Temperature dependence of resistance is a crucial concept in understanding how materials behave electrically under varying conditions. This topic explores how resistance changes with temperature in different materials, from metals to semiconductors to superconductors.

The relationship between resistance and temperature has significant practical implications. It affects the design and operation of electronic devices, sensors, and electrical systems across a wide range of applications, from everyday electronics to advanced scientific instruments.

Resistance and temperature relationship

• Electrical resistance in materials changes with temperature due to atomic vibrations and electron mobility
• Understanding this relationship is crucial for designing and operating electronic devices in various temperature conditions
• Temperature dependence of resistance varies significantly between different types of materials (metals, semiconductors, superconductors)

Positive temperature coefficient

• Resistance increases with rising temperature in materials with positive temperature coefficient (PTC)
• Most metals and some ceramics exhibit PTC behavior
• Caused by increased lattice vibrations at higher temperatures impeding electron flow
• PTC materials used in self-regulating heating elements (car rear window defrosters)

Negative temperature coefficient

• Resistance decreases as temperature rises in materials with negative temperature coefficient (NTC)
• Many semiconductors and certain ceramics display NTC behavior
• Results from increased thermal excitation of charge carriers at higher temperatures
• NTC materials commonly used in temperature sensors and voltage regulators

Temperature coefficient of resistance

Definition and units

• Quantifies the change in resistance per degree of temperature change
• Expressed as $\alpha = \frac{1}{R} \frac{dR}{dT}$
• Units typically given in parts per million per degree Celsius (ppm/°C) or inverse Kelvin (K⁻¹)
• Positive value indicates resistance increases with temperature, negative value indicates decrease

Typical values for materials

• Pure metals have positive coefficients ranging from 3000 to 6000 ppm/°C
• Copper approximately 3900 ppm/°C, aluminum about 3700 ppm/°C
• Semiconductors can have large negative coefficients (-20,000 to -70,000 ppm/°C)
• Certain alloys (constantan, manganin) engineered to have near-zero temperature coefficients

Resistance in conductors

Electron collisions and temperature

• Resistance in conductors primarily arises from electron collisions with lattice vibrations (phonons)
• Higher temperatures increase lattice vibrations, leading to more frequent electron collisions
• Mean free path of electrons decreases with rising temperature
• Matthiessen's rule describes total resistivity as sum of temperature-dependent and temperature-independent components

Linear approximation for metals

• Resistance of most metals increases approximately linearly with temperature over a wide range
• Described by equation $R(T) = R_0[1 + \alpha(T - T_0)]$
• R₀ represents resistance at reference temperature T₀
• Linear approximation breaks down at very low or very high temperatures

Resistance in semiconductors

Band gap and temperature

• Semiconductors have an energy band gap between valence and conduction bands
• Temperature increase provides thermal energy for electrons to cross the band gap
• More charge carriers available at higher temperatures, decreasing resistance
• Band gap narrows slightly with increasing temperature, further enhancing conductivity

Intrinsic vs extrinsic semiconductors

• Intrinsic semiconductors rely solely on thermal excitation for charge carriers
• Resistance in intrinsic semiconductors decreases exponentially with temperature
• Extrinsic semiconductors have added impurities (dopants) to modify carrier concentration
• Temperature dependence in extrinsic semiconductors varies based on doping level and type

Superconductivity

Critical temperature

• Superconductors transition to zero resistance state below a critical temperature (Tc)
• Tc varies widely among different superconducting materials
• Low-temperature superconductors (LTS) have Tc below 30 K (niobium-titanium, Tc ≈ 10 K)
• High-temperature superconductors (HTS) have Tc above 30 K (YBCO, Tc ≈ 93 K)

Zero resistance phenomenon

• Superconductors exhibit exactly zero DC electrical resistance below Tc
• Caused by formation of Cooper pairs of electrons that flow without scattering
• Meissner effect expels magnetic fields from superconductor interior
• Persistence of supercurrents allows creation of powerful electromagnets (MRI machines)

Applications of temperature-dependent resistance

Thermistors and their uses

• Thermistors are temperature-sensitive resistors with large temperature coefficients
• NTC thermistors commonly used for precise temperature measurement and control
• Applications include medical thermometers, automotive temperature sensors, and HVAC systems
• PTC thermistors used for overcurrent protection and self-regulating heating elements

RTDs vs thermocouples

• Resistance Temperature Detectors (RTDs) use metals with predictable resistance-temperature relationship
• RTDs offer high accuracy and stability over a wide temperature range (-200°C to 850°C)
• Thermocouples generate voltage based on temperature difference between two dissimilar metals
• Thermocouples have wider temperature range (-270°C to 1800°C) but lower accuracy than RTDs

Mathematical models

Callendar-Van Dusen equation

• Describes resistance-temperature relationship for platinum RTDs over wide range
• $R(T) = R_0[1 + AT + BT^2 + C(T-100)T^3]$
• A, B, and C are calibration constants specific to the platinum wire used
• Simplifies to linear form for temperatures above 0°C

Steinhart-Hart equation

• Provides accurate model for resistance-temperature relationship in thermistors
• $\frac{1}{T} = A + B\ln(R) + C[\ln(R)]^3$
• A, B, and C are coefficients determined by calibration
• Typically accurate to ±0.02°C over a 200°C range

Experimental methods

Four-wire resistance measurement

• Eliminates lead resistance errors in precise resistance measurements
• Separate current and voltage connections to the sample
• Current applied through outer leads, voltage measured across inner leads
• Particularly important for low-resistance measurements and RTD calibration

Temperature control techniques

• Precise temperature control crucial for accurate resistance-temperature characterization
• Methods include liquid baths, thermoelectric coolers, and temperature-controlled chambers
• Temperature gradients within samples must be minimized
• Thermal equilibration time considered to ensure steady-state measurements

Material-specific behaviors

Metals vs alloys

• Pure metals generally have simple, near-linear resistance-temperature relationships
• Alloys can exhibit more complex behaviors due to impurity scattering
• Some alloys (nichrome) engineered for high, stable resistance over temperature range
• Certain alloys (invar) designed for minimal thermal expansion, affecting resistance properties

Ceramics and polymers

• Ceramics can exhibit wide range of temperature coefficients (both PTC and NTC)
• Some ceramics (barium titanate) show sharp PTC effect at Curie temperature
• Conductive polymers often have NTC behavior due to increased charge carrier mobility
• Carbon-filled polymers used in self-regulating heating cables

Quantum effects

Electron-phonon interactions

• Quantum mechanical description of resistance based on electron scattering by phonons
• Bloch-Grüneisen formula describes temperature dependence of resistivity
• $\rho(T) \propto (\frac{T}{\Theta_D})^5 \int_0^{\Theta_D/T} \frac{x^5}{(e^x-1)(1-e^{-x})} dx$
• Θ_D is the Debye temperature, characteristic of the material

Kondo effect

• Anomalous increase in resistivity at low temperatures in metals with magnetic impurities
• Caused by spin-dependent scattering of conduction electrons by localized magnetic moments
• Leads to resistance minimum at characteristic Kondo temperature
• Observed in systems like copper with iron impurities

Limitations and considerations

High temperature effects

• Linear approximation for metals breaks down at very high temperatures
• Intrinsic semiconductor behavior dominates in heavily doped semiconductors at high temperatures
• Material degradation and phase changes can occur, altering resistance characteristics
• Thermal expansion effects become significant, changing sample geometry

Low temperature anomalies

• Residual resistance ratio (RRR) important metric for material purity at low temperatures
• Superconducting transitions can occur unexpectedly in some materials
• Weak localization effects in disordered systems can modify temperature dependence
• Quantum corrections to conductivity become relevant at very low temperatures