A. V. Dvurechenskii

Effect of electric field on diffusion in disordered materials. II. Two- and three-dimensional hopping transport

In the previous paper [Nenashev et al., arXiv:0912.3161] an analytical theory confirmed by numerical simulations has been developed for the field-dependent hopping diffusion coefficient D(F) in one-dimensional systems with Gaussian disorder. The main result of that paper is the linear, non-analytic field dependence of the diffusion coefficient at low electric fields. In the current paper, an analytical theory is developed for the field-dependent diffusion coefficient in three- and two-dimensional Gaussian disordered systems in the hopping transport regime. The theory predicts a smooth parabolic field dependence for the diffusion coefficient at low fields. The result is supported by Monte Carlo computer simulations. In spite of the smooth field dependences for the mobility and for the longitudinal diffusivity, the traditional Einstein form of the relation between these transport coefficients is shown to be violated even at very low electric fields.

Strain distribution in quantum dot of arbitrary polyhedral shape: Analytical solution

An analytical expression of the strain distribution due to lattice mismatch is obtained in an infinite isotropic elastic medium (a matrix) with a three-dimensional polyhedron-shaped inclusion (a quantum dot). The expression was obtained utilizing the analogy between electrostatic and elastic theory problems. The main idea lies in similarity of behavior of point charge electric field and the strain field induced by point inclusion in the matrix. This opens a way to simplify the structure of the expression for the strain tensor. In the solution, the strain distribution consists of contributions related to faces and edges of the inclusion. A contribution of each face is proportional to the solid angle at which the face is seen from the point where the strain is calculated. A contribution of an edge is proportional to the electrostatic potential which would be induced by this edge if it is charged with a constant linear charge density. The solution is valid for the case of inclusion having the same elastic co...

Role of diffusion in two-dimensional bimolecular recombination

Experiments on carrier recombination in two-dimensional organic structures are often interpreted in the frame of the Langevin model with taking into account only the drift of the charge carriers in their mutual electric field. While this approach is well justified for three-dimensional systems, it is in general not valid for two-dimensional structures where the contribution of diffusion can play a dominant role. We study the two-dimensional Langevin recombination theoretically and find the critical concentration below which diffusion cannot be neglected. For typical experimental conditions, neglecting the diffusion leads to an underestimation of the recombination rate by several times.

Fundamental characteristic length scale for the field dependence of hopping charge transport in disordered organic semiconductors

Using analytical arguments and computer simulations we show that the dependence of the hopping carrier mobility on the electric field μ(F )/μ(0) in a system of random sites is determined by the localization length a, and not by the concentration of sites N . This result is in drastic contrast to what is usually assumed in the literature for a theoretical description of experimental data and for device modeling, where N−1/3 is considered as the decisive length scale for μ(F ). We show that although the limiting value μ(F → 0) is determined by the ratio N−1/3/a, the dependence μ(F )/μ(0) is sensitive to the magnitude of a, and not to N−1/3. Furthermore, our numerical and analytical results prove that the effective temperature responsible for the combined effect of the electric field F and the real temperature T on the hopping transport via spatially random sites can contain the electric field only in the combination eFa.

Hopping conduction in strong electric fields: Negative differential conductivity

Department of Physics and Material Sciences Center, Philipps-University, 35032 Marburg, Germany(Dated: September 23, 2008)Effects of strong electric fields on hopping conductivity are studied theoretically. Monte-Carlocomputer simulations show that the analytical theory of Nguyen and Shklovskii [Solid State Com-mun. 38, 99 (1981)] provides an accurate description of hopping transport in the limit of veryhigh electric fields and low concentrations of charge carriers as compared to the concentration oflocalization sites and also at the relative concentration of carriers equal to 0.5. At intermediateconcentrations of carriers between 0.1 and 0.5 computer simulations evidence essential deviationsfrom the results of the existing analytical theories.The theory of Nguyen and Shklovskii also predicts a negative differential hopping conductivityat high electric fields. Our numerical calculations confirm this prediction qualitatively. Howeverthe field dependence of the drift velocity of charge carriers obtained numerically differs essentiallyfrom the one predicted so far. Analytical theory is further developed so that its agreement withnumerical results is essentially improved.

Why the apparent order of bimolecular recombination in blend organic solar cells can be larger than two: A topological consideration

The apparent order δ of non-geminate recombination higher than δ = 2 has been evidenced in numerous experiments on organic bulk heterojunction (BHJ) structures intensively studied for photovoltaic applications. This feature is claimed puzzling, since the rate of the bimolecular recombination in organic BHJ systems is proportional to the product of the concentrations of recombining electrons and holes and therefore the reaction order δ = 2 is expected. In organic BHJ structures, electrons and holes are confined to two different material phases: electrons to the acceptor material (usually a fullerene derivative) while holes to the donor phase (usually a polymer). The non-geminate recombination of charge carriers can therefore happen only at the interfaces between the two phases. Considering a simple geometrical model of the BHJ system, we show that the apparent order of recombination can deviate from δ = 2 due solely to the topological structure of the system.

Quantum logic gates from time-dependent global magnetic field in a system with constant exchange

We propose a method for implementation of an universal set of one- and two-quantum-bit gates for quantum computation in the system of two coupled electrons with constant non-diagonal exchange interaction. Suppression of the exchange interaction is offered to implement by all-the-time repetition of single spin rotations. Small g-factor difference of electrons allows to address qubits and to avoid strong magnetic field pulses. It is shown by means of numerical experiments that for implementation of one- and two-qubit operations it is sufficient to change the amplitude of the magnetic field within a few Gauss, introducing in a resonance one and then the other electron. To find the evolution of the two-qubit system, we use the algorithms of the optimal control theory.

Elastic strain field due to an inclusion of a polyhedral shape with a non-uniform lattice misfit

An analytical solution in a closed form is obtained for the three-dimensional elastic strain distribution in an unlimited medium containing an inclusion with a coordinate-dependent lattice mismatch (an eigenstrain). Quantum dots consisting of a solid solution with a spatially varying composition are examples of such inclusions. It is assumed that both the inclusion and the surrounding medium (the matrix) are elastically isotropic and have the same Youngs modulus and Poisson ratio. The inclusion shape is supposed to be an arbitrary polyhedron, and the coordinate dependence of the lattice misfit, with respect to the matrix, is assumed to be a polynomial of any degree. It is shown that, both inside and outside the inclusion, the strain tensor is expressed as a sum of contributions of all faces, edges, and vertices of the inclusion. Each of these contributions, as a function of the observation points coordinates, is a product of some polynomial and a simple analytical function, which is the solid angle subt...

Nucleation sites of Ge nanoislands grown on pit-patterned Si substrate prepared by electron-beam lithography

Regular pit-patterned Si(001) substrates were prepared by electron-beam lithography followed by plasma chemical etching. The geometry of the pits was controlled by varying the etching conditions and the electron-beam exposure duration. It was shown that the location of three-dimensional (3D) Ge nanoislands subsequently grown on the pit-patterned Si substrates depends on the shape of the pit bottom. In the case of pits having a sharp bottom, 3D Ge islands nucleate inside the pits. For pits with a wide flat bottom, the 3D Ge island nucleation takes place at the pit periphery. This effect is attributed to the strain relaxation depending not only on the initial pit shape, but also on its evolution during the Ge wetting layer deposition. It was shown by Monte Carlo simulations that in the case of a pit with a pointed bottom, the relaxation is most effective inside the pit, while for a pit with a wide bottom, the most relaxed area migrates during Ge deposition from the pit bottom to its edges, where 3D Ge islands nucleate.Regular pit-patterned Si(001) substrates were prepared by electron-beam lithography followed by plasma chemical etching. The geometry of the pits was controlled by varying the etching conditions and the electron-beam exposure duration. It was shown that the location of three-dimensional (3D) Ge nanoislands subsequently grown on the pit-patterned Si substrates depends on the shape of the pit bottom. In the case of pits having a sharp bottom, 3D Ge islands nucleate inside the pits. For pits with a wide flat bottom, the 3D Ge island nucleation takes place at the pit periphery. This effect is attributed to the strain relaxation depending not only on the initial pit shape, but also on its evolution during the Ge wetting layer deposition. It was shown by Monte Carlo simulations that in the case of a pit with a pointed bottom, the relaxation is most effective inside the pit, while for a pit with a wide bottom, the most relaxed area migrates during Ge deposition from the pit bottom to its edges, where 3D Ge islan...

Approximate analytical description of the elastic strain field due to an inclusion in a continuous medium with cubic anisotropy

We suggest an approach to the analytical calculation of the strain distribution due to an inclusion in elastically anisotropic media for the case of cubic anisotropy. The idea consists in the approximate reduction of the anisotropic problem to a (simpler) isotropic problem. This gives, for typical semiconductors, an improvement in accuracy by an order of magnitude, compared to the isotropic approximation. Our method allows using, in the case of elastically anisotropic media, analytical solutions obtained for isotropic media only, such as analytical formulas for the strain due to polyhedral inclusions. The present work substantially extends the applicability of analytical results, making them more suitable for describing real systems, such as epitaxial quantum dots.