Biperiodic superlattices and the transparent state

We study biperiodic semiconductor superlattices, which consist of alternating cell types, one with wide wells and the other narrow wells, separated by equal strength barriers. If the wells were identical, it would be a simply periodic system of $N=2n$ half-cells. When asymmetry is introduced, an allowed band splits at the Bragg point into two disjoint allowed bands. The Bragg resonance turns into a transparent state located close to the band edge of the lower (upper) band when the first (second) well is the wider. Analysis of this system gives insight into how band splitting occurs. Further, we consider semiperiodic systems having $N=2n+1$ half-cells. Surprisingly, these have very different transmission properties, with an envelope of transmission maxima, which crosses the envelope of minima at the transparent point.

Figure 1

(Color online) Solid (red) line: biperiodic array[1, 2] of three double cells, each of width $2d$; short-dashed (blue) line: an additional half-cell could be added at right, creating an additional narrow well, as discussed in Sec. 4.

Figure 2

Transmission (solid line) for the three-cell biperiodic array of Fig. 1: (a) wide well first and (b) narrow well first. The dashed line is the envelope of transmission minima in the allowed zones and an upper bound in the band gap.

Figure 3

(Color online) Logarithmic derivatives $\gamma$, $\lambda$ of the solutions $g\left(kd\right)$ and $u\left(kd\right)$ for the delta-barrier system for very small ${10}^{-5}$ and moderate 0.10 asymmetries; $\gamma$ is shown as solid (red) line and short-dashed (blue) line; $\lambda$ as long-dashed (green) and dotted (mauve) lines, respectively.

Figure 4

(Color online) $\cos {\varphi }_h$ of the half-cell (solid/red line) and $\cos \varphi$ of the double cell for the delta-barrier system; (a) for very small $\left({10}^{-5}\right)$ asymmetry (long-dashed and/or green line) and (b) for moderate (0.10) asymmetry (short-dashed and/or blue line).

Figure 5

(Color online) ${\tan }^2\varphi ∕2$: long-dashed (green) line, $Z^2$: solid (red) line, and ${\tilde{Z}}^2$: short-dashed (blue) line for the delta-barrier system. (a) Case of very small $\left({10}^{-5}\right)$ asymmetry and (b) moderate (0.10) asymmetry. In (a) the two poles almost coincide and cancel, leaving only a small glitch.

Figure 6

(Color online) Pole region of $Z^2$ and ${\tilde{Z}}^2$ and wave functions $g\left(x\right)$, $g^{\prime }\left(x\right)$ and $u\left(x\right)$, $u^{\prime }\left(x\right)$; (a) case of ${10}^{-5}$ asymmetry and (b) moderate (0.10) asymmetry. Lines identified at upper right.

Figure 7

Transmission of AlGaAs-barrier three-cell biperiodic arrays: (a) zero asymmetry; (b) $2(a-c)=0.1\mathrm{nm}$; and (c) $2(a-c)=-0.1\mathrm{nm}$. Lines as in Fig. 2.

Figure 8

(Color online) Impedance parameter $\mu$ of the three-cell arrays of Fig. 7: solid (red) line, zero asymmetry; short-dashed (blue) line, small asymmetry; and long-dashed (green) line, opposite asymmetry. The arrows imply divergence to $-\infty$.

Figure 9

(Color online) Transmission of seven half-cell array as solid (red) line, (a) showing both split bands and (b) detail of region around the transparent state. Dash-dotted (turquoise) line is envelope of maxima; dotted (mauve)line is envelope of minima up to the transparent point.

Figure 10

(Color online) Transmission of 35 half-cell array showing detail in region of the transparent state in the upper split band. Lines as in Fig. 9.