We report detailed temperature and gate-voltage dependence of 1/f resistance noise in magnetically doped topological insulators (TI) Crx(Bi,Sb)2−xTe3. The noise is remarkably sensitive to the gate voltage, increasing rapidly as the chemical potential is moved towards the charge neutrality point. Unlike in identically prepared (Bi,Sb)2Te3 films, where mobility-fluctuations in the surface states is the dominant mechanism, the noise in the magnetic Crx(Bi,Sb)2−xTe3 originates from transport in the localized band tail of the bulk valence band. A strong increase in noise with decreasing temperature supports this scenario. At higher temperature (≥10 K), we observed large noise peaks at gate voltage-dependent characteristic temperature scales. In line with similar observations in other non-magnetic TI systems, we attribute these peaks to generation-recombination in the Cr-impurity band.

Ferromagnetically doped topological insulators (FMTI) have been at the center for both fundamental1–6 and technological3,7 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-electric3,4 effect, quantum anomalous hall effect1,2,5,8 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 measurements1,2,16–18 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 state21–24 and thus complements information obtained from conventional current voltage characteristics. In this work, we have investigated 1/f noise in magnetically doped TI Crx(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 Crx(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.

FIG. 1.

(a) Schematic of a Crx(Bi,Sb)2−xTe3 Field-Effect Transistor (FET) grown on a STO substrate with an indium backgate and device micrograph. (b) Longitudinal and transverse resistance (ρxx and ρxy) as a function of magnetic field. ρxy was symmetrized to remove the asymmetry introduced by non-ideal symmetry of the Hall bar due to mechanical scratching. (c) Gate voltage (VG) dependence of sheet-resistance (RS) of Crx(Bi,Sb)2−xTe3 at 0.3 K and (Bi,Sb)2Te3 at 5 K. Temperature (T) dependence of RS at various gate voltages for (d) Crx(Bi,Sb)2−xTe3 and (e) (Bi,Sb)2Te3 samples. The inset of (d) shows a transition from insulating T-dependence ρxx to a metallic T-dependence.

FIG. 1.

(a) Schematic of a Crx(Bi,Sb)2−xTe3 Field-Effect Transistor (FET) grown on a STO substrate with an indium backgate and device micrograph. (b) Longitudinal and transverse resistance (ρxx and ρxy) as a function of magnetic field. ρxy was symmetrized to remove the asymmetry introduced by non-ideal symmetry of the Hall bar due to mechanical scratching. (c) Gate voltage (VG) dependence of sheet-resistance (RS) of Crx(Bi,Sb)2−xTe3 at 0.3 K and (Bi,Sb)2Te3 at 5 K. Temperature (T) dependence of RS at various gate voltages for (d) Crx(Bi,Sb)2−xTe3 and (e) (Bi,Sb)2Te3 samples. The inset of (d) shows a transition from insulating T-dependence ρxx to a metallic T-dependence.

Close modal

To demonstrate the FM ordering in our Crx(Bi,Sb)2−xTe3 films, we have plotted the magnetic field (B) dependence of Hall resistance (ρxy) and longitudinal sheet-resistance (ρxx) at 0.3 K in Fig. 1(b) (see supplementary material, Section IV).26 The square-shaped hysteretic Hall traces indicate robust FM order. ρxy can be expressed as ρxy = RAM + RHB, where M is the magnetization of the sample and RA, RH are the anomalous and ordinary Hall coefficient, respectively.17,28 The butterfly-shaped hysteresis in ρxxB 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 (RS) vs gate voltage (VG) data (Fig. 1(c)) indicate that both the samples are intrinsically hole-doped. While the charge neutrality point (VD) is achieved in (Bi,Sb)2Te3 device at 59 V, resistance of Crx(Bi,Sb)2−xTe3 device only shows early signs of saturation at VG ≈ 150 V, indicating that in this sample VD ≳ 150 V.

Figs. 1(d) and 1(e) show the RS vs T data at different gate voltages for Crx(Bi,Sb)2−xTe3 and (Bi,Sb)2Te3 samples, respectively. Higher value of RS 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 VG = −40 V (at comparable number density with VG = 40 V in Crx(Bi,Sb)2−xTe3 device) and VG = 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 (VG = 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 SV(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 SVV2 in all devices in the operating excitation voltage range (Fig. 2(c)). The estimated Hooge parameter (γH = AnfSV/V2, where A = area and n = number density) varies from 0.3 to 3 indicating MBE-grown Crx(Bi,Sb)2−xTe3 samples are highly disordered compared to similarly grown (Bi,Sb)2Te3 samples in which γH was estimated to be ≈ 10−2.

FIG. 2.

(a) Relative voltage fluctuations (δV/V) in time domain in Crx(Bi,Sb)2−xTe3 (0.3 K) and (Bi,Sb)2Te3 (5 K) samples. (b) Typical normalized noise power spectral density (SV/V2) in the two samples obtained from time dependent fluctuations shown in (a), indicating 1/f type of characteristics. (c) Power spectral density (SV) at 0.046 Hz for Crx(Bi,Sb)2−xTe3 sample (0.3 K) and at 0.03 Hz for (Bi,Sb)2Te3 (5 K) as a function of V2. (d) Frequency exponent of power spectral density (α) as a function of gate-voltage (VG) in both Crx(Bi,Sb)2−xTe3 (for two thermal cycles) and (Bi,Sb)2Te3. Gate-voltage (VG) dependence of Area (A) and frequency (f) normalized power spectral density (AfSV/V2) and R in (e) Crx(Bi,Sb)2−xTe3 samples (for two thermal cycles). The inset shows (AfSV/V2) and R as a function of VG for (Bi,Sb)2Te3sample. (f) Area normalized noise (AfSV/V2) as a function of gate voltage (VG) measured from the Dirac point (VD).

FIG. 2.

(a) Relative voltage fluctuations (δV/V) in time domain in Crx(Bi,Sb)2−xTe3 (0.3 K) and (Bi,Sb)2Te3 (5 K) samples. (b) Typical normalized noise power spectral density (SV/V2) in the two samples obtained from time dependent fluctuations shown in (a), indicating 1/f type of characteristics. (c) Power spectral density (SV) at 0.046 Hz for Crx(Bi,Sb)2−xTe3 sample (0.3 K) and at 0.03 Hz for (Bi,Sb)2Te3 (5 K) as a function of V2. (d) Frequency exponent of power spectral density (α) as a function of gate-voltage (VG) in both Crx(Bi,Sb)2−xTe3 (for two thermal cycles) and (Bi,Sb)2Te3. Gate-voltage (VG) dependence of Area (A) and frequency (f) normalized power spectral density (AfSV/V2) and R in (e) Crx(Bi,Sb)2−xTe3 samples (for two thermal cycles). The inset shows (AfSV/V2) and R as a function of VG for (Bi,Sb)2Te3sample. (f) Area normalized noise (AfSV/V2) as a function of gate voltage (VG) measured from the Dirac point (VD).

Close modal

Noise and resistance studied in Crx(Bi,Sb)2−xTe3 sample as a function of VG is shown in Fig. 2(e). The data presented for two different thermal cycles show similar trend although the area-normalized magnitude of noise (AfSV/V2) at same VG 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 VG-dependence, increasing nearly by a factor of ∼50 with decreasing density, as observed previously for graphene22,30 and transition metal dichalcogenides.23,31 The inset of Fig. 2(e) shows noise as a function of VG 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 graphene22,32 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 Crx(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 Crx(Bi,Sb)2−xTe3, we have plotted the area-normalized noise AfSV/V2 as a function of (VD − VG) in log-log scale (Fig. 2(f)) for both Crx(Bi,Sb)2−xTe3 and (Bi,Sb)2Te3 samples. From RSVG data, it was found that VD = 66 V for (Bi,Sb)2−xTe3 sample and VD = 150 V and 187 V (taken as lower bound), respectively, for first and second thermal cycle in Crx(Bi,Sb)2−xTe3 sample. The 1/(VD − VG) 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 Crx(Bi,Sb)2−xTe3 sample shows much steeper dependence with (VDVG) as shown in Fig. 2(f)). The variation of noise with (VDVG) is too strong to be described by two commonly observed processes in 2D metallic or semiconducting films, namely, Hooge mobility fluctuation (∝ 1/(VDVG))22 or Mcwhorter number-fluctuation model (∝ 1/(VDVG)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 Crx(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 regime34–36 

γhexp(B(f)*(T/T0)3),
(1)

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 RS 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 Crx(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 VG, a strong peak appears at a VG-dependent temperature Tp. Third, Tp increases with increasing VG, i.e., as chemical potential is moved towards band tail. In the paragraph below, we proceed to discuss the origin of this temperature dependence.

FIG. 3.

Temperature dependence of area-normalized power spectral density (AfSV/V2) at gate voltages −40 V (solid square), 0 V (solid circle), and 40 V (solid triangle) for Crx(Bi,Sb)2−xTe3 films and at −40 V (open circle) for (Bi,Sb)2Te3 films in log-log scale. Solid lines are guide to the eye. The inset shows area-normalized noise in log-linear scale for Crx(Bi,Sb)2−xTe3 films at T ≤ 2 K.

FIG. 3.

Temperature dependence of area-normalized power spectral density (AfSV/V2) at gate voltages −40 V (solid square), 0 V (solid circle), and 40 V (solid triangle) for Crx(Bi,Sb)2−xTe3 films and at −40 V (open circle) for (Bi,Sb)2Te3 films in log-log scale. Solid lines are guide to the eye. The inset shows area-normalized noise in log-linear scale for Crx(Bi,Sb)2−xTe3 films at T ≤ 2 K.

Close modal

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 exponential34–36 (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 Tp is given by TpEgKBln(f0fc),42 where fc, f0( ∼ 1013) Hz, and Eg 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), Eg increases (Fig. 4(a)), which results in an increase in Tp. Assuming Tp ≈ 20 K and fc ≃ 0.1 Hz (center of the experimental bandwidth), we estimate Eg ∼ 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 Eg 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.

FIG. 4.

Schematic showing density of states (DOS) in (a) Crx(Bi,Sb)2−xTe3 and (b) (Bi,Sb)2Te3. The dashed lines represent the position of the chemical potential as gate voltage is varied, and the arrows in (a) show the generation recombination processes from the localized states.

FIG. 4.

Schematic showing density of states (DOS) in (a) Crx(Bi,Sb)2−xTe3 and (b) (Bi,Sb)2Te3. The dashed lines represent the position of the chemical potential as gate voltage is varied, and the arrows in (a) show the generation recombination processes from the localized states.

Close modal

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.

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