Size effect in the electronic transport of thin films of Bi 2 Se 3

Thin films of a topological insulator (TI) Bi2Se3 of various thicknesses from 20 nm to 75 nm were obtained. The resistivity measurements were carried out according to the conventional 4-contact DC technique. This allows to “separate” the bulk and surface conductivities at different temperatures and magnetic fields. It was suggested that similar effects should be observed in other TIs and systems with inhomogeneous distribution of dc-current on sample cross section.


Introduction
The new functional materials with the unique physical properties are needed for spintronic devices.One of such promising materials are the topological insulators (TIs) [1], which have a nontrivial topological band structure, arising from strong spin-orbital interaction [2].The TI has an energy gap and, hence, is the insulator or semiconductor, in a bulk and has the protected gapless conduction states on its surface.A rigid connection between the directions of the momentum and the electron spin leads to the emergence of spin polarization of charge carriers and the possibility of a spin-polarized current flowing near the TI surface with practically no loss [3].This spin-polarized surface current can be used for spintronic devices.
It is known that Bi2Se3 compound is the TI [4, 5] with a metallic conductivity near surface and the gapless semiconductive one [6] in its bulk.Since the electroconductivity value in a bulk and near surface of such materials can differ substantially, it is of interest to "divide" them experimentally.For this, we can use the results of Ref. [7], where we studied the size effect in the conductivity of pure tungsten single crystals under conditions of the static skin effect (SSE) [8], i.e. a predominant flow of direct electric current near a sample surface.The aim of this paper is to search for and study the size effect in the electronic transport of thin films of TI Bi2Se3.

Samples and measurements
Thin films of Bi2Se3 were grown by the molecular beam epitaxy method on Al2O3 substrates [4,5] with thickness from 20 to 75 nm.The XRD data and the atomic content of elements analysis showed the synthesized films have Bi2Se3 composition (see Refs. [4,5]).The atomic content of elements was measured by a scanning electron microscope equipped with an EDAX X-ray microanalysis attachment.Our examination showed that the deviations from a stoichiometric composition were insignificant in all samples.The measurements of the electroresistivity ρ0 and magnetoresistivity ρxx were carried out by the conventional 4-points method at dccurrent in the temperature range from 4.2 to 80 K and in magnetic fields of up to 10 T. The results are presented in units of conductivities σ 0 ≈ 1/ρ0 and σxx ≈ 1/ρxx.

Results and discussion
Schematic view of the thin film of TI is shown in Fig. 1.The electric current I is passed through the film of thickness d and width c (Fig. 1), and the voltage U is measured between the potential leads located at a distance L. In the near-surface layer of thickness δ, a region with high surface conductivity σ sur appears.
Total conductivity of such a system contains two terms, namely, the surface conductivity σ sur and the bulk conductivity σ bulk .It is quite similar to Refs.[7,8], where the dc-current is concentrated near a sample surface due to the static skin effect (SSE) [8].It is quite easy to show that (1) Thus, a linear dependence of the conductivity on the film thickness should be observed (2) The first term in Eq.( 1) is proportional to surface conductivity σ sur and second one is the bulk conductivity σ bulk , i.e. we can "separate" σ sur and σ bulk .To do this the films of different thickness d were synthesized.Fig. 2 presents the experimental results for conductivity of Bi2Se3 films without magnetic field (Fig. 2a) and in a field of 10T at T=4.2 K (Fig. 2b).One can see that there is a linear dependence on d -1 both for σ0 and σxx.It allowed us to separate the bulk and surface conductivities.Taking into account Refs [11], it was assumed that delta is not more than 1 nm*.
Fig. 3a and Fig. 4a show the temperature dependence of σ0 sur and σ0 bulk without field.σ0 sur decreases with temperature as it should be for protected conductive states on TI surface.σ0 bulk also decreases with T as it was observed for Bi2Se3 in Ref. [6] since Bi2Se3 has relatively small gap in its electron energy spectrum at Fermi level.It leads to that fact the scattering of charge carriers begin to play the main role in formation of its conductivity.As a result, σ0 bulk decreases with temperature due to the scattering processes.
Magnetic field should change a type of bulk conductivity, as it should be in semiconductors with small gaps and in gapless semiconductors.Fig. 3b, 4b present the temperature dependence of the magnetoconductivities σxx sur and σxx bulk .It is seen that σxx sur decreases with temperature, and it is typical for systems with metallic states.σxx bulk increases with temperature, which can be explained by changing the character of current carriers scattering for systems in which the scattering processes play the main role in conductivity.
It should be noted that a similar change in a type of temperature dependence for surface and bulk conductivities was observed in Refs.[9,10], where dc-current was concentrated near a samples surface under the SSE [7,8].

Conclusions
The size effect, i.e. a dependence of the conductivity on the film inverse thickness, was observed in thin films of TI Bi2Se3.This allows us to "separate" the bulk and surface conductivities at different temperatures and magnetic fields.In apparently, similar effects should be observed in other TIs and systems with inhomogeneous distribution of dc-current on sample cross section.

Fig. 1 .
Fig. 1.The schematic view of experiment.Where c is the width of the sample; d is the thickness of the sample; δ is the thickness of the near-surface layer; L is the distance between potential contacts, I is dc current, U is measured voltage.σ ≈ σ sur •(δ/d)+σ bulk(1) Thus, a linear dependence of the conductivity on the film thickness should be observedσ = f(d -1 )(2) The first term in Eq.(1) is proportional to surface conductivity σ sur and second one is the bulk conductivity σ bulk , i.e. we can "separate" σ sur and σ bulk .To do this the films of different thickness d were synthesized.Fig.2presents the experimental results for conductivity of Bi2Se3 films without magnetic field (Fig.2a) and in a field of 10T at T=4.2 K (Fig.2b).One can see that there is a linear dependence on d -1 both for σ0 and σxx.It allowed us to separate the bulk and surface conductivities.Taking into account Refs[11], it was assumed that delta is not more than 1 nm*.According to estimations (evaluations) at T = 4.2 K, σ0 bulk ~ 0.5•10 3 and σ0 sur ~ 9.8•10 4 without magnetic field and σxx bulk ~ 0.2•10 3 and σxx sur ~ 4.8•10 4 in a field of 10T.That means the surface conductivity σ sur is almost 200 times higher than σ bulk , σ0 sur , σxx sur >> σ0 bulk , σxx bulk .The obtained results are in good qualitative agreement with the Ref.[6].Fig.3aandFig.4ashow the temperature dependence of σ0 sur and σ0 bulk without field.σ0 sur decreases with temperature as it should be for protected conductive states on TI surface.σ0 bulk also decreases with T as it was observed for Bi2Se3 in Ref.[6] since Bi2Se3 has relatively small gap in its electron energy spectrum at Fermi level.It leads to that fact the scattering of charge carriers begin to play the main role in formation of its conductivity.As a result, σ0 bulk decreases with temperature due to the scattering processes.Magnetic field should change a type of bulk conductivity, as it should be in semiconductors with small gaps and in gapless semiconductors.Fig.3b, 4bpresent the temperature dependence of the magnetoconductivities σxx sur and σxx bulk .It is seen that σxx sur decreases with temperature, and it is typical for systems with metallic states.σxx bulk increases with temperature, which can be explained by changing the character of current carriers scattering for systems in which the scattering processes play the main role in conductivity.It should be noted that a similar change in a type of temperature dependence for surface and bulk conductivities was observed in Refs.[9,10],where dc-

Fig. 2 .
Fig.2.Size effect in the conductivity of thin films of Bi2Se3 in magnetic fields 0T (a) and 10T (b) at 4.2K.