Computation of the Stark effect in P impurity states in silicon

We compute within the effective-mass theory and without adjustable parameters the Stark effect for shallow P donors in Si with anisotropic band structure. Valley-orbit coupling is taken into account in a nonperturbative way and scattering effects of the impurity core are included to properly describe low-lying impurity states. The ground-state energy slightly decreases with increasing electric field up to a critical value $E_{\mathit{cr}}\sim 25\mathrm{keV}∕\mathrm{cm}$, at which the donor can be ionized by tunneling due to a field-induced mixing of the “$1s$-like” singlet ground state with a “$2p_0$-like” excited state in zero field. The resulting ground-state wave function at high field extends significantly outside the potential barrier surrounding the impurity. Calculations of the hyperfine splitting and of the $A$-shell superhyperfine coupling constants as a function of the electric field complete the work.

Figure 1

(Color online) Bottom panel: the lowest-energy levels of Si:P (solid lines); the black dashed line denotes the minimum of the barrier energy. Top panel: the square modulus of the corresponding wave functions computed at the impurity site and normalized to the zero-field value. Each ${\parallel \Psi \left(\mathbf{0}\right)\parallel }^2$ is denoted with the same gray scale (color) of the corresponding energy level (the black dotted line denotes the ground state).

Figure 2

(Color online) Square modulus of the impurity wave function (solid lines) and of the corresponding envelope function (${\sum }_i{\parallel F_i\left(z\right)\parallel }^2$, dashed lines) for the three lowest-energy states (in black, red, and green) of Si:P (top and middle panel with different vertical scales) and the sum of electric field and screened Coulomb potential (bottom panel) as a function of the Cartesian coordinate $z$. The rapid oscillatory behaviors are fingerprints of interferences between states from different conduction band valleys. $\mathcal{E}=(0,0,24.5)\mathrm{keV}∕\mathrm{cm}$. The vertical dotted line corresponds to (the saddle point at) the minimum of the energy barrier surrounding the impurity. Inset: the same as bottom panel of Fig. 1 on a magnified scale.