One dimensional transport in silicon nanowire junction-less field effect transistors

Junction-less nanowire transistors are being investigated to solve short channel effects in future CMOS technology. Here we demonstrate 8 nm diameter silicon nanowire junction-less transistors with metallic doping densities which demonstrate clear 1D electronic transport characteristics. The 1D regime allows excellent gate modulation with near ideal subthreshold slopes, on- to off-current ratios above 108 and high on-currents at room temperature. Universal conductance scaling as a function of voltage and temperature similar to previous reports of Luttinger liquids and Coulomb gap behaviour at low temperatures suggests that many body effects including electron-electron interactions are important in describing the electronic transport. This suggests that modelling of such nanowire devices will require 1D models which include many body interactions to accurately simulate the electronic transport to optimise the technology but also suggest that 1D effects could be used to enhance future transistor performance.

The values plotted in Fig. 4 of the main paper, were extracted from the fits in Fig. S1 and Fig. S2, that show the scaled conductance as a function of bias energy normalized by thermal energy for all temperatures at different gate voltages.
The scaled normalized conductance traces that were offset for clarity in Fig. 3 of the main paper were calculated using:

II. CALCULATION OF THE NUMBER OF CHANNELS ACCORDING TO DISORDERED MULTI-CHANNEL LUTTINGER LIQUID THEORY
Following Mishchenko's theory 1 the conductance at zero bias as a function of temperature should behave according to the following expression: Using this expression we can fit the data in Fig. 5 from the main paper in order to find T * = 94 ± 14 K. From that we can estimate the number of channels involved in the transport, N according to the following expressions from Mishchenko's theory 1 : where N is the number of channels involved in the transport, k B is Boltzmann's constant and τ is the elastic momentum relaxation time. The mobility in the present nanowires has been measured 2 in the low-field drift regime as 107 cm 2 /Vs and using where q is the electron charge, the effective mass m * = 0.19m 0 [Ref. 3] and therefore τ = 11.4 fs is the elastic momentum relaxation time. p i is defined by the following expression: where d = 26 ± 0.5 nm is the radial distance to the gate, R = 4 ± 0.5 nm is the nanowire radius, ǫ 0 is the vacuum-and ǫ r = 3.9 the relative permittivity of silicon dioxide.
Next we need to estimate the Fermi-velocity using the charge carrier density in one dimension n 1D = 1.62 × 10 6 cm −1 as extracted from the measurements to produce, v F =h 2πn 1D g s g v m * = 6.38 × 10 6 ms −1 (5) where g s = 2 is the spin degeneracy, g v = 2 is the valley degeneracy. From this calculation we get the number of occupied channels N = 3.9 ± 0.9.

III. OTHER SAMPLES
This section presents data taken in measurements on other nominally identical devices fabricated in the same batch as the device analysed in the main text to demonstrate that the reported 1D electronic transport results are reproducible across many devices. Fig. 3 demonstrates the normalized conductance in these devices as a function of drain voltage for different temperatures from 14 K to 300 K. Identical to the device in the main text, we observe the characteristic zero bias feature, that disappears with increasing temperature.

IV. UNCERTAINTIES
The uncertainty in the width of the nanowires was obtained by a number of line scans taken on electron energy loss spectroscopy (EELS) data taken across different diameters of the nanowires. The line scans for both the Si content and O content were used as both demonstrate the interface between the Si nanowire and the SiO2 gate oxide. The measurements from 4 separate line scans at different angles produced an uncertainty of ±0.5 nm.
The error bars in both Figures 4 and 5 were obtained from the 90% confidence bounds of a polynomial fit of the current around the zero bias voltage. A 90% confidence level was used due to the estimated 10% uncertainty in obtaining the normalized conductance for the