Hao Wang

Alexander C Forse

John M Griffin

Nicole M Trease

Lorie Trognko

Pierre-Louis Taberna

Patrice Simon

Clare P Grey

In Situ NMR Spectroscopy of Supercapacitors: Insight into the Charge Storage Mechanism

, and two video files.

INTRODUCTION

The development of efficient energy storage devices is essential to meet the growing demand for portable electronic devices and emerging green transportation technologies. While lithium ion batteries have met a large proportion of our needs over the past 20 years, electrochemical capacitors, also known as supercapacitors, are receiving renewed interest as an alternative and complementary technology. 1 The storage of electrical energy in carbon-based supercapacitors relies on the formation of an electric double-layer at the interface between a solid electrode and a liquid electrolyte solution. This non-faradaic charge storage mechanism allows rapid rates of charge and discharge and essentially unlimited cycle lives, making supercapacitors well-suited to high power applications. 2,3 Supercapacitors typically use organic electrolytes, such as tetraethylammonium tetrafluoroborate (NEt 4 BF 4 ) in acetoni-trile. 1 Aqueous electrolytes are also sometimes used because they are less toxic and inexpensive; however, the decomposition of water at voltages above 1.2 V limits the amount of energy that can be stored. Supercapacitor electrodes are generally constructed using porous carbon owing to its high surface area and conductivity. 4 In particular, activated carbons derived from organic precursors such as coconut shells and wood have been widely used in commercial devices owing to their high surface area and conductivity, and low cost. Although predominantly microporous (with pore sizes less than 2 nm), these materials also exhibit a range of pore diameters in the mesoporous (2-50 nm diameter) and macroporous (greater than 50 nm diameter) regimes. The importance of pore size effects was highlighted by Chmiola et al., who reported that the capacitance of carbon derived from titanium carbide dramatically increased when the average size of carbon micropores was less than the size of the solvated electrolyte ions (typically less than 1.5 nm). 5 This finding could not be explained with traditional concepts of double-layer formation and has triggered considerable efforts toward understanding the nature of ion electrosorption inside the porous carbon. It was hypothesized that ion desolvation allowed a closer approach of charge centers at the electrodeelectrolyte interface. 6 Distortions of cyclic voltammograms in cavity microelectrode studies confirmed that electrolyte ions are at least partially desolvated as they enter the carbon micropores upon charging. 7 However, many questions relating to this phenomenon and the fundamental mechanism of supercapacitance in general remain unanswered.

A deeper understanding of the processes that take place within supercapacitors upon charging and discharging will enable further development and optimization of these important devices. To this end, a number of experimental methods have been employed to further investigate the behavior of ions in micropores. 8 In situ electrochemical quartz crystal microbalance (EQCM) methodologies have been used to study the gravimetric response of porous carbon electrodes in a range of electrolyte systems under dynamic charging conditions. Deviations from theoretical mass changes have been interpreted in terms of distinct charging regimes that are characterized by the compositional differences in the ionic part of the electric double layer. [9][10][11][12] Small-angle neutron scattering measurements have been used to observe changes in the concentration of electrolyte ions in micropores as a function of applied potential, 13 while neutron diffraction has been employed to quantify the diffusion of ions within mesoporous carbons in different charge states. 14 Recently, an in situ infrared spectroscopy method has been developed to study supercapacitor systems. 15 Decreases in electrolyte ion absorbances were interpreted as arising from a net movement of positive and negative ion pairs into carbon micropores upon charging. 15 In addition to experimental work, a large number of theoretical studies have given insight into charging phenomena in supercapacitors. In particular, molecular dynamics simulations [16][17][18][19][20][21][22] have shown the importance of overscreening effects at planar electrode-electrolyte interfaces, whereby the first layer of adsorbed species encompasses more charge than the electrode surface itself, with subsequent layers of electrolyte ions balancing the overall charge at the interface. It has been suggested that the absence of such effects in microporous electrodes allows high capacitances to develop. 23 Further work based on mean field theory models has proposed that image forces arising from charged micropore surfaces facilitate the packing of ions of the same polarity during charging, accounting for the large increases in capacitance experimentally measured for carbons with micropores smaller than the solvated electrolyte ions. [24][25][26] While experimental and theoretical work has advanced the understanding of charge storage in porous carbons, further development is required to directly observe and quantify changes of ion environments in supercapacitor electrodes. Nuclear magnetic resonance (NMR) spectroscopy is a powerful technique for studying electrochemical interfaces, as the observed resonance frequency is sensitive to the local chemical environment. This approach is also element selective, allowing the separate observation of different chemical species in electrochemical systems. The potential of NMR to study adsorption of molecules on carbon was recognized in early work by Harris et al. where distinct resonances were observed for adsorbed and non-adsorbed species within activated carbons. [27][28][29][30] More recent work on adsorption on a variety of different carbon morphologies has revealed similar trends. [31][32][33][34][35][36][37][38][39] Regardless of the molecule or nuclear spin studied, resonances from species adsorbed on a given carbon show similar shifts to low frequencies, relative to the nonadsorbed species. This suggests that the observed shift is a nucleusindependent chemical shift (NICS), arising from aromatic ring current effects associated with delocalized π electrons in the carbon. In addition to experimental NMR methods for probing this phenomenon, computational approaches have been developed to calculate NICSs arising from delocalized systems. 40,41 Such calculations have been applied to a range of carbon systems including coronenes, 42 fullerenes, 43,44 nanotubes, [45][46][47] and graphene. 47 In addition to studies of adsorption within porous carbons with no applied potential, NMR can also be used to study the charging and discharging mechanisms of supercapacitors. Indeed, NMR uniquely allows the direct observation of the charge storing species (adsorbed species). Ex situ NMR [48][49][50] involves holding a supercapacitor at a particular voltage, disassembling it, and then acquiring the NMR spectrum of the electrolyte species that remain inside the electrode film. With this approach, magic angle spinning (MAS) can be used to improve resolution and identify subtle changes in ion environments. 50 However, the rapid evaporation of acetonitrile during disassembly can result in the solidification of electrolyte salt. Desorption of ions (or self-discharge) during disassembly may also make it difficult to know exactly how much charge is stored when an NMR spectrum is acquired.

In situ NMR experiments [51][52][53][54][55] can address some of the issues associated with ex situ approaches, since spectra of the working device are acquired directly. Such experiments offer direct insight into the behavior of working devices but come with some resolution losses due to the incompatibility with MAS techniques. Our previous in situ NMR study demonstrated the utility of NMR for the study of supercapacitors during charging (and discharging), and systematic changes were observed in the spectra. 32 11 B NMR experiments were used to probe the behavior of BF 4 anions at different applied cell potentials. It was shown that the adsorption processes occurring at each electrode could be studied separately by introducing a prototype long supercapacitor design. Changes in populations and resonance frequencies were observed for adsorbed BF 4 anions upon cell charging. However, the electrochemical properties of the long supercapacitor cell were not fully explored, and while clear changes in both the local environments and populations of the adsorbed anions were observed, the link between these and the supercapacitor charging mechanism was not studied in detail.

In this work, in situ NMR is further developed to study supercapacitors comprising activated carbon electrodes and a NEt 4 BF 4 in acetonitrile electrolyte. We first address some of the practical aspects associated with in situ NMR of supercapacitors, using 19 F NMR in this study because of the high sensitivity of this nucleus. We then show how this methodology can provide insight into the charge storage mechanism of supercapacitors by directly tracking changes in the amount of charge storing species as a function of applied voltage and identifying different charge storage regimes in a high surface area, coconut-derived activated carbon. Supercapacitor charging is also shown to have a marked effect on the local magnetic environment experienced by adsorbed species. Real-time NMR experiments with simultaneous cyclic voltammetry show a systematic variation of the observed resonance frequency of adsorbed species, which is directly correlated with the development of charge on the electrodes. This is supported by NICS calculations on a model graphene fragment that show that the addition or removal of electrons from carbon systems has significant effects on the resonance frequencies of adsorbed species.

EXPERIMENTAL SECTION

2.1. Materials. In this work all carbon electrodes were prepared from YP50F carbon (Kuraray Chemical, Japan). Film electrodes were prepared in the standard way 56 by mixing carbon powder (95 wt %) with polytetrafluoroethylene (5 wt %) (Sigma Aldrich, 60 wt % dispersion in water). Prior to sample preparation, carbon film pieces (6.0 mg, approximate dimensions of 10 mm × 4 mm × 0.25 mm) were dried under vacuum at 200 °C overnight and then transferred to an argon glovebox. In initial electrochemical tests of different cell designs, the electrolyte used was a 1.5 M solution of tetraethylammonium tetrafluoroborate (NEt 4 BF 4, Sigma Aldrich, electrochemical grade >99.0%) in acetonitrile (Sigma Aldrich, 99.8% anhydrous). This electrolyte is referred to as NEt 4 BF 4 /ACN. The same electrolyte was used for NMR experiments except that the solvent was replaced by deuterated acetonitrile, D 3 CCN (Euriso-top, 99.96%). We refer to this electrolyte as NEt 4 BF 4 /dACN.

2.2. Sample Preparation. All sample preparation was carried out in an argon glovebox. Model electrodes were prepared for preliminary experiments by hermetically sealing a carbon film piece (6.0 mg) inside a plastic bag (3M packaging film) with the chosen volume of NEt 4 BF 4 /dACN electrolyte, which was added using a microsyringe. All supercapacitors in this study employed plastic bag cell designs, modified from Bellcore's plastic lithium ion battery technology. 53 Two pieces of carbon film (6.0 mg) were adhered to carbon-coated aluminum mesh, and a Cellgard 2500 (monolayer polypropylene, 25 μm thickness) separator was placed between them. These components were placed inside a plastic bag (3M packaging film), which was hermetically sealed on three sides. The cell was then saturated with electrolyte before making the final seal to close the bag. For NMR measurements, the supercapacitor bag cells were held between two plastic inserts in order to provide rigidity and allow better control over the cell orientation inside the NMR coil.

2.3. Electrochemical Tests. All cells were tested on a Bio-logic cycler or an Arbin Instruments battery cycler with a two-electrode configuration. Cyclic voltammetry (CV) was used both to test the performance of different cells and to run real-time in situ NMR experiments. Fixed voltage in situ NMR experiments were performed using chronoamperometry (CA) experiments. A fixed cell voltage is applied and a current flows around the external circuit until equilibrium is achieved before NMR acquisition.

2.4. NMR Details. All NMR experiments were performed using a Bruker Avance spectrometer operating at a magnetic field strength of 7.05 T, corresponding to a 19 F Larmor frequency of 284.2 MHz with a Bruker HX double resonance static probe using a 6.8 mm inner diameter solenoid coil. 19 F NMR spectra were referenced relative to neat hexafluorobenzene (C 6 F 6 ) at -164.9 ppm. The "depth" pulse sequence 57 was used to acquire all 19 F NMR spectra in order to reduce the background signal from the PTFE probe-head housing. The total delay between excitation of transverse magnetization and acquisition of the free induction decay was 60 μs in all cases. Spectral fitting was carried out using dmfit software. 58 The minimum number of components required to obtain a satisfactory fit was found to be four for each electrode. For the positive electrode, deconvolutions were carried out assuming the free electrolyte feature to be made up of three purely Lorentzian components and assuming the strongly adsorbed feature to be made up of a single purely Gaussian component. For the negative electrode, much better fits were obtained assuming two Lorentzians and one Gaussian component for the free electrolyte and assuming a Gaussian line shape for the strongly adsorbed resonance. That more than one resonance is required to model the free electrolyte is ascribed to bulk magnetic susceptibility (BMS) effects and strong variations in local magnetic fields across the bag cell, which results in a range of different local fields and thus shifts for the same chemical species (see Supporting Information, Figure S3). For the spectra of the negative electrode, the shift of the strongly adsorbed resonance was first obtained by fitting spectra recorded with a short recycle interval (0.5 s) which favors observation of the strongly adsorbed resonance due to faster spin-lattice, T 1 , relaxation. The shifts obtained were then fixed in the fits of the quantitative spectra recorded with the longer recycle interval of 30 s. For each electrode, the spectrum obtained at 0 V was fitted first, as this showed the best resolution of the strongly adsorbed resonance. The peak positions and intensities obtained were then used as a starting point to fit the spectrum at the next highest voltage. For the positive and negative electrodes, spectra recorded above 1.5 and 1.25 V, respectively, were not fitted, as unique fits could not be obtained owing to the almost complete overlap of the strongly adsorbed and free electrolyte features. Deconvoluted spectra are shown in Supporting Information, Figure S1.

2.5. In Situ NMR Experiments. Prior to performing NMR experiments, the cell was cycled between 0 and 2.3 V five times by cyclic voltammetry at a sweep rate of 0.5 mV•s -1 . The positively charged electrode was first positioned inside the coil for detection, and the cell was charged sequentially to a series of different voltages (0, 0.25, 0.5, 0.75, 1, 1.25, 1.5, 1.75, and 2 V). The cell was held at each voltage for 40 min to ensure that an equilibrium charge state was attained before 19 F NMR spectra were acquired. After the positively charged electrode was studied, the cell was shifted to position the negatively charged electrode inside the NMR coil for detection. The cell was then charged to the same series of voltages, and 19 F NMR spectra were collected at each voltage after waiting for 40 min. For fixed voltage experiments, a recycle interval of 30 s was sufficient for spectra to be quantitative. 19 F NMR spectra were also collected while simultaneous cyclic voltammetry (CV) was performed on the cell. Unless otherwise stated, 94 NMR spectra were obtained per full CV cycle (0 V → 2.3 V → 0 V) with a recycle interval of 3 s. While this does not yield fully quantitative spectra, it reduces the time required to obtain each spectrum, enabling more spectra to be acquired during the CV cycle.

2.6. Calculations. All geometry optimizations and NMR calculations were performed using Gaussian 03 software. 59 First, geometries were optimized at the B3LYP level of theory with the 6-31G(d) basis set, as in other work. 42 Then an NMR calculation was performed at the same level of theory and with the same basis set to evaluate the chemical shielding tensors for each atom, as well as for several "ghost" atoms introduced around the structure. These ghost atoms have no electrons or nuclear charge and simply serve to act as probes of the local effective magnetic field at their position. The calculations generate the isotropic chemical shielding, σ iso . The isotropic nucleus-independent chemical shift, δ iso NICS , is given by -(σ iso -σ ref ), where σ ref is a reference shielding which has a value of 0 ppm in this case.

RESULTS AND DISCUSSION

3.1. Cell Design for NMR Studies. Here, we considered three different supercapacitor cell designs in order to compare their electrochemical performances. The "conventional" design (Figure 1a) represents a typical supercapacitor, with the two electrodes situated directly on top of each other. With this design, both electrodes must be placed inside the NMR coil during in situ NMR experiments, making it difficult to separate processes occurring at each electrode. 32 In the "long" cell design (Figure 1b) the electrodes are shifted laterally with respect to each other so that a single electrode can be placed in the NMR detection region. In our previous work, this allowed the anion environments to be tracked separately in the positive and negative electrodes, although long times were needed to charge and discharge the device because of the significant ohmic losses associated with this design. 32 The "overlaid" cell design introduced here (Figure 1c) also allows a single electrode to be placed in the NMR detection region, but the current collector of the opposite electrode is situated closer to the electrode under study.

Cyclic voltammograms for conventional, long, and overlaid supercapacitors, each cycled at 0.5 mV•s -1 between 0 and 2.3 V, are shown in parts d, e, and f of Figure 1, respectively. Each cell comprised two YP50F film electrodes (6.0 mg each) and was saturated with NEt 4 BF 4 /ACN electrolyte (∼30 μL for the conventional design and ∼50 μL for the long and overlaid designs). A rectangular cyclic voltammogram is expected for an ideal capacitor, with positive currents flowing in the charge part of the cycle and negative currents flowing in the discharge part of the cycle. Distortion from the rectangular shape arises because of the contribution of resistive components associated with ionic diffusion, as well as the contact between the current collector (carbon coated aluminum mesh in our design) and the carbon electrode. A characteristic near-rectangular shape is observed for the conventional supercapacitor (Figure 1d), whereas the long cell design displays far from ideal behavior (Figure 1e), and a plateau indicative of pure capacitive behavior is not observed. However, a considerable improvement is achieved by switching to the overlaid design, where the observed cyclic voltammogram is much closer to the ideal case (Figure 1f). We also note that the estimated capacitance of YP50F (calculated from the discharge current at 1 V) is 99 F• g -1 , which is in good agreement with the literature value of 95 F•g -1 . 60 Faster scan rates were also studied for the long and overlaid cells, but poorer electrochemical performance was observed (see Figure S2 in Supporting Information). Importantly, the overlaid design introduced here allows a single electrode to be placed inside the NMR detection region while maintaining good electrochemical performance. Therefore, the overlaid design was used for all in situ NMR experiments that follow in this work, and a sweep rate of 0.5 mV•s -1 was used for cyclic voltammetry experiments.

3.2. Effect of Supercapacitor Components on the NMR Spectrum. In this work we focus on 19 F NMR to study the BF 4

-electrolyte anion environments. While 1 H NMR is an attractive option to study the environments of the NEt 4 + cations, the large number of plastic components used in the bag cell assembly results in a broad and intense background 1 H signal that precludes the observation of the relatively low intensity electrolyte resonances. Before carrying out in situ 19 F NMR measurements on working supercapacitor cells at different applied potential differences, the form of the spectrum observed for an overlaid supercapacitor was investigated by comparison with model carbon film electrodes soaked with electrolyte. Figure 2a shows the static 19 F NMR spectrum obtained for a piece of YP50F carbon film (6.0 mg) soaked with 8 μL of NEt 4 BF 4 /dACN electrolyte, sealed inside a plastic bag. This carbon film piece represents a model electrode, with no applied potential difference. As in our previous studies, two main environments are observed. 31,32 The feature at -158.0 ppm is assigned to the electrolyte anions strongly adsorbed on carbon surfaces inside micropores. The feature at -152.4 ppm is assigned to weakly adsorbed anions, situated further from carbon surfaces and occupying spaces in large carbon pores and spaces between primary carbon particles in the carbon film. The -5.6 ppm shift of the strongly adsorbed relative to the weakly adsorbed feature is attributed to ring current effects arising from the delocalized electrons in the carbon. On increasing the amount of electrolyte in the sample to 15 μL (Figure 2b), the intensity of the weakly adsorbed resonance is increased. A small additional feature is also observed at higher frequency due to the presence of free electrolyte external to the carbon film piece. Figure 2c shows the NMR spectrum for an overlaid supercapacitor cell held at a cell voltage of 0 V. Here a single electrode is placed inside the NMR detection region. Again the strongly adsorbed electrolyte anions can be observed, appearing at -156.2 ppm. In addition, several more intense features corresponding to the weakly adsorbed anions and anions in free electrolyte external to the carbon electrode are observed at higher frequencies. This is due to the larger amount of electrolyte used in the supercapacitor cell to ensure good electrochemical performance. We note that the free electrolyte signal appears to comprise at least three superimposed components. Bag cell samples containing just free electrolyte (and current collector and free electrolyte) gave similar spectral features (see Supporting Information Figure S3). We attribute this to bulk magnetic susceptibility (BMS) effects owing to the irregular shape of the sample contained within plastic bag cells. Indeed, NMR spectra of isotropic substances contained within nonspherical samples have been shown to be complex, especially for flat orientated samples. 61,62 It is also noted that all the features in this spectrum are shifted to higher frequency by approximately 2 ppm relative to their counterparts in Figures 2a and2b. Previous studies have shown that anisotropic BMS effects can be significant when studying flat electrode assemblies by NMR. 31,52,63 For the spectra shown in Figure 2, care was taken to ensure all samples were oriented vertically inside the NMR coil; however, the geometry and dimensions of the overlaid supercapacitor cell are intrinsically different to the soaked carbon film samples, and therefore, it is likely that the observed ∼2 ppm difference in chemical shift arises from BMS effects relating to the anisotropic shape of the sample and from the presence of a second current collector in the NMR coil. Importantly, despite the intense feature arising from the large amount of free electrolyte in the plastic bag cell, the spectra show that the strongly adsorbed ions are clearly evident in the NMR spectrum of an overlaid supercapacitor at zero potential and can be observed separately from the weakly adsorbed and free electrolyte features. Thus, with this design, changes in the anion environments may be tracked as a function of applied potential.

3.3. In Situ 19 F NMR Spectroscopy at Fixed Voltages. To investigate changes to the BF 4 anion environments as a function of applied potential, 19 F NMR spectra of a single supercapacitor electrode were acquired while the cell was held at different voltages. Previous studies have indicated that small irreversible changes attributed to ion trapping or surface reactions can take place in supercapacitors during the first few charging cycles. 64 For this reason, the supercapacitor cell was cycled 5 times by cyclic voltammetry prior to being studied by NMR. First the positive electrode (and the surrounding electrolyte) was studied, with spectra shown in Figure 3a. At 0 V, the strongly adsorbed feature is observed at -156.5 ppm and the intense free electrolyte signal is observed between -148 and -150 ppm. As the voltage is increased from 0 to 2 V, the strongly adsorbed feature shows gradual increases in resonance frequency (the peak shifts toward the left-hand side of the spectrum), shown more clearly in the inset in Figure 3a. This feature also appears to show small but significant increases in intensity as the cell voltage is increased. The weakly adsorbed and free electrolyte features do not display significant changes in resonance frequency upon charging. After the positive electrode was studied, the supercapacitor cell was shifted so that the negatively charged electrode was situated in the NMR detection region, and the 19 F NMR spectra at various applied cell voltages are shown in Figure 3b. This was done to maintain the same polarity of the device and reduce any possible ion trapping effects. 64 At 0 V, a spectrum similar to that in Figure 3a is observed, with the strongly adsorbed feature visible at -154.9 ppm. We note that this feature is approximately 1.5 ppm higher in frequency than was observed for the positive electrode at 0 V. This suggests that cycling of the cell prior to acquisition of the NMR spectra results in small differences in the equilibrium states for the two electrodes. BMS effects are also expected to result in small shift differences, since different electrodes (with slightly different sizes and orientations relative to the applied magnetic field) are investigated. As the voltage is increased from 0 to 2 V, the strongly adsorbed feature again shifts to higher frequency, as was observed in the positive electrode (Figure 3a). However, here changes in the intensity of the strongly adsorbed feature appear to be less pronounced as the feature moves to higher frequency.

To aid with the interpretation of the results shown in Figure 3, spectra were deconvoluted to quantify the position and intensity of the strongly adsorbed resonance at each voltage. Details of the deconvolution procedure are given in the Experimental Section and Supporting Information. Deconvolutions were not performed for spectra acquired above 1.5 V (positive electrode) and 1.25 V (negative electrode), as the overlap between the strongly and weakly adsorbed features precluded accurate fitting. Figure 4a and Figure 4b show the fitted intensity of the strongly adsorbed resonance for the positive and negative electrodes, which in each case are normalized to the intensity measured for each electrode at 0 V. For the positive electrode, at cell potentials between 0 and 0.75 V, the intensity of the adsorbed resonance does not change significantly. However, between 0.75 and 1.5 V a relatively sharp increase in intensity is observed. For the negative electrode, the intensity of the adsorbed resonance is found to decrease steadily over the voltage range studied. We note that the exact changes in intensity depend upon the fitting procedure used for the deconvolutions, which is not straightforward owing to the complexity of the spectra. However, deconvolutions performed using a different fitting procedure (see Supporting Information Figures S4 andS5) also show similar trends whereby a pronounced increase in the intensity of the adsorbed resonance is observed for the positive electrode at potentials higher than 0.75 V, and an approximately constant decrease in the intensity of the adsorbed resonance is observed for the negative electrode over the voltage range studied. The deconvoluted integrated intensities of the weakly adsorbed and free electrolyte features were not found to change significantly in either electrode over the voltage range studied (as shown in Supporting Information Figure S6). However, interpretation of the intensities of these features is not straightforward, since only part of the cell is situated inside the NMR detection region. This makes it difficult to distinguish intensity changes resulting from rearrangement of ions within the electrodes and changes resulting from diffusion of ions in and out of the NMR detection region.

From integration of the current relaxation, the total charge stored in the supercapacitor was found to increase approximately linearly with applied voltage (see Supporting Information Figure S7). The quantification of strongly adsorbed BF 4 anions in the positive and negative electrodes therefore offers insight into the fundamental mechanism underlying the charging process in this system. For the positive electrode, two distinct charging regimes can be identified. At voltages lower than 0.75 V, the almost constant intensity of the strongly adsorbed feature indicates that the charge storage process does not involve significant changes in the number of adsorbed anions. This suggests that the mechanism of charge storage in this low-voltage regime is dominated by short-scale ion rearrangement and/or the ejection of NEt 4 cations from the micropores. Indeed, recent molecular dynamics simulations have shown that local rearrangements of ions within porous carbon play an important role in supercapacitor charging. 65 At voltages higher than 0.75 V, the abrupt increase in intensity of the strongly adsorbed feature shows that BF 4

-anions are adsorbed into the micropores and that this plays a more significant role in charge storage in this regime. For the negative electrode, the intensity of the strongly adsorbed feature decreases approximately linearly over the range studied. In principle, the loss of BF 4 anions from the micropores should leave an excess of cations within the micropores to counteract the negative charge developed in the electrode. These results are in qualitative agreement with conclusions drawn from EQCM measurements on a similar porous carbon with a NEt 4 BF 4 /propylene carbonate electrolyte. 10 For this system, mass changes that are lower than expected on the basis of Faraday's law were observed at low charge densities and were explained in terms of anion-cation mixing within the micropores. At intermediate charge densities, mass changes close to the theoretical value were observed, indicating that the adsorption of counterions dominates the charge storage process. Our observation of an apparent "threshold" voltage of 0.75 V, whereby an abrupt increase in the strongly adsorbed resonance intensity is observed, may be related to the different charging regimes identified by the EQCM approach.

Figure 4c shows how the chemical shift of the strongly adsorbed resonance varies in the positively charged electrode as the cell voltage is increased. Clear increases in chemical shift are observed for the strongly adsorbed anions as the voltage is increased. While only small increases are observed between 0 and 0.5 V, at higher voltages an approximately linear trend is seen. As the cell voltage is increased from 0 to 1.5 V, a total increase in chemical shift of 3.9 ppm is measured. At voltages above 1.5 V, the strongly adsorbed peak appears to shift to higher frequencies still (Figure 3a), although this precludes reliable fitting of the spectra because of increased overlap of spectral features. In Figure 4d the chemical shift of the strongly adsorbed resonance for the negatively charged electrode is shown as a function of applied cell voltage. Again, clear increases in chemical shift are observed. Here, the chemical shift appears to vary in a linear fashion as the cell voltage is increased. As the voltage is increased from 0 to 1.25 V, the chemical shift increases from -155.3 to -152.4 ppm, representing a total increase of 2.9 ppm. This change is very similar to the overall change in resonance frequency of 3.0 ppm for the strongly adsorbed resonance in the positive electrode over the same voltage range. However, the change in resonance frequency for the negative electrode appears to be approximately linear over the range studied, whereas the positive electrode displays a nonlinear change in resonance frequency at low voltages.

In both the positive and negative electrodes, clear increases in the chemical shift of the resonances corresponding to strongly adsorbed BF 4 anions are observed as the supercapacitor is charged. This shows that the local environment of the anions changes during supercapacitor charging. We previously reported similar observations in a 11 B in situ NMR study of a highly analogous supercapacitor system. 32 When the BF 4 anion was probed, 11 B in situ NMR experiments revealed increases in the chemical shift of adsorbed species in both electrodes as the supercapacitor was charged. Furthermore, recent 11 B and 13 C ex situ studies of supercapacitors comprising activated carbon electrodes and a NEt 4 BF 4 /ACN electrolyte revealed similar increases in the chemical shift of adsorbed features upon charging. 50 In that study two activated carbons different from the one studied here were investigated, suggesting that the chemical shift variations we have observed are general phenomena. 19 F NMR Spectroscopy. To study the behavior of a working supercapacitor in more detail, in situ NMR spectra were recorded in real time for an overlaid supercapacitor undergoing simultaneous cyclic voltammetry. 19 F NMR spectra were collected while the cell was cycled between 0 and 2.3 V at a rate of 0.5 mV•s -1 , with spectra acquired for four full electrochemical cycles. Since the cell voltage is being changed continuously, small changes of ∼0.05 V occur during the acquisition of one NMR spectrum. Since this change is small (2%) compared to the full cycling window of 2.3 V, we may approximate the voltage to be constant during the acquisition of one NMR spectrum. Figure 5a shows selected 19 F NMR spectra of the positive electrode extracted at different points from the cyclic voltammogram of the second cycle, shown as an inset. Spectra corresponding to voltages of 0, 0.5, 1, 1.5, and 2 V are shown. The spectrum acquired at 0 V shows a form very similar to the spectrum of the positive electrode for a fixed (static) voltage of 0 V shown in Figure 3a, with the strongly adsorbed resonance observed at -156.3 ppm. For spectra obtained at later stages in the charging cycle, the strongly adsorbed feature is observed to shift to higher frequencies, in good agreement with the fixed voltage behavior shown in Figure 3a.

Real-Time in Situ

The NMR spectra shown in Figure 5a represent a small selection of the total acquired NMR data. Indeed, 94 19 F NMR spectra were obtained for each full 0 → 2.3 → 0 V cycle, with four full cycles studied. In Figure 5b, NMR spectra for all four electrochemical cycles are shown in a three-dimensional plot. The same spectra are displayed in the first video file in Supporting Information. This plot reveals a highly periodic variation of the adsorbed resonance, the frequency of which varies in almost perfect correlation with the applied voltage. During the charging part of the cycle, the adsorbed resonance shifts to higher frequency, moving underneath the free electrolyte feature. During the discharging part of the cycle, the adsorbed resonance moves back to lower frequency until it reaches -156.3 ppm at 0 V. Over each 0 V → 2.3 V → 0 V cycle, a total change in frequency of ∼3.5 ppm is observed.

The real-time in situ experiment was repeated with the supercapacitor cell shifted so that the negative electrode occupied the NMR detection region. NMR spectra of the negative electrode extracted from the charge of the second electrochemical cycle are shown in Figure 5c. Again, for the spectrum acquired at 0 V the strongly adsorbed environment is clearly visible at approximately -155.1 ppm. As the cell voltage is successively increased to 2.3 V, the strongly adsorbed peak shifts to higher frequency. In Figure 5d, the full set of real-time in situ NMR spectra for the negative electrode is presented as a three-dimensional plot (spectra can also be seen in the second video file in Supporting Information), where again the periodic shifting of the strongly adsorbed environment is clearly observed.

The highly periodic behavior observed for the shift of the adsorbed resonance in both electrodes indicates that changes in the chemical shift originate from a reversible process, directly related to the charging and discharging of the supercapacitor electrode. Further insight into this phenomenon can be obtained by comparing real-time in situ NMR spectra acquired at different electrochemical cycling rates. Real-time in situ 19 F NMR spectra recorded during electrochemical cycling at rates of 0.5 and 5 mV•s -1 are compared as contour plots in Figure 6, together with the corresponding cyclic voltammograms. At the relatively slow cycling rate of 0.5 mV•s -1 (Figure 6a) the variation of the strongly adsorbed chemical shift (seen to the right-hand side of the spectra) is well synchronized with the applied voltage. Moreover, a capacitive electrochemical signature with a near rectangular cyclic voltammogram is observed. However, when the cycling rate is increased to 5 mV• s -1 (Figure 6b), a considerable lag between the periodic variation of the strongly adsorbed chemical shift and the applied voltage is observed. Indeed, close inspection of the spectra (see inset) shows that between voltages of 0.64 V on charge and 1.66 V of the following discharge, the shift of the strongly adsorbed resonance is found to increase. On the other hand, between voltages of 1.66 V on discharge and 0.64 V on the next charge cycle, the shift of the strongly adsorbed resonance decreases. This differs from the behavior observed at the lower cycling rate of 0.5 mV•s -1 , where the shift of the strongly adsorbed resonance shows only very small lags behind the applied voltage. Inspection of the electrochemistry helps explain this behavior. At the faster cycling rate of 5 mV•s -1 , the cyclic voltammogram is far from ideal, with a nonrectangular shape observed. Importantly, positive currents flow between 0.67 V on the charge and 1.64 V on the next discharge, and during this time the magnitude of the charge stored on each electrode increases. However, between voltages of 1.64 V on discharge and 0.67 V on the next charge, a negative current flows and the magnitude of the charge stored at each electrode decreases. The voltages at which the direction of the current reverses agree well with the voltages at which the direction of the shift variation of the adsorbed resonance changes in the real-time in situ NMR spectrum. This confirms that the changes in resonance frequency of the adsorbed resonances are directly correlated with the magnitude of the charge stored on the electrode and not the potential difference applied to the cell. We note that on this basis, the cyclic voltammogram recorded at the slower cycling rate of 0.5 mV•s -1 indicates that we should also observe a small lag in the real-time in situ NMR spectrum, since the CV curve is not perfectly rectangular. Indeed, close inspection of the NMR data reveals that a small lag is indeed observed; however, this is found to be on the order of the time taken to record one slice of the two-dimensional data set and so is not readily noticeable in Figure 6a.

Interpretation of Chemical Shifts for Strongly

Adsorbed Species. In the 19 F in situ NMR spectra of the BF 4 anions, the resonance corresponding to strongly adsorbed ions is observed to shift to higher frequencies upon charging. This observation is consistent with NMR studies of other nuclei such as 13 C and 11 B probing both cations and anions in similar systems. 32,50 This suggests that the changes in chemical shift are nucleus-independent and are related to changes in the ring current effects experienced by the nuclei during charging and discharging. Assuming that the observed chemical shift changes are dominated by ring current effects, there are several possible factors that may contribute to the changes in the resonance frequencies observed for adsorbed ions. First, addition or removal of electrons from the carbon electronic structure may modify the ring current effects. Second, during supercapacitor charging cations are expected to become more strongly associated with the negative electrode surface, while anions are expected to become less strongly associated with the negative electrode surface (and vice versa at the positive electrode). The resulting increases or decreases in carbon-ion distances may change the observed chemical shift. Finally, we note that there may be small changes in the carbon micropore sizes during charging and discharging. In situ electrochemical dilatometry revealed small macroscopic height changes of porous carbon electrode films during charging and discharging. 66,67 If these height changes arise from dilation of the carbon micropores as proposed, the effective distance between nuclei in adsorbed molecules and carbon surfaces may again be modified. To investigate these effects, DFT calculations were carried out to determine nucleus-independent chemical shifts (NICS) associated with a simple model system. NICS calculations are readily performed using standard quantum mechanics programs whereby isotropic chemical shifts are calculated for "ghost" atoms placed at arbitrary positions above and around carbon surfaces. These calculations provide quantification of the effect of ring currents on the resonance frequencies of nuclei situated close to systems containing delocalized electrons. Diatropic (diamagnetic) ring current effects 68 cause shielding fields above the plane of aromatic rings, while paratropic (paramagnetic) ring current effects 69 cause deshielding fields above the plane of antiaromatic rings. 41 Activated carbons have complex amorphous structures consisting of curved graphene sheets, 70 and it is difficult to accurately model these complicated structures in DFT calculations. Recent studies have employed coronene-and fullerene-based structures as simple models for more extended carbon structures. 42,[71][72][73] To explore the origin of the observed NMR frequencies reported in the current work during charging and discharging of supercapacitor electrodes, NICS calculations were performed on the molecule coronene.

Figure 7 shows the calculated NICS as a function of distance above the coronene ring plane for three different positions in the xy plane. The structure of coronene and the positions in the xy plane (A, B, and C) for which values were calculated are shown as an inset. A lower limit of 2.5 Å above the coronene ring plane was chosen for calculations, since the approach of an adsorbed molecule and the carbon will be limited by the van der Waals radii of atoms in each. Indeed, the van der Waals radius of sp 2 hybridized carbon is ∼1.7 Å and the van der Waals radii of hydrogen and fluorine are expected to be greater than 1 Å. 74 For all three positions (A, B, and C) considered, NICS values were calculated to be negative above the ring plane, consistent with diamagnetic ring currents. Moreover, for the three positions considered, NICS values show good agreement with each other, showing that there is an approximately uniform shielding field above coronene. Thus, resonances from adsorbed molecules should appear shielded relative to the corresponding free species. In Figure 2a above we noted that the resonance corresponding to strongly adsorbed BF 4 anions appears at a chemical shift of 5.6 ppm less than the weakly adsorbed anions. This is in reasonable agreement with the NICS value of approximately -5 ppm calculated at 2.5 Å above coronene.

Figure 7 allows us to consider the effect of a nucleus moving closer to, or further away from, a carbon surface. As a nucleus becomes more distant from the coronene surface, the calculated NICS becomes less negative (higher NMR frequency). This in principle could explain the increases in resonance frequency we have observed upon charging (∼3.5 ppm), corresponding to ions moving roughly 6 Å further away from the coronene molecule considered here. While such a distance change may be reasonable for an ion adsorbed on a carbon surface of like charge, the same behavior is not expected for ions adsorbed on carbon surfaces of opposite charge. In this case, charging is expected to reduce the ion-carbon separation due because of electrostatic attraction which would decrease the observed resonance frequency according to the NICS calculations. This is contrary to what is observed in the NMR experiments, where the resonance frequencies of the strongly adsorbed species are always observed to increase during charging, regardless of the electrode studied. It is therefore unlikely that the changes in NMR frequency observed for adsorbed species during charge and discharge arise solely from changes in carbon-ion distances. This is further substantiated by noting that a molecular dynamics study of a supercapacitor comprising porous carbon electrodes showed only small changes of ioncarbon distances (about 0.2 Å) upon charging to a cell voltage of 1 V, although in this case an ionic liquid electrolyte was studied. 23 We also note that the small reversible electrode height changes revealed by electrochemical dilatometry (1.1% and 0.5% for the positive and negative electrodes respectively using YP17 activated carbon and an NEt 4 BF 4 /ACN electrolyte) 66 are unlikely to have a significant effect on the observed resonance frequencies, as the distance changes would result in a negligible change in the NICS.

To explore the effect of charging on the NICS, calculations were performed on charged coronene molecules. Figure 8 shows NICS plots as a function of distance above the coronene ring plane for three different positions in the xy plane (A, B, and C). NICS calculations were performed for optimized coronene molecules with charges of +1e (Figure 8a) and -1e (Figure 8b), where e is the electronic charge.

For both the positively and negatively charged coronene molecules, positive NICS values were calculated for all positions considered above the carbon surface, consistent with paratropic ring currents. The NICS tends toward zero for large distances away from the carbon surface. The calculated increase in resonance frequency for both positively and negatively charged carbon qualitatively agrees with our in situ NMR results, whereby the strongly adsorbed resonance shifts to higher frequencies upon charging in both the positive and negative electrodes. The differences in the magnitudes of the calculated NICSs close to the carbon surface may explain why slightly different changes in resonance frequency are observed in the positive and negative electrodes experimentally, particularly in the low voltage regime. However, the magnitudes of the shifts predicted by the DFT calculations are significantly larger than what we observe experimentally. We note that the model system studied here is a considerable oversimplification of what is expected to be an extended graphene-like pore system in the activated carbon, and so quantitative agreement between the calculated and experimental results is unlikely. Finally, we also note the analogy with the shifts observed in our system with those seen in an early experimental work on [18]annulene. Following two-electron reduction, the 1 H chemical shifts of the interior protons changed from -3.0 ppm to 28.1 and 29.5 ppm, explained as arising from the introduction of paratropic ring currents. While changes in carbon-ion distances are still predicted to affect the observed resonance frequencies, the calculations presented here suggest that the introduction of paratropic ring currents by the charge imposed on the carbon will play an important role.

CONCLUSIONS

This work has presented a detailed account of the use of NMR for in situ studies of supercapacitors, not only developing the methodology but also revealing fundamental insight into the charge storage mechanism. Specifically, we make two main conclusions for YP50F, an important microporous carbon that is widely used as a supercapacitor electrode material:

(1) Changes in the intensity of the adsorbed resonances in 19 F NMR spectra of a YP50F-NEt 4 BF 4 /dACN supercapacitor system recorded at fixed voltages correspond to changes in the populations of adsorbed BF 4 -species. Analysis of these changes suggests that charging can be divided into at least two distinct regimes. At low cell voltages (less than 0.75 V), the approximately constant anion population in the positive electrode and decreasing anion population in the negative electrode indicate that charging is not dominated by the adsorption of counterions and is instead achieved through short-scale rearrangements and the ejection of co-ions from the micropores. At high cell voltages (greater than 0.75 V), an abrupt increase in the adsorbed anion intensity in the positive electrode, combined with the continued loss of anions from the negative electrode, suggests that both counterion adsorption and co-ion ejection are important in this regime.

(2) Changes in the resonance frequency of the adsorbed species observed during charging arise because of changes in the carbon electronic structure. This effect is external to the electrolyte itself and is therefore independent of the nature of the adsorbed electrolyte species inside the carbon micropores. Indeed, DFT calculations on a simple model system confirm that the addition or removal of electrons to a delocalized system should lead to marked changes in the nucleusindependent chemical shift of nearby species. Importantly, regardless of whether electrons are added or removed from the carbon surface, the DFT calculations predict that a deshielding effect should be observed, in qualitative agreement with our experimental results.

The observation and characterization of these effects open the way for the future study of the supercapacitance phenomenon using NMR. In particular, we envisage that the approaches we have developed will provide insight into the effect of pore size and pore size distribution on the charge storage process. We also note that in principle these methods are applicable to other electrolytes such as ionic liquids, and work is ongoing in our laboratory to investigate these systems.

■ ASSOCIATED CONTENT

Figure 1 .

 1 Figure 1. Supercapacitor cell designs (a-c) and their cyclic voltammograms at 0.5 mV•s -1 (d-f). In each case the cyclic voltammogram of the fourth cycle is shown.

Figure 2 .

 2 Figure 2. 19 F NMR spectrum obtained for pieces of carbon film soaked with (a) 8 μL and (b) 15 μL of NEt 4 BF 4 /dACN electrolyte.(c) Spectrum obtained for an overlaid supercapacitor held at a cell voltage of 0 V. The films were orientated vertically inside the NMR coil to enhance resolution of spectral features.31 All spectra are shown on the same scale.

Figure 3 .

 3 Figure 3. (a)19 F NMR spectra of the positively charged electrode of an overlaid cell at voltages between 0 and 2 V. (b)19 F NMR spectra of the negatively charged electrode of the same cell at voltages between 0 and 2 V. Insets show enlarged sections of the spectra to illustrate changes of the strongly adsorbed feature.

Figure 4 .

 4 Figure 4. Summary of deconvolution results of spectra shown in Figure 3. Panels a and b show intensities of the strongly adsorbed feature in the positive and negative electrodes, each normalized by the intensity of this feature in the 0 V spectrum. Panels c and d give the deconvoluted chemical shift values for the strongly adsorbed feature in the positive and negative electrodes, respectively. It was not possible to obtain unique fits for spectra corresponding to voltages above 1.5 and 1.25 V for the positive and negative electrodes, respectively.

Figure 5 .

 5 Figure 5. (a) Selected 19 F NMR spectra of the positive electrode of an overlaid capacitor cycled by cyclic voltammetry. Spectra are extracted from the charge during the second cycle. (b) Successive NMR spectra from four full cyclic voltammetric cycles are shown back to back. (c, d) Analogous results for the negative electrode of the supercapacitor cell are shown.

Figure 6 .

 6 Figure 6. Contour plots of successive in situ 19 F NMR spectra of the negative electrode of an overlaid supercapacitor cell with simultaneous cyclic voltammetric cycling. Representative cyclic voltammograms are shown beside the spectra. Spectra and cyclic voltammograms are shown for cycling rates of (a) 0.5 mV•s -1 and (b) 5 mV•s -1 . In part b, enlarged spectra are shown for the fifth cycle as in inset. For each two-dimensional spectrum, 18 slices were recorded per charging cycle, with recycle intervals of (a) 30 s and (b) 3 s.

Figure 7 .

 7 Figure 7. Calculated NICS values above a neutral coronene molecule.

Fellowship to C.P.G.) for funding. N.M.T. was supported by the Northeastern Center for Chemical Energy Storage and by Energy Frontier Research Center funded by the U.S. DOE, BES under Award No. DE-SC0001294. P.S. acknowledges the support from the European Research Council (ERC, Advanced Grant, ERC-2011-AdG, Project 291543IONACES). L.T. was supported by the Deĺeǵation Geńeŕale pour l'Armement. A.C.F. and J.M.G. thank the NanoDTC Cambridge for travel funding. We also thank Yury Gogotsi, Volker Presser, Ceĺine Merlet, and Andy Ilott for useful discussions.

Author Contributions

∥ H.W. and A.C.F. contributed equally to this manuscript.

Notes

The authors declare no competing financial interest.

■ ACKNOWLEDGMENTS

A.C.F., J.M.G., H.W., and C.P.G. acknowledge the Sims Scholarship (A.C.F.), EPSRC (via the Supergen consortium, J.M.G. and H.W.), and the EU ERC (via an Advanced


Texturation of lead-free BaTiO3-based piezoelectric ceramics

A Ngueteu-Kamlo, F Levassort, M Pham Thi, Pascal Marchet

To cite this version:

A Ngueteu-Kamlo, F Levassort, M Pham Thi, Pascal Marchet. Nowadays, piezoelectric ceramics are integrated in a wide range of devices, in particular in ultrasonic applications (underwater sonar systems, medical imaging, non-destructive testing…). Most of them use Pb(Zr,Ti)O3 (PZT). However, due to health care and environmental problems, lead content must be reduced in such applications [1]. Recent reviews demonstrated that few lead-free materials families can be considered: the alkaline-niobates (K0.5Na0.5NbO3), the alkaline-bismuth-titanates (Na0.5Bi0.5TiO3) and barium titanate based materials (BaTiO3) [2,3]. One of the limitations of ceramic materials is their isotropic nature. This is the reason why texturation process has been developed in order to improve their properties in particular electromechanical parameters. The aim of the present study is thus to obtain textured BaTiO3 based materials by using the templated grain growth process (TGG) and to measure their piezoelectric properties. Nanosized BaTiO3 powders were prepared by classical solid state route at relatively low temperature. BaTiO3 templates of different morphologies were elaborated by a molten salts process. Dispersing agent, binder and plasticizer, the mixture of the templates and matrix particles was then dispersed in the appropriated solvent using non-aqueous formulation. The slurry was then tape-casted on plastic film. After drying, the green sheet was cut, stacked, thermo compressed and then sintered at the appropriated temperature. This process allowed obtaining highly-oriented materials with a texturation degree around 95% (fig. 1), presenting Kt values around 35-40%, higher than for non-textured BaTiO3 ceramics.

Fig. 1 :

 1 Fig. 1: X-ray diffraction diagram of (tetragonal) textured sample BaTiO3 and (tetragonal) non-textured BaTiO3 ceramic as reference.

Acknowledgements:

The authors kindly acknowledge financial support from the French Research National Agency (ANR), project HYPERCAMPUS


João Bosco

Ferreira Filho
email: joao.ferreirafilho@inria.fr

Mathieu Acher
email: mathieu.acher@irisa.fr

Olivier Barais
email: olivier.barais@irisa.fr

Challenges on Software Unbundling: Growing and Letting Go

Keywords: D.2.9 [SOFTWARE ENGI-NEERING]: Distribution, Maintenance, and Enhancement Unbundling, modularization, features, aspects, reengineering, refactoring, evolution

Unbundling is a phenomenon that consists of dividing an existing software artifact into smaller ones. For example, mobile applications from well-known companies are being divided into simpler and more focused new ones. Despite its current importance, little is known or studied about unbundling or about how it relates to existing software engineering approaches, such as modularization. Consequently, recent cases points out that it has been performed unsystematically and arbitrarily. In this paper, our main goal is to present this novel and relevant concept and its challenges in the light of software engineering, exemplifying it with recent cases. We relate unbundling to standard software modularization, presenting the new motivations behind it, the resulting problems, and drawing perspectives for future support in the area.

Introduction

Software is designed to meet user needs and requirements, which are constantly changing and evolving [START_REF] Pohl | Requirements engineering: fundamentals, principles, and techniques[END_REF]. Meeting these requirements allows software companies to acquire new users. For example, mobile applications compete with each other to gain market share in different domains; they constantly provide new features and services for the end user, growing in size and complexity. In some cases, the software artifact absorbs several distinct features, overloading the application and overwhelming the user and his/her acceptance of the software product [START_REF] Ickin | Factors influencing quality of experience of commonly used mobile applications[END_REF] -he/she has to carry dozens of Swiss Army knives in his smart phone.

A recent phenomenon is to unbundle these dense pieces of software into smaller ones, trying to provide simpler and more focused applications. Unbundling consists of dividing an existing software artifact into smaller ones, each one serving to different end use purposes; it requires an unplanned coarse-grained modularization of mature software.

The main claim and goal of modularization techniques when applied to mature code is to improve software properties like maintainability and understandability [START_REF] Mortensen | Aspect-oriented refactoring of legacy applications: An evaluation[END_REF]. In unbundling, this is not the [Copyright notice will appear here once 'preprint' option is removed.] case, these desired good properties become means, instead of ends of the process. The goal is the division itself, in order to attend to market issues imposed by trends of usage and competition to gain market share; nevertheless, the good properties may work as enablers to accomplish unbundling.

Although the importance of unbundling in today's software industry, no studies have been conducted to conceptualize or analyse these challenges. This paper is a first step to fill this void. Our main contributions are: to define and analyse the unbundling phenomenon (Section 2), to present the main challenges of unbundling (Section 3), showing examples of this current phenomenon and explaining how it relates to existing software engineering approaches for software modularization, and to draw perspectives on how to facilitate and better exploit unbundling (Section 4).

The unbundling phenomenon

Recently, many well-known software companies started to divide their mobile applications into smaller ones. This is the case of Foursquare, Dropbox, LinkedIn, Evernote, Facebook and Google. Some of them, like Foursquare, have split into two, continuing with the original one and separating part of its features into a second brand new application (in this case, Swarm); some others unbundled into several applications, for example, LinkedIn originated Pulse, Connected, Job Search, Recruiter, Sales Navigator and Slide Share. IBM also adopted the strategy of dividing their software and services to better adequate to market and regulatory needs [START_REF] Humphrey | Software unbundling: a personal perspective[END_REF].

In all these cases, unbundling was essentially about identifying parts of the original software that could be isolated in separated applications. For doing this, these parts must be reengineered in a new software, according to a high-level end user purpose. A purpose is a high-level and often subjective end user goal when using the software product, it can be the reason why the user acquired the product (e.g., sharing photos, making todo lists, searching places, etc.), and it can gather a set of requirements or features of it. Ideally, one application should serve one highest purpose, however secondary purposes often start as small features of the application, and then they grow until they are a considerable part of the software, which can now be separated as a self-contained purpose.

Reengineering these parts that serve to different purposes comprehends: (1) decoupling them from other parts that will not be in the same application, so they can exist separately; (2) developing missing parts that were removed during refactoring or that are necessary to make the part usable (e.g., changing a user interface code to show only a group of functionalities). In the case of mobile applications, software companies are trying to follow the principle of having one big purpose per application, so the user knows clearly why and how to use it, avoiding numerous, cluttering and confusing features. In this way, application vendors create a strong identity for their products and link them well-defined purposes.

Figure 1 gives an overview of the unbundling process. Let us consider that an original software artifact Θ is a candidate to be unbundled. Θ contains different parts Θ = {a1, a2, a3, b1, b2, b3, c1, c2, c3} expressed in any unit (e.g., class, method, block, statement). These parts can be associated or intersect each other. Let us also consider that there exists subsets of parts in Θ that implement different high-level purposes, which is a motivation for unbundling Θ into different applications containing these different parts. In the example of Figure 1, the subsets of parts {a1, a2, a3} and {b1, b2, b3} are isolated into two new software ∆1, ∆2 because they serve to two distinct end user purpose. This isolation implies changing the structure of the part to make it decoupled or even adequating its behaviour to work in a different context (a1 is reengineered into a 1 in Figure 1).

As also illustrated in Figure 1, it is possible that the new software artifacts share common parts among them and with the original one (c1, c2, c3). This is the case, for example, of parts responsible for the implementation of crosscutting concerns or of essential functionalities of any application derived from the original one, such as authentication, storage, cryptography modules, etc. It is also possible that the new software artifacts demand new parts (n1, n2, n3, n4, n5, n6) to implement new functionalities or to adapt the existing ones to the new context.

n 2 n 3 n 1 a' 2 a' 1 a' 3 c' 2 c' 3 a' 1 n 4 n 6 n 5 c" 2 c" 3 c" 1 b' 2 b' 3 b' 1 Θ Δ 1 Δ 2 Figure 1. Unbundling.
In Figure 1, ∆1 can be the original software without the secondary purpose parts, as it is of the company's interest to continue their original software product in order to keep its associated market share. However, if the result of the Unbundling δ was only ∆1 without any other new applications, we would call it Reductive Unbundling. From this perspective, unbundling can serve to reduce the original application, removing near-unused/dead code without the preoccupation of making it usable in another application. The difference between actual dead code elimination [START_REF] Knoop | Partial dead code elimination[END_REF][START_REF] Xi | Dead code elimination through dependent types[END_REF] and reductive unbundling is that, before the unbundling, the eliminated code could still be executed at run-time in the original application, which does not fit the definition of dead code.

In resume, if we think of unbundling as a process, we can enumerate key activities that it encompasses:

• Identifying distinct end user purposes when using a given original software; • Identifying the parts of the original software that relates to each identified purpose. • Extracting and reengineering the parts to be placed in the different new software artifacts; • Identifying and extracting common parts so they can be reused.

• Occasionally implementing new parts that will complement the existing ones in the new software artifact.

Challenges

In this section, we present five key challenges of unbundling. In order to efficiently handle unbundling, it is essential to: (1) understand its new causes and objectives, which are different from standard modularization; (2) handle an unplanned need for correcting the software structure, so it can better evolve and grow; (3) manage the software parts from a high-level perspective related to an end user purpose, which is a coarser-grained abstraction when compared to existing software engineering abstractions; (4) handle the division itself and the isolation of existing parts of software to work on different applications; and (5) unbundling efficiently by avoiding code replication.

New causes and objectives

The causes and objectives of unbundling differentiates from the ones of modularizing, or simply componentizing [START_REF] Washizaki | A technique for automatic component extraction from object-oriented programs by refactoring[END_REF], or aspectualizing [START_REF] Ajila | Aspectualization of code clonesan algorithmic approach[END_REF][START_REF] Mortensen | A test driven approach for aspectualizing legacy software using mock systems[END_REF][START_REF] Rebêlo | Modularizing crosscutting contracts with aspectjml[END_REF] software. Since its origins, modularization seeks to improve flexibility and comprehensibility of a given software [START_REF] Parnas | On the criteria to be used in decomposing systems into modules[END_REF]; these goals have been enlarged to also consider maintainability and testability [START_REF] Mortensen | Improving software maintainability through aspectualization[END_REF][START_REF] Munoz | Usage and testability of aop: an empirical study of aspectj[END_REF].

In unbundling, the goal is beyond increasing maintainability, understandability, flexibility or testability of an application; it aims at separating the software artifact and creating new smaller and self-contained ones, which will then be managed by different engineers, used by different clients and placed in different domains. Essentially, the ends become the means: all software good properties are now means for the end of dividing the software, while before, modularizing and dividing the software artifact in modules were the means to reach such good properties.

Therefore, unbundling is triggered by business goals, it creates new opportunities with separable markets [START_REF] Humphrey | Software unbundling: a personal perspective[END_REF]; it is a specialization of software businesses. From the marketing perspective, unbundling is challenging for the company because there is a risk to lose clients in the transition to the new software. On the other hand, if the transition succeeds, it can open a new market and attract more clients.

From the software engineering perspective, modularization is a established good practice and it is driven by developers or stakeholders that are related to the development process. Market competition is often what drives unbundling -maintainability or any other software engineering concepts are less important when faced to the needs of competing to a market share or simply aligning the software to a new trend of use. The way that developers thought the software modularization can be very different from the division imposed by these high-level purposes. However, this does not prevent that software engineering best practices also come to be a trigger for unbundling, probably if the way the application is structured starts to interfere in the company's business.

Unplanned corrective evolution

Two essential characteristics present in software to be unbundled are: it is already mature, and the possibility of splitting it into several other software artifacts was not conceived in earlier stages of its development. Therefore, unlike, for example, Software Product Lines (SPL) [START_REF] Clements | Software Product Lines : Practices and Patterns[END_REF], unbundling is not a long-term strategy that is planned in advance by the company. The fact that the software artifact is already mature and no longer in a conceptual phase implies that unbundling is a corrective action that has to handle code with all its existing good or bad design and implementation choices. Another challenge is that the software is up and running, having numerous clients relying on it, therefore the evolution must not in-terfere in this relationship; this interference is very likely to happen if a bad release of the unbundled software is launched.

Coarser-grained modules

The criteria used when decomposing systems into modules have been studied for decades [START_REF] Parnas | On the criteria to be used in decomposing systems into modules[END_REF]; from sub-routines to features [START_REF] Kästner | Granularity in software product lines[END_REF], the unity of modularity is a primary concern in software design and implementation. These units of modularity are thought by stakeholders directly involved with the code (e.g., developers, architects). As the motivation of unbundling is to separate distinct end user purposes residing in one single software artifact, it is necessary to group the set of software parts in the original software product that meets these purposes.

This coarser modularization raises the challenge of having to categorize the existing fine units and associate them to greater goals, which are not explicitly modeled in the software, but that are rather high level abstractions that are used to sell the software to a client. In the example of Foursquare and Swarm, the purpose of checking in places was extracted from the original application (Foursquare) and placed in a new one (Swarm); checking in places gathers functional and non-functional finer features of the software like: geographic location, map visualization, social network sharing, etc.

Dividing and isolating

Dividing software has always proved challenging; many different software engineering approaches have been proposed to decompose [START_REF] Ossher | Using multidimensional separation of concerns to (re) shape evolving software[END_REF] code or slicing programs [START_REF] Weiser | Program slicing[END_REF].

As explained in the last subsection, unbundling has to handle coarser abstractions, but still actually managing finer ones. The problem is that this is not always intuitive, as classes, components, aspects or features are not perfect modular units, they share dependencies and interactions [START_REF] Apel | Feature-interaction detection based on feature-based specifications[END_REF]. Therefore, parts of the original software product may finish to be present in different purposes, or the functioning of a feature to be affected by the presence or absence of another, which makes the task of dividing and isolating purposes hard.

A great challenge is to make current componentization, apectualization, feature extraction [START_REF] Acher | Extraction and evolution of architectural variability models in pluginbased systems[END_REF] techniques to work seamlessly with an additional level of abstraction: the purpose. These techniques work by analysing the software and evaluating its parts, clustering them in categories according to their structure and semantics as criteria. In unbundling, this division must be enriched with the end user purpose, which will drive the division itself.

Considering the steps of the unbundling process, we can state that the essential challenges of this process are comprehended in dividing and isolating parts of software: identifying, extracting and reengineering existing features. Identifying is challenging because of the complicated matching between one single high-level and often subjective purpose to concrete fine-grained implementations. Extracting the features imposes the difficulty of not breaking the semantics of the code after removing parts of it, thoroughly analysing the dependencies from statement to component levels. And finally, reengineering the parts is challenging because it depends on the two past activities and because it demands knowledge on the new application environment in which the new parts are now placed.

Besides their own challenging nature, these activities can be hardened according to how strong is the relationship between modules of the original software artifact that belong to different purposes (i.e., coupling degree). On the other hand, unbundling may be facilitated if the modularity units belonging to a purpose present a high cohesion (i.e., their function relate only to the given purpose).

Code replication

As dividing and isolating legacy software parts is a hard task, one can be tempted to simply clone and own the original application, but only hiding the secondary purpose features from the end user. However, this leads to lots of replicated and useless code, decreasing the comprehensibility and maintainability of the software artifact, also demanding additional memory space in limited devices.

Figure 2 shows the evolution over time of Foursquare and Swarm mobile applications with respect to their sizes in MB. After the unbundling, the Foursquare user started to need two applications for doing what he/she used to do with only one application, and about 7 MB more of storage, not considering the additional RAM that Swarm consumes. Although these numbers do not clearly prove that there is code replication, it gives us initial insight that unbundling in an efficient way is challenging. This is also true in other unbundling cases and the problem can gain bigger proportions as the number of unbundled application increases.

Perspectives

Following, we present the perspectives for analysing and leveraging unbundling, describing key topics that can be the starting point for a research roadmap.

Unbundling can take advantage from existing modularization approaches. Although we still need experiences on whether these techniques actually facilitate or not unbundling, our intuition is that code with cohesive and decoupled coarse-grained modularity units is easier to be unbundled. Therefore, a first perspective is that refactoring approaches for modularization of legacy code should be adapted to cope with the new challenges of unbundling, such as maximizing cohesion, minimizing coupling and mapping modularity units to usage purposes in order to facilitate the concrete division of the software.

Ideally, unbundling should be as much automated as possible. As a perspective, we envision automated techniques to analyse and execute unbundling. For example, detecting patterns of application usage (as in [START_REF] Böhmer | Falling asleep with angry birds, facebook and kindle: a large scale study on mobile application usage[END_REF][START_REF] Falaki | Diversity in smartphone usage[END_REF]) in order to classify features that are frequently used or not, features that are always used by a profile of user and others by other profiles and so on. These patterns would serve to make explicit the different purposes that the user has when manipulating the software product, better motivating and justifying a division. As for the automation of the division itself, we envision future research on techniques to extract and isolating features in separated applications, going beyond the simple synthesis of feature models after source code [START_REF] She | Usage scenarios for feature model synthesis[END_REF].

Another perspective is to systematize the unbundling process. Nevertheless unbundling is not a desired or planned event in software lifetime, it can be exploited as a first step to move towards a product line paradigm or a software ecosystem [START_REF] Costa | From applications to a software ecosystem platform: an exploratory study[END_REF][START_REF] Messerschmitt | Software ecosystem: understanding an indispensable technology and industry[END_REF]. For this, the original application can be seen as a source of reusable assets that, if efficiently extracted and isolated, could be the basis to construct several different new applications. This systematization is justified when the company desires, as long-term vision, to carry on building a family of applications for a specific domain, sharing commonalities and managing variabilities.

Unbundling can happen in other kinds of software. Particularly, there is the case of big and complex APIs and frameworks, which provide several features for programmers in a specific domain; they sometimes provide much more than the programmer needs or their use can vary according to different purposes. Therefore we can envision unbundling these artifacts to better suit the needs of different end user purposes, reducing the overload of using a given framework or API.

Conclusion

We presented in this paper the novel phenomenon of unbundling software, showing evidences in the domain of mobile applications. In order to better understand it, we explained the process of unbundling, introducing how an original software artifact is transformed into two or more new ones. We then discussed the problems and challenges of unbundling, also relating it to standard planned modularization. Finally, we discussed perspectives on the support of this new phenomenon. Our main conclusion is that unbundling needs special support and the existing modularization techniques can help in this task; because of its unpredicted nature, its highlevel abstractions and its needs for concrete isolation of software parts, unbundling raises new challenges that merit to be further investigated, understood and supported. As long-term future work, we consider the points explained in the perspectives section, while in short-term, we want to proceed on analysing unbundling cases deeper, studying the division and distribution of features from the original software products into the new ones.

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Acknowledgments

The research leading to these results has received funding from the European Commission under the Seventh (FP7 -2007-2013) Framework Programme for Research and Technological Development (HEADS project).

We thank the reviewers of Modularity'15 for insightful feedback on this work; they helped us to vastly improve the concepts related to software unbundling.


S Shihab

H Riahi

L Thevenard

H J Von Bardeleben

A Lemaître

C Gourdon

Systematic study of the spin stiffness dependence on Phosphorus alloying in (Ga,Mn)As ferromagnetic semiconductor

Keywords: numbers: 75.78.Jp, 75.30.Ds, 75.50.Pp

We study the dependence of the spin stiffness constant on the phosphorus concentration in the ferromagnetic semiconductor (Ga,Mn)(As,P) with the aim of determining whether alloying with phosphorus is detrimental, neutral or advantageous to increase the spin stiffness. Time resolved magneto-optical experiments are carried out in thin epilayers. Laser pulses excite two perpendicular standing spin wave modes which are exchange related. We show that the first mode is spatially uniform across the layer corresponding to a k ≈0 wavevector.

From the two frequencies and k-vector spacings we obtain the spin stiffness constant for different phosphorus concentrations using weak surface pinning conditions. The mode assessment is checked by comparison with the spin stiffness obtained from domain pattern analysis for samples with out-of-plane magnetization and with ferromagnetic resonance experiments when more than one spin wave mode is observed. The spin stiffness is found to exhibit little variation with phosphorus concentration in contradiction with ab-initio predictions.

Building complex heterostructures, such as tunnel magnetic junctions, from the same host material is a challenge in order to reduce detrimental interface effects between different parts of a spintronic device. In this framework, diluted magnetic semiconductors (DMS) are a class of materials able to address this challenge 1 . More fundamentally, DMS and more specifically the III-V based (Ga,Mn)As have become in the past decade a benchmark material in order to achieve predictable tuning of magnetic properties. Levers such as the temperature 2 , the carrier concentration 3,4 but also the strain applied on the magnetic layer 5,6 or alloying with phosphorus 7,8 have been used in order to change the micromagnetic properties, e.g. the Curie temperature T C , the saturation magnetization M s , and the magnetic easy axis. Among these properties, the spin stiffness D is perhaps the most difficult to tune, despite theoretical guidelines 9 .

An increase of the spin stiffness keeping constant magnitude of the magnetization M s would mean larger exchange constant A (A = DM s /2) and therefore larger domain wall width and domain wall velocity, this tunability remaining out of reach for metals. It is expected that alloying with phosphorus should increase the spin stiffness owing to an increase of Mn-hole exchange integral J pd , the stronger hybridization of the p-d wave functions arising mainly from the smaller lattice constant of GaMnP 10 . The effect on T C remains theoretically unclear. Whereas an overall increase is predicted from (Ga,Mn)As to (Ga,Mn)P 10,11 , a decrease might occur for P concentration between zero and 25 % 11 . Furthermore it was shown experimentally, that (Ga,Mn)(As,P) suffers a metal-to-insulator transition with increasing Phospho-55 rus concentration with a drop of its T C 12,[START_REF] Cubukcu | A. 318 Lemaître[END_REF] . A modest 56 increase of D between y = 0 and y > 6 % can be in-57 ferred from results on the exchange constant A obtained 58 for (Ga,Mn)(As,P) samples with out-of-plane easy axis 59 using domain pattern analysis 14,15

S(t) = i A i e αi2πfit sin(2πf i t + φ i ) , (1)
where f i is the frequency, φ i the phase, A i the amplitude and α i the effective (inhomogeneous) damping of the ith excited mode. The signal in Fig. 1 is obtained in a y=3.4 % sample at 12 K. Parameters from the fit are

f 1 = 1.11 ± 0.02 GHz, f 2 = 4.03 ± 0.05 GHz , α 1,2 =0.08
and the amplitude ratio between the two modes is 0.3 .

To extract D from the observed frequencies, the magnetization dynamics is modeled in a standard way, starting from the Landau-Lifshitz-Gilbert equation with the exchange field and without damping 6,18 . For in-plane easy-axis, the equations for small precession angle of the The top curve displays the experimental signal (squares) and the associated fit (solid line) for (Ga,Mn)(As,P) with x ef f = 3.6% and y = 3.4% at T=12 K. The two curves below represent the oscillatory components of the experimental signal obtained from the fit of the signal, shifted for clarity. Inset: FMR spectra at T=4 K for different angles of an outof-plane field (θ with respect to the [001] direction). magnetization vector read:

140 δθ = γ -F φφ δφ/M s + D ∂ 2 δφ ∂z 2 δ φ = γ F θθ δθ/M s -D ∂ 2 δθ ∂z 2 , (2)
where the z-axis ( [001]) is perpendicular to the layer 146

F θθ = -2K 2⊥ + K 4 (3 + cos 4φ 0 ) /2 (3) +2K 2 (1 -sin 2φ 0 ) + µ 0 M 2 s F φφ = 2 K 4 cos 4φ 0 -K 2 cos 2φ 0 , (4)
with φ 0 the equilibrium angle with respect to [100] given

147 by sin 2φ 0 = -K 2 /K 4 if K 2 /K 4 <
f (k) = γ 2π F θθ F φφ /M 2 s ± (F θθ + F φφ )Dk 2 /M s + D 2 k 4 , (5)
where γ is the gyromagnetic ratio. The plus sign refers to 153 a bulk mode (cos kz, sin kz) and the minus one to a sur-154 face mode (cosh kz, sinh kz). Note that by setting D = 0, 155 one recovers the Smit-Beljers formula used for FMR 22 .

156

The dispersion curve for a sample with y = 4.3% at 157 T =12 K and H=0 is displayed in Fig. 2(a). The k = 0 158 frequency represents the spatially uniform mode and is related only to the anisotropy constants, while the curvature of the bulk mode dispersion curve depends on D.

The spatial dependence of δθ and δφ is calculated by finding the allowed confined surface-bulk hybridized spin wave wavevectors for a sample thickness L. Following Refs. [6,18], Rado-Weertman symmetric general boundary conditions for an in-plane magnetization are used 23 :

∂δφ ∂z | ±L/2 = 0 ∂δθ ∂z | ±L/2 = ∓ 2K s DM s δθ , (6)
where F s = K s cos 2 θ is the surface anisotropy energy acting as a pinning term hindering the surface spin precession. The natural freedom condition (K s = 0) gives a k-quantification in nπ/L, where even (odd) n correspond to even (odd) spin wave modes. Given the experimental precession frequencies, the spin stiffness and the mode profiles are obtained through an iterative adjustment of D and K s (Fig. 2(a)). D determines the frequency spacing between the two modes, while the pinning constant K s shifts their frequencies.

In order to run this model, a preliminary step is to check the spatial profile of the lowest energy mode. To that end, a wet etching procedure was carried out on a y = 4.3 % sample in order to obtain a staircase pattern (Fig. 2(c)). Successive oxidation and oxide removal sequences lead to a thickness difference of 5 nm between each step. The experimental results for this sample (Fig. 2(c)) show that the first mode frequency does not change with decreasing thickness while the second one varies with the thickness as expected from Eqs. (5, 6). This shows firstly that the sample magnetic properties have a good in-depth and lateral uniformity and, secondly, that we are observing two exchange related PSSWs, where the first mode is a quasi-uniform mode as found from the mode profile calculation (Fig. 2(a) inset).

Indeed we find a very small surface anisotropy constant In a TRMOKE experiment, one is not able to discriminate between an odd or even mode. To determine the k-number, we compared the D value obtained from stripe domain analysis 15 for a y = 8.8% sample (perpendicular

K s = -1.3 µJ m -2 ,

84

  Samples used for this study are epilayers of (Ga 1-x ,Mn x )(As 1-y ,P y ) grown on a (001) GaAs substrate and annealed 1 hour at 250 • C. The thickness obtained from X-rays measurements is in the range 43-50nm. Samples from two sets with effective Mn concentration x ef f around 3.5 % and 5 % were studied. x ef f was determined from the saturation magnetization M s measured by SQUID magnetometry. All samples were characterized beforehand by FMR8 . Typical FMR spectra are displayed in the inset of Fig.1, where only one PSSW mode is observed, as for most samples. This allows the determination of the magnetic anisotropy constants but precludes any estimation of the spin stiffness constant. TRMOKE experiments are carried out on inplane magnetized samples with zero external magnetic field after a 60 mT initial preparation step. For outof-plane samples an in-plane field is applied to pull the magnetization into the layer plane. The laser source is a Ti:Sa laser with pulse width ≈130 fs at wavelength λ=703 nm. The sample is glued on the cold finger of a liquid He flow cryostat. To limit thermal effects, experiments are performed at low pump and probe fluence (F pump =1.1 µJ cm -2 , F probe =0.4 µJ cm -2 ), with a circularly or linearly pump beam, and a probe beam linearly polarized along the magnetization direction. At low temperature a pump induced stationary increase of temperature of 0.5 K is estimated. The magnetization dynamics is monitored through the Kerr rotation of the polarization detected by a balanced optical bridge.A typical dynamical signal is shown in Fig.1. After excitation by the pump beam at t=0, which generates a transient change of the anisotropy constants and therefore of the effective magnetic field, the magnetization is launched into precession and relaxes toward its equilibrium position in a few ns. The dynamical signal exhibits damped oscillations with two frequencies, representing optically generated spin wave modes (in some samples only one frequency is detected). For most of the samples, there is no difference between linearly and circularly polarized pump, except for two samples without P where the second spin-wave frequency was only observed in the helicity dependent signal. The TRMOKE signal is fitted by the relation:

igure 1 .

 1 Typical dynamical signal obtained in TRMOKE experiments.

  141 plane, and θ and φ are the polar and the azimuthal an-142 gles, respectively. F ij = ∂ 2 F ∂i∂j | φ=φ0 are the second deriva-143 tives of the magnetic energy with respect to the spherical 144 coordinates using FMR convention 21 for the anisotropy 145 constants K i :

1 and φ 0 = π/ 4 148

 4 otherwise. For an out-of-plane easy axis and an in-plane 149 applied magnetic field the above equations are modified 150 to include the field. Calculations of the time dependent 151 part gives the magnon dispersion relation:

  152

  Fig.2(b) shows an excellent agreement between these two experiments for the first mode frequency. Such a comparison was shown for Nickel 20 and for (Ga,Mn)As at one temperature only 18 but, to our knowledge, it is the first time that such a good agreement on a large temperature range is demonstrated for DMS. Because two PSSWs are generated, one can obtain D from the frequency spacing. However the correct value of D depends on the identification of the mode k-number (and thus symmetry) for the second excited spin wave.

Figure 2 .

 2 Figure 2. (a) Magnon dispersion relation showing the bulk modes (solid black line) and the surface modes (dashed black line) for a sample with y = 4.3 %, and L=50 nm. Horizontal lines represent the experimental frequencies (T =12 K).Squares (circles) represent the wavevectors of the bulk (surface) components of the spin waves found from the boundary conditions (Eq. 6). Inset: Mode profile (out-of-plane component δθ) across the layer thickness for the first and second excited modes. (b) Frequencies of spin wave modes from TRMOKE (symbols) as a function of temperature, compared with the k = 0 frequency (solid line) calculated from FMR anisotropy constants (inset). (c) Layer thickness dependence of the frequencies of the two spin wave modes observed in TRMOKE experiments for a piece of the sample y=4.3 % (T =12 K). The solid lines are the frequencies calculated from Eq. (5, 6) using D = 3.4 T nm 2 , and the anisotropy constants from FMR (inset of (b)). A sketch of the staircase-like sample after etching is shown below the curves.

Rev. B 83, 245211 (2011).

tained from TRMOKE and FMR are in good agreement 223 for this sample, which shows two modes with both tech-224 niques (Fig. 3). This points to a good homogeneity of

247

From TRMOKE results we obtain D for 248 (Ga,Mn)(As,P) samples with y up to ≈9 % (Fig. 3). It with the lattice cell volume, which decreases with y. In increase of J pd could be counterbalanced by a decrease 266 of the carrier concentration as suggested in Ref. [13].

267

As a conclusion, we have carried out time-resolved 268 magneto-optical experiment for (Ga,Mn)(As,P) samples.