (h = 1) 14

H = k E( k)a † k a k + k ω( k)b † k + ¯ k ǫ( k)c † k c k + H XP + H SI (1)
In the present model polaritons are to be described by boson operators linearly interacting with a large boson bath (excitons). The polariton-polariton interaction term is given by

H XP = k, k ′ , k1, k2 Ṽ ( k, k ′ , k 1 , k 2 )a † k b † k ′ c k1 c k2 + H.C. (2)
where Ṽ ( k, k ′ , k 1 , k 2 ) accounts for the polariton-polariton coupling and H.C. means hermitic conjugate. Since a large population of polaritons can condense at the bottom of the trap in the LP branch, -→ k ∼ 0, a selfinteraction term has to be included as given by Intermediate quasiparticles ("bottleneck" polaritons), can reach a high population due to rapid relaxation processes from high energy states. Thus, the effective interaction between optically active polaritons and excitonlike polaritons in the bath can be assumed to be modulated by the presence of those "bottleneck" quasiparticles. It is well known that for composite environments, where extra degrees of freedom modulate the interaction between a quantum system of interest and a large reservoir, an effective non-Markovian behaviour on the quantum system dynamics arises even when the reservoir itself can be described in the Markovian approximation [START_REF] Budini | [END_REF] .

H SI = V 0 a † 0 a † 0 a 0 a 0 (3)
Following this line of thought, we shall consider that polaritons optically pumped at the UP branch rapidly accumulate into an intermediate or "bottleneck" region of quasiparticles 18 .

The exact quantum evolution of the fast relaxing polariton system is far from describable in simple terms and a detailed theory of its evolution is not yet available. We can, however, fill this gap with the reasonable hypothesis that, under high intensity pumping, the intermediate polaritons can be described by a parametric approximation by using classical fields instead of full quantum field operators. This approximation ignores quantum fluctuations in the intermediate polariton fields. Thus, in H XP , the c k1 c k2 operator is replaced by the c-number < c k1 c k2 >∼ h k1 (t)δ k1, k2 with h k (t) acting as an effective pump on the intermediate polaritons, depending on the actual pumping pulse at the UP branch. In this context, an effective classical intensity sets the systemreservoir coupling strength. Consequently, the polaritonpolariton interaction term adopts the time-dependent effective form

H XP (t) = k, k ′ V ( k, k ′ , t)a † k b † k ′ + k, k ′ V * ( k, k ′ , t)a k b k ′ (4)
where

V ( k, k ′ , t) = k1, k2 Ṽ ( k, k ′ , k 1 , k 2 )h k1 (t)δ k1, k2 (5)
Pump depletion is well accounted for with a simpleminded pulse shape as determined by the shape/length of the excitation laser pulse as well as by proper polariton relaxation mechanisms in a microcavity. We shall return to this point later. The effective Hamiltonian describing the coupling of trapped polaritons (a k modes) with the high energy polariton bath (b k modes), Eq.( 4), corresponds to a coupling strength V ( k, k ′ , t), which is now adjustable experimentally by varying the excitation pulse parameters such as their width and intensity. Thus, it is already clear from the present discussion how an optically controlled mechanism may exist to turn on and off memory effects in a polariton system. Additionally, the above description explains why a Born-Markov based theoretical approach should be invalid in the present context: (i) even if the optically active polariton system is weakly coupled to the high energy polariton reservoir, a time-dependent coupling strength yields to a non-exponential decay; (ii) by continuously increasing the pump pulse intensity a strongly coupled system-reservoir situation should be reached beyond a certain threshold, hence a theory based on a lowest-order perturbation in the coupling strength breaks down; (iii) the idler-polariton bath, as being formed by higher energy polaritons (massive exciton-like particles) with a quadratic dependence of ω(k) on k, constitutes a highly structured continuum, for which a single characteristic correlation time cannot be identified.

It is worth noting that the Hamiltonian in Eq.( 1), with H XP as given by Eq.( 4), corresponds to a non-degenerate parametric amplifier for massive particles, where the parametric gain implies that the LP population, as well as the polariton bath population, are amplified at the expenses of intermediate polariton depletion. We emphasize that we are considering a pulsed excitation, thus no indefinite gain will arise and the validity of the parametric approximation is still guaranteed. Furthermore, for a time-independent system-reservoir coupling, our problem can be solved exactly. We use this as a check on our numerical results later.

Before we address the full numerical solution of the proposed model, an analytically tractable method is to be discussed first. This discussion is presented here because it complements the results of the microscopic treatment to be detailed below and it will also serve as a reference point.

B. Time-convolutionless non-Markovian master equation

The first type of approximation we consider is based in a semi-analytical approach by using a Lindblad-like master equation with time-dependent rates allowing non-Markovian effects to be included. We start by considering a bipartite system conformed on one hand by the optically active polaritons -→ k ∼ 0 and on the other hand by a high energy polariton bath. The underlying physics is analogous to the parametric-down conversion processes in non-linear optics, hence consequently with an antirotating-wave-like coupling between those sub-systems. For an initially empty polariton bath, the master-like equation for the reduced density operator of optically active (signal) polaritons, without any other decay mechanism, is

∂ρS (t) ∂t = - i 2 S(t)[a † 0 a 0 , ρ S (t)] + 1 2 γ(t) -a 0 a † 0 ρ S (t) -ρ S (t)a 0 a † 0 + 2a † 0 ρ S (
t)a 0 (6) Time-convolutionless master equations of the latter form have been largely documented [START_REF] Breuer | Open Quantum Systems[END_REF] . The time-dependent parameters S(t) and γ(t) depend on the reservoir spectral density [START_REF] Breuer | [END_REF] . Eq.( 6) is local in time but contains all the information about memory effects in the time dependent parameter γ(t).

From this master equation is immediate to obtain the equation of motion for n(t) = T r S {ρ S (t)a † 0 a 0 }, the population of optically active polaritons

dn(t) dt = γ(t)(n(t) + 1) (7)
Given that initially the k = 0 mode is empty, n(0) = 0, we found that the number of optically active polaritons grows up as

n(t) = e t 0 γ(t ′ )dt ′ -1 (8)
In the Markov regime for which γ(t) = γ M =constant, the polariton population grows exponentially. However, non-Markovian effects are evident when the relaxation rates in the master equation are time dependent. Up to now, no dissipation effects have been included in our discussion. Thus, the polariton population in the optically active mode does not cease to steadily increase. Polariton-polariton interactions, creating scattered quasiparticles out of the -→ k ∼ 0 zone, cavity losses and recalling that the system-bath coupling is pulsed for a finite period of time (no continuous pumping mechanism replenishing the intermediate polariton zone is present), produce finally that n(t) temporally saturates and then goes to 0. Nevertheless, any oscillation associated with the time-dependent rate γ(t) should manifest itself not only in the rise-but also in the decay-evolution of the optically active polariton density.

One noteworthy feature of this treatment is the possibility of clarifying the growth of n(t) at early times. In the Markov approximation, n(t) = e γM t -1, thus at short times the polariton population starts growing with a linear slope. Since a signature of non-Markovian behavior is a time dependent relaxation rate, which starts growing as γ(t) ∼ gt, the polariton population should behave at short times as n(t) ∼ e gt 2 /2 -1 ∼ gt 2 /2, in qualitative agreement with the experimental results referring to the curvature of the initial time evolution (see below).

Despite the exact solvability of this model a systematic approach for dealing with mechanisms of polariton interactions and losses are not easily implementable, which makes the present formalism inadequate to describe the non-monotonic dependence of tr-PL signals. We consider, therefore, an analytical solution of the microscopic Heisenberg dynamics to investigate in a more detailed way memory effects in polariton systems.

C. Heisenberg-Langevin dynamics

From the previous analysis, it is apparent that there is a large amount of theoretical insight to be gained from a more exhaustive examination of the proposed model. Now, a thorough formal analysis of the proposed polariton dynamics is performed. We solve directly the Heisenberg equation of motion for a given operator A. In particular, for a k and b † k we obtain

i ȧ k = E( k)a k + k ′ V ( k, k ′ , t)b † k ′ + 2V o a † 0 a 0 a 0 δ k,0 (9)
i ḃ † k = -ω( k)b † k - k ′ V * ( k, k ′ , t)a k ′ (10)
In order to solve numerically the coupled equations of motion, Eqs.( 9) and ( 10), we formally integrate b † k (t) in Eq.( 10), then inserted in Eq.( 9), to get

ȧ k (t) = -iE( k)a k -i k ′ V ( k, k ′ , t)b k ′ (0)e iω( k ′ )t + t 0 k ′ , k ′′ V ( k, k ′ , t)V * ( k ′ , k ′′ , t -τ )a k ′′ (t -τ )e iω( k ′ )τ dτ -2iV 0 a † 0 a 0 a 0 δ k,0 (11)
For the sake of simplicity we take k ′′ = k and assume that the effective interaction V can be separated as

V ( k, k ′ , t) = g( k, k ′ )h(t)
, where g( k, k ′ ) accounts for both the Coulomb and Pauli effects in the polariton-polariton scattering while h(t), a dimensionless function of time, represents an effective pump-polariton pulse. As a consequence, Eq. ( 11) can be rewritten as

ȧ k (t) = -i E( k) -iΓ 0 + 2V 0 a † 0 a 0 δ k,0 a k -iη k (t) + t 0 h(t)h(t -τ )a k (t -τ )K k (τ )dτ ( 12
)
in terms of the kernel function

K k (τ ) = k ′ g( k, k ′ )g( k ′ , k)e iω( k ′ )τ (13)
and a polariton-bath noise function

η k (τ ) = k ′ g( k, k ′ )b k ′ (0)e iω( k ′ )τ h(τ ) (14)
Since the memory time of the radiation field outside the microcavity is extremely short, on the order of 1/E(0) ∼ 1 fs for typical II-VI gap energies E(0) ∼ 2 -3 eV, we are justified to treat radiation losses from the microcavity within the Markov approximation with the simple inclusion of a phenomenological damping term Γ 0 in Eq. (12). However, of primary interest here are the non-Markovian effects coming from the strong coupling between signal and idler-bath polaritons. Eq.( 12) embodies the memory effects on the dynamics of the emitting polaritons. The term g( k, k ′ ) describes the strength and spectral form of the signal-idler coupling. We emphasize that, in our model, the bath is formed by high energy polaritons (exciton-like quaiparticles), hence the idler-bath we are considering consists of massive particles. In this sense, the present signal-idler polariton coupled system is very similar to atom-laser systems with a continuous output coupler [START_REF] Breuer | [END_REF]21 . However, the main difference with atom-lasers is that in our case the signalidler outcoupling is of an anti-rotating-wave kind (see Eq.( 4)) instead of the standard rotating-wave approximation usually employed in atomic systems. In order to proceed, we assume a quadratic energy dispersion relation for the idler particles, ω(k) = k 2 /2M X with M X the bare exciton effective mass, and a Gaussian-like profile for the trapped ground-state polariton in k-space. Consequently, the signal-idler coupling g( k, k ′ ) can be written (in the continuum approximation for idler particles) as [START_REF] Breuer | [END_REF]21

g( k, k ′ ) = iΓ 1/2 (2πσ 2 k ) 1/4 e - ( k-k ′ ) 2 4σ 2 k ( 15
)
where Γ 1/2 and σ k settle the strength and width of the system-reservoir coupling, respectively. The kernel term, for the optically active mode k ∼ 0, becomes

K 0 (τ ) = Γ √ 1 -iατ (16)
where α = σ 2 0 Mx . Note that in our model, the bath is assumed to be formed by bosons with a dispersion relation ω(k) = k 2 /2M X . This fact gives rise to distinct features in the spectral bath response and consequently in the reduced system dynamics, as compared with that found for non-massive and structureless baths, such as those corresponding to photons or phonons.

The noise term is the responsible for initiating the polariton relaxation towards the bottom LP states. Thus, we adopt for this term the role of a classical seed and have checked numerically that a very small value for it does not affect the results.

At this point we have completed the presentation of our theoretical framework and the underlying approximations, thus we can now proceed to evaluate the time evolution of the mean number of emitting polaritons at the bottom of the lower polariton branch to compare it with tr-PL experimental data.

III. RESULTS

A. Experiments

The sample under study is a Cd 0.4 Mg 0.6 Te λ-cavity, with top (bottom) distributed Bragg reflectors (DBRs) built with 17.5 (23) pairs of alternating λ/4 thick layers of Cd 0.4 Mg 0.6 Te and Cd 0.75 Mn 0.25 Te. In each of the antinodes of the electromagnetic field confined in-between the DBRs there are two CdTe quantum wells (QWs) of 90 Å thickness. The strong coupling between the excitons confined in the QWs and the photons confined in the cavity yields to a Rabi splitting of ∼10 meV at low temperature. One important parameter to be considered is the detuning δ = E(0) -ω(0). The cavity thickness varies across the wafer, allowing to tune the photon in and out of resonance with the excitons, thus varying the detuning.

The sample is kept inside a cold-finger cryostat at a temperature of 8 K and is resonantly excited with 2 ps-long pulses arriving at the sample at ∼ 3 o , their energy tuned to the UP branch. The time evolution of the PL is obtained by means of a spectrograph coupled to a streak camera (time resolution ∼10 ps). We have selected the emission originating from k ∼ 0 lower polariton states by means of a small pinhole (angular resolution ∼ 1 o ). For polarization-resolved measurements we have used two λ/4-plates to excite and analyze the PL into its co/cross-circularly polarized components, after excitation with σ + -polarized pulses.

Figure 2 shows typical experimental co-circularly polarized tr-PL data for different laser intensities of I = 1, 40, 70 and 110 mW but the same detuning δ = -10 meV. The maximum of the PL signal increases in a nonlinear manner with the pulse laser intensity (note that for the weakest intensity I = 1 mW, the curve has been amplified by a factor of 225), while the temporal widths reduce slightly. New features developing on the decay side of the PL are clearly observed, with a temporal behavior dependent on the pump laser intensity. In particular, the emergence of a revival of the decaying PL signal is markedly evident for high laser intensities. This new peak becomes clearer for high intensity pumping. For positive detunings the extra peak on the decay side of the tr-PL is not present (results not shown). These observed intensity-dependent features can be fitted with good quantitative agreement using our theoretical model with memory effects controlled by the pulsed pump intensity.

B. Discussion

From the theoretical side our aim is to calculate the time evolution of the mean number of polaritons in the optically active -→ k = 0 state, i.e. < a † 0 a 0 > (t), by numerically solving Eq.( 12). The parameters used cor-respond to CdTe microcavities with an exciton mass M X = m e + m h = (0.51 + 0.9)m 0 = 1.41m 0 . We consider only heavy-hole excitons coupled to the cavity mode.

The relaxation of UP-resonantly created polaritons can be qualitatively described as follows. The direct relaxation to the LP states is strongly inhibited and the majority of polaritons scatter to large-k LP states. The situation is equivalent to that obtained after non-resonant excitation, i.e. there is a large polariton population at the LP bottleneck; from there polaritons relax to k ∼ 0 states via polariton-polariton parametric scattering. We assume that the accumulation of intermediate or "bottleneck" polaritons, following the actual laser pulse, takes place for a longer period of time (tens of picoseconds) than the exciting pulse width (∼ 2 ps), corresponding to an asymmetrical shape. Thus, it is to be modeled by an effective pump-polariton field of the form h(t) = At 3 e -βt , where β can be associated with the accumulation rate of intermediate polaritons. This assumption is justified since time-resolved PL in such CdTe microcavities, under pulsed excitation, has a clear asymmetric shape with typical decay times on the order of 10 1 -10 2 ps. We will call as the effective intensity, the square of the time integral of the polariton-pump pulse, I pp ∼ |Aβ -4 | 2 , where dimensionless I pp is assumed to be directly linked to the intensity of the laser pulse, I, and will enable us to connect the real excitation pump intensity with the intermediatepolariton-pump classical field. In all cases reported below, we focus on the analysis of co-polarized signals, i.e. the excitation is σ + -polarized and the emission is analyzed into its σ + -polarized component.

By energy conservation requirements, the parametriclike interaction (see Eq.( 4)), giving rise to the simultaneous creation of a -→ k = 0 polariton and a bath polariton, makes only sense for negative values of δ. Besides that, tr-PL experimental results show unconventional oscillatory features in co-polarized signals only for negative values of δ where a polariton trap is possible 22 . Consequently, we restrict our discussion and comparison with experimental data to the δ < 0 case.

Figure 3 shows the high-intensity experimental tr-PL signals from Fig. 2 (symbols, now independently normalized to their maximum value for each pulse intensity), with the corresponding fits (solid lines) to our memorybased theory. A non-oscillatory decay is obtained in the weak pump limit or when memory effects are neglected (see below). We therefore focus on the high intensity laser pumping cases. The memory kernel K 0 (τ ) depends on the coupling function g( k, k ′ ) through its width (in k-space) σ 0 and strength Γ 1/2 . We use the following fitting parameters: δ = -10 meV, σ 0 = 25.7 × 10 7 m -1 , Γ = 3 × 10 24 s -2 , which together with the bare exciton mass, M X , define a typical signal-idler coupling time scale as t c = e |δ|/α |δ|α/4πΓ 2 ∼ 10 -10 s. In terms of this coupling time, the effective-pump inverse temporal width is taken as β = 6/t c . Other parameters are Γ 0 = 0.45|δ| and V 0 = 0.02|δ|. By adjusting the pump-polariton pulse amplitude A, which together with β determine h(t) (in-sets in Figure 2), in such a way to obtain the same relative ratios between the effective pump-polariton intensities, I pp = 1, 0.64, 0.36 as those in the actual laser pulses I = 110, 70, 40 mW, the computed tr-PL results reproduce both qualitatively and quantitatively the experimental data. A non-linear initial rise of the tr-PL signal, instead of a simple linear one, as well as a temporal width that decreases for increasing pump intensity, are clear signatures of non-Markovian behavior. Clearly, at long times the decays are similar for both high-and low-intensity pump levels, thus memory effects are practically unobservable in that limit. The non-Markovian effects are more visible after the accumulation of intermediate polaritons reaches its maximum value. In order to explore additional consequences of our model, beyond the comparison with measured tr-PL data, we proceed to discuss the effects of the pumppolariton pulse parameters on the tr-PL results. Figure 4 shows simulations of tr-PL, where only variations of the polariton-pump intensities I pp are assumed (I pp = 1 corresponds to the fitting pump intensity for the experimental curve in Fig. 3-a). Coupling parameters σ 0 , Γ and V 0 are identical to those used in Fig. 3. Calculated tr-PL signals are plotted in Figure 4-a on a logarithmic vertical scale to show the sensitivity of memory effects on the pump-polariton intensity or equivalently on the laser pulse intensity. Oscillations in the polariton population develop as the polariton-pump intensity increases. Our results clearly demonstrate that the polariton population dynamics, for a high pump excitation, shows a nonexponential (oscillatory) behavior in contrast with the typical exponential one predicted by simple Markovian approaches. In the low-intensity limit a Markovian approach should be sufficient to explain the non-oscillatory behavior. This is further illustrated on Fig. 4-b, where a vertical linear scale is used to better see the evolution towards a typical Markovian signal for low intensity pumping. Furthermore, by collapsing the memory function or kernel (see Eqs.( 13) and ( 16)) to a delta function, i.e. K(τ ) ∼ δ(τ ), for a high pump intensity case, the peak on the decay side of the tr-PL disappears, as it is demonstrated in the inset of Figure 4(b). This fact brings further support to an intensity-controlled memory mechanism in these CdTe microcavities.

One last issue to address is the sensitivity of our results on the effective pump-pulse parameters, A corresponding to the pulse height and β associated to the inverse decay time. In Figure 5 simulated tr-PL results, for a given effective pump intensity I pp = 1 but different pulse parameters, are depicted. We emphasize that the coupling strength between signal and idler subsystems is time-dependent and goes as Γ 1/2 h(t). The main conclusion from Figure 5 is that non-Markovian signatures are enhanced for an impulse-like effective pump pulse, i.e. for large β or equivalently, for given I pp , a large amplitude A. A non-oscillatory behavior of tr-PL signal occurs for wide pump-pulses but intense and short pump-pulses give rise to oscillatory patterns in tr-PL. On this basis we can conclude that memory effects switch-on when a high intensity pump-pulse has a temporal width such that βt * c ≫ 1, where t * c is an intensity controlled coupling time given by t * c = t c /A 2 . This feature is fully consistent with our basic premises for the existence of memory effects in the sense that a large accumulation rate of intermediate polaritons would lead to a strong coupling between signal and idler (bath) polaritons. Moreover, our results suggest new possibilities for indirectly monitoring the rapid relaxation of UP polaritons to LP states. Non-Markovian effects in tr-PL could be interpreted as a signature of a rapid relaxation dynamics from nonresonantly pumped polaritons to optically active lower branch polaritons. Markovian theories give temporal broader tr-PL signals and no oscillations are observed. Some previous theoretical treatments 12 have attributed those oscillations to the existence of other states as dark excitons or spin-splitted states. Our theoretical analysis shows that oscillations in the tr-PL can be also produced by non-Markovian effects coming from the coupling between the trapped optically active polaritons with a polariton bath via a modulated parametric-like interaction. Furthermore, the wide range of intensities for which a satisfactory agreement is obtained demonstrates that our the-oretical model indeed captures the main physics of the pulsed response of polaritons in such II-VI SM as due to memory effects, without the need of invoking other states as responsible for those oscillations, as required by Markovian theories. Finally, it is worthwhile to mention that spin effects are important in the polariton physics for such II-VI SM. For the cross-polarized case, i.e. excitation σ + -polarized and emission analyzed into its σ -component, the same set of parameters fitting a co-polarized case, for identical detuning and pump intensity, does not fit the experimentally observed results. This evidences that in this case the spin dependent polariton scattering must be properly included to describe the observed data.

IV. CONCLUSIONS

In summary, we have demonstrated that non-Markovian, or memory effects, produce oscillatory features in tr-PL signals in II-VI SM. In particular, we have found that the nonlinear rise and the revival of the decaying PL signal for high laser intensities is explained in terms of a non-Markovian behavior of the optically active polariton system as a consequence of being efficiently coupled to a structured reservoir. These new features are enhanced in the high-laser-power excitation case. The shape and temporal width of a pump laser pulse should lead to control the dynamics of relaxing polaritons from a nonresonant initial distribution in the UP to access optically active states in the LP branch. Experiments to investigate this control effect would provide valuable insight into polariton dynamics and are feasible with current technology.

It is important to note, that while the tr-PL signal does not give a definite answer to the question about the exact intermediate states of the relaxation process from the initially created UP polaritons, the qualitative and quantitative agreement with the tr-PL experimental results, as a function of the excitation intensity, are strong indications of the validity of the present model. A full quantum treatment of the intermediate polariton states would be desirable. These analysis, which require considering a much more complex quantum state space, are left for further investigations.

FIG. 1 :

 1 FIG. 1: Schematic representation of different polariton subsystems. Inset: two systems, signal and idler, indirectly coupled by an intensity dependent strength.

FIG. 2 :

 2 FIG. 2: Experimental co-circularly polarized tr-PL signals for a detuning δ = -10 meV and different pulse laser intensities: I=110 mW (blue circles), I=70 mW (green squares), I=40 mW (red diamonds) and I=1 (mW) (black triangles).

FIG. 3 :

 3 FIG. 3: Experimental tr-PL normalized signal for different pump intensities (symbols) and theoretical fits (continuous lines). Experimental pulse laser intensities: (a) I=110 mW, (b)=70 mW and (c) I=40 mW. Insets represent the effective pump-polariton pulses h(t). For theoretical parameters see the text.

3 FIG. 4 :

 34 FIG. 4: Simulated tr-PL signals for different pump-polariton intensities Ipp. (a) Logarithmic scale. (b) Linear-scale. Inset: Comparison between Markovian (delta kernel) and non-Markovian tr-PL results for a high intensity pulse Ipp = 1.3 corresponding to a laser intensity of I = 143 mW.

FIG. 5 :√ 2 .

 52 FIG. 5: Simulated tr-PL signals for a fixed polariton-pump effective intensity Ipp = 1, or equivalently I = 110 mW, but different pulse-shape parameters: (i) Red curve, A = A1, β = β1, corresponding to those parameters used in the fitting of the experimental data in Fig.3-a. (ii) Green curve, A = 7A1/4, β = (7/4) 1/4 β1. (iii) Black curve A = A1/4, β = β1/ √ 2. The inset shows the pump-pulses h(t).

V. ACKNOWLEDGMENTS

LQ would like to acknowledge MEC(Spain) for a sabbatical grant and the Universidad Autónoma de Madrid-Spain for hospitality. FJR and LQ have been partially supported by research project funds from Facultad de Ciencias-Uniandes. MDM thanks the Ramón y Cajal Program. This work was partially supported by the Spanish MEC (MAT2005-01388, NAN2004-09109-C04-04, and QOIT-CSD2006-00019), and the CAM (S-0505/ESP-0200).


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Walt De Heer

Controlled epitaxial graphene growth within amorphous carbon corrals

Structured growth of high quality graphene is necessary for technological development of carbon based electronics. Specifically, control of the bunching and placement of surface steps under epitaxial graphene on SiC is an important consideration for graphene device production. We demonstrate lithographically patterned evaporated amorphous carbon corrals as a method to pin SiC surface steps. Evaporated amorphous carbon is an ideal step-flow barrier on SiC due to its chemical compatibility with graphene growth and its structural stability at high temperatures, as well as its patternability. The amorphous carbon is deposited in vacuum on SiC prior to graphene growth. In the graphene furnace at temperatures above 1200 • C, mobile SiC steps accumulate at these amorphous carbon barriers, forming an aligned step free region for graphene growth at temperatures above 1330 • C. AFM imaging and Raman spectroscopy support the formation of quality step-free graphene sheets grown on SiC with the step morphology aligned to the carbon grid.

In (b) the mobile steps have accumulated to the lower free energy configuration.

To be technologically viable, graphene must be grown in a well controlled manner. Beyond preparing well ordered 2-dimensional graphene films, growth needs to be tailored at the nanoscale to produce the desired devices at pre-defined locations. Epitaxial graphene on silicon carbide (EG) meets these criteria 1 . EG has excellent electronic transport properties, including record high frequency operation 2 , long spin diffusion lengths 3 , and room temperature ballistic transport [START_REF] Baringhaus | Nature advance online publication[END_REF] . Precise EG growth has been successful on tailored SiC substrates. Single layer nanoribbon growth has been produced on the SiC sidewalls of plasma etch steps [START_REF] Baringhaus | Nature advance online publication[END_REF][START_REF] Sprinkle | [END_REF] and EG thickness control was demonstrated on etched mesas 6 . Selective graphene growth was also achieved by capping flat SiC with a high temperature stable layer to prevent (or enhance) graphene formation [7][8][9] . Progress in graphene growth selectivity was shown using Chemical Vapor Deposited (CVD) methods, where an oxide barrier 10 provides better growth control on Cu and self-assembled nanoribbons on gold surfaces have been demonstrated. 11 Unlike CVD methods, however, the advantage of EG for technological development is to be scalable directly on a high quality semi-insulating production substrate. Further advantages of EG include its integration with silicon 12 and SiC electronics 13 . This Letter focuses on locally controlling the step bunching of SiC during EG growth. Although EG grows like an overlaying carpet on the stepped SiC surface, steps provide graphene nucleation sites 6,14,15 that need to be controlled. Furthermore, underlying SiC steps induce additional electronic scattering in graphene. 16 SiC step spacing depends on the miscut angle of the SiC crystal surface from the basal plane of the Si or C terminated faces, in addition to the thermodynamically favored step height. The smallest miscut angle of commercially available SiC wafers is at most 0.1 • , but it varies throughout the wafer and leads to variations in the step density and orientation. Preparation of well-defined SiC surfaces with regularly spaced steps, by hydrogen etching, can be preserved in a range of growth parameters 17 but generally leads to further step bunching at the graphene growth temperature (T>1300 • C in the confinement controlled sublimation method 18 or higher under atmospheric pressure argon 19 ). These methods, however, cannot provide a precise alignment of the steps. Another way is to get rid of the steps altogether by step flow at the edges of etched mesas 18 .

On unpatterned EG on SiC, the steps tend to bunch into unit cell high steps 20 , which is 1 nm high for 4H-SiC. Their spacing is determined by the local miscut angle, which for on-axis wafers is typically in the range 0.1 • to 0.5 • , giving step spacing of 600nm to 110nm. Under the proper H 2 etching and growth conditions higher step heights give correspondingly wider terraces, often around 1 µm wide. 17 However, a naturally formed surface cannot have an arbitrarirly determined step width, and in addition an arbitrarily determined step location and orientation. This means that given step spacings of this size, steps are likely to interfere with a graphene device.

To successfully tailor growth of EG on SiC requires placement of structures that are stable at EG growth temperatures on the SiC surface. Mesas 18 and trenches 21 etched into SiC are one solution.

Generally, any surface configuration that would prevent step diffusion on SiC will successfully structure step bunching, including on other materials such as silicon 22,23 . Deeply etched grid features in these silicon samples enclose the step-flow 22,23 .

Recognizing that graphene layers hinder step-flow 18 , and that graphitic materials are also high temperature stable, we propose here to enclose the naturally occuring surface steps on SiC with evaporated amorphous carbon (aC). Compared with other refractory materials, amorphous carbon is completely compatible with graphene growth as it will not introduce additional chemical elements.

Annealed aC forms nanocrystalline graphite 24 and is at least as structurally stable as graphene, such that it retains its patterned shape.

Creating a two-dimensional corral-like enclosure confines the diffusion of the atoms on the surface, such that any preferred step-flow direction will organize the steps into fewer larger steps bunched at one end of the enclosure. See Figure 1 for a schematic cutaway of the step bunching controlled growth. We choose a grid with dimensions that is comparable with the expected distance over which steps will bunch, which is typically a few microns. This step-flow control method harnesses the surface free energy minimization such that step bunching is favored at one end of the enclosure. Mobile steps flow by surface diffusion or sublimation of the surface atoms.

Step-flow on Si is mediated by evaporation of the Si atoms from the surface. 22,23 However, with EG on SiC, the step-flow is a separate mechanism from the graphene growth. 15 If graphene is growing, additional step flow due to Si sublimation is possible. Note that unlike in silicon, the step flow on SiC before EG growth conserves mass. Steps accumulate at one end of an individual grid cell according the local miscut angle on the sample surface.

We prepared EG samples using research grade 4H SiC (0001) and (0001) substrates cut to 3.5×4.5 mm 2 dimensions. These chips came from a nominally on-axis wafer from Cree corporation having a nominal miscut of 0 • , with an actual miscut angle of up to 0.1 • . The samples were ultrasonically cleaned for thirty minutes each in acetone and isopropyl alcohol. Three samples were studied: G1, which has monolayer graphene grown on the hydrogen etched Si-face, G2 which is multilayer graphene grown on the polished C-face (see supplemental for images of G2), and B1 which is buffer layer grown on chemical and mechanically polished Si-face. Sample G1 was hydrogen etched in a 40cc/s (3% H 2 )/(97% Ar) flowing forming gas mixture at 1450 • C at one atm. The hydrogen etching furnace is an inductively heated furnace with a tantalum sample holder. The hydrogen etched surface has well defined SiC steps before the step controlled graphene growth. As seen from Figure 2, the initial step height on the hydrogen etched sample is about 1nm, corresponding to 4 SiC unit cells. Amorphous carbon (aC) is deposited using a Cressington 108A carbon deposition system modified for high vacuum use. Deposition occurs by resistively heating a graphite rod point contact junction to a temperature exceeding the temperature of graphite sublimation. A 3 Ω junction evaporates with a current 120 A. The aC was uniformly deposited at 10 -6 mbar over all the sample surface. An in-situ quartz crystal monitors the deposited aC thickness, until a typical a thickness of 20nm is deposited.

The aC is then patterned into a 5×5 µm 2 square grid pattern using standard electron beam lithography. For this, MaN 2403 negative e-beam resist is spin coated onto the samples at 5000 RPM and baked for 60 seconds at 90 • C on a hot plate. E-beam exposure occurs with a cumulative dose of 200 µC/cm 2 . The resist is developed for 5 minutes in MF-319 (3% TMAH) developer.

Oxygen plasma RIE etches the unmasked aC. The oxygen is flowed at 4 sccm and RF power of 16 W is applied to the reaction chamber. A typical etch time is 75 seconds. Once the aC is etched through, any overetch into the SiC will introduce a silicon oxide passivation layer to the SiC, which is maximally a few nanometers thick. The oxide layer thickness is self-limiting and is removed during an anneal step as part of the Confinement Controlled Sublimation (CCS) graphene growth process in the furnace 18 .

Prior to growth, the samples are cleaned overnight at room temperature in an acetone solution to dissolve the remaining resist, and then the samples are rinsed with isopropol alcohol. Samples are grown using Confinement Controlled Sublimation (CCS) in a graphite enclosure in an induction heated furnace. 18 The sample is annealed at 1150 • C for 20 minutes to remove silicon dioxide, and then the graphene growth step proceeds.The Si-face graphene sample G1 was grown at 1600 • C for 40 minutes and the buffer layer sample B1 was grown at 1350 • C for 30 minutes.

Characterization of the surface topography at various stages of process was done using noncontact mode atomic force microscopy (AFM). Before the growth on the unetched surface, there is a nanometer scale roughness in the region within the aC enclosure.(Figure 2 See Figures 2 and3 part (f) for the AFM topography after the sample growth. All samples indicate the step morphology aligns to the aC grid. Larger, nanometer sized aC particles from the aC deposition pin the steps seen in sample B1 away from the gridded area into a pinched shape(Figure 3 (b)). Nanometer sized aC particles are deposited in greater numbers as the carbon rod source is heated to higher temperatures.

Averaging the step widths from the AFM images indicates that the average step width before growth on sample G1 is 0.96 µm. (Figure 2, subfigures (a) and (c)) Away from the grid the average step width is 0.22 µm after growth, and within the grid the step width is increased to 1.9 µm. On sample B1 the average step width before growth is 0.2 µm. After growth the average step width away from the grid is 1.4 µm and within the grid the step width is 3.3 µm. In both hydrogen etched and non-hydrogen etched cases the presence of the step flow barrier increases the average Using Non-negative Matrix Factorization (NMF) spectral decomposition to separate the Raman spectra into its principle components 25 , the Raman spectra of the graphene on sample G1 may be separated from the aC (see Figure 4). The principle component corresponding to the graphene indicates monolayer growth according to the intensity of the peaks. The principle component that corresponds to the aC indicates nanocrystalline graphite, where the ratio of the D to G peak intensity I D I G = 0.86. This ratio corresponds to a graphitic crystallite size of 20 nm. 26 Nanocrystals form upon annealing the aC at graphene growth temperatures. Buffer layer Raman spectra, with a much weaker signal than graphene, is seen on sample B1. 27,28 Understanding the role that the aC grid structure, growth temperature and SiC wafer miscut play is important for optimal step flow control. For instance, the step flow of the buffer sample B1 appears to be better organized than on the graphene sample G1. The reasons for this difference will require further experiments. We can nevertheless conclude firstly that the step flow indeed involves mass transfer within the aC enclosure only. This is because the total mass of SiC has been conserved before and after graphitization as calculated (The volume of silicon carbide is calculated from step heights and positions before and after growth as well as the carbon contribution to graphene.) from the position of the steps in the AFM images of samples G1 and B1. This mass conservation includes the loss of silicon due to graphene growth (roughly 3 SiC bilayer are required to form one graphene layer). Secondly, the 5×5 µm 2 square grid also appears to be smaller than the distance over which steps will rearrange, particularly on sample B1, since one dominate bunched step is present after growth.

Further step-flow experiments suggested include modifying the aC grid, the wafer miscut, and the growth conditions. Different sized and oriented aC step flow enclosures are in principle possible in order to tailor the step orientation according to the boundary conditions set by the grid. Additionally, a suitable insulating alternative to aC, like oxides or diamond-like carbon, may make the step flow control even more desirable from a device perspective, since aC is conductive. Performing step flow control on vicinal cut wafers may yield more consistent step bunching, since the total miscut angle will not be as sensitive to the miscut angle uncertainty, unlike on-axis wafers. The step bunching is likely to be greater as well on vicinal cut wafers, thereby enhancing the step flow control. 20 Finally, by increasing the background pressure with an inert gas in the CCS furnace it is possible to raise the onset temperature of graphene growth 18 . This would increase the step flow rates to possibly reach lower surface free energy before graphene forms and therefore inhibits stepflow during graphitization. Lower surface free energy may also be attained by hydrogen etching the sample prior to patterning the grid 17 . It is possible to tune the growth and hydrogen etch procedures to have even larger or specifically tailored step-free areas, in addition to learning more about SiC surface kinetics and graphene growth.

In conclusion, we have shown that evaporated amorphous carbon is effective for structured EG growth and is completely compatible with the CCS growth method. We have demonstrated that SiC step bunching is pinned by an aC cap that acts as a step flow barrier. The step morphology is aligned to the aC grid. Single layer graphene is thereby grown on up to 4 µm wide step-free areas which locations are predefined by lithography. Further work ought to yield larger step free areas and the possibility of controlling the growth of sidewall graphene on the bunched steps. Careful measurements of step flow control ought to yield deeper information about the graphene growth process on SiC, including mass transport and graphene nucleation rates. Integration of graphene into nanoelectronic processes demands structured growth, such that location and morphology of the graphene are controlled.

We acknowledge financial support from NSF-MRSEC (#DMR-0820382), AFSOR, the W. M.

  FIG. 1. aC step pinning process. Depicted in (a) are the SiC steps occuring before the graphene growth.

FIG. 2 .

 2 FIG. 2. AFM images of the surface of hydrogen etched SiC(0001) (a) before and (b) after graphene growth. The corresponding step profiles taken along the step flow (white line in figures (a) and (b)) (c) before and (d) after graphene growth. The AFM images (e) and (f) and corresponding profiles along step flow (g) and(h), respectively, depict the morphology before and after graphene growth using the amorphous carbon grid.

  Figure 2 part (e) and (f) note that the orientation of the steps is different in each sub figure. This is due to differing local wafer miscut throughout the sample surface.

FIG. 3 .

 3 FIG. 3. As in Fig. (2) but here for sample B1. The SiC surface was chemically and mechanically etched prior to growth. The process of buffer formation without aC grid (a) before and (b) after buffer growth, corresponding profiles (c) and (d). The surface morphology using aC grid (g) before and (h) after buffer growth. Corresponding step profiles along the step flow are shown in figures (g) before and (h) after buffer growth.

FIG. 4 .

 4 FIG.4. Raman scattering spectra (using a λ = 532nm laser) of (a) in red: aC and in black: aC and SiC, (b) in red: Sample G1 single layer graphene and in black: graphene and SiC (c) in red: sample B1 buffer layer on a Si-face 4H-SiC, and in black: SiC and buffer.


John Landers

Frédéric Chérioux

Maurizio De Santis

Nedjma Bendiab

Simon Lamare

Laurence Magaud

Johann Coraux
email: johann.coraux@neel.cnrs.frfrederic.cherioux@femto-st.fr

Convergent Fabrication of a Nanoporous Two-Dimensional Carbon Network from an Aldol Condensation on Metal Surfaces

Keywords: Nanoporous, covalent organic framework, self-assembled monolayer, scanning tunneling microscopy, density functional theory calculation, flat bands, Dirac cone

des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The family of two-dimensional (2D) materials is a rapidly growing one, 1,2 which emerged with the study of graphene. Tailoring the structure of these materials allows engineering their properties. Striking examples are the change in the topology of the electronic band structure of graphene as a function of the number of layers, 3,4 the occurrence of electronic resonances in graphene islands under strong compressive strain, 5,6 and the confinement in graphene nanoislands, [7][8][9] and nanoribbons. 10 Hollow versions of graphene offer new degrees of freedom for structureengineering of the properties. Examples of such hollow 2D materials are graphynes, 11 graphdiynes, 12 and graphene antidot lattices. 13 The first two consist of low-density carbon atomic lattices with sub-nanometer pores, and have been predicted to exhibit exciting electronic properties, arising from multiple and/or anisotropic Dirac cones in their electronic band structure. 14 The latter has been much discussed in the literature, as it is predicted to yield spin qubits, 13 to open sizable band-gaps in an otherwise gapless graphene, 15 and to host dispersion-less electronic bands. 16 Experimental realization of graphene antidot lattices has been mostly achieved with the help of topdown approaches relying on lithography performed on plain graphene sheets. 17,18 An alternative approach would consist of a controlled assembly of well-chosen molecular blocks. Accordingly, covalent nanoporous networks, which are actually graphene antidot lattices with ultimately thin pore walls, have been prepared by interfacial Ullmann coupling reactions at metallic surfaces, consisting of a 2D polymerization of halogenated aromatic monomers. 19 Recently, other 2D reaction schemes yielding nanoporous networks have been discovered, 20,21 which also have the potential to yield conjugated nanoporous networks with electronic properties like those of graphene antidot lattices. In addition to being an ultimately precise version of the antidot lattice, these systems are also of great interest for the prospect of nanosieving or nanosensing, enabled by functionalization of the pores.

An ongoing effort has focused on the development of surface chemistry schemes capable of delivering nanoporous 2D covalent networks. The synthesis of purely carbon networks has been demonstrated recently by a few reactions, for instance the above-mentioned Ullmann coupling, 19 homo-coupling of alkynes, 20 and diyne cyclotrimerization. 21 One promising alternative would be to employ a convergent pathway, by which several molecular monomers react together, promoted by strong energy gains, to form for instance additional aromatic rings. 22 To date such convergent methods have not yet been exploited for achieving carbon-rich networks. We report such a reaction, inspired by previous works formerly implemented in solution by one of us, to achieve fullyconjugated hyper-branched dendrimers. [23][24][25] The reaction proceeds by a triple aldol condensation reaction between three acetyl groups, and is demonstrated here on Au(111). Unlike in previous reports of aldol condensation for the polymerization of methyl pyruvate on Pt(111), [26][27][28] which only yielded disconnected agglomerates, we observe extended 2D covalent nanoporous networks. The reaction is not self-poisoning and proceeds up to a full surface coverage. The only by-product is water, which is readily desorbed from the surface under the explored conditions. Using scanning tunneling microscopy (STM), Raman spectroscopy and density functional theory (DFT) calculations, we identify intermediate states in the growth process. The network is stable up to 500°C under ultra-high vacuum (UHV) and when exposed to ambient conditions, the carbon backbone remains intact.

Results and discussion

We have used 1,3,5-tri(4'-acetylphenyl)benzene (see Scheme 1), referred to in the following as TriAc, as the molecular precursor for the reaction. TriAc molecules were deposited under UHV onto the herringbone-reconstructed (111) surface of Au, between room temperature and 400°C. The molecules form domains which nucleate between the ridges (partial dislocations 29 ) of the herringbone reconstruction of Au(111), where adsorption is expected to be more favorable (Figure S1 in the Supporting Information). 30 The molecules form compact supramolecular networks, giving rise to spotty electron diffraction patterns (Figure S2 in the Supporting Information), and whose degree of order and symmetry vary as a function of the surface coverage. Figure 1a shows one of these supramolecular networks, obtained uniformly across the sample surface, consisting of ordered domains typically extending over 50 nm 2 , obtained for a TriAc dose prior to the formation of the second molecular layer, and consisting of a dense arrangement of interpenetrated molecules. In this arrangement, each oxygen atom of a TriAc interacts with one hydrogen atom on the C 2 ortho position of the lateral benzene ring on a neighboring TriAc molecule (see zoomed-in view of Figure 1b). The STM analysis reveals a 0.2 nm distance between these H and O atoms, which is typical of a hydrogen bond between acetyl and phenyl groups. 31,32 Other types of H-bond stabilized supramolecular networks were found, some of which are seen in the phase diagram of Figure 2 and reported in Figure S3 in the Supporting Information. Noteworthy, the ridges of the Au(111) herringbone reconstruction are still observed in the presence of the supramolecular networks, for instance seen in the form of a brighter line of TriAcs in the diagonal running from bottom left to top right in Figure 1a. No substantial decrease of the surface coverage is observed upon annealing for a few minutes between room temperature and 200°C, unlike other molecules of similar size (see, e.g., Ref. 33), i.e. the bonding of the TriAcs with Au(111) is substantial. Actually, ultraviolet photoelectron spectroscopy of oxygen-containing molecules on Au(111) point to a soft chemisorption, 34 a likely situation in the present case as well. Depending on the preparation of the tip, STM reveals a non-uniform conductance of the molecules (inset of Figure 1a), pointing to variations of the electronic density of states (DOS) (Figure S3 and S4 in the Supporting Information). In the following, unless specified, all experiments consist of TriAc deposition at 200°C for a few minutes, a temperature at which highly ordered supramolecular networks still form without the TriAc molecules reacting with one another, followed by annealing at higher temperatures.

Scheme 1: Aldol condensation surface reaction. Upon moderate heating (300°C) two TriAc molecules react together on Au(111) to form an intermediate dimer while releasing a water molecule (Step 1). At higher temperature (>300°C), the convergent triple aldol condensation reaction sets in, which yields a new benzene ring (Step 2). At higher temperatures (>400°C), the polymerization reaction sets in, yielding a 2D nanoporous polymer, prominently composed of fourand six-sided pores, involving at least one new benzene ring each (here three and two extra benzene rings are formed for six-and four-sided pores).

When longer annealing times are employed at 200°C, consisting of hours instead of a brief period of a few minutes, a progressive disorder begins to develop in the supramolecular network. This disorder is associated with the TriAc molecules forming head-to-head dimers, as observed in STM images (Figure S5 in the Supporting Information). During the period of dimer formation the electron diffraction patterns progressively become dominated by diffuse scattering (see diffraction patterns samples annealed between 300-400°C in Figure S6 in the Supporting Information). After 6 h at a temperature T = 200°C, the fraction of TriAc molecules having formed dimers is 27±5%. As expected, annealing at a higher temperature T = 300°C more efficiently promotes the formation of dimers: a 10 min annealing already results in 36±6% of the TriAc molecules forming dimers (Figure 1c). By estimating the percentage of TriAc dimers formed at these two temperatures, and assuming a ν 0 exp(-E/k B T), Arrhenius-like decay (k B being Boltzmann's constant) with a ν 0 = 10 13 Hz attempt frequency, an activation barrier E between 1.5 and 1.6 eV is derived.

The analysis of the STM data gives a 152±16° angle between the two connected legs of the dimer and a 1.85±0.30 nm distance between the central rings of the TriAc molecules (Figure 1d).

The latter value fits well with the distance of 1.92 nm obtained from DFT calculations (Figure S7) TriAcs evaporated, are achieved through the polymerization process (Figure 1e and Figure S9 in the Supporting Information). The pores of these networks are of various kinds (Figure 1f), in terms of shape, reticulation nodes, and progress of the reaction. The latter can be characterized by the fraction of TriAcs, one acetyl group of which at least has not participated in the aldol condensation reaction. This fraction decreases with increasing surface coverage (Figure 2), from 35-40% for 30% surface coverage (corresponding to network patches of extension of the order of 10 nm) to 15-20% for full-coverage, for similar annealing/deposition temperatures (~400°C). This decrease can be understood as a manifestation of the decrease of the amount of the network's edges upon increasing coverage. The fraction also decreases with annealing and deposition temperatures (Figure 2), for instance by 10%, down to 15-20%, when increasing the deposition temperature from room temperature to 200°C, while with similar high annealing temperatures. Such a decrease is consistent with the expected temperature-promoted surface diffusion of the TriAcs.

The nanoporous network is prominently composed of four-and six-sided pores, usually containing at least one new benzene ring as a reticulation node (Figure 1f-h), each corresponding to the completion of one triple aldol condensation. In some instances two and three benzene rings per pore are formed, for four-and six-sided pores respectively (Figure 1g,h), which corresponds to fully reacted pores, shown as the last step of Scheme 1. In other instances we observe reticulation nodes which are larger than benzene rings (shown as elongated hexagons in Figure 1f, for instance visible at the bottom left of the image in the case of four-sided pores). Such pores could actually be rings comprising eight carbon atoms, which would correspond to a dual aldol condensation between the ketone functions of two species of kind Step 1 shown in Scheme 1. The coexistence of pores of different shapes obviously causes strong distortions in the network, as seen by careful inspection of Figure 1f. The proportion of four-and six-sided pores varies with the coverage of the network. At low coverages (30%, network patches of extension of the order of 10 nm, e.g. point corresponding to 350°C deposition temperature and 450°C temperature in Figure 2), roughly 20% more six-sided pores are obtained, while for full-coverage cases 20% less six-sided pores are obtained. We ascribe this difference to a reduced mobility of the TriAcs prior to reaction at high coverages, which hinders the formation of the more energetically favorable (in view of steric and bond-distortion considerations) six-sided pores. We do not observe a substantial change in the proportion of fourand six-sided pores with annealing and deposition temperatures, presumably because when the polymerization reaction has significantly progressed (boundary between yellow and green domains in Figure 2), the surface mobility of the reactive species has already strongly decayed.

Other types of pores are observed as well, yet in minority, which correspond to situations when the 2D polymerization process is ended at Step 1 of Scheme 1 (see Figure S10 in the Supporting Information), presumably due to a limited mobility of the TriAc molecules and oligomers on the surface. 35 Competing reactions, involving the metallic surface (no competing reaction exist in solution), could also hinder the aldol condensation reaction to fully proceed, as was found in other systems. 36 Such reaction could promote partial graphitization through the formation of extra C-C bonds. Overall, the network is fully covalently bonded. We find that the number of TriAc molecules needed to achieve the 2D covalent carbon network is 40 % smaller than in the dense supramolecular network addressed before, implicating a loss of molecules by desorption.

Figure 1: 2D self-Assembly to a covalent carbon nanoporous network on Au(111), observed with STM. Inset: TriAc molecules exhibiting non uniform conductance after a tip change (a) TriAc molecules deposited at 200°C, leading to a H-bonded supramolecular network. (b) Zoom-in showing the arrangement of the supramolecular network with the hydrogen bonding highlighted by the dashed line. (c) At temperatures between 300 and 400°C, the system becomes disordered and the monomers begin to form TriAc dimers bonded together by covalent C=C bonds. (d) Zoom-in of a dimer. (e) 2D covalent nanoporous network. (f) Schematized view of the network in (e): the reacted TriAc molecules are shown as three-legged stars, green hexagon mark the position of extra carbon rings (see text for details), and disks mark the position of reticulation nods corresponding to

Step 1 in Scheme 1. (g,h) Zoom-in views with the presumed atomic arrangement superimposed onto the image, for six-and four-sided pores marked in dark grey in (f). Scale bares measure 2.5 nm. All STM topographs were obtained at +1.25 V and 0.1 nA.

The 2D covalent carbon network is stable under UHV up to 500°C, a temperature from which the surface coverage decreases, due to desorption of some molecules, which implies the breaking of covalent bonds, presumably the C-C bonds attached to the remaining O atoms. From 550°C, the network undergoes strong modifications. We speculate that these modifications are related to those occurring during the thermal reaction between aromatic quinone molecules, 37 coronene, 38 or waste molecules, 39 and yielding graphene. DFT calculations were performed corresponding to a perfect (thus, ideal) lattice of six-sided pores after full completion of the triple aldol condensation, without the presence of the substrate (Figure 3a) and for the smallest coincidence lattice between an ordered 2D covalent carbon network on Au(111) (Figure 3b). These calculations predict that the benzene rings shall exhibit little distortion and lie flat on Au(111), while unsupported structures are distorted with a rotation along the axis of the intermediate benzene rings (Figure S11 in the Supporting Information). The average molecule substrate distance is predicted to be 0.31 nm, varying only by 4.5 % across the plane of the network. Such a distance is typical of a weakly bond system. As a comparison, graphene grown on gold, a system known for its weak interaction with the substrate, exhibits an experimentally determined separation distance of 0.33 nm. 40,41 The partial charge map (Figure S3 in the Supporting Information) integrated between Fermi level and 0.5 eV above is in good agreement with the STM observations, although the former is better spatially resolved, as is expected since the calculation does not include the effect of the STM tip. No substantial changes are visible between the STM topographs obtained before and after exposure to air, even without annealing the sample after introducing it back into UHV (compare Figures 4a andb). The only feature that arises from exposing the sample to air is that the imaging conditions are less stable likely due to molecules from the atmosphere adsorbed on the sample, while the carbon framework remains intact. These observations establish the high stability of the 2D covalent carbon network, presumably a carbon backbone saturated with C-H bonds, to which molecules from air bond only weakly. This claim is further supported by Raman spectroscopy performed in air. Figure 5 shows the Raman spectra for a 2D network synthesized on Au(111) and for the TriAc monomers drop-casted onto Au(111) then allowed to dry. The spectra for the 2D network displays a peak centered at 1585 cm -1 , and a full-width at half maximum (FWHM) of 30 cm -1 . This peak is akin to the characteristic signature of a sp 2 hybridized carbon-conjugated systems, whose width would stem from disorder. In fact, this peak presumably corresponds to the so-called G vibration mode, which develops in largely extended sp 2 carbon systems. It differs from the mode observed in the case of the drop-casted monomers, which is centered at 1605 cm -1 and as a 7 cm -1 FWHM. Such features most probably relate to the stretching vibration mode of the carbon-carbon bonds in the monomer. Neither for the drop-casted monomer and the 2D network do we observe the characteristic features of the C=O stretching modes, which are expected to show up as peaks at about 1680 and 1820 cm -1 . Though they are indeed not expected for the reacted network, they are expected for the drop-casted monomers. In the latter case, their absence is presumably due to the strong and extended background in the Raman spectrum yielded by the gold substrate. On a quartz substrate on the contrary, we do observe a peak at ~1680 cm -1 for the drop-casted monomers; however this substrate lacks the catalytic activity for synthesizing the reacted network. A perfect isolated 2D network is expected to give rise to a G mode with a narrower peak, appearing at higher wavenumber, than the one we observe. We interpret these observations as an effect of strain, which is known to modify the G peak position and width in a related system, graphene. 42 In this scenario, the G peak of the 2D network actually encloses contributions from regions with varying local strains, which coexist due to the presence of different pore geometries, to the coalescence of growing domains, and to the mismatch in thermal expansion coefficient between the network and Au(111). As such, the G mode width provides an indirect evidence of covalent bond formation. In this light the observed G mode for the reacted network shows strong similarities with that found in polymers. [START_REF] Dekker | Infrared and Raman of Polymers[END_REF] The inset of Figure 5 shows a Raman map, displaying a signal for the peak centered at 1585 cm -1 extending over 80% of the surface, indicating a large extension of the reacted network across the gold. These traits of sp 2 carbon found across nearly the entire surface, clearly indicate a fully-extended covalent network comprised of carbon. Overall, both molecular resolution imaging and vibrational spectroscopy indicate that the covalent bonds of the 2D networks are mostly unaffected by exposure to air, as is also the case in other 2D covalent networks. [START_REF] Cai | [END_REF]45 Such stability is desirable in the prospect of applications outside UHV.

The triple aldol condensation reaction, performed with TriAc molecules, is a 2D polymerization process which is not directional in essence. Using templated surfaces (e.g. vicinal ones) or tailored monomers (e.g. with specific shape and/or reactivity), like was done for other kinds of surface reactions 46,[START_REF] Cai | [END_REF] could allow to achieve one-dimensional polymerized covalent networks, owing to lateral hindering of the monomers' mobility and reaction self-steering respectively.

Methods

Surface reaction of Triac molecules.

TriAc molecules (melting point 252°C, 99.99% purity) were synthesized according to a procedure described in the Supporting Information. Surface reactions were performed in a UHV system with 10 -10 mbar base pressure. The molecules were evaporated with the same flux using two evaporator types, a multi-pocket commercial Kentax one and a home-made one, in which the quartz crucible containing the molecules is resistively heated, in our case at temperature of 200°C. The deposition rate of the TriAc molecules was calibrated with a quartz microbalance prior to deposition assuming a 1 g/cm 3 molecular density, and the deposition rate was set at 0.5 monolayers/min for a full coverage as obtained in Figure 1a. An aligned and polished Au(111) crystal with dimensions of 12×3 mm 2 and 0.5 mm thickness was purchased from Surface Preparation Laboratory. Prior to molecule deposition the Au(111) surface was prepared by repeating cycles of Ar + sputtering at 0.8 keV and annealing at 900 K until no evidence of carbon was detected, as verified by in situ Auger electron spectroscopy, and the herringbone reconstruction was fully developed, as verified by in situ STM.

Characterizations. All STM measurements were performed under UHV and at room temperature, in the same system where the samples were prepared, using a commercial Omicron variable temperature STM/AFM, with electrochemically etched W tips. All images were obtained in constant current mode, with the tunnel bias applied to the tip. Images obtained for angles and distances were corrected from drift due to the piezoelectric elements, which was determined by tentatively recording two successive images at the same location. Micro-Raman spectroscopy was performed with a commercial Witec Alpha 500 spectrometer setup with a dual axis XY piezo stage in a back-scattering/reflection configuration. Grating used has 1800 lines/mm which confer a spectral resolution of 0.01 cm -1 for 100 s integration time. Laser excitation wavelengths of 532 nm (solid state argon diode) was used. Raman spectra were recorded in air with a Nikon ×100 objective (NA = 0:9) focusing the light on a 320 nm diameter spot (532 nm light), and with a Mitutoyo ×50 objective (NA = 0:75). A background, whose intensity decays with increasing wave-number, corresponding to the luminescence from the Au substrate, is observed in Figure 6.

Density functional theory calculations. DFT calculations were performed using the VASP code, with the projector augmented wave (PAW) approach. 47 The exchange correlation interaction is treated within the general gradient approximation parameterized by Perdew, Burke and Ernzerhof (PBE). 48 When the gold substrate was taken into account, long range dispersion corrections (van der Waals interactions) were accounted for by using Grimme corrections. 49 The dispersion coefficient C6 and van der Waals radius R0 for Au are not listed in the original paper by Grimme and thus have been taken from Amft et al. 50 These values for C6 and R0 are 40.62 J×nm/mol and 1.772 Å respectively. The Au(111) surface was modelled using a four-layer slab that was first relaxed. The TriAc covalent network was then added. Gold atoms were kept fixed in the bottom layers, all other atoms were allowed to relax. The lateral periodicity was imposed by the smallest -commensuratesupercell chosen to accommodate a TriAc molecule, therefore a 3√3×3√3 periodicity of the Au(111) surface. Note the DFT calculations were performed for unreconstructed Au(111). The empty space in the direction perpendicular to the surface was chosen equal to 14 Å. After convergence, residual forces were lower than 0.02 eV/Å. Calculations of the isolated TriAc or TriAc dimer were performed in very large supercells to avoid spurious interaction due to periodicity.

  for an unsupported stable structure consisting of two TriAc molecules having formed one covalent C=C double bond between the methyl group of the first TriAc molecule and the carbonyl group of the second TriAc molecule, releasing a water molecule in the process. This product corresponds to the first aldol condensation, i. e. the initiation of the polymerization process (Step 1 of Scheme 1). By extending the annealing time, more branched structures are observed (Figure S8 in the Supporting Information). Some of these structures correspond to the products of Step 2 of Scheme 1, i.e. to the convergent triple aldol condensation reaction of three TriAcs yielding a new benzene ring. The formation of these structures, which are distant one from the other by a few nanometers in the explored experimental conditions, marks the beginning of the propagation of the 2D polymerization reaction. Increasing the annealing and deposition temperatures above 350°C more efficiently promotes this propagation than increasing the annealing time. Disordered nanoporous networks, whose coverage can reach 100% of the Au(111) surface depending on the amount of

  Figure 2 summarizes our observations as a function of annealing and deposition temperature.

Figure 2 :

 2 Figure 2: Diagram representing the four kinds of phases (red, yellow, green, and blue sectors of the diagram) encountered as a function of deposition and annealing temperatures. Instances of both low and high coverage (θ) are shown. The fraction of TriAc molecules having at least one unreacted acetyl group (f) and the ratio in the number of six-to four-sided pours (r 6/4 ) are specified whenever relevant. Annealing time is 10 min. STM topographs are 20 nm-wide and were acquired at +1.25 V and 0.1 nA.

Figure 3 :

 3 Figure 3: Top views of the structure of the periodic 2D covalent nanoporous network, inferred from DFT, without (a) and in the presence (b) of the Au(111) substrate.

Figure 4 :

 4 Figure 4: STM topograph of (a) a 2D covalent nanoporous network synthesized on Au(111) under UHV (deposition at 200°C; annealing at 400°C); and (b) of a network prepared under similar conditions, exposed to air, and introduced back in UHV. Images were acquired at +1.25 V and 0.1 nA.

Figure 5 :

 5 Figure 5: Raman spectrum acquired in air for the 2D covalent nanoporous network on Au(111) (bottom curve) and the monomer drop-casted onto Au(111) (two top curves). The peak at 1590 cm -1 is characteristic of a sp 2 carbon bonded systems. Spectra averaged over two 100 s acquisitions at the same position with a laser spot size of 1 μm. Inset: 30×30 μm 2 map of the peak intensity at 1590 cm -1 for the 2D covalent nanoporous network.

Acknowledgement

JL acknowledges financial support for a research fellowship from the Nanosciences Foundations. We acknowledge Dr. S. Vlaic, Dr. Z. Han and Prof. Dr. F. Palmino for valuable discussions. JL is grateful to Prof. Kocabas for providing the Raman spectra for graphene grown on gold, displayed in the supporting information. 51

Supporting Information

Synthesis details of the TriAc monomer, additional surface characterization and DFT calculations are provided in the Supporting Information.


MD, PhD João Pedro Ferreira

MD, PhD Faiez Zannad

MD Scott D Solomon

MD Bertram Pitt

MD, PhD Patrick Rossignol

Prof Pedro João

Ferreira

Influence of race in the estimation of glomerular filtration rate in patients with a high cardiovascular and renal risk

The estimation of glomerular filtration rate (eGFR) is usually indirectly calculated from serum creatinine in routine clinical practice. This indirect eGFR determination is then used to perform therapeutic decisions. However, serum creatinine may be influenced by GFR, but also by muscle mass and diet; therefore, age, sex, race (Black/African American vs. other), height, and weight have been used to adjust the eGFR. [START_REF] Levey | A new equation to estimate glomerular filtration rate[END_REF] While age, sex or height are biological variables, race/ethnicity certainly has biological/genetic background but also carries a strong social and psychological burden, and is usually a very sensitive topic. As consequence, many medical centres have removed the race adjustment from eGFR despite the lack of studies assessing the potential clinical impact of this decision. [START_REF] Eneanya | Reconsidering the Consequences of Using Race to Estimate Kidney Function[END_REF][START_REF] Vyas | Hidden in Plain Sight -Reconsidering the Use of Race Correction in Clinical Algorithms[END_REF] The Chronic Kidney Disease Epidemiology Collaboration (CKD-EPI) equation to estimate GFR from serum creatinine was developed and validated across 10 studies including people with and without chronic kidney disease (CKD). Race was classified as African American (AA) or other as assigned by the study participants or the investigators. [START_REF] Levey | A new equation to estimate glomerular filtration rate[END_REF] A reassessment of the original CKD-EPI equation excluding the adjustment for race has led to a systematic error in the evaluation of AA participants, leading to an underestimation of the measured GFR and a worse performance of the CKD-EPI equation. [START_REF] Levey | Estimation of Glomerular Filtration Rate With vs Without Including Patient Race[END_REF][START_REF] Diao | Clinical Implications of Removing Race From Estimates of Kidney Function[END_REF][START_REF] Hsu | Race, Genetic Ancestry, and Estimating Kidney Function in CKD[END_REF] New eGFR equations that omit race but incorporate both creatinine and cystatin C together in the equation were found more accurate. [START_REF] Inker | New Creatinine-and Cystatin C-Based Equations to Estimate GFR without Race[END_REF] Given the potential implications of removing race from the CKD-EPI eGFR equation, we aim to replicate previous findings in a broad population of patients with a high cardiovascular and renal risk, including: 1) type 2 diabetes in the ACCORD (NCT00000620) trial, 2) hypertensive without diabetes in the SPRINT (NCT01206062) trial, and 3) heart failure with preserved ejection fraction in the TOPCAT (NCT00094302) trial.

We computed eGFR from serum creatinine using the CKD-EPI equation with and without the race indicator for AA/Black patients. We then compared the distributional eGFR shift, CKD reclassification, and prognostic implications of calculating eGFR with and without race. ACCORD enrolled 10,251 patients of whom 1,947 (19%) were Black/AA. The inclusion of race adjustment in the CKD-EPI equation gave a mean eGFR of 87±19 ml/min/1.73m 2 , when race was excluded from the equation the mean eGFR was 75±17 ml/min/1.73m 2 i.e., a difference of minus 12±3 ml/min/1.73m 2 by excluding race from the equation. Figure 1A. Excluding race from the CKD-EPI equation more than doubled the number of patients classified has having CKD stage 3: 378 excluding race vs. 166 including race. A similar pattern was seen in SPRINT and TOPCAT. In SPRINT 9,361 patients were included of whom 2,928 (31%) were Black/AA. Excluding race from the equation led to a difference of minus 10±3 ml/min/1.73m 2 , with many more patients classified as having CKD stage 3 (680 vs. 1048). Figure 1B. In TOPCAT-Americas 1767 patients were included of whom 293 (17%) were Black/AA. Excluding race from the equation led to a difference of minus 8±3 ml/min/1.73m 2 , and more patients classified has having CKD stage 3 (187 vs.145). Figure 1C.

In this study excluding race from the CKD-EPI equation systematically led to an underestimation of GFR and classified more patients has having worse CKD stages. These findings have important clinical implications and potential unintended consequences, as Black/AA patients could have received lower drug doses or have a label contra-indication for taking certain drug and some could have been excluded from these trials based on low eGFR.

Our findings expands to high CV risk patients previous reports from others, [START_REF] Levey | Estimation of Glomerular Filtration Rate With vs Without Including Patient Race[END_REF][START_REF] Diao | Clinical Implications of Removing Race From Estimates of Kidney Function[END_REF][START_REF] Hsu | Race, Genetic Ancestry, and Estimating Kidney Function in CKD[END_REF][START_REF] Inker | New Creatinine-and Cystatin C-Based Equations to Estimate GFR without Race[END_REF][START_REF] Levey | Kidney Disease, Race, and GFR Estimation[END_REF] which led an NKF-ASN task force to recommend immediate implementation of the new CKD-EPI creatinine equation refit without the race variable. [START_REF] Inker | New Creatinine-and Cystatin C-Based Equations to Estimate GFR without Race[END_REF][START_REF] Delgado | A Unifying Approach for GFR Estimation: Recommendations of the NKF-ASN Task Force on Reassessing the Inclusion of Race in Diagnosing Kidney Disease[END_REF] Further research on GFR estimation aimed at eliminating race and ethnic disparities is warranted. Caption: Calculating eGFR without the inclusion of the race variable ("African American") leads to a leftward shift in eGFR i.e., African American patients have their eGFR underestimated, which may have major implications for treatment, followup, and prognostic assessment.

Figure 1 .

 1 Figure 1. Influence of race in the estimation of glomerular filtration rate

Note: We used the CKD-EPI formular to compute eGFR, hence these results may differ from those presented in the respective trials, using the MDRD formula.

Fax : +33 (0) 3 83 15 73 24 Mail: j.ferreira@chru-nancy.fr Word-count: 600

Disclosures

The authors have no potential conflicts of interests regarding the content of this manuscript.