Introduction

The Gauss’s law Q = CV in a field effect transistor (FET) is generally believed to be , where, e, n2D, and VG are the elementary charge, sheet carrier density of the channel, sheet capacitance of the gate insulator and gate voltage, respectively1. The equation is valid, but only when the channel is an ideal metal, where the gate electric field is completely screened (zero screening length) at the channel surface due to the infinite charge compressibility (μ is the chemical potential). Meanwhile, for finite κ, is replaced by . is called a quantum capacitance2. Total energy of the carriers corresponds to , thus, in general, is positive and . Nevertheless, negative κ, for which , is a long-standing target of research both experimentally3,4,5,6,7,8,9 and theoretically10,11, manifesting itself due to strong exchange interactions between carriers. Especially, in two-dimensional electron system (2DES), the exchange energy is negative and scales as , while the positive (e.g., kinetic) energy scales as n2D; therefore, for sufficiently small n2D, the total energy can be negative. What we demonstrate here is, however, far beyond the classic examples. A quasi-2DES at the channel of SrTiO3 FET shows anomalous enhancement of n2D: ten times as large as . The enhancement cannot be explained only by the exchange interaction, suggesting another mechanism of inducing negative κ.

A schematic cross-section of a standard FET is shown in Fig. 1 with the band diagrams and the relationships between the capacitances following a widely-accepted concept of the accumulation-type metal-oxide-semiconductor FET; thick substrate (channel) of the FET is implicitly grounded in the far distance, which gives zero of the chemical potential. The gate voltage is a sum of the voltage drop in the gate insulator and the band-bending of the channel material . Therefore, VG = Vins + φ means (Fig. 1a). For the metallic channel, the chemical potential (Fermi energy) μ/e substitutes for φ and is replaced by the quantum capacitance (Fig. 1b). It is still possible to consider for the nonmetallic bulk part of the substrate and . But term is usually omitted. The channel material in this study is SrTiO3. It changes from an insulator to metal by gating12, so we use a notation in lieu of both and . SrTiO3 is a classic material for solid-state physics but is a cynosure of modern oxide-electronics researches because of the formation of quasi 2DES at the surface13,14,15 or interface16,17, as well as the large mobility of the confined 2D carriers without a freeze-out18,19,20,21. Both the confinement and the large mobility are originated in or, if not more, influenced by the quantum paraelectricity22 with a large and nonlinear dielectric response23. Furthermore, at the surface and interface, where the inversion symmetry is broken, the charge confinement induces some intriguing electronic properties; for example, the Rashba spin-orbit coupling at the surface of SrTiO3 discussed in refs 24, 25, 26.

Figure 1
figure 1

Schematic pictures of the cross section and the band diagram of FET.

Neither distance nor energy of the picture scales to that of the real device. (a) The channel is an n-type non-metallic material (e.g., non-doped SrTiO3). By differentiating VG = Vins + φ with respect to n2D and by using the Gauss’s law, we obtain . (b) For larger VG, the channel becomes metallic and VG = Vins + μ/e. Same as (a) the relationship is obtained, where is called a quantum capacitance (ref. 2). becomes larger than only when is negative.

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However, it is intensely difficult to fabricate such a high quality FET on SrTiO3 as to reveal the true nature of the exotic phenomena. The band gap of SrTiO3 is nearly 3.2 eV27,28, but it turns to be a good metal by a very tiny electron doping of 8.5 × 1015 cm−3 (corresponding to the removal of a few oxygen atoms out of 107), which is orders of magnitude lower than the threshold of metallicity in Si (3.5 × 1018 cm−3) or Ge (3.5 × 1017 cm−3)29. Thus, the channel of SrTiO3 FET becomes conductive quite easily by the oxygen-defect formation. In other words, the channel current of some SrTiO3 FETs might be rather dominated by electrochemical reaction than purely electrostatic carrier-density modulation30. Therefore, in this paper, we propose an alternative gate insulator: an organic/inorganic bilayer consisting of 6 nm ultra-thin poly-monochloro-para-xylylene (Parylene-C) and 20 nm HfO2, as schematically shown in Fig. 2a. The film of Parylene-C polymer is widely used for coating a variety of material surfaces, because it is highly conformal, pin-hole free, quite inert to any gases and chemicals and sufficiently stable from around 200 °C down to at least 60 mK31. The bilayer gate insulator was deposited on the atomically-flat (100) surface (miscut-angle is less than about 0.03°) of non-doped SrTiO3 single crystals provided by Shinkosha Co., Ltd. The photos of our FETs are shown in Fig. 2 and the cross-section images obtained by the transmission electron microscopy (TEM) are shown in Fig. 3a–d. The step and terrace surface of our SrTiO3 crystals is noticeably insulating with the sheet resistance above our instrumental limit (~1013 Ω) at room temperature and this surface is kept sufficiently insulating after the fabrication process of FET.

Figure 2
figure 2

FET with HfO2 (20 nm)/Parylene-C (6 nm) solid-state bilayer gate insulator studied in this work.

(a) Schematic cross-section image of our three-terminal FET device. (b) Photograph of a 10 mm × 10 mm (100) SrTiO3 substrate with the FET devices fabricated on it. Scanning electron microscopy images of (c) a three-terminal FET device and (d) multi-terminal one. G, S and D stand for gate, source and drain electrodes, respectively. V1–V6 are potential probes.

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Figure 3
figure 3

Characteristics of SrTiO3 FET with HfO2 (20 nm)/Parylene-C (6 nm) bilayer gate insulator.

(a) Cross sectional TEM image of the channel. (b) Cross sectional scanning TEM (STEM) image near the Al electrode (dotted line is a guide to eyes separating Parylene-C and Al). (c) Energy-dispersive x-ray spectroscopy mapping for Hf atom and (d) that for Sr atom. (e) ISD − VSD plots for 3-terminal device with L = 20 μm and W = 80 μm for several VG. (f) for four FETs with different sizes but fixed W/L ratio plotted as a function of L (open circles). Solid line is the least-square fit (WRexp = RO + LR2D) to deduce R2D. (g) R2D vs. plot gives S of 171 mV/decade.

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The high quality of our FET manifested itself in the subthreshold behaviour. Figure 3e shows ISD − VSD plot for fixed VG for the three-terminal device (See Fig. 2c). ISD ∝ VSD for small VG, but it shows upward convex for , which is called the threshold voltage Vth and the region VSD < Vth is dubbed as the subthreshold region. As shown in the ISD − VG curves in Fig. S1 of the Supplementary Materials, it is seen in the subthreshold region that log10 ISD ∝ VG. Indeed, with φ in Fig. 1a, ISD ∝ exp(eφ/kBT), where T is the temperature and kB is the Boltzmann constant, because only the thermally activated carriers of the valence band can contribute to the transport. Since , the subthreshold swing defined as can be expressed as . n(T) is denoted as transport factor and m as body factor32,33. By measuring S, we can obtain ratio.

S is in general estimated from the ISD − VG plot, but it might include a significant contribution of the contact resistance. Therefore, we deduced S in a more comprehensive manner. We carried out ISD − VG measurements in the subthreshold region with fixed VSD = 1 V for the devices with L = 2, 4 and 9 μm and W = 4L. Then, WRexp was plotted against L (Fig. 3f). Here, Rexp = VSD/ISD at . is the cut-off gate voltage, below which ISD is smaller than the noise level of 100 fA (see Supplementary Materials for details). By comparing WRexp with RO + LR2D, the sheet resistance R2D is obtained. We applied this method for dozens of and all the R2D values were plotted in Fig. 3g. We can see log10 R2D is clearly proportional to and deduced S = 171 mV/decade. This is indeed in good coincidence with the values simply estimated from the ISD − VG plot, indicating that the contact resistance of our FET does not contribute to the S value. Since the material-independent transport factor n(T) = (kB T/e)ln10 is 60 mV/decade at 300 K, S = 171 mV/decade of this study is surprisingly small. (It was reported that the value of S was ~100 mV/decade even for Si34, ~250 mV/decade for SrTiO335 and ~1200 mV/decade for KTaO336). From m = 2.8, we deduced . If we assume the dielectric constant of SrTiO3 is 310 at room temperature, the effective thickness for is 0.55 μm. This means in the subthreshold region the gate electric field can penetrate into deep bulk of SrTiO3 (0.55 μm) without a large Thomas-Fermi screening of free carriers possibly originated in the defects of Parylene-C/SrTiO3 interface. Put plainly, the Parylene-C passivation on the defect-prone SrTiO3 surface37,38 works fairly well. This is one of the two important premisses of this study.

The other premise is that the ultra-thin Parylene-C film works not only as a passivation layer protecting SrTiO3 channel from the high-k dielectric HfO2 but also works as a good capacitive layer by itself. We fabricated Ti(10 nm)/Parylene-C (3 nm)/HfO2 (20 nm)/Ti(5 nm)/Au(500 nm) parallel plate capacitors and scrutinised the capacitance by both quasi-static and ac measurement. Details are given in the Supplementary Materials. The deduced sheet capacitance of the gate insulator of our FET, Parylene-C (6 nm)/HfO2 (20 nm), is as well as the dielectric constants of 21.5 and 2.70 for the HfO2 layer and the Parylene-C layer, respectively, consistent to the values of 20 and 3.15 reported in literature. Alternatively, we may also assume the dielectric constants of 20 and 3.15 for HfO2 and Parylene-C, respectively. Then, the film thickness becomes 18.6 nm and 3.5 nm for HfO2 and Parylene-C, respectively, both of which are almost equivalent to the results of TEM.

By using this bilayer gate insulator, we have finally obtained both the fairly clean channel and the continuous electrostatic control of the carrier density on SrTiO3. This achievement, however, has given a new twist to the research of SrTiO3. Figure 4a shows n2D obtained by the Hall effect measurement for the multi-terminal FET device (Fig. 2d: details of the experiments are described in the Supplementary Materials). As mentioned above, n2D = C2D (VG − 1.88)/e, where with and . Thus, n2D = 1.1 × 1012(VG − 1.88) cm−2. (It should be noted here that the 1.88 V offset, above which the accumulation of the carriers in the channel becomes observable by the Hall effect measurements, may be due to the relatively larger contact resistance of the multi-terminal FET device used for the measurements; however, the origin of this offset does not affect to the following discussion). To our surprise, the measured n2D is much larger than this naive estimation; it reaches to around 1 × 1014 cm−2 for VG = 6 V. Even if this extra carriers are provided by the formation of oxygen/cation defects in the SrTiO3 channel during the application of the large VG (though the channel is fairly protected by Parylene-C layer and is actually clean), it should be noted that n2D cannot be modulated without a change of C2D, independent of sources of the carriers.

Figure 4
figure 4

Interpretation of enhancement by a negative κ model.

(a) Top: sheet carrier density n2D (open circles) obtained by the Hall effect measurement for the multi-terminal FET device. Solid line (blue) is a least-square fit of the data to , where , is 0.28 μF/cm2 and is 1.88 V. For , we used a model shown in the bottom panel (See main text for details). Bottom: the dash-dotted line (purple) represents a case that in the subthreshold region with the body factor m of 2.8 changes to of the ideal metal. The solid line (blue) becomes negative which explains the enhancement of n2D. Inset shows the capacitance of the HfO2/Parylene-C gate insulator as a function of time measured while continuously applying the voltage. The variation is less than 2% for one hour even for the application of 8 V which is close to the breakdown voltage. (b) Schematic picture of ISD, and n2D with respect to VG. In the metal region, and comes back from −∞. However, changes continuously, which explains the observed n2D. (c) Negative capacitance means the charge compressibility is negative, i.e., dμ/dn2D is negative. In the general rigid-band model, dμ/dn2D > 0. If the density of states D(ω) is changed by the carrier doping, dμ/dn2D < 0 can be realised. Closure of the correlation gap such as the Mott transition and a band-splitting such as the Rashba effect are the typical examples.

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We have measured the Hall effect for more than ten FET devices on three different SrTiO3 substrates (two results are shown in Fig. S3) and confirmed all of them showed qualitatively same n2D enhancement. In order to explain this large discrepancy, we have assumed a naive model that the channel is a serial connection of a bulk SrTiO3 () and a surface layer . When VG is small, is most dominant to , but as VG increased, accumulated carriers screen the gate voltage; i.e., for VG > Vmin, becomes more dominant. Then, we introduced a tractable model: , where . (α = 0.68 and Vmin = 2.8 V are non-essential parameters). This is an ad-hoc phenomenological model to express that changes smoothly from -dominant to -dominant, thus the mathematical formula is not relevant. If is a large positive number as that of a good metal, and corresponding n2D behave as dash-dotted lines (purple) in Fig. 4a. Deviation is still large. Then, if we assume negative capacitance , the calculated n2D coincides with the measured n2D.

We understand that ISD, and n2D should behave as shown schematically in Fig. 4b. Negative , i.e., negative κ, is inevitable for explaining the large enhancement of n2D. But a question arises. If this is ascribed to the exchange interaction of the quasi-2DES on SrTiO3 as explained in literature3,4,5, averaged distance between the electrons should be much larger than the Bohr radius aB, i.e., and the system may become like the Wigner crystal with negative chemical potential , where ε is a direlectric constant of SrTiO3 (ref. 39). However, n2D in this study is in the order of 1014 cm−2, then the corresponding values and are both unreasonable. It was suggested that negative κ is also realised in electronic systems close to half filling40, but this neither is applicable to our samples. Therefore, the significant enhancement of n2D cannot be explained solely by the negative κ originating in the exchange interactions; we need an alternative idea.

Then, we consider the shift of μ further. In a rigid-band model, where the binding energy of each band shifts monotonously without changing the gaps, μ increases by the electron doping and decreases by the hole doping, always leading to positive dμ/dn2D and thus positive κ as shown in Fig. 4c (top). On the contrary, in strongly correlated electron systems, the carrier doping drives the spectral weight transfer (naively a change of the density of states) from the higher energy incoherent states to the lower energy quasiparticle band to fill the Mott-Hubbard gap. Since the band gap decreases, μ decreases effectively and dμ/dn2D becomes negative41 as shown in Fig. 4c (middle) more interesting is that the carrier confinement at the surface of SrTiO3 with perpendicular gate electric field gives rise to the Rashba effect42. If the Rashba spin-orbit coupling is large, the band structure depends on the gate voltage, i.e., n2D, leading to a non-rigid band structure as well. That is, the coupling lowers the band edge quadratically and thus the negative dμ/dn2D is realised43,44 as depicted in Fig. 4c (bottom). However, the absolute value of our negative capacitance −0.31 μF/cm2, which corresponds to dμ/dn2D = −5.1 × 10−13 eV cm2, is too large. For VG between 4 V and 6 V, Δn2D is around 5 × 1013 cm−2, then , which is difficult to be understood either by the Mott transition10,41 or the Rashba effect43,44,45.

We think a clue to approach this problem is an inhomogeneity of the channel. As shown in Fig. S7c in the Supplementary Materials, we have observed a sudden decrease of the internal voltage distribution in the channel along ISD while increasing VG. This has already been observed in other SrTiO3 -FET, indicating a formation of conducting domains in the insulating matrix, which eventually forms a conducting filament by percolation19. Here we assume that the channel consists of two regions, i.e., the metallic domains with the negative sheet capacitance and the non-metallic matrix with the normal positive sheet capacitance . Then, the channel sheet capacitance , which is given by with the volume fraction ξ can be −0.31 μF/cm2, even if is the value which gives a reasonably small Δμ. Details are given in the Supplementary Materials.

Inhomogeneity of Parylene-C thickness in our gate insulator (~30% at most) and other features such as the one dimensional metallic state at the step edge of SrTiO3 (ref. 46) would be the origins of charge inhomogeneity. Moreover, the large positive κ, which the insulating SrTiO3 substrate holds due to the quantum paraelectricity, can augment the inhomogeneities further. Nevertheless, those “extrinsic inhomogeneities” cannot explain the 1000% enhancement of n2D as observed in this study. Therefore, we made an inference that an electronic phase separation with the spinodal instability may be induced by κ → 0, i.e., in our case47. In fact, the phase separation and the charge segregation are natural consequences of the negative capacitance even in ideally homogeneous 2DES43. The charge segregation may cause a local charge imbalance at finite length scales. The frustration between the electrostatic cost and the energy gain due to the phase separation is a possible mechanism of charge inhomogeneous (stripe) states48. However, in our FET, the local charge on the SrTiO3 surface is balanced by the charge on the gate, thus the frustration may be weaken and the typical size of inhomogeneous regions can be microscopic. We hope that this insight motivate further investigation and brings us better understanding of the intriguing physics still hidden in the SrTiO3 surface.

In summary, n2D of the channel of SrTiO3 FET with Parylene-C (6 nm)/HfO2 (20 nm) hybrid gate insulator showed anomalous enhancement: ten times as large as the expected value , indicating negative κ, i.e., negative . However, if the whole channel is a single metallic state with the negative κ, the chemical potential shift becomes too large. On the other hand, transport behaviour suggests the inhomogeneous carrier distribution of the channel, though the channel is fairly clean as evidenced by the small subthreshold swing S = 171 mV/decade and large carrier mobility ~11 cm2/Vs. An intrinsic electronic inhomogeneity is a natural consequence of the negative κ, thus it can happen on the channel of our SrTiO3 -FET. The missing link among the huge n2D enhancement, the negative κ and the intrinsic inhomogeneity will be elucidated by detailed studies.

Methods Summary

Experimental and data analysis methods with associated references are available in the Supplementary Materials.

Additional Information

How to cite this article: Kumar, N. et al. Anomalous enhancement of the sheet carrier density beyond the classic limit on a SrTiO3 surface. Sci. Rep. 6, 25789; doi: 10.1038/srep25789 (2016).