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 subbands, respectively, valence band edge shifts between strained and unstrained bulk silicon for the three lowest subbands, respectively. The split-off spin-orbit energy of bulk silicon is Δ=

44meV [35], [37].

Using 2D density of states and Fermi-Dirac statistic, the carrier density in inversion layer for each subband can be derived as

⎟⎟⎠ subband energy defined by Equation (4.1) for the three lowest subbands.

*

mdn E_vn

The relationship between gate voltage and is determined from the voltage balance equation

VG F_Si in polysilicon gate due to poly depletion, and is the substrate band bending.

VFB V_ox V_poly

VS

To derive the flat-band voltage for pMOSFETs under arbitrary strain in both polysilicon gate and channel, Fig. 4.3 shows the band diagram of a p⁺ polysilicon/SiO2/n-Si (pMOS) structure with a negative gate voltage. Then, to

establish the energy relation of both side with the help of the band diagram. Next, let and , the expression of flat-band voltage can be derived as for polysilicon gate and channel, respectively. The strain-altered band gap of polysilicon gate is EGG =EG

(

σpoly =0

)

− ΔECG₁

( )

σpoly − ΔEVG₁

( )

σpoly as shown in Fig. 4.3. N_Sub is the substrate doping concentration and NC

( )

σpoly is the conduction effective DOS described in Chapter 3.

The self-consistent procedure for subband calculation is described as following:

(1) given Npoly, tox, Nsub, and VG. (2) Let _ox V_G V_FB N

V = k − where since

the voltage drop of oxide must be between zero to

N increasing the value of N will increase the accuracy of simulated results, but the computation time increases as well. (3) Using the equations summarized in Table 4.1 to calculate subband energies, carrier density of each subband, voltage drop, depletion charge, etc. (4) Examine the calculated results with Poisson equation, or the conservation of electric flux, q

(

Ninv+Ndepl

)

≈εoxFox. Then, find the value of k and the corresponding solution with smallest error between the both sides of the electric flux conservation equation.

4.2.2 Hole Direct Tunneling Current for pMOSFETs

After modifying the subband energy expression and subband calculation

procedure for the strained silicon, we can readily apply the well-developed WKB approximation [3], [4], and [34] to calculate the tunneling current density. The hole direct tunneling current density J_G can be expressed as a sum of the tunneling current contribution of each subbands,

∑ ( )

of the nth subband. The lifetime of an nth subband in the triangular potential well can be expressed as [3], [4] band edge, is the transmission probability of a carrier. Transmission probability can be written in the form of

( )

^E

T

( )

E T

( ) ( )

E T E

T = _WKB ⋅ _R (4.7)

where is the typical WKB approximation of the transmission probability, and is the correction factor taking into account the reflections from boundaries of the oxide. Appling parabolic dispersion relationship in the oxide [3], the T

( )

E T_WKB

TR

WKB(E) can be simplified as

( ) ( ) ( )

barrier height of tunneling hole with quantized energy E_vn at cathode side, and ϕ_an is that at anode side. ϕ_cath =Φ_BV −qV_ox −E_vn and ϕ_an =Φ_BV −E_vn. The reflection

correction factor is expressed as the purely imaginary group velocity of hole at the cathode and anode side within the oxide. υ_Si(E_vn)= 2E_vn/m_zn and υ_ox(E_vn)= 2(Φ_BV −E_vn−qV_ox(z))/m_oxh for the parabolic dispersion relationship.

4.2.3 Electron Direct Tunneling Current for nMOSFETs

Fig. 4.4 illustrates the band diagram of an n⁺ polysilicon/SiO2/p-Si structure biased in channel inversion condition and stressed with uniaxial longitudinal compressive stress. The figure also shows the subband energy confinement in the inversion layer and the electron direct tunneling current from channel to gate. Similar to pMOSFETs, the strain-altered subband energy can be expressed as

( ) ( ) π ( )₂

( )

σ

subband of each valleys. and are the quantization effective mass for the 2 and 4 valleys, respectively. As discussed in Chapter 2 and 3, the

( )Δ2

mz m_{z( )Δ4}

Δ Δ

quantization, 2D DOS, and 3D DOS effective mass are independent of strain for silicon conduction band. The values of the quantization and 2D DOS effective masses are given in Table 2.5. In addition, the strain-altered flat-band voltage for nMOSFETs under arbitrary strain in both polysilicon gate and channel can be derived with the help of the band diagram as shown in Fig. 4.5.

( ) ^{( ) ( )} _{( )}

_⎟⎟ described in Chapter 3.

The remaining calculation procedures are the same as that for pMOSFETs, but with the corresponding physical parameters for nMOSFETs.

4.3 Results and Discussion

4.3.1 Hole Direct Tunneling Current for pMOSFETs

4.3.1.1 Parameters Extraction for the Vertical Electric Field Component

For simplification, we assume that quantization and density of state effective mass are constant and have no significant change within the range of vertical electric field and stress in our calculation.

Using the triangular potential approximation and improved one-band EMA, Equations (4.1) with σ =0, we can extract the quantization effective masses for the three lowest subbands. The quantization effective masses are the only adjustable parameter in Equation (4.1) with σ =0 and the values for best fitting are as follows:

, , and

0 1 0.28m

m_z = m_z2 =0.23m₀ m_z3 =0.21m₀. These values are consistent with

the values used in [4], [35]-[37] for unstressed case ( , , and ) and approximated the value used in [6] for longitudinal uniaxial

compressive case ( , , and ).

0

*

1 0.29m

m_z = m^*_z2 =0.24m₀

0

*

3 0.22m m_z =

0

*

1 0.27m

m_z = m^*_z2 =0.22m₀ m^*_z3 =0.23m₀

Fig. 4.6 shows that the hole subband energy (σ =0) evaluated by the improved one band EMA, which provides better computing efficiency and takes the non-ideal properties of dispersion relationship in channel inversion condition into account, well reproduced the results calculated by accurate six-band k ‧ p method (with the self-consistent Schrödinger-Poisson equation) for the three lowest subbands versus

varying from zero to 2.5MV/cm. Thus, the use of constant values for quantization effective mass is applicable.

FSi

For simplification, we adopt the density of state effective mass suggested by [36]

for channel inversion and unstressed condition. Furthermore, we assume that the density of state effective mass do not have significant change under small stress condition. However, it is noteworthy that for large stress or precise computation, the change of density of state effective mass should be taken into consideration.

4.3.1.2. Calculation for the Stress Component (Valence Band Edge Shifts)

The method for calculating valence band edge shifts has been discussed in Chapter 2. Fig. 4.7 shows the hole subband energies calculated by Equation (4.1) with the quantization effective masses extracted above. The figure also shows the data evaluated by the six-band k‧p method (with the self-consistent Schrödinger-Poisson equation).

It can be observed that our model can well reproduce the data calculated by six-band k‧p method within the range,F_Si varies from 0.5MV/cm to 1.5MV/cm and

the stress varies from zero to 300MPa. Note that comparing to the subband energy and the thermal energy =0.026 eV at room temperature, the error is tolerated for later calculation. Moreover, in usual operation of pMOSFETs, which oxide thickness is thinner than 2nm, the gate voltage is between zero to 1.5V and the corresponding is between around 0.5MV/cm to 1.5MV/cm. Therefore, the assumption, the influences of strain-induced subband energy shift is independent of , is appropriate within the range of and

kT

FSi

FSi

FSi σ in our discussion.

Fig. 4.8 shows the strain-induced band edge shifts in bulk silicon for both the conduction and valence band. The level of zero energy is the conduction and valence band edge without external stress, respectively.

4.3.1.3. Hole Direct Tunneling Current

Our model has been verified that it can well reproduce the experimental data published by previous works, HDT current in Ref. [3], [4] and EDT current in [34], for unstrained silicon with various gate oxide thickness and doping concentration.

Then, using our model and the parameters extracted above, the simulation result, versus stress at =1V, was calculated and plotted in Fig. 4.9(a). The device parameters used in simulation are as following: gate oxide thickness is 1.3nm;

polysilicon and substrate doping concentration are 5

G

G J

ΔJ / V_G

× 10¹⁹ cm^-3 and 10¹⁷ cm^-3, respectively. The figure also shows the experimental data published by former works.

The squares is the data measured at =1V in Ref. [2]. The diamonds are the date

measured by four point bending jig at =1V for the device samples consisting of heavily doped poly-silicon gate, 1.3nm physical thickness SiO

VG

VG

2 gate dielectrics, and

~5×10¹⁷ cm^-3 well doping in Ref. [40]. The circles are the data measured by wafer

bending technique at V_G=1V for the device samples with 10¹⁷ cm^-3 n-type substrate doping and 1.3nm physical thickness nitrided SiO₂ gate insulators in Ref. [6].

In our calculation, the carrier density of third subband is only about 2% of total three subbands. Thus, according to Fermi-Dirac statistic, contribution of higher subbands is negligible [37]. Also note that the assumption of parabolic dispersion relationship in oxide is precise at =1V and oxide thickness is thinner than 2nm with =0.4m

VG

moxh ₀ [3], [4]. Moreover, since the experimental data cited from Ref. [6]

and [40] are measured by the wafer bending technique, the stress type and magnitude of polysilicon gate are set to be the same as that of channel [16]. Furthermore, the doping effects on the tunneling currents have been examined. It was identified that when substrate doping varies from 10¹⁷ to 5×10¹⁷ cm^-3, there is no significant change on the versus stress curve. This result is consistent with the experimental data.

G

G J

ΔJ /

It can be observed that the simulated result of ΔJ /_G J_G versus stress is consistent with the trend of experimental data published in Ref. [2], [6], but with some deviation at larger stress.

Fig. 4.9 also shows the corresponding subband energy, carrier density, and the hole direct tunneling current versus stress for the three lowest subbands. According to simulated results, the strain-induced change of HDT current is primarily results from the carrier repopulation. Under longitudinal uniaxial compressive stress, the first subband energy become lower and the second and third subbands become higher than that in unstressed case as shown in Fig. 6(b). Consequently, according to the Fermi-Dirac statistic, a number of carriers redistribute from second and third subbands into the first subband as shown in Fig. 6(c). Note that the total carrier

density remains almost constant with the stress varying from zero to 300MPa. In addition, the barrier for hole tunneling of first subband is higher and corresponding WKB transmission probability is smaller than other subbands. Therefore, the total tunneling current decreases while the stress increases as shown in Fig. 6(d).

The deviation between the experimental data and simulation results at larger stress may result from the use of constant 2D DOS effective mass in our calculation.

Fig. 2.23, 2.24 and 3.2 shows that the 2D DOS effective mass of the first band indeed increase while that of the second band decreases at larger stress. Consequently, there are indeed more carriers repopulate from higher subband into the first subband than that expected by the model. Thus, the HDT current further reduced.

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 V_G=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

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

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