Mesoscopic Fluctuations of Conductance in a Depleted Built-in Channel of a MOSFET

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1 ISSN , Semiconductors, 2006, Vol. 40, No. 9, pp Pleiades Publishing, Inc., Original Russian Text B.A. Aronzon, A.S. Vedeneev, A.A. Panferov, V.V. Ryl kov, 2006, published in Fizika i Tekhnika Poluprovodnikov, 2006, Vol. 40, No. 9, pp SEMICONDUCTOR STRUCTURES, INTERFACES, AND SURFACES Mesoscopic Fluctuations of Conductance in a Depleted Built-in Channel of a MOSFET B. A. Aronzon a, A. S. Vedeneev b, A. A. Panferov a, and V. V. Ryl kov a,b^ a Kurchatov Institute Russian Research Center, Moscow, Russia b Institute of Radio Engineering and Electronics, Russian Academy of Sciences, Fryazino, Moscow oblast, Russia ^ rylkov@imp.kiae.ru Received January 11, 2006; accepted for publication January 24, 2006 Abstract Mesoscopic fluctuations of the off-diagonal component R xy of the resistance tensor have been observed in macroscopic Si-MOS structures with a built-in p-channel at T = 77 K under the conditions in which the channel is depleted of free holes. It was found that the fluctuations δr xy are related to transition from the 3D conduction by free holes to 2D-percolation in the fluctuation potential of ionized impurities of the p-doped surface layer depleted due to the field effect. From the analysis of data on δr xy, the correlation length L c of the percolation cluster, which describes the spatial scale of electrical nonuniformity in the structure, is obtained as a function of the gate potential V g. The dependence of L c on V g is well described in terms of the concepts of nonlinear screening of the fluctuation potential by holes and of the percolation nature of the hole transport for L c varying from ~10 nm to ~1 µm. PACS numbers: Qv, b DOI: /S INTRODUCTION Mesoscopic phenomena are inherent to disordered electronic systems of finite size [1]. Considering the problems of electron transport, these phenomena are usually observed in samples of size less than, or on the order of, the scale of resistance self-averaging. In systems with metallic conduction, the phase coherence length of the electron wave function usually stands for this scale [2]. In media with percolation conduction, noncoherent mesoscopic effects can appear [3]. In this case, the specific scale of self-averaging (the electrical inhomogeneity) is determined by the correlation length L c of the percolation cluster [3, 4], and the related phenomena are usually observed when the object size is comparable with this scale. Most of experiments on studies of noncoherent mesoscopic phenomena were performed in the hopping conduction mode in structures of small length, L < L c [5 7]. The conductance of these structures is defined by percolation paths with anomalously low resistance, which, however, do not form an infinite cluster, so at L L c they do not contribute to the electrical conductance of a structure [3]. As shown in [8], noncoherent mesoscopic phenomena can be observed also in macroscopic objects with dimensions considerably exceeding L c. For example, they were found in studies of fluctuations of the offdiagonal component R xy of the resistance tensor of quasi-2d objects in the hopping conduction mode [8]. The mechanism of these fluctuations is related to the fact that, even if the configuration of the transverse (Hall) probes is not asymmetric with respect to longitudinal current, a potential difference appears between these probes (~E x L c, where E x is the electric field along the sample) owing to the inhomogeneity of a percolation cluster over the scale ~L c. In turn, this leads to R xy fluctuations of amplitude ~R xx (L c /L), where R xx is the longitudinal resistance [8], in the conditions when a percolation cluster is rearranged upon a change in the external conditions (the transverse electric field, longitudinal voltage, temperature, etc.). The mesoscopic fluctuations of R xy differ from the fluctuations of the longitudinal resistance R xx [3] in that they are determined by conducting chains forming an infinite cluster; therefore, their analysis opens the way for direct experimental estimation of an important parameter of the percolation system, namely, the correlation length L c [8]. Earlier, we observed similar noncoherent mesoscopic phenomena in systems with hopping conductivity. These are Si MIS (metal insulator semiconductor) transistor structures at liquid-helium temperature, in which the cluster rearrangement was stimulated by varying the gate voltage or longitudinal field [8, 9], and also Fe/SiO 2 nanocomposites, in which the current paths randomly changed under the action of a magnetic field and/or temperature in conditions of a thermally induced metal insulator transition [10, 11]. In this paper, we show that mesoscopic fluctuations of R xy are typical also of disordered systems, in which the electronic transport is affected by free carriers under 1055

2 1056 ARONZON et al. G xx, kω M SiO 2 10 p-si n-si µ the conditions of a strong fluctuation potential. As model samples, transistor Si-MOS (metal oxide semiconductor) structures with the built-in p-channel were chosen, in which the fluctuation potential varies at a depleting potential V g of the gate contact via electron screening of ionized acceptors (sources of the fluctuation potential [12]) and owing to a change in their concentration upon a partial freeze-out of impurities. As a result, a considerable rearrangement of a percolation cluster and related fluctuations of R xy are expected in the studied field-effect mode. 2. RESULTS AND DISCUSSION Transistor Si-MOS structures with a built-in p-channel of L = 150 µm length and W = 50 µm width were produced in the double-cross configuration by planar technology on (100) n-si substrates with the donor concentration N d cm 3 [13]. The Si:B doped surface layer (the built-in channel) of thickness D 0.5 µm was formed by ion implantation of boron and insulated from the p + -polysilicon gate contact by thermal oxide of d = 62 nm in thickness. The acceptor concentration determined from the room-temperature Hall effect measurements was N a cm 3. The diagonal (R xx ) and off-diagonal (R xy ) components of the resistance tensor were studied as functions of V g at a constant longitudinal voltage ( 0.1 V) in the liquid-nitrogen temperature range. Figure 1 shows the dependence of conductance G = 1/R xx of the structure on V g at T = 77 K temperature. The increasing of G at V g < 0 (to be more precise, at V g < V FB, where V FB 0 is the flat-band voltage) is related to E a E v V g > V 2D V FB V 2D Fig. 1. The diagonal component of the conduction tensor G xx as a function of the gate voltage at T = 77 K temperature. Inset: the band diagram of a Si-MOS structure with a built-in p-channel. µ, E a, and E v are the energy positions of the Fermi level, acceptor levels, and the valence band top, respectively. the formation of a hole conduction channel at the Si SiO 2 interface in the accumulation mode; and the decreasing of G at positive V g is related to depletion of the Si:B layer of free holes. In the depletion mode, at V g > 0, the thickness of the depleted layer increases as the gate bias increases, so the conducting channel, which is bounded on the other side by the p n junction, is pinched. As a result, a quasi-2d channel is formed at the boundary of the doped p-layer (see the inset in Fig. 1). In this situation, the doping impurity is completely ionized on both sides of the quasi-2d channel, and the free hole density in the channel decreases as V g increases. It is worth noting that a similar situation occurs in channel depletion in a GaAs FET with the Schottky barrier [12]. As shown in [12], in this case, the behavior of the low-frequency capacitance can be satisfactorily described by the generation of a fluctuation potential under conditions of its nonlinear screening by carriers in the quasi-2d channel. In our case, the transition to conduction along the quasi-2d channel is manifested by a change in the G(V g ) behavior after some threshold value V g = V 2D is reached as the gate voltage increases. At V g < V 2D 8 V, the decreasing of conductance follows a power law, whereas at V g V 2D, the decreasing of conductance with the rise of V g becomes exponential. From the physical point of view, this transition (the transition to conduction along the quasi-2d channel) is due to the fact that the thickness of the conducting channel becomes comparable with the correlation length L c of the percolation cluster. 1 In this case, the fluctuation potential strongly distorts the conduction along the quasi-2d channel by forming pinchedoff regions at the potential maxima, which cannot be bypassed via the third dimension. In this situation, the conductance will be determined by thermal activation of carriers to the percolation level via saddle regions of the fluctuation potential, which accounts for the transition to exponential dependence of G(V g ). The gate voltage corresponding to complete pinching-off of the conduction channel is V g = V t 10 V. Figure 2 shows R xy and the ratio R xy /R xx (upper inset) as functions of the gate voltage. In the range of small depleting voltages, V g < V 2D 8 V, R xy is constant and small (the ratio α = R xy /R xx ), which is indicative of the in-plane uniformity of the channel and the virtually symmetrical arrangement of the transverse potential probes on the sample. In contrast, on passing from the 3D to quasi-2d mode of the hole transport (V g V 2D ), R xy exhibits regular fluctuations, which become more pronounced as V g increases. Following [8], we assume that fluctuations of R xy have mesoscopic origin and are related to reconstruc- 1 It is in this sense that we use the term quasi-2d conduction channel, by analogy with the term 2D-film, used in the description of the longitudinal transport in layers with hopping conductivity with the layers thickness less than L c [4].

3 MESOSCOPIC FLUCTUATIONS OF CONDUCTANCE 1057 R xy, kω R xy /R xx ln(r xx, kω) /T, K Fig. 2. The off-diagonal component of the resistance tensor R xy of a Si-MOS structure as a function of the gate voltage at T = 77 K temperature. Upper inset: the ratio R xy /R xx as a function of the gate voltage. Lower inset: temperature dependence of the sample resistance in the range where fluctuations are observed (V g = 9.6 V). tion of the percolation cluster under the action of the field effect. Indeed, if the fluctuation potential is generated by ionized impurities in the depleted Si:B layer and nonlinearly screened by quasi-2d holes with the density p s, the amplitude of the fluctuation potential can be represented as [14, 15]: N s δϕ = A---- e2 (1) ε p s ( D) A ---- e , ε p s ( D+ d) where N s is the concentration of ionized impurities, reduced to the surface; e, the elementary charge; ε, the dielectric constant of Si; and A, the coefficient on the order of unity. If sources of the fluctuation potential lie between the gate and 2D-channel, A = (2π) 1/2 [14]. It is noteworthy that in this situation, as the density of 2D carriers decreases, the amplitude of the fluctuation potential tends to a constant value δϕ g, which is defined by screening of large-scale fluctuations by the metallic gate of the structure [14]: δϕ g = 4πN s ( 1 ln2) e (2) ε In the structures under study, with the concentration of implanted impurity N a cm 3 N d cm 3, sources of the fluctuation potential are N s mainly situated between the gate and quasi-2d hole channel, so in the estimation of the fluctuation potential amplitude we use the results from [14]. The value of N s at V g V 2D we find by considering the structure capacitance C as a series connection of the insulator capacitance C d and the capacitance of the depletion layer in the semiconductor C s = 2C sd, where C sd κ/4πd is the differential capacitance of the depletion layer. At V g = V t 10 V, we obtain N s = 2C d C sd V t /e(c d + 2C sd ) cm 2. According to (2), this value corresponds to δϕ 29 mev, which significantly exceeds the thermal energy kt. The obtained δϕ is in agreement with the activation energy of conductance (~30 mev) in the range where fluctuations of R xy are observed (see the lower inset in Fig. 2). In this situation, it seems natural to expect that the charge transport is performed by thermally activated holes via saddle regions of the fluctuation potential, which have an exponentially wide, to the extent of variation of the exponent δϕ/kt > 1, scatter of the local resistance [4, 14]. Accordingly, the conduction in the system becomes percolative, with the correlation length of the percolation cluster given by [4, 16]: L c a δϕ kt ν a δϕ = kt (3)

4 1058 ARONZON et al. L c, µm L c, µm Fig. 3. The dependence of L R xy /R xx α on the gate voltage at T = 77 K, which reflects the behavior of the correlation length L c of the percolation cluster. Arrows indicate several local maxima of the dependence. The solid line 1 shows the correlation length L cp (V g ) in the nonlinear screening mode, calculated using Eqs. (1), (3), and (4). The horizontal line 2 is the maximum (calculated) value of the correlation length L cg, related to screening of the fluctuation potential by the gate of structure. Inset: points, the dependence of L c on the gate voltage, obtained by averaging over the regions of local maxima of the L R xy /R xx α values; solid line, the calculated L c (V g ) curve. Here, ν is the critical index in the percolation theory (we set ν = 1 as an average between 2D (ν = 1.33) and 3D (ν = 0.83) situations) and a is the characteristic spatial scale of the random potential, which in our case coincides with the screening length of the fluctuation 1/2 potential [14, 15]. At p s > N s /D, a = N s /Dp 2 s. (4) In the opposite case, when the fluctuation potential is screened by the gate of the structure [14], a = 21 ( ln2)d. (5) According to (1), (3), and (4), we must know, in order to calculate the L c (V g ) = L cp (V g ) dependence corresponding to nonlinear screening of the fluctuation potential, the dependence of hole density in the channel, p s, on the gate potential, which can be found from the data on the field effect (Fig. 1) by the method described in [17]. In the range V 2D < V g < V t, the p s (V g ) dependence is determined by the relation p s C d C sd (V t V g )/e(c d + C sd ). Substituting this p s and the value N s = 2C d C sd V t /e(c d + 2C sd ) cm 2 (V t = 10 V) into (1) and (4) and using Eq. (3), we obtain the L cp (V g ) dependence shown in Fig. 3. This dependence diverges at V g V t, and for p s < N s /D the 1/2 screening length of the fluctuation potential is deter- 2 1 mined by the distance from the gate of the structure. As follows from (2), (3), and (5), in this case L c (V g ) = L cg (V g ) = const (the horizontal line in Fig. 3), and at V g 9.7 V the correlation length L c (V g ) tends to the constant value L c = L cg 1.3 µm. Now we analyze the behavior of the correlation length of a percolation cluster L c on the gate potential V g based on the data obtained in the measurements of fluctuations of the off-diagonal component of the resistance tensor R xy. Fluctuations of R xy were studied earlier in Si-MOS transistor structures [8] in the hopping conductivity mode, in the conditions when the cluster correlation length L c remained unchanged at the cluster reconstruction. It was shown that L c ( ) a L , δr xy 2R xx (6) where (δr xy ) a is the characteristic amplitude of a fluctuation, determined from the difference between the minimum and maximum values of R xy. In the conditions under study, the reconstruction of a cluster leads to a strong variation of L c (Fig. 3), so in the experimental estimation of the correlation length we analyze the envelope of the δr xy /R xx ratio magnitude, using Eq. (6). Figure 3 shows the dependence of L R xy /R xx α on V g, which clearly demonstrates local maxima (some of these are marked by arrows). Averaging L R xy /R xx α over these regions, we find the dependence L c (V g ), represented by points in the inset in Fig. 3. The same inset shows the calculated L c (V g ) dependence obtained from the interpolation relation L c (V g ) = L cp L cg /(L cp + L cg ). As can be seen, this dependence well describes the experimental L c (V g ) curve, which was found from the analysis of fluctuations of the off-diagonal component of the resistance tensor R xy for L c varying in the range from ~10 nm to ~1 µm. 3. CONCLUSIONS In conclusion, it may be said that the presented results can be used as a method for experimental study of the spatial scale of electrical inhomogeneities in a wide class of 2D objects in which the conduction channel is formed in the threshold manner. ACKNOWLEDGMENTS The study was supported by the International Science & Technology Center (grant no. 2503). REFERENCES 1. Y. Imry, Introduction to Mesoscopic Physics (Oxford Univ. Press, Oxford, 1997; Fizmatlit, Moscow, 2002). 2. B. L. Al tshuler and B. Z. Spivak, Pis ma Zh. Éksp. Teor. Fiz. 42, 363 (1985) [JETP Lett. 42, 447 (1985)]; B. L. Al tshuler and D. E. Khmel nitskiœ, Pis ma Zh.

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