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The magnetoresistance effect of a p–n junction under an electric field which is introduced by the gate voltage at room temperature is investigated by simulation. As auxiliary models, the Lombardi CVT model and carrier generation-recombination model are introduced into a drift-diffusion transport model and carrier continuity equations. All the equations are discretized by the finite-difference method and the box integration method and then solved by Newton iteration. Taking advantage of those models and methods, an abrupt junction with uniform doping is studied systematically, and the magnetoresistance as a function of doping concentration, SiO_{2} thickness and geometrical size is also investigated. The simulation results show that the magnetoresistance (MR) can be controlled substantially by the gate and is dependent on the polarity of the magnetic field.

In view of the physical interest and potential applications, magnetoresistance (MR) effects have attracted a lot of theoretical and experimental attention. MR effects can be achieved by using non-magnetic materials such as doped silicon,^{[1]} GaAs,^{[2]} and magnetic materials.^{[3,4]} However, compared with magnetic material, non-magnetic material has a large MR ratio and the resistivity increases approximately linearly with the external magnetic field, which makes the non-magnetic material more attractive to random access memory, ultrasensitive magnetic field sensors,^{[5]} logic devices,^{[6,7]} etc. By virtue of long spin coherence^{[8,9]} and compatibility with the current CMOS technology, silicon is a promising material for investigating the magnetoresistance effect.

Many researches based on doped silicon have been carried out to explore the effect. Michael *et al*. have reported that the large positive magnetoresistance effect can be induced by breaking the quasi-neutrality of the space-charge effect.^{[10]} While another mechanism was also proposed, i.e., the magnetoresistive effect can be achieved by shrinking the acceptor wave functions in the direction perpendicular to the magnetic field.^{[11]} Meanwhile, it can also be achieved by the process of impact ionization^{[12]} which is controlled by magnetic field.

However, a few methods are put forward to regulate MR when the magnetic field is fixed. Though varying the geometrical size^{[13]} can achieve it, it is inconvenient for practical applications. In the present work, the magnetoresistance effect of the p–n junction induced by the space-charge effect under an electric field is studied by simulation. The results show that a wide range of magnetoresistance can be controlled, which implies that it is a more potential method to adjust magnetoresistance by using the electric field which is introduced by gate voltage.

Classical p–n theories proposed by Shockley^{[14]} are basic to analyse the properties of the p–n junction. This theory consists of a set of fundamental equations, which need to be modified slightly when they are used to simulate the properties of p–n under the external magnetic field and electric field.

Equations (*n* and *p* are the electron and hole density respectively, *R* represents the net recombination rate (including the Shockley–Read–Hall recombination rate and Auger recombination rate), and *q* denotes the elementary charge. Equation (*φ* is the electric potential and *N* is the ionized net doping concentration, *Q*_{T} represents the charge due to traps and defects with setting *Q*_{T} to be zero). Equations (^{[15]} here, *μ*_{n} and *μ*_{p} are the electron and hole drift mobility, ^{[16]} (*D*_{n} and *D*_{p} denote the electron and hole diffusion coefficient.

To obtain accurate results, the Lombardi CVT Model,^{[17]} which contains the effect of a transverse field, doping- and temperature-dependent carrier mobility, is used. In this model, the mobility is given by three components that are composed according to the Matthiessen rule:

On the right-hand side of Eq. (*μ*_{ac} is the mobility limited by surface phonon scattering, and *μ*_{b} is the mobility determined by both doping concentration and temperature. The final component *μ*_{sr} takes into account the surface roughness scattering. More details are shown in Ref. [17].

For numerical simulation of a semiconductor device, the carrier generation-recombination is the crucial process that must be considered. In this paper, Shockley–Read–Hall recombination^{[18]} and Auger recombination^{[19]} within the bulk of the semiconductor are adopted.

*E*

_{i}and

*E*

_{t}give the intrinsic Fermi level and trap energy level respectively. For simplicity, we assume that the energy level of trap centers overlaps with the intrinsic Fermi level.

*τ*

_{n}and

*τ*

_{p}are the electron and hole lifetime, respectively. Owing to the fact that electron and hole may recombine or be generated at interfaces,

*τ*

_{n}and

*τ*

_{p}need to be modified

^{[20]}when Shockley–Read–Hall recombination is used at interfaces. Equation (

*C*

_{n}and

*C*

_{p}are Auger coefficients for electron and hole,

*n*

_{i}is the intrinsic carrier concentration given by Boltzmann statistics,

*n*and

*p*are the electron and hole densities, which can be given by the Boltzmann approximation.

During numerical simulation, several boundary conditions (ohmic contacts, insulated contacts, Neumann boundaries) must be considered. To implement ohmic contacts, Dirichlet boundary conditions, where surface potential, electron concentration and hole concentrations are fixed, are taken into account. In the oxide region, the zero normal current condition *J*_{n} · *s**J*_{p} · *s*^{[16]} To ensure that the current only flows out of the p–n junction through the contact, Neumann boundaries are adopted.

To discretize those equations, the finite-difference method and the box integration method are used in order to solve them in an appropriate mesh. After discretization, the Newton iteration is used to solve the partial differential equations. The SGFramework,^{[21]} a highly flexible partial differential equations solver, is introduced.

The structure simulated is shown in Fig. _{2} film is formed above the p–n junction surface and then an electrode gate is formed in sequence (shown in Fig.

Figure *I*–*V* characteristics of the p–n junction at *T* = 300 K for various magnetic fields. Curent is gradually suppressed with increasing magnetic field due to the drastic change of the space-charge region. In order to describe the variation of current with applied magnetic field, here the magnetoresistance (MR) ratio is defined as MR(%) = [*R*(*B*) − *R*(0)]/*R*(0) × 100%, where *R*(0) and *R*(*B*) are the resistance (*V*/*I*) at zero and applied magnetic field. The inset in Fig. *V*_{bias} = 0.9 V. It is fitted well with the parabolic relation (MR ∝ (*μB*)^{2}), which is similar to the results reported in Ref. [13].

Figure

Figures *y* = 2.5 μm and *y* = 0.3 μm. As illustrated, there is no apparent distinction from the distribution of net charge concentration at *y* = 2.5 μm for different magnetic fields, however, a dramatic change appears at *y* = 0.3 μm (the difference between the distribution of net charge concentration at *x* = 0 μm and that at *x* = 10 μm is induced by the carriers injected). The width of the space-charge region far from the gate (*y* = 0.3 μm), compared with that under the condition without magnetic fields, increases but near the gate region decreases when we apply a positive magnetic field, the shape of the space-charge region changes from rectangular to trapezoid (Fig. *et al*.^{[13]} Moreover, it is clearly observed that the net charge concentration changes drastically in the p-type but slightly in the n-type. It can be considered to be due to the different properties about hole and electron, which leads to various performances of magnetic field.

In order to discuss the magnetoresistance effect of the p–n junction under an electric field clearly, here, the MR ratio is modified into *M*_{1}(%) = [*R*(*B*,*V*_{gate}) − *R*(0)]/*R*(0) × 100%, where *R*(0) and *R*(*B*,*V*_{gate}) are the resistance (*V*/*I*) at zero applied magnetic field and that at applied magnetic field *B* and gate voltage *V*_{gate}. MR_{1} can also be given by MR_{1} = MR + Δ*R _{E}*/

*R*

_{0}+

*f*(

*B*,

*E*), where Δ

*R*=

_{E}*R*(

*B*= 0 T,

*V*

_{gate}) −

*R*(

*B*= 0 T,

*V*

_{gate}= 0 V) is the resistance variation (

*B*= 0 T). MR and Δ

*R*/

_{E}*R*

_{0}are the resistance variation ratios without gate voltage and without magnetic field respectively,

*f*(

*B*,

*E*) is an interaction term induced by the interaction between magnetic field and gate voltage.

At the zero-magnetic field, the resistance increases with an increase of negative gate voltage applied and decreases with the increase of positive gate voltage. The resistance variation ratio (Δ*R _{E}*/

*R*

_{0}) without magnetic field is shown in Fig.

*R*/

_{E}*R*

_{0}increases and gradually tends to be saturated with negative gate voltage decreasing. When positive gate voltage is applied, Δ

*R*/

_{E}*R*

_{0}decreases linearly with gate voltage increasing, which can be seen from the fact that the ability for Δ

*R*/

_{E}*R*

_{0}to compensate for MR

_{1}is enhanced. The change contributes to the difference in electrical property (

*μ*

_{n}>

*μ*

_{p}) and the variation of hole and electron concentration. When positive voltage is applied to the gate, the electron concentration increases and the hole concentration decreases, resulting in the electron current increasing and the hole current decreasing. Whereas the increase of electron current dominates the decrease of hole current because of

*μ*

_{n}>

*μ*

_{p}, thus the total current increases, which indicates the decrease of resistance. It is opposite to the scenario of applying negative gate voltage.

To explore the variation of *f* (*B*,*E*), the force analysis of hole under magnetic field and gate voltage is shown in Fig. *f* (*B*,*E*) < 0 (compensating for MR). However, for the conditions (b) and (c), the Lorentz force and electric field force compete with each other, the *f* (*B*,*E*) depends on the stronger of the two forces. When electric field force *qE*_{e} dominates Lorentz force *qvB*, the compensating effect occurs.

On the basis of the above discussion, the results shown in Fig. *R _{E}*/

*R*

_{0}=

*f*(

*B*,

*E*) = 0), MR

_{1}shows an approximately quadratic relationship and is symmetrical about

*B*= 0 T. Whereas with the absolute value of gate voltage increasing, the symmetry of MR

_{1}curves is gradually broken because of

*f*(

*B*,

*E*) ≠ 0. It demonstrates that MR is dependent on the polarity of magnetic field, which is different from other results.

^{[10–13]}

In addition, it can also be noted that the MR is about 0.396% with a magnetic field of 0.5 T and an applied bias voltage of 0.9 V, while an 11.107% magnetoresistance ratio is achieved with a negative gate voltage of −1 V at the same bias voltage. The MR is enlarged substantially due to the contribution of Δ*R _{E}*/

*R*

_{0}(

*f*(

*B*,

*E*) is small.) However, the distinction of the magnetoresistance curve between various negative gate voltages disappears with magnetic field increasing and gate voltage decreasing. It can be explained by the fact that the Δ

*R*/

_{E}*R*

_{0}tends to be saturated which is shown in Fig.

*B*= 2 T for a bias of 0.9 V, but the magnetoresistance ratio reaches 4.477% when only magnetic field

*B*= 2 T is applied. The difference is induced by the compensating of Δ

*R*/

_{E}*R*

_{0}and

*f*(

*B*,

*E*) for magnetoresistance, which makes the magnetoresistance reduced. While

*f*(

*B*,

*E*) becomes more important with positive gate voltage increasing, which can be seen from the degree of symmetry breaking.

Figure _{2} thickness, *W*/*L* and doping concentration respectively. As indicated in Fig. _{2} thickness increasing, magnetoresistance increases linearly because of the compensation capabilities of Δ*R _{E}*/

*R*

_{0}reducing when positive gate voltage is applied and decreases slightly with applied negative voltage increasing due to Δ

*R*/

_{E}*R*

_{0}reducing. On account of

*f*(

*B*,

*E*) ≠ 0, the intervals of MR

_{1}curves occur when different polar magnetic fields are applied at the same gate voltage. Figure

*W*/

*L*for different values of

*V*

_{gate}and

*B*. The magnetoresistance tends to be saturated with

*W*/

*L*increasing. It derives from MR tending to saturation

^{[10]}and Δ

*R*/

_{E}*R*

_{0}gradually decreasing to zero due to the fact that the electric field diminishes with W increasing. The variations of magnetoresistance with doping concentration are shown in Fig.

*f*(

*B*,

*E*) gradually decrease to zero.

In this paper, the magnetoresistance effect of the p–n junction under an electric field at room temperature is simulated. The results indicate that it is a useful method to control magnetoresistance by gate voltage. A larger MR can be achieved by negative gate voltage, and it can also be compensated by positive gate voltage. Meanwhile, MR is dependent on the polarity of magnetic field when gate voltage is applied. Furthermore, the doping concentration, SiO_{2} thickness and p–n junction size are also investigated. The results will promote the applications of silicon-based magnetoresistance devices such as a reconfigurable logic device based on magnetoresistance, access memories, and so on.

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