Chapter 4 Strain-induced Change of Gate Direct Tunneling Current
4.2 Physical Model
4.2.3 Electron Direct Tunneling Current for nMOSFETs
4.3.1.4 Uniaxial Transverse Stress
It can be observed that the experimental data measured in Ref. [2], [40] have the same trend and magnitude of the strain-induced change of HDT current under uniaxial longitudinal and transverse stress. This phenomenon can also be expected in our model since the strain-induced valence band edge shifts and the 2D DOS effective masses are equivalent under longitudinal and transverse stress on (001) wafer.
4.3.2 Electron Direct Tunneling Current for nMOSFETs
Fig. 4.10 shows the simulated results for nMOSFETs under uniaxial longitudinal compressive stress. Note that the calculation includes the first two subbands of 2 valleys and the first subband of
Δ Δ 4 valleys since the subband energy of second subband of Δ2 valleys are close to that of the first subband of Δ 4 valleys as shown in Fig4.10(b). The circles are the experimental data measured by four-point bending jig at VG=1V for the device samples consisting of arsenic doped polysilicon gate,
1.3nm nitrided SiO2 gate dielectric, and 1017 cm-3 boron doped p well in Ref. [38].
The diamonds and squares are the data from Ref. [2] and [40] as described in the pMOSFETs case. It can be seen that the simulation results well reproduce the experimental data. The mechanism for the change of EDT current can also be explained by the carrier repopulation. Under uniaxial compressive stress, the lowest subband (Δ2 valleys) shifts up while the second subband (Δ 4 valleys) shifts down as shown in Fig. 4.8(a) and Fig. 4.10(b). Consequently, a number of carriers repopulate from the lowest subband to the second subband, which has lower tunneling barrier. In addition, the barrier of the lowest subband also decreases. Thus, the EDT current increases. Note that this mechanism is reverse to that for pMOSFETs under uniaxial compressive stress. Moreover, the analysis for these two cases can also be applied to explain the change of direct tunneling current under other stress conditions.
Fig. 4.11 shows the simulation results for different substrate doping concentration, 5×1017 cm-3 and 1017 cm-3. It can be seen that the change of EDT current for 1017 cm-3 doping concentration are smaller than that for 5×1017 cm-3. The results are also consistent with the experimental data.
4.4 Conclusion
Using the modified subband energy expression of triangular potential approximation and the WKB approximation, we have demonstrated an efficient and reasonably accurate physical model to calculate HDT and EDT current for longitudinal uniaxial compressive stressed silicon device. The improved one band EMA and the data calculated by six-band k‧p method were used to extract the quantization effective mass. The simulated results correspond to the experimental data,
versus
G
G J
ΔJ / σ , published by former work.
In our model, the subband energy is a function of vertical electric field and stress.
Therefore, this model can be used directly to characterize the - curve and be applied easily to different stress conditions on various wafer orientations with extracting the corresponding physical parameters.
JG VG
Chapter 5
Conclusions
In the chapter 2, the strain tensors have been expressed as a function of normal, longitudinal, and transverse stress on (001), (110), and (111) wafers, respectively.
Then, the strain-altered band structures, band edge shifts, constant energy surface, 2D energy contour, and effective masses for various stress conditions and wafer orientations have been calculated by deformation potential theory and k‧p framework for conduction and valence band, respectively. Utilizing these simulated results as tools to estimate the device performance, we concluded that the best advantageous strains among these stress types and wafer orientations for nMOSFETs have shown to be uniaxial and biaxial tension on (001) wafer while for pMOSFETs they are uniaxial longitudinal compression on both (001) and (110) wafer. Finally, we have examined the influences of additional transverse or normal strain and have found that the additional transverse tensile stress on (001) wafer can further enhance the hole mobility.
In chapter 3, the quantization effective masses, the 2D DOS effective masses, and the 3D DOS effective masses of silicon under uniaxial and biaxial stress on (001) wafer have been extracted. In addition, the strain-altered conduction and valence effective DOS have been derived as a function of the strain-induced band edge shift and strain-altered 3D DOS effective mass. From the calculated results, the effective DOS drops quickly due to the band edge splitting when stress increases from zero to 1GPa, and then tends to a constant determined by 3D DOS effective mass of lowest valley for further increased stress. Furthermore, the Fermi energy of bulk silicon with
non-degenerate doping has also been derived as a function of the strain-induced conduction or valence band edge shifts and the strain-altered effective DOS. The calculated results have shown that the Fermi energy is dominated by band shifts under large stress because the effective DOS tends to constant. Finally, the intrinsic carrier concentration has been derived and expressed as a function of strain-altered effective DOS and strain-induced band gap narrowing. The calculated results have shown that the intrinsic carrier density increases rapidly due to the strain-induced band gap narrowing when the stress is larger than 1GPa.
In Chapter 4, a triangular potential approximation based physical model for HDT current in pMOSFETs and EDT current in nMOSFETs under longitudinal uniaxial compressive stress has been presented. A modified subband energy expression, which comprises the vertical electric field component and stress component, is used to evaluate the subband energy in the inversion layer of MOSFETs under gate bias and stress. Then, an improved one band effective mass approximation and data calculated by six-band k‧p method are used to extract quantization effective mass. Moreover, WKB approximation is utilized to evaluate transmission probability and tunneling current. The simulated results agree with the experimental data published by former works. The primarily reason accounting for the decrease of HDT current for pMOSFETs while increase uniaxial compressive stress is that a number of carriers redistribute from higher subband into the lowest subband due to the stress induced subband energy shift. Since the barrier of the lowest subband for hole tunneling is higher and the corresponding transmission probability is smaller than other subbands, the total tunneling current decreases while stress increases. The reverse mechanism of above can also be used to explain the opposite trend of EDT current for nMOSFETs under uniaxial compressive stress. Thus, the proposed model provides a simple method to assess the influence of external stress for direct tunneling
currents in MOSFETs qualitatively and quantitatively. Moreover, with extracting the corresponding physical parameters, our model can be applied directly to various wafer orientations and different stress conditions. For example, the device with different magnitude and type of stress in poly gate and channel. Alternatively, the longitudinal and transverse stresses exist in the device in the meantime.
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Table 2.1 The deformation potentials, Luttinger parameters, elastic stiffness constants, and split-off energy for Si, Ge, and GaAs.
Si Ge GaAs
a a(eV) 2.46 1.24 1.16
b a (eV) -2.1 -2.9 -2.0
d a (eV) -4.8 -5.3 -4.8
γ1a 4.22 13.4 6.98
γ2a 0.39 4.24 2.06
γ3a 1.44 5.69 2.93
S11 (10-12 m2/N) 7.68 b 9.64 b 11.75 c S12 (10-12 m2/N) -2.14 b -2.6 b -3.65 c S44 (10-12 m2/N) 12.6 b 14.9 b 16.8 c
Δ (eV) 0 0.044 d 0.29 d 0.34 d
aSee Ref. 18.
bSee Ref. 12.
cSee Ref. 43.
dSee Ref 30.
Table 2.2 The normal, longitudinal, and transverse direction for (001), (110), and (111) wafer.
wafer orientation normal direction (out of plane)
longitudinal direction (in-plane)
transverse
direction (in-plane) (001) [001] [110] [110]
(110) [110] [110] [001]
(111) [111] [110] [11 2 ]
Table 2.3 The stress tensor and strain tensor for biaxial stress on (001) wafer, uniaxial stress along [110], [110], [001], [111], and [11 2 ] direction.
Biaxial d [110] [110] [001] [111] [11 2 ]
aThe form of stress tensor is defined by Equation (2.2).
bThe form of strain tensor is defined by Equation (2.6).
c
σk indicates the stress applied along k-direction.
dFor biaxial stress on (001) wafer, k is along [100] or [010] and σ[ ]100 =σ[ ]010 .
Table 2.4 The resultant strain tensors in response to the combination of normal, longitudinal, and transverse stress for the three wafer orientations
Wafer orientation Strain tensor
Biaxial stress on (001) wafer
Uniaxial stress on (001) wafer
Uniaxial stress on (110) wafer
Uniaxial stress on (111) wafer
Table 2.5 Numerical values of effective mass for silicon conduction band in inversion layer given by [27].
surface (100) (110) (111)
valleys lower higher lower higher all
degeneracy 2 4 4 2 6
Normal mass (m0) 0.916 0.190 0.315 0.190 0.258
Conductivity mass (m0) 0.190 0.315 0.283 0.315 0.296 DOS effective mass (m0) 0.190 0.417 0.324 0.417 0.358
Table 2.6 The conductivity, transverse, and quantization effective masses of the top band for bulk silicon with various stress conditions and wafer orientations. The quantization effective masses of the second band are also listed.
wafer Stress type
mc,1st (m0) mtran,1st
(m0)
mnorm,1st
(m0)
mnorm,2nd
(m0)
σ <0 0.12 1.37 0.28 0.22
Uniaxial
longitudinal σ >0 0.46 0.18 0.21 0.24
σ <0 1.37 0.12 0.28 0.22
Uniaxial
transverse σ >0 0.18 0.46 0.21 0.24
σ <0 0.22 0.22 0.29 0.27
(001)
Biaxial
σ >0 0.28 0.28 0.18 0.29
σ <0 0.12 0.28 1.37 0.15
Uniaxial
longitudinal σ >0 0.46 0.21 0.18 0.17
σ <0 0.28 0.18 0.28 0.22
(110)
Uniaxial
transverse σ >0 0.22 0.29 0.22 0.22
σ <0 0.12 0.17 0.47 0.18
Uniaxial
longitudinal σ >0 0.46 0.29 0.19 0.20
σ <0 0.37 0.23 0.68 0.17
(111)
Uniaxial
transverse σ >0 0.19 0.25 0.18 0.19
Table 2.7 Comparison the effective masses between the 1GPa uniaxial longitudinal compressive stress with and without additional 1GPa uniaxial transverse tensile stress for (001) and (110) wafer.
wafer Additional transverse stress
mc,1st
(m0)
mc,2nd
(m0)
mtran,1st
(m0)
mnorm,1st
(m0)
mnorm,2nd
(m0)
Without 0.12 0.59 1.37 0.28 0.22
(001)
With 0.12 0.3 1.88 0.28 0.18
Without 0.12 0.59 0.28 1.37 0.15
(110)
With 0.12 1.21 0.29 1.32 0.12
Table 4.1. Equations for subband calculation.
Description Equation Oxide electric field
ox ox
ox t
F =V
Potential drop due to poly depletion
Substrate band bending
ox
Silicon surface field
Si
Subband energies Equation (4.1) for pMOSFETs Equation (4.10) for nMOSFETs Inversion carrier density per
subband ⎟⎟
Total inversion layer carrier per
area Ninv =
∑
NijTotal inversion QM channel
thickness ox ox
ij Average QM channel thickness =
∑
s
Silicon potential drop
q Ionized impurity density per area
q N Ndepl 2εSiVdepl sub
=
( ) P Δ R
( ) P
Δ V
( ) P
V
tΔ
( ) P
V
sΔ
( ) P
Δ F
Δ A
σ
zσ
yσ
xτ
yxτ
yzτ
zyτ
zxτ
xyτ
xzFig. 2.1. (a) An arbitrary force ΔR
( )
P acting on an infinitesimal area ΔA at point P.The normal component of the force is ΔF
( )
P and the tangential components of the force are ΔVs( )
P and ΔVt( )
P along two orthogonal directions in the plane. (b) Schematic of the nine components defining the stress state at an arbitrary point in three dimensions.σ
yσ
y2 Δ L 2
Δ L
τ
yzτ
zy2 π
θ ′
θ ′
y uz
∂
∂ z
uy
∂
∂
Fig. 2.2. (a) Schematic of the deformation of a body applied to normal stress along y-axis; and (b) schematic the deformation of a body applied to pure shear stress. Dash line indicates the size and shape of the original body before deformation and solid line indicates those of the body after deformation
[100]
[010]
] 10 1 [
] 110 [
] 001 [
Fig. 2.3. Schematic of the surface orientation and the corresponding stress directions for (001) wafer. The shadow region indicates the wafer surface. The surface normal is [001], the longitudinal (channel) direction is [110], and the transverse direction, which is perpendicular to the channel in the plane, is [110].
] 110 [
] 10 1 [
] 1 00 [
Fig. 2.4. Schematic of the surface orientation and the corresponding stress directions for (110) wafer. The shadow region indicates the wafer surface. The surface normal is [110], the longitudinal (channel) direction is [110], and the transverse direction, which is perpendicular to the channel in the plane, is [001].
[100]
[010]
[001]
] 10 1 [
] 111 [
] 2 11 [
Fig. 2.5. Schematic of the surface orientation and the corresponding stress directions for (111) wafer. The shadow region indicates the wafer surface. The surface normal is [111], the longitudinal (channel) direction is [110], and the transverse direction, which is perpendicular to the channel in the plane, is [112].
Fig. 2.6. Silicon valence band structures for (a) unstressed, (b) 1GPa uniaxial longitudinal compressive, (c) 1GPa uniaxial longitudinal tensile, (d) 1GPa uniaxial transverse compressive, and (e) 1GPa uniaxial transverse tensile stress on (001) wafer.
Fig. 2.7. Silicon valence band structures for (a) 1GPa biaxial compressive and (b) 1GPa biaxial tensile stress on (001) wafer.
Fig. 2.8. Silicon valence band structures for (a) unstressed, (b) 1GPa uniaxial longitudinal compressive, (c) 1GPa uniaxial longitudinal tensile, (d) 1GPa uniaxial transverse compressive, and (e) 1GPa uniaxial transverse tensile stress on (110) wafer.
Fig. 2.9. Silicon valence band structures for (a) unstressed, (b) 1GPa uniaxial longitudinal compressive, (c) 1GPa uniaxial longitudinal tensile, (d) 1GPa uniaxial transverse compressive, and (e) 1GPa uniaxial transverse tensile stress on (111) wafer.
Fig. 2.10. Strain-induced hole subband energy shift versus (a) uniaxial longitudinal and (b) uniaxial transverse stress on (001) wafer.
Fig. 2.11. Strain-induced hole subband energy shift versus biaxial stress on (001) wafer.
Fig. 2.12. Comparison between the strain-induced hole subband energy shift calculated by 4×4 and 6×6 Hamiltonian for (a) uniaxial longitudinal and (b) biaxial stress on (001) wafer. The solid line indicates the subband energy calculated by the 6×6 Hamiltonian. The dotted line indicates the subband energy calculated by 4 by 4 Hamiltonian.
Fig. 2.13. Strain-induced hole subband energy shift versus (a) uniaxial longitudinal and (b) uniaxial transverse stress on (110) wafer.
Fig. 2.14. Strain-induced subband energy shift versus (a) uniaxial longitudinal and (b) uniaxial transverse stress on (111) wafer.
Top band
Second band
Third band
Fig. 2.15. Hole constant energy surface of unstressed bulk silicon for three lowest bands.
Top band
Second band
Third band
(a) (b) Fig. 2.16. Hole constant energy surface of silicon under 1GPa uniaxial longitudinal (a)
compressive and (b) tensile stress on (001) wafer for three lowest bands.
Top band
Second band
Third band
(a) (b) Fig. 2.17. Hole constant energy surface of silicon under 1GPa uniaxial transverse (a)
compressive and (b) tensile stress on (001) wafer for three lowest bands.
Top band
Second band
Third band
(a) (b) Fig. 2.18. Hole constant energy surface of Silicon under 1GPa biaxial (a) compressive
and (b) tensile stress on (001) wafer for three lowest bands.
Top band
Second band
Third band
(a) (b)
Fig. 2.19. Hole constant energy surface of silicon under 1GPa uniaxial longitudinal (a) compressive and (b) tensile stress on (110) wafer for three lowest bands.
Top band
Second band
Third band
(a) (b)
Fig. 2.20. Hole constant energy surface of silicon under 1GPa uniaxial transverse (a) compressive and (b) tensile stress on (110) wafer for three lowest bands.
Top band
Second band
Third band
(a) (b)
Fig. 2.21. Hole constant energy surface of silicon under 1GPa uniaxial longitudinal (a) compressive and (b) tensile stress on (111) wafer for three lowest bands.
Top band
Second band
Third band
(a) (b)
Fig. 2.22. Hole constant energy surface of silicon under 1GPa uniaxial longitudinal (a) compressive and (b) tensile stress on (111) wafer for three lowest bands.
Top band
Second band
Third band
Fig. 2.23. Contour map in kx, ky plane (kz=0) of unstressed bulk silicon on (001) wafer for three lowest valence bands.
Top band
Second band
Third band
(a) (b) Fig. 2.24. Contour map in kx, ky plane (kz=0) of silicon under 1GPa uniaxial
longitudinal (a) compressive and (b) tensile stress on (001) wafer for three lowest valence bands.
Top band
Second band
Third band
(a) (b) Fig. 2.25. Contour map in kx, ky plane (kz=0) of silicon under 1GPa uniaxial transverse
(a) compressive and (b) tensile stress on (001) wafer for three lowest valence bands.
Top band
Second band
Third band
(a) (b) Fig. 2.26. Contour map in kx, ky plane (kz=0) of silicon under 1GPa biaxial (a)
compressive (b) tensile stress on (001) wafer for three lowest valence bands.
Top band
Second band
Third band
(a) (b) Fig. 2.27. Contour map in kx, ky plane (kz=0) of silicon under 1GPa uniaxial
longitudinal (a) compressive and (b) tensile stress on (110) wafer for three lowest valence bands.
Top band
Second band
Third band
(a) (b) Fig. 2.28. Contour map in kx, ky plane (kz=0) of silicon under 1GPa uniaxial transverse
(a) compressive and (b) tensile stress on (110) wafer for three lowest valence bands.
Top band
Second band
Third band
(a) (b) Fig. 2.29. Contour map in kx, ky plane (kz=0) of silicon under 1GPa uniaxial
longitudinal (a) compressive and (b) tensile stress on (111) wafer for three lowest valence bands.
Top band
Second band
Third band
(a) (b) Fig. 2.30. Contour map in kx, ky plane (kz=0) of silicon under 1GPa uniaxial transverse
(a) compressive and (b) tensile stress on (111) wafer for three lowest valence bands.
Band structure
Constant energy surface
Energy contour in the surface
(a) (b) Fig. 2.31. Band structures, constant energy surface, and energy contour of bulk
silicon under 1GPa longitudinal compressive stress with additional 1GPa transverse tensile stress on (a) (001) and (b) (110) wafer, respectively.