We investigate the behavior of a mesoscopic one-dimensional ring in an external magnetic field by simulating the time dependent Ginzburg-Landau equations with periodic boundary conditions. We analyze the stability and the different possible evolutions for the phase slip phenomena starting from a metastable state. We find a stability condition relating the winding number of the initial solution and the number of flux quanta penetrating the ring. The analysis of multiple phase slips solutions is based on analytical results and simulations. The role of the ratio of two characteristic times u is studied for the case of a multiple phase slips transition. We found out that if u>>1, consecutive multiple phase slips will be more favorable than simultaneous ones. If u<<1 the opposite is true and we confirm that u>>1 is often a necessary condition to reach the ground state. The influence of the Langevin noise on the kinetics of the phase transition is discussed.
Superconducting vortices and phase slips are primary mechanisms of dissipation in superconducting, superfluid, and cold atom systems. While the dynamics of vortices is fairly well described, phase slips occurring in quasi-one dimensional superconducting wires still elude understanding. The main reason is that phase slips are strongly non-linear time-dependent phenomena that cannot be cast in terms of small perturbations of the superconducting state. Here we study phase slips occurring in superconducting weak links. Thanks to partial suppression of superconductivity in weak links, we employ a w…
The properties of one-dimensional superconductors are strongly influenced by topological fluctuations of the order parameter, known as phase slips, which cause the decay of persistent current in superconducting rings and the appearance of resistance in superconducting wires. Despite extensive work, quantitative studies of phase slips have been limited by uncertainty regarding the order parameter's free energy landscape. Here we show detailed agreement between measurements of the persistent current in isolated flux-biased rings and Ginzburg-Landau theory over a wide range of temperature, magnet…
Using the nonlinear Ginzburg-Landau theory we investigated the dependence of the magnetic coupling between two concentric mesoscopic superconducting rings on their thickness. The size of this magnetic coupling increases with the thickness of the rings.
Experimental results on the phase slip process in superconducting lead nanowires are presented under two different experimental conditions: constant applied current or constant voltage. Based on these experiments we established a simple model which gives us the condition of the appearance of phase slip centers in a quasi-one-dimensional wire. It turns out that the competition between two relaxations times (relaxation time of the absolute value of the order parameter $τ_{|ψ|}$ and relaxation time of the phase of the order parameter in the phase slip center $τ_ϕ$) governs the phase slip process.…
Robust porous silicon substrates were employed for generating interconnected networks of superconducting ultrathin Nb nanowires. Scanning electron microscopy analysis was performed to investigate the morphology of the samples, which constitute of polycrystalline single wires with grain size of about 10 nm. The samples exhibit nonzero resistance over a broad temperature range below the critical temperature, fingerprint of phase slippage processes. The transport data are satisfactory reproduced by models describing both thermal and quantum fluctuations of the superconducting order parameter in t…
Quantum phase slips have received much attention due to their relevance to superfluids in reduced dimensions and to models of cosmic string production in the Early Universe. Their establishment in one-dimensional superconductors has remained controversial. Here we study the nonlinear voltage-current characteristics and linear resistance in long superconducting Al wires with lateral dimensions $\sim$ 5 nm. We find that, in a magnetic field and at temperatures well below the superconducting transition, the observed behaviors can be described by the non-classical, macroscopic quantum tunneling of…
Quantum phase slip (QPS) is the particular manifestation of quantum fluctuations of the order parameter of a current-biased quasi-1D superconductor. The QPS event(s) can be considered a dynamic equivalent of tunneling through conventional Josephson junction containing static in space and time weak link(s). At low temperatures T<<Tc the QPS effect leads to finite resistivity of narrow superconducting channels and suppresses persistent currents in tiny nanorings. Here we demonstrate that the quantum tunneling of phase may result in Coulomb blockade: superconducting nanowire, imbedded in high-Ohm…
We report on the transport properties of an array of N about 30 interconnected Nb nanowires, grown by sputtering on robust porous Si substrates. The analyzed system exhibits a broad resistive transition in zero magnetic field, H, and highly nonlinear V(I) characteristics as a function of H which can be both consistently described by quantum tunneling of phase slips.
Superconducting nanowires undergoing quantum phase-slips have potential for impact in electronic devices, with a high-accuracy quantum current standard among a possible toolbox of novel components. A key element of developing such technologies is to understand the requirements for, and control the production of, superconducting nanowires that undergo coherent quantum phase-slips. We present three fabrication technologies, based on using electron-beam lithography or neon focussed ion-beam lithography, for defining narrow superconducting nanowires, and have used these to create nanowires in niob…
We consider structure of a thermal phase-slip center for a simple microscopic model of a clean one-dimensional superconductors in which superconductivity occurs only within one conducting channel or several identical channels. Surprisingly, the Eilenberger equations describing the saddle-point configuration allow for exact analytical solution in the whole temperature and current range. This solution allows us to derive a closed expression for the free-energy barrier, which we use to compute its temperature and current dependences.
We show the results of two-terminal and four-terminal transport measurements on few-layer NbSe$_2$ devices at large current bias. In all the samples measured, transport characteristics at high bias are dominated by a series of resistance jumps due to nucleation of phase slip lines, the two dimensional analogue of phase slip centers. In point contact devices the relatively simple and homogeneous geometry enables a quantitative comparison with the model of Skocpol, Beasley and Tinkham. In extended crystals the nucleation of a single phase slip line can be induced by mechanical stress of a region…
The discrete shell structure of vortex matter strongly influences the flux dynamics in mesoscopic superconducting Corbino disks. While the dynamical behavior is well understood in large and in very small disks, in the intermediate-size regime it occurs to be much more complex and unusual, due to (in)commensurability between the vortex shells. We demonstrate unconventional vortex dynamics (inversion of shell velocities with respect to the gradient driving force) and angular melting (propagating from the boundary where the shear stress is minimum, towards the center) in mesoscopic Corbino disks.
Solving numerically the 3D non linear Ginzburg-Landau (GL) equations, we study equilibrium and nonequilibrium phase transitions between different superconducting states of mesoscopic disks which are thinner than the coherence length and the penetration depth. We have found a smooth transition from a multi-vortex superconducting state to a giant vortex state with increasing both the disk thickness and the magnetic field. A vortex phase diagram is obtained which shows, as function of the magnetic field, a re-entrant behavior between the multi-vortex and the giant vortex state.
We compute the phase transition curves for mesoscopic superconductors. Special emphasis is given to the limiting shape of the curve when the magnetic flux is large. We derive an asymptotic formula for the ground state of the Schrödinger equation in the presence of large applied flux. The expansion is shown to be sensitive to the smoothness of the domain. The theoretical results are compared to recent experiments.
A phase-slip flux qubit, exactly dual to a charge qubit, is composed of a superconducting loop interrupted by a phase-slip junction. Here we propose a tunable phase-slip flux qubit by replacing the phase-slip junction with a charge-related superconducting quantum interference device (SQUID) consisting of two phase-slip junctions connected in series with a superconducting island. This charge-SQUID acts as an effective phase-slip junction controlled by the applied gate voltage and can be used to tune the energy-level splitting of the qubit. Also, we show that a large inductance inserted in the l…
We describe a method for direct measurement of the magnetic penetration length in thin (10 - 100 nm) superconducting structures having overall dimensions in the range 1 to 100 micrometers. The method is applicable for broadband magnetic fields from dc to MHz frequencies.
The fluctuating superconducting correlations emerging in dirty hybrid structures in conditions of the strong proximity effect are demonstrated to affect the validity range of the widely used formalism of Usadel equations at mesoscopic scales. In superconductor -- ferromagnet (SF) structures these giant mesoscopic fluctuations originating from the interference effects for the Cooper pair wave function in the presence of the exchange field can be responsible for an anomalously slow decay of superconducting correlations in a ferromagnet even when the non-collinear and spin-orbit effects are negli…
A thermodynamically stable vortex-antivortex pattern has been revealed in mesoscopic type I superconducting triangles, contrary to type II superconductors where similar patterns are unstable. The stable vortex-antivortex "molecule" appears due to the interplay between two factors: a repulsive vortex-antivortex interaction in type I superconductors and the vortex confinement in the triangle.
A projected order parameter is used to calculate, not only local minima of the Ginzburg-Landau energy functional, but also saddle points or energy barriers responsible for the metastabilities observed in superconducting mesoscopic disks (Geim et al. Nature {\bf 396}, 144 (1998)). We calculate the local minima magnetization and find the energetic instability points between vortex configurations with different vorticity. We also find that, for any vorticity, the supercurrent can reverse its flow direction on decreasing the magnetic field before one vortex can escape.
We theoretically investigate the appearance of spatially modulated superconducting states in mesoscopic superconducting thin-wall cylinders in a magnetic field at low temperatures. Quantization of the electron motion around the circumference of the cylinder leads to a discontinuous evolution of the spatial modulation of the superconducting order parameter along the transition line T_c(H). We show that this discontinuity leads to the non-monotonic behavior of the specific heat jump at the onset of superconductivity as a function of temperature and field. We argue that this geometry provides an …
We study current-voltage (V-I) characteristics of short superconducting nanowires of length ~ 100 nm exposed to microwave radiation of frequencies between 100 MHz and 15 GHz. The radiation causes a decrease of the average switching current of the wire. This suppression of the switching current is modeled assuming that there is one-to-one correspondence between Little's phase slips and the experimentally observed switching events. At some critical power P* of the radiation a dissipative dynamic superconducting state occurs as an extra step on the V-I curve. It is identified as a phase slip cent…
Response of a mesoscopic superconducting disk to perpendicular magnetic fields is studied by using the multiple-small-tunnel-junction method, in which transport properties of several small tunnel junctions attached to the disk are measured simultaneously. This allows us for the first experimental distinction between the giant vortex states and multivortex states. Moreover, we experimentally find magnetic field induced rearrangement and combination of vortices. The experimental results are well reproduced in numerical results based on the nonlinear Ginzburg-Landau theory.
We measure quantum and thermal phase-slip rates using the standard deviation of the switching current in superconducting nanowires at high bias current. Our rigorous quantitative analysis provides firm evidence for the presence of quantum phase slips (QPS) in homogeneous nanowires. We observe that as temperature is lowered, thermal fluctuations freeze at a characteristic crossover temperature Tq, below which the dispersion of the switching current saturates to a constant value, indicating the presence of QPS. The scaling of the crossover temperature Tq with the critical temperature Tc is linea…
The electronic structure of the giant vortex states in a mesoscopic superconducting disk is studied in a dirty limit using the Usadel approach. The local density of states profiles are shown to be strongly affected by the effect of quasiparticle (QP) tunneling between the states localized in the vortex core and the ones bound to the sample edge. Decreasing temperature leads to a crossover between the edge-dominated and core-dominated regimes in the magnetic field dependence of the tunneling conductance. This crossover is discussed in the context of the efficiency of quasiparticle cooling by th…
Superconducting state in mesoscopic cylinder placed in the external magnetic field is analyzed at general de Gennes boundary condition for the order parameter. The lower and the surface critical fields of the cylinder are calculated at different values of de Gennes "extrapolation length" $b$. For the case of standard boundary condition (corresponding to $b\to \infty $) we use a variational method to calculate the phase transitions between different superconducting states. Based on the trial function for the order parameter, we find an approximate solutions to the Ginzburg-Landau equations for …
The magnetization behavior of mesoscopic superconducting disks can show hysteretic behavior which we explain by using the Ginzburg-Landau (GL) theory and properly taking into account the de-magnetization effects due to geometrical form factors. In large disks the Bean-Livingston surface barrier is responsible for the hysteresis. While in small disks a volume barrier is responsible for this hysteresis. It is shown that although the sample magnetization is diamagnetic (negative), the measured magnetization can be positive at certain fields as observed experimentally, which is a consequence of th…
The theory of current transport in a narrow superconducting channel accounting for thermal fluctuations is revisited. The value of voltage appearing in the sample is found as the function of temperature (close to transition temperature $T-T_{\mathrm{c}}$ $\ll T_{\mathrm{c}}$) and bias current $J<J_{\mathrm{c}}$ ( $J_{\mathrm{c}}$ is a value of critical current calculated in the framework of the BCS approximation, neglecting thermal fluctuations). It is shown that the careful analysis of vortex crossing of the stripe results in considerable increase of the activation energy.
Superconducting thin-film rings smaller than the film penetration depth (the Pearl length) are considered. The current distribution, magnetic moment, and thermodynamic potential ${\cal F}(H,N,v)$ for a flat, washer-shaped annular ring in a uniform applied field $H$ perpendicular to the film are solved analytically within the London approach for a state with winding number $N$ and a vortex at radius $v$ between the inner and outer radii.
We study the thermodynamics of ultrasmall metallic grains with the mean level spacing comparable or larger than the pairing correlation energy in the whole range of temperatures. A complete picture of the thermodynamics in such systems is given taking into account the effects of disorder, parity and classical and quantum fluctuations. Both spin susceptibility and specific heat turn out to be sensitive probes to detect superconducting correlations in such samples.
A numerical scheme to study the mixed states in a mesoscopic type-II superconducting cylinder is described. Steady-state configurations and transient behavior of the magnetic vortices for various values of the applied magnetic field H are presented. Transitions between different multi-vortex states as H is changed is demonstrated by contour plots and jumps in the B vs H plot. Evolving a uniformly-superconducting initial state using the simplest set of relaxation equations shows that the system passes through nearly metastable intermediate configurations, while seeking the final minimum-energy,…
In the present paper we develop an algorithm to solve the time dependent Ginzburg-Landau (TDGL) equations, by using the link variables technique, for circular geometries. In addition, we evaluate the Helmholtz and Gibbs free energy, the magnetization, and the number of vortices. This algorithm is applied to a circular sector. We evaluate the superconduting-normal magnetic field transition, the magnetization, and the superconducting density. Furthermore, we study the nucleation of giant and multi-vortex states for that geometry.
Current steps in the current-voltage characteristics of wide superconducting Sn films exposed to a microwave irradiation were observed in the resistive state with phase slip lines. The behaviour of the magnitude of the steps on the applied irradiation power was found to be similar to that for the current steps in narrow superconducting channels with phase slip centers and, to some extent, for the Shapiro steps in Josephson junctions. This provides evidence for the Josephson properties of the phase slip lines in wide superconducting films and supports the assumption about similarity between the…
We solve the Ginzburg-Landau equation (GLE) for the mesoscopic superconducting thin film of the square shape in the magnetic field for the wide range of the Ginzburg-Landau parameter $0.05<κ_{eff}<\infty $. We found that the phase with the antivortex exists in the broad range of parameters. When the coherence length decreases the topological phase transition to the phase with the same total vorticity and a reduced symmetry takes place. The giant vortex with the vorticity $m=3$ is found to be unstable for any field, $ξ/a$ and $κ_{eff}\ge 0.1$. Reduction of $ κ_{eff}$ does not make the phase wit…
We study thermal fluctuations and capacitive effects on small Josephson Junction Rings (JJR) that mimics the rectification phenomena recently observed in triangle shaped mesoscopic superconductors, due to the superposition of the field induced persistent current with the bias current. At finite temperature we predicted that the amplitude of the rectified signal depends strongly on the current contacts configuration on the JJR, in coincidence with experiments. In addition we analize the range of parameters where a closed loop of capacitive junctions is an appropriate model to explain the experi…
A theoretical analysis based on a numerical solution of the coupled time-dependent Ginzburg-Landau and heat dissipation equations shows a strong dependence of the critical currents on the applied magnetic field in a mesoscopic square with attached contacts. In agreement with experiment we found hysteresis which are caused by a strong heat dissipation in the sample at currents close to the depairing Ginzburg-Landau current and/or the dynamics of the superconducting condensate. The theoretically obtained nonmonotonous dependence of the switching current (from superconducting to the resistive sta…
Our study sample is a superconducting bi-dimensional octagon with different boundary conditions immersed in a magnetic external field H. The boundary conditions are simulated by considering different values of the deGennes extrapolation length b on different surfaces of the sample. Our investigation was carried out by solving the two-band time dependent Ginzburg-Landau equations (TB-TDGL). We analyzed the superconducting electron density and the magnetization curves as functions of H and temperature T in zero field cooling and in zero field cooling processes for different values of b and size …
The dense vortex matter structure and associated magnetization are calculated for type-II superconducting mesoscopic disks. The magnetization exhibits generically first-order phase transitions as the number of vortices changes by one and presents two well-defined regimes: A non-monotonous evolution of the magnitude of the magnetization jumps signals the presence of a vortex glass structure which is separated by a second-order phase transition at $H_{c2}$ from a condensed state of vortices (giant vortex) where the magnitude of the jumps changes monotonously. We compare our results with Hall mag…
We propose a transistor-like circuit including two serially connected segments of a narrow superconducting nanowire joint by a wider segment with a capacitively coupled gate in between. This circuit is made of amorphous NbSi film and embedded in a network of on-chip Cr microresistors ensuring a sufficiently high external electromagnetic impedance. Assuming a virtual regime of quantum phase slips (QPS)in two narrow segments of the wire, leading to quantum interference of voltages on these segments, this circuit is dual to the dc SQUID. Our samples demonstrated appreciable Coulomb blockade volta…
In this paper we review several aspects of mesoscopic hybrid superconducting systems. In particular we consider charge and heat transport properties in hybrid superconducting-metal structures and the effect of charging energy in superconducting nanostructures.
Solutions of Ginzburg-Landau eqns. coupled with three dimensional Maxwell eqns. reveal intriguing magnetic response of small superconducting particles, qualitatively different from the two dimensional approximation but in agreement with recent experiments. Depending on the radius and thickness first or second order transitions are found for the normal to superconducting state. For a sufficient large radius of the disc several transitions in the superconducting phase are obtained which correspond to different angular momentum giant vortex states. The incorporation of the finite thickness in the…
Short-range columnar defects essentially influence the magnetic properties of a mesoscopic superconducting disc.They help the penetration of vortices into the sample, thereby decrease the sample magnetization and reduce the upper critical field. Even the presence of weak defects split a giant vortex state (usually appearing in a clean disc in the vicinity of the transition to a normal state) into a number of vortices with smaller topological charges. In a disc with a sufficient number of strong enough defects vortices are always placed onto defects. The presence of defects lead to the appearan…
We study the rate of quantum phase slips in an ultranarrow superconducting nanowire exposed to weak electromagnetic radiations. The superconductor is in the dirty limit close to the superconducting-insulating transition, where fluxoids move in strong dissipation. We use a semiclassical approach and show that external radiation stimulates a significant enhancement in the probability of quantum phase slips. This can help to outline a new type of detector for microwave to submillimetre radiations based on stimulated quantum phase slip phenomenon.
A microscopic model is constructed which is able to describe multiple magnetic flux transitions as observed in recent ultra-low temperature tunnel experiments on an aluminum superconducting ring with normal metal - insulator - superconductor junctions [Phys. Rev. B \textbf{70}, 064514 (2004)]. The unusual multiple flux quantum transitions are explained by the formation of metastable states with large vorticity. Essential in our description is the modification of the pairing potential and the superconducting density of states by a sub-critical value of the persistent current which modulates the…
Using an approach based on quasiclassical Green's functions we present a theoretical study of transport in mesoscopic S/N structures in the diffusive limit. The subgap conductance in S/N structures with barriers (zero bias and finite bias anomalies) are discused. We also analyse the temperature dependence of the conductance variation $δS(T)$ for a Andreev interferometer. We show that besides the well know low temperature maximum a second maximum near $T_c$ may appear. We present the results of studies on the Josephson effect in 4 terminal S/N/S contacts and on the possible sign reversal of the…
Here we report a novel nitridation technique for transforming niobium into hexagonal Nb2N which appears to be superconducting below 1K. The nitridation is achieved by high temperature annealing of Nb films grown on Si3N4/Si (100) substrate under high vacuum. The structural characterization directs the formation of a majority Nb2N phase while the morphology shows granular nature of the films. The temperature dependent resistance measurements reveal a wide metal-to-superconductor transition featuring two distinct transition regions. The region close to the normal state varies strongly with the f…
We solve the Ginzburg-Landau equation (GLE) for the mesoscopic superconducting thin film of the square shape in the magnetic field for the wide range of the Ginzburg-Landau parameter 0.05< kappa. We focus on the region of the field where formation of the antivortex has been reported previously. We found that the phase with the antivortex exists in the broad range of parameters. When the coherence length decreases the topological phase transition to the phase with the same total vorticity and a reduced symmetry takes place. The giant vortex with the vorticity m=3 is found to be unstable for any…
We analyze theoretically and experimentally vortex configurations in mesoscopic superconducting squares. Our theoretical approach is based on the analytical solution of the London equation using Green's-function method. The potential-energy landscape found for each vortex configuration is then used in Langevin-type molecular-dynamics simulations to obtain stable vortex configurations. Metastable states and transitions between them and the ground state are analyzed. We present our results of the first direct visualization of vortex patterns in micrometer-sized Nb squares, using the Bitter decor…
The appearance of phase-slip lines, induced by an rf field, was observed experimentally in wide superconducting films in which the destruction of superconductivity by a dc current is associated with the formation of phase-slip lines. The characteristics of the separation of the film into superconducting and nonuniform, isothermal, nonequilibrium regions (phase-slip lines) under electromagnetic irradiation were studied.
We study the effect of phase slips in a quasi 1d superconducting channel along which a current flows and report a new phenomenon where an avalanche of phase slips occurs. This limits the critical current in thin films and wires and drives the system to a topological phase transition at a temperature lower than the bulk critical temperature. We describe the mechanism of such a catastrophic phase slip avalanche and, following Kosterlitz and Thouless, we use group renormalization techniques to derive an exact analytical expression for the critical current as a function of film width and temperatu…
In mesoscopic superconducting disks vortices form shell structures as recently observed in Nb disks. We study the dynamics of such vortices, driven by an external current I_0, in a Corbino setup. At very low I_0, the system exhibits rigid body rotation while at some critical current I_c,i vortex shells rotate separately with angular velocities omega_i. This critical current I_c,i has a remarkable non-monotonous dependence on the applied magnetic field which is due to a dynamically-induced structural transition with a rearrangement of vortices over the shells similar to the Coster-Kronig transi…
We consider the steady-state nonequilibrium behavior of mesoscopic superconducting wires connected to normal-metal reservoirs. Going beyond the diffusive limit, we utilize the quasiclassical theory and perform a self-consistent calculation that guarantees current conservation through the entire system. Going from the ballistic to the diffusive limit, we investigate several crucial phenomena such as charge imbalance, momentum-resolved nonequilbrium distributions, and the current-to-superflow conversion. Connecting to earlier results for the diffusive case, we find that superconductivity can bre…
Recent experiments [I.V. Grigorieva et al., Phys. Rev. Lett. 96, 077005 (2006)] on visualization of vortices using the Bitter decoration technique revealed vortex shells in mesoscopic superconducting Nb disks containing up to L=40 vortices. Some of the found configurations did not agree with those predicted theoretically. We show here that this discrepancy can be traced back to the larger disks with radii R ~ 1 to 2.5μm, i.e., R ~ 50-100ξ(0) used in the experiment, while in previous theoretical studies vortex states with vorticity L < 40 were analyzed for smaller disks with R ~ 5-20ξ(0). The p…
Usually, the measurements of electronic and magnetic properties of superconducting samples are carried out under a constant temperature bath. On the other hand, thermal gradients induce local variation of the superconducting order parameter, and the vortex dynamics can present interesting behaviors. In this work, we solved the time-dependent Ginzburg-Landau equations simulating samples under two different thermal gradients, and considering two values of the Ginzburg-Landau parameter, \k{appa}. We find out that both parameters, i.e., \k{appa} and thermal gradients, play an important role on the…
The authors report magnetic measurements of fluxoid transitions in mesoscopic, superconducting aluminum rings. The transitions are induced by applying a flux to the ring so that the induced supercurrent approaches the critical current. In a temperature range near $T_c$, only a single fluxoid enters or leaves at a time, leading to a final state above the ground state. Upon lowering the temperature, several fluxoids enter or leave at once, and the final state approaches the ground state, which can be reached below approximately 0.5 $T_c$. A model based on the widely used time dependent Ginzburg-…
We reply to the recent Comment [cond-mat/9702231] by J.-M. Duan. Our point of view is markedly different on every issue raised. Much of the disagreement can be traced to a different preception of experimentally relevant length scales. i) We explain the difference between our formulation, which rests on a microscopic basis, and the phenomenological one of the author. ii) Our renormalization scheme is fundamentally right, as the "log(log)" interaction appears only in wires of astronomical lengths. iii) The tunneling barrier is profoundly reduced by the kinetic inductance. iv) We do make an appro…
We perform measurements of phase-slip-induced switching current events on different types of superconducting weak links and systematically study statistical properties of the switching current distributions. We employ two types of devices in which a weak link is formed either by a superconducting nanowire or by a graphene flake subject to proximity effect. We demonstrate that, independently on the nature of the weak link, higher moments of the distribution take universal values. In particular, the third moment (skewness) of the distribution is close to -1 both in thermal and quantum regimes. T…
