Dissertation Organization

The purpose of this work focuses on the assessment of the STI stress quantities as well as the physical model establishment. Based on the extracted values and status of stress, we can extend the issues to modeling of strain-induced dopant diffusion and gate oxide integrity. At this point, this dissertation is organized into seven chapters.

Chapter 2 begins with the mechanics of materials. The definitions of stress and strain are first reviewed. Their dependencies on each other are associated with the elasticity. Meanwhile, the theory of strain-induced energy band shift both on conduction and valence bands is introduced and will be used in later chapters.

As for the main parts of the dissertation, first of all, we present a simple method to electrically assess the average mechanical stress in channel region using the gate direct tunneling current changes. In this study, shallow trench isolation-induced mechanical stress can serve as the dominant source in the channel due to the thermal expansion and the layout technique will be utilized to produce a variety of stress. The different approaches to determining stresses in longitudinal and transverse directions were detailed in Chapter 3 and 4, respectively. Confirmative evidence is verified by piezoresistance coefficient. Especially, in the narrowing direction, the delta width effect together with the stress effect is adopted to clarify the anomalous trend of gate tunneling current.

Second, in Chapter 5, we show how to transform the drain subthreshold current change to the source/drain corner stress. With the modeling of edge direct tunneling current, it leads to the underlying gate-to-source/drain extension overlap length. Therefore, a physically-oriented analytic model is successfully established, expressing the lateral diffusion as a function of corner stress.

4

In Chapter 6, low-frequency noise measurement will be conducted to extract the oxide integrity in the channel narrowing direction in Chapter 6. Using the stress extraction technique built in previous chapters, the effect of stress on interface states during the oxidation will be demonstrated.

Finally, Chapter 7 delivers a conclusion to the research work, and also addresses the future work as extension of this dissertation.

5

References

[1.1] H. A. Rueda, “Modeling of mechanical stress in silicon isolation technology and its influence on device characteristics,” Ph.D. Dissertation, University of Florida, 1999.

[1.2] 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.

[1.3] 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, pp. 1790–1797, Nov. 2004.

[1.4] 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. Wang, 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.

[1.5] 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, pp. 731–733, Nov. 2004.

[1.6] 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, pp.

895–900, Apr. 1991.

[1.7] 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

6

strain,” IEEE Electron Device Lett., vol. 26, pp. 410–412, Jun. 2005.

[1.8] 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, pp. 052108, Jan. 2006.

[1.9] 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, pp. 073509, Aug. 2006.

[1.10] M. J. Aziz, Y. Zhao, H.-J Gossmann, S. Mitha, S. P. Smith, and D. Schiferl, “Pressure and stress effects on the diffusion of B and Sb in Si and Si-Ge alloys,” Phys. Rev. B, vol. 73, p. 054101, Feb. 2006.

[1.11] S. T. Dunham, M. Diebel, C. Ahn, and C. L. Shih, “Calculations of effect of anisotropic stress/strain on dopant diffusion in silicon under equilibrium and nonequilibrium conditions,” J. Vac. Sci. Technol. B, vol. 24, pp. 456-461, Jan./Feb.

2006.

[1.12] M. J. Chen and Y. M. Sheu, “Effect of uniaxial strain on anisotropic diffusion in silicon,” Appl. Phys. Lett., vol. 89, p. 161908, Oct. 2006.

[1.13] E. Simoen, G. Eneman, P. Verheyen, R. Delhougne, R. Loo, K. De Meyer, and C.

Claeys, “On the beneficial impact of tensile-strained silicon substrates on the low-frequency noise of n-channel metal-oxide-semiconductor transistors,” Appl.

Phys. Lett., vol. 86, p. 223509, May 2005.

[1.14] M. P. Lu, W. C. Lee, and M. J. Chen, “Channel-width dependence of low-frequency noise in process tensile-strained n-channel metal-oxide-semiconductor transistors,”

Appl. Phys. Lett., vol. 88, p. 063511, Feb. 2006

7

[1.15] A. Stesmans, P. Somers, V. V. Afanas'ev, C. Claeys and E. Simoen, “Inherent density of point defects in thermal tensile strained (100)Si/SiO2 entities probed by electron spin resonance,” Appl. Phys. Lett., vol. 89, p. 152103, Oct. 2006.

8

Chapter 2

Strain Effect on Electronic Band Structure

2.1 Review of Mechanics of Materials

The property of solid materials to deform under the application of an external force and to regain their original shape after the force is removed is referred to as its elasticity. The external force applied on a specified area is known as stress, while the amount of deformation is called the strain. In this section, the theory of stress, strain and their interdependence is briefly discussed.

2.1.1 Stress and Strain

Stress - Stress is the distribution of internal body forces of varying intensity due to

externally applied forces [2.1], [2.2]. Consider a general body subjected to forces acting on its surface: Passing a plane through the body cutting it along surface A and letting the force, which is transmitted through an incremental area ΔA of A by the part on positive side Q, be denoted by ΔF. The force ΔF may be resolved into components ΔFN and ΔFS, as illustrated in Fig. 2.1(a), along unit normal N and unit tangent S, respectively, to the plane Q. The force ΔFN is called normal (perpendicular) stress and ΔFS is shear (tangential) stress on area ΔA.

The magnitude of the average forces per unit area is ΔF/ΔA. The concept of stress at a point is obtained by letting ΔA become an infinitesimal. The limiting ratio ΔF/ΔA as ΔA goes to zero defines the stress vector as given by

lim0

9

Similarly, the limiting ratios of ΔFN/ΔA and ΔFS/ΔA define the normal stress vector

σ

N and the shear stress vector

σ

S that act on a point in the plane Q. These vectors are described by the relations

Three stress vectors acting on three mutually orthogonal planes intersecting at that point can then determine the stress state as shown in Fig. 2.1(b). The stress tensor is composed of the three stress vectors and is sufficient to define the stress state in any element in a body. To illustrate the tensor nature of stress present at point in the continuous body, consider a cubic element of infinitesimal dimensions. For simplicity of notation, let the cube be aligned perpendicular with the system axis. The stress vector T^x acting on the plane normal to the

x-direction is the following:

x xx xy xz

T =σ ⋅ +xr τ ⋅ +ury τ ⋅zr

(2.3)

The nine stress components relative to rectangular coordinate axes may tabulated in array form as follows:

where

σ

ij

represents the stress array called stress tensor, σ

ii

are the normal stress components

acting on the faces perpendicular to i-direction and

τ

ij

are the shear stress components

oriented in the j-direction on the face with normal in the i-direction. At mechanical equilibrium, it can be shown that three pairs of shear stresses are equal that lead to the result

ij ji

τ =τ (2.5)

Hence, a column vector of six independent components can then describe the state of stress at a point:

10

T

xx yy zz xy yz zx

σ = ⎣⎡σ σ σ τ τ τ ⎤⎦ (2.6)

Strain – The application of stress to a body in equilibrium causes it to undergo

deformation or strain. It is the geometrical measure of deformation representing the relative displacement between particles in the material body. Normal strain is defined as the amount of stretch or compression along a material line element while shear strain is a degree of distortion associated with the sliding of plane layers over each other within a deforming body.

Consider a two-dimensional deformation of an infinitesimal rectangular material element with dimensions as shown in Fig. 2.2. From the geometry, we can write

Under the assumption of small displacement, which means ∇u<<1, the length of A B' ' can reduce dx+ ∂

(

ux ^/∂x dx

)

. The normal strain in x - direction of the element is defined as The shear strain is the change of the angle between two originally orthogonal axes. For small rotation (i.e.

α, β

<<1) and infinitesimal approximation, we get

tan , tan By expanding this definition in three dimensions, the strain can be related to the displacements by the following strain components:

11

where u, v, and w are the displacements in the x, y, and z directions, respectively. The results analogous to those of stress theory hold, and therefore the symmetric array of strain tensor (

ε

kl) can be arranged as:

Similar to stress, only six independent components are required to then define the state of strain at a certain point:

T

xx yy zz xy yz zx

ε = ⎣⎡ε ε ε ε ε ε ⎤⎦ (2.13)

2.1.2 Stress-Strain Relationship

The relationship between the stress tensor and the deformation is known as a constitutive relation. All structural materials possess the property of elasticity. When the force is removed, the body will return to its original shape if it is an ideal elastic body and it had not reached its yield stress. For an elastic solid, the stress tensor is linearly proportional to the strain tensor over a specific range of deformation:

σ_ij =c_{ijkl kl}ε (2.14)

where c^ijkl is the tensor of stiffness constants. In order to relate each of the nine elements of the second rank strain tensor to each of the nine elements of the second rank stress tensor, c^ijkl consists of a fourth rank tensor of 81 elements. However, due to the symmetries involved for

12

the stress and strain tensors under equilibrium, c^ijkl can reduce a tensor of 36 elements. Crystal silicon has diamond cubic crystal geometry resulting from its strong directional covalent bonds. For such crystals, cijkl has the following form due to their cubic symmetry:

11 12 12

Thus, for silicon the tensor of elastic stiffness constants reduces to the three independent components: c11, c12, and c44. Of practical interest is the strain arising from a certain stress condition. The strain components can be obtained by inverting Hook's law and utilizing the compliance coefficients,

ij sijkl kl

ε = σ (2.16)

The stiffness and compliance tensors are linked through the above relation. Consequently, the three independent compliance coefficients can be calculated as [2.3],[2.4]

11 12

The compliance coefficients for Si, together with the stiffness coefficients, are listed in Table 2.1 [2.5].

2.2 Strain-induced Energy Splitting

13

The effect of stress on the resistivity of Si was first investigated by Smith [2.6]. This finding was contributed to the modification of the electronic band structure. Microscopically, stress breaks the symmetry of lattice which can then cause the energy shift and band distortion. In the following these effects are discussed in detail.

Deformation potential theory originally developed by Bardeen and Shockley [2.7] was used to investigate the interaction of electrons with acoustic phonons. It was later generalized to include different scattering modes by Herring and Vogt [2.8]. The technique was applied to strained systems by Bir and Pikus [2.9].

Within the framework of this theory, the energy shift of a band extremum l is expanded in terms of the components of the strain tensor

ε

ij.

The coefficients of this expansion are called the deformation potential tensor. This tensor is characteristic of a given non-degenerate band in the solid. The symmetry of the strain tensor is also reflected in that of the deformation potential tensor, giving

( )l ( )l

ij ji

Ξ = Ξ (2.19)

The maximum number of independent components of this tensor is six which can reduce two or three for a cubic lattice. They are usually denoted by

Ξ

u, the uniaxial deformation potential constant, and

Ξ

d, the dilatation deformation potential constant. The deformation potential constants can be calculated using theoretical techniques such as density functional theory [2.10], the non-local empirical pseudo-potential method [2.11], or ab-initio calculations.

However, a final adjustment of the potentials is obtained only after comparing the calculated values with those obtained from measurement techniques [2.12]–[2.14]. The deformation potential constants used in this work are listed in Table 2.1 [2.15].

14

The general form of the strain-induced energy shifts of the conduction band valleys for an arbitrary strain tensor can be written as

( )^{i j,} ( )^j

( )

^{( )j T}

C d u i i

E Tr ε a ε a

Δ = Ξ + Ξ ⋅ ⋅ (2.20)

where ai is a unit vector of the ith valley minimum for the jth valley type. The first term in Eq.

(2.20) shifts the energy level of all the valleys equally and is proportional to the hydrostatic strain. The difference in the energy levels of the valleys arises from the second term in Eq.

(2.20). In this method, strain effect only shifts the band edge while it does not cause the band warping. In this study, the stress along <110> direction on (001) surface can first be transformed into strain. Then, by applying Eq. (2.20), the quantities of band shift for Δ2 and Δ4 valley can be expressed as

It is noteworthy that the approximation is reasonable under moderate stress [2.16] while it may need to include the effect band warping for large stress because of the strong influence of effective mass change.

15

References

[2.1] A. P. Boresi, R. J. Schmidt, and O. M. Sidebottom, “Advanced mechanics of materials,” 5th ed., New York: John Wiley & Son, 1993.

[2.2] H. A. Rueda, “Modeling of mechanical stress in silicon isolation technology and its influence on device characteristics,” Ph.D. Dissertation, University of Florida, 1999.

[2.3] C. Kittel, “Introduction to solid state physics,” 7th ed., New York: John Wiley & Son, 1995.

[2.4] S. Dhar, “Analytical mobility modeling for strained Silicon-based devices” Ph.D.

Dissertation, Vienna University of Technology, 2007.

[2.5] Y. Kanda, “Effect of stress on Germanium and Silicon p-n junctions,” Jpn. J. Appl.

Phys., Vol. 6, No. 4, pp. 475-486, 1967.

[2.6] C. S. Smith, “Piezoresistance effect in Germanium and Silicon,” Phys. Rev., vol. 94, no. 1, pp. 42-49, Apr. 1954.

[2.7] J. Bardeen and W. Shockley, “Deformation potentials and mobilities in non-polar crystals,” Phys. Rev., vol. 80, pp. 72-80, Oct. 1950.

[2.8] C. Herring and E. Vogt, “Transport and deformation-potential theory for many-valley semiconductors with anisotropic scattering,” Phys. Rev., vol. 101, pp. 944-961, Feb.

1956.

[2.9] G. L. Bir and G. E. Pikus, “Symmetry and strain induced effects in semiconductors,'' New York: Wiley, 1974.

[2.10] C. G. Van de Walle, “Theoretical calculations of heterojunction discontinuities in the Si/Ge system,” Phys. Rev. B, vol. 34, pp. 5621-5633, Oct. 1986.

[2.11] M.V. Fischetti and S.E. Laux, “Band structure, deformation potentials, and carrier

16

mobility in strained Si, Ge, and SiGe alloys,” J. Appl. Phys., vol. 80, pp. 2234-2252, Aug. 1996.

[2.12] C. Herring and E. Vogt, “Transport and deformation-potential theory for many-valley semiconductors with anisotropic scattering,” Phys. Rev., vol. 101, pp. 944–961, Feb.

1956.

[2.13] I. Balslev, “Influence of uniaxial stress on the indirect absorption edge in silicon and germanium,” Phys. Rev., vol. 143, pp. 636–647, Mar. 1966.

[2.14] C. G. Van de Walle and R. M. Martin, “Theoretical calculations of heterojunction discontinuities in the Si/Ge system,” Phys. Rev. B, vol. 34, pp. 5621–5634, Oct.

1986.

[2.15] 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, pp. 731–733, Nov. 2004.

[2.16] Y. Sun, S. E. Thompson, and T. Nishida, “Physics of strain effects in semiconductors and metal-oxide-semiconductor field-effect transistors,” J. Appl. Phys., vol. 101, p.

104503, May 2007.

17

Table 2.1 Compliance and stiffness coefficients, Luttinger parameters, deformation potential constants, and split-off energy for silicon.

Stiffness Coefficients

c

11

(10

¹²

dyne/cm

²

)

1.657

c

12

(10

¹²

dyne/cm

²

)

0.639

c

44

(10

¹²

dyne/cm

²

)

0.796

Compliance Coefficients

s

11

(10

^-12

m

²

/Nt)

7.68

s

12

(10

^-12

m

²

/Nt)

-2.14

s

44

(10

^-12

m

²

/Nt)

12.6

Deformation Potential Constants

Ξ

d

(eV)

1.13

Ξ

u

(eV)

9.16

18 A

FN

FS

F

(a)

y

z

x

yy

zz

xx

yx

zy xz

zx

yz xy

(b)

Fig. 2.1 (a) Schematic of an arbitrary force

Δ F acting on an infinitesimal area Δ A, along with

the resolved components: normal

Δ F

N and shear terms

Δ F

S. (b) A cubic element located within a continuous body with stress tensor components shown.

19

Fig. 2.2 Two-dimensional geometric deformation of an infinitesimal material element.

20

Chapter 3

Measurement of Channel Stress Using Gate Direct Tunneling Current in Uniaxially Stressed n-MOSFETs

3.1 Introduction

It is well recognized that the mechanical stress in MOSFETs can significantly affect many electrical properties such as the mobility [3.1]–[3.3], the hot carrier immunity [3.4], the threshold voltage [3.5], and the gate direct tunneling current [3.6]–[3.8]. Thus, the ability to quantitatively determine the magnitude of the underlying mechanical stress, as well as its status (compressive or tensile), is essential. Three fundamentally different methods have been introduced in this direction: (i) wafer bending jig [3.9]; (ii) sophisticated stress simulation [3.10]; and (iii) Raman spectroscopy [3.11]. Obviously, the electrical approach to the mechanical stress was lacking to date. However, it is noteworthy that the gate direct tunneling current has been well studied under externally applied mechanical stress [3.6]–[3.8].

Particularly in the citation [3.8], the deformation potential constants [3.12]–[3.14] have been experimentally determined with the values consistent with theoretical works [3.15]. Therefore, with known deformation potential constants, it is plausible to measure mechanical stress by means of the gate direct tunneling current.

In this chapter, we show how to transform the gate direct tunneling current in stressed devices into the value of the stress, achieved without adjusting any parameters. Confirmative evidence is presented in terms of the piezoresistance coefficient electrically created on the

21

same device.

3.2 Experiment

The n⁺ poly-silicon gate n-MOSFETs were fabricated in a state-of-the-art manufacturing process. The device process flow is depicted in Fig. 3.1. Also plotted in the Fig. 3.2 are the schematic cross section and topside view of the test device. Three key process parameters were obtained by capacitance-voltage (C−V) fitting: n⁺ poly-silicon doping concentration = 1

× 10 ²⁰ cm^-3, gate oxide thickness = 1.27 nm, and substrate doping concentration = 4 × 10 ¹⁷ cm^-3. In this process, the STI induced compressive stress was applied. The gate length along the <110> direction is 1 μm large enough that the following effects can be effectively eliminated: external series resistance and short channel or drain induced barrier lowering (DIBL). The gate width is wide (10 μm), indicating that the transverse stress is relatively negligible. Layout technique was utilized to produce a variety of stress in terms of the gate edge to STI sidewall spacing, designated a, with four values of 10, 2.4, 0.495, and 0.21 μm. A decrease in a means increased magnitude of longitudinal stress. A considerable number of contacts were formed on the source/drain diffusion along the gate width direction, far away from the STI in the <110> direction. The spacing between the diffusion contact and the gate edge is fixed in this work. It has been reported that silicide can introduce stress into channel and its effect can be eliminated by well controlling the silicide formation [3.10]. Thus, the silicide process was fine tuned for the device under study to minimize its effect as compared with STI stress.

The gate direct tunneling current was measured in inversion conditions with the source, drain, and substrate all tied to ground. Also characterized was the mobility on the same device at Vd = 25 mV. The change of the conduction-band electron direct tunneling current at Vg = 1

22

V and the mobility at Vg = 0.5 V, all with respect to a = 10 μm, are plotted in Fig. 3.3 versus gate to STI spacing. It can be seen that a decrease in the gate to STI spacing can produce an increase in both the gate current while degrading the mobility.

3.3 Stress Extraction

Existing direct tunneling models [3.16], [3.17] on the basis of the triangular potential approximation [3.18] in the channel, taking into account the poly-silicon depletion, can readily apply with some slight modifications such as incorporating stress dependencies of the subbands. The electrons in inversion primarily populate the two lowest subbands [3.8]: one of the two-fold valley Δ2 and one of the four-fold valley Δ4. The corresponding stress dependencies are well defined in the literature [3.8], [3.12]–[3.14]:

, 2 32 of Ref. [3.8], were cited here. Stress along <110> direction can be resolved into two different components: normal and shear stress terms in <100> coordination. Shear terms can cause the band distortion, which in turn, influences the effective mass. This effect becomes significant when applied strain approaches 1% and beyond, whose magnitude is much greater than that in our study case. Thus, it is reasonable to assume that effective mass change can be neglected under moderate stress in the subsequent calculation. One of the expressions for the effective

23

electric field Eeff can be found elsewhere [3.8]. With the aforementioned process parameters as input, the two lowest subband levels with respect to the Fermi level Ef can be determined. The stress dependencies of the lowest subbands under different gate voltages were found to be consistent with those in earlier works [3.8]. The inversion-layer carrier density per unit area can further be calculated by N_i =(k T_B /πh²)g m_{i di}ln(1 exp((+ E_f −E_i) /k T_B )) [3.16]–[3.18],

electric field Eeff can be found elsewhere [3.8]. With the aforementioned process parameters as input, the two lowest subband levels with respect to the Fermi level Ef can be determined. The stress dependencies of the lowest subbands under different gate voltages were found to be consistent with those in earlier works [3.8]. The inversion-layer carrier density per unit area can further be calculated by N_i =(k T_B /πh²)g m_{i di}ln(1 exp((+ E_f −E_i) /k T_B )) [3.16]–[3.18],

在文檔中 應變金氧半場效電晶體機械應力萃取與其相關物理模型建立之研究 (頁 19-0)