Abdul Amir Chaaban

Eliane Bruno Lartiges
email: bruno.lartiges@get.omp.eu

El Hayek

Veronique Kazpard

Celia Plisson-Chastang

Patricia Vicendo

Celine Caillet

Eliane El Hayek

Complexes of humic acid with cationic surfactants support the supramolecular view of extracted humic matter

Keywords: Suwannee River Humic Acid, cationic surfactant, DTAC, CTAC, Supramolecular organization, Cryo-transmission electron microscopy

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Humic substances are traditionally extracted by alkali from natural organic matter (NOM) present in soils, sediments and natural waters, and are then further classified into humic acid and fulvic acid according to their solubility in water [START_REF] Thurman | Preparative isolation of aquatic humic substances[END_REF]. Whether those operationally defined substances represent a native state of NOM has been a legitimate and recurrent debate in the literature [START_REF] Burdon | Are the traditional concepts of the structures of humic substances realistic ?[END_REF][START_REF] Kelleher | Humic substances in soils: are they really chemically distinct ?[END_REF]Lehman and Kleber 2015;[START_REF] Baveye | The (Bio)Chemistry of soil humus and humic substances: Why is the "New View" still considered novel after more than 80 years ?[END_REF][START_REF] Olk | Environmental and agricultural relevance of humic fractions extracted by alkali from soils and natural waters[END_REF]. Another controversial issue, also related to the nature of NOM, concerns the organization of extracted matter. Humic substances are generally regarded as macromolecular structures resulting either from the polymerisation of phenolic/peptidic/carbohydrate compounds [START_REF] Stevenson | Humus Chemistry. Genesis, Composition, Reactions[END_REF][START_REF] Swift | Macromolecular properties of soil humic substances: fact, fiction, and opinion[END_REF][START_REF] Gerke | Concepts and misconceptions of humic substances as the stable part of soil organic matter: a review[END_REF] or from the abiotic and/or biotic degradation of original pre-existing biopolymers [START_REF] Hayes | The Search for Structure: Setting the Scene[END_REF][START_REF] Burdon | Are the traditional concepts of the structures of humic substances realistic ?[END_REF]. They have also been described as micelle-like or pseudomicelles aggregates [START_REF] Wershaw | Molecular aggregation of humic substances[END_REF][START_REF] Von Wandruszka | Humic acids: Their detergent qualities and potential uses in pollution remediation[END_REF], and more recently as supramolecular assemblies of small heterogeneous molecules essentially associated through hydrogen bonds and hydrophobic interactions [START_REF] Piccolo | The supramolecular structure of humic substances[END_REF]Simpson et al. 2002, Sutton and[START_REF] Sutton | Molecular structure in soil humic substances: the new view[END_REF].

The argumentation for a macromolecular humic structure is based on the fact that the humic matter retains its structural integrity through the extensive fractionation procedures that precede reliable molecular weight assessments [START_REF] Swift | Macromolecular properties of soil humic substances: fact, fiction, and opinion[END_REF][START_REF] Cameron | Molecular weight and shape of humic acid from sedimentation and diffusion measurements on fractionated extracts[END_REF]). On the other hand, experimental support for a supramolecular structure essentially relies on the disaggregation of humic matter detected after the addition of organic acids to disrupt the hydrogen bonds that hold the humic components within molecular aggregates [START_REF] Piccolo | Conformational arrangement of dissolved humic substances. Influence of solution composition on association of humic molecules[END_REF][START_REF] Piccolo | The supramolecular structure of humic substances[END_REF][START_REF] Simpson | Molecular structures and associations of humic substances in the terrestrial environment[END_REF][START_REF] Smejkalova | Aggregation and Disaggregation of humic supramolecular assemblies by NMR Diffusion Ordered Spectroscopy (DOSY-NMR)[END_REF]. The enhanced solubilisation of non-polar organic compounds in the presence of humic substances, measurements of specific heat capacity, reveal the presence of hydrophobic domains within the humic matter and NOM [START_REF] Carter | Binding of DDT to dissolved humic materials[END_REF][START_REF] Engebretson | The effect of molecular size on humic acid associations[END_REF][START_REF] Von Wandruszka | Humic acids: Their detergent qualities and potential uses in pollution remediation[END_REF]Chilom and Rice 2009;[START_REF] Chilom | Hierarchical self-assembling properties of natural organic matter's components[END_REF]. However, such hydrophobic domains may be accounted for either by the occurrence of micellar aggregates or by the formation of hydrophobic cores upon polymer coiling. Likewise, aggregation studies of extracted humic matter and NOM do not directly identify a macromolecular or supramolecular structure [START_REF] Kerner | Self-organization of dissolved organic matter to micelle-like microparticles in river water[END_REF]Chilom et al. 2009;[START_REF] Wells | Supramolecular architectures of natural organic matter[END_REF]. [START_REF] Chaaban | Probing the organization of fulvic acid using a cationic surfactant[END_REF] recently proposed to probe the organization of extracted humic matter through its interaction with varying concentrations of a cationic surfactant. In the case of macromolecules, the association with the oppositely charged surfactant leads to the formation of complexes that start to aggregate at a Critical Aggregation Concentration (CAC) in surfactant (Fig. 1a). Above CAC, further addition of surfactant proceeds through cooperative binding of supplementary molecules to the complexes that progressively become decorated with micelle-like structures of surfactant (Jain et al. 2004;[START_REF] Bain | Complexes of surfactants with oppositely charged polymers at surfaces and in bulk[END_REF][START_REF] Khan | Intermolecular interactions in polyelectrolyte and surfactant complexes in solution[END_REF]. Eventually, the polyelectrolyte/surfactant complexes re-solubilize. The CAC in macromolecule-surfactant suspensions generally occurs several orders of magnitude lower than the Critical Micellar Concentration (CMC), i.e. the concentration at which free surfactant molecules associate with each other to form micelles [START_REF] Khan | Intermolecular interactions in polyelectrolyte and surfactant complexes in solution[END_REF]. In the case of a supramolecular assembly of small heterogeneous molecules, the interaction with the cationic surfactant may organize new types of self-assemblies such as mixed micelles, vesicles, or lamellae (Fig. 1d), thus achieving much richer phase behaviour than what would be observed with macromolecules [START_REF] Marques | Vesicle formation and general phase behavior in the catanionic mixture SDS-DDAB-Water. The anionic-rich side[END_REF][START_REF] Marques | Vesicle formation and general phase behavior in the catanionic mixture SDS-DDAB-Water. The cationic-rich side[END_REF]. The occurrence of a given self-assembly can reasonably be predicted from the geometrical characteristics of individual components according to the molecular packing concept introduced by Israelachvili et al.

(1976) (Fig. 1b-c-e). In the case of an association between a cationic surfactant and anionic components such as humic matter, the assembly may contain both complexes and free molecules which leads to more complicated expressions for the packing parameter [START_REF] Manohar | Average packing factor approach for designing micelles, vesicles and gel phases in mixed surfactant systems[END_REF]. Applying the above approach to the Suwannee River fulvic acid (SRFA), [START_REF] Chaaban | Probing the organization of fulvic acid using a cationic surfactant[END_REF] showed that the addition of increasing amounts of Dodecyltrimethylammonium chloride (DTAC) successively leads to fulvic acid-rich unilamellar vesicles, aggregates, DTAC-rich unilamellar vesicles, and mixed micelles. Such a sequence, consistent with a supramolecular organization of fulvic matter also suggests that a major component of SRFA has an overall molecular size similar to that of DTAC. This paper extends the previous research by investigating the complexation behaviour of Suwannee River humic acid (SRHA), a reference aquatic humic matter of the International Humic Substances Society (IHSS), in the presence of a series of alkyltrimethylammonium chloride cationic surfactants (C8H17N

+ (CH3)3Cl -(OTAC), C12H25N + (CH3)3Cl -(DTAC), C16H33N + (CH3)3Cl -(CTAC))
. Turbidity, surface tension, and electrophoretic mobility measurements are combined with cryogenic transmission electron microscopy observations to examine the characteristics of complexes formed between SRHA and the homologous series of surfactants under conditions of concentration relevant to natural aquatic systems. The formation of oppositely charged humic-rich and surfactant-rich vesicles supports a supramolecular organization for SRHA.

Experimental

2.1 Materials. Suwannee River Humic acid (SRHA) was purchased from the International Humic Substances Society (IHSS) (Humic Acid Standard 2S101H). The isolation procedure of SRHA includes (i) filtration of the river water through a 0.45µm porous membrane filter, (ii) addition of concentrated HCl to lower the pH of the filtrate to 2.0, (iii) XAD-8 resin adsorption and then elution with 0.1 M NaOH followed by immediate acidification of eluate with HCl to avoid oxidation of the humic matter, (iv) centrifugation, washing with water, and freeze-drying to obtain SRHA in a solid form [START_REF] Thurman | Preparative isolation of aquatic humic substances[END_REF]. According to Thurman and Malcom (1981), the recovery factor is better than 95% for humic substances from the Suwannee River.

The cationic surfactants -Dodecyltrimethyl ammonium chloride (C12H25N + (CH3)3Cl --DTAC ≥ 99%), Octyltrimethyl ammonium chloride (C8H17N + (CH3)3Cl --OTAC ≥ 97%) and Cetyltrimethyl ammonium chloride (C16H33N + (CH3)3Cl --CTAC ≥ 98%) -were purchased from Sigma-Aldrich. The surfactants were used without further purification. Those compounds are quaternary ammonium salts: they completely dissociate in water and possess a permanent positive charge. According to literature data, DTAC is readily soluble in water at room temperature with a CMC of 21.7 mmol//L at 30°C [START_REF] Perger | Thermodynamics of micelle formation of alkyltrimethylammonium chlorides from high performance electric conductivity measurements[END_REF] and a Krafft temperature below 0°C (Lachewsky et al. 2005) -the Krafft temperature of a surfactant is the minimum temperature that ensures the flexibility of the alkyl chain and the formation of micelles. The CTAC Krafft temperature is about 18°C with a CMC of 1.3 mmol/L at 30°C [START_REF] Tan | Rapid determination of surfactant critical micelle concentration in aqueous solutions using fiber-optic refractive index sensing[END_REF]; finally, OTAC possesses a CMC of 250 mmol/L at 28°C with a Krafft temperature below 0°C [START_REF] Lindblom | Interaction between halide ions and amphiphilic organic cations in aqueous solutions studied by nuclear quadrupole relaxation[END_REF]. As the determination of the Critical Micelle Concentration is very sensitive to both the presence of impurities and the method of measurement, the CMCs of commercial surfactants were measured using a Wilhelmy plate (KRUSS K12 Processor Tensiometer). As shown in Figure S1 of Supporting Information, CMCs of 20 mmol/L, 1.1 mmol/L, and 200 mmol/L were then obtained for DTAC, CTAC, and OTAC, respectively. The presence of a minimum in the surface tension curve of DTAC indicates a slight contamination by a non-ionic impurity. In that case, the re-increase in the surface tension vs concentration curve marks the CMC, the surface-active impurities leaving the air/water interface as they incorporate the forming DTAC micelles [START_REF] Martinez-Balbuena | Applicability of the Gibbs adsorption isotherm to the analysis of experimental surface-tension data for ionic and nonionic surfactants[END_REF]. Our experimentally determined values of CMC are used throughout the paper.

Sample preparation.

As this work was initially inspired by the literature on ternary diagrams of oppositely charged surfactants, which are generally conducted in pure water, all the solutions were prepared without any supporting electrolyte. The DTAC and CTAC stock solutions were prepared in deionised water (Millipore, MilliQ, resistivity > 18.2MΩ.cm) at 10 g/L and 200 mg/L, respectively. For OTAC, stock solutions at 10, 35 and 70 g/L were necessary to minimize the aliquot volume of surfactant solution added to the SRHA suspension. The concentration of the SRHA suspension was adapted to the technique of investigation used. For turbidity, surface tension and electrophoretic mobility measurements, a SRHA suspension of 20 mg/L, i.e. ~ 12 mg C/L, was prepared by adding the appropriate mass of humic acid into 500 ml of deionised water. For Cryogenic Transmission Electron Microscopy (Cryo-TEM), a stock solution of 1g/L was used. The SRHA stock solutions were stirred overnight and then vacuum-filtered through a 0.2 µm cellulose acetate membrane (Macherey-Nagel). Fresh SRHA suspensions were prepared every day and the dissolved organic carbon of each stock suspension was checked (Shimadzu TOC-Vcsn). An average value of 12.28 ±1.53 mg C/L was obtained for the 20 mg/L suspensions. The suspensions with too much or too little DOC were discarded.

The SRHA/Surfactants complexes were prepared by adding aliquots of the concentrated surfactant solutions to a total of 10 mL (1 mL for Cryo-TEM) of SRHA suspension. The SRHA/surfactant suspension was gently shaken (3 times) and then placed in a thermostated bath at 30°C. The humic acid suspension and the surfactants stock solutions were also maintained at 30°C during the experiments. Such temperature was selected to be above the Krafft temperature of all surfactants used in the study. The experiments were carried out at pH 4.5 (equilibrium pH of the SRHA stock suspension at a 20 mg/L concentration in pure water) and at pH 6.7. In the latter case, the pH of the SRHA stock suspension was adjusted by dropwise addition of NaOH 0.1M (about 250 µL in 0.5 L stock suspension of SRHA). The pH of SRHA/surfactant mixtures was not readjusted after surfactant addition. The pH was nevertheless measured in the 10 mL vial at the end of the experiment (Fig. S3).

Characterization of SRHA/Surfactants complexes.

The turbidity of SRHA/Surfactants suspensions was measured as a function of time using a HACH LANGE 2100Q turbidimeter.

The conductivity and the pH (WTW Multiline F/SET), the surface tension (KRUSS K12

Processor Tensiometer using a Wihelmy plate), and the electrophoretic mobility of the complexes (Malvern Zetasizer Nano ZS) were measured after 1h of mixing.

The cryo-TEM observations were carried out at the Multiscale Electron Imaging (METI) platform (University of Toulouse III (Paul Sabatier)) using a JEOL 2100 Lab6 transmission electron microscope equipped with a Gatan 626 electron microscope cryo-holder. The samples were prepared by depositing 4 μl of SRHA/Surfactant suspension onto a LACEY/CARBON FILM grid previously ionized using PELCO-easiGlow 91000 glow discharging-cleaning system. After blotting for 0.2-0.8 seconds (depending on the surfactant concentration) with filter paper to remove the excess liquid (using a Leica EM GP controlled vitrification system, chamber at 30 °C and 85% relative humidity to avoid evaporation of the liquid), the grid was immediately plunged into liquid ethane close to its freezing point -183 ° C. The vitrified sample was then transferred to the microscope under liquid nitrogen to prevent any sample perturbation and the formation of ice crystals. The images were then recorded with a Gatan ultrascan1000 and Gatan ultrascan4000 camera under low dose conditions (~ 10-20 electrons per Å 2 ).

RESULTS

Turbidity measurements.

Figure 2 compares the turbidity curves obtained 1h after mixing a suspension of 20 mg/L SRHA with various amounts of surfactants. After 1h, the turbidity measurements, if not stable, vary less than 10% over the following hour (see Figure S2 in Supporting Information). The result of the interaction between SRHA and the homologous series of cationic surfactants is represented as a function of the surfactant concentration normalized to their respective critical micelle concentration (CMC) to account differences in water solubility of the surfactants [START_REF] Cases | Thermodynamic model of ionic and non-ionic surfactant adsorption abstraction on heterogeneous surfaces[END_REF]. Drastic differences are observed in the turbidity curves according to the alkyl chain length of the surfactant: OTAC addition to SRHA leads to a minor increase in turbidity even at high surfactant concentration. In contrast, the addition of DTAC to SRHA suspensions is characterized by a strong initial increase in turbidity that broadly peaks around 0.28 CMCDTAC (5.6 mmol/L), followed by a gradual decrease to reach low turbidity values at high surfactant concentrations. The addition of DTAC to pure water leads to a negligible increase in turbidity (fig. 2a). The SRHA/CTAC turbidity curve exhibits two very noisy peaks at roughly 0.09 and 0.18 CMCCTAC (i.e. 0.1 and 0.19 mmol/L); the noisy pattern mainly results from the presence of clearly visible but sparse aggregates in suspension. At pH 6.7 (Figure 2b), the interaction between SRHA and DTAC leads to two prominent turbidity peaks around 0.15 and 0.45 CMCDTAC (i.e. 3 and 9 mmol/L). In the low concentration range, the base of the first turbidity peak exhibits smaller turbidity values than at pH 4.5. This can be explained by the higher negative charge of SRHA at pH 6.7 [START_REF] Lead | Determination of diffusion coefficients of humic substances by fluorescence correlation spectroscopy: role of solution conditions[END_REF][START_REF] Hosse | Determination of electrophoretic mobilities and hydrodynamic radii of three humic substances as a function of pH and ionic strength[END_REF], which requires higher DTAC concentrations to start forming aggregates.

The formation of two turbidity peaks for the associations SRHA/CTAC at pH 4.5 and SRHA/DTAC at pH 6.7, as well as the limited interaction between SRHA and OTAC at natural pH, reveal complex interactions between the humic acid and the cationic surfactants. Those interactions certainly involve the charge neutralization of SRHA anionic groups by the positive head groups of surfactants but also various hydrophobic effects driven by the alkyl chain length of surfactants. All the surfactants used in this study are 1:1 salts. However, as illustrated in Figure 2 from additions of various NaCl concentrations to the humic suspension, the change in solution ionic strength brought by the surfactant does not play any significant role in determining the turbidity of the suspension in the range of surfactant concentrations investigated. Moreover, the pH of the SRHA/surfactants suspensions remains close to the initial pH (Figure S3 in Supporting Information).

Electrophoretic mobility measurements.

The electrophoretic mobility of SRHA/Surfactant complexes exhibits two different patterns as a function of surfactant concentration (Figure 3).

For SRHA/DTAC complexes at both pH 4.5 and 6.7, the electrophoretic mobility increases rapidly at low surfactant additions, and then more slowly at higher DTAC concentrations.

Interestingly, the point of zero mobility (PZM) reached at about 3.2 mmol/L and 5.5 mmol/L for pHs 4.5 and 6.7, respectively, cannot be related to the turbidity peaks: indeed, the PZM occurs before the turbidity peak at pH 4.5, and is situated in the valley between the two turbidity peaks at pH 6.7 (Figure 3b).

In contrast, for OTAC and CTAC, the electrophoretic mobility becomes more negative at low surfactant concentration, reaches a minimum at about 0.01 CMC for both surfactants, and then increases to reach positive values from 0.06 CMCCTAC in the case of CTAC, or decreases to stabilize at negative values in the case of OTAC. Unlike DTAC, the point of zero mobility of SRHA/CTAC complexes can be related to the first turbidity peak. It is worth noting that the initial electrophoretic mobility of SRHA colloids could not be measured with the Nanosizer and that all the data points reported in the plots identify SRHA/surfactant complexes. Previous capillary electrophoresis measurements carried out by [START_REF] Hosse | Determination of electrophoretic mobilities and hydrodynamic radii of three humic substances as a function of pH and ionic strength[END_REF] led to electrophoretic mobilities of SRHA close to -3.8 and to -4.9 10 8 m 2 V -1 s -1 at pH 4.5 and 6.7, respectively. This suggests that the electrophoretic mobility of OTAC/SRHA and CTAC/SRHA complexes comprises an initial maximum not resolved with our measuring technique.

Surface tension measurements.

Figure 4 shows the surface tension of SRHA/surfactant suspensions as a function of surfactant concentration or surfactant concentration normalized to the corresponding CMC in a semi-log plot. The surface tension of pure surfactant solutions is also shown for comparison. Two main regimes can be distinguished: at very low surfactant concentration, i.e. [surfactant]/CMC < 0.1, the surface tension of SRHA/surfactant system decreases more rapidly with concentration than that of pure surfactant. Such synergistic effect is attributed to the strong electrostatic interaction between the humic anionic groups and the cationic headgroups of surfactant that leads to humic/surfactant complexes of enhanced hydrophobicity [START_REF] Gamboa | Association of cationic surfactants to humic acid. Effect on the surface activity[END_REF]. Following this decrease, the surface tension curve stabilizes or even exhibits a well-defined breakpoint in the case of CTAC. Upon further addition of surfactant, the surface tension of SRHA/surfactant system remains fairly constant, before decreasing again at higher surfactant concentration. Such additional drop in surface tension suggests the presence of non-associated surfactant in addition to the SRHA/surfactant complexes. It can also be noted that, in that concentration range ([surfactant]/CMC ≥ 0.1), the longer the alkyl chain, the higher the surface activity of SRHA/surfactant complexes. At high surfactant concentration, the surface tension of pure surfactant solutions becomes lower than that of SRHA/surfactant systems.

3.4

Cryo-TEM observations. Typical cryo-TEM images at pH 3.1 and at pH 6.7 for the SRHA/DTAC system, and at pH 3.1 for the SRHA/CTAC system, are shown in Figures 5,6 and 7, respectively. At the concentration of the DTAC stock solution (38 mmol/L), i.e. slightly above the CMC, micelles could not be clearly seen in the micrographs (Fig. 5a). In contrast, humic structures in the 3-5 nm range and small linear aggregates of those structures, 30 to 140 nm in length, are found for a SRHA concentration of 1 g/L and an equilibrium pH equal to 3.1 (Fig. 5b). Such observations are consistent with previous fluorescence correlation spectroscopy measurements that suggest a partial aggregation of SRHA at low pH [START_REF] Hosse | Determination of electrophoretic mobilities and hydrodynamic radii of three humic substances as a function of pH and ionic strength[END_REF][START_REF] Duval | Humic substances are soft and permeable: evidence from their electrophoretic mobilities[END_REF]. However, the linear aspect of aggregates likely results from the intense shearing that occurs during the blotting stage while draining the excess liquid deposited on the grid [START_REF] Lee | Shear free and blotless cryo-TEM imaging: a new method for probing early evolution of nanostructures[END_REF].

Addition of a 7.6 10 -3 mmol/L DTAC concentration to the 1 g/L SRHA suspension leads to the formation of slightly irregular globules, about 25-50 nm in diameter, marginally aggregated, and with no internal structure (Fig. 5c2). Such globules may represent the early stages of interaction between DTAC molecules and SRHA and might be attributed to mixed micelles.

The same amount of surfactant diluted in deionised water leads to featureless micrographs (Fig. 5c1). Further additions of cationic surfactant (37.86 10 -3 mmol/L and 188.5 10 -3 mmol/L) reveal a dense packing of unilamellar vesicles often opened and sometimes reminiscent of facetted self-assemblies (Fig. 5d2 and5e) [START_REF] Vautrin | Balance of pH and ionic strength influences on chain melting transition in catanionic vesicles[END_REF]). The vesicular architecture is identified from the higher electron density at the edges of structures. The diameter of the vesicles ranges from 15 to 45 nm whereas the thickness of the vesicles walls is in the 1.2-2.8 nm range. Spongy network structures are also found to coexist with the vesicles (e.g. Fig. 5d1).

According to [START_REF] Zheng | Cryo-TEM imaging the flow-induced transition from vesicles to threadlike micelles[END_REF], those structures correspond to entangled threadlike micelles that form from some severely sheared vesicles during the blotting stage. When the DTAC concentration is increased to 375.2 10 -3 mmol/L (Fig. 5f), irregular monodisperse structures, slightly inhomogeneous and about 16 nm in diameter, are observed. Those structures are partially aggregated and might correspond to almost neutral SRHA/DTAC complexes. Only one DTAC concentration (33 mmol/L) was investigated on the DTAC-rich side of the phase diagram at natural pH (Fig. 5g). Considering the high dilution of the SRHA suspension necessary to achieve such DTAC concentration, only a few well-formed vesicles with diameters ranging10 to 25 nm were identified (Fig. 5g).

At pH 6.7, an addition of 7.6 10 -3 mmol/L of cationic surfactant to the SRHA suspension also promotes the formation of small irregular vesicular structures more complete than at pH 3.1; the size of those vesicles ranges from 10 to 50 nm (Fig. 6a1). Increase of the DTAC concentration to 226 10 -3 mmol/L leads to well-defined spherical vesicles -16 to 30 nm in diameter -that form multilamellar structures with ample spacing between entrapped and outer vesicles (Fig. 6b1). The wall thickness of vesicles remained in the 1.2-2.8 nm range. In both cases, some deformed vesicles are observed, thus indicating that the vesicles may either merge or separate from larger assemblies. Branched spongy networks, with budding vesicles at the higher investigated DTAC concentration, can also be found (Fig. 6a2-b2). The surfactant concentration added to reach the DTAC-rich side of the phase diagram also implies an important dilution of the SRHA suspension. At a DTAC concentration of 33 mmol/L, large (~ 100 nm in diameter) slightly granulated globules are observed to coexist with infrequent vesicles. Such globules represent aggregates of neutralized SRHA/DTAC complexes, whereas an increase of DTAC concentration to 34.9 mmol/L lead to partially aggregated vesicles, about 20 nm in size (Fig. 6d). Therefore, at both investigated pH, a sequence of structures including humic-rich vesicles, aggregated complexes, and DTAC-rich vesicles, can be observed with increasing DTAC concentration.

As illustrated in Figure 7, when a cationic surfactant with a longer alkyl chain is added to the SRHA suspension at natural pH, i.e. CTAC (C16), aggregated globules are the prevailing molecular structures. The size of globules increases with the CTAC concentration, from 12-20 nm in size at 3.12 x10 -5 mmol/L to 20-32 nm at 18.7 x 10 -5 mmol/L, reaches 30-60 nm at 93.6

x 10 -5 mmol/L, and then decreasing to around 40 nm at 311 x 10 -5 mmol/L. The aggregates of globules are ramified at low CTAC concentration, becoming more compact as the surfactant concentration is increased, and appear to redisperse at the highest investigated concentration of CTAC.

The difference in behaviour between SRHA/DTAC and SRHA/CTAC complexes can be related to the length of the surfactant alkyl chain. In the case of CTAC, the temperature of the suspension is only slightly above the Krafft temperature of the surfactant. Upon interaction between CTAC and SRHA, the neutralization of the negatively charged functional groups of the humic structure by the positive ammonium headgroups of the surfactant increases the hydrophobicity of the resulting complex. Near the chain melting temperature of the free surfactant, the additional hydrophobic interactions between the alkyl chain and the humic complex might stiffen the CTAC chain, thus facilitating the formation of a precipitate. The presence of facetted vesicles in the SRHA/DTAC system certainly reflects the coexistence of ordered alkyl chains in the molecular structures generated with the C12 surfactant [START_REF] Vautrin | Balance of pH and ionic strength influences on chain melting transition in catanionic vesicles[END_REF]).

DISCUSSION

Complexation of SRHA with alkyltrimethylammonium cationic surfactants. Previous

studies on the interactions between humic substances and cationic surfactants were interested in two main aspects: (i) the fate of the surfactant molecule introduced in the natural environment either as an organic contaminant through wastewater discharge [START_REF] Koopal | Binding of ionic surfactants to purified humic acid[END_REF][START_REF] Ishiguro | Binding of cationic surfactants to humic substances[END_REF]Hakim and Kobayashi 2019) or as a detergent for the remediation of aquifers and contaminated soils [START_REF] Harwell | Surfactants for ground water remediation[END_REF][START_REF] Mao | Use of surfactants for the remediation of contaminated soils: a review[END_REF]; (ii) the chemical structure of the humic substances, the surfactant being used as a probe of amphiphilic properties of humic matter or as a titrant of the total amount of carboxyl functional groups [START_REF] Tombacz | An X-ray diffraction study of alkylammonium humate complexes[END_REF][START_REF] De Nobili | Alternative method for carboxyl group determination in humic substances[END_REF][START_REF] Yee | Evaluation of amphiphilic properties of fulvic acid and humic acid by alkylpyridinium binding study[END_REF][START_REF] Shang | Investigation of humate-cetyltrimethylammonium complexes by small-angle X-ray scattering[END_REF][START_REF] Chaaban | Probing the organization of fulvic acid using a cationic surfactant[END_REF]. Overall, the reports concur that the binding of the cationic surfactant to the humic substance initially proceeds through strong electrostatic interactions between the carboxylic groups of humic matter and the positively charged head groups of the surfactant. This leads to the formation of a precipitate when the charge neutralization of the humic matter is achieved [START_REF] De Nobili | Investigation of the interactions between humic substances and a cationic detergent (Cetiltrimethylammonium Bromide)[END_REF][START_REF] Koopal | Binding of ionic surfactants to purified humic acid[END_REF]Hakim and Kobayashi 2019). Beyond the point of charge neutralization, supplementary binding of cationic surfactant to the humic/surfactant complexes occurs through cooperative hydrophobic effects. A charge reversal and a restabilization of humic/surfactant complexes can also be observed in this region [START_REF] De Nobili | Alternative method for carboxyl group determination in humic substances[END_REF][START_REF] Ishiguro | Binding of cationic surfactants to humic substances[END_REF]Hakim and Kobayashi 2019). Highly surface-active complexes are formed at very low surfactant concentrations, which suggests that structural rearrangements of the humic substances may occur upon the addition of cationic surfactant [START_REF] Koopal | Binding of ionic surfactants to purified humic acid[END_REF]Gambo and Olea 2006).

Our findings are consistent with the literature, yet they provide further precision about some key features of the complexation of humic matter with cationic surfactants. The contrasting evolution of turbidity-surfactant concentration curves, normalized to the respective solubility of surfactants, suggest that the alkyl chain length plays a significant role in the association between the cationic surfactant and SRHA for the low surfactant concentrations. Indeed, the addition of OTAC to the SRHA suspension only produces a minor increase of turbidity in comparison with that induced by CTAC, even though the electrophoretic mobility curves of resulting complexes indicate similar neutralization behaviours at low surfactant concentrations (Fig. 23). A likely explanation is that the order-disorder transition of surfactant alkyl chain occurs at increased temperature after interaction of surfactant with the hydrophobic moieties of humic matter. In the case of DTAC, the presence of vesicles implies that the alkyl chain maintains a high degree of conformational disorder upon association with SRHA. Likewise, the shorter alkyl chain of OTAC should also remain disordered. However, the C16-alkyl chain of CTAC certainly becomes ordered in the SRHA/CTAC complex as suggested by (i) the early formation of a compact precipitate (Fig. 7), and (ii) the surface tension of SRHA/CTAC suspensions similar to that of pure water in the low surfactant concentration regime unlike the surface tension of SRHA/DTAC and SRHA/OTAC suspensions (Fig. 4). It is indeed peculiar that the complexes formed with the longer-chain surfactant are the less surface active at low surfactant concentrations, thus suggesting that the alkyl chain does not stick out of water under those conditions.

The minimum exhibited in the electrophoretic mobility curves of SRHA/CTAC and SRHA/DTAC complexes is also an intriguing feature. A similar trend has been reported for the Suwannee River fulvic acid/DTAC system. In that case, the minimum is interpreted as the result of a rearrangement of fulvic acid upon neutralization with the cationic surfactant [START_REF] Chaaban | Probing the organization of fulvic acid using a cationic surfactant[END_REF]. Indeed, the initial neutralization of external sites of humic matter by the cationic surfactant enhances the hydrophobicity of the complex, that then undergoes a structural rearrangement turning inside-out some previously unattainable negatively charged sites [START_REF] Sieliechi | Changes in humic acid conformation during coagulation with ferric chloride: Implications for drinking water treatment[END_REF]. Our results are consistent with such interpretation, although, in the case of DTAC, the minimum in electrophoretic mobility certainly occurs at surfactant concentrations slightly lower than those investigated in this study.

Only partial neutralization of SRHA is reached with the addition of OTAC, whereas CTAC and DTAC promote a complete charge compensation of the humic matter. Using the titration data of SRHA reported by [START_REF] Ritchie | Proton-binding study of standard and reference fulvic acids, humic acids, and natural organic matter[END_REF], it can be verified that the amount of CTAC added at the point of charge compensation, i.e. 0.077 mmol/L, is close to the amount of negative charges of SRHA at pH 4.5, i.e. 0.066 meq/L. This indicates that most (around 85%) CTAC molecules are bound to SRHA. Similar calculations indicate the opposite situation for DTAC where a minor amount of surfactant molecules seems to be associated with the humic matter at the point of charge neutralization at either pH 4.5 or 6.7. Such results are in line with the binding isotherms of dodecyl-pyridinium chloride and cetyl-pyridinium chloride to various humic acids reported by [START_REF] Ishiguro | Binding of cationic surfactants to humic substances[END_REF] and [START_REF] Yee | Evaluation of amphiphilic properties of fulvic acid and humic acid by alkylpyridinium binding study[END_REF].

As the neutralization of SRHA is carried out with monovalent cationic surfactants, further binding of surfactant above the point of charge compensation is necessarily driven by cooperative hydrophobic interactions either between interacting alkyl tails of the surfactant or between alkyl chains and hydrophobic moities of the humic matter. This change in surfactant binding mode and the resulting charge reversal of complexes, have been well documented in previous studies [START_REF] Ishiguro | Binding of cationic surfactants to humic substances[END_REF]Hakim and Kobayashi 2019). However, the break in the surface tension curve that defines the CAC is also known to signal the onset of cooperative association of surfactant to the complexes in polyelectrolyte/surfactant systems (Jain et al. 2004;[START_REF] Bain | Complexes of surfactants with oppositely charged polymers at surfaces and in bulk[END_REF]). In our case, CACs of 3.6 mmol/L, 0.4 mmol/L, and 0.022 mmol/L are obtained at pH 4.5 for OTAC, DTAC, and CTAC, respectively; a CAC of 0.7 mmol/L is found for DTAC at pH 6.7 (Fig. 4). The CAC values determined for DTAC and CTAC are therefore much lower than the surfactant concentrations necessary for charge neutralization of SRHA.

Interestingly, The CAC concentrations are in the vicinity of minima of electrophoretic mobility curves. Furthermore, even though the degree of conformational disorder of surfactant alkyl chain may vary in the SRHA/surfactant complexes, the logarithm of CACs can be related to the surfactant chain length just as log(CMC) of cationic surfactants (Fig. S4 in Supporting information). Such dependence implies that micellization is initiated in the process of SRHA/surfactant interaction and well before charge neutralization. Therefore, after noncooperative binding of cationic surfactant to the external carboxyl groups of SRHA, the pooling of alkyl chains of bound surfactant probably triggers a rearrangement of humic matter to make additional negatively charged sites available for binding. A rapid calculation gives a value of 3.2 kJ/mol for the free energy of binding per CH2 group which is consistent with previous studies on humic substances/surfactant binding [START_REF] Yee | Evaluation of amphiphilic properties of fulvic acid and humic acid by alkylpyridinium binding study[END_REF].

Clues to a supramolecular organization of extracted humic matter.

The use of a surfactant as a probe of a given molecular structure, polymeric or self-assembly, primarily relies on the aggregation behaviour upon addition of increasing concentrations of surfactant. In the SRHA/n-alkyl trimethylammonium chloride system, both pH and alkyl chain length of the surfactant, i.e. charge of humic matter and strength of hydrophobic interactions, strongly influence the aggregation. If we only consider DTAC, the turbidity pattern is reminiscent of a polyelectrolyte/surfactant-like interaction at pH 4.5 [START_REF] Bakshi | Surfactant polymer interactions between strongly interacting cationic surfactants and anionic polyelectrolytes from conductivity and turbidity measurements[END_REF], whereas the observation of two distinct turbidity peaks at pH 6.7 is in line with the phase behaviour of a mixture of oppositely charged surfactants [START_REF] Marques | Vesicle formation and general phase behavior in the catanionic mixture SDS-DDAB-Water. The anionic-rich side[END_REF][START_REF] Marques | Vesicle formation and general phase behavior in the catanionic mixture SDS-DDAB-Water. The cationic-rich side[END_REF], i.e. implies a supramolecular structure of humic matter. Obviously, both molecular structures cannot simultaneously apply to describe the organization of SRHA.

Another key feature in identifying the organization of extracted humic matter is the nature of turbidity that results from the SRHA/surfactant interaction. At acid pH, direct imaging using cryo-TEM evidences a sequence of structures comprising globules tentatively attributed to mixed micelles, small unilamellar humic-rich vesicles, presumably aggregated structures, and then DTAC-rich vesicles. At pH 6.7, an equivalent sequence of humic-rich vesicles, SRHA/DTAC aggregated structures, and DTAC-rich vesicles can be observed. Such sequences bear striking similarities with the phase behaviour of aqueous mixtures of anionic and cationic surfactants [START_REF] Herrington | Phase behavior of aqueous mixtures of Dodecyltrimethylammonium bromide (DTAB) and sodium dodecyl sulfate (SDS)[END_REF][START_REF] Marques | Vesicle formation and general phase behavior in the catanionic mixture SDS-DDAB-Water. The anionic-rich side[END_REF][START_REF] Marques | Vesicle formation and general phase behavior in the catanionic mixture SDS-DDAB-Water. The cationic-rich side[END_REF]. Indeed, mixing of oppositely charged surfactants typically leads to mixed micelles, anionic-rich vesicles, equimolar aggregates, cationic-rich vesicles, and mixed micelles, as a function of mixing ratio. Therefore, the sequence of molecular structures resulting from the interaction of SRHA with increasing concentrations of DTAC supports a supramolecular organization of extracted humic matter.

Such conclusion, reached with 1h mixing time, is unlikely to be reversed by using longer mixing times. The SRHA/DTAC turbidity curve at pH 4.5 may then reflect the close vicinity of the two regions of vesicles stability in the phase diagram. Moreover, an increased protonation of SRHA clearly facilitates the formation of a high number of vesicles, thus suggesting a facilitated restructuring.

Synthetic amphiphilic block copolymers have also been shown to assemble into micelles, lamellae, and a variety of complex vesicular structures including flowerlike vesicles or stacked bowls [START_REF] Liu | Electrostatic interaction mediates the formation of vesicular structures from coassembly of PS-b-PAA with quantum dots[END_REF][START_REF] Wang | Tandem self-assembly of block copolymer: from vesicles to stacked bowls[END_REF]. However, in that case, the transition process from one molecular structure to another generally involves shear-induced transformations and the obtained assemblies are kinetically trapped. In our study, all the vesicles were spontaneously formed which is also a supplementary feature of aqueous mixtures of anionic and cationic surfactants [START_REF] Kaler | Phase behavior and structures of mixtures of anionic and cationic surfactants[END_REF][START_REF] Marques | Vesicle formation and general phase behavior in the catanionic mixture SDS-DDAB-Water. The cationic-rich side[END_REF]). In addition, in the case of amphiphilic block copolymers the thickness of vesicle wall is about 10 to 30 nm, whereas less than 2.8 nm thick walls, i.e. less than the expected thickness of a bilayer formed from 12-carbon tail surfactants, was found for the vesicles formed in the SRHA/DTAC system.

The thinness of vesicles walls suggests that the building blocks of humic matter are of a size similar to that of surfactant molecules. Such result is consistent with previous size estimates of humic matter [START_REF] Lead | Determination of diffusion coefficients of humic substances by fluorescence correlation spectroscopy: role of solution conditions[END_REF]. Not knowing the amount of DTAC bound to SRHA, the geometric packing constraints associated with the formation of vesicles cannot be used to better define the size of those humic constituents. Nevertheless, estimates for the mean crosssectional surface area per SRHA/DTAC complex aSRHA/DTAC at the air/water interface can be inferred from the surface tension measurements using the Gibbs adsorption isotherm at the maximum value of dg/dln(C) [START_REF] Eastoe | Micelles of asymmetric chain catanionic surfactants[END_REF][START_REF] Gamboa | Association of cationic surfactants to humic acid. Effect on the surface activity[END_REF]: in which Gmax is the surface excess of SRHA/DTAC complexes, Na Avogadro's number, R the ideal gas constant, T the absolute temperature, g the surface tension, and C the molar concentration of surfactant. The least-squares fits to the pre-CMC dg/dln(C) data lead to aSRHA/DTAC values of 191 ± 32 Å 2 and 69 ± 25 Å 2 at pH 4.5 and at pH 6.7, respectively (The confidence intervals for the aSRHA/DTAC values are obtained using a Student t-test with a significance level of 5%). Such values compare favourably with those obtained by [START_REF] Gamboa | Association of cationic surfactants to humic acid. Effect on the surface activity[END_REF] in the Aldrich humic acid/alkyltrimethyl ammonium Bromide system. In mixtures of oppositely charged surfactants, previous studies have shown that the mean area per molecule of complex is slightly less than the sum of individual components cross-sectional surface areas [START_REF] Rodakiewicz-Nowak | Surface characteristics of some anionic, cationic, and anioniccationic surfactants[END_REF][START_REF] Eastoe | Micelles of asymmetric chain catanionic surfactants[END_REF]. Using a value of 40 Å 2 for aDTAC [START_REF] Weiner | Interfacial properties of antimicrobial long-chain quaternary ammonium salts I. Soluble films at the air water interface[END_REF], aSRHA should then be about 30 Å 2 at pH 7 and 150 Å 2 at natural pH. Those cross-sectional surface areas fall in the 30 to 190 Å 2 range reported in the literature for humic acids (Chen and Schniter 1978;[START_REF] Kawahigashi | Surface-active properties of particle size fractions in two humic acids[END_REF][START_REF] Terashima | Influence of pH on the surface activity of humic acid: micelle-like aggregate formation and interfacial adsorption[END_REF][START_REF] Gamboa | Association of cationic surfactants to humic acid. Effect on the surface activity[END_REF][START_REF] Chaaban | Probing the organization of fulvic acid using a cationic surfactant[END_REF]). Nevertheless, it should be noted that in the pre-CMC linear region, SRHA/DRAC complexes remain slightly charged. As a result, because of electrostatic repulsion, close packing may not be reached and the mean surface area occupied by the complexes at the air/water interface is certainly slightly overestimated. In any case, such molecular area is much smaller than those typically found for polyelectrolytes [START_REF] Caminati | Polystyrene sulfonate adsorption at water-graphon and water-air interfaces[END_REF]. The results presented in this paper shed new light on the interaction mechanisms involved between humic acid and cationic surfactants under environmentally relevant conditions. In particular, we show that the hydrophobic interactions occurring between either alkyl chains of bound surfactants or surfactant alkyl chains and hydrophobic moieties of humic matter, induce a drastic rearrangement of the humic structure at low surfactant concentration. Those hydrophobic interactions may stiffen the alkyl tails of surfactants, i.e. decrease the orderdisorder transition temperature of alkyl chain, which in the case of C16 chain length promotes the formation of precipitates. In contrast, shorter alkyl chains remain flexible and trigger the formation of small nanoscale vesicles in the case of DTAC. Cooperative hydrophobic binding of surfactant molecules also concurs to the neutralization and the charge reversal of humic/surfactant complexes except for the short C8-alkyl chains. The formation of vesicular structures and the high surface activities of complexes formed between humic acid and DTAC, offer renewed perspectives for the use of cationic surfactants in the remediation of contaminated ecosystems. Indeed, because of enhanced adsorption onto negatively charged soil particles and acute toxicity for microorganisms, cationic surfactants are generally not used for the treatment of oil-contaminated environments [START_REF] Mao | Use of surfactants for the remediation of contaminated soils: a review[END_REF]. The possibility of obtaining negatively charged nanosize vesicles from complexes of humic matter and cationic surfactant would certainly improve both transport properties and environmental compatibility of cationic surfactants.

CONCLUSIONS

The phase behaviour of SRHA/DTAC complexes and the spontaneous formation of nanoscale unilamellar vesicles are key features of mixtures of oppositely charged surfactants. Such results support the supramolecular view of humic matter and are completely in line with the amphiphilic nature of humic substances [START_REF] Chen | Surface-Tension of Aqueous-Solutions of Soil Humic Substances[END_REF][START_REF] Guetzloff | Does Humic-Acid Form a Micelle[END_REF][START_REF] Von Wandruszka | Humic acids: Their detergent qualities and potential uses in pollution remediation[END_REF][START_REF] Klavins | Humic substances as surfactants[END_REF]. As humic substances have been shown to be mixtures of heterogeneous compounds [START_REF] Simpson | Molecular structures and associations of humic substances in the terrestrial environment[END_REF][START_REF] Sutton | Molecular structure in soil humic substances: the new view[END_REF], it is possible that only part of the humic building blocks participate to the formation of vesicular structures, the remaining compounds not being detected by our investigation techniques.

Moreover, a self-assembled humic matter suggests that part of NOM might be supramolecular but does not imply that all NOM is supramolecular. Further investigations should assess how the extraction process of humic substances could affect their organization structure.

Declaration of interest:

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

104, 5263-5271.

Acknowledgements: AAC was awarded a PhD fellowship from Azm & Saade Association (Lebanon). AAC, BSL and CPC are grateful to Stephanie Balor and Vanessa Soldan (METI Platform -University of Toulouse III) for precious help.

Data Availability Statement:

The data that support this study are available on request from the corresponding author.


T Huong Dang

D Quang To

E Erina

T L Hoai Nguyen

V I Safarov

H Jaffrès

H.-J Drouhin

Theory of the Anomalous Tunnel Hall Effect at Ferromagnet-Semiconductor Junctions

Keywords: numbers: 72.25.Mk, 75.70.Tj, 75.76.+j

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I -INTRODUCTION

Spinorbitronics is a science that uses the spin degree of freedom together with the spin-orbit interactions (SOI) to generate spin-currents without the need of a ferromagnetic material. Those spin-currents become essential in view to control the magnetization state of a nanomagnet, via the spin-Hall effect of heavy material either semiconductor [1][2][3][4][5] or metals [6,[START_REF] Liu | [END_REF][13][14][15]. The interplay between particle spin and orbital motion is currently at the basis of a new family of effects like the Anomalous Tunnel Hall effect described by the generation of a charge current transverse to a tunneling spin-current [7][8][9][10] or the spin-galvanic effects [START_REF] Ganichev | Handbook of spin-transport & magnetism[END_REF]. SOI at an interface with broken inversion symmetry may lead to the observation of Rashba-split states [16,17] which may be used to convert a flow of spin-current into a transverse charge current by inverse Edelstein effect (IEE) [18,19]. Recent magnetoresistance (MR) measurements with an unidirectional character (UMR) have been evidenced in metallic [20] and semiconductor bilayers [21] as well as with topological insulator (TI) [22]. It was ascribed to the asymmetric scattering of electrons by magnons absorption-emission processes at ferromagnet-TI interfaces. It clearly reveals a novel symmetry in the field of MR explained by a certain carrier scattering asymmetry incorporated in the Boltzmann transport equation developed at the second order. An anisotropy of electronic diffusion on the Fermi surface along the current direction leads to a supplementary non-linear contribution of M R proportional to the current.

In this article, as an extension to our previous work [9], we study unconventional quantum effects resulting in a giant transport asymmetry of carriers in semiconductor heterostructures, interfaces, tunnel barriers or quantum wells. Those are composed of ferromagnets and spin-orbit split electrodes made of semiconductors, e. g. III-V compounds, with magnetizations of opposite direction (AP). The symmetry of the structure allows a difference of transmission upon positive or negative incidence (that we will note ξ = ±k ) with respect to the reflection plane defined by the magnetization and the surface normal. However, this quantum process departs from the effect of a beam deviation by the action of the Lorentz force [23] and, unlike spin-filtering effects [24,25], the scattering asymmetry requires the simultaneous action of both inplane and out-of plane spin-orbit fields in the case of Dresselhaus interaction in the condution band (CB). In a second part of the paper, we emphasize on the perturbation calculation techniques needed to understand this phenomena as well as the case of intrinsic SOI in the valence band (VB).

II -CASE OF THE CONDUCTION BAND OF SEMICONDUCTORS OF T d SYMMETRY GROUP.

We first consider the Dresselhaus interactions [26],

H D = ( γχ) • σ, H D = -γξ 2 k -iγξk 2 iγξk 2 γξ 2 k (1)
in the conduction band of a semiconductor junction made of two magnetic materials in the AP state. We refer the structure to the x, y, z cubic axes (unit vectors x, y, z) and assume that electron transport occurs along the z axis, whereas the magnetization lies along x. One then introduces (0, ξ, k) the electron wavevector; σ the Pauli operator, and χ = 0, ξk 2 , -ξ 2 k the D'yakonov-Perel' (DP) internal field responsible for the spin splitting [26,27]. One introduces the tensor γ = (γ i δ ij ) which characterizes the DP-field strength, with γ x = γ y = γ, γ z = γ, and δ ij the Kronecker symbol. We consider the two cases γ = γ and γ = 0 on switching off the ξ 2 perturbation. We study the transmission asymmetry when the wavevector component along y is changed from ξ to -ξ. Electrons are injected from the first conduction band of material I to the left ( = 1) into the first conduction band of material II to the right ( = -1). Then, the relevant 2 × 2 Hamiltonians respectively write:

H I,II = γ c k 2 + ξ 2 I + wm • σ + H D = γ c (k 2 + ξ 2 ) -γξ 2 k -iγξk 2 + w iγξk 2 + w γ c (k 2 + ξ 2 ) + γξ 2 k ( 2
)
where I is the identity matrix, γ c the conduction effective mass, m is the unit magnetization, 2w the exchange strength.

Transmission from quantum boundary conditions (quantum wavefunction matching).

The two energies in the electrodes are given by E

1 = γ c k 2 1 + ξ 2 -w and E 2 = γ c (k 2 2 + ξ 2 ) + w,
where k 1 (k 2pure imaginary-) is the z-component of the wavevector in the lower (upper) subband. The two eigenvectors write:

u ,1 (ξ, k 1 ) = 1 -2 iµk 2 1 , -(1 -2 µξk 1 ) / √ 2, (3) u ,2 (ξ, k 2 ) = 1 -2 iµk 2 2 , (1 + 2 µξk 2 ) / √ 2, (4)
where µ = γξ/(2w) and µ = γξ/(2w). Note that the form of the eigenvectors does not foresee any tunneling transmission asymmetry in usual tunneling models [9] based on the density of states [28,29]. The asymmetry arises from a full-quantum treatment discussed in terms of chirality. Because k is conserved, we are dealing with states with the same longitudinal kinetic energy E along z and a total kinetic energy E = E + γ c ξ 2 . The boundary conditions are the continuity of the wavefunction and of the current wave JΨ I,II = (1/ ) (∂ H I,II /∂k)Ψ I,II because H I,II contains no more than quadratic k terms [30][31][32][33][34].

The transmission of a pure up-spin incident electron into a pure down-spin state is only possible under oblique incidence via SOI which introduces off-diagonal matrix elements. The spin-orbit field is also responsible for a discontinuity of the spin-current between incident (inc) and transmitted (trans.) waves. Moreover, a non-vanishing diagonal part of SOI is necessary to obtain a non-zero asymmetry although the z component of the DP field along z does not depend on the sign of k [9]. Then, from now on, we take γ = γ. The wavevector k 1 in the lower subband has to be real so that we can define K = k 1 > 0. We introduce the parameter λ > 0 with k 2 = iλK, the reduced longitudinal energy η = E/w = 1 -λ 2 / 1 + λ 2 , and the incidence parameter t = ξ/K. One finally obtains the transmission T (t, η) and its average T upon ±ξ incidence t and asymmetry A:

T (|t|, η) = γ 2 γ 3 c wt 2 (1 + η) 2 4η 2 (1 -η) + t 2 (1 + η)(2η -1) 2 , (5)
A (t, η) = 4tη 1 -η 2 (2η -1) 4η 2 (1 -η) + t 2 (1 + η) (2η -1) 2 . ( 6
)
where

A = [T (t, η) -T (-t, η)] / [T (t, η) + T (-t, η)],
emphasizing the increase of T (t, η) with t and γ. The analytical asymmetry A is plotted in Fig. 2a for several values of t (full lines), where the symbols refer to the 2 × 2 numerical calculations, showing an excellent agreement. It is a remarkable result that A (t, η) does not depend either on the material parameters or on the sign of γ, thus conferring to A a universal character. Reversing the magnetization (changing w into -w) makes transport occur in the k 2 channel leading to a change of A (t, η) into -A (t, η). Our convention is that A is positive, at small energy η (or averaged over the energy band) when (m, ξ, k) forms a direct frame and negative otherwise. Another striking feature is that an arbitrarily small perturbation is able to produce a 100% transport asymmetry i.e., a total quenching of transmission in the CB. Fig. 1b display the 2dimensional map of the electron transmission at a given total energy in the reciprocal space calculated using both a 2 × 2 effective Hamiltonian and a full 14 × 14 band k • p treatment involving odd-potential coupling terms P and ∆ [35][36][37]. These calculations are based on the multiband transfer matrix technique detailed in Refs. [30,31].

Transverse Surface Currents.

The transmitted current summed,

J [t, η] = J ξ [Ψ II (z)] + J -ξ [Ψ II (z)],
originates from incident waves of equal amplitude and opposite k . To the lowest order in γ, we find

J y,z [t, η] = 4 (γ c w) 1/2 (1 + η) 1/2 T (t, η) [A (t, η) t y+ z] (7)
A non-zero A gives rise to a transverse carrier momentum and then to a tunneling surface current (per unit length) j y = J y × mf p ( mf p is the electron mean free path), of the form J C = m × J S,z (C for current and S for spin-current), leading to a potentially large Anomalous Tunnel Hall Effect (AT HE). The ratio of the surface transverse to the longitudinal current density, j y [t, η] /J z [t, η] = tA (t, η) mf p , leads to the AT HE length, or AT HE angles [9], in the spirit of the work dealing with IEE phenomenon [18,19].

III -CASE OF THE CB: PERTURBATION CALCULATIONS INVOLVING SOI.

Using advanced perturbation procedures, one may give a general expression for the change of the transmission amplitude δt σσ of a propagative spin-↑ wave from the left transmitted into a propagative spin-↓ wave to the right, after having experienced a SOI potential V σσ (spin-flip) in a confined region of space. The calculation is based on Ref. [41] and we will demonstrate in fine that,

δΨ 0σ in (z) = a 0 G σσ 0 (z, z )V σσ (z )Ψ 0σ in (z )dz , (8)
and from the expression of G 0 , that t ↑↓ may be written:

δt σσ = -im * 2 k a 0 Ψ 0σ out (z )V σσ (z )Ψ 0σ in (z )dz , (9)
where (in) and (out) refer respectively to the unperturbed incoming wave at left and outgoing wave at right [41]. G σσ 0 is the (spin-diagonal) Green's function (GF) to consider and we are searching for. Such perturbative scattering approach has hardly been employed to investigate the role of the evanescent waves in transport like investigated here. The method is particularly suitable for the case of non-degenerate orbital systems but however could be applied, in a future work, to the case of the valence band (VB). We consider the Green's function (GF) G 0 of an hamiltonian system Ĥ0 =

E -Ĥ0 G σσ 0 (z, z ) = δ (z -z ) , (10)
Green's function without orbital degeneracy.

The strategy to find the GF is then i) to find two different ground states Ψ 0,↑↓ L , Ψ 0,↑↓ R of the homogenous Schrödinger equation E -Ĥ0 Ψ = 0 (L for left and R for right whith characteristic wavevector k I and k II ), ii) to find the relevant linear combinations of Ψ 0 L and Ψ 0 R that make y 1 and y 2 solution of the equation E -Ĥ0 y = 0 finite at z = -∞ [y 1 (z)] and z = +∞ [y 2 (z)] depending on the use of the retarded or advanced quantities, and iii) to define the correct GF by making use of:

G σσ 0 (z, z ) = y σ 1 (z)y σ 2 (z ) W (y1,y2) -∞ < z < z < +∞ y σ 1 (z )y σ 2 (z) W (y1,y2) -∞ < z < z < +∞ , (11)
where

W (y 1 , y 2 ) = 2 2m * y 1 (z ) ∂y2(z ) ∂z -∂y1(z )
∂z y 2 (z ) is the Wronskian. The homogenous Schrödinger equation, E -Ĥ0 Ψ = 0, admits the solutions:

Ψ 0,σ L = e ik σ I z< + r σ L e -ik σ I z< + t σ L e ik σ II z> |σ > Ψ 0,σ R = e -ik σ II z> + r σ R e ik σ II z> + t σ R e ik σ I z< |σ > (12)
where z < and z > stand for z < 0 and z > 0. If we chose y 1 = Ψ 0,σ R and y 2 = Ψ 0,σ L , Eq. 10 admits a particular solution:

G σσ 0 (z, z ) = Ψ 0,σ R (z )Ψ 0,σ L (z)Θ(z -z ) + Ψ 0,σ R (z)Ψ 0,σ L (z )Θ(z -z) W (Ψ 0 R , Ψ 0 L ) , (13)
On the assumption of a same effective mass, the Wronskian

W = 2 2m * 4ik σ I k σ II k σ II +k σ I
is a constant (∂W/∂z = 0) and we recover the retarded GF introduced in Ref. [42] according to:

G0(z<, z > ) = 2m * 2 tL 2ik σ I e -ik σ I z e ik σ II z , G0(z>, z < ) = 2m * 2 tR 2ik σ II e -ik σ I z e ik σ II z , G0(z<, z < ) = 2m * 2 1 2ik σ 1 e ik σ I |z-z | + r σ L e -ik σ I (z+z ) , G0(z>, z > ) = 2m * 2 1 2ik σ II e ik σ II |z-z | + r σ R e -ik σ II (z+z ) (14)
that we will use henceforth. Case of the exchange-step in the CB: perturbation calculations.

We revisit here the issue and results of section II (exchangestep) with Ĥ0 , eigenvectors and eigenvalues given in section II. We recall that the current is along ẑ and magnetizations along x. The incident wavevector is k = (0, ξ, k).

Reflection, transmission and perturbating SOI potential

We consider the electron transmission within an energy window in the exchange step, -w < E < w, where the transmission asymmetry takes place, so that k ↑ I and k ↓ II are real whereas k ↓ I and and k ↑ II are pure imaginary. In the right contact, the spin ↑ state admits a pure propagative character whereas the spin ↓ state is purely evanescent. It is then quite convenient to define

k ↑ I = k ↓ II = k 1 and k ↓ I = k ↑ II = ik 2 .
The two solutions of the homogeneous Schrödinger equation, Ψ 0,↑ L , and Ψ 0,↓ R are given by Eq. [START_REF] Liu | [END_REF] with reflection, r L↑ = r R↓ = k1-ik2 k1+ik2 , and transmission, t

L↑ = t R↓ = 2k1 k1+ik2 (t R↑ = t L↓ = 2k2 k2-ik1
), amplitudes found via the matching conditions at z = 0. This allows possible transmission from propagative to evanescent states (t R↓ and t L↑ ) and vice-versa (t L↑ and t L↓ ).

The SOI, Ĥσσ D , is then introduced as a perturbation potential according to:

ĤD = - ξ 2 σz 2 γk + k + γ(z) - ξσy 2 γk 2 + k + 2 γ = iξ 2 σz 2 γ -→ ∂ ∂z - ← - ∂ ∂z γ - ξσy 2 γ --→ ∂ 2 ∂z 2 + ← - ∂ 2 ∂z 2 γ , ( 15
)
with γ = γ(z). ĤD acquires a pure non-diagonal form like:

V ↑↓ = ↑ | ĤD | ↓ , = ↑ |    iξ 2 σz 2 γ -→ ∂ ∂z - ← - ∂ ∂z γ - ξσy 2   γ --→ ∂ 2 ∂z 2 + ← - ∂ 2 ∂z 2 γ      | ↓ , =   iξ 2 2 γ -→ ∂ ∂z - iξ 2 γ --→ ∂ 2 ∂z 2   -   iξ 2 2 ← - ∂ ∂z γ + iξ 2 ← - ∂ 2 ∂z 2 γ   , (16)
and

V ↓↑ =   iξ 2 2 γ -→ ∂ ∂z + iξ 2 γ --→ ∂ 2 ∂z 2   -   iξ 2 2 ← - ∂ ∂z γ - iξ 2 ← - ∂ 2 ∂z 2 γ   (17)
From Eq. [8] and

W = i 2 k1
m * t R↑ , the correction to the amplitude of transmission is:

δt ↑↓ = - m * i 2 k1 +∞ -∞ Ψ 0↑ R (z ){- iξ 2 2 γ ∂Ψ ↓0 L (z ) ∂z + iξ 2 γ ∂ 2 Ψ ↓0 L (z ) ∂z 2 - - iξ 2 2 γ ∂Ψ 0↑ R (z ) ∂z + iξ 2 γ ∂ 2 Ψ 0↑ R (z ) ∂z 2 }Ψ ↓0 L (z )dz . (18)
We are now going to calculate the properties of the carrier transmission A for the different SOI configurations: at left, at right, and SOI in both contacts for an incoming left electron.

SOI at left for electrons incoming from left.

We first note that the zeroth-order transmission coefficient, t ↑↓ 0 = 0 is zero without spin-mixing interactions. Then, from Eq. 18, the transmission amplitude, t ↑↓ L , with SOI at left is:

δt ↑↓ = - m * i 2 k1 0 -∞ Ψ 0↑ R (z ){- iξ 2 2 γ ∂Ψ ↓0 L (z ) ∂z + iξ 2 γ ∂ 2 Ψ ↓0 L (z ) ∂z 2 - - iξ 2 2 γ ∂Ψ 0↑ R (z ) ∂z + iξ 2 γ ∂ 2 Ψ 0↑ R (z ) ∂z 2 }Ψ ↓0 L (z )dz . (19)
By considering k 1 = K (incoming propagative wavevector) and k 2 = λK (imaginary transmitted wavevector), one then obtains:

t ↑↓ L = - 1 2w γξK 2 (1 + iλ) 2 ξ K (3λ 2 -1) + 2λ λ 2 -1 (20)
SOI at right for electrons incoming from left.

The transmission changes from the previous case by changing the integral from

0 -∞ into +∞ 0 giving t ↑↓ R = t ↑↓ L .
Figure 3. Asymmetry coefficient A vs. reduced energy calculated in the VB of GaMnAs/GaAs/GaMnAs 3 nm thick tunnel junction with an exchange strength of 0.3 eV and k = 0.05 nm -1 . The barrier height is -0.55 eV.

SOI at left and right side for electrons incoming from left.

With SOI in the whole space (left and right), we find from Eq. [18] the transmission amplitude t ↑↓ :

t ↑↓ = - γξK 2 w ξ K (3λ 2 -1) + 2λ λ 2 -1 (1 + iλ) 2 (21)
From T (t, η) = |t ↑↓ | 2 , we recover transmission T (t, η) and asymmetry A derived from the application of the pure matching conditions (Eqs. [6]). This proves the power of this perturbation methodology involving mixed propagative and evanescent electronic states.

Case of a tunnel junction in the CB: perturbation calculations.

We focus now on the case of a tunnel junction, of thickness a, made of two ferromagnetic contacts (in the AP state) and separated by a thin semiconductor belonging to the T dsymmetry. The contacts are free of SOI. The incident energy in the CB lies in the range of the exchange step, -w < E < w, with a single incident propagative wave of a pure spin ↑ character. However, the electrons may scatter, now, at the two different interfaces of the junction and this makes the problem generally different from the previous treatment. One then considers a particular value for the barrier height equal to the exchange potential, V 0 = |w| , so as to prevent any back and forth scattering. The calculation of the most general shape of the GF is given in Ref. [43]. To the first order of perturbation, the transmission, δt ↓↑ , now equals:

δt ↑↓ = m * i 2 k1 a 0 Ψ 0↑ L (z ){- iγξ 2 2 ∂Ψ ↓0 R (z ) ∂z + iγξ 2 ∂ 2 Ψ ↓0 R (z ) ∂z 2 - - iγξ 2 2 ∂Ψ 0↑ L (z ) ∂z + iγξ 2 ∂ 2 Ψ 0↑ L (z ) ∂z 2 }Ψ ↓0 R (z )dz , (22)
The coefficient of the wave functions Ψ 0↓ R , and Ψ 0↑ L , without SOI, are found from the relevant matching condition in a similar way to the case of the exchange step to give: -a) , for z a (23) Detailed calculations give out the transmission coefficient we are searching for:

Ψ 0↑ L = t L↑ e -k2z = 2k 1 k 1 + ik 2 e -k2z , for z > 0 Ψ 0↓ R = t L↑ e k2(z-a) = 2k 1 k 1 + ik 2 e k2(z
δt ↑↓ = e -k2a γ c 2γξk 2 k 1 a (k 1 + ik 2 ) 2 (ξ + k 2 ). ( 24
)
where we remind that a is the barrier thickness. Without SOI perturbation, the transmission coefficient is also zero in the situation of pure spin states, and consequently, T ↑↓ = δt ↑↓ 2 . If one defines again the incidence parameter t = tan θ = ξ/K for and η = 1-λ 2 1+λ 2 = E w the reduced incident kinetic energy, we find the asymmetry of transmission for the tunnel barrier like:

A = |ξ + k 2 | 2 -|-ξ + k 2 | 2 |ξ -k 2 | 2 + |-ξ -k 2 | 2 = 2 (1 -η)(1 + η)t t 2 (1 + η) + (1 -η) . (25)
One obtains a perfect agreement between the perturbative scattering method and our multiband calculations for |t ↑↓ | 2 and A (Fig. 2b). The transmission coefficient for an incoming propagative spin-↑ electron into an outgoing propagative spin-↓ electron is non-zero after SOI is branched on. The transmission vs. incident kinetic energy and incident angle is different from the case of a simple exchange-step. The maximum of transmission depends also on the incidence angle or t parameter. The k • p theory gives a maximum of asymmetry when the evanescent wavevector equals in magnitude the parallel incoming wavevectors in the CB.

IV -CASE OF INTRINSIC CORE SOI IN THE VALENCE BAND: CHIRALITY

We now turn on the case of the VB of a tunnel junction composed of two p-type ferromagnets separated by a thin tunnel barrier (3 nm in the present case). The barrier height have been chosen so as to match with the exchange strength (0.3 eV). The structure is free of any odd-potential k-terms ( H D =0) and only includes core SOI (p-orbitals). Results are displayed in Fig. 1c for the transmission maps and Fig. 2c for the corresponding asymmetry resulting from a multiband k •p treatment. In the 2D-map calculation procedures obtained for a hole kinetic energy of = 0.23 eV, we have checked (Fig. 1c) that the whole numerical approaches (6, 14, 18 and 30-bands models) provide about exact similar data. The transmission scales within the range 15 -45 × 10 -3 with (P = 0 and ∆ = 0. Those results demonstrate that the absence of inversion symmetry (T d ) is not mandatory to observe an asymmetry A. Fig. 2c displays the asymmetry A vs. hole energy E for k = 0.05 nm -1 . The energy range covers the spin-↑, ↓ heavy (HH) and light (LH)-hole subbands whereas the respective spin ↑ and ↓ split-off bands are not represented here. We refer to points (1) to (4) marked by vertical arrows in the following discussion. Here, the energy of the HH ↑ (HH ↓) corresponds to 0.15 eV [-0.15 eV] as indicated by point (1) [( 4)], the energy zero being taken at the top of the VB of the non-magnetic material. Correspondingly, one observes a large negative transmission asymmetry (-60%) in this energy range for predominant majority spin ↑ injection as far as HH ↓ does not contribute to the current. At more negative energy [E < -0.15 eV: point (4)], a sign change of A occurs at the onset of HH ↓ to reach about +20%. From Ref. [9] A changes sign two times at characteristic energy points corresponding to a sign change of the injected particle spin. Also, we have performed similar calculation for a simple contact [9]. Remarkably, A, although smaller, keeps the same trends as for the tunnel junction, except for a change of sign. Without tunnel junctions, A abruptly disappears as soon as SO ↓ contributes to tunneling i.e., when evanescent states disappear. In the case of tunnel junction, A, although small, subsists in this energy range and this should be related to the evanescent character of the wavefunction in the barrier.

V -CONCLUSIONS

We have presented theoretical evidence for large interfacial tunneling asymmetry of carriers (scattering), electrons or holes, vs. their incidence in exchange-split semiconductor structures. The effect of transmission asymmetry occurs in the CB via the SOI Dresselhaus interactions whereas intrinsic SOI of the p-type VB is sufficient. This transmission asymmetry have been revealed by taking into account boundary wavefunctions matching, advanced multiband k • p calculations as well as scattering perturbation theory. After averaging over incoming states, a large surface current parallel to the barrier is results in an Anomalous Tunnel Hall effect.

Figure 1 .

 1 Figure 1. Scheme of the transmission process at an exchange-SOI step with AP magnetizations. k1 are the propagative wavevector whereas ±ξ are the parallel ones. Carriers with +ξ in-plane wavevector component are more easily transmitted than those carrying -ξ. (Top right inset): Energy profile of the exchange step; E is the longitudinal kinetic energy and 2w is the exchange splitting. b) Asymmetry in the CB: 2D-map of the transmission coefficient T in a 2 × 2 and 14 × 14 k • p band model. γ = -24 eV Å 3 . c) Asymmetry in the VB: 2D-map of the transmission coefficient T of a tunnel junction within a 6 × 6, 14 × 14, 18 × 18 and 30 × 30 k • p band models. In both CB and VB, the exchange strength is 0.3 eV and the total kinetic energy E = 0.23 eV. For VB, the barrier thickness is 3 nm. The respective 14-band and 30-band parameters are taken from Refs. [[35]] and [[38]]

  (z) in a homogenous potential U 1 for z < 0, and U 2 for z > 0 satisfying:

Figure 2 .

 2 Figure 2. (a) Universal asymmetry coefficient A vs. reduced energy η = E/w obtained for an exchange-step with different values of t = ξ/K [t = 0.01 (black; circles), t = 0.5 (blue; squares), t = 1 (red; stars), and t = 2 (purple; triangles) by 2-band analytical (full line) and numerical (symbols) calculations. (b) Transmission coefficients and asymmetry coefficient A vs. reduced energy η = E/w obtained for a 3 nm tunnel junction (TJ) with different values of t = ξ/K [t = 1 (black), t = 2 (red), by perturbative scattering (pert.: full lines) method and numerical k • p calculations (Calc.: symbols).

THD acknowledges Idex Paris-Saclay and Triangle de la Physique for funding.