Microelectronics Reliability

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1 Microelectronics Reliability xxx (2) xxx xxx Contents lists available at ScienceDirect Microelectronics Reliability journal homepage: Modeling of enhancement factor of hole mobility for strained silicon under low stress intensity Hongxia Liu, Shulong Wang, Yue Hao Key Lab. of Ministry of Education for Wide Band-Gap Semiconductor Materials and Devices, School of Microelectronics, Xidian University, Xi an 77, China article info abstract Article history: Received 9 August 2 Received in revised form 6 December 2 Accepted 6 December 2 Available online xxxx In order to quantitatively characterize the enhancement of hole mobility of strained silicon under different stress intensity conditions, changes of hole effective mass should be studied. In the paper, strained silicon under in-plane biaxially tensile strain based on ( ) substrate and longitudinal uniaxially compressive strain along h i are investigated thoroughly. By solving the Hamiltonian of valence band using KP model, we can obtain the relationship of density of state effective mass (m DOS ), conductivity effective mass (m C ) and splitting energy in valence band energy with stress intensity for both biaxially tensile strain and uniaxially compressive strain. For the stress intensity less than GPa, the paper presents the models of enhancement factor of hole mobility under the biaxially tensile strain and uniaxially compressive strain. The results show that biaxially tensile strain of silicon cannot enhance hole mobility under low stress intensity, while uniaxially compressive stress of silicon can enhance hole mobility greatly. Ó 2 Elsevier Ltd. All rights reserved.. Introduction Geometric scaling of Si CMOS devices has provided much performance improvement for each new technology generation. However, as CMOS technology progresses into nanometer regime, sustaining the expected performance improvements has become an increasingly complex, expensive problem due to some fundamental physical limits. Several other ways to improve the device performance have been presented. Strained silicon technology, importing proper strain in MOSFET channel to improve the carrier mobility, enhancing the drive ability, and increasing the operation speed, is one of the most advanced technologies in nowadays microelectronics. It has the advantages of low cost, high efficiency and compatible processing technology compared with the traditional silicon technology [ 5]. Investigation of strained silicon structure mainly focused on fabricating strained silicon layer on the relaxed SiGe substrate. The theory of electron mobility improvement has been known in this kind of device structure [6 8]. However, for hole, it is much more complicated because of the anisotropy and non-polarization in the valence band. Using experienced Pseudopotential method, Giles calculated the impact of uniaxially compressive stress on the valence band, and analyzed the reasons of improvement of electron mobility [9]. The impact of scattering on carrier mobility for strained silicon with different surface orientation was investigated by Pham et al. [] and Wang et al. []. Carrier mobility is Corresponding author. Tel.: ; fax: address: hxliu@mail.xidian.edu.cn (H. Liu). significantly influenced by the effective mass, density of state and scattering. However, up to now, the intrinsic theory of the impact of effective mass on hole mobility is not clear, and there is very little quantitative characterization in the variation of hole mobility with the strain intensity. The hole mobility enhancement is mainly determined by the decreased scattering rate and effective mass due to the strain. In the feasible strain range (strain intensity is less than GPa), band splitting is small, and scattering rate only decreases a little, so the hole mobility is mostly influenced by the hole effective mass. In our work, the change of the hole effective mass and its effect on hole mobility under the lower strain were investigated. The paper studies the valence band structure of the in-plane biaxially tensile strained silicon based on ( ) substrate and longitudinal uniaxially compressive strained silicon along hi using KP model [2]. The relationship of density of state (DOS) effective mass, conductivity effective mass and band splitting energy is presented. The models of hole mobility enhancement factor under biaxially tensile strain and uniaxially compressive strain are proposed. 2. Mobility theory and valence model The total Hamiltonian in strained silicon is shown in: H ¼ H KP þ H s:o: þ H strain where H KP is the Hamiltonian taking only heavy hole and light hole in consideration, H s.o. is spin orbit coupling perturbations Hamiltonian, H strain is strain-determined Hamiltonian. ðþ /$ - see front matter Ó 2 Elsevier Ltd. All rights reserved. doi:.6/j.microrel.2.2.5

2 2 H. Liu et al. / Microelectronics Reliability xxx (2) xxx xxx H KP ¼ H 33 x; y; z " 33 H x; y; z " where 2 Lk 2 x þ Mðk2 y þ k2 z Þ Nk 3 xk y Nk z k x x H ¼ h2 Nk 2m x k y Lk 2 y þ Mðk2 z þ k2 x Þ Nk 6 yk z y Nk z k x Nk y k z Lk 2 z þ Mðk2 x þ k2 y Þ z ð3þ 2 3 i x " i i y " H s:o: ¼ D i z " 3 i x # i i 5 y # i z # jx " i, jy " i, jz " i, jx # i, jy # i, jz # i are representations, x, y, z are the three orbits related with the valence band edge, and " and ; are the raising operator and lowering operator respectively. Parameters L, M, N are related to the Luttinger parameters [3], D is the spin orbit splitting energy. The energy band can be calculated by the eigenvalue of H KP. After calculating the energy band using KP method, the effective mass of each direction in k-space can be obtained, as shown in:! m ¼ d 2 E ð5þ h 2 dk 2 k¼ Values of parameters used in calculation are listed in Table. For unstrained and strained silicon, the hole mobility can be expressed as follows: l ¼ p LHu LH þ p HH u HH p LH þ p HH u LH ¼ qs ; u HH ¼ qs where p LH is the light hole concentration, p HH is the heavy hole concentration. u LH is the light hole mobility, u HH is the heavy hole mobility. s is the momentum relaxed time of hole, q is the unit electric charge, is the conductivity effective mass of light hole, is the conductivity effective mass of heavy hole. The hole concentration at the valence band edge is shown in: p ¼ 2 ð2pm p k TÞ 3=2 exp E v E f ð8þ h 3 k T By solving Eqs. (7) and (8), the relationship between mobility and effective mass is shown in Eqs. (9) and () for unstrained silicon and strained silicon respectively. ðm ph Þ3=2 ðm pl Þ3=2 l / þ ðm ph Þ3=2 þðm pl Þ3=2 ðm ph Þ3=2 þðm pl Þ3=2 ðm ph Þ3=2 l / ðm ph Þ3=2 þðm pl Þ3=2 exp DE split k T ðm pl Þ3=2 exp DE split k T þ ðm ph Þ3=2 þðm pl Þ3=2 exp DE split k T ð2þ ð4þ ð6þ ð7þ ð9þ ðþ where l and l are the mobility of unstrained silicon and mobility of strained silicon respectively, m ph, m pl are the density of states effective masses of heavy hole and light hole respectively,, are the conductivity effective masses of light hole and heavy hole respectively, DE split is the splitting energy of energy levels due to strain. Enhancement of the hole mobility is mainly determined by reduction of the scattering rate and decrease of effective mass. However, in the feasible strain range, the strain intensity is less than GPa. The band splitting is small, scattering rate only reduces a little. The change of s in Eq. (7) can be ignored, and the hole mobility is mainly affected by the effective mass. As a result, given density of state effective mass, conductivity effective mass, and splitting energy of energy levels, the enhancement factor f of mobility can be obtained. f ¼ l l ðþ It must be mentioned that similar method was used to calculate the mobility. However the splitting between heavy hole band and light hole band was neglected when calculated the mobility strain intensity is less than GPa. Also, the density of state effective mass in the mobility model was taken as a constant in Ref. [8]. However, the anisotropy in valence band edge makes it difficult to solve the Fermi distributing function of every energy level for the density of state effective mass using the traditional method. Limits calculation with numerical method is adopted in the work [4]. By using this method and accurate calculation, the difference between calculation results is less than % [4]. The calculation expression is shown in: ðm p Þ3=2 E =2 th 2 =2 p 2 h 3 ¼ ð2pþ 3 Z 2p Z p d/ dh sinðhþk 2 a ðe=2; h; aðh; /Þ k¼ka ð2þ where E th ¼ð3=2Þk B T, k B is Boltzmann constant, h ¼ h=2p, h is Planck constant, m p is density of state effective mass, k a represents the wave vector at (h, u) ofa energy band with energy of E th. The m DOS of heavy hole (m ph ) and the m DOS of light hole (m pl )in biaxial strained silicon are shown in Fig. respectively. It must be pointed out that the result in Fig. is transformed from the result in Fig. 2a for the reason that the m DOS in Ref. [4] is given in the form of a k /a.the stress in Fig. is more than GPa and the result agrees well with Ref. [4]. The difference of m pl increases a little with increasing a k /a. The difference is caused by different parameters used in the calculation of the energy band which are not completely given in Ref. [4] and different densities of the energy pot used in the calculation of the effective mass. When the stress is bellow GPa, the difference is so small that it does not influence following research work. Calculation of m DOS includes the energy band distributions in all directions and it verifies the accuracy of the energy band. And the model of m DOS can be extended to uniaxial strained silicon in the following calculation. 3. Simulation results and discussion Strain intensity is used to represent strain level in the model, which makes it possible to apply the model to the strain introduced by different methods. The variations of density of state effective mass and conductivity effective mass with strain intensity Table Values of parameters in KP model. Material Lattice constant (nm) L (h 2 /2m ) M (h 2 /2m ) N (h 2 /2m ) D (ev) Si Ge

3 H. Liu et al. / Microelectronics Reliability xxx (2) xxx xxx Spilt Energy/eV Fig.. Variations of m DOS of heavy and light holes with a k /a for biaxial strained silicon. under biaxially tensile strain are calculated, which are shown in Fig. 2a and b respectively. Fig. 2a shows that the density of state effective mass of heavy hole decreases and density of state effective mass of light hole increases as increasing strain intensity under biaxial strain. It means the first sub-band with heavy hole characteristics tends to be the one with light hole characteristics. The density of state effective mass of light hole increases less under larger tensile strain. Fig. 2b shows that the conductivity effective masses of both the heavy hole and the light hole increase slightly with the increase of the tensile strain under very low strain intensity, because the strain causes slightly distortion of valence band structure. Fig. 3 shows the variation of the splitting energy of the heavy and light holes with strain intensity under biaxially tensile strain. The splitting energy goes up linearly with the increase of the strain intensity, which can be explained using the symmetry theory. The tensile strain introduced by the epitaxy on the relaxed Si X Ge X layer based on ( ) substrate changes the cubic crystal relaxed silicon to tetragonal class strained silicon. The reduction of symmetry partially eliminates the degeneracy of the band structure and results the splitting of energy band. The larger the strain, the more the symmetry reduces, which produces larger splitting. The impact of the change of valence band on hole mobility under biaxially tensile strain can be demonstrated as follows. Firstly, the Fig. 3. Variation of splitting energy of heavy and light holes with strain intensity under biaxially tensile strain. biaxially tensile strain reduces the symmetry of the crystal, and eliminates the degeneracy of valence band edge, which contributes to the splitting of valence band. It means the energy band of heavy hole goes down and the energy band of light hole goes up. The first sub-band presents the characteristic of the light hole gradually instead of the heavy hole, therefore the hole mobility increases. On the other hand, the conductivity effective masses of heavy and light holes increase with increasing strain, which prevent the improvement of hole mobility. The improvement of hole mobility depends on the coupling effects of two impact factors. In addition, suppression of strain on scattering increases with larger splitting between the heavy and light holes energy bands, which is helpful to improve the hole mobility. However, only lower strain is taken account in this work, so the impact can be ignored. It is concluded that the hole mobility cannot be improved obviously and there is even a possibility that the mobility will degrade under lower tensile strain. The hole mobility enhances obviously under higher tensile strain. Combining the results shown in Figs. 2 and 3 with Eq. (), the enhancement factor of hole mobility under biaxially tensile strain can be obtained, as shown in Fig. 4. The hole mobility decreases as the strain increases, and the tendency diminishes after a certain strain value, which is consistent with the change of hole conductivity effective mass shown in Fig.. Using the data in Fig. 4, the mobility enhancement factor can be obtained by polynomial fitting method, as shown in: (a). m ph (b).3.8 m pl.275 m DOS /m.6 m C /m Fig. 2. Variations of effective mass of hole with strain intensity under biaxially tensile strain. (a) Density of state effective mass, (b) conductivity effective mass.

4 4 H. Liu et al. / Microelectronics Reliability xxx (2) xxx xxx f ¼ :45x 3 þ :45x 2 :78x þ :9635 ð < x < Þ ð3þ where f is enhancement factor of mobility, x is the tensile strain intensity, the unit is GPa. Because scattering can be ignored for lower strain, the equation can be used to determine the enhancement factor of hole mobility for the strain being less than GPa. The variation of the density of state effective mass and the conductivity effective mass with strain intensity under uniaxially compressive strain is shown in Fig. 5a and b respectively. Similar to biaxially tensile strain, uniaxially compressive strain reduces the symmetry of valence band. The state effective mass of heavy hole decreases and the state effective mass of light hole increases with increasing strain intensity. Fig. 5a shows that the density of state effective mass of light hole begins to decrease when the strain is larger than 3 GPa. Different from biaxially tensile strain, the uniaxially compressive strain cannot only reduce the symmetry of valence band, it can also provide large valence band distortion. The distortion causes the change of conductivity effective mass of hole. The curve of conductivity effective mass of heavy enhancment factor stress/gpa Fig. 4. Variation of enhancement factor of hole mobility with strain intensity under biaxially tensile strain..8 hole crosses the curve of conductivity effective mass of light hole. The conductivity effective mass of the heavy hole band including more holes decreases with increasing strain intensity, which is even less than that of the light hole band. These changes mentioned above improve the hole mobility significantly when the heavy hole band contains more holes. Fig. 6 shows the variation of the splitting energy of heavy and light holes with the strain intensity under uniaxially compressive strain. Similar to the biaxially tensile strain case, the symmetry of hole energy band diminishes as increasing strain intensity, which contributes to the increasing splitting energy. It also causes the redistribution of holes in the heavy hole band and light hole band. More and more holes shift from the heavy hole band to the light hole band. The conductivity effective mass of light hole increases with increasing strain intensity, as shown in Fig. 5b. The two factors gradually counteract the good effect of the reduced conductivity effective mass of heavy hole on the hole mobility for high strain intensity. Using the results of the effective mass and the splitting energy shown in Figs. 5 and 6 to solve Eq. (), one can obtain the variation of the hole mobility enhancement factor with the uniaxially compressive strain intensity, which is shown in Fig. 7. The hole mobility increases at first, and then tends to be saturated with increasing strain intensity, which is consistent with the above discussion result. Using the data in Fig. 7, the mobility enhancement factor can be obtained using polynomial fitting method, as shown in: f ¼ :446x 3 :47x 2 þ :487x þ :742 ð < x < Þ ð4þ where f represents enhancement factor of hole mobility, x represents strain intensity. It can be concluded that under uniaxially compressive strain, the hole mobility increases rapidly at first, and then tends to be stable when increasing strain intensity. It agrees well with Refs. [5,6]. Fig. 8 shows a comparison study of the model presented in the paper and Ref. [8]. One can see that the results calculated by the model agree well with the reference when the stress intensity less than GPa. It can also verify the accurate of density of state effective mass, conductor effective mass and the energy band calculated in the paper. Above analysis results can be applied in the device design. When the stress intensity is near GPa, the enhancement factor in reference is a little bigger than the result presented in the (a).9.8 m ph m pl (b).2. m DOS /m m C /m Fig. 5. Variation of effective mass of hole with strain intensity under uniaxially compressive strain. (a) Density of state effective mass, (b) conductivity effective mass.

5 H. Liu et al. / Microelectronics Reliability xxx (2) xxx xxx 5 Split Energy/eV Fig. 6. Variation of splitting energy of heavy and light hole with strain intensity under uniaxially compressive strain. enhancment factor stress/gpa Fig. 7. Variation of enhancement factor of hole mobility with strain intensity under uniaxially compressive strain. paper. When the stress intensity is bigger than GPa, the splitting energy of heavy and light hole increase and the scattering reduces. At the very beginning of the curve, there is a little difference between the calculation results presented in the paper and the reference. It is caused by the model of conductor effective mass. There are different models of conductor effective mass in strained silicon especially when the stress is very small. And the model used in the reference has not been presented. 4. Conclusion Using KP theory, the model of valence band structure for strained silicon is proposed, and the Hamiltonian is presented. Under both biaxially tensile strain and uniaxially compressive strain, the effects of the strain intensity on density of state effective mass of hole, conductivity effective mass of hole, and splitting energy of valence band are discussed in the paper. For both biaxially tensile strain and uniaxially compressive strain, the density of state effective mass of heavy hole decreases and the light hole density of state effective mass increases with increasing stress intensity. The valence band splitting energy increases linearly with increasing stress intensity. The effects of these factors on hole mobility are further investigated, and the intrinsic mechanism of the influence of effective mass on hole mobility is discussed. When the strain intensity is less than GPa, the models of enhancement factor of hole mobility under biaxially tensile strain and uniaxially compressive strain are presented. Under the lower strain intensity, scattering rate influences the mobility slightly, and the biaxially tensile strain degenerates the hole mobility instead of improving it. Under uniaxially compressive strain, the high mobility improvement can be obtained. Lower uniaxially compressive strain can diminish the symmetry of the valence band, and cause bigger valence band distortion, which reduces the conductivity effective mass of hole greatly. It is the main reason why the hole mobility can be improved significantly under lower uniaxially compressive strain. Acknowledgments This work was supported in part by the Project of National Natural Science Foundation of China (Grant Nos , 69365), in part by the Cultivation Fund of the Key Scientific and Technical Innovation Project, Ministry of Education of China Program (Grant No. 7883). References [] Watling JR, Yang L, Borici M. Solid-State Electron 24;48:337. [2] Song JJ, Zhang HM, Hu HY, Dai XY, Xuan RX. Chin Phys 27;6:3827. [3] Zhang ZF, Zhang HM, Hu HY. Acta Phys Sinica 29;58:4948. [4] Chattopadhyay S, Driscoll LD, Kwa SK. Solid-State Electron 24;48:47. [5] Smirnov S, Kosina H. Solid-State Electron 24;48:325. [6] Gu WY, Liang RR, Zhang K, Xu J. J Semiconduct 28;29:893. [7] Shima M, Ueno T, Kumise T. VLSI technology digest; 22. p. 94. [8] Sun Y, Thompson SE, Nishida T. J Appl Phys 27;:453. [9] Giles MD, Armstrong M, Auth C. Symposium on VLSI technology; 24. p. 8. [] Pham AT, Jungemann C, Meinerzhagen B. Solid-State Electron 28;52:437. [] Wang E, Matagne P, Shifren S. Proc IEDM 24:47. [2] Sverdlov V, Karlowatz G, Dhar ST. Solid-State Electron 28;52:563. [3] Hinckley JM, Singh J. J Appl Phys 994;76:492. [4] Fischetti MV, Laux SE. J Appl Phys 996;8:2234. [5] Guillaume T, Mouis M. Solid-State Electron 26;5:7. [6] Thompson SE, Sun GY, Choi YS. IEEE Trans Electron Dev 26;53:. Fig. 8. Enhancement factor of hole mobility for biaxial strain silicon, uniaxial strain silicon and the data in Ref. [8].