effects of uniaxial strain on electron effective mass and tunneling capability of direct gap ge1 xsnx alloys

Effects of uniaxial strain on electron effective mass and tunneling capability of direct gap Ge1 xSnx alloys Lei Liu, Renrong Liang, Jing Wang, and Jun Xu Citation AIP Advances 6, 015102 (2016); doi 1[.]

Effects of uniaxial strain on electron effective mass and tunneling capability of direct gap Ge1-xSnx alloys Lei Liu, Renrong Liang, Jing Wang, and Jun Xu Citation: AIP Advances 6, 015102 (2016); doi: 10.1063/1.4939816 View online: http://dx.doi.org/10.1063/1.4939816 View Table of Contents: http://aip.scitation.org/toc/adv/6/1 Published by the American Institute of Physics AIP ADVANCES 6, 015102 (2016) Effects of uniaxial strain on electron effective mass and tunneling capability of direct gap Ge1−x Snx alloys Lei Liu, Renrong Liang,a Jing Wang, and Jun Xu Tsinghua National Laboratory for Information Science and Technology, Institute of Microelectronics, Tsinghua University, Beijing 100084, People’s Republic of China (Received 26 October 2015; accepted 30 December 2015; published online January 2016) Direct gap Ge1−x Sn x alloys under [100] and [110] uniaxial strain are comprehensively investigated by theoretical calculations using the nonlocal empirical pseudopotential method (EPM) It is shown that [100] uniaxial tensile strain aids indirectto-direct gap transition in Ge1−x Sn x alloys The Γ electron effective mass along the optimal direction under [110] uniaxial strain is smaller than those under [100] uniaxial strain and (001) biaxial strain Additionally, the direct tunneling gap is smallest along the strain-perpendicular direction under [110] uniaxial tensile strain, resulting in a maximum direct band-to-band tunneling generation rate An optimal [110] uniaxial tensile strain is favorable for high-performance direct gap Ge1−x Sn x electronic devices C 2016 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4939816] I INTRODUCTION Recently, germanium-tin (Ge1−x Sn x ) alloys have attracted a great deal of research attention The electronic band structures of Ge1−x Sn x alloys are strongly dependent on their Sn composition x.1–4 Incorporating more Sn atoms into Ge1−x Sn x alloys would result in a transition from an indirect gap to a direct gap, making it possible to achieve direct gap semiconductors with Group IV elements.1 Direct gap Ge1−x Sn x alloys are very promising materials for optoelectronic applications.5,6 Additionally, Ge1−x Sn x alloys are expected to exhibit enhanced carrier transport capability, which would allow them to be used as high mobility channel materials for the next generation of metal-oxide-semiconductor field-effect transistors (MOSFETs).7,8 Apart from material innovations to boost the performance of MOSFETs, tunneling FETs (TFETs) have been proposed as a possible solution to the low power challenges of ultra-scaled devices.9,10 The small band gaps and direct gap properties of Ge1−x Sn x alloys are also beneficial for enhancing the band-to-band tunneling (BTBT) rate, which is critical for improving the on-state current of TFETs in practical applications.11 Hence, direct gap Ge1−x Sn x alloys are of great importance for novel high-performance electronic devices Theoretical calculations and experimental results of the direct gap transition compositions of Ge1−x Sn x alloys have already been published in the literature.12–17 Some experimental demonstrations of Ge1−x Sn x MOSFETs and TFETs have also been reported, confirming their superior electronic carrier properties.18–21 The most commonly reported Ge1−x Sn x alloys at present are grown on Ge or Si substrates.22–24 These Ge1−x Sn x alloys are usually biaxially strained owing to the lattice constant mismatch with the substrate,24,25 and the introduced biaxial strain is beneficial for improving Ge1−x Sn x device on-state currents.26 In our previous work, the effects of the biaxial strain on Ge1−x Sn x alloys were analyzed in detail, providing some guidelines for enhancing the performance of Ge1−x Sn x devices.27 In addition to the biaxial strain, the uniaxial strain also plays an important role in device performance engineering.28 The applied uniaxial strain affects the electronic band structure of Ge1−x Sn x alloys in a different way from the biaxial strain, which has significant effects on the performance of Ge1−x Sn x devices However, there is still a lack of understanding about the a Electronic mail: liangrr@tsinghua.edu.cn 2158-3226/2016/6(1)/015102/9 6, 015102-1 © Author(s) 2016 015102-2 Liu et al AIP Advances 6, 015102 (2016) effect of uniaxial strain on the electronic properties of Ge1−x Sn x alloys In this paper, the electron effective mass and the tunneling capability of uniaxially strained Ge1−x Sn x alloys are comprehensively investigated The electronic energy bands of Ge1−x Sn x alloys under an applied uniaxial strain are calculated using the nonlocal empirical pseudopotential method (EPM) The Γ electron effective mass and the direct BTBT generation rate of the uniaxially strained direct gap Ge1−x Sn x alloys are evaluated in detail The findings presented in this work are expected to be useful for the design and optimization of high-performance Ge1−x Sn x field-effect transistor devices II CALCULATION METHOD The nonlocal EPM with spin-orbit interaction is widely used to calculate the electronic band structures of semiconductors.16,29–31 Reasonably accurate band structures and acceptable computation cost can be simultaneously achieved with this method.31 Here, the pseudopotentials of local, nonlocal, and spin-orbit contributions are included in the calculation to simulate the accurate band diagrams of Ge1−x Sn x alloys For the strained conditions, the symmetric form factors Vs at arbitrary wave vector q are needed, and the corresponding values for Ge and α-Sn are interpolated in the following form.30
 ( )  a1 q − a2 a5 − q 1   + , (1) Vs (q) = Ωa exp a3 (q2 − a4) + a6 where Ωa is the atomic volume, and a1 – a6 are the interpolation coefficients The virtual crystal approximation (VCA) can be used to describe the semiconductor alloys within the EPM framework, and the corresponding pseudopotentials and parameters are linearly interpolated in this scheme.31 However, because Ge1−x Sn x alloys commonly exhibit severe nonlinear characteristics, the simple VCA cannot describe their compositional disorder effects.16 Hence, a modified VCA is adopted in this work to account for the random potential fluctuations of Ge1−x Sn x alloys, and the interpolated local form factors can be written as16,32 ΩSn ΩGe VGe (q) + x VSn (q) VGe1−x Sn x (q) = (1 − x) ΩGe1−x Sn x ΩGe1−x Sn x −Ploc x (1 − x) (ΩSnVSn (q) − ΩGeVGe (q)) , ΩGe1−x Sn x (2) where Ploc is a fitting parameter for the targeted direct band gap bowing parameter of Ge1−x Sn x alloys The calculation parameters were calibrated, and some key parameters used in this work are listed in Table I A more detailed parameter set can be found in Ref 27 We extract the carrier effective masses (in units of m0, the free electron mass) from the calculated band diagram by a parabolic line fit The band structures of Ge and α-Sn are accurately reproduced Table II lists the calculated effective masses of Ge (x=0), which are consistent with the reported results.29,31 Calculated deformation potentials of Ge also agree well with the data in the literature.27 The obtained direct gap bowing parameter and the spin-orbit splitting bowing parameter of the Ge1−x Sn x alloys are 1.936 and 0.405, respectively, in accordance with Ref 13 The calculated direct gap transition composition for unstrained Ge1−x Sn x alloy is around 0.062, which is consistent with the results reported in Refs 14–17 All these calculation results confirm the validity of the nonlocal EPM approach employed in this work The uniaxial strain investigated in this work is applied along the [100] and [110] directions of the (001) Ge1−x Sn x alloys Different uniaxial strain configurations are discussed and compared, and the anisotropy on the (001) plane is also analyzed III RESULTS AND DISCUSSION The band structures of Ge1−x Sn x alloys under uniaxial strain were calculated Unstrained Ge0.95Sn0.05 alloy is an indirect gap material, because the Sn composition is lower than the transition point Accordingly, Ge0.95Sn0.05 alloy was chosen as the example used to examine the effect of uniaxial strain on the indirect-to-direct band gap transition Figure illustrates the calculated 015102-3 Liu et al AIP Advances 6, 015102 (2016) TABLE I Local and nonlocal parameters of Ge and Sn used in the calculations (see Ref 27 and its citations) Quantity Symbol Units Ge Sn a0 a1 a2 a3 a4 a5 a6 α0 β0 A2 R0 R2 Å atomic units atomic units atomic units atomic units atomic units atomic units Ry 5.658 22.6517 2.471 0.7853 1.3176 0.3 0 0.309 6.489 25.7873 1.7993 2.5172 1.8811 0.3 0.365 0.71 1.453 Lattice constant Interpolation coefficients s-well depth s-well energy dependence d-well depth s-well radius d-well radius Ry Å Å 1.2788 electronic band structures of Ge0.95Sn0.05 alloy under 2% uniaxial compressive and tensile strain The uniaxial strain breaks the degeneracy of the valance bands at the Γ point The relative position of the conduction band edges of the L and Γ valleys changes with the applied uniaxial strain In Fig 1(b), the Γ valley is located at a lower energy than the L valley, indicating that Ge0.95Sn0.05 alloy under 2% [100] uniaxial tensile strain is a direct gap material However, Ge0.95Sn0.05 alloy is still an indirect gap material under the other uniaxially strained conditions investigated To gain an insight into more general cases, the energy band gaps at the L point EgL and the Γ point Eg Γ were calculated for various uniaxial-strained conditions and Sn compositions The contour plot of the corresponding band gap difference (EgL − Eg Γ) is presented in Fig The direct gap transition point for Ge1−x Sn x alloys is obtained where EgL is equal toEg Γ It can be observed that [100] uniaxial tensile strain is beneficial for the direct gap transition, and the transition composition is lower than that under the unstrained condition For [100] uniaxial compressive strain, the direct gap transition composition increases with the applied strain For [110] uniaxial strain, both the compressive strain and the tensile strain cause an increase in the transition composition Therefore, [100] uniaxial tensile strain can be used to achieve direct gap Ge1−x Sn x alloys with low Sn composition It is obvious that the applied 2% [100] uniaxial tensile strain in Fig 1(b) satisfies the corresponding direct gap transition requirement needed to produce direct gap Ge0.95Sn0.05 alloy The electron effective mass of direct gap Ge1−x Sn x alloys was analyzed from the developed electronic band structures The Γ electron effective masses m∗eΓ were extracted for Ge0.9Sn0.1 alloy under 1% [100] and [110] uniaxial strain It can be seen from Fig that Ge0.9Sn0.1 alloy under these strained conditions is a direct gap material The calculated m∗eΓ results for the (001) plane are presented in Fig The electron effective mass exhibits severe anisotropic characteristics under the uniaxially strained conditions For [100] and [110] uniaxial tensile strain, the smallest m∗eΓ occurs along the strain-perpendicular direction, and the largest value occurs along the strain-parallel direction In contrast, for [100] and [110] uniaxial compressive strain, the smallest m∗eΓ occurs along the strain-parallel direction, and the largest value occurs along the strain-perpendicular direction, which is opposite to the tensile-strained condition Hence, the optimal direction for uniaxial tensile strain TABLE II Calculated carrier effective masses of Ge and corresponding results published in the literature m ∗l h and m ∗hh denote the light hole and heavy hole effective mass, respectively m ∗eΓ denotes the Γ electron effective mass All effective masses are in units of m This work Ref 29 Ref 31 m ∗l h [100] m ∗l h [110] m ∗l h [111] m ∗hh [100] m ∗hh [110] m ∗hh [111] m ∗eΓ 0.0528 0.0529 0.0600 0.0481 0.0476 0.0530 0.0468 0.0463 0.0520 0.2502 0.226 0.251 0.4718 0.439 0.467 0.6357 0.597 0.623 0.0417 0.0420 0.0470 015102-4 Liu et al AIP Advances 6, 015102 (2016) FIG Calculated electronic band structures of Ge0.95Sn0.05 alloy under 2% (a) [100] uniaxial compressive strain, (b) [100] uniaxial tensile strain, (c) [110] uniaxial compressive strain, and (d) [110] uniaxial tensile strain The minimum energy of L valleys under uniaxial strain is compared with the band edge of Γ valleys For [110] uniaxial tensile strain, the L valley minimum is along the [−111] direction is the strain-perpendicular direction, while for uniaxial compressive strain it is the strain-parallel direction It should be noted that the m∗eΓ values along the optimal directions under uniaxial strain are all smaller than the isotropic m∗eΓ obtained for unstrained Ge0.9Sn0.1 alloy The applied optimal uniaxial tensile and compressive strain both contribute to decreasing the Γ electron effective mass of direct gap Ge1−x Sn x alloys Figure plots the uniaxial strain dependence of the m∗eΓ results along the optimal directions for direct gap Ge0.9Sn0.1 alloy Although both uniaxial tensile strain and compressive strain lead to a reduction in the optimal m∗eΓ, for the same magnitude of applied uniaxial strain, the optimal m∗eΓ under tensile strain is smaller than that under compressive strain For instance, the optimal m∗eΓ under 1% [100] and [110] uniaxial tensile strain is around 6.5% and 8.5% lower, respectively, than that under the corresponding compressively strained conditions It is clear that the optimal m∗eΓ 015102-5 Liu et al AIP Advances 6, 015102 (2016) FIG Contour plot of Ge1−x Sn x band gap difference between E gL and E g Γ (E gL − E g Γ) for (a) [100] uniaxial strain and (b) [110] uniaxial strain The band gap energies are in units of eV under [110] tensile strain is smaller than that under [100] tensile strain, indicating that [110] uniaxial tensile strain is more favorable than the [100] counterpart The m∗eΓ results for (001) biaxial strain are also presented in Fig 4, which are isotropic on the (001) plane The optimal m∗eΓ results under uniaxial strain are smaller than the biaxially strained m∗eΓ, especially under compressive strained conditions Hence, uniaxial strain is more preferable than the biaxially strained situation In addition, the Sn composition dependence of m∗eΓ was also examined, and the results for the optimal 1% [100] and [110] uniaxial tensile strains are presented in the inset of Fig The m∗eΓ results of direct gap Ge0.85Sn0.15 are 25.3% and 27.7% lower than those of direct gap Ge0.9Sn0.1 for [100] and [110] uniaxial tensile strain, respectively The combination of high Sn composition and optimal [110] uniaxial tensile strain may be advantageous for the application of direct gap Ge1−x Sn x in nMOSFETs owing to the small electron effective mass Improving the on-state current is a major issue for the development of TFETs.9 Enhancing the BTBT rate of direct gap Ge1−x Sn x alloys is important for high-performance TFETs in application 015102-6 Liu et al AIP Advances 6, 015102 (2016) FIG The Γ electron effective mass m ∗eΓ of direct gap Ge0.9Sn0.1 alloy on the (001) plane under 1% uniaxial strain and unstrained conditions The direct gap BTBT generation rate G can be described by Kane’s model in the form33,34 1/2 3/2  ( )2  1/2 gmr (qe F0)2  F *.− π mr Eg Γ, t +/ ,  G= exp  F qe hF  πh2 Eg1/2 Γ, t  , - (3) where F is the electric field, F0=1 V/cm, g is the degeneracy factor, Eg Γ, t is the tunneling band gap at the Γ point, qe is the elementary charge, and h is Planck’s constant The reduced tunneling mass mr is related to m∗eΓ and the light-hole-like effective mass m∗l h (mr =m∗eΓ m∗l h /(m∗eΓ + m∗l h )) The value of m∗l h can be obtained from the valance band structure The effect of uniaxial strain on direct BTBT was comprehensively examined for Ge1−x Sn x alloys Figure plots the direct BTBT FIG Strain dependence of Γ electron effective mass m ∗eΓ for direct gap Ge0.9Sn0.1 alloy m ∗eΓ was extracted along the optimal strain-perpendicular direction for uniaxial tensile strain, and along the strain-parallel direction for uniaxial compressive strain The in-plane isotropic m ∗eΓ results under (001) biaxial strain are shown for comparison The inset presents the composition dependence of m ∗eΓ under the optimal 1% uniaxial tensile strain 015102-7 Liu et al AIP Advances 6, 015102 (2016) FIG Direct BTBT generation rate for direct gap Ge0.9Sn0.1 alloy under 1% [100] and [110] uniaxial strain generation rate of direct gap Ge0.9Sn0.1 alloy under the 1% [100] and [110] uniaxial strain The direct tunneling along the uniaxial strain-parallel direction and the uniaxial strain-perpendicular direction was compared Under [100] and [110] uniaxial tensile strain, the strain-perpendicular directions exhibit the highest BTBT generation rates, while the lowest generation rates are observed along the strain-perpendicular directions under uniaxial compressive strain For the tensile strainperpendicular directions, the [110] tensile strain is more advantageous than the [100] tensile strain The direct BTBT generation rate is mainly affected by the tunneling gap Eg Γ, t and the reduced tunneling mass mr Table III summarizes the G values obtained at MV/cm electric field and the corresponding m∗l h , m∗eΓ, and Eg Γ, t results for direct gap Ge0.9Sn0.1 alloy under the uniaxially strained conditions analyzed in Fig For both [100] and [110] uniaxial strain, it can be seen that the compressively strained m∗l h values along the strain-parallel directions are the smallest, but the smallest tensile-strained m∗eΓ and Eg Γ, t results along the strain-perpendicular directions contribute more to the BTBT rate Thus, the tensile strain-perpendicular directions exhibit the highest G results The effective masses and tunneling gaps for the strain-perpendicular directions are both the largest under compressive strain, resulting in the lowest G values Comparing the [100] and the [110] uniaxial strain-perpendicular directions, the tunneling gaps and the reduced tunneling masses observed under [110] uniaxial tensile strain are smaller than those under [100] uniaxial tensile strain Consequently, the strain-perpendicular direction under [110] uniaxial tensile strain is the best choice to boost the direct BTBT rate of direct gap Ge1−x Sn x alloys The corresponding results for unstrained Ge0.9Sn0.1 alloy are also listed in Table III Under MV/cm electric field, the TABLE III Direct BTBT generation rates at MV/cm electric field for Ge0.9Sn0.1 alloy under various uniaxial strain conditions The maximum BTBT rate on the (001) plane for the unstrained condition is also presented Uniaxial strain 1% [100] 1% [100] −1% [100] −1% [100] 1% [110] 1% [110] −1% [110] −1% [110] Unstrained Direction m ∗eΓ (m 0) m ∗l h (m 0) E g Γ, t (eV) G (1031 cm−3 s−1) Strain-parallel Strain-perpendicular Strain-parallel Strain-perpendicular Strain-parallel Strain-perpendicular Strain-parallel Strain-perpendicular 0.0296 0.0257 0.0275 0.0313 0.0292 0.0249 0.0272 0.0316 0.0286 0.0438 0.0403 0.0309 0.0498 0.0464 0.0333 0.0269 0.0470 0.0333 0.497 0.438 0.496 0.567 0.489 0.408 0.474 0.595 0.504 1.19 2.14 1.39 0.56 1.26 2.84 1.73 0.44 1.25 015102-8 Liu et al AIP Advances 6, 015102 (2016) value of G for the [110] tensile strain-perpendicular direction is ∼2.3 times higher than that of unstrained Ge0.9Sn0.1 alloy However, the G values obtained for the compressive strain-perpendicular directions are lower than the unstrained results owing to the much larger tunneling gaps and effective masses of the former Optimal uniaxial strain is beneficial for boosting the direct BTBT rate of direct gap Ge1−x Sn x alloys compared with the unstrained situation, but some unfavorable strained conditions may worsen the device performance Therefore, to enhance the performance of Ge1−x Sn x TFETs, the most suitable uniaxial-strained condition should be carefully selected Moreover, it should be noted that the small band gap of strained Ge1−x Sn x alloys may lead to more serious ambipolar effects and higher leakage current, but these drawbacks could be overcome by optimizing the short-gate structures and the hetero-structures at the drain side.35,36 IV CONCLUSION In summary, the effects of uniaxial strain on direct gap Ge1−x Sn x alloys are comprehensively investigated using nonlocal EPM calculations [100] uniaxial tensile strain is beneficial for inducing the direct gap transition of Ge1−x Sn x alloys with low Sn composition The band structures and the carrier effective masses of these alloys are distinctively affected by applied [100] and [110] uniaxial strains Severe anisotropy of the uniaxially strained Γ electron effective mass m∗eΓ is observed on the (001) plane The uniaxial strain-perpendicular and strain-parallel directions present different electronic properties The optimal [110] uniaxial strain is more favorable than [100] uniaxial strain and (001) biaxial strain to achieve a small m∗eΓ value, and tensile strain is more effective than compressive strain The [110] tensile strain-perpendicular direction exhibits the smallest m∗eΓ and the smallest tunneling gap, which cause a significantly enhanced direct BTBT capability in direct gap Ge1−x Sn x alloys An appropriate uniaxial-strained condition may be beneficial for boosting the performance of direct gap Ge1−x Sn x devices compared with unstrained Ge1−x Sn x counterparts The optimal [110] uniaxial tensile strain-perpendicular direction is recommended for the design and optimization of high-performance Ge1−x Sn x MOSFETs and TFETs for practical applications ACKNOWLEDGMENTS This work was supported in part by the National Science and Technology 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(2016) effect of uniaxial strain on the electronic properties of Ge1? ??x Sn x alloys In this paper, the electron effective mass and the tunneling capability of uniaxially strained Ge1? ??x Sn x alloys are... 015102 (2016) Effects of uniaxial strain on electron effective mass and tunneling capability of direct gap Ge1? ??x Snx alloys Lei Liu, Renrong Liang,a Jing Wang, and Jun Xu Tsinghua National Laboratory... rate of direct gap Ge0.9Sn0.1 alloy under the 1% [100] and [110] uniaxial strain The direct tunneling along the uniaxial strain- parallel direction and the uniaxial strain- perpendicular direction