Transport through open quantum dots: Making semiclassics quantitative

We investigate electron transport through clean open quantum dots (quantum billiards). 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: (i) inclusion of pseudopaths which have the topology of linked classical paths resulting from diffraction in addition to classical paths and (ii) a high-level approximation to diffractive scattering. Within this framework of the pseudopath semiclassical approximation (PSCA), 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.

Figure 1

(Color online) Geometry of the circular billiard of radius $\rho$ with perpendicular leads of equal width $d$. In accordance with previous work (Refs. 34, 35, 36) on the circular and the stadium billiard of equal area $a=4+\pi$, we choose $d=0.25$ and $\rho =\sqrt{1+4/\pi }$.

Figure 2

(Color online) Two-dimensional length vs enclosed area distribution of classical paths within the open (black dots) and closed (orange triangles) circular billiard for (a) reflection and (b) transmission. Each point represents one classical path which connects the centers of each lead. The inset of (a) shows typical paths for the path topology of the three first branches.

Figure 3

(Color online) Length-area distribution of first-order pseudopaths (blue dots) for (a) reflection and (b) transmission. Zeroth-order pseudopaths (i.e., the classical paths) are denoted by orange triangles (same as Fig. 2). The number of first-order pseudopaths up to length $L=40$ is larger by more than a factor of 60 compared to the number of classical paths.

Figure 4

(Color online) Sketch of diffraction at an open lead mouth: a path $q$ reaches the orifice under an angle $\theta$ and is backscattered into a path $q^{\prime }$ that leaves with an angle ${\theta }^{\prime }$ (the angles $\theta$ and ${\theta }^{\prime }$ as depicted in the figure have opposite signs). The dashed lines denote that $\vec{r}$ and ${\vec{r}}^{\prime }$ are in the far-field region.

Figure 5

(Color online) Absolute square of the diffraction coefficient ${\left|v\left({\theta }^{\prime },\theta ,k,d\right)\right|}^2$ for diffractive scattering (see Fig. 4) within the FDA, the KDA, the GTD-UTD, and exact quantum-mechanical data for angle of incidence (a) $\theta =\pi /8$, (b) $\theta =\pi /4$, and (c) $\theta =3\pi /8$ at $k=2.5\pi /d$. The vertical lines in each frame mark the angles of specular reflection. The quantum-mechanical (QM) and KDA coefficients are taken from Ref. 10. Inset of Fig. (b): the real and imaginary parts of $v\left({\theta }^{\prime },\theta ,k,d\right)$ within the GTD-UTD.

Figure 6

(Color online) Two-dimensional distribution of the absolute square of the quantum $S$-matrix element ${\left|t_{22}\left(k,B\right)\right|}^2=T_{22}\left(k,B\right)$ as a function of the wave number $k$ and the magnetic field $B$.

Figure 7

(Color online) The absolute square of the Fourier transform ${\left|\tilde{S}\left(L,a\right)\right|}^2$ as a function of length $L$ and enclosed directed area $a$: (a) ${\tilde{R}}_{22}\left(L,a\right)$, (b) ${\tilde{R}}_{11}\left(L,a\right)$, (c) ${\tilde{T}}_{22}\left(L,a\right)$, and (d) ${\tilde{T}}_{11}\left(L,a\right)$. The color shading is determined by $\text{log}\left({\left|\tilde{S}\left(L,a\right)\right|}^2+1\right)$. The insets show the probabilities along the entire resolved length. The integration of Eq. (14) is performed numerically over $k∊\left[2.2,3.45\right]\pi /d$ discretized with 251 points and $B/c∊\left[-3,3\right]$ with 121 points. The spectra are compared to the distribution of classical paths (black dots) at zero $B$ field (the curvature of the trajectories in the present magnetic field and energy regime is negligible).

Figure 8

(Color online) ${\tilde{T}}_{22}\left(L,a\right)$ as a function of length $L$ and enclosed directed area $a$: (a) full quantum $S$-matrix, (b) the PSCA using the GTD-UTD, and (c) the PSCA using the FDA for the diffraction coefficients. Several regions with nonclassical paths are highlighted by red circles.

Figure 9

(Color online) (Conductance) Fluctuations in the transmission $T\left(k,B=0\right)$ and reflection $R\left(k,B=0\right)$ as a function of $k$. The maximal included length is $L_{max}=40$. (a) Reflection and (b) transmission within the SCA (red/green solid line), and quantum mechanics (black dashed line). For the diffraction coefficients $c_m\left(\theta ,k,d\right)$ entering the SCA we have used the GTD-UTD. (c) Reflection and (d) transmission within the PSCA with the GTD-UTD (red/green solid line), and quantum mechanics (black dashed line). The PSCA includes diffractive scattering up to fifth order.

Figure 10

(Color online) Comparison between the PSCA, the SCA, and quantum mechanics (QM) for (a) the averaged transmission ${⟨T⟩}_{\Delta k}$ as a function of the inverse cutoff length $1/L_{max}$ and (b) the averaged variance of the conductance fluctuations $\sigma$. The average is performed in the interval $k∊\left[2.2,2.8\right]\pi /d$. The data within the PSCA are calculated in third order for lengths $L_{max}=10–30$, in fourth order for $L_{max}=40$ and fifth order for $L_{max}=50$. The QM results extend to the exact value $\left(L_{max}=\infty \right)$.

Figure 11

(Color online) Comparison between truncated quantum mechanics, the SCA, and the PSCA for the weak-localization dip in transmission for different $L_{max}$. (a) PSCA using the GTD-UTD for all diffraction coefficients (red solid line). (b) Standard SCA (red solid line) (using the GTD-UTD for the diffraction coefficients $c_m\left(\theta ,k,d\right)$ for entering and exiting the circular cavity.) Black dashed lines: quantum results.

Figure 12

(Color online) Weak localization for $L_{max}=40$ as produced by quantum mechanics: left column (a), (c), and (e), and PSCA with the GTD-UTD: right column (b), (d), and (f).

Figure 13

(Color online) Weak localization for $L_{max}=40$. Left column (a), (c), and (e): quantum calculations, right column (b), (d), and (f): PSCA with diffraction coefficients from FDA.

Figure 14

(Color online) The $k$-averaged Fano factor $F$ as a function of the inverse cutoff length $L_{max}$. The average is performed over the interval $k∊\left[2.2,2.8\right]\pi /d$. Comparison between PSCA with the GTD-UTD, the SCA, and QM. ($L_{max}=\infty$ corresponds to the exact result.)

Figure 15

(Color online) Propagation between two points $\vec{r}$ and ${\vec{r}}^{\prime }$ in a semi-infinite plane in the presence of an open (lead). Contributions from the direct path (black dotted line), from specular reflection (blue solid line), and from scattering at the orifice (red dashed line) are depicted.

Figure 16

(Color online) (a) Diffraction at a wedge. (b) Diffraction at a lead described as diffraction at two wedges. The reference path is marked by a red dashed line. The angle is counted positive for a path lying on the left of the vertical axis, for a path on the right the angle is negative. (In the present example $\theta$ is negative and ${\theta }^{\prime }$ is positive). The points $\vec{r}$ and ${\vec{r}}^{\prime }$ are assumed to be in the far-field region. (c) Higher-order corrections enter via paths scattered between the wedges.

Figure 17

(Color online) Absolute square of the diffraction coefficient ${\left|v^{\text{GTD}}\left({\theta }^{\prime },\theta ,k,d\right)\right|}^2$ (or $v^{\text{GTD-UTD}}$ with $j_{max}=0$) compared with ${\left|v^{\text{GTD-UTD}}\left({\theta }^{\prime },\theta ,k,d\right)\right|}^2$ for $j_{max}=1$, $j_{max}=5$, and $j_{max}=10$ (for the definition of $j_{max}$ see text), $k=2.5\pi /d$. The QM result is taken from Ref. 10.

Figure 18

(Color online) A lead of width $d$ coupled to a half-infinite plane. An incoming wave in mode $m$ can be separated into two rays $\theta =\pm {\theta }_m=\pm \text{arcsin}m\pi /dk$ which diffractively scatter at the lead wedges. The reference path is denoted by a red dashed line.

Figure 19

(Color online) The absolute square of the coupling coefficient ${\left|c_m\left(\theta ,k,d\right)\right|}^2$ multiplied by $k_{x,m}$ for $m=2$ at two different wave numbers (a) $k=2.1\pi /d$ and (b) $k=2.5\pi /d$ within the GTD and the GTD-UTD. Note the breakdown of the GTD for $\theta \to \pm \pi /2$ near $k\approx 2\pi /d$.