Resistance noise in epitaxial thin films of ferromagnetic topological insulators

Ferromagnetically doped topological insulators (FMTI) have been at the center for both fundamental^1–6 and technological^3,7 interests. Interplay of topologically protected states and broken time-reversal symmetry leads to many important physical phenomena like gap opening in surface states,^1–3 topological magneto-electric^3,4 effect, quantum anomalous hall effect^1,2,5,8 magnetic monopoles,⁹ or the inverse spin-galvanic⁶ effect. These make FMTI a suitable candidate for quantum computing, spintronics and magnetic sensing.^6,10,11 A remarkable consequence of magnetic doping of a topological insulator (TI) is the breakdown of the topological protection for surface charge carriers and they become prone to back-scattering and localization.³ In addition, doping TI with ferromagnetic (FM) atoms creates crystal defects, which lead to broadening of surface states,^12,13 emergence of impurity bands in the insulating gap¹⁴ or magnetic scattering of surface carriers,¹⁵ all of which cause deterioration of the mobility of surface states.¹² 

In spite of extensive transport measurements^1,2,16–18 and spectroscopic studies,^12,19,20 a study of the low frequency 1/f noise in FMTI systems has not been carried out so far. Apart from being an important performance marker, 1/f noise is also a sensitive probe to the nature of electronic state^21–24 and thus complements information obtained from conventional current voltage characteristics. In this work, we have investigated 1/f noise in magnetically doped TI Cr_x(Bi,Sb)_2−xTe₃ as a function of chemical potential and temperature (T). Surprisingly, we find that away from the charge-neutrality point, noise in FMTI system originates from transport in the localized states at the band-tail of the bulk valence-band, unlike the nonmagnetic TI films, where noise in the surface transport play the dominant role.²¹ 

The FMTI devices studied here were fabricated from epitaxial films of Cr_x(Bi,Sb)_2−xTe₃ grown by molecular beam epitaxy on $〈 111 〉$ SrTiO₃ (STO) substrates with a metallic back-coating of Indium that is used as back gate electrode²⁵ (Fig. 1(a)). In order to isolate the impact of magnetic doping, we have also carried out similar measurements in (Bi,Sb)₂Te₃ samples grown in the same method and conditions. The channel-thickness for both samples was ≈10 nm (supplementary Fig. S1).²⁶ Devices were defined in Hall bar geometry by mechanical etching. The non-invasive hall bar geometry (Fig. 1(a)) minimizes contribution of contacts in electrical transport and noise measurements. Moreover, the advantage of mechanical etching process over e-beam lithography is that in the former process the film does not come into contact with the chemicals (e-beam resist and solvents), resulting in a higher surface quality. The large dielectric constant of the STO substrate at cryogenic temperatures allows effective electrical back gating for tuning the chemical potential in the sample.²⁷ Transport and noise measurements were conducted in a Helium-3 cryostat down to 300 mK.

To demonstrate the FM ordering in our Cr_x(Bi,Sb)_2−xTe₃ films, we have plotted the magnetic field (B) dependence of Hall resistance (ρ_xy) and longitudinal sheet-resistance (ρ_xx) at 0.3 K in Fig. 1(b) (see supplementary material, Section IV).²⁶ The square-shaped hysteretic Hall traces indicate robust FM order. ρ_xy can be expressed as ρ_xy = R_AM + R_HB, where M is the magnetization of the sample and R_A, R_H are the anomalous and ordinary Hall coefficient, respectively.^17,28 The butterfly-shaped hysteresis in ρ_xx–B curve represents typical magnetoresistance observed in ferromagnetic metals caused by the spin-dependent scattering of carriers by local magnetic ordering.^17,28,29 Sheet-resistance (R_S) vs gate voltage (V_G) data (Fig. 1(c)) indicate that both the samples are intrinsically hole-doped. While the charge neutrality point (V_D) is achieved in (Bi,Sb)₂Te₃ device at 59 V, resistance of Cr_x(Bi,Sb)_2−xTe₃ device only shows early signs of saturation at V_G ≈ 150 V, indicating that in this sample V_D ≳ 150 V.

Figs. 1(d) and 1(e) show the R_S vs T data at different gate voltages for Cr_x(Bi,Sb)_2−xTe₃ and (Bi,Sb)₂Te₃ samples, respectively. Higher value of R_S in the Cr-doped film suggests the presence of strong disorder, which leads to insulating transport at all gate voltages at low temperatures (Fig. 1(d)). For comparison, we note that (Bi,Sb)₂Te₃ shows metallic behavior at V_G = −40 V (at comparable number density with V_G = 40 V in Cr_x(Bi,Sb)_2−xTe₃ device) and V_G = 0 V (Fig. 1(e)). In this device a weak insulator-like behavior (negative T coefficient of R) appears only at much lower number density (V_G = 40 V) and could be the result of an interplay of weak antilocalization and electron-electron interaction.

In Fig. 2(a), typical time-dependent relative fluctuations in the voltage drop (δV/V) (at a fixed biasing current) across the channel is shown for the two devices (see supplementary material, Section V for further details).²⁶ The noise has a normalized power spectral density S_V(f)/V² ∝ 1/f ^α with frequency f in all devices (Fig. 2(b)), where α ≈ 0.8 – 1.3 (Fig. 2(d)). The linear transport regime is verified from S_V ∝ V² in all devices in the operating excitation voltage range (Fig. 2(c)). The estimated Hooge parameter (γ_H = AnfS_V/V², where A = area and n = number density) varies from 0.3 to 3 indicating MBE-grown Cr_x(Bi,Sb)_2−xTe₃ samples are highly disordered compared to similarly grown (Bi,Sb)₂Te₃ samples in which γ_H was estimated to be ≈ 10⁻².

Noise and resistance studied in Cr_x(Bi,Sb)_2−xTe₃ sample as a function of V_G is shown in Fig. 2(e). The data presented for two different thermal cycles show similar trend although the area-normalized magnitude of noise (AfS_V/V²) at same V_G is slightly lower in the second thermal cycle owing to change in disorder configuration from one thermal cycle to the other. The noise in both thermal cycles shows a strong V_G-dependence, increasing nearly by a factor of ∼50 with decreasing density, as observed previously for graphene^22,30 and transition metal dichalcogenides.^23,31 The inset of Fig. 2(e) shows noise as a function of V_G for the sample (Bi,Sb)₂Te₃, where the suppression of noise close to the Dirac point bears close resemblance to the “M”-type behavior commonly observed in graphene^22,32 and thin TI,²¹ attributed to the long-range correlations of coulomb disorders.³³ This also suggests that the noise in (Bi,Sb)₂Te₃ originates from the surface state transport although this cannot be concluded for Cr_x(Bi,Sb)_2−xTe₃ as the highest chemical potential accessed in the latter was not sufficiently close to the charge-neutrality point.

To understand the noise mechanism in Cr_x(Bi,Sb)_2−xTe₃, we have plotted the area-normalized noise AfS_V/V² as a function of (V_D − V_G) in log-log scale (Fig. 2(f)) for both Cr_x(Bi,Sb)_2−xTe₃ and (Bi,Sb)₂Te₃ samples. From R_S–V_G data, it was found that V_D = 66 V for (Bi,Sb)_2−xTe₃ sample and V_D = 150 V and 187 V (taken as lower bound), respectively, for first and second thermal cycle in Cr_x(Bi,Sb)_2−xTe₃ sample. The 1/(V_D − V_G) type dependence observed in (Bi,Sb)₂Te₃ sample indicates that noise in the undoped sample originates from mobility fluctuations in the surface transport caused by fluctuating coulomb disorders from either STO-TI interface or charged vacancies or defects inside the TI-films.^21–23 Surprisingly, noise in Cr_x(Bi,Sb)_2−xTe₃ sample shows much steeper dependence with (V_D – V_G) as shown in Fig. 2(f)). The variation of noise with (V_D – V_G) is too strong to be described by two commonly observed processes in 2D metallic or semiconducting films, namely, Hooge mobility fluctuation (∝ 1/(V_D – V_G))²² or Mcwhorter number-fluctuation model (∝ 1/(V_D – V_G)²).²³ This indicates that metallic surface states are unlikely to be responsible for the observed noise in our FMTI films.

An alternative mechanism may arise from the localized states transport in the bulk of the Cr_x(Bi,Sb)_2−xTe₃ film. In Efros-Schklovskii type variable range hopping (VRH), for example, the noise magnitude is expected to become exponentially sensitive to a small variation of temperature and chemical potential in the VRH regime^34–36 

where B and T₀ are T-independent pre-factor and correlation energy scales, respectively. At low temperature, thermally excited bulk carriers freeze out, and VRH in the impurity band may dominate the transport in the bulk of the TI film, giving rise to a metal-insulator transition (Fig. 1(d)).³⁷ Strong insulating behavior of R_S supports such a possibility.

To elaborate this further, we measure the temperature dependence of noise at three different gate-voltages (−40 V, 0 V, and 40 V) in Cr_x(Bi,Sb)_2−xTe₃ (Fig. 3). We find that, first, at T < 2 K, noise increases with decreasing T (inset, Fig. 3). Second, for all V_G, a strong peak appears at a V_G-dependent temperature T_p. Third, T_p increases with increasing V_G, i.e., as chemical potential is moved towards band tail. In the paragraph below, we proceed to discuss the origin of this temperature dependence.

The increasing trend in noise with decreasing temperature can be attributed to two different mechanisms: (i) Universal conductance fluctuations in the quantum transport regime in a diffusive metallic system (including topological insulator surface states), which gives rise to 1/T type dependence.^38,39 (ii) Variable range hopping, where the T-dependence of noise is expected to be exponential^34–36 (Eq. (1)). From the limited T-range (inset, Fig. 3), it is difficult to distinguish between these two mechanisms. However, UCF noise depends only weakly on carrier density,⁴⁰ suggesting the observed T-dependence to arise from VRH in the Cr-impurity band.

Finally, we note that the noise peaks in temperature have been recently observed in TI noise measurements²¹ and have been attributed to the generation-recombination induced charge fluctuations.^21,41 The peak position T_p is given by $Tp≈EgKBln(f0fc)$⁠,⁴² where f_c, f₀( ∼ 10¹³) Hz, and E_g are the corner frequency, characteristic phonon frequency, and energy separation between chemical potential and nearest energy band with extended quasiparticle states. As the chemical potential is moved away from the valence band maximum (by applying a positive gate voltage), E_g increases (Fig. 4(a)), which results in an increase in T_p. Assuming T_p ≈ 20 K and f_c ≃ 0.1 Hz (center of the experimental bandwidth), we estimate E_g ∼ 50 meV, which is less than half of observed bulk bandgap of (Bi,Sb)₂Te₃, and suggests the possibility of a deep impurity band. We note that E_g could correspond to the difference between the chemical potential and mobility edge in the valence band,⁴³ or that in the Cr-impurity band;¹⁴ however, no quantitative estimate is available to distinguish between these two possibilities.

In summary, we have reported detailed study of 1/f noise in magnetically doped topological insulators. Although hopping transport through localized mid-gap states does not have any clear signature in conventional transport measurements, its presence is uniquely revealed through the variation of resistance fluctuations with temperature and chemical potential (gate voltage). To improve signal to noise ratio and make these devices more suitable for practical applications, optimization in the material and doping protocol is required to eliminate the impact of the impurity band.

The sample growth and characterization effort at Penn State was supported by ONR Grant No. N0014-15-1-2370. S.B. thanks CSIR for senior research fellowship.