nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
Selfconsistent 6band k.p calculations of holes in strained Si/SiGe MOSFETs
Note: This tutorial's copyright is owned by Stefan Birner,
www.nextnano.com.
Authors:
Stefan Birner
If you want to obtain the input files that are used within this tutorial, please contact stefan.birner@nextnano.de.
> 1DstrainedSi_SiGe_OberhuberPRB1998_step1_selfconsistent.in
 selfconsistent singleband Schrödinger equation
1DstrainedSi_SiGe_OberhuberPRB1998_step2_6x6kp_selfconsistent.in 
selfconsistent 6band k.p
Schrödinger equation which reads in potential of singleband results as initial
guess
1DstrainedSi_SiGe_OberhuberPRB1998_6x6kp_selfconsistent.in
 selfconsistent 6band k.p
Schrödinger equation
Selfconsistent 6band k.p calculations of holes in strained Si/SiGe MOSFETs
This tutorial is based on the following paper:
[Oberhuber]
Subband structure and mobility of twodimensional holes in strained
Si/SiGe MOSFET's
R. Oberhuber, G. Zandler, P. Vogl
Physical Review B 58, 9941 (1998)
Structure
We calculate the hole energy levels, wave function and density of the
following pchannel MOS structure:
Gate  SiO_{2}
 strained Si  Si_{0.8}Ge_{0.2}
 Gate (applied voltage: V_{G} = 1.6 V and 1.1 V)
 5 nm SiO_{2}
 15 nm strained Si(001) with respect to Si_{1x}Ge_{x},
x=0.2, homogenously ntype doped 5 * 10^{16} cm^{3}
 500 nm unstrained Si_{1x}Ge_{x} buffer, x=0.2,
homogenously ntype doped 5 * 10^{16} cm^{3}
Both, Si and SiGe are ntype doped.
The method is 6band k.p where we have to solve the Schödinger equation
(6band k.p Hamiltonian) not only for k_{} = 0
but also for a lot of k_{} vectors, i.e. for k_{
}= (k_{x},k_{y}) /= 0.
Our algorithm is the following:

1DstrainedSi_SiGe_OberhuberPRB1998_step1_selfconsistent.in
 selfconsistent singleband Schrödinger equation
First, we solve the singleband Schrödinger equation and the Poisson
equation selfconsistently.
This is useful because we want to obtain a reasonable start value for
the electrostatic potential.
$simulationflowcontrol
...
rawpotentialin = no
! STEP 1 only
$quantummodelholes
...
modelname =
effectivemass ! STEP 1 only
(singleband)

1DstrainedSi_SiGe_OberhuberPRB1998_step2_6x6kp_selfconsistent.in 
selfconsistent 6band k.p
Schrödinger equation which reads in potential of singleband results as
initial guess
Then we read in the electrostatic potential calculated in 1) (which
serves as a start value for the electrostatic potential) and
solve the 6band k.p Schrödinger equation and the Poisson equation
selfconsistently (for both k_{} = 0 and k_{}
/= 0).
$simulationflowcontrol
...
rawpotentialin = yes
! STEP 2 (read in electrostatic
potential)
$quantummodelholes
...
modelname =
6x6kp
! STEP 2 (6band k.p)
1DstrainedSi_SiGe_OberhuberPRB1998_6x6kp_selfconsistent.in
 selfconsistent 6band k.p
Schrödinger equation
In principle, step 1 can be omitted but it helps to save some CPU time for step 2. Note: If you want to avoid step 1, rawpotentialin
must be set to no . This is
then equivalent to this input file.
Note: "3" leads to the same results as "1" + "2"
In agreement with [Oberhuber] (fur the purpose of this tutorial), we do not
allow the wave functions to penetrate into the SiO_{2} layer (i.e. we
assume infinite barriers at the SiO_{2}/Si interface) although nextnano³
is general enough to also include the oxide into the quantum region if desired.
Material parameters
We use the same material parameters as quoted in the [Oberhuber] paper. The
only modification is the band offset.
We use a GeSi valence band offset, i.e. average of the three valence band edges
(E_{v,av}), of 0.58 eV (unstrained) and
for Si we take the following value for the absolute valence band deformation
potential:
absolutedeformationpotentialvb = 2.05d0
! [eV]
This results in a band offset with respect to the topmost valence band edges:
heavy/light hole of Si_{0.8}Ge_{0.2}
(unstrained)  light hole strained Si = 0.046027 eV
Results
The following figure shows the valence band edge profile and the hole density
for two different gate voltages (V_{G} = 1.6 V, V_{G} = 1.1 V).
$poissonboundaryconditions
...
appliedvoltage = 1.6d0 ! V_{G}
= 1.6 V Fig. 2(a) of [Oberhuber]
! appliedvoltage = 1.1d0 ! V_{G}
= 1.1 V Fig. 2(b) of [Oberhuber]
Due to tensile strain, the topmost valence band in the strained Si layer is the
light hole band.
The heavy/light hole splitting for
strained Si on Si_{0.8}Ge_{0.2} substrate: 79.7 meV ([Oberhuber]
80 meV)
In light gray, the squares of the topmost wave functions are indicated.
One can clearly see, that the inversion channel has a width of less than 10 nm.
At room temperature (300 K), in the Si layer only the 34 highest hole subbands
are significantly occupied (not shown).
The small density ("parasitic channel") in the SiGe layer is not due to these
34 states but due to other states.
In our calculations, the influence of the parasitic channel in the Si_{0.8}Ge_{0.2}
layer is negligible whereas [Oberhuber] found that only for high voltages (e.g.
V_{G} = 1.6 V) it is negligible and for V_{G} = 1.1 V almost 30
% of this density is contained in the parasitic channel.
The topmost subband ( 31.3 meV for V_{G} = 1.6 V,
 89.4 meV for V_{G} = 1.1 V) has predominantly lighthole character whereas the
second, heavyhole related subband lies
 65.4 meV below the topmost subband (for V_{G} = 1.6 V).
 54.6 meV below the topmost subband (for V_{G} = 1.1 V).
([Oberhuber] 55 meV)
The inversion layer sheet density (integrated over whole device) is found to
be:
 for V_{G} = 1.6 V: n_{s} = 1.85 * 10^{12}
cm^{2} ([Oberhuber], Fig. 2(a): n_{s} = 1.2 *
10^{12} cm^{2})
 for V_{G} = 1.1 V: n_{s} = 2.08 * 10^{11}
cm^{2} ([Oberhuber], Fig. 2(b): n_{s} = 4
* 10^{11} cm^{2})
Our results are in reasonable agreement with Fig. 2(a) and 2(b) of
[Oberhuber] considering the uncertainty in some assumptions (e.g. k_{}
space resolution, valence band offset, grid resolution).
Density of states (DOS)
The following figure shows the density of states (i.e. all eigenstates are
considered) for the two different gate voltages.
Note:
 At V_{G} = 1.6 V, the valence band edge
maximum is approximately at E_{v} = 0.106 eV.
 At V_{G} = 1.1 V, the valence band edge maximum is approximately
at E_{v} =
 0.010 eV.
The total DOS output is contained in the following file: Schroedinger_kp/DOS_hl_sum_norm.dat
The following figure shows the density of states for the ground state (light
hole state) for the two different gate voltages.
(The DOS for the spin up state is very similar to the DOS of the spin down
state.)
The DOS output for each eigenstate is contained in the following file:
Schroedinger_kp/DOS_hl_norm.dat
The DOS, as calculated within this 6band k.p approach is used for
obtaining the selfconsistent quantum mechanical hole density, thus taking into
account nonparabolicity rather than employing a parabolic energy dispersion E(k_{x},k_{y}).
For more details on the density of states, see this tutorial:
Electron density of states (DOS) of a GaAs quantum well with infinite barriers
and
hole density of states (DOS) of a Si hole channel (triangular potential)
 Please help us to improve our tutorial. Send comments to
support
[at] nextnano.com .
