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J. Semicond. > 2015, Volume 36 > Issue 7 > 072003

SEMICONDUCTOR PHYSICS

Scattering mechanism of hole in (001), (101), (111) biaxially-strained Si and Si1-xGex materials

Lixia Zhao1, 2, Chao Yang2, He Zhu2 and Jianjun Song2,

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 Corresponding author: Jianjun Song, E-mail: jianjun_79_81@xidian.edu.cn

DOI: 10.1088/1674-4926/36/7/072003

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Abstract: Based on Fermi's golden rule and the theory of Boltzmann collision term approximation, the hole scattering mechanism related to stress and orientation in Si-based strained materials was studied in-depth. The results show that: (1) the total hole scattering rates in Si-based strained materials decrease obviously under strain; (2) the turn is Si/(111)Si1-xGex > Si/(101)Si1-xGex > Si1-xGex/(111)Si > Si1-xGex/(101)Si > Si/(001)Si1-xGex > Si1-xGex/(001)Si when Ge fraction is about 0.2; (3) the decreasing total hole scattering rates of in strained materials with the increasing stress is mainly caused the decreasing acoustic phonon scattering rate under strain. The theoretical conclusions obtained could provide important references for researching the hole mobility and the understanding of Si-based materials or other physical strained materials.

Key words: strainorientationscatteringhole

Strained technology can effectively improve the carrier mobility of Si-based materials (Si and Si1xGex), 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.

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

{E1V=2QcosΘ3p3,E2V=2QcosΘ2π3p3,E3V=2QcosΘ+2π3p3,Q=(p23q)/9,p=Δ(a11+a22+a33),R=(2p39pq+27r)/54,Θ=cos1(R/Q3),q=a11a22+a22a33+a33a11a212a213a223(2Δ/3)(a11+a22+a33),r=a11a223+a22a213+a33a212a11a22a332a12a23a13+(Δ/3)(a11a22+a22a33+a33a11a212a213a223),(1)
where aij (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 (md) in Si and Si1xGex strained materials can be obtained, which is the important parameter to calculate the hole scattering rate in Si-based strained materials[11].

md=[(ma)3/2hexp(ΔEV,Split/KBT)]2/3,(2)
where (ma)h and (ma)l are "heavy", "light" average effective mass respectively, ΔEV,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 Si1xGex: the ionize impurities scattering, the acoustic phonon scattering, the non-polar optical phonon scattering and the alloy disorder scattering (denoted by PII, Pac, Pin, Pad respectively). And based on the Fermi golden rule, models of each hole scattering rate in Si-based strained materials are derived.

PII=Nie416π(2m)1/2(ε0ε)2E3/2ln12mdK2BT2ε0εe22Ni,(3)
Pac=m3/2dΞ2KBT(2E)1/2π4cl,(4)
Pop=D20(md)3/221/2π3ρω0(nop+1212)(E±ω0)1/2,(5)
Pad=2m3/2dx(1x)(ΔE)2π4NE1/2,(6)
where Table 1 shows the physical meaning and specific values of the parameters.

Table  1.  Parameter values for calculation[13, 14].
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The total scattering rates of the Si-based strained materials is the sum of each scattering rate:

P=iPi=PII+Pac+Pin+Pad. (7)

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.

Figure  2.  Relationship between the ionized impurity scattering rate and stress in Si-based strained materials when doping is (a) 1 × 1017~cm3 and (b) 1 × 1019 cm3.

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 Si1xGex/(001)Si > strained Si1xGex/(101)Si > strained Si/(001)Si1xGex > strained Si1xGex/(111)Si > strained Si/(101)Si1xGex > strained Si/(111)Si1xGex when Ge fraction is about 0.2. Note that the order result is opposite when the doping concentration is 1019 cm3 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)Si1xGex > strained Si/(101)Si1xGex > strained Si1xGex/(111)Si > strained Si/(001)Si1xGex > strained Si1xGex/(101)Si > strained Si1xGex/(001)Si when Ge fraction is about 0.2.

Figure  3.  Relationship between the acoustic phonon scattering rates and the stress in Si-based strained materials.

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)Si1xGex > strained Si/(101)Si1xGex > strained Si1xGex/(111)Si > strained Si/(001)Si1xGex > strained Si1xGex/(101)Si > strained Si1xGex/(001)Si when Ge fraction is about 0.2.

Figure  4.  Relationship between the non-polar optical phonon scattering rate and the stress in Si-based strained materials.

Alloy disorder scattering rates in strained Si1xGex materials will increase with the increasing stress from Figure~5, and the order is strained Si1xGex/(111)Si > strained Si1xGex/(101)Si > strained Si1xGex/(001)Si when Ge fraction is about 0.2.

Figure  5.  Relationship between the alloy disorder scattering rate and the stress in strained Si1xGex materials.

The order is strained Si/(111)Si1xGex > strained Si/(101)Si1xGex > strained Si1xGex/(111)Si > strained Si1xGex/(101)Si > strained Si/(001)Si1xGex > strained Si1xGex/(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.

Figure  6.  Relationship between the total scattering rate and the stress in strained Si and Si1xGex materials.

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:

μ=q<τ>/mc, (8)
1/μp=qmc[Nie416π(2m)1/2(ε0ε)2E3/2ln12mK2BT2ε0εe22Ni+m3/2Ξ2KBT(2E)1/2π4cl+D20m3/221/2π3ρω0(nop+1212)(E±ω0)1/2]1|E=1.5KBT,T=300K. (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.

Figure  7.  Relationship between the averaged hole mobilities and the stress in Si-based strained materials.

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 1019 cm3) increase with increasing stress. and their order is strained Si1xGex/(001)Si > strained Si1xGex/(101)Si > strained Si/(001)Si1xGex > strained Si1xGex/(111)Si > strained Si/(101)Si1xGex > strained Si/(111)Si1xGex 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)Si1xGex > strained Si/(101)Si1xGex > strained Si1xGex/(111)Si > strained Si/(001)Si1xGex > strained Si1xGex/(101)Si > strained Si1xGex/(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)Si1xGex > strained Si/(101)Si1xGex > strained Si1xGex/(111)Si > strained Si/(001)Si1xGex > strained Si1xGex/(101)Si > strained Si1xGex/(001)Si when Ge fraction is about 0.2.

(4) Alloy disorder scattering rates in strained Si1xGex materials will increase with the increasing stress from Figure 5, and the order is strained Si1xGex/(111)Si > strained Si1xGex/(101)Si > strained Si1xGex/(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.



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Fig. 2.  Relationship between the ionized impurity scattering rate and stress in Si-based strained materials when doping is (a) 1 × 1017~cm3 and (b) 1 × 1019 cm3.

Fig. 3.  Relationship between the acoustic phonon scattering rates and the stress in Si-based strained materials.

Fig. 4.  Relationship between the non-polar optical phonon scattering rate and the stress in Si-based strained materials.

Fig. 5.  Relationship between the alloy disorder scattering rate and the stress in strained Si1xGex materials.

Fig. 6.  Relationship between the total scattering rate and the stress in strained Si and Si1xGex materials.

Fig. 7.  Relationship between the averaged hole mobilities and the stress in Si-based strained materials.

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Table 1.   Parameter values for calculation[13, 14].

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    Lixia Zhao, Chao Yang, He Zhu, Jianjun Song. Scattering mechanism of hole in (001), (101), (111) biaxially-strained Si and Si1-xGex materials[J]. Journal of Semiconductors, 2015, 36(7): 072003. doi: 10.1088/1674-4926/36/7/072003
    L X Zhao, C Yang, H Zhu, J J Song. Scattering mechanism of hole in (001), (101), (111) biaxially-strained Si and Si1-xGex materials[J]. J. Semicond., 2015, 36(7): 072003. doi: 10.1088/1674-4926/36/7/072003.
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    Received: 19 November 2014 Revised: Online: Published: 01 July 2015

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      Lixia Zhao, Chao Yang, He Zhu, Jianjun Song. Scattering mechanism of hole in (001), (101), (111) biaxially-strained Si and Si1-xGex materials[J]. Journal of Semiconductors, 2015, 36(7): 072003. doi: 10.1088/1674-4926/36/7/072003 ****L X Zhao, C Yang, H Zhu, J J Song. Scattering mechanism of hole in (001), (101), (111) biaxially-strained Si and Si1-xGex materials[J]. J. Semicond., 2015, 36(7): 072003. doi: 10.1088/1674-4926/36/7/072003.
      Citation:
      Lixia Zhao, Chao Yang, He Zhu, Jianjun Song. Scattering mechanism of hole in (001), (101), (111) biaxially-strained Si and Si1-xGex materials[J]. Journal of Semiconductors, 2015, 36(7): 072003. doi: 10.1088/1674-4926/36/7/072003 ****
      L X Zhao, C Yang, H Zhu, J J Song. Scattering mechanism of hole in (001), (101), (111) biaxially-strained Si and Si1-xGex materials[J]. J. Semicond., 2015, 36(7): 072003. doi: 10.1088/1674-4926/36/7/072003.

      Scattering mechanism of hole in (001), (101), (111) biaxially-strained Si and Si1-xGex materials

      DOI: 10.1088/1674-4926/36/7/072003
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      • Corresponding author: E-mail: jianjun_79_81@xidian.edu.cn
      • Received Date: 2014-11-19
      • Accepted Date: 2015-01-13
      • Published Date: 2015-01-25

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