SUPPLEMENTARY INFORMATION

Transcription

1 doi: /nature10375 Table of contents 1. The role of SiO 2 layer 2. The role of resistivity of silicon 3. Minority drifting length 4. Geometry effect of the IMR 5. Symmetry of the field dependence of raw MR data 6. A crossover of field dependence from normal parabolic type to linear type 7. The temperature dependence of IMR 8. Reproducibility of the IMR devices * Contact: xzzhang@tsinghua.edu.cn 1

2 1. The role of SiO 2 layer: We found the intrinsic layer of SiO 2 played a crucial role in enhancing IMR effect in silicon. Fig. S1(a) shows the I-V characteristics of samples with and without the SiO 2. The transition from a low resistive region to a higher one was not observed in the samples without SiO 2, indicating no minority injection occurred. The IMR was not observed in these samples as well. We compared the band structures of In/SiO 2 /Si and In/Si electrodes as shown in Fig. S1 (b) and (c). Without loss of generality, we first let Fermi levels of Indium and Silicon equal at V=0 for simplicity. In the samples with SiO 2 at positive bias, the SiO 2 layer undertook a portion of applied voltage, which raised the valence band of silicon as well, making minority injection from indium to silicon feasible when the valence band was elevated above the Fermi level of indium (Fig. S1 (b)). However, the barrier between the Fermi level of indium and the valence band of silicon was kept the same if there was no such a layer of intrinsic SiO 2 (Fig. S1 (c)). It was very hard for holes to conquer the barrier into silicon at room temperature. In the case of Delmo et al. 1,2, the intrinsic SiO 2 layer was etched, which was regarded as the main reason why we realized minority injection while they realized majority injection into silicon. 2

3 Figure S1 influence of intrinsic SiO 2 layer (a) IV characteristics of samples with and without the intrinsic SiO 2 layer. No transition between a low resistive region and a higher resistive region was observed in the sample without SiO 2. (b) The band structure of In/SiO 2 /Si interface at V=0 and V>0. The conduction band (CB) and the valence band (VB) of Si could be elevated by applied voltage because of bending of SiO 2 bands. Holes could tunnel from indium to silicon as long as the VB was higher than Fermi level (E F ) of indium. (c) The barrier between the VB of silicon and E F of indium was unchanged in In/Si electrode at V>0, hampering the hole injection into silicon. 3

4 2. The role of resistivity of silicon: We also measured the samples with resistivity of 10 Ω m, 4 Ω m and 1 Ω m. The abnormal IMR was only observed in the samples of 10 Ω m. We thought a higher carrier density would first significantly decrease minority lifetime, thus shorten minority drifting length and make the p-n boundary hard to detect. Secondly, a higher electron density made the carrier type inversion hard to achieve. These reasons, we thought, caused the IMR effect not detectable at low resistive samples. 3. Minority drifting length: minority, once injected, would drift into the inner of samples until its recombination with majority. For the 1D case, charge continuity equation (2) holds. dp/dt = -μ h Edp/dr +D h d 2 p/dr 2 - p/τ h (2) Here p was minority density, r was the distance away from the belt electrode for current injecting, t was time, E was electric field, D h, μ h and τ h was minority diffusion coefficient, mobility and lifetime, respectively. Carrier drifting was much faster than diffusion in our case. Therefore the second term was ignored. At steady state dp/dt = 0. Then dp/dr=-p/(μ h Eτ h ) and p=p 0 exp[-r/(μ h Eτ h )]. The p 0 was the minority density at the injecting electrodes. In this case, minority mainly localized in the region r 0 =μ h Eτ h. For the 2D case, p=p 0 exp[-(πr 2 H)/(Iρ h μ h τ h )]. The H, I and ρ h was the thickness of samples, applied current and resistivity of minority region, respectively. Minority mainly localized in the region r 0 =[Iρ h μ h τ h /(Hπ)] 1/2. The 1D case was applicable to belt 4

5 electrodes in Sample 15 and 20 while the 2D case was applicable to point electrodes in Sample 40. Our devices worked at about 10 V. Thus E 20 V/cm. The μ h m 2 /V s, the ρ h = 153 Ω m and the τ h 100~200 μs. The working current I 200 μa and the sample thickness H 0.5 mm. Then the r 0 2.0~4.0 mm for the 1D case and r 0 4.4~6.2 mm for the 2D case. The sizes of r 0 in the both cases were comparable with the electrode geometries, the distance D = 3.15 mm for Sample 15 (Fig. 1 (a)) and the W = 5 mm for Sample 40 (Fig. 4 (a)). This comparability demonstrated the minority injection and carrier type inversion phenomenon could be detected in our electrode geometries. 4. Geometry effect of the IMR: As the FEM results (Fig. 3(b, c)) showed, the apparent IMR increased with increasing distortion degree of current trajectory. Here the distortion degree was estimated as E H x /E e x =(dv H /dx)/e e x since a larger E H x could shorten current near the edge of samples more easily and thus induce a more distorted current trajectory. Here, E H x and E e x denoted electric field in x direction induced by Hall voltage and applied voltage, respectively. The V H was Hall voltage. Here we mainly focused on geometry effect. Therefore, for simplicity, we let n=p, μ e = μ h =μ just as in the FEM model. As shown in Fig. S2, the V H (x 0, y 0 ) (R H IBy 0 )/(WH) at a point (x 0, y 0 ) before considering the voltage continuity condition. The V H would decrease when the condition was considered. We estimated the (dv H /dx) near the p-n boundary as [V(-x 0,y 0 )-V(x 0,y 0 )]/2x 0 =jr H By 0 /x 0. Thus E H x /E e x jr H By 0 /ρjx 0 =(μb)(y 0 /x 0 ). When the four electrodes were placed on the corners of samples, E H x /E e x (μb)(w/l). 5

6 We attributed the giant IMR of Sample 40 to this large W/L ratio. However, it was worthy of reminding that if there was no minority injection, the giant MR could not be achieved even though the W/L was still very large. Figure S2 measurement geometry. The width and length of sample was W and L, respectively. The boundary was modulated on the y axis. The pair of voltage detecting electrodes were placed at (x 0,y 0 ) and (-x 0,y 0 ). 5. Symmetry of the field dependence of raw MR data: Intrinsic MR should be evenly dependent on magnetic field. However, the raw MR data might mix with Hall signals to make the MR asymmetric about magnetic field. Thus we took R even (B)=[R(B)+R(-B)]/2 as the real resistance under field and acquired MR data from these R even (B) data. The odd part R odd (B)=[R(B)-R(-B)]/2 of Sample 40, as an example, was shown in Fig. S3. The dr odd /db (B=0) of Sample 40 was inversed from negative at 210 μa to positive at 285 μa, indicating inversion of carrier type occurred. 6

7 Figure S3 The odd part of raw resistance data. The slope of this part at zero field was transited from negative at 210 μa to positive 285 μa. 6. A crossover of field dependence from normal parabolic type to linear type: The field dependence of Sample 20 was observed to experience a crossover from the normal parabolic one to a nearly linear one in the transition region. The crossover field was reduced from about 6.0 T at I = 146 μa to about 1.0 T at I = 226 μa. In order to parameterize this abnormal linear dependence, we linearly fitted the dependence at some field range with formula IMR = k (B-BB0), where k and B 0B was, respectively, a quantity to estimate the enhancement of the IMR effect and the crossover field between the different field dependences. The fitting, as an example, was conducted at 2.7 T B 7.0 T for I = 188 μa. The fitted parameters k and BB0 were shown in Fig. S4. The largest enhancement k max occurred during the transition 7

8 B B doi: /nature10375 region, which coincided with the prediction of the IMR theory that the IMR would be largely enhanced when the density of electrons were nearly equal to that of holes. 3,4 More interestingly, the crossover field B 0 was remarkably reduced during the transition region after which the BB0 increased slightly again at Region 2 (Fig. S4). 5 1 This transition was also observed in such systems as Ag 2 Te/Ag 2 Se and Si. Distinguishing from the former systems, the crossover field B 0 in our case could be modulated by applied current, which not only provided a tool to testify the predictions of the IMR theory 3 but also endowed this kind of IMR effect with more versatility. Parish et al. 3 researched the IMR using a simulation method similar to FEM and found that a crossover phenomenon from the normal MR dependent on <μ> to an abnormal MR dependent on Δμ would indeed occur in large inhomogeneous situations. The crossover field B 0 was proportional to (Δμ) -1. Figure S4 The enhancement of IMR and the crossover field from quadratic normal MR to 8

9 linear IMR. The black dots and the red dots show the crossover field BB0 and the slope k of the fitting results for Fig. 2 (b). The lines were guides for eyes. The inset shows the calculated dependence of B 0B on a ratio α p/n. Error bars, standard error of mean (s. e. m.), are calculated from at least 4 data. In our case, inhomogeneity was induced by minority injection. The size of minority region had already been close to the size of samples. Therefore, it was meaningless to average mobility among the whole sample as elaborated by Stroud et al. 6. This argument was supported by two facts: (I) the IMRs measured in 4-electrode method were much larger than those measured in 2-electrode method, which demonstrated that this IMR effect was sensitive to local potential distribution instead of an averaged one; (II) the FEM results (Fig. 3) showed that potential and current distribution were only severely distorted near the p-n boundary with other place nearly unaffected. Therefore, instead of averaging a whole sample, we only considered a small region near the pair of voltage electrodes where there existed holes and electrons with density p, n and mobility μ h and μ e, respectively. The average mobility <μ> = (nμ e + pμ h )/(n+p)=(μ e +αμ h )/(1+α), where α p/n>0. Thus the (Δμ) 2 = [(μ e -<μ>) 2 +α(μ h -<μ>) 2 ] 1/2-1/2 =α/(1+α) 2 (μ e -μ h ) 2.Δμ=α 1/2 /(1+α) μ e -μ h. Therefore BB0 (α +α ). Without minority -4 injection, the α= p0/n The p0 and n 0 was the equilibrium density of holes and electrons in our silicon samples. When minority was injected by applied current, the α would increase. The final value could be estimated from the Hall coefficient data (Fig. 1) as α=p/n R H (I=120 μa)/r H (I=180 μa)=3.6. The calculated dependence of B 0B on the ratio α was shown in Fig. S4 inset. The calculated dependence, we thought, 9

10 produced the main characteristics of the experimental dependence of BB0 on applied current, including a sharp decrease at the very beginning of injection as well as a following slight increase. This coincidence supported our argument that the abnormal MR was enhanced by inhomogeneity or carrier type inversion induced by minority injection. Further, as discussed in the section Geometry effect of the IMR in the revised supporting information, the IMR effect was mainly determined by a local ratio E x H /E x e =(dv H /dx)/e x e. Severe distortion of current trajectory occurred only when E x H /E x e >1. The ratio was not only dependent on (ΔμB) but also on (y 0 /x 0 ). Therefore not only large Δμ but also large W/L ratio could lead to a small crossover field B 0B. We thought this was the reason why the crossover field could be further reduced in Sample 40, compared with Sample The temperature dependence of IMR: The temperature dependence of the IMR effect was measured in Sample 32 as shown in Fig. S5. The IMR effect was observed in a wide temperature range from 260 K to 350 K with the peak position appearing at 290 K for I=200 μa (Fig. S5 inset). The field dependence, as an example in Fig. S5(b), was parabolic at 350 K. However, the abnormal IMR appeared at T 320 K and the crossover field was reduced with further decreasing temperature. The shift of the crossover field with temperature was also observed in Fig. S5(a). It seemed interesting that the field dependence of the IMR evolved with decreasing temperature (Fig. S5) in nearly the same manner as increasing current (Fig. 2 and Fig. S4). In fact, the variation of mobility Δμ, according to the calculation in our case, was mainly 10

11 determined by a ratio α p/n which would sensitively depend on applied current I and temperature T since a larger I or a lower T could both increase the α. We thought this was the reason why the IMR evolved in a similar way when we increased current or decreased temperature. This agreement also indirectly supported our IMR model. The insets in Fig. S5 show also the temperature dependence of IMR value measured at a fixed magnetic field. The IMR value increased sharply above some temperature until it reached its peak value. Further increasing temperature would lead to a moderate decrease of the IMR value. The temperature range for the IMR 0.7 IMR max was about 30 K in Fig. S5(b) inset, demanding further optimization. It was worth pointing out that the absolute value of slopes of the IMR vs. T curves above the peak temperatures were much smaller than those below the peak temperatures, which meant that heating effect could not severely deteriorate the performance of devices as long as we adjusted the peak temperature at room temperature or slight below. As to the method to adjust the peak temperature at room temperature, we could achieve our goal by geometry designation and/or select suitable applied current. Fig. S5 has already indicated that a larger current leaded to a higher peak temperature. 11

12 Figure S5 The temperature dependence of the IMR effect. (a) The temperature dependence of IMR measured in Sample 32 at I = 200 μa and inset shows the dependence of IMR 4T. (b) The temperature dependence of IMR measured in the same sample at I = 150 μa and inset shows the dependence of IMR 5T. 8. Reproducibility of the IMR devices: We have supplemented a series of experiments to test reproducibility of our devices with the nominally identical W/L 12

13 ratio and positions of the electrodes. We have fabricated more than 30 samples using the method described in our manuscript with 5 ratios of W/L, 25, 65, 97, 100 and 120, respectively. The shape of the samples and the positions of corresponding electrodes are shown in Fig. S6 inset. In order to diminish the influence of the spatial variance of carrier density, we have selected silicon chips at the adjacent regions in the raw wafer for making the MR devices with the same W/L ratio. Because MR properties, e. g. magnitude and sensitivity, of our devices highly depended on measuring current as elaborated in our manuscript, we had to select proper measuring current to measure MR properties of the samples with the same W/L ratio in order to make these MR data comparable among the different samples. Therefore we first measured I-V characteristics of each device and determined the current range for the transition region where the magnitude of the MR was the largest. For convenience to compare MR properties among the different samples with the same W/L ratio, we further selected the critical current I c which separated Region 1 and Transition Region to measure the MR properties. Then we estimated the reproducibility of our devices based on the measured MR data as well as the measuring currents. It is shown in Fig. S6 that the IMR at 1.2 T and the I c increased with increasing the W/L ratio. At the meantime, the error (s. e. m. labeled as the red bar in Fig. S6) also increased with increasing the ratio W/L. The ratio of error to mean (relative error) of the samples with W/L = 25, 65, 97, 100 and 120 was 2.3%, 8.3%, 4.4%, 9.3% and 11.4%, respectively. This indicates that the magnitude of this IMR effect was tested to 13

14 be reproducible within approximately 10% with nominally identical W/L ratios and equivalent positions of the electrodes, i.e., reproducibility of the IMR effect of these devices is very good, we think, considering that the above electrodes were made by hand at the current stage. If lithography technique is applied to more precisely control the geometry of samples, we could expect even better reproducibility. Figure S6 The dependence of error (s. e. m.) on the ratio of W/L. The black dots are the mean values of MR measured at 1.2 T and critical current I c while the red caps indicate the errors among different samples with the same W/L ratio. Inset schematically shows the geometry of samples. The numbers of the samples with the same ratio are also labeled in the figure. As to the reproducibility of crossover field, we could estimate it by two means: measure the deviation of crossover field at the same current or measure the deviation of critical current at the same magnetic field. We adopted the second method. The mean of critical current I c of the samples with W/L=97 was μa. The deviation 14

15 of these samples was 7.2 μa (evaluated with s. e. m.). According to Fig. 4(b) and Fig. 4(c), a variance of 10 μa might lead to a variance of 0.5 T in the crossover field. Thus the reproducibility of the crossover field was not very ideal at the current stage, demanding further optimization. We thought the crossover field was not only related with geometry of samples but also maybe more closely depended on contact reproducibility of the In/SiO 2 /Si electrodes. Reference in supporting information 1. Delmo, M. P., Yamamoto, S., Kaisa, S., Ono, T. & Kobayashi, K. Large positive magnetoresistive effect in silicon induced by the space-charge effect. Nature. 457, (2009). 2. Delmo, M. P., Kaisa, S., Kobayashi, K. & Ono, T. Current-controlled magnetoresistance in silicon in non-ohmic transport regimes. Appl. Phys. Lett (2009). 3. Parish, M. M. & Littlewood, P. B. Non-saturating magnetoresistance in heavily disorder semiconductors. Nature. 426, (2003). 4. Guttal, V & Stroud, D. Model for a macroscopically disordered conductor with an exactly linear high-field magnetoresistance. Phys. Rev. B 71, (R) (2005). 5. Xu, R. et al. Large magnetoresistance in non-magnetic silver chalcogenides. Nature. 390, (1997). 6. Stroud, D. & Pan, F. P. Effect of isolated inhomogeneities on the galvanomagnetic properties of solids. Phys. Rev. B 13, (1976). 15