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

experimental data published by previous works and widely used in industry to date.

Nevertheless, the approach and analysis developed here can be applied directly to other stress conditions and wafer orientations with extracted physical parameters.

3.2 Effective Mass

As discussed in Chapter 2, the conduction band effective mass of bulk silicon remains unchanged under strain. For the valence band, we assume that the constant energy surfaces can be approximated to ellipsoids, that is, the energy dispersion relations along the three axes of the ellipsoid are parabolic-like. Thus, the energy dispersion relation of bulk silicon near the gamma point can be expressed as

* directions, respectively. For uniaxial stress, the directions along [110], the longitudinal direction, along [110], the transverse direction, and along [001], the normal direction, in k-space are selected to be the three orthogonal axes of the ellipsoid due to the symmetry of uniaxial strain as shown in Fig. 2.16 and Fig. 2.24.

On the other hand, for biaxial stress, the direction of [100], [010], and [001] in k-space are selected to be the three orthogonal axes of the ellipsoid due to the symmetry of biaxial strain as shown in Fig. 2.18 and Fig. 2.16. Note that the effective masses along [100], [010], [110], and [110] are the same under biaxial stress due to the symmetry of silicon crystal and biaxial strain. Therefore, for biaxial stress, the energy contour is circle-like near the gamma point under large strain, while for uniaxial stress, the energy contour is ellipse-like.

Also, note that the constant energy surfaces, or 2D energy contours, of the heavy

hole band of unstrained bulk silicon, as shown in Fig. 2.15 and Fig. 2.23, are indeed far from the ellipsoid or ellipse in the plane. Thus, for small strain case, the assumption of elliptic constant energy surface is not suitable. However, the approach used in Ref. [32] for deriving the effective masses of unstrained silicon cannot be applied directly to the strained case due to the complex form of energy dispersion relation under strain. In addition, the analytic solution used in [32] for heavy and light hole band are extracted from the 4×4 Hamiltonian described in Chapter 2, which ignores the mixing effect of split-off band and hence induces significant errors as compared with the 6×6 Hamiltonian. Moreover, the conventional effective masses given by the Ref. [30]-[32] for the unstrained case are extracted from bulk silicon.

They are not applicable to describe the inversion layer of MOSFETs. Thus, the one band effective mass approximation is adopted in Chapter 4 for the small strain case instead of the values extracted here. On the other hand, when the strain is large enough, the crystal symmetry of band structure will be destroyed and forced to the strain symmetry. Thus, the hypothetical elliptic constant energy surface is a good approximation.

Next, the effective masses along the three axes of the ellipsoid can be defined as

1

where E is the energy and the subscript i denotes the ith valleys.

Consequently, based on the assumption of elliptic constant energy surface, the 3D (bulk) DOS effective mass can be derived as

(

ai bi ci

)

³¹

di m m m

m = . (3.3)

On the other hand, assuming energy contour in k_x-k_y plane is ellipse-like, the energy in inversion layer of MOSFETs can be expressed as

inz

where the is the quantization energy along the z direction. The subscript i and n denote the nth subband of ith valley. Then, the 2D DOS effective mass in inversion layer can be derived as

Einz

(

_{ai bi}

)

²¹

di m m

m = (3.5)

Fig. 3.1 shows the effective masses along the three axes of the ellipsoid, m_{[ ]110} , [ ]110

m , and , versus uniaxial longitudinal stress for three lowest valence bands.

Note that some of these effective masses vary significantly from small to large strain while the others remain almost constant. Thus, the influences of strain-altered effective masses cannot be ignored and must be incorporated into our physical model.

It can be seen that the longitudinal (conductivity) effective mass of top band under compressive stress is much smaller than that under uniaxial tensile stress. It is consistent with the analysis in Chapter 2. Especially, the transverse and quantization effective mass increase while the uniaxial compressive stress increases. It implies that introducing larger strain into the channel is beneficial and desirable. Fig. 3.2 shows the 3D (bulk) and 2D DOS effective masses versus uniaxial longitudinal stress for three lowest valence bands. It can be observed that the 3D and 2D DOS effective masses of top band increase significantly while the uniaxial compressive stress increases from zero to 3GPa.

[ ]001

m

Fig. 3.3 shows the effective masses along the three axes of the ellipsoid, , , and , versus biaxial stress for three lowest valence bands. The and are equivalent due to the strain and crystal symmetry. Then, Fig. 3.4 shows the

[ ]100

3D and 2D DOS effective masses versus biaxial stress for three lowest valence bands.

Note that the 3D DOS effective masses for the three lowest valence bands appear to remain constant (about 0.24m₀) due to the reverse trends between _] ( ) and

while the compressive or tensile strain increases form zero to 3GPa.

[100

m m_{[ ]010}

[ ]001

m

3.3 Carrier Density and Effective DOS

3.3.1 Electrons in Conduction Band

To derive the expression of carrier density of bulk silicon with non-degenerate doping as a function of band shifts and 3D DOS effective masses of all valleys, Fig.

3.5(a) shows the strain-induced energy valleys splitting for conduction band under arbitrary stress. The E1, E2, E3 represent, respectively, the energy of conduction band minima along one of the three orthogonal axes, k_x, k_y, or k_z. Note that for uniaxial and biaxial compressive stress on (001) wafer, the E1 and E2 are degenerate and are the valley minima along k_x, and k_y axes while E₃ is the valley minima along k_z axis and higher than E1 and E2. On the other hand, for uniaxial and biaxial tensile stress, the E2

and E₃ are degenerate and are the valley minima along k_x, and k_y axes while E₁ is the valley minima along kz axis and lower than E2 and E3. In addition, the ΔE1,2

represents the band splitting between E₁ and E₂ while ΔE_1,3 represents the band splitting between E1 and E3. Then the carrier density of electron in conduction band can be expressed as

where E_f is the Fermi energy. The N_c1, N_c2, and N_c3 are the effective DOS of the E₁, E₂, and E3, respectively, and can be expressed as

2

Reminding that the strain does not alter the band warping in conduction band, and therefore, the 3D DOS effective masses remain unchanged, leading to the expression:

(

^{*2 *}

)

¹³

Consequently, the N_c1, N_c2, and N_c3 areequivalent. Therefore, the expression of electron carrier density can be further simplified as shown in Equation (3.6) where the N_C^* is the effective conduction band effective DOS under stress:

( ) ( ) ( )

^{( )}

( )

^{( )}

3.3.2 Holes in Valence Band

Similar to the case of conduction band, Fig. 3.5(b) shows the schematic strain-induced energy valleys splitting for the three lowest valence bands. Thus, the carrier density of hole in valence band can be expressed by

( ) [ ( )] [ ( )]

where the Nv,top, Nv,second, and Nv,third are the effective DOS of the three lowest valence bands, respectively, and can be determined by the 3D DOS effective masses shown in

Fig. 3.2 and 3.4 for uniaxial and biaxial stress, respectively. Therefore, one can write

Then, the effective valence band effective DOS under stress, N_V^*, can be expressed as

( ) ( ) ( )

^{( )} V

( )

^{EkT( )} versus uniaxial and biaxial stress, respectively. It can be seen that for both uniaxial and biaxial stress, the N_c and N_v drop very quickly when the stress increases from zero to 1GPa, but almost remain constant above 1GPa. The phenomena can be explained by Equation (3.9) and (3.12). That is, the second and third terms in Equation (3.9) and (3.12) decrease exponentially due to the energy splitting increase when the stress increases (see Fig. 2.10 and 2.11). Ultimately, the second and third terms tend to zero, thus, Nc and Nv are dominated by the first term under large strain. Note that, it is different from the biaxial case, the N_v increases slightly when the uniaxial compressive stress increases. It can be explained by the 3D DOS effective mass of the top valence band under uniaxial stress (see Fig. 3.2), which increases significantly when the uniaxial stress increases from zero to 3GPa while for biaxial stress it remains almost constant (see Fig. 3.4).

3.4 Fermi Energy of Bulk Silicon

Using the strain-induced shift of conduction band edge extracted in Chapter 2 and the strain-altered conduction effective DOS extracted in previous section, the Fermi energy of n-type silicon under strain can be expressed as [30], [31]

( )

σ ⁻

( )

σ ⁼⁻ ln^⎜⎜_⎝^⎛ _*

( )

⁰σ ^⎟⎟_⎠^⎞

Likewise, using the strain-induced shift of valence band edge and the strain-altered valence effective DOS, the Fermi energy of p-type silicon under strain can be expressed as [30], [31]

( )

σ ⁻

( )

σ ⁼⁻ ln^⎜⎜_⎝^⎛ _*

( )

⁰σ ^⎟⎟_⎠^⎞

Finally, the strain-altered intrinsic Fermi energy level, Ei, can be determined by

( ) ( )

Fig. 3.8 and Fig. 3.9 show the Fermi energy of bulk silicon versus uniaxial and biaxial stress, respectively, for various doping concentrations. The figures also show how the intrinsic Fermi level, conduction band edge, and valence band edge vary with stress. Note that the band gap of unstrained silicon is 1.12eV. From Fig. 3.8, Equation (3.13), and Equation (3.14), it can be observed that the strain-induced conduction band edge shift decreases the Fermi energy of n-type silicon while the strain-induced valence band edge shift increases the Fermi energy of p-type silicon when the stress becomes large. On the other hand, the energy difference between conduction band edge and the Fermi energy of n-type silicon, or the energy difference between valence band edge and Fermi energy of p-type silicon, reduces since the DOS effective masses decrease while stress increases. Thus, different from the influences of band edge shift, the effective DOS introduces reverse influences on the strain-altered Fermi energy. Moreover, the strain-altered Fermi energies are primary dominated by the band edge shift under stress above 1GPa since the effective DOS of conduction and valence band both tend to constant when the stress becomes large.

3.5 Intrinsic Carrier Concentration

Utilizing the conduction and valence effective DOS of bulk silicon under strain extracted in Section 3.4 and the strain-induced band gap narrowing, which enhances the intrinsic carrier density, extracted in Chapter 2 and 3.4., the intrinsic carrier density can be obtained as

( ) ( )

_V ^{E ( ) kT}

C i

B

e G

N N

n = ^* σ ^* σ ^{− σ /2} (3.16)

where E_G

( )

σ is the band gap of bulk silicon under strain. k_B is the Boltzmann constant and T is temperature.

B

Fig. 3.10 shows the intrinsic carrier concentration versus uniaxial and biaxial stress. In the figure, it can be seen that the intrinsic carrier density increases slightly under 1GPa and enhances quickly above 1GPa. In addition, the intrinsic carrier density increases faster under biaxial stress than that under uniaxial stress. The phenomena can be understood by Equation (3.16), Fig. 3.6, and Fig. 3.7. For the stress under 1GPa, the conduction and valence effective DOS reduce quickly, thus suppressing the enhancement of intrinsic carrier density due to the band gap narrowing. On the other hand, when the stress becomes large, the effective DOS tends to constant. Thus, the intrinsic carrier density is primary dominated by the band gap narrowing and increases quickly when stress becomes large. Moreover, the strain-induced band gap narrowing is larger under biaxial stress than that under uniaxial stress as shown in Fig. 3.8 and Fig. 3.9. Therefore, the enhancement of intrinsic carrier density is larger under biaxial stress than that under uniaxial stress.

3.6 Conclusion

In this chapter, 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. These extracted effective masses determine the characteristics of inversion layer in MOSFETs such as the quantization subband energy, occupation ratio of each subband, and the Fermi energy. 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.

Chapter 4

Strain-induced Change of Gate Direct Tunneling Current

4.1 Introduction

There are two approaches to study the conduction band electron direct tunneling (EDT) current in unstrained nMOSFETs. One is the self-consistent Schrödinger-Poisson equation [5]. Another is triangular potential approximation [33], [34], [41]. The triangular potential approximation was also applied successfully to the hole direct tunneling (HDT) current of valence band [3], [42]. However, the physical parameters used in above studies were extracted from energy dispersion relationship (band structure) of bulk silicon. The actual dispersion relation of valence band in the channel inversion condition possesses many non-ideal properties such as band mixing, anisotropic, far from parabolic, camel back, vertical electric field-dependent, and the density of state function is deviated from the step-like function [4], [35]-[37] as shown in Fig. 1(a) for unstressed case and (b) for uniaxial longitudinal compressive stress. In Fig. 1, note that the energy of the heavy, light, and split-off spin-orbit hole band are degenerated and zero at gamma point in bulk silicon without stress.

The actual dispersion relation and the corresponding calculation procedure are too complex so that for further applications are impractical. Fortunately, the improved one band effective mass approximation (improved one band EMA) introduced in Ref.

[4], [35]-[37], [42] can resolve this difficulty. This approach can achieve both the reasonably accuracy and computing efficiency.

On the other hand, the dispersion relation of valence band with external stress using six-band k‧p method has been deeply developed [19]. Moreover, using the six-band k‧p method and Self-consistent Schrödinger-Poisson equation to calculate hole tunneling current in pMOSFETs with various type of stress has been report [6].

In particular, there is an analytic expression for subband energy shift of conduction band under longitudinal uniaxial stress, which is a function of vertical electric field and stress [38]. Furthermore, using the analytic expression and measured electron gate direct tunneling current, it has been corroborated that the expression can be used to extract the conduction band deformation potential constant [38] and quantify channel stress in devices [39].

However, there is no available procedure, which is based on the triangular potential approximation and strain-altered dispersion relationship of silicon, to calculate the hole direct tunneling current. Therefore, in this work, model and characterize direct tunneling current in MOSFETs under uniaxial compressive stress by using the modified triangular potential approximation is demonstrated.

The simulation result can provide information for future strain engineering and later calculation. Furthermore, the model can provide an explicit physical picture for the impact of strain on the gate direct tunneling current in MOSFETs.

In this chapter, we focus on the direct tunneling current for pMOSFETs under uniaxial longitudinal compressive stress on (001) wafer, then, extend that for nMOSFETs. The same approach and analysis can be applied to other stress conditions and wafer orientation with corresponding modifications of the model.

4.2 Physical Model

4.2.1 Hole Subband Energy and Carrier Density (pMOSFETs)

Fig. 4.2 shows the band diagram of a pMOSFETs under uniaxial compressive stress (note that the band diagram may be different for other stress conditions) and biased in channel inversoin condition. The figure also illustrates the quantizated subband energies in the inversion layer and HDT current from the channel into the gate. EC(unstressed) and EV(unstressed) are the conduction and valence band edge in bulk silicon without external stress. EF is the Fermi energy. EC(^Δ2) and EC(^Δ4) indicate the strain-induced conduction band edge shift of the Δ 2 and Δ 4 valleys of bulk silicon, respectively. EV1(FSi=0, σ ), EV2(FSi=0, σ ), EV3(FSi=0, σ ) indicate the strain-induced valence band edge shift of the three lowest bands of bulk silicon.

EV1(FSi, σ), EV2(FSi, σ), EV3(FSi, σ) are the quantized energy of the three lowest subbands in the inversion layer. Note that under uniaxial compressive stress, the order of three lowest subbands in the inversion layer are indeed the same as the three lowest bands (valleys) of bulk silicon. Φ_BV is the valence band edge difference between SiO2 and Si without stress. tox is the gate oxide thickness.

In Fig. 4.2, the energy band bending induced by gate voltage in the inversion layer can be approximated by a triangular potential well. The slope of the triangular potential well can be modeled by the silicon surface electric field . Analogizing to Ref. [38], we assume that the impact of strain-induced band edge shift and the electric-field-induced subband energy confinement in the inversion layer are independent. Therefore, we can express the three lowest valence subband energy in the inversion layer by directly adding the triangular potential component, first term in the right hand size in Equation (4.1), and stress component, second term, as

FSi

( )

σ

where , , and are the quantization effective masses of the three lowest

where , , and are the quantization effective masses of the three lowest

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