## Abstract

The high carrier concentrations typically reported for nanowire devices indicate that when Schottky barrier transport is present, it occurs in the thermionic field emission regime with a substantial but not exclusive tunneling component. Analysis by thermionic field emission is difficult due to its multivariate nature. In recent work, we developed a mathematical stability approach that greatly simplified the evaluation of the multivariate thermionic field emission parameters. This is a general method with potentially wide applicability, requiring only the effective mass m* and relative dielectric constant εr for a given semiconductor as inputs. In the present work, we investigate the influence of the materials properties effective mass m* and relative dielectric constant εr on stability for a range of real and simulated semiconductor nanowires. A further investigation of temperature sensitivity and regime trends is presented.

## Introduction

Semiconducting nanowires represent a new class of device building blocks with properties enhanced by their small size and their large aspect ratios. To harness their outstanding device potential, it is urgent to fully understand and to control the contacts. The high carrier concentrations typically reported for nanowire devices indicate that when Schottky barrier transport is present, it involves a significant quantum-tunneling component. This further indicates that transport through the controlling barrier occurs in the Thermionic Field Emission (TFE) transport regime, with its substantial but not exclusive tunneling component. Both the Thermionic Emission (TE: no tunneling) and Field Emission (FE: dominant tunneling, an electron transparent Ohmic contact) descriptions of Schottky barrier transport are mathematically simpler to use for fitting experimental I-V data. The Thermionic Field Emission description is difficult to use due to the multivariate nature of this model, which requires that values for the maximum barrier height qφ B , free carrier concentration n, effective Richardson constant A** and temperature T be known or guessed to within an accuracy that enables convergence to the experimental data. Use of experimentally pre-determined values acquired under, e.g., TE conditions can lead to unstable fitting results. As TFE is often the correct description for nanoFET Schottky barrier transport, it is important to develop methods that increase its ease of use.

In recent work 1 , we developed a mathematical stability approach that greatly simplified the evaluation of the multivariate thermionic field emission parameters and enabled a first-time analysis of changes in the barrier heights, tunneling probabilities and potential drops over time for Schottky barrier transport in gallium nitride (GaN) nanoFETs in an extreme environment. Fits performed on a generated TFE curve using the Levenberg-Marquardt Algorithm are used to identify stable ranges for important parameters before fits to experimental I-V curves are executed. This stability approach is a general method with potentially wide applicability, which requires only the effective mass m* and relative dielectric constant ε r for a given semiconductor as inputs. In the present work, we investigate stability versus these materials properties for a range of real and simulated semiconductor nanowires. We also explore the sensitivity of free carrier concentration n and temperature T to changes in effective mass m* and relative dielectric constant ε r, as these quantities are all synergistically linked through an important ratio !” ! !! that sets the scale of the tunneling contribution.

## Theory

The dependence on the material properties effective mass m* and relative dielectric constant ε r arises naturally when describing the tunneling component of transport through a Schottky barrier. The transmission coefficient T(E) for tunneling through a finite potential barrier of width W and constant barrier height is well known 2 . The equivalent transmission coefficient for tunneling through a finite potential barrier with a slowing varying barrier height such as a Schottky barrier can be found using the WKB approximation 2 to be:

( ) = !! ! ! ! ! ! ! ℏ ! = 2 * ( ! ! − )

which shows the dependence of tunneling on effective mass m*.

The barrier height Eφ (x) and the free carrier energy E can be conveniently normalized to the thermal energy kT:

ℎ = = !(!) = = leaving ! = 1 ℏ 2 * ( − ℎ)

which introduces a dependence on temperature T. Following the approach of Zhang et al 3 , the free carrier charge density is related to the barrier potential using the Poisson equation:

ℰ( ) = − ! ! = ! ! = 1 − !” ! !” = 1 − !” ! = 1 − !!

which shows the dependence on free carrier concentration n and relative dielectric constant ε r .

Integration once by parts shows that the variation of normalized barrier height with distance is given by

= ± 2 ( !! + ) = ! !

and therefore

= ± 2 ( !! + )

The final result for the transmission coefficient for tunneling through a slowly varying Schottky barrier is therefore

(ℎ) = ! !” ! !! !!! !”# !! !! !” !(! !) ! with !! = ℏ 2 ! ! * The ratio !” ! !!

therefore sets the scale for the amount of tunneling. When

!” ! !!

is large, the tunneling coefficient is small and the TE description is accurate. As

!” ! !! decreases, the tunneling coefficient increases. Ratio values of !” ! !!

~ 10, 1 and 0.1 are typically used to indicate the regimes in which TE, TFE and FE transport is dominant, respectively 3, 4 . We therefore see that the high free carrier concentrations n typically reported for nanowire nanoFETs 5 will typically drive Schottky barrier transport into the TFE regime.

## Analysis

As discussed in detail in Refs. [1, 3, 6] , a nanoFET is a metal-semiconductor-metal (MSM) device with reverse and forward biased Schottky barriers. The reverse bias metal-into-nanowire Schottky barrier (Fig. 1) dominates the transport when drain-source voltages are low to moderate, as was the case in our GaN experiments (±5V). In our stability approach, the reverse TFE J-V curve

J = * * !! V + !” cosh ! !! exp − !” ! exp V ′

! = !! coth !! ! = !! !! − tanh !!

is generated using known (generation) values for effective barrier height qφ Bn temperature T, and carrier concentration n. Fits performed on the generated curve using the Levenberg-Marquardt Algorithm are then used to identify stable parameter ranges.

The materials properties effective mass m* and dielectric constant ε r appear together as the combined product m*ε r in in E 00 , as shown in Eq’ns X-Y. The impact of m*ε r product variation on TFE stability was investigated over a realistic range from 0.1 to 8.0. The GaN, InP and PbTe choices were motivated by reported nanoFET investigations 1, 7, 8 . We report a first investigation of the scaling ratio for tunneling