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Electronic structures of the Si(001) thin film under 〈110〉- and 〈010〉-direction uniaxial tensile strains
Computer Physics Communications 180 (2009) 659–663
Contents lists available at ScienceDirect
Computer Physics Communications www.elsevier.com/locate/cpc
Electronic structures of the Si(001) thin film under 110- and 010-direction uniaxial tensile strains J.-Y. Lin, Y.-H. Tang, M.-H. Tsai ∗ Department of Physics, National Sun Yat-Sen University, Kaohsiung, 80424 Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 8 September 2008 Received in revised form 1 December 2008 Accepted 1 December 2008 Available online 6 December 2008 PACS: 73.63.-b 68.35.Gy Keywords: Nanoscale Si(001) thin film Strain
The electronic structures of the Si(001) ultra-thin film under various 110- and 010-direction uniaxial tensile strains have been calculated using the first-principles modified pseudofunction calculation method and a 20-layer single slab model. It can be inferred from calculated effective masses of electrons near the absolute conduction band minimum (CBM) that the 110-direction tensile strain induces enhancement and reduction of the mobility in parallel and perpendicular conduction channels, respectively. As for the 010-direction tensile strain, the effective mass results suggest that tensile strain induces reduction of the mobility in both parallel and perpendicular conduction channels. Under both 110- and 010direction strains, the band gap decreases and near-CBM density of states increases with strain, which suggests strain induced enhancement of thermally excited electron carrier density. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Over the last few decades, silicon has been the basis of manufacturing a majority of semiconductor devices. To meet the need of higher integration densities, speed, and efficiency, device dimensions have been scaled down to the extent where conventional Si approaches its fundamental limits. Strained-Si MOSFET has increasingly been the subject of many investigations in recent years [1–11]. The drive current has been improved due to strained-Si technologies [1–5]. Experimentally, biaxial and uniaxial strains can be achieved by a SiGe substrate and a process induced type of stress, respectively [10]. Previous works showed that an optimum combination of channel direction and the direction of the uniaxial strain enhances mobility [6–10]. The strain induced enhancement of mobility has often been explained by the splitting of the six-fold degenerate conduction band minimum (CBM) valleys of bulk Si crystal and a reduced phonon scattering rate [5–11], though the Si layers in these devices are ultra-thin films. Due to the larger surface to volume ratio of the ultrathin films, their electronic structures are expected to be different from those of thick films or bulk solid. However, the ultra-thin films in the practical devices are still too thick to be represented by nanoscale thin films, whose electronic structures can be obtained by first-principles calculations using slab models. In other words, there is still a lack of suitable structural model to real-
*
Corresponding author. Tel.: +886 7 525 3729. E-mail address: [email protected] (M.-H. Tsai).
0010-4655/$ – see front matter doi:10.1016/j.cpc.2008.12.001
©
2008 Elsevier B.V. All rights reserved.
istically obtain the electronic structures of these films. Nevertheless, in this study, an attempt has been made to calculate the electronic structures of nanoscale Si(001) ultra-thin films under 110- and 010-direction uniaxial tensile strains. It is hoped that based on the calculated trend of strain induced electronic structures, the origin of the enhancement of the mobility can be elucidated. 2. The structural model and the calculation method Si has a diamond structure with a space group of Fd3m. The Si(001) ultra-thin film is modeled by a nanometer-scale Si(001) thin film with 20 atomic layers. Uniaxial 110- and 010-direction tensile strains are simulated by adjusting the two-dimensional basis vectors a1 and a2 as shown in Fig. 1(a). Note that the x- and y-axes of the two-dimensional (001) thin film are in the 110¯ -directions, respectively, and that by cubic symmetry the and 110 010-direction is equivalent to the 100-direction. In this study, the periodicity of the two-dimensional square lattice of the strainfree Si(001) film, |a1 | = |a2 | = a, is determined by energy minimization to be 3.908 Å, which is 2% larger than that of a truncated Si(001) film from bulk Si with an experimental lattice constant of 5.43072 Å√ [12]. (The periodicity of the two-dimensional square lattice is 1/ 2 times the lattice constant of the Si crystal with a diamond structure.) Since semiconductors have a tendency to conserve average bond length [13], the atomic positions are adjusted to conserve the nearest neighbor Si–Si average bond length to be 2.393 Å for each strain. Since each surface atom of the Si(001)
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Fig. 1. Definition of the two-dimensional basis vectors and changes of the basis vectors corresponding to tensile strains along the 110- and 010-directions.
film has two dangling bonds, all surface atoms are attached with two hydrogen atoms to saturate their dangling bonds to simulate the Si atoms at the interfaces which are bonded with neighboring substrate- and capping-layer-atoms and do not have dangling bond. The first-principles calculation method used in this study is the modified pseudofunction (PSF) method [14,15]. In this method, the z-component of the Coulomb potential is calculated numerically by solving the Poisson equation, where the z-coordinate is perpendicular to the Si(001) film. The potential, which includes Coulomb and exchange–correlation potentials within the local density approximation (LDA) of Hedin and Lundqvist [16], was divided into spherical potentials within the muffin-tin spheres and a planewave expanded potential that extends throughout the film. The basis set used in the PSF method contains two sets of Bloch sums of muffin-tin orbitals with exponentially decaying spherical Hankel functions, which describe more localized lower-energy states, and oscillating Neumann tailing functions, which describe higherenergy conduction-band states. This method uses the linear theory
of Andersen [17] to obtain muffin-tin orbitals inside muffin-tin spheres. PSFs are smooth mathematical functions constructed by extending the muffin-tin-orbital tails into muffin-tin spheres. They are simply devised to calculate the interstitial and nonspherical parts of matrix elements efficiently through plane waves using the fast Fourier transform technique. PSF’s are expanded in threedimensional plane waves even for a film [14]. To limit the number of plane waves used to expand PSFs, a criterion, RG max > n, where n = 3, 5 or 7, respectively, for s, p, and d orbitals, is used, where R is the muffin-tin radius of the Si atom and G max is the magnitude of the largest wave vectors along the reciprocal basis vectors and the z-axis. Let b1 and b2 be the two reciprocal basis vectors and b3 = 2π /d, where d is greater than the thickness of the film [14], the total number of plane waves for PSF’s is (2n1 + 1)(2n2 + 1)(2n3 + 1) with ni ∼ G max /bi for i = 1, 2 and 3. Since the charge density is the absolute square of the wavefunction, the charge density as well as the potential is expanded by (4n1 + 1)(4n2 + 1)(4n3 + 1) plane waves. In this study, n1 = n2 = 7 and n3 = 59 are chosen. For the calculation of self-consistent potentials, the special k point scheme of Monkhorst and Pack has been used [18]. The dispersion of a free electron is parabolic, i.e. E (k) = (¯h2 k2 )/2mo , where m0 is the mass of a free electron and k is the wave number. In semiconductors, the dispersion of the conduction band near a local CBM can be approximated as E (k) = (¯h2 k2 )/2me , where E (k) is the state energy relative to the local CBM. The effective mass me can be calculated by the equation: 1/me = 1/¯h2 · ∂ 2 E (k)/∂ k2 . Under the 110-direction tensile strain in the x-axis direction, the two-dimensional first Brillouin zone is rectangular. The Brillouin zone center, i.e. the Γ point, X , Y , and M symmetry points are defined in Fig. 2(a). When the absolute CBM and those local energy minima (local CBM’s) within 1 eV of the absolute CBM are identified, the effective masses at those CBM’s along the x-axis, i.e. the 110-direction, which represents the ¯ -diparallel conduction channel, and along the y-axis, i.e. 110 rection, which is perpendicular to the 110-direction strain and represents the perpendicular conduction channel, are calculated. Under the 010-direction strain, the first Brillouin zone is a polygraph as shown in Fig. 2(b). Γ , B and E are the zone center, and zone edges, which correspond to the two M points shown in Fig. 2(a). The energy levels at Γ , B and E and along the lines joining them are calculated. After the absolute CBM and those local CBM’s within 1 eV of the absolute CBM are identified, the effective masses at those CBM’s along the 010-direction, which represents the parallel conduction channel, and along the 100direction, which is perpendicular to the 010-direction strain and represents the perpendicular conduction channel, are calculated.
Fig. 2. Diagrams of the two-dimensional first Brillouin zone and corresponding high symmetry k points of the Si(001) thin film under (a) 110- and (b) 010-direction
uniaxial tensile strain, in which b 1 and b 2 are the two reciprocal vectors.
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Fig. 4. The differences of lowest conduction-band states between X and Γ points, E ( X − Γ ), and between Y and Γ points, E (Y − Γ ), with respect to the 110direction uniaxial tensile strain, ε110 .
Fig. 3. The band structure of the Si(001) thin film (a) without strain and (b) with ε110 = 5%. VBM is chosen as zero energy.
3. Results and discussion 3.1. Electronic structures of the Si(001) thin film under 110-direction uniaxial tensile strain The electronic structures of the Si(001) thin film under 110direction uniaxial tensile strain, ε110 , with ε110 = 0, 1, 2, 3, 4, and 5% have been calculated. The band structures without strain and with ε110 = 5% are shown in Fig. 3(a) and (b), respectively. Without strain, X , Y , and M points are 2aπ (1, 0), 2aπ (0, 1)
and 2aπ (1, 1), respectively, where a is the periodicity of the twodimensional square lattice of the strain-free Si(001) film. Under the 110-direction tensile strain, X and M become approximately 2π (1 − ε110 , 0) and 2aπ (1 − ε110 , 1), respectively. Since the atomic a arrangements under zero strain are symmetric along x and ydirections, the energy bands are symmetric between Γ → X and Γ → Y directions. For ε110 = 0, the absolute CBM is located at the Γ point and local CBM states at X and Y points are degenerate as shown in Fig. 3(a). (Note that the X point here is different from the X point of the three-dimensional Brillouin zone of a bulk Si crystal.) Under the 110-direction tensile strain, energy levels at X and Y points are no longer symmetric and the local CBM states at X and Y points are no longer degenerate. As shown in Fig. 4, the energy difference between the local CBM states at X and Γ points, E ( X − Γ ), increases with the increase of ε110 , but the difference between the local CBM states at Y and Γ points, E (Y − Γ ), decreases with ε110 . At ε110 = 5%, the local CBM state at Y point becomes lower than that at the Γ point as shown in Fig. 3(b) and the absolute CBM is now located at the Y point instead of the Γ point. The upward shift of the local CBM state at the X point is due to that the x-direction tensile strain increases the bond distance between neighboring Si atoms in the x-direction,
which increases the energy of the conduction-band state at the x-direction Brillouin zone edge, i.e. the X point. By conservation of the average bond length, the increase of the x-direction bond distance due to the x-direction tensile strain will decrease the y-direction bond distance, which lowers the energy of the local CBM state at the Y point. The absolute valence band maximum (VBM) also shifts slightly away from the Γ point to be located on the Γ –Y axis and is about 0.05 eV above the highest occupied energy level at the Γ point. The energy gaps between absolute CBM and VBM, Eg’s, are 1.57, 1.54, 1.50, 1.45, 1.37, and 1.22 eV for ε110 = 0, 1, 2, 3, 4, and 5%, respectively. Note that the ε110 = 0 band gap is larger than the experimental band gap of Si of 1.1 eV for bulk Si, which is due to the nanometer-scale ultra-thin Si(001)film model used in this study. Moreover, the energy differences between the lowest and next-lowest conduction-band states at the Γ point, i.e. E (Γ2c − Γ1c ), are 0.82, 0.83, 0.85, 0.82, 0.77, and 0.72 eV and the corresponding energy differences at the Y point, i.e. E (Y 2c − Y 1c ), are 0.21, 0.22, 0.16, 0.06, 0.05, and 0.06 eV for ε110 = 0, 1, 2, 3, 4, and 5%, respectively. Since these energy differences between the lowest and next-lowest conduction-band states at the Γ and Y points are small, the conduction-band states in the vicinity of these local energy minima will contribute to the carrier density and conductivity for an applied voltage of, say, 1.0 Volt, which is a typical order of magnitude of the bias used in strained Si devices. The calculated effective masses of the conduction-band states parallel with the strain direction 110 at the Γ point along the Γ → X axis and at the Y point along the Y → M axis as functions of ε110 are shown in Fig. 5(a), which shows a trend that the effective mass decreases with the increase of ε110 . Since mobility is inverse proportional to the effective mass, the present result implies that mobility increases with the tensile strain when the conduction channel is parallel with the tensile strain. The enhancement of the mobility in the parallel conduction channel under a 110-direction tensile strain agrees with the experimental measurements of Lime et al. [10]. The calculated effective masses of the conduction-band states perpendicular to the strain direction 110 at the Γ point along the Γ → Y axis and at the Y point along the Y → Γ axis with respect to ε110 are also plotted in Fig. 5(b). In opposite to the parallel case, the effective mass increases with the increase of ε110 , which suggest that the mobility along the perpendicular conduction channel decreases with the increase of ε110 . The general trend of the decrease of E (Γ2c − Γ1c ) and E (Y 2c − Y 1c ) with ε110 indicates tensile strain induced enhancement of the density of states near CBM. This effect in conjunction
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Fig. 6. The band structures of the Si(001) thin film (a) without strain and (b) with ε010 = 5%. VBM is chosen as zero energy.
Fig. 5. The calculated effective masses of the conduction-band states (a) parallel with and (b) perpendicular to the 110 strain direction at Γ and Y points as functions of ε110 .
with the trend of the decrease of Eg indicates a tensile strain induced enhancement of thermally excited carrier density. Since conductivity is proportional to the product of mobility and carrier density, the present result suggests an increase of conductivity along the parallel conduction channel with the increase of the 110-direction tensile strain. However, when the conduction channel is perpendicular to the 110-direction tensile strain, whether the conductivity increases or decreases with ε110 depends on the competition between the effects on the effective mass and the energy level splitting at Y and Γ points. 3.2. Electronic structures of the Si(001) thin film under 010-direction uniaxial tensile strain The electronic structures of the Si(001) thin film for ε010 = 0, 1, 2, 3, 4, and 5% have been calculated. The band structure for ε010 = 5% is shown in Fig. 6. Without strain, I point is equivalent to the X and Y points, which have degenerate energy states. Under the 010-direction tensile strain, B and E points become approximately
ε 2 + (1 − ε010 )2 ε2010 + (1 − ε010 )2 010 ,− √ √ a 2 cos(π /4 − θ) 2 cos(π /4 − θ)
2π
and
ε 2 + (1 − ε010 )2 ε2010 + (1 − ε010 )2 010 , √ , √ a 2 cos(π /4 + θ) 2 cos(π /4 + θ)
2π
respectively, where θ is tan−1 (ε010 /(1 − ε010 )). Unlike the case of the 110-direction uniaxial tensile strain, the absolute CBM is always at the Γ point, and the energy differences between the local CBM states at I and Γ points do not change under the 010-direction tensile strain. Eg’s, are 1.57, 1.56, 1.53, 1.47, 1.41, and 1.36 eV for ε010 = 0, 1, 2, 3, 4, and 5%, respectively. Moreover, the energy differences between the lowest and next-lowest conduction-band states at the Γ point, i.e. E (Γ2c − Γ1c ), are 0.82, 0.83, 0.84, 0.83, 0.80, and 0.76 eV and the corresponding energy differences at the I point, i.e. E ( I 2c − I 1c ), are 0.21, 0.20, 0.18, 0.14, 0.06, and 0.03 eV for ε010 = 0, 1, 2, 3, 4, and 5%, respectively. Both E (Γ2c − Γ1c ) and E ( I 2c − I 1c ) are less than 1 eV within ε010 = 5%. Thus, the conduction-band states in the vicinity of these local energy minima will all contribute to the carrier density and conductivity for an applied voltage of 1.0 Volt. In Fig. 7(a), the calculated effective masses of the conductionband states parallel with the strain direction 010 at Γ and I points both increase with ε010 , which shows that mobility decreases with the increase of the 010-direction uniaxial tensile strain. Fig. 7(b) shows that the calculated effective masses of the conduction-band state perpendicular to the strain direction 010 at Γ and I points also increase with ε010 , which suggests that mobility also decreases with the increase of the 010-direction uniaxial tensile strain. However, the trends of the decrease of Eg, E (Γ2c − Γ1c ), and E ( I 2c − I 1c ) reveal that the thermally excited carrier density can be enhanced by the 010-direction uniaxial tensile strain. Therefore, whether the conductivity increases or decreases with ε010 depends on the competition between the effects on the effective mass and the energy level splitting at the Γ and I points. 4. Conclusion The electronic structures of the 20-layer Si(001) thin film under various 110- and 010-direction uniaxial tensile strains have been calculated using the first-principles pseudofunction (PSF) method. The effective masses of electrons at the absolute CBM and other local CBM’s along the directions in the Brillouin zone parallel with and perpendicular to the 110 strain direction are found to decrease and increase with ε110 , respectively. These trends suggest tensile-strain induced enhancement and reduction of the mobility in parallel and perpendicular conduction channels, respectively. As for the 010-direction uniaxial tensile strain, the effective masses of electrons at the absolute CBM and other local CBM’s are found to increase with the strain, which suggests a 010-direction uniaxial tensile strain induced reduction of the mobility in both parallel and perpendicular conduction channels.
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lowest conduction-band states at the absolute CBM and other local CBM’s decrease with the strain, which suggests a strain induced enhancement of thermally excited electron carrier density. Acknowledgement This work was supported by the National Science Council of Taiwan (Contract No. NSC-95-2120-M-110-003). References
Fig. 7. The calculated effective masses of the conduction-band states (a) parallel with and (b) perpendicular to the 010 strain direction at Γ and I points as functions of ε010 .
Under both 110- and 010-direction uniaxial tensile strains, the band gap and energy differences between the lowest and next-
[1] T. Mizuno, S. Takagi, N. Sugiyama, H. Satake, A. Kurobe, A. Triumi, IEEE Electron Device Lett. 21 (2000) 230. [2] L.-J. Huang, J. Chu, S. Goma, C. Emic, S. Koester, D. Canaperi, P. Mooney, S. Cordes, J. Speidell, R. Anderson, H. Wong, in: VLSI Symp. Tech. Dig., 2001, pp. 57–58. [3] N. Sugii, D. Hisamoto, K. Washio, N. Yokoyama, S. Kimura, in: IEDM Tech. Dig., 2001, pp. 737–740. [4] K. Rim, J. Chu, H. Chen, K. Jenkins, T. Kanarsky, K. Lee, A. Mocuta, H. Zhu, R. Roy, J. Newbury, J. Ott, K. Petrarca, P. Mooney, D. Lacey, S. Koester, K. Chan, D. Boyd, M. Ieong, H. Wong, in: VLSI Symp. Tech. Dig., 2002, pp. 98–99. [5] E. Scott, IEEE Trans. Electron Devices 51 (2004) 1790. [6] A. Hamada, IEEE Trans. Electron Devices 38 (1991) 895. [7] C. Gallon, IEEE Trans. Electron Devices 51 (2004) 1254. [8] H. Irie, IEDM Tech. Dig., 2004, pp. 225–228. [9] M.-N. Tsai, T.C. Chang, P.-T. Liu, O. Cheng, C.T. Huang, Electrochemical and SolidState Letters 9 (2006) g276. [10] F. Lime, F. Andrieu, J. Derix, G. Ghibaudo, F. Boeuf, T. Skotnicki, Solid-State Electronics 50 (2006) 644. [11] S. Takagi, J.L. Hoyt, J.J. Welser, J.F. Gibbons, J. Appl. Phys. 80 (1996) 1567. [12] D.R. Lide (Ed.), CRC Handbook of Chemistry and Physics, 73rd Edition, CRC Press, Boca Raton, 1992–93. [13] M.-H. Tsai, S. Lee, F.C. Peiris, J.K. Furdyna, Phys. Rev. B 65 (2002) 235202. [14] R.V. Kasowski, M.-H. Tsai, T.N. Rhodin, D.D. Chambliss, Phys. Rev. B 34 (1986) 2656. [15] In the modified PSF method, the projections of the tailing functions of the muffin-tin orbitals of surrounding atoms onto a given atom are calculated using two-dimensional structure factors: K. Kambe, Z. Naturf. 22a (1967) 322; Z. Naturf. 22a (1967) 422; Z. Naturf. 23a (1968) 1280; J.B. Pendry, Low Energy Electron Diffraction, Academic Press, London, 1974, p. 137 rather than the partial waves of plane-wave expanded pseudofunctions in the original method [14]. [16] L. Hedin, B.I. Lundqvist, J. Phys. C 4 (1971) 2064. [17] O.K. Andersen, Phys. Rev. B 12 (1975) 3060. [18] H.J. Monkhorst, J.D. Pack, Phys. Rev. B 13 (1976) 5188.
Contents lists available at ScienceDirect
Computer Physics Communications www.elsevier.com/locate/cpc
Electronic structures of the Si(001) thin film under 110- and 010-direction uniaxial tensile strains J.-Y. Lin, Y.-H. Tang, M.-H. Tsai ∗ Department of Physics, National Sun Yat-Sen University, Kaohsiung, 80424 Taiwan
a r t i c l e
i n f o
a b s t r a c t
Article history: Received 8 September 2008 Received in revised form 1 December 2008 Accepted 1 December 2008 Available online 6 December 2008 PACS: 73.63.-b 68.35.Gy Keywords: Nanoscale Si(001) thin film Strain
The electronic structures of the Si(001) ultra-thin film under various 110- and 010-direction uniaxial tensile strains have been calculated using the first-principles modified pseudofunction calculation method and a 20-layer single slab model. It can be inferred from calculated effective masses of electrons near the absolute conduction band minimum (CBM) that the 110-direction tensile strain induces enhancement and reduction of the mobility in parallel and perpendicular conduction channels, respectively. As for the 010-direction tensile strain, the effective mass results suggest that tensile strain induces reduction of the mobility in both parallel and perpendicular conduction channels. Under both 110- and 010direction strains, the band gap decreases and near-CBM density of states increases with strain, which suggests strain induced enhancement of thermally excited electron carrier density. © 2008 Elsevier B.V. All rights reserved.
1. Introduction Over the last few decades, silicon has been the basis of manufacturing a majority of semiconductor devices. To meet the need of higher integration densities, speed, and efficiency, device dimensions have been scaled down to the extent where conventional Si approaches its fundamental limits. Strained-Si MOSFET has increasingly been the subject of many investigations in recent years [1–11]. The drive current has been improved due to strained-Si technologies [1–5]. Experimentally, biaxial and uniaxial strains can be achieved by a SiGe substrate and a process induced type of stress, respectively [10]. Previous works showed that an optimum combination of channel direction and the direction of the uniaxial strain enhances mobility [6–10]. The strain induced enhancement of mobility has often been explained by the splitting of the six-fold degenerate conduction band minimum (CBM) valleys of bulk Si crystal and a reduced phonon scattering rate [5–11], though the Si layers in these devices are ultra-thin films. Due to the larger surface to volume ratio of the ultrathin films, their electronic structures are expected to be different from those of thick films or bulk solid. However, the ultra-thin films in the practical devices are still too thick to be represented by nanoscale thin films, whose electronic structures can be obtained by first-principles calculations using slab models. In other words, there is still a lack of suitable structural model to real-
*
Corresponding author. Tel.: +886 7 525 3729. E-mail address: [email protected] (M.-H. Tsai).
0010-4655/$ – see front matter doi:10.1016/j.cpc.2008.12.001
©
2008 Elsevier B.V. All rights reserved.
istically obtain the electronic structures of these films. Nevertheless, in this study, an attempt has been made to calculate the electronic structures of nanoscale Si(001) ultra-thin films under 110- and 010-direction uniaxial tensile strains. It is hoped that based on the calculated trend of strain induced electronic structures, the origin of the enhancement of the mobility can be elucidated. 2. The structural model and the calculation method Si has a diamond structure with a space group of Fd3m. The Si(001) ultra-thin film is modeled by a nanometer-scale Si(001) thin film with 20 atomic layers. Uniaxial 110- and 010-direction tensile strains are simulated by adjusting the two-dimensional basis vectors a1 and a2 as shown in Fig. 1(a). Note that the x- and y-axes of the two-dimensional (001) thin film are in the 110¯ -directions, respectively, and that by cubic symmetry the and 110 010-direction is equivalent to the 100-direction. In this study, the periodicity of the two-dimensional square lattice of the strainfree Si(001) film, |a1 | = |a2 | = a, is determined by energy minimization to be 3.908 Å, which is 2% larger than that of a truncated Si(001) film from bulk Si with an experimental lattice constant of 5.43072 Å√ [12]. (The periodicity of the two-dimensional square lattice is 1/ 2 times the lattice constant of the Si crystal with a diamond structure.) Since semiconductors have a tendency to conserve average bond length [13], the atomic positions are adjusted to conserve the nearest neighbor Si–Si average bond length to be 2.393 Å for each strain. Since each surface atom of the Si(001)
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Fig. 1. Definition of the two-dimensional basis vectors and changes of the basis vectors corresponding to tensile strains along the 110- and 010-directions.
film has two dangling bonds, all surface atoms are attached with two hydrogen atoms to saturate their dangling bonds to simulate the Si atoms at the interfaces which are bonded with neighboring substrate- and capping-layer-atoms and do not have dangling bond. The first-principles calculation method used in this study is the modified pseudofunction (PSF) method [14,15]. In this method, the z-component of the Coulomb potential is calculated numerically by solving the Poisson equation, where the z-coordinate is perpendicular to the Si(001) film. The potential, which includes Coulomb and exchange–correlation potentials within the local density approximation (LDA) of Hedin and Lundqvist [16], was divided into spherical potentials within the muffin-tin spheres and a planewave expanded potential that extends throughout the film. The basis set used in the PSF method contains two sets of Bloch sums of muffin-tin orbitals with exponentially decaying spherical Hankel functions, which describe more localized lower-energy states, and oscillating Neumann tailing functions, which describe higherenergy conduction-band states. This method uses the linear theory
of Andersen [17] to obtain muffin-tin orbitals inside muffin-tin spheres. PSFs are smooth mathematical functions constructed by extending the muffin-tin-orbital tails into muffin-tin spheres. They are simply devised to calculate the interstitial and nonspherical parts of matrix elements efficiently through plane waves using the fast Fourier transform technique. PSF’s are expanded in threedimensional plane waves even for a film [14]. To limit the number of plane waves used to expand PSFs, a criterion, RG max > n, where n = 3, 5 or 7, respectively, for s, p, and d orbitals, is used, where R is the muffin-tin radius of the Si atom and G max is the magnitude of the largest wave vectors along the reciprocal basis vectors and the z-axis. Let b1 and b2 be the two reciprocal basis vectors and b3 = 2π /d, where d is greater than the thickness of the film [14], the total number of plane waves for PSF’s is (2n1 + 1)(2n2 + 1)(2n3 + 1) with ni ∼ G max /bi for i = 1, 2 and 3. Since the charge density is the absolute square of the wavefunction, the charge density as well as the potential is expanded by (4n1 + 1)(4n2 + 1)(4n3 + 1) plane waves. In this study, n1 = n2 = 7 and n3 = 59 are chosen. For the calculation of self-consistent potentials, the special k point scheme of Monkhorst and Pack has been used [18]. The dispersion of a free electron is parabolic, i.e. E (k) = (¯h2 k2 )/2mo , where m0 is the mass of a free electron and k is the wave number. In semiconductors, the dispersion of the conduction band near a local CBM can be approximated as E (k) = (¯h2 k2 )/2me , where E (k) is the state energy relative to the local CBM. The effective mass me can be calculated by the equation: 1/me = 1/¯h2 · ∂ 2 E (k)/∂ k2 . Under the 110-direction tensile strain in the x-axis direction, the two-dimensional first Brillouin zone is rectangular. The Brillouin zone center, i.e. the Γ point, X , Y , and M symmetry points are defined in Fig. 2(a). When the absolute CBM and those local energy minima (local CBM’s) within 1 eV of the absolute CBM are identified, the effective masses at those CBM’s along the x-axis, i.e. the 110-direction, which represents the ¯ -diparallel conduction channel, and along the y-axis, i.e. 110 rection, which is perpendicular to the 110-direction strain and represents the perpendicular conduction channel, are calculated. Under the 010-direction strain, the first Brillouin zone is a polygraph as shown in Fig. 2(b). Γ , B and E are the zone center, and zone edges, which correspond to the two M points shown in Fig. 2(a). The energy levels at Γ , B and E and along the lines joining them are calculated. After the absolute CBM and those local CBM’s within 1 eV of the absolute CBM are identified, the effective masses at those CBM’s along the 010-direction, which represents the parallel conduction channel, and along the 100direction, which is perpendicular to the 010-direction strain and represents the perpendicular conduction channel, are calculated.
Fig. 2. Diagrams of the two-dimensional first Brillouin zone and corresponding high symmetry k points of the Si(001) thin film under (a) 110- and (b) 010-direction
uniaxial tensile strain, in which b 1 and b 2 are the two reciprocal vectors.
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Fig. 4. The differences of lowest conduction-band states between X and Γ points, E ( X − Γ ), and between Y and Γ points, E (Y − Γ ), with respect to the 110direction uniaxial tensile strain, ε110 .
Fig. 3. The band structure of the Si(001) thin film (a) without strain and (b) with ε110 = 5%. VBM is chosen as zero energy.
3. Results and discussion 3.1. Electronic structures of the Si(001) thin film under 110-direction uniaxial tensile strain The electronic structures of the Si(001) thin film under 110direction uniaxial tensile strain, ε110 , with ε110 = 0, 1, 2, 3, 4, and 5% have been calculated. The band structures without strain and with ε110 = 5% are shown in Fig. 3(a) and (b), respectively. Without strain, X , Y , and M points are 2aπ (1, 0), 2aπ (0, 1)
and 2aπ (1, 1), respectively, where a is the periodicity of the twodimensional square lattice of the strain-free Si(001) film. Under the 110-direction tensile strain, X and M become approximately 2π (1 − ε110 , 0) and 2aπ (1 − ε110 , 1), respectively. Since the atomic a arrangements under zero strain are symmetric along x and ydirections, the energy bands are symmetric between Γ → X and Γ → Y directions. For ε110 = 0, the absolute CBM is located at the Γ point and local CBM states at X and Y points are degenerate as shown in Fig. 3(a). (Note that the X point here is different from the X point of the three-dimensional Brillouin zone of a bulk Si crystal.) Under the 110-direction tensile strain, energy levels at X and Y points are no longer symmetric and the local CBM states at X and Y points are no longer degenerate. As shown in Fig. 4, the energy difference between the local CBM states at X and Γ points, E ( X − Γ ), increases with the increase of ε110 , but the difference between the local CBM states at Y and Γ points, E (Y − Γ ), decreases with ε110 . At ε110 = 5%, the local CBM state at Y point becomes lower than that at the Γ point as shown in Fig. 3(b) and the absolute CBM is now located at the Y point instead of the Γ point. The upward shift of the local CBM state at the X point is due to that the x-direction tensile strain increases the bond distance between neighboring Si atoms in the x-direction,
which increases the energy of the conduction-band state at the x-direction Brillouin zone edge, i.e. the X point. By conservation of the average bond length, the increase of the x-direction bond distance due to the x-direction tensile strain will decrease the y-direction bond distance, which lowers the energy of the local CBM state at the Y point. The absolute valence band maximum (VBM) also shifts slightly away from the Γ point to be located on the Γ –Y axis and is about 0.05 eV above the highest occupied energy level at the Γ point. The energy gaps between absolute CBM and VBM, Eg’s, are 1.57, 1.54, 1.50, 1.45, 1.37, and 1.22 eV for ε110 = 0, 1, 2, 3, 4, and 5%, respectively. Note that the ε110 = 0 band gap is larger than the experimental band gap of Si of 1.1 eV for bulk Si, which is due to the nanometer-scale ultra-thin Si(001)film model used in this study. Moreover, the energy differences between the lowest and next-lowest conduction-band states at the Γ point, i.e. E (Γ2c − Γ1c ), are 0.82, 0.83, 0.85, 0.82, 0.77, and 0.72 eV and the corresponding energy differences at the Y point, i.e. E (Y 2c − Y 1c ), are 0.21, 0.22, 0.16, 0.06, 0.05, and 0.06 eV for ε110 = 0, 1, 2, 3, 4, and 5%, respectively. Since these energy differences between the lowest and next-lowest conduction-band states at the Γ and Y points are small, the conduction-band states in the vicinity of these local energy minima will contribute to the carrier density and conductivity for an applied voltage of, say, 1.0 Volt, which is a typical order of magnitude of the bias used in strained Si devices. The calculated effective masses of the conduction-band states parallel with the strain direction 110 at the Γ point along the Γ → X axis and at the Y point along the Y → M axis as functions of ε110 are shown in Fig. 5(a), which shows a trend that the effective mass decreases with the increase of ε110 . Since mobility is inverse proportional to the effective mass, the present result implies that mobility increases with the tensile strain when the conduction channel is parallel with the tensile strain. The enhancement of the mobility in the parallel conduction channel under a 110-direction tensile strain agrees with the experimental measurements of Lime et al. [10]. The calculated effective masses of the conduction-band states perpendicular to the strain direction 110 at the Γ point along the Γ → Y axis and at the Y point along the Y → Γ axis with respect to ε110 are also plotted in Fig. 5(b). In opposite to the parallel case, the effective mass increases with the increase of ε110 , which suggest that the mobility along the perpendicular conduction channel decreases with the increase of ε110 . The general trend of the decrease of E (Γ2c − Γ1c ) and E (Y 2c − Y 1c ) with ε110 indicates tensile strain induced enhancement of the density of states near CBM. This effect in conjunction
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Fig. 6. The band structures of the Si(001) thin film (a) without strain and (b) with ε010 = 5%. VBM is chosen as zero energy.
Fig. 5. The calculated effective masses of the conduction-band states (a) parallel with and (b) perpendicular to the 110 strain direction at Γ and Y points as functions of ε110 .
with the trend of the decrease of Eg indicates a tensile strain induced enhancement of thermally excited carrier density. Since conductivity is proportional to the product of mobility and carrier density, the present result suggests an increase of conductivity along the parallel conduction channel with the increase of the 110-direction tensile strain. However, when the conduction channel is perpendicular to the 110-direction tensile strain, whether the conductivity increases or decreases with ε110 depends on the competition between the effects on the effective mass and the energy level splitting at Y and Γ points. 3.2. Electronic structures of the Si(001) thin film under 010-direction uniaxial tensile strain The electronic structures of the Si(001) thin film for ε010 = 0, 1, 2, 3, 4, and 5% have been calculated. The band structure for ε010 = 5% is shown in Fig. 6. Without strain, I point is equivalent to the X and Y points, which have degenerate energy states. Under the 010-direction tensile strain, B and E points become approximately
ε 2 + (1 − ε010 )2 ε2010 + (1 − ε010 )2 010 ,− √ √ a 2 cos(π /4 − θ) 2 cos(π /4 − θ)
2π
and
ε 2 + (1 − ε010 )2 ε2010 + (1 − ε010 )2 010 , √ , √ a 2 cos(π /4 + θ) 2 cos(π /4 + θ)
2π
respectively, where θ is tan−1 (ε010 /(1 − ε010 )). Unlike the case of the 110-direction uniaxial tensile strain, the absolute CBM is always at the Γ point, and the energy differences between the local CBM states at I and Γ points do not change under the 010-direction tensile strain. Eg’s, are 1.57, 1.56, 1.53, 1.47, 1.41, and 1.36 eV for ε010 = 0, 1, 2, 3, 4, and 5%, respectively. Moreover, the energy differences between the lowest and next-lowest conduction-band states at the Γ point, i.e. E (Γ2c − Γ1c ), are 0.82, 0.83, 0.84, 0.83, 0.80, and 0.76 eV and the corresponding energy differences at the I point, i.e. E ( I 2c − I 1c ), are 0.21, 0.20, 0.18, 0.14, 0.06, and 0.03 eV for ε010 = 0, 1, 2, 3, 4, and 5%, respectively. Both E (Γ2c − Γ1c ) and E ( I 2c − I 1c ) are less than 1 eV within ε010 = 5%. Thus, the conduction-band states in the vicinity of these local energy minima will all contribute to the carrier density and conductivity for an applied voltage of 1.0 Volt. In Fig. 7(a), the calculated effective masses of the conductionband states parallel with the strain direction 010 at Γ and I points both increase with ε010 , which shows that mobility decreases with the increase of the 010-direction uniaxial tensile strain. Fig. 7(b) shows that the calculated effective masses of the conduction-band state perpendicular to the strain direction 010 at Γ and I points also increase with ε010 , which suggests that mobility also decreases with the increase of the 010-direction uniaxial tensile strain. However, the trends of the decrease of Eg, E (Γ2c − Γ1c ), and E ( I 2c − I 1c ) reveal that the thermally excited carrier density can be enhanced by the 010-direction uniaxial tensile strain. Therefore, whether the conductivity increases or decreases with ε010 depends on the competition between the effects on the effective mass and the energy level splitting at the Γ and I points. 4. Conclusion The electronic structures of the 20-layer Si(001) thin film under various 110- and 010-direction uniaxial tensile strains have been calculated using the first-principles pseudofunction (PSF) method. The effective masses of electrons at the absolute CBM and other local CBM’s along the directions in the Brillouin zone parallel with and perpendicular to the 110 strain direction are found to decrease and increase with ε110 , respectively. These trends suggest tensile-strain induced enhancement and reduction of the mobility in parallel and perpendicular conduction channels, respectively. As for the 010-direction uniaxial tensile strain, the effective masses of electrons at the absolute CBM and other local CBM’s are found to increase with the strain, which suggests a 010-direction uniaxial tensile strain induced reduction of the mobility in both parallel and perpendicular conduction channels.
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lowest conduction-band states at the absolute CBM and other local CBM’s decrease with the strain, which suggests a strain induced enhancement of thermally excited electron carrier density. Acknowledgement This work was supported by the National Science Council of Taiwan (Contract No. NSC-95-2120-M-110-003). References
Fig. 7. The calculated effective masses of the conduction-band states (a) parallel with and (b) perpendicular to the 010 strain direction at Γ and I points as functions of ε010 .
Under both 110- and 010-direction uniaxial tensile strains, the band gap and energy differences between the lowest and next-
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