Transport through open quantum dots: Making semiclassics quantitative

Abstract

We investigate electron transport through clean open quantum dots ͑quantum billiards͒. We present a semiclassical theory that allows to accurately reproduce quantum-transport calculations. Quantitative agreement is reached for individual energy-dependent and magnetic field dependent elements of the scattering matrix. Two key ingredients are essential: ͑i͒ inclusion of pseudopaths which have the topology of linked classical paths resulting from diffraction in addition to classical paths and ͑ii͒ a high-level approximation to diffractive scattering. Within this framework of the pseudopath semiclassical approximation ͑PSCA͒, typical shortcomings of semiclassical theories such as violation of the anticorrelation between reflection and transmission and the overestimation of conductance fluctuations are overcome. Beyond its predictive capabilities the PSCA provides deeper insights into the quantum-to-classical crossover.

Scalar wave scattering in spherical cavity resonator with conical channels

“…The transport properties of several open 2D cavity structures have been studied recently, including the rectangular [16,17], circular [18][19][20][21], stadium [22,23], and elliptic billiards [24]. However, open 3D cavities have not received quite as much attention as their two-dimensional counterparts, and to the best of our knowledge, there is no work regarding the scattering properties of the spherical resonator with attached channels.…”

Semiclassical wave functions for open quantum billiards

“…The derivation can be found in Ref. 16. The diffraction amplitude v GTD (θ , θ, k) for backscattering into the cavity within the GTD [34] with N = 3/2 the exterior angle (in units of π) of a perpendicular wedge.…”

“…5 involves two steps: First, the evaluation of the integrals over the entrance/exit leads by either a stationary phase approximation (SPA) as done in the conventional semiclassical approximation or as a diffractive integral [12] as implemented in the PSCA. We employ for the latter a combination of the geometric theory of diffraction [34] (GTD) and the uniform theory of diffraction [35,36] (UTD), called GTD-UTD, which has been previously successfully applied for the scattering matrix of the circular billiard [16]. The GTD-UTD takes into account the multiple scattering between the edges of a given lead (for details see [16]).…”

“…We employ for the latter a combination of the geometric theory of diffraction [34] (GTD) and the uniform theory of diffraction [35,36] (UTD), called GTD-UTD, which has been previously successfully applied for the scattering matrix of the circular billiard [16]. The GTD-UTD takes into account the multiple scattering between the edges of a given lead (for details see [16]). Second, the quantum propagator is replaced within the PSCA by a semiclassical propagator that contains in addition to classical paths also pseudopaths, i.e.…”

“…We construct the wavefunction in terms of a sum over paths connecting the entrance lead with an arbitrary point r in the interior of the billiard closely following the Feynman path integral prescription. We employ the pseudopath semiclassical approximation (PSCA) [14][15][16][17] to include both classical and diffractive, i.e. non-classical paths into the path sum.…”