Low temperature saturation of phase coherence length in topological insulators

TL;DR

This study investigates the saturation of phase coherence length in topological insulators at low temperatures, revealing the need to identify new dephasing mechanisms affecting quantum coherence in these materials.

Contribution

It systematically rules out known factors affecting phase coherence, highlighting the necessity to find alternative dephasing sources in topological insulators.

Findings

Phase breaking length saturates below certain temperatures.

Saturation is consistent across different measurement methods.

Known dephasing factors are ruled out as causes.

Abstract

Implementing topological insulators as elementary units in quantum technologies requires a comprehensive understanding of the dephasing mechanisms governing the surface carriers in these materials, which impose a practical limit to the applicability of these materials in such technologies requiring phase coherent transport. To investigate this, we have performed magneto-resistance (MR) and conductance fluctuations\ (CF) measurements in both exfoliated and molecular beam epitaxy grown samples. The phase breaking length ($l_{\phi}$) obtained from MR shows a saturation below sample dependent characteristic temperatures, consistent with that obtained from CF measurements. We have systematically eliminated several factors that may lead to such behavior of $l_{\phi}$ in the context of TIs, such as finite size effect, thermalization, spin-orbit coupling length, spin-flip scattering, and surface-bulk coupling. Our work indicates the need to identify an alternative source of dephasing that dominates at low $T$ in topological insulators, causing saturation in the phase breaking length and time.

Tables

Table 1. Table 1: Details of measured devices : The thickness, substrate, composition, saturation value of temperature ( T s ​ a ​ t superscript 𝑇 𝑠 𝑎 𝑡 T^{sat} ), and saturation value of phase breaking length ( l ϕ s ​ a ​ t superscript subscript 𝑙 italic-ϕ 𝑠 𝑎 𝑡 l_{\phi}^{sat} ) for various devices that been investigated is provided in the table.

Sample	Thickness	Substrate	Composition	$T^{sat}$	$l_{\varphi }^{sat}$
F$7$	$7$	SiO2/Si$++$	Bi1.6Sb.4Te2Se	$\sim 2$ K	$200$ nm
F$100$	$100$	SiO2/Si$++$	Bi1.6Sb.4Te2Se	$\sim 2$ K	$150$ nm
F$9$	$9$	SiO2/Si$++$	Bi1.6Sb.4Te2Se	$\sim 2$ K	NA
TBN$11$	$11$	Boron nitride	Bi1.6Sb.4Te2Se	$\sim 2$ K	$190$ nm
M$10$	$10$	STO	(Bi,Sb)2Te3	$\sim 300$ mK	$2000$ nm

Equations

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

$$⟨\delta G^2⟩\simeq \left(\frac{3}{\pi }\right){\left(\frac{e^2}{h}\right)}^2{\left(\frac{l_{\varphi }}{L}\right)}^2$$

$$l_{\varphi }=\frac{1}{(A_0+A_{1}T+A_2{T}^2{)}^{0.5}}$$

$$N(B)/N(0)=\frac{1}{2}+\frac{1}{b^2}\sum _{n=0}^{\infty }\frac{1}{[\left(n+\frac{1}{2}\right)+\frac{1}{b}{]}^3}$$

$$=\frac{1}{2}-\frac{1}{2b^2}{\Psi }^{{}^{\prime \prime }}\left(\frac{1}{2}+\frac{1}{b}\right).$$

Full text

Topological insulators (TIs) Hasan and Kane (2010); Moore (2010); König et al. (2007); Chen et al. (2009) are a new class of materials characterized by the presence of gapless and linearly dispersing metallic surface states present in the bulk band gap due to non-trivial topology of the bulk band structure. The surface carriers are prohibited from back-scattering against non-magnetic impurities and exhibit a plethora of fundamentally important effects such as spin-momentum locking, hosting Majorana fermions in the presence of a superconductor, topological magnetoelectric effect, and quantum anomalous Hall effect Chang et al. (2013); Hasan and Kane (2010). The topological protection of these surface states makes these materials a strong contender for the building blocks of qubits, which require long phase coherence length ($l_{\phi}$) for error tolerant quantum computation. Hence, it is critical to understand the mechanisms responsible for dephasing or decoherence, which is equivalent to loss of information, in the surface states of TIs. The most common dephasing mechanism in TIs at low temperature ($T$) has been known to be electron-electron interaction Kim et al. (2011); Ockelmann et al. (2015); Wang et al. (2011); Zhang et al. (2012a); Kandala et al. (2013), and the coupling of the surface states to localized charged puddles in the bulk Liao et al. (2017). Li et al. have demonstrated that electron-phonon interaction is also required to explain the dependence of $l_{\phi}$ on $T$ Li et al. (2012). Although theoretically, all these mechanisms lead to a diverging $l_{\phi}$ with decreasing $T$ Checkelsky et al. (2011); Lin and Bird (2002); Kim et al. (2011); Ockelmann et al. (2015), experimentally, the increase of $l_{\phi}$ with reducing $T$ is often followed by its saturation for $T$ $\leq 2-5$ K Liao et al. (2017); Steinberg et al. (2011); Li et al. (2012); Islam et al. (2018). The saturation of $l_{\phi}$ at a finite value instead of its divergence for $T\rightarrow 0$ K, which is predicted for electron-electron or electron-phonon interactions, has been a matter of active discourse Fukai et al. (1990); Lin and Giordano (1987); Vranken et al. (1988); Fournier et al. (2000); Pivin Jr et al. (1999); Schopfer et al. (2003); Mohanty et al. (1997); Pierre and Birge (2002); Pierre et al. (2003); Mohanty and Webb (1997); Lin and Bird (2002); Chuang et al. (2013); Huard et al. (2005).

In this report, we have inspected the factors that can lead to the saturation of $l_{\phi}$ in TIs by measuring both gate-voltage ($V_{G}$) and time ($t$)-dependent conductance fluctuations and magneto-resistance (MR) Hikami et al. (1980); Birge et al. (1990); Chuang et al. (2013); Ghosh and Raychaudhuri (2000); Ghosh et al. (2004); Lee and Stone (1985); Lee et al. (1987); Pal et al. (2012); Shamim et al. (2017). Conductance fluctuations result from the quantum interference of different electron trajectories, manifested as sample specific, aperiodic fluctuations in the conductance due to varying disorder configuration, Fermi energy, and magnetic field; such fluctuations have been used as a tool to probe the presence of time-reversal symmetry (TRS) breaking disorders, since the saturation of $l_{\phi}$ at low $T$ is often attributed to spin-flip scattering processes. The magnitude of the conductance fluctuations $\langle\delta G^{2}\rangle$, however, shows a factor of two reduction upon application of a perpendicular magnetic field ($B_{\perp}$), implying that TRS is intrinsically preserved in these systems. Additionally, at different gate voltages, $\langle\delta G^{2}\rangle$ displays a saturation for $T<2$ K, even in the presence of a large $B_{\perp}$ which suppresses spin-spin scattering; this implies that neither magnetic impurities nor the coupling of the surface and the bulk impurity states is responsible for the saturation. Our experiment suggests an unconventional mechanism that saturates $l_{\phi}$ in TIs, possibly arising from unscreened Coulomb fluctuations from the charged disorders present in the bulk Liao et al. (2017).

The field effect devices investigated in this paper were fabricated from both exfoliated and molecular beam epitaxy (MBE) grown TIs. To fabricate the former, the TI Bi1.6Sb*.4Te2Se (BSTS) (purity of the starting elements Bi, Te, Sb, Se $\geq 4$N) was exfoliated from a single crystal onto a SiO2*/Si$++$ substrate with the $285$ nm thick SiO2 acting as the back gate dielectric inside a glove box Taskin et al. (2011). This was followed by standard electron-beam lithography and sputtering of $100$ nm Au to form the source-drain contacts (inset of Fig. 1(a)). The details of the devices measured are provided in Table. I. In sample TBN$11$, the TI flake was transferred onto an atomically flat boron nitride (BN) substrate to reduce the effect of charged traps and dangling bonds of SiO2 on the electrical transport Dean et al. (2010), followed by lithography and metallization. The quarternary alloy BSTS offers a reduced bulk number density due to compensation doping, resulting in a higher percentage of surface transport Taskin et al. (2011). The exfoliated samples were covered with PMMA (poly(methylmethacrylate)) during the entire measurement cycle to prevent oxidation and subsequent degradation of the surface quality. The large area ($0.5$ mm $\times$ $1$ mm) sample (M10) was fabricated from thin (thickness, $d=10$ nm) (Bi,Sb)2Te3 (BST) (purity of starting elements was $5$N for Bi, $5$N for Sb and $6$N for Te),) grown by molecular beam epitaxy (MBE) on $\langle 111\rangle$ SrTiO3 (STO) substrate and mechanically etched into a Hall bars with a metallic coating of Indium at the back, that was used as a back gate electrode Bhattacharyya et al. (2016); Islam et al. (2017). Resistivity measurements were performed in a low-frequency four-probe AC configuration in a pumped He-$3$ system (base $T=320$ mK) and in a dilution refrigerator (base$T=20$ mK).

Preliminary electrical transport characteristics in the exfoliated device TBN$11$ (at $T=20$ mK) and the MBE-grown device (at $20$ mK) are shown in Fig. 1(a). The $R$-$V_{G}$ data indicates that at $V_{G}=0$ V, TBN$11$ is intrinsically electron doped and M$10$ is intrinsically hole doped. Whereas M$10$ shows a clear graphene like ambipolar transport with a Dirac point at $60$ V, which could be achieved due to the high dielectric constant of the STO at low $T$ ($\epsilon_{r}\sim 44000$ at $5$ K) Islam et al. (2017), TBN$11$ shows a clear signature of an electron-hole puddle regime at $V_{G}=-60$ V. The estimated value of intrinsic number density at $V_{G}=0$ V are $-2.9\times 10^{13}$ m*-2* and $9\times 10^{13}$ m*-2* respectively. The quantitative difference of the $V_{G}$-dependent characteristics here indicates dominance of different types of disorder species in the samples, owing to different processes of synthesis, fabrication, and composition. The n-doping in BSTS mostly comes due to Se vacancies, the most likely probable cause of p-doping in the epitaxially grown samples is the presence of anti–site defects.

Fig. 1(b) shows weak anti-localization (WAL) for different samples, which is characterized by a cusp in the quantum correction to conductivity $\triangle\sigma$ at $B=0$ T, and is a result of $\pi$ Berry phase in topological insulators. The magneto-conductance data can be fitted with the Hikami-Larkin-Nagaoka (HLN) expression for diffusive metals with high spin-orbit coupling $(\tau_{\phi}>>\tau_{so},\tau_{e})$ Hikami et al. (1980); Bao et al. (2012):

[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 $l_{\phi}^{MR}=\sqrt{\hbar/4eB_{\phi}}$ and the value of $\alpha$ gives an estimate of the number of independent conducting channels in the sample (See supplementary information).

The magnitude of gate voltage dependent conductance fluctuations $\langle\delta G^{2}\rangle$, has been evaluated by using a method similar to Ref. Gorbachev et al. (2007); Pal et al. (2012); Islam et al. (2018) by varying the chemical potential with the back gate voltage in steps of $5$ mV over a small window of $4$ V. $\langle\delta R^{2}\rangle$ is extracted from $R$ - $V_{G}$ by fitting the data with a smooth polynomial curve. $\langle\delta G^{2}\rangle$ is then obtained using the relation:$\langle\delta G^{2}\rangle=\langle\delta R^{2}\rangle/\langle R\rangle^{4}$ ($\langle\delta R^{2}\rangle$ is extracted from the variance of the residual). As shown in Fig. 2(a) for a typical $4$ V window, the fluctuations are aperiodic yet reproducible but weaken with increasing $T$. The $T$-dependence of the standard deviation $\left\langle\delta G^{2}\right\rangle^{\frac{1}{2}}$, at $V_{G}=0$ V (center of the corresponding window) for F$100$ in Fig. 2(b) shows two distinctly different regions. Above $T>2$ K, $\left\langle\delta G^{2}\right\rangle^{\frac{1}{2}}\propto T^{-0.5}$, which is expected from the $T$-dependence of $l_{\phi}\propto T^{-0.5}$ and the number of active scatterers ($n_{s}\propto T$) Islam et al. (2018). $\left\langle\delta G^{2}\right\rangle^{\frac{1}{2}}$, however, saturates for $T<2$ K.

Conductance fluctuations due to changes in the disorder configuration were detected by measuring the normalized noise magnitude, defined as $\frac{\left\langle\delta G^{2}\right\rangle}{\langle G^{2}\rangle}=\frac{\int S_{V}df}{V^{2}}$ where $S_{V}/V^{2}$ is the normalized power spectral density (P.S.D.) of the time-dependent signal, in an AC-four probe Wheatstone bridge technique (Scofield (1987); Ghosh et al. (2004)). The normalized time-dependent fluctuations in resistance, ($\delta R/R$) for various temperatures ($0.3$ K $\leq T\leq 6$ K) is shown in Fig. 2(c). $\left\langle\delta G^{2}\right\rangle^{\frac{1}{2}}$ extracted from the P.S.D. of time-dependent conductance fluctuations is plotted as a function of $T$ in Fig. 2(d) and is found to show a saturation below $T=2$ K, which is consistent with the behavior of $\left\langle\delta G^{2}\right\rangle^{\frac{1}{2}}$ obtained from the $V_{G}$-dependence. The order of magnitude difference is caused due to integration of the signal over a finite frequency window as well as the sensitivity of resistance changes to individual defect movements Shamim et al. (2017); Birge et al. (1989); Trionfi et al. (2004).

The phase breaking length, $l_{\phi}$ extracted from $\langle\delta G^{2}\rangle^{\frac{1}{2}}$-$T$ data (Fig. 2(a)) using the expression Akkermans and Montambaux (2007); Adroguer et al. (2012)

[TABLE]

is shown in Fig. 3a. Since $\langle\delta G^{2}\rangle\propto l_{\phi}^{2}$, any saturation obtained from time or gate voltage-dependence should also be reflected in the saturation of $l_{\phi}$, obtained directly from MR measurements. The values of $l_{\phi}$ extracted from MR data similar to Fig. 1(b), as a function of $T$ for the exfoliated samples F$100$ and F$7$ is shown in Fig. 3(b). We find that $l_{\phi}$ obtained from two different methods, $l_{\phi}^{MR}$ and $l_{\phi}^{UCF}$ (extracted from Eq. 1 and Eq. 2 respectively) show similar trends with $T$, first increasing with decreasing $T$, followed by a saturation below $T\sim 2$ K, thus discarding the possibility of the saturation to be an artifact. The discrepancies in the values of $l_{\phi}^{MR}$ and $l_{\phi}^{UCF}$ are within uncertainties of the prefactor of Eq. 2 Akkermans and Montambaux (2007); Adroguer et al. (2012); Beenakker and van Houten (1991). The higher value of $l_{\phi}$ for F$7$ compared to $F100$ can be due to enhanced dephasing due to trapping-detrapping processes in the bulk, since the thickness of F$100$ is much larger than that of F$7$.

For a quantitative understanding, $l_{\phi}$-$T$ data (Fig. 3 (a-b)) has been fitted with the expression commonly used to fit the $l_{\phi}$-$T$ data in 2D diffusive systems Lin and Bird (2002); Li et al. (2012).

[TABLE]

Here, $l_{\phi}\propto T^{-0.5},$ and $l_{\phi}\propto T^{-1}$ are the respective contributions from electron-electron (e-e) and electron-phonon (e-ph) scattering. $A_{1}$, $A_{2}$ are fitting parameters and $A_{0}=\frac{1}{l_{\phi 0}^{2}}$, the saturation value of $l_{\phi}$. In 2D systems, although e-e interactions are the dominant source of dephasing at low $T$ and have been adequate to describe dephasing in graphene Morozov et al. (2006); Wu et al. (2007); Lin and Bird (2002), and in some reports of TI Checkelsky et al. (2011); Liu et al. (2011), e-ph interaction cannot be neglected for TI because of the vicinity of the bulk to the surface states Li et al. (2012); Liao et al. (2017). However, instead of saturation, these two mechanisms lead to a diverging $l_{\phi}$ at low $T$. The saturation of $l_{\phi}$ has also been obtained in device TBN$11$, where the TI has been transferred onto a boron nitride substrate (Fig. 3b). The atomically flat boron nitride (thickness $d=14$ nm) prevents trapping-detrapping that is commonly observed between the channel and the SiO2 substrate, thus reducing any dephasing due to electromagnetic fluctuations induced by potential traps present in SiO2.

The saturation of $l_{\phi}$ is often attributed to (a) finite size effects, where $l_{\phi}$ becomes comparable to $L$, the length of the sample, (b) saturation of electron temperature due to heating from external sources Lin and Bird (2002), (c) spin-orbit scattering length becoming comparable to the phase breaking length Fukai et al. (1990), and (d) magnetic impurities or local magnetic moment Pierre and Birge (2002); Pierre et al. (2003); Schopfer et al. (2003). We have systematically explored the possibility of saturation arising from any of the above reasons. To probe the effect of finite size, we have performed MR measurements on M$10$ with a channel length of $1$ mm which is three orders of magnitude more than the saturated value of $l_{\phi}$. The $l_{\phi}$ extracted from magneto-conductance data using Eq. 1 also shows a similar saturation for $T<300$ mK for all $V_{G}$’s. The saturation obtained in the large area sample, indicates that finite-size effects at low $T$ is not the cause of the observed behavior of $l_{\phi}$. Another important factor for the saturation is thermalization, where the electron $T_{e}$ can be much higher than the lattice temperature $T_{L}$. For extracting $T_{e}$ accurately in our He-$3$ system, we have measured and analyzed the $T$ dependence of Shubnikov-de Haas oscillations using GaAs/Al0:33Ga0:66As hetero-structure to ensure that down to $T=0.3$ K, $T_{e}$ matches $T_{L}$. This type of saturation has also been observed in systems where the spin-orbit coupling length ($l_{SO}$) becomes comparable to the phase breaking length Fukai et al. (1990). We have extracted $l_{SO}$ by using the full HLN equation Zhang et al. (2012b); Dey et al. (2014) (See supplementary information). The extracted $l_{SO}$ is much smaller compared to $l_{\phi}$ (as expected for TI systems) ruling out that possibility as well.

The most common reason, however, for the saturation of $l_{\phi}$ is the presence of magnetic impurities or localized spins, which leads to the saturation even in extremely pure systems Pierre and Birge (2002); Pierre et al. (2003); Schopfer et al. (2003). Experimental and theoretical studies have reported the presence of localized spins Nisson et al. (2013) or intrinsic magnetic instabilities in the TI surface Baum and Stern (2012). One manifestation of the presence of magnetic impurities can be long- or short-range magnetic ordering , which can lead to the removal of TRS in the system. To probe this, we have measured $\langle\delta G^{2}\rangle$ as a function of $B_{\perp}$ on TBN$11$. The magnitude of the conductance fluctuations is plotted as $N_{G}(B)/N_{G}(0)$ ($N_{G}=\langle\delta G^{2}\rangle/\langle G^{2}\rangle$ is the normalized variance) in Fig. 4(a). We observe a factor of two reduction in the normalized magnitude when $B_{\perp}>>B_{\phi}$ due to the suppression of the Cooperon contribution in transport and the crossover of the system from symplectic to unitary symmetry class, which indicates that TRS is intact intrinsically in these systems. For a quantitative understanding, the normalized magnitude has been fitted with the expression Stone (1989); Shamim et al. (2017); Lee et al. (1987); Islam et al. (2018)

[TABLE]

[TABLE]

Here $b=8\pi B(l_{\phi})^{2}/(h/e)$, $\Psi^{{}^{\prime\prime}}$ is the double derivative of the digamma function, and $l_{\phi}$ is the fitting parameter. The solid line in Fig. 4 (a) is the fit according to Eq. 4, which captures the variation with normalized magnitude with $B$. The $l_{\phi}$ obtained from the fit is $220$ nm, similar to $l_{\phi}$ obtained from UCF and MR, which confirms the validity of the analysis and the factor of two reduction.

Spin-flip scattering due to the presence of unwanted magnetic impurities, e.g., due to finite purity of the metal components, may also cause dephasing. To investigate this, we have extracted $\langle\delta G^{2}\rangle$ as a function of $T$ for $B_{\perp}=0$ T and $2$ T at different $V_{G}-V_{D}$. The normalized magnitude $\langle\delta G^{2}\rangle/G^{2}$ shows a saturation for both $B_{\perp}=0$ T and $2$ T for $T<2$ K. The presence of a magnetic field $B_{\perp}>>k_{B}T/(g\mu_{B})$ is expected to freeze the magnetic moments and suppress spin-flip scattering in the sample. Here $k_{B}$, $g$, and $\mu_{B}$ are the Boltzmann constant, Land$\acute{e}$ g-factor, and Bohr magneton respectively. The fact that the saturation persists even in the presence of $B_{\perp}$, indicates that it is not due to any magnetic impurities or localized spins in the system. To probe the effect of coupling between the surface states and charged disorders in the bulk, we have measured the conductance fluctuations for more positive gate voltages ($V_{G}-V_{D}=60$ V and $120$ V). The number density at $V_{G}-V_{D}=120$ V is $8.6\times 10^{16}$ m*-2*, which corresponds to bulk transport dominated regime, where the coupling of the two surfaces and the bulk is also higher, compared to that near the Dirac point ($-60$ V) Kim et al. (2012); Liao et al. (2017). However, the nature of saturation also does not change for more positive gate voltages, implying that it is independent of the coupling between surface states and the charged puddles in the bulk (inset of Fig. 4b).

While the exact source of saturation remains unascertained, we discuss some plausible mechanisms that may lead to the saturation of $l_{\phi}$ in TIs. The saturation can arise from the presence of two-level systems as has been explored in Ref. Imry et al. (1999); Aleshin et al. (2001). Such two-level systems can arise from the charge fluctuations in the bulk, which are known to be the dominant source of $1/f$ noise in TI Bhattacharyya et al. (2015); Islam et al. (2017); Bhattacharyya et al. (2016). The relaxation dynamics of these charged defects in the bulk can lead to a very weak dependence of $l_{\phi}$ on $T$. We also note that the temperature where $l_{\phi}$ saturates in M$10$ is $\sim 300$ mK, which is an order of magnitude lower compared to the exfoliated samples (F$7$ and F$10$), and $l_{\phi}^{sat}$ is also an order of magnitude higher. This difference in $l_{\phi}^{sat}$ and $T^{sat}$ between these samples, grown by totally different methods could be indicative of a lower charge impurity driven inhomogeneity in the bulk in case of the MBE sample. Liao et al. have shown that the charge puddles in the bulk lead to a sublinear dependence of $l_{\phi}$ on $T$ Liao et al. (2017). It is possible that at low $T$, these uncompensated charges are strongly localized, leading to reduced screening of electromagnetic fluctuations. This can produce additional dephasing of the surface carriers, which might limit $l_{\phi}$ to a finite value. Recently Väyrynen et al. have proposed back-scattering of electrons by electromagnetic fluctuations from the charge puddles in the bulk in 2D TIs Väyrynen et al. (2018). The effect of such inelastic scattering on $l_{\phi}$ in 3D TIs remains to be seen, and may also provide crucial insight into the factors leading to the saturation of $l_{\phi}$ in topological insulators.

In conclusion, we have measured the Fermi energy and time-dependent conductance fluctuations and magneto-resistance to probe the sources of dephasing of the surface carriers in topological insulators. The phase breaking length obtained from both these techniques show a saturation below some specific temperatures which are sample dependent. We have eliminated several factors that may lead to the saturation of $l_{\phi}$ such as finite-size effects, spin-orbit coupling length, and surface-bulk coupling. The magnetic field dependence of the conductance fluctuations also eliminates the possibility of the saturation arising due to the presence of magnetic impurities or localized spins in the system. Our work suggests an additional dephasing mechanism in TIs which is dominant at low temperatures, and limits the phase breaking length to a finite value at low temperatures.

S.I., S.B., D.S., and A.G. acknowledge support from DST, India. A.R., A.K., and N.S. acknowledge support from The Pennsylvania State University Two-Dimensional Crystal Consortium – Materials Innovation Platform (2DCC-MIP), which is supported by NSF cooperative Agreement No. DMR-1539916.