of a potential U(z), assuming one-dimensional case, is described by wave functions that satisfy Schrödinger’s equation where ħ is the reduced Planck’s constant, z is the position, and me is the electron mass. In the zero-potential regions on two sides of the barrier, the wave function takes on the forms for z < 0, for z > w, where . Inside the barrier, where E < U, the wave function is a superposition of two terms, each decaying from one side of the barrier: for 0 < z < w, where . The coefficients r and t provide measure of how much of the incident electron's wave is reflected or transmitted through the barrier. Namely, of the whole impinging particle current only is transmitted, as can be seen from the probability current expression which evaluates to . The transmission coefficient is obtained from the continuity condition on the three parts of the wave function and their derivatives at z = 0 and z = w (detailed derivation is in the article Rectangular potential barrier). This gives where . The expression can be further simplified, as follows: In STM experiments, typical barrier height is of the order of the material's surface work function W, which for most metals has a value between 4 and 6 eV. The work function is the minimum energy needed to bring an electron from an occupied level, the highest of which is the Fermi level (for metals at T = 0 K), to vacuum level. The electrons can tunnel between two metals only from occupied states on one side into the unoccupied states of the other side of the barrier. Without bias, Fermi energies are flush, and there is no tunneling. Bias shifts electron energies in one of the electrodes higher, and those electrons that have no match at the same energy on the other side will tunnel. In experiments, bias voltages of a fraction of 1 V are used, so is of the order of 10 to 12 nm−1, while w is a few tenths of a nanometer. The barrier is strongly attenuating. The expression for the transmission probability reduces to The tunneling current from a single level is therefore where both wave vectors depend on the level's energy E, and Tunneling current is exponentially dependent on the separation of the sample and the tip, typically reducing by an order of magnitude when the separation