Transformation electronics: Tailoring the effective mass of electrons

The speed of integrated circuits is ultimately limited by the mobility of electrons or holes, which depend on the effective mass in a semiconductor. Here, building on an analogy with electromagnetic metamaterials and transformation optics, we describe a transport regime in a semiconductor superlattice characterized by extreme anisotropy of the effective mass and a low intrinsic resistance to movement—with zero effective mass—along some preferred direction of electron motion. We theoretically demonstrate that such a regime may permit an ultrafast, extremely strong electron response, and significantly high conductivity, which, notably, may be weakly dependent on the temperature at low temperatures. These ideas may pave the way for faster electronic devices and detectors and functional materials with a strong electrical response in the infrared regime.

Figure 1

Transformation electronics and electronic metamaterials: Sketch of the geometry and electronic band diagram of the elements of the superlattice. (a) Geometry of a stratified superlattice formed by alternating layers of semiconductor alloys with band gaps with different signs. (b) Electromagnetic analog of the superlattice for energy levels close to $E-V_{\mathrm{eff}}\approx 0$: In the band gap the semiconductor with a positive (negative) band gap is the electronic analog of an $\varepsilon <0$ ($\mu <0$) electromagnetic material. (c) Detailed energy band structure of each layer of the superlattice, showing the valence band offset $\Lambda$ between the two semiconductors.

Figure 2

Effective parameters (mass and potential) of the semiconductor alloys. Left axis: Dispersive mass as a function of the normalized electron energy $E$; solid lines: exact result taking into account the effect of the split-off bands (see Refs. 18 and 19); dashed lines: linear mass approximation described in the main text. As seen, the effect of the split-off bands is negligible in the energy range of interest. Right axis: Effective potential ($E-V$) as a function of the normalized electron energy $E$.

Figure 3

Electronic band structure of the superlattice electronic metamaterials. (a) Energy dispersion of a superlattice with $v_P=1.06\times {10}^6\mathrm{m}/\mathrm{s}$, $E_{g1}=-E_{g2}=0.30\mathrm{eV}$, $\Lambda =0.40E_{g1}$, $f_1=f_2=0.5$, and $a=6a_s=3.9\mathrm{nm}$. (b) Contours of constant energy in the $x$-$z$ plane. (c) Comparison between the energy dispersion calculated using the Kronig-Penney model [Eq. (2)] and the approximate result given by Eq. (4) for the values of $k_{x}a/\pi =0.0,0.2,0.5,1.0$. (d) Energy dispersion of the superlattice when $f_1=f_2=0.5$ (black solid line), $f_1=0.48$ (green dashed line), and $f_1=0.52$ (blue dotted-dashed line).

Figure 4

Conductivity and density of states of the superlattice. (a) Conductivity of the semiconductor superlattice (SL) at 10 THz as a function of the temperature for different values of the valence band offset: $\Lambda =E_g$ (green lines), $\Lambda =0.67E_g$ (blue lines), and $\Lambda =0.4E_g$ (black lines). The gray arrows indicate the direction of increasing $\Lambda$. The solid lines were calculated using the “exact” energy dispersion of the superlattice, whereas the dashed lines were obtained from Eqs. (5a) and (5b). (b) Conductivity of the semiconductor superlattice as a function of frequency for different values of the valence band offset at 300 K. (c) Similar to (a) but for the normalized intraband conductivity of Hg${}_{0.65}$Cd${}_{0.35}$Te, assuming that the Fermi level lies at the midpoint of the energy band gap. (d). Normalized density of states of the semiconductor superlattice for different values of the valence band offset. In all the calculations it was assumed that $k_{\left|\right|,\max }=0.1\pi /a_s$. Since $g\left(E\right)$ was computed based on the approximate model (4), the results of (d) are meaningful only for $\left|E\right|\ll \Lambda$.