nextnano^{3}  Tutorial
next generation 3D nano device simulator
1D Tutorial
Mobility in twodimensional electron gases (2DEGs)
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.
> 1DInSb_mobility_ShaoFig3.in
> 1DGaAs_mobility_WalukiewiczFig2.in
> 1DGaAs_mobility_WalukiewiczFig3.in
> 1DInGaAs_mobility_WalukiewiczFig8.in
> 1DGaN_mobility_WalukiewiczFig4.in
> 1DGaN_mobility_WalukiewiczFig5.in
Mobility in twodimensional electron gases (2DEGs)
> 1DInSb_mobility_ShaoFig3.in
This tutorial is based on the following paper:
[Shao]
Carrier Mobilities in deltadoped Heterostructures
Y. Shao, S.A. Solin, L.R. RamMohan
(2006)
arXiv:condmat/0602140
Our implementation is based on the equations that are given in this paper
(with the exception of Eq. (3.5) where we added a factor of 1 / (4 pi) because
of SI units).
We calculate the mobility in a 40 nm InSb quantum well that is surrounded by
and strained with respect to Al_{0.15}In_{0.85}Sb barriers.
At z =  20 nm, there is a deltadoping layer with a sheet doping
density of 1 * 10^{12} cm^{2}.
The deltadoping layer is separated from the InSb QW by a 40 nm Al_{0.15}In_{0.85}Sb
spacer layer.
quantumwellwidth =
40d0 ! [nm] [Shao] Fig. 3
spacerwidth
= 40d0 ! [nm] [Shao] Fig. 3
remotedopingsheetdensity = 1d12 ! [cm^2]
[Shao] Fig. 3
We calculate all properties for different temperatures. This can be done as
follows:
$globalparameters
latticetemperature
= 1d0 ! start value T =
1 [K]
temperaturesweepactive
= yes ! 'yes'
/ 'no'
temperaturesweepstepsize =
10d0 ! increase temperature each time
by T = 10 [K]
temperaturesweepnumberofsteps = 31
! increase temperature 31 times
dataouteverynthstep
= 1 ! output all
data for every temperature sweep
$end_globalparameters
All output files are labelled with an index, starting from zero, that
refers to each individual temperature sweep: .../..._ind000....dat
Here, the index runs from 0 (000 ) to 30 (030 ), i.e.
31 output files for each property in total.
We note that band gaps and lattice constants depend on temperature. This is
taken into account automatically for each temperature sweep.
The following figure shows the conduction band edge, the Fermi level and the
square of the lowest two electron wave functions at T = 1 K.
To plot such a figure, the following output files are needed:
 band_structure/cb1D_001_ind000.dat:
1^{st} column: distance [nm]
2^{nd} column: conduction band edge at the Gamma
point [eV]
Note: The index 'ind000 ' refers to the temperature
sweep. Here, index 0 means T = 1 K.
 current/fermi1Del_ind000.dat:
1^{st} column: distance [nm]
2^{nd} column: Fermi level of the electrons [eV]
In this tutorial, the Fermi level is always equal to 0 eV.
 Schroedinger_1band/cb001_ind000_qc001_sg001_deg001_dir_Kz001.dat:
1^{st} column: distance [nm]
n columns: n energies of the eigenstates
[eV]
n columns: n squares of the wave functions (psi^{2})
[eV]
In the figure, we plotted the columns for psi^{2} of the
two lowest states: psi_{1}^{2},
psi_{2}^{2}
The following figure shows the sheet electron density as a function of
temperature.
We considered the two lowest subbands for calculating the 2DEG density (the spin
degeneracy of the subbands is included):
2DEGsheetdensitynumberofsubbands = 2
Our results differ from the results of Fig. 3(b) of the [Shao] paper.
(Some obvious discrepancies are the conduction band offset (We used ~0.15 eV
whereas [Shao] used ~0.25 eV.) and the Schottky barrier height.)
To plot such a figure, the following output file was used:
 Monte_Carlo/mobility_TemperatureSweep.dat:
1^{st} column: temperature [K]
last column: electron sheet density of the lowest
subband(s) [m^{2}]
The following figure shows the calculated 2DEG mobility as a function of
temperature.
The relevant data can be found in this file:
 Monte_Carlo/mobility_TemperatureSweep.dat:
1^{st} column: temperature [K]
2^{nd} column: total mobility [m^{2}/Vs]
3^{rd} column: mobility due to ionized
impurity scattering [m^{2}/Vs]
4^{th} column: mobility due to background
impurity scattering [m^{2}/Vs]
5^{th} column: mobility due to deformation
potential acoustic phonon scattering [m^{2}/Vs]
6^{th} column: mobility due to polar optic
(LO)
phonon scattering [m^{2}/Vs]
We included the following scattering mechanisms:
ionizedimpurityscattering =
yes ! [Shao] (including remote and
background ionized impurity scattering)
acousticphononscattering =
yes ! [Shao]
polaropticalphononscattering = yes !
[Shao]
alloyscattering
= no ! [Shao]
We now discuss the agreement/disagreement compared to Fig. 3(a) of the [Shao]
paper.
 The mobility due to acoustic phonon scattering is in excellent
agreement.
 The mobility due to polar optic LO phonon scattering is in
excellent agreement if one takes into
account that [Shao] forgot to include the factor of 1/(4 pi)
due to SI units.
 The mobility due to ionized and background impurity scattering
differs significantly.
It seems that the disagreement is not only due to the
different sheet density that has been used.
We used the following value:
impuritybackgrounddopingconcentration =
5d15 ! [cm^3] [Shao] Fig. 3
The following InSb material parameters have been used:
conductionbandmasses
= 0.0135d0 0.0135d0 0.0135d0 ! [m0] [Shao]
...
!
staticdielectricconstants =
16.82d0 16.82d0 16.82d0 ! [Shao]
epsilon(0)
opticaldielectricconstants =
15.7d0
! [Shao] epsilon(infinity)
LOphononenergy
= 0.025d0
! [eV] [Shao] (optical phonon energy)
massdensity
= 5.79d3
! [kg/m^3] [Shao]
soundvelocity
= 3.7d3
! [m/s] [Shao]
acousticdeformationpotential = 7.2d0
! [eV] [Shao]
> 1DGaAs_mobility_WalukiewiczFig2.in  (Experiment of Hiyamizu
et al.)
> 1DGaAs_mobility_WalukiewiczFig3.in  (Experiment of DiLorenzo et al.)
Here, we test our algorithm to results on GaAs 2DEGs of another publication.
We note that our algorithm is suitable for deltadoped 2DEGs but the GaAs
examples are not deltadoped.
This tutorial is based on the following paper:
[Walukiewicz]
Electron mobility in modulationdoped heterostructures
W. Walukiewicz, H.E. Ruda, J. Lagowski, H.C. Gatos
Physical Review B 30, 4571 (1984)
The experimental data is based on:
[Walukiewicz, Fig. 2]: [Hiyamizu]
Improved Electron Mobility Higher than 10^6
cm^2/Vs in Selectively Doped GaAs/NAlGaAs Heterostructures grown by MBE
S. Hiyamizu, J. Saito, K. Nanbu
Japan. J. Appl. Phys. 22, L609 (1983)
[Walukiewicz, Fig. 3]: [DiLorenzo]
Material and device considerations for
selectively doped heterojunction transistors
J.V. DiLorenzo, R. Dingle, M. Feuer, A.C.
Gossard, R. Hendel, J.C.M. Hwang, A. Kastalsky, V.G. Keramidas, R.A. Kiehl,
P. O'Connor
IEEEIEDM (International Electron Devices Meeting) 28,
578 (1982)
[Walukiewicz, Fig. 2]: [Hiyamizu] 

[Walukiewicz, Fig. 3]: [DiLorenzo] 



Conduction band profile and
wave functions 

Conduction band profile and
wave functions 
20 nm Al_{0.3}Ga_{0.7}As
spacer
spacerwidth = 20d0 ! 20
[nm] 

23 nm Al_{0.3}Ga_{0.7}As
spacer
spacerwidth = 23d0 ! 23
[nm] 









2DEG sheet density 

2DEG sheet density 









Mobility 

Mobility 
impuritybackgrounddopingconcentration = 9d13
! [cm^{3}]
remotedopingsheetdensity =
3.5d11 ! [cm^{2}]
(to fit experiment)
(remotedopingsheetdensity =
1.948344d11 ! [cm^{2}])
[Walukiewicz] Fig. 2: 8.6 * 10^{16} [cm^{3}]
==> 8.6 * 10^{16} [cm^{3}]^{2/3}
=
1.948344d11 (?)


impuritybackgrounddopingconcentration = 1d15
! [cm^{3}]
remotedopingsheetdensity
= 1d12 ! [cm^{2}]










Differences with respect to
Walukiewicz paper 

Differences with respect to
Walukiewicz paper 


In Walukiewicz's paper, the
background impurity scattering is
dominating the ionized impurity scattering.
We found the opposite. 



Walukiewicz used a 2DEG density of 3 * 10^{11}
cm^{2}. 

Walukiewicz used a 2DEG density of 2.2 
3.8 * 10^{11} cm^{2}. 
conductionbandmasses = 0.067d0 0.067d0 0.067d0
! [m0]
...
! a higher value than for bulk because of nonparabolicity
Here we used 0.067 as this gives better
agreement to the mobility at higher temperatures and
this is the usually accepted material parameter for GaAs
quantumwellwidth = 13d0 ! [nm] (13
nm seems to be a better value than 20 nm.)
alloyscattering =
yes
Alloy scattering is relevant for the part of the wave function that
penetrates into the AlGaAs barrier.
The
squares are experimental values of Fig. 5 in:
Improved Electron Mobility Higher than 10^{6} cm^{2}/Vs
in Selectively Doped GaAs/NAlGaAs Heterostructures Grown by MBE
S. Hiyamizu et al.
Jpn. J. Appl. Phys. 22, L609 (1983) 

conductionbandmasses = 0.076d0 0.076d0 0.076d0 ! [m0] [Walukiewicz]
...
! a higher value than for bulk because of nonparabolicity. This is also
the value used by Walukiewicz.
quantumwellwidth = 20d0 ! [nm] (This
value might be too large. See left where 13 nm was used.)
alloyscattering =
no

The following GaAs material parameters have been used:
staticdielectricconstants =
12.9d0 12.9d0 12.9d0 !
[Walukiewicz] epsilon(0)
opticaldielectricconstants =
10.9d0
! [Walukiewicz] epsilon(infinity)
LOphononenergy
= 0.036d0
! [eV] [Walukiewicz] (optical phonon energy)
massdensity
= 5.318d3
! [kg/m^3] [Davies] p. 411
soundvelocity
= 5.29d3
! [m/s] [X.L. Lei, J. Phys. C 18, L593 (1985)]
acousticdeformationpotential = 7d0
! [eV] [Walukiewicz]
Into the equation for the deformation potential
acoustic phonon scattering, the quantum well width is an input
parameter.
We used a value of 13 nm which corresponds roughly to the extension of the
ground state wave function inside the "triangular" QW.
> 1DInGaAs_mobility_WalukiewiczFig8.in  (Experiment of Kastalsky
et al.)
Here, we test our algorithm to results on InGaAs 2DEGs of another
publication.
We note that our algorithm is suitable for deltadoped 2DEGs but the
InGaAs examples are not deltadoped.
This tutorial is based on the following paper:
[Walukiewicz]
Electron mobility in modulationdoped heterostructures
W. Walukiewicz, H.E. Ruda, J. Lagowski, H.C. Gatos
Physical Review B 30, 4571 (1984)
The experimental data is based on:
[Walukiewicz, Fig. 8]: [Kastalsky]
Twodimensional electron gas at a molecular beam
epitaxialgrown, selectively doped, In_{0.53}Ga_{0.47}AsIn_{0.48}Al_{0.52}As
interface
A. Kastalsky, R. Dingle, K.Y. Cheng, A.Y. Cho
Applied Physics Letters 41, 274 (1982)
[Kastalsky] and [Walukiewicz] have used In_{0.48}Al_{0.52}As
whereas we used In_{0.52}Al_{0.48}As
which is lattice matched to InP and In_{0.53}Ga_{0.47}As.
The conduction band profile is shown in the following figure.
Here, two subbands (psi_{1}^{2},
psi_{2}^{2}) are occupied
although our implementation of calculating the mobility is only applicable to
one occupied subband.
Note that the 2DEG is located in an alloy, i.e. InGaAs. Thus we expect that
alloy scattering has a significant effect on the total mobility.
The following figure shows the subband density of the first subband
and of the first two subbands as a function
of temperature.
Inside the mobility algorithm, only the density of the first subband has been
considered.
The following figure shows the mobility as a function of temperature.
At temperatures below 100 K, the total mobility is dominated by alloy
scattering.
Our results for the mobility are in reasonable agreement with Fig. 8 of the
paper of [Walukiewicz].
The following parameters have been used:
ionizedimpurityscattering
= yes ! (including
remote and background ionized impurity scattering)
acousticphononscattering
= yes !
polaropticalphononscattering
= yes !
alloyscattering
= yes !
quantumwellwidth
= 15d0 ! [nm] 15 nm seems
to be a reasonable approximation for the triangular well
spacerwidth
= 8d0 ! [nm]
impuritybackgrounddopingconcentration = 1.0d16
! [cm^{3}]
remotedopingsheetdensity
= 1.0d12 ! [cm^{2}]
2DEGsheetdensitynumberofsubbands =
1
alloydisorderscatteringpotential =
0.60d0 ! [eV] InGaAs bulk value [J.R.
Hayes et al. (1982)]
!
! In_{0.53}Ga_{0.47}As material parameters
!
massdensity
= 5.5025d3 ! [kg/m^3] InGaAs
[Wen et al., JAP 100, 103516 (2006)]
soundvelocity
= 4.753d3 ! [m/s] In_{0.53}Ga_{0.47}As[111]
[Wen et al., JAP 100, 103516 (2006)]
acousticdeformationpotential = 7d0
! [eV] InGaAs
[Walukiewicz]
> 1DGaN_mobility_WalukiewiczFig4.in
> 1DGaN_mobility_WalukiewiczFig5.in
Here, we test our algorithm to results on GaN 2DEGs of another publication.
We note that our algorithm is suitable for deltadoped 2DEGs but the GaN
examples are not deltadoped.
This tutorial is based on the following paper:
[WalukiewiczGaN]
Electron mobility in Al_{x}Ga_{1x}N/GaN heterostructures
L. Hsu, W. Walukiewicz
Physical Review B 56, 1520 (1997)
Note: To be consistent with the paper of Walukiewicz, no (!) piezo and
pyroelectricity is included.
[WalukiewiczGaN] Fig. 4 

[WalukiewiczGaN] Fig. 5 



Conduction band profile and
wave functions 

Conduction band profile and
wave functions 
20 nm Al_{0.15}Ga_{0.85}N
spacer
spacerwidth = 20d0 ! 20
[nm] 

no spacer
spacerwidth = 0d0 ! 0
[nm] 






Into the equation for the
deformation potential acoustic phonon scattering, the quantum
well width is an input parameter.
We used values which correspond roughly to the extension of the ground
state wave function inside the "triangular" QW. 
quantumwellwidth = 15d0 ! 15
[nm] 

quantumwellwidth = 10d0 ! 10
[nm] 






2DEG sheet density 

2DEG sheet density 



[WalukiewiczGaN] used a 2DEG density of
6.2 * 10^{15} m^{2}. 

[WalukiewiczGaN] used a 2DEG density of
1.59 * 10^{16} m^{2}. 






Mobility 

Mobility 
impuritybackgrounddopingconcentration = 1d14
! [cm^{3}]
remotedopingsheetdensity =
7.883735d11 ! [cm^{2}]
[WalukiewiczGaN] Fig. 4: 7 * 10^{17} [cm^{3}]
==> 7 * 10^{17} [cm^{3}]^{2/3}


impuritybackgrounddopingconcentration = 4d15
! [cm^{3}]
remotedopingsheetdensity
= 1d12 ! [cm^{2}]




alloydisorderscatteringpotential = 2.3d0
! [eV] conduction band offset GaN/AlN 
The following GaN material parameters have been used:
conductionbandmasses
= 0.21d0 0.21d0 0.21d0 ! [m0]
[WalukiewiczGaN]
staticdielectricconstants =
9.5d0 9.5d0 9.5d0 ! [WalukiewiczGaN]
epsilon(0)
opticaldielectricconstants =
5.35d0
5.35d0
5.35d0
! [WalukiewiczGaN] epsilon(infinity)
LOphononenergy
= 0.0905d0 0.0905d0 0.0905d0
! [eV] [WalukiewiczGaN] (optical phonon energy)
massdensity
= 6.1d3
! [kg/m^3] [WalukiewiczGaN]
soundvelocity
= 6.6d3 ! [m/s] [WalukiewiczGaN]
acousticdeformationpotential = 8.5d0 ! [eV] [WalukiewiczGaN]
Here, alloy scattering is only relevant for the part of the wave function that
penetrates into the AlGaN barrier.
Final remark: In principle, the results of these GaN 2DEGs are not reliable
as piezo and pyroelectricity have to be included.
Our results disagree quantitatively with the results of [WalukiewiczGaN].
However, it is not clear, which material parametes he used for the conduction
band offset and the alloy scattering.
Further hints
If two remote doping regions should be taken into account, one can input an
array of values.
spacerwidth
= 20d0 10d0 ! [nm]
spacer width of 1^{st} and 2^{nd} doping region
remotedopingsheetdensity = 1d12
1d11
! [cm^{2}] remote doing density of 1^{st} and 2^{nd}
doping region
 Please help us to improve our tutorial. Send comments to
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[at] nextnano.com .
