Absence of strong localization at low conductivity in the topological surface state of low disorder Sb2Te3

TL;DR

This study investigates the transport properties of a high-mobility, low-disorder topological insulator, revealing the absence of strong localization at low conductivities and providing insights into surface state hybridization and disorder.

Contribution

It demonstrates that strong localization does not occur at low conductivities in topological surface states, challenging conventional expectations and offering a detailed analysis of disorder effects.

Findings

No strong localization at conductivities below e^2/h

Temperature behavior of localization peak differs from conventional models

Quantitative estimates of disorder potential and surface state hybridization

Abstract

We present low-temperature transport measurements of a gate-tunable thin film topological insulator system that features high mobility and low carrier density. Upon gate tuning to a regime around the charge neutrality point, we infer an absence of strong localization even at conductivities well below $e^2/h$, where two dimensional electron systems should conventionally scale to an insulating state. Oddly, in this regime the localization coherence peak lacks conventional temperature broadening, though its tails do change dramatically with temperature. Using a model with electron-impurity scattering, we extract values for the disorder potential and the hybridization of the top and bottom surface states.

Equations

$${\rho }_{xx}(B)={\rho }_{xx}(0){\left[1-2A+2A/\sqrt{1+(\mu B{)}^2}\right]}^{-1}$$

$$\Delta {\sigma }_{xx}(T)=\frac{e^2}{\pi h}\alpha \log \left(T/T_0\right)$$

$$\Delta {\sigma }_{\text{EEI}}(T)=\frac{e^2}{2\pi h}\left(2-\frac{3}{2}{\tilde{F}}_{\sigma }\right)\log \left(T/T_0\right),$$

$$\Delta \sigma =-\frac{e^2}{2\pi h}F\left(\frac{B}{B_{\varphi }}\right),$$

$$\begin{array}{rl}\Delta \sigma =& -\frac{1}{2\pi }\frac{e^2}{h}[F\left(\frac{B}{B_{\varphi }}\right)-2F\left(\frac{B}{B_{\varphi }+B_{\mathrm{\Delta }}}\right)\\ & -F\left(\frac{B}{B_{\varphi }+2B_{\mathrm{\Delta }}}\right)],\end{array}$$

Full text

Time-reversal invariant three-dimensional topological insulators (3D TIs) are gapped materials with inverted bulk bands. At energies within the bulk bandgap, topological surface states (TSS) are guaranteed to exist Fu et al. (2007). Each surface state of \ceBi2Se3 family materials is a single two-dimensional (2D) Dirac cone in which the in-plane spin is correlated with the wave vector Zhang et al. (2009a); Xia et al. (2009). Whereas topologically trivial (conventional) 2D electron systems (2DES) are strongly insulating at low carrier densities because of Anderson (strong) localization Lee and Ramakrishnan (1985), TSS are expected to be impervious to localization, even under strong disorder Nomura et al. (2007). As far as we know, no other time-reversal invariant two-dimensional system has a metallic single-particle description in the presence of disorder – even in graphene, intervalley scattering due to disorder leads to localization Chen et al. (2009). Intuitively, TSS should not localize because large angle scattering is prohibited without a time-reversal symmetry-breaking spin-flip: states with opposite wavevector have opposite spin.

Localization can in principle occur in 3D TI thin films. Tunneling through the thickness of a film hybridizes the top and bottom TSS, opening a surface gap $2\Delta$ around the Dirac point Linder et al. (2009); Lu et al. (2010). The massive Dirac fermions no longer enjoy absolute protection against localization. When hybridization is small, however, large angle scattering should still be suppressed; therefore, strong localization may be suppressed at low densities where conventional 2DES would be expected to localize. Benefiting from gate-tunability and suppressed bulk conduction Chen et al. (2010), thin film 3D TIs provide an opportunity to study the effect of spin texture on localization physics in two dimensions.

Careful study of these systems, however, has been hampered by various materials issues: defects push the Fermi levels of the binary V-VI topological compounds (\ceBi2Se3, \ceBi2Te3, and \ceSb2Te3) far from the Dirac point Zhang et al. (2011), and epitaxial mismatches between the topological insulator and the substrate introduce additional disorder Koirala et al. (2015). Disorder decidedly affects the electrical conduction of TSS: the mobilities of typical thin film V-VI topological materials reach only of order 100 $\text{cm}^{2}/\text{Vs}$. Furthermore, while electrostatic gating can tune the Fermi level to the charge neutrality point (CNP), charged impurities obscure low density transport physics in favor of transport through charge puddles Skinner et al. (2013); Borgwardt et al. (2016); Nandi et al. (2018). Consequently, insulating ($\sigma_{xx}<e^{2}/h$) time-reversal symmetry-protected 3D TI systems have only been seen in the thinnest films, where the clean-limit hybridization gap far exceeds room temperature Kim et al. (2011); Jiang et al. (2012); Lang et al. (2013).

We report transport properties of a top-gated Hall bar of a novel \ceSb2Te3-based thin film. The key components of this platform are 1) an epitaxially matched trivial insulator serves as a virtual substrate for the growth of the topological insulator, reducing defects, and 2) the topological insulator is counter-doped to bring its Fermi level close to the Dirac point, even before electrostatic gating. This materials platform, introduced in more detail in Ref. Salehi et al. (2018), offers a high mobility TSS with small and comparable surface gap and disorder potential, allowing study of the electrical transport near the Dirac point. Like other Dirac metals, this film’s magnetoconductance is negative and agrees with weak anti-localization (WAL) theory at high carrier densities. Strikingly, signatures of WAL persist close to the CNP, although the conductivity $\sigma_{xx}\ll e^{2}/h$. This feature implies the absence of scaling to strong localization, possibly associated with the topological origin of the surface states.

I Methods

\ce

Sb2Te3 was grown by molecular beam epitaxy. Interface engineering reduces disorder stemming from the lattice mismatch between topological insulator and substrate. A 15 quintuple layer (QL; 1 QL $\approx$ 1 nm) \ce(Sb_0.65In_0.35)_2Te3 (a trivial insulator) buffer layer was grown on a sapphire substrate. Growth of the topological insulator, 8 QL \ceSb2Te3 counter-doped by 2% Ti, followed. A 2 QL \ce(Sb_0.65In_0.35)_2Te3 capping layer was deposited in situ to protect the TI. The thickness of the capping layer was chosen as a compromise to protect the \ceSb2Te3 while not impairing the efficacy of the top gate. A 50 $\mu\text{m}$ wide Hall bar was fabricated. A top gate was formed with a 40 nm alumina dielectric atop the \ce(Sb_0.65In_0.35)_2Te3 capping layer. Measurements were made in a \ce^3He/\ce^4He dilution refrigerator (30 mK to 1.2 K) and a \ce^4He cryostat with a variable temperature insert (1.5 K to 30 K) using standard lock-in techniques. To accurately measure the high resistances near the CNP at dilution refrigeration temperatures, some measurements were made using a high-impedance DC current source and a nanovoltmeter. All resistance (conductance) values are obtained through four-terminal measurements and are presented as two dimensional resistivity (conductivity).

Hall measurements at zero gate voltage and $T=50\text{ mK}$ yielded carrier density $n=-2.5\times 10^{12}\text{ cm}^{-2}$, where the negative sign indicates holes rather than electrons, and mobility $\mu=580\text{ cm}^{2}/\text{Vs}$. To account for variation in the Fermi level between different cooldowns, we present gate voltage as $\Delta V_{g}=V_{g}-V_{\text{min}}$ where $V_{\text{min}}$ is the gate voltage at which the conductivity is minimized during that cooldown (between 17 V and 20 V for all cooldowns). $V_{\text{min}}$ is often associated with the CNP; however, given the presence of charge puddles in a disordered potential landscape, the CNP occurs precisely at $V_{\text{min}}$ only if electrons and holes have equal mobility.

II Results

The zero-field resistance of the device is shown as a function of gate voltage in Fig. 1 (a). The carrier density, as extracted from fitting the Hall slope at applied fields $\absolutevalue{B}<0.25\text{ T}$ to a single-carrier model, is shown in Fig. 1 (b) along with the Hall mobility. At gate voltages well below $V_{\text{min}}$, the conductivity saturates at $\sigma_{xx}\approx 6e^{2}/h$, meaning the mobility $\mu\propto|n|^{-1}$. In this high-density limit, the conductivity increases logarithmically with temperature sup . Moving toward $V_{\text{min}}$, hole carriers are depleted and the conductivity drops, reaching $\sigma_{xx}<0.01e^{2}/h$ at $V_{\text{min}}$. As $|n|$ decreases, the temperature dependence of the conductivity evolves to an Arrhenius activation law with activation energy $\Delta_{\text{Arr}}=84$ $\mu$eV at $V_{g}=V_{\text{min}}$ (Fig. 1 (a), inset) sup .

The resistivity in perpendicular applied fields up to 10 T at various gate voltages is shown in the supplement sup . A sharp positive quantum coherence peak in the magnetoconductance at zero field, indicative of WAL, is observed at all gate voltages. At $\Delta V_{g}\leq-4$ V, the coherence peak broadens with increasing temperature (Fig. 2 (a-b)), while at $\Delta V_{g}\geq-4$ V, the magnetoconductance flattens or switches sign as $B$ increases (Fig. 2 (c-d)). The lower the temperature, the lower the field at which the magnetoconductance switches sign. In a disordered system, aside from the coherence peak, there is a positive classical contribution to the magnetoresistance that changes from quadratic at low magnetic fields to linear at high magnetic fields Ramakrishnan et al. (2017); Nandi et al. (2018) and saturates in some experiments Cho and Fuhrer (2008). A two parameter phenomenological model based on such behavior

[TABLE]

fits well at most gate voltages sup . Here, $0\leq A\leq 0.5$ is the quadratic coefficient of magnetoreresistance i.e. $\rho_{xx}=\rho_{xx,0}[1+A(\mu B)^{2}+\cdots]$ and $\mu$ is the carrier mobility. Fig. 3 shows the coherence peaks after subtracting the background from this classical contribution. In all figures, the plotted magnetoconductance is symmetrized with respect to field sup .

Before proceeding, we provide quantitative estimates of the material parameters. From zero field transport and Hall coefficient ($R_{\mathrm{H}}$) data, we extract the density of charged impurities $n_{\mathrm{imp}}=(4.8\pm 2.8)\times 10^{11}\mathrm{cm}^{-2}$ lying an average distance $d=4.7~{}\mathrm{nm}$ from the 2DES plane, the dimensionless interaction parameter $r_{s}=1.3\pm 0.8$, and the characteristic charge density fluctuations $n_{\mathrm{rms}}=(1.2\pm 0.2)\times 10^{11}\mathrm{cm}^{-2}$ sup .

III Discussion

Arrhenius activation of the conductivity at $V_{g}=V_{\text{min}}$ confirms the presence of a surface gap. Gaps of order 100 meV have been observed through angle-resolved photoemission spectroscopy (ARPES) in 3D TI films thinner than 5 QLs Zhang et al. (2010); Neupane et al. (2014). Our measured Arrhenius activation scale of 84 $\mu$eV is surprisingly small in comparison, even considering that our 8 QL film is thicker, and that $\Delta$ should decrease exponentially in film thickness. To explain the small value of $\Delta_{\text{Arr}}$, we note that the clean-limit surface gap $2\Delta$ could be smeared by disorder so that $\Delta_{\text{Arr}}\ll\Delta$ Chang et al. (2016). Disorder smearing may also explain why many prior experiments on thin film 3D TIs do not observe a gap and instead see $\sigma>e^{2}/h$ at the CNP Zhang et al. (2009b); Taychatanapat and Jarillo-Herrero (2010).

At $\Delta V_{g}\lesssim-10$ V, we observed positive logarithmic temperature corrections to the conductivity. From the magnetoconductance, we know that WAL is present. WAL should contribute a negative temperature correction to conductivity:

[TABLE]

with $\alpha=-0.5$ per channel. Our observation of positive logarithmic corrections to conductivity with increasing temperature would naively indicate weak localization (WL), with $\alpha=1$ rather than $=-0.5$. This apparent mismatch between the signs of the temperature and magnetoconductance corrections has been previously observed and resolved by including an electron-electron interaction (EEI) contribution to the conductivity Wang et al. (2011); Lu and Shen (2014a); Choi et al. (2016)

[TABLE]

with screening factor $\tilde{F}_{\sigma}>0$ Lee and Ramakrishnan (1985). The observation of overall positive temperature corrections to the conductivity means that the EEI correction dominates over the localization correction, in agreement with other experiments as well as calculations Lu and Shen (2014b).

In principle, conventional 2DES cannot be metallic. Surprisingly, metallic temperature dependence (higher conductivity at lower temperature, even at milliKelvin temperatures) was found in semiconductor-based 2DES of exceptional cleanliness. This is now understood to result from EEI driving the system into a metallic phase. These systems transition to insulators as carrier density is reduced. Empirically, this transition consistently occurs when the conductivity is of order $e^{2}/h$ in a variety of 2DES platforms including Si MOSFETs Kravchenko et al. (1995), GaAs/AlGaAs heterostructures Hanein et al. (1998); Simmons et al. (1998), graphene Amet et al. (2013), transition metal dichalcogenides Radisavljevic and Kis (2013), and even other 3D TI thin films Liao et al. (2015).

In our system, electron-electron interactions have the opposite effect. As discussed above, EEI causes increasing conductivity with increasing temperature. We therefore never observe metallic temperature dependence in our device, despite the conductivity ranging from $6e^{2}/h$ at high carrier density to $<0.01e^{2}/h$ ($T=35\text{ mK}$) at $V_{\text{min}}$ sup . Nevertheless, unlike conventional 2DES, 3D TIs (in the limit $\Delta=0$) are expected to have metallic single-particle descriptions. Is our system metallic? At the most fundamental level, a metal is characterized by delocalized electronic wavefunctions, not by the temperature dependance of its conductivity. Since here the temperature dependance of the resistivity fails to reflect even the presumed metallicity of the system at high doping, we must turn to the system’s magnetoconductivity to address this question.

In a 3D TI for which hybridization between the top and bottom surfaces is weak, electrons should exhibit weak antilocalization Adroguer et al. (2015). The magnetoconductance is given by the Hikami-Larkin-Nagaoka (HLN) formula Hikami et al. (1980)

[TABLE]

where $F(x)=\ln(x)+\Psi(x+\frac{1}{2})$, $\Psi$ is the digamma function, and $B_{\phi}=\hbar/(4el_{\phi}^{2})$ the characteristic field associated with the electron coherence length $l_{\phi}$. WAL is caused by the $\pi$ Berry phase of the 2D Dirac dispersion, which suppresses backscattering.

We observe conventional quantum transport corrections at substantial hole doping. The WAL peak broadens with increasing temperature (Fig. 2 (a)). In theory, the peak should broaden as $l_{\phi}\propto T^{-p}$ with $p=0.5$ in diffusive two dimensional metals. As shown in Fig. 4, we find that $p=0.39$ at $\Delta V_{g}=-8$ V.

However, quantum corrections near the Dirac point differ from those at finite doping. At gate voltages more positive than $\Delta V_{g}=-4$ V, the magnetoconductivity becomes non-monotonic, qualitatively departing from equation (4). To understand this, recall that TSS hybridization in thin films should generate a Dirac mass $(\Delta)$, and the Berry phase $\phi_{\textrm{b}}$ should deviate from $\pi$ as $\phi_{\textrm{b}}=\pi\left(1-{\Delta}/{E_{F}}\right)$. The Berry phase thus should induce a crossover from perfect WAL ($\phi_{\textrm{b}}=\pi$) in the massless (relativistic) limit to perfect WL ($\phi_{\textrm{b}}=0$) in the large mass (non-relativistic) regime Lang et al. (2013); Zhang et al. (2013), with an associated magnetic field dependence of conductivity described by a modified HLN formula Iordanskii et al. (1994)

[TABLE]

where $B_{\phi,\Delta}=\hbar/(4el_{\phi,\Delta}^{2})$ are the characteristic fields associated with the coherence length $l_{\phi}$ and the crossover length scale $l_{\Delta}$, respectively.

The quality of fits (Fig. 3) of equation (5) is greatly improved from that of equation (4), in exchange for an additional fitting parameter. We extract $l_{\Delta}\sim 40\mathrm{~{}nm}$ at all gate voltages. Unexpectedly, at the lowest field scales, we observe a WAL peak at all gate voltages, indicating that the system does not scale to strong localization, even when $\sigma\ll e^{2}/h$. Furthermore, the temperature dependence of the magnetoconductance peak at $V_{g}\approx V_{\text{min}}$ is unusual, the WAL peak being more pronounced at higher temperatures (Fig. 2).

We note that the quality of fit becomes poor near $\Delta V_{g}=0$, as shown in Fig. 3 (d). Here, the data have quantum corrections only at very small magnetic fields. We cannot make definitive statements about this observation since the quantum corrections are cleanly separable from the classical contribution to magnetoconductivity only for $\sigma\gg e^{2}/h$; thus, the HLN equation becomes invalid when $\sigma<e^{2}/h$. However, if the data at the smallest fields are interpreted as due to quantum corrections, the extracted $l_{\phi}$ still decays with increasing temperature according to a power law, albeit with power roughly half that expected from EEI. At present, we lack an explanation for this discrepancy.

We may use the extracted crossover length scale to estimate the clean-limit surface gap by dimensional analysis as $\Delta\approx\hbar v_{F}/l_{\Delta}=6.7\text{ meV}$. This value is consistent with extrapolation from ARPES measurements of gaps for thinner films; as noted above, our measured transport gap $\Delta_{\text{Arr}}$ is much smaller, presumably because we are not in the clean limit.

An interesting pattern in the literature is that WAL is observed in topological insulators having $\sigma_{xx}>e^{2}/h$, while WL is observed when $\sigma_{xx}\lesssim e^{2}/h$ Yang and Kapitulnik (2018). This observation is explained by noting that WL in a topological insulator requires a mass gap around the Dirac point; since a gapped system insulates, we expect $\sigma_{xx}<e^{2}/h$. Our results contradict this pattern: at the smallest magnetic field scales (and therefore longest length scales), we observe a quantum coherence peak with negative magnetic field corrections at all carrier densities. This signature of WAL implies delocalized electronic states. Yet, this observation holds even at low carrier densities where the longitudinal conductivity falls well below $e^{2}/h$. Traditionally, $\sigma_{xx}\sim e^{2}/h$ is associated with reaching the Ioffe-Regel criterion $k_{F}l\approx 1$, which predicts that metallic 2DES do not exist at lower conductivities. Our results suggest that this device does not scale to strong localization, and instead enters an Ioffe-Regel-violating regime: a consequence of the symplectic character of the system together with disorder scattering. In the supplement, we theoretically justify this conclusion by finding self-consistent solutions in violation of the Ioffe-Regel limit for a low-energy model of massless 2D Dirac fermions for a broader range of the dimensionless parameters $r_{s}$ and $n_{\text{imp}}d^{2}$, using a combination of analytic and numeric results.

Acknowledgements.

The authors thank Eli J. Fox, Hassan Shapourian, and Chi-Te Liang for insightful conversations, and Hava R. Schwartz for developments in our fabrication techniques. We are grateful for the contributions to instrumentation and measurement software by Andrew J. Bestwick, Eli J. Fox, Aaron L. Sharpe, and Menyoung Lee. Device fabrication and measurement was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division, under Contract DE-AC02-76SF00515. Infrastructure and cryostat support were funded in part by the Gordon and Betty Moore Foundation through Grant No. GBMF3429. Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported by the National Science Foundation under award ECCS-1542152. The theoretical work in Singapore was funded by the National University of Singapore Young Investigator Award (R-607-000-094-133), and the Singapore Ministry of Education (MOE2017-T2-1-130). The work at Rutgers was supported by the Gordon and Betty Moore Foundation’ s EPiQS Initiative (GBMF4418) and the National Science Foundation (NSF) (EFMA-1542798).