- Email: [email protected]
Study of charge carrier quantization in strained Si-nMOSFETs
ARTICLE IN PRESS
Materials Science in Semiconductor Processing 8 (2005) 363–366
Study of charge carrier quantization in strained Si-nMOSFETs C.D. Nguyen, A.T. Pham, C. Jungemann, B. Meinerzhagen NST, Technical University Braunschweig, Postfach 33 29, 38023 Braunschweig, Germany Available online 28 October 2004
Abstract In this paper, the size quantization of the electron gas in the channel of a strained Si-MOSFET is investigated based on the Schro¨dinger equation and density gradient model, which is widely used in commercial TCAD software. Since the results of the density gradient model based on standard parameters found in the literature strongly deviate from the more fundamental Schro¨dinger equation, a new parameter model has been developed for strained Si. The improved density gradient model yields good results for a wide range of strain, temperature, and doping concentrations in the Si layer. r 2004 Elsevier Ltd. All rights reserved. Keywords: MOS devices; Quantization effects; Inversion layer; MOSFET
1. Introduction Strained Si is a promising material for the improvement of Si MOS performance because of its enhanced carrier mobilities. The electron mobility enhancement in strained Si inversion layers has been studied theoretically [1,2] and experimentally [3,4]. The tensile strain splits the sixfold-degenerate valleys of the first conduction band into two energetically lowered valleys ‘‘D2 ’’ and four raised valleys ‘‘D4 ’’ (Fig. 1a). This improves the mobility in two ways: (1) The electrons prefer to populate the lower valleys ‘‘D2 ’’ with their lower conductivity mass. (2) Intervalley phonon scattering between the ‘‘D2 ’’-and the ‘‘D4 ’’-valleys is suppressed. The change in the band structure due to strain has a strong impact on size quantization in the inversion layer which has to be taken into account in device modeling in order to obtain the correct threshold voltage and gate capacitance Corresponding author. Tel.: +49 0 531 391 3168.
E-mail addresses: [email protected], [email protected] (C.D. Nguyen).
for MOSFETs. This can be easily done on the level of the Schro¨dinger equation (SE). However, solving the SE in a TCAD device simulator is very CPU intensive and leads to numerical problems [5,6]. Therefore, simple and more efficient approximate quantum correction models have been developed (e.g. density gradient method (DGM) [6]), but only for the case of unstrained Si. In this paper we study the impact of strain on inversion layer modeling. First, the size quantization in an nMOSFET is discussed based on the SE. Second, we examine the accuracy of the DGM, which was developed for unstrained Si [6], and demonstrate how to improve it for the case of strained Si.
2. Simulation model Similar to unstrained Si [7], the dynamic behavior of electrons in the inversion layer of strained Si can be described by a SE: _2 d2 v 2 eV v ðzÞ zvl ðzÞ ¼ E vl zvl ðzÞ ð1Þ 2mz dz with the boundary conditions zvl ðz ¼ 0Þ at the SiO2 /Siinterface and zvl ðz ¼ 1Þ ¼ 0; where E vl and zvl ðzÞ in
1369-8001/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mssp.2004.09.055
ARTICLE IN PRESS C.D. Nguyen et al. / Materials Science in Semiconductor Processing 8 (2005) 363–366
364 Perpendicular ∆ 2 valleys
Eq. (3) does not take into account any density contributions of states above E max : For these states, the classical expression of the charge density is used: X E F eV v ; (4) N vc exp ncl ðzÞ ¼ kB T v
kz
3D
ky In plane ∆ 4 valleys
kx ∆4 EC
∆ ES ∼ ∼ 0.67y eV ∆2
2D
ky
(b)
kx
(a)
strained Si on Si1− y Gey
Fig. 1. (a) The six equivalent valleys of the lowest conduction band in the first Brillouin zone and the projection on the kx ky plane and (b) strained-induced conduction band splitting in strained Si.
Eq. (1) are the lth subband energy and envelope function in vth valley, respectively. The potential V v ðzÞ is the sum of the electrostatic potential V ðzÞ and the conduction band edge in the respective valley (Fig. 1b). The conduction band edge step at the strained Si=Si1y Gey interface is given by [9] DE c ¼ 0:53 eV y 0:2 eV y2 :
(2)
The energy splitting between the ‘‘D2 ’’- and ‘‘D4 ’’-valleys is given by DE s 0:67 eV y [1]. mvz is the electron effective mass for motion in the z-direction. For unstrained Si it is given by mDz 2 ¼ 0:916m0 and mDz 4 ¼ 0:19m0 : The dependence of the mass on the Ge content y was investigated in Ref. [8] and found to be negligible. The SE is self-consistently solved with the Poisson equation [7]. Having obtained all eigensolutions of the SE up to an energy E max for all valleys, the quantum mechanical contribution to the charge density (using Maxwell– Boltzmann statistics) is calculated as kB T X v X v 2 E F E vl qm n ðzÞ ¼ (3) mxy jzl ðzÞj exp kB T p_2 v l (for Fermi statistics, expðxÞ is replaced by ð1 þ expðxÞÞ), where E F is the quasi-Fermi potential, kB the Boltzmann pffiffiffiffiffiffiffiffiffiffiffiffi constant, T the lattice temperature and mvxy ¼ mvx mvy the density-of-states mass in the xy-plane. Similar to mvz ; the dependence of this mass on the Ge content y can be neglected [8] and given by mDxy2 ¼ mt ¼ 0:19m0 ; mDxy4 ¼ pffiffiffiffiffiffiffiffiffiffiffi mt ml ¼ 0:417m0 ; where ml and mt are the longitudinal and transverse electron mass of unstrained Si.
where N vc is the effective density of states in v: Valleys and is given by kB T 3=2 v v v 1=2 ðmx my mz Þ : (5) N vc 2 2p_2 The above-described quantum mechanical model (QM) is used as a reference for the density gradient model. In the case of the density gradient model, the calculation of the density is based on Eq. (4) together with the following approximate quantum correction [6,10]: X E F eV v Fm L ndg ðzÞ ¼ ; (6) N vc exp kB T v where Fm ¼ ð3=2ÞkB T logðm Þ (m a fit factor) and L is obtained by solving a differential equation [10]: ( ) ¯ 1 ¯ 2 _2 g EF F EF F L¼ rr r þ : (7) 12m kB T 2 kB T ¯ ¼ eV v þ Fm þ L is the ‘‘smoothed potential’’, Here, F m the local density-of-states mass and g a fitting parameter. The standard parameters of the DGM determined for unstrained Si are given in Table 1. Note that those parameters were determined by matching the results of our QM for the case of unstrained Si [7]. Hence, they slightly deviate from the parameters given in Ref. [10]. The use of those parameters for strained Si leads to large errors in the threshold region. In order to improve the DGM, the parameters m and g have been determined for a wide range of Ge contents y and strained Si thicknesses by matching the inversion charge Table 1 Parameters of the DGM for unstrained Si m
g
m
1.07
3.6
1.4
Table 2 Parameters of the DGM for strained Si i
ai
bi
ci
di
m g
9:38 101 3.35
5:64 101 0.25
1:01 101 11.62
3:38 101 4:2 101
ARTICLE IN PRESS C.D. Nguyen et al. / Materials Science in Semiconductor Processing 8 (2005) 363–366
365
Fig. 2. Schematic diagram of the subband energies and wave-functions in a MOS inversion layer without (left figure) and with (right figure) strain. Biaxial tension introduces an additional energy separation between the D2 and D4 valleys.
10 10
i ¼ m ; g;
12
(8)
where ai ; bi ; ci and d i are parameters which are shown in Table 2.
Error [%]
with
-2
10
f i ðyÞ ¼ ai þ ðbi þ ci yÞ expðd i yÞ
13
2
10
1
10
10
3. Results In our study, a strained Si-nMOSFET with a Si0:8 Ge0:2 substrate and a 10 nm thick strained Si layer is investigated. The oxide thickness is 5 nm and the channel doping 3 1017 =cm3 : Fig. 2 illustrates the energy subbands and wave-functions calculated by the QM. The larger shift between the subband energies of the two groups of valleys in strained Si compared to the unstrained case is caused by the biaxial strain. In order to investigate the impact of strain on the accuracy of the DGM, the inversion layer density has been calculated by the DGM and QM for the strained Si example described above. This comparison is performed for two versions of the DGM. In Fig. 3, the DGM based on the parameters g and m for unstrained Si (see Table 1) is compared to the QM. The resulting inversion layer density is shown for both models together with the relative error of the DGM in Fig. 3. It can be seen that the error of the DGM without parameter modification is large in the threshold region.
inversion layer density [cm ]
density calculated by the QM. It is found that m and g depend mainly on the Ge content y and can be approximated by the following analytic function of y
10
10
0
DGM Reference (QM)
10
11
-1
0
0.5
1 gate bias [V]
1.5
10
10 2
9
8
Fig. 3. Inversion layer density at room temperature as a function of the gate voltage for zero back bias for the SE and DGM and relative error of the DGM.
In Fig. 4, the DGM with modified parameters gðyÞ and m ðyÞ (see Eq. (8) and Table 2) is compared to the QM for the same strained Si example. Now a good agreement is found between the QM and DGM. To demonstrate that the new gðyÞ and m ðyÞ model yields good results for many different devices, the comparison of the modified DGM with the QM was repeated for other relevant doping levels and Ge contents. The results are summarized in Figs. 5 and 6.
ARTICLE IN PRESS C.D. Nguyen et al. / Materials Science in Semiconductor Processing 8 (2005) 363–366
366 2
Error [%]
10
10
10
10
0
DGM (Mod.) Reference (QM)
10
11
10
9
-2
12
1
inversion layer density [cm ]
10
10
13
-2
10
inversion layer density [cm ]
10
10
13
12
17
NA=5x10 /cm
10
10
11
-1
0
0.5
1 gate bias [V]
10 2
1.5
Si/Si0.5Ge0.5 Si/Si0.6Ge0.4
10
Si/Si0.7Ge0.3 10
10
3
DGM (Mod.) Reference (QM)
9
8
10
Fig. 4. Inversion layer density at room temperature as a function of the gate voltage for zero back bias where the DGM was based on the new gðyÞ and m ðyÞ-parameter.
8
0
0.5
1 gate bias [V]
1.5
2
Fig. 6. Inversion layer density for three different Ge contents y.
References 10
13
17
-2
inversion layer density [cm ]
17
10
3
NA=7x10 /cm 12
3
NA=5x10 /cm
Si/Si0.8Ge0.2
10
11 18
3
NA=1x10 /cm
10
10
10
DGM (Mod.) Reference (QM)
9
8
10 0
0.5
1 gate bias [V]
1.5
2
Fig. 5. Inversion layer density for three different doping concentrations.
The results confirm the universality and reliability of the new gðyÞ and m ðyÞ model.
4. Conclusions We have studied the impact of size quantization on inversion layers in strained Si-nMOSFETs. Similar to the unstrained Si case, quantum effects cannot be neglected. The standard g; m -parameters of the DGM determined for unstrained Si are found to lead to large errors in strained Si, whereas our new gðyÞ; m ðyÞ model yields good results for a large variety of devices.
[1] Vogelsang T, Hofmann RK. Electron transport in strained Si layers on Si1x Gex substrates. Appl Phys Lett 1993; 63:186–8. [2] Takagi S, Hoyt JL, Welser J, Gibbons J. Comparative study of phonon-limited mobility of two-dimensional electrons in strained and unstrained Si metal-oxidesemiconductor field-effect transistors. J Appl Phys 1996; 80:1567–77. [3] Welser J, Hoyt JL, Takagi S, Gibbons JF. Strain dependence of the performance enhancement in strainedSi n-MOSFETs. In: IEDM Technical Digest 1994. p. 373–6. [4] Rim K, Hoyt JL, Gibbons JF. Fabrication and analysis of deep submicron strained-Si N-MOSFETs. IEEE Trans Electron Devices 2000;47(7):1406–15. [5] Jungemann C, Nguyen CD, Neinhu¨s B, Decker S, Meinerzhagen B. Improved modified local density approximation for modeling of size quantization in NMOSFETs. In: MSM’2001, international conference on modeling and simulation of microsystems. (USA): 2001. p. 458–61. [6] Wettstein A, Schenk A, Fichtner W. Quantum devicesimulation with the density-gradient model on unstructured grids. IEEE Trans Electron Devices 2001;48(2): 279–84. [7] Jungemann C, Emunds A, Engl WL. Simulation of linear and nonlinear electron transport in homogeneous silicon inversion layers. Solid-State Electron 1993;36:1540–993. [8] Rieger MM, Vogl P. Electronic-band parameters in strained Si1x Gex alloys on Si1y Gey substrates. Phys Rev B 1993;48:14276–87. [9] Nuernbergk D. Simulation des Transportverhaltens in Si=Si1x Gex =Si-Heterobipolartransistoren. Diss., TU Ilmenau, Ilmenau, 1999. (H. Utz Verlag Wissenschaft, Mu¨nchen: 1999). [10] Wettstein A, Penzin O, Lyumkis E. Integration of the density gradient model into a general purpose device simulator. VLSI Design 2002;15:751–9.
Materials Science in Semiconductor Processing 8 (2005) 363–366
Study of charge carrier quantization in strained Si-nMOSFETs C.D. Nguyen, A.T. Pham, C. Jungemann, B. Meinerzhagen NST, Technical University Braunschweig, Postfach 33 29, 38023 Braunschweig, Germany Available online 28 October 2004
Abstract In this paper, the size quantization of the electron gas in the channel of a strained Si-MOSFET is investigated based on the Schro¨dinger equation and density gradient model, which is widely used in commercial TCAD software. Since the results of the density gradient model based on standard parameters found in the literature strongly deviate from the more fundamental Schro¨dinger equation, a new parameter model has been developed for strained Si. The improved density gradient model yields good results for a wide range of strain, temperature, and doping concentrations in the Si layer. r 2004 Elsevier Ltd. All rights reserved. Keywords: MOS devices; Quantization effects; Inversion layer; MOSFET
1. Introduction Strained Si is a promising material for the improvement of Si MOS performance because of its enhanced carrier mobilities. The electron mobility enhancement in strained Si inversion layers has been studied theoretically [1,2] and experimentally [3,4]. The tensile strain splits the sixfold-degenerate valleys of the first conduction band into two energetically lowered valleys ‘‘D2 ’’ and four raised valleys ‘‘D4 ’’ (Fig. 1a). This improves the mobility in two ways: (1) The electrons prefer to populate the lower valleys ‘‘D2 ’’ with their lower conductivity mass. (2) Intervalley phonon scattering between the ‘‘D2 ’’-and the ‘‘D4 ’’-valleys is suppressed. The change in the band structure due to strain has a strong impact on size quantization in the inversion layer which has to be taken into account in device modeling in order to obtain the correct threshold voltage and gate capacitance Corresponding author. Tel.: +49 0 531 391 3168.
E-mail addresses: [email protected], [email protected] (C.D. Nguyen).
for MOSFETs. This can be easily done on the level of the Schro¨dinger equation (SE). However, solving the SE in a TCAD device simulator is very CPU intensive and leads to numerical problems [5,6]. Therefore, simple and more efficient approximate quantum correction models have been developed (e.g. density gradient method (DGM) [6]), but only for the case of unstrained Si. In this paper we study the impact of strain on inversion layer modeling. First, the size quantization in an nMOSFET is discussed based on the SE. Second, we examine the accuracy of the DGM, which was developed for unstrained Si [6], and demonstrate how to improve it for the case of strained Si.
2. Simulation model Similar to unstrained Si [7], the dynamic behavior of electrons in the inversion layer of strained Si can be described by a SE: _2 d2 v 2 eV v ðzÞ zvl ðzÞ ¼ E vl zvl ðzÞ ð1Þ 2mz dz with the boundary conditions zvl ðz ¼ 0Þ at the SiO2 /Siinterface and zvl ðz ¼ 1Þ ¼ 0; where E vl and zvl ðzÞ in
1369-8001/$ - see front matter r 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mssp.2004.09.055
ARTICLE IN PRESS C.D. Nguyen et al. / Materials Science in Semiconductor Processing 8 (2005) 363–366
364 Perpendicular ∆ 2 valleys
Eq. (3) does not take into account any density contributions of states above E max : For these states, the classical expression of the charge density is used: X E F eV v ; (4) N vc exp ncl ðzÞ ¼ kB T v
kz
3D
ky In plane ∆ 4 valleys
kx ∆4 EC
∆ ES ∼ ∼ 0.67y eV ∆2
2D
ky
(b)
kx
(a)
strained Si on Si1− y Gey
Fig. 1. (a) The six equivalent valleys of the lowest conduction band in the first Brillouin zone and the projection on the kx ky plane and (b) strained-induced conduction band splitting in strained Si.
Eq. (1) are the lth subband energy and envelope function in vth valley, respectively. The potential V v ðzÞ is the sum of the electrostatic potential V ðzÞ and the conduction band edge in the respective valley (Fig. 1b). The conduction band edge step at the strained Si=Si1y Gey interface is given by [9] DE c ¼ 0:53 eV y 0:2 eV y2 :
(2)
The energy splitting between the ‘‘D2 ’’- and ‘‘D4 ’’-valleys is given by DE s 0:67 eV y [1]. mvz is the electron effective mass for motion in the z-direction. For unstrained Si it is given by mDz 2 ¼ 0:916m0 and mDz 4 ¼ 0:19m0 : The dependence of the mass on the Ge content y was investigated in Ref. [8] and found to be negligible. The SE is self-consistently solved with the Poisson equation [7]. Having obtained all eigensolutions of the SE up to an energy E max for all valleys, the quantum mechanical contribution to the charge density (using Maxwell– Boltzmann statistics) is calculated as kB T X v X v 2 E F E vl qm n ðzÞ ¼ (3) mxy jzl ðzÞj exp kB T p_2 v l (for Fermi statistics, expðxÞ is replaced by ð1 þ expðxÞÞ), where E F is the quasi-Fermi potential, kB the Boltzmann pffiffiffiffiffiffiffiffiffiffiffiffi constant, T the lattice temperature and mvxy ¼ mvx mvy the density-of-states mass in the xy-plane. Similar to mvz ; the dependence of this mass on the Ge content y can be neglected [8] and given by mDxy2 ¼ mt ¼ 0:19m0 ; mDxy4 ¼ pffiffiffiffiffiffiffiffiffiffiffi mt ml ¼ 0:417m0 ; where ml and mt are the longitudinal and transverse electron mass of unstrained Si.
where N vc is the effective density of states in v: Valleys and is given by kB T 3=2 v v v 1=2 ðmx my mz Þ : (5) N vc 2 2p_2 The above-described quantum mechanical model (QM) is used as a reference for the density gradient model. In the case of the density gradient model, the calculation of the density is based on Eq. (4) together with the following approximate quantum correction [6,10]: X E F eV v Fm L ndg ðzÞ ¼ ; (6) N vc exp kB T v where Fm ¼ ð3=2ÞkB T logðm Þ (m a fit factor) and L is obtained by solving a differential equation [10]: ( ) ¯ 1 ¯ 2 _2 g EF F EF F L¼ rr r þ : (7) 12m kB T 2 kB T ¯ ¼ eV v þ Fm þ L is the ‘‘smoothed potential’’, Here, F m the local density-of-states mass and g a fitting parameter. The standard parameters of the DGM determined for unstrained Si are given in Table 1. Note that those parameters were determined by matching the results of our QM for the case of unstrained Si [7]. Hence, they slightly deviate from the parameters given in Ref. [10]. The use of those parameters for strained Si leads to large errors in the threshold region. In order to improve the DGM, the parameters m and g have been determined for a wide range of Ge contents y and strained Si thicknesses by matching the inversion charge Table 1 Parameters of the DGM for unstrained Si m
g
m
1.07
3.6
1.4
Table 2 Parameters of the DGM for strained Si i
ai
bi
ci
di
m g
9:38 101 3.35
5:64 101 0.25
1:01 101 11.62
3:38 101 4:2 101
ARTICLE IN PRESS C.D. Nguyen et al. / Materials Science in Semiconductor Processing 8 (2005) 363–366
365
Fig. 2. Schematic diagram of the subband energies and wave-functions in a MOS inversion layer without (left figure) and with (right figure) strain. Biaxial tension introduces an additional energy separation between the D2 and D4 valleys.
10 10
i ¼ m ; g;
12
(8)
where ai ; bi ; ci and d i are parameters which are shown in Table 2.
Error [%]
with
-2
10
f i ðyÞ ¼ ai þ ðbi þ ci yÞ expðd i yÞ
13
2
10
1
10
10
3. Results In our study, a strained Si-nMOSFET with a Si0:8 Ge0:2 substrate and a 10 nm thick strained Si layer is investigated. The oxide thickness is 5 nm and the channel doping 3 1017 =cm3 : Fig. 2 illustrates the energy subbands and wave-functions calculated by the QM. The larger shift between the subband energies of the two groups of valleys in strained Si compared to the unstrained case is caused by the biaxial strain. In order to investigate the impact of strain on the accuracy of the DGM, the inversion layer density has been calculated by the DGM and QM for the strained Si example described above. This comparison is performed for two versions of the DGM. In Fig. 3, the DGM based on the parameters g and m for unstrained Si (see Table 1) is compared to the QM. The resulting inversion layer density is shown for both models together with the relative error of the DGM in Fig. 3. It can be seen that the error of the DGM without parameter modification is large in the threshold region.
inversion layer density [cm ]
density calculated by the QM. It is found that m and g depend mainly on the Ge content y and can be approximated by the following analytic function of y
10
10
0
DGM Reference (QM)
10
11
-1
0
0.5
1 gate bias [V]
1.5
10
10 2
9
8
Fig. 3. Inversion layer density at room temperature as a function of the gate voltage for zero back bias for the SE and DGM and relative error of the DGM.
In Fig. 4, the DGM with modified parameters gðyÞ and m ðyÞ (see Eq. (8) and Table 2) is compared to the QM for the same strained Si example. Now a good agreement is found between the QM and DGM. To demonstrate that the new gðyÞ and m ðyÞ model yields good results for many different devices, the comparison of the modified DGM with the QM was repeated for other relevant doping levels and Ge contents. The results are summarized in Figs. 5 and 6.
ARTICLE IN PRESS C.D. Nguyen et al. / Materials Science in Semiconductor Processing 8 (2005) 363–366
366 2
Error [%]
10
10
10
10
0
DGM (Mod.) Reference (QM)
10
11
10
9
-2
12
1
inversion layer density [cm ]
10
10
13
-2
10
inversion layer density [cm ]
10
10
13
12
17
NA=5x10 /cm
10
10
11
-1
0
0.5
1 gate bias [V]
10 2
1.5
Si/Si0.5Ge0.5 Si/Si0.6Ge0.4
10
Si/Si0.7Ge0.3 10
10
3
DGM (Mod.) Reference (QM)
9
8
10
Fig. 4. Inversion layer density at room temperature as a function of the gate voltage for zero back bias where the DGM was based on the new gðyÞ and m ðyÞ-parameter.
8
0
0.5
1 gate bias [V]
1.5
2
Fig. 6. Inversion layer density for three different Ge contents y.
References 10
13
17
-2
inversion layer density [cm ]
17
10
3
NA=7x10 /cm 12
3
NA=5x10 /cm
Si/Si0.8Ge0.2
10
11 18
3
NA=1x10 /cm
10
10
10
DGM (Mod.) Reference (QM)
9
8
10 0
0.5
1 gate bias [V]
1.5
2
Fig. 5. Inversion layer density for three different doping concentrations.
The results confirm the universality and reliability of the new gðyÞ and m ðyÞ model.
4. Conclusions We have studied the impact of size quantization on inversion layers in strained Si-nMOSFETs. Similar to the unstrained Si case, quantum effects cannot be neglected. The standard g; m -parameters of the DGM determined for unstrained Si are found to lead to large errors in strained Si, whereas our new gðyÞ; m ðyÞ model yields good results for a large variety of devices.
[1] Vogelsang T, Hofmann RK. Electron transport in strained Si layers on Si1x Gex substrates. Appl Phys Lett 1993; 63:186–8. [2] Takagi S, Hoyt JL, Welser J, Gibbons J. Comparative study of phonon-limited mobility of two-dimensional electrons in strained and unstrained Si metal-oxidesemiconductor field-effect transistors. J Appl Phys 1996; 80:1567–77. [3] Welser J, Hoyt JL, Takagi S, Gibbons JF. Strain dependence of the performance enhancement in strainedSi n-MOSFETs. In: IEDM Technical Digest 1994. p. 373–6. [4] Rim K, Hoyt JL, Gibbons JF. Fabrication and analysis of deep submicron strained-Si N-MOSFETs. IEEE Trans Electron Devices 2000;47(7):1406–15. [5] Jungemann C, Nguyen CD, Neinhu¨s B, Decker S, Meinerzhagen B. Improved modified local density approximation for modeling of size quantization in NMOSFETs. In: MSM’2001, international conference on modeling and simulation of microsystems. (USA): 2001. p. 458–61. [6] Wettstein A, Schenk A, Fichtner W. Quantum devicesimulation with the density-gradient model on unstructured grids. IEEE Trans Electron Devices 2001;48(2): 279–84. [7] Jungemann C, Emunds A, Engl WL. Simulation of linear and nonlinear electron transport in homogeneous silicon inversion layers. Solid-State Electron 1993;36:1540–993. [8] Rieger MM, Vogl P. Electronic-band parameters in strained Si1x Gex alloys on Si1y Gey substrates. Phys Rev B 1993;48:14276–87. [9] Nuernbergk D. Simulation des Transportverhaltens in Si=Si1x Gex =Si-Heterobipolartransistoren. Diss., TU Ilmenau, Ilmenau, 1999. (H. Utz Verlag Wissenschaft, Mu¨nchen: 1999). [10] Wettstein A, Penzin O, Lyumkis E. Integration of the density gradient model into a general purpose device simulator. VLSI Design 2002;15:751–9.