Numerical experiments on phonon properties of isotope and vacancy-type disordered graphene

We have studied phonon properties of graphene theoretically with different concentrations of ¹³C isotope and vacancy-type defects. The forced vibrational method, which is based on the mechanical resonance to extract the pure vibrational eigenmodes by numerical simulation, has been employed to compute the phonon density of states (PDOSs) and mode pattern of isotope-disordered graphene as well as a combined isotope and vacancy-type defective graphene structure. We observe a linear reduction of the E_2g mode frequencies with an increase in ¹³C concentration due to the reduced mass variation of the isotope mixture. We find a downshift of the E_2g mode of 65 cm^− 1, which is a very good agreement with the experimental results, and the phonon frequencies described by the simple harmonic oscillator model. The vacancy-type defects break down the phonon degeneracy at the Г point of the LO and TO modes, distort and shift down the phonon density of states significantly. The PDOS peaks for the combined isotope and vacancy-type defects show the remarkable increase in the low-frequency region induced by their defect formations. Due to phonon scattering by ¹³C isotope or vacancies, some graphene phonon wave functions become localized in the real space. Our numerical experiments reveal that the lattice vibrations in the defective graphene show the remarkably different properties such as spatial localization of lattice vibrations due to their random structures from those in the perfect graphene. The calculated typical mode patterns for in-plane K point optical phonon modes indicate that the features of strongly localized state depend on the defect density, and the phonon is localized strongly within a region of several nanometers in the random percolation network structures. In particular, for in-plane K point optical phonon modes, a typical localization length is on the order of ≈ 7 nm for isotope impurities, ≈ 5 nm for vacancy-type defects and ≈ 6 nm for mixed-type defects at high defect concentrations of 30%. Our findings can be useful for the interpretation of experiments on infrared, Raman, and neutron-diffraction spectra of defective graphene, as well as in the study of a wide variety of other physical properties such as thermal conductivity, specific heat capacity, and electron–phonon interaction.     

Heat transport across the metal–diamond interface

Thermal conductivity of the aluminium–diamond (Al–diamond) composites, prepared by the gas pressure infiltration method, is measured by steady state technique. A detailed theoretical investigation on the heat conduction mechanism across the Al–diamond interface is presented. It was confirmed that both electrons and phonons actively take part in the flow of heat at the interface. In the Al side, electrons of Al couple with the phonons and carry the heat up to the interface. This electron–phonon pair which predominantly carries heat in the Al, breaks down at the Al–diamond interface. The coupling between phonons of both Al and diamond takes place at the interface which eventually leads the heat conduction across the interface to the diamond. The phonon–phonon coupling across the interface is discussed by scattering mediated acoustic mismatch model (SMAMM). It is shown that for Al–diamond composite, the implementation of the SMAMM yields an interface thermal resistance (ITR) value of 4.44 × 10^− 9 m²K/W, which is in fairly good agreement with values derived from experimental thermal conductivity values of this composite implemented in the Hasselman–Johnson (HJ) mean field scheme.     

Spectral phonon thermal properties in graphene nanoribbons

This work provides a comprehensive investigation on the spectral phonon properties in graphene nanoribbons (GNRs) by the normal mode decomposition (NMD) method, considering the effects of edge chirality, width, and temperature. We find that the edge chirality has no significant effect on the phonon relaxation time but has a large influence to the phonon group velocity. As a result, the thermal conductivity of around 707 W/(m K) in the 4.26 nm-wide zigzag GNR at room temperature is higher than that of 467 W/(m K) in the armchair GNR with the same width. As the width decreases or the temperature increases, the thermal conductivity reduces significantly due to the decreasing relaxation times. Good agreement is achieved between the thermal conductivities predicted from the Green–Kubo method and the NMD method. We find that optical phonons dominate the thermal transport in the GNRs while the relative contribution of acoustic phonons to the thermal conductivity is only 10.1% and 13% in the zigzag GNR and the armchair GNR, respectively. Interestingly, the ZA mode is found to contribute only 1–5% to the total thermal transport in GNRs, being much lower than that of 30–70% in single layer graphene.     

Electronic thermal conductivity and thermopower of armchair graphene nanoribbons

A theoretical investigation of the diffusion contribution to thermopower, S_d, and the electronic thermal conductivity, κ_e, of semiconducting armchair graphene nanoribbons (GNRs) is made for T ⩽ 300 K. Considering the electrons to be scattered by edge roughness, impurities and deformation-potential coupled acoustic phonons and optical phonons, expressions for S_d and κ_e are obtained. Numerical calculations of S_d and κ_e, as functions of temperature and linear carrier density, bring out the relative importance of the contributing scattering mechanisms. A GNR of width 5 nm, supporting an electron density 2 × 10⁸ m⁻¹, is found to exhibit room temperature values of S_d and κ_e as 42 μV/K and 26.5 W/mK, respectively. A decrease in armchair GNR width, is found to enhance S_d and reduce κ_e. The effect of varying the electron density is to increase their magnitude when Fermi energy moves into the second subband. An analysis of thermopower and thermal conductivity data in clean armchair GNR samples will enable better understanding of the electron–phonon interaction.     

Frequency scaling of ac hopping transport in amorphous carbon nitride

The dynamics of hopping transport in amorphous carbon nitride is investigated in both Ohmic and non-linear regimes. Dc current and ac admittance were measured in a wide range of temperatures (90 K < T < 300 K), electric fields (F < 2 × 10⁵ V cm^− 1) and frequencies (10² < f < 10⁶ Hz).The dc Ohmic conductivity is described by a Mott law, i.e. a linear ln(σ_OHMIC) vs T^− 1/4 dependence. The scaling of field-enhanced conductivity as ln(σ / σ_OHMIC) = ϕ[F^S / T] with S ≈ 2/3, observed for F > 3 × 10⁴ V cm^− 1 over 5 decades in σ(T,F), is explained by band tail hopping transport; the filling rate, Γ_F(E_DL), of empty states at the transport energy is obtained with a “filling rate” method which incorporates an exponential distribution of localized states, with a non-equilibrium band tail occupation probability f(E) parametrized by an electronic temperature T_EFF (F).As the ac frequency and temperature increase, the increase in conductance G is accurately described by Dyre's model for hopping transport within a random spatial distribution of energy barriers. This model predicts a universal dependence of the complex ac conductivity of the form σ_ac = σ(0)[iωτ / ln(1 + iωτ)], where σ(0) is the zero frequency ac conductivity and τ(T,F) is a characteristic relaxation time. We find that the inverse characteristic time 1 / τ can also be described by a Mott law. It is compatible with the filling rate Γ_F(E_DL) at the transport energy, which governs the dc conductivity; this rate increases with increasing dc field, as more empty states become available in the band tail for hopping transitions. This “universal” scaling law for the ac conductance provides a scaling parameter K(T,F) = τ(T,F) σ(T,F,ω = 0) / ɛ which is found to decrease with increasing electric field from 5 to 0.5, depending weakly on temperature. Our band tail hopping model predicts a high-field value of K(T,F) smaller than the Ohmic value, under the condition (eFγ^− 1 / E°) ≤ (kT / E°)^1/4, where γ^− 1 is the localization radius and E° the disorder energy of the band tail distribution.     

Anharmonicity of graphite from UV Raman spectroscopy to 2700 K

First-order Raman spectra of pyrolytic graphite (PG) and highly oriented pyrolytic graphite (HOPG) were recorded in situ up to 2670 K and 2491 K, respectively, using a development of wire-loop heating cell technique attached to a UV-Raman spectrometer (244 nm). Raman shift of the E_2g in-plane stretching mode of graphite (G band) is used to discuss the anharmonicity by a comparison with calculations in the density-functional theory (DFT). High temperature Raman shifts are well described by anharmonic DFT calculations [1] up to 900 K. Anharmonicity is also determined from the temperature dependence of the Raman linewidth. The quartic term of phonon–phonon scattering process dominates at high temperature with respect to electron–phonon coupling that causes a slight decrease of linewidth with increasing temperature below 1000 K. The G band position is determined with a good reproducibility to 2700 K and can be used as a thermometer for in situ studies. Deep UV-Raman proves a viable solution for expanding significantly the temperature range for studying in situ vibrational properties of condensed matter, and particularly the monitoring of carbon-based material processing.     

Theoretical investigation of mechanical and thermal properties of MPO4 (M = Al,Ga)

Minimum lattice thermal conductivities and mechanical properties of polymorphous MPO₄ (M = Al, Ga) are investigated by first principles calculations. The theoretical minimum thermal conductivities are found to be 1.02 W (m K)^?1 for α-AlPO₄, 1.20 W (m K)^?1 for β-AlPO₄, 0.87 W (m K)^?1 for α-GaPO₄ and 0.88 W (m K)^?1 for β-GaPO₄. The lower thermal conductivities in comparison to YSZ can be attributed to the lattice phonon scattering due to the framework of heterogeneous bonds. In addition, the low shear-to-bulk modulus ratio for both β-AlPO₄ (0.38) and β-GaPO₄ (0.30) is observed. Our results suggest their applications as light-weight thermal insulator and damage-tolerant/machinable ceramics.     

Strain modification in epitaxial 2H-AlN layers on 3C-SiC/Si(111) pseudo-substrates

Optical measurements are used to investigate the crystalline quality and the stress in thin AlN layers; these thin films are grown on cubic silicon carbide layers which are in turn grown on silicon (111) substrates. Different Ge amounts were deposited at the silicon substrate in order to reduce the lattice parameters mismatch between Si and SiC grown layers. The residual stress of the hexagonal AlN layers is derived from the phonon frequency shifts of the E₁(TO) phonon mode. The crystalline quality of AlN films is investigated by considering the intensity of E₁(TO) mode of the 2H-AlN and its full width of the half maximum (FWHM). Ge deposition at low temperature 325 °C, before the carbonization process leads to an improved crystalline quality and a reduced residual stress in the AlN/SiC/Si heterostructures. The best crystalline quality and the lowest stress value are found in the case where 1ML Ge amount was predeposited. The E₁(TO) mode, phonon frequency shifts-down by 3 cm^? 1/GPa with respect to an unstrained layer. The obtained values for the phonon deformation are in reasonable agreement with theoretical calculations.     

Raman scattering by defect-induced excitations in boron-doped diamond single crystals

Polarized Raman spectra of the oriented boron-doped diamond with a different content of boron (≤ 200 ppm) were obtained with 514.5 and 1064 nm excitations. The additional bands were found in the region below 1200 cm^− 1. Their intensity increased with doping. It was shown that in polarized spectra these bands were in agreement with the singularities of density of phonon states (DOS) of diamond for the A_1g, E_g and F_2g symmetries. It was assumed that the ~ 900 cm^− 1 band which does not coincide with any DOS peak and has the highest resonance character may be attributed to the localized mode of boron in a diamond lattice. The spectra were accompanied by continuum that had the same symmetry F_2g as optical phonon at 1333 cm^− 1.     

Structural and optical characterization of Zn0.95−xMg0.05AlxO nanoparticles

The structural and optical properties of ZnO nanoparticles doped simultaneously with Mg and Al were investigated. XRD results revealed the hexagonal wurtzite crystalline structure of ZnO. The FE-SEM study confirmed the formation of nano-sized homogeneous grains whose sizes decreased monotonously with increasing doping concentrations of Mg and Al. The absorption spectra showed that band gap increased from 3.20 to 3.31 eV with Mg doping. As the Al concentration changed from x=0.01 to x=0.06 mol% at constant Mg concentration the band gap observed to be decreased. Particle sizes estimated from effective mass approximation using absorption data and these values are in good agreement with the crystallite sizes calculated from XRD data. Raman spectra of ZnO showed a characteristic peak at 436 cm⁻¹ correspond to a non-polar optical phonon E₂ (high). With increase of the Al doping concentrations, E₂ (high) phonon frequency shifted to 439 cm⁻¹ from to 436 cm⁻¹. The origin of E₂ (high) peak shift in ZnO nanoparticles is attributed to optical phonon confinement effects or the presence of intrinsic defects on the nanoparticles. PL spectra indicated that with increase of Al co-doping along with Mg into ZnO, intensity of the peak positioned at 395 nm was initially increased at x=0 and then decreased with increase of the Al concentrations from x=0.01 to x=0.06 mol%.     

Electrical properties of the high quality boron-doped synthetic single-crystal diamonds grown by the temperature gradient method

Temperature dependencies of the resistivity and the Hall coefficient in high-quality boron-doped synthetic single crystal diamonds grown by the high-pressure-high-temperature (HPHT) method with different boron contents have been investigated. The concentration of acceptors was varied in the range of 2 × 10¹⁵ to 3 × 10¹⁷ cm^–3 in (001) cut plates by a change of boron content in a growth mixture in a range from 0.0004 to 0.04 atomic percent. A special sample preparation has been used for precise measurements. Thin rectangular plates with uniform boron content and without linear and planar structure defects have been laser cut after X-ray topography and UV-luminescence mapping. The donor and acceptor concentrations in each sample have been calculated from the Hall effect data and capacitance–voltage characteristics. The concentrations correlate with the boron content in a growth mixture. Minimum donor to acceptor compensation ratio slightly below 1% was observed at 0.002 at.% boron content in a growth mixture, while it increased at an increase and decrease of boron amount. Samples grown at such boron concentration had maximum carrier mobility. It was 2200 cm² / (V × s) at T = 300 K and 7200 cm² / (V × s) at T = 180 K. The phonon scattering of holes dominates in the entire temperature range of 180–800 K, while the scattering by point defects such as neutral and ionized impurity atoms is insignificant. Due to a perfect crystal quality and lattice scattering mechanism bulk diamond crystals grown from the mixture containing 0.0005 to 0.002 at.% of boron may serve as reference semiconductor materials.     

Low-energy excitations of graphene on Ru(0 0 0 1)

The structure and the acoustic phonon branches of graphene on Ru(0 0 0 1) have been experimentally investigated with helium atom scattering (HAS) and analyzed by means of density functional theory (DFT) including Grimme dispersion forces. In-plane interactions are unaffected by the interaction with the substrate. The energy of 16 meV for the vertical rigid vibration of graphene against the Ru(0 0 0 1) surface layer indicates an interlayer effective force constant about five times larger than in graphite. The Rayleigh mode observed for graphene/Ru(0 0 0 1) is almost identical to the one measured on clean Ru(0 0 0 1). This is accounted for by the strong bonding to the substrate, which also explains the previously reported high reflectivity to He atoms of this system. Finally, we report the observation of an additional acoustic branch, closely corresponding to the one already observed by HAS in graphite, which cannot be ascribed to any phonon mode and suggests a possible plasmonic origin.     

High and anisotropic thermal conductivity of body-centered tetragonal C4 calculated using molecular dynamics

The thermal properties of body-centered tetragonal C₄ (bct-C₄), a new allotrope of carbon, were investigated using molecular dynamics (MD) simulations. The calculations gave a high and anisotropic thermal conductivity that is the first of its kind. The cross-plane thermal conductivity is 1209 W/(m K) at room temperature, which is even higher than that of diamond. The thermal conductivity decreases as the temperature increases from 80 to 400 K. The density of states of bct-C₄ was analyzed, which has a prominent peak at 36 THz. The relaxation times were calculated by fitting a heat flux autocorrelation function. The results showed that the acoustic phonons play the dominant role in the heat conduction, with a contribution of more than 99%. The relaxation times decrease with increasing temperature, as does the contribution of the acoustic phonons. Finally, the thermal conductivity based on lattice dynamics agreed well with that from the MD method, with which the group velocity and mean free path were deduced. This outstanding thermal property makes bct-C₄ a promising substitute for diamond, especially as thermal interface materials in microelectronic packaging.     

Large E-field induced strain and polar evolution in lead-free Zr-doped 92.5%(Bi0.5Na0.5)TiO3–7.5%BaTiO3 ceramics

Structure, dielectric permittivity, strain, electric (E) polarization, and piezoelectric responses of (Bi_1/2Na_1/2)_0.925Ba_0.075(Ti_1−xZr_x)O₃ (BNT7.5BT-100xZr; x = 0–0.04) ceramics were investigated as functions of poling E field and temperature. The BNT7.5BT ceramic reveals a phase transition from P4bm nanodomains to long-range-ordered P4mm domains. The Zr-doped BNT7.5BT ceramic reveals a reversible change of unit cell with dynamically fluctuating polar nanoregions, which are responsible for the large strain. The poled BNT7.5BT ceramic displays a depolarization temperature of T_d = 90 °C, which correspond to a phase transition from ferroelectric to relaxor states. The Zr-doped BNT7.5BT ceramics have Burns temperatures (T_B) in the region of 400–435 °C, below which polar nanoregions begin to develop. The Zr-doped BNT7.5BT ceramics display wide diffuse phase transitions, suggesting a transition from R + T to T phases. BNT7.5BT-2Zr ceramic shows a temperature dependent linear large strain of 0.482% at 150 °C and can be a potential candidate for lead-free actuator.     

Dynamics of negatively charged divacancies in neutron irradiated type Ib diamond

Thermal neutron irradiation of synthetic type Ib diamond, followed by annealing, results in a number of paramagnetic point defect centers. One of these is the composite defect designated W29. Previous EPR measurements have shown that the W29 center (S = 3/2) is a charged divacancy [VV]⁻ with spin Hamiltonian parameters similar to those of the R4/W6 neutral divacancy center [VV]⁰ (S = 1) found in irradiated high purity type IIa diamond. The present EPR linewidth and spin-lattice relaxation rate measurements, made as a function of temperature on the W29 center, show that an Orbach two-phonon process, with a gap of ~ 20 meV, plays a dominant role in determining the relaxation behavior above 20 K. Below 20 K a direct single phonon process provides the spin-lattice relaxation mechanism and the relaxation times become long, of the order of milliseconds. The linewidth behavior with temperature is accounted for in terms of changes in the spin dynamics.     

Symmetries and selection rules in the measurement of the phonon spectrum of graphene and related materials

When the phonon spectrum of a material is measured in a scattering experiment, selection rules preclude the observation of phonons that are odd under reflection by the scattering plane. Understanding these rules is crucial to correctly interpret experiments and to detect broken symmetries. Taking graphene as a case study, in this work we derive the complete set of selection rules for the honeycomb lattice, showing that some of them have been missed or misinterpreted in the literature. Focusing on the technique of high-resolution electron energy loss spectroscopy (HREELS), we calculate the scattering intensity for a simple force constant model to illustrate these rules. In addition, we present HREELS measurements of the phonon dispersion for graphene on Ru(0 0 0 1) and find excellent agreement with the theory. We also illustrate the effect of different symmetry breaking scenarios in the selection rules and discuss previous experiments in light of our results. Finally we clarify why the shear horizontal label is not equivalent to odd parity, and how this can be misleading in the identification of selection rules.     

Raman spectra and extended X-ray absorption fine structure characterization of La(2−x)/3NaxTiO3 and Nd(2−x)/3LixTiO3 microwave ceramics

A-site deficient perovskite compounds, La_(2?x)/3Na_xTiO₃ (0.02 ≤ x ≤ 0.5) and Nd_(2?x)/3Li_xTiO₃ (0.1 ≤ x ≤ 0.5) microwave ceramics, were investigated by Raman scattering. Nd_(2?x)/3Li_xTiO₃ (0.1 ≤ x ≤ 0.5) was also investigated by extended X-ray absorption fine structure (EXAFS) measurement. The Raman shifts of the E (239 cm^?1) and A₁ (322 cm^?1) modes of La_(2?x)/3Na_xTiO₃ were found to decrease with x. However, the E (254 cm^?1) and A₁ (338 cm^?1) of Nd_(2?x)/3Li_xTiO₃ were found to blueshift with x, which was caused by Li substitution. The redshift of the A₁ (471 cm^?1) phonon of Nd_(2?x)/3Li_xTiO₃ (0.1 ≤ x ≤ 0.3) indicates that O–Ti–O bonding forces lessen with Li concentration, which is consistent with the EXAFS result that Ti–O bond lengths increase for 0.1 ≤ x ≤ 0.3. For x > 0.3, the EXAFS result shows that Ti–O bond lengths decrease. Moreover, Ti–O bond lengths show strong correlation with the microwave dielectric constants of Nd_(2?x)/3Li_xTiO₃.     

Application of the Weibull distribution function for modeling the kinetics of isothermal dehydration of equilibrium swollen poly (acrylic acid) hydrogel

A theoretical procedure for evaluating the density distribution function (p(E_a)) of apparent activation energies (E_a), based on modeling of the experimental isothermal conversion curves by Weibull probability function of reacting times was established. This procedure was applied on isothermal dehydration of poly (acrylic acid) hydrogel (PAAH) at four different operating temperatures (306, 324, 345 and 361 K). It was established that: (a) the experimental conversion curves at all temperatures can be described by the Weibull probability distribution function of reacting times in a wide range of the degree of conversion (0.14 ? α ? 0.95); (b) the Weibull distribution parameters (β and η) show functional dependences on the operating temperature, and (c) the apparent activation energy (E_a) varies with the degree of conversion (α). It was shown that the density distribution function of activation energies is independent on the temperature, and because of this fact, the considered function is an Eigen characteristic of the dehydration studied. The calculated p(E_a) function is in good agreement with the experimental p(E_a) function (p(E_a)_exp), obtained by the Miura procedure.     

Study of polar relaxation processes in Sr(1−1.5x)LaxTiO3 ceramics by using field-induced thermally stimulated currents

Polar relaxation processes in Lanthanum doped SrTiO₃ (STO) ceramics, with general formulae Sr_(1−1.5x)La_xTiO₃, were studied by undertaking field-induced thermally stimulated currents measurements below room temperature.The experimental results obtained for doped ceramic (x = 0.0133) were analysed by using dipolar and space-charge relaxation thermally stimulated depolarization currents (TSDC) models in order to determine the nature of the relaxation processes involved.Our results reveal the existence of different relaxation processes in the temperature range 60–300 K. Whereas at low temperature, a relaxation mechanism of a dipolar type was disclosed within the temperature interval centred around 100 K, a space-charge relaxation process could be identified in the temperature range 120–300 K. The temperature dependence of the relaxation parameters will be also discussed in detail.     

Mechanical characterization of a polysiloxane-derived SiOC glass

Silicon oxycarbide glass with the composition Si_1.0O_1.6C_0.8 was synthesized from a commercial polysiloxane by polymer pyrolysis. Dense SiOC samples were obtained by cross linking of the polysiloxane followed by warm pressing to form cylindrical samples and subsequent pyrolysis of the shaped polymer at 1100 °C in Ar. Hardness (H), Young's modulus (E) and Poisson's ratio (ν) of the as-prepared SiOC glass were evaluated from indentation studies and from acoustic microscopy. Indentation studies showed that E depends on the applied load and amounts to 90 GPa for low load and to 180 GPa for high load. Average values of 6.4 and 101 GPa were obtained for H and E, respectively, by the Vickers indentation method. Acoustic microscopy analysis yielded values of 96 GPa and 0.11 for E and ν, respectively. Compared to vitreous silica, the Young's modulus of the SiOC glass is about 1.3–1.5 times higher. To the knowledge of the present authors, the measured Poisson's ratio (ν = 0.11) is the lowest reported so far for glasses and polycrystalline ceramics.