## Introduction

Bismuth selenide (Bi2Se3) is a typical three-dimensional topological insulator which has an insulating bulk state with a band gap and a conducting surface state with no band gap1,2. Electron spin at the surface state is helically polarized3, attracting much interest from the perspective of spintronics4,5. However, it is difficult to ascertain the surface conduction because the transport is dominated by bulk conduction owing to the natural deficiency of Se atoms and accumulation of excess electrons in the bulk region. One solution to observe the surface conduction is to replace trivalent Bi atoms with divalent or monovalent atoms. In fact, divalent Ca6,7 or Cd8 atoms work as p dopants when substituted for Bi atoms. Therefore, the Fermi level (EF) decreases from the bulk conduction band (BCB) to the bulk valence band (BVB). Impurity doping is used not only for tuning the EF, but also for achieving ordered states. Superconductivity by doping with Cu9,10, Sr11,12, or Nb13,14 atoms has been observed and discussed in relation to its topological nature. Furthermore, ferromagnetism by doping with Mn15, Cr16, or V17 atoms can lead to novel quantum phenomena, such as the quantum anomalous Hall effect. Thus, impurity doping allows to study a variety of electronic phenomena induced in topological insulators.

Recently, it was observed that a simple monovalent impurity, Ag, provided several intriguing phenomena18,19. First, bulk Ag-doped Bi2Se3 exhibited pressure-induced superconductivity, indicating the possibility of p-wave superconductivity18. Second, ultrathin Ag-doped Bi2Se3 showed a semiconductor–metal (S-M) transition with decreasing temperature19. The semiconducting behavior at high temperature is explained by the decrease in electron density owing to the substitution of Ag for Bi atoms. With decreasing temperature, resistance abruptly dropped at the transition temperature Tcr of approximately 35 K. Recently, a similar temperature dependence of resistance was reported in Cu-doped Bi2Se3 by two groups20,21. The following phenomena are commonly observed in the Cu-doped and Ag-doped Bi2Se3 ultrathin flakes. (i) Tcr was approximately 37 K20 or 30 K21 for Cu-doped Bi2Se3, similar to 35 K19 for Ag-doped Bi2Se3. (ii) Hysteresis in resistance during the cooling and heating processes was observed below Tcr in Cu-doped Bi2Se320 and Ag-doped Bi2Se3 as described later. Thus, the results of this study of the S-M transition in Ag-doped Bi2Se3 may not be characteristic of Ag impurity however a universal phenomenon in doped Bi2Se3. Although the origin of the transition is unknown, hybridization between Cu+ and Cu2+ conduction bands or the crossover of two types of carrier transport has been proposed in Cu-doped Bi2Se3. In this study, extended analyses of Hall effect and electric-field effect are carried out to investigate the origin of the anomalous S-M transition observed in Ag-doped Bi2Se3. Ag impurities can donate or accept electrons, based on the impurity sites, interstitial site through intercalation or substitution of the Bi site, respectively. Because of the dual doping effects, EF is fixed around the bottom of the BCB. This level of the EF plays a key role to induce the S-M transition.

## Results and discussion

### S-M transition in AgxBi2Se3

Figure 1a–c summarize the S-M transition observed in AgxBi2Se3, and Fig. 1d indicates an optical microscope image of a typical device. Figure 1a shows the temperature dependence of the resistivity of Ag0.05Bi2Se3 and Bi2Se3 with various thicknesses. Bi2Se3 exhibits metallic behavior; the resistivity monotonically decreases with decreasing temperature. On the contrary, the resistivity of all Ag0.05Bi2Se3 devices increased with decreasing temperature and decreased below Tcr = 35 K. The resistivity shown in Fig. 1a does not exhibit a systematic thickness dependence. However, when it is normalized by the resistivity at 270 K, clear thickness dependence appears in the semiconducting region above 35 K. As shown in Fig. 1b, the increase in the normalized resistivity above Tcr becomes prominent when the thickness increases. This implies that the semiconducting behavior is caused by the bulk properties of the samples because the influence of the metallic surface state on the total transport property diminishes with increasing the thickness.

The $$\rho \left(T\right)$$ curve of AgxBi2Se3 was hysteretic below Tcr, that is, $$\rho \left(T\right)$$ was different during the cooling or heating process. A device with a thickness of 82 nm, as shown in Fig. 1c, exhibits an example of the hysteretic behavior. The device was cooled from T = 270 to 1.6 K at a rate of 0.50 K/min, maintained at 1.6 K for 2 h, and heated to 0.89 K/min. The resistivity traces different paths below Tcr, although it is identical above Tcr. Because the resistivity measured at a maximum bias current of 1 μA (green symbol) is the same as that measured at 10 nA (red and blue symbols), the hysteresis is not caused by heating of the sample. Thus, hysteresis is not an experimental artifact caused by the disagreement of temperatures between the sample and the environment. This hysteretic behavior indicates a gradual decrease in resistivity below Tcr, because extended time is required to establish an equilibrium electronic state. In fact, the resistance gradually decreased at fixed temperature below Tcr as shown in Fig. S1 in the supplementary information.

### Absence of structural transition

Single-crystal X-ray diffraction (SC-XRD) measurement was performed to explore the possibility that the transition has a structural origin, because some layered materials exhibit CDW transition or lattice distortion22,23,24, which can change the transport property. Figure 2a,b show the SC-XRD patterns measured at 50 and 20 K, i.e., above and below Tcr = 35 K, respectively. As shown in the figure, the two patterns show no significant difference. Therefore, the lattice parameters were calculated using the same rhombohedral structure of the space group: $$R\overline{3 }m$$, which has the same structure as non-doped Bi2Se3. Figure 2c,d show the temperature dependence of the lattice parameters a and c measured during the cooling and heating processes. Within the scattering of the data, neither CDW superlattice nor significant change in the lattice parameters was observed at Tcr, however a smooth decrease in the two parameters was observed as temperature decreased.

Our previous experiment of Ag-doped Bi2Se3 under the pressure showed that the application of pressure induced the superconductivity owing to the structural transition at 8.8 GPa18. With increasing pressure from 0.18 to 8.8 GPa, the lattice parameters a and c decreased from 4.12 to 3.95 Å and from 28.6 to 27.2 Å, respectively, in which the variation of a and c were ∆a =  − 0.17 Å and ∆c =  − 1.4 Å. In this pressure range, the resistance increased by a factor of 4 with decreasing a and c. If the lattice shrinkage at low temperatures causes the same effect as physical pressure, the resistance should increase at low temperatures. Furthermore, the variation in a and c with decreasing temperature shown in Fig. 2 are ∆a ~  − 0.01 Å and ∆c ~  − 0.2 Å, which are much smaller than those with increasing pressure. Therefore, the abrupt change in the transport property at Tcr is ascribed to neither the structural transition nor the lattice shrinkage, assuming that the same order of variation of a and c is expected for the SM transition and the superconducting transition. A more detailed analysis of the crystal structure was difficult because of the twin structure of Ag-doped Bi2Se3, in which the twinning axis b is shared.

### Increase in carrier density and Hall mobility below Tcr

Experimental setup for the Hall effect measurement is shown in Fig. 3a. The transverse voltage Vy was measured to obtain Ryx (= Vy/Ix). The carrier density ns and Hall mobility μH was analyzed from Ryx, which is linear to the magnetic field in the range from − 1 to 1 T (see Fig. S2 in the supplementary information). Figure 3b–d show the temperature dependence of ρs, ns, and μH, respectively, obtained from two devices, #1 and #2. The thicknesses of the AgxBi2Se3 flakes were 82 and 100 nm for #1 and #2, respectively. Part of the data from #1 was used in our previous study19. The measurements were performed at the heating process after waiting enough time at the lowest temperature. The S-M transition at Tcr = 35 K was reproduced in two devices, as shown in Fig. 3b. Figure 3c,d show that decrease in ρs below Tcr is accompanied by an increase in both the carrier concentration and Hall mobility. As shown in Fig. 3c, the carrier concentration decreases with decreasing temperature, whereas it increases below Tcr. Above Tcr, ns decreases almost linearly with decreasing temperature, indicating that the carrier is thermally excited and EF is located near the bottom of the BCB. The minimum carrier concentration at Tcr, $$3.8\times {10}^{13} \;\mathrm{ c}{\mathrm{m}}^{-2}$$, exceeds the carrier concentration at the surface states of Bi2Se3, $$1\times {10}^{13} \; \mathrm{ c}{\mathrm{m}}^{-2}$$25,26. Thus, the dominant carrier arises from the bulk rather than from the surface alone. On the contrary, the Hall mobility is almost constant above Tcr, whereas it abruptly increases below Tcr, as shown in Fig. 3d.

### Position of the Fermi level

Here, the Fermi level was evaluated from the temperature dependence of the carrier density per unit area $${n}_{\mathrm{s}}(T)$$ shown in Fig. 3c. The total carrier density $${n}_{\mathrm{s}}(T)$$ consists of carriers at the surface $${n}_{\mathrm{s}(\mathrm{surf})}$$ and those in the bulk $${n}_{\mathrm{s}(\mathrm{bulk})}$$, in which only $${n}_{\mathrm{s}(\mathrm{bulk})}$$ is assumed to depend on $$T$$; $${n}_{\mathrm{s}}\left(T\right)={n}_{\mathrm{s}(\mathrm{surf})}+{n}_{\mathrm{s}(\mathrm{bulk})}(T)$$. The value of $${n}_{\mathrm{s}(\mathrm{bulk})}(T)$$ is expressed as follows:

$${n}_{\mathrm{s}(\mathrm{bulk})}\left(T\right)={\int }_{0}^{\infty }\frac{D(E)dE}{\mathrm{exp}\left[\left(E-{E}_{\mathrm{F}}\right)/{k}_{\mathrm{B}}T\right]+1} ,$$
(1)

where $$E$$ is measured from the BCB minimum, and $$D(E)\equiv a{E}^{1/2}$$ represents the density of states based on the 3-dimensional free electron model. Substituting $$\varepsilon =E/{E}_{\mathrm{F}}$$ and t $$={k}_{\mathrm{B}}T/{E}_{\mathrm{F}}$$, $${n}_{\mathrm{s}(\mathrm{bulk})}(T)$$ can be simplified as follows:

\begin{aligned} {n}_{\mathrm{s}(\mathrm{bulk})}\left(T\right) & =a{E}_{\mathrm{F}}^\frac{3}{2}{\int }_{0}^{\infty }\frac{{\varepsilon }^\frac{1}{2}d\varepsilon }{\mathrm{exp}\left[\frac{\varepsilon -1}{t}\right]+1} \\ & \equiv a{E}_{\mathrm{F}}^\frac{3}{2}f\left(t\right).\end{aligned}
(2)

Thus, when $${n}_{\mathrm{s}(\mathrm{bulk})}\left(T\right)$$ is normalized by $${n}_{\mathrm{s}(\mathrm{bulk})}\left(0\right)$$, it is expressed by the universal function $$f\left(t\right)/f\left(0\right)$$. Two datasets of $${n}_{\mathrm{s}}\left(T\right)$$ for 82 and 100 nm thick crystals in Fig. 3c were transformed by adjusting the parameter $${n}_{\mathrm{s}(\mathrm{surf})}$$, so that the values

$$\frac{{n}_{\mathrm{s}}\left(T\right)-{n}_{\mathrm{s}(\mathrm{surf})}}{{n}_{\mathrm{s}}\left(0\right)-{n}_{\mathrm{s}(\mathrm{surf})}}\left(=\frac{{n}_{\mathrm{s}(\mathrm{bulk})}\left(T\right)}{{n}_{\mathrm{s}(\mathrm{bulk})}\left(0\right)}=\frac{f\left(t\right)}{f\left(0\right)}\right)$$
(3)

show the same temperature dependence above $$T>{T}_{\mathrm{cr}}$$. The values of $${n}_{\mathrm{s}}\left(0\right)$$ were evaluated by extrapolating each $${n}_{\mathrm{s}}\left(T\right)$$ curve at $$T>{T}_{\mathrm{cr}}$$ to $$T=0$$. Using two fitting parameters, $${n}_{\mathrm{s}(\mathrm{surf})}=1.13\times {10}^{13} \; {\mathrm{cm}}^{-2}$$ and $${E}_{\mathrm{F}}=23.5 \;\; \mathrm{meV}$$, the two data sets of $${n}_{\mathrm{s}}\left(T\right)$$ in Fig. 3c were made to fall on the common black curve, $$f\left(t\right)/f\left(0\right)$$, as shown in Fig. 3e. The above value of $${n}_{\mathrm{s}(\mathrm{surf})}$$ agrees fairly well with the literature value of the maximum electron density at the surface states, $$1\times {10}^{13} \; {\mathrm{cm}}^{-2}$$25,26, indicating the validity of this model. Furthermore, the small $${E}_{\mathrm{F}}$$ of 23.5 meV evaluated by the above model indicates that the $${E}_{\mathrm{F}}$$ is located immediately above the BCB minimum at $$T>{T}_{\mathrm{cr}}$$.

In this section, we discussed temperature dependence of ns shown in Fig. 3c. Furthermore, we can compare temperature dependence of the sheet conductivity σs of the two samples using Fig. 3b. The difference of the sheet conductivity ∆σs approximately gives the bulk conduction alone, which increases below Tcr as shown in Fig. S3a in the supplementary information.

### Different field-effect mobility at positive and negative gate voltages

The carrier mobility was also evaluated from electric-field effect measurements to distinguish between bulk and surface conduction. The Hall effect reflects the entire conduction in the bulk and the surface of the material, whereas the electric field effect is more sensitive to surface conduction. This is because the carrier density only at the bottom surface is modulated by Vg. Based on the two-carrier model, the total conductivity consists of the bulk and the two surfaces at top and bottom: $${\sigma }_{\mathrm{s}}={\sigma }_{\mathrm{s}(\mathrm{top})}+{\sigma }_{\mathrm{s}(\mathrm{bulk})}+{\sigma }_{\mathrm{s}(\mathrm{bottom})}$$. Here, the conductivity is the product of carrier density and mobility: $${\sigma }_{\mathrm{s}\left(i\right)}={n}_{\mathrm{s}(i)}e{\mu }_{(i)}$$ (i = top, bulk, or bottom). When applying Vg, the $${\sigma }_{\mathrm{s}}$$ changes as $$\frac{d{\sigma }_{\mathrm{s}}}{d{V}_{\mathrm{g}}}=\frac{d{\sigma }_{\mathrm{s}(\mathrm{top})}}{d{V}_{\mathrm{g}}}+\frac{d{\sigma }_{\mathrm{s}(\mathrm{bulk})}}{d{V}_{\mathrm{g}}}+\frac{d{\sigma }_{\mathrm{s}(\mathrm{bottom})}}{d{V}_{\mathrm{g}}}\sim \frac{d{\sigma }_{\mathrm{s}(\mathrm{bottom})}}{d{V}_{\mathrm{g}}}={C}_{\mathrm{o}}{\mu }_{(\mathrm{bottom})}$$, where $${C}_{\mathrm{o}}(=11.5 \; \mathrm{ nF } \; {\mathrm{cm}}^{-2})$$ represents the capacitance per unit area for a gate insulator of SiO2 300 nm thick. Thus, we can discuss $${\mu }_{(\mathrm{bottom})}$$ from $$\frac{d{\sigma }_{\mathrm{s}}}{d{V}_{\mathrm{g}}}$$ regardless of the magnitude of the total $${\sigma }_{\mathrm{s}}$$.

Experimental setup for the electric-field effect measurement is shown in Fig. 4a. The $${\sigma }_{\mathrm{s}}({V}_{\mathrm{g}})$$ curves at different temperatures for #2 are shown in Fig. 4b. The curves are offset for clarity, so that the values of $${\sigma }_{\mathrm{s}}(0)$$ are shifted by 0.05 mS. The inverses of $${\sigma }_{\mathrm{s}}(0)$$ (or $${\rho }_{\mathrm{s}}(0)$$) are indicated as a function of T, as shown in Fig. 3b. The slope of all $${\sigma }_{\mathrm{s}}({V}_{\mathrm{g}})$$ curves, $$\mathrm{d}{\sigma }_{\mathrm{s}}/{\mathrm{d}V}_{\mathrm{g}}$$, is positive, indicating that the charge carrier is n-type, which is in agreement with the result of the Hall effect. In this study, the inflection of the curves around the zero gate voltage is noticed. Because the slope of $${\sigma }_{\mathrm{s}}({V}_{\mathrm{g}})$$ curve is proportional to the field-effect mobility, $${\mu }_{\mathrm{FE}}$$, the inflection of the curve implies that $${\mu }_{\mathrm{FE}}$$ is larger at $${V}_{\mathrm{g}}<{V}_{0}$$ than at $${V}_{\mathrm{g}}>{V}_{0}$$, where $${V}_{0}$$ is an inflection point. The $${V}_{\mathrm{g}}$$-dependent mobility is responsible for Ag impurities because such a peculiar $${\sigma }_{\mathrm{s}}({V}_{\mathrm{g}})$$ curve has not been reported in non-doped Bi2Se3. To evaluate the data quantitatively, they were fitted using the following equation

$${\sigma }_{\mathrm{s}}\left({V}_{\mathrm{g}}\right)={-a}_{1}\left|{V}_{\mathrm{g}}-{V}_{0}\right|+{a}_{2}{V}_{\mathrm{g}}+{a}_{3},$$
(4)

where ai (i = 1…3), and V0 represent the fitting parameters. The values of $${\mu }_{\mathrm{FE}}$$ were evaluated by $${{\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} < {V}_{0})}\equiv (a}_{1}+{a}_{2})/{C}_{\mathrm{o}}$$ at $${V}_{\mathrm{g}}<{V}_{0}$$ and $${{\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} > {V}_{0})}\equiv (-a}_{1}+{a}_{2})/{C}_{\mathrm{o}}$$ at $${V}_{\mathrm{g}}>{V}_{0}$$. The temperature dependences of $${\mu }_{\mathrm{FE}}$$ and $${V}_{0}$$ are shown in Fig. 4c,d, respectively. Note that $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} < {V}_{0})}$$ is larger than $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} > {V}_{0})}$$ over the entire temperature range. In particular, $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} < {V}_{0})}$$ significantly increases with decreasing temperature above $${T}_{\mathrm{cr}}$$, which is in contrast to the almost constant $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} > {V}_{0})}$$ and $${\mu }_{\mathrm{H}}$$ above $${T}_{\mathrm{cr}}$$. The inflection point V0 changed from positive to negative with decreasing temperature. As discussed later, the results are explained by the different mobilities at the surface and bulk states.

### Difference between three kinds of mobilities

When EF is adjacent to the BCB minimum, the gate voltage has a significant impact on the electronic state, which can mean the difference between $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} < {V}_{0})}$$ and $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} > {V}_{0})}$$. As shown in the band diagrams in Fig. 4a, electrons with sheet density of $$n\left(\equiv -{C}_{\mathrm{o}}{V}_{\mathrm{g}}/e\right)$$ are removed by applying a negative Vg, whereas those of $$n\left(\equiv {C}_{\mathrm{o}}{V}_{\mathrm{g}}/e\right)$$ accumulated by applying a positive Vg. In general, the depletion depth xd is much longer than the accumulation depth xa27. The existence of the surface state may narrow the difference between $${x}_{\mathrm{d}}$$ and $${x}_{\mathrm{a}}$$. However, because the surface state is too thin to shield the electric field completely, the relation of $${x}_{\mathrm{d}}>{x}_{\mathrm{a}}$$ will still hold.

To explain the different field-effect mobility at $${V}_{\mathrm{g}}<{V}_{0}$$ and $${V}_{\mathrm{g}}>{V}_{0}$$, a two-layer model is considered; the surface layer of thickness $${d}_{\mathrm{s}}$$ is related to the surface mobility of $${\mu }_{\mathrm{s}}\left(={\mu }_{(\mathrm{bottom})}\right)$$, and the other region to the bulk mobility of $${\mu }_{\mathrm{b}} (={\mu }_{\left(\mathrm{bulk}\right)})$$. Assuming the length relation $${x}_{\mathrm{a}}<{d}_{\mathrm{s}}<{x}_{\mathrm{d}}$$, the sheet conductivity approximately changes with Vg as follows:

\begin{aligned} {\sigma }_{\mathrm{s}}^{\left({V}_{\mathrm{g}} < {V}_{0}\right)}\left({V}_{\mathrm{g}}\right)& ={\sigma }_{\mathrm{s}}\left(0\right)-\left(n\frac{{d}_{\mathrm{s}}}{{x}_{\mathrm{d}}}\right)e{\mu }_{\mathrm{s}}-\left(n\frac{{x}_{\mathrm{d}}-{d}_{\mathrm{s}}}{{x}_{\mathrm{d}}}\right)e{\mu }_{\mathrm{b}} \\ & ={\sigma }_{\mathrm{s}}\left(0\right)+\left\{\frac{{d}_{\mathrm{s}}}{{x}_{\mathrm{d}}}{\mu }_{\mathrm{s}}+\left(1-\frac{{d}_{\mathrm{s}}}{{x}_{\mathrm{d}}}\right){\mu }_{\mathrm{b}}\right\}{C}_{\mathrm{o}}{V}_{\mathrm{g}} \;\; \text{at} \;\; {V}_{\mathrm{g}}<{V}_{0}, \;\; \mathrm{ and} \;\; \end{aligned}
(5)
$${\sigma }_{\mathrm{s}}^{\left({V}_{\mathrm{g}} > {V}_{0}\right)}\left({V}_{\mathrm{g}}\right)={\sigma }_{\mathrm{s}}\left(0\right)+ne{\mu }_{\mathrm{s}}={\sigma }_{\mathrm{s}}\left(0\right)+{\mu }_{\mathrm{s}}{C}_{\mathrm{o}}{V}_{\mathrm{g}} \;\; \text{at} \;\; {V}_{\mathrm{g}}>{V}_{0}.$$
(6)

Here, the effective capacitance at the interface may be smaller than Co at Vg < V0 due to the capacitance of the depletion layer connected in series27. This effect was neglected for simplicity, because the reduction of the effective capacitance decreases the slope of the $$\sigma \left({V}_{\mathrm{g}}\right)$$ curves at Vg < V0, which is opposite to the experimental result. From Eqs. (5) and (6), the field-effect mobility $${\mu }_{\mathrm{FE}}\left(\equiv \frac{1}{{C}_{o}}\left|\frac{d\sigma }{d{V}_{\mathrm{g}}}\right|\right)$$ is given by

$${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} <{ V}_{0})}=\frac{{d}_{\mathrm{s}}}{{x}_{\mathrm{d}}}{\mu }_{\mathrm{s}}+\left(1-\frac{{d}_{\mathrm{s}}}{{x}_{\mathrm{d}}}\right){\mu }_{\mathrm{b}} \;\; \text{at} \;\; {V}_{\mathrm{g}}<{V}_{0}, \;\; \text{and}$$
(7)
$${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} > {V}_{0})}={\mu }_{\mathrm{s}} \;\; \text{at} \;\; {V}_{\mathrm{g}}>{V}_{0}.$$
(8)

As shown in Eqs. (7) and (8), $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} < {V}_{0})}$$ obtained by removing the electrons within the depletion depth provides helpful information on the mobility in the bulk, compared with $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} > {V}_{0})}$$ evaluated in the accumulation region. The fact that $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} < {V}_{0})}> {\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} > {V}_{0})}$$ indicates that the electron mobility is higher in the bulk than at the surface. This conclusion is also supported by the comparison between $${\mu }_{\mathrm{H}}$$ and $${\mu }_{\mathrm{FE}}$$. As shown in Figs. 3d and 4c, an inequality, $${\mu }_{\mathrm{H}}>{\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} < {V}_{0})}> {\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} > {V}_{0})}$$ holds at each temperature. This magnitude relation between three mobilities shows that the bulk mobility is larger than the surface mobility because $${\mu }_{\mathrm{H}}$$ and $${\mu }_{\mathrm{FE}}$$ reflect the conduction in the entire film and at the surface, respectively. In addition, it has been reported that the surface mobility is lower than the bulk mobility in undoped Bi2Se328.

Owing to the assumption that $${\mu }_{\mathrm{b}}>{\mu }_{\mathrm{s}}$$, $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} <{ V}_{0})}$$ is always larger than $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} > {V}_{0})}$$. Furthermore, this simple model can explain not only the different field-effect mobilities, but also the temperature dependence of $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} < {V}_{0})}$$ shown in Fig. 4c. With decreasing temperature, $${x}_{\mathrm{d}}$$ is enhanced because the thermally excited carrier density $${n}_{\mathrm{s}}$$ decreases24. As a result, $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} < {V}_{0})}$$ approaches $${\mu }_{\mathrm{b}}$$ from Eq. (7) as shown in Fig. 4c. The slight decrease in $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} < {V}_{0})}$$ below Tcr (from 35 to 20 K, see Fig. 4c), which is contrary to the behavior of $${\mu }_{\mathrm{H}}$$ in Fig. 3d, is qualitatively explained by the increase in the carrier density ns indicated in Fig. 3c. Then, $${x}_{\mathrm{d}}$$ decreases in turn, and $${\mu }_{\mathrm{FE}}^{{(V}_{\mathrm{g}} <{ V}_{0})}$$ decreases according to Eq. (7).

Finally, we discuss the possibility that the SM transition is induced by the band bending at the interface. The inflection point $${V}_{0}$$, which separates the depletion and accumulation regions, is the gate voltage required to flatten the band at the interface (see Fig. 4a). The electron density accumulated at the bottom surface without Vg was estimated as $${{n}_{0}\equiv -C}_{\mathrm{o}}{V}_{0}/e$$ for negative $${V}_{0}$$. As shown in Fig. 4d, $${V}_{0}$$ smoothly varies from + 2.5 to − 10 V with decreasing temperature. That is, the temperature dependence of $${n}_{0}$$ has no singularity at Tcr. The increase in $${n}_{0}$$ from 300 to 25 K is maximum $$9\times {10}^{11} \; {\mathrm{cm}}^{-2}$$, which is much smaller than that of $${n}_{\mathrm{s}}$$ $$(>2\times {10}^{13} \; {\mathrm{ cm}}^{-2})$$ shown in Fig. 3c. Thus, the change in the surface carrier due to the band bending does not affect the transition at Tcr, which suggests that the bottom surface is not related to the transition.

### Roles of Ag atoms for carrier doping

In this section, the roles of Ag dopant in AgxBi2Se3 for the doping of the carriers and the level of the EF are discussed. The EF in AgxBi2Se3 is fixed at around the bottom of the BCB despite of x in the range from 0.05 to 0.219, which we call the pinning of EF. To pin the EF at the same level, the carrier density must be fixed even if the Ag impurity atoms induce carriers in Bi2Se3. When a monovalent Ag atom is substituted for a trivalent Bi atom, two holes are accumulated, or two excess electrons are compensated. On the contrary, when an Ag atom is intercalated into the van der Waals gap between the Se-Se layers, one electron is supplied from the Ag atom. When x is small, excess electrons in Bi2Se3 in the BCB can be removed by the substitution of Ag for Bi. The substitution of Bi with Ag was supported by the observation with X-ray fluorescence holography18. However, once the EF reaches the bottom of the BCB (this can occur at x = 7.2 × 10−4, by assuming a typical electron density of 1 × 1019 cm−3 in Bi2Se3), further electrons must be removed from the BVB, requiring additional energy of 0.3 eV2 owing to the existence of the bulk band gap. The value of x = 0.05 in the sample of this study, is sufficient for the $${E}_{\mathrm{F}}$$ to reach the bottom of the BCB. The fact that the carrier is n-type even at x = 0.05 suggests that a significant portion of Ag atoms are intercalated to provide an electron. Thus, the number of impurity sites occupied by Ag atoms may be determined so that the orbital energy of electrons in the BCB can be minimized. When the ratio of the substituted and intercalated Ag atoms is 1:2, the carrier density is preserved regardless of the value of x. The detailed balance between substitution and intercalation may pin the EF around the bottom of the BCB. In fact, recently both substituted and intercalated Ag atoms have been observed with both X-ray photoemission and fluorescence holographies29.

### Possible origins for the S-M transition

Here, we present two possible mechanisms which can induce the S-M transition. One originates from the bulk state, and the other from the top surface sate. The obtained results from transport measurements are summarized as follows: (i) semiconducting behavior of $${\upsigma }_{\mathrm{s}}$$ above Tcr is due to the bulk (Figs. 1b and S3a), (ii) increase in ns below Tcr exceeds the surface carrier concentration. (Fig. 3c), (iii) carrier mobility is larger in the bulk than at the bottom at all temperatures (Fig. 4c), and (iv) temperature dependence of $${\upsigma }_{\mathrm{s}}$$ is qualitatively similar to that of the bulk (Fig. S3a). These results suggest that the transition originates from the change in the bulk electronic state, but the possibility that the top surface contributes to $${\upsigma }_{\mathrm{s}}$$ below Tcr cannot be excluded.

Assuming the pinning of the EF at T > Tcr, the S-M transition can be explained by the depinning of EF and the increase in ns at T < Tcr, as shown in Fig. 3c. Figure 5 illustrates the scenarios which explain the depinning of EF. Figure 5b shows the change in the energy relation between the EF and the bottom of the BCB. For example, the shrinkage of the crystal lattice at low temperatures may expand the bandwidth to lower the BCB under EF. However, the reduction of the lattice parameters under the pressure increased the resistance18, in contrast to our result. Band splitting owing to electron correlation below Tcr (see Fig. 5c) could also enhance ns. In this case, the increase in ns comes from the split band with lower energy, and the increase in $${\mu }_{\mathrm{H}}$$ is also possible in the correlated states. If band splitting is accompanied by spin correlation, it may take long time to establish the equilibrium state, which can explain the hysteretic behavior observed at low temperatures. The Rashba effect also could cause the spin–orbit band splitting, which has been observed at the surface of the Bi2Se330. The change in the band structure is expected to be observed using the ARPES. The possibility of such novel electronic states will be further explored in the future.

Another mechanism is due to the crossover between the bulk and surface conduction31,32. There was no structural transition at Tcr, and the transition was observed only in ultrathin samples18. These results support the surface effect on the transition, which becomes invisible in a thicker sample. One problem is that the field-effect mobility $${\mu }_{\mathrm{FE}}$$ is lower than the Hall mobility $${\mu }_{\mathrm{H}}$$ below Tcr as shown in Figs. 3d and 4c. This discrepancy can be explained by the different mobility at the top and bottom surfaces. The mobility at the top surface may be larger than the $${\mu }_{\mathrm{FE}}$$ because $${\mu }_{\mathrm{FE}}$$ can be limited by the interfacial scattering due to the substrate. If the mobility at the top surface layer exceeds that in the bulk below Tcr, the transition to high $${\mu }_{\mathrm{H}}$$ states is qualitatively explained by the conduction at the top surface. However, it should be noted that this is not simple crossover between the insulating bulk conduction and the metallic surface conduction, because the bulk conduction becomes metallic below Tcr as seen from Fig. S3a. The significant enhancement of the mobility at the top surface is required.

The two plausible mechanisms for high mobility can be examined by measuring the transport properties at high magnetic fields. The carrier density and the mobility in the bulk or the surface can be distinguished by analyzing the nonlinear Hall effect. Furthermore, if the top surface state gives the high mobility, the Shubnikov-de Haas oscillation will be observed in the resistivity33,34. The mechanism of the high mobility will be further studied in future.

## Conclusion

The transport properties of Ag-doped Bi2Se3 were studied under magnetic and electric fields. Based on the impurity sites, the Ag atom acts as a p or n dopant, causing the Fermi level to be fixed around the bottom of the BCB. The carrier number and carrier mobility are dominated by the bulk region rather than the surface state above Tcr. Below Tcr, the carrier density and mobility significantly increase, suggesting that the S-M transition is caused by the depinning of the EF or the crossover between the bulk and surface conduction. The band splitting is suggested to be the possible origin of the depinning of the EF. The possibility of the surface conduction will be examined by transport measurement at the high magnetic field. These candidates will be investigated in the future. Ag doping is observed to be an effective method for probing the electronic state with low carrier concentration in topological insulators. In particular, the transition to a state with high mobility will attract interest from fundamental research and applications. The abrupt changes in the conductivity and mobility at the specific temperature add a new function to thermoelectric properties of Be2Se3. Furthermore, the inflection of $${\upsigma }_{\mathrm{s}}({V}_{\mathrm{g}})$$ curve will be useful to sense the charge adsorbed at surface. In this way, doping effects on topological insulators are important also in development of novel functional devices.

In the same manner as the sample of this study, AgxBi2Se3, CuxBi2Se3 shows a similar S-M transition. Because of the small ionic radii of the impurity atoms, these materials have two kinds of impurity sites, substituted and intercalated. Note that these different impurity sites have also been observed in SrxBi2Se311,12. If the pinning of $${E}_{\mathrm{F}}$$ owing to the different roles of impurities is essential for the S-M transition, the same S-M transition would be anticipated in SrxBi2Se3. By tuning the $${E}_{\mathrm{F}}$$, the doping method based on dual roles of impurities may pave the way for novel functionalization of topological materials.

## Methods

Bulk crystals of AgxBi2Se3 were grown using a melt-growth method described elsewhere18,19. The samples were prepared from the same bulk crystal used in our previous report19. In this study, the samples are denoted as Ag0.05Bi2Se3 based on the nominal molar ratio of Ag, Bi, and Se mixed. The bulk crystal was exfoliated with adhesive tape, and the ultrathin crystals were transferred onto a SiO2/Si substrate. Using the ultrathin AgxBi2Se3 crystal as a channel material, multi-terminal field-effect transistors (FETs) were prepared. Six electrodes (Cr 5 nm/Au 150–200 nm) were attached to the crystal in the Hall bar geometry as shown in Fig. 1d, and a gate electrode was connected to the Si substrate, allowing a bottom gate voltage Vg to be applied to the crystal through a SiO2 gate insulator. The device was cooled in a 4He cryostat using an Oxford superconducting magnet system. The temperature and magnetic field were controlled using mercury iTC and iPS, respectively. For the Hall effect measurement, longitudinal and transverse voltages (Vx and Vy) were measured by supplying electric current Ix. The current was supplied with a Keithley 220 programmable current source and measured using an Advantest R8240 digital electrometer. The voltage was measured using an Agilent 34420A nanovoltmeter. The longitudinal and transverse resistances (Rxx and Ryx) were determined from dVx/dIx and dVy/dIx, respectively. The resistivity $$\rho$$, sheet resistance $${\rho }_{\mathrm{s}}$$, and sheet conductance $${\sigma }_{\mathrm{s}}$$ were evaluated by $$\rho \equiv {R}_{\mathrm{xx}}wt/l$$, $${\rho }_{\mathrm{s}}\equiv \rho /t$$, and $${\sigma }_{\mathrm{s}}\equiv 1/{\rho }_{\mathrm{s}}$$, where $$l$$, $$w$$, and $$t$$ represent the channel length, width, and thickness, respectively. The transverse resistance $${R}_{\mathrm{yx}}$$ was measured with respect to the perpendicular magnetic field B, and the Hall coefficient $${R}_{\mathrm{H}}$$ was calculated as $${R}_{\mathrm{H}}\equiv t\frac{d{R}_{\mathrm{yx}}}{dB}$$. RH was negative when the charge carrier was an electron. The electron density per unit area, ns, and Hall mobility, $${\mu }_{\mathrm{H}}$$, were evaluated as $${n}_{\mathrm{s}}=-\frac{t}{{eR}_{\mathrm{H}}}$$ and $${\mu }_{\mathrm{H}}\equiv {\sigma }_{\mathrm{s}}\left|{R}_{\mathrm{H}}\right|/t$$, respectively.

The electric-field effect was studied by applying Vg. The field-effect mobility $${\mu }_{\mathrm{FE}}$$ was calculated as $${\mu }_{\mathrm{FE}}\equiv \frac{1}{{C}_{\mathrm{o}}}\left|\frac{d{\sigma }_{\mathrm{s}}}{d{V}_{\mathrm{g}}}\right|$$. The gate voltage was applied using a Keithley 2635A system source meter. After the transport measurements, the thickness of AgxBi2Se3 was measured using atomic force microscopy (SII technology, SPA400-DFM). To examine the possibility of the structural transition at Tcr, single-crystal X-ray diffraction (SC-XRD) of Ag0.05Bi2Se3 was measured down to 20 K by blowing 4He gas, using synchrotron radiation at BL02B1 of SPring-8. The wavelength was 0.300900 Å.