Signature of pseudodiffusive transport in mesoscopic topological insulators

One of the unique features of Dirac Fermions is pseudodiffusive transport by evanescent modes at low Fermi energies when disorder is low. At higher Fermi energies, i.e., charge carrier densities, the electrical transport is diffusive in nature and the propagation occurs via plane waves. In this study, we report the detection of such evanescent modes in the surface states of topological insulator through 1 / f noise for the ﬁrst time. While signatures of pseudodiffusive transport have been seen experimentally in graphene, such behavior is yet to be observed explicitly in any other system with a Dirac dispersion. To probe this, we have studied 1 / f noise in topological insulators as a function of gate voltage, and temperature. Our results show a nonmonotonic behavior in 1 / f noise as the gate voltage is varied, suggesting a crossover from pseudodiffusive to diffusive transport regime in mesoscopic topological insulators. The temperature dependence of noise points towards conductance ﬂuctuations from quantum interference as the dominant source of the noise in these samples.

Topological insulators (TIs), with their spin-polarized, topologically protected, linear, metallic surface states, act as the perfect playground for investigating a plethora of fundamental phenomena [1][2][3][4][5].These surface carriers obey the Dirac equation for massless Fermions, where the Hamiltonian of the system is given by Here v F , σ, and k refer to the Fermi velocity, spin matrices, and momentum respectively.Due to the massless nature of the charge carriers, the screening properties of Dirac materials such as TIs or graphene, are also significantly different from other traditional 2D electron systems, and the potential due to charged disorder remains long-ranged even after screening is taken into account in Dirac materials [6].Another key feature of these materials is that it is possible to reach n = 0, without opening up a band-gap, even though strong carrier inhomogenities in the form of electron-hole puddles might be present around charge neutrality point or the Dirac point [7].The electrical transport properties of these classes of materials near the Dirac point, where the Fermi surface diminishes to a point, has been a matter of intense discourse, and has led to several fascinating discoveries in the context of graphene, such as dissipative quantum Hall effect, minimum conductivity, and pseudo-diffusive transport .Accessing the Dirac point in TIs compared to graphene has been a challenge due to high doping from bulk defects as well as the substrate, thus making it difficult to probe the intriguing properties of Dirac Fermions in TIs including the origin of 1/f noise.Previous investigation of 1/f noise in TIs have revealed the role of bulk disorder-mediated Hooge type mobility fluctuation type noise in 100 nm thick mesoscopic samples and correlated mobility-number density fluctuation model to be the dominant mechanism in large area epitaxially grown samples [8,[31][32][33][34][35][36][37].However, the origin of 1/f noise in TIs in thin (thickness d ∼ 10 nm) mesoscopic samples, especially near the Dirac point, also remains a matter of debate.In this manuscript, we have explored the origin of 1/f noise in mesoscopic samples, where we have access to the Dirac point also.Our investigation has revealed a non-monotonic dependence of 1/f noise magnitude on the carrier number density, which is a strong function of temperature as well, suggesting a crossover from pseudo-diffusive to diffusive transport, and the conductance fluctuations from quantum interference effects as the main source of noise in these types devices at low T .
The devices studied in this paper were fabricated using the TI Bi 1.6 Sb 0.4 Te 2 Se, which was exfoliated from a single crystal onto a SiO 2 /Si wafer using Scotch tape technique [9,10,38].Due to compensation doping, the ) f (Hz) quarternary alloy Bi 1.6 Sb 0.4 Te 2 Se has an insulating bulk, resulting in enhanced surface transport [31,38,39].Here below temperature T = 50 K for samples with thickness, d ≤ 100 nm, the current is essentially carried by the surface carriers [31].The atomically flat boron nitride (BN) substrate (Fig. 1a), significantly reduces the effect of dangling bonds and charged traps of the SiO 2 substrate on the electrical transport in the TI channel [34,40,41].The hetero-structure was then finally assembled onto a pre-patterned heavily hoped SiO 2 /Si substrate, with the 285 nm thick SiO 2 acting as a back gate dielectric, using a home-made transfer technique.The sample contacts were patterned by standard electronbeam lithography, followed by thermal evaporation of 5/40 nm Cr/Au (Fig. 1a).A layer of the polymer PMMA (poly(methylmethacrylate)) was coated on the samples, which ensured that the surface integrity is preserved throughout the measurement cycle.The measurements reported in this manuscript were performed on two identically prepared samples, BT1 and BT2, in a home-built variable temperature cryostat.The resistivity measurements were performed using a low frequency AC-four probe technique with carrier frequency of 18 Hz with an excitation current of 100 nA.The resistance (R) vs temperature (T ) shows metallic behavior, implying the predominance of surface states in the transport, as expected for 10 nm thin TIs flakes (Fig. 1b) [31].Fig. 1c shows the R vs V G , where a maximum in the resistance at V G ≈ −40 V at T = 5 K represents the Dirac point.The asymmetry in the R-V G on the electron and holes sides may arise due to asymmetry in the bandstructure itself [42].The typical mobility extracted from the R − V G graph is ∼ 100 cm 2 V −1 s −1 .Fig. 1d shows magneto-resistance (MR) behavior of BT1 at V G −V D = 0 V, 60 V and 120 V, characterized by a cusp in the quantum correction to conductivity △σ at B = 0 T [43][44][45][46].This demonstrates weak-antilocalization phenomenon, as expected for spin-momentum locked TI surface states, resulting from an additional π Berry phase between the back-scattered, time reversed path of the carriers leading to negative magneto-conductance.The magnetoconductance data can be fitted with the Hikami-Larkin-Nagaoka (HLN) [47,48] equation for diffusive metals with high spin orbit coupling (τ φ >> τ so , τ e ): where τ φ , τ so , τ e are the phase coherence or dephastime, spin-orbit scattering time and elastic scattering time respectively, ψ is the digamma function and B φ is the phase breaking field.Here α and B φ are fitting parameters.The phase coherence length l MR φ can be extracted using l MR φ = /4eB φ .The l φ obtained from the fit was ∼ 180 nm at To extract the magnitude of 1/f noise of the samples accurately, we have utilized a AC four-probe Wheatstone bridge technique [49][50][51].The voltage fluctuations were recorded as a function of time using a 16-bit digitizer.This was followed by digital processing of the time-series data to obtain the power spectral density (PSD, S V ) as a function of frequency (f ) (Fig 2a).In both the devices BT1 and BT2, S V ∝ 1/f α , and the exponent of the frequency, α ≈ 1. S V depends on the the bias (V ) in a quadratic manner, which ensured that all the measurements were performed in the Ohmic regime (Fig 2b).
The V G -dependence of the integrated noise magnitude 3a and Fig. 3b for samples BT1 and BT2.Although these two samples were identically prepared from the same bulk crystal, and show similar average electrical characteristics [34], they demonstrate contrasting behaviors in the V G dependence of noise.Whereas δG 2 G 2 vs V G for sample BT1 displays a M-shaped curve with a dip around the Dirac point (| V G − V D |= 20 V) (Fig. 3a), the identically prepared device BT2 shows a monotonic reduction as V G is tuned away from the Dirac point, as demonstrated in Fig. 2b.The non-monotonic behavior of 1/f noise previously reported in the context of graphene [8] and in TIs [31], has been attributed to the interplay of charge exchange noise (originating due to exchange of carriers between the channel and the surrounding environment) and configuration noise (arising due to potential fluctuations due to reorganization of trapped charges).Incase of graphene, however, this dip in noise across the Dirac point persists till room temperature, while for mesoscopic TI-FETs, this is a very strong function of T , and persists only till T = 14 K in sample BT1.We have fitted the V G -dependence of the normalized noise magnitude data using the framework of correlated mobility-number density fluctuations model [33,52], which is known to be the dominant mechanism of noise in large-area, thin (∼ 10 nm) TIs, where the effect of conductance fluctuations are suppressed to a large L/l φ ratio.The total noise can be expressed as, where J 1 = 1 8α represents a pure number fluctuation, )  2 dx represents combined number and mobility fluctuations (α is the decay constant for the spatially decaying time constant τ T of a typical trapping event and A(x) is the scattering constant) and can be evaluated using phenomenological values [52].D it , k B , W , L, σ, n, x are the areal trapped charge density per unit energy, Boltzmann constant, width of the channel, length of the channel, conductance and number density of charge carriers, axis in the direction perpendicular to the channel respectively, f = 1 Hz frequency and d = 1 nm is the distance over which the tunneling is effective.As is evident from the fit, this framework does not satisfactorily explain the observed nature of 1/f noise in mesoscopic samples, implying that the dominant source of 1/f in mesoscopic samples and large area TI samples are different (Fig. 3a-b).Such behavior of 1/f noise on the number density have been predicted theoretically for Dirac fermions for long-range as well as Gaussian disorder, due to a crossover from pseudo-diffusive to diffusive transition, which we believe is the scenario here [6].In the pseudo-diffusive regime, the transport in the channel occurs through quantum tunneling of evanescent modes.However, due to the presence of disorder, the system is driven into a diffusive metal phase, with the propagation occurring via plane waves.Although signatures of pseudo-diffusive transport has been reported in graphene [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22]53], there is no such clear signature in TIs.In the pseudo-diffusive regime, δG 2 enhances rapidly in magnitude compared to G with increasing n, while in the diffusive regime, δG 2 is almost constant whereas G increases.This leads to a non-monotonic dependence of 1/f noise magnitude on n, which is a generic property of crossover between these two regimes.
To gain further insight into the origin of 1/f noise in mesoscopic TI-FETs, we have performed V G -dependence of noise at various temperatures for both samples.The non-monotonic behavior of 1/f noise in sample BT1 shows a strong T -dependence with the peak almost disappearing for T > 20 K (Fig. 3c).The V G -dependence of noise in sample BT2 shows a monotonic decrease with number density at all temperatures.The T -dependence of noise for sample BT2 at various gate-voltages is shown in Fig. 3e.The magnitude of noise, reduces as the T is increased (Fig. 3e), contrary to what has been observed in MBE grown TIs before, where the noise magnitude increases due to scattering from thermally activated defects [33].The noise magnitude, as shown in Fig. 3e, for BT2, reduces as ∼ T −1 .Such a Tdependence of noise can be explained using the frame- ) n (10 16 m -2 ) φ , where k F , l, L x and L y are the Fermi wave-vector, mean free path and sample dimensions in x and y directions respectively [54][55][56].α(x) represents the change of the phase of electron wave-function due to scattering by a moving impurity at a distance δr, and n s (T ) is the number of active scatteres.For electron-electron interaction mediated dephasing, l 2 φ ∝ 1/T and n s (T ) ∝ T , we have δG 2 ∝ l 4 φ n s (T ) ∝ 1/T , as observed [54][55][56][57].While the overall noise magnitude for sample BT1 reduces at the T is increased, there is no specific trend which is observed, and the noise in the data prevents a conclusive claim in this particular sample.
Taking into consideration these results, we believe that the origin of 1/f noise in thin, mesoscopic samples of TIs can be attributed to universal conductance fluctuations, which arises due to quantum interference effects [32-34, 55, 58, 59] and is schematically shown in Fig. 3f.The charge carriers undergo multiple elastic scatterings from impurities, defects or boundaries, and follow trajectories which are a strong function of disorder configuration, Fermi energy, and magnetic field.Interference between these trajectories, which can involve back-scattered carriers or interference between partial waves between two points having different paths leads to conductance fluctuations, whose noise spectra is 1/f in nature [58].These conductance fluctuations are the dominant source of 1/f noise in mesoscopic topological insulators at low T .
To verify whether this is the dominant mechanism, we have fitted σ-n data (Fig 4a ), where σ = L RW and n = CS (VG−VD ) e , within the framework of charge-impurity limited scattering of Dirac fermions [60] where n * is the residual carrier density in electron and hole puddles, and E is a constant depending on the Wigner-Seitz radius r s .The extracted value of number density of Coulomb traps in sample BT1 is n imp = 1.5×10 16 m −2 , while for BT2, n imp = 5×10 16 m 2 , which matches well with the theoretically predicted values.The density of electron-hole puddles is n * = 7 × 10 14 m −2 and n * = 5×10 15 m −2 for samples BT1 and BT2 respectively.This difference in impurity density is reflected in the the qualitative nature of V G -dependence of noise as seen in Fig. 3a-b, thereby providing further support to the observation of pseudo-diffusive transport in device BT1.
In summary, we have measured time-dependent voltage fluctuations to extract the magnitude of 1/f noise in mesoscopic topological insulators devices as a function of gate-voltage and temperature.The temperature dependence implies that the noise originates from universal conductance fluctuations due to quantum interference effects.More importantly, the non-monotonic dependence of noise on the number density in the low disordered samples signifies a crossover from pseudo-dissusive to diffusive transport regime, a phenomena unique to Dirac Fermions.

Figure 1 .
Figure 1.Device characteristics (a) Schematic of a typical TI-FET (top).Optical micrograph of an actual device (below).(b) Resistance vs temperature of sample BT1, showing a weak dependence on T , indicating metallic behavior and the dominance of surface states in the transport.(c) Resistance vs gate-voltage of sample BT1, at different T , showing ambipolar transport.(d) Weak-antilocalization in TIs, characterized by a cusp in the correction to conductivity at B = 0 T. The solid lines are fits to the data using Eq 1.

Figure 2 .
Figure 2. Noise measurements: (a) Schematic of the experimental setup for measuring 1/f noise.(b) SV vs V 2 at T = 5 K for both samples BT1 and BT2, showing a quadratic behavior, implying that the response is in the linear regime.(c) SV as a function of frequency, showing 1/f behaviour.

Figure 3 . 2 G 2 at T = 9 K 2 G 2 vs
Figure 3. 1/f noise measurements: (a) Integrated noise magnitude δG 2 G 2 as a function of VG for sample BT1.The dashed

Figure 4 . 1 2 2 α 3 x
Figure 4. Impurity density calculation: (a) (a) Conductance (σ = L RW ) vs. calculated number density (n calc = C(V G −V D ) e) at T = 6 K for sample BT1.The black lines are fit of this data according to the Eq. 3 and 4. (b) σ vs n for sample BT2.The solid lines are fits to the data according to Eq. 3 and 4.