X Wang

W Gao

X Li

Q Zhang

S Nanot

E H Hároz

J Kono

W D Rice

Magneto-Transport in Type-Enriched Single-Wall Carbon Nanotube Networks

Single-wall carbon nanotubes (SWCNTs) exhibit a wide range of physical phenomena depending on their chirality. Nanotube networks typically contain a broad mixture of chiralities, which prevents an in-depth understanding of SWCNT ensemble properties. In particular, electronic-type mixing (the simultaneous presence of semiconductor and metallic nanotubes) in SWCNT networks remains the single largest hurdle to developing a comprehensive view of ensemble nanotube electrical transport, a critical step towards their use in optoelectronics. Here, we systematically study temperature-dependent magnetoconductivity (MC) in networks of highly enriched semiconductor and metal SWCNT films. In the semiconductor-enriched network, we observe two-dimensional variable-range hopping conduction from 5 to 290 K. Low-temperature MC measurements reveal a large, negative MC from which we determine the wavefunction localization length and Fermi energy density of states. In contrast, the metal-enriched film exhibits positive MC that increases with decreasing temperature, a behavior attributed to two-dimensional weak localization. Using this model, we determine the details of the carrier phase coherence and fit the temperature-dependent conductivity. These extensive measurements on type-enriched SWCNT networks provide new insights that pave the way for the use of SWCNTs in solid-state devices.

Single-wall carbon nanotubes (SWCNTs) are rolledgraphene, one-dimensional, cylindrical crystals that show remarkable electronic, optical, magnetic, and thermal behaviors, such as ultra-high current carrying capacities [START_REF] Yao | [END_REF] , exciton-dominated absorption [2][3][4] , long quantum coherence lengths 5,6 , large Young's moduli 7 , proximityinduced superconductivity 8 , and giant thermal conductivities 9,10 . These physical properties stem from the rolling orientation of the underlying graphene sheet that forms the nanotube [START_REF] Saito | Physical Properties of Carbon Nanotubes[END_REF] : for certain wrapping angles (chiralities), the nanotubes are semiconducting, while for other orientations, the nanotubes are metallic or narrowbandgap semiconductors. Beyond impressive physical behaviors, SWCNTs have an all-carbon composition and, thus, a very low mass density (∼1.3 g/cm 3 , as compared to ∼9 g/cm 3 for copper). However, the promise of various nanotube-based technologies continues to be hampered by the inability to transition from well-understood, single-nanotube experiments to macroscopic SWCNT networks. At the ensemble scale, nanotube network properties are complicated by defects, accidental dopants, inter-nanotube junctions, physical deformations, mixtures of different electronic types, random orientation, and non-nanotube materials.

Electrical and thermal transport behavior in SWCNT networks differs considerably from conduction in individual nanotubes. For instance, charge carriers in individual nanotubes exhibit quantum effects, such as quantum conductance and conductance fluctuations 6,[START_REF] Liang | [END_REF] , which are not observed in nanotube ensembles. Indeed, since (macro-scopic) translational invariance is absent in SWCNT networks, carriers encounter electrical potentials created via nanotube terminations, different nanotube species, and a lack of crystal orientation (i.e., no alignment). Consequently, it is unsurprising that electrical transport in seemingly similar SWCNT ensembles show a broad array of behaviors, including fluctuation-induced tunneling 13,14 , weak localization 13,[15][16][17] , strong localization 16 , metallicity 18 , and Coulomb gap opening 17,19 . Beyond the absence of translation invariance, SWCNT ensembles also suffer from electronic type mixing: that is, SWCNT ensembles contain both semiconductor and metal nanotubes, which significantly confounds electrical and thermal measurement results. Recently, a new generation of type-enriched nanotube films 17 and heavily doped fibers 20,21 have been used in electrical transport measurements to provide more insight into the conduction mechanism of SWCNT ensembles. Unfortunately, none of these works present a complete study of temperatureand field-dependent conductivity, which prevents a comprehensive understanding of transport in SWCNT ensembles.

In this work, we investigate temperature-(T -) dependent magneto-transport of pristine SWCNT ensembles and semiconductor-and metal-enriched SWCNT films to reveal the underlying electrical transport mechanism in SWCNT networks. Measurements of pristine and semiconductor-enriched SWCNT conductivity, σ, show that electrical conduction follows a two-dimensional (2D) variable-range hopping (VRH) conduction over a broad temperature range. When a magnetic field, B, is applied, we observe a large, negative, T -dependent magnetoconductivity (MC) signal, from which we determine both the localization length of the hopping-like species and the density of states at the Fermi energy. Unlike the strongly localized carrier transport in the semiconductordominated nanotube networks, the metal-enriched film exhibits two-dimensional weak localization (2DWL) behavior. We use this model to describe the electrical transport behavior of metal-enriched SWCNT ensembles. Our extensive measurements and thorough data modeling provide new insights into the carrier transport mechanisms in nanotube ensembles.

For the electrical transport experiments presented here, we prepared a set of three SWCNT networks with different semiconductor-to-metal nanotube ratios. We used nanotubes that either had no electronic-type enrichment ('pristine') or contained high concentrations of semiconductor ('semiconductor-enriched' or s-SWCNTs) or metallic ('metal-enriched' or m-SWCNTs) nanotubes. The pristine sample was prepared using acid-purified, laser-oven SWCNTs that were first sonicated and then centrifuged to form a pellet from which a small piece was cut and then mounted to a glass slide (see 22,23 for details about the nanotube composition). The electrical conductivity was measured in a four-point probe configuration (current source, voltage probe) to mitigate contact resistance effects. We silver painted the ends of the pelleted sample to source current through the entire sample, while gold contacts (spaced ∼1 mm apart) were sputtered on the three sides of the parallelepiped sample for voltage measurements.

To prepare the semiconductor-and metal-enriched nanotube films, we performed density gradient ultracentrifugation 25,26 on arc-discharge SWCNTs, following Yanagi et al. 27 and Hároz et al. 4,28 . Next, both the semiconductor-and metal-enriched suspensions were vacuum-filtered, dried, washed, and then transferred to a sapphire substrate after which gold contacts (spaced ∼3 mm apart) were sputtered onto the thin films; see Zhang et al. 24 for details regarding the film preparation. The s-SWCNT film was 410 nm thick, while the m-SWCNT film was 190 nm thick. X-ray photoelectron spectroscopy of the prepared nanotube films revealed that adsorbed gaseous species, such as nitrogen and oxygen, were present. To remove these molecules, all three nanotube samples were vacuum annealed for several hours.

We evaluated the composition and purity of the semiconductor-and metal-enriched SWCNT networks using several characterization techniques, such as optical absorption, photoluminescence excitation spectroscopy, profilometry, and resonant Raman spectroscopy. Figure 1a shows a scanning electron microscope image of the deposited, enriched nanotube bundles in a film. Images like this one indicate that after processing, the average nanotube length is several hundred nanometers, which is in line with other estimates of enriched SWCNTs 29,30 . 24 . The difference in absorption clearly show the presence of distinct SWCNT electronic types, which in combination with other characterization techniques, allow us to conclude that the films were highly type enriched.

Despite the well-resolved excitonic transitions seen in the films, electron microscopy shows that the SWCNT bundles are in close, intimate contact with one another.

Figure 1b shows the optical absorption of both the semiconductor-and metal-enriched SWCNT films from 0.9 meV to 6 eV 24 . In both cases, the low energy absorption spectrum is dominated by the plasmon resonance 24 , while the high-energy region (>4 eV) shows the so-called M -point singularity created by the saddle-point exciton 31 . For this work, the most important absorption data falls in the region from 0.5 to 3 eV, where nanotube electronic transitions occur 32 . In this regime, the absorption from pristine SWCNTs contains numerous strong excitonic absorption transitions 3,33 ; in contrast, the typeenriched samples have only specific exciton peaks, which reflects the suppression of the metallic (semiconductor) species in the semiconductor-(metal-)enriched fractions.

Four-point probe T -and B-dependent conductivity experiments on the three SWCNT networks were performed in a helium-flow magnet (either a Janis 6 T or an Oxford 10 T system) using dual lock-in amplifiers. To avoid sample heating, we kept the source current at or below 100 µA. As seen in the inset of Fig. 2a, I-V curves were taken at several temperatures from 1.5 to 300 K to ensure that we were in the linear regime: at low temperatures, we observe the I-V curve becomes non-linear above a source current of 2 mA. For all measurements, the field was applied perpendicular to both the plane of the sample and the direction of the current.

We first consider the electrical transport in the pristine SWCNT ensemble. Figure 2a shows σ (T ) for the pristine SWCNT network across a broad temperature range. As with other work on pristine SWCNT ensembles 13,15,16,34 , σ decreases with T from 350 K to 1.5 K, exhibiting a four-fold reduction in this temperature range. Despite the fact that dσ dT >0 for all T , the nature (weak or strong) and source of the localization is complex, having contributions from a lack of long-range order, the presence of metallic and semiconducting nanotubes, SWCNT impurities, dislocations, and defects, and the physics of bundling (i.e., intimately contacted nanotubes).

The T -dependent MC = σ(B,T )-σ(0,T ) σ(0,T ) measurements shown in Figures 2b and2c suggest at least two separate temperature regimes: above ≈7 K the MC is increasingly positive as T decreases, while below ≈7 K, the MC becomes increasingly negative with lower T and higher B. Previous MC measurements in unsorted nanotubes 15,16,35 and intentionally doped semiconductors [36][37][38] have shown very similar T -dependent MC behaviors. Given the complex behavior of the T -dependent MC, a large variety of transport models that have both positive and negative MC components can be used to explain this data.

Instead of attempting to computationally de-convolute the different transport processes occurring in the pristine sample, we eliminated the largest confounding factor (electronic-type mixing) by performing T -dependent MC measurements on type-sorted SWCNT networks. Figure 3a shows the T -dependent conductivities of SWCNT films with different ratios of metallic to semiconducting nanotubes. As expected, we observe that as the SWCNT metal-to-semiconductor ratio decreases, so does the conductivity. The conductivity of the s-SWCNTs drops by two orders of magnitude from 300 to 5 K, which is a much larger decrease than in the pristine film. This large conductivity decrease suggests that the s-SWCNT network contains markedly fewer conducting pathways than the pristine ensemble.

Despite its metallic nature, the m-SWCNT film also exhibits a decrease in conductivity with temperature, which indicates some degree of localization. However, in contrast to the pristine and s-SWCNT films, the m-SWCNT film changes by less than a factor of two when T is varied between 300 and 5 K. Unlike other studies on Tdependent conductivity in SWCNT ensembles 13,16,18,20 , we do not observe a change in the sign of dσ dT as we move from high to low temperatures. This lack of negative

FIG. 2. (a)

T -dependent conductivity for the pristine SWCNT ensemble. The conductivity monotonically decreases with decreasing T . Inset: I-V curve (magenta dots) for the pristine SWCNT sample at 3 K. Only at sourced currents below ∼1.5 mA does the I-V curve behave linearly (black line fit). T -dependent MC for (b) 7.0 to 50 K and (c) 2.0 to 6.0 K. At high T , the MC is positive, transitioning from linear with B to sub-linear as T is lowered. When T drops below 7.0 K, we begin to see that the sub-linear behavior observed at higher temperatures is caused by a negative MC contribution, which dominates the MC at high B and low T . dσ dT in the m-SWCNT film may suggest a high degree of nanotube disorder in our films, which could be caused by the shorter-than-normal nanotubes created by our typeenrichment procedure.

We now turn our attention to the electrical transport in s-SWCNT networks. It should be noted from the outset that defects present in semiconductor SWCNTs, as well as remnant metallic and narrow-bandgap SWC-NTs, can create localized conductive sites that facilitate electrical transport in a nominally insulating film. This situation is similar to electrical measurements made on polymers 39,40 , doped semiconductors 38,41 , and granular media 42 , whereby carriers move between localized states of differing energies spaced at varying distances apart (i.e., transport between non-adjacent sites). This transport behavior, known as VRH, is a competition between energy-state differences and spatial separation that produces a T -dependent conductivity first investigated by Mott 43,[START_REF] Mott | Conduction in Non-Crystalline Materials[END_REF] :

σ(0, T ) = σ 0 exp - T 0 T α . (1)
Here α depends on the particular hopping mechanism, while σ 0 (high-temperature conductivity limit) and T 0 are T -independent variables. T 0 can provide deep insights into the system under study, since T 0 = 13.8 kBξ 2 N (EF) in 2D [START_REF] Shklovskii | Electronic Properties of Doped Semiconductors[END_REF] , where ξ is the wavefunction localization length, N (E F ) is the density of states at the Fermi energy, E F , and k B is the Boltzmann constant. In Mott-like VRH, α = 1/(1 + d), where d is the dimensionality of the system. Given the complexity of Eq. 1 and the difficulty of accurately determining α (and thus, d), we calculate W , which is defined as 16

: W = ∂ ln σ ∂ ln T = α T0 T α .
The slope of ln W as a function of ln T , which is graphically depicted in Fig. 3b, is equal to -α providing us with a route for unambiguously determining the system dimensionality. For the pristine SWCNT ensemble, we find α = 0.370 from 2 to 110 K, which translates into a dimensionality, d, of 1.70, while for the s-SWCNT film, α = 0.369 (d = 1.71) from 4 to 286 K (full temperature range). Given these values for d, we consider conduction for both samples to be 2D VRH. In contrast to the primarily semiconductor SWCNT ensembles, the behavior of ln W for the m-SWCNT film has the opposite sign for α, confirming that the m-SWCNT conductivity cannot be described as VRH conduction; we note that a similar finding using W -parameter analysis was seen by Vavro et al. in doped SWCNT ensembles 16 . Using d = 2, we have plotted and fit the natural log of the conductivity as a function of T -1/3 for both the pristine and s-SWCNT samples in Fig. 3c. As can be seen by the fitting, we find that the data can be very well described by 2D VRH conduction in the low temperature regime with T 0 = 26.6 K and 745 K for pristine and s-SWCNTs, respectively. The excellent agreement of σ (0, T ) with 2D VRH agrees with previous observations by Vavro et al. 16 and Kim et al. 15 , but diverges with data presented by Jaiswal et al. 19 and Yanagi et al. 17 who suggest that a soft Coulomb gap opens around the Fermi energy at lower temperatures. Prompted by these previous results, we closely examined σ (0, T ) at low temperatures (<20 K), but never observed Efros-Shklovskii VRH [START_REF] Efros | [END_REF] (α= [START_REF] Yao | [END_REF] 2 ) for either the pristine or s-SWCNT networks.

A deeper understanding of this strongly localized conduction behavior can be gained from T -dependent MC. For this work, we focus on the low-T (2 to 10 K) behavior of the s-SWCNT films where the most significant MC changes are observed (see Fig. 4a). At very low fields (B<0.5 T), the MC is positive and small. Above 0.5 T, the MC becomes negative for temperatures below 10 K. The negative MC behavior increases with decreasing T , reaching ∼-0.35 at 2 K, 10 T; this MC magnitude is over four times greater than the MC change seen in the pristine SWCNT network. Additionally, unlike MC in the pristine SWCNTs, the MC for the semiconductor-enriched network continually decreases with lower temperature, similar to previous work on annealed nanotube buckypapers 16 and partially semiconductor-enriched nanotube networks 19 .

To more fully understand our MC data, we focus on two key processes within the strong localization paradigm that occur when an external magnetic field is applied. The first term stems from two negative MC contributions created by wavefunction shrinkage [START_REF] Shklovskii | Electronic Properties of Doped Semiconductors[END_REF] and spindependent hopping 47,[START_REF] Kamimura | The Physics of Interacting Electrons in Disordered Systems[END_REF] , both of which approximately scale as B 2 . We expect that the majority of this contribution comes from wavefunction shrinkage, which is expressed as ln ρ(B,T )

ρ(0,T ) orbital = 2 3 B B0 2
, where B 0 = √ 2φ0 πξ 2 T 0 T , φ 0 = h e is the flux quantum, and ρ (B, T ) is the B and T -dependent resistivity. The second MC contribution is from the so-called forward interference between two hopping sites [START_REF] Nguyen | [END_REF]50 . In this model, multiple conduction pathways between two points interfere to increase the probability of going from site 1 to site 2, thus generating positive MC. First pioneered by Nguyen et al. [START_REF] Nguyen | [END_REF]50 , and then later refined by Schirmacher 51 and Raikh et al. 52 , this interference model produces a positive MC that saturates at large B values. In Raikh et al.'s model 52 , ρ(B,T ) ρ(0,T ) interference = 1 -K(A)B/B 0 . Logarithmically adding to the two contributions, ln ρ(B,T ) ρ(0,T ) total = ln ρ(B,T ) ρ(0,T ) orbital + ln ρ(B,T ) ρ(0,T ) interference , and remember-

ing that MC = 1-ρ(B,T ) ρ(0,T ) ρ(B,T ) ρ(0,T )
, we have 52 :

MC VRH = exp -ln 1 -K(A) B B 0 - 2 3 B B 0 2 -1.
(2) Figure 4(a) shows T -dependent MC for the s-SWCNT film along with numerical fits using Eq. 2. The full Tdependent MC data set was simultaneously fit to obtain T -independent values of ξ. For the traces shown in Fig. 4(a), we find ξ = 2.81 ± 0.10 nm best describes the data. In conjunction with our previously determined value of T 0 in 2D = 13.8

kBξ 2 N (EF) , we obtain N (E F ) = (2.7 ± 0.2) × 10 15 eV -1 cm -2 .
In addition to finding ξ and N (E F ), fits using Eq. 2 also give insight into the microscopic nature of the scattering process. In Raikh et al.'s analysis of scatteringinduced MC in the VRH regime, the T -dependent K(A) parameter varies between 0 (no scattering) to 1.1 (strong scattering). Here, K(A) is a description of scattering strength and is a complicated function A, which is a parameter directly proportional to the electrostatic potential between two hopping sites, V 0 52 : A =

π 3/2 2 5/2 N (E F ) V 0 ξ 2 T0
T . As seen in Fig. 4(b), K(A) ranges from ∼0.22 at 10 K to ∼-0.15 at 2 K, which strongly suggests that carriers undergoing 2D VRH conduction in s-SWCNT ensembles are weakly scattered. In the weak scattering regime, K (A) is approximately equal to 16 3 √ π A, which is proportional to 1 T , as defined above 52 . Thus, the dependence of K (A) on T -1/2 shown in Fig. 4(b) adds considerable support for the conclusion that carriers are weakly scattered in this system 52 .

In contrast to the negative MC exhibited by the s-SWCNT film, the m-SWCNT ensemble shows a small, positive MC that increases with decreasing T and increasing B. As seen in Fig. 5a, the change in MC with field at high temperatures is linear and small. When T decreases, the positive MC increases until it begins to saturate. At the lowest temperatures (<10 K), the MC saturates at increasingly smaller applied fields, which suggests other effects, such as wavefunction shrinkage or electron-electron interactions 13 , become significant. This observation of saturated, positive MC in m-SWCNT films has been previously seen by Yanagi et al. 17 at 2 K. However, unlike our work, T -dependent MC was not investigated, leaving several questions unanswered, such as the nature of the dephasing and the magnitude of the coherence length.

For disordered metallic films [53][54][55][56] , conduction is weakly localized by imperfections in the lattice translational invariance 57,58 . In the WL model, phase-coherent carrier wavefunctions constructively interfere along timereversed paths augmenting the probability of returning to their original positions 59 . The increased carrier backscattering thus decreases the electrical conduction in disordered metallic films. Scattering via the presence of other carriers, phonons, impurities, and magnetic scatterers destroys this phase coherence, which provides a means for us to measure WL if we know how the scattering time (or length) scales. Typically, it is assumed that the phase decoherence time, τ φ , is equal to τ

-1 in + 2τ -1 s -1
, where τ in is the inelastic scattering time and is proportional to T -p , τ s is the magnetic scattering time, and p is a fraction that determines the dominant inelastic scattering mechanism [57][58][59] . From these relationships, we can obtain a form for σ (T ) in the WL regime, a subject we will return to later.

Although a power-law temperature scaling of σ (0, T ) is predicted in the WL model, T -dependent MC is a more conclusive way to discern and understand WL. Specifically, MC in the WL regime has several distinguishing traits: the MC is positive, it increases with decreasing T , and it has a distinct functional form. These behaviors arise from the broken time-reversal symmetry between interfered partial waves, which suppresses backscattering when a magnetic field is applied perpendicular to the plane of the interference. We note that positive MC has been seen in variety of carbon-based systems, such as pregraphitic carbon 60,61 , nanofibers 62 , and graphitic and amorphous carbon films 63 . Beyond these studies using carbonaceous materials, 2DWL has been observed in highly doped, aligned nanotube fibers 21 , individual multi-wall nanotubes 59 , doped SWCNT films 13 , and metallic SWCNT bundles 17,64 . Based off of these works and recognizing that the expected B 2 dependence for 3DWL at low fields (and/or high temperatures) does not describe our data 16,65 , we focus our attention on 2DWL, which predicts that MC is quantitatively described as 21,58,64 :

MC 2DWL = C σ (0, T ) e 2 2π 2 ψ 1 2 + B φ B + ln B B φ , (3)
where C is a constant, ψ is the digamma function, and B φ , the phase coherence field, is equal to 4eDτ φ = 4eL 2 φ , where D is the diffusion constant and L φ is the phase coherence length. Although C = 1 in the strict theoretical treatment of 2DWL, we take C be a fit parameter here (i.e., =1) to properly scale the model to our data, since the curvature of the data (determined by B φ ) is the main objective of our analysis. Additionally, it should also be noted that we are not dealing with a traditional 2D system, which exhibits translational invariance across the crystal, but a network of 1D wires collected into bundles that define a quasi-2D cylindrical structure. As such, the appropriate model describing our system may not conform to the traditional form of Eq. 3, thus causing C to deviate from 1.

In Fig. 5a, we use Eq. 3 to fit the MC curves from 70 to 2 K taken on the m-SWCNT film. At high T , the fits describe the data well, which suggests that 2DWL is the dominant mechanism in this range. To some extent, this behavior is expected, since even if our metallic film contains narrow-bandgap semiconductors 66 , thermal activation of carriers across the meV-sized bandgaps will produce metal-like behavior. However, as T drops (<10 K), only the low-field range of the MC can be reasonably fit with the 2DWL model, indicating that the metallicity of the film has decreased enough so that wavefunction shrinkage, which is typically seen when carrier localization is present, or electron-electron interactions, which can occur in low-dimensional systems with slow carrier diffusion 13 , begin to play a significant roles. The discrepancy with Eq. 3 at low T is highlighted by the dotted lines included on the 2 and 5 K MC traces (inset of Fig. 5a), which show how 2DWL behavior would manifest itself at high magnetic fields without the presence of another transport mechanism. Competition between 2DWL (positive MC) and wavefunction shrinkage (negative MC) may arise from the presence of a freezing out of thermally activated carriers from narrow-bandgap semiconductors.

From the fitting of Eq. 3, we obtain the temperature behavior of B φ (inset of Fig. 5), which we use to determine both L φ and the main dephasing mechanism of the 2DWL transport. B φ is assumed to follow a powerlaw scaling with T , such that B φ ∝ T p . Using a singleparameter fit to our values of B φ , we find that p is 0.80, which suggests that electron-electron interactions in either 2D (p=1) or 1D (p=0.67) is the dominant dephasing mechanism. We use B φ (T ) to calculate the phase coherence length, L φ , of the carriers. As shown in Fig. 5b, L φ increases with reduced temperature rising from ≃10 to ≃40 nm as the temperature decreases from 70 to 2 K. These numbers closely agree with results obtained in doped, pristine SWCNT mats 13 , m-SWCNT films 17 , and multi-wall carbon nanotubes 21,67 . Importantly, the temperature scaling (fit without any offset parameter) of L φ is shown in Fig. 5b, which follows from

L φ ∝ 1 B φ and p ≈ 1, so L φ ∝ T -1/2 .
As noted above, the ambiguity of the dimensionality of the electron-electron dephasing mechanism arises from a value of p that falls between 0.67 and 1. Given the magnitude of L φ is on the order of the bundle diameter, we believe that the phase decoherence from electron-electron interactions is two dimensional. If the electron transport is on the surface of a SWCNT bundle, then this picture make sense. However, it is very reasonable to expect that 1D electron-electron interactions may be dominant. In this paradigm, electron transport occurs along individual nanotubes (or closely grouped nanotubes) in a bundle. Regardless of the exact dimensionality of the dephasing mechanism, the observation of electron-electron interactions has precedent in several other closely related structures, such as iodine-doped polyacetylene 39 , pregraphitic nanofibers 62 , individual multi-wall carbon nanotubes 59 , ensembles of multi-wall carbon nanotubes 67 , and doped SWCNTs 13 . This behavior suggests that even at high temperatures, electron-electron interactions play a role in ensemble metallic nanotube electrical transport.

Finally, we return to the T -dependent conductivity of m-SWCNTs shown previously in Fig. 5c. Given we are dealing with a two-dimensional system, we express the σ (0, T ) sheet as a sheet conductivity by multiplying conductivity by film thickness. 2DWL theory predicts that σ (0, T ) sheet should scale as ln (T ) if no other effects are significant 58 :

σ (0, T ) sheet = σ 0 + Γ pe 2 2π 2 ln T T C , (4)
where σ 0 is from impurity scattering, Γ is a fitting constant, and T C is the critical temperature. As seen in Fig. 5c, Eq. 4 describes the low temperature behavior of σ (0, T ) sheet very well. Deviations from the ln (T ) scaling behavior occur when T goes above ∼30 K (indicated by the dashed line) suggesting that a high-T mechanism other than 2DWL may also be affecting carrier transport. We note that the inclusion of a Coulombic interaction correction, often called the Altshuler-Aronov correction 58,68,69 , did not improve our description of the data compared to Eq. 4. Though carrier-carrier interactions are likely involved in the zero field, T -dependent transport behavior, their effect in m-SWCNT ensembles manifests itself most prominently the 2DWL dephasing mechanism.

In summary, we systematically studied T -and Bdependent carrier transport using SWCNT ensembles of varying metallicity. We observed distinct magnetotransport behaviors for the metal-and semiconductorenriched nanotube networks that deepen our understanding of magneto-transport in SWCNT ensembles. Specifically, we ascribe 2D VRH transport to the semiconductor-enriched and pristine SWCNT ensembles. The large, negative MC of the semiconductor SWCNT films at T <10 K reveals contributions from forward interference (positive) and wavefunction shrinkage (negative) from which we estimate that the localization length of the hopping species, ξ, to be 2.81±0.1 nm and the density of states at the Fermi energy as (2.7 ± 0.2) × 10 15 eV -1 cm -2 . In contrast to the large, negative MC seen in the s-SWCNT film, a weak, positive MC was observed in the metal-enriched SWCNT sample. We find that this behavior is well-described by a 2DWL mechanism, whose dominant dephasing process occurs via low-dimensional electron-electron interactions over a wide temperature range. From the T -dependent MC, we establish that L φ is tens of nm and scales as T -1/2 . Importantly, we find that 2DWL is also able to explain σ (0, T ) sheet for the m-SWCNTs at low T . Taken together, our results provide deep insights into the magneto-transport mechanisms of SWCNT ensembles, which are increasingly becoming a technologically important carbon material.

FIG. 1 .

 1 FIG. 1. (a) Scanning electron microscope image of SWCNT networks. The films used for electrical transport experiments were washed to remove surfactants and thermally annealed to remove adsorbed species. (b) Absorption of the postannealed metal-enriched (blue) and semiconductor-enriched (red) SWCNT films24 . The difference in absorption clearly show the presence of distinct SWCNT electronic types, which in combination with other characterization techniques, allow us to conclude that the films were highly type enriched.

FIG. 3 .

 3 FIG. 3. (a) T -dependent conductivity for the pristine (green), metal-(blue), and semiconductor-enriched (red) SWCNT ensembles. Consistent with expectations, as the metallicity of the samples increases, so does the conductivity. (b) ln W as a function of ln T for each of the three samples. The slope, ∂ ln W ∂ ln T , is equal to -α, which we find to be -0.370 (d = 1.70 ≃ 2) for pristine SWCNTs and -0.369 (d = 1.71 ≃ 2) for s-SWCNTs. As seen by positive slope of ln W , the m-SWCNT conductivity clearly does not follow a VRH behavior. (c) ln σ as a function T -1/3 for the pristine and s-SWCNT samples. Two-dimensional VRH describes the pristine and s-SWCNT network conductivity across a broad temperature range.

FIG. 4 .

 4 FIG. 4. (a)T -dependent s-SWCNT MC below 10 K exhibits very large negative MC (on the order of 0.10) that monotonically decreases with decreasing temperature. The black fit lines from Eq. 2 show that interference (positive term) and wavefunction shrinkage (negative term) can explain most of the MC behavior of s-SWCNT ensembles. (b) Temperature dependence of the K(A) ratio scales as T -1/2 . The small value of K(A) (<0.25 for all T ) and the T -1/2 trend strongly suggest the transport occurs in the weak scattering regime. Inset: B0 values for all temperatures. When B>B0, the wavefunction shrinkage MC component dominates the interference term.

FIG. 5 .

 5 FIG. 5. (a) T -dependent MC for the m-SWCNT film. As T decreases, the MC increases and becomes saturated at increasingly lower B values. The 2DWL mechanism (black lines) fits the MC data well from 2 to 70 K. Inset: MC traces with fits for 2 and 5 K. The dotted black lines show the extrapolated 2DWL MC fit. (b) L φ increases with decreasing T , following a T -1/2 behavior from 2 to 70 K. This scaling suggests that the phase decoherence occurs via electron-electron interactions. Inset: The measured values of the coherence field, B φ , show a linear dependence on T . (c) T -dependent sheet conductivity of the m-SWCNT film (blue) follows a 2DWL behavior at low T , as indicated by the black fit line from Eq. 3 (dotted line is the extrapolation out to 100 K).

ACKNOWLEDGMENTS

The authors thank Matthew Fischer, Chris Leighton, and Angela Hight-Walker for stimulating and productive conversations.

XW thanks the National


A Segura

L Artús

R Cuscó

T Taniguchi

G Cassabois

B Gil

Natural optical anisotropy of h-BN: Highest giant birefringence in a bulk crystal through the mid-infrared to ultraviolet range

d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Natural optical anisotropy of h-BN: Highest giant birefringence in a bulk crystal through the mid-infrared to ultraviolet range

A Segura, L. Artus, R. Cusco, Takashi Taniguchi, Guillaume Cassabois,

Bernard Gil

To cite this version:

A Segura, L. Artus, R. Cusco, Takashi Taniguchi, Guillaume Cassabois, et al.. Natural optical anisotropy of h-BN: Highest giant birefringence in a bulk crystal through the mid-infrared to ultraviolet range. Physical Review Materials, 2018, 2, pp.024001. 10.1103/PhysRevMaterials.2.024001. hal-

I. INTRODUCTION

Hexagonal boron nitride (h-BN) is a van der Waals crystal that has been intensively studied over the last few years for the high interest of its numerous potential applications. The wide band gap of this layered material (∼6 eV) [1] and its high emission efficiency [2] make h-BN a material of special interest for deep ultraviolet photonic device applications [3][4][5]. In spite of the fact that ab initio band structure calculations long predicted an indirect band gap for h-BN [6,7], only recently the experimental demonstration of the indirect nature of the h-BN band gap has been provided [8]. Because of the similarity between the in-plane lattice constant of h-BN and graphene, h-BN is ideally suited to form van der Waals stacked heterostructures. The exceptional crystalline quality of such structures has opened up new avenues for exploring novel physics and potential applications of stacked two-dimensional materials [9,10]. By encapsulating graphene in h-BN layers, graphene devices exhibiting extremely high electron mobility at room temperature were demonstrated [11]. Atomically thin graphene-boron nitride lateral heterostructures have been realized that may enable the development of atomically thin integrated circuits [12]. Besides its prominent role in advanced optoelectronics, h-BN has also emerged as a material with unique properties for solid-state neutron detector applications [13].

In spite of the huge amount of research recently reported on h-BN, the knowledge of its optical properties is rather limited, possibly because of the difficulties in growing sizable samples with high crystal quality. Spectroscopic ellipsometry studies on amorphous BN films deposited by rf magnetron sputtering consisting of mixed cubic-BN and h-BN phases reported the dielectric function in a wide energy range [14]. Values of the macroscopic dielectric constant along the c axis (ε ) and perpendicular to the c axis (ε ⊥ ) were reported in infrared reflectivity studies on h-BN polycrystalline samples grown by pyrolytic methods [15]. Ab initio calculations indicate that the layered structure of h-BN induces a remarkably strong optical anisotropy [7,16], which was not confirmed by the experimental work of Geick et al. [15], probably because the polycrystalline nature of the samples precluded an accurate determination of pure ε and ε ⊥ values. To our knowledge, the disparity between theoretical and experimental values of the dielectric function anisotropy has not been elucidated yet.

In recent times, a great deal of effort has been devoted to develop semiconductor nanostructures exhibiting large artificial birefringence for applications in quantum optics [17], polarization sensitive optoelectronics [18], and microphotonic polarization converters applications [19]. Birefringence values as high as n = 0.8 at 633 nm have been reported in ensembles of GaP nanowires [20], which are much larger than the birefringence of the YVO 4 crystal (0.21 at 633 nm) widely used in laser devices and optical components [21].

While the structural anisotropy intrinsic to all van der Waals crystals is presumed to produce a strong optical birefringence, detailed experimental studies of birefringence in these materials are lacking. Even though on account of its layered structure the h-BN crystal is expected to be highly optically anisotropic, and indeed it has emerged as a natural hyperbolic crystal suitable for supporting phonon polariton modes [22], birefringence of h-BN has not received much attention despite the vast variety of technological applications of this material. The deceptively low value of dielectric constant anisotropy reported in Ref. [15] may have contributed to dim the interest of h-BN as a birefringent crystal.

Here we report a determination of the refractive index anisotropy of h-BN in a wide energy range from mid-infrared (MIR) to ultraviolet (UV) from polarized transmittance measurements. The high-frequency dielectric constant was evaluated from fits to reflectivity measurements in the MIR. The ordinary refractive index as a function of wave number n ⊥ (σ ) was then obtained from the transmittance interference fringe pattern measured under normal incidence on the c face. The small thickness of the high quality h-BN samples and the uneven nature of the lateral faces resulting from their layered structure preclude polarized transmittance measurements with the electric field parallel to the c axis. Therefore, the extraordinary refractive index dispersion n (σ ) was determined from the analysis of transmittance measurements on the c face at various incidence angles with polarization in the incidence plane and perpendicular to it.

II. EXPERIMENT

High quality h-BN single crystals were synthesized at 4.5 GPa and 1500 • C using barium boron nitride as a solvent in a modified belt-type high-pressure and high-temperature apparatus [23]. Samples of thickness ranging from 1 to 15 μm were cleaved from h-BN single crystalline platelets. Optical transmittance and reflectance experiments in a wide spectral range from MIR to deep UV were performed in several homemade laboratory setups. Optical reflectance measurements in the MIR range (1800-7000 cm -1 ) were performed by means of Fourier transform infrared spectroscopy (FTIR) using an Interspectrum TEO-400 Michelson interferometer module and a noncommercial all-reflecting microscope optical bench [24]. For transmittance spectra in the near IR, visible, and UV range, halogen and deuterium lamps were used as sources, along with an all-reflecting optical bench, with long focal length concave mirrors, and a series of multichannel spectrometers covering a wavelength range from 1700 to 184 nm. For transmission measurements at various incidence angles, the sample was mounted on a rotation stage equipped with an angle and position centering system. Normal incidence was checked by verifying that transmission spectra for a given angle were identical, irrespective of the rotation direction.

III. RESULTS AND DISCUSSION

Figure 1 (symbols) shows the refractive index spectrum n ⊥ (σ ) of h-BN in the MIR range for polarization perpendicular to the c axis, as obtained from the interference fringe pattern of the reflectance spectrum. The reflectance minima are given by the constructive interference condition 2n ⊥ (σ )d = K, where σ is the wave number, d is the sample thickness, and K is the interference order. The interference order and the sample thickness can be self-consistently determined by matching the interference minima to the refractive index determined from the dielectric function of a polar mode, which is given by the well-known expression

ε ⊥ (σ ) = ε ⊥ ∞ + (ε ⊥ 0 -ε ⊥ ∞ ) σ 2 TO⊥ σ 2 TO⊥ -σ 2 -i ⊥ σ . ( 1
)
Here ε ⊥ 0 and ε ⊥ ∞ are, respectively, the static and electronic contributions to the dielectric constant for light polarization perpendicular to the c axis, σ TO⊥ is the frequency of the TO phonon polarized perpendicular to the c axis (E 1u symmetry), and ⊥ is its damping parameter. The best fit (blue dashed line in Fig. 1) was obtained for the values σ TO⊥ = 1392 cm -1 , ε ⊥ 0 = 6.96, ε ⊥ ∞ = 4.95, ⊥ = 3 cm -1 , and a sample thickness d = 13.5 μm. The σ TO⊥ value is in good agreement with experimental and theoretical values of the E 1u (TO) frequency previously reported [15,25].

Once the refractive index dispersion n ⊥ (σ ) is established in the MIR range, its frequency dependence in the NIR, visible, and UV range can be determined from the transmittance interference fringe pattern of thinner samples (3.4 and 1.7 μm thick). The interference order and sample thickness are evaluated by nonlinear fits in the MIR range and then the refractive index is obtained from the constructive interference condition. Figure 2 shows the refractive index spectrum of h-BN, for polarization perpendicular to the c axis and frequencies above the restrahlen band. Symbols correspond to the experimental values determined by the above method, whereas dashed and solid lines are the results of several simple physical models that were applied to quantitatively account for the refractive index dispersion. The green line was calculated using Eq. ( 1) by assuming a constant electronic contribution to the dielectric function. The fit of this simple model illustrates the fact that below 7000 cm -1 the refractive index dispersion is practically determined by polar phonons.

The blue curve was obtained by assuming that the electronic contribution to the refractive index dispersion obeys a simple Phillips-van Vechten model [26,27]. This model assumes that all valence-to-conduction band transitions that contribute to the material electronic dielectric function can be described by a single transition at an average frequency which is known as Penn gap [28]. Under this assumption, the electronic dielectric FIG. 2. Experimental values of h-BN refractive index for polarization perpendicular and parallel to the c axis obtained from the interference pattern of the transmission spectra (filled dots). For n ⊥ , the solid green line was calculated using Eq. ( 1) assuming a constant electronic dielectric constant. The blue line additionally includes a simple Phillips-van Vechten model [Eq. ( 2)] and the red line is obtained by considering also an additional excitonic term [Eq. ( 3)].

For n , the dashed red line was obtained by fitting Eq. ( 2) to the data. Inset: Squared effective refractive index at σ = 18 000 cm -1 (blue) and σ = 43 000 cm -1 (red) as a function of the squared sinus of the incidence angle for s-(filled squares) and p-polarized (filled dots) light. The slope of the linear fit (dotted lines) to the p-polarization data allows n to be accurately determined.

function is given by

ε ⊥ ∞ (σ ) = 1 + σ 2 P 0⊥ σ 2 0⊥ -σ 2 , ( 2
)
where σ 0⊥ and σ P 0⊥ are, respectively, the Penn gap frequency and an average plasma frequency of valence electrons involved in allowed transitions for polarization perpendicular to the c axis. The best fit to the experimental results was obtained with a Penn gap of 63 000 cm -1 (7.8 eV) and a plasma frequency of 124 800 cm -1 (15.5 eV). This model accounts for the refractive index dispersion up to 30 000 cm -1 , but obviously cannot explain the steep increase of the refractive index for photon energies approaching the fundamental gap of h-BN. This discrepancy is a clear indication that, in spite of the well-established indirect character of h-BN [8], there must be an allowed transition at an energy slightly above the indirect band gap whose contribution to the refractive index cannot be neglected.

While the direct gap contribution to the refractive index can be accurately reproduced in many semiconductors by assuming a steplike absorption onset at the band gap energy [29,30], this model does not seem to fit the electronic structure of h-BN. In agreement with ab initio band-structure calculations [7], which suggest strong excitonic effects in the dielectric function of h-BN, an additional Lorentzian term corresponding to an excitonic transition with central frequency σ 1⊥ , damping parameter 1⊥ , and effective plasma frequency σ P1⊥ was included in Eq. ( 2) in order to obtain a better fit to the experimental refractive index dispersion n ⊥ (σ ). Then, the dielectric function for polarization perpendicular to the c axis is given by

ε ⊥ ∞ (σ ) = 1 + σ 2 P 0⊥ σ 2 0⊥ -σ 2 + σ 2 P 1⊥ (σ 2 1⊥ -σ 2 ) (σ 2 1⊥ -σ 2 ) 2 + 2 1⊥ σ 2 .
(

) 3
Given that the explored range of wave numbers showing interference does not include the refractive index maximum associated with the excitonic term in Eq. ( 3), a degree of uncertainty in the determination of σ 1⊥ exists, and good fits to the experimental data can be obtained with different sets of parameters. We considered as a reference an experimental study of the dielectric constant in pyrolytic h-BN [31], which revealed an intense peak at 6.1 eV in its imaginary part. Then, the best fit to the experimental points (red line in Fig. 2) was obtained with σ 1⊥ = 49 200 cm -1 (6.1 eV), σ P 1⊥ = 53 200 ± 800 cm -1 (6.60 ± 0.10 eV), and 1⊥ = 1560 ± 270 cm -1 (0.193 ± 0.033 eV). Obviously the introduction of the additional excitonic term implies a shift of the Penn gap contribution to higher energy. The new values of the Penn gap parameters we find from Eq. ( 3) are σ 0⊥ = 75 100 ± 1300 cm -1 (9.31 ± 0.16 eV) and σ P 0⊥ = 124 100 ± 1400 cm -1 (15.39 ± 0.17 eV). The excitonic lines contributing to the strong peak in the imaginary part of the dielectric constant were identified as originating from π → π * interband transitions mainly with k points parallel to the c axis in the direction H K [7,31]. It should be stressed that because these excitonic lines involve bands with the same parity with respect to the mirror plane within each layer, they are only active for in-plane polarization.

The refractive index dispersion of h-BN for light polarization perpendicular to the c axis (n ⊥ ) and parallel to the c axis (n ) was determined from the interference fringe pattern of transmission spectra taken at different incidence angles θ i by rotating the sample about an axis perpendicular to the optical axis of the setup. Since the samples were cleaved with faces perpendicular to the c axis, for light polarization perpendicular to the incidence plane (s polarization) only n ⊥ appears in the light propagation equations, as only the ordinary ray propagates in the sample. Then, the interference maxima in the transmission spectrum are given by the constructive interference condition [32] 2n

⊥ d λ 1 - sin 2 θ i n 2 ⊥ = 2d λ n eff,s (θ i ) = K, ( 4
)
where we have defined the effective index n eff,s as

n eff,s = n ⊥ 1 - sin 2 θ i n 2 ⊥ . ( 5
)
Note that n eff,s is a purely instrumental parameter for the determination of the actual refractive index n ⊥ . For p-polarized light, both n ⊥ and n affect the light propagation inside the sample, and the transmission spectrum at a given incidence angle exhibits interference maxima at wavelengths determined by the following equation [32]:

2n ⊥ d λ 1 - sin 2 θ i n 2 = 2d λ n eff,p (θ i ) = K. ( 6
)
As in the case of s-polarized light, an effective index n eff,p for each incidence angle can be defined as

n eff,p = n ⊥ 1 - sin 2 θ i n 2 . ( 7
)
Transmission spectra recorded on the 3.4-μm-thick sample for s and p polarizations for a number of incidence angles are shown in the Supplemental Material [33]. From the interference patterns, with the interference order K and the sample thickness determined from the analysis of the MIR reflectivity, the effective index dispersion for sand p-polarized light can be obtained. Plots of the effective index dispersion are given in the Supplemental Material [33].

According to Eqs. ( 5) and ( 7), plots of the square of the effective indices versus sin 2 θ i at a given wave number should yield straight lines for both polarization conditions. For n eff,s the slope is -1, whereas for n eff,p the slope equals the squared anisotropy ratio (n ⊥ /n ) 2 , from which the value of n can be accurately determined. The inset of Fig. 2 illustrates this method for two wave numbers. While the slope of n eff,s is found to be 1 ± 0.05 for both wave numbers, the slope of n eff,p varies from 2 in the visible range to 4 in the UV range. It should be stressed that the main source of error associated with this method comes from the error in the incidence angle.

Figure 2 also shows the values of the refractive index of h-BN for polarization parallel to the c axis. The dashed red line is a fit of a simple model equivalent to Eq. ( 2) with the parameters for light polarization parallel to the c axis. The best fit is obtained for σ 0 = 90 000 cm -1 (11.2 eV) and σ P 0 = 110 000 cm -1 (13.7 eV), which yields an electronic dielectric constant ε ∞ = 2.86. This is in excellent agreement with the dielectric constant values predicted by ab initio calculations [7,16].

Even in the visible-NIR range h-BN exhibits a giant negative birefringence ( n = n en o ∼ -0.7) which is much higher in absolute value than the birefringence found, for instance, in YVO 4 ( n = 0.21 at 633 nm), a standard example of a birefringent crystal that is widely used in laser devices and optical components [34]. These results are in stark contrast with previously published infrared reflectivity measurements on pyrolytic h-BN [15], which suggested a much smaller anisotropy ( n ∼ -0.2). As pointed out above, the origin of this discrepancy is most likely related to the relatively poor crystalline quality of the h-BN samples available at the time of Ref. [15]. The optical anisotropy of h-BN increases markedly in the UV range as the wave number approaches the band gap because of the dramatic increase of the dielectric function for polarization perpendicular to the c axis. Such a vast increase in ε ⊥ ∞ (σ ) (140% from 3.2 to 5.7 eV), which was modeled by including a Lorentzian excitonic term in Eq. ( 3), arises from the huge excitonic effects taking place in h-BN. As a consequence of the layered two-dimensional character of the h-BN crystal, the charge density inhomogeneity is large along the c axis and the excitons are tightly confined within the layers. Low-lying excitons stem from interband transitions between conduction and valence bands of the same parity with respect to the mirror plane within each layer and they exhibit a large oscillator strength [7,31]. Parity selection rules allow these transitions only for polarization perpendicular to the c axis. Therefore they contribute to ε ⊥ (σ ) but not to ε (σ ), and as a result, the giant birefringence of h-BN shows a sizable increase in the UV range.

In Fig. 3 we compare the birefringence of h-BN measured over the NIR-UV range in this work with birefringence data on a selected group of birefringent semiconductor crystals available in the literature. In the visible range, the h-BN birefringence in absolute value is more than one order of magnitude higher than in other uniaxial semiconductors such as wurtzite-type GaN [35] and ZnO [36] or 6H-SiC [37]. The birefringence in other layered semiconductors such as GaTe [38] is also more than one order of magnitude lower. Highly birefringent photonic metamaterials based on the artificial birefringence of aligned nanowires are being developed for optical applications. For oblique-aligned ZnO nanowires, the birefringence is almost one order of magnitude higher than that of bulk ZnO [39], but still much less than the one measured in h-BN. Only the record values of n = 0.8 at an optical wavelength of 633 nm achieved in GaP ensemble nanowires with very large filling fractions [20] can compare with the optical anisotropy observed at this wavelength in h-BN. And more strikingly, the birefringence of h-BN displays a huge increase in absolute value for increasing wave numbers in the UV range. This behavior is not observed in other uniaxial semiconductors or layered compounds, and is linked to the unique electronic structure of the layered h-BN crystal which underpins the strong electron-hole interaction and huge excitonic effects present in this material.

IV. CONCLUSIONS

In-plane and out-of-plane refractive index dispersion of h-BN has been determined over a wide energy range, from MIR up to UV energies slightly above the band gap, by analyzing the interference patterns in transmission measurements performed at different incidence angles and with polarization parallel and perpendicular to the incidence plane. Our results reveal an extreme variation of the ordinary ray refractive index n ⊥ from 1.5 in the MIR range up to 3.7 in the UV range close to the band gap. A simple Penn gap model fails to account for such a huge increase. The experimental results can be quantitatively explained by considering a strong excitonic peak contribution to the dielectric function for polarization perpendicular to the c axis. This finding is in agreement with previous ab initio calculations that indicate the presence of huge excitonic effects in layered h-BN and predict excitonic transitions with large oscillator strengths that are symmetry allowed only for polarization perpendicular to the c axis. The refractive index for polarization parallel to the c axis (n ) is significantly lower than n ⊥ (n ∼ 1.6-1.7) and shows a much less marked increase as excitonic transitions are not allowed by the polarization selection rules. This results in a giant birefringence that steadily increases towards the UV range, where birefringence values as high as | n| ∼ 2 have been measured. This is about seven times higher than the birefringence measured in calcite at the same wavelength [40]. Such a giant anisotropy has not been reported in other uniaxial semiconductors or layered compounds, nor even in engineered birefringent metamaterials. In the visible range, only high filling fraction GaP nanowire layers exhibit a comparable birefringence. Remarkably, in the UV range the birefringence in h-BN is sizably larger than in any other known material.

From the fits of the dielectric function model we obtain values of the electronic dielectric constants ε ⊥ ∞ = 4.95 and ε ∞ = 2.86. The dielectric constant we find for polarization parallel to the c axis is much lower than previously reported experimental values, but it is in excellent agreement with theoretical calculations. Thus, our measurements reveal that h-BN exhibits a much stronger birefringence than previously thought and therefore pinpoint h-BN as a material of outstanding interest for optical applications that require huge birefringence components.

FIG. 1 .

 1 FIG.1. Experimental values of the refractive index for polarization perpendicular to the c axis in the MIR range as determined from the constructive interference condition at the interference minima of the reflectance spectrum of a 13.5-μm-thick h-BN sample (filled dots). The refractive index calculated using Eq. (1) with the parameter values given in the figure is plotted as a blue dashed line.

FIG. 3 .

 3 FIG. 3. Birefringence of h-BN in the NIR-UV range compared with the birefringence of several representative uniaxial semiconductors reported in the literature. The giant birefringence of GaP and ZnO ensemble nanowires is also displayed for further comparison.

ACKNOWLEDGMENTS

This work has been supported by the Spanish MINECO/FEDER under Contracts No. MAT2015-71035-R and No. MAT2016-75586-C4-1-P. B.G. acknowledges the Russian Megagrant program (Ioffe Institute of RAS, Contract No. 14.W03.31.0011). T.T. acknowledges support from the Elemental Strategy Initiative conducted by the MEXT, Japan and JSPS KAKENHI Grant No. JP15K21722.


Nikolaos Chouliaras

NUTRITIONAL STATUS OF OLIVE GROVES IN MAGNESIA (GREECE)-NUTRITION SURVEY

In 20 groves in Magnesia, a nutrition survey program was applied for 10 years, in order to investigated the nutritional content of leaf dry matter, in relation with productivity of olive groves. It was found that the best olive trees were in calcareous soils, in flat surfaces, with relatively younger age trees.

INTRODUCTION

The olive tree is a plant preferring calcium and potassium, growing on a wide variety of soils, either deep or shallow, even in well fertile hills. That widespread is evident in the olive cultivation in Magnesia (Greece). Magnesia is characterized by soils situated in a variable environment of geomorphology, and the climate is classified either in the orange zone (subtropical Mediterranean), toward the Aegean and in the zone of olives (Mediterranean plus Temperate, [START_REF] Papadakis | The agricultural climate of Greece[END_REF]. The nutritional status of trees depends significantly to field practices, particularly from fertilizing operations. In this paper, the nutritional status of olive groves is studied in relation to their productivity.

METHODS AND MATERIALS

The nutritional status of olive studied in Magnesia area, is based on dry matter content of the leaves (lamina+petiole) in N, P, K, Ca, Mg. The program in Magnesia was applied for 10 years (1974 to 1983) from Soil Mapping Institute of Larissa. Table-1, notes the geographical distribution of the olive groves and the basic characteristics of soils, referring to shallow ENTISOL or INCEPTISOL. Olive groves enumerated from1 to 16, are irrigated and those 17-20 are not irrigated (rain fed). They are edible varieties, grafted on olive seedlings, aged from 9 to 50 years, at the starting of leaf analysis program. Sampling of leaves made during the period of flowering and the estimation of yields of trees took place during the harvest.

The data was processed for statistical evaluation and using the test of TUKEY.

. Achilleio

RESULTS

In Table 2, the data of foliar analysis are listed, expressed as content on % of dry matter, and marked as averages data.

The average content of all groves in nutrients are: N = 1.67 %, P = 0.13, % K = 0,84%, Ca = 1,49%, Mg = 0, 20%

The average yield ( R ) of olive trees during the study are: R = 29,3 kg of fresh fruit per tree. Olive groves, enumerated 14 and 11, indicated respectively the maximum and the minimum average with 56,3 kg/tree and 14,7 kg/ tree. In a work conducted in southern Greece [START_REF] Yassoglou | Soil and leaf analyses of olive groves in Greece. 14th[END_REF], the average content of leaf nutrient, found as follows:  Sub-humid climates, N = 2 %, K =0,5%, Ca=1,7%, Mg=0.18%  Arid climates, N=1.9%, K=0,78%, Ca=1,3%, Mg=0.17% Several researchers, studying the problem of olive nutrition, (as in other plants), seek criteria based on balanced rates of elements, and the relationships formed, among the concentrations of various elements within plant tissues [START_REF] Gavalas | The nutrition of olive trees[END_REF][START_REF] Gonzales | Importance of nutritional factors for production of flowers and fruits in Manzanillo olive trees[END_REF][START_REF] Gonzales | The importance of potassium in the physiologica factors concerned in fruiting of olive[END_REF][START_REF] Mazvelos | Development of physiological equilibrium and their relationships with the metabolism of micro and macronutrients in olive trees[END_REF]). The linear correlations investigated between various criteria recommended by the literature and productivity of olive groves, showed the following significant correlations at 0.05 (*) and 0.01 (**) levels of significance, between the mean values of the following parameters:  among content of leaves in N and P : Positive (*)  among content of leaves in K and Mg : Positive (*)  among the productvity of trees and the content of leaves in Ca : Positive (**)  among the total content of leaves in T=N +P+K+Ca_Mg and the productivity of trees: Positive (**). The last linear correlation is described by the following equation regression: R = 41.8 x ( T ) + 151,3 ( r = 0,880 **), where T = N + P + K + Ca + Mg. Further investigation shows the existence of a positive linear correlation between total nutritional status of the trees and their productivity on an annual basis, r=0,500 (*) for the year 1977 and r = 0,570 (*) for the year 1978. The average content of Ca in leaves related positively and linearly with the average productivity of olive groves, should probably be attributed to the limestone preference of olive. The problem of plant nutrition as pedo-climatic, is complex [START_REF] Eryuge | Seasonal variation of nutrients contents in the leaves if olive variety from the Aynalik district[END_REF], and this study allows to discern only trends. Notwithstanding that, the statistical analysis indicates the sum of the concentration in nutrients in leaves as significant criterion of nutritional status olive grove. For all samples in that study, the average sum is T=4,3 %. The corresponding maximum and minimum noted in olive groves were found at gloves enumerated with s/n:14 is 4.72% and at s/n:11 is 3.90%. Statistical test, shows that there are significant differences (in the level 0.05), between values corresponding to averages of total nutrient content of the leaves. So based on the total content of leaf nutrient (T) and the yields of trees (R) two limit groups of groves are distinguished:  ( A): Improved category, including olive groves enumerated with: 14.15  (B): Degraded category. Includes groves with number: 11 and 12. The groves of Category ( A) are located in alluvial ENTISOL, almost flat lands enriched with CaCO 3 . The olive groves consist of trees aged 10-14 years at the starting of this study. The nutritional status of olive groves is characterized by the nutrition criterion (values of T) between 4.69 to 4.72 % and by the productivity (R) from 54,0 to 56,3 kg / tree. The available K (exchangeable + water soluble), in these groves is always > 105 ppm. The groves of class (B), located on slopes ENTISOL not containing CaCO 3 in surface, consist of trees aged over 50 years. Also, they are characterized by the following sizes: T = from 3.9% to 4.04 % , R = from 14,7 to 18,6 kg / tree. Available K is <80 ppm. Based on the above criteria, the average nutrient content of plant tissues, and the average productivity of olive groves were found as follows: Irrigated: T=4.37%, R =31,16kg/tree, Rainfed:T=4.12%, R =21,65kg/tree Table ( 2 Τ= N+P+K+Ca+Mg, (1): R = Grove's production

The positive linear correlation between the average content of the leaves in K and Mg, shows no apparent competition. This should probably be attributed to the low content of soil in K. But there' isn't any systematic problems of K deficiency. In most groves the available K is always <80 ppm. Regarding the content of soil P is low in all cases, P (OLSEN) is <5 ppm, but the needs of our sample seems warranted. The levels of P in the dry matter of leaves in different territories are not making substantial differences. But in calcareous soils, the fixing P phenomena occurred, with consequences for plants of olive (KhoIi et al. 1979). The Spanish direction, (Gavalas 1878), adopted the sum of percent content N+10P+K, as a criterion for assessing the nutrition category of olive groves.

According to this criterion, the groves in our sample are, always placed outside the area of excellent nutrition. The groves enumerated 14 & 15, belonging to the improved class (A), are placed at a limit area.

CONCLUSIONS

The study of the nutritional status of 20 groves in Magnesia , showed that the total content of Leaf of T = N + P + K + Ca + Mg, is statistically with significant correlation, compared to the productivity of olive groves.

According to this criterion, the improved groves in terms of productivity, are characterized by means values of T, ranging from 4.69 to 4.72 %, and average annual productivity (R) from 54,0 to 56,3 kg/tree, in fresh fruit.

The degraded groves, are respectively characterized by T, ranging from 3.9 to 4.04 % and R, from 14,7 to 18,6 kg / tree The general characteristics of olive groves in accordance with these criteria, indicate better conditions of nutrition and productivity, for groves locating in the small slopes, containing calcareous minerals, and of lower age.

Conversely, olive groves, with inferior nutrition and productivity, are in increased slopes, in the absence of CaCO 3 and with poorer content of available K, even when the age of the trees is more than 50 years. The literature notes and other general characteristics of soil, and even climatic conditions, associated with particular nutrition problems as color, gradient, altitude, (Sierra et al.1974); moreover, in years of extreme drought, trees show chlorosis, according to [START_REF] Matar | Diagnosis and control of lime induced chlorosis on olive trees grown under dry farming conditions by iron chelates[END_REF]).

  ) : Average % composition of dry matter of the leaves

Table ( 1

 ( ):General properties of soils

		Texture	CaCO 3		Organic
	s/n	of soil	(%)	pΗ	Matter	location
					(%)
	1	SL	10,5	7,7	3,1	Agios Lavrentios
	2	SL	1,3	7,7	1,9	Agios Lavrentios
	3 4	SL L	2,9 2,6	7,9 7,6	1,4 , 2,8	Agios Lavrentios Ano Lehonia
	5	SCL	0	7,4	1,9	Ano Lehonia
	6	SCL/L	traces	7,9	2,3	Ano Lehonia
	7	SCL	0,4	7,7	1,9	Ano Lehonia
	8	L	1,5	7,7	2,3	location Ano Lehonia
	9	SCL	2,1	7,7	1,5	Kato Gazea
	10	L	0,8	7,7	2,3	Kala Nera
	11	SCL/L	0	7,1	2,1	Kala Nera
	12	SCL	0	7,2	1,7	Kala Nera
	13	SCL/L:	0	7,1	2,3	Agios Vlassios
	14	S ι L	4,2	7,0	1,4	Platanakia Almyrou
	15 16	SCL/SL CL/SCL 0 2,3	7,3 6,5	1,7 , 1 , 1	Platanakia Almyrou Maloula
	17	SL	0	5,8	1,7	Itia Pteleou
	18	SL	0	5,9	1,5	Pteleos
	19	SCL	0	6,0	1,5	Achilleio
	20	SCL	0	5,9	1,5


Christopher J Fullerton

Ludovic Berthier

Glassy behaviour of sticky spheres: What lies beyond experimental timescales?

We use the swap Monte Carlo algorithm to analyse the glassy behaviour of sticky spheres in equilibrium conditions at densities where conventional simulations and experiments fail to reach equilibrium, beyond predicted phase transitions and dynamic singularities. We demonstrate the existence of a unique ergodic region comprising all the distinct phases previously reported, except for a phase-separated region at strong adhesion. All structural and dynamic observables evolve gradually within this ergodic region, the physics evolving smoothly from well-known hard sphere glassy behaviour at small adhesions and large densities, to a more complex glassy regime characterised by unusually-broad distributions of relaxation timescales and lengthscales at large adhesions.

Steeply repulsive particles with very short-range attractive forces ('sticky spheres') are experimentally realised with colloids [1,2]. When the attraction range is small compared to the particle size, the physics of sticky spheres differs qualitatively from that of atomic liquids [3][4][5]. Sticky spheres thus represent a unique paradigm for the statistical mechanics of soft materials and simple fluids, motivating a large number of theoretical studies and experiments. The phase diagram of sticky spheres is explored by changing the volume fraction and the adhesion strength, showing interesting behaviour at low (clustering and phase separation [6]) and large (crystallisation [7,8], glassy dynamics [9]) volume fractions.

Over the last two decades, the glass transition of sticky spheres received considerable attention. This effort gathered momentum when the mode-coupling theory (MCT) of the glass transition [START_REF] Götze | Complex dynamics of glass-forming liquids: A mode-coupling theory[END_REF] was applied to the square-well potential to predict the phase behaviour and glassy dynamics of sticky spheres [START_REF] Bergenholtz | [END_REF][12][13][14][15]. The predicted existence of two types of glass transition, of reentrant glassy dynamics, and of a glass-glass phase transition line ending at a singular critical point giving rise to non-trivial relaxation patterns triggered massive theoretical [16][17][18][19][20], numerical [21][22][23][24][25][26][27][28][29][30] and experimental [9,[31][32][33][34][35][36][37][38] efforts, which continue to this day.

For this reason, published work is often torn between successes and failures of these MCT predictions. Two recent computational studies [39,40] offer contradicting conclusions even on basic features of glassy sticky spheres and important physical questions are left unanswered which go well beyond the relevance of MCT predictions. There is a broad agreement on the existence of reentrant dynamics along isochores [9,24,32], non-trivial dynamic correlation functions at intermediate adhesion [23,26], and increasingly localised particle motion at large adhesion [9]. On the other hand the existence, nature and physical relevance of the MCT liquid-glass and glass-glass lines, of various phases (equilibrium gel, attractive, repulsive, bonded and non-bonded glasses), and the interplay between gelation, glassiness and phase separation remain debated. Resolving these questions has been technically too difficult so far, as large relaxation timescales plague both computer simulations and experiments, and prohibit the exploration of the equilibrium phase diagram. Informative non-equilibrium aging studies at large densities have been performed instead [27,39,41].

Here we show that the swap Monte Carlo algorithm, which has recently provided an equilibration speedup larger than 10 11 in several three-dimensional model glassformers [42] (including hard spheres [43][44][45]), performs equally well for dense sticky spheres. This decisive computational advance allows us to perform a complete exploration of the equilibrium phase diagram of sticky spheres, including regions at large densities where distinct phases were predicted or numerically reported. Our simulations instead reveal the existence of a broad ergodic fluid phase limited at large adhesions by a phaseseparated region where non-equilibrium gelation may occur. Within the ergodic fluid, the dynamics is reentrant along isochores, and evolves smoothly between the wellknown hard sphere limit to a more complex sticky glassy dynamics characterised by a broad hierarchy of relaxation timescales and lengthscales, but this appears distinct from the predicted MCT phases and singularities, which we do not observe.

We describe sticky spheres using the well-studied system of hard spheres decorated with a short-range attractive square-well. Particles separated by r ij have interaction energy

V (r ij ≤ σ ij ) = ∞, V (σ ij < r ij < λσ ij ) =
-u, and V (r ij > σ ij ) = 0 where (λ -1)σ ij defines the width of the attractive well, and σ ij = (σ i + σ j )/2. To compare with other studies, we use the relative width = (λ -1)/λ = 0.03 of the square well [23,25,27]. This value is often used as for it MCT predicts the existence of an A 3 singularity within the glass phase, close enough to affect the system's dynamics at points where it can be equilibrated on accessable timescales. We use a continuous distribution of particle diameters, P (σ min ≤ σ ≤ σ max ) = A/σ 3 , where A is a normalisation constant. We choose σ min and σ max to provide a polydispersity of ∆ = σ 2 -σ 2 / σ = 23%. This choice prevents crystallisation, makes the swap Monte Carlo algorithm efficient, and does not appear to lead to novel features in the dynamics, when compared to different types of size dispersity [42,43,46]. We use Monte Carlo dynamics to explore the structure and dynamics of the system. Equilibration is achieved using swap Monte Carlo, with details as in [42,47]. To analyse the dynamics, we perform conventional Monte Carlo simulations, which describe glassy dynamics equivalently to Brownian and Molecular Dynamics [48]. We simulate N = 1000 particles in a periodic cubic box of volume V . The packing fraction is φ = πN σ 3 /(6V ). We fix the temperature β = k B T = 1 and vary the well depth and packing fraction to explore the (u, φ) phase diagram. This produces equivalent results to varying the temperature with u fixed, as u/T is the appropriate control parameter. Additional simulations with N = 8000 are performed to investigate the phase separation boundary at large u. Times are measured in units of Monte Carlo steps, where a step represents N attempted Monte Carlo moves (swap or translational), and distances in units of the average particle diameter σ .

To quantify dynamics, calculate the meansquared displacement (MSD) defined as r

2 (t) = (1/N ) i |r i (t) -r i (0)| 2
, where r i (t) is the position of particle i at time t. The MSD is the second moment of the van Hove distribution of single particle displacements: G s (x, t) = δ(x -|x i (t) -x i (0)|) , for displacements along the x-direction (later averaged over all directions). We define the self-part of the incoherent scattering function: f (q, t) = (1/N ) j e iq.(rj (t)-rj (0)) . We perform a spherical average at |q| = 7.8, close to the first peak of the static structure factor, and define the structural relaxation time τ α as f (|q| = 7.8, τ α ) = e -1 . When the system is nearly arrested, we fit these functions using f (q, t)

= f q + h q [B (1) q ln(t/τ ) + B (2)
q ln 2 (t/τ )] [22,26], mainly to extract the non-ergodicity parameter f q .

To ensure efficient equilibration at large φ, we use swap Monte Carlo. At each state point, we define τ swap α via f (q, t) measured in the presence of swap moves. Note that all particles (small and large) need to relax for this function to decay, which ensures full ergodicity. We consider our system as adequately equilibrated if it has been simulated longer than 4τ swap α [42]. We collect independent equilibrium configurations at many state points (u, φ) to study static behaviour, and from these we launch many independent, conventional Monte Carlo simulations lasting up to t s = 5 × 10 8 MC steps to analyse the equilibrium dynamics over a broad time window, including at conditions where the physical relaxation time τ α is larger than t s by many orders of magnitude. This is only possible thanks to the combined use of swap and conventional Monte Carlo. The decisive progress provided by the swap algorithm can be appreciated in Fig. 1(a), which shows the equilibrium (u, φ) phase diagram. We distinguish two regions. The large blue area comprises state points where we achieved thermal equilibrium. This region extends to arbitrarily low φ, and is limited at large u > ∼ 3.5 by a phase-separated region. The ergodic region is limited at large φ by our ability to reach equilibrium, i.e. by the time spent running simulations. With more time, or an algorithm more efficient than swap we expect the ergodic region to extend to higher φ. We empirically define the right-most boundary as the isochrone where τ swap α = 10 7 MC steps, but densities even larger than this φ ≈ 0.65 -0.66 empirical boundary could presumably be explored by performing longer simulations.

There is a single phase transition in Fig. 1(a), not described by MCT. Holding φ constant and increasing u, the system phase separates into two phases with distinct densities. The potential energy after quenching from u = 0 to different points along the isochore φ = 0.5 is shown in Fig. 1(b). The large jump and different time dependence of the energy at long times between u = 3 and u = 3.5 indicates phase separation occurs somewhere between these points. The heterogeneous structure of the system was studied at long times, and the red crosses in Fig. 1(a) placed at points above which phase separation is seen to have occured. This can be done up to large volume fractions near φ = 0.6 [40] but at larger φ, the amount of low-density phase becomes too small and the coarsening too slow to identify the phase separation clearly. At very large u, the system resembles a gel but the slow decrease in energy at long times shows that the system coarsens [49], indicating the gel is not stable.

Everywhere in the blue area of Fig. 1(a), the system is an ergodic fluid. The significance of this conclusion comes when considering the physical dynamics of the system. Increasing φ at constant u, the relaxation time τ α increases very fast and the system becomes arrested on the observational timescales, as shown for u = 3.0 in Fig. 1(c). This figure also illustrates the giant speedup afforded by swap Monte Carlo at high φ for sticky particles. We report in Fig. 1(a) the isochrone τ α = 10 7 MC steps, which marks the limit where conventional simulations equilibrate. The isochrones with and without swap are parallel, but separated by a large gap ∆φ 0.05, nearly independent of u. This wide new territory is explored in equilibrium for the first time here, and provides distinct insight into the physics of sticky spheres at large densities.

Crucially, all distinct phases reported previously for this system belong to the same ergodic fluid phase. We conclude that none of these phases actually exists as such, and the phase diagram is much simpler than anticipated [12,32,39] with only two phases separated by the well-known discontinuous liquid-gas thermodynamic instability. Deep inside the phase separating region, coarsening towards a fully demixed state may become slow, but is not arrested [49]. Given the time window accessible to colloidal experiments, the phase separation is never complete and the system behaves as a colloidal gel [36,[50][51][52], with physical properties that are slowly aging. This represents the non-equilibrium route to colloidal gelation [53].

The ergodic region contains in particular all sharp features theoretically predicted by MCT which we map in Fig. 1(a) by following earlier work fitting our measured relaxation times to MCT power law predictions. The liquid-glass and glass-glass transition lines ending at the A 3 singularity all belong to the ergodic fluid. Therefore, they represent, at best, smooth physical crossovers [19]. Our demonstration that all ideal MCT singularities disappear in physical systems of sticky spheres echoes equivalent findings for molecular glasses [54] and colloidal hard spheres [55], which remain debated in the colloidal con- FIG. 2. f (q, t) at various q measured at a) the black cross (φ = 0.630, u = 2.5) or b) the orange cross (φ = 0.654, u = 2.5) in Fig. 1a. The near-logarithmic decay highlighted in a) for q = 26.3 is no longer present closer to the putative A3 singularity in b).

text [56][57][58][59][60].

Is the concept of an avoided A 3 singularity nevertheless useful? Our model displays the physical behaviour expected for a system with competing attractive and repulsive interactions. The banana-shaped iso-τ α line in Fig. 1(a) implies reentrant glassy dynamics as u varies along isochores. Reentrance is mathematically described by MCT via the existence of two distinct glass transition lines, but these are not required to explain it [19]. Much less trivial is the observation of a transient 'logarithmic' decay of f (q, t) at well-chosen state points approaching the A 3 point [23,26] where the relaxation should become purely logarithmic [14]. In Fig. 2(a), we show f (q, t) at (φ = 0.630, u = 2.5) (black cross in Fig. 1(a)) for a range of wavevectors q. The decay time increases with decreasing q, showing that the system remains mobile on short length scales but is frozen on long length scales. At intermediate q-values, a nearly logarithmic time dependence holds over about 5 decades, a behaviour clearly distinct from the conventional two-step decay observed in most glassy materials [54].

Previous work attributed this unsual dynamics to proximity to the A 3 singularity [23,24,26]. We can test this hypothesis directly by measuring the equilibrium dynamics much closer to the A 3 singularity, as in Fig. 2(b). We find that all hints of logarithmic behaviour are gone, the dynamics now being consistent with a simpler twostep decay. (At timescales much larger than those shown here, structural relaxation will eventually take place.) These data suggest that the existence of an A 3 singularity may not be the best physical way to interpret the unconventional dynamics in Fig. 2(a). It was shown, for instance, that by numerically tuning the strength of competing attractive and repulsive interactions [25,61,62], a near-logarithmic decay may appear or disappear, or be replaced by a simpler multi-step decay.

Our results dispel the possibility that several distinct phases characterize dense sticky spheres [32,39]. No sharp distinction exists between attractive, repulsive, bonded and non-bonded glasses. Instead, we now show that increasing adhesion smoothly changes the physics between two qualitatively-distinct types of glassy dynamics. To see this, we explore the large φ region using several paths in the phase diagram changing either u or φ. Glassy dynamics is encountered for any u < ∼ 3.5 as φ is increased, see Fig. 3. In all cases, the diffusion constant drops by several orders of magnitude as φ increases, until diffusion becomes too slow to be observed. However, interesting differences can be seen between repulsive and sticky particles. When u = 0, the MSD displays a well-defined plateau, whose amplitude decreases smoothly with φ. For u = 3.0 no well-defined plateau can be seen, even for packing fractions as large as φ = 0.66 (remember that all data are taken in equilibrium and so we do not expect a plateau at even larger times). The plateau is replaced by a slow subdiffusive regime that extends over 7 decades in time dramatically distinct from the hard sphere cage physics. This behaviour also differs from ideas of an MCT-inspired attractive [12,25] or a bonded [39] glass suggested from simulations, and is not to be confused with non-equilibrium gelation either [40].

The sharp distinction between attractive and repulsive glasses is nonexistent, but in the regime u ≈ 2.5 -3.5 between phase separation and hard spheres the system exhibits unusual glassy dynamics, uncovered here thanks to swap Monte Carlo. We characterize this regime further in Fig. 4 by changing u along the φ = 0.65 isochore, which crosses the (putative) glass-glass line very close to the A 3 singularity. This isochore lies in the region where equilibration can only be achieved using swap.

In Fig. 4(a), we show the non-ergodicity parameter. At all wavevectors f q is higher at u = 3.5 than it is at u = 0, showing that stronger adhesion means less mobility at all lengthscales. The change in f q is greatest for large q (short lengthscales). When u is small, particles are free to move within the hard sphere cages but are immobilised on long lengthscales. As u increases, the attractive well can trap (or 'bond' [39]) particles at much shorter distances. Attractive interactions also destabilise the hard sphere glass, which results in a slight non-monotonic behaviour of f q at small q near u = 1.5. Again, f q varies smoothly with u (this is true across a range of φ) in contrast to the sharp jump predicted across the MCT glass-glass line.

The marked (but gradual) evolution along the φ = 0.65 isochore is further illustrated in Figs. 4(b,c) showing the time dependence of r 2 (t) and f (q, t). These functions change dramatically in the range u ∈ [0, 3.5]. At small u a well-developed plateau exists: the particles are caged by repulsive interactions with their neighbours. The approach to this long-lived (6 decades in time) plateau is fast. As u increases clear signs of a structural relaxation speedup appear at long times, together with a weakening of the plateau. Increasing u further the fast approach to a plateau gets replaced by a slow sub-diffusion (in r 2 (t) ), or a slow decay (in f (q, t)). This shows that at large u particles are neither caged nor bonded, but instead get arrested over multiple lengthscales, ranging from very short corresponding to the attractive well width to larger than the hard sphere cage size, which is no longer relevant. This differs from the picture of a bonded glass [39], but leaves room for a glass transition where adhesion is relevant, at odds with [40]. Rather they demonstrate that the structure and short-time dynamics of sticky spheres at large u is highly heterogeneous [29,34,37], and involves a very broad hierarchy of timescales and lengthscales long before structural relaxation.

The increasing heterogeneity of the glassy structure of sticky spheres is finally confirmed by the evolution of the van Hove distribution in Fig. 4(d). A near-Gaussian distribution is observed at small u, confirming the per-tinence of a description of the hard sphere glass with a typical cage size [63]. By contrast the van Hove distribution is much broader and strongly non-Gaussian at large u, with both a large peak at very small displacements and a fat non-Gaussian tail at large displacements, suggesting enhanced dynamic heterogeneity [29].

Using swap Monte Carlo, we have explored the complete equilibrium phase diagram of dense sticky spheres. A clarifying physical picture emerges with three distinct regimes of slow dynamics. At large adhesions, u ≥ 3.5, the system phase separates at least up to φ = 0.60 and discontinuously enters a slowly coarsening aging regime leading to non-equilibrium gelation. At small u ≤ 1.5 and large φ the system displays well-known hard sphere glassy dynamics, characterised by a two-step decay of correlation functions and a well-defined cage size at intermediate times. Finally, in the regime u = 1.5 -3.5 and large φ unusual glassy dynamics are observed, characterised by a broad distribution of relaxation timescales and length scales and a short-time dynamics distinct from hard spheres. We are aware of no atomic or molecular experimental analog of this unsual glassy behaviour, which involves multiple (time and length) scales and extended sub-diffusion long before the structural relaxation. The sharp distinction predicted by MCT between two types of glassy dynamics is invalidated by the data, which more importantly do not support the physical relevance of an avoided A 3 singularity to interpret the dynamics. The transient logarithmic time decay has a simpler interpretation and is not seen on approaching the A 3 location. The very unsual time correlation functions we report are instead observed at a much larger adhesion strength, away from the avoided A 3 singularity. The proposed clarification of the phase behaviour and dynamics of dense sticky systems should help reinterpreting past experiments and suggest new ones. Future numerical work could also help understand better the rheological behaviour [21,[64][65][66][67] in adhesive colloidal glasses, by subjecting high density equilibrium states to quasi-static and oscillatory shear.

FIG. 1

 1 FIG.1. a) Equilibrium phase diagram (u, φ) of sticky spheres, with a large ergodic fluid (blue) and a phase separated region (green). The isochrone τ swap α = 10 7 MC steps (full cyan line) limits the ergodic region at large φ, whereas the isochrone τα = 10 7 MC steps (black dashed line) marks the limit of conventional simulations. The avoided MCT singularities are mapped in orange (dashed: fluid to attractive glass, dotdashed: fluid to repulsive glass, solid: glass-glass line) ending at the A3 singularity (orange symbol). Grey areas could not be explored in equilibrium, and the black cross corresponds to Fig.2(a). b) Potential energy as a function of the time after a quench along the isochore φ = 0.5. c) Relaxation times with (blue) and without (red) swap at u = 3.0.

FIG. 3 .

 3 FIG.3. Evolution of the MSD with packing fraction for a) u = 0. and b) u = 3.0. The equilibrium glassy physics at short timescales and lengthscales for the adhesive system is different from that of hard spheres.

FIG. 4 .

 4 FIG. 4. Evolution of a) the non-ergodicity parameter, b) the mean-squared displacement, c) the intermediate scattering function, d) the van Hove function along the isochore φ = 0.65 at equilibrium.

We thank M. Cates, P. Royall and E. Zaccarelli for useful exchanges. This work was supported by a grant from the Simons Foundation (Grant No. 454933, L. B.).