Hexagonal boron nitride（h-BN）features a layered honeycomb structure comprising boron and nitrogen rings，as shown in Fig. 1（a）^［1-6］. The unique crystal structure grants h-BN an impressive array of properties，including wide bandgap，low dielectric constant，high breakdown voltage，exceptional chemical stability，flat surface free from dangling bonds and charged impurities^［7-13］. These remarkable attributes firmly establish h-BN as an exemplary dielectric material，garnering widespread utilization in pioneering micro- and nano- electronic device applications. Furthermore，it's worth noting that h-BN displays opposite signs for its in-plane and out-of-plane relative dielectric constants within the mid-infrared band^［14］. This unique behavior suggests that h-BN can be excited to produce particularly strong phonon resonances in the mid-infrared band. Due to this characteristic，it has been used to design high-performance electro-optic devices，such as electro-optic modulators，absorbers，and detectors^［15-18］. Its potential in optical applications，particularly in the realm of infrared optoelectronics，remains highly promising.

Fig. 1（b）illustrates the energy band diagram of a typical metal/insulator/metal（MIM）structure. When a positive voltage（V＜φ₀）is applied to metal2（M2），the Fermi level（E_F）of M2 decreases（Fig. 1（c）），leading to the flow of electrons across the whole trapezoidal barrier，which corresponds to the direct tunneling（DT）. Conversely，upon increasing the applied bias voltage（V＞φ₀），the Fowler-Nordheim（FN）tunneling becomes prominent，where electrons tunnel through a triangular potential barrier，as illustrated in Fig. 1（d）. While h-BN layers with different thicknesses have been extensively utilized in device fabrication，previous research efforts have primarily focused on investigating the overall electrical properties of these devices^［19-25］. However，the intrinsic physical properties of h-BN layers with different thicknesses have received limited attention.

In this paper，with the combination of experimental and theoretical research methods，we have successfully extracted the effective mass in ultrathin h-BN layers through simulating the current-voltage（I-V）curves of MIM structures. It is found that the effective mass in h-BN exhibits a consistent decrease as the number of atomic layers increases within the nanometer range. This phenomenon underscores the high susceptibility of the physical properties of a few h-BN layers to external factors. Moreover，through effective mass calculations linked to the number of layers，we can precisely determine the physical parameters of the FN tunneling model during electron tunneling in h-BN. This has significant implications for optimizing the design and utilization of gate dielectric and tunneling layers. This work provides valuable insights and hold crucial relevance for promoting applications in the domain of h-BN-based infrared physics and technology.

It has been verified that the tunneling process in h-BN is predominantly governed by the FN tunneling process under high bias voltages. The corresponding tunneling current is nonlinearly expressed as^［19-22］：

where $A=\frac{q^{\mathrm{3}}m}{\mathrm{8}\mathrm{\pi }h{\phi }_{\mathrm{B}}{m}^{\mathrm{*}}}$，$B=\frac{\mathrm{8}\mathrm{\pi }\sqrt[]{\mathrm{2}{m}^{\mathrm{*}}}{\phi }_{\mathrm{B}}^{\frac{\mathrm{3}}{\mathrm{2}}}}{\mathrm{3}hq}$，and the electric field in the insulator $F_{\mathrm{i}\mathrm{n}\mathrm{s}}=\frac{V}{d}$. Thus，formula（1） can be re-expressed in the following form：

where S_eff and φ_B are effective tunneling area and barrier height，respectively，h is Planck's constant，d is the thickness of the insulator h-BN，V is the applied bias voltage，and m^* is the effective mass of the electrons in the conduction band of h-BN.

Prior research has explored the tunneling process of electrons in ultra-thin h-BN with thickness exceeding 4 layers^［22］. However，the impact of layer variation of 2D materials with a limited number of layers on intrinsic physical properties has been underappreciated in the past，such as bandgap and effective mass^［26］. In this work，we adopt an improved research method for FN tunneling. By emulating the experimental the electric properties of MIM structures，we extract the effective mass and S_eff concerning different h-BN layer numbers. In line with conventional methods for processing physical data，we attribute all other non-ideal factors with layer dependency to effective mass. The detailed calculation method and process are outlined below. In order to obtain ln（I/V²）-1/V curve，a mathematical transformation is applied to the experimental I-V curve of a h-BN film，as shown in the inset of Fig. 2（a）. Subsequently，the curve is subjected to fitting analysis to determine the slope value，which can be known according to formula（2）：

Give that the h-BN affinity energy is 2 eV^［27-29］ and the work function of Gold is 5 eV^［30-31］，the barrier height φ_B can be determined. By substituting this value into formula（3），the effective mass of electrons in h-BN can be calculated out. Thereafter，the two essential physical parameters，A and B，within the FN tunneling formula can be derived.

To determine S_eff of h-BN at different layers，current-voltage values are extracted from the I-V curve in Fig. 2（a）. These values are then used in tunneling current formula（1）. Subsequently，the comprehensive average of the tunneling area is obtained. The calculated S_eff reaches a satisfactory agreement with the device size of 25 nm（±10 nm）reported in this experiment. As a consequence，the validity of the h-BN effective mass value calculation is further affirmed from this juncture. Table 1 summarizes the calculated physical quantities for h-BN layers with varying layers from Ref. 22，including the effective mass，S_eff，the parameters A and B of the FN tunneling model. For d_h-BN=1.38 nm，the calculated S_eff is three orders of magnitude smaller than for other thicknesses. This discrepancy may be attributed to the extreme thinness of the h-BN，which can lead to uneven film thickness or irregular gap distances between the h-BN layer and the metal material during the experimental process^［32］.

Fig. 2（b）shows a strong agreement between the experimental（solid line）and calculated（signed line）curves of the current in the MIM structure. As the number of h-BN layers increases，the device requires a higher voltage to meet the FN tunneling conditions. Consequently，the device exhibits a significant enhancement in the breakdown voltage，which is the voltage at which the current in the device reaches 1×10^-11 A.

Fig. 3（a）depicts the layer-dependent effective mass of h-BN. As the thickness of h-BN increases，the effective mass of electrons in its conduction band decreases. Similar phenomena also exist in other 2D materials^［33-34］. When h-BN is in contact with Gold，the variation of the parameters A and B in the FN tunneling effect formula with respect to thickness is shown in Fig. 3（b）. The parameter A varies inversely with the effective mass，which essentially reflects the influence of effective mass. Meanwhile，the parameter B decreases with the increase of h-BN thickness，in accordance with the behavior of the effective mass. From a physical perspective，it is directly proportional to the effect of the electric field. Under the condition of the same voltage，a thinner h-BN thickness results in a stronger internal electric field，thereby enhancing the likelihood of FN tunneling occurrence. Consequently，the parameter B also becomes larger. Therefore，the accurate determination of the effective mass of the insulator material allows for the precise determination of the parameters of the FN tunneling model.

We further investigated the Gold/h-BN/Gold structure using semiconductor Sentaurus TCAD software，with the h-BN thickness set at 7.54 nm. The FN tunneling model and traps model were both incorporated at the Gold/h-BN interface. We conducted simulations with fixed charge concentrations ranging from 1×10¹¹~1×10¹² cm^-2 at the Gold/h-BN interface^［35］ and observed consistent results. Therefore，a single example will be provided for analysis and explanation. According to the simulation results shown in Fig. 4（a），regardless of fixed charge densities of 1×10¹¹ cm^-2 or 0，both the F_ins and the FN tunneling current density（J_FN）remain the same. Since the fixed charge is connected to the power supply through the metal. In an ideal scenario，the power supply is considered a device capable of providing an infinite amount of charge to the device. In contrast，fixed charge defects are very small，and their impact on the device can be neglected. From an alternative perspective，this also serves as evidence for the validity of the method used to extract the effective mass of h-BN from the I-V curve of the MIM structure.

The effect of different metallic materials on FN tunneling effect in h-BN have also been investigated. It is well known that the work function values of nickel（Ni）and tungsten（W）are about 5.2 and 4.6 eV^［36-37］，respectively. Thus，theoretically we can calculate the φ_B of Ni/h-BN and W/h-BN in contacts to be 3.2 and 2.6 eV，respectively. Then，the parameters A and B of the corresponding FN tunneling model can be obtained precisely. Since the metallic material remains consistent on both sides of the h-BN，the I-V characteristics exhibit symmetry under positive and negative bias. As shown in Fig. 4（b），the Ni/h-BN/Ni structure has the highest breakdown voltage. Under an applied voltage of 10 V，the J_FN in the device is two orders of magnitude smaller than that when W is in contact with h-BN. This difference indicates that the Ni/h-BN contact produces a much higher φ_B than the W/h-BN contact，thereby demanding greater energy for electron traversal across the barrier. However，the higher breakdown voltage of the Ni/h-BN device results in a delayed onset of current flow，leading to lower overall current under the same applied voltage. When extracting the effective mass of h-BN layers using the I-V curve of the MIM structure，it is imperative to accurately account for the effect of the metallic material employed. In addition，simulations revealed a direct correlation between the metal work function and device breakdown voltage，with an increase in work function resulting in higher breakdown voltage but lower current. When employing h-BN as a dielectric or tunneling layer，meticulous attention must be paid to the choice of metal materials for contact.

In summary，by employing a research methodology that combines experimental and theoretical approaches，we have successfully studied the effective mass and electrical properties of few-layer 2D h-BN films. The results reveal a distinct correlation that the effective mass consistently decreases with the increasing h-BN film thickness. Furthermore，Sentaurus TCAD simulations have been performed to verify that the fixed charge at the metal/h-BN interface essentially does not influence the FN tunneling current，confirming the reliability and accuracy of our approach to extract the h-BN effective mass through the I-V curve of the MIM structure. Additionally，different metallic materials significantly affect the FN tunneling in h-BN，which is attributed to the difference in barrier height caused by work function. Overall，this work is of great significance in advancing the application of 2D h-BN atomic layers in fields to be expanded in infrared physics and technology.