The space charge limited current and huge linear magnetoresistance in silicon

Huge magnetoresistance in space charge regime attracts broad interest on non-equilibrium carrier transport under high electric field. However, the accurate fitting for the current-voltage curves from Ohmic to space charge regime under magnetic fields has not been achieved quantitatively. We conjecture that the localized intensive charge dynamic should be taken into consideration. Here, by introducing a field-dependent dielectric constant, for the first time, we successfully simulate the current-voltage curves of covalent crystal silicon wafers under different magnetic fields (0–1 Tesla). The simulation reveals that the optical phonon, instead of the acoustic phonon, plays a major role for the carriers transport under magnetic fields in space charge regime.

field. It needs to be emphasized that the indium electrodes were pressed on center of the upper and lower surfaces of the silicon wafer. The lateral size of silicon wafer was not less than 16 mm, therefore, the distance between the contacts L and the width W meets L ≪ W.

Result and Discussion
As shown in Fig. 1a, a typical I-V curve of an intrinsic N-Si exhibits a slow rise (V < 10 V) followed by a sharp, power-law rise at a critical voltage V 0 (black open diamonds). In order to avoid the competition of the surface and bulk parallel paths 18 and the edge effects of a limited-size sample 14 , a symmetric out-of-plane electrode set-up is selected in experiments, as shown in the insert of Fig. 1a. The slow rise is the Ohmic regime, and the sharp rise belongs to the space-charge regime according to space-charge-limited-currents theory [7][8][9][10] . Recently, Zhang and Pantelides evidenced in theory and experiment that the interplay between dopants and traps controls the power-law rise of I-V curves 10 . The prediction of ZP model is drawn in Fig. 1a (blue solid line) with the calculation parameters: the dielectric constant ε r = 12 22 , dopant density N D = 1.2 × 10 17 m −3 , and trap density N t = 2 × 10 17 m −3 16 . It is clear that the predicted current by ZP model significantly lags behind the experiment data at high electric field. In principle, for a high purity semiconductor, the acoustic and optical phonon scatterings dominate the transport of carriers, while the ionized impurity contribution can be neglected at room temperature 22 . In respect to the matters, as presented in Fig. 1b, the ratios of the relaxation time and the effective mass of carrier τ/m* with the external electric fields are evaluated by ZP model for both the optical and the acoustic phonons at room temperature (blue lines in Fig. 1b). Obviously, τ/m* of the optical and acoustic phonons remain a constant with the increase of voltage in ZP model. The acoustic phonon τ a /m a * obeys a power law rise of temperature in a broad temperature range rather than a function of electric field 10,22 . The higher the temperature is, the lower the carrier mobility is, which is homogenous in sample and offers a minor correction for carrier transport if Joule heat is limited.
ZP model may underestimate the growth of carrier density due to intense electro-phonon excitation. We conjecture that the overall dielectric constant ε r−sum can be expressed as 1 + χ + χ n + χ L , where χ, χ n and χ L are the intrinsic, the non-equilibrium and the orientation terms, respectively. χ n is assumed proportional to the density of non-equilibrium carriers. χ n = ηΔn/n 0 , here Δn defines as the density of non-equilibrium carriers, n 0 is the density of thermal-excited carriers and η is an arbitrary parameter for simulation. χ L can be expressed as γln(1 + Δn/n 0 ) with a fitting coefficient γ. Under a high electric field, the uncompensated charge and the concomitant push-back electrostatic field yield the spatial distribution of polarity-conserved charges 14,17,18 . The localized charges are polarized along electric field, and the term of polarization convergence ought to be considered even for a non-polar one. For our experiments, a symmetric out-of-plane electrode set-up with L ≪ W, thus the non-uniform electric field in silicon is fusiform with rotational symmetry. The higher the electric field applies, the stronger convergence of the polarization is. Therefore, we introduce a modified Langevin function 23 in the framework of Mott-Gurney theorem to describe the polaron inhomogeneity under fields (Supplemental Material S1).
Using ε r−sum in ZP model, the I-V curve can be successfully simulated in both Ohmic and space-charge regimes for the intrinsic N-Si, as indicated in Fig. 1a  In the case of a heavily-doped N-Si, the thermal excited carriers dominate its properties. The localized high-energy processes are screened by the strong Coulomb effect and the contribution of LO phonons is therefore negligible 21 . It is understood that PPV, polycrystal SrBi 2 Ta 2 O 9 and nanocrystal CdS thin films, with relative larger density of dopants and traps, can be simulated by ZP model 10,24 . As presented in Fig. 1c, the I-V curve of a heavily-doped N-Si exhibits a slow rise followed by a power-law rise at a critical voltage V 0 , with the Mott-Gurney limit attained asymptotically (black solid squares) [7][8][9][10] . ZP model (blue solid line) fits well with the experiment data as V < 10 V, but the fitting curve slightly lags behind the current increase at high electric field. In principle, the Coulomb screening in heavily-doped N-Si effectively suppresses the emission and trapping of a charge between a lattice site and the free-carrier states. The field-dependent electro-LO phonon interaction compensates the current increase at high electric field, resulting in a better simulation (red solid line). Here, the influence of the electrode configuration is degraded significantly as the sample behaves like a conductor.
Under a high electric field, the virtual phonon emissions are accompanied with the lattice distortions, which alter the charge distributions. The uncompensated charge and the concomitant push-back electrostatic field yield the polarity-conserved charges accumulated on boundaries 10,14 . It is no doubt that the capacitance of sample is modified accordingly. The voltage-induced capacitance is defined as VIC = (C V /C 0 − 1) × 100%, where C 0 and C V are the capacitances under zero and a certain voltage. As shown in Fig. 1d, VICs of both an intrinsic N-Si (black open diamonds) and a heavily doped N-Si (black solid squares) show quasi-linear growth with the increase of the bias voltage. Whereas, for a heavily doped N-Si, VIC has only a ~4% increase from 0 to 16 V (black solid squares), which is much smaller than the ~30% increase in an intrinsic N-Si. Although the spatial dynamic of ionization and filling occurs at a lattice site, the strong electron-LO phonon interaction accompanied with the virtual phonon processes modifies the dielectric responses of sample. Approaching to a metallic limitation, such as a heavily doped one, the dynamic gets weakened significantly and the intense electron-LO phonon interaction is forbidden 21 . Figure 2a shows the typical I-V curves for an intrinsic N-Si measured at different magnetic fields. Here, the magnetic field is parallel to the sample surface and perpendicular to the transport direction, as illustrated in the insert plot of Fig. 1a. The I-V curve is more sensitive to the magnetic field in the space-charge regime than in the Ohmic regime. The magnetoresistance approaches to zero in Ohmic regime, which is exactly consistent with the ScienTific REpoRTs | (2018) 8:775 | DOI:10.1038/s41598-017-19022-1 classical equilibrium magneto-electric transport theory 21 . The turn-on voltage, V 0 , grows quasi linearly, from 14 to 104 V, with the increase of magnetic field from 0 to 1 T. Under a magnetic field, the current suppression constrains the field-induced ionization along the transport direction 13,18,25 , resulting in the postponed of V 0 . In fact, although such phenomena are reported in those pioneer works for silicon wafers 13,15,17,26 , to our best knowledge qualified explanations have not been achieved. Here, we present that I-V curves under different magnetic fields are successfully simulated by LO model with a set of unified fundamental parameters, such as N D , N t and energy levels etc., as presented in Fig. 2a by the solid lines (Supplemental Material S3).
The magnetoresistance is defined as [ρ(H)/ρ(0) − 1] × 100%, with ρ(0) and ρ(H) the resistance at zero and applied magnetic field, respectively. In Fig. 2b, the out-of-plane magnetoresistances measured at a constant current mode I = 10 mA under different θ are shown together. The measurements are performed below the breakdown voltages to ensure the data stability. The inhomogeneous spatial dynamic of ionization and filling is inevitably influenced by a perpendicular magnetic field under a certain voltage, viz., prompting the traverse filling and suppressing the ionization along carrier transport direction 21,25 . In our experiment, the magnetoresistance has an excellent linear relationship with magnetic fields under different θ, as shown in Fig. 2b, exhibiting a pronounced anisotropic behavior. As θ = 90°, the magnetoresistance reaches ~2600% at 1.2 T, which is much larger than those reported in the former works 13,15,26 . In the case of θ = 15°, the magnetoresistance is ~400% under 1.2 T, which is an order of magnitude smaller than the one at θ = 90°.
Numerous theories implicate spatial variation of the carrier mobility as being responsible for such an anomalously huge magnetoresistance 19,20,[27][28][29][30] . The spatial variation can be aroused by several factors, such as macroscopic inclusions 31 , geometric configurations 15,17,18 , defects 16,27,32 , and electric field fluctuations [13][14][15]18,25,26 in nonmagnetic materials. The magneto-induced phonon resonance relays on the inelastic inter-Landau-level scatterings 33 . It is known that the value of the inhomogeneous magnetoresistance follows Δρ/ρ ∝ ω c τ 27,28 , where ω c is the cycling frequency and τ is the relaxation time of carriers. As shown in the inserted plot of Fig. 2b, ω co τ o of optical phonon shows a linear increase with a 600% change as magnetic field ≤1T. Whereas, the ω ca τ a of the acoustic phonon also exhibits a linear increase under magnetic field with only a 20% change. At Ohmic regime, the mobility of carriers is uniform, and the secondary deflection current of Hall electric field offsets that of Lorenz force. But the balance breaks due to the excitation of high-energy electrons at the space-charge regime 21 . As demonstrated in Fig. 2c, the magnetoresistance grows nonlinearly with the increase of θ and reaches the maximum as θ = 90°. The curves of the magnetoresistance and θ can be fitted well by Asinθ with the arbitrary coefficients.
The magnetic-field-induced capacitance (MIC) variations are evaluated experimentally. MIC is defined as MIC = (C H /C 0 − 1) × 100%, where C 0 and C H are the capacitances at zero and a certain magnetic field, as shown in Fig. 2d. C 0 are 59 pF and 81 pF for 0 V and 20 V bias voltages, respectively. Although their absolute values are different, their MICs have a quite similar magnetic field dependence, MIC ∝ −H 2 . The quadratic magnetic field dependence is in line with the classical electromagnetism theory for the dielectric response under magnetic field. A perpendicular magnetic field regulates the inhomogeneity and reduces of LO phonon scattering along carrier transport direction, which subdues the dielectric response along transport direction.
In a heavily-doped semiconductor, the screening field destroys the inhomogeneity, and the huge magnetoresistance disappears 16 . The I-V curves of a heavily-doped N-Si under magnetic field ranging from 0 to 1.2 T are shown in Fig. 3a. There is an undetectable shift of I-V curves under magnetic field in both Ohmic and non-Ohmic regimes. MIC of a heavily-doped silicon only has a 0.4% change from 0 to 1.2 T under 50 V bias voltage, which is much smaller than the change of MIC in an intrinsic one, as presented in Fig. 3b. Obviously, the second order term of Hall and Lorentz deflection currents offset 21 , resulting in a negligible effect on the I-V curves and a small MIC change under magnetic fields in the heavily-doped N-Si. Altering the symmetry of the electrode set-up 14,17,31 or producing doping density fluctuations 16,32 purposely, huge linear magnetoresistances have been reported in heavily-doped N-Si. Utilizing the surface imperfection with an in-plane electrodes configuration, an asymmetric magnetoresistance response has also been realized in heavily-doped Ge 18 . In those methods, the electric-field-induced spatial dynamic of ionization and filling does not play a major role.
Few mechanisms are known to produce a large positive magnetoresistance in P-Si. Schoonus et al. obtained an extremely large magnetoresistance in a Si-SiO 2 -Al structure because the magnetic field raises the acceptor level 34   modulation of the electron-to-hole density ratio under magnetic fields 35 . By the symmetric out-of-plane electrode set-up, as illustrated in the inserted plot of Fig. 1a, we revisit the magnetoresistance for P-Si. Figure 3c shows the I-V curves under different magnetic fields as θ = 90° for an intrinsic P-Si. It is noted that the breakdown voltage only increases from 21 to 23 V between 0 and 1.2 T. Such phenomenon contracts to the intense magnetic-field variation of the breakdown voltages for an intrinsic N-Si, as demonstrated in the same plot by the gray dots. The relaxation of high-energy carriers depends on the intervalley scattering processes due to the different mobility of band valleys 36,37 . For silicon, the density states of heavy-hole band are much larger than that of light-hole and spin-orbit splitting bands 38 , resulting in weak intervalley scatterings and a small variation of hole mobility under electric fields. The spatial inhomogeneity of hole mobility in P-Si is trivial compared to that of electrons in N-Si. The magnetoresistance of P-Si is inconspicuous accordingly. Capacitances under different magnetic fields are presented in the Fig. 3d. The black and red boxes represent the capacitances measured at 0 and 10 V, respectively. The unchanged capacitances under magnetic fields confirm the uniform spatial distribution of the hole mobility in our intrinsic P-Si devices.

Conclusion
In this letter, we reveal that under high electric field, the dynamic of ionization and filling arouses the strong electro-LO phonon interaction in N-type silicon, accompanied with the virtual phonon processes and the large lattice distortion. The dielectric response of silicon is thus a function of fields especially in non-equilibrium regimes, as evidenced by capacitance measurements. Further experiment and theory on carrier transport under the intense spatial charge dynamic are encouraged, which can shed light on applications in novel devices, ranging from energy-harvesting cells to novel magneto-electric devices.