nextnano^{3}  Tutorial
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1D Tutorial
CapacitanceVoltage curve of a "metal"insulatorsemiconductor (MIS)
structure
Authors:
Stefan Birner
> MIS_CV_1nmSiO2_1D_nn3.in / *_nnp.in  input file for the nextnano^{3}
and nextnano++ software
MIS_CV_3nmSiO2_1D_nn3.in / *_nnp.in  input file for the nextnano^{3} and nextnano++ software
MIS_CV_3nmSiO2_metal_1D_nn3.in / *_nnp.in  input file for the nextnano^{3}
and nextnano++ software
MIS_CV_3nmSiO2_metal_2D_nn3.in / *_nnp.in  input file for the nextnano^{3}
and nextnano++ software
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CapacitanceVoltage curve of a "metal"insulatorsemiconductor (MIS)
structure
In this tutorial we calculate the CV (capacitancevoltage) characterstics of
a polySi / SiO_{2} / Si structure.
This tutorial is based on the following paper:
[Richter]
A Comparison of QuantumMechanical CapacitanceVoltage Simulators
C.A. Richter, A.R. Hefner, E.M. Vogel
IEEE Electron Device Letters 22 (1), 35 (2001)
Our structure consists of a heavily ntype doped polysilicon (gate
material), a thin SiO_{2} layer (insulator), and a ptype doped Si
substrate.
Such a structure is similar to a MIS (metalinsulatorsemiconductor) structure
(or MOS = metaloxidesemiconductor).
We apply a voltage of V_{G} =  3 V to the polySi region
and vary this voltage stepwise until V_{G} = +3 V.
Accumulation and inversion
> MIS_CV_3nmSiO2_1D_nn3.in / *_nnp.in
The following two figures show the band profiles and the electron and hole densities
for two different gate voltages:
The left figure shows the conduction (Delta (X) valley) and valence band edges
of this structure at V_{G} =  3 V
where in the ptype Si region the charge density is dominated by holes (accumulation).
The right figure is for V_{G} = +3 V where in the ptype Si region the
charge density is dominated by electrons (inversion).
The band profile for zero bias is indicated here. Inside the quantum region
(0 nm  50 nm), the electron and hole density in the ptype Si is essentially
zero.
From 50 nm to 200 nm the hole density is nonzero (0.21 * 10^{18} cm^{3},
mostly heavy hole contribution).
The following parameters were used:
 SiO_{2} thickness: 3 nm
 doping concentration: ptype: N_{d} = 3
* 10^{17} cm^{3
}ntype: N_{poly} = 5 * 10^{19} cm^{3}
The following figure shows the charge carrier density as a function of bias
voltage.
One can clearly see that in the accumulation regime the hole density dominates
whereas in the inversion regime the electron density dominates the device
behavior.
The electron density can be found in this file: densities/int_el_dens1D.dat
The hole density can be found in this file:
densities/int_hl_dens1D.dat
The relevant column is labeled 'Cl_003 ', i.e.
clusternumber = 3 and
contains the integrated electron (or hole) density of the region no. 3 which
corresponds to the Si material from 0 nm to 50 nm.
This is the area where we solve the Schrödinger equation.
The following figure shows basically the same data but this time the charge
carrier density has been multiplied
with the elementary charge (e =  1.6022 * 1019 As). The red curve is the sum
of electron and hole charge.
Capacitancevoltage curve
> MIS_CV_3nmSiO2_1D_nn3.in / *_nnp.in
MIS_CV_1nmSiO2_1D_nn3.in / *_nnp.in
The following figure shows the capacitancevoltage (CV) curve for the 3
nm thick SiO_{2} layer (left) and for the 1 nm thick SiO_{2}
layer (right).
The CV curve has been calculated with the Origin software. Basically we took
the derivative of the total sheet charge density
in the Si substrate (clusternumber = 3 )
with respect to the bias voltage.
C = d Q / d V
C is the differential capacitance per unit area.
The red line is the calculated capacitance using the
simple approximation of a parallelplate condensator
which cannot account for the complicated behavior of the real CV curve.
(Note that the capacitance is a series capacitance, i.e. 1/C is the sum
of 1/C_{oxide} and 1/C_{semiconductor}.)
In the strong accumulation regime, and in the strong inversion regime, the
space charge layers are very thin and thus the capacity is "approximately equal"
to the insulator capacitance, i.e. it is dominated by the thickness of the
insulating layer (SiO_{2}).
The following parameters were used:
Left figure
SiO_{2} thickness: 3 nm 
Right figure
SiO_{2} thickness: 1 nm 
doping concentration: ptype: N_{d}
= 3 x 10^{17} cm^{3
}ntype: N_{poly} = 5 x 10^{19} cm^{3} 
doping concentration: ptype: N_{d}
= 1 x 10^{18} cm^{3
}ntype: N_{poly} = 1 x 10^{20} cm^{3} 
Our results are in reasonable agreement with the paper of Richter et al.
In his paper, Richter did not take into account wave function penetration into
the SiO_{2} barrier.
Consequently and in order to be able to reproduce his results, we also did not
include this effect.
However, wave function penetration can be included easily by extending the
quantum cluster into the SiO_{2} region.
The following figure shows the CV curve again for the 3 nm SiO_{2}
layer.
Here, we compare the three cases:
 classical
calculation in the pSi channel, metal contact instead of ntype polySi
layer, i.e. the "metal contact region" is excluded from the Poisson equation
 quantum mechanical calculation in the pSi
channel, metal contact instead of ntype polySi layer, i.e. the "metal contact region" is excluded from the Poisson equation
 quantum mechanical calculation in the pSi channel,
including ntype polySi layer depletion
In order to mimick the zero bias condition where we do not have flat bands
(see figure above), we include a workfunction for the metal.
The workfunction of the metal is modeled by a Schottky barrier height of 3.2 V,
so that the minimum of the capacitance alignes with V_{G} = 0 V.
Note: If we simulate a "metal contact", the region from z =  20 nm
to z = %OxideThickness is exempted from the simulation.
Only the boundary condition (boundaryconditiontype =
Schottky ) is included at z =
%OxideThickness .
For the case where we take polySi depletion into account (boundaryconditiontype
= Fermi ), the region from z =  20
nm to z = %OxideThickness is included in the simulation.
All three cases are in good agreement with Fig. 1 of Richter et al.
One can also plot the surface potential vs. applied gate voltage, e.g. as in Fig. 10
(b) of Chapter 4 of S. M. Sze, Physics of Semiconductor Devices (2007).
$outputbandstructure
...
potential
= yes ! output electrostatic potential
outputgridposition = 0.0 !
[nm] output information at x = 0.0
nm, e.g. electrostatic potential vs. voltage
Some comments on the calculations...
 The temperature was set to 300 Kelvin.
 Selfconsistent solution of the 1DSchrödingerPoisson equation within
the singleband effectivemass approximation (using ellipsoidal effective
mass tensors) for the (Delta) conduction band edges and isotropic masses for
the heavy, light and splitoff holes.
The size of the quantum cluster extends from 0 nm to 50 nm.
 The polySi depletion has been taken into account (although only
classically and not quantum mechanically).
 Please help us to improve our tutorial! Send comments to
support
[at] nextnano.com .
