Mott Formula Research Articles

Thermopower in the hopping conductivity region: Transition from Mott’s to Zvyagin’s formula

Zvyagin’s theoretical calculation for the thermopower due to electron hops is confirmed experimentally. It is shown for a-GaSb that in the Mott-law region the thermopower is a square-root function of temperature $$\sqrt T $$ , and at low temperatures T<25 K the hopping contribution to the Seebeck coefficient dominates. A temperature increase induces a transition to conductivity due to hops between nearest centers. The thermopower in this regime is described by the Mott formula modified so as to take into account the higher-order derivatives of the density of states. It is established that in a-GaSb the thermopower at temperatures 4.2 K <T<300 K can be represented as a superposition of two contributions: a hopping contribution and an anomalous contribution presumably due to phonon drag. A model is proposed which gives a quantitative description of the temperature dependence of the hopping thermopower on the basis of a single setup parameters characterizing the density of localized states.

Mechanism of conduction in doped polyaniline

The conductivity of -doped emeraldine polymers is studied as a function of temperature in the range . The dopant concentration of the samples varies between 0.1 M and 1.0 M. The temperature dependence of the dc electrical conductivity gives evidence for a transport mechanism based on variable-range hopping in three dimensions. Using Mott's formula for the dc conductivity, physically meaningful values of the density of states at the Fermi energy, the hopping energy and hopping distance are calculated.

Thermal activation and Andrade creep

Becker and Mott have provided different formulae for the thermal activation energy of a process that occurs under stress. As the applied stress tends towards the critical stress, Becker's formula requires the smaller activation energy. We argue that, while the Mott formula represents the energy required for the system to cross the barrier, the Becker formula represents the energy required for the barrier to be temporarily suppressed. Work of at least the Mott value has to be done to move the system from its position before the barrier to one beyond the barrier, so the Becker formula does not control the process. Andrade creep is a process in which the deformation increases with time with an exponent 1/3. Cottrell provided a model, based on the Becker formula, which explained this result. The Mott formula, applied to a modified physical model, yields an exponent 4/13, experimentally indistinguishable from Andrade's Law.

Unusual beats of the perpendicular-current giant magnetoresistance and thermopower of magnetic trilayers.

Oscillations of the giant magnetoresistance (GMR) and thermo-electric power (TEP) vs. both the thickness of the non-magnetic spacer and also that of the ferromagnetic slabs are studied in the current-perpendicular-to-plane (CPP) geometry of magnetic trilayer systems, in terms of a single-band tight-binding model without impurities. The spin-dependent conductance has been calculated from the Kubo formula by means of a recursion Green's function method and the semi-infinite ideal-lead wires trick. Additionally the TEP is obtained directly from Mott's formula. In general, the thickness oscillations of the GMR and the TEP may have just one or two (short and long) oscillations. The long period, related to spectacular beats, is apparently of non-RKKY type. The TEP oscillations are strongly enhanced with respect to those of the GMR, have the same periods, but different phases and a negative bias.

Electric-field activated variable-range hopping transport in PrBa2Cu3O7- delta.

We demonstrate the transport of charge carriers in ${\mathrm{PrBa}}_{2}$${\mathrm{Cu}}_{3}$${\mathrm{O}}_{7\mathrm{\ensuremath{-}}\mathrm{\ensuremath{\delta}}}$ (PBCO) to be dependent both on the applied electric field and on the temperature. In our measurements we use inert noble-metal contacts on laser ablated and sputtered PBCO films. By applying the transmission line model we are able to separate the contact resistance from the PBCO resistance. The average hopping distance can be found by extending Mott's formula to field activation, and is found to be much greater than the dimensions of the PBCO unit cell. From the measurements in strong electric field a minimum hopping distance in the direction of the applied field of about 14 nm is determined, which we discuss in terms of localized states and intrinsic mixed valence of the Pr atoms in the PBCO film.

High precision measurement of Debye-Waller factors for NiAl

Motivation Quantitative convergent beam electron diffraction (CBED) is increasingly appreciated as a tool to determine bonding charge densities of crystalline materials. Simulated CBED patterns are fitted to experimental ones to derive the structure factors. These are converted by means of the Mott formula to yield the total charge density. Finally a neutral atom total charge density is subtracted and the difference is interpreted as the bonding charge density. Accounting for the temperature by a Debye-Waller factor (DWF) the g-th Fourier coefficient of the bonding charge density is then given by where u denotes the thermal root mean square atomic displacement. Here we have assumed thesimplest case of identical isotropic atomic vibrations for all atoms in the crystal. In order to estimate the error in due to the uncertainty in u we insert the results obtained by Fox and Tabbernor for NiAl at room temperature. They found differences between atomic and measured values of in the order of 2 percent.

The chemical control of high-Tc superconductivity: Metal-superconductor-insulator transition in (Tl1−yPby)Sr2(Ca1−xYx)Cu2O7

The authors have carried out a detailed study of the relationship between the semiconducting, the superconducting, and the metallic region in the system (Tl{sub 0.5}Pb{sub 0.5})Sr{sub 2}(Ca{sub 1{minus}x}Y{sub x})Cu{sub 2}O{sub 7}. The critical density (or concentration), n{sub c}, derived from Hall measurement of the titled series compounds at which the transition from insulator to metal place can be given by the well-known Mott formula: n{sub c}{sup 1/3}a{sub H*}=0.26 where a{sub H*} is the effective Bohr orbit radius of the isolated donor or acceptor state wavefunction. The authors propose that the insulator-metal transition in the cuprates generally occurs at a very high hole density (n{sub c} {approximately}10{sup 20}-10{sup 21}cm{sup {minus}3}) as compared to conventional doped semiconductors (n{sub c} {approximately} 10{sup 17}-10{sup 18}cm{sub {minus}3}) but rather lower than the critical densities for elemental metals (n{sup c} > 10{sup 21}cm{sup {minus}3}).

Calculation of the residual resistivity and the thermoelectric power of sp impurities in silver.

We present ab initio calculations for the residual resistivity and the thermopower of sp impurities in silver. The calculations are based on density-functional theory and the Korringa-Kohn-Rostoker Green's-function method. The linearized Boltzmann equation is solved by means of a Fermi-surface harmonic expansion to calculate the residual resistivity, taking into account the anisotropic electronic properties of the dilute alloy. The thermoelectric power was obtained from the energy dependence of the residual resistivity by means of Mott's formula. The results are in satisfying agreement with highly precise experimental results and confirm experimentally obtained deviations from a quadratic dependence on the valence difference \ensuremath{\Delta}Z for the thermopower of sp impurities in silver.

Structure refinement from systematic row integrated intensities and non-systematic critical voltages

Renewed interest in the quantitative side of electron diffraction may be due to several factors: •convergent beam diffraction is reckognized as a source for precise and useful intensity data for structure determination;•structural features where electron diffraction offers high sensitivity, e.g. disorder, defects, transformation structures and microstructures, charge distributions, receive increasing attention in solid state research. Various approaches and applications of electron diffraction to structure determination and refinement have been reported in recent years,. In our lab we have concentrated on two methods: the non-systematical critical voltage effect applied to a inner reflections from small unit cells; intensity profiles and integrated intensities from reflections in a systematic row. Applications include charge distributions and ionicities, refinement of coordinates and thermal parameters, supplement to X-ray structure determinations.The studies of charge distribution combine two basic features of electron difraction: the high sensitivity of inner reflections to charge transfer, cf the Mott formula

Elastic scattering of fast electrons and positrons by atoms

Elastic scattering of relativistic electrons and positrons by atoms is considered in the framework of the static field approximation. The scattering field is expressed as a sum of Yukawa terms to allow the use of various approximations. Accurate phase shifts have been computed by combining B\"uhring's power-series method with the WKB and Born approximations. This combined procedure allows the evaluation of differential cross sections for kinetic energies up to several tens of MeV. Numerical results are used to analyze the validity of several approximate methods, namely the first- and second-order Born approximations and the screened Mott formula, which are frequently adopted as the basis of multiple scattering theories and Monte Carlo simulations of electron and positron transport.

Instanton approach to the conductivity of a disordered solid

For the ac-conductivity of a one-dimensional gaussian-disordered electron system a supersymmetric field theory is constructed. The saddle-point equation for the limit of deeply localized states is solved exactly. Calculating the fluctuations around this saddle point in lowest order of the frequency, we obtain the Mott formula σ( ω)∼ ω 2In 2 ω also in the limit of weak disorder in contrast to the work of Houghton, Schäfer and Wegner.

Electron-phonon contribution to the thermopower of metals.

A theorem is proved for the thermopower when limited by the electron-phonon interaction. It is shown for simple metals that the thermopower is given by the Mott formula plus a correction term. The correction term is evaluated and shown to be small for simple metals. Thus we are able to show that Mott's formula is valid for phonon scattering by electrons, even when the energy dependence of the phonons is taken into account. This proof extends our earlier result, which only applied to static impurities, and to phonons in the adiabatic approximation.

Optimum collection angle for detection of small catalyst particles in high-angle dark-field Scanning Transmission Electron Microscopy

Since Crewe et al demonstrated the detection of single heavy atoms on thin carbon support films, there has been considerable interest in the use of high angle annular dark field detectors(HADD) to study particles in the subnanometer size range. A recent contribution to this subject is the work of Treacy and Rice.Increased interest in this technique can be attributed to the minimization of diffraction and phase contrast effects in the imaging of small atom catalyst particles .In this study we calculate the expected signal using realistic electron scattering cross sections. For the high angle dark field considered, it is reasonable to assume single scattering and to consider dynamical effects averaged out by the averaging over annular aperture.The electron scattering amplitude used is given by the Mott formula;

Relativistic Hartree–Fock X-ray and electron atomic scattering factors at high angles

An enlarged set (atomic number Z = 2 to 98) of free-atom X-ray atomic scattering (form) factors for high angles [2 ≤ (sin θ)/λ ≤ 6Å-1] has been calculated based on those of Doyle & Turner [Acta Cryst. (1968). A24, 390-397]. Four-parameter 'exponential polynomial' fits for these are presented which give far more accurate estimates of the scattering factors at high angles than the Gaussian fits normally used. The use of the Mott formula in conjunction with these new high-angle X-ray form factors allows the calculation of improved-accuracy high-angle electron scattering factors. The use of these high-accuracy high-angle scattering factors for important applications such as Fourier charge-density analysis and computer simulation of high-resolution electron microscope (HREM) images is discussed.

On the accurate measurement of low-order structure factor amplitudes and phases by quantitative CBED

Over the past four years we have developed the software and computer control of electron microscope data acquisition needed for quantitative electron crystallography. CBED patterns in the systematics orientation (see Fig. 1) are scanned over the slit of a Gatan Model 607 ELS unit, and the energy filtered elastic scattering rocking curves compared with the results of theoretical calculations (see Figs. 2 and 3). The calculations are based on the Bloch wave method, and treat three-dimensional dynamical diffraction from a non-centrosymmetric crystal with absorption and inclined boundary conditions. The Fortran source code has been published, and is available by Bitnet from ZALUZEC @ANLMST. High voltage, sample thickness, absorption coefficients Vg and selected low order structure factors Vg are adjusted(using the most sensitive portion of the pattern for each)for best fit. The resulting Fourier coefficients of crystal potential may be used (via the Mott Formula) to obtain a charge density difference map, which reveals the distribution of valence electrons involved in bonding, as shown for GaAs in figure 4.The partially ionic character of the bond is seen. We find V(111)1/V(111) = 0.05, V1 (220)/V(220) = 0.035, V1 (400)/V (400) = 0.07 and V1 (333)/V(333)=0.08. Results for work on MgO in progress are shown in figs. 2 and 3.

Electrical properties of a poly(meta-phenylene) network

Three-dimensional networks built up by phenylene groups linked in meta positions were prepared by polymerization of 1,3,5-trichlorobenzene according to a Wurtz-Fittig reaction. Compared with linear polyphenylenes, the phenylene network exhibits a high electrical conductivity of the order of 10 −4 ohm −1 cm −1 at 300 K. The temperature dependence of the dc electrical conductivity and the thermopower gives evidence for a transport mechanism based on variable-range hopping (VRH) in three dimensions. Using Mott's formula for the dc conductivity in VRH, physically meaningful values of the density of states at E F, the hopping energy and hopping distance are calculated. The small thermopower values are consistent with the proposed explanation in terms of VRH.

Errors arising from numerical use of the Mott formula in electron image simulation

The Mott formula relating the electron scattering factor to that for X-rays is inaccurate in its numerical form in the small-angle region. Effects of the inaccuracy on both kinematic and dynamical diffraction have been investigated. Some artifacts arising from the inaccuracy have been found in simulated electron microscopy images. A modified form for the Mott formula, which minimizes the error, is proposed. However, for any accurate and reliable calculation only those electron scattering factors derived directly from the atomic potential can be used.

A Modification of the Mott Formula for prediction of the fragment size distribution

AbstractA modification of the Mot fragmentation formula, accounting for a maximum fragment size, is suggested. Twelve experimental data sets are used to test the goodness of fit of the original and the modified formula. Generally, the modified formula is found to give a better representation of the experimental data, and particularly, the overestimation of the number of very large fragments is avoided.

Thermoelectric power of two-dimensional electron gas in heterojunctions at low temperatures

The low-temperature thermopower of two-dimensional electron gas has been calculated by using the expression obtained from the Boltzmann equation. The relevant scattering mechanisms are taken into account. It is found that the calculated thermopower increases almost linearly with temperature in the GaAs-AlGaAs and InGaAs-InP structures, but in all cases the results differ widely from the values given by the Mott formula and also from the experimental values. The results are not influenced by phonon scattering and are slightly affected if the impurity density is changed by the order of magnitude. For high-mobility samples, the experimental values are higher than the theoretical ones and it seems that phonon drag thermopower may account for the difference. For low-mobility samples, however, the experimental values are lower and in some cases decrease with temperature. The present model is then inadequate. Possible refinements of the theory are suggested.

Compensation Effect of Electrical Transport and Thermal Properties in Si:P

Electrical transport and thermal properties were studied on compensated Si:P with special interest in the critical behavior around the metal-nonmetal (MN) transition induced by Anderson localization in disordered system, measuring DC electrical resistivity and specific heat. Rigidity of the band structure and critical behavior of localization length were confirmed from the electronic specific heat and the Mott formula resistivity.