⇦Chinese Physics Letters, 2020, Vol. 37, No. 7, Article code 077301 Giant-Capacitance-Induced Wide Quantum Hall Plateaus in Graphene on LaAlO$_{3}$/SrTiO$_{3}$ Heterostructures Ran Tao (陶然)1,2, Lin Li (李林)1,2*, Li-Jun Zhu (朱丽君)1,2, Yue-Dong Yan (严跃冬)1,2, Lin-Hai Guo (郭林海)1,2, Xiao-Dong Fan (范晓东)1,2, and Chang-Gan Zeng (曾长淦)1,2* Affiliations 1International Center for Quantum Design of Functional Materials (ICQD), Hefei National Laboratory for Physical Sciences at the Microscale, and Synergetic Innovation Center of Quantum Information and Quantum Physics, University of Science and Technology of China, Hefei 230026, China 2CAS Key Laboratory of Strongly-Coupled Quantum Matter Physics, and Department of Physics, University of Science and Technology of China, Hefei 230026, China Received 17 April 2020; accepted 27 May 2020; published online 21 June 2020 Supported by the National Natural Science Foundation of China (Grant Nos. 11974324, 11804326 and U1832151), the Strategic Priority Research Program of Chinese Academy of Sciences (Grant No. XDC07010000), the National Key Research and Development Program of China (Grant No. 2017YFA0403600), Anhui Initiative in Quantum Information Technologies (Grant No. AHY170000), and Hefei Science Center CAS (Grant No. 2018HSC-UE014).
*Corresponding author. Email: lilin@ustc.edu.cn; cgzeng@ustc.edu.cn Citation Text: Tao R, Li L, Zhu L J, Yan Y D and Guo L H et al. 2020 Chin. Phys. Lett. 37 077301    Abstract Hybrid structures of two distinct materials provide an excellent opportunity to optimize functionalities. We report the realization of wide quantum Hall plateaus in graphene field-effect devices on the LaAlO$_{3}$/SrTiO$_{3}$ heterostructures. Well-defined quantized Hall resistance plateaus at filling factors $v=\pm2$ can be obtained over wide ranges of the magnetic field and gate voltage, e.g., extending from 2 T to a maximum available magnetic field of 9 T. By using a simple band diagram model, it is revealed that these wide plateaus arise from the ultra-large capacitance of the ultra-thin LAO layer acting as the dielectric layer. This is distinctly different from the case of epitaxial graphene on SiC substrates, where the realization of giant Hall plateaus relies on the charge transfer between the graphene layer and interface states in SiC. Our results offer an alternative route towards optimizing the quantum Hall performance of graphene, which may find its applications in the further development of quantum resistance metrology. DOI:10.1088/0256-307X/37/7/077301 PACS:73.43.-f, 72.80.Vp, 71.70.Di © 2020 Chinese Physics Society Article Text As a typical two-dimensional (2D) Dirac electron system, graphene has attracted intense interest due to its unique electronic features, such as the bipolar field effect, non-zero Berry phase, and energy-independent Fermi velocity.[1,2] Especially, Dirac-type carriers of this semi-metal lead to exceptionally large cyclotron gaps in the Landau level (LL) spectrum, making graphene competitive for realizing the quantum Hall effect (QHE) at much reduced magnetic fields and/or much higher temperatures.[3–6] As a consequence, there have been extensive efforts devoted to optimizing the QHE performance of graphene towards the development of quantum resistance metrology beyond conventional GaAs systems,[7] and the epitaxial graphene grown on SiC substrates (G/SiC) has proven to be a suitable choice.[8–10] Besides the relatively large size of the as-grown graphene wafer that can sustain a large breakdown current of the QHE, the remarkable wide Hall plateau in G/SiC allows resistance quantization over the entire sample despite local variations of carrier density, enabling the realization of the quantized Hall resistance with high precision.[11] In G/SiC, the acquisition of wide Hall plateau is rooted in the electric-field- or magnetic-field-dependent charge transfer between interface states in SiC and the LL states in graphene.[12,13] However, this charge transfer process normally leads to a strong doping of graphene,[14] and additional carrier density control is thus needed to enter into a well-defined quantized Hall region within a small magnetic field.[7] On the other hand, constructing graphene-based hybrid structures has been demonstrated to be a ready and effective way to tune the electronic transport properties of graphene. A typical example is the graphene field-effect device built on the LaAlO$_{3}$/SrTiO$_{3}$ (LAO/STO) heterostructure, a representative oxide heterostructure hosting two-dimensional electron gas (2DEG) at the interface.[15,16] In this graphene-LaAlO$_{3}$-SrTiO$_{3}$ hybrid device (denoted as G-LAO-STO), an ultra-low operating voltage is needed when utilizing the ultra-thin LAO layer as the dielectric layer.[17] Significant improvement of carrier mobility for graphene is readily achieved and the quantum transport performance also gets significantly enhanced, benefiting from the screening effects of LAO/STO on multiple scattering processes.[18] Our recent work further demonstrated that well-defined QHEs can be realized under relatively modest conditions compared with most other graphene devices.[19] In this study, similar hybrid structures are fabricated, and the detailed characteristics of the QHE in graphene are systematically investigated. Well-defined quantized Hall resistance plateaus at filling factors $v=\pm2$ can be obtained over a wide range of the magnetic field and gate voltage. By analyzing the Hall data via a simple band diagram model,[12,13,20] the realization of wide Hall plateaus is revealed to originate from the ultra-large gate capacitance of the ultra-thin LAO layer. Our results thus reveal a brand-new route to enlarge the operational parameter space of the QHE in graphene. This G-LAO-STO hybrid device also offers an alternative for developing quantum resistance metrology with high precision. The G-LAO-STO devices, as schematically shown in Fig. 1(a), were fabricated following the established procedures detailed in our previous report.[19] LAO/STO (001) heterostructures were fabricated via pulsed laser deposition, with the thickness of the grown LAO layer monitored by reflection high-energy electron diffraction intensity oscillations. Monolayer graphene flakes were mechanically exfoliated from Kish graphite and transferred onto the LAO/STO surface via a completely dry transfer method.[21] Figure 1(c) shows the typical scanning electron microscope image of the finally obtained G-LAO-STO devices. The Hall-bar electrodes were patterned by electron-beam lithography, and Ti/Pd/Au films with the thickness of 0.3/5/40 nm were deposited as electrode materials via e-beam evaporation. Finally, the graphene layer was also etched into the Hall-bar geometry through an oxygen plasma etching process. The good conductivity of the LAO/STO interface in the G-LAO-STO device was carefully verified, with the basic transport properties consisting well with those in our previous studies.[22,23] The BN-encapsulated graphene devices (BN-G-BN, see Fig. 1(b)) were fabricated as a "control sample". The stack was assembled onto the SiO$_{2}$/Si substrate via the typical van der Waals assembly technique,[24] and shaped into a Hall-bar geometry by electron-beam lithography and reactive ion etching (Fig. 1(d)).
cpl-37-7-077301-fig1.png
Fig. 1. [(a),(c)] Schematic illustration and scanning electron microscope image of the G-LAO-STO device, respectively. [(b),(d)] Schematic illustration and the optical microscope image of the BN-G-BN device, respectively. The scale bars in (c) and (d) are 5 µm. [(e),(f)] $|R_{xy}|$ as a function of $B$ and ($V_{\rm g}-V_{\rm d}$) measured at 1.5 K for G-LAO-STO-1 and BN-G-BN-1, respectively. $V_{\rm d}$ is the gate voltage corresponding to the Dirac point of graphene.
The transport measurements were performed in an Oxford Instruments $^{4}$He cryostat with standard AC lock-in techniques. For both of the devices, the carrier polarity and density of graphene can be readily tuned by applying a gate voltage ($V_{\rm g}$). In G-LAO-STO, the ultra-thin LAO layer ($ < 2$ nm for 5-uc LAO) and the conducting LAO/STO interface are utilized as the natural dielectric layer and the gate electrode (Fig. 1(a) and Fig. S1(a) in the supplementary material), respectively. While for BN-G-BN, the 300-nm-thick SiO$_{2}$ layer serves as the back-gate dielectric layer. For G-LAO-STO the typical carrier mobility extracted from Hall measurements (e.g., 27000 cm$^{2}$/V$\cdot$s at a carrier density $n\sim 9.1\times 10^{10}$ cm$^{-2}$ and temperature $T=1.5$ K) is significantly enhanced as compared with previous pristine graphene on SiO$_{2}$, but is still smaller than the typical value of BN-G-BN (up to 68000 cm$^{2}$/V$\cdot$s at $T=1.5$ K). Next, we will focus on the quantum transport performance under a magnetic field applied perpendicular to the graphene plane. Figures 1(e) and 1(f) show the Hall resistance ($R_{xy}$) mapping as a function of $V_{\rm g}$ and the magnetic field ($B$) measured at 1.5 K for G-LAO-STO and BN-G-BN, respectively. It is clear that for G-LAO-STO, a much smaller operating gate voltage is needed, due to the much larger gate capacitance of the ultra-thin LAO layer. Well-defined quantized Hall resistance plateaus can be observed even under $B$ below 2 T in both the devices, consistent with their relatively high carrier mobilities. However, the $v=\pm2$ Hall plateaus (white regions) in G-LAO-STO cover much larger proportions of the parameter space along both axes than those in BN-G-BN. Especially, a wide $v=-2$ Hall plateau extending from $\sim $2 T to a maximum available field of 9 T can be obtained at certain $V_{\rm g}$. This characteristic can also be observed in the simultaneously obtained mapping of the longitudinal resistance ($R_{xx}$) (Fig. S2), manifesting as the wide regions of zero $R_{xx}$ corresponding to the $v=\pm2$ Hall plateaus. Although there has been a lot of research devoted to the quantum transport performance of graphene, the observation of such a wide quantum Hall plateau thus far has been achieved in limited systems, e.g., epitaxial graphene on SiC,[25,26] and InSe covered graphene.[27]
cpl-37-7-077301-fig2.png
Fig. 2. (a) $R_{xy}$ vs ($V_{\rm g}-V_{\rm d}$) curves measured at $B=9$ T and $T=1.5$ K for G-LAO-STO-2 and BN-G-BN-1. [(b),(c)] $R_{xy}$ vs ($V_{\rm g}-V_{\rm d}$) curves under 9 T for G-LAO-STO-2 and BN-G-BN-2 obtained at 10 K and 50 K, respectively.
If we plot the $R_{xy}$–$V_{\rm g}$ curves taken under a quantizing magnetic field, more details can be obtained. Figure 2(a) respectively shows the typical results of G-LAO-STO-2 and BN-G-BN-1 measured at $T=1.5$ K and $B=9$ T. Since different gate voltages are used, these two curves are aligned in $x$-axis according to the position of $R_{xy}=h/4e^{2}$ (green squares) and the Dirac point of graphene. It is evident that in G-LAO-STO the relative width of the $v=\pm2$ Hall plateaus along $V_{\rm g}$ are much larger than those in BN-G-BN. To better demonstrate it, we define the regions corresponding to the $v=-2$ Hall plateau and the $N=-1$ LL as region I (red region) and region II (blue region), respectively. The relative width of the Hall plateau can thus be represented as $w =\Delta V_{\rm I}/\Delta V_{\rm I\!I}$, where $\Delta V_{\rm I}$ and $\Delta V_{\rm I\!I}$ are the gate-voltage ranges of the two regions, respectively. From other comparison results shown in Figs. 2(b) and 2(c), a larger $w$ can be commonly achieved in G-LAO-STO. Note that these wide Hall plateaus presented in Fig. 2 do not arise from the gate hysteresis, since negligible hysteresis behavior in response to the reversal of the sweeping direction is seen (Figs. S1(b)–S1(e)). This observation together with the wide Hall plateau along $B$ (Fig. 1(e)) indicates the substantially enlarged operational parameter space of the QHE in the present G-LAO-STO device. To unveil the underlying mechanism for the obtained wide Hall plateaus in G-LAO-STO, we first develop a simple band diagram model[12,13,20] to account for the tuning effect of $V_{\rm g}$, as schematically illustrated in Fig. 3(a). Note that applying $V_{\rm g}$ not only induces charge carriers in graphene, but also leads to the Fermi energy shift for both graphene and LAO/STO interface. Nevertheless, since the density of states in the LAO/STO interface (on the order of 10$^{15}$ eV$^{-1}$ cm$^{-2})$[28,29] is much larger than that in the graphene layer (on the order of 10$^{13}$ eV$^{-1}$ cm$^{-2}$),[20,30] the Fermi energy shift of LAO/STO is negligible. Thus the effect of $V_{\rm g}$ on the carrier density modulation can be described using the equation $$ e(V_{\rm g}-V_{\rm d})=E_{\rm F}-\frac{e^{2}}{C_{\mathrm{LAO}}}n,~~ \tag {1} $$ where $E_{\rm F}$ is the Fermi energy of graphene, and $C_{\rm LAO}$ is the geometrical capacitance of the LAO layer. In our previous work,[19] the validity of this equation has been demonstrated by satisfactory fits of the nonlinear $V_{\rm g}$ dependence of $n$ determined from the low-field Hall data ($ < 0.5$ T). Moreover, the extracted $C_{\rm LAO}$ is more than two orders larger than that of conventionally used 300-nm-thick SiO$_{2}$, benefiting from the ultra-small thickness and relatively high dielectric constant of the LAO layer (20–25).[31] Considering graphene under a quantizing magnetic field, carriers are ranged in LLs (the right panel in Fig. 3(a)). In this case, Eq. (1) remains valid, but varies in detailed form based on the location of the Fermi level (Fig. 3(b)). If ignoring the impact of LL broadening, a semi-quantitative estimation of the dependence of carrier density can be readily figured out. When the Fermi level is inside the gap between the neighboring $N$th and ($N$+1)th LLs (e.g., region I in Fig. 3(b)), the $N$th LL is fully filled and the carrier density can be expressed as $n = -(4N+2)eB/h$. When the Fermi level is in the $N$th LL (e.g., region II), the Fermi energy equals the energy of the $N$th LL, i.e., $E_{\rm F}=E_{\rm N}={\rm sgn}(N)v_{\rm F}\sqrt {2e\hbar B\vert N\vert}$. Based on Eq. (1) by involving the above expressions (see Note 1 in the supplementary material for details), the main characteristics of the experimental ($V_{\rm g}-V_{\rm d}$)-dependent $n$ curve can be well reproduced (Fig. 3(d)). The parameters used here are $C_{\rm LAO}=1.59$ µF/cm$^{2}$ and $v_{\rm F}=1.17\times 10^{6}$ m/s, according to the values obtained in our previous work.[19] In region I the dominant variable term is the first term on the right side of Eq. (1) corresponding to the quantum Hall plateaus, such that $\Delta V_{\rm I} \sim (E_{0}-E_{-1})/e=v_{\rm F}(2\hbar B/e)^{1/2}$. In region II, the second term on the right side dominates the variation, such that $\Delta V_{\rm I\!I} \sim (1/C_{\rm LAO})4e^{2}B/h$. Therefore, in the $n$ vs ($V_{\rm g}-V_{\rm d}$) curve, $w = \Delta V_{\rm I}/\Delta V_{\rm I\!I} \sim C_{\rm LAO}v_{\rm F}h^{3/2}/(4\pi^{1/2}e^{5/2}B^{1/2}) \propto C_{\rm LAO}$, which should be quite high due to the ultra-large value of $C_{\rm LAO}$. As a result, quantum Hall plateaus over a large proportion of the gate-voltage range are achieved in the present G-LAO-STO device. By utilizing the same model, the typical characteristics of the quantum Hall plateau along the magnetic field could also be reproduced to some extent. It is well known that the intervals of LLs enhance with increasing $B$ (Fig. 3(c)), the variation of the relative position between Fermi level and LLs leads to oscillations of the carrier density.[12] Here we take the $R_{xy}$ vs $B$ data obtained at $V_{\rm g}-V_{\rm d} = -0.048$ V as an example. According to $v = -nh/eB$ and $R_{xy}=h/ve^{2}$, the experimental $n$ vs $B$ data can be obtained, and the result is shown in Fig. 3(e). On the other hand, according to Eqs. (S1) and (S2) in the supplementary material, the $B$ dependence of $n$ profiles can be described as follows: in situations like region I (Eq. (S1)), $n$ increases linearly with $B$, while in situations like region II (Eq. (S2)), $n$ decreases linearly with $B^{1/2}$. Thus the $n$ vs $B$ curve can also be simulated by further computing the lower/upper critical magnetic field of the quantum Hall plateau. For instance, the lower critical field ($B_{\rm IL}$) and upper critical field ($B_{\rm IU}$) of the $v=-2$ Hall plateau (region I) can be represented as $B_{\mathrm{IL}}=(\frac{-2\pi ehD}{\sqrt {\pi eh^{3}v_{\rm F}^{2}C_{\mathrm{LAO}}^{2}-8\pi ^{2}e^{4}hD} +\sqrt {\pi eh^{3}v_{\rm F}^{2}C_{\mathrm{LAO}}^{2}} })^{2}$ and $B_{\mathrm{IU}}=-\frac{h}{2e^{2}}D$, respectively (see Note 2 in the supplementary material for details), where $D = (V_{\rm g}-V_{\rm d})C_{\rm LAO}$ $( < 0)$ is the electric displacement. As depicted in Fig. 3(e), in region I the simulation result agrees well with the experimental data. Note that $B_{\rm IU}$ is constant for a fixed $D$, while the obtained $B_{\rm IL}$ should be quite small due to the ultra-large value of $C_{\rm LAO}$. Thus, the large width of the $v=-2$ Hall plateau along $B$, i.e., $B_{\rm IU}-B_{\rm IL}$, can also be reasonably explained.
cpl-37-7-077301-fig3.png
Fig. 3. (a) Schematic band diagram across the G-LAO-STO hybrid structure. The middle and right panels show the band diagram of graphene under $B=0$ (Dirac cone form) and quantizing $B$ (LL form), respectively. The dashed diagram in the middle panel illustrates a case where the Fermi level of graphene is at the Dirac point. [(b),(c)] Evolution of the Fermi level and the LLs of graphene with varying $V_{\rm g}$ and $B$, respectively. (d) ($V_{\rm g}-V_{\rm d}$)-dependent $n$ under 9 T and the corresponding simulation result. (e) $B$-dependent $n$ with $V_{\rm g}-V_{\rm d}=-0.048$ V and the corresponding simulation result. The experimental data of $n$ in (d) and (e) are calculated via $n = -veB/h = -B/(eR_{xy})$ from the $R_{xy}$ data, while the measurements are conducted at 1.5 K in G-LAO-STO-2.
Here we would like to address that the above analyses of the QHE are applicable in a wide variety of cases. To demonstrate it, $n$ as a function of ($V_{\rm g}-V_{\rm d}$) under different $B$ were simulated using Eq. (1) (dashed lines in Fig. 4(a)), and then compared with the experimental data (solid lines). It is evident that the main characteristics are well reproduced, especially for plateaus at small filling factors. With decreasing $B$, the widths of the Hall plateaus decrease accordingly due to the reduction of energy gaps between adjacent LLs. In addition, the value of $V_{\rm g}$ at which the LL is half-filled ($v=4N$), corresponding to the position of $R_{xx}$ peak, can be estimated by solving Eq. (1) with $E_{\rm F}=E_{\rm N}$ and $n=-4NeB/h$.[13,32] As can be clearly seen from Fig. 4(b), the simulated curves agree well with the experimental data, further demonstrating the validity of our model.
cpl-37-7-077301-fig4.png
Fig. 4. (a) ($V_{\rm g}-V_{\rm d}$)-dependent $n$ under different $B$ and the corresponding simulation results. (b) The values of $V_{\rm g}$ and $B$ corresponding to half-fillings of graphene LLs. The circular points represent the peak positions in $R_{xx}$ vs ($V_{\rm g}-V_{\rm d}$) data (Fig. S3) obtained under different $B$, while the dashed lines represent the simulation results. The experimental data in (a) and (b) are obtained at 1.5 K in G-LAO-STO-2.
Unlike the case of epitaxial graphene on SiC, in which the realization of giant quantum Hall plateaus was mainly attributed to the charge transfer between graphene and interface states in SiC,[12,13] in the present G-LAO-STO system the ultra-large gate capacitance of the LAO layer plays a key role. Normally, interface states will accommodate a portion of charges induced by the gate-voltage while tuning the Fermi level of graphene.[13] As a consequence, the voltage range corresponding to the quantum Hall plateau, e.g., $\Delta V_{\rm I}$, would be significantly enlarged when the effect of interface states dominates the widening of the Hall plateau (see Note 3 and Fig. S4 in the supplementary material for details). However, in G-LAO-STO the experimental $\Delta V_{\rm I}$ agrees well with the value simulated without involving interface states (Figs. 3(d) and S4(c)), revealing the negligible effect of interface states on the widening of the quantum Hall plateau. In summary, quantum Hall plateaus over a wide range of the magnetic field and gate-voltage are successfully realized in G-LAO-STO, a hybrid device consisting of graphene and LAO/STO heterostructure. Note that in terms of the localization model, the lower-quality samples may also present wide Hall plateaus.[33,34] However, the low carrier mobility results in a relatively large critical magnetic field where the quantum Hall plateau emerges, and high accuracy of the quantum Hall resistance is also hard to achieve. In our G-LAO-STO devices, the carrier mobility of graphene is substantially enhanced as compared with previous pristine graphene on SiO$_{2}$, and the obtained wide Hall plateaus originate from the ultra-large gate capacitance of the LAO layer. The good sample quality, together with high tunability, make G-LAO-STO an ideal candidate to obtain quantized Hall resistance with high precision under modest conditions. Such a universal approach utilizing a large-capacitance dielectric layer can also be readily extended to more generalized 2D material systems for optimizing the quantum transport performance.[35,36] References The rise of grapheneThe electronic properties of grapheneTwo-dimensional gas of massless Dirac fermions in grapheneExperimental observation of the quantum Hall effect and Berry's phase in grapheneQuantum Hall effect in grapheneRoom-Temperature Quantum Hall Effect in GrapheneQuantum resistance metrology using grapheneTowards a quantum resistance standard based on epitaxial grapheneQuantum Hall resistance standards from graphene grown by chemical vapour deposition on silicon carbideQuantum Hall resistance standard in graphene devices under relaxed experimental conditionsEpitaxial graphene for quantum resistance metrologyAnomalously strong pinning of the filling factor ν = 2 in epitaxial grapheneImpact of graphene quantum capacitance on transport spectroscopyCharge transfer between epitaxial graphene and silicon carbideA high-mobility electron gas at the LaAlO3/SrTiO3 heterointerfaceTunable Quasi-Two-Dimensional Electron Gases in Oxide HeterostructuresTunnel and electrostatic coupling in graphene-LaAlO 3 /SrTiO 3 hybrid systemsRoom-Temperature Quantum Transport Signatures in Graphene/LaAlO 3 /SrTiO 3 HeterostructuresSubstantially enhanced robustness of quantum Hall effect in graphene on LaAlO 3 /SrTiO 3 heterostructureDirect Measurement of the Fermi Energy in Graphene Using a Double-Layer HeterostructureMulticomponent fractional quantum Hall effect in grapheneNonmonotonically tunable Rashba spin-orbit coupling by multiple-band filling control in SrTiO 3 -based interfacial d -electron gasesOptical Manipulation of Rashba Spin–Orbit Coupling at SrTiO 3 -Based Oxide InterfacesOne-Dimensional Electrical Contact to a Two-Dimensional MaterialGiant quantum Hall plateaus generated by charge transfer in epitaxial graphenePuddle-Induced Resistance Oscillations in the Breakdown of the Graphene Quantum Hall EffectGiant Quantum Hall Plateau in Graphene Coupled to an InSe van der Waals CrystalTwo-dimensional electron gas with universal subbands at the surface of SrTiO3Tuning the Kondo effect via gating-controlled orbital selection in the LaAlO 3 / SrTiO 3 interfacial d -electron systemDensity of States and Zero Landau Level Probed through Capacitance of GrapheneFormation of LaAlO3 films on Si(100) substrates using molecular beam depositionProbing the extended-state width of disorder-broadened Landau levels in epitaxial grapheneEffect of localization on the hall conductivity in the two-dimensional system in strong magnetic fieldsScaling theory of the integer quantum Hall effectAtomic-Layer-Deposition Growth of an Ultrathin HfO 2 Film on GrapheneUniform and ultrathin high-κ gate dielectrics for two-dimensional electronic devices
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