Dissertation Organization

In Chapter 2, a simulator based on a triangular potential approximation, named

“TRP”, is introduced. However, a huge error is accompanied with this method. New algorithm with a corrected coefficient “ηi” embedded in original TRP, is proposed to eliminate this error.

In Chapter 3, the strain and stress are introduced. The band shift caused by strain is considered in TRP. From the fitting of experimental gate direct tunneling current data by TRP, the importance and values of the piezo-effective-mass coefficients are brought out. We also compare those extracted piezo-effective-mass coefficients with those published in the literature.

In Chapter 4, a powerful simulator of fully Schrődinger and Poisson self-consistent solver for n-channel MOSFETs, named “NEP”, is presented. With the outputs such as subband energy level, inversion charge density, wave-function, etc., we can estimate mobility with key scattering mechanisms included. Gate direct tunneling can be gotten form NEP. Once again, stress effect is considered inside NEP.

In Chapter 5, from the fitting of experimental gate direct tunneling current and mobility at the same time, we provide another evidence for the existence and importance of the piezo-effective-mass coefficients.

In Chapter 6, each of (001), (110), and (111) nMOSFETs are discussed under longitudinal, transverse, vertical, and biaxial stress conditions. The double-gate version of the simulator is introduced.

4

Finally, conclusions of the research work and made in Chapter 7.

5

References

[1.1] J. Welser, J. L. Hoyt, and J. F. Gibbons, “NMOS and PMOS transistors fabricated in strained silicon/relaxed silicon-germanium structures,” in IEDM

Tech. Dig., 1992, pp. 1000-1002.

[1.2] C. H. Ge, C. C. Lin, C. H. Ko, C.C. Huang, Y. C. Huang, B. W. Chan, B. C.

Perng, C. C. Sheu, P. Y. Tsai, L. G. Yao, C. L. Wu, T. L. Lee, C. J. Chen, C. T.

Wnag, S. C. Lin, Y. C. Yeo, and C. Hu, “Process-strained Si (PSS) CMOS technology featuring 3D strain engineering,” in IEDM Tech. Dig., 2003, pp.

73-76.

[1.3] S. E. Thompson, M. Armstrong, C. Auth, M. Alavi, M. Buehler, R. Chau, S. Cea, T. Ghani, G. Glass, T. Hoffman, C. H. Jan, C. Kenyon, J. Klaus, K. Kuhn, Z. Ma, B. Mcintyre, K. Mistry, A. Murthy, B. Obradovic, R. Nagisetty, P. Nguyen, S.

Sivakumar, R. Shaheed, L. Shifren, B. Tufts, S. Tyagi, M. Bohr, and Y. El-Mansy,

“A 90-nm logic technology featuring strained-silicon,” IEEE Trans. Electron

Devices, vol. 51, no. 11, pp. 1790-1797, Nov. 2004.

[1.4] J. S. Lim, S. E. Thompson, and J. G. Fossum, “Comparison of threshold-voltage shifts for uniaxial and biaxial tensile-stressed n-MOSFETs,” IEEE Electron

Device Lett., vol. 25, no. 11, pp. 731-733, Nov. 2004.

[1.5] A. Hamada, T. Furusawa, N. Saito, and E. Takeda, “A new aspect of mechanical stress effects in scaled MOS devices,” IEEE Trans. Electron Devices, vol. 38, no.

4, pp. 895-900, Apr. 1991.

[1.6] W. Zhao, A. Seabaugh, V. Adams, D. Jovanovic, and B. Winstead, “Opposing dependence of the electron and hole gate currents in SOI MOSFETs under uniaxial strain,” IEEE Electron Device Lett., vol. 26, no. 6, pp. 410-412, Jun.

2005.

6

[1.7] X. Yang, J. Lim, G. Sun, K. Wu, T. Nishida, and S. E. Thompson,

“Strain-induced changes in the gate tunneling currents in p-channel metal-oxide-semiconductor field-effect transistors,” Appl. Phys. Lett., vol. 88, no.

5, pp. 052108-1-052108-3, Jan. 2006

[1.8] S. H. Lo, D. A. Buchanan, Y. Taur, and W. Wang, “Quantum-mechanical modeling of electron tunneling current from the inversion layer of ultra-thin-oxide nMOSFETs,” IEEE Electron Device Lett., vol. 18, no. 5, pp.

209-211, May 1997.

[1.9] K. N. Yang, H. T. Huang, M. C. Chang, C. M. Chu, Y. S. Chen, M. J. Chen, Y. M.

Lin M. C. Yu, M. Jang, C. H. Yu, and M. S. Liang, “A physical model for hole direct tunneling current in p+ poly-gate pMOSFETs with ultrathin gate oxides,”

IEEE Trans. Electron Devices, vol. 47, no. 11, pp. 2161-2166, Nov. 2000.

[1.10] Y. T. Hou, M. F. Li, Y. Jin, and W. H. Lai, “Direct tunneling hole currents through ultrathin gate oxides in metal-oxide-semiconductor devices,” J. Appl.

Phys., vol. 91, no. 1, pp. 258-264, Jan. 2002.

7

Chapter 2

Triangular Potential based Simulator

2.1 Introduction

A simulator based on a triangular potential approximation, named TRP, is presented in this chapter. Some electrical properties of nMOSFET are calculated by

TRP. However, when comparing with the self-consistent

Schrődinger and Poisson’s equations solver, Schred [2.1], unacceptable error appears. Thus, a new algorithm is proposed and incorporated to correct the error.

2.2 Triangular Potential Approximation

2.2.1 Physical Model

The description below is dedicated to the case of (100) nMOSFET. The energy band diagram of poly-gate MOSFET is given in Fig. 2.1, where V_s, V_ox, and V_poly are the potential drop in the Si substrate, silicon dioxide, and poly gate region, respectively, E_f is the electron Fermi level, and F_s is the surface electric field. This band diagram is characterized by Fs. As we give a value of Fs, the Vox is calculated via continuous electric displacement (i.e., no charge) at the interface:

si

ox s ox

ox

V F ε t

= ε

(2.1) where εsi and εox are the permittivity of silicon and silicon dioxide, respectively. Vpoly

8

in the poly depletion region can be calculated:

where q is the elementary charge of an electron and Npoly is the doping concentration of poly gate. V_s can be expressed as a function of the gate bias Vg:

V

_s

= V

_g

− V

_ox

− V

_poly

+ V

_fb (2.3) where the flat band voltage Vfb is calculated as:

_{fb B}

ln(

^poly2 ^sub

)

i

N N

V k T

= − n

(2.4) where N_sub is the doping concentration of substrate, n_i is the intrinsic carrier density in substrate, kB is the Boltzmann’s constant, and T is the absolute temperature. The solving of the Schrődinger equation in the quantum-confined direction normal to the

SiO₂/Si surface yields the ∆2 subband level i in the absence of the stress [2.2]:

The average depth of the 2DEG is:

2,_i

2

2,_i

/ 3

_s

Z

_∆

= E

_∆

qF

(2.6)

where m_z,∆2 is the ∆2 quantization effective mass,  is the Planck’s constant divided by 2π. Eq. (2.5) can apply to 4-fold case by replacing ∆2 with ∆4. From Fig. 2.1, the Fermi level is related to the surface potential:

9

where Eg is the energy gap of silicon, Nv is the effective density of states in valence band. With Eq. (2.5) and (2.7), we can get the charge density for each subband [2.3]:

_{2/4,i 2/4}( ^{d, 2/4}₂ ^B ) ln(1 exp( ^{f 2/4,i}))

The surface drop due to bulk depletion V_dep and 2D depletion charge density N_dep are [2.2]:

where N_inv is 2D inversion charge density which is equal to the summation of N_∆2/4,i, and Z_qm is the average penetration of the inversion-layer charge from the surface. As we give a gate voltage Vg as input, initial Fs is guessed until it is consistently equal to

( _{inv dep}) / _si

q N

+

N ε

according to Gauss law. The flow chart is presented in Fig. 2.2.

On the other hand, if the device is manufactured by metal gate, the potential drop of metal gate is equal to zero and the flat band voltage relates to work function.

10

2.2.2 Outcome of TRP

In Fig. 2.3, the resulting conduction potential profile is shown, along with five lowest subband levels for (100) nMOSFETs in terms of three of ∆2 subband and two of ∆4 subbnad. We show both cases of poly silicon and metal gate. To examine the validity of the triangular potential approximation, a self-consistent Poisson-Schrödinger equations solver, Schred [2.1], was used and the resulting subband levels are shown in Fig. 2.3. In the figure, the drawback of the conventional triangular potential approximation is clear, especially in the higher energy levels where the corresponding electric field deviates from the surface field F_s, as shown in Fig. 2.1.

2.3 Correction Coefficient Generator

To address this issue, different methods have been proposed previously: (i) the variation approach dedicated to the correction of the lowest subband [2.2], [2.4]; and (ii) the effective field F_eff to replace F_s in Eq. (2.5) [2.5]-[2.10]:

The correction coefficient η in Eq. (2.11) is constant with a spanned range from 0.5 to 1.0: η = 0.75 for ∆2 and 1.0 for ∆4 [2.5], [2.6]; η = 0.5 for all subbands [2.7]; and η = 0.75 for all subbands [2.8]-[2.10]. However, the previous improvements that led to Eq. (2.11) are not enough from the aspect of the direct tunneling: each of the subbands involved in the tunneling should have its own correction coefficient such as

11

to ensure the proper direct tunneling calculation. Obviously, due to different electric fields encountered from level to level as revealed in Fig. 2.1, different correction coefficient values should correspond to different subbands. To take this into account, we suggest the individual correction coefficient η∆2,i for the ∆2 level i and the corresponding effective electric field can be written as:

2, the correction coefficient values, the solver Schred [2.1] was conducted in a MOS system on (001) silicon surface. A wide range of the key process parameters was included: the substrate doping concentration N_sub= 10¹⁵, 10¹⁶, 10¹⁷, and 10¹⁸ cm^-3; the gate oxide thickness tox = 1, 3, and 6 nm; and the different gate stacks in terms of a polysilicon and a metal electrode. By matching the subband levels produced by

Schred with those from Eq. (2.5) (with F

_s replaced by F_∆2,i

for 2-fold valley and F

_∆4,i for 4-fold valley), the values of the η_∆2,i

and

η_∆4,i result. A scatter plot between the correction coefficient values and the corresponding subband levels is given in Fig. 2.4, which is made with the surface field F_s as a parameter. Strikingly, the figure points to two relevant relationships. First, under fixed Fs, all data points fall on or around a straight line, indicating that the correction coefficient depends linearly on the subband level. Second, the straight line appears to shift with Fs. This specific behavior can be modeled by the intercept, designated as ηo, of the extrapolated line at zero subband level. In the inset of the figure, ηo is plotted against F_s, clearly showing another linear relationship, regardless of the N_sub, t_ox, or gate stack material. This is expected from the aspect of the MOS electrostatics. The combination of these two linear

12

relationships therefore leads to a subband-level correction-coefficient generating expression suitable for both ∆2 and ∆4

[2.11]:

2 / 4,_i

0.003 E

2 / 4,_i

(1.01 0.308 F

_s

)

η

_∆

= −

_∆

+ +

(2.13)

The units of E_∆2/4i

and F

s in Eq. (2.13) are meV and MV/cm, respectively. Eq. (2.13) can provide a transparent understanding of the effect of the subband level and surface field on the calculated correction coefficient. Interestingly, Eq. (2.13) is also self-consistent: for those of the subband levels close to the reference point (that is, the classical conduction band edge at the surface), the correction coefficients lie in close proximity of unity and hence the effective electric field approaches the surface one.

To testify to the validity of Eq. (2.13) in the subband level calculation, the results are compared with those from Schred [2.1], as given in Fig. 2.5 for two different gate stacks. Excellent agreements are evident, obtained without adjusting any parameters.

Note that the expression Eq. (2.13) is valid only for (001) substrate. Further study is needed concerning the underlying physical origins as well as its applicability to other substrate orientations. We think that the two linear relationships in Fig. 2.4 may be helpful in this direction.

13

References

[2.1] Schred. [Online]. Available: http://nanohub.org/resources/schred.

[2.2] F. Stern, “Self-consistent results for n-type Si inversion layers,” Phys. Rev. B, vol. 5, no.12, pp.4891-4899, Jun. 1972.

[2.3] F. Stern and W. E. Howard, “Properties of semiconductor surface inversion layers in the electric quantum limit,” Phys. Rev., vol. 163, no. 3, pp. 816–835, Oct. 1967.

[2.4] N. Yang, W. K. Henson, J. R. Hauser, and J. J. Wortman, “Modeling study of ultrathin gate oxides using direct tunneling current and capacitance-voltage measurements in MOS devices,” IEEE Trans. Electron Devices, vol. 46, no. 7, pp. 1464–1471, Jul. 1999.

[2.5] J. S. Lim, X. Yang, T. Nishida, and S. E. Thompson, “Measurement of conduction band deformation potential constants using gate direct tunneling current in n-type metal oxide semiconductor field effect transistors under mechanical stress,” Appl. Phys. Lett., vol. 89, no. 7, pp. 073509-1-073509-3, Aug. 2006.

[2.6] C. Y. Hsieh and M. J. Chen, “Measurement of channel stress using gate direct tunneling current in uniaxially stressed nMOSFETs,” IEEE Electron Device

Lett., vol. 28, no. 9, pp. 818-820, Sept. 2007.

[2.7] C. K. Park, C. Y. Lee, B. J. Moon, Y. H. Byun, and M. Shur, “A unified current-voltage model for long-channel nMOSFET’s,” IEEE Trans. Electron

Devices, vol. 38, no. 2, pp. 399-406, Feb. 1991.

[2.8] Y. Ma, L. Liu, Z. Yu, and Z. Li, “Validity and applicability of triangular potential well approximation in modeling of MOS structure inversion and accumulation layer,” IEEE Trans. Electron Devices, vol. 47, no. 9, pp. 1764-1767, Sept. 2000.

[2.9] Y. T. Hou, M. F. Li, Y. Jin, and W. H. Lai, “Direct tunneling hole currents

14

through ultrathin gate oxides in metal-oxide-semiconductor devices,” J. Appl.

Phys., vol. 91, no. 1, pp. 258–264, Jan. 2002.

[2.10] H. Abebe, E. Cumberbatch, H. Morris, and V. Tyree, “Compact models of the quantized sub-band energy levels for MOSFET device application,” in IEEE

UGIM Proceedings, 2008, pp. 58-60.

[2.11] W. H. Lee and M. J. Chen, “Gate direct tunneling current in uniaxially compressive strained nMOSFETs: a sensitive measure of electron piezo effective mass,”

IEEE Trans. Electron Devices, vol. 58, no. 1, pp. 39-45, Jan.

2011.

15

Fig. 2.1 Silicon energy-band diagram produced by Schred [2.1] (black lines), two calculated subband energy level (pink lines), and the Fermi level (blue line), as well as the red line for the triangular potential approximation under the same surface field.

16

Fig. 2.2 The flow chart of the calculation process inside the TRP.

17

Fig. 2.3 Subband levels calculated by the triangular potential approximation (solid dots) and by Schred (lines) for two cases: (a) n+ poly silicon doping N_poly = 10²⁰ cm^-3,

t

ox = 1 nm, and Nsub = 10¹⁵ cm^-3; and (b) metal gate with zero flat-band voltage, tox = 1nm, and N_sub = 10¹⁸ cm^-3_.

18

Fig. 2.4 The extracted correction coefficient versus the corresponding subband level with the surface field as a parameter. The fitting lines are drawn. The intercept,

η

o, of the extrapolated line at the zero subband level is inserted and plotted versus the surface field. A fitting line is also shown in the inset.

200 400 600 800

19

Fig. 2.5 Repeating the calculation work by the triangular potential approximation based on the new η correction generator.

20

Chapter 3

Strain Altered Electron Gate Direct Tunneling Current

3.1 Introduction

In this chapter, we discuss about the model of conduction band electron direct tunneling (EDT) current. For the silicon nMOSFETs formed on (001) substrate, the quantum confinement effect [3.1] around the inversion layer makes the bulk conduction band split into two distinctive components: 2-fold (∆2) and 4-fold (∆4) valleys. The longitudinal effective mass (ml) and transverse effective mass (mt) associated with those subband valleys essentially remain intact [3.1]. The energetic difference between ∆2 and ∆4 levels can be further changed via the applied mechanical stress as in the state-of-the-art strain engineering. The stress induced subband shift has been thoroughly studied theoretically [3.2] in terms of the deformation potential constants [3.3]-[3.5]. Thus, the change ratio of EDT current under strain can be estimated.

Comparing with the experimental data of EDT current [3.6], [3.7], one important physical phenomenon can be brought out: the effective mass of electron varies with applied stress. Recently, the sophisticated band-structure calculation [3.8]-[3.10] on (001) silicon surface has pointed out that only with the strain dependence of m_l and m_t taken into account can the strain induced mobility change be elucidated. The significance of the strain dependent electron effective masses in (110) case has also been mentioned [3.11]. Thus, in addition to the deformation potential counterparts, the

21

strain dependence of m_l and m_t or equivalently the electron piezo-effective-mass coefficient, πm, should not be absent in the strain altered conduction-band structure.

The mobility measurement method has been constructed to experimentally determine the πm of electrons [3.12]. On the other hand, the effect of the mechanical stress on the electron gate direct tunneling current has been experimentally observed [3.6], [3.7], [3.13]-[3.17]. In the citations [3.6], [3.7], [3.13]-[3.17], however, the impact of the πm on the strained electron gate direct tunneling current has not been noticed. According to the quantum confinement picture [3.1], a change in the electron quantization effective mass due to the stress will produce a change in the subband level and therefore change the transmission probability dramatically. Thus, through the inverse modeling technique, the electron gate direct tunneling current in strained device may serve as a sensitive detector of πm. However, few studies on this subject were done to date..

3.2 Strain-Altered Band Structures

In this section, we make a connection between the strain and the stress. Notice that the temperature-induced strain does not be considered here. Stress is the average force over the area on which the force acts. Thus, the intensity of stress is expressed as function of applied force per area. The force applied on an area can be separated into two directions: out-of-plane direction (normal force) and in-plane direction (shear force). The stress caused by normal/shear force is called normal/shear stress.

For a force F applied on an infinitesimal area A which is normal to the z direction, as show as in Fig. 3.1, the projected quantity of the force along x, y, and z are F_x, F_y,

22 The notation σii refers to the normal stress acting on the plane perpendicular to i-direction, and τij refers to the shear stress component along j-direction acting on the plane perpendicular to i-direction.

Furthermore, we consider the case of an infinitesimal cube whose six surfaces face to ±x, ±y, and ±z. There should be 18 stress components by Eq. (3.1).

However, two conditions are observed: (1) F_x and F_-x are reaction force of each other;

(2) τxy

= τ

yx, τyz

= τ

zy, and τzx

= τ

xz can be derived because of the total applied force and torque on the cube are zero. Thus, stress tensor is simplified to the only 6 terms [3.18]:

23

It is notable that the tensile stress is shown as positive value. On the other hand, the compressive stress is the negative value. With external stress, a deformable body changes its size and shape. In Fig. 3.2(a), a normal tensile force along x-direction σxx

is applied on deformable body and the length along x-direction is increased. The normal strain is defined:

𝜀

_𝑥𝑥

=

^∆𝐿_𝐿^𝑥

𝑥 (3.3) Again, positive ε means the length elongates, a situation called tensile strain.

Negative ε means that the length is contracted, the case compressive strain.

In Fig. 3.2(b), a shear stress τzy is applied on a planar body that causes the change of its shape. The angle varies from π/2 to q. Besides, the lengths of four side lines are unchanged. The shear strain γzy is defined as the change in the angle between two neighbor sidelines of the square on y-z surface [3.18]:

2

^{z y}

zy zy

u u

y z

γ = ε = ^∂ + ^∂

∂ ∂

(3.4)

where uz and uy mean the displacements along z- and y-direction respectively. γzy

presents the shear strain, and εzy is the average shear strain equal to the half of γzy

[3.18]-[3.20].

Similar to stress tensor, the strain tensor is also composed of six independent components:

24

When a stress is applied to a homogeneous and isotropic material, the normal strain has a linear relationship with normal stress, which is the well-known Hooke’s Law [3.18]-[3.20]:

σ = E ε

(3.6) where the constant of proportionality E is the Young’s modulus. Furthermore, while the normal stress is applied on elastic material, the strain transversal to stress usually accompanies. The relationship between normal strain and transverse strain is [3.18]-[3.20]:

ε

_tran

= − v ε

_long (3.7)

where the constant of proportionality v is the Poisson’s ratio. Finally, the Hooke’s law still holds for shear strain [3.18]-[3.20]:

τ = G γ

(3.8)

where the constant of proportionality G is the shear modulus.

With Eq. (3.6)-(3.8), the relation between strain and stress is:

1

[ ( )]

xx xx

v

yy zz

ε = E σ − σ + σ

(3.9a)

25 For simplicity, we usually transfer the strain and stress relationships to the matrix form. With Eq. (3.2), (3.5), and (3.9), the elastic relationship between strain and stress is established [3.18]-[3.22]:

In deformation potential theory, the total Hamiltonian for each energy valleys of silicon conduction band is [3.27]:

26

where k_l and k_t are the wavevectors parallel and perpendicular to the axis where the valleys are located, respectively, Ξd

and Ξ

u are the hydrostatic and shear deformation potential constants, respectively, Tr(εij

) is the trace of the strain tensor. And Ξ

d = 1.13 eV and Ξu = 9.16 eV are given in silicon case [3.26], εl is the longitudinal strain component. Appling Eq. (3.11), the band edge shift for the minima of the six conduction band valleys along the <100> direction is:

∆ E

_{c x,}

= Ξ

_d

( ε

_xx

+ ε

_yy

+ ε

_zz

) + Ξ

_u

ε

_xx (3.12a)

∆ E

_{c y,}

= Ξ

_d

( ε

_xx

+ ε

_yy

+ ε

_zz

) + Ξ

_u

ε

_yy (3.12b)

∆ E

_{c z,}

= Ξ

_d

( ε

_xx

+ ε

_yy

+ ε

_zz

) + Ξ

_u

ε

_zz (3.12c)

Actually, the applied stress is not always along [100], it may be in the direction of [110], [111], and [112], etc. Fortunately, the stress is easily transformed between different coordinates [3.18]. The stress tensors in some cases are listed in Table 3.1.

3.3 Gate Direct Tunneling Current Model

Using both quantum mechanical simulator TRP and a modified WKB [3.28]-[3.30] approximation for transmission probability, the model for calculating the gate direct tunneling currents across ultra-thin gate oxides of MOS structures is discuss here.

The correction coefficient generator via Eq. (2.5) and (2.12) were incorporated into existing strain quantum simulator in our previous works [3.29]-[3.31]. The resulting subband level in the presence of the uniaxial channel stress σ in the <110>

direction can be written with respect to the non-stress conduction-band edge at the

27

Si/SiO₂ interface [3.3]-[3.6]

σ

The carrier repopulation under stress can be calculated accordingly:

'

Finally, the triangular potential based electron gate direct tunneling current density can be computed:

where f represents the electron impact frequency on the Si/SiO2 interface and is equal to (qF_∆2/4,i/2)(2m_z,∆2/4

E

_∆2/4,i)^-1/2; and Pt

(E

^’_∆2/4,i

) is the electron transmission probability

across the SiO2 film. In Fig. 3.3, the energy band diagram of the MOS system under study is schematically shown, where the electron direct tunneling process from the subband level is highlighted. Throughout the work, only five lowest subbands (3 of ∆2

and 2 of ∆4) will be adopted to calculate the gate current.

Here, the electron effective mass in the oxide for the parabolic type dispersion relationship was used with mox = 0.50 mo, which is equivalent to mox

= 0.61 m

o

for the

tunneling electrons in the oxide using the Franz type dispersion relationship [3.32].

The SiO2/Si interface barrier height in the absence of stress is 3.15 eV. Given the situations that the deformation potential constants are known and the channel stress can be determined by other means, there are four variables in using (3.15) to quantify

在文檔中 n型反置層精準量子計算：應變、次能帶、遷移率及三維結構 (頁 22-0)