Semiclassical transport in nearly symmetric quantum dots. I. Symmetry breaking in the dot

We apply the semiclassical theory of transport to quantum dots with exact and approximate spatial symmetries; left-right mirror symmetry, up-down mirror symmetry, inversion symmetry, or fourfold symmetry. In this work—the first of a pair of articles—we consider (a) perfectly symmetric dots and (b) nearly symmetric dots in which the symmetry is broken by the dot’s internal dynamics. The second article addresses symmetry-breaking by displacement of the leads. Using semiclassics, we identify the origin of the symmetry-induced interference effects that contribute to weak localization corrections and universal conductance fluctuations. For perfect spatial symmetry, we recover results previously found using the random-matrix theory conjecture. We then go on to show how the results are affected by asymmetries in the dot, magnetic fields, and decoherence. In particular, the symmetry-asymmetry crossover is found to be described by a universal dependence on an asymmetry parameter ${\gamma }_{\text{asym}}$. However, the form of this parameter is very different depending on how the dot is deformed away from spatial symmetry. Symmetry-induced interference effects are completely destroyed when the dot’s boundary is globally deformed by less than an electron wavelength. In contrast, these effects are only reduced by a finite amount when a part of the dot’s boundary smaller than a lead-width is deformed an arbitrarily large distance.

Figure 1

(Color online) Sketch of a planar quantum dot with a left-right mirror symmetry. On the left we show a classical path (solid line) and its mirror image (dashed line). Each path is piecewise straight and undergoes specular reflections at the boundary of the dot. The path is chaotic (we assume hyperbolic chaos), so its position and momentum after time $t$ are exponentially sensitive to the initial position and momentum. We assume the system is such that a typical path crosses the symmetry axis a large number of times before escaping into a lead (these crossings occur on a time scale of order of the time-of-flight across the system). To simplify the sketches elsewhere in this article, we draw paths as abstract curves (shown on the right) which only show those crossings of the symmetry axis that are crucial to the nature of the path.

Figure 2

(Color online) Contributions to weak localization due to a left-right mirror symmetry. When we fold the paths into one half of the dot (as shown in the right-hand sketches), paths hitting the symmetry axis are specularly reflected; however, we must keep track of whether the path escapes into the right lead. Thus, in those sketches, paths that end on the left lead are marked with solid circles, while those that end on the right lead are marked with open (red) circles. The sketch in (a) is an enhanced forward-scattering contribution, because it enhances transmission through the dot.

Figure 3

(Color online) Two ways in which the left half of a quantum dot can be deformed away from symmetry with the right half. In (a), the left half is deformed over a long range $W_{\text{asym}}⪢{\lambda }_F$ by a gentle ripple of small magnitude $\delta L<{\lambda }_F$. A special case of this is where the whole left half of the dot is deformed outwards by a distance of order $\delta L$, then $W_{\text{asym}}\sim L$. In (b), a small part of the boundary of the left half (of width $W_{\text{asym}}$) is deformed by a distance $\delta L⪢{\lambda }_F$. In the former case, the parameter which controls the deviation from symmetry is $\delta L/{\lambda }_F$ while in the latter case it is $W_{\text{asym}}/L$.

Figure 4

(Color online) Contributions to weak localization due to an inversion symmetry, which is equivalent to symmetry under a $180°$ rotation about the point indicated by $\oplus$. In the folded sketches on the right-hand side, paths that end on the left lead are marked with solid circles, while those that end on the right lead are marked with open (red) circles (as in Fig. 2). When we fold all paths into one half of the dot (see the right-hand sketches) by rotating the other half of the dot by $180°$, any path which crosses the line between the two halves (dashed line) at $\left(r,p\right)$ reappears at $\left(-r,-p\right)$, where $r=0$ is defined as the center of the inversion symmetry (marked $\oplus$). The sketch in (a) is an enhanced forward-scattering contribution similar to that in Fig. 2a.

Figure 5

(Color online) Contributions to weak localization due to an up-down mirror symmetry for a dot with symmetric leads. The sketch in (a) is an enhanced backscattering contribution; there are no enhanced forward-scattering contributions (unlike for left-right and inversion-symmetric dots).

Figure 6

(Color online) (a) A contribution to universal conductance fluctuations in a left-right mirror-symmetric system. On the right we show the same contribution folded into half the dot, such that one can see the two effective encounters (we only show the dashed paths at the encounters). (b) The full list of pairings for path ${\gamma }_1$ and ${\gamma }_2$, with left-right, up-down and inversion symmetry. There are also contributions in which paths are paired with the time-reverse of their images (arrows reversed); indeed, (a) is an example of this for a left-right symmetric system.

Figure 7

(Color online) Effect of a magnetic field $B=\left(\lambda /{\lambda }_c\right)B_c\simeq \lambda B_0$ on the weak-leak localization correction (left panels) and the conductance fluctuations (right panels) for quantum dots of different spatial symmetry. The solid lines show the universal crossover functions (46, 47), derived in this paper based on semiclassical arguments. The data points show the result of numerical computations based on the interpolation formulas Eqs. (41, 42) between the random-matrix ensembles tabulated in Table . In these computations, $M=1000$ and $N=50$, resulting in ${\lambda }_c=1/\sqrt{20}$. Each data point is obtained by averaging over 5000 realizations.

Figure 8

(Color online) Same as Fig. 7, but for transitions induced by the breaking of internal spatial symmetries $\left(B=0\right)$.

Figure 9

(Color online) Same as Fig. 8, but for a finite magnetic field.