Device fabrication and electrical measurement

The transport and 1/f noise measurements were carried out in a dual-gated single-layer graphene (SLG) field-effect transistor with hBN as the top gate and STO as the back gate dielectric. The schematic of a typical dual-gated SLG transistor with the electrical layout is shown in Fig. 1a. The TiO 2 surface termination of the 0.5 mm thick STO (100) substrate (from CrysTec GmbH) was achieved through chemical processes and annealing (Fig. S1 in Supplementary Information). The SLG and hBN flakes were exfoliated on SiO 2 /Si++ substrates, and subsequently transferred onto the TiO 2 −terminated surface of STO through van der Waals epitaxy33,34 (see Methods for more details). The layer number of graphene was verified by Raman spectroscopy as shown in Fig. 1d. The absence of D peak at 1350 cm−1 in the Raman spectrum confirms a defect-free graphene channel. The metal contacts were patterned by e-beam lithography followed by thermal deposition of 5/50 nm of chromium/gold. The optical microscope image of the SLG/hBN heterostructure before transferring onto STO, and the dual-gated transistor after depositing contact pads, are shown in Fig. 1b, c, respectively. All measurements of resistance and noise were carried out using low-frequency AC technique34,35,36,37,38,39,40 in four probe geometry under high vacuum condition (~10−5 torr). The carrier mobility of the graphene channel, similar for both electrons and holes, was found to be ~2000 cm2 Vs−1 at 140 K which increases to ~7300 cm2 Vs−1 at 10 K, in agreement with recent experimental report24 (Fig. S2 in the Supplementary Information). The resistivity of the graphene channel increases with increasing temperature upto ~60 K beyond which it either saturates or decreases marginally (Fig. S3 in the Supplementary Information).

Fig. 1 Structure and electrical characterization of graphene–STO hybrid. a Schematic of a dual-gated single-layer graphene (SLG) transistor on STO substrate with STO and hBN as back and top gate dielectrics, respectively, together with the circuit diagram for electrical transport measurement. Optical microscope images of SLG-hBN stack b on the transfer tape before transferring on STO and c after fabricating dual-gated transistor on STO. Scale bars in both (b, c) are 25 μm. d Raman spectroscopy of graphene layer used to make the dual-gated transistor showing single-layer characteristics. e Transfer characteristics with respect to top gate voltage (V TG ) at 85 K. The solid and dashed lines depict the resistance of SLG channel with forward and reverse sweeps of top gate voltage respectively. f Transfer characteristics of device D1 with respect to V TG at 100 K showing the shift of charge neutrality points (CNP) at different fixed V BG from 126 V to 158 V. g The locus of the charge neutrality points (\(V_{{\mathrm{TG}}}^{{\mathrm{CNP}}}\)) with varying V BG at different temperatures from 5 K to 200 K. h The dielectric constant of STO (\(\varepsilon _{\mathrm{r}}^{{\mathrm{STO}}}\)) with varying temperatures estimated from three different devices D1, D2 and D3. For device D4, \(\varepsilon _{\mathrm{r}}^{{\mathrm{STO}}}\) was measured from Hall measurement at low temperatures Full size image

Measurement of dielectric constant of STO

The resistance of the dual-gated SLG transistor with varying top gate voltage (V TG ) shows conventional bell curve with a charge neutrality point (CNP) or Dirac point at \(V_{\mathrm{TG}} \approx - 1.6\) V (Fig. 1e). The hysteresis in the top gate sweep is nearly negligible, as expected from trap-free interface of SLG and hBN.32,41 The dual-gated geometry allows a direct measurement of the dielectric constant of STO (\(\varepsilon _{\mathrm{r}}^{{\mathrm{STO}}}\)) from the locus of the CNP in the (V TG , V BG ) space, which balances the STO back gate capacitance with the known capacitance of hBN top gate. In Fig. 1f, the resistance (at T = 100 K) of the dual-gated SLG transistor is shown while sweeping V TG at different fixed back gate voltages (V BG ). The CNP (\(V_{{\mathrm{TG}}}^{{\mathrm{CNP}}}\)) shifts expectedly to the left with increasing V BG .42,43 Plotting \(V_{{\mathrm{TG}}}^{{\mathrm{CNP}}}\) vs. V BG (Fig. 1g) gives a straight line which can be fitted with the equation,

$$V_{{\mathrm{TG}}}^{{\mathrm{CNP}}} = - \frac{{C_{\mathrm{B}}}}{{C_{\mathrm{T}}}}V_{{\mathrm{BG}}} - \frac{{n_{\mathrm{0}}e}}{{C_{\mathrm{T}}}},$$ (1)

where C B and C T are the capacitances of the back gate with STO as dielectric and the top gate with hBN as dielectric, respectively, n 0 is the intrinsic carrier density in graphene and e is the electronic charge. Since C T (=2.7 × 10−3 F m−2) is known from the thickness of hBN (≈13 nm, measured from atomic force microscopy), we obtain the dielectric constant \(\varepsilon _{\mathrm{r}}^{{\mathrm{STO}}} = C_{\mathrm{B}}d_{{\mathrm{STO}}}{\mathrm{/}}\varepsilon _0\) (d STO being the thickness of STO and ε 0 the vacuum permittivity) (also see Fig. S4 in the Supplementary Information). The magnitude of \(\varepsilon _{\mathrm{r}}^{{\mathrm{STO}}}\), measured at different temperatures, as shown in Fig. 1h, are in agreement with previous reports7,44,45,46,47 and matches well with the value of \(\varepsilon _{\mathrm{r}}^{{\mathrm{STO}}}\) estimated from Hall measurement at low temperatures. This agreement confirms that both graphene–hBN and graphene–STO interfaces in our device are atomically clean, and free of undesired adsorbates and chemical species.

Anti-hysteretic transfer characteristics

Unlike the top gate, sweeping of the back gate voltage V BG in forward and reverse directions led to strong anti-hysteresis in the transfer characteristics. The extent of anti-hysteresis depends on both temperature and sweep range of V BG . As shown in Fig. 2a, the anti-hysteresis decreases with increasing temperature and vanishes at the temperature range of 150–180 K whereas, it decreases with decreasing sweep range (Fig. 2b). See Fig. S5 in the Supplementary Information for results from different devices. Hysteretic transfer characteristics in graphene FET48 on SiO 2 and other substrates49,50,51 are commonly attributed to slow charge transfer in the presence of impurity states and absorbed water molecules. Although our experiment was performed under high vacuum condition, and the collapse of the anti-hysteretic transfer characteristics is observed at significantly low temperature (~180 K), the possibility of physi/chemisorption of OH− and H+ on individual atomic site52 cannot be ruled out. However, the independence of the anti-hysteretic behaviour to the ramp rate in V BG (Fig. S6 in the Supplementary Information) suggests a fundamentally different physical mechanism in our case.

Fig. 2 Temperature and gate voltage range-dependent transfer characteristics. a Anti-hysteresis in transfer characteristics with respect to back gate voltage (V BG ) at different temperatures showing the decrease in anti-hysteresis with increasing temperature. b Anti-hysteresis in transfer characteristics with respect to V BG at 70 K showing the decrease in anti-hysteresis with decreasing sweep range of V BG . The arrows indicate the direction of V BG sweep in (a, b). c The CNP of forward and reverse sweep directions as a function of overall sweep range of V BG (ΔV BG ) at different temperatures for devices D2 and D1 (inset). d The difference in CNP of forward and reverse sweep directions of V BG as a function of temperature at different sweep ranges for devices D2 and D1 (inset) Full size image

The similarity of the transfer characteristics to that observed in the earlier studies of graphene transistors on STO substrate25,26 and also in the single/multilayer graphene on ferroelectric substrates27,28,29,30,31 strongly suggests that electric polarization at the surface gives rise to quantum confined states that trap, store and release charge from graphene periodically as V BG is swept back and forth. The temperature and sweep range dependence provide crucial insight into the energy and confinement scale of these states. Figure 2c shows the CNPs (V CNP ) for varying sweep range ΔV BG of the back gate. The symmetric positions of the CNPs about V BG = 0 for the forward and reverse sweep directions eliminate oxygen vacancy-mediated anti-hysteretic transfer characteristics.53 In all devices, the (anti-)hysteresis becomes undetectable for ΔV BG ≲ 20 V, i.e., maximum |V BG | = 10 V, which is insensitive to temperature (for T ≲ 200 K). This indicates the energy of the localized trapping state (measured from the Dirac point), \(E_{\mathrm{t}} \sim E_{\mathrm{F}} = \hbar \upsilon_{\mathrm{F}}\sqrt {\pi C_{\mathrm{B}}\left( {V_{{\mathrm{BG}}} - V_{{\mathrm{CNP}}}} \right){\mathrm{/}}e} \sim 0.15\) eV, where υ F is the Fermi velocity. The inset of Fig. 2c confirms similar behaviour in a different device (D1).

The temperature dependence of the hysteresis (Fig. 2a) provides an estimate of the energy barrier to charge exchange between the substrate (STO) and graphene. To quantify this, we have plotted the difference in CNP (ΔV CNP ) for the forward and the reverse sweep directions in Fig. 2d as a function of temperature. For large sweep range ΔV BG ≳ 100 V, the anti-hysteresis in both D1 (inset) and D2 vanishes at \(\approx 150 - 200\) K, suggesting a confinement energy scale \(\Phi _{\mathrm{P}} \simeq 0.02\) eV. At lower ΔV BG , the hysteretic behaviour vanishes at lower T, possibly due to lower magnitude of effective polarization due to remnant domains.

The temperature-dependent anti-hysteretic behaviour can arise from two possible mechanisms. First, the structural transition to the tetragonal phase at low temperatures is known to form domains in the near-surface region, causing rumpling of the surface. This may potentially cause trap states of possibly both structural (domain) and electrostatic (dipole moments) origin. Although the tetragonal domains have been observed to persist up to ~105 K, the temperature dependence of resistance in our case seems to indicate the structural transition to be limited below ~60 K (Fig. S3 in supplementary information). The structural origin is further unlikely because the temperature scale of disappearance of the anti-hysteresis is found to be strongly sweep range dependent, being ~30 K and >200 K (extrapolated) for sweep ranges of ±10 and ±40 V, respectively, in the same device (D2, Fig. 2d). Second, an alternative origin of the hysteretic behaviour can be traced to electrostatically confined trap states arising due to formation of surface dipoles. Our DFT calculations at the graphene–STO interface indicate a possible origin of such surface dipole moments which can be attributed to the movement of Ti and Sr atoms at the surface of STO (Fig. 3a–d). The DFT calculation was performed to estimate the surface polarization, following the formalism adopted by Vanderbilt et al.54 considering slab geometries of paraelectric/ferroelectric bulk compounds.

Fig. 3 Calculation of surface band renormalization and out-of-plane dipole moment at the graphene-STO interface. Schematic representation of atomic displacements in [001] (z-direction)-oriented TiO 2 terminated STO substrate under different conditions: a non-optimized, b optimized and c optimized with single-layer graphene (SLG) on top, as obtained in DFT calculations in the absence of any external electric field. The displacements of Ti atoms in the topmost TiO 2 layer and Sr atoms in the next SrO layer are shown explicitly. The surface polarizations of STO with and without graphene are denoted as \(\bar P_{{\mathrm{STO}} + {\mathrm{SLG}}}\) and \(\bar P_{{\mathrm{STO}}}\), respectively. d Average vertical displacements (Δz) of Ti and Sr atoms at the surface of bare STO and STO with SLG calculated from DFT. e Density of states (DOS) of STO derived from DFT calculation showing the energy band gap of the surface (\(\Delta E_{{\mathrm{gap}}}^{\mathrm{s}}\)) and bulk (\(\Delta E_{{\mathrm{gap}}}\)). f Energy band diagram of STO, SLG and their interface showing the trapped states with potential energy barrier, \(\Phi _{\mathrm{P}}\) appeared due to the presence of surface dipoles. The energy band diagram of the same at g V BG > 0 and h V BG < 0 Full size image

Two key aspects of the DFT calculations can be summarized as below (see Fig. S8 and associated discussions in the Supplementary Information for more details). First, the vertical displacement (Δz, shown in Fig. 3d) of Ti atoms in TiO 2 layer and Sr atoms in SrO layer result in a formation of out-of-plane dipole moment on the surface of STO. The surface dipole moment of bare STO (\(\bar P_{{\mathrm{STO}}} = - 13.89\,\) μC cm−2, in agreement with ref. 54) is significantly enhanced to \(\bar P_{{\mathrm{STO}} + {\mathrm{SLG}}} = - 34.90\,\) μC cm−2 in the presence of graphene. This result is obtained by assuming an epitaxial registration between graphene and STO (model-1). Calculations were also carried out considering model-2, where the lattice parameters of graphene were kept intact. See Supplementary Information section for details. The polarization (\(\bar P_{{\mathrm{STO}} + {\mathrm{SLG}}}\)) calculated for model-1 and model-2 turned out to be −34.9 and −24 μC cm−2, respectively. Thus, the polarization computed from model-2 geometry gives better agreement to experimental value and that obtained from simple phenomenological model. Such enhancement is presumably caused by the rumpling of the surface TiO 2 layer in the presence of graphene, as observed in DFT optimized structure. Second, the band gap at the STO surface \(\Delta E_{{\mathrm{gap}}}^{\mathrm{s}} \sim 0.21\) eV is considerably smaller than the band gap of bulk STO \(\Delta E_{{\mathrm{gap}}} \sim 1.84\) eV (Fig. 3e). Notwithstanding the intrinsic underestimation of band gap in DFT due to over-screening problem,55 this indicates a gradual bending of the bulk bands to surface,56 as shown in the schematic of Fig. 3f.

The competing effects of band bending and electrostatic energy due to polarization at the surface can be combined to develop a phenomenological model for the observed anti-hysteretic behavior (Fig. 3f–h). A quantum well may be formed by the decrease in the band gap and the increase in the electrostatic potential \(\approx\)ΔzP2A cell /2ε 0 ε r at the surface due to the dipolar field, where A cell is the area of the TiO 2 unit cell. Equating the latter to the confinement scale Φ P ≈ 0.02 eV, and assuming air gap between Ti and O atoms (ε r = 1) at the surface, we get P ≈ 13 μC cm−2, which is similar to that obtained from DFT for bare STO, but smaller than expected from graphene–STO hybrid. DFT is well known to overestimate the polarization value even as much as by an order of magnitude, as observed in bulk materials.57 Thus, we consider the agreement between DFT and the value obtained experimentally to be reasonable. The expected trap layer energy \(E_{\mathrm{t}} \approx \Delta E_{{\mathrm{gap}}}^{\mathrm{s}}{\mathrm{/}}2\), where \(\Delta E_{{\mathrm{gap}}}^{\mathrm{s}} \approx 0.21\) eV (Fig. 3e) is the surface band gap, is also close to that (~0.15 eV) estimated experimentally, although the underestimation of the band gap in DFT limits the accuracy of such a comparison.

Figure 3g, h describes the anti-hysteresis process schematically. As the sweep range of V BG (ΔV BG ) increases beyond E t , more charge carriers (electrons or holes) get trapped at the interface quantum well which increase the screening of V BG leading to an increase in anti-hysteresis (Fig. 2c). At higher temperature ≳200 K, the anti-hysteresis decreases as thermal energy of the trapped charge carriers becomes too large to remain confined by Φ P .

Unconventional low-frequency 1/f noise

In addition to the transport measurement, the low-frequency 1/f noise in the channel resistance is also sensitive to the interface dipoles. We assume trapping–detrapping noise to be the dominant mechanism for resistnce fluctuation, which is the case for graphene FETs on conventional substrates.37 In the presence of out-of-plane polarization P at STO surface, the interfacial potential barrier that determines the trapping–detrapping rate of charge across the interface, and hence the 1/f noise, is modified by the local effective electric field (\({\vec{\mathbf E}}_{{\mathbf{eff}}}\)) (Fig. 4a, b). The typical time dependence of resistance fluctuations of the graphene channel is shown in Fig. 4d for two representative V BG , with a 1/f-like power spectral density (S R /R2) (Fig. 4e). The details of the noise measurement technique58,59 are discussed in the Methods section and Supplementary Information. Figure 4c, f correlates the variation in noise magnitude with V BG (Fig. 4f) with the anti-hysteretic behavior in the channel resistance R (Fig. 4c). We find that \(\left\langle {\left( {\Delta R} \right)^2} \right\rangle {/R}^2\) (obtained by integrating S R /R2 over the experimental frequency range) displays a strong (anti-)hysteretic two-state behaviour as a function of V BG . The top gate dependence of noise in the same graphene channel is non-hysteretic and exhibits conventional ‘V’-shaped behavior37,60 (Fig. S7 in the Supplementary Information section), confirming that the anti-hysteretic behaviour is due to surface electrical polarization on STO. Remarkably, the V BG dependence of \(\left\langle {\left( {\Delta R} \right)^2} \right\rangle {/R}^2\) collapses on a single trace as function of density n, irrespective of the sweep directions or temperature (Fig. 4g). Since \(n = |{\vec{\mathbf E}}_{{\mathbf{eff}}}|\varepsilon _0/e\), the monotonic change in noise across the Dirac point (\({\vec{\mathbf E}}_{{\mathbf{eff}}} = 0\)) indicates an unconventional microscopic origin that depends on the direction of \({\vec{\mathbf E}}_{{\mathbf{eff}}}\), rather than just its magnitude.

Fig. 4 Low-frequency 1/f noise. Schematics of the potential energy barrier of STO surface, a V b1 at electron-doped region with positive effective electric field (\({\vec{\mathbf E}}_{{\mathbf{eff}}}\)) and b V b2 at hole-doped region with negative \({\vec{\mathbf E}}_{{\mathbf{eff}}}\) showing V b2 > V b1 , where \({\vec{\mathbf p}}\) represents the interface dipole moment. c The anti-hysteresis in the transfer characteristics of device D1 at 30 K. d The time series of resistance fluctuations at V BG = 90 V and −30 V, showing higher noise for V BG = 90 V. e Power spectral density (S R /R2) of the resistance fluctuations showing 1/f noise characteristics. f Normalized 1/f noise (\(\left\langle {\Delta R^2} \right\rangle {\mathrm{/}}R^2\)) of SLG on STO with forward sweep (FS) and reverse sweep (RS) of V BG at 30 K, showing a very large magnitude on the right side of the charge neutrality point (CNP) and almost two order of magnitude lower value on the left side of CNP for both FS and RS directions of V BG . g Normalized 1/f noise (\(\left\langle {\Delta R^2} \right\rangle {\mathrm{/}}R^2\)) vs. carrier density (n) of the SLG channel at different temperatures from 7 K to 150 K showing a bistable feature in the electron-doped (red background) and hole-doped (blue background) regions for both FS and RS directions of V BG . h The exponential fitting (blue line) of the noise magnitude near CNP which provides the magnitude of interfacial polarization Full size image

When the STO surface is spontaneously polarized, the interface potential barrier \(V_{\mathrm{b}} \approx E_{\mathrm{t}} - {\vec{\mathbf p}}.{\vec{\mathbf E}}_{{\mathbf{eff}}}\) naturally leads to correlated number-mobility fluctuation noise in the graphene channel37,38 that is sensitive to the direction of \({\vec{\mathbf E}}_{{\mathbf{eff}}}\) with respect to the dipole moment \({\vec{\mathbf p}}\) at the surface. Here, \(E_{\mathrm{t}}\left( { \approx \Delta E_{{\mathrm{gap}}}^{\mathrm{s}}{\mathrm{/}}2 \sim 0.1\,{\mathrm{eV}}} \right)\) is the zero-field surface barrier for electron exchange. When the characteristic trapping time scale \(\tau ( = \tau _0exp[2\alpha d])\) is distributed as ~1/τ,61 where \(\alpha = \sqrt {\frac{{2m_{\mathrm{e}}^ \ast V_{\mathrm{b}}}}{{\hbar ^2}}}\), and \(d\) are the tunnelling wave vector and the distance between the channel and surface states of STO substrate, respectively, one obtains

$$\left( {S_{\mathrm{R}}/R^2} \right) \propto exp\left( {\frac{{d\sqrt {2m_{\mathrm{e}}^ \ast } }}{\hbar }\frac{{{\vec{\mathbf p}}.{\vec{\mathbf E}}_{{\mathbf{eff}}}}}{{\sqrt {E_{\mathrm{t}}} }}} \right).$$ (2)

As shown in Fig. 4h, from the exponential fitting of the experimental \(1{\mathrm{/}}f\) noise magnitude with Eq. 2, we obtain \({\vec{\mathbf p}} \approx 3 \times 10^{ - 30}\) C m by assuming \(d \approx 3 \pm 0.5\) nm for the experimental bandwidth. Here, \(m_{\mathrm{e}}^ \ast\) is the effective mass of the electron in STO.62 Estimation of surface polarization through slab calculation14 yields \(P \approx 10\,\) μC cm−2, which is in good agreement to that obtained from the T dependence of anti-hysteresis (Fig. 2a).