nextnano^{3}  Tutorial
next generation 3D nano device simulator
2D Tutorial
DGMOS Structure  Double Gate MOSFET (Metal Oxide Semiconductor Field Effect
Transistor)
Author:
Stefan Birner
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 2DDoubleGateMOSFET_cl.in
> 2DDoubleGateMOSFET_qm.in
5 nm Double Gate MOSFET (Metal Oxide Semiconductor Field Effect
Transistor)
Some of these figures are included in this publication:
Modeling of semiconductor nanostructures with nextnano³
S. Birner, S. Hackenbuchner, M. Sabathil, G. Zandler, J.A. Majewski, T.
Andlauer, T. Zibold, R. Morschl, A. Trellakis, P. Vogl
Acta Physica Polonica A 110 (2), 111 (2006)
Structure
The main idea of a Double Gate MOSFET is to control the Si channel very
efficiently by choosing the Si channel width to be very small and by
applying a gate contact to both sides of the channel. This concept helps to
suppress short channel effects and leads to higher currents as compared with
a MOSFET having only one gate.
The structure consists of an intrinsic Si channel having the length 25 nm and
the width 5 nm.
(The width of 25 nm corresponds to the 65 nm technology node.)
This channel is connected to heavily ntype doped source and drain regions of
length 10 nm each (constant doping profile with a concentration of 1 x 10^{20}
cm^{3}, fully ionized).
The gates have a length of 25 nm and are separated from the Si channel by a 1.5
nm thick oxide layer.
(SiO_{2}, staticdielectricconstants =
3.9d0 )
The Double Gate MOSFET contains the following regions
cluster 
region 

1 
1 
Source contact 
2 
2 
ntype doped source region (Si) 
3 
3 
Si channel (undoped) 
4 
4 
ntype doped drain region (Si) 
5 
5 
Drain contact 
6 
6 
SiO_{2} 
7 
7 
Top gate 
7 
8 
Bottom gate 
Both the regions 7 (top gate) and 8 (bottom gate) form the cluster no. 7.
Along the channel direction (x direction), the grid lines are separated by 1 nm
(48 grid points),
along the y direction, the grid lines have a 0.5 nm separation (21 grid lines).
The total number of grid points is 48 x 21 = 1008.
Contacts
We apply a voltage of V_{D} = 0.5 V to the drain contact.
The gate voltage is varied from 0.3 V to 1.0 V in steps of 0.1 V (=>
$voltagesweep ).
At the gate a Schottky barrier of 3.075 V is assumed to mimick the gate
electrode work function which has been assumed to be 4.1 eV.
$poissonboundaryconditions
poissonclusternumber = 1
! Source
regionclusternumber = 1
appliedvoltage =
0.0d0 ! 0.0 [V]
boundaryconditiontype = ohmic
contactcontrol =
voltage
poissonclusternumber = 2
! Drain
regionclusternumber = 5
appliedvoltage =
0.5d0 ! 0.5 [V]
boundaryconditiontype = ohmic
contactcontrol =
voltage
poissonclusternumber = 3
! Gate
regionclusternumber = 7
appliedvoltage =
0.3d0 ! 0.3 [V] ... 1.0
[V]
boundaryconditiontype = Schottky
schottkybarrier =
3.075d0 ! phi_B
contactcontrol =
voltage
$end_poissonboundaryconditions
$voltagesweep
sweepnumber
= 1
sweepactive
= yes
poissonclusternumber = 3
! Gate: poissonclusternumber = 3
stepsize
= 0.1d0 ! voltage
steps of 0.1 [V]
numberofsteps =
13
dataouteverynthstep = 1
Mobility
The mobility is assumed to depend on temperature (T = 300 K) and on the
doping concentration but is independent of the electric field.
$simpledriftmodels
mobilitymodel = mobilitymodelsimba0
! =>
$mobilitymodelsimba
Thus we have two different electron mobilities:
 ntype doped Si region: 64.47 cm^{2}/Vs
µ = 55.2 cm^{2}/Vs + 1374.0 cm^{2}/Vs
/ [ 1 + ( 1.0*10^{20 }/ 1.072*10^{17})^{0.73} ] =
64.470195 cm^{2}/Vs
 intrinsic Si region: 1429.2 cm^{2}/Vs
µ = 55.2 cm^{2}/Vs + 1374.0 cm^{2}/Vs
= 1429.2 cm^{2}/Vs
Results
Electron density and conduction band profile
The following figure shows a slice through the middle of the device
(x=constant), i.e. through the gate contacts.
The source drain voltage is V_{SD} = 0.5 V, and the gate voltage is V_{G}
= 0.7 eV.
Two results are shown:
a) classical calculation
selfconsistent solution of the twodimensional
Poisson and current equations
The current equation is solved within a driftdiffusion model
based on the classical density.
b) quantum mechanical calculation
selfconsistent solution of the twodimensional
Poisson, Schrödinger and current equations
The current equation is solved within a driftdiffusion model
based on the quantum mechanical density.
The Schrödinger equation has to be solved each time three
times because of the different orientations of the
ellipsoidal electron effective mass tensors (m_{longitudinal} = 0.916 m_{0}
and m_{transversal}
= 0.190 m_{0}).
The Fermi level is almost flat, i.e.
constant (0.249 eV) and very similar in both simulations a) and b).
The conduction band edge in the Si
channel is lower in the case of the quantum mechanical simulation b).
The main difference is attributed to the electron density:
a) The classical density has its
maximum at the Si/SiO_{2} interface
because E_{F,n}  E_{C}
has its maximum there, i.e. the conduction
band edge is farthest below the Fermi level.
b) The quantum mechanical density is
practically zero at the Si/SiO_{2} interface because
due to the SiO_{2} barrier, the wave functions tends
to zero at the Si/SiO_{2} interface.
One can clearly see that the electron density has the highest values in
the middle of the channel
and not at
the Si/SiO_{2} interfaces.
The same figures as above but this time
 left figure: at a gate voltage of 0.3 V (closed channel) and
 right figure: at a gate voltage of 1.0 V (open channel).
The quantum mechanical density has different shapes at different voltages (one
maximum in the middle vs. two maxima offthecenter).
(Note: The axes are scaled differently.)
Electron wave functions
There are three Schrödinger equations that have to be solved each time having
the following mass tensors that enter into the Hamiltonian H(x,y):
 m_{xx} = m_{longitudinal} = 0.916 m_{0}
and m_{yy} = m_{zz} = m_{transversal}
= 0.190 m_{0}
 m_{yy} = m_{longitudinal} = 0.916 m_{0}
and m_{xx} = m_{zz} = m_{transversal}
= 0.190 m_{0}
 m_{zz} = m_{longitudinal} = 0.916 m_{0}
and m_{xx} = m_{yy} = m_{transversal}
= 0.190 m_{0}
Note that m_{zz}(x,y) does not enter the Hamiltonian but m_{zz}(x,y)
is used to calculate the quantum mechanical density (m_{} dispersion).
The quantum mechanical density for such a twodimensional simulation is
proportional to the square root of m_{zz}(x,y).
The quantum mechanical density is obtained for each grid point by
 summation over all eigenstates E_{i}
 evaluation of the square of the wave function psi(x,y)^{2}
 weighting psi(x,y)^{2} with the FermiDirac integral F_{1/2}[ (E_{F}
 E_{i}) / k_{B}T ] (which includes the Gamma prefactor of the
FermiDirac integral)
 multiplication by a factor (spin and valley degeneracy / (surface of
grid point), square root of (m_{zz }k_{B}T / (2 pi h_{bar}^{2}))
Most of the wave functions are located in the source and drain region (not
shown in the figure below).
This figure shows some of the lowest wave functions (psi^{2}) that
contribute to the quantum mechancial density in the region where the 1D slice
was taken (i.e. in the middle of the device (V_{G} = 0.7 V, V_{SD}
= 0.5 V).
The Fermi energy along the 1D slice through the middle of the device lies at
0.249 eV.
The states are labelled from top to bottom:
(Note: They are sorted by energies but their distance is not equivalent to their
energy spacing.)
deg1: 35^{th} state
0.215 eV (psi^{2} is zero at the 1D slice which can be seen in
the right figure)
deg1: 32^{nd} state 0.224 eV
(25 meV above Fermi level)
deg3: 13^{th} state 0.226 eV
(23 meV above Fermi level)
deg2: 32^{nd}
state 0.250 eV (below Fermi level, corresponding
to 2^{nd} subband)
deg2: 25^{th}
state 0.277 eV (below Fermi level,
corresponding to 1^{st} subband)
deg1: These are the states originating from the valleys having
the light, transversal mass perpendicular to the channel
(i.e. these
states have higher energies).
deg3: These are the states originating from the valleys having the
light, transversal mass in the plane of the channel
m_{xx} = m_{yy} = m_{transversal}
= 0.190 m_{0} (high energies due to light masses)
deg2: These
are the states originating from the valleys having the heavy, longitudinal
mass perpendicular to the channel
as is the
case in standard MOSFETs (i.e. these are the states that are occupied because
the energies are the lowest).
Output
 The band structure (conduction and valence bands) and the electrostatic
potential will be saved into the directory band_structure/ .
 The densities (electron densities) will be saved into densities/ .
 Current data (Fermi levels, current density, mobility and IV characteristics) will be saved into current/ .
 The raw data for the potentials and the Fermi levels (can be read in
later in subsequent runs) will be saved into raw_data/ .
IV characteristics (currentvoltage characteristics)
 The currentvoltage (IV) characteristic can be found in the following
file:
current/IV_characteristics2D.dat
The drain voltage has been kept constant at 0.5 V, the gate voltage varied from
0.3 V to 1.0 V.
Due to the influence of quantum mechanics a shift of the threshold voltage
V_{th} to a slightly higher voltage can be seen.
Note that the absolute magnitude of the current is
determined mostly by the mobility model.
By using a more realistic mobility model that takes into account the
dependency of the parallel and perpendicular electric fields, a smaller
current would be obtained.
a) To calculate the IV characteristics for the classical simulation (13 voltage steps from
0.3 V to 1.0 V), it took about 30 minutes.
(Linux executable for 64bit Opteron)
b) To calculate the IV characteristics for the quantum mechanical simulation (13 voltage steps from
0.3 V to 1.0 V), it took about minutes.
(Windows executable for 64bit Opteron)
 Please help us to improve our tutorial. Send comments to
support
[at] nextnano.com .
