Advances in Condensed Matter Physics

Volume 2017 (2017), Article ID 5038462, 12 pages

https://doi.org/10.1155/2017/5038462

## Low Temperature Conductivity in -Type Noncompensated Silicon below Insulator-Metal Transition

^{1}Belarusian State University of Informatics and Radioelectronics, P. Browka 6, 220013 Minsk, Belarus^{2}Belarusian State University, Nezalezhnastsi Av. 4, 220030 Minsk, Belarus^{3}National Research Nuclear University (MEPHI), Kashirskoe Highway 31, Moscow 115409, Russia

Correspondence should be addressed to S. L. Prischepa

Received 17 November 2016; Revised 4 January 2017; Accepted 22 January 2017; Published 14 February 2017

Academic Editor: Da-Ren Hang

Copyright © 2017 A. L. Danilyuk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We investigate the transport properties of -type noncompensated silicon below the insulator-metal transition by measuring the electrical and magnetoresistances as a function of temperature for the interval 2–300 K. Experimental data are analyzed taking into account possible simple activation and hopping mechanisms of the conductivity in the presence of two impurity bands, the upper and lower Hubbard bands (UHB and LHB, resp.). We demonstrate that the charge transport develops with decreasing temperature from the band edge activation (110–300 K) to the simple activation with much less energy associated with the activation motion in the UHB (28–90 K). Then, the Mott-type variable range hopping (VRH) with spin dependent hops occurs (5–20 K). Finally, the VRH in the presence of the hard gap (HG) between LHB and UHB (2–4 K) takes place. We propose the empiric expression for the low density of states which involves both the UHB and LHB and takes into account the crossover from the HG regime to the Mott-type VRH with increasing temperature. This allows us to fit the low experimental data with high accuracy.

#### 1. Introduction

Interest in studies of conductivity mechanisms in semiconductor materials, including traditional doped semiconductors near the metal-insulator transition (MIT), does not stop currently [1, 2]. This is due to both the fundamental problems of electron transport in the vicinity of such transition and applied aspects related to the development of highly sensitive sensors of magnetic and electric fields. In particular, it is not yet fully understood mechanisms of low temperature electrical conductivity in doped semiconductors involving multiply charged localized states, mechanisms of positive and negative magnetoresistance (MR), as well as mechanisms of localization and peculiarities of the energy band structures of impurity and localized states. Due to the above the detailed investigation of the conductivity of doped semiconductors near the MIT in a wide temperature range, influence on it the magnetic field is still relevant. Actually, the MIT in three dimensional (3D) system occurs when the Mott criterion is satisfied, where is the critical concentration of the localized states for the MIT and is the effective Bohr radius of an insulated defect center [3]. The validity of this criterion was confirmed in various experiments [4–6]. However, some ambiguity in correct understanding of the temperature dependence of the conductivity and MR near the critical concentration of the localized sites still does present. The main reason for this is related to the competition between various types of hopping conductivity, mechanisms of weak localization, percolation and metallic and/or impurity bands (Hubbard bands) conductivity. Therefore, a series of crossovers could be observed between different types of conductivity in a wide temperature range. This inevitably leads to the need for their very careful consideration during the processing of the experimental data. In particular, with decreasing temperature, the crossover from band edge activation () to hopping () conductivity is observed in doped semiconductors [6].

Hopping is one of the most likely mechanisms that determines the overall conductivity on the insulating side of the MIT. Among various types of hops, the variable range hopping (VRH) is one of the most relevant. The VRH mechanism, in turn, could be classified as the Mott mechanism, for which the density of states (DOS) on the Fermi level is constant, [3] and the Efros-Shklovskii (ES) mechanism, which implies the soft Coulomb gap (CG), [7, 8]. From a practical point of view, at a constant , the Mott-type VRH occurs when the doping concentration is far less than the value, , and one can neglect the Coulomb interaction [9]. The ES mechanism, in turn, dominates when and the CG has to be taken into account [10, 11]. From the literature a few models of crossovers between the Mott and ES hopping are known quite [12–14].

The temperature dependence of the resistivity for both mechanisms is described by the well-known expressionwhere is the preexponential factor and and are the characteristic temperature and exponent, respectively. Last two quantities depend on the mechanism of hopping. In particular, for 3D systems and for the Mott mechanism, the exponent , while for the ES VRH [3, 6]. The exponent can be obtained from the experiment.

Actually, the concept of hopping between disorder-induced localized electronic states near the MIT is a universal feature of disordered Mott systems. In particular, it was successfully applied to describe transport in amorphous/nanocrystalline silicon hybrids [15], semiconductor nanocrystals [16], ruthenate [17], carbon nanotube fibers [18], as well as transport of single (few) donors in a silicon nanoscale transistor [19, 20]. Known models of crossover between different types of VRH are characterized by two different approaches to the calculation of the exponent from (1). The first approach is based on the analysis of the percolation problem [12, 13, 21], while the second deals with the optimization of the exponent using the interpolation expression for the DOS [14, 22, 23].

In general, the percolation approach can be applied when the spatial correlation length for the random potential is much larger than the phase coherence length [24, 25]. This is a very powerful tool for the description of charge transport in disordered systems with localized states, which occurs due to the electron hops from one site to another [26, 27]. It should be noted that the percolation theory was applied to explain the properties not only of doped semiconductors, but also granular metals [28], manganese oxides [29], quantum Hall plateau transistors [30], high- cuprates [31], and so forth. Moreover, the percolation phase was directly observed in vanadium dioxide close to the Mott transition by means of nanoscale X-ray imaging [32].

The second approach is less stringent and leads to a noticeable overestimation of the width of the crossover region. General description of the crossover from the Mott to ES VRH on the basis of rather complex multivariable integral equation, which cannot be solved analytically and needs rather difficult numerical analysis, was proposed in [13]. In [14, 22, 23] the procedure based on the optimization of the exponent in the expression for the hopping probability and using the interpolation expression for the DOS, (here is integer which is equal to 0 for the Mott mechanism and to 2 for the ES VRH), has been proposed. This approach leads to quite simple analytical expressions but, as we mentioned above, is less stringent than the percolation task.

In addition to the crossover from the Mott to ES VRH, there is another, less studied low crossover, between VRH and a simple activation dependence (SAD) [33]. For the SAD the temperature behavior of resistivity is described by (1) but with the exponent . Actually, the SAD may occur for different reasons. At high it could indicate the nearest neighbor hopping (NNH) [23, 34–36], or a band conductivity [6, 37]. At low , however, the probability of both the NNH and band conductivity becomes negligibly small. Thus, the SAD at low is usually associated with a hard gap (HG) in the DOS [38], that is, in a certain energy range . One of the reason of the HG in doped semiconductors could be the Coulomb interaction [39, 40]. The manifestation of these mechanisms depends not only on the concentration of the main impurities , but also on the degree of compensation of the semiconductor, .

Traditional Mott, ES, and NNH mechanisms are based on hops to the empty impurity centers. Therefore, the availability of sufficient number of empty donor cites is important for such kind of hops in -type semiconductor. At low temperatures, this can only be achieved by compensation of semiconductor. Arisen due to compensation charged donors and acceptors create dispersion of the donor energy levels due to their chaotic potential. This dispersion exceeds significantly the exponentially small splitting of levels of neighbor donors caused by the overlap of the wave functions. The characteristic feature of noncompensated semiconductor () is the rapid decrease of the empty (ionized) donors with decreasing temperature. In weakly compensated semiconductors, in a limited range of concentrations (close to the MIT), in addition to the band and hopping conductivity another activation mechanism develops in the temperature dependence of conductivity, conductivity. This mechanism exhibits in the intermediate between band and hopping conductivity temperature interval [41–43]. It is believed that the conductivity involves migration of electrons on the single occupied neutral donors ( states). They have large radius and, consequently, at the intermediate impurity concentrations are strongly overlapped. The result is a wide band of states. This band is an analogue of the upper Hubbard band (UHB), formed in a disordered systems [6]. The decrease in donor impurity concentration leads to a strong narrowing of a band. On the other hand, in the absence of compensation, when conductivity is zero, there are most favorable conditions for conductivity; that is, the concentration of neutral donors is the highest. The increase of the compensation improves significantly conditions for the conductivity and worsens the conductivity.

Electron hopping ( conductivity) may occur not only on free localized but also on occupied sites via the spin dependent transport [44]. In fact, for doped semiconductors, the spin degree of freedom could play a significant role in the electron hops. If, for example, the width of the distribution function of the energies of the localized sites overcomes the Coulomb repulsion between electrons, double occupancy of the site becomes possible, as was first argued by Kurobe and Kamimura [44]. In this case, two types of hops contribute to the transport. The first occurs between singly occupied (SO) and unoccupied (UO) sites, while the second type of hops involves doubly occupied sites (DO) with opposite electron spins. As a result, the spin dependent charge transport could occur in this case. The UO site is considered as a singly ionized site with the elementary charge +. Consequently, SO site has charge , and DO site has charge −. Developing of a quantitative theory of the conductivity faced with very great difficulties. The main difficulty is that in this concentration range the overlap of the wave functions play a role comparable to the Coulomb interaction of electrons with impurities and with each other. In addition, disorder in the distribution of impurities significantly complicates the analysis of experimental data. Therefore, many problems related to the interpretation of the experimental data for doped semiconductors, in which conductivity is observed, are still unclear. In this regard, for doped semiconductors with a low degree of compensation, in which conductivity is manifested, the study of the mechanisms of low temperature conductivity and crossover between and hopping conductivity is a topical problem.

Therefore, the correct interpretation of the experimental data measured in a wide range can be significantly hampered, because, in order to compare the experimental values of the parameters with the calculated ones, it is important to identify and properly describe the transport mechanisms in different temperature intervals.

In this work we performed a thorough investigation of the low temperature () conductivity of -type noncompensated silicon close to the MIT taking into account possible VRH, SAD, and spin dependent mechanisms. We show that above 30 K the conductivity is of the activation type and is carried out in the UHB, while in the range 5–25 K the Mott-type VRH dominates and is accompanied by the spin dependent charge transport. Below 5 K the crossover from the VRH to SAD occurs. We demonstrate that the reason for the SAD in this range is the HG in the DOS. We developed a model of this crossover based on the percolation task, using the expression for DOS, which takes into account the evolution of the DOS from the HG at K to the Mott-type at K. This approach allowed us to fit the experimental dependence with high accuracy.

#### 2. Samples and Experimental Details

Single crystalline -type Si (100) grown by Czochralski method and doped with Sb was used in this work. Samples of rectangular shape with a width of 1 mm and a thickness of 0.5 mm were covered with 6 indium contacts as electric probes using ultrasonic soldering. Two contacts were for current supply, 2 were for Hall measurements, and, finally, last 2 contacts were for voltage measurements. All contacts were ohmic in the whole studied temperature range which was proved by the linear current-voltage characteristics. A dc current of 10 A generated by multimeter Keithley 6430 was used to bias the sample during the resistivity measurements and to measure the voltage drop down to 5 V. For low resistivity samples we used the two-channel nanovoltmeter Keithley 2182A. Samples were inserted into the cryogen free measuring system (Cryogenic Ltd., London) with the superconducting magnet. The system allowed performing measurements in the range between 2 and 300 K in magnetic fields up to kOe. Lakeshore controller allowed 0.1 K/min sweeping rate of temperature during the versus measurements and stabilizing temperature with the accuracy of K during the sweep of the magnetic field or current-voltage characteristics acquisition. Semiconducting GaAs thermometer was calibrated with an accuracy better than 0.1%. On the basis of measurements of the dependence of the Hall effect the Sb concentration was estimated as , which is 3 times less than the critical Mott concentration for Sb in Si [45, 46]. Therefore, our samples are on the insulating side of the MIT. Actually, a series of samples cut from different Si wafers belonging to the same technological set of fabrication were measured. All the data are in nice agreement with each other. Here we present typical results obtained on the investigated samples.

#### 3. Results

In Figure 1 we show the resistivity versus temperature measured in the range 2–300 K at zero magnetic field and at the maximum applied field of 80 kOe. The strong change in (4 orders of magnitude) is evident. The dependence at is linear in the Arrhenius coordinates in the temperature range 28–300 K, as is clearly seen from the inset to Figure 1. Actually, in this temperature interval there are two significantly different activation energies: meV for the range 110–300 K and meV for 28–90 K. Resistivity in the first range is increased by two times, while in the second interval it is increased by only 1.2 times.