4.4 Resistivity

Definition of resistivity

Resistivity quantifies how strongly a material opposes the flow of electric current. It's a property of the material itself, not of any particular object made from that material. This distinction matters: a long thin copper wire has more resistance than a short thick one, but both have the same resistivity because they're both copper.

Resistance vs resistivity

Resistance depends on both the material and the object's shape. Resistivity depends only on the material.

They connect through this equation:

$R = \rho \frac{L}{A}$

• $R$ = resistance (ohms, $\Omega$)
• $\rho$ = resistivity (ohm-meters, $\Omega \cdot m$)
• $L$ = length of the conductor
• $A$ = cross-sectional area

Think of it this way: making a wire longer increases resistance (more material for electrons to push through), while making it thicker decreases resistance (more room for electrons to flow). Resistivity captures the material's contribution, separate from geometry.

Units of resistivity

Resistivity is measured in ohm-meters ($\Omega \cdot m$). You can see why from rearranging the formula: $\rho = \frac{RA}{L}$, which gives $\Omega \cdot m^2 / m = \Omega \cdot m$.

Because resistivity spans a huge range across materials, you'll often see prefixes:

• Metals: micro-ohm-meters ($\mu\Omega \cdot m$)
• Insulators: mega-ohm-meters ($M\Omega \cdot m$) or higher

The inverse of resistivity is conductivity ($\sigma$), measured in siemens per meter (S/m).

Factors affecting resistivity

Temperature dependence

For most metals, resistivity increases as temperature rises. Higher temperature means atoms vibrate more, and those vibrations (called phonons) scatter electrons more frequently, slowing current flow.

Semiconductors often behave the opposite way: heating them frees more charge carriers, so resistivity decreases with temperature.

The temperature coefficient of resistivity ($\alpha$) captures this relationship and is covered in more detail below.

Material composition

• Pure metals tend to have lower resistivity than alloys. Mixing metals disrupts the regular atomic arrangement, creating more scattering sites for electrons.
• Impurities and crystal defects also raise resistivity for the same reason.
• In semiconductors, adding tiny amounts of impurities (doping) can change resistivity by many orders of magnitude.

Crystal structure

A material's atomic arrangement affects how easily electrons move through it. Single crystals with highly ordered structures generally have lower resistivity than polycrystalline materials, where grain boundaries act as additional scattering sites. Some crystals, like graphite, conduct much better in one direction than another (this anisotropy is discussed later).

Resistivity in different materials

Resistivity spans roughly 24 orders of magnitude from the best conductors to the best insulators.

Metals and alloys

Metals have the lowest resistivity, typically $10^{-8}$ to $10^{-6} \; \Omega \cdot m$. Their abundance of free electrons explains this. Some key values:

• Copper: $1.68 \times 10^{-8} \; \Omega \cdot m$
• Aluminum: $2.82 \times 10^{-8} \; \Omega \cdot m$
• Stainless steel (an alloy): $6.9 \times 10^{-7} \; \Omega \cdot m$

Notice that stainless steel, an alloy, has resistivity roughly 40 times higher than copper. Alloying increases electron scattering.

Semiconductors

Semiconductors fall in a wide intermediate range, roughly $10^{-4}$ to $10^{8} \; \Omega \cdot m$, depending heavily on temperature and doping.

• Pure silicon: $2.3 \times 10^{3} \; \Omega \cdot m$
• Pure germanium: $0.46 \; \Omega \cdot m$

Doping can push semiconductor resistivity down toward metallic values, which is what makes them so useful in electronics.

Insulators

Insulators have resistivity above $10^{8} \; \Omega \cdot m$, with very few free charge carriers.

• Glass: $10^{10}$ to $10^{14} \; \Omega \cdot m$
• Rubber: $\sim 10^{13} \; \Omega \cdot m$
• Air: $\sim 1.3 \times 10^{16} \; \Omega \cdot m$

Mathematical representation

Resistivity formula

Resistivity is defined as:

$\rho = \frac{RA}{L}$

This applies to a uniform conductor with constant cross-sectional area. If you know three of the four quantities ($\rho$, $R$, $A$, $L$), you can find the fourth.

Example: A 2.0 m copper wire with cross-sectional area $1.0 \times 10^{-6} \; m^2$ has a resistance of:

$R = \rho \frac{L}{A} = (1.68 \times 10^{-8}) \frac{2.0}{1.0 \times 10^{-6}} = 0.034 \; \Omega$

Relationship to conductivity

Conductivity ($\sigma$) is simply the reciprocal of resistivity:

$\sigma = \frac{1}{\rho}$

It connects current density ($J$) to the electric field ($E$) inside a material:

$J = \sigma E$

This is actually the microscopic form of Ohm's law. High conductivity means a small electric field can drive a large current density.

Measurement techniques

Four-point probe method

This is the standard technique for measuring resistivity of semiconductor wafers and thin films. It uses four probes in a line on the sample surface.

1. Press four equally spaced collinear probes onto the sample.
2. Pass a known current $I$ through the two outer probes.
3. Measure the voltage $V$ across the two inner probes.
4. Calculate resistivity using $\rho = \frac{2\pi s V}{I}$, where $s$ is the spacing between adjacent probes.

The key advantage: because the voltage probes carry essentially no current, contact resistance at the probes doesn't affect the measurement. A two-probe method can't avoid that error.

Van der Pauw method

This technique works on thin, flat samples of any shape, as long as four small contacts are placed on the sample's edge. Multiple measurements with different probe configurations are combined using the Van der Pauw equation to extract resistivity. It's especially useful when you can't cut your sample into a standard geometry.

Applications of resistivity

Circuit design

Resistivity determines what materials to use for each part of a circuit. Copper's low resistivity makes it ideal for wiring, while nichrome's higher resistivity makes it useful for heating elements. In integrated circuits, knowing resistivity precisely is essential for calculating voltage drops and power dissipation.

Material characterization

Resistivity measurements reveal a lot about a material's internal state. Changes in resistivity can indicate impurities, defects, or phase transitions. In semiconductor manufacturing, resistivity measurements verify that doping levels are correct.

Geophysical exploration

Geologists use resistivity surveys to map underground structures. By injecting current into the ground and measuring voltage at the surface, they can identify rock types, locate water tables, and find mineral deposits. Electrical resistivity tomography (ERT) builds 2D and 3D images of subsurface resistivity.

Resistivity in thin films

When a film's thickness drops below the mean free path of its electrons (the average distance an electron travels between collisions), resistivity increases noticeably compared to the bulk material.

Size effects

In a thin film, electrons scatter not just off atoms and impurities but also off the film's surfaces and grain boundaries. As thickness decreases, these surface and boundary effects become a larger fraction of total scattering. The Fuchs-Sondheimer model describes this thickness-dependent resistivity increase for metal films.

Surface scattering

Surface roughness amplifies the scattering effect. A smoother film surface means more specular (mirror-like) reflection of electrons, which preserves their momentum and keeps resistivity lower. Careful control of deposition conditions can minimize surface roughness and reduce this effect.

Temperature coefficient of resistivity

The temperature coefficient of resistivity ($\alpha$) tells you how much resistivity changes per degree of temperature change, relative to the original resistivity:

$\alpha = \frac{1}{\rho_0} \frac{d\rho}{dT}$

For small temperature changes, you can use the linear approximation:

$\rho = \rho_0 [1 + \alpha (T - T_0)]$

where $\rho_0$ is the resistivity at reference temperature $T_0$.

Positive vs negative coefficients

• Metals have positive $\alpha$: resistivity goes up with temperature. Copper has $\alpha \approx 3.9 \times 10^{-3} \; /°C$.
• Semiconductors often have negative $\alpha$: resistivity drops as temperature rises because more carriers are thermally excited.
• Special alloys like constantan and manganin are engineered to have $\alpha \approx 0$, making them ideal for precision resistors that shouldn't drift with temperature.

Superconductivity

Below a material-specific critical temperature ($T_c$), certain materials lose all electrical resistance. Resistivity drops to exactly zero, not just very small. This enables lossless current flow and is used in MRI magnets, particle accelerators, and other applications requiring strong, stable magnetic fields. High-temperature superconductors (like cuprate ceramics) have higher $T_c$ values, making them easier to cool to their superconducting state.

Anisotropic resistivity

In some materials, resistivity depends on the direction of current flow. This is described mathematically using a resistivity tensor rather than a single scalar value.

Directional dependence

Anisotropic materials have different resistivity values along different crystal axes. This can be exploited in devices that need direction-sensitive conduction. Measuring anisotropic resistivity requires specialized techniques like the Montgomery method, which accounts for the different resistivity components.

Examples in crystals

Graphite is the classic example. Its layered structure means electrons move easily within the carbon planes but poorly between them. The in-plane resistivity is roughly $3.5 \times 10^{-5} \; \Omega \cdot m$, while the perpendicular resistivity is about 100 times higher. Liquid crystals also display anisotropic resistivity, which plays a role in display technology.

Resistivity in semiconductors

Semiconductor resistivity is uniquely tunable, which is what makes semiconductors the foundation of modern electronics.

Doping effects

Doping means intentionally adding impurity atoms to a semiconductor:

• N-type doping adds atoms with extra valence electrons (donors), increasing free electron concentration and decreasing resistivity.
• P-type doping adds atoms with fewer valence electrons (acceptors), increasing hole concentration and also decreasing resistivity.

The dopant concentration directly controls resistivity. Even parts-per-million levels of doping can reduce silicon's resistivity by several orders of magnitude.

Intrinsic vs extrinsic

• Intrinsic semiconductors are pure. Their resistivity depends on the band gap and temperature, since only thermal energy creates charge carriers.
• Extrinsic semiconductors are doped. At normal operating temperatures, the dopant atoms dominate the carrier concentration.

At very high temperatures, even extrinsic semiconductors behave intrinsically because thermally generated carriers overwhelm the dopant contribution. The temperature where this crossover happens is called the intrinsic temperature.

Resistivity in composite materials

Composites combine two or more materials, and their resistivity depends on what's mixed, how much of each component is present, and how the components are arranged.

Effective medium theory

Mathematical models like the Maxwell-Garnett and Bruggeman models estimate composite resistivity by averaging the properties of the constituents. These models account for factors like particle shape, orientation, and the nature of interfaces between components.

Percolation threshold

If you mix conductive particles into an insulating matrix, the composite stays insulating until the conductive filler reaches a critical volume fraction called the percolation threshold. At that point, conductive particles form continuous pathways through the material, and resistivity drops sharply. The exact threshold depends on filler geometry: long, thin fillers (like carbon nanotubes) percolate at much lower volume fractions than spherical particles.