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
Key Terms to Review (21)
Arrhenius Equation: The Arrhenius Equation is a mathematical formula that describes how the rate of a chemical reaction depends on temperature. It expresses the relationship between temperature and reaction rate through an exponential function, indicating that as temperature increases, the rate of reaction also increases due to higher energy collisions among molecules. This equation is essential for understanding the temperature dependence of resistance in materials, particularly in conductors and semiconductors.
Bloch-Grüneisen Formula: The Bloch-Grüneisen formula is a mathematical expression that describes how the electrical resistivity of metals varies with temperature, particularly at low temperatures. This formula highlights the contributions of electron scattering by phonons, which are quantized vibrations in the lattice of a solid, and is crucial for understanding the temperature dependence of resistance in conductive materials.
Callendar-Van Dusen Equation: The Callendar-Van Dusen equation is a mathematical expression that describes how the resistance of a conductor changes with temperature. This equation is significant in understanding the temperature dependence of resistance, particularly for metals, and is instrumental in precision temperature measurements using resistance thermometers.
Cryogenic Measurements: Cryogenic measurements refer to the techniques and methods used to measure physical properties of materials at extremely low temperatures, typically below -150°C. These measurements are critical for understanding the behavior of materials as they approach absolute zero, where quantum effects become significant and properties such as electrical resistance and thermal conductivity can change dramatically.
Electron-phonon interactions: Electron-phonon interactions refer to the coupling between electrons and phonons, which are quantized modes of vibrations within a crystal lattice. This interaction plays a crucial role in understanding how the electrical resistance of materials changes with temperature, as it can influence the scattering of electrons, thereby affecting their mobility and the overall conductivity of the material.
Four-point probe method: The four-point probe method is a technique used to measure the electrical resistance of a material with high precision by using four equally spaced probes. This method minimizes the effects of contact resistance and allows for an accurate measurement of the material's intrinsic resistivity. By applying a current through the outer probes and measuring the voltage between the inner probes, this method provides valuable information about the material's conductive properties, which relates to both its basic resistance and how temperature affects that resistance.
Joule heating: Joule heating, also known as resistive heating, is the process by which electrical energy is converted into heat due to the resistance in a conductor when an electric current flows through it. This phenomenon is significant because it relates to the flow of electric current, how current density affects heating, the role of resistance, temperature changes in conductors, and the calculation of electrical power in circuits.
Kondo Effect: The Kondo effect refers to the increase in electrical resistance of metals at low temperatures due to the scattering of conduction electrons by localized magnetic moments, typically from impurities or defects in the material. This phenomenon is significant because it highlights how the interactions between electrons and magnetic impurities can lead to unusual and non-intuitive behavior in materials as temperature changes.
Matthiessen's Rule: Matthiessen's Rule states that the total electrical resistivity of a metal is the sum of its temperature-dependent resistivity and its resistivity due to impurities and defects. This concept highlights how both intrinsic and extrinsic factors contribute to the overall resistance of a material, making it essential for understanding the behavior of conductive materials under varying temperatures.
Negative temperature coefficient: A negative temperature coefficient refers to a property of certain materials where their electrical resistance decreases as the temperature increases. This behavior is typical in semiconductors and some metallic materials, allowing them to conduct electricity more efficiently at higher temperatures. Understanding this concept is crucial for applications in electronics and temperature sensors.
Ohm's Law: Ohm's Law states that the current flowing through a conductor between two points is directly proportional to the voltage across the two points and inversely proportional to the resistance of the conductor. This fundamental principle connects voltage, current, and resistance, allowing for a clear understanding of electrical circuits and components.
Positive Temperature Coefficient: A positive temperature coefficient refers to a property of materials, particularly conductors and semiconductors, where the resistance of the material increases as the temperature rises. This behavior is significant because it helps to explain how materials respond to changes in temperature, affecting their electrical conductivity and overall performance in various applications.
R(t) = r0[1 + α(t - t0)]: This equation describes how the resistance of a material changes with temperature, where 'r(t)' is the resistance at time 't', 'r0' is the resistance at a reference temperature 't0', 'α' is the temperature coefficient of resistance, and '(t - t0)' represents the change in temperature. This formula highlights the linear relationship between resistance and temperature, which is crucial for understanding how materials conduct electricity under varying thermal conditions.
Resistivity: Resistivity is a fundamental property of materials that quantifies how strongly they resist the flow of electric current. It is represented by the symbol $$\rho$$ and is dependent on the material's intrinsic properties, including its composition and structure, which distinguishes conductors from insulators. The concept of resistivity connects deeply to current density, resistance, and how temperature changes can affect a material's ability to conduct electricity.
Semiconductors: Semiconductors are materials that have electrical conductivity between conductors and insulators. They are essential in modern electronics as they can be manipulated to control the flow of electric current, making them crucial for devices like transistors and diodes. Their unique properties allow them to conduct electricity under certain conditions, especially when influenced by temperature and impurities.
Steinhart-Hart Equation: The Steinhart-Hart equation is a mathematical model used to describe the relationship between temperature and the resistance of thermistors. This equation is particularly valuable because it provides a more accurate representation of the nonlinear behavior of thermistors over a wide temperature range compared to simpler linear approximations. It consists of a cubic polynomial equation that incorporates coefficients specific to the thermistor being analyzed, allowing for precise temperature measurements based on resistance changes.
Superconductors: Superconductors are materials that can conduct electricity with zero resistance when cooled below a certain critical temperature. This unique property allows them to carry electric current without any energy loss, which is a significant contrast to normal conductive materials where resistance leads to energy dissipation in the form of heat. Superconductors also exhibit the Meissner effect, expelling magnetic fields and allowing for applications like magnetic levitation.
Temperature coefficient of resistance: The temperature coefficient of resistance is a measure of how much a material's electrical resistance changes with temperature. Specifically, it quantifies the change in resistance per degree change in temperature, usually expressed in units of ohms per degree Celsius (Ω/°C). Understanding this coefficient is crucial for predicting how materials behave under different thermal conditions and is essential for applications involving electric current and resistance.
Temperature sensors: Temperature sensors are devices that measure temperature and convert the detected temperature into a readable output signal. These sensors often rely on materials whose resistance changes with temperature, making them integral in various applications, from everyday thermometers to complex industrial systems. Understanding how resistance varies with temperature is key to their operation, linking them to the principles of resistance and the temperature dependence of resistance.
Thermal agitation: Thermal agitation refers to the random motion of particles in a substance due to thermal energy. This motion increases with temperature, causing particles to vibrate, rotate, and translate, which impacts the physical properties of materials, including electrical resistance.
Thermistors: Thermistors are temperature-sensitive resistors that exhibit a significant change in resistance with varying temperatures. They are typically made of ceramic materials and are used in various applications, including temperature sensing and circuit protection. These components can be classified into two main types: NTC (Negative Temperature Coefficient) thermistors, which decrease in resistance as temperature increases, and PTC (Positive Temperature Coefficient) thermistors, which increase in resistance with rising temperature.