COMSOL 6.4 - Surface-Trap-Induced Hysteresis in an InAs Nanowire FET

Surface-Trap-Induced Hysteresis in an InAs Nanowire FET — a Density-Gradient Analysis

This tutorial analyzes the hysteresis of the conductance–gate-voltage (G–Vg) curves of an InAs nanowire FET, using the density-gradient theory to add the effect of quantum confinement to the conventional drift–diffusion formulation, without a large increase of computational costs. The hysteresis is caused by the dynamic charging effects of fast and slow semiconductor–oxide interface traps of continuous energy distributions and of both donor and acceptor types. The capture probability is modeled as thermally activated with a barrier height that varies with the trap energy level. The qualitative behavior and the order of magnitude of the computed G–Vg curves under various voltage ramping conditions agree well with simulation and experiment results found in the literature.

Introduction

The effect of surface states plays an important role on the electronic behavior of nanowires. This tutorial follows the approach described in the reference paper (Ref. 1) to compute the conductance as a function of the gate voltage history of an InAs nanowire FET. Instead of solving Schrödinger–Poisson equations, the density-gradient theory (as reviewed in Ref. 2) is used to add the effect of quantum confinement in a computationally efficient manner.

Model Definition

The model computes the conductance and the Fermi level relative to the conduction band edge, to be compared with Fig. 3 in Ref. 1.

The Semiconductor interface is used to describe the governing physics. This approach is not exactly the same but is in close correspondence with the one described in Ref. 1 (more details later). Some parameters for our approach have to be guessed. In particular, the bulk material properties of InAs are taken from Ref. 3 and the density-gradient oxide barrier parameters from Ref. 4. The section Modeling Instructions lists all the parameters used in the model. The density-gradient parameters are directly entered in the physics features.

In the experiment reported in Ref. 1, the gate voltage is held at an initial voltage for a long time for the system to reach equilibrium. Then the gate voltage (Vg) is ramped to a peak or valley voltage and then immediately ramped back to the initial voltage. During this voltage ramp, the conductance (G) is measured and recorded. Different voltage ramp rates are used. The G–Vg curves show different degrees of hysteresis, with slower ramps showing larger hysteresis.

Three representative gate voltage ramping conditions are simulated in this model, as shown in the table below, to be compared with Fig. 3 in Ref. 1.

Table 1: Gate voltage ramping conditions simulated in the model
Condition	1	2	3
Ramp rate (dVdt)	44 uV/s	27 mV/s	27 mV/s
Initial voltage (V0)	6 V	-4 V	6 V
Peak/Valley voltage (V1)	-4 V	6 V	-4 V

In Ref. 1, the hexagonal InAs nanowire is approximated as an infinitely long cylinder to be modeled in 1D axial symmetric geometry. Comparing to the time scales of the voltage ramps, the carriers in the bulk are assumed to be instantaneously equilibrated, while the dynamics of the surface traps is simulated using time-dependent equations. In the tutorial model, on the other hand, everything is treated as time dependent and a grounded metal contact is placed at the center axis of the cylinder. This achieves the same instantaneous equilibrium for the carriers in the bulk.

In Ref. 1, the conductance is computed from the electron density divided by a weighted sum of the ionized surface trap densities to account for their scattering effects. The electron density is computed by summing quantized 1D subbands and the hole density using the bulk 3D formula. In the tutorial model, both carrier types are treated as bulk 3D, and the density-gradient formulation is used to include the quantum confinement effect. In addition, the zero-density-gradient boundary condition of the density-gradient formulation at metal contacts helps maintaining a reasonable carrier density profile at the center axis of the cylinder.

The gate contact boundary condition in the model incorporates the effective oxide relative permittivity and effective oxide thickness using the formulas given in Ref. 1. The properties of the four different types of surface traps are all provided by the reference paper and entered into the model accordingly.

See the comments in the section Modeling Instructions for more detailed discussions on the model construction, solution processes, and result visualization.

Results and Discussion

Figure 1 shows the simulated conductance as a function of the gate voltage, to be compared with Fig. 3(a) and (b) in Ref. 1. The hysteresis and order of magnitude match reasonably well with the experiment and model results shown in the reference paper.

Figure 1: Conductance versus gate voltage curves.

Figure 2 plots the Fermi level relative to the conduction band edge versus the gate voltage, to be compared with Fig. 3(c) in the reference paper. The general trend and order of magnitude match reasonably well.

Figure 2: Fermi level versus gate voltage.

References

1. D. Lynall and others, “Surface state dynamics dictating transport in InAs nanowires,” Nano Letters, vol. 18, no. 2, p. 1387, 2018.

2. M.G. Ancona, “Density-gradient theory: a macroscopic approach to quantum confinement and tunneling in semiconductor devices,” J. Comput. Electron., vol. 10, p. 65, 2011.

3. M. Levinshtein, S. Rumyantsev, and M. Shur, Handbook Series on Semiconductor Parameters, vol. 1, World Scientific, 1996, reprinted 2000.

4. S. Jin, Y.J. Park, and H.S. Min, “Simulation of Quantum Effects in the Nano-scale Semiconductor Device,” J. Semicond. Tech. Sci., vol. 4, no. 1, p. 32, 2004.

Semiconductor_Module/Transistors/inas_nanowire_traps_hysteresis_density_gradient

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, Approximate the nanowire as an infinitely long cylinder using the 1D axial symmetric geometry.

click

In the tree, select > .

Click .

Use the Semiconductor Equilibrium study step to simulate the initial condition of the experiment after the sample has been biased at a fixed gate voltage for a long time.

Click

In the tree, select > .

Click

Geometry 1

The Model Wizard exits and opens the Model Builder desktop environment at the Geometry node. Set the length scale here right away.

In the window, under click .

In the window for , locate the section.

From the list, choose .

Global Definitions

Next, enter model parameters. Use separate Parameters nodes to organize the parameters in different categories, beginning with the nanowire parameters: the radius, length, temperature, and gate oxide parameters are from Ref. 1. The semiconductor material parameters are guess values using bulk material properties. The series conductance is also a guess value which in the paper was a fitting parameter to account for other effects such as the contact resistances.

Parameters 1 - Nanowire FET

In the window, under click .

In the window for , type Parameters 1 - Nanowire FET in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
R0	23[nm]	2.3E-8 m	Radius of nanowire
L0	1.6[um]	1.6E-6 m	Length of nanowire FET channel
mun0	4e4[cm^2/V/s]	4 m²/(V·s)	Electron mobility
mup0	5e2[cm^2/V/s]	0.05 m²/(V·s)	Hole mobility
epsr0	15.15	15.15	Relative permittivity
Eg00	0.354[V]	0.354 V	Band gap
chi00	4.9[V]	4.9 V	Electron affinity
Nc0	8.7e16[cm^-3]	8.7E22 1/m³	Conduction band effective density of state
Nv0	6.6e18[cm^-3]	6.6E24 1/m³	Valence band effective density of state
T0	300[K]	300 K	Temperature
d_ox0	100[nm]	1E-7 m	Thickness of gate oxide
d_ox	d_ox0*acosh((d_ox0+R0)/R0)	2.361E-7 m	Effective thickness of gate oxide
epsr_ox	2.12	2.12	Effective relative permittivity of gate oxide
G_s	50[uS]	5E-5 S	Series conductance

Hide these parameters from parameter selections, such as for parametric sweeps, to make it easier to set up the studies later with a smaller number of parameters to choose from.

Click to expand the section. Clear the checkbox.

Next, enter the trap parameters from Ref. 1 in a new Parameters node and hide also these ones from study settings.

Parameters 2 - Surface Traps

In the toolbar, click

and choose > .

In the window for , type Parameters 2 - Surface Traps in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
Ds_tot	4e13[cm^-2]	4E17 1/m²	Total trap density
Ds_f	1.6e12[cm^-2]	1.6E16 1/m²	Fast trap density
Ds_s	Ds_tot-Ds_f	3.84E17 1/m²	Slow trap density
Ea0_f	0[mV]	0 V	Emission barrier parameter of fast trap
Ea0_s	700[mV]	0.7 V	Emission barrier parameter of slow trap
gamma0	1e6[1/s]	1E6 1/s	Capture rate constant
Et0_a	-400[mV]	-0.4 V	Center of acceptor trap energy level distribution below Ec (negative value means above Ec, that is, inside the conduction band)
Et0_d	300[mV]	0.3 V	Center of donor trap energy level distribution below Ec
sig0	200[mV]	0.2 V	Width of trap energy level distribution
gD0	2	2	Degeneracy factor

Locate the section. Clear the checkbox.

Before entering the remaining parameters, define a triangle function to be used in the subsequent parameter definition for the ramping of the gate voltage.

Triangle 1 (tri1)

In the toolbar, click

and choose > .

In the window for , locate the section.

In the text field, type 0.

In the text field, type 1.

Click to expand the section. Clear the checkbox.

Now define the parameters for the gate voltage bias conditions of the experiment. Use the triangle function to define the voltage sweep as a function of time: the gate voltage Vg is swept from V0 to V1 and then back to V0 at the ramp rate of dVdt. Leave this set of parameters visible for study settings.

Parameters 3 - Bias Conditions

In the toolbar, click

and choose > .

In the window for , type Parameters 3 - Bias Conditions in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Value	Description
dVdt	44[uV/s]	4.4E-5 V/s	Voltage ramp rate
V0	6[V]	6 V	Initial and final voltage of triangular voltage ramp
V1	-4[V]	-4 V	Peak (or valley) voltage of triangular voltage ramp
t	0[s]	0 s	Time parameter
t_max	2*abs(V1-V0)/dVdt	4.5455E5 s	Duration of voltage ramp
Vg	V0+(V1-V0)*tri1(t/t_max)	6 V	Gate voltage

To approximate the nanowire as an infinitely long cylinder using the 1D axial symmetric geometry, simply add a line interval with the length set to the radius of the cylinder R0.

Geometry 1

Interval 1 (i1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Coordinates (nm)
0
R0

Click

Add a blank material and enter the parameters for the semiconductor material properties defined earlier.

Materials

Material 1 (mat1)

In the window, under right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Electron mobility	mun	mun0	m²/(V·s)	Semiconductor material
Hole mobility	mup	mup0	m²/(V·s)	Semiconductor material
Relative permittivity	epsilonr_iso ; epsilonrii = epsilonr_iso, epsilonrij = 0	epsr0	1	Basic
Band gap	Eg0	Eg00	V	Semiconductor material
Electron affinity	chi0	chi00	V	Semiconductor material
Effective density of states, conduction band	Nc	Nc0	1/m³	Semiconductor material
Effective density of states, valence band	Nv	Nv0	1/m³	Semiconductor material

Semiconductor (semi)

Now set up the physics. Enter the channel length parameter as the out-of-plane thickness. Choose Fermi–Dirac statistics and use the density-gradient formulation to add quantum-confinement effect to the drift–diffusion equation system.

In the window, under click .

In the window for , locate the section.

In the d text field, type L0.

Locate the section. From the list, choose .

Click to expand the section. From the list, choose .

Enter the temperature and some guess values for the density-gradient effective masses.

Semiconductor Material Model 1

In the window, under > click .

In the window for , locate the section.

In the T text field, type T0.

Locate the section. In the meDG text field, type me_const*0.03.

In the mhDG text field, type me_const*0.4.

Use a metal contact boundary condition at the center of the cylinder to capture the effect of the source and drain contacts used in the experiment, which are located out-of-plane in the 1D axisymmetric geometry for this model. The zero-gradient boundary condition for the carrier concentrations at the metal contact as given by the density-gradient formulation is particularly suitable in the case to approximate the carrier density profile near the origin.

Metal Contact 1

In the toolbar, click

and choose .

Select Boundary 1 only.

Add a gate contact boundary condition at the circumference of the cylinder, using the nanowire gate parameters defined earlier and some guess values for the density-gradient formulation of the potential barrier at the semiconductor–oxide interface.

Thin Insulator Gate 1

In the toolbar, click

and choose .

Select Boundary 2 only.

In the window for , locate the section.

In the V0 text field, type Vg.

Locate the section. In the εins text field, type epsr_ox.

In the dins text field, type d_ox.

Locate the section. From the list, choose .

In the meOx text field, type me_const*0.5.

In the meOx* text field, type me_const*0.22.

In the ΦnOx text field, type 3[V].

In the ΦpOx text field, type 3[V].

Ref. 1 used four kinds of continuous trap energy distributions to simulate the effect of traps at the semiconductor–oxide interface. This can be done in the model using a physics node with four subnodes. First add the parent physics node. Use the formulation which works better for time-dependent studies.

Trap-Assisted Surface Recombination 1

In the toolbar, click

and choose .

Select Boundary 2 only.

In the window for , locate the section.

From the list, choose .

Locate the section. From the list, choose .

Select the checkbox.

From the list, choose .

Next, add the subnodes using the trap parameters for the fast acceptors defined earlier, except for the electron capture probability. Since the latter varies with the trap energy level, enter it later after having defined some relevant variables. In Ref. 1, the hole capture probability is assumed to be zero. Discretize the continuous trap energy distribution using 100 levels as was done in Ref. 1. In this model the 100 levels are distributed within three standard deviations of the Gaussian distribution to efficiently cover the majority of the traps.

Continuous Energy Levels - Fast Acceptor

In the toolbar, click

and choose .

In the window for , type Continuous Energy Levels - Fast Acceptor in the text field.

Locate the section. From the list, choose .

Locate the section. In the Nt text field, type Ds_f.

From the list, choose .

In the Et,0 text field, type Et0_a.

In the σ text field, type sig0.

In the gD text field, type gD0.

In the Nxdelem text field, type 100.

In the Et,min text field, type semi.tasr1.Ec0t-Et0_a-3*sig0.

In the Et,max text field, type semi.tasr1.Ec0t-Et0_a+3*sig0.

Locate the section. From the list, choose . From the list, choose . In the Cp text field, type 0[cm^3/s].

Duplicate the subnode to reuse most settings for the fast donors. Enter the settings that are different for the fast donors: the trap type and the parameter for the center of the trap distribution.

Continuous Energy Levels - Fast Donor

Right-click and choose .

In the window for , type Continuous Energy Levels - Fast Donor in the text field.

Locate the section. From the list, choose .

Locate the section. In the Et,0 text field, type Et0_d.

In the Et,min text field, type semi.tasr1.Ec0t-Et0_d-3*sig0.

In the Et,max text field, type semi.tasr1.Ec0t-Et0_d+3*sig0.

Continuous Energy Levels - Slow Acceptor

In the window, right-click and choose .

In the window for , type Continuous Energy Levels - Slow Acceptor in the text field.

Locate the section. In the Nt text field, type Ds_s.

Continuous Energy Levels - Slow Donor

In the window, right-click and choose .

In the window for , type Continuous Energy Levels - Slow Donor in the text field.

Locate the section. In the Nt text field, type Ds_s.

Definitions (comp1)

Now define variables for the electron capture probability, which depends on the trap energy level. The dependence is based on an energy barrier between the trap level and the conduction band edge as formulated in Ref. 1. Since the density-gradient formulation reduces the electron density at the semiconductor–oxide interface as the result of quantum confinement, but the actual electron flux should not be reduced by the same factor, a factor Nfac with a guess value of 10 is introduced to compensate for the reduced electron density.

Variables 1 - Capture Probabilities

In the window, under right-click and choose .

In the window for , type Variables 1 - Capture Probabilities in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
dE_fa	semi.tasr1.ctb1.Vxd-semi.tasr1.Ec0t	V	Fast acceptor trap energy level relative to conduction band edge
dE_fd	semi.tasr1.ctb2.Vxd-semi.tasr1.Ec0t	V	Fast donor trap energy level relative to conduction band edge
dE_sa	semi.tasr1.ctb3.Vxd-semi.tasr1.Ec0t	V	Slow acceptor trap energy level relative to conduction band edge
dE_sd	semi.tasr1.ctb4.Vxd-semi.tasr1.Ec0t	V	Slow donor trap energy level relative to conduction band edge
Eb_fa	dE_fa+max(-dE_fa,Ea0_f)	V	Fast acceptor trap capture barrier
Eb_fd	dE_fd+max(-dE_fd,Ea0_f)	V	Fast donor trap capture barrier
Eb_sa	dE_sa+max(-dE_sa,Ea0_s)	V	Slow acceptor trap capture barrier
Eb_sd	dE_sd+max(-dE_sd,Ea0_s)	V	Slow donor trap capture barrier
Nfac	10		Compensation factor for electron flux
Cn_fa	gamma0/Nc0exp(-Eb_fae_const/k_B_const/T0)*Nfac	m³/s	Fast acceptor trap electron capture probability
Cn_fd	gamma0/Nc0exp(-Eb_fde_const/k_B_const/T0)*Nfac	m³/s	Fast donor trap electron capture probability
Cn_sa	gamma0/Nc0exp(-Eb_sae_const/k_B_const/T0)*Nfac	m³/s	Slow acceptor trap electron capture probability
Cn_sd	gamma0/Nc0exp(-Eb_sde_const/k_B_const/T0)*Nfac	m³/s	Slow donor trap electron capture probability

Semiconductor (semi)

Enter the electron capture probabilities defined above into the physics settings.

Continuous Energy Levels - Fast Acceptor

In the window, under > > click .

In the window for , locate the section.

In the Cn text field, type Cn_fa.

Continuous Energy Levels - Fast Donor

In the window, click .

In the window for , locate the section.

In the Cn text field, type Cn_fd.

Continuous Energy Levels - Slow Acceptor

In the window, click .

In the window for , locate the section.

In the Cn text field, type Cn_sa.

Continuous Energy Levels - Slow Donor

In the window, click .

In the window for , locate the section.

In the Cn text field, type Cn_sd.

Definitions (comp1)

Define variables for the conductance using the formula given in Ref. 1 to account for the ionized impurity scattering effect. First define coupling operators to access the trap variables at the gate and to integrate the computed conductance over the modeling domain. Then define the variables.

Integration 1 - Access Trap Variables at the Gate

In the toolbar, click

and choose .

In the window for , type Integration 1 - Access Trap Variables at the Gate in the text field.

Locate the section. From the list, choose .

Select Boundary 2 only.

Locate the section. Clear the checkbox.

Integration 2 - Integrate Over the Cylinder

In the toolbar, click

and choose .

In the window for , type Integration 2 - Integrate Over the Cylinder in the text field.

Locate the section. From the list, choose .

Variables 2 - Conductance

In the window, right-click and choose .

In the window for , type Variables 2 - Conductance in the text field.

Locate the section. In the table, enter the following settings:


Name	Expression	Unit	Description
NAm	-intop1(semi.tasr1.ctb1.rho_tr+semi.tasr1.ctb3.rho_tr)/e_const	1/m²	NA- : Ionized acceptor trap density
NDp	intop1(semi.tasr1.ctb2.rho_tr+semi.tasr1.ctb4.rho_tr)/e_const	1/m²	ND+ : Ionized donor trap density
G_0	intop2(mun0semi.N)e_constDs_tot/(NDp+30NAm)/L0	S	Channel conductance
G	1/(1/G_0+1/G_s)	S	Total conductance

Study 1 - Two Fast and One Slow Gate Voltage Ramps

The default mesh works fine for this model so you can proceed directly to setting up the studies. Use an outer Parametric Sweep to run two fast and one slow gate voltage ramps as in the abstract panel of Ref. 1.

In the window, click .

In the window for , type Study 1 - Two Fast and One Slow Gate Voltage Ramps in the text field.

Parametric Sweep

In the toolbar, click

In the window for , locate the section.

Click

three times.

In the table, enter the following settings:


Parameter name	Parameter value list	Parameter unit
dVdt (Voltage ramp rate)	44[uV/s] 27[mV/s] 27[mV/s]	mV/s
V0 (Initial and final voltage of triangular voltage ramp)	6 -4 6	V
V1 (Peak (or valley) voltage of triangular voltage ramp)	-4 6 -4	V

Note that the list in the column is populated only with the last set of parameters, as set up earlier.

Add a time-dependent step to run the transient study.

Step 2: Time Dependent

In the toolbar, click

In the window for , locate the section.

In the text field, type range(0,0.01,1)*t_max.

In the toolbar, click

Results

Net Dopant Concentration (semi)

Because the model has no doping profiles defined, the default plot of net dopant concentration is not needed.

In the window, under right-click and choose .

Create a G–Vg plot to compare with the simulation and experimental results of Ref. 1. We find a good match of both the qualitative hysteresis behavior and the order of the magnitude of the conductance. Use the model-tree node label as the plot title.

G-Vg curves

In the toolbar, click

In the window for , type G-Vg curves in the text field.

Locate the section. From the list, choose .

Click to expand the section. From the list, choose .

Locate the section. From the list, choose .

Global 1

Right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Expression	Unit	Description
G	uS	Total conductance

Locate the section. From the list, choose .

In the text field, type Vg.

Click to expand the section. Find the subsection. Clear the checkbox.

In the toolbar, click

Create an Ef–Vg plot to compare with the simulation results of Ref. 1. We again find a good match of both the qualitative hysteresis behavior and the order of the magnitude of the nonequilibrium Fermi level relative to the conduction band edge.

Ef-Ec vs. Vg

In the toolbar, click