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Research Paper

Applied Science and Convergence Technology 2023; 32(2): 41-44

Published online March 30, 2023


Copyright © The Korean Vacuum Society.

Prediction of Absolute Hall Effect Sensitivity and Minimum Magnetic Resolution for Two-Dimensional Rhenium Disulfide Multilayer Magnetic Sensors without Magnetic Fields

Min-Kyu Jooa , b , ∗

aDepartment of Applied Physics, Sookmyung Women’s University, Seoul 04310, Republic of Korea
bInstitute of Advanced Materials and Systems, Sookmyung Women’s University, Seoul 04310, Republic of Korea

Correspondence to:mkjoo@sookmyung.ac.kr

Received: February 20, 2023; Revised: March 15, 2023; Accepted: March 15, 2023

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/3.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

Absolute Hall-effect sensitivity (SA) and minimum magnetic resolution (Bmin) of two-dimensional (2D) van der Waals Hall elements are predicted without magnetic fields by considering the drain voltage-dependent transconductance and current power spectrum density (PSD). The measured drain-bias-dependent PSD of rhenium disulfide multilayers is suitably described by the carrier number fluctuation noise model, indicating that the effects of carrier trapping/de-trapping into oxide traps dominate the observed current variations. To achieve high currentnormalized Hall sensitivity and SA with a low Bmin at a specific current value, the contact resistance and oxide trap density should be further optimized. Our discussion provides an effective approach for the optimization of 2D multilayer-based Hall elements.

Keywords: Two-dimensional materials, Hall sensor, Magnetic resolution, Analytical model, Contact resistance, Carrier mobility

A Hall element is a magnetic sensor that detects the presence of magnetic flux and converts this magnetic signal to a Hall voltage (VH) proportional to the magnetic intensity when the current flows perpendicularly to the magnetic fields [13]. Based on this simple yet powerful relationship between VH and the intensity of magnetic fields, numerous applications of Hall elements have been demonstrated for tachometers, machinery switches, speedometers, and acceleration sensors [25]. In addition, three-dimensional magnetic sensor arrays provide useful information, such as real-time angle and position variations, for mechanics, robotics, weapons, and micro/ nanofluidic systems [3,5]. Thus far, silicon and III-V compounds with two-dimensional (2D) electron gas have governed the magnetic sensor industry with a high magnetic field resolution [6]. However, further improvement is necessary to not only extend their functionality with a high magnetic sensitivity but also embed them into flexible and transparent substrates.

According to the conventional working principle of Hall sensors, high conductivity (or high carrier mobility at a given carrier density) with a thin material is an essential prerequisite for achieving high magnetic sensor performance [13]. Because of this requirement, Hall elements based on 2D van der Waals (vdW) monolayers have garnered significant interest, particularly those in graphene and molybdenum disulfide (MoS2) [1,2,79]. However, because the carrier mobility (or effective mass) increases (or decreases) with increasing thickness (or number of layers), 2D vdW multilayers are used as alternatives despite their greater thickness relative to monolayers. Furthermore, the feasibility of using 2D vdW multilayers has not yet been systematically investigated.

In the following section, the magnetic sensor limitations of 2D vdW multilayers for Hall elements are demonstrated. Analytical expressions for the absolute Hall-effect sensitivity (SA), currentnormalized Hall sensitivity (SI), and magnetic resolution (Bmin) of 2D rhenium disulfide (ReS2) multilayers are proposed by considering the drain voltage-dependent transconductance and current power spectrum density (PSD) (Fig. 1) [2,8]. Furthermore, the obtained lowfrequency (LF) noise data are suitably interpreted by the carrier number fluctuation (CNF) noise model regardless of the drain bias, implying that the effects of carrier trapping/de-trapping into oxide traps dominate the observed current variations. The effects of contact resistance and oxide trap density on SA, SI, and Bmin are also discussed.

Figure 1. Prerequisites for Bmin estimation. General trend of (a) B-dependent VH, (b) PSD of VH, and (c) Bmin, respectively. VH, B, SV, and f denote the Hall-effect voltage, magnetic fields, voltage PSD obtained from Hall bar electrodes for VH measurement, and frequency, respectively.

micromechanically exfoliated ReS2 flake with a distorted octahedral (1T′) structure (purchased from a 2D semiconductor) was first transferred to a 90 nm thick SiO2/p+-Si substrate to establish a conventional back-gate device configuration [Fig. 2(a)]. To clearly define the channel and contact regions, conventional electron-beam lithography (MIRA3, TESCAN) was conducted, and a 100 nm thick Au film was deposited using an electron-beam evaporator (INFOVION). The observed various intrinsic vibrational Raman modes of Eg (≈ 154.7 cm−1), A1g-like (≈ 214.7 cm−1), and Eg-like (≈ 311.4 cm−1) clearly manifest the presence of 1T′ ReS2 multilayer, as demonstrated in [Fig. 2(b)] [10,11]. The thickness of ReS2 (~8.8 nm) and the geometrical channel length and width ratio (W/L = 1.14 μm/4.23 μm) were confirmed via atomic force microscopy (AFM, Park Systems, NX10) and optical microscopy (BX53M, Olympus) [Fig. 2(c)]. Static and LF noise measurements were performed under high-vacuum conditions (<10−6 Torr) using a commercial semiconductor analyzer (B1500A Keysight). The homemade LF noise measurement system consists of a battery box, low-noise current-to-voltage pre-amplifier (SR570, Stanford Research Systems), and data acquisition system (DAQ-4431, National Instruments).

Figure 2. (a) Cross-sectional view for the fabricated 1T′ ReS2 transistor. (b) Representative Raman spectrum of ReS2 multilayers. (c) Device image (bottom panel) and thickness profile along the white line (top panel) confirmed by AFM. (d) Output and (e) transfer characteristic curves of ReS2 multilayers under a high vacuum condition (<10−6 Torr).

The drain current–voltage (IDVD) output characteristic curves as a function of gate bias (VG) clearly demonstrate a pseudo-ohmic behavior, indicating suitable contact quality between the deposited Au metal and the ReS2 semiconductor from low to high VD regimes [Fig. 2(d)] [12,13]. In addition, the VD-dependent transfer curves (IDVG) manifest as a conventional n-type transistor, in which ID increases with VG at a given VD [Fig. 2(e)]. Furthermore, these observations rationalize the use of a linear approximation drain current model (ID ≈ µ⋅COX⋅(W/L)⋅(VGVT) VD, where µ, COX, and VT denote the carrier mobility, oxide capacitance per unit area, and turn-on voltage, respectively) for multilayer ReS2.

Previous studies have reported that the magnetic-flux-induced VH can be defined as VH = rH⋅α⋅BID/(en2D), where rH, α, B, e, and n2D are the Hall factor, geometrical correction factor, incident perpendicular magnetic field, elementary unit charge, and carrier density per unit area, respectively [13,8]. Notably, n2DCOX⋅(VGVT)/e and transconductance (gm) = ∂ID/∂VG = µ⋅COX⋅(W/L) VD when VGVT [14]. The absolute Hall effect sensitivity and current normalized SI can be described as SA = ΔVH/ΔB ∝ (rH⋅α)⋅ID/COX⋅(VGVT) ∝ gm/COX and SI = SA/ID ∝ 1/COX⋅(VGVT) [Fig. 1(a)] [2,8]. Here, rH⋅α is assumed 1 based on previous reports on graphene and MoS2 Hall sensors, in addition to our geometrical active channel ratio of L ≥ 3W [1,2]. Consequently, the electrostatic VD and VG bias conditions corresponding to the maximum gm result in the highest SA. Furthermore, to determine Bmin, the VH PSD (SV) should be obtained at a given ID in addition to SA [Figs. 1(b) and 1(c)].

Based on the analytical definition above, the ID-dependent SA and SI as a function of VD were numerically calculated by considering the obtained transconductance curves of the ReS2 multilayers [Figs. 3(a) and 3(b)]. The observed SA that is directly proportional to gm is enhanced by the VD at a given ID [Fig. 3(a)]. This is mainly because of gm dependence on VD in the linear approximation of the drain current model (gm = µ⋅COX⋅(W/L)⋅VDVD) [8,14]. To exclude the effect of ID on SA, VD-dependent SI (= SA/ID) curves are displayed in Fig. 3(b). As ID (or VG) increases, SI decreases. The expected maximum SI of the ReS2 multilayers is ≈ 900 V⋅A−1⋅T−1 at ID = 100 nA and VD = 3.0 V. In principle, SI should be identical regardless of VD. More specifically, SI directly connects not to VD but to [COX⋅(VGVT)]−1. The observed notable SI disagreement in this figure can be attributed to the following: the enhancement of carrier mobility [Fig. 3(c)], downshifted VT, and suppressed interlayer resistance along the c-axis of ReS2 in addition to the reduction of contact resistance with increasing VD.

Figure 3. Numerically calculated (a) SA and (b) SI of multilayer ReS2 as a function of VD. (c) VD-dependent maximum field-effect carrier mobility [µFE = gm⋅L/(COXW VD)].

To estimate SV and Bmin, LF noise measurements were performed under different VG conditions as a function of VD. Although SV should be obtained from the voltage probes between the transverse Hall bars at a given ID, SV is directly converted to drain current noise PSD (SID) according to the noise equivalent relation (SID/ID 2 = SV/VD 2 → SV = SID × R2, where R (= VD/ID) is the total resistance) [15]. Notably, the calculated SV represents the maximum amplitude of the ReS2 multilayer, revealing the worst case of Bmin (= SV0.5/SA). The thermal noise PSD (SV = 4⋅kBTR, where kB and T are the Boltzmann constant and absolute temperature, respectively) can be regarded as the minimum SV, representing the best case of Bmin [15]. Figure 4(a) shows the LF noise characteristics of multilayer ReS2 as a function of VG at VD = 1.0 V. The clear observance of the 1/f trend from 5 Hz to 5 kHz indicates uniformly distributed trap sites in space and the energy density of VG [16,17]. To gain further insight into the drain current variation mechanism with the oxide surface trap density (NST) at the quasi-Fermi level, the current-normalized SID curves obtained at frequency (f) = 10 Hz for different VD were fitted to the CNF (= SID/ID2 = e2kBTNST/(LWCOX 2f)⋅(gm/ID)2) noise model [Fig. 4(b)] [16,17]. A clear correlation can be observed between the LF noise data and CNF from the subthreshold to the high electron accumulation region, indicating that the dominant charge scattering mechanism is carrier trapping/de-trapping into oxide trap sites. The determined NST ranges from 3.6 × 1011 to 5.5 × 1011 cm−2⋅eV−1 in the observed VD regimes.

Figure 4. (a) VG-dependent SID of the ReS2 multilayer at VD = 1.0 V. (b) Currentnormalized SID as a function of VD (symbols) and the corresponding CNF fitting curves (dotted lines) obtained at f = 10 Hz.

Figure 5(a) shows the computed VG (or ID)-dependent Bmin (=SV 0.5/SA) at VD = 1.0 V based on the corresponding SA [Fig. 3(a)] and SV as shown in Fig. 5(a). Here, the total SV (= SV_Thermal + SV_Flicker) mainly originates from thermal (SV_Thermal) and 1/f flicker (SV_Flicker) noise, as described in the following form [8]:

Figure 5. (a) VG-dependent Bmin of the ReS2 multilayer at VD = 1.0 V. (b) Thermal (closed symbols) and flicker (open symbols) noise contribution to currentdependent Bmin at f = 300 Hz for different VD values. (c) Corresponding relative Bmin ratio of Bmin_Thermal/Bmin_Total (closed symbols) and Bmin_Flicker/Bmin_Total (open symbols), respectively.

Bmin=SVSA=SV_Thermal+SV_FlickerSA=4kBTR+VD2e2kB TNST fWLCOX2 gm ID 2rHαgm COX =Bmin_Thermal+Bmin_Flicker.

Based on this final expression, the contribution of SV_Thermal and SV_Flicker to the total Bmin (Bmin_Total = Bmin_Thermal + Bmin_Flicker) [Fig. 5(b)] and relative Bmin ratio of Bmin_Thermal/Bmin_Total and Bmin_Flicker/Bmin_Total [Fig. 5(c)] were calculated. When electrons are accumulated with increasing ID (or VG), the expected Bmin is inversely proportional to ID and Bmin gradually saturates, mainly owing to the enhanced contact resistance effects at high ID (or Vg) regimes, regardless of VD [Fig. 5(b)]. In addition, the estimated Bmin_Flicker is at least 100 times higher than Bmin_Thermal, implying the increasing importance of NST at a given conductivity. This insight can be further rationalized by the fact that the lowest Bmin_Flicker of the ReS2 multilayers (3–4 mG/Hz0.5 at f = 300 Hz) with a moderate carrier mobility (12−15 cm2⋅V−1⋅s−1) is almost similar to that of CVD graphene on SiO2 (0.5–90 mG/Hz0.5 at f = 300 Hz) with a relatively high mobility (1,600–6,900 cm2⋅V−1⋅s−1). This manifests the optimization of NST of the supporting substrate, which is crucial for suppressing Bmin_Flicker [1,79,18].

We propose various analytical expressions for the absolute SA, current-normalized SI, and Bmin of 2D ReS2 multilayers to investigate the magnetic-sensor limitations of 2D vdW multilayers for Hall elements. The drain-voltage-dependent transconductance and current PSD are considered to determine the minimum Bmin. From a practical perspective, because high carrier mobility at a given conductivity is the key requirement for achieving the best Hall element, the choice of an active channel material with low contact resistance and surface trap density should be considered carefully to achieve a high-performance magnetic sensor. Our findings provide an effective approach for the optimization of 2D multilayer-based Hall elements.

This study was supported by an NRF grant funded by the Korean Government (MSIT) (NRF-2022R1A2C4001245). The authors also thank Minji Chae, Yeongseo Han, Dahyun Choi, and Sooyeon Kim for their assistance with the Raman and electrical measurement data collection of the 2D ReS2 multilayers.

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