Temperature gradient and Fourier's law in gradient-mass harmonic systems

Heat flow and thermal profile in a 1D harmonic lattice with coordinate-dependent masses has been calculated in the thermodynamic limit. It is shown in the particular example of a 1D harmonic lattice with linearly increasing masses that in standard Langevin conditions of contact, a temperature gradient can form, and Fourier's law can be obeyed.Comment: http://link.aps.org/doi/10.1103/PhysRevE.87.05210

Dielectric Constant and Charging Energy in Array of Touching Nanocrystals

We calculate the effective macroscopic dielectric constant $\varepsilon_a$ of a periodic array of spherical nanocrystals (NCs) with dielectric constant $\varepsilon$ immersed in the medium with dielectric constant $\varepsilon_m \ll \varepsilon$. For an array of NCs with the diameter $d$ and the distance $D$ between their centers, which are separated by the small distance $s=D-d \ll d$ or touch each other by small facets with radius $\rho\ll d$ what is equivalent to $s < 0$, $|s| \ll d$ we derive two analytical asymptotics of the function $\varepsilon_a(s)$ in the limit $\varepsilon/\varepsilon_m \gg 1$. Using the scaling hypothesis we interpolate between them near $s=0$ to obtain new approximated function $\varepsilon_a(s)$ for $\varepsilon/\varepsilon_m \gg 1$. It agrees with existing numerical calculations for $\varepsilon/\varepsilon_m =30$, while the standard mean-field Maxwell-Garnett and Bruggeman approximations fail to describe percolation-like behavior of $\varepsilon_a(s)$ near $s = 0$. We also show that in this case the charging energy $E_c$ of a single NC in an array of touching NCs has a non-trivial relationship to $\varepsilon_a$, namely $E_c = \alpha e^2/\varepsilon_a d$, where $\alpha$ varies from 1.59 to 1.95 depending on the studied three-dimensional lattices. Our approximation for $\varepsilon_a(s)$ can be used instead of mean field Maxwell-Garnett and Bruggeman approximations to describe percolation like transitions near $s=0$ for other material characteristics of NC arrays, such as conductivity

Accumulation, inversion, and depletion layers in SrTiO3_3

We study potential and electron density depth profiles in accumulation, inversion and depletion layers in crystals with large and nonlinear dielectric response such as $\mathrm{SrTiO_3}$. We describe the lattice dielectric response using the Landau-Ginzburg free energy expansion. In accumulation and inversion layers we arrive at new nonlinear dependencies of the width $d$ of the electron gas on applied electric field $D_0$. Particularly important is the predicted electron density profile of accumulation layers (including the $\mathrm{LaAlO_3/SrTiO_3}$ interface) $n(x) \propto (x+d)^{-12/7}$, where $d \propto D_0^{-7/5}$ . We compare this profile with available data and find satifactory agreement. For a depletion layer we find an unconventional nonlinear dependence of capacitance on voltage. We also evaluate the role of spatial dispersion in the dielectric response by adding a gradient term to the Landau-Ginzburg free energy

Surface roughness scattering in multisubband accumulation layers

Accumulation layers with very large concentrations of electrons where many subbands are filled became recently available due to ionic liquid and other new methods of gating. The low temperature mobility in such layers is limited by the surface roughness scattering. However theories of roughness scattering so far dealt only with the small-density single subband two-dimensional electron gas (2DEG). Here we develop a theory of roughness-scattering limited mobility for the multisubband large concentration case. We show that with growing 2D electron concentration $n$ the surface dimensionless conductivity $\sigma/(2e^2/h)$ first decreases as $\propto n^{-6/5}$ and then saturates as $\sim(da_B/\Delta^2)\gg 1$, where $d$ and $\Delta$ are the characteristic length and height of the surface roughness, $a_B$ is the effective Bohr radius. This means that in spite of the shrinkage of the 2DEG width and the related increase of the scattering rate, the 2DEG remains a good metal. Thus, there is no re-entrant metal-insulator transition at high concentrations conjectured by Das Sarma and Hwang [PRB 89, 121413 (2014)].Comment: A few corrections to the version published in PRB are included here in this versio

Collapse of electrons to a donor cluster in SrTiO3_3

It is known that a nucleus with charge $Ze$ where $Z>170$ creates electron-positron pairs from the vacuum. These electrons collapse onto the nucleus resulting in a net charge $Z_n<Z$ while the positrons are emitted. This effect is due to the relativistic dispersion law. The same reason leads to the collapse of electrons to the charged impurity with a large charge number $Z$ in narrow-band gap semiconductors and Weyl semimetals as well as graphene. In this paper, a similar effect of electron collapse and charge renormalization is found for donor clusters in SrTiO$_3$ (STO), but with a very different origin. At low temperatures, STO has an enormously large dielectric constant. Because of this, the nonlinear dielectric response becomes dominant when the electric field is not too small. We show that this leads to the collapse of surrounding electrons into a charged spherical donor cluster with radius $R$ when its total charge number $Z$ exceeds a critical value $Z_c\simeq R/a$ where $a$ is the lattice constant. Using the Thomas-Fermi approach, we find that the net charge $Z_ne$ grows with $Z$ until $Z$ exceeds another value $Z^*\simeq(R/a)^{9/7}$. After this point, $Z_n$ remains $\sim Z^*$. We extend our results to the case of long cylindrical clusters. Our predictions can be tested by creating discs and stripes of charge on the STO surface

Electron gas induced in SrTiO3_3

This mini-review is dedicated to the 85th birthday of Prof. L. V. Keldysh, from whom we have learned so much. In this paper we study the potential and electron density depth profiles in surface accumulation layers in crystals with a large and nonlinear dielectric response such as SrTiO$_3$ (STO) in the cases of planar, spherical and cylindrical geometries. The electron gas can be created by applying an induction $D_0$ to the STO surface. We describe the lattice dielectric response of STO using the Landau-Ginzburg free energy expansion and employ the Thomas-Fermi (TF) approximation for the electron gas. For the planar geometry we arrive at the electron density profile $n(x) \propto (x+d)^{-12/7}$, where $d \propto D_0^{-7/5}$. We extend our results to overlapping electron gases in GTO/STO/GTO multi-heterojunctions and electron gases created by spill-out from NSTO (heavily $n$-type doped STO) layers into STO. Generalization of our approach to a spherical donor cluster creating a big TF atom with electrons in STO brings us to the problem of supercharged nuclei. It is known that for an atom with nuclear charge $Ze$, where $Z > 170$, electrons collapse onto the nucleus resulting in a net charge $Z_n < Z$. Here, instead of relativistic physics, the collapse is caused by the nonlinear dielectric response. Electrons collapse into the charged spherical donor cluster with radius $R$ when its total charge number $Z$ exceeds the critical value $Z_c \simeq R/a$, where $a$ is the lattice constant. The net charge $eZ_n$ grows with $Z$ until $Z$ exceeds $Z^* \simeq (R/a)^{9/7}$. After this point, the charge number of the compact core $Z_n$ remains $\simeq Z^*$, with the rest $Z^*$ electrons forming a sparse Thomas-Fermi electron atmosphere around it. We extend our results to the case of long cylindrical clusters as well.Comment: mini-review dedicated to the 85th birthday of Prof. L. V. Keldys

Anomalous conductivity, Hall factor, magnetoresistance, and thermopower of accumulation layer in SrTiO3\text{SrTiO}_3

We study the low temperature conductivity of the electron accumulation layer induced by the very strong electric field at the surface of $\text{SrTiO}_3$ sample. Due to the strongly nonlinear lattice dielectric response, the three-dimensional density of electrons $n(x)$ in such a layer decays with the distance from the surface $x$ very slowly as $n(x) \propto 1/x^{12/7}$. We show that when the mobility is limited by the surface scattering the contribution of such a tail to the conductivity diverges at large $x$ because of growing time electrons need to reach the surface. We explore truncation of this divergence by the finite sample width, by the bulk scattering rate, or by the crossover to the bulk linear dielectric response with the dielectric constant $\kappa$. As a result we arrive at the anomalously large mobility, which depends not only on the rate of the surface scattering, but also on the physics of truncation. Similar anomalous behavior is found for the Hall factor, the magnetoresistance, and the thermopower

Theory of a field effect transistor based on semiconductor nanocrystal array

We study the surface conductivity of a field-effect transistor (FET) made of periodic array of spherical semiconductor nanocrystals (NCs). We show that electrons introduced to NCs by the gate voltage occupy one or two layers of the array. Computer simulations and analytical theory are used to study the array screening and corresponding evolution of electron concentrations of the first and second layers with growing gate voltage. When first layer NCs have two electrons per NC the quantization energy gap between its 1S and 1P levels induces occupation of 1S levels of second layer NCs. Only at a larger gate voltage electrons start leaving 1S levels of second layer NCs and filling 1P levels of first layer NCs. By substantially larger gate voltage, all the electrons vacate the second layer and move to 1P levels of first layer NCs. As a result of this nontrivial evolution of the two layers concentrations, the surface conductivity of FET non-monotonically depends on the gate voltage. The same evolution of electron concentrations leads to non-monotonous behaviour of the differential capacitance

Hopping conductivity and insulator-metal transition in films of touching semiconductor nanocrystals

This paper is focused on the the variable-range hopping of electrons in semiconductor nanocrystal (NC) films below the critical doping concentration $n_c$ at which it becomes metallic. The hopping conductivity is described by the Efros-Shklovskii law which depends on the localization length of electrons. We study how the localization length grows with the doping concentration $n$ in the film of touching NCs. For that we calculate the electron transfer matrix element $t(n)$ between neighboring NCs for two models when NCs touch by small facets or just one point. We study two sources of disorder: variations of NC diameters and random Coulomb potentials originating from random numbers of donors in NCs. We use the ratio of $t(n)$ to the disorder-induced NC level dispersion to find the localization length of electrons due to the multi-step elastic co-tunneling process. We found three different phases at $n<n_c$ depending on the strength of disorder, the material, sizes of NCs and their facets: 1) "insulator" where the localization length of electrons increases monotonically with $n$ and 2) "oscillating insulator" when the localization length (and the conductivity) oscillates with $n$ from the insulator base and 3) "blinking metal" where the localization length periodically diverges. The first two phases were seen experimentally and we discuss how one can see the more exotic third one. In all three the localization length diverges at $n=n_c$. This allows us to find $n_c$

Photoluminescence in array of doped semiconductor nanocrystals

We study the dependence of the quantum yield of photoluminescence of a dense, periodic array of semiconductor nanocrystals (NCs) on the level of doping and NC size. Electrons introduced to NCs via doping quench photoluminescence by the Auger process, so that practically only NCs without electrons contribute to the photoluminescence. Computer simulation and analytical theory are used to find a fraction of such empty NCs as a function of the average number of donors per NC and NC size. For an array of small spherical NCs, the quantization gap between 1S and 1P levels leads to transfer of electrons from NCs with large number of donors to those without donors. As a result, empty NCs become extinct, and photoluminescence is quenched abruptly at an average number of donors per NC close to 1.8. The relative intensity of photoluminescence is shown to correlate with the type of hopping conductivity of an array of NCs