Signature of pseudo-diffusive transport in mesoscopic topological insulators

TL;DR

This paper detects pseudo-diffusive transport in topological insulators via 1/f noise measurements, revealing a crossover from pseudo-diffusive to diffusive regimes and highlighting quantum interference effects.

Contribution

First experimental observation of pseudo-diffusive transport signatures in topological insulators using 1/f noise analysis.

Findings

Non-monotonic 1/f noise behavior with gate voltage indicating transport regime crossover

Temperature dependence suggests conductance fluctuations from quantum interference

Evidence of evanescent modes in surface states of topological insulators

Abstract

One of the unique features of Dirac Fermions is pseudo-diffusive transport by evanescent modes at low Fermi energies when the disorder is low. At higher Fermi energies i.e. 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. While signatures of pseudo-diffusive 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 non-monotonic behavior in 1=f noise as the Fermi energy is varied, suggesting a crossover from pseudo-diffusive 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.

Equations

$$△\sigma =-\alpha \frac{e^2}{\pi h}\left[\psi \left(\frac{1}{2}+\frac{B_{\phi }}{B}\right)-\ln \left(\frac{B_{\phi }}{B}\right)\right]$$

$$\frac{S_V}{V^2}=\frac{D_{it}{k}_{B}T}{dWL}{\left(\frac{d\sigma }{dn}\right)}^2\left(\frac{J_1}{{\sigma }^2}+\frac{J_2}{\sigma }+J_3\right)$$

$$\displaystyle\sigma\sim E\left|\frac{n}{n_{imp}}\right|\left[e^{2}/h\right]\ \$$

$$\displaystyle\sigma\sim E\left|\frac{n^{*}}{n_{imp}}\right|\left[e^{2}/h\right]\ \$$

Full text

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 Hasan and Kane (2010); Hsieh et al. (2008); Roushan et al. (2009); König et al. (2007); Zhang et al. (2009). These surface carriers obey the Dirac equation for massless Fermions, where the Hamiltonian of the system is given by $H=\hbar v_{F}\overrightarrow{\sigma}\cdot\overrightarrow{k}$. Here $v_{F}$, $\vec{\sigma}$, and $\vec{k}$ 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 Rossi et al. (2012). Another key feature of these materials is that it is possible to reach $\langle n\rangle=0$, 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 Beidenkopf et al. (2011). 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 Pal et al. (2011); Novoselov et al. (2004, 2005); Neto et al. (2009); Tworzydło et al. (2006); Katsnelson (2006); Cuevas and Yeyati (2006); Akhmerov and Beenakker (2007); Titov and Beenakker (2006); DiCarlo et al. (2008); Danneau et al. (2008); Miao et al. (2007); Heersche et al. (2007); Du et al. (2008); Kumaravadivel and Du (2016); Abanin et al. (2007, 2011); Peres (2010); Tan et al. (2007); Checkelsky et al. (2009); Jiang et al. (2007); Feldman et al. (2009); Du et al. (2009). 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 Bhattacharyya et al. (2015, 2016); Islam et al. (2017, 2018); Ghatak et al. (2011); Pal et al. (2011); Ghatak et al. (2014); Shamim et al. (2016). However, the origin of $1/f$ noise in TIs in thin (thickness $d\sim 10$ 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 Bi1.6Sb0.4Te2Se, which was exfoliated from a single crystal onto a SiO2/Si wafer using Scotch tape technique Taskin et al. (2011); Novoselov et al. (2004, 2005). Due to compensation doping, the quarternary alloy Bi1.6Sb0.4Te2Se has an insulating bulk, resulting in enhanced surface transport Taskin et al. (2011); Ren et al. (2011); Bhattacharyya et al. (2015). Here below temperature $T=50$ K for samples with thickness, $d\leq 100$ nm, the current is essentially carried by the surface carriers Bhattacharyya et al. (2015). The atomically flat boron nitride (BN) substrate (Fig. 1a), significantly reduces the effect of dangling bonds and charged traps of the SiO2 substrate on the electrical transport in the TI channel Dean et al. (2010); Karnatak et al. (2016); Islam et al. (2018). The hetero-structure was then finally assembled onto a pre-patterned heavily hoped SiO2/Si substrate, with the $285$ nm thick SiO2 acting as a back gate dielectric, using a home-made transfer technique. The sample contacts were patterned by standard electron-beam 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, BT$1$ and BT$2$, 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) Bhattacharyya et al. (2015). Fig. 1c shows the $R$ vs $V_{G}$, where a maximum in the resistance at $V_{G}\approx-40$ 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 band-structure itself Adam et al. (2012). The typical mobility extracted from the $R-V_{G}$ graph is $\sim 100$ cm2V*-1s-1*. Fig. 1d shows magneto-resistance (MR) behavior of BT$1$ at $V_{G}-V_{D}=0$ V, $60$ V and $120$ V, characterized by a cusp in the quantum correction to conductivity $\triangle\sigma$ at $B=0$ T Zhang et al. (2013); He et al. (2011); Kim et al. (2011); Bansal et al. (2012). This demonstrates weak-antilocalization phenomenon, as expected for spin-momentum locked TI surface states, resulting from an additional $\pi$ Berry phase between the back-scattered, time reversed path of the carriers leading to negative magneto-conductance. The magneto-conductance data can be fitted with the Hikami-Larkin-Nagaoka (HLN) Hikami et al. (1980); Bao et al. (2012) equation for diffusive metals with high spin orbit coupling $(\tau_{\phi}>>\tau_{so},\tau_{e})$:

[TABLE]

where $\tau_{\phi}$, $\tau_{so}$, $\tau_{e}$ are the phase coherence or dephasing time, spin-orbit scattering time and elastic scattering time respectively, $\psi$ is the digamma function and $B_{\phi}$ is the phase breaking field. Here $\alpha$ and $B_{\phi}$ are fitting parameters. The phase coherence length $l_{\phi}^{MR}$ can be extracted using $l_{\phi}^{MR}=\sqrt{\hbar/4eB_{\phi}}$. The $l_{\phi}$ obtained from the fit was $\sim 180$ nm at $T=5$ K for $V_{G}-V_{D}=0$ V.

To extract the magnitude of $1/f$ noise of the samples accurately, we have utilized a AC four-probe Wheatstone bridge technique Scofield (1987); Weissman (1996); Dutta and Horn (1981). 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, $S_{V}\propto 1/f^{\alpha}$, and the exponent of the frequency, $\alpha\approx 1$. $S_{V}$ 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 ($\frac{\langle\delta G^{2}\rangle}{G^{2}}=\int\frac{S_{v}}{V^{2}}df$) at $T=5$ K, shown in Fig. 3a and Fig. 3b for samples BT$1$ and BT$2$. Although these two samples were identically prepared from the same bulk crystal, and show similar average electrical characteristics Islam et al. (2018), they demonstrate contrasting behaviors in the $V_{G}$ dependence of noise. Whereas $\frac{\langle\delta G^{2}\rangle}{G^{2}}$ vs $V_{G}$ for sample BT$1$ displays a M-shaped curve with a dip around the Dirac point ($\mid V_{G}-V_{D}\mid=20$ V) (Fig. 3a), the identically prepared device BT$2$ 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 Pal et al. (2011) and in TIs Bhattacharyya et al. (2015), 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 BT$1$. We have fitted the $V_{G}$-dependence of the normalized noise magnitude data using the framework of correlated mobility-number density fluctuations model Jayaraman and Sodini (1989); Islam et al. (2017), which is known to be the dominant mechanism of noise in large-area, thin ($\sim 10$ nm) TIs, where the effect of conductance fluctuations are suppressed to a large $L/l_{\phi}$ ratio. The total noise can be expressed as,

[TABLE]

where $J_{1}=\frac{1}{8\alpha}$ represents a pure number fluctuation, $J_{3}=\int A^{2}(x)\frac{\tau_{{}_{T}}}{1+\left(2\pi f\tau_{{}_{T}}\right)^{2}}dx$ represents pure mobility fluctuations and $J_{2}=\int 2A(x)\frac{\tau_{{}_{T}}}{1+\left(2\pi f\tau_{{}_{T}}\right)^{2}}dx$ represents combined number and mobility fluctuations ($\alpha$ is the decay constant for the spatially decaying time constant $\tau_{{}_{T}}$ of a typical trapping event and $A(x)$ is the scattering constant) and can be evaluated using phenomenological values Jayaraman and Sodini (1989). $D_{it}$, $k_{B}$, $W$, $L$, $\sigma$, $n$, $x$ 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 Rossi et al. (2012). 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 Pal et al. (2011); Novoselov et al. (2004, 2005); Neto et al. (2009); Tworzydło et al. (2006); Katsnelson (2006); Cuevas and Yeyati (2006); Akhmerov and Beenakker (2007); Titov and Beenakker (2006); DiCarlo et al. (2008); Danneau et al. (2008); Miao et al. (2007); Heersche et al. (2007); Du et al. (2008); Kumaravadivel and Du (2016); Dufouleur et al. (2017), there is no such clear signature in TIs. In the pseudo-diffusive regime,$\langle\delta G^{2}\rangle$ enhances rapidly in magnitude compared to $\langle G\rangle$ with increasing $n$, while in the diffusive regime, $\langle\delta G^{2}\rangle$ is almost constant whereas $\langle G\rangle$ 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 BT$1$ shows a strong $T$-dependence with the peak almost disappearing for $T>20$ K (Fig. 3c). The $V_{G}$-dependence of noise in sample BT$2$ shows a monotonic decrease with number density at all temperatures. The $T$-dependence of noise for sample BT$2$ 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 Islam et al. (2017). The noise magnitude, as shown in Fig. 3e, for BT$2$, reduces as $\sim T^{-1}$. Such a $T$-dependence of noise can be explained using the framework of universal conductance fluctuations. For $T\rightarrow 0$, UCF magnitude $\left\langle\delta G^{2}\right\rangle^{\frac{1}{2}}\rightarrow e^{2}/h$, while at finite temperature $\langle\delta G^{2}\rangle\simeq\left(\frac{e^{2}}{h}\right)^{2}\alpha(k_{F}\delta r)\frac{1}{k_{F}l}\frac{L_{y}}{L_{x}^{3}}n_{s}(T)l_{\phi}^{4}$, 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 Birge et al. (1990); Feng et al. (1986); Shamim et al. (2017). $\alpha(x)$ represents the change of the phase of electron wave-function due to scattering by a moving impurity at a distance $\delta r$, and $n_{s}(T)$ is the number of active scatteres. For electron-electron interaction mediated dephasing, $l_{\phi}^{2}\propto 1/T$ and $n_{s}(T)\propto T$, we have $\langle\delta G^{2}\rangle\propto l_{\phi}^{4}n_{s}(T)\propto 1/T$, as observed Birge et al. (1990); Feng et al. (1986); Shamim et al. (2017); Altshuler et al. (1982). While the overall noise magnitude for sample BT$1$ 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 Lee and Stone (1985); Lee et al. (1987); Feng et al. (1986); Islam et al. (2018, 2017); Bhattacharyya et al. (2016) 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 Lee and Stone (1985). 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 $\sigma$-$n$ data (Fig 4a), where $\sigma=\frac{L}{RW}$ and $n=\frac{C_{S}\left(V_{G}-V_{D}\right)}{e}$, within the framework of charge-impurity limited scattering of Dirac fermions Kim et al. (2012), where

[TABLE]

and

[TABLE]

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 BT$1$ is $n_{imp}=1.5\times 10^{16}$ m*-2*, while for BT$2$, $n_{imp}=5\times 10^{16}$ m2, which matches well with the theoretically predicted values. The density of electron-hole puddles is $n^{*}=7\times 10^{14}$ m*-2* and $n^{*}=5\times 10^{15}$ m*-2* for samples BT$1$ and BT$2$ 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 BT$1$.

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.