1. Introduction

Strained technology can effectively improve the carrier mobility of Si-based materials (Si and Si$_{1-x}$Ge$_{x})$, so it is widely used in high speed and performance semiconductor devices, also in integrated circuit^{[1, 2, 3]}. The hole mobility enhancement is closely related to the decreasing hole scattering rate in Si-based strained materials^{[4, 5, 6]}. The model establishment will make a contribution to the understanding the hole mobility enhancement, and the material design and its application.

At present, some problems still exist regarding the hole scattering mechanism in Si-based strained materials: (1) although the results for the hole mobility of the inversion layer of a Si-based strained device have been widely reported, their calculations cover a little part of the discussion of the hole scattering rate^{[7, 8]}. And the results for the hole scattering rate can only be applicable for the strained MOS inversion layer, not for the Si-based strained materials. (2) Since the hole mobility is generally obtained by using the Monte Carlo Simulation Method, there is no reported quantization results for the hole scattering rate in Si-based strained materials. This is disadvantageous to the understanding of how the effect of the decreasing hole scattering rate on the hole mobility enhancement in the Si-based strained materials. (3) The hole scattering rate in Si-based strained materials is related to the orientation of the strained epitaxial layer growth. Their relationships have not been discussed yet.

Therefore, based on Fermi's golden rule and the theory of Boltzmann collision term approximation, we will establish a theoretical relational model of the hole scattering rate and stress in order to study the hole scattering mechanism in-depth in Si-based strained materials. Our research could provide theoretical reference to the understanding, design and application of Si-based or other physical strained materials.

2. Model establishment

The $E$-$k$ relationship of the valence band structure in Si-based strained materials is given before we discuss the hole scattering mechanism^[9],

$\begin{cases} E_{\rm V}^{\rm 1} =2\sqrt Q \cos \frac{\Theta}{3}-\frac{p}{3}, E_{\rm V}^2 =2\sqrt Q \cos \frac{\Theta -2\pi }{3}-\frac{p}{3}, \\[2mm] E_{\rm V}^3 =2\sqrt Q \cos \frac{\Theta +2\pi }{3}-\frac{p}{3}, \\[2mm] Q={(p^2-3q)}/9, p=\Delta -(a_{11} +a_{22} +a_{33}), \\[2mm] R={(2p^3-9pq+27r)}/{54}, \Theta =\cos ^{-1}(-R/{\sqrt {Q^3} }), \\[2mm] q=a_{11} a_{22} +a_{22} a_{33} +a_{33} a_{11} -a_{12}^2 -a_{13}^2 -a_{23}^2 \\[2mm] \quad \quad -\, ({2\Delta }/3)(a_{11} +a_{22} +a_{33} ), \\[2mm] r=a_{11} a_{23}^2 +a_{22} a_{13}^2 +a_{33} a_{12}^2 -a_{11} a_{22} a_{33} -2a_{12} a_{23} a_{13}\\ \; +\, (\Delta/3)\left( {a_{11} a_{22} +a_{22} a_{33} +a_{33} a_{11} -a_{12}^2 -a_{13}^2 -a_{23}^2 } \right), \\ \end{cases}$

(1)

where $a_{ij}$ ($i, j =$ 1, 2, 3) are the elements in the Hamilton matrix $H'+H''$^[10].

On the basis of the $E$-$k$ relationship equations, taking the splitting energy of the valence band into consideration, the hole density of states effective masses at the top of the valence band ($m_{\rm d}^\ast)$ in Si and Si$_{1-x}$Ge$_{x}$ strained materials can be obtained, which is the important parameter to calculate the hole scattering rate in Si-based strained materials^[11].

$m_{\rm d}^\ast =\left[{(m_{\rm a}^\ast )_{\rm h}^{3/2} \exp ({-\Delta E_{\rm V, Split} }/{K_{\rm B} T})} \right]^{2/3},$

(2)

where $(m_{\rm a}^\ast )_{\rm h}$ and $(m_{\rm a}^\ast )_{\rm l}$ are "heavy", "light" average effective mass respectively, $\Delta E_{\rm V, Split}$ is the splitting energy between the "heavy" and the "light" hole bands. Its relationship with the stress is shown below.

There are the following hole scattering mechanisms^[12] in strained Si or Si$_{1-x}$Ge$_{x}$: the ionize impurities scattering, the acoustic phonon scattering, the non-polar optical phonon scattering and the alloy disorder scattering (denoted by $P_{\rm II}$, $P_{\rm ac}$, $P_{\rm in}$, $P_{\rm ad}$ respectively). And based on the Fermi golden rule, models of each hole scattering rate in Si-based strained materials are derived.

$P_{\rm II} =\frac{N_{\rm i} e^4}{16\pi (2m^\ast )^{1/2}(\varepsilon _0 \varepsilon )^2E^{3/2}}\ln \frac{12m_{\rm d}^\ast K_{\rm B}^2 T^2\varepsilon _0 \varepsilon }{e^2\hbar ^2N_{\rm i}},$

(3)

$P_{\rm ac} =\frac{m_{\rm d}^{\ast 3/2}\Xi ^2K_{\rm B} T(2E)^{1/2}}{\pi \hbar ^4c_{\rm l}},$

(4)

$P_{\rm op} =\frac{D_0^2 (m_{\rm d}^\ast )^{3/2}}{2^{1/2}\pi \hbar ^3\rho \omega _0 }\left( {n_{\rm op} +\frac{1}{2}\mp \frac{1}{2}} \right)(E\pm \hbar \omega _0 )^{1/2},$

(5)

$P_{\rm ad} =\frac{\sqrt 2 m_{\rm d}^{\ast 3/2}x\left( {1-x} \right)\left( {\Delta E} \right)^2}{\pi \hbar^4N} E^{1/2},$

(6)

where Table 1 shows the physical meaning and specific values of the parameters.

The total scattering rates of the Si-based strained materials is the sum of each scattering rate:

$$P=\sum\limits_i {P_{\rm i} } =P_{\rm II} +P_{\rm ac} +P_{\rm in} +P_{\rm ad} .$$

(7)

3. Results and discussions

The results for the hole scattering rates in Si-based strained materials as the function of stress can be seen in Figures 2-6. On the basis of the results, we turn to discuss the relevant laws about the hole scattering mechanism in Si-based strained materials.

The hole ionized impurity scattering rates in Si-based strained materials increases with increasing stress is clearly shown in Figure 2(a), and their order is strained Si$_{1-x}$Ge$_{x}$/(001)Si $>$ strained Si$_{1-x}$Ge$_{x}$/(101)Si $>$ strained Si/(001)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(111)Si $>$ strained Si/(101)Si$_{1-x}$Ge$_{x}$ $>$ strained Si/(111)Si$_{1-x}$Ge$_{x}$ when Ge fraction is about 0.2. Note that the order result is opposite when the doping concentration is 10$^{19}$ cm$^{-3}$ in Si-based strained materials (shown in Figure 2(b)).

As shown in Figure 3, the acoustic phonon scattering rates of the hole decrease significantly with the increasing stress, and their sequence is strained Si/(111)Si$_{1-x}$Ge$_{x}$ $>$ strained Si/(101)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(111)Si $>$ strained Si/(001)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(101)Si $>$ strained Si$_{1-x}$Ge$_{x}$/(001)Si when Ge fraction is about 0.2.

As shown from Figure 4, the hole non-polar optical phonon scattering declines dramatically with the increasing stress in Si-based strained materials, and the sequence is strained Si/(111)Si$_{1-x}$Ge$_{x}$ $>$ strained Si/(101)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(111)Si $>$ strained Si/(001)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(101)Si $>$ strained Si$_{1-x}$Ge$_{x}$/(001)Si when Ge fraction is about 0.2.

Alloy disorder scattering rates in strained Si$_{1-x}$Ge$_{x}$ materials will increase with the increasing stress from Figure~5, and the order is strained Si$_{1-x}$Ge$_{x}$/(111)Si $>$ strained Si$_{1-x}$Ge$_{x}$/(101)Si $>$ strained Si$_{1-x}$Ge$_{x}$/(001)Si when Ge fraction is about 0.2.

The order is strained Si/(111)Si$_{1-x}$Ge$_{x}$ $>$ strained Si/(101)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(111)Si $>$ strained Si$_{1-x}$Ge$_{x}$/(101)Si $>$ strained Si/(001)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(001)Si (Ge fraction is about 0.2).

From Figure 6 we can see the relationship between the total scattering rate and the stress in Si-based strained materials. The total hole scattering rates in Si-based strained materials decrease with the increasing stress.

Now we discuss the contribution percent of each scattering rate to the total scattering rate in Si-based strained materials. It is found that the decrease of the total hole scattering rate is caused mainly by decreasing the acoustic phonon scattering rate under strain.

In order to verify the correction of our model, this paper calculated the averaged hole mobilities on the basis of our model. The expressions are as below:

$$\mu =q<\tau >/m_{\rm c},$$

(8)

$$\begin{split} 1/\mu _{\rm p} {}& =\frac{q}{m_{\rm c}} \left[ \frac{N_{\rm i} e^4}{16\pi (2m^\ast)^{1/2}(\varepsilon _0 \varepsilon)^2E^{3/2}} \ln \frac{12m^\ast K_{\rm B}^2 T^2\varepsilon _0 \varepsilon }{e^2\hbar^2N_{\rm i}}\right. \\[2mm]& +\frac{m^{\ast 3/2}\Xi ^2K_{\rm B} T(2E)^{1/2}}{\pi \hbar ^4c_l} +\frac{D_0 ^2m^{\ast 3/2}}{2^{1/2}\pi \hbar ^3\rho \omega _0 } \\[2mm]& \left.\left. \left( {n_{\rm op} +\frac{1}{2}\mp \frac{1}{2}} \right) \left( E\pm \hbar \omega _0 \right)^{1/2} \right]^{-1} \right|_{E=1.5K_{\rm B} T}, \\& T=300\, {\rm K}. \\ \end{split}$$

(9)

Figure 7 shows the average hole mobilities increase with the increasing stress. Note that the result for the hole mobility of unstrained Si obtained by our model (when Ge fraction is 0) is consistent with the literature results, which indirectly verify our model.

4. Conclusions

Based on Fermi's golden rule and the theory of Boltzmann collision term approximation, the quantitative results for the hole scattering rates related to the stress in Si-based strained materials were studied in-depth, including the ionized impurity scattering rate, acoustic phonon scattering rate, non-polar optical phonon scattering rate, alloy disorder scattering and the total scattering rate.

(1) The hole ionized impurity scattering rates in Si-based strained materials (doping 10$^{19}$ cm$^{-3})$ increase with increasing stress. and their order is strained Si$_{1-x}$Ge$_{x}$/(001)Si $>$ strained Si$_{1-x}$Ge$_{x}$/(101)Si $>$ strained Si/(001)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(111)Si $>$ strained Si/(101)Si$_{1-x}$Ge$_{x}$ $>$ strained Si/(111)Si$_{1-x}$Ge$_{x}$ when Ge fraction is about 0.2.

(2) The acoustic phonon scattering rates of the hole decrease significantly with the increasing stress, and their sequence is strained Si/(111)Si$_{1-x}$Ge$_{x}$ $>$ strained Si/(101)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(111)Si $>$ strained Si/(001)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(101)Si $>$ strained Si$_{1-x}$Ge$_{x}$/(001)Si when Ge fraction is about 0.2.

(3) The hole non-polar optical phonon scattering declines dramatically with the increasing stress in Si-based strained materials, and the sequence is strained Si/(111)Si$_{1-x}$Ge$_{x}$ $>$ strained Si/(101)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(111)Si $>$ strained Si/(001)Si$_{1-x}$Ge$_{x}$ $>$ strained Si$_{1-x}$Ge$_{x}$/(101)Si $>$ strained Si$_{1-x}$Ge$_{x}$/(001)Si when Ge fraction is about 0.2.

(4) Alloy disorder scattering rates in strained Si$_{1-x}$Ge$_{x}$ materials will increase with the increasing stress from Figure 5, and the order is strained Si$_{1-x}$Ge$_{x}$/(111)Si $>$ strained Si$_{1-x}$Ge$_{x}$/(101)Si $>$ strained Si$_{1-x}$Ge$_{x}$/(001)Si when Ge fraction is about 0.2.

(5) The total hole scattering rates in Si-based strained materials decrease with the increasing stress. The decrease of the total hole scattering rate is caused mainly by decreasing the acoustic phonon scattering rate under strain.