Figure compares the field-dependent two-terminal resistivities of the aluminium- and the nickel silicide-contacted Hall bars. The small resistance peak in Figure originates from weak localization in the phosphorus δ
-doped layer, where electrons become locked into phase coherent loops [25
]. These loops are broken with the application of a perpendicular magnetic field, making the carriers available for transport and reducing the resistivity of the δ
-doped layer for B
> 0. The magnetoresistance can be well described by the Hikami model for weak localization in a disordered 2D system [26
] as shown in Figure , where the phase coherence length of the system (i.e. the distance electrons travel between phase randomizing scattering events) can be obtained as a fitting parameter. For the fit in Figure , we obtain a phase coherence length of 450 nm, in agreement with previous studies [27
]. In contrast, the magnetoresistance of the aluminium-contacted Hall bar in Figure is dominated by a large peak near B
= 0 spanning B
= ±10.5 mT, preventing fitting to the underlying weak localization peak. This magnetic field range is consistent with the critical field BC
for aluminium [7
], confirming that the origin of the peak is related to the BCS superconducting gap.
Figure 1 The two-terminal magnetoresistance at base temperature (T≈50 mK) for aluminium and nickel silicide contact metallizations to the Si:P δ-layers. Figure 1a shows a small peak resulting from weak localization within the δ-layer, and (more ...)
To further study the nature of this anomalous resistance peak, we have performed temperature dependence measurements as shown in Figure . The magnitude of the peak is seen to rapidly increase as the temperature is reduced. Whilst the BCS gap is known to increase towards a limiting value of 3.52 kTc
as the temperature is reduced (≈ 360 μ
eV for aluminium), it changes only weakly in the temperature range shown here (≈10%) [28
]. This is therefore unlikely to cause the exponential increase in resistance shown in Figure . Instead we attribute this trend to the reduction of thermal energy for carrier activation over the BCS energy gap. The resistive peak continues to grow until T
< 200 mK, at which point the electron temperature begins to saturate.
Figure 2 Temperature dependence of the two-terminal magnetoresistance for the aluminium contacted Hall bar from base temperature to 800 mK. The inset illustrates the exponential increase in the magnitude of the resistance peak, suggesting thermal activation over (more ...)
Both the mobility and phase coherence length can be extracted from four-terminal resistivity measurements, which eliminate contact resistance and are therefore unaffected by the two terminal resistance peaks at B
= 0. The mobility, μ
, is calculated directly from the measured zero-field resistivity according to the relation
. For highly disordered 2D systems, the phase coherence length, lϕ
, can be extracted by fitting the weak-localization peak to the Hikami model near B
= 0, as demonstrated in Figure [26
]. Figure shows the temperature dependence of both μ
and the fitted values of lϕ
, and can be seen to be independent of the choice of contact metallization. The obtained values are commensurate with previous studies of δ
-doped silicon [27
]. In this temperature regime, the mobility is dominated by weak localization and electron-electron interactions, which both result in a ln(T
) dependence [29
]. Electron dephasing is dominated by Nyquist scattering, resulting in a T -0.5
dependence for the phase coherence length [29
]. The nickel silicide Hall bar has a higher mobility by ≈ 30%, which can be attributed to inhomogeneities in the initial δ
-layer. For both samples, the mobility and phase coherence length are observed to saturate below T
= 200 mK, confirming that the saturation of the resistive peak observed in Figure is simply a consequence of the limiting electron temperature. Importantly, the fact that both samples saturate at the same temperature indicates that it is the refrigerator and not the metallization which limits thermal equilibration of carriers.
Figure 3 Low-temperature magnetotransport properties of the 2D δ-layers as a function of temperature. Figure 3a shows the phase coherence length as calculated from Hikami fitting while 3b shows the mobility trend. The phase coherence length is dominated (more ...)
Whilst pure nickel is ferromagnetic, previous theoretical study has concluded that transition metal silicides including NiSi are diamagnetic [30
]. However previous experimental results have indicated ambiguity in the magnetic properties of NiSi for fields below 200 mT at low temperatures [31
]. It is therefore important to determine whether the nickel silicide contacts used here have any influence on the measured magnetic field hysteresis.
We have measured the four-terminal magnetoresistance for both metallizations as a function of magnetic field for different magnetic field sweep rates as shown in Figure . Particular care was taken to ensure that the magnetic environment of each sample was identical. To this end, the samples were measured sequentially (several days apart) using the same package in the same dilution refrigerator configuration. Magnetic hysteresis is seen for both samples with fast sweep rates of 0.2 T/min, cooling the sample as the field sweeps towards B = 0 and heating as the field sweeps away from B = 0. This is characteristic of adiabatic demagnetization of a ferromagnetic material, where thermal and magnetic energies are exchanged faster than the cryostat can equilibrate. Figure shows that the level of hysteresis is similar in both samples, suggesting that it is the ferromagnetic impurities in the immediate environment rather than the ohmic contacts that are responsible for this effect. For both samples, the hysteresis can be eliminated by decreasing the magnetic field sweep rate to < 0.1 T/min to allow sufficient time for the system to equilibrate. We note that the slight difference in noise between Figure is because of the different measurement electronics used for the second series of measurements. Within each measurement set the noise levels were comparable between the samples.
Figure 4 Dynamic hysteresis in the magnetoresistance measured at base temperature. Figure 4a shows the hysteresis in the magnetoresistance of the aluminium contacted Hall bar as a function of magnetic field sweep rate. At a fast sweep rate of 0.2 T/min clear hysteresis (more ...)