Anass Bakour

Omar Bajjou
email: bajjou.omar@gmail.com

Mohamed Khenfouch

Mimouna Baitoul

Bakang Mothudi

Malik Maaza

Jany Wéry

Eric Faulques

Optical absorption and electron dynamics in reduced graphene oxidenanostructured porphyrin for active solar cell layers

Keywords: Reduced graphene oxide, Porphyrin, XRD, Luminescence, Absorption, Exciton lifetime

This work outlines a study to gain further understanding of the interactions of two interesting materials, the reduced graphene oxide, and porphyrin nanostructured. We seek to address the changes occurring in morphological and opto-structural properties of the resultant composite material using different spectro-scopies and by changing pH values. The structural analysis by X-ray diffraction suggests an enhancement of the composite structure, which is further confirmed by the scanning electron microscopy. The acidity effect on the resulting properties was investigated by preparing multiple samples with different pH val-ues. The optical properties highlighted the photoluminescence intensities quenching in the composite for optimum pH values, which is a favourable property for inserting in energy applications. Furthermore, the study of electron-hole pair dynamics was conducted by means of Time resolved photoluminescence. Obtained results prove that the excitons exhibit a longer lifetime for the composite sample and therefore confirming the electron transfer from porphyrin nanostructured to reduced graphene.

Introduction

The two dimension allotrope of carbon, presents one of the most widespread and studied materials in this field since its discovery in 2004 [START_REF] Geim | The rise of graphene[END_REF], thanks to the its outstanding physical and opti-cal properties as well as its potential of use in a multitude of electronic and medical fields [START_REF] Liu | Structure, energy, and structural transformations of graphene grain boundaries from atomistic simulations[END_REF][START_REF] Brownson | Graphene electrochemistry: fundamental concepts through to prominent applications[END_REF][START_REF] Zeng | Electroactive cellulose-supported graphene oxide interlayers for Li-S batteries[END_REF][START_REF] Yen | Preparation of graphene/multi-walled carbon nanotube hybrid and its use as photoanodes of dye-sensitized solar cells[END_REF].

The graphene synthesis techniques has known great develop-ment in the last years, depending on the desired application varying from epitaxial growth, chemical vapor deposition, and chemical methods that include the synthesis via Hummers method, that we used in this work [START_REF] Wang | Preparation of graphene nanosheets through detonation[END_REF][START_REF] Omar | venturini photoluminescence quenching and structure of nanocomposite based on graphene oxide layers decorated with nanostructured porphyrin[END_REF][START_REF] Singh | Microwave absorbing properties of a thermally reduced graphene oxide/nitrile butadiene rubber composite[END_REF]. The tuning of the final properties of graphene is achieved by functionalization and mixing with other organic and inorganic materials [START_REF] Tang | Structure-stability relationships for graphene-wrapped fullerene-coated carbon nanotubes[END_REF][START_REF] Einarsson | 2 -Synthesis of carbon nanotubes and graphene for photonic applications[END_REF].

Reduced graphene oxide (RGO) is considered as an efficient and easy-synthesis substitute for graphene, thanks to the possibility of production via a chemical reduction of graphene oxide (GO) already prepared by oxidation of pristine graphite using Hummer's method [START_REF] Bajjou | Structural and photophysical studies of few layers of reduced graphene oxide functionalized with Sn(IV) tetrakis (4-pyridyl)porphyrin dichloride[END_REF][START_REF] Anass | Improving optical properties of in situ reduced graphene oxide/poly(3-hexylthiophene) composites[END_REF].

Porphyrins present a kind of conjugated organic molecules based on macrocycle pyrroles linked via methine bonds [START_REF] Wang | Porphyrin nanotubes by ionic self-assembly[END_REF][START_REF] Xu | Synthesis, characterization and properties of covalently linked porphyrin-naphthalimide pentamer and its metal complexes[END_REF]. They have been widely used in composites because of their trans-ferable properties to carbon nanomaterials such as carbon nan-otubes and graphene, via functionalization. The strong conjugation of porphyrin comes from the high number of delocal-ized electrons offering great photophysical properties allowing successful interactions with other conjugated materials [START_REF] Radivojevic | Self-organized nanofibers and nanorods of porphyrins bearing hydrogen bonding motifs[END_REF][START_REF] Serpone | Photoluminescence and transient spectroscopy of free base porphyrin aggregates[END_REF][START_REF] Angaridis | Functionalized porphyrin derivatives for solar energy conversion[END_REF].

The porphyrin, with an extended p-bond system, was mixed with RGO samples that plays the role of electron acceptors thanks to its low bandgap [START_REF] Murali Krishna | Synthesis and structural, spectroscopic and nonlinear optical measurements of graphene oxide and its composites with metal and metal free porphyrins[END_REF], has given rise to a composite with enhanced optical properties translated with higher absorption intensities and photoluminescence (PL) quenching. Electron-hole dynamics were further investigated by time resolved photolumi-nescence (TRPL), and showed the enhancement of transport and relaxation of the photogenerated excitons. pH effect was investi-gated to better determine the factors affecting the optical and elec-tronics processes. In this paper, we report also the modification of structure and surface by means of X-Ray Diffraction (XRD) and Scanning electron microscopy (SEM).

Setup

We proceed by synthesizing graphene oxide from graphite powder using Hummer's method [START_REF] Bajjou | Structural and photophysical studies of few layers of reduced graphene oxide functionalized with Sn(IV) tetrakis (4-pyridyl)porphyrin dichloride[END_REF]. After the GO solution is prepared, a reduction step is fulfilled by using hydrazine hydrate. The obtained RGO solution submitted a sonication process in order to better exfoliate the graphene sheets. Porphyrin nanostructured, synthesis procedure is described in our early works [START_REF] Omar | venturini photoluminescence quenching and structure of nanocomposite based on graphene oxide layers decorated with nanostructured porphyrin[END_REF][START_REF] Khenfouch | Photoluminescence and dynamics of excitation relaxation in graphene oxide-porphyrin nanorods composite[END_REF]. Afterwards the final nanocomposite was further sonicated at room tem-perature for 30 min to ensure homogeneity, and drop casted finally on glass substrates.

X-ray diffraction (XRD) measurements were performed with Bruker's D8 advanced X-ray diffractometer using CuKa radiation (k = 1.5418° Å). SEM images were obtained using a JEOL JSM 7600F. UV-Vis spectra were measured with a PerkinElmer LAMBDA 950/1050 UV/VIS/NIR spectrophotometers are dual-beam, dual-monochromator instruments equipped with an optional 150 mm integrating sphere accessory. The steady-state photoluminescence spectra were recorded using a Jobin-Yvon flu-orolog 3 with kexc = 420 nm and luminescence dynamics spectra were obtained at room temperature using a Jobin-Yvon Fluorolog spectrometer with a CCD detector.

Results and discussion

X-Ray diffraction

XRD is used to understand the structural order and its modifica-tion in the RGO-PN composite samples. Fig. 1. shows the XRD pat-terns of RGO, PN, and RGO-PN composite films. We can see that the characteristic diffraction peak of RGO arises at 2h = 23.4°, which is attributed to the diffraction from the (0 0 2) plane [START_REF] Cheng | Synthesis of graphene paper from pyrolyzed asphalt[END_REF][START_REF] Del Pino | Laser-induced chemical transformation of graphene oxide-iron oxide nanoparticles composites deposited on polymer substrates[END_REF][START_REF] Yeon | Enhanced anode performance of micro/ meso-porous reduced graphene oxide prepared from carbide-derived carbon for energy storage devices[END_REF]. This diffraction peak is wide and thus indicating a good reduction of GO, giving therefore rise to the irregularity in the interlayer spac-ing at the edges or inside the RGO sheets. In the RGO-PN composite pattern we can see the characteristic (0 0 2) of the RGO at 2h = 23.4° (dspacing = 3.78 nm) interfering with the PN peaks. The RGO diffraction peaks intensity exhibit a slight decrease in the composite sample, meaning that the presence of PN changes the structure of RGO which is proved by the increase in d-spacing between the RGO sheets. The increase in the inter-planar distance can be attributed to the effect of PN trapped in RGO layers leading to their expansion and eventually their exfoliation. These results will be better conceived by scanning electron microscopy results.

Morphological analysis

Scanning Electron Microscopy images of the RGO, PN and RGO-PN composite films are presented in Fig. 2. The RGO sample pre-sented in Fig. 2a exhibits the wrinkled sheet structure generally found in literature [START_REF] Remyamol | Synthesis and nonlinear optical properties of reduced graphene oxide covalently functionalized with polyaniline[END_REF].

The PN sample is presented in Fig. 2b shows a grain structure agglomerated with a good homogeneity in the surface. The com-posite exhibited in Fig. 2c shows a modified RGO sheet structure with small PN particles embedded within. Brighter zones in the composite image could be attributed to the arising edges or entire RGO sheets, proving the increase in d-spacing between the RGO sheets already discussed. These results can be explained by the presence of electronic interactions between reduced graphene oxide sheets and PN that will be further investigated by optical studies.

UV-Visible absorption

The UV-Visible absorption spectroscopy is an important tool to gain a better insight in the optical processes taking place within the composite and the resulting interaction as well as their effect on the optical properties. We conducted a study based on the variation of pH value as a key factor, and observed the absorption by means of UV-Vis absorption spectroscopy on the RGO and RGO-PN solutions at different pH values. In Fig. 3a we can observe in the acidic range from pH 2 to 5 that the characteristic peak of RGO is more pronounced and better resolved, however in the basic range we can see the rise of a shoulder around 300 nm especially when reaching pH 10, as well as a widening of the characteristic band of RGO. The change of the characteristic band at 250 nm is explained by the restoration of RGO's conjugated structure [START_REF] Oldfield | Multilayered graphene films prepared at moderate temperatures using energetic physical vapour deposition[END_REF][START_REF] Wang | ZnO disks loaded with reduced graphene oxide for the photodegradation of methylene blue[END_REF][START_REF] Tian | Optical and electronic properties of thin films and solutions of functionalized forms of graphene and related carbon materials[END_REF]. The absorption intensity of the RGO increases gradually with pH value. The RGO UV-Vis absorption spectra presented a depen-dency of the pH value, which can be attributed to p-p stacking property which can play an important role in the absorption process.

The RGO-PN composite spectra for different pH values are shown in Fig. 3b. For the acidic range from pH 2 to 7 we can always see the RGO characteristic peak around 240 nm and the rise of a new peak and a small shoulder attributed to the PN. Remarkably, the same observations made for the 240 nm peak in the RGO can be attributed to the composite in the acidic range, while an increase in the 420 nm is noted due to both, the PN component and the pH higher levels.

In Fig. 3b in the basic range we observe a declining trend of the intensity of the PN characteristic peak that goes in reverse to the increase trend already observed in the acidic region. This behavior can allow to fix an optimal level of pH value for a maximum level of absorption for the composite which lays around the pH 8. This can be explained by the restoration of the conjugated structure in addition to the improvement of the interaction between the RGO sheets and PN when pH value is increasing up to a pH 8 level.

Steady state photoluminescence

The PL spectroscopy is an effective way to probe the electronic structure of materials. Steady state PL measurements were conducted in order to observe the evolution of PL spectra of the RGO and PN samples and how it relates to the generated electronhole pairs properties. Fig. 4 shows the PL spectra of RGO, PN and the PN-RGO composite.

The Fig. 4 exhibits PN spectrum, that is characterized by two key bands at 650 and 714 nm [START_REF] Omar | venturini photoluminescence quenching and structure of nanocomposite based on graphene oxide layers decorated with nanostructured porphyrin[END_REF], while the RGO sample shows a large emission over the red and near infrared region, rather than distinct bands. The compound PL spectrum present two main peaks at 620 and 690 nm, these emissions can be attributed to the presence of isolated sp2 groups in the sp3 carbon matrix, as well as resulting from the electron recombination already generated on the PN, with the holes generated and concentrated in the RGO surface [START_REF] Bajjou | Structural and photophysical studies of few layers of reduced graphene oxide functionalized with Sn(IV) tetrakis (4-pyridyl)porphyrin dichloride[END_REF][START_REF] Matsuda | 1 -Fundamental optical properties of carbon nanotubes and graphene[END_REF]. The composite PL spectrum exhibits a more distinct emission peak at 650 nm which is attributed to higher absorption in the composite as well as the interface interaction between the RGO and the PN which facilitates the emission of the already absorbed energy. This intensity of the composite is obtained for pH7 which makes it a perfect value for both application in absorption and emission purposes.

The quenching observed for the composite PL spectrum can be attributed to the interaction between RGO and PN. This extinction can be explained by an efficient charge/energy transfer between the two components.

3D PL mapping

The 3D photoluminescence mapping of the RGO, PN and RGO-PN composite is shown in Fig. 5. the false color images show qual-itatively that the PL emission intensity presents some changes when RGO is inserted in the PN matrix. These results confirm the quenching effect already observed in the steady state PL in the previous section. The photogenerated excitons decline spectra are subsequently extracted from these 3D PL mapping images.

TRPl

TR-PL spectroscopy is a strong tool to extract useful information about the dynamics and lifetimes of photogenerated excitons as well as the mechanisms taking place between RGO and PN compo-nents in the excited state. The PL decay measurements of PN and the RGO-PN composite in solutions were extracted from the false color image in Fig. 5. which are excited at a wavelength of 400 nm and a probe wavelength of 620 nm Fig. 6.

The PN component exhibits two different exciton lifetimes obtained for s1 = 16 ps and s2 = 315 ps mentioned in Table 1. The key changes in exciton dynamics created in PN after their interac-tion with RGO was perceived in the increase in the larger compo-nent s2, which increased from 315 to 396 ps Table 1. This can be interpreted by the alteration of their energy band structure due to the role of RGO related to the size of the sp2 clusters, leading to a retardation in recombination of the interband carriers [START_REF] Velasco-Soto | Selective band gap manipulation of graphene oxide by its reduction with mild reagents[END_REF].

The TRPL measurements bring into light the important effect of PN particle size and their aggregation on the excitons lifetime. Consequently we can infer a considerable change of the composite optical properties. Furthermore, the results of this work showed notable emission extinction and an increase in the average lifetime of excitons. These results constitute a proof to the presence of energy/ charge transfer between PN and RGO sheets, which is very appealing for optical applications [START_REF] Liang | Novel D-p-A structured porphyrin dyes containing various diarylamino moieties for dye-sensitized solar cells[END_REF][START_REF] Mongwaketsi | Synthesis and characterization of porphyrin nanotubes/rods for solar radiation harvesting and solar cells[END_REF].

Conclusion

To sum up, our results show that RGO-PN composite presents an enhanced structural order as well as better absorption and emission properties. XRD spectroscopy results imply the improve-ment of the overall structure after inserting RGO in the PN Matrix. The pH effect study suggests an enhanced optical absorption for higher pH up to a slightly basic value. Furthermore, the results of the PL measurements in steady state and time resolved showed a notable emission quenching and an increase in the photogenerated excitons lifetime. This study have provided further evidence of an overall enhancement of the optical properties that can be achieved by monitoring the pH value in order to obtain electron transfer between the two components of the composite.

Declaration of Competing Interest

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

Fig. 1 .

 1 Fig.1. XRD patterns of RGO, PN, and RGO-PN composite films.

Fig. 2 .

 2 Fig. 2. SEM images of: (a) RGO, (b) PN and (c) RGO-PN composite films.

Fig. 3 .

 3 Fig. 3. (a) UV-Vis absorption of RGO (b) RGO-PN composite.

Fig. 4 .

 4 Fig. 4. PL spectra of RGO, PN and the PN-RGO composite.

Fig. 5 .

 5 Fig. 5. False color images of RGO, PN and RGO-PN.

Fig. 6 .

 6 Fig. 6. The PL decay time spectra of PN and the RGO-PN composite.

Table 1

 1 Excitons lifetime of PN and RGO-PN.

	Sample	s1(a1 )	s2 (a2)	smoy(ps)
	PORPHYRIN	16(2.20)	315(0.23)	217
	RGO-PN	27(3.12)	396(0.41)	270

Acknowledgements

This work was supported by UNISA Africa Chair in Nanosciences-Nanotechnology, College of Graduate Studies, University of South Africa, Muckleneuk ridge, PO Box 392, Pretoria, South Africa, as well as the Institut des Matériaux Jean Rouxel, Nantes, 2 rue de la Houssinière, BP32229, 44322 Nantes, Cedex3, France.


Alexandre Bucamp

Christophe Coinon

David Troadec

Sylvie Lepilliet

Gilles Patriarche

Xavier Wallart

Ludovic Desplanque

TABLE OF CONTENTS (TOC) Gate length dependent transport properties of inplane core-shell nanowires with raised contacts

Keywords: Molecular Beam Epitaxy, core-shell nanowire, selective area growth, effective electron mobility

published or not. The documents may come

Gate length dependent transport properties of in-plane

core-shell nanowires with raised contacts

Introduction

During the last fifteen years, many research efforts have targeted the fabrication of efficient MOSFETs based on low effective mass III-V materials that could reduce the power consumption of CMOS devices. Among these developments, metal-organic based nanoscale selective area epitaxy has demonstrated its efficiency for the integration of III-Vs on silicon [START_REF] Waldron | Digest of Technical Papers -Symposium on VLSI Technology[END_REF][START_REF] Borg | [END_REF][3], for the gate stack engineering [START_REF] Tomioka | Device Research Conference -Conference Digest[END_REF], for self-aligned contact definition [START_REF] Egard | [END_REF] or for the elaboration of quantum confined nanostructures [6][7][8]. More recently, molecular beam epitaxy (MBE) has also proven its usefulness for fabricating InAs or InSb in-plane nanowires (NW) [9][10][11][12][13][14] that combined with the in-situ deposition of an aluminum superconducting shell could produce hybrid mesoscopic devices for quantum bit manipulation [15]. Thus, even if silicon has remained the material of choice for CMOS, the technological breakthroughs that have been developed offer an interesting playground for the fabrication of nanoscale devices based on III-V materials with strong spin orbit coupling for quantum technologies. In this frame, MBE grown InGaAs presents an interesting trade-off between electron effective mass, spin orbit coupling and structural quality because it can be associated with lattice matched InP barrier and semi-insulating substrate and can be grown in the form of in-plane nanostructures with good electrical properties [16]. In this paper, we show that surrounding the InGaAs in-plane nanowire core by an InP shell results in a low interface trap density at the oxide/semiconductor interface without any ex-situ passivation treatment. Together with raised source and drain Si doped InGaAs contacts the resulting nanostructure turns out to be an efficient MOSFET with a scalable gate length down to 30 nm. Electrical characterization of the transistors is performed down to 4K and we particularly investigate the impact of the gate length on the gate coupling efficiency and the effective electron mobility.

Experimental

The fabrication of the three-dimensional (3D) architectures of InGaAs/InP in-plane NWs requires a two-step selective area MBE growth. Semi-insulating InP (001) substrates are first patterned with a silicon dioxide mask following the process described in [17]. 3 µm long and 1 µm down to 50 nm wide openings in SiO2 oriented both in [110] and [START_REF] Waldron | Digest of Technical Papers -Symposium on VLSI Technology[END_REF][START_REF] Borg | [END_REF][3][START_REF] Tomioka | Device Research Conference -Conference Digest[END_REF][START_REF] Egard | [END_REF][6][7][8][9][10] directions are defined for the first epitaxy. After in-situ III-V native oxides removal by annealing the sample at 500°C under atomic hydrogen and arsenic flux, in-plane InGaAs NWs are grown selectively with respect to the silicon dioxide mask using atomic hydrogen assisted MBE [16]. An InP shell with a 10 nm nominal thickness is then added after transfer of the sample under ultra-high vacuum in another MBE system equipped with a phosphine thermal cracker cell. Details about the selective area MBE growth conditions for the InGaAs and InP layers can be found in references [13] and [18] respectively. The nominal deposited thickness of undoped InGaAs and InP are respectively 30 and 10nm. Scanning Electron Micrograph (SEM) of an array of in-plane coreshell nanowires grown inside 100 nm openings can be seen on figure 1a. After growth, a part of the nanostructures is masked using hydrogenosilsesquioxane resist (HSQ) ribbons (figure 1b). These 1 µm down to 30 nm wide ribbons are patterned with the spin-coating of an 80 nm thick layer of Dow Corning XR-1541 resist, e-beam exposure and tetramethylammonium hydroxide based development. A subsequent 200°C annealing under N2/H2 is performed before the introduction of the sample into the outgazing chamber where it is annealed under ultra-high vacuum for 90 minutes. After III-V native oxides removal as for the first epitaxy, the HSQ ribbons serve as a dummy gate for the subsequent selective area MBE growth of silicon doped InGaAs contacts with nominal thickness and doping of 30nm and 2x10 19 cm -3 respectively. After source and drain metal contact deposition using standard e-beam lithography, HSQ ribbons are removed using a buffered hydrofluoric acid solution (figures 1c-e). A 7 nm thick Al2O3 layer is then deposited using Atomic Layer Deposition at 300°C in a Benecq TFS200 system. Eventually a Ti/Au gate metallization is performed to obtain MOSFET devices (figures S1a and S1b). The longitudinal dimensions of the devices are schemed in figure 1f. Transmission Electron Microscope cross sections are performed after the whole process by means of Focus Ion Beam (FIB) etching. The STEM studies are carried out on a Titan Themis 200 (FEI) microscope equipped with a spherical aberration corrector on the probe and the EDX analysis system super X (0.7 sr solid angle of collection). The accelerative voltage was 200 kV. The (half) convergence angle of the probe was 17.6 mrad and the probe current 85 pA.

Results and discussion

The scanning transmission electron micrograph (STEM) and EDX chemical analysis of the cross sections taken off in the channel area of transistors grown inside 100 nm wide openings reveal that along [110] direction (figure 2), the InGaAs core exhibits a half cylinder shape with a radius of approximately 50 nm whereas the InP shell is roughly delimited by a semi-hexagonal contour formed by (1-10), ( 111) and (001) facets. The composition profile of III and V elements constituting the structure and deduced from EDX analysis indicates that the indium content in the InGaAs core is about 60% at the basis but evolves to 70-75% at the extreme top of the nanostructure (figure 2b). The rounded shape of the InGaAs core with the absence of a well-defined top (001) facets is quite different than the one observed for shell-free In0.53Ga0.47As in-plane nanowires grown with the same conditions [16]. This difference is probably resulting from strain induced indium migration from side facets to the top one.

In the [1-10] direction (supporting information SI-1, figure S1.1), the shape of the InGaAs core is quite similar but the composition grading is reversed since a slight decrease of the indium content is observed from the bottom to the top of the core (figure S1.1b). Concerning the InP shell, it forms mainly (111) facets with a preferential deposition of InP on the top leading to a triangular shape for the nanostructure. The prevalence of these facets can be attributed to the different InP surface energies between (111)A, (001) and (111)B facets [18]. For larger openings (w=500 nm and w=1 µm), we assume that InGaAs exibits mainly a (001) surface (supporting information SI-1, figure S1.2). The MOSFET transfer characteristics measured at 300K for a drain to source voltage of 50 mV are displayed on figure 3 for single NWs with a 500 nm gate length grown inside 1 µm down to 50 nm wide openings and oriented in the [110] azimuth. The characteristics for the other azimuth can be found in supplementary information SI-2. As can be seen, the geometry of the opening and thus the shape of the nanostructure deeply impacts the threshold voltage of the transistor which is about -0.8V for large opening and shifted toward positive voltage when the size of the (001) surface of the nanowires is shrinking with respect to the lateral facets. This evolution is quite similar for both crystal directions although a little more pronounced for the [110] oriented devices which also exhibit an improvement of the subthreshold slope when the width is reduced. However, for the 50 nm wide opening, a degradation of the conductance slope is observed above the threshold probably because of poor ohmic contacts. In the following, we focus on nanowires grown inside 100 nm wide openings in the [110] direction.

From these nanowires, transistors with different gate length (defined by the HSQ ribbon width) from 1 µm down to 30 nm have been fabricated and their electrical properties above and below the threshold voltage are investigated. Figure 4a-d displays the ID=f(VDS) characteristics of these devices. In the linear regime (at VDS=10mV), we extract the ON resistance for a fixed gate voltage 0.4 V above the threshold and different temperatures (figure 4e). The linear interpolations of these data enable the identification of the mean linear resistance of the channel and the mean access resistance with respect to the temperature (figure 4f). One can observe that for short channels, the ON state characteristics are mainly governed by the access resistances which exhibit some variations from one device to another. One can also notice that the mean channel resistance decreases when the temperature is reduced, probably related to an improvement of the electron mobility at low temperature. 1) and (2) using the parameters µlong and K described on (d) with respect to the temperature.

In order to extract the effective electron mobility µ0 for each device independently from the access resistance, we used the model proposed by Ghibaudo et al [19] involving a mobility correction factor  taking into account series resistance and gate voltage impact to fit our experimental transfer characteristics (figure 5a). This model is based on the following equations [19]:

𝐼 𝐷 = 𝑊 𝐺 .𝐶 𝑖 .𝜇 0 𝐿 𝐺 (1+𝜃(𝑉 𝐺𝑆 -𝑉 𝑇 )) (𝑉 𝐺𝑆 -𝑉 𝑇 ). 𝑉 𝐷𝑆 Eq. ( 1
)
with 𝜃 = 𝜃 0 + 𝑅 𝑆 . 𝐶 𝑖 . 𝜇 0 .

𝑊 𝐺 𝐿 𝐺 Eq. ( 2
)
where WG is the gate development (length of the transverse core/shell interface), Ci is the gate capacitance per unit area formed by the 7 nm thick Al2O3 layer (supporting information SI-3) and the InP shell. Assuming a mean thickness of about 15 nm for the shell, a value of 4.56 mF/m² is estimated for Ci. Rs is the series resistance and 𝜃 0 is the intrinsic mobility reduction factor. The values of  used for the fit as well as their evolution with the temperature are summarized in the supplementary materials SI-4. Regarding the evolution of the electron mobility µ0 with the gate length, a striking degradation can be noticed for short gate length. The highest mobility is indeed obtained for the device with a 1 µm gate length increasing from about 1500 cm 2 .V -1 .s -1 at room temperature to reach more than 2700 cm 2 .V -1 .s -1 at 4K. Although reduced, the electron mobility for the device with a 500 nm gate length exhibits a similar behavior. However, for devices with a gate length of 100 nm and below, a much lower value is achieved and no more evolution with the temperature is observed. On figure 5c, this evolution is fitted using the Matthiessen rule involving two different mobility [19]: a diffusion limited classical one µlong independent of the gate length which prevails for long gate length, and a term µshort proportional to the gate length. The relationship between µ0, µlong and µshort is given by:

µ 0 = ( 1 µ 𝑙𝑜𝑛𝑔 + 1 µ 𝑠ℎ𝑜𝑟𝑡 ) -1
Eq. ( 3) with µshort = K.LG Eq. ( 4)

The values of µlong and K corresponding to the dashed lines of figure 5c are summarized in figure 5d. We can notice that the electron mobility is expected to reach 4500 cm 2 -1 .s -1 at 4K for long NWs and that the K parameter is almost constant (about 6.5 cm 2 .V -1 .s - 1 .nm -1 ) with the temperature. A quasi-linear dependency of the effective electron mobility with the gate length for nanoscale devices has already been observed in different field effect transistors with various interpretations [20][21][START_REF] Karatsori | IEEE Int. Eletron. Dev. Meeting (IEDM)[END_REF].

Shur et al interpret this phenomenon as an expression of ballistic transport between source and drain in the case of AlGaAs/GaAs HEMTs with submicron gate length [20]. They introduce a term of effective "ballistic" mobility µbal proportional to the gate length (µbal= Kbal.LG) by substituting the electron mean free path by the gate length in the expression of the mean scattering time. The value of the ballistic coefficient Kbal is thus only dependent of the injection velocity of electrons from the source, taken as the thermal velocity vth. Kbal can then be expressed from the electron effective mass m*, the temperature T, the Bolztmann constant kB and the elementary charge q according to the following equations:

𝐾 𝑏𝑎𝑙 = 2.𝑞 𝜋.𝑚 * .𝑣 𝑡ℎ
Eq. ( 5) and 𝑣 𝑡ℎ = √ 8.𝑘 𝐵 .𝑇 𝜋.𝑚 * Eq.( 6) However, considering an electron effective mass of InGaAs of about 0.04 m0, the calculated values of Kbal using these expressions are much higher than the one deduced experimentally from our K coefficient. The discrepancy is about one order of magnitude at 300K and even much higher at low temperature (Supporting information SI-5). Even if the electron mobility obtained for a gate length of 1 µm is compatible with a ballistic transport over a length of 30 nm, it seems that another scattering phenomenon related to raised source and drain contacts affects the electron mean free path for short gate length. A similar interpretation was given by Karatsori et al in the case of strained or fully depleted SOI MOSFET and InGaAs on insulator FinFET with comparable electron mobility values [START_REF] Karatsori | IEEE Int. Eletron. Dev. Meeting (IEDM)[END_REF]. The diffusion of neutral defects from source and drain areas during the process was suspected. Such a mechanism is unlikely to happen in our case considering the quite low thermal budget of fabrication and the low diffusion of Si in InGaAs. More reasonably, elastic scattering from randomly distributed ionized impurities in the heavily doped source and drain extensions is probably the main mechanism inhibiting the ballistic transport [START_REF] Fiori | IEEE Int. Eletron. Dev. Meeting (IEDM)[END_REF]. Adding some spacers separating the channel from heavily doped access should limit this phenomenon. Let's now turn to the subthreshold regime. Figure 6a displays the evolution of the conductance G of the device with a 100 nm gate length with respect to the temperature. A clear improvement of the subthreshold slope can be noticed at low temperature. By plotting the evolution of G/T versus 1000/T, one can extract the potential barrier height Vb at the source/channel interface for different gate voltage as well as the active cross-sectional area 𝑆 𝑎 of the channel following the expression [START_REF] Tettamanzi | [END_REF]:

𝐺 = 𝑆 𝐴 𝐴 * 𝑇 𝑞 𝑘 𝐵 𝑒 - 𝑞𝑉 𝑏 𝑘 𝐵 𝑇
Eq. (7) Where A* is the Richardson constant for In0,6Ga0.4As (A * =4,582 A.cm -2 .K -2 considering an electron effective mass of 0.041m0 [25]). The same extraction has been performed for the other devices with gate length of 30nm, 500nm and 1 µm (Supporting information SI-6). These data are summarized in figure 6c and6d. The slope  of the Vb=f(VGS-VT) curves of figure 6c indicates the gate coupling efficiency. A gate coupling efficiency of 1 corresponds to a complete transfer of the gate voltage variation on the source/channel conduction band bending. The decrease of  for small gate length is typically due to short channel effects (SCE), arising when the gate potential is not able to fully deplete the channel. This is confirmed by figure 6d where a larger active cross-sectional area is obtained for LG=30nm meaning that the bulk current is mainly contributing to the OFF-state current. For larger gate length, the small active cross-sectional area indicates that the leakage current is probably located at the channel/gate interface. The density of traps at this interface can be estimated from the values of the subthreshold slope using the following expression [26]:

𝑆𝑆 = 𝐶 𝑖 +𝐶 𝑖𝑡 𝐶 𝑖 𝑘 𝐵 𝑇 𝑞 ln 10
Eq. ( 8)

Where Cit the capacitance related to the defects in the gate stack with the density Dit (Cit=q Dit). We can thus deduce Dit from the data of figure 6e (figure 6f). Owing to SCE limiting the subthreshold slope, Dit is largely over-estimated for the device with a 30 nm gate length. For the other devices, a Dit rather independent of the gate length is obtained with a value of about 4x10 12 cm -2 .eV -1 at room temperature and lower than 2x10 12 cm -2 .eV -1 below 250K. These results are close to the best reported values for III-V MOSFET using a similar extraction method [26]. This is a clear indication of the positive impact of InP shell since no passivation of the surface has been required before the gate oxide deposition.

Conclusion

Selective area molecular beam epitaxy is a useful technique to shape III-V semiconductors at the nanoscale. Field effect transistors based on in-plane InGaAs NWs surrounded with an InP shell exhibit a low density of interface traps in the gate stack and quite high effective electron mobility for long channel devices. The regrowth process based on a HSQ dummy gate is useful to contact the nanowire and define very short gate length down to 30 nm but it leads also to a severe effective mobility degradation preventing the ballistic regime to be attained at low temperature. This degradation may come from scattering from source and drain impurities, whose density should be carefully optimized to reduce their impact on electron transport in the channel while keeping low series resistances. With such improvement, we believe that this bottom-up process will be able to break through the barriers for the fabrication of nanoscale reliable quantum circuits. Where vth is the thermal velocity. m* is the electron effective mass of InGaAs, T is the temperature, kB the Bolztmann constant and q the elementary charge q.

STEM analysis of the channel cross section

Figure 1 .

 1 Figure 1. SEM images after different steps of the device fabrication: after selective area MBE of in-plane InGaAs/InP core-shell nanowires inside 100 nm wide openings in the SiO2 layer (a), after the deposition of a HSQ ribbon on the array of in-plane InGaAs/InP nanowires (b), after regrowth of InGaAs:Si raised contacts, source and drain pads deposition and HSQ removal (c and d), zoom on the channel part after removing the 30 nm wide HSQ ribbon (e), and gate metal deposition on multi-(f) or single-NW (g) transistors . Scheme of the longitudinal cross-section of the devices (hf).

Figure 2 .

 2 Figure 2. High Angle Annular Dark field (HAADF) image of the channel crosssection of an InGaAs/InP core-shell nanowire grown inside a 100 nm wide opening along [110] (a). The dash line indicates the position of the line scan of the composition profile given in graph (b). Detailed chemical composition of the NW: Indium (c), Gallium (d), Aluminum (e), Phosphorus (f), Arsenic (g) and Oxygen (h). The scale bar is 20 nm.

Figure 3 .

 3 Figure 3. Transfer characteristics at VDS=50mV of single in-plane InGaAs/InP NW MOSFET grown inside mask openings oriented in the [110] crystal direction with different width w and a gate length of 500nm in logarithmic (a) and linear (b) scales. Evolution of the threshold voltage with respect to the width of the opening for both orientations (c).

Figure 4 .

 4 Figure 4. Drain current normalized either by the width w of the mask opening (left scale) or by the gate development WG (corresponding to the length of the InGaAs/InP interface deduced from STEM; Right scale) versus drain to source voltage for LG=1 µm (a), 500nm (b), 100nm (c) and 30 nm (d). Resistance of the device for VGS-VT=0.4V and VDS=10mV at different temperature versus gate length (e), squares correspond to experimental data and dash lines to linear interpolations. Evolution of the linear resistance of the channel and of the access resistance deduced from these interpolations with respect to the temperature (f).

Figure 5 .

 5 Figure 5. Transfer characteristics measured at different temperature on a single in-plane NW transistor grown inside a 100 nm wide opening and with a gate length of 1 µm (a): solid line corresponds to the measurement and squares to the corrected mobility model with µ0 values of figure 5b. Evolution of the electron mobility µ0 with the gate length (c): square corresponds to the data deduced from the fit of the transfer characteristics and dashed lines to the model based on Eq. (1) and (2) using the parameters µlong and K described on (d) with respect to the temperature.

Figure 6 .

 6 Figure 6. Conductance versus gate voltage of the transistor for VDS=10mV and various temperatures (a). Arrhenius plot of G/T values extracted along the vertical dashed lines of figure 6a (b): squares correspond to the experimental data and the dashed lines correspond to the model described with equation 3. Potential barrier Vb (c) and active cross-sectional area SA (d) deduced from the previous model. The dashed lines of figure 6c correspond to the linear regression of the experimental points obtained for different gate length and authorizing the deduction of the gate coupling efficiency . Subthreshold slope of the devices with different gate length measured at VDS=10mV and different temperatures (e). The red dashed line represents the theoretical thermionic limit of the slope for a perfect gate coupling. Interface trap density versus temperature and gate length deduced from subthreshold slopes (f).

Figure S2. 1 .

 1 Figure S2.1. TEM analysis of the channel cross-section of an InGaAs/InP core-shell nanowire grown inside a 100 nm wide opening along [1-10] direction (a). The dashed line indicates the position of the line scan of the composition profile given in graph (b). High Angle Annular Dark field (HAADF) image (c) and detailed chemical composition of the NW: Gallium (d), Arsenic (e), Aluminum (f), Indium (g), oxygen (h), Phosphorus (i) and Titanium (j). The scale bar represents 20 nm.

Figure S2. 2 .

 2 Figure S2.2. FIB-STEM cross-section images at 30 keV of 30 nm InGaAs deposited by selective area MBE on InP (001) substrate without InP shell inside SiO2 openings with a width of 1 µm and oriented along [110] (a) and along [1-10] (b) and with a width of 500nm and oriented along [110] (c) and [1-10] (d). InGaAs exhibits mainly a top (001) facet delimited by (111)B facets for openings along [110] and delimited by (113)A facets for openings along [1-10].

Figure S5 .

 S5 Figure S5. Transfer characteristics measured at different temperature on a single in-plane NW transistor grown inside a 100 nm wide opening and with a gate length of 500 nm (a), 100 nm (b), 50 nm (c) and 30 nm (d): solid line corresponds to the measurement and squares to the corrected mobility model with the µ0 values of figure 5b. The correction factors  used for the fit are plotted with respect to the temperature (e).

6 .

 6 Calculation of Kbal coefficient versus temperature for InGaAs:

Figure S6 .

 S6 Figure S6. Calculated values of Kbal versus temperature.

Figure S6 .

 S6 Figure S6. Device conductance G versus gate voltage measured at different temperature, Arrhenius plot of G/T and extracted source to channel barrier voltage Vb and channel section for single in-plane NW transistor grown inside a 100 nm wide mask opening and with a gate length of 1µm (a, b, c), 500 nm (d, e, f), 100 nm (g, h, i) and 30 nm (j, k, l).

Address correspondence to L. Desplanque, ludovic.desplanque@univ-lille.fr

Acknowledgements

The authors would like to acknowledge financial support from the French Technological Network Renatech, and the Région Hauts de France.

Electronic Supplementary Material: Supplementary material (additional information about STEM analysis of nanostructures, electrical characterization, effective electron mobility and ballistic model) is available in the online version of this article.

Electronic Supplementary Material

Gate length dependent transport properties of in-plane core-shell nanowires with raised contacts

STEM evaluation of the gate oxide thickness:


Yi Wu
email: yi.wu@insa-rennes.fr

Erwan Fourn
email: erwan.fourn@insa-rennes.fr

Philippe Besnier
email: philippe.besnier@insa-rennes.fr

Novel Sext-Band Band-Pass Filter with Non-Coupled Structure Based on Multi-Shorted-Stub Resonators

Keywords: Sext-band band-pass filter, shorted-stub resonators, Chebyshev filter, star-like topology

In this paper, we present a simple way to design sextband band-pass filter based on multi-shorted-stub resonators. Each sext-band resonator consists in the parallel association of five stepped-impedance resonators in a non-coupled structure and allows the control of transmission zeros and bandwidths. A second-order sext-band band-pass filter is synthesized and realized in microstrip technology, the final circuit successfully integrating five stubs per resonators into a single planar circuit. Compared with coupled structures, the solution proposed here can achieve good in-band responses, band-to-band isolation with a simple design procedure. It also needs less optimization. All of these merits make this filter an attractive solution for multi-band wireless communication systems.

I. INTRODUCTION

Multi-band band-pass filters (BPFs) with high performance are more and more needed in modern wireless communication systems. Nevertheless, their design is highly challenging when the number of bands increases. This paper focuses on the case of sext-band BPF for which only few papers were already published [START_REF] Hsu | Compact sext-band bandpass filter with sharp rejection response[END_REF]- [START_REF] Li | Sext-band bandpass filter with compactness based on cascading stage of SIRs[END_REF]. In [START_REF] Hsu | Compact sext-band bandpass filter with sharp rejection response[END_REF], a sext-band BPF was constructed by using various quarter-wavelength stepped-impedance resonators (SIRs). By controlling the combinations of SIRs with different impedance ratios and electrical length ratios, several resonant frequencies were achieved to design the filter. In [START_REF] Tu | Design of sext-band bandpass filter and sextaplexer using semilumped resonators for system in a package[END_REF], authors developed a sext-band BPF by using semi-lumped resonators, which consist of two identical microstrip lines and a chip inductor at the center. Compared with half-wavelength resonators, these semi-lumped resonators are compact and allow a better control of the harmonic frequencies. In [START_REF] Ai | Compact sext-band bandpass filter based on single multimode resonator with high band-to-band isolations[END_REF], a single multi-mode resonator was proposed to design a sextband BPF with wide upper stop-band and high band-to-band isolation. Moreover, thanks to these multi-mode resonators, reflection zeros (RZ) and transmission zeros (TZ) can be controlled by adjusting the electrical lengths of two openended stubs. In [START_REF] Li | Sext-band bandpass filter with compactness based on cascading stage of SIRs[END_REF], the use of SIR cascading stages, leads to a miniaturized sext-band BPF. However, the band-to-band isolation and pass-band ripple level are not satisfying. All these sext-band BPFs are synthesized by coupled structures. However, multi-band (BPFs) based on coupled structures have some drawbacks. It is therefore difficult to obtain an adequate adjustment of the filter responses with most design methods. As a result, one may rarely meet all the design requirements for all pass-bands simultaneously. Such multi-band BPF also suffer from in-band or band-to-band poor performance and numerous optimization steps are often needed. Most of these drawbacks are also true for the rare published works on BPF with seven or more bands [START_REF] Ai | Miniaturized Quint-Band Bandpass Filter Based on Multi-Mode Resonator and λ/4 Resonators With Mixed Electric and Magnetic Coupling[END_REF]- [START_REF] Firmansyah | Hepta-band bandpass filter based on folded crossloaded stepped impedance resonator[END_REF].

In this paper, we propose a non-coupled structure to design sext-band BPF based on Multi-Shorted-Stub Resonators (MSSR) implemented in a star-like topology. A simple but efficient method is then proposed to design the sext-band filter. Measurements validate the proposed theory and designh techniques. Good simulation results and show that this MSSR topology is a good candidate for synthesizing sext-band BPFs.

II. THEORY OF NON-COUPLED MULTI-SHORTED-STUB

RESONATORS The proposed MSSR is constructed by combining several SIRs. As shown in Fig. 1, such a resonator is composed of five short-ended branches on a star-like implementation. For simplicity, we assume that the electrical length of each part of the branch i (i ∈ [1; 5]) of the MSSR equals to λ/4 (or θ i = π/2 in Fig. 1) at its resonant frequency f zi which will, at the end, correspond to a transmission zero. So the total input admittance can be written as:

Y tot = 5 i=1 Y i = j 5 i=1 Z i2 tan 2 ( f f zi π 2 ) -Z i1 Z i1 (Z i1 + Z i2 ) tan( f f zi π 2 ) (1)
where Y tot denotes the total input characteristic admittance of the sext-band resonator, f is the frequency and Z i1 and Z i2 are the characteristic impedance of the first and the second parts of the i th short-ended stub (i ∈ [1; 5]). It is easy to show that the TZ are obtained when f = f zi . In this condition, the input admittance goes to infinity and each transmission zero f zi determines the length of the associated branch i. So, firstly, five TZ are produced, one per SIR. In fact, one can also take into account two additional TZ: one at the origin (f z0 = 0 Hz) and another, called f z6 , which is the first harmonic of f z1 . Of course, these two additional TZ cannot be fixed independently.

One reflection zero can be obtained between each TZ leading to a total of six pass-bands. These RZ appear when Y tot = 0.

One should notice that this equation is non-linear as the one associated to the slope parameter that can be easily deduced from [START_REF] Hsu | Compact sext-band bandpass filter with sharp rejection response[END_REF]. So to simplify the filter conception, an optimization was made to determine the impedances Z i1 and Z i2 , knowing that the firsts need to be higher than possible and the second relatively low to obtain narrow rejected bands between each pass-band frequency.

An example of such a sext-band resonator was designed with the following input parameters: Its frequency response is presented in Fig. 2.It was obtained from (1) after solving for Z i1 and Z i2 and exhibits the required performance of a sext-band filter with such narrow rejection bands.

• Transmission zeros: f z1 = 1.8 GHz, f z2 = 2.1 GHz, f z3 = 2.4 GHz, f z4 = 2.
An N -order sext-band band-pass filter could then be designed using N MSSR and J-inverters as presented schematically in Fig. 3.

III. DESIGN AND CHARACTERIZATION OF A SECOND-ORDER SEXT-BAND FILTER

As an example, a second-order sext-band band-pass filter has been designed. The two sext-band resonators used here are identical and are based on the one sizing in the previous section. The admittance value of the inverters are J 01 = 0.0125S and J 12 = 0.00862S respectively. The ideal frequency responses are presented in Fig. 4 Fig. 6 shows the simulated and measured S parameters of the sext-band BPF. A good correlation is obtained by EM simulation and measurement except in terms of insertion losses. Nevertheless, the important measured losses (between 3.3 and 5.5 dB according to the band) are mainly due to the internal low-cost laser etching realization process which alters the substrate characteristics. The adjacent band-to-band isolation are above -36.1dB, -41.1dB, -38.4dB, -41.7dB and -39.2dB respectively which shows we can achieve sharp and deep rejections between the adjacent pass-bands.

IV. CONCLUSION

In this paper, a non-coupled sext-band BPF with a star-shape Multi-Shorted-Stub Resonator has been presented. The design procedure is simple and fast and high band-to-band isolation can be easily obtained. The filter had been implemented in microstrip technology to validate the proposed theory. Good agreements have been achieved between simulation and measurements. The proposed Multi-Shorted-Stub Resonator can be used as a building block of high-performance multi-band band-pass filters.

Fig. 1 .

 1 Fig. 1. Proposed Multi-Shorted-Stub Resonators (MSSR) topology.

7

 7 GHz, f z5 = 3.0 GHz and so f z6 = 2f z1 = 3.6 GHz; • Reflection zeros: f 01 = 1.650 GHz, f 02 = 1.935 GHz, f 03 = 2.245 GHz, f 04 = 2.520 GHz, f 05 = 2.815 GHz, f 06 = 3.115 GHz.

Fig. 2 .

 2 Fig. 2. Single resonator's S parameters with Z 11 = 138Ω, Z 12 = 23.1Ω, Z 21 = 137.8Ω, Z 22 = 30Ω, Z 31 = 136Ω,Z 32 = 22Ω, Z 41 = 24Ω, Z 42 = 23Ω, Z 51 = 138Ω, Z 52 = 22Ω.

Fig. 3 .

 3 Fig. 3. The schematic of a N -order multi-band BPF based on the proposed MSSR.

  and again fulfill the initial requirements. All the pass-bands return losses are better than -20 dB and the bandwidths are equal to 56 MHz, 59 MHz, 55 MHz, 50 MHz, 48 MHz, 47 MHz respectively. This filter was then implemented in microstrip technology. It was fabricated on Rogers RO4003 substrate of height h = 0.508 mm, relative permittivity ε r = 3.55 and loss tangent tan δ = 0.0027. The sext-band band-pass filter was designed using ADS 2017 from Keysight Technologies©. Mitigation of cross-coupling was realized by appropriate arrangement of all the resonators. A photograph of the fabricated filter is shown in Fig.5. For each sext-band resonators, the dimensions of the filter are the following (all in millimetres): L s = 20.00, W s = 1.13, L 01 = 18.55, W 01 = 0.47, L 02 = 19.00, W 02 = 0.18, L 11 = 26.64, W 11 = 0.10, L 12 = 23.81, W 12 = 3.35, L 21 = 22.83, W 21 = 0.10, L 22 = 20.73, W 22 = 2.38, L 31 = 0.11, W 31 = 19.96, L 32 = 17.60, W 32 = 3.55, L 41 = 17.73, W 41 = 0.12, L 42 = 15.60, W 42 = 3.36, L 51 = 15.98, W 51 = 0.10, L 52 = 13.8, W 52 = 3.55.

Fig. 4 .

 4 Fig. 4. S-parameters of the proposed sext-band BPF.

Fig. 5 .

 5 Fig. 5. Layout of the fabricated sext-band filter.

Fig. 6 .

 6 Fig. 6. Simulation (solid lines) and measurements (dashed lines) of the proposed sext-band BPF -S 11 in red and S 21 in blue; The photograph of the fabricated sext-band filter.


Weiya Chen
email: weiyachen@hust.edu.cn

Nicolas Ladevèze

Wei Hu

Shiqi Ou
email: oushiqi@isyslab.org

Patrick Bourdot
email: patrick.bourdot@limsi.fr

Comparison Between the Methods of Adjustment and Constant Stimuli for the Estimation of Redirection Detection Thresholds

Keywords: Navigation, Redirected Walking, Detection Threshold, Method of Adjustment

Redirection Detection Thresholds (RDTs) are defined to represent the limits of undetectable gains and serve as important input parameters for redirected walking algorithms. However, it is not trivial to get a user's RDT estimation in a few trials with existing methods such as the commonly used Method of Constant Stimuli (MCS). In aim to achieve efficient RDT estimation, we chose a classic psychophysical method -the Method of Adjustment (MoA), and compared it against MCS with a series of within-subject experiments respectively on translation, rotation and curvature gains. The results show that MoA gets overall similar RDT estimations with MCS over the same population, except for some systematic differences on translation and rotation gains. Moreover, MoA (with 20 trials) saves about 33% experiment time when compared against MCS, and has the potential to save more as the results of MoA remain relatively stable when the number of trials decreases. Further studies are needed to compare MoA with adaptive methods and to discover the potential relationship between RDTs and perception traits at individual level.

Introduction

Exploring large virtual worlds by natural walking within limited physical workspace is still an open research problem in Virtual Reality and related fields. Unlike walk-in-place [START_REF] Templeman | Virtual locomotion: Walking in place through virtual environments[END_REF] or other motion-based steering metaphors [START_REF] Bourdot | Polyvalent display framework to control virtual navigations by 6dof tracking[END_REF], natural walking is considered to yield best subjective presence [START_REF] Usoh | Walking¿ walking-in-place¿ flying, in virtual environments[END_REF].

Besides sophisticated walking hardware, many software solutions were developed to enable natural infinite walking with lower cost and easier hardware deployment [START_REF] Suma | A taxonomy for deploying redirection techniques in immersive virtual environments[END_REF]. Most existing techniques either have limited applicable virtual scenes, or somehow interrupt users' ongoing task which may lead to the break of presence. In searching for redirection methods that minimize artificial interventions during walking, Razzaque et al. [START_REF] Razzaque | Redirected walking[END_REF] developed another method called redirected walking. It allows users to walk through a large virtual world while keeping their real-world path inside a much smaller region than the virtual counterpart. It leverages users' tolerance of subtle visual-vestibular conflicts to decouple their virtual and physical paths while a user walks in the virtual environment. In general, three types of gains -translation, rotation and curvature gains [START_REF] Steinicke | Taxonomy and implementation of redirection techniques for ubiquitous passive haptic feedback[END_REF] -can be allocated at specific moments by algorithms using reactive [START_REF] Hodgson | Comparing four approaches to generalized redirected walking: Simulation and live user data[END_REF] or predictive control strategies [START_REF] Nescher | Planning redirection techniques for optimal free walking experience using model predictive control[END_REF].

Redirected walking has been actively developed and is pushed forward by advances from both virtual reality and science of perception for more than ten years [START_REF] Nilsson | 15 years of research on redirected walking in immersive virtual environments[END_REF]. Besides designing algorithms for optimal gain distribution during walking, the main question to be answered for redirected walking is -how much can we redirect the users before they notice the gains? To answer this question, the Redirection Detection Thresholds (RDTs) are defined to represent the limits of undetectable gains and serve as important input parameters for redirected walking algorithms. Any type of redirection gain applied at a given moment shall not go over the corresponding RDT in order to remain imperceptible (although being perceptible will not necessarily lead to the break of presence as shown recently [START_REF] Schmitz | You spin my head right round: Threshold of limited immersion for rotation gains in redirected walking[END_REF]).

To understand the mechanism of gain detection, existing studies begun by measuring RDTs with psychophysical experiments [START_REF] Razzaque | Redirected Walking[END_REF]. Since the comprehensive study by Steinicke et al. [START_REF] Steinicke | Estimation of detection thresholds for redirected walking techniques[END_REF], many follow-up studies repeated the measurement of RDTs with the same estimation method, but under different experimental conditions [START_REF] Langbehn | Redirected walking in virtual reality[END_REF]. The mainstream estimation method used in these experiments is the Method of Constant Stimuli combined with a Two-Alternative Forced Choice task (MCS-2AFC). The MCS can help us to draw the whole psychometric curve for a given subject, but at the cost of large number of trials.

The use of MCS as a threshold estimation method leads to several problems due to its high time cost: first, it is difficult to scale up the experiments by involving more subjects or testing different factors that may have potential influence on RDTs. Second, it is difficult, even impossible to collect data from people having trouble with long-exposure VR experience (e.g. cybersickness). Moreover, as the RDTs appear to be user-dependent [START_REF] Azmandian | Design and Evaluation of Adaptive Redirected Walking Systems[END_REF], a time-saving RDT evaluation method can help to provide users with personalized RDTs through calibration instead of the average thresholds from all users. As a consequence, we need a more time-efficient evaluation method that allows us to make quick estimation of RDTs.

In this paper, we present a series of experiments using a classical psychophysical method -the Method of Adjustment (MoA) to make quick assessment of RDTs for redirected walking. The reason for choosing MoA is that it is intu-itively appealing, easy to set up, and in general requires much fewer test trials than other classical or adaptive psychophysical methods.

We conducted a group of experiments using MoA and another group with MCS-2AFC on the same group of subjects. Through the comparison, the goal is not to find the "better" method for RDT estimation, but rather to see whether a time-efficient method like MoA yields similar results with MCS over the same population, despite their inherent methodological differences. The experimental design and preliminary results were briefly reported in [START_REF] Chen | Quick estimation of detection thresholds for redirected walking with method of adjustment[END_REF]. Here we detail all the experimental settings and conditions, along with results and in-depth discussions, that shall serve as a basis for future studies. Moreover, with the aim of understanding the between-user variability of RDTs, we report a series of tests we conducted to collect users' perception traits related to space and motion. We conclude that further correlation analyses between such perception traits and RDT identification tests are necessary.

Related Work

The study of RDTs actually concerns the measurement of subjective sensation in response to physical stimuli, so we can use psychophysical theories and tools to get estimations of RDTs.

In fact, the estimation of RDTs is more complex than other perceptionrelated tasks (e.g. size or color discrimination) as we are comparing the stimuli from visual and non-visual channels -at least three different sensory channels (visual, vestibular and proprioceptive) of human sensorimotor system are involved in the walking process.

Here we first briefly present the three classical psychophysical methods, then describe in detail how they have been applied for the estimation of RDTs, and finally some existing observations on the between-subject variation of RDTs.

Threshold Estimation in Psychophysics

Threshold estimation is one of the fundamental tasks in Psychophysics. Two types of thresholds are defined: Absolute Threshold (or Absolute Limen) -the intensity at which a stimulus becomes detectable, and the Difference Threshold (or Difference Limen) -the difference in intensities at which an observer can notice the differences.

No matter the type of threshold that we are targeting, threshold estimation experiments are composed of tasks and sampling methods [START_REF] Fechner | Elements of Psychophysics[END_REF]:

-The task: it can be either a detection ("yes/no" questions) or discrimination (e.g. forced choice) task. In general forced choice is considered more robust as compared to "yes/no" questions by eliminating response bias. -The sampling methods: they define the way that stimuli intensities are presented to the subject. The three classical sampling methods are the Method of Constant Stimuli (MCS), the Method of Limits (MoL) and the Method of Adjustment (MoA).

MCS is characterized by large number of trials with equally spaced levels of stimulus intensities in a randomized order, whereas MoL uses alternate ascending and descending trials and the threshold is taken as the average of the transition points from each series. MoA is quite similar with MoL in term of intensity sampling, except that the subjects are given direct control of the stimulus intensity [START_REF] Gescheider | Psychophysics: the fundamentals[END_REF]. Being the most intuitive and efficient method among the three, MoA is often used to get initial estimates of thresholds.

It is known that threshold values depend in part on the employed experimental method [START_REF] Guilford | Psychometric methods[END_REF]. Many studies [START_REF] Podlesek | Comparison of three psychophysical methods for measuring displacement in frontal plane motion[END_REF][START_REF] Wier | A comparison of method-of-adjustment and forced-choice procedures in frequency discrimination[END_REF] reported the difference of threshold values obtained with these methods for different research purposes, but were careful about their recommendations on choosing the "best" method because there was no common rule found for the thresholds and the results seemed to be experiment-dependent.

RDT Estimation Overview

With the inherent complexity of walking, it is unlikely that one could get a clear picture of the whole process from a single experiment, our knowledge about RDTs had grown incrementally as more experiments were conducted.

Razzaque [START_REF] Razzaque | Redirected Walking[END_REF] conducted several initial experiments on RDT estimation, mostly on scene rotation detection with adaptive 2-track staircase methods (20 trials). These experiments were conducted with limited number of subjects and trials, and the results turned out to have wide confidence interval.

Steinicke et al. [START_REF] Steinicke | Estimation of detection thresholds for redirected walking techniques[END_REF] conducted the first formal study on RDT evaluation by a series of psychophysical experiments in which RDTs for translation, rotation and curvature gains were tested separately. The evaluation method used was a combination of MCS and 2AFC. They describe the experiments as 2AFC experiments for simplicity (as commented by Grechkin et al. [START_REF] Grechkin | Revisiting detection thresholds for redirected walking: Combining translation and curvature gains[END_REF], the 2AFC used here is actually a kind of pseudo-2AFC task since subjects didn't choose between a test signal and a reference signal as in traditional 2AFC tasks), and found that the detection thresholds for translation gain were [0.86, 1.26], similarly the detection thresholds for rotation gains were [0.67, 1.24] and a curvature gain of no more than 0.045 was considered unnoticeable.

Many following experiments employed the MCS-2AFC method for RDT estimation, but instead of studying all three kinds of gains, they often focused on one type of redirection gain and found that RDTs could vary under different testing conditions [START_REF] Walker | Redirected walking in virtual environments[END_REF][START_REF] Langbehn | Redirected walking in virtual reality[END_REF]. For example, the threshold for translation gain was smaller with the presence of virtual feet [START_REF] Kruse | I can see on my feet while walking: Sensitivity to translation gains with visible feet[END_REF], the threshold for rotation gains was different when users were sitting in a wheelchair [START_REF] Bruder | Redirecting walking and driving for natural navigation in immersive virtual environments[END_REF], or exposed to different level of visual densities [START_REF] Paludan | Disguising rotational gain for redirected walking in virtual reality: Effect of visual density[END_REF]. For the curvature gains, the corresponding threshold could interact with factors such as ambient noise [START_REF] Razzaque | Redirected Walking[END_REF], walking speed [START_REF] Neth | Velocity-dependent dynamic curvature gain for redirected walking[END_REF] and cognitive load [START_REF] Bruder | Cognitive resource demands of redirected walking[END_REF], etc.

It would be interesting to scale up the experiments mentioned above for all types of redirection gains, or to check the interaction between these factors. However, the MCS-2AFC method is too time consuming for such purpose. So researchers started to work with more light-weight adaptive methods. For example, Grechkin et al. [START_REF] Grechkin | Revisiting detection thresholds for redirected walking: Combining translation and curvature gains[END_REF] studied the interaction between translation and curvature gains by using Green's maximum likelihood procedure, which is a variant of the adaptive method combined with a "yes/no" task. Nguyen et al. [START_REF] Nguyen | Individual differences and impact of gender on curvature redirection thresholds[END_REF] used a two interleaved 2AFC task combined with a three-down/one-up staircase method with fixed step size for left and right directions.

The adaptive methods are surely more efficient than MCS as they require less trials, but it is more complex to make design choices with numerous variants and parameters as compared to MCS and MoA. Another potential bias using adaptive methods is that participants are less likely to detect large gains when the gain value increases progressively from trial to trial, because it is known that users adapt to increasing redirection gains to become less sensitive to the manipulation over time [START_REF] Grechkin | Revisiting detection thresholds for redirected walking: Combining translation and curvature gains[END_REF]. Considering the two drawbacks that we just highlighted for adaptive methods, we choose to study the usability of MoA for the estimation of RDTs.

The Between-subject Variability of RDTs

An observation can be made from existing studies that the reported RDTs to some extent all differ from one to another, which leads researchers to look into the exact cause of such diversity. Besides methodological differences, Nguyen et al. [START_REF] Nguyen | Individual differences and impact of gender on curvature redirection thresholds[END_REF] proposed to put different factors into two groups: intrinsic and extrinsic. Intrinsic factors are specific to a person's perception trait (e.g. visual dependence, sensitivity to visual-vestibular conflict, etc.), while extrinsic factors include user-related factors (engagement, cognitive load, etc.), environment design (e.g. presence of virtual feet, visual density, etc.) and hardware setup (HMD intrinsics, tracking fidelity, etc.) [START_REF] Azmandian | Design and Evaluation of Adaptive Redirected Walking Systems[END_REF].

Recent research work has just begun to study the between-subject variability of RDTs. For example, Nguyen et al.'s experiment [START_REF] Nguyen | Individual differences and impact of gender on curvature redirection thresholds[END_REF] shows that on average women have higher detection thresholds for curvature gains than men. Rothacher et al. [START_REF] Rothacher | Visual capture of gait in redirected walking[END_REF] find that the threshold for curvature gains is positively correlated with a user's visual dependency, though they did not test other types of redirection gains in this experiment.

The study of the relationship between intrinsic factors and RDTs is not only useful for building a user-specific profile in order to provide better user experience, but can also help us to understand the underlying model of sensory fusion during redirected walking.

Experiments

Since MoA is an efficient classical sampling method in psychophysics, but has not yet been formally used in the context of RDT estimation, we designed the following experiments to see how its results differ from those of MCS-2AFC (which is believed to be the most reliable method, and hereafter referred as MCS for simplicity) on the same population. In addition, we examined the correlation between each user's perception trait and their RDTs in order to see how RDTs differ between individuals. Experimental protocol and results for all experiments are presented in this section.

Participants

In total 24 graduate students participated our experiments (17 male and 7 female) with an average age of 23 years old (+/-0.8). All participants have normal or corrected to normal vision. Besides age and sex, we collected their familiarity with video game and virtual reality devices using questions with a 4-point rating scale, ranging from 1 (none), 2 (a little), 3 (sometimes), 4 (often). The average score of video game experience is relatively high (mean=3.04, sd=1.04) since only two of them have never played video games. On the contrary, most participants have little experience with virtual reality devices (mean=1.67, sd=0.87).

All participants had no experience with redirected walking and were naive about the purpose of the experiments. We used a within-subject design for the comparison between results from experiments with MCS and MoA. These two groups of experiments were counterbalanced to reduce order effects and were arranged on different days. Each group of experiments contained three sub-experiments (translation, rotation and curvature) that were also counterbalanced. Subjects participated a training session of about 10 minutes to get familiar with the hardware setup and experiment procedures, they were allowed to take breaks at any time during the experiments. Although no official research ethics committee is yet available in our establishment, all participants were recruited and treated in accordance with the principles outlined in the Declaration of Helsinki.

Experimental Setup

The experiments were carried out in a lab room with a standard commercial VR setup (HTC Vive Pro) that contains an HMD (1400 × 1600 pixels per eye, 90Hz refresh rate, 110 field of view) and two handheld controllers. During the experiments, subjects wore the HMD and held one controller as input device (see Figure 1).

The two base stations used for tracking were placed at a distance of 5.5m (slightly larger than the specified 5m limit) forming a valid rectangular tracking zone of 3m × 4m with a safety margin of 0.3m to the walls. Users were able to cover the whole tracking area with a cable (5m) attached to the headset from a workstation positioned on one side of the workspace.

The workstation (Intel Core i7 7700K CPU, 32GB RAM, GeForce GTX 1080Ti) was fully VR compatible and was able to run the simulation at 90Hz. The virtual scene was a futuristic city3 rendered by Unity 3D. Only a small part of the whole city was used for our experiments (see Figure 2), its asymmetric configuration and many animation effects in the surroundings allowed users to quickly locate themselves with respect to the landmarks around. We coded the redirection control and the experiment process with C# scripts on top of SteamVR components.

Measures

RDT estimation In existing literature, RDTs were mostly reported in form of a tuple of lower and upper detection limits. This form is intuitive as it directly shows the usable interval of undetectable gains for the redirection controller.

Here we took a different representation of the threshold: a combination of the Point of Subjective Equality (PSE) and the Interval of Uncertainty (IU) [START_REF] Colman | A dictionary of psychology[END_REF].

The RDT for a user could be written in form of [PSE-IU/2, PSE+IU/2]. As found in previous studies, the PSE seldom meets exactly the theoretical equivalence point (i.e. gain=1) which leads to a bias (positive or negative) on gain perception, for example, a user with a PSE higher than 1 tends to be more sensitive to smaller gains (< 1) than larger ones (> 1). The IU is another important factor characterizing user's gain detection ability, for example, a user with high IU provides more room for maneuver to the redirection system than those who have lower ones.

MCS and MoA employ very different ways to compute PSE and IU. The former gets PSE and IU indirectly from the fitted psychometric curve. We used the same logistic function as described in [START_REF] Steinicke | Estimation of detection thresholds for redirected walking techniques[END_REF]:

p = f (x) = 1 1 + e ax+b (1)
Here x is the applied redirection gain and p is the probability that the user considers the amplitude of real locomotion to be greater than the virtual counterpart. This probability is computed by counting the "real is greater" trials The PSE corresponds to the gain value of f -1 (0.5). IU is bounded by a lower detection threshold f -1 (0.25) and a upper bound f -1 (0.75), which is two times the Just Noticeable Difference (JND). Here are the equations for computing PSE (2) and IU (3):

P SE = f -1 (0.5) = - b a (2)
IU = f -1 (0.75) -f -1 (0.25) = 2 • JN D = - 2 • ln 3 a (3)
Unlike MCS, MoA directly asks users to manually adjust the magnitude of redirection gain from a random starting value till no difference can be detected between visual and non-visual stimuli.

MoA contains equal number of ascending and descending trials that are tested alternatively in order to reduce estimation error. The computation of PSE (Equation 4) and IU (Equation 5) for MoA is described as follows:

P SE = µ(g) (4
)

IU = 2 • JN D = 2 • z(0.75) • σ(g) = 1.349 • σ(g) (5)
g is the gain value submitted at the end of each trial, µ and σ are the mean and standard deviation of submitted gain values. JND is given by multiplying the standard deviation of PSEs by a z score of probability of 0.75 (about 0.6745) [START_REF] Kuroda | The very first step to start psychophysical experiments[END_REF].

In addition to PSE and IU, we also collected each subject's mean walking speed during trials for translation and curvature gains. Since Neth et al. [START_REF] Neth | Velocity-dependent dynamic curvature gain for redirected walking[END_REF] found that the sensitivity to walking on a curved path was significantly lower for slower walking speeds. The walking speed data may provide cues to help us understand the obtained detection thresholds.

Perception Trait Since all subjects were tested under the same experimental protocol and condition, their RDTs should be mostly influenced by intrinsic factors such as each individual's perception trait. So we conducted a series of tests on each subject to collect their spatial and motion perception ability before starting experiments for RDTs identification.

Here we conducted tests respectively for egocentric distance and orientation perception and several other tests recommended by Ngoc et al. [START_REF] Ngoc | Estimation of individual redirected walking thresholds using standard perception tests[END_REF] (i.e. the Rod & Frame, Romberg and vection test) to measure a user's visual dependence during sensory integration, which are listed in Table 1. Here are some implementation details of these tests:

Group A We assume that the error of users' egocentric distance and orientation estimation is positively correlated with RDT for translation and rotation. For distance perception, we employed a visually directed action method -blind walking [START_REF] Matsushima | The psychophysics of visually directed walking[END_REF], to assess subject's distance perception accuracy. Each subject walked 5 times with each time a random distance ranging from 2.5m to 4.5m (interval of 0.5m). The final score is the root mean square error (RMSE) of all trials.

The spatial orientation test was conducted in a similar way. Each subject was tested 8 times with angles ranging from 30 to 120 degrees (with interval of 30 degrees and left/right symmetry).

Group B The Rod & Frame test experiment followed the standard procedure of the original test [START_REF] Witkin | Studies in space orientation. iv. further experiments on perception of the upright with displaced visual fields[END_REF] except that the rod and frame were displayed through the HMD. We tested five different angles for the frame: 0 • , 3 • , 15 • , 27 • , 42 • , each tested twice in a random order. The final score is also the RMS error of all trials.

In the Romberg's test [START_REF] Romberg | A manual of the nervous diseases of man[END_REF], the subject was asked to stand with feet together on the Wii balance board4 , hands by the sides, eyes open and then closed for respectively 20 seconds. The final score is the ratio between deviations of a subject's center of pressure (CoP) with eyes open and eyes closed. We also noted the deviations of CoP with eyes closed as an additional score (put in Group A) to describe a subject's profile regarding non-visual balance control.

The vection test was performed by asking the subject to stand inside a rotating optokinetic drum with constant speed (shown through an HMD). We tested three rotation speed -20, 30 and 40 deg/s -each for twice in a random order. The time from the beginning till the subject reported self-motion was logged (a limit of 45s was used if no self-motion was reported).

Experiment MCS

In this experiment, we tried to identify users' RDTs for different types of gainstranslation, rotation and curvature. This experiment primarily serves as a baseline for the other experiment with MoA since MCS is considered to be more robust against various bias.

Procedure

We followed similar procedure of Steinicke et al.'s previous experiment [START_REF] Steinicke | Estimation of detection thresholds for redirected walking techniques[END_REF].

Figure 3 shows a subject's first person view of the virtual street used for RDT evaluation. In the translation and curvature sub-experiments, for each trial, the subject begun by standing on the starting point, then walked towards the target (a green floating sphere) till it changed color. In the rotation sub-experiment, the subject started from a fixed orientation and then rotated in place towards the target sphere till it changed color. After each trial, a question was shown in a pop-up window that allowed the subject to make the forced choice.

The detailed settings of each sub-experiment is shown in Table 2. It should be noted that the curvature gain is expressed by angles, here it is actually the amount of direction change after walking 3m straight in the virtual world. The actual curvature gain can be computed by Equation 6:

g c = 1 r ≈ π • θ 180 • d ( 6
)
where r is the radius of the corresponding circle, θ is the direction change expressed in degree and d is the virtual walking distance (not exactly the arc length).

For the curvature sub-experiment, we used a small step (5 • interval) at gains in range [0 • , 20 • ] and a bigger one (10 • ) for gains larger than 20 • . We also measured left and right turns separately so each angle was tested five times. This experiment differs from the one of Steinicke et al. on two aspects: First, since the physical workspace available was quite small, in the translation subexperiment we fixed the virtual travel distance to be 2.5m and the real walking distance varied according to the gain value. No "redirection-free" pre-walking was possible due to the space limitation. Second, Steinicke et al. got estimates of RDTs from the average samples of all subjects instead of fitting individual psychometric functions. Here we fitted the psychometric curve for each subject so we can get a per-subject RDT estimation.

Results

We employed the logistic function in Equation 1 to fit the psychometric curve of each subject. The mean and standard deviation of RDTs, experiment duration and walking speed are presented in Table 3.

All 24 subjects finished all the trials, however, two of them got data unable to be fitted by a logistic function under all three sub-experiments. The rotation and curvature sub-experiments had respectively three more subjects that were not included in the final results shown in Table 3. The data for these subjects either resulted in abnormal fitted thresholds (e.g. P SE < 0 for translation and rotation) or were even unable to be fitted by a sigmoid curve.

The detection threshold DT c (lef t) = -0.14 and DT c (right) = 0.17 correspond respectively to a radius of 7.1m and 5.9m, or 8.02 • /m and 9.74 • /m (mean=8.88 • /m).

Experiment MoA

The goal of this experiment was to collect users' RDTs with MoA. Similar to the experiments with MCS, the experiment with MoA also contains three randomly ordered sub-experiments: translation, rotation and curvature.

Procedure

In Experiment MoA, subjects were not constrained in a routine starting position and walking path as imposed by MCS. The settings for this experiment are shown in Table 4. The task for the subjects to accomplish was to adjust gain values until they can no longer feel the discrepancy between the virtual and real paths.

During the test, the subject was given a handheld controller with buttons to adjust the gain in two directions, they can adopt fine tuning with the step value described in Table 4 by clicking the button, or apply quick modifications by pressing and holding the button. There was no imposed starting location for all trials, so the next trial begun immediately after each gain submission. For translation and curvature tests, the subject was instructed to walk back and forth between two target objects. For rotation, a random rotation direction (left or right) was chosen by the program and the subject should follow the given direction.

The angle notation used for the curvature sub-experiment was with respect to an arc length of 3m, which means the step of gains was 0.33 • direction change per meter, and the curvature gain can vary from -0.29 to 0.29 according to Equation 6.

For the translation and rotation sub-experiments, ascending (g init < 1) and descending (g init > 1) trials were presented alternatively, where g init was the starting gain value. For the curvature sub-experiment, we did not force left-right alternative distribution of the starting gain g init to prevent learning effect. In all sub-experiments, g init was a random value chosen to be away from the neutral zone and the lower or upper limits of gains. There was no time limit or predefined path for all subjects during the experiment. They can submit the final choice for a trial by pulling the trigger button on the same controller.

Results

All 24 subjects finished successfully all the trials in this experiment, though we put aside the results of the subjects who failed to get valid data with MCS so we can compare the two methods over the same group of subjects. A brief summary of the results for the remaining subjects can be found in Table 5. The PSE and IU was computed using Equation 4 and 5. The detection threshold DT c (lef t) = -0.08 and DT c (right) = 0.06 correspond respectively to a radius of 12.5m and 16.7m, or 4.58 • /m and 3.44 • /m.

Since it was difficult to choose a threshold to determine whether a subject was walking, the walking speed shown in Table 5 was the average speed from the start till the end of the experiment for a given subject.

Analyses and Discussions

In this section, we thoroughly analyzed the data collected in the previous section. All the analyses were performed with R5 . The results presented were considered statistically significant when p ≤ 0.05.

Comparison between MCS & MoA

We first used Shapiro-Wilk test for normality checking. Several RDT variables from different sub-experiments failed the normality test, therefore we employed Wilcoxon Signed-rank test to compare dependent variables like RDTs, experiment durations and walking speeds of two conditions -MCS and MoA, the results are summarized in Table 6. We denote M as the median value for a given data set.

Regarding RDTs, the PSE for translation gains obtained with MCS (M = 0.938) was smaller than that (M = 1.013) of MoA (W = 65, Z = -1.9966, p < Regarding walking speed during experiments, subjects on average took quicker moves with MCS than MoA in the translation (M M CS = 0.62, M M oA = 0.51, W = 185, Z = 1.8996, p = 0.05, r = 0.28) and the curvature (M M CS = 0.6, M M oA = 0.4, W = 189, Z = 3.7831, p < 0.01, r = 0.61) sub-experiments (see Figure 5). This difference is possibly due to the uncontrolled walking behavior with MoA, as we observed that some subjects switched constantly between gain adjusting and walking during the sub-experiments with MoA. Overall, subjects walked relatively slow compared to the study of Neth et al. [START_REF] Neth | Velocity-dependent dynamic curvature gain for redirected walking[END_REF] (possibly due to our relatively small physical workspace), which may explain why the detection threshold for curvature gain we got with MCS (v = 0.61m/s, g = 8.88 • /m) was higher than what they found (v = 0.75m/s, g = 5.42 • /m). However, their finding (i.e. lower walking speed corresponds to higher DT for curvature gain) did not apply to the results of Experiment MoA. In fact, if we compare the results provided by these two methods at subject level, no apparent rules can be seen because some subjects got very close RDTs in these two experiments while others not. However, when we run statistical analyses on the whole population, significant effects can be found regarding the PSE for translation gains and the IU for rotation gains (see Table 6). The latter might be linked to the initial experiment by Razzaque [START_REF] Razzaque | Redirected Walking[END_REF], in which he used MoA for scene rotation detection, but abandoned at last because the pilot subjects found the rotation to be more noticeable while turning the knob (for gain adjustment) in either direction.

It is also interesting to see that the between-subject variances of PSE and IU obtained with MoA were quite small despite the diversity of gain submission strategies among the subjects (some were really quick while others were always in hesitation). Since we were comparing MCS and MoA with the same population, virtual environment and hardware setup, it is safe to claim that the difference of results shown in Table 6 came from the sampling methods or the conducted tasks, but no further conclusions can be drawn from the current data. An important question arises when we use MoA is to decide the number of trials. More trials theoretically lead to more robust results as the chance of random choice reduces. However, increasing the number of trials also makes the experiment more time-consuming and burdensome for the participants, which deviates from our initial purpose of using MoA. Here we picked the first 2 and 10 trials per subject respectively, and compared their results with the full-trials (i.e. 20 trials) MoA and MCS. We denote MoA20 as the previously mentioned MoA to facilitate the illustration of its comparison with MoA10 and MoA2.

Since the RDTs did not follow a normal distribution, we applied a withinsubject repeated-measures non-parametric test -Friedman test to assess the influence of testing method (MCS, MoA20, MoA10, MoA2) on RDTs. For posthoc comparisons, we did pairwise comparison with Wilcoxon rank sum tests with Bonferroni correction.

For the translation sub-experiment, no significant effect of testing method on PSE (χ 2 (3) = 1.78, p = 0.62) or IU (χ 2 (3) = 2.02, p = 0.57) was found. The significant difference between MCS and MoA20 found in section subsection 6.1 disappeared with the application of Bonferroni correction, so conclusions on the influence of testing method on PSE should be carefully reconsidered.

For the rotation sub-experiment, there was no effect of testing method on PSE (χ 2 (3) = 0.79, p = 0.85), but significant effect on IU (χ 2 (3) = 21.88, p < 0.01). A post-hoc test showed the significant differences between MCS and MoA20 (p < 0.01, r = 0.57), between MCS and MoA10 (p < 0.01, r = 0.51) and between MCS and MoA2 (p < 0.01, r = 0.41).

Regarding the curvature sub-experiment, no significant effect of testing method on PSE (χ 2 (3) = 3.76, p = 0.29) and IU (χ 2 (3) = 7.61, p = 0.055) was found. We still conducted a post-hoc test for testing method on IU since the p-value was close to 0.05, but no significant effects were found.

The above results show that the number of trials for MoA did not seem to drastically influence the obtained results, except that the between-subject variances (especially for PSE) tend to become smaller as the number of trials increases (see Figure 6).

It is interesting to see that the significant difference between MCS and MoA on IU for rotation gains (Table 6) still holds even when MoA only contains two tests (one ascending trial paired with one descending trial). This means more time can be saved with MoA by reducing the number of trials.

The large variance of experiment duration in the curvature sub-experiment was actually not only due to the diversity of gain submission strategy, but also the difficulties for some subjects to detect the curvature gain. This is possibly because the participants tend to walk in a triangle rather than on an arc when the walking route is short [START_REF] Steinicke | Estimation of detection thresholds for redirected walking techniques[END_REF].

Perception Trait Correlation Test

Our hypothesis is that users' IUs for redirection are positively related to the scores of tests in group A (in Table 1) since larger non-vision estimation errors should lead to higher tolerance for visual and non-visual sensory conflicts. We also assume that IUs for redirection are positively associated with scores of tests in group B because users with higher visual dependence should be easier to be redirected by visual manipulations.

We used Pearson's product-moment correlation test to examine the correlation between subjects' perception traits and their RDTs given by MCS and MoA. We also created two additional variables -diffPSE and diffIU -to check the potential relationship between the difference of RDTs (given by MCS and MoA) and other existing variables.

The correlation tests showed that subjects' perception traits were neither correlated with their RDTs nor the difference of RDTs in all sub-experiments, which means our hypothesis of a simple linear correlation between the two groups of perceptual abilities and RDTs does not hold.

Conclusion

Based on the reported experiments and associated analyses, our primary conclusion is that MoA can provide similar RDT estimates over the same population with MCS except for slightly higher PSE for translation gains and smaller IU for rotation gains. The main advantage of MoA over MCS is that it is highly time-efficient, which can save about 33% and even more experiment time as the results of MoA remain relatively stable when the number of trials decreases. So MoA is a good option if we want to quickly assess averaged RDTs for a given population, which can serve as a useful tool for pre-walking calibration or to evaluate the impact of different factors on RDTs in an affordable way.

However, it is still unclear how to explain the difference between results from MoA and MCS at individual level, and how they are correlated with a user's perception trait. The interaction between RDTs and users' intrinsic characteristics such as perception trait should be tested on a bigger yet more heterogeneous population to attempt a personalized model for redirection detection.

Although we had put aside the theoretical difference between MCS and MoA in this paper, we should keep in mind that the underlying psychophysical evaluation method do have an impact on the collected RDTs. Moreover, the RDTs are intrinsically more complex to be explained than other problems studied in the literature of Psychophysics since these usually focus on stimuli getting from the same sensory channel (e.g. the perception of sound, pain, etc.). So inputs from the multi-sensory integration community may help to shed light on this question.

In the future, experiments can be designed using MoA and adaptive methods to clarify the influence of more extrinsic factors (e.g. cognitive load, configuration of the virtual world, real-world ambient noise, etc.) on RDTs. The comparison between MoA and popular adaptive methods can be conducted in a separate study (as there are many variants and parameters for adaptive methods) in order to provide more useful information about the impacts of methodology on RDTs.

Fig. 1 .

 1 Fig. 1. A user walked in the physical workspace with an HMD and a handheld controller.

Fig. 2 .

 2 Fig. 2. Overview of the T-formed street in a futuristic city used for walking in our experiments.

Fig. 3 .

 3 Fig. 3. The subject's first-person view in Experiment MCS. Left: the orange cylinder is the starting point for the subject to step on; Right: the green target sphere.

Fig. 4 .

 4 Fig. 4. Comparisons between MCS and MoA on PSE, IU and experiment duration in the translation, rotation and curvature sub-experiments.

Fig. 5 .

 5 Fig. 5. Comparisons between MCS and MoA on average walking speed in the translation and curvature sub-experiments.

Fig. 6 .

 6 Fig. 6. Further comparisons between MCS and MoA (the number of trials varies from 20 to 2) on PSE and IU.

Table 1 .

 1 A summary of tests for perception trait.

	group	test description
		(1) ego-distance: blind walking
	A -non-visual perception	(2) ego-orientation: blind rotation
		(3) balance eye-closed: control of CoP
		(4) Rod & Frame: vertical alignment
	B -visual dependence	(5) Romberg's ratio: control of CoP
		(6) vection: on-set time for self-motion

Table 2 .

 2 Detailed settings used in Experiment MCS.

	Parameters	Translation Rotation Curvature
	Lower limit of gains	0.6	0.5	0 •
	Upper limit of gains	1.4	1.5	50 •
	Step of gains	0.1	0.1	5-10 •
	Trials per gain value	10	10	5
	Total number of trials	90	110	75
	Virtual walking distance/rotation angle	2.5m	±90 •	3m

Table 3 .

 3 The mean and standard deviation for the results of Experiment MCS.

		Translation Rotation Curvature
	Number of subjects	22	19	19
	PSE	0.95±0.13 1.03±0.22 0.01±0.06
	IU	0.23±0.16 0.5±0.34 0.31±0.48
	Upper & lower bounds [0.83, 1.06] [0.78, 1.28] [-0.14, 0.17]
	duration (minutes)	26.8±5.3	25.5±3.8	24.3±4.5
	walking speed (m/s) 0.62±0.17	/	0.61±0.12

Table 4 .

 4 Detailed settings used in Experiment MoA.

	Parameters	Translation Rotation Curvature
	Lower limit of gains	0.6	0.5	0 •
	Upper limit of gains Step of gains	1.4 0.01	1.5 0.01	±50 • 1 •
	Total number of trials	20	20	20

Table 5 .

 5 The mean and standard deviation for the results of Experiment MoA (only subjects with valid results for MCS were included).

		Translation Rotation Curvature
	Number of subjects	22	19	19
	PSE	1.01±0.09 0.98±0.10 -0.01±0.03
	IU	0.22±0.08 0.26±0.10 0.15±0.07
	Upper & lower bounds [0.90, 1.12] [0.85, 1.11] [-0.08, 0.06]
	duration (minutes)	17.0±5.9	14.4±6.3	19.8±8.2
	walking speed (m/s) 0.54±0.11	/	0.39±0.07

https://assetstore.unity.com/packages/3d/environments/sci-fi/pbr-sky-city-62261

https://en.wikipedia.org/wiki/Wii Balance Board

https://www.r-project.org/