Antoine Tiberj
email: antoine.tiberj@univ-montp2.fr

Nicolas Camara

Philippe Godignon

Jean Camassel

Micro-Raman and micro-transmission imaging of epitaxial graphene grown on the Si and C faces of 6H-SiC

Micro-Raman and micro-transmission imaging of epitaxial graphene grown on the Si and C faces of 6H-SiC

Antoine Tiberj, Nicolas Camara, Philippe Godignon, Jean

Introduction

Since the first report by C.V. Raman in 1928 [START_REF] Raman | A new class of spectra due to secondary radiation[END_REF][START_REF] Raman | A change of wave-length in light scattering[END_REF], Raman spectroscopy has become increasingly popular in materials science and, especially, in semiconductor physics and microelectronics. Basically, Raman scattering probes the inelastic scattering of a monochromatic light (photons) by the lattice vibrations (phonons) in a solid. Since in a crystalline solid, the phonons are very sensitive to the internal and external perturbations, like doping and stress, the frequency of the scattered light (photons) is a local probe of the perturbation experienced (or not) by the medium. For a more detailed introduction and description of Raman and micro-Raman spectroscopy in materials science (and especially in semiconductors) please refer to [START_REF] Yu | Fundamentals of Semiconductors: Physics and Materials Properties[END_REF][START_REF] Weber | Raman Scattering in Materials Science[END_REF][START_REF] Nakashima | Raman imaging of semiconductor materials: characterization of static and dynamic properties[END_REF].

Today numerous applications exist that cover the whole development of modern electronic and optoelectronic devices. They run from basic inspection of asgrown semiconductors to advanced device inspection tools. For instance, Raman mapping enables to check the crystalline quality [START_REF] Richter | The one phonon Raman spectrum in microcrystalline silicon[END_REF][START_REF] Villeneuve | Raman spectroscopy study of damage and strain in (001) and (011) Si induced by hydrogen or helium implantation[END_REF], the composition [START_REF] Nakashima | Characterization of Si/Ge x Si 1-x structures by micro-Raman imaging[END_REF][START_REF] Tiberj | Stress metrology: the challenge for the next generation of engineered wafers. High-Mobility Group-IV Materials and Devices[END_REF], the doping level [START_REF] Burton | Spatial characterization of doped SiC wafers by Raman spectroscopy[END_REF][START_REF] Wellmann | Optical mapping of aluminum doped p-type SiC wafers[END_REF][START_REF] Harima | Raman scattering from anisotropic LO-phonon-plasmon-coupled mode in n-type 4H-and 6H-SiC[END_REF][START_REF] Perlin | Investigation of longitudinal-optical phonon-plasmon coupled modes in highly conducting bulk GaN[END_REF], or the uniformity of as-grown semiconductor materials. Along this line one on the most popular applications in microelectronics is strain measurements, either at the device or at the full wafer scale [START_REF] Tiberj | Stress metrology: the challenge for the next generation of engineered wafers. High-Mobility Group-IV Materials and Devices[END_REF][START_REF] Wolf | Micro-Raman spectroscopy to study local mechanical stress in silicon integrated circuits[END_REF][START_REF] Wolf | The investigation of microsystems using Raman spectroscopy[END_REF][START_REF] Poborchii | Study of stress in a shallow-trenchisolated Si structure using polarized confocal near-UV Raman microscopy of its cross section[END_REF][START_REF] Ogura | Measurement of in-plane and depth strain profiles in strained-Si substrates[END_REF]. Raman measurements can also be used for final device inspection, through the temperature mapping of operating devices like FETs, lasers, and actuators [START_REF] Ochalski | Complementary thermoreflectance and micro-Raman analysis of facet temperatures of diode lasers[END_REF][START_REF] Beechem | Invited article: simultaneous mapping of temperature and stress in microdevices using micro-Raman spectroscopy[END_REF][START_REF] Pu | Temperature dependence of Raman scattering in hexagonal indium nitride films[END_REF][START_REF] Pomeroy | Thermal mapping of defects in AlGaN/GaN heterostructure field-effect transistors using micro-Raman spectroscopy[END_REF]. In this case, thanks to the use of recent turnkey Raman systems, one can perform fast mapping with spatial resolution down to 300 nm. Combining with the use of several laser wavelengths, one can also probe the in-depth profile of multilayer systems and device. All together, these features confirm the unique versatiliy and potentialities of micro-Raman imaging in microelectronics. As a consequence, and because of its contactless and nondestructive nature, micro-Raman spectrosocopy (μRS) has become an attractive characterization tool in industrial clean-room facilities.

In this field, graphene is a new comer. Because of its outstanding electronic, thermal, optical, and mechanical properties [START_REF] Neto | The electronic properties of graphene[END_REF][START_REF] Nair | Fine structure constant defines visual transparency of graphene[END_REF][START_REF] Ferralis | Probing mechanical properties of graphene with Raman spectroscopy[END_REF] it can be considered as a promising candidate for future carbon-based electronics [START_REF]The latest International Technology Roadmap for Semi-conductors (ITRS) is available[END_REF]. However, and because of the so-called Kohn anomaly (which is nothing but the failure of the usual adiabatic Born-Oppenheimer approximation in zero-gap semiconductors [START_REF] Malard | Raman spectroscopy in graphene[END_REF]) it is also a perfect example to illustrate all the applications of μRS that have been mentioned before. Raman spectroscopy on few layers graphene (FLG) not only can evaluate the crystalline quality but also, the thickness, the stacking order of graphene sheets, the doping level, and finally the residual strain.

In this study, we review some recent Raman imaging results collected on epitaxial graphene grown on the Cand Si-faces of 6H-SiC substrates [START_REF] Camara | Anisotropic growth of long isolated graphene ribbons on the C face of graphite-capped 6H-SiC[END_REF][START_REF] Camara | Current status of self-organized epitaxial graphene ribbons on the C face of 6H-SiC substrates[END_REF][START_REF] Jouault | Probing the electrical anisotropy of multilayer graphene on the Si face of 6H-SiC[END_REF]. In the first section, we briefly describe the growth techniques and the experimental set-up used for micro-Raman and micro-transmission imaging. In the second section, we discuss results collected on self-organized graphene ribbons grown on the C-face of 6H-SiC substrates and we show how thicknesses, stacking order, and a rough estimate of doping level can be obtained. Finally, in the third section, FLG grown on the Si-face are investigated and we show how the compressive stress experienced by such FLGs can be estimated.

Experimental details

Growth technology

All samples were grown using the processes described in [START_REF] Camara | Anisotropic growth of long isolated graphene ribbons on the C face of graphite-capped 6H-SiC[END_REF][START_REF] Camara | Current status of self-organized epitaxial graphene ribbons on the C face of 6H-SiC substrates[END_REF][START_REF] Jouault | Probing the electrical anisotropy of multilayer graphene on the Si face of 6H-SiC[END_REF]. We used 1 × 1 cm 2 pieces of on-axis 6H-SiC substrates cut from, either, the Si-terminated (0001) face or the C-terminated (000 1) face of 6H-SiC wafers. Before cutting, polishing was done by Novasic to get Epiready ® morphology [30]. Then, a sacrificial oxide was thermally grown and chemically etched in HF to remove any trace of sub-surface damage from the polishing process. Finally, standard RCA treatments were done to remove any trace of surface contamination. All treatments were clean-room compatible and similar to the one used for SiC before thermal oxidation or postimplantation annealing. In this way, atomically flat surfaces were systematically obtained.

For sublimation, we used a high temperature furnace from Jipelec [31] previously dedicated to post-implantation annealing. It was rf-induction heated and fitted with a turbo-molecular pump. The vacuum limit reached in this way was 10 -6 Torr. Before sublimation, the samples were heated at 1150°C for 10 min to remove any trace of native oxide. During the growth, the samples were covered by a graphite cap to increase the C and Si partial pressures over the SiC surface. Such graphite coverage lowers the Si out-diffusion process during the growth and enables to perform FLG growth at higher temperature. This promotes better SiC surface reconstruction.

In this way, on the C-face of 6H-SiC SiC substrates after 15 min annealing at 1700°C in a secondary vacuum, the growth of long (self-organized) graphene ribbons can be reached [START_REF] Camara | Anisotropic growth of long isolated graphene ribbons on the C face of graphite-capped 6H-SiC[END_REF]. These ribbons are 5-μm wide and 150-μm long, but a longer 1700°C annealing results in longer ribbons (up to 600-μm) with the same width. The width does not depend on the annealing time because the ribbons fully occupy a single terrace of the heavily reconstructed (step-bunched) SiC surface [START_REF] Camara | Current status of self-organized epitaxial graphene ribbons on the C face of 6H-SiC substrates[END_REF].

On the Si-face, to increase the FLG anisotropy, a modified growth process was used. The growth was done at 1750°C for 20 min under argon with a graphite cap covering the sample [START_REF] Jouault | Probing the electrical anisotropy of multilayer graphene on the Si face of 6H-SiC[END_REF]. In this way, a full graphene coverage was obtained with a similar (pronounced) step-like morphology of the SiC substrate. The average terrace width was again 5 -μm and the average step height 10 nm. Optical microscopy showed that the terraces had a remarkable homogeneity of width and orientation over a scale of 1 cm 2 .

Coupling micro-Raman spectroscopy with microtransmission measurements

Raman spectra were collected at room temperature, using a Jobin-Yvon Horiba T64000 spectrometer operated in the confocal mode. The 514-nm line of an Ar-Kr ion laser was used for excitation. With a × 100 microscope objective, the spot diameter was about approximately 1 μm with, typically 1-mW power focussed on the sample. To combine micro-Raman spectroscopy with micro-transmission experiments, a low noise photodiode was inserted between the SiC substrate and the XYZ piezoelectric stage. For details, see [START_REF] Camara | Current status of self-organized epitaxial graphene ribbons on the C face of 6H-SiC substrates[END_REF]. In this way, it was possible to measure at the same time (using the same laser beam as probe) the power transmitted through the sample and the associated micro-Raman spectrum. The true FLG's spectra were obtained by subtracting the SiC reference signal from the experimental results.

The graphene extinction was deduced using the following equation:

η = T 0 -T T 0 (1)
in which T 0 is the bare SiC transmittance value and T the modified one with epitaxial FLGs on top. From the work of Ref. [START_REF] Stauber | Optical conductivity of graphene in the visible region of the spectrum[END_REF] it can be expressed as:

T = 4n 1 + n + Nσ ε 0 c 2 in which σ = π e 2 2h (2)
In this equation, n = 2.68 is the SiC refractive index, N is the number of graphene layers, and s is the optical conductivity of a single (isolated) graphene sheet that was confirmed experimentally [START_REF] Nair | Fine structure constant defines visual transparency of graphene[END_REF]. The relative (theoretical) extinction of a monolayer and a bilayer graphene on top of a SiC substrate is then 1.23 and 2.44%, respectively.

Raman imaging of isolated graphene ribbons grown on the C-face

In Figure 1, we show the results of a large (20 × 100 μm 2 ) map collected on two neighboring graphene ribbons. The step size was 0.5 μm for the X direction and 2 μm for the Y direction. Six individual maps are shown. The first one corresponds to the extinction values, the second one to the integrated intensity of the G band normalized to the HOPG peak. The third one gives the normalized integrated intensity of the 2D band, while the fourth and fifth ones give the shift of the G and 2D bands, respectively. Finally, the last one corresponds to the absolute value of the Fermi level computed from the previous results. Of course, because of the limited range of the XY piezostage (100 × 100 μm 2 ) the two ribbons could not be completely probed. But a first point to be noticed is that, on both ribbons, no D band map could be given. This is shown in greater details in Figures 2 and3 and demonstrates the excellent crystalline quality of these graphene samples.

Let us now consider everything more in detail. The first (extinction) map shows that both ribbons have an excellent thickness uniformity. However, and because the relative extinction is different, it shows also that the left one is thicker than the right one. Concerning the absolute values, from these extinction maps complemented by additional point by point measurements, we find that the thinner (right) ribbon corresponds with relative extinction values h ranging from 1.2 to 1.4%. This shows that we deal with a true epitaxial monolayer graphene (MLG) ribbon. To ascertain this result, a Raman spectrum collected at the center of ribbon is shown in Figure 2 (lower spectrum). It is clearly similar to the one reported in the literature [START_REF] Ferrari | Raman spectrum of graphene and graphene layers[END_REF] for monolayers graphene exfoliated on top of an oxidized silicon substrate and all spectra collected on the same ribbon exhibited the same Raman fingerprint. Typically, the G band falls between 1583 and 1587 cm -1 , with a FWHM of the order of 13 cm -1 , while the 2D band ranges from 2682 to 2688 cm -1 with a FWHM around 25 cm -1 . This means that the graphene ribbons are strain free, unlike epitaxial graphene grown on the Si-face of SiC. This is confirmed by the presence of wrinkles evidenced by AFM [START_REF] Camara | Anisotropic growth of long isolated graphene ribbons on the C face of graphite-capped 6H-SiC[END_REF]. For such graphene monolayers, the absolute value of the Fermi level (and the doping level) can be extracted from the ratio I G /I 2D between the integrated intensities of the G and 2D band a according to [START_REF] Basko | Electron-electron interactions and doping dependence of the two-phonon Raman intensity in graphene[END_REF]:

E f = γ ep 0.06 I G I 2D 1 0.26 -1 (3) |n| = 1 π E f hv 0 2 (4)
In these equations, g ep = 21 meV is proportional to the electron-phonon scattering rate [START_REF] Casiraghi | Doping dependence of the Raman peaks intensity of graphene close to the Dirac point[END_REF], v 0 = 10 6 ms -1 is the electron velocity. The 0.06 factor is deduced from the function f(r S ) [START_REF] Basko | Electron-electron interactions and doping dependence of the two-phonon Raman intensity in graphene[END_REF] by using the effective dielectric constant ε eff = 5.33 for our graphene layer comprised between air and 6H SiC (ε 6H-SiC = 9.66 [START_REF] Patrick | Static dielectric constant of SiC[END_REF]). The absolute value of the Fermi level was then evaluated for all data points collected on the right ribbon (see Figure 1), giving absolute values between 200 and 350 meV. It corresponds to a doping level between 3 × 10 12 and 9 × 10 12 cm -2 with an average of 6 × 10 12 cm -2 . We have not checked directly on the same ribbon but transport measurements performed on few similar ones [START_REF] Camara | Current status of self-organized epitaxial graphene ribbons on the C face of 6H-SiC substrates[END_REF] gave a hole concentration of 5 × 10 12 cm -2 . According to the work of [START_REF] Das | Phonon renormalization in doped bilayer graphene[END_REF] such concentration is also in excellent agreement with the G and 2D band positions.

Concerning the second ribbon (on the left side), as already said we found a twice larger relative extinction. Ranging from 2.6 to 2.8%, this indicates a bilayer system. The relative extinction and the G band intensity both indicate that this ribbon is a bilayer with an excellent thickness uniformity. On the contrary, the 2D band intensity map in Figure 1 reveals sharp variations. Basically, the ribbon can be divided into three different domains, the top and bottom part having a less intense 2D band intensity than the middle one. Since these variations are correlated with shifts of the 2D and G bands positions, we assume that there are some doping level fluctuations. Indeed, it has been recently demonstrated that the G Raman band depends strongly on its electrostatic environment [START_REF] Gava | Probing the electrostatic environment of bilayer graphene using Raman spectra[END_REF][START_REF] Ando | Field effects on optical phonons in bilayer graphene[END_REF]. If the top graphene sheet has a different doping level than the bottom one, the doping difference changes the Raman shift and intensity of the G band. It also breaks the inversion symmetry and activates antisymmetric modes (that are usually Raman inactive). This results in a splitting of the G band which was observed experimentally [START_REF] Bruna | Observation of Raman G -band splitting in top-doped few-layer graphene[END_REF][START_REF] Yan | Optical phonon mixing in bilayer graphene with a broken inversion symmetry[END_REF][START_REF] Malard | Observation of distinct electron-phonon couplings in gated bilayer graphene[END_REF]. Our interpretation is strengthened by the two Raman spectra shown in Figure 3. In both cases, the 2D band exhibits the characteristic shape of AB (Bernal) stacking [START_REF] Ferrari | Raman spectrum of graphene and graphene layers[END_REF] but the G band is different. On the upper spectrum a single G peak is observed while on the lower one (collected in the bottom part of the ribbon) a clear G band splitting shows that both layers are not evenly doped.

G band 2D band

Figure 2 Raman spectra of different ribbons: a monolayer, a misoriented bilayer, and a turbostratic multilayer with their corresponding relative extinction. A misoriented bilayer has a similar Raman spectrum than a monolayer with a twice intensity.

The multilayer corresponds to eight graphene sheets that are all disoriented with respect to each other. Therefore, the single Lorentzian shape of the 2D band cannot be used as a proof to assert the monolayer character of a FLG sample. Bernal stacking is not that usual for epitaxial graphene grown on the C-face of SiC substrates. Most of the time the graphene planes are slightly misoriented, corresponding to turbostratic stacking. In this work, we also found misoriented ribbons (not shown). The presence of rotational stacking faults between the two (or more) successive graphene planes results in Raman spectra similar to the monolayer one as shown in Figure 2. The line shape is not modified. Simply the intensity increases as the number of graphene layers increases. In Figure 2, we mentioned the relative extinction measured on these ribbons. We found 2.5 and 9.7% that corresponds, respectively, to a bilayer and a 7 or 8 misoriented layers stack. These spectra correspond to FLG where all graphene sheets are disoriented with respect to each other. Therefore, the fact that the 2D band has a single Lorentzian shape can definitively not be used as a proof to assert the monolayer character of FLG flakes. The combination of μRS with micro-transmission measurements appears then as a most necessary tool to discriminate (without any ambiguity) between true MLG and misoriented multilayers. Of course, to perform such reproducible intensity measurements, any laser power fluctuation has to be carefully corrected. In this work, this was done using an additional low noise photodiode that measured continuously the laser power during the Raman map acquisitions.

Coming back to the maps in Figure 1, some correlation exists between the extinction and the G band intensity. From this observation one could conclude that the G band intensity can be used to evaluate the number of graphene layers. This is not that simple. In Figure 4, we plot the normalized integrated intensities of the G and 2D band measured on many mono and bilayer graphene (not shown here) versus the corresponding extinction values. The theoretical extinctions values are indicated as vertical dashed lines. Despite the correlation existing between the G band intensity and the relative extinction, the scattering of the G band intensity is too large when compared to the difference between a mono-and bilayer. We can even find bilayers that has the same G band intensity than monolayers. Therefore, the G band intensity cannot be used as an absolute thickness measurement but rather as a first guess if the relative extinction cannot be measured. The relative extinction is indeed not measurable in several cases: (i) if no bare SiC exists on the sample (for instance as graphene fully covers the SiC surface), (ii) if impurities (dust) or metal contacts cover the graphene and/or the SiC. In this particular case, we can evaluate the thickness by assuming that the average G band normalized intensity of a monolayer is between 0.025 and 0.03. Beware that these values depend on the experimental configuration and must be calibrated. For thin FLG (¡5 layers) the error can be of 1 layer and, for thicker FLGs, the estimated thickness can have a factor two error. Unlike the G band, the 2D band intensity cannot be used to evaluate the thickness for several reasons. For monolayers, the 2D band intensity strongly depends on the Fermi Level [START_REF] Basko | Electron-electron interactions and doping dependence of the two-phonon Raman intensity in graphene[END_REF]. We used this dependance to evaluate the absolute value of the doping in the monolayer ribbon. Moreover in Figure 4, we can clearly see that the 2D band of an AB bilayer is as intense as the one of a monolayer.

Raman imaging of FLG grown on the Si face

In Figure 5, we show a 30 × 40 μm 2 micro-Raman map collected on an epitaxial graphene stack grown on the Si-face of a 6H-SiC substrate. The normalized integrated intensity of the G band is compared to its Raman shift and to an optical microscopy image (OM) of the same area. On the OM image, dark areas correspond to central part of the terraces, while the bright areas correspond to the edges of the step bunched SiC reconstructed surface. As already said, the terraces are 5 μm wide and 10 nm high and, from the G band intensity, we find that graphene covers all the SiC surface. It is then impossible to measure directly the relative extinction and (consequently) the FLG thickness. Of course, a thickness estimate can still be done from the G band intensity. We found about 5 layers in the center of terraces (green-blue areas in Figure 5) and about 11 layers on the stripes close to the edge of terraces. On these stripes, we could also distinguish some black A correlation exists between the G band intensity and the relative extinction. Two cloud of points can be easily distinguished between mono-and bilayers thanks to the relative extinction. The scattering of the G band intensity is too large to use it as an absolute thickness measurement. The 2D band scattering is even higher especially for bilayer. 2D band of AB bilayer is as intense as monolayers whereas misoriented bilayers are twice intense than a monolayer.

points on the OM image. At these points, the G band intensity is much intense which suggests that they correspond to thick graphitic pits. The Raman spectra of these pits exhibit a strong D band, characteristic of a bad crystalline quality. Such pits are probably induced by an increased growth rate coming from the presence of structural defects, like threading dislocations, in the SiC wafer.

In Figure 6, we gathered Raman spectra collected in the middle of the terraces and on the stripes. Unlike the graphitic pits, no D band can be observed. This shows that most of the grown FLG have an excellent crystalline quality. The 2D band are broad with a lower frequency shoulder. This shoulder is more pronounced for the thickest FLG which have a 2D band shape similar to the HOPG one. This asymmetric 2D band is a clear indication of Bernal stacking even for the thinner FLG where the low-frequency shoulder is known to become less visible [START_REF] Graf | Spatially resolved Raman spectroscopy of single-and few-layer graphene[END_REF].

These Raman spectra also reveal that both G and 2D bands are shifted to high frequencies. For the thinnest FLG the G band falls between 1590 and 1600 cm -1 and the 2D band between 2750 and 2760 cm -1 . Such high blueshift cannot be explained by doping but rather by a compressive stress experienced by the FLGs. This stress comes from the differential dilatation coefficient between the FLG and the SiC substrate when cooling down the sample after the growth. Thanks to the Grüneisen parameters that have been recently measured [START_REF] Ferralis | Probing mechanical properties of graphene with Raman spectroscopy[END_REF][START_REF] Mohiuddin | Uniaxial strain in graphene by Raman spectroscopy: G peak splitting, Grüneisen parameters, and sample orientation[END_REF], this biaxial stress/strain can be estimated from the coefficients listed in Table 1. We find strain values between -0.3 and -0.4%, corresponding to local stress values ranging from -3 and -5 GPa. However, for thicker Raman spectra collected in the middle of the terraces (5 to 6 layers) and on the stripes close to the step edges (11 layers). No D band can be observed confirming the excellent crystalline quality of these FLG. The asymmetric shape of the 2D band (that is more pronounced for the thicker FLG) reveals a Bernal stacking of the graphene planes. Finally, both bands are blueshifted. Such shift can only be explained by a compressive strain of the graphene lattice coming from the differential dilatation during the cooling down of the sample. A partial strain relaxation occurs for thicker FLG since the thicker the less shifted.

FLGs, the G and 2D band are less shifted. This reveals a partial strain relaxation as the graphene thickness increases, which can be easily seen in Figures 5 and6.

The thicker the FLG, the less blue shifted the spectrum.

From these results, we confirm that there is a strong difference between graphene grown on the C-face and graphene grown on the Si-face of SiC substrates. On the Si-face graphene strongly interacts with the underlying SiC lattice. This interaction leads to the formation of the so-called buffer layer, which is covalently bound to the SiC lattice [START_REF] Qi | Epitaxial graphene on SiC(0001): More than just honeycombs[END_REF][START_REF] Varchon | Electronic structure of epitaxial graphene layers on SiC: effect of the substrate[END_REF]. This buffer layer interacts also with the graphene layers that are grown subsequently. It induces a downward shift (E D = -0.4 eV) of the K point corresponding to a n-type doping. This interaction causes also deviations from the linear band dispersion leading to a parabolic dispersion with an apparent gap of approximately 0.25 eV. This strong interaction is also responsible for the thermal stress experienced by these graphene layers. This is no longer true on the C-face, on which it has been shown that graphene interacts very weakly with the underlying substrate [START_REF] Hiebel | Atomic and electronic structure of monolayer graphene on 6H-SiC (3 × 3): a scanning tunneling microscopy study[END_REF][START_REF] Hiebel | Graphene-substrate interaction on 6H-SiC: a scanning tunneling microscopy study[END_REF]. It is the weakness of this interaction that explains why rotational stacking faults can easily occur and why the graphene sheets can relax the thermal stress by forming wrinkles or pleats.

Finally, on the G band intensity map we can see several points marked in blue. These blue points correspond to area where crystalline silicon clusters were found. One of the corresponding uncorrected Raman spectrum is shown in Figure 7. The presence of these crystalline silicon (c-Si) clusters is evidenced by the sharp and intense band around 532 cm -1 blue shifted compared to bulk silicon. This high blue shift is again due to a strong compressive stress induced by the SiC substrate (-2 GPa or a strain of -1.3%). First-order Raman scattering of the SiC substrate -1 and the corresponds to the two TO modes of E 2 symmetry at 764 and 786 cm -1 and the A 1 (LO) phonon at 964 cm -1 . Its second overtone with its characteristic fingerprint [START_REF] Burton | Micro-Raman and micro-transmission imaging of epitaxial graphene grown on the Si and C faces of 6H-SiC[END_REF] falls between 1400 and 2000 cm -1 under the sharp G band of FLG around 1590 cm -1 . The 2D band is around 2780 cm -1 . No D band can be seen on these points. Moreover, we can see that no significant variations of the G band intensity and Raman shift can be observed close to these Si clusters. This means that these Si clusters do not alter the graphene growth. These clusters are located close to the step edges, like the graphitic pits. There might be a link between the presence of different defects at the step edges like these clusters, the higher growth rate and the clear electrical anisotropy that has been evidenced by magnetoresistance experiments performed on several Hall bars with different orientations [START_REF] Jouault | Probing the electrical anisotropy of multilayer graphene on the Si face of 6H-SiC[END_REF].

Conclusions

Reviewing recent Raman imaging experiments performed on epitaxial graphene grown on the C and Si face of 6H SiC substrates, we have shown the benefits of combining Raman spectroscopy with micro-transmissions measurements. Provided the relative extinction of FLGs can be obtained, this enables to determine (without any ambiguity) the thickness, homogeneity, and stacking order (Bernal or turbostratic) of FLGs. On the C-face of SiC substrates we have shown that long, self-ordered, graphene ribbons can be grown. These ribbons have excellent crystalline quality and are strain relaxed. They are up to 600-μm long and 5-μm wide. They are mainly monolayers and Bernal stacked bilayers but turbostratic bi and multilayer areas have also been found. Finally for monolayers graphene, we also illustrated how the absolute value of the Fermi level can be found, in good agreement with electrical results.

On the Si face, on the opposite, a full graphene coverage of the SiC surface has been found. The surface is still heavily step-bunched but a high compressive thermal strain and n-type doping has been observed. It confirms that on the Si face a strong interaction exists Table 1 Frequency shifts of the G and 2D bands for a biaxial strain of 1% or a biaxial stress of 1 GPa [START_REF] Mohiuddin | Uniaxial strain in graphene by Raman spectroscopy: G peak splitting, Grüneisen parameters, and sample orientation[END_REF][START_REF] Ferralis | Probing mechanical properties of graphene with Raman spectroscopy[END_REF] between the graphene layers and the underlying SiC substrate. FLG on the Si-face exhibits Bernal stacking with thickness inhomogeneity. Thin (5 layers) FLGs were grown in the middle of terraces, while thicker graphene stripes grew close to the step edges. In the vicinity of these steps disordered graphite pits and crystalline Si clusters were found. There might be a link between the presence of these defects, the thickness inhomogeneity and the clear electrical anisotropy that has been recently evidenced by magnetoresistance experiments.

Endnote

a The ratio between the integrated intensities of the 2D and G band depends on the experimental setup. In our case I 2D = I G = 1.02 for our HOPG reference sample.

For the lack of a better knowledge, we assume that our setup is similar to the ones of Refs. [START_REF] Basko | Electron-electron interactions and doping dependence of the two-phonon Raman intensity in graphene[END_REF][START_REF] Casiraghi | Doping dependence of the Raman peaks intensity of graphene close to the Dirac point[END_REF].

Figure 1

 1 Figure 1 20 × 100 μm 2 maps of two graphene ribbons grown on the C-face of 6H-SiC. The step sizes are 0.5 and 2 μm for the X-and Y-axes, respectively. The relative extinction, the normalized intensities, and Raman shifts of the G and 2D band are shown. The right (left) ribbon corresponds to a monolayer (bilayer) graphene. The absolute value of the Fermi level is evaluated from the ratio between the intensities of the 2D and G bands only for the monolayer. It corresponds to a doping level between 3 × 10 12 and 9 × 10 12 cm -2 with an average of 6 × 10 12 cm -2 .

Figure 3

 3 Figure[START_REF] Yu | Fundamentals of Semiconductors: Physics and Materials Properties[END_REF] Raman spectra of different area of the AB bilayer ribbon. One is extracted from the bottom, the second from the middle of the ribbon. The bottom part of the ribbon exhibits a double G peak that can correspond to a different doping level between the top and bottom graphene sheet.

Figure 4

 4 Figure 4 Plot of the normalized integrated intensities of the G and 2D band against the relative extinction for several monoand bilayer graphene. The G and 2D band integrated intensities are normalized with the integrated intensities of an HOPG reference sample. Theoretical values of the relative extinction are indicated by two vertical dashed lines.A correlation exists between the G band intensity and the relative extinction. Two cloud of points can be easily distinguished between mono-and bilayers thanks to the relative extinction. The scattering of the G band intensity is too large to use it as an absolute thickness measurement. The 2D band scattering is even higher especially for bilayer. 2D band of AB bilayer is as intense as monolayers whereas misoriented bilayers are twice intense than a monolayer.

Figure 5

 5 Figure 5 30 × 40 μm 2 optical image of the graphene surface and the corresponding Raman maps of the G band intensity and Raman shift.The intensity of the G band is integrated and normalized by the G band of an HOPG reference sample. A full graphene coverage of the surface is observed with thickness inhomogeneities. FLG are thicker at the step edges (about 11 layers) than in the middle of the terraces (about 5 layers). On the edges we can clearly observed stripes: bright areas on the OM image and red areas on the G band intensity map. On the OM image, we can also see black points that correspond to C-rich graphite pits induced by an increased growth rate due to the presence of crystalline defects. On the G band intensity map, blue points mark the presence of Si clusters where the Raman fingerprint of silicon was observed. Finally, the G band is shifted to higher frequencies indicating that FLG are compressively stressed. This stress is progressively relaxed as FLG are thicker.

Figure 6

 6 Figure[START_REF] Richter | The one phonon Raman spectrum in microcrystalline silicon[END_REF] Raman spectra collected in the middle of the terraces (5 to 6 layers) and on the stripes close to the step edges(11 layers). No D band can be observed confirming the excellent crystalline quality of these FLG. The asymmetric shape of the 2D band (that is more pronounced for the thicker FLG) reveals a Bernal stacking of the graphene planes. Finally, both bands are blueshifted. Such shift can only be explained by a compressive strain of the graphene lattice coming from the differential dilatation during the cooling down of the sample. A partial strain relaxation occurs for thicker FLG since the thicker the less shifted.

Figure 7

 7 Figure7Uncorrected Raman spectrum extracted from the Raman mapping that corresponds to one of the blue point in the G band intensity map. The first-order Raman scattering of SiC correspond to the bands at 764, 786, and 964 cm -1 . Its second overtone falls between 1400 and 2000 cm -1 with the sharp G band around 1590 cm -1 . The 2D band is around 2780 cm -1 . No D band can be seen on this point. The sharp and intense band around 532 cm -1 correspond to a crystalline Si cluster that is highly compressively stressed by the SiC substrate.

Acknowledgements

We greatly acknowledge for partial support the EC through the RTN "ManSiC" Project (Grant MRTN-CT-2006-035735) and the French ANR through the Graph-SiC Project (Grant No. ANR-07-Blanc-0161). Finally, one of us (N.C.) acknowledges the Spanish Government for a Grant "Juan de la Cierva 2006" during his stay in Barcelona.

Author details

Competing interests

The authors declare that they have no competing interests.


Tak-San Ho
email: tsho@princeton.edu

Herschel Rabitz
email: hrabitz@princeton.edu

Gabriel Turinici
email: gabriel.turinici@dauphine.fr

T.-S Ho

Critical points of the optimal quantum control landscape: a propagator approach

Keywords: quantum control, singular control, landscape analysis in quantum control PACS 32.80.Qk, 02.30.Yy Mathematics Subject Classification (2000) 15A57, 49J20, 65Z05

. For bilinear quantum control problems, no general results are available to fully determine when this mapping is singular or not. In this paper we give sufficient conditions, in terms of elements of the evolution semigroup, for a trajectory to be non-singular. We identify two lists of "way-points" that, when reached, ensure the non-singularity of the control trajectory. It is found that under appropriate hypotheses one of those lists does not depend on the values of the coupling operator matrix.

Introduction

Manipulating the evolution of physical systems at the quantum level has been a longstanding goal from the very beginnings of laser technology. Investigations in this area accelerated after the introductions of (optimal) control theory tools [START_REF] Judson | Teaching lasers to control molecules[END_REF], which greatly contributed to the first positive experimental results, see [START_REF] Assion | Control of chemical reactions by feedback-optimized phase-shaped femtosecond laser pulses[END_REF][START_REF] Levis | Selective bond dissociation and rearrangement with optimally tailored, strong-field laser pulses[END_REF][START_REF] Weinacht | Controlling the shape of a quantum wavefunction[END_REF][START_REF] Bardeen | Feedback quantum control of molecular electronic population transfer[END_REF][START_REF] Bardeen | Quantum control of population transfer in green flourescent protein by using chirped femtosecond pulses[END_REF][START_REF] Hornung | Optimal control of molecular states in a learning loop with a parameterization in frequency and time domain[END_REF][START_REF] Kunde | Adaptive feedback control of ultrafast semiconductor nonlinearities[END_REF] and references herein. Consider a quantum system in the presence of a control field ǫ(t) ∈ R, t ≥ 0. In the electric dipole approximation and in the density matrix formulation, the underlying time-dependent density matrix ρ(t) satisfies the Liouville-von Neumann equation

i ∂ ∂t ρ(t) = [H 0 -ǫ(t)µ, ρ(t)] (1)
ρ(t = 0) = ρ 0 where H 0 is the field-free Hamiltonian (including the potential) and µ is the dipole moment of the system. In the analysis, we will suppose that the system contains N levels, thus, H 0 , µ and ρ(t) are all N × N Hermitian matrices. In the following, for simplicity, we further assume that H 0 and µ are real and symmetric matrices. Moreover, to avoid trivial settings, the dipole moment operator µ is assumed to have zero trace, i.e.,

T r(µ) = 0.

(2)

In the following we adopt the notation isu(N ) to denote the set of all complex N ×N Hermitian matrices with zero trace. Recall that ρ(t) = U (t, 0)ρ(0)U * (t, 0), where the unitary propagator U (t, 0) ∈ U (N ), t ≥ 0 is the solution of the timedependent Schödinger equation

i ∂ ∂t U (t, 0) = H 0 -ǫ(t)µ U (t, 0) (3)
U (t = 0, 0) = I N .
The control goal can be expressed in terms of a (self-adjoint) operator O in that the corresponding functional to be optimized (maximized) is O (T ) = Tr ρ(T )O . Accordingly, a maximum value of O (T ) indicates that the control is of optimal quality and vice-versa.

At the heart of a good understanding of any quantum control problem, including the efficiency, accuracy, and stability of quantum control experiments as well as simulations, lies essential features of the control landscape, depicting the functional dependence of the observable on the control field, i.e. the mapping: ǫ(t) → O (T ). The slopes (gradient) and the curvatures (Hessian) are key characteristics of the control landscape. In particular, the topologies at and around the critical points, in which the slopes vanish, can shed important light on the questions, for example, of what makes quantum control experiments (simulations) apparently "easy to perform" and why quantum control beats the "curse of dimensionality" (i.e., overcomes the anticipated exponentially growing effort required when searching over increasing numbers of control variables) [START_REF] Ho | Why do effective quantum controls appear easy to find?[END_REF]. A thorough study of the underlying control landscape should give insight into the classes of future feasible quantum control experiments, especially those involving complex molecules and materials.

Motivation: landscape analysis and beyond

The search procedures optimize the control quality O (T ) with respect to variations in the control ǫ(t). The mapping ǫ(t) → O (T ) is the composition of two maps: a control map ǫ(t) → U (T, 0) and a "kinematic" map U (T, 0) → O (T ). A key issue is whether there exist critical points that are suboptimal solutions. To analyse the critical points it is expedient to consider these two maps separately.

Previous analyses [START_REF] Rabitz | Quantum Optimally Controlled Transition Landscapes[END_REF] have investigated the mapping from the propagator to the observable i.e. U (T, 0) → O (T ); the critical points correspond to the equation

δ O (T ) δU (T, 0) = 0. ( 4
)
Assuming controllability hypothesis [START_REF] Altafini | Controllability of quantum mechanical systems by root space decomposition of su(N )[END_REF][START_REF] Turinici | Wavefunction controllability in quantum systems[END_REF], this equation is satisfied if and only if [U * (T, 0)OU (T, 0), ρ(0)] = 0,

i.e, we obtain a necessary and sufficient condition for the kinematic critical points [START_REF] Girardeau | Kinematical bounds on optimization of observables for quantum systems[END_REF]. The optimization "lanscape", when looked upon in terms of this so called "kinematic" mapping, is therefore exempt of suboptimal critical points.

To extend the landscape analysis beyond the kinematic setting, i.e. to investigate the dependence on the control itself, we need to analyse the solutions of the following critical point condition [START_REF] Ho | Why do effective quantum controls appear easy to find?[END_REF] δ O (T )

δǫ(t) = 0. (6)
The solutions of this equation are among the critical points of the control map ǫ(t) → U (T, 0); it was shown in [START_REF] Wu | Singularities of Quantum Control Landscapes[END_REF] that this mapping is nonsingular when all matrix entries of the dipole moment μ(t) = U * (t, 0)µU (t, 0), [START_REF] Girardeau | Kinematical bounds on optimization of observables for quantum systems[END_REF] as functions of t ∈ [0, T ], are linearly independent (to the extent possible), or equivalently if the dipole moment matrices μ(t)'s span the space of all zerotraced Hermitian matrices isu(N ).

In previous works, it has always been assumed that "full controllability" implies "fully linear independence" of the dipole moment matrix entries. An assertion of the linear independence of dipole moment matrix entries is also essential for understanding the landscape of unitary transformation control problem [START_REF] Ho | Landscape of unitary transformations in controlled quantum dynamics[END_REF]. In this paper, we present a rigorous analysis of this assertion and identify properties of the trajectory, in terms of the propagators, that are sufficient to ensure the linear independence of entries of μ(t). In addition, the analysis of which controls correspond to a non-singular mapping ǫ(t) → U (T, 0) or ǫ(t) → O (T ) is also relevant, e.g., for stabilization purposes, error cancellation, maintaining coherence, etc.

Dipole dependent way-points

Denote by L the Lie algebra generated by -iH 0 and -iµ as a sub-algebra of u(N ). We will suppose that the system is density matrix controllable, or equivalently

L = su(N ) or L = u(N ). ( 8
)
Note that when the system is density matrix controllable, then it is also wave function controllable, cf [START_REF] Albertini | Notions of controllability for multilevel bilinear quantum mechanical systems[END_REF][START_REF] D'alessandro | Introduction to quantum control and dynamics[END_REF].

Theorem 1 Under assumptions (2) and (8) there exists a set {U 1 , ..., U 2N 2 -2N } of unitary propagators (that we will call the set of "way-points") such that if U (t, 0) visits all of them i.e. U (t k , 0) = U k for some times t k , k = 1, ..., 2N 2 -2N the components f ij (t) of the matrix μ(t) in (7) are linearly independent as functions of time.

Consequently, under the hypotheses (2) and (8) for T sufficiently large a control field ǫ(t), t ∈ [0, T ] exists such that functions f ij (t) are linearly independent (or equivalently such that μ(t) spans the set of all zero-traced Hermitian matrices).

Remark 1 Note that here there is no information required on the structure of the matrices H 0 or µ.

Proof Since the Hermitian matrix µ has zero trace it cannot be a constant and thus it has at least two different eigenvalues λ 1 and λ 2 .

Let us begin by some general conventions: for any i = j and any matrix M we denote by M <i,j> its 2 × 2 sub-matrix formed by the i-th and j-th rows and columns:

M <i,j> = M ii M ij M ji M jj . (9)
Moreover, when we alter the N × N identity matrix by putting in the i-th and j-th rows and columns a given 2 × 2 matrix D, we will denote the resultant N × N matrix by D <i,j> :

D <i,j> = i j             1 0 0 0 0 0 1 0 0 0 0 0 D 11 D 21 0 0 0 D 12 D 22 0 0 0 0 0 1             i j (10)
We choose U k in the following way: let l = 1, ..., N (N -1) 2 index the couples (i, j), i, j = 1, ..., N with i < j.

-U 4l+1 is such that μ(t 4l+1 ) <i,j> = λ 1 0 0 λ 2 ; -U 4l+2 = U 4l+1 0 1 1 0 <i,j>
. Then μ(t 4l+2 )

<i,j> = λ 2 0 0 λ 1 and all other entries are identical to those of μ(t 4l+1 ).

-

U 4l+3 = 1 2 U 4l+1 1 1 -1 1 <i,j> . Then μ(t 4l+3 ) <i,j> = 1 2 λ 1 + λ 2 λ 1 -λ 2 λ 1 -λ 2 λ 1 + λ 2 (11)
and all other entries are identical to those of μ(t 4l+1 ).

-

U 4l+4 = 1 2 U 4l+1 i 1 1 i <i,j>
. Then

μ(t 4l+4 ) <i,j> = 1 2
λ 1 + λ 2 (λ 2 -λ 1 )i (λ 1 -λ 2 )i λ 1 + λ 2 (12)
and all other entries are identical to those of μ(t 4l+1 ). Note that all μ(t k ) = U * k µU k are Hermitian (and of zero trace). Let us take a zero trace Hermitian matrix Z such that

Z, μ(t k ) = 0, ∀k = 1, ..., 4N. ( 13
)
where we used the canonical scalar product of Hermitian matrices A, B = T r(A * B) = T r(AB). We recall also the definition of the norm induced by this scalar product A = A, A .

We denote x y y * z = Z <i,j> ; since Z, μ(t k ) = 0 for k = 4l + 1, ..., 4l + 4 it follows that Z <i,j> , (μ(t k )) <i,j> is the same for k = 4l + 1, ..., 4l + 4. The consequence is that λ 1 x+λ 2 z = λ 2 x+λ 1 z which together with λ 1 = λ 2 implies x = z. Moreover we also have λ 1 x + λ 2 x = 1 2 2(λ 1 + λ 2 )x + (λ 1 -λ 2 )(y + y * ) thus y + y * = 0. Additionally, we have λ 1 x + λ 2 x = 1 2 2(λ 1 + λ 2 )x + (λ 1λ 2 )i(y * -y) , thus y = y * . From y + y * = 0 and y = y * we infer that y = 0 and obtain Z <i,j> = xI 2 (I 2 is the 2 × 2 unit matrix). Since this is true for arbitrary i < j we obtain Z = xI N . But, since T r(Z) = 0 it follows Z = 0, q.e.d.

For the second part of the conclusion we use the fact that under the hypotheses (2) and (8) the system is controllable thus for T sufficiently large a control field ǫ(t), t ∈ [0, T ] exists such that functions f ij (t) are linearly independent.

Dipole independent way-points

The result of the previous section can be interpreted as follows: as long as the control field ensures that the propagator will visit some specific unitary transformations, then the required linear independence of the time-dependent elements of μ(t) is satisfied. Note that this set of unitary transformations (i.e., propagators) explicitly depends on the dipole moment. In the laboratory, procedures have been designed that are able to control the quantum evolution even in the absence of precise information on the dipole [START_REF] Judson | Teaching lasers to control molecules[END_REF]. It is therefore possible in principle to experimentally find a control passing through a specified list of propagators when this list does not depend on the dipole entries. A universal procedure can therefore be implemented that will ensure the nonsingularity of the mapping ǫ(t) → U (T, 0) even when the dipole is unknown. Beyond the question of the landscape analysis, such a procedure can be useful in additional circumstances when the mapping is required to be non-singular, e.g. for stabilization purposes, etc. We will therefore investigate the following circumstance: suppose that the system is controllable and thus one can experimentally implement arbitrary propagators U ∈ SU (N ). Can we find a list of propagators (as in the previous section) which are independent of the precise values of entries of µ ? Theorem 2 Under assumptions (2) and (8) there exists a set of "way-points" W ⊂ SU (N ) independent of µ such that if U (t, 0) visits all propagators in W (i.e. for all U k ∈ W there exists t k with U (t k , 0) = U k ) the components f ij (t) of the matrix μ(t) in (7) are linearly independent as functions of time.

Proof We denote for any U ∈ SU (N ):

C U = {(Z, µ) ∈ (isu(N )) 2 | Z = µ = 1, T r(ZU * µU ) = 0}. ( 14
)
This set is open because its complement is the solution of a linear equation in the entries of the matrices Z and µ.

We prove now that for any Z, µ ∈ isu(N ) of unit norm there exists a U ∈ SU (N ) such that T r(ZU * µU ) = 0. To see this, diagonalize Z and µ: ) N k=1 there are some elements that are strictly negative and some strictly positive. We will also suppose that the diagonalization is done in such a way that the diagonal elements are ordered from the lowest (that is strictly negative) to the highest, which is strictly positive. Then,

Z = U * 1 D 1 U 1 and µ = U * 2 D 2 U 2 with U 1 ,

T r(ZU

* µU ) = T r(U * 1 D 1 U 1 U * U * 2 D 2 U 2 U ) = T r(D 1 V * D 2 V ) (15)
with

V = U 2 U U * 1 ∈ SU (N ). Take now U such that V is a permutation matrix corresponding to permutation σ. Then T r(D 1 V * D 2 V ) = N k=1 d 1 k d 2 σ(k) . (16)
If

N k=1 d 1 k d 2 σ(k) = 0 for any permutation σ then it follows in particular that N k=1 d 1 k d 2 k = N k=1 d 1 k d 2 N -k
, but the first member is always superior to the second with the equality implying that one of the vectors d a k is constant which leads to a contradiction with the zero-trace hypothesis.

We therefore obtain that ∪ U∈SU(N ) C U is a covering of the compact set {(Z, µ) ∈ (isu(N )) 2 | Z = µ = 1} and thus one can extract a finite covering i.e. a finite set W such that

∪ U∈W C U = {(Z, µ) ∈ (isu(N )) 2 | Z = µ = 1}. ( 17
)
This means that for any Z and µ in isu(N ) (note that there is no need to impose the norm condition) there exists U ∈ W such that T r(ZU * µU ) = 0. Expressed otherwise, for any µ ∈ isu(N ) there does not exist Z ∈ isu(N ) such that T r(ZU * µU ) = 0 for any U ∈ W. This implies that {U * µU | U ∈ W} form an independent set of vectors in isu(N ), q.e.d.

Remark 2 Theorem 2 only assures that such a set W exists but does not give yet precise information on its size (which can be large) nor does it provide a constructive approach to identify in practice W. The following result addresses these questions upon imposing an additional constraint on the dipole µ.

Theorem 3 Under assumptions (2) and (8) and

µ ij = 0, ∀i = j. ( 18
)
there exists a set of "way-points" S ⊂ SU (N ) such that if U (t, 0) visits all propagators in S (i.e. for all U k ∈ S there exists t k with U (t k , 0) = U k ) the components f ij (t) of the matrix μ(t) in (7) are linearly independent as functions of time. Moreover the set S can be chosen to be the same for all coupling operators µ satisfying the hypotheses (2), ( 8), (18).

Proof Suppose that the entries of μ(t) are not linearly independent i.e. a matrix Z, T r(Z) = 0 exists such that Z is orthogonal to all matrices U * µU , ∀U ∈ S, i.e., T r(ZU * µU ) = 0 ∀U ∈ S.

Let us compute U * µU : for µ = µ 11 µ 12 µ 21 µ 22 (here µ 11 is a 2 × 2 matrix, µ 22 a N -2 × N -2 matrix, etc.) and

U = W <1,2> = W 0 0 I N -2 U * µU = W * µ 11 W W * µ 12 µ 21 W µ 22 . (19)
A typical example of W is cos(φ) sin(φ) sin(φ) -cos(φ) . The entries of W * µ 11 W will contain terms of second order in cos(φ) and sin(φ): cos 2 (φ) = cos(2φ)+1 2 and sin 2 (φ) = 1-cos(2φ) 2 , sin(φ) cos(φ) = sin(2φ) 2 ; the entries of W * µ 12 and µ 21 W will contain only first order terms in cos(φ) and sin(φ) while µ 22 does not depend on φ at all.

We recall that for any θ ∈ R the functions 1, cos(θ), sin(θ), cos(2θ), sin(2θ) are linearly independent, which will be used in the following fashion Lemma 1 There exist five real values of θ, for instance Θ = {0, π/3, π/2, π, 3π/3} such that

α 1 + α 2 cos(θ) + α 3 sin(θ) +α 4 cos(2θ) + α 5 sin(2θ) = 0 ∀θ ∈ Θ (20) implies α 1 = ... = α 5 = 0. ( 21
)
We now introduce into the set S the matrices

U θ,i;j = 0 cos(θ) + i sin(θ) cos(θ) -i sin(θ) 0 <i,j> , (22)
for any i = j and θ ∈ Θ. Since T r(ZU * θ,i;j µU θ,i;j ) = 0 ∀θ ∈ Θ by Lemma 1, the coefficients of sin 2θ and cos 2θ in T r(ZU * µU ) = 0 will be zero. The coefficient of sin 2θ turns out to be µ ij (Z ij + Z ji ) and that of cos 2θ is iµ ij (Z ij -Z ji ). It follows from (18) that Z ij = Z ji = 0.

Let us add to the set S the matrices V θ,i;j = cos(θ) sin(θ) sin(θ) -cos(θ)

<i,j> ,

for any i = 1, ..., N -1, j = i+1 and θ ∈ Θ. We use again T r(ZV * θ,i;j µV θ,i;j ) = 0 ∀θ ∈ Θ and identify the coefficients of sin 2θ and cos 2θ which are respectively µ ij (Z ii -Z jj ) + µii-µjj 2

(Z ij + Z ji ) and -µ ij (Z ij + Z ji ) + µii-µjj 2
(Z ii -Z jj ). We obtain Z ii = Z jj . Since this is true for any i = 1, ..., N -1 from T r(Z) = 0 it follows that Z ii = 0 ∀i ≤ N .

Remark 3 Although the hypothesis (18) may seem strong we believe that it is rather a technical requirement which could hopefully be relaxed in the future.

Conclusion and discussion

Following previous works in [START_REF] Rabitz | Quantum Optimally Controlled Transition Landscapes[END_REF] which study the properties of the control input-output map we investigate in this paper the singularity of the control to propagator map through the study of the linear independence of entries of the time-dependent coupling operator μ(t) (cf. [START_REF] Girardeau | Kinematical bounds on optimization of observables for quantum systems[END_REF]). We provide several sufficient conditions in terms of the evolution semigroup points that are reached during the propagation of Eq. (1). We expect that the criterion in (18) can be relaxed to accommodate even more general coupling operators.

  U 2 unitary and D 1 , D 2 diagonal; we will denote by d a k the k-th diagonal entry of D a . Recall that since Z, µ are zero-traced N k=1 d a k = 0 for a = 1, 2. Therefore, in each set (d a k

Acknowledgements

GT acknowledges support from the ANR-06-BLAN-0052 program and the MicMac project of INRIA Rocquencourt. TSH and HR were supported by U.S. Department of Energy.


H T J I Leest

K S S Steen

W F Lems

M A F J Laar

A M Huisman

S W Kadir

H H M L Houben

P J Kostense

E J Kuipers

Leest Htji De

Steen Kss

Lems Wf

MAFJ Van De Laar

Huisman Am

Kadir Sw

Houben Hhml

Kostense Pj

Kuipers Ej

Dijkmans Bac

Debets-Ossenkopp Yj

MD H T J I De Leest

Efficacy of serology driven "test

come   L'archive ouverte pluridisciplinaire

INTRODUCTION

H. pylori eradication is strongly recommended in all patients with atrophic gastritis and peptic ulcer disease, but may also benefit subgroups of patients with dyspepsia, and patients who start with NSAID therapy [START_REF] Kuipers | Helicobacter pylori and the risk and management of associated diseases: gastritis, ulcer disease, atrophic gastritis and gastric cancer[END_REF][START_REF] Chan | Randomised trial of eradication of Helicobacter pylori before nonsteroidal anti-inflammatory drug therapy to prevent peptic ulcers[END_REF][START_REF] Chan | Eradication of Helicobacter pylori and risk of peptic ulcers in patients starting long-term treatment with non-steroidal anti-inflammatory drugs: a randomised trial[END_REF][START_REF] Labenz | Primary prevention of diclofenac associated ulcers and dyspepsia by omeprazole or triple therapy in Helicobacter pylori positive patients: a randomised, double blind, placebo controlled, clinical trial[END_REF][START_REF] Moayyedi | Eradication of Helicobacter pylori for non-ulcer dyspepsia[END_REF][6]. H. pylori eradication therapy is an important component of guidelines concerning these patients [7;8].

Currently, non-invasive management strategies and the widespread shortage in endoscopic capacity make that many patients with H. pylori are managed without upper gastrointestinal endoscopy. The American College of Gastroenterology recommends that when an endoscopy is not performed, a serological test, which is the least expensive means of evaluating for evidence of H. pylori infection, should be done [START_REF] Chey | American College of Gastroenterology guideline on the management of Helicobacter pylori infection[END_REF]. When endoscopy is indicated, biopsy specimens can be taken for microscopic demonstration of the organism, culture, histology or urease testing. Nowadays, in the Netherlands, biopsies are not routinely sent for culture and susceptibility testing of the infecting strain because of the high costs.

Apart from patient compliance, resistance of Helicobacter pylori to antibiotics can decrease the success of H. pylori eradication therapy. Regimens of choice for eradication of H. pylori should be guided by local antibiotic resistance rates. In the Netherlands, the overall prevalence of resistance to clarithromycin and metronidazol was lower than in some surrounding countries possibly due to restrictive use of antimicrobials [START_REF] Debets-Ossenkopp | Prevalence of Helicobacter pylori resistance to metronidazole, clarithromycin, amoxycillin, tetracycline and trovafloxacin in The Netherlands[END_REF][START_REF] Van Der Wouden | Rapid increase in the prevalence of metronidazole-resistant Helicobacter pylori in the Netherlands[END_REF][START_REF] Janssen | Helicobacter pylori antibiotic resistance in a Dutch region: trends over time[END_REF] . The advised treatment in the Netherlands consists of a proton pump inhibitor (PPI)-triple therapy for 7 days without prior susceptibility testing. An increase of resistance rates to antimicrobial agents is however expected because increasing number of patients treated and increasing consumption of antibiotics, in particular macrolides, was observed in recent years [START_REF] Swab | NethMap 2005 -Consumption of antimicrobial agents and antimicrobial resistance among medically important bacteria in the Netherlands[END_REF].

The aim of the present study was firstly, to determine the efficacy of 7-day PPI-triple therapy for H. pylori in a well defined group of patients with a rheumatic disease and serologic evidence of H. pylori infection who were on long term NSAID therapy and secondly, to get insight in the prevalence of antibiotic resistance of H. pylori in the studied population.

METHODS

This study was part of a placebo-controlled randomized clinical trial of which the clinical results have been described elsewhere [START_REF] De Leest | Eradication of Helicobacter pylori does not reduce the incidence of gastroduodenal ulcers in patients on long-term NSAID treatment: doubleblind, randomized, placebo-controlled trial[END_REF],wherein we described that H. pylori eradication has no beneficial effect on the incidence of gastroduodenal ulcers or occurrence of dyspepsia in patients on long term NSAIDs treatment. Between May 2000 and June 2002, patients were recruited from 8 rheumatology outpatient departments in 6 cities in the Netherlands. Patients with a rheumatic disease were eligible for inclusion if they were between 40 and 80 years of age, were positive for H. pylori on serological testing and were on long term NSAID treatment. Forty-eight percent used a gastroprotective drug (7% H2 receptor antagonists (H2RA), 37% proton pump inhibitors (PPI), 7% misoprostol, 3% used a combination of these). Exclusion criteria were: previous eradication therapy for H. pylori, known allergy for the study medication or presence of severe concomitant disease.

Serologic testing for H. pylori IgG-antibodies was performed with a commercial enzymelinked immunosorbent assay (Pyloriset new EIA-G, Orion Diagnostica, Espoo, Finland) according to the manufacturer's instructions. A serum sample was considered positive for IgG antibodies to H. pylori if the test result was 250 International Units (IU). This assay has been assessed, in a population similar to the population in the presented trial, and has proven a sensitivity and specificity in the Netherlands of 98-100% and 79-85%, even in patients on acid suppressive therapy [START_REF] Lerang | Accuracy of seven different tests for the diagnosis of Helicobacter pylori infection and the impact of H2-receptor antagonists on test results[END_REF][START_REF] Meijer | Evaluation of eight enzyme immunoassays for detection of immunoglobulin G against Helicobacter pylori[END_REF][START_REF] Van De Wouw | Comparison of three commercially available enzyme-linked immunosorbent assays and biopsydependent diagnosis for detecting Helicobacter pylori infection[END_REF]. The study protocol was approved by research and medical ethics committees of all participating centers and all patients gave written informed consent.

After stratification by concurrent use of gastroprotective agents (proton pump inhibitors, H2 receptor antagonists or misoprostol, but not prokinetics, or antacids), patients were randomly assigned to receive either H. pylori eradication therapy with omeprazole 20 mg, amoxicillin 1000 mg, and clarithromycin 500 mg (OAC) twice daily for 7 days or placebo. Patients with an allergy for amoxicillin were treated with omeprazole 20 mg, metronidazole 500 mg and clarithromycin 250 mg (OMC) or placebo therapy twice daily for one week in a distinct stratum. All study personnel and participants were blinded to treatment assignment for the duration of the study. The study protocol was approved by research and medical ethics committees of all participating centres and all patients gave written informed consent.

At the 2-week follow-up visit, unused study medication was returned and remaining tablets were counted in order to check compliance. Patients were considered to be noncompliant if ≤ 6 days (85%) of study medication were used. Three months after baseline, and additionally if clinically indicated, patients underwent endoscopy of the upper gastrointestinal tract. A total of eight biopsies were taken during each endoscopy. Four samples, two from the antrum and two from the corpus were used for histology. All biopsies were stained with haematoxylineosin. The slides were scored independently by an experienced gastrointestinal pathologist and the investigator (HdL), blinded to treatment assignment and clinical data, according to the updated Sydney classification [START_REF] Dixon | Classification and grading of gastritis. The updated Sydney System. International Workshop on the Histopathology of Gastritis, Houston[END_REF]. In case of discrepant results, the specimen was discussed until agreement was reached. The remaining four biopsies were sent to a microbiological laboratory for culture and storage at -70°C. A patient was considered H. pylori-negative when histology as well as culture was negative. All isolated strains were assessed for susceptibility to clarithromycin, metronidazole, tetracycline and amoxicillin at the central laboratory.

Both biopsy specimens of corpus and antrum were streaked on Columbia agar (CA) (Becton Dickinson, Cockeysville, MD, USA) with 10% lysed horse blood (Bio Trading, Mijdrecht, The Netherlands), referred to as Columbia agar plates, and on CA with H. pylori selective supplement (Oxoid, Basingstoke, UK). Plates were incubated for 72 h at 37°C in a micro-aerophilic atmosphere (5% O 2 , 10% CO 2 , 85% N 2 ). Identification was carried out by Gram's stain morphology, catalase, oxidase, and urea hydrolysis measurements.

Inocula were prepared from an H. pylori culture grown on CA plates. MICs of metronidazole, clarithromycin, tetracycline and amoxicillin were determined by E-test (AB Biodisk, Solna, Sweden) on CA plates essentially as described by Glupczynski et al [START_REF] Glupczynski | Evaluation of the E test for quantitative antimicrobial susceptibility testing of Helicobacter pylori[END_REF]. CA plates were inoculated with a bacterial suspension with a turbidity of a 3 McFarland standard (2 x 10 8 CFU/ml ). CLSI (tentative) breakpoints 2009 for susceptibility (S) and resistance (R ) were applied (metronidazole MIC <= 8 mg/l (S) and >= 16 mg/l (R ), amoxicillin MIC <= 0,5 mg/l (S) and >= 2 mg/l (R); tetracycline MIC <= 2 mg/l (S) and >= 8 mg/l (R ), and clarithromycin MIC <= 0,25 mg/l.(S) and >=1 mg/l (R ) [START_REF]Performance Standards for antimicrobial susceptibility testing. 19 th Informational Supplement[END_REF].

Measurements with a Gaussian distribution were expressed at baseline as mean and SD, and measures with a non-Gaussian distribution were expressed as the median and interquartile range (IQR; expressed as the net result of 75 th percentile -25 th percentile). An additional analysis compared outcomes (presence of H. pylori after H. pylori eradication therapy or placebo) between stratum (patients on gastroprotective drugs (n=165) and not on gastroprotective drugs (n=182) by computing the homogeneity of the common odds ratio.

SPSS software (version 17.0.0) was used to perform these analyses. Differences in the proportions of patients with susceptible and resistant H. pylori strains and for compliant and non-compliant patients were analyzed with 95% confidence interval using the Confidence Interval Analysis software for Windows (version 2.2.0). The level of significance was set at p<0.05, two sided.

RESULTS

A total of 347 patients consented to be randomly assigned to eradication therapy (172 patients) or placebo (175 patients). Anti-H. pylori IgG antibodies were present in all patients (median titre 1689 (IQR 700-3732).The treatment groups were similar in terms of demographic, rheumatic disease, NSAID and other drug use. Our eligibility criteria resulted in a study group with mainly inflammatory rheumatic diseases (rheumatoid arthritis 61%, spondylarthropathy 8%, psoriatic arthritis 7%, osteoarthritis 9%, other 15%). The most commonly used NSAIDs were diclofenac (29%), naproxen (18%), and ibuprofen (13%), most at full therapeutic doses (median relative daily dose 1 (IQR 0.5-1).The mean age was 60 years (SD 10), 61% was female. Twenty-two patients had a known allergy for amoxicillin and received metronidazole instead (10 patients) or placebo (12 patients).

Of these 347 patients, data on culture and histology of 304 patients were available (Table 1).

In two cases only culture data were available and in 1 case only histology result was available all three cases met the criteria for H. pylori-positivity and were found in the placebo group, but for clarity purposes were left out of Table 1. A total of 32 patients (with no significant differences between eradication and placebo groups) refused the 3-month endoscopy, withdrew informed consent, or could not undergo endoscopy because of adverse events.

Seven patients used anticoagulant therapy ruling out biopsy sampling according the protocol, and in one patient no biopsy specimens could be obtained because of discomfort requiring early completion of the procedure.

At follow-up after 3 months, 79% (120 /152; 95% CI 72-85%) of the patients in the placebo group were H. pylori-positive by histology or culture of biopsy specimen. In the eradication group, this number was 13% (20/152; 95% CI 9-20%) (Table 1).

Patients in the placebo group who were H. pylori negative at 3 months as assessed by culture and histology had significant lower titers of H. pylori anti IgG antibodies at baseline than those who were H. pylori culture-and or histology-positive (mean difference -1582, 95% CI -2637 to -527, p=0.004). There were no differences between strata according to the use of gastroprotective drugs for the presence of H. pylori by culture and or histology (p= 0.454).

Compliance was 89 % in patients in the eradication group and 98% in the placebo group with the assigned regimen (p<0.001). In the eradication group, H. pylori could not be demonstrated in 91% in patients with full compliance (n=136). In patients who did not take all 7 days of eradication therapy (n=16) H. pylori was found in 50% (difference of 41%; 95% CI 18-63%).

No differences were found in the placebo group.

Antibiotic resistance rates

A total of 105 clinical isolates of H. pylori were available for susceptibility testing (one isolate per patient; 95 isolates from the placebo group, 10 from the eradication group) from the 6 participating laboratories in The Netherlands. The rates of resistance are summarized in Table 2.

In the placebo group (n=95), resistance was found in 4% (4/95) to clarithromycin (MIC >= 1 mg/l), in 1% (1/95) and 2% (2/95) intermediate susceptibility to amoxicillin (MIC: 1 mg/l) and tetracycline (MIC: 4 mg/l), respectively, and in 19% (18/95) resistance to metronidazole. Amongst these 95 isolates 2 were resistant to metronidazole in combination with intermediate susceptible to tetracycline and one strain was resistant to metronidazole and clarithromycin.

The placebo group had an MIC90 for clarithromycin of 0,085 mg/l, for metronidazole >256 mg/l, for tetracycline 0,341 mg/l and for amoxicillin 0,16 mg/l. One H. pylori strain was resistant to clarithromycin and metronidazole, and intermediate susceptible to tetracycline and amoxicillin. No difference was found in H. pylori resistance rates between man and women (P=0.217) or between patients who used gastroprotective agents and who did not (p=0.25). In the eradication group, 2 strains were resistant to clarithromycin and 3 to metronidazole. No strains were resistant to tetracycline or amoxicillin.

CONCLUSION AND DISCUSSION

We report the results of a study on the efficacy of a test and treat strategy for H. pylori in rheumatology patients of the Netherlands who were positive for anti-H. pylori IgG-antibodies.

The main findings in the studied patient population were: (1) a 7-day PPI-triple eradication therapy either with clarithromycin or metronidazole was efficacious with eradication rates of 87% (95% CI 80-91%) without prior testing for susceptibility of the infecting strain ; (2) in 21% of the patients in the placebo group, the positive H. pylori serology test could not be confirmed by positive culture or histology. (3) compliance was an important factor for successful eradication of H. pylori; (4) prevalence of antibiotic resistance in H. pylori was low.

The main reason for not performing endoscopy at baseline was that this was not feasible in everyday rheumatology practice, therefore serology was done to test for H. pylori. The reliability of serological kits for H. pylori infection has been widely confirmed [START_REF] Monteiro | Diagnosis of Helicobacter pylori infection[END_REF], contributing to the reputation of serology as a simple, minimally invasive and inexpensive diagnostic and screening test. The best available serology test at the time of the study was the Pyloriset an EIA-G from Orion Diagnostica, Espoo, Finland, with a specificity of 79-91% as assessed in previous studies in the Netherlands, including patients on acid suppressive therapy [START_REF] Lerang | Accuracy of seven different tests for the diagnosis of Helicobacter pylori infection and the impact of H2-receptor antagonists on test results[END_REF][START_REF] Meijer | Evaluation of eight enzyme immunoassays for detection of immunoglobulin G against Helicobacter pylori[END_REF][START_REF] Van De Wouw | Comparison of three commercially available enzyme-linked immunosorbent assays and biopsydependent diagnosis for detecting Helicobacter pylori infection[END_REF].This specificity correlates well with our finding that in the placebo group, H. pylori could not be confirmed by culture or histology in 21% of the IgG positive patients. PPI usage (in this study 37% of the population) can result in false negative invasive and noninvasive diagnostic tests, such as culture, histology and13-C urea breath, and should be stopped two weeks before testing [START_REF] Kokkola | Diagnosis of Helicobacter pylori infection in patients with atrophic gastritis: comparison of histology, 13C-urea breath test, and serology[END_REF]. This does not apply for serology. 13-C urea breath test have better accuracy (>90%), but the serology test used in this study was less expensive and in all study centres easily available [START_REF] Lindsetmo | Accuracy of Helicobacter pylori serology in two peptic ulcer populations and in healthy controls[END_REF]. On the other hand we must not overlook that conditions during transport of biopsies are critical for successful isolation of H. pylori.

Possibly, the antibacterial effect of NSAIDs, as has been suggested in in vitro studies, might also partly explain a false positive rate of serology of 21% [START_REF] Gu | Effect of cyclo-oxygenase inhibitors on Helicobacter pylori susceptibility to metronidazole and clarithromycin[END_REF][START_REF] Wang | Aspirin inhibits the growth of Helicobacter pylori and enhances its susceptibility to antimicrobial agents[END_REF][START_REF] Park | Effect of high-dose aspirin on Helicobacter pylori eradication 5[END_REF]. However, in a randomized clinical trial of 122 patients, aspirin in combination with a standard 7-days course OAC eradication was not significantly different compared to the standard therapy [START_REF] Park | Effect of high-dose aspirin on Helicobacter pylori eradication 5[END_REF]. Based on culture and histology findings we conclude that one fifth of our patients was treated superfluously, with possible risk of side-effects of the eradication medication. Fortunately, both regimens were generally well tolerated [START_REF] De Leest | Eradication of Helicobacter pylori does not reduce the incidence of gastroduodenal ulcers in patients on long-term NSAID treatment: doubleblind, randomized, placebo-controlled trial[END_REF].

Resistance to antibiotics in H. pylori is of particular concern because it is one of the major determinants in the failure of eradication regimens. Resistance rates for metronidazole and clarithromycin found in this study were similar as previously observed in other studies in the Netherlands in the years 1997-98 [10] and 1997-2002[11;12;30;31]. To our knowledge, there are no recent data available on H. pylori antibiotic primary resistance rates in the Netherlands [START_REF] Megraud | H pylori antibiotic resistance: prevalence, importance, and advances in testing[END_REF]. In addition, in this study, compliance played a crucial role in success of eradication of H. pylori: Treatment failure was as high as 50% in the non compliant group of patients. Possibly, the high number of tablets that has to be consumed during H. pylori eradication therapy is a contributing factor for non-compliance in this group of elderly patients who were also on other medications.

In conclusion, serology driven test and treat strategy eradication of H. pylori with a 7-day PPI-triple therapy is successful in the majority of patients. Success of eradication is, also in

Table 1

 1 Results of culture and histology on H. pylori

	Total no. of patients	Eradication group (n=152)	Placebo group (n=152)
	n=304
				Histology
			H. pylori	H. pylori	H. pylori	H. pylori
			positive	negative	positive	negative
	Culture	positive H. pylori	10 (6%)	1 (1%)	89 (57%)	9 (6%)
		H. pylori	9 (6%)	132 (87%)	22 (15%)	32 (21%)
		negative

Table 2

 2 Antibiotic resistance of H. pylori isolates All patients in the eradication group who were still H. pylori positive were assigned OAC and not OMC *Intermediate susceptible.

		Placebo group	Eradication group
		n=95	n = 10
		Resistance rate	Resistance rate
	Clarithromycin	4%	20%
	Metronidazole	19 %	30%
	Tetracyclin*	2%	0
	Amoxicillin*	1%	0

ACKNOWLEDGEMENTS.

We thank the nursing staff members, the medical staff members, and all other personnel who cooperated in this study of all contributing centres for their generous support and invaluable contribution. We would like to thank all laboratory assistants and other members of staff of the medical microbiology laboratories for collecting the strains and especially the VU University Medical Center Amsterdam lab for performing the E-tests.

Keywords:

Helicobacter pylori antibiotic resistance eradication therapy serology this group of rheumatology patients, to a great extent determined by compliance.

Conflict of interest statement

We declare that we have no conflict of interest.