Research Article E-k Relation of Valence Band in Arbitrary Orientation/Typical Plane Uniaxially Strained

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1 Advances in Condensed Matter Physics Volume 14, Article ID 68633, 9 pages Research Article E-k Relation of Valence Band in Arbitrary Orientation/Typical Plane Uniaially Strained Zhang Chao, Xu Da-Qing, Liu Shu-Lin, and Liu Ning-Zhuang School of Electrical and Control Engineering, Xi an University of Science and Technology, Xi an 7154, China Correspondence should be addressed to Zhang Chao; kkyy6677 1@16.com Received 13 January 14; Revised 6 February 14; Accepted 18 March 14; Published 16 April 14 Academic Editor: Jörg Fink Copyright 14 Zhang Chao et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Uniaial strain technology is an effective way to improve the performance of the small size CMOS devices, by which carrier mobility can be enhanced. The E-k relation of the valence band in uniaially strained Si is the theoretical basis for understanding and enhancing hole mobility. The solving procedure of the relation and its analytic epression were still lacking, and the compressive results of the valence band parameters in uniaially strained Si were not found in the references. So, the E-k relation has been derived by taking strained Hamiltonian perturbationinto account. And then the valence band parameters were obtained, including the energy levels at Γ point, the splitting energy, and hole effective masses. Our analytic models and quantized results will provide significant theoretical references for the understanding of the strained materials physics and its design. 1. Introduction Uniaially strained Si technique is the preferred option to the design and application for the high-speed and highperformance nanoscale CMOS devices and circuits, which can enhance significantly the carrier mobility [1 7]. It is theoretically essential to study the E-k relation of valence band in uniaially strained Si for the understanding and improvement of hole mobility enhancement. Although wide reports [8 13] aboutthevalencebandof uniaially strained Si can be found, there still have been some questions to discuss carefully. (1) The E-k relation is key to obtain the physical parameters of valence band. However, reports about how to derive the relation and what its analytical epression is are not found. () The cell structures and physical properties are different when the uniaial stress is applied on Si materials along different orientation/different planes. So, it is very important to give the valence band parameters under different strained situation for the device design. Only some valence band structures in uniaially strained Si are reported and overall analysis is still needed. Inviewofthose,weaimtodiscusshowtosolvethe eigenvalue matri of the valence band structure in uniaially strained Si and establish the analytical E-k model for the valence band. Moreover, the valence band parameters inarbitrary orientation/typical plane uniaially strained Si will be studied, including the energy levels at Γ point, the splitting energy, and the hole effective mass. Our analytic models and quantized results will provide significant theoretical references for the understanding of the strained materials physics and its design.. The E-k Relations.1. Strain Tensor. Strain tensor model is the basis of the calculation of the deformation potential energy in uniaially strained Si materials [14, 15]. The strain tensors for arbitrary orientation/(1), (11), and (111) planes uniaially strained Si are, respectively, shown as follows: ε = cos θc 11 + cos θc 1 c 11 +c 11c 1 c 1 c 1 ε zz = c11 +c 11c 1 c1 T, T, ε yy = sin θc 11 cos θc 1 c11 +c 11c 1 c1 T, ε y = sin θ T, c 44

2 Advances in Condensed Matter Physics ε z =ε yz = ε = cos θc 11 +(1 4sin θ) c 1 3(c11 +c 11c 1 c1 ) T, ε yy = sin θc 11 (3 4sin θ) c 1 3(c11 +c 11c 1 c1 ) T, ε zz = 1 3(c 11 +c 1 ) T, ε y = 1 3 ε z = cos θ 3 T, ε c yz = 44 3 sin θ T, c 44 ε = cos θc 11 +(1 4sin θ) c 1 3 (c 11 +c 11c 1 c 1 ) T, sin θ T c 44 ε yy = sin θc 11 (3 4sin θ) c 1 3(c11 +c 11c 1 c1 ) T, ε zz = 1 3(c 11 +c 1 ) T, ε y = 1 3 ε z = cos θ 3 T, ε c yz = 44 3 sin θ T, c 44 sin θ T, c 44 where c 11, c 1,andc 44 are the elastic stiffness coefficients of Si materials; T is the applied uniaial stress; the azimuth angle θ is shown in Figure 1... Strained kp Hamilton. The degeneracy case of valence band in unstrained Si is Γ 5 threefold degenerate states without the coupling of spin-orbit taken into consideration. And the corresponding kp Hamilton can be written as follows: (1) H = [ Lk +M(k y +k z ) Nk k y Nk z k Nk k y Lk y +M(k z +k ) Nk ] yk z. () [ Nk z k Nk y k z Lk z +M(k +k y ) ] Theapplieduniaialstresswillleadtothedisplacement of wave vector k and the corresponding variation can be described by the strained tensors in Section.1. And thus the matri for the strained Hamilton is similar to epression (): H = [ lε +m(ε yy +ε zz ) nε y nε z nε y lε yy +m(ε +ε zz ) nε yz ], (3) [ nε z nε yz lε zz +m(ε yy +ε ) ] where the values for L, M, N, l, m, andn can be found in [17, 18]. In order to obtain the finerband structures, we must take the coupling of spin-orbit into consideration. And we need to change () and(3) totheepressionforthedoublegroup firstly: H 3 3 H KP = [ 3 3 H ] H 3 3, H Str = [ 3 3 H ] [ ] [ ]. (4) Thenwealsotakethefollowingspin-orbitHamilton (H SO ) into consideration [17]. Solving the eigenvalues of the matri (H KP +H Str +H SO ) and the results we desired will be obtained. The final E-k relationsofthevalencebandin uniaially strained Si are shown as follows: where a ij (i, j = 1,, 3) is the element of the matri H +H : Q= (p 3q), p = Δ (a a +a 33 ), R= (p3 9pq+7r), Θ = cos 1 ( R 54 Q ), 3 q=a 11 a +a a 33 +a 33 a 11 a 1 a 13 a 3 ( Δ 3 )(a 11 +a +a 33 ) r=a 11 a 3 +a a 13 +a 33a 1 a 11a a 33 a 1 a 3 a 13 +( Δ 3 )(a 11a +a a 33 +a 33 a 11 a 1 a 13 a 3 ). (6) V = Q cos (Θ 3 ) p 3, V V = Qcos (Θ π) p 3 3, = Qcos (Θ+π) p 3 3, (5) 3. Results and Discussion 3.1.Energy Levels at Γ Point. For unstrained Si, the maimum of valence band energy level is at the Γ point; that is,

3 Advances in Condensed Matter Physics 3 z z z z θ y T (1) y T z (11) T z θ y y θ y (111) y Figure 1: Schematic description of the applied uniaial stress along arbitrary orientation/(1), (11), and (111) (1) (1) θ= (a) θ= (b) θ=π/ (c) T=+1GPa (tensile stress) (d) T= 1GPa (compressive stress) Figure : The levels at Γ point versus stress and angle in uniaially strained Si (1).

4 4 Advances in Condensed Matter (11) (11) θ= (a) θ= (b) θ=π/ (c) T=+1GPa (tensile stress) (d) T= 1GPa (compressive stress) Figure 3: The levels at Γ point versus stress and angle in uniaially strained Si (11). k = (,, ). The relations between the energy levels of the first valence band ( Heavy Hole Band, HH ), the second valenceband( LightHoleBand, LH ),andthethirdvalence band ( Spin-Orbit Coupling Band, SO ) and the stress can be given by (5). The energy levels at the Γ point of the HH, LH, and SO in uniaially strained Si (1), (11), and (111) are shown in Figures, 3, and 4, respectively. As indicated in the figures, splitting between the first valence band and the second valence band in all the strained Si occurs and the degeneracy at the top of the valence band lifts, which are obviously different from one of the unstrained Si. Under the tensile strain, the energy levels of the first and the second bands increase with the increasing stress. The energy level of the first band also increases with the increasing compressive stress while the level of the second band decreases with the increasing compressive stress. Moreover, the change in the energy levels with the angle of applied stress can also be seen in Figures, 3,and Splitting Energy. The splitting energy is closely related to the hole distribution in strained Si, which is the basis to establish the model of the density of states in valence band. The relationship between the splitting energy and the magnitude of the stress T and the direction of the applied stress θ in uniaially strained Si (1), (11), and (111) are shown in Figure 5. Ascanbeseenfromthefigure,the function relations between the splitting energy and the compressivetressarenearlylinear.andtheslopeofthe

5 Advances in Condensed Matter Physics (111) (111) θ=π/ (a) θ= (b) θ=π/ (c) T=+1GPa (tensile stress) (d) T= 1GPa (compressive stress) Figure 4: The levels at Γ point versus stress and angle in uniaially strained Si (111). relation curve is greatest when θ=π/4. The splitting energies increase with the increasing tensile stress and they tend to be gradual when the magnitude of the stress is beyond 1 GPa. Moreover, the effect of the applied stress θ on the splitting energy can be neglected under tensile stress, which can be indicated from the nearly coincide in the curves for θ =, π/,andθ=π/4. In summary, the splitting energy under compressive strain is much greater than the one under tensile strain. That is, the change in splitting energy for the compressive case is more sensitive. This is one of the most important reasons why to choose compressive film to improve the pmosperformance.ourresultsshownintherightpartof Figure 6 are consistent with the results from literature [16]is shown in the left part of Figure 6.Itiswithagreementinour resultshownintherightpartoffigure Directional Effective Mass. Equienergy surface can directly reflect the directional effective mass. The D and 3D equienergy surfaces of the first valence bands in [11]/(1), [11 1]/(11), and[1 1]/(111) uniaially compressive strained Si (1 GPa) are shown in Figures 7, 8, and9. The values of the D equienergy are 1, 5, 5, 75, and 1 mev, respectively, and the value of the 3D equienergy is 4 mev. For instance, [11]/(1) uniaially compressive strained Si (1 GPa) is shown in Figure 7. As can be seen, the protruding parts in equienergy surface of unstrained Si significantly shrink when [11]/(1) uniaially compressive stress is applied on the unstrained Si. And the shape of the equienergy surface of [11]/(1) uniaially compressive strained Si turnstobeellipsoidspherical. It is worthy to note that the anisotropy of the valence band in strained Si is more significant compared with the one in unstrained Si.For eample,[ 11] direction and [11] direction are equivalent directions in unstrained Si and their corresponding effective masses are also equal. However, their values are not coincident in [11]/(1) uniaially compressive strained Si, which can be seen from the D

6 6 Advances in Condensed Matter Physics θ θ=π/4.6.4 θ=,π/ (a) (1) plain θ θ=π/4.6 θ =, π/ (b) (11) plain θ=π/4 6 4 θ θ =, π/ (c) (111) plain Figure 5: The splitting energy versus stress and angle in uniaially strained Si. equienergy surface in Figure 7 (the long ais along the [ 11] direction and the minor ais along the [11] direction). In order to verify the accuracy of our results, the equienergy surface in [11]/(1) uniaially compressive strained Si reported from [16]isshowninFigure 1.Itisthe same as our results in Figure 7, whichprovesthevalidityof our model Physical Interpretation. Now we turn to discuss the reasonwhythevalencebandstructuresofsimaterialswill change under strain from two aspects. Strained Si has different valence band structure and hole mass from that of relaed Si. These may be understood by their thin film cell structures. The cubic cell structure of relaed Si goes triangular or tetragonal or monoclinic due

7 Advances in Condensed Matter Physics 7 1 Biaial stress E.1.9 Energy splitting (mev) LH ΔE LH-HH k HH SO Si: biaial tension Ge: uniaial compression Si: uniaial compression Stress (GPa) Figure 6: The splitting energy under [11]/(1) compressive strain from [16] and our result (a) 3D (E =4meV) (b) D (E =1, 5, 5, 75, and 1 mev) Figure 7: Equienergy surface in [11]/(1) uniaially compressive strained Si. totheuniaialstress.thelowersymmetryofcellstructure lifts the degeneracy of valence band edge and leads to the splitting of the first and the second valence band in uniaially strained Si. And hence, their mutual coupling force caused by their splitting and relative moving is different from what is in relaed Si. For this reason, the significant change in the valence band structures along different directions and the corresponding hole effective masses with the increasing stress occur. Second, the strain perturbation mies the (3/), (1/) state with the (1/), (1/) state and the (3/), (1/) state with the (1/), (1/) state at k =.Forthecaseof uniaial stress along <1>/(1) substrate orientation, no other strain induced miing occurs. The miing of states is clarified by considering a 9 rotation of each basis state aboutthesubstratenormal.theeffecton j, m j of a rotation through an angle φ about the ais of angular momentum quantization, is the multiply the state by an overall phase factor of e im jφ. Eigenstates of the strain perturbed system must retain the same relative phase between their constituent basisstatesunderallsymmetryoperations.onesuchsymmetry operation is a 9 rotation about the substrate normal. Only basis states with the same value of m j may be mied by the perturbation. The valence band edge, under tensile

8 8 Advances in Condensed Matter Physics (a) 3D (E =4meV) (b) D (E =1, 5, 5, 75, and 1 mev) Figure 8: Equienergy surface in [11 1]/(11) uniaially compressive strained Si (a) 3D (E =4meV) (b) D (E =1, 5, 5, 75, and 1 mev) Figure 9: Equienergy surface in [1 1]/(111) uniaially compressive strained Si. strain, is characterized by the (3/), ±(3/) states. Under compressivestrain,thebandedgeischaracterizedbyastrain dependent miture of (3/), ±(1/) and (1/), ±(1/) states, being dominantly (3/), ±(1/) for low values of strain. For the other cases, strain will mi the states j, m j in a nonsimple way. 4. Conclusions In this paper, we establish the models of strain tensor and strain Hamilton firstly and derive the E-k relation of the valence band in uniaially strained Si by kp method. Finally the valence band parameters were obtained, including the energy levels at Γ point, the splitting energy, and hole effective masses. The results show that (1) splitting between the first valence band and the second valence band in all the strained Si occurs and the degeneracy at the top of the valence band lifts, which are obviously different from one of the unstrained Si. () The splitting energy under compressive strain is much greater than the one under tensile strain and the change in splitting energy for the compressive case is more sensitive. (3) The anisotropy of the valence band in strained Si is more significant compared with the one in unstrained Si.

9 Advances in Condensed Matter Physics 9 k y. Energy: 1, 5, 5, 75 mev.. k Figure 1: Equienergy surface in [11]/(1) uniaially compressive strained Si (E =1, 5, 5, 75, and 1 mev) [16]. Our analytic models and quantized results will provide significant theoretical references for the understanding of the strained materials physics and its design. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. References [1] J. Kim and M. V. Fischetti, Electronic band structure calculations for biaially strained Si, Ge, and III-V semiconductors, Applied Physics, vol. 18, no. 1, Article ID 1371, 1. [] O. Weber, M. Takenaka, and S.-I. Takagi, Eperimental determination of shear stress induced electron mobility enhancements in Si and Biaially Strained-Si metal-oidesemiconductor field-effect transistors, Japanese Applied Physics,vol.49,no.7,5pages,1. [3] J.J.Song,H.M.Zhang,X.Y.Dian,R.X.Xuan,andH.Y.Hu, Density of state of strained Si in different crystal system, Acta Physica Sinica,vol.6,no.4,ArticleID4716,11. [4] J.-J. Song, H.-M. Zhang, H.-Y. Hu, X.-Y. Wang, and G.-Y. Wang, Hole scattering mechanism in tetragonal strained Si, Acta Physica Sinica,vol.61,no.5,ArticleID5734,1. [5] K. Uchida, A. Kinoshita, and M. Saitoh, Carrier transport in (11) nmosfets: subband structures, non-parabolicity, mobility characteristics, and uniaial stress engineering, in Proceedings of the International Electron Devices Meeting (IEDM 6), December 6. [6] J.-J. Song, H.-M. Zhang, X.-Y. Dai, H.-Y. Hu, and R.-X. Xuan, Band structure of strained Si/(111)Si 1 Ge : a first principles investigation, Acta Physica Sinica,vol.57,no.9,pp , 8. [7] Y.Tan,X.Li,L.Tian,andZ.Yu, Analyticalelectron-mobility model for arbitrarily stressed silicon, IEEE Transactions on Electron Devices,vol.55,no.6,pp ,8. [8] J. R. Courtesy, Mobility enhancement, IEEE Circuits & Magazine,vol.9,p.18,5. [9]E.Ungersboeck,V.Sverdlov,H.Kosina,andS.Selberherr, Electron inversion layer mobility enhancement by uniaial stress on (1) and (11) oriented MOSFETs, in Proceedings of the International Conference on Simulation of Semiconductor ProcessandDevices(SISPAD 6), pp , September 6. [1] S. E. Thompson, M. Armstrong, C. Auth et al., A 9-nm logic technology featuring strained-silicon, IEEE Transactions on Electron Devices,vol.51,no.11,pp ,4. [11] D. J. Paul, Si/SiGe heterostructures: from material and physics to devices and circuits, Semiconductor Science and Technology, vol. 19, no. 1, pp. R75 R18, 4. [1] S. Chakraborty, M. K. Bera, S. Bhattacharya, P. K. Bose, and C. K. Maiti, Determination of the valence band offset and minority carrier lifetime in Ge-rich layers on relaed-sige, Thin Solid Films,vol.54,no.1-,pp.73 76,6. [13] T. Guillaume and M. Mouis, Calculations of hole mass in [1 1 ]-uniaially strained silicon for the stress-engineering of p- MOS transistors, Solid-State Electronics, vol. 5, no. 4, pp , 6. [14] J.J.Song,H.M.Zhang,H.H.Hu,andQ.Fu, Calculationof band structure in (11)-biaially strained Si, Science in China G: Physics, Mechanics and Astronomy, vol.5,no.4,pp , 9. [15] H. Zhang, Y. Wang, and Z. Song, Absolute stabilization of singular systems with ferromagnetic hysteresis nonlinearity, Science China Information Sciences,vol.56,no.7,pp.1 14,13. [16] S. E. Thompson, G. Sun, K. Wu, J. Lim, and T. Nishida, Key differences for process-induced uniaial vs. substrate-induced biaial stressed Si and Ge channel MOSFETs, in Proceedings of the IEEE International Electron Devices Meeting (IEDM 4),pp. 1 4, 4. [17] J.-J. Song, H.-M. Zhang, H.-Y. Hu, X.-Y. Dai, and R.-X. Xuan, Determination of conduction band edge characteristics of strained Si/Si 1 Ge, Chinese Physics,vol.16,no.1,pp , 7. [18] J.J.Song,H.M.Zhang,H.Y.Hu,X.Y.Dai,andR.X.Xuan, Valence band structure of strained Si/(111)Si 1 Ge, Science China: Physics, Mechanics and Astronomy,vol.53,no.3,pp , 1.

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