Band Structures

Applying Table 2.4 and Equations (2.12)-(2.16), the calculations of strain-altered band structures on the three wafer orientations with various stress conditions are straightforward.

As discussed before, the band warping of silicon conduction band remains unchanged (band edge shift will be discussed in next section). The valence band structures with various strain conditions including unstressed bulk silicon, 1GPa

uniaxial longitudinal compression, 1GPa uniaxial longitudinal tension, 1GPa uniaxial transverse compression, and 1GPa uniaxial transverse tension, are shown in Fig. 2.6, 2.8, and 2.9 for (001), (110), and (111) wafer, respectively. Fig. 2.7 also shows the 1GPa biaxial compressive and tensile stress on (001) wafer. The right hand side of the figures is along the out-of-plane direction, and the left hand counterpart is along the channel direction. Moreover, the band structures are plotted in electron energy, that is, more positive value at energy axis represents smaller hole energy. The effective masses along normal and channel direction for the three lowest bands are also marked in the figures. The three lowest valence bands in Fig. 2.6-2.9 are denoted as top, second, and third band since the designations, heavy, light, and split-off, lose their meanings under stress. For example, Fig. 2.6(b) shows the band structure of bulk silicon under uniaxial compressive stress on (001) wafer. The top band along [001]

direction is “heavy-hole like” and second band is “light-hole like,” but along [110]

direction, the situation is reverse: the top band becomes “light-hole like” and second band is “heavy-hole like.”

2.5.2 Strain-induced Band Edge Shift

In order to give insight into the trends of strain-altered band structures from small to large strain, there are two characteristics of band structures that should be considered. One is band edge shift discussed in this section, and the other is band warping which will be modeled into effective masses and extracted in Chapter 3.

Using Equation (2.12) and Table 2.4, the band edge shift of the six conduction band minima along the <100> directions can be obtained. Note that the first term of Equation (2.13) shifts the six valleys in the same magnitude while the second term splits the Δ4 valleys (the conduction band minima along [100], [100], [010], and [010] directions) and Δ2 valleys (the minima along [001] and [001] directions) since

zz yy

xx ε ε

ε = ≠ for all stress types in the discussion. Also note that associated with

compressive (tensile) strain in one direction, there are generally a tensile (compressive) strain in the other two directions. Thus, the signs of ε_xx (ε ) and _yy ε are opposite _zz for a single uniaxial or biaxial stress. On the other hand, with the combination of the normal, longitudinal, and transverse stress, the ε_xx (ε ) and _yy ε may be produced _zz with the same sign. However, it indeed favors opposing sign for increasing the population in the lowest subband and enhancing mobility.

Using Table 2.4 and Equation (2.14)-(2.16) with k_x =k_y =k_z =0 , the strain-induced valence subband energy shift can be obtained since the valance band minima for bulk silicon are all at gamma point (see Fig. 2.6-2.9). The three lowest valence band edges versus stress with uniaxial longitudinal, uniaxial transverse stress are shown in Fig. 2.10, 2.13, and 2.14 for (001), (110), and (111) wafer, respectively.

Fig. 2.11 also shows the biaxial stress case on (001) wafer. The negative value of stress indicates compressive stress and the positive value indicates tensile stress. Note that the band edge shift under uniaxial longitudinal stress on (001), (110), and (111) wafer, and uniaxial transverse stress on (001) wafer are the same since these stress directions have the same symmetry in silicon crystal.

The figures also label the quantization effective mass, which is along the direction normal to the surface, for top two bands. The “hh” denotes the effective mass of the corresponding band as “heavy-hole like” while “lh” is “light-hole like”

among the two bands. It is interesting that no matter whether the stress is uniaxial longitudinal, uniaxial transverse, or biaxial on the (001) wafer, under the compressive stress the first band is heavy-hole like and the second band is light-hole like.

Nevertheless, under tensile stress, the situation is reverse. This is the main mechanism

accounting for the reverse trends of the change of direct tunneling current between compressive and tensile stress. It will be discussed thoroughly in Chapter 5. The analysis can be applied to (110) and (111) wafer as well.

Moreover, the subband energy splitting between top and second band is larger under uniaxial compression (biaxial tension) than uniaxial tension (biaxial compression). The influences will be discussed in chapter 3, 4, and 5.

Next, let us compare the difference between results calculated by the 6 6 Hamiltonian and the 4×4 Hamiltonian. The latter is widely used in the previous works for device modeling [16], [22] since its solutions have very simple form for biaxial stress on (001) wafer [Equation (2.19)] and uniaxial stress along [110] [Equation (2.20)]. The solutions can be obtained through Equation (2.18) and Table 2.4. Note that the band edge is proportional to the stress.

×

On the other hand, the analytic solution of 6×6 Hamiltonian is complex for uniaxial stress along [110] and not listed here due to space limit. However, the solution of that for biaxial stress [Equation (2.21)] is relatively simple since biaxial stress have no shear strain terms and withε_xx =ε_yy, thus, L_ε =M_ε =0 in Equation

Fig. 2.12 shows the comparison between the band edge shift calculated by 6×6 (solid line) and 4×4 (dotted line) Hamiltonian under uniaxial and biaxial stress on (001) wafer. It can be seen that under uniaxial or biaxial stress, one of the top two bands can be approximated by a straight line, hence the use of Equations (2.19) and

(2.20) can get fairly good results. However, the other band is far from the linear line calculated by 4×4 Hamiltonian and even the lowest energy is not located at zero stress, that is, it first decreases and then increases while the stress increases. Therefore, the usage of analytic solutions derived from 4×4 Hamiltonian can only serve for very small stress and induces significant error under moderate and large stress as shown in Fig. 2.12. For this reason, we will apply the 6× 6 Hamiltonian throughout our simulation work in Chapter 2, 3, and 4.

2.5.3 Constant Energy Surface

Constant energy surface in k-space is also an important tool for estimating the influences of strain and can be obtained from Equation (2.12) for conduction band and (2.14) for valence band. Fig. 2.16, 2.17, and 2.18 show the constant energy surface in k-space of bulk silicon for three lowest valence bands with 1GPa uniaxial longitudinal, uniaxial transverse, biaxial stress on (001) wafer, respectively. For comparison, Fig.

2.15 also shows the case of unstressed bulk silicon (the results are consistent with Ref.

[23]). The three coordinate axes are along kx, ky, and kz. The figures also label the effective masses along normal, longitudinal, transverse, and other principal directions.

In addition, constant energy surface for bulk silicon under 1GPa uniaxial longitudinal and uniaxial transverse on (110) and (111) are shown in Fig. 2.19-2.22. Note that, in these figures, the three coordinate axes are along the normal, longitudinal, and transverse directions.

2.5.4 Two-dimensional Energy Contour in the Plane of Wafer Surface

The two-dimensional energy contour in the plane of wafer surface can help us determine the characteristics of inversion layer of MOSFETs including the conductivity effective mass, transverse effective mass, density of states, and the

symmetry of E-k relation under various stress conditions.

The energy contour of valence band can be obtained by Equation (2.14) and setting the wavevector along normal direction at zero. The results are plotted in Fig.

2.23-2.30 for various stress conditions and wafer orientations as discussed above.

Note that the horizontal and vertical axes are along k_x and k_y for (001) wafer and, contrary to that, they are along the longitudinal and transverse directions for (110) and (111) wafers.

2.5.5 Advantageous Strains and Wafer Orientations

The general expression of conductivity for n- or p-MOSFETs operating in inversion condition can be described by

⎥⎦

where q, n, τ , and mc are the elementary charge, carrier density, scattering relaxation time, and conductivity effective mass along channel direction, respectively. The subscript denotes the first and second subband in the inversion layer of MOSFETs.

For high performance and low power requirements, advantageous strains need to meet following criteria [2], [24]-[26]: (1) small conductivity effective mass of the lowest subband, m_c1, for enhancing the mobility since most of carriers occupy the lowest subband; (2) large quantization effective mass along the out-of-plane direction of the lowest subband, which enhances the carrier population by lowering the quantization energy in the inversion layer; (3) large 2D DOS effective mass, or large transverse effective mass, of the lowest subband which also increases the carrier population of the lowest subband; (4) large energy splitting of the two lowest subbands for lowering the intervalley (optical phonon) scattering; and (5) the strain-induced subband shift and confinement effect in inversion layer are additive,

that is, the band shifted down by strain must also have a larger quantization effective mass, whereas the band shifted up by strain must have a smaller quantization effective mass. The requirement not only enhances mobility due to increased carrier population in lowest subband which have small conductivity effective mass, but also reduces the power dissipation due to decreased gate direct tunneling current (details will be discussed in Chapter 4).

Let us first examine the potential stress types and wafer orientations with these criteria for nMOSFETs, then, for pMOSFETs. The quantization, conductivity, and DOS effective masses of the lowest subbands for nMOSFETs operating in inversion conditions are given by [27]-[29] and listed in Table. 2.5. For conservative reason, we assume the stress is not large enough to perturb significantly the original system described in [27]-[29]. Under this assumption, the effective masses keep constant under strain, that is, strain has no influences on the criteria 1-3. In addition, the total carrier density in inversion layer does not change significantly when the carriers repopulate from one subband to another subband due to the strain-induce subband energy shift.

For criterion 5, uniaxial longitudinal, uniaxial transverse, and biaxial tension are advantageous strains for (001) wafer since these strains lift the Δ4 valleys, which have smaller quantization effective mass, and shift down the Δ2 valleys, which have larger quantization effective mass [see Equation (2.13) and Table 2.4]. On the other hand, the uniaxial longitudinal compression are advantageous strains for (110) wafer since these strains lift the 2 valleys, which have smaller quantization effective mass, and shift down the 4 valleys, which have larger quantization effective mass. Note that the 4 valleys are the conduction band minima along [100], [

Δ Δ

Δ 100], [010], and

[010] directions while Δ2 valleys are the minima along [001] and [001] directions on both (001) and (110) wafer [27]. For the (111) wafer, the six valleys are degenerate

in inversion layer and have the same conductivity effective mass, that is, the strain-induced subband energy shift does not provide additional benefits for the conductivity.

For comparing the (001) and (110) wafer, let us consider the same carriers concentration in inversion layer on (001) and (110) wafer. The quantization effective mass of the lower valleys on (001) wafer is much larger than that of (110) wafer while for higher valleys, it remains the same. That is, the occupation ratio of the lower valleys is larger on (001) wafer than that on (110) wafer due to the much lower subband energy of lower valleys compared to higher valleys on (001) wafer. In addition, the conductivity effective mass of the lower valleys is smaller on (001) wafer than that on (110) wafer while for the higher valleys it is equivalent on both wafers. Moreover, the magnitudes of strain-induced subband energy shift are equivalent since the directions of uniaxial longitudinal stress on (001) and (110) wafers have the same crystal symmetry. Therefore, the conductivity on (001) wafer is better than that on (110) wafer. However, experiments and accurate numerical simulations must be conducted to corroborate this argument.

Next, let us examine these stress types and wafer orientations for pMOSFETs using the criteria, the simulation results, Fig. 2.6-2.30, and the effective masses summarized in Table 2.6. For criterion 5, the disadvantageous strains producing smaller quantization effective mass for top band and larger quantization effective mass for second band are marked with a strikethrough on the quantization effective mass. Then, for criteria 1-3, the advantageous strains producing smallest conductivity effective mass, largest transverse effective mass, and best quantization effective mass among these stress types and wafer orientations are emphasized with bold effective mass.

In Table 2.6, it can be seen that the uniaxial longitudinal compression on both

(001) and (110) wafers is better among all advantageous strains. For uniaxial longitudinal compression on (001) wafer, it can provide smallest conductivity effective mass and largest transverse effective mass of the top band, but the quantization effective masses are not as desirable as that on (110) wafer. On the other hand, uniaxial longitudinal compression on (110) wafer can provide the smallest conductivity effective mass as that on (100) wafer, the largest quantization effective mass of the top band, and the smallest quantization effective mass of the second band, which not only increases the carrier population in top band, but also reduces the gate direct tunneling current. However, the transverse effective masses are small comparing with that on (001) wafer. Moreover, the magnitudes of strain-induced band edge shift on both wafers are equivalent as shown in Fig. 2.10(a) and Fig. 2.13(a).

Indeed, there are reported simulation results [26] indicating that the mobility on (110) wafer is larger than that on (001) wafer below about 1.3GPa, but the situation is reverse above 1.3GPa. Nevertheless, the conductivity and total drive current, which relate to the carrier density and occupation ratio of each subband, were not reported in the work. Therefore, there are advantages and disadvantages on each wafer orientation, but for low power application, (110) wafer may be better than (001) wafer.

2.5.6 Influences of Additional Transverse or Normal Stress

In Section 2.5.5, we concluded that uniaxial and biaxial tensile stresses on (001) wafer favor the conductivity enhancement for nMOSFETs while it is the uniaxial longitudinal compressive stress on both (001) and (110) wafer for pMOSFETs. Then, in this section, we focus on the influences on these advantageous stress with additional uniaxial transverse stress, or normal stress, which is existent in process such as capping layer or STI stressor when the dimension of channel width is comparable with channel length.

Let us first consider an additional transverse stress on (001) wafer, or an additional normal stress on (110) wafer with the same sign, that is, compressive stress, and magnitude of the longitudinal stress. It is possible in process such as capping layer or STI stressor. Table 2.4 shows that the shear strain term is canceled while the normal strain term is doubled. Thus, the strain tensors reduce to the form as biaxial compressive stress on (001) wafer (a pure normal stress). It is not desirable for pMOSFETs since the benefits of longitudinal compressive stress is degraded.

Next, let us consider additional transverse stress on (001) wafer, or normal stress on (110) wafer, with the opposing sign, that is, tensile stress, and the same magnitude of longitudinal stress. [Note that uniaxial longitudinal compressive and transverse tensile stresses are both advantageous strains on (001) wafer as shown in Fig. 2.6(e), Fig. 2.17(b), Fig. 2.25(b), and Table 2.6.] Table 2.4 shows that the normal strain terms are canceled while the shear strain term doubles. Thus, the strain tensors readily reduce to a pure shear strain. It is not desirable in nMOSFETs since there is no energy splitting between the 2 and 4 valleys due to the normal strain terms being zero.

Thus, the mobility enhancement of nMOSFETs by uniaxial longitudinal stress is degraded. For pMOSFETs, Fig. 2.31 shows the band structures, constant energy surfaces, and 2D energy contours of bulk silicon with addition transverse stress on (001), and (110) and the effective masses are summarized in Table. 2.7. It can be found that with additional transverse tensile stress on (001) wafer, the conductivity effective mass remains 0.12m

Δ Δ

0 for top band, but that reduced from 0.59m₀ to 0.3m₀ for second band. In addition, the transverse effective mass of top band increases from 1.37m₀ to 1.88m₀. Note that the simulated results for additional normal tensile stress on (110) wafer are similar to that for an additional transverse tensile stress on (001) wafer, but the normal and transverse direction are exchanged. Thus, additional transverse tensile stress on (001) wafer can further enhance the hole mobility while

additional normal tensile stress on (110) wafer can further reduce the gate direct tunneling current (the quantization effective mass of top band increase from 1.37m0 to 1.88m₀). On the other hand, additional transverse tensile stress have no apparent benefits on (110) wafer as shown in Table 2.7.

To introduce the additional transverse tensile stress on (001) wafer for enhancing hole mobility, it is possible to be achieved without additional costs by modifying slightly the standard strained CMOS logic technology process flow [1]. The undertaken technology enhances the electron and hole mobility on the same wafer by first using SiGe source/drain to introduce longitudinal compressive stress in the channel of pMOSFETs and then introduces longitudinal tensile stress in the channel of nMOSFETs by applying nitride capping layer on both nMOSFETs and pMOSFETs.

The disadvantage of this process flow is that it needs additional step for neutralizing the capping layer strain on pMOSFETs. However, instead of the longitudinal tensile stress with the nitride capping layer, if the tensile stress is incorporated along the transverse direction during the same step, which not only enhances the election mobility in the same order of magnitude, but also introduces additional hole mobility enhancement. (Remind that the longitudinal and transverse directions are symmetry in silicon crystal. Thus, the energy splitting of the Δ2 and Δ4 valleys are equivalent under these two type stresses. Moreover, effective masses remain unchanged in conduction band).

2.6 Conclusion

In this chapter, 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 band and valence band, respectively. Utilizing these simulated results as tools to estimate the device performance, 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.

Chapter 3

The Properties of Bulk Silicon in the Presence of Strain

3.1 Introduction

In order to model the characteristics of strained MOSFETs such as the change of gate direct tunneling current (Chapter 4), there are two important features of strain-altered band structure that should be take into consideration. One is the strain-induced band edge shift, which has been discussed and extracted in Chapter 2.

The other is the strain-induced band warping, which will be incorporated into our physical model (developed in Chapter 4) via the effective masses extracted in this chapter such as the quantization effective mass, the 2D density of state (DOS) effective mass, and 3D DOS effective mass. In Chapter 4, we will verify qualitatively and quantitatively that the strain-induced change of gate direct tunneling current can be attributed to these two features of strain-altered band structure. Moreover, utilizing the extracted 3D DOS effective masses and band edge shifts of all valleys, the conduction band effective DOS, N_c, and valence band effective DOS, N_v, can be determined. Then, following the approach of conventional “Semiconductor Device Physics,” [30], [31] the strain-altered Fermi energy level of bulk silicon, which is an important physical parameter for device modeling, can be determined. Finally, the strain-altered intrinsic carrier density of bulk silicon will be calculated as well.

Note that in Chapter 3 and 4, it is primarily focused on the silicon under uniaxial longitudinal stress and biaxial stress on (001) wafer since there are adequate