A Hall element is a magnetic sensor that detects the presence of magnetic flux and converts this magnetic signal to a Hall voltage (V_H) proportional to the magnetic intensity when the current flows perpendicularly to the magnetic fields [1–3]. Based on this simple yet powerful relationship between V_H and the intensity of magnetic fields, numerous applications of Hall elements have been demonstrated for tachometers, machinery switches, speedometers, and acceleration sensors [2–5]. 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 [1–3]. 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) [1,2,7–9]. 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 (S_A), currentnormalized Hall sensitivity (S_I), and magnetic resolution (B_min) 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 S_A, S_I, and B_min are also discussed.

Figure 1. Prerequisites for B_min estimation. General trend of (a) B-dependent V_H, (b) PSD of V_H, and (c) B_min, respectively. V_H, B, S_V, and f denote the Hall-effect voltage, magnetic fields, voltage PSD obtained from Hall bar electrodes for V_H 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 E_g (≈ 154.7 cm⁻¹), A1g-like (≈ 214.7 cm⁻¹), and E_g-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 (I_D−V_D) output characteristic curves as a function of gate bias (V_G) clearly demonstrate a pseudo-ohmic behavior, indicating suitable contact quality between the deposited Au metal and the ReS₂ semiconductor from low to high V_D regimes [Fig. 2(d)] [12,13]. In addition, the V_D-dependent transfer curves (I_D−V_G) manifest as a conventional n-type transistor, in which I_D increases with V_G at a given V_D [Fig. 2(e)]. Furthermore, these observations rationalize the use of a linear approximation drain current model (I_D ≈ µ⋅C_OX⋅(W/L)⋅(V_G − V_T) V_D, where µ, C_OX, and V_T 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 V_H can be defined as V_H = r_H⋅α⋅B⋅I_D/(e⋅n_2D), where r_H, α, B, e, and n_2D are the Hall factor, geometrical correction factor, incident perpendicular magnetic field, elementary unit charge, and carrier density per unit area, respectively [1–3,8]. Notably, n_2D ≈ C_OX⋅(V_G−V_T)/e and transconductance (g_m) = ∂I_D/∂V_G = µ⋅C_OX⋅(W/L) V_D when V_G ≥ V_T [14]. The absolute Hall effect sensitivity and current normalized S_I can be described as S_A = ΔV_H/ΔB ∝ (r_H⋅α)⋅I_D/C_OX⋅(V_G−V_T) ∝ g_m/C_OX and S_I = S_A/I_D ∝ 1/C_OX⋅(V_G−V_T) [Fig. 1(a)] [2,8]. Here, r_H⋅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 V_D and V_G bias conditions corresponding to the maximum g_m result in the highest S_A. Furthermore, to determine B_min, the V_H PSD (S_V) should be obtained at a given I_D in addition to S_A [Figs. 1(b) and 1(c)].

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

Figure 3. Numerically calculated (a) S_A and (b) S_I of multilayer ReS₂ as a function of V_D. (c) V_D-dependent maximum field-effect carrier mobility [µ_FE = g_m⋅L/(C_OX⋅W V_D)].

To estimate S_V and B_min, LF noise measurements were performed under different V_G conditions as a function of V_D. Although S_V should be obtained from the voltage probes between the transverse Hall bars at a given I_D, S_V is directly converted to drain current noise PSD (S_ID) according to the noise equivalent relation (S_ID/I_D 2 = S_V/V_D 2 → S_V = S_ID × R2, where R (= V_D/I_D) is the total resistance) [15]. Notably, the calculated S_V represents the maximum amplitude of the ReS₂ multilayer, revealing the worst case of B_min (= S_V^0.5/S_A). The thermal noise PSD (S_V = 4⋅k_B⋅T⋅R, where k_B and T are the Boltzmann constant and absolute temperature, respectively) can be regarded as the minimum S_V, representing the best case of B_min [15]. Figure 4(a) shows the LF noise characteristics of multilayer ReS₂ as a function of V_G at V_D = 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 V_G [16,17]. To gain further insight into the drain current variation mechanism with the oxide surface trap density (N_ST) at the quasi-Fermi level, the current-normalized S_ID curves obtained at frequency (f) = 10 Hz for different V_D were fitted to the CNF (= S_ID/I_D² = e²⋅k_B⋅T⋅N_ST/(L⋅W⋅C_OX ²⋅f)⋅(g_m/I_D)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 N_ST ranges from 3.6 × 10¹¹ to 5.5 × 10¹¹ cm⁻²⋅eV⁻¹ in the observed V_D regimes.

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

Figure 5(a) shows the computed V_G (or I_D)-dependent B_min (=S_V ^0.5/S_A) at V_D = 1.0 V based on the corresponding S_A [Fig. 3(a)] and S_V as shown in Fig. 5(a). Here, the total S_V (= S_{V_Thermal} + S_{V_Flicker}) mainly originates from thermal (S_{V_Thermal}) and 1/f flicker (S_{V_Flicker}) noise, as described in the following form [8]:

Figure 5. (a) V_G-dependent B_min of the ReS₂ multilayer at V_D = 1.0 V. (b) Thermal (closed symbols) and flicker (open symbols) noise contribution to currentdependent B_min at f = 300 Hz for different V_D values. (c) Corresponding relative B_min ratio of B_{min_Thermal}/B_{min_Total} (closed symbols) and B_{min_Flicker}/B_{min_Total} (open symbols), respectively.

Based on this final expression, the contribution of S_{V_Thermal} and S_{V_Flicker} to the total B_min (B_{min_Total} = B_{min_Thermal} + B_{min_Flicker}) [Fig. 5(b)] and relative B_min ratio of B_{min_Thermal}/B_{min_Total} and B_{min_Flicker}/B_{min_Total} [Fig. 5(c)] were calculated. When electrons are accumulated with increasing I_D (or V_G), the expected B_min is inversely proportional to I_D and B_min gradually saturates, mainly owing to the enhanced contact resistance effects at high I_D (or Vg) regimes, regardless of V_D [Fig. 5(b)]. In addition, the estimated B_{min_Flicker} is at least 100 times higher than B_{min_Thermal}, implying the increasing importance of N_ST at a given conductivity. This insight can be further rationalized by the fact that the lowest B_{min_Flicker} of the ReS₂ multilayers (3–4 mG/Hz^0.5 at f = 300 Hz) with a moderate carrier mobility (12−15 cm²⋅V⁻¹⋅s⁻¹) is almost similar to that of CV_D graphene on SiO₂ (0.5–90 mG/Hz^0.5 at f = 300 Hz) with a relatively high mobility (1,600–6,900 cm²⋅V⁻¹⋅s⁻¹). This manifests the optimization of N_ST of the supporting substrate, which is crucial for suppressing B_{min_Flicker} [1,7–9,18].

We propose various analytical expressions for the absolute S_A, current-normalized S_I, and B_min 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 B_min. 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.

The authors declare no competing financial interest.