Analytic Theory of Wannier-Stark Quantization in Two Dimensions

For the first time, one-particle theory of Wannier-Stark quantization for a $a(N$$-$1)-long chain affected by a homogeneous electric field $E_e$ is extended to the 2D case of $L_N$=$a(N$$-$1)-long and $L_{\cal N}$=$a$$({\cal N}$$-$1)-wide conductor, which is modeled by the atomic square lattice with the electron site-energy change from atom to atom in the direction parallel to $\cal N$ axis by the amount of electric field parameter (efp) $\varepsilon \equiv aeE_e/|t|$ in units of $|t|$. It is shown that each field-provoked $\mu$-level in the chain spectrum gives birth to the $\mu$-subband of field-independent levels due to the electron-transfer interaction in the direction perpendicular to $E_e$. The level spacing and hence, the width of $\mu$-subbands $E^0_{\rm bw}$, is dictated by the conductor length, the hopping integral $t$, and by the lattice constant $a$. Another principal result is that the levels, which are within the energy interval $0.5E^0_{\rm bw}$ on either sides of the spectrum (that is the edge spectrum), correspond to the delocalized states which are extended in the direction perpendicular to the electric field. In this direction, the electron spectrum width $E^{\perp}_{\rm bw}$ is larger by $E^0_{\rm bw}$ than the spectrum width in the field direction $E^{\parallel}_{\rm bw}$. Several special cases of the interrelation between the electron energy versus applied voltage are identified, when the spacing between the $\mu$-subband centers is equal either to integer or fraction ($>1$) portion of dimensionless $\varepsilon$. Otherwise, the level spacing is shown to be irregular and, depending on $N$, $\cal N$, and $\varepsilon$, it can be equal to any portion of $\varepsilon$. It is argued that one of straightforward applications of the presented theory is the quantum-mechanical explanation of Hall effect in two dimensions.