Introduction

Topological insulator (TI) is a state of quantum matter characterized by Z2 invariance1,2,3,4. It is an insulator in the bulk state but manifests conducting helical states at the boundary. The exotic boundary states of TIs are expected to form a playground of various topological quantum effects and show great potential in spintronics and quantum computation. Since the first experimental realization of TI in CdTe/HgTe/CdTe quantum well structures5, an extensive effort has been put up in search for new TI systems. To date, CdTe/HgTe/CdTe5 and InAs/GaSb6 have been experimentally confirmed to be two-dimensional TIs while strained HgTe and many Bi based compounds such as BixSb1-x, Bi2Se3 and Bi2Te3 have been realized as three-dimensional TIs7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26. To explore the possible application of TI in spintronics, ideal TI systems are expected to exhibit high mobility and gate electric field tunability. However, most of the TI materials show very small surface contribution due to defect-induced large bulk contribution and therefore electric field tunability cannot be realized. On the other hand, very few TIs that manifest electric field ambipolar effect such as Bi2-xSbxTe3-ySey27,28,29 and thin Bi2Se3 flakes21,25, show low mobility. On this basis, it is important to identify new TIs that show large gate electric field tunability and high surface mobility for the application of future TI based spintronic devices.

The β-Ag2Te is known for its unusual large and non-saturating quasi-linear magnetoresistance (MR) in the field range of 10–105 Oe and temperature range of 5–300 K30,31,32. The origin of this unusual property has generated much debate since its discovery and may be associated with its 3D-TI nature. Theoretical calculation suggests that β-Ag2Te is a TI with anisotropy and the unusual MR is largely originated from electrical transport on the topological surface states33. Recently, Aharonov-Bohm (AB) oscillations have been observed in a nanowire of β-Ag2Te which indicates the existence of surface states34,35. However, the topological nature of the surface state still remains to be confirmed.

In this paper, we report further experimental evidence of the existence of topological surface states in β-Ag2Te. Especially, the topological surface state of narrow Ag2Te nanoplate exhibits the largest electric field ambipolar effect in TI so far (~2500%). We fabricated nanoplate devices as shown in the inset of Fig. 1(a) and Fig. 1(b). Both devices show pronounced two dimensional (2D) SdH oscillations. Using the measured resistivity ρxx and Hall resistivity ρxy of the wide nanoplate as shown in Fig. 1(a), the conductivity σxx is calculated and the Berry phase is determined to be near π by Landau Level (LL) fan diagram. This reveals that the SdH oscillation originates from the topological surface states. Utilizing the large ambipolar field effect of the narrow nanoplate as shown in Fig. 1(b), the evolution of SdH oscillations with applied gate voltage (Vg) was investigated and also strongly indicates a Dirac cone composed surface state. Moreover, the mobility of the surface transport of the narrow β-Ag2Te nanoplate is determined to be several thousand cm2s−1V−1. Finally, the MR variation with Vg of the narrow nanoplate indicates a correlation effect between the bulk electrons and Dirac fermions.

Figure 1
figure 1

Temperature and voltage dependence of resistance.

(a) and (b) show the temperature dependence of resistance of β-Ag2Te wide and narrow nanoplates, respectively. The insets show the corresponding device images. The red bars are 5 μm and 10 μm in (a) and (b), respectively. The arrow in (b) points to the narrow nanoplates. (c) and (d) show the Vg dependence of resistance of β-Ag2Te at 2 K.

Results

Temperature and gate voltage dependence of resistance

The typical transport behavior of two devices fabricated from a relatively wide nanoplate and a narrow nanoplate grown by CVD methods are shown in Fig. 1(a)–(d). It should be emphasized that similar results have been repeated in several devices fabricated using wide and narrow nanoplates. The wide β-Ag2Te nanoplates tend to be slightly n-type. The inset of Fig. 1(a) and Fig. 1(b) show the fabricated devices using heavily p-doped Si with 300 nm thick of SiO2 dielectric layer. The thicknesses and widths for the two nanoplate devices are 120 nm (thickness), 5 μm (width) and 98 nm (thickness), 395 nm (width), respectively. As shown in Fig. 1(a), the resistance (Rxx) of the relatively wide nanoplate decreases slightly with temperature from T = 300 K to 250 K, then increases about several times from 250 K to 30 K and finally decreases to 10 K. The special temperature dependence of Rxx of the nanoplate is due to the light impurity doping, which has been observed in many doped semiconductor systems36. The metallic behavior at low temperature of the nanoplate is attributed to the conduction in the impurity band, while the metallic behavior when T > 250 K can be explained by the thermal excitation of electrons from the Anderson localized states to extended states above the mobility edge. A better stoichiometry is achieved in the narrow nanoplate and hence we observed much sharper increase of resistance with decreasing temperature as shown in Fig. 1(b). The gradually saturating behavior at low temperatures is due to the surface states. To probe the existence of surface states, we have performed the gate-tuned resistance measurement. As shown in Fig. 1(c), the Rxx of the wide nanoplate only changes slightly with Vg in view of electron doping. Although the Vg dependence of Rxx shows a peak near 0 V, the voltage dependence of Hall resistance (Rxy) and SdH oscillations indicates that electric field ambipolar gate effect cannot be realized in wide nanoplate devices due to charge doping. The resistance change with an applied gate voltage may originate from the variation of the density of states at the Fermi level in the bulk states when the Fermi level is slightly shifted by the gate electric field. We observed the phenomenon in several wide nanoplate samples. In contrast to wide nanoplates, the narrow nanoplates shows a huge ambipolar type electric field effect under a back Vg, which further confirms the high stoichiometry as indicated by the temperature dependence of resistivity curve. Fig. 1(d) shows that the gate induced resistance change (Rpeak/R+50 V) in the narrow β-Ag2Te nanoplate is ~2500%, a value which is much larger than that obtained in any other TIs. Here, the Rpeak and R+50 V is the resistance at the peak of the ambipolar curve and Vg = +50 V, respectively. The sharp transition at highest resistance point (the charge neutrality point) and the gradual change of resistance when the Vg deviates from the charge neutrality point indicate that the ambipolar behavior could be due to topological surface states of Dirac cone. However, it should be emphasized that the ambipolar electric field effect can also be realized in semimetal or semiconductor with a very narrow band gap. In order to probe the topological nature of the surface state, we have performed two experiments. We first determine the Berry phase from the SdH oscillation of the conductivity of the wide nanoplate and then investigate the evolution of SdH oscillation with Vg since the narrow nanoplate shows huge Vg dependence of resistance of Dirac cone type.

Berry phase obtained from SdH oscillations

Figure 2(a) shows the Rxy of the wide nanoplate device under tilted magnetic field (B) as a function of the component of the magnetic field perpendicular to the sample surface (B). The angle (θ) is defined between the B field and sample surface. The Rxy indicates an n-type carrier and a clear deviation from the linear relationship with B field. This demonstrates that there are more than one transport channel with different mobility values. At higher field, the Rxy oscillates periodically in 1/B, which is the standard behavior of SdH oscillation. More importantly, we observe that the positions of maxima and minima do not change with B field. This clearly indicates that the oscillations originate from 2D transport behavior. No oscillations are observed in Rxx vs B curve (not shown here) when the applied magnetic field is parallel to the sample surface, which further supports the 2D transport and rules out the oscillation from bulk in this sample. The lack of the SdH oscillations from bulk electrons indicates low mobility of the bulk transport in this device.

Figure 2
figure 2

SdH oscillations and fan diagram analysis.

(a) Hall resistance (Rxy) as a function of the B field measured at various tilted angles (θ); (b) Rxx and Rxy as functions of the B field with θ = 90°. The measurements were performed using the same sample for (a), which had been stored for three days in vacuum after the measurements in (a); (c), SdH fan diagrams for measured 1/B with the filling factor n. The blue and red data and fitting lines are for the device before the storage and after the storage, respectively. The squares and circles are the maxima and minima, respectively. The inset shows the Δσxx vs. 1/B plot for the device after the storage; (d) Rxx vs. B field curves at various temperatures. The inset shows the SdH oscillation after subtracting the background MR. (e) The temperature dependence of relative amplitude of SdH oscillation in ΔRxx(B) for the 3rd LL. The solid line is a fit to ; (f) ln(ΔRB sinh(αTEN)) is plotted as a function of 1/B.

A prominent property of Dirac fermions is that they carry the Berry phase of π. The observation of a π phase shift in SdH oscillation would clearly demonstrate that the 2D transport is indeed due to the topological surface transport4,37. SdH oscillations originate from successive emptying of Landau Levels (LL) with increasing magnetic field. The LL index n is related to the cross section area SF of the Fermi surface by

where γ = 0 or 1/2 for topological trivial electrons and Dirac Fermions, respectively, e is the electron charge, h is the Planck constant () and B is the magnetic flux density. Fig. 2(b) is the Rxy and Rxx of the wide nanoplate with a storage in vacuum for three days after the measurement in Fig. 2(a). As shown in Fig. 2(b), the SdH oscillation of Rxy changes after the storage in vacuum, which indicates the doping of surface or bulk states. To obtain the value of Berry phase of the system, we calculate the conductivity using the formula σxx = ρxx/(ρxx2 + ρxy2) for both measurements in Fig. 2(a) and Fig. 2(b). Thereafter the Δσxx is obtained through a smooth back ground subtraction. The Δσxx vs 1/B curves for both conditions show oscillations of a single period which indicates only one transport channel in the surface states of the sample contributing to the oscillations. The inset of Fig. 2(c) is the Δσxx vs 1/B curve obtained from Fig. 2(b). The LL fan diagrams based on the oscillations of the conductivity for both situation are shown in Fig. 2(c), where their minima are identified to signify the integer n (indicated by arrows), while the half integers n + 1/2 are assigned to the positions of maxima (indicated by arrows). The interception at 1/B = 0 should be at n = 0 (for the topological trivial surface states) or n = 1/2 (for the topological surface states). As shown in Fig. 2(c), linear fits to the data give interception of 0.52 (the blue fitting line) and 0.54 (the red fitting line) for the sample before the storage and after the storage, respectively. The values are close to the value of 0.5 expected for Dirac fermions. This result strongly supports that the SdH oscillation is indeed originated from the topological surface states. The doping due to the storage can shifts the Fermi level, but it does not change the topological characteristics.

Figure 2(d) displays the temperature dependence of SdH oscillations of Rxx. The inset shows the ΔRxx obtained from Rxx by subtracting a polynomial fit to the background. The amplitude of the SdH oscillations decreases with increasing temperature due to thermal agitation of electrons on the Landau levels. We have used the standard Lifshitz-Kosevich theory

to fit the temperature dependence of SdH oscillations, where ΔEN = heB/2πm*c is the energy gap between Nth and (N + 1)th Landau Level, TD = h/4π2τkB is the Dingle temperature and α = 2π2kB. The B, h, m*, kB and c are the magnetic field, Planck constant, the effective mass of carriers, Boltzmann constant and speed of light, respectively. The temperature dependence of ΔR/ΔR(0 K) for the 3rd Landau level is plotted in Fig. 2(e). The solid line is a fit to . The ΔR (0 K) is the ΔR at 0 K obtained from the fitting and m* can be calculated using the fitted value of EN. We averaged the value obtained for different B to get m* = 0.12 me. From the slope of the semi-log plot of ΔRB sinh(αTEN) vs 1/B at T = 2 K, the TD is determined to be 13.6 K and the carrier life-time is calculated to be 8.9 × 10−14 s. The mobility is then determined to be 1310 cm2s−1V−1.

SdH oscillations with gate

The existence of Dirac cone composed topological surface states can be further investigated by examining the evolution of the SdH oscillation in the narrow nanoplate through back gating. From Eq. (1), we know that the period of Rxx vs 1/B is determined by the cross section of Fermi surface as shown in Eq. (3),

Assuming a circular 2D Fermi surface or a spherical 3D Fermi surface, the value of Fermi wave vector kF can be calculated using . Thereafter, the carrier density of 2D surface states and 3D bulk states can be calculated by and , respectively. The dependence of Rxx on B field under various Vg was performed and the results are shown in Fig. 3(a)–(h). To analyze the SdH oscillations, ΔRxx is obtained from Rxx by subtracting a polynomial fit to the background. The ΔRxx vs 1/B curves with various Vgs are shown in Fig. 4(a)–(f). The curves at −12 V and −50 V do not show reasonable SdH oscillations, which may contribute to the very small Fermi surface and large noise, respectively. From Fig. 4, it is clearly evident that the period of SdH oscillation rises when Vg varies from +50 V to −8 V. This demonstrates that the area of Fermi surface decreases when Vg changes from +50 V to −8 V. With further increasing amplitude of Vg to the negative direction, the magnitude of SdH oscillations gradually diminishes. The SdH oscillations can still be clearly observed and the period of the oscillations decreases when Vg varies from −8 V to −25 V. This demonstrates that the area of Fermi surface increases in the procedure. This correlates well with the scenario that the area of the Fermi surface increases with the Fermi level shifting away from the Dirac point. In order to understand the band structure of β-Ag2Te, we fit the ΔRxx vs 1/B curve with the theoretical expression for SdH oscillations. The formula can be written as

where BF is the frequency of the SdH oscillation, A is the amplitude, μ is the mobility of carriers and β is the Berry phase22,23. Equation (4) is in fact a zero temperature formula, which considers the effect of finite relaxation time but ignores the temperature effect38. As the accurate phase analysis should use the value of conductivity σxx = ρxx/(ρxx2 + ρxy2), especially for systems like β-Ag2Te that has similar values of ρxx and ρxy, the β value obtained in the fitting to ΔRxx cannot provide the information of topological nature4. Here, we focus on two fitting parameters, the BF and μ. The experimental data (red circles) and fitting curves (green lines) are shown in Fig. 4. In the fitting process, the BF values are first roughly determined by inspection and FFT transformation. Thereafter, accurate BFs and μs are fitted. Three transport channels are employed to fit the SdH oscillations at Vg = +50 V and Vg = +25 V. One transport channel is utilized to fit the oscillations at other applied Vgs. The SdH oscillations at −8 V (Fig. 4(e)) show an abrupt increase at the large field, which probably originates from the deviation of background subtraction. A large deviation at large field can appear when the number of oscillations is small. The phase deviation of the fitting at large field (1/B < 0.15 T−1 or B > 6.7 T) at −25 V (Fig. 4(f)) may result from the large noise of the data and error in background subtraction. The three peaks and three valleys in low field range of 0.15 T−1 < 1/B < 0.25 T−1 clearly indicates a larger oscillation period compared with that at Vg = 0 V and −8 V. Although the fittings deviate in some features, our fitting do reveal the main features of the experimental data. The fitted BFs and μs are shown in table 1. Based on the BF values, the values of kFs are calculated as listed in the table. The Vg dependence of kF is depicted in Fig. 5(a), in which both the negative and positive value of kF is plotted as the red squares. For the SdH at +50 V and +25 V, we choose the largest BF to calculate the kF. The green dashed line is drawn to clearly depict the cone structure. As previously discussed, the ambipolar effect can originate from the topological surface or bulk state of a semimetal. For the same Fermi surface area perpendicular to B field, the bulk state has a much larger carrier density. Using the formulae as aforementioned, the calculated carrier densities at +50 V are n3D = 4.3 × 1018 cm−3 for the bulk state of semimetal and n2D = 4.0 × 1012 cm−2 for the Dirac cone composed surface state. At 0 V, the corresponding values for n3D and n2D are 3.0 × 1017 cm−3 and 6.8 × 1011 cm−2, respectively. We note that the charge tunability of a 300 nm SiO2 dielectric layer is about 3.6 × 1012 cm−2/50 V. Considering the sample thickness of 98 nm, if the ambipolar behavior was due to the bulk state, the tuned charge would be 4.2 × 1013 cm−2 which is 11 times larger than the tuning ability of a SiO2 dielectric layer. Thus, our results clearly rule out the possibility of bulk originated ambipolar behavior and provide further strong evidence of the existence of surface states of Dirac cone type. As for the other two channels showing SdH oscillations at Vg = +50 V and +25 V, we speculate that they originate from the Rashba splitting induced topological trivial surface states, which have shown by the Altshuler-Aronov-Spivak oscillations in the β-Ag2Te nanowire34.

Table 1 The fitting parameter BF and μ for SdH oscillations and the calculated kF from the fitted BF at various Vgs
Figure 3
figure 3

Rxx of the narrow plate as a function of B field measured under various Vg.

(a) +50 V, (b) +25 V, (c) +10 V, (d) 0 V, (e) −8 V, (f) −12 V, (g) −25 V and (h) −50 V.

Figure 4
figure 4

ΔRxx as a function of 1/B field with various Vg.

(a) +50 V, (b) +25 V, (c) +10 V, (d) 0 V, (e) −8 V, (f) −25 V. The red circles are the experimental data. The green lines are the fitting curves.

Figure 5
figure 5

Vg dependence of kF and μ.

(a) The variation of kF with Vg. The red squares are the calculated results based on the SdH oscillations. The green lines are plotted to depict the cone structure. (b) The Vg dependence of the carrier mobility μ.

The Vg dependence of the mobility is shown in Fig. 5(b), the carrier mobility is in the range between ~1 × 103 to ~2 × 103 cm2s−1V−1 when the gate voltage varies from +50 V to +10 V. It increases to ~4 × 103 cm2s−1V−1 at Vg = 0 V and then the mobility decreases to ~2 × 103 cm2s−1V−1 at Vg = −25 V. The obtained mobility values are decent compared with that in other topological insulators11,19,20,21,39,40. It also agrees with the mobility value (1310 cm2s−1V−1) obtained for the doped β-Ag2Te nanoplate. The shape of the oscillations at −8 V is different from that of the standard SdH oscillation, which indicates a large artefact from subtraction. Because the fitted value of μ is closely related to the shape of the SdH oscillation (the decay of the oscillation amplitude), the very large mobility value (3 × 104 cm2s−1V−1 as shown in table 1) at Vg = −8 V is probably not reliable. Using the width of the narrow nanoplate (395 nm) and the interval between the contact (7.2 μm), we can calculate the sheet resistance and then calculate the mobility using μ = σ/en. Considering the devices have top and bottom surfaces, the calculated mobility at Vg = 0 is ~5 × 103 cm2s−1V−1, which also agrees well with the result obtained from SdH oscillation. For Vgs very near the charge neutrality point, we cannot get accurate kF due to the lacking of enough oscillations. The variation of MR with Vg in the low field regime also shows an interesting feature. As depicted in Fig. 3(d), (e), (f), the MR presents a relatively sharp increase in the low field regime (circled by green dash lines) at Vg = 0 V, −8 V and −12 V, near the charge neutrality point and does not appear in the MR curves under other applied Vg. This may suggest that the π Berry phase induced weak anti-localization can be manifested when the EF is far away from the bulk band. In all, it is strongly indicated that β-Ag2Te is a TI from the Berry phase of π obtained from LL fan diagram as shown in Fig. 2(c) and Dirac cone type evolution of SdH oscillations under Vg as depicted in Fig. 3, 4, 5.

The voltage dependence of MR

As aforementioned, the origin of the unusual MR of β-Ag2Te is still under debate, which may originate from the bulk effect or surface effect28,29,30,31,32. To distinguish the bulk from the surface effect, high quality samples with tunable Fermi level are indispensable. It has not been realized prior to this work. Figure 6 shows the comparison of ambipolar effect and MR under applied Vg. The MR is defined as (R(9T) − R(0T))/R(0T), which depicts a clear correlation with the EF position. The MR displays a lowest value of 78% at Vg of the charge neutrality point. When the Vg deviates from the charge neutrality point, the MR gradually increases in both negative and positive directions and reaches a maximum value at the voltages when the electric field ambipolar effect saturates. Thereafter, the resistance decreases with increasing Vg magnitude in both negative and positive directions. Based on the relationship between the EF position and electric field ambipolar curve, we can clearly describe the MR behavior in β-Ag2Te. At the charge neutrality point, the EF is located at the Dirac point and MR presents the lowest value. When the EF in the bulk band gap (but on the Dirac type surface states) is moved to the conduction band or valence band, the MR increases continuously until the EF touches the bulk band. Thereafter, any further manipulation to the bulk causes MR to decrease. Other than the bulk or surface origin of large quasi-linear MR, our experiment provides the possibility of controlling the interplay of bulk and topological surface state41. As shown in Fig. 4, the gated MR demonstrates that the largest MR appears near the top of the valence band and bottom of the conduction band, where the bulk electrons and Dirac Fermions can have a strong correlation. We still do not fully understand the correlation effect between the bulk electrons and Dirac Fermions, which deserves further theoretical and experimental investigations.

Figure 6
figure 6

The MR measured under various applied Vg.

The Rxx vs Vg curve is plotted for comparison.

Discussion

In conclusion, using LL fan diagram obtained from the oscillation of σxx and the variation in the period of SdH oscillation with applied Vg, we provide strong evidence of the topological nature of surface states in β-Ag2Te. The topological surface states of highly stoichiometric narrow β-Ag2Te nanoplate exhibit the largest ambipolar effect in TI so far (~2500%) and decent mobility (a few thousand cm2s−1V−1). This indicates that β-Ag2Te has the potential to become an important material for TI investigation. Moreover, the first report of Vg dependence of MR in β-Ag2Te suggests that the interplay between the bulk electrons and surface Dirac fermions has a large effect on MR for this material.

Methods

Using standard chemical vapor deposition (CVD) method, we have successfully obtained high quality β-Ag2Te nanoplates. The details of growth method and characterization can be found in Ref. 31. The thickness of all the nanoplates varies from ~100 nm to ~300 nm, while the width varies from ~100 nm to ~20 μm. It is experimentally observed that the thickness of the β-Ag2Te nanostructures does not change much with increasing growth time (from 10 mins to 3 hours). Longer growth time only results in the increase of width and length of the nanostructures. Standard photolithography technique was employed to pattern electrodes on the nanoplates. Cr/Au (5 nm/120 nm) contacts were deposited in a magnetron sputtering system with a base pressure of 1 × 10−8 torr. Standard lock-in technique was utilized to perform four-terminal magnetoresistance measurements in two Quantum Design PPMS systems with 9 Tesla and 14 Tesla magnets respectively. All the measurements in this manuscript were carried out from −14 T to 14 T or from −9 T to 9 T and then the Rxx (H) and Rxy (H) is calculated using formula

to eliminate the effect of the non-symmetric contacts.