A Sub kBT/q Semimetal Nanowire Field Effect Transistor

Supporting Material:

  1. Electron transmission

Applying the gate electric field modifies the channel band structure such that positive gate bias builds up the transmission due to self-consistent change in the channel’s charge density.Fig. S1 shows the transmission characteristics of the TFET device at different gate bias voltages corresponding to OFF- and ON-states for ND1 and ND2 doping profiles. The conductance in quantum regime which is defined as G(E) = 2e2/h T(E) is also illustrated in the inset of Fig. S1 in semi-log scale. As can be seen in the figure, the conductance is suppressed in the OFF-state condition (VGS = −0.12 V); however, by applying positive gate voltage (VGS = 0.12 V), conductivity increases more than 5 orders of magnitude at EF.

Fig. S1 Electron transmission versus energy at different gating fields of the TFET at VDS = 0.3 V. The inset shows the quantum conductance in semi-log scale versus energy. Dashed and solid lines correspond to ND1 and ND2, respectively.

  1. Drain-Source Current Calculations

Upon obtaining the transmission, the drain-source current is calculated using the Landauer formula as follows [S1]:

(S-1)

wherefS(E) and fD(E) are Fermi distribution functions in the source and drain electrodes, respectively:

(S-2)

The source-drain current is determined by the transmission probability within the bias window defined by the left and right electrode chemical potentials.

  1. Subthreshold Slope

Subthreshold slope is defined as the inverse of the derivative of the logarithm of the drain current versus gate voltage: . Essentially, the subthreshold slope determines how efficiently a device can be switched from the ‘OFF’ to the ‘ON’ state and therefore sets power consumption limits for integrated circuits.

  1. Mulliken population analysis

The SS of the TFET device with higher drain doping concentration, i.e., ND2, is lower than the device with the lower doping concentration ND1. This improvement can be explained by considering the lower conductance corresponding to the black curve in Fig. S1. In order to further analyze the physics behind this improvement in the OFF-current we have investigated the effective Mulliken population [S2] along the channel as well as cross-sectional charge difference density at two different locations along the channel at VGS = −0.12 V and VDS = 0.3 V, as shown in Fig. S2. The effective Mulliken population in the middle of the channel is positive, i.e., charge depletion, for both ND1 and ND2doped. Note the drain doping influences the band alignment in the OFF state resulting in a larger potential for the higher drain doping.

Fig. S2. Effective Mulliken charge along nanowire transport direction of the two TFET devices with doping concentrations of ND1 and ND2at(a) OFF-state (VGS = −0.12 V & VDS = 0.3 V) (b) ON-state (VGS = 0.12 V & VDS = 0.3 V).

Note that in both doping cases the potential barrier at the source channel is eliminated in the ON state.In the manuscript, the formation of a channel resonance is discussed and how the state in the channel hybridizes with the source to produce effectively an inverted channel at higher gate voltages. The fact that the potential barrier ‘collapses’ is related to i) the lack of a chemical interface, i.e. the junction is formed in a monomaterial,and ii) the fact that the band gap emerging from quantum confinement results in a relatively small difference between the work function in the semimetallic and semiconducting regions. These two points are related effects.

At equilibrium, the charge transfer associate with the formation of the source/channel junction results in an interfacial dipole results in a voltage difference between the source and channel. Estimating the voltage at the interface from the Mulliken charges directly at the interface with the thinner nanowire section and by treating the surface dipole as a parallel plate capacitor yields an interface voltage of 0.4 V, a value consistent with the alignment of the Fermi level in the semiconducting region at equilibrium (zero external bias) leading to an n-type alignment.Unlike in a conventional metal-semiconductor junction, this is a relatively small interfacial dipole due to chemical bonding between the Sn electrode and the Sn channel atoms – quantum confinement induces a relatively small change to the electron affinity in the channel region. The charge transfer between the contacts and the channel can be seen as MIGs in the channel and the resultant n-type alignment of the semiconducting region to the semimetal. This is in contrast to a typical Schottky n-type barrier where charge from the semiconducting region is transferred to the metallic region but is expected for the band alignment of a metal with an intrinsic semiconductor.

In Fig. S3 just below 1 nm is seen the change in the interface dipole between the semimetal contact and the intrinsic channel as gate voltage is applied.As the device is driven to the OFF state by the gate voltage, the interfacial dipole is reduced. The reduction in the dipole is represented by the black curve in Fig. S3 whereby the charge transfer at the interface is reversed pushing the alignment of thesource Fermi level to midgap with respect to the channel. Current is strongly suppressed. As the device turns ON, the interface dipole is increased as represented by the red curve in Fig. S3 driving the Fermi level of the source above the channel’s conduction band, leading to a barrierless device and the charge profile in the channel of Fig. S2.

Fig. S3. Charge difference density map for charge averaged over planes normal to the transport axis. The change to the surface dipole at the source/channel junction is seen as the charge dipoles just below 1 nm in the graph. Black curve- VGS=0.12 V. Red curve- VGS=+0.12 V.

As the device is driven to the OFF state by the gate voltage, the interfacial dipole is reduced and the Fermi level in the channel shifts toward midgap – the reduction in the dipole is represented by the black curve whereby the charge transfer at the interface is reversed. As the device turns ON, the interface dipole is increased as represented by the red curve driving the Fermi level of the source above the channel’s conduction band, leading to a barrierless device.

It should be noted there are is no large interfacial density of states to pin the Fermi level as the bonding between the channel and the source is between similar atoms, i.e. there is no formation of an interfacial chemical dipole. This is a significant point, as to reverse the effect of the strong interfacial dipole on the Schottky barrier would entail overcoming the much larger charge transfer due to chemical bonding and overcome the effects of Fermi level pinning.

  1. Local DoS and energy-resolved current

Here Figs. 3 and 4 of the manuscript are presented in a slightly different format to allow for the direct comparison of the LDoS in the device with the energy resolved current.

Fig. S4. Energy resolved local density (LDoS) and energy-resolved current profilesat VDS=0.3 V and varying VGS for the device with drain doping ND2at (a) VGS = −0.12 V (b)VGS = −0.06 V (c) VGS = 0 V (d) VGS = 0.12 V.

Supporting References:

[S1]J.-P. Colinge and J. C. Greer, Nanowire Transistors: Physics of Devices and Materials in One Dimension, Cambridge Univ. Press, Cambridge(2016).

[S2] R. S. Mulliken, J. Chem. Phys.23, 1833 (1955).