Hall-effect sensors have dominated most markets of magnetic sensors over the past several decades because of their high performance, small physical size and low cost. Their various applications such as electronic compass, position detecting, current sensing, and contactless switching have made them the most popular type of magnetic devices. This is even truer since silicon Hall sensors are easily integrated on complementary metal oxide technology (CMOS) wafers with the readout electronics^[1-4]. Generally, the Hall sensors enjoy wide detection range from 10 μT to 20 T, high spatial resolution smaller than 1 μm, and a wide band-width from DC to over 1 MHz^[5]. However, with the increased use of Hall-effect sensors both in industry and research fields, studying higher performance materials and fabrication process for Hall-effect sensors becomes super important.

We have previously demonstrated a high-performance Hall sensor in CMOS technology with its integrated readout circuit^[2]. According to our experience on CMOS hall sensors, the mobility is a critical material parameter of Hall-effect device, because large mobility Hall-effect device always means large sensitivity and low power consumption. Table 1 shows the electron mobility and energy gap of four common materials of Hall-effect devices and gallium nitride (GaN) studied recently for Hall-effect devices^{[6, 7]}. In mobility, Si < GaN < GaAs < InAs < InSb ^[8]. According to the positive correlation between the drift velocity of carriers and the electron mobility, higher carrier mobility can bring higher magnetic sensitivity of the Hall device. Although indium arsenide (InAs) and indium antimonide (InSb) have ultra-high mobility, their narrow energy gap leads to bad temperature performance and limits their applications. GaAs is a superior material for Hall-effect device, as it has large mobility and energy gap. GaAs Hall-effect sensors have emerged in previous decades, but they are not popular as silicon Hall-effect sensor because of their high price. However, the technology has attracted a lot of attention in recent years, because it is an outstanding material for radio frequency (RF) and microwave components, and 5G communication technology which promotes its development. The studying of high-performance GaAs Hall-effect sensors are becoming gradually burning issues at present. A GaAs Hall-effect sensor with a compact design－an amplifier consisting of 5 pseudomorphic high electron mobility transistors (pHEMT) and bias circuits consisting of a pHEMT and two diodes－can obtain the performance better than mature commercial low-noise CMOS Hall-effect sensor, and it can detect the magnetic field as low as 177 nT^[9]. After combining the noise suppression circuits, GaAs Hall-effect sensor is able to detect the magnetic field as low as 30 nT^[10].

In addition, other benefits of GaAs based Hall sensors include: (a) a high Hall constant about 6250 cm³/C with an electron concentration of 10¹⁵ cm⁻³; (b) a high thermal conductivity about 0.81 W/(cm·K), utilizing an advantage of the heat released in the epitaxial layer; (c) almost any custom shape can be achieved in photolithographic nanofabrication process; (e) an excellent temperature stability due to the 1.4 eV energy gap of GaAs at 300 K; (d) a high resistivity about 10⁵ Ω∙cm; and (e) an ultra-thin thickness without changing desired properties of the bulk materials.

Considering there is no specified modelling of GaAs Hall-effect device, in this paper, a constant voltage bias modelling of GaAs Hall-effect device is presented by using commercial finite element method (FEM) software Silvaco TCAD^® to guide and further study the GaAs Hall-effect sensor. The device has been fabricated, and its experimental results of sensitivity performance and temperature performance are compared with simulation results. The important thing is that this model can also be used for the simulation of Hall sensors made of materials such as Si, Ga and InSb. In the design process, we can choose different types of materials, and we can also choose the corresponding electron mobility according to different manufacturing processes.

The layer structure of our Hall magnetic sensor is shown in Fig. 1. The 300 nm N⁺ GaAs with 10¹⁷ cm⁻³ uniform Si doping is the active layer to induce Hall output, and it is also used to form ohmic-contact with metal wire. AlGaAs has a larger band gap than GaAs, which can form a barrier to prevent the migration of electrons. The 20 nm Al_0.24Ga_0.76As is used as the barrier layer to make electron be restricted in N⁺ GaAs space. Periodic Al_0.24Ga_0.76As and GaAs form a superlattice structure which is used as a buffer layer and to produce periodic barriers for further preventing current flowing away from the substrate. The bottom layer is the GaAs substrate which is to provide mechanical strength and grow high quality GaAs epilayer.

The Hall-effect sensor is based on the physical effect produced by a current-carrying semiconductor in a magnetic field. Its output voltage is directly proportional to the magnetic field strength. Currently, the cross-shaped model is widely employed as an optimum geometry to achieve the highest sensitivity and lowest offset^[2]. In addition, such symmetric geometry can dramatically decrease the mismatch resulting from the fabrication process^[11]. In our model, a conventional constant voltage bias Hall sensor is adopted, which can reduce the design complexity of bias circuit^[7]. A constant voltage supply (V_bias) is utilized to bias the Hall-effect sensor and then an output voltage (V_hall) as a result of an external magnetic field is measured at terminals. The Hall voltage with a voltage biasing can be expressed as^[11]

$ V_{{\rm{hall}}} \cong \mu_{\rm{H}} G_{\rm{H}} \frac{W}{L}B V_{\rm{bias}} , $  (1)

where μ_H denotes the Hall mobility of majority carriers, W and L represent the width and length of the sensor respectively, B is the external magnetic field vertical to the device surface, G_H is the geometrical correction factor of Hall voltages. G_H is usually determined by the actual sensor structure and the shape of the contact pole, and the value range is 0 < G_H < 1. For a sensor structure that is shaped like a crisscross type and whose rotating 90° structure remains unchanged, the structure factor can be approximated by the following Eq. (2):

$ \mathop G\nolimits_{\rm{H}} = 1 - 1.045\exp \left[ - \frac{{\pi (L - W)}}{{2W}}\right]\frac{{\mathop \theta \nolimits_{\rm{H}} }}{{\mathop {{\rm{tan}} \theta }\nolimits_{\rm{H}} }} , $  (2)

where θ_H denotes the Hall angle. Reducing the width-to-length ratio W/L of the Hall device can improve the device geometrical correction factor, which will also improve the sensor performance. However, as the width of the Hall device increases, the area of the sensor increases accordingly. Therefore, it is necessary to compromise the aspect ratio of the device within an appropriate range. For a cross-type horizontal Hall sensor, G_H value can usually reach approximately 1. In addition, the absolute sensor sensitivity can be defined from the Eq. (3). The sensitivity depending on the voltage output quantity can be given by^[12]:

$ S_{Av} = \frac{{V_{\rm{Hall}}}}{B} , $  (3)

which represents the capability of the sensor to convert the magnetic field into a Hall voltage. For eliminating the influence of bias voltage on device sensitivity, the relative sensitivity is finally defined as the Eq. (4) dividing by the bias voltage, shown below^[12]:

$ S_{Vv} = \frac{{V_{\rm{Hall}}}}{{V_{\rm{bias}} B}}$  (4)

In FEM, the quantity relationship of Hall-effect is defined in Eq. (5)^[13]:

$ \left[\begin{array}{c}{J}_{x}\\ {J}_{y}\\ {J}_{z}\end{array}\right]=\frac{1}{1+{\mu }_{\rm{H}}^{2}{B}_{z}^{2}}\left[\begin{array}{ccc}1 & -{\mu }_{\mathrm{H}}{B}_{z} & 0\\ {\mu }_{\rm{H}}{B}_{z} & 1 & 0\\ 0 & 0 & 1+{\mu }_{\mathrm{H}}^{2}{B}_{z}^{2}\end{array}\right]\left[\begin{array}{c}{J}_{x0}\\ {J}_{y0}\\ {J}_{z0}\end{array}\right]. $  (5)

When an external magnetic field is applied, the changes of current density are equal to multiplying a tensor, where are the instinct current with no magnetic field. The tensor is consisted with magnetic field B_z and Hall mobility μ_H which is equal to the Hall scattering factor multiplying carrier mobility. The classical value of Hall scattering factor is about 1.1 in this situation^[13]. 8500 cm²/(V·s) electronic mobility of GaAs can be adopted in the modelling. However, concerning mobility is a doping and temperature dependent parameter, its specified physical model should be considered for accuracy modelling. In order to simplify the calculation, the mobility approximation is used to replace the Hall mobility in most cases. Both Hall mobility and mobility are related to temperature and doping. The Caughey-Thomas model is shown in Eq. (6) represents the change of Hall mobility with temperature and doping.

$ \mu_{\rm{n}} = \mu_{\rm{min}} \cdot {\left( {\dfrac{{T_{\rm{L}}}}{{300}}} \right)^\alpha } + \dfrac{{\mu_{\rm{max}} \cdot {{\left( {\dfrac{{T_{\rm{L}}}}{{300}}} \right)}^{ - \beta }} - \mu_{\rm{max}}\cdot {{\left( {\dfrac{{T_{\rm{L}}}}{{300}}} \right)}^\alpha }}}{{1 + \mu_{\rm{max}}\cdot {{\left( {\dfrac{{T_{\rm L}}}{{300}}} \right)}^\gamma } \cdot {{\left( {\dfrac{N}{{N_{\rm 0}}}} \right)}^{ - \delta }}}}, $  (6)

where μ_min and μ_max represent the mobility saturation at the highest and lowest doping concentrate respectively, N₀ denotes the doping concentrate at which mobility reduces to half of μ_max value, N denotes the concentration of ionized donors. T_L denotes the lattice temperature.α, β, γ, δ are adjustment parameters^[14].

In Caughey-Thomas mobility model of Silvaco, μ_min = 500 cm²/(V·s), μ_max = 8600 cm²/(V·s), α = 0, β = –2.1, γ = –1.182, δ = 0.394, N₀ = 6 × 10¹⁶ cm^–3. In Ref. [14], μ_min = 500 cm²/(V·s), μ_max = 9400 cm²/(V·s), α = 0, β = −2.1, γ = −1.182, δ = 0.394, N₀ = 6 × 10¹⁶ cm^–3, meaning they use the Hall mobility and assume μ ~ μ_H .

COMSOL Multiphysics is a multiphysics simulation software based on finite element analysis methods, which can simulate and analyze actual engineering problems involved in many fields including electromagnetics, structural mechanics, acoustics, fluid mechanics, and thermals. Because the COMSOL finite element simulation software has a parameter scanning function, COMSOL is used to explore the geometric factors of the Hall element. The shape of the fully symmetrical cross Hall element is shown in Fig. 2(a). The input and output ports have the same length L and width W. When studying this structure, the ratio of W and h is more commonly used instead of length L and width W as the abscissa of the coordinate axis. The simulation result is shown in Fig. 2(b). At the beginning, the Hall voltage increases with W/h. When W/h is about 3, the sensitivity of the fully symmetrical cross-shaped Hall disk is the greatest. When W/h is greater than 3, the Hall voltage tends to decrease. The narrow cross-shaped Hall element studied in this paper is shown in Fig. 3(a), which is similar in shape to a fully symmetrical cross-shaped Hall plate, but the output interface is narrower. Because the width and length of the output port have a small effect on the input resistance, it has a small effect on the static operating point of the Hall element. Therefore, this paper studies the influence of the changes in the width W’ and length L’ of the output port of the narrow cross-shaped Hall element on the Hall voltage. Based on the fully symmetrical cross-shaped Hall plate when the W/h is about 3 and the Hall voltage output is the maximum. Take the fully symmetrical cross-shaped Hall element at the maximum W/h of 3 Hall voltage output as the reference. By changing the width of the output port of the reference Hall element, the Hall voltage of the narrow cross Hall element in Fig. 3(b) varies with the width of the output port. It can be concluded that the smaller the output port, the greater the Hall voltage. When W’/W = 0.5, that is, when the output port is half of the input port, the Hall voltage output growth tends to slow down as W’/W decreases.

Numerical simulation of the Hall-effect sensor is performed using Silvaco TCAD^®. Previous simulations of Hall sensors are summarized in Ref. [15]. Fig. 4 shows our proposed structure of the Hall-effect sensor. The color legend indicates the surface potential distribution by applying a magnetic field of 100 mT. The model geometry of the Hall sensor has been simulated without any mismatch. The ground and supply voltage are added to cathode and anode of Hall-effect sensor.

The 2D and 1D electrons distribution of vertical cut-plane are demonstrated in Figs. 5(a) and 5(b). Although Al_0.24Ga_0.76As is just below the N⁺ doping epilayer, it almost has no carrier because of its larger energy gap barrier which stops carriers in GaAs flowing to Al_xGa_1–xAs. So the electrons are inside the potential well. Electrons are restricted in the thin N⁺ GaAs channel, which forms the two-dimensional electron gas (2DEG) structure confined in the potential well near Al_xGa_1–xAs/GaAs interface. The final output voltage of the Hall sensor is demonstrated in Fig. 5(c). The value of y-axis represents the Hall voltage. It is noted that the output is linearly increased by changing the magnetic field from 0 to 100 mT. The voltage response is very linear, and the maximum output voltage of 0.25 V is obtained. The sensitivity of the Hall effect device is 0.51 V/(V·T) .

Fig. 6 illustrates the microphotograph of the Hall-effect sensor. The red line cross-shape represents the active region of dthe evice, but the effective region of the device doesn’t include pads in white color, and its maximum width and length are 110 and 90 μm, respectively. The sensor manufacture is compatible with pHEMT processing technology^[9]. There are 4 main steps in its manufacture. Firstly, isolating the device through implant Boron. Secondly, Au/Pt/Ti metals are grown on N⁺ GaAs epilayer, which is used to form Ohmic contact—a classic contact for avoiding forming nonlinear I–V relationship between the metal and the semiconductor. Thirdly, Si₃N₄ is deposited on the wafer surface and etched in the pad to protect the device. Finally, a metal wire is grown for its connection to circuits.

During the experiment, electromagnet Helmholtz coil is used inducing magnetic field. Hall-effect sensor is put in the center of Helmholtz coil for obtaining a relatively constant magnetic field, and there has no consideration for shielding geomagnetic, because it induces the Hall voltage less than 0.1 mV. Guassmeter HT208 (0.1 μT resolution) from Hengtong Inc. is used to measure the induced magnetic fields from the coils. In terms of temperature performance, temperature chamber CST842T from Hhtesting Inc. is used to generate temperature from −75 to 200 °C, providing a constant temperature environment. The measured sensitivity at room temperature is shown in Fig. 7(a). The average is 0.28 V/(V·T), which is in the same order of magnitude as the simulation result, but it still has large mismatch, so the model resolution should be improved, which will be further discussed.

Moreover, the temperature dependence of the final output voltages is shown in Fig. 7(b), with the temperature ranging from −40 to 150 °C at 50 mT vertical magnetic field. The highest measured value of the Hall voltage is V_H = 72.5 mV at −40 °C. In addition, it is clearly noticed that the output signal slightly drops as the temperature increases. It is worth mentioning that the increasing temperature brings about the raise of carrier scattering, which leads the decrease of carrier mobility of Hall sensor. Therefore, the higher the temperature, the smaller the carrier mobility, which results in the lower Hall output voltage.

Finally, Fig. 7(c) performs the offset with different misalignment of output contacts from −300 to 300 nm to find the lowest offset. The measurement is at room temperature and the external applied magnetic field is zero. Ideally, non-misalignment of output contacts will obtain zero output voltage. However, non-ideal factors of piezoelectric effect and discontinuity of material will result in an asymmetry of the device property and then generate offset voltage. It can be alleviated by misaligning output contacts. The offset is equal to 5 mV when the misalignment is 0. When there is 100 nm mismatch, the device approximately has the lowest zero offset. Subsequently, a standard CMOS-based analogue front-end circuit comprises of an operational amplifier and a low-pass filter could achieve on-chip noise and offset suppression.

If only using the Shockley-Read-Hall Recombination model, a common model in semiconductor, it can obtain 0.44 V/(V·T) sensitivity, which leads to large error compared with 0.28 V/(V·T) in experimental results. Aiming at narrowing the difference between the simulation and experimental results, models need to be improved. Considering the Eq. (5), carrier mobility μ must be corrected. It is higher in previous simulation work. Carrier mobility is always related with the temperature and the doping concentrate, so analytic physical model is used to correct the device modelling. The quantity relationship of analytic physical model is defined in 6, where μ₁, μ₂, α, β, γ, δ, N₀ are constant parameters related to material property and they can be obtained in the datasheet. T_L and N are lattice temperature and carrier concentration, respectively, which are related to the user defined doping concentration and temperature. On the other hand, the simulation model of Hall device is specified into 3 sub-models according to its epilayers. Single GaAs layer simulation, GaAs and Al_xGa_1−xAs two-layer simulation, and all epilayers simulation are built for studying the model pithily, effectively and accurately.

Figs. 8(a) and 8(b) show the new simulation results of Hall outputs. In order to explore the difference between GaAs Hall sensor and Al_0.24Ga_0.76As/GaAs Hall sensors, we further conducted some tests on individual GaAs Hall devices to explore the influence of Al_0.24Ga_0.76As layer on the sensor. The experimental results are also added for comparison. The two different epilayer simulations show slight differences. That means only GaAs channel layer needs to be considered, which can effectively reduce the simulation complexity and time. It also shows Al_xGa_1−xAs is a suitable barrier material to form 2DEG. After considering analytic physical model, the simulation results are highly in consistent with the experimental results. The simulated sensitivity is also 0.28 V/(V·T). Furthermore, simulation results are shown in Fig. 8(b), which are in consistent with the experimental results after using analytic physical mode.

Table 2 summaries and compares simulation results using the Caughey-Thomas model with different parameters and our experimental results at 300 K. Regarding the mobility and Hall mobility, Refs. [16, 13] are relatively low. We used the Caughey-Thomas mobility model provided in Ref. [13] and got a voltage sensitivity of 0.24 V/(V·T). We used another Caughey-Thomas mobility model provided by Dr Hadi Heidari in Ref. [16] and got a voltage sensitivity of 0.26 V/(V·T). If only using the common model in semiconductor, it can obtain 0.44 V/(V·T) sensitivity, but it leads to large error compared with 0.28 V/(V·T) in experimental results. It can be seen that we have obtained more accurate sensitivity after we have corrected the mobility model.

Table 3 summaries specifications of our fabricated Al_0.24Ga_0.76As/GaAs Hall-effect sensor and compares with other works in terms of the sensitivity, offset, and input/output resistances. From the experimental results of our developed Hall sensor, four times higher sensitivity has been achieved compared to the state-of-the-art Hall sensors based on Si^[15] and GaN^[7]. It operates at the highest absolute sensitivity of 1.38 V/T. Also, it achieves a relatively low offset compared with the GaAs Hall-effect device in Ref. [18].

The simulation results are highly consistent with the experimental output. μ_min = 500 cm²/(V·s), μ_max = 8600 cm²/(V·s),α = 0, β = −2.1, γ = −1.182, δ = 0.394, N₀ = 6 × 10¹⁶ cm^–3 are adopted for fitting the experimental data.

We have developed a GaAs based Hall sensor and its constant voltage-bias modelling. We used the COMSOL multiphysics simulation tool to study the influence of geometry on the Hall voltage. It is found that the sensitivity of the fully symmetrical Hall element is optimal when W/h is equal to 3. Studies have found that the sensitivity of the narrow cross is better than that of a fully symmetrical Hall element, and it is optimal when the length of the long contact port is twice the length of the short contact port. The cross-shaped Hall sensor has been simulated using Silvaco TCAD^® to investigate the sensitivity and temperature performance. Simulation results are highly in consistent with the experimental results. The high sensitivity of 0.28 V/(V·T) is obtained from the experiment with a low offset of 5 mV. A high absolute sensitivity of 1.38 V/T is achieved at 5 V bias voltage. Moreover, the temperature dependence of the Hall sensor is investigated for the maximum Hall voltage of 72.5 mV and the sensitivity of 0.275 V/(V·T) at −40 °C. The future work will focus on the design and implementation of a whole Hall microsystem with readout circuits using standard CMOS technology.

The work of Hua Fan was supported by the National Natural Science Foundation of China (NSFC) under Grant 61771111, supported by Sichuan Provincial Science and Technology Important Projects under Grant 22ZDYF2805, supported by the Open Foundation of the State Key Laboratory of Electronic Thin Films and Integrated Devices under Grant KFJJ202006, and supported by Intelligent Terminal Key Laboratory of Sichuan Province under Grant SCITLAB-1001.

The work of Quanyuan Feng was supported by Major Project of the National Natural Science Foundation of China under Grant 62090012.