field at a distance from the surface, and the number of electrons per unit volume. Then the current is given has two contributions, one due to drift and the other due to diffusion: When is the electrons mobility and the diffusion coefficient. Laplace's equation gives for the field: Hence, eliminating , we have: After integrating, making use of the Einstein relation and neglecting the term we obtain for the electric field: where is a constant. We may neglect the term because we are supposing that and . Since, at , , we have: It follows that the potential drop across the crystal is: Making use of () and () we can write in terms of . For small , is small and , so that: Thus the current increases as the square of . For large , and we obtain: As an application example, the steady-state space-charge-limited current across a piece of intrinsic silicon with a charge-carrier mobility of 1500 cm2/V-s, a relative dielectric constant of 11.9, an area of 10−8 cm2 and a thickness of 10−4 cm can be calculated by an online calculator to be 126.4 μA at 3 V. Note that in order for this calculation to be accurate, one must assume all the points listed above. In the case where the electron/hole transport is limited by trap states in the form of exponential tails extending from the conduction/valence band edges, the drift current density is given by the Mark-Helfrich equation, where is the elementary charge, with being the thermal energy, is the effective density of states of the charge carrier type in the semiconductor, i.e., either or , and is the trap density. Low voltage regime In the case where a very small applied bias is applied across the single-carrier device, the current is given by: Note that the equation describing the current in the low voltage regime follows the same thickness scaling as the Mott–Gurney law, , but increases linearly with the applied voltage. Saturation regimes When a very large voltage is applied across the semiconductor, the current can transition into a saturation regime. In the velocity-saturation regime, this equation takes the following form Note the different dependence of on between the Mott–Gurney law and the equation describing the current in the velocity-saturation regime. In the ballistic case (assuming no collisions), the Mott–Gurney equation takes the form of the more familiar Child–Langmuir law. In the