K. Inagaki and S. Tanda Department of Applied Physics, Hokkaido University, Sapporo 060, Japan K. Sajiki Research Division, Komatsu Ltd., 1200 Manda, Hiratsuka-shi, Kanagawa 254, Japan (Received 16 July 1996; accepted for publication 4 September 1996) Hall coefficients and resistivities of a series of Bi 2+ x Sr 2 − x CuO x single crystals were measured to investigate the relation between the carrier concentration and the disorder of the system in the superconductor-insulator transition. Excimer-laser cutting technique was applied to fabricate the Hall-bar shaped samples in order to determine precisely the transport properties of the single crystals. The Hall coefficients were found to be independent of the temperature, which suggests Coulomb interaction was negligible in the localized regime of Bi 2+ ,Sr 2 _ ,CuO v . We found the Ioffe–Legel condition k pl ～ 1  laid around n 0 ～ 8 × 10 20  cm − 3 in this system, and the carriers in the localized regime were essentially ''metallic.''  © 1996 American Institute of Physics. [S003-6951(96)02946-4] The normal state of the superconducting cuprate can be explained as a two-dimensional doped Mott insulator. The CuO 2 plane becomes metallic when the sufficient amount of carriers are doped on it, that is, the superconductor-insulator (S-I) transition occurs at a certain threshold carrier concen- tration. In the cuprates, however, the doped carriers also in- troduce disorder into the CuO 2 plane. It is well known that the nature of transport phenomena is changed by the disorder of system: in noninteracting two-dimensional electron sys- tem, any amount of disorder can prevent the system from being metallic. 1 Consequently, in investigating the transport phenomena in S-I transition of cuprates, it is necessary to distinguish the effects due to varying of carrier concentration and those of disorder. Bi 2 + xSr2 - xCuO y is an ideal system to observe the S-I transition in cuprates, since it has the properties of lower superconducting transition temperature ( T c ～ 8 K) and aniso- tropic electric conduction (σαb/σς> 10 4 ). 2 This system changes from superconductor to insulator by doping excess Bi and Sr sites into stoichiometric Bi 2 Sr 2 CuO 6 . 3,4 The ex- cess Bi 3+ ions are thought to compensate carriers in the CuO 2 planes. They are also regarded as the source of random potential.5−7 To date, the relation between the carrier concen- tration and the disorder remains unknown. In this letter, we present the experimental results of re- sistivity and Hall coefficient in the S-I transition of Bi 2 + xSr2 – x CuO y single crystals, aiming to investigate the relation between the disorder and the carrier concentration. To determine both of sheet resistance and Hall coefficient precisely, it is necessary to know accurate dimensions of the sample. Recently, excimer laser become available to apply for material processing, such as drilling and cutting. We used laser cutting technique to fabricate the Hall bars of Bi2+rSr2_rCuO x single crystals. We, thereby, obtained ac- curate control of the dimensions of the single crystals. We were able to observe the S-I transition in a series of crystals, by varying the Bi/Sr ratio. We found the relation between the disorder and the carrier concentration in the insulator side of S-I transition of Bi 2 + x Sr 2 − x CuO v . The Bi 2 + x Sr 2 – x CuO x single crystals used in this study were synthesized by the self-flux method. We changed the ratio of starting powders Bi 2 O 3 and SrCO 3 to control Bi/Sr substitution. A composite powder consisting of Bi 2 O 3 , SrCO 3 , and CuO was placed in an MgO crucible and the mixture was reacted in a furnace at 1000 °C for 4 h. The furnace was cooled at the rate of 2 °C/h. All the processes were carried out under standard atmosphere. The resulting crystals were black and slab-shaped. The samples were cut into the shape of the Hall bar by repeating step-to-step laser drilling. The wavelength of the laser we used to fabricate to the Hall bars was 248 nm. The laser passed through an attenuator and a lens ( f = 1 : 300). The beam was focused to a 50×50 μm² spot. The sample was fixed on a computer-controlled x-y stage. The x-y stage was moved along the programmed position with an accuracy of 2 μm. Bi 2+ xSr 2 − x CuO x crystals of 10−200 μm thickness were able to be pierced by 50–1000 shots of laser pulse. Figure 1 shows a picture of the fabricated Hall bar of FIG. 1. A Hall-bar shaped Bi 2+x Sr 2 -x CuO 7 single crystal. The sample was cut by repeating step-by-step laser drilling. Sample size was 2.7 mm in length and 0.5 mm in width. Bi 2 + xSr 2 – x CuO y single crystal. The size of sample shown in Fig. 1 is 2.7 mm in length, and 0.5 mm in width. The sample has two potential terminals and two Hall terminals, the size of which is 0.3×0.4 mm². Sample thickness t was calculated from sample area and density. Weight of the sample was measured by a microbalance (Mettler M3) up to the order of 0.1 µg. The typical thickness was ~10 µm. The resistivity of sample was measured by the standard four probe method. The passing current was less than 100 μA. The voltage drop across the sample was piked up by a nanovoltmeter (Keithley 182). In the Hall measurement, magnetic fields up to 4 T were applied perpendicular to the CuO 2 plane. The voltages across the Hall terminals were measured by the nanovoltmeter. We measured the Hall volt- ages of both direction in order to eliminate the offset voltage at zero field. The passing current through the samples was 5 mA for all Hall measurement. We used a battery-operated current generator to minimize the line noise induced by the power lines. Before we proceed to the experimental results, we briefly discuss the required accuracy of sample dimensions. Magnetoresistance will appear on Hall terminals, if the ter- minals are not positioned on the same equipotential line. When the Hall terminals have misalignment of the length Δl , the voltage drop across the Hall terminals occurs:
where ρ is resistivity, t is thickness, and w is width. Magne- toresistance effect may change ρ in Eq. (1) as much as a few percent. Since the Hall voltage is represented as VH =RHIB/t, we compared it to the voltage drop across the Hall terminals:
In cuprates, the typical resistivity ρ ~ 10 − 6 Ω m, and the Hall coefficient RH~ 10~ 9 C~ 1 m 3 . To make AV/VH compa- rable, thus Δl/wB～10− 3 T− 1 is required. Assuming the magnetic field of a several teslas, and sample width w ～ 1 mm, one should align the Hall terminals with Δ l < 10 − 3 mm to satisfy the above conditions. We attained this accuracy by means of the laser cutting technique described above. We also estimated Δ l by the voltage drop across the Hall probes at zero magnetic field. The electrically-estimated misalign- ment Δl was, however, the order of 10 –2 mm. Typically, △V/VH~ 100 was obtained. Figure 2 shows that the temperature dependence of the resistivity in Bi 2 + x Sr 2 − x CuO v single crystals with x = 0.0 [one is metallic (A) and the other weakly-localized (B)], 0.2, 0.3, 0.4 (hereafter C, D, and E, respectively). It is shown that the resistivities are dependent systematically of the Bi/Sr ra- tio. Samples A and B showed superconducting transition at 8 K, whereas C, D, and E showed no superconducting transi- tion down to 1.8 K. We found that the S-I transition lays samples B and C in between the the observed Bi 2 + x Sr 2 - x CuO x system. We calculated the sheet resistance per CuO 2 plane, considering the separation between CuO 2 planes is 12 Å. The sheet resistance of 1.0 kΩ corresponds to
the resistivity 1.2×10− 4 Ω cm. It is proposed that the superconductor-insulator transition has a universal critical resistance h/4e 2 (6.45 kΩ). 8 The critical sheet resistance ob- tained here was about two times of this value. We evaluated the disorder of the crystals from the re- sidual resistivity. Randomness of a system is characterized by a metallic parameter k Fl , where k F is a fermi wave vector and l is an elastic mean-free path. The metallic parameter kfl is related with residual resistivity:
We obtained k Fl = 4.0 for sample B, 2.5 for sample C, and 1.5 for sample D. This result clearly shows that the Bi/Sr substitution enlarges the degree of disorder of the system. If much Bi/Sr substitution is carried out up to the Ioffe–Legel limit k Fl ~ 1, the carriers cannot extend over the sample due to disorder, hence the sample changes to insulator. Sample B exhibits ln T upturn in the resistivity in the temperature regime from superconducting transition tem- perature to 50 K. In the weak localized regime, the conduc- tivity obeys the following logarithmic form: 1 σ(T)= σ 0 + ap(e 2 /2π 2 ћ )ln T , where α is a parameter of the order of unity, and p is determined as τ i α T − P . We obtained σ 0 =9.24×10~5 Ω~1 and αp(e 2 /2π hence αρ = 0.58. Consequently, it is shown that the behavior of sample B is described in terms of the weak localization. The residual resistivities of samples C, D, and E are larger than the weakly-localized sample B. The carriers in these samples are, hence, thought to affect the stronger disorder due to much Bi/Sr substitution. Instead of the logarithmic behavior, the electric conduction in these sample exhibited the ''variable range hopping'' behavior: σ ☞ exp( − T o /T) α , where α = 1/( d + 1), d is the dimension of system.5 Figure 3 shows the carrier concentration of samples B, C, D, and E, determined as n = 1/RHe , where RH is the Hall coefficient. We found the two characteristics in the Hall co- efficients as follows: the carrier concentration became smaller when much Sr sites were substituted by excess Bi atoms; and the temperature dependence of each sample has
the same behavior. It is surprising that the Hall coefficients are temperature-independent even in the ''insulating'' crys- tals studied here. 9 We suggest that the Coulomb interaction is negligible in the localized regime of Bi 2 + x Sr 2 _ x CuO x , as well as the weak localized Bi 2 Sr 2 CuO, which we previously reported. 7 It is known that the Anderson localization gives no change in the Hall coefficient. 10 On the contrary, Hall coef- ficient can be influenced by the resistivity change, if other mechanism is dominant. The Coulomb interaction gives the correction for the Hall resistance RH , which is proportional to the temperature dependence of resistance: 11 δRH(T)/RH =2δR(T)/R. We observed the temperature independent Hall coefficients in the samples B, C, and D (see Fig. 4). The result of Hall measurement shows that the resistivity rise at the low temperatures should be attributable to the Anderson localization. We demonstrated that Bi 2 + x Sr 2 − x CuO y changes from superconductor to insulator by Bi/Sr substitution. Both of the disorder of the system and the carrier concentration were subject to the Bi/Sr substitution. Figure 5 shows the carrier concentration as a function of the metallic parameter 1/k Fl . By extrapolating the n vs 1/k fl plot, we show the Ioffe— Legel limit k fl ～ 1  lays around n 0 ～ 8 × 10 20  cm − 3 . The av- erage spacing between carriers at n 0 is calculated as 10 Å, corresponding to ～3 times as long as the spacing between Cu sites (3.8 Å). The system may lose metallic behavior at n 0 since the wavefunction cannot extend over the system. In this sense, our result implies that the carrier concentration of samples B, C, D, and E are larger than n 0 and still remain ''metallic,'' even the carriers are localized by the disorder. The authors would like to thank Professor K. Yamaya for useful discussions and Y. Niwatsukino for experimental support. One of the authors (K. I.) wishes to thank the Japan Society for the Promotion of Science for Young Scientists for financial support. This work was partly supported by a Kurata Research Grant.