Download A Sub kBT/q Semimetal Nanowire Field Effect

A Sub kBT/q Semimetal Nanowire Field Effect Transistor
Supporting Material:
a. 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 OFFand 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.
b. Drain-Source Current Calculations
Upon obtaining the transmission, the drain-source current is calculated using the Landauer formula as
follows [S1]:
I DS   i( E )dE ,
(S-1)
e
i( E )  T ( E ,V )[ f S ( E )  f D ( E )]
h
where fS(E) and fD(E) are Fermi distribution functions in the source and drain electrodes, respectively:

 E  Ef
f S ( E )  1  exp 
 kT

1

 ,


 E  E f  qVDS 

f D ( E )  1  exp 
kT



1
(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.
c. Subthreshold Slope
Subthreshold slope is defined as the inverse of the derivative of the logarithm of the drain current
versus gate voltage: 𝑆𝑆 = (
𝑑(𝑙𝑜𝑔10 (𝐼𝐷 ))
𝑑𝑉𝐺
−1
) . 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.
d. 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 ND2 doped. 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 ND2 at (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 the source
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.
e. 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 profiles at VDS = 0.3 V and varying VGS for the
device with drain doping ND2 at (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).