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 to .
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
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Resistance increases with rising temperature in materials with (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 (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 and voltage regulators
Temperature coefficient of resistance
Definition and units
Quantifies the change in resistance per degree of temperature change
Expressed as α=R1dTdR
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
describes total 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)=R0[1+α(T−T0)]
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
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)=R0[1+AT+BT2+C(T−100)T3]
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
T1=A+Bln(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
describes temperature dependence of resistivity
ρ(T)∝(ΘDT)5∫0ΘD/T(ex−1)(1−e−x)x5dx
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