4.6 Temperature dependence of resistance

Resistance and Temperature Relationship

Electrical resistance in materials changes with temperature. In metals, higher temperatures cause atoms in the crystal lattice to vibrate more, which makes it harder for electrons to flow through. In semiconductors, the story is different: higher temperatures free up more charge carriers, which actually makes current flow easier. The direction and magnitude of this change depends on the type of material.

Positive Temperature Coefficient

Materials with a positive temperature coefficient (PTC) have resistance that increases as temperature rises. Most metals behave this way. As temperature goes up, lattice vibrations (phonons) intensify, and electrons collide with the lattice more frequently. That increased scattering is what drives resistance up.

• Most metals and some ceramics exhibit PTC behavior
• PTC materials are used in self-regulating heating elements, like car rear window defrosters: as the element heats up, its resistance rises, which naturally limits the current and prevents overheating

Negative Temperature Coefficient

Materials with a negative temperature coefficient (NTC) have resistance that decreases as temperature rises. Many semiconductors and certain ceramics behave this way. The reason is that thermal energy excites more electrons into the conduction band, increasing the number of available charge carriers. Even though scattering also increases, the surge in carrier concentration wins out, and resistance drops.

• NTC materials are commonly used in temperature sensors and voltage regulators

Temperature Coefficient of Resistance

Definition and Units

The temperature coefficient of resistance ($\alpha$) quantifies how much a material's resistance changes per degree of temperature change, relative to its resistance at a reference temperature. It's defined as:

$\alpha = \frac{1}{R} \frac{dR}{dT}$

• Units are typically ppm/°C (parts per million per degree Celsius) or $\text{K}^{-1}$
• A positive $\alpha$ means resistance increases with temperature; a negative $\alpha$ means it decreases

Typical Values for Materials

• Pure metals have positive coefficients, typically 3000 to 6000 ppm/°C. Copper is about 3900 ppm/°C, and aluminum is about 3700 ppm/°C.
• Semiconductors can have large negative coefficients, ranging from -20,000 to -70,000 ppm/°C.
• Specialty alloys like constantan and manganin are engineered to have near-zero temperature coefficients, making them useful in precision resistors where you don't want resistance drifting with temperature.

Resistance in Conductors

Electron Collisions and Temperature

Resistance in metals comes from conduction electrons scattering off lattice vibrations (phonons). When temperature rises, the lattice vibrates more intensely, so electrons collide more often and the mean free path (average distance an electron travels between collisions) gets shorter.

Matthiessen's rule says you can treat total resistivity as the sum of two parts:

• A temperature-dependent part from phonon scattering (goes up with temperature)
• A temperature-independent part from defects and impurities (stays roughly constant)

$\rho_{\text{total}} = \rho_{\text{phonon}}(T) + \rho_{\text{impurity}}$

Linear Approximation for Metals

Over a moderate temperature range, the resistance of most metals increases approximately linearly with temperature:

$R(T) = R_0[1 + \alpha(T - T_0)]$

• $R_0$ is the resistance at a chosen reference temperature $T_0$ (often 20°C)
• $\alpha$ is the temperature coefficient of resistance
• This linear model works well for everyday temperature ranges but breaks down at very low temperatures (where quantum effects matter) and very high temperatures (where higher-order terms become significant)

Resistance in Semiconductors

Band Gap and Temperature

Semiconductors have an energy band gap separating the valence band (where electrons are bound) from the conduction band (where electrons can move freely). At low temperatures, very few electrons have enough energy to jump across this gap, so resistance is high.

As temperature increases, thermal energy promotes more electrons into the conduction band, creating more charge carriers. The band gap also narrows slightly with increasing temperature, which further boosts conductivity. The net result: resistance drops as temperature rises.

Intrinsic vs. Extrinsic Semiconductors

• Intrinsic semiconductors (pure, undoped) rely entirely on thermal excitation for their charge carriers. Their resistance decreases exponentially with temperature.
• Extrinsic semiconductors have been doped with impurities to add extra electrons or holes. At moderate temperatures, the dopant atoms dominate the carrier concentration and resistance changes more gradually. At high enough temperatures, thermal excitation overwhelms the doping, and the material starts behaving like an intrinsic semiconductor again.

Superconductivity

Critical Temperature

Below a material-specific critical temperature ($T_c$), superconductors transition to a state of zero electrical resistance. This isn't just very low resistance; it's exactly zero.

• Low-temperature superconductors (LTS) have $T_c$ below 30 K. Niobium-titanium, for example, has $T_c \approx 10 \text{ K}$.
• High-temperature superconductors (HTS) have $T_c$ above 30 K. YBCO (yttrium barium copper oxide) has $T_c \approx 93 \text{ K}$, which is above the boiling point of liquid nitrogen (77 K), making it much more practical to cool.

Zero Resistance Phenomenon

The zero-resistance state arises from the formation of Cooper pairs: pairs of electrons that couple through interactions with the lattice and travel through the material without scattering. This is a quantum mechanical effect with no classical analog.

• The Meissner effect is a related phenomenon where a superconductor expels all magnetic fields from its interior when cooled below $T_c$
• Supercurrents can persist indefinitely, which is why superconducting electromagnets (used in MRI machines and particle accelerators) can maintain enormous magnetic fields with no energy loss

Applications of Temperature-Dependent Resistance

Thermistors and Their Uses

Thermistors are resistors specifically designed to have large temperature coefficients, making them very sensitive to temperature changes.

• NTC thermistors are widely used for precise temperature measurement: medical thermometers, automotive engine sensors, and HVAC systems
• PTC thermistors are used for overcurrent protection (resistance spikes when the device overheats, limiting current) and self-regulating heating elements

RTDs vs. Thermocouples

Resistance Temperature Detectors (RTDs) use a metal element (usually platinum) whose resistance changes predictably with temperature.

• RTDs offer high accuracy and stability over a range of about -200°C to 850°C
• They work because metals have a well-characterized, nearly linear resistance-temperature relationship

Thermocouples work on a completely different principle: they generate a voltage based on the temperature difference between a junction of two dissimilar metals and a reference point.

• Thermocouples cover a wider range (-270°C to 1800°C) but are generally less accurate than RTDs
• They're preferred in extreme-temperature or harsh environments

Mathematical Models

Callendar-Van Dusen Equation

For platinum RTDs, the linear approximation isn't accurate enough over wide temperature ranges. The Callendar-Van Dusen equation provides a more precise model:

$R(T) = R_0[1 + AT + BT^2 + C(T - 100)T^3]$

• $A$, $B$, and $C$ are calibration constants specific to the platinum element
• The $C$ term only applies below 0°C; above 0°C, the equation simplifies to a quadratic

Steinhart-Hart Equation

For thermistors, whose resistance changes exponentially, the Steinhart-Hart equation provides an accurate model:

$\frac{1}{T} = A + B\ln(R) + C[\ln(R)]^3$

• $T$ is absolute temperature in Kelvin, and $R$ is resistance
• $A$, $B$, and $C$ are coefficients found by calibrating at three known temperatures
• Typically accurate to $\pm 0.02°\text{C}$ over a 200°C span

Experimental Methods

Four-Wire Resistance Measurement

When measuring small resistances, the resistance of the connecting wires themselves can introduce significant error. The four-wire (Kelvin) measurement technique eliminates this problem:

1. Connect two outer leads to pass a known current through the sample
2. Connect two separate inner leads directly across the sample to measure the voltage drop
3. Because the voltage leads carry negligible current, their resistance doesn't affect the reading
4. Calculate resistance using $R = V/I$ from the measured voltage and known current

This technique is essential for RTD calibration and any low-resistance measurement.

Temperature Control Techniques

Accurate resistance-temperature characterization requires precise temperature control:

• Liquid baths provide uniform temperature with good thermal contact
• Thermoelectric coolers (Peltier devices) allow fine electronic control of temperature
• Temperature-controlled chambers work for larger samples or assemblies
• Minimizing temperature gradients within the sample and allowing sufficient thermal equilibration time are both critical for reliable data

Material-Specific Behaviors

Metals vs. Alloys

Pure metals generally follow a simple, nearly linear resistance-temperature relationship because phonon scattering dominates. Alloys are more complex because impurity scattering adds a large temperature-independent component to the resistivity.

• Nichrome (nickel-chromium alloy) is engineered for high, stable resistance across a wide temperature range, making it ideal for heating elements
• Constantan and manganin are designed for near-zero temperature coefficients, used in precision measurement equipment

Ceramics and Polymers

• Some ceramics like barium titanate show a dramatic PTC effect near their Curie temperature, where resistance can jump by several orders of magnitude
• Conductive polymers often show NTC behavior as increased temperature enhances charge carrier mobility
• Carbon-filled polymers are used in self-regulating heating cables: as the polymer heats up, it expands, breaking conductive pathways and increasing resistance to limit further heating

Quantum Effects

Electron-Phonon Interactions

At a deeper level, the temperature dependence of resistivity in metals is described by the Bloch-Grüneisen formula:

$\rho(T) \propto \left(\frac{T}{\Theta_D}\right)^5 \int_0^{\Theta_D/T} \frac{x^5}{(e^x - 1)(1 - e^{-x})} \, dx$

Here $\Theta_D$ is the Debye temperature, a material-specific constant related to the maximum phonon frequency. At high temperatures ($T \gg \Theta_D$), this reduces to the familiar linear dependence. At very low temperatures ($T \ll \Theta_D$), resistivity drops as $T^5$.

Kondo Effect

In metals containing small amounts of magnetic impurities (like iron atoms in copper), something unexpected happens at low temperatures: instead of resistance continuing to drop, it reaches a minimum and then rises again. This is the Kondo effect.

It's caused by conduction electrons scattering off the localized magnetic moments of the impurity atoms in a spin-dependent way. The temperature at which the resistance minimum occurs is called the Kondo temperature, and it depends on the strength of the coupling between the conduction electrons and the magnetic impurity.

Limitations and Considerations

High Temperature Effects

• The linear approximation for metals breaks down at very high temperatures, where higher-order polynomial terms become necessary
• In heavily doped semiconductors, intrinsic behavior takes over at high temperatures as thermally generated carriers outnumber the dopant-provided ones
• Material degradation, oxidation, and phase changes can permanently alter resistance characteristics
• Thermal expansion changes the sample's geometry (length and cross-sectional area), which affects resistance independently of resistivity changes

Low Temperature Anomalies

• The residual resistance ratio (RRR), defined as $R(300\text{ K})/R(4.2\text{ K})$, is an important measure of material purity. Higher RRR means fewer impurities and defects.
• Unexpected superconducting transitions can occur in some materials
• Weak localization effects in disordered systems can cause resistance to increase slightly at very low temperatures, deviating from the expected behavior
• Quantum corrections to conductivity become relevant when the electron's thermal wavelength approaches the mean free path