Band Structure Of Silicon Research Articles

Photoluminescence lifetimes of Si quantum dots

We present a continuum model for the calculation of the electron states in Si dots that accounts for the effects of size, shape, and crystallographic orientation of the dots. This formalism has been used to study the behavior of the photoluminescence (PL) lifetime in Si quantum dots. This is due to the anisotropy of the silicon band structure and the confinement in quantum dots, which result in a cluster of energy levels from the different valleys of Si. Although these levels are very close in energy, they have very different recombination rates. Hence, there are (relatively) fast and slow levels at approximately the same energy. This feature causes a temperature dependence of PL in Si nanostructures, hence it is suggested that dispersion in the magnitude of the PL lifetimes in Si dots is at the origin of the observed stretched exponential behavior of PL lifetime in porous Si. Both zero phonon and phonon-assisted recombinations have been included in the calculations. Zero phonon recombination dominates in small dots (∼2nm) and the lifetime is ∼10μs. In larger dots, of a size of ∼4nm and above, phonon-assisted transitions become dominant and PL lifetimes are of the order of 1–10ms.

The One Phonon Raman Spectrum of Silicon Nanostructures

Porous silicon is a structurally complex material, in which effects of the pore topology on its physical properties are even controversial. We use the Born potential and the Green's function, both applied to a supercell model, in order to analyze the Raman response and the phonon band structure of porous silicon. In this model, the pores are simulated by empty columns of atoms, in direction [001], produced in a crystalline silicon structure. An advantage of this model is the interconnection between silicon nanocrystals, then, all the states are extended. The results show that the Raman spectra are sensitive to the pore topology. Moreover, a shift of the main Raman peak towards lower frequencies is found, in agreement with experimental data

Physics of Hole Transport in Strained Silicon MOSFET Inversion Layers

A comprehensive quantum anisotropic transport model for holes was used to study silicon PMOS inversion layer transport under arbitrary stress. The anisotropic band structures of bulk silicon and silicon under field confinement as a twodimensional quantum gas are computed using the pseudopotential method and a six-band stress-dependent k.p Hamiltonian. Anisotropic scattering is included in the momentum-dependent scattering rate calculation. Mobility is obtained from the Kubo-Greenwood formula at low lateral field and from the fullband Monte Carlo simulation at high lateral field. Using these methods, a comprehensive study has been performed for both uniaxial and biaxial stresses. The results are compared with device bending data and piezoresistance data for uniaxial stress, and device data from strained Si channel on relaxed SiGe substrate devices for biaxial tensile stress. All comparisons show a very good agreement with simulation. It is found that the hole band structure is dominated by 12 wings, where mechanical stress, as well as the vertical field under certain stress conditions, can alter the energies of the few lowest hole subbands, changing the transport effective mass, density-of-states, and scattering rates, and thus affecting the mobility

PBG properties of three-component 2D photonic crystals

In this paper we analyze theoretically how the introduction of the third component into the two-dimensional photonic crystal influences the photonic band structure and the density-of-states of the system. We consider the periodic array of cylindrical air rods in a dielectric, and the third medium is introduced as a ring-shaped intermediate layer of thickness d and dielectric constant ɛ i between the air pores and the dielectric background. Using the plane wave method, we have obtained the band structures for the 2D triangular lattice photonic crystals. The dependencies of TE and TM band gaps’ widths and gaps’ edges position on the interlayer dielectric constant and interlayer thickness were analyzed. In the framework of this approach, we have estimated the influence of the surface oxide layer on the band structure of macroporous silicon. We observed the shift of the gaps’ edges to the higher or lower frequencies, depending on the interlayer thickness and dielectric constant. We have shown that the existence of a native oxide surface layer should be taken into consideration to understand the optical properties of 2D photonic crystals, particularly in macroporous silicon structures.

Band structure of silicon as measured in extended momentum space

Direct measurement of the wave function (or at least the modulus squared of the wave function, the spectral function) is an important goal in electron spectroscopy. This requires a state-selective (i.e., energy resolved) measurement of the momentum density in all of momentum space, not just the reduced Brillouin zone. Photoemission has been used very successfully to measure dispersion, mainly in the reduced zone scheme. Compton measurements determine a projection of the momentum density in the full momentum space, but do not contain energy information. Here we present electron momentum spectroscopy measurements of extremely thin silicon single crystals, that resolve both energy and momentum, not just the reduced momentum. Measurements were done along different lines in extended momentum space, that are equivalent within the reduced zone scheme. For different lines different bands dominate, resulting in dramatic different spectral momentum densities. The observed intensities compare well to the spectral function as obtained by linear muffin tin band structure calculations. The results show a unified picture that forms a bridge between Compton measurements determining densities and photoemission measurements determining dispersion.

Valence band structure of ultrathin silicon and germanium channels in metal-oxide-semiconductor field-effect transistors

The ultrathin body (UTB) silicon-on-insulator metal-oxide-semiconductor field-effect transistor (MOSFET) is promising for sub-50-nm complementary metal-oxide semiconductor technologies. To explore a high-mobility channel for this technology, this paper presents an examination of Si and Ge hole sub-band structure in UTB MOSFETs under different surface orientations. The dependence of the hole subband structure on the film thickness (TBody) was also studied in this work. We found that the valence-band mixing in the vicinity of the zone center Γ is strongly dependent on TBody for both Si and Ge, particularly for the ⟨110⟩ surface orientation. This gives rise to the following two phenomena that crucially affect the electrical characteristics of p-MOSFETs: (1) an anomalous increase of quantization mass for ⟨110⟩ Si and Ge surfaces as TBody is scaled below 5nm. (2) The dependence of energy dispersion and anisotropy on TBody especially for the ⟨110⟩ surface, which advantageously increases hole velocity along the [011] channel as TBody is decreased. The density of states for different surface orientations are also calculated, and show that—for any given surface orientation—Ge has a smaller density of states than Si. The Ge ⟨110⟩ surface has the lowest density of states among the surface orientations considered.

Poisson-Schrödinger andab initiomodeling of doped Si nanocrystals: Reversal of the charge transfer between host and dopant atoms

We present ab initio density functional calculations that show P (Al) dopant atoms in small hydrogen-terminated Si crystals to be negatively (positively) charged. These signs of the dopant charges are reversed relative to the same dopants in bulk Si. We have therefore developed a self-consistent Poisson-Schr\"odinger model that allows us to bridge these two regimes of different charge character. Our Poisson-Schr\"odinger model is based on a nonorthogonal tight-binding model that reproduces the band structure of silicon very well, and we have also developed parameters for P, Al, and H. Using this model, we predict this reversal of the dopant charge to occur at crystal sizes of the order of 100 Si atoms. We explain it as a result of the competition between fundamental principles governing charge transfer in bulk semiconductors and molecules. Based on these general considerations, we expect it to occur in nanocrystals of most semiconductors. We also calculate band-edge energies and dopant-level energies for a number of crystallites containing 29--888 Si atoms.

Silicon-based Molecular Electronics

Molecular electronics on silicon has distinct advantages over its metallic counterpart. We describe a theoretical formalism for transport through semiconductor-molecule heterostructures, combining a semi-empirical treatment of the bulk silicon bandstructure with a first-principles description of the molecular chemistry and its bonding with silicon. Using this method, we demonstrate that the presence of a semiconducting band-edge can lead to a novel molecular resonant tunneling diode (RTD) that shows negative differential resistance (NDR) when the molecular levels are driven by an STM potential into the semiconducting band-gap. The peaks appear for positive bias on a p-doped and negative for an n-doped substrate. Charging in these devices is compromised by the RTD action, allowing possible identification of several molecular highest occupied (HOMO) and lowest unoccupied (LUMO) levels. Recent experiments by Hersam et al. [1] support our theoretical predictions.

High-Speed Light Modulation in Avalanche Breakdown Mode for Si Diodes

The light emission process from a p-n junction in the forward-bias region is slow to respond to modulation signals due to the indirect band structure of silicon. Experimental results for a reverse-bias region showing light modulation in the range of tens of gigahertz are observed for the first time. For such a light emitter, the limiting speed of light modulation is shown to be determined by the transit time of the minority carriers across the junction during the filament formation of breakdown currents, which has been demonstrated by simulation of the propagation of a shockwave-like pattern in the breakdown field.

Analysis of quantum ballistic electron transport in ultrasmall silicon devices including space-charge and geometric effects

A two-dimensional device simulation program which self consistently solves the Schrödinger and Poisson equations with current flow is described in detail. Significant approximations adopted in this work are the absence of scattering and a simple six-valley, parabolic band structure for silicon. A modified version of the quantum transmitting boundary method is used to describe open boundary conditions permitting current flow in device solutions far from equilibrium. The continuous energy spectrum of the system is discretized by temporarily imposing two different forms of closed boundary conditions, resulting in energies which sample the density-of-states and establish the wave function normalization conditions. These standing wave solutions (“normal modes”) are decomposed into their traveling wave constituents, each of which represents injection from only one of the open boundary contacts (“traveling eigencomponents”). These current-carrying states are occupied by a drifted Fermi distribution associated with their injecting contact and summed to form the electron density in the device. Holes are neglected in this calculation. The Poisson equation is solved on the same finite element computational mesh as the Schrödinger equation; devices of arbitrary geometry can be modeled. Computational performance of the program including characterization of a “Broyden+Newton” algorithm employed in the iteration for self consistency is described. Device results are presented for a narrow silicon resonant tunneling diode (RTD) and many variants of idealized silicon double-gate field effect transistors (DGFETs). The RTD results show two resonant conduction peaks, each of which demonstrates hysteresis. Three 7.5 nm channel length DGFET structures with identical intrinsic device configurations but differing access geometries (straight, taper and “dog bone”) are studied and found to have differing current flows owing to quantum-mechanical reflection in their access regions. Substantial gate-source overlap (10 nm) in these devices creates the possibility that the potential in the source can precipitously decrease for sufficiently high gate drive, which allows electron tunneling backwards through the channel from drain to source. A 7.5 nm gate length zero gate overlap taper device with 3 nm thick silicon channel is analyzed and internal distributions of device potential, electron density, velocity and current density are presented. As this device is scaled to 5 nm gate length, channel current is restricted due to the insufficient number of current-carrying states in the now 2 nm thick silicon channel. This restriction in current flow is removed by increasing the source and drain doping. A simple theory is presented to estimate the maximum current which can be carried by the ground state two-dimensional subband, and explains this restriction in current flow. Finally, the presence of circulating flow around vortices in individual subband states is demonstrated in both RTD and DGFET devices, including taper and dog bone DGFETs, a straight DGFET including a roughened Si–SiO2 interface, and a “bent” RTD.

Effective spin susceptibility, electron mass and Landé factor in two-dimensional (2D) Si inversion layers

We have studied the silicon (Si) band-structure, electron–electron and electron-ionized donor interaction effects on our accurate and approximate results (AcR and ApR) for renormalized effective spin susceptibitity (RESS), electron mass (EEM), Landé factor and spin polarization in the impure 2D Si (electron system), showing that: (i) our ApR, being strongly deviated from our AcR, reproduces approximately all the data obtained recently by Pudalov et al. (Phys. Rev. Lett. 88 (2002) 196404) [in particular, RESS =4.7 at the critical value of Wigner–Seitz radius r s : r s =r c ≈8.5 at which occur the “apparent” metal–insulator transition (MIT)] and can also be compared with other ApRs found in the recent literature, (ii) both the RESS and EEM produce physical singularities at the same critical value: r s= r c≅11.0566⪢1 (weakly disordered samples) at which occurs the “true” MIT; the existence of such two “apparent and true” critical values in this impure system agrees with a recent discussion by Abrahams et al. (Rev. Mod. Phys. 73 (2001) 251), and (iii) at r s= r c=8.5, at which occurs the “apparent” MIT, our AcR for effective spin polarization and the corresponding result, obtained using a disordered Hubbard model and a determinant quantum Monte Carlo method by Denteneer and Scalettar (Phys. Rev. Lett. 90 (2003) 246401), both give the same result: ξ eff.= ξ c≅0.31 at B≅0.4 T , which is found to be lower than the critical parallel magnetic field for full spin polarization, B c =1.29 T , supporting thus the existence of such an “apparent” MIT.

Quasiparticle band structure and screening in silicon and carbon clathrates

We present an accurate first-principles calculation of the quasiparticle electronic band structure of Si-34 and C-34 clathrates. Within the $\mathrm{GW}$ approximation, we find that Si-34 is a 1.85 direct band gap material. The reciprocal and real space dependence of the static dielectric function of Si-34 are provided. Further, we show by studying ${\mathrm{Xe}}_{8}@\mathrm{Si}\ensuremath{-}46$ that doping of Si-46 can lead to green-light semiconducting compounds, confirming that Si-clathrates are promising materials for optoelectronic applications. We finally predict the electronic properties of the hypothetical C-34 and C-46 phases within the same quasiparticle approach.

Optical study of the full photonic band gap in silicon inverse opals

An optical study of the band structure of both silicon–silica composite and silicon inverse opals is presented. The study is aimed at demonstrating the development of a full photonic band gap for a system already revealed as paradigmatic. The characterization is based on the comparison between the band structure calculations and optical reflectance spectroscopy experiments. This study is carried out for various symmetry points in the Brillouin zone, some never explored before as K, (110) and W, (210). The results show that, in accordance with the band structure, there is a certain frequency range that produces a reflectance peak regardless of orientation and can be assigned to the band gap. Similarly all other reflectance peaks can be accounted for by other band structure features.

Voltage- and temperature-dependent gate capacitance and current model: application to ZrO/sub 2/ n-channel MOS capacitor

Based on the energy-dispersion relation in each region of the gate-dielectric-silicon system, a tunneling model is developed to understand the gate current as a function of voltage and temperature. The gate capacitance is self-consistently calculated from Schrodinger and Poisson equations subject to the Fermi-Dirac statistics, using the same band structure in the silicon as used for tunneling injection. Franz two-band dispersion is assumed in the dielectric bandgap. Using a Wentzel-Kramer-Brillouin (WKB)-based approach, direct and Fowler-Nordheim (FN) tunneling and thermionic emission are considered simultaneously. The model is implemented for both the silicon conduction and valence bands and both gate- and substrate-injected currents. ZrO/sub 2/ NMOSFETs were studied through temperature-dependent C/sub g/-V/sub g/ and I/sub g/-V, simulations. The extracted band gaps and band offsets of the ZrO/sub 2/- and interfacial-Zr-silicate-layer are found to be comparable with the reported values. The gate currents in ZrO/sub 2/-NMOSCAPs are found to be primarily contributed from the silicon conduction band and tunneling appears to be the most probable primary mechanism through the dielectric. Oscillations of gate currents and kinks of gate capacitance were observed near the flat-band in the experiments. These phenomena might be caused by the interface states.

Optical second-harmonic interferometric spectroscopy ofSi(111)−SiO2interface in the vicinity ofE2critical points

The direct electron transitions at the $\mathrm{Si}(111)\ensuremath{-}{\mathrm{SiO}}_{2}$ interface are studied by combined second-harmonic (SH) intensity and phase spectroscopy with the SH photon energy range from 3.5 eV to 5 eV. A strong resonant behavior of the quadratic optical response is observed in the vicinity of the ${E}_{2}(X,\ensuremath{\Sigma})$ critical points (CP's) of the silicon band structure. Good representation of the experimental SH intensity and phase spectra is provided by a line shape of the quadratic susceptibility calculated both with excitonic and two-dimensional (2D) types of ${E}_{2}(X,\ensuremath{\Sigma})$ CP's. The line-shape dependence of resonance energies, namely, repulsion of nearby ${E}_{2}(X,\ensuremath{\Sigma})$ resonances in the case of a 2D CP in comparison with the excitonic ones, is observed.

Spontaneous hot-carrier photon emission rates in silicon: Improved modeling and applications to metal oxide semiconductor devices

The recent publication of controversial experimental evidence on the origin of hot-carrier currents in 4\char21{}10 nm tunnel metal oxide semiconductor capacitors renewed the interest in improving hot-carrier luminescence models for silicon devices. This work presents several such improvements, aimed at making possible a physically based analysis of the hot-carrier luminescence effects taking place during tunneling experiments in relatively thick ${\mathrm{SiO}}_{2}$ layers. To this purpose, silicon band structure and scattering rate calculations have been extended well above 10 eV by considering eight conduction bands, instead of the usual four, so as to allow for a detailed description of the high-energy carriers injected from silicon into silicon dioxide during tunneling experiments. The absolute contributions of the direct and phonon-assisted, interband and intraband transitions of electrons and holes to the total photon emission rate are analyzed, so the results can be directly compared with the experimental data. To the best of our knowledge, it is for the first time that results for valence-to-valence band transitions of holes are presented and compared with those of conduction-to-conduction band transitions of electrons. Results can be directly compared with experimental data. Template results obtained with a variety of carrier distributions (Maxwellian, Gaussian, and Dirac's delta-like) are shown and implications for device analysis are discussed.

Penetration of electronic states from silicon substrate into silicon oxide

The depth profiling of O 1s energy loss in silicon oxide near the SiO 2/Si interface was performed using extremely small probing depth. As a result, the energy loss of O 1s photoelectrons with threshold energy of 3.5 eV was found. This value of 3.5 eV is much smaller than the SiO 2 bandgap of 9.0 eV, but quite close to direct interband transition at Γ point in energy band structure of silicon. This can be explained by considering the penetration of electronic states from silicon substrate into silicon oxide up to 0.6 nm from the interface. In addition, the penetrating depth is larger than the thickness of the compositional transition layer.

Structural, electronic, and effective-mass properties of silicon and zinc-blende group-III nitride semiconductor compounds

The electronic band structures of silicon and the zinc-blende-type III-N semiconductor compounds BN, AlN, GaN, and InN are calculated by using the self-consistent full potential linear augmented plane wave method within the local-density functional approximation. Lattice constant, bulk modulus, and cohesive energy are obtained from full relativistic total-energy calculations for Si and for the nitrides. Band structures and total density of states (DOS) are presented. The role played by relativistic effects on the bulk band structures and DOS is discussed. In order to provide important band structure-derived properties, such as effective masses and Luttinger parameters, the ab initio band structure results are linked with effective-mass theory. Electron, heavy-, light-, and split-off-hole effective masses, as well as spin-orbit splitting energies are extracted from the band-structure calculations. By using the Luttinger-Kohn $6\ifmmode\times\else\texttimes\fi{}6$ effective-mass Hamiltonian we derive the corresponding Luttinger parameters for the materials. A comparison with other available theoretical results and experimental data is made.

Probe of the vicinal [formula omitted] surface by second harmonic phase spectroscopy

The intrinsic surface sensitive technique of optical second harmonic (SH) phase and intensity measurements is used to probe a slightly miscut natively oxidized Si(111) surface. The experiments have been performed in a spectral interval covering the E2 resonance of silicon band structure. The phase and intensity of the SH wave measured at several azimuthal angular positions allow to separate the vicinal contributions.

Temperature dependence analysis of the thermo-optic effect in silicon by single and double oscillator models

The thermo-optic coefficient (dn/dT) of crystalline silicon has been critically analyzed in the temperature range 300–600 K, at the fiber optic communication wavelength of 1.5 μm. The temperature dependence has been attributed to the variation of the interband transition energies at some critical points of the silicon band structure. The experimental data have been fitted using single and double oscillator models. In particular, the double oscillator model, which is physically correlated to the silicon band structure, has been exploited to extrapolate the temperature dependence of the interband transition energies at some points (critical points) of the combined density of states. The extracted parameters are in good agreement with the data reported in the literature. Finally, in connection with both of the oscillator approximations, an analysis based on thermodynamic considerations is carried out, and electron–hole formation entropy and specific heat are calculated. The consistency of the obtained results validate the reliability of the proposed analysis.