There are two major mechanisms for current flow in physical systems: Drift current is motion of charge due to an electric field (\mathcal{E} field). Diffusion current is random motion of charge that follows a concentration gradient, i.e. it flows from areas of high concentration toward areas of low concentration.
An illustration of drift current is shown below. An electric field \mathcal{E} is applied from left to right across the material. Mobile h^+ are accelerated in the direction of the field, and mobile e^- are accelerated in the opposite direction. By convention, when e^- flow runs from the right-to-left, the electric current flows from left-to-right. So both the e^- and h^+ contribute to a current in the same direction, oriented with the electric field.
In free space, a particle can be accelerated continuously toward the velocity of light. In a solid material, the particle reaches a lesser velocity v_{\text{drift}} as determined by the material’s mobility \mu, which has units of velocity per electric field. The mobility describes the terminal velocity of a particle as it moves through a solid, sort of like the terminal velocity of an object falling through air:
v_{\text{drift}}=\mu\mathcal{E}
As a particle moves through a solid, it tends to follow an irregular path disturbed by collisions with other particles, and with atoms in the crystal lattice. We usually consider mobility to be a constant, but it does have some dependency on temperature and on the total charge density.
Given the average e^- velocity, the drift current density J_{\text{drift}} is the product of v_{\text{drift}} with the carrier density (n or p) and the elementary charge per carrier (q). To obtain the total current, we add together the contributions from both e^- and h^+, then multiply the current density into the total cross-sectional area (A_{cs}) through which the current flows:
\begin{aligned} J_{\text{drift}} &= q(\mu_n n + \mu_p p)\mathcal{E} & \text{A~per~cm}^2\\ I_{\text{drift}} &= J_{\text{drift}}A_{CS} & \text{A} \end{aligned}
Resistivity is a property of drift current. If the material is homogeneous, meaning it has the same composition throughout its volume, then the drift current should usually be proportional to an applied voltage. Suppose a voltage V is applied across a section of material with length L. This will induce a constant electric field in the material, \mathcal{E}=V/L. Combining this with the expression for I_{\rm drift}, we obtain
\begin{aligned} I_{\textrm{\small drift}} &= A_{cs}q \mathcal{E} \left(\mu_p p + \mu_n n\right)\\ &= A_{cs}q\frac{V}{L}\left(\mu_p p + \mu_n n\right)\\ \Rightarrow R = \frac{V}{I_{\rm drift}} &= \frac{L}{A_{CS}}\frac{1}{q\left(\mu_p p + \mu_n n\right) } \end{aligned}
Finally we can see that the expression has the same form as classical resistance, R=\rho L/A_{CS}, where the \rho is the material’s resistivity defined as
\rho = \frac{1}{q\left(\mu_p p + \mu_n n\right)}.
Conversely, the material’s conductivity \sigma is the inverse of resistivity: \sigma=1/\rho.
Resistivity can be expressed in units of \Omega/\text{cm}, but in semiconductor applications it also appears in units of “Ohms per square,” written \Omega/\square. This is useful for calculating the resistance of wire traces with a constant thickness t_w. For a square piece of material, the total resistance is
R_\square=\rho \frac{t_w W}{W} = \rho t_w.
Then if we make a wire by placing N squares in series, the total resistance is NR_\square.
Diffusion is the process by which randomly moving particles establish a net flow along a concentration gradient. To visualize diffusion, consider a box with three chambers as shown below. The box is full of particles represented by little flies that move randomly. The left-most chamber has the most flies, and the right-most has the least. We suppose that the concentration of flies, N(x) [particles per \,\text{cm}^3], changes uniformly so that there is a constant \Delta N \triangleq N(x) - N(x+1). Also imagine that the chambers continue indefinitely to the left and to the right, so we are looking somewhere in the middle of a large system.
If the partitions are suddenly removed then the flies will freely migrate between chambers. Suppose that after a short time \Delta t all flies have moved either -\Delta x (to the left) or +\Delta x (to the right). Each fly has equal probability of moving either direction. So half the flies from N(0) move to the right into chamber 1, and half of N(1) move to the left into chamber 0. So after \Delta t, the net movement across the boundary is
\begin{aligned} N_{0\rightarrow 1} &= \frac{N(0)}{2}-\frac{N(1)}{2}\\ &= \frac{\Delta N}{2} \end{aligned}
The net particle flux is then equal to the net number of transferred particles times their average velocity, \Delta x / \Delta t:
\begin{aligned} \phi_{\rm diff} &= \left(\frac{\Delta N}{2}\right) \times \left(\frac{\Delta x}{\Delta t}\right)\\ &= \left(\frac{\Delta N}{\Delta x}\right) \times \left(\frac{\left(\Delta x\right)^2}{2\Delta t}\right)\\ &= \nabla N(x) D \end{aligned}
Here we have introduced two important concepts: the first is the concentration gradient \nabla N(x) [particles per cm], defined as the spacial derivative of concentration with respect to position:
\nabla N(x) \triangleq \frac{\partial N(x)}{\partial x}.
The second new concept is the diffusivity D, with units \,\text{cm}^2/s. Diffusivity indicates the speed with which particles move under thermal excitation. It is a property of Brownian motion, an important branch of thermodynamics that was illuminated in one of Einstein’s first papers.
Diffusivity is strongly related to both temperature and mobility, and is quantified in the famous Einstein relations:
\begin{aligned} D &= \mu \frac{k_B T}{q}.\\ &= \mu U_T, \end{aligned}
where U_T is the highly important “thermal voltage’ ’ that arises in many device equations. At room temperature, U_T=27\,\text{mV}. This quantity is ubiquitous in electronics since most devices rely in some way on diffusion, and U_T determines the speed of diffusion via the Einstein relations.
Now let’s suppose that the flies are electrons. To obtain the current density from the particle flux, we include the charge -q. In a semiconductor, holes are also able to diffuse with charge +q. So there are two components of diffusion current:
\begin{aligned} J_{\rm diff}~(e^-) &= -q\nabla n(x) D_n\\ J_{\rm diff}~(h^+) &= q\nabla p(x) D_p \end{aligned}
In these expressions we included the position x since the gradient can change at different positions in the material.
From the analyses above, we know there are two mechanisms of current flow (drift and diffusion) and two species of charge carriers to carry current (electrons and holes). The total current in a semiconductor is the superposition of all four:
\begin{aligned} J_{\rm drift} &= q\mathcal{E}\left(\mu_p p + \mu_n n\right)\\ J_{\rm diff} &= q\left(\nabla p D_p - \nabla n D_n\right)\\ J &= J_{\rm diff} + J_{\rm drift}\\ I &= A_{CS} J \end{aligned}
In thermal equilibrium these current mechanisms can all occur simultaneously, but the net equilibrium current must balance out to zero. We can treat this as a definition of thermal equilibrium for electronic devices. To have a non-zero net current, there must be an excess in at least one mechanism.