Par Madame

Camille Paliniewicz Née

Monsieur Le Professeur

Olivier Guérin

Directeur De Thèse

Monsieur Le Docteur

Guillaume Sacco
email: guillaume.sacco@univ-cotedazur.fr

Madame Le Docteur

PhD Aurélie Mouton

Madame Le Docteur

Emeline Michel

Camille Paliniewicz

Alexandre Derreumaux Msc

MSc Roxane Fabre

MD, MSc Éric Ettore

MD François-Xavier Couchoud

MD Michel Emeline

PhD Robert Philipe

MSc Sacco Guillaume

THESE D'EXERCICE EN MEDECINE

Keywords: 2, Figure count: 3, Appendix count: 1, neurocognitive disorder, cognitive decline, elderly, Alzheimer disease, vascular disease

Reference count: 9 1.6±2.1 respectively for AD, VE and Mi). An early diagnosis at a minor stage of neurocognitive disorder, with higher initial MMSE score and an isolated vascular encephalopathy seemed associated with a slower cognitive decline.

Conclusions: etiological diagnosis and early entrance in care pathway seemed closely related with cognitive evolution. This study seems to identify a possible slower decline in a subgroup of patient with isolated microvascular encephalopathy. Further researches were necessary to confirm this finding.

A l'équipe des Sources ; A Caroline, merci pour ton humanité, pour tout ce chocolat et ces thés, pour ton écoute et pour ta patience. Tu m'as beaucoup appris et je te remercie. A Williams, merci pour toutes les connaissances que tu as partagées avec moi, qu'elles soient médicales ou littéraires, et merci pour la confiance que tu m'as toujours accordée. A Sylvie, pour ton énergie et ta bonne humeur.

A Anne-Marie pour tout ce que tu m'apporté et tout ce que tu m'as appris. A votre soutien sans faille dans les moments difficiles. A toute l'équipe soignante, pour vos conseils et votre aide précieuse.

A l'équipe du Centre Mémoire, merci pour votre accueil, votre bonne humeur constante et votre écoute. Merci pour tout ce que vous m'avez tous appris.

Au Docteur Anne Charpentier, merci de m'avoir fait découvrir et aimer la gériatrie. Je ne serais pas ici sans vous.

A mes parents ; je n'ai pas les mots pour vous remercier pour votre soutien, pour vos encouragements, pour la foi que vous avez en moi. J'espère être un jour à la hauteur de votre fierté et de votre amour. Merci. Je suis si fière d'être votre fille.

A mon frère ; je t'aime Clément, que la force soit toujours avec toi. A ma soeur, mon microbe. Je suis si fière de toi. 사랑해요 A Caroline, ma moitié de cerveau, ma colloc préférée. La liste des choses pour lesquelles je dois te remercier est plus longue que cette thèse. Je te souhaite tout le bonheur du monde, même si parfois, Caroline, tu crains.

A Thomas ; à notre amitié avec son début tellement improbable. A tous ces moments qu'on a partagés et qu'on partagera ensemble. Souviens-toi : science sans conscience n'est que ruine de l'âme. A FX, pour nos voyages, nos fous rires et pour les moments tristes aussi, merci de les avoir passés avec moi. Results: data from 7033 patients, 61.8% of female, mean age 73.6 ± 13 years old were available in the period. The mean at first consult was 22 for the mini mental state examination (MMSE) and 13/18 for the Frontal Assessment Battery. 29.2% of our patients were diagnosed with a major neurocognitive disorder (NCD). 791 patients were included in the subgroup analyses. Among them, the mean cognitive decline was around 1.5 points loss for the MMSE. There seemed to be a steeper annual cognitive decline when Alzheimer's disease was involved in the pathological process (-2.0±2.0 VS -0.7±1.8 VS -

Introduction

Major neurocognitive disorders (NCD) are the seventh leading cause of death and one of the major cause of disability and dependency among older adults worldwide [START_REF]Dementia is the major cause of functional dependence in the elderly: 3-year follow-up data from a population-based study -PubMed[END_REF]. In France, the reported number of people diagnosed with major NCD is estimated around 1.2 million with Alzheimer's disease (AD) responsible for two third of the cases [START_REF]Les chiffres clés[END_REF]. Major NCD are both underdiagnosed and untreated when symptomatic treatment is possible [START_REF]Journey through the diagnosis of dementia[END_REF]. Today, the diagnosis remains often lately established when preventive action and treatment plan are less effective. The delay leads to a massive financial and psychological impact on the patients and their family [START_REF] Helmer | Dementia in subjects aged 75 years or over within the PAQUID cohort: prevalence and burden by severity[END_REF]. Both are often eager to have clear answers as to how the cognitive and functional decline will evolve. Answering this question is made difficult by the heterogeneity of cognitive declines [START_REF] Nichols | Estimation of the global prevalence of dementia in 2019 and forecasted prevalence in 2050: an analysis for the Global Burden of Disease Study 2019[END_REF] found in even one country. To expand our knowledge of the disease we rely on epidemiological tools, such as La Banque Nationale Alzheimer, an organization dedicated to systematic data collection of patients attending French memory Clinic. However, those tools tend to be underexploited and/or in need of improvement [START_REF] Boise | Dementia assessment in primary care: results from a study in three managed care systems[END_REF]. A few initiatives have been launched, among them a prospective study in the city of Lyon (France) [START_REF] Dauphinot | A multicenter cohort study to investigate the factors associated with functional autonomy change in patients with cognitive complaint or neurocognitive disorders: the MEMORA study protocol[END_REF] and a proposal to pool together the data collected by a number of French Memory Clinics with a view to identify and study different cognitive profiles.

Our aim was to describe the characteristic of the patients attending Nice Memory Clinic with a focus on the evolution of disorder related impairment in patients with AD, vascular encephalopathy (VE) and mixed (Mx) NCD. We tried to identify factors that influences the speed of cognitive decline.

Materials and methods

Design and setting

Participants

The population studied was the patients attending the Nice Memory Clinic. Its main activity was the diagnosis and management of patient with NCD. The patients were usually referred to the Memory Clinic by a general practitioner or a medical specialist. The clinic also had a research and resources purpose.

Physicians and neuropsychologists could be solicited to offer their expertise on the diagnosis or management of patients by medical specialists, who might refer them for a specific inquiry. As a result, some patients only consulted once or came for specialized care that might not include a cognitive assessment, which had an impact on the number of entries in our database and should be taken into account.

Inclusion criteria was: at least one cognitive assessment using the Cobtest software coupled with a least one first level diagnosis (i.e. subjective memory complaint (SMC), minor or major NCD according to the DSM-V diagnosis criteria ( 8)) as entered into The Dossier Mobile Alzheimer (local record of the national Alzheimer database). Patients were informed of data collection and could object if they wished to (noninclusion criteria).

Data collected

The CoBTest software was a computerized tool allowing its user to enter patient information's, clinical notes, to access a number of tests and evaluation scales in order to assess cognitive and functional states in real time. The consultations consisted of either a medical examination and/or standardized neuropsychological assessment to evaluate cognitive performance and functional autonomy. The interval between each consult as well as the number of consultations per year were determined by the physician in charge of the patient and thus might vary from one patient to another.

The data collected concerned sociodemographic characteristics of patients (age, gender, years of schooling or last diploma), neuropsychological assessment (including initial and last known Mini Mental Examination score (MMSE) [START_REF] Derouesne | Mini-Mental State Examination:a useful method for the evaluation of the cognitive status of patients by the clinician. Consensual French version[END_REF] and Frontal Assessment Battery score (FAB) [START_REF] Dubois | The FAB: a Frontal Assessment Battery at bedside[END_REF]) and the most recent known functional (SMC, minor or major NCD) and etiological diagnosis (AD, VE excluding stoke aftereffect, Mx) according to the DSM-V criteria. The total number of consultations and the follow-up duration were also reported for each patient.

We excluded patients according to their number of evaluation (<2) or aberrant data as reported in the Flow chart (Figure 1).

Statistical analyses

Data were reported on Excel ® . Categorical variables were described using absolute frequencies and percentages. Quantitative variables were described using means and standard deviations as appropriate.

Initially, the characteristics of the patients and the scores at the first consultation were described across the whole population and based on diagnosis. Categorical data were compared using Chi2 and quantitative variable were compared using ANOVA test.

To estimate the speed of cognitive decline, we excluded patients who had less than two MMSE test scores reported into our database. Progression score was calculated by subtracting the test score results between first and last tests for each patient. Univariate analyses were performed according to these two groups. Categorical data were compared using Chi2 test as appropriate. Quantitative data were compared using the Student t test as appropriate. To test the MMSE evolution between the 3 diagnosis groups, the progression MMSE score (first MMSE-last MMSE) and the number of points lost per year ((first MMSE-last MMSE)/ time between first and last MMSE) were compared using ANOVA test and Tukey's Post-hoc. We defined the "slow cognitive decline" group as the population whose decline was slower than the mean MMSE loss and "fast" group as those whose rate was faster compared to our mean loss. Aberrant data were excluded based on boxplots, which lead to the exclusion of data inferior or superior to quartiles Q1 or Q3 more or less the standard deviation between the two quartiles.

Each statistic test used an alpha risk of 5%. All statistical analyses were performed with R 4.0.1 software.

Results

There were 7033 patients entered into our database. 61.8% of those patients were women and the mean age of the population at first consult was 73.6 (±13.0) years old. The mean MMSE score first entered into our database was 22.3 (± 6.4) and the mean FAB score was 13.0 (± 3.7). The majority of our patients (29.7%) had a high level of education (beyond baccalaureate). Our most reported diagnosis was major NCD (29.2%), followed by minor NCD (27.0%) and SMC (10.8%) (Table 1).

Patients had an average of 1.8±1.4 consultations spaced 0.7±1.5 years apart (eight month). If we excluded patients who came only once (representing 4324 patients), the mean follow-up was 1.8±1.9 years.

Out of the 791 patients studied for the estimation of cognitive speed decline, 46.3% were diagnosed with AD, 28.5% with VE and 25.3% with Mx NCD. That population was largely female (54.6%), and had an age at first consult of 77.0±7.9 years old. 34.2% of our patients had a low level of education and 25.4% stopped attending school after 8 years. The vast majority (72.7%) had a major NCD and a quarter a minor one (25.4%). The complete characteristics of the population was reported in Table 1.

The mean difference between first and last MMSE was 3.6±4.8 points for the whole population with a yearly decline of 1.5±2.1 points. For AD patients, the mean difference was 4.5±5.0 points with a yearly decline of 2.0±2.0 points, whereas VE patients had a mean difference of 1.5±3.6 points with a yearly loss of 0.7 (±1.8) and Mx patients had a mean difference of 4.1±4.8 points and a yearly decline of 1.6±2.1 points. There was a significant difference for both parameters between AD, VE and Mx NCD (Table 1 and Figure 2).

When searching for factors influencing the speed of cognitive decline, we find a difference in initial MMSE, functional status and etiological diagnoses. The patients with the slower cognitive decline had a higher initial MMSE and were predominantly diagnosed with VE NCD. Both groups (slow and fast cognitive decline) had a majority of major NCD but minor NCD was overrepresented in the slower decline group. Age, gender or level of education had no influence on the speed of cognitive decline in our study (Table 2).

Discussion

Understanding and managing major neurocognitive disorder is a crucial 21 th century's public health stake, and studying patients' characteristics is one way of expanding our knowledge of the disorder.

Our aim was to describe the characteristics of the population attending the Nice Memory Clinic. We also attempted to estimate a speed of cognitive decline through the number of points lost between first and last MMSE tests, with a particular focus on the factors influencing cognitive speed decline.

There were 7033 patients included in our database, mostly women, which was expected given that female gender is a risk factor for major neurocognitive disorder (11) (12) [START_REF] Matthews | Racial and ethnic estimates of Alzheimer's disease and related dementias in the United States (2015-2060) in adults aged ≥65 years[END_REF]. It is generally agreed upon that the increased prevalence of major neurocognitive disorder is due to the greater life span of women relative to men. The mean age at first consultation, which will be approximated as the mean age of diagnosis, was 73.6 years old, in range or slightly younger than what is usually found in the literature (14) (15) [START_REF] Matthews | A two-decade comparison of prevalence of dementia in individuals aged 65 years and older from three geographical areas of England: results of the Cognitive Function and Ageing Study I and II[END_REF]. The minor discrepancy in age might be due to the specific referral of patients with early onset NCD to our research memory center. It might also be explained by the public health campaigns raising awareness about major NCD and encouraging people to seek a professional assessment when in doubt about memory loss, which might lead younger people to attend to a Memory Clinic (17). Most of the patients registered into our database had a high level of education, as defined by years of schooling superior to 13 years, which is discordant with the globally accepted hypothesis that a lower level of education is a risk factor for the development of major neurocognitive disorder [START_REF] Contador | Impact of literacy and years of education on the diagnosis of dementia: A population-based study[END_REF] [START_REF] Zhou | Prevalence of dementia in rural China: impact of age, gender and education[END_REF] [START_REF] Hoenig | Level of education mitigates the impact of tau pathology on neuronal function[END_REF].

The discrepancy might also be linked to our Memory Clinic being a research center. People with a higher socioeconomic level might have an easier access to specialized medical resources. They might also be more likely to be sensitive to the early memory loss symptoms. In accordance to the literature, a significant portion of the population studied had a low level of education (as defined as 5 years of schooling). A quarter of our patients at diagnosis had a disorder impacting at least one of the items of the Lawton Instrumental Activities of Daily Living scale simplified to 4 items in our consultations (management of treatment, use of public transport or personal vehicle, use of a phone, management of taxes and budget). What this finding, which is concordant with the literature [START_REF] Giezendanner | Early diagnosis and management of dementia in general practicehow do Swiss GPs meet the challenge?[END_REF], underlines is how difficult diagnosing major NCD in early stage is, especially since the first symptoms are non-specific and easily attributed to another disorder or mistaken as the physiological signs of ageing by the general public (22) (23) [START_REF] Leung | Early detection of dementia: The knowledge and attitudes of primary care physicians in Hong Kong[END_REF]. However, an early diagnosis is all the more important since symptomatic treatment and preventive action are more effective in the first stages of the disease. An emphasis on education of primary care physicians and the general population is needed to remedy this situation. The MMSE and the FAB were the two tests chosen to estimate the cognitive function of our patients because on one hand, they were the most reported tests we had, and on the other hand because they both have a significant clinical meaning, in global cognitive state for the MMSE and executive functions for the FAB.

The first MMSE registered in the database was 22.3/30, marginally superior to what is described in the literature (25) (26) [START_REF] Tariq | Comparison of the Saint Louis University mental status examination and the mini-mental state examination for detecting dementia and mild neurocognitive disorder--a pilot study[END_REF]. The studied population also had a somewhat higher Frontal Assessment Battery than what is usually found in the literature (28) (29) (30) (31) even when the population seemed fairly similar. It might be explained by the outpatient nature of our consultations which demands a preserved level of executive and cognitive functions (e. g. to plan how and when to attend the consult). The population attending the Nice Memory Clinic is thus moderately comparable to the population described in the literature studying major NCD. The main discrepancies might be explained by the specific research purpose of our clinic and its implications in terms of patient referral.

We also aimed to study major NCD impairment evolution in our patients and estimate a speed of cognitive decline. We decided to focus on three groups of patients attending our center (AD, VE and Mx NCD) [START_REF]Prévention de la maladie d'Alzheimer et des maladies apparentées[END_REF], and represented 931 patients out of the 7035 into our database. We then excluded people with aberrant data in point loss which left us with a population of 791 patients. In comparison of the three groups, Mx NCD patients (mostly associating AD and VE disorder) were older and had both lower MMSE and FAB. The age difference between groups is coherent with what has been reported in the literature (3). There was a higher number of women in all groups, expected since female gender is a known risk factor, except for the VE patients who were predominantly male. The difference might be explained by the higher prevalence of cardiovascular risk factors in men [START_REF] Colafella | Sex-specific differences in hypertension and associated cardiovascular disease[END_REF]. A majority of patients in all group had a low level of education (as defined as 5 years of schooling), which was also expected given the data reported in the current literature [START_REF] Clouston | Education and Cognitive Decline: An Integrative Analysis of Global Longitudinal Studies of Cognitive Aging[END_REF]. In both AD and Mx NCD group, the patients had a functional diagnosis of major NCD. In VE patients, the functional diagnosis was minor MCD. This is coherent with our finding which showed a slower cognitive decline in VE NCD group.

There was a significant difference in total and yearly number of points lost between AD, VE and Mx patients. All of this seems to indicate a steeper decline in cognitive function when AD is present compared to VE disorder alone.

While the yearly MMSE loss of points is fairly consistent with what can be found in the literature [START_REF] Morris | The Consortium to Establish a Registry for Alzheimer's Disease (CERAD). Part I. Clinical and neuropsychological assessment of Alzheimer's disease[END_REF] (36) in relation with AD, the loss in VE patients largely varies depending on the studies. Some studies reported a lower point loss [START_REF] Smits | Trajectories of cognitive decline in different types of dementia[END_REF], some a number concordant with ours [START_REF] Incidence | Manifestations, and Predictors of Worsening White Matter on Serial Cranial Magnetic Resonance Imaging in the Elderly | Stroke[END_REF] and some a higher number of MMSE score point loss [START_REF] Agüero-Torres | Prognostic factors in very old demented adults: a seven-year follow-up from a population-based survey in Stockholm[END_REF]. Concerning the point loss in patients with Mx NCD, most of the literature showed a somewhat lower number compared to ours (39) (40) (41) [START_REF] Brodaty | Patients in Australian Memory Clinics: baseline characteristics and predictors of decline at six months[END_REF] although one study found a greater point loss [START_REF] Corey-Bloom | Clinical features distinguishing large cohorts with possible AD, probable AD, and mixed dementia[END_REF].

The statistically significant difference between some groups our study showed is not in concordance with what is known about the disorders. The literature shows that there is no difference in cognitive decline between AD and vascular major NCD (36) [START_REF] Ballard | Cognitive decline in patients with Alzheimer's disease, vascular dementia and senile dementia of Lewy body type[END_REF]. There are also a number of studies reporting a more severe cognitive decline in Mx NCD compared with AD alone [START_REF] Toledo | Contribution of cerebrovascular disease in autopsy confirmed neurodegenerative disease cases in the National Alzheimer's Coordinating Centre[END_REF] [START_REF] Jellinger | The enigma of mixed dementia[END_REF] [START_REF]Pathological correlates of late-onset dementia in a multicentre, community-based population in England and Wales[END_REF]. One explanation might be the exclusion of stroke after-effect NCD with a sole focus on vascular encephalopathy-related neurocognitive disorder. The cognitive and functional decline may vary depending on the localization or the severity of the stroke. This leads to a different evolution in cognitive abilities compared to vascular encephalopathy lesions alone [START_REF] Gorelick | Vascular contributions to cognitive impairment and dementia: a statement for healthcare professionals from the american heart association/american stroke association[END_REF].

Since the neurodegenerative part of the disorder seems to be the one accelerating the cognitive decline, it is one more argument in favor of furthering the etiological diagnosis. This might be done by screening for the amyloid biomarker with either a lumbar puncture, an amyloid PET scan or soon a blood sample.

Three factors seemed to influence cognitive speed decline in our population: the initial MMSE, the functional status and etiological diagnoses. In literature, age is associated with a faster decline (49) [START_REF] Piccinin | Coordinated analysis of age, sex, and education effects on change in MMSE scores[END_REF] but, as previously discussed, our patients are slightly younger which might have an influence on our findings. There were very few studies that showed gender to impact cognitive speed decline (50) (51) [START_REF] Levine | Sex Differences in Cognitive Decline Among US Adults[END_REF], and those that show an effect of the level education are even rarer (53) (49) [START_REF] Piccinin | Coordinated analysis of age, sex, and education effects on change in MMSE scores[END_REF]. Better performance at baseline neuropsychological tests was associated with a slower decline both in our study and the literature (54) (55) [START_REF] Lopez | Predicting cognitive decline in Alzheimer's disease: an integrated analysis[END_REF]. Each of our groups consisted of a high number of major NCD patients but the slow decline group had an increased minor NCD diagnoses in accordance to what is known of the cognitive function progression. (57) [START_REF] Morris | The Consortium to Establish a Registry for Alzheimer's Disease (CERAD). Part I. Clinical and neuropsychological assessment of Alzheimer's disease[END_REF]. That patients diagnosed with a minor NCD at first consultation seemed to evolve more slowly is a strong argument to encourage patients to consult and be assessed as early in disease stage as possible.

Most studies show a faster cognitive decline in patients with mixed neurocognitive disorder (58) [START_REF] Campbell | Risk Factors for the Progression of Mild Cognitive Impairment to Dementia[END_REF]. As previously discussed, this discordance with our finding might be due to our focus on vascular encephalopathy and our exclusion of cerebral infarction which have a major impact on cognitive performance [START_REF] Regan | Relationship of vascular risk to the progression of Alzheimer disease[END_REF]. This finding should obviously be further studied to test our hypothesis.

The majority of VE in the slow progression group might also be linked with the modifiable and treatable factors involved in the disease [START_REF] Tariq | Dementia risk and prevention by targeting modifiable vascular risk factors[END_REF]. Our study has several limitations, most of them linked to its retrospective nature. There were very little means to verify the accuracy of diagnosis entered into our database. However, since the physicians (either a neurologist, a psychiatrist or a geriatric doctor) diagnosing the patient were part of a memory reference center, they could be trusted to be as accurate as can be with a clinical diagnosis.

Data were analyzed retrospectively and thus without any way of standardizing the dataset entered into the database for each consult, although a large part of what we collected was fairly standard (e. g. age, gender, level of education…). As is usually the case with retrospective study, there were a number of missing data, mostly dealing with level of education and diagnosis. Another limit of our study was the low number of years of attendance to Memory Clinic. The mean number of consultations was 1.8 with 0.7 mean years between first and last consult. For the patients who attended at least twice, the mean years between first and last consult was 1.8. This finding shows that for a large majority of the patients attending Nice Memory Clinic we have a limited hindsight as to the evolution of the disorder diagnosed.

As to why most of our patients attended a limited number of consultations, it might depend on the initial evaluation result. There were also at least 10.8% of our population who had no significant alteration of their neuropsychological tests score and who might have felt a follow-up was unnecessary. There might also have been a number of patients who had a too altered neuropsychological test score at first consult and for whom repeated neuropsychological evaluations were not recommended. It should also be underlined that some of our patients were referred by their usual physicians for an expert opinion and thus were only seen once. The Memory Clinic also play a big part in specific management of major NCD which did not require multiple neuropsychological assessments or medical examinations and so might not be reported into our database.

Conclusions

Our study found that in the population attending Nice Memory Clinic, the diagnosis was predominantly a major NCD with a mean MMSE of 22/30. Yearly MMSE decline was 2 points for AD, 0.7 for VE and 1.6 points for Mx NCD patients. Our findings emphasized how crucial improving the quality of systematic data collected during Memory Clinics consultations is [START_REF] Ramirez Diaz | The need for a consensus in the use of assessment tools for Alzheimer's disease: the Feasibility Study (assessment tools for dementia in Alzheimer Centres across Europe), a European Alzheimer's Disease Consortium's (EADC) survey[END_REF]. The French initiative of creating a national memory observatory might enhance the data collected and help better understand cognitive decline and its impacts on our patients.

10 Remerciements

 10 Aux membres du jury : -Professeur Olivier Guérin Pour avoir accepté de présider ce jury, pour votre enseignement et pour votre accueil et votre écoute -Docteur Guillaume Sacco Pour avoir accepté de diriger ce travail, pour ta patience et tes nombreux conseils -Docteur Aurélie Mouton Pour avoir accepté de juger ce travail, pour l'analyse que tu y as apporté et pour aide précieuse face à mes questions pendant mon stage -Docteur Emeline Michel Pour avoir accepté d'apporter ton expertise à ce travail et ton engagement dans notre enseignement A toute l'équipe médicale de l'hôpital de Cimiez ; A Cyrielle et Marine mes premières « cheffes » de gériatrie à Cimiez, merci pour votre accueil, pour tout ce que vous m'avez appris et pour votre confiance grandissante au fur et à mesure du stage. A Jihene, pour ta bonne humeur, ta gentillesse et la confiance que tu as pour moi. A Déborah, pour ton écoute, tes conseils et ton enseignement qui me reste précieux. A toute l'équipe paramédicale, merci pour votre soutien et votre bonne humeur.

23 A

 23 Aux copains ; A Julie, Yiannis, Norman, Florence, Gauthier, Madeline, Charles, Marine, Cécile. Merci pour tous ces moments passés à vos côtés, pour votre soutien et vos conseils. Merci pour votre amitié qui m'est précieuse. A nos prochaines vacances, hivernale ou estivale. A mes cointernes de gériatrie ; A Pauline, Imene, Axelle, Damien, Serena, Maxime et les autres. Merci pour l'aide que vous êtes toujours prêt à m'apporter, et pour votre accompagnement dans l'aventure de l'internat ! Table des matières Introduction ............................................................................................................................................ Matériels et Méthode ............................................................................................................................. Résultats ................................................................................................................................................ Discussion ............................................................................................................................................. Conclusion……………………………………………………………………………………………………….descriptive study of the population attending Nice Memory Clinic: the Cobtest database analysis Introduction: Understanding the progression of neurocognitive disorders in our population remain essential for current care in memory consultation. The main objective of this study was to describe the evolution of cognitive decline in patient attending in a memory clinic. Method: A 7 years longitudinal retrospective and monocentric study was conducted using the neuropsychological assessment software developed in the memory clinic of Nice university hospital. Patients with Alzheimer's disease (AD), vascular encephalopathy (VE, related with microvascular etiology and excluding stroke sequelae) or mixed neurocognitive disorder (Mx) and at least two neuropsychological assessment were included in the subgroup analyzes.

  This retrospective study analyzed data from outpatients consulting in the memory clinic of the university hospital of Nice between 2014 and 2021. The data collection was performed through the homemade neuropsychological assessment software (CoBTest) in accordance with the 74 th article of the Public Health Code and the law of 1978 concerning digital data and liberty protection. The database was registered by the CIL (liberty and informatics correspondent) on June 26 th 2014 under the number 198.

  Our study's strengths include our large sample of patients which was sometimes more than twice what could be found in the literature concerning cognitive decline (25) (26) (27). The diagnosis we used were based on either the DSM-IV or the DSM-V criteria, and we reclassified the DSM-IV diagnosis according to the DSM-IV nomenclature to harmonize our data. The DSM-V was published in 2013 but the Dossier Mobile Alzheimer based its diagnosis on the previous version until 2018 which led us to recategorize obsolete diagnosis with their closest equivalent in the DSM-V nomenclature.

FiguresFigure 1 .

 1 Figures Figure 1.Flow chart

  Figure 1. Flow chart

Figure 2 .

 2 Figure 2. Annual MMSE test score point loss of patients with Alzheimer's disease, vascular and mixed neurocognitive disorder

Table 1 .

 1 Baseline characteristics of the whole population (N=7033) and of patients diagnosed with Alzheimer's disease, vascular and mixed neurocognitive disorders (n = 791)

	General characteristics	Whole population N=7033	AD+VD+ MD (N=791) AD (n=366) VD (n=225) MD (n=200)	p-value
	Age, mean ± SD	73.6 ± 13.0	76.2 ± 8.1	76.0 ± 8.2	79.8 ± 6.4	<0.001**
	Female gender, n (%)	4345 (61.8)	214 (58.5)	106 (47.1)	112 (56.0)	0.024**
	MMSE score, mean ± SD	22.3 ± 6.4	19.7 ± 5.9	23.8 ± 4.6	19.0 ± 5.1	<.001**
	FAB score, mean ± SD	13.0 ± 3.7	11.7 ± 3.6	13.0 ± 3.2	10.8 ± 3.2	<0.001**
	Level of education, n (%)					0.044**
	None	151 (3.1)	1 (0.6)	3 (2.4)	4 (4.7)
	Lower level	1078 (22.4)	41 (22.7)	38 (29.9)	21 (24.4)
	Upper level	962 (20)	30 (16.6)	24 (18.9)	15 (17.4)
	Primary education	1195 (24.8)	60 (33.2)	32 (25.2)	35 (40.7)
	Higher education	1430 (29.7)	49 (27.1)	30 (23.6)	11 (12.8)
	Missing data	2217	185	98	114
	Functional diagnosis, n (%)					<0.001**
	Other	268 (3.8)	2 (0.6)	0 (0)	0 (0)
	SMC	758 (10.8)	2 (0.6)	12 (5.3)	1 (0.5)
	mNCD	1900 (27.0)	59 (16.1)	122 (54.2)	18 (9.0)
	MNCD	2055 (29.2)	303 (82.8)	91 (40.4)	181 (90.5)
	MMSE point loss, mean ± SD
	Total, mean ± SD			3.6 ± 4.8
			4.5 ± 5.0	1.5 ± 3.6	4.1 ± 4.8	< .001**
	Annual, mean ± SD			1.5 ± 2.1
			2.0 ± 2.0	0.7 ± 1.8	1.6 ± 2.1	< .001**

Notes: AD stands for Alzheimer disease, VD for vascular neurocognitive disorder and MD for mixed neurocognitive disorder. The MMSE and FAB scores were those reported at first consult. Functional diagnoses were as follow: other case, subjective memory complaint, minor neurocognitive disorder and major neurocognitive disorder:

Table 2 .

 2 Comparison of the population diagnosed with Alzheimer's disease, vascular and mixed

	neurocognitive disorders (n=791) according to cognitive speed decline
		Annual MMSE point loss
		< 1.5 (n=424)	>= 1.5 (n=367)
	General characteristics			p-value
	Age, mean ± SD	77.0 ± 7.8	77.1 ± 8.0	0.997
	MMSE score, mean ± SD	21.1 ± 5.8	20.2 ± 5.6	0.016**
	Female gender, n (%)	228 (53.8)	204 (55.6)	0.610
	Level of education, n (%)			0.962
	None / primary	75 (34.1)	60 (34.5)
	Lower level	54 (24.6)	46 (26.4)
	Upper level	39 (17.7)	30 (17.2)
	Higher education	52 (23.6)	38 (21.8)
	Functional diagnosis, n (%)			<0.001**
	Other	1 (0.2)	1 (0.3)
	SMC	10 (2.4)	5 (1.4)
	mNCD	143 (33.7)	56 (15.3)
	MNCD	270 (63.7)	305 (83.1)
	Etiological diagnosis, n (%)			<0.001**
	AD	158 (37.3)	208 (56.7)
	VD	162 (38.2)	63 (17.2)
	MD	104 (24.5)	96 (26.2)

Notes: AD stands for Alzheimer disease, VD for vascular neurocognitive disorder and MD for mixed neurocognitive disorder.

The MMSE and FAB scores were those reported at first consult. Functional diagnoses were as follow: other case, subjective memory complaint, minor neurocognitive disorder and major neurocognitive disorder

Acknowledgments

The authors would like to thank all the people belonging to the Nice Memory Clinic for their hard work to improve the life and care of our patients.

Funding

Not applicable.

Availability of data and materials

There are no linked research data sets for this paper.

Ethics approval and consent to participate

Not applicable.

Competing interests

All authors state that they have no conflicts of interest with this paper. The authors have no relevant personal financial interest in this manuscript.

Authors contributions

-Camille PALINIEWICZ has full access to all of the data in the study, takes responsibility for the data, the analyses and interpretation and has the right to publish any and all data, separate and apart from the attitudes of the sponsors. All authors have read and approved the manuscript.

-Study concept and design: Guillaume SACCO, MD, PhD Mon premier souci sera de rétablir, de préserver ou de promouvoir la santé dans tous ses éléments, physiques et mentaux, individuels et sociaux.

Je respecterai toutes les personnes, leur autonomie et leur volonté, sans aucune discrimination selon leur état ou leurs convictions. J'interviendrai pour les protéger si elles sont affaiblies, vulnérables ou menacées dans leur intégrité ou leur dignité. Même sous la contrainte, je ne ferai pas usage de mes connaissances contre les lois de l'humanité. J'informerai les patients des décisions envisagées, de leurs raisons et de leurs conséquences. Je ne tromperai jamais leur confiance et n'exploiterai pas le pouvoir hérité des circonstances pour forcer les consciences.

Je donnerai mes soins à l'indigent et à quiconque me le demandera. Je ne me laisserai pas influencer par la soif du gain ou la recherche de la gloire.

Admise dans l'intimité des personnes, je tairai les secrets qui me seront confiés. Reçue à l'intérieur des maisons, je respecterai les secrets des foyers et ma conduite ne servira pas à corrompre les moeurs.

Je ferai tout pour soulager les souffrances. Je ne prolongerai pas abusivement les agonies. Je ne provoquerai jamais la mort délibérément.

Je préserverai l'indépendance nécessaire à l'accomplissement de ma mission. Je n'entreprendrai rien qui dépasse mes compétences. Je les entretiendrai et les perfectionnerai pour assurer au mieux les services qui me seront demandés. J'apporterai mon aide à mes confrères ainsi qu'à leurs familles dans l'adversité.

Que les hommes et mes confrères m'accordent leur estime si je suis fidèle à mes promesses ; que je sois déshonorée et méprisée si j'y manque. »


Ulises Acevedo-Salas

Boris Croes

Yide Zhang

Olivier Cregut

Kokou Dodzi Dorkenoo

Benjamin Kirbus

Ekta Singh

Henrik Beccard

Michael Rüsing

Lukas M Eng

Riccardo Hertel

Eugene A Eliseev

Anna N Morozovska
email: anna.n.morozovska@gmail.com

Salia Cherifi-Hertel

Dodzi Kokou

Riccardo Dorkenoo

Hertel

Impact of 3D curvature on the polarization orientation in non-Ising domain walls

Keywords: Ferroelectric domain walls, curvature, non-Ising domain walls, secondharmonic generation, phase-field simulations, lithium niobate

Ferroelectric domain boundaries are quasi-two-dimensional functional interfaces with high prospects for nanoelectronic applications. Despite their reduced dimensionality, they can exhibit complex non-Ising polarization configurations and unexpected physical properties.

Here, the impact of the three-dimensional (3D) curvature on the polarization profile of nominally uncharged 180° domain walls in LiNbO 3 is studied using second-harmonic generation microscopy and 3D polarimetry analysis. Correlations between the domain wall curvature and the variation of its internal polarization unfold in the form of modulations of the Néel-like character, which we attribute to the flexoelectric effect. While the Néel-like character originates mainly from the tilting of the domain wall, the internal polarization adjusts its orientation due to the synergetic upshot of dipolar and monopolar bound charges and their variation with the 3D curvature. Our results show that curved interfaces in solid crystals may offer a rich playground for tailoring nanoscale polar states.

In ferroic materials, the sample geometry can significantly influence the order parameter because of boundary conditions and charge distributions at the surface. A correlation between the topology of the sample surface and the ferroic order occurs in many systems 1 , which opens a smart pathway towards engineering specific materials properties by tailoring the surface morphology. In the case of ferromagnetic materials, such situations have been investigated in detail over the past years for several complex geometries, including rolled-up structures 2 , hemispherical shells 3 , and several shapes therealike 4 . In studies of such artificially structured materials, the role of surface curvature has become a topic of growing interest, as theory predicted that curvature might lead to a chiral symmetry breaking of the order parameter 5,6 and, as a result, to surprising effects like non-reciprocal wave propagation and asymmetric domain-wall dynamics 7,8 . While the topic of curvilinear magnetism appears to be in full bloom 9 , identifying and understanding the role of curvature in ferroelectric materials is still in its infancy.

Owing to the recent progress in the fabrication of curvilinear crystalline ferroelectric oxides and the modeling of their properties, it has become clear that curvature can provide a new pathway towards the polarization control at surfaces 10 , and for tailoring exotic polarization textures such as bubble domains 11,12 . Functionality enhancement and variation of the phase transition temperature mediated by curvature are evidenced in ferroelectric nanocylinders 13 . Dong et al. 14 demonstrated that the ultraflexibility of freestanding crystalline barium titanate membranes is directly connected to the dynamic variation of the ferroelectric domain structure with curvature. In a similar context, Zhou et al. 15 reported periodic domain patterns in a rippled BaTiO 3 freestanding film in which the ferroelectric domain structure is linked to the local curvature and can be controlled by applying a pressure through flexoelectric coupling.

Interfaces with irregular shapes in ferroelectric materials can also result in unexpected local physical properties. Recent studies have shown that curved interfaces can naturally develop in ferroelectric/dielectric heterostructures through epitaxial strain and strain gradients, resulting in complex three-dimensional (3D) domain structures 16 , or periodic dipole waves [START_REF] Gong | Atomic Mapping of Periodic Dipole Waves in Ferroelectric Oxide[END_REF] owing to the complex balance between elastic and electrostatic energies. The control of elastic effects in epitaxial heterostructures is, however, limited by the substrate clamping.

The engineering of competing electrostatic and strain forces, which can lead to exotic polar topological structures, is better enabled by setting the epitaxial film free [START_REF] Li | Electrostatically Driven Polarization Flop and Strain-Induced Curvature in Free-Standing Ferroelectric Superlattices[END_REF] through complex growth procedures [START_REF] Pesquera | Beyond Substrates: Strain Engineering of Ferroelectric Membranes[END_REF] .

Bent and crumpled homo-interfaces formed by domain boundaries are also expected to show enhanced functional properties. In fact, a ferroelectric domain wall can be regarded as a homo-interface that can be programmed, (re)moved, or bent quasi at will, simply by applying an external field. Zhang et al. [START_REF] Zhang | Intrinsic Conductance of Domain Walls in BiFeO3[END_REF] reported a p-n junction within a single domain wall resulting from the wall bending in BiFeO 3 thin films. Maksymovych et al. [START_REF] Maksymovych | Tunable Metallic Conductance in Ferroelectric Nanodomains[END_REF] established direct correlations between the metallic conductivity of domain walls and the conical shape of their adjacent ferroelectric domains in otherwise insulating PbZrTiO 3 thin films. Both theory [START_REF] Eliseev | Static Conductivity of Charged Domain Walls in Uniaxial Ferroelectric Semiconductors[END_REF][START_REF] Eliseev | Domain Wall Conduction in Multiaxial Ferroelectrics[END_REF] and experimental studies [START_REF] Godau | Enhancing the Domain Wall Conductivity in Lithium Niobate Single Crystals[END_REF][START_REF] Lu | Electrical Tunability of Domain Wall Conductivity in LiNbO 3 Thin Films[END_REF] confirm that the local charge and the related conductivity can be affected by the tilt angle of the walls with respect to the polar axis. Clear correlations between the domain wall conductivity and the variation of bound charges due to the tilt angle in lithium niobate crystals have been evidenced by combining 3D noncolinear Cherenkov SHG and electrical conductivity measurements [START_REF] Godau | Enhancing the Domain Wall Conductivity in Lithium Niobate Single Crystals[END_REF][START_REF] Esin | Tilt Control of the Charged Domain Walls in Lithium Niobate[END_REF][START_REF] Kirbus | Real-Time 3D Imaging of Nanoscale Ferroelectric Domain Wall Dynamics in Lithium Niobate Single Crystals under Electric Stimuli: Implications for Domain-Wall-Based Nanoelectronic Devices[END_REF] . Based on nanoscale studies combining focused-ion-beam-assisted scanning electron microscopy and tomographic piezo-response force microscopy, Roede et al. [START_REF] Roede | The Third Dimension of Ferroelectric Domain Walls[END_REF] showed the same effect in hexagonal menganites.

Despite the intensive research on the morphology-structure-functionality relation in ferroelectrics, the effect of curvature on the internal structure of ferroelectric domain walls has hitherto not been addressed. Experimental studies on the variation of the internal polarization as a function of the domain wall curvature might uncover new physical aspects on, e.g., the propagation of domain walls in disordered media [START_REF] Giamarchi | Dynamics of Disordered Elastic Systems[END_REF] in which pinning [START_REF] Paruch | Nanoscale Studies of Ferroelectric Domain Walls as Pinned Elastic Interfaces[END_REF] and roughening [START_REF] Rapin | Dynamic Response and Roughening of Ferroelectric Domain Walls Driven at Planar Electrode Edges[END_REF] are dominant, and it bears new prospects for next-generation domain wallbased nanotechnology [START_REF] Sharma | Roadmap for Ferroelectric Domain Wall Nanoelectronics[END_REF][START_REF] Meier | Ferroelectric Domain Walls for Nanotechnology[END_REF][START_REF]Domain Walls: From Fundamental Properties to Nanotechnology Concepts[END_REF][START_REF] Nataf | Domain-Wall Engineering and Topological Defects in Ferroelectric and Ferroelastic Materials[END_REF] .

In this work, the impact of curvature on the internal polarization of a nominally uncharged 180° ferroelectric domain walls is investigated in a z-cut LiNbO 3 bulk crystal using noninvasive 3D nonlinear optical microscopy and polarimetry analysis. The ferroelectric polarization profile of an "arc-shaped" domain wall segment is derived through colinear 3D SHG microscopy in back-reflection geometry and polarimetry analysis (Figure 1). The variation of the 3D SHG response with the light polarization is analyzed pixel-by-pixel throughout the domain wall profile. The analysis of the local second-harmonic emission and its polarization dependence [START_REF] Cherifi-Hertel | Shedding Light on Non-Ising Polar Domain Walls: Insight from Second Harmonic Generation Microscopy and Polarimetry Analysis[END_REF] reveals an overall Néel-like internal structure that follows the mean curvature of the wall. Nevertheless, clear local deviations of the internal polarization with respect to the surface normal of the domain wall are observed. This result is discussed in terms of the synergetic action of monopolar bound charges due to the wall tilting, and dipolar charges due to the Néel-like character. The polarization arrangement at the homointerface formed by a wall separating two ferroelectric domains is affected by the complex interplay between the electrostatic and elastic forces [START_REF] Vasudevan | Domain Wall Geometry Controls Conduction in Ferroelectrics[END_REF][START_REF] Morozovska | Anisotropic Conductivity of Uncharged Domain Walls in BiFeO 3[END_REF] , in a similar way as in crystalline heterointerfaces. The balance of the different forces at play in non-Ising domain walls, namely electrostatic forces and flexoelectric coupling, enhances the sensitivity to morphology variations, thereby providing a new degree of freedom to control the polarization.

RESULTS AND DISCUSSION

Because of their trigonal symmetry and stoichiometry, uniaxial ferroelectrics such as LiNbO 3 or LiTaO 3 crystal families are expected to show rather simple domain structures of hexagonal or triangular shapes [START_REF] Scrymgeour | Phenomenological Theory of a Single Domain Wall in Uniaxial Trigonal Ferroelectrics: Lithium Niobate and Lithium Tantalate[END_REF] . Nonetheless, a large variety of complex polygonal shapes can be obtained by non-equilibrium switching conditions due to incomplete or ineffective screening of the depolarization field during the poling procedure, as explained by Shur et al. [START_REF] Shur | Domain Shapes in Bulk Uniaxial Ferroelectrics[END_REF] . This can be caused, e.g., by a fast sweeping of the applied electric field, or by pyroelectric effects induced by a local laser heating. To create curved walls, the uniform polarization of a z-cut congruent Mg:LiNbO 3 crystal was locally switched under UV-laser irradiation, following the procedure detailed by Haussmann et al. [START_REF] Haussmann | Ferroelectric Lithography: Bottom-up Assembly and Electrical Performance of a Single Metallic Nanowire[END_REF] . We obtain a large domain of rounded shape using laser-assisted switching (see Figure 1b). This domain extends in the z-direction throughout the whole crystal thickness (200 µm), and it contains in its core a small conventional hexagon-shaped domain with reversed polarity. This small inner domain serves as a reference object for the crystal orientation since the characteristic Y-walls surrounding the hexagonal-shape domain are oriented along three equivalent crystallographic axes

([1120], [2110], [1210]) 42 .
Owing to its quadratic dependence on the electric field and its high sensitivity to local symmetry breaking, nonlinear optical microscopy based on the second-harmonic process allows for the detection of nanostructures that are a priori incompatible with the resolution limit of optical methods [START_REF] Bonacina | Harmonic Generation at the Nanoscale[END_REF] . In this context, SHG microscopy has recently emerged as a method of choice to probe the local ferroelectric polarization at domain walls [START_REF] Cherifi-Hertel | Shedding Light on Non-Ising Polar Domain Walls: Insight from Second Harmonic Generation Microscopy and Polarimetry Analysis[END_REF][START_REF] Cherifi-Hertel | Non-Ising and Chiral Ferroelectric Domain Walls Revealed by Nonlinear Optical Microscopy[END_REF][START_REF] De Luca | Domain Wall Architecture in Tetragonal Ferroelectric Thin Films[END_REF][START_REF] Yokota | Optical Second-Harmonic Generation Microscopy as a Tool for Ferroelastic Domain Wall Exploration[END_REF] . The acquisition of the localized nonlinear optical emission at the domain walls allows a noninvasive 3D profiling of domain walls embedded in the volume of ferroelectric crystals 47- 49 , while the polarization analysis of the second-harmonic emission has proven its efficiency in detecting the local symmetry lowering at the domain wall regions [START_REF] Cherifi-Hertel | Shedding Light on Non-Ising Polar Domain Walls: Insight from Second Harmonic Generation Microscopy and Polarimetry Analysis[END_REF] . This capability made it possible to unveil a non-Ising and chiral internal structure of the wall [START_REF] Cherifi-Hertel | Non-Ising and Chiral Ferroelectric Domain Walls Revealed by Nonlinear Optical Microscopy[END_REF] with an overall in-plane polarization component along the domain wall (Bloch-like) or perpendicular to it (Néel-like).

Local SHG polarimetry hence provides an unambiguous signature of the internal polarization structure of the domain wall as demonstrated by the complementarity of the polar plots displayed in Figure 1c for Néel and Bloch-like configurations.

Local 2D pixel-by-pixel polarimetry analysis of ferroelectric domains [START_REF] Kaneshiro | SHG Microscopic Observation of Pb(Zn 1/3 Nb 2/3 )O 3 -9%PbTiO 3 Single Crystal[END_REF] and polar domain walls [START_REF] Yokota | Direct Evidence of Polar Nature of Ferroelastic Twin Boundaries in CaTiO 3 Obtained by Second Harmonic Generation Microscope[END_REF] was previously reported by Uesu's group. In the present study, we use colinear SHG microscopy with a 3D pixel-by-pixel polarimetry analysis to probe the variation of the internal polarization of a ferroelectric domain wall with its morphology throughout the sample volume.

It is worth noting that the effect of the domain wall morphology is not limited to curvature.

According to theoretical predictions, a change in the orientation of the wall with respect to the crystal axes is also expected to affect the internal structure and chiral character of ferroelectric domain walls [START_REF] Scrymgeour | Phenomenological Theory of a Single Domain Wall in Uniaxial Trigonal Ferroelectrics: Lithium Niobate and Lithium Tantalate[END_REF][START_REF] Yudin | Bichiral Structure of Ferroelectric Domain Walls Driven by Flexoelectricity[END_REF] . To distinguish the effect of curvature from other geometrical effects, we consider here only small changes (on the order of ten degrees) in the orientation of the wall. In addition, we take great care in positioning the curved domain wall portion along the x-axis to avoid unwanted effects related to the beam focus which may result in distortions of the SHG polar plots, as discussed elsewhere [START_REF] Zhang | Focusing Characteristics of Polarized Second-Harmonic Emission at Non-Ising Polar Domain Walls[END_REF] .

Figure 1d displays local SHG polar plots measured at three different points labeled A, B, and C along the "arc-shaped" domain wall. These three points show similar SHG polar plots that can be fitted very well with a Néel model (continuous lines in Figure 1c) resulting from an overall polarization aligned perpendicular to the domain wall. This observation is further confirmed by the 3D variation of the SHG polar plots throughout the domain wall profile (Figure 1e). As a general trend, the internal polarization seems to follow the mean curvature of the domain wall, while preserving an overall Néel-like character (i.e., showing an overall in-plane polarization component perpendicular to the domain wall) throughout the sample volume.

Let us now examine the domain wall profile in terms of the local polarization orientation and morphology. Information on the morphology of the domain wall (Figure 2b) is obtained based on the 3D SHG image (Figure 2a), while the local orientation of the internal polarization angle  P is derived by fitting each polar plot, pixel-by-pixel, as explained in Supplementary note S2.

By repeating the pixel-by-pixel fitting procedure at different focusing depths, ranging from 50 µm to 170 µm below the surface of the sample, we obtain a full 3D SHG analysis of the curved domain wall segment. The combination of the x,y,z coordinates of the wall morphology derived from the 3D SHG image (see Figure S2), and the angle  P obtained as an output parameter of the pixel-by-pixel fitting of the SHG polarimetry response, allows us to represent the internal polarization profile (Figure 2c). Selected results (fitting factor R 2 > 0.96) are presented in the form of bidirectional arrows. Comparing the polarization angle with the surface normal, i.e.,  P (Figure 2c) versus  n (Figure 2d), demonstrates deviations up to  15° of the internal polarization with respect to the domain wall normal vector. To better understand the origin of this deviation angle, we analyze the charge density distribution and study its variation with the domain wall curvature. Let us first consider the bound charge density distribution  bound induced at tilted domain walls with respect to the polar axis. The schematic representation shown in Figure 3a illustrates the polarization bound charges resulting from the head-to-head or tail-to-tail polarization arrangement at domain walls. This type of charge develops as a result of a positive or negative tilt angle of the domain wall with respect to the polar axis. The bound charge density can be estimated as 𝜌 𝑏𝑜𝑢𝑛𝑑 ≅ 2𝑃 𝑆 cos(𝑃 ⃗ , 𝑛 ⃗ ) [START_REF] Eliseev | Static Conductivity of Charged Domain Walls in Uniaxial Ferroelectric Semiconductors[END_REF] . The angle between the domain wall normal 𝑛 ⃗ and the polarization of the adjacent domain (along z-axis) 𝑃 ⃗ is extracted from the 3D SHG experiments (Figure 2). This allows us to derive  bound and to represent this charge density in a color map superimposed to the surface of the curved wall (Figure 3b). In addition to the dominant bound charges arising from a tilt of the domain wall with respect to the polar axis, complex internal polarization textures and topological structures are also expected to contribute to the electrostatic charge density distribution. The effect of these additional contributions can be particularly perceptible in the case of nominally uncharged domain walls showing a small tilt (Figure S6), and giving rise to a relatively small bound charge density ( 5 10 -2 P s ) such as the example considered for this study. In this case, additional contributions can play a role equally important as the monopolar bound charge and may significantly affect the overall charge distribution at the domain wall region. Among possible additional charge contributions, we consider the contribution of the local dipole at intrinsically charged Néel-like domain walls. The curvature of bent domain walls leads in this case to a local excess or depletion of the dipolar charge density depending on the sign of the curvature, as outlined in Figure 3c. Hence, the variation of the relative angle between adjacent dipoles (Figure 2c) is a good quantity to estimate the charge density excess related to the Néel-like internal structure. This additional term is calculated at each site i of the domain wall 𝜌 𝑒𝑥𝑐𝑒𝑠𝑠 ≅ 𝑃 𝑡 sin (𝜃 𝑃 ⃗ 𝑖+1 -𝜃 𝑃 ⃗ 𝑖 ) and plotted as a color map in units of the in-plane polarization P t , as displayed in Figure 3d. The monopolar and dipolar bound charge densities seams to both vary with the 3D curvature of the domain wall. Curvature can be defined in different ways to highlight different geometrical aspects such as to describe the degree by which a surface is concave or convex, or whether it is concavely bent in a specific direction but convexly bent in another. To unveil the detailed features in the 3D morphology of the domain wall, we use two different intrinsic measures of the curvature, known as mean curvature (H), and Gaussian curvature (K), as detailed in Supplementary note S3. Mean curvature expresses the local concavity or convexity of the surface, while Gaussian curvature is sensitive to the presence of saddle-or dome-shaped regions, irrespective of the concave or convex local character of the surface (see Figure S4).

Based on the morphology of the domain wall obtained from 3D SHG microscopy, Gaussian and mean curvatures are derived and displayed in panels a and f of Figure 4. We observe that the electrostatic charge density shows clear correlations with the domain wall curvature. This comparison is automated by calculating the Pearson coefficient r as a quantitative measure of the statistical correlation between the domain wall curvature and the electrostatic charge density based on the covariance method (see Supporting note S4). This coefficient provides information about the correlation strength (no correlation for r = 0, and maximum correlation for r = 1) and direction (correlation for r > 0 and anticorrelation for r < 0). The variation of the charge density distribution due to the Néel-like character clearly varies with the mean curvature (99% correlations, Figure4d), while the overall charge distribution accounting for the synergetic effect of monopolar and dipolar bound charges correlates with the Gaussian curvature of the domain wall (80% correlations, Figure 4i). This result shows that the dependence of the electrostatic charge density on curvature has a direct effect on the orientation of the local polarization angle and its variation throughout the wall profile. Moreover, the domain wall curvature may also affect the electronic conductivity paths linked to the charge excess lines (see, e.g., Figure 4d,g), providing an additional degree of freedom to design 2D resistor networks in the domain wall [START_REF] Wolba | Resistor Network Modeling of Conductive Domain Walls in Lithium Niobate[END_REF] .

Let us now compare the charge density distribution to the internal polarization variation (Figure 4c,h). In the same way as described earlier, this comparison is automated by calculating the Pearson coefficient r. The exhaustive statistical correlation results are summarized in Table S1. Best correlations have been observed: (i) between the in-plane variation of the polarization angle and the dipole charge density due to the Néel character, and (ii) between the out-of-plane polarization variation and the joint action of the dipolar and monopolar bound charges (Figure 4e,j).

Analytical calculations in the framework of Landau-Ginzburg-Devonshire (LGD) approach and finite element modeling (FEM) of the polarization distribution in the vicinity of domain walls in uniaxial ferroelectrics (see Methods and problem formulation in the Supplementary note S5), in particular in LiNbO 3 22 , lead to the (trivial) conclusion that straight and uncharged Ising-type domain walls can be stabilized only in a bulk ferroelectric. This is valid only in the absence of pinning centers (such as lattice barriers, or charged and/or elastic defects), and without significant spatial gradients of elastic, electric and temperature fields. The violation of any of these conditions can lead to a curvature of the domain wall, and very often in these cases the Néel-type polarization appears at the wall due to the omnipresent flexoelectric coupling [START_REF] Eliseev | Domain Wall Conduction in Multiaxial Ferroelectrics[END_REF] . In particular, curvature can strongly affect the domain wall structure as well as the domain wall profile (see Figure S7 .

While the monopolar term if of electrostatic origin, and it depends only on the domain wall tilt with respect to the polar axis, the dipolar term is of elastic origin. It is directly related to the flexoelectric effect and it varies with the mean curvature 𝐻 = 1 2 (𝑘 1 + 𝑘 2 ) (see Supplementary note S3), as corroborated by the experimental results displayed in panels a,b,d of Figure 4.

In the simplest case of a cylindrical nanodomain, we have 𝑅 = 𝑟 𝑐 , where 𝑟 𝑐 is the correlation radius referred to the bulk sample [START_REF] Eliseev | Domain Wall Conduction in Multiaxial Ferroelectrics[END_REF] (see details and 3D FEM results in Supporting note S6).

The absence of a domain wall tilt with respect to the polar axis leads to a week Néel-type polarization component at the domain wall regions, and closure domains at the top and bottom surfaces (see Figure S8 and Figure S9). The monopolar and dipolar components can have very different magnitudes, and, as a rule, the dipolar component depends on the mean curvature and the correlation radius, while the monopolar component depends on the wall tilt.

Yet, in particular cases, in which the domain wall tilt is small such as in the present study, the internal polarization orientation of the domain wall can be affected by the synergetic influence of the domain wall tilting and the intrinsically charged character of the domain wall.

CONCLUSION

The tomography of a curved domain wall segment was conducted by means of secondharmonic microscopy and its internal polarization structure was derived based on volumetric pixel-by-pixel polarimetry analysis. This allowed us to analyze the impact of curvature on the domain wall's polarization structure. The rotation of the local polarization angle by up to 15° with respect to the wall normal is explained by polarization reordering due to the local variation of the electrostatic monopolar and dipolar charges with curvature. This work sheds new light onto the intrinsically charged character of Néel-type domain walls and the effect of curvature on the residual dipolar charge, thereby demonstrating that domain wall tilting (resulting in monopolar bound charges) is not the only parameter at play at domain boundaries. These findings pave the way towards the design of energy-friendly nanoelectronic device concepts based on the control of the internal polarization of domain walls and the related functionalities with curvature.

METHODS

Poling Procedure. As a uniaxial ferroelectric crystal, we use the archetypal z-cut congruent lithium niobite crystal. More specifically, a commercially available (Yamaju Ceramics) 200 μm-thick Mg(5%)-doped LiNbO 3 is used to engineer a curved domain wall by means of a laser-assisted poling procedure. A HeCd laser (λ = 325 nm, P = 10 μW) was focused through a 10x objective onto the -z-side of the crystal. A voltage of 0.8 kV well below the forward coercive voltage of 1.3 kV was applied parallel to the z-axes, with the polarity chosen to support polarization inversion. Under these conditions, the domain nucleation starts after a about five seconds of exposure to the UV light. We then blocked the UV laser and immediately applied a 1.2 kV bias voltage using liquid electrodes (5wt%-NaCl in deionized H 2 O). The voltage was then continuously decreased to zero within 2 s. Following this protocol, individual domains of a roundish shapes having mostly straight domain walls with inclination angles of ~ 0.2° can be prepared reproducibly. The scrambled morphology of the domain wall is obtained following the enhancement procedure described in ref. [START_REF] Godau | Enhancing the Domain Wall Conductivity in Lithium Niobate Single Crystals[END_REF] . This produces a bent domain wall with an average tilt angle of 0.3° (see Figure S6) and a local tilt angle up to 1°.

Second-Harmonic Microscopy and 3D Polarimetry analysis.

The second-harmonic generation measurements are carried out using an inverted scanning confocal microscope in reflection geometry. The fundamental wave is generated with a Spectra Physics Ti:Al 2 O 3 laser (Millenium-Tsunami combination), producing 100 fs pulses with a repetition rate of 80 MHz and a wavelength centered at 800 nm. The laser beam is directed at normal incidence to the sample and focused with a 60x magnification objective lens. The SHG intensity is scanned using computer-controlled stepping motors with 100 nm step size and by recording the SHG signal at each scan step with an exposure time of 20 ms per step, at a power of 20 mW. The output intensity is spectrally filtered and collected into a photomultiplier. Series of SHG images are recorded at different α analyzer angles at a laser polarization along the xaxis (φ = 0) or y-axis (φ = 90°). Each image of the recorded stack is subdivided into a checkerboard of pixels. An individual polar plot is obtained in each pixel by integrating the local SHG intensity over the stack of images. All the polar plots are then displayed at their pixel center position, yielding a polar plot map (see Figure S1). The local polarization orientation in each pixel is derived by fitting the local polar plots to the analytic form of SHG while accounting for the non-Ising character of the domain wall (see Supplementary note S1). More than 50% of the selected results exhibit a fitting factor R 2 > 0.96. The threedimensional SHG analysis is obtained by repeating the procedure described above at different focal depths, ranging from 50 µm to 170 µm below the surface of the sample.

Theory and computation Methods. We conducted finite-element modeling (FEM) based on the Landau-Ginzburg-Devonshire (LGD) theory. FEM is performed using COMSOL@MultiPhysics software, with electrostatics, solid mechanics, and general math (PDE toolbox) modules. The LGD free energy functional of a uniaxial ferroelectric and LGDtype equation are listed in the Supplementary Materials. The ferroelectric, dielectric, and elastic properties of the LiNbO 3 core are given in Table S2.

S1 Local polarimetry analysis

In the following, we present the general analytic form of the second-harmonic polarimetry accounting for the non-Ising character of ferroelectric domain walls and their respective nonlinear optical susceptibility tensors. The SHG intensity variation with the polarization 𝜑 of the fundamental wave (FW) and analyzer angle 𝛼 is given by 𝐼 𝑆𝐻𝐺

where 𝐸𝑖(𝜑) is the electric field of the FW (𝐸 0 cos 𝜑, 𝐸 0 sin 𝜑, 0), and 𝑑 𝑖𝑗 represent the elements of the nonlinear optical susceptibility tensor written following the Voigt notation: 2𝑑 𝑖𝑗 = 𝜒 (2) 𝑖𝑗𝑘 . The indices i, j, k refer to the Cartesian laboratory coordinates (x, y, z). The complete dependence of the SHG response on the analyzer angle 𝛼 and the input polarization 𝜑 is obtained using the Jones formalism accounting for the rotation 𝛼 of a linear polarizer as follows:

The orientation of the local polarization P is identified with respect to the laboratory coordinates system by the angle θ P (taken with respect to the x-axis). The related susceptibility tensor is derived using rotation and transformation matrices and by assuming a Néel or Bloch character of the domain wall, as explained in Refs. [1,2]:

𝒅 𝑩𝒍𝒐𝒄𝒉 =
The measurements are conducted by fixing the polarizer angle along the x-axis and rotating the analyzer angle (i.e., 𝛼-scan at 𝜑 = 0). The obtained results are fitted using the analytic form of the SHG (see above). The fitting function takes the following form for a Néel-type internal domain wall structure:

𝐼 𝑆𝐻𝐺 = {[Acos 2 α + B cos α sin α] 2 + [A cos α sin α + B𝑠𝑖𝑛 2 𝛼] 2 } + C (5)
where C is a constant accounting for a background signal, and the coefficients A and B correspond to the effective nonlinear optical anisotropy factors of the domain wall, defined as: ) sin 𝜃 𝑃 (7)

A = -(

S2 Determination of the domain wall curvature and polarization profile based on the 3D SHG data

The position of the pixels forming the domain wall (red region in Figure S2a) is vectorized, and (x, y) coordinates are then associated to each pixel for a given measurement depth (zposition), as depicted in Figure S2b. A polynomial function is then used to fit the resulting domain wall geometry (see the red line in Figure S2c). The pixels forming the core of the domain wall are then selected by their proximity to the fitted curve (see square dots in Figure S2d). We use typically 95% confidence bands, as highlighted in light blue color in Figure S2d. The 3D morphology of the domain wall is then reconstructed in the z-direction using a cubic-spline interpolation of the polynomial functions obtained at each z-position (see Figure S2e).

The local polar plots forming the domain wall are fitted using the analytic form of the SHG (given by equation ( 5)). The local polarization orientation is derived at each pixel as a fit

parameter. An average polarization angle θ P accounting for the first nearest neighbors is calculated and assigned to the selected pixels (see the blue square in Figure S2d). This procedure is repeated for the data sets measured at each z-position. The 3D distribution of θ P is determined by an interpolation of the results in the z-direction. The red spheres are discrete points of the fitted polynomial curves at each analyzed depth, while the gray spheres represent the interpolated 3D surface.

S3 Mean curvature and Gaussian curvature definition

Let us consider a given point p of a regular two-dimensional surface, described in the Euclidean space as shown in Figure S3. The intersection of the normal planes and the surface (depicted in blue in Figure S3) forms curved sections (displayed as dashed lines in Figure S3). Their maximum and minimum curvature is characterized by the principal curvatures 𝑘 1 and 𝑘 2 , expressed by the eigenvalues of the shape operator at the point p. The local mean curvature and Gaussian curvature are defined by the two principal curvatures, 𝑘 1 and 𝑘 2 , as explained in the following. The mean curvature (H) is defined as the mean value of the principal curvatures 𝑘 1 and 𝑘 2 (see Supplementary Figure S3)

𝐻 = 𝑘 1 +𝑘 2 2 (8)
It can be determined from the first and second fundamental forms of the surface (i.e., the eigenvalues of the shape operator) as:

𝐻 = 𝐺𝐿-2𝐹𝑀+𝐸𝑁 2(𝐸𝐺-𝐹²) , (9)
where E, F and G are the coefficients of the first, and L, M and N are the coefficients of the second fundamental form that describe the full geometry of the surface. These coefficients are numerically derived from discrete points of the reconstructed surface by calculating the corresponding partial derivatives [3-5]:

𝐸 = | 𝜕𝑘 1 𝜕𝑢 | ², 𝐹 = 𝜕𝑘 1 𝜕𝑢 • 𝜕𝑘 2 𝜕𝑣 , 𝐺 = | 𝜕𝑘 2 𝜕𝑣 | ², Eq (3) 𝐿 = ∑ 𝑘 𝑖 𝜕²𝑘 𝑖 𝜕𝑢 𝑖 , 𝑀 = ∑ 𝑘 𝑖 𝜕²𝑘 𝑖 𝜕𝑢𝜕𝑣 𝑖 , 𝑁 = ∑ 𝑘 𝑖 𝜕²𝑘 𝑖 𝜕𝑣 𝑖 . ( 10
)
In the same way, Gaussian curvature (K) is defined as the product of the principal curvatures as 𝐾 = 𝑘 1 𝑘 2 . Alternatively, it can be expressed by the eigenvalues of the shape operator as 𝐾 =

(𝐿𝑁-𝑀²) (𝐸𝐺-𝐹²) . ( 11)

S4 Statistical correlations

To establish correlations between parameters A (e.g., K or H curvatures) and B (e.g., θ P ), the data set of A and B is first normalized (intensity ranging from 0 to 255). Then, a sampling grid sub-dividing each dataset into 𝑖 × 𝑗 super-pixels is defined. Each of these super-pixels contains 𝑚 × 𝑛 pixels, where 𝑖, 𝑚 are the indexes for the rows, and 𝑗, 𝑛 are the indexes for the columns, respectively (see Figure S5). The grid elements of the datasets 𝐴 and 𝐵 are then converted into their respective vector representation 𝑎 𝑖𝑗 and 𝑏 𝑖𝑗 containing N elements (𝑁 = 𝑚 × 𝑛), as depicted in Figure S5.

The linear relationship between two parameters can be characterized by a correlation coefficient. The Pearson's coefficient r measures the statistical correlation between two continuous variables based on the covariance method. It provides information about the correlation and the direction, i.e., it distinguishes between correlations and anticorrelations, depending on the positive or negative sign of the coefficient.

Pearson's coefficient is determined at each element of A and B, by calculating the correlation between the vectors 𝑎 𝑖𝑗 and 𝑏 𝑖𝑗 as:

𝑟 𝑖𝑗 (𝑎 𝑖𝑗 , 𝑏 𝑖𝑗 ) = 1 𝑁-1 ∑ ( 𝑎 𝑖𝑗𝑘 -𝜇 𝑎 𝜎 𝑎 ) 𝑁 𝑘=1 ( 𝑏 𝑖𝑗𝑘 -𝜇 𝑏 𝜎 𝑏 ), (12)
where 𝜇 𝑎 , 𝜇 𝑏 , 𝜎 𝑎 and 𝜎 𝑏 are the local mean and the standard deviation of the vectors 𝑎 and 𝑏. A correlation value is obtained at each grid element, which is then mapped into a new correlation dataset. The values of r range from -1 to 1, where a value of -1 indicates a perfect negative linear correlation (i.e., anticorrelation), whereas +1 indicates a perfect positive linear correlation. A value of r = 0 means that there is no correlation [6,7].

Figure S5

. Schematic representation of the correlation method between two parameters A and B corresponding in our study to the curvature, K or H, and the polarization angle θ P .

Table S1. Summary of the statistical correlations between the domain wall curvature (either K or H), the charge density distribution (ρ excess and ρ bound ) and the variation of the polarization angle. The percentage (%) represents the domain wall surface fraction showing correlations.

We denote strong correlations (maximum positive correlation factor) as r p , and strong anticorrelations (maximum negative correlation factor) as r n . Intermediate and zero correlations are labeled r i and r 0 , respectively.

Strong correlations

S5 Problem formulation and FEM details

The bulk part of the LGD free energy functional G of a uniaxial ferroelectric LiNbO 3 includes a Landau energyan expansion on powers of 2-4-6 of its ferroelectric polarization 𝑃 3 , 𝐺 𝐿𝑎𝑛𝑑𝑎𝑢 , a polarization gradient energy, 𝐺 𝑔𝑟𝑎𝑑 ; an electrostatic energy, 𝐺 𝑒𝑙 ; an elastic, electrostriction contribution 𝐺 𝑒𝑠 , and a flexoelectric contribution, 𝐺 𝑓𝑙𝑒𝑥𝑜 . It has the form:

𝐺 = 𝐺 𝐿𝑎𝑛𝑑𝑎𝑢 + 𝐺 𝑔𝑟𝑎𝑑 + 𝐺 𝑒𝑙 + 𝐺 𝑓𝑙𝑒𝑥𝑜 + 𝐺 𝑓𝑙𝑒𝑥𝑜 , (13)
𝐺 𝐿𝑎𝑛𝑑𝑎𝑢 = ∫ 𝑑 3 𝑟 𝑉𝑐 [ 𝛼 2 𝑃 3 2 + 𝛽 4 𝑃 3 4 + 𝛾 6 𝑃 3 6 ], (14)
𝐺 𝑔𝑟𝑎𝑑 = ∫ 𝑑 3 𝑟 𝑉𝑐 𝑔 𝑖𝑗 2 𝜕𝑃 3 𝜕𝑥 𝑖 𝜕𝑃 3 𝜕𝑥 𝑗 , ( 15
)
𝐺 𝑒𝑙 = -∫ 𝑑 3 𝑟 𝑉𝑐 (𝑃 3 𝐸 3 + 𝜀 0 𝜀 𝑏 2 𝐸 𝑖 𝐸 𝑖 ), (16)
𝐺 𝑒𝑠 = -∫ 𝑑 3 𝑟 𝑉𝑐 ( 𝑠 𝑖𝑗𝑘𝑙 2 𝜎 𝑖𝑗 𝜎 𝑘𝑙 + 𝑄 𝑖𝑗33 𝜎 𝑖𝑗 𝑃 3 2 ), (17)
𝐺 𝑓𝑙𝑒𝑥𝑜 = -∫ 𝑑 3 𝑟 𝑉𝑐 𝐹 𝑖𝑗3𝑙 2 (𝜎 𝑖𝑗 𝜕𝑃 3 𝜕𝑥 𝑙 -𝑃 3 𝜕𝜎 𝑖𝑗 𝜕𝑥 𝑙 ). (18)
Here 𝑉 𝐶 is the volume of LiNbO 3 sample. The coefficient 𝛼 linearly depends on temperature T, 𝛼(𝑇) = 𝛼 𝑇 [𝑇 -𝑇 𝐶 ], where 𝛼 𝑇 is the inverse Curie-Weiss constant and 𝑇 𝐶 is the ferroelectric Curie temperature. The component 𝛽 and 𝛾 are regarded as temperatureindependent. The gradient coefficients 𝑔 𝑖𝑗 are positively defined and regarded as temperature-independent. In Eq.( 17), 𝜎 𝑖𝑗 is the stress tensor, 𝑠 𝑖𝑗𝑘𝑙 is the elastic compliances tensor, and 𝑄 𝑖𝑗𝑘𝑙 is the electrostriction tensor. In the Lifshitz invariant, Eq.( 18), 𝐹 𝑖𝑗𝑘𝑙 is the flexoelectric tensor.

In order to find the spatial distribution of the out-of-plane ferroelectric polarization component

𝑃
Here 𝜀 0 is the universal electric constant, 𝜀 𝑏 is a background permittivity [ 8 ]. There may be free charge is LNO. To account for it, the concentration of the ionized donors, 𝑁 𝑑 + , and free electrons, 𝑛, inside the semiconducting channel, which depend on the electric potential 𝜙 in the conventional way [ 9 ]:

𝑁 𝑑 + (𝜙) = 𝑁 𝑑 0 𝑓 ( -𝐸 𝑑 +𝑒𝑍 𝑑 𝜙+𝐸 𝑓 𝑘 𝐵 𝑇 ), (21)
𝑛(𝜙) = 𝑁 𝐶 𝐹 1 2 ⁄ ( 𝑒𝜙+𝐸 𝑓 -𝐸 𝐶 𝑘 𝐵 𝑇 ) (22)
Here 𝐸 𝑑 is the donor energy levels respectively, T is the absolute temperature, 𝑘 𝐵 is a Boltzmann constant, 𝐸 𝐶 is the position of conduction band bottom. The effective density of states in the conductive band 𝑁 𝐶 = (

𝑚 𝑛 𝑘 𝐵 𝑇 2𝜋ℏ 2 ) 3/2
, 𝑚 𝑛 is the effective mass of electrons. 𝐸 𝐹 is the Fermi energy level in equilibrium, which is determined from the electroneutrality condition 𝑍 𝑑 𝑁 𝑑 + -𝑛 = 0 at zero potential. In Eq.

Equations ( 23) should be solved along with equations of mechanical equilibrium 𝜕𝜎 𝑖𝑗 (𝒙) 𝜕𝑥 𝑖 ⁄ = 0, The finite element modeling (FEM) is performed in a COMSOL@MultiPhysics software, using electrostatics, solid mechanics, and general math (PDE toolbox) modules. The ferroelectric, dielectric, and elastic properties of the LiNbO 3 core are given in Table S2. * The estimation is based on the values of the spontaneous polarization and permittivity at room temperature ** the order of magnitude is estimated from the uncharged domain wall width [12].

To illustrate the effect of curvature on the polarization profile across the curved domain wall, and to take into account all above effects in a self-consistent way, we performed the FEM (see Figure S7). We consider the situation when the curved wall separates a nanodomain induced by the electric field of a biased heated tip, which is placed near the unscreened surface of LiNbO 3 . The inhomogeneous laser heating of the LiNbO 3 can stabilize the curved wall after the electric bias is switched off. We observed the evident correlation between the wall width, its bound charge and its curvature. As expected, the correlation is strongest under the absence of free screening charges. Namely, the strongly curved and charged part of wall is thick, the slightly curved part is thinner, and the straight uncharged domain wall is the thinnest, respectively. Moreover, we see that the strong flexoelectric coupling and high concentration of the screening charges significantly help the wall to maintain the curvature without making the wall thicker. So, it may seem that the curvature does not to act noticeably on the wall structure. However, it is not exactly so, because the screening by free charges compensates the monopolar and dipolar components of the bound charge, while the flexoelectric coupling defines the Neel-type dipolar component only and changes the wall elastic energy. The wall width correlates with its curvature, and the correlation is the strongest under the absence of screening charges (see Fig. S7). Plots S7a, b, c illustrate the shape of stable domains calculated by FEM for increasing tip voltage. Plots S7d, e, f show zoomed portions of the strongly curved and thick, slightly curved and thinner, and the thinnest straight domain walls, respectively. The polarization-dependent density G b can be written as a Taylor series expansion of the polarization components P i as:

Δ𝐺 𝑏 = 𝑎 1 (𝑃 )

𝜕𝑃 3 𝜕𝑥 2 ) ( 29
)
Table S3. Polarization expansion coefficients of LiNbO3 used in these simulations.

Table S4. Electrostriction coefficients and elastic stiffness coefficients of LiNbO3 used in these simulations.

FIG. S8.

Color map of the three polarization components P 1 , P 2 and P 3 in a non-tilted cylindrical-shape domain. The FEM calculations where conducted using a tetrahedral mesh for both "Open" (top row) and "closed" (bottom row) boundary conditions. Clear in-plane polarization is observed around the cylindrical-shape domain with more or less broadening, depending on the chosen boundary conditions.

FIG. S9.

Distribution of the in-plane polarization components P 1 , P 2 and the out-of-plane polarization P 3 in a vertical cross-section. Closure domains are observed at the near "open" surface and a weaker in-plane polarization is observed at the domain wall regions. This distribution is almost independent of the anisotropy of the LGD coefficients and the mesh type, but it strongly depends on the surface screening length . For instance, the closure domains practically vanish in the limit 0 ("closed" surface).

Figure1.

  Figure1. 3D SHG polarimetry analysis. (a) Schematic representation of the SHG microscope. (b) Isotropic SHG image displaying a rounded polygonal-shape c-domain in congruent Mg:LiNbO 3 . A small concentric hexagonal domain with reversed polarity is used as a reference for the crystallographic orientation of the Y-axis with respect to the (x,y,z) laboratory coordinates system. (c) Simulated SHG polarimetry responses expected in the case of straight Néel and Bloch-type domain walls oriented parallel to x-axis. (d) SHG polarimetry response measured at three positions (A, B, C in panel b) along the "arc-shaped" domain wall segment at a depth of z = 160 m below the surface. The experimental data (scattered dots) can be fitted very well by assuming a Néel model (continuous line). The presented simulations and experiments are obtained for a fundamental beam polarized along x-axis ( = 0°), as depicted by the pink arrow. (e) Pixel-by-pixel 3D polarimetry mapping of the curved domain wall portion connecting points A,B,C at different depths z (z 1 =52.5 m, z 2 =73.5 m, z 3 =107.1 m, z 4 =136.5 m, z 5 =168 m).

Figure 2 .

 2 Figure 2. (a) 3D SHG image measured in a sample volume of 20 × 8 × 125 μm 3 displaying the profile of a curved domain wall segment. The image was recorded at polarizer and analyzer angles  = 0° and α = 90°, respectively. (b) Reconstruction of the wall morphology based on the experimental data illustrating the internal polarization vector 𝑃 ⃗ and the normal vector of the domain wall surface 𝑛 ⃗ . (c) 3D polarization profile of the domain wall, derived from the pixel-by-pixel fitting of the 3D SHG polarimetry data as detailed in Supplementary note S2. As a comparison, the surface normal vector 𝑛 ⃗ is displayed in panel (d). The color map (purple-white-orange) in panels c-d represents the orientation angle of the vectors 𝑛 ⃗ and 𝑃 ⃗ taken with respect to the x-axis.

Figure 3 .

 3 Figure 3. (a) Schematic representation showing locally head-to-head or tail-to-tail bound charges due to the local tilting of the domain wall with respect to the polar axis. (b) 2D map of the bound charge density of the curved domain wall in units of the saturation polarization P S . (c) Schematic representation of the internal dipole distribution corresponding to an intrinsically charged Néel-like curved domain wall as viewed by SHG. (d) 2D representation of the variation of the dipole charge density depletion (red) or excess (blue) with the domain wall morphology given in units of the transverse in-plane polarization component P t .

Figure 4 .

 4 Figure 4. (a) Mean curvature H of the domain wall, (b) dipolar charge excess ( excess ) in units of the transverse polarization P t of the Néel-type domain wall, and (c) variation of the in-plane polarization angle. Local distribution of the statistical correlations (d) between  excess and H, and (e)  excess and the variation of the in-plane polarization angle. (f) Gaussian curvature K of the domain wall, (g) combined monopolar ( bound ) and dipolar ( excess ) charge density in units of P S P t , and (h) variation of the out-of-plane polarization angle. Local distribution of the statistical correlations (i) between K and the total charge density combining  excess and  bound , and (j) the total charge density and the variation of the out-of-plane polarization angle. In panels d,e,i, and j, the absence of correlations is characterized by a zero Pearson coefficient

  and the related text). The magnitude of the Néel-like dipolar component can be estimated as 𝑀 𝑑𝑖𝑝 ≈ (expressed as a function of the flexoelectric coefficient F 12 , the electrostriction coefficients Q ij , and the elastic compliances s ij ), and be written as a function of the principal curvatures 𝑘 1

Figure S1 .

 S1 Figure S1. (a) Stack of images recorded as a function of the analyzer angle  at a given depth (here, z = 162.5 µm below the top surface). The checkerboard-type pixel-by-pixel resolution is indicated with gray grid lines in (a). The polar-plots representing the variation of the SHG intensity over the stack of images at each pixel are displayed in panel (b).

Figure S2 .

 S2 Figure S2. 3D Domain wall geometry analysis obtained based on (a) SHG images obtained at a given depth (here, z = 162.5 µm below the surface of the sample). (b) The domain wall region is then vectorized, so that each pixel can be labeled by its (x, y) coordinates. The red line displayed in panel (c) is a polynomial fit of the central domain wall position 𝑦 = 𝑓(𝑥) in the xy-plane. (d) The wide blue line represents a 95% confidence band around the fitted domain wall positions. The 3D reconstruction of the domain wall geometry is obtained by repeating the above mentioned procedure (a-c) at each measurement depth, labeled z 1-5 . (e)

Figure S3 .

 S3 Figure S3. Principal curvatures (𝑘 1 and 𝑘 2 ) and principal directions (given by the unit vectors 𝑢 ̂ and 𝑣 ̂) at a given point p of the surface.

Figure S4 .

 S4 Figure S4. Visual comparison of Gaussian curvature (K) and mean curvature (H).

Figure S6 .

 S6 Figure S6. Tilt angle of the domain wall as derived from the 3D SHG image. The average value of the angle as derived from the tangent lying in the yz-plane at each point of the domain wall's surface is of 0.3°. The absolute value of the tilt angle varies locally from 0° to 1°.

  (21-22) we introduced the Fermi-Dirac distribution function and the Fermi integral, 𝑓(𝜉) of state for the strains 𝑢 𝑖𝑗 follow from the variation of the LGD free energy with respect to elastic stresses 𝜎 𝑖𝑗 : 𝑢 𝑖𝑗 = 𝑠 𝑖𝑗𝑘𝑙 𝜎 𝑘𝑙 + 𝛽 𝑖𝑗 (𝑇 -𝑇 0 ) + 𝐹 𝑖𝑗𝑘𝑙 𝜕𝑃 𝑙 𝜕𝑥 𝑘 + 𝑄 𝑖𝑗𝑘𝑙 𝑃 𝑘 𝑃 𝑙 , 0 ≤ 𝑧 ≤ ℎ.

  , and compatibility equations, 𝑒 𝑖𝑘𝑙 𝑒 𝑗𝑚𝑛 𝜕 2 𝑢 𝑙𝑛 (𝒙) 𝜕𝑥 𝑘 𝜕𝑥 𝑚 ⁄ = 0, which are equivalent to the continuity of elastic displacements 𝑈 𝑖 [ 1 0].

FIG. S7 .

 S7 FIG. S7. Spatial distributions (𝑥𝑧 cross-sections) of the polarization 𝑃 3 in a thick LiNbO 3 layer. Plots (a, b, c) illustrate the shape of stable domains calculated by FEM for increasing tip voltage. Plots (d, e, f) show zoomed pieces of the strongly curved, slightly curved and straight domain walls, respectively. Before the domain formation the layer was homogeneously polarized. Material parameters are listed in Table SI, 𝑇 =293 K.

  3 inside a uniaxial FE and the acting electric field 𝐸 𝑖 one should solve a coupled problem consisting of the LGD-type equation for 𝑃 3 and Poisson equation for electric potential 𝜙:

	[𝛼 𝑇 (𝑇 -𝑇 𝐶 ) -𝑄 𝑖𝑗33 𝜎 𝑖𝑗 ]𝑃 3 + 𝛽𝑃 3 3 + 𝛾𝑃 3 5 -𝑔 11	𝜕 2 𝑃 3 𝜕𝑧 2 -𝑔 44 ( 𝜕 2 𝑃 3 𝜕𝑥 2 +	𝜕 2 𝑃 3 𝜕𝑦 2 ) = 𝐸 3 -𝐹 𝑖𝑗𝑘3	𝜕𝜎 𝑖𝑗 𝜕𝑥 𝑘	.	(19)
	𝜀 0 𝜀 𝑏 ( 𝜕𝑥 2 + 𝜕 2	𝜕 2 𝜕𝑦 2 +	𝜕 2 𝜕𝑧 2 ) 𝜙 =	𝜕𝑃 3 𝜕𝑧 -𝑒(𝑍 𝑑 𝑁 𝑑 + -𝑛),

Table S2 .

 S2 LGD coefficients and other material parameters of a bulk LiNbO 3 𝑠 12 = 1.0110 12 , 𝑠 13 = 1.4710 12 , 𝑠 33 = +5.0210 12 , 𝑠 14 = 1.0210 12 , 𝑠 44 = 17.1010 12 [13] 𝑄 𝑖𝑗 m 4 /C 2 Q 33 = +0.016, Q 13 = 0.003 𝐹 𝑖𝑗 m 3 /C F 11 =1.010 11 , F 12 =0.910 11 , F 44 =310 11 (estimate)

	Parameter Dimension	Values for LiNbO 3 collected from Refs.[11-16]
	 b	1	4.6 [9]
	 T	m/(F K)	1.56910 6 [10]
	T C	K	1477 Ref. [11]
	𝛽	C -4 •m 5 J	2.3110 9*
	𝛾	C -6 •m 9 J	1.7610 9*
	𝑔 𝑖𝑗	m 3 /F	𝑔 44 =7.9610 -11 ** [12]
	𝑠 𝑖𝑗	1/Pa	𝑠 11 =5.7810 12 ,

  The electrostriction contribution to the free energy is𝛥𝐺 strict = -𝑄 11 (𝜎 1 𝑃 1 2 + 𝜎 2 𝑃 2 2 ) -𝑄 33 𝜎 3 𝑃 3 2 -𝑄 12 [𝜎 1 𝑃 2 2 + 𝜎 2 𝑃1 2 ] -𝑄 13 (𝑃 1 2 + 𝑃 2 2 )𝜎 3 -𝑄 31 𝑃 3 2 (𝜎 1 + 𝜎 2 ) -𝑄 44 (𝜎 4 𝑃 2 𝑃 3 + 𝜎 5 𝑃 3 𝑃 1 ) -𝑄 51 𝑃 3 (𝑃 1 (𝜎 1 -𝜎 2 ) -2𝑃 2 𝜎 6 ) -𝑄 66 𝑃 1 𝑃 2 𝜎 6 (27), 𝑠 12 𝜎 1 𝜎 2 -𝑠 13 (𝜎 2 𝜎 3 + 𝜎 3 𝜎 1 ) -

			1 2 + 𝑃 2 2 ) + 𝑎 3 𝑃 3 2 + 𝑎 11 (𝑃 1 2 + 𝑃 2 2 ) 2 + 𝑎 13 (𝑃 1 2 + 𝑃 2 2 )𝑃 3 2 + 𝑎 15 𝑃 1 (𝑃 1 2 -3𝑃 2 2 )𝑃 3 + 𝑎 33 𝑃 3 4 +
	𝑎 333 𝑃 3 6																			(26)
	and the elastic energy is
	𝛥𝐺 elast = -	1 2 𝑠 11 (𝜎 1 2 + 𝜎 2 2 ) -	1 2 𝑠 33 𝜎 3 2 -1 2 𝑠 44 (𝜎 4 2 + 𝜎 5 2 ) -
	𝑠 15 ((𝜎 1 -𝜎 22 )𝜎 5 -2𝜎 4 𝜎 6 ) -	1 2	𝑠 66 𝜎 6 2			(28)
	The polarization gradient contribution to free energy (Ginzburg term) is
	𝛥𝐺 inhomo =		𝑔 11 2	(( 𝜕𝑥 1 𝜕𝑃 1	) 2	+ ( 𝜕𝑥 2 𝜕𝑃 2	) 2	+ ( 𝜕𝑥 3 𝜕𝑃 3	) 2	) + 𝑔 12 ( 𝜕𝑥 1 𝜕𝑃 1	𝜕𝑃 2 𝜕𝑥 2	+	𝜕𝑃 1 𝜕𝑥 1	𝜕𝑃 3 𝜕𝑥 3	+	𝜕𝑃 3 𝜕𝑥 3	𝜕𝑃 2 𝜕𝑥 2	) +	𝑔 44 2	((	𝜕𝑃 1 𝜕𝑥 2	+
	𝜕𝑃 2 𝜕𝑥 1	) 2	+ ( 𝜕𝑥 3 𝜕𝑃 2	+	𝜕𝑃 3 𝜕𝑥 2	) 2	+ ( 𝜕𝑥 1 𝜕𝑃 3	+	𝜕𝑃 1 𝜕𝑥 3	) 2	) + 𝑔 15 ({ 𝜕𝑥 1 𝜕𝑃 1	-	𝜕𝑃 2 𝜕𝑥 2	}	𝜕𝑃 3 𝜕𝑥 1	-( 𝜕𝑥 2 𝜕𝑃 1	+	𝜕𝑃 2 𝜕𝑥 1

ACKNOWLEDGMENTS

This work was supported by the French National Research Agency (ANR) through the "TOPELEC" project (ANR-18-CE92-0052) cofounded by the Deutsche Forschungsgemeinschaft (DFG) (EN-434/41-1). S.C.H. acknowledges the Interdisciplinary Thematic Institute EUR QMat (ANR-17-EURE-0024), as part of the ITI 2021-2028 program supported by the IdEx Unistra (ANR-10-IDEX-0002) and SFRI STRAT'US (ANR-20-SFRI-0012) through the French Programme d'Investissement d'Avenir. A.N.M. acknowledges EOARD project 9IOE063 and related STCU partner project P751. E.S., H.B., M.R., and L.M.E. acknowledge financial support by the DFG within the FOR5044 (ID: 426703838) and the Würzburg-Dresden Cluster of Excellence on "Complexity and Topology in 409 Quantum Matter"ct.qmat (EXC 2147; ID: 39085490) as well as support by the Light Microscopy Facility, a Core Facility of the CMCB Technology Platform at TU Dresden.

ASSOCIATED CONTENT

Supporting Information

Detailed information on the pixel-by-pixel polarimetry analysis; the determination of the domain wall curvature and polarization profile based on 3D SHG microscopy data; definition of Gaussian and mean curvatures; description of the statistical correlations and summary of the results; theoretical aspects (problem formulation and FEM details) as well as 3D simulation results.

AUTHOR INFORMATION

Corresponding Authors

Salia Cherifi-Hertel Université de Strasbourg, CNRS, Institut de Physique et Chimie des Matériaux de Strasbourg, UMR 7504, 67034, Strasbourg, France ; orcid.org/0000-0002-9617-2098; Email: salia.cherifi@ipcms.unistra.fr

Supporting Information

Impact of 3D curvature on the polarization orientation in non-

Ising domain walls

Ulises Acevedo-Salas 1 , Boris Croes 1 , Yide Zhang