Electronic structure and transport calculations of group-IV ...

Electronic structure and transport calculations of group-IV semiconductors H.-S. Lan1, P.-S. Chen2, H.-Y. Ye1, and C. W. Liu1,2,3,4,* 1

Graduate Institute of Electronics Engineering, Department of Electrical Engineering, National Taiwan University,

Taipei 10617, Taiwan 2

Graduate Institute of Photonics and Optoelectronics, National Taiwan University, Taipei 10617, Taiwan

3

Center for Condensed Matter Sciences, National Taiwan University, Taipei 10617, Taiwan

4

National Nano Device Laboratories, Hsinchu 30078, Taiwan

*

E-mail: [email protected]

Advanced Silicon Device and Process Lab: http://nanosioe.ee.ntu.edu.tw

Abstract In this project, we plan to calculate electronic structures and transport properties of group-IV semiconductors. Electronic structure properties including bandgap, effective masses of conduction band (the Γ, L, and ∆ valleys) and valence bands (heavily hole, light hole, and split-off hole), nonparabolic band structures, and band alignments at heterointerfaces of group-IV semiconductors are studied. The developed in-house MATLAB programs of several methods including deformation potential method, empirical pseudopotential method, model-solid theory, and 6×6 k·p model are used to study the electronic structures. The one- and two- dimensional Schrödinger and Poisson equations are calculated iteratively to solve quantized subbands and envelope functions in inversion layers of field effect transistors through in-house programs and TCAD simulators. Several possible scattering mechanisms such as phonon, interface roughness, and coulomb scatterings are considered by developing in-house programs. The Kubo-Greenwood formula and the calculated injection velocity is used to calculate mobility and ballistic current, respectively. Index Terms—group-IV semiconductor, first principle calculation, deformation potential method, empirical pseudopotential method, model-solid theory, band alignment, 6×6 kp method, self-consistently solution of Schrödinger and Poisson equations, mobility, injection velocity, ballistic current, phonon scattering, roughness scattering, coulomb scattering, Kubo-Greenwood

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I. Review Group-IV materials (Si, Ge, and α-Sn) attract great attentions in the electronic and optoelectronic device applications [1-3]. For electronic devices, replacing Si with high-mobility channel materials (e.g. Ge/GeSn alloys) is one near-term approach to improving performance. However, replacing Si and SiO2 is not a trivial task, and complex “integration” issues must be overcome. The lattice mismatch between new channel materials and Si substrates can generate dislocations at the interface, which degrades performance and yield. To mitigate these effects, grade relaxed buffers [4, 5] and dislocation removal [2, 6] etc. have been explored to reduce the dislocation density in active areas. The source/drain (S/D) resistance reportedly contributes ∼46% of the total resistance for the 7 nm technology node [7]. The heavily doped S/D and low contact resistance are the keys to reduce the S/D resistance. Laser annealing can effectively remove the defects and achieve the high electron concentration of ∼3×1020 cm−3 of phosphorus doped Ge on Si grown by the chemical vapor deposition (CVD) [8]. Moreover, ZrO2 is considered to be preferable to HfO2 on Ge for EOT scaling owing to the IL suppression after annealing. We demonstrated an IL-free ZrO2/Ge gate stack with an EOT of 0.39 nm by using NH3/H2 remote plasma treatment (RPT) on GeO2/Ge surface [9]. A post-gate fluorine incorporation by CF4 plasma into tetragonal ZrO2 ultrathin EOT of 0.41 nm was also demonstrated to improve the hysteresis [10]. The Zr-F bond formation to remove the midgap states calculated by the density-functiontheory (DFT) is the origin of passivation. Advanced complementary metal oxide semiconductor (CMOS) is boosted by the stressors. In addition to S/D stressors, the stress field produced by TSV [11] or edge dislcoation [12] can add additional strain in transistors. To further enhance mobilities, the tensile strain and compressive strain is applied to Ge n-type field-effect transistors (NFETs) [13] and p-type FETs [14], respectively, by mechanical stress. The record high two-dimensional electron gas (2DEG) mobility has reached 2.4×106 cm2/V-s [4, 5] in a SiGe/strained Si/SiGe 2

quantum well near 0 K. Higher Ge content in the structures leads to a larger threading dislocation density and larger interface roughness, resulting in the degradation of the 2DEG mobility. The ultrathin strained Si0.2Ge0.8 quantum well channel (~5 nm) directly grown on Si substrates was also demonstrated with low defect density and high hole mobility at room temperature [15]. Moreover, recently, the CVD-grown GeSn channels with low thermal budget of 400oC significaly outperforms the Ge channel processed at high theraml budget of 550oC. The record high mobility (428 cm2/V-s) of the CVD-grown GeSn PFETs was achieved [3]. Beside the advantage of high mobility, new device architectures such as gate-all-around (GAA) and ultra-thin body FETs are needed to improve the electrostatic control for sub-10 nm nodes. For 3D transistors, high performacne Ge junctionless GAA NFETs with Ion = 1235 µA/µm at Vov = VDS = 1 V, and high Ion/Ioff = 2×106 was demonstrated [2]. The operating characteristics of junctionless Ge PFETs with Ion ~ 390 µA/µm and high Ion/Ioff = 1.2×106 was aslo reported [16]. While III-V devices appear to be good candidates for n-type field-effect transistors (NFETs), Ge/GeSn alloys appear to be superior for p-type FETs. The improvement of interface trap density (Dit) passivation/low EOT, doping concentraion/activation, contact resistance, and surface roughness/mobility of Ge/GeSn FETs is in progress and sheds light on Ge/GeSn complementary metal oxide semiconductor (CMOS) applications. For optoelectronic devices, strain can be an extra factor to enhance the direct transition of Ge [17, 18]. The progressive improvement of the radiative recombination makes it possible to have Ge-based light emitting devices and chip-to-chip optical interconnect for practical applications. We also affirm that the defects in the Ge-on-Si are responsible for the weak indirect transition and relatively strong direct transition. The scattering of electrons by roughness at Ge/oxide interface can provide extra momentum of the indirect band transition of Ge, and thus enhance the indirect radiative transition [19]. Moreover, the electroluminescence (EL) emission peak of GeSn can be 3

tuned by the Sn content in the metal-insulator-semiconductor (MIS) tunneling diodes without the Ge cap. The high quality GeSn layer grown by CVD reveals good EL and photoluminescence (PL) and can serve the photonic application on Si platform [1]. To rapidly and efficiently address the group-IV electronic and optoelectronic devices, physicsbased modeling techniques must be employed: electronic structures calculations such as densityfunctional theory (DFT) [20] and EPM etc. (Sec. II), the Boltzmann Transport Equation (BTE), and a semi-empirical and quantum transport solvers (Sec. III) are considered as material pieces. After benchmarking with experimental data, they can be used to accurately simulate group-IV transistors and photonic devices through in-house programs or TCAD simulators.

II. Electronic structures Alloying Sn into Ge makes GeSn as a direct band gap materials due to the faster decreasing energy of the direct Γ valley than indirect L valleys with increasing Sn content. The indirect to direct crossover point is the critical issue to enhance the mobility/ballistic current of metal-oxidesemiconductor field-effect transistor (MOSFETs) because the large energy difference between direct and indirect valleys make more carriers occupying in the direct Γ valley which owes the smaller transport mass (mc) than that of indirect L valleys [21]. The direct bandgap can be measured directly as compared to the indirect bandgap because the indirect-gap recombination should be assisted by additional phonons to maintain the momentum conservation. Therefore, the determination of crossover point become a debatable issue due to the uncertain values of indirect bandgaps of GeSn alloys. GeSn bandgaps are usually predicted roughly by a quadratic polynomial with a coefficient bi named the bowing factor, which are determined by the above direct and indirect bandgap measurements at discrete Sn content. Several extracted bowing factors from measurements were reported [22-29]. We use the reported bi to plot the energy bandgaps (EgL and 4

EgΓ) of Ge1-xSnx as a function of Sn content (Fig. 1). The predicted crossover point has a wide range of 6~11% at room temperature. Recently, the nonlinear bi dependence of Sn content was also reported from PL data.

Fig. 1 The energy band gap of Ge1-xSnx vs. Sn content at 300 K [22-31]. The crossover point is around 6~11%. The indirect-direct transition point affects the amount of the carrier population of direct Γ valley which determine the enhancement of transport properties of GeSn channel transistors. In the following Sec. II A, we use the nonlocal empirical pseudopotential method (EPM) to calculate the GeSn band structure properties including bandgaps, effective mass, and non-parabolicity effect of Γ valley. A. Empirical pseudopotential method. Four terms of the one-electron pseudo-Hamiltonian: the kinetic energy, atomic local pseudopotential form factors (Vloc), nonlocal correction term (Vnloc), and the spin-orbit interactions (Vso) are derived from Ref. [32]. The solved eigenvalues from the Hamiltonian matrix correspond to the band energies. The detailed EPM calculation for SiGeSn used here is referred to Ref. [33]. The used same EPM parameters of Vloc(q) (Vloc √3 , Vloc √4 , Vloc √8 , and Vloc √11 ), spin-orbit interactions (ζ and µ), and a fast cut-

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off “tanh” part [34] (b5 and b6) of Ge and α-Sn are also listed and the number of element plane

ur wave basis set {G} is 339. The parameters of nonlocal correction terms are obtained from Ref. [32]. Linear interpolation of the elastic constants (C11, C12, and C44) [35] and a bowing of 0.047 Å [36] for the lattice constant (a0) of GeSn alloys are used. B. Deformation potential method. The splitting valence bands by biaxial strain on a (001) substrate are calculated using the reported DFM. [37, 38] The scheme of DFM for SiGeSn alloys is referred to refs. [38, 39]. The spin-orbit splitting ∆0 for SiGeSn is obtained by linear interpolation of Si, Ge, and Sn. The shear deformation potential b of -1.88 eV is used for Ge, Sn, and GeSn. [38] C. Model solid theory. A correction term ∆(x) [39] are used to modify the linear interpolation of ∆Ev,av. ∆Ev,av = 0.85 eV between Sn and Ge, where Ev,av = 0 eV for Ge, is obtained from reported ab initio calculations.[40] avGe = 2.23 eV and avSn = 1.58 eV are the hydrostatic valence band deformation potentials for Ge and Sn, respectively [41]. Note that for ∆Ev,avVCA, the hydrostatic contribution of the strain to the additional shift of Ev,av is defined as avGeSn∆ΩGeSn/ΩGeSn, where avGeSn is obtained by linear interpolation of avGe and avSn, and ∆ΩGeSn/ΩGeSn = εxx + εyy + εzz, where εxx, εyy, and εzz represent the strain components on a cubic system. D. 6 × 6 k·p model. The 6×6 k·p model has been widely used to calculate the valence band properties of bulk [17, 18] and the inversion layer of SiGe PFETs [42, 43] through the linear interpolation of the reported Luttinger-like and deformation potential parameters. Nevertheless, the knowledge of valence band structure properties in GeSn inversion layers is very limited due to unknown Luttinger parameters and deformation potential parameters of 6×6 k·p model of GeSn alloys. The fitted Luttinger parameters and deformation potential parameters of 6×6 k·p model derived from EPM is adopted [44].

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E. Importing Strain. We use TCAD tools to design how to induce strain effectively into channel of transistors through the CESL, dislocation stressor [12], the changes on Sn content of S/D stressors, TSV [11, 45, 46], and the changes on structures.

III. Transport properties Ge is a promising alternative channel material to Si because of its high mobility and compatibility with current Si technologies [13, 14, 47]. For hole channels, Ge outperforms other semiconductors [48], whereas III-V semiconductors (e.g. InAs, InSb) offer better electron mobility [49]. To ensure feasible integration with the current Si platforms, the Ge-based electron channel is highly desired in the fin field-effect transistor (FinFET) architecture [50]. Two concepts reportedly boost the Ge channel’s electron performance: orientation effects, such as the best performance on the (111) surface orientation due to the largest confined mass mz and smallest transport mass mc of indirect L valleys [6, 51, 52], and external strain effects that change the valley separation and degeneracy [13, 53]. However, the key factor in the lower electron mobility of indirect-gap Ge compared to direct-gap III-V semiconductors is the indirect L valleys with higher mc than the direct Г valley [49]. GeSn has recently started to attract a lot of attention since it could eventually work as both n- and p-type transistor. This following sections (Sec. III A and B) are focused on impacts of dimensions, structures, sidewall orientations, strain, and novel potential materials such as groupIV alloys by Sn incorporation on transport properties in scaled transistors. In this topic, we develop self-consistent Poisson-Schrödinger calculations that accounts for the different dimensions of fin height and width to solve the confined inversion layers of the double gate (DG), tri-gate, and GAA FinFET structures. The ballistic current or mobilites (electron and hole) of transistor is calculated based on virtual source approximation and the Kubo-Greenwood formula.

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A. Self-consistently solution. The Poisson and Schrödinger equations of electron and hole are developed by MATLAB program and TCAD simulators [43, 47]; the different shapes of FinFET structures such as DG [54], rectangle tri-gate, trapezoidal tri-gate, and Gate-all-around (GAA) [55] are considered. Then, the impacts of the sidewall orientations and the top gate will also be studied. B. Scattering mechanisms. The semi-classical transport-Boltzmann transport is used to calculate the electron and hole mobility, and the scattering mechanisms including Coulomb, phonon, interface roughness and remote phonon scattering are considered as compared to our experimental results (planar [43, 47, 56] and 3D transistors [55]). Carrier screening on Coulomb charges is calculated using Lindhard’s screening theory.

In summary, through advanced material and device simulations supported by experimental data, we investigate the potential group-IV materials in future electronic and optoelectronic devices. This project will allow for a better understanding of this group-IV materials, determine whether it can replace Si, and provide the semiconductor industry with improved simulation tools to design post-Si ultra-scaled devices and optoelectronics with lower power consumption and better performance.

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Acknowledgement This work was supported by Ministry of Science and Technology, Taiwan, R.O.C. The support of high-performance computing facilities by the Computer and Information Networking Center, National Taiwan University, is also highly appreciated.

Acknowledgements to high-performance computing facilities by the Computer and Information Networking Center, NTU. 1.

H.-S. Lan and C. W. Liu, “Hole Effective Mass of Strained Ge1-xSnx Alloys P-Channel Quantum-

Well MOSFETs on (001), (110), and (111) Ge Substrates,” ECS Transactions (also presented at the ECS SiGe, Ge, and Related Compounds Symposium), Vol. 75, no. 8, pp.571-578, 2016. (EI) 2.

H.-S. Lan and C. W. Liu, “Ballistic electron transport calculation of strained germanium-tin fin field-

effect transistors”, Appl. Phys. Lett., Vol. 104, 192101, 2014. (SCI) 3.

H.-S. Lan, and C. W. Liu, “Electron Ballistic Current Enhancement of Ge1-xSnx

FinFETs,” International Symposium on VLSI Technology, Systems and Applications (VLSI-TSA), Hsinchu, Taiwan, 2014. 4.

H.-S. Lan, Y.-T. Chen, J.-Y. Lin, and C. W. Liu, “Hole mobility boost of Ge p-MOSFETs by

composite uniaxial stress and biaxial strain”, ECS Transactions (also presented at the ECS SiGe, Ge, and Related Compounds Symposium), vol. 50, no. 9, pp.151-155, 2013. (EI) 5.

H.-S. Lan, Y.-T. Chen, Hung-Chih Chang, J.-Y. Lin, William Hsu, W. -C. Chang, and C. W. Liu,

"Electron scattering in Ge metal-oxide-semiconductor field-effect transistors," Appl. Phys. Lett., Vol. 99, 112109, 2011. (SCI)

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