Analog Sauter-Schwinger effect in semiconductors for spacetime-dependent fields

The Sauter-Schwinger effect predicts the creation of electron-positron pairs out of the quantum vacuum via tunneling induced by a strong electric field. Unfortunately, as the required field strength is extremely large, this fundamental prediction of quantum field theory has not been verified experimentally yet. Here, we study under which conditions and approximations the interband tunneling in suitable semiconductors could be effectively governed by the same (Dirac) Hamiltonian, especially for electric fields which depend on space and time. This quantitative analogy would allow us to test some of the predictions (such as the dynamically assisted Sauter-Schwinger effect) in this area by means of these laboratory analogs.

Figure 1

Electron dispersion relations in the two systems under consideration, without an external electric field $\left(A=0\right)$. (a) Dirac case: the two branches of the relativistic energy-momentum relation. (b) Semiconductor case: example for an electronic two-band structure in the first Brillouin zone (reduced zone scheme). We assume throughout this paper that the semiconductor has a direct band gap at the center of the Brillouin zone, measuring ${\mathcal{E}}_g=\mathrm{\Delta }\mathcal{E}\left(0\right)$.

Figure 2

Two energy bands separated by a gap $\chi$ are bent by a localized, space-dependent electric field centered at $x=0$ (schematically, $tanh$ profiles). The solid curves are the lower edge of the upper energy band and the upper edge of the lower band, respectively. We have $\chi =2m$ in QED and $\chi ={\mathcal{E}}_g$ in the semiconductor analog. (a) For field profiles which give rise to a large potential difference $q\mathrm{\Delta }\mathrm{\Phi }\gg \chi$, there are many different states between which tunneling is possible (e.g., along the dashed line). (b) For $q\mathrm{\Delta }\mathrm{\Phi }\searrow \chi$, the number of possible tunneling transitions approaches zero and the spatial turning points $x_{\pm }^{★}$ diverge. In this $\mathrm{\Delta }\mathrm{\Phi }$ range, the tunneling rate can significantly be increased via additional electric low-frequency pulses according to Ref. [11]. (c) If the energy step $q\mathrm{\Delta }\mathrm{\Phi }$ is smaller than the gap $\chi$, the bands are separated energetically as indicated by the dashed constant-energy line, so tunneling becomes impossible.

Figure 3

Threshold ${\mathrm{CO}}_2$-laser-beam intensity (61) for dynamical assistance of tunneling as a function of the width $L=2\pi /k$ of the static Sauter pulse $E_1/{cosh}^2\left(kx\right)$ in GaAs. The parameter values in this plot are $E_1=E_{\mathrm{crit}}^{\mathrm{GaAs}}/10\approx 60\mathrm{MV}/\mathrm{m},\omega =0.117\mathrm{eV}$ (so ${\gamma }_{\omega }=1.56)$, and $I_{\mathrm{crit}}(L\to \infty )=I_{\mathrm{crit}}(k=0)=47\mathrm{kW}/{\mathrm{cm}}^2$ (see Sec. 2f3). Tunneling vanishes in the limit ${\gamma }_k\nearrow 1$ [cf. Fig. 2], which corresponds to $L\searrow L_0=76\mathrm{nm}$ here.